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--- abstract: | The universal order 1 invariant $f^U$ of immersions of a closed orientable surface into ${{{\Bbb R}^3}}$, whose existence has been established in [o]{}, takes values in the group ${{\Bbb G}}_U = K \oplus {{\Bbb Z}}/2 \oplus {{\Bbb Z}}/2$ where $K$ is a countably generated free Abelian group. The projections of $f^U$ to $K$ and to the first and second ${{\Bbb Z}}/2$ factors are denoted $f^K, M, Q$ respectively. An explicit formula for the value of $Q$ on any embedding has been given in [a]{}. In the present work we give an explicit formula for the value of $f^K$ on any immersion, and for the value of $M$ on any embedding. address: 'Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel.' author: - Tahl Nowik date: 'May 4, 2003' title: | Formulae for order one invariants\ of immersions and embeddings of surfaces --- \[section\] \[thm\][Lemma]{} \[thm\][Proposition]{} \[thm\][Corollary]{} \[thm\][Definition]{} \[thm\][Remark]{} \[thm\][Question]{} [^1] introduction {#intro} ============ Finite order invariants of stable immersions of a closed orientable surface into ${{{\Bbb R}^3}}$ have been defined in [o]{}, where all order 1 invariants have been classified. In [h]{} all higher order invariants have been classified, and it has been shown that they are all functions of order 1 invariants. This brings the attention back to order 1 invariants, and to the problem of finding explicit formulae for them. In [o]{}, the existence of a universal order 1 invariant $f^U$ has been established, which takes values in a group ${{\Bbb G}}_U = K \oplus {{\Bbb Z}}/2 \oplus {{\Bbb Z}}/2$ where $K$ is a countably generated free Abelian group. The existence proof, however, gave no clue for computing the invariant. We will denote the projections of $f^U$ to $K$ and to the first and second ${{\Bbb Z}}/2$ factors of ${{\Bbb G}}_U$ by $f^K, M, Q$ respectively. (The geometric meaning of $M$ and $Q$ will be explained in Section [inv]{}.) In [a]{}, an explicit formula has been given for $Q(i\circ h) - Q(i)$ where $h:F \to F$ is a diffeomorphism such that $i$ and $i\circ h$ are regularly homotopic, and for $Q(e')-Q(e)$ where $e,e'$ are any two regularly homotopic embeddings. In the present work we give an explicit formula for: 1. The value of $f^K$ on all immersions. 2. $M(i\circ h) - M(i)$ where $h:F \to F$ is a diffeomorphism such that $i$ and $i\circ h$ are regularly homotopic. 3. $M(e')-M(e)$ for any two regularly homotopic embeddings. Note that the invariant $f^U$ is specified only up to an order 0 invariant, i.e. up to an additive constant in each regular homotopy class, and so the same is true for $f^K,M,Q$. For $M$ and $Q$ we will not have a specific choice of constants, and so as in (2),(3) above, we will speak only of the difference of the value of $M$ and $Q$ on regularly homotopic immersions. The structure of the paper is as follows: In Section [back]{} we give the necessary background. Note that in the present work we deviate from [o]{},[h]{} in our procedure for defining order one invariants, and accordingly we deviate in our choice of generators for ${{\Bbb G}}_U$. This is of no consequence in the abstract setting of [o]{},[h]{}, but will greatly effect the simplicity of the explicit formula for $f^K$ that we will find in the present work. In Section [inv]{} we explain the geometric meaning of the invariants $M$ and $Q$. In Section [st]{} we present the formulae that will be proved in this paper. In Section [fk]{} we prove the formula for $f^K$. In Section [ap]{} we give two applications. In Section [m]{} we prove the formula for $M$. Background {#back} ========== In this section we summarize the background needed for this work. Given a closed oriented surface $F$, ${{Imm(F,{{{\Bbb R}^3}})}}$ denotes the space of all immersions of $F$ into ${{{\Bbb R}^3}}$, with the $C^1$ topology. A CE point of an immersion $i:F\to {{{\Bbb R}^3}}$ is a point of self intersection of $i$ for which the local stratum in ${{Imm(F,{{{\Bbb R}^3}})}}$ corresponding to the self intersection, has codimension one. We distinguish twelve types of CEs which we name $E^0, E^1, E^2, H^1, H^2, T^0, T^1, T^2, T^3, Q^2, Q^3, Q^4$. Their precise description appears in the proof of Proposition [p1]{} below. This set of twelve symbols is denoted ${{\mathcal C}}$. A co-orientation for a CE is a choice of one of the two sides of the local stratum corresponding to the CE. All but two of the above CE types are non-symmetric in the sense that the two sides of the local stratum may be distinguished via the local configuration of the CE, and for those ten CE types, permanent co-orientations for the corresponding strata are chosen once and for all. The two exceptions are $H^1$ and $Q^2$ which are completely symmetric. In fact, there does not exist a consistent choice of co-orientation for $H^1$ and $Q^2$ CEs since the global strata corresponding to these CE types are one sided in ${{Imm(F,{{{\Bbb R}^3}})}}$ (see [o]{}). We fix a closed oriented surface $F$ and a regular homotopy class ${{\mathcal A}}$ of immersions of $F$ into ${{{\Bbb R}^3}}$ (that is, ${{\mathcal A}}$ is a connected component of ${{Imm(F,{{{\Bbb R}^3}})}}$). We denote by $I_n{\subseteq}{{\mathcal A}}$ ($n\geq 0$) the space of all immersions in ${{\mathcal A}}$ which have precisely $n$ CE points (the self intersection being elsewhere stable). In particular, $I_0$ is the space of all stable immersions in ${{\mathcal A}}$. Given an immersion $i\in I_n$, a *temporary co-orientation* for $i$ is a choice of co-orientation at each of the $n$ CE points $p_1, \dots , p_n$ of $i$. Given a temporary co-orientation ${{\mathfrak{T}}}$ for $i$ and a subset $A{\subseteq}\{p_1,\dots,p_n\}$, we define $i_{{{\mathfrak{T}}},A} \in I_0$ to be the immersion obtained from $i$ by resolving all CEs of $i$ at points of $A$ into the positive side with respect to ${{\mathfrak{T}}}$, and all CEs not in $A$ into the negative side. Now let ${{\Bbb G}}$ be any Abelian group and let $f:I_0\to{{\Bbb G}}$ be an invariant, i.e. a function which is constant on each connected component of $I_0$. Given $i\in I_n$ and a temporary co-orientation ${{\mathfrak{T}}}$ for $i$, $f^{{\mathfrak{T}}}(i)$ is defined as follows: $$f^{{\mathfrak{T}}}(i)=\sum_{ A {\subseteq}\{p_1,\dots,p_n\} } (-1)^{n-|A|} f(i_{{{\mathfrak{T}}},A})$$ where $|A|$ is the number of elements in $A$. The statement $f^{{\mathfrak{T}}}(i)=0$ is independent of the temporary co-orientation ${{\mathfrak{T}}}$ so we simply write $f(i)=0$. An invariant $f:I_0\to{{\Bbb G}}$ is called *of finite order* if there is an $n$ such that $f(i)=0$ for all $i\in I_{n+1}$. The minimal such $n$ is called the *order* of $f$. The group of all invariants on $I_0$ of order at most $n$ is denoted $V_n$. From now on our discussion will reduce to order 1 invariants only. The more general setting may be found in [o]{},[h]{}. For an immersion $i:F\to{{{\Bbb R}^3}}$ and any $p\in{{{\Bbb R}^3}}$, we define the degree $d_p(i) \in {{\Bbb Z}}$ of $i$ at $p$ as follows: If $p \not\in i(F)$ then $d_p(i)$ is the (usual) degree of the map obtained from $i$ by composing it with the projection onto a small sphere centered at $p$. If on the other hand $p \in i(F)$ then we first push each sheet of $F$ which passes through $p$, a bit into its preferred side determined by the orientation of $F$, obtaining a new immersion $i'$ which misses $p$, and we define $d_p(i)=d_p(i')$. If $i\in I_1$ and the unique CE of $i$ is located at $p \in {{{\Bbb R}^3}}$, then we define $C(i)$ to be the expression $R^a_m$ where $R^a\in{{\mathcal C}}$ is the symbol describing the configuration of the CE of $i$ at $p$ (one of the twelve symbols above) and $m=d_p(i)$. We denote by ${{\mathcal C}}_1$ the set of all expressions $R^a_m$ with $R^a\in{{\mathcal C}}, m\in{{\Bbb Z}}$. The map $C:I_1 \to {{\mathcal C}}_1$ is surjective. Let $f \in V_1$. For $i\in I_1$, if the CE of $i$ is of type $H^1$ or $Q^2$ and ${{\mathfrak{T}}}$ is a temporary co-orientation for $i$, then $2f^{{\mathfrak{T}}}(i)=0$ ([o]{} Proposition 3.5), and so in this case $f^{{\mathfrak{T}}}(i)$ is independent of ${{\mathfrak{T}}}$. This fact is used to extend any $f\in V_1$ to $I_1$ by setting for any $i\in I_1$, $f(i) = f^{{\mathfrak{T}}}(i)$, where if the CE of $i$ is of type $H^1$ or $Q^2$ then ${{\mathfrak{T}}}$ is arbitrary, and if it is not of type $H^1$ or $Q^2$ then the permanent co-orientation is used for the CE of $i$. We will always assume without mention that any $f \in V_1$ is extended to $I_1$ in this way. For $f\in V_1$ and $i,j\in I_1$, if $C(i)=C(j)$ then $f(i)=f(j)$ ([o]{} Proposition 3.7), so any $f\in V_1$ induces a well defined function $u(f):{{\mathcal C}}_1\to{{\Bbb G}}$. The map $f\mapsto u(f)$ induces an injection $u:V_1 / V_0 \to {{\mathcal C}}_1^*$ where ${{\mathcal C}}_1^*$ is the group of all functions from ${{\mathcal C}}_1$ to ${{\Bbb G}}$. The main result of [o]{} is that the image of $u:V_1 / V_0 \to {{\mathcal C}}_1^*$ is the subgroup $\Delta_1 = \Delta_1({{\Bbb G}}) {\subseteq}{{\mathcal C}}_1^*$ which is defined as the set of functions in ${{\mathcal C}}_1^*$ satisfying relations which we write as relations on the symbols $R^a_m$, e.g. $T^0_m = T^3_m$ will stand for $g(T^0_m) = g(T^3_m)$. The relations defining $\Delta_1$ are: - $E^2_m = - E^0_m = H^2_m$,    $E^1_m = H^1_m$. - $T^0_m = T^3_m$,    $T^1_m = T^2_m$. - $2H^1_m =0 $,   $H^1_m = H^1_{m-1}$. - $2Q^2_m =0 $,   $Q^2_m = Q^2_{m-1}$. - $H^2_m - H^2_{m-1} = T^3_m - T^2_m$. - $Q^4_m - Q^3_m = T^3_m - T^3_{m-1}$,    $Q^3_m - Q^2_m = T^2_m - T^2_{m-1}$. Let ${{\Bbb B}}{\subseteq}{{\Bbb G}}$ be the subgroup defined by ${{\Bbb B}}=\{ x\in {{\Bbb G}}: 2x=0\}$. To obtain a function $g\in\Delta_1$ one may assign arbitrary values in ${{\Bbb G}}$ for the symbols $\{T^2_m\}_{m\in{{\Bbb Z}}}$, $\{H^2_m\}_{m\in{{\Bbb Z}}}$ (here is where we deviate from [o]{},[h]{}) and arbitrary values in ${{\Bbb B}}$ for the two symbols $H^1_0 , Q^2_0$. Once this is done then the value of $g$ on all other symbols is uniquely determined, namely: 1. $E^1_m = H^1_m = H^1_0$ for all $m$. 2. $E^2_m = -E^0_m = H^2_m$ for all $m$. 3. $T^3_m = T^2_m + H^2_m - H^2_{m-1}$ 4. $T^0_m = T^3_m$,   $T^1_m = T^2_m$ for all $m$. 5. $Q^2_m = Q^2_0$ for all $m$. 6. $Q^3_m (= Q^2_m + T^2_m - T^2_{m-1}) =Q^0_m + T^2_m - T^2_{m-1}$ for all $m$. 7. $Q^4_m (= Q^3_m + T^3_m - T^3_{m-1}) = Q^0_m + 2T^2_m - 2T^2_{m-1} + H^2_m - 2H^2_{m-1} + H^2_{m-2}$ for all $m$. In the sequel we will refer to this procedure as the “7-step procedure”. The Abelian group ${{\Bbb G}}_U$ is defined as follows (again note the difference from [o]{},[h]{}): $${{\Bbb G}}_U = \left< \{t^2_m\}_{m\in{{\Bbb Z}}}, \{h^2_m\}_{m\in{{\Bbb Z}}}, h^1_0, q^2_0 \ | \ 2h^1_0 = 2q^2_0 = 0 \right>.$$ The universal element $g^U\in\Delta_1({{\Bbb G}}_U)$ is defined by $g^U(T^2_m) = t^2_m$, $g^U(H^2_m)=h^2_m$, $g^U(H^1_0)=h^1_0$, $g^U(Q^2_0)=q^2_0$ and the value of $g^U$ on all other symbols of ${{\mathcal C}}_1$ is determined by the 7-step procedure. In [o]{} the existence of an order 1 invariant $f^U:I_0\to{{\Bbb G}}_U$ with $u(f^U)=g^U$ is proven. (Note that this is the same $g^U$ as in [o]{} only presented via different generators). The invariant $f^U$ is a *universal* order 1 invariant, meaning the following: \[uni\] A pair $({{\Bbb G}},f)$ where ${{\Bbb G}}$ is an Abelian group and $f:I_0 \to {{\Bbb G}}$ is an order $n$ invariant, will be called a *universal order $n$ invariant* if for any Abelian group ${{\Bbb G}}'$ and any order $n$ invariant $f':I_0 \to {{\Bbb G}}'$ there exists a unique homomorphism $\varphi:{{\Bbb G}}\to {{\Bbb G}}'$ such that $f' - \varphi \circ f$ is an invariant of order at most $n-1$. In [h]{} all higher order invariants are classified, and for every $n$ a universal order $n$ invariant is constructed as ${{\mathcal F}}_n \circ f^U$ where ${{\mathcal F}}_n:{{\Bbb G}}_U \to M_n$ is an explicit function (not homomorphism) into a certain Abelian group $M_n$. The invariants {#inv} ============== In this section we introduce the three invariants $f^K,M,Q$ that interest us. We define $K {\subseteq}{{\Bbb G}}_U$ to be the subgroup generated by $\{t^2_m\}_{m\in{\Bbb Z}} \cup \{h^2_m\}_{m\in{\Bbb Z}}$ (this is the same as the subgroup $K_1$ in [h]{}) and define $f^K:I_0 \to K$ to be the projection of $f^U$ to $K$. Similarly we define $M : I_0 \to {{\Bbb Z}}/2$ (respectively $Q: I_0 \to {{\Bbb Z}}/2$) to be the projection of $f^U$ to ${{\Bbb Z}}/2$ sending all generators of ${{\Bbb G}}_U$ to 0 except $h^1_0$ (respectively except $q^2_0$). Then $f^U = f^K \oplus M \oplus Q$. Note that $f^U$ is defined only up to an additive constant in each regular homotopy class, and so the same is true for $f^K,M,Q$. More in detail, the invariants $Q$ and $M$ are defined as follows: $M:I_0 \to {{\Bbb Z}}/2$ is the order 1 invariant defined by $u(M)(H^1_0)=1$, $u(M)(Q^2_0)=0$ and $u(M)(T^2_m)=u(M)(H^2_m)=0$ for all $m$. By the 7-step procedure, this extends to $u(M)(H^1_m)=u(M)(E^1_m)=1$ for all $m$, $u(M)(H^a_m)=u(M)(E^a_m)=0$ for $a\neq 1$ and any $m$ and $u(M)(T^a_m)=u(M)(Q^a_m)=0$ for all $a,m$. That is, if $i_+,i_- \in I_0$ are the two immersions obtained from $i \in I_1$ by resolving its CE, then $M(i_+)-M(i_-) = 1 \in {{\Bbb Z}}/2$ iff the CE of $i$ is a “matching tangency” i.e. tangency of two sheets of the surface where the orientations of the two sheets match at time of tangency. (Thus the name $M$ for this invariant). And so for any $i,j \in I_0$, $M(j)-M(i) \in {{\Bbb Z}}/2$ is the number mod 2 of matching tangencies ocurring in any regular homotopy between $i$ and $j$. Similarly, $Q:I_0 \to {{\Bbb Z}}/2$ is the ${{\Bbb Z}}/2$ valued order 1 invariant satisfying $u(Q)(Q^2_0) = 1$, $u(Q)(H^1_0)=0$ and $u(Q)(T^2_m)=u(Q)(H^2_m)=0$ for all $m$. By the 7-step procedure, we have $u(Q)(Q^a_m)=1$ for all $a,m$ and $u(Q)(T^a_m)=U(Q)(E^a_m)=u(Q)(H^a_m)=0$ for all $a,m$. That is, $Q$ is the invariant such that for any $i,j \in I_0$, $Q(j)-Q(i)\in{{\Bbb Z}}/2$ is the number mod 2 of quadruple points occurring in any regular homotopy between $i$ and $j$. This invariant has been studied in [q]{} and [a]{}. In [a]{} an explicit formula has been given for $Q(i \circ h) - Q(i)$ for any diffeomorphism $h:F \to F$ such that $i$ and $i \circ h$ are regularly homotopic, and for $Q(e')-Q(e)$ for any two regularly homotopic embeddings. In the present work we will do the same for $M$, leaving open the interesting problem of finding an explicit formula for $Q$ and $M$ on *all* immersions. For $f^K$ however, we will indeed give a formula for all immersions. Statement of results {#st} ==================== Let $i \in I_0$. For every $m\in{{\Bbb Z}}$ let $U_m = U_m(i) = \{ p \in {{{\Bbb R}^3}}-i(F) \ : \ d_p(i)=m \}$. This is an open set in ${{{\Bbb R}^3}}$ which may be empty, and may be non-connected or unbounded, but in any case, the Euler characteristic $\chi(U_m)$ is defined. Denote by $N_m = N_m(i)$ the number of triple points $p\in{{{\Bbb R}^3}}$ of $i$ having $d_p(i)=m$. The following formula for $f^K : I_0 \to K {\subseteq}{{\Bbb G}}_U$ will be proved in Section [fk]{}: $$f^K(i)= \sum_{m\in{{\Bbb Z}}} \chi(U_m) \bigg(\sum_{ -{{1 \over 2}}< k < {\lfloor}{m \over 2}{\rfloor}+ {{1 \over 2}}} h^2_{m-2k}\bigg) + \sum_{m\in{{\Bbb Z}}} {{1 \over 2}}N_m \bigg( t^2_m - \sum_{ -{{1 \over 2}}< k < m-{{1 \over 2}}} h^2_k \bigg)$$ where for $a \in{{\Bbb R}}$, ${\lfloor}a {\rfloor}$ denotes the greatest integer $\leq a$, and for $a,b \in {{\Bbb R}}$ the sum $\sum_{a<k<b}$ means the following: If $a<b$ then it is the sum over all integers $a<k<b$, if $a=b$ then the sum is 0, and if $a>b$ then $\sum_{a<k<b} = - \sum_{b<k<a}$. For $i,j \in I_0$ let $M(i,j)=M(j)-M(i)$. The following two formulae for $M$ will be proved in Section [m]{}: For any diffeomorphism $h:F\to F$ such that $i$ and $i \circ h$ are regularly homotopic: $$M(i,i\circ h)=\bigg({{\mathrm{rank}}}(h_*-Id)\bigg)\bmod{2}$$ where $h_*$ is the map induced by $h$ on $H_1(F,{{\Bbb Z}}/2)$. If $e:F\to{{{\Bbb R}^3}}$ is an embedding then $e(F)$ splits ${{{\Bbb R}^3}}$ into two pieces, one compact and one non-compact, which we denote $D^0(e)$ and $D^1(e)$ respectively. By restriction of range, $e$ induces maps $e^k : F \to D^k(e)$, $k=0,1$. Let $e^k_* : {{H_1(F,{{\Bbb Z}}/2)}}\to H_1(D^k(e),{{\Bbb Z}}/2)$ be the map induced by $e^k$. Then for two regularly homotopic embeddings $e,e':F \to {{{\Bbb R}^3}}$, $M(e,e')$ is computed as follows: 1. Find a basis $a_1,\dots,a_n,b_1,\dots,b_n$ for ${{H_1(F,{{\Bbb Z}}/2)}}$ such that $e^0_*(a_i)=0$, $e^1_*(b_i)=0$ and $a_i \cdot b_j = \delta_{ij}$ (where $a \cdot b$ denotes the intersection form in ${{H_1(F,{{\Bbb Z}}/2)}}$). 2. Find a similar basis $a'_1,\dots,a'_n,b'_1,\dots,b'_n$ using $e'$ in place of $e$. 3. Let $m$ be the dimension of the subspace of ${{H_1(F,{{\Bbb Z}}/2)}}$ spanned by: $$a'_1 - a_1 \ , \ \dots \ , \ a'_n - a_n \ , \ b'_1 - b_1 \ , \ \dots \ , \ b'_n - b_n.$$ Then $M(e,e') = m\bmod{2} \in {{\Bbb Z}}/2$. Proof of formula for $f^K$ {#fk} ========================== We define the group ${{\Bbb O}}$ to be the free Abelian group with generators $\{x_n\}_{n\in {{\Bbb Z}}} \cup \{y_n\}_{n\in{{\Bbb Z}}}$. For $i\in I_0$ we define $k(i) \in {{\Bbb O}}$ as follows (the terms are defined in Section [st]{} and the sums are always finite): $$k(i)= \sum_{m\in {{\Bbb Z}}} \chi(U_m) x_m + \sum_{m \in {{\Bbb Z}}} {{1 \over 2}}N_m y_m .$$ Indeed this is an element of ${{\Bbb O}}$ since as we shall see below, $N_m$ is always even. In the mean time say $k$ attains values in the $\Bbb Q$ vector space with same basis. \[p1\] The invariant $k$ is an order 1 invariant, with $u(k)$ given by: - $u(k)(E^a_m) = u(k)(H^a_m) = x_{m+a-2} - x_{m-a}$ - $u(k)(T^a_m) = x_{m+a-3} + x_{m-a} + y_m$ - $u(k)(Q^a_m) = x_{m+a-4} - x_{m-a} + (a-2)y_m + (2-a)y_{m-1}$ We use the explicit description of the CE types, as appearing in [o]{}, where more details may be found. A model in 3-space for the different sheets involved in the self intersection near the CE, is given. The CE is obtained at the origin when setting ${{\lambda}}=0$. We will show that for any $i \in I_1$, if $i_+ \in I_0$ is the immersion on the positive side of $i$ with respect to the permanent co-orientation for the CE of $i$, (if such exists, otherwise an arbitrary side is chosen) and $i_- \in I_0$ is the immersion on the other side, then indeed $k(i_+) - k(i_-)$ depends on $C(i)$ as in the statement of this proposition. By showing in particular, that this change depends *only* on $C(i)$, we show that $k$ is indeed an invariant of order 1. Model for $E^a_m$:  $z=0$,  $z=x^2+y^2+{{\lambda}}$. The positive side is that where ${{\lambda}}< 0$, where there is a new 2-sphere in the image of the immersion, which is made of two 2-cells, and bounds a 3-cell in ${{{\Bbb R}^3}}$. The superscript $a$ is then the number of 2-cells (0, 1 or 2) whose prefered side determined by the orientation of the surface, is facing away from the 3-cell, (and $m$ is the degree at the CE at time ${{\lambda}}=0$). The degree of points in the new 3-cell is seen to be $m+a-2$, and its $\chi$ is 1, and so the term $x_{m+a-2}$. The second change ocurring, is that the region just above the plane $z=0$, has a 2-handle removed from it, so its $\chi$ is reduced by 1, and the degree in this region is seen to be $m-a$, and so the term $- x_{m-a}$. Model for $H^a_m$:   $z=0$,  $z=x^2-y^2+{{\lambda}}$. The positive side for $H^2$ is that where both sheets have their preferred side facing toward the region that is between them near the origin. For $H^1$ a positive side is chosen arbitrarily. By rotating the configuration if necessary, say the positive side is where ${{\lambda}}<0$. The superscript $a$ then denotes the number of sheets (1 or 2) whose preferred side is facing toward the region that is between the two sheets near the origin, when ${{\lambda}}<0$. The changes ocurring in the neighboring regions when passing from ${{\lambda}}> 0$ to ${{\lambda}}< 0$ are that a 1-handle is removed from the region $X$ just above the $x$ axis, and a 1-handle is added to the region $Y$ just below the $y$ axis. The degree of $X$ is seen to be $m+a-2$ and since a 1-handle is removed, $\chi(X)$ increases by 1 and thus the term $x_{m+a-2}$. The degree of $Y$ is seen to be $m-a$, and since a 1-handle is added, $\chi(Y)$ decreases by 1 and thus the term $-x_{m-a}$. Model for $T^a_m$:  $z=0$,  $y=0$,  $z=y+x^2+{{\lambda}}$. The positive side for this configuration is when ${{\lambda}}< 0$, where there is a new 2-sphere in the image of the immersion, which is made of three 2-cells, and bounds a 3-cell in ${{{\Bbb R}^3}}$. The superscript $a$ is the number of 2-cells (0, 1, 2 or 3) whose prefered side is facing away from the 3-cell. The degree in the new 3-cell is $m+a-3$ and its $\chi$ is 1 and so the term $x_{m+a-3}$. The second change ocurring is that a 1-handle is removed from the region near the $x$ axis having negative $y$ values and positive $z$ values. The degree of this region is $m-a$ and since a 1-handle is removed, $\chi$ is increased by 1 and so the term $x_{m-a}$. The last change that effects the value of $k$ is that two triple points are added, each of degree $m$. This increases ${{1 \over 2}}N_m$ by 1 and so the term $y_m$. Model for $Q^a_m$:  $z=0$,  $y=0$,  $x=0$,  $z=x+y+{{\lambda}}$. On both the positive and negative side there is a simplex created near the origin, and the positive side is that where the majority of the four sheets are facing away from the simplex (and for $Q^2$ a positive side is chosen arbitrarily). The superscript $a$ denotes the number of sheets (2,3 or 4) facing away from the simplex created on the positive side, its degree thus seen to be $m+a-4$. The simplex on the negative side has $4-a$ sheets facing away from it and so its degree is $m-a$. So when moving from the negative to the positive side, a 3-cell ($\chi=1$) of degree $m-a$ is removed and a 3-cell of degree $m+a-4$ is added, and so the terms $x_{m+a-4} - x_{m-a}$. In addition to that, the degree of the four triple points of the simplex changes. On the positive side there are $a$ triple points with degree $m$, (namely, the triple points which are opposite the faces which are facing away from the simplex), and $4-a$ triple points with degree $m-1$. On the negative side the situation is reversed, i.e. there are $4-a$ triple points with degree $m$ and $a$ triple points with degree $m-1$. So the total change in $N_m$ is $a-(4-a) = 2a-4$ and the total change in $N_{m-1}$ is $(4-a)-a = 4-2a$ and so the terms $(a-2)y_m + (2-a)y_{m-1}$. We can now verify that indeed the values of $k$ are in ${{\Bbb O}}$ i.e. no half integer coefficients appear (which means $N_m$ is always even). From Proposition [p1]{} we see that the change in the value of $k$ is in ${{\Bbb O}}$ along any regular homotopy, and so it is enough to show that the value is in ${{\Bbb O}}$ for one immersion in any given regular homotopy class. Indeed, we show a bit more: \[l1\] Let $g$ be the genus of $F$. Any immersion $i:F\to{{{\Bbb R}^3}}$ is regularly homotopic to an immersion $j$ with $k(j) = (2-g)x_0 + (1-g)x_{-1}$. By [p]{}, any immersion $i:F\to {{{\Bbb R}^3}}$ is regularly homotopic to an immersion whose image is of one of two standard forms, either a standard embedding, or an immersion obtained from a standard embedding by adding a ring to it. (For what we mean by a “ring” see [a]{}.) For an embedding $e$, $k(e)$ is either $(2-g)x_0 + (1-g)x_{-1}$ or $(2-g)x_0 + (1-g)x_1$, depending on whether the preferred side of $e(F)$, determined by the orientation of $F$, is facing the compact or the non-compact side of $e(F)$ in ${{{\Bbb R}^3}}$, respectively. Now take an orientation reversing diffeomorphism $h:F\to F$ such that $e\circ h$ is regularly homotopic to $e$, to see that both values are attained. (Such $h$ exists by [p]{}, take e.g. an $h$ that induces the identity on $H_1(F, {{\Bbb Z}}/2)$.) Now, a ring added to such embedding bounds a solid torus, whose $\chi$ is 0, and the topological type and degree of the other two components remains the same, and so by the same argument as for an embedding, the two values are attained in this case too. We define a homomorphism $\varphi:{{\Bbb G}}_U \to {{\Bbb O}}$ on generators as follows: - $\varphi(h^2_m) = x_m - x_{m-2}$ - $\varphi(t^2_m) = x_{m-1} + x_{m-2} + y_m$ - $\varphi(h^1_0) = \varphi(q^2_0) = 0$ By Proposition [p1]{}, $u(k) = u(\varphi \circ f^U)$ and so $k=\varphi \circ f^U + c$ where $c \in {{\Bbb O}}$ is a constant. We define the following homomorphism $F:{{\Bbb O}}\to K$ satisfying that $F \circ \varphi$ is the projection of ${{\Bbb G}}_U$ onto $K$, and so $F \circ k = F \circ \varphi \circ f^U + F(c) = f^K + F(c)$. By redefining $f^U$ as $f^U + F(c)$ we have $F \circ k = f^K$. We define $F$ on generators of ${{\Bbb O}}$ as follows (the notation involved is defined in Section [st]{}): $$F(x_m) = \sum_{ -{{1 \over 2}}< k < {\lfloor}{m \over 2}{\rfloor}+ {{1 \over 2}}} h^2_{m-2k} \ \ \ \ \ \ \ \ \ \ \ \ \ \ F(y_m) = t^2_m - \sum_{ -{{1 \over 2}}< k < m-{{1 \over 2}}} h^2_k$$ One checks directly that indeed $F \circ \varphi$ maps each generator of $K$ to itself. Since $\varphi$ is not surjective, there was a certain choice in the construction of $F$. Indeed the image of $\varphi$ is the subgroup of ${{\Bbb O}}$ of all elements $\sum A_m x_m + \sum B_m y_m$ with $A_m,B_m \in {{\Bbb Z}}$ satisfying $\sum_m A_{2m} = \sum_m A_{2m+1} = \sum_m B_m$. And so any two generators $x_i , x_j$ with $i$ even and $j$ odd, generate a subgroup in ${{\Bbb O}}$ which is a direct summand of the image of $\varphi$. Our choice for $F$ was that $F(x_{-2}) = F(x_{-1}) = 0$. Note that by Lemma [l1]{}, the image of $k:I_0 \to {{\Bbb O}}$ is contained in a non trivial coset of the image of $\varphi$ in ${{\Bbb O}}$, (and so the constant $c$ appearing above is non-zero, regardless of an additive constant for $f^U$). Composing the formula for $F$ with the formula for $k$ we obtain our formula for $f^K$: $$f^K(i)= \sum_{m\in{{\Bbb Z}}} \chi(U_m) \bigg(\sum_{ -{{1 \over 2}}< k < {\lfloor}{m \over 2}{\rfloor}+ {{1 \over 2}}} h^2_{m-2k}\bigg) + \sum_{m\in{{\Bbb Z}}} {{1 \over 2}}N_m \bigg( t^2_m - \sum_{ -{{1 \over 2}}< k < m-{{1 \over 2}}} h^2_k \bigg).$$ The choice of constants for $f^K$ here may be characterized by saying that in each regular homotopy class, $ f^K(j) = (2-g)h^2_0 $ for $j$ of Lemma [l1]{}. Since $f^U$ is universal, the image of $f^K:I_0\to K$ is not contained in any coset of a proper subgroup of $K$, yet the image of $f^K$ is far from being the whole group $K$, since as we see from the formula, the coefficients of all generators $t^2_m$ are always non-negative. It would be interesting to determine the precise image of $f^U:I_0 \to {{\Bbb G}}_U$. Applications {#ap} ============ We give the following two applications. The first will be used in the second and the second will be used in Section [m]{}. We will use the fact that $\varphi:{{\Bbb G}}_U \to {{\Bbb O}}$ is not surjective to obtain identities on immersions: Let $\theta_0,\theta_1 : {{\Bbb O}}\to {{\Bbb Z}}$ be the homomorphisms defined by: $\theta_0(x_{2m})=1, \theta_0(x_{2m+1})=0, \theta_0(y_m)=-1$ for all $m$ and $\theta_1(x_{2m})=0, \theta_1(x_{2m+1})=1, \theta_1(y_m)=-1$ for all $m$ and so $\theta_0 \circ \varphi = \theta_1 \circ \varphi = 0$. It follows that $\theta_0 \circ k$ and $\theta_1 \circ k$ are constant invariants, which are given explicitly by $\theta_0 \circ k(i) = \sum \chi(U_{2m}) - {1 \over 2}N$ and $\theta_1 \circ k(i) = \sum \chi(U_{2m+1}) - {1 \over 2}N$ where $N=N(i)=\sum N_m(i)$ is the total number of triple points of $i$. To find the value of these constants we need to evaluate them on a single immersion in every regular homotopy class. For the immersion $j$ of Lemma [l1]{}, $\theta_0 \circ k(j)=2-g$ and $\theta_1 \circ k(j)=1-g$, so we get the following two identities: For any $i \in I_0$, $$\sum_m \chi(U_{2m}) - {1 \over 2}N = 2-g \ \ \ \ \ \ \text{and} \ \ \ \ \ \ \sum_m \chi(U_{2m+1}) - {1 \over 2}N = 1-g.$$ For our second application, let $U:I_0 \to {{\Bbb Z}}$ be the order one invariant defined by $u(U)(H^2_m)=1$, $u(U)(T^2_m)=0$ for all $m$ and $u(U)(H^1_0)=u(U)(Q^2_0)=0$. By the 7-step procedure we will have $u(U)(H^2_m)=u(U)(E^2_m)=-u(U)(E^0_m)=1$ for all $m$, $u(U)(H^1_m)=u(U)(E^1_m)=0$ for all $m$, and $u(U)(T^a_m)=u(U)(Q^a_m)=0$ for all $a,m$. That is, for any $i,j \in I_0$, $U(j)-U(i) \in {{\Bbb Z}}$ is the signed number of *un*-matching tangencies occurring in any regular homotopy from $i$ to $j$ (thus the name $U$ for this invariant) where each such tangency is counted as $\pm 1$ according to its permanent co-orientation and the prescription $u(U)(H^2_m)=u(U)(E^2_m)=-u(U)(E^0_m)=1$. Following the definition of $U$ we define $\eta : K \to {{\Bbb Z}}$ on generators as follows: $\eta(h^2_m)=1$ and $\eta(t^2_m)=0$ for all $m$. Then $u(U) = u(\eta \circ f^K)$ and so (up to choice of constants) $U=\eta \circ f^K$. So from our formula for $f^k$ we get an explicit formula for $U$: $$U(i)=\sum_{m\in{{\Bbb Z}}} \chi(U_m) {\lfloor}{m+2 \over 2} {\rfloor}- \sum_{m \in{{\Bbb Z}}} {1\over 2}m N_m .$$ Again we may characterize the choice of constants by saying that $U(j)=2-g$ for $j$ of Lemma [l1]{} We denote $U(i,j)=U(j)-U(i)$. For two regularly homotopic embeddings $e,e':F \to {{{\Bbb R}^3}}$ we would like to compute $U(e,e')$. For $e:F \to {{{\Bbb R}^3}}$ an embedding let $c(e)\in {{\Bbb Z}}$ be the degree of the points in the compact side of $i(F)$ in ${{{\Bbb R}^3}}$, so $c(e)=\pm 1$. We have $U(e) = (2-g) + (1-g) {\lfloor}{c(e) + 2 \over 2} {\rfloor}$ and so $$U(e,e') = U(e')-U(e) = (1-g)\bigg({\lfloor}{c(e') + 2 \over 2} {\rfloor}- {\lfloor}{c(e) + 2 \over 2} {\rfloor}\bigg) = (1-g){{\epsilon}}(e,e')$$ where ${{\epsilon}}(e,e')$ is $0$ if $c(e)=c(e')$, is $1$ if $c(e)=-1,c(e')=1$ and is $-1$ if $c(e)=1,c(e')=-1$. Now for $i \in I_0$ and $h:F \to F$ a diffeomorphism such that $i$ and $i \circ h$ are regularly homotopic, we would like to compute $U(i,i\circ h)$. If $h$ is orientation preserving then from the formula we have for $U(i)$ it is clear that $U(i)=U(i \circ h)$ and so $U(i,i \circ h)=0$. Now let $h:F \to F$ be orientation reversing. If $p \in {{{\Bbb R}^3}}- i(F)$ then $d_p(i \circ h) = -d_p(i)$ and if $p\in{{{\Bbb R}^3}}$ is a triple point of $i$ then $d_p(i \circ h) = 3-d_p(i)$ and so we get: $$U(i \circ h) - U(i)= \sum_m \chi(U_m(i)) ({\lfloor}{-m+2 \over 2} {\rfloor}- {\lfloor}{m+2 \over 2} {\rfloor}) - \sum_m {1\over 2}(3-m - m) N_m(i) .$$ Using the two identities from the beginning of this section and the fact that ${\lfloor}{-m+2 \over 2} {\rfloor}- {\lfloor}{m+2 \over 2} {\rfloor}= -2{\lfloor}{m+2 \over 2} {\rfloor}+ k(m)$ where $k(m)$ is 2 for $m$ even and 1 for $m$ odd, we get: $$U(i, i\circ h) = (1-g) + 2\bigg(2-g-U(i)\bigg).$$ Note that the $U(i)$ appearing here on the right, stands for our specific formula for the invariant $U$, and not for the abstract invariant which is defined only up to a constant. This equality for $h$ orientation reversing can be interpreted as $U(i,i\circ h) = U(j,j\circ h) + 2U(i,j)$ for $j$ of Lemma [l1]{}, offering another way for proving the equality. Let $\widehat{U}:I_0\to {{\Bbb Z}}/2$ be the mod 2 reduction of $U$. The reduction mod 2 of the above results reads as follows: For embeddings $e,e':F \to {{{\Bbb R}^3}}$, $\widehat{U}(e,e')=(1-g)\widehat{{{\epsilon}}}(e,e')$ where $\widehat{{{\epsilon}}}(e,e')\in{{\Bbb Z}}/2$ is 0 if $c(e)=c(e')$ and is 1 if $c(e) \neq c(e')$. For $h:F\to F$ a diffeomorphism such that $i$ and $i\circ h$ are regularly homotopic, $\widehat{U}(i,i\circ h)=(1-g){{\epsilon}}(h)$ where ${{\epsilon}}(h) \in {{\Bbb Z}}/2$ is 0 if $h$ is orientation preserving and is 1 if $h$ is orientation reversing. Proof of formula for $M$ {#m} ======================== For $i \in I_0$ and $h:F \to F$ a diffeomorphism such that $i$ and $i \circ h$ are regularly homotopic, let $M'(i,i\circ h)$ denote our proposed formula for $M(i,i\circ h)$ presented in Section [st]{}. So we must show that indeed $M(i,i\circ h)=M'(i,i\circ h)$. Similarly, for regularly homotopic embeddings $e,e':F \to {{{\Bbb R}^3}}$, let $M'(e,e')$ denote the proposed value for $M(e,e')$ presented in Section [st]{}, so we must show $M(e,e')=M'(e,e')$. In [a]{} it is shown that $Q(i,i\circ h) = M'(i,i\circ h) + (1-g){{\epsilon}}(h)$ and $Q(e,e')= M'(e,e') + (1-g)\widehat{{{\epsilon}}}(e,e')$. In view of the concluding paragraph of Section [ap]{}, this means $Q(i,i\circ h) = M'(i,i\circ h) + \widehat{U}(i,i\circ h)$ and $Q(e,e')= M'(e,e') + \widehat{U}(e,e')$. So showing $M=M'$ in these two settings is equivalent to showing $Q=M+\widehat{U}$ in these settings, which means that the number mod 2 of quadruple points occurring in any regular homotopy between such two immersions or embeddings, is equal to the number mod 2 of all tangencies occurring (matching and un-matching). So, it remains to prove the following: \[p2\] Let $i,j \in I_0$ such that either there is a diffeomorphism $h:F\to F$ such that $j= i \circ h$ or $i,j$ are both embeddings. Then in any regular homotopy between $i$ and $j$, the number mod 2 of quadruple points occurring, is equal to the number mod 2 of tangencies occurring. For a closed 3-manifold $N$ and stable immersion $f:N \to {{\Bbb R}}^4$, there is defined a closed surface $S_f$ and immersion $g:S_f \to {{\Bbb R}}^4$ such that the image $g(S_f) {\subseteq}{{\Bbb R}}^4$ is precisely the multiple set of $f$. It is shown in [ec]{} that the number mod 2 of quadruple points of $f$ is equal to $\chi(S_f) \bmod 2$. Now let $i,j:F \to {{{\Bbb R}^3}}$ be as in the assumption of this proposition and let $H_t:F \to {{{\Bbb R}^3}}$, $0 \leq t \leq 1$, be a regular homotopy with $H_0=i$, $H_1=j$. We define an immersion $f:F \times [0,1] \to {{{\Bbb R}^3}}\times [0,1]$ by $f(x,t) = (H_t(x) , t)$. In case $i,j$ are embeddings we continue $f$ into ${{\Bbb R}}^4 = {{{\Bbb R}^3}}\times {{\Bbb R}}$ and construct a closed 3-manifold $N$ by attaching two handle bodies to $F \times [0,1]$, glued so that $f$ can be extended to embeddings of these handle bodies into ${{{\Bbb R}^3}}\times (-\infty ,0]$ and ${{{\Bbb R}^3}}\times [1,\infty)$. We thus obtain an immersion $\bar{f}:N\to{{\Bbb R}}^4$ with self intersection being precisely the original self intersection of $F \times [0,1]$. The projection ${{{\Bbb R}^3}}\times [0,1] \to [0,1]$ induces a Morse function on $S_{\bar{f}}$ with singularities precisely wherever a tangency CE occurs in the regular homotopy $H_t$, and so by Morse theory $\chi(S_{\bar{f}})$ is equal mod 2 to the number of tangencies. By [ec]{} then, the number mod 2 of quadruple points of $H_t$ which is the number mod 2 of quadruple points of $\bar{f}$ is equal to the number mod 2 of tangencies. In case $j = i \circ h$, let $N$ be the 3-manifold obtained from $F \times [0,1]$ by gluing its two boundary components to each other via $h$ so that there is induced an immersion $\bar{f}:N \to {{{\Bbb R}^3}}\times S^1$. Composing $\bar{f}$ with an embedding of ${{{\Bbb R}^3}}\times S^1$ in ${{\Bbb R}}^4$, we see again that the number of quadruple points of $H_t$ is equal mod 2 to $\chi(S_{\bar{f}})$ which is equal mod 2 to the number of tangencies of $H_t$. We remark that one can prove the formulae for $M$ presented in Section [st]{} directly, without resorting to the result of [ec]{}, by going along the lines of [a]{}. Proposition [p2]{} would then be obtained as a corollary. [StoPC]{} M.A. Asadi-Golmankhaneh and P.J. Eccles: “Double point self-intersection surfaces of immersions.” *Geometry and Topology* 4 (2000) 149-170. T. Nowik: “Quadruple points of regular homotopies of surfaces in 3-manifolds.” *Topology* 39 (2000) 1069-1077. T. Nowik: “Automorphisms and embeddings of surfaces and quadruple points of regular homotopies.” *Journal of Differential Geometry* 57 (2001) 421-455. T. Nowik: “Order one invariants of immersions of surfaces into 3-space.” - Preprint - (may be viewed at: http://www.math.biu.ac.il/$\sim$tahl). T. Nowik: “Higher order invariants of immersions of surfaces into 3-space.” - Preprint - (may be viewed at: http://www.math.biu.ac.il/$\sim$tahl). U. Pinkall: “Regular homotopy classes of immersed surfaces.” *Topology* 24 (1975) No.4, 421–434. [^1]: Partially supported by the Minerva Foundation

--- abstract: | ola We derive necessary and sufficient conditions for local unitary (LU) operators to leave invariant the set of 1-qubit reduced density matrices of a multi-qubit state. LU operators with this property are tensor products of [*cyclic local*]{} operators, and form a subgroup, the centralizer subgroup of the set of reduced states, of the Lie group $SU(2)^{\otimes n}$. The dimension of this subgroup depends on the type of reduced density matrices. It is maximum when all reduced states are maximally mixed and it is minimum when none of them is maximally mixed. For any given multi-qubit state, pure or mixed, we compute the LU operators that fix the corresponding reduced density matrices and determine the equivalence class of the given state.\ \ PACS number(s) 03.67.Mn, 03.65.Aa, 03.65.Ud. author: - | A. M. Martins\ Instituto Superior Técnico, 1049-001 Lisboa, Portugal title: Invariance of reduced density matrices under Local Unitary operations --- Introduction ============ Measuring and classifying quantum entanglement has been the object of extensive research work. The motivations are related to applications in quantum information and computation tasks [@Bennett1992; @Bennett1993] as well as to the foundations of quantum physics [@Ariano2010; @Pusey2012]. An exhaustive bibliography about these different aspects can be found in a recent review article by Horodecki and al. [@Horodecki2009] A very fruitful approach to understand entanglement, was launched by the seminal work of Linden and al. [@Popescu1997; @Popescu1998] who first used group-theoretic methods to classify entanglement in multi-qubit systems through their classes of local unitary (LU) equivalent states. Two quantum states that can be transformed into each other by LU operations, have the same amount of entanglement and are characterized by local polynomial invariants [@Grassl1998; @Sudbery2001]. Among all possible local unitary operations that can be applied to a subsystem of a quantum system, there are the [*cyclic operations*]{} [@Fu2005], that fix the corresponding reduced state. These operations originate nonlocal effects in the global quantum state of the system that may distinguish product states from classically correlated states. Based on these operations new entanglement measures have been proposed [@Gharibian2008; @Illuminati2011]. LU operations that fix reduced states, leave also invariant local measurements (LM). We say that two quantum states are LM-equivalent when they have the same set of 1-party reduced density matrices. In this work we answer the following question: Given an $n$-qubit input state $\rho$, pure or mixed, what is the set of states to which it can be converted by LU operations that leave invariant the corresponding reduced states? This is, what is the set of states that are LU and LM equivalent to $\rho$? We say that a state, $\rho_U$ is LU-equivalent to $\rho$ if $\rho_U =U \rho U^{\dag}$, with $U= \otimes_j^{n} U_j ( \in G )$, where $U_j \in SU(2)$, is the unitary operator acting in qubit $j$ and $G= SU(2)^{\otimes n}$, is the local unitary group. Each equivalence class of LU-equivalent states is an orbit of this group. We say that $\rho_U$ is LM-equivalent to $\rho$ when their set $S$, of reduced states, is the same, i.e, ${\bf T}_{(j)} (\rho_U) = {\bf T}_{(j)} (\rho) =\rho_j \,\ ( i=1,..,n)$, where ${\bf T}_{(j)}$ is the partial trace over all qubits except qubit $j$. The set of local operators $U $ that fix each of the $n$ reduced states $\rho_j$, is the centralizer subgroup of the set $S$. We derive necessary and sufficient conditions for an LU operator to belong to the centralizer subgroup of the set $S$ and identify all possible types of centralizers subgroups. We show that their dimension is directly related with the number of maximally mixed 1-qubit states. We also prove that the operator $U_i$ that fixes any non-maximally mixed reduced state $\rho_i$, is a 1-parameter unitary operator completely determined by the Bloch vector of $\rho_i$. The partial trace operator play a central role in the derivation of the above mentioned results and deserve a place of their own right in this work. We explore the isomorphism existent between the orthogonal complement of the kernel of ${\bf T}_{(j)}$ and the Hilbert space of qubit $j$, to identify the vectors representing the LM-equivalence classes. The paper is organized as follows. In Section 2, we use the partial trace operator to decompose the Hilbert space of the whole system in pairs of complementary subspaces. In Section 3, we define an isomorphism between the reduced density matrices and vectors of the Hilbert space of the $n$-qubits and derive the necessary and sufficient conditions obeyed by a local unitary operator that fix the corresponding reduced state. In Section 4, we compute all possible centralizer subgroups of a set of reduced states and give the explicit form of the LU/LM operators. Finally we conclude in Section 5. The partial trace ================== A suitable choice of the basis set to develop the density matrices may simplify considerably solving specific physical problems, or may help to identify new properties of the system. In this work, where systems are formed by $n$ similar $2$-level constituents, and where the partial trace operators play a determinant role, the natural choice of basis set is the generalized Bloch vector basis. Let ${\cal V}_j$ denote the $4$-dimensional Hilbert space of $2 \times 2$ Hermitian matrices. A convenient basis for ${\cal V}_j$ is ${\cal B}_j= \{\sigma_{\alpha_j} ; {\alpha_j}=0,1,2,3 \}$, where $ \sigma_{\alpha_j}( \alpha_j=1,2,3)$ represents the usual Pauli matrices, and $\sigma_0 = {\bf 1}$, is the $2 \times 2$ identity matrix. Using in ${\cal V}_j$ the Hilbert-Schmidt inner product $(\sigma_{\alpha_i }, \sigma_{\alpha_k}) = Tr\{ \sigma_{\alpha_i }\sigma_{\alpha_k } \} = 2 \delta_{ij} $, then ${\cal B}_j $ is an orthogonal basis set. We are going to consider the set $ {\cal B}_{{\cal V}^{\otimes n}} = \{ \sigma_{\vec \alpha} \}$, where $$\label{vector2} \sigma_{\vec \alpha} = \otimes_{j=1}^{n} \sigma_{\alpha_j}$$ The vector index ${\vec \alpha} =(\alpha_1, \alpha_2,... ,\alpha_n)$ is a $n$-tuple containing the $n$ indices $\alpha_j$. There exist $4^{n}$ such matrices all being traceless, except for $\sigma_{\vec 0 }= {\otimes}_{j=1}^n {\bf 1}_j$, which corresponds to the $2^n \times 2^n $ identity matrix with trace $Tr \{ \sigma_{\vec 0 } \} = 2^{n}$. $ {\cal B}_{{\cal V}^{\otimes n}} $ is an orthogonal basis set of the complex $4^{n}$-dimensional Hilbert-Schmidt vector space ${\cal V}^{\otimes n} =\otimes_{j=1}^n {\cal V}_j $. Every complex square matrix, $(2^{n} \times 2^{n})$, can be seen as a vector $\bf v$, uniquely written in the form $$\label{vector} {\bf v} = \sum_{\vec \alpha} v_{\vec \alpha} \,\ \sigma_{\vec \alpha}$$ where the components $v_{\vec \alpha} $ are given by $$\label{components} v_{\vec \alpha} = \frac{1}{2^n} Tr\{ \sigma_{\vec \alpha} \,\ {\vec v} \}$$ Any $n$-qubit quantum state $\rho=\sum_{\vec \alpha} r_{\vec \alpha} \sigma_{\vec \alpha} \in {{\cal V}^{\otimes n}} $, must be hermitian, $\rho = \rho^{\dag}$, definite positive $\rho \geq 0$, and normalized $Tr \{ \rho \} =1$. These requirements on $\rho$ impose certain constrains to the components $ r_{ \vec \alpha} $: (a) $\forall_{\vec \alpha}, r_{\vec \alpha} \in \Re$, (b) $r_{\vec 0} = \frac{1}{2^n} $, (c) $r_{\vec \alpha} = \frac{1}{2^n} Tr\{ \sigma_{\vec \alpha} \,\ \rho \}$ and (d) $\sum_{ \vec \alpha} r_{ \vec \alpha}^{2} \leq 1 $, the equality is attained for pure states. The translated vector, ${\bar \rho }= \rho - {\bf 1}^{ \otimes n} /2^{n}$,( ${\bf 1}^{ \otimes n} = \otimes_{j=1}^{n} {\bf 1}_j$), characterizes completely the quantum state $\rho$ and is the well known [*generalized Bloch vector representation*]{} of dimension $(4^{n} - 1)$. Let $D_n$ be the set of the $n$-qubit density matrices $\rho$ and let ${\bf T}_{(i)} : {\cal V}^{\otimes n} \rightarrow {\cal V}_i $ be the linear transformation defined by $$\label{partial} {\bf T}_{(i)} (\rho) =Tr_{n/ \{i \}} \{ \rho \} = \rho_i \,\,\,\,\,\,\,\,\,\,\,\,\ ( i=1,...,n)$$ where $Tr_{n/ \{i \}} \{ . \}$ is the partial trace operator over $(n-1)$ qubits, except qubit $i$. This is a surjective map of ${\cal V}^{\otimes n}$, (the vector space of the $n$ qubits), onto ${\cal V}_i \equiv Im({\bf T}_{(i)} ) $, (the vector space of qubit $i$), where $Im({\bf T}_{(i)} )$ is the image space of ${\bf T}_{(i)} $. Let $ {\cal K}_i $ be the kernel of ${\bf T}_{(i)} $ and let ${\cal Q}_i$ be the orthogonal complement of $ {\cal K}_i $, i.e., $$\label{partial} {\cal V}^{\otimes n} = {\cal Q}_i \oplus {\cal K}_i \,\,\ ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ \,\,\ (i=1,...,n)$$ The subspace $ {\cal Q}_i $ is a $4$-dimensional space isomorphic to ${\cal V}_i $. Applying the map ${\bf T}_{(i)} $ to the vectors of the basis set $$\label{basis2} {\cal B}_{{\cal Q}_i} = \{ {\bf b}_{\alpha_i}= \otimes_{k=1}^{i-1} {\bf 1}_k \otimes \sigma_{\alpha_i} \otimes_{k'=i+1}^n {\bf 1}_{k'} ; \,\ \alpha_i = 0,1,2,3\}$$ where $ {\cal B}_{{\cal Q}_i} \subset {\cal B}_{{\cal V}^{\otimes n}} $, we obtain $$\label{correspondance} {\bf T}_{(i)} ({\bf b}_{\alpha_i} )=2^{n-1} \sigma_{\alpha_i}$$ Let ${\cal B}_{{\cal K}_i} = {\cal B}_{{\cal V}^{\otimes n}} \setminus {\cal B}_{{\cal Q}_i} $. The image of any vector of ${\cal B}_{{\cal K}_i} $ under ${\bf T}_{(i)}$ is the zero vector $0_i \in {\cal V}_i $. We conclude that ${\cal B}_{{\cal Q}_i}$ and ${\cal B}_{{\cal K}_i}$ are orthogonal basis sets for the subspaces ${\cal Q}_i$ and ${\cal K}_i$, such that, ${\cal B}_{{\cal V}^{\otimes n}} = {\cal B}_{{\cal Q}_i} \cup {\cal B}_{{\cal K}_i} $. Any density matrix $\rho \in D_n$ can be written in a unique way as $$\label{direct} \rho =\rho_{{\cal Q}_i} + \rho_{{\cal K}_i}$$ where $\rho_{{\cal Q}_i}$ and $ \rho_{{\cal K}_i} $ are the projections of $\rho$ on the subspaces ${\cal Q}_i $ and $ {\cal K}_i $. The projection operator $\pi_{{\cal Q}_i} : {\cal V}^{\otimes n} \rightarrow {\cal Q}_i$ is defined by $$\label{rhoq1} \pi_{{\cal Q}_i} (\rho) = \rho_{{\cal Q}_i} = \sum_{{\alpha_i}=0}^3 \frac{(\rho , {\bf b}_{\alpha_i})}{\parallel {\bf b}_{\alpha_i} \parallel^2 } {\bf b}_{\alpha_i}= \sum_{{\alpha_i}=0}^3 r_{0_1 ... 0_{i-1} \alpha_i 0_{i+1}...0_n}{\bf b}_{\alpha_i}$$ The action of ${\bf T}_{(i)}$ on both sides of eq.(\[direct\]), gives $$\label{rhoq2} {\bf T}_{(i)}(\rho ) ={\bf T}_{(i)}(\rho_{{\cal Q}_i} )= \rho_i$$ Any 1-qubit density matrix $\rho_i \in {\cal V}_i$ can be written in the basis ${\cal B}_i $ in the form $$\label{rhoi} \rho_i = \frac{1}{2} ({\bf 1}_i+ {\vec r}_i . {\vec \sigma}(i)) \,\,\,\,\,\,\,\,\,\,\,\,\ ( i=1,...,n)$$ where, $ {\vec r}_i . {\vec \sigma}(i) = 2^n \sum_{a_i =1}^3 r_{0_1 ... 0_{i-1} a_i 0_{i+1}...0_n} \sigma_{a_i}$, and ${\vec r}_i $ is the Bloch vector of qubit $i$, such that $\| {\vec r}_i \| \leq 1 $. Substituting (\[rhoi\]) in the l.h.s. of eq.(\[rhoq1\]), we obtain the following explicit one-to-one correspondence between the vectors $\rho_i \in {\cal V}_i $ and the vectors $\rho_{{\cal Q}_i} \in {\cal Q}_i$, $$\label{rhoq5} \rho_{{\cal Q}_i} = \frac{1}{2^{n-1}} \otimes_{k=1}^{i-1} {\bf 1}_k \otimes \rho_i \otimes_{k'=i+1}^n {\bf 1}_{k'}$$ The translated vector $$\label{rhoqj2} {\bar \rho}_{{\cal Q}_i} = \rho_{{\cal Q}_i} - \frac{1}{ 2^n}{\bf 1}^{ \otimes n}= \otimes_{k=1}^{i-1} {\bf 1}_k \otimes [ {\vec r}_i . {\vec \sigma}(i)] \otimes_{k'=i+1}^n {\bf 1}_{k'}$$ contains the same quantum information as the reduced density matrix $\rho_i$, this is, the quantum state of qubit $i$, is fully represented by the vector ${\bar \rho}_{{\cal Q}_i} \in {\cal V}^{\otimes n}$. Moreover, $$\label{orto} ( {\bar \rho}_{{\cal Q}_i} ,{\bar \rho}_{{\cal Q}_j} ) = \| {\bar \rho}_{{\cal Q}_i} \|^2 \delta_{ij}$$ i.e., vectors ${\bar \rho}_{{\cal Q}_i} $ associated to different qubits are orthogonal to each other and to any other vectors of the basis set $ {\cal B}_{{\cal V}^{\otimes n}} $. The norm, $\| {\bar \rho}_{{\cal Q}_i} \| = [ \lambda_{-}^2 (i) +\lambda_{+}^2 (i) ]^{1/2} $, where, $\lambda_{\mp}$, are the eigenvalues of ${\bar \rho}_{{\cal Q}_i}$. It is now obvious that any quantum state $\rho \in D_n$ can be written in the form, $$\label{rho4} \rho = \frac{1}{2^n} {\bf 1}^{ \otimes n} + \sum_{i=1}^{n} {\bar \rho}_{{\cal Q}_i} + \Delta$$ where $\Delta$ refers to the terms of $\rho$ containing all possible $k$-partite correlations ($ 2 \leq k \leq n $) existing between the $n$-qubits. Reduced states and equivalence classes ======================================= The possible outcomes of the measurement of any local observable $ {\hat A}_j \in {\cal V}_j $, performed on qubit $j$, are given by the eigenvalues $a_k(j)$ of the operator ${\hat A}_j $. The expectation value of this measurement, when the system is in the state $\rho$, is given by $$\label{expectation} \langle {\hat A}_j \rangle = Tr \{{\hat A}_j \rho \} = Tr_j \{ {\hat A}_j \rho_j \}$$ where $Tr_j \{ \} $ is the trace in qubit $j$ and $Tr \{ \}$ is the trace in all qubits. This equality shows that measurements performed on qubit $j$ give the same result as if it would be in the reduced state $\rho_j = {\bf T}_{ (j) } (\rho) $. A imediate consequence of eq.(\[expectation\]) is that different global quantum states $\rho$ with equal reduced states $ \rho_j $ have equal $1$-qubit expectation values $\langle {\hat A}_j \rangle $. When two states $\rho $ and $\rho^{'} $ have the same image $\rho_j$, under the map ${\bf T}_{ (j) }$, their difference belong to the kernel ${\cal K}_j$, i.e., they are congruent modulo ${\cal K}_j$. The set of all states with reduced state $\rho_j$ forms a LM$_j$-equivalence class $ C_j$, this is, $$\label{class} C_j = \{ \rho \in D_n : {\bf T}_{ (j) } (\rho) =\rho_j \}$$ The set of all LM$_j$-equivalence classes is the quotient space ${\cal V}^{\otimes n}/ {\cal K}_j$. We have shown that for any quantum state $\rho$, there is a one-to-one correspondence between its reduced state $\rho_j$ and its projection $\rho_{{\cal Q}_j }$. This enables us to define a linear map $\psi_j: {\cal V}^{\otimes n}/ {\cal K}_j \rightarrow {\cal Q}_j $, such that $$\label{class} \psi (C_j )= \rho_{{\cal Q}_j }$$ assigns to each class $C_j \in {\cal V}^{\otimes n}/ {\cal K}_j$ the vector $\rho_{{\cal Q}_j } \in {\cal Q}_j$, we say that the vector $\rho_{{\cal Q}_j } \in {\cal Q}_j$, is the representative state of the class $C_j $ and we may write [@Martins2008] $$\label{class1} C_j = \{ \rho \in D_n : \rho =\rho_{{\cal Q}_j } + {\cal K}_j \}$$ The set of all $n$-qubit density matrices, such that their reduced density matrices belong to $S= \{ \rho_i = {\bf T}_{ (i) } (\rho), \,\ ( i=1,...,n ) \}$, is given by the intersection of the equivalence classes $C_i$, i.e., $$\label{class2} {\bar C} = \bigcap_{i=1}^{n} C_i = \{ \rho \in D_n : \rho =\rho_{{\cal Q}_i } + {\cal K}_i \,\ ; i=1,...,n \}$$ saying it in another way, quantum states in the set ${\bar C}$ have their $\rho_{{\cal Q}_i}$ projections in the set $$\label{set3} {\bar S} = \{ \rho_{{\cal Q}_i}= \pi_{{\cal Q}_i}(\rho) \,\ ; i=1,...,n \}$$ The set ${\bar S}$ is isomorphic to the set $S$ therefore, they have the same content of quantum information. This isomorphism is particularly useful when we are studying local properties of the qubits because, instead of working with the $n$ Hilbert spaces ${\cal V}_i$, we can use the original Hilbert space ${\cal V}^{\otimes n}$ of the $n$-qubits. The unitary transformation $U \in G$ acts on a $n$-qubit state $\rho$ via the adjoint action, $$\label{LU} \rho_U = ad \,\ U [ \rho] = U \rho U^{\dag} = \left( \otimes_{j=1}^{n} U_j \right) \rho \left( \otimes_{j=1}^{n} U_j^{\dag} \right)$$ where $G= SU(2)^{\otimes n}$ is a $3n$-dimensional Lie group and ${\cal L} = su(2) \oplus su(2) \oplus ... \oplus su(2) $ is the corresponding Lie algebra. The set ${\cal B}_{\cal L} =\{ {\bf b}_{a_i} \in {\cal B}_{{\cal Q}_i}; \,\ a_i =1,2,3 $ and $ i=1,...,n \}$ is a basis set for ${\cal L}$ whose elements are the generators of $G$. In this work we are looking for all quantum states $\rho_U $, LU equivalent to $\rho$, such that measurements of any local observable ${\hat A}_i$ are not able to distinguish between $\rho$ and $\rho_U$. Having in mind eq.(\[expectation\]), we are looking for states $\rho_U $ with the same set $S$ of 1-qubit reduced density matrices. This is, $$\label{trace2} {\bf T}_{(i)} (\rho_U) = {\bf T}_{(i)} (\rho) =\rho_i ; \,\,\,\,\ i=1,...,n$$ or, given the one-to-one correspondence between $\rho_i$ and $\rho_{{\cal Q}_i}$, the LU equivalent states are such that $$\label{trace3} \pi_{{\cal Q}_i} (\rho_U) = \pi_{{\cal Q}_i} (\rho) = \rho_{{\cal Q}_i} ; \,\,\,\,\ i=1,...,n$$ i.e., the states $ \rho_U $ belong to the set ${\bar S}$. Not all adjoint actions of local unitary operators $U$ on $\rho$ obey this condition, however all local unitary operators leave the subspaces ${\cal Q}_j$ and ${\cal K}_j$ invariant, as we prove in the next Theorem. [**Theorem 1:**]{} [*The subspaces ${\cal Q}_j$ and ${\cal K}_j$ are invariant under LU transformations.*]{} [**Proof:**]{} Any vector ${\bf v}_{{\cal Q}_j} \in {\cal Q}_j$ has the form ${\bf v}_{{\cal Q}_j} = \otimes_{k=1}^{i-1} {\bf 1}_k \otimes [ \sum_{\alpha_i=0}^{3}v _{\alpha_i } \sigma_{\alpha_i} ] \otimes_{k'=i+1}^n {\bf 1}_{k'} $. The adjoint action of $U $ on $\rho$ is $$\label{invariant} ad \,\ U [ {\bf v}_{{\cal Q}_j} ] = U {\bf v}_{{\cal Q}_j} U^{\dag} =\otimes_{k=1}^{i-1} {\bf 1}_k \otimes [ \sum_{\alpha_i=0}^{3}v _{\alpha_i } U_i \sigma_{\alpha_i} U_i^{\dag} ] \otimes_{k'=i+1}^n {\bf 1}_{k'}$$ As $$\label{rotation} \sum_{\alpha_i=0}^{3}v _{\alpha_i } U_i \sigma_{\alpha_i} U_i^{\dag} = \sum_{\alpha_i=0}^{3}v _{\alpha_i }^{'} \sigma_{\alpha_i}$$ then $ U {\bf v}_{{\cal Q}_j} U^{\dag} \in {\cal Q}_j$. When ${\cal Q}_j$ (or ${\cal K}_j$ ) is invariant under a unitary transformation so is the complementary subspace ${\cal K}_j$ (or ${\cal Q}_j$) [@Halmos1987]. $\Box$ [**Corollary 1:**]{} [*The subspace ${\bar {\cal K}} = \cap_{j=1}^{n} {\cal K}_j$ is invariant under the local adjoint action.*]{} [**Corollary 2:**]{} [*The projection operator $\pi_{{\cal Q}_i}$ commutes with any LU transformation, i.e.,*]{} $$\label{comutador2} \pi_{{\cal Q}_i} (U \rho U^{\dag}) = U [ \pi_{{\cal Q}_i} ( \rho ) ] U^{\dag} = U \rho_{{\cal Q}_i} U^{\dag}$$ [**Proof:** ]{} If a subspace is invariant under a linear transformation $U$ then $U$ commutes with every projection operator on that subspace [@Halmos1987]. $\Box$ This corollary shows that $\pi_{{\cal Q}_i} ( \rho_U) =U \rho_{{\cal Q}_i} U^{\dag}$. Imposing now the constrain of eq.(\[trace3\]), i.e., that $ \rho_U $ has the same set of 1-qubit reduced density matrices as $\rho$, we conclude that the LU transformations we are looking for, are such that $$\label{inv} U \rho_{{\cal Q}_i} U^{\dag} = \rho_{{\cal Q}_i} \,\,\ ; \,\,\,\ i=1,...,n$$ $\rho_{{\cal Q}_i}$ is invariant under LU transformations. Local unitary operators $U \in G$ obeying condition (\[inv\]), for all elements of the set ${\bar S} $, belong to the centralizer subgroup $C_G ({\bar S})$ of the set ${\bar S}$, i.e. $$\label{centralizergroup} C_G ({\bar S}) = \{U \in G : U \rho_{{\cal Q}_i} U^{\dag} = \rho_{{\cal Q}_i}, \forall_{\rho_{{\cal Q}_i} \in {\bar S}} \}$$ Next theorem sets the conditions obeyed by the local unitary transformations $U_i $ in order that equality (\[inv\]) holds. [**Theorem 2 :**]{} [*A state $\rho_U $, LU equivalent to $\rho$, has the same set $S$ of reduced density matrices as $\rho$, iff each local unitary operator $U_i \in SU(2)$ commutes with $\rho_i$, i.e.,*]{} $$\label{comutador} [U_i , \rho_i]=0 \,\,\ , \,\,\ i=1,...,n$$ [**Proof:** ]{} By Corollary 1, $$\label{trans} \pi_{{\cal Q}_i} (U \rho U^{\dag}) =U \rho_{{\cal Q}_i} U^{\dag} = \otimes_{k=1}^{i-1} {\bf 1}_k \otimes U_i \rho_i U_i^{\dag} \otimes_{k'=i+1}^n {\bf 1}_{k'}$$ The condition (\[inv\]) is verified when, $U_i \rho_i U_i^{\dag} = \rho_i $ for each qubit $i$. This is equivalent to equality (\[comutador\]). $ \Box$ Theorem 2 refers to these multi-qubit LU operations and proves that the cyclic property is a necessary and sufficient condition for invariance of any number of reduced states. Moreover, local unitary operators acting in different qubits $i$ and $j$, commute with each other, i.e., $ [ U_i , U_j ] = 0$. In conclusion, the general form of any quantum state $\rho_U$, LU equivalent to $\rho$, and with the same set $S$ of 1-qubit reduced density matrices is given by $$\label{flu} \rho_U = \frac{1}{2^n} {\bf 1}^{ \otimes n} + \sum_{i=1}^{n} {\bar \rho}_{{\cal Q}_i} + U \Delta U^{\dag}$$ where the operators $U$ belong to the centralizer subgroup $C_G ({\bar S})$. The problem of finding $\rho_U$ in the last equation is solved when the centralizer subgroup of a state $\rho$ is known. We call $\rho $-family and denote by ${\cal F}_{\rho}$, the set of states $\rho_U$ given by eq.(\[flu\]). The elements of this family have the same type of entanglement but are not distinguishable by local measurements. Not all states in the LU-orbit of $\rho$ belong to ${\cal F}_{\rho}$. Next proposition is a criterium to decide wether a state $\rho^{'}$, is not in the family ${\cal F}_{\rho}$. [**Proposition 1:**]{} A state $\rho^{'}$, LM-equivalent to the state $\rho$, does not belong to the family ${\cal F}_{\rho}$, if $$\label{cond} Tr\{ \rho^{' 2} \} \neq Tr\{ \rho^{2} \} \,\,\,\,\ \mbox{or if} \,\,\,\,\ Tr\{ \Delta^{' 2} \} \neq Tr\{ \Delta^{2} \}$$ Centralizers subgroups and LU/LM-equivalence ============================================ In this section we show that the translated vectors $ {\bar \rho}_{{\cal Q}_i}$, present in eq.(\[rho4\]), determine the centralizer subgroup $C_G ({\bar S})$ and the set of states LU/LM equivalent to each quantum state $\rho$. Any generic local unitary operator $U_j \in SU(2)$ is a three real continuous parameter operator and can be written in the form $$\label{uni} U_j ( \phi_j , \theta_j , \omega_j )= e^{i {\vec s}_j .{\vec \sigma} (j) } = \cos ( \omega_j ) {\bf 1}_j +i \sin (\omega_j ) {\hat n}_{ {\vec s}_j} . {\vec \sigma} (j)$$ where ${\hat n}_{ {\vec s}_j} = {\vec s}_j / \parallel {\vec s}_j \parallel \equiv ( \cos \phi_j \sin \theta_j , \sin \phi_j \sin \theta_j, \cos \theta_j )$ is a unit vector in the 3-dimensional Euclidian space (Bloch space of qubit $j$), parametrized by the azimuthal angle, $0 \leq \phi_j \leq 2 \pi $, and the polar angle, $0 \leq \theta_j \leq \pi $. The third parameter is $\omega_j = \parallel {\vec s}_j \parallel$ ($ 0 \leq \omega_j \leq \pi /2$) is the length of the vector ${\vec s}_j$. Any 1-qubit density matrix can be written in the form (\[rhoi\]). When ${\vec r}_j =0$, then the 1-qubit density matrix is maximally mixed, i.e., $\rho_j^{*} =\frac{1}{2}{\bf 1}_j $, and any local unitary operation $U_j = e^{i {\vec s}_j .{\vec \sigma} (j) }$ commutes with $\rho_j$. When ${\vec r}_j \neq 0$, then condition (\[comutador\]) is verified when ${\vec s}_j = \xi_j {\vec r}_j $ (see Appendix), with $\xi_j \in \Re$. The corresponding [*local cyclic*]{} operator is $$\label{uni2} U_j (\xi_j )= e^{i \xi_j {\vec r}_j .{\vec \sigma} (j) } = \cos ( \omega_j ) {\bf 1}_j + i \sin (\omega_j ) {\hat n}_{ {\vec r}_j} . {\vec \sigma} (j)$$ a single parameter unitary operator, where $ \omega_j = \xi_j \parallel {\vec r}_j \parallel $ is the continuous parameter. The direction ${\hat n}_{ {\vec r}_j}$ is fixed by the cyclic condition (\[comutador\]). Varying continuously the parameter $\xi_j $, in eq.(\[uni2\]), between, $0$, and, $\pi /2 \parallel {\vec r}_j \parallel $, then $U_j $ varies between $ {\bf 1}_j $ and $ ( i {\hat n}_{ {\vec r}_j} . {\vec \sigma} (j) )$. Invoking the local isomorphism between SU(2) and SO(3) we see that the unitary operator $U_j$, of eq.(\[uni2\]), represents a rotation of an angle $\omega_j$ around the vector ${\vec r}_j $ of the Bloch sphere of qubit $j$, which leaves this vector and the corresponding $\rho_j$ invariants. In the generalized $(4^{n}-1)$ Bloch vector space, the vectors ${\bar \rho}_{{\cal Q}_j}$, of different qubits, are orthogonal to each other. Local unitary operations of $SU(2)^{\otimes n}$ of the type $$\label{uni3} U = \otimes_{j=1}^{m} e^{i \xi_j {\vec r}_j .{\vec \sigma} (j) } \otimes_{l=m+1}^{n} {\bf 1}_l \,\,\,\, (m=1,...,n)$$ correspond to $m$ independent rotations of the the group $SO(3)$ around each vector ${\bar \rho}_{{\cal Q}_j}$. These results reveal the intimate relation between the set $S$ of the 1-qubit reduced density matrices and the centralizer subgroup $C_G ({\bar S})$ of the set ${\bar S}$. Different cases are possible. [**Case 1:**]{} When $ {\bar \rho}_{{\cal Q}_i} =0 $, for all $i=1,...,n$, then eq.(\[rho4\]) reduces to $$\label{rho5} \rho = \frac{1}{2^n} {\bf 1}^{ \otimes n} + \Delta$$ and $\rho$ have maximally mixed 1-qubit reduced states, i.e., $$\label{set1} S= \{ \rho_i^{*}= \frac{1}{2} {\bf 1}_i \,\ ; \,\ i=1,...,n\}$$ The centralizer subgroup $C_G ({\bar S})$ is the entire $G$ whose dimension is $3n$. The states $\rho_U $, are given by $$\label{rho6} \rho_U = \frac{1}{2^n} {\bf 1}^{ \otimes n} + U \Delta U^{\dag}$$ where $ U =\otimes_{j=1}^{n} e^{i{\vec s}_j .{\vec \sigma} (j) } $. As each $U_j$ only acts on qubit $j$ then $U \Delta U^{\dag}$ can be easily computed by replacing each Pauli matrix $\sigma_{\alpha_j}$ in $\Delta$ by $U_j \sigma_{\alpha_j} U_j^{\dag}$. Any n-qubit Werner state $\rho^{\cal W}$, has maximally mixed 1-qubit reduced states. All states in the LU-orbit of a $\rho^{\cal W}$ are LM-equivalent. The maximally mixed state $\rho^{*}= \frac{1}{2^n} {\bf 1}^{ \otimes n} $ is a special type of Werner state. When all local unitary operators are equal, i.e., $U_j =e^{i{\vec s} .{\vec \sigma} (j) }$ (independent o $j$) the state $\rho^{\cal W}$ is transformed into itself. The $n$-GHZ entanglement class has maximally mixed 1-qubit density matrices. All states in this class are LM-equivalent. [**Case 2:**]{} When, in eq.(\[rho4\]), there are $m < n$ operators $ {\bar \rho}_{{\cal Q}_k} \neq 0, (k=1,...,m) $ and the remaining $(n-m)$ operators are $ {\bar \rho}_{{\cal Q}_l} =0, (l=m+1,...,n) $, then $$\label{rho7} \rho = \frac{1}{2^n} {\bf 1}^{ \otimes n} + \sum_{i=1}^{m} {\bar \rho}_{{\cal Q}_i} + \Delta$$ The corresponding 1-qubit reduced density matrices belong to the set $$\label{set2} S=\{ \rho_k =\frac{1}{2}{\bf 1}_k + {\vec r}_k . {\vec \sigma} (k) ; \,\ ( k=1,...,m) \wedge \rho_l^{*} =\frac{1}{2}{\bf 1}_l ; ( l=m+1,...,n) \}$$ The centralizer subgroup $C_G ({\bar S})$ is $$\label{centralizergroup2} C_G ({\bar S}) = \{U \in G : U = \otimes_{k=1}^{m}e^{i\xi_l {\vec r}_l .{\vec \sigma} (l) } \otimes_{l=m+1}^{n} e^{i {\vec s}_k .{\vec \sigma} (k) } \}$$ Different values of $m$ give rise to different centralizer subgroups with dimension $dim[ C_G ({\bar S})] =3n -2m$, the same as the number of independent continuous parameters. When $m=n$, $dim[ C_G ({\bar S})] =n$, this number is minimum. When $m=0$, then $dim[ C_G ({\bar S})] =3n $ is maximum and the centralizer subgroup is the entire $G$ (Case 1). The states $\rho_U $, of the family ${\cal F}_{\rho}$ are obtained by replacing $\Delta$ in eq.(\[rho7\]) by $U \Delta U^{\dag}$, where $U $ belong to the stabilizer subgroup of eq.(\[centralizergroup2\]). Biseparable states $$\label{by} \rho = \rho^{(m)} \otimes \rho^{(n-m)}$$ where $\rho^{(m)}$ is any state of $m$-qubits and the remaining $(n-m)$-qubits are in a state $\rho^{(n-m)}$, (for instance, $(n-m)$-GHZ or $(n-m)$-Werner states), have centralizers subgroups of the form (\[centralizergroup2\]). A product state $$\label{product} \rho = \otimes_{k=1}^{m} \rho_k \otimes_{l=m+1}^{n} \rho_l^{*}$$ is a special case of biseparable state. The $\rho$-family of a product state is the product state itself. Concluding remarks ================== We have investigated a special type of local unitary operations that fix the set of reduced states of a pure or mixed multi-qubit state. We have shown that the possible forms of the [*cyclic local*]{} transformations is determined by the 1-qubit reduced density matrices. The dimension of the centralizer subgroups of the set reduced states is minimum when no 1-qubit reduced matrix is maximally mixed and it is maximum when all 1-qubit reduced states are maximally mixed. We have shown that local [*cyclic*]{} unitary operations of multiqubit states, with non maximally mixed 1-qubit reductions, are a subgroup of $SU(2)^{\otimes n}$ whose elements are given by the tensorial product of $n$-single parameter unitary operators, this suggests that it is possible to analyze in a continuous way the measures of entanglement and of non-classicality [@Gharibian2012], by varying independently these parameters. Simultaneous application of these local [*cyclic*]{} operations to different qubits goes beyond the bipartite studies [@Fu2005; @Gharibian2008] and may reveal new nonlocal effects. Appendix ======== [**Theorem 3**]{}: [*The commutation relation $ [U_j , \rho_j ]=0 $, where the 1-qubit reduced state is $\rho_j = \frac{1}{2} {\bf 1}_j+ {\vec r}_j . {\vec \sigma}(j)$, with ${\vec r}_j \neq 0$, is verified iff* ]{} $$\label{uni5} U_j (\xi_j )= e^{i \xi_j {\vec r}_j .{\vec \sigma} (j) } = \cos ( \omega_j ) {\bf 1}_j + i \sin (\omega_j ) {\hat n}_{ {\vec r}_j} . {\vec \sigma} (j)$$ [**Proof**]{}: Any local unitary operator has the general form (\[uni\]). The cyclic condition is equivalent to $$\label{comutador2} [ { {\vec s}_j} . {\vec \sigma} (j) , {\vec r}_j . {\vec \sigma}(j) ] = 0$$ Computing the above commutator, we obtain $$\label{comutador3} [ {\vec s}_j . {\vec \sigma} (j) , {\vec r}_j . {\vec \sigma}(j) ] = \sum_{k=1}^{3} \sum_{l=1}^{3} s_{jk} r_{jl} [ \sigma_k (j) , \sigma_l (j) ] = 2 i \{ \sum_{k=1}^{3} \sum_{l=1}^{3} s_{jk} r_{jl} \epsilon_{klu} \} Ê\sigma_r (j)$$ where $\epsilon_{klu}$ is the Levi-Civita symbol. After some straightforward calculations we show that the commutator will be zero iff, ${\vec s}_j = \xi_j {\vec r}_j $. [99]{} C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett.[**69**]{}, 2881 (1992). C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. [**70**]{}, 1895 (1993). G. Chiribella, G. M. D’Ariano, P. Perinotti, Phys. Rev. A [**81**]{}, 062348 (2010). M. F. Pusey, J. Barrett, and T. Rudolph, Nature Phys., [**8**]{}, 476 (2012). R. Horodecki, P. Horodecki, M. 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[**A model of anomalous production of strange baryons in nuclear collisions**]{} [Roman Lietava${}^{a)d)}$, Ján Pišút${}^{a)b)}$, Neva Pišútová${}^{a)}$ and Petr Závada${}^{c)}$]{} [*a) Department of Physics MFF UK, Comenius University, SK-84215 Bratislava, Slovakia\ b) Laboratoire de Physique Corpusculaire, Université Blaise Pascal, Clermont- Ferrand, F-63177 Aubière, Cedex, France\ c) Institute of Physics, Czech Academy of Science, Na Slovance 2, CZ-18040 Praha 8, Czech Republic\ d) CERN,European Laboratory for Particle Physics,Geneva,Switzerland*]{} Introduction {#intro} ============ Enhanced production of strange hadrons has been suggested [@RAF82; @KOCH86] as a signature of Quark- Gluon Plasma (QGP) formation in heavy ion collisions. Enhancement of strangeness production - with respect to pp interactions - has been observed [@NA35AA] in pS, pAg, SS and SAg interactions, but it is in our opinion rather unlikely that QGP has been formed in these collisions. Strangeness enhancement observed in these interactions has been analysed by Werner [@wer], Sorge [@sor] and Tai An and Sa Ben-Hao [@an] in models with collective string interactions and by Capella et al. [@cap] in the dual parton model. When analyzing the production of strange hadrons in Pb+Pb interactions one has to disentangle that part of strangeness enhancement which is given by the extrapolation of trends observed already in collisions of lighter ions from genuine “anomalous” strangeness enhancement due to the production of new form of matter in Pb+Pb collisions. In Refs. [@wer; @sor; @an] the increase of strangeness due to interactions of final state hadrons has been taken into account and found insufficient to describe the observed enhancement without string-string effects. In contradistinction to these authors we use phenomenological parametrization of the production of $u\bar u$, $d\bar d$, $s\bar s$ pairs which will appear in final state hadrons as valence quarks and antiquarks. This parametrization is based on [@LIET97] and on the data analysis by Wroblewski [@WRO85]. We have also taken into account a possible production of anomalous strangeness rich matter in the spirit of Refs. [@BLAIZ; @KHARa; @KHARb] In the present paper we shall study a simple model of the production of strange baryons and antibaryons in proton- nucleus (pA) and nucleus- nucleus (AB) interactions in the CERN SPS energy range. The model is based on the assumption that yields of particles of different types are roughly given by the recombination of quarks and antiquarks formed during the first stage of the nuclear collision. According to this assumption the yields of particles of different type are not strongly influenced by the stage of interacting gas of hadrons. Models based on the idea of recombination have been intensively studied during past 15 years, see e.g. Refs.[@BIRO95]-[@BIZ83]. An essential ingredient of any recombination model is the number of quarks and anti- quarks (future valence quarks) just before the recombination. We shall estimate the number of recombining quarks and antiquarks of the three flavours $u,d,s$ from the analysis of data on production of strange and non- strange hadrons in pA and AB interactions. Compilations of data can be found in Refs. [@LIET97],[@JEON97]-[@MAL83]. In what concerns the strange quarks and antiquarks we shall use the parametrization of our recent work [@LIET97], see also Refs. [@WRO85; @BIALK92], the number of created $u\bar u$ and $d\bar d$ pairs is described by a parametrization of the type used in Ref.[@LIET97] and results of Wroblewski [@WRO85]. A rapid recombination differs from models based on the assumption of a formation of a thermalized system in AB interactions, see e.g. Refs. [@LET95], although the free parameters in models permit to obtain similar results. In particular the baryon chemical potential and parameter of partial chemical thermalization of strange hadrons have their counterparts in the ratio of quarks and antiquarks and the ratio of strange to non-strange quarks before the thermalization. Assuming a rapid recombination implies a rather short pre-hadronic stage in heavy ion collisions, that means that hadrons are formed within about 2fm/c, which is a time interval too short to permit a transfer of quarks and antiquarks over distances in the transverse plane comparable with the dimensions of the system - about 10fm. An indirect support of a rather short pre- hadronic stage in nuclear collisions is provided by the phenomenological success of models by Blaizot and Ollitrault [@BLAIZ] and by Kharzeev, Lourenco, Nardi and Satz [@KHARa; @KHARb] in describing $J/\Psi$ suppression, observed by NA50 collaboration [@NA50a] in Pb+Pb collisions at the CERN SPS. In models of Refs.[@BLAIZ; @KHARa; @KHARb] it is assumed that Quark- Gluon Plasma (QGP) is formed in a part of impact parameter plane, separated from the rest where only hadronic matter is present. Since the separation can hardly remain there for a time of the order of 10 fm/c the existence of the boundary indicates that the existence of QGP or, more generally, of the form of matter which is strongly absorbing $J/\Psi $ is present only for a short time of the order of 1-2 fm/c. The present note is structured as follows. In the Sect.2 we shall describe our parametrization of the numbers of produced pairs $u\bar u$, $d\bar d$ and $s\bar s$ - in addition of valence quarks present in incoming nuclei - in pA and AB interactions. In Sect.3 the rapid recombination scheme is considered. A possible modification of the rapid recombination scenario is briefly mentioned in Sect.4. In Sect.5 we discuss the modifications of the number of created quark pairs due to the presence of “anomalous”, strangeness richer, matter. The last Sect.6 contains comments and conclusions. Parametrization of the production of $<u\bar u>$, $<d\bar d>$ and $<s\bar s>$ pairs in pA and AB interactions {#para} ============================================================================================================= In our work [@LIET97] we have parametrized the production of $<s\bar s>$ pairs in pA and AB interactions as $$N_s^{coll}=<s\bar s>= <s\bar s>_{nn}\sum _{i,j} (1-\beta_s)^{i-1}(1-\beta_s)^{j-1} \label{eq1}$$ where the superscript “coll” indicates that $s\bar s$ pairs are produced in nucleon- nucleon collisions and $$<s\bar s>_{nn}(1-\beta_s)^{i-1}(1-\beta_s)^{j-1}$$ is the number of $<s\bar s>$ pairs produced in the nucleon- nucleon collision which is i-th for the nucleon coming from the left and j-th for the nucleon coming from the right. Values of parameters $<s\bar s>_{nn}$ and $\beta_s$ are according to data excluding those for Pb+Pb collisions [@LIET97] $$\beta_s \approx 0.13,\qquad <s\bar s>_{nn}\approx 0.63\pm 0.08$$ In Ref.[@WRO85] Wroblewski has analyzed the production of future valence quarks and antiquarks in hadron- hadron interactions, obtaining for the total number of valence quarks and antiquarks in pp interactions at 205 GeV/c the result $<N_{q\bar q}> = 7.4\pm 0.6$. Subtracting from that the number of $s\bar s$ pairs and dividing by two to get separately $<u\bar u>_{nn}$ and $<d\bar d>_{nn}$ we obtain $$<d\bar d>_{nn}\approx <u\bar u>_{nn}\approx 3.4\pm 0.4$$ Making use of data on $<h^->$ production in pA and AB interactions compiled by Gazdzicki and Röhrich [@GAZD96] and assuming that the number of quark- antiquark pairs is proportional to $<h^->$ we have obtained $\beta_u =\beta_d \approx 0.4$. In this way we have $$N_u^{coll}=<u\bar u>=<d\bar d>\approx <u\bar u>_{nn}\sum _{i,j}(1-\beta_u)^{i-1}(1-\beta_u)^{j-1} \label{eq2}$$ where $$\beta_u=\beta_d\approx 0.4,\qquad <u\bar u>_{nn}=<d\bar d>_{nn}\approx 3.4\pm 0.4$$ According to the scheme described above we can now write down formulas for the average number of quark- antiquark pairs produced in pA and AB interactions. We shall start with pA collisions at a fixed value of the impact parameter $b$. The number of $s\bar s$ pairs denoted as $N^{coll}_{s\bar s}(pA;b)$ is given as $$N^{coll}_{s\bar s}(pA;b)=<s\bar s>_{nn}\sum _{i=1}^{\mu} (1-\beta_s)^{i-1} =$$ $$=<s\bar s>_{nn}\frac {1-(1-\beta_s)^{\mu}}{\beta_s} \label{eq3}$$ where $$\mu = \sigma \rho 2L_A(b); \qquad L_A(b)=\sqrt {R_A^2 -b^2}$$ Here $R_A$ is the radius of the nucleus A, $b$ is the impact parameter of the collision, $\sigma $ is the inelastic nucleon- nucleon cross- section and $\rho $ is the nuclear density. Similarly for $u\bar u$ and $d\bar d$ pairs $$N^{coll}_{u\bar u}(pA;b)= N^{coll}_{d\bar d}(pA;b)= <u\bar u>_{nn}\sum_{i=1}^{\mu} (1-\beta_u)^{i-1} =$$ $$=<u\bar u>_{nn}\frac {1-(1-\beta_u)^{\mu}}{\beta_u} \label{eq4}$$ where the notation is the same as in Eq.(3). In addition to quark- antiquark pairs created in the interactions there are also valence quarks of incoming nucleons. Averaging over $d$ and $u$-quarks we have 1.5 of $d$-quark and 1.5 of $u$-quark for each participating nucleon. This gives $$N^{val}_{u}= N^{val}_{d}=1.5(\mu +1); \qquad N^{val}_{s} =0 \label{eq5}$$ The total number of quarks and antiquarks taking part in the recombination is given as the sum of contributions from collisions and from valence quarks $$N_u= N^{val}_{u}+N^{coll}_{u\bar u}$$ $$N_d= N^{val}_{d}+N^{coll}_{d\bar d};\qquad N_s=N_{\bar s}= N^{coll}_{s\bar s} \label{eq6}$$ In the case of AB interactions we shall start with considering the number of quarks and antiquarks due to the collisions of nucleons present in two colliding tubes, each having cross section $\sigma$. The impact parameter of the collision is denoted as $\vec b$. In the transverse plain the position of the tube in nucleus A is specified by the vector $\vec s=\vec s_A$ Within the nucleus B, the transverse position of the second tube is given by the vector $\vec s_B=\vec b-\vec s$. The average numbers of nucleons in tubes in A and B are respectively given as $$\nu \equiv \nu _A= \sigma \rho 2L_A(s); \qquad L_A(s)=\sqrt {R_A^2-s^2}$$ $$\mu \equiv \mu _B=\sigma \rho 2L(\vec b,\vec s); \quad 2L_B(\vec b,\vec s)= \sqrt {R_B^2-b^2-s^2+2bs.cos(\theta )} \label{eq7}$$ Numbers of future valence quarks and antiquarks created in such a tube- on -tube collision are given as $$N^{coll}_{s\bar s}(AB;b,s,\theta)= <s\bar s>_{nn}\sum _{i=1}^{\nu } \sum _{j=1}^{\mu } (1-\beta _s)^{i-1}(1-\beta _s)^{j-1}=$$ $$=<s\bar s>_{nn}{\frac {1-(1-\beta_s)^{\nu}}{\beta _s}} {\frac {1-(1-\beta_s)^{\mu}}{\beta _s}}$$ $$N^{coll}_{u\bar u}(AB;b,s,\theta)=N^{coll}_{d\bar d}(AB;b,s,\theta)=$$ $$=<u\bar u>_{nn}\sum _{i=1}^{\nu } \sum _{j=1}^{\mu } (1-\beta _u)^{i-1}(1-\beta _u)^{j-1}=$$ $$=<u\bar u>_{nn}{\frac {1-(1-\beta_u)^{\nu}}{\beta _u}} {\frac {1-(1-\beta_u)^{\mu}}{\beta _u}} \label{eq8}$$ The total number of future valence quarks and antiquarks produced by interaction within these tubes then becomes $$N_u(AB;b,s,\theta)=N_d(AB;b,s,\theta)= 1.5(\mu +\nu)+ N^{coll}_{u\bar u}(AB;b,s,\theta)$$ $$N_s(AB;b,s,\theta)=N_{\bar s}(AB;b,s,\theta)= N^{coll}_{s\bar s}(AB;b,s.\theta)$$ $$N_{\bar u}(AB;b,s,\theta)=N_{\bar d}(AB;b,s,\theta)= N^{coll}_{u\bar u}(AB;b,s,\theta) \label{eq9}$$ In other to obtain a qualitative feeling of ratios of different flavours of quarks and antiquarks we plot in Table 1 values of $N^{coll}_s$, $N^{coll}_u$, $N_u=N^{coll}_u$, and their ratios for a set of values of $\mu $ and $\nu $. [**Table 1.**]{} Production of quarks and antiquarks in tube- on tube collision; $\mu$ and $\nu$ are numbers of nucleons in the tubes. $\mu$,$\nu$ $N_s=N_s^{coll} $ $N_u^{coll} $ $N_u=N_u^{coll}+N_u^{val} $ $N_s/N_u^{coll}$ $N_s/N_u$ ------------- ------------------- --------------- ----------------------------- ------------------ ----------- 1,1 0.63 3.4 6.4 0.185 0.1 1,2 1.18 5.44 9.9 0.22 0.12 1,3 1.65 6.66 12.7 0.25 0.13 1,4 2.07 7.40 14.9 0.28 0.14 1,5 2.43 7.84 16.84 0.31 0.144 2,2 2.2 8.7 14.7 0.253 0.15 2,3 3.1 10.7 18.2 0.29 0.17 2,4 3.87 11.83 20.83 0.33 0.19 2,5 4.54 12.54 23.04 0.36 0.20 3,3 4.33 13.06 22.1 0.33 0.20 3,4 5.43 14.5 25.0 0.37 0.22 3,5 6.37 15.36 27.4 0.415 0.23 4,4 6.8 16.1 28.1 0.42 0.24 4,5 8.0 17.0 30.5 0.47 0.26 5,5 9.36 18.1 33.1 0.52 0.28 As seen in the Table 1. the ratio $N_s/N_u$ increases by a factor of 2.8 when going from the case of $\mu=1,\nu=1$ to that of $\mu=5,\nu=5$. The increase of the production of strange to non- strange hadrons will be, of course, smaller since in the nuclear collisions one always integrates over region of nuclei overlap in the impact parameter plane and the influence of the central region is suppressed by geometry. The NA49 Collaboration has recently presented [@BOR97] results on $K^+/K^-$ and $\bar \Lambda/\Lambda $ ratios in the central Pb+Pb collisions at 158 GeV per nucleon. The resulting numbers $${{K^+}\over {K^-}}\approx 1.8; \qquad {{\bar \Lambda }\over {\Lambda}} \approx 0.2$$ give a hint on whether the recombination models has a chance. Since $K^+$ consists of $s\bar u$, $K^-$ of $\bar u s$, $\Lambda$ of $s,d,u$ and $\bar \Lambda$ of $\bar s,\bar d,\bar u$ we expect in a recombination model $${{K^-}\over {K^+}}\approx {{N_u}\over {N_{\bar u}}}. {{N_{\bar s}}\over {N_s}} ; \quad {{\bar \Lambda}\over {\Lambda}}\approx {{N_{\bar s}}\over {{N_s}}}. {{N_{\bar u}}\over {{N_u}}}. {{N_{\bar d}}\over {{N_d}}} \label{eq10}$$ Table 1 shows that for most of combinations of $\mu$ and $\nu$ it holds $N_u\approx 2N_{\bar u}$ and by assumption $N_s=N_{\bar s}$ so both of ratios come out roughly correct. Note that in Table 1 we include all the quarks and antiquarks just before recombination independently of their rapidity. This may be adequate for the results of NA49 with a large acceptance. Production of hadrons via fast recombination {#stran} ============================================ In this section we shall assume that the recombination is so fast that quarks and antiquarks produced in a given tube- on - tube collision can recombine mutually and what happens in a given tube- on tube system is independent of what happens to systems produced by other tube- on- tube collisions. We shall use the recombination model suggested by Biró and Zimányi [@KOCH86; @BIRO95; @ZIM93; @BIZ83]. According to this model the number of pions $N_{\pi}$, kaons $N_K$, $\phi$-mesons $N_{\phi}$, baryons $N_B$, $Y$-hyperons $N_Y$, $\Xi$-hyperons $N_{\Xi}$, $\Omega$’s and corresponding antibaryons are given by the following relations: $$N_{\pi}= \alpha (N_u+N_d)(N_{\bar u}+N_{\bar d}), \qquad N_K=\alpha (N_u+N_d)N_{\bar s},\quad$$ $$N_{\bar K}=\alpha (N_{\bar u}+N_{\bar d}),\qquad N_{\phi} = \alpha N_sN_{\bar s},$$ $$N_B=\beta {1\over {3!}}(N_u+N_d)^3,\qquad N_Y=\beta {1\over {2!}}(N_u+N_d)^2 N_s,$$ $$N_{\Xi}=\beta {1\over {2!}} N_s^2(N_u+N_d),\qquad N_{\Omega}=\beta {1\over {3!}}N_s^3. \label{eq11}$$ The constants $\alpha $ and $\beta $ are obtained from the consistency conditions requiring that the number of quarks and antiquarks is equal to the corresponding number of valence quarks and antiquarks in hadrons formed by recombination. In this way one finds [@KOCH86; @BIRO95; @ZIM93; @BIZ83] $$\alpha= \frac {Q+\bar Q}{Q^2+Q\bar Q+\bar Q^2}; \quad \beta = \frac {2}{Q^2+Q\bar Q +\bar Q^2} \label{eq12}$$ where $$Q=N_u+N_d+N_s; \qquad \bar Q = N_{\bar u}+N_{\bar d}+N_{\bar s}$$ The yield of a certain particle is calculated in the following way. Numbers of quarks and antiquarks of all flavours in a tube- on- tube collision at given $(b,s,\theta)$ are obtained via Eqs.(8,9). This is inserted into Eqs.(11,12). In these equations we have used a short hand notation like $N_{\pi}$. The full notation should be $N_{\pi}(b,s,\theta)$. The total yield of $Y$-hyperons is then obtained from the expression $$N_Y(AB;b)={1\over {\sigma}}\int_0^{R_A}sds \int _0^{2\pi}d\theta N_Y(AB;b,s,\theta) \label{eq13}$$ We have calculated the yields $N_Y(AB,b)$ by two independent numerical methods which gave very similar results. The expression $N_Y(AB;b) $ corresponds to the sum of hyperons $\Lambda$, $\Sigma ^-$, $\Sigma ^0$ and $\Sigma ^-$. Taking into account the decay $\Sigma ^0 \to \Lambda +\gamma$ and the decay $\Xi ^0 \to \Lambda \pi^0$ we have $$<\Lambda >={1\over 2}N_Y+{1\over 2}N_{\Xi}$$ Although we shall make in this paper no attempts at comparison with the data, let us point out that previous relationship corresponds to the situation in NA49 data analysis, but not to the one in WA97. Similarly the experimentally observed number of $\Xi^-$ is given as $$<\Xi^->={1\over 2} N_{\Xi}$$ The results for Pb+Pb and S+S are presented in Tables 2 and 3. [**Table 2.**]{} Yields of strange baryons and antibaryons in Pb+Pb collisions as a function of the impact parameter $b$ in the model of rapid recombination. ------ -------- ----------- -------------- -------------- ---------------- -------------------- b $N_Y$ $N_{\Xi}$ $N_{\Omega}$ $N_{\bar Y}$ $N_{\bar \Xi}$ $N_{\bar \Omega} $ 0.0 141.4 18.62 0.84 43.6 10.27 0.84 1.0 136.1 18.02 0.82 41.64 9.91 0.82 2.0 125.83 16.68 0.75 38.1 9.14 0.75 3.0 113.1 14.9 0.67 33.9 8.15 0.67 4.0 98.9 12.92 0.58 29.6 7.05 0.58 5.0 84.13 10.83 0.47 25.1 5.92 0.47 6.0 69.36 8.74 0.38 20.8 4.79 0.38 7.0 55.1 6.76 0.28 16.63 3.72 0.28 8.0 41.8 4.94 0.20 12.74 2.74 0.20 9.0 29.8 3.37 0.13 9.22 1.88 0.13 10.0 19.4 2.07 0.075 6.11 1.16 0.075 11.0 11.0 1.09 0.036 3.54 0.62 0.036 12.0 4.93 0.43 0.013 1.6 0.25 0.013 ------ -------- ----------- -------------- -------------- ---------------- -------------------- [**Table 3.**]{} Yields of strange baryons and antibaryons in S+S collisions as a function of the impact parameter $b$ in the model of rapid recombination. ----- ------- ----------- -------------- -------------------- ---------------- -------------------- b $N_Y$ $N_{\Xi}$ $N_{\Omega}$ $N_{\bar \Lambda}$ $N_{\bar \Xi}$ $N_{\bar \Omega} $ 0.0 16.44 1.41 0.041 5.71 0.83 0.041 1.0 14.85 1.28 0.038 5.12 0.75 0.038 2.0 11.97 1.02 0.030 4.08 0.60 0.030 3.0 8.72 0.72 0.020 2.93 0.42 0.020 4.0 5.59 0.44 0.012 1.85 0.26 0.012 5.0 2.94 0.21 0.0053 0.96 0.12 0.0053 6.0 1.07 0.068 0.0015 0.33 0.038 0.0015 ----- ------- ----------- -------------- -------------------- ---------------- -------------------- The translation from the impact parameter $b$ to the number of nucleon- nucleon collisions $N_c(b)$ and to the number of participating (wounded) nucleons $N_p(b)$ is given by the standard relations $$N_p(b)={1\over {\sigma}}\int _0^{R_A}sds\int _0^{2\pi}d\theta \Theta (R_B^2-b^2-s^2+2bs.cos\theta)$$ $$\left(\rho_A\sigma 2L_A(s)[1-e^{-\rho_B\sigma 2L_B(b,s,\theta)}]+ \rho_B\sigma 2L_B(b,s,\theta)[1-e^{-\rho_A\sigma 2L_A(s)}]\right)$$ $$N_c(b)={1\over {\sigma}}\int_0^{R_A}sds\int_0^{2\pi}d\theta \rho_A\sigma 2L_A(s)\rho_B\sigma 2L_B(b,s,\theta)$$ where $$L_A(s)=\sqrt{R_A^2-s^2},\quad L_B(b,s,\theta)=\sqrt{R_B^2-b^2-s^2 +2bs.cos(\theta)}$$ The average values of strange baryons are calculated by using the relations $$<\Lambda>={1\over 2} (N_Y+N_{\Xi}),\quad <\Xi^->={1\over 2}N_{\Xi}, \quad <\Omega>=N_{\Omega}$$ and similarly for antibaryons. In this way we obtain results presented in Tables 4 and 5. [**Table 4.**]{} Yields of strange baryons and antibaryons in Pb+Pb collisions as a function of the impact parameter $b$ in the model of rapid recombination within the tubes. ------ ------------- ----------- -------------------------- ------------------ ---------------- b $<\Lambda>$ $<\Xi^->$ $<\Omega>=<\bar \Omega>$ $<\bar \Lambda>$ $<\bar \Xi^+>$ 0.0 80.01 9.31 0.84 26.95 5.13 1.0 77.07 9.01 0.82 25.78 4.95 2.0 71.26 8.34 0.75 23.97 4.57 3.0 63.99 7.46 0.67 21.04 4.07 4.0 55.90 6.46 0.58 18.3 3.53 5.0 47.48 5.42 0.47 15.53 2.96 6.0 39.05 4.37 0.38 12.79 2.40 7.0 30.92 3.38 0.28 10.18 1.86 8.0 23.23 2.97 0.20 7.74 1.37 9.0 16.57 1.68 0.129 5.55 0.94 10.0 10.73 1.033 0.075 3.64 0.582 11.0 6.054 0.542 0.036 2.08 0.308 12.0 2.68 0.216 0.013 0.926 0.124 ------ ------------- ----------- -------------------------- ------------------ ---------------- In Fig.1 we present the dependence of the production of of $<\Lambda>+<\bar \Lambda>$, $<\Xi^->+<\bar \Xi^+>$ and $<\Omega>+<\bar \Omega>$ on the number of nucleon- nucleon collisions. All yields are normalized to 1 at the impact parameter value of b=10, corresponding to the number of nucleon- nucleon collisions $N_c(b=10)$=104.5. In the next section these results will be compared with the situation when anomalous and strangeness richer matter is present. In order to permit a comparison with earlier work we present in Table 5. the yields of strange baryons and antibaryons in S+S collisions. [**Table 5.**]{} Yields of strange baryons and antibaryons in S+S interactions as a function of the impact parameter $b$ ----- ------------- ----------- --------------------------- ------------------ ---------------- -- b $<\Lambda>$ $<\Xi^->$ $<\Omega>=<\bar \Omega> $ $<\bar \Lambda>$ $<\bar \Xi^+>$ 0.0 8.93 0.71 0.041 3.27 0.42 1.0 8.07 0.64 0.038 2.94 0.38 2.0 6.50 0.51 0.030 2.34 0.30 3.0 4.72 0.36 0.020 1.68 0.21 4.0 3.01 0.22 0.0118 1.055 0.128 5.0 1.58 0.107 0.0053 0.54 0.062 6.0 0.567 0.034 0.0015 0.185 0.019 ----- ------------- ----------- --------------------------- ------------------ ---------------- -- Results for central S+S interactions can be compared with the data [@NA35AA] and with the calculations performed within the ALCOR model by Biró, Lévai and Zimányi [@BIRO95]. Biró et al. obtain $<\Lambda>$=10.37, the data give $<\Lambda>=9.4\pm 1.0$ and our value in Table 5. is $<\Lambda>$=8.93. For $<\Xi^->$ Biró et al. find 1.15 whereas our value is 0.73. The largest discrepancy between Ref.[@BIRO95] and our results appears in $\Omega $ production where the authors of Ref.[@BIRO95] find $<\Omega>$=0.14, whereas our value is 0.041. In the case of central Pb+Pb interactions Biró et al. have obtained $<\Lambda>$=82.4 and $<\Lambda>$=111.3 in the two versions of their model. Our result in Table 5 is $<\Lambda>$=80.0. Production of hadrons by a slow recombination {#slow} ============================================= In the the previous section we have considered a model in which both the formation of future valence quarks and antiquarks and their recombination to hadrons takes place within tubes of the cross- section equal to the inelastic nucleon- nucleon cross- section $\sigma $. What happens in one tube is in this model completely independent of what happens in another tube. In order to have a qualitative feeling of the effects of this assumption we shall discuss in this section a model in which all quarks and antiquarks formed in an A+B interaction at a given value of the impact parameter $b$ can recombine with each other. The calculations proceed as above but the order is reversed. By using Eqs.(8,9) we compute numbers of future valence quarks produced in individual tube- on- tube interactions. In the next step we calculate total numbers of quarks and antiquarks formed in the A+B collision at given value of the impact parameter $b$ according to equations like $$N^T_u(AB;b)= {1\over {\sigma}}\int _0^{R_A} sds \int _0^{2\pi}d\theta N_u(AB;b,s,\theta ) \label{eq17}$$ The obtained total numbers of quarks and antiquarks are then recombined via the Biró - Zimányi scheme as given by Eqs.(11) and (12). The results are presented in Table 6. They are very close to those shown in Table 2. This fact is easyly understood for the case of $Y$ hyperons. Looking in Table 1. we see that with very good approximation $N_u\approx 2N_{\bar u} >> N_s $ and $N_u=N_d$. Then parameter $\beta$ in Eq. (12) is $\beta \propto 1/N_u^2$ and $N_Y\propto N_s$ Now the number of hyperons in rapid recombination we get by calculating number of $N_Y$ in each row and summing over all rows. $$N_Y^{rapid}=\sum_{rows} N_Y^{row}=const.\sum_{rows} N_s^{row} =const. N_s=N_Y^{slow}$$. Similar arguments can be done for $\Xi$ and $\Omega$ hyperons. [**Table 6.**]{} Yields of strange baryons and antibaryons in Pb+Pb collisions as a function of the impact parameter $b$ in the model of slow recombination. ------ ------- ----------- --------------- -------------- ---------------- -------------------- b $N_Y$ $N_{\Xi}$ $N_{\Omega }$ $N_{\bar Y}$ $N_{\bar \Xi}$ $N_{\bar \Omega }$ 0.0 142.3 18.14 0.77 44.34 10.13 0.77 1.0 136.9 17.63 0.76 42.18 9.78 0.76 2.0 126.6 16.37 0.71 38.46 9.02 0.71 3.0 113.8 14.67 0.63 34.19 8.04 0.63 4.0 99.5 12.72 0.54 29.71 6.95 0.54 5.0 84.7 10.67 0.45 25.22 5.82 0.45 6.0 69.9 8.63 0.36 20.83 4.71 0.36 7.0 55.5 6.67 0.27 16.64 3.65 0.27 8.0 42.1 4.88 0.19 12.72 2.69 0.19 9.0 30.00 3.33 0.12 9.19 1.84 0.12 10.0 19.5 2.05 0.07 6.08 1.14 0.07 11.0 11.1 1.08 0.03 3.51 0.60 0.03 12.0 4.97 0.43 0.01 1.59 0.24 0.01 ------ ------- ----------- --------------- -------------- ---------------- -------------------- Possible presence of anomalous matter and thresholds in strange baryon and antibaryon production {#anomaly} ================================================================================================= Data on the multiplicity of negative secondary hadrons and on the total transverse energy in Pb+Pb interactions [@NA49a] do not indicate a presence of some thresholds connected with the formation of a new “anomalous” form of matter. It rather seems that the multiplicity of secondary hadrons in Pb+Pb and transverse energy can be obtained as the extrapolation of results obtained in collisions of lighter ions. On the other hand recent data of the WA97 Collaboration indicate that the production of strange baryons within the acceptance region of the experiment is increased [@WA97a; @KRAL97; @WA98]. The WA97 experiment takes data only in a small part of the total phase- space. The accepted events cover a region near the central rapidity in the c.m.s. and transverse momenta of baryons above 0.6 GeV/c. In order to disentangle the extrapolated strangeness content one would need to use data from lighter ion collisions in the same experiment to determine the values of parameters $\beta_u$, $\beta_s$, $<u\bar u>_{nn}$ and $<s\bar s>_{nn}$ corresponding to the experimental acceptance. Presence of a threshold in the production of strange baryons together with approximately no increase in the total multiplicity lead to the assumption that the total number of quark- antiquark pairs in tube- on- tube collisions in Pb+Pb interactions is approximately the same as calculated by the formulas given above, but starting with tube- on- tube collisions which satisfy a certain criticality condition the matter is in some sense “melted” and a part of $u\bar u$ and $d\bar d$ pairs is transformed to $s\bar s$ pairs. In order to permit a comparison with the description of the anomalous $J/\Psi$ suppression in Pb+Pb in models of Blaizot and Ollitrault [@BLAIZ] and of Kharzeev et al. [@KHARa] we shall use the criticality condition of Ref.[@KHARa]. In their description of $J/\Psi$ suppression Kharzeev et al. [@KHARa; @KHARb] assume that QGP is formed only in a limited region of the transverse plane. Taking that view we shall introduce the parameter $$\kappa (b,s,\theta)= \frac {\sigma_{nn}\rho_A 2L_A(s).\sigma_{nn}\rho_B 2L_B(b,s,\theta)} {\sigma_{nn}\rho_A 2L_A(s)+\sigma_{nn}\rho_B 2L_B(b,s,\theta)} \label{eq14}$$ The parameter $\kappa(b,s,\theta)$ is roughly proportional in the “tube- on -tube picture” to the ratio of the number of nucleon- nucleon collisions to the longitudinal dimension of the system formed by the two colliding tubes. It is further assumed that for $$\kappa(b,s,\theta) \ge \kappa_{crit} \label{eq15}$$ QGP is formed, whereas in the opposite case the system remains in the normal state. In Ref.[@KHARa] the authors introduce two values of $\kappa_{crit}$. Above $\kappa_{crit}\approx $2.3 QGP is formed and the $\chi$ meson responsible for about 40% of $J/\Psi$ production is dissolved. For $\kappa_{crit}$ above 2.9 also $J/\Psi$ is completely dissolved. In the present work we shall use a single threshold, corresponding to the onset of a “new” or “anomalous” form of matter. According to Refs.[@KHARa; @KHARb] we expect this threshold at about $\kappa_{crit}$=2.3. We assume that above this threshold the matter goes into a new form characterized by a higher value of the strangeness abundance, that means that above $\kappa _{crit}$ a part of $u\bar u$ and $d\bar d$ pairs is transformed to $s\bar s$ ones. To simulate this effect, we go back to the Eqs.(8,9) and for $\kappa \ge \kappa_{crit}$ make the replacement $$N^{coll}_{s\bar s} \to N^{coll}_{s\bar s} [1+(\xi -1)\Theta (\kappa-\kappa_{crit})]$$ $$N^{coll}_{u\bar u} \to N^{coll}_{u\bar u} - 0.5(\xi -1)N^{coll}_{s\bar s}\Theta (\kappa-\kappa_{crit})$$ $$N^{coll}_{d\bar d} \to N^{coll}_{s\bar s}- 0.5(\xi -1)N^{coll}_{s\bar s}\Theta (\kappa-\kappa_{crit}) \label{eq16}$$ and then continue as within the scheme of the rapid recombination model in Sect.3. In this way we obtain for the two cases considered the results presented in Tables 7 and 8. In order to see the effects caused by the presence of the anomalous matter these results should be compared with those given in Table 2. [**Table 7.**]{} Yields of strange baryons and antibaryons in Pb+Pb collisions as a function of the impact parameter $b$ within the model of rapid recombination. Anomalous matter present: $\xi=2.0$, $\kappa_c$ =2.1. ------ ------- ----------- -------------- -------------- ---------------- -------------------- b $N_Y$ $N_{\Xi}$ $N_{\Omega}$ $N_{\bar Y}$ $N_{\bar \Xi}$ $N_{\bar \Omega} $ 0.0 186.8 52.6 5.59 45.95 24.96 5.59 1.0 179.8 50.7 5.40 43.85 24.08 5.40 2.0 166.6 47.2 5.00 40.17 22.30 5.00 3.0 149.6 42.00 4.41 35.88 19.87 4.41 4.0 130.3 35.9 3.72 31.35 17.04 3.72 5.0 110.1 29.4 2.98 26.77 14.05 2.98 6.0 89.6 22.8 2.24 22.23 11.04 2.24 7.0 69.5 16.5 1.55 17.82 8.12 1.55 8.0 50.7 10.8 0.94 13.6 5.42 0.94 9.0 33.9 5.9 0.44 9.68 3.07 0.44 10.0 19.4 2.07 0.08 6.11 1.16 0.08 11.0 11.0 1.09 0.04 3.54 0.62 0.04 12.0 4.93 0.43 0.01 1.6 0.25 0.01 ------ ------- ----------- -------------- -------------- ---------------- -------------------- [**Table 8.**]{} Yields of strange baryons and antibaryons in Pb+Pb collisions as a function of the impact parameter $b$ within the model of rapid recombination. Anomalous matter present: $\xi$=2.0; $\kappa_c$=2.5. ------ ------- ----------- -------------- -------------- ---------------- -------------------- b $N_Y$ $N_{\Xi}$ $N_{\Omega}$ $N_{\bar Y}$ $N_{\bar \Xi}$ $N_{\bar \Omega} $ 0.0 167.7 40.04 3.99 44.13 19.17 3.99 1.0 161.8 38.89 3.88 42.16 18.59 3.88 2.0 149.4 35.71 3.53 38.59 17.06 3.53 3.0 133.4 31.20 3.03 34.44 14.96 3.03 4.0 115.1 25.77 2.42 30.03 12.46 2.42 5.0 95.91 19.98 1.78 25.55 9.80 1.78 6.0 76.30 14.02 1.11 21.20 7.06 1.11 7.0 57.21 8.33 0.50 16.75 4.41 0.50 8.0 41.8 4.94 0.2 12.74 2.74 0.2 9.0 29.8 3.37 0.13 9.22 1.88 0.13 10.0 19.4 2.07 0.08 6.11 1.16 0.08 11.0 11.0 1.09 0.04 3.54 0.62 0.04 12.0 4.93 0.43 0.01 1.6 0.25 0.01 ------ ------- ----------- -------------- -------------- ---------------- -------------------- Proceeding as below the Table 2, we obtain the yields of $<\Lambda>+<\bar \Lambda>$, $<\Xi_->+<\bar \Xi^+>$ and $<\Omega>+<\bar \Omega>$ normalized to 1 at $b=10$ that means $N_c$=104,5 as presented in Figs.2 and 3. The rapid increase of the yields of $<\Lambda>+<\bar \Lambda>$, $<\Xi^->+<\bar \Xi^+>$ and in particular of $<\Omega>+<\bar \Omega>$ seen in Figs.2 and 3. in comparison with Fig.1 shows that the presence of anomalous matter increases the yields of strange hyperons significantly. The two parameters $\kappa_c$ and $\xi $ regulate the position of the onset of the increase as a function of the impact parameter or alternatively as a function on the number of nucleon- nucleon collisions or the number of participating nucleons. These two parameters should be determined by the comparison of model predictions with the data. We shall not attempt to make this comparison here. Let us remark that a similar calculation was done also for slow recombination scenario. Comparison of the results obtained in both recombination schemes shows greater differences between them for the case of the anomalous enhanced strangeness production. This behaviour is natural since the difference in final hyperon multiplicities between slow and fast recombination depends mainly on the ratio of strange to nonstrange quarks. We shall not discuss this issue in more detail since we consider the rapid recombination scheme as a more realistic model. Summary, comments and conclusions {#comments} ================================= We have presented here a recombination model of hadron formation in nuclear collisions, which permits to calculate hadron yields as a function of the impact parameter. The model is based on the phenomenological parametrization of the number of quarks and antiquarks just before the hadronization. A few parameters are determined by comparison with data on hadron production in interactions of lighter ions and extrapolated to the case of Pb+Pb collisions. The second ingredient is the Biró - Zimányi recombination scheme. The model contains several simplifications. In its present form it does not contain fluctuations in the number of produced quarks and antiquarks which may be important in estimating yields of multistrange baryons in pA interactions and in collisions of lighter ions. For this reason we have not normalized the strange baryon yields to pA interactions. The model does not analyse rapidity and $p_T$ distributions of quarks and antiquarks before the recombination and therefore gives only predictions for the total numbers of final state hadrons. Because of that, when comparing model predictions with the data in a small part of the phase space, one should rather determine the values of input parameters by data on hadron production in the corresponding acceptance region. The model neglects modifications of the chemical composition of final state hadrons due to interactions in the hadronic stage of the nuclear collisions. The changes in the chemical composition in the hadronic phase are known to be slow, and we do not expect that they will modify substantially the yields of strange baryons. On the other hand, the model includes a phenomenological description of the influence of the presence of anomalous, strangeness rich, matter. The parametrization of the effects due to the anomalous matter (perhaps QGP) is similar to models used by Blaizot and Ollitrault, and Kharzeev and Satz to describe the anomalous $J/\Psi$ suppression in Pb+Pb interactions. The analysis of data on strange baryon production within this model can thus contribute to the understanding of the anomalous $J/\Psi$ suppression as observed by the NA 50 collaboration. We have made here no attempt to compare our results with the data of the WA97 collaboration covering the midrapidity region. This would require a specification of what fraction of valence quarks participate in the recombination to strange baryons in this region. We shall return to this question in the near future. We are indebted to W.Geist, R.C.Hwa, I.Králik, M.Mojžiš, E.Quercigh, L.Šándor and other members of the WA97 Collaboration for numerous useful discussions. One of the authors (J.P.) would like to thank G.Roche and B.Michel for the hospitality at the Laboratoire de Physique Corpusculaire at the Blaise Pascal University at Clermont- Ferrand, where a part of this work has been done. This work was supported in part by the US- Slovak Science and Technology Joint Fund under grant No. 003-95 and by the Slovak Grant Agency under grants No. V2F13-G and V2F18-G. Yields of $<\Lambda>+<\bar \Lambda>$, $<\Xi^->+<\bar \Xi^+>$ and $<\Omega>+<\bar \Omega>$ as a function of the number of nucleon- nucleon collisions $N_c$ in the case when no anomalous matter is present. All yields normalized to 1 at $b=10$ that means at $N_c$=104,5. 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--- abstract: 'Probabilistic Latent Variable Models (LVMs) provide an alternative to self-supervised learning approaches for linguistic representation learning from speech. LVMs admit an intuitive probabilistic interpretation where the latent structure shapes the information extracted from the signal. Even though LVMs have recently seen a renewed interest due to the introduction of Variational Autoencoders (VAEs), their use for speech representation learning remains largely unexplored. In this work, we propose Convolutional Deep Markov Model (ConvDMM), a Gaussian state-space model with non-linear emission and transition functions modelled by deep neural networks. This unsupervised model is trained using black box variational inference. A deep convolutional neural network is used as an inference network for structured variational approximation. When trained on a large scale speech dataset (LibriSpeech), ConvDMM produces features that significantly outperform multiple self-supervised feature extracting methods on linear phone classification and recognition on the Wall Street Journal dataset. Furthermore, we found that ConvDMM complements self-supervised methods like Wav2Vec and PASE, improving on the results achieved with any of the methods alone. Lastly, we find that ConvDMM features enable learning better phone recognizers than any other features in an extreme low-resource regime with few labelled training examples.' address: | $^1$MIT - CSAIL, Cambridge, USA\ $^2$LIUM - Le Mans University, France\ $^3$University of Wrocław, Poland $^4$NVIDIA Corporation, Poland $^5$LIS - University of Toulon, France bibliography: - 'mybib.bib' title: | A Convolutional Deep Markov Model\ for Unsupervised Speech Representation Learning --- **Index Terms**: Neural Variational Latent Variable Model, Structured Variational Inference, Unsupervised Speech Representation Learning Introduction ============ One of the long-standing goals of speech and cognitive scientists is to develop a computational model of language acquisition [@goldwater2009bayesian; @lee2012nonparametric; @ondel2016variational; @harwath2016unsupervised]. Early on in their lives, human infants learn to recognize phonemic contrasts, frequent words and other linguistic phenomena underlying the language [@dupoux2018cognitive]. The computational modeling framework of generative models is well-suited for the problem of spoken language acquisition, as it relates to the classic analysis-by-synthesis theories of speech recognition [@halle1962speech; @liberman1967perception]. Although, generative models are theoretically elegant and informed by theories of cognition, most recent success in speech representation learning has come from self-supervised learning algorithms such as Wav2Vec [@schneider2019wav2vec], Problem Agnostic Speech Encoding (PASE) [@pascual2019learning], Autoregressive Predictive Coding (APC) [@chung2019unsupervised], MockingJay (MJ) [@liu2019mockingjay] and Deep Audio Visual Embedding Network (DAVENet) [@harwath2020learning]. Generative models present many advantages with respect to their discriminative counterparts. They have been used for disentangled representation learning in speech [@hsu2017unsupervised; @khurana2019factorial; @li2018disentangled]. Due to the probabilistic nature of these models, they can be used for generating new data and hence, used for data augmentation [@hsu2017unsuperviseda; @hsu2018unsupervised] for Automatic Speech Recognition (ASR), and anomaly detection [@Grathwohl2020Your]. In this paper, we focus solely on designing a generative model for low-level linguistic representation learning from speech. We propose Convolutional Deep Markov Model (ConvDMM), a Gaussian state-space model with non-linear emission and transition functions parametrized by deep neural networks and a Deep Convolutional inference network. The model is trained using amortized black box variational inference (BBVI) [@ranganath2013black]. Our model is directly based on the Deep Markov Model proposed by Krishnan et. al [@krishnan2017structured], and draws from their general mathematical formulation for BBVI in non-linear Gaussian state-space models. When trained on a large speech dataset, ConvDMM produces features that outperform multiple self-supervised learning algorithms on downstream phone classification and recognition tasks, thus providing a viable latent variable model for extracting linguistic information from speech. We make the following contributions: 1) Design a generative model capable of learning good quality linguistic representations, which is competitive with recently proposed self-supervised learning algorithms on downstream linear phone classification and recognition tasks. 2) Show that the ConvDMM features can significantly outperform other representations in linear phone recognition, when there is little labelled speech data available. 3) Lastly, demonstrate that by modeling the temporal structure in the latent space, our model learns better representations compared to assuming independence among latent states. The Convolutional Deep Markov Model {#sec:2} =================================== ConvDMM Generative Process {#sec:2.1} -------------------------- Given the functions; ${f_{\theta}(\cdot), u_{\theta}(\cdot)}$ and ${t_{\theta}(\cdot)}$, the ConvDMM generates the sequence of observed random variables, ${x}_{1:T} = {(x}_1, \ldots,{x}_T)$, using the following generative process $$\begin{aligned} {z}_1 &\sim {\mathcal{N}(0, I)}\\ {z}_\tau|{z}_{\tau-1} &\sim {\mathcal{N}(t^{\mu}_{\theta}(z}_{\tau-1}),{ t^{\sigma}_{\theta}(z}_{\tau-1}))& \tau=2,\ldots,L \label{eq:2}\\ {e}_{1:T} &= {u_\theta(z}_{1:L}) \\ {\mu}_{1:T} &= {f_{\theta}(e}_{1:T})\\ {x}_t|{e}_t &\sim {\mathcal{N}(\mu}_t, \gamma)& t=1,\ldots, T \label{eq:5}\end{aligned}$$ where $T$ is a multiple of $L$, $T = k\cdot L$ and ${z}_{1:L}$ is the sequence of latent states. We assume that the observed and latent random variables come from a multivariate normal distribution with diagonal covariances. The joint density of latent and observed variables for a single sequence is $${p(z}_{1:L}, {x}_{1:T}) = {p(x}_{1:T}|{z}_{1:L}) {p(z}_1)\prod\limits_{\tau=2}^{L} {p(z}_\tau|{z}_{\tau-1}).$$ For a dataset of i.i.d. speech sequences, the total joint density is simply the product of per sequence joint densities. The scale ${\gamma}$ is learned during training. **The transition function** ${t_\theta : z}_{\tau-1} \rightarrow {(\mu}_\tau, {\sigma}_\tau) \big|_{\tau=2}^{L}$ estimates the mean and scale of the Gaussian density over the latent states. It is implemented as a Gated Feed-Forward Neural Network [@krishnan2017structured]. The gated transition function could capture both linear and non-linear transitions. **The embedding function** ${u_{\theta} : z}_{1:L} \rightarrow {e}_{1:T}$ transforms and up-samples the latent sequence to the same length as the observed sequence. It is parametrized by a four layer CNN with kernels of size 3, 1024 channels and residual connections. We use the activations of the last layer of the embedding CNN as the features for the downstream task. This is reminiscent of kernel methods [@hofmann2008kernel] where the raw input data are mapped to a high dimensional feature space using a user specified feature map. In our case, the CNN plays a similar role, mapping the low-dimensional latent vector sequence, ${z}_{1:L} \in \mathbb{R}^{L \times 16}$, to a high dimensional vector sequence, ${e}_{1:T}\in \mathbb{R}^{T\times 1024}$, by repeating the output activations of the CNN $k$ times, where $k = T/L$. In our case, $k$ is 4 which is also the downsampling factor of the encoder function (§ \[sec:inf\]). A similar module was used in Chorowski et. al [@chorowski2019unsupervised], where they used a single CNN layer after the latent sequence. **The emission function** ${f_{\theta} : e}_t \rightarrow {(\mu}_t)\big|_{t=1}^{T}$ (a decoder) estimates the mean of the likelihood function. It is a two-layered MLP with 256 hidden units and residual connections. We employ a low capacity decoder to avoid the problem of posterior collapse [@bowman2015generating], a common problem with high capacity decoders. ConvDMM Inference {#sec:inf} ----------------- The goal of inference is to estimate the posterior density of latent random variables given the observations ${p(z | x)}$. Exact posterior inference in non-conjugate models like ConvDMM is intractable, hence we turn to Variational Inference (VI) for approximate inference. We use VI and BBVI interchangeably throughout the rest of the paper. In VI, we approximate the intractable posterior ${p(z|x)}$ with a tractable family of distributions, known as the variational family ${q_{\phi}(z|x)}$, indexed by $\phi$. In our case, the variational family takes the form of a Gaussian with diagonal covariance. Next, we briefly explain the Variational Inference process for ConvDMM. Given a realization of the observed random variable sequence ${x}_{1:T} = {x}_1, \ldots, {x}_T$, the initial state parameter vector ${\hat{z}_1}$, and the functions ${g_{\phi}(\cdot)}$ and ${c_{\phi}(\cdot)}$, the process of estimating the latent states can be summarized as: $$\begin{aligned} {h}_{1:L} &= {g_{\phi}(x}_{1:T}) \\ {\hat{z}}_\tau|{\hat{z}}_{\tau-1}, {x} &\sim {\mathcal{N}(c^{\mu}_{\phi}(h}_\tau, {\hat{z}}_{\tau-1}),{ c^{\sigma}_{\phi}(h}_\tau,{ \hat{z}}_{\tau-1}))& \tau=2\ \text{to}\ L .\end{aligned}$$ Let $T = k*L$, where $k$ is the down-sampling factor of the encoder, ${g_{\phi} : x}_{1:T} \rightarrow {h}_{1:L}$ is the encoder function, ${c_{\phi}}$ is the combiner function that provides posterior estimates for the latent random variables. We parameterize the encoder ${g_\phi}$ using a 13-layer CNN with kernel sizes (3, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 3, 3), strides (1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1) and 1024 hidden channels. The encoder down-samples the input sequence by a factor of four. The last layer of the encoder with ${h}_{1:L}$ as its hidden activations has a receptive field of approximately 50. This convolutional architecture is inspired by [@chorowski2019unsupervised], but other acoustic models such as Time-Depth Separable Convolutions [@hannun2019sequence], VGG transformer [@mohamed2019transformers], or ResDAVENet [@harwath2020learning] could be used here. We leave this investigation for future work. **The combiner function** ${c_\phi}$ provides structured approximations of the variational posterior over the latent variables by taking into account the prior Markov latent structure. The combiner function follows [@krishnan2017structured]: $$\begin{aligned} {h_{\text{combined}}} &= {\frac{1}{2} (\text{tanh}(W\hat{z}}_{t-1} + {b) + h}_t)\\ {\mu}_t &= {W_{\mu} h_{\text{combined}} + b_{\mu}}\\ {\sigma}_t &= {\text{softplus}(W_{\sigma}h_{\text{combined}} + b_{\sigma})}.\end{aligned}$$ It uses tanh non-linearity on ${z_{t-1}}$ to approximate the transition function. Future work could investigate sharing parameters with the generative model as in Maal[ø]{}e et. al’s Bidirection inference VAE (BIVA) [@maaloe2019biva]. We note that structured variational inference in neural variational models is an important area of research in machine learning, with significant recent developemnts [@johnson2016composing; @lin2018variational]. Structured VAE has also been used for acoustic unit discovery [@ebbers2017hidden], which is not the focus of this work. ![An illustration of the ConvDMM.[]{data-label="fig:dmm"}](dmm.pdf){width="\linewidth"} ConvDMM Training ---------------- ConvDMM like other VAEs is trained to maximize the bound on model likelihood, known as the Evidence Lower Bound (ELBO): $${\mathcal{L(\theta, \phi)}} = {E_{q_{\phi}(z|x)} [\text{log}\ p_{\theta}(x|z)] - \text{KL} (q_{\phi}(z|x) || p_{\theta}(z))}$$ where ${p_\theta (x|z)} = \prod\limits_{t=1}^{T} {p(x}_t|{e}_t)$ is the Gaussian likelihood function and ${p(x}_t|{e}_t)$ is given by the ConvDMM generative process in Section \[sec:2.1\]. The Gaussian assumption lets us use the *reparametrization trick* [@kingma2013auto] to obtain low-variance unbiased Monte-Carlo estimates of the expected log-likelihood, the first term in the R.H.S of the ELBO. The KL term, which is also an expectation can be computed similarly and its gradients can be obtained analytically. In our case, we use the formulation of Equation 12, Appendix A., in Krishnan et. al [@krishnan2017structured], to compute the KL term analytically. The model is trained using the Adam optimizer with a learning rate of 0.001 for 100 epochs. We reduce the learning rate to half of its value if the loss on the development set plateaus for three consecutive epochs. L2 regularization on model parameters with weight 5e-7 is used during training. To avoid latent variable collapse we use KL annealing [@bowman2015generating] with a linear schedule, starting from an initial value of 0.5, for the first 20 epochs of training. We use a mini-batch size of 64 and train the model on a single NVIDIA Titan X Pascal GPU. Experiments =========== Evaluation Protocol and Dataset ------------------------------- We evaluate the learned representations on two tasks; phone classification and recognition. For phone classification, we use the ConvDMM features, the hidden activations from the last layer of the embedding function, as input to a softmax classifier, a linear projection followed by a softmax activation. The classifier is trained using Categorical Cross Entropy to predict framewise phone labels. For phone recognition the ConvDMM features are used as input to a softmax layer which is trained using Connectionist Temporal Classification (CTC) [@graves2006connectionist] to predict the output phone sequence. We do not fine-tune the ConvDMM feature extractor on the downstream tasks. The performance on the downstream tasks is driven solely by the learned representations as there is just a softmax classifier between the representations and the labels. The evaluation protocol is inspired by the unsupervised learning works in the Computer Vision community [@tian2019contrastive; @henaff2019data], where features extracted from representation learning systems trained on ImageNet are used as input to a softmax classifier for object recognition. Neural Networks for supervised learning have always been seen as feature extractors that project raw data into a linearly separable feature space making it easy to find decision boundaries using a linear classifier. We believe that it is reasonable to expect the same from unsupervised representation learning methods and hence, we compare all the representation learning methods using the aforementioned evaluation protocol. We train ConvDMM on the publicly available LibriSpeech dataset [@panayotov2015librispeech]. To be comparable with other representation learning methods with respect to the amount of training dataset used, we train another model on a small 50 hours subset of LibriSpeech. For evaluation, we use the Wall Street Journal (WSJ) dataset [@paul1992design]. ![PER on WSJ eval92 dataset using features extracted from different models.[]{data-label="box"}](box.pdf){width="\linewidth"} Results & Discussion -------------------- Table \[tab:wsj\_probes\] presents framewise linear phone classification (FER) and recognition (PER) error rates on the WSJ eval92 dataset for different representation learning techniques. ConvDMM is trained on Mel-Frequency Cepstral Coefficients (MFCCs) with concatenated delta and delta-delta features. ConvDMM-50 and PASE-50 are both trained on 50 hours of LibriSpeech, ConvDMM-960, Wav2Vec-960 and MockingJay-960 are trained on 960 hours. ConvDMM-360 is trained on the 360 hours of the clean LibriSpeech dataset. RDVQ is trained on the Places400k spoken caption dataset [@harwath2016unsupervised]. We do not train any of the representation learning systems that are compared against ConvDMM on our own. We use the publicly available pre-trained checkpoints to extract features. The linear classifiers used to evaluate the features extracted from unsupervised learning systems are trained on different subsets of WSJ train dataset, ranging from 4 mins (0.1%) to 40 hours (50%). To study the effect of modeling temporal structure in the latent space as in ConvDMM, we train a Gauss VAE which is similar to the ConvDMM except that it does not contain the transition model and hence, is a traditional VAE with isotropic Gaussian priors over the latent states [@kingma2013auto]. To generate the numbers in the table we perform the following steps. Consider, for example, the column labelled 1% as we describe how the numbers are generated for different models (rows). We randomly pick 1% of the speech utterances in the WSJ train dataset. This is performed three times with different random seeds, yielding three different 1% data splits of labelled utterances from the WSJ train dataset. We then train linear classifiers on the features extracted using different representation learning systems, on each of the three splits five times with different random seeds. This gives us a total of 15 classification and recognition error rates. The final number is the mean of these numbers after removing the outliers. Any number greater than $q_3 + 1.5*iqr$ or less than $q_1 - 1.5*iqr$, where $q_1$ is the first Quartile, $q_3$ is the third Quartile and $iqr$ is the inter-quartile range, is considered an outlier. We follow the same procedure to create different training splits, 2%, 5%, 10%, 50%, from the WSJ train dataset and present classification error rates in the table for all splits. Figure \[box\] shows the box plot for the PER on WSJ eval92 dataset using features extracted from different models. In terms of PER, ConvDMM-50 outperforms PASE by 23.4 percentage points (pp), MockingJay by 15.4pp and RDVQ by 6.3pp under the scenario when 1% of labeled training data is available to the linear phone recognizer, which corresponds to approximately 300 spoken utterances ($\approx$ 40 mins). Compared to Wav2Vec, ConvDMM lags by 0.2pp, but the variance in Wav2Vec results is very high as can be seen in Figure \[box\]. Under the 50% labeled data scenario, ConvDMM-50 outperforms MockingJay by 14.4pp, PASE by 19.1pp, RDVQ by 11.5pp and lags Wav2Vec by 6.2pp. The gap between ConvDMM-50 and RDVQ widens in the 50% labeled data case. ConvDMM-960 similarly outperforms all the methods under the 1% labeled data scenario, outperforming Wav2Vec, the second best method, by 5.1pp. Also the variance in the ConvDMM-960 results is much lower than Wav2Vec (See Figure \[box\]). ConvDMM systematically outperforms the Gauss VAE which does not model the latent state transitions, showing the value of prior structure. ConvDMM-PASE which is the ConvDMM model built on top of PASE features instead of the MFCC features, outperforms PASE features by 25.8pp under the 1% labeled data scenario. A significant gap exists under all data scenarios. Similar results can be observed with ConvDMM-Wav2Vec model, but the improvements over Wav2Vec features is not as drastic, probably due to the fact that Wav2Vec already produces very good features. For low shot phone recognition with 0.1% labeled ($\approx$ 4 mins), ConvDMM-960 significantly outperforms all other methods. Surprisingly, RDVQ shows excellent performance under this scenario. ConvDMM-Wav2Vec-960 performs 10pp better than ConvDMM-960 trained on MFCC features and 38pp better than Wav2Vec features alone. We could not get below 90% PER with MockingJay and hence, skip reporting the results. Lastly, we compare the performance of features extracted using unsupervised learning systems trained on LibriSpeech vs features extracted using the fully supervised system neural network acoustic model trained on the task of phone recognition on 960 hours of labeled data (See the row labeled Supervised Transfer-960). The supervised system has the same CNN encoder as the ConvDMM. There is a glaring gap between the supervised system and all other representation learning techniques, even in the very few data regime (0.1%). This shows there is still much work to be done in order to reduce this gap. Related Work {#sec:5} ============ Another class of generative models that have been used to model speech but not explored in this work are the autoregressive models. Autoregressive models, a class of explicit density generative models, have been used to construct speech density estimators. Neural Autoregressive Density Estimator (NADE) [@uria2016neural] is a prominent earlier work followed by more recent Wavenet [@oord2016wavenet], SampleRNN [@mehri2016samplernn] and MelNet [@vasquez2019melnet]. An interesting avenue of future research is to probe the internal representations of these models for linguistic information. We note that, Waveglow, a flow based generative model is recently proposed as an alternative to autoregressive models for speech [@prenger2019waveglow]. Conclusions =========== In this work, we design the Convolutional Deep Markov Model (ConvDMM), a Gaussian state-space model with non-linear emission and transition functions parametrized by deep neural networks. The main objective of this work is to demonstrate that generative models can reach the same, or even better, performance than self supervised models. In order to do so, we compared the ability of our model to learn linearly separable representations, by evaluating each model in terms of PER and FER using a simple linear classifier. Results show that our generative model produces features that outperform multiple self-supervised learning methods on phone classification and recognition task on Wall Street Journal. We also find out that these features can achieve better performances than all other evaluated features when learning the phone recogniser with very few labelled training examples. Another interesting outcome of this work is that by using self-supervised extracted features as input of our generative model, we produce features that outperforms every other one in the phone recogniser task. Probably due to enforcing temporal structure in the latent space. Lastly, we argue that features learned using unsupervised methods are significantly worse than features learned by a fully supervised deep neural network acoustic model, setting the stage for future work.

--- abstract: 'We study expansive properties for the geodesic and horocycle flows on Riemann surfaces of constant negative curvature. It is well-known that the geodesic flow is expansive in the sense of Bowen-Walters and the horocycle flow is positive and negative separating in the sense of Gura. In this paper, we give a new proof for the expansiveness of the geodesic flow and show that the horocycle flow is positive and negative kinematic expansive in the sense of Artigue as well as expansive in the sense of Katok/Hasselblatt but not expansive in the sense of Bowen-Walters. We also point out that the geodesic flow is neither positive nor negative separating.' author: - | [Huynh Minh Hien]{}\ Department of Mathematics,\ Quy Nhon University,\ 170 An Duong Vuong, Quy Nhon, Vietnam;\ e-mail: huynhminhhien@qnu.edu.vn title: Expansiveness for the geodesic and horocycle flows on compact Riemann surfaces of constant negative curvature --- [expansiveness; geodesic flow; horocycle flow]{} Introduction ============ The study of expansive flows started in 1972 with the works of Bowen/Walters [@bw] and Flinn [@flinn]. In [@bw], the authors generalized the definition of expansive homeomorphisms to introduce a reasonable definition of expansiveness for flows that is called ‘expansive in the sense of Bowen and Walters’ (or shortly BW–expansive). Since then, there have been different varieties of expansive flows introduced. In 1984, Komuro [@komuro] gave the notions ‘C–expansive’ (as the same to ‘BW–expansive’), ‘K–expansive’ (as the same to ‘expansive’ in the same of Flinn) and ‘K$^*$–expansive’ to investigate geometric Lorenz attractors. In general, ‘K–expansive’ is weaker than ‘BW–expansive’ but stronger than ‘K$^*$–expansive’. In the case of fixed-point-free flows on compact metric spaces, the three notions are equivalent (see [@artigue2; @oka]). A different and very interesting kind of expansiveness called ‘separating’ was discovered by Gura [@gura] in 1984. The author showed that the horocycle flow on a compact surface with negative curvature is positive and negative separating. His definition in [@gura] requires to separate every pair of points in different orbits. The author also proved a remarkable result: every global time change of such flow is positive and negative separating. In 1995, Katok/Hasselblatt [@kh] gave another kind of expansiveness (also called KH–expansiveness) which is weaker than BW–expansiveness but implies separation. It then was showed by Artigue [@artigue2] that a flow is KH–expansive if and only if it is separating and the set of its fixed points is open. Recently, in 2016, Artigue [@artigue] used the term ‘geometric expansive’ as [*K*]{}–expansive and introduced the term ‘kinematic expansive’ which is a stronger property than separation and weaker than BW–expansiveness. The author also considered the forms of ‘strong kinematic expansive’, ‘geometric separating’, ‘strong separating’ and ‘separating’ flows. Examples are given to analyze the relationships among the above definitions. Some interesting properties are proved in different contexts: surfaces, suspension flows and compact metric spaces. Regarding properties of the geodesic flow, in 1967, Anosov [@anosov] showed that the geodesic flow on compact Riemannian manifolds with negative curvature is hyperbolic. In 1972, it was proved by Bowen [@bowen-pe] that hyperbolic flows are BW–expansive and consequently the geodesic flow on compact Riemannian manifolds of negative curvature is BW–expansive. As mentioned above, in 1984, Gura [@gura] showed that the horocycle on compact surface of negative curvature is positive and negative separating. The paper is organized as follows. In the next section, we introduce the necessary background material which is well-known in principle [@bedkeanser; @einsward; @ratcliff]. Section \[sec3\] is devoted to consider expansive properties mentioned above for the geodesic and horocycle flows on compact factors of the hyperbolic plane. A new detailed proof for BW–expansiveness of the geodesic flow (Theorem \[geoex\]) via a property of the injectivity radius is given. The horocycle flow on this model is not only positive/negative separating but also positive/negative kinematic expansive (Theorem \[pne\]) as well as KH–expansive (Theorem \[khthm\]). In the end, we point out that the horocycle flow is not BW–expansive and the geodesic flow is neither positive nor negative separating; see Remark \[rm\]. Preliminaries ============= We consider the geodesic and horocyle flows on compact Riemann surfaces of constant negative curvature. It is well-known that any compact orientable surface with constant negative curvature is isometric to a factor $\Gamma\backslash \H^2=\{\Gamma z, z\in\H^2 \}$, where $\H^2=\{z=x+iy\in \C:\, y>0\}$ is the hyperbolic plane endowed with the hyperbolic metric $ds^2=\frac{dx^2+dy^2}{y^2}$ and $\Gamma $ is a Fuchsian group that is discrete subgroup of the group $\PSL(2,\R)={{\rm SL}}(2,\R)/\{\pm E_2\}$; where ${{\rm SL}}(2,\R)$ is the group of all real $2\times 2$ matrices with unity determinant, and $E_2$ denotes the unit matrix. In the hyperbolic plane model, geodesics are vertical lines and semi-circles centered on the real axis. The group $\PSL(2,\R)$ acts transitively on $\H^2$ by Möbius transformations $z\mapsto \frac{az+b}{cz+d}$. If the action is free of fixed points, then the factor $\Gamma\backslash\H^2$ has a Riemann surface structure that is a closed Riemann surface of genus at least $2$ and has the hyperbolic plane $\H^2$ as the universal covering. The unit tangent bundle $T^1\H^2$ is isometric to the group $\PSL(2,\R)$ and as a consequence, the unit tangent bundle $T^1(\Gamma\backslash\H ^2)$ is isometric to the quotient space $\Gamma\backslash \PSL(2,\R)=\{\Gamma g,g\in\PSL(2,\R)\}$, which is the system of right co-sets of $\Gamma$ in $\PSL(2,\R)$, by an isometry $\Xi$. Since ${\rm PSL}(2, \R)$ is connected, also $\Gamma\backslash {\rm PSL}(2, \R)$ is connected. Furthermore, $X=\Gamma\backslash {\rm PSL}(2, \R)$ is a three-dimensional real analytic manifold. The geodesic flow on $T^1\H^2$ can be described as the flow $\varphi_t^\G(g)=ga_t$ on $\G:=\PSL(2,\R)$, where $a_t\in\G$ denotes the equivalence class obtained from the matrix $A_t=\scriptsize\big(\begin{array}{cc} e^{t/2} & 0\\ 0 & e^{-t/2} \end{array}\big)$, and whence the geodesic flow $(\varphi_t^\X)_{t\in\R}$ on $\X=T^1(\Gamma\backslash\H^2)$ can be described as the ‘quotient flow’ $$\varphi^X_t(\Gamma g)=\Gamma g a_t$$ on $X=\Gamma\backslash\PSL(2,\R)$ by the conjugate relation $$\label{ce} \varphi_t^\X=\Xi^{-1}\circ\varphi_t^X\circ\Xi.$$ A horocycle is a (euclidean) circle tangent to real axis or a horizontal line. The stable and unstable horocycle flows on $T^1\H^2$ can be described as the flows: $\theta_t^\G(g)=gb_t,\, \eta_t^\G(g)=gc_t$ on $\G$; where $b_t,c_t\in\PSL(2,\R)$ denote the equivalence classes obtained from the matrices $B_t=\scriptsize\big(\begin{array}{cc} 1 & t\\ 0 & 1 \end{array}\big),C_t=\scriptsize\big(\begin{array}{cc}1 &0\\ t&1 \end{array} \big)\in{{\rm SL}}(2,\R)$. Therefore the stable and unstable horocycle flows $(\theta_t^\X)_{t\in\R}$, $(\eta_t^\X)_{t\in\R}$ on $\X=T^1(\Gamma\backslash\H^2)$ can be equivalently described as the flows $$\theta^X_t(\Gamma g)=\Gamma g b_t, \quad \eta^X_t(\Gamma g)=\Gamma g c_t$$ on $X=\Gamma\backslash\PSL(2,\R)$ by the conjugate relations $$\label{cr} \theta_t^\X=\Xi^{-1}\circ\theta_t^X\circ\Xi,\quad \eta_t^\X=\Xi^{-1}\circ\eta_t^X\circ\Xi \quad \mbox{for all}\quad t\in\R.$$ There are some more advantages to work on $X=\Gamma\backslash\PSL(2,\R)$ rather than on $\X=T^1(\Gamma\backslash\H^2)$. For example, one can calculate explicitly the stable and unstable manifolds at a point $x$ to be $$W^s_X(x)=\{\theta^X_t(x),t\in\R\} \quad \mbox{and}\quad W^u_X(x)=\{\eta^X_t(x), t\in\R\}.$$ The flow $(\varphi^X_t)_{t\in\R}$ is hyperbolic, that is, for every $x\in X$ there exists an orthogonal and $(\varphi_t^X)_{t\in\R}$-stable splitting of the tangent space $T_xX$ $$T_x X= E^0(x)\oplus E^s(x)\oplus E^u(x)$$ such that the differential of the flow $(\varphi_t^X)_{t\in \R}$ is uniformly expanding on $E^u(x)$, uniformly contracting on $E^s(x)$ and isometric on $E^0(x)=\langle \frac{d}{dt}\varphi_t^X(x)|_{t=0}\rangle$. One can choose $$\begin{aligned} E^s(x) = \Big\langle\frac{d}{dt}\,\theta^X_t(x)\Big|_{t=0}\Big\rangle \quad\mbox{and}\quad E^u(x) = \Big\langle\frac{d}{dt}\,\eta^X_t(x)\Big|_{t=0}\Big\rangle. \end{aligned}$$ The horocycle flows $(\theta^\X)_{t\in\R}$ and $(\eta^\X)_{t\in\R}$ are ergodic [@markus]. If the space $\Gamma\backslash\H^2$ has a finite volume, each orbit is either periodic or dense. In the case that the space $\Gamma\backslash\H^2$ is compact, there are no periodic orbits for the horocycle flows. General references for this section are [@bedkeanser; @einsward], and these works may be consulted for the proofs to all results which are stated above. In what follows, we will drop the superscript $X$ from $(\varphi^X_t)_{t\in\R},(\theta^X_t)_{t\in\R},(\eta^X_t)_{t\in\R}$ to simplify notation. We consider the stable horocycle flow only and use the term ‘horocycle flow’ for it. In the whole present paper, we always assume the action of $\Gamma$ on $\H^2$ to be free (of fixed points) and the factor $\Gamma\backslash\H^2$ to be compact. Note that $\Gamma\backslash\H^2$ is compact if and only if $\Gamma\backslash\PSL(2,\R)$ is compact. In the rest of this section we collect some notions and useful technical results. \[at\] There is a natural Riemannian metric on $\G=\PSL(2,\R)$ such that the induced metric function $d_\G$ is left-invariant under $\G$ and $$d_\G(a_t,e)=\frac{1}{\sqrt 2}|t|, \quad d_\G(b_t,e)\leq |t|,\quad d_\G(c_t,e)<|t|\quad\mbox{for all}\quad t\in\R$$ where $e=\pi(E_2)$ is the unity of $\G$. We define a metric function $d_{X}$ on $X=\Gamma\backslash\PSL(2,\R)$ by $$d_{X}(x_1, x_2) =\inf_{\gamma_1, \gamma_2\in\Gamma} d_{\G}(\gamma_1 g_1, \gamma_2 g_2) =\inf_{\gamma\in\Gamma} d_{\G}(g_1, \gamma g_2),$$ where $x_1=\Gamma g_1 $, $x_2=\Gamma g_2$. In fact, if $X$ is compact, one can prove that the infimum is a minimum: $$d_{X}(x_1, x_2)=\min_{\gamma\in\Gamma} d_{\G}(g_1, \gamma g_2).$$ It is possible to derive a uniform lower bound on $d_{\G}(g, \gamma g)$ for $g\in \PSL(2,\R)$ and $\gamma\in\Gamma\setminus\{e\}$. \[sigma\_0\] If the space $X=\Gamma\backslash\PSL(2,\R)$ is compact, then there exists $\sigma_0>0$ such that $$d_\G(\gamma g, g)>\sigma_0\quad\mbox{for all}\quad \gamma\in\Gamma\setminus\{e\}.$$ The number $\sigma_0$ is called an injectivity radius. See [@ratcliff Lemma 1, p. 237] for a similar result. For $g=\pi(G)\in\PSL(2,\R), G=\big(\scriptsize\begin{array}{cc}a&b\\c&d\end{array} \big)$, the trace of $g$ is defined by $$\tr(g)=|a+d|.$$ If the action of $\Gamma$ on $\H^2$ is free and the factor $\Gamma\backslash\H^2$ is compact then all elements $g\in\Gamma\setminus\{e\}$ are hyperbolic [@ratcliff Theorem 6.6.6], i.e. $\tr(g)>2$. Furthermore, one gets a stronger result: \[hy\] If the factor $\Gamma\backslash\H^2$ is compact, then there exists $\eps_*>0$ such that $$\tr(g)\geq 2+\eps_*\quad \mbox{for all}\quad g\in \Gamma\setminus \{e\}.$$ Here are some more auxiliary results. \[konvexa\] (a) For every $\delta>0$ there is $\rho>0$ with the following property. If $G=\Big(\begin{array}{cc}g_{11}&g_{12}\\ g_{21}&g_{22} \end{array}\Big)\in{{\rm SL}}(2,\R)$ satisfies $|g_{11}-1|+|g_{12}|+|g_{21}|+|g_{22}-1|<\rho$ then $d_{\G}(g, e)<\delta$ for $g=\pi(G)$, where $\pi: {\rm SL}(2, \R)\to {\rm PSL}(2, \R)$ is the natural projection. \(b) For every $\eps>0$ there is $\delta>0$ with the following property. If $g,h\in\G$ satisfying $d_\G(g,e)<\delta$ then there are $$G=\Bigg(\begin{array}{cc}g_{11}&g_{12}\\ g_{21}&g_{22} \end{array}\Bigg)$$ such that $g=\pi(G), h=\pi(H)$ and $$|g_{11}-1|+|g_{12}|+|g_{21}|+|g_{22}-1|<\eps.$$ [**Proof:**]{} (a) See [@HK Lemma 2.17 (a)] for a proof. \(b) Indeed, suppose on contrary that $$\label{hodett} |g^j_{11}-1|+|g^j_{12}|+|g^j_{21}|+|g^j_{22}-1|\ge\eps_0$$ for some sequence $d_{\G}(g^j, e)\to 0$ and all $G^j\in {\rm SL}(2, \R)$ such that $g^j=\pi(G^j)$. For $j\in\N$ take any $G^j\in{{\rm SL}}(2,\R)$ so that $g^j=\pi(G^j)$. From (a) we deduce that $|g^j_{12}|+|g^j_{21}|\to 0$, $|g^j_{11}|\to 1$, $|g^j_{22}|\to 1$, and $g^j_{11} g^j_{22}\to 1$. Thus, along a subsequence which is not renamed, either $g^j_{11}\to 1$, $g^j_{22}\to 1$ or $g^j_{11}\to -1$, $g^j_{22}\to -1$. The first case is impossible in view of (\[hodett\]). In the second case we consider $\tilde{G}^j=-G^j$ which also has $g^j=\pi(\tilde{G}^j)$. But then (\[hodett\]) implies $$|g^j_{11}+1|+|g^j_{12}|+|g^j_{21}|+|g^j_{22}+1|\ge\eps_0,$$ and once more this is impossible. Let $\phi:\R\times M\to M$ be a flow. 1. A point $x\in M$ is called a [*fixed point*]{} (or [*singular point*]{}) if $$\phi_t(x)=x\quad\mbox{for all}\quad t\in\R.$$ 2. A point $x\in M$ is called a [*periodic point*]{} if there is $T>0$ such that $$\phi_T(x)=x.$$ \[fixedpoint\] Assume that $X=\Gamma\backslash\PSL(2,\R)$ is compact. Then the flow $(\theta_t)_{t\in\R}$ does not have a periodic point. In particular, it has no fixed points. [**Proof:**]{} Suppose in contrary that $x=\Gamma g$ is a periodic point of $(\theta_t)_{t\in\R}$, i.e. $\theta_T(x)=x$ for some $T>0$. Then $g^{-1}\gamma g=b_T$ for some $\gamma\in\Gamma$ implies $\tr(\gamma)=\tr(b_T)=2$. It follows from Lemma \[hy\] that $\gamma=e$. Therefore $g=gb_T$ yields $T=0$ which is a contradiction. The latter assertion is obvious. [$\Box$]{} Two continuous flows $\phi:\R\times X\to X$ and $\psi: \R\times Y\to Y$ is said to be [*equivalent*]{} if there is a homeomorphism $h:X\to Y$ such that $\phi_t=h^{-1}\psi_th$ for all $t\in\R$. Via and , the flows $(\varphi_t)_{t\in\R}$ and $(\varphi^\X_t)_{t\in\R}$ are equivalent, and so are $(\theta^\X_t)_{t\in\R}$ and $(\theta_t)_{t\in\R}$. It is easy to see that all the expansive properties introduced in the next section are invariant under equivalence. Expansive properties {#sec3} ==================== In this section we study BW–expansive, kinematic expansive, separating, and KH–expansive properties for the geodesic flow $(\varphi_t^\X)_{t\in\R}$ and the horocycle flow $(\theta^\X_t)_{t\in\R}$ on $\X=T^1(\Gamma\backslash\H^2)$. We reprove that the geodesic flow is BW–expansive. The horocycle flow is positive/negative kinematic expansive as well as KH–expansive but not BW–expansive. BW–expansiveness ---------------- This subsection provides a new detailed proof of the expansiveness in the sense of Bowen-Walters for the geodesic flow $(\varphi_t^\X)_{t\in\R}$ owing to a characteristic property of the injectivity radius. Let $(M,d)$ be a compact metric space. A continuous flow $\phi:\R\times M\longrightarrow M$ is called [*BW–expansive*]{} if for each $\eps>0$, there exists $\delta>0$ with the following property. If $s:\R\rightarrow \R$ is a continuous function with $s(0)=0$ and $$d(\phi_t(x),\phi_{s(t)}(y))<\delta \quad \mbox{for all}\quad t\in \R$$ then $y=\phi_\tau(x)$ for some $\tau\in (-\eps,\eps)$. It was showed in [@bw Theorem 3] that ‘continuous function’ in the above definition can be replaced by ‘increasing homeomorphism’ in the case of fixed-point-free flows. Then this definition becomes the one in [@flinn], ‘K–expansive’ [@komuro], and ‘geometric expansive’ [@artigue]. \[geoex\] The geodesic flow $(\varphi^\X_t)_{t\in\R}$ is BW–expansive. [**Proof:**]{} Since BW–expansiveness is an invariant under equivalence [^1], it follows from that it suffices to show that the flow $(\varphi_t)_{t\in\R}$ is BW–expansive. Let $\eps>0$ be given, $\eps_0=e^{\eps/2}-e^{-\eps/2}>0$ and set $\delta=\delta(\eps_0)<\sigma_0/4$ as in Lemma \[konvexa\](b); here $\sigma_0$ is from Lemma \[sigma\_0\]. Let $x,y\in X$ and $s:\R\to\R$ be continuous with $s(0)=0$ such that $$d_X(\varphi_{s(t)}(y),\varphi_t(x))<\delta\quad \mbox{for all}\quad t\in\R.$$ Write $x=\Gamma g, y=\Gamma h$ for $g,h\in\G$. For every $t\in \R$, there is $\gamma(t)\in\Gamma$ so that $$\begin{aligned} \label{gammat} d_X(\varphi_{s(t)}(y),\varphi_t(x)) =d_X(\Gamma ha_{s(t)}, \Gamma ga_t) =d_\G(ha_{s(t)},\gamma(t) ga_t)<\delta. \end{aligned}$$ We claim that $\gamma(t)=\gamma(0)=:\gamma$ for all $t\in\R$. For any $t_1,t_2\in\R$, we have $$\begin{aligned} \lefteqn{d_\G(\gamma(t_2)^{-1}\gamma(t_1)ga_{t_1},ga_{t_1})} \\ &=& d_\G(\gamma(t_1)ga_{t_1},\gamma(t_2)ga_{t_1}) \\ &\leq& d_\G(\gamma(t_1)ga_{t_1},ha_{s(t_1)}) +d_\G(ha_{s(t_1)},ha_{s(t_2)}) +d_\G(ha_{s(t_2)},\gamma(t_2)ga_{t_2}) \\ &&+\ d_\G(\gamma(t_2)ga_{t_2},\gamma(t_2)ga_{t_1}) \\ &=& d_\G(\gamma(t_1)ga_{t_1},ha_{s(t_1)}) +d_\G(a_{s(t_1)},a_{s(t_2)}) +d_\G(ha_{s(t_2)},\gamma(t_2)ga_{t_2}) +d_\G(a_{t_2},a_{t_1} ) \\ &\leq& 2\delta+\frac{1}{\sqrt 2}|s(t_1)-s(t_2)|+\frac{1}{\sqrt 2}|t_1-t_2|,\end{aligned}$$ due to Lemma \[at\]. For given $L>0$, we verify that $\gamma(t)=\gamma(0)$ for all $t\in [-L,L]$. Indeed, since $s:[-L,L]\to\R$ is uniformly continuous, there is $0<\rho=\rho(L,\delta)<\delta$ such that if $t_1,t_2\in [-L,L]$ and $|t_1-t_2|<\rho$ then $|s(t_1)-s(t_2)|<\delta$. For $t_1,t_2\in [0,\rho/2]$, then $|t_1-t_2|<\rho$ implies $|s(t_1)-s(t_2)|<\delta$. This yields $$d_\G(\gamma(t_2)^{-1}\gamma(t_1)c_1(t_1),c_1(t_1))<4\delta <\sigma_0.$$ From the property of $\sigma_0$ in Lemma \[sigma\_0\], it follows that $\gamma(t_2)=\gamma(t_1)$ for $|t_1-t_2|<\rho$. Here if we specialize this to $t_1=0$ and $t_2\in [0,\rho/2]$, then $\gamma(t_2)=\gamma(0)$ for all $t_2\in [0,\rho/2]$. Then we repeat the argument for $t_1=\rho/2$ and $t_2\in [\rho/2, \rho]$, we deduce that $\gamma(t)=\gamma(0)$ for all $t\in [0,\rho]$, which upon further iteration leads to $\gamma(t)=\gamma(0)$ for all $t\in [0,L]$ and similarly $\gamma(t)=\gamma(0)$ for all $t\in [-L,0]$. Therefore, $$\label{st} d_X(\varphi_{s(t)}(y),\varphi_t(x))=d_\G(a_{-t}g^{-1}\gamma ha_{s(t)},e)<\delta \quad\mbox{for all}\quad t\in\R.$$ Write $g^{-1}\gamma h=\pi(K)$ for $K=\big(\scriptsize\begin{array}{cc} a&b\\c&d\end{array}\big)\in{{\rm SL}}(2,\R)$. Thus $$A_{-t}KA_{s(t)}=\Big(\begin{array}{cc} ae^{\frac{s(t)-t}2}&be^{-\frac{s(t)+t}2}\\c e^{\frac{s(t)+t}2}&d e^{\frac{t-s(t)}2}\end{array}\Big)$$ together with implies $$\label{abcdeq} \big||a|e^{\frac{s(t)-t}2}-1\big|+|b|e^{-\frac{s(t)+t}2} +|c| e^{\frac{s(t)+t}2}+ \big||d| e^{\frac{t-s(t)}2}-1\big|<\eps_0\quad \mbox{for all}\quad t\in\R,$$ using Lemma \[konvexa\](b). Then there is $M>0$ such that $|s(t)-t|\leq M$ for all $t\in\R$ and hence $s(t)+t\to +\infty$ as $t\to +\infty$ and $s(t)+t\to-\infty$ as $t\to -\infty$. Together with this yields $b=c=0$. Since $ad=1$ we can assume that $a>0, d>0$ and $a=e^{\tau/2},d=e^{-\tau/2}$ for some $\tau\in\R$. This implies that $g^{-1}\gamma h=a_\tau$ or $y=\varphi_\tau(x)$. Finally, using $\big||a|-1\big|+\big||d|-1\big|<\eps_0$, we have $e^{|\tau|/2}-e^{-|\tau|/2}<\eps_0=e^{\eps/2}-e^{-\eps/2}$, consequently $|\tau|< \eps$ which completes the proof. [$\Box$]{} Kinematic expansiveness, separation ----------------------------------- This subsection is devoted to demonstrate the kinematic expansiveness for the horocycle flow. It is also showed that the horocycle flow is not BW–expansive while the geodesic flow is not positive/negative separating. \[dnke\]Let $(M,d)$ be a compact metric space. A continuous flow $\phi:\R\times M\longrightarrow M$ is called [*kinematic expansive*]{} if for each $\eps>0$, there exists $\delta>0$ with the following property. If $$\label{exdn}d(\phi_t(x),\phi_{t}(y))<\delta \quad \mbox{for all}\quad t\in \R$$ then $y=\phi_\tau(x)$ for some $\tau\in(-\eps,\eps)$. If the inequality in holds for $t\in [0,\infty)$ (resp. $t\in (-\infty, 0]$) then the flow is called ‘positive kinematic expansive’ (resp. ‘negative kinematic expansive’). If the condition $\tau\in(-\eps,\eps)$ is ignored, the flow is called separating in the sense of Gura. Let $(M,d)$ be a compact metric space. A continuous flow $\phi:\R\times M\longrightarrow M$ is called [*separating*]{} if there exists $\delta>0$ with the following property. If $$\label{sdn} d(\phi_t(x),\phi_t(y))<\delta \quad \mbox{for all}\quad t\in \R$$ then $y=\phi_\tau(x)$ for some $\tau\in\R$; i.e. $x$ and $y$ lie on the same orbit. The number $\delta$ is called a ‘separating constant’. If the inequality in holds only for $t\in [0,\infty)$ (resp. $t\in (-\infty, 0]$) then the flow is called ‘positive separating’ (resp. ‘negative separating’). It is showed in [@gura] that the horocyle flow on a compact surface of negative curvature is positive and negative separating. The next result gives a stronger property of the horocyle flow on $\Gamma\backslash\H^2$ which is a compact Riemann surface with constant negative curvature. \[pne\] The horocycle flow $(\theta^\X_t)_{t\in\R}$ is positive and negative kinematic expansive. [**Proof:**]{} We consider the positive kinematic expansiveness only. Since the positive kinematic expansiveness is invariant under equivalence, it follows from that it suffices to show that the flow $(\theta_t)_{t\in\R}$ is positive expansive. Let $\eps>0$ be given and set $\delta=\delta(\eps)$ as in Lemma \[konvexa\](b). Let $x,y\in X$ be such that $$d_X(\theta_t(x),\theta_t(y))<\delta\quad\mbox{for all}\quad t\geq 0.$$ Write $x=\Gamma g, y=\Gamma h$ for $g,h\in\PSL(2,\R)$. For every $t\geq 0$, there is $\gamma(t)\in\Gamma$ so that $$\begin{aligned} d_X(\theta_t(y),\theta_t(x)) =d_X(\Gamma hb_t, \Gamma gb_t) =d_\G(\gamma(t) hb_t,gb_t)\leq \delta. \end{aligned}$$ Analogously to the proof of Theorem \[geoex\], we can check that $\gamma(t)=\gamma(0)=:\gamma$ for all $t\in\R$ (here we do not need the uniform continuity of $s(t)=t$). It follows from that $$\label{dd}d_X(\theta_{t}(y),\theta_t(x))=d_\G(b_{-t}g^{-1}\gamma hb_{t},e)<\delta\quad\mbox{for all}\quad t\geq 0.$$ Write $g^{-1}\gamma h=\pi(K)$ for $K=\big(\scriptsize\begin{array}{cc} g_{11}&g_{12}\\g_{21}&g_{22}\end{array}\big)\in{{\rm SL}}(2,\R)$. Then $$B_{-t}KB_{t}=\Big(\begin{array}{cc} g_{11}-tg_{21}&g_{11}t+g_{12}-t(g_{21}t+g_{22})\\g_{21} &g_{21}t+g_{22}\end{array}\Big)$$ together with imply that for all $t\geq 0$, $$\label{abcdeq2} \big|\,|g_{11}-tg_{21}|-1\big|+\big|(g_{11}-g_{22})t+ g_{21}t^2+g_{12}\big|+|g_{21}|+\big|\,|g_{21}t+g_{22}|-1\big|<\eps_0;$$ here $\eps_0=\eps_0(\delta)<\eps$ obtained from Lemma \[konvexa\](b). Letting $t\to +\infty$ yields $g_{21}=0, g_{11}=g_{22}$ and $|g_{12}|<\eps_0$. Since $g_{11}g_{22}-g_{12}g_{21}=1$, we get $g_{11}=g_{22}=1$ or $g_{11}=g_{22}=-1$ that leads to $K=\big(\scriptsize\begin{array}{cc} 1&g_{12}\\0&1\end{array}\big)$ or $K=\big(\scriptsize\begin{array}{cc} -1&g_{12}\\0&-1\end{array}\big)$ and hence $g^{-1}\gamma h=b_{\tau}$ with $\tau=g_{12}$, $|\tau|<\eps$. This completes the proof. [$\Box$]{} \[rm\] \(a) The horocycle flow is not BW–expansive. Indeed, for any $\delta>0$, we need to find $x, y\in X$ and $s:\R\to \R$ continuous with $s(0)=0$ such that $d_X(\theta_t(x),\theta_{s(t)}(y))<\delta\quad\mbox{for all}\quad t\in\R$ but the orbits of $x$ and $y$ do not coincide. Take $\rho=\rho(\delta)$ as in Lemma \[konvexa\](a) and choose any $x=\Gamma g$ and $y=\Gamma h$ with $h,g\in\PSL(2,\R), h\ne g, h^{-1}g=\pi(K)$, $K=\Big(\begin{array}{cc}a&0\\ 0& d\end{array}\Big)$ such that $ad=1$, $|a-1|<\rho, |d-1|<\rho$ and $\tr(h^{-1}g)=|a+d|<2+\eps_*$; recall $\eps_*>0$ in Lemma \[hy\], we have $d_\G(h^{-1}g,e)<\delta$ due to Lemma \[konvexa\](a). Setting $s(t)=\frac{d}{a}t$, we have $$\begin{aligned} d_X(\theta_{s(t)}(x),\theta_t(y)) &=&d_X(\Gamma gb_{s(t)},\Gamma hb_t)\leq d_\G(gb_{s(t)},hb_t )=d_\G(b_{-t}h^{-1}gb_{s(t)},e)\\ &=&d_\G(h^{-1}g,e)<\delta\quad\mbox{for all}\quad t\in\R; \end{aligned}$$ using $b_{-t}h^{-1}gb_{s(t)}=h^{-1}g$. It remains to verify that $x$ and $y$ are not in the same orbit. Indeed, otherwise there would exist $\tau\in\R$ such that $y=\theta_\tau(x)$, then $\gamma h= g b_\tau$ for some $\gamma\in\Gamma$ implies $\tr(\gamma)=\tr(gb_\tau h^{-1})=\tr(b_\tau h^{-1}g)=|a+d|<2+\eps_*$. It follows from Lemma \[hy\] that $\gamma=e$. This yields $b_{-\tau}=h^{-1}g=\pi(K)$ and hence $\tau=0, h=g$ which contradicts to $h\ne g$. \(b) The flow geodesic flow is neither positive nor negative separating. Indeed, we consider the equivalent flow $(\varphi_t)_{t\in\R}$. Since the group $\Gamma$ is discrete, for every $\delta>0$, there is an $s\in (-\delta,\delta)$ such that $a_tb_{-s}\notin\Gamma$ for all $t\in\R$. Set $x=\Gamma e$ and $y=\Gamma b_{s}$ to have $$\begin{aligned} d_X(\varphi_{t}(x),\varphi_t(y)) =d_X(\Gamma a_{t},\Gamma b_{s}a_t)\leq d_\G(a_{t},b_sa_t )\leq |s|e^{-t} <\delta\quad \mbox{for all}\quad t\geq 0. \end{aligned}$$ However, if $y=\varphi_\tau(x)$ then $\Gamma b_s=\Gamma a_\tau$ implies that there is $\gamma=a_\tau b_{-s}\in\Gamma$ which is a contradiction, whence $(\varphi_t)_{t\in\R}$ is not positive separating. In the same manner one obtains that the flow $(\varphi_t)_{t\in\R}$ is not negative separating. \(c) It is worth to recall that the geodesic flow is BW–expansive but neither positive nor negative kinematic expansive while the horocycle flows are positive and negative kinematic expansive but not BW–expansive. [$\diamondsuit$]{} KH– expansiveness ----------------- In [@kh] Katok and Hasselblatt introduce the following expansiveness: Let $(M,d)$ be a compact space. A continuous flow $\phi_t:M\longrightarrow M$ is called [*KH–expansive*]{} if there exists $\delta>0$ with the following property. If $x\in X, s:\R\to\R$ is continuous, $s(0)=0$ and $d(\varphi_t(x),\varphi_{s(t)}(x))<\delta $ for all $t\in\R, y\in X$ is such that $d(\varphi_t(x),\varphi_{s(t)}(y))<\delta$ for all $t\in\R$ then $x$ and $y$ lie on the same orbit. It is clear that [KH]{}–expansiveness is weaker than BW–expansiveness but implies separation. Furthermore, one has the following result: \[KHthm\] A flow on a compact metric space is KH–expansive if and only if it is separating and the set of its fixed points is open. It follows immediately from propositions \[fixedpoint\] and \[KHthm\] that the flow $(\theta_t)_{t\in\R}$ is KH–expansive, and hence the horocycle flow $(\theta_t^\X)_{t\in\R}$ is KH–expansive owing to . Nevertheless we can verify it directly. \[khthm\] The flow horocycle flow $(\theta^\X_t)_{t\in\R}$ is KH–expansive. [**Proof:**]{} If $x,y\in X, s:\R\to\R$ is continuous, $s(0)=0$ and $$\label{s1} d_X(\theta_t(x),\theta_{s(t)}(x))<\delta \quad\mbox{for all}\quad t\in\R$$ and $$\label{s2} d_X(\theta_t(x),\theta_{s(t)}(y))<\delta\quad\mbox{for all}\quad t\in\R.$$ Analogously to the proof of Theorem \[geoex\], using we can show that there is $M>0$ such that $$|s(t)-t|<M\quad\mbox{for all}\quad t\in\R.$$ This means that $$\label{s3} s(t)\to +\infty\quad \mbox{as}\quad t\to +\infty.$$ It follows from and that $$d_X(\theta_{s(t)}(x), \theta_{s(t)}(y))<2\delta\quad\mbox{for all}\quad t\in\R.$$ Together with , this follows in the same manner of the proof of Theorem \[pne\]. [$\Box$]{} [**Acknowledgments:**]{} This work is supported by Vietnam National Foundation for Science and Technology Development (Grant No. 101.02-2017.304). I would like to thank an anonymous referee for carefully reading my paper and useful suggestions. I enjoyed many fruitful discussions with Alfonso Artigue. [99]{} D. V. Anosov, [*Geodesic flows on closed Riemannian manifolds of negative curvature*]{}, Trudy Mat. Inst. Steklov. [**90**]{} (1967) , Discrete Contin. Dyn. Syst. Ser. A [**33(2)**]{} (2013), 505-525 , Topol. Appl. [**165**]{} (2014), 121-132. , Ergod. Th. & Dynam. Sys. [**36**]{} (2016), 390-421. , Discrete Contin. Dyn. Syst. Ser. A [**38(9)**]{} (2018), 4433-4447. , Oxford University Press, Oxford 1991 , Amer. J. Math. [**Vol. 94**]{} (1972) 1-30. , [J. Differential equations]{} [**12**]{} (1972), 180-193. , [*Ergodic Theory with a View towards Number Theory*]{}, Springer, Berlin-New York 2011 L. Flinn, [*Expansive Flows*]{}, PhD thesis, Warwick University 1972 A. Gura, Horocycle flow on a surface of negative curvature is separating, [*Mat. Zametki*]{} [**36**]{} (1984), 279-284. , Nonlinearity [**28**]{} (2015), 593-623. A. Katok and B. Hasselblatt, [*Introduction to the Modern Theory of Dynamical Systems*]{}, Cambridge University Press, Cambridge 1995. M. Komuro, [*Expansive properties of Lorenz attractors*]{}, The Theory of Dynamical Systems and its Applications to Nonlinear Problems (Toyoto, 1984), World Scientific, Singapore (1984), 4-26 B. Marcus, [*Unique ergodicity on the horocycle flows: the variable curvature case*]{}, Israel J. Math [**21**]{} (1975), 133-144. M. Oka, [*Expansiveness of real flows*]{}, Tsukuba J. Math [**14**]{} (1990), no. 1, 1–8. , 2nd edition, Springer, Berlin-Heidelberg-New York 2006 [^1]: It is showed in [@bw Corollary 4] that BW–expansiveness is an invariant under conjugacy that is weaker than equivalence. Recall hat the flows $(\phi_t)_{t\in\R}$ on $X$ and $(\psi_t)_{t\in\R}$ on $Y$ is said to be conjugate if there is a homeomorphism from $X$ to $Y$ mapping the orbits of $(\phi_t)_{t\in\R}$ onto orbits of $(\psi_t)_{t\in\R}$.

--- author: - 'Dai Aoki$^{1,2}$[^1], Kenji Ishida$^3$[^2], and Jacques Flouquet$^1$[^3]' title: 'Review of U-based Ferromagnetic Superconductors: Comparison between UGe$_2$, URhGe, and UCoGe' --- Introduction: Experimental Probes, Crystal Structure ==================================================== Introduction ------------ The interplay between ferromagnetism and superconductivity (SC) is a challenging problem in the coupling between the two major ground states of condensed matter. This problem was theoretically posed six decades ago in 1957, [@GinzburgJETP1957]. Early experiments in 1958 showed that ferromagnetic (FM) impurities, such as Gd, dissolved in La ($T_{\rm SC}$ = 5.7 K) destroy SC with 2% doping[@MatthiasPRL1958]. Exchange interactions put stringent limits on the occurrence of SC. However, SC can easily coexist with antiferromagnetic (AF) sublattices of localized rare earth (RE) atoms. The first discovered cases were the Chevrel phases REMo$_6$S$_8$ [@IshikawaSSC1977], and soon after, in 1977, another example of RERh$_4$B$_4$ was discovered [@FertigPRL1977]. Basically, on the scale of the SC coherence length, which is larger than the magnetic one, the Cooper pairs go through zero exchange interaction. Two singular cases were ErRh$_4$B$_4$ and HoMo$_6$S$_8$, where SC and ferromagnetism are in competition. Despite the SC temperature ($T_{\rm SC}$) being higher than the Curie temperature ($T_{\rm Curie}$ ) of ferromagnetism, the ground state ends up in the FM ground state with the collapse of the singlet SC. For example, ErRh$_4$B$_4$ is a superconductor below $T_{\rm SC}=8.7\,{\rm K}$ [@SinhaPRL1982]; up on cooling to $T = 1\,{\rm K}$, a compromise between ferromagnetism and SC is realized by forming a modulated structure with a domain of alternating magnetic moments. The period of $100\,{\rm \AA}$ is smaller than the SC coherence length. Upon further cooling below 0.8 K, ferromagnetism becomes the ground state and SC disappears. Here the energy gained by the FM atoms exceeds that of the Cooper pair formation at $T_{\rm SC}$ as the number of quasiparticles involved, $k_{\rm B}T_{\rm SC} \times \rho (E_{\rm F})$, is much lower than 1, where $\rho(E_{\rm F})$ is the electronic density of states [@FlouquetBuzdin]. The occurrence of the modulated structure is discussed theoretically in Refs. [@AndersonPR1959] and . An exotic observation was the detection of magnetic-field-induced SC in 1984 in Eu$_{0.75}$Sn$_{0.25}$Mo$_6$S$_{7.2}$Se$_{0.8}$. [@MeulPRL1984]. This is in agreement with a theoretical prediction given by Jaccarino and Peter in 1962 [@JaccarinoPRL1962], which stresses that the compensation of the exchange internal field by the opposite external magnetic field can overcome the bare Pauli limit of the upper critical field $H_{\rm c2}$. In these localized magnetic SC compounds, $T_{\rm SC}$ is higher than $T_{\rm Curie}$ or $T_{\rm N}$ (for antiferromagnets). There are two types of electrons: localized ones that carries the magnetic moment and itinerant ones that are paired via electron-phonon coupling (see Refs.  for a review). In the reported U compounds UGe$_2$, URhGe, and UCoGe, the $5f$ electrons participate both in the magnetic coupling and in the formation of heavy quasiparticles; quasiparticles with high effective mass $m^*$ ($\sim 20\,m_0$, $m_0$: bare mass of an electron) are detected on the orbit at the Fermi surface (FS). The suggestion of unconventional SC and itinerant FM was given many decades ago on the basis of a Cooper pairing generated by FM spin fluctuation[@FayPRB1980], in which the expression for $T_{\rm SC}$ vanishes at the FM critical pressure because of the poor approximation of the theory. [@MonthouxPRB2001] For a Fermi liquid, a well-known example is liquid $^3$He [@OsheroffPRL1972; @LeggettRMP1975]; however, the system is very far from FM instability[@Flo06_review]. In bulk electronic materials, the first observation was made on UGe$_2$[@SaxenaNature2000], SC emerges under pressure ($P$) near the switch at $P_x \sim 1.2\,{\rm GPa}$ between two FM phases (FM2, FM1) [@HuxleyPRB2001], upon entering the paramagnetic (PM) phase at a higher pressure, no SC is detected. A breakthrough in research on the domain was realized with the discovery of two new cases, URhGe[@AokiNature2001] and UCoGe[@HuyPRL2007], which show FM and SC transitions at ambient pressure. In both examples, the singular feature is that a transverse magnetic field with respect to the easy FM magnetization axis ($H\perp M_0$) leads to the suppression of $T_{\rm Curie}$, enhancing the SC pairing via the enhancement of FM fluctuations. We will see that in the particular case of UCoGe, the weakness of the FM interaction is associated with a strong decrease in $H$ of the SC pairing in a longitudinal field scan ($H \parallel M_0$). An important approach is to restore the normal phase by $H$ above the upper critical field $H_{\rm c2}$. The magnetic field can destroy singlet-pairing SC in two ways. The first one, called the orbital limit, is a manifestation of the Lorentz force on electrons, and the second one, called the Pauli limit, occurs when $H$ breaks the spin antialignment and orients the spin along the field direction due to the Zeeman effect. The critical field of the Pauli limit, $H_{\rm P}$, is expressed as $\mu_0 H_{\rm P}/T_{\rm SC} \sim 1.86\,{\rm T/K}$ on the assumption that the electronic $g$ factor is 2 and $2\Delta(0) = 3.53 k_{\rm B}T_{\rm SC}$. In contrast to most unconventional singlet superconductors, where AF fluctuations are the main glue, in UGe$_2$, URhGe, and UCoGe[@SheikinPRB2001; @Har05; @HuyPRL2007], the upper critical field $H_{\rm c2}$(0) in the three main directions ($a$, $b$, $c$) of these orthorhombic structures exceeds the Pauli limiting field $H_{\rm P}$, as shown in Fig. \[fig1\] (see Sec. \[sec:UCoGe\] for UCoGe), (and the necessity to take into account the strong $H$ dependence of the pairing). Thus, it seems established that a triplet pairing with equal spin pairing (ESP) is realized. In the three cases, the coexistence of SC with FM has been directly verified by neutron scattering, $\mu$SR, and NQR experiments. [@AokiJPSJ2012Rev; @AokiJPSJ2014Rev; @Aok14_actinide; @HattoriJPSJ2014Rev; @HuxleyPhysicaC2015] Previous experimental reviews can be found in Refs. , and . Reviews concerning the theory are given in Refs.  and . Basically, ESP between the same effective spin components seems to be established here. For the PM phase of $^3$He, a beautiful clear case is the A phase of superfluid $^3$He [@OsheroffPRL1972; @LeggettRMP1975]; in the B phase of $^3$He, mixing occurs between spin-down and spin-up components. In our case of FM superconductors, the general form of the order parameter can be more complex with mixing of the magnetic component (see Ref. ). However, owing to the strength of FM exchange coupling in the FM phase, the realization of ESP seems to be achieved. The situation may change on entering the PM phase. The aim of this review paper is to show our new experimental progress with the strong interplay between macroscopic and microscopic properties revealed recently via various NMR experiments (see Ref.  for the strengths of NMR experiments in the study of SC). Special focus will be made on combined magnetic field, pressure, and uniaxial stress ($\sigma$) scans to cross the FM instability and observe their effect on SC. Our report mainly concerns results achieved in Grenoble, Oarai, Kyoto, and Tokai. The discovery of SC in UGe$_2$ was made through a collaboration between Grenoble and Cambridge [@SaxenaNature2000]. The SC of URhGe was discovered in Grenoble [@AokiNature2001] and that of UCoGe was discovered in Amsterdam [@HuyPRL2007]. Special attention on the duality between the local and itinerant character of the 5$f$ electron notably in UGe$_2$, was stressed in the work of Wroclaw[@TrocPRB2012], and the effect of doping with Ru on UCoGe was studied in Prague[@Val15], as well as analyses of SC in both FM and PM phases[@VejpravovaPRB2010]. {width="0.8\hsize"} Experimental probes ------------------- Of course, thermodynamic measurements \[specific heat ($C$), thermal expansion ($\alpha$), magnetization ($M$)\] as well as transport measurements \[resistivity ($\rho$), Hall effect ($R_{\rm H}$), thermoelectric power (TEP, $S$), thermal conductivity ($\kappa$)\] are basic experiments to establish the normal and SC properties. In the crude frame chosen to relate to the physical parameters in the normal and SC phases, we take the view that the SC is driven by the effective mass enhancement due to magnetic fluctuations over the renormalized band mass $m_{\rm B}$ induced by local fluctuations connected with Kondo phenomena. The basic experiments on the normal phase allow the evaluation of the band mass $m_{\rm B}$ and the additional contribution $m^{**}$ given by FM fluctuation to the effective mass $m^\ast = m_{\rm B} + m^{\ast\ast} = (1+\lambda) m_{\rm B}$, where $\lambda \equiv m^{\ast\ast}/m_{\rm B}$ is the so-called mass enhancement factor due to the many-body effect. The renormalized band mass ($m_{\rm B}$) in heavy fermions is the consequence of complex electronic couplings which cannot be restricted to a single impurity Kondo effect. [@Flo06_review] Short-range interactions modify the Kondo temperature of a single U atom; The estimation of the Kondo temperature must be performed in the intermediate-temperature domain, for example, just above $T_{\rm Curie}$ or by decoupling the interaction by the magnetic field. For a large number of heavy-fermion compounds, the relation $\gamma^2 \propto A$ between the Sommerfeld coefficient $\gamma$ of the linear $T$ term of the specific heat $C \sim \gamma T$ and the $A$ coefficient of the $T^2$ term of the resistivity [@Kad86] is used to follow the $P$ and $H$ variations of $m^*$, despite the fact that such a relation is not valid near FM instabilities. [@Mor95; @Mor03] In our studies, the $\gamma^2 \propto A$ relation is roughly obeyed even close to $P_c$; the “hidden” responsible for this cutoff of FM quantum criticality may indicate a sign of the strong first-order nature of $P_c$ and/or the associated change in the FS. Thermal conductivity experiments are a supplementary tool for revealing spin fluctuation phenomena and for trying to derive the anisotropic gap structure in the SC phase. TEP is a very sensitive probe for detecting topological changes in the FS in these complex multiband systems, where classical quantum oscillation techniques as well as photoemission often fail to resolve the full FS structure. Elastic and inelastic neutron scattering are powerful probes, mainly for clarifying the FM transition. Here, we will stress the strengths of NMR experiments. NMR has already been quite successful in the study of conventional [@Slichtertextbook] and unconventional [@Asayamatextbook] superconductors. A major interest is the anisotropy of the spin-lattice ($T_1$) and spin-spin ($T_2$) relaxation times as well as its field dependence. We will see how unique responses are obtained between longitudinal and transverse fluctuations with respect to the FM magnetization axes. Along the three $x$, $y$, and $z$ crystallographic axes, $1/T_1$ is related to the transverse fluctuation via the dynamical susceptibility $\chi''({\bm q},\omega_0)$ as [@Slichtertextbook], $$\begin{aligned} \lefteqn{\left(\frac{1}{T_1T}\right)_{x}}\\ & = & \frac{\gamma_n^2 k_{\rm B}}{(\gamma_e\hbar)^2} \sum_{{\bm q}} \left[ |A^{y}_{\rm hf}|^2\frac{\chi''_{y}({\bm q},\omega_0)}{\omega_0} + |A^{z}_{\rm hf}|^2\frac{\chi''_{z}({\bm q},\omega_0)}{\omega_0}\right], \label{eq:T1}\end{aligned}$$ while $1/T_2$ (spin-spin relaxation rate) is mainly sensitive to the longitudinal magnetic fluctuation with an extra contribution originating from $1/T_1$ and is expressed as $$\begin{aligned} \lefteqn{\left(\frac{1}{T_2}\right)_{x} =\frac{\gamma_n^2 k_{\rm B}T}{(\gamma_e\hbar)^2} \lim_{\omega \rightarrow 0} \sum_{{\bm q}} \left[ |A^{x}_{\rm hf}|^2\frac{\chi''_{x}({\bm q},\omega_0)}{\omega_0}\right]} \\ & & +\left[I(I+1)-m(m+1)-1/2\right]\left(\frac{1}{T_1}\right)_x.\end{aligned}$$ In an uncorrelated condition, the product of $T_1T$ and the spin part of the Knight shift $K_s$ is a constant, $T_1TK_s^2 = R_0$. This product $R$ will differ strongly in correlated systems ($R > R_0$ for FM, $R < R_0$ for AF), and the characteristic of magnetic correlations can be established from the value of $R/R_0$. In the SC phase, pairing in a spin singlet or spin triplet is related to a decrease or invariance of the Knight shift across $T_{\rm SC}$. The temperature dependence of $1/T_1$ below $T_{\rm SC}$ reflects the gap structure. $1/T_1$ varies as $\exp{( -\Delta/T)}$ for a finite gap, as $T^3$ for a line-node gap, and as $T^5$ for a point-node gap. [@Asayamatextbook] In the presence of the residual density of states induced by impurities and the inhomogeneity of samples, $1/T_1$ deviates from the above temperature dependences and approaches the Korringa relation. Crystal structure ----------------- UGe$_2$ crystallizes in the ZrGa$_2$-type orthorhombic structure with the space group $Cmmm$ ([\#]{}65, $D_{2h}^{19}$), which belongs to the symmorphic space group, as shown in Fig. \[fig:structure\](a). In early studies [@Mak59], the crystal structure was determined to be the orthorhombic ZrSi$_2$-type with the space group $Cmcm$, but after the refinement of the structure, the ZrGa$_2$-type structure was confirmed [@Oik96]. The U atoms form zigzag chains along the $a$-axis and the FM moment is directed along the $a$-axis. The distance of the first nearest neighbor of the U atom is 3.854 Å, which is larger than the so-called Hill limit. URhGe and UCoGe have the TiNiSi-type orthorhombic structure with the space group $Pnma$ ([\#]{}62, $D_{2h}^{16}$), which belongs to the nonsymmorphic space group, as shown in Fig. \[fig:structure\](b). The U atoms form the zigzag chains along the $a$-axis, and the distances between two U atoms are $d_1 = 3.497$ and $3.481\,{\rm \AA}$ for URhGe and UCoGe, respectively, which are close to Hill limit. Most U$T$Ge ($T$: transition metal) compounds crystallize in the TiNiSi-type structure or its non-ordered variant CeCu$_2$-type structure. As shown in Fig. \[fig2A\], URhGe and UCoGe are located between the PM ground state and the AF ground state, as a function of the next-nearest-neighbor distance, and have relatively large $\gamma$-values. The global inversion symmetry is preserved in the TiNiSi-type structure; however, the local inversion symmetry at the sites of the U atoms is broken because of the zigzag chain. Theoretically, it has been proposed that a spatially inhomogeneous antisymmetric spin-orbit interaction and peculiar physical properties that depend on the energy scale of the band structure might appear. In URhGe, a small AF component along the $a$-axis as a result of the small relativistic Dzyaloshinskii–Moriya interaction has been theoretically proposed. [@Min06] However, it has not been observed experimentally in a high-quality single crystal, although the early studies using polycrystalline powder samples revealed a small AF component [@Tra98]. ![(Color online) Crystal structures of (a) UGe$_2$ and (b) URhGe/UCoGe. (c), (d) Projections of URhGe/UCoGe from $b$- and $a$-axes, respectively.[]{data-label="fig:structure"}](Fig2_structure.pdf){width="\hsize"} Another important point is that the TiNiSi-type orthorhombic structure is derived from the distorted AlB$_2$-type hexagonal structure. In fact, the U atoms of URhGe are almost located in the $bc$-plane ($a$-plane), but their alignment is slightly corrugated owing to the $x$ parameter of the atomic coordinate ($x$ = 0.0041), which corresponds to a quite small displacement of $0.028\,{\rm \AA}$ from the $bc$-plane. If we neglect this small corrugation, the U atoms form a distorted hexagon or successive triangles, as shown in Fig. 2(d). The distances $d_2$ and $d_3$ between the U atoms are $3.746$ and $4.327\,{\rm \AA}$, respectively, and the ratio $d_2/d_3$ is 0.866 in URhGe. If $d_2/d_3$ is 1, the U atoms form equilateral triangles, and the magnetic anisotropy between the $b$- and $c$-axes will be very small because the exchange interactions due to $d_2$ and $d_3$ are almost equivalent. As described later, URhGe shows spin reorientation from the $c$- to $b$-axis at low temperatures when a field is applied along the $b$-axis, indicating that the magnetic anisotropy between the $b$- and $c$-axes is relatively small in spite of the Ising magnetic character. The small anisotropy between the $b$- and $c$-axes and the hard-magnetization $a$-axis are a general trend in the U$T$Ge system. The key parameters are the $x$-value of the atomic coordinate of the U atom and the ratio $d_2/d_3$. The small $x$-value and the large $d_2/d_3$ close to 1 are preferable conditions for small anisotropy between the $b$- and $c$-axes, leading to the spin reorientation at low fields. Figure \[fig2B\] shows $d_2/d_3$ plotted against the $x$-value for different U$T$Ge compounds. URhGe satisfies the preferable conditions for spin reorientation, while UCoGe has a larger $x$-value, increasing the difficulty of spin reorientation. In Fig. \[fig2B\], one can recognize that spin reorientation is more likely to occur in UIrGe and UPtGe. In fact, the antiferromagnet UIrGe shows metamagnetic transitions from an AF state to a polarized PM state at 21 and 14 T for $H \parallel b$ and $H \parallel c$, respectively, while the hard-magnetization axis ($a$-axis) shows no metamagnetism up to $50\,{\rm T}$. [@Yos06] In UPtGe, an incommensurate cycloidal magnetic structure is found in the $bc$-plane for the TiNiSi-type structure [@Man00], although the structure is refined to the EuAuGe type from the TiNiSi type with small modifications. Upon 10 % Ir doping of URhGe, the spin reorientation field is significantly reduced to $\mu_0 H_{\rm R}$ = 9.4 T with $T_{\rm Curie}$ = 9.3 K. A similar trend is also observed for 2% Pt-doped URhGe with $\mu_0 H_{\rm R}$ = 10 T and $T_{\rm Curie}$ = 8.6 K. ![(Color online) Sommerfeld coefficient and magnetic ordering temperature as a function of the distance of uranium atoms from the first nearest neighbors in U$T$Ge ($T$: transition metal). UCoGe and URhGe are located at the border between paramagnetism and antiferromagnetism.[]{data-label="fig2A"}](Fig3_gamma_Tord.pdf){width="0.8\hsize"} ![(Color online) $d_2/d_3$ vs atomic coordinate $x$ of uranium atom. A large value of $d_2/d_3$ close to 1 and small $x$ are preferable for spin reorientation.[]{data-label="fig2B"}](Fig4_d_x.pdf){width="0.8\hsize"} Common Features and Particularities =================================== The goal is to present the domain of existence of the different phases in $(T,P,H)$ space, to see the consequences of the self-induced vortex (SIV) created in the FM phase, and to stress the unique opportunity for the modification of the SC pairing by $H$ acting on the FM interaction. ($P$, $T$) phase diagram ------------------------ {width="80.00000%"} Figure \[fig3\] shows the ($P, T$) phase diagram of the three compounds. As shown in Table \[tab:1\], the FM moments at ambient pressure are $1.5$, $0.4$, and $0.06\,\mu_{\rm B}$ in UGe$_2$, URhGe, and UCoGe, respectively. Thus, the duality between the localized and itinerant character of the 5$f$ electron is strong in UGe$_2$, while an itinerant description of 5$f$ electrons seems to be justified in UCoGe. UGe$_2$ has a rather high Curie temperature $T_{\rm Curie} \sim 52\,{\rm K}$ with a large magnetic moment $M_0 \sim 1.5\,\mu_{\rm B}$ at $P = 0$ in the FM2 ground state. On cooling below $T_{\rm Curie}$, the competition between the low-magnetic-moment ($M_0 \sim 0.9 \mu_{\rm B}$) phase FM1 and the FM2 ground state is marked by a large $T$ crossover. [@Har09_UGe2] At $P \sim 1$ GPa, a first-order transition between FM2 and FM1 appears at $P_{\rm CEP}$ and $T_{\rm CEP}$, which will end up at $P_x \sim$ 1.2 GPa. Specific attention is given to the FM-PM change from a second order to first-order transition at the tricritical point (TCP) at $T_{\rm TCP}$ and $P_{\rm TCP}$; this change is directly associated with the observation of the FM wings created by the restoration of ferromagnetism in magnetic fields[@TaufourPRL2010]. The particularities of SC in UGe$_2$ are that (1) it exists only in the FM domain with the maximum $T_{\rm sc}$ at $P_x$, and (2) $P_x$ is coupled with a drastic change of the FS on switching from FM2 to FM1. In URhGe, $T_{\rm Curie}$ ($=9.5\,{\rm K}$) and $M_0$ ($=0.4\,\mu_{\rm B}$) at ambient pressure are much lower than those of UGe$_2$ at $P_x$ where SC emerges. Under pressure, the FM becomes more stable and $T_{\rm Curie}$ still increases even at 13 GPa[@HardyPhysicaB2005]. As $T_{\rm Curie}$ increases under pressure, the magnetic fluctuations become weaker and $T_{\rm SC}$ decreases. The collapse of SC will occur at around $P_{\rm S} \sim$ 4 GPa. UCoGe has a small FM moment, $M_0 \sim 0.06 \mu_{\rm B}$, with a small magnetic entropy release at $T_{\rm Curie}$, indicating a clear example of itinerant FM. Here, pressure drives the system towards FM instabilities. FM disappears at around 1 GPa[@SlootenPRL2009; @HassingerJPSJ2008; @BastienPRB2016], while SC survives deep inside the PM phase up to $P_{\rm S} \sim 4\,{\rm GPa}$. [@BastienPRB2016] The characteristic values for the three compounds are shown in Table \[tab:1\]. The internal field $H_{\rm int}$ created by $M_0$ below $T_{\rm Curie}$ as well as the FM molecular field $H_{\rm mol}$ are shown in Table \[tab:2\]. --------- ------ ----------------- ----------------- ------------------------------- ------------- ------------------ Easy $T_{\rm Curie}$ $M_0$ $\gamma$ $P_{\rm C}$ $T_{\rm SC}$ axis (K) $(\mu_{\rm B})$ (${\rm mJ\,mol^{-1} K^{-2}}$) (GPa) (K) UGe$_2$ a 52 1.48 34 1.6 0.8 at $P = P_x$ URhGe c 9.5 0.4 163 $> 13$ 0.25 at $P$ = 0 UCoGe c 2.7 0.06 57 $\sim 1$ 0.8 at $P$ = 0 --------- ------ ----------------- ----------------- ------------------------------- ------------- ------------------ \[tab:1\] $H_{\rm int}$ (T) $H_{\rm mol}$ (T) --------- ------------------- ------------------- UGe$_2$ 0.28 50 URhGe 0.08 10 UCoGe 0.01 2.5 : Internal field $H_{\rm int}$ and molecular field $H_{\rm mol}$ for $T\to 0$. \[tab:2\] SC depairing and self-induced vortex state ------------------------------------------ The relative variation of $T_{\rm SC}$ as a function of the residual resistivity ($\rho_0$) (Fig. \[fig4\]) shows that, as expected in unconventional SC, $T_{\rm SC} / T^0_{\rm SC}$ depends strongly on $\rho_0$, which is inversely proportional to the electronic mean free path. ![(Color online) Relative dependence of $T_{\rm SC}$ as a function of residual resistivity $\rho_0$ in URhGe and UCoGe. The solid line is obtained from the Abrikosov–Gor’kov pair-breaking function.[]{data-label="fig4"}](Fig6_Tsc_rho0.pdf){width="0.6\hsize"} Another singularity is that as $H_{\rm int}$ is much higher than $H_{\rm c1}$ (as shown later in magnetization curves for UCoGe), thus SIVs exist at $H = 0$. The creation of self induced vortex below $T_{\rm SC}$ will lead to the residual contribution $\gamma_1 \sim H_{\rm int} /H_{\rm c2}$ to the linear $T$ term. In addition, the phenomenon is enhanced by the additional contribution $\gamma_2$ given by the Volovik effect in the inter vortex phase[@MineevPUsr2017; @VolovikJETPL1993]. Figure \[fig:Cp\] shows the SC anomaly of the three compounds; the SC jump at $T_{\rm SC}$ is directly related to the weakness of $M_0$. Even the residual $\gamma$ term in the SC phase follows a $\sqrt{M_0}$ dependence as predicted for the main origin of the Volovik effect (Fig. \[fig6\]) [@MineevPUsr2017]. ![(Color online) Specific heat in UGe$_2$, URhGe, and UCoGe.[]{data-label="fig:Cp"}](Fig7_Cp.pdf){width="\hsize"} ![(Color online) Relative variation of the residual value $\gamma_0$ normalized by $\gamma_{\rm N}^{\mathstrut}$ as a function of $\sqrt{M_0}$. This graph proves that strong Volovik vortices already contribute at $H = 0$.[]{data-label="fig6"}](Fig8_gamma0.pdf){width="0.6\hsize"} Transverse and longitudinal $H$ scan: consequences on SC pairing ---------------------------------------------------------------- Figure \[fig7\] shows the low-field susceptibility $\chi$ of the three compounds measured along their principal axes. At room temperature, the $a$-axis is already the easy magnetization axis in UGe$_2$, and the $c$-axis becomes the easy axis in UCoGe. [@HuyPRL2008] However, in URhGe, almost no anisotropy appears between the $b$- and $c$-axes at room temperature, and the $c$-axis can be differentiated from the $b$-axis only below $50\,{\rm K}$. UGe$_2$ and UCoGe are considered to be Ising ferromagnets. For URhGe, the duality between FM along the $c$- and $b$-axes is at the core of its extremely high field-reentrant superconductivity (RSC) for $H\parallel b$. ![(Color online) Susceptibilities of (a) UGe$_2$, (b) URhGe, and (c) UCoGe. In URhGe, owing to the weak magnetocrystalline term, $\chi_c$ clearly becomes the easy magnetization axis only below 50 K. The susceptibility in UGe$_2$ is replotted from Ref. .[]{data-label="fig7"}](Fig9_chi.pdf){width="0.6\hsize"} Figure \[fig8\] shows the magnetizations of UGe$_2$, URhGe, and UCoGe at low temperatures along their $a$-, $b$-, and $c$-axes. [@Sakon07; @Har11; @HuyPRL2008] In URhGe, the large value of the initial slope of magnetization, $dM/dH\equiv \chi$, for the $b$-axis compared with that for the $c$-axis indicates that under a magnetic field of $\mu_0 H_{\rm R} \sim 12\,{\rm T}$, the $b$-axis will become the easy magnetization axis. On the other hand, $\chi$ for the $c$-axis always exceeds that for the $b$-axis in UCoGe; thus the Ising character will be preserved up to a very high magnetic field. The strong curvature of the magnetization curve for the $c$-axis in UCoGe implies that the contribution of the spin fluctuation will drastically decrease with increasing $H$; this unusual $M(H)$ curve has major consequences on the SC properties for $H \parallel c$, particularly on the $H_{\rm c2}$ curvature (see below). ![(Color online) Magnetization of (a) UGe$_2$ [@Sakon07], (b) URhGe, and (c) UCoGe in their normal phase. In URhGe, the magnetization for $H \parallel b$ shows “metamagnetic”-like transition at $H = H_{\rm R}$, indicating a switch of the easy magnetization axis from the $c$-axis to the $b$-axis. [@Har11] Note the high value of $\chi_b = dM_b /dH$ up to $H_{\rm R}$ and the weak curvature of $M(H)$ for $H \parallel c$. In UCoGe, the strong curvature of $M(H)$ for $H \parallel c$ is directly linked with the strong decrease in $m^{**}$ with increasing $H$ and that in $\lambda$, the relation, $\chi_c > \chi_b \sim \chi_a$, is preserved regardless of the magnetic field. [@HuyPRL2008; @KnafoPRB2012][]{data-label="fig8"}](Fig10_mag.pdf){width="0.6\hsize"} In UGe$_2$ for $H \parallel a$ \[Fig. \[fig:Hc2\](a)\], the sudden enhancement of $H_{\rm c2}$ for $P$ just above $P_x$ has been taken to be a consequence of the $H$-switch from the FM1 to FM2 phases, since these two phases have different SC temperatures at $H$ = 0. [@SheikinPRB2001] ![(Color online) (a) $H_{\rm c2}$ versus $T$ for $H \parallel a$ (longitudinal field scan with respect to $H \parallel M_0$) in UGe$_2$ [@SheikinPRB2001], providing evidence that the $T_{\rm SC}$ dependence close to $P_x$ is sharp. $H_{\rm c2}$ versus $T$ for $H \parallel b$ (transverse field scan $H \perp M_0$) in (b) URhGe [@LevyScience2005] and (c) UCoGe [@AokiJPSJ2009]. In these cases, the evidence of $H$ reinforcement of the pairing is connected with the collapse of $T_{\rm Curie}$. []{data-label="fig:Hc2"}](Fig11_Hc2.pdf){width="\hsize"} Considerably more interesting in Figs. \[fig:Hc2\](b) and \[fig:Hc2\](c) are the cases of URhGe and UCoGe, where reentrant SC in URhGe and field-reinforced SC in UCoGe occur for a transverse magnetic field scan ($H\parallel b$) with respect to the initial FM direction ($M_0 \parallel c$). The key origin of this singular behavior is that a transverse $H$ scan leads to the collapse of the Ising FM along the $c$-axis and thus gives a unique elegant opportunity to cross the FM-PM instability with the enhancement of FM fluctuations. Proof of the collapse of ferromagnetism has been observed in thermodynamic and transport experiments as well as in NMR measurements. Figure \[fig10\] shows the FM and SC domains of URhGe and UCoGe. Table \[tab:mag\] [@Har09_UGe2] summarizes the different values of the low-temperature susceptibility at $H \rightarrow 0$ and the estimated critical magnetic fields $H^\ast_a$, $H^\ast_b$, and $H^\ast_c$, where each magnetization $M_a$, $M_b$, and $M_c$ reaches $M_0$. [@Har11] Note that for URhGe, $\chi_b > \chi_c > \chi_a$ and that $\chi_b$ in URhGe is even larger than $\chi_c$ in UCoGe. The weakness of the magnetic anisotropy leads to FM instabilities with $M_0 \parallel c$ and $M_0 \parallel b$ for URhGe occuring at $H_{\rm R}$. For UCoGe, $\chi_c$ is much greater than $\chi_b$; this property will reflect only the Ising FM proximity with $M_0 \parallel c$. These different behaviors will be demonstrated in NMR experiments. ![(Color online) Overlap between the SC and FM domains in the ($H, T$) plane at $P = 0$ for $H \parallel b$. For URhGe (a), data are taken from transport, magnetization, and thermal expansion measurements and for UCoGe (b), data are taken from transport, thermal expansion, and NMR measurements.[]{data-label="fig10"}](Fig12_HT_phase.pdf){width="\hsize"} --------- ------------------- ------------------- ------------------- --------- --------- --------- $\chi_a$ $\chi_b$ $\chi_c$ $H^*_a$ $H^*_b$ $H^*_c$ ($\mu_{\rm B}/T$) ($\mu_{\rm B}/T$) ($\mu_{\rm B}/T$) (T) (T) (T) UGe$_2$ 0.006 0.0055 0.011 230 250 122 URhGe 0.006 0.03 0.01 66 13 40 UCoGe 0.006 0.0055 0.011 29 12 2.5 --------- ------------------- ------------------- ------------------- --------- --------- --------- : Field derivative magnetizations at $H \to 0$ and the estimated critical field along each axis. [@Har09_UGe2] \[tab:mag\] Now we will describe the singular features of each compound.\ For UGe$_2$: - FM transition switches from second order to first order at $T_{\rm TCP}$, $P_{\rm TCP}$ - Detection of FM wings in the $(T,P,H)$ phase diagram for $H \parallel a$ (easy magnetization axis) up to the quantum critical end point - Drastic change of the FSs on entering the three different phases FM2, FM1, and PM. - Consequences on the SC domain with the interplay between FS instability and FM spin fluctuations. For URhGe: - Appearance of FM wings upon tilting the field direction from the $b$- to $c$-axis around $H_{\rm R}$. - Duality between FM along the $c$- and $b$-axes: concomitant longitudinal and transversal fluctuations detected by NMR near $H_{\rm R}$. - Link between RSC and $H$ dependence of $m^{**}$. - Pressure, uniaxial stress dependences of RSC: evidence of scaling in $m^\ast (H_{\rm R})/m^\ast (0)$ supporting Lifshitz transition. For UCoGe: - Precise magnetization measurements at very low temperature: hierarchy between $H_{\rm c1}$, $H_{\rm int}$ /strong $H$ curvature of $M$($H$) for $H \parallel c$. - Observation by NMR of mainly longitudinal spin fluctuations along $c$-axis, regardless whether the field direction is along the $a$-, $b$-, or $c$-axis. - Huge decrease in $m^{\ast\ast}$ with increasing $H$ in longitudinal scan and strong increase in $m^{**}$ in transverse magnetic field on approaching $H^\ast_b$. - Description of $H_{\rm c2}$ curve via the field dependence of the parameter $\lambda$ defined by $m^{**} / m_{\rm B}$. - ($P$, $H$) phase diagram with collapse of FM at $P_C \sim 1\,{\rm GPa}$, persistence of SC up to $P_{\rm S} \sim 4\,{\rm GPa}$, link between FM collapse and $H_{\rm c2}$ singularities for $H \parallel M_0$ and $H \perp M_0$. The different theoretical approaches will be presented with focus on the interplay between magnetism and unconventional SC, and on additional phenomena related to the Lifshitz transition. Special attention will be given later to the present knowledges of FSs referring to band structure calculations. Properties of UGe$_2$ {#sec:UGe2} ===================== Contrary to the canonical example of SC around $P_{\rm c}$ driven by spin fluctuations, for UGe$_2$, the singularity is that SC appears close to $P_{\rm x}$ where the system switches from FM2 to FM1 phases. The clear feature is that FS reconstruction at $P_x$ must be considered to evaluate the SC pairing. Two FM ground states, FM wing, and FS instability ------------------------------------------------- The first determination of the ($P$, $T$) phase diagram of UGe$_2$ was realized in 1993, showing a collapse of ferromagnetism between 1.5 and 2 GPa[@TakahashiPhysicaB1993]. Evidence of an anomaly at $T_x$ (signature of the competition between FM2 and FM1) was reported in 1998[@OomiJAlloyComp1998]. The next breakthrough was the discovery of SC in the FM domain ($P < 1.5$ GPa) in 2000[@SaxenaNature2000]. The key role of the switch from FM2 to FM1 in the SC onset was pointed out in 2001 [@HuxleyPRB2001; @BauerJPCM2001; @TateiwaJPSJ2001; @MotoyamaPRB2001]. Above $P_x$, $M_0 = 0.9\,\mu_{\rm B}$/U in the FM1 phase and $M_0 = 1.5\,\mu_{\rm B}$/U in the FM2 phase (see Fig. \[fig11\]) [@TateiwaJPSJ2001; @MotoyamaPRB2001; @PfleidererPRL2002; @HuxleyJPCM2003]. Above $P_x$, FM2 will be reached at a magnetic field $H_x$ for $H\parallel a$ (easy magnetization axis). At a field $H_c$ above $P_{\rm C}$, the PM phase will switch to FM1; increasing the field to $H_x$ leads to a transition to FM2. Figure \[fig11\] shows the $P$ dependences of $M_0(P)$, $H_x$, and $H_c$ [@PfleidererPRL2002]. Complementary studies can be found in Refs. , and . The jump of $M_0$ at $P_x$ and $P_c$ clearly shows that both transitions at $P_x$ and $P_c$ are of the first order. The transition line $T_{\rm Curie}(P)$ between FM and PM at zero field changes from second order to first order at a TCP of $T_{\rm TCP}\sim 24\,{\rm K}$ and $P_{\rm TCP} = 1.42\,{\rm GPa}$. The range of the first-order transition is quite narrow $(P_c-P_{\rm TCP})/P_{\rm TCP} \sim 0.05$ [@TaufourPRL2010]. However, for $H \parallel M_0$ along the easy axis, the first-order FM wing appears up to the quantum critical end point (QCEP) at $P_{\rm QCEP}\sim 3.5\,{\rm GPa}$ and $\mu_0 H_{\rm QCEP}\sim 18\,{\rm T}$, as shown in Fig. \[fig12\]. [@KotegawaJPSJ2011; @TaufourPhD] Note the large separation between $P_{\rm QCEP}$ and $P_c$ directly linked to the large jump of $M_0$ at $P_c$. Figure \[fig13\] shows how the step like jump of the resistivity coefficient $A$ below $P_{\rm QCEP}$ is replaced by a maximum above $P_{\rm QCEP}$. [@TaufourPhD] The phase transition at $T_x$ between FM1 and FM2 starts at the critical end point (CEP) equal to $T_{\rm CEP} = 7\,{\rm K}$ and $P_{\rm CEP}$ =1.16 GPa. At $T = 0$ K, it collapses at $P_x \sim 1.2\,{\rm GPa}$. At $P = 0$, the specific heat and thermal expansion show a crossover between FM1 and FM2 ground states. [@Har09_UGe2]. Note that drastic changes are observed in the normal component of the Hall effect[@TranPRB2004] as well as in the thermoelectric power (Fig. \[fig14\])[@MoralesPRB2016]. These low-pressure effects are precursors of the drastic changes in the FS on entering the different ground states through $P_x$. ![(Color online) (a) Variation of the FM $M_0$ component at $H$ = 0 for the FM2 and FM1 states in UGe$_2$ at $2.3\,{\rm K}$. [@PfleidererPRL2002] Extrapolation of $M_0$ above $H_{\rm X}$. (b) Field variation of $H_{\rm X}$ and $H_c$ as a function of pressure ($P$). []{data-label="fig11"}](Fig13_UGe2_M_HP_phase.pdf){width="0.6\hsize"} ![(Color online) Three-dimensional ($T, H, P$) phase diagram of UGe$_2$, indicating evidence of the FM wings extending far above $P_{\rm C}$ up to the QCEPs ($H_{\rm QCEP}\sim 18\,{\rm T}$, $P_{\rm QCEP}\sim 3.5\,{\rm GPa}$). [@KotegawaJPSJ2011; @TaufourPhD][]{data-label="fig12"}](Fig14_UGe2_TPH_phase.pdf){width="\hsize"} ![(Color online) Variation of the $A$ coefficient of the $AT^2$ resistivity term upon crossing $H_{\rm X}$ at different pressures through the QCEP of UGe$_2$. [@TaufourPhD][]{data-label="fig13"}](Fig15_UGe2_Acoef.pdf){width="\hsize"} ![(Color online) Variation of the Hall constant $R_{\rm H}$ and the thermoelectric power through the crossover temperature $T^*$, which is a characteristic of the choice of FM2 as the ground state of UGe$_2$ at low pressures.[]{data-label="fig14"}](Fig16_Hall_TEP.pdf){width="0.6\hsize"} Measurements of the specific heat under pressure (Fig. \[fig15\]) indicate that the $\gamma$ coefficient jumps at $P_x$ while the $\beta T^3$ term also unexpectedly has a maximum at $P_x$ [@PhilippsPrivateComm]. There is no maximum of $\gamma$ at $P_x$, suggesting that the additional effect, rather than spin fluctuation, should occur for the establishment of SC. In addition, the $P$ dependence of the $A$ coefficient obeys $A \propto \gamma^2$ for the current $I \parallel b$, but has a pronounced maximum for $I \parallel a$ [@TerashimaPRB2006; @KobayashiJPCM2002; @SettaiJPCM2002]. ![(a) Pressure dependence of $\gamma$ in UGe$_2$. (b) Note that the jump of $\gamma$ at $P_x$ is associated with a maximum of $\beta$ for the $T^3$ term of the specific heat. [@Flo06_review][]{data-label="fig15"}](Fig17_UGe2_Cp_pressure.pdf){width="0.9\hsize"} The new feature is that quantum-oscillation measurements demonstrate drastic changes in the FS through $P_{x}$ and $P_c$. For $H\parallel b$, corresponding to the hard-magnetization axis, the three phases, FM1, FM2, and PM, are not affected by the magnetic field. The main dHvA branches in FM1, namely $\alpha$, $\beta$, and $\gamma$, which might be due to the nearly cylindrical FSs [@SettaiJPCM2002], disappear in the PM state, and new branches are detected [@TerashimaPRL2001], as shown in Fig. \[fig16\]. In the PM state, a heavy electronic state is realized with large effective masses of up to $64\,m_0$, which is consistent with the large Sommerfeld coefficient ($100\,{\rm mJ\,K^{-2}mol^{-1}}$) measured under pressures above $P_{\rm c}$. For $H\parallel a$ (easy-magnetization axis), FM1, FM2, and PM are separated by metamagnetic transitions. Thus, the results are more complicated. Figure \[fig:UGe2\_Freq\_pressure\] shows the pressure dependence of the dHvA frequencies. [@Hag02; @TerashimaPRB2002] The observed frequencies, which are less than approximately $1\times 10^7\,{\rm Oe}$, correspond to the small FSs, revealing the relatively strong pressure dependence in the FM2 phase. It is also clear that the FS changes with the transition from FM2 to FM1. ![Pressure dependence of FFT spectra obtained from dHvA experiments for $H\parallel b$ in UGe$_2$. The FSs are drastically changed above $P_{\rm c}$. [@TerashimaPRL2001][]{data-label="fig16"}](Fig18_UGe2_FFT.pdf){width="0.8\hsize"} ![Pressure dependences of the dHvA frequencies for $H\parallel a$ in UGe$_2$ in the (a) FM2 and (b) FM1 phases. [@Hag02][]{data-label="fig:UGe2_Freq_pressure"}](Fig19_UGe2_Freq_pressure.pdf){width="0.8\hsize"} One interesting theoretical proposal is that the transition at $T_x$ may be the signature of a charge density wave (CDW) onset linked to supplementary nesting of the FS[@WatanabeJPSJ2002], but attempts to detect a CDW have failed[@AsoJPSJ2006]. Band-structure calculation indicates that over a range of pressures, the two FM states are nearly degenerate with different orbital and spin moments on the U sites[@ShickPRB2004]. The ($P$, $T$) phase diagram was qualitatively explained by a phenomenological model with two initial maxima in the density of states[@SandemanPRL2003]. In contrast to the case of AF-PM instability, which is often of the second order, a first-order collapse in clean FM itinerant materials is observed; theoretical arguments to justify this were given through the nonanalytic term in the Landau expression for the free energy[@BelitzPRL1999; @BelitzPRL2005] and through the feedback with magnetoelastic coupling. [@GehringEPL2008; @MineevPUsr2017; @MineevCRP2011]. To summarize the normal properties of UGe$_2$, the main features are as follows. - $T_{\rm Curie}$ is suppressed with increasing $P$, and a switch to a first order transition occurs at the TCP. - FM wings exist far above $P_c$ up to $P_{\rm QCEP} \sim 3.5\,{\rm GPa}$. - There is a drastic change in the FS at $P_x$ and $P_c$ SC phase: interplay of FS instability and FM fluctuations --------------------------------------------------------- The SC domain was first determined by resistivity measurements; the two important points are that optimum of $T_{\rm SC}$ coincides with $P_x$ and that the collapse of SC occurs at $P_{\rm SC} \sim P_c$. Unexpectedly, the specific heat jump at $T_{\rm SC}$ (Fig. \[fig18\]) can only be observed near $P_x$[@TateiwaPRB2004]. A series of supplementary results derived from the $ac$ susceptibility measurements[@NakaneJPSJ2005; @BanJMMM2007; @KabeyaPhysicaB2009] suggests that SC will exist only in the FM1 phase (Fig. \[fig19\]). The different behavior of SC in FM2 and FM1 has been confirmed by the fast broadening in the SC transition below $P_x$ (see Fig. \[fig20\]). The behavior of the intrinsic SC region remains an open question. One difficulty is the narrow $P$ width of SC (0.3 GPa) compared with the pressure inhomogeneity close to the first-order transition with a volume change of approximately $10^{-3}$. The unusual field dependence of $H_{\rm c2}$ reported in Fig. \[fig:Hc2\] with pressure slightly above $P_x$ corresponds to the field switching between FM1 and FM2 at $H_x$. The initial claim that the maximum of $T_{\rm SC}$ occurs in the FM2 phase just below $P_x$ must be verified. A sharp structure of $T_{\rm SC}$ in FM1 with a maximum upon approaching $P_x$ may give an alternative explanation. ![Unexpectedly, a clear jump $\Delta C/\gamma_{\rm N}T_{\rm SC}$ can be detected in UGe$_2$ only at around $P_x$ but with a very weak value compared with the conventional BCS value (1.4) (see Fig. \[fig:Cp\]).[]{data-label="fig18"}](Fig20_UGe2_TP_SC.pdf){width="0.7\hsize"} ![$T_{\rm SC}$ of UGe$_2$ detected in susceptibility measurements from the peak in the $\chi''(T)$ curve. The volume fraction is obtained from $-4\pi\chi'(T)$ at 60 mK. A double structure in $\chi''(T)$ is detected for $P > P_x$. [@NakaneJPSJ2005; @BanJMMM2007; @KabeyaPhysicaB2009][]{data-label="fig19"}](Fig21_UGe2_SC_volume.pdf){width="0.8\hsize"} ![(Color online) Determination of $T_{\rm SC}$ by the achievement of zero resistivity. Note the huge broadening of the $\rho$ anomaly on entering FM2. [@TaufourPhD][]{data-label="fig20"}](Fig22_UGe2_Tsc_res_pressure.pdf){width="\hsize"} Unique information is given by NQR-NMR experiments using the $^{73}$Ge isotope. At low pressures, in agreement with the neutron scattering experiments[@HuxleyPRL2003], the Ising FM character of the fluctuations is clearly observed[@NomaJPSJ2018]. Figure \[fig:UGe2\_T1\] illustrates the temperature dependence of $1/T_1$ [@KotegawaJPSJ2005; @HaradaJPSJ2005]. In the FM phase (FM2 or FM1), $T_{\rm Curie}$ and $T_{\rm SC}$ are clearly detected; above $T_{\rm SC}$, $1/T_1$ follows a $T$-linear Korringa dependence linked with the value of $\gamma$, and below $T_{\rm SC}$, a $T^3$ term is observed and regarded as an indication of a line-node SC gap. At $P_c$, because of the first-order nature of the transition, both FM and PM signals are detected with the noteworthy feature that SC ($T_{\rm SC} \sim 0.2\,{\rm K}$) is only detected in the FM1 phase and not in the PM phase[@HaradaJPSJ2005]. ![(Color online) Temperature dependence of $1/T_1$ of $^{73}$Ge at (a) 1.15, (b) 1.2, (c) 1.3, and (d) 1.5 GPa. $1/T_1$ in panels (a), (b), and (c) was measured at the peak of the Ge1 (4$i$) site for FM1. The solid curves in (a), (b), and (c) are the results of calculations based on an unconventional superconducting model with a line-node gap. The identification of the phase transitions into both SC and FM ensures a phase with their uniform coexistence. $T_{\rm Curie}$ was determined by ac-$\chi$ measurement. The inset in (c) shows the frequency dependence of $1/T_1T$ at $P$ = 1.2 GPa and $f$ = 7.75, 8.5, and 9.12 MHz. The observation of a similar $T$ dependence of $1/T_1T$ ensures the onset of SC over the whole sample. $1/T_1$ in (d) was measured at the peak of PM (open squares) and FM1 (solid squares). The long component in $1/T_1$ for FM1 indicates that SC sets in at $T_{\rm SC} \sim 0.2\,{\rm K}$, but the short components for PM do not. [@KotegawaJPSJ2005; @HaradaJPSJ2005][]{data-label="fig:UGe2_T1"}](Fig23_UGe2_T1.pdf){width="\hsize"} Note that the results of muon experiments [@Sak10_UGe2] emphasize the duality between the localized and itinerant character of the 5$f$ electrons (see also Ref.  and recent neutron data in Ref. ). The common point with our previous consideration is that the FM1 phase has an electronic density larger than the FM2 phases, and hence multiband effects must be taken into account. The conduction electron carries a quasi isotropic magnetic moment considerably lower than $M_0$. The statement that $P_x$ collapses close to $P_c$ is in contradiction to all previous data; this can be evidence that the muon signal is not directly linked with the switch of magnetism at $P_x$ or a indicator of nonhydrostaticity, which happens in the chamber of a pressure cell fixed at a constant volume but not at a constant pressure (the reaction of the pressure cell causing the deformation of the crystal depends on the specific arrangement). This puzzle should be solved. The main attempt to derive the SC properties from the normal ones was made by changing the FS topology in two critical peaks of density of states of the PM phase [@SandemanPRL2003]. By comparison with a later consideration on URhGe and UCoGe, the pressure dependences of $m_{\rm B}$ and $m^{**}$ were considered to have singularities at $P_x$ and $P_c$. Using the parameter in Ref. , the pressure variation of $T_{\rm SC}$ is shown in Fig. \[fig22\]. In agreement with Figs. \[fig19\]-\[fig:UGe2\_T1\], a sharp SC singularity occurs at $P_x$. The prediction of SC in the PM region is not verified by the experiments; it is difficult to predict SC quantitatively in the complex case of UGe$_2$. ![(Color online) For UGe$_2$, $T_{\rm SC}$ versus $P$ or $I / I_c$ the Stoner factor using the parameter of the phenomenological model of Ref. .[]{data-label="fig22"}](Fig24_UGe2_Tsc_theory.pdf){width="0.8\hsize"} Recently, a mechanism of spin-triplet pairing was proposed [@Kad18], as an alternative to FM spin-fluctuation pairing. Robust SC is predicted to exist only in the FM1 phase by the combination of the FM exchange based on Hund’s rule and the interelectronic Coulomb interaction (see Fig. 4 in Ref. ). A sound proof will be microscopic evidence of orbital-selective Mott-type delocalization of the 5$f$ electrons in the transition from FM2 to FM1. Although the effect is small (see Fig. 5 in Ref. ), the effect on SC pairing is predicted to be large. Properties of URhGe {#sec:URhGe} =================== We focus on the RSC observed in the transverse field scan along the $b$ axis ($H_b$), and its link with the FM wing structure detected by tilting the field angle $\theta$ from the $b$ to $c$-axis. The possibility of crude modeling through the field dependence of $m^{\ast\ast}(H)$ and its collapse under pressure is shown. The additional new possibility of boosting SC via uniaxial stress is also presented. Reentrant superconductivity, FM wing, QCEP, FS instabilities ------------------------------------------------------------ In a transverse field scan ($H\parallel b$), RSC appears in the field range from 8 to $13\,{\rm T}$, and the easy magnetization axis switches from the $c$- to $b$ axis at $H_{\rm R}=12\,{\rm T}$. [@LevyScience2005; @LevyThesis]. Figure \[fig23\](a) shows the evolution of the total magnetization $M_{\rm tot}$ with $H$ and its component along the $b$-axis in an $H$ scan along the $b$-axis at 2 K. Figure \[fig23\](b) shows the resistivity measurements in this $H$ sweep revealing a sharp maximum at $H_{\rm R}=12\,{\rm T}$ for $500\,{\rm mK}$, and zero resistivity in the field range from 8 to $13\,{\rm T}$ at $40\,{\rm mK}$. [@LevyScience2005] Misalignment of $H$ by $5^{\circ}$ towards the $c$-axis leads to a weak maximum of resistivity at 500 mK and a narrowing of RSC from $12$ to $14\,{\rm T}$ at 40 mK. Based on the evidence that in the $H_b$-$H_c$ plane, FM wings appear upon adding an extra $H_c$ component, a QCEP at $14\,{\rm T}$ tilted $\theta \sim6^\circ$ to the $c$-axis was proposed (Fig. \[fig:FM\_wing\]) [@LevyThesis; @HuxleyJPSJ2007]. The wing structure has been investigated extensively by angle-resolved magnetization measurements [@NakamuraPRB2017]; the schematic $T$, $H_b$, and $H_c$ phase diagram is shown in Fig. \[fig:FM\_wing\](b) [@NakamuraPRB2017]. The QCEP is located at $H_{\rm QCEP}$ = 13.5 T and $H_c$ = 1.1 T, in good agreement with the previous estimation of $H_{\rm QCEP}$ = 14 T [@LevyNatPhys2007]. Confirmation of the TCP emerges from Hall effect,[@AokiJPSJ2014] ac susceptibility[@AokiJPSJ2014], TEP,[@GourgoutPRL2016; @GourgoutThesis] and NMR[@KotegawaJPSJ2015] measurements. For example, as shown in Fig. \[fig:URhGe\_NMR\], NMR spectra show both FM and PM signals at around $4\,{\rm K}$ at $12\,{\rm T}$ close to $H_{\rm R}$. ![(a) Magnetization of URhGe determined from neutron scattering experiments. $M_{\rm total}$ is the total magnetization contributed from both the $M_b$ and $M_c$ components, where $M_b$ is the magnetization of the $b$-axis component for $H \parallel b$ with perfect alignment ($\theta$ = 0). Spin reorientation occurs at $H_{\rm R}$ = 11.8 T. (b) Consequence of 5$^{\circ}$ misalignment on resistivity curve at 40 mK, for $\theta = 0^{\circ}$. SC ($\rho = 0$) extends from 8 to 13 T, while a sharp maximum in $\rho$ can be found in the normal phase ($T \geq$ 500 mK). A weak misalignment of 5$^{\circ}$ towards the $c$-axis leads to an increase in $\mu_0 H_{\rm R} to \sim$ 13 T, a broadening of the maximum $\rho$ in the normal phase, and a shrinking of the SC domain ($12 - 14$ T). For $\theta$ = 5$^{\circ}$, we are already close to the QCEP indicated in Fig. 26. [@LevyScience2005; @LevyThesis][]{data-label="fig23"}](Fig25_URhGe_mag.pdf){width="0.8\hsize"} ![(Color online) (a) Existence of FM wings in the ($T$, $H_b$, $H_c$) phase in URhGe reported in Refs. . The QCEP is located at $H_c \sim$1.1 T and $H_{\rm R} \sim$13.5 T, corresponding to a misalignment of $\theta \sim5^{\circ}$. (b) Confirmation of the wings in URhGe by precise magnetization measurements. [@NakamuraPRB2017][]{data-label="fig:FM_wing"}](Fig26_FM_wing.pdf){width="\hsize"} ![(Color online) NMR spectra of the central transitions for $H \parallel b$ in URhGe. At 9 T, the peak position shifts to a higher frequency down to $\sim8$ K and then returns to a slightly lower frequency. This temperature of $\sim8$ K corresponds to $T_{\rm Curie}$ under $H \parallel b$. At 12 T, the spectrum is composed of two components at around $T_{\rm Curie}$, indicating that the transition is of the first order and accompanied by phase separation[@KotegawaJPSJ2015].[]{data-label="fig:URhGe_NMR"}](Fig27_URhGe_NMR.pdf){width="0.8\hsize"} $^{59}$Co-NMR experiments on URhGe doped with 10% clearly show the strong increase in $1/T_2$ in a field scan along the $b$-axis towards $H_{\rm R}$[@TokunagaPRL2015; @TokunagaPRB2016]. The huge increase in the longitudinal fluctuations at $H_{\rm R}$ shown by the $1/T_2$ measurement coincides with a concomitant increase in $1/T_1$, which is sensitive to transverse fluctuations. The transition towards an FM instability along the $b$-axis has been indicated already by the huge value of $\chi_b$ compared with $\chi_c$ in a low field. Because of the weakness of the magnetocrystalline coupling, the specific feature in URhGe is the competition between FM along two axes ($c$ and $b$): the transverse fluctuations of one mode become longitudinal fluctuations for the other mode. Figure \[fig27\] shows the variation of $1/T_2$ in the $H_b$-$H_c$ plane. In contrast to UCoGe[@IharaPRL2010], where, in low fields, $(1/T_1)_b/(1/T_1)_c$ is larger than 20, [@IharaPRL2010] the ratio in URhGe is small, $(1/T_1)_b/(1/T_1)_c \sim2.5$[@KotegawaJPSJ2015; @TokunagaPRL2015]. ![(Color online) Map of the magnetic fluctuations detected via $1/T_2$ at 1.6 K for URh$_{0.9}$Co$_{0.1}$Ge with $H \parallel b$ and field misalignment along the $c$-axis ($H_c$). $\theta$ is varied from 0 to 11$^{\circ}$. The open squares indicate where RSC occurs at low temperatures. The solid triangles show the variation of $H_{\rm R}$. [@TokunagaPRL2015; @TokunagaPRB2016][]{data-label="fig27"}](Fig28_URhGe_T2.pdf){width="0.8\hsize"} An additional effect detected from the TEP[@GourgoutPRL2016; @GourgoutThesis] is that FS instability occurs at $H_{\rm R}$, as demonstrated by the change in the sign of $S$ (Fig. \[fig28\]). The reconstruction of the FS has already been reported on the basis on Shubnikov$-$de Haas oscillations at $\theta \sim12^{\circ}$ from the $b$- to $c$-axis in order to escape from RSC. [@YellandNatPhys2011] However, the FM wing is never crossed in field at $\theta \sim12^\circ$ as it is beyond the QCEP. We will discuss how a Lifshitz transition enhances the electronic correlation in Section 6. Contrary to the case of UGe$_2$ mentioned before, in URhGe as well as in UCoGe, at first glance, it seems that the SC pairing is driven mainly by the strength of the FM fluctuation (constant $m_{\rm B}$) without the necessity of invoking FS reconstruction. ![(Color online) Results of thermoelectric power (TEP) experiments on URhGe down to $T$ = 0.25 K. $T_{\rm CR}$ represents the crossover line between the PM and polarized PM phase above $T_{\rm Curie}$. The transition widths observed in the TEP around $H_{\rm R}$ are shown by horizontal lines. The TCP is located close to 2 K. [@GourgoutPRL2016; @GourgoutThesis][]{data-label="fig28"}](Fig29_URhGe_TEP.pdf){width="0.8\hsize"} Modeling by considering field enhancement of $m^{\ast\ast}$ at $H_{\rm R}$ -------------------------------------------------------------------------- The proximity of the FM instabilities at $H_{\rm R}$ is indicated by the enhancement of the Sommerfeld coefficient (Fig. \[fig29\]), the enhancement of the $A$ coefficient in the $T^2$ dependence of resistivity, and the concomitant increase in $1/T_1$ and $1/T_2$. The $H_{\rm c2}$ curve for $H \parallel b$ can almost be quantitatively explained by a crude model, where sweeping $H$ drives an enhancement of $m^{**}$ linked to the approach of the FM instabilities[@MiyakeJPSJ2008; @MiyakeJPSJ2009]. In the normal FM phase, the $H$ and $P$ dependence of $m^{**}$ and $m_{\rm B}$ can be estimated from the temperature dependence of $C/T$ through $T_{\rm Curie}$[@Aok11_CR] or the $H$ dependence of the $A$ coefficient. ![(Color online) Field variation of the Sommerfeld coefficient $\gamma$ as a function of $H$. [@Har11][]{data-label="fig29"}](Fig30_URhGe_gamma.pdf){width="0.8\hsize"} The feedback on $H_{\rm c2}$ (Fig. \[fig30\]) was to boost the reference zero field $T_{\rm SC}$ corresponding to $m_H^{**}$ and also act on the slope of $H_{\rm c2}(T)$ by increasing the total effective mass $m^{*}(H) = m^{**}(H) + m_{\rm B}$. Thus, $T_{\rm SC}(m_H^{**})$ was estimated to vary with the McMillan-type formula $$T_{\rm SC} = T_0 \exp \left(-\frac{\lambda + 1}{\lambda}\right)\\ \hspace{2em} \mbox{with} \hspace{3mm}\lambda \equiv \frac{m^{**}}{m_{\rm B}},$$ where $T_0$ is the renormalized electronic energy related to $m_{\rm B}$. The orbital limit gives the $H_{\rm c2}$ dependence $H_{\rm c2} (0)\sim (m^\ast T_{\rm SC})^2$. From the specific heat measurements, $\lambda =0.5$ at $H = 0$ and $\lambda = 1$ at $H_{\rm R}$. ![(a) Field dependence of the SC transition $T_{\rm SC}^0$($m_H^*$) for URhGe,evaluated at zero field assuming $m^{**}$($H$) is equal to the $H = 0$ effective mass. (b) Calculation of $H_{\rm c2}^2$(0)$ \propto (m_H^\ast T_{\rm SC})^2$ taking into account the variation of $T_{\rm SC}^0$ assuming with the hypothesis of the invariance of $m_{\rm B}$. [@MiyakeJPSJ2008; @MiyakeJPSJ2009][]{data-label="fig30"}](Fig31_URhGe_Hdep_Tsc_Hc2.pdf){width="0.6\hsize"} Another way to evaluate the field dependence of $\lambda$ [@WuNatComm2017; @WuThesis] is to use the $H_{\rm c2}(T)$ dependence and then verifies its agreement with the field variation of $\gamma$ according to the relation: $$\lambda(H) = \frac{\gamma(H)}{\gamma(0)}\left(1+\lambda(0)\right) - 1.$$ A series of $H_{\rm c2}$ curves with a fixed $\lambda$ are drawn via a conventional treatment for strong coupling SC and are adjusted to extract $\lambda(H)$. The modified McMillan formula $T_{\rm SC} \propto \exp{\left(-1/(\lambda-\mu^*)\right)}$ was used with the Coulomb repulsion parameter $\mu^* = 0.1$. In this analysis, $\lambda(H = 0) = 0.75$ and $\lambda(H_{\rm R}) \sim1.4$. Effects of pressure and uniaxial stress {#sec:URhGe_pressure_uniaxial} --------------------------------------- The $P$ dependence of RSC predicted by the first approaches indicates that RSC disappeared at a pressure of $P_{\rm RSC} \sim1.5\,{\rm GPa}$, [@MiyakeJPSJ2009] i.e., much lower than the pressure $P_S$ where SC collapses (Fig. \[fig31\]). Under pressure, $T_{\rm Curie}$ increases with $H_{\rm R}$, while $m^\ast (H_{\rm R})$ decreases. ![(Color online) Pressure ($P$) dependence of the ($T, H$) phase diagram of URhGe for $H\parallel b$ with the shrinkage of RSC. RSC collapses for $P_{\rm RSC}$ = 1.5 GPa, which is roughly $P_s$/2. [@MiyakeJPSJ2008][]{data-label="fig31"}](Fig32_URhGe_RSC_pressure.pdf){width="0.8\hsize"} Note that the quasi-coincidence of the collapse of RSC to $H_{\rm QCEP}$ with increasing $\theta$. The increase in $H_{\rm R}$ along the $b$-axis with the $H_c$ component is a direct consequence of the wing structure. As $\theta$ increases in the ($H_b$, $H_c$) plane, $H_{\rm R}$ increases, whereas the effective mass $m^{**}$ and $\chi_b$ decrease. The butterfly SC shape given by $H_{\rm c2}$ (see Fig. \[fig41\]) is a direct consequence of not only adding an $H_c$ component with increasing of $H_{\rm R}$ but also decreasing $T_{\rm SC}^0$($m^*$) as $m_{\rm H}^\ast (\theta)$ decreases. RSC will collapse for a critical value of $T_{\rm SC}^0 (\theta)$. The RSC domain appears to be quite robust, at least for $H_{\rm R} < H_{\rm QCEP}$ . As shown in Fig. \[fig34N\] for a moderately clean crystal, diamagnetic shielding of the low-pressure phase is negligible while a clear diamagnetic signal is observed in the RSC domain. The RSC domain above $H_{\rm R}$ is narrow close to 0.2 T. This experimental observation provides evidence of a change in the order parameter at $H_{\rm R}$. An interesting point is that $m^{\ast\ast}(H)/m^{\ast\ast}(0)$ depends only on the ratio $H/H_{\rm R}$. [@MiyakeJPSJ2009] A decrease in $m^{**}(0)$ leads to a decrease in $m^{**}$($H_{\rm R}$). This situation is reminiscent of the case of CeRu$_2$Si$_2$, where a sharp pseudo-metamagnetic crossover at $H_{\rm M}$ occurs from a nearly AFM phase at $H$ = 0 to a polarized PM phase at $H_{\rm M}$ with $M(H_{\rm M})=\chi_0 H_{\rm M}$ ($\chi_0$: initial-low field susceptibility). [@Flo06_review] Scaling of $H/H_{\rm M}$ is observed under pressure of $m^*(H)/m^*(0)$. Then $P$ motion of $H_{\rm M}$ occurs for the critical value of magnetization, $M(H_{\rm M})$ [@AokiJPSJ2011; @Flo06_review]. A change in magnetic correlations is associated with a drastic change in the FS; when the magnetic polarization reaches a critical value, [@Flo06_review] the low-field PM FS becomes unstable. As we will see in Sect. 6, a Lifshitz transition at $H_{\rm R}$ can strongly enhance $m^*_H$ as it will occur qualitatively through the crossing of the FM instability. Scaling of $H/H_{\rm R}$ is an additional key signature of the Lifshitz transition. There is an additional mechanism to FM spin fluctuations with the assumption of invariance of the FS for RSC. ![(Color online) Diamagnetic shielding response in URhGe detected from AC susceptibility ($\chi_{\rm ac}$) as a function of $H$ for $H \parallel b$. The inset shows the temperature dependence of $H_{\rm c2}$.[]{data-label="fig34N"}](Fig33_URhGe_AC_chi.pdf){width="0.8\hsize"} A novel possibility for driving the FM instability between the $c$- and $b$-axes is to apply uniaxial stress $\sigma$ along the $b$-axis, as thermal-expansion experiments demonstrate that $T_{\rm Curie}$ strongly decreases with increasing $\sigma$. The target is to reach the FM quantum criticality along the $c$-axis and even to switch to FM along the $b$-axis by changing the sign of the magnetocrystalline energy[@BraithwaitePRL2018]. At least the tendency to reach the FM instability along the $b$-axis is clear from the increase in the susceptibility $\chi_b$ \[Fig. \[fig32\](a)\]. Furthermore, as $H_{\rm R}$ would occur when $\chi_b H_{\rm R} \approx M_0$, the expected concomitant effect is a decrease in $H_{\rm R}$. Figure \[fig32\](b) shows the uniaxial stress dependence of $T_{\rm Curie} (H = 0)$, $T_{\rm SC}(H = 0)$, $M_b$ (the magnetization at 5 T) and the increase of $H_{\rm R}^{-1}$. Let us emphasize the major boost of $T_{\rm SC}(\sigma)$ at zero field associated with the increasing of $\chi_b$. At $\sigma = 1.2\,{\rm GPa}$, the maximum $T_{\rm SC}$ reaches $1\,{\rm K}$ at $H_{\rm R}=4\,{\rm T}$, while $T_{\rm SC}$ at zero field increases to $0.5\,{\rm K}$. ![(Color online) (a) Effect of uniaxial stress $\sigma$ on the URhGe magnetization curve with $H \parallel b$. (b) Uniaxial stress dependence of the zero field $T_{\rm SC}$(0), the inverse $H_{\rm R}$, and the inverse $T_{\rm Curie}$ at $H = 0$. [@BraithwaitePRL2018][]{data-label="fig32"}](Fig34_URhGe_uniaxial_M_etc.pdf){width="\hsize"} In the $\sigma$ experiments, it was not possible to quantitatively derive the normal-phase parameter. presumably because of insufficient $\sigma$ homogeneity. However, as shown in Fig. \[fig33\](a), drastic changes occur in the behavior of $H_{\rm c2}$, RSC is replaced by upward enhancement of SC. Analysis of $H_{\rm c2}$ leads to the field dependence of $\lambda(H)$ at different stresses $\sigma$ \[Fig. \[fig33\](b)\]. Furthermore, scaling of $\lambda(H)/\lambda(0)$ as a function of $H/H_{\rm R}$ is obeyed. Approaching the FM instability at $H = 0$ under uniaxial stress result in the marked enhancement of SC at $H_{\rm R}$. ![(Color online) (a) Phase diagram of URhGe at different $\sigma$ with $H\parallel b$. (b) Dependence of $\lambda$ on field in URhGe at different $\sigma$ with $H \parallel b$[]{data-label="fig33"}](Fig35_URhGe_uniaxial.pdf){width="0.8\hsize"} Properties of UCoGe {#sec:UCoGe} =================== UCoGe offers the opportunity to study in more detail the interplay of FM and SC with the decrease in $T_{\rm Curie}$ down to 2.5 K and the increase in $T_{\rm SC}$ up to 0.6 K at ambient pressure. Furthermore, in contrast to URhGe, a moderate pressure ($P \sim1$ GPa) drives the FM-PM instability. Special attention is given to NQR and NMR results. NQR view of FM and SC transition -------------------------------- Figure \[Fig-T1\] shows the temperature dependence of $1/T_1$ measured by $^{59}$Co-NQR in single-crystal UCoGe down to 70 mK[@OhtaJPSJ2010]. $1/T_1$ in the single crystal remains nearly constant down to $T^* \simeq40$ K and gradually decreases below $T^*$. The magnetic susceptibility deviates from the Curie-Weiss behavior and the electrical resistivity along the $c$-axis shows metallic behavior below $T^*$; $T^*$ is regarded as the characteristic temperature below which the U-$5f$ electrons become itinerant with relatively heavy electron mass. Below 10 K, $1/T_1$ increases and shows a large peak at $T_{\rm Curie} \simeq2.5$ K owing to the presence of FM critical fluctuations. However, the $^{59}$Co-NQR spectrum gives evidence of first-order transition behavior at $T_{\rm Curie}$. As shown in Fig. \[CoNQRSpectrum\][@OhtaJPSJ2010], with decreasing temperature, the intensity of the 8.3 MHz NQR signal arising from the PM region decreases below $\simeq$3.7 K, while an 8.1 MHz signal from the FM region, which shifts as result of the presence of the internal field ($H_{\rm int}$) at the Co site, appears below 2.7 K. The two NQR signals coexist between 1 and 2.7 K, but the PM signal disappears below 0.9 K. This indicates that although the phase separation between the PM and FM regions occurs at $T_{\rm Curie}$, the single-crystal UCoGe is in the homogeneous FM state, which is proof of the absence of the PM signal below 1 K. Also note that the frequency of the FM signal, 8.1 MHz, is nearly unchanged from its first appearance. The experimental results of the discontinuous appearance of the FM signal and of the coexistence of FM and PM signals around $T = 2$ K show that the FM transition is of the first order. However, when the temperature variation of the NQR intensity of the PM and FM signals was recorded in cooling and warming processes, no hysteresis behavior was observed, the energy difference between the PM and FM phases was very small. The results are consistent with the previous discussion on UGe$_2$ that the low-temperature transition in itinerant ferromagnets is generally of the first order (see Sect. \[sec:UGe2\] on UGe$_2$) [@BelitzPRL1999; @BelitzPRL2005]. The FM transition of UCoGe is close to the TCP. In the SC state, the fast component of $1/T_1$ in the FM signal is roughly proportional to $T$, indicating that it originates from non-superconducting regions. In contrast, the slow component in the FM signal decreases rapidly below $T_{\rm SC}$, roughly as $T^3$, suggestive of line nodes on a SC gap. The red broken line in Fig. \[Fig-T1\] shows a fit using the line-node model $\Delta (\theta) = \Delta_0 \cos \theta$ with $\Delta_0 = 2.3 k_{\rm B} T_{\rm SC}$. The detection of the SC gap via the FM signal provides unambiguous evidence for the microscopic coexistence of ferromagnetism and superconductivity. In addition, the results below $T_{\rm SC}$ provide some new insight on the nature of the superconductivity in UCoGe. From the relaxation in the FM signal, nearly half of the sample volume remains non-SC even at 70 mK, wheseas the sample is in a homogeneous FM state below 1 K. The inhomogeneous SC state is expected to be related with the SIV state, as discussed below. Co-NQR measurements of the reference compound YCoGe were performed[@KarubeJPSJ2011]. YCoGe has the same TiNiSi crystal structure and similar lattice constants to UCoGe but has no $f$ electrons. The band calculation suggests that the contribution of Co-3$d$ electrons to the density of states is similar in UCoGe and YCoGe. As shown in Fig. \[Fig-T1\]$, 1/T_1$ of Co follows the $T$-linear relation below 250 K down to 0.5 K; this is a typical metallic behavior, and neither ferromagnetism nor superconductivity was observed down to 100 mK[@KarubeJPSJ2011]. These results prove that the ferromagnetism and unconventional superconductivity in UCoGe originate from U-5$f$ electrons. ![(a) (Color online) Temperature dependence of $^{59}$Co NQR $1/T_1$ in single-crystal UCoGe. $1/T_1$ was measured at the PM ($8.3$ MHz) frequency above $2.3$ K, shown by blue circles. Below $2.3$ K, $1/T_1$ was measured at the FM ($8.1$ MHz) frequency. Two $1/T_1$ components were observed in the SC state: the faster (slower) component denoted by red solid (open) squares. The red broken curve below $T_{\rm SC}$ represents the temperature dependence calculated assuming a line-node gap with $\Delta_0 /k_{\rm B} T_{\rm SC} = 2.3$[@OhtaJPSJ2010]. The inset shows the Co-NQR spectra corresponding to the $E_{\pm 5/2}\leftrightarrow E_{\pm 7/2} (\nu_3)$ transitions above and below $T_{\rm Curie}$.[]{data-label="Fig-T1"}](Fig36_UCoGeNQRT1.pdf){width="0.8\hsize"} ![(Color online) (a) $^{59}$Co NQR spectrum of single-crystal UCoGe in PM state. (b) Temperature variation of the $^{59}$Co NQR spectrum from the $\pm5/2 \Leftrightarrow \pm 7/2$ transitions ($\nu_3$) in the single-crystal sample[@OhtaJPSJ2010]. The inset (c) shows the FM signal at 8.1 MHz at $T$ = 2.5 K.[]{data-label="CoNQRSpectrum"}](Fig37_CoNQR.pdf){width="0.9\hsize"} Self-induced-vortex (SIV) state ------------------------------- Since the temperature where the second component of $1/T_1$ emerges coincides with $T_{\rm SC}$ in UCoGe, the two-relaxation behavior of $1/T_1$ is considered to be intrinsic. Furthermore, from recent pressure NQR measurements on the same single crystal, the non-SC component of $1/T_1$ disappears in the high-pressure SC state, where the FM state is suppressed. This strongly indicates that the non-SC component is not an extrinsic effect, such as impurities or inhomogeneity of the sample, but an intrinsic effect induced by the presence of the FM moments. The plausible origin of the non-SC component is ascribed to the SIV, as pointed out in Sect. 2. The SIV state in UCoGe has also been indicated by a muon experiment[@VisserPRL2009], and is now clearly observed in magnetizatiion data. It is worth examining the response of the magnetization of UCoGe to the interplay between FM and SC[@DeguchiJPSJ2010; @PaulsenPRL2012]. A rough estimation of the local critical field $H_{c1}$ from $H_{\rm c2}$ and $H_c$ (thermodynamic critical field estimated from specific heat) gives $H_{c1} \sim3$ G along the $c$-axis and approximately 0.1 G along the $a$- and $b$-axes. Clearly, owing to the strength of the internal field near 100 G (see table II), SIVs already exist at $H = 0$. In Fig. \[fig44\], it is worth observing the change in the hysteresis cycle above 500 mK and below $T_{\rm SC}$ at 75 mK: the coercive field is 6 G at 500 mK and increases 16 G at 75 mK[@HykelPRB2014]. Expulsion of the flux is shown in Fig. \[fig44\]. For $H \parallel c$, the flux expulsion is directly related to the bulk magnetization; it operates on each FM domain. Scanning SQUID microscopy [@HykelPRB2014] helps to clarify the macroscopic figures but the vortex lattice has not yet been observed. No shrinkage of FM domains has been detected, as proposed theoretically[@Fau05; @Dao11]. Recent calculations of the magnetization in the FM-SC phase confirm slight magnetization expulsion in the frame of two FM bands with equal spin pairing. [@Min18] The regime near the vortex core may be the origin of the fast component of $T_1$. However, the number of vortices that can be derived from specific heat measurement (Fig. \[fig:Cp\]) cannot quantitatively explain the large relaxation component detected by NQR. Further improvement of the crystal purity may help clarify the SIV phase. ![(Color online) Magnetization curve of UCoGe in SC and normal phases for $H \parallel c$. (a and b) Hysteresis cycle measured in the PM and SC domains. (c) Field-cooled magnetization measured along the c-axis at 30 Oe, showing flux expulsion below 500 mK. The dashed line is the extrapolation to $T=0\,{\rm K}$ in the case of normal-phase behavior. (d) $c$-axis field-cooled data after deduction of normal-phase contribution. [@PaulsenPRL2012][]{data-label="fig44"}](Fig38_UCoGe_mag.pdf){width="0.6\hsize"} Ising fluctuation with strong $H$ dependence in longitudinal and transverse $H$ scan NMR studies ------------------------------------------------------------------------------------------------ The weakness of $M_0$ and $T_{\rm Curie}$ leads to the new feature that the FM Ising-type interaction in the FM and PM ground states will collapse rapidly under a longitudinal magnetic field $H \parallel c \parallel M_0$[@HuyPRL2008; @AokiJPSJ2009]. Further neutron inelastic experiments,[@StockPRL2011] as well as measurements of $T_1$[@IharaPRL2010; @HattoriPRL2012] by NMR with $H \parallel a, \parallel b$, and $\parallel c$, show that the magnetic fluctuations are of the Ising type with fluctuations of the magnetism parallel to $M_0$. Figure \[fig34\] shows the temperature dependences of $1/T_1T$ in the three main directions. This longitudinal fluctuation is strongly affected strongly by the strength of the $H^c$ component of $H$ along the $c$ axis, as shown in Fig. \[fig35\](a), where $\theta$ is the angle from the $b$-axis in the $bc$-plane. Although the data follow the classical equation of $$\frac{1}{T_1}(\theta) = \frac{1}{T_1^b} \cos^2{\theta} +\frac{1}{T_1^c} \sin^2{\theta} \label{eq:T1vstheta}$$ with constant values of $T_1^{b}$ and $T_1^{c}$ at 20 K, the low-temperature data below 4.2 K do not follow the relation at all, since $T_1^b$ depends on $H^c$. As shown in Fig. \[fig35\](c), $H^c$ is the key parameter[@HattoriPRL2012]. To investigate how the longitudinal FM fluctuations along the $c$-axis $\langle (\delta H^{c} )^2 \rangle$ couple to the external field, $\langle (\delta H^{c} )^2 \rangle$ is derived as a function of $1/T_1 (\theta)$ by combining the equation of $1/T_1T$ and eq. (\[eq:T1vstheta\]) assuming that the magnetic fluctuations in the $ab$-plane are isotropic in the low-field region ($\langle (\delta H^a)^2\rangle \sim \langle (\delta H^b)^2\rangle$): $$\langle (\delta H^c)^2\rangle \propto \frac{1}{\cos^2{\theta}}\left(\frac{1}{T_1}(\theta)-\frac{(1+\sin^2{\theta})}{2}\frac{1}{T_1^c}\right).\label{eq:fluctuation}$$ Figure \[fig35a\] is a plot of $\left<(\delta H^c)^2\right>$ at 1.7 and 0.6 K against $H^c$. When $H_{\rm c2}$ along the $b$-axis is drawn in the same figure, superconductivity is observed in this narrow field region where the longitudinal FM spin fluctuations are active. The longitudinal FM fluctuations $\left<(\delta H^c)^2\right>$, which are coupled with superconductivity, are different from the ordinary spin-wave excitation observed in the FM ordered state. In the conventional FM state, the low-lying spin excitation is a transverse mode corresponding to the Nambu–Goldstone mode, but the FM fluctuations observed in UCoGe are an longitudinal mode of the U-5$f$ moment. In addition, to point out the link between the angle dependence of $H_{\rm c2}(\theta)$ and that of the low-field FM fluctuations seen via $T_1(\theta)$, Fig. \[fig41\] shows the difference in $H_{\rm c2}$ between UCoGe and URhGe as a function of the angle $\theta$ from the $b$-axis: the link between the Ising character of the fluctuation along the $c$-axis is sensitive in UCoGe, but rather insensitive in URhGe. It is noteworthy that there are two SC regions in UCoGe as well as in URhGe: one is in the SC state, which has an extremely large $H_{\rm c2}$ and is sensitive to $\theta$ for $H \parallel b$, and the other is in the SC state, which has a small $H_{\rm c2}$ and is weakly sensitive to deviation from $\theta = 90^\circ$ for $H\parallel c$. The former SC state is considered to be induced by critical FM fluctuations.  was measured. []{data-label="fig34"}](Fig39_UCoGe_T1.pdf){width="0.8\hsize"} ![(Color online) (a) Angular dependence of $1/T_1$ at various temperatures. $\theta$ is the angle from the $b$-axis in the $bc$-plane. The curves show the equation $1/T_1(\theta) = 1/T_1^b \cos^2{\theta} +1/T_1^c \sin^2{\theta}$, with $1/T_1^{b,c}$ a constant value. This curve can consistently explain the smooth variation at 20 K but not the sharp angle dependence observed below 4.2, 1.7, and 0.6 K, which shows a cusp centered at $\theta = 0^{\circ}$. (b) Angular dependences of $1/T_1$ in the $bc$-plane measured at three different magnetic fields at $T$ = 1.7 K. (c) Plot of $1/T_1$ against the $c$-axis component of the field $H^c = H \sin{\theta}$.[]{data-label="fig35"}](Fig40_UCoGe_T1_AngDep.pdf){width="\hsize"} ![(Color online) $H^c$ dependence of magnetic fluctuations along the $c$-axis $\left<(\delta H^c)^2\right>$ at 1.7 and 0.6 K, extracted using Eq. (\[eq:fluctuation\]). $H_{\rm c2}$ determined from $\chi_{\rm ac}$ is plotted against $H^c = H_{\rm c2}\sin{\theta}$, and the $H^c$ region where superconductivity is observable is shown by the yellow area. The relation $\left<(\delta Hc)^2\right> \propto 1/\sqrt{H^c}$ is shown by the dotted line []{data-label="fig35a"}](Fig41_UCoGe_T1_HcDep.pdf){width="0.8\hsize"} ![(Color online) Angular singularity in $H_{\rm c2}(\theta)$ in (a) URhGe and (b) UCoGe for $H \parallel b \perp M_0$; note the collapse of the RSC of URhGe for $\theta = 5^{\circ}$. (c) Angular dependence of $1/T_1$ measured in UCoGe and URhGe for $H \parallel b$ in low-field scan. []{data-label="fig41"}](Fig42_UCoGe_URhGe_Hc2_angdep.pdf){width="0.8\hsize"} The strong reduction of the FM fluctuation for $H \parallel c$ is also clearly detected in the specific heat measurements shown in Fig. \[fig36\], as well as in the field dependence of $A$. Comparing the field dependence of $\gamma$ between UGe$_2$,[@Har09_UGe2] URhGe,[@AokiJPSJ2014Rev] and UCoGe[@AokiJPSJ2014Rev; @WuNatComm2017] (Fig. \[fig37\]), we see that the relative decrease depends roughly on the ratio $a H/T_{\rm Curie}$. For UCoGe, in agreement with the NMR data, the $H$ dependence of $\gamma$ is large for $H < 0.3$ T. ![(Color online) (a) $T$ dependence of $C/T$ in UCoGe at different fields for $H \parallel c$. (b) $H$ decrease of $A$ in longitudinal scan ($H\parallel M_0 \parallel c$) and transverse scan ($H\parallel b$ and $H \parallel a$) []{data-label="fig36"}](Fig43_UCoGe_Cp_Acoef.pdf){width="0.8\hsize"} ![(Color online) Decrease in $\gamma(H)$ as a function of $H$ for UGe$_2$, URhGe, and UCoGe for longitudinal field scan ($H \parallel H_0$. [@Har09_UGe2; @Har11; @WuThesis] The decrease in $\gamma(H)$ in UCoGe will lead to a decrease in $\lambda(H) $ and the universal upward curvature of $H_{\rm c2}$ for $H \parallel b$. []{data-label="fig37"}](Fig44_UCoGe_gamma_Hdep.pdf){width="0.8\hsize"} Consequence on superconductivity: $H_{\rm c2}$ data and modeling ---------------------------------------------------------------- The direct consequence on SC is that the field dependence of $\gamma(H_c)$ will also lead to a drastic decrease in $\lambda(H_c)$, driving the usual upward curvature of $H_{\rm c2}(H \parallel c)$ due to the term $(dT_{\rm SC}/d\lambda)(d\lambda/dH)$ in the expression for $H_{\rm c2}$[@WuNatComm2017] $$\left( \frac{dT_{\rm SC}}{dH_{\rm c2}} \right) = -\frac{1}{\alpha_0 T_{\rm SC} {m^\ast}^2} + \frac{dT_{\rm SC}}{d\lambda}~\frac{d\lambda}{dH}$$ Figure \[fig38\](a) shows how the $H_{\rm c2}$ data for $H \parallel c$ can be parameterized taking into account the field dependence of $\lambda (H)$ on $T_{\rm SC}(m_H^*)$ and $H_{\rm c2}$ caused by the decrease in $m^*_H$. Figure \[fig38\](b) shows $\lambda$ ($H \parallel c$) derived from the analysis of $H_{\rm c2}$, from the direct determination of $\gamma$, and from the phenomenological model described later in Sect. 6. From NMR measurements above $T_{\rm SC}$ at 1.5 K, it was proposed that the FM fluctuation $\left<\delta H_c\right>^2$ decreased as $\sqrt{H_c}$; this $H_c$ dependence will lead to an infinite derivative of $\lambda$ for $H \rightarrow 0$ from the relation $\left<(\delta Hc)^2\right> \propto 1/\sqrt{H^c}$[@HattoriPRL2012]. Measurements of the specific heat down to $T_{\rm SC} \sim0.7$ K show that $\gamma$ and thus $\lambda$ initially decrease linearly with increasing $H$. Thus, the $H$ dependence of $1/T_1$ at intermediate temperatures is markedly enhanced, probably owing to the strong coupling with the critical FM fluctuations. ![(Color online) (a) Universal upward curvature of $H_{\rm c2}$($T$) for $H \parallel M_0 \parallel c$ in UCoGe. The $H_{\rm c2}(T)$ curve is adjusted by selecting $\lambda(H)$ and $T_{\rm SC}$($\lambda (H)$) point by point. [@WuNatComm2017; @WuThesis] (b) Analysis of the field dependence of the pairing strength for $H \parallel c$ in UCoGe[@WuNatComm2017; @WuThesis]. Filled (open) circles were estimated from the experimental $H_{c2}$ curve (from specific-heat measurement). The solid line is a calculation of $\lambda(H)$ derived from the magnetization measurements. [@MineevPUsr2017; @HassingerJPSJ2008] The low-field regime for $H\parallel b$ is presented by blue squares.[]{data-label="fig38"}](Fig45_Hc2_lambda.pdf){width="\hsize"} As already underlined, for $H \parallel b$ the consequence is that the long-range FM along the $c$-axis will collapse for $H^*_b \sim12\,{\rm T}$. This estimation is in excellent agreement with the strong $H$ shift of the maximum of $1/T_1T$ on approaching 12 T, which is shown in Fig. \[fig40\]. Furthermore, the strong increase in $1/T_1T$ at $T$ = 2 K by a factor of 2 between $H = 0$ and $12$ T \[the inset of Fig. \[fig40\](c)\][@HattoriJPSJ2014] is also in good agreement with the increase in $A$ by a factor of 1.8 (in a crude electronic model with $1/T_1T \sim\gamma^2 \sim A$). Parameterization of $\lambda$ via $H_{\rm c2}$ ($H \parallel b$) (see Fig. \[fig39\]) leads to an increase from $\lambda \simeq0.57$ at $H = 0$ to $\lambda \simeq0.68$ at $H^*_b$; the estimation of $\gamma(H^*_b)/\gamma(0) \sim1.06$ is much lower than the value of 1.4 expected from $1/T_1T$ or $A$. [@WuNatComm2017; @WuThesis] The strength of the ratio $T_{\rm Curie}/T_{\rm SC}$ must be related to the switch from the weak to the strong coupling condition. Between URhGe and UCoGe, this ratio differs by one order of magnitude (40 in URhGe compared with 3.8 in UCoGe). As the magnitude of $\lambda$ must be connected to the proximity of the FM-SC instability ($P_{\rm c}$), one may expect at $H=0$ that $\lambda$ of UCoGe is greater than that of URhGe. The derivation of $\lambda$ via $H_{\rm c2}$ gives the opposite result ($\lambda = 0.75$ in URhGe and $\lambda=0.57$ in UCoGe). We remark that the quasi-invariance of $T_{\rm SC}$ against pressure in UCoGe through $P_{\rm c}$ is not compatible with the expected variation of $\lambda (P \to P_{\rm c})$ or a McMillan-type dependence. We again discuss the origin of this discrepancy in Sect. 6. ![(Color online) Temperature dependence of $^{59}$Co-NMR $1/T_1T$ in various fields along the (a) $a$-axis and (b) $b$-axis. (c) Field dependence of $T_{\rm Curie}$ determined by the peak of $1/T_1T$ against temperature in the fields along the $a$- and $b$-axes. The inset shows the field dependences of $1/T_1T$ measured by $^{59}$Co-NMR at $T =$ 2.0 K ($< T_{\rm Curie}$). The dotted lines in the inset are guides for the eye. While the field along the $a$-axis does not change the magnetic properties, the magnetic field along the $b$-axis enhances the magnetic fluctuations. []{data-label="fig40"}](Fig46_UCoGe_T1.pdf){width="0.8\hsize"} ![(Color online) Field dependence of $\lambda(H)$ in UCoGe, for $H \parallel c$ longitudinal mode and $H \parallel b$ and $a$ transverse modes derived from the $H_{\rm c2}$ analysis with the variation of $\lambda$ as shown in the inset. See in Refs. .[]{data-label="fig39"}](Fig47_lambda_Hdep.pdf){width="0.8\hsize"} Attempt to determine the order parameter ----------------------------------------- One of the reliable methods of determining the order parameter is to measure the measurement of Knight shift in the SC state. To estimate the spin susceptibility related to superconductivity, the Knight shift at the Co and Ge sites was measured. Figure \[fig40A\] shows the $^{59}$Co and $^{73}$Ge Knight shift along three directions in normal-state UCoGe[@ManagoPRB2018]. The Knight shift along the $i$ direction ($i$ = $a$, $b$, and $c$) at the Co and Ge sites is described as\ $$\begin{aligned} \label{eq:knightshift} ^{m}K_{i} = {}^{m}A_{i} \chi_{\text{spin},i} + {}^{m}K_{\text{orb},i} \\ \mbox{($m$ = 59 for $^{59}$Co and 73 for $^{73}$Ge)},\end{aligned}$$ where $^{m}A_{i}$ is the hyperfine coupling constant, $\chi_{\text{spin},i}$ is the spin susceptibility, and $^{m}K_{\text{orb},i}$ is the orbital part of the Knight shift. The latter part is usually independent of temperature, and $\chi_{\text{spin}}$ is no longer pure spin in an $f$ electron system because of the strong spin-orbit interaction, but we use the term “spin susceptibility” for simplicity. ![(Color online) $^{73}$Ge (closed symbols) and $^{59}$Co (open symbols) Knight shifts measured at a central line ($1/2 \leftrightarrow -1/2$) with the field of 3 T parallel to the $a$ (squares)-, $b$ (circles)-, and $c$ (triangles)-axes. The inset shows the result along the $c$ direction on a different scale. []{data-label="fig40A"}](Fig48_KnigftShift.pdf){width="0.8\hsize"} When the field is parallel to the $b$- or $c$- axis, the Knight shift at two sites shows the same behavior in a wide temperature range. This indicates that the dominant temperature dependence of the Knight shift can be attributed to the single component of the spin susceptibility from the U-$5f$ electrons, and that the simple treatment of the Knight shift described above is valid even in a $5f$ electron system since the system has a large spin susceptibility and the temperature dependence of $K_\text{orb}$ is relatively small. The hyperfine coupling constants of $^{73}$Ge are estimated from the linear relations and are $\sim0.9$ times those at $^{59}$Co, suggesting that the U-$5f$ electrons couple to the $^{59}$Co and $^{73}$Ge nuclei almost equally. When the field is parallel to the $a$-axis, the temperature dependence of the Knight shift at both sites is relatively small. This result suggests that the spin susceptibility along the $a$-axis is much smaller than those along the $b$- and $c$- axes since $^{m}A_i$ is considered to be isotropic in this system. Note that the magnitude of the $^{59}$Co Knight shift along the $a$-direction in URh$_{0.9}$Co$_{0.1}$Ge at low temperatures \[$^{59}K_a \sim3.5~(2.8) $% in URh$_{0.9}$Co$_{0.1}$Ge (UCoGe)\] is a similar value to that of UCoGe, although the difference in $^{59}K_b$ is huge \[$^{59}K_b \sim18~(4.1) $% in URh$_{0.9}$Co$_{0.1}$Ge (UCoGe)\]. This suggests that the spin susceptibility along the $a$ axis in URhGe is also negligibly smaller than those along the $b$- and $c$-axes, in good agreement with the susceptibility and $M(H)$ data shown in Figs. \[fig7\] and \[fig8\]. The Knight shift in the SC state was measured in various fields along the $a$- and $b$-axes. Figure \[fig40M\] shows the temperature dependence of $^{59}K$ and the Meissner signal below 1 K. The deviation from $^{59}K$ at $T = 1$ K $[\Delta K \equiv K - K(1~{\rm K})]$ is plotted since, as pointed out above, the NQR measurement shows that the whole region of the same single-crystal sample is in the FM state below 1 K[@OhtaJPSJ2010]. In the normal state, $\Delta K$ increases with decreasing temperature, following the development of the FM moments. At $\mu_0 H = 1$ T for $H \parallel a$ and $\mu_0 H = 0.5$ T for $H \parallel b$, the increase in $^{59}K$ looks small or saturates around the SC transition temperature, below which the diamagnetic signal appears (vertical dotted lines in Fig. \[fig40M\]). The extrapolation of $^{59}K$ is determined from the linear fit of $^{59}K$ from 1 K to $T_{\rm SC}$, which is thought to give the upper limit of $^{59}K$ at $T = 0$ K. Therefore, the derivation from the linear extrapolation of $^{59}K$, $\delta K^{a,b}$, which is the maximum value of the suppression of $^{59}K$ due to the occurrence of superconductivity, is estimated to be less than 0.05%. The tiny amount or absence of $^{59}K$ suppression below $T_{\rm SC}$ excludes the spin-singlet pairing state, since an appreciable decrease in the Knight shift, which is of the order of $10^{-1} \sim2$%, is expected in $K_b$ when the spin-singlet pairing is formed. Actually, a clear decrease in the Knight shift was reported in the U-based superconductor UPd$_2$Al$_3$, which is a spin-singlet superconductor coexisting with the antiferromagnetism. [@Kyogaku93; @Kitagawa18] For a spin-triplet superconductor, the spin component of the Cooper pair is expressed by the SC $\mbox{\boldmath $d$}$ vector, which is defined to be perpendicular to the spin component of the Cooper pair, and the Knight shift decreases below $T_{\rm SC}$ when $H$ is applied parallel to the $\mbox{\boldmath $d$}$ vector fixed along a certain crystal axis. However, the situation is not so simple in the present FM superconductors. It was pointed out that the decrease in the Knight shift will be reduced when there is spontaneous magnetization ($M_c$), which splits the up-spin and down-spin bands significantly [@MineevPRB2010]. Thus, Knight shift measurements in the SC state without the FM ordering, which is achieved under pressure, are crucial to determine the SC $\mbox{\boldmath $d$}$ vector. These measurements are now in progress. ![(Color online) Temperature dependences of $^{59}$Co NMR Knight shift and Meissner signal for $H \parallel a, b$ below 1 K. The change from the value at 1 K $\left[\Delta K \equiv K - K(1~{\rm K}) \right]$ is shown. The blue lines in the top two figures are the extrapolations of the linear fit between $T_{\rm SC}$ and 1 K. The vertical dotted lines indicate the onset of the superconductivity probed by the measurements of the ac susceptibility $\chi_{\rm ac}$.[]{data-label="fig40M"}](Fig49_KnightShift_lowT.pdf){width="\hsize"} Thermal conductivity in the normal and SC phases of FM UCoGe [@TaupinPRB2014] was carefully measured in a situation far from very clean material conditions. In the normal phase, the strong anisotropy of $\kappa$ is in excellent agreement with the Ising nature of the magnetic fluctuations and thus provides extra proof of the strong itinerant character of the magnetism of UCoGe. In the SC phase, rather isotropic behavior of $\kappa /T$ was observed, presumably governed by the dominant isotropic impurity effect. More fascinating behavior [@Wu18] was recently reported in the determination of $H_{\rm c2}$ detected by from the resistivity ($\rho$) and thermal conductivity in a transverse field $H_b$ scan. As shown in Fig. \[fig:UCoGe\_Hc2\_rho\_kappa\], close to the field $H^*_b$ where $T_{\rm Curie}$ collapses, the anomalous difference between $\kappa$ and $\rho$ is associated with a drastic decrease in the SC resistivity broadening detected from a gap. The concomitant features have led to the proposal of a field-induced vortex liquid phase coupled to a change in the SC order parameter. ![(Color online) $H_{\rm c2}$ in $H\parallel b$ scan for UCoGe probed by thermal conductivity ($\kappa$) (red circles) and resistivity (blue triangles for $T_{\rho=0}$ and purple triangles for the onset of SC). [@Wu18] The results indicate a possible change in the vortex liquid phase for $H >7$ T associated with a rotation of the SC order parameter. Note also that the broadening of the resistive transition decreases by a factor of approximately 3 after the crossing of $\kappa$ and $\rho$ curves. []{data-label="fig:UCoGe_Hc2_rho_kappa"}](Fig50_UCoGe_Hc2_rho_kappa.pdf){width="0.8\hsize"} Entering the PM regime under pressure ------------------------------------- The application of pressure puts the UCoGe system in the PM region when $T_{\rm Curie} = T_{\rm SC}$[@HassingerJPSJ2008; @SlootenPRL2009; @BastienPRB2016; @BastienPhD]. SC survives far above $P_c \sim$1 GPa up to $P_s \sim$4 GPa. [@BastienPRB2016] In a large $P$ window close to $P_c$ up to $P_s$, the Fermi liquid regime ($AT^2$ in resistivity) is masked by the occurrence of SC (Fig. \[fig42\]); the characteristic spin fluctuation energy appears to remain low. Analysis of $H_{\rm c2}$ under pressure shows that the strong enhancement of $H_{\rm c2}$ for $H \parallel b$ collapses at $P_c$. The unusual field dependence of $\lambda$ for $H \parallel c$ decreases slowly with increasing $P$ (Fig. \[fig43\]). The upward curvature of $H_{\rm c2}$ for $H\parallel c$ survives very close to $P_s$ and far from $P_c$. The corresponding variation of $\lambda (H)$ with $p$ for $H \parallel c$ is shown in Fig. \[fig43\]. [@BastienPRB2016; @BastienPhD] The variation of $H_{\rm c2}$ through $P_{\rm c}$ does not support the proposal [@Kus13] that in the vicinity of $P_{\rm c}$, a switch from type II SC to type I SC may occur. The collapse of $M_0$, and thus of $H_{\rm int}$, leads to the disappearance of the SIV state, as observed already in NQR experiment. Note that no Pauli depairing effect was observed even in the pressure region above $P_c$ along the $b$ axis (Fig. \[fig43\]). This implies that the SC $\mbox{\boldmath $d$}$ vector is perpendicular to the $b$-axis, and thus along the $a$-axis with high possibility, at least when the magnetic field is near $H_{\rm c2}$ along the $b$-axis. To confirm this and determine the SC $\mbox{\boldmath $d$}$ vector thoroughly, it is important to measure $H_{\rm c2}$ along the $a$-axis above $P_c$ and the Knight shift under pressure. ![(Color online) Contour plot of the resistivity exponent $n$ of UCoGe as a function of $T$ and $P$ in the SC and FM domains. When SC collapses, Fermi liquid behavior ($n \sim2$) is recovered below $T_{\rm FL}$. [@BastienPRB2016; @Wu18][]{data-label="fig42"}](Fig51_UCoGe_res_contour.pdf){width="0.8\hsize"} {width="0.8\hsize"} Theory ====== Progress has been realized in clarifying the link between FM fluctuations, the order of quantum criticality, and SC pairing. The difficult issue remaining is the description of the strong-coupling case in UCoGe when $T_{\rm Curie}$ becomes comparable to $T_{\rm SC}$. Special focus is given to different theoretical approaches to explaining the RSC of URhGe. The main approaches are based on the field dependence of the magnetic coupling. A recent attempt involved determining how the shift of the electronic sub-bands with $H$ modifies the SC pairing. Finally, particular phases may enter the class of topological superconductors. Quantum criticality: TCP, QCEP ------------------------------ Different theoretical approaches to studying AFM and FM quantum critical points (QCPs) assuming a second-order transition at $P_c$ ($M_0$ will collapse continuously at $P_c$) can be found in Refs. . However, taking into account the nonanalytic term in the free energy, the compressibility, and the interaction with acoustic phonons, a phase transition can switch the QCP to the TCP[@MillisPRB1993; @LarkinJETP1969]. A generalization of the idea given by Larkin and Pikin has been proposed recently,[@Cha18] with the conclusion that for a first-order transition, quantum criticality will be restored by zero-point fluctuations. It was shown for UGe$_2$ that under pressure[@MineevCRP2011], a TCP exists, in agreement with the experimental results. This is also the case for URhGe as $H$ approaches $H_{\rm R}$ in an $H \parallel b$ scan, and it has already occurred for UCoGe at $P = 0$. At least, the theory for a clean three-dimensional itinerant ferromagnet predicts $P_{\rm TCP} < P_c$[@MillisPRL2002]. The possibility of an intermediate state below the TCP [@Chu09; @Ped13] has been stressed if a drastic FS reconstruction drives a change in the nature of the interaction itself. Figure \[fig45\] shows the data with the prediction [@BelitzPRL2005] based on long-wavelength correlation effects. Poor agreement with the predictions [@MillisPRL2002] has been previously obtained. Recently, a microscopic approach to UGe$_2$[@WysokinskiPRB2015] led to the main features of the CEP, $H_x$, $H_c$, TCP and QCEP being obtained even a good evaluation of the QCEP. [@WysokinskiPRB2015]. A key ingredient to obtain two FM2 and FM1 phases is to start with a two-band mode; quantum critical fluctuations are coupled to the instability of the FS topology. Changes in the topology of the FS via the Lifshifz transition were considered first in Ref. . ![(Color online) Measured and theoretically calculated $T_{\rm CEP}$ in UGe$_2$. [@KotegawaJPSJ2011; @WysokinskiPRB2015; @BelitzPRL2005; @Kad18].[]{data-label="fig45"}](Fig53_T_CEP.pdf){width="0.8\hsize"} The feedback between the localized and itinerant duality of $4f$ or $5f$ electrons leads to spin dynamics different from those in $3d$ itinerant systems. [@HuxleyPRL2003; @StockPRL2011] It was demonstrated that the residual damping detected in an inelastic neutron scattering experiment on UGe$_2$ is a mark of the duality. [@Chu14; @Min13] Superconductivity ----------------- In a magnetic medium, an attractive pairing between electrons at sites $\mbox{\boldmath$r$}$ and $\mbox{\boldmath$r'$}$[@RoussevPRB2001] can be mediated by FM interactions. In the case of a triplet state, the interaction is $$V(\mbox{\boldmath$r$}-\mbox{\boldmath$r'$}) = -C \sum_{\alpha, \beta} \mbox{\boldmath$S$}_{j, \alpha} \chi_{\alpha, \beta}(r - r') \mbox{\boldmath$S$}_{i, \beta},$$ where $\chi_{\alpha, \beta}$ is the static susceptibility, and $\mbox{\boldmath$S$}_i$ and $\mbox{\boldmath$S$}_j$ are the spins at sites $\mbox{\boldmath$r$}$ and $\mbox{\boldmath$r'$}$, respectively [@deGennes]. The coexistence of $p$-state superconductivity with itinerant ferromagnetism[@FayPRB1980] was discussed in the framework of a Hubbard-type exchange interaction $I$ corresponding to a Stoner enhancement factor of $S = (1 - I)^{-1}$; a maximum of $T_{\rm SC}$ was found on both sides of $I = 1$ (at $P_c$), but $T_{\rm SC}$ decreases to zero at $P_c$. It was pointed out that this collapse is an artificial consequence of the approximation; $T_{\rm SC}(P_c)$ is determined by low- but fixed-energy spin fluctuations. [@Shen03] Breakdown of the coexistence of singlet superconductivity and itinerant ferromagnetism for the same electrons was emphasized in Ref. . The possible existence of triplet superconductivity in an almost localized Fermi liquid frame was pointed out in Ref. . The possibility of maintaining a high $C/T$ term below $T_{\rm SC}$ was stressed in Ref. . Pairing gaps near an FM QCP were discussed in Ref.  for two-dimensional itinerant FM systems: a superconducting quasi-long-range order is possible according to an Ising-like hypothesis but will be destroyed in Heisenberg ferromagnets (the case of ZrZn$_2$ is now accepted to be non superconductivity [@Yel05]). To justify that superconductivity in UGe$_2$ occurs only in the FM phase, it was proposed that a coupling with FM magnons will boost $T_{\rm SC}$ in the FM state; [@Kirkpatrick2001PRL; @Kir03; @Kar03] however, experiments failed to detect any magnons. Magnetically mediated superconductivity with AFM and FM coupling near their magnetic instability was calculated in the quasi-two and three dimensional models[@MonthouxPRB2001]. The results are shown in Fig. \[fig46\], where $\xi_m$ is the magnetic coherence length, which diverges at $P_c$. For large $\xi_m$, we are very far from the McMillan formula used in the previous analysis of the experimental results. However, the collapse of FM through a first-order transition leads to a finite value of $\xi_m$ at $P_c$, and thus a good description of $T_{\rm SC}$ is recovered with the McMillan formula $T_{\rm SC} \propto \exp{(-1/\lambda)}$ when $\xi_m$ approaches a distance of a few atomic distances. Just after the discovery of SC in UGe$_2$, a model where localized spins are the source of singlet pairing for the quasiparticles was presented. [@Suh01; @Abr01] However, two decades of experiments rule out this possibility. ![Variation of $T_{\rm SC}/T_{\rm sf}$ as a function of $\xi_m$ for two and three dimensional models for $p$-wave pairing provided by FM fluctuation ($T_{\rm sf}$: spin-fluctuation temperature) []{data-label="fig46"}](Fig54_Tsc_calc.pdf){width="\hsize"} After the discovery of superconductivity in UGe$_2$ and URhGe, the symmetry of the triplet SC states in these orthorhombic materials was proposed. If the $S_z$ = 0 component is negligible (equal spin pairing), the two superconducting states are the A state with a point node along the $z$ easy magnetization $M_0$ axis and the B state with a line node in the ($x, y$) plane. [@MineevPUsr2017] Experimentally, there is no convincing proof of the A and B states. One of the difficulties is the already discovered SIV state. In addition, the purity of crystals may be too low to avoid dominant impurity effects at very low temperatures. The main aspects of FM superconductivity are well described in the framework of BCS weak-coupling theory [@MineevPUsr2017] with the pairing interaction expressed through the static magnetic susceptibility of an FM medium with orthorhombic anisotropy. Such an interaction in anisotropic ferromagnet was discussed in Ref. . The key results are: 1. the magnetic field dependence of the pairing interaction[@MineevPRB2011], 2. the prediction that $T_{\rm Curie}$ decreases as $H^2$ in $H$ transverse scan [@MineevPRB2014], 3. RSC near TCP with $T_{\rm SC}(H_{\rm R}) \sim (1/2) T_{\rm TCP}$[@MineevPRB2015], 4. the drop of TCP with $\sigma$ associated with that of $H_{\rm R}$[@MineevPRB2017_96]. The proposal of a change of the order parameter around $P_c$ (see Ref. ) is actually very difficult to verify experimentally. Special focus on $H_{\rm c2}$ and $H$ dependence of the pairing strength ------------------------------------------------------------------------ We will present the different models developed to explain the unusual shape of $H_{\rm c2}$. The first three proposal deal with the magnetism: the SC pairing is given by spin interactions. The last one shows the effect of FS reconstruction on the pairing. As indicated in the sections on URhGe and UCoGe, comparisons have been made between the $\lambda$ values derived from $H_{\rm c2}$ analysis [@WuNatComm2017; @WuThesis] and those predicted theoretically in the previous Landau approach [@MineevPRB2011]. The calculated variations of $\lambda(H_c)$ and $\lambda(H_b)$ are $$\lambda(H_c) = \lambda(0) \frac{\left[1+(\xi_{\rm mag} k_{\rm F})^2\right]^2}{\left[\frac{3M_z^2}{2M_0^2}-\frac{1}{2}+(\xi_{\rm mag} k_{\rm F})^2\right]^2}$$ and $$\lambda(H_b) = \lambda(0) \frac{\left[1+(\xi_{\rm mag} k_{\rm F})^2\right]^2}{\left[\frac{T_{\rm SC}-T_{\rm Curie}(H)}{T_{\rm SC}-T_{\rm Curie}(0)}+(\xi_{\rm mag} k_{\rm F})^2\right]^2},$$ respectively, where $\xi_{\rm mag}$ is the magnetic coherence length and $k_{\rm F}$ is the Fermi wave vector. When the $5f$ itinerant character dominates, $\xi_{mag} k_F > 1$. Excellent agreement is found between the field dependence of the magnetization (see Fig. \[fig38\]) at low temperatures and $\lambda(H \parallel c)$ determined by comparison with the experiments. The respective values of $\xi_{\rm mag} k_{\rm F}$ for URhGe and UCoGe are linked to the size of $M_0$. For $H \perp M_0$ in UCoGe, the derivation of $\lambda$ (Fig. \[fig39\]) does not agree with the data derived in the normal phase from NMR and transport measurements (see Sect. 5). [@WuNatComm2017; @WuThesis] When $T_{\rm Curie} \leq T_{\rm SC}$, a strong-coupling approach is necessary and, a McMillan representation of $T_{\rm SC}$ is certainly not correct. The agreement between experiments and theory for $H\parallel c$ indicates that in longitudinal $H$ scan, no drastic change in the coupling conditions occurs, that is, the perturbation is weak. The RSC in URhGe was explained by a microscopic model, [@HattoriPRB2013URhGe] where in a transverse magnetic field, soft magnons generate a strong attractive interaction on approaching $H_{\rm R}$ with the change in the spin components of the Cooper pair[@HattoriPRB2013URhGe]. The FM XXZ model describes the coupling of localized moments, and the interaction of magnons and conduction electrons is mediated via anisotropic AFM coupling. The lowest-order fluctuations yield a magnon quasiparticle interaction. In agreement with the experiments, two domes are found for the SC domain. Figure \[fig47\] shows the prediction of $T_{\rm SC}$ with the pairing interaction expressed through the static susceptibility of the FM medium with the orthorhombic anisotropy (see Chapter IV in Ref.  for isotropic ferromagnets). As shown in Fig. \[fig31\](a), the RSC domain above $H_{\rm R}$ is quite narrow in experiments. This is due to the suppression of FM fluctuations by $H_b$. This effect is not taken into account in the model. ![Appearance of two SC domains in the duality between $c$ and $b$ easy magnetization axes described in the framework of soft magnons, which generate a strong attractive interaction on approaching $H_{\rm R}$ [@HattoriPRB2013URhGe]. However, inconsistency is observed in the field region above $H_{\rm R}$.[]{data-label="fig47"}](Fig55_Tsc_RSC.pdf){width="0.8\hsize"} On the basis of mainly on NMR results, the superconductivity in UCoGe was analyzed in a model calculation when longitudinal FM fluctuations induce the spin-triplet pairing[@HattoriPRL2012; @TadaJPCS2013]. The Ising FM fluctuations are described by the susceptibility $$\chi_z=\left(\delta(H_c) +q^2+\frac{\omega}{\gamma_q}\right)^{-1},$$ where $\gamma$ is the Fermi velocity and $\delta(H_c)$ indicates the fluctuations taken at this time, given as $1+c\sqrt{H_c/H}$, from the NMR data[@HattoriPRL2012]. Good agreement was found for the A state, as shown in Fig. \[fig48\]. Experimentally, we have already pointed out that the $\sqrt{H_c}$ dependence is not obeyed at low temperatures. ![(Color online) Calculation of $H_{\rm c2}$ in UCoGe for $H \parallel a$ and $H \parallel c$ using NMR results. [@TadaJPCS2013][]{data-label="fig48"}](Fig56_calc_Tada.pdf){width="0.8\hsize"} Magnetism and superconductivity in UCoGe were recently revisited with aim of explaining the $a$-$b$ anisotropy of $H_{\rm c2}$ with special focus on the Dyaloshinskii-Moriya interaction produced by the zigzag chain structure of UCoGe[@TadaPRB2016]. In agreement with the phenomenological model, the unusual $S$ shape of $H_{\rm c2}$ is linked with the enhancement of FM fluctuations owing to the decrease in $T_{\rm Curie}$. Rotation of the $\mbox{\boldmath $d$}$ vector of the order parameter is a noteworthy feature. Figure \[fig:UCoGe\_Hc2\_calc\] shows the $H_{\rm c2}$ predictions for $H \parallel a$ and $H \parallel b$. ![(Color online) Calculation of $H_{\rm c2}$ in UCoGe for $H \parallel b$ and $H \parallel a$, taking into account the Dyaloshinski–Moriya interaction created by the zigzag chain structure. [@TadaPRB2016][]{data-label="fig:UCoGe_Hc2_calc"}](Fig57_calc_Tada2.pdf){width="0.8\hsize"} To see the effect of FS instability on phenomena such as RSC, the first step is to develop a two-band model with a field-induced instability for RSC in URhGe. Quite recently, such an approach was made to derive the effect of a Lifshitz transition on SC. [@She18] Attention was given to the $H$-induced Lifshitz transition inside a minimum frame of two bands with unequal dispersions and band minima. The band shifts with the field, as shown in Fig. \[fig58\]. A Lifshitz transition occurs at $H_{\rm L}$ when the spin-up branch of band 2 reaches the chemical potential; the relative motions of bands 1 and 2 lead to the maximum of $\chi_0$ and $m^*$ being located at $H_{\rm R} = 1.5 H_{\rm L}$. Eliashberg treatment for the superconductivity shows that the $H$ enhancement of $m^*_H$ drives the increase in $T_{\rm SC}(m^*_H)$ without an orbital effect; upon adding pair-orbital breaking at $H_{\rm c2}$, an RSC domain appears as shown in Fig. \[fig58\] (see Fig. \[fig30\] in Sect. \[sec:URhGe\]). This inconsistency was seen in the field range above $H_{\rm R}$, as in the case shown in Fig. \[fig47\] ![(Color online) Phase diagram of URhGe for $H \parallel b$. The experimental data are shown by red triangles. The solid line is the fitting by the model. The inset shows the schematic description of two electron bands separated by $K_0$, which are shifted up increasing the Zeeman energy. [@She18][]{data-label="fig58"}](Fig58_URhGe_Lifshitz.pdf){width="0.8\hsize"} From comparison with the experiments, the FS topology should be known. A few quantum oscillation experiments have been reported for $H \parallel b$. In the first one, a misalignment of $\theta \sim12^{\circ}$ allows us to escape from RSC; [@YellandNatPhys2011] a frequency of 600 T was observed, which collapses exactly at $H_{\rm R}$. This disappearance was the first evidence of a possible Lifshitz transition. It was hypothesized that it leads to reduction of the SC coherence length as the Fermi velocity vanishes, and thus to the recovery of SC; no proof was given that this selected orbit plays a key role in the SC pairing. In a rigid frame, the contribution of this orbit is only 2% of the total FS volume. However, in this highly correlated multiband system, the reconstruction of one subband reacts to that of the other. In a second experiment with the crystal perfectly aligned along the $b$ axis, one frequency was detected at around 200 T below $H_{\rm R}$ and two frequencies were detected, at 600 and 1200 T above $H_{\rm R}$. A detailed experimental detection of the FS below and above $H_{\rm R}$ is missing. TEP measurements (see Fig. \[fig28\]) indicate that a Lifshitz transition occurs in a sharp window through $H_{\rm R}$ (not in a large domain). Specific heat measurements show that the increase in $m^*$ starts near $H_{\rm R}/1.5$. [@Har11] The interplay between FS and FM instability is obviously stronger than that described in the model. Clearly, this frame is valid in the sense that for a given magnetic polarization (reaching a critical value of the magnetization), a Lifshitz transition drives the enhancement of $m^*$, in good agreement with the scaling observed in pressure and uniaxial experiments (see Sect.. \[sec:URhGe\_pressure\_uniaxial\]). Possible topological cases -------------------------- The symmetrical and topological properties of FM superconductors were stressed in Refs.  for the SC-PM phase of UCoGe with the remark that the superconductivity here will be the electronic analog of the B phase of $^3$He. It was emphasized that UCoGe is an excellent candidate for unconventional SC with hidden protected line nodes. [@Nom17] It was proposed that the SC-PM phase is a promising candidate for Z$_4$ nontrivial topological nonsymmorphic crystalline SC. [@Dai18] A comparison with $^3$He was also made in Ref. . Fermi Surface and Band Structure ================================ An open question is why FM triplet superconductivity has so far been restricted to U compounds. A key point of interest is whether future experiments and band-structure calculation will converge towards a clear view of the interplay between FS topology and SC pairing. There are similarities among Ce, Yb, and U intermetallic compounds. However, major differences exist in the Coulomb repulsion ($U$), spin orbit ($H_{\rm SO}$), crystal field (CF), and hybridization directly linked to the localization of the $f$ electrons. [@Fournier85] Typical values for these parameters for a lanthanide (La) and actinide (Ac) are listed in Table \[table2\] with the unit of eV. $H_{\rm ex}$ is the order of magnitude of exchange coupling. In the case of $U$, as the hybridization is strong like Ce and Yb, Kondo coupling pushes the system close to magnetic instability. Up to now, no SC has been found in FM Ce or Yb compounds. The specificity of U compounds as well as the role of the zigzag configuration in these three compounds must to be clarified. From the rather high values of $T_{\rm SC}$ of the actinide heavy-fermion compounds \[PuCoGa$_5$ ($T_{\rm SC}= 18.5\,{\rm K}$) [@Sar02] and NpPd$_5$Al$_2$ ($T_{\rm SC}= 4.9\,{\rm K}$) [@Aok07_NpPd5Al2]\], the specificity of the U band structure may favor the observation of SC at a moderate temperature. La Ac -------------- ------------- ---------- $U$ 20 10 $H_{\rm SO}$ 0.1 0.3 CF $\leq 0.01$ 0.01–0.1 $H_{\rm ex}$ 0.001–0.01 0.01 : Typical values for Coulomb repulsion ($U$), spin orbit ($H_{\rm SO}$), crystal as above field (CF), and exchange coupling ($H_{\rm ex}$) for lanthanide (La) and actinide (Ac). The unit of these energies is eV. \[table2\] Up to now, the main features of the electronic structure have been revealed by soft X-ray photoelectron spectroscopy[@FujimoriJPSJ2016]. Despite the debate between the localized and itinerant description of the 5$f$ electrons[@TrocPRB2012], the itinerant treatment of the 5$f$-U state is required to analyze the experiments focusing at the Fermi level. Similarities and differences between ARPES experiments and LDA band calculations are discussed in Ref. . Because of the low symmetry of the orthorhombic crystals and the different sites of the U atoms in the lattice, the calculation of the FS topology is notoriously difficult. Figure \[fig50\] shows the FS topology obtained with relativistic linear augmented plane waves within the LDA. UGe$_2$ is a model with a large carrier concentration, while URhGe and UCoGe can be considered semimetals. As already pointed out, the FS of UGe$_2$ has been investigated via elegant quantum oscillation experiments, notably for the FM2 and PM phases. The agreement with band structure calculations remains poor[@ShickPRL2001; @MoralesPRB2016]. Measurements of the magnetic form factor[@KuwaharaPhysicaB2002; @HuxleyJPCM2003] conclude that the ratio $m_{\rm L} /m_{\rm S}$ of the orbital moments to the spin moments increases on switching from FM2 to FM1, whereas the theory predicts a decrease. At least in UGe$_2$, a sufficient numbers of frequencies of the orbits on the FS have been determined to improve the calculations. The band structure in the FM phase of URhGe was first determined by the local spin density approximation (LSDA)[@ShickPRB2002] with the proposal of an additional AFM component. However, neutron scattering experiments ruled out this possibility[@LevyScience2005]. ARPES measurements above and below $T_{\rm Curie}$ suggest a small change in the band structure between the PM and FM phases[@FujimoriPRB2014]. Clearly, experimental progress must be made in this area to give a sound basis for future calculations. In the case of UCoGe, the first study was on combining density functional theory (DFT) and the Kohn–Sham equation with fully relativistic self-consistent resolution, [@DivisPhysicaB2008] but the proposal of a large moment carried by the Co sites disagrees with the NMR results[@KarubeJPSJ2011] and with other measurements[@VisserPRL2009; @TaupinPRB2014]. A similar DFT approach[@MoraJPCS2009] gives an FM moment of 1.35 $\mu_{\rm B}$/U, far above the experimental value of 0.06 $\mu_{\rm B}$/U. Another calculation was done by resolving the Kohn–Sham–Dirac equation[@CzekalaJPCM2010]. The predicted FM moment (0.47 $\mu_{\rm B}$/U ) is basically one order of magnitude higher than the measured one; an interesting feature is the large difference between FM and PM FSs. An interesting feature in UCoGe is the cascade of field-induced Lifshitz transitions for $H \parallel c$ much higher than $H_{\rm c2}$. [@Bas16] For $H \parallel b$ at $\mu_0 H_b^\ast \sim12\,{\rm T}$, preliminary TEP measurements also suggest also a Lifshitz transition in this configuration[@MalonePRB2012]; confirmation is required using a high-quality crystal. Note that dichroism measurement confirms that in UCoGe[@TaupinPRB2014], in agreement with ARPES, the U 5$f$ electron count is close to 3; the Co 3$d$ moment induced by the U moment is only 0.1 $\mu_{\rm B}$ at 17 T, In URhGe, no evidence has been found for a change in orbital and spin components across $H_{\rm R}$[@WilhelmPRB2017]. ![(Color online) Fermi surfaces (FSs) of (a) UGe$_2$, (b) URhGe, and (c) UCoGe in their PM phase according to Ref. []{data-label="fig50"}](Fig59_FS.pdf){width="0.5\hsize"} Conclusion ========== In the U compounds discussed in this review, equal spin-triplet pairing has been established via a large variety of experiments. The important difference between the three materials is the magnitude of ordered moments, leading to different $(T,P)$ phase diagrams. All materials show strong Ising anisotropy and a switch from a second- to first-order transition at the TCP. In spite of the first order transition, the relatively low temperature of the TCP preserves the FM fluctuation and the interplay with the FS instability. This may play an important role in the realization of superconductivity. Note that URhGe is a special case of an “Ising” ferromagnet because of the weak anisotropy between the $b$- and $c$-axes, retaining only hard magnetization along $a$-axis. Taking into account the fact that the combined FS topology and anisotropy of the local contribution of the magnetism governs the Ising character, for URhGe at a low field, a specific subband may play a dominant role in the Ising character a SC pairing, and another subband may dominate the physical properties around $H_{\rm R}$ (a key role of the Lifshitz transition). In UCoGe, owing to the low energies involved in the renormalization of the quasiparticle, the magnetic field modifies the strength of the FM fluctuation with a strong decrease for $H \parallel M_0 \parallel c$ and an increase for $H = H_b^* \perp M_0$. In both UCoGe and URhGe, a transverse field scan ($H \perp M_0$) leads to the collapse of $T_{\rm Curie}$ for $H \sim H_b^*$. URhGe exhibits a singular situation realized by the switch of the easy-magnetization $c$- axis to the $b$-axis at $H = H_b^* =H_{\rm R}$. RSC is a remarkable phenomenon directly linked to the TCP and to the wing structure in the ($H_b, H_c$) plane. RSC disappears exactly when the QCEP is reached. Quantum oscillation and TEP experiments indicate that Lifshitz transition is coupled to the variation of the FM fluctuation. Scaling in $m^\ast (H_{\rm R})/m^\ast (0)$ as a function of $H/H_{\rm R}$ is a signature that the $H$-induced magnetization along the $b$-axis reaches a critical value at $H_{\rm R}$. URhGe belongs to a large class of compounds with an $H$-induced transition driven by a Lifshitz transition such as CeRu$_2$Si$_2$, for example. Further experiments are required to confirm that the Lifshitz transition occurs close to $H_b^*$ in UCoGe. UGe$_2$ is an example of a clean superconductor with simultaneous FS, FM, and PM instabilities. An experimental paradox is that despite the realization of very clean crystals, the intrinsic boundary of SC in the FM domain remains poorly determined. In pressure experiments, the details of the various instabilities when $T_{\rm Curie} < T_{\rm SC}$ are yet unresolved owing to the strong decrease in $T_{\rm Curie}$ towards $P_c$. It is a major experimental challenge to select an accurate tool at pressure in the vicinity of $P_{\rm c}$ to monitor changes in $T_{\rm Curie}$, $M_0$ and $T_{\rm sc}$. A transverse field scan is an elegant way to reach this boundary. The theory has been focused on explaining the experiments. Many properties are explained by a phenomenological weak-coupling approach without consideration of the FS reconstruction even the FM criticality. Up to now, there has been no global consideration combining the FM wing structure, the additional Lifshitz transition, and the consequence on superconductivity, notably above the QCEP. A sound viewpoint may be given by remarks on unconventional quantum criticality (see Ref. ). Recently for URhGe, a crude two-band model has shown the key role of a field induced-Lifshitz transition in the enhancement of $m^\ast$ and in RSC. Combined progress in band structure calculation and in FS determination should be realized, Here, we have not discussed the selection of a given SC order parameter (between A and B) since a new generation of experiments must be performed to resolve the choice. Experimental observation suggests a change in the SC order parameter with $H$ in the transverse field scan of URhGe as well as in a similar scan of UCoGe at $H_{\rm R}$ and $H_b^\ast$. A major interest in these three FM-SC uranium compounds is that the origin of the pairing (FM fluctuations) is well established and that its strength can be tuned easily in $H$, $P$ and $\sigma$ scans. In comparison with other SCs in the class of strongly correlated electronic systems (cuprates, pnictides,, etc.), a negative point is the complexity of their multiband structures. High-$T_c$ cuprates with their initial single-band calculations appear to have a simple band structure. [@Dam03] However, it took two decades to show that the observed frequency at 530 T occupies 2% of the first Brillouin zone in the underdoped regime [@Seb15] and four years of active research to establish that this nodal electron Fermi pocket is created by a charge order. [@WuT2011; @Ghi12] In iron-based superconductors, as three orbitals, $d_{xy}$, $d_{yz}$, and $d_{zx}$, contribute to the electronic states at the FS [@Chu15], the multiband character plays an important role in the pairing. Often quantum oscillation studies give a full view of the FS. [@Ghi12] An illustrative result is the report of orbital-selective Cooper pairing in FeSe. An interesting development is the creation of spin-triplet Cooper pairs at SC interfaces with FM materials and their use in superconductivity spintronics. [@Esc15] Despite the fact that studies on FM-SC have been restricted to a few groups, key results have been discovered that impact on considerably different materials including high-$T_{\rm c}$, pnictide, organic superconductors and a quantum liquid ($^3$He). A basis has been given for the interplay between FS instability and FM fluctuations. The microscopic coexistence of FM and SC has so far only been found in uranium compounds (UGe$_2$, URhGe, UCoGe, UIr [@AkazawaJPCM2004]) despite a long search for new materials. This is probably due to the unique characteristics of the 5$f$ electron in the U atom. It is known that the 5$f$ electron has an intermediate nature between 3$d$ and 4$f$ electrons, in other words, duality of the itinerant/localized character, leading to strong electronic correlations. Furthermore, the strong spin-orbit coupling favors the Ising anisotropy of FM. These two factors might be quite important for FM superconductors. If one can tune the strong correlation and the strong spin-orbit coupling by, for example, the crystal structure, 5$d$ electrons, or other artificial controls, a new FM superconductor with novel properties might be discovered. Looking at the number of publications, the subject of FM superconductivity in U compounds appears to be marginal (350 publications) compared with cuprate superconductors (65000 publications) and Fe-based superconductors (already 5000 publications). In the class of heavy-fermion superconductors, the numbers of publications are similar for CeCoIn$_5$, UPt$_3$, and CeCu$_2$Si$_2$, which have, respectively 321, 576, and 230 articles with the compound name plus superconductivity in the title. We hope that the reported data will clarify key issues in the general understanding of unconventional SC in strongly correlated electronic systems, which should be further examined by investigating the physical properties of other materials. The case of FM triplet superconductivity has often omitted and AF coupling as a source of SC pairing has mostly been discussed (see, for example, Ref. ). Research should meet the interests of researchers. The complexity of 5-$f$ electron behavior commonly observed in low-symmetry crystals may repel researches who only like pure initial conditions. Fortunately, using few renormalized parameters, the complexity vanishes and new territory is reached. The hope is that an ideal material that is easy to grow, easy to purify, easy to cleave, and easy to model with a unique U atom and a high-symmetry structure will be discussed.. Acknowledgements {#acknowledgements .unnumbered} ================ D. A. and J. F. thank their colleagues in Grenoble: J. P. Brison, D. Braithwaite, A. Huxley, G. Knebel, V. Mineev, and A. Pourret. Many results were obtained in the Ph. D. work carried out in Grenoble by I. Sheikin, F. Hardy, F. Levy, V. Taufour, E. Hassinger, L. Howald, M. Taupin, G. Bastien, A. Gourgout, and B. Wu. We also thank A. Miyake, H. Kotegawa, and Y. Tokunaga for their works on high pressure and NMR experiments. Comments on the draft by G. Knebel, K. Miyake, S. Kambe were a great help in improving the manuscript. NMR/NQR studies on UCoGe and YCoGe were done in collaboration with the group of N. K. Sato in Nagoya university. K. I. thanks N. K. Sato, K. Deguchi, Y. Tada, S. Fujimoto, Y. Yanase, H. Ikeda, Y. Ihara, N. Nakai, T. Hattori, S. Kitagawa, M. Manago, for the collaborations, and A. de Visser, S. Yonezawa, and Y. Maeno for valuable discussions. The initial discovery of SC in UGe$_2$ by G. Lonzarich and S. Saxena opened the field and their continuous feedback was very stimulating. We would like to express our gratitude to K. Asayama, Y. Kitaoka and Y. Ōnuki for their interest in our studies. This work was supported by ERC (NewHeavyFermion) and KAKENHI (JP15H05882, JP15H05884, JP15K21732, JP15H05745, JP16H04006, JP15K05156). Note added in proof {#note-added-in-proof .unnumbered} ------------------- Just during reading of our final proof, a stimulating discovery of the superconductivity in the dichalcogenide UTe$_2$ with $T_{\rm SC} \sim 1.5\,{\rm K}$ at ambient pressure was reported [@Ran2018], which is located on the PM verge of the FM instability. A large residual Sommerfeld coefficient far below $T_{\rm SC}$, approximately half of the normal-state value, was observed, which pushes to the proposal of a spontaneous half gapped superconductivity. This statement deserves to carry out careful measurements on the dependence of the specific heat with the material purity. 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--- abstract: 'We investigate the onset of parity-time ($\mathcal{PT}$) symmetry breaking in non-Hermitian tight-binding lattices with spatially-extended loss/gain regions in presence of an advective term. Similarly to the instability properties of hydrodynamic open flows, it is shown that $\mathcal{PT}$ symmetry breaking can be either absolute or convective. In the former case, an initially-localized wave packet shows a secular growth with time at any given spatial position, whereas in the latter case the growth is observed in a reference frame moving at some drift velocity while decay occurs at any fixed spatial position. In the convective unstable regime, $\mathcal{PT}$ symmetry is restored when the spatial region of gain/loss in the lattice is limited (rather than extended). We consider specifically a non-Hermitian extension of the Rice-Mele tight binding lattice model, and show the existence of a transition from absolute to convective symmetry breaking when the advective term is large enough. An extension of the analysis to ac-dc-driven lattices is also presented, and an optical implementation of the non-Hermitian Rice-Mele model is suggested, which is based on light transport in an array of evanescently-coupled optical waveguides with a periodically-bent axis and alternating regions of optical gain and loss.' author: - Stefano Longhi title: 'Convective and absolute $\mathcal{PT}$ symmetry breaking in tight-binding lattices' --- Introduction ============ Non-Hermitian Hamiltonian models are often encountered in a wide class of quantum and classical systems [@Moiseyev]. They are introduced, for example, to model open systems and dissipative phenomena in quantum mechanics (see, for instance, [@Moiseyev; @Rotter; @ob1; @ob2; @vi; @kor]). In optics, non-Hermitian models naturally arise owing to the presence of optical gain and loss regions in dielectric or metal-dielectric structures [@Siegman]. A special class of non-Hermitian Hamiltonians is provided by complex potentials having parity-time ($\mathcal{PT}$) symmetry [@Bender_PRL_98; @Bender_RPP_2007], that is invariance under simultaneous parity transform ($\mathcal{P}$: $\hat{p} \rightarrow -\hat{p}$, $\hat{x} \rightarrow -\hat{x}$, where $\hat{p}$ and $\hat{x}$ stand for momentum and position operators, respectively) and time reversal ($\mathcal{T}$: $\hat{p} \rightarrow -\hat{p}$, $\hat{x} \rightarrow \hat{x}$, $i\rightarrow-i$). An important property of $\mathcal{PT}$ Hamiltonians is to admit of an entirely real-valued energy spectrum below a phase transition symmetry-breaking point, a property that attracted great attention in earlier studies on the subject owing to the possibility to formulate a consistent quantum mechanical theory in a non-Hermitian framework [@Bender_PRL_98; @Bender_RPP_2007; @Mos1; @Ben]. Indeed, $\mathcal{PT}$-symmetric Hamiltonians are a special case of pseudo-Hermitian Hamiltonians, which can be mapped into Hermitian ones [@Mos2]. $\mathcal{PT}$-symmetric Hamiltonians have found interest and applications in several physical fields, including magnetohydrodynamics [@Guenther_JMP_2005], cavity quantum electrodynamics [@Plenio_RMP_1998], quantum-field-theories [@Ben; @BenQFT], and electronics [@Kottos]. More recently, great efforts have been devoted to the study and the experimental implementation of optical structures possessing $\mathcal{PT}$ symmetry (see, for instance, [@Muga; @El-Ganainy_OL_07; @Makris_PRL_08; @Klaiman_PRL_08; @Mostafazadeh_PRL_09; @Longhi_PRL_09; @Guo09; @Ruter_NP_10; @Longhi10; @Feng2011; @Kivshar12; @Regensburger_Nature_12; @Feng12; @uffa] and references therein). The huge interest raised by the introduction of $\mathcal{PT}$ optical media is mainly motivated by their rather unique properties to mold the flow of light in non-conventional ways, with the possibility to observe, for example, double refraction and nonreciprocal diffraction patterns [@Makris_PRL_08], unidirectional Bragg scattering and invisibility [@Longhi_PRL_09; @Lin_PRL_2011; @Regensburger_Nature_12; @LonghiPRA10; @Longhi_JPA_2011; @Graefe11; @Feng12; @MosIN], non-reciprocity [@nonlinearPT], giant Goos-Hänchen shift [@Longhi_PRA_2011], and simultaneous perfect absorption and laser behaviour [@Longhi_PRA_2010; @Chong_PRL_2011]. So far, $\mathcal{PT}$ quantum and classical systems have been mainly investigated in the unbroken $\mathcal{PT}$ phase, where the energies are real-valued, or at the symmetry breaking point, where exceptional points or spectral singularities appear in the underlying Hamiltonian (see, for instance, [@Klaiman_PRL_08; @Longhi10; @MosRes]). In the broken $\mathcal{PT}$ phase, complex-conjugate energies appear. In the context of spatially-extended dissipative dynamical systems and hydrodynamic flows [@Cross], breaking of the $\mathcal{PT}$ phase indicates a bifurcation from a marginally-stable phase to an unstable phase. This means that, while an initially localized wave packet can not secularly grow in the unbroken $\mathcal{PT}$ phase, it does in the broken $\mathcal{PT}$ phase owing to the emergence of modes with complex energies. In hydrodynamics, an unstable open flow can be classified as either [*absolutely*]{} or [*convectively*]{} unstable [@rev1; @flow1; @flow2]. A one-dimensional flow described by an order parameter $\psi(x,t)$ is unstable if, for any given localized perturbation $\psi(x,0)$ at initial time $t=0$, $\psi(x,t) \rightarrow \infty$ as $t \rightarrow \infty$ along at least one ray $x/t=v={\rm const}$. The instability is said to be absolute if $\psi(x,t) \rightarrow \infty$ along the ray $x/t=0$, whereas it is convective if $\psi(x,t) \rightarrow 0$ along the ray $x/t=0$ [@rev1]. Physically, in the convectively unstable regime the initial perturbation grows when observed along the trajectory $x=vt$ at some drift velocity $v$, whereas it decays when observed at a fixed position. Convectively unstable flows generally arise in the presence of an advective (drift) term in the system, in such a way that the growing perturbation drifts in the laboratory reference frame and eventually escapes from the system. Originally introduced in hydrodynamic contexts, the concepts of convective and absolute instabilities have found interest and applications in other physical fields, for example in the study of dissipative optical patterns and noise-sustained structures in nonlinear optics [@Santa]. Inspired by the properties of hydrodynamic unstable flows [@rev1], in this work we introduce the concepts of convective and absolute $\mathcal{PT}$ symmetry breaking for spatially-extended Hamiltonian systems. Specifically, we investigate the symmetry-breaking properties of a tight-binding lattice model with spatially-extended alternating gain and loss regions, and show that the presence of an advective term can change the symmetry breaking from absolute to convective. The lattice model that we consider is a non-Hermitian extension of the famous Rice-Mele Hamiltonian, originally introduced to model conjugated diatomic polymers [@Rice]. In the convectively $\mathcal{PT}$ symmetry breaking regime, the $\mathcal{PT}$ symmetry can be restored when the gain/loss region becomes spatially confined. A physical implementation of the non-Hermitian Rice-Mele lattice model is proposed using arrays of coupled optical waveguides in a zig-zag geometry with periodically-bent axis and alternating optical gain and loss. The paper is organized as follows. In Sec.II the Rice-Mele tight-binding lattice model with non-Hermitian and advective terms is presented, and a physical implementation based on light transport in arrays of coupled optical waveguides is suggested. In Sec.III the concepts of absolute and convective $\mathcal{PT}$ symmetry breaking are introduced for periodic potentials, and the transition from absolute to convective symmetry breaking for the Rice-Mele lattice model is studied by application of asymptotic (saddle point) methods. The concepts of convective and absolute symmetry breaking are also discussed for ac-dc driven lattice models, where the quasi-energy bands . In Sec.IV the main conclusions and future developments are outlined. Finally, in two Appendixes some technical details on Floquet analysis of the ac-dc driven lattice model and saddle point calculations for the Rice-Mele Hamiltonian are presented. The model ========= Extended Rice-Mele Hamiltonian and $\mathcal{PT}$ symmetry breaking ------------------------------------------------------------------- We consider transport of classical or quantum waves on a $\mathcal{PT}$-invariant tight-binding dimerized superlattice with nearest and next-nearest neighborhood hopping schematically shown in Fig.1(a). The evolution of the amplitude probabilities $a_n(t)$, $b_n(t)$ at the two sites of the $n$-th unit cell in the lattice is governed by the following coupled-mode equations $$\begin{aligned} i \frac{d a_n}{d t} & = & -\kappa b_n-\sigma b_{n-1}-\rho \exp(i \varphi) a_{n+1} \nonumber \\ & - & \rho \exp(-i \varphi) a_{n-1}+ig a_n \\ i \frac{d b_n}{d t} & = & -\kappa a_n-\sigma a_{n+1}-\rho \exp(i \varphi) b_{n+1} \nonumber \\ & - & \rho \exp(-i \varphi) b_{n-1}-ig b_n \end{aligned}$$ ![(Color online). (a) Schematic of the non-Hermitian extension of the Rice-Mele tight-binding lattice model. The lattice unit cell contains two sites (a dimer), one with gain and the other with loss. Next-nearest neighborhood hopping occurs at a rate $\rho \exp(i \varphi)$. Convective transport at the $\mathcal{PT}$ symmetry breaking point is obtained for $\rho \neq 0$ and $\varphi \neq 0,\pi$. (b) Optical realization of the Rice-Mele model in a zig-zag array of optical waveguides with alternating optical gain and loss \[cross section in the transverse $(x,y)$ plane\]. The optical axis of the array is bent along the paraxial propagation distance $t$. Axis bending realizes an effective combined ac-dc driving of the lattice with forces $F_x(t)$ and $F_y(t)$ along the two transverse directions $x$ and $y$.](Fig1){width="8cm"} where $\kappa, \sigma >0$ are the nearest-neighborhood modulated hopping rates within the unit cell, $\rho \exp(i \varphi)$ is the complex-valued hopping rate of next nearest sites with controlled phase $\varphi$, and $g$ is the gain/loss rate at alternating sites. The coupled-mode equations (1) and (2) are derived from the tight-binding Hamiltonian $$\begin{aligned} \hat{H} & = & -\sum_n \left( \kappa \hat{a}_n^{\dag} \hat{b}_{n}+\sigma \hat{a}_n^{\dag} \hat{b}_{n-1} + H.c. \right) \nonumber \\ & - & \sum_n \left[ \rho \exp(i \varphi) \left( \hat{a}_n^{\dag} \hat{a}_{n+1} +\hat{b}_n^{\dag} \hat{b}_{n+1} \right) + H.c. \right] \;\;\;\;\; \\ & +& ig \sum_n \left( \hat{a}_n^{\dag} \hat{a}_{n}-\hat{b}_n^{\dag} \hat{b}_{n} \right) \nonumber\end{aligned}$$ which is Hermitian in the limiting case $g=0$ or after replacing $g \rightarrow ig$. The lattice Hamiltonian (3) is invariant under simultaneous parity transformation and time reversal, and can be regarded as a non-Hermitian extension of the Rice-Mele Hamiltonian [@Rice; @note], originally introduced to model conjugated diatomic polymers [@Rice] and found in other physical systems as well, for example in cold atoms moving in one-dimensional optical superlattices [@Bloch]. A possible physical implementation of this Hamiltonian will be discussed in the following subsection. We note that tight-binding lattice models with non-Hermitian terms have been introduced and studied in several recent works [@TB1; @TB2; @TB3; @TB4]. In particular, the limiting case $\rho=0$ and $\kappa_1=\kappa_2$ of the Hamiltonian (3) was previously considered in Refs.[@Longhi_PRL_09; @TB1], where for the infinitely-extended system $\mathcal{PT}$ symmetry breaking was shown to occur at $g=g_{th}=0$. As discussed in the next section, this kind of $\mathcal{PT}$ symmetry breaking is always absolute.\ For the general case $\kappa_1 \neq \kappa_2$ and $\rho \neq 0$, the onset of $\mathcal{PT}$ symmetry breaking can be readily determined by analytical calculation of the energy spectrum of the Hamiltonian (3). To this aim, let us search for a solution to Eqs.(1) and (2) in the form of Bloch-Floquet states $$\left( \begin{array}{c} a_{n}(t) \\ b_{n}( t) \end{array} \right) = \left( \begin{array}{c} A \\ B \end{array} \right) \exp(-iEt+iqn) \;\;\;$$ where $q$ is the quasi-momentum, which is assumed to vary in the interval $(0, 2 \pi)$, and $E=E(q)$ the corresponding energy. Substitution of the Ansatz (4) into Eqs.(1) and (2) yields the following homogeneous linear system for the complex amplitudes $A=A(q)$ and $B=B(q)$ $$\begin{aligned} \left[ E+2 \rho \cos( q+\varphi) -ig \right] A +[\kappa + \sigma \exp(-iq)] B & = & 0 \nonumber \\ \left[ \kappa+\sigma \exp(iq) \right] A+ \left[ E+2 \rho \cos( q+\varphi) +ig \right] B & = & 0 \;\;\;\;\end{aligned}$$ which is solvable provided that the determinantal equation $$\left| \begin{array}{cc} E + 2 \rho \cos (q+\varphi)-ig & \kappa+\sigma \exp(-iq ) \\ \kappa+\sigma \exp(iq ) & E + 2 \rho \cos (q+\varphi) +ig \end{array} \right|=0 \;\;\;$$ is satisfied. This yields the following dispersion relations $E=E_{\pm}(q)$ for the two superlattice minibands $$E_{\pm}(q)=-2 \rho \cos(q+\varphi) \pm \sqrt{-g^2+\kappa^2+\sigma^2+2 \kappa \sigma \cos q }$$ and the following expressions for the amplitudes $A$, $B$ of Bloch-Floquet eigenmodes $$\left( \begin{array}{c} A_{\pm}(q) \\ B_{\pm}(q) \end{array} \right)= \left( \begin{array}{c} \kappa+\sigma \exp(-iq) \\ ig-E_{\pm}(q)-2 \rho \cos(q+\varphi) \end{array} \right).$$ From Eq.(7) it follows that the energy spectrum is entirely real-valued for $g<g_{th}$ with $g_{th} \equiv |\sigma-\kappa|$. In this case, corresponding to the unbroken $\mathcal{PT}$ phase, the energy spectrum comprises two minibands which do not cross. In particular, at $q=\pi$ the two minibands are separated by an energy gap of width $2 \sqrt{g_{th}^2-g^2}$. As $g \rightarrow g_{th}^-$ the gap at $q= \pi$ shrinks and the two minibands touch at $q=\pi$; as $g$ overcomes $g_{th}$, complex-conjugate energies appear near $q=\pi$, which is the signature of $\mathcal{PT}$ symmetry breaking; see Fig.2. It is worth noticing that the group velocity $v_g$ of Bloch modes near $q= \pi$ at the symmetry breaking point, defined by $v_g=(d {\rm {Re}} (E_{\pm})/dq)$, is given by $$v_g=-2 \rho \sin \varphi$$ which does not vanish provided that $\rho \neq 0$, i.e. in the presence of next-nearest neighborhood hopping, and $\varphi \neq 0, \pi$. As it will be shown in Sec.III.B, a non-vanishing and sufficiently large group velocity can cause the $\mathcal{PT}$ symmetry breaking to change from absolute to convective. {width="14cm"} Ac-dc driven lattice model and optical realization of the non-Hermitian Rice-Mele Hamiltonian --------------------------------------------------------------------------------------------- Before discussing the nature of the $\mathcal{PT}$ symmetry breaking for the extended non-Hermitian Rice-Mele Hamiltonian (3), it is worth suggesting possible physical implementations of this model. To realize the Hamiltonian (3), in addition to the non-Hermitian (gain and loss) terms one needs to implement next-nearest neighborhood hoppings with controlled phase $\varphi$. Rather generally, tight-binding lattice models with controlled phase of hopping rates can be realized by combined ac-dc forcing. Here we briefly propose a photonic realization of the extended Rice-Mele model, based on light transport in a superlattice of evanescently-coupled optical waveguides. The Rice-Mele Hamiltonian (3) can be basically obtained as a limiting case of an ac-driven tight-binding lattice at high modulation frequencies. Another possible physical system where the combined ac-dc driven lattice model could be implemented is provided by cold atoms trapped in optical superlattices [@Bloch], where gain is introduced via atom injection at alternating sites [@kor; @BEC]. However, in spite of several theoretical proposals, experimental realizations of $\mathcal{PT}$-symmetric Hamiltonians using ultracold atoms is still missing, and hence we limit here to briefly discuss the photonic system. The optical structure that we consider is shown in Fig.1(b) and is basically composed by a sequence of evanescently-coupled optical waveguides in a zig-zag geometry with alternating optical amplification (gain) and loss. The waveguides are displaced in the horizontal ($x$) and vertical ($y$) directions by the distances $d_x$ and $d_y$, respectively. In the zig-zag geometry, non-negligible evanescent coupling occurs for nearest and next-nearest waveguides [@Felix], with coupling constants (hopping rates) $\kappa_1$, $\kappa_2$ for adjacent guides and $\kappa_3$ for next-nearest guides, as indicated in Fig.1(b). The values of the coupling constants $\kappa_1$, $\kappa_2$ and $\kappa_3$ are determined by certain overlapping integrals of the optical modes trapped in the waveguides, and they are usually exponentially-decaying functions of waveguide separation. For dielectric waveguides, the coupling constants take real and positive values. The difference of couplings $\kappa_1$ and $\kappa_2$ can be controlled by changing the horizontal ($d_x$) and vertical ($d_y$) distances of waveguides, with $\kappa_1=\kappa_2$ for $d_x \simeq d_y$. For straight waveguides, the array of Fig.1(b) thus realizes the extended Rice-Mele model of Fig.1(a) with $\kappa=\kappa_1$, $\sigma=\kappa_2$, $\rho=\kappa_3$ and $\varphi=0$. To realize an effective complex-valued amplitude for the hopping rate between next-nearest neighborhood guides, i.e. $\varphi \neq 0$, we bend the waveguide axis in both $x$ and $y$ directions along the paraxial propagation distance $t$, so that the optical axis of the array describes a curved path with parametric equations $x=x_0(t)$ and $y=y_0(t)$. Arrays of waveguides with arbitrarily curved axis in three-dimensions can be realized, for example, by the technique of femtosecond laser writing in optical glasses (see, for instance, [@Crespi]). In the tight-binding and paraxial approximations, light transport in the superlattice with a bent axis is governed by the following coupled-mode equations (see, for instance, [@LonghRev]) $$\begin{aligned} i \frac{dA_n}{dt} & = & -\kappa_1 B_n -\kappa_2 B_{n-1}-\kappa_3 (A_{n+1}+A_{n-1}) \nonumber \\ & - & [F_x(t)+F_y(t)]n A_n+igA_n \\ i \frac{dB_n}{dt} & = & -\kappa_1 A_n -\kappa_2 A_{n+1}-\kappa_3 (B_{n+1}+B_{n-1}) \nonumber \\ & - & [F_x(t)+F_y(t)]nB_n-F_x(t) B_n-igB_n \;\;\;\end{aligned}$$ where $A_n$, $B_n$ are the mode amplitudes of light trapped in the alternating waveguides with optical gain and loss, respectively, $g$ is the optical gain/loss coefficient, and $$F_x(t)=-\frac{2 \pi n_sd_x}{ \lambda} \frac{d^2x_0}{dt^2} \;, \; \; F_y(t)=\frac{2 \pi n_sd_y}{ \lambda} \frac{d^2y_0}{dt^2}. \;\;\;$$ account for the axis bending in the horizontal ($x$) and vertical ($y$) directions [@LonghRev; @LonghiPRL06]. In Eq.(12), $\lambda$ is the wavelength of the propagating light and $n_s$ is the substrate refractive index at wavelength $\lambda$. Note that Eqs.(10) and (11) describe a dimerized lattice with external forcing, with $F_x(t)$ and $F_y(t)$ playing the role of the external forces. Note also that, in the absence of axis bending, i.e. for $F_x=F_y=0$, Eqs.(10) and (11) reproduce the extended Rice-Mele model \[Eqs.(1) and (2)\] with $\varphi=0$. The equivalence of the driven lattice model \[Eqs.(10) and (11)\] with the static Rice-Mele lattice model \[Eqs.(1) and (2)\] with $\varphi \neq 0$ can be established as follows. Let us tailor the axis bending profiles $x_0(t)$ and $y_0(t)$ in the horizonatl and vertical directions to realize the following ac-dc forces $F_x(t)$ and $F_y(t)$ $$\begin{aligned} F_x(t) & = & U-(\Gamma \omega) \cos(\omega t + \phi) \nonumber \\ F_y(t) & = & -U-(\Gamma \omega) \cos(\omega t-\phi), \end{aligned}$$ where $U$, $\Gamma$ and $\omega$ are real-valued positive parameters. In our optical waveguide system, the combined ac-dc forcing corresponds to a sinusoidal axis bending with spatial frequency $\omega$ superimposed to a parabolic path [@Dignam]. Note that the sinusoidal bending is not in phase for the horizontal and vertical directions owing to the phase term $\phi$. Let us further assume that the following resonance condition $$M \omega=U$$ is satisfied for some integer $M$, and let us introduce the amplitudes $a_n$, $b_n$ via the gauge transformation $$\begin{aligned} A_n (t)& = & a_n (t) \exp[i \varphi n+i n \Phi(t)] \\ B_n (t) & = & b_n (t) \exp[i \varphi n +i \beta + i n \Phi(t)+i \Theta(t)]\end{aligned}$$ where we have set $$\Phi (t)=\int_0^t dt' [F_x(t')+F_y(t')] \; , \; \; \Theta(t)=\int_0^t dt' F_x(t'),$$ $\beta=M \phi-\Gamma \sin \phi$, and $$\varphi=2 M \phi + M \pi$$ Substitution of Eqs.(15) and (16) into Eqs.(10,11) yields a system of coupled-equations for the amplitudes $a_n(t)$ and $b_n(t)$ with time-periodic coefficients of period $T=2 \pi / \omega$. As shown in the Appendix A, if the system is observed at discrete times $\tau=0,T,2T,3T,...$, the evolution of the amplitudes $a_n(\tau)$, $b_n(\tau)$ can be mapped into the dynamics of an effective static lattice (i.e. with time-independent hopping rates) which sustains two minibands with dispersion relations $E_{\pm}(q)$ given by the quasi-energies of the original time-periodic system. In particular, in the large modulation limit $\omega \gg \kappa_{1}, \kappa_2, \kappa_3,g$, i.f. for $T \rightarrow 0$, it can be shown (see Appendix A) that the a-dc driven lattice model exactly reproduces the Rice-Mele static model \[Eqs.(1) and (2)\] with effective hopping rates given by $$\begin{aligned} \kappa & = & \kappa_1J_M(\Gamma) \\ \sigma & = & \kappa_2 J_M(\Gamma) \\ \rho & = & \kappa_3 J_0(2 \Gamma \cos \phi)\end{aligned}$$ and with the phase $\varphi$ given by Eq.(18), where $J_n$ is the Bessel function of first kind and order $n$. Therefore, the zig-zag waveguide array of Fig.1(b) with alternating optical gain and loss and with a suitable axis bending effectively realizes the extended Rice-Mele lattice model of Fig.1(a) with a non-vanishing advective term $\varphi$ and with controlled hopping rates $\kappa$, $\sigma$, $\rho$. Convective and absolute $\mathcal{PT}$ symmetry breaking ======================================================== In this section we introduce the notion of convective and absolute $\mathcal{PT}$ symmetry breaking, inspired by the concepts of convective and absolute unstable flows in hydrodynamics [@rev1; @flow1; @flow2], and then we apply such concepts to the non-Hermtiian Rice-Mele and ac-dc driven models presented in Sec.II. Definition of absolute and convective $\mathcal{PT}$ symmetry breaking for a periodic potential ----------------------------------------------------------------------------------------------- In this subsection we present the rather general definition of convective and absolute $\mathcal{PT}$ symmetry breaking for a continuous system in one spatial dimension $x$, described by a $\mathcal{PT}$-invariant Hamiltonian $\hat{H}=-\partial^2_x+V(x)$ with a potential $V(x)=V_R(x)+i g V_I(x)$, where $V_R(-x)=V_R(x)$ and $V_I(-x)=-V_I(x)$ are the real and imaginary parts of the potential and $g \geq 0$ is a real-valued parameter that measures the strength of the non-Hermitian part of the potential. The concept of convective and absolute $\mathcal{PT}$ symmetry breaking is meaningful in case where at the symmetry breaking point complex-conjugate energies emanate from the continuous spectrum of $\hat{H}$, i.e. the corresponding eigenstates are not normalizable. In fact, if the symmetry breaking arises because of the appearance of pairs of normalizable states with complex-conjugate energies, the $\mathcal{PT}$ symmetry breaking is always absolute and can not be convective, according to the hydrodynamic definitions of absolute and convective unstable flows briefly mentioned in the introduction section and formally defined below. An important case where $\mathcal{PT}$ symmetry breaking arises because of the emergence of extended (non-normalizable) states with complex conjugate energies is the one of a periodic potential, $V(x+d)=V(x)$. In this case, the energy spectrum is absolutely continuous and composed by energy bands. We assume that the energy spectrum of $\hat{H}$ is entirely real-valued for $g \leq g_{th}$, corresponding to the unbroken $\mathcal{PT}$ phase, whereas complex-conjugate energies appear for $g>g_{th}$, where $g_{th} \geq 0$ determines the symmetry breaking point. For example, for the potential $V_R(x)=\cos(2 \pi x/d)$ and $V_I(x)=\sin (2 \pi x /d)$ $\mathcal{PT}$ symmetry breaking is attained at $g_{th}=1$ [@Makris_PRL_08; @LonghiPRA10; @Longhi_JPA_2011; @Graefe11]. Let us then consider an initial wave packet $\psi(x,0)$ at time $t=0$, and let $\psi(x,t)=\exp(-i \hat{H}t) \psi(x,0)$ be the evolved wave packet at successive time $t$. In the unbroken $\mathcal{PT}$ phase, one has $\psi(x,t) \rightarrow 0$ as $t \rightarrow \infty$ at any fixed position $x$ owing to delocalization of the wave packet in the lattice. However, in the broken $\mathcal{PT}$ phase, i.e. for $g>g_{th}$, owing to the appearance of complex energies the wave packet $\psi(x,t)$ is expected to secularly grow as $t \rightarrow \infty$. According to the definitions of unstable flows in hydrodynamic systems [@rev1; @flow1], the $\mathcal{PT}$ symmetry breaking is said to be [*absolute*]{} if $\psi(x,t) \rightarrow \infty$ at $x=0$ (or at any fixed position $x=x_0$), whereas it is said to be convective if $\psi(x,t) \rightarrow \infty$ along the ray $x=vt$ for some drift velocity $v$, but $\psi(x,t) \rightarrow 0$ at $x=0$ (or at any fixed position $x=x_0$). The physics behind the definition of absolute and convective unstable flows is rather simple and is visualized in Fig.3. In the convectively unstable regime, an initial wave packet (perturbation) drifts in the laboratory reference frame with some velocity $v$, and along the ray $x=vt$, i.e. in the reference frame moving with the wave packet, the perturbation secularly grows with time. The drift velocity $v$ is basically determined by the wave packet group velocity at the quasi-momentum $k=k_s$ where the maximum growth rate (i.e. largest imaginary part of the energy) occurs. However, at a fixed position $x=x_0$ (e.g. $x_0=0$), the perturbation $\psi(x_0,t)$ can grow only transiently, but finally it vanishes as $t \rightarrow \infty$ owing to the (possibly fast) drift of the growing wave packet \[see Fig.3(a)\]. Conversely, in the absolutely unstable regime the perturbation grows so fast that, even in the presence of an advective term (a drift), at a fixed spatial position $x_0$ the perturbation $\psi(x_0,t)$ grows indefinitely with time \[see Fig.3(b)\]. To determine whether the $\mathcal{PT}$ symmetry breaking is convective or absolute, let us consider the Hamiltonian $\hat{H}$ with $g>g_{th}$, and let us consider an initial wave packet given by a superposition of Bloch-Floquet modes $\phi_k(x)=u_k(x) \exp(ikx)$ with energy $E=E(k)$, i.e. $\hat{H} \phi_k(x)=E(k) \phi_k(x)$, with $u_k(x+d)=u_k(x)$ and which the quasi-momentum $k$ that varies from $-\infty$ to $\infty$ to account for all the lattice bands (extended band representation). The wave packet then evolves according to the relation $$\psi(x,t)= \int_{-\infty}^{\infty} dk F(k) u_k(x) \exp[ikx-iE(k)t]$$ where $F(k)$ is the spectrum of excited Bloch-Floquet modes. Along the ray $x=vt$ one has $$\psi(t)= \int_{-\infty}^{\infty} dk F(k) u_k(vt) \exp[ikvt-iE(k)t].$$ The determination of the nature (absolute or convective) of the $\mathcal{PT}$ symmetry breaking entails the estimation of the asymptotic behavior of $\psi(t)$ as $ t \rightarrow \infty$. Since $u_k(x)$ is a limited and periodic function of $x$, we can study the asymptotic behavior of the associated wave packet $$\psi_1(t)= \int_{-\infty}^{\infty} dk F(k) \exp[ikvt-iE(k)t]$$ obtained by dropping the term $u_k(vt)$ under the integral in Eq.(23). In fact, it can be readily shown that ${\rm lim \; sup}_{t \rightarrow \infty } |\psi (t)| \rightarrow \infty$ ($ \rightarrow 0$) if and only if ${\rm lim \; sup}_{t \rightarrow \infty } |\psi_1(t)| \rightarrow \infty$ ($ \rightarrow 0$). Note that for the determination of the asymptotic behavior of $\psi_1(t)$ we only need to evaluate the integral on the right hand side of Eq.(24) for those values of $k$ for which ${\rm Im} \{ E(k) \} \geq 0$, the other modes giving no contribution (they are surely decaying).The asymptotic behavior of $\psi_1(t)$ as $t \rightarrow \infty$ can be determined, under certain conditions which are generally satisfied, by the saddle-point (or steepest descend) method [@rev1]. This entails analytic continuation of the function $E(k)$ is the complex $k$ plane and, using the Cauchy theorem, the deformation of the path of the integral along a suitable contour which crosses a (dominant) saddle point $k_s$ of $E(k)-kv$ in the complex plane, along the direction of the steepest descent [@rev1; @flow1; @flow2]. The asymptotic behavior of the integral is then given by the value of the exponential part of the integrand calculated at the saddle point. More precisely, for a saddle point of order $n \geq 2$, i.e. for which $E(k)=E(k_s)+v(k-k_s)+(d^n E/dk^n)_{k_s}(k-k_s)^n+o((k-k_s)^n)$, for $t \rightarrow \infty$ one has [@steep] $$\begin{aligned} \psi_1(t) & \sim & \frac{F(k_s)}{|t (d^n E/dk^n)_{k_s}|^{1/n}} (n!)^{1/n} \Gamma \left( \frac{1}{n} \right) \nonumber \\ & \times & \exp [it vk_s \pm i \pi /(2n) ] \exp[-it E(k_s)]\end{aligned}$$ where the saddle point $k_s$ in the complex plane is determined from the equation $$\left( \frac{dE}{dk} \right)_{k_s}=v.$$ The decay or secular growth of $\psi_1(t)$ thus depends on the sign of the imaginary part of $E(k)$ at the saddle point $k=k_s$. It can be readily shown that, for $g>g_{th}$, there is always a velocity $v=v_s$ for which the solution $k_s$ to Eq.(26) is real-valued and corresponds to the maximum growth rate \[i.e. the maximum of ${\rm Im} (E(k))>0$\], so that along the ray $x=v_s t$ the amplitude $\psi_1(t)$ shows a secular growth. To determine whether the symmetry breaking is either convective or absolute, we should consider the asymptotic behavior of $\psi_1(t)$ for $v=0$, which is determined by the sign of the imaginary part of $E(k)$ at the saddle point $k=k_s$ obtained from Eq.(26) with $v=0$. Hence, the $\mathcal{PT}$ symmetry breaking is absolute if ${\rm Im} \{ E(k_s) \}>0$, whereas it is convective if ${\rm Im} \{ E(k_s) \} \leq 0$, where the saddle point $k_s$ is determined from the equation $(dE/dk)_{k_s}=0$. As a general rule of thumb, for $g$ larger but close the $\mathcal{PT}$ symmetry breaking threshold, indicating by $k_s$ the quasi momentum on the real axis with maximum growth rate, i.e. that maximizes ${\rm Im}(E(k))$ for $k$ real, the $\mathcal{PT}$ symmetry breaking is absolute if the group velocity $v_g$ at $k=k_s$, given by $v_s=(d {\rm Re}(E) /dk)_{k_s}$, vanishes, whereas is it expected to be convective for a nonvanishing (and possibly large) value of $v_s$. Physically, the latter regime corresponds to the case where, owing to a non-vanishning group velocity, the unstable growing Bloch-Floquet mode is advected away, for an observer at rest, fast enough that it decays in time when observed at a fixed spatial position. {width="8.3cm"} {width="8.3cm"} Absolute and convective $\mathcal{PT}$ symmetry breaking for the non-Hermitian Rice-Mele Hamiltonian ---------------------------------------------------------------------------------------------------- In this subsection we describe in details the nature of the $\mathcal{PT}$ symmetry breaking for the extended Rice-Mele Hamiltonian defined by Eq.(3). As shown in Sec.II.A, the superlattice comprises two minibands, with dispersion relations $E_{\pm}(q)$ and corresponding Bloch-Floquet modes defined by Eqs.(7) and (4,8), respectively. After setting $\psi(n,t)=(a_n(t),b_n(t))^T$, let us consider the propagation of an initial wave packet $\psi(n,0)$ in the lattice, which is assumed to be given by a superposition of Bloch-Floquet modes belonging to the two minibands with spectral functions $F_{\pm}(q)$. The evolved wave packet at time $t$ is then given by $$\begin{aligned} \psi(n,t) & = & \int_{0}^{2 \pi} dq F_+(q) \phi_+(q) \exp[iqn-iE_{+}(q)t] \nonumber \\ & + & \int_{0}^{2 \pi} dq F_-(q) \phi_-(q) \exp[iqn-iE_{-}(q)t] \;\;\;\;\;\;\end{aligned}$$ where we have set $\phi_{\pm}(q)=(A_{\pm}(q),B_{\pm}(q))^T$. As shown in Sec.II.A, $\mathcal{PT}$ symmetry breaking occurs when the gain/loss parameter $g$ is increased to overcome the threshold value $g_{th}=|\kappa-\sigma|$. Correspondingly, complex conjugate energies appear for a wave number $q$ close to $q_0=\pi$ \[see Fig.2(c)\]. Note that, since ${\rm Im} \{ E(q) \} \geq 0$ for one miniband and ${\rm Im} \{ E(q) \} \leq 0$ for the other miniband, one of the two integrals on the right hand side of Eq.(27) decays toward zero as $t \rightarrow \infty$, and therefore we can limit to consider the contribution arising from the other integral involving unstable modes. Assuming, for the sake of definiteness, ${\rm Im} \{ E_+(q) \} \geq 0$ and ${\rm Im} \{ E_-(q) \} \leq 0$, one has $$\psi(n,t) \sim \int_{0}^{2 \pi} dq F_+(q) \phi_+(q) \exp[iqn-iE_{+}(q)t]$$ as $t \rightarrow \infty$. The asymptotic form of the integral on the right hand side of Eq.(28) along the ray $n=vt$ can be estimated by the saddle point method and takes a form similar to the one given by Eq.(25). According to the analysis presented in Sec.III.A, the $\mathcal{PT}$ symmetry breaking is thus convective if ${\rm{Im}} \{ E_+(q_s)\} \leq 0$ , whereas it is absolute for ${\rm{Im}} \{ E_+(q_s)\}>0$, where $q_s$ is the dominant saddle point obtained from the equation $(dE_+/dq)_{q_s}=0$, i.e. $$\frac{2 \rho}{\kappa \sigma} \left (\cos \varphi \sin q_s \sin \varphi \cos q_s \right)=\frac{\sin q_s}{-\epsilon^2+2 \kappa \sigma(1+ \cos q_s)}.$$ In Eq.(29) we have set $\epsilon^2=g^2-g_{th}^2$, which provides a measure of the distance from the $\mathcal{PT}$ symmetry breaking point. To simplify our analysis, let us consider the case where the gain/loss parameter $g$ is larger but close to its threshold value $g_{th}$, so that $\epsilon^2$ is a small quantity. In this case the solutions to Eq.(29) can be determined analytically by an asymptotic analysis in the small parameter $\epsilon$. The calculations are detailed in the Appendix B. The main result of the calculations is that the $\mathcal{PT}$ symmetry breaking is [*convective*]{} for $$|v_g|> \sqrt{\sigma \kappa}$$ ![(Color online). Numerically-computed quasi-energy minibands $E_{\pm}(q)$ for the ac-dc driven lattice model \[Eqs.(10) and (11)\] for increasing values of the modulation frequency $\omega$: (a) $\omega=6$, (b) $\omega=15$, and (c) $\omega=150$. The other parameter values are given in the text. In (d) the energy minibands of the static Rice-Mele lattice are shown, that correspond to the asymptotic limit $\omega \rightarrow \infty$. Solid curves refer to the real part of $ E_{\pm}(q)$, whereas the thin dotted curves to the imaginary part of $ E_{\pm}(q) $. For the sake of clearness, the imaginary part of $ E_{\pm}(q) $ has been multiplied by a factor of 10.](Fig5){width="8.3cm"} {width="8.3cm"} whereas it is [*absolute*]{} in the opposite case $|v_g| \leq \sqrt{\sigma \kappa}$, where $v_g=-2 \rho \sin \varphi$ is the group velocity at the symmetry breaking point of the most unstable mode with wave number $q=\pi$ \[see Eq.(9)\]. Hence, as expected, a sufficiently large advective term in the extended Rice-Mele Hamiltonian can change the $\mathcal{PT}$ symmetry breaking from absolute to convective. Note that for $\rho=0$ or $\rho \neq$ but real-valued, the symmetry breaking is always absolute. As discussed in the next subsection, an important physical implication of the convective (rather than absolute) $\mathcal{PT}$ symmetry breaking is that the unbroken $\mathcal{PT}$ phase can be restored in the convectively regime by making the region of alternating gain and loss sites in the lattice [*spatially limited*]{} rather than extended. Numerical results ----------------- We checked the predictions of the theoretical analysis and the transition form absolute to convective $\mathcal{PT}$ symmetry breaking induced by advection for both the static Rice-Mele lattice of Fig.1(a) and the ac-dc driven lattice of Fig.1(b) by direct numerical simulations. As an example, in Fig.4 we depict the evolution of a wave packet in the Rice-Mele lattice with advective term ($\rho \neq 0$, $\varphi \neq 0, \pi$), showing the transition from convective \[Fig.4(a)\] to absolute \[Fig.4(b)\] $\mathcal{PT}$ symmetry breaking. The numerical results are obtained by solving the coupled-mode equations (1) and (2) using an accurate fourth-order variable-step Runge-Kutta method assuming as an initial condition a Gaussian wave packet with carrier wave number $q_0=\pi$ at lattice sites $a_n$ solely, namely $a_n(0)=\exp[-2(n/w)^2+iq_0n] $ and $b_n(0)=0$, where $w$ is the size of the wave packet. Such an initial condition mainly excites (unstable) Bloch-Floquet modes with imaginary energy at wave numbers $q$ close to the most critical one $q=q_0=\pi$. Parameter values used in the simulations are $\kappa=\sigma=1$ (corresponding to $g_{th}=0$), $\varphi=\pi/2$, $g=0.05$ and $\rho=0.7$ in Fig.4(a), and $\rho=0.3$ in Fig.4(b). In Fig.4(a), the condition $|v_g|>\sqrt{\sigma \kappa}$ is satisfied and, according to the analysis of Sec.III.B, the symmetry breaking is of convective nature. In fact, while the wave packet $|\psi (t)|^2$ secularly grows along the ray $n=v_g t$, it decays when observed at a fixed spatial position (e.g. $n=0$), as shown in the lower panel of Fig.4(a). Conversely, in Fig.4(b) the advective term in the Rice-Mele Hamiltonian is lowered so that $|v_g|$ is smaller than $\sqrt{\sigma \kappa}$: in this case the symmetry breaking is absolute, as clearly shown in the lower panel of Fig.4(b).\ A similar transition from absolute to convective $\mathcal{PT}$ symmetry-breaking for increasing advection is observed in the ac-dc driven lattice model of Fig.1(b) presented in Sec.II.B. As shown in the Appendix A, the dynamical properties of the ac-dc driven lattice at discretized times $\tau=0,T,2T,...$ can be mapped into the ones of a static lattice with an energy band structure that is determined by the quasi-energy spectrum $E(q)$ of the ac-dc driven lattice. In particular, at large modulation frequencies the driven lattice model, defined by Eqs.(A1) and (A2), exactly reproduces the Rice-Mele model with effective hopping rates $\kappa$, $\sigma$, $\rho$ and phase $\varphi$ given by Eqs.(18-21). As an example, in Figs.5(a-c) we show the numerically-computed quasi energies of the two minibands (real and imaginary parts) for the ac-dc driven lattice above the $\mathcal{PT}$ symmetry breaking point for parameter values $\kappa_1=\kappa_2=2.1124$, $\kappa_3=1.4784$, $M=1$, $\Gamma=1.109$, $\phi=-\pi/4$, $g=0.05$ and for increasing values of the modulation frequency $\omega$. Parameter values have been chosen such as to reproduce, at large modulations frequencies, the static Rice-Mele lattice with parameters as in Fig.4(b). The quasi-energies have been obtained by numerical computation of the Floquet exponents for the eigenvalue problem defined by Eqs.(A6) and (A7) given in the Appendix. For comparison, in Fig.5(d) the minibands of the static Rice-Mele lattice with parameters of Fig.4(b) are also depicted. According to the theoretical analysis, in the high modulation regime the quasi-energy spectrum of the driven lattice asymptotically reproduces the spectrum of the static Rice-Mele model \[compare Fig.5(c) and (d)\]. At low or moderate values of the modulation frequency $\omega$, deviations from the two models can be clearly appreciated \[compare Figs.5(a) and (b) with Fig.5(d)\]. In particular, the driven lattice model at low modulation frequencies shows a wider range of wave numbers with complex energies, and the real part of the quasi energies for the two minibands are not degenerate. Nevertheless, the transition from convective to absolute symmetry breaking, which is basically related to the value of the group velocity (the derivative of the real part of the quasi-energy) of the unstable mode at the symmetry breaking point, can be observed even at moderate modulation frequencies. This is shown, as an example, in Fig.6, where we depict the numerically-computed evolution of the same initial Gaussian wave packet as in Fig.4 but in the ac-dc driven lattice for a modulation frequency $\omega=15$ and for $\kappa_3=1.4784$ \[Fig.6(a)\], corresponding to a convective symmetry breaking, and $\kappa_3=0.6336$ \[Fig.6(b)\], corresponding to absolute $\mathcal{PT}$ symmetry breaking. {width="8.3cm"} As a final comment, it is worth discussing a physically relevant implication of convective versus absolute $\mathcal{PT}$ symmetry breaking. In the convectively unstable regime, the growing wave packet drifts in the laboratory reference frame fast enough that locally (i.e. at a fixed spatial position) it is observed to decay in spite of its growth in a moving reference frame (see Figs.3 and 4). Let us now consider a lattice with a spatially confined (rather than infinitely extended) region of unit cells with gain and loss. In the convective regime, advection pushes the wave packet far from the “non-Hermitian” region of the lattice, and hence after a transient the wave packet ceases to grow. Conversely, in the absolute symmetry breaking regime it is expected to grow indefinitely even for a spatially-finite extension of unit cells with gain and loss. Such a simple physical picture suggests that in the convectively unstable regime $\mathcal{PT}$ symmetry (i.e. an entirely real-valued energy spectrum) may be restored when the gain/loss region in the lattice is spatially limited. We checked such a prediction by considering the Rice-Mele lattice Hamiltonian (3) with a spatially-dependent gain/loss term vanishing at infinity, namely $$\begin{aligned} \hat{H} & = & -\sum_{n=-\infty}^{\infty} \left( \kappa \hat{a}_n^{\dag} \hat{b}_{n}+\sigma \hat{a}_n^{\dag} \hat{b}_{n-1} + H.c. \right) \nonumber \\ & - & \sum_n \left[ \rho \exp(i \varphi) \left( \hat{a}_n^{\dag} \hat{a}_{n+1} +\hat{b}_n^{\dag} \hat{b}_{n+1} \right) + H.c. \right] \;\;\;\;\; \\ & +& i \sum_{n=-\infty}^{\infty}g_n \left( \hat{a}_n^{\dag} \hat{a}_{n}-\hat{b}_n^{\dag} \hat{b}_{n} \right) \nonumber\end{aligned}$$ where $g_n \rightarrow 0$ as $n \rightarrow \infty$. In particular, we numerically computed the energy spectrum of $\hat{H}$ by considering a square-wave profile of $g_n$, i.e. $g_n=g$ for $|n| \leq N_g/2$ and $g_n=0$ otherwise. This case corresponds to a central lattice section comprising $(N_g+1)$ dimers with gain and loss (i.e. locally non-Hermitian), and and abrupt transition to two outer lattice sections with locally-Hermitian dimers (i.e. $g_n=0$). In the numerical simulations, the total number of unit cells $(N+1)$ is finite, corresponding to truncation of the outer lattice sections. As an example, in Fig.7 we show the numerically-computed energies of the truncated lattice described by the Hamiltonian $$\begin{aligned} \hat{H} & = & -\sum_{n=-N/2}^{N/2} \left( \kappa \hat{a}_n^{\dag} \hat{b}_{n}+\sigma \hat{a}_n^{\dag} \hat{b}_{n-1} + H.c. \right) \nonumber \\ & - & \sum_{n=-N/2}^{N/2} \left[ \rho \exp(i \varphi) \left( \hat{a}_n^{\dag} \hat{a}_{n+1} +\hat{b}_n^{\dag} \hat{b}_{n+1} \right) + H.c. \right] \;\;\;\;\; \\ & +& ig \sum_{n=-N_g/2}^{N_g/2} \left( \hat{a}_n^{\dag} \hat{a}_{n}-\hat{b}_n^{\dag} \hat{b}_{n} \right) \nonumber\end{aligned}$$ for $N_g=10$, $N=300$ and for parameter values corresponding to absolute \[Fig.7(a)\] and convective \[Fig.7(b)\] $\mathcal{PT}$ symmetry breaking in the extended (i.e. $N,N_g \rightarrow \infty$) limit. Note that, within numerical accuracy, the energy spectrum is entirely real-valued in the convective regime \[Fig.7(b)\], whereas pairs of complex-conjugate energies persist in the absolute regime \[Fig.7(a)\]. It should be noted, however, that restoring of the $\mathcal{PT}$ symmetry in the convective regime is not a strict rule, since the interfaces from the outer lattice regions to the inner (non-Hermitian) lattice section might sustain localized (interface) modes with imaginary energies, which can not be predicted by our simple picture. Moreover, it is expected that restoring of the $\mathcal{PT}$ symmetry depends on the choice of the profile $g_n$; for example a smooth (rather than sharp) transition from the inner (locally non-Hermitian) to the outer (locally Hermitian) regions is expected to avoid the appearance of interface states. Symmetry breaking in case of inhomogeneous gain/loss parameter $g_n$ would require a further study, however this goes beyond the scope of the present work. Conclusions =========== In this work we have introduced the concepts of convective and absolute $\mathcal{PT}$ symmetry breaking for wave transport in periodic complex potentials, inspired by the hydrodynamic concepts of convective and absolute instabilities in open flows. In particular, we have investigated analytically and numerically the transition from absolute to convective $\mathcal{PT}$ symmetry breaking in two tight-binding lattice models: a non-Hermitian extension of the Rice-Mele dimerized lattice, originally introduced to model conjugated diatomic polymers, and an ac-dc driven lattice, which reproduces the Rice-Mele model in the large modulation frequency limit. In the context of spatially-extended dissipative dynamical systems, $\mathcal{PT}$ symmetry breaking can be viewed as a phase transition from a marginally stable state (the unbroken $\mathcal{PT}$ phase) to an unstable state (the broken $\mathcal{PT}$ phase). The instability arises because of the appearance of pairs of complex-conjugate energies in the broken $\mathcal{PT}$ phase. The distinction between convective and absolute $\mathcal{PT}$ symmetry breaking arises when considering the evolution of a wave packet in the broken $\mathcal{PT}$ phase: while in the absolute symmetry breaking case the wave packet amplitude observed at a fixed spatial position secularly grows in time, in the convective symmetry breaking case the amplitude grows in a reference frame moving at some drift velocity, however it decays when observed at a fixed spatial position, i.e. for an observer at rest. A convective regime is generally found when the unstable modes have a group (drift) velocity large enough that at a fixed spatial position the wave packet decay due to the drift overcomes the growth due to the instability. The nature (either absolute or convective) of the $\mathcal{PT}$ symmetry breaking is basically determined by the sign of the imaginary part of the energy (for static lattices) or quasi-energy (for periodically-driven lattices) at the dominant band saddle point in complex plane. An interesting application of the concepts of convective and absolute symmetry breaking is found when considering a spatially-limited region of gain/loss in the system, i.e when the periodicity of the system is broken and the imaginary part of the potential is confined to a limited region of space. Owing to the fast drift of a wave packet in the convective regime, after a transient the wave packet escapes from the imaginary potential region and thus it ceases to grow. This means that the instability is only transient, i.e. we expect that $\mathcal{PT}$ symmetry is restored in the convective regime when the imaginary potential is spatially confined. This is not the case of the absolute symmetry breaking regime, where the broken $\mathcal{PT}$ phase is expected to persist even for a spatially-limited imaginary potential. Other possible applications and developments of the hydrodynamic concepts of convective and absolute instabilities can be foreseen into the rapidly growing field of wave transport in $\mathcal{PT}$-symmetric quantum and classical systems. For example, like for hydrodynamic and dissipative optical systems [@rev1; @Santa], interesting effects (like the appearance of noise-sustained structures [@Santa]) might be envisaged for convective $\mathcal{PT}$ symmetry breaking in presence of classical or quantum noise [@uffa]. Floquet analysis of the ac-dc driven lattice and effective static lattice model =============================================================================== In this Appendix we present a Floquet analysis of the driven lattice model defined by Eqs.(10) and (11) with time-periodic coefficients and show that, at discretized times, it behaves like an effective static lattice with a band structure that is determined by the quasi-energy spectrum of the driven lattice. To this aim, let us note that, after the gauge transformation defined by Eqs.(15) and (16) given in the text, the evolution of the amplitudes $a_n(t)$, $b_n(t)$ is governed by the following linear system of equations $$\begin{aligned} i \frac{da_n}{dt} & = & -\kappa_1 F(t) b_{n}-\kappa_2 G(t) b_{n-1}-\kappa_3H(t) a_{n+1} \nonumber \\ & - & \kappa_3 H^*(t) a_{n-1}+iga_n \\ i \frac{db_n}{dt} & = & -\kappa_1 F^*(t) a_{n}-\kappa_2 G^*(t) a_{n+1}-\kappa_3H(t) b_{n+1} \nonumber \\ & - & \kappa_3 H^*(t) b_{n-1}-igb_n\end{aligned}$$ with time-dependent coefficients $F(t)$, $G(t)$ and $H(t)$ given by $$\begin{aligned} F(t) & = & \exp \left[ i \beta +i \Theta (t) \right] \nonumber \\ G(t) & = & \exp \left[ i \beta -i \varphi +i \Theta(t)-i \Phi(t) \right] \\ H(t) & = & \exp \left[ i \varphi + i \Phi(t) \right]. \nonumber\end{aligned}$$ In the previous equations, the functions $\Theta(t)$ and $\Phi(t)$ and constant parameters $\varphi$ and $\beta$ are defined by Eqs.(17) and (18) given in the text. For the driving terms $F_x$, $F_y$ defined by Eq.(13), one has explicitly $$\begin{aligned} F(t) & = & \exp \left[ iM \phi +i M \omega t -i \Gamma \sin(\omega t + \phi) \right] \nonumber \\ G(t) & = & \exp \left[ -i M (\phi+ \pi) +iM \omega t +i \Gamma \sin (\omega t - \phi) \right] \; \; \; \; \; \; \; \\ H(t) & = & \exp \left[ i M ( 2 \phi+\pi) -2 i \Gamma \cos \phi \sin (\omega t) \right]. \nonumber\end{aligned}$$ where we assumed the resonance condition $U= M \omega$. Since the coefficients $F(t)$, $G(t)$ and $H(t)$ are periodic in time with period $T=2 \pi / \omega$, the solution to Eqs.(A1) and (A2) can be obtained from Floquet theory of liner periodic systems. Specifically, the general solution to Eqs.(A1) and (A2) is given by an arbitrary superposition of Bloch-Floquet states $$\left( \begin{array}{c} a_n(q,t) \\ b_n(q,t) \end{array} \right)= \left( \begin{array}{c} A(q,t) \\ B(q,t) \end{array} \right) \exp \left[ iqn-i E(q) t \right]$$ where $q$ is the wave number (quasi-momentum), which varies in the range $(0, 2 \pi)$, $E(q)$ is the quasi-energy, with $-\omega/2 \leq {\rm {E}}(q)< \omega/2$, and $A(q,t)$, $B(q,t)$ are periodic in time with period $T$. The quasi-energy $E(q)$ and corresponding Floquet states $(A(q,t),B(q,t))^T$ are found by solving the eigenvalue problem $$\begin{aligned} E(q) A & = & -i \frac{dA}{dt}-\kappa_3[H \exp(iq)+H^* \exp(-iq)]A \nonumber \\ & + & igA-[\kappa_1 F + \kappa_2 G \exp(-iq)]B \\ E (q) B & = & -i \frac{dB}{dt}-\kappa_3[H \exp(iq)+H^* \exp(-iq)]B \nonumber \\ & - & igB-[\kappa_1 F^* + \kappa_2 G^* \exp(iq)]A \end{aligned}$$ in the interval $(0,T)$ with the periodic boundary conditions $A(q,T)=A(q,0)$ and $B(q,T)=B(q,0)$. Floquet theorem ensures that the quasi-energy spectrum comprises two branches $E(q)= E_{\pm}(q)$, like for the static lattice model discussed in Sec.II.A, with corresponding Floquet states $ \phi_{\pm}(q,t)=(A_{\pm}(q,t), B_{\pm}(q,t))^T$. Note that, if the dynamics of the system defined by Eqs.(A1) and (A2) is observed at discretized times $\tau=0,T,2T,...$, from Eq.(A5) and owing to the periodicity of the functions $A(q,t)$ and $B(q,t)$ it follows that it is equivalent to the dynamics of a static lattice with two minibands whose dispersion relations $E_{\pm}(q)$ are given by the quasi-energies of the periodic system. In fact, after setting $\psi(n,t)=(a_n(t),b_n(t))^T$, an initial wave packet, obtained from the superposition of Bloch-Floquet states with arbitrary spectra $F_{\pm}(q)$, evolves in time according to the relation $$\begin{aligned} \psi(n,t) & = & \int_{0}^{2 \pi}dq F_+(q) \phi_+(q,t) \exp[iqn-iE_{+}(q)t] \nonumber \\ & + & \int_{0}^{2 \pi}dq F_-(q) \phi_-(q,t) \exp[iqn-iE_{-}(q)t]. \;\;\;\;\;\end{aligned}$$ If the evolution of the wave packet is observed at discretized times $\tau =l T$ with $l=0,1,2,3,...$, since $\phi_{\pm}(q, l T)=\phi_{\pm}(q, 0)$ is independent of $\tau$, from Eq.(A8) it follows that $\psi(n,\tau)$ shows the same evolution as the one of a wave packet in a static dimerized lattice with energy band dispersion given by the quasi-energies $E_{\pm}(q)$ of the time-periodic lattice \[compare Eq.(A8) with Eq.(27) given in the text\]. Therefore, all the dynamical aspects of the time-periodic system defined by Eqs.(A1) and (A2), including the onset of $\mathcal{PT}$ symmetry breaking and its convective or absolute nature, can be derived from an equivalent static lattice with a band structure given by the quasi-energy band structure of the original time-periodic system. In particular, as discussed in Sec.III the convective or absolute nature of the symmetry breaking will be determined by the imaginary part of the quasi-energies at the dominant saddle point. The determination of the quasi-energy spectrum $E_{\pm}(q)$ generally requires to resort to a numerical analysis of Eqs.(A6) and (A7). An approximate analytical form the quasi energies can be obtained, however, in the large frequency limit. In fact, assuming $\kappa_{1}, \kappa_2, \kappa_3,g \ll \omega$, the change of the amplitudes $A$ and $B$ over one oscillation cycle are small, so that in Eqs.(A6) and (A7) we may neglect the derivative terms $(dA/ dt)$, $(dB/dt)$ and replace the functions $F(t)$, $G(t)$, $H(t)$ with their average values over the oscillation cycle (rotating-wave approximation), namely one can set $$\begin{aligned} E(q) A & \simeq & -\kappa_3[ \langle H \rangle \exp(iq)+\langle H^* \rangle \exp(-iq)]A \nonumber \\ & + & igA-[\kappa_1 \langle F \rangle + \kappa_2 \langle G \rangle \exp(-iq)]B \\ E (q) B & \simeq & -\kappa_3[\langle H \rangle \exp(iq)+\langle H^* \rangle \exp(-iq)]B \nonumber \\ & - & igB-[\kappa_1 \langle F^* \rangle + \kappa_2 \langle G^* \rangle \exp(iq)] A . \; \; \; \; \end{aligned}$$ where $\langle ... \rangle$ denotes the time average. Using the identity of Bessel functions $\exp(i \Gamma \sin x)=\sum_{n=-\infty}^{\infty}J_n(\Gamma) \exp(inx)$, from Eq.(A4) one readily obtains $$\langle F \rangle = \langle G \rangle =J_M(\Gamma), \;\; \langle H \rangle =J_0(2 \Gamma \cos \phi) \exp(i \varphi) \;\;\;\;$$ where $\varphi=M(\pi+2 \phi)$. A comparison of Eqs.(A9),(A10) with Eq.(5) given in the text shows that, in the large modulation limit, the ac-dc driven lattice described by Eqs.(A1) and (A2) effectively describes the static Rice-Mele lattice \[Eqs.(1) and (2) given in the text\], where the effective hopping rate $\kappa$, $\sigma$ and $\rho$ are given by Eqs.(19-21) and the phase $\varphi$ by Eq.(18). Determination of the saddle point for the extended Rice-Mele lattice model ========================================================================== In this Appendix we calculate the saddle points $q_s$, i.e. the roots $q_s$ of Eq.(29) given in the text, which determine the convective or absolute nature of the $\mathcal{PT}$ symmetry breaking for the non-Hermitian Rice-Mele Hamiltonian (3). To this aim, it is worth introducing the variables $X_s= \cos q_s$ and $Y_s=\sin q_s$. After some algebra, from Eq.(29) it follows that $X_s$ and $Y_s$ are the roots (in the complex plane) of following system of algebraic equations $$\begin{aligned} X_s^2+Y_s^2 & = & 1 \\ \left( \frac{\kappa \sigma}{2 \rho} \right)^2 Y_s^2 & = & \left( \cos^2 \varphi Y_s^2+ \sin^2 \varphi X_s^2-2 \cos \varphi \sin \varphi X_s Y_s\right) \;\;\; \nonumber \\ & \times & (-\epsilon^2+2 \kappa \sigma +2 \kappa \sigma X_s).\end{aligned}$$ To simplify the analysis, let us consider the case where the gain/loss parameter $g$ is larger but close to its threshold value $g_{th}$, so that $g^2-g_{th}^2=\epsilon^2$ is a small quantity. Note that, for $\epsilon \rightarrow 0$, a solution to Eqs.(B1) and (B2) is $X_s=-1$ and $Y_s=0$, corresponding to $q_s=\pi$, i.e. to the wave number where the most unstable mode arises at the $\mathcal{PT}$ symmetry breaking threshold. For $\epsilon^2>0$, we look for a solution to Eqs.(B1) and (B2) in the form of power series $$\begin{aligned} X_s & = & -1+\frac{\alpha^2}{2}-\frac{\alpha^4}{4 !} + ... \\ Y_s & = & -\alpha +\frac{\alpha^3}{3 !} -\frac{\alpha^5}{5!}+ ... \; ,\end{aligned}$$ where $\alpha=q_s-\pi$ is a small amplitude of order $\epsilon^ \gamma$ with $\gamma>0$ to be determined. Note that, at leading order in $\alpha$, the energy $E_{+}(q_s)$ is given by $$E_+(q_s) \simeq -2 \rho \cos \varphi - 2 \rho \sin \varphi \alpha +\sqrt{\kappa \sigma \alpha^2-\epsilon^2}.$$ Note also that, with the Ansatz (B3) and (B4), Eq.(B1) is automatically satisfied for any $\alpha$. The small complex amplitude $\alpha$ can be determined by substitution of Eqs.(B3) and (B4) into Eq.(B2) and letting equal the terms of lowest order on the left and right hands of the equations so obtained. Three cases should be distinguished.\ \ (1) $|v_g| \neq \sqrt{\sigma \kappa}$, where $v_g=-2 \rho \cos \varphi$.\ In this case Eq.(B2) is satisfied at leading order for $\alpha \sim \epsilon $ (i.e. $ \gamma=1$), namely one obtains $$\alpha^2=\frac{\epsilon^2 v_g^2}{\kappa \sigma (v_g^2-\kappa \sigma)}.$$ For $v_g^2 > \sigma \kappa$, the two roots $ \alpha$ of Eq.(B6) are real-valued, and correspondingly the imaginary part of $E_{+}(q_s)$, with $q_s=\pi+\alpha$, vanishes \[see Eq.(B5)\]. Therefore, for $v_g^2 > \sigma \kappa$ one has $\psi(n,t) \rightarrow 0$ as $t \rightarrow \infty$ along the ray $n/t=0$, i.e. the $\mathcal{PT}$ symmetry breaking is convective. Conversely, for $v_g^2 < \sigma \kappa$ according to Eq.(B6) the amplitude $\alpha$ is purely imaginary, and correspondingly for one of the two roots the imaginary part of $E_{+}(q_s)$ is positive according to Eq.(B5). In this case $|\psi(n,t)| \rightarrow \infty $ as $t \rightarrow \infty$ along the ray $n/t=0$, i.e. the $\mathcal{PT}$ symmetry breaking is absolute.\ \ (2) $|v_g| = \sqrt{\sigma \kappa}$ and $\varphi \neq \pm \pi/2$. In this case one obtains $\alpha \sim \epsilon^{2/3}$, i.e. $\gamma=2/3$, and $\alpha$ satisfies the cubic equation $$\alpha^3= - \frac{\sin \varphi \epsilon^2}{2 \kappa \sigma \cos \varphi}.$$ Two of the three roots of such an equation are complex-valued, and correspondingly one can readily shown from Eq.(B5) that a positive imaginary part for the energy $E_+(q_s)$ arises from one of the two complex roots. In fact, since $\epsilon^2$ is of higher order than $\alpha^2$ and $2 \rho \cos \varphi = \sqrt{\kappa \sigma}$, from Eq.(B5) one has ${\rm Im} \{ E_+(q_s)\} \simeq 2 v_g {\rm Im} (\alpha)$. Therefore in this case the $\mathcal{PT}$ symmetry breaking is absolute.\ \ (3) $|v_g| = \sqrt{\sigma \kappa}$ and $\varphi = \pm \pi/2$. 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--- abstract: 'What are the main drivers of activity in the local universe? Observations have been instrumental in identifying the mechanisms responsible for fueling activity in galaxy nuclei. In this context we summarize the main results of the NUclei of GAlaxies (NUGA) survey. The aim of NUGA is to map, at high resolution and high sensitivity, the distribution and dynamics of the molecular gas in the central kiloparsec region of 25 galaxies, and to study the different mechanisms responsible for gas fueling of low-luminosity AGNs (LLAGN). Gas flows in NUGA maps reveal a wide range of instabilities. The derived gravity torque maps show that only $\sim$1/3 of NUGA galaxies show evidence of ongoing fueling. Secular evolution and dynamical decoupling are seen to be key ingredients to understand the AGN fueling cycle. We discuss the future prospects for this research field with the advent of instruments like the Atacama Large Millimeter Array (ALMA).' address: - '$^1$, OAN, Observatorio de Madrid, Alfonso XII, 3, 28014, Madrid, Spain' - '$^2$, OAN, Observatoire de Paris, LERMA, CNRS, 61, Av de l«Observatoire, 75014, Paris, France' author: - 'Santiago García-Burillo$^1$, Francoise Combes$^2$' title: 'The feeding of activity in galaxies: a molecular line perspective' --- The fueling problem =================== Active Galactic Nuclei (AGN) must be fueled with material which lies originally far away from the gravitational influence of the supermassive black hole. On its way to the sphere of influence of the central engine the gas must lose virtually all of its angular momentum during the process. The feeding requirements, $dM/dt$ in M$_\odot$ yr$^{-1}$ units, are markedly different in high-luminosity AGNs (QSOs: 10–100M$_\odot$ yr$^{-1}$) and low-luminosity AGNs (LLAGNS: Seyferts and LINERs: 10$^{-5}$–10$^{-2}$M$_\odot$ yr$^{-1}$) \[1\]. However, observational studies have shown that while in high-luminosity AGNs, the most demanding in terms of feeding requirements, kiloparsec-scale perturbations produced by bars and galaxy interactions are clearly related to the onset of activity \[2\], such correlation is marginal if any in the case of LLAGNs \[3\] \[4\] \[5\]. The search of a ÔuniversalÕ feeding agent in LLAGNs is seen to be challenging. Large-scale bars, bars within bars, lopsided $m=1$ instabilities, warps, nuclear spirals, and winds from stars, among other mechanisms, have been invoked as possible feeding agents in LLAGNs. On the other hand, there is growing evidence, based on theoretical models and state-of-the-art numerical simulations that several mechanisms, and not a single mechanism, might be at work and that these operate at different spatial scales \[6\] \[7\] \[8\]. To further complicate this picture in the case of LLAGNs, the duty cycle of activity is expected to be very short and the associated feeding event could be of chaotic/intermittent nature \[9\]\[6\]. While various modelsÕ predictions are still debated in the literature, there is still ample room for improvement in the picture drawn from observations. {width="7.75cm"} {width="8cm"} An observational perspective: The NUclei of GAlaxies (NUGA) survey ================================================================== The NUclei of GAlaxies (NUGA) survey is a high-spatial resolution ($\sim$0.5“-1Õ”) and high-dynamic range interferometer CO survey of 25 nearby LLAGNs conducted with the IRAM array \[10\]\[11\]. The aim of this survey is to probe the critical scales for angular momentum transfer ($<$10–100 pc at $D$$\sim$5–30 Mpc) in a significant sample of galaxies that include Seyferts, LINERs and transition objects. The high sensitivity/resolution of these observations make possible a detailed study of the distribution and kinematics of molecular gas in the circumnuclear disks of these galaxies. Molecular line maps are used to track down evidence of ÔongoingÕ feeding. Together with the CO NUGA maps, we also have access to high-resolution NIR maps obtained by HST, Spitzer and/or ground-based telescopes for these galaxies. The NIR imaging is used to derive the stellar potentials, which are combined with the CO maps, to derive the gravity torque budget in the circumnuclear disks of the NUGA targets. We implicitly assumed that NIR maps are good estimates of mass distribution and that the overall torque budget on the gas is mostly determined by the stellar potential. Two-dimensional torque maps are used to estimate the torques averaged over the azimuth $t(r)$ using the gas column density $N(x,y)$ derived from CO as weighting function \[12\]. The final output of this analysis is the radial profile of the angular momentum transfer efficiency in the disks ($\Delta L$/$L$). With this information at hand we can quantitatively evaluate if there is ongoing AGN fueling in the disks of the galaxies analyzed, down to the spatial resolution of our observations. NUGA results: statistics ======================== About one third of the LLAGNs analyzed in NUGA show negative torques $t(r)$$<$0, indicative of inflow, down to typical radial distances $r$$<$25–100 pc. Among these galaxies, which can be considered as those showing ’smoking gun’ evidence of ongoing fueling, we distinguish two types of objects: ![ The radial profile of the angular momentum transfer $\Delta L$/$L$ in the disk of NGC 3627, computed with the HST-NICMOS F160W image is plotted for $^{12}$CO(1–0) (left) and $^{12}$CO(2–1) (right). The (red) dashed region identifies the radial range imposed by the resolution limit of observations. Torques are systematically negative inside $r$$\sim$400 pc. Figure adapted from \[15\]. ](ngc3627.eps){width="16cm"} - [**]{}: These include nuclear bars - within - bars/ovals (e.g.; NGC 2782) and nuclear ovals - within -bars (e.g.; NGC 4579). The stellar potential of NGC 2782, analyzed by \[13\] shows two embedded bars: an outer (weak) oval of $\sim$6 kpc diameter and a (strong) nuclear bar of $\sim$1.5 kpc diameter. The nuclear bar shows signs of decoupling. This configuration of the stellar potential has facilitated the inflow of molecular gas inside the Inner Lindblad Resonance (ILR) of the oval. The derived gravity torques $t(r)$ are systematically negative down to $\sim$100–200 pc. The stellar potential of NGC 4579, studied by \[14\], shows two embedded bars: an outer bar of $\sim$12 kpc diameter and an embedded weak nuclear oval of $\sim$0.3 kpc diameter. In the outer disk, the decoupling of the spiral arms allows the gas to efficiently populate the Ultra Harmonic Resonance (UHR) region of the large-scale bar. This favors net gas inflow on intermediate scales. Furthermore, closer to the AGN, gas feels negative torques due to the combined action of the outer bar and the nuclear oval. The combination of the two $m=2$ modes produces net gas inflow down to $r$$\sim$50 pc, providing inward gas transport on short dynamical timescales ($\sim$1–3 rotation periods) (Fig. 1). - [**]{}: These galaxies are characterized by the apparent absence of gravity torque barriers in their nuclei (e.g.; NGC 3627). The stellar potential of NGC 3627, analyzed by \[15\] shows one large-scale bar of $\sim$6 kpc diameter. The bar has no ILR barrier. Molecular gas is concentrated along the leading edges of the bar and shows no ring feature. Down to the spatial resolution of these observations ($\sim$25 pc), gravity torques $t(r)$ are systematically negative in the circumnuclear disk of NGC 3627 (Fig. 2). This scenario suggests that the bar in this galaxy is young and rapidly rotating, and thus has not yet formed an ILR. On the contrary, about two thirds of the LLAGNs analyzed in NUGA show positive torques $t(r)$$>$0, indicative of outflow, down to typical radial distances $r$$<$300 pc. This [*puzzling*]{} gravity torque budget is found in two categories of objects in our sample: - [**]{}: In these galaxies, nuclear bars/ovals - within - bars, like those found in NGC 4321 and NGC 6951 are not always conducive to gas inflow at present \[12\]\[16\]\[17\]. The stellar potential of NGC 6951, analyzed by \[12\] and \[16\] shows two embedded bars: a large-scale bar of $\sim$8 kpc diameter and an inner oval of $\sim$0.4 kpc diameter. Molecular gas is stalled in the ILR ring of the bar. While molecular gas is also detected at the AGN locus \[12\]\[17\]\[18\], gas is gaining angular momentum inside the ILR due to oval forcing (Fig. 3). - [**]{}: The absence of a clear non-axisymmetric feature in the stellar potential makes the angular momentum transfer in the nucleus a very inefficient process in some galaxies at present (e.g.; NGC 4826 \[11\]; NGC 7217 \[16\]\[19\]; NGC 5953 \[20\]). ![a) ([*Upper panel*]{}) Overlay of the $^{12}$CO(2–1) NUGA map (contours) on the J-band HST image (grey scale) of NGC 6951; images have been deprojected onto the galaxy plane. The orientations of the large-scale bar-BAR-and of the nuclear oval-OVAL(n)-are shown. b) ([*Lower panels*]{}) We plot the strengths ($Q_i$, $i$=1,2) and phases ($\Phi_i$, $i$=1,2) of the $m=1$ and $m=2$ perturbations inside the image field-of-view. Figure adapted from \[12\]. ](ngc6951.eps){width="8cm"} In summary, the overall results of NUGA indicate that high spatial resolution is instrumental in quantifying angular momentum transfer processes at critical scales ($\sim$10–100 pc). Gas flows in NUGA maps reveal a wide range of large-scale and embedded $m=2$, $m=1$ instabilities in the circumnuclear disks of LLAGNs. The derived gravity torque maps indicate that gas is frequently stalled in rings, which are the signposts of gravity torque barriers. Only $\sim$1/3 of NUGA galaxies show negative torques down to $\sim$50 pc. Several short-lived ($<$a few 10$^7$ yr) mechanisms are at work to drain angular momentum. These various fueling mechanisms are related to bar cycles \[21\]\[22\]. Secular evolution and dynamical decoupling are key ingredients to understand the AGN fueling cycle. Future surveys with ALMA ======================== To confirm the different feeding scenarios we must improve the spatial resolution and sensitivity of millimeter line observations. The advent of the Atacama Large Millimeter Array (ALMA) will allow us to boost the spatial resolution of observations of galaxy nuclei. The full ALMA capabilities will make possible to map the emission of a set of molecular gas tracers in the inner $\sim$1–10 pc of a significant sample of $\sim$50 nearby ($D<$5–30 Mpc) AGNs in about 100 hrs of observing time with an order of magnitude higher sensitivity compared to the current NUGA survey. Improving the statistics will allow us to better explore the evolutionary sequence in AGN fueling. The CO and NIR maps of the observed galaxies, obtained at a common spatial resolution of $\sim$0.05“–0.1” with ALMA and HST/VLT, will provide a sharp view of the gravity torque budget in the circumnuclear disks. The efficiency of new feeding mechanisms will be observationally tested at unprecedented scales $\sim$1–10 pc (see F. Combes’ contribution to this symposium). In particular, the role of $m=1$ modes, gas self-gravity, granularity of the stellar potential, and dynamical friction of GMCs in AGN fueling will be thus tested. Another key ingredient in galaxy evolution is AGN feedback. ALMA enhanced capabilities will also allow us to simultaneously search for the signature of molecular gas outflows and high-velocity winds in nearby AGNs. 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--- abstract: 'We propose that a non-thermal X-ray arc inside the remnant of Tycho’s supernova (SN) represents the interaction between the SN ejecta and the companion star’s envelope lost in the impact of the explosion. The X-ray emission of the remnant further shows an apparent shadow casted by the arc in the opposite direction of the explosion site, consistent with the blocking of the SN ejecta by the envelope. This scenario supports the single degenerate binary origin of Tycho’s SN. The properties of the X-ray arc, together with the previous detection of the companion candidate and its space velocity by Ruiz-Lapuente et al. (2004) and Hernández et al.(2009), enables us to further infer 1) the progenitor binary has a period of 4.9$^{+5.3}_{-3.0}$ days, 2) the companion gained a kick velocity of 42$\pm$30 km s$^{-1}$, and 3) the stripped envelope mass is about 0.0016($\leq0.0083$) $M_{\sun}$. However, we notice that the nature of the companion candidate is still under debate, and the above parameters need to be revised according to the actual properties of the companion candidate. Further work to measure the proper motion of the arc and to check the capability of the interaction to emit the amount of X-rays observed from the arc is also needed to validate the current scenario.' author: - 'F.J. Lu, Q.D. Wang, M.Y. Ge, J.L. Qu, X.J. Yang, S.J. Zheng, and Y. Chen' title: 'The Single-degenerate Binary Origin of Tycho’s Supernova as Traced by the Stripped Envelope of the Companion' --- Introduction ============ While Type Ia supernovae (Ia SNe) play a fundamental role in cosmology and chemical evolution of the Universe, their exact origin remains greatly uncertain. One of the leading mechanisms for such a SN is the thermonuclear explosion of a white dwarf when its mass reaches a critical value via accretion from a normal stellar companion in a binary, which is called the single-degenerate scenario [@Whelan1973; @Wheeler1975; @Nomoto1982; @Hillebrandt2000]. This scenario is supported by the possible identification of the survived stellar companion of the Ia SN observed by Tycho Brahe in 1572 [@Ruiz2004], although little is yet known about the putative progenitor binary. A specific prediction of the single-degenerate scenario is that up to 0.5 solar mass can be stripped by the impact of Ia SN from its companion star [@Wheeler1975; @Fryxell1981; @Taam1984; @Chugaj1986; @Marietta2000; @Meng2007]. So far, however, no direct observational evidence for the stripped envelope has been reported; only an upper limit of 0.01 solar mass has been set for two extragalactic Ia SNe [@Leonard2007]. In comparison, stellar remnants are expected to be completely dispersed if a SN is due to the merger of two white dwarfs (the double-degenerate scenario; Iben & Tutukov 1984; Whelan & Iben 1973). We select the Tycho supernova remnant (SNR) to search for the stripped material motivated by the detection of its candidate companion star (Tycho G, a G0-G2 type subgiant), which shows a large peculiar velocity [@Ruiz2004; @Hern2009]. This remnant is one of the two identified Galactic type Ia SNRs, as revealed by the light curve, radio emission, and X-ray spectra [@Baade1945; @Baldwin1957; @Hughes1995; @Ruiz2004]. It is young (437 years), nearby (3$\pm$1 kpc; de Vaucouleurs 1985), and of high X-ray brightness (Cassam-Chenaï et al. 2007). With the available deep observations from the Chandra X-ray Observatory (CXO), this remnant is an ideal laboratory to search for the stripped mass entrained in the ejecta. Observation and Data Reduction ============================== Table 1 lists the 12 CXO observations of the Tycho SNR used in this study. These observations were carried out by the ACIS-I, the imagine array of the Advanced CCD Imaging Spectrometer (ACIS), with a field of view large enough to cover the whole remnant. The data were calibrated with the Chandra Interactive Analysis of Observations (CIAO V4.1) software package following the standard procedure to correct for charge-transfer inefficiency (CTI) effects and the time-dependence of the gain, to clean bad pixels, and to remove time intervals of background flares. The final total effective exposure for these observations is a little bit longer than 1 Ms. Results ======= Fig. 1 shows the intensity images of the remnant in different energy bands. In the 4-6 keV band image (Fig. 1 (a)), which is dominated by non-thermal X-ray emission sensitive to shocks, which accelerate particles, there appears an intriguing arc (as marked in the figure), only about half way from the SN site (RA (2000) =00:25:23.8; DEC (2000) =64:08:04.7; Ruiz-Lapuente et al. 2004a) and is as bright, narrow, and sharp as those filaments at the outer boundaries of the remnant. This unusual arc was first noticed by Warren et al. (2005). They suggested that it may still be part of the SNR rim seen in projection. In comparison, Figs. 1 (b), (c), and (d) show the remnant in 1.6-2.0, 2.2-2.6, and 6.2-6.8 keV energy bands, representing the intensity distributions of Si, S, and Fe emission lines. We find that the Fe K$\alpha$ line emission is unusually faint in a cone just outside of the arc (in the direction away from the SN site) , in comparison with the other regions beyond the same radius. The opening angle of this cone (about 20$^{\circ}$) is similar to that of the arc relative to the SN site. In the same cone, especially at the outer rims of the SNR, the 4-6 keV continuum emission as well as the Si and S line intensities also appear to be relatively deficient, though not as obvious as in the Fe K$\alpha$ band. These relative intensity contrasts are shown quantitatively in Fig. 2 (a). On the other hand, Fig 2(b) shows the same intensity contrasts for region within the arc radius, where local peaks appear at nearly the same angular positions of the dips of the profiles outside the arc. Observing the images we find that the enhancements are very close to the arc. We interpret the intensity deficiency in the cone outside the arc and the local enhancements inside the arc as the blocking of the SN ejecta by the arc. The spectrum of the arc is shown in Fig 3. The on-source spectrum is extracted from the region defined by the inner polygon in Fig. 1(a), while the background is from the region between the inner and outer polygons. Fig 3 shows obvious dips at the positions of the Si, S and Fe lines in the source spectrum obtained. Since the background thermal emission dominates the nonthermal emission at these energies, these dips are probably due to the clumpy distribution of the SN ejecta and so highly variable background thermal emission surrounding the arc. Fitting the source spectrum with a power law model gives a photon index of 2.45$\pm$0.09 (90% confidence errors), an absorption column density of (8.7$\pm$0.9)$\times$10$^{21}$ cm$^{-2}$, an unabsorbed 0.5-10 keV flux of 7.8$\times$10$^{-13}$ erg cm$^{-2}$ s$^{-1}$, and $\chi^2$ of 291 for 200 degrees of freedom. If we ignore the data in 1.6-2.0, 2.2-2.6 and 6.0-7.0 keV, which are probably effected by the over-subtraction of the background line emission, the fitted parameters are then photon index 2.47$^{+0.16}_{-0.08}$, absorption column density 9.0$^{+1.4}_{-0.7}\times$10$^{21}$ cm$^{-2}$, and unabsorbed 0.5-10 keV flux 8.4$\times$10$^{-13}$ erg cm$^{-2}$ s$^{-1}$, as well as a significantly improved $\chi^2$ of 176 for 171 degrees of freedom. Therefore, we conclude that the arc is nonthermal. The nature and origin of the X-ray arc ====================================== We find that the X-ray arc is most probably in the interior of the SNR instead of a (morphologically) unusual feature of the outer rim projected well inside the remnant. The sharp and bright appearance of the X-ray arc is similar to those of the filaments at the outer rim of the SNR (Cassam-Chenaï et al. 2007), implying that the arc is observed almost edge on. If it is in the out layer of the SNR and the observed small angular distance from the geometric centre is due to the projection effect, then the arc should be much more diffuse as it is observed half face on. In the 4-6 keV map, there exist some relatively bright features within the blast wave boundary. However, most of them are substantially more diffuse and coincide with bright thermal structures spatially, and none of them is as far away from the boundary as the arc is (see also Warren et al. 2005). The arc is convexed toward the SN site, which is opposite to those of the filaments at the outer boundary. Furthermore, larger and fainter filaments tend to run northward from the southeast (Warren et al. 2005) and bent outward. The morphology of the arc can be naturally produced by the interaction between the SN ejecta and a cloud. As will be discussed in the following, the arc most probably represents the materials stripped from the companion star by the SN explosion and is in the interior of the SNR. First, the arc must be related to the progenitor system of the SN. If it is not related to the progenitor system, it should then represent a dense cold molecular cloud that has survived for long time. Actually, the milli-meter and optical observations suggest that the Tycho SNR is possibly interacting with molecular clouds at the northeastern and the southwestern rims (Lee et al. 2004; Ghavamian et al. 2000). These regions show strong 4-6 keV emission. However, neither CO nor optical emission enhancement has been detected to be spatially coincident with the X-ray arc. It is unlikely to be a molecular cloud. Second, the arc cannot arise from the materials ejected by the progenitor binary system of the SN. Since the mass donor is suggested to be very similar to the Sun but a slightly evolved one (Ruiz-Lapuente et al. 2004), we don’t expect that it could contribute to the cloud (e.g., via a stellar wind). One might think that a planetary nebula surrounding the exploded white dwarf could be a source of the cloud material. However, planetary nebulae always show a spherically or axially symmetric morphology. The singleness of the X-ray arc makes this possibility very unlikely. Finally, and most probably, the matter generating the X-ray arc is stripped from the companion star during the SN explosion. Such a mechanism has been suggested by many theoretical works (Wheeler et al. 1975; Fryxell & Arnett 1981; Taam & Fryxell 1984; Chugai 1986; Marietta et al. 2000; Meng et al. 2007) although no direct observational evidence is currently available (Leonard 2004). Arguments for this mechanism are as follows: (1) The stripped materials are confined in a small angular range, which can naturally explain the singleness of the arc; (2) The opening angle (about 20$\degr$) of the arc relative to the SN site (Ruiz-Lapuente et al. 2004), the absence of the X-ray line emission in the cone away from the arc, and the local enhancements of X-ray emission immediately within the arc radius are all well consistent with the ejecta blocking scenario of the stripped stellar envelope (Marietta et al. 2000); (3) The gas in the envelope of the companion star is expected to have a temperature of a few $10^3$ K, it can not cool down to the CO emitting temperature in 400 years, especially in a hot environment; (4) The impact of the SN blast wave and ejecta on the stripped envelope will generate a shock wave, just like those represented by the nonthermal X-ray filaments at the outer rim, but with a smaller velocity because the envelope is denser than the surrounding ISM. The shock wave can accelerate electrons to emit the nonthermal X-rays. (5) The angle between the direction of the arc to the explosive centre and the proper motion velocity of Tycho G is well consistent with the theoretical predictions and simulations (Marietta et al. 2000; Ruiz-Lapuente et al. 2004; Meng et al. 2007), as detailed in the following. Constraints on the progenitor binary system =========================================== The above interpretation together with the existing measurement of the companion’s present velocity gives tight constraints on properties of the progenitor binary of Tycho SN (see Fig. 4 for an illustration). Because of the SN impact, the companion star should receive a kick as well as the envelope stripping in the same direction and in the orbital plane of the progenitor binary. The direction should be perpendicular to the orbiting velocity of the companion as it is expected to be in a circular orbit just before the SN [@Marietta2000; @Meng2007]. For Tycho G, the measured radial velocity is -50$\pm$10 km s$^{-1}$ \[with the projected Galactic rotation contribution (-30$\pm10$ km s$^{-1}$) subtracted; Hernández et al. 2009\], while the tangential one is -94$\pm27$ km s$^{-1}$ [@Ruiz2004]. Taking the uncertainty in the distance (3$\pm$1 kpc; de Vaucouleurs 1985) into account, the tangential velocity is -94$\pm41$ km s$^{-1}$. The projected angle between the stripping direction and the current velocity of Tycho G relative to the SN site is $\alpha$=63$\pm$13$\degr$, where the error accounts for the uncertainties in both the proper motion measurement and the arc/shadow central line (about 2$\degr$). Because the X-ray arc/shadow is viewed almost edge on (with an assumed uncertainty of 10$\degr$), the stripped velocity should be nearly perpendicular to the line of sight. Therefore, the real angle between the stripping (or the kick) and the velocity of Tycho G is $\beta$ =67$\pm$16$\degr$. Numerical simulation by Marietta et al. (2000) shows that the ratio between the orbital velocity and kick velocity is from 2.3 to 11.2, and so the real angle between the kick and the space velocity of the stellar remnant is between 67$\degr$ to 85$\degr$. Our result is consistent with these predictions. We can then infer that the companion had an orbital velocity ($V_o$) of 98$\pm$36 km s$^{-1}$ in the progenitor binary, received a kick velocity ($v_k$) of 42$\pm$30 km s$^{-1}$, and the inclination angle of the orbital plane is 31$\pm13\degr$. We can also constrain the separation between the two stars in the progenitor binary with the formula $a=\frac{M_{1}^{2}G}{(M_{1}+M_{2})V_{o}^{2}}$, where $M_{1}$ is the mass of the SN progenitor and is assumed to be 1.4 solar mass — the Chandrasekhar mass of a white dwarf, $M_{2}$ the mass of the companion star, and $G$ the gravitational constant. Tycho G is similar to the Sun spectroscopically [@Ruiz2004; @Hern2009] and thus has a mass of about one solar or slightly higher (depending on the luminosity type; see below). We then estimate $a=\frac{1}{2.4+\varepsilon}(2.7\pm2.0)\times10^{7}$ km, where $\varepsilon$ is the mass stripped from the companion and should be considerably smaller than one, hence negligible. The corresponding orbital period is then 4.9$^{+5.3}_{-3.0}$ days. These orbital parameters and kick velocity are summarized in Table 2. They are well consistent with the theoretical predictions [@Marietta2000; @Meng2007]. Pakmor et al. (2008) simulated the impact of type Ia SN on main sequence binary companions. They find that the kick velocity of the companion star after the impact of the SN ejecta varies from 17 to 61 km s$^{-1}$ for different models. Our results are also consistent with their simulations. We may also check the evolutionary state of the companion, assuming that it filled the Roche lobe when the SN took place. Because the Roche lobe radius is [@Paczynski1971] $r_{r}=[0.38+0.2\log(\frac{1+\varepsilon}{1.4})]\times[\frac{1}{2.4+\varepsilon}(2.7\pm2.0)]\times10^{7}\approx(4.0\pm2.7)\times10^{6}$ km, about 5.7 times the solar radius, the companion should be a subgiant, fully consistent with the luminosity classification of Tycho G [@Ruiz2004] and [@Hern2009]. The angular size (about 20$\degr$) of the X-ray arc relative to the SN site is smaller than that subtended by the Roche lobe (40$\degr$), which may be expected from the compression and stripping of the envelope in the bow-shocked SN ejecta material (Fig. 4). The angular separation between the SN site and the arc is about half of the outer radius of the remnant, which presumably reflects the difference in their velocities. A type Ia SN explosion releases a typical kinetic energy of (1-1.4)$\times10^{51}$ erg, and the mean ejecta velocity ($v_{ej}$) for a Chandrasekhar mass of 1.4 $M\odot$ is 8500-10000 km s$^{-1}$. The momentum conservation for the stripped mass ($M_{str}$), the kicked companion star, and the ejecta in the solid angle subtended by the envelope can be expressed as: $M_{ej}v_{ej}$=$(M_{ej}+M_{str})\frac{v_{ej}}{2}+M_{c}v_k$. Assuming a spherically symmetric SN and using a companion star mass of 1 solar, an ejecta velocity of 9230 km s$^{-1}$, and $v_k$=42 km s$^{-1}$,as well as the 20$\degr$ wide solid angle, we estimate the mass of the envelope to be $0.0016$ $M\odot$. Taking the uncertainty of $v_k$ ($\pm30$ km s$^{-1}$) into account, the upper limit of the stripped mass is 0.0083 $M_{\odot}$. The mass outside the assumed solid angle (e.g., to account for the full Roche lobe) should be negligible because of the expected highly-concentrated radial mass profile of a subgiant star [@Meng2007]. This estimation of the stripped mass is consistent with that observed for two extragalactic Ia SNe (Leonard 2007), significantly lower than the theoretical predictions by Marrieta et al. (2000) and Meng et al. (2007), and is close to (though still lower than) 1 to several percent that simulated by Pakmor et al. (2008). Although the stripped mass is small, it is enough to produce the X-ray arc. Katsuda et al. (2010) estimated that the mean ambient density of Tycho’s SNR is 0.0015 cm$^{-3}$, or $<$0.2 cm$^{-3}$. The mass of the ambient ISM in a cone of 20$\degr$ radius should then be about 4$\times$10$^{-5}$ $M_{\sun}$, or $<$ 5$\times$10$^{-3}$ $M_{\sun}$, smaller than or at most comparable to the stripped mass. Since bright nonthermal filaments have been observed all along the rim of the remnant (e.g., Cassam-Chenai et al. 2007), the interaction between SN ejecta (probably denser at the arc position) and the stripped mass should be strong enough to be responsible of the X-ray arc emission. Recently, Kerzendorf et al. (2009) have reported the new space velocity and mass measurements of Tycho G, which are different from those by Ruiz-Lapuente et al. (2004a). We list the new measurements and the corresponding binary parameters in Table 2. The new orbital radius and period are several times as those derived from [@Ruiz2004] and [@Hern2009]. The angle between the kick and the space velocity of the stellar remnant is therefore about 82 degrees, only marginally consistent with the numerical simulations (Marietta et al. 2000). In the scheme of momentum conservation, the amount of the stripped mass is derived as 0.0087$\pm0.0017$ $M_{\sun}$. It is higher than that from [@Ruiz2004] and [@Hern2009], but still consistent with the simulations by Pakmor et al. (2008) and the observations by Leonard (2007). As pointed out by Kerzendorf et al. (2009), there is a simple relationship between the companion’s rotation velocity ($v_{rot}$) and its orbital velocity ($v_{orb, 2})$: $v_{rot}=\frac{M_1+M_2}{M_1}f(q)V_{orb,2}$, where $f(q)$ is the ratio of the companion’s Roche-lobe radius to the orbital separation and $q=M_1/M_2$ is the mass ratio of the primary to the companion at the time of the explosion. If $M_2$ is 0.3-0.5 $M_{\sun}$ (Kerzendorf et al. 2009), $v_{rot}$ of the companion’s surface should be about 24 km s$^{-1}$, much higher than 7.5 km s$^{-1}$, the upper limit of Tycho G’s rotation velocity ($v_{rot}\rm{sin}\it{i}$), where $i$ is the inclination angle of the the orbital velocity. Using the inclination angle that we obtained in this paper, the upper limit of $v_{rot}$ for Tycho G is about 10 km s$^{-1}$, still significantly lower than 24 km s$^{-1}$. Kerzendorf et al. (2009) proposed that a red giant scenario where the envelope’s bloating has significantly decreased the rotation could be consistent with their observation of the low rotation velocity. Since the effect of the inclination angle is small, Tycho G remains a stripped giant if it is the mass donor, as suggested by Kerzendorf et al. (2009). If the companion is about 1 $M_{\sun}$, its surface velocity was about 60 km s$^{-1}$ at the SN explosion, as derived from the binary parameters listed in Table 2 and that it filled the Roche lobe, which has a radius of about 5.7 $R_{\sun}$. However, the companion may have a radius of 1-3 $R_{\sun}$ now [@Ruiz2004; @Hern2009], and so the surface velocity of the stellar remnant should be about 10-31 km s$^{-1}$, marginally consistent with the upper limit of $v_{rot}$ observed by [@kerz2009] and the inclination angle that we obtained. We speculate that the decrease of the radius is possibly due to the destruction of the white dwarf. Before the SN explosion, the strong radiation of the accreting white dwarf inflated the envelope of the companion star to fill the Roche-lobe, and the radius of the companion star shrinks to 1-3 $R_{\sun}$ now because the heating of the white dwarf does not exist. In addition, we note here that the shrink did not accelerate the rotation significantly, because the inflated envelope only contributes a small fraction of the total mass of the companion star and most of it was stripped away by the SN. Using the position of the arc and the age of the remnant, we obtained a mean velocity of the arc outward from the SN site as about 0$\farcs$28 yr$^{-1}$. As measured by Katsuda et al. (2010), the Tycho’s remnant is expanding at a proper motion velocity around 0$\farcs$3 yr$^{-1}$, in contrast to the mean expansion speed of about 0$\farcs$55 yr$^{-1}$ from the radius and age of the remnant. This shows that the remnant are in a deceleration phase due to the interaction with the ISM. If the arc is a projected feature that is in the outer layer of the remnant, it should have a proper motion velocity of $\sim0\farcs$15 yr$^{-1}$. If the arc is actually about half way from the SN site and represents the interaction of the SN explosion and the stripped companion envelope, the proper motion should be quite close to the mean velocity $\sim$0$\farcs$28 yr$^{-1}$, as discussed below. On one hand, the arc is unlikely in an accelerating phase. The binary separation is tiny compared to the distance of the X-ray arc from the SN site. Even the Fe ejecta, which is expected to have the lowest velocity and was measured as $\sim3000$ km s$^{-1}$ currently (Furuzawa et al. 2009), can pass such a separation within several hours. Most of the impact of the ejecta on the companion envelope (and thus the acceleration of the stripped mass) should take place soon after the explosion. On the other hand, the arc can not be significantly decelerated. Since the distance of the X-ray arc from the SN site is about half of the remnant radius, using the ISM density given by Katsuda et al. (2010), the ISM mass in the cone between the SN site and the X-ray arc is $\sim$5$\times$10$^{-6}$ $M_{\sun}$, much smaller than the stripped envelope mass. Although the ISM decelerates the motion of the stripped envelope material, it is quite insignificant given the small mass. Therefore, the stripped envelope went through a very short accelerating phase in the beginning of the SN explosion, and has remained in an almost free expansion state since then. We have attempted to measure the proper motion of the X-ray arc so as to check the above scenario, since the expected proper motion may be detectable at [*Chandra*]{}’s resolution. Unfortunately, the arc typically fallen more or less at a gap between two ACIS-I chips, especially in three early observations. In addition, the arc is heavily contaminated by strong thermal emission in the low energy band. In 4-6 keV, where nonthermal emission dominates, the counting statistics of the arc are typically not sufficient in early observations. As a result, we cannot yet get a reliable multi-epoch measurements of the arc positions to allow for a reliable determination of the proper motion. Future observations with more careful positioning of the arc in the detector and with a total exposure time comparable to the ACIS-I observations in 2009 will make such measurements feasible. Summary ======= We have shown that a self-consistent single-degenerate binary model provides a natural and unified interpretation of the observed unique X-ray arc/shadow in the Tycho’s SNR. Two sets of parameters of the progenitor binary system have also been presented using the optical observation results of the candidate companion star (Tycho G) obtained by [@Ruiz2004] and [@kerz2009] respectively. The main points in favor of our interpretation are: (1) Although the nonthermal X-ray arc is half way from the remnant center, the high brightness show that it is viewed almost edge on and so unlikely a projected feature in the outer layer of the remnant. Together with its sharp inward convex shape, the arc most probably represents the interaction between the ejecta and a bulk of materials in the interior of the remnant. (2) This bulk of materials can not be due to a pre-existing molecular cloud or materials ejected by the progenitor binary system. The impact generating the X-ray arc is most likely between SN ejecta and the stripped envelop of the companion star. (3) The X-ray emission of the remnant shows an apparent shadow casted by the arc in the opposite direction of the explosion site, and there are local enhancements in the same direction immediately within the X-ray arc, consistent with the blocking of the SN ejecta by the envelope. (4) We obtained a stripped mass of $\leq$ 0.0083 $M_{\sun}$, which is consistent with that observed for two extragalactic Ia SNe (Leonard 2007) and close to the recent simulations by Pakmor et al. (2008). (5) The angle between the motion of the companion candidate and the direction of the arc as well as the derived kick velocity of the companion star are well consistent with the theoretical predictions and the numerical simulation results. However, we note that there are still several points that can not be well interpreted by the current scenario, and further work is needed to reveal the physical processes related to the nonthermal X-ray arc. (1) The properties of Tycho G and whether it is the stellar remnant of Tycho’s SN are under debate [@Ruiz2004; @Hern2009; @kerz2009]. If Tycho G is not the stellar remnant, the binary parameters and kick velocity obtained in this paper are unreliable anymore. Also, if Tycho G is the stellar remnant but has properties as obtained by [@kerz2009], the obtained angle between the kick and the space velocity of the stellar remnant is only marginally consistent with the numerical simulations [@Marietta2000]. (2) It has not been quantitatively estimated whether the interaction between the stripped envelope and the ejecta can produce shock wave strong enough to produce the nonthermal X-ray arc. (3) We failed to obtain a precise proper motion of the X-ray arc, which is an important criterion to differentiate the X-ray arc as inside the remnant from a projected feature in the outer layer. Further studies of the stellar remnant as well as the measurement of the proper motion velocity of the X-ray arc are therefore urged to check the validity of the scenario proposed in this paper. Acknowledgments {#acknowledgments .unnumbered} =============== We thank the referee for many insightful comments and suggestions that helped us greatly in the revision of the paper. This work is supported by National Basic Research Program of China (973 program, 2009CB824800) and by National Science Foundation of China (10533020 and 10903007). QDW acknowledges the support by CXC/NASA under the grant GO8-9047A and NNX10AE85G. Baade, W.B. 1945, , 102, 309 Baldwin, J.E., & Edge, D.O. 1957, [*Observatory*]{}, 77, 139 Cassam-Chenaï, G., Hughes, J. P., Ballet, J., & Decourchelle, A. 2007, , 665, 315 Chugaj, N. N.  1986, , 63, 951 de Vaucouleurs, D.  1985, ,289, 5 Fryxell, B. A. & Arnett, W. D. 1981, , 243, 994 Furuzawa, A. et al. 2009, , 693, L61 Ghavamian, P., Raymond, J., Hartigan, P., & Blair, W.P. 2000, , 535, 266 Hernández, J. I., Ruiz-Lapuente, P., Filippenko, A. V., et al.  2009, , 691, 1 Hillebrandt, W. & Niemeyer, J. C. 2000, , 38, 191 Hughes, J. P. et al. 1995, , 444, L81 Iben, I. Jr. & Tutukov, A. V.  1984, , 54, 335 Katsuda, S. et al. 2010, , 709, 1387 Kerzendorf, W.E.et al. 2009, , 701, 1665 Lee, J.J., Koo, B.C., & Tatematsu, K.C. 2004, , 605, L113 Leonard, D. C.  2007, , 670, 1275 Marietta, E., Burrows, A. & Fryxell, B. A.  2000, , 128, 615 Meng,X., Chen, X. & Han, Z.  2007, , 59, 835 Nomoto, K. 1982, , 253, 798 Paczynski, B.  1971, , 9, 183 Pakmor, R., Röpke, F.K., Weiss, A., & Hillebrandt, W.  2008, , 489, 943 Ruiz-Lapuente, P. et al. 2004a, Nature, 431, 1069 Ruiz-Lapuente, P.  2004b, , 612, 357 Taam, R. E. & Fryxell, B. A.  1984, , 279, 166 Warren, J.S. et al.  2005, , 634, 376 Whelan, J. & Iben, I. Jr. 1973, , 186, 1007 Wheeler, J. C., Lecar, M. & McKee, C. F.  1975, , 200, 145 [ccc]{} 3837 & 2003-04-29 & 145\ 7639 & 2007-04-23 & 109\ 8551 & 2007-04-26 & 33\ 10093& 2009-04-13 & 118\ 10094& 2009-04-18 & 90\ 10095& 2009-04-23 & 173\ 10096& 2009-04-27 & 106\ 10097& 2009-04-11 & 107\ 10902& 2009-04-15 & 40\ 10903& 2009-04-17 & 24\ 10904& 2009-04-13 & 35\ 10906& 2009-05-03 & 41\ [lcc]{} Proper motion (mas yr$^{-1}$) &$\mu_l$=-2.6$\pm1.34$& $\mu_l$=-1.6$\pm2.1$\ &$\mu_b$=-6.11$\pm1.34$ &$\mu_b$=-2.7$\pm1.6$\ Tangential velocity (km/s)&94$\pm$41 & 51$\pm28$\ Radial velocity (km/s) &50$\pm10$ & 49$\pm10$\ Companion mass ($M\odot$) &1.0 & 0.3-0.5\ Orbital velocity (km/s) &98$\pm36$&71$\pm16$\ Orbital period (Day) & 4.9$^{+5.3}_{-3.0}$ & 28$\pm26$\ Separation (10$^{7}$ km) & 1.1$\pm0.7$& 3.2$\pm0.6$\ Inclination angle ($\degr$) & 31$\pm13$ & 47$\pm20$\ Kick velocity (km/s) & 42$\pm30$ & 23$\pm20$\ Stripped Mass ($M\odot$) & 0.0016 ($<$0.0083) & 0.0087$\pm$0.0017\

--- abstract: 'We solve connection problem between fundamental solutions at singular points $0$ and $1$ for the generalized hypergeometric function, using analytic continuation of the integral representation. All connection coefficients are products of the sine and the cosecant.' address: - 'YM: School of Mathematics and Physics, Kanazawa University, Kanazawa, Ishikawa 920-1192, Japan' - 'HN: School of Mathematics and Physics, Kanazawa University, Kanazawa, Ishikawa 920-1192, Japan' author: - 'Y. Matsuhira and H. Nagoya' --- [Connection problem for the generalized hypergeometric function]{} Introduction ============ Let $n \in {{\mathbb Z}}$ ($n>0$), $\alpha_1, \dots, \alpha_{n+1}, \beta_1, \dots, \beta_n \in {{\mathbb C}}$. The generalized hypergeometric series is $$\begin{aligned} _{n+1} F _n \left( \begin{matrix} \alpha_1 , \dots , \alpha_{n+1} \\ \beta_1 , \dots , \beta_n \\ \end{matrix} ; z \right) =\sum_{k=0}^\infty \frac{(\alpha_1)_k \cdots (\alpha_{n+1})_k} {(\beta_1)_k \cdots (\beta_n)_k k!}z^k, \end{aligned}$$ where $(a)_k = a(a+1) \cdots (a+k-1)$. The generalized hypergeometric series converges on $|z|<1$ and satisfies the Fuchsian differential equation with three singular points $0$, $1$, $\infty$: $$\left\{ \frac{d}{dz} \prod_{k=1}^{n}\left(\frac{d}{dz} + \beta_k -1\right) -z \prod_{k=1}^{n+1}\left(\frac{d}{dz}+\alpha_k\right) \right\}F=0.$$ We call this differential equation the generalized hypergeometric equation and a solution to the equation a generalized hypergeometric function. The Riemann scheme, which is the table of the characteristic exponents, of the generalized hypergeometric equation is $$\begin{aligned} \left\{ \begin{array}{ccc} z=0 & z=1 & z=\infty \\ 0 & 0 & \alpha_1 \\ 1-\beta_1 & 1 & \alpha_2 \\ \vdots & \vdots & \vdots \\ 1-\beta_{n-1} & n-1 & \alpha_n \\ 1-\beta_n & \sum_{i=1}^{n}\beta_i-\sum_{i=1}^{n+1}\alpha_i & \alpha_{n+1} \end{array} \right\}. \end{aligned}$$ Assume $\alpha_i-\beta_j\notin{{\mathbb Z}}$ and $\beta_i-\beta_j\notin{{\mathbb Z}}$ ($i\neq j$). Then the generalized hypergeometric equation admit a fundamental system of solutions at $z=0$ given by $$\begin{aligned} f_i^{(0)}(z)=(-z)^{1-\beta_i} {}_{n+1} F _n \left( \begin{matrix} \alpha_1-\beta_i+1 , \alpha_2-\beta_i+1, \dots , \alpha_{n+1}-\beta_i+1 \\ \beta_1-\beta_i+1 , \dots , \widehat{\beta_i-\beta_i+1} , \dots , \beta_{n+1}-\beta_i+1 \\ \end{matrix} ; z \right)\quad (i=1,\ldots,n+1), \end{aligned}$$ where the symbol $\widehat{A}$ means omitting $A$ and $\beta_{n+1}=1$, and a fundamental system of solutions at $z=\infty$ given by $$\begin{aligned} f_i^{(\infty)}(z)=(-z)^{-\alpha_i} {}_{n+1} F _n \left( \begin{matrix} \alpha_i-\beta_1+1 , \alpha_i-\beta_2+1, \dots , \alpha_i-\beta_{n+1}+1 \\ \alpha_i-\alpha_1+1 , \dots , \widehat{\alpha_i-\alpha_i+1} , \dots , \alpha_i-\alpha_{n+1}+1 \\ \end{matrix} ; \frac{1}{z} \right)\quad (i=1,\ldots,n+1). \end{aligned}$$ Fundamental systems of solutions at $z=1$ consist of one non-holomorphic solution with the characteristic exponent $\sum_{i=1}^{n}\beta_i-\sum_{i=1}^{n+1}\alpha_i$ and $n$ holomorphic solutions. Connection problem between fundamental systems of solutions of the generalized hypergeometric equation has been solved by various authors by several methods [@Kawabata; @1], [@Kawabata; @2], [@Kawabata; @3] [@Mimachi; @intersection; @numbers; @for; @n+1Fn], [@Norlund], [@Okubo; @Takano; @Yoshida], [@S], [@Winkler]. The choice of the fundamental systems of solutions $$X_0=\left\{ f_1^{(0)}(z),\ldots, f_{n+1}^{(0)}(z)\right\},\quad X_\infty=\left\{ f_1^{(\infty)}(z),\ldots, f_{n+1}^{(\infty)}(z)\right\}$$ at $z=0$ and $z=\infty$ is canonical. All connection coefficients associated with $X_0$ and $X_\infty$ are products of the Gamma function and the inverse of the Gamma function. We note that if we multiply $f_i^{(0)}(z)$ and $f_i^{(\infty)}(z)$ by suitable scalars, then connection coefficients become products of the sine and the cosecant. On the other hand, there is no canonical choice of fundamental systems of solutions at $z=1$. A connection matrix depends on choice of the fundamental system of solutions and connection coefficients are not necessarily products of the Gamma function and the inverse of the Gamma function. If we take the fundamental system of solutions at $z=1$ by $(z-1)^i+O((z-1)^n)$ ($i=0,1,\ldots,n-1$), $(z-1)^a(1+O(z-1))$ with $a=\sum_{i=1}^{n}\beta_i-\sum_{i=1}^{n+1}\alpha_i$, then the connection coefficients with $X_0$ or $X_\infty$ involve values of the generalized hypergeometric series at $z=1$. Kawabata reported in [@Kawabata; @3] that there is a fundamental system of solutions at $z=1$ such that non-diagonal elements of the connection matrix with $X_0$ or $X_\infty$ are products of the sine and the cosecant and diagonal elements of that are one minus products of the sine and the cosecant. To the author’s knowledge, connection problem between fundamental systems of solutions at singular points $0$ and $1$ for the generalized hypergeometric equation has not been solved [*completely*]{}. The aim of the present note is to do that. Namely, we give connection matrices whose elements are products of the sine and the cosecant. In order to obtain the connection formula expressing a fundamental system of solution at $z=1$ in terms of the integral representations corresponding to $X_0$, we calculate analytic continuation of the connection formula expressing the integral representations corresponding to $X_\infty$ in terms of the integral representations corresponding to $X_0$ [@Mimachi; @intersection; @numbers; @for; @n+1Fn]. We deform the domains of integration loaded with the integrand of the generalized hypergeometric function. The technique dealing with deformation for the loaded cycles associated with Selberg type integrals is elaborated in [@Mimachi; @Iwahori-Hecke]. Unfortunately, by using analytic continuation of the connection formula expressing the integral representations corresponding to $X_0$ in terms of the integral representations corresponding to $X_\infty$ [@Mimachi; @intersection; @numbers; @for; @n+1Fn], we have not found a way to obtain the connection formula expressing the integral representations corresponding to $X_0$ in terms of the fundamental system of solution at $z=1$. Instead, we present directly the inverse matrix of the connection matrix expressing a fundamental system of solution at $z=1$ in terms of the integral representations corresponding to $X_0$, and give a proof by the residue calculus. There had been two methods deriving connection formulas of rigid Fuchsian systems by hypergeometric integrals. One is to use the Cauchy’s integral theorem for obtaining linear relations among loaded cycles [@Aomoto; @1], [@Aomoto; @2]. Such method was used for obtaining the connection formulas of the Simpson’s Even four [@Haraoka; @Mimachi], ${}_3F_2$ [@Mimachi; @3F2], sixth order rigid Fuchsian systems derived from conformal field theory [@BHS]. However, in the case of a general $n$-multiple hypergeometric integrals, there are too many linear relations among loaded cycles to obtain connection formulas between fundamental systems of solutions. Another method is to compute intersection numbers of loaded cycles [@Mimachi; @intersection; @numbers; @for; @n+1Fn], which avoids to solve too many linear relations among loaded cycles. We believe that to compute analytic continuation of known connection formulas is also useful for obtaining another connection formulas. Especially because the connection coefficients for multiplicity-free case, such as the connection coefficients for elements of $X_0$ and $X_\infty$, is solved in [@Oshima]. The plan of the paper is as follows. In Section 2, we prepare notations and recall the connection matrix of the generalized hypergeometric function between $X_0$ and $X_\infty$ in [@Mimachi; @intersection; @numbers; @for; @n+1Fn]. In Section 3, we introduce domains of integration for a fundamental system of solutions at $z=1$, and prove the connection formulas between the fundamental system of solutions at $z=1$ and $z=0$. At the end, we remark on periodicity of the connection matrices. Preliminary =========== We consider a multi-valued function $$\begin{aligned} u(t)=\prod_{i=1}^n t_i^{\lambda_i} \prod_{i=1}^{n} (t_{i}-t_{i-1})^{\mu_i} (t_n-z)^{\mu_{n+1}}\end{aligned}$$ for $t=(t_1,\ldots, t_n)\in{{\mathbb C}}^n$ with parameters $\lambda_i,\mu_i\in{{\mathbb C}}$ defined on $$\begin{aligned} T_z={{\mathbb C}}^n-\bigcup_{i=1}^n \{t_i=0\} \cup \bigcup_{i=1}^{n+1} \{t_{i-1}-t_i=0\}, \end{aligned}$$ where $$\begin{aligned} t_0=1,\quad t_{n+1}=z. \end{aligned}$$ The function $u(t)$ is the integrand of an integral representation of the generalized hypergeometric series ${}_{n+1}F_n(z)$. Namely, we have $$\begin{aligned} \label{eq_integral_representation_for_GHS} {}_{n+1}F_n\left( \begin{matrix} \alpha_1 , \dots , \alpha_{n+1} \\ \beta_1 , \dots , \beta_n \\ \end{matrix} ; z \right)=&\prod_{1\le i\le n}B(\alpha_i,\beta_i-\alpha_i) \int_{D^{(0)}_{n+1}} u(t)dt,\end{aligned}$$ where $B(\alpha,\beta)$ is the Beta function and $D^{(0)}_{n+1}=\{t\in T_z\mid 1<t_1<t_2\cdots<t_n\}$ with $$\label{eq_transformation_lambda_alpha_beta} \lambda_i=\alpha_{i+1}-\beta_i\quad (1 \leq i \leq n), \quad \mu_i=\beta_i-\alpha_i-1\quad (1 \leq i \leq n+1).$$ Here, we suppose $$\begin{aligned} \mathrm{Re}(\alpha_i)>0,\quad \mathrm{Re}(\beta_i-\alpha_i)>0 \quad (1\le i\le n)\end{aligned}$$ for the convergence of the integral and fix the arguments as $$\begin{aligned} \mathrm{arg}(t_i)=\mathrm{arg}(t_i-t_{i-1})=0\quad (1\le i\le n), \quad |\mathrm{arg}(t_n-z)|<\frac{\pi}{2}. \end{aligned}$$ The formula implies that the domain $D^{(0)}_{n+1}$ of integration describes the asymptotic behaviour of the holomorphic solution at $z=0$. There are domains of integration yielding the asymptotic behaviours of the non-holomorphic solutions at $z=0$ with the characteristic exponents $1-\beta_i$ ($1\le i\le n$), and the non-holomorphic solutions at $z=\infty$ with the characteristic exponents $1-\beta_i$ ($1\le i\le n+1$) [@Mimachi; @intersection; @numbers; @for; @n+1Fn]. In order to define integrals for domains of integration, we should fix branches of $u(t)$. For the convenience, suppose $z \in {{\mathbb R}}-\{0,1\}$. We first fix branches of $u(t)$ for real $z$, and then we consider analytic continuation of $u(t)$ for general $z$. For a simply connected domain $D$ in the real part $T_{{\mathbb R}}$ of $T_z$, let us define $u_D(t)$ as $$\begin{aligned} u_D(t)=\prod_{i=1}^n (\epsilon_it_i)^{\lambda_i} \prod_{i=1}^{n+1} (\eta_i(t_{i-1}-t_i))^{\mu_i}, \end{aligned}$$ where $\epsilon_i,\eta_i\in\{1,-1\}$ such that $\epsilon_it_i>0$ and $\eta_i(t_{i-1}-t_i)>0$ on $D$. We fix the arguments of all $\epsilon_i t_i$ and $\eta_i(t_{i-1}-t_i)$ as $0$. The function $u_D(t)$ is a branch of $u(t)$ multiplied by a scalar. In what follows, for convenience we use $$\begin{aligned} &e(A) =\exp (\pi \sqrt{-1} A), \quad s(A) =\sin (\pi A) \quad (A \in {{\mathbb C}}), \\ &\alpha_{i,j}=\sum_{s=i}^j\alpha_s,\quad \beta_{i,j}=\sum_{s=i}^j\beta_s\quad (i<j), \end{aligned}$$ and for the exponents $\lambda_i, \mu_i$ of $u(t)$ $$\begin{aligned} \lambda_{i, j} &= \begin{cases} \lambda_i + \cdots +\lambda_j & (i \leq j) , \\ 0 & (i=j+1) , \\ -(\lambda_{j+1}+ \cdots +\lambda_{i-1}) & (i \geq j+2) , \end{cases} \\ \mu_{i, j} &= \begin{cases} \mu_i + \cdots +\mu_j & (i \leq j) , \\ 0 & (i=j+1) , \\ -(\mu_{j+1}+ \cdots +\mu_{i-1}) & (i \geq j+2) , \end{cases} \\ e_{i, j} &= e(\lambda_{i, j}), \\ \tilde{e}_{i, j} &= e(\mu_{i, j}). \end{aligned}$$ We note that for all $i$ we have $$\begin{aligned} \lambda_{i}=\lambda_{i, i}, \quad \mu_{i}=\mu_{i, i}.\end{aligned}$$ Connection problem between fundamental solutions at $z=0$ and $z=\infty$ ------------------------------------------------------------------------ In this subsection, we recall the results in [@Mimachi; @intersection; @numbers; @for; @n+1Fn]. We fix $z \in {{\mathbb C}}$ such that $z < 0$. Set $$\begin{aligned} D_i^{(0)}=&\{t\in T_{{\mathbb R}}\mid z<t_n< \cdots <t_i <0,\ 1<t_1< \cdots < t_{i-1} \} \quad (1 \leq i \leq n+1), \\ D_i^{(\infty)}=&\{t\in T_{{\mathbb R}}\mid t_i< \cdots <t_n <z,\ 0<t_{i-1}< \cdots < t_1< 1 \} \quad (1 \leq i \leq n+1).\end{aligned}$$ The domains of integration in the case of $n=2$ are pictured as follows. (-3,0)–(3.5,0)node\[anchor=west\][$t_2=0$]{}; (-3,-1.5)–(3.5,-1.5)node\[anchor=west\][$t_2=z$]{}; (0,3.5)–(0,-3)node\[anchor=north\][$t_1=0$]{}; (2,3.5)–(2,-3)node\[anchor=north\][$t_1=1$]{}; (-3,-3)–(3.5,3.5)node\[anchor=west\][$t_2=t_1$]{}; node at (2.5,3) [$D_1^{(0)}$]{}; node at (2.5,-0.75) [$D_2^{(0)}$]{}; node at (1.2,0.5) [$D_3^{(\infty)}$]{}; node at (1,-2) [$D_2^{(\infty)}$]{}; node at (-0.5,-1) [$D_3^{(0)}$]{}; node at (-2.5,-2) [$D_1^{(\infty)}$]{}; The orientation of the domains $D_i^{(0)}$ or $D_i^{(\infty)}$ of integration is fixed to be natural one induced from $T_{{\mathbb R}}$. In [@Mimachi; @intersection; @numbers; @for; @n+1Fn], it was shown that the domains $D_i^{(0)}$ of integration give a fundamental system of solutions at $z=0$, and $D_{i}^{(\infty)}$ gives a fundamental system of solutions at $z=\infty$. \[[@Mimachi; @intersection; @numbers; @for; @n+1Fn], Proposition 2.1\] \[\[M2\], prop 2.1\] (1) For a fixed $i$ such that $1 \leq i \leq n+1$, if ${\rm Re}(\alpha_i-\beta_s+1)>0$ and ${\rm Re}(\beta_s-\alpha_s)>0$ for $1 \leq s \leq n+1$ with $s \neq i$ and $|z|>1$, then we have $$\begin{aligned} &\int _{D_i^{(\infty)}} u_{D_i^{(\infty)}}(t)dt_1 \cdots dt_n \label{sol_infty} =\prod_{1 \leq s \leq n+1, s \neq i} B(\alpha_i-\beta_s+1, \beta_s-\alpha_s)f_i^{(\infty)}(z),\end{aligned}$$ where $$\begin{aligned} f_i^{(\infty)}(z)&=(-z)^{-\alpha_i} {}_{n+1} F _n \left( \begin{matrix} \alpha_i-\beta_1+1 , \alpha_i-\beta_2+1, \dots , \alpha_i-\beta_{n+1}+1 \\ \alpha_i-\alpha_1+1 , \dots , \widehat{\alpha_i-\alpha_i+1} , \dots , \alpha_i-\alpha_{n+1}+1 \\ \end{matrix} ; \frac{1}{z} \right). \end{aligned}$$ \(2) For a fixed $i$ such that $1 \leq i \leq n+1$, if ${\rm Re}(\alpha_s-\beta_i+1)>0$, ${\rm Re}(\beta_s-\alpha_s)>0$ for $1 \leq s \leq n+1$ with $s \neq i$ and $|z|<1$, then we have $$\begin{aligned} &\int _{D_i^{(0)}} u_{D_i^{(0)}}(t)dt_1 \cdots dt_n \label{sol_0} =\prod_{1 \leq s \leq n+1, s \neq i} B(\alpha_s-\beta_i+1, \beta_s-\alpha_s)f_i^{(0)}(z),\end{aligned}$$ where $$\begin{aligned} f_i^{(0)}(z)&=(-z)^{1-\beta_i} {}_{n+1} F _n \left( \begin{matrix} \alpha_1-\beta_i+1 , \alpha_2-\beta_i+1, \dots , \alpha_{n+1}-\beta_i+1 \\ \beta_1-\beta_i+1 , \dots , \widehat{\beta_i-\beta_i+1} , \dots , \beta_{n+1}-\beta_i+1 \\ \end{matrix} ; z \right). \end{aligned}$$ Let $F_D(z)=\int_D u_D(t)dt$. \[\[M2\], prop 2.5\] For $i$ and $j$ such that $1\leq i,j\leq n+1$, suppose that $$\begin{aligned} &\mathrm{Re}(\alpha_i-\beta_j+1)>0,\quad \mathrm{Re}(\beta_j-\alpha_j)>0 \quad (i\neq j), \\ &\alpha_i-\beta_j\notin{{\mathbb Z}},\quad \beta_i-\beta_j\notin{{\mathbb Z}}\quad (i\neq j). \end{aligned}$$ Then we have $$\label{eq_connection_formula_i0} F_{D_i^{(\infty)}}(z)=\sum_{1 \leq j \leq n+1} \frac{s(\beta_i-\alpha_i)}{s(\beta_j-\alpha_i)} \prod_{1 \leq s \leq n+1,\atop s \neq j} \frac{s(\alpha_s-\beta_j)}{s(\beta_s-\beta_j)} \times F_{D_j^{(0)}}(z)$$ for $1\leq i \leq n+1$. Connection problem between fundamental system of solutions at $z=0$ and $z=1$ ============================================================================= Asymptotic behaviour -------------------- We fix $z \in {{\mathbb C}}$ such that $0<z < 1$. Set $$\begin{aligned} \tilde{D}_i^{(0)}&=\{t\in T_{{\mathbb R}}\mid 0<t_i< \cdots <t_n <z,\ 1<t_1< \cdots < t_{i-1} \} \quad (1 \leq i \leq n+1), \label{domain0_0<z<1} \\ \tilde{D}_i^{(1)}&=\{t\in T_{{\mathbb R}}| t_i< \cdots <t_n <0,\ 0<t_{i-1}< \cdots < t_1< 1 \} \quad (1 \leq i \leq n), \label{holo_domain1_0<z<1} \\ \tilde{D}_{n+1}^{(1)}&=\{t\in T_{{\mathbb R}}| z<t_n< \cdots < t_1< 1 \}. \label{nonholo_domain1_0<z<1}\end{aligned}$$ The domains of integration in the case of $n=2$ are pictured as follows. (-3,0)–(3.5,0)node\[anchor=west\][$t_2=z$]{}; (-3,-1.5)–(3.5,-1.5)node\[anchor=west\][$t_2=0$]{}; (-1.5,3.5)–(-1.5,-3)node\[anchor=north\][$t_1=0$]{}; (2,3.5)–(2,-3)node\[anchor=north\][$t_1=1$]{}; (-3,-3)–(3.5,3.5)node\[anchor=west\][$t_2=t_1$]{}; node at (2.5,3) [$\tilde{D}_3^{(0)}$]{}; node at (2.5,-0.75) [$\tilde{D}_2^{(0)}$]{}; node at (1.25,0.5) [$\tilde{D}_3^{(1)}$]{}; node at (-1,-0.5) [$\tilde{D}_1^{(0)}$]{}; node at (0.25,-2) [$\tilde{D}_2^{(1)}$]{}; node at (-2.5,-2) [$\tilde{D}_{1}^{(1)}$]{}; The orientation of the domains $\tilde{D}_i^{(0)}$ or $\tilde{D}_i^{(1)}$ of integration is fixed to be natural one induced from $T_{{\mathbb R}}$. In [@Mimachi; @intersection; @numbers; @for; @n+1Fn], it was shown that the domains $\tilde{D}_i^{(0)}$ of integration give a fundamental system of solutions at $z=0$, and $\tilde{D}_{n+1}^{(1)}$ gives the non-holomorhic solution at $z=1$. \[\[M2\], prop 3.1\] (1) For a fixed $i$ such that $1 \leq i \leq n+1$, if ${\rm Re}(\alpha_s-\beta_i+1)>0$ and ${\rm Re}(\beta_s-\alpha_s)>0$ for $1 \leq s \leq n+1$ with $s \neq i$ and $|z|<1$, then we have $$\begin{aligned} &\int _{\tilde{D}_i^{(0)}} u_{\tilde{D}_i^{(0)}}(t)dt_1 \cdots dt_n =\prod_{1 \leq s \leq n+1,\atop s \neq i} B(\alpha_s-\beta_i+1, \beta_s-\alpha_s)f_i^{(0)}(z),\end{aligned}$$ where $$\begin{aligned} f_i^{(0)}(z)&=z^{1-\beta_i} {}_{n+1} F _n \left( \begin{matrix} \alpha_1-\beta_i+1 , \alpha_2-\beta_i+1, \dots , \alpha_{n+1}-\beta_i+1 \\ \beta_1-\beta_i+1 , \dots , \widehat{\beta_i-\beta_i+1} , \dots , \beta_{n+1}-\beta_i+1 \\ \end{matrix} ; z \right). \end{aligned}$$ \(2) If ${\rm Re}(\beta_{1,s}-\alpha_{1,s})>0$ for $1 \leq s \leq n$ and ${\rm Re}(\beta_s-\alpha_s)>0$ for $1 \leq s \leq n+1$, and $|1-z|<1$, then we have $$\begin{aligned} &\int _{\tilde{D}_{n+1}^{(1)}} u_{\tilde{D}_{n+1}^{(1)}}(t)dt_1 \cdots dt_n \label{nonholosol_1} =\prod_{s=1}^{n} B(\beta_{1,s}-\alpha_{1,s}, \beta_{s+1}-\alpha_{s+1})f_{n+1}^{(1)}(z), \end{aligned}$$ where $$\begin{aligned} f_{n+1}^{(1)}(z)=&(1-z)^{\beta_{1,n}-\alpha_{1,n+1}} \sum_{i_1, \dots, i_n \geq 0} \prod_{s=1}^{n} \frac{(\beta_s-\alpha_{s+1})}{i_s!}\prod_{s=1}^{n}\frac{(\sum_{k=1}^{s} (\beta_k-\alpha_k))_{i_1+\cdots+i_s}} {(\sum_{k=1}^{s+1}(\beta_k-\alpha_k))_{i1+\cdots+i_s}}(1-z)^{i_1+\cdots+i_n}.\end{aligned}$$ \[holo\_sols\_at\_1\] For a fixed $i$ such that $1 \leq i \leq n$, if $$\begin{aligned} &\mathrm{Re}(\alpha_i-\beta_s+1)>0\quad (1\leq s\leq n+1,\ s\neq i), \\ &\mathrm{Re}(\beta_s-\alpha_s)>0\quad (1\leq s\leq n,\ s\neq i), \\ &\mathrm{Re}(\alpha_{n+1}-\beta_i+1)>0, \end{aligned}$$ and $|1-z|<1$, then we have $$\begin{aligned} \label{eq_holosol_1} \int _{\tilde{D}_{i}^{(1)}} u_{\tilde{D}_{i}^{(1)}}(t)dt_1 \cdots dt_n =&\prod_{s=1,s\neq i}^{n} B(\alpha_i-\beta_s+1, \beta_s-\alpha_s) B(\alpha_i, \alpha_{n+1}-\beta_i+1) f_{i}^{(1)}(z), \end{aligned}$$ where $$\begin{aligned} f_{i}^{(1)}(z)=&\sum_{m_1, m_2 \geq 0}\frac{(\alpha_{n+1})_{m_1} (\alpha_i-\beta_i+1)_{m_2} (\alpha_1)_{m_1+m_2}}{m_1!(\alpha_i+\alpha_{n+1}-\beta_i+1)_{m_1+m_2}} \sum_{m_3=0}^{m_2} \frac{(-1)^{m_3}}{ m_3 ! (m_2-m_3)!} \prod_{s=1,s\neq i}^{n}\frac{(\alpha_i-\beta_s+1)_{m_3}} {(\alpha_i-\alpha_s+1)_{m_3}} (1-z)^{m_1}.\end{aligned}$$ We change the integration variables as $$\begin{aligned} t_s&=u_1u_2 \cdots u_s \ (1 \leq s \leq i-1) , \\ t_s&=u_s^{-1}u_{s+1}^{-1} \cdots u_n^{-1}(u_n-1) \ (i \leq s \leq n).\end{aligned}$$ Its Jacobian is $$\begin{aligned} \frac{\partial(t_1, \dots , t_n)}{\partial(u_1, \dots, u_n)}=u_1^{i-2}u_2^{i-3}\cdots u_{i-2}^{1}u_{i-1}^{0}u_{i}^{-2}u_{i+1}^{-3}\cdots u_{n-1}^{i-n-1}u_n^{i-n-2}(1-u_n)^{n-i}. \end{aligned}$$ Hence, we have $$\begin{aligned} &\int _{\tilde{D}_{i}^{(1)}} u_{\tilde{D}_{i}^{(1)}}(t)dt_1 \cdots dt_n \\ =&\int_{(0, 1)^n} \prod_{s=1}^{i-1}u_s^{\lambda_{s, i-1}+\mu_{s+1, i-1}+i-s-1}\prod_{s=i}^{n}u_s^{-\lambda_{i, s}-\mu_{i, s+1}+i-s-2}\prod_{s=1}^{i-1}(1-u_s)^{\mu_s} \prod_{s=i}^{n-1}(1-u_s)^{\mu_{s+1}} (1-u_n)^{\lambda_{i, n}+\mu_{i+1, n}+n-i} \\ & \times \left( 1-u_n(1-u_1\cdots u_{n-1}) \right)^{\mu_j} \left( 1-(1-z)u_n \right)^{\mu_{n+1}}du_1\cdots du_n \\ =&\sum_{m_1, m_2 \geq 0} \sum_{m_3=0}^{m_2} \frac{(-\mu_{n+1})_{m_1}(-\mu_i)_{m_2}}{m_1 ! m_3 ! (m_2-m_3)!}(1-z)^{m_1} \prod_{s=1,s\neq i}^{n} \int_{0}^{1} u_s^{\lambda_{s, i-1}+\mu_{s+1, i-1}+i-s+m_3-1}(1-u_s)^{\mu_s}du_s \\ &\times \int_{0}^{1} u_n^{-\lambda_{i, n}-\mu_{i, n+1}+i-n+m_1+m_2-2}(1-u_n)^{\lambda_{i, n}+\mu_{i+1, n}+n-i}du_n\end{aligned}$$ by the binomial theorem $$\begin{aligned} (1-(1-z)u_n)^{\mu_{n+1}}&= \sum_{m_1 \geq 0} \frac{(-\mu_{n+1})_{m_1}}{m_1 !} u_n^{m_1}(1-z)^{m_1} , \\ (1-u_n(1-u_1\cdots u_{n-1}) )^{\mu_i} &=\sum_{m_2 \geq 0} \sum_{m_3=0}^{m_2} \frac{(-\mu_i)_{m_2}(-1)^{m_3}}{m_3 !(m_2-m_3)!}u_1^{m_3}\cdots u_{n-1}^{m_3}u_n^{m_2}. \end{aligned}$$ Using the formula $$\begin{aligned} B(\alpha+m,\beta)=\frac{(\alpha)_m}{(\alpha+\beta)_m}B(\alpha,\beta), \end{aligned}$$ we obtain $$\begin{aligned} &\int _{\tilde{D}_{i}^{(1)}} u_{\tilde{D}_{i}^{(1)}}(t)dt_1 \cdots dt_n \\ =&\sum_{n_1, n_2 \geq 0} \sum_{n_3=0}^{n_2} \frac{(-\mu_{n+1})_{n_1}(-\mu_i)_{n_2}}{n_1 ! n_3 ! (n_2-n_3)!}(1-z)^{n_1} \prod_{s=1,s\neq i}^{n} B(\lambda_{s, i-1}+\mu_{s+1, i-1}+i-s+n_3, \mu_s +1) \\ &\times B(-\lambda_{i, n}-\mu_{i, n+1}+i-n+n_1+n_2-1, \lambda_{i, n}+\mu_{i+1, n}+n-i+1) \\ =&\prod_{s=1,s\neq i}^{n} B(\lambda_{s, i-1}+\mu_{s+1, i-1}+i-s, \mu_s +1) B(-\lambda_{i, n}-\mu_{i, n+1}+i-n-1, \lambda_{i, n}+\mu_{i+1, n}+n-i+1) \\ &\times \sum_{n_1, n_2 \geq 0} \sum_{n_3=0}^{n_2} \frac{(-\mu_{n+1})_{n_1}(-\mu_i)_{n_2}(-\lambda_{i, n}-\mu_{i, n+1}-n-1)_{n_1+n_2}}{n_1 !n_3 ! (n_2-n_3)!(-\mu_j-\mu_{n+1})_{n_1+n_2}} \prod_{s=1,s\neq i}^{n}\frac{(\lambda_{s, i-1}+\mu_{s+1, i-1}+i-s)_{n_3}}{(\lambda_{s, i-1}+\mu_{s, i-1}+i-s+1)_{n_3}} (1-z)^{n_1}.\end{aligned}$$ Finally the transformation yields . Main theorem ------------ In this subsection, we solve connection problem between fundamental systems of solutions at singular points $0$ and $1$ for the generalized hypergeometric equation. \[thm\_connection\_formula\_10\] Assume for $1 \leq i, j \leq n+1$ $$\begin{aligned} &\mathrm{Re}(\alpha_i-\beta_j+1)>0\quad (j\neq i), \\ &\mathrm{Re}(\beta_j-\alpha_j)>0, \\ &\alpha_i-\beta_j \notin {{\mathbb Z}}, \quad \beta_i-\beta_j \notin {{\mathbb Z}}\quad (j\neq i). \end{aligned}$$ Then we have $$\label{eq_connection_formula_10} F_{\tilde{D}_i^{(1)}}(z)=\sum_{1\le j\le n+1}\frac{s(\beta_i-\alpha_i)s(\alpha_{n+1})} {s(\beta_j-\alpha_i)s(\beta_j-\alpha_{n+1})} \prod_{1\le k\le n+1,\atop k\neq j}\frac{s(\alpha_k-\beta_j)}{s(\beta_k-\beta_j)} \times F_{\tilde{D}_j^{(0)}}(z)$$ for $1\le i\le n$ and $$\label{eq_connection_formula_10_n+1} F_{\tilde{D}_{n+1}^{(1)}}(z)= \sum_{j=1}^{n+1} \prod_{1 \leq k \leq n+1, \atop k \neq j} \frac{s(\alpha_k-\beta_j)}{s(\beta_k-\beta_j)} \times F_{\tilde{D}_j^{(0)}}(z).$$ By induction. If $n=1$, the connection formulas and are for the Gauss hypergeometric function. Suppose the identity is true for $k<n$. Our strategy to prove the theorem is to calculate analytic continuation of the connection formula along the following path $\gamma$ on the plane of the variable $z$. (-3,0)–(3,0); (-1.5,0) to \[out=270, in=180\] (0,-1); (0,-1) to \[out=0, in=180\] (1.75,0); (0, 0)node\[anchor=south\][$0$]{} circle \[radius=2pt\]; (2.5, 0)node\[anchor=south\][$1$]{} circle \[radius=2pt\]; (-1.5, 0)node\[anchor=south\][$z$]{} circle \[radius=2pt\]; First we compute analytic continuation of the integral representations along the path $\gamma$. In what follows, we abbreviate $F_D(z)$ as $D$, and $\{t\in T_{{\mathbb R}}\mid \cdots\}$ as $\{\cdots\}$. \[lem\_analytic\_continuation\_of\_domain\] By analytic continuation along the path $\gamma$, we have $$\begin{aligned} \gamma^{*}\left(D_j^{(0)}\right)=&(-1)^{n-j+1}e_{j, n}\tilde{e}_{j+1, n+1}\tilde{D}_j^{(0)}\quad (1 \leq j \leq n+1), \\ \label{eq_analytic_continuation_infinity} \gamma^{*}\left(D_j^{(\infty)}\right)=&\tilde{D}_{j}^{(1)} +\sum_{k=j+1}^{n}e_{k, n} \biggl\{ \begin{array}{c} t_j< \cdots <t_{k-1} <0, \\ 0<t_k< \cdots < t_n<z,\ 0<t_{j-1}< \cdots <t_1<1 \end{array} \biggr\} \\ &+e_{j, n} \{ 0<t_j< \cdots <t_n <z,\ 0<t_j< \cdots < t_1<1\}\nonumber \\ &+e_{j, n} \tilde{e}_{j} \{ 0<t_{j-1}<t_j< \cdots <t_n <z, \ 0<t_{j-1}< \cdots < t_1< 1\}\quad (1 \leq j \leq n), \nonumber \\ \gamma^{*}\left(D_{n+1}^{(\infty)}\right)=&\tilde{D}_{n+1}^{(1)} +\tilde{e}_{n+1}\{ t_n<z,\ 0<t_n<\cdots <t_1<1\}, \label{eq_analytic_continuation_infinity_n+1}\end{aligned}$$ where $\gamma^{*}(D)$ stands for the result of analytic continuation along the path $\gamma$ for $D$. By the definition of $D_j^{(0)}$, analytic continuation along the path $\gamma$ increases the arguments of $t_k$ ($j \leq k \leq n$), $t_{k-1}-t_k$ ($j+1 \leq k \leq n+1)$ by $\pi$. Hence, we have $$\begin{aligned} \gamma^{*}(D_j^{(0)})=e_{j, n}\tilde{e}_{j+1, n+1} \{ \overleftarrow{0< t_j< \cdots <t_n <z},\ 1<t_1< \cdots < t_{j-1}\}. \end{aligned}$$ Here, $ \overleftarrow{0< t_j< \cdots <t_n <z}$ means that the orientation is inverse. Hence we obtain $$\begin{aligned} \gamma^{*}(D_j^{(0)})&=e_{j, n}\tilde{e}_{j+1, n+1} \{ \overleftarrow{0< t_j< \cdots <t_n <z},\ 1<t_1< \cdots < t_{j-1}\} \\ &=(-1)^{n-j+1}e_{j, n}\tilde{e}_{j+1, n+1} \{ 0< t_j< \cdots <t_n <z,\ 1<t_1< \cdots < t_{j-1} \}. \end{aligned}$$ Next we compute analytic continuation of $D_j^{(\infty)}$. By the definition, after analytic continuation along the path $\gamma$, the line $\{ t_n <z\}$ divides into two lines $\{ t_n <0\}$ and $\{0<t_n<z\}$ on the plane of the variable $t_n$. Since the argument of $t_n$ increases $\pi$ on $\{0<t_n<z\}$, the line $\{0<t_n<z\}$ is multiplied by $e_n$. Similarly, $\{t_{n-1}<t_n,\ 0<t_n<z\}$ divides into two lines $\{t_{n-1}<0\}$ and $\{0<t_{n-1}<t_n\}$ on the plane of the variable $t_{n-1}$. Since the argument of $t_{n-1}$ increases $\pi$ on $\{0<t_{n-1}<t_n\}$, the line $\{0<t_{n-1}<t_n\}$ is multiplied by $e_{n-1}$. By repeating this procedure from $t_{n-2}$ plane to $t_j$ plane, we have $$\begin{aligned} \gamma^{*}(D_j^{(\infty)})=&\tilde{D}_{j}^{(1)} +\sum_{k=j+1}^{n}e_{k, n} \biggl\{ \begin{array}{c} t_j< \cdots <t_{k-1} <0, \\ 0<t_k< \cdots < t_n<z,\ 0<t_{j-1}< \cdots <t_1<1 \end{array} \biggr\} \\ &+e_{j, n} \left\{ 0<t_{j}< \cdots <t_n <z,\ 0<t_{j-1}< \cdots < t_1<1\right\}.\end{aligned}$$ The last domain is actually divided as $$\begin{aligned} \{ 0<t_j< \cdots <t_n <z,\ 0<t_{j-1}< \cdots < t_1<1 \} =&\tilde{e}_{j} \{ 0<t_{j-1}<t_j< \cdots <t_n <z,\ 0<t_{j-1}<t_{j-2}< \cdots < t_1<1 \} \\ &+ \{ 0<t_j< \cdots <t_n <z,\ t_j<t_{j-1}< \cdots < t_1<1 \}, \end{aligned}$$ because the line $\{0<t_{j-1}<t_{j-2}\}$ splits into two lines $\{0<t_{j-1}<t_j\}$ and $\{0<t_j<t_{j-1}<t_{j-2}\}$ on the plane of the variable $t_{j-1}$, and the argument of $(t_{j-1}-t_j)$ increases $\pi$ on $\{0<t_{j-1}<t_j\}$. Therefore, we obtain . In the same way, we obtain . We back to the proof of the theorem. The connection formula in Proposition \[\[M2\], prop 2.5\] reads as $$\begin{aligned} D_j^{(\infty)}=\sum_{ k =1}^{n+1} (-1)^{n+k-j} \frac{s(\mu_j)}{s(\lambda_{j, k-1}+\mu_{j, k})} \prod_{1 \leq l \leq n+1,\atop l \neq k} \frac{s(\lambda_{l, k-1}+\mu_{l, k})}{s(\lambda_{l, k-1}+\mu_{l+1, k})} D_k^{(0)}\end{aligned}$$ for the parameters $\lambda_i$, $\mu_i$ by the relations . Due to Lemma \[lem\_analytic\_continuation\_of\_domain\], analytic continuation of this for $j=1,\ldots,n$ along the path $\gamma$ is equal to $$\begin{aligned} \label{eq_annalytic_continuation_connection_formula_1} \tilde{D}_{j}^{(1)} =&\sum_{ k=1}^{ n+1} (-1)^{n+k-j} \frac{s(\mu_j)}{s(\lambda_{j, k-1}+\mu_{j, k})} \prod_{1 \leq l \leq n+1,\atop l \neq k} \frac{s(\lambda_{l, k-1}+\mu_{l, k})}{s(\lambda_{l, k-1}+\mu_{l+1, k})} (-1)^{n-k+1}e_{k, n}\tilde{e}_{k+1, n+1}\tilde{D}_k^{(0)}{\nonumber}\\ &-\sum_{k=j+1}^{n}e_{k, n} \{ -\infty<t_j< \cdots <t_{k-1} <0,\ 0<t_k< \cdots < t_n<z,\ 0<t_{j-1}< \cdots <t_1<1\}\nonumber \\ &-e_{j, n} \{ 0<t_j< \cdots <t_n <z,\ 0<t_j< \cdots < t_1<1\}\\ &-e_{j, n} \tilde{e}_{j} \{ 0<t_{j-1}<t_j< \cdots <t_n <z,\ 0<t_{j-1}< \cdots < t_1< 1\}. \nonumber\end{aligned}$$ In the integral associated with $\{ t_j< \cdots <t_{k-1} <0,\ 0<t_k< \cdots < t_n<z,\ 0<t_{j-1}< \cdots <t_1<1\}$ in the second term of , the integral with respect to the variables $t_1,\ldots, t_{k-1}$ corresponds to the integral defined for $\tilde{D}_{j}^{(1)}$ of ${}_kF_{k-1}(t_k)$ because of $0<t_k<1$. Hence, we can apply the induction hypothesis to $\{ t_j< \cdots <t_{k-1} <0,\ 0<t_k< \cdots < t_n<z,\ 0<t_{j-1}< \cdots <t_1<1\}$ and obtain a sum of integrals associated with $\{ 0<t_k< \cdots < t_n<z,\ 0<t_m<\cdots<t_k,\ 1<t_1<\cdots<t_{m-1}\}$ ($m=1,\ldots,k$), which is nothing but $\tilde{D}_m^{(0)}$ of ${}_{n+1}F_n(z)$. In the same way, we can regard the third term $\{ 0<t_j< \cdots <t_n <z,\ 0<t_j< \cdots < t_1<1\}$ in as $\tilde{D}_j^{(1)}$ of ${}_jF_{j-1}(t_j)$. Hence, we can apply the induction hypothesis to $\{ 0<t_j< \cdots <t_n <z,\ 0<t_j< \cdots < t_1<1\}$ and obtain a sum of integrals associated with $\{ 0<t_j< \cdots < t_n<z,\ 0<t_k<\cdots<t_j,\ 1<t_1<\cdots<t_{k-1}\}$ ($k=1,\ldots,j$), which is nothing but $\tilde{D}_k^{(0)}$ of ${}_{n+1}F_n(z)$. We can also regard the fourth term $\{ 0<t_{j-1}<t_j< \cdots <t_n <z,\ 0<t_{j-1}< \cdots < t_1< 1\}$ in as $\tilde{D}_{j-1}^{(1)}$ of ${}_{j-1}F_{j-2}(t_{j-1})$. Hence, we can apply the induction hypothesis to $\{ 0<t_{j-1}<t_j< \cdots <t_n <z,\ 0<t_{j-1}< \cdots < t_1< 1\}$ and obtain a sum of integrals associated with $\{ 0<t_{j-1}< \cdots < t_n<z,\ 0<t_k<\cdots<t_{j-1},\ 1<t_1<\cdots<t_{k-1}\}$ ($k=1,\ldots,j-1$), which is nothing but $\tilde{D}_k^{(0)}$ of ${}_{n+1}F_n(z)$. Since the identities and read as $$\begin{aligned} \tilde{D}_j^{(1)}=&(-1)^{j-1} \sum_{k=1}^{n} \frac{s(\mu_j)s(\mu_{n+1})}{s(\lambda_{j, k-1}+\mu_{j, k}) s(\lambda_{n+1, k-1}+\mu_{n+2, k})} \prod_{l=1, l\neq k}^{n}\frac{s(\lambda_{l, k-1}+\mu_{l, k})} {s(\lambda_{l, k-1}+\mu_{l+1, k})} \tilde{D}_{k}^{(0)} \\ &+(-1)^{j-1}\frac{s(\mu_j)}{s(\lambda_{j, n}+\mu_{j, n+1})} \prod_{l=1}^{n}\frac{s(\lambda_{l, n}+\mu_{l, n+1})} {s(\lambda_{l, n}+\mu_{l+1, n+1})} \tilde{D}_{n+1}^{(0)} \end{aligned}$$ for the parameters $\lambda_i$, $\mu_i$ by the relations , for $j=1,\ldots, n$ we obtain $$\begin{aligned} \label{eq_connection_formula_proof_j=1-n} \tilde{D}_{j}^{(1)} =&\sum_{ k=1}^{ n+1} (-1)^{-j+1} e_{k, n}\tilde{e}_{k+1, n+1} \frac{s(\mu_j)}{s(\lambda_{j, k-1}+\mu_{j, k})} \prod_{1 \leq l \leq n+1, l \neq k} \frac{s(\lambda_{l, k-1}+\mu_{l, k})} {s(\lambda_{l, k-1}+\mu_{l+1, k})} \tilde{D}_k^{(0)} \\ &-\sum_{k=j+1}^{n} e_{k, n} \left\{ (-1)^{j-1} \sum_{m=1}^{k-1} \frac{s(\mu_j)s(\mu_{k})}{s(\lambda_{j, m-1}+\mu_{j, m}) s(\lambda_{k, m-1}+\mu_{k+1, m})} \prod_{l=1, l\neq m}^{k-1}\frac{s(\lambda_{l, m-1}+\mu_{l, m})}{s(\lambda_{l, m-1}+\mu_{l+1, m})} \tilde{D}_{m}^{(0)} \right. {\nonumber}\\ &\left.+(-1)^{j-1}\frac{s(\mu_j)}{s(\lambda_{j, k-1}+\mu_{j, k})} \prod_{l=1}^{k-1}\frac{s(\lambda_{l, k-1}+\mu_{l, k})} {s(\lambda_{l, k-1}+\mu_{l+1, k})} \tilde{D}_{k}^{(0)} \right\}{\nonumber}\\ &-e_{j, n}\sum_{k=1}^{j}(-1)^{j-1} \prod_{l=1, l \neq k}^{j} \frac{s(\lambda_{l, k-1}+\mu_{l, k})}{s(\lambda_{l, k-1}+\mu_{l+1, k})}\tilde{D}_{k}^{(0)} -e_{j, n}\tilde{e}_{j}\sum_{k=1}^{j-1}(-1)^{j} \prod_{l=1, l \neq k}^{j-1} \frac{s(\lambda_{l, k-1}+\mu_{l, k})} {s(\lambda_{l, k-1}+\mu_{l+1, k})}\tilde{D}_{k}^{(0)} {\nonumber}\\ =&\sum_{k=1}^{j-1} \left(A_k-\sum_{m=j+1}^{n}B_{k, m}-E_k-F_k \right) \tilde{D}_{k}^{(0)}+\sum_{k=j}^{n} \left( A_k-\sum_{m=k+1}^{n+1}B_{k, m}-C_k \right) \tilde{D}_{k}^{(0)},{\nonumber}\end{aligned}$$ where $$\begin{aligned} A_k&=(-1)^{j-1}e_{k, n}\tilde{e}_{k+1, n+1}\frac{s(\mu_j)}{s(\lambda_{j, k-1}+\mu_{j, k})}\prod_{l=1, l \neq k}^{n+1} \frac{s(\lambda_{l, k-1}+\mu_{l, k})}{s(\lambda_{l, k-1}+\mu_{l+1, k})}\quad (1\leq k\leq n+1) , \\ B_{k, m}&=e_{m, n} (-1)^{j-1} \frac{s(\mu_j)s(\mu_{m})}{s(\lambda_{j, k-1}+\mu_{j, k}) s(\lambda_{m,k-1}+\mu_{m+1,k})} \prod_{l=1, l \neq k}^{m-1} \frac{s(\lambda_{l, k-1}+\mu_{l, k})}{s(\lambda_{l, k-1}+\mu_{l+1, k})} \quad (1\leq m\leq n,\ 1\leq k\leq m), \\ C_k&=e_{k, n}(-1)^{j-1}\frac{s(\mu_j)}{s(\lambda_{j, k-1}+\mu_{j, k})} \prod_{l=1}^{k-1}\frac{s(\lambda_{l, k-1}+\mu_{l, k})}{s(\lambda_{l, k-1}+\mu_{l+1, k})} \quad (j\leq k \leq n+1) , \\ E_k&=e_{j, n}(-1)^{j-1}\prod_{l=1, l \neq k}^{j} \frac{s(\lambda_{l, k-1}+\mu_{l, k})}{s(\lambda_{l, k-1}+\mu_{l+1, k})} \quad (1\leq k\leq j-1) , \\ F_k&=e_{j, n}\tilde{e}_{j}(-1)^{j} \prod_{l=1, l \neq k}^{j-1} \frac{s(\lambda_{l, k-1}+\mu_{l, k})}{s(\lambda_{l, k-1}+\mu_{l+1, k})} \quad (1\leq k\leq j-1).\end{aligned}$$ First, we compute the first term in the right hand side of as follows. For $k=1,\ldots, j-1$ we have $$\begin{aligned} A_k-\sum_{m=j+1}^{n}B_{k, m}-E_k-F_k=&(-1)^{j-1} \prod_{l=1, l \neq k}^{j-1} \frac{s(\lambda_{l, k-1}+\mu_{l, k})}{s(\lambda_{l, k-1}+\mu_{l+1, k})} \\ &\times \left( e_{k, n}\tilde{e}_{k+1, n+1}\frac{s(\mu_j)}{s(\lambda_{j, k-1}+\mu_{j, k})}\prod_{l=j}^{n+1} \frac{s(\lambda_{l, k-1}+\mu_{l, k})}{s(\lambda_{l, k-1}+\mu_{l+1, k})} \right. \\ &-\sum_{m=j+1}^ne_{m, n} \frac{s(\mu_j)s(\mu_{m})}{s(\lambda_{j, k-1}+\mu_{j, k}) s(\lambda_{m,k-1}+\mu_{m,k})} \prod_{l=j}^{m-1} \frac{s(\lambda_{l, k-1}+\mu_{l, k})}{s(\lambda_{l, k-1}+\mu_{l+1, k})} \\ &\left.-e_{j, n} \frac{s(\lambda_{j, k-1}+\mu_{j, k})} {s(\lambda_{j, k-1}+\mu_{j+1, k})}+e_{j, n}\tilde{e}_{j} \right). \end{aligned}$$ The sum of the last term and the second last term above is computed as $$\begin{aligned} &-e_{j, n} \frac{s(\lambda_{k, j-1}+\mu_{k+1, j-1})}{s(\lambda_{k, j-1}+\mu_{k+1, j})}+e_{j, n}\tilde{e}_{j} \\ &=\frac{1}{s(\lambda_{k, j-1}+\mu_{k+1, j})} \frac{-e_{j, n}(e_{k, j-1}\tilde{e}_{k+1, j-1}-e_{k, j-1}^{-1}\tilde{e}_{k+1, j-1}^{-1})+e_{j, n}\tilde{e}_{j}(e_{k, j-1}\tilde{e}_{k+1, j}-e_{k, j-1}^{-1}\tilde{e}_{k+1, j}^{-1})}{2\sqrt{-1}} \\ &=e_{k, n}\tilde{e}_{k+1, j}\frac{s(\mu_j)} {s(\lambda_{k, j-1}+\mu_{k+1, j})}.\end{aligned}$$ Suppose for $i\geq j$ that it holds $$\begin{aligned} &-\sum_{m=j+1}^ie_{m,n} \frac{s(\mu_j)s(\mu_{m})}{s(\lambda_{j, k-1}+\mu_{j, k}) s(\lambda_{m,k-1}+\mu_{m+1,k})} \prod_{l=j}^{m-1} \frac{s(\lambda_{l, k-1}+\mu_{l, k})}{s(\lambda_{l, k-1}+\mu_{l+1, k})}+e_{k, n}\tilde{e}_{k+1, j}\frac{s(\mu_j)} {s(\lambda_{k, j-1}+\mu_{k+1, j})} \\ &=e_{k, n}\tilde{e}_{k+1, i}\frac{s(\mu_j)} {s(\lambda_{k,j-1}+\mu_{k+1,j-1})}\prod_{l=j}^{i} \frac{ s(\lambda_{k, l-1}+\mu_{k+1, l-1})} {s(\lambda_{k, l-1}+\mu_{k+1, l})}.\end{aligned}$$ Then we have $$\begin{aligned} &-\sum_{m=j+1}^{i+1}e_{m,n} \frac{s(\mu_j)s(\mu_{m})}{s(\lambda_{j, k-1}+\mu_{j, k}) s(\lambda_{m,k-1}+\mu_{m+1,k})} \prod_{l=j}^{m-1} \frac{s(\lambda_{l, k-1}+\mu_{l, k})}{s(\lambda_{l, k-1}+\mu_{l+1, k})}+e_{k, n}\tilde{e}_{k+1, j}\frac{s(\mu_j)} {s(\lambda_{k, j-1}+\mu_{k+1, j})} \\ &=\frac{s(\mu_j)}{s(\lambda_{k, j-1}+\mu_{k+1, j-1})} \frac{\prod_{l=j}^{i} s(\lambda_{k, l-1}+\mu_{k+1, l-1})} {\prod_{l=j}^{i+1}s(\lambda_{k, l-1}+\mu_{k+1, l})} \\ &\times \frac{-e_{i+1,n}\left(\tilde{e}_{i+1}-\tilde{e}_{i+1}^{-1}\right) +e_{k, n}\tilde{e}_{k+1, i}\left(e_{k,i}\tilde{e}_{k+1,i+1} -e_{k,i}^{-1}\tilde{e}_{k+1,i+1}^{-1}\right) } {2\sqrt{-1}} \\ &=e_{k, n}\tilde{e}_{k+1, i+1}\frac{s(\mu_j)} {s(\lambda_{k,j-1}+\mu_{k+1,j-1})}\prod_{l=j}^{i+1} \frac{ s(\lambda_{k, l-1}+\mu_{k+1, l-1})} {s(\lambda_{k, l-1}+\mu_{k+1, l})}. \end{aligned}$$ Hence we obtain $$\begin{aligned} -\sum_{m=j+1}^{n}B_{k, m}-E_k-F_k =e_{k, n}\tilde{e}_{k+1, n}\frac{s(\mu_j)} {s(\lambda_{k,j-1}+\mu_{k+1,j-1})}\prod_{l=j}^{n} \frac{ s(\lambda_{k, l-1}+\mu_{k+1, l-1})} {s(\lambda_{k, l-1}+\mu_{k+1, l})}.\end{aligned}$$ Therefore, we have $$\begin{aligned} A_k-\sum_{m=j+1}^{n}B_{k, m}-E_k-F_k =&\frac{(-1)^{j-1}e_{k,n}\tilde{e}_{k+1,n}s(\mu_j)} {s(\lambda_{k,j-1}+\mu_{k+1,j-1})} \frac{\prod_{l=1, l \neq k}^{n}s(\lambda_{l, k-1}+\mu_{l, k})} {\prod_{l=1, l \neq k}^{n+1}s(\lambda_{l, k-1}+\mu_{l+1, k})} \\ &\times \frac{-\tilde{e}_{n+1}(e_{k,n}\tilde{e}_{k+1,n}-e_{k,n}^{-1}\tilde{e}_{k+1,n}^{-1}) +e_{k,n}\tilde{e}_{k+1,n+1}-e_{k,n}^{-1}\tilde{e}_{k+1,n+1}^{-1}}{2\sqrt{-1}} \\ =&\frac{(-1)^{j-1}s(\mu_j)s(\mu_{n+1})} {s(\lambda_{k,j-1}+\mu_{k+1,j-1})} \frac{\prod_{l=1, l \neq k}^{n}s(\lambda_{l, k-1}+\mu_{l, k})} {\prod_{l=1, l \neq k}^{n+1}s(\lambda_{l, k-1}+\mu_{l+1, k})}.\end{aligned}$$ Similarly, we can compute the second term in the right hand side of . This completes the proof for the connection formula . The connection formula for $j=n+1$, namely, the connection formula can be proved in the same way above. By analytic continuation of the connection formula in Proposition \[\[M2\], prop 2.5\] and Lemma \[lem\_analytic\_continuation\_of\_domain\], we get $$\begin{aligned} \tilde{D}_{n+1}^{(1)}=& \sum_{ k=1}^{ n+1} (-1)^{n} \frac{s(\mu_{n+1})}{s(\lambda_{n+1, k-1}+\mu_{n+1, k})} \prod_{1 \leq l \leq n+1,\atop l \neq k} \frac{s(\lambda_{l, k-1}+\mu_{l, k})}{s(\lambda_{l, k-1}+\mu_{l+1, k})} e_{k, n}\tilde{e}_{k+1, n+1}\tilde{D}_k^{(0)}{\nonumber}\\ &-\tilde{e}_{n+1}\{ t_n<z,\ 0<t_n<\cdots <t_1<1\}. \end{aligned}$$ Then, in the integral associated with $\{ t_n<z,\ 0<t_n<\cdots <t_1<1\}$, the integral with respect to the variables $t_1,\ldots, t_{n-1}$ corresponds to the integral defined for $\tilde{D}_n^{(1)}$ of ${}_nF_{n-1}(t_n)$ because $0<t_n<1$. Hence we can apply the induction hypothesis to $\{ t_n<z,\ 0<t_n<\cdots <t_1<1\}$ and obtain a sum of integrals associated with $\{0<t_n<z,\ 0<t_k<\cdots<t_n,\ 1<t_1<\cdots<t_{k-1} \}$ ($k=1,\ldots,n$), which is nothing but $\tilde{D}_k^{(0)}$ of ${}_{n+1}F_n(z)$. Therefore, we obtain $$\begin{aligned} \tilde{D}_{n+1}^{(1)} =&\sum_{k=1}^{n} \left\{ (-1)^ne_{k, n}\tilde{e}_{k+1, n+1} \frac{s(\mu_{n+1})}{s(\lambda_{n+1, k-1}+\mu_{n+1, k})} \prod_{l=1, l \neq k}^{n+1} \frac{s(\lambda_{l, k-1}+\mu_{l, k})} {s(\lambda_{l,k-1}+\mu_{l+1, k})} \right. \\ &\left.+(-1)^n\tilde{e}_{n+1} \prod_{l=1, l \neq k}^{n} \frac{s(\lambda_{l, k-1}+\mu_{l, k})} {s(\lambda_{l, k-1}+\mu_{l+1, k})} \right\}\tilde{D}_k^{(0)} \\ &+(-1)^{n-1}\frac{s(\mu_{n+1})}{s(\lambda_{n+1, n}+\mu_{n+1, n+1})} \prod_{l=1}^{n} \frac{s(\lambda_{l, n}+\mu_{l, n+1})}{s(\lambda_{l, n}+\mu_{l+1, n+1})}\tilde{D}_{n+1}^{(0)}. \end{aligned}$$ Straightforward computation yields the connection formula . The connection formulas and for $n=2$ was computed in [@Mimachi; @3F2]. The connection formula was obtained in [@Kawabata; @1], [@Mimachi; @intersection; @numbers; @for; @n+1Fn], [@Okubo; @Takano; @Yoshida]. From Theorem \[thm\_connection\_formula\_10\], we have the connection matrix $C^{(10)}$ given by $$\left(\tilde{D}_1^{(1)},\ldots, \tilde{D}_{n+1}^{(1)}\right)= \left(\tilde{D}_1^{(0)},\ldots,\tilde{D}_{n+1}^{(0)}\right)C^{(10)}.$$ The elements of $C^{(10)}$ are products of the sine and the cosecant: $$\begin{aligned} \left(C^{(10)}\right)_{i,j}=&\frac{s(\beta_j-\alpha_j)s(\alpha_{n+1})}{s(\beta_i-\alpha_j)s(\beta_i-\alpha_{n+1})} \prod_{k=1, k \neq i}^{n+1}\frac{s(\alpha_k-\beta_i)}{s(\beta_k-\beta_i)}\quad (1\leq i\leq n+1,\ 1\leq j\leq n), \\ \left(C^{(10)}\right)_{i,n+1}=&\prod_{k=1, k \neq i}^{n+1} \frac{s(\alpha_k-\beta_i)}{s(\beta_k-\beta_i)}\quad (1\leq i\leq n+1). \end{aligned}$$ We expect that there exists the inverse matrix $C^{(01)}$ of $C^{(10)}$, which implies the set $\left\{\tilde{D}_1^{(1)},\ldots, \tilde{D}_{n+1}^{(1)}\right\}$ of the domains of integration gives a fundamental system of solutions at $z=1$, and the elements of $C^{(01)}$ are products of the sine and the cosecant. Recall that $C^{(10)}$ are derived from analytic continuation of the connection formulas in Proposition \[\[M2\], prop 2.5\], which relates the fundamental system of solutions at $z=\infty$ to that at $z=0$. However, we have not found a way to use the induction hypothesis to the analytic continuation along the path $\gamma$ of the connection formula relating the fundamental system of solutions at $z=0$ to that at $z=\infty$. Instead, we present the inverse matrix $C^{(01)}$ below and give a proof by direct calculation. Let a square matrix $C^{(01)}$ of order $n+1$ be defined by $$\begin{aligned} \left(C^{(01)}\right)_{i,j}=&- \frac{s(\alpha_i)s(\beta_{1, n} -\alpha_{1, n+1}-\beta_j +\alpha_i)s(\beta_j-\alpha_j)}{s(\alpha_{n+1})s(\beta_{1, n}-\alpha_{1, n+1})s(\beta_j-\alpha_i)} \prod_{k=1, k \neq i}^{n} \frac{s(\beta_k - \alpha_i)}{s(\alpha_k-\alpha_i)}\end{aligned}$$ for $1\le i\le n$, $1\le j\le n+1$, and $$\left(C^{(01)}\right)_{n+1,i}=-\frac{s(\beta_i-\alpha_i)}{s(\beta_{1, n}-\alpha_{1, n+1})}$$ for $1\le i\le n+1$. \[thm\_connection\_matrix\_01\] Assume for $1\leq i,j \leq n+1$, $\alpha_i-\beta_j,\alpha_i-\alpha_j,\beta_i-\beta_j,\beta_{1, n} -\alpha_{1, n+1}\not\in{{\mathbb Z}}$. Then the matrix $C^{(01)}$ is the inverse of the connection matrix $C^{(10)}$. We compute the elements $\left(C^{(01)} C^{(10)}\right)_{i, j}$ of the product directly. We consider separately the following cases: (i) $1\leq i,j\leq n$, (ii) $1\leq i\leq n$, $j=n+1$, (iii) $i=n+1$, $1\leq j\leq n$, (iv) $i=j=n+1$. In the case of (i), we have $$\begin{aligned} \left(C^{(01)} C^{(10)}\right)_{i, j}=&-\sum_{k=1}^{n+1}\frac{s(\alpha_i)s(\beta_{1, n}-\alpha_{1, n+1}-\beta_k+\alpha_i)s(\beta_k-\alpha_k)}{s(\alpha_{n+1})s(\beta_{1, n}-\alpha_{1, n+1})s(\beta_k-\alpha_i)}\\ & \times \frac{s(\beta_j-\alpha_j)s(\alpha_{n+1})}{s(\beta_k-\alpha_j)s(\beta_k-\alpha_{n+1})}\prod_{l=1, l \neq i}^{n} \frac{s(\beta_l-\alpha_i)}{s(\alpha_l-\alpha_i)} \prod_{l=1, l \neq k}^{n+1}\frac{s(\alpha_l-\beta_k)}{s(\beta_l-\beta_k)} \\ =&-\frac{s(\alpha_i)s(\beta_j-\alpha_j)}{s(\beta_{1, n}-\alpha_{1, n+1})}\prod_{l=1, l \neq i}^{n} \frac{s(\beta_l-\alpha_i)}{s(\alpha_l-\alpha_i)} \\ & \times \left( \sum_{k=1}^{n+1} \frac{s(\beta_{1, n}-\alpha_{1, n+1}-\beta_k+\alpha_i)s(\beta_k-\alpha_k)}{s(\beta_k-\alpha_i)s(\beta_k-\alpha_j)s(\beta_k-\alpha_{n+1})}\prod_{l=1, l \neq k}^{n+1}\frac{s(\beta_k-\alpha_l)}{s(\beta_k-\beta_l)} \right). \end{aligned}$$ Put $a_i=e^{\pi \sqrt{-1} \alpha_i}, b_i=e^{\pi \sqrt{-1} \beta_i}$. Then by definition of the sine, we get $$\begin{aligned} \label{eq_1_ab} &\sum_{k=1}^{n+1}\frac{s(\beta_{1, n}-\alpha_{1, n+1}-\beta_k+\alpha_i)s(\beta_k-\alpha_k)}{s(\beta_k-\alpha_i)s(\beta_k-\alpha_j)s(\beta_k-\alpha_{n+1})}\prod_{l=1, l \neq k}^{n+1}\frac{s(\beta_k-\alpha_l)}{s(\beta_k-\beta_l)} \\ &=- 2\sqrt{-1}\frac{ a_j a_{n+1} }{a_1^2 \cdots a_{n+1}^2}\sum_{k=1}^{n+1} \frac{(b_1^2 \cdots b_n^2 a_i^2 - a_1^2 \cdots a_{n+1}^2 b_k^2)}{(b_k^2-a_i^2)(b_k^2-a_j^2)(b_k^2-a_{n+1}^2)} \frac{\prod_{l=1}^{n+1} (b_k^2-a_l^2)}{\prod_{l=1, l \neq k}^{n+1} (b_k^2-b_l^2)}. {\nonumber}\end{aligned}$$ It is sufficient to prove that the right hand side above is equal to zero as a rational function of $a_i$ and $b_i$. Let the rational function $f_1(x)$ be defined by $$\begin{aligned} f_1(x)=& -2\sqrt{-1} \frac{ a_i a_{n+1} }{a_1^2 \cdots a_{n+1}^2} \frac{(b_1^2 \cdots b_n^2 a_i^2 - a_1^2 \cdots a_{n+1}^2 x)}{(x-a_i^2)(x-a_j^2)(x-a_{n+1}^2)} \prod_{l=1}^{n+1} \frac{(x-a_l^2)}{ (x-b_l^2)} . \end{aligned}$$ In the case of $j\neq i$, $f_1(x)$ has only simple poles $x=b_k^2$ ($k=1,\ldots, n+1$). The residues are given as $$\begin{aligned} \underset{x=b_k^2}{\rm Res}f_1(x)dx&=-2\sqrt{-1} \frac{ a_i a_{n+1} }{a_1^2 \cdots a_{n+1}^2} \frac{(b_1^2 \cdots b_n^2 a_i^2 - a_1^2 \cdots a_{n+1}^2 b_k^2)}{(b_k^2-a_i^2)(b_k^2-a_j^2)(b_k^2-a_{n+1}^2)} \frac{\prod_{l=1}^{n+1} (b_k^2-a_l^2)}{\prod_{l=1, l \neq k}^{n+1} (b_k^2-b_l^2)}.\end{aligned}$$ Since the sum of all residues of a rational function on $\mathbb{P}$ is equal to zero, the right hand side of is zero. In the case of $i=j$, $f_1(x)$ has only simple poles $x=a_i^2$, and $x=b_k^2$ ($k=1,\ldots, n+1$). The sum of the residues at $x=b_k^2$ ($k=1,\ldots, n+1$) is equal to the right hand side of . The residues at $x=a_i^2$ is computed as $$\begin{aligned} \underset{x=a_i^2}{\rm Res}f_1(x)dx=& -2\sqrt{-1} \frac{ a_i a_{n+1} }{a_1^2 \cdots a_{n+1}^2} \frac{(b_1^2 \cdots b_n^2 a_i^2 - a_1^2 \cdots a_{n+1}^2 a_i^2)}{(a_i^2-a_{n+1}^2)} \frac{\prod_{l=1, l \neq i}^{n+1} (a_i^2-a_l^2)}{\prod_{l=1}^{n+1} (a_i^2-b_l^2)} \\ =& 2\sqrt{-1} \frac{ a_i^2 b_i }{a_1 \cdots a_{n+1} b_1 \cdots b_n} \frac{(b_1^2 \cdots b_n^2 - a_1^2 \cdots a_{n+1}^2)}{(b_i^2-a_i^2)(a_i^2-1)} \prod_{l=1, l \neq i}^{n} \frac{b_l (a_i^2-a_l^2)}{a_l (a_i^2-b_l^2)} \\ =& \frac{s(\beta_{1, n}-\alpha_{1, n+1})}{s(\beta_i -\alpha_i)s(\alpha_i)}\prod_{l=1, l \neq i}^{n} \frac{s(\alpha_i -\alpha_l)}{s(\alpha_i - \beta_l)}.\end{aligned}$$ Hence, we have $$\begin{aligned} \text{R.H.S. of \eqref{eq_1_ab}} =&\sum_{k=1}^{n+1} \underset{x=b_k^2}{\rm Res} f_1(x)dx \\ =& -\underset{x=a_i^2}{\rm Res}f_1(x)dx \\ =& -\frac{s(\beta_{1, n}-\alpha_{1, n+1})}{s(\beta_i -\alpha_i)s(\alpha_i)}\prod_{l=1, l \neq i}^{n} \frac{s(\alpha_i -\alpha_l)}{s(\alpha_i - \beta_l)}. \end{aligned}$$ Therefore, we obtain $\left(C^{(01)} C^{(10)}\right)_{i, i}=1$ for $1\le i, j\le n$. In the case of (ii), we have $$\begin{aligned} \left(C^{(01)} C^{(10)}\right)_{i, n+1}=&-\sum_{k=1}^{n+1} \frac{s(\alpha_i) s(\beta_{1, n}-\alpha_{1, n+1}-\beta_{k}+\alpha_{i})s(\beta_k-\alpha_k)}{s(\alpha_{n+1})s(\beta_{1, n}-\alpha_{1, n+1})s(\beta_k -\alpha_i)} \prod_{l=1, l \neq i}^{n} \frac{s(\beta_l-\alpha_i)}{s(\alpha_l-\alpha_i)}\prod_{l=1, l \neq k}^{n+1}\frac{s(\alpha_l-\beta_k)}{s(\beta_l-\beta_k)} \\ =&-\frac{s(\alpha_i)}{s(\alpha_{n+1})s(\beta_{1, n}-\alpha_{1, n+1})}\prod_{l=1, l \neq i}^{n} \frac{s(\beta_l-\alpha_i)}{s(\alpha_l-\alpha_i)} \\ &\times \sum_{k=1}^{n+1} \frac{s(\beta_{1, n}-\alpha_{1, n+1}-\beta_{k}+\alpha_{i})s(\beta_k-\alpha_k)}{s(\beta_k -\alpha_i)} \prod_{l=1, l \neq k}^{n+1}\frac{s(\beta_k-\alpha_l)}{s(\beta_k-\beta_l)}. \end{aligned}$$ Rewriting the parameters $\alpha_i$, $\beta_i$ to $a_i$, $b_i$, we get $$\begin{aligned} &\sum_{k=1}^{n+1} \frac{s(\beta_{1, n}-\alpha_{1, n+1}-\beta_k +\alpha_i)s(\beta_k -\alpha_k)}{s(\beta_k -\alpha_i)} \prod_{l=1, l \neq k}^{n+1} \frac{s(\beta_k -\alpha_l)}{s(\beta_k -\beta_l)} \\ &=\sum_{k=1}^{n+1} \frac{1}{2 \sqrt{-1}} \frac{1}{a_1 \cdots a_{n+1} b_k^2} \frac{(b_1^2 \cdots b_n^2 a_i^2 - a_1^2 \cdots a_{n+1}^2 b_k^2)}{(b_k^2-a_i^2)} \frac{\prod_{l=1}^{n+1} (b_k^2-a_l^2)}{\prod_{l=1, l \neq k}^{n+1} (b_k^2 -b_l^2)}. \end{aligned}$$ Let the rational function $f_2(x)$ be defined by $$\begin{aligned} f_2(x)=\frac{1}{2 \sqrt{-1}} \frac{1}{a_1 \cdots a_{n+1} x} \frac{(b_1^2 \cdots b_n^2 a_i^2 - a_1^2 \cdots a_{n+1}^2 x)}{(x-a_i^2)} \prod_{l=1}^{n+1} \frac{(x-a_l^2)}{(x -b_l^2)}.\end{aligned}$$ Then $f_2(x)$ has only simple poles $x=0$, $x=\infty$, and $x=b_k^2$ ($k=1,\ldots, n+1$). The residues are given as $$\begin{aligned} \underset{x=0}{\rm Res}f_2(x)dx =& - \frac{1}{2 \sqrt{-1}}, \\ \underset{x=\infty}{\rm Res}f_2(x)dx=&\frac{1}{2 \sqrt{-1}}, \\ \underset{x=b_k^2}{\rm Res}f_2(x)dx=&\frac{1}{2 \sqrt{-1}} \frac{1}{a_1 \cdots a_{n+1} b_k^2} \frac{(b_1^2 \cdots b_n^2 a_i^2 - a_1^2 \cdots a_{n+1}^2 b_k^2)}{(b_k^2-a_i^2)} \frac{\prod_{l=1}^{n+1} (b_k^2-a_l^2)}{\prod_{l=1, l \neq k}^{n+1} (b_k^2 -b_l^2)}. \end{aligned}$$ Hence, we have $$\begin{aligned} \sum_{k=1}^{n+1}\underset{x=b_k^2}{\rm Res}f_2(x)dx =&-\underset{x=0}{\rm Res}f_2(x)dx-\underset{x=\infty}{\rm Res}f_2(x)dx=0.\end{aligned}$$ Therefore, we obtain $\left(C^{(01)} C^{(10)}\right)_{i, n+1}=0$ for $1\leq i\leq n$. In the case of (iii), we have $$\begin{aligned} \left(C^{(01)} C^{(10)}\right)_{n+1, j}=& -\frac{s(\beta_j -\alpha_j)s(\alpha_{n+1})}{s(\beta_{1, n}-\alpha_{1, n+1})}\left( \sum_{k=1}^{n+1} \frac{s(\beta_k- \alpha_k)}{s(\beta_k -\alpha_j)s(\beta_k -\alpha_{n+1})} \prod_{l=1, l \neq k}^{n+1} \frac{s(\beta_k-\alpha_l)}{s(\beta_k- \beta_l)} \right). \end{aligned}$$ Rewriting the parameters $\alpha_i$, $\beta_i$ to $a_i$, $b_i$, we get $$\begin{aligned} &\sum_{k=1}^{n+1} \frac{s(\beta_k- \alpha_k)}{s(\beta_k -\alpha_j)s(\beta_k -\alpha_{n+1})} \prod_{l=1, l \neq k}^{n+1} \frac{s(\beta_k-\alpha_l)}{s(\beta_k- \beta_l)} \\ &= \sum_{k=1}^{n+1} 2 \sqrt{-1} \frac{b_1 \cdots b_n a_j a_{n+1}}{a_1 \cdots a_{n+1}} \frac{1}{(b_k^2-a_j^2)(b_k^2-a_{n+1}^2)} \frac{\prod_{l=1}^{n+1} (b_k^2-a_l^2)}{\prod_{l=1, l \neq k}^{n+1} (b_k^2-b_l^2)}. \end{aligned}$$ Let the rational function $f_3(x)$ be defined by $$\begin{aligned} f_3(x)= 2 \sqrt{-1} \frac{b_1 \cdots b_n a_j a_{n+1}}{a_1 \cdots a_{n+1}} \frac{1}{(x-a_j^2)(x-a_{n+1}^2)} \prod_{l=1}^{n+1} \frac{ (x-a_l^2)}{ (x-b_l^2)}. \end{aligned}$$ Then $f_3(x)$ has only simple poles $x=b_k^2$ ($k=1,\ldots, n+1$). The residues are given as $$\begin{aligned} \underset{x=b_k^2}{\rm Res}f_3(x)dx =2 \sqrt{-1} \frac{b_1 \cdots b_n a_j a_{n+1}}{a_1 \cdots a_{n+1}} \frac{1}{(b_k^2-a_j^2)(b_k^2-a_{n+1}^2)} \frac{\prod_{l=1}^{n+1} (b_k^2-a_l^2)}{\prod_{l=1, l \neq k}^{n+1} (b_k^2-b_l^2)}. \end{aligned}$$ Hence we obtain $\left(C^{(01)} C^{(10)}\right)_{n+1, j}=0$ for $1\leq j\leq n$. In the case of (iv), we have $$\begin{aligned} \left(C^{(01)} C^{(10)}\right)_{n+1, n+1}=-\frac{1}{s(\beta_{1, n}-\alpha_{1, n+1})} \sum_{k=1}^{n+1} s(\beta_k -\alpha_k) \prod_{l=1, l \neq k}^{n+1} \frac{s(\beta_k -\alpha_l)}{s(\beta_k -\beta_l)}. \end{aligned}$$ Rewriting the parameters $\alpha_i$, $\beta_i$ to $a_i$, $b_i$, we get $$\begin{aligned} &\sum_{k=1}^{n+1} s(\beta_k -\alpha_k) \prod_{l=1, l \neq k}^{n+1} \frac{s(\beta_k -\alpha_l)}{s(\beta_k -\beta_l)} = -\sum_{k=1}^{n+1} \frac{b_1 \cdots b_n}{2 \sqrt{-1} a_1 \cdots a_{n+1} b_k^2} \frac{\prod_{l=1}^{n+1} (b_k^2 -a_l^2)}{\prod_{l=1, l \neq k}^{n+1} (b_k^2 -b_l^2)}. \end{aligned}$$ Let the rational function $f_4(x)$ be defined by $$\begin{aligned} f_4(x)=-\frac{b_1 \cdots b_n}{2 \sqrt{-1} a_1 \cdots a_{n+1}x} \prod_{l=1}^{n+1} \frac{x-a_l^2}{x-b_l^2}. \end{aligned}$$ Then $f_4(x)$ has only simple poles $x=0$, $x=\infty$, and $x=b_k^2$ ($k=1,\ldots, n+1$). The residues are given as $$\begin{aligned} \underset{x=0}{\rm Res}f_4(x)dx =& - \frac{a_1 \cdots a_{n+1}}{2 \sqrt{-1} b_1 \cdots b_n}, \\ \underset{x=\infty}{\rm Res}f_4(x)dx =& \frac{b_1 \cdots b_n}{2 \sqrt{-1} a_1 \cdots a_{n+1}}, \\ \underset{x=b_k^2}{\rm Res}f_4(x)dx =& -\sum_{k=1}^{n+1} \frac{b_1 \cdots b_n}{2 \sqrt{-1} a_1 \cdots a_{n+1} b_k^2} \frac{\prod_{l=1}^{n+1} (b_k^2 -a_l^2)}{\prod_{l=1, l \neq k}^{n+1} (b_k^2 -b_l^2)}. \end{aligned}$$ Hence, we have $$\begin{aligned} \sum_{k=1}^{n+1}\underset{x=b_k^2}{\rm Res}f_4(x)dx =& -\underset{x=0}{\rm Res}f_4(x)dx -\underset{x=\infty}{\rm Res}f_4(x)dx \\ =& -s(\beta_{1, n}-\alpha_{1, n+1}). \end{aligned}$$ Therefore, we obtain $\left(C^{(01)} C^{(10)}\right)_{n+1,n+1}=1$. Assume for $1 \leq i, j \leq n+1$ $$\begin{aligned} &\mathrm{Re}(\alpha_i-\beta_j+1)>0\quad (j\neq i), \\ &\mathrm{Re}(\beta_j-\alpha_j)>0, \\ &\alpha_i-\alpha_j\notin {{\mathbb Z}}\quad (j\neq i),\quad \alpha_i-\beta_j \notin {{\mathbb Z}}, \quad \beta_i-\beta_j \notin {{\mathbb Z}}\quad (j\neq i). \end{aligned}$$ Then we have $$\begin{aligned} F_{\tilde{D}_{i}^{(0)}}(z)=&-\sum_{j=1}^{n}\frac{s(\alpha_j)s(\beta_{1, n} -\alpha_{1, n+1}-\beta_i +\alpha_j)s(\beta_i-\alpha_i)}{s(\alpha_{n+1})s(\beta_{1, n}-\alpha_{1, n+1})s(\beta_i-\alpha_j)} \prod_{l=1, l \neq j}^{n} \frac{s(\beta_l - \alpha_j)}{s(\alpha_l-\alpha_j)}F_{\tilde{D}_{j}^{(1)}}(z) \\ &-\frac{s(\beta_i-\alpha_i)}{s(\beta_{1, n}-\alpha_{1, n+1})}F_{\tilde{D}_{n+1}^{(1)} }(z).\end{aligned}$$ Moreover, $\left\{ F_{\tilde{D}_1^{(1)}}(z),\ldots,F_{\tilde{D}_{n+1}^{(1)}}\right\}$ forms a fundamental system of solutions to the generalized hypergeometric equation. We note that together with Proposition \[\[M2\], prop 2.1\] and Theorem \[thm\_connection\_formula\_10\], we have the connection matrix $C^{(1\infty)}$ expressing the fundamental system of solutions $\left\{ F_{\tilde {D}^{(1)}_1}(z),\ldots, F_{\tilde{D}^{(1)}_1}(z)\right\}$ at $z=1$ in terms of the fundamental system of solutions $\left\{ F_{D^{(\infty)}_1}(z),\ldots, F_{D^{(\infty)}_1}(z)\right\}$ at $\infty$ as a product of the connection matrices. It is observed that the elements of the connection matrix $C^{(1\infty)}$ for $n\le 3$ are products of the sine and cosecant. However, there are elements of the inverse of $C^{(1\infty)}$ for $n\le 3$ which are not products of the sine and cosecant. If we want to have connection matrices between the fundamental system of solutions of singular points $z=1$ and $z=\infty$ whose elements are products of the sine and cosecant, then we should perform the change of variable $z=1/w$ to the connection formulas and . Due to Theorem \[thm\_connection\_matrix\_01\], all elements of both the connection matrix and its inverse are products of the sine and cosecant. Periodicity ----------- In this subsection, we point out that the connection matrices multiplied by diagonal matrices in the previous sections are invariant under integer shifts of the characteristic exponents $\alpha_i$, $\beta_i$ of the generalized hypergeometric equation. This is true for the connection matrix with the fundamental systems $X_0$ and $X_\infty$ of solutions at $z=0$ and $z=\infty$, and it was used for the construction of the monodromy-invariant general solutions of Fuchsian systems of rank $2$ in [@Iorgov; @Lisovyy; @Teschner] and rank $N\geq 2$ in [@GIL2] with help of the $W_N$ conformal field theory with central charge $c=N-1$. When $N=2$, $W_N$-algebra is the Virasoro algebra. As a result, for $N=2$ the tau function of the sixth Painlevé equation is obtained in terms of the Virasoro conformal blocks with central charge $c=1$ [@Iorgov; @Lisovyy; @Teschner], and for $N\ge 2$ the isomonodromic tau function of the so-called Fuji-Suzuki-Tsuda system [@FS], [@Suzuki], [@Tsuda1] is obtained in terms of semi-degenerate conformal blocks of $W_N$-algebra with central charge $c=N-1$, which are equivalent to the Nekrasov partition functions [@Nekrasov] by AGT correspondence [@AGT], [@Wyllard]. The series representation of the tau function of the Fuji-Suzuki-Tsuda system is also derived by expanding the Fredholm determinant [@GIL3]. The periodicity of the connection matrix of $q$-hypergeometric series was used in [@Jimbo; @Nagoya; @Sakai] in order to obtain a fundamental solution to the connection-preserving deformation of the $2$ by $2$ $q$-difference linear system associated with the $q$-difference Painlevé VI equation. We take $$\begin{aligned} D_i^{(0)}=&\{t\in T_{{\mathbb R}}\mid z<t_n< \cdots <t_i <0,\ 1<t_1< \cdots < t_{i-1} \} \quad (1 \leq i \leq n+1), \\ D_i^{(\infty)}=&\{t\in T_{{\mathbb R}}\mid t_i< \cdots <t_n <z,\ 0<t_{i-1}< \cdots < t_1< 1 \} \quad (1 \leq i \leq n+1).\end{aligned}$$ as domains of integration giving the fundamental systems of solutions at $z=0$ and $z=\infty$, and $$\begin{aligned} \tilde{D}_i^{(1)}&=\{t\in T_{{\mathbb R}}| t_i< \cdots <t_n <0,\ 0<t_{i-1}< \cdots < t_1< 1 \} \quad (1 \leq i \leq n), \\ \tilde{D}_{n+1}^{(1)}&=\{t\in T_{{\mathbb R}}| z<t_n< \cdots < t_1< 1 \}\end{aligned}$$ as domains of integration giving the fundamental system of solutions at $z=1$. From Proposition \[\[M2\], prop 2.5\], we have the connection matrix $C^{(\infty 0)}$ given by $$\left(D_1^{(\infty)},\ldots, D_{n+1}^{(\infty)}\right)= \left(D_1^{(0)},\ldots,D_{n+1}^{(0)}\right)C^{(\infty 0)}.$$ The elements of $C^{(\infty 0)}$ read as $$\begin{aligned} \left( C^{(\infty 0)}\right)_{i,j}= \frac{s(\beta_j-\alpha_j)}{s(\beta_i-\alpha_j)} \prod_{k=1, k \neq i}^{n+1} \frac{s(\alpha_k-\beta_i)}{s(\beta_k-\beta_i)} \end{aligned}$$ for $1\leq i,j\leq n+1$. Let the diagonal matrices $N_0$, $N_1$, $N_\infty$ of order $n+1$ be defined by $$\begin{aligned} N_0=&{\rm diag} (e(\alpha_1),\ldots,e(\alpha_{n+1})), \\ N_1=&e(\beta_{1,n}-\alpha_{1,n+1}) {\rm diag}(e(\beta_1),\ldots,e(\beta_{n+1})), \\ N_\infty=&N_1. \end{aligned}$$ The connection matrices associated with the fundamental systems of solutions $$\begin{aligned} \left(D_1^{(0)},\ldots,D_{n+1}^{(0)}\right)N_0, \quad \left(\tilde{D}_1^{(1)},\ldots, \tilde{D}_{n+1}^{(1)}\right)N_1, \quad \left(D_1^{(\infty)},\ldots, D_{n+1}^{(\infty)}\right)N_\infty \end{aligned}$$ are invariant under integer shifts of the characteristic exponents of the generalized hypergeometric equation. It is sufficient to consider the shifts $\alpha_i\mapsto \alpha_i\pm 1$ and $\beta_i\mapsto \beta_i\pm 1$ for $1\leq i\leq n+1$. 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--- abstract: 'Attitude control systems naturally evolve on nonlinear configuration spaces, such as ${\ensuremath{\mathsf{S}}}^2$ and ${\ensuremath{\mathsf{SO(3)}}}$. The nontrivial topological properties of these configuration spaces result in interesting and complicated nonlinear dynamics when studying the corresponding closed loop attitude control systems. In this paper, we review some global analysis and simulation techniques that allow us to describe the global nonlinear stable manifolds of the hyperbolic equilibria of these closed loop systems. A deeper understanding of these invariant manifold structures are critical to understanding the global stabilization properties of closed loop control systems on nonlinear spaces, and these global analysis techniques are applicable to a broad range of problems on nonlinear configuration manifolds.' author: - 'Taeyoung Lee, Melvin Leok, and N. Harris McClamroch[^1][^2][^3][^4] [^5]' bibliography: - 'CDC11.bib' title: | **Stable Manifolds of Saddle Points\ for Pendulum Dynamics on ${\ensuremath{\mathsf{S}}}^2 $ and ${\ensuremath{\mathsf{SO(3)}}}$** --- Introduction ============ Global nonlinear dynamics of various classes of closed loop attitude control systems have been studied in recent years. An overview of results on attitude control of a rotating rigid body is given in  [@ChSaMcCSM11]. Closely related results on attitude control of a spherical pendulum (with attitude an element of the two-sphere ${\ensuremath{\mathsf{S}}}^2$) and of a 3D pendulum (with attitude an element of the special orthogonal group ${\ensuremath{\mathsf{SO(3)}}}$) are given in  [@ChMcIJRNC07; @ChMcBeAut08; @ChaMcCITAC09]. These and other similar publications address the global closed dynamics of smooth vector fields. Assuming that the closed loop vector field has an asymptotically stable equilibrium, as desired in attitude stabilization problems, additional hyperbolic equilibria necessarily exist. The domain of attraction of the asymptotically stable equilibrium is contained in the complement of the union of the stable manifolds of the hyperbolic equilibria. These geometric factors motivate the current paper, in which new analytical and computational results on the stable manifolds of the hyperbolic equilibria are obtained. To make the development concrete, the presentation is built around two specific closed loop vector fields: one for the attitude dynamics of a spherical pendulum and one for the attitude dynamics of a 3D pendulum. In analyzing these two cases, we introduce new analytical and computational tools that are broadly applicable to studying the geometry of more general attitude control systems. Spherical Pendulum ================== A spherical pendulum is composed of a mass $m$ connected to a frictionless pivot by a massless link of length $l$. It is acts under uniform gravity, and it is subject to a control moment $u$. The configuration of a spherical pendulum is described by a unit-vector $q\in{\ensuremath{\mathbb{R}}}^3$, representing the direction of the link with respect to a reference frame. Therefore, the configuration space is the two-sphere ${\ensuremath{\mathsf{S}}}^2=\{q\in{\ensuremath{\mathbb{R}}}^3\,|\, q\cdot q =1\}$. The tangent space of the two-sphere at $q$, namely ${\ensuremath{\mathsf{T}}}_q{\ensuremath{\mathsf{S}}}^2$, is the two-dimensional plane tangent to the unit sphere at $q$, and it is identified with ${\ensuremath{\mathsf{T}}}_q{\ensuremath{\mathsf{S}}}^2\simeq\{\omega\in{\ensuremath{\mathbb{R}}}^3\,|\, q\cdot \omega =0\}$, using the following kinematics equation: $$\begin{aligned} \dot q= \omega\times q,\end{aligned}$$ where the vector $\omega\in{\ensuremath{\mathbb{R}}}^3$ represents the angular velocity of the link. The equation of motion is given by $$\begin{gathered} \dot\omega = \frac{g}{l}q\times e_3 + \frac{1}{ml^2} u,\end{gathered}$$ where the constant $g$ is the gravitational acceleration, and the vector $e_3=[0,0,1]\in{\ensuremath{\mathbb{R}}}^3$ denotes the unit vector along the direction of gravity. The control moment at the pivot is denoted by $u\in{\ensuremath{\mathbb{R}}}^3$. Control System -------------- Several proportional-derivative (PD) type control systems have been developed on ${\ensuremath{\mathsf{S}}}^2$ in a coordinate-free fashion [@BulMurN95; @BulLew05]. Here, we summarize a control system that stabilizes a spherical pendulum to a fixed desired direction $q_d\in{\ensuremath{\mathsf{S}}}^2$. Consider an error function on ${\ensuremath{\mathsf{S}}}^2$, representing the projected distance from the direction $q$ to the desired direction $q_d$, given by $$\begin{aligned} \Psi(q,q_d)=1-q\cdot q_d.\end{aligned}$$ The derivative of $\Psi$ with respect to $q$ along the direction $\delta q =\xi\times q$, where $\xi\in{\ensuremath{\mathbb{R}}}^3$ and $\xi\cdot q=0$, is given by $$\begin{aligned} {\ensuremath{\mathbf{D}}}_q \Psi(q,q_d)\cdot\delta q = -(\xi\times q)\cdot q_d = (q_d\times q)\cdot \xi.\end{aligned}$$ For positive constants $k_q,k_\omega$, the control input is chosen as: $$\begin{aligned} u = ml^2(- k_\omega \omega - k_q q_d\times q -\frac{g}{l}q\times e_3).\end{aligned}$$ The corresponding closed loop dynamics are given by $$\begin{gathered} \dot \omega = -k_\omega\omega - k_q q_d\times q,\label{eqn:dotw}\\ \dot q= \omega\times q.\label{eqn:dotq}\end{gathered}$$ This yields two equilibrium solutions: (i) the desired equilibrium $(q,\omega)=(q_d,0)$; (ii) additionally, there exists another equilibrium $(-q_d,0)$ at the antipodal point on the two-sphere. It can be shown that the desired equilibrium is asymptotically stable by using the following Lyapunov function: $$\begin{aligned} \mathcal{V} = \frac{1}{2} \omega\cdot\omega + k_q \Psi(q,q_d).\end{aligned}$$ In this paper, we analyze the local stability of each equilibrium by linearizing the closed loop dynamics to study the equilibrium structures more explicitly. In particular, we develop a coordinate-free form of the linearized dynamics of [(\[eqn:dotw\])]{}, [(\[eqn:dotq\])]{}, in the following section. Linearization ------------- A variation of a curve $q(t)$ on ${\ensuremath{\mathsf{S}}}^2$ is a family of curves $q^\epsilon(t)$ parameterized by $\epsilon\in{\ensuremath{\mathbb{R}}}$, satisfying several properties [@BulLew05]. It cannot be simply written as $q^\epsilon(t)=q(t)+\epsilon\delta q(t)$ for $\delta q(t)$ in ${\ensuremath{\mathbb{R}}}^3$, since in general, this does not guarantee that $q^\epsilon(t)$ lies in ${\ensuremath{\mathsf{S}}}^2$. In [@LeeLeoIJNME08], an expression for a variation on ${\ensuremath{\mathsf{S}}}^2$ is given in terms of the exponential map as follows: $$\begin{aligned} q^\epsilon(t) = \exp(\epsilon \hat\xi(t))q(t),\label{eqn:qe}\end{aligned}$$ for a curve $\xi(t)$ in ${\ensuremath{\mathbb{R}}}^3$ satisfying $\xi(t)\cdot q(t)=0$ for all $t$. The *hat map* $\hat\cdot:{\ensuremath{\mathbb{R}}}^3\rightarrow{\ensuremath{\mathfrak{so}(3)}}$ is defined by the condition that $\hat x y =x\times y$ for any $x,y\in{\ensuremath{\mathbb{R}}}^3$. The resulting infinitesimal variation is given by $$\begin{aligned} \delta q (t) = \frac{d}{d\epsilon}\bigg|_{\epsilon=0} q^\epsilon(t) = \xi(t)\times q(t). \label{eqn:delq}\end{aligned}$$ The variation of the angular velocity can be written as $$\begin{aligned} \omega^\epsilon(t) = \omega(t) + \epsilon \delta \omega(t),\label{eqn:we}\end{aligned}$$ for a curve $\delta w(t)$ in ${\ensuremath{\mathbb{R}}}^3$ satisfying $q(t)\cdot w(t)=0$ for all $t$. Hereafter, we do not write the dependency on time $t$ explicitly. The time-derivative of $\delta q$ can be obtained either from [(\[eqn:delq\])]{} or by substituting [(\[eqn:qe\])]{}, [(\[eqn:we\])]{} into [(\[eqn:dotq\])]{}, and considering the first order terms of $\epsilon$. In either case, we have $$\begin{aligned} \delta\dot q= \dot\xi\times q + \xi\times(\omega\times q) = \delta\omega\times q + \omega\times(\xi\times q).\end{aligned}$$ Using the vector cross product identity $a\times (b\times c)=(a\cdot c)b-(a\cdot b) c$ for any $a,b,c\in{\ensuremath{\mathbb{R}}}^3$, this can be written as $$\begin{gathered} \dot\xi\times q + (\xi\cdot q)w -(\xi\cdot\omega) q = \delta\omega\times q +(\omega\cdot q)\xi -(\omega\cdot\xi)q.\end{gathered}$$ Since $\xi\cdot q=0$, $\omega\cdot q=0$, this reduces to $$\begin{gathered} \dot\xi\times q = \delta\omega\times q.\end{gathered}$$ Since both sides of the above equation are perpendicular to $q$, this is equivalent to $q\times(\dot\xi\times q) = q\times(\delta\omega\times q)$, which yields $$\begin{gathered} \dot \xi - (q\cdot\dot\xi) q = q\times(\delta\omega\times q).\end{gathered}$$ Since $\xi\cdot q =0$, we have $\dot\xi\cdot q +\xi\cdot\dot q=0$. Using this, the above equation can be rewritten as $$\begin{aligned} \dot \xi & = -(\xi\cdot(\omega\times q))q+ q\times(\delta\omega\times q)\nonumber\\ & = (qq^T\hat\omega) \xi +(I-qq^T)\delta\omega.\label{eqn:dotxi}\end{aligned}$$ This corresponds to the linearized equation of motion for [(\[eqn:dotq\])]{}. Similarly, by substituting [(\[eqn:delq\])]{}, [(\[eqn:we\])]{} into [(\[eqn:dotw\])]{}, we obtain $$\begin{aligned} \delta\dot\omega &= -k_\omega\delta\omega -k_qq_d\times(\xi\times q)\nonumber\\ &=-k_\omega\omega +k_q\hat q_d \hat q\,\xi,\label{eqn:dotdelw}\end{aligned}$$ which is the linearized equation for [(\[eqn:dotw\])]{}. Equations [(\[eqn:dotxi\])]{}, [(\[eqn:dotdelw\])]{} can be written in a matrix form as $$\begin{aligned} \dot x = \begin{bmatrix}\dot\xi \\ \delta\dot\omega \end{bmatrix} =\begin{bmatrix} qq^T\hat\omega & I-qq^T\\k_q\hat q_d\hat q & -k_wI\end{bmatrix} \begin{bmatrix}\xi \\ \delta\omega \end{bmatrix}=Ax,\label{eqn:xdot}\end{aligned}$$ where the state vector of the linearized controlled system is $x=[\xi;\delta\omega]\in{\ensuremath{\mathbb{R}}}^6$. A spherical pendulum has two degrees of freedom, but this linearized equation of motion evolves in ${\ensuremath{\mathbb{R}}}^6$ instead of ${\ensuremath{\mathbb{R}}}^4$. Since $q\cdot\omega=0$ and $q\cdot\xi=0$, we have the following two additional constraints on $\xi,\delta\omega$: $$\begin{aligned} Cx= \begin{bmatrix} q^T & 0 \\ -\omega^T\hat q & q^T\end{bmatrix} \begin{bmatrix}\xi \\ \delta\omega \end{bmatrix} =\begin{bmatrix} 0 \\ 0 \end{bmatrix}.\label{eqn:con}\end{aligned}$$ Therefore, the state vector $x$ should lie in the null space of the matrix $C\in{\ensuremath{\mathbb{R}}}^{2\times 4}$. However, this is not an extra constraint that should be imposed when solving [(\[eqn:xdot\])]{}. As long as the initial condition $x(0)$ satisfies [(\[eqn:con\])]{}, the structure of [(\[eqn:dotw\])]{}, [(\[eqn:dotq\])]{}, and [(\[eqn:xdot\])]{}, guarantees that the state vector $x(t)$ satisfies [(\[eqn:con\])]{} for all $t$, i.e. $\frac{d}{dt}C(t)x(t) =0$ for all $t\geq 0$ when $C(0)x(0)=0$. This means that the null space of $C$ is a flow-invariant subspace. Equilibrium Solutions --------------------- We choose the desired direction as $q_d=e_3$. The equilibrium solution $(q_d,0)=(e_3,0)$ is referred to as the hanging equilibrium, and the additional equilibrium solution $(-q_d,0)=(-e_3,0)$ is referred to as the inverted equilibrium. We study the eigen-structure of each equilibrium using the linearized equation [(\[eqn:xdot\])]{}. To illustrate the ideas, the controller gains are selected as $k_q=k_\omega=1$. ### Hanging Equilibrium The eigenvalues $\lambda_i$, and the eigenvectors $v_i$ of the matrix $A$ at the hanging equilibrium $(e_3,0)$ are given by $$\begin{gathered} \lambda_{1,2}=(-1\pm\sqrt{3}i)/2,\; \lambda_{3,4}=\lambda_{1,2},\;\lambda_5=0,\;\lambda_6=-1,\\ v_{1,2}= e_1 + (-1\pm\sqrt{3}i)e_4/2,\; v_{3,4}= e_2 + (-1\pm\sqrt{3}i)e_5/2,\\ v_5=e_3,\quad v_6=e_6,\end{gathered}$$ where $e_i\in{\ensuremath{\mathbb{R}}}^6$ denotes the unit-vector whose $i$-th element is one, and other elements are zeros. Note that there are repeated eigenvalues, but we obtain six linearly independent eigenvectors, i.e., the geometric multiplicities are equal to the algebraic multiplicities. The basis of the null space of the matrix $C$, namely $\mathcal{N}(C)$ is $\{e_1,e_2,e_4,e_5\}$. The solution of the linearized equation can be written as $x(t)=\sum_{i=1}^6 c_i \exp(\lambda_it) v_i$ for constants $c_i$ that are determined by the initial condition: $x(0)=\sum_{i=1}^6 c_i v_i$. But, the eigenvectors $v_5,v_6$ do not satisfy the constraint given by [(\[eqn:con\])]{}, since they do not lie in $\mathcal{N}(C)$. Therefore, the constants $c_5,c_6$ are zero for initial conditions that are compatible with [(\[eqn:con\])]{}. We have $\mathrm{Re}[\lambda_i]<0$ for $1\leq i\leq 4$. Therefore, the equilibrium $(q,\omega)=(e_3,0)$ is asymptotically stable. ### Inverted Equilibrium The eigenvalues $\lambda_i$, and the eigenvectors $v_i$ of the matrix $A$ at the inverted equilibrium $(-e_3,0)$ are given by $$\begin{gathered} \lambda_{1,2}=-(\sqrt{5}+1)/2,\lambda_{3,4}=(\sqrt{5}-1)/2,\lambda_5=0,\lambda_6=-1,\nonumber\\ v_{1}= e_1 -(\sqrt{5}+1)e_4/2,\, v_2=e_2-(\sqrt{5}+1)e_5/2,\label{eqn:v1v2}\\ v_3=(\sqrt{5}+1)e_1/2 +e_4,\, v_4=(\sqrt{5}+1)e_2/2+e_5,\nonumber\\ v_5=e_3,\, v_6=e_6.\nonumber\end{gathered}$$ The basis of $\mathcal{N}(C)$ is $\{e_1,e_2,e_4,e_5\}$. Hence, the eigenvectors $v_5,v_6$ do not lie in $\mathcal{N}(C)$. Therefore, the solution can be written as $x(t)=\sum_{i=1}^4 c_i \exp(\lambda_it) v_i$ for constants $c_i$ that are determined by the initial condition. We have $\mathrm{Re}[\lambda_{1,2}]<0$, and $\mathrm{Re}[\lambda_{3,4}]>0$. Therefore, the inverted equilibrium $(q,\omega)=(-e_3,0)$ is a hyperbolic equilibrium, and in particular, a saddle point. Stable Manifold for the Inverted Equilibrium {#sec:SM} -------------------------------------------- ### Stable Manifold The saddle point $(-e_3,0)$ has a stable manifold $W^s$, which is defined to be $$\begin{aligned} W^s(-e_3,0) &= \{ (q,\omega)\in {\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{S}}}^2\,|\, \lim_{t\rightarrow\infty} \mathcal{F}^t(q,\omega) = (-e_3,0)\},\end{aligned}$$ where $\mathcal{F}^t:(q(0),\omega(0))\rightarrow(q(t),\omega(t))$ denotes the flow map along the solution of [(\[eqn:dotw\])]{}, [(\[eqn:dotq\])]{}. The existence of $W^s(-e_3,0)$ has nontrivial effects on the overall dynamics of the controlled system. Trajectories in $W^s(-e_3,0)$ converge to the antipodal point of the desired equilibrium $(e_3,0)$, and it takes a long time period for any trajectory near $W^s(-e_3,0)$ to asymptotically converge to the desired equilibrium $(e_3,0)$. According to the stable and unstable manifold theorem [@Kuz98], a local stable manifold $W^s_{loc}(-e_3,0)$ exists in the neighborhood of $(-e_3,0)$, and it is tangent to the stable eigenspace $E^s(-e_3,0)$ spanned by the eigenvectors $v_1$ and $v_2$ of the stable eigenvalues $\lambda_{1,2}$. The (global) stable manifold can be written as $$\begin{aligned} W^s(-e_3,0) & = \bigcup_{t>0} \mathcal{F}^{-t} ( W^s_{loc}(-e_3,0)),\label{eqn:Ws}\end{aligned}$$ which states that the stable manifold $W^s$ can be obtained by globalizing the local stable manifold $W^s_{loc}$ by the backward flow map. This yields a method to compute $W^s(-e_3,0)$ [@KraOsiIJBC05]. We choose a small ball $B_\delta\subset W^s_{loc}(-e_3,0)$ with a radius $\delta$ around $(-e_3,0)$, and we grow the manifold $W^s(-e_3,0)$ by evolving $B_\delta$ under the flow $\mathcal{F}^{-t}$. More explicitly, the stable manifold can be parameterized by $t$ as follows: $$\begin{aligned} W^s(-e_3,0) =\{ \mathcal{F}^{-t} (B_\delta)\}_{t>0}.\label{eqn:Wc}\end{aligned}$$ We construct a ball in the stable eigenspace of $(-e_3,0)$ with sufficiently small radius $\delta$, i.e. $B_\delta\subset E^s_{loc}(-e_3,0)$. From the stable eigenvectors $v_1,v_2$ at [(\[eqn:v1v2\])]{}, $E^s_{loc}(-e_3,0)$ can be written as $$\begin{aligned} E^s_{loc}& (-e_3,0) = \{ (q,\omega)\in {\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{S}}}^2\,|\, q=\exp(\alpha_1\hat e_1+\alpha_2\hat e_2)(-e_3),\nonumber\\ & \omega=-\hat q^2(-(\sqrt{5}+1)/2)(\alpha_1 e_1+\alpha_2e_2)\text{ for $\alpha_1,\alpha_2\in{\ensuremath{\mathbb{R}}}$}\},\label{eqn:Esloc}\end{aligned}$$ where $-\hat q^2$ in the expression for $\omega$ corresponds to the orthogonal projection onto the plane normal to $q$, as required due to the constraint $q\cdot\omega=0$. We define a distance on ${\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{S}}}^2$ as follows: $$\begin{aligned} d_{{\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{S}}}^2} ((q_1,\omega_1),(q_2,\omega_2)) = \sqrt{\Psi(q_1,q_2)} + \|\omega_1-\omega_2\|.\label{eqn:dis}\end{aligned}$$ For $\delta>0$, the subset $B_\delta$ of $E^s_{loc}(-e_3,0)$ is parameterized by $\theta\in{\ensuremath{\mathsf{S}}}^1$ as $$\begin{aligned} B_\delta & = \{ (q,\omega)\in {\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{S}}}^2\,|\, q=\exp(\alpha_1\hat e_1+\alpha_2\hat e_2)(-e_3),\nonumber\\ & \omega=-\hat q^2(-(\sqrt{5}+1)/2)(\alpha_1 e_1+\alpha_2e_2),\text{ where}\nonumber\\ & \text{$\alpha_1=\frac{\delta}{1/\sqrt{2}+(\sqrt{5}+1)/2}\cos\theta,$\;}\nonumber\\ & \text{$\alpha_2=\frac{\delta}{1/\sqrt{2}+(\sqrt{5}+1)/2}\sin\theta$,} \text{ for $\theta\in{\ensuremath{\mathsf{S}}}^1$} \}.\label{eqn:Bdelta}\end{aligned}$$ The given choice of the constants $\alpha_1,\alpha_2$ guarantees that any point in $B_\delta$ has a distance $\delta$ to $(-e_3,0)$ according to the distance metric [(\[eqn:dis\])]{}. ### Variational Integrators The parameterization of the stable manifold $W_s$ in [(\[eqn:Wc\])]{} requires the computation of the backward flow map $\mathcal{F}^{-t}$. However, general purpose numerical integrators may not preserve the structure of the two-sphere or the underlying dynamic characteristics, such as energy dissipation rate, accurately, and they may yield qualitatively incorrect numerical results in simulating a complex trajectory over a long-time period [@HaiLub00]. Geometric numerical integration is concerned with developing numerical integrators that preserve geometric features of a system, such as invariants, symmetry, and reversibility. In particular, variational integrators are geometric numerical integrators for Lagrangian or Hamiltonian systems, constructed according to Hamilton’s principle. They have desirable computational properties of preserving symplecticity and momentum maps, and they exhibit good energy behavior [@MarWesAN01]. A variational integrator is developed for Lagrangian or Hamiltonian systems evolving on the two-sphere in [@LeeLeoIJNME08]. It preserves both the underlying symplectic properties and the structures of the two-sphere concurrently. A variational integrator on ${\ensuremath{\mathsf{S}}}^2$ for the controlled dynamics of a spherical pendulum can be written in a backward-time integration form as follows: $$\begin{aligned} q_{{k}} & = -{\ensuremath{\left( h\omega_{k+1} - \frac{h^2}{2ml^2} M_{k+1} \right)}}\times q_{k+1}\nonumber\\ &\quad + {\ensuremath{\left( 1-{\ensuremath{\left\| h\omega_{k+1} - \frac{h^2}{2ml^2} M_{k+1} \right\|}}^2 \right)}}^{1/2} q_{k+1},\label{eqn:qk}\\ \omega_{{k}} & = \omega_{k+1} - \frac{h}{2ml^2} M_k - \frac{h}{2ml^2} M_{k+1},\label{eqn:wk}\end{aligned}$$ where the constant $h>0$ is time step, the subscript $k$ denotes the value of a variable at the time $t_k=kh$, and $M_k = ml^2(-k_\omega \omega_k - k_q q_d\times q_k)$. For given $(q_{k+1},\omega_{k+1})$, we first compute $M_{k+1}$. Then, $q_k$ is obtained by [(\[eqn:qk\])]{}, followed by $M_k$, and $\omega_k$ is computed by [(\[eqn:wk\])]{}. This yields an explicit, discrete inverse flow map $\mathcal{F}_d^{-h}((q_{k+1},\omega_{k+1}))=(q_{k},\omega_{k})$. ### Visualization We choose 100 points on the surface of $B_\delta$ with $\delta=10^{-6}$, and each point is integrated backward using [(\[eqn:qk\])]{}, [(\[eqn:wk\])]{} with timestep $h=0.002$. The resulting trajectories are illustrated in [Fig. \[fig:Ws\]]{} for several values of $t$. Each colored curve on the sphere represents a trajectory on ${\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{S}}}^2$, since at any point $q$ on the curve, the direction of $\dot q=\omega\times q$ is tangent to the curve at $q$, and the magnitude of $\dot q$ is indirectly represented by color shading. We observe the following characteristics of the stable manifold $W_s(-e_3,0)$ of the inverted equilibrium: - The boundary of the stable manifold $W_s(-e_3,0)\subset {\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{S}}}^2$ parameterized by $t$ is circular when projected onto ${\ensuremath{\mathsf{S}}}^2$. - Each trajectory in $W_s(-e_3,0)$ is on a great circle, when projected onto ${\ensuremath{\mathsf{S}}}^2$. According to the closed loop dynamics [(\[eqn:dotw\])]{}, and the given initial condition at the surface of $B_\delta$, the direction of $\dot\omega$ is always parallel to $\omega$. Therefore, the direction of $\omega$ is fixed, and the resulting trajectory of $q$ is on a great circle. This also corresponds to the fact that the eigenvalue $\lambda_1$ for the first mode representing the rotations about the first axis is equal to the eigenvalue $\lambda_2$ for the second mode representing the rotations about the second axis at [(\[eqn:v1v2\])]{}, i.e. the convergence rates of these two rotations are identical. - The angular velocity decreases to zero as the direction of the pendulum $q$ converges to $-e_3$. - The stable manifold $W_s(-e_3,0)$ may cover ${\ensuremath{\mathsf{S}}}^2$ multiple times if $t$ is sufficiently large, as illustrated at [Fig. \[fig:Ws9\]]{}. Therefore, at any point $q\in{\ensuremath{\mathsf{S}}}^2$, we can choose $\omega$ such that $(q,\omega)$ lies in the stable manifold $W^s(-e_3,0)$ (the corresponding value of $\omega$ is not unique, since if it is sufficiently large, $q$ can traverse the sphere several times before converging to $-e_3$). This is similar to *kicking* a damped spherical pendulum carefully such that it converges to the inverted equilibrium. 3D Pendulum =========== A 3D pendulum is a rigid body supported by a frictionless pivot acting under a gravitational potential. This is a generalization of a planar pendulum or a spherical pendulum, as it has three rotational degrees of freedom. It has been shown that a 3D pendulum may exhibit irregular maneuvers [@ChaLeeJNS11]. We choose a reference frame, and a body-fixed frame. The origin of the body-fixed frame is located at the pivot point. The attitude of a 3D pendulum is the orientation of the body-fixed frame with respect to the reference frame, and it is described by a rotation matrix representing the linear transformation from the body-fixed frame to the reference frame. The configuration manifold of a 3D pendulum is the special orthogonal group, ${\ensuremath{\mathsf{SO(3)}}}=\{R\in{\ensuremath{\mathbb{R}}}^{3\times 3}\,|\, R^T R=I,\mathrm{det}[R]=1\}$. The equations of motion for a 3D pendulum are given by $$\begin{gathered} J\dot\Omega + \Omega\times J\Omega = mg \rho\times R^T e_3 + u,\label{eqn:Wdot2}\\ \dot R = R\hat\Omega,\label{eqn:Rdot}\end{gathered}$$ where the matrix $J\in{\ensuremath{\mathbb{R}}}^{3\times 3}$ is the inertia matrix of the pendulum about the pivot, and $\rho\in{\ensuremath{\mathbb{R}}}^3$ is the vector from the pivot to the center of mass of the pendulum represented in the body-fixed frame. The control moment at the pivot is denoted by $u\in{\ensuremath{\mathbb{R}}}^3$. Control System -------------- Several control systems have been developed on ${\ensuremath{\mathsf{SO(3)}}}$ [@BulLew05; @ChaMcCITAC09; @LeePACC11]. Here, we summarize a control system to stabilize a 3D pendulum to a fixed desired attitude $R_d\in{\ensuremath{\mathsf{SO(3)}}}$. Consider an attitude error function given by $$\begin{aligned} \Psi(R,R_d)=\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ (I-R_d^TR)G \right]}}}},\end{aligned}$$ for a diagonal matrix $G=\mathrm{diag}[g_1,g_2,g_3]\in{\ensuremath{\mathbb{R}}}^{3\times 3}$ with $g_1,g_2,g_3>0$. The derivative of this attitude error function with respect to $R$ along the direction of $\delta R= R\hat\eta$ for $\eta\in{\ensuremath{\mathbb{R}}}^3$ is given by $$\begin{aligned} {\ensuremath{\mathbf{D}}}_R & \Psi(R,R_d)\cdot\delta R = -\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ R_d^TR\hat\eta G \right]}}}} \\ & = \frac{1}{2}(GR_d^TR -R^TR_d G)^\vee \cdot\eta\equiv e_R\cdot \eta,\end{aligned}$$ where we use the property that $\mathrm{tr}[\hat x A]=-x\cdot(A-A^T)^\vee$ for any $x\in{\ensuremath{\mathbb{R}}}^3,A\in{\ensuremath{\mathbb{R}}}^{3\times 3}$. The *vee map*, $\vee:{\ensuremath{\mathfrak{so}(3)}}\rightarrow{\ensuremath{\mathbb{R}}}^3$, denotes the inverse of the hat map. An attitude error vector is defined as $e_R = \frac{1}{2}(GR_d^TR -R^TR_d G)\in{\ensuremath{\mathbb{R}}}^3$. For positive constants $k_\Omega,k_R$, we choose the following control input: $$\begin{aligned} u = -k_R e_R -k_\Omega \Omega -mg \rho\times R^T e_3.\end{aligned}$$ The corresponding closed loop dynamics are given by $$\begin{gathered} J\dot\Omega =- \Omega\times J\Omega -k_R e_R -k_\Omega \Omega,\label{eqn:Wdot2}\\ \dot R = R\hat\Omega.\label{eqn:Rdot}\end{gathered}$$ This system has four equilibria: in addition to the desired equilibrium $(R_d,0)$, there exist three other equilibria at $(R_d\exp (\pi\hat e_i,0),0)$ for $i\in\{1,2,3\}$, which correspond to the rotation of the desired attitude by $180^\circ$ about each body-fixed axis. The existence of additional, undesirable equilibria is due to the nonlinear topological structure of ${\ensuremath{\mathsf{SO(3)}}}$, and it cannot be avoided by constructing a different control system (as long as it is continuous). It has been shown that it is not possible to design a continuous feedback control stabilizing an attitude globally on [$\mathsf{SO(3)}$]{} [@BhaBerSCL00; @KodPICDC98]. The stability of the desired equilibrium can be studied by using the following Lyapunov function, $$\begin{aligned} \mathcal{V} =\frac{1}{2}\Omega\cdot J\Omega + k_R \Psi(R,R_d).\end{aligned}$$ In this paper, we analyze the stability of each equilibrium by linearizing the closed loop dynamics to study the equilibrium structures more explicitly. Linearization ------------- A variation in ${\ensuremath{\mathsf{SO(3)}}}$ can be expressed as  [@LeeLeoPICCA05]: $$\begin{aligned} R^\epsilon=R\exp(\epsilon\hat\eta),\quad \Omega^\epsilon=\Omega +\epsilon\delta\Omega,\label{eqn:delRdelW}\end{aligned}$$ for $\eta,\delta\Omega\in{\ensuremath{\mathbb{R}}}^3$. The corresponding infinitesimal variation of $R$ is given by $\delta R = R\hat\eta$. Substituting this into [(\[eqn:Rdot\])]{}, $$\begin{aligned} R\hat\Omega\hat\eta + R\hat{\dot\eta}= R\hat\eta\hat\Omega + R\delta\hat\Omega.\end{aligned}$$ Using the property $\hat x \hat y -\hat y\hat x=\widehat{x\times y}$ for any $x,y,\in{\ensuremath{\mathbb{R}}}^3$, this can be rewritten as $$\begin{aligned} \dot\eta = \delta\Omega -\hat\Omega\eta.\label{eqn:etadot}\end{aligned}$$ Similarly, by substituting [(\[eqn:delRdelW\])]{} into [(\[eqn:Wdot2\])]{}, we obtain $$\begin{aligned} J\delta\dot\Omega & = -\delta\Omega\times J\Omega -\Omega\times J\delta\Omega\nonumber\\ &\quad -\frac{1}{2}k_R (GR_d^T R\hat\eta +\hat\eta R^T R_d G) -k_\Omega\delta\Omega,\nonumber\\ & = (\widehat{J\Omega}-\hat\Omega J -k_\Omega I) \delta\Omega -\frac{1}{2}k_RH\eta,\label{eqn:delWdot}\end{aligned}$$ where $H=\mathrm{tr}[R^T R_d G]I-R^T R_d G\in{\ensuremath{\mathbb{R}}}^{3\times 3}$, and we used the property, $\hat x A + A^T\hat x = \mathrm{tr}[A]I-A$ for any $x\in{\ensuremath{\mathbb{R}}}^3, A\in{\ensuremath{\mathbb{R}}}^{3\times 3}$. Equations [(\[eqn:etadot\])]{},[(\[eqn:delWdot\])]{} can be written in matrix form as $$\begin{aligned} \dot x & = \begin{bmatrix}\dot\eta \\ \delta\dot\Omega \end{bmatrix} =\begin{bmatrix} -\hat\Omega & I\\ -\frac{1}{2}k_R J^{-1}H & J^{-1}(\widehat{J\Omega}-\hat\Omega J -k_\Omega I) \end{bmatrix} \begin{bmatrix}\eta \\ \delta\Omega \end{bmatrix}\nonumber\\ & = Ax.\label{eqn:xdotSO3}\end{aligned}$$ This corresponds to the linearized equation of motion of [(\[eqn:Wdot2\])]{}, [(\[eqn:Rdot\])]{}. Equilibrium Solutions --------------------- We choose the desired attitude as $R_d=I$. In addition to the desired equilibrium $(I,0)$, there are three additional equilibria, namely $(\exp(\pi\hat e_1),0)$, $(\exp(\pi\hat e_2),0)$, $(\exp(\pi\hat e_3),0)$. We study the eigen-structure of each equilibrium using the linearized equation [(\[eqn:xdotSO3\])]{}. We assume that $$\begin{aligned} J=\mathrm{diag}[3,2,1]\,\mathrm{kgm^2},\; G=\mathrm{diag}[0.9,1,1.1],\; k_R=k_\Omega=1.\end{aligned}$$ ### Equilibrium $(I,0)$ The eigenvalues of the matrix $A$ at the desired equilibrium $(I,0)$ are given by $$\begin{gathered} \lambda_{1,2}=-0.1667\pm0.5676i,\\ \lambda_{3,4}=-0.25\pm 0.6614i,\\ \lambda_{5,6}=-0.5\pm 0.8367i.\end{gathered}$$ This equilibrium is an asymptotically stable focus. ### Equilibrium $(\exp(\pi\hat e_1),0)$ At this equilibrium, the eigenvalues and the eigenvectors of $A$ are given by $$\begin{gathered} \lambda_1=-0.7813,\quad v_1= e_1-0.7813 e_4,\nonumber\\ \lambda_2=-0.5854,\quad v_2=e_2-0.5854e_5,\nonumber\\ \lambda_3=-1.0477,\quad v_3=e_3-1.0477e_6,\label{eqn:v3_SO31}\\ \lambda_4=0.4480,\quad v_4=e_1+0.4480e_4,\nonumber\\ \lambda_5=0.0854,\quad v_5=e_2+0.0854e_5,\nonumber\\ \lambda_6=0.0477,\quad v_6=e_3+0.0477e_6.\nonumber\end{gathered}$$ Therefore, this equilibrium is a saddle point, where three modes are stable, and three modes are unstable. ### Equilibrium $(\exp(\pi\hat e_2),0)$ At this equilibrium, the eigenvalues and the eigenvectors of $A$ are given by $$\begin{gathered} \lambda_1=-0.3775,\quad v_1= e_1-0.3775 e_4,\nonumber\\ \lambda_2=-1,\quad v_2=e_2-e_5,\label{eqn:v2_SO32}\\ \lambda_3=-0.9472,\quad v_3=e_3-0.9472e_6,\nonumber\\ \lambda_4=-0.0528,\quad v_4=e_3-0.0528e_6,\nonumber\\ \lambda_5=0.0442,\quad v_5=e_1+0.0442e_4,\nonumber\\ \lambda_6=0.5,\quad v_6=e_2+5e_5.\nonumber\end{gathered}$$ Therefore, this equilibrium is a saddle point, where four modes are stable, and two modes are unstable. ### Equilibrium $(\exp(\pi\hat e_3),0)$ At this equilibrium, the eigenvalues and the eigenvectors of $A$ are given by $$\begin{gathered} \lambda_1=-0.0613,\quad v_1= e_1-0.0613 e_4,\nonumber\\ \lambda_2=-0.2721,\quad v_2= e_1-0.2721 e_4,\nonumber\\ \lambda_3=-0.1382,\quad v_3= e_2-0.1382 e_5,\nonumber\\ \lambda_4=-0.3618,\quad v_4= e_2-0.3618 e_5,\nonumber\\ \lambda_5=-1.5954,\quad v_5= e_3-1.5954 e_6,\label{eqn:v5_SO33}\\ \lambda_6= 0.5954,\quad v_6= e_2+0.5954 e_6.\nonumber\end{gathered}$$ Therefore, this equilibrium is a saddle point, where five modes are stable, and one mode is unstable. Stable Manifolds for the Saddle Points -------------------------------------- The eigen-structure analysis shows that there exist multi-dimensional stable manifolds for each saddle point. They have zero measure as the dimension of stable manifold is less than the dimension of ${\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{SO(3)}}}$. But, the existence of these stable manifolds may have nontrivial effects on the attitude dynamics. We numerically characterize these stable manifolds using backward time integration, as discussed in Section \[sec:SM\]. The stable eigenspace for each saddle point can be written as $$\begin{aligned} &E^s_{loc} (\exp(\pi\hat e_1),0) = \{ (R,\Omega)\in {\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{SO(3)}}}\,|\,\nonumber\\ &\; R=\exp(\pi\hat e_1)\exp(\alpha_1\hat e_1+\alpha_2\hat e_2+\alpha_3\hat e_3),\\ &\; \Omega=-0.7813\alpha_1e_1-0.5854\alpha_2e_2-1.0477\alpha_3e_3\text{ for $\alpha_i\in{\ensuremath{\mathbb{R}}}$}\},\\ &E^s_{loc} (\exp(\pi\hat e_2),0) = \{ (R,\Omega)\in {\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{SO(3)}}}\,|\,\nonumber\\ &\; R=\exp(\pi\hat e_2)\exp(\alpha_1\hat e_1+\alpha_2\hat e_2+(\alpha_3+\alpha_4)\hat e_3),\\ &\; \Omega=-0.37\alpha_1e_1-\alpha_2e_2-(0.94\alpha_3+0.05\alpha_4)e_3\text{ for $\alpha_i\in{\ensuremath{\mathbb{R}}}$}\},\\ &E^s_{loc} (\exp(\pi\hat e_3),0) = \{ (R,\Omega)\in {\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{SO(3)}}}\,|\,\nonumber\\ &\; R=\exp(\pi\hat e_3)\exp((\alpha_1+\alpha_2)\hat e_1+(\alpha_3+\alpha_4)\hat e_2+\alpha_5\hat e_3),\\ &\; \Omega=-(0.06\alpha_1+0.27\alpha_2)e_1-(0.13\alpha_3+0.36\alpha_4)e_2\\ &\;\quad -1.59\alpha_5e_3\text{ for $\alpha_i\in{\ensuremath{\mathbb{R}}}$}\},\end{aligned}$$ We define a distance on ${\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{SO(3)}}}$ as follows: $$\begin{aligned} d_{{\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{SO(3)}}}} ((R_1,\Omega_1),(R_2,\Omega_2)) = \sqrt{\Psi(R_1,R_2)} + \|\Omega_1-\Omega_2\|.\end{aligned}$$ A variational integrator for the attitude dynamics of a rigid body on ${\ensuremath{\mathsf{SO(3)}}}$ is developed in [@LeeLeoPICCA05; @LeeLeoCMDA07]. It can be rewritten in a backward integration form as follows: $$\begin{gathered} h (\Pi_{k+1}-\frac{h}{2}M_{k+1})^\wedge = J_dF_k - F_k^TJ_d,\label{eqn:Fk}\\ R_k = R_{k+1}F_k^T,\label{eqn:Rk}\\ \Pi_k = F_k\Pi_{k+1}-\frac{h}{2}F_kM_{k+1}-\frac{h}{2} M_k,\label{eqn:Pik}\end{gathered}$$ where $M_{k}=u_k+mg\rho\times R^T e_3\in{\ensuremath{\mathbb{R}}}^3$ is the external moment, $\Pi_k=J\Omega_k\in{\ensuremath{\mathbb{R}}}^3$ is the angular momentum. The matrix $J_d\in{\ensuremath{\mathbb{R}}}^{3\times 3}$ denotes a non-standard inertia matrix given by $J_d = \frac{1}{2}\mathrm{tr}[J]I-J$, and the rotation matrix $F_k\in{\ensuremath{\mathsf{SO(3)}}}$ represent the relative attitude update between two integration time steps. For given $(R_{k+1},\Pi_{k+1})$, we first compute $M_{k+1}$, and solve [(\[eqn:Fk\])]{} for $F_k$. Then, $R_k$ is obtained by [(\[eqn:Rk\])]{}, and $\Pi_k$ is computed by [(\[eqn:Pik\])]{}. This yields a discrete inverse flow map, $\mathcal{F}^{-h}_d(R_{k+1},\Pi_{k+1})\rightarrow(R_{k},\Pi_{k})$. ### Visualization of $W_s(\exp(\pi\hat e_1),0)$ In [@LeeLeoPICDC08], a method to visualize a function or a trajectory on ${\ensuremath{\mathsf{SO(3)}}}$ is proposed. Each column of a rotation matrix represents the direction of a body-fixed axis, and it evolves on ${\ensuremath{\mathsf{S}}}^2$. Therefore, a trajectory on ${\ensuremath{\mathsf{SO(3)}}}$ can be visualized by three curves on a sphere, representing the trajectory of three columns of a rotation matrix. The direction of the angular velocity should be chosen such that the corresponding time-derivative of the rotation matrix is tangent to the curve, and the magnitude of angular velocity can be illustrated by color shading. An example of visualizing a rotation about a single axis is illustrated in [Fig. \[fig:visSO3\_demo\]]{}. We choose 112 points on the surface of $B_\delta\subset E^s_{loc} (\exp(\pi\hat e_1),0)$ with $\delta=10^{-6}$, and each point is integrated backward using [(\[eqn:qk\])]{}, [(\[eqn:wk\])]{} with timestep $h=0.002$. The resulting trajectories are illustrated in [Fig. \[fig:SM1\]]{} for several values of $t$. In each figure, three body-fixed axes of the desired attitude $R_d=[e_1,e_2,e_3]$, and three body-fixed axes of the additional equilibrium attitude $\exp(\pi\hat e_1)=[e_1,-e_2,-e_3]$ are shown. From these computational results, we observe the following characteristics on the stable manifold $W_s(\exp(\pi\hat e_1),0)$: - When $t\leq 15$, the trajectories in $W_s(\exp(\pi\hat e_1),0)$ are close to rotations about the third body-fixed axis $e_3$ to $\exp(\pi\hat e_1)$. This is consistent with the linearized dynamics, where the eigenvalue of the third mode, corresponding to the rotations about $e_3$, has the fastest convergence rate, as seen in [(\[eqn:v3\_SO31\])]{}. - When $t\geq 15$, the first mode representing the rotations about $e_1$ starts to appear, followed by the second mode representing the rotation about $e_2$. This corresponds to the fact that the first mode has a faster convergence rate than the second mode, i.e. $|\lambda_1|>|\lambda_2|$. - As $t$ is increased further, the third body-fixed axis leaves the neighborhood of $-e_3$, and it exhibit the following pattern: {width="0.32\columnwidth"} - The stable manifold $W_s(\exp(\pi\hat e_1),0)$ covers a certain part of ${\ensuremath{\mathsf{SO(3)}}}$, when projected on to it. So, when an initial attitude is chosen such that its third body-fixed axis is sufficiently close to $-e_3$, there possibly exist multiple initial angular velocities such that the corresponding solution converges to $\exp(\pi\hat e_1)$ instead of the desired attitude $R_d=I$. ### Visualization of $W_s(\exp(\pi\hat e_2),0)$ We choose 544 points on the surface of $B_\delta\subset E^s_{loc} (\exp(\pi\hat e_2),0)$ with $\delta=10^{-6}$, and each point is integrated backward using [(\[eqn:qk\])]{}, [(\[eqn:wk\])]{} with timestep $h=0.002$. The resulting trajectories are illustrated in [Fig. \[fig:SM2\]]{} for several values of $t$. In each figure, three body-fixed axes of the desired attitude $R_d=[e_1,e_2,e_3]$, and three body-fixed axes of the additional equilibrium attitude $\exp(\pi\hat e_2)=[-e_1,e_2,-e_3]$ are shown. From these computational results, we observe the following characteristics on the stable manifold $W_s(\exp(\pi\hat e_2),0)$: - When $t\leq 12$, the trajectories in $W_s(\exp(\pi\hat e_2),0)$ is close to the rotations about the second body-fixed axis $e_2$. As $t$ increases, rotations about $e_3$ starts to appear. This corresponds to the linearized dynamics where the second mode representing rotations about $e_2$ has the fastest convergence rate, followed by the third mode at [(\[eqn:v2\_SO32\])]{}. - As $t$ is increased further, nonlinear modes become dominant. The trajectories in $W_s(\exp(\pi\hat e_2),0)$ almost cover [$\mathsf{SO(3)}$]{}. This suggests that for any initial attitude, we can choose several initial angular velocities such that the corresponding solutions converges to $\exp(\pi\hat e_2)$. ### Visualization of $W_s(\exp(\pi\hat e_3),0)$ Similarly, we choose 976 points on the surface of $B_\delta\subset E^s_{loc} (\exp(\pi\hat e_3),0)$ with $\delta=10^{-6}$, and each point is integrated backward using [(\[eqn:qk\])]{}, [(\[eqn:wk\])]{} with timestep $h=0.002$. The resulting trajectories are illustrated in [Fig. \[fig:SM3\]]{} for several values of $t$. At each figure, three body-fixed axes of the desired attitude $R_d=[e_1,e_2,e_3]$, and three body-fixed axes of the additional equilibrium attitude $\exp(\pi\hat e_3)=[-e_1,-e_2,e_3]$ are shown. From these computational results, we observe the following characteristics on the stable manifold $W_s(\exp(\pi\hat e_3),0)$: - When $t\leq 8$, the trajectories in $W_s(\exp(\pi\hat e_3),0)$ are close to the rotations about the third body-fixed axis $e_3$. This corresponds to the linearized dynamics where the fifth mode representing rotations about $e_3$ has the fastest convergence rate given in [(\[eqn:v5\_SO33\])]{}. - The rotations about $e_3$ are still dominant, even as $t$ is increased further. For the given simulation times, all trajectories in $W_s(\exp(\pi\hat e_3),0)$ are close to rotations about $e_3$. Conclusions =========== Stable manifolds of saddle points that arise in the closed-loop dynamics of two pendulum models are characterized numerically, and several properties are observed. Although the analytical and computational results have been presented for a spherical pendulum and a 3D pendulum, the methods presented naturally extend to any closed loop attitude control system with configurations in either ${\ensuremath{\mathsf{S}}}^2$ or ${\ensuremath{\mathsf{SO(3)}}}$. [^1]: Taeyoung Lee, Mechanical and Aerospace Engineering, Florida Institute of Technology, Melbourne, FL 39201 [taeyoung@fit.edu]{} [^2]: Melvin Leok, Mathematics, University of California at San Diego, La Jolla, CA 92093 [mleok@math.ucsd.edu]{} [^3]: N. Harris McClamroch, Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109 [nhm@umich.edu]{} [^4]: ^$*$^This research has been supported in part by NSF under grants CMMI-1029551. [^5]: ^$\dagger$^This research has been supported in part by NSF under grants DMS-0726263, DMS-1001521, DMS-1010687, and CMMI-1029445.

--- abstract: | Orthogonal Frequency Division Multiple Access (OFDMA) is a multi-user version of the Orthogonal Frequency Division Multiplexing (OFDM) transmission technique, which divides a wideband channel into a number of orthogonal narrowband subchannels, called subcarriers. An OFDMA system takes advantage of both *frequency diversity* (FD) gain and *frequency-selective scheduling* (FSS) gain. A FD gain is achieved by allocating a user the subcarriers distributed over the entire frequency band whereas a FSS gain is achieved by allocating a user adjacent subcarriers located within a subband of a small bandwidth having the most favorable channel conditions among other subbands in the entire frequency band. Multi-User Multiple Input Multiple Output (MU-MIMO) is a promising technology to increase spectral efficiency. A well-known MU-MIMO mode is Space-Division Multiple Access (SDMA) which can be used in the downlink direction to allow a group of spatially separable users to share the same time/frequency resources. In this paper, we study the gain from FSS in SDMA-OFDMA systems using the example of WiMAX. Therefore, a complete SDMA-OFDMA MAC scheduling solution supporting both FD and FSS is proposed. The proposed solution is analyzed in a typical urban macro-cell scenario by means of system-level packet-based simulations, with detailed MAC and physical layer abstractions. By explicitly simulating the MAC layer overhead (MAP) which is required to signal every packed data burst in the OFDMA frame we can present the overall performance to be expected at the MAC layer. Our results show that in general the gain from FSS when applying SDMA is low. However, under specific conditions, small number of BS antennas or large channel bandwidth, a significant gain can be achieved from FSS. author: - bibliography: - 'IEEEabrv.bib' - 'literature.bib' title: 'On Frequency-Selective Scheduling in SDMA-OFDMA Systems' --- =5 OFDMA, SDMA, frequency-selective scheduling, frequency diversity gain, WiMAX, 802.16

--- abstract: 'We show that if $A$ is a finite $K$-approximate subgroup of an $s$-step nilpotent group then there is a finite normal subgroup $H\subset A^{K^{O_s(1)}}$ modulo which $A^{O_s(\log^{O_s(1)}K)}$ contains a nilprogression of rank at most $O_s(\log^{O_s(1)}K)$ and size at least $\exp(-O_s(\log^{O_s(1)}K))|A|$. This partially generalises the close-to-optimal bounds obtained in the abelian case by Sanders, and improves the bounds and simplifies the exposition of an earlier result of the author. Combined with results of Breuillard–Green, Breuillard–Green–Tao, Gill–Helfgott–Pyber–Szabó, and the author, this leads to improved rank bounds in Freiman-type theorems in residually nilpotent groups and certain linear groups of bounded degree.' address: 'Pembroke College, Cambridge, CB2 1RF, United Kingdom' author: - 'Matthew C. H. Tointon' title: Polylogarithmic bounds in the nilpotent Freiman theorem --- Introduction ============ This paper concerns *sets of small doubling* and *approximate groups* in non-abelian groups. This topic has been extensively covered in the recent mathematical literature; the reader may consult the author’s forthcoming book [@book] or the surveys [@bgt.survey; @app.grps; @ben.icm; @helf.survey; @sand.survey; @raconte-moi] for detailed background to the topic and examples of some of its many applications. Given sets $A$ and $B$ in a group $G$ we define the *product set* $AB$ by $AB=\{ab:a\in A,b\in B\}$, and define $A^n$ recursively for $n\in{\mathbb{N}}$ by setting $A^1=A$ and $A^{n+1}=A^nA$. We also write $A^{-1}=\{a^{-1}:a\in A\}$ and $A^{-n}=(A^{-1})^n$. If $G$ is abelian we often use additive notation instead, for example writing $A+B$ or $nA$ in place of $AB$ or $A^n$, respectively. By the *doubing* of a finite set $A$ we mean the ratio $|A^2|/|A|$, and when we say that a set has ‘small’ or ‘bounded’ doubling we mean that there is some constant $K\ge1$ such that $|A^2|\le K|A|$. Of course, this always holds for $K=|A|$, so $K$ should be thought of as being substantially smaller than $|A|$ in order for this to be meaningful. One of the central aims in the theory of sets of small doubling is to describe the algebraic structure of such sets. The first result in this direction was Freiman’s theorem [@freiman], which describes sets of small doubling in terms of objects called *progressions*. Given elements $x_1,\ldots,x_r$ in an abelian group $G$ and reals $L_1,\ldots,L_r\ge0$, the *progression* $P(x;L)$ is defined via $P(x;L)=\{\ell_1x_1+\cdots+\ell_rx_r:|\ell_i|\le L_i\}$. Freiman’s theorem states that if a subset $A\subset{\mathbb{Z}}$ satisfies $|A+A|\le K|A|$ then there exists a progression $P$ of rank at most $r(K)$ and size at most $h(K)|A|$ such that $A\subset P$. This was subsequently generalised to an arbitrary abelian group by Green and Ruzsa [@green-ruzsa], where one must replace the progression with a *coset progression*, which simply means a set of the form $H+P$, with $H$ a finite subgroup and $P$ a progression. The best bounds currently available in this theorem are due to Sanders (although Schoen [@schoen] had previously obtained similar bounds in the special case of subsets of integers). Sanders’s main result is the following variant of Freiman’s theorem; in it and elsewhere we write $\log^mK$ to mean $(\log K)^m$. \[thm:sanders\] Let $A$ be a finite subset of an abelian group such that $|A+A|\le K|A|$. Then there exists a coset progression $H+P$ of rank at most $O(\log^{O(1)}2K)$ such that $H+P\subset 2A-2A$ and $|H+P|\ge\exp(-O(\log^{O(1)}2K))|A|$. Combining \[thm:sanders\] with the so-called *covering argument* of Chang [@chang]—which we present in \[lem:chang\], below—one obtains the following bounds in Freiman’s theorem. \[cor:sanders.ext\] Let $A$ be a finite subset of an abelian group such that $|A+A|\le K|A|$. Then there exists a coset progression $H+P$ of rank at most $O(K\log^{O(1)}2K)$ satisfying $|H+P|\le\exp(O(K\log^{O(1)}2K))|A|$ such that $A\subset H+P$. These bounds are close to best possible, as can be seen by considering, for example, an appropriate union of $K$ translates of a finite subgroup or a rank-$1$ progression. It is worth remarking that using a simpler covering argument due to Ruzsa [@ruzsa], on which Chang’s argument is based, one can also deduce the following variant of Theorem \[thm:sanders\]; we give details in Section \[sec:prelim\]. \[cor:sanders\] Let $A$ be a finite subset of an abelian group such that $|A+A|\le K|A|$. Then there exists a coset progression $H+P\subset 4A-4A$ of rank at most $O(\log^{O(1)}2K)$ satisfying $|H+P|\le K^8|A|$, and a set $X\subset A$ of size at most $\exp(O(\log^{O(1)}2K))$ such that $A\subset X+H+P$. In this paper we are concerned with generalisations of these results to non-abelian groups, and specifically to nilpotent groups. The basic properties of nilpotent groups that we use can be found in [@hall Chapter 10] or [@book §5.2]. In the non-abelian setting it is usual for technical reasons to replace the small-doubling assumption $|A+A|\le K|A|$ with a slightly stronger assumption. This is usually either a ‘small-tripling’ assumption $|A^3|\le K|A|$, or the qualitativey even stronger assumtion that $A$ is a *$K$-approximate group*. Given $K\ge1$, a subset $A$ of a group $G$ is said to be a *$K$-approximate subgroup of $G$*, or simply a *$K$-approximate group*, if $A^{-1}=A$ and $1\in A$, and if there exists $X\subset G$ with $|X|\le K$ such that $A^2\subset XA$. The reasons for making these stronger assumptions are explained at length in [@tao.product.set; @book], but let us highlight the fact that a set $A$ satisfying $|A^2|\le K|A|$ is contained in the union of a few translates of a relatively small $O(K^{O(1)})$-approximate group [@tao.product.set Theorem 4.6], so there is no great loss of generality in doing so. Note, conversely, that if $A$ is a finite $K$-approximate group then $|A^m|\le K^{m-1}|A|$ for every $m\in{\mathbb{N}}$, a fact we will use on a number of occasions without further mention. There are a number of ways to formulate the appropriate generalisation of a coset progression to non-abelian groups. The easiest to define is probably a *coset nilprogression*. Given elements $x_1,\ldots,x_r$ in a group $G$ and $L_1,\ldots,L_r\ge0$, the *nonabelian progression* $P(x;L)$ is defined to consist of all those elements of $G$ that can be expressed as words in the $x_i$ and their inverses in which each $x_i$ and its inverse appear at most $L_i$ times between them. We define $r$ to be the *rank* of $P(x;L)$. If the $x_i$ generate an $s$-step nilpotent group then $P(x;L)$ is said to be a *nilprogression* of step $s$, and in this instance we write $P_{\text{\textup{nil}}}(x;L)$ instead of $P(x;L)$. A set $P$ is said to be a *coset nilprogression* of rank $r$ and step $s$ if there exists a finite subgroup $H\subset P$, normalised by $P$, such that the image of $P$ in $\langle P\rangle/H$ is a nilprogression of rank $r$ and step $s$. Another useful formulation is a closely related object called a *nilpotent progression*. Again, a nilpotent progression $\overline P(x;L)$ is defined using elements $x_1,\ldots,x_r$ in a nilpotent group $G$ and reals $L_1,\ldots,L_r\ge0$, but its definition is a little more involved than that of a nilprogression, so we refer the reader to any of [@bg; @nilp.frei; @book]. Nilpotent progressions have tripling bounded in terms of their rank and step, as do nilprogressions if the reals $L_1$ are large enough [@bt Corollary 3.16]. For technical reasons, it is also convenient to define a third type of progression in a non-abelian group, although in general this one will not have bounded doubling. Given $x_i$ and $L_i$ as above, the *ordered progression* $P_{\text{\textup{ord}}}(x;L)$ is defined to be $P_{\text{\textup{ord}}}(x;L)=\{x_1^{\ell_1}\cdots x_r^{\ell_r}:|\ell_i|\le L_i\}$. The following result shows that it does not matter too much which of the above versions of progression we use. \[prop:nilprog.equiv\] Let $G$ be an $s$-step nilpotent group, let $x_1,\ldots,x_r\in G$, and let $L_1,\ldots,L_r\in{\mathbb{N}}$. Then $P_{\text{\textup{ord}}}(x;L)\subset P_{\text{\textup{nil}}}(x;L)\subset\overline P(x;L)\subset P_{\text{\textup{ord}}}(x;L)^{(96s)^{s^2}r^s}$. The bounds we state here are written more explicitly than in [@nilp.frei Proposition C.1], but bounds of the type we claim here can easily be read out of the argument there. \[prop:nilprog.equiv\] is also proved exactly as stated above in [@book Proposition 5.6.4]. The author has previously extended \[cor:sanders.ext\] to nilpotent groups, proving the following result. \[thm:old\] Let $s\in{\mathbb{N}}$ and $K\ge1$. Let $G$ be an $s$-step nilpotent group $s$, and suppose that $A\subset G$ is a finite $K$-approximate group. Then there exist a subgroup $H$ of $G$ normalised by $A$ and a nilprogression $P_{\text{\textup{nil}}}(x;L)$ of rank at most $K^{O_s(1)}$ such that $$A\subset HP_{\text{\textup{nil}}}(x;L)\subset H\overline P(x;L)\subset A^{K^{O_s(1)}}.$$ In particular, $|H\overline P(x;L)|\le\exp(K^{O_s(1)})|A|$. The aim of the present paper is to show that, like in the abelian case, if we ask for $HP$ to be dense in $A$, rather than the other way around, we can replace most of the polynomial bounds of \[thm:old\] with polylogarithmic bounds, as follows. \[thm:new.gen\] Let $s\in{\mathbb{N}}$ and $K\ge1$. Let $G$ be an $s$-step nilpotent group, and suppose that $A\subset G$ is a finite $K$-approximate group. Then there exist a subgroup $H\subset A^{K^{e^{O(s)}}}$ normalised by $A$ and an ordered progression $P_{\text{\textup{ord}}}(x;L)\subset A^{e^{O(s^2)}\log^{O(s)}2K}$ of rank at most $e^{O(s^2)}\log^{O(s)}2K$ such that $$P_{\text{\textup{ord}}}(x;L)\subset P_{\text{\textup{nil}}}(x;L)\subset\overline P(x;L)\subset A^{e^{O(s^3)}\log^{O(s^2)}2K}$$ and $$|HP_{\text{\textup{ord}}}(x;L)|\ge\exp\left(-e^{O(s^2)}\log^{O(s)}2K\right)|AH|.$$ The proof of \[thm:old\] proceeds by an induction on the step $s$, in which \[thm:sanders\] features both in the base case $s=1$ and in the proof of the inductive step. The original proof used an earlier version of \[thm:sanders\], due to Green and Ruzsa, in which the bounds are polynomial rather than polylogarithmic. Let us emphasise, though, that losses elsewhere in the argument overwhelmed the bounds of \[thm:sanders\] to the extent that it made no difference to the shape of the final bounds to use the Green–Ruzsa result instead. In particular, proving \[thm:new.gen\] is not merely a case of substituting \[thm:sanders\] for the Green–Ruzsa result in the original proof: we also need to make the rest of the argument more efficient. The one bound that is still polynomial in \[thm:new.gen\] is the bound $H\subset A^{K^{e^{O(s)}}}$; it appears that a new idea would be required to improve this any further (see \[rem:poly.bound\], below, for further details). Note, though, that in the case where the ambient group has no torsion the subgroup $H$ is automatically trivial, leaving only the polylogarithmic bounds, as follows. \[thm:new.tf\] Let $s\in{\mathbb{N}}$ and $K\ge1$. Let $G$ be a torsion-free $s$-step nilpotent group, and suppose that $A\subset G$ is a finite $K$-approximate group. Then there exist an ordered progression $P_{\text{\textup{ord}}}(x;L)\subset A^{e^{O(s^2)}\log^{O(s)}2K}$ of rank at most $e^{O(s^2)}\log^{O(s)}2K$ such that $$P_{\text{\textup{ord}}}(x;L)\subset P_{\text{\textup{nil}}}(x;L)\subset\overline P(x;L)\subset A^{e^{O(s^3)}\log^{O(s^2)}2K}$$ and $$|P_{\text{\textup{ord}}}(x;L)|\ge\exp\left(-e^{O(s^2)}\log^{O(s)}2K\right)|A|.$$ As in the abelian case, Ruzsa’s covering argument combines with \[thm:new.gen\] to give the following variant. \[cor:ruzsa\] Let $s\in{\mathbb{N}}$ and $K\ge1$. Let $G$ be an $s$-step nilpotent group, and suppose that $A\subset G$ is a finite $K$-approximate group. Then there exist a subgroup $H\subset A^{K^{e^{O(s)}}}$ normalised by $A$, a nilprogression $P_{\text{\textup{nil}}}(x;L)$ of rank at most $e^{O(s^2)}\log^{O(s)}2K$ such that $$P_{\text{\textup{nil}}}(x;L)\subset\overline P(x;L)\subset A^{e^{O(s^3)}\log^{O(s^2)}2K},$$ and a subset $X\subset G$ of size at most $\exp(e^{O(s^2)}\log^{O(s)}2K)$ such that $A\subset XHP_{\text{\textup{nil}}}(x;L)$. In particular, $|H\overline P(x;L)|\le\exp(K^{e^{O(s)}})|A|$. In the torsion-free setting the subgroup $H$ is again trivial, and in that case we may conclude instead that $|\overline P(x;L)|\le\exp(e^{O(s^3)}\log^{O(s^2)}2K)|A|$. Chang’s covering argument also allows us to recover \[thm:old\] with much more precise bounds, as follows. \[cor:chang.ag\] Let $s\in{\mathbb{N}}$ and $K\ge1$. Let $G$ be an $s$-step nilpotent group, and suppose that $A\subset G$ is a finite $K$-approximate group. Then there exist a subgroup $H\subset A^{K^{e^{O(s)}}}$ normalised by $A$ and a nilprogression $P_{\text{\textup{nil}}}(x;L)$ of rank at most $e^{O(s^2)}K\log^{O(s)}2K$ such that $$A\subset HP_{\text{\textup{nil}}}(x;L)\subset H\overline P(x;L)\subset HA^{e^{O(s^3)}K^{s+1}\log^{O(s^2)}2K}.$$ In particular, $|H\overline P(x;L)|\le\exp(K^{e^{O(s)}})|A|$, or $|P(x;L)|\le\exp(e^{O(s^3)}K^{s+1}\log^{O(s^2)}2K)|A|$ in the torsion-free setting. We deduce these corollaries in \[sec:covering\]. <span style="font-variant:small-caps;">Applications to other groups.</span> A theorem of Breuillard, Green and Tao [@bgt] states, in one form, that an arbitrary finite $K$-approximate group $A$ is contained in a union of at most $O_K(1)$ translates of a coset nilprogression of rank and step $O(K^2\log K)$ and size at most $K^{11}|A|$. This result is powerful enough to have some quite general applications, such as those contained in [@bgt §11] and [@bt; @tao.growth; @tt], but its usefulness is slightly lessened by the fact that it does not give an explicit bound on the number of translates needed to contain $A$. Partly for this reason, various papers by several different authors have given explicit versions of this theorem for certain specific classes of groups. The approach taken in these results is generally first to reduce to the nilpotent case, and then to apply \[thm:old\] (or an earlier result of Breuillard and Green [@bg] valid only in the torsion-free setting) to obtain the nilprogression. Unsurprisingly, using \[thm:new.gen\] or one of its corollaries in place of \[thm:old\] in these arguments leads to better bounds in a number of cases. In \[sec:non-nilp\] we present such better bounds for linear groups over ${\mathbb{F}}_p$ or fields of characteristic zero, and in residually nilpotent groups. <span style="font-variant:small-caps;">Acknowledgement.</span> I was prompted to revisit the bounds in \[thm:old\] by a question from Harald Helfgott. Standard tools {#sec:prelim} ============== In this section we record various standard results relating to sets of small doubling and approximate groups. This material is likely to be familiar to experts in the subject, who may therefore decide to skip straight to \[sec:details\]. \[lem:slicing\] Let $K,L\ge1$ and let $G$ be a group. Let $A\subset G$ be a $K$-approximate group and $B\subset G$ an $L$-approximate group. Then for every $m,n\ge2$ the set $A^m\cap B^n$ is covered by at most $K^{m-1}L^{n-1}$ left translates of $A^2\cap B^2$. In particular, $A^m\cap B^n$ is a $K^{2m-1}L^{2n-1}$-approximate group. \[lem:fibre.pigeonhole\] Let $k\in{\mathbb{N}}$. Let $G$ be a group with a subgroup $H$, let $A\subset G$, and suppose that $A$ is contained in a union of $k$ left cosets of $H$. Then $A$ is contained in a union of $k$ left translates of $A^{-1}A\cap H$. Let $x_1,\ldots,x_m\in A$ be representatives of the left cosets of $H$ containing at least one element of $A$, noting that $m\le k$ by hypothesis. If $a$ is an arbitrary element of $A\cap x_iH$ then there exists $h\in H$ such that $a=x_ih$. It follows that $h=x_i^{-1}a\in A^{-1}A$, and hence $h\in A^{-1}A\cap H$ and $a\in x_i(A^{-1}A\cap H)$. \[thm:plun\] Let $G$ be an abelian group, and let $A$ be a finite subset of $G$. Suppose that $|A+A|\le K|A|$. Then $|mA-nA|\le K^{m+n}|A|$ for all non-negative integers $m,n$. \[lem:covering\] Let $A$ and $B$ be finite subsets of some group and suppose that $|AB|/|B|\le K$. Then there exists a subset $X\subset A$ with $|X|\le K$ such that $A\subset XBB^{-1}$. Let $H$ and $P$ be as given by \[thm:sanders\]. Then we have $$\begin{aligned} \frac{|A+H+P|}{|H+P|}&\le\exp(O(\log^{O(1)}2K))\frac{|A+H+P|}{|A|}\\ &\le K^5\exp(O(\log^{O(1)}2K))&\text{(by \cref{thm:plun})}\\ &\le\exp(O(\log^{O(1)}2K)),\end{aligned}$$ and so \[lem:covering\] gives a set $X\subset A$ of size at most $\exp(O(\log^{O(1)}2K))$ such that $A\subset X+H+2P$. Now $2P$ is also a progression of the same rank as $P$. Moreover, since $H+P\subset 2A-2A$, we have $H+2P\subset 4A-4A$, and hence $|H+2P|\le K^8|A|$ by \[thm:plun\]. This completes the proof. \[lem:chang\] Let $K,C\ge1$ and $m\in{\mathbb{N}}$. Let $G$ be a group, and suppose that $A\subset G$ is a finite $K$-approximate group. Suppose that $B\subset A^m$ is a set with $|B|\ge|A|/C$. Then there exist $t\ll\log C+m\log K$ and sets $S_1,\ldots,S_t\subset A$ satisfying $|S_i|\le2K$ such that $A\subset S_{t-1}^{-1}\cdots S_1^{-1}B^{-1}BS_1\cdots S_t$. Let $k\in{\mathbb{N}}$, let $G$ be a group, and let $A$ be a subset of a group. Then a map $\varphi:A\to G$ is a *Freiman $k$-homomorphism*, or simply a *$k$-homomorphism*, if whenever $x_1,\ldots,x_k,y_1,\ldots,y_k\in A$ satisfy $$x_1\cdots x_k=y_1\cdots y_k$$ we have $${\varphi}(x_1)\cdots{\varphi}(x_k)={\varphi}(y_1)\cdots{\varphi}(y_k).$$ If $1\in A$ and ${\varphi}(1)=1$ then we say that ${\varphi}$ is *centred*. \[lem:fr.hom.ap.grp\] Let $K\ge1$. Let $A$ be a $K$-approximate group, let $G$ be a group. Suppose that ${\varphi}:A\to G$ is a centred Freiman $3$-homomorphism. Then ${\varphi}(A)$ is a $K$-approximate group. The set ${\varphi}(A)$ contains the identity by definition of a centred Freiman homomorphism. Moreover, for every $a\in A$ we have ${\varphi}(a^{-1}){\varphi}(a){\varphi}(1)={\varphi}(1)^3$, and hence $$\label{eq:centred} {\varphi}(a^{-1})={\varphi}(a)^{-1},$$ so ${\varphi}(A)$ is symmetric. Finally, by definition there is a set $X$ of size at most $K$ such that $A^2\subset XA$. We may assume that $X$ is minimal satisfying this property, and hence that $X\subset A^3$. For each $x\in X$ there therefore exist elements $\alpha_1(x),\alpha_2(x),\alpha_3(x)\in A$ such that $x=\alpha_1(x)\alpha_2(x)\alpha_3(x)$. Set $Y=\{{\varphi}(\alpha_1(x)){\varphi}(\alpha_2(x)){\varphi}(\alpha_3(x)):x\in X\}$, noting that $|Y|\le K$. We claim that ${\varphi}(A)^2\subset Y{\varphi}(A)$, which will complete the proof. To prove this claim, fix $a_1,a_2\in A$, and let $x\in X$ and $a_3\in A$ be such that $a_1a_2=xa_3$. It follows from that ${\varphi}(a_1){\varphi}(a_2){\varphi}(a_3)^{-1}={\varphi}(\alpha_1(x)){\varphi}(\alpha_2(x)){\varphi}(\alpha_3(x))$, and so $a_1a_2\in Y{\varphi}(A)$ as claimed. Proof of the main result {#sec:details} ======================== Before we prove \[thm:new.gen\], let us remark that at various points we make the seemingly unnecessary assumption that $K\ge2$. The reason for this is purely notational: it allows us to replace bounds such as $O(\log^{O(1)}2K)$ or $O(K^{O(1)})$ with the slightly more succinct $\log^{O(1)}2K$ or $K^{O(1)}$, respectively. Note that if $K<2$ then a $K$-approximate group is an actual subgroup, in which regime all of our main results become trivial, so we lose nothing in making this assumption. We start the proof of \[thm:new.gen\] with the following result, a version of which with worse bounds was also central to the original proof of \[thm:old\]. \[prop:key.general\] Let $m>0$ and $s\ge\tilde s\ge2$ be integers, and let $K,\tilde K\ge2$. Let $G$ be an $s$-step nilpotent group generated by a finite $K$-approximate group, and let $\tilde A\subset A^m$ be a $\tilde K$-approximate group that generates an $\tilde s$-step nilpotent subgroup $\tilde G$ of $G$. Then there exist a normal subgroup $N\lhd G$ with $N\subset A^{K^{e^{O(s)}m}}$, an integer $r\le\log^{O(1)}2\tilde K$, and $\tilde K^{O(1)}$-approximate groups $A_0,\ldots,A_r\subset\tilde A^{O(1)}$ such that $$|A_0\cdots A_r|\ge\frac{|\tilde A|}{\exp(\log^{O(1)}2\tilde K)},$$ and such that, writing $\rho:G\to G/N$ for the quotient homomorphism, each group $\langle\rho(A_i)\rangle$ has step less than $\tilde s$. The main ingredients in the proof of \[prop:key.general\] are the next three results. \[prop:pre-chang.tor.free\] Let $s\ge2$ and $K\ge2$. Let $G$ be an $s$-step nilpotent group, and write $\pi:G\to G/[G,G]$ for the quotient homomorphism. Suppose that $A\subset G$ is a finite $K$-approximate group. Then there exist an integer $r\le\log^{O(1)}2K$, elements $x_1,\ldots,x_r\in\pi(A^4)$, and a subgroup $H\subset\pi(A^4)$ such that $$\left|\left(A^{18}\cap\pi^{-1}(H)\right)\prod_{i=1}^r\left(A^{24}\cap\pi^{-1}(\langle x_i\rangle)\right)\right|\ge\frac{|A|}{\exp(\log^{O(1)}2K)}.$$ We prove \[prop:pre-chang.tor.free\] shortly. \[lem:x\[G,G\]\] Let $s\ge2$. Let $G$ be an $s$-step nilpotent group, write $\pi:G\to G/[G,G]$ for the quotient homomorphism, and let $x\in G/[G,G]$. Then the group $\pi^{-1}(\langle x\rangle)$ has step at most $s-1$. This is implicitly shown in the proofs of [@nilp.frei Propositions 4.2 & 4.3]; it is also proved explicitly in [@book Lemma 6.1.6 (i)]. \[prop:lower.step.in.quotient\] Let $m>0$ and $s\ge\tilde s\ge2$ be integers, and let $K\ge2$. Let $G$ be an $s$-step nilpotent group generated by a finite $K$-approximate group $A$, and let $\tilde G$ be an $\tilde s$-step nilpotent subgroup of $G$. Write $\pi:\tilde G\to\tilde G/[\tilde G,\tilde G]$ for the quotient homomorphism, and suppose that $H\subset\pi(A^m\cap\tilde G)$ is a finite group. Then there is a normal subgroup $N\lhd G$ such that $[\,\pi^{-1}(H),\ldots,\pi^{-1}(H)\,]_{\tilde s}\subset N\subset A^{K^{e^{O(s)}m}}$. The bounds stated here are more precise than those stated in [@nilp.frei Proposition 7.1], but the bounds claimed here can be read out of the argument there. Alternatively, \[prop:lower.step.in.quotient\] is proved exactly as stated here in [@book Proposition 6.6.2]. Combine Proposition \[prop:pre-chang.tor.free\] with Lemmas \[lem:slicing\] and \[lem:x\[G,G\]\] and \[prop:lower.step.in.quotient\]. Before we prove \[prop:pre-chang.tor.free\] we isolate the following lemma, which is inspired by a lemma of Tao [@tao.product.set Lemma 7.7]. \[lem:splitting\] Let $G$ be a group, let $N\lhd G$ be a normal subgroup, and let $\pi:G\to G/N$ be the quotient homomorphism. Let $A$ be a symmetric subset of $G$, and define a map $\varphi:\pi(A)\to A$ by choosing, for each element $x\in\pi(A)$, an element $\varphi(x)\in A$ such that $\pi(\varphi(x))=x$. Then 1. \[eq:split.inv\] for every $a\in A$ we have $a\in\left(A^2\cap N\right)\varphi(\pi(a))$; and 2. \[eq:split.hom\] for every $x,y\in G/N$ with $x,y,xy\in\pi(A)$ we have $\varphi(xy)\in\varphi(x)\varphi(y)\left(A^3\cap N\right)$. This is essentially just an observation: by definition of $\varphi$ we have $a\varphi(\pi(a))^{-1}\in A^2\cap N$ and $\varphi(y)^{-1}\varphi(x)^{-1}\varphi(xy)\in A^3\cap N$. \[lem:pullback.large\] Let $G$ be a group, let $N\lhd G$ be a normal subgroup, and let $\pi:G\to G/N$ be the quotient homomorphism. Let $A$ be a finite symmetric subset of $G$, and let $P\subset\pi(A^m)$. Suppose that $|P|\ge c|\pi(A)|$. Then $|\pi^{-1}(P)\cap A^{m+2}|\ge c|A|$. \[lem:fibre.pigeonhole\] implies that $|N\cap A^2|\ge|A|/|\pi(A)|$, which in turn implies that $|\pi^{-1}(x)\cap A^{m+2}|\ge|A|/|\pi(A)|$ for every $x\in\pi(A^m)$. In particular, $|\pi^{-1}(P)\cap A^{m+2}|\ge|A||P|/|\pi(A)|\ge c|A|$, as desired. Write $\pi:G\to G/[G,G]$ for the quotient homomorphism, and note that $\pi(A)$ is a finite $K$-approximate subgroup of the abelian group $G/[G,G]$. Theorem \[thm:sanders\] therefore implies that there exists a finite subgroup $H\subset G/[G,G]$, and a progression $P=\{x_1^{\ell_1}\cdots x_r^{\ell_r}:|\ell_i|\le L_i\}$ with $r\le\log^{O(1)}2K$ such that $HP\subset\pi(A^4)$ and $|HP|\ge\exp(-\log^{O(1)}2K)|\pi(A)|$. Lemma \[lem:pullback.large\] then implies that $$\label{eq:pullback.large} |\pi^{-1}(HP)\cap A^6|\ge\exp(-\log^{O(1)}2K)|A|.$$ Now let $\varphi:\pi(A^6)\to A^6$ be an arbitrary map such that $\pi(\varphi(x))=x$ for every $x\in\pi(A^6)$. Suppose that $a\in\pi^{-1}(HP)\cap A^6$, so that there exist $h\in H$ and $\ell_1,\ldots,\ell_r\in{\mathbb{Z}}$ such that $\pi(a)=hx_1^{\ell_1}\cdots x_r^{\ell_r}$. It follows from \[lem:splitting\] \[eq:split.inv\] that $$\begin{aligned} a&\in\left(A^{12}\cap[G,G]\right)\varphi(\pi(a))\\ &=\left(A^{12}\cap[G,G]\right)\varphi(hx_1^{\ell_1}\cdots x_r^{\ell_r}),\end{aligned}$$ and hence by repeated application of \[lem:splitting\] \[eq:split.hom\] that $$\begin{aligned} a&\in\left(A^{12}\cap[G,G]\right)\varphi(h)\prod_{i=1}^r\varphi(x_i^{\ell_i})\left(A^{18}\cap[G,G]\right)\\ &\subset\left(A^{18}\cap\pi^{-1}(H)\right)\prod_{i=1}^r\left(A^{24}\cap\pi^{-1}(\langle x_i\rangle)\right).\end{aligned}$$ Since $a$ was an arbitrary element of $\pi^{-1}(HP)\cap A^6$, the proposition then follows from . It is at this point that we diverge from the original proof of \[thm:old\]. \[prop:ind.tor-free.post.chang\] Let $m>0$ and $s\ge\tilde s\ge2$ be integers, and let $K,\tilde K\ge2$. Let $G$ be an $s$-step nilpotent group generated by a finite $K$-approximate group $A$, and let $\tilde A\subset A^m$ be a $\tilde K$-approximate group that generates an $\tilde s$-step nilpotent subgroup $\tilde G$ of $G$. Then there exist a normal subgroup $N\lhd G$ with $N\subset A^{K^{e^{O(s)}m}}$; an integer $r\le\log^{O(1)}2\tilde K$; finite $\tilde K^{O(1)}$-approximate groups $A_1,\ldots,A_r\subset\tilde A^{O(1)}$ such that, writing $\rho:G\to G/N$ for the quotient homomorphism, each group $\langle\rho(A_i)\rangle$ has step less than $\tilde s$; and a set $X\subset\tilde A$ of size at most $\exp(\log^{O(1)}2\tilde K)$ such that $\tilde A\subset XA_1\cdots A_r$. This is immediate from \[prop:key.general,lem:covering\]. Using \[prop:ind.tor-free.post.chang\] to induct on the step, we arrive at the following result. \[prop:tor-free.post.induc\] Let $m>0$ and $s\ge\tilde s\ge1$ be integers, and let $K,\tilde K\ge2$. Let $G$ be an $s$-step nilpotent group generated by a finite $K$-approximate group $A$, and let $\tilde A\subset A^m$ be a $\tilde K$-approximate group that generates an $\tilde s$-step nilpotent subgroup $\tilde G$ of $G$. Then there exist integers $r,\ell\le e^{O(\tilde s^2)}\log^{O(\tilde s)}2\tilde K$; a normal subgroup $N\lhd G$ satisfying $$\label{eq:tor-free.post.induc.N} N\subset A^{e^{O(\tilde s^2)}K^{e^{O(s)}m}\log^{O(\tilde s)}2\tilde K};$$ finite $\tilde K^{e^{O(\tilde s)}}$-approximate groups $A_1,\ldots,A_r\subset\tilde A^{e^{O(\tilde s)}}$ such that, writing $\pi:G\to G/N$ for the quotient homomorphism, each group $\langle\pi(A_i)\rangle$ is abelian; and sets $X_1,\ldots,X_\ell\subset\tilde A^{e^{O(\tilde s)}}$ of size at most $\exp(e^{O(\tilde s)}\log^{O(1)}2\tilde K)$ such that $$\tilde A\subset N\prod\{A_1,\ldots,A_r,X_1,\ldots,X_\ell\},$$ with the product taken in some order. Here, and throughout this paper, given an ordered set $X=\{x_1,\ldots,x_m\}$ of subsets and/or elements in a group $G$, we write that a product $\Pi$ of the members of $X$ is *equal to $\prod X$ with the product taken in some order* to mean that there is a permutation $\xi\in{\text{\textup{Sym}}}(m)$ such that $\Pi=\prod_{i=1}^m x_{\xi(i)}$. If $Y=\{y_1,\ldots,y_m\}$ is another ordered set of the same number subsets and/or elements of $G$, then we say that products $\prod X$ and $\prod Y$ are *taken in the same order* if $\prod X=\prod_{i=1}^m x_{\xi(i)}$ and $\prod Y=\prod_{i=1}^m y_{\xi(i)}$ for the same permutation $\xi$. If $\tilde A$ is abelian then the proposition is trivially true with $r=1$, $\ell=0$, $A_1=\tilde A$ and $N=\{1\}$. We may therefore assume that $s\ge\tilde s\ge2$ and, by induction, that the proposition holds for all smaller values of $\tilde s$. We start by rewriting the part of the statement we are trying to prove as $$N\subset A^{e^{O(\tilde s^2)}K^{e^{O(s)+O(\tilde s)}m}\log^{O(\tilde s)}2\tilde K}.$$ This is exactly equivalent to , but writing the bound in this way makes it slightly easier to keep track of through the induction. For the same reason, at various points in the argument we use the trivial observation that any quantity bounded by $O(1)$ is also bounded by $e^{O(1)}$. Applying Proposition \[prop:ind.tor-free.post.chang\], we obtain a normal subgroup $N_0\lhd G$ with $N_0\subset A^{K^{e^{O(s)}m}}$; an integer $r_0\le\log^{O(1)}2\tilde K$; finite $\tilde K^{e^{O(1)}}$-approximate groups $\tilde A_1,\ldots,\tilde A_{r_0}\subset\tilde A^{O(1)}\subset\tilde A^{e^{O(1)}}\subset A^{e^{O(1)}m}$ such that, writing $\rho:G\to G/N$ for the quotient homomorphism, each group $\langle\rho(\tilde A_i)\rangle$ has step less than $\tilde s$; and a set $X\subset\tilde A$ of size at most $\exp(\log^{O(1)}2\tilde K)$ such that $$\label{eq:induction.step} \tilde A\subset X\tilde A_1\cdots\tilde A_r.$$ Since $G/N_0$ is generated by the $K$-approximate group $\rho(A)$, we may apply the induction hypothesis to each approximate subgroup $\rho(\tilde A_i)$ of $G/N_0$ to obtain, for each $i=1,\ldots,r_0$, integers $$\begin{aligned} r_i,\ell_i&\le e^{O((\tilde s-1)^2)}\log^{O(\tilde s-1)}(2\tilde K^{e^{O(1)}})\\ & \le e^{O(\tilde s(\tilde s-1))}\log^{O(\tilde s-1)}2\tilde K;\end{aligned}$$ a normal subgroup $N_i\lhd G$ containing $N_0$ and satisfying $$\begin{aligned} N_i&\subset A^{e^{O((\tilde s-1)^2)}K^{e^{O(s)+O(\tilde s-1)}(e^{O(1)}m)}(e^{O(1)}\log2\tilde K)^{O(\tilde s-1)}}N_0\\ &\subset A^{e^{O(\tilde s(\tilde s-1))}K^{e^{O(s)+O(\tilde s)}m}\log^{O(\tilde s-1)}2\tilde K}N_0;\end{aligned}$$ finite $\tilde K^{e^{O(\tilde s)}}$-approximate groups $A_1^{(i)},\ldots,A_{r_i}^{(i)}\subset\tilde A_i^{e^{O(\tilde s-1)}}\subset\tilde A^{e^{O(\tilde s)}}$ such that, writing $\pi_i:G\to G/N_i$ for the quotient homomorphism, each group $\langle\pi_i(A_j^{(i)})\rangle$ is abelian; and sets $X_1^{(i)},\ldots,X_{\ell_i}^{(i)}\subset\tilde A_i^{e^{O(\tilde s-1)}}\subset\tilde A^{e^{O(\tilde s)}}$ satisfying $$\begin{aligned} |X_j^{(i)}|&\le\exp(e^{O(\tilde s-1)}\log^{O(1)}(2\tilde K^{e^{O(1)}}))\\ &\le\exp(e^{O(\tilde s)}\log^{O(1)}2\tilde K)\end{aligned}$$ such that $$\label{eq:induction.hyp} \tilde A_i\subset N_i\prod\{A_1^{(i)},\ldots,A_{r_i}^{(i)},X_1^{(i)},\ldots,X_{\ell_i}^{(i)}\},$$ with the product taken in some order. Defining $N=N_1\cdots N_{r_0}$, we then have $$\begin{aligned} N&\subset A^{r_0e^{O(\tilde s(\tilde s-1))}K^{e^{O(s)+O(\tilde s)}m}\log^{O(\tilde s-1)}2\tilde K}\cdot N_0\\ &\subset A^{e^{O(\tilde s(\tilde s-1))}K^{e^{O(s)+O(\tilde s)}m}\log^{O(\tilde s)}2\tilde K}\cdot N_0\\ &\subset A^{e^{O(\tilde s(\tilde s-1))}K^{e^{O(s)+O(\tilde s)}m}\log^{O(\tilde s)}2\tilde K}\cdot A^{K^{e^{O(s)}m}}\\ &\subset A^{e^{O(\tilde s^2)}K^{e^{O(s)+O(\tilde s)}m}\log^{O(\tilde s)}2\tilde K}.\end{aligned}$$ Moreover, and imply that $$\tilde A\subset N\prod\{A_1^{(1)},\ldots,A_{r_1}^{(1)},\ldots,A_1^{(r_0)},\ldots,A_{r_{r_0}}^{(r_0)},X_1^{(1)},\ldots,X_{\ell_1}^{(1)},\ldots,X_1^{(r_0)},\ldots,X_{\ell_{r_0}}^{(r_0)},X\}$$ with the product taken in some order. We also have $$\begin{aligned} (r_1+\ldots+r_{r_0})\,,\,(\ell_1+\ldots+\ell_{r_0}+1)&\le r_0e^{O(\tilde s(\tilde s-1))}\log^{O(\tilde s-1)}2\tilde K+1\\ &\le e^{O(\tilde s(\tilde s-1))}\log^{O(\tilde s)}2\tilde K+1\\ &\le e^{O(\tilde s^2)}\log^{O(\tilde s)}2\tilde K.\end{aligned}$$ Finally, since every $\langle\pi_i(A_j^{(i)})\rangle$ is abelian, every $\langle\pi(A_j^{(i)})\rangle$ certainly is, so the proof is complete. \[prop:grp.in.normal\] Let $s\in{\mathbb{N}}$ and $K\ge1$. Let $G$ be an $s$-step nilpotent group generated by a finite $K$-approximate group $A$. Let $H\subset A^m$ be a subgroup of $G$. Then there exists a normal subgroup $N$ of $G$ such that $H\subset N\subset A^{K^{e^{O(s)}m}}$. The bounds stated in [@nilp.frei Proposition 7.3] are less explicit than the ones claimed here; as usual, the bounds claimed here can be read out of the argument there, or alternatively found explicitly in [@book Corollary 6.5.2]. \[prop:prod.of.progs.and.small\] Let $s\in{\mathbb{N}}$ and $K\ge2$. Let $G$ be an $s$-step nilpotent group, and suppose that $A\subset G$ is a finite $K$-approximate group. Then there exist $k,\ell\le e^{O(s^2)}\log^{O(s)}2K$, ordered progressions $P_1,\ldots,P_k\subset A^{e^{O(s)}}$ of rank at most $e^{O(s)}\log^{O(1)}2K$, sets $X_1,\ldots,X_\ell\subset A^{e^{O(s)}}$ of size at most $\exp(e^{O(s)}\log^{O(1)}2K)$, and a subgroup $H<G$ normalised by $A$ satisfying $H\subset A^{K^{e^{O(s)}}}$ such that $$A\subset H\prod\{P_1,\ldots,P_k,X_1,\ldots,X_\ell\},$$ with the product taken in some order. We may assume that $A$ generates $G$. Applying \[prop:tor-free.post.induc\] with $\tilde A=A$, we obtain integers $r,t\le e^{O(s^2)}\log^{O(s)}2K$; a normal subgroup $N\lhd G$ satisfying $N\subset A^{K^{e^{O(s)}}}$; finite $K^{e^{O(s)}}$-approximate groups $A_1,\ldots,A_r\subset A^{e^{O(s)}}$ such that, writing $\pi:G\to G/N$ for the quotient homomorphism, each group $\langle\pi(A_i)\rangle$ is abelian; and sets $X_1,\ldots,X_t\subset A^{e^{O(s)}}$ of size at most $\exp(e^{O(s)}\log^{O(1)}2K)$ such that $$A\subset N\prod\{A_1,\ldots,A_r,X_1,\ldots,X_t\}$$ with the product taken in some order. For each $i=1,\ldots,r$, apply \[cor:sanders\] to the set $\pi(A_i)$ to obtain a subgroup $H_i\subset A_i^8N\subset A^{e^{O(s)}}N$ containing $N$, an ordered progression $P_i\subset A_i^8\subset A^{e^{O(s)}}$ of rank at most $e^{O(s)}\log^{O(1)}2K$, and a set $Y_i\subset A_i\subset A^{e^{O(s)}}$ of size at most $\exp(e^{O(s)}\log^{O(1)}2K)$, such that $A_i\subset Y_iH_iP_i$. Since $G/N$ is gererated by the $K$-approximate group $\pi(A)$, applying Proposition \[prop:grp.in.normal\] in $G/N$ implies that for each $i$ there is a normal subgroup $N_i\lhd G$ such that $H_i\subset N_i\subset A^{K^{e^{O(s)}}}N$. The subgroup $H=N_1\cdots N_r$ is then normal in $G$, and satisfies $$\begin{aligned} H&\subset A^{rK^{e^{O(s)}}}N\\ &\subset A^{K^{e^{O(s)}}}\end{aligned}$$ and $$A\subset H\prod\{P_1,\ldots,P_r,Y_1,\ldots,Y_r,X_1,\ldots,X_t\},$$ with the product taken in some order. This completes the proof. Note that if $K<2$ then $A$ is a finite subgroup of $G$, and so the theorem holds with $A=H$. We may therefore assume that $K\ge2$. Let $k,\ell\le e^{O(s^2)}\log^{O(s)}2K$, $$\label{eq:P.S.contained} P_1,\ldots,P_k,X_1,\ldots,X_\ell\subset A^{e^{O(s)}},$$ and $H\subset A^{K^{e^{O(s)}}}$ be as coming from \[prop:prod.of.progs.and.small\], noting in particular that $$\label{eq:prod.order} AH\subset H\prod\{P_1,\ldots,P_k,X_1,\ldots,X_\ell\}.$$ The pigeonhole principle therefore implies that there exist elements $u_1,\ldots,u_\ell$ with $u_i\in X_i$ such that the product $\prod\{P_1,\ldots,P_k,u_1,\ldots,u_\ell\}$, taken in the same order as the product in , satisfies $$\begin{aligned} \left|H\prod\{P_1,\ldots,P_k,u_1,\ldots,u_\ell\}\right|&\ge\frac{|AH|}{|X_1|\cdots|X_\ell|}\\ &\ge\frac{|AH|}{\exp(e^{O(s)}\log^{O(1)}2K)^\ell}\\ &\ge\frac{|AH|}{\exp(e^{O(s^2)}\log^{O(s)}2K)}.\end{aligned}$$ In particular, setting $Q_i=\{u_i^{-1},1,u_i\}$ for $i=1,\ldots,\ell$, we have $$\left|H\prod\{P_1,\ldots,P_k,Q_1,\ldots,Q_\ell\}\right|\ge\frac{|AH|}{\exp(e^{O(s^2)}\log^{O(s)}2K)},$$ with the product again taken in the same order. Now $\prod\{P_1,\ldots,P_k,Q_1,\ldots,Q_\ell\}$ is an ordered progression, say $P_{\text{\textup{ord}}}(x;L)$. The ranks of the progressions $P_i$ coming from \[prop:prod.of.progs.and.small\] are at most $e^{O(s)}\log^{O(1)}2K$, and hence that the rank of $P_{\text{\textup{ord}}}(x;L)$ is at most $ke^{O(s)}\log^{O(1)}2K+\ell$, which is at most $e^{O(s^2)}\log^{O(s)}2K$. Furthermore, the containment implies that $$\begin{aligned} P_{\text{\textup{ord}}}(x;L)&\subset A^{(k+\ell)e^{O(s)}}\\ &\subset A^{e^{O(s^2)}\log^{O(s)}2K},\end{aligned}$$ and \[prop:nilprog.equiv\] therefore implies that $$P_{\text{\textup{ord}}}(x;L)\subset P_{\text{\textup{nil}}}(x;L)\subset\overline P(x;L)\subset A^{e^{O(s^3)}\log^{O(s^2)}2K}.$$ This comletes the proof. \[rem:poly.bound\]The polynomial bound on the product set of $A$ required to contain $H$ in \[thm:new.gen\] comes from our applications of Propositions \[prop:lower.step.in.quotient\] and \[prop:grp.in.normal\]. These propositions are themselves both applications of the same result, namely [@nilp.frei Proposition 7.2], and so the polynomial bound in \[thm:new.gen\] can be traced to this result. It appears that a new idea would be required to improve this result in such a way as to remove the polynomial bound from \[thm:new.gen\]. Covering arguments {#sec:covering} ================== In this section we use covering arguments to prove \[cor:ruzsa,cor:chang.ag\]. \[cor:ruzsa\] follows from \[thm:new.gen\] and a straightforward application of Ruzsa’s covering lemma, as follows. We may assume that $A$ generates $G$. Let $H$ and $P=P_{\text{\textup{ord}}}(x;L)$ be as given by \[thm:new.gen\], noting that $H\lhd G$. Let $\pi:G\to G/H$ be the quotient homomorphism, and note that $$\begin{aligned} \frac{|\pi(AP)|}{|\pi(P)|}&=\frac{|APH|}{|PH|}\\ &\le\exp(e^{O(s^2)}\log^{O(s)}2K)\frac{|APH|}{|AH|}\\ &=\exp(e^{O(s^2)}\log^{O(s)}2K)\frac{|\pi(AP)|}{|\pi(A)|}\\ &\le\exp(e^{O(s^2)}\log^{O(s)}2K),\end{aligned}$$ the last inequality coming from the fact that $\pi(A)$ is a $K$-approximate group and $\pi(AP)\subset\pi(A)^{e^{O(s^2)}\log^{O(s)}2K}$. Applying \[lem:covering\] in the quotient $G/H$ therefore gives a set $X\subset A$ of size at most $\exp(e^{O(s^2)}\log^{O(s)}K)$ such that $A\subset XHPP^{-1}$. Now $PP^{-1}\subset A^{e^{O(s^2)}\log^{O(s)}2K}$ is an ordered progression of rank double that of $P$, which is still at most $e^{O(s^2)}\log^{O(s)}2K$. The corollary therefore follows from \[prop:nilprog.equiv\]. We may assume that $A$ generates $G$. Let $H$ and $P_0=P_{\text{\textup{ord}}}(x;L)$ be as given by \[thm:new.gen\], noting that $H\lhd G$. Let $\pi:G\to G/H$ be the quotient homomorphism, noting that $$\frac{|\pi(P_0)|}{|\pi(A)|}=\frac{|P_0H|}{|AH|}\ge\exp(-e^{O(s^2)}\log^{O(s)}2K).$$ Applying and \[lem:chang\] in the quotient $G/H$, we therefore have $$t\le e^{O(s^2)}\log^{O(s)}2K$$ and sets $S_1,\ldots,S_t\subset A$ with $|S_i|\le2K$ such that $$A\subset S_{t-1}^{-1}\cdots S_1^{-1}P_0^{-1}P_0S_1\cdots S_tH.$$ Enumerating the elements of each $S_i$ as $s_{1,i},\ldots,s_{r_i,i}$ and writing $$Q_i=\{s_{1,i}^{\epsilon_1}\cdots s_{r_i,i}^{\epsilon_{r_i}}:\epsilon_j,\in\{-1,0,1\}\},$$ the set $P=Q_{t-1}\cdots Q_1P_0^{-1}P_0Q_1\cdots Q_t$ is therefore an ordered progression of rank at most $$e^{O(s^2)}K\log^{O(s)}2K$$ satisfying $$A\subset PH\subset A^{4Kt+e^{O(s^2)}\log^{O(s)}2K}H\subset A^{e^{O(s^2)}K\log^{O(s)}2K}H.$$ The corollary therefore follows from \[prop:nilprog.equiv\]. Applications to non-nilpotent groups {#sec:non-nilp} ==================================== In this section we use our results to improve the bounds on the ranks of the coset nilprogressions appearing in various Freiman-type theorems in non-nilpotent groups. As in \[sec:details\], at various points we separate the trivial case $K<2$ from the meaningful case $K\ge2$ so as to avoid the need for multiplicative constants. Our first corollary improves an earlier result of the author for residually nilpotent groups [@resid Corollary 1.4], and partially improves on \[cor:ruzsa\] for large values of $s$. \[cor:resid\] Let $K\ge1$. Let $G$ be a residually nilpotent group, and suppose that $A\subset G$ is a finite $K$-approximate group. Then $A$ is contained in the union of at most $\exp(K^{O(1)})$ left translates of a coset nilprogression $P\subset A^{O_K(1)}$ of rank at most $\exp(O(K^{12}))$ and step at most $K^6$. This compares with the bound of $\exp(\exp(K^{O(1)}))$ on the rank of $P$ obtained previously by the author using \[thm:old\]. It follows from [@resid Theorem 1.2] that there exist subgroups $H\lhd N<G$ such that $H\subset A^{O_K(1)}$, such that $N/H$ is nilpotent of step at most $K^6$, and such that $A$ is contained in a union of at most $\exp(K^{O(1))}$ left cosets of $N$. \[lem:fibre.pigeonhole\] then implies that $A$ is contained in a union of at most $\exp(K^{O(1))}$ left translates of $A^2\cap N$, which is a $K^3$-approximate group by \[lem:slicing\]. The desired result therefore follows from applying \[cor:chang.ag\] to the image of $A^2\cap N$ in $N/H$. \[cor:resid\] gives a better rank bound than \[cor:ruzsa\] if the step of the ambient group is greater than $K^6$. It gives a better bound on number of translates of $P$ required to cover $A$ as soon as the step is logarithmic in $K$. Our next corollary applies to linear groups over fields of prime order, and arises from combining \[cor:ruzsa\] with a result of Gill, Helfgott, Pyber and Szabó [@gill-helf Theorem 3]. \[cor:ghps\] Let $n\in{\mathbb{N}}$ and $K\ge1$, and let $p$ be a prime. Suppose that $A\subset GL_n({\mathbb{F}}_p)$ is a finite $K$-approximate group. Then there is a coset nilprogression $P\subset A^{K^{O_n(1)}}$ of rank at most $e^{O(n^2)}\log^{O(n)}2K$ and step at most $n$ such that $A$ is contained in the union of at most $\exp(K^{O_n(1)})$ left translates of $P$. This compares with the bound of $K^{O_n(1)}$ on the rank of $P$ obtained by Gill, Helfgott, Pyber and Szabó using \[thm:old\]. If $K<2$ then $A$ is a finite subgroup and the corollary is trivial, so we may assume that $K\ge2$. It follows from [@gill-helf Theorem 3] that there exist subgroups $H\lhd N<GL_n({\mathbb{F}}_p)$ such that $H\subset A^{K^{O_n(1)}}$, such that $N/H$ is nilpotent of step at most $n$, and such that $A$ is contained in a union of at most $\exp(K^{O_n(1))}$ left cosets of $N$. \[lem:fibre.pigeonhole\] then implies that $A$ is contained in a union of at most $\exp(K^{O_n(1))}$ left translates of $A^2\cap N$, which is a $K^3$-approximate group by \[lem:slicing\]. The desired result therefore follows from applying \[cor:ruzsa\] to the image of $A^2\cap N$ in $N/H$. One can obtain a similar result in characteristic zero by combining \[cor:chang.ag\] with a result of Breuillard, Green and Tao [@bgt.lin Theorem 2.5], as follows. \[cor:bgt\] Let $n\in{\mathbb{N}}$ and $K\ge1$, and let $\Bbbk$ be a field of characterisic zero. Suppose that $A\subset GL_n(\Bbbk)$ is a finite $K$-approximate group. Then there is a coset nilprogression $P_1\subset A^{K^{O_n(1)}}$ of rank at most $e^{O(n^2)}\log^{O(n)}2K$ such that $A$ is contained in the union of at most $\exp(\log^{O_n(1)}2K)$ left translates of $P_1$, and a coset nilprogression $P_2\subset A^{K^{O_n(1)}}$ of rank at most $e^{O(n^2)}K^3\log^{O(n)}2K$ such that $A$ is contained in the union of at most $K^{O_n(1)}$ left translates of $P_2$. If $K<2$ then $A$ is a finite subgroup and the corollary is trivial, so we may assume that $K\ge2$. It then follows from [@bgt.lin Theorem 2.5] that $A$ is contained in a union of at most $K^{O_n(1)}$ left cosets of a nilpotent subgroup $N$ of $GL_n(\Bbbk)$ of step at most $n-1$, and hence from \[lem:fibre.pigeonhole\] that $A$ is contained in a union of at most $K^{O_n(1)}$ left translates of $A^2\cap N$. \[lem:slicing\] implies that $A^2\cap N$ is a $K^3$-approximate group, and so the existence of $P_1$ follows from \[cor:ruzsa\] and the existence of $P_2$ follows from \[cor:chang.ag\]. In the special case in which $\Bbbk={\mathbb{C}}$, an argument of Breuillard and Green shows that the coset nilprogression appearing in \[cor:bgt\] can be replaced with simply a nilprogression, as follows. \[cor:bg\] Let $n\in{\mathbb{N}}$ and $K\ge1$. Suppose that $A\subset GL_n({\mathbb{C}})$ is a finite $K$-approximate group. Then there is a nilprogression $P_1\subset A^{e^{O(n^3)}\log^{O(n^2)}2K}$ of rank at most $e^{O(n^2)}\log^{O(n)}2K$ such that $A$ is contained in a union of at most $\exp(e^{O(n^2)}\log^{O(n)}2K)$ left translates of $P_1$, and a nilprogression $P_2\subset A^{{e^{O(n^3)}K^{3n+3}\log^{O(n^2)}2K}}$ of rank at most $e^{O(n^2)}K^3\log^{O(n)}2K$ such that $A$ is contained in a union of at most $K^{O_n(1)}$ left translates of $P_2$. For the convenience of the reader we reproduce the Breuillard–Green argument giving \[cor:bg\]. The argument is facilitated by the following two general results about complex linear groups, in which we write ${\text{\textup{Upp}}}_n({\mathbb{C}})$ to mean the group of upper-triangular $n\times n$ complex matrices. \[thm:malcev\] Let $n\in{\mathbb{N}}$, and suppose that $G<GL_n({\mathbb{C}})$ is a soluble subgroup. Then $G$ contains a normal subgroup $U$ of index at most $O_n(1)$ that is conjugate to a subgroup of ${\text{\textup{Upp}}}_n({\mathbb{C}})$. \[prop:red.tf\] Let $n,s\in{\mathbb{N}}$, and let $N$ be an $s$-step nilpotent subgroup of ${\text{\textup{Upp}}}_n({\mathbb{C}})$. Then there is a torsion-free $s$-step nilpotent group $\Gamma$ such that $N$ embeds into ${\mathbb{R}}^n/{\mathbb{Z}}^n\times\Gamma$. Although [@bg.sol Proposition 3.2] does not include the statement that $\Gamma$ has the same step as $N$, one can easily obtain this by replacing ${\mathbb{R}}^n/{\mathbb{Z}}^n\times\Gamma$ with $({\mathbb{R}}^n/{\mathbb{Z}}^n)N$. We follow part of the proof of [@bg.sol Corollary 1.5]. It follows from [@bgt.lin Theorem 2.5] that $A$ is contained in a union of at most $K^{O_n(1)}$ left cosets of a nilpotent subgroup $N$ of $GL_n({\mathbb{C}})$ of step at most $n-1$. By \[thm:malcev\] we may assume that $N\subset{\text{\textup{Upp}}}_n({\mathbb{C}})$, and then by \[prop:red.tf\] we may assume that there exists a torsion-free $(n-1)$-step nilpotent group $\Gamma$ such that $N={\mathbb{R}}^n/{\mathbb{Z}}^n\times\Gamma$. \[lem:fibre.pigeonhole\] then implies that $A$ is contained in a union of at most $K^{O_n(1)}$ left translates of $A^2\cap({\mathbb{R}}^n/{\mathbb{Z}}^n\times\Gamma)$. Set $B=A^2\cap\left(\left[-\frac{1}{8},\frac{1}{8}\right]^n\times\Gamma\right)$. The set $\left[-\frac{1}{16},\frac{1}{16}\right]^n\times\Gamma$ is a $2^n$-approximate group; since $(\left[-\frac{1}{16},\frac{1}{16}\right]^n\times\Gamma)^2=\left[-\frac{1}{8},\frac{1}{8}\right]^n\times\Gamma$ and $(\left[-\frac{1}{16},\frac{1}{16}\right]^n\times\Gamma)^8={\mathbb{R}}^n/{\mathbb{Z}}^n\times\Gamma$, \[lem:slicing\] therefore implies that $B$ is a $2^{3n}K^3$-approximate group, and that $A^2\cap({\mathbb{R}}^n/{\mathbb{Z}}^n\times\Gamma)$ is contained in a union of at most $2^{7n}K$ left translates of $B$. Let ${\varphi}:{\mathbb{R}}^n/{\mathbb{Z}}^n\times\Gamma\to\left(-\frac{1}{2},\frac{1}{2}\right]^n\times\Gamma\subset{\mathbb{R}}^n\times\Gamma$ be the obvious lift. The restriction of ${\varphi}$ to $\left[-\frac{1}{8},\frac{1}{8}\right]^n\times\Gamma$ is a Freiman $3$-homomorphism, so \[lem:fr.hom.ap.grp\] implies that ${\varphi}(B)$ is a $2^{3n}K^3$-approximate subgroup of the torsion-free $s$-step nilpotent group ${\mathbb{R}}^n\times\Gamma$. \[cor:ruzsa,cor:chang.ag\] therefore imply the existence of a nilprogression $Q_1\subset{\varphi}(B)^{e^{O(n^3)}\log^{O(n^2)}2K}$ of rank at most $e^{O(n^2)}\log^{O(n)}2K$ such that ${\varphi}(B)$ is contained in a union of at most $\exp(e^{O(n^2)}\log^{O(n)}2K)$ left translates of $Q_1$, and a nilprogression $Q_2\subset{\varphi}(B)^{{e^{O(n^3)}K^{3n+3}\log^{O(n^2)}2K}}$ of rank at most $e^{O(n^2)}K^3\log^{O(n)}2K$ such that ${\varphi}(B)\subset Q_2$. Write $\pi:{\mathbb{R}}^n\times\Gamma\to{\mathbb{R}}^n/{\mathbb{Z}}^n\times\Gamma$ for the quotient homomorphism, and note that $\pi({\varphi}(B))=B$, so that $A$ is contained in a union of at most $K^{O_n(1)}$ left translates of $\pi({\varphi}(B))$. 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--- abstract: 'The integers $n=\prod_{i=1}^r p_i^{a_i}$ and $m=\prod_{i=1}^r p_i^{b_i}$ having the same prime factors are called exponentially coprime if $(a_i,b_i)=1$ for every $1\le i\le r$. We estimate the number of pairs of exponentially coprime integers $n,m\le x$ having the prime factors $p_1,...,p_r$ and show that the asymptotic density of pairs of exponentially coprime integers having $r$ fixed prime divisors is $(\zeta(2))^{-r}$.' author: - '[László Tóth]{} (Pécs, Hungary)' date: 'Pure Math. Appl. (PU.M.A.), 15 (2004), 343-348' title: '**On exponentially coprime integers**' --- =6.truein =9.truein =-.5truein =-.8truein \[section\] \[section\] \[section\] Mathematics Subject Classification (2000): 11A05, 11A25, 11N37 [**1. Introduction**]{} Let $n>1$ be an integer of canonical form $n=\prod_{i=1}^r p_i^{a_i}$. The integer $d$ is called an [*exponential divisor*]{} of $n$ if $d=\prod_{i=1}^r p_i^{c_i}$, where $c_i | a_i$ for every $1\le i \le r$, notation: $d|_e n$. By convention $1|_e 1$. This notion was introduced by [M. V. Subbarao]{} [@Su72]. The smallest exponential divisor of $n>1$ is its squarefree kernel $\kappa(n):=\prod_{i=1}^r p_i$. Let $\tau^{(e)}(n)= \sum_{d|_e n} 1$ and $\sigma^{(e)}(n)=\sum_{d|_e n} d$ denote the number and the sum of exponential divisors of $n$, respectively. Properties of these functions were investigated by several authors, see [@FaSu89], [@KaSu2003], [@PeWu97], [@SmWu97], [@Su72], [@Wu95]. Two integers $n,m >1$ have common exponential divisors iff they have the same prime factors and for $n=\prod_{i=1}^r p_i^{a_i}$, $m=\prod_{i=1}^r p_i^{b_i}$, $a_i,b_i\ge 1$ ($1\le i\le r$), the [*greatest common exponential divisor*]{} of $n$ and $m$ is $$(n,m)_e:=\prod_{i=1}^r p_i^{(a_i,b_i)}.$$ Here $(1,1)_e=1$ by convention and $(1,m)_e$ does not exist for $m>1$. The integers $n,m >1$ are called [*exponentially coprime*]{}, if they have the same prime factors and $(a_i,b_i)=1$ for every $1\le i\le r$, with the notation of above. In this case $(n,m)_e=\prod_{i=1}^r p_i$. $1$ and $1$ are considered to be exponentially coprime. $1$ and $m>1$ are not exponentially coprime. Exponentially coprime integers were introduced by [J. Sándor]{} [@Sa96]. Let $p_i$ ($1\le i\le r$) be fixed distinct primes and let $P^{(e)}(p_1,...,p_r;x)$ denote the number of pairs $\langle n,m \rangle$ of exponentially coprime integers such that $\kappa(n)=\kappa(m)=\prod_{i=1}^r p_i$ and $n,m\le x$. In this note we estimate $P^{(e)}(p_1,...,p_r;x)$ and show that the asymptotic density of pairs of exponentially coprime integers having $r$ fixed prime divisors is $(\zeta(2))^{-r}$. As an open problem we formulate the following: What can be said on the asymptotic density of pairs of exponentially coprime integers if their prime divisors are not fixed ? For a real $x\ge 1$ and an integer $n\ge 1$ consider the Legendre-type function $L^{(e)}(x,n)$ defined as the number of integers $k\le x$ such that $k$ and $n$ are exponentially coprime. The following estimate holds: Let $N(p_1,...,p_r;x)$ denote the number of integers $n\le x$ having the kernel $\kappa(n)=p_1\cdots p_r$. Taking $a_1=\cdots =a_r=1$ we obtain from Theorem 1 the following known estimate, cf. for ex. [@Te95], Ch. III.5 regarding integers free of large prime factors. The proofs of Theorems 1 and 2 are by induction on $r$, while Corollary 3 follows from Theorem 2 and Corollary 1. First we prove the following lemma. We will use the well-known estimate: if $s\ge 0$, then $$\phi_s(z,a):=\sum_{n\le z \atop{(n,a)=1}} n^s =\frac{z^{s+1}\phi(a)}{(s+1)a}+O(z^s\theta(a)), \leqno(5)$$ uniformly for $z\ge 1$ and $a\ge 1$. Induction on $r$. For $r=1$ (4) follows from (5) applied for $s=0$. Suppose formula (4) is valid for $r-1$ and prove it for $r$. $$\sum_{k_1t_1+\cdots +k_rt_r\le z \atop{(k_1,a_1)=\cdots =(k_r,a_r)=1 \atop{k_1,...,k_r\ge 1}}} 1 = \sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{(k_r,a_r)=1 \atop{k_r\ge 1}}} \sum_{k_1t_1+\cdots +k_{r-1}t_{r-1}\le z-k_rt_r \atop{(k_1,a_1)=\cdots =(k_{r-1},a_{r-1})=1 \atop{k_1,...,k_{r-1}\ge 1}}} 1$$ $$=\sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{(k_r,a_r)=1 \atop{k_r\ge 1}}} \left(\frac1{(r-1)!} \left( \prod_{i=1}^{r-1} \frac{\phi(a_i)}{a_i t_i}\right) (z-k_rt_r)^{r-1} + O\left( z^{r-2} \sum_{i=1}^{r-1} \theta(a_i) \right) \right)$$ $$= \frac1{(r-1)!} \left( \prod_{i=1}^{r-1} \frac{\phi(a_i)}{a_i t_i}\right) \sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{(k_r,a_r)=1 \atop{k_r\ge 1}}} (z-k_rt_r)^{r-1}+ O\left( z^{r-1} \sum_{i=1}^{r-1} \theta(a_i) \right).$$ Using the binomial formula and estimate (5) the sum appearing here is $$\sum_{j=0}^{r-1} (-1)^j {r-1 \choose j} z^{r-1-j} t_r^j \sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{(k_r,a_r)=1 \atop{k_r\ge 1}}} k_r^j$$ $$= \sum_{j=0}^{r-1} (-1)^j {r-1 \choose j} z^{r-1-j} t_r^j \left( \frac{(z-t_1-\cdots -t_{r-1})^{j+1}\phi(a_r)}{(j+1)t_r^{j+1} a_r} +O(z^j \theta(a_r))\right)$$ $$=\frac{\phi(a_r)}{t_ra_r} z^r \sum_{j=0}^{r-1} (-1)^j {r-1 \choose j} \frac1{j+1} + O( z^{r-1}\theta(a_r))$$ $$=\frac{\phi(a_r)}{rt_ra_r} z^r + O(z^{r-1}\theta(a_r))$$ and the proof is complete. Apply Lemma 1 for $z=\log x$, $t_1=\log p_1,...,t_r=\log p_r$. In order to prove Theorem 2 we need Induction on $r$, similar to the proof of Lemma 1. We use the well-known estimate: let $s\ge -1$ be a real number, then for $z\ge 3$, $$\sum_{n\le z} \phi(n)n^s =\frac{z^{s+2}}{(s+2)\zeta(2)} + O(z^{s+1}\log z). \leqno(7)$$ For $r=1$ (6) follows from (7) applied for $s=-1$. Suppose formula (6) is valid for $r-1$ and prove it for $r$. $$\sum_{k_1t_1+\cdots +k_rt_r\le z \atop{k_1,...,k_r\ge 1}} \prod_{i=1}^r \frac{\phi(k_i)}{k_i} = \sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{k_r\ge 1}} \frac{\phi(k_r)}{k_r} \sum_{k_1t_1+\cdots +k_{r-1}t_{r-1}\le z-k_rt_r \atop{k_1,...,k_{r-1}\ge 1}} \prod_{i=1}^{r-1} \frac{\phi(k_i)}{k_i}$$ $$=\sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{k_r\ge 1}} \frac{\phi(k_r)}{k_r} \left(\frac1{(r-1)! (\zeta(2))^{r-1}} \left( \prod_{i=1}^{r-1} \frac1{t_i} \right) (z-k_rt_r)^{r-1} + O\left( z^{r-2} \log z\right) \right)$$ $$= \frac1{(r-1)! (\zeta(2))^{r-1}} \left( \prod_{i=1}^{r-1} \frac1{t_i}\right) \sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{k_r\ge 1}} \frac{\phi(k_r)}{k_r} (z-k_rt_r)^{r-1}+ O\left( z^{r-1} \log z \right).$$ The sum appearing here is, applying (7), $$\sum_{j=0}^{r-1} (-1)^j {r-1 \choose j} z^{r-1-j} t_r^j \sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{k_r\ge 1}} \phi(k_r)k_r^{j-1}$$ $$= \sum_{j=0}^{r-1} (-1)^j {r-1 \choose j} z^{r-1-j} t_r^j \left( \frac{(z-t_1-\cdots -t_{r-1})^{j+1}}{(j+1)t_r^{j+1} \zeta(2)} +O(z^j \log z)\right)$$ $$=\frac1{t_r\zeta(2)} z^r \sum_{j=0}^{r-1} (-1)^j {r-1 \choose j} \frac1{j+1} + O(z^{r-1}\log z)$$ $$=\frac1{r t_r \zeta(2)} z^r + O(z^{r-1}\log z),$$ which completes the proof. Using estimate (4), $$\sum_{k_1t_1+\cdots +k_rt_r\le z \atop{j_1t_1+\cdots +j_rt_r\le z \atop{(k_1,j_1)=\cdots =(k_r,j_r)=1 \atop{k_1,j_1,...,k_r,j_r\ge 1}}}} 1 = \sum_{k_1t_1+\cdots +k_rt_r\le z \atop{k_1,...,k_r\ge 1}} \sum_{j_1t_1+\cdots +j_rt_r\le z \atop{(j_1,k_1)=\cdots =(j_r,k_r)=1 \atop{j_1,...,j_r\ge 1}}} 1$$ $$=\sum_{k_1t_1+\cdots +k_rt_r\le z \atop{k_1,...,k_r\ge 1}} \left(\frac1{r!} \left( \prod_{i=1}^r \frac{\phi(k_i)}{k_i t_i} \right) z^r + O\left(z^{r-1} \sum_{i=1}^r \theta(k_i) \right)\right)$$ $$=\frac{z^r}{r!\prod_{i=1}^r t_i} \sum_{k_1t_1+\cdots +k_rt_r\le z \atop{k_1,...,k_r\ge 1}} \prod_{i=1}^r \frac{\phi(k_i)}{k_i} + O\left( z^{r-1} \sum_{i=1}^r \sum_{k_1t_1+\cdots +k_rt_r\le z} \theta(k_i) \right)$$ here the $O$-term is $O(z^{r-1}z^{r-1}z\log z)=O(z^{2r-1}\log z)$ and applying Lemma 2 to the main term finishes the proof. Apply Lemma 3 for $z=\log x$, $t_1=\log p_1,...,t_r=\log p_r$. This is a direct consequence of Theorem 2 and Corollary 1. The considered asymptotic density is $$\lim_{x\to \infty} P^{(e)}(p_1,...,p_r;x) (N(p_1,...,p_r;x))^{-2}= %\qquad ( \sum_{p_1^{a_1}\cdots p_r^{a_r}\le x \atop{ %p_1^{b_1}\cdots p_r^{b_r}\le x \atop{ %(a_1,b_1)=...=(a_r,b_r)=1}}} 1 ) \cdot %( \sum_{p_1^{a_1}\cdots p_r^{a_r}\le x} 1 )^{-2} = (\zeta(2))^{-r}.$$ [99]{} and [M. V. Subbarao]{}, The maximal order and the average order of multiplicative function $\sigma^{(e)}(n)$, [*Théorie des nombres. Proc. of the Int. Conf. Québec, 1987*]{}, de Gruyter, Berlin – New York, 1989, 201-206. and [M. V. Subbarao]{}, On the distribution of exponential divisors, [*Annales Univ. Sci. Budapest., Sect. Comp.*]{}, [**22**]{} (2003), 161-180. and [J. Wu]{}, On the sum of exponential divisors of an integer, [*Acta Math. Acad. Sci. Hung.*]{}, [**77**]{} (1997), 159-175. , On an exponential totient function, [*Studia Univ. Babeş-Bolyai, Math.*]{}, [**41**]{} (1996), 91-94. and [J. Wu]{}, On the exponential divisor function, [*Publ. Inst. Math. (Beograd) (N. S.)*]{}, [**61**]{} (1997), 21-32. , On some arithmetic convolutions, in [*The Theory of Arithmetic Functions*]{}, Lecture Notes in Mathematics No. [**251**]{}, 247-271, Springer, 1972. , [*Introduction to Analytic and Probabilistic Number Theory*]{}, Cambridge Univ. Press, 1995. , Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré, [*J. Théor. Nombres Bordeaux*]{}, [**7**]{} (1995), 133-141. [[**László Tóth**]{}\ University of Pécs\ Institute of Mathematics and Informatics\ Ifjúság u. 6\ 7624 Pécs, Hungary\ ltoth@ttk.pte.hu]{}

--- author: - | Qianru Sun$^{1,3}$ Yaoyao Liu$^{2}$ Tat-Seng Chua$^{1}$ Bernt Schiele$^{3}$\ \ $^{1}$National University of Singapore $^{2}$Tianjin University$\thanks{Yaoyao Liu did this work during his internship at NUS.}$\ $^{3}$Max Planck Institute for Informatics, Saarland Informatics Campus\ \ bibliography: - 'egbib.bib' title: 'Meta-Transfer Learning for Few-Shot Learning' ---

--- abstract: 'A persistent challenge in practical classification tasks is that labeled training sets are not always available. In particle physics, this challenge is surmounted by the use of simulations. These simulations accurately reproduce most features of data, but cannot be trusted to capture all of the complex correlations exploitable by modern machine learning methods. Recent work in weakly supervised learning has shown that simple, low-dimensional classifiers can be trained using only the impure mixtures present in data. Here, we demonstrate that complex, high-dimensional classifiers can also be trained on impure mixtures using weak supervision techniques, with performance comparable to what could be achieved with pure samples. Using weak supervision will therefore allow us to avoid relying exclusively on simulations for high-dimensional classification. This work opens the door to a new regime whereby complex models are trained directly on data, providing direct access to probe the underlying physics.' author: - 'Patrick T. Komiske' - 'Eric M. Metodiev' - Benjamin Nachman - 'Matthew D. Schwartz' bibliography: - 'highweaklett.bib' title: 'Learning to Classify from Impure Samples with High-Dimensional Data' --- Data analysis methods at the Large Hadron Collider (LHC) rely heavily on simulations. These simulations are generally excellent and allow us to explore the mapping between truth information (particles from collisions) and observables (reconstructed momenta and energies). In particular, simulations let us train complex algorithms to extract the truth information from the observables. Machine learning methods trained on low-level inputs have been developed for collider physics [@Larkoski:2017jix] to identify boosted $W/Z/$Higgs bosons [@Cogan:2014oua; @deOliveira:2015xxd; @Baldi:2016fql; @Barnard:2016qma; @Louppe:2017ipp; @Datta:2017rhs; @Komiske:2017aww], top quarks [@Almeida:2015jua; @Kasieczka:2017nvn; @Pearkes:2017hku; @Butter:2017cot; @Egan:2017ojy], $b$-quarks [@CMS-DP-2017-005; @ATL-PHYS-PUB-2017-003; @Sirunyan:2017ezt], and light quarks [@Komiske:2016rsd; @CMS-DP-2017-027; @ATL-PHYS-PUB-2017-017; @Bhimji:2017qvb], for removing noise [@Komiske:2017ubm], and for emulating particle interactions with calorimeters [@deOliveira:2017pjk; @Paganini:2017hrr; @Paganini:2017dwg]. These new methods achieve excellent performance by exploiting subtle features of the simulations, which are presumed to be similar to the features in the data. Unfortunately, the simulations are known to be imperfect. This is particularly true for subtle features in high-dimensions, as illustrated clearly for boosted $W$ bosons in Ref. [@Barnard:2016qma] and by the need for non-negligible corrections (“scale factors”) to be applied to multivariate classifiers used by the current LHC experiments (see e.g. ). Thus it is natural to question the performance of machine learning algorithms trained on simulations as we know that if a model is trained on unphysical artifacts, this is what the model will learn. This objection certainly has merit, as the power of these methods for physics applications stems precisely from their ability to find features that we do not fully understand and cannot easily interpret. Data-driven approaches avoid the pitfalls of relying on simulations in experimental analyses. For simple observables, such as the invariant mass of a photon pair, a traditional experimental approach has been to perform sideband fits directly to the data. This avoids relying on the simulation altogether. Unfortunately, most of the sophisticated discrimination techniques developed in recent years use [*full supervision*]{}, where truth information is needed in order to train the classifier. However, real data generally consist only of mixed samples without truth information, arising from underlying statistical or quantum mixtures of two classes (henceforth referred to as “signal" and “background"). Occasionally one can find a small region of phase space where the signal or background is pure, but these regions are generally sparsely populated and may not produce representative distributions. Recent work on *weak supervision* [@hernandez2016weak] allows classifiers to be trained using only the information available from mixed samples. Two weakly supervised paradigms tailored to physics applications are Learning from Label Proportions (LLP) [@Dery:2017fap] and Classification Without Labels (CWoLa) [@Metodiev:2017vrx]. considered the problem of discriminating the radiation pattern of quark from gluons ($q$/$g$) using three standard observables and showed how to achieve fully supervised discrimination power by using LLP with two samples of different but known quark fractions. In , it was shown that the proportions are not necessary for training since the likelihood ratio of the mixed samples is monotonically related to the signal/background likelihood ratio, the optimal binary classifier for signal vs. background. One potential objection to the weak-learning demonstrations in is that the dimensionality of the inputs used is small. Indeed, for a one-dimensional discriminant one can extract the exact pure distributions from mixed samples using the fractions. It is not obvious that weak supervision will succeed when trained on high-dimensional inputs where the feature space may be sparsely populated. Indeed, the most powerful modern methods are trained on high-dimensional, low-level inputs, where numerically approximating and weighting the probability distribution is completely intractable. These deep learning techniques can expose subtle correlations in many dimensions which are also much harder to model than simple low-dimensional features. In this paper, we demonstrate that weak supervision can approach the effectiveness of full supervision on complex models with high-dimensional inputs. As a concrete illustration, we use an image representation to distinguish the radiation pattern from high energy quarks from gluons (“jet images" [@Cogan:2014oua]). Convolutional neural networks (CNNs) are applied to the quark and gluon jet images, where the dimensionality of the inputs is $\mathcal{O}(1000)$ and simulation mis-modeling issues are a challenge [@Aad:2014gea; @ATLAS-CONF-2016-034; @CMS-PAS-JME-13-002; @CMS-DP-2016-070; @CMS-PAS-JME-16-003; @ATL-PHYS-PUB-2017-009]. We find that CWoLa more robustly generalizes to learning with high-dimensional inputs than LLP, with the latter requiring careful engineering choices to achieve comparable performance. Though we use a particle physics problem as an example, the lessons about learning from data using mixtures of signal and background are applicable more broadly. We begin by establishing some notation and formulating the problem. Let ${{\bf{x}}}$ represent a vector of observables (*features*) useful for discriminating two classes we call *signal* ($S$) and *background* ($B$). For example, ${{\bf{x}}}$ might be the momenta of observed particles, calorimeter energy deposits, or a complete set of observables [@Datta:2017rhs; @Komiske:2017aww]. In fully supervised learning, each training sample is assigned a truth label such as 1 for signal and 0 for background. Then the fully supervised model is trained to predict the correct labels for each training example by minimizing a loss function. For a sufficiently large training set, an appropriate model parameterization, and a suitable minimization procedure, the learned model should approach the optimal classifier defined by thresholding the likelihood ratio. Data collected from a real detector do not come with signal/background labels. Instead, one typically has two or more *mixtures* $M_a$ of signal and background with different signal fractions $f_a$, such that the distribution of the features, $p_{M_a}({{\bf{x}}})$, is given by: $$\label{eq:decomp} p_{M_a}({{\bf{x}}}) = f_a \,p_S({{\bf{x}}}) + (1 - f_a)\, p_B({{\bf{x}}}),$$ where $p_S$ and $p_B$ are the signal and background distributions, respectively. Weak supervision assumes [*sample independence*]{}, that Eq. \[eq:decomp\] holds with the same distributions $p_S({{\bf{x}}})$ and $p_B({{\bf{x}}})$ for all mixtures. Although in most situations sample independence does not hold perfectly (see e.g. ), it is often a very good approximation (cf. Table \[tab:sampledep\] below). LLP uses any fully supervised classification method and modifies the loss function to globally match the signal fraction predicted by the model on a batch of training samples to the known truth fractions $f_a$. Breaking the training set into batches, normally done to parallelize training, takes on a new significance with LLP since the loss function is evaluated globally on each batch. The batch size, which for LLP we define as the number of samples drawn from each mixture during one update of the model, is a critical hyperparameter of LLP. The loss functions we use for LLP differ from those in . Analogous to the mean squared error (MSE) loss function for fully supervised (or CWoLa) training, we introduce the weak MSE (WMSE) loss for the LLP framework: $$\label{eq:wmse}\ell_{\rm WMSE} = \sum_{a} \left(f_a - \frac{1}{N}\sum_{i=1}^{N}h({{\bf{x}}}_i)\right)^2,$$ where $N$ is the batch size, $a$ indexes the mixed samples, and $h$ is the model. Analogous to the crossentropy, we also introduce the weak cross entropy (WCE) loss: $$\label{eq:wce}\ell_{\rm WCE} = \sum_{a} \text{CE}\left(f_a,\, \frac{1}{N} \sum_{i=1}^{N}h({{\bf{x}}}_i)\right),$$ where $\text{CE}(a, b) = - a \log b - (1 - a) \log (1 - b)$. One caveat we discovered while exploring LLP is that the range of $h({{\bf{x}}})$ must be restricted to $[0,1]$, otherwise the model falls into trivial minima of the loss function. We also observe the effect of model outputs becoming effectively binary at 0 and 1, necessitating additional care to avoid numerical precision issues. CWoLa classifies two mixtures, $M_1$ and $M_2$, from each other using any fully supervised classification method. The resulting classifier is then used to directly distinguish the original signal and background processes. Amazingly, the CWoLa classifier asymptotically (as the amount of training data increases) approaches an ideal classifier trained on pure samples [@Metodiev:2017vrx; @scott2013; @Cranmer:2015bka]. CWoLa does not require that the fractions $f_a$ are known for training (the fractions on smaller test sets can be used to calibrate the classifier operating points). The CWoLa framework has the nice property that as the samples approach complete purity ($f_1\to0,\,f_2\to1$) it smoothly approaches the fully supervised paradigm. CWoLa presently only works with two mixtures; if more than two are available they can be pooled at the cost of diluting their purity. The key features of CWoLa and LLP are compared in . Note that no learning is possible with either method as $f_1\rightarrow f_2$. ------------------------------------- ----------------------- -- -------------------------   [**LLP**]{} [**CWoLa**]{} Compatible with any trainable model No training modifications needed Training does not need fractions Smooth limit to full supervision Works for $>2$ mixed samples **?** ------------------------------------- ----------------------- -- ------------------------- : The essential pros (), cons (), and open questions (**?**) of the CWoLa and LLP weak supervision paradigms.[]{data-label="tab:comparison"} To explore weak supervision methods with high-dimensional inputs, we simulate $Z+q/g$ events at $\sqrt{s}=13$ TeV using Pythia 8.226 [@Sjostrand:2007gs] and create artificially mixed samples with various quark (signal) fractions. Jets with transverse momentum $p_T^{\text{jet}}\in[250,275]\text{ GeV}$ and rapidity $|y|\le2.0$ are obtained from final-state, non-neutrino particles clustered using the anti-$k_t$ algorithm [@Cacciari:2008gp] with radius $R=0.4$ implemented in FastJet 3.3.0 [@Cacciari:2011ma]. Single-channel, $33\times33$ jet images [@Cogan:2014oua; @deOliveira:2015xxd; @Komiske:2016rsd] are constructed from a patch of the pseudorapidity-azimuth plane of size $0.8\times0.8$ centered on the jet, treating the particle $p_T$ values as pixel intensities. The images are normalized so the sum of the pixels is 1 and standardized by subtracting the mean and dividing by the standard deviation of each pixel as calculated from the training set. All instantiations and trainings of neural networks were performed with the python deep learning library Keras [@keras] with the TensorFlow [@tensorflow] backend. A CNN architecture similar to that employed in was used: three 32-filter convolutional layers with filter sizes of $8\times 8$, $4\times 4$, and $4\times 4$ followed by a 128-unit dense layer. Maxpooling of size $2\times2$ was performed after each convolutional layer with a stride length of 2. The dropout rate was taken to be 0.1 for all layers. Keras VarianceScaling initialization was used to initialize the weights of the convolutional layers. Due to numerical precision issues caused by the tendency of LLP to push outputs to 0 or 1, a softmax activation function was included as part of the loss function rather than the model output layer. Validation and test sets were used consisting each of 50k 50%-50% mixtures of quark and gluon jet images. Training was performed with the Adam algorithm [@adam] with a learning rate of 0.001 and a validation performance patience of 10 epochs. Each network was trained 10 times and the variation of the performance was used as a measure of the uncertainty. Unless otherwise specified, the following are used by default: Exponential Linear Unit (ELU) [@clevert2015fast] activation functions for all non-output layers, the CE loss function for CWoLa, and the WCE loss function for LLP. The performance of a binary classifier can be captured by its receiver operating characteristic (ROC) curve. To condense the classifier performance into a single number, we use the area under the ROC curve (AUC). The AUC is also the probability that the classifier output is higher for signal than for background. Random classifiers have $\text{AUC}=0.5$ and perfect classifiers have $\text{AUC}=1.0$. We also confirmed that our conclusions are unchanged when using the background mistag rate at 50% signal efficiency as a performance metric instead. ![The AUC and training time of CWoLa (solid) and LLP (dashed) as the batch size is varied. Training times are measured on an NVIDIA Tesla K80 GPU using CUDA 8.0, TensorFlow 1.4.1, and Keras 2.1.2. AUC is a measure of classifier performance and is 1 for a perfect classifier and 0.5 for a completely random one.[]{data-label="fig:batchsweep"}](figures/BatchSize_Timing_Sweep){width="\columnwidth"} As previously noted, the LLP paradigm works by matching the predicted fraction of signal events to the known fraction for multiple mixed samples. In , the averaging took place over the entire mixed sample. Averaging over the entire training set at once is effectively impossible for high-dimensional inputs such as jet images because the graphics processing units (GPUs) that are needed to train the CNNs in a reasonable amount of time typically do not have enough memory to hold the entire training set at one time. Hence, the ability to train with batches is highly desirable for using LLP with high-dimensional inputs. There are many tradeoffs inherent with choosing the LLP batch size. Smaller batch sizes are susceptible to shot noise in the sense that the actual signal fraction on that batch may differ significantly from the fraction for the entire mixed sample, an effect which decreases as the batch size increases. Smaller batch sizes result in longer training times per epoch (because the full parallelization capabilities of the GPU cannot be used) but often require fewer epochs to train. Larger batch sizes have shorter training times per epoch but typically require more epochs to train. For CWoLa, the batch size plays the same role as in full supervision, with the performance being largely insensitive to it but the total training time varying slightly. These tradeoffs are captured in , which shows both the performance and training time for CWoLa and LLP models as the batch size is swept in powers of two from 64 to 16384, trained on two mixtures with $f_1 = 0.2$ and $f_2 = 0.8$. The expected independence of CWoLa performance and the degradation of LLP performance for low batch sizes can clearly be seen. The training time curves are concave with optimum batch sizes toward the middle of the swept region. Based on this figure, we choose default batch sizes of $4000$ for LLP and 400 for CWoLa. In order to explore a slightly more realistic scenario than artificially mixing samples from the same distribution of quarks and gluons, we generate $Z+\text{jet}$ and dijet events with the same generation parameters and cuts as described previously. These “naturally” mixed samples have quark fractions $f_{Z+\text{jet}}=0.88$ and $f_{\text{dijets}}=0.37$. The signal and background fractions have been systematically explored for these and many other processes in . As indicated by Table \[tab:sampledep\], there is no significant difference in performance on the naturally mixed or artificially mixed samples. Hence, artificially mixed samples are used in the rest of this study in order to evaluate weak supervision performance at different quark purities. **Learning** -------------- -------------------------- --------------------- -- $Z$+jet vs. dijets 0.8626 $\pm$ 0.0020 Artificial $Z$ + $q$/$g$ 0.8621 $\pm$ 0.0019 $Z$+jet vs. dijets 0.8544 $\pm$ 0.0019 Artificial $Z$ + $q$/$g$ 0.8549 $\pm$ 0.0018 : AUCs for training with CWoLa and LLP on $Z+\text{jet}$ and dijet samples as well as on artificial mixtures of $Z+g$ and $Z+q$ samples. The error given is the interquartile range. There is no significant difference in classifier performance between the naturally mixed ($Z$+jet vs. dijets) samples and the artificially mixed ($Z + q/g$) samples with the same signal fractions.[]{data-label="tab:sampledep"} compares CWoLa and LLP performance for various quark/gluon purities as a function of the number of training samples. Each network is trained using two samples, one with quark fraction $f_1$ and the other with quark fraction $f_2=1-f_1$. Each point in the figure is the median of 10 independent network trainings and the error bars show the $25^{\text{th}}$ and $75^{\text{th}}$ percentiles. Full supervision performance corresponds to CWoLa with $f_1=0$. The most important takeaway from is that we have achieved good performance with both weak supervision methods over a large variety of sample purities and training sample sizes. We also see that CWoLa consistently outperforms LLP and continues to get better as additional training samples are used, likely a result of the increasingly-populated feature space, whereas LLP performance tends to level off. It should be noted that given the binary output nature of LLP models, classifiers trained in this way effectively come with a working point and sweeping the threshold to produce a ROC curve may not be ideal. The purity/data tradeoff analysis of can provide valuable information for practical applications of weak supervision methods in physics, particularly in cases where more data can be acquired at the expense of worsening sample purity. The sensitivity of LLP to different choices of loss function and activation function was examined. We studied the choices of the symmetric squared loss of and the weak crossentropy loss of with Rectified Linear Unit (ReLU) [@nair2010rectified] and ELU activation functions. We found a significant improvement in LLP classification performance in using ELU activations instead of ReLU activations, particularly at high signal efficiencies. The choice of loss function was found to be less important than the choice of activation function, but minor improvements in AUC were observed with the WCE loss function over WMSE. We also studied the dependence of CWoLa on the choice of activation function and found consistent performance between ELU and ReLU activations. These results justify our default choices of ELU activation and WCE loss functions. With the choice of ELU activation, LLP achieves almost the same performance to our CWoLa-trained network near the operating point with equal signal and background efficiencies. We suspect this is a result of the tendency of LLP to output binary predictions (near 0 or 1) rather than a continuous output that can be easily thresholded. Lastly, LLP has the potential advantage over the present implementation of CWoLa that it can naturally encompass multiple mixed samples with different purities. While in principle adding more samples should help, it is not obvious whether the network will effectively take advantage of them. Indeed, we did not find significant improvement to LLP when adding additional samples with intermediate purities, even after significant, dedicated architecture engineering. ![Classifier performance (AUC) shown for both CWoLa (solid) and LLP (dashed) trained on two mixed samples with various signal fractions $f_1,\,1-f_1$ as the number of training data is varied between 100k and 1M. Each training is repeated 10 times and the $25^{\text{th}}$, $50^{\text{th}}$, and $75^{\text{th}}$ percentiles are shown. The $f_1 = 0.0$ CWoLa curve corresponds to full supervision. CWoLa outperforms LLP by this metric, though both methods work quite well.[]{data-label="fig:fracdatasweep"}](figures/Fraction_NumData_Sweep){width="\columnwidth"} In conclusion, we have shown that machine learning approaches using very high-dimensional inputs can be trained directly on mixtures of signal and background, and therefore on data. This addresses one of the main objections to the use of modern machine learning in jet tagging: sensitivity to untrustworthy simulations. We have implemented and tested weakly supervised learning with both LLP and CWoLa, finding that for the quark/gluon discrimination problem considered here CWoLa outperforms LLP and is less sensitive to particular hyperparameter choices. We have developed a method for training LLP with high-dimensional inputs in batches and demonstrated that the batch size is a critical hyperparameter for both performance and training time. Given any fully supervised classifier, CWoLa works “out-of-the-box” whereas LLP requires additional engineering to achieve good performance and is generally harder to train. Nonetheless, the success in using both of these weak supervision approaches on high-dimensional data is encouraging for the future of modern machine learning techniques in particle physics and beyond. The authors would like to thank Lucio Dery and Francesco Rubbo for collaboration in the initial stages of this work. We are grateful to Jesse Thaler for helpful discussions. PTK and EMM would like to thank the MIT Physics Department for its support. Computations for this paper were performed on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University. This work was supported by the Office of Science of the U.S. Department of Energy (DOE) under contracts DE-AC02-05CH11231 and DE-SC0013607, the DOE Office of Nuclear Physics under contract DE-SC0011090, and the DOE Office of High Energy Physics under contract DE-SC0012567. Cloud computing resources were provided through a Microsoft Azure for Research award. Additional support was provided by the Harvard Data Science Initiative.

--- abstract: 'The expansion of programmatically-accessible materials data has cultivated opportunities for data-driven approaches. Highly-automated frameworks like  not only manage the generation, storage, and dissemination of materials data, but also leverage the information for thermodynamic formability modeling, such as the prediction of phase diagrams and properties of disordered materials. In combination with standardized parameter sets, the wealth of data is ideal for training machine learning algorithms, which have already been employed for property prediction, descriptor development, design rule discovery, and the identification of candidate functional materials. These methods promise to revolutionize the path to synthesis and, ultimately, transform the practice of traditional materials discovery to one of rational and autonomous materials design.' author: - Corey Oses - Cormac Toher - Stefano Curtarolo title: 'Autonomous data-driven design of inorganic materials with AFLOW' --- Introduction {#introduction .unnumbered} ============ Density functional theory implementations [@kresse_vasp; @VASP4_2; @vasp_cms1996; @vasp_prb1996; @quantum_espresso_2009; @gonze:abinit; @Blum_CPC2009_AIM] offer a reasonable compromise between cost and accuracy [@Haas_PRB_2009], stimulating rapid development of automated frameworks and corresponding data repositories. Prominent examples include  (the utomatic Framework for Materials Discovery) , [NoMaD]{} (vel terials iscovery Laboratory) [@nomad], Materials Project [@APL_Mater_Jain2013], [OQMD]{} (pen uantum aterials atabase) [@Saal_JOM_2013], Computational Materials Repository and its associated scripting interface [ASE]{} (tomic imulation nvironment) [@cmr_repository], and [AiiDA]{} (utomated nteractive nfrastructure and tabase for Computational Science) [@Pizzi_AiiDA_2016]. Such repositories house an abundance of materials data. For instance, the  database contains over 1.8 million compounds, each characterized by about 100 different properties [@aflowlibPAPER; @curtarolo:art92; @curtarolo:art104; @aflux]. Investigations employing this data have not only led to advancements in modeling electronics [@curtarolo:art77; @curtarolo:art94; @curtarolo:art124; @ceder:nature_1998], thermoelectrics [@curtarolo:art120; @curtarolo:art129], superalloys [@curtarolo:art113], and metallic glasses [@curtarolo:art112], but also the synthesis of two new magnets — the first discovered by computational approaches [@curtarolo:art109]. Further advancements and discoveries are contingent on continued development and expansion of these materials repositories. New entries are generated both by **i.** calculating the properties of previously observed compounds from sources such as the Inorganic Crystal Structure Database [@ICSD], and **ii.** decorating structural prototypes [@aflowANRL] to predict new materials. Accurate computation of materials properties — including electronic, magnetic, chemical, crystallographic, thermomechanical, and thermodynamic features — demands a combination of **i.** reliable calculation parameters/thresholds [@curtarolo:art104] and **ii.** robust algorithms that scale with the size/diversity of the database. For example, convenient definitions for the primitive cell representation [@aflowPAPER] and high-symmetry Brillouin Zone path [@curtarolo:art58] have both optimized and standardized electronic structure calculations. Moreover, careful treatment of spatial tolerance and proper validation schemes have finally enabled accurate and fully autonomous determination of the complete symmetry profile of crystals [@curtarolo:art135], which is essential for elasticity [@curtarolo:art115] and phonon [@aflowPAPER; @curtarolo:art114; @curtarolo:art119; @curtarolo:art125] calculations. Beyond descriptions of simple crystals, exploration of complex properties [@curtarolo:art96; @curtarolo:art115] and materials [@curtarolo:art110; @curtarolo:art112] typically warrants advanced (and expensive) characterization techniques [@Hedin_GW_1965; @GW; @ScUJ]. Fortunately, state-of-the-art workflows [@curtarolo:art96; @curtarolo:art110; @curtarolo:art115] and careful descriptor development [@curtarolo:art112] have enabled experimentally-validated modeling within a density functional theory framework. Furthermore, the combination of plentiful and diverse materials data and its programmatic accessibility [@curtarolo:art92; @aflux] justify the application of data-mining techniques. These methods can quantitatively resolve subtle trends and correlations among materials and their properties [@curtarolo:art94; @Ghiringhelli_PRL_2015; @curtarolo:art124; @curtarolo:art129; @curtarolo:art135], as well as motivate the formulation of novel property descriptors [@curtarolo:art112; @Lederer_HEA_2018]. These “black-box” models are surprisingly accurate and quite valuable, particularly when few practical alternative modeling schemes exist — as is the case for predicting superconducting critical temperatures [@curtarolo:art94; @curtarolo:art137]. Ultimately, the power in  lies in the speed of its predictions, which out-paces density functional theory calculations by orders of magnitude [@Isayev_ChemSci_2017]. Given that the number of currently characterized materials pales in comparison to the full space of hypothetical structures, methods to filter/screen the most interesting candidate materials [@Walsh_NChem_2015] — powered by  models — will undoubtedly become integral to future materials discovery workflows. Thermodynamic formability modeling {#thermodynamic-formability-modeling .unnumbered} ================================== {width="100.00000%"} **Prediction of phase diagrams.** Descriptions of thermodynamic stability and structural/chemical disorder are resolved through statistical analyses of aggregate sets of structures. Thermodynamic stability largely governs synthesizability, which can be determined by an analysis of how structures of similar compositions compete energetically, *i.e.*, determination of the minimum Gibbs free energy surface. The procedure is algorithmically equivalent to finding the lower-half convex hull of all the relative free energy minima [@curtarolo:art20] as illustrated in Figure \[fig:GFA\_descriptor\]. Composition and energy information from relevant  calculations are plotted, and the phases defining the minimum energy surface are identified [@qhull]. Assuming sufficient sampling, the ground state structures on the minimum energy surface form the low-temperature phase diagram [@monsterPGM]. The convex hull construction offers a wealth of related thermodynamic properties. For near-hull structures, the energetic distance from the minimum energy surface is treated as a metric for synthesizability, as only small perturbations in temperature or pressure may be needed for it to be realized. In fact, this distance is equivalent to the amount of energy driving the decomposition of an unstable state to a linear combination of nearby ground state structures. A similar distance — that of a stable phase from the pseudo-hull formed by neglecting it — quantifies the impact of a structure on the minimum energy surface and characterizes the robustness of stable structures, *i.e.*, the stability criterion.  offers a module for autonomous calculation of the convex hull, which retrieves the set of relevant structure calculations from the repository [@curtarolo:art92; @aflux] and delivers a thorough thermodynamic characterization for each. Filtering schemes based on these thermodynamic properties, including the stability criterion and tie-line construction, played key roles in the discovery of new magnets [@curtarolo:art109] and modeling superalloys [@curtarolo:art113]. The module powers an online web application for enhanced visualization of two/three-dimensional hulls available at [aflow.org/aflow-chull]{}. **Modeling disordered materials.** Incorporating the effects of disorder is a necessary, albeit difficult, step in materials modeling. Not only is disorder intrinsic to all materials, but it also offers a route to enhanced and even otherwise inaccessible functionality. Disordered materials range from chemically disordered high entropy materials and solid solutions, in which sites on a periodic crystal lattice are randomly occupied, to structurally disordered amorphous glasses, with no crystalline periodicity. Materials such as high entropy alloys [@Gao_HEA_book_2015; @Miracle_HEAs_NComm_2015] containing four to five metallic elements in equi-composition are being investigated for their enhanced thermomechanical properties [@HEAapp1; @MoNbTaW; @Gludovatz_hea_mech_properties; @MoNbTaW2; @HEAprop2], and have also been reported to display superconductivity [@HEAprop1]. Research interest has recently expanded beyond metallic alloys to include high entropy ceramics such as entropy stabilized oxides [@curtarolo:art99; @curtarolo:art122] and high-entropy borides [@Gild_borides_SciRep_2016], which display promising behavior including colossal dielectric constants [@Beradan_2016_PSSA_ESO_Colossal] and superionic conductivity [@Beradan_2016_JMCA_ESO_superionic]. *Ab-initio* modeling of chemical/substitutional disorder — including vacancies and random site occupations — is a notoriously formidable problem, since it results in systems that cannot be described directly by a single unit cell with periodic boundary conditions. Rigorous statistical treatment of chemical disorder leverages a set of representative ordered supercells in thermodynamic competition. System-wide properties are resolved through ensemble averages of these supercells. The approach has been implemented in  [@curtarolo:art110] for autonomous characterization, and successfully validated for a number of technologically significant systems, recovering characteristic trends as a function of composition and offering additional insight into underlying physical mechanisms. The module determines the smallest superlattice size that accommodates the required stoichiometry to within a user-defined tolerance, and then generates the corresponding superlattices using Hermite Normal Form matrices [@gus_enum1]. All allowed decoration permutations are considered for each superlattice variant, generating the full set of possible supercell configurations. Degeneracies are rapidly identified by comparing approximate structure energies calculated with the Universal Force Field method [@Rappe_1992_JCAS_UFF]. Only unique supercells are individually characterized using standard *ab-initio* packages [@kresse_vasp; @VASP4_2; @vasp_cms1996; @vasp_prb1996; @quantum_espresso_2009; @gonze:abinit; @Blum_CPC2009_AIM]. The ensemble average values of properties such as the electronic band gap, density of states, and the magnetic moment — weighted according to a Boltzmann distribution for a particular temperature — are then calculated to resolve the behavior of the disordered material. {width="100.00000%"} Metallic glasses lack an ordered lattice, and its associated defects, which endow them with a unique combination of superb mechanical properties [@chen2015does] and plastic-like processability [@schroers2006amorphous; @Schroers_blow_molding_2011; @kaltenboeck2016shaping], rendering them of great interest for several potential commercial and industrial applications [@Schroers_Processing_BMG_2010; @johnson2016quantifying; @ashby2006metallic]. To predict the lass orming bility () of metal alloy systems [@curtarolo:art112], statistical approaches have been employed that blend the concept of thermodynamically competing ordered structures with the large quantities of pre-calculated data available in the  repository. The proposed physical mechanism is that ordered phases which have similar energies, but are structurally distinct, will compete against each other during solidification, frustrating crystal nucleation and thus promoting glass formation, as illustrated in Figure \[fig:GFA\_descriptor\]. The energy distribution of the different structures can be considered as forming a thermodynamic density of states: a narrow distribution indicates a high , while a wider distribution implies a low . Atomic environment [@daams_villars:environments_2000; @daams:cubic_environments] comparisons determine the similarity of ordered crystalline phases, enabling the formulation of a quantitative descriptor that can be applied to the entire  database. The different structures are weighted according to a Boltzmann distribution to create the  descriptor. The model is found to successfully predict 73% of the glass forming compositions for a set of 16 experimentally well-characterized alloy systems, and also indicates that about 17% of binary alloy systems should be capable of glassification. By exploiting the pre-calculated data in the  repository, this model can be leveraged to rapidly predict  as a function of composition for thousands of alloy systems, demonstrating the power of applying intelligently constructed descriptors to computational materials data. The  formation energy data is also employed to train cluster expansion models [@atat2], which can be combined with thermodynamic modeling to predict the order-disorder transition temperature for solid-solutions in high-entropy alloys [@Lederer_HEA_2018]. Order-disorder transitions in the form of spinodal decomposition have also been proposed as a mechanism to **i.** embed topologically-protected conducting interface states in an insulating matrix [@curtarolo:art134] and **ii.** self-assemble nanostructures (such as thermoelectric devices [@curtarolo:art107]). The boundaries between different layers act as phonon-scatterers, suppressing the thermal conductivity and thus improving efficiency. Exploiting machine learning algorithms {#exploiting-machine-learning-algorithms .unnumbered} ====================================== **Model development.**  is rapidly emerging as a powerful tool for computational materials design [@Bhadeshia_ISIJ_ML_1999; @Ghiringhelli_PRL_2015; @PyzerKnapp_AdFM_2015; @Guzik_NMat_2016]. Given sufficient training data, algorithms such as neural networks [@sumpter_nnetworks_review_1996], random forests [@randomforests], gradient boosting decision trees [@Friedman_AnnStat_2001] and support vector machines [@Cortes_ML_SVM_1995] can learn to **i.** identify the structures that are thermodynamically accessible for a given composition [@Ghiringhelli_PRL_2015] and **ii.** accurately predict materials properties, such as the electronic band gap [@curtarolo:art124], elastic moduli [@curtarolo:art124; @deJong_SR_2016], vibrational energies [@curtarolo:art129], and lattice thermal conductivity [@curtarolo:art84]. The successful training of  models depends crucially on the set of features characterizing the material, *i.e.*, the set of descriptors that form the feature vector [@Ghiringhelli_PRL_2015]. Such representations include electronic structure fingerprints [@curtarolo:art94] and crystal graphs [@curtarolo:art124; @Xie_ML_CNN_2017]. Optimal descriptors are resolved by exploring different linear and non-linear combinations of properties, and extracting the most efficient feature vector via compressive sensing [@curtarolo:art135]. Compressive sensing finds the sparse solution ($\ell_0$-norm minimization) of the underdetermined system of linear equations mapping the set of observable materials properties to the large set of possible test features — effectively reducing the dimensionality of the problem. The algorithm also filters for physically meaningful combinations of properties, based on dimensional analysis, to maximize interpretability of the final descriptor set. Several different  frameworks are leveraging data from the  repository. The materials fingerprinting model [@curtarolo:art94] codifies aspects of the electronic structure [@curtarolo:art58] to serve as unique markers for each material. In particular, the number of bands that intersect high-symmetry Brillouin Zone points at discretized energy values form the band structure fingerprint (illustrated in Figure \[fig:ML\_fingerprints\]), while simple discretization of the density of states form the density of states fingerprint. The Tanimoto coefficient — a distance metric [@Bajusz_JCheminfo_2015] — between fingerprint vectors quantifies the similarity of the electronic structure between different materials. These fingerprints are employed for the construction of networks, *i.e.*, materials cartography [@curtarolo:art94], where materials are represented by nodes and similarity correlates with relative positioning. When applied to compounds in the Inorganic Crystal Structure Database, significant clustering and structure can be identified for these networks, particularly with respect to material complexity (binaries *versus* ternaries, *etc.*), type (metal *versus* insulator), and, surprisingly, superconducting critical temperature [@curtarolo:art94]. In the case of high-temperature superconductors, significant clustering suggests strong correlations among the electronic structure of these materials; although, as expected, these features alone are not enough to quantitatively resolve critical temperatures. Indeed, modeling improves with integration of more experimental observations [@Supercon; @curtarolo:art137] and properties, such as structural features and partial charges [@bader3]. Incorporating additional relevant and physically meaningful training data, such as the phonon spectra, should offer an applicability domain expansion and higher fidelity predictions. Thermomechanical properties calculated using the elastic constants [@curtarolo:art115] and Debye-Gr[ü]{}neisen [@curtarolo:art96] modules of  have been employed to train a gradient boosting decision trees framework [@curtarolo:art124] to predict quantities such as the bulk and shear modulus, Debye temperature, and heat capacity. Indicative of its versatility, the same model [@curtarolo:art124] has also been trained on  electronic structure data to classify materials as metals or insulators, and to predict the electronic band gap for compounds identified as non-metals. Model development is based on a fragment construction approach: each crystal is represented by a graph where nodes are decorated with corresponding atomic properties and connectivity is dictated by distance and Voronoi polyhedra adjacency. Path and circular fragments — representative of linear geometry and coordination polyhedra within the crystal — form the basis for feature development. To train the models, the gradient boosting decision tree algorithm is employed, which amalgamates a series of weak, easily constructed prediction rules to resolve a single, highly predictive function. The resulting models have been thoroughly validated with simulated and real tests sets, showing predictive metrics at 90% or higher against existing calculated and experimental measurements. Beyond property value prediction, feature-importance analyses of the models recovers meaningful ways to tune the band gap and Debye temperature, offering practical design rules for device engineering. The development of such models achieves the greatest impact on thermomechanical properties, where characterizations demand many single *ab-initio* calculations, and thus presents a substantial boost in prediction speed at a fraction of the resources. **Workflow integration.**  approaches are expected to become indispensable in two specific scenarios, prediction of complex properties and screening of large sets of materials. Unfortunately, widespread exploitation of  techniques in materials science has been hindered by the difficulty of setting up and interfacing the models with materials design infrastructures. To streamline this process, the   [@aflowmlapi] has been created to provide programmatic access to the  models described in Refs.  and , with plans to extend it with additional models as they are developed. By posting a structure file to the , users can retrieve  predictions of thermal, mechanical, and electronic properties in the  data format. In this way, all technical details of the  algorithms are abstracted away, rendering a simple interface no more complicated than that of a standard . This procedure can be easily incorporated into materials design workflows, due to its use of ubiquitous HTTP commands, along with the  format that is easily interpreted by a wide range of modern programming languages. Source code and online forum {#source-code-and-online-forum .unnumbered} ============================ Since 2018, the  software (V3.1.193) has been made available for download/redistribution under the terms of the GNU License [http://www.gnu.org/licenses]{}. The source code/license/readme files can be found by following the links in the  project website. Though some of the aforementioned modules are conveniently interfaced through the website, only the executable offers full and unabridged functionality. Additionally, the Forum ([aflow.org/forum]{}) advertises updates and new functionality, as well as hosts discussion boards for registered members to post questions.\ Conclusion {#conclusion .unnumbered} ========== Broad scale thermodynamic formability modeling and exploitation of  algorithms represent current frontiers in computational materials design. Recent progress in these fields has been enabled by large, programmatically-accessible materials databases generated by automated computational infrastructures. Ensembles of ordered phases are being successfully employed to **i.** construct phase diagrams and **ii.** formulate descriptors and models to predict the formation and properties of disordered materials.  models have the potential to rapidly accelerate materials design as tools for predicting properties and identifying subtle/hidden trends — thus leading to enhanced understanding of the physical mechanisms underlying materials behavior. 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--- abstract: 'With the explosion of online news, personalized news recommendation becomes increasingly important for online news platforms to help their users find interesting information. Existing news recommendation methods achieve personalization by building accurate news representations from news content and user representations from their direct interactions with news (e.g., click), while ignoring the high-order relatedness between users and news. Here we propose a news recommendation method which can enhance the representation learning of users and news by modeling their relatedness in a graph setting. In our method, users and news are both viewed as nodes in a bipartite graph constructed from historical user click behaviors. For news representations, a transformer architecture is first exploited to build news semantic representations. Then we combine it with the information from neighbor news in the graph via a graph attention network. For user representations, we not only represent users from their historically clicked news, but also attentively incorporate the representations of their neighbor users in the graph. Improved performances on a large-scale real-world dataset validate the effectiveness of our proposed method.' author: - Suyu Ge - Chuhan Wu - Fangzhao Wu - Tao Qi - Yongfeng Huang bibliography: - 'main.bib' title: | Graph Enhanced Representation Learning\ for News Recommendation --- =4 &lt;ccs2012&gt; &lt;concept&gt; &lt;concept\_id&gt;10002951.10003317.10003347.10003350&lt;/concept\_id&gt; &lt;concept\_desc&gt;Information systems Recommender systems&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010147.10010178.10010179&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computing methodologies Natural language processing&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;/ccs2012&gt;

--- abstract: | Massive neutrinos demand to ask whether they are Dirac or Majorana particles. Majorana neutrinos are an irrefutable proof of physics beyond the Standard Model. Neutrinoless double electron capture is not a process but a virtual $\Delta L=2$ mixing between a parent $^AZ$ atom and a daughter $^A(Z-2)$ excited atom with two electron holes. As a mixing between two neutral atoms and the observable signal in terms of emitted two-hole X-rays, the strategy, experimental signature and background are different from neutrinoless double beta decay. The mixing is resonantly enhanced for almost degeneracy and, under these conditions, there is no irreducible background from the standard two-neutrino channel. We reconstruct the natural time history of a nominally stable parent atom since its production either by nature or in the laboratory. After the time periods of atom oscillations and the decay of the short-lived daughter atom, at observable times the relevant “stationary" states are the mixed metastable long-lived state and the short-lived excited state, as well as the ground state of the daughter atom. Their natural population inversion is most appropriate for exploiting the bosonic nature of the observed X-rays by means of stimulating X-ray beams. Among different observables of the atom Majorana mixing, we include the enhanced rate of stimulated X-ray emission from the long-lived metastable state by a high-intensity X-ray beam. A gain factor of 100 can be envisaged in a facility like European XFEL. address: 'Departament de Física Teòrica & IFIC, Universitat de València - CSIC, C/ Dr. Moliner 50, E-46100 Burjassot (Spain)' author: - A Segarra and J Bernabéu title: 'Stimulated X-rays in resonant atom Majorana mixing' --- Introduction ============ The experimental evidence of neutrino oscillations [@1; @2] implies that neutrinos are massive particles and that the three flavor neutrinos are mixtures of the neutrinos with definite masses. The existence of non-vanishing neutrino masses opens up the most fundamental question of whether neutrinos are Dirac or Majorana particles, which cannot be answered by neutrino oscillation experiments. Up to now, there is a consensus that the highest known sensitivity to small Majorana neutrino masses can be reached in experiments on the search of the $L$-violating neutrinoless double-$\beta$ decay process ($0\nu\beta\beta$) $$\label{eq:0nbb} ^AZ \to\, ^A(Z+2) + 2e^-\,,$$ where $^AZ$ is a nucleus with atomic number $Z$ and mass number $A$. This observable is proportional to the key parameter $$\label{eq:mbb} m_{\beta\beta} \equiv \sum_i U^2_{ei}\, m_{\nu_i}\,,$$ which is a coherent combination of the three neutrino masses. There is an alternative to $0\nu\beta\beta$ by means of the mechanism of neutrinoless double electron capture ($0\nu\text{ECEC}$), $$\label{eq:0necec} ^AZ + 2 e^- \to\, ^A(Z-2)^* \,.$$ This is actually a mixing between two states of two different neutral atoms differing in the total lepton number $L$ by two units, and the same baryonic number $A$, and not a process conserving energy and momentum in general. The daughter atom is in an excited state with two electron holes and its decay provides the signal for (\[eq:0necec\]). We study the implications of this mixing between parent and daughter atom. The evolution Hamiltonian ========================= In the basis of the ${\left| ^AZ\, \right>}$ and ${\left| ^A(Z-2)^* \right>}$ states, which we’ll refer to as 1 and 2, the dynamics of this two-state system of interest is governed by the Hamiltionian $$\dsH = \dsM - \frac{i}{2}\, \dsGamma = \left[ \begin{aligned} M_1 &\hspace{0.2cm} M_{21}^* \\ M_{21} &\hspace{0.2cm} M_2 \end{aligned} \right] - \frac{i}{2} \left[ \begin{aligned} 0 &\hspace{0.2cm}0 \\ 0 &\hspace{0.2cm}\Gamma \end{aligned} \right] \,, \label{eq:H}$$ with a Majorana $\Delta L = 2$ mass mixing $M_{21}$ from Eq.(\[eq:0necec\]). The anti-Hermitian part of this Hamiltonian is due to the instability of ${\left| ^A(Z-2)^* \right>}$, which de-excites into ${\left| ^A(Z-2)_\text{g.s.} \right>}$, external to the two-body system in Eq.(\[eq:H\]), emitting its two-hole characteristic X-ray spectrum. Besides being non-Hermitian, $\dsH$ is not a normal operator, i.e. $[\dsM,\, \dsGamma] \neq 0$. As a consequence, $\dsM$ and $\dsGamma$ are not compatible. The states of definite time evolution, eigenstates of $\dsH$, have complex eigenvalues and are given in non-degenereate perturbation theory [@galindo] by $$\begin{aligned} \nonumber {\left| \lambda_L \right>} &= {\left| 1 \right>} + \alpha {\left| 2 \right>}, \hspace{2cm} \lambda_L \equiv E_L -\frac{i}{2}\, \Gamma_L = M_1 + {\left| \alpha \right|}^2 \left[ \Delta - \frac{i}{2}\, \Gamma \right] , \\ \label{eq:eigen} {\left| \lambda_S \right>} &= {\left| 2 \right>} - \beta^* {\left| 1 \right>}, \hspace{1.9cm} \lambda_S \equiv E_S -\frac{i}{2}\, \Gamma_S = M_2 -\frac{i}{2}\,\Gamma - {\left| \alpha \right|}^2 \left[ \Delta - \frac{i}{2}\, \Gamma \right],\end{aligned}$$ with $\Delta = M_1 - M_2$. As seen in Eq.(\[eq:eigen\]), $\Gamma_{L,S}$ are **not** the eigenvalues of the $\dsGamma$ matrix. The eigenstates are modified at first order in $M_ {21}$, $$\label{eq:alpha} \alpha = \frac{M_{21}}{\Delta + \frac{i}{2}\,\Gamma}\,,\hspace{2cm} \beta = \frac{M_{21}}{\Delta - \frac{i}{2}\,\Gamma}\,,$$ so the “stationary” states of the system don’t have well-defined atomic properties: both the number of electrons and their atomic properties are a superposition of $Z$ and $Z-2$. Also, these states are *not* orthogonal—their overlap is given by $${\left< \lambda_S | \lambda_L \right>} = \alpha - \beta = -i\frac{M_{21}\Gamma}{\Delta^2 + \frac{1}{4}\,\Gamma^2}\,,$$ with its non-vanishing value due to the joint presence of the mass mixing $M_{21}$ and the decay width $\Gamma$. Notice that Im$(M_{21})$ originates a real overlap. As seen in Eq.(\[eq:eigen\]), the modifications in the corresponding eigenvalues appear at second order in ${\left| M_{21} \right|}$ and they are equidistant with opposite sign. Since these corrections are small, from now on we will use the values $$\begin{aligned} \nonumber E_L &\approx M_1\,, \hspace{2.375cm} E_S \approx M_2\,,\\ \label{eq:GammaLS} \Gamma_L &\approx {\left| \alpha \right|}^2\, \Gamma\,,\hspace{2cm} \Gamma_S \approx \Gamma\,.\end{aligned}$$ The only relevant correction at order ${\left| \alpha \right|}^2$ is the one to $\Gamma_L$, since ${\left| 1 \right>}$ was a stable state—even if it’s small, the mixing produces a non-zero decay width. This result shows that, at leading order, the Majorana mixing becomes observable through $\Gamma_L \propto {\left| \alpha \right|}^2$. The value of $\alpha$ in Eq.(\[eq:alpha\]) emphasizes the relevance of the condition $\Delta \sim \Gamma$, which produces a Resonant Enhancement [@17] of the effect of the $\Delta L=2$ mass mixing $M_{21}$. Natural time history for initial $^AZ$ ====================================== As seen in Eq.(\[eq:eigen\]), the states ${\left| ^AZ\, \right>}$ and ${\left| ^A(Z-2)^* \right>}$ are not the stationary states of the system. For an initially prepared ${\left| ^AZ\, \right>}$, the time history is far from trivial and the appropriate language to describe the system short times after is that of *Atom Oscillations* between ${\left| ^AZ\, \right>}$ and ${\left| ^A(Z-2)^* \right>}$ due to the interference of the amplitudes through ${\left| \lambda_S \right>}$ and ${\left| \lambda_L \right>}$ in the time evolution. The time-evolved ${\left| ^AZ\, \right>}$ gives raise to the appearance probability $$\label{eq:t1} {\left| {\left< ^A(Z-2)^* | ^AZ(t) \right>} \right|}^2 = {\left| \alpha \right|}^2 \left\{ 1 + e^{-\Gamma t} - 2e^{-\frac{1}{2} \Gamma t}\cos(\Delta \cdot t) \right\}\,,$$ with an oscillation angular frequency ${\left| \Delta \right|}$. The characteristic oscillation time $\tau_\text{osc} = {\left| \Delta \right|}^{-1}$ is the shortest time scale in this system. For $t \ll \tau_\text{osc}$, one has $${\left| {\left< ^A(Z-2)^* | ^AZ(t) \right>} \right|}^2 \approx {\left| M_{21} \right|}^2\, t^2$$ induced by the mass mixing. The next shortest characteristic time in this system is the decay time $\tau_S = \Gamma^{-1}$, associated to the ${\left| \lambda_S \right>}$ state. For $\tau_\text{osc} \ll t \ll \tau_S$, the only change with respect to Eq.(\[eq:t1\]) is that the interference region disappears, and the two slits ${\left| \lambda_L \right>}$ and ${\left| \lambda_S \right>}$ contribute incoherently, $${\left| {\left< ^A(Z-2)^* | ^AZ(t) \right>} \right|}^2 \approx {\left| \alpha \right|}^2\,\left( 2-\Gamma t \right)\,.$$ For $t\gg \tau_S$, the contribution of ${\left| \lambda_S \right>}$ disappears and the appearance probability simply becomes $${\left| {\left< ^A(Z-2)^* | ^AZ(t) \right>} \right|}^2 = {\left| \alpha \right|}^2\,.$$ In other words, the initially prepared ${\left| ^AZ\, \right>}$ state evolves towards the stationary metastable state ${\left| \lambda_L \right>}$, $$\label{eq:tl} {\left| ^AZ(t) \right>} \to e^{-i\lambda_L t}{\left| \lambda_L \right>}\,,$$ with the long lifetime $\tau_L = \Gamma_L^{-1}$ from Eq.(\[eq:GammaLS\]). For a realistic time resolution $\delta t$ in an actual experiment, this regime is the interesting one, with the behavior in Eq.(\[eq:tl\]). The different time scales involved in this problem are thus $$\tau_\text{osc} \ll \tau_S \ll \delta t \ll t \ll \tau_L \,,$$ where $t$ refers to the elapsed time since the production of ${\left| ^AZ\, \right>}$, either by nature or in the lab—given the smallness of the mixing, the metastability of the state (\[eq:tl\]) is valid even for cosmological times. Therefore, for any time between the two scales $\tau_S$ and $\tau_L$, the populations of the three states involved are given by the probabilities\ $$\begin{aligned} \mbox{}\\ \hspace{3cm}\tau_S \ll t \ll \tau_L \hspace{0.5cm} \Longrightarrow \hspace{0.5cm} \left\{\begin{aligned} P_L(t) &\approx 1-\Gamma_L\, t\\ P_S(t) &\approx 0\\ P_\text{g.s.}(t) &\approx {\left| \alpha \right|}^2\, \Gamma\, t \end{aligned} \right. \\ \mbox{} \end{aligned}$$ \[eqs:P\] $$\begin{aligned} \mbox{} \label{eq:PL}\\ \mbox{} \label{eq:PS}\\ \mbox{} \label{eq:Pgs} \end{aligned}$$ \ where $P_\text{g.s.}(t)$ refers to the population of the ground state of the $^A(Z-2)$ atom after the decay of the unstable “stationary” state ${\left| \lambda_S \right>}$, with rate $\Gamma$. No matter whether $t$ refers to laboratory or cosmological times, the linear approximation in $t$ is excellent. Observables =========== With this spontaneous evolution of the system, an experiment beginning its measurements a time $t_0$ after the $^AZ$ was produced will probe the three-level system with relative populations $P_L \approx 100\%,\, P_S \approx 0,\, P_\text{g.s.}\approx {\left| \alpha \right|}^2\Gamma\, t_0$. We discover different methods, involving the third state beyond the mixed states, to be sensitive to the resonant Majorana mixing of atoms: - [**Spontaneous Emission from the metastable state to the daughter atom ground state.**]{} The population in the upper level ${\left| \lambda_L \right>}$, as shown in Eq.(\[eq:PL\]), decreases with time as $P_L(\Delta t) \approx 1 - \Gamma_L\, \Delta t$, where $\Delta t = t - t_0$, due to the decay of the metastable “stationary” state ${\left| \lambda_L \right>}$ to ${\left| ^A(Z-2)_\text{g.s.} \right>}$. This process is associated to the spontaneous emission of X-rays with a rate $\Gamma_L$, considered in the literature after the concept of Resonant Mixing was introduced in Ref.[@17]. For one mole of $^{152}$Gd, the X-ray emission rate would be of order $10^{-12}$ s$^{-1}\sim 10^{-5}$ yr$^{-1}$. Its unique signature is the total energy of the two-hole X-ray radiation, displaced by $\Delta$ with respect to the characteristic ${\left| ^A(Z-2)^* \right>}\to \, {\left| ^A(Z-2)_\text{g.s.} \right>}$ X-ray spectrum, i.e. its energy release is the $Q$-value between the two atoms in their ground states. - [**Stimulated Emission from ${\left| \lambda_L \right>}$ to ${\left| ^A(Z-2)_\text{g.s.} \right>}$.**]{} The natural population inversion between the ground state and the metastable “stationary” state ${\left| \lambda_L \right>}$ gives raise to the possibility of stimulating the decay ${\left| \lambda_L \right>} \to {\left| ^A(Z-2)_\text{g.s.} \right>}$. The experimental signature of this process would be the emission of X-rays with total energy equal to the $Q$-value of the process, just like in the first observable of spontaneous emission. When compared with the spontaneous rate, one finds a gain factor $$G = \hbar\, (\hbar c)^2\, \frac{\pi^2}{(\hbar \omega)^3}\, \frac{\dd N}{\dd{t}\dd{S}}\, \left[ \frac{\dd \omega}{\omega} \right]^{-1}\,, \label{eq:Gain}$$ where $\dd N/\dd t\dd S$ is the luminosity $\cal L$ of the beam and $\omega$ the transition angular frequency. At XFEL, a sound simulation of the conditions of the machine [@altarelli] gives, for typical energies of tens of keV, the expected number of photons per pulse duration $dN/dt = 10^{10}$ fs$^{-1}$ and the spectral width $\dd\omega/\omega = 1.12\times 10^{-3}$. Nanofocusing of this X-ray FELs has been contemplated [@gain]; using a beam spot of the order of 100 nm would lead to a gain factor from (\[eq:Gain\]) of $G\sim 100$. Applying this beam on a target of one mole of $^{152}$Gd would provide an X-ray emission rate of order $10^{-10}$ s$^{-1}\sim 10^{-3}$ yr$^{-1}$. - [**Daughter Atom Population.**]{} The presence of the daughter atom in the parent ores (see Eq.\[eq:Pgs\]), can be probed e.g. by geochemical methods. For one mole of the nominally stable $^{152}$Gd isotope produced at the time of the Earth formation, one would predict an accumulated number of order $10^4$ $^{152}$Sm atoms. This observable could be of interest for cosmological times $t_0$ since, contrary to $\beta\beta$-decay, in the ECEC case there is no irreducible background from a $2\nu$ channel for a resonant atom mixing. In the presence of a light beam, the daughter atom population would absorb those characteristic frequencies corresponding to its energy levels, which would then de-excite emitting light of the same frequency. In the case of the one mole $^{152}$Gd ore that we mentioned in the previous section, all $10^4$ Sm atoms could be easily excited to any of its $\sim 1$ eV levels using a standard pulsed laser of order $100$ fs pulse duration, with a mean power of $5$ W and a pulse rate of $100$ MHz. It is worth noting from the results of this section that the bosonic nature of the atomic radiation is a property that can help in getting observable rates of the Atom Majorana Mixing, including the stimulated X-ray emission from the parent atom as well as the detection of the presence of the daughter atoms by means of its characteristic absorption lines. The actual values correspond to the specific case of $^{152}$Gd $\to$ $^{152}$Sm, which is still off the Resonance Condition by at least a factor 30, implying a factor $10^3$ in the rates. On this regard, there are many experimental searches \[7-11\] of candidates with better fulfillment of the resonance condition, using the trap technique for precission measurement of atomic masses. Conclussions ============ Neutrinoless double electron capture in atoms is a quantum mixing mechanism between the neutral atoms $^AZ$ and $^A(Z-2)^*$ with two electron holes. It becomes allowed for Majorana neutrino mediation responsible of this $\Delta L = 2$ transition. This Majorana Mixing leads to the X-ray de-excitation of the $^A(Z-2)^*$ daughter atomic state which, under the resonance condition, has no Standard Model background from the two-neutrino decay. The intense experimental activity looking for atomic candidates satisfying the resonance condition by means of precise measurements of atomic masses, thanks to the trapping technique, has already led to a few cases of remarkable enhancement effects and there is still room for additional adjustements of the resonance condition. With this situation, it is important to understand the complete time evolution of an atomic state since its inception and whether one can find, form this information, different signals of the Majorana Mixing, including the possible enhancement due to the bosonic nature of atomic transition radiation. These points have been addressed in this work. The effective Hamiltonian for the two mixed atomic states leads to definite non-orthogonal states of mass and lifetime, each of them violating Global Lepton Number, one being metastable with long lifetime and the other being short timelived. For an initial atomic state there are time periods of Atom Oscillations, with frequency the mass difference, and the decay of the short lived state, which are not observable. For observable times, the system of the two atoms has three relevant states for discussing transitions: one highly populated state with long lifetime, one empty state with short lifetime and the ground state of the daughter atom with a small population as a result of the past history. As a consequence, this is a case of natural population inversion suggesting the possibility of stimulated radiation transitions besides the natural spontaneous X-ray emission. The actual results obtained in this work demonstrate that the gain factors which could be obtained for double electron capture signals, by using the strategy of stimulating the relevant transitions, are significant. Taking into account the ongoing searches for new atomic candidates with a better fulfillment of the resonance condition, these processes could become realistic alternatives in the quest for the Dirac/Majorana nature of neutrinos. This research has been supported by MINECO Project FPA 2014-54459-P, Generalitat Valenciana Project GV PROMETEO II 2013-017 and Severo Ochoa Excellence Centre Project SEV 2014-0398. AS acknowledges the MECD support through the FPU14/04678 grant. References {#references .unnumbered} ========== [9]{} Super-Kamiokande Collaboration (Fukuda Y et al.) 1998 *Phys. Rev. Lett.* [**81**]{} 1562 SNO Collaboration (Ahmad Q R et al.) 2002 *Phys. Rev. Lett.* [**89**]{} 011301 Galindo A and Pascual P 1990 *Quantum Mechanics* (Berlin: Springer) Bernabeu J, De Rujula A and Jarlskog C 1983 *Nucl. Phys.* B [**223**]{} 15. Schneidmiller E and Yurkov M, DESY note, private communication from Altarelli M Yamauchi K, Yabashi M, Ohashi H, Koyama T and Ishikawa T 2015 *J. Synchrotron Rad.* [**22**]{} 592, Eliseev S et al 2011 *Phys. Rev. Lett* [**106**]{} 052504 Kolhinen V S et al 2010 *Phys. Lett.* B [**684**]{} 17 Barabash A S et al 2008 *Nucl. Phys.* A [**807**]{} 269 Goncharov M et al 2011 *Phys. 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-1.cm 22.0cm [DE-FG05-92ER40717-7]{} [**Optimization of $R_{e^+e^-}$\ and\ “Freezing” of the QCD Couplant at Low Energies**]{}\ [A. C. Mattingly and P. M. Stevenson]{}\ [*T.W. Bonner Laboratory, Physics Department,\ Rice University, Houston, TX 77251, USA*]{}\ [**Abstract:**]{} The new result for the third-order QCD corrections to $R_{e^+e^-}$, unlike the old, incorrect result, is nicely compatible with the principle-of-minimal-sensitivity optimization method. Moreover, it leads to infrared fixed-point behaviour: the optimized couplant, $\alpha_s/\pi$, for $R_{e^+e^-}$ does not diverge at low energies, but “freezes” to a value $0.26$ below about 300 MeV. This provides some direct theoretical evidence, purely from perturbation theory, for the “freezing” of the couplant – an idea that has long been a popular and successful phenomenological hypothesis. We use the “smearing” method of Poggio, Quinn, and Weinberg to compare the resulting theoretical prediction for $R_{e^+e^-}$ with experimental data down to the lowest energies, and find excellent agreement. Introduction ============ The calculation [[@old; @new]]{} of the third-order (next-to-next-to-leading order) QCD corrections to $R_{e^+e^-}$: $$R_{e^+e^-} \equiv \sigma_{tot}(e^+e^- \rightarrow {\rm hadrons})/ \sigma(e^+e^- \rightarrow \mu^+ \mu^-),$$ provides valuable empirical information on the behaviour of perturbation theory in QCD. This paper is concerned with “optimized perturbation theory” (OPT) [[@OPT]]{}, and is motivated by three questions which the $R_{e^+e^-}$ calculation can answer: [[@letter]]{} \(1) [*Does perturbation theory seem to be well behaved?*]{} Is the third-order “optimized” result in reasonable agreement with the second-order “optimized” result? What can we learn about the error estimate? \(2) [*Is the optimized couplant, $a \equiv \alpha_s/\pi$, smaller in third order than in second?*]{} The “induced-convergence” picture [[@optult]]{} suggests that the optimized couplant $\bar{a}^{(n)}$, as determined by the $n$th-order optimization equations, will tend to decrease as the order $n$ increases. In this way “optimization” could lead to a convergent sequence of perturbative approximations, even if the perturbation series in any [*fixed*]{} renormalization scheme is divergent [[@optult]]{}. \(3) [*Does one find infrared fixed-point behaviour?*]{} A third-order calculation is a prerequisite for addressing this question in “optimized” perturbation theory, and the answer basically depends upon whether the invariant $\rho_2$ (defined below) is negative or not [[@KSS]]{}. It is striking that, with the originally published third-order result [[@old]]{}, the answer to all three questions was “No” — while, with the new, corrected, result [[@new]]{} the answer to all three questions is “Yes”. The purpose of this paper is to elaborate on these three points, and especially to discuss the infrared fixed-point behaviour [@chyla2; @us]. We do not share the pessimistic attitude of Chýla [*et al*]{} [@chyla2; @higgs] to the infrared results. If one believes in OPT, the infrared results – though quantitatively uncertain – are qualitatively unequivocal: We propose to take them at face value and compare them to experimental data [@us]. The plan of this paper is as follows: Sect. 2 reviews OPT, and applies it to $R_{e^+e^-}$ in third-order, with particular emphasis on the infrared limit. Sect. 3 compares the predicted $R_{e^+e^-}$ with experimental data, using Poggio-Quinn-Weinberg (PQW) smearing [@PQW]. Sect. 4 briefly discusses the phenomenology of a “frozen” couplant. Conclusions are summarized in Sect. 5. Some technical matters are relgated to the appendices. Optimized Pertubation Theory and Fixed-Point Behaviour ====================================================== The principle of minimal sensitivity ------------------------------------ We begin with a few words about the principle of minimal sensitivity, upon which OPT is based. It deals with any situation where an exact result is known to be independent of certain variables, but where the corresponding approximate result depends upon those variables, and hence is ambiguous. (In the QCD context, physical quantities are Renormalization Group (RG) invariant [@SP], but perturbative approximations to them are not, due to truncation of the perturbation series.) The philosophy is that such a non-invariant approximant is most believable where it is least sensitive to small variations in the extraneous variables, because this is where it best approximates the exact result’s vital property of being completely insensitive to the extraneous variables. A simple example is perhaps the best way to convey this idea. Consider the quantum-mechanical problem of computing the eigenvalues, $E_k$, of the quartic-oscillator Hamiltonian: $$\label{QHO} H = {\scriptstyle \frac{1}{2}} p^2 + \lambda x^4,$$ where $[x,p] = i$. Suppose we do standard perturbation theory, but with [@CK]: $$H_0 = {\scriptstyle \frac{1}{2}}(p^2 + \Omega^2 x^2), \quad\quad H_{{\rm int}} = \lambda x^4 - {\scriptstyle \frac{1}{2}} \Omega^2 x^2.$$ This introduces an “extraneous variable” $\Omega$, and the approximate eigenvalues so calculated will be $\Omega$-dependent. For example, first-order for the $k$th eigenvalue gives: $$\label{E1} E^{(1)}_k = \frac{1}{2} (k + {\scriptstyle \frac{1}{2}}) \Omega + \frac{3 \lambda}{4 \Omega^2}(2 k^2 + 2 k + 1).$$ However, we [*know*]{} that the exact eigenvalues are $\Omega$-independent. Therefore, it is sensible to choose $\Omega$ so that the approximant, $E^{(1)}_k$, is minimally sensitive to $\Omega$; [*i.e.*]{}, $$\label{Omega} \bar{\Omega} = \left[ 3 \lambda \, \frac{(2k^2 + 2k + 1)} {(k+{\scriptstyle \frac{1}{2}})} \right]^{\frac{1}{3}}.$$ (Quite generally, we shall use an overbar to denote an “optimized” value.) This gives the “optimized” result: $$\label{E1opt} E_k^{(1)}({\rm opt.}) = \frac{3}{4} (k+{\scriptstyle \frac{1}{2}}) \left[ 3 \lambda \, \frac{(2k^2 + 2k + 1)} {(k+{\scriptstyle \frac{1}{2}})} \right]^{\frac{1}{3}}.$$ This simple formula fits the ground-state energy to 2%, and [*all*]{} other energy levels to within 1%. The secret of this success is the “optimal” choice of $\Omega$, which is different for different levels. One may proceed to the calculation of higher-order corrections for some specific eigenvalue ([*e.g.*]{}, the ground state, $k\!=\!0$). For any fixed $\Omega$ the perturbation series would diverge, but if $\Omega$ is chosen in each order according to the “minimal sensitivity” criterion (which gives an $\bar{\Omega}$ that gradually increases with order), one finds quite nice convergence [@CK]. This is an example of the “induced convergence” mechanism [@optult; @indcon]. One may also use the same method to obtain accurate approximate wavefunctions, $\psi_k(x)$, from first-order perturbation theory [@KP]. Here the “optimal” $\Omega$ will be a function of $x$; in particular, it will be proportional to $|x|$ at large $|x|$, thereby converting the Gaussian dependence $\exp(- \frac{1}{2} \Omega x^2)$ into the correct large-$|x|$ behaviour. A variety of other examples and applications can be found in Refs. [@OPT; @indcon; @examples]. Some examples of QCD applications can be found in Refs. [@duke; @aurenche]. RG invariance and optimization ------------------------------ We next review “optimized perturbation theory” (OPT) [@OPT] as applied to the QCD corrections to the $R_{e^+e^-}$ ratio. Ignoring quark masses for the present, we may write $R_{e^+e^-} = 3 \sum q_i^2 (1 + {\cal R})$, where ${\cal R}$ has the form: $${\cal R} = a(1 + r_1 a + r_2 a^2 + \ldots),$$ and depends upon a single kinematic variable, $Q$, the [*cm*]{} energy. OPT is based on the fundamental notion of RG invariance [@SP], which means that a physical quantity is independent of the renormalization scheme (RS). Symbolically, we can express this by: $$0 = \frac{d{\cal R}}{d(RS)} = \frac{\partial {\cal R}}{\partial (RS)} + \frac{da}{d(RS)} \frac{\partial {\cal R}}{\partial a}, \label{rg}$$ where the total derivative is separated into two pieces corresponding to RS dependence from the series coefficients, $r_i$, and from the couplant, respectively. A particular case of Eq. (\[rg\]) is the familiar equation expressing the renormalization-scale independence of ${\cal R}$: $$\left( \mu \frac{\partial}{\partial \mu} + \beta(a) \frac{\partial}{\partial \mu} \right) {\cal R} = 0,$$ where $$\beta(a) \equiv \mu \frac{da}{d\mu} = - ba^2(1+ca + c_2a^2+\ldots).$$ The first two coefficients of the $\beta$ function are RS invariant and, in QCD with $N_f$ massless flavours, are given by: $$b = \frac{(33-2N_f)}{6}, \quad\quad c= \frac{153 - 19 N_f}{2(33-2N_f)}.$$ When integrated, the $\beta$-function equation can be written as: $$\label{beta} \int_0^{a} \frac{d a^{\prime}}{ \beta( a^{\prime})} + {\cal C} = \int_{\tilde{\Lambda}}^{\mu} \frac{d \mu^{\prime}}{\mu^{\prime}} = \ln(\mu/{\tilde{\Lambda}}).$$ where ${\cal C}$ is a suitably infinite constant and $\tilde{\Lambda}$ is a constant with dimensions of mass. The particular definition of $\tilde{\Lambda}$ that we use corresponds to choosing [@OPT] $${\cal C} = \int_0^{\infty} \frac{d a^{\prime}} { b {a^{\prime}}^2(1 + c a^{\prime})}$$ (where it to be understood that the integrands on the left of (\[beta\]) are to be combined before the bottom limit is taken). This $\tilde{\Lambda}$ parameter is related to the traditional definition [@BBDM] by an RS-invariant, but $N_f$-dependent factor: $$\ln (\Lambda/\tilde{\Lambda}) = (c/b) \ln (2c/b).$$ The $\Lambda$ parameter is scheme dependent, but the $\Lambda$’s of different schemes can be related exactly by a 1-loop calculation [@CG; @fncg]. As is usual, we shall regard $\Lambda_{\overline{\rm MS}}$ (for 4 flavours) as the free parameter of QCD [@fnlam]. From Eq. (\[beta\]) it is clear that $a$ depends on RS only through the variables $\mu/\tilde{\Lambda}$ and $c_2, c_3, \ldots$, the scheme-dependent $\beta$-function coefficients. The coefficients of ${\cal R}$ can depend on RS only through these same variables, because of RG invariance, Eq. (\[rg\]). Therefore, these variables provide a complete RS parametrization, as far as physical quantities are concerned [@OPT]. Thus, we may write: $$a=a({\rm RS})=a(\tau,c_2,c_3,\ldots),$$ where $$\tau \equiv b \ln(\mu/\tilde{\Lambda}).$$ The $\tau$ variable is convenient and also serves to emphasize the very important point that RS dependence involves only the [*ratio*]{} of $\mu$ to $\tilde{\Lambda}$. “Optimization” does [*not*]{} determine an “optimum $\mu$”, but it will determine an optimum $\tau$. The dependence of $a$ on the set of RS parameters $\tau$ and $ c_j$ [@OPT] is most easily obtained [@pol] by taking partial derivatives of Eq. (\[beta\]), varying one parameter while holding the others constant. This yields: $$\frac{\partial a}{\partial \tau} = \beta(a)/b,$$ $$\label{betaj} \frac{\partial a}{\partial c_j} \equiv \beta_j(a)= -b \beta(a) \int_0^a dx \; \frac{x^{j+2}}{[\beta(x)]^2}.$$ Note that the $\beta_j$ functions begin at order $a^{j+1}$. The symbolic RG-invariance equation (\[rg\]) can now be written out explicitly as the following set of equations: $$\begin{aligned} \label{rga} \left( \left. \frac{\partial}{\partial \tau} \right|_a + \frac{\beta(a)}{b} \frac{\partial}{\partial a} \right) {\cal R} \, = 0, & & \\ \label{rgb} \left( \left. \frac{\partial}{\partial c_j} \right|_a + \beta_j(a) \frac{\partial}{\partial a} \right) {\cal R} = 0 & \quad\quad {\scriptstyle (j = 2,3,\ldots)}. &\end{aligned}$$ These equations determine how the coefficients $r_i$ of ${\cal R}$ must depend on the RS variables. Thus, $r_1$ depends on $\tau$ only, while $r_2$ depends on $\tau$ and $c_2$ only, etc., with $$\frac{\partial r_1}{\partial \tau} = 1,$$ $$\frac{\partial r_2}{\partial \tau} = 2 r_1 + c, \quad\quad \frac{\partial r_2}{\partial c_2} = - 1,$$ etc.. Upon integration one will obtain $r_i = f(\tau,c_2,\ldots,c_i) + \mbox{const.}$, where $f$ is a known function and the constant of integration is an RS invariant. Thus, certain combinations of series coefficients and RS parameters: $$\label{rho1} \rho_1(Q) \equiv \tau - r_1,$$ $$\label{rho2} \rho_2 \equiv r_2 + c_2 -(r_1+\frac{1}{2}c)^2,$$ etc., are RS invariant [@OPT; @fnterho2]. In the $e^+e^-$ case, $\rho_1$ is a function of the [*cm*]{} energy $Q$, while $\rho_2$ and the higher-order invariants are pure numbers, dependent only on the number of flavours, $N_f$. Although the exact ${\cal R}$ is RG-invariant, the truncation of the perturbation series spoils this invariance. The $n$th-order approximant ${\cal R}^{(n)}$, defined by truncating ${\cal R}$ and the $\beta$ function to only $n$ terms, depends on the RS variables $\tau,\ldots,c_{n-1}$. OPT corresponds to choosing an “optimal” RS in which the approximant ${\cal R}^{(n)}$ is stationary with respect to RS variations; [*i.e.*]{}, the RS in which ${\cal R}^{(n)}$ [*exactly*]{} satisfies the RG-invariance equations, (\[rga\], \[rgb\]). (Note that only the first $(n-1)$ equations will be nontrivial in $n$th order.) The second order approximant is $${\cal R}^{(2)} = a(1 + r_1 a),$$ where $a$ here is short for $a^{(2)}$, the solution to (\[beta\]) with $\beta$ truncated at second order. ${\cal R}^{(2)}$ depends on RS only through the variable $\tau$. The optimization equation, from (\[rga\]), is $$\bar{a}^2 - \bar{a}^2(1 + c \bar{a})(1 + 2 \bar{r}_1 \bar{a}) = 0.$$ This equation, together with the $\rho_1$ definition and the second-order integrated $\beta$-function equation, (\[beta\]), uniquely determines the optimized result. (For details, see [@OPT; @duke].) The third order approximant is: $${\cal R}^{(3)} = a(1 + r_1 a + r_2 a^2),$$ where now $a$ is short for $a^{(3)}$, the solution to (\[beta\]) with $\beta$ truncated at third order. ${\cal R}^{(3)}$ depends on RS through two parameters $\tau$ and $c_2$, so there are two optimization equations, coming from (\[rga\], \[rgb\]). These can be reduced to [@OPT]: $$\label{eq1} (3\bar{r}_2 + 2\bar{r}_1c + \bar{c}_2) + (3\bar{r}_2 c + 2 \bar{r}_1 \bar{c}_2) \bar{a} + 3 \bar{r}_2 \bar{c}_2 \bar{a}^2 = 0,$$ $$\label{eq2} I(1+(c+2\bar{r}_1)\bar{a}) - \bar{a} = 0, \;\;\;\;\;\; \mbox{where} \;\;\;\; I= \int_0^{\bar{a}} \frac{dx}{(1+cx + \bar{c}_2 x^2)^2}.$$ This integral can be done analytically, and is given by: $$I = \frac{1}{\Delta^2} \left[ \frac{\bar{a}(c^2 -2 \bar{c}_2 + c \bar{c}_2 \bar{a})} {(1+c \bar{a} + \bar{c}_2 \bar{a}^2)} - 4 \bar{c}_2 f(\bar{a},\bar{c}_2) \right],$$ with $$\label{f} f(\bar{a},\bar{c}_2) = \frac{1}{2\Delta} \ln \left[ \frac{1+ {\scriptstyle \frac{1}{2}} \bar{a}(c+\Delta)} {1+ {\scriptstyle \frac{1}{2}} \bar{a}(c-\Delta)} \right],$$ where $\Delta^2 \equiv c^2-4\bar{c}_2$. (This assumes that $\Delta^2 > 0$, which proves to be true here.) The procedure for solving these optimization equations is discussed further in subsection 2.4, but we next discuss the infrared limit. Infrared limit and fixed-point behaviour ---------------------------------------- Suppose we consider ${\cal R}$ at lower and lower [*cm*]{} energy, $Q$. Since $c$ is positive for $N_f \! \le \! 8$, the second-order $\beta$ function has no non-trivial zero. Thus, in any RS, the couplant $a^{(2)}$ and approximant ${\cal R}^{(2)}$ must become singular at some $Q$ of order $\Lambda_{\overline{\rm MS}}$. In third order this may or may not happen, depending on whether the RS has a positive or negative $c_2$. If $c_2$ is negative then the couplant remains finite and tends to a “fixed-point” value, $a^*$, which is the non-trivial zero of the third-order $\beta$ function; [*i.e.,*]{} the positive root of $$\label{betaz} 1 + c a^* + c_2 a^{*2} = 0.$$ Since fixed-point behaviour hinges on $c_2$, which is scheme dependent, it is vital to have a sensible choice of RS [@KSS]. In OPT the optimal $c_2$ is determined by the optimization equations, and depends on $Q$ somewhat. If $\bar{c}_2$ is negative as $Q \to 0$, then OPT will give fixed-point behaviour. The infrared limit of the optimization process was analyzed in Ref. [@KSS] and we briefly review the relevant results. Since $\beta$ vanishes at a fixed point, the $\tau$ optimization equation, corresponding to (\[rga\]), reduces to $$1 + (2 \bar{r}_1 + c)\bar{a}^* = 0.$$ Then, just by differentiating (\[betaz\]) with respect to $c_2$, one obtains: $$\lim_{a \rightarrow a^*} \beta_2^{(3)}(a) = \frac{\partial a^*}{\partial c_2} = \frac{- a^{*2}}{(c + 2 c_2 a^*)}.$$ (This can also be obtained, more laboriously, as the limit of (\[betaj\]).) Thus, the $c_2$ optimization equation, corresponding to (\[rgb\]), becomes $$\bar{a}^* + \frac{(1+ 2 \bar{r}_1 \bar{a}^* + 3 \bar{r}_2 \bar{a}^{*2})} {(c+2 \bar{c}_2 \bar{a}^*)} =0.$$ The two optimization equations yield: $$\label{fpr} \bar{r}_1 = - \frac{1}{2} \frac{(1+c \bar{a}^*)}{\bar{a}^*}, \quad\quad \bar{r}_2 = - \frac{2}{3} \bar{c}_2.$$ Using the expression for the invariant $\rho_2$, (\[rho2\]), one obtains: $$\label{fpeqc2} \bar{c}_2 = 3(\rho_2 + \frac{1}{4 \bar{a}^{*2}}).$$ Finally, substituting into the fixed-point condition (\[betaz\]), one finds [@KSS]: $$\label{fpeq} \frac{7}{4} + c \bar{a}^* + 3 \rho_2 \bar{a}^{*2} = 0,$$ which determines $\bar{a}^*$ in terms of the RS-invariant quantities $c$ and $\rho_2$. A positive $\bar{a}^*$ exists if $\rho_2$ is negative, and the more negative $\rho_2$ is, the smaller $\bar{a}^*$ will be. Implementing the optimization procedure --------------------------------------- Returning to finite $Q$, we now consider how to obtain the third-order optimized approximant $\bar{{\cal R}}^{(3)}$ numerically as a function of $Q$. As input, we need the values of $\rho_1$ and $\rho_2$. Being invariants, they can be obtained from calculations performed in any computationally convenient RS. The calculations in the literature have used the “modified minimal subtraction” ($\overline{\rm MS}$) convention, with the renormalization point $\mu$ chosen to be $Q$. The ${\cal R}$ coefficients are [@r1; @new]: $$r_1({\scriptstyle \overline{\rm MS}; \; \mu = Q}) = 1.9857 - 0.1153 N_f,$$ $$r_2({\scriptstyle \overline{\rm MS}; \; \mu = Q}) = -6.6368 - 1.2001 N_f - 0.0052 N_f^2 - 1.2395 \: {( {\textstyle \sum} q_i )}^2 \! / (3 {\textstyle \sum} q_i^2 ).$$ The RS parameters of the $\overline{\rm MS}(\mu \! =\! Q)$ scheme are: $$\tau({\scriptstyle \overline{\rm MS}; \; \mu = Q}) = b \ln (Q/\tilde{\Lambda}_{\overline{\rm MS}}) = b \ln (Q/\Lambda_{\overline{\rm MS}}) + c \ln (2c/b),$$ $$c_2({\scriptstyle \overline{\rm MS}}) = \frac{3}{16} \frac{1}{(33-2N_f)} \left[ \frac{2857}{2} - \frac{5033}{18} N_f + \frac{325}{54} N_f^2 \right].$$ (The latter was first calculated in Ref. [@c2], and has recently been confirmed independently [@c2new].) Substitution of these results into (\[rho1\], \[rho2\]) gives the invariants. One can see explicitly that $\rho_1$ depends logarithmically on the [*cm*]{} energy $Q$, and on the free parameter of QCD, $\Lambda_{\overline{\rm MS}}$ [@fnlam]. However, $\rho_2$ depends only on $N_f$. Since $\rho_2$ turns out to be negative, one will find fixed-point behaviour in the “optimum” scheme [@fac; @fna]. Table 1 gives the fixed-point couplant values, determined from Eq. (\[fpeq\]), for various $N_f$ values [@fnb]. Consider a world with $N_f$ massless quarks, ignoring complications due to quark thresholds for the present. For simplicity we assume that the value of $\Lambda_{\overline{\rm MS}}$ is given, and our numerical results use $\Lambda_{\overline{\rm MS}} = 230$ MeV for four flavours [@fnrep]. For any chosen $Q$ we then have definite numerical values for the invariant quantities $\rho_1$, $\rho_2$ (and $b$, $c$). We need to solve for the optimum couplant, $\bar{a}$, and the optimized coefficents, $\bar{r}_1$, $\bar{r}_2$, and this will involve determining the RS parameters $\bar{\tau}$, $\bar{c}_2$ of the optimal RS. These five variables are related by five equations; the two optimization equations, (\[eq1\], \[eq2\]), the $\rho_1$, $\rho_2$ equations, (\[rho1\], \[rho2\]), and the integrated $\beta$-function equation, (\[beta\]), which for the $\beta$-function truncated at third order becomes explicitly: $$\label{k3} \tau = \frac{1}{a} + c \ln (ca) - \frac{1}{2} c \ln (1+ca+c_2 a^2) - (c^2 - 2 c_2) f(a,c_2),$$ where $f(a,c_2)$ is given by (\[f\]). By substituting this last equation into the $\rho_1$ equation we can obtain $\bar{r}_1$ explicitly as a function of $\bar{a}$, $\bar{c}_2$. We can then rearrange the $\rho_2$ equation to give $\bar{r}_2$ explicitly as a function of $\bar{a}$, $\bar{c}_2$. This leaves $\bar{a}$, $\bar{c}_2$ to be solved for from the two optimization equations. Starting from an initial guess for $a$ and $c_2$, our procedure was to solve (\[eq2\]) numerically for a new $\bar{a}$; and then (with the new $\bar{a}$) to solve (\[eq1\]) for a new $\bar{c}_2$. We then iterated this procedure until the difference between successive solutions reached a specified tolerance. Further details are given in Appendix A. At very low $Q$ we encountered technical problems with slow convergence of the iteration scheme. These are discussed in Appendix A. Nevertheless, with care it was possible to obtain accurate solutions at low energies. In Fig. 1 we show the optimized solution in the $a$, $c_2$ plane as it smoothly approaches the fixed-point solution, which lies on the infrared boundary $1+ca+c_2 a^2 = 0$. The figure shows two cases, $N_f=3$ and $N_f=2$. (In the real-world case we must switch from 3 flavours to 2 when we cross the strange-quark threshold. This requires a matching of $\Lambda$ parameters, as discussed in Appendix B.) The optimized couplant $\bar{a}$ is shown as a function of $Q$ in Fig. 2. Note that the effective couplant below 300 MeV is nearly constant at about 0.263, which is the $N_f=2$ fixed-point value. Fig. 2 also shows the second- and third-order optimized results for ${\cal R}$. The second-order result diverges at $Q \approx 400$ MeV, where $\rho_1(Q)$ vanishes. However, $\bar{{\cal R}}^{(3)}$ remains finite, rising only to 0.33 at $Q=0$. Illustrative results -------------------- We pause for a moment to consider a comparison between the second- and third-order optimized results at moderately high $Q$. This exercise was performed by several authors [@max; @oldopt] when the ‘old’ third-order result [@old] was first published, and the results were disquieting. However, the new result [@new] has transformed the situation, which is now very satisfactory. In Table 2 we give details for the two illustrative cases considered by Maxwell and Nicholls [@max], namely $N_f = 5$, $Q = 34$ GeV, with either ${\tilde{\Lambda}}_{\overline{\rm MS}} = 100$ MeV or ${\tilde{\Lambda}}_{\overline{\rm MS}} = 500$ MeV. \[Note, though, that the results depend only on the ratio of $Q$ to ${\tilde{\Lambda}}_{\overline{\rm MS}}$.\] From Table 2 we see that between second and third order the optimized prediction $\bar{{\cal R}}$ decreases only a few percent. With the ‘old’ result there had appeared to be a disconcertingly large increase [@max; @oldopt]. The new situation is much more satisfactory in other ways, too: In both examples the coefficient $\bar{r}_2$ now has a more reasonable magnitude, and the $\bar{r}_1$ coefficient has not changed so drastically from second to third order. The optimized couplant $\bar{a}$ now shows a marked decrease from second to third order. This is just what one would expect in the “induced convergence” picture of Ref. [@optult]. In that picture “optimization” induces convergence through a mechanism in which the effective expansion parameter, $\bar{a}$, shrinks from one order to the next. Note that the ‘old’ result gave the opposite behaviour, with $\bar{a}$ apparently increasing from second to third order. The results also shed some light on the error-estimation question: If we knew just $\bar{{\cal R}}^{(2)} = \bar{a}(1 + \bar{r}_1 \bar{a})$, how might we estimate the error? Two estimates suggest themselves: (i) $n \bar{a}^3$, where $n$ is an order-one number, which presumes a well-behaved converging series, or (ii) $| \bar{r}_1 \bar{a}^2 |$, the magnitude of the last calculated term, which is a typical error estimate for an asymptotic series. Knowing $\bar{{\cal R}}^{(3)}$ we can, presumably, get a much better idea of the actual error in $\bar{{\cal R}}^{(2)}$ from the difference $\delta \equiv \bar{{\cal R}}^{(3)} - \bar{{\cal R}}^{(2)}$. We have compared $\delta$ with estimates (i) and (ii) over a wide range of $Q/\Lambda_{\overline{\rm MS}}$ values. Estimate (i), if we had assumed $n \approx 1$ or 2, would have been rather too optimistic. In fact, $| \delta |$ is between 7 and 14 times $\bar{a}^3$ (for $N_f = 4$ and $Q/\Lambda_{\overline{\rm MS}} \ge 5$). This is directly related to the size of the invariant $\rho_2$ (which is about $-14$ for 4 flavours). \[One can show analytically that $\delta = \rho_2 \bar{a}^3 + {\cal O}(\bar{a}^4)$ in the large-$Q$ limit.\] Of course, we could not know $\rho_2$ until a third-order calculation was done. Arguably, though, 14 can still be considered an “order-one” number, especially in a theory that naturally involves numbers such as 4 (flavours), 3 (colours), 8 (gluons), etc.. Estimate (ii), based on the last calculated term, agrees with $| \delta |$ to within a factor of two either way for $Q/\Lambda_{\overline{\rm MS}}$ between 5 and 1000. At higher $Q/\Lambda_{\overline{\rm MS}}$ values this estimate would be overly pessimistic. However, we think that for present energies the estimate (ii) is perhaps the safest way to estimate the error. We suggest that it be used in QCD applications where only second-order results are known. Credibility of the infrared results ----------------------------------- We have stressed that OPT yields finite results for ${\cal R}$ down to $Q=0$. The crucial question is, of course: How meaningful are these results? We would like to explain why, in contrast to other authors [@chyla2; @higgs; @chyla], we take a positive attitude on this issue. Firstly, suppose we adopt the philosophy that the last calculated term in the “optimized” perturbation series is a measure of the error. As we saw in this last section, this proved to be reasonable in the second-order case. In third order this gives $|\bar{r}_2 \bar{a}^3|$ as the error estimate. Since $\bar{{\cal R}} \approx \bar{a}$, this implies a fractional error of $|\bar{r}_2 \bar{a}^2|$. In Table 3 we show some illustrative results at low energies, together with their estimated error. From this one can see that the behaviour of the series is quite satisfactory above $Q=1$ GeV. The situation undoubtedly deteriorates at lower energies; by the time we reach $Q=0$ we have a series of the form $0.26(1-0.76+1.01)$ in which the higher-order terms are comparable to the leading term. While this is hardly a good situation, it is not completely disastrous; the corrections alternate in sign, and they do not dwarf the leading term. We believe that our error estimate, which grows to 100% at $Q=0$, is not unreasonable: the result may well be off by a factor of 2, but is unlikely to be off by an order of magnitude. The qualitative conclusion, that ${\cal R}$ remains small (say, $0.3 \pm 0.3$) at low energies is hard to escape. Secondly, it is instructive to view the use of QCD perturbation theory in the infrared limit as an extrapolation away from $N_f = 33/2$ [@banks]. At $N_f = 33/2$ the leading $\beta$-function coefficient, $b$, vanishes (and hence $c$ goes to $- \infty$). For $N_f = 33/2 - \epsilon$, with $\epsilon$ small and positive, there must be an infrared fixed point at $a^* \sim - 1/c = {\cal O}(\epsilon)$ [@banks]. Perturbative calculations, even in the infrared, should then be meaningful if $\epsilon$ is sufficiently small. Furthermore, one could naturally expect that the more orders in perturbation theory one has, the further one can extrapolate from $N_f = 33/2$. With sufficient orders one should be able to get infrared results down to $N_f =0$, unless there is some unknown reason for the behaviour of the theory to change fundamentally at some critical $N_f$ between $33/2$ and 0. What does happen? Well, at second order, of course, one finds fixed-point behaviour, with $a^* = -1/c$, provided $c$ is negative, which requires $N_f > 153/19 \approx 8$, though $N_f$ needs to be still larger if $a^*$ is to be reasonably small. In third order our results imply that, in the $R_{e^+e^-}$ case, fixed-point behaviour — with moderately small $a^*$ values — does extend to $N_f=0$. In the $\epsilon \to 0$ limit, $a^*$ tends to $-1/c$, and hence to $(8/321) \epsilon$. The small coefficient suggests that the natural expansion parameter of an extrapolation from $N_f = 33/2$ is not $\epsilon$ but approximately $\epsilon/40$. One can verify that the third-order OPT results smoothly approach the limiting form as $\epsilon \to 0$. For $N_f = 16$ one has $\rho_2 = -1724.4$, and one gets a series of the form $0.012(1 - 0.03 + 0.04)$. As $N_f$ decreases, the behaviour of the series deteriorates, but it does so quite steadily; there is no dramatic change around $N_f =8$ or any other $N_f$. In conclusion, our view is that the $N_f = 2$ infrared results, while quantitatively uncertain, are qualitatively credible. Having made this case in theoretical terms, let us now see what experiment has to say. Comparing Theory to Experiment ============================== $R_{e^+e^-}$ including quark masses ----------------------------------- In this section we construct the theoretical prediction for $R_{e^+e^-}$ (allowing for quark masses) and discuss its comparison with experiment using the PQW smearing method. We limit ourselves to the region below 6 GeV, and we shall be particularly interested in the region below 1 GeV. To allow for quark masses in $R_{e^+e^-}$, we used the following approximate formula [@PQW]: $$R_{e^+e^-} = 3 \sum_i q_i^2 \; T(v_i)[ 1 + g(v_i) {\cal R}],$$ where the sum is over all quark flavours that are above threshold ([*i.e.*]{}, whose masses, $m_i$, are less than $Q/2$), and $$\begin{aligned} v_i & = & (1-4 m_i^2/Q^2)^{\frac{1}{2}}, \nonumber \\ T(v) & = & v(3-v^2)/2, \\ g(v) & = & \frac{4 \pi}{3} \left[ \frac{\pi}{2v} - \frac{(3+v)}{4} \left( \frac{\pi}{2} - \frac{3}{4 \pi} \right) \right]. \nonumber\end{aligned}$$ The coefficient $T(v_i)$ is the parton-model mass dependence and $g(v_i)$ is a convenient approximate form for the mass dependence of the leading-order QCD correction [@PQW; @schwinger]. The higher-order corrections have been calculated only for massless quarks, so we simply evaluate ${\cal R}$ with $N_f$ equal to the the number of above-threshold flavours. In our numerical results we used standard values for the current-quark masses [@PDG]: $m_u = 5.6$ MeV, $m_d = 9.9$ MeV, $m_s = 199$ MeV, $m_c = 1.35$ GeV. For $\Lambda_{\overline{\rm MS}}$ we used a 4-flavour value of 230 MeV above charm threshold ($Q > 2 m_c$). Then, each time a flavour threshold was crossed as we decreased $Q$, we reduced $N_f$ by 1 and computed the new $\Lambda_{\overline{\rm MS}}$ parameter appropriate to the new $N_f$. The matching of $\Lambda$’s is discussed in Appendix B. In this way we obtained the “raw” theoretical prediction for $R_{e^+e^-}$ shown in Fig. 3. For comparison, the figure also shows the parton-model result ([*i.e.,*]{} with the QCD correction term ${\cal R}$ set to zero). PQW smearing ------------ A [*direct*]{} comparison of the theoretical prediction with the experimental data is not possible, because there is no direct correspondence between the perturbative quark-antiquark thresholds and the hadronic thresholds and resonances of the data. However, a meaningful comparison is possible if some kind of “smearing” procedure is used [@PQW; @barnett]. We used the smearing method of Poggio, Quinn, and Weinberg (PQW) [@PQW], who define the “smeared” quantity: $$\label{PQWeq} \bar{R}_{\scriptscriptstyle{PQW}}(Q;\Delta) = \frac{\Delta}{\pi} \int_0^{\infty} ds^{\prime} \frac{R_{e^+e^-}(\sqrt{s^{\prime}})} {(s^{\prime} - Q^2)^2 + \Delta^2}.$$ In terms of the vacuum-polarization amplitude $\Pi$, one can write $\bar{R}_{\scriptscriptstyle{PQW}}$ as [@PQW]: $$2 i \bar{R}_{\scriptscriptstyle{PQW}}(Q;\Delta) = \Pi(Q^2 + i \Delta) - \Pi(Q^2 - i \Delta).$$ In the limit $\Delta \to 0$ this reduces to $2 i R_{e^+e^-}$, which is the discontinuity of $\Pi$ across its cut. However, a finite $\Delta$ keeps one away from the infrared singularities and nonperturbative effects that lurk close to the cut. The idea is to apply this smearing to both the theoretical and experimental $R_{e^+e^-}$’s and then compare them. In principle, the more orders in perturbation theory one has, the smaller one can take $\Delta$ [@PQW]. However, this requires the full mass dependence of the higher-order corrections, which we do not know. In their leading-order study of the charm-threshold region, PQW used a value $\Delta = 3$ GeV$^2$, and we shall use values of the same order of magnitude. We take a pragmatic view: the best choice of $\Delta$ is the smallest value that will smooth out any rapid variations in either the experimental or the theoretical $R_{e^+e^-}$. It turns out that this depends upon the energy region one is interested in. Around charm threshold a $\Delta$ of 3 GeV$^2$ or more is necessary, while in the lowest energy region a $\Delta$ as small as 1 GeV$^2$ can be used. The integral in Eq. (\[PQWeq\]) was evaluated by numerical integration, after first making a change of variables $s^{\prime} - Q^2 = \Delta \tan\theta$. The computer routine was designed to take an input $R_{e^+e^-}$, specified over a range 0 to $Q_{{\rm max}}$ and to evaluate the integral over this range. A term was then added to account for the contribution from $Q_{{\rm max}}$ to $\infty$, assuming that $R_{e^+e^-}$ remained constant above $Q_{{\rm max}}$. The accuracy of the numerical-integration routine was tested against analytic results for several simple input functions. Experimental data and resonances -------------------------------- The experimental data we used comes from a variety of sources: $e^+e^- \to \pi^+\pi^-$ data in the $\rho$ region and above from the OLYA/CMD and DM2 collaborations [@barkov; @dm2]; Adone $\gamma\gamma 2$ data from 1.4 to 3 GeV [@bacci]; SLAC Mark I data from 3 to 6 GeV [@siegrist]; and Crystal Ball data above 5 GeV [@crystal]. For useful compilations and reviews see Ref. [@reviews]. We used simple fits to the data in some regions, particularly when the data had a lot of structure and/or had large statistical errors. This was more convenient for the numerical integration routine and made it easier for us to examine the effect of the experimental uncertainties on the smeared result. Fig. 4 shows our data compilation, up to 6 GeV, excluding narrow resonances. In fact, we used data going well beyond $b$ threshold, but they have no real effect on the results we present. The sharp resonances $\omega$, $\phi$, $J/\psi$, $\psi^{\prime}$, and $\psi(3770)$ were not included in the data compilation so that their contribution to $\bar{R}_{\scriptscriptstyle{PQW}}$ could be put in analytically. They have a relativistic Breit-Wigner form [@collider]: $$R_{\scriptscriptstyle{res}} = \frac{9}{\alpha^2} B_{\it ll} B_h \frac{M^2 \Gamma^2}{(s-M^2)^2 + M^2\Gamma^2}, \label{eq:rres}$$ where $M$, $\Gamma$, $B_{\it ll}$, and $B_h$ are, respectively, the mass, width, leptonic branching fraction, and hadronic branching fraction of the resonance. The parameters for the resonances were taken from the 1992 Review of Particle Properties [@PDG]. For $B_{\it ll}$ we used the weighted average of the $ee$ and $\mu\mu$ branching ratios. The contribution of such a Breit-Wigner resonance to the smearing integral (\[PQWeq\]) can be evaluated analytically using partial fractions. \[The resulting expression is too cumbersome to quote, but we may note that the narrow width approximation: $$\frac{1}{(s-M^2)^2 + M^2\Gamma^2} \approx \frac{\pi}{M\Gamma} \delta(s-M^2),$$ which gives a contribution to $\bar{R}_{\scriptscriptstyle{PQW}}$ of $$\bar{R}_{\scriptscriptstyle{res}} \approx \frac{9 B_{\it ll} B_h \Delta M \Gamma}{\alpha^2[(s-M^2)^2 + \Delta^2]},$$ is a pretty good approximation.\] In Fig. 5 we show, for two different $\Delta$ values, the contributions of the various resonances to the experimental $\bar{R}_{\scriptscriptstyle{PQW}}$. The $\rho$’s contribution is shown by a dotted line. However, since the $\rho$ is rather wide and asymmetric, it was actually treated, not in this manner, but by numerical integration, using the data points from Ref. [@barkov] as part of the data compilation (Fig. 4). Results and uncertainty estimates --------------------------------- The results obtained by applying PQW smearing to both theory and experiment are shown in Fig. 6. For the smaller $\Delta$ (1 GeV$^2$) there is good agreement between theory and experiment below 1 GeV, but in the charm-threshold region there is clearly insufficient smearing for the comparison to be meaningful. Increasing $\Delta$ to 3 GeV$^2$ smooths out the experimental curve almost completely. The agreement between theory and experiment is excellent below 2 GeV. In the charm region the agreement is less good, but this can be attributed mainly to the sizable systematic normalization uncertainty (10 – 20%) in the data in this region, which produces an uncertainty of about $\pm 0.4$ in the experimental $\bar{R}_{\scriptscriptstyle{PQW}}$ at around $Q = 4$ GeV. For comparison, Fig. 6(b) also includes the naive parton-model prediction. One can see from this that the QCD correction term ${\cal R}$ provides about a 20% increase which is vital to the good agreement with the data. Using $Q^2$, rather than $Q$ as the variable, we can continue $\bar{R}_{\scriptscriptstyle{PQW}}(Q^2)$ into the negative $Q^2$ region (Cf. Ref. [@adler]). As shown in Fig. 7, for $\Delta = 1$ GeV$^2$, the good agreement persists. To quantify the good agreement at low energies, we discuss how various uncertainties would affect $\bar{R}_{\scriptscriptstyle{PQW}}$ at $Q=0$. First we discuss the experimental uncertainties. There is about a 5% uncertainty in the $\rho$, $\omega$ and $\phi$ contributions, due to the uncertainty in their total and leptonic widths. For $\Delta=1$ (3) GeV$^2$ this gives an error in $\bar{R}_{\scriptscriptstyle{PQW}}(0)$ of about $\pm 0.04$ ($\pm 0.02$). Uncertainties in the $\psi$ resonance parameters affect $\bar{R}_{\scriptscriptstyle{PQW}}(0)$ by $\pm 0.01$ or less. We considered the effect of a 15% normalization change in the continuum data in the 1.5 – 3 GeV region: The effect on $\bar{R}_{\scriptscriptstyle{PQW}}(0)$ was about $\pm 0.03$ ($\pm 0.06$) for $\Delta=1$ (3) GeV$^2$. We also allowed for a 15% normalization change in the 3 – 5 GeV region. The effect on $\bar{R}_{\scriptscriptstyle{PQW}}(0)$ was about $\pm 0.02$ ($\pm 0.065$) for $\Delta=1$ (3) GeV$^2$. Combining these four distinct sources of error in quadrature, we estimate an overall uncertainty in the experimentally determined $\bar{R}_{\scriptscriptstyle{PQW}}(0)$ of $\pm 0.06$ for $\Delta=1$ GeV$^2$ and $\pm 0.08$ for $\Delta=3$ GeV$^2$. On the theoretical side, errors arise from two sources: (i) uncertainty in the input parameters (quark masses and $\Lambda_{\overline{\rm MS}}$), and (ii) truncation of the perturbation series. We varied each quark mass by its quoted error [@PDG]. Varying the $u$ and $d$ masses had negligible effect. $\bar{R}_{\scriptscriptstyle{PQW}}(0)$ changed by $\pm 0.004$ ($\pm 0.001$) on varying the $s$ mass, and by $\pm 0.005$ ($\pm 0.013$) on varying the $c$ mass for $\Delta=1$ (3) GeV$^2$. Changing $\Lambda_{\overline{\rm MS}}$ by 50 MeV to 280 MeV increased $\bar{R}_{\scriptscriptstyle{PQW}}(0)$ by 0.019 (0.014) for $\Delta=1$ (3) GeV$^2$. The series-truncation error can reasonably be estimated from the last term in the optimized series, as we argued earlier. At 1 GeV this suggests that ${\cal R}$ is accurate to about 10%, and is considerably more accurate at larger energies. This is corroborated by the good agreement between second- and third-order results. The theoretical uncertainty in ${\cal R}$ above 1 GeV contributes an error in $\bar{R}_{\scriptscriptstyle{PQW}}(0)$ of less than $\pm 0.006$ ($\pm 0.009$) for $\Delta=1$ (3) GeV$^2$. Below 1 GeV the prediction for ${\cal R}$ is much more uncertain. However, as discussed in Subsect. 2.6, we think that even at $Q=0$ the result is reliable to within a factor of 2. Conservatively, we considered the effect of increasing the predicted ${\cal R}$ by a factor of 2 over the whole range, $0 < Q < 1$ GeV. This affects the low-energy $\bar{R}_{\scriptscriptstyle{PQW}}$ by 0.033 (0.011) for $\Delta=1$ (3) GeV$^2$. If we linearly add all the above-mentioned uncertainties we get a total uncertainty of $\pm 0.07$ for $\Delta=1$ GeV$^2$ and $\pm 0.05$ for $\Delta=3$ GeV$^2$. Thus, the theoretical uncertainties are comparable to the experimental uncertainties. Significance of the results --------------------------- We can now discuss the significance of the agreement between theory and experiment. We first ask: How restrictive is the data? To quantify the discussion we define a ‘straw-man’ model for ${\cal R}$ in which ${\cal R}$ is the same as the OPT result down to 2 GeV, but then follows the one-loop, 3-flavour form, $(12/27)(1/\ln Q^2/\Lambda_0^2)$, with $\Lambda_0 \approx 0.2$ GeV, until it reaches a value $H$, at which it remains frozen down to $Q=0$. If the “freeze-out” value, $H$, is about 0.3, then this ‘H model’ is essentially equivalent to the OPT result. If $H$ is much larger then this model gives a result for $\bar{R}_{\scriptscriptstyle{PQW}}(0)$ that is too large by more than the uncertainties just estimated. We find that $H$’s above 2 are disfavoured by the data. (As an illustration Fig. 8 shows the result with $H = 4.6$, which is clearly ruled out.) At the other extreme, the data disfavour an $H$ less than 0.09. Thus, although a wide range of $H$ values can be tolerated, the data do imply that the couplant cannot grow very large in the infrared region; nor can it remain too small. Next we ask: How predictive is the theory? Because of the need for smearing, the theory tells us almost nothing about the shape or structure of the $e^+e^-$ data in the region below 1 GeV. However, it does tell us something about the average magnitude of the cross section. The low-energy data is, in fact, dominated by the $\rho$ peak. After smearing with $\Delta = 1 $ GeV$^2$, this contributes about 0.7 to $\bar{R}_{\scriptscriptstyle{PQW}}$ below 1 GeV. Thus a 10% change in the area under the $\rho$ peak would change $\bar{R}_{\scriptscriptstyle{PQW}}$ by the $\pm 0.07$ estimated uncertainty in the theoretical prediction. We conclude that perturbative QCD can tell us, at least crudely, the size of the $\rho$ resonance. The smeared derivative ---------------------- As an extension of PQW’s ideas we also considered a quantity: $$\label{smder} D(Q,\Delta) = \frac{2 \Delta}{\pi} \int_0^{\infty} ds^{\prime} \frac{R_{e^+e^-}(\sqrt{s^{\prime}}) (s^{\prime}-Q^2)} { \{(s^{\prime}-Q^2)^2 +\Delta^2 \}^2 },$$ which represents a “smeared derivative”, in the sense that $$\lim_{\Delta \to 0} D(Q,\Delta) = {\rm d} R_{e^+e^-} / {\rm d} Q^2.$$ This provides a somewhat different test, though obviously not an independent test, of the relationship between theory and experiment. Its calculation requires only straightforward modifications to the procedures used to calculate $\bar{R}_{\scriptscriptstyle{PQW}}$. In Fig. 8 we compare the smeared derivatives from theory and experiment for $\Delta = 2$ and 4 Gev$^2$. For $\Delta = 2$ GeV$^2$ there is good agreement at low energies, and the theory qualitatively gives the first peak just below 3 GeV. However, there is clearly insufficient smearing in the charm region. Increasing $\Delta$ to 4 GeV$^2$ greatly smooths out both curves and gives quite good agreement. Phenomenological Virtues of a Frozen Couplant ============================================= The idea that the strong coupling constant, $\alpha_s(Q^2)$, “freezes” at low energies has long been a popular and successful phenomenological hypothesis. We first note that a freezing of $\alpha_s(Q^2)$ is a natural consequence of a picture where the gluon aquires an effective, dynamical mass $m_g$ [@mg]. Naively, this would modify the leading-order, 3-flavour couplant to: $$\label{mgform} \frac{\alpha_s(Q^2)}{\pi} = \frac{12}{27} \, \frac{1}{\ln[(Q^2 + 4m_g^2)/\Lambda_0^2]},$$ a form that has been used in many phenomenological papers. For $m_g$ a little larger than $\Lambda_0$ this gives a zero-$Q$ value comparable to ours. Note, though, that the variation with $Q$ at low energies is somewhat different from ours in Fig. 2. Another commonly used form is the “hard-freeze” form in which: $$\label{Hform} \frac{\alpha_s(Q^2)}{\pi} = \left\{ \begin{array}{ll} (12/27)(1/\ln Q^2/\Lambda_0^2), \quad & {\mbox{\rm for}} \,\, Q^2 \ge Q_0^2, \\ {\mbox{\rm constant}} \equiv H, & {\mbox{\rm for}} \,\, Q^2 \le Q_0^2, \end{array} \right.$$ with $H = (12/27)(1/\ln Q_0^2/\Lambda_0^2)$. This is the “$H$ model” that we mentioned in Subsect. 3.5. For $H \approx 0.26$ ([*i.e.,*]{} $Q_0/\Lambda_0 \approx 2.3$) it is a close approximation to our $\alpha_s(Q^2)/\pi$ shown in Fig. 2. We now briefly survey some of the phenomenological literature in order to make two points: (i) a frozen $\alpha_s$ provides a way to understand many important facts in hadronic physics, and (ii) the values extracted phenomenologically are very much in accord with our low-$Q$ value $\alpha_s/\pi = 0.26$. \[Note that we quote $\alpha_s/\pi$ rather than $\alpha_s$ values.\] \(a) [*Total hadron-hadron cross sections*]{}, although slowly rising at very high energies, are remarkably constant over a wide energy range, and their relative sizes correlate with their quark content in a very suggestive way. A simple and succesful description is provided by the 2-gluon-exchange model [@gunion], based on the Low-Nussinov model of the Pomeron [@lownuss]. This model requires a finite couplant at low momentum transfer, and Ref. [@gunion] found a value $\alpha_s/\pi \approx 0.17$. A recent version of this model, framed in terms of a dynamical gluon mass ($m_g = 0.37$ GeV, for $\Lambda_0 = 0.3$ GeV), is given in Ref. [@hkn]. Another recent version of this model [@nik] uses the ‘H-model’ form of $\alpha_s(Q^2)/\pi$. In order to fit the absolute magnitude of the $\pi$-nucleon cross section, $Q_0$ needs to be about 0.44 GeV [@nik] if $\Lambda_0 = 0.2$ GeV. This corresponds to $H = 0.28$. The same frozen couplant has been used successfully in subsequent work on deriving nucleon structure functions from the constituent-quark model [@barone]. \(b) [*Hadron spectroscopy*]{} also points to a low-energy couplant of around 0.2 – 0.25 [@barnes]. Godfrey and Isgur [@isgur] provide a unified description of light- and heavy-meson properties in a “relativized” potential model with a universal one-gluon-exchange-plus-linear-confinement potential. For the model to work for light mesons it is crucial to incorporate relativistic effects, and to employ a form of the couplant that freezes at low energies. Their fits yield a form of $\alpha_s(Q^2)/\pi$ that freezes to about 0.19, and has a shape similar to ours. In a fully relativistic treatment Zhang and Koniuk [@zhang] can naturally explain why the $\pi$ is so much lighter than the $\rho$. The $\pi/\rho$ mass ratio is a steeply falling function of the strong couplant, and the experimental value occurs at $\alpha_s/\pi= 0.265$ [@zhang]. \(c) [*Hadron form factors*]{} at low energies can be successfully treated assuming a frozen couplant, as shown in Ref. [@ji], which used the form (\[mgform\]) with $m_g \approx 0.1$ to 0.5 GeV, for $\Lambda_0 \approx 0.1$ GeV. \(d) [*Chiral soliton models of the nucleon*]{} can fit a wide variety of nucleon properties if one includes one-gluon exchange corrections with an $\alpha_s/\pi$ of about 0.2 [@stern; @duck]. Ref. [@stern] finds that the experimental deviation from the Gottfried sum rule, the $\Delta$-nucleon mass difference, the first moment of the polarized proton structure function, and the neutron-proton mass difference all require a common $\alpha_s/\pi$ value. (However, the actual value found, 0.2, could be re-scaled by making a different choice for another parameter in the model [@stern].) Other nucleon properties are consistent with an $\alpha_s/\pi$ of this size [@duck]. \(e) [*The $p_T$ spectrum in $W, Z$ production*]{} in $pp$ or $p\bar{p}$ collisions can be successfully predicted by QCD right down to $p_T=0$ if multiple gluon radiation effects are appropriately re-summed [@WZpt]. However, it is essential in the low-$p_T$ region to invoke a freezing of $\alpha_s(p_T^2)$. The form (\[mgform\]) has been used, with $4 m_g^2/\Lambda_0^2$ denoted by ‘$a$’. Unfortunately, the results are very insensitive to the parameter $a$; anything in the range 3 – 100 gives an acceptable fit to current data [@halzen; @fletcher]. This corresponds to a range 0.1 – 0.4 for the zero-$Q$ couplant. Perhaps, future data will make it possible to narrow this range. \(f) [*Jet properties*]{} can be quite successfully described by the “modified leading log approximation” [@dok; @khoze], but to obtain predictions at small momenta it is necessary to invoke a freezing of the couplant. Fits to data on heavy-quark-initiated jets give zero-$Q$ values of $\alpha_s/\pi$ around 0.22 [@khoze]. This value depends somewhat on the form of $\alpha_s(Q^2)$ assumed in the fit, but it was found empirically that the result for the integral: $$\int_0^{1 {\mbox{{\small \rm GeV}}}} \! {\rm d} k \, \frac{\alpha_s^{{\rm eff}}(k^2)}{\pi} \approx 0.2 \, {\mbox{\rm GeV}}$$ was [*fit invariant*]{} [@khoze]. Integrating our $\alpha_s/\pi$ in Fig. 2 leads to precisely 0.2 GeV. \(g) [*Hadron-hadron scattering at very high energies*]{} where the cross sections rise asymptotically, but must satisfy unitarity, seems to call for the ‘critical Pomeron’ picture [@critpom], at least as a first approximation. It seems that one could only hope to derive such a picture from QCD if there is an infrared fixed point [@white]. In fact, White has argued for additional quarks, or colour-sextet quarks, in order to have $N_f$ effectively equal to 16 [@white]. (See the discussion in Subsect. 2.6 above.) However, our results imply that the infrared fixed point persists down to low $N_f$. This may mean that one can have all the virtues of White’s picture without the need for more quarks. Summary and Conclusions ======================= We have applied OPT to the third-order QCD calculation of $R_{e^+e^-}$. At energies above about 1 GeV there is every sign that the approximation is healthy: the perturbation series in the “optimized” scheme is well behaved, and there is good agreement between second- and third-order results. This was not true of the situation created by the old, incorrect $R_{e^+e^-}$ calculation [@old; @max; @oldopt] (see Table 2). The contrast between the ‘old’ and ‘new’ results emphasizes the point that the third-order $R_{e^+e^-}$ calculation provides a very real, empirical test of “optimization” ideas. At the time, the statements [@oldopt] that the ‘old’ third-order results [@old] tended to cast doubt on the usefulness of “optimization” were perfectly fair comment. Because of this history we take especial satisfaction in the transformed situation produced by the new result [@new]. Furthermore, contrary to the old situation, the optimized couplant now shows a marked decrease from second to third order. This is in accord with the “induced convergence” conjecture that “optimization” naturally cures the divergent-series problem [@optult; @indcon]. The third-order OPT results remain finite down to $Q = 0$, with the optimized couplant, $\alpha_s/\pi$, “freezing” to a value 0.26 below 300 MeV. No [*ad hoc*]{} assumption was used to obtain this result: it is the direct consequence of using the calculated $R_{e^+e^-}$ and $\beta$ series coefficients as inputs to the “optimization” procedure specified in Ref.[@OPT]. It must be admitted that, at very low energies, the prediction for ${\cal R}$ (the QCD correction term in $R_{e^+e^-}$) has a large uncertainty. Since third order is the lowest order at which it is even possible to get finite infrared results, one should not be surprised if the approximation is somewhat crude. Nevertheless, as we discussed in Subsect.2.6, the qualitative conclusion that ${\cal R}$ remains small (say, $0.3 \pm 0.3$ at $Q=0$) is inescapable in the context of OPT. The OPT prediction is supported by the data. As we showed in Sect. 3, the PQW-smeared $R_{e^+e^-}$ data is consistent with a perturbative QCD description, provided that the couplant freezes to a modest value at low energies. The hypothesis that the couplant freezes at low energies has been used very successfully in a wide variety of phenomenological work, where the low-energy couplant is treated as a free parameter to be fitted to experiment. The values that emerge are quite comparable to ours. There are some other theoretical indications of a freezing of the couplant [@mg; @gribov], but our evidence is remarkable in that it comes solely from perturbation theory and RG invariance. The predicted value, $\alpha_s/\pi = 0.26$, for the frozen couplant is a purely theoretical number. It does not depend on knowing the value of $\Lambda_{\overline{\rm MS}}$, but only on knowing the number of light quarks. We thank A. Kataev and S. Larin for correspondence, and Ian Duck, Robert Fletcher, Francis Halzen, Nathan Isgur, Valery Khoze, Chris Maxwell, and John Ralston for helpful comments.\ This work was supported in part by the U.S. Department of Energy under Contract No. DE-FG05-92ER40717. Appendix A: Numerical solution of the optimization equations {#appendix-a-numerical-solution-of-the-optimization-equations .unnumbered} ------------------------------------------------------------ After expressing $\bar{r}_1$ and $\bar{r}_2$ in terms of $\bar{a}$ and $\bar{c}_2$, using the $\rho_1$, $\rho_2$ definitions and (\[k3\]), we have to simultaneously solve the optimization equations, (\[eq1\], \[eq2\]). These define two curves in the $a$, $c_2$ plane whose intersection point we seek. In what we call the ‘spiralling’ method, (\[eq2\]) is first solved for $\bar{a}$; then, with this $\bar{a}$, (\[eq1\]) is solved for $\bar{c}_2$; then, with this $\bar{c}_2$, (\[eq2\]) is solved for $\bar{a}$; and so on. Which equation is solved for which variable is crucial; the other choice would ‘spiral out’ from the desired solution. (The standard “secant method” [@nrecipe] was generally sufficient for solving the individual optimization equations.) A convenient starting point for this iterative procedure was provided by an approximate solution to the optimized equations due to Pennington, Wrigley, and Mignaco and Roditi (PWMR) [@PWMR]. This approximation expands the optimization equations (\[eq1\], \[eq2\]) as a series in $\bar{a}$ and keeps only the lowest non-trivial term. Noting that $I = \bar{a}(1 - c \bar{a} + \ldots)$, it is easy to check that this gives: $$\bar{r}_1 \approx 0 \quad\quad \bar{r}_2 \approx -{\scriptstyle \frac{1}{3}} \bar{c}_2.$$ One can improve this approximation by writing each $\bar{r}_i$ as a series in $\bar{a}$ and successively equating coefficients of different orders in $\bar{a}$ to zero. To next order this gives: $$\bar{r}_1 \approx {\scriptstyle \frac{1}{3}} \bar{c}_2 \bar{a}, \quad\quad \bar{r}_2 \approx -{\scriptstyle \frac{1}{3}} \bar{c_2} + {\scriptstyle \frac{1}{9}} c \bar{c_2} \bar{a}.$$ \[One may note that in $\bar{{\cal R}}^{(3)}$ there is a near cancellation between the second and third order terms, $\bar{r}_1 \bar{a}$ and $\bar{r}_2 \bar{a}^2$. Thus, $\bar{{\cal R}}^{(3)}$ turns out to be closely equal to $\bar{a}^{(3)}$.\] The ‘spiralling’ method worked well for $Q > 0.3$ GeV starting from the PWMR solution. However, at lower energies the PWMR approximation breaks down, and does not provide a satisfactory initial guess. In fact, at the infrared fixed point one has instead, from (\[fpr\]): $$\bar{r}_1 = {\scriptstyle \frac{1}{2}} \bar{c}_2 \bar{a}^*, \quad\quad \bar{r}_2 = -{\scriptstyle \frac{2}{3}} \bar{c}_2.$$ We therefore proceeded to low $Q$ in successive stages, utilizing the solution at the previous $Q$ as the initial guess for the next lower $Q$. We also encountered a ‘creep’ problem: At low $Q$ the two curves representing the “optimization” equations become almost parallel (each being almost parallel to the infrared boundary line $1 + c a + c_2 a^2 = 0$) and they cross at a very small angle. Thus, instead of ‘spiralling in’ to the solution, one creeps towards it stepwise. The convergence is very slow and the danger is that the solution can appear to have converged within the specified tolerance, when in fact it still has a considerable way to go. To avoid this pitfall we would repeat the procedure from a different starting point, so as to creep towards the solution from the other side. In this way we could bracket the true solution, and hence ensure reliable accuracy. We also tried the ‘intersection’ method as an alternative. Taking an initial guess for $\bar{c}_2$, one solves for $\bar{a}$ in each of the two optimized equations. For each $\bar{a}$ one then solves the other equation for $\bar{c}_2$. This gives a pair of points on each of the two curves. The straight lines that join up each pair should approximate the curves themselves, and hence their intersection should approximate the desired solution. The procedure can then be iterated. This method also worked well for $Q > 0.3$ GeV starting from the PWMR solution. At lower energies, where the two curves become nearly parallel, this method did not suffer from the ‘creep’ problem, but it had the opposite vice: it tended to make such a large extrapolation in each iteration that it would become unstable and erratic. Appendix B: Flavour thresholds {#appendix-b-flavour-thresholds .unnumbered} ------------------------------ Since ${\cal R}$ has been calculated only with massless quarks, we are really approximating “full QCD” with a [*set*]{} of effective theories, each with a different number of massless quarks. The $\Lambda_{\overline{\rm MS}}$ parameters of these theories need to be appropriately matched, so that they correspond to a single, underlying “full QCD” theory. The point is well explained by Marciano [@marciano], who provides explicit formulas for matching $\Lambda_{\overline{\rm MS}}$ across thresholds. Unfortunately his analysis uses a truncated expansion of $a(\mu)$ in powers of $1/\ln(\mu/\Lambda)$, which would not be a valid approximation at low energies; in particular at $s$-quark threshold. Our procedure was simply to require the optimized $\bar{{\cal R}}^{(3)}$ to be continuous at a threshold. This was done numerically by running our optimization program at the threshold energy ($Q = 2 m_q$) with both values of $N_f$ and adjusting one of the $\Lambda_{\overline{\rm MS}}$ parameters until the two $\bar{{\cal R}}^{(3)}$ results agreed. Starting with $\Lambda_{\overline{\rm MS}}^{(4)} = 230$ MeV for 4 flavours, we found $\Lambda_{\overline{\rm MS}}^{(3)} = 281$ MeV, and $\Lambda_{\overline{\rm MS}}^{(2)} = 255$ MeV. \[In terms of $\tilde{\Lambda}$ the corresponding values are: 257, 308, and 277 MeV for 4, 3, 2 flavours, respectively.\] Essentially the same results were obtained if we required instead that $\bar{a}$ be continuous. We checked that this procedure agreed very closely with Marciano’s formulas at both $c$- and $b$-quark thresholds. It is noteworthy that we find $\Lambda_{\overline{\rm MS}}^{(2)}$ to be smaller than $\Lambda_{\overline{\rm MS}}^{(3)}$, contrary to the pattern at the higher thresholds. Our final results are very insensitive to the $\Lambda_{\overline{\rm MS}}^{(2)}$ value, however, because at energies below $s$ threshold the $\bar{{\cal R}}$ results are essentially governed by the infrared fixed point. [99]{} S. G. Gorishny, A. L. Kataev, and S. A. Larin, Phys. Lett. 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D [**29**]{}, 580 (1984). $N_f$ $\rho_2$ $\alpha_s^*/\pi$ ------- ----------- ------------------ 0 $ -8.410$ 0.313 1 $ -9.997$ 0.280 2 $-10.911$ 0.263 3 $-12.207$ 0.244 4 $-13.910$ 0.224 5 $-15.492$ 0.208 6 $-17.665$ 0.191 : $\rho_2$ invariants and fixed-point couplants for $N_f = 0$ to 6. $N_f=5$, $Q=34$ GeV Order $\bar{a}$ $\bar{r}_1$ $\bar{r}_2$ $\bar{\cal R}$ change --------------------------------------------- ------- ----------- ------------- ------------- ---------------- --------- 2nd 0.0415 $-0.599$ — 0.0404 — $\tilde{\Lambda}_{\overline{MS}} = 100$ MeV 3rd 0.0394 $-0.301$ 7.64 0.0394 $-2.4$% ($\Lambda_{\overline{MS}} = 87$ MeV) old 0.0452 +1.363 $-29.48$ 0.0453 12% 2nd 0.0569 $-0.588$ — 0.0550 — $\tilde{\Lambda}_{\overline{MS}}= 500$ MeV 3rd 0.0526 $-0.405$ 7.71 0.0526 $-4.4$% ($\Lambda_{\overline{MS}} = 436$ MeV) old 0.0690 +1.988 $-27.59$ 0.0694 26% : Comparison of second- and third-order optimized results: ‘old’ refers to third order with the old, incorrect result. $Q$ (GeV) $N_f$ $\bar{a}$ $\bar{r}_1$ $\bar{r}_2$ $\bar{\cal R}$ error ----------- ------- ----------- ------------- ------------- ---------------- ------- 3.0 4 0.076 $-0.53$ 6.9 0.076 4% 1.0 3 0.126 $-0.79$ 6.3 0.126 10% 0.4 3 0.221 $-1.77$ 8.8 0.229 43% 0 2 0.263 $-2.89$ 14.6 0.330 100% : Illustrative third-order optimized results at low energies. $\Lambda_{\overline{MS}} {\mbox{\rm (4 flavours)}} = 230$ MeV. The estimated fractional error is $|\bar{r}_2 \bar{a}^2|$. Figure Captions {#figure-captions .unnumbered} =============== Fig. 1. The optimized solutions in the $a, c_2$ plane for 2 and 3 quark flavours, in the low-energy region. The open squares represent the fixed-point solution, Eqs. (\[fpeq\], \[fpeqc2\]), which lies on the infrared boundary $1+ca+c_2 a^2 = 0$. The boundary is shown by the solid line ($N_f = 2$) or the dashed line ($N_f = 3$) at the right. The dotted vertical lines are to indicate $s\bar{s}$ threshold at $Q=0.40$ GeV where $N_f$ changes from 3 to 2. (The $\Lambda$ parameters are matched so that $\bar{R}$ is continuous (see Appendix B.), but there are then slight discontinuities in $\bar{a}$ and $\bar{c}_2$.) The dotted line shows the solution for a 3 flavor world down to $Q=0$, while the dashed line shows a 2 flavour world extending up towards 1 GeV. The points shown are spaced at 0.05 GeV intervals from $Q=0.40$ GeV. Fig. 2. The optimized third-order results for $\bar{a} = \alpha_s/\pi$ and $\bar{{\cal R}}^{(3)}$. Also shown is the second-order result, $\bar{{\cal R}}^{(2)}$. Quark thresholds are indicated by the vertical lines. Fig. 3. The perturbative QCD prediction for $R_{e^+e^-}$ from third-order OPT (solid line). The inset shows the region around $u$ and $d$ quark thresholds. The dashed line is the parton-model prediction. Fig. 4. Compilation of experimental $R_{e^+e^-}$ data (excluding narrow resonances). A few representative statistical error bars are shown. The solid line represents an ‘eyeball fit’. Fig. 5. The contributions of narrow resonances to $\bar{R}_{\scriptscriptstyle{PQW}}$ for two values of the smearing parameter $\Delta$. Fig. 6. Comparison of “smeared” theoretical and experimental results. The parton-model result is shown by the dotted line in (b). Fig. 7. Comparison of “smeared” results extended to spacelike $Q^2$. The dotted line shows a ‘straw-man’ model in which the couplant becomes large at low energies (see subsect. 3.5). Fig. 8. Comparison of theoretical and experimental results for the “smeared derivative” (Eq. (\[smder\])) for two values of $\Delta$.

--- abstract: 'We study the growth of massive black holes (BH) in galaxies using smoothed particle hydrodynamic simulations of major galaxy mergers with new implementations of BH accretion and feedback. The effect of BH accretion on gas in its host galaxy is modeled by depositing momentum at a rate $\sim \tau L/c$ into the ambient gas, where $L$ is the luminosity produced by accretion onto the BH and $\tau$ is the wavelength-averaged optical depth of the galactic nucleus to the AGN’s radiation (a free parameter of our model). The accretion rate onto the BH is relatively independent of our subgrid accretion model and is instead determined by the BH’s dynamical impact on its host galaxy: BH accretion is thus self-regulated rather than “supply limited.” We show that the [final]{} BH mass and total stellar mass formed during a merger are more robust predictions of the simulations than the [time dependence]{} of the star formation rate or BH accretion rate. In particular, the latter depend on the assumed interstellar medium physics, which determines when and where the gas fragments to form star clusters; this in turn affects the fuel available for further star formation and BH growth. Simulations over a factor of $\sim 30$ in galaxy mass are consistent with the observed $M_{BH}-\sigma$ relation for a mean optical depth of $\tau \sim 25$. This requires that most BH growth occur when the galactic nucleus is optically thick to far-infrared radiation, consistent with the hypothesized connection between ultra-luminous infrared galaxies and quasars. We find tentative evidence for a shallower $M_{BH}-\sigma$ relation in the lowest mass galaxies, $\sigma \lesssim 100 {{\rm \, km \, s^{-1}}}$. Our results demonstrate that feedback-regulated BH growth and consistency with the observed $M_{BH}-\sigma$ relation do not require that BH feedback terminate star formation in massive galaxies or unbind large quantities of cold gas.' author: - | Jackson DeBuhr,$^{1,2}$ Eliot Quataert,$^{1,2}$ and Chung-Pei Ma$^2$\ $^1$Department of Physics, University of California, Berkeley, CA 94720, USA\ $^2$Department of Astronomy and Theoretical Astrophysics Center, University of California, Berkeley, CA 94720, USA bibliography: - 'paper\_bib.bib' - 'main\_bib.bib' title: The Growth of Massive Black Holes in Galaxy Merger Simulations with Feedback by Radiation Pressure --- galaxies: evolution – galaxies: active Introduction ============ Feedback from an active galactic nucleus (AGN) has been invoked to resolve a number of observational problems in galaxy formation: (1) to explain the tight observed [@ferrarese2000; @gebhardt2000; @haring2004] correlations between central black hole (BH) and galaxy properties such as the $M_{BH}-\sigma$ and $M_{BH}-M_*$ relations and the BH “fundamental plane” [@silk1998; @king2003; @murray2005; @di-matteo2005; @sazonov2005; @hopkins2007], (2) to shut off star formation in elliptical galaxies (e.g., by blowing gas out of the galaxy), thereby explaining how ellipticals become “red and dead” (e.g., @springel2005b [@ciotti10]), (3) to heat the hot intracluster plasma (ICM) in groups and clusters, thereby suppressing cooling and star formation in these environments (e.g., @tabor1993 [@ciotti1997; @croton2006]), and (4) to help explain “cosmic downsizing,” namely the fact that both star formation and AGN activity reside in progressively lower mass halos at lower redshifts (e.g., @scannapieco2005). It is plausible that AGN perform the roles desired of them, but this is by no means certain. Understanding whether this is indeed the case requires developing more sophisticated theoretical models that can be compared quantitatively to observations. There are several key theoretical problems that must be addressed in order to better understand the role of massive BHs in galaxy formation, and to understand the properties of massive BHs themselves. The first is the problem of AGN fueling, i.e., how is gas transferred from galactic scales ($\sim 0.1-1$ kpc) to the vicinity of the massive BH ($\lesssim 0.1$ pc)? A second key problem is the problem of AGN feedback: how do energy and momentum generated by accretion onto a central BH – in the form of radiation and outflows – couple to the surrounding gas, and how does this affect star formation and the growth of the BH itself? Much of the recent work addressing the impact of BHs on galaxy formation has used qualitatively similar physics (e.g., @2005MNRAS.361..776S [@2009ApJ...690..802J]). For example, many calculations assume that a BH of mass $M_{BH}$ will accrete mass at a rate proportional to the Bondi rate [@1952MNRAS.112..195B]: $$\dot{M}_{Bondi} = \frac{4 \pi f G^2 M_{BH}^2 \rho}{c_s^3} \label{EquationBondi}$$ where $\rho$ is the density of the surrounding gas, $c_s$ is the sound speed of that gas, and $f \sim 10-100$ is a factor taking into account the possible multi-phase structure of the gas and that the sphere of influence of the BH is often not resolved [@2009MNRAS.398...53B]. There is, however, little justification for using equation \[EquationBondi\]. The Bondi accretion rate estimate assumes that the gas surrounding the BH is spherically symmetric. When the gas is not spherically distributed, the rate of angular momentum transport determines the BH accretion rate (e.g., @shlosman1990). It is generally believed that the progenitors of todays $\gtrsim L^{*}$ ellipticals are gas-rich disk galaxies, the mergers of which lead to luminous starbursts and the growth of the central massive BHs [@1988ApJ...325...74S; @2005ApJ...630..705H]. Most of the gas in disk galaxies, merging galaxies, luminous starbursts [@1998ApJ...507..615D; @2006ApJ...640..228T], and nearby luminous AGN [@ho2008] appears to reside in a rotationally supported disk. There is thus no reason to expect that the spherically symmetric Bondi rate provides a good estimate of the BH accretion rate in gas rich galaxies. Even in the central $\sim$ parsec of own galaxy, where the ambient gas [*is*]{} hot and pressure supported, the Bondi accretion rate fails by orders of magnitude to predict the accretion rate onto the central BH [@sharma07]. There are a number of ways that an AGN can strongly influence its surroundings (e.g., @ostriker10b). Relativistic jets inject energy into intracluster plasma and may be the primary mechanism suppressing cooling flows in galaxy clusters [@mcnamara2007], even though the details of how the energy in the jet couples to the plasma in a volume filling way are not fully understood [@vernaleo2006]. On galactic scales, a wind from an accretion disk around the BH can drive gas out of the galaxy (e.g., @king2003) as could cosmic-ray protons produced by a radio loud AGN [@socrates10]. In addition, the AGN’s radiation can strongly affect the surrounding gas, both by Compton heating/cooling (e.g., @sazonov2005) and by the momentum imparted as UV radiation is absorbed by dust grains [@chang1987; @1988ApJ...325...74S; @murray2005]. This diversity of feedback mechanisms can be roughly separated into two broad classes: energy and momentum injection. We believe that momentum injection is the dominant mode of feedback for most of the gas in a galaxy, largely because of the very short cooling times of dense gas. For example, if a BH radiates at $\sim 10^{46}$ erg $\rm{s}^{-1}$ with a typical quasar spectrum, only gas with $n \lesssim 1 \, \rm{cm}^{-3}$ can be heated to the Compton temperature within $\sim 100$ pc. However, the mean gas densities in the central $\sim 0.1-1 \, \rm{kpc}$ of luminous star forming galaxies are $\sim 10^{3-5} \rm{cm}^{-3}$ [@1998ApJ...507..615D; @2006ApJ...640..228T]. At these densities, the cooling time of gas is sufficiently short that it is unable to retain much injected energy – be it from the AGN’s radiation or from shocks powered by AGN outflows. Thus it is largely the momentum imparted by AGN outflows and by the absorption and scattering of the AGN’s radiation that dominates the impact of the AGN on dense gas in galaxies. Since it is the dense gas that fuels star formation and the growth of the BH itself, it is critical to understand the impact of momentum feedback on this gas.[^1] In this paper, we present simulations of major mergers of spiral galaxies using a model for the growth of BHs that includes (1) a BH accretion rate prescription motivated by the physics of angular momentum transport and (2) AGN feedback via momentum injection (e.g., radiation pressure). Some results of this model appear in a companion Letter [@2009arXiv0909.2872D]. The remainder of this paper is organized as follows. Section \[sectionMethods\] presents a summary of our methods, including a description of the model galaxies (§\[sectionICs\]), the model for star formation and the interstellar medium (§\[sectionSFR\]), our BH accretion and feedback model (§\[sectionBlackHoles\]) and a summary of our parameter choices (§\[sectionParam\]). Section \[sectionResults\] shows the results of applying this model to BH growth and star formation in major mergers of gas-rich galaxies. In section \[sectionGalaxyBH\] we show that our model of BH growth and feedback produces a reasonably tight $M_{BH}-\sigma$ correlation similar to that observed. Finally, in section \[sectionDiscussion\] we discuss our results and compare our approach to previous models in the literature. Appendix \[resolution\] presents resolution tests for our fiducial simulation while Appendix \[sectionAppendix\] presents some of the tests used to verify the BH accretion and feedback models that we have implemented. Methodology {#sectionMethods} =========== We use a non-public update of the TreeSPH code GADGET-2 [@2005MNRAS.364.1105S] provided by V. Springel to perform simulations of equal-mass mergers of galaxies. This version of the code includes the effective star formation model of [@2003MNRAS.339..289S] but contains no AGN feedback physics. We modified the code further to implement models for massive BH growth and AGN feedback. The details of the simulations are described in the following subsections. The Appendices present resolution tests and some of the tests we performed to verify our implementation of the BH accretion and feedback model. Initial Conditions and Galaxy Parameters {#sectionICs} ---------------------------------------- Each model galaxy used in our major merger simulations is similar to those in [@2005MNRAS.361..776S]. They include a spherical halo of collisionless dark matter, a centrifugally supported disk of gas and stars, a stellar bulge, and a central point mass representing a black hole. The code used to generate the initial conditions was provided by V. Springel and is identical to that used in [@2005MNRAS.361..776S] except for one change that will be described below. Table 1 lists the relevant galaxy and simulation parameters for the key merger simulations we focus on in this paper. The simulations are all major mergers of equal mass galaxies. The fiducial simulation (top entry) assumes a mass of $1.94\times 10^{12} M_\odot$ for each merging galaxy, of which 4.1% is assigned to the gas and stars in the disk, 1.36% is assigned to the stars in the bulge, and the rest is in a dark matter halo. The initial mass fraction of gas in the disk is $f_g=0.1$. This run uses a total of $N_p=1.6\times 10^6$ particles with $6\times 10^5$ dark matter particles, $2\times 10^5$ particles each in the gaseous and stellar disk, and $10^5$ particles for the stellar bulge. This run has a Plummer equivalent gravitational force softening of $\epsilon = 47$ pc. [lcccccccccccc]{} Run Name & $M_{tot}$ & $f_{g,0}$ & $\frac{M_b}{M_d}$ & $N_p$ & $\epsilon$ & $\frac{R_{acc}}{\epsilon}$ & $\alpha$ & $\tau$ & $M_{*,new}$ & $M_{BH,f}$ & $M_{BH,p}$ & $\sigma_f$\ & \[$M_{fid}$\]${}^{a}$ & & & \[$10^6$\] & \[$\rm{pc}$\] & & & & \[$10^{10} M_{\sun}$\] & \[$10^{8} M_{\sun}$\] & \[$10^{8} M_{\sun}$\] & \[$\rm{km} \rm{s}^{-1}$\]\ fid & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 4 & 0.05 & 10 & 1.34 & 1.49 & 1.33 & 169\ fidNof$^{b}$ & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 4 & 0.15 & 0 & 1.36 & 18.1 & 13.5 & 170\ fid3a & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 4 & 0.15 & 10 & 1.34 & 1.03 & 0.90 & 168\ fid6a & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 4 & 0.3 & 10 & 1.35 & 0.86 & 0.77 & 167\ fidTau & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 4 & 0.05 & 3 & 1.36 & 5.05 & 4.31 & 163\ fidt25 & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 4 & 0.05 & 25 & 1.35 & 0.39 & 0.35 & 169\ fid8eps & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 8 & 0.05 & 10 & 1.35 & 2.70 & 1.76 & 163\ fidafg & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 4 & \*$^{c}$ & 10 & 1.32 & 1.21 & 1.02 & 169\ fidq2$^d$ & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 4 & 0.05 & 10 & 1.30 & 1.40 & 1.16 & 168\ fidq07$^e$ & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 4 & 0.05 & 10 & 1.32 & 1.52 & 1.36 & 164\ big & 3.0 & 0.1 & 0.33 & 1.6 & 68 & 4 & 0.05 & 10 & 3.08 & 6.24 & 5.27 & 232\ big6a & 3.0 & 0.1 & 0.33 & 1.6 & 68 & 4 & 0.3 & 10 & 4.17 & 7.86 & 5.15 & 227\ mid & 0.3 & 0.1 & 0.33 & 1.6 & 32 & 4 & 0.05 & 10 & 0.39 & 0.38 & 0.26 & 115\ small & 0.1 & 0.1 & 0.33 & 1.6 & 22 & 4 & 0.05 & 10 & 0.13 & 0.24 & 0.13 & 82\ small6a & 0.1 & 0.1 & 0.33 & 1.6 & 22 & 4 & 0.3 & 10 & 0.13 & 0.25 & 0.24 & 84\ smallq07$^e$ & 0.1 & 0.1 & 0.33 & 1.6 & 22 & 4 & 0.05 & 10 & 0.12 & 0.06 & 0.05 & 81\ fg & 1.0 & 0.3 & 0.33 & 2.4 & 47 & 4 & 0.05 & 10 & 4.41 & 7.10 & 5.53 & 159\ smallfg & 0.1 & 0.3 & 0.33 & 2.4 & 22 & 4 & 0.05 & 10 & 0.36 & 0.31 & 0.23 & 98\ bulge & 1.0 & 0.1 & 0.20 & 1.6 & 47 & 4 & 0.05 & 10 & 1.38 & 1.44 & 1.25 & 161\ LRfid & 1.0 & 0.1 & 0.33 & 0.16 & 102 & 4 & 0.05 & 10 & 1.34 & 1.65 & 0.93 & 164\ MRfid & 1.0 & 0.1 & 0.33 & 0.48 & 70 & 4 & 0.05 & 10 & 1.35 & 2.92 & 2.40 & 168\ MRfidNof$^{b}$ & 1.0 & 0.1 & 0.33 & 0.48 & 70 & 4 & 0.15 & 0 & 1.34 & 13.5 & 11.4 & 167\ LRfidNof$^{b}$ & 1.0 & 0.1 & 0.33 & 0.16 & 102 & 4 & 0.15 & 0 & 1.31 & 13.1 & 11.4 & 175\ fidvol & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 8.62 & 0.05 & 10 & 1.39 & 3.22 & 2.45 & 164\ MRfidvol & 1.0 & 0.1 & 0.33 & 0.48 & 70 & 5.97 & 0.05 & 10 & 1.36 & 3.30 & 1.92 & 164\ Columns are defined as follows: $M_{tot}$ is the total mass in the simulation, $f_{g,0}$ is the initial gas fraction of the disk, $M_b/M_d$ is the bulge to disk mass ratio, $N_p$ is the total number of particles used in the simulation, $\epsilon$ is the Plummer equivalent gravitational force softening, $R_{acc}$, $\alpha$ and $\tau$ are the parameters of the BH accretion and feedback model (§\[sectionBlackHoles\]), $M_{*,new}$ is the total mass of new stars formed during the simulation, $M_{BH,f}$ and $M_{BH,p}$ are the masses of the BH at the end of the simulation and after the peak of accretion (defined to be when the accretion rate drops to one tenth its maximum value), respectively, and $\sigma_f$ is the stellar velocity dispersion of the merger remnant (§\[sectionGalaxyBH\]). To test the dependence of the results of our fiducial simulation on the model and simulation parameters, we have run a number of additional simulations, varying the gas fraction ($f_g=0.3$ vs 0.1), bulge-to-disk mass ratio (0.2 vs 0.33), total galaxy mass (from 0.1 to 3 of the fiducial value), simulation particle number (from $N_p=1.6\times 10^5$ to $2.4\times 10^6$), force softening ($\epsilon=22$ to 102 pc), as well as the parameters in the black hole model (described in § 2.4 below). We use a [@1990ApJ...356..359H] density profile for the structure of the dark matter halo: $$\rho_{halo}(r) = \frac{M_{halo}}{2\pi} \frac{a}{r(r+a)^3}. \label{hernqprof}$$ The scale length $a$ of the halo is set by requiring that the halo enclose the same mass within the virial radius as an NFW profile, and that the densities match at small radii. These conditions yield a relationship among the halo scale length, $a$, the corresponding NFW scale length, $r_s$, and the concentration of the NFW halo, $c$ [@1996ApJ...462..563N; @2005MNRAS.361..776S]: $a = r_s \{2[\ln{(1 + c)}-c/(1+c)]\}^{1/2}$. The halos used in this work all have a concentration of $c = 9$. The stellar and gaseous disks both initially have exponential surface density profiles: $$\Sigma(R) = \frac{M_i}{2 \pi R_d^2} \exp \left(-\frac{R}{R_d}\right) \label{EquationExponentialSurfDens}$$ where $M_i$ is the total mass of the component of interest and $R_d$ is the disk scale length, which is initially the same for the stellar and gaseous disks. The disk scale length for the fiducial simulation is $R_d = 3.5$ kpc, which corresponds to the disk having approximately the same angular momentum per unit mass as a halo with a spin parameter of $0.033$. For simulations with different disk masses, we use $R_d \propto M_d^{1/3}$, which is consistent with the observed relation [@2003MNRAS.343..978S]. The stellar disk’s vertical structure is given by the standard ${\rm sech}^2(z/z_0)$ profile, where the vertical scale height $z_0$ is initially set to $z_0 = R_d/5$ at all radii. Unlike the stellar disk, the gaseous disk’s vertical structure is determined by hydrostatic equilibrium given the assumed sound speed/equation of state of the gas (discussed below). Setting up this initial vertical hydrostatic equilibrium requires an iterative procedure that is described in [@2005MNRAS.361..776S]. The stellar bulges also have Hernquist density profiles. The scale length of the bulge $R_b$ is specified as a fraction of the disk scale length, $R_d$. In the fiducial simulation, $R_b = R_d/5$. For different bulge masses, we use the scaling relation $R_b \propto M_b^{1/2}$, which is motivated by the observed mass-radius relation of elliptical galaxies [@2003MNRAS.343..978S]. In our simulations, two galaxies with identical structure are placed on a prograde orbit. For simulations at our fiducial mass of $1.94\times 10^{12} M_\odot$ (for each galaxy), the initial separation of the two galaxies’ centers is $142.8$ kpc. The orbit has approximately zero total energy, which corresponds to an initial velocity for each galaxy of 160 km s$^{-1}$; the velocity is directed at an angle of $28$ degrees from the line connecting the centers of the two galaxies. In order to break the symmetry of the problem, the individual spin axes of the galaxies have a relative angle of about $41$ degrees, with one galaxy of the pair having an inclination with respect to the orbital plane of $10$ degrees. For the simulations with different overall masses, the orbital parameters are scaled by $M^{1/3}$, so that the time to first passage and the time to final merger are similar to those in the fiducial run. Interstellar Medium Model {#sectionSFR} ------------------------- The version of GADGET we use includes [@2003MNRAS.339..289S]’s sub-resolution model for the interstellar medium (ISM). This model treats the gas as a two phase medium of cold star forming clouds and a hot ISM. When cooling and star formation are rapid compared to the timescale for adiabatic heating and/or cooling (which is nearly always the case in our calculations), the sound speed of the gas is not determined by its true temperature, but rather by an effective sound speed that averages over the multi-phase ISM, turbulence, etc. The effective sound speed as a function of density can be interpolated freely between two extremes using a parameter ${q_{\rm eos}}$. At one extreme, the gas has an effective sound speed of $10 \, {\rm km\,s^{-1}}$, motivated by, e.g., the observed turbulent velocity in atomic gas in nearby spirals; this is the “no-feedback” case with ${q_{\rm eos}}=0$. The opposite extreme, ${q_{\rm eos}}=1$, represents the “maximal feedback” sub-resolution model of @2003MNRAS.339..289S, motivated by the multiphase ISM model of @mckee77; in this case, $100\%$ of the energy from supernovae is assumed to stir up the ISM. This equation of state is substantially stiffer, with effective sound speeds as high as $\sim200\,{\rm km\,s^{-1}}$. Varying ${q_{\rm eos}}$ between these two extremes amounts to varying the effective sound speed of the ISM, with the interpolation $$c_{s} = \sqrt{{q_{\rm eos}}\,c_{s}^{2}[q=1] + (1-{q_{\rm eos}})\,c_{s}^{2}[q=0]}\ . \label{eqn:qeos}$$ In addition to this effective equation of state, GADGET models star formation by stochastically converting gas particles into star particles at a rate determined by the gas density, $$\dot{\rho}_{SF} = \frac{1-\beta}{t_{*}^0 \rho_{th}^{1/2}} \rho^{1/2} \rho_c \propto \rho^{3/2} \label{equationSFRModel}$$ where $\beta = 0.1$ is the fraction of the mass of a stellar population returned to the ISM by stellar evolution. The parameter $t^0_{*}$ is the characteristic timescale for gas to be converted into stars at the threshold density $\rho_{th} = 0.092$ cm$^{-3}$; $\rho_c \approx \rho$ is the density of the cold clouds, which is related to the density of the SPH particle by equations (17) and (18) of [@2003MNRAS.339..289S]. For a given gas equation of state, the parameters in equation \[equationSFRModel\] can be adjusted to produce a global star formation law similar to the observed Kennicutt-Schmidt relations [@2005MNRAS.361..776S]. For parameters in the equation of state model that have been used in previous work [@2005MNRAS.361..776S] – $T_{SN} = 4 \times 10^8$ K, $A_0 = 4000$, $t^0_{*} = 8.4$ Gyr and $q_{EOS} = 0.5$ – we find that the model overpredicts the sound speed relative to the observed “turbulent” velocities of galaxies, i.e., the non-thermal line widths (see Fig. 1 of @hq10 for a compilation of relevant data). For instance, the above model parameters imply $c_s \sim 30$ km s$^{-1}$ at $n \sim 1$ cm$^{-3}$ and $c_s \sim 110$ km s$^{-1}$ at $n \sim 10^3$ cm$^{-3}$. These values are too large by a factor of $\sim 2-3$ compared to the random velocities inferred from atomic and molecular line observations [@1998ApJ...507..615D]. To account for this, we set ${q_{\rm eos}}= 0.5$ and then modified GADGET by reducing the pressure everywhere by a factor of $10$. This reduces the effective sound speed by a factor of $\sim 3$ and is thus more consistent with observations. This reduction in ISM pressure is also used in the initial conditions when setting up vertical hydrostatic equilibrium for the gas. Changing the pressure requires changing the equation of state parameters to $T_{SN} = 6.6 \times 10^8$ K, $A_0 = 6600$, and $t_{*}^0 = 13.86$ Gyr to maintain an average star formation rate of $1 \, M_{\sun}$ yr$^{-1}$ for an isolated galaxy with our fiducial Milky Way like mass. In §\[sec:ISM\] we compare our fiducial calculations with this reduction in pressure to models with smaller values of ${q_{\rm eos}}$, $0.07$ and $0.2$; these also have smaller “sound speeds” more comparable to the observed random velocities of galaxies. The reduction in the sound speed decreases the Jeans length and mass, making it numerically more prohibitive to resolve these critical scales. For the simulations presented here, we are careful to use sufficient numbers of particles so that the Jeans length and mass are always adequately resolved. The higher gas fraction simulations require a higher particle number as a result (see Table \[TableRunParam\]). The reduction in sound speed also makes it more likely that the gas will fragment by gravitational instability into clumps (ala molecular clouds), as we shall discuss in detail later. This fragmentation is real, not numerical; artificially increasing the sound speed to eliminate it is not necessarily physical and could give incorrect results. On the other hand, we do not include sufficient physics in our ISM model to describe the formation and disruption of molecular clouds so our treatment of the resulting clumping is also not correct. In §\[sec:ISM\] we discuss which of our results are the most sensitive to uncertainties related to local gravitational instability in the ISM. Black Hole Accretion and Feedback {#sectionBlackHoles} --------------------------------- ### Black Hole Accretion Model We include a BH as an additional collisionless particle at the center of each galaxy. We model the accretion of the surrounding gas onto the BH, via the transport of angular momentum, using $$\dot{M}_{visc} = 3 \pi \alpha \Sigma \frac{c_s^2}{\Omega} \label{mdotvisceqn}$$ where $\Sigma$ is the mean gas surface density, $\Omega$ is the rotational angular frequency, and $\alpha$ is the dimensionless viscosity (a free parameter of our model). We compute $\Sigma$ and $c_s$ by taking an average of the properties of the SPH particles in a sphere of radius $R_{acc}$ centred on the BH. The radius $R_{acc}$ is typically set equal to four times the gravitational force softening length, i.e., $R_{acc} = 4 \epsilon$, although we explore alternate choices as well. We find that estimating the rotation rate using $\Omega^2 \simeq GM(<R_{acc})/R_{acc}^3$ is more numerically robust than actually calculating the rotation and angular momentum of the gas particles within $R_{acc}$. Although equation (\[mdotvisceqn\]) is reminiscent of the alpha prescription of , in our formulation $\alpha$ characterizes not only the efficiency of angular momentum transport, but also the uncertainty due to the fraction of the inflowing gas that is turning into stars vs. being accreted onto the central BH. The physical mechanisms driving gas from $\sim$ kpc to $\sim 0.1$ pc are not fully understood, but non-axisymmetric gravitational torques are likely responsible [@1989Natur.338...45S; @hq10]. Using numerical simulations that focus on the nuclei of galaxies (from $\sim 0.1-100$ pc) @hq10 simulate the conditions under which there is significant gas inflow to $\lesssim 0.1$ pc. They argue that the net accretion rate is not a strong function of the gas sound speed (unlike [*both*]{} eqns \[EquationBondi\] and \[mdotvisceqn\]) because non-axisymmetric gravitational perturbations produce orbit crossing and strong shocks in the gas. The resulting inflow rate depends primarily on the non-axisymmetry in the potential, rather than the thermodynamics of the gas. Nonetheless, equation (\[mdotvisceqn\]) evaluated at $\sim 100$ pc and with $\alpha \sim 0.1$ approximates the accretion rate at small radii in their simulations, albeit with substantial scatter (factor of $\sim 10$). Given that one of our key results discussed in §\[sectionResults\] is that the accretion rate is not sensitive to the exact value of $\alpha$, we believe that equation (\[mdotvisceqn\]) is sufficient for the exploratory calculations in this paper. ### Mass of the Black Hole Particle In our galaxy merger simulations, the two BHs are initially far apart but approach each other in the late stages of the merger. When the BH particles have a separation of less than $R_{acc}$ we consider them to have merged. When this occurs, we sum the individual masses of the two BH particles and set one of the particles to have this mass. This particle is then moved to the center of mass of the two BH system and given the velocity of the center of mass frame. The other BH particle is removed from the region. The BH particles are subject to stochastic motion due to interaction with the stellar and gaseous particles, which leads to inaccuracy in the position of the BH and noise in the estimate of the accretion rate. To reduce this numerical “Brownian” motion, the BH particles are given a large “tracer” mass of $2 \times 10^8 M_{\sun}$ for the fiducial simulation, and scaled with the overall mass for other simulations. As a result, the BH particle is a factor $\sim 100$ more massive than the halo particles, and a factor $\sim 10^4$ more massive than the stellar and gaseous particles. We artificially increase the BH particle mass solely to reduce numerical relaxation effects. This does not result in spurious dynamical effects on the central stars, gas, and dark matter since the BH’s sphere of gravitational influence extends to $\lesssim 10$ pc for the fiducial simulation, which is significantly smaller than our typical force softening of $\sim 50$ pc. For the results presented below, the “real” mass of the BH ($\equiv M_{BH}$) is computed by integrating the accretion rate of equation (\[mdotvisceqn\]) in time. The gas particles are not removed as the BH mass increases. Instead, the gas particles have an additional label that tracks whether or not they have been “consumed.” We track how much mass the BH should have consumed via accretion at a given time, and the mass of gas that has been consumed. When there is a mis-match, we tag a number of gas particles within $R_{acc}$ (chosen at random) as “consumed” until the total mass accreted by the BH is correct. Particles that have been consumed no longer contribute to the accretion rate estimate, even if they are inside $R_{acc}$. This implementation prevents any gas particle from providing more than its mass to the integrated mass of the BH. ### Feedback from the Black Hole In our simulations, the AGN is assumed to couple to the surrounding gas by depositing momentum into the gas, directed radially away from the BH. This crudely approximates the effects of (1) strong outflows and/or cosmic-ray pressure produced by the AGN [@king2003; @socrates10] and (2) radiation pressure produced by the absorption and scattering of the AGN’s radiation by dust in the ISM [@murray2005]. We focus on the latter when motivating the parameters used in our models. To accurately account for the impact of the AGN’s radiation on gas in its host galaxy would require a radiative transport calculation, which is beyond the scope of the current work. Instead, we model this radiation pressure by depositing a total momentum per unit time of $$\dot{p} = \tau \frac{L}{c} \quad \mbox{ where } L = {min}\left(\eta \dot{M}_{visc} c^2, L_{Edd}\right) \label{momdepeqn}$$ radially away from the BH into the SPH particles within a distance of $R_{acc}$ of the BH particle. This momentum is equally distributed among the particles so that each particle experiences the same acceleration. We use a radiative efficiency of $\eta = 0.1$ in all simulations. The physical picture behind our feedback model in equation (\[momdepeqn\]) is that the feedback is produced by the absorption of the ultraviolet light from the AGN by dust in the surrounding gas, and the subsequent reemission of infrared radiation that must diffuse its way out of the nuclear region. As described shortly, the parameter $\tau$ is the total infrared optical depth of the nuclear region. To motivate equation (\[momdepeqn\]) in more detail, we note that AGN radiate most of their radiation in the ultraviolet. The opacity of dusty gas to UV radiation is $\kappa_{UV} \sim 10^3$ cm$^2$ g$^{-1}$, so that only a surface density of $\sim 10^{-3}$ g cm$^{-2}$ is required to absorb the UV radiation. This is far less than the typical radial column density of gas in the central $\sim 0.1-1$ kpc of luminous star forming galaxies, galaxy mergers, or our simulations (see Fig.  \[FigureSigmaFiducial\] below). As a result, the UV radiation is efficiently absorbed, except perhaps along polar lines of sight. The absorption and scattering of the UV radiation deposits a momentum per unit time of $L/c$ into the ambient gas, assuming for simplicity that all of the UV radiation is absorbed. If the infrared optical depth is $\gtrsim 1$, the infrared radiation re-emitted by the dusty gas must diffuse out through the nuclear region; doing so deposits an additional momentum per unit time of $\tau L/c$, where $\tau \sim \kappa_{IR} \Sigma$ is the infrared optical depth and $\kappa_{IR} \sim$ few-10 cm$^2$ g$^{-1}$ is the infrared opacity for the radiation temperatures of interest $\sim 100-1000$ K. The net force due to the UV and infrared radiation is thus $ \dot p \sim (1 + \tau ) L/c \simeq \tau L/c$, i.e. equation (\[momdepeqn\]), for $\tau \gtrsim 1$, which is valid in our calculations near the peak of activity when the BH gains most of its mass. In our calculations we use a constant value of $\tau$ rather than a time variable $\tau$ given by $\tau = \kappa_{IR} \Sigma$. Given the simplicity of our feedback model relative to a true radiative transfer calculation, this is not an unreasonable approximation. It is also easier to isolate the effects of varying $\tau$ when it is constant in time. As noted above, we apply the force in equation (\[momdepeqn\]) to all particles within a distance $R_{acc}$ of the BH. A more accurate treatment would be to apply the force out to the point where the column is $\sim \kappa_{IR}^{-1}$, i.e., to where the optical depth to infinity is $\sim 1$. At many times, however, this radius is unresolved. Moreover, it is possible that the photons diffuse primarily along the rotation axis of the gas, rather than in the orbital plane. As a result, the radiation pressure force will be applied primarily at small radii. This is why we apply the force only within $R_{acc}$. One consequence of this is that the number of SPH particles experiencing the feedback, $N$, will change as gas moves in and out of $R_{acc}$. Thus, the strength of feedback felt by an individual particle will change with time. However, because the SPH particles are collisional, they readily share this momentum with neighboring gas particles. In test problems described in Appendix B the effects of our feedback model are essentially independent of $N$ and $R_{acc}$. The results are not quite so clean in our full simulations (see §\[sec:BHmodel\] and Appendix \[resolution\]), but nonetheless none of our major results depend sensitively on the region over which the feedback force is applied. One might worry that if the number of particles within $R_{acc}$ were too small, the momentum supplied to a single particle would become large enough to artificially accelerate the particle to the escape velocity. The minimum $N$ required to avoid this is actually quite modest for the range of luminosities in our calculations, and for the simulations presented here this concern is never an issue (although it is for some of the test problems in Appendix B). Parameter Choices for the Black Hole Model {#sectionParam} ------------------------------------------ Our model for BH growth and feedback contains three free parameters: (1) $\alpha$ determines the magnitude of the accretion rate onto the BH; (2) $\tau$ determines the total radiation pressure force produced by accretion onto the BH; it is roughly the optical depth to the far IR in the nuclear region; and (3) $R_{acc}$ is the radius of the spherical region within which the accretion rate is determined and the feedback is applied. Our fiducial values for these parameters are $\alpha = 0.05$, $\tau = 10$, and $R_{acc} = 4 \, \epsilon$ (where $\epsilon$ is the gravitational force softening). We now motivate these particular choices. The fiducial value of the viscosity used in this work is $\alpha = 0.05$, motivated by the rough consistency between the resulting $\dot M$ and @hq10’s numerical simulations of gas inflow from $\sim 100$ pc to $\sim 0.1$ pc (although there is factor of $\sim 10$ scatter in the latter that is not captured here). @hq10’s calculations in fact require a more complicated subgrid accretion model that depends on additional parameters such as the bulge to disk ratio of the galaxy (because this influences the strength of non-axisymmetric torques); this will be explored in more detail in future work. In addition to $\alpha = 0.05$, we also carried out simulations with $\alpha = 0.15$ and $\alpha = 0.3$, and found no significant differences, for reasons explained below. We use a constant value (with time) of $\tau = 10$ in most of our simulations. This is motivated by far infrared opacities of $\kappa_{IR} \sim 3-10$ cm$^{2}$ g$^{-1}$ and surface densities of $\Sigma \sim 1-10$ g cm$^{-2}$ within $R_{acc}$ during the peak of activity in our simulations. These surface densities are also consistent with those directly measured in the nuclei of ultra-luminous infrared galaxies [@1998ApJ...507..615D]. Given the uncertainties associated with the radiative transfer of far infrared photons in galactic nuclei, it is not possible to more accurately estimate the effective value of $\tau$ without detailed radiative transfer calculations. As we shall demonstrate explicitly, however, the exact value of $\tau$ is also not that critical for the qualitative effects of AGN feedback; the value of $\tau$ does, however, strongly affect the final value of the BH mass. In choosing a value for $R_{acc}$, we must satisfy $R_{acc} > \epsilon$ in order to avoid numerical artifacts. In addition, we find that the BH particle remains within $4 \epsilon$ of the centre of mass of the system at nearly all times, but it can wander around within this region. As a result, $4 \epsilon$ is the smallest we can make $R_{acc}$ without having noise induced by the BHs motion. This choice corresponds to several hundred pc in our typical simulation. Larger values of $R_{acc}$ are unphysical because (1) the accretion rate should only depend on the gas close to the BH; i.e., the transport of gas from, for example, $\sim 8 \epsilon$ to $\sim 2 \epsilon$ is presumably adequately described by our simulations so we should not try to also account for this in our subgrid model, and (2) the radiation pressure force produced by the AGN (and the re-radiated infrared photons) is likely concentrated at relatively small radii, for the reasons described in §\[sectionBlackHoles\]. Galaxy Merger Simulations {#sectionResults} ========================= Table \[TableRunParam\] summarizes the simulations we focus on in this paper, including the resolution, the parameters that specify the initial conditions for the merging galaxies, the parameters that specify the BH accretion and feedback models, and the final properties of the merger remnants (stellar and BH mass and velocity dispersion). We begin by describing the results from our fiducial simulation (top row in Table \[TableRunParam\]) and then discuss simulations that vary a single parameter of the feedback model relative to the fiducial run. We have also performed simulations at different overall galactic mass scales, initial gas fractions, and numerical resolution. The latter resolution tests are presented in Appendix A. The Fiducial Simulation {#sec:fid} ----------------------- ![ *Top:* The separation of the black hole particles as a function of time in the fiducial simulation. The blue circles label the times of the images shown in Figure \[figureSFRClump\]. *Middle:* The star formation rate as a function of time for the fiducial simulation (black) and for the run with no feedback (red; run fidNof). *Bottom:* The viscous accretion rate, $\dot{M}_{visc}$ (black), and Eddington rate (grey), as functions of time for the fiducial simulation. The critical $\dot{M}_c$ at which radiation pressure balances gravity (eq. \[EquationMCrit\]) is shown within a radius of $R_{acc}$ (red; solid). The increase in star formation and BH accretion after first passage ($t \sim 0.75$ Gyr) is due to the fragmentation and inspiral of large gaseous/stellar clumps (Fig. \[figureSFRClump\]), while the much larger increase at final coalescence is due to inflow of diffuse gas caused by non-axisymmetric torques. The latter physics dominates the total stellar and BH mass formed during the merger.[]{data-label="FigureFiducialMegaPlot"}](Figure1.eps){width="84mm"} The top panel of Fig.  \[FigureFiducialMegaPlot\] shows the separation of the BH particles for the fiducial simulation, while the middle panel shows the total star formation rate (in both galaxies) for simulations with (black) and without (red) BH feedback. The first close passage of the two galaxies is around $t = 0.33$ Gyr and the system then undergoes a few short oscillations as the BHs finally settle into a merged state around $t = 1.65$ Gyr. The star formation rate increases following the first passage, with a much larger increase in the star formation rate during the final merger of the galaxies. The bottom panel of Fig.  \[FigureFiducialMegaPlot\] shows the BH accretion rate determined from equation \[mdotvisceqn\] (black) and the Eddington accretion rate (grey; $\dot M_{edd} \equiv L_{edd}/0.1c^2$); the initial BH mass is $1.4 \times 10^5 M_\odot$ but as long as it is not too large $\gtrsim 10^8 M_\odot$, the precise initial BH mass is unimportant for our conclusions. In this and similar plots throughout the paper, the value of $\dot{M}$ plotted before the BHs merge is for the BH in the galaxy with the smaller initial inclination relative to the orbital plane; the BH accretion rate for the other galaxy is comparable to that shown here. The evolution of the accretion rate is similar in many of the simulations we have carried out, with an initial period of activity after the first passage of the merging galaxies, and another period of even higher $\dot M$ after the final coalescence of the galaxies and BHs. The latter active episode is when the merged BH gains most of its mass. In particular, the BH reaches the Eddington limit, allowing the mass of the BH to grow exponentially for a few hundred Myr. [@2009arXiv0909.2872D] showed that the BH accretion and feedback model presented in this work leads to self-regulated BH growth, due to a competition between the (inward) gravitational force produced by the galaxy as a whole and the (outward) radiation pressure force produced by the central AGN (eq. \[momdepeqn\]) [@murray2005]. For a spherically symmetric system, equating these two forces leads to $\tau L / c = 4 f_g \sigma^4 / G$, where $\sigma^2 = G M_t / 2 R_{acc}$, $M_t$ is the total mass inside $R_{acc}$, and we have evaluated these expressions within $R_{acc}$, where our accretion rate is determined and feedback is implemented. Equivalently, there is a critical accretion rate $\dot{M}_{c}$, analogous to the Eddington rate, at which the two forces balance: $$\dot{M}_c = \frac{4 f_g}{\eta \tau G c} \sigma^4. \label{EquationMCrit}$$ The bottom panel of Fig. \[FigureFiducialMegaPlot\] shows $\dot{M}_c$ for our fiducial simulation, evaluated within $R_{acc}$ of the BH (solid red). Comparing $\dot M_c$ to the BH accretion rate $\dot M_{visc}$ demonstrates that during the peak episodes of accretion $\dot{M}_{visc} \sim \dot{M}_c$, so that radiation pressure becomes dynamically important. Although it is certainly possible to have accretion rates smaller than $\dot{M}_c$ when there is insufficient gas to fuel the AGN, the accretion rate is limited to a maximum value of $\sim \dot{M}_c$. Fig. \[FigureSigmaFiducial\] shows the surface density of gas within $R_{acc} = 4 \, \epsilon = 0.19$ kpc for the fiducial simulation and for a higher gas fraction simulation with $f_g = 0.3$. As implied by Fig. \[FigureFiducialMegaPlot\], there are two main epochs during which significant gas is driven into the nuclei of the galaxies: after first passage and at final coalescence. The physical origin of these high nuclear gas densities are, however, somewhat different. ![The mean gas surface density $\Sigma$ interior to the accretion radius $R_{acc} = 4\epsilon = 0.19$ kpc for the fiducial simulation with initial gas fraction $f_g = 0.1$ (solid) and for the simulation with $f_g = 0.3$ (dashed; run fg).[]{data-label="FigureSigmaFiducial"}](Figure2.eps){width="84mm"} {width="170mm"} @1996ApJ...464..641M showed that the presence of a bulge like that in our simulation suppresses a nuclear starburst after first passage during galaxy mergers, because the bulge inhibits the non-axisymmetric modes that drive inflow. In our fiducial simulation, the majority of the increase in star formation after first passage is due to gravitational instability and fragmentation of the gas, which produces dense regions of rapid star formation. Fig. \[figureSFRClump\] (left panel) shows the gas density in the vicinity of one of the incoming black holes at $t = 0.74$ Gyr, midway through the first peak in star formation; the companion galaxy is well outside of this image. Two knots of dense gas are clearly seen, both of which will soon enter $R_{acc}$, the BH accretion and feedback region. These two clumps are not the only ones that form after first passage, but they are the only clumps that survive to enter the central region surrounding the BH.[^2] Fig. \[figureSFRClump\] (right panel) also shows an image of the gas density in the nuclear region at $t = 1.71$ Gyr, near the peak of star formation and BH accretion and after the galaxies and BHs have coalesced. At this time, the gas density in the nuclear region is significantly higher than at first passage (see also Fig. \[FigureSigmaFiducial\]) and most of the gas resides in a $\sim 1$ kpc diameter disk. This nuclear gas concentration is the diffuse ISM driven in from larger radii by non-axisymmetric stellar torques during the merger (e.g., @1996ApJ...464..641M). The galaxies in our fiducial simulation are stable when evolved in isolation. The merger itself drives the gas to fragment by locally exceeding the Jeans/Toomre mass. In reality, the gas in such clumps might disperse after $\sim$ a Myr because of stellar feedback not included in our calculations [@murray10]. This would probably not significantly change our estimate of the star formation rate since we are already normalized to the observed Kennicutt relation; however, such dispersal would lead to little inflow of gas associated with the inspiral of stellar clusters and thus would suppress the first peak in BH accretion (see @hq10 for a more detailed discussion). In §\[sec:ISM\] we will return to these issues and show that the total stellar mass and BH mass formed during the merger are relatively insensitive to the details of our assumed ISM model. Fig. \[FigureSigmaofRFiducial\] shows the surface density of gas in the fiducial simulation (top panel) and for the run without feedback (bottom panel) as a function of distance from the BH at four times: the initial condition ($t = 0$), shortly after the first close passage of the two galaxies ($t = 0.85$ Gyr), near the peak of accretion ($t = 1.71$ Gyr) and at the end of the simulation ($t = 2.85$ Gyr). Once $\dot M \sim \dot M_c$ at first passage $\sim 0.85$ Gyr, gas is driven out of the nuclear region by the AGN’s radiation pressure. Since at the same time gravitational torques continue to drive gas inwards, the gas begins to pile up at $\sim R_{acc}$. The particular radius at which the pile up occurs of course depends on our choice of $R_{acc}$, and so the particular size of the evacuated region should not be taken too seriously. Qualitatively, however, the behavior in Fig. \[FigureSigmaofRFiducial\] is reasonable: the AGN pushes on the gas in its neighborhood until it deprives itself of fuel. Near the peak of activity at $t = 1.71$ Gyr, the gas surface density in the central $R_{acc} \simeq 0.19$ kpc is a factor of $\sim 10-30$ larger in the simulations without feedback (bottom panel of Fig. \[FigureSigmaofRFiducial\]). However, the gas density at large radii $\sim 0.5$ kpc is not that different. The radiation pressure force from the BH thus largely affects gas in its immediate environment, rather than the entire gas reservoir of the galaxy. Another indication of this is that the star formation rate is very similar in the simulations with and without feedback (middle panel of Fig. \[FigureFiducialMegaPlot\]). ![Comparison of gas surface density ($\equiv M_g[<r]/\pi r^2$) versus distance from the BH in the fiducial simulation with feedback (top) and without feedback (bottom). Four times are shown: $t = 0$, 0.85 Gyr (first passage), 1.71 Gyr (peak accretion), and 2.85 Gyr (end of simulation). Note that the gas tends to pile up at $R_{acc} = 0.190$ kpc (shown by the vertical line) in the top panel.[]{data-label="FigureSigmaofRFiducial"}](Figure4.eps){width="84mm"} Dependence on Parameters of the BH Model {#sec:BHmodel} ---------------------------------------- {width="180mm"} The models for BH accretion and feedback used here contain uncertain parameters. We have defined the three relevant parameters $\alpha$, $\tau$, and $R_{acc}$ in §\[sectionParam\] and motivated our fiducial values, but it is important to explore how our results change with variations about our fiducial parameters. The value of $\alpha$ parameterizes the efficiency with which gas accretes from $\sim R_{acc} \sim 190$ pc to smaller radii, encapsulating both the efficiency of angular momentum transport and the effects of star formation on unresolved scales. Naively, a higher value of $\alpha$ would lead to a more massive BH. This is, however, not the case, because during the epochs when the BH gains most of its mass, the accretion rate is set by the efficiency of feedback (eq. \[EquationMCrit\]) not by the available mass supply (see Figs \[FigureFiducialMegaPlot\] & \[FigureSigmaofRFiducial\]). To demonstrate this more explicitly, the top left panel of Fig. \[FigureFourPanel\] compares the BH accretion rates for three simulations with feedback, but differing values of $\alpha$ (0.05, 0.15, and 0.3), to the simulation with no feedback, which has $\alpha = 0.15$. The accretion histories for the three values of $\alpha$ are nearly identical. By contrast, the accretion rate is in general much larger in simulations that neglect feedback (and is $\propto \alpha$). In addition to the constant $\alpha$ runs, we tested a model in which $\alpha$ was time variable, set by the local gas fraction near the BH (fidafg2 in Table \[TableRunParam\]): $\alpha = 3 f_g^2$, with $f_g$ determined within $R_{acc}$ (in practice $\alpha$ varied from $\sim 2 \times 10^{-4}-0.3$). Although this precise functional form is somewhat arbitrary, such a variation is motivated by analytic arguments and numerical simulations which show that instabilities due to self-gravity dominate the transport of gas from $\sim 100$ pc inward [@shlosman1990; @hq10]. For our $\alpha = 3 f_g^2$ simulation, we find that the peak accretion rates and final BH mass are very similar to the constant $\alpha$ simulations. This is consistent with our conclusion that in the limit of large fuel supply, feedback, rather than the efficiency of angular momentum transport, sets the rate at which the BH grows. The parameter $\tau$ describes the efficacy of the feedback for a given AGN luminosity. The bottom left panel of Fig. \[FigureFourPanel\] compares the BH accretion rate for the fiducial run with $\tau = 10$ (black) and a simulation with a smaller value of $\tau = 3$ (orange). To the extent that the accretion rate is feedback limited and set by $\dot M_c$ in equation \[EquationMCrit\], $\dot M$ should decrease with increasing $\tau$. Physically, this is because larger $\tau$ leads to a larger feedback force, which then requires a smaller accretion rate to provide the luminosity necessary to drive away the surrounding gas. This expectation is borne out by the simulations. To compare the numerical results with the scaling in equation \[EquationMCrit\], the bottom left panel of Fig. \[FigureFourPanel\] also shows $\dot M$ for the fiducial simulation scaled by a factor of $10/3$ (dashed line). This scaled $\dot M$ of the fiducial simulation is in reasonably good agreement with the $\tau = 3$ simulation, particularly at the first and second peaks in $\dot M$, when most of the BHs mass is accumulated. This demonstrates that the value of $\tau$ does not significantly affect any of the qualitative behavior of how the BH grows, although it does determine the overall value of the BH mass. In the majority of the simulations presented here, the size of the region over which we apply the feedback and average the gas properties to calculate $\dot M$, $R_{acc}$, is set to $4 \, \epsilon$. The rationale for this choice was given in §\[sectionParam\], but it is important to consider the effects of changing this value. The top right panel of Fig. \[FigureFourPanel\] shows the mass accretion rate for the fiducial simulation and a simulation with $R_{acc} = 8 \epsilon = 380$ pc. The peak values of $\dot M$ and the time of the first and second peaks are reasonably similar in the two cases. The principle difference is that in the simulations with the larger value of $R_{acc}$, the feedback is less effective at clearing gas out of the nuclear region (because the force is distributed over a larger number of particles); this allows a higher level of $\dot M$ to be maintained after the first passage and final coalescence. We suspect that the fiducial simulation better approximates what a higher resolution calculation with radiative transfer would find, but this remains to be demonstrated. ![The star formation rate for the run with no feedback (red) and for runs with various values of the BH accretion and feedback parameters: $\alpha = 0.05, 0.15, 0.3$ (black, green, blue), $\alpha = 3 f_g^2$ (grey), $\tau = 3$ (orange), and $R_{acc} = 8 \epsilon$ (magenta). All of these models have very similar star formation histories.[]{data-label="FigureSFRAlphas"}](Figure6.eps){width="84mm"} The bottom right panel of Fig. \[FigureFourPanel\] shows the integrated BH mass as a function of time for the fiducial simulation and for the variations in the feedback/accretion model considered in this subsection that have the same value of $\tau$ (but different values of $\alpha$ and/or $R_{acc}$). The key result is that in the presence of feedback (all but the top curve), there is only a factor of $\simeq 3$ change in the BH mass due to differences in how we treat BH accretion and feedback. A factor of $6$ change in $\alpha$ leads to only a $42 \%$ change in the final BH mass. This is because most of the BH mass is gained during the final coalescence of the two galaxies, at which point the BH accretion self-regulates and reaches the Eddington-like value in equation (\[EquationMCrit\]). The run without feedback (top curve), by contrast, has a factor of $\sim 10$ larger BH mass and the BH mass would scale linearly with the assumed value of $\alpha$. The star formation rates for the simulations with different BH feedback parameters are all shown in Fig. \[FigureSFRAlphas\] (this includes the fiducial simulation with and without feedback and the runs with $\alpha = 0.15, 0.3, 3 f_g^2$, $\tau = 3$, and $R_{acc} = 8\epsilon$). This figure demonstrates that the precise parameters of the BH feedback model have little effect on the galaxy-wide properties such as the star formation rate: the total mass of stars formed in simulations with different BH feedback parameters differ by less than $5\%$. In previous simulations of BH growth and feedback, AGN feedback acting on dense gas in galaxies has been invoked to quench star formation [@springel2005b]. Our results demonstrate, however, that this is by no means guaranteed (we refer here to ‘quasar’ feedback on cold dense gas, not the effect of AGN on hot dilute gas in galaxy groups and clusters). In our calculations BH growth is self-regulated and closely connected to the properties of the surrounding galaxy (e.g., eq. \[EquationMCrit\]). However, the BHs dynamical influence is centered in the galactic nucleus ($\lesssim 300$ pc); as a result, the BH does not significantly alter the star formation history during a merger. In this scenario, the merger remnant can nonetheless be relatively quiescent (“red and dead”) because the burst of star formation uses up much of the available gas. Effects of the ISM Model {#sec:ISM} ------------------------ Motivated by observations (e.g., @1998ApJ...507..615D), we have reduced the effective sound speed in GADGET’s subgrid ISM model (see §\[sectionSFR\]). There is nonetheless considerable uncertainty in the accuracy of this (or any other) subgrid model. To study in more detail the effects of the ISM model on our results, we performed two additional simulations at our fiducial galaxy mass with the subgrid interpolation parameter ${q_{\rm eos}}= 0.2$ and $0.07$ (see eq. \[eqn:qeos\]), and without the factor of 10 reduction in pressure used in our fiducial simulation (an additional simulation with ${q_{\rm eos}}= 0.07$ at a lower galaxy mass will be discussed in §\[sectionGalaxyBH\]).[^3] The three different ISM models have $c_s$ and $Q$ within a factor of $\sim 2$ of one another at all radii, with the ${q_{\rm eos}}= 0.2$ model having the largest values of $c_s$ and Q, and our fiducial model having the smallest values. The parameter $Q$ is initially $\sim 3$ for our fiducial simulation at the disk scale length $R_d$, which is why the merger can induce significant fragmentation of the gas (Fig. \[figureSFRClump\]). Given the limited physics included in the subgrid model, we do not believe that it is feasible to unambiguously conclude which of these ISM models is more realistic. These models thus provide an indication of the systematic uncertainty introduced by our treatment of the ISM. ![Comparison of three simulations that differ only in the ISM models: fiducial (black), $q_{EOS} =0.2$ (red), and $q_{EOS} = 0.07$ (blue). The panels show the viscous accretion rate (top), star formation rate (middle), and the integrated black hole mass and mass of new stars formed (bottom). The three different ISM models have $c_s$ and Toomre $Q$ within a factor of $\sim 2$ of one another at all radii; the ${q_{\rm eos}}= 0.2$ model has the largest values of $c_s$ and Q and our fiducial model has the smallest values. []{data-label="figureISMSeries"}](Figure7.eps){width="84mm"} Fig. \[figureISMSeries\] compares the BH accretion history (top panel), the star formation rate (middle), and the integrated BH mass and mass of new stars formed during the merger (bottom) for the three runs with differing ISM models. For both the fiducial run and the run with $q_{EOS} = 0.07$ there is significant fragmentation after first passage, which generates the first peak in star formation and BH accretion. By contrast, the run with $q_{EOS} = 0.2$ shows no evidence for gas fragmentation or a pronounced peak in activity at first passage. Despite these differing initial histories, the final result of the merger is very similar in all three cases: the large star formation rates and BH accretion rates coincident with the final coalescence of the two galaxies are not due to fragmentation, but are instead largely due to the inflow of diffuse gas to smaller radii. Moreover, the final BH mass and the total amount of new stars formed during the merger are similar in all three cases. Thus, despite uncertainties in the model of the ISM, we find relatively robust integrated quantities (as did the earlier calculations of @1995ApJ...448...41H). The precise time dependence of the star formation and BH accretion (i.e., the lightcurves) are, however, significantly more uncertain and sensitive to the details of the model. Galaxy Parameters {#sec:gal} ----------------- Having shown that the final BH mass and new stellar mass do not depend strongly on the uncertain parameters in our accretion, feedback and ISM models, we now examine how our results vary with galaxy properties such as the total mass, gas fraction, and bulge-to-disk ratio. ![Comparison of four simulations that differ only in the total galaxy mass: $M_{fid}=3.88\times 10^{12}M_\odot$ (fiducial; black), $3 M_{fid}$ (red), $0.3 M_{fid}$ (blue), and $0.1 M_{fid}$ (magenta). The three panels show the viscous accretion rate (top), star formation rate (middle), and the integrated black hole mass (bottom). The same parameters are used in the BH accretion and feedback models. []{data-label="FigureMassScale"}](Figure8.eps){width="84mm"} Fig. \[FigureMassScale\] shows the BH accretion histories (top panel), star formation rate (middle), and integrated BH mass (bottom) for four runs with different total galaxy mass. The models cover a factor of 30 in galaxy mass, from 0.1-3 times our fiducial mass. The BH and star formation parameters are identical in the four simulations, while the gravitational force softening and structural parameters (e.g., disk scale length, bulge radius) change with the total mass (see §\[sectionICs\]). Fig. \[FigureMassScale\] shows that the BH accretion rates and integrated BH masses increase with galaxy mass as expected from equation \[EquationMCrit\]. However, there is a clear difference between the lower and higher mass simulations: the two higher mass simulations show evidence for the first peak in star formation and BH growth that we have shown is due to fragmentation near first passage, while the lower mass runs do not. This is largely a consequence of the fact that observed disks have $R_D \propto M^{1/3}$ [@2003MNRAS.343..978S], so that more massive galaxies have higher surface densities and are thus more susceptible to gravitational instability (our ISM model counteracts this slightly, but not enough to stabilize the higher mass disks). It is important to reiterate, however, that modest changes to the subgrid sound speed can change whether or not the gas fragments near first passage (§\[sec:ISM\]) so it is not clear if the difference as a function of mass in Fig. \[FigureMassScale\] is robust. In addition to the systematic change in the importance of fragmentation near first passage, Fig. \[FigureMassScale\] also shows differences in the late-time BH accretion between the low and high mass simulations. In particular the two smaller mass runs each show a period of increased accretion after the main peak during the merger. In these cases the new stars formed around final coalescence develop a bar in the inner $\sim R_{acc}$ of the galaxy. This helps drive some of the remaining gas into the accretion region leading to the increased accretion at late times. There is a milder version of this late-time accretion in the fiducial mass ${q_{\rm eos}}= 0.2$ model without fragmentation in Fig. \[figureISMSeries\]. Interestingly, there is no analogous late-time inflow of gas to within $R_{acc}$ in our low mass galaxy simulations without BH feedback. The late-time activity is also particularly sensitive to the accretion model at a time when the non-axisymmetry produced by the merger has died away (so that $\alpha$ may in reality decrease significantly). For these reasons, we regard the late time growth in Fig. \[FigureMassScale\] as an interesting deviation from self-similarity in the dynamics, but not necessarily a particularly robust one. One important point that this highlights, however, is that because our implementation of BH growth and feedback does not unbind a significant amount of cold gas at late times (unlike calculations by @springel2005b), the predictions of our model are more sensitive to the post galaxy coalescence physics. In addition to the fiducial gas fraction ($f_g = 0.1$) simulations that we have largely focused on, we performed simulations with an initial gas fraction of $f_g = 0.3$ for our fiducial galaxy mass and at one tenth this mass. The qualitative difference in behavior with galaxy mass in Fig. \[FigureMassScale\] persists in the higher gas fraction runs. In particular, in the low mass $f_g = 0.3$ simulation, the gas does not fragment, while it does in the higher mass $f_g = 0.3$ simulation. Fig.  \[FigureSigmaFiducial\] – discussed in §\[sec:fid\] – explicitly shows the increase in the gas surface density within $R_{acc}$ produced by this at early times. A final property of the galaxy model that we varied was the bulge to disk mass ratio. The majority of our runs include a bulge with one third the mass of the disk; we also ran one simulation with an initial bulge of one fifth the disk mass, at the fiducial galaxy mass. The final BH mass and total mass of stars formed differ by less than $3\%$ each compared to the fiducial simulation. The $M_{BH}-\sigma$ Correlation {#sectionGalaxyBH} =============================== Previous numerical studies using models of BH growth and feedback different from those considered here have reproduced a number of the observed correlations between massive BHs and their host galaxies (e.g., @di-matteo2005 [@sazonov2005; @younger08]). @younger08 argue that the galaxy-BH correlations in simulations (in particular, the BH fundamental plane) are relatively independent of the trigger of BH growth, be it minor mergers, major mergers, or global instabilities of galactic disks. Based on the calculations to date, however, it is unclear to what extent the simulated BH-galaxy correlations depend on the details of the BH feedback or accretion models. In this section we assess this question by quantifying the $M_{BH}$ - $\sigma$ relation produced in our models. We define $\sigma$ of our model galaxies using a method analogous to that of observers: we first project the mass density of the stellar particles into cylindrical bins, and compute the half-mass(light) radius $R_e$. We then compute the velocity dispersion weighted by the surface brightness via $$\sigma^2 = \frac{\int_{R_{min}}^{R_e} \sigma_{los}^2(R) I(R) R dR}{\int_{R_{min}}^{R_e} I(R) R dR} \label{EquationSigmaLOS}$$ where $I(R)$ is the [projected 2-d stellar]{} mass profile, $\sigma_{los}$ is the line of sight velocity dispersion, and $R_{min} = 2 \epsilon$ to ensure that there are that no artificial effects introduced by the force softening. We repeat this calculation along $1000$ lines of sight with random viewing angles through the center of mass of the merger remnant. The $\sigma$ quoted in this paper and listed in Table 1 is the median value over the 1000 lines of sight. Fig. \[FigureSigmaPlot\] shows the correlation between the final BH mass $M_{BH,f}$ and the $\sigma$ of the merged galaxy for most of the simulations in Table \[TableRunParam\]: different total galaxy masses (black), different values of the accretion parameter $\alpha$ (red circle), alternate ISM models (red x), higher gas fraction (blue square), alternate bulge mass (red square), different values of $\tau$ (blue circle), and the resolution studies in Appendix A (grey). The solid line indicates the mean relation from the compilation of observational results in [@2009ApJ...698..198G] while the dotted lines are the $1-\sigma$ error bars. We have linearly rescaled all of our final BH masses to a value of $\tau = 25$, using the fact that both the analytic and numerical results are consistent with $\dot M_{visc}$ and $M_{BH,f}$ being $\propto \tau^{-1}$. The value of $\tau = 25$ is chosen so that the rescaled fiducial simulation lies approximately on the $M_{BH}-\sigma$ relation of [@2009ApJ...698..198G]. For our fiducial simulation carried out with $\tau = 3$ and $\tau = 10$, a linear scaling of $M_{BH,f}$ with $\tau^{-1}$ is accurate to about $2 \%$ (e.g., Table \[TableRunParam\] and Fig. \[FigureFourPanel\]). We also carried out our fiducial simulation with $\tau = 25$; this is consistent with a linear scaling of $M_{BH,f}$ from $\tau = 3$ to $\sim 50 \%$ (Table \[TableRunParam\]). For nearly all of our simulations, rescaling to $\tau = 25$ amounts to dividing the final BH mass by a factor of 2.5. ![The $M_{BH,f}$-$\sigma$ relation for the simulations in this paper, along with the observed relation (solid) and one sigma scatter (dotted) from @2009ApJ...698..198G. The final BH mass $M_{BH,f}$ in all of the simulations has been linearly scaled to $\tau = 25$ from the value used in the simulation (typically $\tau = 10$). The simulations are generally quite consistent with observations; we do find indications of a slight flattening in $M_{BH,f}-\sigma$ at low BH masses.[]{data-label="FigureSigmaPlot"}](Figure9.eps){width="86mm"} ![The ratio of the final BH mass to the BH mass at the peak of accretion for the simulations in Fig. \[FigureSigmaPlot\], using the same symbol types. This quantifies the extent to which late-time accretion increases the BH mass. The late-time increase in BH mass for many of the lower mass systems produces the slight flattening in $M_{BH}-\sigma$ in Fig. \[FigureSigmaPlot\] at low masses; see the text for a discussion of the robustness of this result.[]{data-label="FigureSigmaPlotPeak"}](Figure10.eps){width="86mm"} Previous analytic arguments were able to reproduce the $M_{BH}-\sigma$ relation with $\tau \sim 1$, rather than requiring $\tau \sim 25$ as we do here (e.g., @king2003 [@murray2005]). These calculations, however, assumed $f_{g} = 0.1$. While perhaps appropriate on average, this is not appropriate in galactic nuclei where the gas densities are higher. The analytic derivations also assumed that BH growth terminated when the system reached the luminosity (accretion rate) at which radiation pressure balances gravity (eq. \[EquationMCrit\]). In reality, however, the luminosity must exceed this critical value by a factor of several in order for gas to be efficiently pushed around in the galactic nucleus (as shown explicitly in the test problems in the Appendix). Moreover, the BH continues to accrete some mass even after reaching $\dot M_c$. Fig. \[FigureSigmaPlotPeak\] shows this explicitly via the ratio of the final BH mass to the BH mass at the peak of activity for all of the simulations in Fig. \[FigureSigmaPlot\].[^4] The net effect of the differences between our simulations and the simple analytic calculations is that a much larger feedback force per unit BH mass ($\tau \sim 25$, not $\sim 1$) is required for consistency with the observed $M_{BH}-\sigma$ relation. The physical implications of this larger value of $\tau$ for models of AGN feedback will be discussed in § \[sectionDiscussion\]. The scatter in BH mass in Fig. \[FigureSigmaPlot\] at our fiducial mass scale of $\sigma \sim 175 {{\rm \, km \, s^{-1}}}$ is reasonably consistent with the observed scatter. In the simulations we have varied the BH accretion model ($\alpha$), the ISM model, numerical resolution, size of the feedback/accretion region $R_{acc}$, and galaxy properties such as the total mass, gas fraction, and bulge to disk ratio. It is encouraging that all of these simulations produce BH masses within a factor of few of each other. The largest BH mass at $\sigma \sim 175 {{\rm \, km \, s^{-1}}}$ is the simulation with an initial gas fraction of $f_g = 0.3$; since this run has a larger gas density at small radii close to the BH (Fig. \[FigureSigmaFiducial\]), it should probably also have a larger $\tau$, which would reduce the BH mass further, in better agreement with the data. It is difficult to make this comparison to the observed scatter more quantitative given the limitation that our simulations are all equal-mass non-cosmological binary mergers on the same orbit. The numerical results in Fig. \[FigureSigmaPlot\] suggest a slight flattening of the $M_{BH}-\sigma$ relation at $\sigma \lesssim 100 {{\rm \, km \, s^{-1}}}$. This is in large part a consequence of the additional mass gained by the lower mass BHs after their peak of activity (see Figs. \[FigureMassScale\] & \[FigureSigmaPlotPeak\], in particular the fiducial simulations labeled by black squares in Fig. \[FigureSigmaPlotPeak\]). This change in behavior at lower masses is primarily due to the fact that the lower mass galaxies are less prone to fragmentation than the more massive galaxies (§\[sec:gal\]). Without the fragmentation after first passage, more gas is available to feed the BH at late times leading to the slightly higher BH mass. As discussed in §\[sec:gal\], it is unclear how robust this late time accretion is. In fact, a low mass galaxy simulation with an alternate ISM model (${q_{\rm eos}}= 0.07$) does not show significant late-time accretion, leading to a BH mass in good agreement with the extrapolation from higher $\sigma$ (red x at low mass in Figs. \[FigureSigmaPlot\] & \[FigureSigmaPlotPeak\]). We thus regard the case for flattening of $M_{BH}-\sigma$ at low masses in our models as somewhat tentative; our results may instead indicate enhanced scatter at low masses rather than a change in the mean relation. More comprehensive numerical studies of these lower mass systems will be needed to distinguish these two possibilities. Discussion and Conclusions {#sectionDiscussion} ========================== We have presented a new method for simulating the growth of massive BHs in galaxies and the impact of AGN activity on gas in its host galaxy (see also our related Letter; @2009arXiv0909.2872D). In this method, we use a local viscous estimate to determine the accretion rate onto a BH given conditions in the surrounding galaxy (eq. \[mdotvisceqn\]), and we model the effect of BH feedback on ambient gas by depositing momentum radially away from the BH into the surrounding gas (eq. \[momdepeqn\]). Our accretion model qualitatively takes into account the angular momentum redistribution required for accretion of cold gas in galaxies and is thus more appropriate than the spherical accretion estimate that has been used extensively in the literature. In our feedback model, the applied force is given by $\tau L/c$, where the AGN’s luminosity $L$ is determined by our BH accretion model, and the net efficiency of the feedback is determined by the total optical depth $\tau$ of the galactic gas to the AGN’s radiation, which is a free parameter of our model. Previous calculations have demonstrated that only when the gas fraction in a galaxy decreases to $\lesssim 0.01$ can the AGN’s radiation Compton heat matter to high temperatures [@sazonov2005]. More generally, the cooling times in gas-rich galaxies are so short that the primary [dynamical]{} impact of the AGN on surrounding gas is via the momentum imparted by the AGN’s outflows or radiation. It is thus not physically well-motivated to model AGN feedback by depositing energy, but not momentum, into surrounding gas, as many calculations have done (e.g., @di-matteo2005 [@springel2005b; @kawata2005]); see [@ostriker10b] for related points. Throughout this paper, we have focused on BH growth during major mergers of spiral galaxies. As demonstrated in @2009arXiv0909.2872D, our model leads to a self-regulated mode of BH accretion in which the BH accretion rate is relatively independent of the details of the BH accretion model (see Fig. \[FigureFourPanel\]). This is because the accretion rate self-adjusts so that the radiation pressure force is comparable to the inward gravitational force produced by the host galaxy (see eq. \[EquationMCrit\]). This self-regulated mode of BH accretion is a robust feature of all of our simulations during periods of time when there is a significant nuclear gas reservoir – it thus applies precisely when the BH gains most of its mass. One important consequence of this self-regulated accretion is that AGN feedback does not drive significant large-scale outflows of gas (in contrast to the models of @springel2005b). For example, the surface density profiles in Fig. \[FigureSigmaofRFiducial\] show that AGN feedback causes gas to pile up at a few hundred pc rather than being completely unbound from the galaxy – this precise radius should not be taken too literally since it is a direct consequence of the fact that we implement feedback and determine the BH accretion rate only within a radius $R_{acc} \sim$ few hundred pc. Nonetheless, we believe that this general result may well be generic: because the BH accretion rate is determined by the gas content close to the BH, the AGN can shut off its own accretion before depositing sufficient energy to unbind all of the gas in the galaxy. If we artificially hold the luminosity of the AGN constant in time at a value exceeding the critical value in equation (\[EquationMCrit\]), then the AGN [ *does*]{} eventually unbind all of the surrounding gas (see, e.g., Figs. \[FigureShellTestEta\] & \[FigIsothermalEta00\] in Appendix B). However, both our isothermal sphere test problem (Fig. \[FigureShellFull\]) and our full merger calculations show that when the BH accretion rate is self-consistently determined by the gas properties in the central $\sim 100$ pc of the galaxy, the AGN simply never stays ‘on’ long enough to unbind all the gas. Our results do not, of course, preclude that AGN drive galactic winds. For example, some gas may be unbound by a high speed wind/jet produced by the central accretion disk (which is not in our simulations). In addition, at later stages of a merger or at large radii the gas fraction can be sufficiently low ($\lesssim 0.01$) that gas can be Compton heated to high temperatures and potentially unbound (e.g., @ciotti10). This may in fact be sufficient to quench star formation at late times, but only once most of the gas has already been consumed into stars (so that $f_g \lesssim 0.01$). Our results do suggest that AGN feedback does not quench star formation by unbinding a significant fraction of the cold dense gas in a galaxies interstellar medium (in contrast to, e.g., @springel2005b). In future work it will be important to assess whether variability in the accretion rate on smaller scales than we can resolve (e.g., @hq10 [@levine10]) modifies this conclusion; such variability could produce some epochs during which AGN feedback has a significantly larger effect on the surrounding gas. Another improvement would be to carry out radiative transfer calculations and assess what fraction of the AGN’s radiation is absorbed at large radii in a galaxy ($\sim$ kpc) where the gas has a lower surface density and is thus easier to unbind. Our simulations cover a factor of $\sim 30$ in galaxy mass. The final BH mass in our calculations is $\propto \tau^{-1}$ since a larger value of $\tau$ corresponds to a larger momentum deposition per unit BH mass. We find reasonable consistency with the normalization of the observed $M_{BH}-\sigma$ relation for $\tau \sim 25$. To compare this result to previous work by @di-matteo2005, we note that a momentum deposition of $\dot P$ corresponds to an energy deposition rate of $\dot E \simeq \dot P \sigma$ when the feedback is able to move the gas at a speed comparable to the velocity dispersion $\sigma$ (which is required for efficient self-regulation of the BH growth). For $\tau \simeq 25$, our results thus correspond to $\dot E \simeq 25 \, L \, \sigma/c \simeq 0.02 \, L \, (\sigma/200 \, {\rm km \, s^{-1}})$. This is similar to the results of @di-matteo2005, who found that depositing $\sim 5 \%$ of the BH accretion energy in the surrounding gas was required to explain the $M_{BH}-\sigma$ relation. It is encouraging that these two different sets of simulations, with different BH accretion and feedback models, agree at the factor of $\sim 2-3$ level on the energetics required to reproduce the $M_{BH}-\sigma$ relation. The value of $\tau \sim 25$ required to explain the normalization of the $M_{BH}-\sigma$ relation has strong implications for the dominant physics regulating BH growth. The simplest models of super-Eddington winds from an accretion disk close to the BH are ruled out because they typically have $\tau \sim 1$, i.e., a momentum flux comparable to that of the initial radiation field [@king2003]. Similarly, the radiation pressure force produced solely by the scattering and absorption of the AGN’s UV radiation by dust corresponds to $\tau \sim 1$ [@murray2005] and is thus not sufficient to account for the level of feedback required here. Rather, our results suggest that most BH growth happens when the nuclear regions are optically thick to the re-radiated dust emission in the near and far-infrared, so that $\tau \gg 1$. This is consistent with observational evidence in favor of a connection between BH growth, quasars and luminous dust-enshrouded starbursts such as ULIRGs and sub-mm galaxies (e.g., @1988ApJ...325...74S [@dasyra2006; @alexander2008]). Quantitatively, the observed [*stellar*]{} densities at radii $\sim 1-100$ pc in elliptical galaxies reach $\sim 20$ g cm$^{-2}$ [@hopkins10b], implying $\tau \sim 100$ if a significant fraction of the stars were formed in a single gas-rich epoch. It is encouraging that this is within an order of magnitude of (and larger than!) the value of $\tau$ we find is required to explain the observed $M_{\rm BH}-\sigma$ relation. A fixed value of $\tau \sim 25$ independent of galaxy mass produces an $M_{BH}-\sigma$ relation with a slope and scatter in reasonable agreement with observations (see Fig. \[FigureSigmaPlot\]). Assessing the scatter more quantitatively will require a wider survey of merger orbits. We do find some tentative evidence for a shallower slope in the $M_{BH}-\sigma$ relation at the lowest galaxy masses, corresponding to $\sigma \lesssim 100 {{\rm \, km \, s^{-1}}}$. This range of masses is precisely where the observational situation is particularly unclear, with, e.g., possible differences between the BH-galaxy correlations in classical bulges and pseudo-bulges [@greene08]. It is also unclear whether major mergers are the dominant mechanism for BH growth in these lower mass galaxies (e.g., @younger08). Our simulations show that fragmentation of a galactic disk into clumps can be efficiently [*induced*]{} by a merger (e.g., Fig. \[figureSFRClump\]), even when an isolated galaxy with same properties is Toomre stable (see, e.g., @2007MNRAS.375..805W for related ideas in the context of dwarf galaxy formation in tidal tails). As Figure \[figureISMSeries\] demonstrates, this fragmentation can produce a significant increase in star formation during the first close passage of galaxies even when there is little inflow of the diffuse ISM (because such inflow is suppressed by a bulge until later in the merger; @1996ApJ...464..641M). In our simulations we often see a corresponding increase in the BH accretion rate due to the inspiral of dense gas-rich clumps (Fig. \[figureSFRClump\]). The inflow of [gas]{} by this process may, however, be overestimated: stellar feedback not included in our simulations can unbind the gas in star clusters on a timescale of $\sim$ a Myr, returning most of the gas to the diffuse ISM (e.g., @murray10 [@hq10]). Our calculations use subgrid sound speeds motivated by the observed turbulent velocities in galaxies (§\[sectionSFR\]). We thus believe that our ISM model is physically well-motivated, even though the use of a subgrid sound speed necessarily introduces some uncertainty. Overall, the presence/absence of large-scale clumping of the ISM does not significantly change the final BH mass or the mass of new stars formed in our simulations. It can, however, change the star formation rate and BH accretion rate as a function of time, particularly near the first close passage during a merger. The tentative change in the $M_{BH}-\sigma$ relation we find for lower mass galaxies is largely due to our treatment of the ISM, rather than our BH feedback or accretion model. For a given gas fraction, lower mass galaxies have a lower gas surface density and thus the ISM is less prone to fragmentation (§\[sec:gal\] and Fig. \[FigureMassScale\]). Without the fragmentation after first passage, more gas is available to feed the BH at late times leading to somewhat higher BH mass (Fig. \[FigureSigmaPlotPeak\]). The BH accretion and feedback models used in this paper can be significantly improved in future work, allowing a more detailed comparison to observations. For example, @hq10 carried out a large number of simulations of gas inflow in galactic nuclei from $\sim 100$ pc to $\lesssim 0.1$ pc (see, e.g., @levine10 for related work). These can be used to provide a more accurate estimate of the BH accretion rate given conditions at larger radii in a galaxy (Hopkins & Quataert, in prep). Another important improvement would be to use a radiative transfer calculation to self-consistently determine the infrared radiation field produced by a central AGN (and distributed star formation). This could then be used to calculate the radiation pressure force on surrounding gas, eliminating the need for our parameterization of the force in terms of the optical depth $\tau$. Acknowledgments {#acknowledgments .unnumbered} =============== We thank Phil Hopkins and Yuval Birnboim for useful conversations. JD and EQ were supported in part by NASA grant NNG06GI68G and the David and Lucile Packard Foundation. Support for EQ was also provided in part by the Miller Institute for Basic Research in Science, University of California Berkeley. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This work was partially supported by the National Center for Supercomputing Applications under AST080048 and utilized the Intel 64 cluster Abe. The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper. Resolution Studies {#resolution} ================== {width="180mm"} In this section, we describe some of our resolution tests both with and without BH feedback. In the absence of feedback, the well-posed questions for resolution studies include both how the gas properties as a function of radius and time depend on the resolution and how integrated properties of the galaxy (e.g., the star formation rate) depend on resolution. However, the feedback, when present, has a nontrivial dependence on the resolution and it is by no means clear that the nonlinear system will in fact converge in a simple way with increasing resolution. Physically, e.g., the AGN’s radiation pressure has the strongest effect on the gas that contributes the most to the optical depth, which is largely determined by the column density (the dust opacity being only a relatively weak function of temperature for the conditions of interest). Higher resolution simulations can resolve higher volume and column densities, largely at smaller radii close to the BH, and thus may change some of the details of the BH feedback. Indeed, Fig. \[FigureSigmaofRFiducial\] shows that the column density increases towards smaller radii in our simulations. We first consider the question of how the nuclear gas properties depend on numerical resolution *in the absence of feedback*. To this end, the top left panel of Fig. \[FigureResoFourPanel\] shows the BH accretion rate $\dot M_{visc}$ calculated for three different particle numbers $N_p = 1.6 \times 10^5, 4.8 \times 10^5,$ and $1.6 \times 10^6$, with the gravitational force softening $\epsilon \propto N_p^{1/3}$.[^5] To make a fair comparison, the accretion rate is evaluated within a fixed volume ($R = 406$ pc) and for $\alpha = 0.05$ for all of the simulations. This choice corresponds to $R = 4 \epsilon$ for the lowest resolution run, but is $R \simeq 8.6 \epsilon$ for our fiducial resolution simulation. Fig. \[FigureResoFourPanel\] shows that the lowest resolution simulation (red) does not adequately resolve the fragmentation of the gas, and the resulting peak in the accretion rate, near first passage. The medium and higher (= our fiducial) resolution simulations, however, agree reasonably well, except for a slight difference in the slope of $\dot M_{visc}(t)$ at late times. Computed over a larger volume ($\sim$ kpc), the agreement between these runs improves. To assess the convergence in the presence of feedback, the top right panel of Fig. \[FigureResoFourPanel\] shows the BH accretion rate $\dot M_{visc}$ evaluated just as in the top left panel, i.e., using a fixed $R_{acc} = 406$ pc, in simulations with the same three particle numbers and force softening. Again the lowest resolution (red) simulation is clearly not adequate, but the medium (blue) and high (black) resolution simulations agree well; the integrated BH mass differs only by 2% in the latter two simulations. As a final resolution test, the bottom left panel of Fig. \[FigureResoFourPanel\] shows the BH accretion rate as a function of time in simulations with the same three resolutions and force softening, but in which $R_{acc} = 4 \epsilon$. Thus in this case the accretion rate is determined, and the feedback applied, on increasingly small spatial scales in the higher resolution simulations. This is probably the most physically realistic (see §\[sectionParam\]). This panel shows that the large peak of accretion at final coalescence ($t \sim 1.8$ Gyr) is quite similar in all three cases. This is set by the physics of feedback by momentum deposition and is a robust property of all of our simulations. A corollary of this is that the final BH mass, as shown in the bottom right panel of Fig. \[FigureResoFourPanel\], is the same to within a factor of $\sim 2$ for the three different resolutions. However, the results in the lower left panel of Fig. \[FigureResoFourPanel\] also clearly demonstrate that the detailed evolution of the accretion rate is sensitive to the resolution. This is not particularly surprising: at fixed resolution, Fig. \[FigureFourPanel\] has already demonstrated that the details of $\dot M_{visc}(t)$ depend on the value of $R_{acc}$ – although, again, neither the integrated BH mass or star formation rate do. One implication of these results is that it is difficult for current simulations of BH growth to make quantitative predictions about the light curves of AGN activity triggered by mergers. Code Verification {#sectionAppendix} ================= We have tested our modifications to GADGET on a number of simplified problems that have answers that can be easily obtained through other methods. §B1 describes tests of the additional momentum feedback force applied to a thin spherical shell of gas. §B2 describes tests in which the force is applied to the gas particles in the central regions of an isothermal sphere. Two ways of implementing the force are tested: to a fixed number of particles around the BH, and to all particles within a fixed region $R_{acc}$ around the BH. As we are concerned with the performance of our BH accretion and feedback model, in all of the tests presented in this appendix, the multiphase equation of state and star formation model of [@2003MNRAS.339..289S] are *not* used; instead we use an adiabatic equation of state with $\gamma = 5/3$. Gas shells ---------- To test that the code is applying the radiation pressure force in equation (\[momdepeqn\]) correctly, we have run the code for a simple system containing a black hole particle with a large mass and a thin spherical shell of gas with negligible temperature, pressure and mass. As this gas resides in a thin shell, this problem is more well-posed if we apply the radiation force to a fixed number, $N$, of gas particles. This system has a critical luminosity defined by the point at which the radiation force balances the inward pull of gravity. As the gas shell is of low temperature and pressure, we can ignore pressure forces. For a black hole of mass $M_{BH}$ and a gas shell of mass $m$ at a radius $r_0$ the critical luminosity $L_C$ satisfies (we take $\tau = 1$ for simplicity) $$L_C = G \frac{M_{BH} m}{r_0^2} c. \label{ShellCritLumo}$$ When the luminosity is set to this value, the gas shell should experience no net force. For other values of the luminosity, the expected behaviour can easily be calculated by noting that the gas shell, in the absence of any pressure forces, should have a radius, $r(t)$, that satisfies $$m\frac{d^2 r(t)}{dt^2} = -\frac{G M_{BH} m}{r(t)^2} + \frac{L}{c}. \label{ShellDiffEq}$$ This is easily integrated to give the expected behavior of the gas shell. ![Time evolution of the radius of the test shell for three values of radiation force: $\lambda = 0.5, 1.0, 2.0$ (dashed curves). The results match closely with the solutions from integrating eq. (\[ShellDiffEq\]) (superposed grey curves). Here the force is applied to the 25000 innermost gas particles of the $5\times10^4$ that make up the shell. Time is in units of $t_0 = \sqrt{r_0^3/G M_{BH}}$ and the radius is in units of $r_0$, where $r_0$ is the initial radius of the gas shell.[]{data-label="FigureShellTestEta"}](FigureB1.eps){width="84mm"} A number of tests of this system were performed with varying luminosities, parameterized by the ratio of the luminosity applied to the critical luminosity, $$\lambda = \frac{L}{L_C}.$$ Fig. \[FigureShellTestEta\] shows the exact result in grey, with the numerical solution from the modified version of GADGET in black, for runs with $\lambda = 0.5, 1.0 \mbox{ and } 2.0$. For these tests the number of particles in the shell is $N_{shell} = 50000$, and the force was applied to $N = 25000$ of them. In all cases, the numerical solution appears indistinguishable from the exact solution of eq. (\[ShellDiffEq\]). ![Time evolution of the radius of the test shell for three values of $N / N_{shell}$: 0.5 (solid), 0.25 (dashed), and 0.1 (dot-dashed). The numerical solutions are normalized by the exact solution from eq. (\[ShellDiffEq\]). The radiation force is fixed to be $\lambda = 2.0$. The radius $r(t)$ changes by only about 1% as $N$ is changed, indicating that our results are insensitive to the exact number of particles to which the radiation force is applied.[]{data-label="FigureShellTestXis"}](FigureB2.eps){width="84mm"} We have also tested the dependence of the results on the value of $N/N_{shell}$, the fraction of particles that receives the radiation force. Fig. \[FigureShellTestXis\] shows the ratio of the numerical solution from our code to the exact solution for $N / N_{shell}= 0.5, 0.25$, and 0.1 for the $\lambda = 2.0$ model. This demonstrates that even though the magnitude of the force on an individual particle increases as $N$ decreases, the overall dynamics of the shell is the same, with the radii differing by only $\sim 1$% in the three cases. This is primarily due to the fact that the SPH particles are collisional and can thus transfer their motion to their neighbors via pressure forces. The extra momentum imparted to the subset of particles is transferred in part to the outer region of the shell, leading to the overall motion that agrees well with the exact solution. By extension, if $N$ were to vary over the duration of the simulation, the results would also not depend strongly on the particular value. Isothermal Sphere ----------------- We have performed a second set of tests of the feedback model using an isothermal background given by a King model. The mass of the system is split into two parts. The bulk of the mass makes up the collisionless background that is drawn from the full phase space distribution of the King model. A small fraction of the mass, $f_g=0.05$, is assigned as collisional SPH particles. These gas particles follow the same spatial profile as the collisionless background but are given zero initial velocities and a very low temperature. Both components are realized with $10^5$ particles. Finally, a black hole particle with a small mass is placed at rest at the center of the distribution. {width="180mm"} In the absence of feedback, the SPH particles are not in equilibrium by construction and should flow toward the center of the potential provided by the collisionless background. When the feedback is switched on in the isothermal King potential near the center, the feedback will again have a critical value set by force balance: $$\frac{L_c}{c} = 4 \frac{f_g \sigma^4}{G}. \label{IsoThermCritLEqn}$$ When the luminosity is below this value, we expect the extra momentum to be insufficient to clear the gas out of the center. When the luminosity exceeds this value, the feedback should be strong enough to clear the central regions of the distribution. To test this, we apply feedback with a constant luminosity. Again, we parameterize the strength of the feedback as $\lambda = L / L_c$. We have tested two ways of assigning the radiation force. In the first case, the force is shared (equally) by a fixed number of gas particles nearest to the black hole. In the second case, the force is shared by all gas particles within a fixed radius of the black hole. We discuss the results separately below. ### Fixed $N$ For the tests in this subsection, the radiation force is applied to a fixed number of gas particles: $N=500$. The King model has $\sigma = 100 \rm{km} \rm{s}^{-1}$, $\Psi/\sigma^2 = 12$ and a total mass of $10^{12} M_{\sun}$. Fig. \[FigIsothermalEta00\] compare the density and pressure profiles of three runs with $\lambda=0$ (i.e. no feedback; left panels), 1 (middle), and 2 (right). Four timesteps are shown: $t=0$ (black), 0.16 (red), 0.32 (green), and 0.48 Gyr (blue). As expected, the gas flows to the center in the absence of feedback, increasing the density and pressure as the gas begins to equilibrate in the background potential. The middle and right panels show that the feedback clearly has an effect on the gas at the center, providing some support for the incoming gas, allowing the gas to have a lower pressure. For the case with $\lambda=2$, the feedback is strong enough to effectively clear out the central region. The nature of the feedback allows a calculation of how the size of the evacuated region should grow with time. Ignoring the thickness of the shell swept up as matter begins to be driven out by the feedback, momentum conservation gives $$\frac{d}{dt}\left[M_{shell}(r) dr/dt \right] = \frac{L}{c} - \frac{G M_{bg}(r) M_{shell}(r)}{r^2} \label{SnowPlowEqn}$$ where $M_{shell}(r)$ is the initial mass distribution of gas and $M_{bg}(r)$ is the mass distribution of the background. Near the center of the initial distribution, both the gas and background have an isothermal distribution, for which the mass increases linearly with the distance from the centre. This makes the right hand side of Eq. (\[SnowPlowEqn\]) a constant. In this case, the size of the evacuated region, $r(t)$, depends linearly on time: $$r(t) = \sqrt{2 (\lambda - 1) (1 - f_g)} \sigma t + C \label{SnowPlowSoln}$$ where $C$ is a constant of integration to account for the finite time required to form the shell of swept up gas. Fig. \[FigIsothermalRwithT\] shows the size of the evacuated region as a function of simulation time for a run with $\lambda = 4$, and the exact solution for a shell moving in an isothermal background (Eq. \[SnowPlowSoln\]) with $C$ chosen to match the position of the shell at $t = 0.1$ Gyr. The size of the evacuated region is defined by the position of the gas particle closest to the black hole. The agreement is very good with only slight deviation at the latest times. For the model employed, the potential is only isothermal near the origin, so when the shell expands sufficiently, the potential shallows and the shell should move faster than the prediction. This is indeed seen at late time in Fig. \[FigIsothermalRwithT\]. ![Time evolution of the size of the evacuated region for the isothermal sphere test. The $\lambda = 4$ simulation results shown (solid) match well with the analytic solution (Eq. \[SnowPlowSoln\]; dashed).[]{data-label="FigIsothermalRwithT"}](FigureB4.eps){width="84mm"} ### Fixed $R_{acc}$ For the galaxy merger simulations, we apply the force inside a fixed $R_{acc}$ throughout the simulation. In this section, we run a similar set of tests as in the previous subsection but we hold $R_{acc}$ fixed. When the number of particles inside $R_{acc}$ becomes small, however, the feedback force exerted on individual particles becomes spuriously large. We therefore impose an additional condition of minimum $N$ on the feedback. For the tests in this subsection, the feedback is applied to those particles inside $R_{acc}$, or to the innermost $100$ gas particles if there are fewer than this inside $R_{acc}$. For the simulations in the main paper, however, there were always enough particles inside the accretion and feedback region to avoid the need for such a lower bound on $N$. Our first test uses a constant $L = 4 L_c$, and holds $R_{acc}$ fixed. We use a King model as in the previous section, but with slightly different parameters to connect more closely to the our fiducial simulation: $\sigma = 160 \rm{km} \rm{s}^{-1}$, $\Psi/\sigma^2 = 12$ and a total mass of $10^{12} M_{\sun}$. We tested this model for three different sizes of the accretion and feedback region: $R_{acc} = 0.7, 1.4$ and $2.8$ kpc. The smallest region has initially $N \sim 500$. Note that the values of $R_{acc}$ used here are larger than those used in our galaxy merger simulations in the main text. These values of $R_{acc}$ were necessary to ensure that $R_{acc}$ contains a reasonable number of particles. In the galaxy merger calculations, the overall larger number of particles in the simulation and the high gas density in the central regions imply that smaller values of $R_{acc}$ can be reliably used. They are also more physical, as we argued in §\[sectionParam\]. Fig. \[FigureShellFixed\] shows the position of the shell of swept up material for the three runs with $R_{acc} = 0.7, 1.4$ and $2.8$ kpc in black, red and blue respectively. Initially, all the gas inside $R_{acc}$ experiences the extra force. As the region becomes more evacuated and the number of particles inside $R_{acc}$ drops, we transition to applying the force to the $N = 100$ particles closest to the BH. The evolution of the shells in this case is quite similar to the evolution in the last section. The model used in this section is smaller in size and so the shell expands past the isothermal core of the King model earlier. As a result, it begins to accelerate outward sooner. However, the tests with different $R_{acc}$ have essentially identical evolution. ![Radius of the swept up shell for the isothermal sphere test with $\lambda = 4$ and fixed $R_{acc}$: 0.7 (black), 1.4 (red), and 2.8 kpc (blue). To avoid numerical problems, the feedback was always applied to at least $N \sim 100$ particles. The numerical results agree well with the dashed curve, which shows a numerical integration of the analytic equation for the shell radius (eq. \[SnowPlowEqn\]).[]{data-label="FigureShellFixed"}](FigureB5.eps){width="84mm"} ![The black hole accretion rate for isothermal sphere simulations in which the full black hole accretion and feedback model are used ($\alpha = 0.1$ and $\tau = 1$). Three different values of $R_{acc}$ are shown: $R_{acc} = 0.7$ (black), 1.4 (red), and 2.8 kpc (blue). All three agree well with each other.[]{data-label="FigureShellFull"}](FigureB6.eps){width="84mm"} Finally, we run a test in which we determine the luminosity from the accretion rate as in Eq. (\[momdepeqn\]), and increase the BH mass in time accordingly. This test thus employs the full feedback and accretion model of our galaxy merger simulations. We use the same $\sigma = 160$ km s$^{-1}$ King model, and took $\alpha = 0.1$ and $\tau = 1$ for the feedback parameters. The initial mass of the black hole was $M_{BH,i} = 10^5 M_{\sun}$. Fig. \[FigureShellFull\] shows the accretion history of the BH for the runs with $R_{acc} = 0.7, 1.4$, and 2.8 kpc. In each test, the feedback is initially Eddington limited and it is not until about $t = 0.3$ Gyr that the luminosity approaches that required to evacuate the gas out of $R_{acc}$. At this point, the gas begins to move out of $R_{acc}$ and form a shell of material at $R \sim R_{acc}$. This shell then remains fairly steady as the accretion rate self-regulates around the critical luminosity. As the three values of $R_{acc}$ are all inside the isothermal core of the King model, the critical luminosities (eq. \[IsoThermCritLEqn\]) are the same, and we would thus expect the accretion rate to self-adjust to the same value at late times. This is indeed borne out in the simulations shown in Fig. \[FigureShellFull\]. Of these three runs, only the calculation with $R_{acc} = 0.7$ kpc spends a significant amount of time with fewer than 100 particles inside $R_{acc}$. Despite the large change in the size of the feedback region, Fig. \[FigureShellFull\] shows that the evolution of the gas is quite similar. The black hole masses for these three runs differ by only a factor of $\sim 2$ at the end of the simulation. [^1]: These conclusions do not apply to dilute plasma in the intracluster or intragroup medium. The densities there are sufficiently low that the plasma can be efficiently heated by an AGN. [^2]: In the simulation with a higher initial gas density ($f_g = 0.3$), so many fragments form at large radii and spiral into $R_{acc}$ that the surface density in the central region remains elevated from first passage until the merger completes at $t \sim 1.8$ Gyr (see Fig. \[FigureSigmaFiducial\]). [^3]: We used $T_{SN} = 4 \times 10^8$ K, $A_0 = 4000$ and $t_*^0 = 8.4$ Gyr for these calculations; these values are different from those in our fiducial simulation, and are chosen to fix the total star formation rate for our isolated fiducial galaxy at $1 M_{\sun}$ yr$^{-1}$. [^4]: To account for the fluctuating nature of the BH accretion rate in some of the simulations, we define the BH mass at “peak” to be the mass when $\dot M$ drops by a factor of 10 from its peak value. [^5]: In Fig. \[FigureResoFourPanel\], $\dot M_{visc}$ for the simulations without feedback (upper left) is calculated from the simulation snapshots and the accretion rate is not Eddington limited. The data outputs were relatively infrequent and attempting to integrate the BH mass over such large timesteps was inaccurate.

--- abstract: 'We present a modified adaptive matched filter algorithm designed to identify clusters of galaxies in wide-field imaging surveys such as the Sloan Digital Sky Survey. The cluster-finding technique is fully adaptive to imaging surveys with spectroscopic coverage, multicolor photometric redshifts, no redshift information at all, and any combination of these within one survey. It works with high efficiency in multi-band imaging surveys where photometric redshifts can be estimated with well-understood error distributions. Tests of the algorithm on realistic mock SDSS catalogs suggest that the detected sample is $\sim 85\%$ complete and over $90\%$ pure for clusters with masses above $1.0\times10^{14} h^{-1}$ M$_\odot$ and redshifts up to $z=0.45$. The errors of estimated cluster redshifts from maximum likelihood method are shown to be small (typically less that 0.01) over the whole redshift range with photometric redshift errors typical of those found in the Sloan survey. Inside the spherical radius corresponding to a galaxy overdensity of $\Delta=200$, we find the derived cluster richness $\Lambda_{200}$ a roughly linear indicator of its virial mass $M_{200}$, which well recovers the relation between total luminosity and cluster mass of the input simulation.' author: - 'Feng Dong , Elena Pierpaoli , James E. Gunn , Risa H. Wechsler' bibliography: - 'ms.bib' title: 'Optical Cluster-Finding with An Adaptive Matched-Filter Technique: Algorithm and Comparison with Simulations ' --- Introduction {#sec:Introduction} ============ Clusters of galaxies are the most massive virialized systems in the Universe and have been extensively used to study galaxy population and evolution [@Dre84; @Dre92], to trace the large-scale structure of the universe [@Bah88; @Pos92], and to constrain cosmology [@Evr89; @Bah99; @Hen00; @Pierpa01; @Pierpa03]. Given the important roles clusters of galaxies play in the studies of both astrophysics and cosmology, tremendous efforts have been made during the past several decades to search for these systems. The first large samples of clusters were identified by looking for projected galaxy overdensities through visual inspection of photographic plates [@Abe58; @Abe89; @Zwi68]. These catalogs made pioneering contributions to our understanding of the extragalactic universe and since their generation have opened many new frontiers in the studies of galaxy clusters. However, the compilation of a relatively complete and pure sample of galaxy clusters has remained far from trivial. To date the Abell catalog, which contains about 4000 rich clusters to a redshift of $z\sim0.2$, is still the most widely used cluster catalog in the field, though it was realized early that visually-constructed catalogs suffer from projection effects, subjectivity, and large uncertainties in estimated properties [@Sut88]. It is difficult to apply these catalogs for statistical studies in cosmology because of these uncertainties, in addition to the fact that the selection function and false positive rates of such cluster samples are hard to quantify. To relieve some of these concerns, other approaches for identifying clusters have also been designed and implemented, such as reconstructing the full 3-D structures in complete redshift surveys [@Huc82; @Gel83; @Ram97], detecting clusters in X-ray surveys [@Gio90; @Edg90; @Ebe98; @Ros98; @Rom00; @Sch00; @Boh01; @Mul03; @Boh04], and utilizing the Sunyaev-Zeldovich effect [@Car00; @Moh02; @Pierpa05] and weak gravitational lensing [@Sch96; @Wit01] in search for clusters. Moreover, the realization of large and deep galaxy surveys in recent years has revived optical cluster-finding endeavors and prompted the development of more automated and rigorous algorithms to select clusters from imaging surveys. Using multi-color photometric data from which photometric redshifts can be estimated, it is now possible to mitigate the problems of projection effects, and quantitative analysis of the selection bias is also now possible. Automated peak-finding techniques in optical cluster searches were attempted by @She85 and later used in the Edinburgh/Durham survey [ED @Lum92] as well as the Automatic Plate Measurement Facility survey [APM @Dal94; @Dal97]. In the construction of the cluster catalog from the Palomar Distant Cluster Survey [@Pos96], a matched filter algorithm was developed to select clusters from a photometric galaxy sample. It was widely used in subsequent surveys and several variants have been put forward [@Kaw98; @Sch98; @Kep99; @Kim02; @Whi02]. Meanwhile with the knowledge of the existence of the “E/S0 ridgeline” of cluster galaxies in color-magnitude space and the aid of multi-color CCD photometry, several color-based cluster-finding techniques were also investigated [@Gla00; @Got02; @Gla05; @Mil05]. Some of these have already been successfully applied to select clusters from the Sloan Digital Sky Survey (SDSS) data [@Got02; @Ann02; @Bah03; @Mil05; @Koe07]. The Sloan Digital Sky Survey [@Yor00] is a five-band CCD imaging survey of about 10$^4$ deg$^{2}$ in the high latitude North Galactic Cap and a smaller deeper region in the South, followed by an extensive multi-fiber spectroscopic survey. The imaging survey is carried out in drift-scan mode in five SDSS filters ($u$, $g$, $r$, $i$, $z$) to a limiting magnitude of $r\sim22.5$ [@Fuk96; @Gun98; @Lup01; @Smi02]. The spectroscopic survey targets $\sim$10$^{6}$ galaxies to $r\sim17.7$, with a median redshift of $z\sim0.1$ [@Str02], and a smaller deeper sample of $\sim$10$^{5}$ Luminous Red Galaxies out to $z\sim0.5$ [@Eis01]. In this paper we discuss a modified adaptive matched filter technique incorporating several new features over previous algorithms and designed to detect clusters using both the SDSS imaging and spectroscopic data; it could readily be adapted to other similar multi-band, large-area galaxy surveys for construction of optically-selected cluster samples. It is the first of a series of papers that will explore the application of the technique to select clusters from the Sloan Digital Sky Survey. The general idea of the matched filter method relies on the fact that clusters show on average a typical density profile, now widely assumed to be the “NFW” form suggested first by Navarro, Frenk and White [@Nav96]. Assuming that galaxies trace the dark matter, we expect galaxies within clusters to be distributed according to such profile. The algorithm selects regions in the sky where the distribution of galaxies corresponds to the projection of average cluster density profile. In addition, it is possible to specify the galaxy redshift information inside clusters, and to use prior knowledge on the galaxy luminosity function. The combination of these matched subfilters thus enables us to extract a quantitative signal corresponding to the existence of a cluster at a given location in the surveyed sky area. The modified matched filter technique presented in this paper can fully adapt to imaging surveys with spectroscopic measurements, multicolor photometric redshifts, no redshift information at all, and any combination of these within one survey. In the Sloan Digital Sky Survey where photometric redshifts can be estimated with well-understood error distributions from the five-band ($u$,$g$,$r$,$i$,$z$) multi-color photometry, the matched filter technique described here utilizes not only the spectroscopic coverage for the bright main sample galaxies and Luminous Red Galaxies (LRGs) but also the photometric redshift information for most of the galaxies detected in the imaging survey. This greatly expands the input galaxy sample to feed into the cluster-finding algorithm compared to pure spectroscopic methods [e.g. @Mil05]. The obtained composite cluster catalog can also go much deeper in redshift ($z\sim0.4-0.5$ in this case) than the typical $z\sim0.2$ limit for spectroscopic samples due to the lack of availability of spectroscopic measurements for faint, deep galaxies. Since the matched filter technique does not explicitly use the information about the red sequence to select clusters as is done in some color-based cluster-finding methods [@Ann02; @Mil05; @Koe07], it can theoretically detect clusters of any type in color, and is not restricted only to old, red E/S0 galaxies. Such clusters likely dominate the cluster population, but may not constitute all of it especially as one probes systems of lower richness and at higher redshifts. The use of both spectroscopic and photometric redshift information largely eliminates the projection effects and removes most of the phantom clusters. The matched filter also generates accurate quantitative estimates of derived cluster properties, such as redshift, scale, richness, and concentration, and produces quantitative detection likelihoods, indicative of the combined information for both red and blue galaxies identified as cluster members. These facilitate further studies of detected systems and makes easier the comparison to clusters selected by other methods. One major concern for the matched filter technique is the fact that determination of these parameters depends on the specific cluster model we put in to build the relevant filters. However, these effects can be minimized by careful assumptions about the shape and evolution of luminosity function, and by the fact that our density filter is self-adaptive to different cluster scales and concentration. The clusters selected by the algorithm will provide us the necessary sample on which we then apply an iterative procedure aimed at refining the constraints on clusters’ properties. More details will be discussed in section §\[sec:Algorithm\] and subsequent work following this paper. The new algorithm presented here differs from previous matched filter implementations [@Kep99; @Kim02] in several ways. We use a uniform Poisson likelihood analysis, which is only the second step in the approach by @Kep99 following a first pass using Gaussian statistics for pre-selection of clusters. This avoids the common problem for high-redshift clusters of having too few galaxies in any cell of interest for Gaussian statistics to apply, and the adopted approach yields correct likelihoods even at the detection stage. In addition, both the core radius and virial radius of the matched filter are adaptive over the typical observed dynamical range for clusters, in contrast to most previous cluster-finding techniques that set the cluster core radius or search radius to be fixed. For each individual cluster, a best-fit core radius is found to maximize the likelihood match, as well as an outer radius inside which the galaxy overdensity reaches $\Delta$=200. The cluster richness is then normalized to be the light contained within this virial radius, which we find correlates better with the mass of gravitational systems whose extent is defined by density contrast as is widely adopted in theoretical studies. The new features of our modified algorithm will be further discussed in §\[sec:Algorithm\]. In order to understand the biases and the selection functions of our algorithm, we test it on a mock SDSS catalog which has been constructed from the Hubble Volume Simulation [@Evr02] by assigning luminosities and colors to the dark matter particles in a manner which reproduces many characteristics of the galaxy population from SDSS observations. The “observations” of the simulations have then been further modified so that the redshift scatter of those galaxies which have photometric but no spectroscopic redshifts correspond to that of the photometric redshift errors in actual SDSS data. The comparison of the detected cluster sample with halos in the simulation provides the only rigorous way to assess how the observed cluster properties relate to the real masses, and how the cluster sample can be used to derive cosmological constraints. In section §\[sec:Algorithm\] we describe the modified adaptive matched filter technique and how it is used to extract the cluster sample. Section §\[sec:Simulation\] presents the basic features of the simulated catalog we adopted for the testing purpose. In section §\[sec:Results\] we show results on the completeness and purity of our cluster sample, and the expected scaling relations inferred from runs on the simulations. We conclude in section §\[sec:Conclusion\]. A flat $\Lambda$CDM model with $\Omega_m=0.3$ and $\Omega_{\Lambda}=0.7$ is used throughout this work, and we assume a Hubble constant of $H_0=100 h$ km s$^{-1}$ Mpc$^{-1}$ if not specified otherwise. The Cluster-Finding Algorithm {#sec:Algorithm} ============================= The matched filter technique introduced here is a likelihood method which identifies clusters by convolving the optical galaxy survey with a set of filters based on a modeling of the cluster and field galaxy distributions. A cluster radial surface density profile, a galaxy luminosity function, and redshift information (when available) are used to construct filters in position, magnitude, and redshift space, from which a cluster likelihood map is generated. The peaks in the map thus correspond to candidate cluster centers where the matches between the survey data and the cluster filters are optimized. The algorithm automatically provides the probability for the detection, best-fit estimates of cluster properties including redshift, radius and richness, as well as membership assessment for each galaxy. The modified algorithm can be fully adaptive to current and future galaxy surveys in 2-D (imaging), 2$\half$-D (where multi-color photometric redshifts and their errors can be estimated), and 3-D (with full spectroscopic redshift measurements). Usage of the apparent magnitudes and, where applicable, the redshift estimates instead of simply searching for projected galaxy overdensities effectively suppresses the foreground-background contamination, and the technique has proven to be an efficient way of selecting clusters of galaxies from large multi-band optical surveys. In what follows, we first provide a general introduction on how the likelihood function is constructed and how we detect clusters with the matched filter method. This gives us an overview about how the cluster catalog is derived. Then we discuss in more detail the density models and subfilters used to construct the likelihood. More specifically, we assume an NFW density profile, a general Schechter luminosity function and a Gaussian model for BCGs to model clusters, and use the spectroscopic measurements and obtained error distributions of galaxy photometric redshifts from the Sloan Digital Sky Survey to incorporate redshift uncertainties. In the end we describe how to determine the set of best-fit parameters on cluster properties that maximize the likelihood at a given position over a range of redshift, scale, concentration, and richness. Likelihood Function {#sec:Likelihood} ------------------- The likelihood function used here is based on the assumption that the probability of finding galaxies in an infinitesimal bin in angular position, apparent magnitude and redshift space is given by a Poisson distribution. Under this assumption, the total likelihood of many of such bins, which we take to be centered in the location of the galaxies in the survey, is (see appendix C2 in @Kep99 for a full derivation): $$ln {\cal L} = -N_f -\sum_{k=1}^{N_c}N_k +\sum_{i=1}^{N_g}\ln[P(i)],$$ where $N_f$ is the total number of field galaxies expected within the searching area, $N_g$ is the total number of galaxies and $\sum_{k=1}^{N_c}N_k$ is normalized to be the number of galaxies brighter than $L^*$ as members of the $N_c$ clusters assumed in the model. $P(i)$ represents the predicted probability density of galaxies in a given bin, which includes both probabilities of field galaxies ($P_f$) and of cluster members ($P_c$), $$P(i) = P_f(i) + \sum_{k=1}^{N_c} P_{c}(i,k).$$ These probabilities are the expected number densities for a given location and magnitude. The cluster catalog is constructed with an iterative procedure similar to the one used in @Koc03. We start our process from a density model of a smooth background with no clusters. For each galaxy position, we then evaluate the likelihood increment we would obtain by assuming that there is in fact a cluster centered on that galaxy. The likelihood is then optimized by varying the cluster galaxy number $N_k$, the redshift and cluster scale length. At each iteration, we retain the cluster candidate which resulted in the greatest likelihood increase. We incorporate it in our density model and restart the procedure. The function for finding the $k^{th}$ cluster in the whole surveyed area therefore is $$\begin{array}{ll} \Delta\ln{\cal L}(k) = -N_k + \sum_{i=1}^{N_g}\ln[{P_f(i)+\sum_{j=1}^{k}P_c(i,j) \over P_f(i)+\sum_{j=1}^{k-1}P_c(i,j)}]. \end{array}$$ A list of cluster candidates then becomes available in decreasing order of detection likelihoods. For each candidate one has derived properties, including best-fit position, scale, richness, and estimated redshift. The initial cluster catalog allows us to further inspect each individual candidate for exploration of substructure and better constraints on previously fitted quantities. Density Model {#sec:Density} ------------- As both field and cluster galaxies are found in the survey, the probability of finding a galaxy in a given bin depends on the density of both these populations (see eq.(2)). For galaxy $i$ with angular position $\vec{\theta}_i$, $r$-band apparent magnitude $m^r_i$ and redshift $z_i$ (when available), the background number density $P_f(i)$ can be directly extracted from the global number counts of the galaxy survey, $$P_f(i) = { d N \over dm~ dz }(m^r_i, z_i),$$ and it has to be modified to account for the effects of galaxy redshift uncertainties if photometric redshift estimates are used. For cluster $k$ located at $\vec{\theta}_k$ with proper scale length $r_{ck}$, redshift $z_k$ and galaxy number $N_k$, the probability of galaxy $i$ being a member of it, $P_c(i,k)$, is just the product of a surface density profile $\Sigma_c$ and a luminosity function $\phi_c$ at the cluster’s redshift, times a distribution function $f(z_i- z_k)$ that expresses redshift uncertainties: $$P_c(i,k) = N_{k}\ \Sigma_c\left[D_A(z_k)\theta_{ik}\right]\ \phi_c\left[m^r_i-{\cal D}(z_k); \right]\ f(z_i-z_k),$$ where ${\cal D}(z_k)$ is defined through $$M^r_i = m^r_i - 5 \log (D_L(z_k)/10\mbox{pc}) - k(z_k) = m^r_i - {\cal D}(z_k),$$ and where $D_A(z_k)$ and $D_L(z_k)$ are the angular diameter and luminosity distance at the cluster’s redshift $z_k$, and $k(z_k)$ is the $k$-correction. The conversion of units in luminosity and distance is conducted by performing proper $k$-corrections for galaxies of different spectral types and choosing the proper cosmology (see §\[sec:Introduction\]). Subfilters {#sec:Subfilters} ---------- Based on current observational studies as well as findings from dark matter halos, and for convenient comparisons to theoretical models widely used in analytical studies and N-body simulations, we assume the density profile of galaxies within a cluster follows the form of a NFW profile [@Nav96], which in three dimensions is given by $$\rho_c(r) = {1 \over 4\pi r_c^3 F(c)}{1 \over {r \over r_c}(1+{r \over r_c})^2},$$ where $c$ is the concentration parameter and $F(c)$ is the typical normalization factor for galaxies inside the virial radius of the cluster, $r_v=cr_c$. The 3-D profile is then integrated along the line of sight to derive a projected surface density profile $\Sigma_c(r)$ which is expressible as a much more complicated analytical form (see @Bar96). The profile is normalized so that $\int_0^{cr_c} 2\pi r \Sigma_c(r) dr = 1$. The search radius for galaxies belonging to the cluster is set to be the virial radius of the cluster, or more specifically here, the radius inside which the mass overdensity is 200 times the critical density, i.e., 200$\Omega_M^{-1}$ times the average background [@Evr02]. Since it is hard to directly measure the cluster mass overdensity in observations, we instead determine the virial radius inside which the space density of cluster galaxies is 200$\Omega_M^{-1}$ times the mean field, assuming that the galaxy distribution in a halo traces the overall dark matter distribution (see discussions in @Han05), which has been suggested by recent observations and simulations [@Lin04a; @Nag05; @Lin07a], and is supported by weak lensing measurements [@She04]. For simplicity, we use $r_{200}$ throughout this work to denote the cluster virial radius determined by galaxy overdensities. The cluster richness is then defined to be the total luminosity in units of $L^*$ inside $r_{200}$. As has been discussed before in matched-filter studies [@Pos96; @Kim02] and also shown by our own numerical experiments, the efficiency of the filter is usually much more sensitive to the overall filter cutoff radius than to the details of its shape. Therefore the determination of appropriate values for the scale length in the cluster model is of particular importance, as it may have significant impact on the detection efficiency of the cluster-finding algorithm. Most of the previous matched filter methods have used a carefully chosen fixed value for the model cluster cutoff radii, and they compute the galaxy number or the richness of clusters within such a fixed radius in physical units. @Pos96 concludes that a fixed search radius of 1 Mpc $h^{-1}$ is a near-optimal choice in their radial filter, and this value has been also adopted by @Kep99 [@Kim02] in their method which assumes a modified Plummer law model for the surface density profile. In @Whi02 and @Koc03, the authors set a fixed core radius of $r_c=200$ kpc $h^{-1}$ and concentration parameter of $c=4$ for the NFW profile in the cluster detection and mass estimates. Although we find from observations and simulations that these choices are reasonable values for typical rich clusters, a single fixed scale length for all clusters over a wide range of masses and concentrations will certainly degrade the signal-to-noise ratio, bias detection probabilities, and be responsible for at least part of the large scatter observed in previous cluster mass-richness scaling relations. In our modified adaptive matched filter algorithm, we optimize the core radius for each individual cluster over the dynamical range for typical galaxy clusters. For the core radius value that maximizes the likelihood, we then compute the normalized cluster richness according to the NFW profile with best-fit parameters within a cluster virial radius $r_{200}$ determined from galaxy overdensities. We believe this procedure is more similar to and comparable with the virial mass defined by density contrast in most theoretical studies and analyses of simulations. For the magnitude filter, we adopt a luminosity profile described by a central galaxy plus a standard Schechter luminosity function [@Sch76] $$\phi(M) = {dn \over dM} = 0.4 \ln10\ n^* \left({L \over L^*} \right)^{1+\alpha} \exp(-L/L^*);$$ the integrated luminosity function is $$\Phi(M) = \int_{-\infty}^M \phi(M) dM = n^* \Gamma[1+\alpha, L/L^*].$$ Parameters for the global luminosity function are obtained from the SDSS spectroscopic sample at the redshift of $z=0.1$ [@Bla03]. To account for the evolutionary effects at higher redshifts, we allow a passive evolution of $L^*$ which brightens about 0.8 magnitudes from $z=0$ to $z=0.5$ [@Lov92; @Lil95b; @Nag01; @Bla03; @Lov04; @Bal05; @Ilb05]. We assume that $L^*$ does not vary as a function of cluster richness, which is supported by the results of @Han05. Because the matched filter algorithm uses both a cluster galaxy luminosity function and a field galaxy luminosity function, which are expected to be different due to the morphology-density relation [@Dre80] and the observed dependence of luminosity function on galaxy over-densities [@Chr00; @Mo04; @Cro05], it would be desirable to model these separately. It would also be desirable to further model the luminosity distributions according to galaxy spectral types [@Fol99; @Lin99; @Hog03]. At this stage, however, only a single function is adopted since the work on precise luminosity functions for cluster galaxies of different types has just been started. We hope to investigate this further on the basis of the first catalog we produce. Once a cluster catalog is available for galaxies in all redshift ranges, we can go back and examine the impact of our assumptions about the galaxy luminosity functions as well as their evolution for different environments and spectral types. In order to use the same range in the luminosity function at all distances and therefore avoid bias associated with errors in the assumed form of the luminosity function, we cut off the luminosity function at one magnitude below $L^*$. We can still calculate total luminosities by integrating the assumed form, and we use this in our richness calculation, described below. The existence of Brightest Cluster Galaxies (BCGs) near the cluster centers is incorporated into our cluster galaxy luminosity model as a separate component from the main Schechter function for satellites, as this distinction has been clearly seen in clusters over a range of richness [@RT77; @Han05]. We assume a Gaussian distribution for the luminosities of these objects and adopt the results from @Lin04b for correlations between the BCG luminosity and host cluster properties. More specifically, the BCG luminosity is assumed to follow a single power law with the cluster richness, $L_{BCG} \sim \Lambda_{200}^{1\over4}$, and we take the width of the Gaussian to be $\sim0.5$ mag [@Lin04b; @Zhe05; @Han05]. The luminosity of BCGs is assumed to evolve in the same way as $L^*$ does, [*i.e. *]{} the luminosity at the mean of the gaussian has a constant ratio to $L^*$. This is almost certainly incorrect in detail, but will be explored in follow-up work once the catalog is constructed. This modification of the general Schechter function enhances the detectability of typical clusters with BCGs, especially those at higher redshifts with only few galaxies other than the BCG to be included in the apparent magnitude-limited galaxy sample. Thanks to the accurate five-band ($u$,$g$,$r$,$i$,$z$) multi-color photometry in the SDSS [@Yor00], as well as the associated redshift survey for the bright main sample galaxies [@Str02] and Luminous Red Galaxies (LRGs, @Eis01), it is now also possible to retrieve redshift information for most of the galaxies that we are going to use in construction of the SDSS cluster catalog, either photometrically or spectroscopically. For real SDSS data currently available from DR5, we find that galaxies with valid photometric redshift estimates make up more than $96\%$ of the whole sample in the imaging data, within which about $1\%$, mostly bright, red galaxies, have matched spectroscopic measurements from redshift surveys. Not surprisingly, the inclusion of galaxy redshift estimates greatly improves the accuracy of the cluster redshift determinations and significantly mitigates projection effects, thus allowing the detection of much poorer systems than possible in previous work with no redshift measurements. The uncertainties of galaxy redshifts are assumed to follow Gaussian distributions in the 2$\half$-D and 3-D cases, where in terms of the $f(z)$ function in equation (5) we have $$f(z_k) = {\exp\left[-(z_i-z_k)^2/2\sigma_{z_i}^2\right] \over \sqrt{2\pi}\sigma_{z_i}}.$$ For galaxies with computed photometric redshifts (described below), we add to the cluster galaxy density model a third subfilter based on the distribution of derived redshift uncertainties in the form of a combination of multiple Gaussian modes. These error estimates are obtained by calibrating photometric redshifts with the real redshifts in the SDSS spectroscopic galaxy sample and redshifts for other fainter (but smaller) overlapping surveys. The analysis is done for red and blue galaxies separately using the color separator by @Strv01, and it is found that a model using Gaussian modes with proper weights assigned generally provides a good description of the bias and scatter in the photometric redshifts for galaxies of both spectral types and in different apparent magnitude bins. Some of the results are shown in §\[sec:Simulation\]. In the 3-D case where spectroscopic redshifts of galaxies are measured, we smooth them in Gaussians with assigned cluster velocity dispersions that vary in the range from $400$ km s$^{-1}$ (proper) for poorer clusters to $1200$ km s$^{-1}$ (proper) for the richest systems in the selected cluster sample, according to several discrete estimated richness classes. The same procedure as outlined in the previous paragraph for photometric redshifts is applied to include this redshift filter in the galaxy density model. In addition, there are galaxies we find that either have invalid photometric redshifts computed or fall into the redshift and magnitude range where no good calibrations are available. Such galaxies, which are currently about $3\%$-$5\%$ of the whole sample, are assumed to have no redshift estimates and therefore no constraining filter. Hence we set up for each galaxy the appropriate scenario that adapts the matched filter algorithm to galaxy redshift estimates with varied accuracy. Finally, of course, we fit an overall amplitude, which represents the cluster richness. Since its size, shape and redshift are all determined at this point, we can express the amplitude however we like in physical terms. We have chosen to use the total luminosity within $r_{200}$ expressed as a multiple of $L^*$ (evolved to the relevant redshift using 1.6 mags of luminosity evolution per unit redshift), which we denote as $\Lambda_{200}$. Implementation {#sec:Implementation} -------------- Implementation of the matched filter algorithm starts with reading the galaxy catalog. For each galaxy $i$ in the sample, we read in the positions $\alpha_i$, $\delta_i$, the extinction-corrected five-band apparent magnitudes and their errors, and the redshift $z_i$ if it has a matched spectrum. Using the flux and color information, we compute a photometric redshift estimate using a neural network technique by [@Lin07] as well as $k$-corrections and estimated rest-frame colors for each galaxy, which we add as input to the cluster-finding algorithm. The next step is to define the cluster model we adopt for the filters, including the surface density profile $\Sigma_c(r)$, the luminosity function $\phi(M)$, and the assumed Gaussian modes of photometric redshift uncertainties. The field density model $P_f(m,z)$ is constructed from global number counts of the surveyed background galaxy distributions as a function of magnitude and redshift, as shown in equation (4). We then incorporate these models into the Poisson likelihood functions as discussed above. To map the likelihood distributions of the surveyed area, we grid the sky using the Healpix package of [@Gor05] which provides a useful hierarchical pixelization scheme of equal-area pixels. In @Kep99, the authors choose galaxy positions on an adaptive grid in calculating the likelihoods instead of the uniform grids used in the previous matched filter codes [@Pos96], so that sufficient resolution in the high density regions is ensured while saving computational time and memory for less dense regions. We follow this procedure and evaluate the likelihood functions at every galaxy position to locate the peaks in the map as possible cluster centers. The cluster richness is optimized over the whole redshift range of our search at intervals that finally adapt to $\delta z=0.001$, and for a set of trial cluster scale radii ($r_c$) at $10$ kpc $h^{-1}$ steps. The derived quantities for best fit cluster richness, redshift and scale length thus correspond to the parameters that maximize the likelihood function at the grid position or candidate cluster center. This algorithm possesses several new features. First, the cluster algorithm is fully adaptive to 2-D, 2$\half$-D and 3-D case in the optical surveys, and can deal with data with these different attributes simultaneously. It can easily accommodate the galaxy redshifts with uncertainties in any forms and distributions, from purely single-band imaging data to a complete spectroscopic redshift survey, and works well for the intermediate case where photometric redshifts are estimated from multi-band color information. Projection effects from foreground–background contamination, which have been a long-standing problem for optically-selected clusters, are largely suppressed. This allows the detection of even poorer systems at high redshift, and shows great potential for current and future large, deeper surveys in the optical band. Second, the current adaptive matched filter used a single Poisson statistics in the likelihood analysis, compared to the two-step approach in @Kep99, which uses a “coarse” filter based on Gaussian likelihood for pre-selection of clusters. We write our code in Fortran-90 and by careful arrangement in computations and setting up the quick link search, the optimization of the Poisson likelihood through the whole process is now affordable in the sense of execution time and memory. For a survey field of $\sim300$ deg$^2$, which is comparable to a typical SDSS stripe [@Yor00], the modified adaptive matched filter algorithm requires around 900 megabytes of memory and takes about 30 hours for a single run using one dual-processor node in a Linux Beowulf cluster with 3.06 GHz clock speed each. With no assumption necessary about sufficiently many galaxies inside each virtual bin as is necessary in the Gaussian case, the Poisson statistics remains robust in the common situation where there are too few galaxies in each cell for Gaussian statistics to apply. Third, as discussed in @Whi02 and @Koc03, the current density model explicitly includes the effect of previously found clusters on the global likelihood function. The procedure automatically separates overlapping clusters and avoids multiple detections of the same system in the overdensity regions, somewhat similar to the CLEAN method used in radio astronomy to produce maps [@Hog74; @Sch78]. We do not need to do extra cluster de-blending work afterwards. Finally, as discussed earlier, our approach to maximizing the likelihood differs from most previous cluster-finding techniques that choose a fixed cluster scale or search radius. We optimize the core radius for each individual cluster, and the cluster richness is computed within a virial radius which is determined from galaxy overdensities. This provides insights about the virial mass of such gravitational systems defined by density contrast and better corresponds to what is done in theoretical treatments. Tests on Mock Galaxy Catalogs {#sec:Simulation} ============================= To evaluate the completeness and purity (false positive rate) of our cluster sample, as well as to assess the how well our measured cluster properties correspond to the properties of the underlying dark matter halos, we have run the matched-filter algorithm on a mock galaxy catalog generated from a realistic cosmological N-body simulation. Because of the large redshift range we are trying to probe, it is important to do this with as large a simulation volume as possible. In addition, because we seek here to test the behavior of our algorithm using a combination of spectroscopic and photometric redshifts, it is useful to have a realistic galaxy population in both clusters and the field, with luminosities, colors, and the relation between these quantities and environment that are a good match to SDSS data. Here we have used a mock catalog based on a method namely ADDGALS (Adding Density-Determined Galaxies to Lightcone Simulations) (@Wec04 and in preparation, 2007), which is designed to model relatively bright galaxies in large volume simulations. The underlying dark matter simulation used here tracks $10^9$ particles of mass $2.25\times10^{12} h^{-1} M_\odot$ in a periodic cubic volume with side length of $3 h^{-1}$ Gpc, using a flat $\Lambda$CDM cosmology with $\Omega_m=0.3$, $\sigma_8=0.9$, and $h=0.7$ [the Hubble Volume simulation; @Evr02]. Halos are identified for masses above $2.7\times10^{13} h^{-1} M_\odot$. Data are collected on the past light cone of an observer at the center of the volume. The size of the simulation enables the creation of a full-sky survey out to redshift of $z=0.58$, and is thus suited to testing our cluster-finding algorithm out to high redshifts using the SDSS imaging data. Galaxies are connected to individual dark matter particles on this simulated light-cone, subject to several empirical constraints. The resolution of the simulation allows the mock catalog to include galaxies brighter than about 0.4$L^*$; the number of galaxies of a given brightness placed within the simulation is determined by drawing galaxies from the SDSS galaxy luminosity function [@Bla03], with 1.6 mags of luminosity evolution assumed per unit redshift (the same assumption is made by our cluster finding algorithm). The choice of which particle these galaxies are assigned to is determined by relating the particle overdensities (on a mass scale of $\sim 1e13 M_{\odot}$) to the two-point correlation function of the particles; these particles are then chosen to reproduce the luminosity-dependent correlation function as measured in the SDSS by @Zeh04. Finally, colors are assigned to each galaxy by measuring their local galaxy density (here, the fifth nearest neighbor within a redshift slice), and assigning to them the colors of a real SDSS galaxy with similar luminosity and local density. The local density measure for SDSS galaxies is taken from a volume-limited sample of the CMU-Pitt DR4 Value Added Catalog. This method produces mock galaxy catalogs that reproduces the luminosity and color correlation function of the real sky. The created mock galaxy sample therefore provides a unique tool to assess the performance of the SDSS cluster-finding algorithms in terms of completeness and purity, as well as how the observables of the detected clusters correspond to dark matter halos assuming galaxy clusters do trace the underlying halo population in the universe. Since precise spectroscopic redshift measurements are only available for the SDSS main sample galaxies [@Str02] and LRGs [@Eis01], we must use photometric redshift estimates for most of the galaxies. In order to accurately reproduce this scenario in the simulations, we scatter the given redshifts of mock galaxies according to the error distributions of photometric redshift estimates, which are obtained by calibrating a sample of $\sim$140,000 SDSS photometric redshifts to their known corresponding spectroscopic measurements coming from the SDSS spectroscopic survey and various other sources such as CNOC2 [@Yee00], CFRS [@Lil95a], DEEP [@Wei05], and 2SLAQ LRG [@Pad05]. The photometric redshifts were computed using a neural network technique by [@Lin07] and in preparation; see also the short discussion in the SDSS DR5 data release paper, [@AMc07]. The comparison between calculated photometric redshifts and measured spectroscopic redshifts is shown in Figure 1 for both the red and blue galaxy samples. The distributions of sampled redshift uncertainties are derived for different magnitude and redshift bins, and found to be well described by a combination of multiple Gaussian fits as shown in Figure 2 for examples. The resulted fitting parameters are used for the scattering of mock galaxy redshifts in the simulation. In the case of applying the cluster-finding technique to the real SDSS data, however, instead of deriving “empirical” error estimates collectively, we would use the photo-z errors that are computed based on the Nearest Neighbor Error estimate method (NNE) [@Lin07], which makes it possible to get an estimate of the error for each individual object. This would better constrain the photometric redshift uncertainty, especially for galaxy samples with photo-z errors depending strongly on magnitudes and the actual redshifts. We find the computed errors correspond reasonably well with the empirical ones derived from statistics, with exceptions only for the catastrophic objects. More details would be discussed in a subsequent paper on the application of the modified adaptive matched-filter technique with SDSS data. To summarize, the implementation of simulating the observed galaxy redshifts in the mock sample proceeds as follows: for galaxies that satisfy the SDSS spectroscopic target selection criteria we take the given galaxy redshifts as spectroscopic measurements, while for the rest of the sample we use the scattered redshifts to mimic the photometric redshift estimates. As discussed above in §\[sec:Algorithm\], there are a few percent of such galaxies that fall into the redshift and magnitude ranges where we find no good calibrations are available. For these galaxies we just treat them as if there is no redshift information at all to put into the algorithm. We also impose to the mock galaxy catalog an apparent magnitude cut ($r<21$) as we intend to adopt in the SDSS imaging sample. The procedure described above thus provides the a mock catalog with the most similar characteristics to the SDSS survey and it will allow us to explore the performance of the cluster-finding algorithm on real SDSS data. The modified matched filter algorithm is then run on the mock galaxy catalog, and the detected clusters are compared with matched known halos given in the simulation. We find that the matches are generally robust against details of the matching techniques, as pointed out by @Mil05 [although see also the discussion of various matching algorithms in @Roz07]. Here we adopt a matching criterion of projected separation between the detection and the candidate halo within the virial radius $r_{200}$ and redshift difference $\Delta z<0.05$. To evaluate completeness of the cluster sample, we match each dark halo to the nearest detected cluster within the projected cluster $r_{200}$ and $\Delta z$ of 0.05, while in measurement of purity, we match clusters to their corresponding halos applying the same criteria. In the case of multiple matches which are possible for above matching algorithms, we simply assign the most massive halo within the searching space as the real match. Other methods have also been tried in efforts to refine the matching process, but no significant changes are found in the final results. Results and Discussions {#sec:Results} ======================= In this section we present the results of running the modified adaptive matched-filter algorithm on the simulation-based mock catalogs. These include the completeness and purity check of the detected cluster sample, the derived cluster properties such as estimated redshift and richness, and the expected scaling relations that would link the observed clusters to true halo distributions. Completeness and Purity Check {#sec:Completeness} ----------------------------- We define the completeness $C$ of the selected cluster sample as a cumulative function of $M_{200}$, the mass within the virial radius inside which the overdensity is 200 times the critical density: $$C(M_{200}) = {N_{found} \over N_{total} }$$ where $N_{found}$ is the number of halos with mass greater than $M_{200}$ matched to clusters and $N_{total} $ is the total number of halos above that mass. Figure 3 shows the completeness of the detected cluster sample as a function of redshift and the virial mass of matched dark matter halos, respectively. The cluster sample, which has a richness cut at $\Lambda_{200} > 20$, is over $95\%$ complete for objects with $M_{200} > 2.0\times10^{14} h^{-1} M_\odot$ and $\sim85\%$ complete for objects with masses above $1.0\times10^{14} h^{-1} M_\odot$ in the redshift range of $0.05<z<0.45$. As we will find in the subsequent discussion of cluster scaling relations, the richness cut we impose on the cluster sample contributes to some of the incompleteness for less massive objects because of the large scatter in the cluster richness-mass relation; many of the matched clusters at $\sim 1.0\times10^{14} h^{-1} M_\odot$ are simply scattered below the richness cut and thus not counted to compute the completeness. This can be for sure relieved by lowering the richness cut of the cluster sample, although we choose to stick to this cut for the purity considerations below. Also from Figure 3a, the completeness level of the cluster sample remains almost flat out to $z\sim0.45$, beyond which it suffers a significant decline. This is at least partly due to the volume limit of the mock catalog which only extends to $z=0.58$. When we scatter the given galaxy redshifts with photometric redshift errors, which become large around $z\sim0.5$, many of the galaxies near the far edge of the light cone are scattered away while fewer galaxies would be shifted into that range, since they are absent from the simulation. The apparent magnitude cut we have applied to the mock galaxy sample may also contribute to incompleteness at high redshift. Taking into consideration the necessary $k$-corrections, the galaxy sample is no longer complete down to the luminosity of $0.4 L^*$, which is the limit assumed throughout the simulation tests. The matched filter therefore loses some power in detecting less rich systems at redshifts of $z\sim0.5$ and beyond since many fewer galaxies would be bright enough to be observable at that distance in the current survey. We have not investigated these effects in detail, though the onset of clear incompleteness corresponds well to the distance at which they become important. We similarly define the purity P of the selected cluster sample as a cumulative function of cluster richness $\Lambda_{200}$ which is the total cluster luminosity in units of $L^*$ inside its virial radius $r_{200}$ $$P(\Lambda_{200}) = { N_{match} \over N_{tot,\Lambda}},$$ where $ N_{match} $ is the number of clusters with richness greater than $\Lambda_{200}$ matched to halos and $N_{tot,\Lambda}$ is the total number of clusters with richness above $\Lambda_{200}$. The results of the purity check for the obtained cluster catalog are shown in Figure 4. The sample is over $95\%$ pure for clusters with $\Lambda_{200} > 30$ and around $90\%$ pure for clusters with $\Lambda_{200} > 20$ over the whole redshift range out to $z\sim0.45$. As will be shown in the richness-mass relationship below, these two thresholds in richness correspond to $M_{200} \sim 6.0\times10^{13} h^{-1} M_\odot$ and $M_{200} \sim 4.0\times10^{13} h^{-1} M_\odot$, respectively. It is worth to be noted that the lower purity for $\Lambda_{200} > 20$ is clearly going to be affected by halo incompleteness in the simulation, since some of the matched halos for this richness will fall below the mass resolution of the halo catalog, which means the purity we have derived above is in fact probably a lower limit, in similar logic to the completeness arguments. To ensure a reasonably high purity of selected clusters, we therefore apply a $\Lambda_{200} > 20$ cut for the cluster catalog, which is used for analysis of completeness as well as cluster derived properties and scaling relations. The purity measurement shows a slight but notable uptrend in the last redshift bin of $z\sim0.45-0.5$, which could be similarly explained by the arguments above in the completeness discussions. This reflects a shift in the richness-mass scaling relation at high redshift end where clusters with the same richness measurements may correspond to actually richer and more massive systems because of the under-representation of galaxies that are observable in that redshift range. It is therefore wise to limit the current cluster catalog to a redshift of $z=0.45$ in order to extract a uniform sample for statistical use, though the catalog using real SDSS data may well go deeper reliably. Derived Cluster Properties and Scaling Relations {#sec:Scaling} ------------------------------------------------ As is discussed in §\[sec:Algorithm\], for each selected cluster a redshift estimate is found for the system by the matched filter that optimizes the detection likelihood at the given galaxy position as cluster center. This measurement is then taken as the estimated redshift for the cluster. Since all the halos have known redshifts in the simulation, by matching the detected clusters to halos following the procedure described in §\[sec:Simulation\] we can compare the derived cluster redshifts with the true redshifts of associated halos. Figure 5 illustrates the comparison between estimated cluster redshifts and known halo redshifts. For clusters with redshifts below $z=0.25$ where spectroscopic redshift measurements are often available for member galaxies, the derived cluster redshift estimates precisely reproduce the true redshifts of corresponding dark halos. The inclusion of spectroscopic information of input galaxies markedly sharpens the cluster detection likelihood in the line-of-sight dimension and thus provides accurate measurements of the cluster redshifts. In the higher redshift range where spectroscopic measurements become rare and photometric estimates dominate, the plot illustrates a larger dispersion while the matched filter still gives robust determinations of cluster redshifts even with only photometric galaxy redshift information for inputs. We find that the accuracy of the redshift estimates does incease with cluster richness as expected, which is albeit mostly accounted by higher fraction of cluster galaxy members with spectroscopic measurements inside these systems. There is a slight uptrend bias seen at the redshift of $z\sim0.45$, which we see as a similar indication of incompleteness of the input galaxy sample near the high end of the redshift range for this mock catalog because of the volume limit and magnitude cut. The estimated cluster redshift determined from maximum likelihood tends to drift towards smaller values in some cases since the detection probability at higher redshift is suppressed by such effects. We also note the existence of a few serious outliers, which probably represent the occasional scenario when there exists a mismatch between relevant clusters and dark halos due to the projection effects or false positive detections. The normalized cluster richnesses $\Lambda_{200}$ are also compared with the virial mass $M_{200}$ of matched halos. The results are shown in Figure 6. We find that the richness-mass scaling relation follows $$\Lambda_{200} = (47.2\pm4.1)\times \left( M_{200} \over {10^{14} h^{-1} M_\odot} \right )^{1.03\pm0.04} ,$$ which is roughly a linear fit. Whether this is correct or not, clearly, depends upon the details of the simulation input, and the way the simulation was constructed gives no easy clue to what the results should be. What is important in this test, however, is that we recover what is present in the simulations, not what might or might not be present in the real universe. To that end, we have constructed three more plots. The first, Figure 7, compares the cluster richness determined by the present algorithm with the total three dimensional luminosity of the matched halos; the agreement is very good, with no bias evident at either the sparse or the rich end. Given this agreement and the results of Figure 6, the next plot, Figure 8, of the 3-D halo luminosity vs the 3-D halo mass, contains no surprises. The simulated halo mass is, in fact, linear with its total luminosity, and we recover this relationship. Figure 9 compares the derived cluster virial radius $r_{200}$ from the cluster-finding algorithm and the $r_{200}$ determined from 3-dimensional galaxy overdensities. The agreement is excellent at small virial radii, though there is a strong hint that the algorithm slightly overestimates large virial radii, by seven percent or thereabouts. This is almost certainly due to the assumption of a single NFW profile to describe the cluster; neighboring halos have rather different effects in the cylinder to which the algorithm is sensitive and the corresponding sphere in the simulations, but it is gratifying that the effects are this small. These results further justify our choice to refer our richness measurements to the commonly-used virial radius determined from galaxy overdensities. It is, however, clear that the scatter in the richness–mass relation derived from the cluster finding algorithm (Figure 6) is somewhat larger than that of the intrinsic richeness-mass relation in the simulations (Figure 8), which can be read as an indication of complications in the cluster-halo matching process, e.g., the inevitable difference between the cluster finder and halo finder regarding fragmentation and merging, differing shapes between the galaxy and mass distributions, and, even further, the variable mass-to-light ratios inside the systems incorporated in the current dark matter simulations. Despite these intrinsic dispersions, the richness-mass scaling relation shows a strong linear correspondence between the observables and the mass, and thus makes it possible to extract the true halo distribution in the Universe from the observed cluster abundance and correlation functions. It is important to note that the simulation from which the catalog was made is a dark-matter-only simulation, and thus effects which may well exist in real clusters and can affect the baryon fraction in the intracluster gas and galaxies (see, for example, [@Kra05]) as a function of cluster mass are absent here, but the fact that we recover the relation found from input 3-D simulations, here just linear, indicates that we should be able to investigate a possibly more complex relationship in the real universe. Conclusions {#sec:Conclusion} =========== We present a modified matched filter algorithm which is designed to construct a comprehensive cluster catalog from the Sloan Digital Sky Survey, but is applicable to any deep photometric survey. The technique is fully adaptive to 2-D, 2$\half$-D and 3-D optical surveys, as well as to various cluster scales and substructures. The cluster-finding algorithm has been tested against a realistic mock SDSS catalog from a large N-body simulation. The results suggest that the selected cluster sample is $\sim 85\%$ complete and over $90\%$ pure for systems more massive than $1.0\times10^{14} h^{-1}$ M$_\odot$ with redshifts up to $z=0.45$. The estimated cluster redshifts derived from maximum likelihood analysis show small errors with $\Delta z < 0.01$, and the normalized cluster richness measurements fit linearly with the virial mass of matched halos, the correct relation in this simulation. This offers hope that the (very likely nonlinear) relation between richness and halo mass which exists in the real universe can be investigated with these techniques. F.D. thanks H. Lin, H. Oyaizu, and the SDSS photo-$z$ group for providing the photometric redshifts which allowed us to derive the statistics of the photo-$z$ calibration to the spectroscopic redshifts. E.P. is an ADVANCE fellow (NSF grant AST-0649899), also supported by NASA grant NAG5-11489. RHW was supported in part by the U.S. Department of Energy under contract number DE-AC02-76SF00515. This research used computational facilities supported by NSF grant AST-0216105.

--- abstract: 'Superfluidity and Bose-Einstein condensation are usually considered as two closely related phenomena. Indeed, in most macroscopic quantum systems, like liquid helium, ultracold atomic Bose gases, and exciton-polaritons, condensation and superfluidity occur in parallel. In photon Bose-Einstein condensates realized in the dye microcavity system, thermalization does not occur by direct interaction of the condensate particles as in the above described systems, i.e. photon-photon interactions, but by absorption and re-emission processes on the dye molecules, which act as a heat reservoir. Currently, there is no experimental evidence for superfluidity in the dye microcavity system, though effective photon interactions have been observed from thermo-optic effects in the dye medium. In this work, we theoretically investigate the implications of effective thermo-optic photon interactions, a temporally delayed and spatially non-local effect, on the photon condensate, and derive the resulting Bogoliubov excitation spectrum. The calculations suggest a linear photon dispersion at low momenta, fulfilling the Landau’s criterion of superfluidity . We envision that the temporally delayed and long-range nature of the thermo-optic photon interaction offer perspectives for novel quantum fluid phenomena.' author: - Hadiseh Alaeian - Mira Schedensack - Clara Bartels - Daniel Peterseim - Martin Weitz bibliography: - 'ref.bib' title: 'Thermo-optical interactions in a dye-microcavity photon Bose-Einstein condensate' --- Introduction ============ For a gas sufficiently cold and dense that the thermal de Broglie wavelength exceeds the interparticle spacing, quantum statistical effects come into play. Specifically, for massive bosonic particles, Bose-Einstein condensation into a macroscopically populated ground state minimizes the free energy, as has been experimentally demonstrated in the gaseous regime with ultracold atoms more than 20 years ago [@Ketterle02]. More recently, Bose-Einstein condensation has also been observed with exciton-polaritons, mixed states of matter and light, and with photons [@Carusotto13; @Klaers101; @Klaers11; @Marelic15]. Unlike particles with a non-vanishing rest mass, photons usually do not show Bose-Einstein condensation. The thermal photons of blackbody radiation have no chemical potential, corresponding to a non-conserved particle number upon temperature variation. Therefore, blackbody radiation photons vanish at low temperature instead of showing a phase transition to a condensate. This difficulty was overcome in 2010 by confining photons in a dye-solution filled optical resonator made of two mirrors spaced in the micrometer regime [@Klaers101; @Marelic15]. The short mirror spacing effectively imprints a low-frequency cutoff for the photon gas, and in the presence of a mirror curvature the problem becomes formally equivalent to a two-dimensional system of harmonically confined massive bosons. By repeated absorption and re-emission processes, photons thermalize to the (rovibrational) temperature of the dye solution at room temperature. Experimentally, both the thermalization of the photon gas to the dye temperature [@Klaers10] and Bose-Einstein condensation of photons above a critical particle number have been observed [@Klaers101; @Marelic15]. For larger condensate fractions, the size of the condensate increases, which is attributed to a weak repulsive interaction mainly due to the thermo-optic effects. So far, it is not known if the photon gas, in addition to exhibiting Bose-Einstein condensation, is a superfluid [@Snoke13]. The existence of superfluidity is believed to require direct interparticle interactions, as e.g. present for polaritons, for which superfluidity has been established [@Carusotto04; @Amo09; @Nardin11; @Sanvitto11; @Amo11; @Grosso11]. On the theoretical side, the concept of a nonlinear photon fluid was first introduced by Brambilla et al. and Staliunas [@Brambilla91; @Staliunas93], who used hydrodynamic equations to describe electromagnetic fields in a cavity. Chiao et al. subsequently proposed to generate a photon superfluid in a nonlinear optical cavity using the Kerr-effect and furthermore predicted sound-like modes at the low-momentum part of the Bogoliubov dispersion [@Chiao99; @Chiao00]. In this paper, we examine the effect of thermo-optic interactions on the dispersion of a photon gas trapped inside an optical microcavity. The thermo-optic effect, also known as thermal lensing, was first introduced by J. Gordon while studying transient effects of the output power and the beam size upon inserting a liquid cell inside the laser cavity [@Gordon65]. In a propagating configuration in stationary conditions, the temporal delay of the thermo-optic effect does not play a role. The signatures of superfluidity and non-local effects from the associated thermo-optic nonlinearity have been experimentally observed in this configuration [@VOCKE15; @Vocke16]. In another work, Strinati and Conti have theoretically studied the stationary state of dye microcavity photon condensate subject to a non-local thermo-optic nonlinearity [@Strinati14]. We report a theoretical study on the effects of a thermo-optic nonlinearity giving rise to a temporally delayed and non-local effective photon interaction for the photon condensate in the dye-microcavity system. Assuming a plane mirror microcavity geometry, we derive the Bogoliugov dispersion for such a system subject to small perturbations. For a suitable parameter range, we find a linear dispersion at low momentum, corresponding to a phonon-type dispersion. We discuss the possibility of superfluidity of the photon condensate based on such temporally delayed and long-range interactions. In the following, Chapter II discusses some general properties of the photon gas subject to the thermo-optic interactions, and section III gives steady-state solution. In section IV we derive Bogoliubov modes of the system and the resulting elementary excitation dispersion. Finally, section V concludes the manuscript. Thermo-optic interactions in a photon gas ========================================= We begin by discussing some general formulas describing the system of a photon gas trapped in a dye microcavity subject to thermo-optic interactions. In the experimental scheme of [@Klaers101; @Klaers11], Bose-Einstein condensation of photons is achieved in a dye microcavity (see Fig. \[Fig1\]) with the mirror spacing in the wavelength regime. This leads to a large frequency spacing between longitudinal modes, comparable to the emission width of the dye molecules. Therefore, to good approximation only photons of a fixed longitudinal mode order $q$ are found in the resonator, and the two remaining transverse mode quantum numbers make the system two-dimensional. The transverse $TEM_{00}$-mode has the lowest allowed frequency, which imposes a low-frequency cutoff. Moreover, the photon dispersion becomes quadratic, i.e. massive particle-like, and the mirror curvature imposes a harmonic confinement on the photon gas. One can show that the photon gas in the cavity is formally equivalent to a two-dimensional gas of massive bosons with effective mass $m_{ph}=\hbar k_zn_0/c = \hbar\omega_c (n_0/c)^2$ , where $\omega_c$ denotes the cutoff frequency, $k_z$ the longitudinal wavevector, and $n_0$ the refractive index of the solution. The photon energy in the paraxial limit is: $$\label{photon energy} E_{ph}\simeq m_{ph}c^2 + \frac{(\hbar k_r)^2}{2m_{ph}}+V_{trap}(x,y)+E_{int}~,$$ where $k_r$ denotes the transverse momentum and $V_{trap} (x,y)= \frac{1}{2}m_{ph}\Omega^2(x^2 + y^2)$ is the trapping potential. The trapping frequency of the harmonic potential is given by $\Omega = (c/n_0)/\sqrt{LR/2}$, where $L$ and $R$ are the mirror spacing and curvature, respectively. Finally $E_{int}$ is the effective photon interaction energy, as will be discussed below. Thermal equilibrium of the photon gas is achieved by repeated absorption and re-emission processes by the dye molecules. For the described two-dimensional, harmonically confined system it is known that a Bose-Einstein condensate exists at finite temperature. Accounting for the two-fold polarization degeneracy of photons, one finds the critical particle number: $$\label{photon critical number} N_c = \frac{\pi^2}{3}(\frac{k_BT}{\hbar\Omega})^2~.$$ For $\Omega = 2\pi \times 3.6\times 10^{10}~Hz$, as derived for $L=2~\mu m$, $R=1~m$, and $n_0=1.33$, at room temperature ($T=300^o~K$) one obtains $N_c\approx99000$ for the critical particle number. During the course of the absorption re-emission processes of the dye molecules, a small fraction of inelastic processes due to dye’s finite quantum efficiency ($\eta\simeq 95\%$ for the case of rhodamine dye solution) causes local heating of the solvent. Due to the temperature dependence of the solution refractive index, the optical distance between the mirrors is decreased at the corresponding transverse position in the cavity. This is equivalent to a local rise of the photon gas potential. In other words, the heating with a corresponding decrease of the refractive index results in a smaller optical wavelength, hence a higher photon energy is required to locally match the mirror boundary conditions. The resulting interaction energy is: $$\label{interaction energy} E_{int}\simeq -m_{ph}(\frac{c}{n_0})^2\frac{\Delta n}{n_0}~,$$ where $\Delta n =\beta \Delta T$ , with $\beta = \partial n/\partial T$ as the thermo-optic coefficient of the solution. The photon gas is well described by an equilibrium state, if thermalization by coupling to the dye molecules is faster than both loss and pump processes. In this case, photons can relax towards the ground state of the harmonic trapping potential and form a Bose-Einstein condensate before they are lost through mirror transmission or inelastic processes in the dye [@Kirton13; @Schmitt15]. Accounting for the thermo-optic effective photon interactions, the condensate dynamics can be described with a generalized time-dependent Gross-Pitaevskii equation: $$\label{generalized GPE} i\hbar\frac{\partial\psi(\vec{r},t)}{\partial t}=\Bigg(-\frac{\hbar^2\nabla^2}{2m_{ph}} - m_{ph}(\frac{c}{n_0})^2\frac{\beta}{n_0}\Delta T(\vec{r},t)\Bigg) \psi(\vec{r},t)~.$$ In the above equation $\psi(\vec{r},t)$ denotes the slowly varying envelope of the condensate wavefunction (in the mean-field treatment) in the rotated frame of $\omega_c$. Note that the above equation is more general and captures the full 3D behavior of the wavefunction as ($\psi(x,y,z,t)$). Also instead of using the effective trapping potential of eq. \[photon energy\], we impose Dirichlet boundary conditions on the spherical mirror surfaces. Similar as in cold atom physics literature [@Brachet12; @Salazar13], here we do not consider interaction effects of the thermal cloud, due to its much lower density. The time evolution of the relative local temperature is determined by the heat transport equation as: $$\label{heat transport equation} \frac{\partial \Delta T(\vec{r},t)}{\partial t} = \frac{\kappa}{C_v}\nabla^2\Delta T(\vec{r},t) + \frac{\alpha_{in} c \hbar\omega}{C_v n_0}|\psi(\vec{r},t)|^2~,$$ The first term on the right hand side accounts for the heat diffusion and the second term for heating through inelastic processes in the dye solution. The parameters $\kappa$ and $C_v$ are the thermal conductivity and volume heat capacity, respectively. Further, $\alpha_{in}$ is the inelastic absorption coefficient and is related to the absorption coefficient $\alpha$ via the quantum efficiency of the dye $\eta$, as $\alpha_{in} = (1-\eta)\alpha$. Moreover, the following normalization condition for the wavefunction is held: $$\label{WF normalization condition} \int_V dv|\psi(\vec{r})|^2= N_{BEC}~,$$ where $N_{BEC}$ is the total number of photons in the condensate. Table \[Table1\] gives relevant parameters of a typical dye-filled cavity setup with methanol solvent [@Schmitt15], [@Lusty87]. \[parameter table\] $L (\mu m$) $\alpha_{in} (m^{-1})$ $n_0$ $\beta=\frac{dn}{d\tilde{T}} (K^{-1})$ $C_v (J K^{-1} m^{-3})$ $\kappa (W m^{-1} K^{-1})$ ------------- ------------------------ ------- ---------------------------------------- ------------------------- ---------------------------- -- -- -- -- -- 2 0.63 1.33 -4.8$\times 10^{-4}$ 1.9$\times 10^6$ 0.168 : \[Table1\] List of physical parameters of a dye-filled cavity setup used in this paper. The properties of the solvent, methanol, are from. [@Lusty87] The value of the inelastic absorption coefficient is for 1mM solution of R6G in methanol solvent and is calculated from the experimental data in [@Schmitt15], assuming a quantum yield of $95\%$ for the dye [@Bindhu99]. Steady-State ============ In the steady state, the temperature is settled to a stationary value $T_{ss}(\vec{r})$, and the the time evolution of the condensate wavefunction is given by $\psi(\vec{r};t)=\psi_{ss}(\vec{r})e^{-i\mu t}$. The stationary forms of eq. \[generalized GPE\] and \[heat transport equation\] are: \[steady state equations\] $$\begin{aligned} \hbar\mu\psi_{ss}(\vec{r})=\Bigg(-\frac{\hbar^2\nabla^2}{2m_{ph}}-m_{ph}(\frac{c}{n_0})^2\frac{\beta}{n_0} T_{ss}(\vec{r})\Bigg) \psi_{ss}(\vec{r})~,\\ % \nabla^2T_{ss}(\vec{r})+\frac{\alpha_{in} c \hbar\omega}{n_0\kappa}|\psi_{ss}(\vec{r})|^2=0~.\end{aligned}$$ The mirrors ($M_{1,2}$ in Fig. \[Fig1\]) are macroscopically large. Moreover, their thermal conductivity exceeds that of the dye solution, so in all of the calculations we assume $\Delta T=0$ on the mirror surfaces. Numerical solution ------------------ In the presence of a mirror curvature, as required to obtain a trapping potential, the coupled sets of eq. \[steady state equations\] can only be solved numerically. Without a thermo-optic non-linearity (i.e. for $\beta=0$), the ground state of the harmonic trap is the lowest energy state. Following the usual convention we normalize the chemical potential to yield $\mu = 0$ in this interaction-less case. For earlier work investigating the steady state properties of a thermo-optic interaction in the harmonically trapped condensate inside a microcavity, see Ref. [@Strinati14]. Using a fully numerical algorithm we solved the above coupled non-linear equations to investigate the effect of non-linearity on the interacting condensate. The results for a symmetric cavity with mirrors of $R=1~m$ radius of curvature are shown by the solid lines in Fig. \[Fig2\]. In the absence of the thermo-optic interaction the Gaussian condensate radius is $r_{BEC} = \sqrt{1 \hbar/2m_{ph}\Omega} \simeq 6~\mu m$ for the quoted values of the cavity length and mirror radius of curvature. Panel (a) shows the variation of the condensate radius from the interaction-less case $\Delta r$ and the chemical potential $\Delta \mu$ as a function of number of photons in the condensate. As can be seen both of these parameters linearly increase as the photon number becomes larger, implying a larger condensate with a higher energy. This is consistent with the physical consequence of a repulsive interaction mediated by thermo-optic non-linearity. To clarify the behavior further, Fig. \[Fig2\](b) shows the maximum value of the temperature increase in the dye microcavity as a function of condensate photon number $N_{BEC}$. As can be seen the temperature monotonically increases with increasing number of photons, giving rise to a larger change of the refractive index, hence a stronger effective photon interaction. A comparison with experimental results is not straightforward since the condensate mode diameter increase reported in [@Klaers10] corresponds to the accumulative stationary value observed for a pulsed pump with certain repetition rate. The values here are nevertheless smaller than the experimentally observed ones. We attribute this discrepancy to the boundary conditions used for the temperature distribution, which imposes that $\Delta T$ vanishes at the mirror surfaces. A more realistic model in the presence of the thermo-optic non-linearity needs to account for the finite thermal-conductivity of the mirrors and include the thermal properties of both mirror layers as well as the substrate. However, we expect the dynamic properties of the condensate, as discussed in the following, to be less affected by the thermal properties of the mirrors since the local properties of the heat transfer will dominate in that case. Green’s function approach ------------------------- Aside from being computationally expensive, the fully numerical method fails to provide one with more physical insight about the problem. In this section we propose an alternative method by employing the Green’s function of the heat diffusion problem, making the analysis more efficient. Moreover, this form could be used further for elementary excitation studies as will be discussed in the next sections. The coupled eigenvalue problem of eq. \[steady state equations\] can be efficiently solved using Green’s function of heat diffusion problem when proper boundary conditions are applied. The Green’s function of the heat transfer problem will be determined as: \[Green’s function of heat\] $$\begin{aligned} \nabla^2G_{NL}(\vec{r};\vec{r}')=\delta(\vec{r}-\vec{r}')~,\\ G_{NL}(\vec{\rho}, z=\pm L/2;\vec{r}')=0~.\end{aligned}$$ For simplicity in our analytic calculations the mirrors are assumed to be flat at $z=\pm L/2$, and extended to infinity in the transverse plane ($xy$-plane). As eq. \[Green’s function of heat\] are isomorphic to an electrostatic problem, image theory can be used to find a closed-form Green’s function. Using proper positive and negative images at $(\pm 2nL+z')$ and $(\pm (2n+1)L-z')$ respectively, we obtain: $$\label{Green's function closed form} G_{NL}(\vec{r};\vec{r}')=-\frac{1}{4\pi}\sum_{n=-\infty}^{+\infty}(\frac{1}{\sqrt{|\vec{\rho}-\vec{\rho}'|^2+(z-z'+2nL)^2}} -\frac{1}{\sqrt{|\vec{\rho}-\vec{\rho}'|^2+(z+z'+(2n+1)L)^2}})~,$$ where $\rho=\sqrt{x^2+y^2}$ and $\rho'=\sqrt{x'^2+y'^2}$. When combined with the eigenvalue problem of the wavefunction in eq. \[steady state equations\] one obtains the following equation for the steady-state: $$\label{GPE-steady state} \hbar\mu\psi_{ss}(\vec{r})=\Bigg(-\frac{\hbar^2\nabla^2}{2m_{ph}}+m_{ph}(\frac{c}{n_0})^3\frac{\beta\alpha_{in} \hbar \omega}{n_0\kappa} \int_{V_c} dv G_{NL}(\vec{r};\vec{r}')|\psi_{ss}(\vec{r}')|^2\Bigg) \psi_{ss}(\vec{r})~.$$ Equation \[GPE-steady state\] has the form of a Gross-Pitaevskii equation, while the interaction term is different from the usual contact form. Here, the interaction potential has an integral form describing a non-local interaction given by the Green’s function, where the strength of the interaction potential at point $\vec{r}$ is related to the whole condensate wavefunction distribution. Figure \[Fig3\] shows the variation of the Green’s function $G_{NL}(\vec{r};0)$ as a function of $\rho$ in the transverse plane for different mirror separations. As can be seen at a fixed radial point the value of the Green’s function increases as the cavity length becomes larger. This behavior implies that the temperature changes from a 2D distribution to 3D in thicker cavities. In the limit of very large mirror spacing the Green’s function approaches a Coulomb-type long-range interaction given by $G_{NL}(\vec{r};\vec{r}')\approx1/|\vec{r}-\vec{r}'|$. This behavior can be clearly observed in Fig. \[Fig3\] where the thick cavity Green’s function ($L=10~\mu m$) and the asymptotic Columbic form (black dashed line) are in a good agreement. To compare the predictions of these two approaches, we have used the generalized Gross-Pitaevskii eq. \[GPE-steady state\] to calculate the stationary features of the condensate. Unlike the numerical method however, the Green’s function approach only allows for a treatment of a flat mirror geometry. The red dashed line with squares in Fig. \[Fig2\](a) shows the chemical potential variation for the homogeneous problem calculated with this method. Similarly, the dashed line with circles in Fig. \[Fig2\](b) shows the corresponding variation of the maximum temperature with the number of photons in the condensate. The obtained results from the Green’s function approach are in approximate agreement with the numerical results for the problem with curved mirrors, and show the same trend. We point out that an exact agreement is not expected here due to the different mirror geometries, with correspondingly different condensate mode profiles. In contrary to the case of atomic BECs where the range of interaction is strictly limited by the interaction type, the photon fluid shows a unique feature of possessing a tunable interaction range from a local form to a highly non-local gravitational type interaction in relatively thin and thick cavities, respectively. In practice, the requirement of photon Bose-Einstein condensation for a low-frequency cutoff imposes a maximum usable cavity length for corresponding experiments. Small perturbations and Bogoliubov modes ======================================== For weak perturbations the dynamics of the system can be found by assuming small fluctuations around the stationary solutions. We use the following ansatz to determine the Bogoliubov modes, where $\Omega$ denotes the frequency of the perturbations. \[small perturbation 1\] $$\begin{aligned} \psi(\vec{r};t)=(\psi_{ss}(\vec{r})+u(\vec{r})e^{-i\Omega t}+v^*(\vec{r})e^{+i\Omega^* t})e^{-i\mu t}~,\\ T(\vec{r};t)=T_{ss}(\vec{r})+\delta t(\vec{r})e^{-i\Omega t}+\delta t^*(\vec{r})e^{+i\Omega^* t}~.\end{aligned}$$ Inserting this ansatz into the original non-linear equations and neglecting the terms with orders higher than one in the perturbation, one can derive the following linear coupled equations for the small fluctuations $u(\vec{r}), v(\vec{r})$, and $\delta t(\vec{r})$: $$\label{small perturbation 2} \Omega\begin{bmatrix} u(\vec{r})\\ v(\vec{r})\\ \delta t(\vec{r}) \end{bmatrix} = \mathcal{L} \begin{bmatrix} u(\vec{r})\\ v(\vec{r})\\ \delta t(\vec{r}) \end{bmatrix}~,$$ where: $$\label{Bogoliubov matrix} \mathcal{L}= \begin{bmatrix} [(-\frac{\omega}{2}-\mu)-\frac{\omega}{n_0}\beta T_{ss}-\frac{c^2}{2\omega n_0^2}\nabla^2] & 0 & -\frac{\omega}{n_0}\beta\psi_{ss} \\ 0 & -[(-\frac{\omega}{2}-\mu)-\frac{\omega}{n_0}\beta T_{ss}-\frac{c^2}{2\omega n_0^2}\nabla^2] & \frac{\omega}{n_0}\beta\psi^*_{ss}\\ i\frac{\alpha n_0}{2C_vZ_0}\psi^*_{ss} & i\frac{\alpha n_0}{2C_vZ_0}\psi_{ss} & i\frac{\kappa}{C_v}\nabla^2\\ \end{bmatrix}~.$$ To begin with, we derive a solution that neglects the temporal delay of the temperature and assumes that it follows the time dependency of the condensate density $|\psi(\vec{r},t)|^2$. With this assumption the temperature change $\Delta T(\vec{r},t)$ is given by a diffusion-type equation, as for $T_{ss}$ in eq. \[steady state equations\], which yields a non-local, but instantaneous effective interaction. Later in this section we will modify this assumption to take into account the temporal delay of the temperature distribution due to the finite heat conductivity. For a translationally invariant problem, a plane wave ansatz of the form $e^{i\vec{k}\cdot\vec{r}}$ would be well suited to describe the spatial part of the excitations. More suited for this problem, given that the condensate wavefunction $\psi(\vec{r})$ is not spatially uniform, is the Green’s function approach, building upon the stationary eigenfunctions discussed in the previous section. The mirrors impose Dirichlet boundary conditions and break the translational invariance along the $z$-axis. Therefore, we approximate the longitudinal variation of the wavefunction as $\sqrt{2/L}~ sin(q\pi z/L)$, and define an effective 2D Green’s function in the transverse plane: $$\label{2D GF} G_{eff}^{2D}(\vec{\rho},\vec{\rho}') = \int dz dz' sin^2(\frac{q\pi z}{L}) sin^2(\frac{q\pi z'}{L}) G_{NL}^{3D}(\vec{r},\vec{r'}) %\\ \nonumber % =-\frac{1}{\pi L}\sum_{n_{odd}} (\frac{8q^2}{n\pi(n^2-4q^2)})^2 K_0(\frac{n\pi}{L}|\vec{\rho}-\vec{\rho}'|)~,\$$ where $K_0$ is the $0^{th}$-order modified Bessel function. As can be inferred, this effective 2D potential is translationally invariant in the transverse plane, and leads to a well-defined transverse momentum $\vec{k}$ for the elementary excitation. Therefore, the Bogoliubov dispersion of this system is properly defined for frequency $\Omega$ and the transverse momentum $\vec{k}$, and from eq. \[small perturbation 2\] is determined as: $$\label{non-local dispersion1} \Omega^2=\frac{k^2}{2m_{ph}}\bigg(\frac{\hbar^2k^2}{2m_{ph}} + (2\hbar\omega)^2|\psi_{ss}(0)|^2\frac{\alpha\beta c}{n_0^2\kappa}\hat{G}_{NL}(k)\bigg)~,\\$$ with $$\label{non-local dispersion2} \hat{G}_{NL}(k) = -\frac{2}{L}\sum_{n_{odd}} (\frac{8q^2}{n\pi (n^2-4q^2)})^2 (\frac{L}{n\pi})^2 \frac{1}{1+(\frac{L}{n\pi})^2k^2}~.$$ Here $\hat{G}_{NL}(k)$ is the Fourier transform of $G^{2D}_{eff}(\vec{\rho})$, and in the first formula $|\psi_{ss}(0)|^2$ can be approximated as: $$\label{psi_ss0} |\psi_{ss}(0)|^2\approx \frac{N_{BEC}}{\pi {r_{BEC}}^2L}~.$$ Equation \[non-local dispersion1\] gives the dispersion in the presence of the thermo-optic interaction. At large momenta, the first term on the right hand side dominates, yielding the usual particle-like quadratic dispersion of photons, see also eq. \[photon energy\]. In other words, when the momentum is large the interactions have negligible effect. The thermo-optic interactions impact the low-momentum part of the dispersion, and the corresponding effect is predominantly determined by the function $\hat{G}_{NL}(k)$, i.e. the Fourier transform of the Green’s function. For wavevectors $k\le k_z = q\pi/L$, within the range that the paraxial limit is fulfilled, $\hat{G}_{NL}(k)$ becomes almost $k$-independent, hence a linear tendency for low-momentum excitations is expected. So far we have ignored the explicit dynamics of the delayed temperature given by the $\dot{T}$-term in the heat equation. This effect can be taken into account by employing the proper Green’s function of the heat equation in eq. \[heat transport equation\] which depends on time as well as the position. This time-dependent Greens’ function modifies the aforementioned dispersion of eq. \[non-local dispersion1\] to a transcendental equation for the dispersion (i.e. $\Omega(k)$) after substituting $\hat{G}_{NL}(k)$ with $\hat{G}_{NLD}(\Omega,k)$ as: $$\label{delayed, non-local dispersion} \hat{G}_{NLD}(\Omega,k) = -\frac{2}{L}\sum_{n_{odd}} (\frac{8q^2}{n\pi (n^2-4q^2)})^2\times \frac{1}{(\frac{n\pi}{L})^2-i\Omega} \frac{1}{1+k^2\frac{1}{(\frac{n\pi}{L})^2-i\Omega}}$$ Equation \[delayed, non-local dispersion\] together with eq. \[non-local dispersion1\] gives the final results for the quasi-particle dispersion, including the effects of both non-locality and the delay in such thermo-optic interactions. As the form of the transcendental equation suggests, the delayed nature of the interaction leads to complex frequencies $\Omega$, implying an instability of the condensate. Compared to the non-local case only, this is the main qualitative modification of the temporal effect. We notice this behavior can be compared with the dynamical instability predicted and observed in polariton condensates [@Wouters07]. While the instability of the latter is due to the coupling of the condensate with the exciton reservoir, in this dye microcavity system the condensate instability occurs due to the thermal coupling to the dye solutions. At longer times, we expect that the thermo-optic interaction destroys the photon condensate for larger interaction strengths. Figures \[Fig4\](a),(b) show the real and imaginary part of the dispersion respectively for various number of photons in the condensate $N_{BEC}$. The thick, dotted black line in panel (a) gives the (quadratic) free-particle dispersion in the absence of interactions. As stated above, at large momenta this free particle behavior is approached, and the imaginary part asymptotically approaches zero (Fig. \[Fig4\](b)), leading to a stable condensate with a quadratic dispersion [^1]. The low-momentum behavior however, noticeably deviates from the non-interacting dispersion. The difference between the interacting photon fluid dispersion and the ideal one increases for larger photon numbers. Figure \[Fig4\](c) and (d) show the zoomed-in real and imaginary part of Bogoliubov dispersion at very low momenta. The linear dispersion behavior of Fig. \[Fig4\](c), accompanied with low imaginary value as in Fig. \[Fig4\](d), means that the low momentum quasi-particles behave like phonons, and move with a constant velocity $v_c$ in the photon condensate. This feature suggests that the photon BEC can potentially be a superfluid, a feature that can be better understood using the non-local effect of the Green’s function. In eqs. \[non-local dispersion1\] the effect of non-locality is implicit in $\hat{G}_{NL}(k)$ and $\hat{G}_{NLD}(k,\Omega)$. A good physical intuition can be established by studying two extreme cases for this effect. As demonstrated in Fig. \[Fig3\], the interaction range decreases in thin cavities. In such cavities the interaction ultimately reduces to a local one with $G_{NL}(\vec{r};\vec{r}')\approx \delta(\vec{r} - \vec{r}')$, hence $\hat{G}_{NL}(k) = 1$, leading to a linear dispersion. Therefore, for a contact interaction the dispersion is separated to two distinct forms, one for free particles at large momenta ($\Omega \propto k^2$) and one for sonic modes for small momenta ($\Omega \propto k$). The other extreme happens for thick cavities where the fluid becomes fully 3D with a long range, gravitational-type interaction where $G_{NL}(\vec{r};\vec{r}')=1/|\vec{r}-\vec{r}'|$. With $\hat{G}_{NL}(k)=1/k^2$, this Green’s function would lead to a constant, $k$-independent dispersion at low momenta. To provide a better understanding of the dependency of the dispersion in terms of physical parameters such as $L,N_{BEC}$, and $\alpha_{in}$, we define a critical momentum $k_{critical}$, as the largest momentum at which the dispersion is significantly modified by the interaction. Using the above given dispersion relations this parameter is determined as: $$\label{critical momentum} k_{critical} \le \frac{2\omega}{n_0}|\psi_{ss}(0)|\sqrt{2\frac{\alpha_{in} \beta c m_{ph}}{\kappa}\hat{G}(k_{critical})}~.$$ Where $\hat{G}$ is a general representation for the Green’s function and could be either of $\hat{G}_{NL}(k)$ for the non-local case, or $\hat{G}_{NLD}(\Omega,k)$ for the non-local and delayed one. Figure \[Fig5\](a)-(c) shows the behavior of the critical momentum as a function of some experimental parameters. The solid blue lines correspond to the predictions considering both delay and non-locality given by the Green’s function of eq. \[delayed, non-local dispersion\]. For comparison, the dashed red lines are obtained when only considering the non-locality as given by the Green’s function of eq. \[non-local dispersion2\]. While the critical wavevectors have similar dependencies to the experimental parameters, the inclusion of the temporal delay decreases the values. As discussed earlier, the inclusion of the temporal delay also results in complex eigenfrequencies $\Omega$ (Fig. \[Fig4\](b)), leading to condensate instabilities. The results of Fig. \[Fig5\](a) and (c) show that the critical wavevector increases with both inelastic absorption coefficient $\alpha_{in}$ and photon number $N_{BEC}$ with corresponding dependency $\sqrt{\alpha_{in}}$ and $\sqrt{N_{BEC}}$, respectively. As shown in Fig. \[Fig5\](b) the critical wavevector decreases for larger mirror separations $L$, when the interaction becomes more non-local. According to Fig. \[Fig4\](c), the thermo-optic interaction leads to a linear dispersion at low momenta, indicating the existence of sonic modes in this regime, which fulfills the Landau’s criterion of superfluidiy. The slope of this line at low energies determines the velocity of sound in the condensate. Figure \[Fig5\](d)-(f) shows the dependency of this critical velocity $v_c$ on the inelastic absorption coefficient, the cavity length, and the number of photons in the condensate, respectively, showing that the critical velocity increases with an increase in the inelastic absorption and photon number and decreases with increasing the cavity length. conclusions =========== In this work we investigated the effect of a thermo-optic interaction, a temporally delayed and spatially non-local interaction, on a photon Bose-Einstein condensate in a dye-filled microcavity system. We derived the general form of the dispersion in such systems, calculated the spectrum of Bogoliubov modes in a plane-mirror microcavity, and identified a linear scaling for the low-energy modes. At larger transverse momenta, the usual quadratic free-particle dispersion of cavity photons is restored. The derived linear dispersion, corresponding to sonic modes, fulfills Landau’s criterion for superfluidity. We envision several experimental and theoretical follow up studies based on the reported results here. To obtain accurate quantitative predictions, it would be important to investigate the implications of a finite heat conductivity of the mirrors on both the stationary and the dynamical features of the condensate subject to the thermo-optic non-linearity. For an experimental test of superfluidity based on the thermo-optic non-linearity a possible setup could use a photon fluid in a wedge-shaped cavity composed of two tilted flat mirrors, allowing the photon droplet to flow freely. By intentionally making a perturbation on one of the mirrors in the flowing path of the photon fluid, scattering would be observed if the fluid is dissipative. However, in the superfluid phase the photon droplet passes the perturbation without being scattered. By directly monitoring the photon condensate-defect interaction one should be able to distinguish between these two different phases. Another follow up work could search for long-lived vortices of the photon condensate in a curved mirror microcavity. Along these lines, the implications of such a trapping potential on the Bogoliubov modes and the spectrum of elementary excitations should be studied. Another fascinating topic for future investigations is the search for analogies between the long-range thermo-optic interaction and gravitational physics and its consequences, including possible black-hole physics . acknowledgment {#acknowledgment .unnumbered} ============== We thank Julian Schmitt, Sebastian Diehl, Iacopo Carusotto, Axel Pelster, and Jan Klaers for insightful discussions. H. A. acknowledges financial support from the Alexander von Humboldt foundation in terms of a postdoctoral fellowship. Partial financial support of the research from the DFG (CRC 185) and the ERC (INPEC) is appreciated as well.      [^1]: For a momentum $k$, both $\Omega$ and $-\Omega^*$ are solutions of the dispersion equations. However as the behavior of these two branches are not different here we only show the results for the solution with positive real part.

--- abstract: 'In this work we study the fine-tuning problem in a general gauge theory with scalars and fermions. Then we apply our results to the Standard Model and its extension with additional singlet scalar field. The correlation between the Higgs mass and the scale at which new physics is expected to occur, is studied based on a fine-tuning arguments such as the Veltman condition.' author: - 'Aleksandra A. Drozd, University Of Warsaw' date: 'MSc Thesis, Warsaw, August 2010' title: 'RGE and the Fine-Tuning Problem' --- Introduction {#introduction .unnumbered} ============ The aim of this work is to investigate the fine-tuning problem. We will start with the RGE and quadratic divergences for a generic gauge theory and then apply our results to the Standard Model (SM) and the minimal SM extension. Then we will adopt the fine-tuning argument to estimate the range of allowed Higgs boson mass as a function of the UV cut-off $\Lambda$.\ Renormalization group equations {#renormalization-group-equations .unnumbered} =============================== The idea of the renormalization group is based on the arbitrariness of renormalization prescription. Renormalization procedure is based on expressing the parameters of our theory with help of physical quantities obtained from experiments. Unfortunately, Quantum Field Theory does not properly describe physics at very short distances, which results in divergences at almost every step of calculations at higher orders of perturbative expansion. To interpret such theory, one can introduce a procedure for regularization of divergences. There are very different renormalization and regularization schemes which give the same results, up to the specific order of perturbative calculations. A particularly useful type of changing the renormalization prescription is changing the mass scale parameter $\mu$. For example, the parameter could be the renormalization point at which we define the value of the 1PI Green’s function. As a consequence of RGE, we have for a given physical theory, a definite values of coupling parameters as functions of the energy scale $\mu$. These are called running coupling constants and can be derived from specific differential equations (see section \[Calculating\_beta\_functions\]). Results of calulations of renormalization group functions are very useful and can be found in literature up to several loops order. Standard Model and its problems {#standard-model-and-its-problems .unnumbered} =============================== Physicists are able to describe the fundamental particles and their mutual interactions, with increasing accuracy. As for today, the Standard Model of particle physics is the best theory we have. It has passed almost all of the experimental challenges (except for neutrino oscillations) and is an excellent description of fundamental particles. It has been verified for example in LEP and SLC experiments. But the SM also contains very important gaps and problems. The main issue is the very existence of Higgs boson. It is not proven yet, but there is a lot of hope towards experiments in Large Hadron Collider, Geneva. Higgs boson existence would explain a fundamental problem of masses. Higgs mechanism, which is based on generating masses through a non-zero vacuum expectation value of a specific field, is a very simple and beautiful way of obtaining massive vector particles through symmetry breaking. There exists lots of variations to this idea, but the beauty of this basic concept challenges many scientists to look for a Higgs or Higgs-like particle in experiments. Higgs mass $m_H$ is the most commonly pointed out unknown parameter of Standard Model, but not the only one. If we assume that Standard Model is only a low-energy limit of a more fundamental theory (which does not necessarily have to be a quantum field theory) that could for example explain why the electroweak symmetry is broken. Other problems with the SM are the combined issues of fine-tuning and naturalness. In theoretical physics, fine-tuning refers to circumstances when the parameters of a model must be adjusted very precisely in order to agree with observations [^1]. The requirement of a fine-tuning in a theory is generally unwelcome by physicists, permissible with a presence of a mechanism to explain the precisely needed values. A so called, *little hierarchy problem* is a problem of fine-tuning of the Higgs boson mass corrections. For the SM energy scale much larger than the W boson mass, $\Lambda \gg m_{W}$, corrections to the Higgs mass should cancel each other to a very high precision in order for the mass to be in order of electroweak scale. A simple extension of Standard Model {#a-simple-extension-of-standard-model .unnumbered} ==================================== Standard Model is known to be a good approximation of fundamental interactions, but there are many attempts to extend this theory and get rid of the aforementioned problems. The very simplest extension of Standard Model is an addition of singlet scalar particle. Assuming interactions of $N_{\phi}$ singlet scalar $\phi_{n}$ particles and Higgs, a potential with a discrete $Z_{2}$ symmetry $\phi \rightarrow - \phi$ can be introduced: $$\begin{aligned} V(H, \phi_{n}) = -\mu^2 H ^\dagger H + \lambda (H ^\dagger H)^2 + \sum _{i} ^{N_{\phi}} \frac{\mu ^{i}_{\phi}}{2} \phi _ i ^2 + \sum _{i,j}^{N_{\phi}} \lambda_{\phi}^{ij} \phi ^2 _{i} \phi ^2 _{j} + \sum _{i}^{N_{\phi}} \lambda_{x}^{i} (H ^\dagger H) \phi _{i} ^ 2 \label{scalar_potential}\end{aligned}$$ If $N_{\phi} = 1$ then this extension leaves us with three additional parameters $\lambda_{x}$, $\lambda_{\phi}$ and the additional particle mass.\ In this work we will discuss theoretical constraints on $m_H$ and $\Lambda$ due to the fine-tuning argument. The letter is organized as follows. The first chapter is about 1-loop renormalization of a general gauge theory with fermions and scalars. We will use the dimensional regularization scheme and calculate the RGE beta functions of such theory. In the second chapter we will concentrate on 1-loop corrections in cut-off regularization scheme to the general gauge theory. Third chapter is to present higher order corrections of the perturbative expansion using previously obtained results. In fourth chapter we will concentrate on the Higgs mass corrections and estimation of this parameter using the ’Veltman condition’ and the 2-loop fine-tuning. The fifth chapter presents results in a presence of an additional singlet scalar field in the model. Derivation of beta functions in a generic gauge theory with fermions and scalars ================================================================================ Lagrangian and the counterterms {#general_theory} ------------------------------- We will start our calculation with analysing the most general case: a gauge invariant Lagrangian of a theory with a number of real scalar fields $ \phi _{i}, i = 1,..., N_{\phi} $ and spin-$\frac{1}{2}$ fields $ \psi _{n} , n = 1,..., N_{\psi}$, with a single gauge symmetry and corresponding hermitian gauge fields $ A^{a}_{\mu} $. We adopt the $R_\xi$ gauge and $\eta_{a}$ stands for the ghost fields. Everywhere summation over repeated indices is assumed. $$\begin{aligned} \mathcal{L} & = & - \frac{1}{4} F^{a}_{\mu \nu} F^{a \mu \nu} - \frac{1}{2\xi} \left( \partial_{\mu} A^{\mu}_{a} \right)^{2} + \frac{1}{2} (D_{\mu} \phi)_{i}(D^{\mu} \phi)^{i} + \overline{\psi} ( \imath \gamma^{\mu} D_{\mu} - M ) \psi - \nonumber\\ & & + \partial _{\mu} \eta^{*}_{a} \left( \delta_{a b} \partial ^{\mu} \eta_{b} + g f_{a b c} \eta_{b} A_{c}^{\mu} \right) - \overline{\psi} \kappa^{i} \psi \phi_{i} - V(\phi) \label{lagrangian_general}\end{aligned}$$ where $$F^{a}_{\mu \nu} = \partial_{\mu} A^{a}_{\nu} - \partial_{\nu} A^{a}_{\mu} - g f_{a b c} A^{b}_{\mu} A^{c}_{\nu}$$ The covariant derivative of a field $ \phi $ and $\psi$ can be written as $$D_{\mu}\phi = \left( \partial_{\mu}+ \dot{\imath} g \textbf{T}^{a} A^{a}_{\mu} \right) \phi$$ $$D_{\mu}\psi = \left( \partial_{\mu}+ \dot{\imath} g \overline{\textbf{T}}^{a} A^{a}_{\mu} \right) \psi$$ where $ g $ is the group constant. In general, scalar and fermion fields can transform under different representations of the gauge group, so there are two different sets of generators $\textbf{T}^{a}$ and $\overline{\textbf{T}}^{b}$ for scalar and fermion fields respectively. For each $\phi _{i}$ scalar field and $\psi _{n}$ fermion field one can write the covariant derivative in form of: $$\begin{aligned} & & (D_{\mu}\phi ) _{i} = \partial_{\mu} \phi _{i} + \dot{\imath} g \textbf{T}^{a}_{i j} A^{a}_{\mu} \phi _{j} \\ & & (D_{\mu}\psi )_{n} = \partial_{\mu} \psi _{n} + \dot{\imath} g \overline{\textbf{T}}^{a}_{n m} A^{a}_{\mu} \psi_{m}\end{aligned}$$ Generators fulfil the following relations: $$\begin{aligned} & & [ \textbf{T}^{a}, \textbf{T}^{b} ] = i f_{a b c} \textbf{T}^{c} \\ & & C_{1} \delta_{a b} = f_{acd} f_{cdb} \label{C1}\\ & & \textbf{T}^{a}_{i j} \textbf{T}^{b}_{j i} = \textbf{Tr} (\textbf{T}^a \textbf{T}^b) = C_{2} (R) \delta_{a b} \label{C2} \\ & & (\textbf{T}^{a} \textbf{T}^{a})_{m n} = C_{3} \delta_{m n} \label{C3}\end{aligned}$$ where $f_{a b c}$ are the structure constants, group factors $C_{1}$ and $C_{3}$ depends only on the group we consider, while $C_{2}$ depends on specific representation $R$. All above equations can be simply written in terms of $\overline{\textbf{T}}^{a}$ generators.\ There are also some constraints on the couplings and generators which result from the hermiticity and gauge invariance of the Lagrangian, some of them will be discussed later.\ The potential $V(\phi)$ to consider is no more than quartic in $\phi$. We omit cubic and quadratic terms as they are not relevant hereafter. $$V(\phi) = \frac{1}{4!} h_{i j k l} \ \phi_{i} \phi_{j} \phi_{k} \phi_{l} + \textit{lower-order terms}$$ We will only consider real scalar fields case. For complex scalars it is always possible to rewrite the Lagrangian in terms of real degrees of freedom and re-evaluate the result. To proceed with the renormalization we write for the bare Lagrangian $$\mathcal{L}_{B} = \mathcal{L} + \Delta \mathcal{L}$$ where the $ \Delta \mathcal{L} $ is for the counter terms. We assume the form of the bare Lagrangian to be the same as in the renormalised Lagrangian but with the bare fields (like $\psi _{B}$ or $\phi _{B}$ ) and coupling constants (like $g_{B}$) replaced by the corresponding renormalised quantities.\ We will assume such relationships between bare and renormalised fields: $$\begin{aligned} (A^{a}_{\mu})_{B} & = & \left( 1+ \Delta Z_{A} \right) ^{1/2} A^{a}_{\mu} = Z^{1/2}_{A} A^{a}_{\mu} \label{niediagonalne_wektory}\\ (\eta^{a})_{B} & = & \left( 1+ \Delta Z_{\eta} \right)^{1/2} \eta^{a} = Z^{1/2}_{\eta} \eta^{a} \label{niediagonalne_duchy}\\ (\psi _{n})_{B} & = & \left( (1+ \Delta Z _{\psi})^{1/2} \right) _{n m} \psi _{m} = \left( Z^{1/2}_{\psi} \right) _{n m} \psi _{m} \label{niediagonalne_fermiony}\\ (\phi _{i})_{B} & = & \left((1+ \Delta Z_{\phi})^{1/2}\right)_{i j} \phi_{j} = \left( Z^{1/2}_{\phi} \right) _{i j} \phi _{j} \label{niediagonalne_scalary}\end{aligned}$$ Because Yukawa couplings in our considerations are hermitian, renormalization constant $Z_{\psi}$ is generally a complex matrix. $Z_{\phi}$ and $Z_{A}$ must be real for real scalar and real vector fields. Even at the 1-loop order renormalization, one needs to consider that there can be non-diagonal corrections to the propagators (see [@bouzas]). It was done in (\[niediagonalne\_fermiony\]) and (\[niediagonalne\_scalary\]). In the case of the gauge field, as one can see in later discussions, it happens that 1-loop corrections are purely diagonal and we assumed this in (\[niediagonalne\_wektory\]). We will not discuss later the corrections for the ghost field, but they are diagonal too, as in (\[niediagonalne\_duchy\]). Now we can write the counter terms for the Lagrangian (below only terms important for our calculations): $$\begin{aligned} \Delta \mathcal{L} & = & - \frac{1}{4} \Delta Z_{A} (\partial_{\mu} A_{a \nu} - \partial_{\nu} A_{a \mu}) (\partial^{\mu} A_{a}^{\nu} - \partial^{\nu} A_{a}^{\mu}) - \frac{K_{\xi}}{2\xi} \left( \partial_{\mu} A^{\mu}_{a} \right)^{2} \nonumber\\ & & + \overline{\psi} \Delta Z_{\psi} \ i \gamma^{\mu} \partial_{\mu} \psi - \overline{\psi} \gamma^{\mu} \Delta g \overline{\textbf{T}}^{a} \psi A^{a}_{\mu} - \overline{\psi} \Delta \kappa _{i} \psi \phi_{i} + \frac{1}{2} (\partial_{\mu} \phi) \Delta Z_{\phi} (\partial^{\mu} \phi) \nonumber\\ & & + \Delta Z_{\eta} \partial _{\mu} \eta^{*}_{a} \partial ^{\mu} \eta_{a} - \frac{1}{4!} \Delta h_{i j k l} \ \phi_{i} \phi_{j} \phi_{k} \phi_{l} + \ldots \label{counter}\end{aligned}$$ $\Delta g$, $\Delta \kappa_{i}$ and $\Delta h_{i j k l}$ will be specified further while $K_{\xi}$ is defined by the following relation: $$\begin{aligned} (\xi)_{B}^{-1} & = & \left( 1 + K_{\xi} \right) Z^{-1}_{A} \xi ^{-1} \label{Zetxi}\end{aligned}$$ Bare coupling constants dependence on the renormalized quantities and the renormalization constants in general have a complicated form, because of the previously mentioned non-diagonality of the corrections to the propagator. There are different expressions for $g_{B}$, depending on the vertex we consider, and they result in some relationships between renormalization constants. In later discussion we will consider only the $\overline{\psi} \psi A_{\mu} $ vertex to calculate the beta function of $g$ coupling. The formula for the bare coupling in terms of renormalized quantities is: $$\begin{aligned} g_{B} \overline{T}_{nm} ^{a} & = & (Z^{-1/2}_{\psi})^{\dagger}_{n n'} \left( (g + \Delta g)\overline{T}^{a} \right)_{n' m'} (Z^{-1/2}_{\psi})_{m' m}Z^{-1/2}_{A} \label{counter_g}\end{aligned}$$ If we expand the formula using (\[niediagonalne\_wektory\]) and (\[niediagonalne\_fermiony\]) we can get a relation as follows $$\begin{aligned} g_{B} \overline{T}_{nm} ^{a} &=& g \overline{T}_{n m} ^{a} - \frac{1}{2} (\Delta Z^{\dagger}_{\psi})_{n n'} g \overline{T}_{n' m} ^{a} - \frac{1}{2} g \overline{T}_{n n'} ^{a} (\Delta Z _{\psi})_{n' m} \nonumber\\ & & - \frac{1}{2} g \overline{T}_{n m} ^{a} \Delta Z_{A} + \left( \Delta g \overline{T}^{a} \right)_{n m} + \ldots \label{gB_all}\end{aligned}$$ As one can see, the general and complete relation between the bare and renormalized coupling is complicated - it includes not only the non-diagonal propagator corrections, but also the group generators. In section \[renormalization\_of\_ffb\] we show that after specific calculations all the non-diagonal contributions cancel each other at the 1-loop accuracy. We can use this fact to simplify our result. We define $\Delta Z_{\psi}$ as the non-cancelling part of the $\left( \Delta Z_{\psi} \right)_{m n}$, which (as we can see in section \[renormalization\_of\_ffb\]) happens to be a number multiplying the generator $\overline{T}^{a}$. Similarly, $\Delta \tilde{g} \, \overline{T}^{a}$ is the non-cancelling diagonal part of $\left( \Delta g \overline{T}^{a} \right)_{n m}$ from $\overline{\psi} \psi A_{\mu}$ vertex renormalization. And while the non-diagonal contributions cancel, the simplified equation for $g_{B}$ takes the form $$\begin{aligned} g_{B} = g - \frac{1}{2} g \, \Delta Z_{A} - g \, \Delta Z_{\psi} + \Delta \tilde{g} + \ldots \label{easy_gB}\end{aligned}$$ where $\Delta \tilde{g} = K_2 \, g$ with the renormalization constant $K_2$. For the beta function $\beta(g)$ calculations in section \[Calculating\_beta\_functions\] we will use the simplified formula. For the Yukawa coupling constant and quadrilinear couplings, non-diagonal terms are present in the calculations and don’t vanish. $$\begin{aligned} (\kappa^{i} _{n m})_{B} &=& \sum _{i' n' m'} \left( (Z^{\dagger}_{\psi})^{-1/2} \right)_{n n'} \left( Z^{-1/2}_{\psi} \right)_{m' m} \left( Z^{-1/2}_{\phi} \right)_{i' i} (\kappa^{i'}_{n' m'} + \Delta \kappa^{i'}_{n' m'})\label{ZetKappa1} \\ (h_{i j k l})_{B} &=& \sum _{i' j' k' l'} \left( Z^{-1/2}_{\phi} \right)_{i i'} \left( Z^{-1/2}_{\phi} \right)_{j j'} \left( Z^{-1/2}_{\phi} \right)_{k k'} \left( Z^{-1/2}_{\phi} \right)_{l l'} (h_{i' j' k' l'} + \Delta h_{i' j' k' l'}) \label{Zeth} \nonumber\\\end{aligned}$$ where we can express the $\Delta \kappa^{i} _{n m}$ and $\Delta h_{i j k l}$ as follows: $$\begin{aligned} \Delta \kappa^{i} _{n m} = \sum _{i' j' k' l'} K_{i n m}^{i' n' m'} \kappa^{i'} _{n' m'} \label{ZetKappa} \\ \Delta h_{i j k l} = \sum _{i' j' k' l'} L_{i j k l}^{i' j' k' l'} h_{i' j' k' l'} \label{ZetH}\end{aligned}$$ Renormalization of propagators ------------------------------ We will regularise the divergent integrals adopting the dimensional regularization (we set the number of dimensions to be $d = 4 - \epsilon $). Feynmann diagrams for a general gauge theory can be found in the Appendix A and remarks on calculating the symmetry factors can be found in [@palmer]. At every step of the calculation we mention only the diagrams that contribute in dimensional regularization. ### Two point function for a gauge field One can draw the one particle irreducible (OPI) diagrams contributing in the dimensional regularization to the gauge boson propagator as in fig. \[OPI\_gauge\_propagator\].\ We will do all the calculations step by step starting with the gauge fields loop (fig. \[2point\_boson\_1\]). Here the symmetry factor is $\frac{1}{2}$, so the boson self-energy contribution takes the form: $$(\texttt{diagram \ref{2point_boson_1}}) = \frac{1}{2} \mu^{\epsilon} g^{2} f_{acd} f_{bcd} \int \frac{d^{d} q}{(2\pi)^{d}} \tilde{D}_{F}^{\sigma \tau}(p+q) \tilde{D}^{\lambda \rho}_{F}(q) J_{\sigma \mu \rho \tau \lambda\nu} (p,q) \label{2point_boson_integral}$$ where $\tilde{D}^{\lambda \rho}_{F}$ is a gauge boson propagator and $$\begin{aligned} J_{\sigma \mu \rho \tau \lambda \nu} (p,q) & = & [(p-q)_{\sigma} g_{\mu \rho} + (2q+p)_{\mu} g_{\rho \sigma} -(2p + q)_{\rho} g_{\mu} \sigma] \times \nonumber\\ & & [(p-q)_{\tau} g_{\lambda \nu} - (2p+q)_{\lambda} g_{\tau \nu} + (2q + p)_{\nu} g_{\tau \lambda}] \end{aligned}$$ From (\[2point\_boson\_integral\]) after some calculations one can get the divergent term. $$(\texttt{diagram \ref{2point_boson_1}}) = \frac{i g^2 C_{1} \delta_{ab}}{16 \pi^2 \epsilon} \left[ \left( -\frac{11}{3} - 2\eta \right) p_{\mu} p_{\nu} + \left( \frac{19}{6} + \eta \right) p^2 g_{\mu \nu} \right]$$ where the group theory factor is defined in (\[C1\]) and $\eta = 1 - \xi $. For calculating the ghost fields loop, the symmetry factor is 1 and there is a minus sign because of the closed loop of Grassmann fields.\ $$(\texttt{diagram \ref{2point_boson_2}}) = (-1) g^{2} f_{dca} f_{cdb} \int \frac{d^{d} q}{(2\pi)^{d}} (p+q)_{\mu} q_{\nu} \tilde{G}_{F} (p+q) \tilde{G}_{F}(q)$$ where $\tilde{G}_{F}$ is a ghost propagator. Using (\[C1\]) the contribution from fig. \[2point\_boson\_2\] takes a form $$(\texttt{diagram \ref{2point_boson_2}}) = - g^{2} C_{1} \delta_{a b} \int \frac{d^{d} q}{(2\pi)^{d}} \frac{(p+q)_{\mu} q_{\nu}}{(p+q)^2 q^2}$$ with the final result of $$(\texttt{diagram \ref{2point_boson_2}}) = \frac{i g^2 C_{1} \delta_{ab}}{16 \pi^2 \epsilon} \left( \frac{1}{3} p_{\mu} p_{\nu} + \frac{1}{6} p^2 g_{\mu \nu} \right)$$\ For calculating the contribution from fermion fields, one has as before loop symmetry factor 1 and a minus sign for the closed loop of Grassmann fields.\ $$(\texttt{diagram \ref{2point_boson_3}}) = - \mu^{\epsilon} g^{2} \overline{C}_{2} \delta_{a b} \int \frac{d^{d} q}{(2\pi)^{d}} \textbf{Tr} \left( \gamma_{\mu} \tilde{S}_{F}(q) \gamma_{\nu} \tilde{S}_{F} (p+q) \right)$$ where (\[C2\]) was adopted and $\tilde{S}_{F}$ denotes the fermion propagator. To extract the pole term the easiest way, one can put fermion masses to zero and then obtain $$(\texttt{diagram \ref{2point_boson_3}}) = - \frac{i g^2 \overline{C}_{2} \delta_{ab}}{16 \pi^2 \epsilon} \frac{8}{3} \left( - p_{\mu} p_{\nu} + p^2 g_{\mu \nu} \right)$$ Calculating the scalar fields loop include symmetry factor $\frac{1}{2}$. $$(\texttt{diagram \ref{2point_boson_4}}) = - \frac{1}{2} g^{2} \textbf{T}^{a}_{i j} \textbf{T}^{b}_{j i} \int \frac{d^{d} q}{(2\pi)^{d}} (2q+p)_{\mu} (2q+p)_{\nu} \tilde{D}_{F}(q)\tilde{D}_{F}(p+q)$$ Using the group theory factor from (\[C2\]) and simplifying, one can get the formula $$(\texttt{diagram \ref{2point_boson_4}}) = \frac{1}{2} g^{2} C_{2} \delta_{ab} \int \frac{d^{d} q}{(2\pi)^{d}} \frac{(2q+p)_{\mu} (2q+p)_{\nu}}{(p+q)^2 q^2}$$ and a final result of: $$(\texttt{diagram \ref{2point_boson_4}}) = - \frac{i g^2 C_{2} \delta_{ab}}{16 \pi^2 \epsilon} \frac{1}{3} \left( -p_{\mu} p_{\nu} + p^2 g_{\mu \nu} \right)$$ From those results we can calculate the $\Delta Z_{A}$ and $K_{\xi}$ renormalization constants: $$\begin{aligned} && \dot{\imath}\Delta Z_{A} (-p^2 g_{\mu \nu} + p_{\mu} p_{\nu}) - \dot{\imath} \frac{1}{\xi} K_{\xi} p_{\mu} p_{\nu} = \frac{- \dot{\imath} g^2}{16 \pi^2 \epsilon} \times \nonumber\\ && \times \left[(- p^2 g_{\mu \nu} + p_{\mu} p_{\nu})\left[\frac{8}{3}\overline{C}_{2}+(-\frac{10}{3} - \eta) C_{1} + \frac{1}{3} C_{2} \right] -(\eta C_{1} + 4 C_{2} ) p_{\mu} p_{\nu} \right] \nonumber\end{aligned}$$ $$\begin{aligned} \Delta Z_{A} &=& \frac{- g^2 }{16 \pi^2 \epsilon} \left[(-\frac{10}{3} - \eta) C_{1} +\frac{8}{3}\overline{C}_{2} + \frac{1}{3} C_{2} \right] \label{ZA} \\ K_{\xi} &=& - \frac{\xi g^2}{16 \pi^2 \epsilon} \left( \eta C_{1} + 4 C_{2} \right)\end{aligned}$$ ### Two point function for a fermion field Only two diagrams contribute to the renormalised propagator at the 1-loop accuracy. They are shown in fig.\[OPI\_fermion\_propagator\]. $$(\texttt{diagram \ref{2point_fermion_1}}) = - g^{2} \overline{C}_{3} \delta_{m n} \int \frac{d^{d} q}{(2\pi)^{d}} \left( \gamma_{\mu} \tilde{S}_{F}(p+q) \gamma_{\nu} \right)_{ \beta \alpha} \tilde{D}_{F}^{\mu \nu}(q)$$ To calculate the pole term in (diagram \[2point\_fermion\_1\]) we put $m = 0$ in denominator and use the following identities: $$\begin{aligned} \gamma^{\mu} \gamma _{\mu} = d \textbf{I} \\ \gamma^{\mu} \gamma _{\rho} \gamma_{\mu} = (2-d) \gamma_{\rho}\end{aligned}$$ And after some simple calculations one can obtain the result of: $$(\texttt{diagram \ref{2point_fermion_1}}) = \frac{2 i g^2 \overline{C}_{3} \delta_{m n}(1 - \eta) }{16 \pi^2 \epsilon} \left( \slashed{p} \right) _{\beta \alpha}$$ For the scalar contribution we have no additional factors, so the pole term can be calculated from: $$(\texttt{diagram \ref{2point_fermion_2}}) = - \kappa^{i}_{m n'} \kappa^{i}_{n' n} \int \frac{d^{d} q}{(2\pi)^{d}} \left( \tilde{S}_{F}(p+q) \right)_{ \beta \alpha} \tilde{D}_{F}(q)$$ with the result of: $$(\texttt{diagram \ref{2point_fermion_2}}) = \frac{i (\kappa^{i} \kappa^{i})_{m n}}{16 \pi^2 \epsilon} \left( \slashed{p} \right)_{\beta \alpha}$$ Hence, using the Feynman rules for the counterterms from the Appendix \[fermion\_propagator\_counter\], one can evaluate the fermion propagator counterterm: $$\begin{aligned} \frac{1}{2} \left( \Delta Z_{\Psi}^{\dagger} + \Delta Z_{\Psi}\right)_{m n} = - \frac{2 g^2 \overline{C}_{3} \delta_{m n}(1 - \eta) }{16 \pi^2 \epsilon} - \frac{ (\kappa ^{i} \kappa ^{i})_{m n}}{16 \pi^2 \epsilon} \label{Zpsi}\end{aligned}$$ ### Two point function for a scalar field In diagram \[2point\_scalar\_1\] one has to include a $(-1)$ factor from the fermion closed loop. $$(\texttt{diagram \ref{2point_scalar_1}}) = \textbf{Tr} \left( \kappa^{i} \kappa^{j} \right) \int \frac{d^d q}{(2 \pi)^d} \textbf{Tr} \left( \tilde{S}_{F}(p+q) \tilde{S}_{F}(q) \right)$$ The important term for beta function calculations is the pole term proportional to the $p^2$, so by putting $m$ equal to zero, one can evaluate the pole term simpler and get the result of: $$(\texttt{diagram \ref{2point_scalar_1}}) = \frac{4 i p^2 \, \textbf{Tr} \left( \kappa^{i} \kappa^{j} \right)}{16 \pi ^2 \epsilon}$$ For the gauge boson contribution one gets: $$(\texttt{diagram \ref{2point_scalar_3}}) = - g^2 \textbf{T}^a_{ii'} \textbf{T}^a_{i'j} \int \frac{d^d q}{(2 \pi)^d} \left( 2p+q \right)_{\mu} \left( 2p+q \right) _{\nu} \tilde{D}_{F}(p+q) \tilde{D}^{\mu \nu}_{F}(q)$$ Using (\[C3\]) and simplifying one can get the result of: $$(\texttt{diagram \ref{2point_scalar_3}}) = - \frac{\dot{\imath} g^2 p^2}{16 \pi^2 \epsilon} \left( 4 + 2 \eta \right) C_{3} \delta_{ij}$$ Using those results one can calculate the $\Delta Z_{\phi}$ renormalization constants. $$\begin{aligned} \left( \Delta Z_{\phi} \right) _{i j} = - \frac{4 \textbf{Tr} \left( \kappa^{i} \kappa^{j} \right)}{16 \pi ^2 \epsilon} + \frac{ g^2 }{16 \pi^2 \epsilon} \left( 4 + 2 \eta \right) C_{3} \delta_{i j} \label{Zphi}\end{aligned}$$ Renormalization of fermion-fermion-vector boson coupling {#renormalization_of_ffb} -------------------------------------------------------- For all diagrams in fig. \[3point\_ffA\] there are no additional symmetry factors. In most cases, the evaluation of the pole term can be done the easiest way with masses and the momentum carried by the gauge boson equal to zero (it can be done when the counter terms for vertices have no momentum or mass dependence). The full expresion for the first diagram is: $$(\texttt{diagram \ref{3point_1}}) = i \sum _{n' m'} g \, \kappa ^{i}_{m m'}\overline{\textbf{T}}^{a}_{m' n'} \kappa ^{i}_{n' n} \int \frac{d^{d} k}{(2\pi)^{d}} \left( \tilde{S}_{F}(p+k + q) \gamma_{\mu} \tilde{S}_{F}(p+k) \right) \tilde{D}_{F}(k)$$ After some simplifications, the integral we are interested in, takes the form: $$(\texttt{diagram \ref{3point_1}}) = g \, \sum _{n' m'} g \, \kappa ^{i}_{m m'} \overline{\textbf{T}}^{a}_{m' n'} \kappa ^{i}_{n' n} \int \frac{d^{d} k}{(2\pi)^{d}} \frac{\left( p+k \right)^{\alpha} \left( p+k \right)^{\beta} }{k^2 (p+k)^2 (p+k)^2} \gamma_{\alpha} \gamma_{\mu} \gamma_{\beta}$$ Using the equality $$\begin{aligned} \gamma_{\mu} \gamma_{\alpha} \gamma^{\mu} = (2 - d) \gamma_{\alpha} \end{aligned}$$ one can get the final result $$(\texttt{diagram \ref{3point_1}}) = - \frac{\dot{\imath} g \, (\kappa ^{i} \overline{\textbf{T}}^{a} \kappa ^{i})_{m n} }{16 \pi^2 \epsilon} \gamma_{\mu}$$ To evaluate contribution from diagram \[3point\_2\] one needs to simplify the group theory factor. $$(\texttt{diagram \ref{3point_2}}) = g^2 g (\overline{\textbf{T}}^{b} \overline{\textbf{T}}^{a} \overline{\textbf{T}}^{b})_{m n} \int \frac{d^{d}k}{(2\pi)^d} \tilde{D}_{F}^{\nu \rho} (k) \left[ \gamma_{\rho} \tilde{S}_{F}(p+q+k) \gamma_{\mu} \tilde{S}_{F}(p+k) \gamma _{\nu} \right]$$ $$\begin{aligned} (\overline{\textbf{T}}^{b} \overline{\textbf{T}}^{a} \overline{\textbf{T}}^{b})_{n m} & = & \frac{1}{2} \left( \overline{\textbf{T}}^{b} \overline{\textbf{T}}^{b} \overline{\textbf{T}}^{a} + \dot{\imath} f_{abc} \overline{\textbf{T}}^{b} \overline{\textbf{T}}^{c}\right)_{n m} + \frac{1}{2} \left( \overline{\textbf{T}}^{a} \overline{\textbf{T}}^{b} \overline{\textbf{T}}^{b} - \dot{\imath} f_{abc} \overline{\textbf{T}}^{c} \overline{\textbf{T}}^{b}\right)_{n m} = \nonumber\\ & = & \left( \overline{C}_{3} - \frac{1}{2} C_{1}\right) \overline{\textbf{T}}^{a}_{n m} \end{aligned}$$ We use the previously mentioned simplification to calculate the pole term. With some help of the identity $$\gamma_{\rho} \gamma_{\lambda} \gamma_{\mu} \gamma_{\nu} \gamma^{\rho} = -2 \gamma_{\mu} \gamma_{\nu} \gamma_{\lambda} + (2-d)\gamma_{\lambda} \gamma_{\mu} \gamma_{\nu}$$\ one can obtain the result $$(\texttt{diagram \ref{3point_2}}) = - \frac{2 \dot{\imath} g^3 (\overline{C}_{3} - \frac{1}{2}C_{1})(1-\eta) }{16 \pi^2} \overline{\textbf{T}}^{a}_{m n} \gamma _{\mu}$$ To evaluate contribution from diagram \[3point\_3\] we need to calculate the following expression: $$\begin{aligned} (\texttt{diagram \ref{3point_3}}) &=& \dot{\imath} g^3 \sum _{b, c} f_{abc}(\overline{\textbf{T}}^{c} \overline{\textbf{T}}^{b})_{m n} \int \frac{d^{d}k}{(2\pi)^d} \tilde{D}_{F}^{\tau \rho} (q - k) \tilde{D}_{F}^{\nu \sigma} (k) \left[ \gamma_{\tau} \tilde{S}_{F}(p+k) \gamma_{\sigma} \right] \times \nonumber\\ & &\times \left[ (2k-q)_{\mu}g_{\nu \rho} - (q+k)_{\rho}g_{\mu \nu} +(2q - q)_{\nu} g_{\rho \mu} \right]\end{aligned}$$ One can express the group theory factor using $C_{1}$ as follows: $$\begin{aligned} \sum _{b, c} f_{abc} \overline{\textbf{T}}^{c} \, \overline{\textbf{T}}^{b} = - i \sum _{b} [\overline{\textbf{T}}^{a}, \overline{\textbf{T}}^{b}] \, \overline{\textbf{T}}^{b} = -\frac{\dot{\imath}}{2} C_{1} \overline{\textbf{T}}^{a}\end{aligned}$$ With the same procedure as before we find the pole term: $$\begin{aligned} (\texttt{diagram \ref{3point_3}}) = - \frac{3}{2} \frac{\dot{\imath} g^3 C_{1}(1+\xi)}{16 \pi ^2 \epsilon} (\overline{\textbf{T}}^{a})_{m n} \gamma_{\mu}\end{aligned}$$ To evaluate contribution from diagram \[3point\_4\] one needs to extract the pole term from the following expression: $$\begin{aligned} (\texttt{diagram \ref{3point_4}}) &=& i g \textbf{T}^{a} _{i j} \kappa ^{j} _{m n'} \kappa ^{i} _{n' n} \times \nonumber\\ & & \times \int \frac{d^{d}k}{(2\pi)^d} \tilde{S}_{F} \left( p+k \right) \tilde{D}_{F} \left( k \right) \tilde{D}_{F} \left( q-k \right) \left( 2k - q \right) ^{\mu}\end{aligned}$$ which simplifies to the form $$\begin{aligned} (\texttt{diagram \ref{3point_4}}) &=& \textbf{T}^{a} _{i j} \kappa ^{j} _{m n'} \kappa ^{i} _{n' n} \frac{i g }{16 \pi ^2 \epsilon} \gamma ^{\mu} \label{row2}\end{aligned}$$ As we have previously mentioned in section \[general\_theory\], all the 1-loop non diagonal corrections that occur in the equation (\[gB\_all\]) cancel each other out. Cancelling diagrams are shown in fig. \[non\_OPI\]. We will write partially the equation (\[gB\_all\]) - only with contributions from diagrams in fig. (\[non\_OPI\]). Assuming that the renormalization matrix $Z_{\psi}$ is hermitian we get $$\begin{aligned} g_{B} \overline {\textbf{T}}^{a} _{n m} &=& g \overline {\textbf{T}}^{a} _{n m} + \frac{1}{16 \pi^2 \epsilon} ( \frac{1}{2} \kappa ^{i} _{m m'} \kappa ^{i} _{m' n'} \overline {\textbf{T}}^{a} _{n' n} + \frac{1}{2} \overline {\textbf{T}}^{a} _{m m'} \kappa ^{i} _{m' n'} \kappa ^{i} _{n' n} - \kappa ^{i} _{m m'} \overline {\textbf{T}}^{a} _{m' n'} \kappa ^{i} _{n' n} \nonumber\\ & &+ \textbf{T}^{a} _{i j} \kappa ^{j} _{m n'} \kappa ^{i} _{n' n} ) + \ldots \label{kappy}\end{aligned}$$ To show the cancellation, one needs to consider how fermion and scalar fields change under infinitesimal gauge transformation. $$\begin{aligned} \psi _{n}' = \psi _{n} - i g \overline{\textbf{T}}^{a}_{n m } \Lambda ^{a} \psi _{m} \\ \overline{\psi} _{n}' = \overline{\psi} _{n} + i g \overline{\psi} _{m} \overline{\textbf{T}}^{a}_{m n} \Lambda ^{a} \\ \phi _{i}' = \phi _{j} - i g \textbf{T}^{a}_{i j } \Lambda ^{a} \phi _{j}\end{aligned}$$ From the invariance of the Yukawa term under gauge symmetry one can get a relation between the Yukawa coupling and gauge transformation generators. $$\begin{aligned} \textbf{T}^{a} _{j i} \kappa ^{j} _{n m} = \overline {\textbf{T}}^{a} _{n n'} \kappa ^{i} _{n' m} - \kappa ^{i} _{n n'} \overline {\textbf{T}}^{a} _{n' m} = [ \overline {\textbf{T}}^{a} , \kappa ^{i} ] _{n' m} \label{tozsamosc_kapp}\end{aligned}$$ which guarantees that $$\begin{aligned} \frac{1}{2} \kappa ^{i} _{m m'} \kappa ^{i} _{m' n'} \overline {\textbf{T}}^{a} _{n' n} + \frac{1}{2} \overline {\textbf{T}}^{a} _{m m'} \kappa ^{i} _{m' n'} \kappa ^{i} _{n' n} - \kappa ^{i} _{m m'} \overline {\textbf{T}}^{a} _{m' n'} \kappa ^{i} _{n' n} + \textbf{T}^{a} _{i j} \kappa ^{j} _{m n'} \kappa ^{i} _{n' n} = 0\end{aligned}$$ Now we can simplify the equation (\[gB\_all\]) to the form (\[easy\_gB\]), where $K_{2}$ and $\Delta Z_{\psi}$ are as follows: $$\begin{aligned} K_{2} &=& - \left(\frac{3}{2} C_{1} + \frac{1}{2} C_{1} \xi + 2 \overline{C} _{3} \xi \right) \frac{\dot{\imath} g^2}{16 \pi ^2 \epsilon} \label{Z2} \\ \Delta Z_{\psi} &=& \frac{ g^2 }{16 \pi^2 \epsilon} \left( 4 + 2 \eta \right) C_{3} \label{Zpsi2}\end{aligned}$$ Renormalization of $\phi^4$ interaction --------------------------------------- All contributing diagrams to the 1-loop renormalization of $\phi^4$ interaction are shown in fig. \[4point\_scalar\]. In diagram \[4point\_1\] there is a symmetry factor $\frac{1}{2}$, and one should sum over all $i', j'$ scalar fields. $$\begin{aligned} (\texttt{diagram \ref{4point_1}}) = - \frac{1}{2} \sum _{i' j'} h_{i j i' j'} h_{i' j' k l} \int \frac{d^d q}{(2 \pi)^d} \tilde{D}_{F} (p_{1} + p_{2} + q) \tilde{D}_{F} ( q)\end{aligned}$$ For two similar diagrams, but with differently connected scalar lines, the expressions are analogous. Summing them together result in: $$\begin{aligned} (\texttt{diagram \ref{4point_1} + 2 other}) = \frac{ \dot{\imath} }{16 \pi ^2 \epsilon } \sum _{i' j'} \left( h_{i j i' j'} h_{i' j' k l} + h_{i k i' j'} h_{i' j' j l}+ h_{k j i' j'} h_{i' j' i l} \right)\end{aligned}$$ There are 6 diagrams of the type shown on \[4point\_3\]. Evaluating the contribution from \[4point\_3\] one can get: $$\begin{aligned} (\texttt{diagram \ref{4point_3}}) &=& i g^2 \sum_{i' j'} \sum_{a} \textbf{T} ^{a} _{i i'} \textbf{T} ^{a} _{j j'} h_{i' j' k l} \int \frac{d^d q}{(2 \pi)^d} (2 p_{1} + q)_{\mu} (2 p_{2} - q)_{\nu} \times \nonumber\\ & & \times \tilde{D}_{F} ^{\mu \nu} ( q) \tilde{D}_{F} ( p_{2} - q) \tilde{D}_{F} (p_{1} + q) \end{aligned}$$ $$\begin{aligned} (\texttt{diagram \ref{4point_3}}) &=& \sum_{i', j'} \sum_{a} \textbf{T} ^{a} _{i i'} \textbf{T} ^{a} _{j j'} h_{i' j' k l} \frac{2 i g^2 (1 - \eta)}{16 \pi^2 \epsilon}\end{aligned}$$ For a full contribution we sum all the diagrams of this type. $$\begin{aligned} (\texttt{diagram \ref{4point_3} + 5 other}) &=& \frac{2 i g^2 (1 - \eta)}{16 \pi^2 \epsilon} \times \nonumber\\ & & \times \sum_{a} \sum_{b, c} ( \textbf{T} ^{a} _{i b} \textbf{T} ^{a} _{j c} h_{b c k l} + \textbf{T} ^{a} _{i b} \textbf{T} ^{a} _{k c} h_{b j c l} + \textbf{T} ^{a} _{i b} \textbf{T} ^{a} _{l c} h_{b j k c} + \nonumber\\ & & + \textbf{T} ^{a} _{j b} \textbf{T} ^{a} _{k c} h_{i b c l} + \textbf{T} ^{a} _{j b} \textbf{T} ^{a} _{l c} h_{i b k c} + \textbf{T} ^{a} _{k b} \textbf{T} ^{a} _{l c} h_{i j b c} ) \end{aligned}$$ To simplify this expression we will use an identity obtained from the quadrilinear term invariance under infinitesimal gauge transformation. $$\begin{aligned} \textbf{T} ^{a} _{i i'} h_{i' j k l} + \textbf{T} ^{a} _{j j'} h_{i j' k l} + \textbf{T} ^{a} _{k k'} h_{i j k' l} + \textbf{T} ^{a} _{l l'} h_{i j k l'} = 0 \label{tozsamosc_h}\end{aligned}$$ and write the factor containing generators in a form: $$\begin{aligned} & & \textbf{T} ^{a} _{i i'} \textbf{T} ^{a} _{j j'} h_{i' j' k l} + \textbf{T} ^{a} _{i i'} \textbf{T} ^{a} _{k k'} h_{i' j k' l} + \textbf{T} ^{a} _{i i'} \textbf{T} ^{a} _{l l'} h_{i' j k l'} + \nonumber\\ & & \textbf{T} ^{a} _{j j'} \textbf{T} ^{a} _{k k'} h_{i j' k' l} + \textbf{T} ^{a} _{j j'} \textbf{T} ^{a} _{l l'} h_{i j' k l'} + \textbf{T} ^{a} _{k k'} \textbf{T} ^{a} _{l l'} h_{i j k' l'} = \\ & & = \frac{1}{2} \left( \textbf{T} ^{a} _{i i'} \textbf{T} ^{a} _{j j'} h_{i' j' k l} + \textbf{T} ^{a} _{i i'} \textbf{T} ^{a} _{k k'} h_{i' j k' l} + \textbf{T} ^{a} _{i i'} \textbf{T} ^{a} _{l l'} h_{i' j k l'} + \textbf{T} ^{a} _{j j'} \textbf{T} ^{a} _{i i'} h_{i' j' k l} + \right. \nonumber\\ & & + \textbf{T} ^{a} _{j j'} \textbf{T} ^{a} _{k k'} h_{i j' k' l} + \textbf{T} ^{a} _{j j'} \textbf{T} ^{a} _{l l'} h_{i j' k l'} + \textbf{T} ^{a} _{k k'} \textbf{T} ^{a} _{i i'} h_{i' j k' l} + \textbf{T} ^{a} _{k k'} \textbf{T} ^{a} _{j j'} h_{i j' k' l} + \nonumber\\ & & \left. + \textbf{T} ^{a} _{k k'} \textbf{T} ^{a} _{l l'} h_{i j k' l'} + \textbf{T} ^{a} _{l l'} \textbf{T} ^{a} _{i i'} h_{i' j k l'} + \textbf{T} ^{a} _{l l'} \textbf{T} ^{a} _{j j'} h_{i j' k l'} + \textbf{T} ^{a} _{l l'} \textbf{T} ^{a} _{k k'} h_{i j k' l'} \right) \\ & & = - \frac{1}{2} \textbf{T} ^{a} _{i i'} \textbf{T} ^{a} _{i' i''} h_{i'' j k l} - \frac{1}{2} \textbf{T} ^{a} _{j j'} \textbf{T} ^{a} _{j' j''} h_{i j'' k l} - \frac{1}{2} \textbf{T} ^{a} _{k k'} \textbf{T} ^{a} _{k' k''} h_{i j k'' l} - \frac{1}{2} \textbf{T} ^{a} _{l l'} \textbf{T} ^{a} _{l' l''} h_{i j k l''} \nonumber\\ & & = - 2 C_{3} h_{i j k l}\end{aligned}$$ Now we can write the result in a simpler form: $$\begin{aligned} (\texttt{diagram \ref{4point_3}} + \textit{5 others} ) &=& - \frac{4 i g^2 (1 - \eta) C_{3} \delta_{i j}}{16 \pi^2 \epsilon} \end{aligned}$$ Diagram \[4point\_4\] has to be considered with $(-1)$ factor from a closed fermion loop. $$\begin{aligned} (\texttt{diagram \ref{4point_4}}) &=& - \textbf{Tr} (\kappa _j \kappa _i \kappa_k \kappa _l) \int \frac{d^d q}{(2 \pi)^d} \times \nonumber\\ &\times & \textbf{Tr} \left( \tilde{S}_{F} (p_{1} + p_{2} + p_{4} + q) \, \tilde{S}_{F} (p_{1} + p_{2} + q) \, \tilde{S}_{F} (p_{1} + q) \, \tilde{S}_{F} (q) \right) \nonumber\\\end{aligned}$$ To extract the pole term from this integral one can use the following identity. $$\begin{aligned} \textbf{Tr} (\gamma_{\alpha} \gamma_{\beta} \gamma_{\mu} \gamma_{\nu}) = 4 \left( g^{\alpha \beta} g^{\mu \nu} - g^{\alpha \mu } g^{\beta \nu} + g^{\alpha \nu } g^{\mu \beta} \right)\end{aligned}$$ There are 5 other diagrams similar to \[4point\_4\]. To simplify the result including all of them, we will introduce such quantity: $$\begin{aligned} A_{ijkl} = \textbf{Tr} \left( \kappa_{i} \kappa_{j} \lbrace \kappa_{l}, \kappa_{k} \rbrace + \kappa_{i} \kappa_{k} \lbrace \kappa_{j}, \kappa_{l} \rbrace + \kappa_{i} \kappa_{l} \lbrace \kappa_{j}, \kappa_{k} \rbrace \right)\end{aligned}$$ Then the final contribution is: $$\begin{aligned} (\texttt{diagram \ref{4point_4} + other})= - \frac{8 \dot{\imath} A_{ijkl} }{16 \pi ^2 \epsilon}\end{aligned}$$ Diagram \[4point\_5\] has a symmetry factor $\frac{1}{2}$. $$\begin{aligned} (\texttt{diagram \ref{4point_5}}) = & & - \frac{1}{2} g^4 \left( \textbf{T}^a_{ni}\textbf{T}^b_{nj} + \textbf{T}^a_{nj}\textbf{T}^b_{ni} \right) \left( \textbf{T}^a_{mk}\textbf{T}^b_{ml} + \textbf{T}^a_{ml}\textbf{T}^b_{mk} \right) \times \nonumber\\ & &\times \int \frac{d^d q}{(2 \pi)^d} \tilde{D}^{\alpha \beta}_{F} (p_{1} + p_{2} + q) \tilde{D}^{\mu \nu}_{F} ( q) g_{\mu \alpha} g_{\beta \nu}\end{aligned}$$ To calculate the contribution from this kind of diagrams one needs to perform the following integration: $$\begin{aligned} \int \frac{d^d q}{(2 \pi)^d} \tilde{D}^{\mu \nu}_{F} (p + q) \tilde{D}_{\mu \nu \, F} ( q) = - \frac{2 \dot{\imath} (4 - 2 \eta + \eta ^2) }{16 \pi^2 \epsilon}\end{aligned}$$ To simplify the result including other similar diagrams, it is convenient to introduce the following constant $$\begin{aligned} B_{ijkl} &=& \lbrace \textbf{T}^a , \textbf{T}^b \rbrace _{ij} \lbrace \textbf{T}^a , \textbf{T}^b \rbrace _{kl} + \lbrace \textbf{T}^a , \textbf{T}^b \rbrace _{ik} \lbrace \textbf{T}^a , \textbf{T}^b \rbrace _{jl} + \nonumber\\ & & \lbrace \textbf{T}^a , \textbf{T}^b \rbrace _{il} \lbrace \textbf{T}^a , \textbf{T}^b \rbrace _{jk}\end{aligned}$$ where $$\begin{aligned} \textbf{T}^a_{ni}\textbf{T}^b_{nj} + \textbf{T}^a_{nj}\textbf{T}^b_{ni} = - \lbrace \textbf{T}^a , \textbf{T}^b \rbrace _{ij}\end{aligned}$$ Using this notation one can write the result as follows $$\begin{aligned} (\texttt{diagram \ref{4point_5} + other}) = \frac{\dot{\imath} g^4 B_{ijkl}}{16 \pi ^2 \epsilon} \left(4 - 2 \eta + \eta^2 \right) = \frac{\dot{\imath} g^4 B_{ijkl}}{16 \pi ^2 \epsilon} \left(3 + (1 - \eta )^2 \right)\end{aligned}$$ There are 6 diagrams of type \[4point\_6\] to include in our calculations. Contribution from diagram \[4point\_6\] takes the form: $$\begin{aligned} & & (\texttt{diagram \ref{4point_6}}) = g^4 \left( \textbf{T}^a_{in}\textbf{T}^b_{jn} \textbf{T}^a_{km}\textbf{T}^b_{lm} \right) \times \nonumber\\ & & \times \int \frac{d^d q}{(2 \pi)^d} \tilde{D}^{\alpha \beta}_{F} (p_{1} + p_{2} + q) \tilde{D}^{\mu \nu}_{F} ( q) \tilde{D}_{F} (p_{1} + p_{2} + p_{4} + q) \tilde{D}_{F} (p_{1} + q) \times \nonumber\\ & & \times \left( 2 p_{1} + q \right)_{\mu} \left( p_{3} - p_{1} - p_{2} - p_{4} - q \right)_{\nu} \left( 2p_{4} + p_{1} + p_{2} + q \right)_{\beta} \left( p_{2} - p_{1} - q \right)_{\alpha} \nonumber\\ \end{aligned}$$ Being interested only in extracting the pole of this integral one can get after some simplifications the following form $$\begin{aligned} (\texttt{diagram \ref{4point_6}}) &=& - g^4 (1-\eta)^2 \left( \textbf{T}^a_{in}\textbf{T}^b_{jn} \textbf{T}^a_{km}\textbf{T}^b_{lm} \right) \times \nonumber\\ & & \times \int \frac{d^d q}{(2 \pi)^d} \tilde{D}_{F} (p_{1} + p_{2} + p_{4} + q) \tilde{D}_{F} (p_{1} + q)\end{aligned}$$ With the result $$\begin{aligned} (\texttt{diagram \ref{4point_6}}) = \left( \textbf{T}^a_{in}\textbf{T}^b_{jn} \textbf{T}^a_{km}\textbf{T}^b_{lm} \right) \frac{2 i g^4 (1-\eta)^2 }{16 \pi^2 \epsilon}\end{aligned}$$ One needs to consider other similar diagrams with permutations of the $i,j,k,l$ indices. The final result reads: $$\begin{aligned} (\texttt{diagram \ref{4point_6} + other}) &= \frac{i g^4 B_{ijkl}}{16 \pi ^2 \epsilon} \left( 1 - \eta \right)^2\end{aligned}$$ There are 6 diagrams of type \[4point\_7\] to include in the calculations. Symmetry factor for these diagrams is $1$. $$\begin{aligned} (\texttt{diagram \ref{4point_7}}) &=& -i g^4 g^{\alpha \mu} \lbrace \textbf{T}^a , \textbf{T}^b \rbrace _{i k} \textbf{T}^a_{j i'}\textbf{T}^b_{i' l} \int \frac{d^d q}{(2 \pi)^d} \tilde{D}^{\mu \nu}_{F} ( q) \tilde{D}_{F} (p_{1} + p_{2} + p_{3} - q) \nonumber\\ &\times & \tilde{D}^{\alpha \beta}_{F} (p_{1} + p_{2} - q) (p_{1} + 2 p_{2} + p_{3} - q) _{\beta} (p_{1} + p_{2} + p_{3} - p_{4} - q) _{\nu} \nonumber\\\end{aligned}$$ After considering other similar diagrams with permutations of the $i,j,k,l$ indices we have $$\begin{aligned} (\texttt{diagram \ref{4point_7} + other})= - \frac{2 i g^4 B_{ijkl}}{16 \pi ^2 \epsilon} ( 1 - \eta )^2\end{aligned}$$ The final result for the renormalization constant is below. As we can see, the gauge fixing parameter cancels within 6th to 10th diagram (see figure \[4point\_scalar\]) and only the term proportional to $C_{3}$ (originating from the 4th diagram) depends on the gauge choice. $$\begin{aligned} & & i L_{i j k l} ^{i' j' k' l'} h_{i' j' k' l'} = \frac{3 \dot{\imath} g^4 B_{ijkl}}{16 \pi ^2 \epsilon} - \frac{8 \dot{\imath} A_{ijkl}}{16 \pi ^2 \epsilon} + \frac{ \dot{\imath} }{16 \pi ^2 \epsilon } \sum _{i' j'} ( h_{i j i' j'} h_{i' j' k l} + \nonumber\\ & & + h_{i k i' j'} h_{i' j' j l}+ h_{k j i' j'} h_{i' j' i l} ) - \frac{4 i g^2 (1 - \eta) C_{3} \delta_{i j}}{16 \pi^2 \epsilon} \end{aligned}$$ Renormalization of Yukawa interaction {#renormalization_of_yukawa} ------------------------------------- Diagrams contributing to the renormalization of Yukawa interaction are shown in fig. \[3point\_Yukawa\]. In diagram \[yukawa\_1\] the symmetry factor is equal to 1 and one should sum over all $n'$ and $m'$ indices for fermion fields and over $a$ for gauge fields. $$\begin{aligned} (\texttt{diagram \ref{yukawa_1}}) &=& i g^2 \sum _{n', m', a} \left( \overline{\textbf{T}}^a _{m m'} \kappa ^{i} _{m' n'} \overline{\textbf{T}}^a _{n' n} \right) \times \nonumber \\ & & \times \int \frac{d^d k}{(2 \pi)^d} \tilde{D} ^{\mu \nu}_{F} (k) \gamma_{\mu} \tilde{S}_{F} (p + k + q) \tilde{S}_{F} (p + k) \gamma_{\nu}\end{aligned}$$ After performing the integral one can write the pole term as follows $$\begin{aligned} (\texttt{diagram \ref{yukawa_1}}) = \sum _{a} \left( \overline{\textbf{T}}^a \kappa ^{i} \overline{\textbf{T}}^a \right) _{m n} \frac{2 i g^2 (-4 + \eta) }{16 \pi^2 \epsilon}\end{aligned}$$ In diagram \[yukawa\_2\] one also sums over $n'$ and $m'$ indices of the fermion fields and $i$ index for the scalar field. $$\begin{aligned} (\texttt{diagram \ref{yukawa_2} })= i\sum _{n', m', j} \left( \kappa ^{j} _{m m'} \kappa ^{i} _{m' n'} \kappa ^{j} _{n' n} \right) \int \frac{d^d k}{(2 \pi)^d} \tilde{D} _{F} (k) \tilde{S}_{F} (p + k + q) \tilde{S}_{F} (p + k) \nonumber\\\end{aligned}$$ The pole term contribution reads $$\begin{aligned} (\texttt{diagram \ref{yukawa_2} })= \sum _{ j} \left( \kappa ^{j} \kappa ^{i} \kappa ^{j} \right) _{m n} \frac{2 i}{16 \pi ^2 \epsilon}\end{aligned}$$ The last two diagrams to consider are very similar to each other and do not require any new calculation tricks. $$\begin{aligned} (\texttt{diagram \ref{yukawa_3}})= - i g^2 \sum _{n', j, a} \overline{\textbf{T}}^{a} _{m n'} \kappa ^{j} _{n' n} \textbf{T}^{a}_{ij} \int \frac{d^d k}{(2 \pi)^d} \gamma _{\nu} \tilde{S}_{F} (p + k) \tilde{D} _{F} (k) (q + k)_{\mu} \tilde{D} _{F} ^{\mu \nu} (q-k) \nonumber\\\end{aligned}$$ The result is: $$\begin{aligned} (\texttt{diagram \ref{yukawa_3}})= - g^2 \sum _{n', j, a} \overline{\textbf{T}}^{a} _{m n'} \kappa ^{j} _{n' n} \textbf{T}^{a}_{ij} \frac{i (-2 + 2 \eta)}{16 \pi^2 \epsilon}\end{aligned}$$ Now we can add contribution from the second look-alike diagram, receiving: $$\begin{aligned} (\texttt{diagram \ref{yukawa_3} + other}) = g^2 \sum _{n', j, a} \left( - \overline{\textbf{T}}^{a} _{m n'} \kappa ^{j} _{n' n} \textbf{T}^{a}_{ij} + \kappa ^{j} _{m n'} \overline{\textbf{T}}^{a} _{n' n} \textbf{T}^{a}_{ij} \right) \frac{i (-2 + 2 \eta)}{16 \pi^2 \epsilon} \nonumber\\\end{aligned}$$ This way one can calculate $\Delta \kappa$ (see (\[ZetKappa1\]) and (\[ZetKappa\])) $$\begin{aligned} i K_{i m n}^{i' m' n'} \kappa^{i'} _{m' n'} &=& \sum _{a} \left( \overline{\textbf{T}}^a \kappa ^{i} \overline{\textbf{T}}^a \right) _{m n} \frac{2 i g^2 (-4 + \eta) }{16 \pi^2 \epsilon} + \sum _{j} \left( \kappa ^{j} \kappa ^{i} \kappa ^{j} \right) _{m n} \frac{2 i}{16 \pi ^2 \epsilon} \nonumber\\ & & + g^2 \sum _{n', j, a} \left( \kappa ^{j} _{m n'} \overline{\textbf{T}}^{a} _{n' n} - \overline{\textbf{T}}^{a} _{m n'} \kappa ^{j} _{n' n} \right) \textbf{T}^{a}_{ij} \frac{i (-2 + 2 \eta)}{16 \pi^2 \epsilon}\end{aligned}$$ Calculating beta functions {#Calculating_beta_functions} -------------------------- To calculate beta functions in our generic gauge theory, we need relations between bare and renormalized coupling constants (see equations (\[Zetxi\]), (\[easy\_gB\]), (\[ZetKappa1\]) or (\[Zeth\])). We will start from finding the expression for the beta function of the $g$-coupling. We repeat the relation between $g_{B}$ and renormalized coupling $g$ from equation (\[easy\_gB\]), also including the previously omitted renormalization scale $\mu$ factor, coming from the consistence of units in the dimensional regularization scheme. $$\begin{aligned} g_{B} = \left(g - \frac{1}{2} \Delta Z_{A} g - \Delta Z_{\psi} g + K_{2} g + \ldots \right) \mu^{\epsilon/2}= Z_{g} g \mu^{\epsilon/2}\end{aligned}$$ Since $g_{B}$ does not depend on the scale $\mu$, one gets $$\begin{aligned} & & \mu \frac{d g_{B}}{d \mu} = 0 = \mu \frac{d}{d \mu} \left( Z_{g} g \mu^{\epsilon/2} \right) \\ & & 0 = \frac{\epsilon}{2} Z_{g} g + \mu g \frac{d Z_{g}}{d \mu} + \mu Z_{g} \frac{d g}{ d \mu}\end{aligned}$$ We here have the $\mu$ dependence written explicitly, so $\frac{\partial Z_{g}}{\partial \mu} = 0$. Using the expansion of $Z_{g}$ in terms of the coupling we get the expression for $\mu \frac{dg}{d \mu} $. $$\begin{aligned} & & Z_{g} = 1 + Z_{g}^{(2)} g^2 + \ldots \\ & & \mu \frac{d g}{ d \mu} = \frac{- \frac{\epsilon}{2} Z_{g} g}{g \frac{\partial Z_{g}}{\partial g} + Z_{g}} = - \frac{\epsilon}{2}g + \epsilon g^3 Z_{g}^{(2)}\end{aligned}$$ The beta function is defined as $$\begin{aligned} \beta \left( \mu \right) = \lim_{\epsilon \to 0} \left( \mu \frac{d g}{ d \mu} \right)\end{aligned}$$ Beta function expanded in terms of $g$ gives us the $\beta_{0}$ function we are interested in $$\begin{aligned} \beta \left( \mu \right) = \beta_{0} g^3 + \ldots \\ \beta_{0} = \lim_{\epsilon \to 0} \left( \epsilon Z_{g}^{(2)} \right)\end{aligned}$$ In our case we will calculate the $\beta_{0}$ function for $g$-coupling with help of the (\[ZA\]), (\[Zpsi2\]) and (\[Z2\]). $$\begin{aligned} Z_{g} &=& 1 + K_{2} - \Delta Z_{\psi} - \frac{1}{2} \Delta Z_{A} + \ldots \\ \beta \left( g \right) &=&\left( - \frac{11}{3} C_{1} + \frac{4}{3} \overline{C}_{2} + \frac{1}{6} C_{2} \right) \frac{g^3}{16 \pi^2} + \ldots\end{aligned}$$ For other couplings the renormalization constants are described by matrices with non-zero mixing terms. For a general coupling constant $f_{\alpha}$ relationships between bare and renormalised quantities can be simply written in the form: $$\begin{aligned} (f_{\alpha})_{B} = \sum_{\beta} \mu ^{- \omega} Z_{\alpha \beta} \times (f_{\beta}) _{R} \label{f_alpha}\end{aligned}$$ where $\omega = \epsilon$ for quadrilinear coupling constant and $\omega = \frac{\epsilon}{2}$ for the Yukawa and gauge coupling. As before, with the differentiation of (\[f\_alpha\]) we will get an expression for the beta function. But now the renormalization constant depends in general on all the couplings from the model we are considering. So we obtain a more complicated result.\ We will skip the $R$ index to make the expressions shorter. $$\begin{aligned} \mu \frac {d \, f_{\beta}}{d \mu} & = & \sum _{\alpha, \gamma} \left(X ^{-1} \right) _{\beta \alpha} \left( - \omega Z_{\alpha \gamma} f_{\gamma} \right) \nonumber\\ X_{\alpha \beta} & = & Z_{\alpha \beta} + \sum_{\gamma}\frac{\partial Z_{\alpha \gamma}}{\partial f_{\beta}} f_{\gamma}\end{aligned}$$ Then one can expand the $Z_{\alpha \beta}$ matrix as a delta function with a small correction $$\begin{aligned} Z_{\alpha \beta} & = & \delta _{\alpha \beta} + \Delta Z_{\alpha \beta} \nonumber\\ X_{\alpha \beta} & = & \delta _{\alpha \beta} + \Delta Z_{\alpha \beta} + \sum_{\gamma}\frac{\partial \Delta Z_{\alpha \gamma}}{\partial f_{\beta}} f_{\gamma} \end{aligned}$$ For small values of $\Delta Z_{\alpha \beta} $ we can easily write the inverse matrix of $X _{\alpha \beta}$ $$\begin{aligned} \left( X ^{-1} \right) _{\alpha \beta} & = & \delta _{\alpha \beta} - \Delta Z_{\alpha \beta} - \sum_{\gamma} \frac{\partial \Delta Z_{\alpha \gamma}}{\partial f_{\beta}} f_{\gamma} \end{aligned}$$ We consider first two terms expanding in $\Delta Z_{\alpha \beta}$ and $\epsilon$ with the result of $$\begin{aligned} \beta (f_{\beta}) = \lim_{\epsilon \to 0} \mu \frac {d \, f_{\beta}}{d \mu} = \lim_{\epsilon \to 0} \left( - \omega f_{\beta} + \sum_{\gamma ,\mu}\omega \frac{\partial \Delta Z_{\beta \gamma}}{\partial f_{\mu}} f_{\gamma} f_{\mu} \right)\end{aligned}$$ We will first consider $\beta (\kappa^{i}_{m n})$. Using \[ZetKappa\] one can write $$\begin{aligned} \beta (\kappa ^{i}_{m n}) = \frac{\epsilon}{2} \sum _{i' n' m'} \sum _{i'' n'' m''} \frac{\partial \Delta \tilde{Z}_{i m n}^{i' m' n'}}{\partial \kappa ^{i''}_{m'' n'' }} \kappa ^{i'}_{m' n'} \kappa ^{i''}_{m'' n''} + \frac{\epsilon}{2} \sum _{i' n' m'} \frac{\partial \Delta \tilde{Z}_{i m n}^{i' m' n'}}{\partial g} \kappa ^{i'}_{m' n'} g \nonumber\\\end{aligned}$$ where $$\begin{aligned} \tilde{Z}_{i m n}^{i' m' n'} &=& \sum _{a b c} \left( Z^{-1/2}_{\psi} \right)^{*} _{b m} \left( Z^{-1/2}_{\psi} \right)_{c n} \left( Z^{-1/2}_{\phi} \right)_{a i} (\delta_{a i'} \delta_{b n'} \delta_{c m'} + K_{a b c}^{i' m' n'}) \nonumber\\ &=& \delta_{i i'} \delta_{n n'} \delta_{m m'} + K_{i m n}^{i' m' n'} - \frac{1}{2} \left( \Delta Z _{\psi} \right)_{n' n} \delta_{m m'} \delta_{i i'} \nonumber\\ & & - \frac{1}{2} \left( \Delta Z _{\psi} \right)^{*}_{m' m} \delta_{n n'} \delta_{i i'} - \frac{1}{2} \left( \Delta Z _{\phi} \right)_{i i'} \delta_{n n'} \delta_{m m'} + \ldots\end{aligned}$$ Below summation over repeated indices is assumed. $$\begin{aligned} && \Delta \tilde{Z}_{i m n}^{i' m' n'} \kappa ^{i'}_{m' n'} = \left( \overline{\textbf{T}}^a \kappa ^{i} \overline{\textbf{T}}^a \right) _{m n} \frac{2 g^2 (-4 + \eta)}{16 \pi^2 \epsilon} + \left( \kappa ^{j} \kappa ^{i} \kappa ^{j} \right) _{m n} \frac{2}{16 \pi ^2 \epsilon} \nonumber\\ & & + \left( \kappa ^{j} _{m n'} \overline{\textbf{T}}^{a} _{n' n} - \overline{\textbf{T}}^{a} _{m n'} \kappa ^{j} _{n' n} \right) \textbf{T}^{a}_{ij} \frac{2 g^2 (- 1 + \eta)}{16 \pi^2 \epsilon} - \frac{1}{2} \left( - \frac{2 g^2 \overline{C}_{3} (1-\eta)}{16 \pi^2 \epsilon} \kappa ^{i}_{m n} \right) \nonumber\\ & & - \frac{1}{2} \left( - \frac{\left(\kappa^{j} \kappa^{j} \right)_{n' n} }{16 \pi^2 \epsilon} \kappa^{i}_{m n'} \right) - \frac{1}{2} \left( - \frac{2 g^2 \overline{C}_{3} (1-\eta)}{16 \pi^2 \epsilon} \kappa ^{i}_{m n} \right) - \frac{1}{2} \left( - \frac{\left(\kappa^{j} \kappa^{j}\right)_{m m'}}{16 \pi^2 \epsilon} \kappa^{i}_{m' n} \right) \nonumber\\ & & - \frac{1}{2} \left( - \frac{4 \textbf{Tr} ( \kappa^{i} \kappa^{i'} )}{16 \pi^2 \epsilon} \kappa^{i'} _{m n} \right) - \frac{1}{2} \left( \frac{g^2 (4+ 2\eta) C_{3}}{16 \pi^2 \epsilon} \kappa^{i} _{m n} \right) \nonumber\\\end{aligned}$$ $$\begin{aligned} && 16 \pi^2 \beta (\kappa ^{i}_{m n}) = 2 g^2 (-4 + \eta) \left( \overline{\textbf{T}}^a \kappa ^{i} \overline{\textbf{T}}^a \right) _{m n} + 2 \left( \kappa ^{j} \kappa ^{i} \kappa ^{j} \right) _{m n} + 2 g^2 \overline{C}_{3} (1-\eta) \kappa ^{i}_{m n} \nonumber\\ & & + 2 g^2 (- 1 + \eta) \left( \kappa ^{j} _{m n'} \overline{\textbf{T}}^{a} _{n' n} - \overline{\textbf{T}}^{a} _{m n'} \kappa ^{j} _{n' n} \right) \textbf{T}^{a}_{i j} + \frac{1}{2} \left( \kappa^{i} \kappa^{j} \kappa^{j} + \kappa^{j} \kappa^{j} \kappa^{i} \right)_{m n} \nonumber\\ & & + 2 \textbf{Tr} ( \kappa^{i} \kappa^{i'} ) \kappa^{i'} _{m n} - g^2 (2+ \eta) C_{3} \kappa^{i} _{m n} \label{row1}\end{aligned}$$ Now using (\[tozsamosc\_kapp\]) one can find the following two relations $$\begin{aligned} & & C_{3} \delta _{i j} \kappa^{j} _{m n} = \textbf{T}^{a}_{i i'} \textbf{T}^{a}_{i' j} \kappa^{j} _{m n} = 2 \overline{C}_{3} \kappa^{i} _{m n} - 2 \left( \overline{\textbf{T}}^a \kappa ^{i} \overline{\textbf{T}}^a \right) _{m n} \\ & & \left( \kappa ^{j} _{m n'} \overline{\textbf{T}}^{a} _{n' n} - \overline{\textbf{T}}^{a} _{m n'} \kappa ^{j} _{n' n} \right) \textbf{T}^{a}_{i j} = 2 \overline{C}_{3} \kappa^{i} _{m n} - 2 \left( \overline{\textbf{T}}^a \kappa ^{i} \overline{\textbf{T}}^a \right) _{m n}\end{aligned}$$ Substituting those results to (\[row1\]) one can get $$\begin{aligned} 16 \pi^2 \beta (\kappa ^{i}_{m n}) &=& - 6 g^2 \overline{C}_{3} \kappa ^{i}_{nm} + 2 \left( \kappa ^{j} \kappa ^{i} \kappa ^{j} \right) _{n m} \nonumber\\ & & + \frac{1}{2} \left( \kappa^{i} \kappa^{j} \kappa^{j} + \kappa^{j} \kappa^{j} \kappa^{i} \right)_{n m} + 2 \textbf{Tr} ( \kappa^{i} \kappa^{j} ) \kappa^{j} _{n m} \label{kappa_result}\end{aligned}$$ Now we will consider $\beta (h_{i j k l})$. Using \[ZetH\] one can write: $$\begin{aligned} \beta (h_{i j k l}) &=& \epsilon \sum _{i' j' k' l'} \sum _{i'' j'' k'' l''} \frac{\partial \Delta \tilde{Z}_{i j k l}^{i' j' k' l'}}{\partial h_{i'' j'' k'' l''}} h_{i' j' k' l'} h_{i'' j'' k'' l''} \nonumber\\ & & + \frac{\epsilon}{2} \sum _{i' n' m'} \sum _{i'' n'' m''} \frac{\partial \Delta \tilde{Z}_{i j k l}^{i' j' k' l'}}{\partial \kappa ^{i''}_{m'' n'' }} \kappa ^{i''}_{m'' n''} \, h_{i' j' k' l'} + \frac{\epsilon}{2} \sum _{i' n' m'} \frac{\partial \Delta \tilde{Z}_{i j k l}^{i' j' k' l'}}{\partial g} g h_{i' j' k' l'} \nonumber\\\end{aligned}$$ where $$\begin{aligned} \tilde{Z}_{i j k l}^{i' j' k' l'}&=& \sum _{a b c d} \left( Z^{-1/2}_{\phi} \right)_{i a} \left( Z^{-1/2}_{\phi} \right)_{j b} \left( Z^{-1/2}_{\phi} \right)_{k c} \left( Z^{-1/2}_{\phi} \right)_{l d} (\delta_{a i'} \delta_{b j'} \delta_{c k'} \delta_{d l'} + L_{a b c d}^{i' j' k' l'}) = \nonumber\\ & = & \delta_{i i'} \delta_{j j'} \delta_{k k'} \delta_{l l'} + L_{i j k l}^{i' j' k' l'} - \frac{1}{2} \left( \Delta Z _{\phi} \right)_{i i'} \delta_{j j'} \delta_{k k'} \delta_{l l'} - \frac{1}{2} \left( \Delta Z _{\phi} \right)_{j j'} \delta_{i i'} \delta_{k k'} \delta_{l l'} \nonumber\\ & & - \frac{1}{2} \left( \Delta Z _{\phi} \right)_{k k'} \delta_{i i'} \delta_{j j'} \delta_{l l'} - \frac{1}{2} \left( \Delta Z _{\phi} \right)_{l l'} \delta_{i i'} \delta_{j j'} \delta_{k k'} + \ldots\end{aligned}$$ Below summation over repeated indices is assumed $$\begin{aligned} & & \Delta \tilde{Z}_{i j k l}^{i' j' k' l'} h_{i' j' k' l'} = \frac{3 g^4 B_{ijkl}}{16 \pi ^2 \epsilon} - \frac{8 A_{ijkl}}{16 \pi ^2 \epsilon} - \frac{4 g^2 (1 - \eta) C_{3} }{16 \pi^2 \epsilon} h_{i j k l} \nonumber\\ & & + \frac{ \dot{\imath} }{16 \pi ^2 \epsilon } \left( h_{i j i' j'} h_{i' j' k l} + h_{i k i' j'} h_{i' j' j l}+ h_{k j i' j'} h_{i' j' i l} \right) \nonumber\\ & & - \frac{1}{2} \frac{(- 4)}{16 \pi ^2 \epsilon} ( \textbf{Tr}( \kappa^{i} \kappa^{i'} ) h_{i' j k l} + \textbf{Tr}( \kappa^{j} \kappa^{j'} ) h_{i j' k l} + \textbf{Tr}( \kappa^{k} \kappa^{k'} ) h_{i j k' l} + \textbf{Tr}( \kappa^{l} \kappa^{l'} ) h_{i j k l'} ) \nonumber\\ & & - \frac{1}{2} \frac{g^2 }{16 \pi^2 \epsilon} \left( 4 + 2 \eta \right) C_{3} h_{i j k l} \times 4 + \ldots\end{aligned}$$ Then one gets $$\begin{aligned} && 16 \pi^2 \beta ( h_{i' j' k' l'} ) = 3 g^4 B_{ijkl} - 8 A_{ijkl} + \left( h_{i j i' j'} h_{i' j' k l} + h_{i k i' j'} h_{i' j' j l}+ h_{k j i' j'} h_{i' j' i l} \right) \nonumber\\ & & + 2 ( \textbf{Tr}( \kappa^{i} \kappa^{i'} ) h_{i' j k l} + \textbf{Tr}( \kappa^{j} \kappa^{j'} ) h_{i j' k l} + \textbf{Tr}( \kappa^{k} \kappa^{k'} ) h_{i j k' l} + \textbf{Tr}( \kappa^{l} \kappa^{l'} ) h_{i j k l'} ) \nonumber\\ & & - 12 g^2 C_{3} h_{i j k l}\end{aligned}$$ Beta functions we have calculated are expressed in terms of general group theory factors. To derive the expressions in particular models further analysis is required. For example, if the gauge group is a group product, like in the Standard Model, it is necessary to modify the results. The beta function found here were published for example in [@cheng]. We confirm the result and point out the misprint: in equation (2.7) in [@cheng] the group theory factor $S_2(S)$ (which in our notation is $C_{3}$) should be replaced by $S_2(F)$ (which in our notation stands for $\overline{C}_{3}$). Beta functions for the Standard Model and its extension ======================================================= We would like to apply our general result to the Standard Model and the Minimal Standard Model (MSM) cases. Standard Model result --------------------- The SM[^2] has a $U(1) \times SU(2) \times SU(3)$ gauge symmetry. Following [@machacek3], if a gauge group is a direct product $G_{1} \times ... \times G_{N}$ of simple groups with corresponding gauge constants $g_{1},...,g_{N}$ then the group factors we used in our general theory should be replaced as follows: $$\begin{aligned} g^2 C_{i}(R) &\longrightarrow & \sum_{n} g_{n}^2 C_{i}(R) \\ g^4 B_{ijkl} &\longrightarrow & \sum_{n,m} g_{n}^2 g_{m}^2 \tilde{B}^{n m}_{ijkl}\end{aligned}$$ The factor $\tilde{B}^{n m}_{ijkl}$ is expressed by the group generators of different simple groups $\textbf{T}^a_{n},\textbf{T}^b_{m}$ ($n,m$ - simple group indices, $a,b$ - indices numbering the generators of each group) $$\begin{aligned} \tilde{B}^{n m}_{ijkl} &=& \sum \lbrace \textbf{T}^a_{n} , \textbf{T}^b_{m} \rbrace _{i,j} \lbrace \textbf{T}^a_{n} , \textbf{T}^b_{m} \rbrace _{k,l} + \lbrace \textbf{T}^a_{n} , \textbf{T}^b_{m} \rbrace _{i,k} \lbrace \textbf{T}^a_{n} , \textbf{T}^b_{m} \rbrace _{j,l} + \nonumber\\ & & \lbrace \textbf{T}^a_{n} , \textbf{T}^b_{m} \rbrace _{i,l} \lbrace \textbf{T}^a_{n} , \textbf{T}^b_{m} \rbrace _{j,k}\end{aligned}$$ Second problem that occurs while adapting the general result to the Standard Model case is that left- and right-handed fermion fields attribute to different gauge group representations. In the SM couplings we have an additional operator $P_{L}$ or $P_{R}$ of chiral projections. If we’d like to repeat our calculations in the case of right- or left-handed fields, then there occur some additional factors. For example while integrating over a closed fermion loop, there is an additional factor $\frac{1}{2}$ from the projections, so one has to be very careful. While calculating the final result, we will confine ourselves to the most relevant SM constants: gauge couplings $g_1, g_2, g_3$, top quark Yukawa coupling $y_t$ and quadrilinear Higgs coupling $\lambda$. We will also skip parts of the beta functions calculations, analysing only the group theory factors. For the SU(N) we have $C_{1}^{SU(N)} = N$ and $C_{2}^{SU(N)}(R_{F}) = \frac{1}{2}$ for a fundamental representation $R_{F}$. For U(1) the $C_{1}^{U(1)} = 0$. To calculate $C_{2}^{U(1)}$ one needs to add the squares of scalar hypercharges, and for the $\overline{C}_{2}^{U(1)}$ the fermion hypercharges. In all these calculations we need to remember that there are 3 generations of fermions and 3 colours of quarks. The $C_{3}$ factor in the beta function for the quartic coupling contributes only from U(1) and SU(2). Once again we add the squares of hypercharges in a case of U(1) symmetry, and the $\textbf{T}^a \textbf{T}^a$ for the SU(2), where the generators are half the Pauli matrices $\textbf{T}^a = \frac{1}{2} \sigma^{a}$ $$\begin{aligned} C_{3}^{U(1)} = \frac{1}{4}, \, \, C_{3}^{SU(2)} = \frac{3}{4}\end{aligned}$$ The $\overline{C}_{3}$ factor in the beta function for the Yukawa coupling can be easily calculated in case of SU(2) and SU(3). $$\begin{aligned} \overline{C}_{3}^{SU(2)} = \frac{3}{4}, \, \, \overline{C}_{3}^{SU(3)} = \frac{4}{3}\end{aligned}$$ For the U(1) gauge symmetry one has to consider only the hypercharges of the top quark left- and right- handed part, which give a result $$\begin{aligned} \overline{C}_{3}^{U(1)} = \frac{1}{2} \left(\left(\frac{1}{6}\right)^2 + \left(\frac{2}{3}\right)^2 \right) = \frac{1}{6} \frac{17}{12}\end{aligned}$$ Now we can present final expressions for the SM 1-loop beta functions: $$\begin{aligned} 16 \pi^2 \beta(\lambda) & = & \frac{3}{8} g_{1}^4 + \frac{9}{8} g_{2}^4 + \frac{3}{4} g_{1}^2 g_{2}^2 - 6 y_{t}^4 + 24 \lambda^2 +12 y_{t}^2 \lambda -3 g_{1}^2 \lambda - 9 g_{2}^2 \lambda \nonumber\\ \\ 16 \pi^2 \beta(g_{1}) & = & \frac{41}{6} g_{1}^3 \\ 16 \pi^2 \beta(g_{2}) & = & -\frac{19}{6} g_{2}^3 \\ 16 \pi^2 \beta(g_{3}) & = & -7 g_{3}^3 \\ 16 \pi^2 \beta(y_{t}) & = & \left( -\frac{17}{12} g_{1}^2 -\frac{9}{4} g_{2}^2 - 8 g_{3}^2 \right) y_{t} + \frac{9}{2} y_{t}^3\end{aligned}$$ The results agree with those from the literature, see e.g. [@pirogov] Standard Model plus scalar singlets ----------------------------------- We’d like to consider now a model with additional scalar singlet fields. The general scalar potential with the SM doublet of scalars $H$ and $N_{\phi}$ scalar singlets $\phi_{i}$ is: $$\begin{aligned} V(H, \phi_{n}) = -m^2 H ^\dagger H + \lambda (H ^\dagger H)^2 + \frac{1}{2} \sum _{i} ^{N_{\phi}} \mu ^{i}_{\phi} \phi _ i ^2 + \sum _{i,j}^{N_{\phi}} \lambda_{\phi}^{ij} \phi ^2 _{i} \phi ^2 _{j} + \sum _{i}^{N_{\phi}} \lambda_{x}^{i} (H ^\dagger H) \phi _{i} ^ 2 \nonumber\\\end{aligned}$$ All the previously mentioned problems occur here as well. Additional calculations to make are rather simple and do not require a special comment. Resulting scalar sector beta functions for the SM with $N_{\phi}$ scalar singlets are: $$\begin{aligned} 16 \pi^2 \beta(\lambda) & = & \frac{3}{8} g_{1}^4 + \frac{9}{8} g_{2}^4 + \frac{3}{4} g_{1}^2 g_{2}^2 - 6 y_{t}^4 + 24 \lambda^2 +12 y_{t}^2 \lambda -3 g_{1}^2 \lambda - 9 g_{2}^2 \lambda \nonumber\\ & & + 2 N_{\phi} \lambda_{x}^2 \\ 16 \pi^2 \beta(\lambda_{\phi}) & = & (64 + 8 N_{\phi}) \lambda_{\phi} ^2 + 2 \lambda_{x} ^2 \\ 16 \pi^2 \beta(\lambda_{x}) & = & 12 \lambda \lambda_{x} + 24 \lambda_{\phi} \lambda_{x} + 8 \lambda_{x}^2 + 6 y_{t}^2 \lambda_{x} - \frac{3}{2} g_{1}^2 \lambda_{x} - \frac{9}{2} g_{2}^2 \lambda_{x}\end{aligned}$$ The above results agree with [@davoudiasl]. Right-handed neutrinos {#right_neutrinos_sec} ---------------------- After adding singlet scalar fields to the theory, it is very natural to include also right-handed Majorana neutrino singlets (see [@casas] or [@akhmedov]) and their couplings to scalar singlets: $$\begin{aligned} L_{\nu} = - \frac{1}{2} \overline{(\nu_{R})^{c}} Y_{\phi} \nu_{R} \phi + h.c.\end{aligned}$$ where $(\,)^c$ denotes the charge conjugation operator acting on a fermion field. The coupling $Y_{\phi}$ contributes to the $\beta \left( \lambda_x \right)$ and $\beta \left( \lambda_{\phi} \right)$. To calculate those corrections we need to consider a scalar singlet propagator correction from right neutrinos. For diagram \[2point\_scalar\_majorana\_diag\] we need to include a standard combinatorial factor 1/2 for such loop with self-conjugate particles. A $(-1)$ factor originates from a fermion loop. Feynman rules for the Majorana neutrinos can be found in Appendix C. $$\begin{aligned} \texttt{diagram \ref{2point_scalar_majorana_diag}} &=& - \frac{1}{2} \sum_{a b}\int \frac{d^4 k}{(2 \pi)^4} Tr \left( \left( - \tilde{S}_{F} (p - k) \hat{C} \right) (-i) \hat{C} Y_{i} ^{a b} \left( - \tilde{S}_{F} (k) \hat{C} \right) (- i ) \hat{C} Y_{j} ^{a b} \right) \nonumber\\ &=& \frac{2 i p^2 Tr(Y_{i} Y_{j})}{16 \pi^2 \epsilon}\end{aligned}$$ where we use the fact that $\hat{C}^2 = 1$.\ This diagram contributes to the general beta function formula in the following way: $$\begin{aligned} 16 \pi^2 \beta(h_{ijkl}) &=& \ldots + \left( Tr( Y_{i} Y_{i'} ) h_{i'jkl} + Tr(Y_{j} Y_{j'}) h_{ij'kl}+ Tr(Y_{k} Y_{k'}) h_{ijk'l} \right. \nonumber\\ && \left. + Tr(Y_{l} Y_{l'}) h_{ijkl'} \right)\end{aligned}$$ Now one can calculate the contribution to the $\lambda_x$ and $\lambda_\phi$ beta function for the one singlet SM extension (we also include top-quark Yukawa interaction contribution for comparison) $$\begin{aligned} & & 16 \pi^2 \beta ( \lambda_x )= 4 ( 3 \times \textbf{Tr} ( Y_{t} Y_{t} ) \lambda_x ) + 2 \textbf{Tr} ( Y_{\phi} Y_{\phi} ) \lambda_x + \ldots \\ & & 16 \pi^2 \beta(\lambda_{\phi}) = 4 \textbf{Tr} ( Y_{\phi} Y_{\phi} ) \lambda_{\phi} + \ldots\end{aligned}$$ To have a full $\beta(\lambda_{\phi})$ from the right neutrino coupling one has to consider also 1-loop correction to the $\phi^4$ vertex. As we do not need the $\beta(\lambda_{\phi})$ for our purposes, we will skip this calculation. 1-loop quadratic divergences in a generic gauge theory ====================================================== In this section we will find the quadratically divergent contributions to scalar 2-point Green’s function in a general gauge theory with scalar and fermion fields (as introduced in section \[general\_theory\]). We will adopt the cut-off regularization (see the Appendix B for necessary integrals). For all the loops we assume the same cut-off $\Lambda$ and we keep the $\Lambda$ contributions and $\log (\Lambda)$ for scalar loops, as they will be relevant in later discussion. Below there are mentioned only the diagrams that contribute in this regularization. Below we list all the contributions from diagrams in figure \[1loop\_propagator\].\ $\texttt{Diagram 1}$: symmetry factor $\frac{1}{2}$ $$\begin{aligned} - \frac{1}{2} \int \frac{d^4 k}{(2 \pi)^4} \frac{i}{k^2 - m_{i'}^2} i h_{i j i' i'} = - \frac{1}{2} h_{i j i' i'} \frac{i}{16 \pi^2} \left( \Lambda ^2 - m_{i'}^2 \log \left( \frac{m_{i'}^2 + \Lambda^2}{m_{i'}^2} \right)\right) \nonumber\\\end{aligned}$$ $\texttt{Diagram 2}$: symmetry factor 1, (-1) factor from a fermion loop $$\begin{aligned} (-1) \int \frac{d^4 k}{(2 \pi)^4} Tr \left( i \kappa ^i _{m n} \tilde{S}_{F} (p + k) i \kappa ^j _{n m} \tilde{S}_{F} (k) \right) = Tr (\kappa^i \kappa^j) \frac{4 i \Lambda ^2}{16 \pi^2}\end{aligned}$$ $\texttt{Diagram 3}$: symmetry factor 1, summing over gauge fields $$\begin{aligned} - g^2 T^a_{i i'} T^a_{i' j} \int \frac{d^4 k}{(2 \pi)^4} (2p - k)_{\mu} \tilde{D}_{F} (p-k) (2p - k)_{\nu} \tilde{D}_{F}^{\mu \nu} (p-k) = g^2 ( T^a T^a ) _{i j} \frac{i \Lambda ^2}{16 \pi^2} (1 - \eta) \nonumber\\\end{aligned}$$ $\texttt{Diagram 4}$: symmetry factor $\frac{1}{2}$, $$\begin{aligned} \frac{1}{2} \int \frac{d^4 k}{(2 \pi)^4} i g^2 g_{\mu \nu} 2 (T^a T^a)_{i j} \tilde{D}_{F}^{\mu \nu} = g^2 ( T^a T^a ) _{i j} \frac{i \Lambda ^2}{16 \pi^2} (-4 + \eta) \end{aligned}$$ Now one can write an expression for a 1-loop correction to the scalar particle mass in generic gauge theory (summing over primed indices): $$\begin{aligned} \delta m^2 _{ij} = \frac{1}{16 \pi^2} \left( - \frac{1}{2} h_{i j i' i'} \left( \Lambda ^2 - m_{i'}^2 \log \left( \frac{m_{i'}^2+\Lambda^2}{m_{i'}^2} \right)\right) + 4 Tr (\kappa^i \kappa^j) \Lambda^2 - 3 g^2 ( T^a T^a ) _{i j} \Lambda ^2 \right) \nonumber\\ \label{general_1loop}\end{aligned}$$ Standard Model with scalar singlets case ---------------------------------------- We’d like to calculate a 1-loop correction to the Higgs mass in a case of a SM Higgs doublet and $N_{\phi}$ singlet scalar fields (for the potential see equation (\[scalar\_potential\])) with the common mass $m_{\phi}$. Using $m_{h}^2 = - \mu^2 + 3 \lambda v^2 = 2 \mu^2 $ (where $v$ is the vacuum expectation value of the Higgs field) and (\[general\_1loop\]) one can calculate the Higgs boson mass correction $$\begin{aligned} & & \delta m^2_{h} = \frac{\Lambda ^2}{16 \pi^2} \left( 12 \lambda + 2 N_{\phi} \lambda_x - 12 y_t^2 + \frac{3}{2} g_1^2 + \frac{9}{2} g_2^2 \right) \nonumber\\ & & - \frac{1}{16 \pi^2} \left( 6 \lambda m_{h}^2 \log \left( \frac{m_{h}^2+\Lambda^2}{m_{h}^2} \right)+ 2 \lambda \sum_{I = 1,2,3} m_{I}^2 \log \left( \frac{m_{I}^2+\Lambda^2}{m_{I}^2} \right) \right. \nonumber\\ && \left. + 2 \lambda_{x} N_{\phi} m_{\phi}^2 \log \left( \frac{m_{\phi}^2+\Lambda^2}{m_{\phi}^2} \right) \right)\end{aligned}$$ where $m_{I}$ stands for the masses of Goldstone bosons and $m_{\phi}$ is the mass of singlet scalar fields $m_{\phi}^2 = \mu _{\phi} ^2 + \lambda_{x} v ^2$ Leading quadratic divergences in higher orders ============================================== In this chapter we’d like to show how to calculate quadratic divergences in two ways. As in previous chapter, we’re interested in divergences within general gauge theory with scalar and fermion fields, in a cut-off regularization scheme. We mention only the diagrams that contribute in cut-off regularization scheme. 2-loop Higgs effects in a generic theory ---------------------------------------- The most common approach to calculate 2-loop divergences is a straightforward computation with help of Feynman diagrams. In the figure \[2loop\_propagator\] we drew contributing diagrams in a 2-loop calculation that originate quartic scalar coupling. $\texttt{Diagram 1}$: symmetry factor $\frac{1}{2}$\ $$\begin{aligned} \frac{1}{2} \int \frac{d^4 k}{(2 \pi)^4} \frac{i}{k^2} i h_{i j i' i'} = \frac{i \Lambda ^2}{16 \pi^2} \frac{1}{2} h_{i j i' i'} \end{aligned}$$ $\texttt{Diagram 2}$: symmetry factor $\frac{1}{2}$\ $$\begin{aligned} \frac{1}{2} \int \frac{d^4 k}{(2 \pi)^4} \frac{i}{k^2} \left(i \Delta h_{i j i' i'} \right) = \frac{i \Lambda ^2}{16 \pi^2} \frac{1}{2} \left(i \Delta h_{i j i' i'} \right) \end{aligned}$$ $\texttt{Diagram 3}$: symmetry factor $\frac{1}{2}$\ $$\begin{aligned} \frac{1}{2} \int \frac{d^4 k}{(2 \pi)^4} \frac{i}{k^2} \left( i \Delta Z_{i' j'} \right) \frac{i}{k^2} i h_{i j i' j'} = \frac{i}{16 \pi^2} \frac{1}{2} \Delta Z_{i' j'} h_{i j i' j'} \log (\frac{\Lambda ^2}{m_{i} m_{j}})\end{aligned}$$ $\texttt{Diagram 4}$: symmetry factor $\frac{1}{4}$\ $$\begin{aligned} \frac{1}{4} \int \frac{d^4 k}{(2 \pi)^4} \int \frac{d^4 q}{(2 \pi)^4} \frac{i}{k^2} \frac{i}{k^2} \frac{i}{q^2} i h_{i j i' j'} i h_{i' j' i'' i''} = \frac{i}{(16 \pi^2)^2} \frac{1}{4} h_{i j i' j'} h_{i' j' i'' i''} \Lambda ^2 \log (\frac{\Lambda ^2}{m_{i'} m_{j'}}) \nonumber\\\end{aligned}$$ $\texttt{Diagram 5}$: symmetry factor $\frac{1}{6}$\ $$\begin{aligned} \frac{1}{6} \int \frac{d^4 k}{(2 \pi)^4} \int \frac{d^4 q}{(2 \pi)^4} \frac{i}{k^2} \frac{i}{(k - q)^2} \frac{i}{q^2} i h_{i i' j' k'} i h_{j i' j' k'} = \frac{1}{(16 \pi^2)^2} \frac{i \Lambda ^2}{3} h_{i i' j' k'} h_{j i' j' k'} \nonumber\\\end{aligned}$$ From the results above we can determine the 1-loop counterterms: $$\begin{aligned} & & \Delta Z_{i j} = - \frac{1}{2} h_{i j i' i'} \frac{\Lambda ^2}{16 \pi^2} \\ & & \Delta h_{i j k l} = - \frac{1}{2} \frac{1}{16 \pi^2} \log (\frac{\Lambda ^2}{m_{i} m_{j}}) \left( h_{i j i' j'} h_{k l i' j'} + h_{i k i' j'} h_{j l i' j'} + h_{i l i' j'} h_{j k i' j'} \right)\nonumber\\\end{aligned}$$ And write the final result of the mass correction leading scalar contributions $$\begin{aligned} \texttt{1-loop correction} &=& - \frac{\Lambda ^2}{16 \pi^2} \frac{1}{2} h_{i i i' i'} \label{result_1loop} \\ \texttt{2-loop correction} &=& - \frac{1}{(16 \pi^2)^2} \Lambda ^2 \log \left( \frac{\Lambda^2}{m^2} \right) \frac{1}{4} \left( h_{i j i' j'} h_{i' j' k' k'} + 2 h_{i i' j' k'} h_{j i' j' k'} \right) \nonumber\\ & & + \frac{\Lambda ^2}{(16 \pi ^2)^2} \frac{1}{3} h_{i i' j' k'} h_{j i' j' k'} \label{2_loop_scalar_coefficent}\end{aligned}$$ One can neglect the result proportional to the $\frac{\Lambda ^2 h^2}{(16 \pi ^2)^2}$ as small in comparison to the 1-loop term[^3]. Alternatively, one can obtain the leading higher order quadratic divergences indirectly, with some help of beta functions. Following [@einhorn], in a theory with many couplings $\lambda_{i}$ the leading (containing the highest power of $\log (\Lambda)$ ) quadratic divergences can be written as $$\begin{aligned} \delta m^2 = \Lambda^2 \sum_{n = 0} ^{\infty} f_{n} (\lambda_{i}) \log ^{n} \left( \frac{\Lambda}{\mu}\right) + \ldots \label{einhorn_eq}\end{aligned}$$ where $n+1$ is the number of loops considered, $\mu$ is the renormalization scale and the coefficients $f_{n}$ satisfy $$\begin{aligned} (n+1)f_{n+1} = \mu \frac{\partial}{\partial \mu} f_{n} = \beta_{i} \frac{\partial}{\partial \lambda_{i}} f_{n} \label{recursion}\end{aligned}$$ This method allows to determine only terms proportional to the $\Lambda^2 \log ^{n} \left( \frac{\Lambda}{\mu}\right)$. Terms with the logarithm power less than $n$ are not controlled within this method. The results (\[result\_1loop\]) and (\[2\_loop\_scalar\_coefficent\]) could be used to verify the recursion (\[recursion\]). With $f_{0}$ from (\[result\_1loop\]) and the beta function for $h_{ijkl}$ we get: $$\begin{aligned} & & f_{0} = - \frac{1}{16 \pi^2} \frac{1}{2} h_{iji'i'} + ... \\ & & \beta (h_{ijkl}) = ... + \frac{1}{16 \pi^2}( h_{iji'j'}h_{kli'j'} + h_{iki'j'}h_{jli'j'} + h_{ili'j'}h_{jki'j'}) + ... \\ & & f_{1} = \beta (h_{a b c d}) \frac{\partial}{\partial h_{a b c d}} f_{0} = - \frac{1}{(16 \pi^2)^2} \frac{1}{2}( h_{ijj'k'}h_{i'i'j'k'} + 2 h_{i i' j' k'}h_{j i' j' k'}) + ...\end{aligned}$$ which is the same as in (\[2\_loop\_scalar\_coefficent\]) (watch the form of the logarithm). In (\[2\_loop\_scalar\_coefficent\]) and (\[recursion\]) we use interchangeably $m \longleftrightarrow \mu$ in the logarithm, because the difference from this change is sub-leading. 2-loop Higgs mass corrections in the scalar singlets case --------------------------------------------------------- For the SM with a single scalar extension we have $$\begin{aligned} f_{0} = \frac{1}{16 \pi ^2} \left( 12\lambda + 2\lambda _{x} - 12 y_{t}^2 + \frac{3}{2} g_{1}^2 + \frac{9}{2} g_{2}^2 \right)\end{aligned}$$ That let us calulate the $f_{1}$ coefficient $$\begin{aligned} f_{1} = \frac{1}{16 \pi ^2} \left( 12 \beta(\lambda)+ 2 \beta( \lambda _{x} ) - 24 y_{t} \beta(y_{t}) + 3 g_{1} \beta(g_{1}) + 9 g_{2} \beta(g_{2}) \right)\end{aligned}$$ Inserting the beta functions from chapter 2, we obtain: $$\begin{aligned} f_{1} &=& \frac{1}{(16 \pi ^2)^2} ( 25 g_1^4 + 9 g_1^2 g_2^2 - 15 g_2^4 + 34 g_1^2 y_t^2 + 54 g_2^2 y_t^2 + 192 g_3^2 y_t^2 - 180 y_t^4 \nonumber\\ & & - 36 g_1^2 \lambda - 108 g_2^2 \lambda + 144 y_t^2 \lambda + 288 \lambda^2 - 3 g_1^2 \lambda_x - 9 g_2^2 \lambda_x + 12 y_t^2 \lambda_x \nonumber\\& & + 24 \lambda \lambda_x + 40 \lambda_x^2 + 48 \lambda_x \lambda_{\phi}+ 4 \lambda_x \textbf{Tr}\left( Y_{\phi} Y_{\phi} \right) )+ \ldots \label{f1}\end{aligned}$$ Standard Model result can be easily reproduced by putting all the singlet parameters to zero. 2-loop fine-tuning in the Standard Model ======================================== There are several classical theoretical constraints on the Higgs boson mass: unitarity, triviality, vacuum stability and fine-tuning. For a summary discussion of all these constraints see [@kolda], here we will concentrate on the triviality and the fine-tuning. Triviality bound {#Triviality} ---------------- A constraint traditionally called ’triviality’, is basically a constraint coming from the scale $\Lambda_{\infty}$ at which the value of a theory running parameter tends to infinity. If couplings increase monotonically with the momentum scale (running constants), the theory becomes non-perturbative near the pole (Landau Pole). The name of this effect comes from the fact, that only trivial (non-interacting) theory with vanishing quartic interactions is allowed if one tries to shift location of the pole to infinity. Similar effect is also present in QED. If the only allowed value for the renormalized charge is zero, theory is called non-interacting or ’trivial’. While the triviality problem in QED can be considered minor because the Landau pole scale is far beyond any observable energies, the Higgs boson’s Landau pole appears for much smaller energies and an acceptable solution is to make sure that the pole is above the value of the SM cut-off. This is used to set the ’triviality bound’ on the Higgs mass and the energy scale allowed for the SM. To evaluate location of the pole as a function of the Higgs mass, we will use the beta functions for the SM. In general, one has to solve the set of equations for all of the parameters in the SM. For our purposes, we will approximate the result by considering only the evolution of $\lambda$. $$\begin{aligned} \mu \frac{d \lambda}{d \mu} &=& \frac{3}{8} g_{1}^4 + \frac{9}{8} g_{2}^4 + \frac{3}{4} g_{1}^2 g_{2}^2 - 6 y_{t}^4 + 24 \lambda^2 + 12 y_{t}^2 \lambda -3 g_{1}^2 \lambda - 9 g_{2}^2 \lambda \label{triv_eq}\end{aligned}$$ We need also a specification of the initial conditions and we assume a given value of $\lambda$ at the energy scale $80$ GeV. $$\begin{aligned} \lambda(\mu = 80 \, \mathrm{GeV}) = \lambda_{0} \label{initial}\end{aligned}$$ Solutions of (\[triv\_eq\]) for specific values of $\lambda_ 0$ are shown in the left panel of fig. \[SM\_running\]. The condition for the Landau pole $\Lambda_{\infty}$ is the following: $$\begin{aligned} \lambda(\mu) |_{\mu \rightarrow \Lambda_{\infty}} \rightarrow \infty \label{landau_pole}\end{aligned}$$ Equation (\[landau\_pole\]) can be solved with respect to $\lambda_{0}$ and then the function $\lambda_{0} \left( \Lambda_{\infty} \right)$ leads to the triviality bound. For each $\lambda_{0}$ we want the Landau pole to be above the value of the SM cut-off, so the values of $\Lambda$ beyond $\Lambda_{\infty}$ are forbidden. The result is shown in the right panel of fig. \[SM\_running\] in terms of the Higgs mass $m_{h} = v \sqrt{2 \lambda_{0}}$. We obtained the solution $\lambda_{0} \left( \Lambda_{\infty} \right)$ shown in fig. \[SM\_running\] using numerical solving of the differential equation (\[triv\_eq\]) with initial condition (\[initial\]) in *Wolfram Mathematica 7*. The numerical solution procedure builds a so-called Interpolating Function Grid (see [@wolfram]) - a grid of points at which data is specified while solving the differential equation. The algorithm for a sufficiently large sampling range breaks down at a certain value, which in our case is the pole of the function $\lambda(\mu)$. We can extract the value of the breaking point $\Lambda_{\infty}$ from the Interpolating Function Grid for each initial parameter $\lambda_{0}$, which gives us $\lambda_{0} \left( \Lambda_{\infty} \right)$ - the triviality bound. In the language of *Mathematica*, the function looks as follows: $$\begin{aligned} & &\texttt{Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"]}; \nonumber\\ & &\Lambda \texttt{infinity[}\lambda_0 \texttt{]} := \texttt{Last[InterpolatingFunctionGrid[First[} \lambda /. \, \texttt{NDSolve[} \lbrace \nonumber\\ & & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \beta \, \texttt{[} \lambda\texttt{[}\mu\texttt{]]} == \mu \, \lambda'\texttt{[}\mu\texttt{]]}, \nonumber\\ & & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \lambda \, \texttt{[}80\texttt{]} == \lambda_0 \rbrace, \nonumber\\ & & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \lambda, \lbrace \mu, 1,1000000 \rbrace \texttt{]]]]} \, \texttt{[[}1\texttt{]]};\end{aligned}$$ where the number $1000000$ corresponds to the optional value of an upper bound of the sampling range in GeV, $\beta \, \texttt{[} \lambda\texttt{[}\mu\texttt{]]}$ is defined as the RHS of (\[triv\_eq\]). The function $\Lambda \texttt{infinity[}\lambda_0 \texttt{]}$ has to be inverted. The fine-tuning --------------- As we have mentioned before in the introduction, the fine-tuning is a very precise adjustment of parameters and we would like our theory not to require such procedures. The mass of Higgs boson has quadratically divergent corrections. For a large SM cut-off $\Lambda$, the mass of the Higgs particle should be of an order of $\Lambda$. To get an acceptable Higgs masses not larger than $1$ TeV, the self-energy corrections should be cancelled by the counterterms to a relatively small value of the Higgs boson mass[^4]. If $\Lambda$ is large, the fine-tuning between counterterms and quadratically divergent terms is needed. We would like to avoid such a fine-tuning. A solution for this problem was proposed at first by Veltman (see [@veltman] or [@kundu]). If the corrections to the Higgs self-energy at 1-loop accuracy are zero, the fine-tuning problem vanishes at the 1-loop order: $$\begin{aligned} m_{h}^2 + m_{Z}^2 + 2m_{W}^2 - 4 m_{t}^2 \simeq 0 \label{veltman_condition}\end{aligned}$$ By presenting such condition we assume an underlying theory that can explain the zero value of the divergence coefficient. Such theory may include an additional symmetry and should explain the relationship between the Higgs mass and masses of other particles obtained from (\[veltman\_condition\]). We’d like to estimate the cut-off $\Lambda$ by requiring the following: $$\begin{aligned} \left| \frac{\delta m_{h}^2}{m_{h}^2} \right| \leq \Delta_{h} \label{D_SM_1loop}\end{aligned}$$ Knowing the expression for $\delta m_{h}$ at 1-loop accuracy (here we take only the leading $\Lambda^2$ part) $$\begin{aligned} \delta m_{h \, 1loop \, SM}^{2} = \frac{\Lambda^2}{16 \pi^2} \left( 12 \lambda - 12 y_{t}^2 + \frac{3}{2} g_{1}^2 + \frac{9}{2} g_{2}^2 \right)\end{aligned}$$ one can impose the condition (\[D\_SM\_1loop\]) which gives us a fine-tuning allowed region in a plane $(\lambda, \Lambda)$ for specified values of $\Delta_{h}$. The plot shown in fig. \[FT\_SM\_1loop\] was obtained with the help of a simple RegionPlot function (see [@wolfram2]) in *Wolfram Mathematica 7* $$\begin{aligned} \texttt{RegionPlot[FineTuning[}m_{H}, \Lambda\texttt{]} \leq \Delta_{h} , \lbrace\Lambda, 1000, 100000\rbrace, \lbrace m_{H}, 1, 600 \rbrace \texttt{]}\end{aligned}$$ where the numbers $\lbrace\Lambda, 1000, 100000\rbrace$ correspond to the $\Lambda$ range in GeV, $\lbrace m_{H}, 1, 600 \rbrace$ is the $m_{h}$ range also in GeV and the function $\texttt{FineTuning[}m_{H}, \Lambda\texttt{]}$ is the LHS of (\[D\_SM\_1loop\]). The $\Delta_{h} = 0$ is fulfilled for $m_{h} \sim 310$ GeV. One can assume the fine-tuning cancellation to be very precise ($\Delta_{h} \sim 0.1$) or just quite good ($\Delta_{h} \sim 100$). Even the assumption of $\Delta_{h} \sim 100$ is very useful, because it reduces the arbitrariness of Higgs mass choice. The Veltman condition is sufficient to cancel quadratically divergent contributions to the Higgs mass only at the 1-loop order. A general form of leading higher order contributions, as in equation (\[einhorn\_eq\]) is $$\begin{aligned} m_{h}^2 \longrightarrow m_{h}^2 + \Lambda^2 \sum_{n = 0}^{\infty} f_{n}(\lambda_{i}) \log ^n \left(\frac{\Lambda}{\mu} \right)\end{aligned}$$ The coefficient $f_{1}$ for Standard Model can be deduced from (\[f1\]). We will concentrate on the 2-loop accuracy corrections, because 3-loop corrections are not relevant up to $\sim 50$ TeV scale. $$\begin{aligned} \delta m^2_{h \,2loops \, SM} &=& \frac{\Lambda ^2}{(16 \pi ^2)^2} \log \left( \frac{\Lambda}{\mu} \right) ( 25 g_1^4 + 9 g_1^2 g_2^2 - 15 g_2^4 + 34 g_1^2 y_t^2 + 54 g_2^2 y_t^2 + 192 g_3^2 y_t^2 \nonumber\\ & & - 180 y_t^4 \lambda - 36 g_1^2 - 108 g_2^2 \lambda + 144 y_t^2 \lambda + 288 \lambda^2 )\end{aligned}$$ where we put the renormalization scale to be the vacuum expectation value for the Higgs field $\mu = v = 246$ GeV. As before we can use the estimation of corrections for different $\Delta_{h}$ $$\begin{aligned} \left| \frac{\delta m_{h \, 1loop \, SM}^2 + \delta m_{h \, 2loops \, SM}^2}{m_{h}^2} \right| \leq \Delta_{h} \label{D_SM_2loop}\end{aligned}$$ As a result of this constraint we have a forbidden region in a plane $(\lambda,\Lambda)$ (or $(m_{h},\Lambda)$), which one can see in fig. \[FT\_SM\_2loop\]. The plot was obtained in the same way as fig. \[FT\_SM\_1loop\]. 2-loop fine-tuning in the scalar singlet extension of the Standard Model ========================================================================= So far we presented the SM extension with $N_{\phi}$ singlet scalar fields and singlet right-handed massive neutrinos. We would like now to show, why this particular SM extension is a useful idea to the particle physics and, as in the previous chapter, discuss classical Higgs mass constraints: triviality and fine-tuning. The model with one singlet was presented in [@grzadkowski]. It is the most economic extension of the SM for which the fine-tuning problem is improved while preserving all the successes of the SM. Other advantages of the model are the presence of the Dark Matter candidate, neutrino masses and mixing or possible lepton asymmetry, however in this work we concentrate only on moderating the quadratic divergences of the Higgs mass. The Lagrangian for the model with a single new scalar field $\phi$ with the gauge invariant coupling to the Higgs doublet and three singlet right-handed Majorana neutrinos reads: $$\begin{aligned} L &=& L_{SM} + \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi + \frac{1}{2} \mu _{\phi} \phi ^2 + \lambda_{\phi} \phi ^4 + \lambda_{x} (H ^\dagger H) \phi ^ 2 + \nonumber\\ & & + \overline{\nu_{R}} i \slashed{\partial} \, \nu_{R} - \frac{1}{2} \left( \overline{(\nu_{R})^{c}} M \nu_{R} + h.c.\right) - \frac{1}{2} \left( \overline{(\nu_{R})^{c}} Y_{\phi} \nu_{R} \phi + h.c. \right) \label{lagrangian_scalarandneutrinos}\end{aligned}$$ Through this renormalizable extension, we would like to generate additional radiative corrections to the Higgs boson mass that can soften the little hierarchy problem. The SM contributions to the quartic divergence are dominated by the top quark. Therefore introducing an extra scalar (different statistics) can suppress the SM result leading to a theory with ameliorated hierarchy problem. We will show that this leads also to constraints for the mass of the Higgs boson. The triviality bound -------------------- As mentioned before in section \[Triviality\], for the full triviality constraint, one has to solve the set of equations for all of the parameters in the SM extension. For our purposes, we will approximate the result by considering only the evolution of $\lambda$ and $\lambda_x$. $$\begin{aligned} \mu \frac{d \lambda}{d \mu} &=& \frac{3}{8} g_{1}^4 + \frac{9}{8} g_{2}^4 + \frac{3}{4} g_{1}^2 g_{2}^2 - 6 y_{t}^4 + 24 \lambda^2 + 12 y_{t}^2 \lambda -3 g_{1}^2 \lambda - 9 g_{2}^2 \lambda + 2\lambda_{x}^2 \nonumber\\ \label{diff_set1}\\ \mu \frac{d \lambda_{x}}{d \mu} &=& 12 \lambda \lambda_{x} + 24 \lambda_{\phi} \lambda_{x} + 8 \lambda_{x}^2 + 6 y_{t}^2 \lambda_{x} - \frac{3}{2} g_{1}^2 \lambda_{x} - \frac{9}{2} g_{2}^2 \lambda_{x} + 2 \textbf{Tr} ( Y_{\phi} Y_{\phi} ) \lambda_x \nonumber\\ \label{diff_set2}\end{aligned}$$ The solution for this set of differential equations, with initial conditions $$\begin{aligned} \lambda(\mu = 80 \, \mathrm{GeV})&=& \lambda_0 \label{initial1}\\ \lambda_{x}(\mu = 80 \, \mathrm{GeV})&=& \lambda _{x \, 0} \label{initial2}\end{aligned}$$ are functions $\lambda(\mu)$ and $\lambda_{x}(\mu)$ that have a pole for a specific value $\Lambda_{\infty}$ depending on (\[initial1\]) and (\[initial2\]). As in the previous chapter, if we want to make sure that the Landau pole is above the SM cut-off, then we receive a constraint on $m_{h}$ and $\Lambda$. The region in $(m_{h},\Lambda)$ plane, forbidden due to this constraint, depends on the initial parameter $\lambda_{x \, 0}$ and the matrix $Y_{\phi}$ in (\[lagrangian\_scalarandneutrinos\]). We assume $\lambda_{\phi} \sim 0.1$ and therefore $\lambda_{\phi}$ effects do not influence the result much. We will also assume the form of $Y_{\phi}$ matrix as it is in [@grzadkowski] (which is a consequence of the $Z_{2}$ symmetry of the singlet scalar field): $$\begin{aligned} Y_{\phi} = \left( \begin{array}{ccc} 0 & 0 & b_1 \\ 0 & 0 & b_2 \\ b_1 & b_2 & 0 \\ \end{array} \right)\end{aligned}$$ We will assume $b_1 = b_2 = b$ and choose $b$ such that the 1-loop corrections to the singlet scalar mass $m_{\phi}$ cancel assuming small $\lambda_{\phi}$ (see [@grzadkowski] and [@grzadkowski2] for details). From (\[general\_1loop\]) we can determine the correction to the scalar singlet mass $$\begin{aligned} \delta m_{\phi}^2 = \frac{\Lambda^2}{16 \pi^2 } \left( - \frac{\Lambda^2}{2} h_{\phi \phi i i} + 2 \textbf{Tr}( Y_{\phi} Y_{\phi} ) \right) = \frac{1}{16 \pi^2 } \left( - 4 \lambda_{x} - 12\lambda_{\phi} + 8 b^2 \right) \simeq 0\end{aligned}$$ which gives us $b \simeq \sqrt{\frac{\lambda_{x}}{2}}$. The triviality bound on $m_{h}$ as a function of $\Lambda$, for different values of the initial parameter $\lambda_{x \, 0}$, can be seen in fig. \[triv\_lambdax\]. As one can see, a point $(m_{h},\Lambda)$ that is prohibited for $\lambda_{x \, 0} = 5$ can be allowed if $\lambda_{x \, 0} = 0$. The allowed region shrinks as $\lambda_{x \, 0}$ grows. Therefore, we will take the intersection of the prohibited regions as the triviality bound for $m_{h}$, which corresponds to the $\lambda_{x \, 0} = 0$. We should not forget that also the $\lambda_{x}(\mu)$ function has the Landau divergence. Location of the pole depends on the initial values $\lambda_{x \, 0}$ and $\lambda_{0}$. Growing $\lambda_{x \, 0}$ implies a shift of the pole position towards smaller energies. For every initial condition $\lambda_{0}$ we should specify a certain range of $\lambda_{x \, 0}$ that the Landau pole of $\lambda_{x}(\mu)$ is above a given value of $\Lambda$. Therefore, not every value of $\lambda_{x \, 0}$ parameter is allowed for each Higgs mass and the cut-off $\Lambda$. The maximum $\lambda_{x \, 0}$ one can see in the figure \[lambdaMAX\]. The results in figures \[triv\_lambdax\] and \[lambdaMAX\], were both obtained through the same numerical procedure in *Mathematica* as introduced before in section \[Triviality\]. ![The “Triviality bound” dependence on $\Lambda$ for fixed values $\lambda_{x \, 0} = 0.1, 1, 2, 5$ (starting with the upper most). The region above each curve is forbidden by the triviality constraint for the specific set of parameters.[]{data-label="triv_lambdax"}](Graph/Triv_lambdaxFF.jpg){height="9"} ![Maximum $\lambda_{x \, 0}$ allowed by the triviality bound, as a function of $m_h$ and $\Lambda$. We assume also $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="lambdaMAX"}](Graph/Lambda_X_MAX1.jpg){height="9"} The fine-tuning --------------- To discuss the fine-tuning in the SM extension with a singlet scalar field and right singlet neutrinos, we need the full result for 1-loop and 2-loops corrections to the Higgs mass: $$\begin{aligned} \delta m^2_{h \,1loop} &=& \frac{\Lambda ^2}{16 \pi^2} \left( 12 \lambda + 2 \lambda_x - 12 y_t^2 + \frac{3}{2} g_1^2 + \frac{9}{2} g_2^2 \right) \nonumber\\ & & - \frac{1}{16 \pi^2} \left( 6 \lambda m_{h}^2 \log \left( \frac{m_{h}^2+\Lambda^2}{m_{h}^2} \right) + 2 \lambda_x m_{\phi}^2 \log \left( \frac{m_{\phi}^2+\Lambda^2}{m_{\phi}^2} \right) \right)\\ \delta m^2_{h \,2loops} &=& \frac{\Lambda ^2}{(16 \pi ^2)^2} \log \left( \frac{\Lambda}{\mu} \right) \left( 25 g_1^4 + 9 g_1^2 g_2^2 - 15 g_2^4 + 34 g_1^2 y_t^2 + 54 g_2^2 y_t^2 \right. \nonumber\\ & & + 192 g_3^2 y_t^2 - 180 y_t^4 \lambda - 36 g_1^2 - 108 g_2^2 \lambda + 144 y_t^2 \lambda + 288 \lambda^2 \nonumber\\ & & - 3 g_1^2 \lambda_x - 9 g_2^2 \lambda_x + 12 y_t^2 \lambda_x + 24 \lambda \lambda_x + 40 \lambda_x^2 + 48 \lambda_x \lambda_{\phi} \nonumber\\ & & \left. + 4 \lambda_x \textbf{Tr}\left( Y_{\phi} Y_{\phi} \right) \right.)\end{aligned}$$ where the logarithmic terms in the 1-loop correction are kept as relevant because of the high value of $m_{\phi}$ parameter. As before, the corrections should be relatively small in comparison with the Higgs mass, so we again introduce the fine-tuning parameter $\Delta_{h}$ $$\begin{aligned} \left| \frac{\delta m_{h \, 1loop}^2 + \delta m_{h \, 2loops}^2}{m_{h}^2} \right| \leq \Delta_{h} \label{D_sinlet_2loop}\end{aligned}$$ We would like to repeat the assumptions from the previous section: $\lambda_{\phi} \sim 0.1$, $Y_{\phi}$ should roughly cancel the 1-loop correction to the scalar singlet mass $m_{\phi}$. Higgs coupling to the singlet scalar has to satisfy the following condition for every $m_{h}$ and $\Lambda$ $$\begin{aligned} 0.1 \leq \lambda_{x \, 0} \leq \lambda_{x \, 0}^{MAX} (m_{h},\Lambda) \leq 5 \label{lambda_condition}\end{aligned}$$ where $\lambda_{x \, 0}^{MAX}(m_{h},\Lambda)$ is the triviality constraint (see fig. \[lambdaMAX\]). We would like the singlet scalar mass $m_{\phi}$ to be in a range 500 - 5000 GeV and, in order to satisfy $<\phi>=0$, it must also fulfil the inequality $$\begin{aligned} m_{\phi}^2 - \lambda_{x \, 0} v^2 > 0 \label{mphi_condition}\end{aligned}$$ where $v = 246$ GeV is the Higgs field vacuum expectation value (see [@grzadkowski] for details). With all these assumptions we can now consider allowed values of $m_{h}$ and $\Lambda$ for different $\Delta_{h}$. For each point in the allowed by triviality part of the $(m_{h},\Lambda)$ plane we have a set of parameters $\lambda_{x \, 0}$ and $m_{\phi}$ such that they satisfy all of the just mentioned conditions. If there is no such a set of $\lambda_{x \, 0}$ and $m_{\phi}$ that the fine-tuning inequality (\[D\_sinlet\_2loop\]) is fulfilled for a specified value of $\Delta_{h}$, then the point $(m_{h},\Lambda)$ belongs to the forbidden by $\Delta_{h}$ fine-tuning region. We can solve these numerically using *Mathematica*. A simplified program that minimizes the LHS of (\[D\_sinlet\_2loop\]) in terms of allowed $\lambda_{x \, 0}$ and $m_{\phi}$ and compares it with $\Delta_{h}$, obtaining plots such as in figs. \[wszystkieNaRaz1\] and \[wszystkieNaRaz2\], is the following: $$\begin{aligned} & &\texttt{Figure[}\Delta_{h}\texttt{] := RegionPlot[First[NMinimize[}\lbrace \nonumber\\ & & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \texttt{FineTuning[}m_{H}, \lambda_{x \, 0}, b, \Lambda, m_{\phi}\texttt{]}, \nonumber\\ & & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 0.1 \leq \lambda_{x \, 0} < \texttt{LambdaXMAX[}m_{h}, \Lambda\texttt{]}, \nonumber\\ & & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 500 < m_{\phi} < 5000, \nonumber\\ & & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, b == \sqrt{( \lambda_{x \, 0} / 2)}, \nonumber\\ & & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, m_{\phi}^2 - \lambda_{x \, 0} v^2 > 0 \rbrace, \nonumber\\ & & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \lbrace \lambda_{x \, 0}, m_{\phi} \rbrace \texttt{]]} > \Delta_{h} , \lbrace \Lambda, 1000, 100000 \rbrace, \lbrace m_{h}, 1, 600 \rbrace\ \texttt{]}\end{aligned}$$ where $\texttt{FineTuning[}m_{H}, \lambda_{x \, 0}, b, \Lambda, m_{\phi}\texttt{]}$ is the LHS from (\[D\_sinlet\_2loop\]), $\texttt{LambdaXMAX[}m_{h}, \Lambda\texttt{]}$ is the function from (\[lambda\_condition\]), the $m_{\phi}$ range 500 to 5000 is in GeV, such as the ranges $\lbrace \Lambda, 1000, 100000 \rbrace$ and $\lbrace m_{h}, 1, 600 \rbrace $. In the right panel of figs. \[wszystkieNaRaz1\] and \[wszystkieNaRaz2\] allowed regions of $m_{h}$ and $\Lambda$ are shown in the singlet extended model in comparison with the SM results (left panel). What we can observe, is that the part for low Higgs mass which is forbidden in the SM fine-tuning plots, is allowed in the singlet scalar extension. This happens because, for low Higgs mass the leading contribution to the mass correction comes from the Yukawa top quark coupling. In the extended model it cancels with the contributions from the singlet scalar, as they come with opposite sings (different statistics). For large Higgs masses, the mass correction coming from the Higgs quartic coupling dominates over the correction from the top quark. As all the scalar contributions are of the same sign, they can’t cancel each other. Increasing the additional couplings coming from the presence of the singlet scalar only worsen the fine-tuning condition. That is also why the upper bound difference between models is negligible - for the large Higgs masses we have $\lambda_x \simeq 0$. ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 100$ and $\Delta_{h} = 10$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz1"}](Graph/SM_napis.jpg "fig:"){height="0.3"}  ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 100$ and $\Delta_{h} = 10$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz1"}](Graph/SMS_napis.jpg "fig:"){height="0.3"}\ ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 100$ and $\Delta_{h} = 10$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz1"}](Graph/m_h_GeV.jpg "fig:"){height="4cm"} ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 100$ and $\Delta_{h} = 10$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz1"}](Graph/SM100a.jpg "fig:"){height="6cm"} ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 100$ and $\Delta_{h} = 10$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz1"}](Graph/Singlet100a.jpg "fig:"){height="6cm"} ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 100$ and $\Delta_{h} = 10$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz1"}](Graph/D100_napis.jpg "fig:"){height="4cm"}\ ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 100$ and $\Delta_{h} = 10$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz1"}](Graph/m_h_GeV.jpg "fig:"){height="4cm"} ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 100$ and $\Delta_{h} = 10$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz1"}](Graph/SM10a.jpg "fig:"){height="6cm"} ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 100$ and $\Delta_{h} = 10$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz1"}](Graph/Singlet10a.jpg "fig:"){height="6cm"} ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 100$ and $\Delta_{h} = 10$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz1"}](Graph/D10_napis.jpg "fig:"){height="4cm"}\ ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 100$ and $\Delta_{h} = 10$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz1"}](Graph/Lambda_TeV.jpg "fig:"){height="0.4"} ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 100$ and $\Delta_{h} = 10$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz1"}](Graph/Lambda_TeV.jpg "fig:"){height="0.4"} ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 1$ and $\Delta_{h} = 0.1$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz2"}](Graph/SM_napis.jpg "fig:"){height="0.3"}  ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 1$ and $\Delta_{h} = 0.1$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz2"}](Graph/SMS_napis.jpg "fig:"){height="0.3"}\ ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 1$ and $\Delta_{h} = 0.1$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz2"}](Graph/m_h_GeV.jpg "fig:"){height="4cm"} ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 1$ and $\Delta_{h} = 0.1$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz2"}](Graph/SM1a.jpg "fig:"){height="6cm"} ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 1$ and $\Delta_{h} = 0.1$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz2"}](Graph/Singlet1a.jpg "fig:"){height="6cm"} ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 1$ and $\Delta_{h} = 0.1$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz2"}](Graph/D1_napis.jpg "fig:"){height="4cm"}\ ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 1$ and $\Delta_{h} = 0.1$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz2"}](Graph/m_h_GeV.jpg "fig:"){height="4cm"} ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 1$ and $\Delta_{h} = 0.1$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz2"}](Graph/SM01a.jpg "fig:"){height="6cm"} ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 1$ and $\Delta_{h} = 0.1$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz2"}](Graph/Singlet01a.jpg "fig:"){height="6cm"} ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 1$ and $\Delta_{h} = 0.1$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz2"}](Graph/D01_napis.jpg "fig:"){height="4cm"}\ ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 1$ and $\Delta_{h} = 0.1$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz2"}](Graph/Lambda_TeV.jpg "fig:"){height="0.4"} ![Allowed regions (white) for $m_h$ and $\Lambda$ resulting from the fine-tuning in the SM and the SM singlet extension for $\Delta_{h} = 1$ and $\Delta_{h} = 0.1$. Dark grey region is excluded by the triviality argument for any value of $\lambda_{x \, 0}$ in the range $0.1 \leq \lambda_{x \, 0} \leq 5$.[]{data-label="wszystkieNaRaz2"}](Graph/Lambda_TeV.jpg "fig:"){height="0.4"} Summary and conclusions ======================= There are two main results of this work. First result are the derived 1-loop equations for beta functions in general gauge theory with scalars and fermions and a single gauge symmetry and the 1- and 2-loop quadratic corrections to scalar masses, including contributions from Dirac and Majorana fermions. In the second part of the work we studied the theoretical constraints on the Higgs mass and new physics scale coming from triviality and fine-tuning. In the SM, the fine-tuning condition gives a significant constraint on the Higgs boson mass and on the scale of new physics beyond the SM. **However, the one scalar singlet SM extension opens a window for the low Higgs masses without significant constraint on the new physics scale.** There are still more questions to be answered about the singlet scalar Standard Model extension. Is the new particle a good Dark Matter candidate? Can it explain the leptogenesis? What about multi-singlet SM extensions? Feynman rules for general gauge theory ====================================== The Feynman rules for the propagators for the general gauge theory with scalar, gauge boson, ghost and fermion fields, with no mass for the scalar and gauge fields (for the full Lagrangian see \[lagrangian\_general\])\ [XX]{} & $\tilde{D}_{F}(p) = \frac{i}{p^2 - m^2 + i \epsilon }$\ & $\tilde{D}_{F}^{\mu \nu}(p) = \frac{i (- g^{\mu \nu} + (1- \xi) \frac{p^{\mu} p^{\nu}}{p^2} )}{p^2 - m^2 + i \epsilon }$\ & $\tilde{G}_{F}(p) = \frac{i }{p^2 + i \epsilon }$\ & $\tilde{S}_{F}(p) = \frac{i (\slashed{p} + m)}{p^2 - m^2 + i \epsilon }$\ Wave-function renormalization counterterms contribution to propagators: [XX]{} & $ i \left( \Delta Z_{\phi} \right)_{i j} p^2 $\ & $ i \Delta Z_{A} \left(- p^2 g_{\mu \nu} + p_{\mu} p_{\nu} \right)\delta_{a b} - \frac{i}{\xi} K_{\xi} p_{\mu} p_{\nu} \delta_{a b} $\ & $ i \Delta Z_{\eta} p^2 \delta_{a b} $\ & $ i \frac{1}{2} \left( \Delta Z_{\psi} ^{\dagger} + \Delta Z_{\psi} \right)_{n m} \slashed{p} $ \[fermion\_propagator\_counter\]\ The Feynman rules for the vertices:\ [XX]{} & $-ig \overline{\textbf{T}}^a_{mn} \gamma_{\mu}$\ & $-i \kappa^{i}_{mn}$\ \ The following diagram is symmetric under interchanges $i, j$, which must be included in the vertex coupling. Considering the fact that $T^{a}_{i,j}$ is hermitian and imaginary, the vertex coupling simplifies to: [XX]{} & $ ig (p^{\mu} - q^{\mu}) \textbf{T}^a _{ij} $\ \ Fol term is symmetric under interchanges $(a, \mu),(b, \nu)$ and $i, j$. To have an expression which treats all of the interacting in the vertices fields the same, we need to include all the interchanges. [XX]{} & $ -i g^2 g^{\mu \nu} (\textbf{T}^a _{kj} \textbf{T}^b _{ki} + \textbf{T}^a _{ki} \textbf{T}^b _{kj}) $\ \ The quadrilinear term is symmetric under interchanges $(a, \mu),(b, \nu),(c, \rho),(d, \sigma)$. To have an expression which treats all of the interacting in the vertices gauge fields the same, we need to include all the interchanges. [XX]{} & $$\begin{aligned} -i g^2 ( f_{eab} f_{ecd} (g^{\mu \rho} g^{\nu \sigma} - g^{\mu \rho} g^{\nu \sigma}) \nonumber\\ f_{eac} f_{ebd} (g^{\mu \nu} g^{\rho \sigma} - g^{\mu \sigma} g^{\nu \rho}) \nonumber\\ f_{ead} f_{ebc} (g^{\mu \nu} g^{\rho \sigma} - g^{\mu \rho} g^{\nu \sigma}) ) \nonumber\end{aligned}$$\ The trilinear term is totally antisymmetric under interchanges $(k, \mu),(q, \nu),(p, \rho)$. To have an expression which treats all of the interacting in the vertices gauge fields the same, we need to include all the interchanges. [XX]{} & $ g f_{abc} (g^{\mu \nu}(k-q)^{\rho}+g^{\nu \rho}(q-p)^{\mu}+g^{\rho \mu}(p-k)^{\nu}) $\ [XX]{} & $ - g f_{abc} q^{\mu}$\ [XX]{} & $ -i h _{ijkl}$\ Feynman rules for the counterterms relevant in the work: [XX]{} & $ - i \gamma ^\mu \left( \Delta g \overline{\textbf{T}}^a \right)_{m n} $\ [XX]{} & $ - i \Delta \kappa^{i}_{m n}$\ [XX]{} & $ -i \Delta h _{ijkl}$\ Table of Integrals ================== Integrals in the dimensional regularization\ $$\begin{aligned} & & \mu^{\epsilon} \int \frac{d^{d} q}{(2\pi)^{d}} \frac{1}{q^2 (p+q)^2} = \frac{2 i}{16 \pi^2 \epsilon} \\ & & \mu^{\epsilon} \int \frac{d^{d} q}{(2\pi)^{d}} \frac{q^{\mu}}{q^2 (p+q)^2} = - \frac{ i p^{\mu}}{16 \pi^2 \epsilon} \\ & & \mu^{\epsilon} \int \frac{d^{d} q}{(2\pi)^{d}} \frac{q^{\mu} q^{\nu}}{q^2 (p+q)^2} = \frac{i}{16 \pi^2 \epsilon} \left( \frac{2}{3} p^{\mu} p^{\nu} - \frac{1}{6} g^{\mu \nu} p^2 \right) \\ & & \mu^{\epsilon} \int \frac{d^{d} q}{(2\pi)^{d}} \frac{q^{\mu} q^{\nu}}{q^2 q^2 (p+q)^2} = \frac{i g^{\mu \nu}}{32 \pi^2 \epsilon} \\ & & \mu^{\epsilon} \int \frac{d^{d} q}{(2\pi)^{d}} \frac{q^{\mu} q^{\nu} q^{\alpha} q^{\beta}}{q^2 q^2 q^2 (p+q)^2} = \frac{1}{12} \frac{i}{16 \pi^2 \epsilon} (g^{\mu \nu} g^{\alpha \beta} + g^{\mu \alpha} g^{\nu \beta} + g^{\mu \beta} g^{\nu \alpha})\end{aligned}$$ Integrals in the cut-off regularization (following [@inami], [@varin] )\ $$\begin{aligned} & & \int \frac{d^{4} q}{(2\pi)^{4}} \frac{i}{q^2 - m^2} = \frac{1}{16 \pi^2} \left( \Lambda ^2 - m^2 \log \left( \frac{m^2 + \Lambda^2}{m^2} \right) \right) \\ & & \int \frac{d^{4} q}{(2\pi)^{4}} \frac{i}{(q^2 - m_{a}^2) (q^2 - m_{b}^2)} = \frac{1}{16 \pi^2} \log \left( \frac{\Lambda ^2}{m_{a} m_{b}} \right)+ \ldots \\ & & \int \int \frac{d^{4} q}{(2\pi)^{4}} \frac{d^{4} k}{(2\pi)^{4}} \frac{1}{(q^2 - m_{a}^2) ((q+k)^2 - m_{b}^2) (k^2 - m_{c}^2) } = - \frac{1}{(16 \pi^2)^2} 2 \Lambda ^2 + \ldots\end{aligned}$$ Feynman Rules for Majorana Fermions =================================== Feynman rules for Majorana neutrinos can be found for example in [@denner] or [@gluza]. In this appendix $\psi$ denotes a Majorana fermion field and $\varphi$ a scalar field. We are interested in the following Lagrangian: $$\begin{aligned} L = \overline{\psi} i \slashed{\partial} \psi - \frac{1}{2} M \left( \overline{\psi^{c}} \psi + \overline{\psi} \psi^{C} \right) - \frac{1}{2} \left( \overline{\psi^c} Y_{\varphi} \psi \varphi + \overline{\psi} Y_{\varphi} \psi^c \varphi\right)\end{aligned}$$ where $(\,)^{c}$ denotes the charge conjugation operator, $\overline{\psi^{c}} = \psi^{T} \hat{C}$, $\hat{C}$ is an antisymmetric charge conjugation matrix. We define $a^{\dagger}$ and $b^{\dagger}$ as the creation operator of fermion and antifermion, respectively. Similarly $a$ and $b$ are the annihilation operators. $d^{\dagger}$ and $d$ are creation and annihilation operators of the scalar particle $\varphi$. $|(k,\lambda)\rangle$ is a state of a single Majorana fermion of momentum $k$ and helicity $\lambda$. $|k\rangle$ denotes a one scalar particle state of momentum $k$. $$\begin{aligned} & & |(k,\lambda)\rangle = a^{\dagger}_{k,\lambda} |0\rangle \\ & & |k \rangle = d^{\dagger}_{k} |0\rangle\end{aligned}$$ where $|0\rangle$ is the vacuum state. We would like to determine the Feynman rule for a Yukawa interaction vertex with two Majorana fermions, as in diagram \[majorana\_yukawa\]. Below $\textbf{T}$ denotes the time-order operator. $$\begin{aligned} \texttt{diagram \ref{majorana_yukawa}} &=& \langle 0 \vert a_{k_1, \lambda_1} a_{k_2, \lambda_2} \textbf{T} \left[ \int d^4 x \left( - i \frac{1}{2} \varphi (x) \psi (x) ^T \hat{C} Y_{\varphi} \psi (x) \right) \right] b^{\dagger}_{k_3} \vert 0 \rangle = \nonumber\\ &=& - i v^T_{k_1, \lambda_1} \,\frac{1}{2}(\hat{C} Y_{\varphi} - Y_{\varphi} \hat{C} ^T ) \, v _{k_2, \lambda_2} \, \delta^4 (k_1 + k_2 - k_3) \nonumber\\ &=& - i v^T_{k_1, \lambda_1} \, \hat{C} Y_{\varphi} \, v _{k_2, \lambda_2} \, \delta^4 (k_1 + k_2 - k_3)\end{aligned}$$ Therefore, the Feynman rule for a Yukawa interaction for Majorana fermion vertex with fermion lines as in diagram \[majorana\_yukawa\] is simply $(-i \hat{C} Y_{\varphi})$. The Feynman rule for Majorana fermion propagator can be obtained for example from [@gluza]:\ [XX]{} & $ \langle 0 \vert T \left[ \psi ^T (x) \psi (y) \right] \vert 0 \rangle = - i \left( S(x - y) \hat{C} \right) $\ \ [99]{} A. O. Bouzas, *Mixing-matrix renormalization revisited*, Eur. Phys. J. C20, (2001), 239-252 C. D. Palmer and M. E. Carrington *A general expression for Symmetry Factors of Feynman Diagrams*, arXiv:hep-th/0108088 D. Bailin and A. Love, *Introduction to Gauge Field Theory*, IOP Publishing, (1993) T. P. Cheng *et al.* *Higgs phenomena in asymptotically free gauge theories*, Phys. Rev. D, Vol 9, No 8, (1974) S. Pokorski, *Gauge Field Theories*, Cambridge University Press, (2000) M. E. Machacek and M. T. Vaughn *Two-loop renormalization group equations in a general quantum field theory,I. Wave function renormalization*, Nucl. Phys. B222, (1983), 83-103 M. E. Machacek and M. T. Vaughn *Two-loop renormalization group equations in a general quantum field theory,II. 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Murayama *The Higgs Mass and New Physics Scales in the Minimal Standard Model*, arXiv:hep-ph/0003170 Wolfram Research, *Mathematica Tutorial*, Utility Packages for Numerical Differential Equation Solving,\ http://reference.wolfram.com/mathematica/tutorial/NDSolvePackages.html Wolfram Research, *Mathematica*, Visualisation And Graphics, Function Visualisation, http://reference.wolfram.com/mathematica/ref/RegionPlot.html M. Veltman *The Infrared - Ultraviolet Connection*, Acta Phys. Pol. B12,(1981), 437 A. Kundu and S. Raychaudhuri *Taming the scalar mass problem wit a singlet Higgs boson*, arXiv:hep-ph/9410291 B. Grzadkowski and J. Wudka *Pragmatic approach to little hierarchy problem*, arXiv:hep-ph/0902.0628v3 B. Grzadkowski *A Natural Two-Higgs Doublet Model*, arXiv:hep-ph/0910.4068v1 T.Inami *et al.* *Cancellation of quadratic divergences and uniqueness of softly broken supersymmetry*, Phys. Lett. B, Volume 117, Issue 3-4, p. 197-202 (1982) T.Varin *et al.* *How to preserve symmetries with cut-off regularized integrals?*, arXiv:hep-ph/0611220v1 A. Denner *et al.* *Feynman rules for fermion-number-violating interactions*, Nucl. Phys. B387, (1992), 467-484,\ A. Denner *et al.* *Compact Feynman rules for Majorana fermions*, Phys. Lett. B291, (1992), 278-280 J. Gluza and M. Zralek *Feynman rules for Majorana-neutrinos interactions*, Phys. Rev. D45, (1992), 1693-1700 [^1]: There are some discussions in the literature over the definition of fine-tuning and the degrees of precision in adjustments of parameters. The definition of fine-tuning adopted here will be specified later [^2]: A Lagrangian for the Standard Model can be found in many places in literature, see for example [@bailin], [@pokorski]. [^3]: In the SM with singlets the ratio of the $\Lambda ^2$ (no $\log (\Lambda)$ ) term in 2-loop correction and the 1-loop correction is $\frac{0.15 \lambda^2 + 0.05 \lambda_{x}^2}{6 \lambda + \lambda_{x}}$ which for $\lambda \sim 1$ and $\lambda_{x} < 5$ is negligible. [^4]: Radiative corrections for fermion and vector boson masses do not contain quadratic divergences

--- abstract: 'We consider a real, massive scalar field on $\PAdS_{d+1}$, the Poincaré domain of the $(d+1)$-dimensional anti-de Sitter (AdS) spacetime. We first determine all admissible boundary conditions that can be applied on the conformal boundary, noting that there exist instances where “bound states” solutions are present. Then, we address the problem of constructing the two-point function for the ground state satisfying those boundary conditions, finding ultimately an explicit closed form. In addition, we investigate the singularities of the resulting two-point functions, showing that they are consistent with the requirement of being of Hadamard form in every globally hyperbolic subregion of $\PAdS_{d+1}$ and proposing a new definition of Hadamard states which applies to $\PAdS_{d+1}$.' author: - Claudio Dappiaggi - 'Hugo R. C. Ferreira' date: Revised October 2016 title: | Hadamard states for a scalar field in anti-de Sitter spacetime\ with arbitrary boundary conditions --- Introduction ============ Quantum field theory on curved backgrounds is nowadays a well-established, thriving branch of mathematical and theoretical physics. In the past decade, especially thanks to the algebraic approach [@Benini:2013fia; @Brunetti:2015vmh], not only several specific models, including those with perturbative interactions, were thoroughly studied, but also foundational and structural aspects, such as renormalization or local gauge invariance, were analyzed. A cornerstone of most of the recent papers is the assumption that the underlying background is globally hyperbolic. This condition on the geometry of the spacetime guarantees that solutions to wave like operators, such as the Klein-Gordon, the Dirac or the Proca equation, can be found in terms of an initial value problem. As a consequence, whenever one considers a free field theory, one can follow a well-established quantization scheme, to associate with any such systems an algebra of observables, encompassing the information on structural properties such as dynamics, locality and causality. The only choice consists in the selection of a quantum state, but, also in this respect, it is nowadays universally accepted that a physically acceptable criterion lies in the so-called Hadamard condition. This is a technical requirement which guarantees on the one hand that the singular behavior of the two-point function $G^+(x,x^\prime)$ of the underlying free field theory mimics in the ultraviolet regime that of the Poincaré vacuum, while, on the other hand, the quantum fluctuations of all observables are finite, [@Kay:1988mu; @Khavkine:2014mta]. As a consequence one can give a covariant definition of Wick polynomials, extending the standard one on Minkowski spacetime, and, as a by-product, interactions can be introduced at a perturbative level. In other words, if one focuses the attention on quasi-free/Gaussian states, selecting a physically acceptable state boils down to the construction of a positive, two-point function $G^+(x,x^\prime)$. This is a solution of the equation of motion in both entries, with a prescribed singular behavior. The most famous examples of Hadamard states are the Poincaré vacuum and the Bunch-Davies state on de Sitter spacetime [@Allen; @BD], but several construction schemes are nowadays known, especially on black hole [@Kay:1988mu; @Dappiaggi:2009fx; @Sanders:2013vza; @Gerard] and cosmological spacetimes [@Olbermann:2007gn; @Them:2013uka; @Dappiaggi:2007mx; @Dappiaggi:2008dk]. The situation changes drastically the moment we drop the assumption of the spacetime being globally hyperbolic. Already at a classical level we face additional difficulties since we cannot construct and characterize the solutions of the equation of motion just in terms of an initial value problem. It is therefore tempting to take the easy path of concluding that these scenarios are not of interest since they bear no physical information. This attitude is not justified as there are renowned, experimentally verified effects, e.g. the Casimir force, which are described by field theoretical models whose underlying geometry is not that of a globally hyperbolic spacetime, since the manifold possesses boundaries [@Dappiaggi:2014gea]. Another relevant instance of a manifold which is not globally hyperbolic, while being central in several, important physical models is the $(d+1)$-dimensional anti de Sitter $\AdS_{d+1}$ spacetime, $d \geq 2$. This is a maximally symmetric solution of the vacuum Einstein equations with a negative cosmological constant $\Lambda$ whose underlying manifold $M\simeq\bS^1\times\bR^d$ is such that the time coordinates runs along $\bS^1$, hence yielding closed timelike curves [@HawkingEllis §5.2]. In this paper, we will focus on this class of backgrounds, more precisely on the so-called Poincaré fundamental domain $\PAdS_{d+1}$ which covers only a portion of the full $\AdS_{d+1}$ spacetime and which is extensively used in the prominent AdS/CFT correspondence — see for example the recent monograph [@Ammon:2015wua]. Contrary to $\AdS_{d+1}$, $\PAdS_{d+1}$ can be described as the subset $\bR^d\times(0,\infty)$ of $\bR^{d+1}$ endowed with the metric $\dd s^2=\frac{\ell^2}{z^2}(-\dd t^2 + \dd z^2 + \delta^{ij} \dd x_i \dd x_j)$, $i,j=1,\ldots,d-1$, where $\ell^2=-\frac{d(d-1)}{\Lambda}$ and where $(t,z,x_i)$ are standard Cartesian coordinates with $z$ ranging only over the half line. One can realize per direct inspection that we can attach to $\PAdS_{d+1}$ a conformal, timelike boundary at $z=0$. From the point of view of the matter content, we will consider a real, massive scalar field, with a possibly non minimal coupling to scalar curvature. Although the dynamics is ruled by the Klein-Gordon operator, its smooth solutions cannot be constructed only starting from suitable initial data, but one needs also to prescribe boundary conditions at $z=0$. This additional input has dramatic effects at the level of quantum theory, both in the construction of the collection of all possible observables and in the identification of a physically acceptable quantum state. In this paper, we will be focusing on the second problem. As a matter of fact we will be asking ourselves two questions. The first is if one can construct the two-point function $G^+(x,x^\prime)$ for the ground state. This must be invariant under all isometries of $\PAdS_{d+1}$, a solution of the equations of motion in both entries and it must also encode the choice of boundary conditions. The second question is whether such $G^+(x,x^\prime)$ is the two-point function for a physically acceptable state. In this respect, the notion of Hadamard states cannot be invoked since it is strongly tied to spacetimes which are globally hyperbolic. When such a requirement is missing, there is no universally accepted replacement and actually this is an interesting open problem. Nonetheless, a minimal request, which one can ask, is that at least the restriction of the two-point function to any globally hyperbolic subregion of $\PAdS_{d+1}$ is of Hadamard form, a condition which can be traced back to [@Kay:1992es]. As a first step, we will be showing that for a certain range of the mass and curvature coupling no boundary condition at $z=0$ is required, whereas for its complement a whole one-parameter family of boundary conditions can be considered. As a second step, we will show that a two-point function $G^+(x,x^\prime)$ with the desired characteristics exists and it encodes in particular the choice of boundary conditions. It is important to stress that, while Dirichlet and Neumann are always unproblematic choices (whenever admissible), the Robin boundary conditions are rather tricky. As a matter of fact, we will prove that there exist instances where, upon choosing Robin boundary conditions, “bound states” solutions, which are exponentially suppressed for large $z$, appear. This is a very troublesome feature, first of all since it destroys the invariance under the $\PAdS_{d+1}$ isometry group. This peculiar scenario is drastically different from the usual free field theories and for this reason we will highlight its existence, leaving a more detailed analysis to future works. In terms of the singular behavior of $G^+(x,x^\prime)$, we will show that singularities occur whenever $x$ and $x^\prime$ are connected by a null geodesic, possibly reflected at the boundary. This behavior is consistent not only with the requirement that $G^+(x,x^\prime)$ be of Hadamard form in every globally hyperbolic subregion, but also with the construction via the method of images of the two-point function, associated with the Casimir effect [@Dappiaggi:2014gea], which corresponds to one of the particular cases considered here: a massless, conformally coupled scalar field. It is important to stress that we are not the first ones to study the quantization of a real, massive scalar field in anti-de Sitter, since a first analysis appeared already in the late 1970s in [@Avis:1977yn]. Also the construction of a maximally symmetric two-point function was tackled before, see [@Allen:1985wd; @Burges:1985qq]. Other recent works for this and other matter fields on AdS include [@Kent:2014nya; @Belokogne:2016dvd]. Yet, these works considered only the special case of the Dirichlet boundary condition, which corresponds to the Friedrichs extension of the Helmholtz operator built out of the $\PAdS_{d+1}$ metric at constant time. In [@Ishibashi:2004wx], the Friedrichs extension was shown to be only one of the possible self-adjoint extensions of the Helmholtz operator, which correspond to different Robin boundary conditions. In this paper, we use an alternative method to determine all these possible boundary conditions [^1] and, in addition, we construct the associated two-point functions for a ground state, obtaining their singular behavior. The paper is organized as follows: In Section \[sec:AdS\], we will recall the basic structural, geometric properties of $\AdS_{d+1}$ and in particular of the associated Poincaré fundamental domain. In Section \[sec:KG\], we will consider the Klein-Gordon equation on $\PAdS_{d+1}$ and, by means of a conformal rescaling, we will transform it to a wave equation with a singular potential on $\mathring{\bH}^{d+1}$, the subset of Minkowski spacetime with $z>0$. After a Fourier transform in the directions orthogonal to $z$ we will reduce the dynamics to a one-dimensional ordinary differential equation of Sturm-Liouville type. This is a well studied topic, in particular with reference to the assignment of boundary conditions at the endpoint of the domain of definition of the equation. As a matter of fact, at the point $z=0$, where we want to prescribe boundary conditions, the potential of the differential equation is singular. Hence, it fails the usual idea that Dirichlet, Neumann or Robin boundary conditions are nothing but a prescription at the boundary of the behavior of a linear combination between a solution of the differential equation and its derivatives. We will outline how this obstruction can be circumvented in the language of a Sturm-Liouville problem. In Section \[sec:2-pt\], we will construct via a mode expansion the two-point functions for all admissible boundary conditions. We will show that for a certain class of Robin boundary conditions “bound states” solutions appear, while, in all other cases, one can push the analysis to the very end obtaining a closed form expression for the two-point function. In addition, we will show invariance under all isometries of the background, hence proving that we have constructed a maximally symmetric state. Finally, we will study the singular behavior of the two-point function, unveiling its consistence with the standard Hadamard prescription in all globally hyperbolic subregions and proposing a new definition of Hadamard states which apply to $\PAdS_{d+1}$. Anti-de Sitter and the Poincaré domain {#sec:AdS} ====================================== In this paper, our starting point is the anti-de Sitter spacetime, $\AdS_{d+1}$, the maximally symmetric solution to the $(d+1)$-dimensional Einstein’s equation ($d\geq 2$) with a negative cosmological constant $\Lambda$. It can be constructed starting from the embedding space $\bM^{2,d}$, that is, $\bR^{d+2}$ endowed with metric $$\dd s^2 = \tilde{\eta}^{AB} \dd X_A \dd X_B = -\dd X^2_0 - \dd X^2_1 + \sum\limits_{i=2}^{d+1} \dd X^2_i \, ,$$ where $(X_0,...,X_{d+1})$ are the standard Cartesian coordinates, and considering only the region identified by the relation $$\label{eq:covering_space} -X^2_0-X^2_1+\sum_{i=2}^{d+1}X^2_i=-\ell^2 \, , \qquad \ell^2 \doteq -\frac{d(d-1)}{\Lambda} \, .$$ For our purposes and in many physical applications, we do not work directly on $\AdS_{d+1}$, but rather on the [*Poincaré fundamental domain*]{}, $\PAdS_{d+1}$, which is identified via the coordinate transformation $$\label{eq:Poincare_chart} \arraycolsep=1.4pt\def\arraystretch{2} \left\{\begin{array}{l} X_0 = \dfrac{\ell}{z}t \, , \\ X_i = \dfrac{\ell}{z} x_i \, , \quad i=1,...,d-1,\\ X_d=\ell\left(\dfrac{1-z^2}{2z}+\dfrac{-t^2+\delta^{ij}x_i\,x_j}{2z}\right) \, , \\ X_{d+1}=\ell\left(\dfrac{1+z^2}{2z}-\dfrac{-t^2+\delta^{ij}x_i\,x_j}{2z}\right) \, , \end{array}\right.$$ where both $t$ and all $x_i$ are ranging over the whole $\bR$, whereas $z\in (0,\infty)$. This translates the constraint which descends from the identity $X_d+X_{d+1}=\frac{\ell}{z}$, hence showing that $\PAdS_{d+1}$ covers only half of the full $\AdS_{d+1}$ (see Fig. \[fig:AdS\]). In addition, the metric of the Poincaré domain becomes $$\label{eq:Poincare_metric} \dd s^2 = \frac{\ell^2}{z^2} \left(-\dd t^2+\dd z^2+\delta^{ij} \dd x_i \dd x_j\right) \, , \qquad i=1,...,d-1 \, ,$$ where $\delta^{ij}$ stands for the Kronecker delta. Hence, $\PAdS_{d+1}$ is conformal to a portion of Minkowski spacetime, the “upper-half plane” $$\mathring{\bH}^{d+1}\doteq\{(t,x_1, \ldots, x_{d-1},z)\in\bR^{d+1}\;|\;z>0\} \, ,$$ where we adopted the same Cartesian coordinates as in . If we endow $\mathring{\bH}^{d+1}$ with the standard Minkowskian metric $\eta$, then $\eta=\Omega^2 g=\frac{z^2}{\ell^2}g$ where $g$ is the metric of $\PAdS_{d+1}$ and $\Omega=\frac{z}{\ell}$ is the conformal factor. ![\[fig:AdS\]Conformal diagram of $\AdS_{d+1}$ and the Poincaré domain and the representation with one spatial dimension restored.](Figures/ads-diagrams-latex.png) To finish this short introduction on the geometric aspects of the background, let us briefly describe the notion of invariant distance in AdS. One can proceed in two distinct, albeit equivalent ways. Intrinsically, one can define the geodesic distance $s$ on $\PAdS_{d+1}$ between two arbitrary points $x$ and $x'$ and Synge’s world function $\sigma$ given by $\sigma(x,x') \doteq \frac{1}{2}s(x,x')^2$. In view of , one can start instead from the chordal distance $s_{\rm e}$ between $x$ and $x'$ through the embedding space $\bM^{2,d}$ and from Synge’s world function defined on $\bM^{2,d}$ as $$\label{eq:sigmae} \sigma_{\rm e}(x,x') \doteq \frac{1}{2}s_{\rm e}(x,x')^2 = \frac{1}{2} \tilde{\eta}^{AB} (X_A - X'_A) (X_B - X'_B) \, ,$$ with the constraint that $x$ and $x^\prime$ are two points constrained by , hence lying in $\AdS_{d+1}$. These two notions are related by $$\label{eq:relationsigmas} \cosh \left(\frac{s}{\ell}\right) = 1 + \frac{s_{\rm e}^2}{2 \ell^2} \, , \qquad \cosh \left(\frac{\sqrt{2\sigma}}{\ell}\right) = 1 + \frac{\sigma_{\rm e}}{\ell^2}$$ (see e.g. Section 2.4 of [@Kent:2013]). In the rest of the paper, we set $\ell \equiv 1$. Massive scalar field on AdS {#sec:KG} =========================== Klein-Gordon equation --------------------- We consider a real, massive scalar field $\phi: \PAdS_{d+1}\to\bR$ such that $$\label{eq:dynamicsP} P\phi=\left(\Box_g - m_0^2 - \xi R \right)\phi=0 \, ,$$ where $\Box_g$ is the D’Alembert wave operator built out of , $m_0$ is the mass of the scalar field, $\xi$ is the scalar-curvature coupling constant and $R = - d(d+1)$ is the Ricci scalar. In order to study the solutions of this equation, we follow a slightly unconventional strategy which relies on the observation made previously that $\PAdS_{d+1}$ is conformal to $\mathring{\bH}^{d+1}$ and it consists in translating into a partial differential equation intrinsically defined on $\mathring{\bH}^{d+1}$. This is a standard procedure, see e.g. Appendix D of [@Wald]. Let $\phi:\PAdS_{d+1}\to\bR$ be any solution of and let $\Phi\doteq\Omega^{\frac{1-d}{2}}\phi$. The latter can be read as a scalar field $\Phi:\mathring{\bH}^{d+1}\to\bR$, solution of the equation $$\label{eq:conformally_rescaled_dynamics} P_\eta \Phi = \left(\Box_\eta-\frac{m^2}{z^2}\right) \Phi(z)=0 \, ,$$ in which $\Box_\eta$ is the standard wave operator built out of the Minkowski metric $\eta$ and we define [^2] $m^2 \doteq m_0^2+(\xi-\frac{d-1}{4d})R$. In other words, the Klein-Gordon equation in $\PAdS_{d+1}$ is transformed to a wave equation on $\mathring{\bH}^{d+1}$ with a potential, singular at $z=0$. In order to construct solutions of , in view of the invariance of the metric under translations along the directions orthogonal to $z$, we take the Fourier transform, $$\label{eq:Fouriertransf} \Phi(\underline{x},z) = \int_{\bR^d} \frac{\dd^d\underline{k}}{(2\pi)^{\frac{d}{2}}} \, e^{i\underline{k}\cdot \underline{x}} \, \widehat{\Phi}_{\underline{k}}(z) \, ,$$ where $\underline{x} \doteq (t, x_1, \ldots, x_{d-1})$, $\underline{k} \doteq (\omega, k_1, \ldots, k_{d-1})$ and $\widehat{\Phi}_{\underline{k}}$ are solutions of $$\label{eq:STeq} L \, \widehat{\Phi}_{\underline{k}} \doteq \left(- \frac{\dd^2}{\dd z^2} +\frac{m^2}{z^2} \right) \widehat{\Phi}_{\underline{k}}(z) = \lambda \, \widehat{\Phi}_{\underline{k}}(z) \, , \qquad \lambda \equiv q^2 \doteq \omega^2 - \displaystyle\sum_{i=1}^{d-1} k_i^2 \, .$$ This is a *singular Sturm-Liouville equation* [^3] on $z \in (0,+\infty)$ with spectral parameter $\lambda$. To fully specify a well-posed Sturm-Liouville problem, we need to add at most two boundary conditions at the endpoints 0 and $+\infty$. The required number and the form of the boundary conditions depend on the classification of the endpoints, as explained in the next section. After specifying appropriate boundary conditions, it is known that there is a continuous spectrum contained in $(0, \infty)$ and, for some boundary conditions, there is also a point spectrum with negative eigenvalues, which is indicative of the existence of “bound states” in the space of solutions of , that is, exponentially decaying solutions in $z$. To the best knowledge of the authors of this paper, these solutions have not been discussed so far in the literature of scalar field theory on AdS. The next step is therefore to study the possible ($\lambda$-independent) boundary conditions that can be applied to this problem. For that purpose, as a preliminary step, we note that two linearly independent solutions of are $\sqrt{z} \, J_{\nu} \big(\sqrt{\lambda} z\big)$ and $\sqrt{z} \, Y_{\nu} \big(\sqrt{\lambda} z\big)$, where $J_{\nu}$ and $Y_{\nu}$ are the Bessel functions of the first and second kinds, respectively, and $$\label{eq:defnu} \nu \doteq \frac{1}{2} \sqrt{1+4m^2} \, .$$ We assume that $\nu \in [0,\infty)$ or, equivalently, $m^2 \in [-\frac{1}{4},\infty)$ (the lower bound is the Breitenlohner-Freedman bound [@Breitenlohner:1982jf]). Endpoint classification ----------------------- The types of boundary conditions that are allowed at a given endpoint depend on the integrability of the solutions near the endpoint. This classification of endpoints for a Sturm-Liouville problem has its origins with Weyl’s classical limit-point and limit-circle theory [@Weyl]. A modern overview may be found e.g. in [@Zettl:2005]. Here, we summarize the main results. For the Sturm-Liouville problem the two endpoints are 0 and $+\infty$. ### Endpoint 0 Concerning the endpoint 0, it is classified as 1. *regular* if the “potential term” $z \mapsto \frac{m^2}{z^2} \in L^1(0,z_0)$ for some $z_0 \in (0,+\infty)$, i.e. if $m^2=0$ ($\nu = \frac{1}{2}$); otherwise, it is *singular*, i.e. if $m^2 \neq 0$ ($\nu \neq \frac{1}{2}$); 2. *limit-circle* (notation LC) if, for some $\lambda \in \bC$, all solutions of the equation are in $L^2(0,z_0)$ for some $z_0 \in (0,+\infty)$; otherwise, it is *limit-point* (notation LP). Given that, for any $\lambda > 0$, $z \mapsto \sqrt{z} \, J_{\nu} \big(\sqrt{\lambda} z\big) \sim_{z \to 0} z^{\nu+\frac{1}{2}}$ is in $L^2(0,z_0)$ for all $\nu \in [0,\infty)$ and that $z \mapsto \sqrt{z} \, Y_{\nu} \big(\sqrt{\lambda} z\big) \sim_{z \to 0} z^{-\nu+\frac{1}{2}}$ is in $L^2(0,z_0)$ only if $\nu \in [0,1)$, for any $z_0 \in (0,+\infty)$, we may conclude that the endpoint 0 is LC for $\nu \in [0,1)$ (in particular, it is regular for $\nu = \frac{1}{2}$) and LP for $\nu \in [1,\infty)$. ### Endpoint $+\infty$ As for the endpoint $+\infty$, it is classified as 1. *singular*, as it is not finite; 2. *limit-circle* (notation LC) if, for some $\lambda \in \bC$, all solutions of the equation are in $L^2(z_0, +\infty)$ for some $z_0 \in (0,+\infty)$; otherwise, it is *limit-point* (notation LP). For any $\lambda \in \bC$, we see that the solutions $z \mapsto \sqrt{z} \, J_{\nu} \big(\sqrt{\lambda} z\big) \sim_{z \to +\infty} \cos \big(\sqrt{\lambda} z- \frac{\nu\pi}{2} - \frac{\pi}{4}\big)$ and $z \mapsto \sqrt{z} \, Y_{\nu} \big(\sqrt{\lambda} z\big) \sim_{z \to +\infty} \sin \big(\sqrt{\lambda} z- \frac{\nu\pi}{2} - \frac{\pi}{4}\big)$ are not in $L^2(z_0, +\infty)$ for all $\nu \in [0,\infty)$. There is a solution, given by $z \mapsto \sqrt{z} \, H^{(1)}_{\nu} \big(\sqrt{\lambda} z\big) \sim_{z \to +\infty} \exp \big[i \big(\sqrt{\lambda}z- \frac{\nu\pi}{2} - \frac{\pi}{4}\big)\big]$, called the first Hankel function, which is in $L^2(z_0, +\infty)$ when ${\rm Im}(\lambda) \neq 0$, but any other linearly independent solution is not. Hence, the endpoint $+\infty$ is always LP. Boundary conditions ------------------- In this section, we identify the $\lambda$-independent boundary conditions that may be assigned to the endpoints of the Sturm-Liouville problem . Their necessity and type essentially depend on the classification of the endpoints given in the previous section. We note that in [@Ishibashi:2004wx] the boundary conditions that can be applied to the conformal boundary of AdS were determined by finding all self-adjoint extensions of the Helmholtz operator built out of the $\PAdS_{d+1}$ metric. Here, we give an alternative method that is consistent and complements that of [@Ishibashi:2004wx] and, in addition, it gives an account of the “bound states” solutions that occur for a class of boundary conditions. We pick as a fundamental pair of solutions $\big\{ \widehat{\Phi}_{\underline{k}}^1, \, \widehat{\Phi}_{\underline{k}}^2 \big\}$, with $$\begin{aligned} \widehat{\Phi}_{\underline{k}}^1(z) &= \sqrt{\frac{\pi}{2}} \, q^{-\nu} \sqrt{z} \, J_{\nu}(qz) \, , \label{eq:fundamentalsolutions1} \\ \widehat{\Phi}_{\underline{k}}^2(z) &= \begin{cases} - \sqrt{\dfrac{\pi}{2}} \, q^{\nu} \sqrt{z} \, J_{-\nu}(qz) \, , & \nu \in (0,1) \, , \\ - \sqrt{\dfrac{\pi}{2}} \sqrt{z} \left[ Y_{0}(qz) - \dfrac{2}{\pi} \log(q) \right] \, , & \nu = 0 \, . \end{cases} \label{eq:fundamentalsolutions2}\end{aligned}$$ \[eq:fundamentalsolutions\] For future reference, we note that $\widehat{\Phi}_{\underline{k}}^1$ is the *principal solution* at the endpoint 0, as it is the unique solution (up to scalar multiples) such that $\lim_{z \to 0^+} \widehat{\Phi}_{\underline{k}}^1(z)/\widehat{\Psi}_{\underline{k}}(z) = 0$ for every solution $\widehat{\Psi}_{\underline{k}}$ which is not a scalar multiple of $\widehat{\Phi}_{\underline{k}}^1$. The other solution $\widehat{\Phi}_{\underline{k}}^2$ is called a non-principal solution and is not unique, as it may be given by a linear combination of the principal solution with any linearly independent solution. A general solution may then be written as $$\widehat{\Phi}_{\underline{k}}(z) = \mathcal{N}_{\underline{k}} \left[ \cos(\alpha) \, \widehat{\Phi}_{\underline{k}}^1(z) + \sin(\alpha) \, \widehat{\Phi}_{\underline{k}}^2(z) \right] \, ,$$ where $\mathcal{N}_{\underline{k}}$ and $\alpha \in [0,\pi)$ are independent of $z$. The fundamental solutions were chosen such that $\alpha$ is in addition independent of $\underline{k}$. We note that this Sturm-Liouville problem is discussed in Section 4.11 of the classical work of Titchmarsh [@Titchmarsh], in the context of Fourier-Bessel expansions, and instead of $\alpha$, it is used a constant $c \in \bR$, also independent of $\underline{k}$, which is related to $\alpha$ by $c = \cot(\alpha)$. We consider separately the following cases for different values of $\nu$. ### Case $\nu = \frac{1}{2}$ This corresponds to the massless, conformally coupled scalar field. The endpoint 0 is regular in this case, whereas the endpoint $+\infty$ is singular. The fundamental pair of solutions $\big\{ \widehat{\Phi}_{\underline{k}}^1, \, \widehat{\Phi}_{\underline{k}}^2 \big\}$ reduces to $$\widehat{\Phi}_{\underline{k}}^1(z) = \sqrt{\frac{\pi z}{2q}} \, J_{\frac{1}{2}}(qz) = \frac{\sin(qz)}{q} \, , \qquad \widehat{\Phi}_{\underline{k}}^2(z) = - \sqrt{\frac{\pi qz}{2}} \, J_{-\frac{1}{2}}(qz) = - \cos(qz) \, .$$ Since the endpoint 0 is regular, the most general homogeneous boundary condition that may be applied is a *Robin boundary condition* in its regular form $$\label{eq:BCregular} \cos(\alpha) \, \widehat{\Phi}_{\underline{k}}(0) + \sin(\alpha) \, \widehat{\Phi}'_{\underline{k}}(0) = 0 \, , \qquad \alpha \in [0,\pi) \, .$$ The particular case which selects the principal solution $\widehat{\Phi}_{\underline{k}}^1$, i.e. $\alpha = 0$, is called the *Friedrichs boundary condition* and it corresponds to the standard homogeneous *Dirichlet boundary condition* $\widehat{\Phi}_{\underline{k}}(0) = 0$. Other common examples are the homogeneous *Neumann boundary condition*, $\widehat{\Phi}'_{\underline{k}}(0) = 0$, which corresponds to $\alpha = \frac{\pi}{2}$, and the *transparent boundary conditions*, [^4] which corresponds to $\alpha = \frac{\pi}{4}$. An important feature occurs when we impose a Robin boundary condition with $c > 0$ or, equivalently, $\alpha \in (0, \frac{\pi}{2})$. In this case, it can be shown (Section 4.11 of [@Titchmarsh]) that the spectrum of the eigenvalue problem associated with the Sturm-Liouville problem does not consist purely of the continuous spectrum but it also includes a negative eigenvalue $\lambda_{\rm bs} = - c^2 = -\cot^2(\alpha)$. This indicates the existence of a “bound state” solution, that is, of a mode solution which exponentially decays with $z$, given by $e^{-c z} = e^{-\cot(\alpha) z}$, and which satisfies trivially the boundary condition. This eigenvalue implies that the Fourier transform does not represent the full solution to the equation when $c > 0$. A general solution for a boundary condition of this type needs to include this “bound state”, besides the usual propagating modes. ### Case $\nu \in [0,1) \setminus \{\frac{1}{2}\}$ {#sec:bcnusmall} In this case, the endpoint 0 is singular and limit-circle. Hence, both solutions $\widehat{\Phi}_{\underline{k}}^1$ and $\widehat{\Phi}_{\underline{k}}^2$ are square integrable near the origin and may be used to construct a general solution. However, a Robin boundary condition written in the regular form is no longer valid, as for instance $\lim_{z \to 0} \widehat{\Phi}_{\underline{k}}^2(z)$ diverges. To motivate a natural way to implement a Robin boundary condition for a singular endpoint, note that in the regular case may be equivalently written as $$\label{eq:BCsing} \lim_{z \to 0} \left\{ \cos(\alpha) \, W_z \big[\widehat{\Phi}_{\underline{k}},\widehat{\Phi}_{\underline{k}}^1\big] + \sin(\alpha) \, W_z \big[\widehat{\Phi}_{\underline{k}},\widehat{\Phi}_{\underline{k}}^2\big] \right\} = 0 \, ,$$ since $\widehat{\Phi}_{\underline{k}}^1(0) = 0$, $\big(\widehat{\Phi}_{\underline{k}}^1\big)'(0) = 1$, $\widehat{\Phi}_{\underline{k}}^2(0) = -1$ and $\big(\widehat{\Phi}_{\underline{k}}^1\big)'(0) = 0$ when $\nu = \frac{1}{2}$. In the expression, $W_z [u,v] \doteq u(z)v'(z) - v(z)u'(z)$ is the Wronskian of two differentiable functions $u$ and $v$. However, is also valid in the singular case as the limit exists. [^5] Hence, one may take as the form of a Robin boundary condition when the endpoint 0 is singular and limit-circle, a natural generalization of the regular case. Therefore, is the most general boundary condition that can be applied for all $\nu \in (0,1)$. The important particular example of the Friedrichs boundary condition, which selects the principal solution at 0, corresponds to $\alpha = 0$ and we use it to define the *generalized Dirichlet boundary condition*. When $\nu \in (0,1)$, we also define the *generalized Neumann boundary condition* to correspond to $\alpha = \frac{\pi}{2}$, which selects the non-principal solution $\widehat{\Psi}_{\underline{k}}^2$. However, note that, given the non-uniqueness of non-principal solutions, there is no unique way to define a generalized Neumann boundary condition in the singular case which reduces to the standard definition in the regular one, $\widehat{\Phi}'_{\underline{k}}(0) = 0$. For instance, since $J_{-\frac{1}{2}}(z) = - Y_{\frac{1}{2}}(z)$, the boundary condition obtained by replacing $\widehat{\Phi}_{\underline{k}}^2$ in by a solution proportional to $ q^{\nu} \sqrt{z} \, Y_{\nu}(qz)$ and setting $\alpha = \frac{\pi}{2}$ is not equivalent to the generalized Neumann boundary defined above. Finally, when $\nu=0$, we define these examples of generalized boundary conditions similarly. [^6] As in the regular case, it can be shown (Section 4.11 of [@Titchmarsh]) that, if $c>0$, or, equivalently, $\alpha \in (0, \frac{\pi}{2})$, there is a negative eigenvalue in the spectrum of the eigenvalue problem associated with the Sturm-Liouville problem , $\lambda_{\rm bs} = - c^{1/\nu} = -\cot^{1/\nu}(\alpha)$ if $\nu \in (0,1)$ and $\lambda_{\rm bs} = - e^{-\pi c} = - e^{\pi \cot(\alpha)}$ if $\nu=0$. Hence, there is a “bound state” solution of the form $\sqrt{z} \, K_{\nu}(\sqrt{|\lambda_{\rm bs}|} \, z) \sim_{z \to \infty} e^{-\sqrt{|\lambda_{\rm bs}|} \, z}$, where $K_{\nu}$ is the modified Bessel solution of the second kind. This negative eigenvalue implies once more that the Fourier transform does not represent the full solution to the equation when $c > 0$. A general solution for a boundary condition of this type needs to include this “bound state” solution, besides the usual propagating modes. ### Case $\nu \in [1,\infty)$ In this case, the endpoint 0 is singular and limit-point. Among the two fundamental solutions , only the principal solution $\widehat{\Phi}_{\underline{k}}^1$ is square integrable near the origin and, hence, no boundary condition is required. In practice, this is as if one had chosen the generalized Dirichlet boundary condition. Furthermore, there are no eigenvalues in the spectrum of the eigenvalue problem associated with the Sturm-Liouville problem and thus there is no “bound state” solution. All the cases analyzed above and the allowed boundary conditions, when necessary, are summarized in Table \[tab:BCsummary\]. $\nu = \frac{1}{2}\sqrt{1+4m^2}$ Classification of $z=0$ Boundary condition at $z=0$ ------------------------------------------ ------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- $\nu = \frac{1}{2}$ Regular (R) $\cot(\alpha) \, \widehat{\Phi}_{\underline{k}}(0) + \widehat{\Phi}_{\underline{k}}'(0) = 0$ $\nu \in [0,1), \, \nu \neq \frac{1}{2}$ Limit-circle (LC) $\cot(\alpha) \, W_z\big[\widehat{\Phi}_{\underline{k}}, \widehat{\Phi}_{\underline{k}}^1 \big] + W_z\big[\widehat{\Phi}_{\underline{k}}, \widehat{\Phi}_{\underline{k}}^2 \big] = 0$ $\nu \in [1,\infty)$ Limit-point (LP) Not required : Allowed boundary conditions at $z=0$, with $\alpha \in [0,\pi)$ and $\widehat{\Phi}_{\underline{k}}^1$ and $\widehat{\Phi}_{\underline{k}}^2$ defined in . \[tab:BCsummary\] Two-point function {#sec:2-pt} ================== In this section we calculate the two-point or Wightman function [^7] $G^+$ for a massive scalar field on $\PAdS_{d+1}$ $$G^+(x,x') \doteq \langle \psi | \Phi(x) \Phi(x') | \psi \rangle \, ,$$ for the ground state $|\psi\rangle$. We perform the calculation in two ways: by a mode expansion and by closed form solutions of the differential equation satisfied by $G^+$. We show that both approaches coincide in Appendix \[app:2pfcomputation\]. Mode expansion {#sec:modeexpansions} -------------- In order to construct the two-point function for the ground state, we first use a mode expansion, a procedure already advocated in previous works, e.g. [@Danielsson:1998wt], but always in the special case of Dirichlet boundary conditions. Here, we want to consider all admissible boundary conditions discussed in the previous section. We perform the calculation for the two-point function $G^+_{\bH}$ in $\mathring{\bH}^{d+1}$, but we can immediately obtain the two-point function on $\PAdS_{d+1}$, by using the relation $G^+(x,x') = (zz')^{\frac{d-1}{2}} G^+_{\bH}(x,x')$. Starting from , we look for $G^+_{\bH}$ satisfying $$\label{eq:defining_Green} \left(P_\eta\otimes\mathbb{I}\right) G^+_{\bH} = \left(\mathbb{I}\otimes P_\eta\right) G^+_{\bH} = 0 \, . \\$$ where $P_\eta$ is the operator defined in . We consider the Fourier transform [^8] $$\label{eq:FouriertransG} G^+_{\bH}(\underline{x},z;\underline{x}^\prime,z^\prime) = \int_{\bR^d} \frac{\dd^d\underline{k}}{(2\pi)^{\frac{d}{2}}} \, e^{i\underline{k}\cdot(\underline{x}-\underline{x}^\prime)} \, \widehat{G}^+_{\underline{k}}(z,z^\prime) \, .$$ The remaining unknown $\widehat{G}^+_{\underline{k}}(z,z^\prime)$ is a solution of $$(L\otimes\mathbb{I}) \, \widehat{G}^+_{\underline{k}} = (\mathbb{I}\otimes L) \, \widehat{G}^+_{\underline{k}} = \lambda \, \widehat{G}^+_{\underline{k}} \, ,$$ and where appropriate boundary conditions are applied at $z=0$ and $z'=0$ when $\nu \in [0,1)$. Given that $G^+_{\bH}$ is radially symmetric in the $(d-1)$ spatial directions excluding the $z$-direction, instead of a Fourier transform along those directions we consider instead a Hankel or Fourier-Bessel transform $$\begin{gathered} G^+_{\bH}(x,x') = \lim_{\epsilon \to 0^+} \int_{0}^{\infty} \frac{\dd \omega}{\sqrt{2\pi}} \, e^{-i\omega (t-t'-i\epsilon)} \int_0^{\infty} \dd k \, k \, \left(\frac{k}{r}\right)^{\frac{d-3}{2}} \! J_{\frac{d-3}{2}}(kr) \, \widehat{G}^+_{\underline{k}}(z,z') \, .\end{gathered}$$ where $r \doteq \sum_{i=1}^{d-1} (x^i-{x'}^i)$, only positive frequencies are taken for the ground state and $i\epsilon$ was introduced to regularize the two-point function [@Fulling:1987]. Finally, a change of integration variables $q^2 \doteq \omega^2 - k^2$ leads to $$\begin{gathered} G^+_{\bH}(x,x') = \lim_{\epsilon \to 0^+} \int_0^{\infty} \dd q \, q \int_0^{\infty} \dd k \, k \,\frac{e^{-i\sqrt{k^2+q^2} (t-t'-i\epsilon)}}{\sqrt{2\pi(k^2+q^2)}} \left(\frac{k}{r}\right)^{\frac{d-3}{2}} \! J_{\frac{d-3}{2}}(kr) \, \widehat{G}^+_{\underline{k}}(z,z') \, .\end{gathered}$$ At this point, some comments are in order. The *antisymmetric* part of the two-point function is given by $i G(x,x')$, where $G(x,x') = \langle \psi | \left[\Phi(x), \Phi(x')\right] | \psi \rangle$ is the *commutator function*. In addition to satisfying as the two-point function, it also satisfies $$\label{eq:commutatorconditions} G(x,x')\big|_{t=t'} = 0 \, , \qquad \partial_{t} G(x,x')\big|_{t=t'} = \partial_{t'} G(x,x')\big|_{t=t'} = \prod_{i=1}^{d-1} \delta(x^i-{x'}^i) \delta(z-z') \, .$$ We can then write it as $$\begin{gathered} G(x,x') = \lim_{\epsilon \to 0^+} \sqrt{2} \int_0^{\infty} \dd q \, q \int_0^{\infty} \dd k \, k \,\frac{\sin\left(\sqrt{k^2+q^2} (t-t'-i\epsilon)\right)}{\sqrt{\pi(k^2+q^2)}} \left(\frac{k}{r}\right)^{\frac{d-3}{2}} \! J_{\frac{d-3}{2}}(kr) \, \widehat{G}^+_{\underline{k}}(z,z') \, .\end{gathered}$$ The second condition in implies that $$\begin{gathered} \sqrt{\frac{2}{\pi}} \int_0^{\infty} \dd q \, q \int_0^{\infty} \dd k \, k \, \left(\frac{k}{r}\right)^{\frac{d-3}{2}} \! J_{\frac{d-3}{2}}(kr) \, \widehat{G}^+_{\underline{k}}(z,z') = \prod_{i=1}^{d-1} \delta(x^i-{x'}^i) \delta(z-z') \, .\end{gathered}$$ If we assume that $\widehat{G}_{\underline{k}}$ does not depend on $k$ (as it is the case), then by using the identity derived in Appendix \[app:deltafunction\] $$\begin{gathered} \int_0^{\infty} \dd k \, k \, \left(\frac{k}{r}\right)^{\frac{d-3}{2}} \! J_{\frac{d-3}{2}}(kr) = 2^{\frac{d-3}{2}} \, \Gamma\left(\frac{d-1}{2}\right) \, \frac{\delta(r)}{r^{d-2}} = \frac{(2\pi)^{\frac{d}{2}}\Gamma\left(\frac{d-1}{2}\right)}{\sqrt{2} \, \Gamma\left(\frac{d}{2}\right)} \prod_{i=1}^{d-1} \delta(x^i-{x'}^i) \, ,\end{gathered}$$ we obtain the one-dimensional delta distribution representation $$\begin{gathered} \frac{(2\pi)^{\frac{d}{2}}\Gamma\left(\frac{d-1}{2}\right)}{\sqrt{\pi} \, \Gamma\left(\frac{d}{2}\right)} \int_0^{\infty} \dd q \, q \, \widehat{G}^+_{\underline{k}}(z,z') = \delta(z-z') \, .\end{gathered}$$ In other words we are looking for a resolution of the identity in terms of eigenfunctions of $L$. This problem has its roots in the theory of eigenfunction expansions and, for the case in hand, it has been tackled in Section 4.11 of [@Titchmarsh]. We present the results below and leave the details of their derivation to Appendix \[app:eigenfunctionexpansion\]. ### Case $\nu \in [1,\infty)$ When $\nu \in [1,\infty)$, as we have seen in the previous section, no boundary conditions are required at the endpoint 0. The delta distribution expanded in terms of eigenfunctions of $L$ is given by $$\delta(z-z') = \sqrt{zz'} \int_0^\infty \dd q \, q \, J_\nu(qz) J_\nu(qz') \, .$$ Hence, the two-point function is $$\label{eq:2pfnularge} G^+_{\bH}(x,x') = \lim_{\epsilon \to 0^+} \mathcal{N} \sqrt{zz'} \int_0^\infty \dd k \, k \left(\frac{k}{r}\right)^{\frac{d-3}{2}} \! J_{\frac{d-3}{2}}(kr) \int_0^\infty \dd q \, q \, \frac{e^{-i\sqrt{k^2+q^2}(t-t^\prime-i\epsilon)}}{\sqrt{2\pi(k^2+q^2)}} \, J_\nu(qz) J_\nu(qz') \, ,$$ where $\mathcal{N}$ is a normalization constant. ### Case $\nu \in (0,1)$ This case, which includes the $\nu = \frac{1}{2}$ example, requires a Robin boundary condition of the form to be applied to the solutions of the field equation. The delta distribution, expanded in terms of eigenfunctions of $L$ which satisfy the boundary condition with $c<0$, is given by $$\label{eq:deltaexpansionnu01} \delta(z-z') = \sqrt{zz'} \int_0^\infty \dd q \, q \, \frac{\left[cJ_\nu(qz)-q^{2\nu}J_{-\nu}(qz)\right] \left[cJ_\nu(qz^\prime)-q^{2\nu}J_{-\nu}(qz^\prime)\right]}{c^2-2cq^{2\nu}\cos(\nu\pi)+q^{4\nu}} \, .$$ Hence, for $c<0$, the two-point function is given by $$\begin{aligned} G^+_{\bH}(x,x') &= \lim_{\epsilon \to 0^+} \mathcal{N} \sqrt{zz'} \int_0^\infty \dd k \, k \left(\frac{k}{r}\right)^{\frac{d-3}{2}} \! J_{\frac{d-3}{2}}(kr) \int_0^\infty \dd q \, q \, \frac{e^{-i\sqrt{k^2+q^2}(t-t^\prime-i\epsilon)}}{\sqrt{2\pi(k^2+q^2)}} \notag \\ &\quad \times \frac{\left[cJ_\nu(qz)-q^{2\nu}J_{-\nu}(qz)\right] \left[cJ_\nu(qz^\prime)-q^{2\nu}J_{-\nu}(qz^\prime)\right]}{c^2-2cq^{2\nu}\cos(\nu\pi)+q^{4\nu}} \, .\end{aligned}$$ If we denote $G^{+({\rm D})}_{\bH} \doteq G^+_{\bH}|_{\alpha=0}$ and $G^{+({\rm N})}_{\bH} \doteq G^+_{\bH}|_{\alpha=\frac{\pi}{2}}$, we verify that the two-point function satisfies the following boundary conditions at $z=0$ and $z'=0$ \[eq:BCsing2pf\] $$\lim_{z \to 0} \left\{ \cos(\alpha) \, W_z \Big[G^+_{\bH},G^{+({\rm D})}_{\bH}\Big] + \sin(\alpha) \, W_z \Big[G^+_{\bH},G^{+({\rm N})}_{\bH}\Big] \right\} = 0 \, ,$$ $$\lim_{z' \to 0} \left\{ \cos(\alpha) \, W_{z'} \Big[G^+_{\bH},G^{+({\rm D})}_{\bH}\Big] + \sin(\alpha) \, W_{z'} \Big[G^+_{\bH},G^{+({\rm N})}_{\bH}\Big] \right\} = 0 \, .$$ In the particular case $\nu=\frac{1}{2}$, these reduce to $$\cos(\alpha) \, G^+_{\bH}(0,z') + \sin(\alpha) \, \left.\frac{\dd G^+_{\bH}(z,z')}{\dd z}\right|_{z=0} = 0 \, ,$$ $$\cos(\alpha) \, G^+_{\bH}(z,0) + \sin(\alpha) \, \left.\frac{\dd G^+_{\bH}(z,z')}{\dd z'}\right|_{z'=0} = 0 \, .$$ For $c>0$, the existence of a “bound state” solution with spectral parameter $\lambda = -c^{1/\nu}$ adds a contribution to the delta distribution $$\begin{aligned} \delta(z-z') &= \sqrt{zz'} \int_0^\infty \dd q \, q \, \frac{\left[cJ_\nu(qz)-q^{2\nu}J_{-\nu}(qz)\right] \left[cJ_\nu(qz^\prime)-q^{2\nu}J_{-\nu}(qz^\prime)\right]}{c^2-2cq^{2\nu}\cos(\nu\pi)+q^{4\nu}} \\ &\quad + 2\sqrt{zz'} \, c^{1/\nu} \, \frac{\sin(\pi \nu)}{\pi \nu} K_{\nu}\big(c^{1/(2\nu)}z\big) K_{\nu}\big(c^{1/(2\nu)}z'\big) \, .\end{aligned}$$ Hence, for $c>0$, the two-point function is given by $$\begin{aligned} G^+_{\bH}(x,x') &= \lim_{\epsilon \to 0^+} \mathcal{N} \sqrt{zz'} \int_0^\infty \dd k \, k \left(\frac{k}{r}\right)^{\frac{d-3}{2}} \! J_{\frac{d-3}{2}}(kr) \left\{ \int_0^\infty \dd q \, q \left[ \frac{e^{-i\sqrt{k^2+q^2}(t-t^\prime-i\epsilon)}}{\sqrt{2\pi(k^2+q^2)}} \right. \right. \notag \\ &\quad \times \left. \frac{\left[cJ_\nu(qz)-q^{2\nu}J_{-\nu}(qz)\right] \left[cJ_\nu(qz^\prime)-q^{2\nu}J_{-\nu}(qz^\prime)\right]}{c^2-2cq^{2\nu}\cos(\nu\pi)+q^{4\nu}} \right] \notag \\ &\quad + \left. 2c^{1/\nu} \, \frac{e^{-i\sqrt{k^2-c^{1/\nu}}(t-t^\prime-i\epsilon)}}{\sqrt{2\pi(k^2-c^{1/\nu})}} \, \frac{\sin(\pi \nu)}{\pi \nu} K_{\nu}\big(c^{1/(2\nu)}z\big) K_{\nu}\big(c^{1/(2\nu)}z'\big) \right\} \, .\end{aligned}$$ The extra term is not invariant under the isometries of AdS, as it is not a function of the geodesic distance (the first term is in fact invariant, as it is shown in the next section). Therefore, it is not the two-point function for the ground state. Note, however, that it is still invariant under translations along the directions orthogonal to $z$ and $z'$, and hence the Fourier transform still makes sense. ### Case $\nu = 0$ This case also requires a Robin boundary condition of the form to be applied to the solutions of the field equation. The delta distribution, expanded in terms of eigenfunctions of $L$ which satisfy the boundary condition, is given by $$\begin{aligned} \delta(z-z') &= \sqrt{zz'} \int_0^\infty \dd q \, q \frac{\left[(c+\frac{2}{\pi}\log(q))J_0(qz)-Y_0(qz)\right] \left[(c+\frac{2}{\pi}\log(q))J_0(qz')-Y_0(qz')\right]}{(c+\frac{2}{\pi}\log(q))^2+1} \notag \\ &\quad + 2 \sqrt{zz'} \, e^{-\pi c} \, K_0\big(e^{-\pi c/2}z\big) K_0\big(e^{-\pi c/2}z'\big) \, .\end{aligned}$$ For any $c \in \bR$ there is an extra contribution from a “bound state” solution. The two-point function is given by $$\begin{aligned} G^+_{\bH}(x,x') &= \lim_{\epsilon \to 0^+} \mathcal{N} \sqrt{zz'} \int_0^\infty \dd k \, k \left(\frac{k}{r}\right)^{\frac{d-3}{2}} \! J_{\frac{d-3}{2}}(kr) \left\{ \int_0^\infty \dd q \, q \left[ \frac{e^{-i\sqrt{k^2+q^2}(t-t^\prime-i\epsilon)}}{\sqrt{2\pi(k^2+q^2)}} \right. \right. \notag \\ &\quad \times \left. \frac{\left[(c+\frac{2}{\pi}\log(q))J_0(qz)-Y_0(qz)\right] \left[(c+\frac{2}{\pi}\log(q))J_0(qz')-Y_0(qz')\right]}{(c+\frac{2}{\pi}\log(q))^2+1} \right] \notag \\ &\quad + \left. 2e^{-\pi c} \, \frac{e^{-i\sqrt{k^2-e^{-\pi c/2}}(t-t^\prime-i\epsilon)}}{\sqrt{2\pi(k^2-e^{-\pi c/2})}} \, K_0\big(e^{-\pi c/2}z\big) K_0\big(e^{-\pi c/2}z'\big) \right\} \, . \label{eq:2pfmenuzero}\end{aligned}$$ Similar to the case $\nu \in (0,1)$, the extra term is not invariant under the isometries of AdS, as it is not a function of the geodesic distance. Therefore, when $\nu=0$, we conclude that we are unable to construct a ground state. This may be seen as the counterpart of a massless, minimally coupled scalar field on four-dimensional de Sitter spacetime, for which there is also no ground state [@Kirsten:1993ug]. Closed form expression {#sec:2pfclosedform} ---------------------- The two-point function $G^+(x,x')$ for a scalar field in AdS${}_{d+1}$ on the ground state, or more generally in any maximally symmetric state, may also be given in closed form. Because of the maximal symmetry of the spacetime and of the state, it depends only on the geodesic distance between $x$ and $x'$. In Ref. [@Allen:1985wd] it was shown that $G^+$ satisfies an ordinary differential equation of hypergeometric type, $$\label{eq:hypeq} \left\{u(1-u) \frac{\dd^2}{\dd u^2} + \left[c-(a+b+1)u\right] \frac{\dd}{\dd u} - ab \right\} G^+(u) = 0 \, ,$$ where $$a = \frac{d}{2} - \nu \, , \qquad b = \frac{d}{2} + \nu \, , \qquad c = \frac{d+1}{2} \, ,$$ and $$u = u(\sigma) \doteq \cosh^2 \left(\frac{\sqrt{2\sigma}}{2}\right)$$ is an invariant quantity which depends only on Synge’s world function $\sigma$ defined in section \[sec:AdS\]. In the Poincaré domain, using and , $u$ may be written as $$\label{eq:udef} u = \cosh^2 \left(\frac{\sqrt{2\sigma}}{2}\right) = 1 + \frac{\sigma_{\bM}}{2zz'} = \frac{\sigma_{\bM}^{(-)}}{2zz'} \, ,$$ where, with $i=1,\ldots,d-1$, $$\begin{aligned} \sigma_{\bM} &= \frac{1}{2} \left[ - (t-t')^2 + \delta^{ij} (x_i - x'_i)(x_j - x'_j) + (z-z')^2 \right] \, , \\ \sigma_{\bM}^{(-)} &= \frac{1}{2} \left[ - (t-t')^2 + \delta^{ij} (x_i - x'_i)(x_j - x'_j) + (z+z')^2 \right] \, .\end{aligned}$$ Note that $u \in [0,1)$ for timelike separation and $u \in (1,\infty)$ for spacelike separation. Two independent solutions of when $\nu > 0$ are \[eq:Ghyp\] $$\begin{aligned} G^+_1(u) &= \lim_{\epsilon \to 0^+} u_{\epsilon}^{-\frac{d}{2}-\nu} \, \frac{F \big(\tfrac{d}{2}+\nu, \tfrac{1}{2}+\nu; 1+2\nu; u_{\epsilon}^{-1}\big)}{\Gamma(1+2\nu)} \, , \label{eq:solution1} \\ G^+_2(u) &= \lim_{\epsilon \to 0^+} u_{\epsilon}^{-\frac{d}{2}+\nu} \, \frac{F \big(\tfrac{d}{2}-\nu, \tfrac{1}{2}-\nu; 1-2\nu; u_{\epsilon}^{-1}\big)}{\Gamma(1-2\nu)} \, , \label{eq:solution2}\end{aligned}$$ where $u_{\epsilon} \doteq u(\sigma + 2i \epsilon(t-t')+\epsilon^2)$ implements the regularization of the two-point function. The function $F(a,b;c;z)/\Gamma(c)$ (known as the regularized hypergeometric function) is an entire function of its parameters $a$, $b$ and $c$ (see e.g. $\S 9.4$ of [@Lebedev:1972]). Hence, the solutions above are defined for all $\nu \geq 0$. However, they are identical for $\nu=0$, and thus a second linearly independent solution needs to be found. In this case, two independent solutions are \[eq:Ghypnu0\] $$\begin{aligned} G^+_1(u) &= \lim_{\epsilon \to 0^+} u_{\epsilon}^{-\frac{d}{2}} \, F \big(\tfrac{d}{2}, \tfrac{1}{2}; 1; u_{\epsilon}^{-1}\big) \, , \\ G^+_2(u) &= \lim_{\epsilon \to 0^+} F \big(\tfrac{d}{2}, \tfrac{d}{2}; \tfrac{d+1}{2}; u_{\epsilon} \big) \, .\end{aligned}$$ The second independent solution may equivalently be written as $$\begin{aligned} G^+_2(u) = \lim_{\epsilon \to 0^+} \Gamma\left(\frac{d+1}{2}\right) (-u_{\epsilon})^{-\tfrac{d}{2}} \sum_{j=0}^{\infty} \frac{\Gamma\left(\tfrac{d}{2}+j\right) \Gamma\left(\tfrac{1}{2}+j\right)}{(j!)^2} \left[ \log(-u_{\epsilon}) + h(j) \right] u_{\epsilon}^{-j} \, ,\end{aligned}$$ where $$h(j) \doteq 2\psi(j+1) - \psi \left(\tfrac{d}{2}+j\right) - \psi \left(\tfrac{1}{2}-j\right) \, ,$$ and $\psi(w) \doteq \Gamma'(w)/\Gamma(w)$ is the digamma function. These closed form expressions for the two-point functions coincide with the mode expansions obtained in the previous section. In Appendix \[app:2pfcomputation\], we show that for $\nu > 0$ $$\begin{aligned} G^+_1(u) &\propto \lim_{\epsilon \to 0^+} (zz')^{\frac{d}{2}} \int_0^\infty \dd k \, k \left(\frac{k}{r}\right)^{\frac{d-3}{2}} \! J_{\frac{d-3}{2}}(kr) \int_0^\infty \dd q \, q \, \frac{e^{-i\sqrt{k^2+q^2}(t-t^\prime-i\epsilon)}}{\sqrt{2\pi(k^2+q^2)}} \, J_\nu(qz) J_\nu(qz^\prime) \, , \\ G^+_2(u) &\propto \lim_{\epsilon \to 0^+} (zz')^{\frac{d}{2}} \int_0^\infty \dd k \, k \left(\frac{k}{r}\right)^{\frac{d-3}{2}} \! J_{\frac{d-3}{2}}(kr) \int_0^\infty \dd q \, q \, \frac{e^{-i\sqrt{k^2+q^2}(t-t^\prime-i\epsilon)}}{\sqrt{2\pi(k^2+q^2)}} \, J_{-\nu}(qz) J_{-\nu}(qz^\prime) \, .\end{aligned}$$ We see that, up to normalization, $G^+_1$ is the two-point function for $\nu \in [1,\infty)$ and for $\nu \in (0,1)$ when Dirichlet boundary conditions are applied, $G^+_1 \propto G^{+({\rm D})}$, whereas $G^+_2$ is the two-point function for $\nu \in (0,1)$ when Neumann boundary conditions are applied, $G^+_2 \propto G^{+({\rm N})}$. Since they are linearly independent, we conclude that the two-point function for $\nu \in (0,1)$ and Robin boundary conditions of the form is $$\label{eq:2pfBCexact} G^+(x,x') = \mathcal{N} \left[ \cos(\alpha) \, G^+_1(u) + \sin(\alpha) \, G^+_2(u) \right] \, , \qquad \alpha \in (\tfrac{\pi}{2},\pi) \, .$$ where $\mathcal{N}$ is a normalization constant. For Robin boundary conditions with $\alpha \in (0,\tfrac{\pi}{2})$, the contributions from the “bound state” solutions obtained in Section \[sec:modeexpansions\] need to be added. Observe in particular that the admissible range for $\alpha$ includes also $\frac{\pi}{4}$, which corresponds to transparent boundary conditions. In this case, thus, we expect that bound states must be taken into account, a feature which was not highlighted previously in the literature. In the case $\nu=0$, the first result in is still valid, $$\begin{aligned} G^+_1(u) &\propto \lim_{\epsilon \to 0^+} (zz')^{\frac{d}{2}} \int_0^\infty \dd k \, k \left(\frac{k}{r}\right)^{\frac{d-3}{2}} \! J_{\frac{d-3}{2}}(kr) \int_0^\infty \dd q \, q \, \frac{e^{-i\sqrt{k^2+q^2}(t-t^\prime-i\epsilon)}}{\sqrt{2\pi(k^2+q^2)}} \, J_0(qz) J_0(qz') \, ,\end{aligned}$$ and thus $G^+_1$ is still the two-point function when Dirichlet boundary conditions are applied, $G^+_1 \propto G^{+({\rm D})}$, minus the contribution coming from the “bound states”. We were unable to show explicitly that $G^+_2 \propto G^{+({\rm N})}$, but given that it is a non-principal solution of it must be given by a linear combination of $G^{+({\rm D})}$ and $G^{+({\rm N})}$, as given in . However, we note once more that, when $\nu=0$, the two-point function obtained above is not that of the ground state, given the lack of maximal symmetry. Hadamard condition ================== In this section, we verify that the states for which the two-point functions were obtained in the previous section satisfy a natural generalization of the Hadamard condition for $\PAdS_{d+1}$. First, recall that, for globally hyperbolic spacetimes, a quantum state is said to satisfy the *local Hadamard condition* if its two-point function is of Hadamard form. A two-point function is of the Hadamard form if it is given by $$\begin{aligned} G^+(x,x') &= H^{(d+1)}(\sigma(x,x'))+ \mathcal{O}(\sigma^0) \notag \\ &\doteq \lim_{\epsilon \to 0^+} \frac{\Gamma(\frac{d-1}{2})}{2(2\pi)^{\frac{d+1}{2}}} \left[ \frac{U(x,x')}{\left(\sigma_{\epsilon}(x,x')\right)^{\frac{d-1}{2}}} + V(x,x') \log\left(\sigma_{\epsilon}(x,x')\right) + \mathcal{O}(\sigma^0) \right] \, , \label{eq:Hadamardform}\end{aligned}$$ where $\sigma_{\epsilon} \doteq \sigma + 2i \epsilon(t-t')+\epsilon^2$ and $U$ and $V$ are smooth biscalars which are uniquely determined and only depend on the geometric features of the spacetime and on the parameter $m^2$ [@Decanini:2005eg]. The biscalar $V$ is identically zero for odd $d+1$ spacetime dimensions. $H^{(d+1)}(\sigma(x,x'))$ is the so-called $(d+1)$-dimensional *Hadamard parametrix*. It is important to keep in mind that, having set $\ell=1$ in , $\sigma(x,x')$ is a dimensionless quantity. Hence, although in the standard version of the local Hadamard form of the two-point function the argument of the logarithm is divided by a reference scale length, in our case this is not necessary as this length has been fixed *a priori*. In globally hyperbolic spacetimes, it follows that the local Hadamard condition is equivalent to the global Hadamard one [@Radzikowski:1996pa; @Radzikowski:1996ei], which entails in addition that the only singularity of the two-point function is at $\sigma=0$ and that it is of Hadamard form. A more rigorous definition requires the tools of microlocal analysis and may be found in [@Brunetti:2015vmh Ch.5]. Even though AdS is not globally hyperbolic, we can still verify if the two-point functions obtained above are of Hadamard form for every globally hyperbolic subregion. If that is the case, we say that the maximally symmetric state in AdS satisfies the local Hadamard condition. However, it does not follow that the state satisfies a global Hadamard condition, as the standard definition, adopted in globally hyperbolic spacetimes, does not apply. A novel analysis is required and we plan to address it in future work [@DappiaggiFerreira], also in view of the investigation in [@Vasy]. Here, we verify that the two-point functions in $\PAdS$ have a richer singularity structure [^9] than those in globally hyperbolic spacetimes, while at the same time satisfying the local Hadamard condition in any globally hyperbolic subregion. We will focus on the study of the singularities of the two-point functions obtained for the ground state in the cases of $d=2$ and $d=3$. Analogous comments can be made for larger $d$, as we discuss briefly below. We start with $d=3$, the physically relevant case, and assume $\nu > 0$. The two-point function is a linear combination of the solutions and we know that the hypergeometric functions in those solutions have only three singular points: $u=0, \, 1, \infty$. The latter, $u \to \infty$, occurs when either $z \to 0$ or $z' \to 0$, which takes one of the points $x, \, x'$ to the boundary and, therefore, does not belong to the spacetime. The singularity $u=1$ corresponds to $\sigma = 0$, cf. . If we expand the solutions with $d=3$ in $\sigma$, such that $x$ and $x'$ belong to a globally hyperbolic subregion of $\AdS_4$, $$\begin{aligned} G^+_1(u_{\epsilon}) &= \frac{2^{1+2\nu} \, \Gamma \left(1+\nu\right)}{\sqrt{\pi} \, \Gamma \left(\frac{3}{2}+\nu\right) \Gamma \left(1+2\nu\right)} \left[ \frac{1}{\sigma_{\epsilon}} + \frac{1}{2} \left(\nu^2 - \frac{1}{4}\right) \log(\sigma_{\epsilon}) + \mathcal{O}(\sigma^0) \right] \, , \\ G^+_2(u_{\epsilon}) &= \frac{2^{1-2\nu} \, \Gamma \left(1-\nu\right)}{\sqrt{\pi} \, \Gamma \left(\frac{3}{2}-\nu\right) \Gamma \left(1-2\nu\right)} \left[ \frac{1}{\sigma_{\epsilon}} + \frac{1}{2} \left(\nu^2 - \frac{1}{4}\right) \log(\sigma_{\epsilon}) + \mathcal{O}(\sigma^0) \right] \, .\end{aligned}$$ This expansion is exactly the Hadamard expansion , up to normalization constants, presented for $d=3$ in [@Decanini:2005eg] for a globally hyperbolic subregion of $\AdS_4$. Hence, in view of , the two-point function reads $$G^+(x,x') \propto \left[\cos(\alpha) + \sin(\alpha)\right] H^{(4)}(\sigma(x,x'))+\mathcal{O}(\sigma^0) \, .$$ By choosing a suitable $\alpha$-dependent normalization constant, we can make $G^+(x,x')$ satisfy the local Hadamard property , except if $\alpha = \frac{3\pi}{4}$. The singularity $u=0$ corresponds to $\sigma^{(-)} = 0$, where $\sigma^{(-)}$ is such that, cf. Eq. , $$u \doteq - \sinh^2 \left(\frac{\sqrt{2\sigma^{(-)}}}{2} \right) = \frac{\sigma_{\bM}^{(-)}}{2zz'} \, .$$ Two points $x$ and $x'$ are such that $\sigma^{(-)}(x,x') = 0$ if there is a null geodesic starting at $x$ that is “reflected” at the boundary and ends at $x'$ (see Fig. \[fig:wavefrontset\]). More rigorously, if we consider the conformally related spacetime $\mathring{\bH}^4$ and allow $z$ to take all real values, $\sigma^{(-)}(x,x')$ vanishes if $\sigma_{\bM}^{(-)}(x,x') = 0$, or equivalently if $\sigma_{\bM}(x^{(-)},x') = 0$, where $x^{(-)} \doteq x|_{z \mapsto -z}$. Note that there is no globally hyperbolic subregion of $\AdS_4$ in which $\sigma^{(-)}(x,x') = 0$, hence this singularity is not present for a two-point function on a globally hyperbolic submanifold. ![\[fig:wavefrontset\]Singularity structure of the two-point function.](Figures/singularities.pdf) If we now expand the two solutions in $\sigma^{(-)}$, we obtain $$\begin{aligned} G^+_1(u_{\epsilon}) &= i (-1)^{\nu} \frac{2^{1+2\nu} \, \Gamma \left(1+\nu\right)}{\sqrt{\pi} \, \Gamma \left(\frac{3}{2}+\nu\right) \Gamma \left(1+2\nu\right)} \left[\frac{1}{\sigma_{\epsilon}^{(-)}} + \frac{1}{2} \left(\nu^2 - \frac{1}{4}\right) \log(\sigma_{\epsilon}^{(-)}) + \mathcal{O}\big((\sigma^{(-)})^0\big) \right] \, , \\ G^+_2(u_{\epsilon}) &= i (-1)^{-\nu} \frac{2^{1-2\nu} \, \Gamma \left(1-\nu\right)}{\sqrt{\pi} \, \Gamma \left(\frac{3}{2}-\nu\right) \Gamma \left(1-2\nu\right)} \left[\frac{1}{\sigma_{\epsilon}^{(-)}} + \frac{1}{2} \left(\nu^2 - \frac{1}{4}\right) \log(\sigma_{\epsilon}^{(-)}) + \mathcal{O}\big((\sigma^{(-)})^0\big) \right] \, . \end{aligned}$$ This has exactly the same Hadamard form, up to normalization constants, but with respect to $\sigma^{(-)}$. Hence, in view of , the two-point function reads $$G^+(x,x') \propto \left[\cos(\alpha) + (-1)^{-2\nu} \sin(\alpha)\right] H^{(4)}(\sigma^{(-)}(x,x'))+\mathcal{O}\big((\sigma^{(-)})^0\big) \, .$$ The singular contribution vanishes for $\nu=\frac{1}{2}$ and $\alpha = \frac{\pi}{4}$, for which there are no singularities along reflected null geodesics. This justifies why $\alpha = \frac{\pi}{4}$ is referred to as transparent boundary conditions for the massless, conformally coupled scalar field. We analyze now the two-point function for $d=2$ and $\nu > 0$. The two solutions with $d=2$ have the same two singularities at $u=1$ and $u=0$. If we expand them in $\sigma$, such that $x$ and $x'$ belong to a globally hyperbolic subregion of $\AdS_3$, we obtain $$G^+_1(u_{\epsilon}) = \frac{2^{\frac{1}{2}+2\nu}}{\Gamma \left(1+2\nu\right)} \frac{1}{\sqrt{\sigma_{\epsilon}}} + \mathcal{O}(\sigma^0) \, , \qquad G^+_2(u_{\epsilon}) = \frac{2^{\frac{1}{2}-2\nu}}{\Gamma \left(1-2\nu\right)} \frac{1}{\sqrt{\sigma_{\epsilon}}} + \mathcal{O}(\sigma^0) \, .$$ Again, this is of the same Hadamard form as presented in [@Decanini:2005eg] for $d=2$ in any globally hyperbolic subregion of $\AdS_3$. Hence, in view of , the two-point function reads $$G^+(x,x') \propto \left[\cos(\alpha) + \sin(\alpha)\right] H^{(3)}(\sigma(x,x'))+\mathcal{O}(\sigma^0) \, .$$ If we instead expand them in $\sigma^{(-)}$, we obtain $$G^+_1(u_{\epsilon}) = i (-1)^{\nu} \frac{2^{\frac{1}{2}+2\nu}}{\Gamma \left(1+2\nu\right)} \frac{1}{\sqrt{\sigma^{(-)}_{\epsilon}}} + \mathcal{O}(\sigma^0) \, , \quad G^+_2(u_{\epsilon}) = i (-1)^{-\nu} \frac{2^{\frac{1}{2}+2\nu}}{\Gamma \left(1+2\nu\right)} \frac{1}{\sqrt{\sigma^{(-)}_{\epsilon}}} + \mathcal{O}(\sigma^0) \, .$$ Hence, in view of , the two-point function reads $$G^+(x,x') \propto i (-1)^{\nu} \left[\cos(\alpha) + (-1)^{-2\nu} \sin(\alpha)\right] H^{(3)}(\sigma^{(-)}(x,x'))+\mathcal{O}\big((\sigma^{(-)})^0\big) \, .$$ Therefore, we shall also call a quantum state *Hadamard* in $\PAdS_3$ a state whose two-point function has the singularity structure described above. Similar investigations can be made for larger $d$. However, it becomes increasingly impractical to perform the expansions in $\sigma$ and $\sigma^{(-)}$ since we would have to resort to a case by case analysis. There are recursive methods to obtain the expansions of $U$ and $V$ in $\sigma$ for any fixed $d$, but they get significantly more complex for larger $d$ (detailed expressions for $d+1$ up to 6 may be found in Ref. [@Decanini:2005eg]). Nevertheless, using the tools of microlocal analysis, it is possible to show that the singularity structure observed for $d=2$ and $d=3$, with which we defined the notion of a Hamadard state on $\PAdS_3$ and $\PAdS_4$, is verified for any $d$. We leave this proof to a forthcoming work [@DappiaggiFerreira]. In view of the above analysis, we define a *Hadamard* quantum state in $\PAdS_{d+1}$, $d \geq 2$, to be any state whose two-point function $G^+(x,x')$ is such that $$G^+(x,x') - H^{(d+1)}(\sigma(x,x')) - i (-1)^{-\nu} \frac{\cos(\alpha) + (-1)^{-2\nu} \sin(\alpha)}{\cos(\alpha) + \sin(\alpha)} \, H^{(d+1)}(\sigma^{(-)}(x,x'))$$ is a smooth function on $\PAdS_{d+1}\times\PAdS_{d+1}$. In particular, if $\alpha = \frac{3\pi}{4}$, we cannot find a Hadamard state satisfying this definition. Notice that although a ground state does not exist for $\alpha \in (0,\frac{\pi}{2})$, on account of the presence of “bound state” solutions, the proposed definition still applies to these cases. Conclusions =========== In this paper, we have considered a real, massive scalar field on $\PAdS_{d+1}$, the Poincaré domain of the $(d+1)$-dimensional AdS spacetime. In particular, we have determined all admissible boundary conditions that can be applied on the conformal boundary and we have constructed the two-point function associated with the ground state, finding ultimately an explicit closed form. In addition, we have investigated its singular structure, showing consistency with the minimal requirement of being of Hadamard form in every globally hyperbolic subregion of $\PAdS_{d+1}$. As a consequence we propose a new definition of Hadamard states which applies to $\PAdS_{d+1}$. To conclude our work, we would like to highlight two open issues which we deem appropriate of further investigations. The first concerns the choice of boundary conditions. As we have shown, there are instances where “bound state” solutions appear in the construction of the two-point and of the commutator functions. The main direct consequence of this unexpected feature is the lack of a ground state for the underlying system, as invariance under the action of certain isometries is broken. On the one hand, we can observe that this poses no obstruction to the existence of Hadamard states, but, on the other hand, there is no clear physical interpretation why such “bound state” solutions appear and what are the concrete consequences of their existence. The second open problem lies in the investigation of the notion of Hadamard states for a real, massive scalar field on $\PAdS_{d+1}$. In the last section we have given a local definition, which exploits ultimately the existence of a global coordinate chart on $\mathring{\bH}^{d+1}$. If one aims at generalizing these results to asymptotically AdS spacetimes or even to manifolds with timelike boundaries, we cannot expect that a local construction becomes practical. Hence, following a similar path to the one taken by those who investigated Hadamard states on globally hyperbolic spacetimes, we expect that a necessary step is to translate our analysis in the language of microlocal analysis. In this way, we hope to be able to give a global definition of Hadamard states and to formulate a version of the work of Radzikowski [@Radzikowski:1996ei; @Radzikowski:1996pa] in the context of asymptotically AdS spacetimes. Acknowledgments {#acknowledgments .unnumbered} =============== We are grateful to Jorma Louko for enlightening discussions and for pointing out Ref. [@Titchmarsh]. We are also grateful to Nicolò Drago, Gabriele Nosari, Pedro Lauridsen Ribeiro, Nicola Pinamonti and Micha[ł]{} Wrochna for useful comments and discussions. The work of C.D. was supported by the University of Pavia. The work of H. F. was supported by the INFN postdoctoral fellowship “Geometrical Methods in Quantum Field Theories and Applications”. Two-point function computation for $\nu>0$ {#app:2pfcomputation} ========================================== The two-point function for a massive scalar field in $\mathring{\bH}^{d+1}$ for $\nu \in [1,\infty)$ or in the case of Dirichlet boundary conditions for $\nu \in (0,1)$ is given by , $$\label{eq:tpf1} G^+_{\bH}(x,x') = \mathcal{N} \sqrt{zz'} \int_0^\infty \dd k \, k \left(\frac{k}{r}\right)^{\frac{d-3}{2}} \! J_{\frac{d-3}{2}}(kr) \int_0^\infty \dd q \, q \, \frac{e^{-i\sqrt{k^2+q^2}(t-t^\prime-i\epsilon)}}{\sqrt{2\pi(k^2+q^2)}} \, J_\nu(qz) J_\nu(qz') \, ,$$ where we omit the limit $\epsilon \to 0^+$ for presentation simplicity. The two-point function in the case of Neumann boundary conditions when $\nu \in (0,1)$ can also be obtained from by allowing $\nu \in (-1,0)$ (see section \[sec:bcnusmall\]). In this appendix, we compute explicitly the integrals in and obtain the two-point function in closed form, as presented in Section \[sec:2pfclosedform\]. Using Eqs. (6.737.5) and (6.737.6) of [@Gradshteyn], for $d=2, \, 3$, we obtain $$\begin{aligned} G^+_{\bH}(x,x') &= \mathcal{N} \sqrt{zz'} \int_0^\infty \dd q \, q^{\frac{d}{2}} \, J_{\nu}(qz) J_\nu(qz^\prime) \left\{ \frac{1}{\pi} \, \Theta(r-(t-t')) \frac{K_{\frac{d}{2}-1} \left(q \sqrt{\chi_{\epsilon}^2}\right)}{\left(\sqrt{\chi_{\epsilon}^2}\right)^{\frac{d}{2}-1}} \right. \notag \\ &\quad \left. -\frac{i}{2} \Theta(t-t'-r) \frac{J_{1-\frac{d}{2}} \left(q \sqrt{-\chi_{\epsilon}^2}\right) - i \, Y_{1-\frac{d}{2}} \left(q \sqrt{-\chi_{\epsilon}^2}\right)}{\left(\sqrt{-\chi_{\epsilon}^2}\right)^{\frac{d}{2}-1}} \right\} \notag \\ &= \mathcal{N} \, \frac{\sqrt{zz'}}{\pi} \int_0^\infty \dd q \, q^{\frac{d}{2}} \, \frac{K_{\frac{d}{2}-1} \left(q \sqrt{\chi_{\epsilon}^2}\right)}{\left(\sqrt{\chi_{\epsilon}^2}\right)^{\frac{d}{2}-1}} \, J_{\nu}(qz)J_\nu(qz^\prime) \, , \label{eq:2pfintegralstep1}\end{aligned}$$ where $\chi_{\epsilon}^2 \doteq r^2-(t-t'-i\epsilon)^2$, $\Theta$ is the Heaviside function, $K_{\frac{d}{2}-1}$ is the modified Bessel function of the second kind and we used (see e.g. [@Lebedev:1972 §5.6-5.7]) $$\begin{gathered} J_{\alpha}(w) - i Y_{\alpha}(w) = H_{\alpha}^{(2)}(w) = e^{i\pi\alpha} H_{-\alpha}^{(2)}(w) \, , \qquad w \notin (-\infty,0] \, , \\ K_{\alpha}(w) = - \frac{i\pi}{2} e^{-\frac{i\pi\alpha}{2}} H_{\alpha}^{(2)}(-iw) \, , \qquad \arg(w) \in \left[-\frac{\pi}{2},\pi\right] \, ,\end{gathered}$$ where $H_{\alpha}^{(2)}$ is the second Hankel function. Even though the calculation leading to is valid for $d=2, \, 3$, the result can be analytically continued to $d \geq 2$. At this point, it is convenient to consider even and odd $d$ separately. Let $d = 2n+1$, $n=1, 2, \ldots$. Then, $$\begin{aligned} G^+_{\bH}(x,x') &= \mathcal{N} \, \frac{\sqrt{zz'}}{\pi} \int_0^\infty \dd q \, q^{n+\frac{1}{2}} \, \frac{K_{n-\frac{1}{2}} \left(q \chi_{\epsilon}\right)}{\chi_{\epsilon}^{n-\frac{1}{2}}} \, J_{\nu}(qz)J_\nu(qz') \notag \\ &= \mathcal{N} \, \frac{\sqrt{zz'}}{\pi} \left. \left(\frac{1}{\chi} \frac{\dd}{\dd \chi} \right)^n \int_0^\infty \dd q \, \frac{K_{-\frac{1}{2}} \left(q \chi\right)}{\left(q\chi\right)^{-\frac{1}{2}}} \, J_{\nu}(qz)J_\nu(qz') \right|_{\chi=\chi_{\epsilon}} \notag \\ &= \mathcal{N} \, \frac{\sqrt{zz'}}{\sqrt{2\pi}} \left. \left(\frac{1}{\chi} \frac{\dd}{\dd \chi} \right)^n \int_0^\infty \dd q \, e^{-q \chi} \, J_{\nu}(qz)J_\nu(qz') \right|_{\chi=\chi_{\epsilon}} \notag \\ &= \mathcal{N} \, \frac{1}{\sqrt{2\pi^3}} \left. \left(\frac{1}{\chi} \frac{\dd}{\dd \chi} \right)^n Q_{\nu-\frac{1}{2}}\left( \frac{z^2+{z'}^2+\chi^2}{2zz'} \right) \right|_{\chi=\chi_{\epsilon}} \notag \\ &= \mathcal{N} \, \frac{1}{\sqrt{2\pi^3}(zz')^n} \, \left. \frac{\dd^{n}}{\dd \eta^{n}} Q_{\nu-\frac{1}{2}}\left(\eta\right) \right|_{\eta=\eta_{\epsilon}} \notag \\ &= \mathcal{N} \, \frac{1}{\sqrt{2\pi^3}(2zz')^n} \, \left. \frac{\dd^{n}}{\dd u^{n}} Q_{\nu-\frac{1}{2}}\left(2u-1\right) \right|_{u=u_{\epsilon}} \notag \\ &= \mathcal{N} \, \frac{1}{\sqrt{2\pi^3}(zz')^n} \, \left. \left(\frac{1}{\sinh(s)} \frac{\dd}{\dd s}\right)^n Q_{\nu-\frac{1}{2}}\left(\cosh(s)\right) \right|_{s=s_{\epsilon}} \, , \label{eq:2pfQ}\end{aligned}$$ where $$\eta_{\epsilon} \doteq \frac{z^2+{z'}^2+r^2-(t-t'-i\epsilon)^2}{2zz'} = 2u_{\epsilon} - 1 \doteq \cosh(s_{\epsilon}) \, ,$$ $Q_{\nu-\frac{1}{2}}$ is the Legendre function of the second kind and where we used Eq. (6.612.3) of [@Gradshteyn] and the relation $$\label{eq:derbesselK} \left(\frac{1}{\chi} \frac{\dd}{\dd \chi} \right)^n \left( \chi^{-\alpha} K_{\alpha}(\chi) \right) = \chi^{-\alpha-n} K_{\alpha+n}(\chi) \, .$$ Note that is valid for $\nu > -\frac{1}{2}$ but is not defined for $\nu = -1/2$, hence it cannot be used as currently written for the case of Neumann boundary conditions. However, we can extend it analytically to $\nu > -1$ as follows. From Eq. (14.10.4) of [@NIST], $$\label{eq:Qderrec} (1-\eta_{\epsilon}^2) \, Q'_{\nu-\frac{1}{2}}(\eta_{\epsilon}) = \left(\nu+\frac{1}{2}\right) \left[\eta_{\epsilon} \, Q_{\nu-\frac{1}{2}}(\eta_{\epsilon}) - Q_{\nu+\frac{1}{2}}(\eta_{\epsilon}) \right] \, .$$ $Q_{\nu-\frac{1}{2}}$ is not defined for $\nu = -1/2$, as for Eq. (14.3.7) of [@NIST] one has $$Q_{\nu-\frac{1}{2}}(\eta) = \frac{\sqrt{\pi} \, \Gamma\left(\nu+\frac{1}{2}\right)}{2^{\nu-\frac{1}{2}} \, \eta^{\nu+\frac{1}{2}} \, \Gamma(\nu+1)} \, F\left(\tfrac{\nu}{2}+\tfrac{3}{4}, \tfrac{\nu}{2}+\tfrac{1}{4}; \nu+1; \tfrac{1}{\eta^2}\right) \, ,$$ for $\nu \notin -\frac{2\mathbb{N}+1}{2}$ and $\eta>1$. Nevertheless, one can analytically continue to $\nu > -1$ as $$(1-\eta_{\epsilon}^2) \, Q'_{\nu-\frac{1}{2}}(\eta_{\epsilon}) = \frac{\sqrt{\pi} \, \Gamma\left(\nu+\frac{3}{2}\right)}{2^{\nu-\frac{1}{2}} \, \eta_{\epsilon}^{\nu-\frac{1}{2}} \, \Gamma(\nu+1)} \, F\left(\tfrac{\nu}{2}+\tfrac{3}{4}, \tfrac{\nu}{2}+\tfrac{1}{4}; \nu+1; \tfrac{1}{\eta_{\epsilon}^2}\right) - \left(\nu+\frac{1}{2}\right) Q_{\nu+\frac{1}{2}}(\eta_{\epsilon}) \, .$$ Using the same notation for the extended function, may be used for the Neumann boundary conditions with $\nu \in (-1,0)$. Let $d = 2n$, $n=1, 2, \ldots$. Then, $$\begin{aligned} G^+_{\bH}(x,x') &= \mathcal{N} \, \frac{\sqrt{zz'}}{\pi} \int_0^\infty \dd q \, q^{n} \, \frac{K_{n-1} \left(q \chi_{\epsilon}\right)}{\chi_{\epsilon}^{n-1}} \, J_{\nu}(qz)J_\nu(qz') \\ &= \mathcal{N} \, \frac{\sqrt{zz'}}{\pi} \left. \left(\frac{1}{\chi} \frac{\dd}{\dd \chi} \right)^{n-1} \int_0^\infty \dd q \, q \, K_0 \left(q \chi\right) J_{\nu}(qz)J_\nu(qz') \right|_{\chi=\chi_{\epsilon}} \\ &= \mathcal{N} \, \frac{\sqrt{zz'}}{\pi} \left. \left(\frac{1}{\chi} \frac{\dd}{\dd \chi} \right)^{n-1} \frac{\left(\frac{\sqrt{\chi^2+(z+z')^2}+\sqrt{\chi^2+(z-z')^2}}{\sqrt{\chi^2+(z+z')^2}-\sqrt{\chi^2+(z-z')^2}}\right)^{-\nu}}{\sqrt{(\chi^2+(z+z')^2)(\chi^2+(z-z')^2)}} \right|_{\chi=\chi_{\epsilon}} \\ &= \mathcal{N} \, \frac{2^{\nu-1}}{\pi(zz')^{n-\frac{1}{2}}} \left. \frac{\dd^{n-1}}{\dd \eta^{n-1}} \frac{\left(\eta+\sqrt{\eta^2-1}\right)^{-\nu}}{\sqrt{\eta^2-1}} \right|_{\eta=\eta_{\epsilon}} \\ &= \mathcal{N} \, \frac{2^{\nu-2}}{\pi(zz')^{n+\frac{1}{2}}} \left. \frac{\dd^{n-1}}{\dd u^{n-1}} \frac{\left(2u-1+\sqrt{u(u-1)}\right)^{-\nu}}{\sqrt{u(u-1)}} \right|_{u=u_{\epsilon}} \\ &= \mathcal{N} \, \frac{2^{\nu-1}}{\pi(zz')^{n-\frac{1}{2}}} \left. \left(\frac{1}{\sinh(s)} \frac{\dd}{\dd s}\right)^{n-1} \frac{e^{-\nu s}}{\sinh(s)} \right|_{s=s_{\epsilon}} \, ,\end{aligned}$$ where we used Eq. (6.522.3) of [@Gradshteyn] and . To prove that these results are equivalent to the ones written in terms of the hypergeometric functions , we show that they satisfy the same initial conditions, since they are all solutions of the same differential equation. First, we verify the claim for $d=2, \, 3$. In terms of the invariant quantity $u$, for $d=2$, let $$\begin{aligned} g_1^{d=2}(u) &= u_{\epsilon}^{-1-\nu} F\left(1+\nu, \tfrac{1}{2}+\nu; 1+2\nu; u_{\epsilon}^{-1} \right) \, , \\ g_2^{d=2}(u) &= 4^{\nu} \, \frac{\left(2u_{\epsilon}-1+2\sqrt{u_{\epsilon}(u_{\epsilon}-1)}\right)^{-\nu}}{\sqrt{u_{\epsilon}(u_{\epsilon}-1)}} \, .\end{aligned}$$ Then, $$g_1^{d=2}(u) = g_2^{d=2}(u) = u_{\epsilon}^{-1-\nu} \left(1 + \frac{1+\nu}{2u} + \mathcal{O}(u^{-2}) \right) \, .$$ Hence, $g_1^{d=2} = g_2^{d=2}$. For $d=3$, let $$\begin{aligned} g_1^{d=3}(u) &= u_{\epsilon}^{-\frac{3}{2}-\nu} F\left(\tfrac{3}{2}+\nu, \tfrac{1}{2}+\nu; 1+2\nu; u_{\epsilon}^{-1} \right) \, , \\ g_2^{d=3}(u) &= -\frac{4^{1+\nu}}{\sqrt{\pi}} \frac{\Gamma(1+\nu)}{\Gamma(\frac{3}{2}+\nu)} \, Q'_{\nu-\frac{1}{2}}(2u_{\epsilon}-1) \, .\end{aligned}$$ Then, $$g_1^{d=3}(u) = g_2^{d=3}(u) = u_{\epsilon}^{-\frac{3}{2}-\nu} \left(1 + \frac{\frac{3}{2}+\nu}{2u} + \mathcal{O}(u^{-2}) \right) \, .$$ Hence, $g_1^{d=3} = g_2^{d=3}$. For arbitrary $d$, we give a proof by induction. For even $d$, let it be true for a fixed $d=2n$. For $d=2(n+1)$, let $$\begin{aligned} g_1^{d=2n+2}(u) &= u_{\epsilon}^{-n-1-\nu} F\left(n+1+\nu, \tfrac{1}{2}+\nu; 1+2\nu; u_{\epsilon}^{-1} \right) \, , \\ g_2^{d=2n+2}(u) &= \mathcal{N}^{n+1} \, \frac{\dd^n}{\dd u^n} \frac{\left(2u_{\epsilon}-1+2\sqrt{u_{\epsilon}(u_{\epsilon}-1)}\right)^{-\nu}}{\sqrt{u_{\epsilon}(u_{\epsilon}-1)}} \, ,\end{aligned}$$ for some constant $\mathcal{N}^{n+1}$. We know that $$g_1^{d=2n}(u) = g_2^{d=2n}(u) = \mathcal{N}^{n} \, \frac{\dd^{n-1}}{\dd u^{n-1}} \frac{\left(2u_{\epsilon}-1+2\sqrt{u_{\epsilon}(u_{\epsilon}-1)}\right)^{-\nu}}{\sqrt{u_{\epsilon}(u_{\epsilon}-1)}}$$ and that $$\begin{aligned} g_2^{d=2n+2}(u) &= \frac{\mathcal{N}^{n+1}}{\mathcal{N}^{n}} \, \frac{\dd}{\dd u} g_2^{d=2n}(u) = \frac{\mathcal{N}^{n+1}}{\mathcal{N}^{n}} \, \frac{\dd}{\dd u} g_1^{d=2n}(u) \\ &= - \frac{\mathcal{N}^{n+1}}{\mathcal{N}^{n}} (n+\nu) u_{\epsilon}^{-n-\nu} \left(1 + \frac{n+1+\nu}{2u} + \mathcal{O}(u^{-2}) \right) \, .\end{aligned}$$ Comparing with $$\begin{aligned} g_1^{d=2n+2}(u) = u_{\epsilon}^{-n-\nu} \left(1 + \frac{n+1+\nu}{2u} + \mathcal{O}(u^{-2}) \right) \, ,\end{aligned}$$ we conclude that $g_1^{d=2n+2} = g_2^{d=2n+2}$ with $$\mathcal{N}^{n+1} = - \frac{\mathcal{N}^{n}}{n+\nu} = (-1)^n \frac{\mathcal{N}^{1}}{\Gamma(n+1+\nu)} = \frac{(-1)^n \, 4^{\nu}}{\Gamma(n+1+\nu)} \, .$$ Similarly, for odd $d$, let it be true for a fixed $d=2n+1$. For $d=2n+3$, let $$\begin{aligned} g_1^{d=2n+3}(u) &= u_{\epsilon}^{-n-\frac{3}{2}-\nu} F\left(n+\tfrac{3}{2}+\nu, \tfrac{1}{2}+\nu; 1+2\nu; u_{\epsilon}^{-1} \right) \, , \\ g_2^{d=2n+3}(u) &= \mathcal{N}^{n+1} \, \frac{\dd^{n+1}}{\dd u^{n+1}} Q_{\nu-\frac{1}{2}}(2u_{\epsilon}-1) \, ,\end{aligned}$$ for some constant $\mathcal{N}^{n+1}$. We know that $$g_1^{d=2n+1}(u) = g_2^{d=2n+1}(u) = \mathcal{N}^{n} \, \frac{\dd^{n}}{\dd u^{n}} Q_{\nu-\frac{1}{2}}(2u_{\epsilon}-1)$$ and that $$\begin{aligned} g_2^{d=2n+3}(u) &= \frac{\mathcal{N}^{n+1}}{\mathcal{N}^{n}} \, \frac{\dd}{\dd u} g_2^{d=2n+1}(u) = \frac{\mathcal{N}^{n+1}}{\mathcal{N}^{n}} \, \frac{\dd}{\dd u} g_1^{d=2n+1}(u) \\ &= - \frac{\mathcal{N}^{n+1}}{\mathcal{N}^{n}} \left(n + \frac{1}{2}+\nu\right) u_{\epsilon}^{-n-\frac{3}{2}-\nu} \left(1 + \frac{n+\frac{3}{2}+\nu}{2u} + \mathcal{O}(u^{-2}) \right) \, .\end{aligned}$$ Comparing with $$\begin{aligned} g_1^{d=2n+3}(u) = u_{\epsilon}^{-n-\frac{3}{2}-\nu} \left(1 + \frac{n+\frac{3}{2}+\nu}{2u} + \mathcal{O}(u^{-2}) \right) \, ,\end{aligned}$$ we conclude that $g_1^{d=2n+3} = g_2^{d=2n+3}$ with $$\mathcal{N}^{n+1} = - \frac{\mathcal{N}^{n}}{n+\nu} = (-1)^n \frac{\mathcal{N}^{1}}{\Gamma(n+\frac{3}{2}+\nu)} = \frac{4^{1+\nu}}{\sqrt{\pi}} \frac{(-1)^{n+1} \, \Gamma(1+\nu)}{\Gamma(\frac{3}{2}+\nu)\Gamma(n+\frac{3}{2}+\nu)} \, .$$ This concludes the proof. Delta distribution representation {#app:deltafunction} ================================= In this appendix, we prove the identities $$\int_0^{\infty} \dd k \, k \, \left(\frac{k}{r}\right)^{\frac{d-3}{2}} \! J_{\frac{d-3}{2}}(kr) = 2^{\frac{d-3}{2}} \, \Gamma\left(\frac{d-1}{2}\right) \, \frac{\delta(r)}{r^{d-2}} = \frac{(2\pi)^{\frac{d}{2}}\Gamma\left(\frac{d-1}{2}\right)}{\sqrt{2} \, \Gamma\left(\frac{d}{2}\right)} \prod_{i=1}^{d-1} \delta(x^i-{x'}^i) \, ,$$ where $d \geq 2$ is an integer and $r>0$. We start with a standard representation of the delta distribution (Eq. (1.17.13) of [@NIST]), $$\delta(r-r') = r \int_0^{\infty} \dd k \, k \, J_{\mu}(kr) J_{\mu}(kr') \, ,$$ with ${\rm Re}(\mu) > -1$ and $r, r' > 0$. Given that $\delta(r-r')$ is zero when $r \neq r'$, we may write $$\delta(r-r') = \frac{r^{\mu+1}}{r'^{\mu}} \int_0^{\infty} \dd k \, k \, J_{\mu}(kr) J_{\mu}(kr') \, .$$ Using $$J_{\mu}(kr') = \frac{\left(\frac{1}{2}kr'\right)^{\mu}}{\Gamma(\mu+1)} + \mathcal{O}({r'}^{\mu+1}) \, ,$$ and letting $r' \to 0$, we get $$\delta(r) = \frac{r^{\mu+1}}{2^{\mu} \Gamma(\mu+1)} \int_0^{\infty} \dd k \, k^{\mu+1} \, J_{\mu}(kr) \, .$$ Letting $\mu = \frac{d-3}{2}$, this allows us to obtain $$\int_0^{\infty} \dd k \, k \, \left(\frac{k}{r}\right)^{\frac{d-3}{2}} \! J_{\frac{d-3}{2}}(kr) = 2^{\frac{d-3}{2}} \, \Gamma\left(\frac{d-1}{2}\right) \, \frac{\delta(r)}{r^{d-2}} \, .$$ Finally, by changing from the Cartesian coordinates $x^i$, $i=1, \ldots, d-1$, to spherical coordinates, $$\prod_{i=1}^{d-1} \delta(x^i-{x'}^i) = \frac{\delta(r)}{A_{d-1} r^{d-2}} = \frac{\Gamma\left(\frac{d}{2}\right)}{2\pi^{\frac{d}{2}}} \frac{\delta(r)}{r^{d-2}} \, ,$$ where $A_{d-1}$ is the area of a $(d-1)$-sphere. Hence, we obtain the desired identity $$\int_0^{\infty} \dd k \, k \, \left(\frac{k}{r}\right)^{\frac{d-3}{2}} \! J_{\frac{d-3}{2}}(kr) = \frac{(2\pi)^{\frac{d}{2}}\Gamma\left(\frac{d-1}{2}\right)}{\sqrt{2} \, \Gamma\left(\frac{d}{2}\right)} \prod_{i=1}^{d-1} \delta(x^i-{x'}^i) \, .$$ Eigenfunction expansion of the delta distribution {#app:eigenfunctionexpansion} ================================================= In this appendix, we show how to compute the expansion of the Dirac delta distribution in terms of the eigenfunctions of the operator $L$ defined in in an efficient way, as presented e.g. in Chapter 7 of Ref. [@Stakgold]. These expansions can be found in Section 4.11 of [@Titchmarsh], but the computation presented there involves convoluted and old-fashioned methods. We present the computation of the expansion in terms of the eigenfunctions of $L$ which satisfy Robin boundary conditions when $\nu \in (0,1)$. The others can be obtained in a similar way. First, we compute the *Green’s function* $\mathcal{G}(z,z';\lambda)$ associated with the Sturm-Liouville problem , which satisfies $$(L\otimes\mathbb{I} - \lambda) \, \mathcal{G} = (\mathbb{I}\otimes L - \lambda) \, \mathcal{G} = \delta(z-z') \, ,$$ and appropriate boundary conditions at $z=0$ and $z'=0$, if necessary. This can be done as follows. For $z < z'$, $\mathcal{G}(z,z';\lambda)$ is the solution of the homogeneous equation in the first entry, $u(z;\lambda)$, satisfying the boundary condition at $z=0$, whereas for $z > z'$, $\mathcal{G}(z,z';\lambda)$ is the solution of the homogeneous equation, $v(z;\lambda)$, which is $L^2(z_0,\infty)$ for some $z_0>0$ and for some $\lambda \in \bC$. Then, ensuring continuity at $z=z'$, one has $$\mathcal{G}(z,z';\lambda) = \mathcal{N}_{\lambda} \, u(z_<;\lambda) \, v(z_>;\lambda) \, ,$$ where $z_< \doteq \min\{z,z'\}$ and $z_> \doteq \max\{z,z'\}$. The jump condition, $$\left.\frac{\dd}{\dd z}\mathcal{G}(z,z';\lambda)\right|_{z=z'^+} - \left.\frac{\dd}{\dd z}\mathcal{G}(z,z';\lambda)\right|_{z=z'^-} = -1 \, ,$$ fixes the normalization constant $$\mathcal{N}_{\lambda} = - \frac{1}{W_z \big[u(\cdot;\lambda), v(\cdot;\lambda)\big]} \, .$$ The Green’s function can also be obtained as an expansion in terms of the eigenfunctions of $L$ which satisfy the same boundary conditions. If the operator $L$ only had a point spectrum with real eigenvalues $\lambda_n$ and corresponding eigenfunctions $\psi_n$, it is easy to show that $\mathcal{G}(z,z';\lambda)$ would be written as $$\mathcal{G}(z,z';\lambda) = - \sum_n \frac{\psi_n(z) \overline{\psi}_n(z')}{\lambda-\lambda_n} \, .$$ As a function of the complex parameter $\lambda$, $\mathcal{G}$ has simple poles at $\lambda = \lambda_n$ and corresponding residues $-\psi_n(z) \overline{\psi}_n(z')$. Hence, one can write $$- \frac{1}{2\pi i} \int_{C_{\infty}} \dd \lambda \, \mathcal{G}(z,z';\lambda) = \sum_n \psi_n(z) \overline{\psi}_n(z') = \delta(z-z') \, ,$$ where $C_{\infty}$ is an infinitely large circle in the $\lambda$ plane and the integral is taken counterclockwise. If there is also a continuous spectrum (as it happens in our case), the Green’s function has a branch cut and the integral above, besides the sum of the residues at the eigenvalues, includes a branch-cut integral over a portion of the real axis, $$\label{eq:deltarepintG} - \frac{1}{2\pi i} \int_{C_{\infty}} \dd \lambda \, \mathcal{G}(z,z';\lambda) = \sum_n \psi_n(z) \overline{\psi}_n(z') + \int \dd \lambda \, \psi_{\lambda}(z) \overline{\psi}_{\lambda}(z') = \delta(z-z') \, .$$ Eq.  allows us to obtain the expansion of the delta distribution in terms of the eigenfunctions of $L$ by performing the integral of the Green’s function $\mathcal{G}$, which we obtained above, over the spectral parameter $\lambda$. In the case at hand, we obtain the expansion in terms of eigenfunctions of the operator $L$ defined in when $\nu \in (0,1)$, satisfying the boundary conditions . The solution of the homogeneous equation satisfying the boundary condition at $z=0$ may be written as $u(z;\lambda) = \sqrt{z} \, \big[ c J_{\nu}\big(\sqrt{\lambda} z\big) - \lambda^{\nu} J_{-\nu}\big(\sqrt{\lambda} z\big) \big]$, whereas $v(z;\lambda) = \sqrt{z} \, H_{\nu}^{(1)}\big(\sqrt{\lambda} z\big)$ is in $L^2(z_0,\infty)$ for any $z_0>0$ if $\lambda \notin [0,\infty)$. Thus, the Green’s function is given by $$\mathcal{G}(z,z';\lambda) = -\frac{i\pi}{2} \frac{\sqrt{z_<} \, \Big[ c J_{\nu}\big(\sqrt{\lambda} z_<\big) - \lambda^{\nu} J_{-\nu}\big(\sqrt{\lambda} z_<\big) \Big] \sqrt{z_>} \, H_{\nu}^{(1)}\big(\sqrt{\lambda} z_>\big)}{c- e^{-i\pi\nu} \lambda^{\nu}} \, ,$$ with $\lambda \notin [0,\infty)$ for all $c \in \bR$ and additionally with $\lambda \neq -c^{1/\nu}$ if $c>0$, which is a pole of $\mathcal{G}$. This is the negative eigenvalue in the spectrum with corresponding “bound state” eigenfunction of the form $\sqrt{z} \, K_{\nu} \big(c^{1/(2\nu)} z\big)$. Consider first the case $c<0$, for which there is no point spectrum and the continuous spectrum is $[0,\infty)$. Then, $$\begin{aligned} \delta (z-z') &= - \frac{1}{2\pi i} \int_{C_{\infty}} \dd \lambda \, \mathcal{G}(z,z';\lambda) \\ &= \frac{1}{2\pi i} \int_0^{\infty} \dd |\lambda| \lim_{\epsilon \to 0^+} \big[\mathcal{G}(z,z';|\lambda|+i\epsilon) - \mathcal{G}(z,z';|\lambda|-i\epsilon) \big] \\ &= \sqrt{zz'} \int_0^{\infty} \dd |\lambda| \, \frac{\Big[ c J_{\nu}\big(\sqrt{|\lambda|} z\big) - |\lambda|^{\nu} J_{-\nu}\big(\sqrt{|\lambda|} z\big) \Big] \Big[ c J_{\nu}\big(\sqrt{|\lambda|} z'\big) - |\lambda|^{\nu} J_{-\nu}\big(\sqrt{|\lambda|} z'\big) \Big]}{c^2 - 2c |\lambda|^{\nu} \cos(\pi \nu) + |\lambda|^{2\nu}} \\ &= \sqrt{zz'} \int_0^{\infty} \dd q \, q \, \frac{\left[ c J_{\nu}(qz) - q^{2\nu} J_{-\nu}(qz) \right] \left[ c J_{\nu}(qz') - q^{2\nu} J_{-\nu}(qz') \right]}{c^2 - 2c q^{2\nu} \cos(\pi \nu) + q^{4\nu}} \, ,\end{aligned}$$ which is Eq. . 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[^4]: The transparent boundary conditions were used in [@Avis:1977yn] for the quantization of the massless, conformally coupled scalar field. [^5]: In fact, the Wronskians in are independent of $z$, but this formula remains valid if instead of the solutions $\widehat{\Phi}_{\underline{k}}^1$, $\widehat{\Phi}_{\underline{k}}^2$ we pick two functions $u$ and $v$ whose Wronskian limit is non-zero and $Lu$ and $Lv$ are square integrable near the origin (see more details in [@Zettl:2005]). [^6]: The definition of Neumann boundary conditions for the non-regular cases varies from author to author, given the non-uniqueness of non-principal solutions. For instance, in [@Ishibashi:2004wx], when $\nu=0$ it coincides with the Dirichlet boundary condition. [^7]: In the literature of algebraic quantum field theory, the two-point function associated with a given algebraic state $\omega$ is denoted by $\omega_2$. Also, note that $G^+$ is sometimes reserved for the advanced propagator. [^8]: The Fourier transforms exists, as we are performing the computation for the ground state, which is maximally symmetric on AdS. For the two-point function of any other quantum state, it is sufficient to add a smooth, positive and symmetric bisolution of . [^9]: As a side comment, this feature resembles what happens in de Sitter spacetime when one considers the so-called $\alpha$-vacua [@Brunetti:2005pr], although, in this case, the additional singularities are pathological, being the underlying background globally hyperbolic.

--- abstract: 'The classical Szegő polynomial approximation theorem states that the polynomials are dense in the space $L^2(\rho)$, where $\rho$ is a measure on the unit circle, if and only if the logarithmic integral of the measure $\rho$ diverges. In this note we give a quantitative version of Szegő’s theorem in the special case when the divergence of the logarithmic integral is caused by deep zeroes of the measure $\rho$ on a sufficiently rare subset of the circle.' author: - 'Alexander Borichev [^1]\' - 'Anna Kononova [^2]\' - 'Mikhail Sodin [^3]' title: | Notes on the Szegő minimum problem.\ I. Measures with deep zeroes --- Introduction ============ Denote by ${\mathcal P}$ the linear space of algebraic polynomials, and by ${\mathcal P}_n$ its subspace of polynomials of degree $n$. Given a finite positive measure $\rho$ on the unit circle ${\mathbb T}$, put $$e_n(\rho) = \min_{q_0, \ldots , q_{n-1}}\, \sqrt{\int_{{\mathbb T}} \bigl| q_0+q_1t + \ldots +q_{n-1}t^{n-1} +t^n \bigr|^2\, {\rm d}\rho(t)} = \text{dist}_{L^2(\rho)}(t^n, {\mathcal P}_{n-1}).$$ Then $$\lim_{n\to\infty} e_n(\rho) = \exp\Bigl( \frac12\int_{\mathbb T}\log\rho'\, {\rm d}m \Bigr),$$ where $m$ is the Lebesgue measure on ${\mathbb T}$ normalized by condition $m({\mathbb T})=1$, and $\rho' = {\rm d}\rho/{\rm d}m$ is the Radon–Nikodym derivative. This is a classical result, first, proven by Szegő for absolutely continuous measures $\rho$, and then, independently, by Verblunsky and Kolmogorov in the general case [@GS Section 3.1] and [@Simon Chapters 1 and 2]. Noting that for $j{\geqslant}0$, $e_{n+j}(\rho)$ coincides with the distance in $L^2(\rho)$ from $ t^{-j}$ to the linear span of $\{ t^k\colon -j+1 {\leqslant}k {\leqslant}n \}$, and recalling that the trigonometric polynomials are dense in $L^2(\rho)$, one sees that the density of algebraic polynomials ${\mathcal P}$ in $L^2(\rho)$ is equivalent to the condition $ \displaystyle\lim_{n\to\infty} e_n(\rho) = 0$, and therefore, to the divergence of the logarithmic integral $$\int_{\mathbb T}\log\rho'\, {\rm d}m = -\infty\,.$$ In these notes we will be occupied by the following question: \[quest\] Suppose $\rho$ is a measure on ${\mathbb T}$ with divergent logarithmic integral. Estimate the rate of decay of the sequence $e_n(\rho)$. Our interest to this question came from the linear prediction for stationary processes. If $\xi\colon {\mathbb Z}\to{\mathbb C}$ is a stationary random sequence with spectral measure $\rho$, then, according to Kolmogorov and Wiener, $e_n(\rho)$ is the error of the best mean-quadratic linear prediction of $\xi(n)$ by $\xi(0), \ldots , \xi(n-1)$; i.e., $$e_n^2(\rho) = \min_{q_0, \ldots , q_{n-1}} {\mathbb E}\Bigl[ \Bigl| \xi(n) - \sum_{0{\leqslant}j {\leqslant}n-1} q_j \xi(j) \Bigr|^2 \Bigr].$$ In the case when the logarithmic integral converges, $e_n(\rho)$ has a positive limit $e_\infty(\rho)$, and dependence of the rate of convergence on the smoothness of the density of $\rho$ is well-understood [@Gol; @Ibr]. In the case of divergent logarithmic integral the situation is quite different and not much is known. If the closed support of $\rho$ is not the whole circle, then it is not difficult to show that $e_n(\rho)$ tends to zero at least exponentially. In the other direction, a version of the classical result of Erdős and Turán says if $\rho' >0$ $m$-a.e. on ${\mathbb T}$, then the measure $\rho$ is regular, i.e., $e_n(\rho)^{1/n}\to 1$. Later, stronger criteria for regularity of $\rho$ were found by Widom, Ullman, and Stahl and Totik, see [@ST Chapter 4]. In these notes we show that in several special but interesting situations it is not difficult to estimate decay of the sequence $e_n(\rho)$ using only simple classical tools. Here, we consider the case when the divergence of the logarithmic integral is caused by deep zeroes of the measure $\rho$ on a sufficiently rare subset of ${\mathbb T}$. The results presented in this note extend Theorems 8 and 9 from [@BSW]. Our main idea is that in the case when the measure ${\rm d}\rho = \Phi\, {\rm d}m$ has divergent logarithmic integral (i.e., $\displaystyle \int_{\mathbb T}\log\Phi\, {\rm d}m = -\infty $), the value $|\log e_n(\rho)|$ can be controlled by the integral $$\int_{\mathbb T}\min\Bigl\{ \log\Bigl(\frac1{\Phi} \Bigr), A \Bigr\}\, {\rm d}m$$ of the cut-off of $\log\Phi^{-1}$ on an appropriate large level $A$ depending on $n$. This can be viewed as a quantitative version of the regularization of the weight $\Phi$ by $\Phi_{\varepsilon}= \Phi+{\varepsilon}$ with $0<{\varepsilon}\ll 1$ used by Szegő in the proof of his theorem. We succeeded to make this work only under additional regularity assumptions on $\Phi$. The toy example is the absolutely continuous measure ${\rm d}\rho=e^{-H}\, {\rm d}m$, where $H(e^{2\pi{\rm i}\theta})=h(\theta)$, $h\colon {\mathbb R}\to [0, +\infty]$ is a $1$-periodic even function, continuous and decreasing on $(0, \frac12]$, and such that $\displaystyle \int_0 h(\theta)\,{\rm d}\theta = +\infty$. Then, under mild assumptions on $h$, we obtain $$| \log e_n(\rho) | \simeq \int_{\mathbb T}h_A\, {\rm d}m\,,$$ where $h_A=\max(h, A)$, and $A=A(n)$ is a solution to the equation $n h^{-1}(a)=a$, $h^{-1}$ is the inverse to the restriction of $h$ on $(0, \frac12]$. Throughout the paper we use the following notation: for positive $A$ and $B$, $A \lesssim B $ means that there is a positive numerical constant $C$ such that $A {\leqslant}CB $, while $A \gtrsim B $ means that $B \lesssim A $, and $A \simeq B $ means that both $A \lesssim B $ and $B \lesssim A $. In the forthcoming second note, we will consider the opposite case when the bulk of the measure $\rho$ is concentrated on a rare subset of ${\mathbb T}$. ### Acknowledgements {#acknowledgements .unnumbered} We thank Sergei Denisov, Fedor Nazarov, and Eero Saksman for several enlightening discussions. Preliminaries ============= Here and elsewhere, $H\colon {\mathbb T}\to [0, +\infty]$ is a measurable function with $$\int_{\mathbb T}H\, {\rm d}m = +\infty.$$ By ${\lambda}_H (a) = m\{H>a\}$ we denote the distribution function of $H$. For $A{\geqslant}1$, we put $H_A(t)=\min(H(t), A)$. To estimate from below and above $\log e_n$, we will use the integrals $$\int_{\mathbb T}H_A\, {\rm d}m = \int_0^A {\lambda}_H (a)\, {\rm d}a = A{\lambda}_H(A) + \int_{\{H{\leqslant}A\}} H\, {\rm d}m$$ with some $A=A(n)$. We record several simple observations, which we frequently use throughout the paper. First, we note that under mild regularity assumptions one of the two terms on the RHS can be discarded. If ${\lambda}_H (a)$ satisfies $$\limsup_{a\to\infty} \frac{{\lambda}_H(a)}{{\lambda}_H(2a)} < 2$$ (i.e., decays not faster than $a^{-p}$ with some $p<1$), then $$\int_{\mathbb T}H_A\, {\rm d}m \simeq A{\lambda}_H(A).$$ On the other hand, if the function $a\mapsto a{\lambda}_H^2(a)$ does not increase (i.e., ${\lambda}_H(a)$ decays as $1/\sqrt a$, or faster), then $$\int_{\mathbb T}H_A\, {\rm d}m \simeq \int_{\{H{\leqslant}A\}} H\, {\rm d}m,$$ provided that ${\lambda}_H(A)$ is separated from $1$ (i.e., $A$ is sufficiently large). To see this, denote by $H^*\colon [0, 1]\to [0, +\infty]$ the decreasing rearrangement of $H$, that is, the function inverse to ${\lambda}_H$. Then, the function $s\mapsto s^2H^*(s)$ does not decrease. Letting $\alpha={\lambda}_H(A)$ (i.e., $A=H^*(\alpha)$), we obtain $$\begin{gathered} A{\lambda}_H(A) = \alpha H^*(\alpha) \lesssim \alpha^2 H^*(\alpha)\, \int_\alpha^1 \frac{{\rm d}s}{s^2} \\ {\leqslant}\int_\alpha^1 \frac{s^2 H^*(s)}{s^2}\, {\rm d}s = \int_\alpha^1 H^*(s)\, {\rm d}s = \int_{\{H{\leqslant}A\}} H\, {\rm d}m.\end{gathered}$$ Furthermore, if $A/{\lambda}_H(A) \simeq B/{\lambda}_H(B)$, then $$\int_{\mathbb T}H_A\, {\rm d}m \simeq \int_{\mathbb T}H_B\, {\rm d}m.$$ Assume, for instance, that $A{\leqslant}B$ and that $B/{\lambda}_H(B) {\leqslant}C\cdot A/{\lambda}_H(A)$. Since ${\lambda}_H$ does not increase, $B{\leqslant}C\cdot A$. Then, $$\begin{gathered} \int_{\{H{\leqslant}B\}} H\, {\rm d}m {\leqslant}\int_{\{H{\leqslant}A\}} H\, {\rm d}m + \int_{\{A < H{\leqslant}C\cdot A\}} H\, {\rm d}m \\ {\leqslant}\int_{\{H{\leqslant}A\}} H\, {\rm d}m + C\cdot A{\lambda}_H(A) {\leqslant}C\, \int_{\mathbb T}H_A\, {\rm d}m\,,\end{gathered}$$ and similarly, $$B{\lambda}_H(B) = \frac{B}{{\lambda}_H(B)}\cdot {\lambda}_H^2(B) {\leqslant}C\cdot \frac{A}{{\lambda}_H(A)}\cdot {\lambda}_H^2(A) = C\cdot A{\lambda}_H(A).$$ Thus, $$\int_{\mathbb T}H_B\, {\rm d}m {\leqslant}C\, \int_{\mathbb T}H_A\, {\rm d}m.$$ Since $B{\geqslant}A$, the opposite estimate is obvious. Our last remark concerns regularity of $H$. Since we will be interested only in rather crude lower and upper bounds for $e_n(\rho)$ under conditions ${\rm d}\rho {\geqslant}e^{-H}\, {\rm d}m$ (for lower bounds) or $\displaystyle \int_{\mathbb T}e^H\, {\rm d}\rho <\infty$ (for upper bounds), our estimates will not distinguish between the sequences $e_n(\rho)$ and $C e_n(\rho)$, and we can always replace the function $H$ by any function $\widetilde{H}$ with $|\widetilde{H}-H|{\leqslant}1$ without affecting our estimates. Keeping this in mind, we always assume that, for any positive $C$, the equation $C{\lambda}_H(A)=A$ has a unique solution. In Section \[sec5\] (Theorem \[the9\]) we will be using the same tacit assumption for the equation $C{\lambda}^*_H(A)=A$, where ${\lambda}_H^*(a)$ is the length of the longest open interval within the set $\{H>a\}$. In the same way, we can always assume that $e_n (\rho)<1/2$. The lower bound for $e_n$ via the Remez-type inequality ======================================================= \[thm:LB\] Suppose ${\rm d}\rho {\geqslant}e^{-H}{\rm d}m$. Then $$| \log e_n (\rho)| \lesssim \int_0^A {\lambda}_H(a)\,{\rm d}a,$$ where $A=A(n)$ solves the equation $n {\lambda}_H(A)=A$. \[cor:LB\] Suppose $H$ belongs to the weak $L^1(m)$-space, i.e., ${\lambda}_H(a)\lesssim 1/a$ for $a{\geqslant}1$. Then $e_n(\rho)$ does not decay to zero faster than a negative power of $n$. Similarly, if $H$ belongs to the weak $L^p(m)$-space with $0<p<1$, that is ${\lambda}_H(a) \lesssim a^{-p}$, then $$| \log e_n(\rho) | \lesssim n^{\frac{1-p}{1+p}}\,.$$ If ${\lambda}_H(a) \lesssim (\log a)^{-1}$ for $a{\geqslant}e$ (in particular, if $\log H$ is integrable), then $$| \log e_n(\rho) | \lesssim \frac{n}{\log^2 n}\,,$$ and so on, until we arrive at the classical Erdős-Turán theorem, which states that $$|\log e_n(\rho)|=o(n), \quad n\to\infty,$$ provided that $H<+\infty$ a.e. on ${\mathbb T}$ (that is, ${\lambda}_H(a)\to 0$ as $a\to\infty$). Proof of Theorem \[thm:LB\] --------------------------- Let $P$ be an extremal algebraic polynomial of degree $n$ such that $P(0)=1$ and $$\int_{\mathbb T}|P|^2\, {\rm d}\rho = e_n(\rho)^2.$$ Then $$\begin{aligned} 0 &{\leqslant}\int_{\mathbb T}\log|P|^2\, {\rm d}m \\ &=\int_{\{H>A\}} \log|P|^2\, {\rm d}m + \int_{\{H{\leqslant}A\}} \log\bigl( |P|^2 e^{-H} \bigr)\, {\rm d}m + \int_{\{H{\leqslant}A\}} H\, {\rm d}m.\end{aligned}$$ Estimating the first and the second integrals on the RHS we let $n$ be so large that $ \int_{\{H{\leqslant}A\}} H\, {\rm d}m {\geqslant}1$ and $$e^A > \rho({\mathbb T}). \label{ko1}$$ By Jensen’s inequality, $$\begin{aligned} \int_{\{H{\leqslant}A\}} \log\bigl( |P|^2 e^{-H}\bigr)\, {\rm d}m &{\leqslant}\frac1e + \log\Bigl( \int_{\{H{\leqslant}A\}} |P|^2 e^{-H}\, {\rm d}m \Bigr) \\ &{\leqslant}\frac1e + \log\Bigl( \int_{\mathbb T}|P|^2 \, {\rm d}\rho \Bigr) = \frac1e + 2 \log e_n(\rho),\end{aligned}$$ and similarly, $$\begin{aligned} \int_{\{H>A\}} \log|P|^2\, {\rm d}m &= {\lambda}_H(A) \Bigl( \frac1{{\lambda}_H(A)}\,\int_{\{H>A\}} \log |P|^2\, {\rm d}m \Bigr) \\ &{\leqslant}{\lambda}_H(A) \log\Bigl( \frac1{{\lambda}_H(A)}\,\int_{\{H>A\}} |P|^2\, {\rm d}m \Bigr) \\ &{\leqslant}\frac1e + {\lambda}_H(A)\log\Bigl( \int_{\mathbb T}|P|^2\, {\rm d}m \Bigr).\end{aligned}$$ Next, applying the $L^2$-version of the classical Remez inequality (which follows, for instance, from a more general Nazarov’s result [@Nazarov]), we obtain $$\begin{aligned} \int_{\mathbb T}|P|^2\, {\rm d}m &{\leqslant}e^{Cn{\lambda}_H(A)}\int_{\{H{\leqslant}A\}} |P|^2\, {\rm d}m \\ &{\leqslant}e^{Cn{\lambda}_H(A)+A}\int_{\{H{\leqslant}A\}} |P|^2e^{-H}\, {\rm d}m \\ &{\leqslant}e^{Cn{\lambda}_H(A)+A}\int_{{\mathbb T}} |P|^2\, {\rm d}\rho \\ &{\leqslant}e^{Cn{\lambda}_H(A)+A}\rho({\mathbb T}) \qquad \qquad \bigl( {\rm by\ extremality\ of\ } P, \int_{{\mathbb T}} |P|^2\, {\rm d}\rho {\leqslant}\rho({\mathbb T})\, \bigr) \\ &{\leqslant}e^{CA} \quad\qquad\qquad\qquad\qquad ({\rm by\ \eqref{ko1}\ and\ the\ equality\ }n{\lambda}_H(A)=A),\end{aligned}$$ whence, $${\lambda}_H(A)\, \log\Bigl( \int_{\mathbb T}|P|^2\, {\rm d}m \Bigr) {\leqslant}C A{\lambda}_H(A).$$ Therefore, $$0 {\leqslant}\frac2e + C A{\lambda}_H(A) + 2\log e_n(\rho) + \int_{\{H{\leqslant}A\}} H\, {\rm d}m,$$ and finally, $$| \log e_n(\rho)| \lesssim A{\lambda}_H(A) + \int_{\{H{\leqslant}A\}} H\, {\rm d}m = \int_{\mathbb T}H_A\, {\rm d}m,$$ proving Theorem \[thm:LB\]. $\Box$ The upper bound for $e_n$ via Taylor polynomials of an outer function ===================================================================== We give two upper bounds for $e_n(\rho)$. Both of them are based on the construction of monic polynomials of large degree with a good estimate for the $L^2(\rho)$-norm. The first bound uses Taylor polynomials of an outer function $F$ such that $1/F$ mimics the behaviour of $\rho$. It is better adjusted to the case when the distribution function ${\lambda}_H(a)$ decays relatively fast as $a\to\infty$. The second bound uses classical Chebyshev’s polynomials and starts working only when ${\lambda}_H (a)$ decays at infinity slower than $1/a$. Let ${\varphi}\colon [0, \frac12]\to (0, +\infty]$ be a continuous decreasing function, ${\varphi}(0)=\infty$, ${\varphi}(\tfrac12){\leqslant}\inf_{{\mathbb T}} H$. Given $\tau\in [-\frac12, \frac12]$, denote by $\theta_\tau$ solution to the equation ${\varphi}(\theta_\tau) = H(e^{2\pi{\rm i}\tau})$. We call the function $H$ [*subordinated to*]{} ${\varphi}$ if, for any $\tau$, $$\begin{aligned} H(e^{2\pi{\rm i}\theta}) &{\leqslant}{\varphi}(\theta+\theta_\tau-\tau), \qquad \tau-\theta_\tau < \theta {\leqslant}\tau, \\ H(e^{2\pi{\rm i}\theta}) &{\leqslant}{\varphi}(\tau+\theta_\tau-\theta), \qquad \tau {\leqslant}\theta < \tau+\theta_\tau.\end{aligned}$$ Note that an equivalent way to express the ${\varphi}$-subordination is to say that the function $({\varphi}^{-1}\circ H)(e^{2\pi{\rm i}\theta})$ is a non-negative Lipschitz function on $[-\frac12, \frac12]$ with the Lipschitz constant at most one. We call the unbounded continuous decreasing function ${\varphi}$ on $(0, \frac12]$ [*regular*]{} if it satisfies at least one of the following two conditions: $${\rm the\ function\ } \theta\mapsto \theta{\varphi}(\theta) \ {\rm does\ not\ decrease \ and\ } {\varphi}(\theta)\gtrsim \log\frac1{\theta} \eqno({\rm Reg}1)$$ $${\varphi}(\theta/2)\lesssim {\varphi}(\theta) \ {\rm and\ } {\varphi}(\theta) \gtrsim 1/\theta. \eqno({\rm Reg}2)$$ \[thm:UB1\] Suppose that $$\int_{\mathbb T}e^H\, {\rm d}\rho <\infty,$$ with $H$ subordinated to a regular function ${\varphi}$. Then $$| \log e_n (\rho) | \gtrsim \int_0^A {\lambda}_H(a)\,{\rm d}a,$$ where $A$ solves the equation $n{\varphi}^{-1}(A)=A$ when ${\varphi}$ satisfies condition [(Reg1)]{}, and $A=\sqrt{n}$ when ${\varphi}$ satisfies condition [(Reg2)]{}. \[cor:UB1\] In the assumptions of Theorem \[thm:UB1\], suppose that $\log {\varphi}(\theta)\gtrsim \log\frac1{\theta}$. If $$\liminf_{a\to\infty} a{\lambda}_H(a) > 0,$$ then $e_n(\rho)$ decay to zero at least as a negative power of $n$. Furthermore, $e_n(\rho)$ decay to zero faster than any negative power of $n$, provided that $$\lim_{a\to\infty} a{\lambda}_H(a) = \infty.$$ Note that we need to impose the additional condition $\log{\varphi}(\theta) \gtrsim \log\tfrac1{\theta}$ in this Corollary only in the case when ${\varphi}$ satisfies the first regularity condition (Reg1). Taylor polynomials ------------------ Denote by ${\mathbb P}_r$ the Poisson kernel for the unit disk evaluated at the point $r\in [0, 1)$. \[lemma\_Taylor\] Let $H$ be a weight such that $$H_A * {\mathbb P}_{1-\delta} \lesssim H + 1 \qquad {\rm everywhere\ on\ } {\mathbb T},$$ with $ \log \delta^{-1} \lesssim A $. Suppose that $$\int_{\mathbb T}e^H\, {\rm d}\rho < \infty.$$ Then there exists a positive constant $C$ such that, for $n{\geqslant}CA/\delta$, we have $$|\log e_n(\rho) | \gtrsim \int_{\mathbb T}H_A\, {\rm d}m.$$ ### Proof of Lemma \[lemma\_Taylor\] Let $M$ be a positive constant such that $ H_A * {\mathbb P}_{1-\delta} {\leqslant}M(H + 1)$, and let $F_A$ be an outer function in ${\mathbb D}$ with the boundary values $|F_A|^2 = e^{H_A/M}$, i.e., $$\log |F_A(rt)| = \frac1{2M} \bigl( H_A * {\mathbb P}_r \bigr) (t).$$ We expand $F_A((1-\delta)z)$ into the Taylor series $$F_A((1-\delta)z) = \sum_{k{\geqslant}0} f_k z^k,$$ and consider the Taylor polynomials $$P_A(z) = \sum_{k=0}^n f_k z^k.$$ Then, $$e_n(\rho)^2 {\leqslant}|P_A(0)|^{-2}\, \int_{\mathbb T}|P_A|^2\, {\rm d}\rho.$$ First, we note that $$|P_A(0)| = |F_A(0)| = \exp\Bigl( \int_{\mathbb T}\log |F_A|\, {\rm d}m \Bigr) = \exp\Bigl( \frac1{2M}\, \int_{\mathbb T}H_A \, {\rm d}m \Bigr),$$ i.e., $$e_n(\rho)^2 {\leqslant}\int_{\mathbb T}|P_A|^2\, {\rm d}\rho \cdot \exp\Bigl( - \frac1{M}\, \int_{\mathbb T}H_A \, {\rm d}m \Bigr).$$ Next, $$\int_{\mathbb T}|F_A((1-\delta)t)|^2\, {\rm d}\rho(t) = \int_{\mathbb T}\exp \Bigl( \frac1{M} H_A * {\mathbb P}_{1-\delta} \Bigr)\, {\rm d}\rho {\leqslant}\int_{\mathbb T}\exp\bigl( H + 1\bigr)\, {\rm d}\rho \lesssim 1,$$ so it remains to estimate the remainder $$| F_A((1-\delta)z) - P_A(z) | {\leqslant}\sum_{k>n} |f_k|.$$ By Cauchy’s estimates, $$|f_k| {\leqslant}(1-\delta)^{k} \max_{{\mathbb T}} \, |F_A| {\leqslant}(1-\delta)^{k} e^{A/(2M)},$$ whence, $$\sum_{k>n} |f_k| \lesssim \delta^{-1} e^{A/(2M)-n\delta} \lesssim 1\,,$$ provided that $A/\delta \lesssim n$ (here, we use that $\log \delta^{-1} \lesssim A$). Thus, $$\int_{\mathbb T}|P_A|^2\, {\rm d}\rho \lesssim 1,$$ which proves the lemma. $\Box$ Estimates of the Poisson integral --------------------------------- Put $p_\delta (\theta) = {\mathbb P}_{1-\delta}(e^{2\pi{\rm i}\theta})$ and recall that $p_\delta(\theta) \lesssim \min(\delta^{-1}, \delta\theta^{-2})$. \[lemma\_Poisson\_A\] Let ${\varphi}\colon (0, \frac12]\to [0, \infty)$ be an unbounded continuous decreasing function, let $\widetilde{\varphi}$ be its even $1$-periodic extension on ${\mathbb R}$, and $\widetilde{\varphi}_A=\min(\widetilde{\varphi}, A)$. Then $$\widetilde{\varphi}_A * p_\delta \lesssim \widetilde{\varphi}+ 1 \qquad {\rm everywhere\ on\ } {\mathbb R},$$ provided that at least one of the following holds: $\rm (i)$ the function $\theta\mapsto \theta^2{\varphi}(\theta)$ does not decrease, and $\delta\lesssim{\varphi}^{-1}(A)$; $\rm (ii)$ ${\varphi}(\theta/2)\lesssim {\varphi}(\theta)$, ${\varphi}(\theta)\gtrsim 1/\theta$, and $\delta\lesssim 1/A$. Note that condition (i) is weaker than condition (Reg1) in Theorem \[thm:UB1\], i.e., the lemma is a bit stronger than what we will use for the proof of Theorem \[thm:UB1\]. We need this version of Lemma \[lemma\_Poisson\_A\] for the proof of Theorem \[thm:DeepZero\]. We also note that condition (i) yields estimate ${\varphi}(\theta/2)\lesssim{\varphi}(\theta)$ from condition (ii). ### Proof of Lemma \[lemma\_Poisson\_A\] We take a sufficiently small $\tau_0>0$ so that ${\varphi}(\tau_0){\geqslant}1$, fix $\tau\in (0, \tau_0]$, and estimate the convolution $(\widetilde{\varphi}_A *p_\delta)(\tau)$. There is nothing to prove if $A{\leqslant}{\varphi}\bigl( \frac12 \tau\bigr)$ since in this case $$(\widetilde{\varphi}_A * p_\delta)(\tau) {\leqslant}\max_{[-\frac12, \frac12]} \widetilde{\varphi}_A = A {\leqslant}{\varphi}\bigl( \tfrac12 \tau \bigr) \lesssim {\varphi}(\tau)$$ for any $\delta>0$. Hence, in what follows, we assume that $A{\geqslant}{\varphi}\bigl( \frac12 \tau\bigr)$, i.e., $\tau{\geqslant}2{\varphi}^{-1}(A)$. First, we note that for any $\theta\in [0, \frac12]$, we have $\widetilde{\varphi}_A(\tau+\theta) {\leqslant}\widetilde{\varphi}_A(\tau-\theta)$ (to see this, one needs to consider three cases: $0{\leqslant}\theta{\leqslant}\tau$, $\tau{\leqslant}\theta{\leqslant}\frac12-\tau$, and $\frac12-\tau{\leqslant}\theta{\leqslant}\frac12$). Therefore, $$\begin{gathered} \int_{-\frac12}^{\frac12} \widetilde{\varphi}_A(\tau-\theta) p_\delta(\theta)\, {\rm d}\theta = \int_0^{\frac12} \bigl[ \widetilde{\varphi}_A(\tau-\theta) + \widetilde{\varphi}_A(\tau+\theta) \bigr] p_\delta(\theta) \, {\rm d}\theta \\ {\leqslant}2A\, \int_{|\tau-\theta|{\leqslant}{\varphi}^{-1}(A)} p_\delta(\theta)\, {\rm d}\theta + 2\, \int_ {\substack{|\theta-\tau|{\geqslant}{\varphi}^{-1}(A), \\ 0{\leqslant}\theta{\leqslant}\frac12}} {\varphi}(\tau-\theta) p_\delta(\theta)\, {\rm d}\theta = I+ II.\end{gathered}$$ Before we start estimating integrals on the RHS, observe that $\delta\lesssim\tau$. In the case (i) it is obvious since $\delta\lesssim {\varphi}^{-1}(A){\leqslant}\tau/2$, in the case (ii) it is also obvious since then $\delta\lesssim 1/A \lesssim {\varphi}^{-1}(A) {\leqslant}\tau/2$. Therefore, in the first integral $\theta {\geqslant}\tau-{\varphi}^{-1}(A) {\geqslant}\tau/2 \gtrsim \delta$. Recalling the standard estimate of the Poisson kernel $p_\delta(\theta) \lesssim \min (\delta^{-1}, \delta \theta^{-2}) $, we get $$I \lesssim A\delta\, \int_{\tau-{\varphi}^{-1}(A)}^{\tau+{\varphi}^{-1}(A)} \frac{{\rm d}\theta}{\theta^2} \lesssim \frac{A\delta{\varphi}^{-1}(A)}{\tau^2}.$$ In both cases (i) and (ii) the RHS is $\lesssim{\varphi}(\tau)$. Indeed, if (i) holds, then it is bounded by $ {\varphi}^{-1}(A)^2 A/\tau^2 {\leqslant}\tau^2{\varphi}(\tau)/\tau^2 = {\varphi}(\tau) $. If (ii) holds, then it is bounded by $ {\varphi}^{-1}(A)/\tau^2 \lesssim 1/\tau \lesssim {\varphi}(\tau) $. We split the second integral into four parts $$\int_{\substack{|\theta-\tau|{\geqslant}{\varphi}^{-1}(A), \\ 0{\leqslant}\theta{\leqslant}\frac12}} = \int_0^{\min(\delta, \frac12 \tau)} + \int_{\min(\delta, \frac12 \tau)}^{\frac12 \tau} + \int_{{\varphi}^{-1}(A){\leqslant}|\theta-\tau| {\leqslant}\frac12 \tau} + \int_{\frac32 \tau}^{1/2}$$ and estimate them one by one. We have $$\begin{gathered} \int_0^{\min(\delta, \frac12 \tau)} {\varphi}(\tau-\theta) p_\delta(\theta)\, {\rm d}\theta \lesssim \frac1{\delta}\, \int_0^{\min(\delta, \frac12 \tau)} {\varphi}(\tau-\theta)\, {\rm d}\theta \\ {\leqslant}\frac1{\delta}\, \min(\delta, \tfrac12 \tau) \cdot {\varphi}(\tfrac12 \tau) \lesssim {\varphi}(\tau),\end{gathered}$$ and $$\begin{gathered} \int_{\min(\delta, \frac12 \tau)}^{\frac12 \tau} {\varphi}(\tau-\theta) p_\delta(\theta)\, {\rm d}\theta \lesssim \delta\, \int_{\min(\delta, \frac12 \tau)}^{\frac12 \tau} \frac{{\varphi}(\tau-\theta)}{\theta^2} \, {\rm d}\theta \\ \lesssim \delta \cdot {\varphi}(\tfrac12 \tau)\, \int_{\min(\delta, \frac12 \tau)}^\infty \frac{{\rm d}\theta}{\theta^2}\, \stackrel{\delta\lesssim\tau}\lesssim \, {\varphi}(\tau).\end{gathered}$$ Next, $$\begin{gathered} \int_{{\varphi}^{-1}(A){\leqslant}|\theta-\tau| {\leqslant}\frac12 \tau} {\varphi}(\tau-\theta) p_\delta(\theta)\, {\rm d}\theta \\ \lesssim \delta\, \int_{{\varphi}^{-1}(A){\leqslant}|\theta-\tau| {\leqslant}\frac12 \tau} \frac{{\varphi}(\tau-\theta)}{\theta^2}\, {\rm d}\theta \simeq \frac{\delta}{\tau^2}\, \int_{{\varphi}^{-1}(A)}^{\frac12 \tau} {\varphi}(\xi)\, {\rm d}\xi.\end{gathered}$$ In the case (i), the integral on the RHS equals $$\frac{\delta}{\tau^2}\, \int_{{\varphi}^{-1}(A)}^{\frac12 \tau} \frac{\xi^2 {\varphi}(\xi)}{\xi^2}\, {\rm d}\xi {\leqslant}\frac{\delta}{\tau^2}\, \frac{(\tau/2)^2 {\varphi}(\tau/2)}{{\varphi}^{-1}(A)} \, \stackrel{\delta\lesssim {\varphi}^{-1}(A)}\lesssim\, {\varphi}(\tau),$$ while in the case (ii), it does not exceed $$\frac{\delta}{\tau^2} \cdot A\tau/2 \, \stackrel{\delta\lesssim 1/A}\lesssim \, \frac1{\tau} \, \stackrel{\tau{\varphi}(\tau)\gtrsim 1}\lesssim \, {\varphi}(\tau).$$ At last, $$\int_{\frac32 \tau}^{1/2} {\varphi}(\tau-\theta) p_\delta(\theta)\, {\rm d}\theta \lesssim \delta \int_{\frac32 \tau}^{\frac12} \frac{{\varphi}(\theta - \tau)}{\theta^2}\, {\rm d}\theta \lesssim \delta {\varphi}(\tau/2) \cdot \frac1\tau \, \stackrel{\delta\lesssim\tau}\lesssim\, {\varphi}(\tau),$$ completing the proof of the lemma. $\Box$ ### The Poisson integral of $H_A$ \[lemma:Poisson\] Let $H$ be subordinated to a regular function ${\varphi}$, and $H_A=\min (H, A)$. Then $$H_A * {\mathbb P}_{1-\delta} \lesssim H+1 \quad{\rm everywhere\ on\ } {\mathbb T},$$ provided that $\delta\lesssim{\varphi}^{-1}(A)$ when ${\varphi}$ satisfies condition [(Reg1)]{}, and $\delta\lesssim A^{-1}$ when ${\varphi}$ satisfies condition [(Reg2)]{}. Clearly, Lemma \[lemma\_Taylor\] and Lemma \[lemma:Poisson\] combined together yield Theorem \[thm:UB1\]. #### Proof of Lemma \[lemma:Poisson\] We write $H(e^{2\pi{\rm i}\theta})=h(\theta)$, fix the point $\tau\in [-\frac12, \frac12]$ with $h(\tau)<\infty$ at which we will estimate the convolution $(h_A * p_\delta)(\tau)$, and choose $\theta_\tau$ so that ${\varphi}(\theta_\tau)=h(\tau)$. Similarly to the proof of the previous lemma, we assume that $A {\geqslant}{\varphi}(\frac12 \theta_\tau)$, i.e., that ${\varphi}^{-1}(A) {\leqslant}\frac12 \theta_\tau$; otherwise, $$(h_A * p_\delta)(\tau) {\leqslant}A {\leqslant}{\varphi}(\tfrac12 \theta_\tau) \lesssim {\varphi}(\theta_\tau) = h(\tau),$$ and we are done. Now, $$(h_A * p_\delta)(\tau) = \Bigl(\, \int_{|\theta-\tau|{\geqslant}\theta_\tau-{\varphi}^{-1}(A)} + \, \int_{|\theta-\tau|{\leqslant}\theta_\tau-{\varphi}^{-1}(A)} \, \Bigr)\, h_A(\theta) p_\delta(\tau-\theta)\, {\rm d}\theta = I + II.$$ To estimate the first integral, we note that, since $\theta_\tau-{\varphi}^{-1}(A){\geqslant}\theta_\tau/2$, we have $$I {\leqslant}A \int_{|\theta-\tau|{\geqslant}\theta_\tau/2} \, p_\delta(\tau-\theta)\, {\rm d}\theta {\leqslant}2A \int_{\theta_\tau/2}^{1/2} p_\delta (\theta)\, {\rm d}\theta \lesssim A\delta \int_{\theta_\tau/2}^\infty \frac{{\rm d}\theta}{\theta^2} \lesssim \frac{A\delta}{\theta_\tau}\,.$$ In the first case, the RHS is $$\, \stackrel{\delta\lesssim {\varphi}^{-1}(A)}\lesssim\, \frac{A{\varphi}^{-1}(A)}{\theta_\tau} \, \stackrel{{\varphi}^{-1}(A){\leqslant}\frac12 \theta_\tau}\lesssim \, \frac{\frac12 \theta_\tau {\varphi}(\frac12 \theta_\tau)}{\theta_\tau} \lesssim {\varphi}(\theta_\tau).$$ In the second case, $A\delta/\theta_\tau \lesssim 1/\theta_\tau \lesssim {\varphi}(\theta_\tau)$. Therefore, in both cases, the first integral is $\lesssim {\varphi}(\theta_\tau)=h(\tau)$. To estimate the second integral, we note that, by the subordination to ${\varphi}$, it is bounded by $ 2 (\widetilde{\varphi}_A * p_\delta)(\theta_\tau) $, which, by the previous lemma, is $\lesssim {\varphi}(\theta_\tau)+1 = h(\tau)+1$. $\Box$ The upper bound for $e_n$ via Chebyshev polynomials {#sec5} =================================================== Here we assume that the function $H$ is lower semicontinuous; i.e., the sets $\{H\!>\!a\}$ are open, and denote by ${\lambda}^*_H(a)$ the length of the longest open interval within $\{H\!>\!a\}$. \[thm:UB2\]\[the9\] Suppose that $$\int_{\mathbb T}e^H\, {\rm d}\rho <\infty.$$ Then $$| \log e_n (\rho) | \gtrsim A {\lambda}_H^*(A)\,,$$ where $A=A(n)$ solves the equation $n{\lambda}_H^*(A)=A$. The following Corollary combines Theorem \[thm:UB2\] with Theorem \[thm:LB\] (and takes into account Observations 2.1 and 2.3) \[cor:UB2\] Let ${\rm d}\rho = e^{-H}\, {\rm d}m $. Suppose that the set $\{H>a\}$ contains an interval with the length comparable to the total length of $\{H>a\}$ (i.e. ${\lambda}_H^* \gtrsim {\lambda}_H$), and that the function ${\lambda}_H$ satisfies $$\limsup_{a\to\infty} \frac{{\lambda}_H(a)}{{\lambda}_H(2a)} <2.$$ Then $$| \log e_n (\rho)| \simeq A\lambda_H(A)\,,$$ where $A=A(n)$ is a solution to the equation $n{\lambda}_H(a)=a$. Proof of Theorem \[thm:UB2\] ---------------------------- We will use the following classical lemma (cf., for instance, [@MR]): \[lemma:Chebyshev-Maergoiz\] For any $\alpha\in (0, \frac{\pi}2 ]$ and for any $n\in 2{\mathbb N}$, there exists a monic polynomial $T_{n,\alpha}$ of degree $n$ such that $$\max\{|T_{n,\alpha}(e^{i\theta})|\colon |\theta| {\geqslant}\alpha \} = 2\cos^n(\alpha/2)\,.$$ For the reader’s convenience, we recall its proof. Put $$T_{2m,\alpha}(e^{i\theta}) = 2\cos^{2m}(\alpha/2)e^{im\theta} \cos\left( 2m\arccos\left(\frac{\cos(\theta/2)}{\cos(\alpha/2)} \right)\right)\,.$$ We only need to show that this is a monic polynomial of degree $n=2m$. Recall that $\cos(2m\arccos x) = 2^{2m-1}Q_m(x^2)$, where $Q_m$ is a monic polynomial of degree $m$. Then $$\begin{aligned} T_{2m,\alpha}(e^{i\theta}) &= 2^{2m} \cos^{2m}(\alpha/2)e^{im\theta} Q_m\left(\frac{\cos^2\theta/2}{\cos^2\alpha/2}\right) \\ &= 2^{2m}\cos^{2m}(\alpha/2)e^{im\theta} Q_m\left(\frac{e^{i\theta}+e^{-i\theta}+2}{4\cos^2\alpha/2}\right)\end{aligned}$$ and it is easily seen that the RHS is a monic polynomial of degree $2m$, proving the lemma. $\Box$ Now, we turn to the proof of Theorem \[thm:UB2\]. Without loss of generality, we assume that $n$ is an even number. Let $J\subset{\mathbb T}$ be the longest arc in the set $\{H>a\}$. We assume that $J=\{t=e^{{\rm i}\varphi}\colon |\varphi|{\leqslant}\alpha \}$, $\alpha = \pi{\lambda}_H^*(a)$. Let $T=T_{n, \alpha}$ be a monic polynomial of degree $n$ as in Lemma \[lemma:Chebyshev-Maergoiz\]. Then, by a straightforward computation (or by the classical Remez inequality) $$\max_J |T| = \max_{|\varphi|{\leqslant}\alpha} \left| T(e^{i\varphi})\right| \lesssim e^{Cn\alpha} \, \bigl( \cos\frac{\alpha}{2} \bigr)^n\,.$$ Noting that $ \cos^n\tfrac{\alpha}{2} {\leqslant}e^{-cn\alpha^2}$, we get $$\begin{gathered} \int_{\mathbb T}|T|^2\, {\rm d}\rho {\leqslant}\max_{{\mathbb T}} \bigl( |T|^2e^{-H} \bigr)\, \int_{\mathbb T}e^H\, {\rm d}\rho \\ \lesssim e^{-a}\, \max_J |T|^2 + \max_{{\mathbb T}\setminus J} |T|^2 \lesssim \Bigl( e^{Cn{\lambda}_H^*(a) - a} + 1 \Bigr) e^{-cn{\lambda}_H^*(a)^2}.\end{gathered}$$ Letting $A_C$ be the unique solution to the equation $Cn {\lambda}_H^*(A_C)=A_C$, we obtain $$| \log e_n(\rho) | \gtrsim n {\lambda}_H^*(A_C)^2 \simeq A_C{\lambda}_H^*(A_C) \simeq A{\lambda}_H^*(A),$$ completing the proof of Theorem \[thm:UB2\]. $\Box$ Examples ======== To illustrate our results, we consider the function $H=h\circ d_K$, where $h\colon (0, \frac12]\to (0, +\infty)$ is a $C^1$-smooth decreasing function, $h(0)=+\infty$, and $d_K(t)=\operatorname{dist}(t, K)$, where $K\subset {\mathbb T}$ is a compact set of zero length (recall that we identify ${\mathbb T}$ with ${\mathbb R}/{\mathbb Z}$). Denote by $K_{+s}=\{t\colon d_K(t)<s \}$ the $s$-neighbourhood of $K$ and by $\psi_K(s) = m(K_{+s})$ its length. Then $${\lambda}_H(a) = m\{H>a\} = \begin{cases} \psi_K(h^{-1}(a)), &a{\geqslant}h(\tfrac12) \\ 1, &a<h(\tfrac12), \end{cases}$$ and $$\int_0^A {\lambda}_H(a)\, {\rm d}a= \Bigl(\int_0^{h(1/2)}+\int_{h(1/2)}^A\Bigr) {\lambda}_H(a)\, {\rm d}a= \int_{h^{-1}(A)}^{1/2} \psi_K |h'| + h(\tfrac12),$$ provided that $A>h(\frac12)$. To estimate the function $\psi_K$ it is convenient to use that $ \psi_K(s) \simeq s N_K(s) \simeq sP_K(s) $, where $N_K(s)$ is the covering number of $K$ and $P_K(s)$ is the packing number of $K$, see, for instance, [@Falc Chapter 3]. We call the set $K$ $\gamma$-regular if $\psi_K(s) \simeq s^{1-\gamma}$. For instance, the set $e^{2\pi {\rm i}\mathcal C}$, where $\mathcal C$ is the standard ternary Cantor set is $\gamma$-regular with $\gamma = \frac{\log 2}{\log 3}$, while the set $\{t=\exp(2\pi {\rm i} n^{-\nu})\colon n\in{\mathbb N}\} \cup \{1\}$ is $\gamma$-regular with $\gamma=(\nu+1)^{-1}$. Two corollaries --------------- We get straightforward corollaries to our results taking $h(s)=s^{-p}$. \[cor:d\_K2\] Let $K\subset{\mathbb T}$ be a $\gamma$-regular compact set with some $\gamma\in [0, 1)$. Suppose that ${\rm d}\rho = \exp \bigl( - d_K^{\gamma-1} \bigr)\, {\rm d}m$. Then $ | \log e_n(\rho)| \simeq \log n$. The second corollary pertains to the case when the length of the longest interval in the set $\{d_K<s\}$ is comparable with the length of the whole set $\{d_K<s\}$. Then Corollary \[cor:UB2\] applies. \[cor:d\_K3\] Let $\nu>0$, $K=\{t=\exp(2\pi {\rm i}n^{-\nu})\colon n\in{\mathbb N}\} \cup \{1\}$, and ${\rm d}\rho = \exp \bigl( - d_K^{-p} \bigr)\, {\rm d}m$ with $p>\vartheta = \frac{\nu}{\nu+1}$. Then $$|\log e_n (\rho)| \simeq n^{\frac{p-\vartheta}{p+\vartheta}}.$$ Measures with deep zero at one point ------------------------------------ The last illustration to our estimates pertains to the simplest case when the measure $\rho$ has a deep zero at one point and is symmetric with respect to this point. In this case, our estimates yield a relatively complete result. \[thm:DeepZero\] Let $h\colon (0, \frac12]\to [0, +\infty)$ be a continuous decreasing function such that $$\int_0 h(a)\,{\rm d}a = +\infty.$$ Suppose that $h$ satisfies at least one of the following two conditions: - $$\theta\mapsto \theta^2 h(\theta) \quad {\rm does\ not\ decrease},$$ and $$|\log \theta| = O(h(\theta)),\qquad \theta\to 0;$$ - $$\limsup_{a\to\infty} \frac{h^{-1}(a)}{h^{-1}(2a)} < 2$$ Let $\rho$ be an absolutely continuous measure on ${\mathbb T}$ with density $e^{-h(|\theta|)}$. Then $$|\log e_n (\rho)| \simeq \int_0^{1/2} h_A(a)\,{\rm d}a$$ where $A$ solves the equation $n h^{-1}(A)=A$ and $h_A=\min(h, A)$. The lower bound for $e_n$ (i.e., the upper bound for $|\log e_n|$) follows from Theorem \[thm:LB\] and does not need any regularity assumptions on $h$. Conditions (i) and (ii) are needed for the proof of the upper bound for $e_n$. In the case (i), it is a consequence of Lemma \[lemma\_Taylor\] combined with the first case of Lemma \[lemma\_Poisson\_A\]. In the case (ii), it follows from Theorem \[thm:UB2\]. Note that these two cases overlap, e.g., the function $h(\theta)=\theta^{-p}$ with $1<p{\leqslant}2$ satisfies both of them. The following corollary gives an idea about the rate of decay of $e_n(\rho)$ for several explicitly written functions $h$. Let $\rho$ be an absolutely continuous measure on ${\mathbb T}$ with density $e^{-h(|\theta|)}$. Then for $n{\geqslant}4$ we have - If $h(\theta)\simeq \theta^{-1}\log^{-1}(1/\theta)$, then $|\log e_n (\rho)| \simeq \log\log n$, - If $p>-1$ and $h(\theta)\simeq \theta^{-1}\log^p(1/\theta)$, then $|\log e_n (\rho)| \simeq (\log n)^{p+1}$, - If $p>1$ and $h(\theta)\simeq \theta^{-p}$, then $|\log e_n (\rho)| \simeq n^{(p-1)/(p+1)}$, - If $p>0$ and $h(\theta)\simeq \exp(\theta^{-p})$, then $|\log e_n (\rho)| \simeq n(\log n)^{-2/p}$. Our last remark is that, plausibly, the technique based on the potential theory in the external field developed by Mhaskar–Saff, Rakhmanov, Levin–Lubinsky, Totik and others should allow one to obtain more precise estimates of $e_n$ in the situation considered in Theorem \[thm:DeepZero\]. See for instance, Theorem 1.22 and Examples 3 and 4 in Section 1.6 in [@LL] which contain similar results for orthogonal polynomials on the real line. On the other hand, likely, this will require much stronger regularity assumptions on the function $h$ and more technical proofs. [A]{} A. Borichev, M. Sodin, B. Weiss, Spectra of stationary processes on ${\mathbb Z}$. In: $50$ years with Hardy spaces, 141–157, Oper. Theory Adv. Appl., [**261**]{}, Birkhäuser-Springer, 2018. K. Falconer, Fractal geometry. Mathematical foundations and applications. Third edition. John Wiley & Sons, 2014. B. L. Golinskii, The asymptotic behavior of the prediction error. Teor. Verojatnost. i Primenen. [**19**]{} (1974), 724–739. U. Grenander, G. Szegő, Toeplitz forms and their applications. Univ. California Press, 1958. I. A. Ibragimov, On the asymptotic behaviour of the prediction error. Teor. Verojatnost. i Primenen. [**9**]{} (1964), 695–703. E. Levin, D. Lubinsky, Orthogonal polynomials for exponential weights. CMS Books in Mathematics, 4, Springer–Verlag, New York, 2001. L. S. Maergoiz, N. N. Rybakova, Chebyshev polynomials with zeros on the circle and related problems. St. Petersburg Math. J. [**25**]{} (2014), 965–979. F. Nazarov, Complete version of Turán’s lemma for trigonometric polynomials on the unit circumference. In: Complex analysis, operators, and related topics, 239–246, Oper. Theory Adv. Appl., [**113**]{}, Birkhäuser, 2000. B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory. Amer. Math. Soc., 2005. H. Stahl, V. Totik, General Orthogonal Polynomials. Cambridge Univ. Press, 1992. [A.B.: Institut de Mathématiques de Marseille, Aix Marseille Université, CNRS, Centrale Marseille, I2M, Marseille, France A.K.: Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia M.S.: School of Mathematics, Tel Aviv University, Tel Aviv, Israel ]{} [^1]: Supported by a joint grant of Russian Foundation for Basic Research and CNRS (projects 17-51-150005-NCNI-a and PRC CNRS/RFBR 2017-2019) and by the project ANR-18-CE40-0035. [^2]: Supported by a joint grant of Russian Foundation for Basic Research and CNRS (projects 17-51-150005-NCNI-a and PRC CNRS/RFBR 2017-2019). [^3]: Supported by ERC Advanced Grant 692616 and ISF Grant 382/15.

--- abstract: 'A current-controlled negative differential resistance has been revealed in the $I$-$V$ characteristics of single crystal La$_{2}$CuO$_{4+\delta}$ in the low temperature region. The non-linear behavior of conductivity is accompanied by a transition from positive to negative magnetoresistance when the current is growing. Possible reasons for the effect observed are discussed.' author: - 'B. I. Belevtsev' - 'N. V. Dalakova' title: 'Negative differential resistance in single crystal La$_{2}$CuO$_{4}$ at low temperature' --- La$_{2}$CuO$_{4+\delta}$ is a mother compound for one of the family of high-$T_c$ superconductors (HTSC). The stoichiometric La$_{2}$CuO$_{4}$ ($\delta = 0$) is an antiferromagnetic (AFM) Mott insulator with the Néel temperature $T_N\approx 320$ K. On doping it with oxygen ($\delta \neq 0$), charge carriers (oxygen holes) appear in the system [@kremer; @tranq], which leads to destruction of the AFM order and brings the system into the metallic (superconducting) state. The excess oxygen resides between the LaO planes [@chai] and thus determines the three-dimensional (3D) character of conductivity in La$_{2}$CuO$_{4+\delta}$. Because of its high mobility, the excess oxygen forms a favorable condition for chemical (impurity-induced) phase separation. Indeed, as neutron-diffraction data show [@jorg], below 320 K crystalline La$_{2}$CuO$_{4+\delta}$ separates into two phases which are crystallographically close to each other. One of the phases has stoichiometry similar to that of La$_{2}$CuO$_{4}$. The other is rich in oxygen and becomes superconducting below $T_c\approx 38$ K. More evidence supporting phase separation in this material was obtained by quadrupole and nuclear magnetic resonance techniques [@ueda]. The phase separation and its investigation are topical problems of HTSC physics. The structural and stoichiometric inhomogeneities caused by phase separation can affect significantly the behavior of the transport properties of copper oxides. This study is concerned with the effect of current upon the conductive and magnetoresistive properties of single crystal La$_{2}$CuO$_{4+\delta}$ ($T_N = 182$ K). The dc resistivity in the direction parallel to the CuO$_2$ planes was measured using the Montgomery method at different pre-assigned current. The magnetic field was parallel to the tetragonal $\vec {c}$-axis of the crystal and perpendicular to the transport current. ![The dependencies of $\lg \rho_{a}$ [*vs.*]{} $T^{-1/4}$ at different transport currents ($\vec{J}\parallel\vec{a}$)](fig1.eps){width="0.9\linewidth"} The temperature dependencies of the resistivity $\rho _{a}$ measured along the $\vec {a}$-axis for the different amplitudes of the measuring currents $J$ are shown in Fig. 1. It is seen that at $J \leq 1$ $\mu$A the resistance is only slightly dependent on current in the whole interval of temperatures (4.2–300 K). The Mott’s law for variable-range hopping (VRH) is well obeyed in the region 20–200 K for $J \leq 1$ $\mu$A: $$\label{eq1} R \approx R_{0} \exp\left( {\frac{{T_{0}} }{{T}}} \right)^{1/4},$$ In this region the Ohm’s law is obeyed well. The exponent (1/4) in Eq. (\[eq1\]) corresponds to the behavior of a 3D system. For $T < 10$ K and $J\leq 1$ $\mu$A the resistance grows with lowering temperature more rapidly than it is predicted by Eq. (\[eq1\]). This behavior is typical for crystal La$_{2}$CuO$_{4+\delta}$ in the low temperature region [@boris1]. The effect may be produced by the isolated superconducting inclusions that appear in the dielectric matrix on phase separation when the volume fraction of the superconducting phase is much smaller than the percolation threshold. {width="0.9\linewidth"} At $J > 1$ $\mu$A there is a significant deviation of $\rho_{a}(T)$ from the Mott’s law in low temperature region. When the current increases, the resistance drops drastically, and the temperature at which $\rho _{a}(T)$ starts to deviate from Eq. (\[eq1\]) shifts towards higher temperatures (Fig. 1). This behavior accounts for the non-linear effects in the conductivity. The non-linear $I$-$V$ curves are illustrated in Fig. 2. At $T < 8$ K some regions with negative differential resistance (NDR) can be seen where $dV/dI < 0$. Earlier, a voltage-controlled NDR effect at low temperatures ($T<10$ K) was observed in single crystal La$_{2}$CuO$_{4}$ with inhomogeneous distribution of oxygen [@boris2]. Here we report for the first time a current-controlled NDR in single-crystal La$_{2}$CuO$_{4}$ with more homogeneous distribution of oxygen. According to Ref. , the influence of electric field on resistance under the VRH condition is described by $$\label{eq2} R\left( {T,E} \right) = R_{0} \left( {T} \right)\exp\left( { - \frac{{eEr_{h} \gamma} }{{kT}}} \right),$$ where $R_{0}(T)$ is the resistance for $E \to 0$ described by Eq. 1, $r_{h}$ is the mean hopping distance, $\gamma$ is a factor of the order of unity. It is evident from Eq. (\[eq2\]) that in rather low fields ($E \ll kT/er_{h}\gamma$), resistance is field independent, i.e the Ohm’s law is obeyed. As follows from estimation, this is true for the sample studied even in the highest fields of the experiment. In this context the non-linear behavior of the $I$-$V$ curves (Fig. 2) can hardly be related to the influence of the electric field on hopping conduction. {width="0.9\linewidth"} There may be another reason for the non-linearity, namely, electron overheating with rather high currents. If the charge carriers do not have enough time to give up quickly the energy received from the field to the lattice, their temperature rises and exceeds that of the phonons. The overheating affects the mobility of the carriers and leads to violation of the Ohm’s law. The theory of “hot” electrons was applied successfully to explain the violation of the Ohm’s law in experiments on doped semiconductors [@con]. In Ref. the non-linearity of experimental $I$-$V$ characteristics of doped Ge with hopping conduction was described quantitatively taking into account electron overheating and the “thermal model” of electron-phonon energy transfer. It was assumed that the resistance of the sample was determined only by the electron temperature $T_{e}$ irrespective of the value of current. In this case the nonlinearity of $I$-$V$ curves was due to a decrease in the sample resistance $R(T_{e})$ caused by the heating of the charge carriers to $T_{e}$. As a result, the voltage over the sample $V = IR(T_{e})$ can decrease when the current increases. Below a certain critical temperature $T_{x}$ an extreme point $dV/dI = 0$ appears in the $I$-$V$ curves, which is followed by a NDR region. This is a region of instability, current and resistance oscillations, and non-equilibrium transitions. The known theories attribute NDR, among other things, to a non-uniform distribution of impurities and defects over the crystal, which produce regions with electric fields of different intensities. In the sample studied, NDR can be caused by phase separation into superconducting and dielectric regions. Qualitatively, the $I$-$V$ curves in Fig. 2 correspond to those calculated in Ref. taking into account the overheating effect. For the sample studied critical temperature transition to NDR is about 6 K (Fig. 2); whereas estimations made in the frame of the “thermal model” [@stef] give the value close to 1 K. This discrepancy may be attributed with phase separation into superconducting and dielectric regions. The model in Refs. was developed for semiconductors and did not allow for superconducting inclusions as factors of inhomogeneities. Nevertheless, the basic concepts of the model [@wang; @stef] account on the whole for the results obtained. The observed current - controlled NDR effect can be interpreted as NDR typical for percolation systems [@ridley] in which increasing electric fields (currents) lead to elongation of the existing high-conductivity percolation paths or even to the formation of new ones. However, the results obtained are not sufficient to analyze comprehensively or to draw conclusions about particular mechanisms of this effect in the investigated sample. {width="0.9\linewidth"} The behavior of magnetoresistance (MR) in the single crystal La$_{2}$CuO$_{4+\delta}$ studied is also strongly dependent on transport current and sensitive to electron overheating. The effect of current is particularly evident in the low temperature region (Fig. 3). For rather low currents $J \leq 1$ $\mu$A (with conductivity close to the Ohmic one), MR is positive in the low temperature region ($T \leq 10$ K) (Fig. 4). We can attribute this positive MR to the influence of superconducting inclusions, like in La$_{2}$CuO$_{4+\delta}$ sample with much higher $T_{N}$ [@boris2]. An increase in the current produces the Joule heating and corresponding pair-breaking effect. As a result, the positive MR disappears. When the current reaches $J \approx 10$ $\mu$A, MR becomes negative. The possible sources of the negative MR in La$_{2}$CuO$_{4+\delta}$ at $T > 10$ K was considered in details in Ref. . In fields above $\approx 5$ T, MR is to a large extent determined by the metamagnetic AFM - weak FM transition. The competition of two different MR mechanisms and the transition from positive to negative MR under electron overheating are illustrated in Fig. 3. This corresponds to the temperature behavior of MR at low currents (Fig. 4). The results of the MR investigation thus attest to the effect of electron overheating, which in turn stimulates NDR at high currents in the low temperature region. The latter effect evolves from the inhomogeneous composition of the sample: because of phase separation typical for this system, superconducting inclusions are produced in the dielectric matrix at low temperatures. [00]{} R. K. Kremer, A. Simon, E. Sigmund, and V. Hizhnyakov, in: E. Sigmund and K.A. Müller (Eds.), Phase Separation in Cuprate Superconductors, Springer, Heidelberg, 1994, p. 66. J. M. Tranquada, S. M. Heald, and A. R. Moodenbaugh, Phys. Rev. B [**36**]{}, 5236 (1987). C. Chaillout, J. Chenavas, S.-W. Cheong et al., Physica C [**170**]{}, 87 (1990). J. D. Jorgensen, B. Dabrowski, S. Pei et al., Phys.Rev. B. [**38**]{}, 11337 (1988). K. Ueda, T. Sugata, Y. Kohori et al., Solid. St. Comm. [**73**]{}, 49 (1990); T. Kobayashi, S. Wada, K. Shibutani et al., J. Phys. Soc. Jpn. [**58**]{}, 3497 (1989). B. I. Belevtsev, N. V. Dalakova, and A. S. Panfilov, Low Temp. Phys. [**23**]{} 274 (1997). B. I. Belevtsev, N. V. Dalakova, A. S. Panfilov, Low Temp. Phys. [**24**]{}, 816 (1998). N. F. Mott and E. A. Davis, Electron Processes in Noncrystalline Materials, Clarendon Press, Oxford (1979). E. M. Conwell, High field transport in semiconductors, Academic Press, NY, 1967. N. Wang, F. C. Wellstood, B. Sadoulet et al., Phys. Rev. B [**41**]{}, 3761 (1990). P. Stefanyi, C. C. Zammit, P. Fozooni et al., J. Phys.: Condens. Matter [**9**]{}, 881 (1997). B. K. Ridley, in: “Negative Differential Resistance and Instabilities in 2-D Semiconductors”, ed. by N. Balkan et.al., New York: Plenum Press, 1993. P.1. B. I. Belevtsev, N. V. Dalakova, V. N. Savitsky V.N. et al., Low Temp. Phys. [**30**]{}, 411 (2004).

--- author: - Makoto Yamaguchi and Tetsuo Ogawa title: 'Equilibrium to nonequilibrium condensation in driven-dissipative semiconductor systems' --- Introduction {#sec:intro} ============ In a semiconductor system, it is known that electron-hole (e-h) bound pairs can be formed by their Coulomb attraction when the conduction and valence band effectively reach an equilibrium state after the carriers are generated e.g. by laser excitation (Fig. \[RelaxationRedistribution\]). An exciton polariton is a quasi-bosonic particle composed of such a Coulomb-bound e-h pair (exciton) and a photon [@Weisbuch92; @Bloch98], the behaviors of which have attracted much attention due to their potential apprications through the Bose-Einstein condensation (BEC) [@Imamoglu96; @Deng02; @Kasprzak06], i.e. a macroscopic occupation of a single exciton-polariton state by a thermodynamic phase transition. ![ Excitation and thermalization process in a semiconductor. Electrons and holes generated by laser excitation subsequently undergo immediate intraband relaxations and redistributions in the conduction band (C.B.) and valence band (V.B.) to effectively reach an equilibrium state. Coulomb-bound e-h pairs (excitons) are formd when the equilibrium state is at sufficiently low temperature and low carrier density. []{data-label="RelaxationRedistribution"}](RelaxationRedistribution.eps){width="0.50\linewidth"} A typical exciton-polariton system is shown in Fig. \[System\]. The system basically consists of semiconductor quantum wells (QWs) and a microcavity, the same structure as a vertical cavity surface emitting laser (VCSEL). In this context, a conventional lasing phase[^1] is involved in this system as well as the exciton-polariton BEC [@Snoke12]. At high densities, moreover, the Bardeen-Cooper-Schrieffer (BCS) -like ordered phase can potentially be caused where electrons and holes form the “Cooper pairs” [@Keldysh65; @Littlewood04], as is discussed in the BCS-BEC crossover in cold atom systems with Feshbach resonances [@Ohashi02; @Ohashi03]. These ordered phases are schematically shown in Fig. \[BEC-BCS-LASER\]. ![ Schematic illustration of a typical exciton-polariton system. Exciton-polaritons are formed by the electrons and holes in the QWs and the photons confined between the two mirrors (microcavity). []{data-label="System"}](System.eps){width="0.50\linewidth"} However, the BEC and BCS phases are in equilibrium, the situation of which is quite different from the semiconducotor laser in nonequilibrium. As a result, approaches for describing the BEC and BCS phases based on equilibrium statistical mechanics, e.g. the BCS theory [@Kamide10; @Byrnes10], are not applicable to the semiconductor laser because any nonequilibrium effects cannot be taken into account, such as pumping and loss. Conversely, past theories for describing the lasing operation, e.g. the Maxwell-Semiconductor-Bloch equations (MSBEs) [@Chow02; @Kamide11], cannot recover such equilibrium statistical approaches.[^2] The difficulty shown here has been one of problems to understand the underlying physics in exciton-polariton systems. In such a situation, we have recently proposed a framework which can treat the phases of the BEC, BCS and laser in a unified way [@Yamaguchi12; @Yamaguchi13]. This framework is an extention of a nonequilibrium Green’s function approach developed in Ref. [@Szymanska06; @Szymanska07; @Keeling10] in which excitons are simply modeled by localized noninteracting two-level systems without internal e-h structures. Our formalism results in the BCS theory when the system can be regarded as in equilibrium, while it recovers the MSBE when nonequilibrium features become important. The internal e-h structures as well as the Coulomb interactions can also be taken into account within the mean-field approximation. In this contribution, we would like to give an introduction to such a “BEC-BCS-LASER crossover theory”. ![ Schematic illustration of several ordered phases involved in the exciton-polariton system. []{data-label="BEC-BCS-LASER"}](BEC-BCS-LASER.eps){width="0.90\linewidth"} BCS theory and MSBE for exciton-polariton systems {#sec:BCS and MSBE} ================================================= In exciton-polariton systems, the equilibrium phases (the BEC and BCS phases) can be described by the BCS theory while the nonequilibrium phase (the lasing phase) can be described by the MSBE. In this Section, we give an overview of the BCS theory and the MSBE to highlight their similarities and differences. For simplicity, we set $\hbar=k_{\mathrm{B}}=1$ in the followings. Model {#subsec:Model} ----- We first describe the Hamiltonian for the exciton-polariton system where electrons and holes in the QWs and photons in the microcavity are taken into account. The system Hamiltonian $\hat{H}_{\mathrm{S}}$ is then given by $\hat{H}_{\mathrm{S}} = \hat{H}_{0} + \hat{H}_{\mathrm{Coul}} + \hat{H}_{\mathrm{dip}}$. Here, $\hat{H}_{\mathrm{0}}$ is the Hamiltonian for free particles without interactions and written as $$\hat{H}_{\mathrm{0}} = \sum_{{\textbf{\itshape{k}}}} \left( {\epsilon_{\mathrm{e},{\textbf{\itshape{k}}}}}\oed_{{\textbf{\itshape{k}}}}\ope_{{\textbf{\itshape{k}}}} + {\epsilon_{\mathrm{h},{\textbf{\itshape{k}}}}}\ohd_{{\textbf{\itshape{k}}}}\oh_{{\textbf{\itshape{k}}}} + {\epsilon_{\mathrm{ph},{\textbf{\itshape{k}}}}}\oad_{{\textbf{\itshape{k}}}}\oa_{{\textbf{\itshape{k}}}} \right), \label{eq:HS}$$ where $\ope_{{\textbf{\itshape{k}}}}$, $\oh_{{\textbf{\itshape{k}}}}$, and $\oa_{{\textbf{\itshape{k}}}}$ are annihilation operators for electrons, holes, and photons with in-plane wave number ${\textbf{\itshape{k}}}$, respectively. ${\epsilon_{\mathrm{e(h)},{\textbf{\itshape{k}}}}} = k^2/2m_{\mathrm{e(h)}} + E_{\mathrm{g}}/2$ is the energy dispersion of electrons (holes) with an effective mass $m_{\mathrm{e(h)}}$, while ${\epsilon_{\mathrm{ph},{\textbf{\itshape{k}}}}} = k^2/2m_{\mathrm{cav}} + E_{\mathrm{cav}}$ is that of photons with an effective mass $m_{\mathrm{cav}}$. $E_{\mathrm{g}}$ is the bandgap and $E_{\mathrm{cav}}$ is the energy of the cavity mode for ${\textbf{\itshape{k}}}=0$ [@Deng10Review]. In contrast, $\hat{H}_{\mathrm{Coul}}$ and $\hat{H}_{\mathrm{dip}}$ denote the Coulomb interaction and the light-matter interaction within the dipole approximation, respectively written as $$\begin{aligned} \hat{H}_{\mathrm{Coul}} &=& \frac{1}{2}\sum_{{\textbf{\itshape{k}}},{\textbf{\itshape{k}}}',{\textbf{\itshape{q}}}}U'_{{\textbf{\itshape{q}}}} \left( \oed_{{\textbf{\itshape{k}}}+{\textbf{\itshape{q}}}}\oed_{{\textbf{\itshape{k}}}'-{\textbf{\itshape{q}}}}\ope_{{\textbf{\itshape{k}}}'}\ope_{{\textbf{\itshape{k}}}} + ( \ope \leftrightarrow \oh ) -2\oed_{{\textbf{\itshape{k}}}+{\textbf{\itshape{q}}}}\ohd_{{\textbf{\itshape{k}}}'-{\textbf{\itshape{q}}}}\oh_{{\textbf{\itshape{k}}}'}\ope_{{\textbf{\itshape{k}}}} \right), \label{eq:HCoul}\\ \hat{H}_{\mathrm{dip}} &=& -\sum_{{\textbf{\itshape{k}}},{\textbf{\itshape{q}}}} \left( g\oad_{{\textbf{\itshape{q}}}}\oh_{-{\textbf{\itshape{k}}}}\ope_{{\textbf{\itshape{k}}}+{\textbf{\itshape{q}}}} + g^{*}\oa_{{\textbf{\itshape{q}}}}\oed_{{\textbf{\itshape{k}}}+{\textbf{\itshape{q}}}}\ohd_{-{\textbf{\itshape{k}}}} \right), \label{eq:Hdip}\end{aligned}$$ where $U'_{{\textbf{\itshape{q}}}}=U'_{-{\textbf{\itshape{q}}}}$ and $U'_{{\textbf{\itshape{q}}}=0} \equiv 0$. Note that $[\hat{H}_{\mathrm{S}},\hat{N}_{\mathrm{S}}] = 0$ is satisfied when an excitation number of the system $\hat{N}_{\mathrm{S}}$ is defined as $\hat{N}_{\mathrm{S}} \equiv \sum_{{\textbf{\itshape{k}}}} [\oed_{{\textbf{\itshape{k}}}}\ope_{{\textbf{\itshape{k}}}}/2 + \ohd_{{\textbf{\itshape{k}}}}\oh_{{\textbf{\itshape{k}}}}/2 + \oad_{{\textbf{\itshape{k}}}}\oa_{{\textbf{\itshape{k}}}}]$. For later convenience, therefore, we redefine $\hat{H}_{\mathrm{S}}-\mu\hat{N}_{\mathrm{S}}$ as $\hat{H}_{\mathrm{S}}$. This means that a grand canonical ensemble is assumed with a chemical potential $\mu$ if we are interested in equilibrium phases. In contrast, for time-dependent problems, this means that dynamics of physical quantities is captured on a rotating frame with the frequency $\mu$. Thus, $\mu$ is a given parameter identical to the chemical potential for the BEC and BCS phases (Subsection \[subsec:BCS\]), whereas it becomes a unknown variable equivalent to the lasing frequency for the semiconductor laser in a steady state (Subsection \[subsec:MSBE\]). Mean-field approximation {#subsec:MF} ------------------------ The Hamiltonians shown in Subsection \[subsec:Model\] give a starting point for theories of the exciton-polariton system. However, in practice, it is difficult to exactly treat $\hat{H}_{\mathrm{Coul}}$ and $\hat{H}_{\mathrm{dip}}$ because these Hamiltonians cause many-body problems. In this Subsection, therefore, we discuss the mean-field (MF) approximation in order to reduce the problems to single-particle problems. In general, the MF approximation is performed by writing a specific operator $\hat{O}$ as $\hat{O} = {\langle \hat{O} \rangle} + \mathnormal{\delta}\hat{O}$ and by neglecting quadratic terms with respect to $\mathnormal{\delta}\hat{O}$ in the Hamiltonians.[^3] Here, ${\langle \hat{O} \rangle} \equiv \mathrm{Tr}[\hat{O}\hat\rho]$ denotes the expectation value for the density operator $\hat\rho$ and the operator $\mathnormal{\delta}\hat{O}$ corresponds to a fluctuation around the expectation value. In our case, the interaction Hamiltonians of $\hat{H}_{\mathrm{Coul}}$ and $\hat{H}_{\mathrm{dip}}$ can easily be reduced to a single-particle problem by employing $\hat{O} \in \{ \oa_{{\textbf{\itshape{k}}}}, \oh_{-{\textbf{\itshape{k}}}}\ope_{{\textbf{\itshape{k}}}'}, \oed_{{\textbf{\itshape{k}}}}\ope_{{\textbf{\itshape{k}}}'}, \ohd_{{\textbf{\itshape{k}}}}\oh_{{\textbf{\itshape{k}}}'} \}$. As a result, with definitions of the photon field ${\langle \oa_{{\textbf{\itshape{k}}}} \rangle} \equiv \delta_{{\textbf{\itshape{k}}},0}\a0$, the polarization function ${\langle \oh_{-{\textbf{\itshape{k}}}}\ope_{{\textbf{\itshape{k}}}'} \rangle} \equiv \delta_{{\textbf{\itshape{k}}},{\textbf{\itshape{k}}}'}\pk$, and the distribution functions of electrons ${\langle \oed_{{\textbf{\itshape{k}}}}\ope_{{\textbf{\itshape{k}}}'} \rangle} \equiv \delta_{{\textbf{\itshape{k}}},{\textbf{\itshape{k}}}'}\nek$ and holes ${\langle \ohd_{{\textbf{\itshape{k}}}}\oh_{{\textbf{\itshape{k}}}'} \rangle} \equiv \delta_{{\textbf{\itshape{k}}},{\textbf{\itshape{k}}}'}\nhk$, the mean-field Hamiltonian $\hat{H}_{\mathrm{S}}^{\mathrm{MF}}$ is obtained as $$\begin{aligned} \hat{H}_{\mathrm{S}}^{\mathrm{MF}} &=& \sum_{{\textbf{\itshape{k}}}} \left( {\tilde{\xi}_{\mathrm{e},{\textbf{\itshape{k}}}}}\oed_{{\textbf{\itshape{k}}}}\ope_{{\textbf{\itshape{k}}}} + {\tilde{\xi}_{\mathrm{h},{\textbf{\itshape{k}}}}}\ohd_{{\textbf{\itshape{k}}}}\oh_{{\textbf{\itshape{k}}}} - [{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}\oed_{{\textbf{\itshape{k}}}}\ohd_{-{\textbf{\itshape{k}}}} + \mathrm{H.c.}] \right) \nonumber\\ &&+\sum_{{\textbf{\itshape{k}}}} \left( {\xi_{\mathrm{ph},{\textbf{\itshape{k}}}}}\oad_{{\textbf{\itshape{k}}}}\oa_{{\textbf{\itshape{k}}}} - [g\pk\oad_0 + g^{*}\pk^{*}\oa_0] \right). \label{eq:HMF}\end{aligned}$$ Here, constants are ignored because the following discussion is not affected. ${\tilde{\xi}_{\mathrm{e(h)},{\textbf{\itshape{k}}}}}$ and ${\xi_{\mathrm{ph},{\textbf{\itshape{k}}}}}$ are respectively defined as ${\tilde{\xi}_{\mathrm{e(h)},{\textbf{\itshape{k}}}}} \equiv {\tilde\epsilon_{\mathrm{e(h)},{\textbf{\itshape{k}}}}} - \mu/2$ and ${\xi_{\mathrm{ph},{\textbf{\itshape{k}}}}} \equiv {\epsilon_{\mathrm{ph},{\textbf{\itshape{k}}}}} - \mu$, where ${\tilde\epsilon_{\mathrm{e(h)},{\textbf{\itshape{k}}}}} \equiv {\epsilon_{\mathrm{e(h)},{\textbf{\itshape{k}}}}} - \sum_{{\textbf{\itshape{k}}}'} U'_{{\textbf{\itshape{k}}}-{\textbf{\itshape{k}}}'} n_{\mathrm{e(h)},{\textbf{\itshape{k}}}'}$ denotes the energy dispersion of electrons (holes) renormalized by the repulsive electron-electron (hole-hole) Coulomb interaction, the first (second) term in Eq. (\[eq:HCoul\]). The well-known bandgap renormalization (BGR) in semiconductor physics is included in ${\tilde\epsilon_{\mathrm{e(h)},{\textbf{\itshape{k}}}}}$. In contrast, ${\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}} \equiv g^{*}a_0 + \sum_{{\textbf{\itshape{k}}}'}U'_{{\textbf{\itshape{k}}}-{\textbf{\itshape{k}}}'}p_{{\textbf{\itshape{k}}}'}$ results from the attractive electron-hole Coulomb interaction, the third term in Eq. (\[eq:HCoul\]), and is called the generalized Rabi frequency [@HaugKoch]. ${\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}$ has a role in forming e-h pairs as can be seen in Eq. (\[eq:HMF\]). The mean-field Hamiltonian is thus obtained. However, note that the expectation values of ${\langle \hat{O} \rangle}$ (i.e. $\a0$, $\pk$, $\nek$, and $\nhk$) are included in $\hat{H}_{\mathrm{S}}^{\mathrm{MF}}$. For self-consistensy, therefore, the following relation should be satisfied : $${\langle \hat{O} \rangle} = {\mathrm{Tr}[\hat{O}\hat\rho^{\mathrm{MF}}({\langle \hat{O} \rangle})]}. \label{eq:SCE}$$ Here, $\hat\rho^{\mathrm{MF}}$ is the density operator determiend by using $\hat{H}_{\mathrm{S}}^{\mathrm{MF}}$. The BCS theory and the MSBE shown below are obtained from this self-consistent equation. BCS theory for exciton-polariton condensation {#subsec:BCS} --------------------------------------------- First, we assume that the exciton-polariton system is in equilibrium. According to the equilibrium statistical mechanics, the density operator $\hat\rho^{\mathrm{MF}}$ at temperature $T$ can be described as $$\begin{aligned} \hat\rho^{\mathrm{MF}}=\hat\rho_{\mathrm{eq}}^{\mathrm{MF}}\equiv\frac{1}{Z}\exp(-\beta\hat{H}_{\mathrm{S}}^{\mathrm{MF}}), \label{eq:equilibrium}\end{aligned}$$ where $Z \equiv \mathrm{Tr}[\exp(-\beta\hat{H}_{\mathrm{S}}^{\mathrm{MF}})]$ and $\beta \equiv 1/T$. In this case, $\mu$ is a given parameter equivalent to the chemical potential, as mentioned above. With ${\epsilon_{\mathrm{e},{\textbf{\itshape{k}}}}} = {\epsilon_{\mathrm{h},{\textbf{\itshape{k}}}}}$ for simplicity, the self-consistent equations obtained from Eqs. (\[eq:HMF\])-(\[eq:equilibrium\]) are $$\begin{aligned} \a0=\sum_{{\textbf{\itshape{k}}}'}\frac{g}{\xi_{\mathrm{ph},0}}p_{{\textbf{\itshape{k}}}'},\ \pk=\frac{{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}}{2{E_{{\textbf{\itshape{k}}}}}}\tanh \left( \frac{\beta {E_{{\textbf{\itshape{k}}}}}}{2} \right), \label{eq:ap}\\ \nek=\nhk=\frac{1}{2} \left\{ 1-\frac{{\tilde{\xi}_{\mathrm{eh},{\textbf{\itshape{k}}}}}^{+}}{{E_{{\textbf{\itshape{k}}}}}}\tanh \left( \frac{\beta {E_{{\textbf{\itshape{k}}}}}}{2} \right) \right\}, \label{eq:number}\end{aligned}$$ where ${\tilde{\xi}_{\mathrm{eh},{\textbf{\itshape{k}}}}}^{\pm} \equiv ({\tilde{\xi}_{\mathrm{e},{\textbf{\itshape{k}}}}} \pm {\tilde{\xi}_{\mathrm{h},{\textbf{\itshape{k}}}}})/2 $ and ${E_{{\textbf{\itshape{k}}}}} \equiv [({\tilde{\xi}_{\mathrm{eh},{\textbf{\itshape{k}}}}}^{+})^2 + |{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}|^2]^{1/2}$. In the derivation, Bogoliubov transformations of $\ope_{{\textbf{\itshape{k}}}}$ and $\oh_{{\textbf{\itshape{k}}}}$ can be applied to the first line in Eq. (\[eq:HMF\]) for diagonalization, while a displacement of $\oa_{0}$ to the second line, because the Hilbert space of the first (second) line of Eq. (\[eq:HMF\]) is spanned only by the electron and hole (photon) degrees of freedom. The gap equation, which is formally equivalent to the BCS theory for superconductors, can then be obtained by substituting Eq. (\[eq:ap\]) into the definition of ${\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}$: $$\displaystyle {\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}} = \sum_{{\textbf{\itshape{k}}}'}U_{{\textbf{\itshape{k}}}',{\textbf{\itshape{k}}}}^{\mathrm{eff}}\frac{{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}{'}}}}{2{E_{{\textbf{\itshape{k}}}{'}}}} \tanh \left( \frac{\beta {E_{{\textbf{\itshape{k}}}{'}}}}{2} \right). \label{eq:gap}$$ In this context, ${\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}$ is an order parameter in the exciton-polariton system as well as in the superconducting system. $U_{{\textbf{\itshape{k}}}',{\textbf{\itshape{k}}}}^{\mathrm{eff}} \equiv |g|^2/\xi_{\mathrm{ph},0} + U'_{{\textbf{\itshape{k}}}'-{\textbf{\itshape{k}}}}$ represents an effective attractive e-h interaction, from which one can find that photon-mediated process also contributes the attractive interaction. Notice that Eqs. (\[eq:number\]) and (\[eq:gap\]) are simultaneous equations with the unknown variables $\nek (=\nhk)$ and ${\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}$. Especially for $T = 0$, this treatment is known to cover the equilibrium phases from the BEC to the BCS states [@Kamide10; @Byrnes10; @Comte82]. ![ A schematic picture of the model including environments. The e-h system is excited and thermalized by the pumping baths by exchanging carriers. Photons in the system are lost into the vacuum. []{data-label="Model"}](Model.eps){width="0.50\linewidth"} MSBE for semiconductor lasers {#subsec:MSBE} ----------------------------- Next, a treatment based on the MSBE is explained for the discussion of the semiconductor laser, which is characterized by nonequilibrium. In contrast to the BCS theory, therefore, the effects of environments (Fig. \[Model\]) cannot be neglected for lasing; the excitation and thermalization of the e-h system and the loss of photons from the microcavity. For this reason, the dynamics of the total density operator $\hat\rho^{\mathrm{MF}}$ is discussed by writing the total mean-field Hamiltonian $\hat{H}^{\mathrm{MF}} \equiv \hat{H}_{\mathrm{S}}^{\mathrm{MF}}+\hat{E}$ with the couplings to the environments $\hat{E}$. Since $\ii\partial_t \hat\rho^{\mathrm{MF}} = [\hat{H}^{\mathrm{MF}},\hat\rho^{\mathrm{MF}}]$ in the Schrödinger picture, a time drivative of Eq. (\[eq:SCE\]) yields $$\ii\partial_t{\langle \hat{O} \rangle} = {\mathrm{Tr}[[\hat{O},\hat{H}_{\mathrm{S}}^{\mathrm{MF}}]\hat\rho^{\mathrm{MF}}]}+{\mathrm{Tr}[[\hat{O},\hat{E}]\hat\rho^{\mathrm{MF}}]}, \label{eq:SCE2}$$ where ${\mathrm{Tr}[\hat{A}\hat{B}]}={\mathrm{Tr}[\hat{B}\hat{A}]}$ is used. The MSBE is then obtained when the first term is derived from Eq. (\[eq:HMF\]) and the second term is replaced by phenomenological relaxation terms: $$\begin{aligned} \partial_t\a0&=&-\ii \xi_{\mathrm{ph},0} \a0 + \ii g \textstyle{\sum_{{\textbf{\itshape{k}}}}} \pk - \kappa \a0, \label{eq:MSBE_a0} \\ \partial_t\pk&=&-2\ii{\tilde{\xi}_{\mathrm{eh},{\textbf{\itshape{k}}}}}^{+}\pk-\ii{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}} N_{{\textbf{\itshape{k}}}}-2\gamma (\pk - \pk^{0}), \label{eq:MSBE_pk} \\ \partial_t n_{\mathrm{e(h)},{\textbf{\itshape{k}}}}&=&-2\mathrm{Im}[{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}} p_{{\textbf{\itshape{k}}}}^{*}]-2\gamma(n_{\mathrm{e(h)},{\textbf{\itshape{k}}}}-n_{\mathrm{e(h)},{\textbf{\itshape{k}}}}^{0}), \label{eq:MSBE_nehk}\end{aligned}$$ where the last term in each equation is the relaxation term and $N_{{\textbf{\itshape{k}}}} \equiv \nek + \nhk -1$ denotes the degree of the population inversion.[^4] $\pk^{0}$ and $n_{\mathrm{e(h)},{\textbf{\itshape{k}}}}^{0}$ are defined as $$\pk^{0} \equiv 0, \hspace{5pt} n_{\mathrm{e(h)},{\textbf{\itshape{k}}}}^{0} \equiv f_{\mathrm{e(h)},{\textbf{\itshape{k}}}} \label{eq:RTA}$$ where $f_{\mathrm{e(h)},{\textbf{\itshape{k}}}} \equiv [1 + \exp \{ \beta ( {\tilde\epsilon_{\mathrm{e(h)},{\textbf{\itshape{k}}}}}-\mu^{\mathrm{B}}_{\mathrm{e(h)}}) \} ]^{-1}$ is the Fermi distribution with the chemical potential $\mu^{\mathrm{B}}_{\mathrm{e(h)}}$ of the electron (hole) pumping bath. The phenomenological approximation shown here is called the relaxation approximation [@Henneberger92]. Each relaxation term suggests that the photon field $a_{0}$ decays with a rate of $\kappa$, the distribution function $n_{\mathrm{e(h)},{\textbf{\itshape{k}}}}$ is driven to approach the Fermi distribution $f_{\mathrm{e(h)},{\textbf{\itshape{k}}}}$ (Fig. \[Energy\]), i.e. thermalization (Fig. \[RelaxationRedistribution\]), and $\pk$ decays due to thermalization-induced dephasing. Solutions for the laser action can then be obtained by determining the unknown variables $\a0$, $\pk$, $\nek$, $\nhk$, and $\mu$ in Eqs. (\[eq:MSBE\_a0\])-(\[eq:MSBE\_nehk\]) under a steady-state condition $\partial_t{\langle \hat{O} \rangle} = 0$. Again, we emphasize that $\mu$ is a unknown variable corresponding to the laser frequency in the steady-state MSBE, in contrast to the BCS theory. This is equivalent to find an appropriate frequency with which the lasing oscillation of $\a0$ and $\pk$ seems to remain stationary on the rotating frame. ![ Energy dispersions of electrons and holes. The distribution functions $n_{\mathrm{e},{\textbf{\itshape{k}}}}$ and $n_{\mathrm{h},{\textbf{\itshape{k}}}}$ are driven to approach the Fermi distributions by the respective pumping baths. []{data-label="Energy"}](EnergyDispersion.eps){width="0.50\linewidth"} BEC-BCS-LASER crossover theory {#sec:BBL} ============================== In the exciton-polariton system, as shown in Section \[sec:BCS and MSBE\], the BCS theory and the MSBE are theoretical frameworks starting from the common Hamiltonians with the same mean-field approximation. However, the difference is the way of deriving the self-consistent equations. In the case of the BCS theory, $\hat\rho^{\mathrm{MF}}$ is directly described by $\hat{H}_{\mathrm{S}}^{\mathrm{MF}}$ (Eq. (\[eq:equilibrium\])). In contrast, in the case of the MSBE, Eq. (\[eq:SCE2\]) is used to introduce the phenomenological relaxation terms. We note, however, that any assumption is not used for $\hat\rho^{\mathrm{MF}}$ in Eq. (\[eq:SCE2\]), which indicates that the MSBE may incorporate the BCS theory at least in principle. In this context, an approach to derive the BCS theory from the MSBE should be discussed briefly. We first consider a situation where the effects of the environments are completely neglected, which is equivalent to set $\kappa = \gamma = 0$ in the MSBE. However, in this case, the BCS theory cannnot be derived because there is no term to drive the system into equilibrium in the MSBE.[^5] A natural condition to consider is physically a limit of $\gamma \rightarrow 0^{+}$ after $\kappa \rightarrow 0$ because the system should be thermalized even though the effects of environments are decreased. Unfortunately, however, the MSBE does not recover the BCS theory even by taking this limit. The relationship between the BCS theory and the MSBE is thus discontinuous in spite of the similarities of the two frameworks. Obviously, the phenomenological relaxation approximation causes such a problem. In regard to this problem, we have recently constructed a unified framework [@Yamaguchi12] by using a nonequilibrium Green’s function approach [@Szymanska06; @Szymanska07; @Keeling10]. The framework, at first, takes an integral form of simultaneous equations and seems quite different from the MSBE (see also Appendix II). However, by rearranging the equations with particular attention to the problem mentioned above, all of important changes can successfully be incorporated in the relaxation terms in the MSBE [@Yamaguchi13]. The result simply replaces Eq. (\[eq:RTA\]) by $$\begin{aligned} &\pk^{0} \equiv \ii\int\frac{\mathrm{d}\nu}{2\pi}\left[ {G^{\mathrm{R}}_{\mathrm{12},{{\textbf{\itshape{k}}}}}}(\nu)\{1-f^{\mathrm{B}}_{\mathrm{h}}(-\nu)\} - {G^{\mathrm{R*}}_{\mathrm{21},{{\textbf{\itshape{k}}}}}}(\nu) f^{\mathrm{B}}_{\mathrm{e}}(\nu) \right], \nonumber\\ &n_{\mathrm{e(h)},{\textbf{\itshape{k}}}}^{0} \equiv \int\frac{\mathrm{d}\nu}{2\pi}f^{\mathrm{B}}_{\mathrm{e(h)}}(\nu)A_{\mathrm{11(22)}}(\pm\nu;{\textbf{\itshape{k}}}), \tag{\ref{eq:RTA}$'$} \label{eq:RTA'}\end{aligned}$$ where $f^{\mathrm{B}}_{\mathrm{e(h)}}(\nu) \equiv [\exp \{ \beta(\nu - \mu^{\mathrm{B}}_{\mathrm{e(h)}} + \mu/2) \} + 1]^{-1}$ is the Fermi distribution of the electron (hole) pumping bath. ${G^{\mathrm{R}}_{\mathrm{\alpha\alpha'},{{\textbf{\itshape{k}}}}}}(\nu)$ is called the retarded Green’s function and described by elements of a matrix $$\begin{aligned} G^{\mathrm{R}}_{{\textbf{\itshape{k}}}}(\nu)=\left( \begin{array}{cc} \nu - \tilde{\xi}_{\mathrm{e},{\textbf{\itshape{k}}}} + \ii\gamma & {\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}} \\ {\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}^{*} & \nu + \tilde{\xi}_{\mathrm{h},{\textbf{\itshape{k}}}} + \ii\gamma \\ \end{array}\right)^{-1}. \label{eq:GR}\end{aligned}$$ On the other hand, $A_{11(22)}(\nu;{\textbf{\itshape{k}}})$ is called the single-particle spectral function and defined as $$A_{\alpha\alpha'}(\nu;{\textbf{\itshape{k}}}) \equiv \ii (G^{\mathrm{R}}_{\alpha\alpha',{\textbf{\itshape{k}}}}(\nu) - G^{\mathrm{R}*}_{\alpha'\alpha,{\textbf{\itshape{k}}}}(\nu)). \label{eq:A}$$ Here, $A_{11(22)}(\nu;{\textbf{\itshape{k}}})$ means the density of states for electron-like (hole-like) quasi-particles with the energy $\nu$ and wave number ${\textbf{\itshape{k}}}$. Some readers might feel difficult to understand the formalism because the above definitions are unique to the Green’s function approach. However, all we have to do is the replacement of Eq. (\[eq:RTA\]) by Eq. (\[eq:RTA’\]). The unknown variables are still $\a0$, $\pk$, $\nek$, $\nhk$, and $\mu$, that is, the same as the MSBE. In this sense, the obtained equations are quite simple, which is one of strong points of this formalism. From the viewpoint of the Green’s funciton, it is relatively easy to understand the physical meaning of Eq. (\[eq:RTA’\]) due to the clear form; the energy integral of (distribution)$\times$(density of states).[^6] We refer to such a formalism as the BEC-BCS-LASER crossover theory. Now, this formalism enables us to cleary understand the standpoint of the BCS theory. For this purpose, let us discuss the limit of equilibrium, based on the idea described above. In the followings, however, ${\epsilon_{\mathrm{e},{\textbf{\itshape{k}}}}} = {\epsilon_{\mathrm{h},{\textbf{\itshape{k}}}}}$ and a charge neutrality $\mu^{\mathrm{B}}_{\mathrm{e}} = \mu^{\mathrm{B}}_{\mathrm{h}}$ are assumed for simplicity. First, in the limit of $\kappa \rightarrow 0$, one can prove $\mu = \mu_{\mathrm{B}} (\equiv \mu^{\mathrm{B}}_{\mathrm{e}} + \mu^{\mathrm{B}}_{\mathrm{h}})$. This is the same as treating $\mu$ as a given parameter, and physically, means that the system reaches in chemical equilibrium with the pumping baths because there is no photon loss. The BCS theory is then derived after taking the limit of $\gamma \rightarrow 0^{+}$, where the integrals in Eq. (\[eq:RTA’\]) can be performed analytically. In this derivation, $\gamma \neq 0$ is required to be canceled down even though $\gamma$ does not appear in the final expression. This means that thermalization is essential to recover the equilibrium theory. Thus, the BCS theory can be derived from the presented theory in the equilibrium limit. However, in some sense, this situation is physically trivial; the situation is not limited to such a trivial one for the system to be in equilibrium. Even under a condition where photons are continuously lost, it may be still possible to identify the system as being in equilibrium (quasi-equilibrium) as long as the e-h system is excited and thermalized. A true advantage of the above-presented framework becomes obvious in such a situation rather than in the trivial one. In this case, $\mu$ is still equivalent to the chemical potential but $\mu_{\mathrm{B}} > \mu$ because the system is influenced by the photon loss. As a result, $\mu$ becomes a unknown variable again. Furthermore, such a quasi-equilibrium condition can easily be obtained from Eqs. (\[eq:RTA’\])-(\[eq:A\]) as[^7] (I) $\min[2{E_{{\textbf{\itshape{k}}}}}] \gtrsim \mu_{\mathrm{B}} - \mu + 2\gamma + 2T$. Here, $\min[2{E_{{\textbf{\itshape{k}}}}}]$ is the minimum energy required for breaking e-h bound pairs and $\mu_{\mathrm{B}}-\mu > 0$ suggests that there is continuous particle flow from the pumping baths into the system.[^8] We can then interpret the condition (I); this is a condition that the particle flux, thermalization-induced dephasing (= $2\gamma$), and temperature effect (= $2T$), do not contribute to the dissociations of the e-h pairs. However, the system can no longer be in quasi-equilibrium when nonequilibrium effect becomes significant. Let us therefore consider a situation where the MSBE, i.e. the physics of the semiconductor laser, becomes important. Such a condition can be found from Eqs. (\[eq:RTA’\])-(\[eq:A\]) as (II) $\mu_{\mathrm{B}} - \mu \gtrsim \min[2{E_{{\textbf{\itshape{k}}}}}] + 2\gamma + 2T$, because $f^{\mathrm{B}}_{\mathrm{e(h)}}(\pm\nu) \simeq f^{\mathrm{B}}_{\mathrm{e(h)}}({\tilde{\xi}_{\mathrm{eh},{\textbf{\itshape{k}}}}}^{+})$ turns out to be a good approximation in Eq. (\[eq:RTA’\]) for ${\textbf{\itshape{k}}}$-resions satisfying (II$'$) $~\mu_{\mathrm{B}} - \mu \gtrsim 2{E_{{\textbf{\itshape{k}}}}} + 2\gamma + 2T$. Note that there are such ${\textbf{\itshape{k}}}$-resions whenever the condition (II) is fulfilled. As a result, we can obtain $p_{{\textbf{\itshape{k}}}}^0 \cong 0$ and $n_{\mathrm{e(h)},{\textbf{\itshape{k}}}}^0 \cong f_{\mathrm{e(h)},{\textbf{\itshape{k}}}}$ which recovers the MSBE. However, we stress that the condition (II$'$) depends on the wave number ${\textbf{\itshape{k}}}$; there remain ${\textbf{\itshape{k}}}$-regions still described by the BCS theory. The MSBE and the BCS theory are, thus, coupled with each other in a strict sense. In this context, the lasing can be referred to as the BCS-coupled lasing when this viewpoint is emphasized. At the same time, the physical meaning of $\mu$ changes into the oscillating frequency of the laser action. Second thresholds, band renormalization, and gain spectra {#sec:Results} ========================================================= Figure \[Results\] shows the number of coherent photons in the cavity $|a_{0}|^2$ and the frequancy $\mu$ as a function of $\mu_{\mathrm{B}}$ calculated by our formalism.[^9] Plots are colour coded by red (blue) when the quasi-equilibrium condition (I) (the lasing condition (II)) is satisfied, while by green when neither of the conditions is satisfied. In Fig. \[Results\](a), $|a_{0}|^2$ arises with increasing $\mu_{\mathrm{B}}$, the point of which is called the first threshold. In this situation, the system is in quasi-equilibrium regime (red) and $\mu$ is around the lower polariton level[^10] $E_{\mathrm{LP}}$ in Fig. \[Results\](b). The first threshold therefore means that the exciton-polariton BEC is caused because the chemical potential of the system reaches the lowest energy of the exciton polariton, $E_{\mathrm{LP}}$. With further increase of $\mu_{\mathrm{B}}$, the system changes from the quasi-equilibrium regime (red) into the lasing regime (blue) through a crossover regime (green). Around the crossover regime in Fig. \[Results\](a), a second threshold can be seen where the number of coherent photons grows rapidly again. $\mu$ is then blue-shifted from $E_{\mathrm{LP}}$ into the bare cavity level $E_{\mathrm{cav}}$. Furthermore, the kinetic hole burning can be seen in the distribution function of electrons $\nek$ (the blue arrow in the inset to Fig. \[Results\](a)). These resuls demonstrate that the exciton-polariton BEC has smoothly changed into the semiconductor laser with the second threshold. In experiments [@Balili09; @Nelsen09; @Dang98; @Tempel12-1; @Tempel12-2; @Tsotsis12; @Kammann12], the second threshold and the blue shift has been reported since more than 10 years ago, the mechanism of which has been attributed to a shift into the weak coupling regime due to dissociations of Coulomb-bound e-h pairs (excitons); the lasing phase is then achieved as a result. However, there is no convincing discussion why such dissociations lead to nonequilibration essential for lasing. ![ Numerical results of (a) the coherent photon number in the cavity $|a_{0}|^2$ and (b) the frequency $\mu$ as a function of $\mu_{\mathrm{B}}$. Plots are colour coded by red and blue when satisfying the quasi-equilibrium condition (I) and the lasing condition (II), respectively. Green colours are used when neither of them are satisfied. $\mu$ represents the chemical potential in the quasi-equilibrium regime (red) but the laser frequency in the lasing regime (blue). Inset: the distribution function of electrons $n_{\mathrm{e},{\textbf{\itshape{k}}}}$ (black) and the polarization $\pk$ (red). In the lasing regime (B), a characteristic dip can be seen in the distribution (the bule arrow), which is known as one of the signatures of lasing and called the kinetic hole burning. []{data-label="Results"}](Results.eps){width="0.70\linewidth"} According to our formalism, this empirical picture can be investigated and shown to be incorrect. This is because, even in the lasing regime, there are gaps around $\pm\mu/2$ in the renormalized band structure as shown in the left of Fig. \[Gain\]. An analytical form of $A_{11(22)}(\nu;{\textbf{\itshape{k}}})$, obtained by Eqs. (\[eq:GR\]) and (\[eq:A\]), enables us to conveniently study the renormalized band: $$A_{11(22)}(\nu;{\textbf{\itshape{k}}}) = 2|u_{{\textbf{\itshape{k}}}}|^2\frac{\gamma}{(\nu-{\tilde{\xi}_{\mathrm{eh},{\textbf{\itshape{k}}}}}^{-} \mp {E_{{\textbf{\itshape{k}}}}})^2+\gamma^2} +2|v_{{\textbf{\itshape{k}}}}|^2\frac{\gamma}{(\nu-{\tilde{\xi}_{\mathrm{eh},{\textbf{\itshape{k}}}}}^{-} \pm {E_{{\textbf{\itshape{k}}}}})^2+\gamma^2}. \label{eq:A2}$$ Here, $u_{{\textbf{\itshape{k}}}}$ and $v_{{\textbf{\itshape{k}}}}$ are the Bogoliubov coefficients defined as $$u_{{\textbf{\itshape{k}}}} \equiv \sqrt{\frac{1}{2}+\frac{{\tilde{\xi}_{\mathrm{eh},{\textbf{\itshape{k}}}}}^{+}}{2{E_{{\textbf{\itshape{k}}}}}}}, \hspace{0.5cm} v_{{\textbf{\itshape{k}}}} \equiv e^{\ii\theta_{{\textbf{\itshape{k}}}}}\sqrt{\frac{1}{2}-\frac{{\tilde{\xi}_{\mathrm{eh},{\textbf{\itshape{k}}}}}^{+}}{2{E_{{\textbf{\itshape{k}}}}}}}, \label{eq:Bogoliubov}$$ with $\theta_{{\textbf{\itshape{k}}}} \equiv \arg({\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}})$. These equations have remarkable similarities to the BCS theory in superconductors [@Abrikosov75; @Yamaguchi13]. Therefore, it is clear that the gaps are opened around $\pm\mu/2$ with the magnitude of $\min[2{E_{{\textbf{\itshape{k}}}}}]$ when ${\tilde{\xi}_{\mathrm{eh},{\textbf{\itshape{k}}}}}^{-}=0$ i.e. ${\epsilon_{\mathrm{e},{\textbf{\itshape{k}}}}} = {\epsilon_{\mathrm{h},{\textbf{\itshape{k}}}}}$ with $\mu^{\mathrm{B}}_{\mathrm{e}} = \mu^{\mathrm{B}}_{\mathrm{h}}$. Note, however, that the unknown variables contained in Eqs. (\[eq:A2\]) and (\[eq:Bogoliubov\]) are determined by the BEC-BCS-LASER crossover theory (Eqs. (\[eq:MSBE\_a0\])-(\[eq:MSBE\_nehk\]) with Eqs. (\[eq:RTA’\])-(\[eq:A\])) rather than the BCS theory. In the BCS phase, the existence of the gap around $\pm\mu/2$ means the formation of Cooper pairs around the Fermi level because $\pm\mu/2$ is equivalent to the Fermi level. In contrast, in the lasing phase, $\pm\mu/2$ corresponds to the laser frequency.[^11] Thus, the gap indicates the formation of bound e-h pairs by mediating photons around the laser frequency. The semiconductor laser in Fig. \[BEC-BCS-LASER\] is drawn along this picture, where the e-h pairs are explicitly depicted. ![Left; A renormalized band structure in a lasing phase (the point B in Fig. \[Results\]). The gaps are opened around $\pm\mu/2$ with the magnitude of $\min[2{E_{{\textbf{\itshape{k}}}}}]$. Right; Optical gain spectra for the exciton-polariton BEC (A) and for the lasing phase (B). Panels (A) and (B) correspond to the point A and B in Fig. \[Results\], respectively. Aqua (pink) represents the gain (absorption). Panels (A) and (B) are reproduced from [@Yamaguchi13]. []{data-label="Gain"}](Gain.eps){width="0.80\linewidth"} Such a “lasing gap” is, at least in principle, measureable in the optical gain spectrum $G(\omega)$ by irradiating probe light with frequency $\omega$ because $G(\omega)$ is strongly affected by the renormalized band structure in general. As a result, in the gain spectrum of the lasing phase (Fig. \[Gain\](B)), there appears a transparent region originating from the gap. The optical gain spectrum is thus one of important ways for the verification of the lasing gap. In addition, we note that behaviors of the gain spectra vary drastically when the exciton-polariton BEC is changed to the laser phase. By comparing Figs. \[Gain\](A) and (B), for example, one can find only absorption but no gain in Fig. \[Gain\](A). This is mainly because there is no ${\textbf{\itshape{k}}}$-region with inverted population $N_{k}>0$ $(\Leftrightarrow n_{\mathrm{h},k} = n_{\mathrm{e},k} > 0.5)$ in Fig. \[Results\](A). In contrast, optical gain is caused in Fig. \[Gain\](B) because there are ${\textbf{\itshape{k}}}$-regions with $N_{{\textbf{\itshape{k}}}} > 0$ in Fig. \[Results\](B). Thus, the existence of the gain after the second threshold gives us important information to identify the phases in the system.[^12] Conclusions and perspectives {#sec:Conclusions} ============================ In this contribution, we have presented a brief explanation of the BCS theory and the MSBE in the exciton-polariton system, to highlight their similarities and differences. We have then shown a framework of describing the BCS theory (the BEC and BCS phases) and the MSBE (the semiconductor laser) in a unified way. As a result, the existence of bound e-h pairs in the lasing phase as well as the lasing gap have been pointed out. The results presented here are the physics elucidated for the first time by considering the BEC, BCS, and Laser phases in a unified way. However, for example, effects of spontaneous emission [@Scully97] and pure dephasing [@Yamaguchi12-2] are still unclear. In this respect, further studies are needed for a full understanding of this system. Experimantal studies are also important, in particular, in a high density regime [@Balili09; @Nelsen09; @Dang98; @Tempel12-1; @Tempel12-2; @Tsotsis12; @Kammann12; @Horikiri13]. Although we have focused on the exciton-polariton system in this contribution, we finally would like to emphasize that this system has a close relationship with superconductors and the Feshbach resonance in cold atom systems because interacting Fermi and Bose particles play important roles in the formation of ordered phases. In this sense, it would be interesting to study the lasing gap by terahertz pulses in a manner similar to superconductors [@Matsunaga13; @Papenkort07]. Inclusions of the e-h center-of-mass fluctuations with mass imbalance are also important, as discussed in the cold atom systems [@Hanai13-1], because these effects cannot be taken into account within the mean-field approximation. We further note that fundamental problems of the nonequilibrium statistical physics are also included in this system in the sense of providing a bridge between the equilibrium and the nonequilibrium phases. We hope that our approach also stimulates new studies in a wide range of such fields. The authors are grateful to K. Kamide, R. Nii, Y. Yamamoto, T. Horikiri, Y. Shikano, Y. Matsuo, T. Yuge, and M. Bamba for fruitful discussions. This work is supported by the JSPS through its FIRST Program, and DYCE, KAKENHI No. 20104008. Appendix I: Excitonic effects in the low density limit {#appendix-i-excitonic-effects-in-the-low-density-limit .unnumbered} ====================================================== In semiconductor exciton-polariton systems, the excitonic effects play quite important roles in the formation of Coulomb-bound e-h pairs (excitons) and exciton-polaritons. In this Appendix, we, therefore, confirm the excitonic effects in our formalism [^13]. For this purpose, we now assume that the density of electrons and holes are sufficiently low ($n_{\mathrm{e},{\textbf{\itshape{k}}}}, n_{\mathrm{h},{\textbf{\itshape{k}}}} \ll 1$ or $N_{{\textbf{\itshape{k}}}} \cong -1$) with no pumping and loss ($\gamma = 0$ and $\kappa = 0$). Under this condition, $2\tilde\xi^{+}_{\mathrm{eh},{\textbf{\itshape{k}}}}$ in Eq. (\[eq:MSBE\_pk\]) can be written as $$\begin{aligned} 2\tilde\xi^{+}_{\mathrm{eh},{\textbf{\itshape{k}}}} &= \epsilon_{\mathrm{e},{\textbf{\itshape{k}}}} + \epsilon_{\mathrm{h},{\textbf{\itshape{k}}}} - \mu = \frac{k^2}{2m_{\mathrm{r}}}+E_g-\mu, \label{eq:temp1}\end{aligned}$$ where $1/m_{\mathrm{r}}=1/m_{\mathrm{e}}+1/m_{\mathrm{h}}$. Then, Eqs. (\[eq:MSBE\_a0\]) and (\[eq:MSBE\_pk\]) can be described as $$\begin{aligned} 0&=-(\epsilon_{\mathrm{ph},0}-\mu)a_0+ g \sum_{{\textbf{\itshape{k}}}} p_{{\textbf{\itshape{k}}}}, \label{eq:a0_}\\ 0&=-\left( \frac{k^2}{2m_{\mathrm{r}}}+E_g-\mu \right) p_{{\textbf{\itshape{k}}}} + g^{*}a_0 + \sum_{{\textbf{\itshape{k}}}'}U'_{{\textbf{\itshape{k}}}'-{\textbf{\itshape{k}}}}p_{{\textbf{\itshape{k}}}'}, \label{eq:pk_}\end{aligned}$$ where the definition of ${\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}} \equiv g^{*}a_0 + \sum_{{\textbf{\itshape{k}}}'}U'_{{\textbf{\itshape{k}}}-{\textbf{\itshape{k}}}'}p_{{\textbf{\itshape{k}}}'}$ is used. Especially, for $g = 0$ in Eq. , we obtain $$\begin{aligned} \frac{k^2}{2m_{\mathrm{r}}} p_{{\textbf{\itshape{k}}}} - \sum_{{\textbf{\itshape{k}}}'}U'_{{\textbf{\itshape{k}}}'-{\textbf{\itshape{k}}}}p_{{\textbf{\itshape{k}}}'}=-(E_g-\mu) p_{{\textbf{\itshape{k}}}}, \label{eq:Wannier}\end{aligned}$$ which is nothing but the Schr$\ddot{\mathrm{o}}$dinger equation in ${\textbf{\itshape{k}}}$-space for the single exciton bound state [@Yamaguchi12; @Comte82; @HaugKoch]. This means that the Coulomb-bound e-h pairs (excitons) can be formed in the low density limit in the presented formalism. In such a case, $p_{{\textbf{\itshape{k}}}}$ can be described by the bound state e-h pair wave-function $\phi_{{\textbf{\itshape{k}}}}$ ($p_{{\textbf{\itshape{k}}}} = \eta \phi_{{\textbf{\itshape{k}}}}$ with $\sum_{{\textbf{\itshape{k}}}}|\phi_{{\textbf{\itshape{k}}}}|^2 = 1$) with $\mu = E_{\mathrm{ex}}$, where $E_{\mathrm{ex}}$ is the energy level of the exciton and the binding energy corresponds to $E_{\mathrm{g}} - E_{\mathrm{ex}}$. The formation of the exciton is, thus, includeed in the theory. Although $p_{{\textbf{\itshape{k}}}}$ is changed from the exciton wave-function $\phi_{{\textbf{\itshape{k}}}}$ by the photon-mediated attraction in the case of $g \neq 0$, it is instructive to consider the case where such an effect is not so large. In this limit, by substituting $p_{{\textbf{\itshape{k}}}} \cong \eta\phi_{{\textbf{\itshape{k}}}}$ into Eqs.  and , we obtain $$\begin{aligned} 0&=(\mu - E_{\mathrm{cav}})a_0 + g_{\mathrm{ex}}\eta, \\ 0&=(\mu - E_{\mathrm{ex}})\eta + g_{\mathrm{ex}}^{*}a_0, \label{eq:eigen}\end{aligned}$$ where $g_{\mathrm{ex}} \equiv g\sum_{{\textbf{\itshape{k}}}}\phi_{{\textbf{\itshape{k}}}}=g\phi_{\mathrm{ex}}({\textbf{\itshape{r}}}=0)$ is the coupling constant renormalized by the exciton wave-function. Then, $\mu$ is given by one of the eigenvalues of these two coupled equations, which are the eigen-energies of the upper and lower polaritons: $$\begin{aligned} E_{\mathrm{UP/LP}} = \frac{E_{\mathrm{cav}}+E_{\mathrm{ex}} \pm \sqrt{(E_{\mathrm{cav}}-E_{\mathrm{ex}})^2+4|g_{\mathrm{ex}}|^2}}{2}. \label{eq:UPLP}\end{aligned}$$ Here, $E_{\mathrm{UP}}$ and $E_{\mathrm{LP}}$ in Eq.  are the well-known expressions obtained when the excitons are treated as simple bosons [@HaugKoch]. This means that the formation of exciton-polaritons are also included in the theory. The excitonic effects are, thus, taken into account in our formalism within the mean-field approximation. We note that the procedure shown here is basically the same as Section 2.1.2 in Ref. [@Yamaguchi12]. Appendix II: Proof of equivalence {#appendix-ii-proof-of-equivalence .unnumbered} ================================= In the main text, we have mentioned that the formalism in Ref. [@Yamaguchi12] seems quite different from the MSBE. This formalism can be described by the follwoing simultaneous equations with the unknown variables of ${\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}$, $\nek$, $\nhk$, and $\mu$: $$\begin{gathered} {\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}} = \sum_{{\textbf{\itshape{k}}}'} U^{\mathrm{eff},\kappa}_{{\textbf{\itshape{k}}}',{\textbf{\itshape{k}}}} {\mathnormal{\Delta}_{{\textbf{\itshape{k}}}{'}}} \int^{\infty}_{-\infty} \frac{\dd\nu}{2\pi} L_{{\textbf{\itshape{k}}}'}(\nu) \\ \times \left \{ (F_{\mathrm{e}}^{\mathrm{B}}(\nu) + F_{\mathrm{h}}^{\mathrm{B}}(\nu)) (\nu-\tilde\xi^{-}_{\mathrm{eh},{\textbf{\itshape{k}}}'}) + (F_{\mathrm{e}}^{\mathrm{B}}(\nu) - F_{\mathrm{h}}^{\mathrm{B}}(\nu)) (\tilde\xi^{+}_{\mathrm{eh},{\textbf{\itshape{k}}}'} + \ii\gamma) \right\}, \label{eqa:Delta}\end{gathered}$$ $$n_{\mathrm{e(h)},{\textbf{\itshape{k}}}} = \frac{1}{2} \mp \int^{\infty}_{-\infty} \frac{\dd\nu}{2\pi} L_{{\textbf{\itshape{k}}}}(\nu) \left\{ F_{\mathrm{e(h)}}^{\mathrm{B}}(\nu)[(\nu\pm\tilde\xi_{\mathrm{h(e)},{\textbf{\itshape{k}}}})^2+\gamma^2] + F_{\mathrm{h(e)}}^{\mathrm{B}}(\nu)|{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}|^2 \right\}, \label{eqa:nehk}$$ where $U_{{\textbf{\itshape{k}}}',{\textbf{\itshape{k}}}}^{\mathrm{eff},\kappa} \equiv |g|^2/(\xi_{\mathrm{ph},0}-\ii\kappa)+U'_{{\textbf{\itshape{k}}}'-{\textbf{\itshape{k}}}}$ and $$L_{{\textbf{\itshape{k}}}}(\nu) \equiv \frac{\gamma}{[(\nu-\tilde\xi^{-}_{\mathrm{eh},{\textbf{\itshape{k}}}}-{E_{{\textbf{\itshape{k}}}}})^{2}+\gamma^2][(\nu-\tilde\xi^{-}_{\mathrm{eh},{\textbf{\itshape{k}}}}+{E_{{\textbf{\itshape{k}}}}})^{2}+\gamma^2]}. \label{eqa:L}$$ $F_{\mathrm{e}}^{\mathrm{B}}(\nu)$ and $F_{\mathrm{h}}^{\mathrm{B}}(\nu)$ are respectively defined as $$\begin{aligned} F_{\mathrm{e}}^{\mathrm{B}}(\nu) & \equiv & \tanh \left( \frac{\beta[\nu - \mu^{\mathrm{B}}_{\mathrm{e}} + \mu/2]}{2} \right) = 1-2f^{\mathrm{B}}_{\mathrm{e}}(\nu),\\ \label{eqa:Fe} F_{\mathrm{h}}^{\mathrm{B}}(\nu) & \equiv & \tanh \left( \frac{\beta[\nu + \mu^{\mathrm{B}}_{\mathrm{h}} - \mu/2]}{2} \right) = 2f^{\mathrm{B}}_{\mathrm{h}}(-\nu)-1. \label{eqa:Fh}\end{aligned}$$ In this Appendix II, therefore, we prove that Eqs. (\[eqa:Delta\])-(\[eqa:Fh\]) are equivalent to Eqs. (\[eq:MSBE\_a0\])-(\[eq:MSBE\_nehk\]) with Eqs. (\[eq:RTA’\])-(\[eq:A\]) under the steady-state condition $\partial_t{\langle \hat{O} \rangle} = 0$. For this purpose, we here note that the following sum rule is satisfied for the single-partice spectral function: $$\int^{\infty}_{-\infty}\frac{\dd\nu}{2\pi}A_{\alpha\alpha'}(\nu;{\textbf{\itshape{k}}}) = \delta_{\alpha,\alpha'}. \label{eqa:SumRule}$$ This relation can be confirmed by the direct integration of $A_{\alpha\alpha'}(\nu;{\textbf{\itshape{k}}})$ described by elements of a matrix $$\begin{aligned} A(\nu;{\textbf{\itshape{k}}}) &= \frac{-2}{|D_{{\textbf{\itshape{k}}}}(\nu)|^2}\left( \begin{array}{cc} \operatorname{Im}[D^{*}_{{\textbf{\itshape{k}}}}(\nu)(\nu+\tilde\xi_{\mathrm{h},{\textbf{\itshape{k}}}}+\ii\gamma)] & \operatorname{Im}[D_{{\textbf{\itshape{k}}}}(\nu)]{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}} \\ \operatorname{Im}[D_{{\textbf{\itshape{k}}}}(\nu)]{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}^{*} & \operatorname{Im}[D^{*}_{{\textbf{\itshape{k}}}}(\nu)(\nu-\tilde\xi_{\mathrm{e},{\textbf{\itshape{k}}}}+\ii\gamma)] \\ \end{array}\right) \nonumber \\ &= \frac{-2\gamma}{|D_{{\textbf{\itshape{k}}}}(\nu)|^2}\left( \begin{array}{cc} -[(\nu+\tilde\xi_{\mathrm{h},{\textbf{\itshape{k}}}})^2+\gamma^2+|{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}|^2] & 2{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}(\nu-\tilde\xi^{-}_{\mathrm{eh},{\textbf{\itshape{k}}}}) \\ 2{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}^{*}(\nu-\tilde\xi^{-}_{\mathrm{eh},{\textbf{\itshape{k}}}}) & -[(\nu - \tilde\xi_{\mathrm{e},{\textbf{\itshape{k}}}})^2+\gamma^2+|{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}|^2] \\ \end{array}\right), \label{eqa:A}\end{aligned}$$ which is obtained from the definition of Eq. (\[eq:A\]) with Eq. (\[eq:GR\]): $$G^{\mathrm{R}}_{{\textbf{\itshape{k}}}}(\nu) = \frac{1}{|D_{{\textbf{\itshape{k}}}}(\nu)|^2}\left( \begin{array}{cc} D^{*}_{{\textbf{\itshape{k}}}}(\nu)(\nu+\tilde\xi_{\mathrm{h},{\textbf{\itshape{k}}}}+\ii\gamma) & -D^{*}_{{\textbf{\itshape{k}}}}(\nu){\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}} \\ -D^{*}_{{\textbf{\itshape{k}}}}(\nu){\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}^{*} & D^{*}_{{\textbf{\itshape{k}}}}(\nu)(\nu-\tilde\xi_{\mathrm{e},{\textbf{\itshape{k}}}}+\ii\gamma) \\ \end{array}\right), \label{eqa:GR}$$ where $$D_{{\textbf{\itshape{k}}}}(\nu) \equiv (\nu-\tilde\xi^{-}_{\mathrm{eh},{\textbf{\itshape{k}}}} + {E_{{\textbf{\itshape{k}}}}} + \ii\gamma)(\nu-\tilde\xi^{-}_{\mathrm{eh},{\textbf{\itshape{k}}}} - {E_{{\textbf{\itshape{k}}}}} + \ii\gamma).$$ The diagonal element $A_{11(22)}(\nu;{\textbf{\itshape{k}}})$ can then be described as Eq. (\[eq:A2\]). In the following, by using thse expressions, Eqs. (\[eqa:Delta\]) and (\[eqa:nehk\]) are derived from Eqs. (\[eq:MSBE\_a0\])-(\[eq:MSBE\_nehk\]) with Eqs. (\[eq:RTA’\])-(\[eq:A\]). First, we discuss $\nu$-integral forms of the population inversion $N_{{\textbf{\itshape{k}}}} \equiv \nek + \nhk -1$ and the polarization function $\pk$ because Eqs. (\[eqa:Delta\]) and (\[eqa:nehk\]) are described by the integration with respect to $\nu$. From Eqs. (\[eq:MSBE\_nehk\]) and (\[eq:RTA’\]) with $\partial_t n_{\mathrm{e(h)},{\textbf{\itshape{k}}}}=0$, we obtain $$N_{{\textbf{\itshape{k}}}} = -\frac{2}{\gamma}\operatorname{Im}[{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}\pk^{*}] + \int^{\infty}_{-\infty}\frac{\dd\nu}{2\pi} \left\{ f^{\mathrm{B}}_{\mathrm{e}}(\nu)A_{11}(\nu;{\textbf{\itshape{k}}}) -(1-f^{\mathrm{B}}_{\mathrm{h}}(-\nu))A_{22}(\nu;{\textbf{\itshape{k}}}) \right\}. \label{eqa:PopInv1}$$ where Eq. (\[eqa:SumRule\]) is used. In a similar manner, from Eqs. (\[eq:MSBE\_pk\]) and (\[eq:RTA’\]) with $\partial_t \pk=0$, $$\begin{aligned} \pk = &- \frac{{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}}{2(\tilde\xi^{+}_{\mathrm{eh},{\textbf{\itshape{k}}}} - \ii\gamma)}N_{{\textbf{\itshape{k}}}} \nonumber \\ &+ \frac{\gamma}{\tilde\xi^{+}_{\mathrm{eh},{\textbf{\itshape{k}}}} - \ii\gamma} \int^{\infty}_{-\infty}\frac{\dd\nu}{2\pi}\left\{ [1-f^{\mathrm{B}}_{\mathrm{h}}(-\nu)]G^{\mathrm{R}}_{12}(\nu;{\textbf{\itshape{k}}}) -f^{\mathrm{B}}_{\mathrm{e}}(\nu)G^{\mathrm{R}*}_{21}(\nu;{\textbf{\itshape{k}}}) \right\}. \label{eqa:pk1}\end{aligned}$$ Therefore, with Eqs. (\[eqa:A\]) and (\[eqa:GR\]), substitution of Eq. (\[eqa:pk1\]) into Eq. (\[eqa:PopInv1\]) yields $$\begin{gathered} N_{{\textbf{\itshape{k}}}}=\int^{\infty}_{-\infty}\frac{\dd\nu}{2\pi}\frac{2\gamma}{|D_{{\textbf{\itshape{k}}}}(\nu)|^2} \left \{ f^{\mathrm{B}}_{\mathrm{e}}(\nu) [(\nu+\tilde\xi_{\mathrm{h},{\textbf{\itshape{k}}}})^2-|{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}|^2+\gamma^2] \right. \\ \left. +[f^{\mathrm{B}}_{\mathrm{h}}(-\nu)-1] [(\nu-\tilde\xi_{\mathrm{e},{\textbf{\itshape{k}}}})^2-|{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}|^2+\gamma^2] \right \}. \label{eqa:PopInv2}\end{gathered}$$ By substituting Eq. (\[eqa:PopInv2\]) into Eq. (\[eqa:pk1\]), we also find $$\pk={\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}} \int^{\infty}_{-\infty}\frac{\dd\nu}{2\pi} \frac{2\gamma}{|D_{{\textbf{\itshape{k}}}}(\nu)|^2} \left \{ [f^{\mathrm{B}}_{\mathrm{h}}(-\nu)-1](\nu - \tilde\xi_{\mathrm{e},{\textbf{\itshape{k}}}} - \ii\gamma) - f^{\mathrm{B}}_{\mathrm{e}}(\nu)(\nu + \tilde\xi_{\mathrm{h},{\textbf{\itshape{k}}}} + \ii\gamma) \right \}. \label{eqa:pk2}$$ Although the derivation of Eqs. (\[eqa:PopInv2\]) and (\[eqa:pk2\]) is straightforward, the following equations would be useful in the derivation: $$\begin{aligned} \pm|{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}|^2\operatorname{Im}[D_{{\textbf{\itshape{k}}}}(\nu)(\tilde\xi^{+}_{\mathrm{eh},{\textbf{\itshape{k}}}}-\ii\gamma)] + &([\tilde\xi^{+}_{\mathrm{eh},{\textbf{\itshape{k}}}}]^2+\gamma^2)\operatorname{Im}[D^{*}_{{\textbf{\itshape{k}}}}(\nu)(\nu \mp \tilde\xi_{\mathrm{e(h)},{\textbf{\itshape{k}}}}+\ii\gamma)] \nonumber \\ &= -\gamma({E_{{\textbf{\itshape{k}}}}}^2 + \gamma^2)((\nu \mp \tilde\xi_{\mathrm{e(h)},{\textbf{\itshape{k}}}})^2-|{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}|^2+\gamma^2), \\ (\nu+\tilde\xi_{\mathrm{h}})^2 - |{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}|^2 + \gamma^2 &-D_{{\textbf{\itshape{k}}}}(\nu) = 2 (\tilde\xi^{+}_{\mathrm{eh},{\textbf{\itshape{k}}}} - \ii\gamma)(\nu + \tilde\xi_{\mathrm{h}} + \ii\gamma), \\ (\nu-\tilde\xi_{\mathrm{e}})^2 - |{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}|^2 + \gamma^2 &-D^{*}_{{\textbf{\itshape{k}}}}(\nu) = -2 (\tilde\xi^{+}_{\mathrm{eh},{\textbf{\itshape{k}}}} - \ii\gamma)(\nu - \tilde\xi_{\mathrm{e}} - \ii\gamma). \end{aligned}$$ The $\nu$-integral forms of $N_{{\textbf{\itshape{k}}}}$ and $\pk$ are thus obtained as Eqs. (\[eqa:PopInv2\]) and (\[eqa:pk2\]), respectively. These expressions are helpful to find the $\nu$-integral forms of ${\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}$, $\nek$, and $\nhk$, which turn out to be the same as Eqs. (\[eqa:Delta\])-(\[eqa:Fh\]), as shown below. From the definition of ${\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}} \equiv g^{*}a_0 + \sum_{{\textbf{\itshape{k}}}'}U'_{{\textbf{\itshape{k}}}-{\textbf{\itshape{k}}}'}p_{{\textbf{\itshape{k}}}'}$ and Eq. (\[eq:MSBE\_a0\]) with $\partial_t\a0 = 0$, ${\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}$ can be described as $${\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}} = \sum_{{\textbf{\itshape{k}}}'} \left\{ \frac{|g|^2}{\xi_{\mathrm{ph},0}-\ii\kappa}+U'_{{\textbf{\itshape{k}}}'-{\textbf{\itshape{k}}}} \right\}p_{{\textbf{\itshape{k}}}'} = \sum_{{\textbf{\itshape{k}}}'} U_{{\textbf{\itshape{k}}}',{\textbf{\itshape{k}}}}^{\mathrm{eff},\kappa} p_{{\textbf{\itshape{k}}}'}. \label{eqa:temp2}$$ Therefore, after the substitution of Eq. (\[eqa:pk2\]) into Eq. (\[eqa:temp2\]), we obtain $$\begin{gathered} {\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}} = \sum_{{\textbf{\itshape{k}}}'} U_{{\textbf{\itshape{k}}}',{\textbf{\itshape{k}}}}^{\mathrm{eff},\kappa} \Delta_{{\textbf{\itshape{k}}}'} \int^{\infty}_{-\infty}\frac{\dd\nu}{2\pi} L_{{\textbf{\itshape{k}}}'}(\nu) \left \{ [F^{\mathrm{B}}_{\mathrm{h}}(\nu)-1][\nu - \tilde\xi_{\mathrm{e},{\textbf{\itshape{k}}}'} - \ii\gamma] \right. \\ \left. + [F^{\mathrm{B}}_{\mathrm{e}}(\nu)-1][\nu + \tilde\xi_{\mathrm{h},{\textbf{\itshape{k}}}'} + \ii\gamma] \right\},\end{gathered}$$ where the definitions of Eqs. (\[eqa:L\])-(\[eqa:Fh\]) are used with $L_{{\textbf{\itshape{k}}}}(\nu) = \gamma/|D_{{\textbf{\itshape{k}}}}(\nu)|^2$. This equation can be rewritten as $$\begin{gathered} {\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}} = \sum_{{\textbf{\itshape{k}}}'} U^{\mathrm{eff},\kappa}_{{\textbf{\itshape{k}}}',{\textbf{\itshape{k}}}} {\mathnormal{\Delta}_{{\textbf{\itshape{k}}}{'}}} \int^{\infty}_{-\infty} \frac{\dd\nu}{2\pi} L_{{\textbf{\itshape{k}}}'}(\nu) \left \{ (F_{\mathrm{e}}^{\mathrm{B}}(\nu) + F_{\mathrm{h}}^{\mathrm{B}}(\nu) - 2) (\nu-\tilde\xi^{-}_{\mathrm{eh},{\textbf{\itshape{k}}}'}) \right. \\ + \left. (F_{\mathrm{e}}^{\mathrm{B}}(\nu) - F_{\mathrm{h}}^{\mathrm{B}}(\nu)) (\tilde\xi^{+}_{\mathrm{eh},{\textbf{\itshape{k}}}'} + \ii\gamma) \right\}. \label{eqa:Delta2}\end{gathered}$$ By noting $\int \frac{\dd\nu}{2\pi} L_{{\textbf{\itshape{k}}}'}(\nu) (\nu-\tilde\xi^{-}_{\mathrm{eh},{\textbf{\itshape{k}}}'}) = 0$, we thus find that Eq. (\[eqa:Delta2\]) is equivalent to Eq. (\[eqa:Delta\]). Our remaining task is now to derive the $\nu$-integral forms of $\nek$ and $\nhk$. By multiplying ${\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}$ by the complex conjugate of Eq. (\[eqa:pk2\]), $$\frac{1}{\gamma}\operatorname{Im}[{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}\pk^{*}] = |{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}|^2 \int^{\infty}_{-\infty}\frac{\dd\nu}{2\pi} \frac{2\gamma}{|D_{{\textbf{\itshape{k}}}}(\nu)|^2} \{ f^{\mathrm{B}}_{\mathrm{e}}(\nu) + f^{\mathrm{B}}_{\mathrm{h}}(-\nu) -1 \}, \label{eqa:temp1}$$ can be obtained. The substitution of Eq. (\[eqa:temp1\]) into Eq. (\[eq:MSBE\_nehk\]) with $\partial_t n_{\mathrm{e(h)},{\textbf{\itshape{k}}}} = 0$, then, yields $$\begin{gathered} n_{\mathrm{e(h)},{\textbf{\itshape{k}}}} = \int^{\infty}_{-\infty} \frac{\dd\nu}{2\pi} \frac{2\gamma}{|D_{{\textbf{\itshape{k}}}}(\nu)|^2} \left \{ [(\nu \pm \tilde\xi_{\mathrm{h(e)},{\textbf{\itshape{k}}}})^2 + \gamma^2]f^{\mathrm{B}}_{\mathrm{e(h)}}(\pm\nu) \right. \\ \left. -|{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}|^2[ f^{\mathrm{B}}_{\mathrm{h(e)}}(\mp\nu) -1] \right\},\end{gathered}$$ which can be rewritten as $$\begin{gathered} n_{\mathrm{e(h)},{\textbf{\itshape{k}}}} = \int^{\infty}_{-\infty} \frac{\dd\nu}{2\pi} L_{{\textbf{\itshape{k}}}}(\nu) \left \{ [(\nu \pm \tilde\xi_{\mathrm{h(e)},{\textbf{\itshape{k}}}})^2 + \gamma^2 + |{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}|^2] \right. \\ \left. \mp [(\nu \pm \tilde\xi_{\mathrm{h(e)},{\textbf{\itshape{k}}}})^2 + \gamma^2]F^{\mathrm{B}}_{\mathrm{e(h)}}(\nu) \mp |{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}|^2F^{\mathrm{B}}_{\mathrm{h(e)}}(\nu) \right\}. \label{eqa:nehk2}\end{gathered}$$ We then find that Eq. (\[eqa:nehk2\]) is identical to Eq. (\[eqa:nehk\]) because, from Eqs. (\[eqa:SumRule\]) and (\[eqa:A\]), $$\int^{\infty}_{-\infty} \frac{\dd\nu}{2\pi} L_{{\textbf{\itshape{k}}}}(\nu)[(\nu \pm \tilde\xi_{\mathrm{h(e)},{\textbf{\itshape{k}}}})^2 + \gamma^2 + |{\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}|^2] = \frac{1}{2}.$$ Thus, we have shown that Eqs. (\[eqa:Delta\]) and (\[eqa:nehk\]) are derived from Eqs. (\[eq:MSBE\_a0\])-(\[eq:MSBE\_nehk\]) with Eqs. (\[eq:RTA’\])-(\[eq:A\]). This means that the formalism in Ref. [@Yamaguchi12] is equivalent to Eqs. (\[eq:MSBE\_a0\])-(\[eq:MSBE\_nehk\]) with Eqs. (\[eq:RTA’\])-(\[eq:A\]). [^1]: In this contribution, the terms ‘lasing’ and ‘laser’ are used only when the condensation is inherently governed by non-equilibrium kinetics, according to [@Bajoni08; @Deng03]. In other words, thermodynamic variables of the system, such as temperatures, cannot be defined for lasing phases. However, we note that these terms are occasionally used even for a condensation dominated by the thermodynamics of the electron-hole-photon system [@Kasprzak08] if the interest is in fabricating a device. [^2]: We note that theories for describing dynamics of equilibrium phases, e.g. the Gross-Pitaevskii equation, can asymptotically be derived from Maxwell-Bloch equations [@Berloff13] even though these theories still do not recover equilibrium statistical approaches. [^3]: We note, however, that physical guesses are required for the determination of what operator(s) should be chosen as $\hat{O}$, e.g. from experiments. [^4]: Here, $-1 \le N_{{\textbf{\itshape{k}}}} \le +1$ because $0 \le n_{\mathrm{e(h)},{\textbf{\itshape{k}}}} \le 1$. Population inversion is formed in ${\textbf{\itshape{k}}}$-resions with $N_{{\textbf{\itshape{k}}}}>0$. [^5]: For $\kappa = \gamma = 0$, the steady state of the MSBE becomes identical to the BCS theory if the solution of the BCS theory is chosen as an initial condition in the MSBE because $[\hat{\rho}_{\mathrm{eq}}^{\mathrm{MF}},\hat{H}_{\mathrm{S}}^{\mathrm{MF}}]=0$. However, this is a special case. [^6]: The retarded Green’s funcition is also seen as a kind of density of states. [^7]: Under the condition (I), $f^{\mathrm{B}}_{\mathrm{e(h)}}(\nu)$ in Eq. (\[eq:RTA’\]) can be approximated by the values at $\nu = \pm {E_{{\textbf{\itshape{k}}}}}$ because $A_{\alpha\alpha'}(\nu;{\textbf{\itshape{k}}})$ and $G^{\mathrm{R}}_{\alpha\alpha',{\textbf{\itshape{k}}}}(\nu)$ have peaks around $\nu = \pm {E_{{\textbf{\itshape{k}}}}}$, as seen in Eqs. (\[eq:A2\]) and (\[eqa:GR\]). [^8]: This means that the system is chemically non-equilibrium with the pumping baths even if the system is in quasi-equilibrium. [^9]: In the numerical calculations, the ${\textbf{\itshape{k}}}$-dependence of ${\mathnormal{\Delta}_{{\textbf{\itshape{k}}}}}$ is eliminated by using a contact potential $U'_{{\textbf{\itshape{q}}} \neq 0} = U = 2.66 \times 10^{-10}$ eV with cut-off wave number $k_{\mathrm{c}} = 1.36 \times 10^{9}$ $m^{-1}$. The other parameters are $m_{\mathrm{e}} = m_{\mathrm{h}} = 0.068m_0$ ($m_0$ is the free electron mass), $\mu^{\mathrm{B}}_{\mathrm{e}} = \mu^{\mathrm{B}}_{\mathrm{h}}$, $T = 10 $K, $g = 6.29 \times 10^{-7}$eV, $\gamma = 4$meV, and $\kappa = 100 \mathrm{\mu eV}$. In this context, our calculations are not quantitative but qualitative even though the parameters are taken as realistic as possible. In this situation, the exciton level ($\equiv E_{\mathrm{ex}}$) is formed at 10 meV below $E_{\mathrm{g}}$ ($E_{\mathrm{ex}} = E_{\mathrm{g}} - 10$ meV) and the lower polariton level $E_{\mathrm{LP}}$ is created at 20 meV below $E_{\mathrm{g}}$ ($E_{\mathrm{LP}} = E_{\mathrm{g}} - 20$ meV) under the resonant condition $E_{\mathrm{cav}} = E_{\mathrm{ex}}$ [@Yamaguchi13]. [^10]: For $E_{\mathrm{LP}}$ and $E_{\mathrm{ex}}$, see also Appendix I. Excitonic effects are discussed in the low density limit. [^11]: The origin of the gap is analogous to the Rabi splitting in resonance fluorescence [@Scully97; @Schmitt-Rink88; @Henneberger92; @Yamaguchi13]. [^12]: In fact, the second threshold and the blue-shift can also be caused by a different mechanism even if the system remains in quasi-equilibrium. In this situation, however, the gain spectrum shows only absorption [@Yamaguchi12]. [^13]: Discussion in Appendix I is reproduced from the supplemental material in Ref. [@Yamaguchi13].

--- address: - 'Department of Physics and Astronomy MSC07 4220, 1 University of New Mexico,Albuquerque NM 87131-0001' - 'Physics Division, Los Alamos National Laboratory MS H803, P-23, Los Alamos, NM, 87545, USA' author: - 'N. McFadden' - 'S. R. Elliott' - 'M. Gold' - 'D.E. Fields' - 'K. Rielage' - 'R. Massarczyk' - 'R. Gibbons' bibliography: - 'mybibfile.bib' title: 'Large-Scale, Precision Xenon Doping of Liquid Argon' --- neutrinoless double beta decay, xenon doping, liquid argon, Birk’s constant, Geant4

--- abstract: 'We describe the implementation of total angular momentum dependent pseudopotentials in a plane wave formulation of density functional theory. Our approach thus goes beyond the scalar–relativistic approximation usually made in [*ab initio*]{} pseudopotential calculations and explicitly includes spin–orbit coupling. We outline the necessary extensions and compare the results to available all–electron calculations and experimental data.' author: - 'G. Theurich' - 'N. A. Hill' bibliography: - '../../PhDThesis/MyBib/mybib.bib' title: | Spin–Orbit Coupling in the\ [*ab initio*]{} Pseudopotential Framework --- The [*ab initio*]{} pseudopotential method [@Ihm79p4409; @Payne92p1045; @Kresse96p15] has become a standard tool in many areas of electronic structure calculation. Even magnetic compounds containing $3d$ transition metal ions lie in the realm of the plane wave pseudopotential approach of density functional theory [@Sasaki95p12760; @Moroni97p15629]. In order to obtain high precision results it is necessary to include relativistic effects when calculating the electronic structure of materials containing third row elements [@Bachelet82p2103]. Hence it is now standard procedure to create scalar–relativistic pseudopotentials that include the kinematic relativistic effects (mass–velocity and Darwin term) from the fully relativistic all–electron solution of the atom [@Bachelet82p4199; @Hamann89p2980; @Rappe90p1227; @Troullier91p1993; @Vanderbilt90p7892]. The spin–orbit interaction, however, is only effectively taken into account by the construction of $j$–averaged pseudopotentials for each angular momentum $l$. Thus no spin–orbit splitting is present in the resulting band structure. Although the scalar–relativistic approximation is acceptable in many situations it becomes insufficient in cases where the observed quantities, such as hole effective masses or spin relaxation times, are a direct consequence of the spin–orbit splitting [@Cardona88p1806]. In this paper we report on the implementation of spin–orbit coupling in the pseudopotential scheme. We give the equations required to program the formalism within a generalized spinor approach, and compare the results to experimental data and to all-electron calculations. This is, to our knowledge, the first zeroth order implementation of spin–orbit coupling in the [*ab initio*]{} pseudopotential scheme. Prior publications on this matter always relied on a second variation of the scalar–relativistic zeroth order eigenstates, including spin–orbit coupling to first order in perturbation theory [@Hybertsen86p2920; @Surh91p4286; @Hemstreet93p4238]. Although the fully relativistic treatment of the problem would require a four–current formulation with Dirac spinors it has been shown by Kleinman that a Pauli–like Schrödinger equation captures all relativistic effects to order $\alpha^2$, where $\alpha$ is the fine structure constant [@Kleinman80p2630]. The total ionic pseudopotential to be used is $$\label{j_psp} V_{PS} =\sum_{l,j,m_j}{{| \Phi^{l,j}_{m_j} \rangle}}V_{l,j}{{\langle \Phi^{l,j}_{m_j} |}}\ ,$$ where the ${{| \Phi^{l,j}_{m_j} \rangle}}$ are the total angular momentum eigenfunctions which can be written in terms of the spherical harmonics, $Y^m_l$, and the eigenfunctions of the $z$–component of the Pauli spin operator, ${{| \uparrow \rangle}}$ and ${{| \downarrow \rangle}}$. For $j=l+\frac{1}{2}$, $m_j=m+\frac{1}{2}$ the ${{| \Phi^{l,j}_{m_j} \rangle}}$ equal $$\left(\frac{l+m+1}{2l+1}\right)^\frac{1}{2}{{| Y^m_l \rangle}}{{| \uparrow \rangle}}+ \left(\frac{l-m}{2l+1}\right)^\frac{1}{2}{{| Y^{m+1}_l \rangle}}{{| \downarrow \rangle}}$$ and for $j=l-\frac{1}{2}$, $m_j=m-\frac{1}{2}$ have the form $$\left(\frac{l-m+1}{2l+1}\right)^\frac{1}{2}{{| Y^{m-1}_l \rangle}}{{| \uparrow \rangle}}- \left(\frac{l+m}{2l+1}\right)^\frac{1}{2}{{| Y^m_l \rangle}}{{| \downarrow \rangle}} \label{j_minus}\ .$$ Hence the operator $V_{PS}$ acts in both orbital and spin space. Note that there is only one radial pseudopotential component $V_{l,j}$ with $j=\frac{1}{2}$ for $l=0$ but two with $j=l+\frac{1}{2}$ and $j=l-\frac{1}{2}$ for each $l>0$. The index $m_j$ in equation (\[j\_psp\]) runs from $-j$ to $+j$. It is computationally more efficient to transcribe each term of the semi–local pseudopotential operator $V_{PS}$ into the fully separable Kleinman–Bylander (KB) form [@Kleinman82p1425] $$\label{relKB} V_{KB} = \sum_{i_s, i_a}\sum_{l,j,m_j}\frac{{{| \delta V^{i_s,i_a}_{l,j}\phi^{i_s,i_a}_{l,j,m_j} \rangle}}{{\langle \phi^{i_s,i_a}_{l,j,m_j}\delta V^{i_s,i_a}_{l,j} |}}}{{{\langle \phi^{i_s}_{l,j,m_j} |}}\delta V^{i_s}_{l,j}{{| \phi^{i_s}_{l,j,m_j} \rangle}}}$$ using the solutions of the atomistic pseudopotential problem $${{| \phi^{i_s,i_a}_{l,j,m_j} \rangle}}={{| R^{i_s,i_a}_{l,j} \rangle}}{{| \Phi^{l,j}_{m_j} \rangle}}\ ,$$ where ${{| R^{i_s,i_a}_{l,j} \rangle}}$ is the radial part of the pseudo eigenfunction of atom species $i_s$ at position $r_{i_s,i_a}$. The potential $\delta V^{i_s,i_a}_{l,j}$ is defined as the difference $$\label{minus_local} \delta V^{i_s,i_a}_{l,j}(r) = V_{l,j}(r-r_{i_s,i_a}) - V_{loc}(r-r_{i_s,i_a})\ ,$$ where $V_{loc}(r)$ is an arbitrary local potential that needs to be chosen such that the remaining $\delta V$’s are short ranged. The complete KB pseudopotential operator is thus given as the sum of the local part and the non–local KB operator. To our knowledge all previous pseudopotential calculations that included spin–orbit coupling did so by using a second variation step on the scalar–relativistic zeroth order wave functions, thus including the spin–orbit term to first order perturbation theory [@Hybertsen86p2920; @Surh91p4286; @Hemstreet93p4238]. In contrast we solve directly for general two–component spinor Bloch wave functions expanding in a plane wave spinor basis $$\label{plane_wave} {{| \psi_{nk} \rangle}} = \sum_{G,\sigma}c^{n,k}_{G,\sigma}{{| k+G \rangle}}{{| \sigma \rangle}},$$ where $G$ are reciprocal lattice vectors and $\sigma$ sums over up and down spin. In the basis of equation (\[plane\_wave\]) the action of the KB operator is as follows: $$\label{KBelement} {{\langle \sigma |}}{{\langle k+G |}}V_{KB}{{| \psi_{nk} \rangle}}=\sum_{i_s, i_a}\sum_{l,j,m_j}D^{KB}_{i_s,l,j}\ \varphi^{i_s,i_a}_{k+G}\ M^{KB, \sigma}_{i_s,l,j,m_j,k+G}\ f^{KB}_{i_s,i_a,k,n,l,j,m_j}\ ,$$ where $$D^{KB}_{i_s,l,j} = \left(\frac{4\pi}{V}\right)\frac{1}{\int dr\ r^2 R^{*,i_s}_{l,j}(r)\ \delta V^{i_s}_{l,j}(r)\ R^{i_s}_{l,j}(r)}$$ and $$\varphi^{i_s,i_a}_{k+G} = e^{-i(\vec k+\vec G)\cdot {\vec r}_{i_s, i_a}}$$ is a phase factor associated with the atomic position. The spin dependent factor $M^{KB}$ of equation (\[KBelement\]) can be written as a spinor and for $j=l+\frac{1}{2}$ is $$\label{M_plus} M^{KB, \sigma}_{i_s,l,j,m_j,k+G} = { \sqrt{l+m+1}\ F^{KB}_{i_s,l,j,m,k+G} \choose \sqrt{l-m}\ F^{KB}_{i_s,l,j,m+1,k+G} }$$ and $$\label{M_minus} M^{KB, \sigma}_{i_s,l,j,m_j,k+G} = { \sqrt{l-m+1}\ F^{KB}_{i_s,l,j,m-1,k+G} \choose -\sqrt{l+m}\ F^{KB}_{i_s,l,j,m,k+G} }$$ for $j=l-\frac{1}{2}$. Also the last factor of equation (\[KBelement\]), $f^{KB}_{i_s,i_a,k,n,l,j,m_j}$, depends on $j$ as follows $$\begin{aligned} &\sum_{G'} \varphi^{* i_s,i_a}_{k+G}& (c^{n,k}_{G',\uparrow}\sqrt{l+m+1}\ F^{* KB}_{i_s,l,j,m,k+G'}\nonumber\\ &&+c^{n,k}_{G',\downarrow}\sqrt{l-m}\ F^{* KB}_{i_s,l,j,m+1,k+G'})\end{aligned}$$ for $j=l+\frac{1}{2}$ and $$\begin{aligned} &\sum_{G'} \varphi^{* i_s,i_a}_{k+G}& (c^{n,k}_{G',\uparrow}\sqrt{l-m+1}\ F^{* KB}_{i_s,l,j,m-1,k+G'}\nonumber\\ &&-c^{n,k}_{G',\downarrow}\sqrt{l+m}\ F^{* KB}_{i_s,l,j,m+1,k+G'})\end{aligned}$$ for $j=l-\frac{1}{2}$. Finally the KB factors $F^{KB}$ appearing in equations (\[M\_plus\]) and (\[M\_minus\]) are defined as $$\begin{aligned} F^{KB}_{i_s,l,j,m,k+G} =\sqrt{\frac{4\pi}{2l+1}}Y^m_l(\theta, \varphi)\times\nonumber\\ \int dr\ r^2\ j_l(|k+G|r)\ \delta V^{i_s}_{l,j}(r)\ R^{i_s}_{l,j}(r),\end{aligned}$$ where $Y^m_l$ are the spherical harmonics, the polar angles $\theta$ and $\varphi$ are determined by the vector $\vec k + \vec G$ and $j_l$ are the spherical Bessel functions. The KB factors are calculated once and stored in memory. The contribution of a state ${{| \psi_{nk} \rangle}}$ to the non–local KB part of the total energy is thus given by $$E^{KB}=\sum_{i_s, i_a}\sum_{l,j,m_j}D^{KB}_{i_s,l,j}\left| f^{KB}_{i_s,i_a,k,n,l,j,m_j} \right|^2\ .$$ In order to test our implementation we calculated the properties of GaAs and compared with all–electron calculations and available experimental data. Self–consistency was achieved by direct minimization of the total energy via a conjugate gradient method [@Payne92p1045]. The gallium and arsenic pseudopotentials were created following the Troullier–Martins scheme [@Troullier91p1993], and both contained $s$ and $p$ components. Care must be given to the local part of the pseudopotential entering in equation (\[minus\_local\]) to ensure good transferability. We used the $j$–average of the unbound $4d$ state in case of gallium and likewise the $j$–average of the $p$ states for the local part of the arsenic pseudopotential. S-FKKR$^a$ S-FLAPW$^b$ R-PWPP Exp.$^c$ ------------- ------------ ------------- -------- ---------- $a_0$ (Å) 5.56 5.620 5.642 5.653 $B_0$ (GPa) 77 74 72.2 74.8 : Equilibrium lattice constant and bulk modulus determined in this work (R-PWPP) compared to all–electron calculations $^a$ Scalar relativistic FKKR, M. Asato et al., PRB [**60**]{}, 5202 (1999), $^b$ Scalar relativistic FLAPW, C. Filippi et al., PRB [**50**]{}, 14947 (1994) and experiment $^c$ Landolt–Börnstein, Vol 22 (1987)[]{data-label="tab_structural"} The results for lattice constant and bulk modulus for GaAs are shown in table \[tab\_structural\]. The good agreement between our fully relativistic pseudopotential results and the scalar–relativistic all–electron values in table \[tab\_structural\] confirms our approach and reaffirms the notion that spin–orbit splittings have little effect on the structural properties of semiconductors [@Bachelet85p879]. Our calculated bandstructure in figure \[fig\_band\] on the other hand shows clear evidence of spin–orbit coupling. The top of the valence band splits into the light hole, heavy hole manifold and the split off band, separated by 350meV. A similar split is also observed in the upper conduction bands at the Brillouin zone center. In table \[tab\_splitting\] we compare the characteristic spin–orbit splittings we obtained for GaAs at the experimental lattice constant with values from two fully relativistic all–electron calculations found in the literature. The agreement with both all–electron calculations is excellent. Splitting R-FKKR$^a$ R-FLAPW$^b$ R-PWPP ---------------------------- ------------ ------------- -------- $\Delta_0(\Gamma_{15}^v)$ 0.35 0.34 0.35 $\Delta'_0(\Gamma_{15}^c)$ 0.20 – 0.20 $\Delta_1(L_{3}^v)$ 0.09 0.09 0.09 $\Delta(X_{5}^v)$ 0.21 0.20 0.22 : Spin–orbit splittings for GaAs obtained in this work (R-PWPP) compared to the results of two relativistic all–electron calculations: $^a$ Fully relativistic FKKR, S. Bei der Kellen, A. J. Freeman, PRB [**54**]{}, 11187 (1996) and $^b$ Scalar relativistic FLAPW + 2nd variation, C. Filippi, et al., PRB [**50**]{}, 14947 (1994)[]{data-label="tab_splitting"} For completeness we compare in table \[tab\_eigenvalues\] the eigenvalue spectrum at three special $k$ points of the Brillouin zone with the eigenvalues obtained from the two fully relativistic all–electron calculations cited in table \[tab\_splitting\]. The zero of energy was chosen to coincide with the top of the valence band. Despite the generally good agreement there are two obvious discrepancies at the Brillouin zone center that need some clarification. First the direct band gap of our pseudopotential calculation at $\Gamma$ is more than 4 times larger than the gap resulting from the all–electron calculations. Second the valence band width of our approach is slightly smaller compared to the all–electron results. Both discrepancies result from the fact that the gallium $3d$ orbitals were placed in the frozen core in our calculation but are free to change in the all–electron approaches. Due to the well known self–interaction problem of the local density approximation to density functional theory [@Perdew81p5048] these fairly localized states will lie too high in energy when not frozen. The symmetry of the $d$ states in the zincblende lattice at $\Gamma$ only allows hybridization with $p$ states, e.g. the top of the valence band. Hence the top of the valence band will shift upwards, leading to a reduced band gap and at the same time an increase in the valence band width. Due to the mixed character of the band states away from the Brillouin zone center the effect of the gallium $3d$ states is most pronounced at $\Gamma$. [c|rr|r]{} Level & R-FKKR$^a$ & R-FLAPW$^b$ & R-PWPP\ $\Gamma_6^v$ & -12.94 & -12.91 & -12.67\ $\Gamma_7^v$ & -0.35 & -0.34 & -0.35\ $\Gamma_8^v$ & 0.00 & 0.00 & 0.00\ $\Gamma_6^c$ & 0.12 & 0.17 & 0.69\ $\Gamma_7^c$ & 3.46 & & 3.40\ $\Gamma_8^c$ & 3.66 & & 3.60\ \ $X_6^v$ & -10.42 & -10.41 & -10.35\ $X_7^v$ & -7.02 & -7.00 & -6.83\ $X_6^v$ & -2.88 & -2.85 & -2.74\ $X_7^v$ & -2.79 & -2.76 & -2.65\ $X_6^c$ & 1.17 & 1.23 & 1.23\ $X_7^c$ & 1.39 & & 1.44\ \ $L_6^v$ & -11.18 & -11.14 & -11.06\ $L_6^v$ & -6.83 & -6.82 & -6.63\ $L_6^v$ & -1.38 & -1.36 & -1.32\ $L_{4,5}^v$ & -1.17 & -1.16 & -1.10\ $L_6^c$ & 0.71 & 0.73 & 0.97\ $L_6^c$ & 4.38 & & 4.34\ $L_{4,5}^c$ & 4.46 & & 4.44\ Compared to calculations that do not include the spin–orbit term we find that the inclusion of spin–orbit coupling worsens the short coming of the local density approximation of underestimating the band gap. The reason for this observation simply lies in the fact that the top of the valence band splits and the light and heavy hole states move closer to the bottom of the conduction band. In conclusion, we have implemented spin–orbit coupling in the well established [*ab initio*]{} pseudopotential approach of density functional theory. This paper gives the necessary expressions in a two–component spinor plane wave basis and demonstrates the applicability of the method for bulk GaAs. Our results compare very well to relativistic all–electron calculations. Since our direct approach is based on a complete spinor plane wave basis it can easily be extended to systems that show exchange splitting and exhibit non–collinear spin arrangements. The code will be available under the GNU General Public License [@GPL] at <http://www.mrl.ucsb.edu/~theurich/Spinor/>. This work was supported by the ONR grant number N00014-00-10557, by NSF-DMR under the grant 9973076 and by ACS PRF under the grant 33851-G5. G.T. acknowledges fellowship support from the UCSB Materials Research Lab., funded by the Corning Foundation.

--- abstract: 'For a compact set $K\subset \mathbb{R}^1$ and a family $\{C_\lambda\}_{\lambda\in J}$ of dynamically defined Cantor sets sufficiently close to affine with $\text{dim}_H\, K+\text{dim}_H\, C_\lambda>1$ for all $\lambda\in J$, under natural technical conditions we prove that the sum $K+C_\lambda$ has positive Lebesgue measure for almost all values of the parameter $\lambda$. As a corollary, we show that generically the sum of two affine Cantor sets has positive Lebesgue measure provided the sum of their Hausdorff dimensions is greater than one.' address: - 'Anton Gorodetski University of California, Irvine' - 'Scott Northrup University of California, Irvine' author: - '[A. Gorodetski]{}' - '[S. Northrup]{}' title: On Sums of Nearly Affine Cantor Sets --- [^1] Introduction and Main results {#sec:intro} ============================= Questions on the structure and properties of sums of Cantor sets appear naturally in dynamical systems [@n1; @n2; @n3; @PaTa], number theory [@CF; @Mal; @Moreira], harmonic analysis [@BM; @BKMP], and spectral theory [@EL06; @EL07; @EL08; @Y]. J.Palis asked whether it is true (at least generically) that the arithmetic sum of dynamically defined Cantor sets either has measure zero, or contains an interval (see [@PaTa]). This claim is currently known as the “Palis’ Conjecture”. The conjecture was answered affirmatively in [@MY] for generic dynamically defined Cantor sets. For sums of generic [*affine*]{} Cantor sets Palis’ Conjecture is still open. Even for the simplest case of middle-$\alpha$ Cantor sets these questions are non-trivial and not completely settled. By a middle-$\alpha$ Cantor set we mean the Cantor set $$\label{e.ddcs} C=\cap_{n=0}^{\infty}I_n, \ I_{n+1}=\cup_{i=1}^{m}\varphi_i(I_n), \ \varphi_i(I_0)\cap \varphi_j(I_0)=\emptyset \ \text{for}\ i\ne j,$$ where $I_0=[0,1]$, $m=2$, $\varphi_1(x)=ax$, $\varphi_2(x)=(1-a)+ax$, $a=\frac{1}{2}(1-\alpha)$. Let us denote this Cantor set by $C_a$. It is easy to show (using dimensional arguments, e.g. see Proposition 1 in Section 4 from [@PaTa]) that if $\frac{\log 2}{\log 1/a}+\frac{\log 2}{\log 1/b}<1$ then $C_a+C_b$ is a Cantor set. On the other hand, Newhouse’s Gap Lemma (e.g. see Section 4.2 from [@PaTa], or [@n1]) implies that if $\frac{a}{1-2a}\frac{b}{1-2b}>1$ then $C_a+C_b$ is an interval. This still leaves a “mysterious region” $R$ in the space of parameters, see Figure \[f.1\], and Solomyak [@So97] showed that for Lebesgue a.e. $(a, b)\in R$ one has $Leb(C_a+C_b)>0$. \[f.1\] ![The region $R$ studied by B.Solomyak in [@So97]](mysterious.png){height="2in"} A description of possible topological types of $C_a+C_b$ was provided in [@MO]. It is still an open question whether $C_a+C_b$ contains an interval for a.e. $(a,b)\in R$. Solomyak’s result was generalized to families of homogeneous self-similar Cantor sets (i.e. Cantor sets given by (\[e.ddcs\]) where all contractions $\{\varphi_i\}_{i=1, \ldots, m}$ are linear with the same contraction coefficient) by Peres and Solomyak [@PeSo]. They showed that for a fixed compact set $K\subseteq \mathbb{R}$ and a family $\{C_\lambda\}$ of homogeneous Cantor sets parameterized by a contraction rate $\lambda$ (i.e. all contractions have the form $\varphi_i(x)=\lambda x +D_i(\lambda)$, $D_i\in C^1$) one has $$\begin{gathered} \label{e.hdsone} \text{dim}_H(C_\lambda+K)= \text{dim}_HC_\lambda + \text{dim}_HK \ \text{for a.e.}\ \lambda\in (\lambda_0, \lambda_1) \\ \text{if} \ \ \ \text{dim}_HC_\lambda + \text{dim}_HK<1\ \ \text{for all}\ \lambda\in (\lambda_0, \lambda_1), \ \text{and} \end{gathered}$$ $$\begin{gathered} \label{e.hdlone} Leb(C_\lambda+K)>0 \ \text{for a.e.}\ \lambda\in (\lambda_0, \lambda_1) \\ \text{if} \ \ \ \text{dim}_HC_\lambda + \text{dim}_HK>1\ \ \text{for all}\ \lambda\in (\lambda_0, \lambda_1). \end{gathered}$$ In the case when $K$ is a non-linear $C^{1+\varepsilon}$-dynamically defined Cantor set, the set of exceptional parameters in (\[e.hdlone\]) in fact has zero Hausdorff dimension, see [@Shm Theorem 1.4]. For a more general case of sums of dynamically defined Cantor sets $C$ and $K$ on the first glance the mentioned above results by Moreira and Yoccoz [@MY] provide the complete answer. But in practice in many cases one has to deal with a finite parameter families of Cantor sets, or even with a specific fixed Cantor sets $C$ and $K$, and [@MY] does not provide specific genericity assumptions that could be verified in a particular given setting. Specific conditions that would allow to claim that $$\label{e.hs} \text{dim}_H(C+K)=\min\left(\text{dim}_HC + \text{dim}_HK, 1\right)$$ are currently known [@HS; @NPS; @PeShm], but the case $\text{dim}_HC + \text{dim}_HK>1$ turned out to be more subtle. In this paper we address this question in the case of affine (all $\varphi_i$ in (\[e.ddcs\]) are affine contractions, not necessarily with the same contraction coefficients) and close to affine dynamically defined Cantor sets. \[t.1\] Suppose $J\subseteq \mathbb{R}$ is an interval and $\lbrace C_\l \rbrace_{\lambda\in J}$ is a family of dynamically defined Cantor sets generated by contracting maps $$\label{e.start}\lbrace f_{i, \lambda}(x)=c_{i}(\lambda)x+b_{i}(\lambda)+g_{i}(x, \lambda)\rbrace_{i=1}^m$$ such that the following holds: $$c_i(\lambda), b_i(\lambda)\ \text{are $C^1$-functions of}\ \lambda;$$ $$\label{e.c} \frac{d|c_i|}{d\lambda} \le -\delta<0\ \text{for all $\lambda\in J$ and some uniform $\delta>0$};$$ $$\label{e.g} g_i(x,\lambda)\in C^2, \ \text{with small (based on $\{c_i(\lambda)\}, \{b_i(\lambda)\}$) $C^2$-norm}.$$ Then for any compact $K\subset \mathbb{R}$ with $$\label{e.condition} \dim_H(K)+\dim_H(C_{\lambda}) > 1\ \text{ for all }\ \l\in J,$$ the sumset $K+C_\lambda$ has positive Lebesgue measure for a.e. $\l\in J$. Theorem \[t.1\] can be generalized in a straightforward way to a larger class of nearly affine Cantor sets where topological Markov chains are allowed instead of the full Bernoulli shift in the symbolic representation (see [@MY] or [@PaTa] for detailed definitions). We restrict ourselves to the case of the full shift only to keep the exposition more transparent. We strongly believe that the assumption on $C_\lambda$ being close to affine is an artefact of the proof, and that a similar statement should hold in a more general setting, for a family of non-linear dynamically defined Cantor sets without any smallness assumptions on non-linearity. We plan to address this question in a future publication. Consider now the non-homogeneous affine case, that is a Cantor set generated by (\[e.ddcs\]), where $\varphi_k(x)=\lambda_kx+d_k$. Moreover, let us include it into a family $\{K_{\Lambda}\}$, where $$\Lambda=(\lambda_1, \ldots, \lambda_m), \ \lambda_k\in J_k\subset (-1, 0)\cup (0,1), \ \text{and} \ d_k=d_k(\Lambda)\ \text{is $C^1$}.$$ The last condition in (\[e.ddcs\]) implies that $$[d_i(\Lambda)+\l_i K_\Lambda]\cap [d_j(\Lambda)+\l_j K_\Lambda] = \emptyset\ \text{ for}\ i\neq j,$$ which is sometimes called [*strong separation condition*]{}, e.g. see [@PeSo]. Fubini’s theorem together with Theorem \[t.1\] gives the following statement. \[c.1\] Suppose $K$ is a compact subset of the real line, and a family $\{K_\Lambda\}$ of affine Cantor sets as above is given such that $$\dim_H K + \dim_H K_\Lambda > 1 \ \text{ for all}\ (\l_1,\dots,\l_m)\in J_1\times\dots\times J_m.$$ Then for a.e. $(\l_1,\dots,\l_m)\in J_1\times\dots\times J_m$ the set $K+K_\Lambda$ has positive Lebesgue measure. Notice that in this setting any affine Cantor set is completely determined by $2m$ parameters, namely $(\lambda_1, \ldots, \lambda_m)\in \Lambda$ and $(d_1, \ldots, d_m)$. Admissible $2m$-tuples of the parameters (i.e. such that $\varphi_k([0,1])\subseteq [0,1]$ for each $k=1, 2, \ldots, m$, and the strong separation condition holds) form a subset in $\mathbb{R}^{2m}$. As an immediate consequence of Corollary \[c.1\] we have \[c.2\] Generically (for Lebesgue almost all admissible tuples of the parameters) the sum of two affine Cantor sets has positive Lebesgue measure provided the sum of their Hausdorff dimensions is greater than one. It is interesting to compare these results with Theorem E from [@ShmS] that claims that for any two affine Cantor sets $C_1$ and $C_2$ with sum of dimensions greater than one, $\text{dim}_H\,\{u\in \mathbb{R}\ |\ Leb(C_1+uC_2)=0\}=0$. The idea of proof of Theorem \[t.1\] is to find some measures supported on $K$ and $C_\lambda$ whose convolution is absolutely continuous with respect to the Lebesgue measure. Since support of a convolution of two measures is the sum of their supports, this would prove that $Leb(K+C_\lambda)>0$. In Section \[s.acc\] we provide the statement of a result from [@DGS] on absolute continuity of convolutions of singular measures under certain conditions. Then in Section \[sec:main\] we verify those conditions for some specific measures supported on $K$ and $C_\lambda$. Absolute continuity of convolutions {#s.acc} =================================== Let $\Omega=\mathcal{A}^{\Z_+}$ with $|\mathcal{A}| = m \geq 2$ be the standard symbolic space, equipped with the product topology. Let $\mu$ be a Borel probability measure on $\Omega$. Let $J$ be a compact interval and assume we are given a family of continuous maps $\Pi_\l:\Omega\to\R$, for $\l \in J$, such that $C_\l = \Pi_\l(\Omega)$ are the Cantor sets, and let $\nu_\l = \Pi_\l (\mu)$. For a word $u\in\mathcal{A}^{n}$, $n\geq 0$, denote by $|u| = n$ its length and by $[u]$ the cylinder set of elements of $\Omega$ that have $u$ as a prefix. For $\omega,\tau\in\Omega$ we write $\omega\wedge\tau$ for the maximum common subword in the beginning of $\omega$ and $\tau$ (empty if $\omega_0\neq\tau_0$; we set the length of the empty word to be zero). For $\omega,\tau\in \Omega$, let $\pwt(\l):=\Pi_\l(\omega)-\Pi_\l(\tau)$. We will need the following statement. \[BlackBox\] Let $\eta$ be a compactly supported Borel probability measure on $\R$ of exact local dimension $d_\eta$. Suppose that for any $\varepsilon > 0$ there exists a subset $\Omega_\varepsilon \subset \Omega$ such that $\mu(\Omega_\varepsilon) > 1 -\varepsilon$ and the following holds; there exist constants $C_1,C_2,C_3,\alpha,\beta,\gamma>0$ and $k_0\in\Z_+$ such that $$\label{BlackBox0} d_\eta+\dfrac{\gamma}{\beta} > 1 \text{ and } d_\eta > \dfrac{\beta -\gamma}{\alpha},$$ $$\label{BlackBox1} \max_{\l\in J} |\pwt(\l)| \leq C_1 m^{-\alpha|\omega\wedge\tau|} \text{ for all }\omega,\tau\in\Omega_\varepsilon, \omega\neq\tau,$$ $$\label{BlackBox2} \sup_{v\in\R} Leb(\lbrace \l\in J: |v+\pwt(\l)|\leq r\rbrace )\leq C_2 m^{|\omega\wedge\tau|\beta}r$$ for all $\omega,\tau\in\Omega_\varepsilon, \omega\neq\tau$ such that $|\omega\wedge\tau| \geq k_0$, and $$\label{BlackBox3} \max_{u\in \mathcal{A}^{n}, [u]\cap \Omega_\varepsilon \neq \emptyset} \mu([u])\leq C_3m^{-\gamma n} \text{ for all } n\geq 1.$$ Then the convolution $\eta\ast\nu_\l$ is absolutely continuous with respect to the Lebesgue measure for a.e. $\l\in J$. In fact, in Proposition \[BlackBox\] the condition on exact dimensionality of the measure $\eta$ can be replaced by the following condition (and this is the only consequence of exact dimensionality of $\eta$ that was used in the proof of Proposition \[BlackBox\] in [@DGS]): $\eta$ is a compactly supported Borel probability measure on the real line, such that $$\label{e.exdim} \eta[B_r(x)] \leq Cr^{d_\eta}, \text{ for all }x\in\R \text{ and } r>0.$$ Proofs {#sec:main} ====== Here we construct the measure $\eta$ supported on $K$ and a family of measures $\nu_\lambda$ with $supp\, \nu_\l=C_\l$ such that Proposition \[BlackBox\] can be applied. Since absolute continuity of the convolution $\eta * \nu_\l$ implies that $Leb(C_\l+K)>0$, this will prove Theorem \[t.1\]. Let us start with construction of the measure $\eta$. The compact set $K\subset \mathbb{R}$ satisfies the condition (\[e.condition\]), i.e. $\dim_H(K)+\dim_H(C_{\lambda}) > 1\ \text{ for all }\ \l\in J$. Take any constant $d\in (0, \dim_H(K))$ that is sufficiently close to $\dim_H(K)$ to guarantee that $d+\dim_H(C_{\lambda}) > 1\ \text{ for all }\ \l\in J$. By Frostman’s Lemma (see, e.g., [@Mattila Theorem 8.8]), there exists a Borel measure $\eta$ supported on $K$ such that (\[e.exdim\]) holds with $d_\eta=d$. Let $\{C_\lambda\}_{\l\in J}$ be a family of dynamically defined Cantor sets generated by contractions $f_{k, \lambda}:[0,1]\to [0,1]$, $k=1, \ldots, m$, given by (\[e.start\]). Define the map $\xi: C_\lambda\to\mathbb{R}$ by $$\xi(x)=\log|f_{k, \lambda}'(f_{k, \lambda}^{-1}(x))|\ \ \text{if}\ \ x\in f_{k, \lambda}([0,1]).$$ Due to [@Man], there is an ergodic Borel probability measure $\mu_\l$ on $C_\l$ (namely, the equilibrium measure for the potential $(\text{dim}_H\,C_{\lambda})\xi(x)$) that satisfies the condition $-h_{\mu_\l}/\mu_\l (\xi) = \dim_H (C_\l)$. This is also a measure on $C_\l$ such that $\dim_H(\mu_\l) = \dim_H (C_\l)$ (i.e. [*the measure of maximal dimension*]{}). Sometimes it is convenient to consider one expanding map $$\Phi_\lambda:\cup_{k=1}^m f_{k, \lambda}([0,1])\mapsto [0,1], \ \text{where}\ \Phi_\lambda(x)=f^{-1}_{k,\lambda}(x)\ \text{ for}\ x\in f_{k, \lambda}([0,1]),$$ instead of the collection of contractions $\{f_{1, \lambda}, \ldots, f_{m,\lambda}\}$. Notice that $\Phi_\lambda(C_\lambda)=C_\lambda$, and the Lyapunov exponent of $\Phi_\lambda$ with respect to the invariant measure $\mu_\lambda$ is equal to $-\mu_\l (\xi)$. We will denote this Lyapunov exponent by $Lyap^u(\mu_\lambda)$. Since $\mu_\l$ is a measure of maximal dimension, we have $$\dim_H(\mu_\l) = \frac{h_{\mu_\l}(\Phi_\l)}{Lyap^u(\mu_\l)}=\dim_H(C_\l).$$ For each $\omega\in\Omega$ let $F_{\l}^n(\omega) = f_{\omega_0,\l}\circ\dots\circ f_{\omega_n,\l}(x)$, for a fixed $x\in [0,1]$. Then we can define the map $\Pi_\lambda : \Omega \to \R$ given by $$\Pi_\lambda (\omega) = \lim_{n\to\infty} F_{\l}^n(\omega),$$ where in fact the limit does not depend on the initial point $x\in [0,1]$. For any $\l_1, \l_2\in J$ the map $h_{\l_1, \l_2}:C_{\l_2}\to C_{\l_1}$ defined by $h_{\l_1, \l_2}=\Pi_{\l_1}\circ \Pi^{-1}_{\l_2}$ is a homeomorphism. It is well known (e.g. see Section 19 in [@KH]) that this homeomorphism must be Hölder continuous. Moreover, due to [@PV] the following statement holds. \[l.help1\] For any $\lambda_0\in J$ and any $\tau\in (0,1)$ there exists a neighborhood $V\subseteq J$, $\lambda_0\in V$, such that for any $\l\in V$ the conjugacy $h_{\l, \l_0}:C_{\l_0}\to C_{\l}$ as well as its inverse $h_{\l_0, \l}:C_{\l}\to C_{\l_0}$ are Hölder continuous with Hölder exponent $\tau$. Define the measure $\mu$ on $\Omega$ by $\mu:=\Pi_{\l_0}^{-1}(\mu_{\l_0})$, and set $$\nu_\lambda:=\Pi_{\l}(\mu)=\Pi_{\l}(\Pi^{-1}_{\l_0}(\mu_{\l_0}))=h_{\l, \l_0}(\mu_{\l_0})=h_{\l, \l_0}(\nu_{\l_0}).$$ If both $h_{\l, \l_0}$ and $h_{\l_0, \l}$ are Hölder continuous with Hölder exponent $\tau$ then $$\tau\dim_HC_{\l_0}=\tau\dim_H\nu_{\l_0}\le \dim_H\nu_\l\le \frac{1}{\tau}\dim_H\nu_{\l_0}=\frac{1}{\tau}\dim_HC_{\l_0}.$$ Since in Lemma \[l.help1\] the value of $\tau$ can be taken arbitrarily close to one, we get the following statement. \[l.help2\] For any $\lambda_0\in J$ there exists a neighborhood $W\subseteq J$, $\lambda_0\in W$, such that for any $\l\in W$ we have $$d_\eta + \dim_H \nu_\l > 1.$$ It is clear that in order to prove Theorem \[t.1\] it is enough to prove that for each $\l_0\in J$ there exists a neighborhood $W, \l_0\in W$, such that the sum $K+C_{\l}$ has positive Lebesgue measure for a.e. $\l$ from $W$. For a given $\l_0\in J$ we can choose positive $\varepsilon,\alpha, \beta$, and $\gamma$ in such a way that $$d_\eta + \dim_H \nu_{\l_0} > 1+\varepsilon,$$ $$\alpha < \frac{Lyap^u(\nu_{\l_0})}{\log m} < \beta,$$ $$\gamma < \frac{h_{\nu_{\l_0}}(\Phi_{\l_0})}{\log m}.$$ If $\alpha, \beta$ are sufficiently close to $\frac{Lyap^u(\nu_{\l_0})}{\log m}$, and $\gamma$ is sufficiently close to $\frac{h_{\nu_{\l_0}}(\Phi_{\l_0})}{\log m}$, then we also have $$d_\eta + \frac{\gamma}{\beta} > 1,$$ which is one of the conditions (\[BlackBox0\]) of Proposition \[BlackBox\], and also $$\frac{\beta}{\alpha} < 1+\frac{\varepsilon}{2}.$$ Decreasing if needed the neighborhood $W$ given by Lemma \[l.help2\] we can guarantee that for all $\l\in W$ the following property holds: $$\frac{h_{\nu_\l}(\Phi_\l)}{Lyap^u(\nu_\l)} -\frac{\gamma}{\alpha} < \frac{\varepsilon}{2}.$$ Therefore $$d_\eta > 1+\varepsilon - \dim_H(\nu_\l) > \frac{\beta}{\alpha} - \frac{\gamma}{\alpha}$$ for $\l\in W$, which implies another part of the condition (\[BlackBox0\]) of Proposition \[BlackBox\], namely, $$d_\eta > \frac{\beta - \gamma}{\alpha}.$$ Finally let us notice that if $W$ is small, then we have $$\alpha < \frac{Lyap^u(\nu_\l)}{\log m} < \beta,$$ $$\gamma < \frac{h_{\nu_\l}(\Phi_\l)}{\log m}$$ for all $\l\in W$. In order to verify the conditions (\[BlackBox1\]), (\[BlackBox2\]), and (\[BlackBox3\]) of Proposition \[BlackBox\], we will try to mimic the proof of Theorem 3.7 from [@DGS]. We will show that for a given small $\varepsilon>0$ there are subsets $\Omega_1$ and $\Omega_2$ in $\Omega$ such that $\mu(\Omega_i)>1-\frac{\varepsilon}{2}$, $i=1,2$, and properties (\[BlackBox1\]) and (\[BlackBox2\]) hold for all $\omega,\tau\in\Omega_1$, and (\[BlackBox3\]) holds for all $\omega,\tau\in\Omega_2$. This will imply that all these conditions hold for all $\omega,\tau\in\Omega_{\varepsilon}=\Omega_1\cap\Omega_2$ with $\mu(\Omega_\varepsilon)>1-\varepsilon$, i.e. justify application of Proposition \[BlackBox\], and therefore prove Theorem \[t.1\]. For $\omega\in \Omega$, $\omega=\omega_0\omega_1\ldots\omega_n\ldots$, set $p(\lambda)=\Pi_\lambda(\omega)$ and $$\label{e.ls} l^{(s)}=\frac{df_{\omega_{s-1}, \lambda}}{dx}(\Phi^s_{\lambda}(p(\lambda))).$$ We will also write $l^{(s)}(\lambda)$ or $l^{(s)}_\omega$ if we need to emphasize the dependence of $l^{(s)}$ on $\lambda$ or $\omega$. Notice that $\{l^{(s)}\}$ is a sequence of multipliers of the contractions along the orbit of point $p(\lambda)$ under the map $\Phi_\lambda$, and if Lyapunov exponent at $p(\lambda)$ exists then $$Lyap^u(p(\lambda))=-\lim_{n\to \infty} \frac{1}{n}\sum_{s=1}^n\log\left|l^{(s)}\right|.$$ \[Egorov\] Given $\epsilon > 0$, there exists a set $\Omega_1\subset \Omega$ with $\mu(\Omega_1) > 1-\frac{\epsilon}{2}$ and $N\in\N$ such that $$\label{e.alphabeta}\alpha\log m < -\frac{1}{n}\sum_{s=1}^{n} \log \left|l^{(s)}(\l)\right|<\beta\log m$$ for every $\l\in W$, $n\ge N$, and all $p\in \Pi_\l(\Omega_1)$. Let us start with the first part of the inequality (\[e.alphabeta\]). First we will show that for a fixed $\l\in W$ and a given $\varepsilon'>0$, there exists $\Omega'$ with $\mu (\Omega') > 1-\varepsilon'$ and $N\in \N$ such that $$\alpha\log m+\xi < -\frac{1}{n}\sum_{s=1}^{n} \log \left|l^{(s)}(\l)\right|,$$ where $0 < \xi < Lyap^u(\mu_\l) - \alpha\log m$, for all $n\ge N$ and all $p\in \Pi_\l(\Omega')$. By the Birkhoff Ergodic Theorem, $$\begin{aligned} Lyap^u(\mu_\l) & =\\ & = \int \log \| D\Phi_\l (\Pi_\l(\omega)\|\,d\mu(\omega) \\ & = \lim_{n\to\infty} \frac{1}{n}\sum_{s=1}^n \log \| D\Phi_\l(\Phi_\l^s(\Pi_\l(\omega))\| \\ & = \lim_{n\to\infty}-\frac{1}{n}\sum_{s=1}^n \log \left|l_\omega^{(s)}(\l)\right|\end{aligned}$$ for $\mu$-a.e. $\omega\in\Omega$. Thus by Egorov’s theorem, there exists $\Omega' \subset \Omega$ with $\mu(\Omega') > 1-\varepsilon'$ such that the convergence is uniform on $\Omega'$. Thus there exists $N\in \N$ such that $\alpha\log m+\xi < -\frac{1}{n}\sum_{s=1}^{n} \log \left|l^{(s)}(\l)\right|$ for all $n\ge N$ and all $p\in \Pi_\l(\Omega')$. Next we will show that $N$ can be chosen uniformly in $\l \in W$. Let $\varepsilon >0$ be given. Consider the family of functions $$L_\omega (\l) = -\log \|D\Phi_\l(\Pi_\l(\omega)\|.$$ We can treat the elements of this family as functions of $\l$ with parameter $\omega$. Then $\lbrace L_\omega (\l)\rbrace_{\omega\in\Omega}$ is an equicontinuous family of functions and there exists $t>0$ such that if $|\l_1 - \l_2 | \leq t$, then $|L_\omega(\l_1) - L_\omega(\l_2)| < \frac{\xi}{100}$ for any $\omega\in\Omega.$ Consider a finite $t$-net $\lbrace y_1,\dots,y_M\rbrace$ in $W$, containing $M=M(W,t)$ points. For each point $y_j$ we can find a set $\Omega^{(j)}\subset \Omega$, $\mu(\Omega^{(j)}) > 1 - \frac{\epsilon}{4M}$, and $N_j\in \N$ such that for every $n \geq N_j$ and every $\omega\in \Omega^{(j)}$, we have $$\frac{1}{n}\sum_{s=1}^{n} L_{\sigma^s(\omega)}(y_j) = -\frac{1}{n}\sum_{s=1}^{n} \log \left|l_\omega^{(s)}(y_j)\right| > \alpha\log m +\xi.$$ Take $\Omega_1 = \cap_{j=1}^M \Omega^{(j)}$. We have $$\mu (\Omega_1) > 1 - M\frac{\varepsilon}{4M} = 1-\frac{\varepsilon}{4},$$ and for every $\l\in W$ there exists $y_j$ with $|y_j-\l| \leq t$. So for every $\omega\in\Omega_1\subseteq\Omega^{(j)}$ and every $n \ge N =\max \lbrace N_1,\dots,N_M\rbrace$, we have $$\begin{aligned} -\frac{1}{n}\sum_{s=1}^{n} \log \left|l_\omega^{(s)}(\l)\right| & = \frac{1}{n}\sum_{s=1}^{n} L_{\sigma^s(\omega)}(\l)\\ & \geq \frac{1}{n}\sum_{s=1}^{n} L_{\sigma^s(\omega)}(y_j) - \left|\frac{1}{n}\sum_{s=1}^{n} L_{\sigma^s(\omega)}(y_j) - \frac{1}{n}\sum_{s=1}^{n} L_{\sigma^s(\omega)}(\l)\right| \\ & \geq \alpha\log m + \xi -\frac{\xi}{100} \\ & > \alpha\log m +\frac{\xi}{2} \\ & > \alpha\log m,\end{aligned}$$ which proofs the first part of the inequality (\[e.alphabeta\]). The proof of the second part is analogous. This concludes the proof of Lemma \[Egorov\]. Notice that Lemma \[Egorov\] directly implies that for $p\in \Pi_\l(\Omega_1)$ and $n\ge N$ we have $$\label{e.abproduct} m^{-\beta n} < \left|\prod_{s=1}^n l^{(s)}\right| < m^{-\alpha n}.$$ The next statement is a simple partial case of Lemma 3.12 from [@DGS]. \[l.333simple\] There is a constant $C >0$ such that for any word $\omega_0\omega_1\ldots\omega_n\in\mathcal{A}^{n+1}$, any $\lambda\in J$, and any $x,y\in I_0=[0,1]$ the following holds. Set $$p=f_{\omega_0, \lambda}\circ f_{\omega_1, \lambda}\circ\ldots \circ f_{\omega_n, \lambda}(x),$$ and define $\{l^{(s)}\}$ by (\[e.ls\]). Denote $$q=f_{\omega_0, \lambda}\circ f_{\omega_1, \lambda}\circ\ldots \circ f_{\omega_n, \lambda}(y).$$ Then $$\frac{1}{C}\left|\prod_{s=1}^n l^{(s)}\right| \leq \frac{|p-q|}{|x-y|} \leq C\left|\prod_{s=1}^n l^{(s)}\right|.$$ The property (\[BlackBox1\]) for all $\omega, \tau\in \Omega_1$ follows now from (\[e.abproduct\]) and Lemma \[l.333simple\]. In order to check (\[BlackBox2\]) for some $\omega, \tau\in \Omega$ it is enough to show that $$\label{e.ineq} \left|\frac{d}{d\lambda}\phi_{\omega, \tau}(\lambda)\right|>C' m^{-\beta |\omega \wedge \tau|}$$ for some uniform (independent of $\omega, \tau\in \Omega$) constant $C'>0$. Let us consider some $\omega, \tau\in \Omega$, and set $n=|\omega \wedge \tau|$. Let us denote $$P_0(\lambda)=\Pi_{\lambda}(\omega), \ Q_0(\lambda)=\Pi_{\lambda}(\tau), \ \$$ and $$\ P_s=\Phi^s_{\lambda}(P_0), \ Q_s=\Phi_{\lambda}^s(Q_0), \ s=0, \ldots, n.$$ Notice that the distance between $P_n$ and $Q_n$ is uniformly bounded away from zero. Indeed, since $n=|\omega \wedge \tau|$, $P_n$ and $Q_n$ belong to different elements of Markov partition of $C_\lambda$. Let us also denote $$k^{(s)}_\lambda=f_{\omega_{s-1}} \ \ \ \text{and}\ \ \ l^{(s)}(\lambda)=\frac{\partial k^{(s)}_\lambda}{\partial x}(P_{s}(\lambda))\ \ \text{for}\ \ s=1, 2, \ldots, n.$$ We have $$\label{e.ineqnext} k^{(s)}_\lambda(x)=k^{(s)}_\lambda(P_s(\lambda))+l^{(s)}\cdot (x-P_s(\lambda)) + O((x-P_s(\lambda))^2)$$ and $$\label{e.ineqnext} \frac{\partial k^{(s)}_\lambda}{\partial x}(x)=l^{(s)}+ O(x-P_s(\lambda)).$$ Notice that $P_0=k^{(1)}_\lambda\circ \ldots \circ k^{(n)}_\lambda(P_n)$, and $Q_0=k^{(1)}_\lambda\circ \ldots \circ k^{(n)}_\lambda(Q_n)$. To prove (\[e.ineq\]) we need to find a bound on $$\begin{aligned} \frac{d}{d\lambda}\phi_{\omega, \tau}(\lambda)=\frac{d}{d\l}(P_0(\l)-Q_0(\l))=\end{aligned}$$ $$\begin{aligned} & = \sum_{i=1}^n \left(\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))\right)\frac{\partial k^{(i)}_\l}{\partial\l}(P_{i}(\l)) + \left(\prod_{s=1}^n \frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))\right)\frac{\partial P_n}{\partial\l}(\l) \\ & - \sum_{i=1}^n \left(\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))\right)\frac{\partial k^{(i)}_\l}{\partial\l}(Q_{i}(\l)) - \left(\prod_{s=1}^n \frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))\right)\frac{\partial Q_n(\l)}{\partial\l} \\ & = \sum_{i=1}^n \left(\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))\right)\left(\frac{\partial k^{(i)}_\l}{\partial\l}(P_{i}(\l))-\frac{\partial k^{(i)}_\l}{\partial\l}(Q_{i}(\l))\right) \\ & + \sum_{i=1}^n \frac{\partial k^{(i)}_\l}{\partial \l}(Q_{i}(\l))\left(\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))-\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))\right) \\ & + \left(\left(\prod_{s=1}^n \frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))\right)\frac{\partial P_n}{\partial\l}(\l) - \left(\prod_{s=1}^n \frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))\right)\frac{\partial Q_n(\l)}{\partial\l}\right)\\ & = S_1 + S_2 + S_3 \end{aligned}$$ Let us estimate $S_1$. We have $$\begin{aligned} S_1 = & \sum_{i=1}^n \left(\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))\right)\left(\frac{\partial k^{(i)}_\l}{\partial\l}(P_{i}(\l))-\frac{\partial k^{(i)}_\l}{\partial\l}(Q_{i}(\l))\right)\\ & = \sum_{i=1}^n \left(\prod_{s=1}^{i-1} l^{(s)}\right)\frac{\partial^2 k^{(i)}_\l}{\partial x\partial\l}(W_{i}(\l))(P_{i}(\l) - Q_{i}(\l))\end{aligned}$$ where $W_{i}(\l)$ is a point between $P_{i}(\l)$ and $Q_{i}(\l)$. Since we have $$\frac{\partial^2 k^{(i)}_\l}{\partial x\partial\l}=\frac{\partial c_{\omega_{i-1}}}{\partial\l}+\frac{\partial^2 g_{\omega_{i-1}}(x,\l)}{\partial x\partial\l},$$ the assumption (\[e.c\]) implies that $\frac{\partial^2 k^{(i)}_\l}{\partial x\partial\l}$ has sign opposite to the sign of $l^{(i)}$. Also, it is easy to see that $P_{i}(\l) - Q_{i}(\l)$ has the same sign as $$\left(\prod_{s=i+1}^n l^{(s)}\right)(P_{n}(\l) - Q_{n}(\l)).$$ Therefore all terms in the sum $S_1$ have the same sign as $$-\left(\prod_{s=1}^n l^{(s)}\right)(P_{n}(\l) - Q_{n}(\l)).$$ Using Lemma \[l.333simple\], assumption (\[e.c\]), and the fact that $|P_{n}(\l) - Q_{n}(\l)|$ is bounded away from zero, this implies that $$\begin{aligned} \label{e.last} |S_1| = & \sum_{i=1}^n \left|\prod_{s=1}^{i-1} l^{(s)}\right|\left|\frac{\partial^2 k^{(i)}_\l}{\partial x\partial\l}(W_{i}(\l))\right|\left|P_{i}(\l) - Q_{i}(\l)\right|\ge nC^{*}\left|\prod_{s=1}^{n} l^{(s)}\right|\end{aligned}$$ for some constant $C^*>0$. Let us now estimate $S_2$. Let us remind that $k_{\l}^{(s)}(x) = f_{\omega_{s-1}, \l} (x)= c_{\omega_{s-1}}({\l})x+b_{\omega_{s-1}}{(\l)} + g_{\omega_{s-1}}{(x, \l)},$ where the $C^2$-norm of $g_{\omega_{s-1}}{(x,\l)}$ is small. $$\begin{aligned} |S_2| &= \left|\sum_{i=1}^n \frac{\partial k^{(i)}_\l}{\partial \l}(Q_{i}(\l))\left(\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))-\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))\right)\right| \\ & \leq \sum_{i=1}^n \left|\frac{\partial k^{(i)}_\l}{\partial \l}(Q_{i}(\l))\right|\cdot\left|\left(\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))-\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))\right)\right| \\ & \leq \sum_{i=1}^n C\left|\left(\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))-\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))\right)\right| \\ & = C \sum_{i=1}^n \left|\prod_{s=1}^{i-1} l^{(s)}\right| \left|1-\frac{\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))}{\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))}\right|\end{aligned}$$ \[l.33\] $$\left|1-\frac{\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))}{\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))}\right| \leq C^{\prime}\left|\prod_{s=i}^{n} l^{(s)}\right|\max_{t=1,\dots,m}\| g_t{(x,\l)}\|_{C^2}$$ for some $C^{\prime} > 0$. Note that if $A$ is near 1 and $B$ is much smaller than 1, we have that $$|\log A| < B \text{ implies }|A-1| \leq 2B.$$ Indeed, $$\begin{aligned} |\log A| < B & \Rightarrow e^{-B}-1 < A-1 < e^B-1 \\ & \Rightarrow -B+O(B^2) < A-1 < B+O(B^2)\\ & \Rightarrow |A-1| < 2B\end{aligned}$$ for small $B$. To prove Lemma \[l.33\], we will show that $\left| \log \frac{\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))}{\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))}\right|$ is small. By the mean value theorem and using Lemma \[l.333simple\] we get $$\begin{aligned} \left| \log \frac{\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))}{\prod_{s=1}^{i-1}\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))}\right| & = \left|\sum_{s=1}^{i-1} \log \frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l)) - \sum_{s=1}^{i-1} \log \frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))\right| \\ & \leq C \sum_{s=1}^{i-1} \left|\frac{\partial k^{(s)}_\l}{\partial x}(Q_{s}(\l))-\frac{\partial k^{(s)}_\l}{\partial x}(P_{s}(\l))\right| \\ & = C\sum_{s=1}^{i-1} \left| \frac{\partial g_{\omega_{s-1}}}{\partial x}(Q_{s}(\l)) - \frac{\partial g_{\omega_{s-1}}}{\partial x}(P_{s}(\l))\right| \\ & = C\sum_{s=1}^{i-1} \left|\frac{\partial^2 g_{\omega_{s-1}}}{\partial x^2}(V_{s+1})\right|\left|Q_{s}(\l) - P_{s}(\l)\right| \\ & \leq \widetilde{C}\max_{t=1,\dots,m}\| g_t{(x,\l)}\|_{C^2} \sum_{s=1}^{i-1} \left|\prod_{j=s+1}^{n} l^{(s)}\right| \\ & \leq \widetilde{\widetilde{C}}\max_{t=1,\dots,m}\| g_t{(x,\l)}\|_{C^2} \left| \prod_{s=i}^n l^{(s)}\right|\end{aligned}$$ since the terms of the last sum are bounded by a geometrical progression. This proves Lemma \[l.33\]. Therefore we have $$\label{e.eqnew1} |S_2| \leq n{C^{\prime\prime}}\left|\prod_{s=1}^n l^{(s)}\right|\max_{t=1,\dots,m}\| g_t{(x,\l)}\|_{C^2}.$$ Notice that Lemma \[l.33\] implies also that for come constant $\hat{C}>0$ we have $$\label{e.eqnew2} |S_3|\le \hat{C}\left|\prod_{s=1}^n l^{(s)}\right|.$$ Now combining (\[e.last\]), (\[e.eqnew1\]), and (\[e.eqnew2\]) we get $$\begin{aligned} \left| \frac{d}{d\lambda}\phi_{\omega, \tau}(\lambda)\right|=|S_1+S_2+S_3|\ge \left(nC^{*}-n{C^{\prime\prime}}\max_{t=1,\dots,m}\| g_t{(x,\l)}\|_{C^2}-\hat{C}\right)\left|\prod_{s=1}^{n} l^{(s)}\right|.\end{aligned}$$ Therefore one can choose smallness of the $C^2$ norms of $\{g_i\}_{i=1, \ldots, m}$ in (\[e.g\]) so that for some $\delta^*>0$ and all large enough values of $n\in \mathbb{N}$ we have $$\begin{aligned} \label{e.estfinal} \left| \frac{d}{d\lambda}\phi_{\omega, \tau}(\lambda)\right|\ge n\delta^*\left|\prod_{s=1}^{n} l^{(s)}\right|\end{aligned}$$ for any $\omega, \tau\in \Omega$ with $|\omega\wedge\tau|=n$. In particular, if $\omega, \tau\in \Omega_1$ then (\[e.estfinal\]) together with (\[e.abproduct\]) imply that $$\left| \frac{d}{d\lambda}\phi_{\omega, \tau}(\lambda)\right|\ge n\delta^*m^{-\beta n}=n\delta^*m^{-\beta |\omega\wedge\tau|},$$ which implies (\[e.ineq\]) and hence verifies the assumption (\[BlackBox2\]). Finally, the Shannon-McMillan-Breiman Theorem implies that $$-\frac{1}{n}\log \mu([\omega]_n)\to h_{\mu}(\sigma)$$ for $\mu$-a.e. $\omega \in \Omega$. By Egorov’s theorem, there exists a set $\Omega_2\subset \Omega$ with $\mu(\Omega_2) > 1-\varepsilon/2$ such that this convergence is uniform in $\omega\in \Omega_2$. Thus we have $$-\frac{1}{n}\log \mu([\omega]_n)\to h_{\mu}(\sigma) > \gamma\log m$$ uniformly for $\omega\in\Omega_2$. 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--- abstract: | A half-tree is an edge configuration whose superimposition with a perfect matching is a tree. In this paper, we prove a half-tree theorem for the Pfaffian principal minors of a skew-symmetric matrix whose column sum is zero; introducing an explicit algorithm, we fully characterize half-trees involved. This question naturally arose in the context of statistical mechanics where we aimed at relating perfect matchings and trees on the same graph. As a consequence of the Pfaffian half-tree theorem, we obtain a refined version of the matrix-tree theorem in the case of skew-symmetric matrices, as well as a line-bundle version of this result.\ **Keywords**: Pfaffian, half-trees, perfect matchings, Matrix-tree theorem. author: - 'Béatrice de Tilière [^1]' bibliography: - 'survey.bib' title: 'Principal minors Pfaffian half-tree theorem' --- Introduction ============ We prove a half-tree theorem for the Pfaffian principal minors of a skew-symmetric matrix whose column sum is zero. This is a Pfaffian version of the classical matrix-tree theorem [@Kirchhoff], see also [@Chaiken] and references therein. Introducing an explicit algorithm, we give a constructive proof of our result and a full characterization of half-trees involved. A precise statement of our main theorem, as well as consequences for the determinant, are given in Section \[sec:11\] of the introduction. An outline of the paper is provided in Section \[sec:13\]. Motivations for proving such a result come from statistical mechanics and are exposed in Section \[sec:12\]. Statement of main result {#sec:11} ------------------------ Let $V^R=V\cup R$, where $V=\{1,\dots,n\}$, $R=\{n+1,\dots,n+r\}$ and $n$ is even. Let $A^R=(a_{ij})_{\{i,j\,\in V^R\}}$ be a skew-symmetric matrix whose column sum is zero, *i.e.* satisfying $ \forall\,i\in V^R,\; \sum_{j\in V^R}a_{ij}=0. $ Denote by $A=(a_{ij})_{\{i,j\,\in V\}}$ the matrix obtained from $A^R$ by removing the $r$ last lines and columns. The matrix $A$ is also skew-symmetric and the *Pfaffian* of $A$, denoted $\Pf(A)$, is defined as: $$\Pf(A)=\frac{1}{2^{\frac{n}{2}}\bigl(\frac{n}{2}\bigr)!} \sum_{\sigma\in\Sn}\sgn(\sigma)a_{\sigma(1)\sigma(2)}\dots a_{\sigma(n-1)\sigma(n)},$$ where $\Sn$ is the set of permutations of $\{1,\dots,n\}$. Using the skew-symmetry of the matrix $A$ it is possible to avoid summing over all permutations. Let $\Pn$ be the set of partitions of $\{1,\dots,n\}$ into $n/2$ unordered pairs, also known as the set of *pairings*. A permutation $\sigma\in\Sn$ is a *description* of a pairing $\pi\in\Pn$ if $\{\sigma(1)\sigma(2),\dots,\sigma(n-1)\sigma(n)\}$ represents the pairing. A pairing $\pi$ is described by $2^{\frac{n}{2}}\bigl(\frac{n}{2}\bigr)!$ permutations: there are $2^{\frac{n}{2}}$ ways of ordering elements of the pairs and $\bigl(\frac{n}{2}\bigr)!$ ways of ordering pairs among themselves. Because of the skew-symmetry of the matrix $A$, the quantity: $$\sgn(\sigma)a_{\sigma(1)\sigma(2)}\dots a_{\sigma(n-1)\sigma(n)},$$ is independent of the choice of permutation $\sigma$ describing a given pairing $\pi$. Indeed, choosing another permutation amounts to exchanging elements of a pair or exchanging pairs. The first operation changes the sign of the permutation, which is compensated by the change of sign in the corresponding matrix element. The second operation does not change the sign, and only changes the order of the matrix elements. As a consequence, the Pfaffian can be rewritten as: $$\begin{aligned} \Pf(A)=\sum_{\pi\in\Pn}\sgn(\sigma_\pi)a_{\sigma_\pi(1)\sigma_\pi(2)}\dots a_{\sigma_\pi(n-1)\sigma_\pi(n)},\end{aligned}$$ where $\sigma_\pi$ is any of the $2^{\frac{n}{2}}\bigl(\frac{n}{2}\bigr)!$ permutations describing the pairing $\pi$. If $n$ is odd, then by convention $\Pf(A)=0$. To the matrix $A^R$ one associates the graph $G^R=(V^R,E^R)$, where $E^R=\{ij:\,i,j\in V^R,\,a_{ij}\neq 0\}$. Every oriented edge $(i,j)$ of $G^R$ is assigned a weight $a_{ij}$, thus defining a skew-symmetric weight function on oriented edges. The matrix $A^R$ is the *weighted adjacency matrix* of the graph $G^R$. A *spanning forest* of $G^R$ is an oriented edge configuration of $G^R$, spanning vertices of  $V$, such that each connected component is a tree containing exactly one vertex of $R$. This vertex is taken to be the root and edges of the component are oriented towards it. Equivalently, a spanning forest of $G^R$ is an oriented edge configuration containing no cycle, such that each vertex of $V$ has exactly one outgoing edge of the configuration. A *leaf* of a spanning forest is a vertex with no incoming edge. Let $G=(V,E)$ be the graph naturally associated to the matrix $A$. A *perfect matching* $M_0$ of $G$ is a subset of edges such that each vertex of $V$ is incident to exactly one edge of $M_0$. Note that a perfect matching of $G$ contains exactly $|V|/2$ edges. In the whole of this paper, we suppose that $G$ has at least one perfect matching; if this is not the case, then $\Pf(A)=0$ (see also Section \[sec:sec21\]). We let $\M$ denote the set of perfect matchings. Let $F$ be a spanning forest of $G^R$, then $F$ is said to be *compatible* with $M_0$ if it consists of the $|V|/2$ edges of $M_0$ and of $|V|/2$ edges of $E^R\setminus M_0$. The oriented edge configuration $F\setminus M_0$ is referred to as a *half-spanning forest*. In the specific case where $R$ is reduced to a point, then $F$ is a tree and $F\setminus M_0$ is referred to as a *half-tree*. <span style="font-variant:small-caps;">Example</span>. Let $V^R=\{1,2,3,4,5\}$, $V=\{1,2,3,4\}$, $R=\{5\}$. Consider the graphs $G^R$ and $G$ pictured in Figure \[fig:fig6\] below. A choice of perfect matching $M_0$ of $G$ is pictured in white, and $F_1,F_2,F_3$ are examples of spanning trees of $G^R$ compatible with $M_0$. ![Spanning trees compatible with $M_0$.[]{data-label="fig:fig6"}](fig6.pdf){height="2.5cm"} Here is the statement of our main theorem. \[thm:main\] Let $A^R$ be a skew-symmetric matrix of size $(n+r)\times (n+r)$, whose column sum is zero, such that $n$ is even; and let $A$ be the matrix obtained from $A^R$ by removing the $r$ last lines and columns. Let $G^R$ and $G$ be the graphs naturally constructed from the matrices $A^R$ and $A$, respectively. For every perfect matching $M_0$ of $G$, the Pfaffian of $A$ is equal to: $$\Pf(A)=\sum_{F\in \F(M_0)}\sgn(\sigma_{M_0(F\setminus M_0)})\prod_{e\in F\setminus M_0}a_{e},$$ where $a_e$ is the coefficient of the matrix $A^R$ corresponding to the oriented edge $e$; $\sgn(\sigma_{M_0(F\setminus M_0)})$ is the sign of the permutation $\sigma_{M_0(F\setminus M_0)}$ of Definition \[def:def2\] below; $\F(M_0)$ is the set of spanning forests of $G^R$ compatible with $M_0$, satisfying Condition *(C)* of Definition \[def:def0\] below. \[def:def2\] Let $F$ be a spanning forest of $G^R$ compatible with $M_0$. The orientation of $F$ induces an orientation of edges of the perfect matching $M_0$, and we let $(i_1,i_2),\dots,(i_{n-1},i_n)$ be a description of the oriented matching. Then, $\sigma_{M_0(F\setminus M_0)}$ is the permutation: $$\sigma_{M_0(F\setminus M_0)}= \begin{pmatrix} 1&2&\dots&n\\ i_1&i_2&\dots&i_n \end{pmatrix}.$$ Note that the interchange of two pairs does not change the sign of the permutation. Here is the algorithm used to characterize half-spanning forests which contribute to $\Pf(A)$. **Trimming algorithm** : a spanning forest $F$ of $G^R$ compatible with $M_0$. : $F_1=F$. Since vertices of the graph $G^R$ are labeled by $\{1,\dots,n+r\}$, there is a natural order on vertices of any subset of $V^R$. We let ${\ell}_1^i$ be the leaf of $F_i$ with the largest label, and consider the connected component containing ${\ell}_1^i$. Start from ${\ell}_1^i$ along the unique path joining ${\ell}_1^i$ to the root of this component, until the first time one of the following vertices is reached: - the root vertex, - a fork, that is a vertex with more than one incoming edge, - a vertex which is smaller than the leaf ${\ell}_1^i$. This yields a loopless path $\lambda_{{\ell}_1^i}$ starting from ${\ell}_1^i$, of length $\geq 1$. Let $F_{i+1}=F_i\setminus \lambda_{{\ell}_1^i}$. If $F_{i+1}$ is empty, stop; else go to Step $i+1$. : since edges are removed at every step, and since $F$ contains a finite number of edges, the trimming algorithm ends in finite time $N$. \[def:def0\] A spanning forest $F$ compatible with $M_0$ is said to satisfy *Condition* (C) if each of the paths $\lambda_{{\ell}_1^1},\dots,\lambda_{{\ell}_1^N}$ obtained from the trimming algorithm, starts from an edge of $M_0$ and has even length. We let $\F(M_0)$ be the set of spanning forests compatible with $M_0$, satisfying Condition (C). <span style="font-variant:small-caps;">Example</span>. Applying the trimming algorithm to each of the spanning forests $F_1,F_2,F_3$ of Figure \[fig:fig6\] yields: $$\begin{aligned} F_1:&\quad\text{Step 1: }{\ell}_1^1=2,\lambda_{2}=2,3,1. \quad\text{Step 2: }{\ell}_1^2=1,\lambda_1=1,4,5.\\ F_2:&\quad\text{Step 1: }{\ell}_1^1=2,\lambda_2=2,3,5. \quad\text{Step 2: }{\ell}_1^2=1,\lambda_1=1,4,5.\\ F_3:&\quad\text{Step 1: }{\ell}_1^1=4,\lambda_4=4,1. \quad\quad\text{Step 2: }{\ell}_1^2=1,\lambda_1=1,2,3,5.\end{aligned}$$ The spanning trees $F_1$ and $F_2$ satisfy Condition (C) but not $F_3$.\ $\,$ - It is interesting to note that taking different perfect matchings $M_0$ yields different families of half-spanning forests. It is not clear a priori, without using the Pfaffian half-tree theorem, that these families should have the same total weight. - Suppose that we change the labeling of the vertices. Let $\tilde{A^R}$ be the corresponding skew-symmetric adjacency matrix, and $\tilde{A}$ be the matrix obtained by removing the $r$ last lines and columns. As long as the re-labeling does not affect vertices of $R$, the matrix $\tilde{A}$ is obtained from the matrix $A$ by exchanging lines and columns, so that $\Pf(\tilde{A})=\pm\Pf(A)$. Applying the Pfaffian half-tree theorem to the matrices $A$ and $\tilde{A}$ nevertheless yields a different set of half-spanning forests and again, it is not clear a priori that they should have the same total weight in absolute value. Note that taking other principal minors amounts to changing the labeling of the vertices. - In the paper [@MasbaumVaintrob], Masbaum and Vaintrob assign to a weighted 3-uniform hypergraph a skew-symmetric matrix whose column sum is zero, and prove that the Pfaffian of any principal minor of this matrix enumerates signed spanning trees of the 3-uniform hypergraph. The matrix considered by Masbaum and Vaintrob satisfies the assumptions of Theorem \[thm:main\], implying that the Pfaffian half-tree theorem can also be used. This naturally raises the question of possible connections between spanning trees of 3-graphs and half-spanning trees of Theorem \[thm:main\]. A detailed account of this question, illustrated by examples, is provided in Appendix \[App:AppendixA\]. Our conclusion is that both theorems can be seen as related to half-spanning trees, but the latter are of a very different nature. The Pfaffian half-tree theorem takes its full meaning for (regular) graphs. It can also be applied for 3-graphs, but the result obtained in that case is rather different from the one of Masbaum and Vaintrob, and not naturally connected to spanning trees of 3-graphs. Using the fact that the determinant of a skew-symmetric matrix is the square of the Pfaffian, we obtain the following corollary. \[cor:main\] Let $A^R$ be a skew-symmetric matrix of size $(n+r)\times (n+r)$, whose column sum is zero, such that $n$ is even; and let $A$ be the matrix obtained from $A^R$ by removing the $r$ last lines and columns. Let $G^R$ and $G$ be the graphs naturally constructed from the matrices $A^R$ and $A$ respectively. The determinant of the matrix $A$ is equal to: $$\det(A)=\sum_{M_0\in\M}\sum_{F\in\F(M_0)}\prod_{e\in F}a_e,$$ where $a_e$ is the coefficient of the matrix $A^R$ corresponding to the oriented edge $e$, and $\F(M_0)$ is the set of spanning forests compatible with $M_0$, satisfying Condition *(C)*. $\,$ - The fact that principal minors of a skew-symmetric matrix whose column sum is zero, count spanning forests is also a consequence of the more general all-minors matrix-tree theorem (which holds for any matrix whose column sum is zero). A combinatorial way of proving this result is to use the explicit expansion of configurations due to Chaiken [@Chaiken]. This method is not satisfactory in our context, since it does not shed a light on how spanning forests are obtained from double perfect matchings, which is what we aim for, see Section \[sec:12\]. Indeed, the idea of Chaiken’s proof is to expand terms on the diagonal of the matrix and show that only spanning forests remain. In the case of skew-symmetric matrices, since diagonal terms are $0$ this amounts to ‘artificially’ creating configurations which do not exist. As a result of our proof, we explicitly construct spanning forests from double perfect matchings, and identify a specific family of spanning forests counted by principal minors. In particular, this implies that in the case of skew-symmetric matrices, specific cancellations occur within spanning forests of the general matrix-tree theorem, a fact hard to establish without using Corollary \[cor:main\]. - An intrinsic definition of $\cup_{M_0\in\M}\F(M_0)$, not using reference perfect matchings, is given in Remark \[rem:main\] of Section \[sec:sec32\]. - A line bundle version of this result, in the spirit of [@Forman] and [@KenyonVectorBundle], is proved in Section \[sec:sec33\], see Corollary \[thm:linebundle\]. Outline of the paper {#sec:13} -------------------- - . In Section \[sec:sec21\], we state the interpretation of the Pfaffian as counting signed perfect matchings of the graph $G$. Fixing a reference perfect matching $M_0$, we then introduce an explicit algorithm, which constructs from the superimposition of $M_0$ and a generic perfect matching $M$ counted by the Pfaffian, a family of half RC-spanning forests whose connected components are trees rooted on vertices of $R$, or on cycles of even length $\geq 4$; and whose total weight is equal to the contribution of $M$ to the Pfaffian. The main tool of the algorithm is the ‘opening’ of doubled edges procedure, described in Section \[sec:sec23\]. Step 1 of the algorithm is exposed in Section \[sec:sec24\], and the complete algorithm is the subject of Section \[sec:sec25\]. A characterization of configurations obtained is given in Section \[sec:sec26\]. - . Section \[sec:sec31\] consists in the proof of Theorem \[thm:main\]. The idea is to show that the contribution of half RC-spanning forests constructed above, having connected components rooted on cycles of length $\geq 4$ cancel, and that only the contribution of spanning forests (rooted on vertices of $R$) remains. The characterization of configurations obtained from the algorithm is also simplified in the case of spanning forests, yielding the trimming algorithm of Section \[sec:11\] of the introduction. The proof of Corollary \[cor:main\] is the subject of Section \[sec:sec32\]. Finally, in Section \[sec:sec33\], Corollary \[thm:linebundle\] proves a line bundle version of the matrix-tree theorem for skew-symmetric matrices of Corollary \[cor:main\]. A question from statistical mechanics {#sec:12} ------------------------------------- As stated in the introduction, the Pfaffian half-tree theorem \[thm:main\] is a Pfaffian version of the classical matrix-tree theorem of Kirchhoff. One of its interesting features is that half-trees involved satisfy specific conditions characterized by the trimming algorithm, allowing for a refinement of the matrix-tree theorem in the case of skew-symmetric matrices. As such, the Pfaffian half-tree theorem is a standalone result. It nevertheless answers a question raised when working on the paper [@deTiliereCRSF] in the field of statistical mechanics. In the paper [@deTiliereCRSF] we prove an explicit relation, on the level of configurations, between two models of statistical mechanics: the dimer model on the Fisher graph corresponding to the low temperature expansion of the critical Ising model (through Fisher’s correspondence [@Fisher]), and spanning forests. The question raised does not rely on the Fisher graph and can be rephrased in the following, more general framework. In the setting of statistical mechanics, a perfect matching of a graph is known as a *dimer configuration*. Assigning non-negative weights to edges of the graph naturally defines a weight for each dimer configuration (by taking the product of the edge-weights present in the configuration) and a probability measure on all dimer configurations of $G$, thus yielding a statistical mechanics model. The dimer model on planar graphs has been the subject of extensive studies in the last 50 years, and of huge progresses in the last 15 years, see [@KenyonLectures] for an overview. A *double dimer configuration* is the superimposition of two dimer configurations. It consists of a collection of disjoint cycles covering all vertices of the graph. This is because, by definition of a dimer configuration, each vertex is incident to exactly one edge of each of the two dimer configurations, so that in the superimposition, each vertex has degree exactly two. Our goal is to explicitly construct spanning forests from double dimer configurations when the model is critical, and to do so on the same graph, thus proving an unexpected relation, on the level of configurations, between two models of statistical mechanics. This relation is unexpected because configurations of the first model consist of cycles, and those of the second contain no cycle, so that they appear to be of a very different nature. When the graph is planar, dimer configurations are counted by the Pfaffian of the *Kasteleyn matrix* [@Kasteleyn; @TemperleyFisher], which is a weighted adjacency matrix of an oriented version of the graph; this matrix is skew-symmetric by construction. It is a general fact that the Pfaffian of an adjacency matrix counts signed perfect matchings. Signs of perfect matchings come from coefficients of the matrix and from the signs of permutations naturally assigned to matchings, see Section \[sec:sec21\]. The contribution of [@Kasteleyn; @TemperleyFisher] is to prove that the orientation of the graph can be chosen so that signs cancel, implying that all perfect matchings appear with the same sign. The square of the Pfaffian of the Kasteleyn matrix, which is the determinant of the matrix, counts double dimer configurations: when expanding the product, each term consists of two dimer configurations, their superimposition is a double dimer configuration. In the case of the dimer model corresponding to the critical Ising model, the column sum of the Kasteleyn matrix is zero (when the graph is embedded on the torus), a fact related to the model being *critical*. Let us give a little hint at what criticality is. The Ising model is a model of ferro-magnetism: a magnet is represented by a graph, vertices of the graph can take two possible values $\pm1$, and an external temperature influences the system. When the temperature is zero, all spins are equal to $+1$ or $-1$; and when the temperature is very high, the configuration is completely random. At a specific temperature, referred to as the *critical* one, the system undergoes a phase transition and has a very interesting and rich behavior, see [@ChelkakSmirnov:ising]. In the dimer interpretation of the Ising model [@Fisher], being critical is related to the fact that a certain polynomial in two complex variables has zeros on the unit torus [@Li:critical; @CimasoniDuminil]. This polynomial is the determinant of a modified weight Kasteleyn matrix, and it has zeros on the unit torus precisely when the column sum of the matrix is zero. This motivates our choice of taking column-sum equal to zero. Our initial question which was constructing spanning forests from double dimer configurations when the model is critical thus translates into: given a Kasteleyn matrix whose column sum is zero, how are spanning forests obtained from double dimer configurations counted by the determinant of the matrix. It turned out that the only feature required of the Kasteleyn matrix is that of being skew-symmetric, the specific orientation of the graph did not play a role, thus taking us away from the setting of statistical mechanics. The question thus transformed into: how are spanning forests obtained from the signed superimpositions of perfect matchings counted by the determinant of a skew-symmetric matrix whose column sum is zero. We obtained more than what we expected, since we have a result on the Pfaffian. Theorem \[thm:main\] proves that principal minors of the Pfaffian of a general skew-symmetric matrix whose column sum is zero count a specific family of half-spanning forests, and half-spanning forests are explicitly constructed from perfect matchings. Corollary \[cor:main\] proves that principal minors of the determinant of such a matrix count a family of spanning forests, and the latter are explicitly constructed from superimposition of perfect matchings. Specifying this result to the case of planar graphs or graphs embedded on the torus, and Kasteleyn matrices, answers our initial question. The main drawback of our result in the context of statistical mechanics is that, even when the matrix is Kasteleyn and perfect matchings all have positive weights, corresponding spanning forests might have negative weights. To close this section on statistical mechanics, let us also mention the work of Temperley [@Temperley], Kenyon, Propp and Wilson [@KPW] proving that spanning trees of planar graphs are in bijection with dimer configurations of a related bipartite graph. The proof consists in a one-to-one correspondence between configurations. Although their result involves the same kind of objects, the two are quite different in spirit. In our case, perfect matchings and trees live on the same graph, the graph must not be bipartite nor even planar, the weight function on edges of the graph must not be positive, but the column sum must be zero. From matchings to half $RC$-rooted spanning forests {#sec:sec2} =================================================== Let us recall the setting: $A^R$ is a skew-symmetric matrix of size $(n+r)\times (n+r)$, whose column sum is zero, such that $n$ is even; and $A$ is the matrix obtained from $A^R$ by removing the $r$ last lines and columns; $G^R$ and $G$ are the graphs naturally constructed from the matrices $A^R$ and $A$ in Section \[sec:11\] of the introduction. \[def:def1\] An *$RC$-rooted spanning forest*, referred to as an RCRSF is an oriented edge configuration of $G^R$ spanning vertices of $G$, such that each connected component is, either a tree rooted on a vertex of $R$, or a tree rooted on a cycle of $G$, which we refer to as a *unicycle*. Edges of each of the components are oriented towards its root, and edges of the cycles are oriented in one of the two possible directions. Let $M_0$ be a reference perfect matching of $G$. An RCRSF $F$ is said to be *compatible with $M_0$*, if it consists of the $|V|/2$ edges of $M_0$, and of $|V|/2$ edges of $E^R\setminus M_0$. Moreover cycles of uni-cycles have even length $\geq 4$, and alternate between edges of $M_0$ and $F\setminus M_0$. The oriented edge configuration $F\setminus M_0$ is referred to as a *half-RCRSF*. In Section \[sec:sec21\], we give the graphical interpretation of the Pfaffian of the matrix $A$ as counting signed perfect matchings of $G$. Let $M$ be a generic perfect matching counted by the Pfaffian and $M_0$ be a fixed reference perfect matching of $G$. In Sections \[sec:sec24\] and \[sec:sec25\], we introduce an explicit algorithm which constructs, from the superimposition of $M_0$ and $M$, a family of half $RC$-rooted spanning forests compatible with $M_0$, whose total weight is equal to the contribution of $M$ to the Pfaffian. In Section \[sec:sec26\], we characterize $RC$-spanning forests obtained. Notations used are given in Section \[sec:sec22\]. The main graphical idea of the algorithm is the subject of Section \[sec:sec23\]. Graphical interpretation of the Pfaffian {#sec:sec21} ---------------------------------------- Recall that $\Pn$ denotes the set of pairings of $\{1,\dots,n\}$, and let $\M$ be the set of perfect matchings of $G$. Observing that every perfect matching of $G$ corresponds to a pairing of $\Pn$, and that pairings of $\Pn$ which do not correspond to perfect matchings of $G$ contribute $0$ to the Pfaffian, we can rewrite $\Pf(A)$ as: $$\Pf(A)=\sum_{M\in \M}\sgn(\sigma_M)a_{\sigma_M(1)\sigma_M(2)}\dots a_{\sigma_M(n-1)\sigma_M(n)},$$ where $\sigma_{M}$ is a permutation such that $\{\sigma_M(1)\sigma_M(2),\dots,\sigma_M(n-1)\sigma_M(n)\}$ is a description of the perfect matching $M$. Choosing the permutation $\sigma_M$ amounts to choosing an order for the $n/2$ pairs and an order for the two elements of each of the pairs, meaning that there are $(\frac{n}{2})! 2^{\frac{n}{2}}$ choices. Exchanging two pairs does not change the sign nor the corresponding coefficients of the matrix, whereas changing two elements of a pair changes the sign of the permutation and the sign of the corresponding element of the matrix. As a consequence, the global sign is unchanged, and fixing the sign of the permutation amounts to choosing an orientation of edges of the perfect matching. We now specify the choice of sign of the permutation $\sigma_M$ by choosing an orientation of edges of $M$. Let $M_0$ be a fixed reference matching of $G$. The superimposition of $M_0$ and $M$, denoted by $M_0\cup M$, consists of disjoint doubled edges (covered by both $M_0$ and $M$) and alternating cycles of even length $\geq 4$, covering all vertices of $G$. Let us, for the moment, consider doubled edges as cycles of length 2. The orientation of the superimposition $M_0\cup M$ is fixed by the following rule: for each cycle, the orientation is compatible with that of the edge $({\ell}_1,{\ell}_1')$, where ${\ell}_1$ is the smallest vertex of the cycle, and ${\ell}_1'$ is its partner in $M_0$. This yields an orientation of edges of $M$, and thus a choice of $\sigma_M$. This procedure also gives an orientation of edges of $M_0$, and thus a choice of $\sigma_{M_0}$. Let us also denote by $M_0\cup M$ the oriented superimposition. <span style="font-variant:small-caps;">Example (Figure \[fig:fig8\]</span>). In white is a choice of reference perfect matching $M_0$ and in black are the three possible matchings $M_1,M_2,M_3$ of the graph $G$. Edges of the respective superimpositions are oriented according to the rule described above. ![Oriented superimposition.\[fig:fig8\]](fig8.pdf){height="2.5cm"} Let $\sigma_{M_0\cup M}$ be the permutation whose cyclic decomposition corresponds to cycles of the superimposition $M_0\cup M$. Then, $$\sgn(\sigma_{M_0\cup M})=(-1)^{|\D(M_0\cup M)|}(-1)^ {|\C(M_0\cup M)|},$$ where $\D(M_0\cup M)$ is the set of doubled edges of $M_0\cup M$ and $\C(M_0\cup M)$ is the set of alternating cycles of length $\geq 4$ of $M_0\cup M$. Note that this sign does not depend on the orientation of the cycles. Following Kasteleyn [@Kasteleyn], the signs of the permutations $\sigma_{M_0}$ and $\sigma_M$ are related as follows: $$\begin{aligned} \sgn(\sigma_M)&=\sgn(\sigma_{M_0})\sgn(\sigma_{M_0\cup M})\\ &=\sgn(\sigma_{M_0})\,(-1)^{|\D(M_0\cup M)|}(-1)^{|\C(M_0\cup M)|}.\end{aligned}$$ Writing $\sigma_{M_0}$ as $\sigma_{M_0(M)}$ to remember that our choice of orientation of edges of $M_0$ depends on $M$, the Pfaffian of $A$ can be expressed as: $$\label{equ:pfaffian} \Pf(A)=\sum_{M\in\M} w_{M_0}(M),$$ where $$\begin{aligned} w_{M_0}(M)&=\sgn(\sigma_{M_0(M)})(-1)^{|\D(M_0\cup M)|}(-1)^{|\C(M_0\cup M)|} \prod_{e\in M}a_{e},\end{aligned}$$ and $a_e$ is the coefficient of the matrix $A$ corresponding to the oriented edge $e$ of $M$. Notations {#sec:sec22} --------- Let $M_0$ be a fixed reference perfect matching of the graph $G$, $M$ be a generic perfect matching, and $M_0\cup M$ be the oriented superimposition of $M_0$ and $M$ constructed in Section \[sec:sec21\]. In order to shorten notations, we write $\D$ instead of $\D(M_0\cup M)$ for the set of doubled edges of the superimposition, $\C$ instead of $\C(M_0\cup M)$ for the set of cycles of length $\geq 4$, and $w(M)$ instead of $w_{M_0}(M)$. We now introduce definitions and notations used in the algorithm of Sections \[sec:sec24\] and \[sec:sec25\]. Let $V^{\C}$ denote the set of vertices of $V$ which belong to a cycle of $\C$. For every subset $\D'$ of doubled edges of $\D$, let $V^{\D'}$ denote the set of vertices of $V$ which belong to doubled edges of $\D'$. Every vertex $i\in V^{\D}$ belongs to a doubled edge covering vertices $i$ and $i'$ of $\D$. We denote by $\eb_i$ (or $\eb_i'$) this doubled edge and define $V_{i}$ to be the set of vertices in the full graph $G^R$, adjacent to $i'$ other than $i$. For every subset $\D'$ of $\D$, denote by $V_{i}^{\D'}$ the set of vertices of $V_{i}$ which belong to doubled edges of $\D'$, and by $(V_i^{\D'})^c$ those which don’t. Then $V_{i}$ can be partitioned as: $V_{i}=V_{i}^{\D'}\cup (V_i^{\D'})^c.$ Idea of the algorithm {#sec:sec23} --------------------- The idea of the algorithm is to use the reference configuration $M_0$ as a skeleton for opening up doubled edges of the superimposition  $M_0~\cup~M$. Indeed, because of the condition $\sum_{j\in V^R} a_{ij}=0$, configurations of Figure \[fig:fig7\] have opposite weights. ![‘Opening’ of doubled edges procedure: $a_{i'i}=-\sum\limits_{j\in V_i}a_{i'j}$.\[fig:fig7\]](fig7.pdf){width="10.5cm"} There are two main difficulties in realizing this procedure: the first is that there is, a priori, no natural way of deciding whether to ‘open’ up the doubled edge at the vertex $i$ or at the vertex $i'$. The second is that we want to keep track of configurations constructed, show that we obtain $RC$-rooted spanning forests, characterize them and prove that only spanning forests remain. It turns out that the ‘opening’ procedure depends strongly on the labeling of vertices. Algorithm: Step 1 {#sec:sec24} ----------------- Recall that the goal of the algorithm is to construct, from the superimposition $M_0\cup M$ of a reference perfect matching $M_0$ and a generic perfect matching $M$, a family of half-$RC$-rooted spanning forests of $G^R$ compatible with $M_0$, whose total weight is equal to the contribution of $M$ to the Pfaffian. In this section, we introduce the first step of the algorithm, setting rules for the opening up of doubled edges of $M_0\cup M$. The complete algorithm, which in essence consists of iterations of Step 1, is the subject of Section \[sec:sec25\]. : oriented superimposition $M_0\cup M$. : if the superimposition $M_0\cup M$ consists of cycles only, that is if the set $\D$ is empty, let ${\cal O}_0=\{M\}$ and stop. Else, let ${\cal O}_0=\{\emptyset\}$ and go to the first iteration. <span style="font-variant:small-caps;">Example (Figure \[fig:fig1\]).</span> Consider Figure \[fig:fig1\] as input of the algorithm. The algorithm will be explicitly performed on this example, throughout Sections \[sec:sec24\] and \[sec:sec25\]. ![Input $M_0\cup M$ of Step 1.\[fig:fig1\]](fig1.pdf){height="2cm"} Since $M_0\cup M$ contains doubled edges, the output ${\cal O}_0$ is $\{\emptyset\}$. - Define ${\ell}_1=\min\{i\in V:\text{ $i$ belongs to a doubled edge of $\D$} \}$. Then ${\ell}_1$ is the partner of a vertex ${\ell}_1'$ in $M_0$ and $M$. By our choice of orientation for $M_0$ and $M$, the edge ${\ell}_1{\ell}_1'$ is oriented from ${\ell}_1$ to ${\ell}_1'$ in $M_0$ and from ${\ell}_1'$ to ${\ell}_1$ in $M$. For every ${\ell}_2\in V_{{\ell}_1}$, define: $$\begin{aligned} M_{{\ell}_1,{\ell}_2}&=\{M\setminus ({\ell}_1',{\ell}_1)\}\cup \{({\ell}_1',{\ell}_2)\}\\ w(M_{{\ell}_1,{\ell}_2})&=\sgn(\sigma_{M_0(M)})(-1)^{|\D|-1}(-1)^{|\C|}\prod_{e \in M_{{\ell}_1,{\ell}_2}}a_{e}.\end{aligned}$$ <span style="font-variant:small-caps;">Example (Figure \[fig:fig2\])</span>: ${\ell}_1=1$, $\ell_1'=4$. By definition, see Section \[sec:sec22\], $V_{\ell_1}$ consists of vertices incident to $\ell_1'=4$ other than $\ell_1=1$, that is, $V_{{\ell}_1}=V_1=\{2,3,5\}$. This yields configurations $M_{1,2}$, $M_{1,3}$, $M_{1,5}$. ![From left to right: black edges are the oriented edge configurations $M_{1,2}$, $M_{1,3}$, $M_{1,5}$.\[fig:fig2\]](fig2.pdf){height="2.5cm"} - Let $D_{{\ell}_1}$ be the set of doubled edges $\D\setminus\{\eb_{{\ell}_1}\}$. Then, the set $V_{{\ell}_1}$ can be partitioned as the set of vertices of $V_{{\ell}_1}$ which belong to a doubled edge of $\D_{{\ell}_1}$ and the set of those which don’t. Using notations of Section \[sec:sec22\], this can be rewritten as: $V_{{\ell}_1}=V_{{\ell}_1}^{\D_{{\ell}_1}}\cup (V_{{\ell}_1}^{\D_{{\ell}_1}})^c$. The output of Iteration 1 is the set of configurations $M_{{\ell}_1,{\ell}_2}$ such that ${\ell}_2$ does not belong to a doubled edge of $\D_{{\ell}_1}$: $$\begin{aligned} {\cal O}_1&=\bigcup_{{\ell}_{2}\in (V_{{\ell}_{1}}^{\D_{{\ell}_1}})^c} M_{{\ell}_1,{\ell}_2},\\ w({\cal O}_1)&=\sum_{M_{{\ell}_1,{\ell}_2}\in {\cal O}_1}w(M_{{\ell}_1,{\ell}_2}).\end{aligned}$$ where by convention, if $(V_{{\ell}_{1}}^{\D_{{\ell}_1}})^c =\emptyset$, then ${\cal O}_1=\emptyset$ and $w({\cal O}_1)=0$. - The algorithm continues with configurations $M_{{\ell}_1,{\ell}_2}$ where ${\ell}_2$ belongs to a doubled edge of $\D_{{\ell}_1}$. Formally we have: if $V_{{\ell}_{1}}^{\D_{{\ell}_1}}=\emptyset$, then stop; else, go to Iteration $2$. <span style="font-variant:small-caps;">Example</span>: the set $\D_{{\ell}_1}=\D_{1}$ consists of the doubled edge $23$. As a consequence, the set $V_{{\ell}_1}=V_1=\{2,3,5\}$ is partitioned as $V_1=\{2,3\}\cup\{5\}$, and the output of Iteration $1$ is ${\cal O}_1=\{M_{1,5}\}$. The algorithm continues with $M_{1,2}$ and $M_{1,3}$. For every ${\ell}_2\in V_{{\ell}_1}^{\D_{{\ell}_1}},\dots, {\ell}_{k}\in V_{{\ell}_{k-1}}^{\D_{{\ell}_1,\dots,{\ell}_{k-1}}}$, do the following. - The vertex ${\ell}_k$ is the partner of a vertex ${\ell}_{k}'$ in $M_0$ and $M$ (since $\D_{{\ell}_1,\dots,{\ell}_{k-1}}$ is a subset of $\D$). If ${\ell}_k<{\ell}_k'$, then by our choice of orientation, the edge ${\ell}_k{\ell}_k'$ is oriented from ${\ell}_k$ to ${\ell}_k'$ in $M_0$ and from ${\ell}_k'$ to ${\ell}_k$ in $M_{{\ell}_1,\dots,{\ell}_{k}}$. If ${\ell}_k>{\ell}_k'$, then we change the orientation of this edge in $M_0$ and in $M_{{\ell}_1,\dots,{\ell}_{k}}$. Let us also denote by $M_{{\ell}_1,\dots,{\ell}_{k}}$ this new configuration. This change of orientation has the effect of changing the permutation assigned to $M_0$, and we denote by $\sigma_{M_0(M_{{\ell}_1,\dots,{\ell}_k})}$ this new permutation. It also negates the contribution of $M_{{\ell}_1,\dots,{\ell}_k}$ so that the global contribution remains unchanged. For every ${\ell}_{k+1}\in V_{{\ell}_k}$, define: $$\begin{aligned} M_{{\ell}_1,\dots,{\ell}_{k+1}}&=(M_{{\ell}_1,\dots,{\ell}_k}\setminus ({\ell}_{k}',{\ell}_{k}))\cup ({\ell}_k',{\ell}_{k+1})\nonumber\\ w(M_{{\ell}_1,\dots,{\ell}_{k+1}})&=\sgn(\sigma_{M_0(M_{{\ell}_1,\dots,{\ell} _k})})(-1)^ { |\D|-k } (-1)^ { |\C| } \prod_{e\in M_{{\ell}_1,\dots,{\ell}_{k+1}}}a_{e} \label{equ:weight}\end{aligned}$$ <span style="font-variant:small-caps;">Example (Figure \[fig:fig3\])</span>. Recall that $V_{{\ell}_1}^{\D_{{\ell}_1}}=V_1^{\{23\}}=\{2,3\}$, so that ${\ell}_2\in\{2,3\}$. If ${\ell}_2=2$, then $V_{{\ell}_2}=V_2=\{1,4,5\}$, yielding configurations $M_{1,2,1}$, $M_{1,2,4}$, $M_{1,2,5}$. If ${\ell}_2=3$, then $V_{{\ell}_2}=V_3=\{1,4\}$, yielding configurations $M_{1,3,1}$, $M_{1,3,4}$. ![First line, from left to right, black edges consists of the configurations $M_{1,2,1}$, $M_{1,2,4}$, $M_{1,2,5}$. Second line, from left to right, black edges consists of the configurations $M_{1,3,1}$, $M_{1,3,4}$.\[fig:fig3\]](fig3.pdf){height="5cm"} - Let $\D_{{\ell}_1,\dots,{\ell}_k}$ be the set of doubled edges $\D_{{\ell}_1,\dots,{\ell}_{k-1}}\setminus\{\eb_{{\ell}_k}\}$. Then, the set $V_{{\ell}_{k}}$ can be partitioned as: $ V_{{\ell}_{k}}=V_{{\ell}_{k}}^{\D_{{\ell}_1,\dots,{\ell}_k}} \bigcup (V_{{\ell}_{k}}^{\D_{{\ell}_1,\dots,{\ell}_k}})^c, $ and the output of Iteration $k$ is the set of configurations $M_{{\ell}_1,\dots,{\ell}_{k+1}}$ such that ${\ell}_{k+1}$ does not belong to a doubled edge of $\D_{{\ell}_1,\dots,{\ell}_k}$. $$\begin{aligned} {\cal O}_k&=\bigcup_{{\ell}_2\in V_{{\ell}_1}^{\D_{{\ell}_1}}}\dots \bigcup_{{\ell}_{k}\in V_{{\ell}_{k-1}}^{\D_{{\ell}_1,\dots,{\ell}_{k-1}}}} \bigcup_{{\ell}_{k+1}\in (V_{{\ell}_{k}}^{\D_{{\ell}_1,\dots,{\ell}_k}})^c} M_{{\ell}_1,\dots,{\ell}_{k+1}},\\ w({\cal O}_k)&=\sum_{M_{{\ell}_1,\dots,{\ell}_{k+1}}\in {\cal O}_k}w(M_{{\ell}_1,\dots,{\ell}_{k+1}}).\end{aligned}$$ - If $V_{{\ell}_{k}}^{\D_{{\ell}_1,\dots,{\ell}_k}}=\emptyset$, then stop. Else, go to Step $k+1$. <span style="font-variant:small-caps;">Example</span>: when ${\ell}_2=2$, the set $\D_{1,2}$ is empty so that $V_{2}$ is partitioned as $\{\emptyset\}\cup\{1,4,5\}$ and the contribution to the output ${\cal O}_2$ of Iteration 2 is $M_{1,2,1}$, $M_{1,2,4}$, $M_{1,2,5}$. When ${\ell}_2=3$, the set $\D_{1,3}$ is also empty, implying that $V_{3}$ is partitioned as $\{\emptyset\}\cup\{1,4\}$ and the contribution to the output ${\cal O}_2$ of Iteration 2 is $M_{1,3,1}$, $M_{1,3,4}$. After Iteration 2, for every ${\ell}_2\in V_{{\ell}_1}^{\D_{{\ell}_1}}$, the set $V_{{\ell}_2}^{\D_{{\ell}_1,{\ell}_2}}$ is empty, so that the algorithm stops. Step 1 of the algorithm stops at time $m$ for the first time, if it hasn’t stopped at time $m-1$, and if for every ${\ell}_{2}\in V_{{\ell}_{1}}^{\D_{{\ell}_1}},\dots, {\ell}_{m}\in V_{{\ell}_{m-1}}^{\D_{{\ell}_1,\dots,{\ell}_{m-1}}}$; $V_{{\ell}_{m}}^{\D_{{\ell}_1,\dots,{\ell}_m}}=\emptyset$. This implies in particular that $(V_{{\ell}_{m}}^{\D_{{\ell}_1,\dots,{\ell}_m}})^c=V_{{\ell}_m}$. Since the number of doubled edges decreases by $1$ every time an iteration of the algorithm occurs, and since the number of doubled edges in $\D$ is finite, we are sure that Step 1 of the algorithm stops in finite time. ### Output of Step 1, geometric properties of configurations {#sec:output} The output of Step 1 of the algorithm is the set of configurations $\S_1=\bigcup_{k=0}^{m} {\cal O}_k$. The weight of this set is defined to be $w(\S_1)=\sum_{k=0}^m w({\cal O}_k)$. If the initial superimposition $M_0\cup M$ consists of cycles only, *i.e.* if the set $\D$ is empty, then $m=0$ and $\S_1=\{M\}$. In all other cases, the set $\S_1$ can be rewritten in a more compact way as: $$\displaystyle \S_1= \bigcup_{\gamma_{{\ell}_1}\in\Gamma_{{\ell}_1}}M_{\gamma_{{\ell}_1}},$$ where: $$\begin{aligned} \Gamma_{{\ell}_1}=\Big\{&\gamma_{{\ell}_1}:\,\gamma_{{\ell}_1} \text{ is a path of length $2k$ for some $k\in\{1,\dots,m\}$}:\, \gamma_{{\ell}_1}={\ell}_1,{\ell}_1',\dots,{\ell}_k,{\ell}_k',{\ell}_{k+1},\\ &{\ell}_1=\min\{i\in V:\,i\text{ belongs to a doubled edge of $\D$} \}, \\ &\forall j\in\{2,\dots,k\},\,{\ell}_j\in V_{{\ell}_{j-1}}^{\D_{{\ell}_1,\dots,{\ell}_{j-1}}} \text{ and } {\ell}_j'\text{ is the partner of ${\ell}_j$ in $M_0$ and $M$},\\ &{\ell}_{k+1}\in (V_{{\ell}_{k}}^{\D_{{\ell}_1,\dots,{\ell}_k}})^c\Big\}.\\ M_{\gamma_{{\ell}_1}}=\,&M_{{\ell}_1,\dots,{\ell}_{k+1}}.\end{aligned}$$ Let $\gamma_{{\ell}_1}={\ell}_1,{\ell}_1',\dots,{\ell}_k,{\ell}_k',{\ell}_{k+1}$ be a generic path of $\Gamma_{{\ell}_1}$ for some $k\in\{1,\dots,m\}$, and let $F_{\gamma_{{\ell}_1}}$ denote the superimposition $M_0\cup M_{\gamma_{{\ell}_1}}$. The configuration $F_{\gamma_{{\ell}_1}}$ and the path $\gamma_{{\ell}_1^1}$ satisfy the following properties. $\bullet$ The oriented edge configuration $F_{\gamma_{{\ell}_1}}$: 1. has one outgoing edge at every vertex of $V$, and contains the path $\gamma_{{\ell}_1}$. 2. has $k$ doubled edges less than $M_0\cup M$. $\bullet$ The oriented path $\gamma_{{\ell}_1}$: 1. has even length $2k$, is alternating (meaning that edges alternate between $M_0$ and $M_{\gamma_{{\ell}_1}}$). It starts from the vertex ${\ell}_1$ followed by an edge of $M_0$. 2. The vertex ${\ell}_1$ is the smallest vertex belonging to a doubled edge of $\D$. The $2k$ first vertices of $\gamma_{{\ell}_1}$ are all distinct and the last vertex ${\ell}_{k+1}$ belongs to $(V_{{\ell}_k}^{\D_{{\ell}_1,\dots,{\ell}_k}})^c$. Observing that: $$(V_{{\ell}_k}^{\D_{{\ell}_1,\dots,{\ell}_k}})^c= V_{{\ell}_k}\cap(R\cup V^{\C}\cup\{{\ell}_1,{\ell}_1',\dots,{\ell}_k,{\ell}_k'\}),$$ we deduce that one of the following holds. 3. If ${\ell}_{k+1}\in R$, then $\gamma_{{\ell}_1}$ is a loopless oriented path from ${\ell}_1$ to one of the root vertices of $R$, and ${\ell}_1$ is a leaf of $F_{\gamma_{{\ell}_1}}$. Since $R=\{n+1,\dots,n+r\}$, ${\ell}_1$ is smaller than all vertices of $\gamma_{{\ell}_1}$. 4. If ${\ell}_{k+1}\in V^{\C}$, then $\gamma_{{\ell}_1}$ is a loopless oriented path ending at a vertex of one of the cycles of $\C$ that is, the connected component containing ${\ell}_1$ is a unicycle with a unique branch. The vertex ${\ell}_1$ is a leaf of $F_{\gamma_{{\ell}_1}}$ and is smaller than the $2k$ first vertices of the path, but cannot be a priori compared to vertices of the cycle of $\C$. By construction of the orientation of $M_0\cup M$, see Section \[sec:sec21\], the orientation of the cycle is compatible with that of the edge $(i_1,i_2)$, where $i_1$ is the smallest vertex of the cycle and $i_2$ is its partner in $M_0$. 5. If ${\ell}_{k+1}\in \{{\ell}_1,{\ell}_1',\dots,{\ell}_k,{\ell}_k'\}$, then $\gamma_{{\ell}_1}$ contains a loop of length $\geq 3$. If ${\ell}_{k+1}={\ell}_i$ for some $i\in\{1,\dots,k\}$, then the loop has even length and is alternating and the part of $\gamma_{{\ell}_1}$ to the loop also has even length, is alternating and starts with an edge of $M_0$. Moreover, the orientation of the loop is compatible with the orientation of the edge $({\ell}_i,{\ell}_i')$, and the vertex ${\ell}_1$ is smaller than all vertices of the path to the cycle and smaller than all vertices of the cycle. Note that if ${\ell}_{k+1}\neq {\ell}_1$, then ${\ell}_1$ is a leaf and the connected component containing ${\ell}_1$ is a unicycle with a unique branch. Else, if ${\ell}_{k+1}={\ell}_1$, the connected component is a cycle. If ${\ell}_{k+1}={\ell}_i'$ for some $i\in\{1,\dots,k\}$, then the loop has odd length with two edges of $M$ incident to the vertex ${\ell}_i'$. Observing that the loop in both directions is obtained from Step 1 of the algorithm, and using the fact that the matrix $A$ is skew-symmetric, we deduce that the contributions of these configurations cancel and we remove them from the output of Step 1. Thus we only consider configurations such that ${\ell}_{k+1}={\ell}_i$ for some $i\in\{1,\dots,k\}$. <span style="font-variant:small-caps;">Example</span>. The output $\S_1$ of Step 1 is: $ \S_1=\{M_{1,5},M_{1,2,1}, M_{1,2,4},M_{1,2,5},M_{1,3,1},M_{1,3,4}\}. $ Configurations $M_{1,5},\,M_{1,2,5}$ are in Case (IV)(1), configurations $M_{1,2,1},\,M_{1,3,1}$ are in Case (IV)(3) with even cycles created, and configurations $M_{1,2,4},\,M_{1,3,4}$ are in Case (IV)(3) with odd cycles created. Contributions of $M_{1,2,4}$ and $M_{1,3,4}$ cancel so that they are removed from the output. As a consequence the final output of Step 1 is, see also Figure \[fig:fig9\]: $$\S_1=\{M_{1,5},M_{1,2,1},M_{1,2,5},M_{1,3,1}\},$$ ![Output of Step $1$ of the algorithm.[]{data-label="fig:fig9"}](fig9.pdf){width="8cm"} ### Weight of configurations {#sec:weight} As a consequence of the next two lemmas, we obtain that Step 1 of the algorithm is weight preserving *i.e.* $w(\S_1)=w(M)$, see Corollary \[cor:1\]. \[lem:1\] We have: - $\displaystyle w(M)=\sum_{{\ell}_2\in V_{{\ell}_1}}w(M_{{\ell}_1,{\ell}_2})$.\ - If $m\geq 2$, then for every $k\in\{2,\dots,m\}$ and every ${\ell}_2\in V_{{\ell}_1}^{\D_{{\ell}_1}},\dots, {\ell}_{k}\in V_{{\ell}_{k-1}}^{\D_{{\ell}_1,\dots,{\ell}_{k-1}}}$: $$w(M_{{\ell}_1,\dots,{\ell}_{k}})=\sum_{{\ell}_{k+1}\in V_{{\ell}_{k}}}w(M_{{\ell}_1,\dots,{\ell}_{k+1}}).$$ Suppose that $m\geq 2$, the proof in the other case being similar. For every ${\ell}_{k+1}\in V_{{\ell}_k}$, $M_{{\ell}_1,\dots,{\ell}_{k+1}}=\{M_{{\ell}_1,\dots,{\ell}_k}\setminus({\ell} _k', {\ell}_k)\}\cup\{({\ell}_k',{\ell}_{k+1})\}$, thus: $$\prod_{e\in M_{{\ell}_1,\dots,{\ell}_{k+1}}}a_{e}= \frac{a_{{\ell}_k',{\ell}_{k+1}}}{a_{{\ell}_k',{\ell}_k}}\prod_{e\in M_{{\ell}_1,\dots, {\ell}_k}}a_{e}.$$ By assumption, coefficients of each line of the matrix $A^R$ sum to $0$. Returning to the definition of $V_{{\ell}_k}$, this implies that $\displaystyle\sum_{{\ell}_{k+1}\in V_{{\ell}_k}}a_{{\ell}_k',{\ell}_{k+1}}=-a_{{\ell}_k',{\ell}_k}$. Thus, $$\label{equation:1} \sum_{{\ell}_{k+1}\in V_{{\ell}_k}}\prod_{e\in M_{{\ell}_1,\dots,{\ell}_{k+1}}}a_{e} =-\prod_{e\in M_{{\ell}_1,\dots, {\ell}_k}}a_{e}.$$ Combining Equation with the definition of the weight of configurations given in Equation yields: $$\begin{aligned} \sum_{{\ell}_{k+1}\in V_{{\ell}_k}}w(M_{{\ell}_1,\dots,{\ell}_{k+1}}) &= \sgn(\sigma_{M_0(M_{{\ell}_1,\dots,{\ell}_k})})(-1)^{|\C|}(-1)^{|\D|-k}\sum_{{ \ell}_{k+1} \in V_{{\ell}_k}}\prod_{e \in M_ { {\ell}_1 , \dots,{\ell}_{k+1}}}a_{e}\\ &=\sgn(\sigma_{M_0(M_{{\ell}_1,\dots,{\ell}_k})})(-1)^{|\C|}(-1)^{|\D|-(k-1)} \prod_{e\in M_{{\ell}_1,\dots,{\ell}_k}}a_{e}\\ &=w(M_{{\ell}_1,\dots,{\ell}_k}). \qedhere\end{aligned}$$ \[lem:2\] Suppose $m\geq 2$. Then for every $k\in\{2,\dots,m\}$, $$\sum_{i=k}^{m}w({\cal O}_i)=\sum_{{\ell}_2\in V_{{\ell}_1}^{\D_{{\ell}_1}}}\dots \sum_{{\ell}_{k}\in V_{{\ell}_{k-1}}^{\D_{{\ell}_1,\dots,{\ell}_{k-1}}}} w(M_{{\ell}_1, \dots , {\ell}_ { k} }) .$$ In order to simplify notations let us write, only in this proof, $V_{{\ell}_k}^{\D}$ instead of $V_{{\ell}_k}^{\D_{{\ell}_1,\dots,{\ell}_{k}}}$. Lemma \[lem:2\] is proved by backward induction on $k$. Suppose $k=m$. By definition of the last step of the algorithm, $V_{{\ell}_{m}}^{\D}=\emptyset$, so that $(V_{{\ell}_m}^{\D})^c=V_{{\ell}_m}$ and: $$\begin{aligned} w({\cal O}_m)&=\sum_{{\ell}_2\in V_{{\ell}_1}^\D}\dots \sum_{{\ell}_{m}\in V_{{\ell}_{m-1}}^\D} \sum_{{\ell}_{m+1}\in V_{{\ell}_{m}}}w(M_{{\ell}_1,\dots,{\ell}_{m+1}})\\ &=\sum_{{\ell}_2\in V_{{\ell}_1}^\D}\dots \sum_{{\ell}_{m}\in V_{{\ell}_{m-1}}^\D}w(M_{{\ell}_1,\dots,{\ell}_{m}}),\text{ (by Lemma \ref{lem:1})},\end{aligned}$$ thus proving the case $k=m$. Suppose that the statement is true for some $k\in\{3,\dots,m\}$. By Iteration $k-1$ of Step 1 of the algorithm, we know that: $$w({\cal O}_{k-1})=\sum_{{\ell}_2\in V_{{\ell}_1}^\D}\dots \sum_{{\ell}_{k-1}\in V_{{\ell}_{k-2}}^\D} \sum_{{\ell}_{k}\in (V_{{\ell}_{k-1}}^{\D})^c}w(M_{{\ell}_1,\dots,{\ell}_{k}})$$ Combining this with the induction hypothesis yields: $$\begin{aligned} \sum_{i=k-1}^{m}w({\cal O}_i)&=w({\cal O}_{k-1})+\sum_{i=k}^{m}w({\cal O}_i)\\ &=\sum_{{\ell}_2\in V_{{\ell}_1}^\D}\dots \sum_{{\ell}_{k-1}\in V_{{\ell}_{k-2}}^\D} \Bigl( \sum_{{\ell}_{k}\in(V_{{\ell}_{k-1}}^{\D})^c}+ \sum_{{\ell}_{k}\in V_{{\ell}_{k-1}}^{\D}}\Bigr) w(M_{{\ell}_1,\dots,{\ell}_{k}})\\ &= \sum_{{\ell}_2\in V_{{\ell}_1}^\D}\dots \sum_{{\ell}_{k-1}\in V_{{\ell}_{k-2}}^\D} \sum_{{\ell}_{k}\in V_{{\ell}_{k-1}}}w(M_{{\ell}_1,\dots,{\ell}_{k}})\\ &=\sum_{{\ell}_2\in V_{{\ell}_1}^\D}\dots \sum_{{\ell}_{k-1}\in V_{{\ell}_{k-2}}^\D}w(M_{{\ell}_1,\dots,{\ell}_{k-1}}) \text{ (by Lemma \ref{lem:1})},\end{aligned}$$ proving the statement for $k-1$ and ending the proof of Lemma \[lem:2\]. \[cor:1\] $$w(\S_1)=w(M).$$ Suppose $m\geq 1$. Then, $$\begin{aligned} w(\S_1)&=w({\cal O}_1)+\sum_{k=2}^m w({\cal O}_k)\\ &=w({\cal O}_1)+\sum_{{\ell}_2\in V_{{\ell}_1}^{\D_{{\ell}_1}}}w(M_{{\ell}_1,{\ell}_2}),\text{ (by Lemma \ref{lem:2})}\\ &=\sum_{{\ell}_2\in (V_{{\ell}_1}^{\D_{{\ell}_1}})^c}w(M_{{\ell}_1,{\ell}_2})+\sum_{{\ell}_2\in V_{{\ell}_1}^{\D_{{\ell}_1}}} w(M_{{\ell}_1,{\ell}_2}), \text{ (by definition of ${\cal O}_1$)}\\ &=\sum_{{\ell}_2\in V_{{\ell}_1}}w(M_{{\ell}_1,{\ell}_2})\\ &=w(M), \text{ (by Lemma \ref{lem:1})}.\end{aligned}$$ When $m=0$, $\S_1=\{M\}$, and the conclusion is immediate. Complete algorithm {#sec:sec25} ------------------ Let $M_0$ be a reference perfect matching of the graph $G$ and let $M$ be a generic one. Recall that $\C$ denotes the set of cycles of length $\geq 4$ of the superimposition $M_0\cup M$, and $\D$ denotes its set of doubled edges. In Section \[sec:sec24\], we established Step 1 of the algorithm, starting from the superimposition $M_0\cup M$, yielding a set of oriented edge configurations $\S_1$ through the opening of doubled edges procedure, whose total weight is equal to the contribution of $M$ to the Pfaffian. In this section, we introduce the complete algorithm, which in essence consists of iterations of Step 1 performed until no doubled edges of $\D$ remain. Let us directly handle the following trivial case. If $M_0\cup M$ consists of cycles only, that is, if the set $\D$ is the empty, then the opening of edges procedure does not start, and recall that the output of Step 1 is $\S_1=\{M\}$. The same holds for the complete algorithm and its output is $\T=\{M\}$. ### Step 1 of the complete algorithm Assume that the initial superimposition contains at least one doubled edge, *i.e.* $\D\neq\emptyset$. Notations are complicated by the fact that the algorithm depends on the labeling of the vertices and that iterations of Step 1 depend on previous steps. We thus need many indices to keep track of everything rigorously, but one should keep in mind that, in essence, we are iterating Step 1. Let us add sub/superscripts to Step 1 of Section \[sec:sec24\]. That is, ${\ell}_1$ becomes ${\ell}_1^1$, Iteration $k$ becomes $k_1$ and Step 1 ends at time $m_1$. The set of configurations obtained from Step 1 is $ \S_{1}=\bigcup_{\gamma_{{\ell}_1^1}\in \Gamma_{{\ell}_1^1}}M_{\gamma_{{\ell}_1^1}}$, and its weight is $w(\S_1)=\sum_{\gamma_{{\ell}_1^1}\in\Gamma_{{\ell}_1^1}}w(M_{\gamma_{{\ell}_1^1 }})$. For every $\gamma_{{\ell}_1^1}\in\Gamma_{{\ell}_1^1}$, let $\D_{\gamma_{{\ell}_1^1}}$ be the set of doubled edges of the superimposition $M_0~\cup~ M_{\gamma_{{\ell}_1^1}}$. If $\D_{\gamma_{{\ell}_1^1}}=\emptyset$, then stop; else go to Step $2$. . It is the subset $\T_1$ of $\S_1$, consisting of configurations $M_{\gamma_{{\ell}_1^1}}$ where $\gamma_{{\ell}_1^1}\in\Gamma_{{\ell}_1^1}$, and $\D_{\gamma_{{\ell}_1^1}}$ is empty. Formally, $$\T_1=\bigcup_{\gamma_{{\ell}_1^1}\in(\widetilde{\Gamma}_{{\ell}_1^1})^c} M_{\gamma_{{\ell}_1^1}},$$ where $\widetilde{\Gamma}_{{\ell}_1^1}$ is the set of paths $\gamma_{{\ell}_1^1}$ of $\Gamma_{{\ell}_1^1}$ such that $\D_{\gamma_{{\ell}_1^1}}$ is non-empty. If for all $\gamma_{{\ell}_1^1}\in \Gamma_{{\ell}_1^1}$, the set $\D_{\gamma_{{\ell}_1^1}}$ is non-empty, then $\T_1=\emptyset$. <span style="font-variant:small-caps;">Example</span>. Recall that the output of Step 1 is $\S_1=\{M_{1,5},M_{1,2,1},M_{1,2,5},M_{1,3,1}\}$. The set of doubled edges of the superimposition of $M_0$ and $M_{1,2,1},M_{1,2,5},M_{1,3,1}$ is empty, so that the output of Step 1 of the complete algorithm is $\T_1=\{M_{1,2,1},M_{1,2,5},M_{1,3,1}\}$, and the algorithm continues with the configuration $M_{1,5}$. ### Step $j$ of the complete algorithm, $j\geq 2$ For every $\gamma_{{\ell}_1^1}\in\widetilde{\Gamma}_{{\ell}_1^1},\dots, \gamma_{{\ell}_1^{j-1}}\in\widetilde{\Gamma}_{{\ell}_1^{j-1}}(\gamma_{{\ell}_1^1 },\dots, \gamma_{{\ell}_1^{j-2}})$, perform Step 1 of the algorithm with the initial superimposition $M_0\cup M_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j-1}}}$. That is, define ${\ell}_1^j=\min\{i\in V:\,i \text{ belongs to a doubled edge of } \D_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j-1}}}\}$, and iterate until the algorithm ends at some time $m_j$. Everything works out in the same way because $\D_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j-1}}}$ is a subset of $\D$. The output is the set of oriented edge configurations: $$\S_j(\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j-1}}) = \bigcup_{\gamma_{{\ell}_1^j}\in \Gamma_{{\ell}_1^{j}}(\gamma_{{\ell}_1^1}, \dots,\gamma_{{\ell}_1^{j-1}})}M_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{ j}}},$$ where $$\begin{aligned} \label{equ:gamma1} \Gamma_{{\ell}_1^j}(\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j-1}}) &=\Bigg\{\gamma_{{\ell}_1^j}:\,\gamma_{{\ell}_1^j}\text{ is a path }{\ell}_1^j,{\ell}_1^{j'},\dots,{\ell}_{k_j}^j,{\ell}_{k_j}^{j'},{\ell}_{k_j+1} ^j, \text{ for some $k_j\in\{1,\dots,m_j\}$},\nonumber\\ &\text{such that } {\ell}_1^j=\min\{i\in V:\,i \text{ belongs to a doubled edge of } \D_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j-1}}}\}\nonumber\\ &\forall i\in\{2,\dots,k_j\},\,{\ell}_i^j\in V_{{\ell}_{i-1}^j}^{ \D_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j-1}},{\ell}_1^j,\dots,{\ell}_ {i-1}^j} },\nonumber\\ &\text{ and }{\ell}_i^{j'}\text{ is the partner of ${\ell}_i^j$ in $M_0$ and $M$},\nonumber\\ & {\ell}_{k_j+1}^j\in \Bigl(V_{{\ell}_{k_j}^j}^{ \D_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j-1}},{\ell}_1^j,\dots,{\ell}_ {k_j}^j}} \Bigr)^c \Bigg\}.\\ M_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j}}} =\,&M_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j-1}},{\ell}_1^j, \dots,{\ell}_{k_j+1}^j},\nonumber\\ w(M_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j}}})&= \sgn(\sigma_{M_0(M_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j-1}},{\ell} _1^j,\dots,{\ell}_ { k_j}^j}) })(-1)^{|\D|-(k_1+\dots+k_j)}(-1)^{|\C|} \prod_{e\in M_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j}}}}a_{e}.\nonumber\end{aligned}$$ For every $\gamma_{{\ell}_1^j}\in \Gamma_{{\ell}_1^j}(\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j-1}})$, do the following: if $\D_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j}}$ is empty, stop; else go to Step $j+1$. . Let $\T_j(\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j-1}})$ be the subset of $\S_j(\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j-1}})$, consisting of configurations $M_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j}}$ such that $\D_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j}}$ is empty. Formally, $$\T_j(\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j-1}})= \bigcup_{ \gamma_{{\ell}_1^j}\in(\widetilde{\Gamma}_{{\ell}_1^j}(\gamma_{{\ell}_1^1}, \dots, \gamma_{{\ell}_1^{j-1}}))^c }M_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j}},$$ where $\widetilde{\Gamma}_{{\ell}_1^j}(\gamma_{{\ell}_1^1},\dots, \gamma_{{\ell}_1^{j-1}})$ is the subset of paths $\gamma_{{\ell}_1^j}$ of $\Gamma_{{\ell}_1^j}(\gamma_{{\ell}_1^1},\dots, \gamma_{{\ell}_1^{j-1}})$ such that $\D_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j}}$ is non-empty. If $\D_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j}}$ is non- empty, then $\T_j(\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j-1}})=\emptyset$. Then, the output $\T_j$ of Step $j$ of the complete algorithm is: $$\T_j=\bigcup_{\gamma_{{\ell}_1^1}\in \widetilde{\Gamma}_{{\ell}_1^1}} \dots \bigcup_{\gamma_{{\ell}_1^{j-1}}\in\widetilde{\Gamma}_{{\ell}_1^{j-1}}(\gamma_{{ \ell}_1^1}, \dots, \gamma_{{\ell}_1^{j-2}} )} \bigcup_{\gamma_{{\ell}_1^j}\in (\widetilde{\Gamma}_{{\ell}_1^j}(\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{ j-1}}))^c} M_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{j}}}.$$ For convenience, we shall also use the notation $\Gamma_j$ for the paths $(\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j})$ involved in $\T_j$, *i.e*: $$\T_j= \bigcup_{(\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j})\in\Gamma_j}M_{\gamma_{ {\ell}_1^1 } ,\dots,\gamma_{{\ell}_1^j}}.$$ The weight of $\T_j$ is the sum of the weights of the configurations it contains.\ <span style="font-variant:small-caps;">Example</span>. In Step 2, we perform Step $1$ of the algorithm starting from the initial superimposition $M_0\cup M_{1,5}$. The latter contains one doubled edge $23$, thus the vertex ${\ell}_1^2$ is the smallest of 2 and 3, that is 2. The output $\S_2$ consists of the configurations $M_{1,5;2,1}, M_{1,5;2,4},M_{1,5;2,5}$, depicted in Figure \[fig:fig4\] below. ![Output of Step $2$ of the algorithm.[]{data-label="fig:fig4"}](fig4.pdf){height="2.5cm"} The superimposition of $M_0$ and the above three configurations contains no doubled edges. As a consequence, the complete algorithm stops and the output $\T_2$ of Step $2$ is: $$\T_2=\{M_{1,5;2,1}, M_{1,5;2,4},M_{1,5;2,5}\}.$$ ### End and output of the complete algorithm The algorithm stops at Step $T$ for the first time, if it hasn’t stopped at time $T-1$, and if for every $\gamma_{{\ell}_1^1}\in\widetilde{\Gamma}_{{\ell}_1^1},\dots, \gamma_{{\ell}_1^{T-1}}\in\widetilde{\Gamma}_{{\ell}_1^{T-1}}(\gamma_{{\ell}_1^1 },\dots, \gamma_ { {\ell}_1^ { T-2 } } )$ , $\gamma_{{\ell}_1^T}\in\Gamma_{{\ell}_1^{T}}(\gamma_{{\ell}_1^1},\dots,\gamma_ {{\ell}_1^{T-1}})$, the superimposition $M_0\cup M_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^T}}$ contains no doubled edge. This implies in particular that $(\widetilde{\Gamma}_{{\ell}_1^{T}}(\gamma_{{\ell}_1^1},\dots,\gamma_ {{\ell}_1^{T-1}}))^c=\Gamma_{{\ell}_1^{T}}(\gamma_{{\ell}_1^1},\dots,\gamma_ {{\ell}_1^{T-1}})$. Since the number of doubled edges decreases at every step and since $\D$ is finite, we are sure that this happens in finite time. The output $\T$ of the complete algorithm is : $$\T=\bigcup\limits_{j=1}^T \T_j.$$ <span style="font-variant:small-caps;">Example</span>. The output of the complete algorithm is: $$\T=\T_1\cup\T_2=\{M_{1,2,1},M_{1,2,5},M_{1,3,1}, M_{1,5;2,1}, M_{1,5;2,4},M_{1,5;2,5}\},$$ summarized in Figure \[fig:fig5\] below. ![Output of the complete algorithm.[]{data-label="fig:fig5"}](fig5.pdf){width="11cm"} The *weight* of $\T$ is the sum of the weights of the configurations it contains. If the initial superimposition $M_0\cup M$ consists of cycles only, *i.e.* the set $\D$ is empty, then $\T=\{M\}$, and $$w(\T)=w(M)=\sgn(\sigma_{M_0(M)})(-1)^{|\C|}\prod_{e\in M}a_e.$$ In all other cases: $$w(\T)=\sum_{j=1}^T \sum_{ (\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j}) \in \Gamma_j} w(M_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j}}).$$ Since for every $j\in\{1,\dots,T\}$, and for every $(\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j})\in\Gamma_j$, the superimposition $M_0\cup M_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j}}$ contains no doubled edge of $\D$, we have: $$\label{equ:weight1} w(M_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j}})=\sgn(\sigma_{M_0(M_{ \gamma_{{\ell}_1^1 },\dots,\gamma_{{\ell}_1^j}})})(-1)^{|\C| } \prod_{e\in M_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j}}}a_{e}.$$ By iterating the argument of Section \[sec:weight\], we obtain the following: \[cor:2\] The complete algorithm is weight preserving, that is: $$w(\T)=w(M).$$ Geometric characterization of configurations {#sec:sec26} -------------------------------------------- Consider the superimposition $M_0\cup M$, recall that $\C$ denotes the set of cycles of length $\geq 4$ of $M_0\cup M$, and that $\D$ denotes its set of doubled edges. Consider the complete algorithm with initial superimposition $M_0\cup M$ in the case where $M_0\cup M$ contains doubled edges, that is, when $\D\neq\emptyset$. Let $j\in\{1,\dots,T\}$, $(\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j})\in\Gamma_j$, and $M_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j}}\in\T$ be a generic output; and denote by $F_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j}}$ the superimposition $M_0 \cup M_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j}}$. In order to simplify notations, we introduce: $$\label{equ:F} \forall i\in\{1,\dots,j\},\quad F_i:= F_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^i}}.$$ One should keep in mind that the index $j$ refers to the last step of the algorithm, and that indices $i\in\{1,\dots,j-1\}$ refer to intermediate steps. As a consequence of the algorithm, see , the configuration $F_j$ and the paths $\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j}$ satisfy the following properties. $\bullet$ The oriented edge configuration $F_j$: 1. has one outgoing edge at every vertex of $V$. It consists of the paths $\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j}$ and of the cycles $\C$ of the initial superimposition $M_0\cup M$; 2. has no doubled edge of $\D$ since the complete algorithm precisely stops when this is the case. For every $i\in\{1,\dots,j\}$, the path $\gamma_{{\ell}_1^i}$ satisfies the following. 1. It has even length $2k_i$ for some $k_i\in\{1,\dots,m_i\}$ and is alternating. It starts from the vertex ${\ell}_1^{i}$, followed by an edge of $M_0$. 2. The vertex ${\ell}_1^i$ is the smallest vertex belonging to a doubled edge of $F_{i-1}$ (understood as $M_0\cup M$ when $i=1$). The $2k_i$ first vertices are all distinct and the last vertex ${\ell}_{k_i+1}^i$ belongs to: $$\Bigl(V_{{\ell}_{k_i}^i}^{\D_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{i-1 }},{\ell}_1^i, \dots , {\ell}_{k_i}^i}}\Bigr)^c= V_{{\ell}_{k_i}^i}\cap\bigl\{R\cup V^{\C}\cup V^{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1}^{i-1}}\cup \{{\ell}_1^i,({\ell}_1^i)',\dots,{\ell}_{k_i}^i,({\ell}_{k_i}^i)'\}.$$ As a consequence, one of the following 5 cases holds. $\bullet$ If ${\ell}_{k_i+1}^i\in R\cup V^{\C}\cup\{{\ell}_1^i,({\ell}_1^i)',\dots,{\ell}_{k_i}^i,({\ell}_{k_i}^i)'\} \bigr\}$, then $\gamma_{{\ell}_1^i}$ consists of a new connected component, and we recover the three cases obtained after Step 1 of the algorithm, replacing $\gamma_{{\ell}_1}$ by $\gamma_{{\ell}_1^i}$, see Section \[sec:output\]. For convenience of the reader, we repeat these cases here. 1. If ${\ell}_{k_i+1}^i\in R$, then $\gamma_{{\ell}_1^i}$ is a loopless oriented path from ${\ell}_1^i$ to one of the root vertices of $R$. Since $R=\{n+1,\dots,n+r\}$, ${\ell}_1^i$ is smaller than all vertices of  $\gamma_{{\ell}_1^i}$. The vertex ${\ell}_1^i$ is a leaf of a connected component of $F_i$, which consists of the path $\gamma_{{\ell}_1^i}$. 2. If ${\ell}_{k_i+1}^i\in V^{\C}$, then $\gamma_{{\ell}_1^i}$ is a loopless oriented path ending at a vertex of one of the cycles of $\C$. The vertex ${\ell}_1^i$ is smaller than the $2k_i$ first vertices of the path, but cannot be compared to vertices of the cycle of $\C$. By construction of the orientation of $M_0\cup M$, see Section \[sec:sec21\], the orientation of the cycle is compatible with that of the edge $(i_1,i_2)$, where $i_1$ is the smallest vertex of the cycle and $i_2$ is its partner in $M_0$. The vertex ${\ell}_1^i$ is a leaf of a connected component of $F_i$, which is a unicycle with $\gamma_{{\ell}_1^i}$ as unique branch and a cycle of $\C$ as cycle. 3. When ${\ell}_{k_i+1}^i\in \{{\ell}_1^i,({\ell}_1^i)',\dots,{\ell}_{k_i}^i,({\ell}_{k_i}^i)'\}$, then $\gamma_{{\ell}_1^i}$ contains a loop of length $\geq~3$. Recall that configurations with odd cycles cancel because of the skew-symmetry of the matrix, so that we only consider configurations where ${\ell}_{k_i+1}={\ell}_s^i$ for some $s\in\{1,\dots,k_i\}$. In this case, the part of the path $\gamma_{{\ell}_1^i}$ to the loop has even length, is alternating and start with an edge of $M_0$. The loop has even length $\geq 4$, is alternating and its orientation is compatible with the orientation of the edge $({\ell}_s^i,{{\ell}_s^i}^{'})$. The vertex ${\ell}_1^i$ is smaller than all vertices of the path to the cycle and smaller than all vertices of the cycle. If ${\ell}_{k_i+1}^i\neq {\ell}_1^i$, then ${\ell}_1^i$ is a leaf of a connected component of $F_i$ which is a unicycle with a unique branch, consisting of the path $\gamma_{{\ell}_1^i}$. If ${\ell}_{k_i+1}^i={\ell}_1^i$, then ${\ell}_1^i$ is the smallest vertex of a connected component of $F_i$ which is a cycle, consisting of the path $\gamma_{{\ell}_1^i}.$ $\bullet$ If ${\ell}_{k_i+1}^i\in V^{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^{i-1}}}$ then, the path $\gamma_{{\ell}_1^i}$ attaches itself to a connected component of $F_{i-1}$, this can only occur when $i\geq 2$, and one of the following happens. 1. The path $\gamma_{{\ell}_1^i}$ attaches itself to a leaf of $F_{i-1}$, that is, ${\ell}_{k_j+1}={\ell}_1^t$ for some $t\in\{1,\dots,i-1\}$. Then $\gamma_{{\ell}_1}^i$ is a loopless oriented path from ${\ell}_1^i$ to ${\ell}_1^t$. The vertex ${\ell}_1^i$ is smaller than the $2k_i$ following ones, but greater than than ${\ell}_1^t$. Indeed ${\ell}_1^t$ is the starting point of a previous step of the algorithm. *This allows to identify the ending vertex of the path $\gamma_{{\ell}_1}^i$*. The vertex ${\ell}_1^i$ is a leaf of $F_i$. 2. The path $\gamma_{{\ell}_1^i}$ creates a new branch of the component. Then $\gamma_{{\ell}_1^i}$ is a loopless oriented path. The vertex ${\ell}_1^i$ is smaller than the $2k_i$ following ones, but we have no a priori information on the last vertex of the path. The last vertex of the path $\gamma_{{\ell}_1^i}$ is nevertheless identified as being a fork. The vertex ${\ell}_1^i$ is a leaf of $F_i$. Note that the component of ${\ell}_1^i$ might be a unicycle with a unique branch. If this is the case, the branch is the path $\gamma_{{\ell}_1^i}$ and the cycle was created by Case (IV)(3) in a previous step of the algorithm. The vertex ${\ell}_1^i$ is thus larger than the smallest vertex of the cycle. \[lem:N1\] The oriented edge configuration $F_j$ is an RCRSF compatible with $M_0$. By definition of the algorithm, the oriented edge configuration $F_j$ contains as many edges as $M_0\cup M$, that is $|V|$ edges. By definition, it contains all edges of $M_0$, that is $|V|/2$ edges, and by the algorithm no doubled edges of $\D$, that is $|V|/2$ edges not in $M_0$. By Point (I) the oriented edge configuration $F_j$ has one outgoing edge at every vertex of $V$, which is equivalent to saying that it is an RCRSF such that edges of each component are oriented towards its root, and cycles are oriented in one of the two possible directions. It thus remains to show that cycles of unicycles are alternating, and have even length $\geq 4$. By Point (II), the oriented edge configuration $F_j$ has no doubled edge of $\D$, thus if $F_j$ has a cycle, it either comes from Point (IV)(2) meaning that it is a cycle of $\C$ implying that it is even, alternating and has length $\geq 4$; or from Point (IV)(3), when it is created by the algorithm. Returning to the description of Point (IV)(3) and recalling that the contribution of configurations with odd cycles cancel, we know that it has the same properties in this case. For every $i\in\{1,\dots,j\}$, and for every connected component of $F_i$ which is a cycle $C$ created by the algorithm (*i.e.* not a cycle of the initial superimposition), denote by $m_C$ the smallest vertex of $C$. Define $x_i$ to be: $$x_i= \begin{cases} \max\{m_C:\,C \text{ is a cycle-connected component of } F_i,\text{ but not of }\C\}&\text{ if $\{\}\neq\emptyset$}\\ -\infty&\text{ otherwise}. \end{cases}$$ If $F_i$ has at least one leaf, let $y_i$ be the maximum leaf of $F_i$, else let $y_i=-\infty$. If both $x_i$ and $y_i$ are $-\infty$, then $F_i$ has no leaves and only contains cycles of the initial superimposition $M_0\cup M$. This means that the set $\D$ is empty, and that $F$ is the initial superimposition $M_0\cup M$. This has been excluded here, since the complete algorithm doesn’t even start the opening of edges procedure in this case. Thus $\max\{x_i,y_i\}>-\infty$. \[lem:charactInitial\] For every $i\in\{1,\dots,j\}$, the initial vertex ${\ell}_1^i$ of Step $i$ is the maximum of $x_i$ and $y_i$. By Point (IV) above, the vertex ${\ell}_1^i$ is either a leaf of $F_i$ or the smallest vertex of a connected component of $F_i$ which is a cycle created by the algorithm, meaning that it is not a cycle of $\C$ *i.e.* not a cycle of the initial superimposition $M_0\cup M$. Arguing by induction, all leaves and smallest vertices of cycle-components of $F_i$ which are not present in $\C$, must be initial vertices of steps $i$ of the algorithm for some $i\in\{1,\dots,j\}$. Moreover by construction, the vertex ${\ell}_1^i$ is larger than all previous initial steps of the algorithm, thus proving the lemma. Properties described in Point(IV) also characterize the path $\gamma_{{\ell}_1^i}$ once the initial vertex ${\ell}_1^i$ is fixed. This can be summarized in the following lemma. \[lem:charactPath\] Let ${\ell}_1^i$ be the initial vertex of Step $i$. Then: - Suppose that ${\ell}_1^i$ is a leaf of a connected component of $F_i$. When the connected component is a unicycle rooted on a cycle created by the algorithm, we assume moreover that it contains more than one branch. Then, we are in Cases (IV)(1)(2) or (5) and the path $\gamma_{{\ell}_1^i}$ is characterized as the subpath of the unique path from ${\ell}_1^i$ to the root of the connected component, stopping the first time one visits a vertex which: belongs to $R$ or to the cycle of the component; is a fork; is smaller than ${\ell}_1^i$. - Suppose that ${\ell}_1^i$ is the leaf of a unicycle of $F_i$ rooted on a cycle created by the algorithm and containing a unique branch. If ${\ell}_1^i$ is larger than the smallest vertex of the cycle, then we are in Case (IV)(5) and the path $\gamma_{{\ell}_1^i}$ is the path from ${\ell}_1^i$ to the cycle, stopping when the cycle is reached. Else, if ${\ell}_1^i$ is smaller than the smallest vertex of the cycle, we are in Case (IV)(3) and the path $\gamma_{{\ell}_1^i}$ is the path from ${\ell}_1^i$ to the cycle, followed by the cycle, with the orientation specified in (IV)(3). - If ${\ell}_1^i$ is the smallest vertex of a connected component of $F_i$ which is a cycle created by the algorithm, then we are in Case (IV)(3) and the path $\gamma_{{\ell}_1^i}$ is the cycle, with the orientation specified in (IV)(3). \[rem:rem1\] If the initial superimposition $M_0\cup M$ consists of cycles only, that is, if the set $\D$ is empty, then the output of the complete algorithm is $F=M_0\cup M$, which consists of alternating cycles of even length $\geq 4$. The orientation of cycles is specified in Section \[sec:sec21\], and cycles are oriented in one of the possible two directions. In this case also, $F$ is an RCRSF compatible with $M_0$. Proofs and corollaries {#sec:sec3} ====================== We now prove Theorem \[thm:main\], Corollary \[cor:main\] and state and prove the line bundle version of the result. Proof of Theorem \[thm:main\] {#sec:sec31} ----------------------------- In this section, we prove Theorem \[thm:main\]. Let $M_0$ be a reference perfect matching of $G$, and let $F$ be an RCRSF compatible with $M_0$, containing $k_F$ unicycles. In Lemma \[lem:alpha\], we suppose that $F$ is an output of the complete algorithm and identify $2^{k_F}$ possible perfect matchings $M$ for the initial superimposition $M_0\cup M$. Then, we introduce a partial reverse algorithm used to define Condition (C) for RCRSFs compatible with $M_0$. In Proposition \[thm:thm2\], we prove that an RCRSF compatible with $M_0$ is an output of the complete algorithm if and only if it satisfies Condition (C), and if this is the case, it is obtained $2^{k_F}$ times. The remainder of the proof consists in showing that contribution of RCRSFs containing unicycles cancel, and that only spanning forests remain with the appropriate weight, thus proving Theorem \[thm:main\]. Let $F$ be an RCRSF compatible with $M_0$, and let $k_F$ denote the number of unicycles it contains. If $k_F\neq 0$, we let $\{C_1,\dots,C_{k_F}\}$ be its set of cycles. For every $(\eps_1,\dots,\eps_{k_F})\in\{0,1\}^{k_F}$, define the edge configuration $M^{(\eps_1,\dots,\eps_{k_F})}$ as follows: $$M^{(\eps_1,\dots,\eps_{k_F})}= \begin{cases} \text{edges of $M_0$ on branches of $F$}\\ \text{edges of $M_0$ on the cycle $C_j$, when $\eps_j=0$}\\ \text{edge of $F\setminus M_0$ on the cycle $C_j$, when $\eps_j=1$}. \end{cases}$$ If $k_F=0$, then the set of cycles of $F$ is $\{\emptyset\}$, and we set $M^{(\eps_1,\dots,\eps_{k_F})}=M_0$. \[lem:epsilonDimer\] For every *RSCRSF* $F$ compatible with $M_0$, and every $(\eps_1,\dots,\eps_k)\in\{0,1\}^{k_F}$, the edge configuration $M^{(\eps_1,\dots,\eps_{k_F})}$ is a perfect matching of $G$. If $F$ contains no unicycles, $M^{(\eps_1,\dots,\eps_{k_F})}=M_0$ and this is immediate. Suppose $k_F~\neq~0$. Since $M_0$ is a perfect matching, and since the restriction of $M_0$ and the restriction of $M^{(\eps_1,\dots,\eps_k)}$ to branches of $F$ are the same, all vertices of $V\setminus\{V(C_1),\dots,V(C_{k_F})\}$ are incident to exactly one edge of $M^{(\eps_1,\dots,\eps_{k_F})}$. Moreover, by assumption for every $j$, the cycle $C_j$ is alternating, implying that each vertex of $V(C_j)$ is incident to exactly one edge of the restriction of $M_0$ and one edge of the restriction of $F\setminus M_0$ to $C_j$. As a consequence, every vertex of $V$ is incident to exactly one edge of $M^{(\eps_1,\dots,\eps_{k_F})}$, proving that it is a perfect matching of $G$. \[lem:alpha\] Let $F$ be the superimposition of $M_0$ and of an output of the complete algorithm. Then, the perfect matching $M$ of the initial superimposition $M_0\cup M$, must be equal to $M^{(\eps_1,\dots,\eps_{k_F})}$ for some $(\eps_1,\dots,\eps_{k_F})\in\{0,1\}^{k_F}$. If $F$ is an output of the complete algorithm, then by Lemma \[lem:N1\] and Remark \[rem:rem1\], it is an RCRSF compatible with $M_0$, so that $\forall (\eps_1,\dots,\eps_{k_F})\in\{0,1\}^{k_F}$, the perfect matching $M^{(\eps_1,\dots,\eps_{k_F})}$ is well defined. Suppose that $F$ is an output of the complete algorithm with initial superimposition $M_0\cup M$, where $M$ is not $M^{(\eps_1,\dots,\eps_{k_F})}$ for some $(\eps_1,\dots,\eps_{k_F})\in\{0,1\}^{k_F}$. Then, $M_0\cup M$ contains at least one cycle $C$ which is not $C_1,\dots,C_{k_F}$. Returning to the definition of the algorithm, we know that cycles present in the initial superimposition are also present in the output $F$. This yields a contradiction since $F$ contains exactly the cycles $C_1,\dots,C_{k_F}$. : an RCRSF $F$ compatible with $M_0$ not consisting of cycles only. : $F_1=F$. Let $\bar{\ell}_1^i$ be the largest leaf of $F_i$, and consider the connected component containing $\bar{\ell}_1^i$. Start from $\bar{\ell}_1^i$ along the unique path joining $\bar{\ell}_1^i$ to the root or the cycle of the component, until the first time one of the following vertices is reached: - the root vertex if the component is a tree, or the cycle if it is a unicycle; - a fork; - a vertex which is smaller than the leaf $\bar{\ell}_1^i$. This yields a loopless path $\lambda_{\bar{\ell}_1^i}$ starting from $\bar{\ell}_1^i$, of length $\geq 1$. Let $F_{i+1}=F_{i}\setminus\lambda_{\bar{\ell}_1^i}$. If $F_{i+1}$ is empty or contains cycles only, then stop. Else, go to Step $i+1$. : since edges are removed at every step and since $F$ contains finitely many edges, the algorithm ends in finite time $N$. \[def:N2\] An RCRSF $F$ compatible with $M_0$ is said to satisfy *Condition* (C) if either $F$ consists of cycles only, or if each of the paths $\lambda_{\bar{\ell}_1^1},\dots,\lambda_{\bar{\ell}_1^N},$ obtained from the partial reverse algorithm has even length and starts from an edge of $M_0$. \[thm:thm2\] Let $F$ be an RCRSF compatible with $M_0$.\ Then, for every $(\eps_1,\dots,\eps_{k_F})\in\{0,1\}^{k_F}$, $F$ is the superimposition of $M_0$ and of an output of the complete algorithm, with initial superimposition $M_0~\cup~ M^{{(\eps_1,\dots,\eps_{k_F})}}$, if and only if $F$ satisfies Condition *(C)*. The orientation of cycles of $F$ is specified by the proof. Let $F$ be an RCRSF compatible with $M_0$ containing $k_F$ unicycles, and denote by $\{C_1,\dots,C_{k_F}\}$ its set of cycles. For every $ (\eps_1,\dots,\eps_{k_F})\in\{0,1\}^{k_F}$, the edge configuration $M^{{(\eps_1,\dots,\eps_{k_F})}}$ is well defined, and by Lemma \[lem:epsilonDimer\] is a perfect matching. We now fix $(\eps_1,\dots,\eps_{k_F})\in\{0,1\}^{k_F}$. Recall that if $k_F=0$, then the perfect matching $M^{{(\eps_1,\dots,\eps_{k_F})}}$ is simply $M_0$. In the case where $k_F\neq 0$, $(\eps_1,\dots,\eps_{k_F})=(1,\dots,1)$, and $F$ consists of cycles only, the superimposition $M_0\cup M^{(\eps_1,\dots,\eps_{k_F})}$ consists of cycles only, and $F=M_0\cup M^{(\eps_1,\dots,\eps_{k_F})}$ is an output of the complete algorithm. The orientation of the cycles is specified by the choice of orientation of Section \[sec:sec21\], thus proving Proposition \[thm:thm2\]. Assume that we are not in the above case. Then $F$ is an output of the algorithm with initial superimposition $M_0\cup M^{(\eps_1,\dots,\eps_{k_F})}$ if and only if there exists a positive integer $j$ and a sequence of paths $(\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j})\in\Gamma_j$ such that $F=M_0\cup M_{\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j}}^{(\eps_1,\dots,\eps_{k_F})} $. Lemmas \[lem:charactInitial\] and \[lem:charactPath\] give a characterization of ${\ell}_1^i$ and $\gamma_{{\ell}_1^i}$ at every step of the algorithm. This allows us to define a complete reverse algorithm. : an RCRSF $F$ compatible with $M_0$, $(\eps_1,\dots,\eps_{k_F})\in\{0,1\}^{k_F}$ as above, and the corresponding perfect matching $M^{(\eps_1,\dots,\eps_{k_F})}$. If $k_F=0$, then $M^{(\eps_1,\dots,\eps_{k_F})}=M_0$. : $F_1=F$. . Since $F_i$ is either $F_1$ or is obtained from $F_{i-1}$ by removing edges, the set of cycles of $F_i$ is included in the set of cycles $\{C_1,\dots,C_{k_F}\}$ of $F_1$. For every connected component of $F_i$ which is a cycle $C_{\alpha}$ such that $\eps_\alpha=0$ (meaning that $C_\alpha$ is not a cycle of the initial superimposition $M_0\cup M^{(\eps_1,\dots,\eps_{k_F})}$), let $m_{C_\alpha}$ be the smallest vertex of $C_\alpha$. Define $$x_i= \begin{cases} \max\{m_{C_\alpha}:\,C_\alpha\text{ is a cycle-connected component of }F_i \text{, and }\eps_\alpha=0\}&\text{ if $\{\}\neq\emptyset$}\\ -\infty&\text{ else}. \end{cases}$$ If $F_i$ has at least one leaf, let $y_i$ be the maximum leaf, else let $y_i=-\infty$. Note that by assumption, we do not have $x_i=y_i=-\infty$. We let ${\ell}_1^i=\max\{x_i,y_i\}$, and $\gamma_{{\ell}_1^i}$ be the oriented path as characterized in Lemma \[lem:charactPath\]. Let $F_{i+1}=F_i\setminus\gamma_{{\ell}_1^i}$. If the oriented edge configuration $F_{i+1}$ is empty, or if it consists of cycles of the superimposition $M_0\cup M^{(\eps_1,\dots,\eps_{k_F})}$ only, then stop; else go to Step $i+1$. : since edges are removed at every step and since $F$ contains finitely many edges, the algorithm ends in finite time $j$, for some integer $j$. This defines for every RCRSF $F$ compatible with $M_0$, a sequence of paths $\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j}$ such that $F$ is the union of these paths and of cycles of the initial superimposition $M_0\cup M^{(\eps_1,\dots,\eps_{k_F})}$. As a consequence, the oriented edge configuration $F$ satisfies Properties (I), (II), (IV)(1)$-$(5). We are thus left with proving that $F$ satisfies Property (III) if and only if it satisfies Condition (C), *i.e.* we need to show that the paths $(\gamma_{{\ell}_1^1},\dots,\gamma_{{\ell}_1^j})$ all have even length, are alternating and start from an edge of $M_0$ if and only if $F$ satisfies Condition (C). Observe that initial vertices $({\ell}_1^1,\dots,{\ell}_1^j)$ of the complete reverse algorithm consist of initial vertices $( {\bar{\ell}}_1^1,\dots,{\bar{\ell}}_1^{N})$ of the partial reverse algorithm, interlaced with smallest vertices of components which are cycles. Indeed, the only difference in the partial reverse algorithm is that cycles are not removed, but this does not change the characterization of largest leaf. If ${\ell}_1^i$ is the smallest vertex of a component of $F_i$ which is a cycle, that is, if ${\ell}_1^i=x_i$, then $\gamma_{{\ell}_1^i}$ is a cycle $C_\alpha\in\{C_1,\dots,C_{k_F}\}$ such that $\eps_\alpha=0$. Since $F$ is compatible with $M_0$, the cycle has even length and is alternating. The orientation is fixed by the algorithm and $\gamma_{{\ell}_1^i}$ always satisfies Property (III). If ${\ell}_1^i$ is the largest leaf of $F_i$, that is, if ${\ell}_1^i=y_i$, then in all cases except one, which we treat below, the path $\gamma_{{\ell}_1^i}$ is exactly the path $\lambda_{\bar{\ell}_1^{i'}}$ of the partial reverse algorithm, for some $i'\leq i$. Condition (C) says that $\lambda_{\bar{\ell}_1^{i'}}$ has even length and starts from an edge of $M_0$. In order to show that this is equivalent to satisfying Property (III), we are left with showing that, by construction, the path $\lambda_{\bar{\ell}_1^{i'}}$ is always alternating. Suppose that this is not the case, then there are at least two edges of the same kind (either in $M_0$ or not in $M_0$) which follow each other. This implies that there is a vertex $v$ of the path incident to two edges of the same kind. Since $M_0$ is a perfect matching, every vertex is incident to exactly one edge of $M_0$, so that we cannot have two edges of $M_0$ following each other. Thus these two edges do not belong to $M_0$. Again, since $M_0$ is a perfect matching, the vertex $v$ is also incident to an edge of $M_0$, implying that $v$ is the end of a branch. By construction of the path $\lambda_{\bar{\ell}_1^{i'}}$, the path must stop at $v$, implying that one of the two edges is not in $\lambda_{\bar{\ell}_1^{i'}}$, yielding a contradiction. We now treat the last case. If ${\ell}_1^i$ is a leaf of a connected component of $F_i$ which is a unicycle rooted on a cycle $C_\alpha$ such that $\eps_\alpha=0$, with a unique branch, and such that ${\ell}_1^i$ is smaller than the smallest vertex of the cycle. Then the path $\gamma_{{\ell}_1^i}$ is the path $\lambda_{\bar{\ell}_1^{i'}}$ followed by the cycle with the appropriate orientation. We have to show that $\gamma_{{\ell}_1^i}$ satisfies Property (III) if and only if $\lambda_{\bar{\ell}_1^{i'}}$ satisfies Condition (C). By Property (III), we know that the part of $\gamma_{{\ell}_1^i}$ stopping when the cycle is reached, which is precisely $\lambda_{\bar{\ell}_1^{i'}}$, has even length and starts from an edge of $M_0$. This is exactly Condition (C), since by the same argument as above, the path $\lambda_{\bar{\ell}_1^{i'}}$ is alternating. We conclude by observing that since $F$ is compatible with $M_0$, the cycle part of $\gamma_{{\ell}_1^i}$ is alternating, and starts from an edge of $M_0$ by construction of the orientation of the cycle. Thus $\gamma_{{\ell}_1^i}$ satisfies Property (III) if and only if $F$ is compatible with $M_0$ and satisfies Condition (C). We denote by $\GG(M_0)$ the set of RCRSFs compatible with $M_0$ satisfying Condition (C). Let $F$ be an RCRSF of $\GG(M_0)$, and let $k_F$ be its number of unicycles. If $k_F\neq 0$, then for every $(\eps_1,\dots,\eps_{k_F})\in\{0,1\}^{k_F}$, denote by $F^{(\eps_1,\dots,\eps_{k_F})}$ the version of $F$ obtained from the complete algorithm with initial superimposition $M_0 \cup M^{(\eps_1,\dots,\eps_{k_F})}$, with the orientation of cycles given by Proposition \[thm:thm2\]. If $k_F=0$, then $F$ is obtained exactly once from the complete algorithm with initial superimposition $M_0 \cup M_0$. Since $M_0 \cup M^{(\eps_1,\dots,\eps_{k_F})}$ has exactly $\sum_{i=1}^{k_F}\eps_i$ cycles, and since $M_0\cup M_0$ has none, we have as a consequence of the complete algorithm, see Equation , that the weight $w_{M_0}(F^{(\eps_1,\dots,\eps_{k_F})}\setminus M_0)$ is equal to: $$\label{equ:final1} \sgn(\sigma_{M_0(F^{(\eps_1,\dots,\eps_{k_F})}\setminus M_0)})\cdot \begin{cases} (-1)^{\sum_{i=1}^{k_F}\eps_i} \prod\limits_{e\in F^{(\eps_1,\dots,\eps_{k_F})}\setminus M_0}a_e& \text{ if $k_F\neq 0$}\\ \prod\limits_{e\in F^{(\eps_1,\dots,\eps_{k_F})}\setminus M_0}a_e& \text{ if $k_F=0$}. \end{cases}$$ Recall that $\T$ denotes the output of the complete algorithm with initial superimposition $M_0\cup M$ for a fixed reference perfect matching $M_0$ and a generic perfect matching $M$ of $G$. Since we now aim at taking the union over all perfects matchings $M$, we write $\T$ as $\T_{M_0}(M)$. As a consequence of Proposition \[thm:thm2\], we have that $\bigcup_{M\in \M}\T_{M_0}(M)$ is equal to: $$\label{equ:final} \Bigl( \bigcup\limits_{\{F\in \GG(M_0):\,k_F\neq 0\}} \bigcup\limits_{(\eps_1,\dots,\eps_k)\in\{0,1\}^{k_F}} F^{(\eps_1,\dots,\eps_k)}\setminus M_0 \Bigr) \bigcup \Bigl( \bigcup\limits_{\{F\in \GG(M_0):\,k_F=0\}} F\setminus M_0 \Bigr).$$ Returning to the definition of the Pfaffian of Equation , using Corollary \[cor:2\] and Equation in the last line, we deduce that: $$\begin{aligned} \Pf(A)&=\sum_{M\in\M}w_{M_0}(M), \text{ (Definition of Equation \eqref{equ:pfaffian})}\\ &=\sum_{M\in \M}w(\T_{M_0}(M)),\text{ (by Corollary \ref{cor:2})}\\ &=\underbrace{\sum_{\{F\in \GG(M_0):\,k_F\neq 0\}} \sum_{(\eps_1,\dots,\eps_{k_F})\in\{0,1\}^{k_F}} w_{M_0}(F^{(\eps_1,\dots,\eps_{k_F})}\setminus M_0)}_{(I)}+ \underbrace{\sum_{\{F\in \GG(M_0):\,k_F= 0\}} w_{M_0}(F\setminus M_0)}_{(II)}.\end{aligned}$$ Let us show that $(I)$ is equal to zero. As a consequence of Equation , it is equal to: $$\begin{aligned} (I)&=\sum\limits_{\{F\in \GG(M_0):\,k_F\neq 0\}} \sum_{(\eps_1,\dots,\eps_{k_F})\in\{0,1\}^{k_F}} \sgn(\sigma_{M_0(F^{(\eps_1,\dots,\eps_{k_F})}\setminus M_0)})(-1)^{\sum_{i=1}^{k_F} \eps_i} \prod_{e\in F^{(\eps_1,\dots,\eps_{k_F})}\setminus M_0} a_{e}.\end{aligned}$$ Observing that the term $\sgn(\sigma_{M_0(F^{(\eps_1,\dots,\eps_{k_F})}\setminus M_0)})\prod_{e\in F^{(\eps_1,\dots,\eps_{k_F})}\setminus M_0} a_{e}$ is independent of $(\eps_1,\dots,\eps_{k_F})$, we conclude that: $$\begin{aligned} (I)&= \sum\limits_{\{F\in \GG(M_0):\,k_F\neq 0\}} \Bigl(\sgn(\sigma_{M_0(F^{(\eps_1,\dots,\eps_{k_F})}\setminus M_0)}) \prod_{e\in F^{(\eps_1,\dots,\eps_{k_F})}\setminus M_0} a_{e} \Bigr) \sum_{(\eps_1,\dots,\eps_{k_F})\in\{0,1\}^{k_F}} (-1)^{\sum_{i=1}^{k_F}}\\ &=\sum\limits_{\{F\in \GG(M_0):\,k_F\neq 0\}} \Bigl(\sgn(\sigma_{M_0(F^{(\eps_1,\dots,\eps_{k_F})}\setminus M_0)}) \prod_{e\in F^{(\eps_1,\dots,\eps_{k_F})}\setminus M_0} a_{e} \Bigr)(1-1)^{k_F}\\ &=0.\end{aligned}$$ Thus, $$\begin{aligned} \Pf(A)&=\sum_{\{F\in \GG(M_0):\,k_F= 0\}} w_{M_0}(F\setminus M_0) \\ &=\sum_{\{F\in \GG(M_0):\,k_F= 0\}} \sgn(\sigma_{M_0(F\setminus M_0)}) \prod_{e\in F\setminus M_0}a_e.\end{aligned}$$ The set $\{F\in\GG(M_0):\,k_F=0\}$ consists of RCRSFs compatible with $M_0$ containing no unicycles, and satisfying Condition (C). Observing that: - the set of RCRSFs compatible with $M_0$, containing no unicycle is exactly the set of spanning forests of $G^R$ compatible with $M_0$ of Section \[sec:11\] of the introduction, - in the case of spanning forests, the partial reverse algorithm is exactly the trimming algorithm of Section \[sec:11\], - Condition (C) of Definition \[def:def0\] and Condition (C) of Definition \[def:N2\] are the same in the case of spanning forests, - the permutation $\sigma_{M_0(F\setminus M_0)}$ obtained from the algorithm is the permutation of Definition \[def:def2\], we deduce that $\{F\in\GG(M_0):\,k_F=0\}=\F(M_0)$, and thus conclude the proof of Theorem \[thm:main\]. <span style="font-variant:small-caps;">Example</span>. Let us take the reference matching $M_0$ which followed us throughout the paper, and consider the three possible perfect matchings $M_1,M_2,M_3$ of the graph $G$ given in Figure \[fig:fig8\]. Figure \[fig:fig5\] shows the output of the complete algorithm with initial superimposition $M_0\cup M_1$. Since the superimpositions $M_0\cup M_2$ and $M_0\cup M_3$ contain doubled edges only, the output of the algorithm with these respective initial superimpositions, are the configurations themselves. By the Theorem \[thm:main\], configurations $M_2$ and $M_{1,2,1}$ have opposite weights, and configurations $M_3$ and $M_{1,3,1}$ as well, so that their contributions cancel in the Pfaffian. As a consequence, signed weighted half-spanning trees counted by the Pfaffian of the matrix $A$ are those of Figure \[fig:fig10\] below. ![Black edges of the above configurations are half-spanning trees counted by the Pfaffian of the matrix $A$.[]{data-label="fig:fig10"}](fig10.pdf){width="8cm"} Proof of Corollary \[cor:main\] {#sec:sec32} ------------------------------- Let us recall the setting: $A^R$ is a skew-symmetric matrix of size $(n+r)\times (n+r)$, whose column sum is zero, with $n$ even; $A$ is the matrix obtained from $A^R$ by removing the $r$ last lines and columns; $G^R$ and $G$ are the graphs naturally constructed from $A^R$ and $A$ in the introduction. Recall, see Section \[sec:sec21\], that the sign of the permutation $\sigma_{M}$ assigned to a perfect matching $M$ counted by the Pfaffian of $A$, depends on the ordering of the two elements of pairs involved in the perfect matching, but not on the ordering of the pairs themselves. Choosing the sign of $\sigma_M$ thus amounts to choosing an orientation of edges of the perfect matching $M$. The Pfaffian of $A$ can thus be written as: $$\Pf(A)=\sum_{M\in\M}\sgn(\sigma_{M})\prod_{e\in M}a_e,$$ where the product is over coefficients corresponding to a choice of orientation of edges of $M$, specifying a choice of permutation $\sigma_M$. Now, it is a known fact that the determinant of a skew-symmetric matrix is equal to the square of the Pfaffian: $$\begin{aligned} \det(A)&=\Bigl(\sum_{M_0\in \M}\sgn(\sigma_{M_0})\prod_{e\in M_0} a_e\Bigr) \Bigl(\sum_{M\in \M}\sgn(\sigma_{M})\prod_{e\in M} a_e\Bigr).\end{aligned}$$ As in Section \[sec:sec21\], for every $M_0\in\M$, we choose the permutation $\sigma_M$ using the superimposition $M_0\cup M$. Equation thus yields: $$\det(A)=\sum_{M_0\in \M}\sgn(\sigma_{M_0})\prod_{e\in M_0} a_e \Bigl(\sum_{M\in \M}\sgn(\sigma_{M_0(M)})(-1)^{|\D(M_0\cup M)|}(-1)^{|\C(M_0\cup M)|}\prod_{e\in M}a_e \Bigr).$$ As a consequence of Theorem \[thm:main\], this can be rewritten as: $$\begin{aligned} \det(A)&=\sum_{M_0\in \M}\sgn(\sigma_{M_0})\prod_{e\in M_0} a_e \Bigl(\sum_{F\in \F(M_0)}\sgn(\sigma_{M_0(F\setminus M_0)}) \prod_{e\in F\setminus M_0}a_e \Bigr),\\ &=\sum_{M_0\in \M}\sum_{F\in \F(M_0)}\Bigl(\sgn(\sigma_{M_0})\prod_{e\in M_0} a_e\Bigr)\sgn(\sigma_{M_0(F\setminus M_0)}) \prod_{e\in F\setminus M_0}a_e.\end{aligned}$$ where $\sigma_{M_0(F\setminus M_0)}$ is defined in Definition \[def:def2\]. We have not yet chosen the permutation $\sigma_{M_0}$ assigned to $M_0$, we do so now. For every $F\in \F(M_0)$, we chose the orientation of $M_0$ to be the orientation of edges induced by the spanning forest $F$: this is precisely $\sigma_{M_0(F\setminus M_0)}$. Combining the product of coefficients $a_e$ over oriented edges in $M_0$ and in $F\setminus M_0$ yields: $$\begin{aligned} \det(A)&=\sum_{M_0\in \M}\sum_{F\in \F(M_0)} \sgn(\sigma_{M_0(F\setminus M_0)})^2 \prod_{e\in F} a_e.\\ &=\sum_{M_0\in \M}\sum_{F\in \F(M_0)} \prod_{e\in F} a_e,\end{aligned}$$ thus proving Corollary \[cor:main\]. \[rem:main\]$\,$ 1. We now give an intrinsic characterization of $\cup_{M_0\in\M}\F(M_0)$, not using reference perfect matchings. Consider the trimming algorithm of Section \[sec:11\] applied to general spanning forests of $G^R$ (not assuming that they are compatible with a reference perfect matching $M_0$). Since the reference perfect matching is not used in the algorithm, everything works out in the same way, and the algorithm yields a sequence of paths $\lambda_{{\ell}_1}^1,\dots,\lambda_{{\ell}_1}^N$. This yields the following more general form of Definition \[def:def0\]. A spanning forest $F$ of $G^R$ is said to satisfy *Condition* (C) if each of the paths $\lambda_1^1,\dots,\lambda_1^N$ obtained from the trimming algorithm has even length. Let $\F$ denote the set of spanning forests of $G^R$ satisfying Condition (C). $$\F=\bigcup_{M_0\in\M}\F(M_0).$$ By definition, we have the following immediate inclusion: $\bigcup_{M_0\in\M}\F(M_0)~\subset~\F$. If $M_0$ and $M_0'$ are two distinct perfect matchings of $G$, then $\F(M_0)\cap \F(M_0')=\emptyset$. Indeed suppose there exists a spanning forest $F$ in the intersection. Then, it must be compatible with $M_0$ and $M_0'$, meaning that it contains all edges of $M_0\cup M_0'$. Since $M_0$ and $M_0'$ are distinct, the superimposition $M_0\cup M_0'$ must contain a cycle, yielding a contradiction with the fact that $F$ is a spanning forest. Thus it remains to show that given a spanning forest $F$ satisfying Condition (C) there exists a perfect matching $M_0$ such that $F\in\F(M_0)$, meaning that $F$ is compatible with $M_0$ and satisfies Condition (C) of Definition \[def:def0\]. Let $F$ be a spanning forest satisfying Condition (C), and let $\lambda_{{\ell}_1^1},\dots,\lambda_{{\ell}_1^N}$ be the sequence of paths obtained from the trimming algorithm. For every $i\in\{1,\dots,N\}$, let $M_0(\lambda_{{\ell}_1^i})$ consist of half of the edges of $\lambda_{{\ell}_1^i}$ such that $\lambda_{{\ell}_1^i}$ alternates between edges of $M_0(\lambda_{{\ell}_1^i})$ and edges of $\lambda_{{\ell}_1^i}\setminus M_0(\lambda_{{\ell}_1^i})$, starting from an edge of $M_0(\lambda_{{\ell}_1^i})$. Let $M_0=\cup_{i=1}^N M_0(\lambda_{{\ell}_1^i})$. Then $F$ is compatible with $M_0$ and satisfies Condition (C) of Definition \[def:def0\]. It remains to show that $M_0$ is a perfect matching. The edge configuration consists of $|V|/2$ edges, since by construction, it consists of half of the edges of a spanning forest. Moreover, since each of the paths $\lambda_{{\ell}_1^1},\dots,\lambda_{{\ell}_1^N}$ has even length, no vertex is incident to two edges of $M_0$, thus proving that $M_0$ is a perfect matching. As a consequence, Corollary \[cor:main\] can be rewritten in the simpler form: \[cor:main2\] $$\det(A)=\sum_{F\in\F}\prod_{e\in F}a_e.$$ 2. Let $\Xi$ be the set of cycle coverings of the graph $G$ by cycles of even length: a typical element $\xi\in\,\Xi$ is of the form $\xi=(C_1,\dots,C_k)$ for some $k$. Then, since the matrix $A$ is skew-symmetric, the determinant of $A$ is equal to: $$\begin{aligned} \det(A)&=\sum\limits_{\xi=(C_1,\dots,C_k)\in\Xi}\; \prod_{\{i:|C_i|\geq 4\}}(-1) \Bigl(\prod_{e\in \overrightarrow{C_i}}a_{e}+ \prod_{e\in \overleftarrow{C_i}}a_e\Bigr) \prod_{\{i:|C_i|=2\}}(-1)a_e a_{-e}\\ &=\sum\limits_{\xi=(C_1,\dots,C_k)\in\Xi}\; \prod_{\{i:|C_i|\geq 4\}}(-2) \bigl(\prod_{e\in \overrightarrow{C_i}}a_{e}\bigr) \prod_{\{i:|C_i|=2\}}a_e^2.\end{aligned}$$ It is also possible to prove Corollary \[cor:main2\] directly, without passing through the Pfaffian, by applying the complete algorithm to doubled edges of configurations counted by the determinant, and by taking into account all edges instead of half of them. Line-bundle matrix-tree theorem for skew-symmetric matrices {#sec:sec33} ----------------------------------------------------------- In the whole of this section, we change notations slightly, and we let $A$ be a skew-symmetric matrix of size $n\times n$, whose column sum is zero, with $n$ even; $G=(V,E)$ denotes the graph associated to the matrix $A$. We now state a line-bundle version of the matrix-tree theorem for skew-symmetric matrices of Corollary \[cor:main\], in the spirit of what is done for the Laplacian matrix in [@Forman], [@KenyonVectorBundle], but first we need a few definitions. A $\CC$-bundle is a copy $\CC_v$ of $\CC$ associated to each vertex $v\in V$. The *total space* of the bundle is the direct sum $W=\oplus_{v\in V}\CC_v$. A *connection* $\Psi$ on $W$ is the choice, for each oriented edge $(i,j)$ of $G$ of linear isomorphism $\psi_{i,j}:\CC_i\rightarrow\CC_j$, with the property that $\psi_{i,j}=\psi_{j,i}^{-1}$; that is, we associate to each oriented edge $(i,j)$ a non-zero complex number $\psi_{i,j}$ such that $\psi_{i,j}=\psi_{j,i}^{-1}$. We say that $\psi_{i,j}$ is the *parallel transport* of the connection over the edge $(i,j)$. The *monodromy* of the connection around an oriented cycle $\vec{C}$ is the complex number $\omega_{\vec{C}}=\prod_{e\in \vec{C}} \psi_{e}$. We consider the matrix $A^{\psi}$ constructed from the matrix $A$ and the connection $\psi$: $$(A^{\psi})_{i,j}=a_{i,j}^{\psi}=a_{i,j}\psi_{i,j}.$$ A *cycle-rooted spanning forest* of $G$, also denoted $CRSF$, is an oriented edge configuration spanning vertices of $G$ such that each connected component is a tree rooted on a cycle. In all that follows, we assume that cycles have length $\geq 3$. Edges of branches of the trees are oriented towards the cycle, and the cycle is oriented in one of the two possible directions. Consider the partial reverse algorithm of Section \[sec:sec31\] applied to a general CRSF $F$. Since the reference perfect matching plays no role in this algorithm, everything works out in the same way, and the algorithm yields a sequence of paths $\lambda_{{\ell}_1^1},\dots,\lambda_{{\ell}_1^N}$, whose union corresponds to branches of $F$. A CRSF of $G$ is said to satisfy *Condition* (C) if all of the paths $\lambda_{{\ell}_1}^1,\dots,\lambda_{{\ell}_1^N}$ obtained from the partial reverse algorithm have even length. Let us denote by $\GG$ the set of CRSFs satisfying Condition (C). Then, Then, for a generic CRSF $F$ of $G$, let us denote by $(C_1,\dots,C_k)$ its cycles. \[thm:linebundle\] $$\begin{aligned} \det(A^{\psi})=\sum_{F\in \GG} \Bigl(\prod_{\{e\in \text{\rm branch}(F)\}}a_e\Bigr) \Bigl(\prod_{\{i:|C_i|\text{\rm is odd}\}} &\prod_{e\in \overrightarrow{C_i}}a_e[\omega_{\overrightarrow{C_i}}-\omega_{\overrightarrow{ C_i }}^{-1} ] \Bigr)\cdot\\ &\cdot\Bigl(\prod_{\{i:|C_i|\text{\rm is even}\}} \prod_{e\in \overrightarrow{C_i}}a_e[2-\omega_{\overrightarrow{C_i}}-\omega_{\overrightarrow {C_i}}^{-1}]\Bigr).\end{aligned}$$ We expand the determinant of $A^{\Psi}$ using cycle decompositions, as we have done for the determinant of $A$ in Point 2 of Remark \[rem:main\]. Since the matrix $A^{\Psi}$ is not skew-symmetric, we cannot omit odd cycles, and we let $\Xi$ be the set of cycle decompositions of the graph $G$, that is, the set of coverings of the graph by disjoint cycles. A typical element of $\Xi$ can be written as $\xi=\{C_1,\dots,C_k\}$, for some positive integer $k$. Then, the determinant of the matrix $A^{\psi}$ is: $$\det(A^{\psi})= \sum\limits_{\xi=(C_1,\dots,C_k)\in\Xi}\; \prod_{\{i:|C_i|\geq 3\}}(-1)^{|C_i|+1} \Bigl(\prod_{e\in \overrightarrow{C_i}}a_{e}\psi_e + \prod_{e\in \overleftarrow{C_i}}a_e\psi_e\Bigr) \prod_{\{i:|C_i|=2\}}\Bigl((-1)a_e\psi_e a_{-e}\psi_{-e}\Bigr).$$ Using the skew-symmetry of the matrix $A$ and the fact that $\psi_{-e}=\psi_e^{-1}$ this yields: $$\begin{aligned} \det(A^{\psi})= \sum_{\xi=(C_1,\dots,C_k)\in \Xi}\;& \prod_{\{i:|C_i|\text{ is odd}\}} \Bigl(\prod_{e\in \overrightarrow{C_i}}a_e[\omega_{\overrightarrow{C_i}}-\omega_{\overrightarrow{ C_i }}^{-1} ]\Bigr)\cdot\\ &\cdot\prod_{\{i:|C_i|\text{ is even}\geq 4\}}(-1) \Bigl(\prod_{e\in \overrightarrow{C_i}}a_e[\omega_{\overrightarrow{C_i}}+\omega_{\overrightarrow{ C_i}}^{-1} ]\Bigr)\cdot \prod_{\{i:|C_i|=2\}} \Bigl(\prod_{e\in \overrightarrow{C_i}}a_e^2\Bigr).\end{aligned}$$ Note that in a given covering there is always an even number of odd cycles, since otherwise there is no covering of the remaining graph by even cycles. We now fix a partial covering of the graph by odd cycles, and sum over coverings of the remaining graph by even cycles. Since the contribution of the parallel transport to doubled edges cancels out, and since the matrix $A$ has columns summing to zero, we then ‘open’ doubled edges according to the complete algorithm, using Remark \[rem:main\]. Everything works out in the same way, with the role of $R$ played by odd cycles. In this case though, because of the parallel transport, the contributions of RCRSFs do not cancel, but looking at the proof of Theorem \[thm:main\], we know precisely what those are. Summing over all partial coverings by odd cycles yields the result. Theorem \[thm:linebundle\] can then be specified in the case of bipartite graphs, in which case there are no odd cycles, in the case of planar graphs or of graphs embedded on the torus etc. seccntformat\#1[Appendix the\#1:]{} Pfaffian matrix-tree theorem for 3-graphs and Pfaffian half-tree theorem for graphs {#App:AppendixA} =================================================================================== In the paper [@MasbaumVaintrob], Masbaum and Vaintrob prove a Pfaffian matrix-tree theorem for spanning trees of 3-uniform hypergraphs. We start by giving an idea of their result. A *3-uniform hypergraph*, or simply *3-graph* consists of a set of vertices and a set of *hyper-edges*, hyper-edges being triples of vertices. Consider the complete 3-graph $K^{(3)}_{n+1}$ on the vertex set $\{1,\dots,n+1\}$, where $n$ is even; hyper-edges consist of the $\binom{n+1}{3}$ possible triples of points. Suppose that hyper-edges are assigned anti-symmetric weights $y=(y_{ijk})$, that is, $y_{ijk}=-y_{jik}=y_{jki}$, and $y_{iij}=0$. Note that considering other 3-graphs amounts to setting some of the hyper-edge weights to zero. A *spanning tree* of $K^{(3)}_{n+1}$ is a sub-3-graph spanning all vertices and containing no cycle; let us denote by $\T^{(3)}$ the set of spanning trees of $K^{(3)}_{n+1}$. To apprehend spanning trees of 3-graphs, it is helpful to use their bipartite representation: a hyper-edge is pictured as a Y, where the end points are black and correspond to vertices of the hyper-edge, and the degree three vertex is white. Then a sub-3-graph is a spanning tree of $K^{(3)}_{n+1}$ if and only if its bipartite representation is a spanning tree of the corresponding bipartite graph, see Figure \[fig:Spanning\] for an example. ![Bipartite graph representation of the following 5 spanning trees of $K^{(3)}_5$: $\{123,145\}$, $\{124,235\}$, $\{134,235\}$, $\{234,145\}$, $\{145,235\}$. The graph $K^{(3)}_5$ has a total of 15 spanning trees.[]{data-label="fig:Spanning"}](figSpanning.pdf){width="\linewidth"} Define the $(n+1)\times(n+1)$ matrix $A^{n+1}=(a_{ij})$ by: $$\forall\,i,j\,\in\{1,\dots,n+1\},\quad a_{ij}=\sum_{k=1}^{n+1}y_{ijk}.$$ Then, Masbaum and Vaintrob [@MasbaumVaintrob] prove that Pfaffian of the matrix $A$, obtained from the matrix $A^{n+1}$ by removing the last line and column, is a signed $y$-weighted sum over spanning trees of $K^{(3)}_{n+1}$: $$\label{equ:MasbaumVaintrob} \Pf(A)=\sum_{T\in \T^{(3)}} \sgn(T)\prod_{(i,j,k)\in T}y_{ijk},$$ where the product is over all hyper-edges of the spanning tree. We refer to the original paper [@MasbaumVaintrob] for the definition of $\sgn(T)$. A combinatorial proof of this result is given by Hirschman and Reiner [@HirschmanReiner] and yet another proof using Grassmann variables is provided by Abdesselam [@Abdesselam]. Using Sivasubramanian’s result [@sivasubramanian], spanning trees of $K^{(3)}_{n+1}$ can be related to half-spanning trees of the (usual) complete graph $K_{n+1}$. Sivasubramanian introduces an analog of the Prüfer code for 3-graphs, allowing him to establish a bijection between spanning trees of $K^{(3)}_{n+1}$ and pairs $(\gamma,M)$, where $\gamma\in\{1,\dots,n+1\}^{\frac{n}{2}-1}$ and $M$ is a perfect matching of the (usual) complete graph $K_{n}$ on the vertex set $\{1,\dots,n\}$. This bijection is also very clearly explained in the paper [@DeMierGoodall] by Goodall and De Mier. Writing $\M(K_n)$ for the set of perfect matchings of $K_n$, the set of spanning trees $\T^{(3)}$ can thus be written as $\cup_{M\in\M(K_n)}\T^{(3)}(M)$, where $\T^{(3)}(M)$ consists of the spanning trees corresponding to $M$ in the bijection. Equation then becomes: $$\label{equ:MasbaumVaintrob2} \Pf(A)=\sum_{M\in \M(K_n)}\sum_{T\in \T^{(3)}(M)} \sgn(T)\prod_{(i,j,k)\in T}y_{ijk}.$$ <span style="font-variant:small-caps;">Example</span>. When $n+1=5$, spanning trees of $K^{(3)}_5$ are in bijection with pairs $(\gamma,M)$, where $\gamma\in\{1,\dots,5\}$, and $M$ is a perfect matching of $K_4$. Returning to the ‘Prüfer code’ of [@sivasubramanian], one sees that the five spanning trees of Figure \[fig:Spanning\] are in bijection with the perfect matching $M=\{14,23\}$, and $\gamma=1,\dots,\gamma=5$, respectively. We now fix a perfect matching $M$ of $K_n$ and let $T_M$ be one of the $(n+1)^{\frac{n}{2}-1}$ corresponding spanning trees of $K^{(3)}_{n+1}$. From $T_M$, we construct a half-spanning tree of $K_{n+1}$ compatible with $M$ as follows. By the bijection, for every hyper-edge $ijk$ of $T_M$, exactly one of the pairs $ij,ik,jk$ belongs to $M$; without loss of generality, let us assume it is $ij$ and that $i<j$. To this hyper-edge, assign the edge configuration of $K_{n+1}$ consisting of the edge $ij$ and of the edge $jk$. Repeating this procedure yields a half-tree of $K_{n+1}$ compatible with $M$. It seems that for different $\gamma$’s, the corresponding half-spanning trees are different. <span style="font-variant:small-caps;">Example</span>. Recall that Figure \[fig:Spanning\] consists of the spanning trees of $K^{(3)}_{5}$ corresponding to the perfect matching $M=\{14,23\}$ through the ‘Prüfer code’. Figure \[fig:Spanning1\] pictures the half-spanning trees of $K_5$ compatible with $M$ obtained by the above construction. ![Half-spanning trees assigned to spanning trees of $K^{(3)}_{5}$ of Figure \[fig:Spanning\].[]{data-label="fig:Spanning1"}](figSpanning1.pdf){width="10cm"} It is interesting to note that not all half-spanning trees compatible with $M$ are obtained, and that they do not all satisfy Condition (C) of Definition \[def:def0\] (the third one does not satisfy it, see also Figure \[fig:fig10\]). A new family of half-spanning trees compatible with $M$ is constructed; it has $(n+1)^{\frac{n}{2}-1}$ elements, and could probably be characterized using the ‘Prüfer code’ and the construction of the half-spanning trees. This implies that the Pfaffian of the matrix $A$, written using the ‘Prüfer code’ of [@sivasubramanian] as in Equation , can be expressed as a sum over all perfect matchings $M$ of $K_n$ of a sum over a new family of half-spanning trees compatible with $M$. Now, by the anti-symmetry of the $y$-weights, the matrix $A^{n+1}$ constructed from the $y$-weights is skew-symmetric and has column sum equal to 0. It thus satisfies the hypothesis of Theorem \[thm:main\]. Let $M_0$ be a fixed perfect matching of $K_{n}$. Since the root $R$ consists of a single vertex $n+1$, the theorem involves half-spanning trees instead of forests, and we denote by $\T(M_0)$ the set of half-spanning trees compatible with $M_0$ of $K_n$, satisfying Condition (C) of Definition \[def:def0\]. By Theorem \[thm:main\], we have: $$\label{equ:thmmain} \Pf(A)=\sum_{T\in\T(M_0)}\sgn(\sigma_{M_0(T\setminus M_0)})\prod_{e\in T\setminus M_0}a_e.$$ Replacing $a_e$ by its definition using $y$-variables, yields $$\Pf(A)=\sum_{T\in\T(M_0)}\sgn(\sigma_{M_0(T\setminus M_0)}\prod_{e\in T\setminus M_0}(\sum_{k=1}^{n+1}y_{ek}).$$ This time, the Pfaffian of $A$ is written as a sum over half-spanning trees compatible with a *single* fixed perfect matching $M_0$, satisfying Condition (C). The term corresponding to a specific half-spanning tree is not a single *spanning tree* of $K^{(3)}_{n+1}$, but a sum over 3-subgraphs which are not necessarily trees. To recover the form of , there must be cancellations involved. <span style="font-variant:small-caps;">Example</span>. Take $M_0=\{14,23\}$, and consider the leftmost half-tree compatible with $M_0$ of Figure \[fig:fig10\]. Not taking into account signs, its contribution to $\Pf(A)$ is $a_{42}a_{35}$. Replacing with the $y$-weights, and using the fact that $y_{iij}=0$ gives a contribution of: $$(y_{421}+y_{423}+y_{425})(y_{351}+y_{352}+y_{354})= y_{421}y_{351}+y_{421}y_{352}+\dots+y_{425}y_{354}.$$ Each term corresponds to a 3-subgraph of $K^{(3)}_5$, but not necessarily a tree: as soon as a pair of triples of points has more than one index in common, it is not a tree, for example $y_{425}y_{354}$. Summarizing, using the ‘Prüfer code’ of [@sivasubramanian], the Pfaffian matrix-tree theorem of [@MasbaumVaintrob] can be written as a sum over a new family of half-spanning trees, and to each half-spanning tree corresponds a single spanning tree of $K^{(3)}_{n+1}$. When applied to 3-graphs, our Pfaffian half-tree theorem \[thm:main\] can be written as a sum over half-spanning trees compatible with a *single* perfect matching $M_0$, satisfying Condition (C). To each half-spanning tree corresponds a family of 3-subgraphs of $K^{(3)}_{n+1}$, not all of which are trees, there are cancellations involved. The Pfaffian half-tree theorem can be applied in the context of 3-graphs, but the result in this case is not naturally related to spanning trees of 3-graphs; this theorem takes its full meaning for (regular) graphs. [^1]: [Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, Université Pierre et Marie Curie, 4 place Jussieu, F-75005 Paris.]{} [`beatrice.de_tiliere@upmc.fr.`]{} [Supported by the ANR Grant 2010-BLAN-0123-02.]{}

--- abstract: | Wind energy has gained a huge interest in the recent years in various countries due to the high demand of energy and the shortage of traditional electricity sources. This is because it is cost effective and environmentally friendly source that could contribute significantly to the reduction of the ever-increasing carbon emissions. Wind energy is one of the fastest growing green technology worldwide with a total generation share of 564 GW as the end of 2018. In Malaysia, wind energy has been a topic of interest in both academia and green energy industry. In this paper, the current status of wind energy research in Malaysia is reviewed. Different contributing factors such as potentiality and assessments, wind speed and direction modelling, wind prediction and spatial mapping, and optimal sizing of wind farms are extensively discussed. This paper discusses the progress of all wind studies and presents conclusions and recommendations to improve wind research in Malaysia.\ author: - | Fuad Noman[^1], Gamal Alkawsi, Dallatu Abbas, Ammar Alkahtani, Seih Kiong Tiong,\ Janaka Ekanyake bibliography: - 'references.bib' title: 'A Comprehensive Review of Wind Energy in Malaysia: Past, Present and Future Research Trends' --- [Shell : Bare Demo of IEEEtran.cls for Journals]{} renewable sources, sizing optimization, Wind energy, wind farm design, wind mapping, wind potentiality, wind prediction. Introduction {#sec:introduction} ============ The increased demand for energy across the globe has led to Malaysia recognizing the significance of renewable energy as a supplement to conventional sources in generating electricity. Malaysia has established several energy policies that made the country one of the leading in energy production in South East Asia. The first National Energy Policy was coined in 1979 and 2011, the Renewable Energy Act was put into existence. The policy framework has the objectives to identify the energy needs properly, conserving resources and environment, promote sustainable development, and low carbon technology [@Oh2018; @Shaikh2017]. Malaysia pledged to the United Nations Climate Change Conference 2009 (COP15) to decrease the carbon emissions by 40% [@CommissionEnergy2017] and committed to reducing its Green House Gas emissions by 45% by 2030 with respect to the year 2005 [@Kardooni2016; @Mah2019]. The new Net Energy Metering (NEM) mechanism updated in 2019, proposes equal tariff for buying and selling electricity for NEM members. This recent process is expected to engage more customers to adopt renewable technology for example wind turbines and solar panels; however, the current NEM covers only the energy generated from solar panels [@Abdullah2019]. Currently, renewable energy in Malaysia are mainly comes from biomass, biogas, solar, wind, mini hydro [@Abdullah2019]. Figure \[Fig:Fig1\] summarizes the installed electricity supply capacity mix for the entire Malaysia (Peninsular Malaysia, Sabah, and Sarawak) as of December 2018 [@MalaysianEnergyCommission2019; @MalaysianEnergyCommission2019a; @SarawakEnergy2018]. Figure \[Fig:Fig1\](a) shows the total installed non-renewable sources dominated by fossil fuels, gas (43%) and coal (37%) with hydro plants larger than 100 MW (19%) and 1% of diesel generators. On the other hand, the renewable sources (grid-tied only), as shown in Figure \[Fig:Fig1\](b), covers solar (67%), biomass (11%), biogas (10%), LSS & NEM (6%), mini-hydro (5%), and solid waste (1%). The renewable capacity provides a total of 625 MW representing 2% of total energy capacity in Malaysia. However, Malaysian green technology master plan (GTMP) targets to increase the capacity of renewable sources mix to 20% by 2025. {width="0.7\linewidth"} Most of the wind power studies in Malaysia focus on the features and characteristics of wind speed. That is due to Malaysia faces challenges in wind energy production as Malaysia is located at 5$^\circ$ on the north side of the equator, also the sea and land air influence the wind circulation system. It was observed that wind changing by area and month and does not blow uniformly [@Islam2011]. As per the statistics, Malaysia is known to have low wind speeds in comparison to countries with high wind speeds such as Denmark and Netherland. However, several locations in Malaysia have good wind upon a particular time in the year, especially during monsoon seasons. Malaysia has two monsoons, the southwest monsoon (May-September) and northeast monsoon (November-March). The average of the wind speed is usually below 3 m/s, which make it less potential for generating countinuous wind energy output, however, the maximum wind speed falls in the range 6-12 m/s indicating the possibility to generate wind power at certain times only [@Masseran2012]. With this speed, there have been several attempts to install wind turbines across the country. A few installed wind turbines as pilot projects were found in the literature. The first wind turbine of 150 kW was installed at Pulau Terumbu Layang-Layang (Sabah), and the findings confirmed the success of the project [@Najid2009]. In contrast, the turbines of 100 kW installed within a hybrid system at Perhentian Island were failed. The on-site observations noticed that the turbines did not rotate continuously [@Albani2017a]. Besides, an earlier pilot project was implemented in a shrimp farm in Setiu (Terengganu) with a wind turbine of 3.3 kW. The turbine was connected directly to water pump systems and the aeration on the site , which was estimated to cover almost 40% of the farm’s energy consumption with energy cost of 0.98 RM /kWh each year [@Ibrahim2014c]. There are few review papers about wind energy in Malaysia that appeared in the literature. However, these reviews are not recent and most of them concentrate on renewable sources in general and do not discuss the literature in depth [@Ashnani2014; @Azman2011; @Foo2015; @Hossain2015; @Hosseini2014; @Ong2011; @Shafie2011]. Among the mentioned papers, only a few reviews concentrate on wind energy. However, these reviews investigate specific factors such as wind speed distribution and spatial models [@Lawan2013], particular aspects like political and regularity support [@Ho2016], and technical issues [@Didane2016; @Palanichamy2015]. None of these reviews conducted a comprehensive review of all related factors to wind potentialities and other wind energy aspects such as wind forecasting, mapping, turbines designs, and optimal sizing of hybrid renewable sources. Therefore, this paper presents a comprehensive review of wind energy research progress in Malaysia extensively. It comprises all potentiality factors, wind predictions, techno-economic, and design, which are further discussed in detail in the following sections. In this paper, search engines of Sciencce-Direct, Scopus, and Web of Sciences (WoS) were used to find the wind-related studies in Malaysia. From the search queries, a total of 177 articles were found. We first perform titles check, where a total of 27 entries were removed as duplicated articles. The rest of the articles were then subjected to full abstract read. During this stage, a total of 46 articles were removed as out of scope. The remaining 104 entries were entitled for full text read which were categorised as follows: feasibility and potentiality (54); mapping (6); design (7); prediction (8); sizing (18); and reviews (11). Figure \[Fig:Fig2\] shows the connections and links of the most occurred keywords in the processed articles in this review. The connectivity map was created using full counting method of all title-abstract keywords [@Leydesdorff2016]. {width="0.7\linewidth"} Potentiality ============ Learning the wind behaviours is very important to assess the performance of wind farms. The location plays an essential role on wind frequency and speed. Therefore, wind turbines are located only in particular regions globally. To evaluate wind eligibility for the management and design of wind energy system of specific location, various statistical measures and analysis are required to be performed using historical wind data extracted from the intended geographical site [@Li2010]. This section reviews energy potential and the statistical evaluation of wind characteristics studies in Malaysia. As shown in Table \[tab1\], the studies categorized based on their objectives. Wind direction, wind speed distribution models, and geographical aspects are presented. Moreover, the Techno-economic effects are reported. \[tab1\] Wind Direction --------------- Shamshad et al., (2007) [@Shamshad2007] conducted statistical and power density analysis of wind direction and speed in Mersing. They studied the effects of area topography and roughness on the wind energy using various wind turbines. The results show that the wind direction analysis presents almost 27% of wind direction occurs in the south-west (240 degrees). However, the analysis of projected wind speed at varying heights indicates that the 240 degrees’ direction has the lowest wind speed. On the other hand, different distributions models were applied to recognize the best model to determine wind direction in Malaysia. For instance, [@Masseran2013a] found that the finite mixture model of the von Mises (vM) distribution with $H$ amount of components considered the most suitable distribution model. In further investigations, by taking advantage of the flexibility of von Mises distribution model in addressing wind directional data with several modes. Nurulkamal Masseran (2015) [@Masseran2015] compared the circular model with the suitability of von Mises model based on nonnegative trigonometric sums. The findings show that the Finite Mixture of von Mises Fisher (mvMF) model with number of $H$ components is $\geq 4$ is the most appropriate model. Furthermore, Nurulkamal Masseran & Razali, (2016) [@Masseran2016] investigated the directions of the wind through the monsoon seasons. They used the data collected from five stations between 1st January 2007 and 30th November 2009 to fit the Finite Mixture model of von Mises distribution to data of wind-direction in hourly basis. The results demonstrated the ability of the model to fit all different data sets with all R2 values $\geq$ 0.9. Similarly, [@Satari2015] employed von Mises distribution. They noticed remarkable shifts in the direction of the wind in ten years. However, they stated that the wind direction not impacting the wind speed. Uniquely, Sanusi et al., (2018) [@Sanusi2018a] used Gama and a finite mixture model of von Mises(vM) distributions for wind speed and direction respectively to assess energy potential in Kuala Terengganu. Analysis of wind power density reveals that the existed uncertainty of using a bivariate model compared to univariate is explaining that the wind with the high speed does not usually concordant with the most common direction of the wind. The study concluded that the combination of Gama and a finite mixture of von Mises distributions was discovered to be the perfect bivariate model to clarify wind data of Kuala Terengganu. Likewise, Sanusi et al., (2017) [@Sanusi2017] investigated the effects of wind direction on producing wind energy in Mersing. The finite mixture of the von Mises and Weibull distributions was applied for modelling wind speed and wind direction and speed data respectively. The results show that within the studded time interval (2007-2013), the approximate power density (power equation) ranges from 18.2 to 25 W/m$^2$. Analysis of wind direction also concluded that the south-southwest direction is the dominant wind direction in Mersing. Wind speed distribution models ------------------------------ Saberi et al., (2019) [@Saberi2019] used the Weibull Distribution model to assess the quality of wind data gathered in the year 2017 in Kuala Terengganu. Likewise, Weibull Distribution model has been used as a method to determine the potentiality of wind energy by many studies as shown in Table \[tab2\] such as [@Daut2011; @Deros2017; @Rizeei2018; @Sanusi2018a; @Sanusi2017; @Syafawati2011]. It was noticed that the use of Malaysia Metrological Data (MMD) was dominant in most of these studies. The MMD data were collected from stations at 10 m height and mostly, extrapolated to higher heights using the standard 1/7 power law. In Butterworth, analysis of the Weibull distribution function performed using five different wind turbines to calculate the energy density. The calculated yield energy density was 288.23 and 315.10 kWh/m$^2$/year and 315.104 kW/m$^2$/year at hub height of 100 m for 2008 and 2009 respectively [@Ahmadian2013]. Likewise, at the height of 100 m and according to the Weibull variables, the mean speed of wind is calculated in hourly basis and then simulated using a joint Sequential Monte Carlo Simulation (SMCS) method and obtained a power densities of 84.59 W/m$^2$, 79.28 W/m$^2$ and 33.36 W/m$^2$ for Mersing, Kudat and Terengganu respectively [@Kadhem2019]. For self-collected data, [@Albani2018] gathered wind data at four locations Kudat, Mersing, Kijal, and Langkawi. The authors installed masts then embedded the sensors at different heights. The collected data were used to generate maps of wind resources and analyse power law indexes (PLIs). The results show that the wind speed for a feasible project 5 m/s can be reached only if the tower is greater than 60 meter above ground level in all locations, excluding Kudat, as it is noticed that the average wind speed surpasses 5 m/s at 50 meter above ground level. Furthermore, they assessed the accuracies of exponential fits via matching the findings with the 1/7 law through the capacity factor (CF) discrepancies. As a results Mersing and Kudat show potentiality to adopt medium-scale wind farms, however Kajil and Langkawi are suitable for small-scale wind farms. Differently, other studies examined various distribution functions. Tiang & Ishak (2012) [@Tiang2012] employed the function of Rayleigh distribution to investigate wind speed and density in Penang. They found that the annual average of power density is 24.54 W/m2, also predicted the monthly annual mean of wind energy which is at 17.98 kW/m2. In southern Sabah, Masseran et al., (2013) [@Masseran2013] used goodness of fit analysis to indicate Gamma and Burr distributions which most of the analysed stations show a good fit. However, it was noticed that Gamma distribution is the best for all stations. Similarly, [@Najid2009] applied three different probability distributions (Weibull, Gamma, burr) to evaluate the potentiality of wind energy in Kuala Terengganu. The study used a self-collected wind speed database recorded daily at 18 m hub-height for two years period (2005-2006) with an average speed of 2.78 and 2.81 m/s respectively. They concluded that among the proposed three distributions, Burr distribution provides the best fit of the analysed wind speed data. In another study, Masseran et al., (2012) [@Masseran2012a] evaluated wind speed persistence (stationarity and variability) in 10 wind stations (Kuala Terengganu ,Alor Setar, Bayan Lepas, Kuantan , Chuping, Ipoh, Kota Bahru, Mersing , Malacca, and Cameron Highlands) during the period from 2007 to 2009 using data obtained from MMD. Wind speed duration curve (WSDC) was used to generate an accumulative distribution of wind speed through a specific duration. The findings demonstrated that the wind speed form Chuping has the lowest variations and the most persistent regardless of the wind speed values (average of 1.03 m/s). However, considering the wind speed values and energy potential, Mersing was recognized to have the most potential site for energy generation with 18.2% of three years wind speed exceeds 4 m/sin comparison with other sites. In more investigation about Modelling and statistical evaluation of wind power density, [@Masseran2015a] utilized wind data from Cameron Highlands, Mersing, Malacca, Sandakan, Kudat, and Putrajaya stations obtained from MMD to evaluate several modelling methods including, Gamma, Weibull, and inverse Gamma density functions. The study proposed Monte Carlo integration method to better estimate the statistical parameters from the power density which was calculated from distribution models. The findings show the efficiency and reliability of the integrated method of Monte Carlo in delivering a good solution for defining the statistical attributes of wind power density. The results show that the mean power per unit-area ranges between 2 and 24 W/m$^2$, and Kudat and Mersing sites have the greatest value compared to other stations. Also, Kudat and Mersing stations have the greatest standard deviation with the range from 4 to 45 W/m$^2$. \[tab2\] Geographical aspects --------------------- It was noticeable that many studies were considering some geographical aspects of determining wind potentiality in Malaysia. Lawan et al. (2015) [@Lawan2015a] proposed a methodology to evaluate wind energy without using direct wind speed measurements. The study covers some terrain areas in Sarawak (Kuching, Samarahan, Serian, Lundu). A topological feed-forward neural network (T-FFNN) method was used to model the monthly wind speed profile using region-specific geographical, metrological, synthesized topological parameters. The study concluded that the wind power of all areas falls within the lower power density class ($\leq$100 W/m$^2$) with annual energy output in the range of 4–12 MWh/year. In another study, Lawan et al., (2017)[@Lawan2017] performed a study of wind power generation at Miri in Sarawak by analysing the characteristics of wind speed at the height of 10–40 m using the method of ground wind station and T-FFNN. With nine meteorological, geographical and topographical data as input parameters, the model uses monthly wind speed as its output variables. The value of 5800-13,622 kWh/year for the annual energy output (AEO) obtained from their micro sitting analysis shows the possibility of harnessing the wind energy for small scale applications. In Peninsular Malaysia, Yanalagaran & Ramli, (2018) [@Yanalagaran2018] investigated the relationship between the wind flow and coastal erosion over the period from 1984 to 2016 using the image layering technique on wind data acquired from MMD and coastal erosion data collected from newspaper articles, news videos, social media, and other relevant media. Using the equilibrium beach profile theory, the wind and coastal erosion analysis were investigated with the use of wind direction data. The findings demonstrate the good correlation between the wind direction and erosion direction. A few researchers employed satellite data to examine wind potentiality in offshore and onshore locations. Uti et al., (2013) [@Uti2013] used a multi-mission satellite altimeter to study offshore wind potential locations. They extracted the wind speed data from nine different satellites between 1993 and 2016. Meanwhile, they gathered data from buoys at the sea to compare it with satellite data for reliability validation purposes. The findings of their study indicated that the wind speed measures from the satellites are a little greater in comparison with the wind speed measures from the buoys. However, validation analysis using satellite track analysis and the root mean square error (RMSE) computation proofs a good correlation with positive values 0.6148 and 0.7976 for Sarawak and Sabah respectively. Another study by [@Albani2014], investigated offshore wind resources in Kijal (Terengganu) using QuickSCAT satellite data obtained from WindPRO software in the period from 2000 to 2008 with a horizontal resolution of 12.5km$^2$. WindPRP and WaSP software were used to conduct the potentiality analysis of wind data along South China Sea in Kijal. The optimum Feed in Tariff (FiT) rates as proposed for offshore wind projects in Kijal was found to be about RM 0.81–RM 1.38. The findings concluded that Malaysia has potential for adopting medium and small-scale wind turbines. In onshore areas, Rizeei et al., (2018)[@Rizeei2018] used the satellite data to derive solar irradiance estimation, and MMD data to provide daily wind over Peninsular Malaysia. Also, they applied the method of simple additive weighting in the geographic information system (GIS) platform to build the hybrid renewable energy suitability model. The result of the analysis shows that coastal areas at Hulu Terengganu have a good potential for the PV-WT hybrid system. Techno-economic --------------- From the techno-economic perspective, many studies have been accomplished by finding the FiT rates to determine the feasibility of wind farms in Malaysia. For instance, Ibrahim & Albani, (2014) [@Ibrahim2014a] selected 22 kW rated power wind turbines for the annual energy production analysis in Kudat. The analysis identified the best site to install wind turbines with a potential production ranging from 37.5–43.1 MWh/Year with FiT rates around RM 0.46-RM 0.80 per kWh. Moreover, Albani et al., (2017) [@Albani2017] proposed a method to calculate the optimal FiT suitable for wind energy generation in Malaysia. They selected Kudat as a case study with real data collected at different heights with an average speed of 5 m/s when the hub-height exceeds 30m. The base case FiT results indicate 0.9245–1.1313 RM/kWh for small-scale turbines and between 0.7396 and 0.9050 RM/kWh for utility-scale wind turbines. The analysis of the capacity factor by using different wind turbines was conducted by [@Akorede2013]. The findings show that at a height of 36.6 m Mersing achieved the maximum capacity factor (CF) of 4.39% with wind turbine of capacities 20 kW and 50 kW. Mersing also has the highest energy production of 378 MWh/Year then Chuping with an annual yield of 254 MWh. Mersing achieved the lowest production cost, which about 21 -35 cents followed by Chuping with the highest cost of production at 70 cents for 600 kW turbines. On the other hand, offshore wind resources in Kijal (Terengganu) were assessed by Albani et al., (2014) [@Albani2014]. Wind turbine capacity factor analysis shows that the turbine with a rated power of 850 kW was the most suitable for installation in Kijal with capacity factor of 26.8% (annual energy production of 3.653 MWh with PBP from 7 to 10 years) compared to other evaluated turbines. The analysis shows optimal FiT rates to be around RM 0.81 to RM 1.38. In another offshore site (near to Terengganu coast), a model of 2000 kW wind turbine was taken for gross energy and economic feasibility analysis. The sensitivity analysis of FiT ratio is found to be between 1 and 3 [@Mekhilef2011]. Notably, the FiT rate is changed when the level of economic parameters changes. In another avenue, Izadyar et al., (2016) [@Izadyar2016] selected a list of potential locations to set the optimal configuration of the hybrid renewable systems. Other researchers have considered a techno-economic factor in assessing wind potential locations and determining the optimal configurations of hybrid systems [@Hossain2017; @Muda2018; @Shezan2016; @Shezan2015; @Wahid2019]. Based on Net Present Cost (NPC), Langkawi was found to be the most potential location for the PV-Wind configuration (NPC of 696,083 USD). Next, Tioman island and Borneo island were found to be fit for the PV-WT- MHP and PV-MHP configuration respectively [@Izadyar2016]. By considering the cost of energy (COE), Ngan & Tan (2012)[@Ngan2012] conducted potential energy and economic analysis on using hybrid PV-WT-DG power with and without a battery storage system in Johor Bahru using HOMER software. Hybrid WT-DG without battery system shows a very high excess electricity from which they finally suggested to add battery storage. The hybrid WT-DG with battery storage shows COE values ranging from RM 1.365/kWh to RM 1.638/kWh. Likewise, Muda Et Al., (2018) [@Muda2018] focused on evaluating the wave energy system in term of its feasibility to provide electricity to Terengganu (Pulau Perhentian island) by proposing and analysing hybrid WT-PV-hydro-DG power system. Findings demonstrate that a hybrid wave energy system achieved a lower cost of electricity (COE) which is RM 1.303/kWh in comparison to the hybrid PV-battery system which achieve COE of RM 1.262/kWh. Moreover, energy analysis shows a total of 712,813 kWh/year (81 kWh) of generated power with only a 2.4% share from wind energy and 0.76% share from wave energy. In another study conducted in Terengganu, a hybrid system (grid-PV-WT) connected to the grid for households was evaluated. The findings show that the most feasible configuration is the PV connected to the grid hybrid system (2 kW PV panels) with a cost of energy at 0.001 \$/kWh. In other configurations, the total electricity produced by wind turbines is only 14% and 22% of total electricity production in PV-WT-grid and wind-grid systems respectively. The analysis shows that the best configuration is PV-WT-grid with an average wind speed of 3.43 m/s and FiT rate of 0.5169 \$/kWh [@Muda2016]. In contrast, Anwari et al., (2012)[@Anwari2012] studied the case of Pemanggil Island (near Mersing) that has annual wind speed around 1.7 to 6.7 m/s. They found that the (WT-DG) hybrid system can be feasible in case that the wind speed is above 3 m/s and the diesel price is greater than \$ 1.02/Litter. Elsewhere, at five locations in Malaysia (Kota Belud, Kudat, Langkawi, Gebeng, Kerteh) Nor et al., (2014)[@Nor2014] conducted the economic-feasibility analysis of installing 77 Leitwind wind turbines of 1,000 kW each . The study shows that the Kota Belud achieves the highest internal rate of return of 21% and the lowest payback period of 4.25 years [@Nor2014]. In contrast, Suboh et al. ( 2019) [@Suboh2019] proposed a dispatch strategy micro-wind turbine farm with storage using monthly average wind data for Mersing in 2009. The study considered a 10 $\times$ 300 W wind turbines (Infiniti 300) with the cut-in speed of 2 m/s. The study concluded that, according to the proposed strategy, the payback period requires 20 years to reimburse capital expenditure. As shown in Table \[tab3\] the PV-WT-DG configuration is the optimal setting for Berjaya, Kuala lumper International Airport KLIA Sepang, and Johor. However, few studies found wind energy not feasible to be utilized in several locations in Sarawak and Terengganu for large-scale turbines as the average wind speed falls below 4 m/s [@Arief2019; @Ahmadian2013a]. [p[2cm]{}|m[4cm]{}|p[4cm]{}|p[4cm]{}]{} Article & Location & Optimal Configuration & COE-Cost\ [@Muda2018] &Pulau Perhentian Kecil &PV-DG &RM 1.262/kWh\ [@Izadyar2016]&Langkawi &PV-WT & NA\ [@Izadyar2016]&Tioman island &PV-WT-MHP &NA\ [@Wahid2019]&Kerteh &PV-WT &USD 0.474/kWh\ [@Shezan2016]&KLIA Sepang (Selangor) &PV-WT-DG &USD 0.625/kWh\ [@Shezan2015]&KLIA Sepang (Selangor) &PV-WT-DG &USD 5.126 /kWh\ [@Hossain2017]&Berjaya Tioman &PV-WT-DG &USD 0.279 /kWh\ [@Ngan2012]&Johor &PV-WT-DG &RM 1.365–1.638/kWh\ \[tab3\] Wind prediction ================ Optimal prediction of wind speed is an open problem of research due to the need to foresee the feasibility of harnessing wind energy form specific sites. Wind speed forecasting has been extensively studied with hundreds of articles found in the literature. However, wind speed prediction methods can be categorized into five groups: persistent method, physical models, statistical models, artificial intelligence methods, and hybrid methods [@Khodayar2017]. A breif summary of these methods is as follows, 1. The persistent method is the most common baseline method which often used as a benchmark method for comparisons with other methods. It assumes that the wind speed will remain constant over time such that the wind speed value in the future is similar to the current value when the forecast is made. However, due to the intermittent nature of wind speed, this assumption fails when the prediction horizon increases. 2. The physical model is based on weather forecast data (e.g. temperature, pressure, surface roughness, and obstacles) and numerical weather prediction (NWP) models. NWP utilizes complex mathematical models to create a terrain-specific weather condition which are then used for wind speed predictions. NWP requires large computational resources with high time consumption; hence, it is not suitable for short-term wind speed predictions [@Eze2019; @Khodayar2017]. 3. The statistical methods learn the correlations of historical wind speed data and/or other explanatory variables (i.e. the common metrological parameters or NWPs). Statistical models are usually less complex than the physical models and show accurate performance when used for short-term forecasting tasks. The conventional statistical techniques can be categorized into linear stationary models, nonlinear stationary models, and linear non-stationary models. Generally, the statistical models are limited by data underlying dynamics constrained by a strict assumption of normality, linearity or variable independence [@Mishra2018]. 4. The artificial intelligence methods and supervised mechanisms are the dominant fields of research in recent years which can represent any complex relationship between the data regardless of the underlying dynamics [@Chang2014]. Recent advances in deep learning methods have seen significant performance improvements in many practical forecasting tasks, sometimes surpassing the known strong statistical models’ performance. This is owing to the capability of deep learning methods to perform both feature extraction and modeling which allows learning of complex data representations with the privilege of hierarchical levels of semantic abstraction via multiple stacked hidden layers and hence the robust and accurate forecasting even based on raw data from multiple exogenous or heterogeneous sources. 5. Hybrid models are based on the combination or integration of two or more models of different methodologies to benefit from the advantages of each model to improve prediction performance [@Khodayar2017; @Chang2014]. In this paper, we limit the review of wind speed forecasting methods to those studies that have been carried out in Malaysia using data collected in Malaysia too. A limited number of research articles found in the literature which tried to improve the prediction performance mostly using metrological station data. We conduct a thorough review of eight published articles related to wind data prediction in Malaysia during the period from 2005 to 2018 in the analysed database. Four prediction methods were used artificial neural network (ANN) [@Kadhem2017; @Lawan2015; @Shamshad2005; @Shukur2015], two deployed adaptive neuro-fuzzy inference system (ANFIS) [@Hossain2018; @Sarkar2019], and two used seasonal autoregressive integrated moving average (SARIMA) [@Deros2017], and Mycielski algorithm [@Goh2016]. The first article of wind speed prediction for Malaysia was published by Shamshad et al. (2005) [@Shamshad2005]. The authors implemented a stochastic synthetic generation of wind data using Markov chain with wind data obtained from the MMD for two stations (Mersing and Kuantan) during the period from 1995 to 2001. A first and second-order matrix of the Markov process with 12 states were estimated from the wind speed data which are then reversely used to predict future wind speed data. To assess the performance of Markov chain matrices, the Weibull distribution was estimated from observed and predicted wind data. Both transition matrices show comparable distribution parameters of k and c. Autocorrelation and spectral power density were also used for performance evaluation when using different correlation time lags. The RMSE of the first and second-order was 0.12 and 0.03 for Mersing and 0.03 and 0.03 for Kuantan. Markov model is one type of statistical model that heavily relies on probabilistic theory and higher orders show better performance with more state-space complexity. Lawan et al. (2015) [@Lawan2015] proposed a topological ANN method for monthly wind speed prediction. The geographical and metrological data are obtained from MMD for eight regions in Sarawak for a period of ten years from 2003–2012 (Kuching, Miri, Sibu, Bintulu, and Sri Aman) and five years from 2008 to 2012 (Kapit, Limbang, and Mulu). Nine parameters are used as input to the ANN model including Latitude, Longitude, Altitude, Terrain Elevation, Terrain Roughness, month, temperature, atmospheric pressure, and relative humidity. For each station, the data were split into 70%, 20%, 10% for train, test, and validation respectively. All data were normalized to the interval of \[-1, 1\]. Monthly prediction results show the minimum and maximum values of correlation coefficient $R$ were 0.8416–0.9120, and the maximum MAPE of 6.46%. Another study by Shukur & Lee, (2015) [@Shukur2015] suggested a hybrid method of ANN and Kalman filter (KF) to improve the performance of ANN models. However, the authors used daily wind speed data obtained from MMD for Muar station during the period from 2006 to 2010. Unlike the previous studies which mostly use data-driven prediction models, state-space model parameters were estimated using the ARIMA model and then used as input to ANN to predict the wind speed future data. ARIMA (0,1,2) (0,1,1)$_12$ model was selected for state-space parameters estimation with MAPE of 19.27, the Hybrid ARIMA-KF achieved 17.20 MAPE, and Hybrid KF-ANN showed 11.29 MAPE. The results indicate that KF-ANN improved the prediction performance by almost 8% compared to the direct use of the ARIMA model. However, the state-space models are usually of high computational complexity and sensitive to the dynamics of data, especially that of the high sampling rate. Kadhem et al. (2017) [@Kadhem2017] proposed a short-term wind speed prediction hybrid model using Weibull distribution and ANN (HANN). Hourly wind speed data were obtained from MDD from three stations (Mersing, Kudat, and Kuala Terengganu) for three years 2013–2015. Data were normalized in the interval \[-1, 1\] and then partitioned into 60% train, 20% validation, and 20% test. Five parameters are used as input to the ANN model including wind speed of Weibull model, direction, hour, day, month. Results show the proposed HANN method achieved a MAPE and RMSE of 6.06% and 0.048 respectively for Mersing data which improves the prediction performance for 10.34% MAPE compared to the ANN model alone. The suitability of using a hybrid ANFIS system and optimization methods for long-term (weekly and monthly) wind prediction in Malaysia was investigated by Hossain et al. (2018) [@Hossain2018]. Wind speed data were obtained from MMD of several locations Mersing, Kuala Terengganu, Pulau Langkawi and Bayan Lepas during the period 2004 to 2014. The data were adjusted to 50m hub-height. Three optimization techniques namely, particle swarm optimization (PSO), genetic algorithm (GA), and differential evaluation (DE) were used to optimize the ANFIS membership function parameters. Method performance was analyzed when using different train-test data partitioning \[70%,30%\], \[60%,40%\], and \[80%,20%\]. Both ANFIS-PSO and ANFIS-GA seem to be able to provide an overall good power density prediction performance. The proposed hybrid ANFIS model was also used for daily wind data extrapolation of Tioman island where no MMD station is available. Even though ANFIS is a combination of ANN and fuzzy inference system (FIS) which supposed to overcome the limitations of both methods and provide better performance. Sahin & Erol, (2017) [@Sahin2017] conducted a comparative analysis of both ANN and ANFIS concluding that the ANN approach performed better than the ANFIS model on predictions. Sarkar et al. (2019) [@Sarkar2019] also conducted a comparative study of wind power prediction using ANFIS model in Kuala Lumpur and Melaka. Wind data were obtained from MMD over the period 2013 to 2015. Four input variables are used for prediction including humidity, pressure, wind speed, and temperature. With one year of test-data, results show satisfactory results when using ANFIS model to predict wind power data of Kuala Lumpur compared to Melaka with (RMSE=1.82, MAE=1.98, MAPE=13.23 and R2=0.944) for Melaka and (RMSE=1.65, MAE=1.87, MAPE=12.35 and R2 = 0.968) for Kuala Lumpur. Mycielski algorithm is a simple non-parametric alternative approach to the complex parametric models which is mainly based on pattern matching [@Croonenbroeck2015]. Goh et al. (2016) [@Goh2016] proposed a wind energy assessment comparing two methods (Mycielski algorithm and K-means clustering) for wind prediction in Kudat, Malaysia. Wind data were obtained from MMD over the period 2002 to 2010, the data from 2002–2009 were used for training while 2010 for testing. Results show that K-means clustering provides more accurate predictions compared to the Mycielski algorithm. Weibull distribution was used to assess the comparison of predicted wind speeds with actual ones with prediction RMSE of Mycielski was 1.875 and 1.391 for K-means. Wind profile in Malaysia is highly dependent on monsoon seasons, where four main seasons take turns throughout the year producing different wind speed and direction behaviours. Deros et al. (2017) [@Deros2017] proposed a wind speed seasonal forecasting model using SARIMA with wind data obtained from MMD during the period 2000 to 2015 for Langkawi. Four monsoons were analysed including Northeast Monsoon, April Inter-monsoon, Southwest Monsoon, and October Inter-monsoon. For variations analysis and determination of seasonal wind distribution, three different distribution models were investigated including Gamma, Lognormal and Weibull distribution. Forecasting results show that SARIMA achieves RMSE=0.3186, MAE=0.265, and MAPE=11.644%. Among the three tested distribution models, the Log-normal distribution model produced the lowest gap value among all the seasonal wind speed data in each monsoon season. The study concluded that the northeast (NE) monsoon acquires the highest mean wind speed with an average between 1.8 to 2.3 m/s, followed by October inter-monsoon, Southwest monsoon and the lowest wind speed was estimated to be in April inter-monsoon between 0.9 to 1.3 m/s. According to the reviewed articles on wind speed prediction, we noticed that the lack of standardization between the proposed methods and wind database makes it difficult for technical or numerical performance comparisons. It’s important to understand the performance of these models and provide some measure of confidence. Fare analysis requires the prediction experiments to be conducted under similar environments and conditions, such as data sampling rate, train-test partition, prediction horizons, input variables, target area, and hub-heights. Hence, a benchmark study is required to test the generalization, universality, and transferability of these prediction models. Turbines design ================ In the continuous search for the possibility of generating power from wind energy in a low wind speed region, a considerable amount of research works has been reported in that respect. The most prevailing is to design wind turbines specifically for low wind speed region. (Wen et al., 2019a) [@Wen2019a] applied blade element momentum theory (BEMT) to design and study the performance analysis of six constant speed horizontal axis wind turbines (CS-HAWT). On the bases of Airfoils low Reynolds number that will fit small wind turbines blade design for low wind speed region application, Six Airfoils: BW-3, FX63-137, NACA4412, S822, SG6040 and SG6043 with a Reynolds number ($<$ 500,000) characteristics and a tip speed ratio of 5 and 6 were chosen for the design. Their results showed that by increasing the Reynold’s number the aerodynamic performance of the Airfoils will rapidly increase as well and will, in turn result in lower cut-in speed as well as a higher power coefficient. A blade of 1.78 m/s cut-in wind speed and a higher power coefficient of 0.52 is obtained at the TSR of 5 and 1.8m/s cut-in wind speed, 0.51 power coefficient at the TSR of 6 with Airfoils SG6043 and FX63-137. A lower cut-in speed of 1.5m/s was obtained with Airfoils BW-3 and FX63-137. The lowest cut-in speed was recorded at the TSR of 5 with the Airfoils. Their results of computed annual energy generation (AEG) when the designed turbine power curve are employed with the wind speed distribution of Kangar, shows that out of the six airfoiled turbine selected, SG6043 has outperformed all with AEG of 64kWh and 66kWh for the 5 and 6 TSR respectively. Similar research was conducted by Wen et al., (2019b) [@Wen2019] to design a small wind turbine’s blade as partial solution to the low wind speed region of Malaysia. Based on Malaysia’s average wind speed of 2-3 m/s, they chose NACA 4412 Airfoilss profile with lower Reynold numbers of Re0.2, Re0.25, Re0.3, Re0.35 and a tip speed ratio (TSR) of 5, 6 and 7 to design a small wind turbines of 4 m/s rated wind speed. A turbine blade that starts rotating at a wind speed of 1.5 m/s and 2.0 m/s with the efficiency of about 49.31% was realized. Their result of the AEG using the designed turbine’s power curve and the Weibull distribution function with wind profiles of Kuching, Miri, Kangar, Labuan and Kudat shows a better performance of SWT\_re0.25 and SWT\_re0.3. Misaran et al. (2017) [@Misaran2017] designed and developed a hybrid vertical axis wind turbine that can start rotating at the wind speed of 1.0 m/s and can generate a useful power from a low and intermittently high wind speed region like Malaysia. The blade designed was carryout with a CAD software and utilizes airfoils tools 14 to generate the NACA0012 profile. In another perspective Johari et al., (2018) [@Johari2018] used CATIA software to design and compare the performance of three-blade horizontal axis wind turbines (HAWT) and Darrius type vertical axis wind turbine (VAWT). The result of the HAWT and VAWT performance analysis test shows that the HAWT produced higher voltage at stable wind condition but the voltage dies up as the wind direction changes. As for the VATW even though its output voltage is not as high as that of HAWT but it remained unaffected by the change in the wind direction. Relatively, a vertical axis wind turbine’s blade based on Aeolos-V1k was modified and redesigned using ANSYS software by [@Yi2018]. The modification which is performed with teak wood material reduces the blade’s weight and cost and improves its efficiency. Several articles to further explore the possibility of generating electricity from the wind energy sources in Malaysia have been reported. Awal et al. (2015) [@Awal2015] designed a vehicle-mounted wind turbine (VMWT) to utilize the speed of the wind that passes the vehicle as it moves and to generates electricity. It uses a permanent magnet DC motor as a generator (PMDC) and produces 150-200W of power as the vehicle moves. Likewise, Khai et al. (2018) [@Khai2018] modified the rooftop ventilator to functions as wind turbine and Harvest wind energy from the steady wind speed of the ventilation channel that is created due to the temperature difference between the indoor and outdoor. The result of their testing with a stand fan shows that the modified wind turbine can produce a minimum of 4.63 V and 21.44 mW power which can light up a 5 V LED bulb. Moreover, the design and implementation of a VAWT that utilizes the waste exhaust coming out from the cooling tower was presented by Rahman et al. (2015). It comprises of the drag blade (C-type), a guide-vane to increase the inlet wind speed and an enclosure to avoid the wind coming from the opposite direction. They have recorded a voltage and a current of 1.97 V and 0.0041 A and average power of 0.0058 W and concluded that their system is capable of generating 0.0081 W of electricity. Optimal sizing of hybrid energy systems {#sec:sizing} ======================================== The integration of conventional energy sources with renewable energy sources and energy storage devices is common in developing hybrid energy systems to fulfill a particular load demand. However, the involvement energy storage system is essential to moderate the supply-demand mismatch. Moreover, conventional sources, like diesel generators or fuel cells can also be added to the renewable sources to get a better energy balance. The hybrid systems begin to be common nowadays as they merge the advantages of both AC and DC- as well as flexibility, cost-effectiveness and to integrate loads and sources based on their characteristics [@Hosseinalizadeh2016; @Sanajaoba2016]. However, there is no one configuration can fit all solutions, therefore, the most suitable configuration must be considered for a particular application and site. The literature of wind energy in Malaysia shows two software tools that have been used in optimizing hybrid renewable energy systems. The first software tool is HOMER that has been used mostly as shown in Table \[tab4\], and the second is iHOGA that was applied by Fadaeenejad et al. (2014)[@Fadaeenejad2014]. In contrast, other studies [@Khatib2015; @Khatib2012; @DeAn2011] used MATLAB simulation. Energy systems connected to the grid or in standalone mode involving conventional renewable sources and storage systems can be optimized and simulated using HOMER. The optimization can be executed using historic meteorological data of the location. The HOMER software was used to find the optimal configuration of hybrid energy systems for different locations in Malaysia. Researchers’ main objective was to reduce NPC subjecting it to a few constraints such as reliability and environmental. Table \[tab4\] lists the studies from different locations in Malaysia that used the HOMER software tool for size optimization for hybrid energy systems. Majority of the studies found that PV-WT-DG energy resources are feasible as it provides the minimum NPC and COE with lower CO2 emissions [@Haidar2011; @Hossain2017; @Khatib2012; @Mohamed2013; @Ngan2012; @Shezan2016]. Nevertheless, Khatib et al., (2013) [@Khatib2013] examined the power sources in nine locations in Malaysia to determine the cost of energy for every power system as a standalone system. They found that the average cost of the wind energy system is 1.6–7.29 USD/kWh. In comparison, the solar, diesel generator and grid power average cost are 0.35–0.5 USD/kWh, 0.27-0.30 USD/kWh and 0.11 USD/kWh respectively. The findings concluded that using wind energy as a standalone system is not feasible in Malaysia. In concurrent with the previous study, Haidar et al. (2011) [@Haidar2011], used HOMER software to study the feasibility of wind energy within a hybrid system in four locations; Pinang, Johor Baharu, Sarawak, and Selangor. The results show that the cost of wind energy is 1.054–1.457 USD/kWh. The findings lead to the fact that PV-DG is the optimal configuration in Malaysia. \[tab4\] Wind mapping ============= A comprehensive onshore or offshore wind resources mapping is necessary for assessing wind feasibility especially for Malaysia with a low wind profile. It is well known that wind speed varies according to terrain conditions, such as temperature, wind direction, and height from the ground, surface roughness, day time, and year seasons [@Laban2019]. Wind resource geographical maps can be generated by using either spatial prediction or altimetry wind data for the unobserved or uncovered locations by metrological stations. Development of wind energy in Malaysia is still in its early stages, with majority studies use secondary metrological wind data which were collected by weather stations mainly located in airports [@Ibrahim2014b]. These wind data were obtained from MMD, which were mainly used in airports for the weather forecasting. Hence, few studies highlighted this issue and conducted nationwide wind energy planning by using spatial interpolation methods to create wind maps for Malaysia [@Ibrahim2015]. Masseran et al., (2012) [@Masseran2012] conducted a study to generate wind resources map for entire Malaysia using statistical and geostatistical analysis including semivariogram and inverse distance weighted (IDW) methods for spatial correlation, estimation, and interpolation. A total of 10 years of wind speed data from 2000 to 2009 for 67 stations were obtained from MMD to create a nationwide spatial map. The study first determines the best suitable wind speed distribution. Nine types of wind speed statistical distributions are examined, namely Weibull, Burr, Gamma, Inverse Gamma, Inverse Gaussian, Exponential, Rayleigh, Lognormal, and Erlang. Goodness-of-fit tests indicate only five distributions can be used for each station, Gamma, Burr, Weibull, Erlang and Inverse Gamma to derive the power distribution for each station. Raw moment and Monte Carlo approach were used to determine the mean power. The spatial correlations and dependencies were investigated using the semivariogram method, while the IDW method was used to estimate/interpolate the wind power in the spatial dimension for mapping. Finally, the spatial maps were created. Spatial Mappings of mean power density reveals that the northeast, northwest, and southeast of peninsular Malaysia and southern region of Sabah are the most potential locations for wind energy development. In 2014, another wind mapping study was conducted by [@Ibrahim2014b] covering only nine sites (Mersing, Kuala Terengganu, Pulau Langkawi, Sandakan, Kudat, Kota Kinabalu, Bintulu, Kuching, and Tawau) with only one-year wind data obtained from MMD for 2009. Weibull distribution and IDW interpolation methods were used for mapping. The study concluded that Mersing and Kudat are considered the best potential sites showing an average of 3 m/s above 60 m high. The IDW spatial wind mapping method showed that the area located at the southern part of Peninsular Malaysia has better wind resources. These conclusions were proved by Weibull distributions of each area. The study conducted further analysis of wind speed distribution and direction. However, it has less spatial analysis in terms of data and locations compared to the previous study of Masseran et al. (2012) [@Masseran2012]. Same authors in cite[Ibrahim2014]{} studied the spatial interpolation and mapping of wind distribution in Malaysia with wind data obtained from MMD considering more sites of 12 stations (Langkawi, Kuala Terengganu, Mersing, Kuching, Bintulu, Kota Kinabalu, Kudat, Sandakan, Tawau, Cameron Highland, Kuantan, and Kapit). For mapping, the wind extrapolation, power-law, and the Digital Elevation Model (DEM) were together used to create the onshore spatial wind distribution for Malaysia. Then the Ordinary Kriging method was used for spatial wind data were interpolation which is then combined with DEM maps via the power law to produce the final wind map. Results show monthly maps variations with an accuracy of the Kriging interpolation and wind map of $\pm$1.1143 and $\pm$1.4949 m/s respectively. The study was limited to the selection of few sites for mapping, the wind direction was excluded in the analysis, and the produced wind map was limited to 10 m height only. The authors published another article in [@Ibrahim2015] following similar methodology and wind data of their previous study in [@Ibrahim2014b]. Power density Spatio-temporal maps were generated with extrapolated wind data to different heights from 10 m to 100 m. The study concluded that the wind power of Malaysia is falling into Class 1, showing 1.5232 MW/m$^2$ at 100 m height. {width="0.75\linewidth"} Partial wind map of the eastern part of Malaysia (i.e. Sarawak) was studied by Lawan et al. (2015) [@Lawan2015] using wind data obtained from MMD for four stations. The proposed topological ANN wind speed prediction method was utilized to predict the wind data for the uncovered areas in Sarawak to generate a more precise spatial map. Two ready-made software (i.e. GEPlot and Google Earth) were utilized to build a DEM model. The target site sampled locations including the longitudes and latitudes were generated using the world geodetic system (WGS 84). These sampled locations were then used to extract the DEM using a terrain zonum solution (TZS) tool. The last element, the surface roughness class was developed using GEPlot software. The study for the first time proposed isovent wind atalas mal of Sarawak at heights from 10 to 40 m. The study reveals that the northeast and southwest and coastal regions of Sarawak have better wind potentials. The main focus of previously reviewed articles was wind resources mapping to explore the wind potential in onshore sites. Ahmad Zaman et al., (2019) [@AhmadZaman2019] studied the mapping and sectorization of offshore wind energy potential locations in Malaysia using multi-mission satellite altimetry data. The satellite altimetry (Jason-1, Jason-2, and Envisat) data was obtained from Radar Altimeter Database Systems located at GNSS and Geodynamics Laboratory, University Technology Malaysia (UTM) for a period of 19 years, from 1993 to 2011. In addition, buoy measurements from two offshore sites were used for validation. Comparing the satellite with buoy data, the average error was found to be 14% with a correlation of 0.835 and RMSE of 1.99. ArcMap 10.2.2 software using Raster Interpolation with the IDW function was used to generate the annual wind energy maps. Figure \[Fig:Fig4\] shows the produces offshore wind energy density map for Malaysia. The study concluded that areas C4 (Terengganu coastline), C9 (Labuan), and D11 (Sabah) have high annual wind speed and wind energy density. {width="0.8\linewidth"} Future research recommendations ================================ 1. or onshore, several studies have recommended a few potential sites for wind energy harvesting in Malaysia. However, the majority of these studies use the MMD database along with interpolation and extrapolation methods to measure the wind energy feasibility. In order to assess the actual wind speed in these sites, it is important to collect wind data from these sites by installing masts with different heights. 2. Although few studies have conducted offshore wind energy-related analysis mainly using satellite data, the lack of actual offshore wind data makes it difficult to prove the recommendations of these studies. Hence, we also recommended installing wind data collection instruments at the previously highlighted sites in the literature. 3. For optimal sizing of the hybrid energy systems, most of the studies used HOMER software tool, and a rising trend is noticed against the use of modern optimal sizing methods as they offer realistic and promising size optimization. Thus, it is recommended to use such modern methods. For example, Teaching-learning-based optimization (TLBO), the hybrid FPA/SA algorithm, and Non dominated sorting particle swarm optimization (NSPSO). 4. The height of the turbines is noticed to be significantly influencing the wind speed and optimization results. Based on this factor, the constraints in turbines installment, designs, and optimization problems should be considered, instead of using wind speed extrapolation to higher heights which may not reflect the correct wind speeds for different terrains and provide wrong estimates of turbines power output. 5. The majority of studies concentrate on reliability and cost in the hybrid system, as discussed previously. Few studies focus on the environmental factors such as CO2 emissions. Environmental factors should be considered especially if the energy system consists of a conventional source. 6. According to the analysed MMD wind data in the previous works, the low wind profile in Malaysia requires a customized wind turbine designs that can operate at low cut-in wind speeds. Therefore, researchers can concentrate more on designing rotor blades that will reach higher rated power at lower wind speed which on the other hand will increase the AEG. 7. The shortage of wind turbines in producing the expected power was probably because of the inappropriate technology used in choosing the wind turbine. Therefore, it is recommended to conduct in-depth research about the wind turbine generator selection process. 8. There is a need to examine the effect of a different national advisory committee on Airfoils (NACA) profile on the performance of Vatical axis wind turbines (VAWT) since current studies have only considered a single NACA 0012 profile. Conclusion ========== In this review paper, we discussed all wind-related literature studies in Malaysia. Several factors have been ascertained during the analysis, including wind potentiality and assessments, wind speed and direction modelling, wind prediction and spatial mapping, and optimal sizing of wind farms. Despite that the majority of literature studies have paid more attention to the wind potentiality in Malaysia, results analysed are mostly based on theoretical studies shows conflicting conclusions and sometimes grossly inaccurate. However, few studies have shown valuable efforts on studying wind energy even though when using metrological data with limited capabilities. According to the wind assessment studies and due to the equatorial location of Malaysia, the wind profile is relatively low in general. However, three potential sites in Malaysia show an acceptable wind profile and could be further investigated for the actual development of wind farms. On average, the reported wind speed is within 2-8 m/s and varies according to the monsoon and site of measurements. Lacking standardization and representation of wind data has led the researchers to use hybrid power systems including other renewable energy sources (i.e. solar and hydropower) along with wind which was highlighted in this review as well. Several studies have been conducted on wind power energy conversion using various methods viz., power law, computer software (e.g. HOMER), wind turbine power curve, or by using wind power or speed density distributions. Different statistical distributions have been reported and evaluated for wind data in Malaysia with most of the studies suggests the Weibull distribution for wind speed. Mersing (2.4309$^{\circ}$ N, 103.8361$^{\circ}$ E), in many studies, showed the highest wind speed profile with an average above 2.5 m/s which suggests involving small-scale wind turbines with low cut-in speed. Nevertheless, wind power potential for obtained from wind data collected over 10 years, Mersing achieved a power density above 50 W/m$^2$ [@Akorede2013; @Sopian1995]. Kudat (6.8831$^{\circ}$ N, 116.8466$^{\circ}$ E), however, in other studies (e.g [@Albani2017a]) showed a slightly better wind profile compared to that of Mersing, with average above 2.8 m/s. Persistent analysis of wind speed by Masseran, et al. (2012) form Chuping (6.4985$^{\circ}$ N, 100.2580$^{\circ}$ E) showed the lowest variations and the most persistent regardless of the wind speed values. It is worth noting that, some studies consider the annual average of wind speed which doesn’t correctly reflect the actual power output as it is heavily dependent on the wind speed fluctuations. Wind prediction was also a subject of interest in general with a limited number of studies proposed for wind prediction in Malaysia. The conducted review of these studies showed that the proposed methods achieved different degrees of uncertainty lacking to generalized prediction approach which can tackle seasonal and annual wind variations for both short and long-term horizons. On the other hand, the extrapolation of wind speed to different heights could impose significant errors producing falsely estimations of power profiles. The topographic parameters should be considered for both vertical extrapolation and spatial interpolation to produce more accurate wind potentiality assessment analysis. Spatial prediction and mapping of entire or part of Malaysia have been studied in the literature, offering valuable preliminary maps highlighting several potential sites. This is much beneficial for future investigations where before any actual wind turbine installation decisions, the wind data need to be measured to confirm the feasibility of the identified site. Though most of the previous studies concluded wind power generation for commercial purposes is not feasible in Malaysia, different methodologies were proposed and suggested further analysis of turbine designs, optimal sizing, and hybrid power systems to overcome the significant barriers of wind energy development for low potential wind areas. Most of the wind turbines in the market have cut-in speed above 3 m/s which is beyond the average annual wind speed in Malaysia. The vertical axis wind turbines were highlighted and analyzed in several studies since it can operate regardless of any sudden changes in wind directions. With the rapid growth of technological advancements in wind turbine designs, a fully customized wind turbine designs are still required for such wind profile of Malaysia. In this context, a hybridization of two or more renewable power sources seemed to provide a partial solution to this issue, many studies have investigated this option and concluded the effectiveness of these hybrid systems. [^1]: Institute of Sustainable Energy, Universiti Tenaga National, Jalan Ikram-Uniten, 43000 Kajang, Selangor, Malaysia (e-mail: fuad.noman@uniten.edu.my)

--- abstract: 'We characterize a convex subset of entanglement witnesses for two qutrits. Equivalently, we provide a characterization of the set of positive maps in the matrix algebra of $3 \times 3$ complex matrices. It turns out that boundary of this set displays elegant representation in terms of $SO(2)$ rotations. We conjecture that maps parameterized by rotations are optimal, i.e. they provide the strongest tool for detecting quantum entanglement. As a byproduct we found a new class of decomposable entanglement witnesses parameterized by improper rotations from the orthogonal group $O(2)$.' author: - | Dariusz Chruściński and Filip A. Wudarski\ Institute of Physics, Nicolaus Copernicus University,\ Grudziadzka 5/7, 87–100 Toruń, Poland title: '**Geometry of entanglement witnesses for two qutrits**' --- Introduction ============ One of the most important problems of quantum information theory [@QIT; @HHHH; @Guhne] is the characterization of mixed states of composed quantum systems. In particular it is of primary importance to test whether a given quantum state exhibits quantum correlation, i.e. whether it is separable or entangled. For low dimensional systems there exists simple necessary and sufficient condition for separability. The celebrated Peres-Horodecki criterium [@Peres; @PPT] states that a state of a bipartite system living in $\mathbb{C}^2 {{\,\otimes\,}}\mathbb{C}^2$ or $\mathbb{C}^2 {{\,\otimes\,}}\mathbb{C}^3$ is separable iff its partial transpose is positive. Unfortunately, for higher-dimensional systems there is no single universal separability condition. The most general approach to separability problem is based on the notion of an entanglement witness. Recall, that a Hermitian operator $W \in \mathcal{B}(\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B)$ is an entanglement witness [@Horodeccy-PM; @Terhal1] iff: i) it is not positively defined, i.e. $W \ngeq 0$, and ii) $\mbox{Tr}(W\sigma) \geq 0$ for all separable states $\sigma$. A bipartite state $\rho$ living in $\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$ is entangled if and only if there exists an entanglement witness $W$ detecting $\rho$, i.e. such that $\mbox{Tr}(W\rho)<0$. The separability problem may be equivalently formulated in terms of linear positive maps: a linear map $\Phi : \mathcal{B}(\mathcal{H}_A) \longrightarrow \mathcal{B}(\mathcal{H}_A)$ is positive if $\Phi(X) \geq 0 $ for all $X \geq 0$ from $\mathcal{B}(\mathcal{H}_A)$. Now, a bipartite state $\rho$ living in $\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$ is separable if and only if $({\rm id}_A {{\,\otimes\,}}\Phi)\rho$ is positive for any positive map $\Phi$ from $\mathcal{H}_B$ into $\mathcal{H}_A$. Positive maps play important role both in physics and mathematics providing generalization of $*$-homomorphism, Jordan homomorphism and conditional expectation. Normalized positive maps define an affine mapping between sets of states of $\mathbb{C}^*$-algebras. Unfortunately, in spite of the considerable effort (see e.g. [@Arveson]–[@Justyna3]), the structure of positive maps (and hence also the set of entanglement witnesses) is rather poorly understood. In the present paper we analyze an important class of positive maps in ${M_3(\mathbb{C})}$ introduced in [@Cho-Kye] (${M_n(\mathbb{C})}$ denotes a set of $n \times n$ complex matrixes). This class provides natural generalization of positive maps in ${M_3(\mathbb{C})}$ defined by Choi [@Choi]. Interestingly, the celebrated reduction map belongs to this class as well. We study the geometric structure of the corresponding convex set. It turns out that part of its boundary defines an elegant class of positive maps parameterized by proper rotations from $SO(2)$. This class was already proposed in [@Kossak1] and generalized in [@kule]. Both Choi maps and reduction map corresponds to particular $SO(2)$ rotations. Equivalently, we provide the geometric analysis of the corresponding convex set of entanglement witnesses of two qutrits. Interestingly, a convex set of positive maps displays elegant $\mathbb{Z}_2$–symmetry. We show that maps which are $\mathbb{Z}_2$–invariant are self-dual and decomposable. All remaining maps are indecomposable and hence may be used to detect bound entangled states of two qutrits. We conjecture that maps/entanglement witnesses belonging to the boundary are optimal, i.e. they provide the strongest tool to detect quantum entanglement. This conjecture is supported by the following observations: i) both Choi maps and reduction map are optimal, ii) all maps from the boundary support another conjecture [@SPA1; @SPA2] stating that so called structural physical approximation to an optimal entanglement witness defines a separable state. As a byproduct we constructed a new class of maps parameterized by improper rotations from $O(2)$. It is shown that all maps from this class are decomposable. A class of positive maps in ${M_3(\mathbb{C})}$ =============================================== Let us consider a class of positive maps in ${M_3(\mathbb{C})}$ defined as follows [@Cho-Kye] $$\label{MAPS} \Phi[a,b,c] = N_{abc} ( D[a,b,c] - {\rm id})\ ,$$ where $D[a,b,c]$ is a completely positive linear map defined by $$\label{} D[a,b,c](X) = \left( \begin{array}{ccc} (a+1) x_{11} + bx_{22} + cx_{33} & 0 & 0 \\ 0 & cx_{11} + (a+1) x_{22} + bx_{33} & 0 \\ 0 & 0 & bx_{11} + cx_{22} + (a+1) x_{33} \end{array} \right)\ ,$$ with $x_{ij}$ being the matrix elements of $X \in {M_3(\mathbb{C})}$, and ‘${\rm id}$’ is an identity map, i.e. ${\rm id}(X)=X$ for any $X \in {M_3(\mathbb{C})}$. The normalization factor $$\label{} N_{abc} = \frac{1}{a+b+c} \ ,$$ guarantees that $\Phi[a,b,c]$ is unital, i.e. $\Phi[a,b,c](\mathbb{I}_3) = \mathbb{I}_3$. Note, that $N_{abc}D[a,b,c]$ is fully characterized by the following doubly stochastic circulant matrix $$\label{DS} D= N_{abc} \left( \begin{array}{ccc} a & b & c \\ c & a & b \\ b & c & a \end{array} \right)\ .$$ This family contains well known examples of positive maps: note that $D[0,1,1](X) = {\mathrm{Tr}}X\, \mathbb{I}_3$, and hence $$\label{R} \Phi[0,1,1](X) = \frac 12 ( {\mathrm{Tr}}X\, \mathbb{I}_3 - X)\ ,$$ which reproduces the reduction map. Moreover, $\Phi[1,1,0]$ and $\Phi[1,0,1]$ reproduce Choi map and its dual, respectively [@Choi]. One proves the following result [@Cho-Kye] \[TH-korea\] A map $\Phi[a,b,c]$ is positive but not completely positive if and only if 1. $0 \leq a < 2\ $, 2. $ a+b+c \geq 2\ $, 3. if $a \leq 1\ $, then $ \ bc \geq (1-a)^2$. Moreover, being positive it is indecomposable if and only if $$\label{ind} bc < \frac{(2-a)^2}{4}\ .$$ Note, that for $a\geq 2$ the map $\Phi[a,b,c]$ is completely positive. In this paper we analyze a class $\Phi[a,b,c]$ satisfying $$\label{abc=2} a+b+c=2 \ .$$ Both reduction map (\[R\]) and Choi maps belong to this class. It is clear that maps satisfying (\[abc=2\]) belong to the boundary of the general class satisfying $a+b+c \geq 2$. Assuming (\[abc=2\]) a family of maps (\[MAPS\]) is essentially parameterized by two parameters $$\Phi[b,c] := \Phi[2-b-c,b,c]\ .$$ Let us observe that condition [*3.*]{} of Theorem \[TH-korea\] defines a part of the boundary which corresponds to the part of the following ellipse $$\label{ellipse} \frac 94 \left(x - \frac 43 \right)^2 + \frac 34\, y^2 = 1\ ,$$ where we introduced new variables $$x= b+c \ , \ \ \ y=b-c \ ,$$ Note, that condition for indecomposability (\[ind\]) simplifies to $b \neq c$. Hence, $\Phi[b,c]$ is decomposable iff $b=c$ which shows that decomposable maps lie on the line in $bc$–plane. This line intersects the ellipse (\[ellipse\]) in two points: $b=c=1$ which corresponds to the reduction map, and $b=c=1/3$. A convex set of positive maps $\Phi[b,c]$ is represented on the $bc$–plane on Figure 1. \[fig\] ![A convex set of positive maps $\Phi[b,c]$. Red line $b=c$ corresponds to decomposable maps. Special points: (i) and (ii) Choi maps, (iii) reduction map, (v) is completely positive map, (iv) decomposable map with $b=c=1/3$.](elipsa_2 "fig:") Let us observe that this set is closed under simple permutation $(b,c) \rightarrow (c,b)$. Now, recall that for any map $\Phi : {M_3(\mathbb{C})}\rightarrow {M_3(\mathbb{C})}$ one defines its dual $\Phi^\# : {M_3(\mathbb{C})}\rightarrow {M_3(\mathbb{C})}$ by $${\rm Tr}[ X \Phi(Y)] = {\rm Tr}[\Phi^\# (X) Y ]\ ,$$ for all $X,Y \in {M_3(\mathbb{C})}$. One easily finds $$\label{} \Phi^\# [b,c] = \Phi[c,b]\ ,$$ that is, dual map to $\Phi[b,c]$ corresponds to permutation of $(b,c)$. This way we proved A map $\Phi[b,c]$ is decomposable if and only if it is self-dual. The above class of positive maps gives rise to the class of entanglement witnesses $$\label{} W[a,b,c] = ({\rm id} {{\,\otimes\,}}\Phi[a,b,c]) P^+\ ,$$ where $P^+$ denotes a projector onto the maximally entangled state in $\mathbb{C}^3 {{\,\otimes\,}}\mathbb{C}^3$. One finds the following matrix representation $$\label{} W[a,b,c]\, =\, \frac{N_{abc}}{3}\, \left( \begin{array}{ccc|ccc|ccc} a & \cdot & \cdot & \cdot & -1 & \cdot & \cdot & \cdot & -1 \\ \cdot& b & \cdot & \cdot & \cdot& \cdot & \cdot & \cdot & \cdot \\ \cdot& \cdot & c & \cdot & \cdot & \cdot & \cdot & \cdot &\cdot \\ \hline \cdot & \cdot & \cdot & c & \cdot & \cdot & \cdot & \cdot & \cdot \\ -1 & \cdot & \cdot & \cdot & a & \cdot & \cdot & \cdot & -1 \\ \cdot& \cdot & \cdot & \cdot & \cdot & b & \cdot & \cdot & \cdot \\ \hline \cdot & \cdot & \cdot & \cdot& \cdot & \cdot & b & \cdot & \cdot \\ \cdot& \cdot & \cdot & \cdot & \cdot& \cdot & \cdot & c & \cdot \\ -1 & \cdot& \cdot & \cdot & -1 & \cdot& \cdot & \cdot & a \end{array} \right)\ ,$$ where to make the picture more transparent we replaced zeros by dots. Interestingly, all indecomposable witnesses $W[a,b,c]$ may be identified using the following family of PPT entangled (unnromalized) states: $$\label{eps} \rho_\epsilon = \sum_{i,j=1}^3 |ii{\rangle}{\langle}jj| + \epsilon \sum_{i=1}^3 |i,i+1{\rangle}{\langle}i,i+1| + \frac 1\epsilon \sum_{i=1}^3 |i,i+2{\rangle}{\langle}i,i+2|\ ,$$ where $\epsilon \in (0,\infty)$. It is well known that $\rho_\epsilon$ is PPT for all $\epsilon$ and entangled for $\epsilon \neq 1$. One easily finds $${\rm Tr}\, (\rho_\epsilon W[a,b,c]) = N_{abc}\, \frac 1\epsilon\, ( b\epsilon^2 + [a-2]\epsilon + c) \ ,$$ and hence $ {\rm Tr}\, (\rho_\epsilon W[a,b,c]) < 0$ might be satisfied only if the corresponding discriminant $$(a-2)^2 - 4bc > 0\ ,$$ which is equivalent to condition (\[ind\]). A subclass parameterized by the rotation group ============================================== Consider now positive maps $\Phi[b,c]$ belonging to the ellipse (\[ellipse\]), i.e. satisfying $bc=(1-b-c)^2$. We show that these maps are uniquely characterized by the rotation group $SO(2)$. Let $f_\alpha$ ($\alpha=0,1,\ldots, n^2-1$) be an orthonormal basis in ${M_n(\mathbb{C})}$ such that $f_0 = \frac{1}{\sqrt{n}}\, \mathbb{I}_n$, and $f_\alpha^*= f_\alpha$. One has $$\label{} {\rm Tr}\, ( f_k f_l) = \delta_{kl}\ , \ \ \ k,l=1,\ldots, n^2-1\ ,$$ and ${\rm Tr} f_k = 0$ for $k=1,\ldots, n^2-1$. The following formula [@Kossak1] $$\label{tw} \Phi_R(X)=\frac{1}{n}\, \mathbb{I}_n {\mathrm{Tr}}X + \frac{1}{n-1}\,\sum_{k,l=1}^{n^2-1} f_k R_{kl} {\mathrm{Tr}}( f_l X) \ ,$$ where $R_{kl}$ is an orthogonal matrix from $O(n^2-1)$, defines a family of unital positive maps in ${M_n(\mathbb{C})}$ (for a slightly more general construction see [@kule]). It is not difficult to construct an orthonormal basis $f_\alpha$. One may take for example the generalized Gell-Mann matrices defined as follows: let $|1{\rangle},\ldots,|n{\rangle}$ be an orthonormal basis in $\mathbb{C}^n$ and define $$\begin{aligned} d_l &=& \frac{1}{\sqrt{l(l+1)}}\Big( \sum_{k=1}^l |k{\rangle}{\langle}k|-l |l+1{\rangle}{\langle}l+1|\Big)\ ,\ \ \ l=1,\ldots,n-1 \\ u_{kl} &=&\frac{1}{\sqrt{2}}(|k{\rangle}{\langle}l|+|l{\rangle}{\langle}k|)\ , \\ v_{kl} &=&\frac{-i}{\sqrt{2}}(|k{\rangle}{\langle}l|-|l{\rangle}{\langle}k|)\ ,\end{aligned}$$ for $k<l$. It is easy to see that $n^2$ Hermitian matrices $(f_0,d_l,u_{kl},v_{kl})$ define a proper orthonormal basis in ${M_n(\mathbb{C})}$. Now, let us take $n=3$ and let $$\label{} R = \left(\begin{array}{c|c} T & 0 \\ \hline 0 & - \mathbb{I}_6 \end{array} \right)\ ,$$ where $T \in O(2)$. An orthogonal group $O(2)$ has two connected components. Let us consider a proper rotation $$\label{T} T(\alpha) = \left(\begin{array}{cc} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{array} \right) \ ,$$ for $\alpha \in [0,2\pi)$. It turns out that $\Phi[\alpha] := \Phi_R$ belongs to the class $\Phi[a,b,c]$. Indeed, one shows [@Kossak1] that $\alpha$-dependent coefficients $a,b,c$ are defined as follows $$\begin{aligned} \label{abc} a(\alpha) &=&\frac{2}{3}\,(1+\cos\alpha)\ ,\nonumber\\ b(\alpha) &=&\frac{2}{3}\left(1-\frac{1}{2}\cos\alpha-\frac{\sqrt{3}}{2}\sin\alpha\right) \ ,\\ c(\alpha) &=&\frac{2}{3}\left(1-\frac{1}{2}\cos\alpha+\frac{\sqrt{3}}{2}\sin\alpha\right)\ ,\nonumber\end{aligned}$$ and hence $$a(\alpha) + b(\alpha)+ c(\alpha)=2\ .$$ Now comes the crucial observation. It is easy to show that $$\label{ABC} b(\alpha)c(\alpha) = [1-a(\alpha)]^2\ ,$$ for each $ \alpha \in [0,2\pi)$. Interestingly, one has $$\label{} a(\alpha)b(\alpha) = [1-c(\alpha)]^2\ , \ \ \ \ a(\alpha)c(\alpha) = [1-b(\alpha)]^2\ ,$$ that is, there is a perfect symmetry between parameters $(a,b,c)$. Hence, all maps $\Phi[\alpha]$ parameterized by $SO(2)$ belong to the characteristic ellipse (\[ellipse\]) forming a part of the boundary of the simplex of $\Phi[b,c]$ (see Fig. 1). Note, that for $\alpha = \pm \pi/3$ one obtains two Choi maps ((i) and (ii) on Fig. 1), for $\alpha = \pi$ one obtains reduction map (point (iii) on Fig. 1) and for $\alpha=0$ one obtains decomposable map (point (iv) on Fig. 1). Let us observe that $\Phi^\#[\alpha]= \Phi[-\alpha]$, and hence $\Phi[\alpha]$ is self-dual if and only if $\alpha=0$ or $\alpha=\pi$. The map $\alpha \rightarrow - \alpha$ realizes $\mathbb{Z}_2$ symmetry of our class of maps. Self-dual maps are $\mathbb{Z}_2$–invariant. Structural physical approximation ================================= It is well known that three points from the part of the boundary formed by the ellipse (\[ellipse\]) define optimal positive maps (optimal entanglement witnesses): $(1,0)$ and $(0,1)$ corresponding to Choi maps, and $(1,1)$ corresponding to the reduction map. In terms of $\Phi[\alpha]$ they correspond to $\alpha=\frac\pi 3,\frac{5}{3\pi}$ and $\alpha=\pi$, respectively. Now, for any entanglement witness $W$ in $\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$ such that ${\rm Tr}\, W=1$, one defines its structural physical approximation (SPA) $$\label{} \mathbf{W}(p) = (1-p)W + \frac{p}{d_Ad_B} \mathbb{I}_A {{\,\otimes\,}}\mathbb{I}_B\ ,$$ with $p \geq p^*$, where $p^*$ is the smallest value of $p$ such that $\mathbf{W}(p) \geq 0$. Hence SPA of $W$ defines a legitimate quantum state $\mathbf{W}(p)$ in $\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$. It was conjectured in [@SPA1] (se also recent paper [@SPA2]) that if $W$ is an optimal entanglement witness in $\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$, then its SPA defines a separable state. This conjecture was supported by several examples of optimal entanglement witnesses (see e.g. [@Justyna1; @Justyna2; @Justyna3]). Now comes a natural question concerning optimality of other entanglement witnesses belonging to the boundary $\frac\pi 3 \leq \alpha \leq \frac{5}{3\pi}$. Let us recall a simple sufficient condition for optimality [@Lew]: if there exists a set product vectors $|\psi {{\,\otimes\,}}\phi{\rangle}\in \mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$ such that $${\langle}\psi {{\,\otimes\,}}\phi|W|\psi {{\,\otimes\,}}\phi{\rangle}=0\ ,$$ and vectors $|\psi {{\,\otimes\,}}\phi{\rangle}$ span the entire Hilbert space $\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$, then $W$ is optimal. Now, one can check that $W[0,1,1]$ corresponding to $\alpha=\pi$ admits the full set (i.e. 9) of such vectors. For the rest points the problem is much more complicated [@Gniewko] (for $W[1,1,0]$ and $W[1,0,1]$ it was already shown in [@SPA1] that there are only 7 vectors). Nevertheless, as we show all these points supports the conjecture of [@SPA1]. We propose the following For $\frac\pi 3 \leq \alpha \leq \frac{5}{3\pi}$ positive maps $\Phi[\alpha]$ are optimal. Actually, it turns out that SPA for a large class of $W[a,b,c]$ defines a separable state. Let us consider $$\label{} \mathbf{W}(p) = (1-p)W[a,b,c] + \frac{p}{9}\, \mathbb{I}_3 {{\,\otimes\,}}\mathbb{I}_3\ .$$ Now, $\mathbf{W}(p) \geq 0$ for $p\geq p^*$, where the critical value $p^*$ is given by $$\label{} p^* = \frac{3(2-a)}{2 + 3(2-a)}\ .$$ One easily finds $$\label{} \mathbf{W}(p^*) = \frac{1}{3[2 + 3(2-a)]}\, \left\{ \sum_{i=1}^3 \Big( 2 |ii{\rangle}{\langle}ii| + (2b+c) |i,i+1{\rangle}{\langle}i,i+1| + (2c+b)|i,i+2{\rangle}{\langle}i,i+2| \Big) - \sum_{i\neq j} |ii{\rangle}{\langle}jj| \right\} \ ,$$ where we have used $a+b+c=2$. Note, that $\mathbf{W}(p^*)$ may be decomposed as follows $$\label{} \mathbf{W}(p^*) = \frac{1}{3[2 + 3(2-a)]}\, \Big( \sigma_{12} + \sigma_{13} + \sigma_{23} + \sigma_d \Big) \ ,$$ where $$\label{} \sigma_{ij} = |ij{\rangle}{\langle}ij| + |ji{\rangle}{\langle}ji| + |ii{\rangle}{\langle}ii| + |jj{\rangle}{\langle}jj| - |ii{\rangle}{\langle}jj| - |jj{\rangle}{\langle}ii| \ ,$$ and the diagonal $\sigma_d$ reads as follows $$\label{} \sigma_d = \sum_{i=1}^3 \Big( (2b+c-1) |i,i+1{\rangle}{\langle}i,i+1| + (2c+b-1)|i,i+2{\rangle}{\langle}i,i+2| \Big) \ .$$ Now, $\sigma_{ij}$ are PPT and being supported on $\mathbb{C}^2 {{\,\otimes\,}}\mathbb{C}^2$ they are separable. Clearly, $\sigma_d$ is separable whenever it defines a legitimate state, that is, $2b+c \geq 1$ and $2c + b \geq 1$. It defines a region in our simplex bounded by the part of the ellipse and two lines: $$c=1-2b\ , \ \ \ b = 1- 2c\ .$$ Interestingly, these lines intersect at $b=c= \frac 13$, i.e. point (iv) on Fig. 1. Decomposable maps parameterized by improper rotations ===================================================== Consider now a second component of $O(2)$ represented by the following family of matrices $$\label{T} \widetilde{T}(\alpha) = \left(\begin{array}{cc} \cos\alpha & \sin\alpha \\ \sin\alpha & -\cos\alpha \end{array} \right) \ ,$$ for $\alpha \in [0,2\pi)$. Note, that ${\rm det}\, T(\alpha)=1$, whereas ${\rm det}\, \widetilde{T}(\alpha)=-1$. One easily shows that in this case $\Phi[\alpha]$ leads to the following map $$\label{MAPS-tilde} \widetilde{\Phi}[a,b,c] = N_{abc} ( \widetilde{D}[a,b,c] - {\rm id})\ ,$$ where $\widetilde{D}[a,b,c]$ is a completely positive linear map defined by $$\label{} \widetilde{D}[a,b,c](X) = \left( \begin{array}{ccc} (a+1) x_{11} + bx_{22} + cx_{33} & 0 & 0 \\ 0 & bx_{11} + (c+1) x_{22} + ax_{33} & 0 \\ 0 & 0 & cx_{11} + ax_{22} + (b+1) x_{33} \end{array} \right)\ ,$$ and $\alpha$-dependent coefficients $a,b,c$ are defined by $$\begin{aligned} \label{abc-new} a(\alpha)&=&\frac{2}{3} \left(1+ \frac 12 \cos\alpha+\frac{\sqrt{3}}{2}\sin\alpha \right)\ ,\nonumber \\ b(\alpha)&=&\frac{2}{3}(1-\cos\alpha)\ ,\\ c(\alpha)&=& \frac{2}{3} \left( 1+ \frac 12 \cos\alpha-\frac{\sqrt{3}}{2}\sin\alpha \right) \ . \nonumber\end{aligned}$$ Note, that $$a(\alpha)+b(\alpha) + c(\alpha)=2\ ,$$ and hence $\widetilde{D}[a,b,c]$ is fully characterized by the following doubly stochastic matrix $$\label{DS-tilde} \widetilde{D} = \frac 12 \left( \begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b \end{array} \right)\ .$$ Now, contrary to $D$ defined in (\[DS\]) it is no longer circulant. Interestingly, new parameters (\[abc-new\]) satisfy the same condition (\[ABC\]) as $a,b,c$ defined in (\[abc\]), that is one has: $$b(\alpha)c(\alpha) = [1-a(\alpha)]^2\ , \ \ \ \ a(\alpha)b(\alpha) = [1-c(\alpha)]^2\ , \ \ \ \ a(\alpha)c(\alpha) = [1-b(\alpha)]^2\ .$$ It shows that $a,b,c$ defined in (\[abc-new\]) belong to the same characteristic ellipse. It is therefore clear that points from the interior of this ellipse defines positive maps as well. This way we proved the following The linear map $\widetilde{\Phi}[a,b,c]$ defined by (\[MAPS-tilde\]) with - $ a,b,c \geq 0\ , $ - $a+b+c=2\ , $ - $ bc \geq (1-a)^2\ , $ is positive. Equivalently, we constructed a new family of entanglement witnesses $$\label{ew1} \widetilde{W}[a,b,c] = \frac 16 \left(\begin{array}{ccc|ccc|ccc} a & \cdot & \cdot & \cdot & -1 & \cdot & \cdot & \cdot & -1 \\ \cdot & b & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & c & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\\hline \cdot & \cdot & \cdot & b & \cdot & \cdot & \cdot & \cdot & \cdot \\ -1 & \cdot & \cdot & \cdot & c & \cdot & \cdot & \cdot & -1 \\ \cdot & \cdot & \cdot & \cdot & \cdot & a & \cdot & \cdot & \cdot \\\hline \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & c & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot &a & \cdot \\ -1 & \cdot & \cdot & \cdot & -1 & \cdot & \cdot & \cdot & b \end{array}\right)\ .$$ Let us observe that $$\label{} {\rm Tr}(\rho_\epsilon \widetilde{W}[a,b,c])= 0 \ ,$$ where $\rho_\epsilon$ is defined in (\[eps\]). Hence, this family of states does not detect indecomposability of $\widetilde{W}[a,b,c]$. Actually, one has the following All entanglement witnesses $\widetilde{W}[a,b,c]$ are decomposable. Proof: it is enough to prove this theorem from maps parameterized by points belonging to the ellipse $bc=(1-a)^2$, i.e. $a,b,c$ defined by (\[abc-new\]). Note, that $$\widetilde{W}[a,b,c] = \frac 16 \, (P + Q^\Gamma)\ ,$$ where $$P=\left(\begin{array}{ccc|ccc|ccc} a & \cdot & \cdot & \cdot & b-1 & \cdot & \cdot & \cdot & c-1 \\ \cdot & 0 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & 0 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\\hline \cdot & \cdot & \cdot & 0 & \cdot & \cdot & \cdot & \cdot & \cdot \\ b-1 & \cdot & \cdot & \cdot & c & \cdot & \cdot & \cdot & a-1 \\ \cdot & \cdot & \cdot & \cdot & \cdot & 0 & \cdot & \cdot & \cdot \\\hline \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 0 & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot &0 & \cdot \\ c -1 & \cdot & \cdot & \cdot & a-1 & \cdot & \cdot & \cdot & b \end{array}\right),\quad Q= \left(\begin{array}{ccc|ccc|ccc} 0 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & b & \cdot & -b & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & c & \cdot & \cdot & \cdot & -c & \cdot & \cdot \\\hline \cdot & -b & \cdot & b & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & 0 & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & a & \cdot & -a& \cdot \\\hline \cdot & \cdot & -c & \cdot & \cdot & \cdot & c & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & -a & \cdot &a & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 0 \end{array}\right) ,$$ and $Q^\Gamma$ denotes partial transposition of $Q$. It is clear that $Q \geq 0$. Now, to prove that $P \geq 0$ let us observe that the principal submatrix $$\left(\begin{array}{ccc}a & b-1 & c-1 \\b-1 & c & a-1 \\c-1 & a-1 & b\end{array}\right)\ ,$$ is positive semi-definite for $a,b,c$ defined by (\[abc-new\]). Actually, the corresponding eigenvalues read $\{2,0,0\}$, i.e. they do not depend upon $\alpha$. $\Box$ Let us observe that $W[a,b,c]= \widetilde{W}[a,b,c]$ if and only if $a=b=c=\frac 23$, that is, these two classes of entanglement witnesses have only one common element. Actually, this common point lies in the center of the ellipse, i.e. in the middle between point (iii) and (iv) on the Fig. 1. Note, that entanglement witnesses $W[a,b,c]$ and $\widetilde{W}[a,b,c]$ differ by simple permutation along the diagonal. Let us define the following unitary matrix $$\label{} U = \left( \begin{array}{ccc} 1 & . & . \\ . & . & 1 \\ . & 1 & . \end{array} \right) \ ,$$ which corresponds to permutation $(x,y,z) \rightarrow (x,z,y)$ and define $$\label{} {{W}}_U[a,b,c] := (U {{\,\otimes\,}}\mathbb{I}_3) W[a,b,c] (U {{\,\otimes\,}}\mathbb{I}_3)^\dagger\ .$$ Since $U {{\,\otimes\,}}\mathbb{I}_3$ is a local unitary operator, ${W}_U[a,b,c]$ defines an entanglement witness. One easily finds $$\label{ew2} {{W}}_U[a,b,c] = \frac 16 \left(\begin{array}{ccc|ccc|ccc} a & \cdot & \cdot & \cdot & \cdot & -1 & \cdot & -1 & \cdot \\ \cdot & b & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & c & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\\hline \cdot & \cdot & \cdot & b & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & c & \cdot & \cdot & \cdot & \cdot \\ -1 & \cdot & \cdot & \cdot & \cdot & a & \cdot & -1 & \cdot \\\hline \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & c & \cdot & \cdot \\ -1 & \cdot & \cdot & \cdot & \cdot & -1 & \cdot &a & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & b \end{array}\right)\ ,$$ which has the same diagonal as $\widetilde{W}[a,b,c]$ but the off-diagonal ‘$-1$’ are distributed according to a different pattern. We stress that ${{W}}_U[a,b,c]$ is an indecomposable entanglement witness for $b \neq c$, whereas $\widetilde{W}[a,b,c]$ is decomposable one. Conclusions =========== We analyzed a geometric structure of the convex set of positive maps in ${M_3(\mathbb{C})}$ (or equivalently a set of entanglement witness of two qutrits). Interestingly, its boundary is characterized by proper rotations form $SO(2)$. It turns out that a positive map $\Phi[b,c]$ is decomposable if and only if it is self-dual. Hence maps which are not self-dual may be used as a tool for detecting PPT entangled states. As a byproduct we constructed a convex set of decomposable entanglement witnesses. The boundary of this set is now parameterized by improper rotations form $O(2)$. It is clear that a convex combination of $W[\alpha]$ and $\widetilde{W}[\beta]$ defines an entanglement witness as well. 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--- abstract: 'The IceCube Neutrino Observatory is the world’s largest high energy neutrino telescope, using the Antarctic ice cap as a Cherenkov detector medium. DeepCore, the low energy extension to IceCube, is an infill array with a fiducial volume of around 30 MTon in the deepest, clearest ice, aiming for an energy threshold as low as 10 GeV and extending IceCube’s sensitivity to indirect dark matter searches and atmospheric neutrino oscillation physics. We will discuss the analysis of the first year of DeepCore data, as well as ideas for a further extension of the particle physics program in the ice with a future PINGU detector.' address: | Department of Physics, Pennsylvania State University, University Park, PA 16802, U.S.A.\ E-mail: deyoung@psu.edu author: - | Tyce DeYoung\ [for the IceCube collaboration]{} title: Particle Physics in Ice with IceCube DeepCore --- =1 astroparticle physics; neutrino oscillations; dark matter Introduction {#sec:intro} ============ The IceCube neutrino telescope, now fully operational at depths of 1450-2450 m below the surface of the Antarctic ice cap, was designed to detect high energy neutrinos from astrophysical accelerators of cosmic rays. Although the energy threshold of a large volume neutrino detector is not a sharp function, the original IceCube design focused on efficiency for neutrinos at TeV energies and above. Recently, the IceCube collaboration decided to augment the response of the detector at lower energies with the addition of DeepCore, a fully contained subarray aimed at improving the sensitivity of IceCube to neutrinos with energies in the range of 10’s of GeV to a few hundred GeV. This energy range is of interest for several topics related to particle physics, including measurements of neutrino oscillations and searches for neutrinos produced in the annihilation or decay of dark matter. DeepCore consists of an additional eight strings of photosensors (Digital Optical Modules, or DOMs) comprising 10” Hamamatsu photomultiplier tubes and associated data acquisition electronics housed in standard IceCube glass pressure vessels. For most of the DeepCore DOMs, the standard IceCube R7081 PMTs were replaced with 7081MOD PMTs with Hamamatsu’s new super-bialkali photocathode. These PMTs provide approximately 35% higher quantum efficiency (averaged over the detected Cherenkov spectrum) than the standard bialkali PMTs. Sited at the bottom center of the IceCube array, DeepCore benefits from the high optical quality of the ice at depths of 2100-2450 m, with an attenuation length of approximately 50 m in the blue wavelengths at which most Cherenkov photons are detected in ice. DeepCore also benefits from the ability of the standard IceCube sensors to detect atmospheric muons penetrating the ice from cosmic ray air showers above the detector, allowing substantial reduction in the background rate by vetoing events where traces of penetrating muons are seen. Each DeepCore string bears 50 DOMs in the fiducial region, with an additional 10 DOMs deployed at shallower depths to improve the vetoing efficiency for steeply vertical muons. In addition to the new DeepCore strings, the DeepCore fiducial volume for analysis includes 12 standard IceCube strings, chosen so that the fiducial region is shielded on all sides by a veto region consisting of three rows of standard IceCube strings, as shown in Fig. \[fig:layout\]. The random noise rate of IceCube DOMs is quite low (around 500 Hz, on average) due to the low temperatures and radiopurity of the ice cap. This permits DeepCore to be operated with a very low trigger threshold, demanding that 3 DOMs within the DeepCore fiducial region detect light in “local coincidence” within a period of no more than 2500 ns. The local coincidence criterion counts DOMs as being hit (i.e., having detected light) only if one of the four neighboring DOMs on a string (two above and two below) also registers a hit within $\pm1 \;\mu$s. Most of the resulting 185 Hz of triggers are due to stray light from muons which simultaneously satisfy the main IceCube trigger condition of 8 DOMs hit in local coincidence within 5 $\mu$s, but the DeepCore trigger contributes an additional (exclusive) rate of around 10 Hz. ![Schematic layout of DeepCore within IceCube. The shaded region indicates the fiducal volume of DeepCore, at the bottom center of IceCube, plus the extra veto cap of DOMs deployed at shallower depths to reinforce the veto against vertically-downgoing atmospheric muons. This schematic depicts both the DeepCore configuration used in 2010, when 79 IceCube strings were operational, and the final DeepCore layout and fiducial region used in the 2011 run. \[fig:layout\]](icecube_deepcore_pingu_koskinen_new.pdf){width="0.8\columnwidth"} The vast majority of the events which trigger DeepCore, irrespective of whether they also trigger IceCube, are due to either penetrating atmospheric muons or random coincidences of dark noise. Immediately after data acquisition, events triggering DeepCore are subjected to an online data rejection algorithm which calculates a characteristic time and location for the activity observed in the DeepCore fiducial region, as an initial estimate of the putative neutrino vertex. The estimated location is the average position of the hit DOMs, and the time is determined by subtracting the time of flight $dn/c$ of an unscattered photon emitted from that location from the observed arrival time of the first photon to hit each DOM. After outliers due to dark noise or scattered light are removed, the average inferred emission time is used as the estimated time of the underlying physics event. Based on this estimated time and location, every locally coincident hit recorded in the veto region prior to the vertex time is examined to determine whether it lies on the light cone connecting it with the estimated event vertex. The distributions of the inferred speed required to connect hits in the veto region to the DeepCore vertex, for both simulated atmospheric muons and simulated neutrinos, is shown in Fig. \[fig:veto\]; positive speeds indicate hits occuring in the veto region prior to the DeepCore vertex time. If any hits are found with inferred speeds between +0.25 and +0.4 m/ns, the event is rejected as being most likely due to an atmospheric muon. This algorithm reduces the event rate by more than two orders of magnitude, to 18 Hz, while retaining over 99% of simulated triggered events due to neutrinos interacting within the fiducial volume. Additional background rejection criteria are applied offline, depending on the goals of each physics analysis making use of these data. ![Distribution of probabilities of observing hits leading to a given inferred particle speed, for simulated atmospheric muons (dashed line) and atmospheric neutrinos (solid). Positive speeds indicate activity in the veto region prior to that in the DeepCore volume, and a peak around $c =$ 0.3 m/ns is visible for penetrating muons. The integral of each distribution corresponds to the mean number of hits observed in the veto region for the given class of events. \[fig:veto\]](ParticleSpeedProbabilities_v22.pdf){width="\columnwidth"} The effective volume of the DeepCore detector for detection low energy muon neutrinos, accounting for this online data filter, is shown in Fig. \[fig:nuMuVolume\]. It should be stressed that this effective volume curve does *not* include losses due to later background rejection or event quality criteria. The contribution of DeepCore to low energy analysis is evident in the fact that despite its relatively small geometric volume, around 3% that of IceCube, the overall sample of neutrino events below 100 GeV consists primarily of those detected by DeepCore. This energy range is of considerable interest for several topics in particle physics, including searches for dark matter and measurements of neutrino oscillations. While DeepCore does not have a sharp energy threshold, it retains around 7 megatons of effective volume at energies as low as 10 GeV. Further details regarding DeepCore’s instrumentation and performance are available in Ref. [@Collaboration:2011ym]. ![Effective volume of DeepCore for muon neutrinos at trigger level (solid) and after application of the online veto algorithm described in the text (dot-dashed line). The effective volume of IceCube as originally proposed is shown for comparison. \[fig:nuMuVolume\]](effectiveVolume_IC86_numu_GENIE_effVolumes_logScale_prelim.pdf){width="\columnwidth"} Observation of Neutrino-Induced Cascades {#sec:cascades} ======================================== Using the first year of data recorded with DeepCore, from May 2010 to April 2011, we have observed cascades induced by atmospheric neutrinos interacting in the DeepCore volume. These cascades include charged current (CC) interactions of electron neutrinos, as well as neutral current (NC) interactions of neutrinos of all flavors. (The background rejection criteria used in this analysis result in an energy threshold of around 40 GeV, so only a negligible contribution from atmospheric muon neutrinos oscillating to tau is expected.) Previous searches for neutrino-induced cascades in AMANDA and IceCube [@Ahrens:2002wz; @Ackermann:2004zw; @Achterberg:2007qy; @Abbasi:2011zz; @Abbasi:2011ui] have focused on higher energies, to avoid the background of bremsstrahlung produced by atmospheric muons. In this analysis, we instead rely on the active veto provided by IceCube to reduce the background of penetrating muons, and exploit the high flux of atmospheric neutrinos at energies of a few hundred GeV to observe a set of 1,029 cascade-like neutrino events in 281 days of the 2010 data run. One such event is shown in Fig. \[fig:cascadeEvent\]. ![Candidate neutrino-induced cascade observed in DeepCore in the 2010 data run. Each black dot indicates a DOM. Colored dots represent DOMs that detected light during the event, with the size of the dot proportional to the amount of light detected. The color indicates the relative arrival time of the first photon detected by that DOM, running through the spectrum from red (earliest) to purple (latest). \[fig:cascadeEvent\]](Event_06_2.pdf){width="\columnwidth"} For this data set, recorded with the incomplete 79-string configuration of IceCube, the smaller DeepCore fiducial volume shown in Fig. \[fig:layout\] was used. This initial configuration consisted of only the central seven standard strings, plus 6 additional DeepCore strings. Based on Monte Carlo simulations, we estimate that approximately 60% of the 1,029 events in the final sample are truly neutrino-induced cascades, while around 40% are in fact $\nu_\mu$ CC events with muon tracks too short to be distinguished in the current analysis; efforts to further reduce this background are underway. The level of background due to atmospheric muons is still under investigation but appears to be small. The rates of observed neutrinos are consistent with simulations of atmospheric neutrinos using the leading atmospheric neutrino flux models from the Bartol and Honda groups, although we are still in the process of assessing our systematic uncertainties. It should be noted that the predictions based on the two atmospheric flux models differ for this event set by approximately 10%, due mainly to the modeling of production of higher energy electron neutrinos by kaons. Work is in progress to lower the energy threshold of the analysis, which would permit observation of neutrino oscillations using the atmospheric neutrino flux. For baselines comparable to the Earth’s diameter, the first maximum of the $\nu_\mu \rightarrow \nu_\tau$ oscillation probability occurs at approximately 25 GeV, well within the energy range accessible to DeepCore [@Mena:2008rh]. Searches for Dark Matter ======================== In addition to studies of atmospheric neutrinos, DeepCore’s reduced energy threshold facilitates indirect searches for evidence of dark matter using IceCube. Searches are underway for neutrinos produced in the annihilation or decay or dark matter captured in the gravitational potential wells of the Earth, Sun [@Abbasi:2009uz; @Abbasi:2009vg], and Galaxy [@Abbasi:2011eq]. Because the WIMP mass must be relatively low compared to the energy range of IceCube, additional sensitivity to lower energy neutrinos substantially extends IceCube’s reach, especially for the lower part of the allowed WIMP mass range or for models where the neutrino spectrum produced is relatively soft. ![Limits on the spin-dependent WIMP-nucleon scattering cross section from various direct and indirect search experiments, and the projected sensitivity of IceCube with DeepCore for a “hard” neutrino spectrum arising from neutralino annihilation in the Sun. The shaded region indicates the possible cross sections in supersymmetric models not already ruled out by direct detection experiments’ limits on the spin-independent cross section. \[fig:WIMPs\]](WIMP_limits.pdf){width="\columnwidth"} The potential of IceCube including DeepCore for detecting evidence of dark matter annihilation in the Sun is shown in Fig. \[fig:WIMPs\]. The shaded region indicates the allowed MSSM parameter space, for models where the WIMP is a neutralino. Direct detection experiments have already probed substantial parts of the allowed supersymmetric parameter space, primarily in regions where there is a substantial spin-independent neutralino-nucleon scattering cross section, so that coherent scattering from heavy nuclei in the detector target enhances the cross section considerably. For models in which the scattering cross section is primarily spin-dependent, indirect searches exploiting the Sun’s mass as a scattering target have an advantage, although the results depend on the branching ratios for neutralino-neutralino annihilation channels. For WIMP masses below roughly 100 GeV, DeepCore provides the bulk of the sensitivity to the neutrinos arising from Solar neutralino annihilation. Future Prospects: PINGU {#sec:PINGU} ======================= Encouraged by the initial success of DeepCore, the IceCube collaboration and other participants are developing a proposal for a Phased IceCube Next Generation Upgrade (PINGU), an extension of IceCube and DeepCore which would further increase the density of instrumentation in the central volume and further reduce the energy threshold. The proposal would augment DeepCore with perhaps 18 to 20 additional strings, of which the majority would be similar to those in DeepCore. Several strings might also include specialized prototypes of novel sensors, perhaps similar to those incorporating a number of 3” PMTs rather than a single 10” PMT, now being developed for the proposed KM3NeT detector. One layout of the additional strings under discussion is shown in Fig. \[fig:PINGUlayout\]. ![Top view of one PINGU configuration now under study, including 16 strings of DeepCore-like instrumentation. Additional strings of prototype next-generation instrumentation are envisioned but not shown. This layout would significantly improve the effectiveness of DeepCore at energies below a few 10’s of GeV. \[fig:PINGUlayout\]](PINGU_2test_geometry_nolabels.pdf){width="\columnwidth"} Such an extension would considerably increase the effective volume of DeepCore at energies below about 30 GeV, with the potential to detect neutrinos as low as a few GeV. The effective volume of the detector for events contained within the geometrical volume is shown in Fig. \[fig:PINGUnuEVolume\], as compared with that of the existing DeepCore detector. Improvements of nearly an order of magnitude can be seen for low energy neutrinos. These effective volumes do not include efficiency losses due to event reconstruction and analysis criteria, which will reduce the effective volume achievable in final physics analysis. ![Preliminary estimate of the effective volume of PINGU for electron neutrino events at trigger level, as compared to that of the completed DeepCore configuration. PINGU would retain considerable effective volume down to energies as low as a few GeV. Analysis and reconstruction efficiencies are not included. The geometry used for this estimate is similar to that shown in Figure \[fig:PINGUlayout\] but with a slightly larger mean spacing between strings, so the effective volume at the lowest energies may be underestimated. \[fig:PINGUnuEVolume\]](effectiveVolume_PINGU_nue_Compare_prelim.pdf){width="\columnwidth"} Summary ======= The effectiveness of IceCube at energies below 100 GeV has been significantly enhanced by the addition of DeepCore, which extends IceCube’s reach to energies of 10’s of GeV. This range is of interest for observations of neutrino oscillations, as well as searches for dark matter. As a first step toward these studies, we have observed a significant sample of atmospheric neutrino-induced cascades, enabled by the ability of the IceCube detector to identify and veto atmospheric muons penetrating to the DeepCore volume. We are also investigating the potential for a further reduction in the energy threshold of IceCube with an additional extension known as PINGU, which could extend IceCube’s reach to energies as low as a few GeV References {#references .unnumbered} ========== [99]{} R. Abbasi *et al.* \[The IceCube Collaboration\], arXiv:1109.6096 \[astro-ph.IM\]. J. Ahrens [*et al.*]{} \[The AMANDA Collaboration\], Phys. Rev.  D [**67**]{}, 012003 (2003). \[arXiv:astro-ph/0206487\]. M. Ackermann [*et al.*]{} \[The AMANDA Collaboration\], Astropart. Phys.  [**22**]{}, 127 (2004). \[arXiv:astro-ph/0405218\]. 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--- abstract: 'We revisit the momentum independent many-body t-matrix approach for boson systems developed by Shi and Griffin [@sg] and Bijlsma and Stoof [@bs]. Despite its popularity, simplicity, and expected advantage of being its applicability to both normal and superfluid phases, we find that the theory breaks down in the normal phase of bosons. We conjecture that this failure is due to neglecting of momentum dependence on the t-matrix.' author: - 'Theja N. De Silva' title: 'Evidence for the breakdown of momentum independent many-body t-matrix approximation in the normal phase of Bosons' --- The impressive experimental realization of weakly interacting Bose-Einstein condensation (BEC) in dilute alkali gases [@becE1; @becE2; @becE3] has renewed the theoretical and experimental interest in studies of Bose systems. The BEC of alkali gases were identified by the existence of a sharp peak in the momentum distribution below a certain critical temperature. Unlike the intensively studied strongly interacting liquid $^4$He system, the momentum distribution of cold atomic systems is easily accessible due to the inhomogeneity of the trapped atoms. While, the focus of the first generation of cold gas experiments has been on BEC of weakly interacting bosons, the center of research has now shifted towards the study of both normal and superfluid phases in strongly interacting bosons. The main advances in this direction have been achieved by increasing the interaction through controlling the two-body scattering length ($a$) or exposing the atoms to optical lattices [@sl1]. Impressive theoretical efforts are currently devoted to study the effect of interactions on Bose-Einstein condensation [@th1; @th2; @th3]. Until recently, there was no consensus on the sign of the shift of critical temperature in the presence of strong interactions. Now it is believed that the shift can be described as $\Delta T_c = 1.3 a n^{1/3} T_c^0$, where $n$ is the density and $T_c^0$ is the critical temperature of an ideal Bose gas [@ct1; @ct2]. Further, it has been shown that none of the mean field theories correctly predict the expected second-order BEC-normal phase transition [@lovr]. These mean field theories include, Hartree-Fock (HF) [@th3; @hf], Popov [@pop], Yukalov-Yukalova (YY) [@yy], and t-matrix approximations [@sg; @bs]. Recently, a self-consistent mean field theory was proposed as the *sole* mean-field theory that explains the correct second-order transition [@tmn]. All theories mentioned above have been applied to study either the properties of the BEC phase or the phase transition. Little attention has been paid to the normal phase. One of the most popular and intensively applied theories in literature was proposed by Bijlsma and Stoof [@bs] and Shi and Griffin [@sg]. This t-matrix approximation (TM) was developed based on Popov theory. The authors in references [@sg] and [@bs] however go beyond the contact interaction and include the many-body effects by taking into account higher order scattering. Then neglecting the momentum dependence on the t-matrix, these authors have derived a simple analytical expression for density in both normal and superfluid phases. The expected advantage of TM theory is that it is applicable for both superfluid BEC and normal phases. In this opinion article, we show that this t-matrix approximation based theory breaks down in the normal phase and the expected advantage and simplicity of the theory is no longer valid. ***TM theory:*** First we summarize the TM approach for the normal phase of bosons, following the reference [@sg]. The density of the normal bosons is $n = g_{3/2}[\beta \Delta]/\lambda^3$, where $\lambda = [2 \pi \hbar^2/(mk_BT)]^{1/2}$ is the thermal deBroglie wavelength and $g_{n}(x)$ is the well known Bose integral. The parameter $\Delta$ is related to the density $n$, chemical potential $\mu$, many-body t-matrix $T = U/(1+ \alpha U$), and an additional parameter $\alpha$ through the expression $\mu = \Delta+ (2nU)/(1+\alpha U)$, where $$\begin{aligned} \alpha = \sum_k\biggr(\frac{1}{2E_k} coth (\beta E_k/2)-\frac{1}{2\epsilon_k}\biggr).\end{aligned}$$ Here $U = 4\pi \hbar^2a/m$ is the free-space scattering amplitude, $\hbar$ is the plank constant, $k_B$ is the Boltzmann constant, $\beta = 1/(k_BT)$ is the inverse temperature, and $m$ is the mass of a boson. The quasi particle energy is given by $E_k = \epsilon_k-\Delta$ with $\epsilon_k = \hbar^2 k^2/2m$. At the normal-BEC transition, $\Delta \rightarrow 0$, while $\alpha \rightarrow \infty$ yielding a chemical potential $\mu_c =0$ as in the case of an ideal Bose gas. In order to demonstrate the problematic behavior of the theory, we solve these set of equations for the density profile of a harmonically trapped Bose gas. Using local density approximation, replacing $\mu \rightarrow \mu_0 - m\omega^2r^2/2$, the density profile of the trapped gas is shown in FIG.\[SG\] as a function of spatial coordinate $r$. Here $\mu_0$ is the chemical potential at the center of the trap and $\omega$ is the trapping frequency. The density profile is shown at the onset of the BEC. In other words, the trap center $r =0$ is set to be at the BEC where we set central chemical potential to be $\mu_0 = \mu_c$. ***HF theory:*** As a comparison, we show HF density profile for the same parameters in gray color in FIG.\[SG\]. HF approach is a self-consistent approach that simplifies the N-particle interacting states into effectively one-particle non-interacting states whose energy spectrum depends self-consistently on both density and the interaction. The HF normal density is given by $n = g_{3/2}[\beta (\mu-2nU)]/\lambda^3$. At the BEC transition, the chemical potential is $\mu_c = 2Ug_{3/2}[0]/\lambda^3$. It is worth pointing out that each of the mean field theories, HF, Popov, and YY are equivalent in the normal phase but they differ in the BEC phase. Notice that the critical density at the transition is the same as that of the TM approach and the non-interacting Bose gases, but the critical chemical potential is different. The non-interacting density profile at the onset of BEC is also shown as a dashed line. ![The scaled density profiles of a harmonically trapped Bose gas at the onset of BEC. While the center of the trap ($\check{r} =0$) is set to be at BEC transition, the entire tail of the density profile is in normal state. The interaction strength and temperature are fixed to be $U =\hbar \omega$ and $k_B T =\hbar \omega$. The dashed-line represents a non-interacting Bose gas, the black and gray lines represent t-matrix and Hartree-Fock theories respectively. The density $\tilde{n} = n l^3$ and the radial coordinate $\check{r} = r/l$ are scaled with the oscillator length $l = \sqrt{\hbar/m\omega}$.[]{data-label="SG"}](SGvsHF.eps){width="\columnwidth"} The homogenous density in the normal phase is expected to be a monotonic function of chemical potential. Likewise, the density profile in a trap is expected to be a monotonically decreasing function of the spatial coordinate $r$. However, as opposed to the HF theory, the TM approach gives non-monotonic density variation. The density at the vicinity of BEC (density of the gas at the trap center is at criticality) however, agrees within the two approaches. The results are shown for representative values of interaction and temperature. The qualitative behavior is very similar for all finite interactions and temperatures. Even though this TM approach is widely used in literature, this problematic behavior has not been previously reported. This may be due to the fact that most studies focus on the BEC phase (where the density is monotonic as expected), but not on the normal phase. Though we use the local density approximation for a harmonically trapped boson system to demonstrate this ill behavior, the same behavior exists as a function of chemical potential. In other words, the density is not a monotonically increasing function of chemical potential. Therefore, it is not the local density approximation, but the approximation made to the t-matrix approach that must be responsible for this problematic behavior. In conclusion, we have revisited the normal state bosons using a momentum independent t-matrix approach and discovered that the expected validity of the theory breaks down in the normal phase. Although this theory has been applied in numerous previous studies, this ill behavior has not been previously reported as to the best of our knowledge. We anticipate that the theory can be recovered by the inclusion of momentum dependence on the t-matrix. This will be a non trivial task due to the infrared divergences and including these will destroy the simplicity of the theory. ***Acknowledgements:*** We are grateful to Joseph Newton for critical comments on the manuscript. H. Shi and A. Griffin, Physics Reports, **304**, 1, (1998). M. Bijlsma and H. T. C. Stoof, Phys. Reev. A **55**, 498, (1997). M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, E. A. Cornell, Science **269**, 198, (1995). K. B. Davis, M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. **75**, 3969, (1995). C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett. **75**, 1687 (1995). For example, see N. Navon, S. Piatecki, K. Gunter, B. Rem, T. C. Nguyen, F. Chevy, W. Krauth, and C. Salomon, Phys. Rev. Lett. **107**, 135301 (2011) and S. B. Papp, J. M. Pino, R. J. Wild, S. Ronen, C. E. Wieman, D. S. Jin, and E. A. Cornell, Phys. Rev. Lett. **101**, 135301 (2008). T. D. Lee and C. N. Yang, Phys. Rev. **105**, 1119 (1957). Kerson Huang and C. N. Yang, Phys. Rev. **105**, 767 (1957). For example, see the book *Bose-Einstein Condensation* by A. Griffin, D. W. Snoke, and S. Stringari, Cambridge University press. V. A. Kashurnikov, N. V. Prokof’ev, and B. V. Svistunov, Phys. Rev. Lett. **87**, 120402 (2001). Peter Arnold and Guy Moore, Phys. Rev. Lett. **87**, 120401 (2001). L. Olivares-Quiroz and V. Romero-Rochin, J. Phys. B: At. Mol. Opt. Phys. **43**, 205302 (2010). For example, see the book *Bose-Einstein Condensation in Dilute Gases*, by C. J. Pethic and H. Smith, Cambridge University press. N. N. Bogoliubov, J. Phys. **11**, 23, (1947). V. I. Yukalov and E. P. Yukalova, Phys. Rev. A `76`, 013602, (2007); V. I. Yukalov and E. P. Yukalova, Phys. Rev. A **74**, 063623, (2006). V.I. Yukalov, E.P. Yukalova, J. Phys. B **47**, 095302, (2014).

--- abstract: 'Suppose that $\F_p$ is coloured with $r$ colours. Then there is some colour class containing at least $c_r p^2$ quadruples of the form $(x, y , x + y, xy)$.' author: - 'Ben Green[^1]' - Tom Sanders title: Monochromatic sums and products --- Introduction and notation ========================= The following beautiful question was asked on numerous occasions by Hindman (see, for example, [@hindmanleaderstrauss Question 3]) and is very well-known. \[q1\] Suppose that the natural numbers $\N$ are finitely coloured. Do there exist $x$ and $y$ such that $x, y, x+y$ and $xy$ all have the same colour? It follows from Schur’s theorem [@schur Hilfssatz] that the answer is affirmative if either $x+y$ or $xy$ is omitted from the list. (In the latter case it is an observation of Graham [@hindman p1] that we can consider the colouring of $\N$ induced on $\{2^n: n \in \N\}$.) It is also known that the answer to Question \[q1\] is affirmative for 2-colourings; in fact, every 2-colouring of $\{1,2,\dots, 252\}$ contains a monochromatic quadruple of the stated type, as established by Graham [@hindman Theorem 4.3]. In general, however, Question \[q1\] is quite open, as indeed is the following considerably weaker statement. (See the remarks following [@hindmanleaderstrauss Question 3].) \[q2\] Suppose that the natural numbers $\N$ are finitely coloured. Do there exist $x$ and $y$, not both $2$, such that $x+y$ and $xy$ all have the same colour? In this paper we are concerned with so-called finite field analogues of the above questions where we replace $\N$ by the finite fields $\F_p$. Schur’s theorem was originally designed for application to $\F_p^*$ so it should be of little surprise that its finite field analogue is routine. Shkredov seems to have been the first to address the finite field analogue of Question \[q2\] in [@shkredov Theorem 1.2] (later generalised by Cilleruelo to any finite field [@cilleruelo Corollary 4.2]) and in fact he proves rather more, namely the following [@shkredov Theorem 1.2]. \[shkredov\] Suppose that $A\subset \F_p$ has size $\alpha p$. Then there are at least $c_\alpha p^2$ triples $(x,x+y,xy)$ in $A^3$, for some $c_{\alpha} > 0$ which does not depend on $p$. (Technically, Shkredov only states that there is at least one such triple, not ${\geqslant}c_{\alpha} p^2$, but it is obvious that his proof gives this stronger statement.) It follows trivially that if $\F_p$ is $r$-coloured and $p$ is sufficiently large in terms of $r$ then there are non-zero elements $x$ and $y$ such that $x,x+y$ and $xy$ all have the same colour. A corresponding result for colourings over $\Q$ (or any countable field) was obtained by Bergelson and Moreira [@bergelson-moreira Theorem 1.2] using ergodic-theoretic techniques. In this paper we solve the finite field analogue of Question \[q1\] by establishing the following. \[mainthm\] Suppose that $\F_p$ is $r$-coloured. Then there are at least $c_r p^2$ monochromatic quadruples $(x,y,x+y,xy)$, where $c_r > 0$ does not depend on $p$. Note that while Theorem \[shkredov\] is a *density result*, there is no such version of Theorem \[mainthm\], as can be seen by considering the set $\{ x \in \F_p : \frac{p}{3} < x < \frac{2p}{3}\}$. This has density roughly $\frac{1}{3}$ but does not even contain a set of the form $\{x, y, x+y\}$. Before diving into the remains of the paper it may be useful to say that we discuss the outline of the main argument in §\[sec4\], after setting up some notation in §\[sec3\]. §\[sec2\] also includes some notation but more importantly develops the tools for counting the configurations we are interested in. After that the remaining sections fill out the details of §\[sec4\]. *Notation.* We use fairly standard asymptotic notation such as $O()$, $o()$, $\gg$ and $\ll$. We write $o_{M; p \rightarrow \infty}(1)$ to mean a quantity tending to $0$ as $p \rightarrow \infty$, but in a manner that may depend on the parameter $M$. Similarly, for example, $O_{\eps}(1)$ denotes a constant which may depend on some parameter $\eps$. Throughout the paper $\F$ will denote the finite field with $p$ elements (we do not explicitly indicate the prime $p$). Occasionally we shall write $\mu_{\F}$ for the uniform probability measure on $\F$. As is standard in additive combinatorics we write $\E_{x \in X} = \frac{1}{|X|} \sum_{x \in X}$ for averages over some finite set $X$. Counting quadruples {#sec2} =================== Given functions $f_1,f_2,f_3,f_4:\F \rightarrow \C$ we write $$T(f_1,f_2,f_3,f_4):=\E_{x,y \in \F}{f_1(x)f_2(y)f_3(x+y)f_4(xy)},$$ so that if $A \subset \F$ then $p^2T(1_A,1_A,1_A,1_A)$ is the number of quadruples $(x,y,x+y,xy) \in A^4$. *The additive Fourier transform and counting sums.* The quantity $p^2T(1_A,1_A,1_A,1_\F)$ is the number of triples $(x,y,x+y)\in A^3$ and this is well understood through the *additive* Fourier transform. We write $\widehat{\F}$ for the dual group of the additive group of $\F$, and given $f:\F \rightarrow \C$ and $\gamma \in \widehat{\F}$ define the (additive) Fourier transform of $f$ by $$\widehat{f}(\gamma):=\E_{x \in \F}{f(x)\overline{\gamma(x)}}.$$ Writing $e_p(x):=\exp(2\pi i x/p)$, we know that the elements of $\widehat{\F}$ are just the maps of the form $x \mapsto e_p(rx)$ as $r$ ranges over $\F$ and so we shall frequently identify $\widehat{\F}$ with $\F$ and write $\widehat{f}(r)$ for $\widehat{f}(\gamma)$ where $\gamma(x)=e_p(rx)$ for all $x \in \F$. As usual the key tools are the inversion formula $$\label{eqn.ainv} f(x) =\sum_{r \in \F}{\widehat{f}(r)e_p(rx)} \text{ for all } x \in \F,$$ and Parseval’s theorem $$\label{eqn.apars} \|f\|_{2}^2=\E_{x \in \F}{|f(x)|^2} = \sum_{r \in \F}{|\widehat{f}(r)|^2}.$$ The convolution of two functions $f,g:\F \rightarrow \C$ is defined to be $$f \ast g(y):=\E_x{f(x)g(y - x)} \text{ for all } y \in \F.$$ It is well-known and easy to prove that $$\widehat{f \ast g}(r) = \widehat{f}(r)\widehat{g}(r) \text{ for all }r \in \F.$$ Finally, define the $u^+_2$-norm of a function $f:\F \rightarrow \C$ by $$\|f\|_{u_2^+}:=\sup\{|\langle f,\gamma\rangle| : \gamma \in \widehat{\F}\}=\sup\{|\E_x{f(x)\overline{e_p(rx)}}|:r \in \F\}.$$ Usually, the $u_2^+$-norm is simply denoted $u_2$; we include the plus sign because we shall shortly need the multiplicative analogue. It is easy to see that $\|\cdot\|_{u_2^+}$ is a norm and that it is dominated by $\|\cdot \|_{1}$. This norm is important because of the following lemma. \[prop.add\] Suppose that $f_1,f_2,f_3:\F \rightarrow \C$ are such that $\|f_1\|_{2},\|f_2\|_{2},\|f_3\|_{2}{\leqslant}1$. Then $$|T(f_1,f_2,f_3,1_{\F})| {\leqslant}\inf_i\|f_i\|_{u_2^+}.$$ This is very standard. By the inversion formula we have $$T(f_1, f_2, f_3, 1_{\F}) = \sum_r \hat{f}_3(r) \hat{f}_1(-r) \hat{f}_2(-r).$$ The stated inequality now follows from the Cauchy-Schwarz inequality and Parseval’s theorem . For example, $$\begin{aligned} |T&(f_1, f_2, f_3, 1_{\F})| {\leqslant}\sup_r |\hat{f}_3(r)| \left(\sum_r |\hat{f}_1(-r)| |\hat{f}_2(-r)|\right) = \Vert f_3 \Vert_{u_2^+} \sum_r |\hat{f}_1(-r)| |\hat{f}_2(-r)| \\ & {\leqslant}\Vert f_3 \Vert_{u_2^+} \big( \sum_r |\hat{f}_1(-r)|^2 \big)^{1/2} \big( \sum_r |\hat{f}_2(-r)|^2 \big)^{1/2} = \Vert f_3 \Vert_{u_2^+} \Vert f_1 \Vert_2 \Vert f_2 \Vert_2 {\leqslant}\Vert f_3 \Vert_{u_2^+},\end{aligned}$$ with almost identical proofs being available to bound the left hand side by $\|f_1\|_{u_2^+}$ and $\|f_2\|_{u_2^+}$. *Multiplicative characters and counting products.* The quantity $p^2 T(1_A,1_A,1_\F,1_A)$ counts the number of triples $(x,y,xy) \in A^3$ and this is well understood through the *multiplicative* Fourier transform. If $\chi : \F^* \rightarrow \C$ is a character, we extend $\chi$ to all of $\F$ by setting $\chi(0) = 1$. We write $\widehat{\F^*}$ for the set of all such extended characters and then define the $u_2^\times$-semi-norm of a function $f:\F \rightarrow \C$ to be $$\|f\|_{u_2^\times}:=\sup \{ |\langle f, \chi \rangle|: \chi \in \widehat{\F^*}\}$$ where $\langle f, \chi \rangle := \E_{x \in \F}{f(x)\overline{\chi(x)}}$. Note that this is only a semi-norm because functions $f$ with $f(0)=-f(1)$ and $f(x)=0$ elsewhere have $\|f\|_{u_2^\times}=0$. The analogue of Proposition \[prop.add\] is then the following. \[prop.mult\] Suppose that $g_1,g_2,g_4:\F \rightarrow \C$ are functions such that - $\Vert g_1 \Vert_{\infty}, \Vert g_2 \Vert_{\infty}, \Vert g_4 \Vert_{\infty} {\leqslant}K$ for some $K {\geqslant}1$ and - $\Vert g_1 \Vert_2, \Vert g_2 \Vert_2, \Vert g_4 \Vert_2 {\leqslant}1$. Then $$|T(g_1,g_2,1_{\F},g_4)| {\leqslant}\inf_i{\|g_i\|_{u_2^\times}} + \frac{4K^3}{p}.$$ Introduce the auxiliary quantity $$\tilde T(g_1, g_2,g_4) := \E_{x,y \in \F^*} g_1(x) g_2(y) g_4(xy).$$ By exactly the same analysis as in the proof of Proposition \[prop.add\], but using the (multiplicative) Fourier transform on $\F^*$ instead of the additive one, we obtain $$\label{mult-an} |\tilde T(g_1, g_2, g_4) | {\leqslant}\frac{p}{p-1}\sup_{\chi} |\E_{x \in \F^*} g_i(x) \overline{\chi(x)}|$$ for each $i \in\{ 1,2,4\}$. The extra factor of $p/(p-1)$ comes from the fact that $$\|g_j\|_{L_2(\mu_{\F^*})}^2{\leqslant}\frac{p}{p-1}\|g_j\|_{L_2(\mu_\F)}^2 \text{ for all } j \in \{1,2,4\}.$$ Now we have $$\begin{aligned} T(g_1, g_2, 1_{\F}, g_4) & = -\frac{1}{p^2} g_1(0) g_2(0) g_4(0) + g_1(0) g_4(0) \frac{1}{p^2}\sum_y g_2(y)\\ & \qquad + g_2(0) g_4(0) \frac{1}{p^2}\sum_x g_1(x)+ \left(\frac{p-1}{p}\right)^2 \tilde T(g_1, g_2, g_4).\end{aligned}$$ The first three terms may be estimated somewhat trivially using the bound $\Vert g_i \Vert_{\infty} {\leqslant}K$; in magnitude they total at most $\frac{3K^3}{p}$. From this and we obtain $$|T(g_1, g_2, 1_{\F}, g_4)| {\leqslant}\frac{3K^3}{p} + \frac{p-1}{p} \sup_{\chi}|\E_{x \in \F^*} g_i(x) \overline{\chi}(x)|,$$ for each $i = 1,2,4$. Noting that $$\E_{x \in \F^*} g_i(x) \overline{\chi(x)} = \frac{p}{p-1}\langle g_i,\chi\rangle - \frac{1}{p-1} g_i(0),$$ the result follows easily. *Counting sums and products.*\[subsec.csp\] In the light of Proposition \[prop.add\] and Proposition \[prop.mult\] one might hope that $|T(f_1,f_2,f_3,f_4)|$ is controlled by the combination of $\|f_i\|_{u_2^+}$ and $\|f_i\|_{u_2^\times}$. Unfortunately this is not the case, as the following example shows. Given $\gamma \in \widehat{\F}$ and $\chi \in \widehat{\F^*}$ let $$f_1(t):=\gamma(t^2)\chi(t), f_2(t):=\gamma(t^2)\chi(t),$$ and $$f_3(t):=\gamma(-t^2), f_4(t):=\gamma(2t)\overline{\chi(t)}.$$ Then $$\begin{aligned} T(f_1,f_2,f_3,f_4) & = &\E_{x,y \in \F}{\gamma(x^2 + y^2 - (x+y)^2+2xy)\chi(x)\chi(y)\overline{\chi(xy)}}\\ & = & \frac{(p-1)^2+1}{p^2} = 1+o_{p\rightarrow \infty}(1).\end{aligned}$$ On the other hand if $\gamma$ and $\chi$ are non-trivial then it may be checked using character sums estimates of the type discussed at the beginning of §\[sec8\] that $\Vert f_i \Vert_{u_2^+}, \Vert f_i \Vert_{u_2^{\times}} \ll p^{-1/2}$. (As this is only a motivating example, the exact details of the proof of this need not concern us.) Write $Q(\F)$ for the set of quadratic phases on $\F$, that is to say $$Q(\F):=\{x \mapsto e_p(rx^2+sx): r,s \in \F\}.$$ In the above example, these quadratic phases mixed with multiplicative characters. This suggests the following definition, which is a key definition in our paper. \[def.qmnorm\] Suppose that $f : \F \rightarrow \C$ is a function. Then we define $$\|f\|_{{\operatorname{QM}}}:=\sup \{ |\langle f, \phi\chi \rangle |: \phi \in Q(\F), \chi \in \widehat{\F^*}\},$$ which is easily seen to be a norm. The reason this is such an important definition for us is the following fact, which is the main result of the section. It says that in a sense these quadratic-multiplicative examples are the *only* ones affecting the count $T(f_1, f_2,f_3,f_4)$. \[gvn-nonlinear\] Suppose that $f_1,f_2,f_3,f_4:\F \rightarrow \C$ are such that $\|f_1\|_\infty,\|f_2\|_\infty,\|f_3\|_\infty,\|f_4\|_\infty {\leqslant}1$. Then $$|T(f_1,f_2,f_3,f_4)| = O(\inf_i\max (p^{-1/64},\|f_i\|_{{\operatorname{QM}}}^{1/5})).$$ We shall begin the proof of this shortly, but first we must introduce one final norm. *The $u^+_3$-norm.* Higher-order variants of the $u_2^+$-norm examining correlation with quadratic phases have been widely-studied since the ground-breaking paper [@gowers-4ap] of Gowers. We shall only need a fragment of his ideas here. We define the $u^+_3$-norm by $$\|f\|_{u^+_3} :=\sup\{|\langle f,\phi \rangle| : \phi \in Q(\F)\}=\sup\{|\E_{x \in \F}{f(x)\overline{\phi(x)}}|:\phi \in Q(\F)\}.$$ We have the chain of inequalities $$\label{chain} \Vert \cdot \Vert_{u_2^+} {\leqslant}\Vert \cdot \Vert_{u_3^+} {\leqslant}\Vert \cdot \Vert_{{\operatorname{QM}}} {\leqslant}\Vert \cdot \Vert_1.$$ A key ingredient in the proof of Proposition \[gvn-nonlinear\] is the following result which, in view of , is in fact stronger than that result when $i = 3$. \[gvn-3-l2\] Suppose that $f_1,f_2,f_3,f_4 : \F \rightarrow \C$ are such that $\Vert f_1 \Vert_{\infty}, \Vert f_2 \Vert_{\infty}, \Vert f_4 \Vert_{\infty} {\leqslant}1$, and such that $f_3$ satisfies the slightly weaker bounds $\Vert f_3 \Vert_2 {\leqslant}1$ and $\Vert f_3 \Vert_{\infty} {\leqslant}p^{1/16}$. Then $$|T(f_1,f_2, f_3, f_4) |^8 {\leqslant}\|f_3\|_{u^+_3}^2+ O(p^{-1/2}).$$ *Remark.* Of course, the conclusion is valid under the stronger assumption that $\Vert f_3 \Vert_{\infty} {\leqslant}1$, but it will be helpful to allow weaker bounds when applying Lemma \[decomposition-quadratic\] below. The proof follows the ideas of [@gowers-4ap], although the additional multiplicative structure makes the argument considerably simpler. In particular we make no use of the Balog-Szemerédi-Gowers theorem or any results in the direction of Freiman’s theorem. Following Gowers we introduce some notation. For any $f:\F \rightarrow \C$, we write $$\Delta_hf(x):=f(x+h)\overline{f(x)} \text{ for all }x,h \in \F.$$ The operator acts as a difference operator in the exponent so that if $\phi \in Q(\F)$ then $\Delta_h \phi$ is (a constant times) a linear character, and that character itself depends linearly on $h$. As in [@gowers-4ap], we use a fairly straightforward converse of this fact. \[lem.tim\] Suppose that $f:\F \rightarrow \C$ has $\|f\|_{2}{\leqslant}1$. Then $$\sup_{h \in \F^*, r \in \F} \E_{z \in \F} |\widehat{\Delta_{zh}f}(zr)|^2 {\leqslant}\Vert f \Vert_{u_3^+}^2.$$ Let $h \in \F^*$ and $r \in \F$ be arbitrary. Since $h \neq 0$ we can write $g(x):=f(x)e_p(-rx^2/2h)$ and note that $\Vert g \Vert_2 = \Vert f \Vert_2 {\leqslant}1$ and that $\Vert g \Vert_{u_3^+} = \Vert f \Vert_{u_3^+}$. Applying the basic facts of additive Fourier analysis we have $$\begin{aligned} \E_{z \in \F} |\widehat{\Delta_{zh}f}(zr)|^2 & = \E_{z}{\E_{x,y}{f(x)\overline{f(x+zh)}e_p(zrx) \overline{f(y)}f(y+zh)e_p(-zry)}}\\ & = \E_{z}{\E_{x,y}{g(x)\overline{g(x+zh)}\overline{g(y)}g(y+zh)}}\\ & = \E_{z'}{\E_{x,y}{g(x)\overline{g(x+z')}\overline{g(y)}g(y+z')}} \;\; \mbox{(since $h \neq 0$)}\\ & = \sum_s{|\widehat{g}(s)|^4} {\leqslant}\|g\|_{u_2^+}^2\|g\|_{2}^2 {\leqslant}\|g\|_{u_2^+}^2 {\leqslant}\| g \|^2_{u_3^+} = \| f \|^2_{u_3^+}.\end{aligned}$$ The result follows. If $z \in \F^*$ then, as $(x,y)$ ranges over $\F \times \F$, so does $(xz^{-1}, zy)$. It follows that $$T(f_1, f_2, f_3, f_4) = \E_{x,y \in \F} f_1(xz^{-1})f_2(zy)f_3(xz^{-1}+zy)f_4(xy).$$ Averaging over all $z$, we have $$T(f_1,f_2,f_3,f_4) = \E_{x,y \in \F} f_4(xy) \E_{z \in \F^*} f_1(xz^{-1})f_2(zy)f_3(xz^{-1}+zy).$$ By Cauchy-Schwarz and the inequality $\|f_4\|_{\infty}{\leqslant}1$ it follows that $$\begin{aligned} \nonumber |T(f_1, f_2, f_3, f_4)|^2 & {\leqslant}\E_{x,y \in \F} |\E_{z \in \F^*} f_1(xz^{-1})f_2(zy)f_3(xz^{-1}+zy) |^2 \\ \nonumber & = \E_{z_0, z_1 \in \F^*} \E_{x \in \F} f_1(xz_0^{-1})\overline{f_1(xz_1^{-1})}\\ & \qquad \qquad \qquad \times \E_{y \in \F} f_2(z_0y)\overline{f_2(z_1y)} f_3(xz_0^{-1}+z_0y)\overline{f_3(xz_1^{-1}+z_1y)} .\label{1st-cauchy}\end{aligned}$$ For fixed $z_0, z_1$, we apply Cauchy-Schwarz to the inner average over $x$. We obtain $$\begin{aligned} \big| & \E_{x \in \F} f_1(xz_0^{-1})\overline{f_1(xz_1^{-1})} \E_{y \in \F} f_2(z_0y)\overline{f_2(z_1y)}f_3(xz_0^{-1}+z_0y)\overline{f_3(xz_1^{-1}+z_1y)} \big|^2 \\ & {\leqslant}\E_{x \in \F} |f_1 (x z_0^{-1}) f_1(x z_1^{-1})|^2 \cdot \E_{x \in \F} \big| \E_{y \in \F} f_2(z_0y)\overline{f_2(z_1y)}f_3(xz_0^{-1}+z_0y)\overline{f_3(xz_1^{-1}+z_1y)} \big|^2.\end{aligned}$$ Since $\Vert f_1 \Vert_{\infty} {\leqslant}1$ (in fact $\Vert f_1 \Vert_4 {\leqslant}1$ would be enough) this is bounded above by $$\begin{aligned} \E_{x \in \F} & \big| \E_{y \in \F} f_2(z_0y)\overline{f_2(z_1y)}f_3(xz_0^{-1}+z_0y)\overline{f_3(xz_1^{-1}+z_1y)} \big|^2 \\ & = \E_{x,y_0,y_1 \in \F} f_2(z_0y_0)\overline{f_2(z_1y_0)}\overline{f_2(z_0y_1)}f_2(z_1y_1) \times \\ & \times f_3(xz_0^{-1}+z_0y_0)\overline{f_3(xz_1^{-1}+z_1y_0)} \overline{f_3(xz_0^{-1}+z_0y_1)}f_3(xz_1^{-1}+z_1y_1).\end{aligned}$$ Substituting back in to , we see that $$\begin{aligned} & |T(f_1, f_2, f_3, f_4)|^4 {\leqslant}\E_{z_0, z_1 \in \F^*, x, y_0, y_1 \in \F} f_2(z_0y_0)\overline{f_2(z_1y_0)}\overline{f_2(z_0y_1)}f_2(z_1y_1) \times \\ & \qquad \times f_3(xz_0^{-1}+z_0y_0) \overline{f_3(xz_1^{-1}+z_1y_0)} \overline{f_3(xz_0^{-1}+z_0y_1)}f_3(xz_1^{-1}+z_1y_1).\end{aligned}$$ Write $y = y_0$ and $h = y_1 - y_0$. The pair $(y,h)$ ranges uniformly over $\F \times \F$ as $(y_0, y_1)$ does. Evidently $$f_2(z_0y_0)\overline{f_2(z_1y_0)} \overline{f_2(z_0y_1)}f_2(z_1y_1) = f_2(z_0y)\overline{f_2(z_1y)}\overline{f_2(z_0(y + h))}f_2(z_1(y+h));$$ moreover $$f_3(xz_0^{-1}+z_0y_0)\overline{f_3(xz_1^{-1}+z_1y_0)} \overline{f_3(xz_0^{-1}+z_0y_1)}f_3(xz_1^{-1}+z_1y_1) =$$ $$\overline{\Delta_{z_0h}f_3(xz_0^{-1}+z_0y)}\Delta_{z_1h}f_3(xz_1^{-1}+z_1y) .$$ Thus $$|T(f_1, f_2, f_3, f_4)|^4 {\leqslant}\E_{z_0, z_1 \in \F^*, y, h \in \F} f_2(z_0y)\overline{f_2(z_1y)}\overline{f_2(z_0(y + h))}f_2(z_1(y+h)) \times$$ $$\label{to-cauchy-2} \qquad \qquad \times \E_{x \in \F}\overline{\Delta_{z_0h}f_3(xz_0^{-1}+z_0y)}\Delta_{z_1h}f_3(xz_1^{-1}+z_1y) .$$ If $z \in \F^*$, we write $m_z : L^{\infty}(\F) \rightarrow L^{\infty}(\F)$ for the operator defined by $(m_z g)(x) := g(zx)$ for all $x \in \F$. Then $$\overline{\Delta_{z_0h}f_3(xz_0^{-1}+z_0y)}\Delta_{z_1h}f_3(xz_1^{-1}+z_1y) = \overline{m_{z_0^{-1}}\Delta_{z_0 h} f _3(x + z_0^2 y)}m_{z_1^{-1}} \Delta_{z_1 h} f_3 (x + z_1^2 y).$$ Therefore $$\E_{x \in \F}\overline{\Delta_{z_0h}f_3(xz_0^{-1}+z_0y)}\Delta_{z_1h}f_3(xz_1^{-1}+z_1y) = (F_{z_0, h} \ast G_{z_1, h} )((z_0^2 - z_1^2) y),$$ where $$F_{z_0, h}(t) := \overline{m_{z_0^{-1}}\Delta_{z_0 h}f_3(t)}$$ and $$G_{z_1, h}(t) := m_{z_1^{-1}}\Delta_{z_1 h} f_3(-t).$$ Therefore by we have $$|T(f_1, f_2, f_3, f_4)|^4 {\leqslant}\E_{z_0, z_1 \in \F^*, y, h \in \F} f_2(z_0y)\overline{f_2(z_1y)} \overline{f_2(z_0(y + h))}f_2(z_1(y+h)) (F_{z_0, h} \ast G_{z_1, h} )((z_0^2 - z_1^2) y).$$ By Cauchy-Schwarz once more (and the fact that $\Vert f_2 \Vert_{\infty} {\leqslant}1$) we have $$|T(f_1, f_2, f_3, f_4)|^8 {\leqslant}\E_{h \in \F} \E_{z_0, z_1 \in \F^*} \E_{y \in \F}|(F_{z_0, h} \ast G_{z_1, h} )((z_0^2 - z_1^2) y)|^2.$$ We have arranged the three averages in this way to make the next step clearer. If $z_0^2 \neq z_1^2$ then the inner average over $y$ is precisely $\Vert F_{z_0, h} \ast G_{z_1, h} \Vert_2^2$. If $z_0^2 = z_1^2$ then the inner average is of course simply $|(F_{z_0, h} \ast G_{z_1, h} )(0)|^2$. There are at most $2(p-1)$ pairs satisfying the second condition, so we have $$\begin{aligned} \nonumber |T(f_1, f_2, f_3, f_4)|^8 & {\leqslant}\E_{h \in \F}\E_{z_0, z_1 \in \F^*} \Vert F_{z_0, h} \ast G_{z_1, h} \Vert_2^2 + \\ \nonumber & \qquad\qquad + \frac{2}{p-1} \sup_{z_0, z_1}|(F_{z_0, h} \ast G_{z_1, h} )(0)|^2 \\ \nonumber & {\leqslant}\E_{h \in \F, z_0, z_1 \in \F^*} \Vert F_{z_0, h} \ast G_{z_1, h} \Vert_2^2 + O(p^{-1/2}) \\ \label{use-9} & = \E_{h \in \F, z_0, z_1 \in \F^*} \sum_{r \in \F}| \widehat{F}_{z_0,h}(r)|^2 |\widehat{G}_{z_1,h}(r)|^2 + O(p^{-1/2}).\end{aligned}$$ where in the second line we used the inequality $\Vert f_3 \Vert_{\infty} {\leqslant}p^{1/16}$, and in the third (the additive) Parseval’s identity and the fact that convolution goes to multiplication. Now observe that for any $g \in L^{\infty}(\F)$ and $z \in \F^*$ we have $$(m_{z^{-1}} g)^{\wedge}(r) = \E_{x \in \F} g(z^{-1} x) e_p(-rx) = \E_{x \in \F} g(x) e_p(-rzx) = \hat{g}(zr).$$ Thus $$\widehat{F}_{z_0, h}(r) = \overline{\widehat{\Delta_{z_0 h} f_3}(-z_0 r)} \text{ and } \widehat{G}_{z_1, h}(r) = \widehat{\Delta_{z_1 h} f_3}(-z_1 r).$$ It follows from that $$\begin{aligned} |T(f_1, f_2, f_3, f_4)|^8 & {\leqslant}\E_{h \in \F, z_0, z_1 \in \F^*} \sum_{r \in \F} |\widehat{\Delta_{z_0h}f_3}(z_0r)|^2| \widehat{\Delta_{z_1h}f_3}(z_1r)|^2+O(p^{-1/2}).\\ & = \E_{h \in \F} \sum_{r \in \F} \bigg( \E_{z \in \F^*} | \widehat{\Delta_{zh} f_3}( zr) |^2 \bigg)^2 + O(p^{-1/2}) .\end{aligned}$$ Using Parseval’s identity, we see that the contribution to this average from $h = 0$ is $$\begin{aligned} \frac{1}{p} \sum_{r \in \F} \bigg( \E_{z \in \F^*} |\widehat{\Delta_0 f_3}(zr) |^2 \bigg)^2 & = \frac{1}{p}\sum_{r \in \F} \bigg( \E_{z \in \F^*} \big| \widehat{|f_3|^2}(zr) \big| \bigg)^2 \\ & {\leqslant}\frac{4}{p^3} \sum_{r \in \F} \bigg( \sum_{z \in \F} \big| \widehat{|f_3|^2}(zr) \big|^2 \bigg)^2 \\ & {\leqslant}\frac{4}{p} \big| \widehat{ |f_3|^2 }(0) \big|^4 + \frac{4}{p^2} \bigg( \sum_{s \in \F} \big| \widehat{|f_3|^2}(s) \big|^2 \bigg)^2 \\ & = \frac{4}{p} \big| \widehat{ |f_3|^2 }(0) \big|^4 + \frac{4}{p^2} \bigg( \E_{x \in \F} \big ||f_3|^2(x) \big|^2\bigg)^2 \\ & {\leqslant}\frac{4\Vert f_3 \Vert_{\infty}^8}{p} + \frac{4\Vert f_3 \Vert_{\infty}^8}{p^2} = O(p^{-1/2}).\end{aligned}$$ Thus by Parseval’s identity (since $z$ ranges over $\F^*$) and interchanging the order of summation we have $$\begin{aligned} |T(& f_1, f_2, f_3, f_4)|^8 \\ & {\leqslant}\E_{h \in \F} 1_{\F^*}(h) \sum_{r \in \F} \bigg( \E_{z \in \F^*} | \widehat{\Delta_{zh} f_3}( zr) |^2 \bigg)^2 + O(p^{-1/2}) \\ & {\leqslant}\bigg(\sup_{h \in \F^*,r \in \F } \E_{z \in \F^*} | \widehat{\Delta_{zh} f_3}( zr) |^2\bigg) \cdot \E_{h \in \F} \sum_{r \in \F} \E_{z \in \F^*} | \widehat{\Delta_{zh} f_3}( zr) |^2 + O(p^{-1/2}) \\ & = \bigg(\sup_{h\in \F^*,r \in \F } \E_{z \in \F^*} | \widehat{\Delta_{zh} f_3}( zr) |^2\bigg) \cdot \E_{z \in \F^*,h \in \F} \Vert \Delta_{zh} f_3 \Vert_2^2 + O(p^{-1/2}).\end{aligned}$$ However, for $z \in \F^*$ we have $$\E_{h \in \F} \Vert \Delta_{zh} f_3 \Vert_2^2 = \E_h \E_x |f_3(x+zh) \overline{f_3(x)}|^2 = \E_x|f_3(x)|^2\E_h{|f_3(x+zh)|^2} = \|f_3\|_2^4 {\leqslant}1,$$ and hence $$\label{to-tim} |T(f_1, f_2, f_3, f_4)|^8 {\leqslant}\sup_{h \in \F^*,r \in \F } \E_{z \in \F^*} | \widehat{\Delta_{zh} f_3}( zr) |^2 + O(p^{-1/2}).$$ We are now almost ready to apply Lemma \[lem.tim\] to (at last) bound this in terms of the $u_3^+$-norm of $f_3$. To do this, we need only extend the $z$-average to include $z = 0$. We have $$\E_{z \in \F^*} |\widehat{\Delta_{zh}f_3}(zr)|^2 {\leqslant}\frac{p}{p-1}\E_{z \in \F} |\widehat{\Delta_{zh} f_3}(zr)|^2 {\leqslant}\E_{z \in \F} |\widehat{\Delta_{zh}(f_3)}(zr)|^2 + \frac{2\Vert f_3 \Vert_{\infty}^4}{p}.$$ Now we use $\|f_3\|_\infty {\leqslant}p^{1/16}$, and Lemma \[lem.tim\] to obtain $$|T(f_1, f_2, f_3, f_4)|^8 {\leqslant}\sup_{h \in \F^*,r \in \F} \E_{z \in \F} | \widehat{\Delta_{zh} f_3}( zr) |^2 + O(p^{-1/2}) {\leqslant}\Vert f_3 \Vert_{u_3^+}^2 + O(p^{-1/2}),$$ which is precisely the stated result. Now we deduce Proposition \[gvn-nonlinear\] itself. A key ingredient in this process is the following decomposition result, reminiscent of results of “Koopman von Neumann” type. There are closely-related results in [@gowers-decomp; @gowers-wolf1; @gowers-wolf2; @gowers-wolf3; @trevisan-et-al]. However, in our particular setting it is not too hard to establish what we need quite directly. \[decomposition-quadratic\] Suppose that $f : \F \rightarrow \C$ has $\|f\|_{2}{\leqslant}1$ and that $\eps {\geqslant}4p^{-1/8}$ is a parameter. Then there are complex numbers $\lambda_{\phi}$, $\phi \in Q(\F)$, such that $$\big\| f - \sum_{\phi \in Q(\F)}{\lambda_{\phi} \phi} \big\|_{u^+_3} {\leqslant}\eps, \;\; \big\|\sum_{\phi \in Q(\F)}{\lambda_{\phi} \phi}\big\|_{2}^2{\leqslant}3 \;\; \mbox{and} \;\sum_{\phi \in Q(\F)} |\lambda_{\phi}| {\leqslant}\frac{4}{\eps}.$$ Define $$\lambda_{\phi}:=\begin{cases} \langle f,{\phi}\rangle & \text{ if } |\langle f,{\phi}\rangle| {\geqslant}\eps/2\\ 0 & \text{ otherwise}.\end{cases}$$ Write $I$ for the set of $\phi$ such that $\lambda_{\phi} \neq 0$, and let $I' \subset I$. By Cauchy-Schwarz we have $|\lambda_{\phi}|{\leqslant}1$ for all $\phi$. Using this and the Gauss sum estimate $$|\langle \phi, \phi' \rangle | {\leqslant}\frac{1}{\sqrt{p}} \text{ whenever } \phi \neq \phi',$$we have $$\label{eq771} \big \|\sum_{\phi \in I'}{\lambda_\phi\phi}\big\|_{2}^2 {\leqslant}\sum_{\phi \in I'}{|\lambda_\phi|^2} + \sum_{\phi\neq \phi'; \phi,\phi' \in I'}{|\langle \phi,{\phi'}\rangle|} {\leqslant}\sum_{\phi \in I'}{|\lambda_\phi|^2} + \frac{|I'|^2}{\sqrt{p}}.$$ On the other hand from the definition of the $\lambda_\phi$s and the Cauchy-Schwarz inequality we have $$\sum_{\phi \in I'} |\lambda_{\phi}|^2 = \big\langle f, \sum_{\phi \in I'} \lambda_{\phi} \phi \big\rangle {\leqslant}\Vert f \Vert_2 \big\Vert \sum_{\phi \in I'} \lambda_{\phi} \phi \big\Vert_2 .$$ Comparing this with (and using $\Vert f \Vert_2 {\leqslant}1$) gives $$\big(\sum_{\phi \in I'}{|\lambda_\phi|^2}\big)^2 {\leqslant}\sum_{\phi \in I'}{|\lambda_\phi|^2} + \frac{|I'|^2}{\sqrt{p}};$$ This implies that $$\label{eq772} \sum_{\phi \in I'}{|\lambda_\phi|^2} {\leqslant}1+ \frac{|I'|^2}{\sqrt{p}} .$$ This is true for all $I' \subset I$. Since $|\lambda_{\phi}| {\geqslant}\eps/2$ for all $\phi \in I$, it follows that $$\frac{\eps^2 m}{4} {\leqslant}1 + \frac{m^2}{\sqrt{p}} \text{ whenever } 0 {\leqslant}m {\leqslant}|I| \text{ and } m \in\N.$$ With the assumption that $\eps {\geqslant}4 p^{-1/8}$ if follows that if $|I| {\geqslant}8/\eps^2$ then we can take $m \in \N$ with $8/\eps^2{\leqslant}m < 8/\eps^2+1$ contradicting the above inequality. Therefore $$\label{ibound} |I| {\leqslant}\frac{8}{\eps^2} {\leqslant}p^{1/4}.$$ Taking $I' = I$ in and , then using , we have $$\label{eq773} \big \|\sum_{\phi \in I}{\lambda_\phi\phi}\big\|_{2}^2 {\leqslant}1 + 2\frac{|I|^2}{\sqrt{p}} {\leqslant}3.$$ Taking $I' = I$ in , then using , we have $$\label{eq774} \sum_{\phi \in Q(\F)} |\lambda_{\phi}| {\leqslant}\frac{2}{\eps} \sum_{\phi \in I} |\lambda_{\phi}|^2 {\leqslant}\frac{4}{\eps}.$$ It follows from this and the Gauss sum estimate that if $\phi' \in I$ then $$\label{eq776} \big|\langle f - \sum_\phi \lambda_\phi \phi , {\phi'}\rangle\big| = \big|\sum_{\phi \neq \phi'}{\lambda_\phi\langle \phi,\phi'\rangle}\big| {\leqslant}\frac{4}{\eps\sqrt{p}} {\leqslant}\eps,$$ whilst if $\phi' \in Q(\F) \setminus I$ then $$\label{eq777} \big|\langle f - \sum_\phi \lambda_\phi \phi , {\phi'}\rangle\big| = \big|\langle f,\phi'\rangle - \sum_{\phi}{\lambda_\phi\langle \phi,\phi'\rangle}\big| {\leqslant}\frac{\eps}{2} + \frac{4}{\eps\sqrt{p}} {\leqslant}\eps.$$ Taken together, , , and cover all the statements we claimed, and the proof is complete. We are finally ready for the proof of Proposition \[gvn-nonlinear\], the main result of this section. Set $\delta := |T(f_1,f_2,f_3,f_4)|$. If $\delta < 4 p^{-1/64}$ then the result is trivial, so suppose this is not the case. If $\inf_{i \in \{ 1,2,3,4\}} \Vert f_i \Vert_{{\operatorname{QM}}} = \Vert f_3 \Vert_{{\operatorname{QM}}}$ then the result follows immediately from Proposition \[gvn-3-l2\]. It therefore suffices to show that under the assumptions just stated we have $$\label{to-show-final} \inf_{i \in\{ 1,2,4\}} \Vert f_i \Vert_{{\operatorname{QM}}} \gg \delta^5.$$ Apply Lemma \[decomposition-quadratic\] with $f = f_3$ and with a parameter $\eps := 2^{-13}\delta^4$. This gives us coefficients $(\lambda_\phi)_{\phi \in Q(\F)}$ such that $$\big\| f_3 - \sum_{\phi \in Q(\F)}{\lambda_{\phi} \phi} \big\|_{u^+_3} {\leqslant}\eps, \;\; \big\|\sum_{\phi}{\lambda_{\phi} \phi}\big\|_{2}^2{\leqslant}3 \; \; \mbox{and} \; \sum_{\phi \in Q(\F)} |\lambda_{\phi}| {\leqslant}\frac{4}{\eps}.$$ Set $$g_3:=f_3 - \sum_{\phi \in Q(\F)}{\lambda_{\phi} \phi};$$ thus $\|g_3\|_{2} {\leqslant}1+\sqrt{3} < 4$, $\|g_3\|_{\infty} {\leqslant}1+ \frac{4}{\eps} {\leqslant}\frac{8}{\eps}$ and $\|g_3\|_{u^+_3}{\leqslant}\eps$. By Proposition \[gvn-3-l2\] we have, since $\eps {\geqslant}8p^{-1/16}$, $$|T(f_1, f_2, g_3, f_4)|^8 {\leqslant}2^{16}\big(\eps^2 + O(p^{-1/2})\big) {\leqslant}2^{17}\eps^2,$$ provided $p$ is sufficiently large absolutely which we may certainly assume. Thus $$\big | T(f_1, f_2, f_3, f_4) - \sum_{\phi} \lambda_{\phi} T(f_1, f_2, \phi, f_4) \big|^8 {\leqslant}2^{17} \eps^2 < \left(\frac{\delta}{2}\right)^8.$$ It follows that $$\big| \sum_{\phi} \lambda_{\phi} T(f_1, f_2,\phi, f_4) \big| {\geqslant}\frac{\delta}{2},$$ and hence that there is some $\phi \in Q(\F)$ for which $$|T(f_1, f_2,\phi, f_4)| {\geqslant}\frac{\delta\eps}{8} \gg \delta^5.$$ Suppose that $\phi(t)=e_p(at^2+bt)$, and write $g_1(t):=f_1(t)e_p(at^2+bt)$, $g_2(t):=f_2(t)e_p(at^2+bt)$ and $g_4(t):=f_4(t)e_p(2at)$. Then $$T(g_1, g_2, 1_{\F}, g_4) = T(f_1, f_2,\phi, f_4),$$ and hence $$|T(g_1, g_2, 1_{\F}, g_4)| \gg \delta^5.$$ By Proposition \[prop.mult\] it follows that $$\inf_{i \in\{ 1,2,4\}} \Vert g_i \Vert_{u_2^{\times}} + O\left(\frac{1}{p}\right) \gg \delta^5,$$ and so $$\inf_{i \in\{ 1,2,4\}} \Vert g_i \Vert_{{\operatorname{QM}}} \gg \delta^5.$$ Since $\Vert f_i \Vert_{{\operatorname{QM}}} =\Vert g_i \Vert_{{\operatorname{QM}}}$, we have $$\inf_{i \in\{ 1,2,4\}} \Vert f_i \Vert_{{\operatorname{QM}}} \gg \delta^5.$$ This is precisely , so the proof is concluded. To conclude this section we state Proposition \[gvn-nonlinear\] in a qualitatively equivalent form, more useful for our later applications. \[gvn-cor\] There is a monotonically increasing function $\nu:(0,1] \rightarrow (0,1]$ with the following property. Suppose $h_1,h_2,h_3,h_4 : \F \rightarrow \C$ are such that $\|h_1\|_\infty,\|h_2\|_\infty,\|h_3\|_\infty,\|h_4\|_\infty {\leqslant}1$ and $T(h_1,h_2,h_3,h_4) {\geqslant}\delta$. Then either $p {\leqslant}1/\nu(\delta)$ or $\Vert h_i \Vert_{{\operatorname{QM}}} {\geqslant}\nu(\delta)$ for all $i \in \{ 1,2,3,4\}$. In fact, $\nu(\delta)$ can be chosen to have the shape $\nu(\delta) \sim c \delta^C$. [${\operatorname{QM}}$]{}-systems and related concepts {#sec3} ====================================================== We begin by defining the notion of a ${\operatorname{QM}}$-system, an important concept in our paper. As is fairly standard, we write $$S^1:=\{z \in \C: |z| = 1\}.$$ We shall also require a non-standard piece of notation: define $$\G := \R/\Z \times \R/\Z \times S^1.$$ The group $\G$ is of course abelian, but it is convenient to use juxtaposition for the group operation; thus $(\theta_1, \theta_2, z) (\theta'_1, \theta'_2, z') = (\theta_1 + \theta'_1 , \theta_2 + \theta'_2, z z')$. We shall often be considering the product group $\G^d$, and we use the same convention there. Let $d \in \N$. A ${\operatorname{QM}}$-system of dimension $d$ is a map $\Psi : \F \rightarrow \G^d$ of the form $$\Psi( x)= (a_i x^2/p, 2a_i x/p, \psi_i(x))_{i = 1}^d,$$ where $a_i \in \F$ and $\psi_i \in \widehat{\F^*}$. Recall from §\[sec2\] that we extend $\psi_i$ to all of $\F$ by setting $\psi_i(0) = 1$. Given a ${\operatorname{QM}}$-system $\Psi$ it will be helpful to write $\Psi' \supset \Psi$ to mean that $\Psi'$ extends $\Psi$ in the sense that $\Psi'$ has dimension $d' {\geqslant}d$ and $\Psi = \Pi \circ \Psi'$ where $\Pi:\G^{d'} \rightarrow \G^d$ is the projection onto the first $d$ co-ordinates. An important fact for us is that the “orbit” $\{ \Psi(x) : x \in \F\}$ is highly equidistributed inside a certain closed subgroup of $\G^d$. To explain what this group is, we need to make some definitions. Let $\Psi$ be a ${\operatorname{QM}}$-system as above. Then we associate to $\Psi$ the sublattices of $\Z^d$ $$\Lambda_{\Psi}^+ := \{ \xi \in \Z^d : \xi_1 a_1 + \dots + \xi_d a_d \equiv 0 \pmod{p}\}$$ and $$\Lambda_{\Psi}^{\times} := \{ \xi \in \Z^d: \psi_1^{\xi_1} \cdots \psi_d^{\xi_d} \; \mbox{is the trivial character}\}.$$ Note, incidentally, that $p \Z^d \subset \Lambda_{\Psi}^+$ and $(p-1)\Z^d \subset \Lambda_{\Psi}^{\times}$, and so both $\Lambda_{\Psi}^+$ and $\Lambda_{\Psi}^{\times}$ have full rank. With these lattices defined we introduce the closed subgroups $$G_{\Psi}^+ := \{ g \in (\R/\Z)^d : \xi \cdot g = 0 \; \mbox{for all $\xi \in \Lambda_{\Psi}^{+}$}\}$$ and $$G_{\Psi}^{\times} := \{ z \in (S^1)^d : z^{\xi} = 1 \; \mbox{for all $\xi \in \Lambda_{\Psi}^{\times}$}\},$$ where here $\xi \cdot g := \xi_1 g_1 + \dots + \xi_d g_d$ and $z^{\xi} := z_1^{\xi_1} \cdots z_d^{\xi_d}$. These two groups $G_{\Psi}^+$ and $G_{\Psi}^{\times}$ are both closed subgroups of compact groups ($(\R/\Z)^d$ and $(S^1)^d$ respectively) and as such they carry natural Haar probability measures $\mu_{G_{\Psi}^+}$ and $\mu_{G_{\Psi}^{\times}}$. Define $$H_{\Psi} := G_{\Psi}^+ \times G_{\Psi}^+ \times G_{\Psi}^{\times},$$ and put the natural probability measure $$\mu_{H_{\Psi}} := \mu_{G_{\Psi}^+} \times \mu_{G_{\Psi}^+} \times \mu_{G_{\Psi}^{\times}}$$ on this group. By abuse of notation, we regard this as a probability measure on $\G^d$ as well (which is permissible, as $H_{\Psi}$ is a subgroup of $\G^d$). Note that $\Psi(\F) \subset H_{\Psi}$. It turns out that $\Psi(\F)$ is close to being equidistributed in $H_{\Psi}$. To formulate this fact in a convenient form (Proposition \[distribution\] below), we need a further definition. \[deftrig\] Let $F : \G^d \rightarrow \C$ be a function. We write $\Vert F \Vert_{{\operatorname{trig}}}$ for the smallest $M \in [0,\infty]$ such that $F$ has a Fourier expansion $$F(\theta_1, \theta_2, z) = \sum_{\|\xi_1\|_1, \|\xi_2\|_1, \|\xi_3\|_1 {\leqslant}M} \hat{F}(\xi_1, \xi_2,\xi_3) e(\xi_1 \cdot \theta_1 + \xi_2 \cdot \theta_2) z^{\xi_3}$$ with $\sum_{\xi_1,\xi_2,\xi_3} |\hat{F}(\xi_1, \xi_2,\xi_3)| {\leqslant}M$. We call this a norm, although “measure of complexity” would be more accurate. We have $$\label{subadd} \Vert F_1 + F_2 \Vert_{{\operatorname{trig}}} {\leqslant}\Vert F_1 \Vert_{{\operatorname{trig}}} + \Vert F_2 \Vert_{{\operatorname{trig}}},$$ $$\Vert F_1F_2 \Vert_{{\operatorname{trig}}} {\leqslant}\max\left\{\Vert F_1 \Vert_{{\operatorname{trig}}} + \Vert F_2 \Vert_{{\operatorname{trig}}},\|F_1\|_{{\operatorname{trig}}}\|F_2\|_{{\operatorname{trig}}}\right\},$$ and $$\label{scale}\Vert \lambda F \Vert_{{\operatorname{trig}}} {\leqslant}\max(1, |\lambda|) \Vert F \Vert_{{\operatorname{trig}}} \text{ for all } \lambda \in \C,$$ as well as the shift property that if we define $T_hF(g) := F(h^{-1}g)$ then $$\Vert T_h F \Vert_{{\operatorname{trig}}} = \Vert F \Vert_{{\operatorname{trig}}} \text{ for all }h \in \G^d.$$ We call those functions $F : \G^d \rightarrow \C$ with finite trig norm *trigonometric polynomials*. They are dense in $C(\G^d)$ (with the sup norm). This follows from the Stone-Weierstrass theorem: the trigonometric polynomials form an algebra and contain the characters, hence separate points. (This could also be established directly using harmonic analysis.) We turn now to the promised equidistribution statement. We call it the “baby counting lemma” as it is a simpler cousin of one of the key ingredients of our work which we shall call the counting lemma. \[distribution\] Let $\Psi$ be a ${\operatorname{QM}}$-system of dimension $d$, and let $F : \G^d \rightarrow \C$ be a trigonometric polynomial. Then we have $$\E_{x \in \F} F(\Psi(x)) = \int F d\mu_{H_{\Psi}} + o_{p \rightarrow \infty}(\Vert F \Vert_{{\operatorname{trig}}}).$$ The proof of Proposition \[distribution\] relies on estimates for character sums twisted by quadratic phases. We recall what we need on this topic at the beginning of §\[sec8\], and give the proof of Proposition \[distribution\] later in that same section. It is useful to have some “absolute values” associated to the groups we are interested in. For $\theta \in \R/\Z$ we write $$\label{eqn.nj} | \theta |:=\|\theta\|_{\R/\Z}:=\min\{|\delta| : \delta \in \R, \theta + \delta \in \Z\}.$$ Since $S^1$ already has a notion of absolute value inherited from $\C$, we have to be slightly careful and for $z \in S^1$ write $$\|z\|_{S^1}:=\left\|\frac{1}{2\pi i}\log z\right\|_{\R/\Z}:=\left| \frac{1}{2\pi i}\log z\right|,$$ where $\frac{1}{2\pi i}\log z$ naturally takes values in $\R/\Z$ and so we can use the notation in (\[eqn.nj\]). These combine in the obvious way for $\G$ so that $$|(\theta_1,\theta_2,z)| := \max \{|\theta_1|,|\theta_2|,\|z\|_{S^1}\}.$$ Finally for $(\R/\Z)^d$ and $\G^d$ we put $$| (g_1,\dots, g_d) | := \max_i |g_i|.$$ Thus $|\cdot|$ is used in several different ways in our paper and in any given situation its meaning must be inferred from context. This should not be difficult. The last piece of notation we need is for boxes in $\G^d$: given $\eps > 0$ define $$\label{eqn.box} X(\eps) := \{x \in \G^d : |x| {\leqslant}\eps\}.$$ The group $H_{\Psi}$ may look rather complicated with respect to $|\cdot |$ (on $\G^d$). (Informally, it may “wind around” $\G^d$ a very large number of times). However, we do at least have some control, as shown in Lemma \[box-pigeon\] below. We deduce that lemma from the following more general fact which will be useful in §\[sec7\]. \[pigeon-projection\] Suppose that $G$ is a compact abelian group with Haar measure $\mu_G$, and that $\pi : G \rightarrow (\R/\Z)^d$ is a homomorphism. Then $\mu_G (\{ x : |\pi(x)| {\leqslant}\delta\}) {\geqslant}\delta^d$. For $\theta \in (\R/\Z)^d$, write $B_{\delta}(\theta) := \prod_{i = 1}^d [\theta_i, \theta_i + \delta]$. For any fixed $x \in (\R/\Z)^d$ we have $$\int 1_{B_{\delta}(\theta)}(x) d\theta = \delta^d.$$ Taking $x = \pi(g)$ and integrating over $G$, we obtain $$\int 1_{B_{\delta}(\theta)} (\pi(g))d\mu_G(g) d\theta = \delta^d.$$ Hence there is some $\theta$ such that $$\mu_G (\{ g : \pi(g) \in B_{\delta}(\theta)\}) {\geqslant}\delta^d.$$ Write $S := \{ g : \pi(g) \in B_{\delta}(\theta)\}$, thus $\mu_G(S) {\geqslant}\delta^d$. Note that if $g_1, g_2 \in S$ then $|\pi(g_1 - g_2) | {\leqslant}\delta$. Thus the result follows from the fact that $\mu_G(S - S) {\geqslant}\mu_G(S - s) = \mu_G(S)$, where $s \in S$ is any element. Note that this proof is really just the same as that of [@taovu Lemma 4.19]. \[box-pigeon\] We have $\mu_{H_{\Psi}}(X(\eps)) {\geqslant}\eps^{3d}$. Apply the previous lemma with $\pi$ being the restriction to $H_{\Psi}$ of the natural homomorphism from $\G^d$ to $(\R/\Z)^{3d}$. A useful corollary of Proposition \[distribution\] and Lemma \[box-pigeon\] is the following assertion. \[cor3.3\] There is a function $p_1:\Z_{{\geqslant}0} \times (0,1] \rightarrow \N$ with $p_1(d',\eps') {\geqslant}p_1(d,\eps)$ whenever $\eps' {\leqslant}\eps$ and $d' {\geqslant}d$, such that if $p {\geqslant}p_1(d,\eps)$ then the following is true. For every $d$-dimensional ${\operatorname{QM}}$-system $\Psi$ and $h \in H_{\Psi}$ we have $$\mu_{\F} (\{x : |h^{-1}\Psi(x)| {\leqslant}\eps\}) {\geqslant}\frac{1}{8}\left(\frac{\eps}{4}\right)^{3d}.$$ We use the fact that the trigonometric polynomials are dense in $C(\G^d)$. Thus if $\eps > 0$ then there is some $F_0: \G^d \rightarrow \R$ such that - $F_0 {\geqslant}-\eps'$ pointwise, where $\eps' := \frac{1}{2}\left(\frac{\eps}{2}\right)^{3d}$; - $F_0 {\leqslant}0$ outside $X(\eps)$, the box defined in (\[eqn.box\]); - $F_0 {\geqslant}1$ on $X(\eps/2)$; - $F_0 {\leqslant}2$ on $\G^d$ and - $\Vert F_0 \Vert_{{\operatorname{trig}}} = O_{\eps, d}(1)$. Now apply Proposition \[distribution\] with $F = T_hF_0$. We have $\Vert T_h F_0 \Vert_{{\operatorname{trig}}} = \Vert F_0 \Vert_{{\operatorname{trig}}} = O_{\eps, d}(1)$, uniformly in $h$. By Proposition \[distribution\], $$\E_{x \in \F} T_hF_0(\Psi(x)) = \int T_h F_0 d\mu_{H_{\Psi}} + o_{p \rightarrow \infty}(\Vert F_0 \Vert_{{\operatorname{trig}}} ) = \int F_0 d\mu_{H_{\Psi}} + o_{\eps, d;p\rightarrow \infty}(1),$$ the second equality being a consequence of the invariance of Haar measure. Since $F_0 {\geqslant}1$ on $X(\eps/2)$ and $F_0 {\geqslant}-\eps'$ everywhere, it follows from Lemma \[box-pigeon\] that $$\int F_0 d\mu_{H_{\Psi}} {\geqslant}\left(\frac{\eps}{2}\right)^{3d} - \eps' = \frac{1}{2}\left(\frac{\eps}{2}\right)^{3d} .$$ Thus $$\E_{x \in \F} T_hF_0(\Psi(x)) {\geqslant}\frac{1}{2}\left(\frac{\eps}{2}\right)^{3d} - o_{\eps,d;p \rightarrow \infty}(1) {\geqslant}\frac{1}{4}\left(\frac{\eps}{2}\right)^{3d}$$ provided $p {\geqslant}p_0(d,\eps)$. However, $T_h F_0(y) {\leqslant}0$ unless $h^{-1} y \in X(\eps)$ and $F_0 {\leqslant}2$ pointwise, so $T_hF_0(y) {\leqslant}2 \cdot 1_{X(\eps)}(h^{-1}y)$. It follows that $$\mu_{\F} (\{x : |h^{-1}\Psi(x)| {\leqslant}\eps\}) {\geqslant}\frac{1}{8}\left(\frac{\eps}{2}\right)^{3d}$$ provided $p {\geqslant}p_0(d,\eps)$. It remains to set $$p_1(d,\eps):=\max\left\{ p_0(d',2^{-j}): 2^{-j} {\geqslant}\eps/2, d' {\leqslant}d,\text{ and } d,j \in \Z_{{\geqslant}0}\right\}.$$ This function $p_1$ is monotonic in the desired sense, and the result follows since there is some $j \in \Z_{{\geqslant}0}$ with $\eps {\geqslant}2^{-j} {\geqslant}\eps/2$ and $$\mu_{\F} (\{x : |h^{-1}\Psi(x)| {\leqslant}\eps\}) {\geqslant}\mu_{\F} (\{x : |h^{-1}\Psi(x)| {\leqslant}2^{-j}\}) {\geqslant}\frac{1}{8}\left(\frac{\eps}{4}\right)^{3d}$$ whenever $p {\geqslant}p_1(d,\eps)$. *Monotonic functions.* By invoking the Stone-Weierstrass theorem we have taken a soft approach to approximating intervals by trigonometric polynomials in Corollary \[cor3.3\]. This adds a slight technical complexity because the constant behind the $O_{\eps,d}(1)$ term in the proof could, in principle, depend in a very peculiar way on $\eps$ and $d$. This complexity will crop up in other parts of the argument and we shall deal with it by ensuring that “universal functions” (in the above case $p_1$) are monotonic in a suitable sense. We take the following convention: all our “universal functions” will be of the form $f:D_1\times \dots \times D_r \rightarrow D_0$ where the $D_i$s are one of the sets $(0,1],\Z_{{\geqslant}0}, \N,\R_{{\geqslant}0}$ or $\R_{{\geqslant}1}$. If $D_i=(0,1]$ then we write $x \preccurlyeq_i y$ whenever $x {\leqslant}y$ ; otherwise $x \preccurlyeq_i y$ whenever $x {\geqslant}y$. We shall say that $f$ is *monotone* if $f(x) \preccurlyeq_0 f(y)$ whenever $x_i \preccurlyeq_i y_i$ for all $1 {\leqslant}i {\leqslant}r$. The two places we have encountered monotonicity so far are in Corollary \[cor3.3\] where $p_1$ is monotonic in the above sense and Corollary \[gvn-cor\] where $\nu$ is monotonic. Finally we come to a crucial definition in the paper. Suppose that $\Psi$ is a ${\operatorname{QM}}$-system of dimension $d$. Then we write $B(\Psi, \eps) := \Psi^{-1}(X(\eps))$, and call this a ${\operatorname{QM}}$-Bohr set of dimension $d$ and width $\eps$. That is, $B(\Psi, \eps) := \{x \in \F : |\Psi(x)| {\leqslant}\eps\}$. It should not come as a surprise that we have an analogue of the usual lower bound for the size of Bohr sets [@taovu Lemma 4.19]. \[bohr-lower\] There is a monotonic function $\beta : \Z_{{\geqslant}0} \times (0,1] \rightarrow (0,1]$ such that the following is true. For all $d$-dimensional ${\operatorname{QM}}$-systems $\Psi$ and parameters $\eps \in (0,1]$ we have $$\mu_\F(B(\Psi, \eps)) {\geqslant}\beta(d,\eps).$$ This is immediate from Corollary \[cor3.3\] by taking $h = {\operatorname{id}}_{\G^d} = ((0,0,1), \dots , (0,0,1))$, the identity element in $\G^d$, and then putting $$\beta(d,\eps):=\min\left\{ \frac{1}{p_1(d,\eps)}, \frac{1}{8}\left(\frac{\eps}{4}\right)^{3d}\right\}.$$ Since ${\operatorname{id}}_{\G^d} \in B_{{\operatorname{QM}}}(\Psi, \eps)$ for all $\eps \in (0,1]$ the result follows. Structure of the main argument {#sec4} =============================== We turn now to a discussion of the basic form of the rest of the argument. The basic strategy is to work by induction on the number of colours, but to get this to work effectively we must establish a more general statement. It may be useful to recall the notion of monotone function we are using from the discussion after Corollary \[cor3.3\]. \[main-prop-refined\] There are monotonic functions $\eta,\zeta : (0,1] \times \Z_{{\geqslant}0} \times \Z_{{\geqslant}0} \rightarrow (0,1]$ and $p_0 : (0,1] \times \Z_{{\geqslant}0} \times \Z_{{\geqslant}0} \rightarrow \N$ with the following property. Suppose that $B \subset \F$ is a ${\operatorname{QM}}$-Bohr set of dimension $d$ and width $\delta$, and that we have an $r$-colouring $c: \tilde B \rightarrow [r]$ for some set $\tilde B \subset B$ with $|\tilde B| {\geqslant}(1 - \eta(\delta,d,r))|B|$. Then, provided $p{\geqslant}p_0(\delta,d,r)$ there are $\zeta(\delta,d,r)p^2$ pairs $(x, y)$ for which $x,y, x+y, xy \in \tilde B$ and $c(x) = c(y) = c(x+y) = c(xy)$. *Remarks.* This immediately gives our main result. One may think of $c$ as an “almost” colouring (or more precisely a $(1 - \eta(\delta,d,r))$-almost $r$-colouring). The nature of our inductive arguments necessitates the consideration of almost colourings in addition to true colourings. Apply Proposition \[main-prop-refined\] with $d=0$, $\delta=1/2$ and $\tilde B = B = \F$. Since $(0,0,0+0,0 \cdot 0)$ is always monochromatic we see that every colouring contains at least $1$ monochromatic quadruple. It follows that the total number of monochromatic quadruples is at least $$\min\big\{\frac{1}{p_0(\frac{1}{2},0,r)^2}, \zeta(\frac{1}{2},0,r)\big\}p^2 \gg_r p^2.$$ The result is proved. The proof of Proposition \[main-prop-refined\] combines three fairly substantial pieces: a regularity lemma, a counting lemma, and a Ramsey-theory result. These three ingredients and their proofs can be understood more-or-less independently, the only common features being the language of §\[sec3\]. *Regularity lemma.* The basic principle of the regularity lemma is that it allows one to replace an arbitrary colouring of $\F$ by one that is induced from a “nice” colouring of $\G^d = \big( \R/\Z \times \R/\Z \times S^1 \big)^d$ by pullback under a ${\operatorname{QM}}$-map $\Psi : \F \rightarrow \G^d$. This is a well-trodden idea which of course goes back to Szemer[é]{}di [@sze]. Perhaps closer to our particular instance is the arithmetic development in [@green-reg Theorem 5.2]; the probabilistic framing in [@tao Theorem 2.11]; and the combination in [@green-tao-regularity Theorem 1.2]. \[regularity\] Suppose that $\Omega : \Z_{{\geqslant}0} \times \Z_{{\geqslant}0} \times (0,1] \times \R_{{\geqslant}0} \times \Z_{{\geqslant}0} \rightarrow \R_{{\geqslant}1}$ is monotonic. Then there are monotonic functions $M_\Omega,D_\Omega$, and $p_\Omega$ mapping $\Z_{{\geqslant}0} \times \Z_{{\geqslant}0} \times (0,1] \times (0,1]$ to $\R_{{\geqslant}1}$, $\Z_{{\geqslant}0}$, and $\N$ respectively such that the following holds. Suppose that $\Psi$ is a $d$-dimensional ${\operatorname{QM}}$-system of width $\delta$, $B:=B(\Psi,\delta)$, $\tilde B \subset B$ has $|\tilde B| {\geqslant}(1 - \frac{\eps^2}{100})|B|$ for some parameter $\eps \in (0,1]$, and $c : \tilde B \rightarrow [r]$ is an $r$-colouring of $\tilde B$. Then there is a ${\operatorname{QM}}$-system $\Psi' \supset \Psi$ of some dimension $d'$, functions $F_1,\dots, F_r : \G^{d'} \rightarrow \R_{{\geqslant}0}$, and functions $g_1,\dots, g_r : \F \rightarrow [-1,1]$, such that 1. $\Vert F_i \Vert_{{\operatorname{trig}}} {\leqslant}M_{\Omega}(r,d,\delta,\eps)$ for all $i \in\{ 1,\dots, r\}$; 2. $d' {\leqslant}D_{\Omega}(r,d,\delta,\eps)$; 3. $\Vert F_i \circ \Psi' - g_i \Vert_2 {\leqslant}\eps$ for all $i \in\{ 1,\dots, r\}$; 4. $\Vert 1_{c^{-1}(i)} - g_i\Vert_{{\operatorname{QM}}} {\leqslant}1/\Omega(r,d,\delta,\Vert F_i \Vert_{{\operatorname{trig}}}, d')$; 5. $\sum_{i = 1}^r F_i \circ \Psi' {\geqslant}1$ pointwise on $B(\Psi, \frac{\delta}{2})$; 6. $\|F_i \circ \Psi'\|_{4} {\leqslant}2$ for all $i \in \{1,\dots,r\}$; provided $p {\geqslant}p_\Omega(r,d,\delta,\eps)$. The proof of the regularity lemma involves an “energy increment” argument of a type familiar to experts, but some of the technical details are a little tricky. The proof is given in §\[sec5\]. *Counting lemma.* As usual we complement the regularity lemma with a counting lemma. There is always a trade-off between the effort needed to prove a regularity lemma and its companion counting lemma; in our work the former contains essentially all of the difficulties. The basic idea of the counting lemma is that if we have a colouring on $\F$ pulled back from a colouring of $\G^d$ under a ${\operatorname{QM}}$-map (as provided by the regularity lemma) then the number of monochromatic quadruples $x,y,x+y, xy$ in $\F$ is related to the number of monochromatic copies of a certain type of *linear* configuration in $\G^d$: specifically, if $y$ is constrained to lie in a small ${\operatorname{QM}}$-Bohr set then $\Psi(x), \Psi(x+y), \Psi(xy)$ correspond to triples of the form $(t,u,v), (t + u', u, v'), (t', u', v)$. To get an idea of why this is so consider the case when $d=1$. In this case we can put $$\Psi:\F \rightarrow \G;x \mapsto (ax^2/p, 2ax/p,\chi(x)),$$ and the colouring of $\F$ is pulled back from a colouring of $\G$. If $\Psi(y) \approx {\operatorname{id}}_{\G} = (0,0,1)$ then the remaining three points $\Psi(x)$, $\Psi(x+y)$, and $\Psi(xy)$ have constraints resulting from the identities $$x^2 + 2xy + y^2 = (x + y)^2,\; x + y = (x + y) \; \text{ and } \; \chi(x)\chi(y)\; \mbox{``$=$''}\; \chi(xy)$$ (where the quotation marks in the last are because our characters $\chi$ are not $0$ at $0$), which imply that $$\frac{ax^2}{p} + \frac{2axy}{p} \approx \frac{a(x + y)^2}{p},\;\; \frac{2ax}{p} \approx \frac{2a(x+y)}{p} \; \text{and} \; \chi(x) \approx \chi(xy).$$ This tells us that $\Psi(x), \Psi(x+y), \Psi(xy)$ have approximately the form $(t,u,v)$, $(t + u',u,v')$, and $(t', u',v)$ respectively. The counting lemma is a much stronger statement than this, essentially asserting that the above is the *only* type of constraint that occurs. \[counting-lem\] Suppose $\Psi$ is a $d$-dimensional ${\operatorname{QM}}$-system, $F : \G^d \rightarrow \C$ is a trigonometric polynomial, and that $S \subset B(\Psi,\eps)$. Then $$\begin{aligned} & T(F \circ \Psi, 1_S, F \circ \Psi, F \circ \Psi) \\ & \qquad = \mu_{\F}(S) \int F(t,u,v) F(t + u', u, v') F(t', u', v) d\mu_{H_{\Psi}}(t,u,v) d\mu_{H_{\Psi}}(t',u',v') \\ & \qquad \qquad + O(\epsilon \mu_{\F}(S)\Vert F \Vert_{{\operatorname{trig}}}^4) + o_{p \rightarrow \infty}(\Vert F \Vert_{{\operatorname{trig}}}^{9d}).\end{aligned}$$ The counting lemma is established using harmonic analysis in §\[sec8\]. It relies on bounds for certain character sums (mixing quadratic phases and shifts of multiplicative characters), which in turn use some fairly deep number-theoretic inputs. These issues are discussed at the beginning of §\[sec8\]. *Ramsey lemma.* Finally, we need a result of a Ramsey-theoretic nature. The counting lemma above transfers our nonlinear question about quadruples $x, y, x+y,xy$ to a question about linear configurations, but we must still solve this linear problem. A toy version of the result we need is the following: if $G$ is a sufficiently large abelian group and if $c : G \times G \times G \rightarrow [r]$ is an $r$-colouring, we may find $t,t',u,u',v,v'$ with $c(t,u,v) = c(t + u', u, v') = c(t',u',v)$. We do indeed prove such a statement, but again we need something a little more complicated for the purposes at hand, designed to dovetail with the conclusions of the counting lemma. \[ramsey-prop\] There is a monotonic function $\rho : \Z_{{\geqslant}0} \times \Z_{{\geqslant}0} \times (0,1] \rightarrow (0,1]$ with the following property. Suppose that $X$ and $Y$ are compact Abelian groups with Haar probability measures $\mu_X, \mu_Y$, that $\pi_X : X \rightarrow (\R/\Z)^{d}$ and $\pi_Y : Y \rightarrow (\R/\Z)^{d}$ are continuous homomorphisms, and that $F_1,\dots, F_r : X \times X \times Y \rightarrow \R_{{\geqslant}0}$ are continuous functions with $\sum_{i=1}^r F_i(x_1, x_2, y) {\geqslant}1$ whenever we have $| \pi_X(x_1)|,|\pi_X(x_2)|,|\pi_Y(y)| {\leqslant}\delta/4$. Let $\mu = \mu_X \times \mu_X \times \mu_Y$. Then $${\int F_i(t,u,v) F_i(t + u', u, v') F_i(t', u',v)} d\mu(t,u,v) d\mu(t',u',v') {\geqslant}\rho(r,d,\delta)$$ for some $i \in [r]$. This result is established in §\[sec7\]. We again proceed by induction on the number of colours (thus, taken as a whole, our paper has two nested inductions on the number of colours). The basic scheme of the argument is inspired by work of Cwalina and Schoen [@cwalina-schoen] on Rado’s theorem, but the details are quite different. We make critical use of the “dependent random choice” technique pioneered by Gowers [@gowers-4ap]. *Proposition \[main-prop-refined\].* We shall shortly give the proof of Proposition \[main-prop-refined\], assuming the regularity, counting and Ramsey lemmas. First we note that the counting and Ramsey lemmas may be combined to give the following statement. \[prop99\] There are monotonic functions $\kappa : \Z_{{\geqslant}0} \times \Z_{{\geqslant}0} \times (0,1] \times \R_{{\geqslant}0} \rightarrow (0,1]$ and $p_2:\R_{{\geqslant}0} \times \Z_{{\geqslant}0}\times \Z_{{\geqslant}0}\times (0,1] \times (0,1]\rightarrow \N$ with the following property. Suppose that $d' {\geqslant}d$ are integers, $M {\geqslant}1$ is real, $\sigma \in (0,1]$; that $\Psi' \supset \Psi$ are ${\operatorname{QM}}$-systems of dimension $d',d$, and that $F_1,\dots, F_r : \G^{d'} \rightarrow \R_{{\geqslant}0}$ are functions with $\Vert F_i \Vert_{{\operatorname{trig}}} {\leqslant}M$ and $\sum_{i = 1}^r F_i \circ \Psi' {\geqslant}1$ pointwise on $B(\Psi, \frac{1}{2}\delta)$. Then for any sets $S_1,\dots,S_r$ with $S_i \subset B(\Psi',\kappa(r,d,\delta,\|F_i\|_{{\operatorname{trig}}}))$ and $\mu_{\F}(S_i) {\geqslant}\sigma$ there is some $i$ such that $$T(F_i \circ \Psi', 1_{S_i}, F_i \circ \Psi', F_i \circ \Psi') {\geqslant}\textstyle\frac{1}{16}\displaystyle\mu_{\F}(S_i)\rho(r,d,\delta)$$ provided that $p {\geqslant}p_2(M,d',r,\sigma,\delta)$. (Where $\rho$ is as in the Ramsey lemma, Proposition \[ramsey-prop\].) *Remark.* A crucial point here is that neither $\kappa$ nor $\rho$ depends on $d'$. If they did, our arguments would be circular. Apply the counting lemma (Proposition \[counting-lem\]) to each of the $F_i$s and the ${\operatorname{QM}}$-system $\Psi'$ to get that $$\begin{aligned} & T(F_i \circ \Psi', 1_{S_i}, F_i \circ \Psi', F_i \circ \Psi') \\ & \qquad = \mu_{\F}(S) \int F_i(t,u,v) F_i(t + u', u, v') F_i(t', u', v) d\mu_{H_{\Psi'}}(t,u,v) d\mu_{H_{\Psi'}}(t',u',v') \\ & \qquad \qquad + O(\kappa \mu_{\F}(S)\|F_i\|_{{\operatorname{trig}}}^4) + o_{p \rightarrow \infty}(M^{9d'})\\ & \qquad {\geqslant}\mu_{\F}(S) \int F_i(t,u,v) F_i(t + u', u, v') F_i(t', u', v) d\mu_{H_{\Psi'}}(t,u,v) d\mu_{H_{\Psi'}}(t',u',v')\\ & \qquad \qquad - \textstyle\frac{1}{16}\displaystyle\mu_{\F}(S_i)\rho(r,d,\delta)\end{aligned}$$ provided $p {\geqslant}p_3(M,r,d',\delta,\sigma)$ (where $p_3$ can be taken monotone increasing in its arguments) and $\kappa=c\rho(r,d,\delta)\min ( 1, \|F_i\|_{{\operatorname{trig}}}^{-4})$ for some absolute $c>0$, which satisfies the relevant monotonicity properties since $\rho$ does. Let $\eps:=\delta/12 \pi rd'(1+M)^2$; the reason for this choice will become clear later. We should like to apply the Ramsey lemma to which end we put $X:=G_{\Psi'}^+$ and $Y:=G_{\Psi'}^\times$ then $H_{\Psi'} = X \times X \times Y$. Moreover, if we write $\pi_X:G_{\Psi'}^+ \rightarrow G_{\Psi}^+$, and $\pi_Y:\G_{\Psi'}^\times \rightarrow G_{\Psi}^\times$ for the respective projections onto the first $d$-coordinates then these are well-defined continuous homomorphisms, as is $\pi:H_{\Psi'} \rightarrow H_{\Psi}$ defined by $\pi(x_1,x_2,y) = (\pi_X(x_1),\pi_X(x_2),\pi_Y(y))$. If $h=(x_1,x_2,y) \in H_{\Psi'}$, then by Corollary \[cor3.3\] (provided $p {\geqslant}p_1(d',\eps)$) there is some $z \in \F$ with $|h^{-1}\Psi'(z)| {\leqslant}\eps$. Two things follow from this. First, $|\pi(h)^{-1}\pi(\Psi'(z))| {\leqslant}\eps$, so if $|\pi(h)| {\leqslant}\delta/4$ then $|\Psi(z)| {\leqslant}\frac{1}{4}\delta + \eps {\leqslant}\frac{1}{2}\delta$ by the triangle inequality *i.e.* $z \in B(\Psi,\frac{1}{2}\delta)$. Secondly, it is easy to see that the functions $F_i$ are Lipschitz in the sense that $$|F_i(w)-F_i(v)| {\leqslant}6\pi d'M^2|w^{-1}v| \text{ for all } w,v \in \G^{d'},$$ so they are continuous, but we also have $$\big|\sum_{i}{F_i(h)} - \sum_i{F_i(\Psi'(z))}\big| {\leqslant}6 \pi rd'M^2|h^{-1}\Psi'(z)| {\leqslant}6\pi \eps rd'M^2 {\leqslant}\textstyle\frac{1}{2}\displaystyle.$$ We conclude from these two facts and the hypothesis that $\sum_i{F_i\circ \Psi'} {\geqslant}1$ on $B(\Psi,\frac{1}{2}\delta)$, that if $h \in H_{\Psi'}$ has $|\pi(h)| {\leqslant}\frac{1}{4}\delta$ then $$\sum_{i}{F_i(h)} {\geqslant}\sum_i{F_i(\Psi'(z))}- \textstyle\frac{1}{2}\displaystyle {\geqslant}\textstyle\frac{1}{2}\displaystyle.$$ We now apply the Ramsey lemma (Proposition \[ramsey-prop\]) to the functions $2F_i$ to get that $$\int F_i(t,u,v) F_i(t + u', u, v') F_i(t', u', v) d\mu_{H_{\Psi'}}(t,u,v) d\mu_{H_{\Psi'}}(t',u',v') {\geqslant}\textstyle\frac{1}{8}\displaystyle\rho(r,d,\delta)$$ for some $i \in [r]$. This gives the result provided $$p {\geqslant}p_2(M,d',r,\sigma,\delta):=p_3(M,r,d',\delta,\sigma)+p_1(d',\delta/12 \pi rd'M^2)$$ which is easily seen to be monotone. The proposition is proved. Finally, we record a fairly simple application of the Cauchy-Schwarz inequality. \[simple-lem\] Suppose that $S \subset \F$ and $f_1,f_3,f_4:\F \rightarrow [-K,K]$ are functions with $\|f_1\|_4,\|f_3\|_4,\|f_4\|_4 {\leqslant}3$. Then $$|T(f_1, 1_S, f_3, f_4)| {\leqslant}\frac{K^3}{p} + 9\mu_{\F}(S)\min_i \Vert f_i \Vert_{2}.$$ By the Cauchy-Schwarz inequality we have $$\begin{aligned} |T(f_1, 1_S, f_3, f_4)| & = |\E_{x,y} f_1(x)1_S(y) f_3(x+y) f_4(xy)| \\ & {\leqslant}\frac{K^3}{p} + |\E_{x,y} 1_{\F^*}(y)f_1(x)1_S(y) f_3(x+y) f_4(xy)|\\ & {\leqslant}\frac{K^3}{p} + \mu_{\F}(S) \sup_{y \in \F^*} \E_x |f_1(x) f_3(x+y) f_4(xy)| \\ & {\leqslant}\frac{K^3}{p} + \mu_{\F}(S) \sup_{y \in \F^*} (\E_x f_1(x)^2)^{1/2} (\E_x f_3(x+y)^2 f_4(xy)^2 )^{1/2}\\ & {\leqslant}\frac{K^3}{p} + \mu_{\F}(S) \sup_{y \in \F^*} (\E_x f_1(x)^2)^{1/2} (\E_x f_3(x+y)^4)^{1/4}(\E_xf_4(xy)^4 )^{1/4}\\ & {\leqslant}\frac{K^3}{p} + 9\mu_{\F}(S) \Vert f_1 \Vert_{2}.\end{aligned}$$ The proofs for $i \in \{ 3,4\}$ are very similar. We are now ready for the proof of Proposition \[main-prop-refined\]. We proceed by induction on $r$, setting $\eta( \cdot, \cdot , 0), \zeta(\cdot,\cdot,0) = \frac{1}{2}$ and $p_0(\cdot,\cdot,0) =1$. All of these functions are monotone and vacuously satisfy the proposition since there is no $0$-colouring of a non-empty set, and $|\tilde B| {\geqslant}\frac{1}{2}|B| >0$ since every ${\operatorname{QM}}$-Bohr set contains $0$. Suppose we have established the existence of $\eta( \cdot, \cdot, r-1),\zeta(\cdot,\cdot,r-1)$ and $p_0(\cdot,\cdot,r-1)$ satisfying the desired conclusion; we shall show how to define $\eta( \cdot, \cdot, r),\zeta(\cdot,\cdot,r)$ and $p_0(\cdot,\cdot,r-1)$. The argument we present is a little delicate and so we shall be quite explicit. We shall make use of the following functions: 1. $\rho$ from Proposition \[ramsey-prop\], the Ramsey lemma; 2. $\kappa$ and $p_2$ from Proposition \[prop99\], the combined counting and Ramsey lemmas; 3. $\beta$ from Lemma \[bohr-lower\] (our lower bound for the density of a ${\operatorname{QM}}$-Bohr set); 4. $M_{\Omega}$, $D_{\Omega}$ and $p_\Omega$ from Proposition \[regularity\], the regularity lemma, depending on some soon-to-be-defined growth function $\Omega$; 5. $\nu$ from Corollary \[gvn-cor\], the generalised von Neumann theorem. We use these functions to define a growth function $\Omega$ by $$\Omega_{r,d,\delta}(R,d'):= 1/\nu\bigg( \textstyle\frac{1}{372}\displaystyle \eta\big(\min\{\kappa(r,d,\delta,R),\delta\},d',r-1\big) \rho(r,d,\delta) \beta\big(d',\min\{\kappa(r,d,\delta,R),\delta\}\big) \bigg)$$ The monotonicity of $\kappa$, $\rho$, $\beta$ and $\eta(\cdot,\cdot,r-1)$ ensures that $\Omega$ is monotonic (as a function from $\Z_{{\geqslant}0} \times \Z_{{\geqslant}0} \times (0,1] \times \R_{{\geqslant}0} \times \Z_{{\geqslant}0} \rightarrow \R_{{\geqslant}1}$), and so we may apply the regularity lemma (Proposition \[regularity\]). Doing so gives up monotonic functions $M_\Omega$ and $D_\Omega$, and with these in hand we make some definitions: $$\eps := \textstyle\frac{1}{1728}\displaystyle \rho(r,d,\delta), M_{r,d,\delta}:=M_\Omega(r,d,\delta,\eps) \text{ and } D_{r,d,\delta}:=D_\Omega(r,d,\delta,\eps),$$ all of which values depend monotonically on the values of $r$, $d$, and $\delta$ by the monotonicity of $\rho$, $M_\Omega$ and $D_\Omega$. Finally we write $$\delta':=\min\{\kappa(r, d,\delta,M_{r,d,\delta}),\delta\},$$ which depends monotonically on $r$, $d$, and $\delta$ by the monotonicity of $\kappa$ and $M_{r,d,\delta}$. We then define the functions $\eta( \cdot, \cdot, r),\zeta(\cdot,\cdot,r)$ and $p_0(\cdot,\cdot,r)$ as follows: $$\eta(\delta,d,r) := \min \bigg\{ \textstyle\frac{1}{100}\displaystyle \eps^2, \textstyle\frac{1}{2}\displaystyle\beta(D_{r,d,\delta},\delta') \eta\big(\delta',D_{r,d,\delta} , r-1\big) ,\eta(\delta,d,r-1)\bigg\},$$ $$\zeta(\delta,d,r) := \min \bigg\{ \zeta(\delta,d,r-1), \zeta(\delta',D_{r,d,\delta},r-1), \textstyle\frac{1}{128}\displaystyle\eta(\delta',D_{r,d,\delta},r-1) \rho(r,d,\delta) \beta(D_{r,d,\delta},\delta')\bigg\},$$ and $$\begin{aligned} p_0(\delta,d,r) &:= \left\lceil\max \bigg\{ p_2\left(M_{r,d,\delta},D_{r,d,\delta},r,\textstyle\frac{1}{2}\displaystyle\eta(\delta',D_{r,d,\delta},r-1)\beta(D_{r,d,\delta},\delta'),\delta\right),\right.\\ & \qquad \qquad \qquad 1+1/\nu\left(1/\Omega_{r,d,\delta}(M_{r,d,\delta},D_{r,d,\delta})\right),p_0(\delta',D_{r,d,\delta},r-1),p_\Omega(r,d,\delta,\eps),\\ & \left.\qquad \qquad \qquad 384(1+M_{r,d,\delta})^3\eta(\delta',D_{r,d,\delta},r-1)^{-1}\beta(D_{r,d,\delta},\delta')^{-1}\rho(r,d,\delta)^{-1}\bigg\}\right\rceil.\end{aligned}$$ These functions inherit the appropriate monotonicity properties from the monotonicity of $\eps$, $D_{r,d,\delta}$, $\delta'$, $\beta$, $\eta(\cdot,\cdot,r-1)$, $\zeta(\cdot,\cdot,r-1)$, $\rho$, $p_2$, $p_0(\cdot,\cdot,r-1)$, $M_{r,d,\delta}$, and $\nu$. Now suppose that $B$ is a $d$-dimensional ${\operatorname{QM}}$-Bohr set of width $\delta$ with underlying ${\operatorname{QM}}$-system $\Psi$, and $c : \tilde B \rightarrow [r]$ is a colouring of $\tilde{B}$ where $|\tilde B| {\geqslant}(1 -\eta(\delta,d,r) )|B|$. Since $\eta(\delta,d,r) {\leqslant}\frac{1}{100}\eps^2$, the regularity lemma tells us that (since $p {\geqslant}p_\Omega(r,d,\delta,\eps)$) there is a ${\operatorname{QM}}$-system $\Psi' \supset \Psi$, functions $F_1,\dots, F_r : \G^{d'} \rightarrow \R_{{\geqslant}0}$ and functions $g_1,\dots, g_r : \F \rightarrow [-1,1]$ such that: 1. \[1\] $\Vert F_i \Vert_{{\operatorname{trig}}} {\leqslant}M_\Omega(r,d,\delta,\eps)=M_{r,d,\delta}$, $i \in \{ 1,\dots, r\}$; 2. \[2\] $d' {\leqslant}D_\Omega(r, d,\delta,\eps)=D_{r,d,\delta}$; 3. \[3\] $\Vert F_i \circ \Psi' - g_i \Vert_2 {\leqslant}\eps$ for $i \in \{1,\dots, r\}$; 4. \[4\] $\Vert 1_{c^{-1}(i)} - g_i\Vert_{{\operatorname{QM}}} {\leqslant}1/\Omega_{r,d,\delta}(\Vert F_i \Vert_{{\operatorname{trig}}}, d')$; 5. \[5\] $\sum_{i = 1}^r F_i \circ \Psi' {\geqslant}1$ pointwise on $B(\Psi, \frac{1}{2}\delta)$; 6. \[6\] $\|F_i \circ \Psi'\|_4 {\leqslant}2$ for all $i \in \{1,\dots,r\}$. For each $i \in [r]$ write $\delta_i := \min\{\kappa(r, d,\delta,\|F_i\|_{{\operatorname{trig}}}),\delta\}$ (so that $\delta_i {\geqslant}\delta'$ for all $i$), $B_i:=B(\Psi',\delta_i)$ and $S_i:=c^{-1}(i)\cap B_i$. (On a first pass it may seem like we could take $\delta_i=\delta'$ for all $i$, but we cannot because in (\[4\]) we only have an upper bound in terms of $\|F_i\|_{{\operatorname{trig}}}$ rather than $M_{r,d,\delta}$, and the former might be much smaller than the latter.) We consider two cases. Suppose first that $|S_i| {\leqslant}\frac{1}{2}\eta(\delta_i,d',r-1) |B_i|$ for some $i$. Then by the lower bound for the density of ${\operatorname{QM}}$-Bohr sets (Lemma \[bohr-lower\]); the definition of $\eta$; the fact that $d' {\leqslant}D_{r,d,\delta}$ and $\delta_i {\geqslant}\delta'$; and the monotonicity of $\eta(\cdot,\cdot,r-1)$ and $\beta$, we have $$\begin{aligned} |B_i \setminus \tilde B| {\leqslant}|B \setminus \tilde B| {\leqslant}\eta(\delta, d, r) |B| & {\leqslant}\frac{\eta(\delta,d, r)}{\beta(d',\delta_i)} |B_i|\\ & {\leqslant}\frac{\eta(\delta',D_{r,d,\delta},r-1)\beta(D_{r,d,\delta},\delta')}{2\beta(d',\delta_i)} |B_i|\\ & {\leqslant}\textstyle\frac{1}{2}\displaystyle\eta(\delta_i, d', r-1) |B_i|.\end{aligned}$$ It follows that $c$ restricts to an $(r-1)$-colouring of $B_i\setminus (\tilde{B}\cup S_i)$ where, by the triangle inequality, we have $$|B_i\setminus (\tilde{B}\cup S_i)| {\geqslant}|B_i| - |B_i\setminus \tilde{B}| - |S_i| {\leqslant}\eta(\delta_i,d',r-1) |B_i|.$$ By the inductive hypothesis and monotonicity of $\zeta(\cdot,\cdot,r-1)$ we conclude that there are at least $$\zeta(\delta_i,d',r-1)p^2 {\geqslant}\zeta(\delta',D_{r,d,\delta},r-1)p^2$$ pairs $(x,y)$ with $x,y,x+y,xy$ all the same colour. Thus in this case the result is proved. The second case is that $|S_i| {\geqslant}\frac{1}{2}\eta(\delta_i,d',r-1) |B_i|$ for all $i$. In this case we have $$\begin{aligned} \nonumber \mu_\F(S_i) & {\geqslant}\textstyle\frac{1}{2}\displaystyle\eta(\delta_i,d',r-1)\mu_\F(B_i) \\ \nonumber & {\geqslant}\textstyle\frac{1}{2}\displaystyle\eta(\delta_i,d',r-1)\beta(d',\delta_i) \\ \label{e1} & {\geqslant}\textstyle\frac{1}{2}\displaystyle\eta(\delta',D_{r,d,\delta},r-1)\beta(D_{r,d,\delta},\delta')\end{aligned}$$ by the lower bound for the density of ${\operatorname{QM}}$-Bohr sets (Lemma \[bohr-lower\]), the monotonicity of $\eta(\cdot,\cdot,r-1)$ and $\beta$, and the fact that $d' {\leqslant}D_{r,d,\delta}$ and $\delta_i {\geqslant}\delta'$. By Proposition \[prop99\] (for which application we need (\[5\])) there is some $i \in \{1,\dots,r\}$ such that $$T(F_i \circ \Psi', 1_{S_i}, F_i \circ \Psi', F_i \circ \Psi') {\geqslant}\textstyle\frac{1}{16}\displaystyle\mu_{\F}(S_i) \rho(r,d,\delta),$$ provided $$p{\geqslant}p_2\left(M_{r,d,\delta},d',r, \textstyle\frac{1}{2}\displaystyle\eta(\delta',D_{r,d,\delta},r-1)\beta(D_{r,d,\delta},\delta'),\delta\right),$$ which follows from the definition of $p_0$ and the monotonicity of $p_2$. Note that by (\[6\]) and the triangle inequality we have $$\|g_i\|_4 {\leqslant}1, \|F_i\circ \Psi'\|_4 {\leqslant}2, \text{ and }\|g_i - F_i \circ \Psi'\|_4 {\leqslant}3.$$ Thus by the triangle inequality, Lemma \[simple-lem\] and item (\[3\]) above we have $$\begin{aligned} & |T(g_i, 1_{S_i}, g_i, g_i) - T(F_i \circ \Psi', 1_{S_i}, F_i \circ \Psi', F_i \circ \Psi')|\\ & {\leqslant}|T(g_i, 1_{S_i}, g_i, g_i) - T(F_i \circ \Psi', 1_{S_i}, g_i,g_i )|\\ & \qquad + |T(F_i \circ \Psi', 1_{S_i}, g_i, g_i) - T(F_i \circ \Psi', 1_{S_i}, F_i \circ \Psi',g_i)|\\ & \qquad + |T(F_i \circ \Psi', 1_{S_i}, F_i \circ \Psi', 1_{S_i}, g_i) - T(F_i \circ \Psi', 1_{S_i}, F_i \circ \Psi',F_i \circ \Psi')|\\ & {\leqslant}\frac{3(1+\|F_i\|_\infty)^3}{p} + 27\mu_{\F}(S_i)\eps {\leqslant}\frac{3(1+M_{r,d,\delta})^3}{p} + 27\mu_{\F}(S_i)\eps.\end{aligned}$$ By the choice of $\eps$, and provided $$p {\geqslant}3(1+M_{r,d,\delta})^3 \cdot 64 \mu_\F(S_i)\rho(r,d,\delta)^{-1}$$ this tells us that $$\label{eqn.s} |T(g_i, 1_{S_i}, g_i, g_i) - T(F_i \circ \Psi', 1_{S_i}, F_i \circ \Psi', F_i \circ \Psi')| {\leqslant}\textstyle\frac{1}{32}\displaystyle\mu_{\F}(S_i)\rho(r,d,\delta).$$ Of course by we see $$\begin{aligned} & 3(1+M_{r,d,\delta})^3 \cdot 64 \mu_\F(S_i)\rho(r,d,\delta)^{-1} \\ & \qquad \qquad {\leqslant}384(1+M_{r,d,\delta})^3\eta(\delta',D_{r,d,\delta},r-1)^{-1}\beta(D_{r,d,\delta},\delta')^{-1}\rho(r,d,\delta)^{-1} {\leqslant}p_0(r,d,\delta),\end{aligned}$$ and so holds provided $p {\geqslant}p_0(r,d,\delta)$. It follows from the hypothesis on the size of the $S_i$s, and the lower bound on the density of ${\operatorname{QM}}$-Bohr sets that $$\begin{aligned} T(g_i, 1_{S_i}, g_i, g_i) & {\geqslant}\textstyle\frac{1}{32}\displaystyle \mu_{\F}(S_i) \rho(r,d,\delta)\\ & {\geqslant}\textstyle\frac{1}{64}\displaystyle\mu_{\F}(B_i) \eta(\delta_i,d',r-1) \rho(r,d,\delta)\\ & {\geqslant}\textstyle\frac{1}{64}\displaystyle \eta(\delta_i,d',r-1) \rho(r,d,\delta) \beta(d',\delta_i).\end{aligned}$$ By the generalised von Neumann theorem (Corollary \[gvn-cor\]), monotonicity of $\nu^{-1}$, the choice of $\Omega$ and (\[4\]) above, this implies that $$\begin{aligned} & |T(g_i, 1_{S_i}, g_i, g_i) -T(1_{c^{-1}(i)}, 1_{S_i}, 1_{c^{-1}(i)}, 1_{c^{-1}(i)})|\\ & \qquad {\leqslant}|T(g_i-1_{c^{-1}(i)}, 1_{S_i}, g_i, g_i)|\\ & \qquad \qquad \qquad + |T(1_{c^{-1}(i)}, 1_{S_i}, g_i-1_{c^{-1}(i)}, g_i)| + |T(1_{c^{-1}(i)}, 1_{S_i}, 1_{c^{-1}(i)},g_i-1_{c^{-1}(i)})|\\ & \qquad {\leqslant}3\nu^{-1}(\|g_i-1_{c^{-1}(i)}\|_{{\operatorname{QM}}})\\ & \qquad {\leqslant}3\nu^{-1}(1/\Omega_{r,d,\delta}(\|F_i\|_{{\operatorname{trig}}},d')) = \textstyle\frac{1}{128}\displaystyle\eta(\delta_i,d',r-1) \rho(r,d,\delta) \beta(d',\delta_i),\end{aligned}$$ provided $$p > 1/\nu\left(1/\Omega_{r,d,\delta}(\|F_i\|_{{\operatorname{trig}}},d')\right).$$ By monotonicity of $\Omega_{r,d,\delta}$ and the fact that $d' {\leqslant}D_{r,d,\delta}$ and $\|F_i\|_{{\operatorname{trig}}} {\leqslant}M_{r,d,\delta}$, this follows from $p {\geqslant}p_0(\delta,d,r)$ as defined. By the monotonicity of $\eta(\cdot,\cdot,r-1)$ and $\beta$, and the pointwise inequality $1_{S_i} {\leqslant}1_{c^{-1}(i)}$ we conclude that $$\begin{aligned} T(1_{c^{-1}(i)}, 1_{S_i}, 1_{c^{-1}(i)}, 1_{c^{-1}(i)}) & {\geqslant}\textstyle\frac{1}{128}\displaystyle \eta(\delta_i,d',r-1) \rho(r,d,\delta) \beta(d',\delta_i)\\ & {\geqslant}\textstyle\frac{1}{128}\displaystyle\eta(\delta',D_{r,d,\delta},r-1) \rho(r,d,\delta) \beta(D_{r,d,\delta},\delta').\end{aligned}$$ The result is proved given the choice of $\zeta$. *Remark.* The above argument is not straightforward, in particular with regard to checking that the parameters do not depend on one another in a circular manner. A different way to arrange the arguments might be via the use of an ultraproduct. However, this introduces a considerable amount of additional language, which propagates out to other sections as well. Therefore, even though on some conceptual level an ultraproduct formulation could be the “right” way to phrase the argument, we have chosen not to follow this route. Proof of the regularity lemma {#sec5} ============================= In this section we prove the regularity lemma, Proposition \[regularity\]. The reader may wish to recall its statement. The proof proceeds using an “energy-increment” argument of a type that will be familiar to experts. However, it takes some effort to sort out the technical details specific to our situation. Let $d$ be a non-negative integer (which $d$ we are talking about at any given point will be clear from context). We begin by describing a partition of $\G^d$ into certain boxes. Let $R > 0$ be a power of $2$. Suppose that $t,u,v \in \{0,1,\dots, R-1\}^d$. Then we define *generalised intervals* $$\begin{aligned} I_{R;t,u,v} & := \big\{ (\theta, \phi, z) \in \G^d : \theta_j \in \big[\frac{t_j}{R} + \sqrt{2}, \frac{t_j+1}{R} + \sqrt{2}\big) +\Z, \phi_j \in \big[\frac{u_j}{R} + \sqrt{2}, \frac{u_j+1}{R} + \sqrt{2}\big)+\Z ,\\ & \frac{1}{2\pi i}\log z_j \in \big[\frac{v_j}{R} + \sqrt{2}, \frac{v_j+1}{R} + \sqrt{2}\big)+\Z \quad \mbox{for}\quad j \in \{1,\dots,d\}\big\} .\end{aligned}$$ It is worth making a few remarks about these sets. 1. For each $R$ the set $\mathcal{I}_R:=\{I_{R;t,u,v}: t,u,v \in \{0,1,\dots, R-1\}^d\}$ is a partition of $\G^d$ into $R^{3d}$ sets. 2. We restrict $R$ to be a power of $2$ as a technical convenience, to ensure that if $R'>R$ then $\mathcal{I}_{R'}$ is a refinement of $\mathcal{I}_{R}$. 3. The $\sqrt{2}$ here is present as a technical device to help control edge effects later on; it has the usual property of being poorly approximated by rationals. The only point in the argument at which it is relevant is in the proof of estimate , a technical point in the proof of Lemma \[lem5.4\]. Let $\Psi = (a_i x^2/p, 2a_i x/p, \psi_i(x))_{i = 1}^d$ be a ${\operatorname{QM}}$-system of dimension $d$. Suppose that $R > 0$ is a power of $2$. Where $\Psi$ is clear like this we write $$A_{R;t,u,v}:=\Psi^{-1}(I_{R;t,u,v}) \text{ for each } t,u,v \in \{0,\dots,R-1\}^d.$$ The sets $\{A_{R;t,u,v}: t,u,v \in \{0,\dots,R-1\}^d\}$ form a partition of $\F$ (possibly with the addition of the empty set) since $\mathcal{I}_R$ is a partition of $\G^d$ and, in particular, generate a $\sigma$-algebra $\mathcal{B}$. We define the associated *projection operator* at *resolution* $R$ to be $$\Pi_R^{\Psi} : L^{2}(\F) \rightarrow L^{2}(\F); f \mapsto \E(f | \mathcal{B}),$$ so that $$\Pi_R^{\Psi} f(x) = \frac{1}{|A(x)|} \sum_{x' \in A(x)} f(x')$$ where $A(x)$ is the atom containing $x$. As with any conditional expectation operator, $\Pi_R^{\Psi}$ is self-adjoint. Indeed $$\langle f, \E(g |\mathcal{B}) \rangle = \E_x f(x) \frac{1}{|A(x)|} \sum_{x' \in A(x)} g(x') = \frac{1}{p} \sum_{x, x'} f(x) g(x') 1_{A(x)}(x') \frac{1}{|A(x)|},$$ and the kernel $1_{A(x)}(x') \frac{1}{|A(x)|}$ is symmetric in $x$ and $x'$ so this also equals $\langle \E(f | \mathcal{B}), g\rangle$. \[lem5.2\] Suppose that $f$ has $\|f\|_{\infty} {\leqslant}1$ and $\Vert f \Vert_{{\operatorname{QM}}} {\geqslant}\delta$. Let $R {\geqslant}C\delta^{-1}$ be a power of $2$. Then there is a ${\operatorname{QM}}$-system $\Phi$ of dimension at most 2 and a function $g \in L^{\infty}(\F)$ with $\Vert g \Vert_{\infty} {\leqslant}1$, such that $|\langle f, \Pi_R^{\Phi} g \rangle| \gg \delta$. *Remark.* The functions of the form $\Pi^{\Phi}_R g$ where $\Vert g \Vert_{\infty} {\leqslant}1$, are precisely those functions which are bounded by $1$ in modulus and are constant on atoms $A_{R;t,u,v}$. It follows from the definition of the ${\operatorname{QM}}$-norm (Definition \[def.qmnorm\]) that, under the hypotheses on $f$, there are $a_1, a_2$ and a multiplicative function $\psi$ such that $$|\E_x f(x) \overline{e_p(a_1x^2 + 2a_2x) \psi(x)}| {\geqslant}\delta.$$ Consider the ${\operatorname{QM}}$-system $\Phi = (a_i x^2/p, 2a_i x/p, \psi_i(x))_{i = 1,2}$ in which $\psi_1 = \psi_2 = \psi$. Then $$\E_x f(x) \overline{e_p(a_1x^2 + 2a_2x) \psi(x)} = \langle f, g\rangle,$$ where $g = F \circ \Phi$ with $F : \G^2 \rightarrow \C$ defined by $$F(\theta_1, \theta'_1, z_1; \theta_2, \theta'_2, z_2) := e(\theta_1 + \theta'_2)z_1.$$ Thus $$\label{eq521} |\langle f, F \circ \Phi \rangle| {\geqslant}\delta.$$ Now since $F$ is quite a smooth function, we have $g \approx \Pi_R^{\Psi} g$. More precisely, $$\Vert \Pi_R^{\Phi} g - g\Vert_{\infty} {\leqslant}\sup_{x' \in A(x)} |F(\Phi(x')) - F(\Phi(x))| \ll R^{-1},$$ since the Lipschitz constant of $F$ is $O(1)$. It follows from this and that if $R > C\delta^{-1}$ with $C$ large enough then $|\langle f, \Pi_R^{\Phi} g\rangle| \gg \delta$, and the result follows. The next lemma is a result of “Koopman von Neumann” type. In establishing this we shall need the following property of the projection operators $\Pi^{\Psi}_R$: if $\Psi' \supset \Psi$ and $R | R'$, then $$\label{nesting} \Pi^{\Psi}_R \Pi^{\Psi'}_{R'} f = \Pi^{\Psi}_{R} f.$$ This is because each atom associated to $\Pi_R^{\Psi}$ is a union of atoms associated to $\Pi_{R'}^{\Psi'}$. \[kvn\] Suppose that $f_1,\dots, f_r : \F \rightarrow \C$ are such that $\|f_i\|_\infty {\leqslant}1$ for all $i \in \{1,\dots,r\}$. Let $\Psi$ be a ${\operatorname{QM}}$-system, and let $R {\geqslant}C\delta^{-1}$ be a power of $2$. Then there is a ${\operatorname{QM}}$-system $\Psi' \supset \Psi$ with $\dim\Psi'{\leqslant}\dim\Psi + O( r\delta^{-2})$ such that $\Vert f_i - \Pi_{R}^{\Psi'} f_i\Vert_{{\operatorname{QM}}} {\leqslant}\delta$ for all $i \in\{ 1,\dots, r\}$. Define a nested sequence $\Psi =: \Psi_0 \subset \Psi_1 \subset \Psi_2 \subset \dots$ of ${\operatorname{QM}}$-systems with $\dim \Psi_j = \dim \Psi + 2j$ in the following manner. For $j = 0,1,2,3,\dots$, proceed as follows. Set $f_{i,j} := f_i - \Pi_R^{\Psi_j} f_i$. If $\Vert f_{i,j} \Vert_{{\operatorname{QM}}} {\leqslant}\delta$ for $i \in \{ 1,\dots, r\}$ then stop; otherwise, by Lemma \[lem5.2\], there is an $i \in \{1,\dots,r\}$, some ${\operatorname{QM}}$-system $\Phi$ of dimension $2$ and a function $g \in L^{\infty}(\F)$ with $\Vert g \Vert_{\infty} {\leqslant}1$, such that $|\langle f_{i,j}, \Pi_R^{\Phi} g \rangle| \gg \delta$. In this case, set $\Psi_{j+1} := \Psi_j \cup \Phi$ and note by idempotence of $\Pi_R^\Phi$ and (\[nesting\]) we have $$\langle f_i - \Pi_R^{\Psi_{j+1}}f_i,\Pi_R^{\Phi} g\rangle = \langle \Pi_R^{\Phi} f_i - \Pi_R^{\Phi}\Pi_R^{\Psi_{j+1}}f_i, \Pi_R^{\Phi} g\rangle= 0,$$ and hence $$\langle \Pi_R^{\Psi_{j+1}}f_i - \Pi_R^{\Psi_{j}}f_i,\Pi_R^{\Phi} g\rangle = \langle f_{i,j},\Pi_R^{\Phi} g\rangle - \langle f_i - \Pi_R^{\Psi_{j+1}}f_i,\Pi_R^{\Phi} g\rangle \gg \delta.$$ By the Cauchy-Schwarz inequality we conclude $$\Vert \Pi_R^{\Psi_{j+1}} f_i - \Pi_R^{\Psi_{j+1}} f_i \Vert_2 \gg \delta,$$ and so (expanding out the $L_2$-norm square and using (\[nesting\])) it follows that $$\Vert \Pi_R^{\Psi_{j+1}}f_i \Vert_2^2 - \Vert \Pi_R^{\Psi_j} f_i\Vert_2^2 = \Vert \Pi_R^{\Psi_{j+1}}f_i - \Pi_R^{\Psi_j}f_i \Vert_2^2 \gg \delta^{2}.$$ Hence, defining the energy $$E_j := \sum_{i = 1}^r \Vert \Pi_R^{\Psi_j} f_i \Vert_2^2,$$ we have $$E_{j+1} - E_j \gg \delta^{2}.$$ As long as this process continues, we obviously have the trivial bound $E_j {\leqslant}r$. It follows that the process terminates after at most $O(r\delta^{-2})$ steps, and the result follows. For $f:\F \rightarrow [0,1]$, the function $\Pi_R^{\Psi} f$ is, by definition, constant on the atoms $A_{R;t,u,v}$. In particular it factors through $\G^d$ so we have $\Pi_R^{\Psi} f(x) = F_0 \circ \Psi(x)$, for a certain function $F_0 : \G^d \rightarrow [0,1]$. We shall need control of the trig-norm of $F_0$ in applications which, as things stand, may be infinite. Since the characters of $\G^d$ form a (Schauder) basis for the functions on $\G^d$, the characters composed with $\Psi$ span all functions $\F \rightarrow [0,1]$. This space of functions is finite dimensional and so the $F_0$s above may be taken to have finite trig-norm. Making this bound on the trig-norm uniform in the function requires an additional argument, and we need a bound uniform in the function *and* $p$. This is a routine, if technical, endeavour. \[lem5.4\] There are monotonic functions $M_0:(0,1]\times \Z_{{\geqslant}0} \times \R_{>0}\rightarrow \R_{{\geqslant}1}$, and $p_3:(0,1] \times \R_{>0}\times \Z_{{\geqslant}0} \rightarrow \N$ such that if $p {\geqslant}p_3(\eps,d,R)$ then the following holds. For all $f : \F \rightarrow [0,1]$, $d$-dimensional ${\operatorname{QM}}$-systems $\Psi$, and parameters $R > 0$ a power of $2$, and $\eps \in (0,1]$ there is a function $F : \G^d \rightarrow \R_{{\geqslant}0}$ such that 1. $F \circ \Psi {\geqslant}\Pi_R^{\Psi}f$ pointwise; 2. $\Vert F \Vert_{{\operatorname{trig}}} {\leqslant}M_0(\eps, d, R)$; 3. $\Vert F \circ \Psi - \Pi_R^{\Psi}f \Vert_2 {\leqslant}\eps/2$, and 4. $\Vert F\circ \Psi\Vert_4 {\leqslant}2$. Set $$\label{eta-def} \eta := \left(\frac{\eps}{100 (1+2R^{3d})^4R d}\right)^4.$$ For each $t,u,v \in \{0,1,\dots, R-1\}^d$ we define an “$\eta$-enlargement” and an “$\eta$-reduction” of $I_{R;t,u,v}$ by $$\begin{aligned} & I_{R;t,u,v}^\pm := \big\{ (\theta, \phi, z) \in \G^d : \big\|\theta_j -\frac{2t_j+1}{2R} - \sqrt{2}\big\|_{\R/\Z} < \frac{1}{2R} \pm \eta, \big\|\phi_j -\frac{2u_j+1}{2R} - \sqrt{2}\big\|_{\R/\Z} < \frac{1}{2R} \pm \eta, \\ & \qquad \big\| \frac{1}{2\pi i}\log z_j -\frac{2v_j+1}{2R} - \sqrt{2}\big\|_{\R/\Z} < \frac{1}{2R} \pm \eta \big\} \quad \mbox{for} \quad j \in \{1,\dots,d\} \big\}.\end{aligned}$$ We then define the “boundary” to be $$E:=\bigcup\{I_{R;t,u,v}^+ \setminus I_{R;t,u,v}^{-}: t,u,v \in \{0,\dots,R-1\}^d\},$$ and make two claims: **Claim A:** If $p {\geqslant}C\eta^{-4}$ then $$|\Psi^{-1}(E)| {\leqslant}\eps^2 p/10(1+2R^{3d})^4;$$ **Claim B:** If $z \in I_{R;t,u,v}^-\cap I_{R;t',u',v'}^+$ then $(t',u',v')=(t,u,v)$. We shall establish these claims later, but give the rest of the proof first. Since the functions on $\G^d$ with bounded trig norm are dense in $C(\G^d)$ we see that there are functions $F_{R;t,u,v} : \G^d \rightarrow \R$ and a function $M_*$ with the following properties: 1. $0 {\leqslant}F_{R;t,u,v} {\leqslant}1+\eps/10R^{3d}$ pointwise; 2. $F_{R;t,u,v}(z) {\geqslant}1$ for all $z \in I_{R;t,u,v}$; 3. $|F_{R;t,u,v}(z)| {\leqslant}\eps/10R^{3d}$ for all $z \not \in I_{R;t,u,v}^+$; 4. $\Vert F_{R;t,u,v} \Vert_{{\operatorname{trig}}} {\leqslant}M_*(\eps, d, R)$. The function $M_*$ need not be monotonic but this can be quickly fixed. For reasons that will become clear later we shall in fact put $$M_0(\eps,d,R) := \max\{1,R^{3d}\max\{M_*(2^{-l}, d_*, R_*): l,d_*,R_* \in \N, 2^{1-l} {\geqslant}\eps, d_* {\leqslant}d, R_* {\leqslant}R\}\},$$ which *is* monotonic. The function $\Pi_R^{\Psi} f$ is constant on atoms $A_{R;t,u,v}$. If $A_{R;t,u,v}$ is non-empty then write $\lambda_{R;t,u,v}$ for the value of $\Pi_R^{\Psi} f$ on this set and if $A_{R;t,u,v}=\emptyset$ we set $\lambda_{R;t,u,v}=0$, so that the $\lambda_{R;t,u,v}$s are all non-negative. Then we define $$F := \sum_{t,u,v \in \{0,1,\dots,R-1\}^d } \lambda_{R;t,u,v} F_{R;t,u,v},$$ and $$F_0 := \sum_{t,u,v \in \{0,1,\dots, R-1\}^d} \lambda_{R;t,u,v} 1_{I_{R;t,u,v}},$$ It follows that $$\Pi_R^{\Psi} f = F_0 \circ \Psi,$$ and since the $F_{R;t,u,v}$s and the $\lambda_{R;t,u,v}$s are non-negative, by (2) above we have $$F = \sum_{t,u,v \in \{0,1,\dots,R-1\}^d } \lambda_{R;t,u,v} F_{R;t,u,v} {\geqslant}\sum_{t,u,v \in \{0,1,\dots, R-1\}^d} \lambda_{R;t,u,v} 1_{I_{R;t,u,v}} = F_0.$$ We now show that $F$ satisfies the relevant properties. 1. This follows immediately since $F {\geqslant}F_0$ pointwise and so $F\circ \Psi(x) {\geqslant}F_0\circ \Psi(x) =\Pi_R^{\Psi}f(x)$. 2. Since $\Vert \cdot \Vert_{{\operatorname{trig}}}$ satisfies (\[subadd\]) and (\[scale\]), and $|\lambda_{R;t,u,v}| {\leqslant}1$, (4) tells us that $$\|F\|_{{\operatorname{trig}}} {\leqslant}\sum_{t,u,v}{\max\{1,|\lambda_{R;t,u,v}|\}\|F_{R;t,u,v}\|_{{\operatorname{trig}}}} {\leqslant}R^{3d}M_*(\eps,d,R) {\leqslant}M_0(\eps,d,R)$$ as required. 3. Suppose that $z \not \in E$. Then, since $\{ I_{R;t,u,v}^+: t,u,v\in \{0,\dots, R-1\}^d\}$ covers $\G^d$ we see that $z \in I_{R;t,u,v}^-$ for some $t,u,v \in \{0,\dots,R-1\}^d$, and so $$F_0(z)=\lambda_{R;t,u,v} 1_{I_{R;t,u,v}}(z).$$ By Claim B we have that $z \not \in I_{R;t',u',v'}^+$ for any $(t',u',v')\neq (t,u,v)$. Hence by (3) we have $|F_{R;t',u',v'}(z)| {\leqslant}\eps/10R^{3d}$ whenever $(t',u',v') \neq (t,u,v)$, and so $$\begin{aligned} |F(z) - F_0(z)| & {\leqslant}|\lambda_{R;t,u,v}(F_{R;t,u,v}(z)- 1_{I_{R;t,u,v}}(z))|\\ & \qquad + \big|\sum_{(t',u',v') \neq (t,u,v) } \lambda_{R;t',u',v'} F_{R;t',u',v'}(z)\big| {\leqslant}\textstyle\frac{1}{10}\displaystyle \eps.\end{aligned}$$ On the other hand, if $z \in E$ then we just have the trivial bound $$|F(z)-F_0(z)| {\leqslant}\big|\sum_{(t,u,v) } \lambda_{R;t,u,v} F_{R;t,u,v}\big| +|F_0(z)| {\leqslant}2R^{3d}+1.$$ It follows that $$\|F\circ \Psi - f\|_{2}^2 = \|F\circ \Psi - F_0 \circ \Psi\|_{2}^2 {\leqslant}\E_{x \in \F}{\left((1+2R^{3d})^21_E(\Psi(x)) + (\eps/10)^2\right)},$$ and we have (iii) by Claim A. 4. Finally, using the above we have $$\begin{aligned} \|F\circ \Psi \|_{4} & {\leqslant}\|F_0\circ \Psi\|_4 + \|F\circ \Psi-F_0\circ \Psi\|_4\\ & {\leqslant}1 + \left( \E_{x \in \F}{(1+2R^{3d})^41_E(\Psi(x))} + (\eps/10)^4\right)^{1/4} {\leqslant}2,\end{aligned}$$ from which (iv) follows. A suitable choice of $p_3$ can be made given the definition of $\eta$ and the hypothesis of Claim A. We now turn to establishing the claims. If $x \in \Psi^{-1}(E)$ then there are some $t,u,v \in \{0,\dots,R-1\}^d$ such that $$\Psi(x) \in I_{R;t,u,v}^+ \setminus I_{R;t,u,v}^-,$$ and so it is enough to establish the following statements for any $a \in \F^*$, non-trivial $\psi \in \widehat{\F^*}$ and for any interval $J \subset \R/\Z$ of the form $J = \frac{j}{R} + \sqrt{2} + (-\eta, \eta)+\Z$, $j \in \{0,1\dots, R-1\}$, we have $$\label{edge-bound-1} \# \{ x \in \F : ax^2/p \in J \} {\leqslant}\frac{\eps^2 p}{100Rd(1+2R^{3d})^4},$$ $$\label{edge-bound-2} \# \{ x \in \F : 2ax/p \in J \} {\leqslant}\frac{\eps^2 p}{100Rd(1+2R^{3d})^4},$$ and $$\label{edge-bound-3} \# \{ x \in \F : \frac{1}{2\pi i}\log \psi(x) \in J \} {\leqslant}\frac{\eps^2 p}{100Rd(1+2R^{3d})^4}.$$ The claim then follows from allowing $j,a,\psi$ to range over all choices from the sets $\{0,\dots, R-1\}$, $\{a_1,\dots, a_d\}$ and $\{\psi_1,\dots, \psi_d\}$ respectively. Of course we must still establish , and . *Proof of .* This is straightforward, as $2ax/p$ takes all the values $r/p$, $r \in \{ 0,1,\dots, p-1\}$, precisely once and so we have $$\# \{x \in \F : 2ax/p \in J\} = p|J| + O(1).$$ follows immediately provided $p {\geqslant}C\eta^{-1}$. *Proof of .* This follows in more-or-less the same way as , using instead the fact that $ax^2/p$ takes each value $r/p$, $r \in \{0,1,\dots, p-1\}$, at most twice. *Proof of .* The image of $\F$ under $\frac{1}{2\pi i}\log \psi$ is $\{0, \frac{1}{Q},\dots, \frac{Q-1}{Q}\}$ for some $Q$, and each point is hit the same number $\frac{p-1}{Q}$ of times as $x$ ranges over $\F^*$. The number of the points $\{0,\frac{1}{Q},\dots, \frac{Q-1}{Q}\}$ lying in $J$ is at most $1 + 2\eta Q$, and so $$\label{first} \# \{x \in \F: \frac{1}{2\pi i} \log \psi(x) \in J\} {\leqslant}\frac{p-1}{Q}(1 + 2\eta Q) + 1.$$ Without a lower bound on $Q$, this is useless. To obtain a lower bound on $Q$, we assume that there exists at least one $x \in \F^*$ for which $\frac{1}{2\pi i}\log \psi(x) \in J$. (Otherwise really is trivial, provided $p {\geqslant}C\eta^{-1}$.) Suppose that for this point we have $\frac{1}{2\pi i}\log \psi(x) = \frac{q}{Q}$. Then we have $\Vert\frac{q}{Q} - \frac{j}{R} - \sqrt{2}\Vert {\leqslant}\eta$, and so $\Vert QR \sqrt{2} \Vert_{\R/\Z} {\leqslant}\eta QR$. On the other hand we have the well-known and elementary bound $\Vert m \sqrt{2} \Vert_{\R/\Z} {\geqslant}\frac{1}{3m}$ for all positive integers $m$, and thus $Q {\geqslant}\frac{1}{3R\sqrt{\eta}}$. Substituting back into gives $$\# \{x : \frac{1}{2\pi i} \log \psi(x) \in J\} {\leqslant}2\eta p +3Rp\sqrt{\eta} + 1,$$ and now follows because of the choice of $\eta$ in provided $p {\geqslant}C\eta^{-1}$. *Proof of Claim B:* We apply the triangle inequality for each $j \in \{1,\dots,d\}$ to get $$\big\|\frac{t_j-t_j'}{R}\big\|_{\R/\Z} {\leqslant}\big\|\theta_j -\frac{2t_j+1}{2R} - \sqrt{2}\big\|_{\R/\Z}+\big\|\theta_j -\frac{2t_j'+1}{2R} - \sqrt{2}\big\|_{\R/\Z}<\frac{1}{2R}+\eta + \frac{1}{2R}-\eta = \frac{1}{R}.$$ Thus $$\min\big\{ \frac{|t_j-t_j'|}{R}, \frac{|R+t_j-t_j'|}{R}, \frac{|t_j-t_j'-R|}{R}\big\} <\frac{1}{R}.$$ Since $-(R-1) {\leqslant}t_j-t_j' {\leqslant}R-1$ it follows that $t_j=t_j'$ for all $j \in \{1,\dots,d\}$ and similarly for $u$ and $u'$, and $v$ and $v'$. The claim is proved, and this concludes the proof of Lemma \[lem5.4\]. We are now ready for the proof of the regularity lemma itself. First we have to define $M_\Omega$ and $D_\Omega$. Suppose that $r,d \in \Z_{{\geqslant}0}$, and $\delta \in (0,1]$ and define sequences $$\textstyle\frac{1}{10}\displaystyle\delta = \delta_0 > \delta_1 > \dots \text{ and } d = d_0^{\max} < d_1^{\max} < \dots$$ and auxiliary sequences $$R_0 < R_1 < \dots, \qquad M_0^{\max}< M_1^{\max} < \dots, \qquad \text{ and } \; \; p_0^{\max} < p_1^{\max} < \dots$$ defined by $$R_{j}:=2^{\lceil \log_2C\delta_{j}^{-1}\rceil}, M^{\max}_{j} := M_0(\eps,d_j^{\max},R_j)\text{ and }p_j^{\max}:=p_3(\eps,d_j^{\max},R_j),$$ (where $M_0$ and $p_3$ are the functions appearing in Lemma \[lem5.4\]) and the following recursive rules: $$\delta_{j+1} := \min\{1/\Omega\big(r,d,\delta,M^{\max}_j, d^{\max}_j \big),\delta_0\} \text{ and }d^{\max}_{j+1} := d^{\max}_j + \lceil Cr \delta_{j+1}^{-2}\rceil.$$ By induction on $j$ and monotonicity of $\Omega$ and $M_0$ we can inductively check that the functions $\delta_j$, $R_j$, $d_j^{\max}$, $M_j^{\max}$, $p_j^{max}$ are monotonic functions of $r,d,\delta,\eps,j$ in the sense introduced at the end of Section \[sec3\]. Set $$\label{J-def} J := \lceil Cr/\eps^2 \rceil$$ and define $$M_\Omega(r,d,\delta,\eps):=M_J^{\max}, D_\Omega(r,d,\delta,\eps):=d_J^{\max} \text{ and } p_\Omega(r,d,\delta,\eps):=p_J^{\max}.$$ These are also monotone functions. With these definitions in hand we are ready for the proof. Suppose that $p {\geqslant}p_\Omega(r,d,\delta,\eps)$. Suppose, further, that $\Psi$ is a $d$-dimensional ${\operatorname{QM}}$-system of width $\delta$, that $B:=B(\Psi,\delta)$, that $\tilde B \subset B$ has $|\tilde B| {\geqslant}(1-\frac{\eps^2}{100})|B|$, and finally that $c:\tilde{B} \rightarrow [r]$ is an $r$-colouring. We extend $c : \tilde B \rightarrow [r]$ to a full colouring $\overline{c} : B \rightarrow [r]$ in some arbitrary way such that $$\label{eqn.c} \Vert 1_{c^{-1}(i)} - 1_{\overline{c}^{-1}(i)} \Vert_2 {\leqslant}\frac{\eps}{10} \text{ for all }i \in [r].$$ (This could be done, for example, by defining $\overline{c} \equiv 1$ identically on $B \setminus \tilde B$.) We shall apply the Koopman von Neumann lemma (Lemma \[kvn\]) $J$ times (where $J$ is given by ), on each occasion with functions $f_i := 1_{\overline{c}^{-1}(i)}$, for $i \in [r]$, extended to functions on all of $\F$ by setting $f_i(x) = 0$ if $x \notin B$. On the $j$th such application we apply Lemma \[kvn\] with parameter $R_j$, obtaining nested ${\operatorname{QM}}$-systems $\Psi = :\Psi_0 \subset \Psi_1 \subset \Psi_2 \subset \dots$ with $$\label{dim-growth} \dim \Psi_{j} {\leqslant}d^{\max}_{j}$$ such that $$\label{eq686} \Vert f_i - \Pi_{R_j}^{\Psi_j} f_i \Vert_{{\operatorname{QM}}} {\leqslant}\delta_j$$ for all $i \in \{ 1,\dots, r\}$. It is convenient to write $P_j := \Pi^{\Psi_j}_{R_j}$ for short; it is also convenient to write $\mathcal{B}_j$ for the $\sigma$-algebra on $\F$ generated by $\{\Psi_j^{-1}(I_{R_j; t, u, v}):t,u,v \in \{0,\dots,R-1\}^d\}$. Thus $P_j = \E( \cdot | \mathcal{B}_j)$. Note that, since $\Psi_j \subset \Psi_{j+1}$ and $R_j | R_{j+1}$ (since $R_j$ and $R_{j+1}$ are powers of $2$), $\mathcal{B}_{j+1}$ is a refinement of $\mathcal{B}_j$ and hence $P_j = P_j P_{j+1}$. By Pythagoras’ theorem we have $$\Vert P_{j+1} f_i\Vert_2^2 - \Vert P_j f_i \Vert_2^2 = \Vert P_{j+1} f_i - P_j f_i \Vert_2^2,$$ thus $$\sum_{j= 1}^J\sum_{i = 1}^r \Vert P_{j+1} f_i - P_j f_i \Vert_2^2 {\leqslant}r,$$ so by the pigeonhole principle and the choice of $J$ there is some $j {\leqslant}J$ such that $$\label{eq477} \sum_{i = 1}^r \Vert P_{j+1} f_i - P_j f_i \Vert_2^2 {\leqslant}\frac{\eps^2}{100}.$$ Fix this value of $j$, and set $d_j := \dim \Psi_j$ (thus $d_j {\leqslant}d^{\max}_j$). Let $l \in \N$ be minimal such that $2^{-l} {\leqslant}\eps$, and apply Lemma \[lem5.4\] to $f_i$ for each $i \in \{1,\dots,r\}$. Since $$p {\geqslant}p_J^{\max} {\geqslant}p_j^{\max}= p_3(\eps,d_j^{\max},R_j) {\geqslant}p_3(\eps,d_j,R_j),$$ we get functions $F^j_i : \G^{d_j} \rightarrow [0,1]$ such that 1. $F^j_i \circ \Psi_j {\geqslant}P_j f_i$ pointwise; 2. $\Vert F^j_i \Vert_{{\operatorname{trig}}} {\leqslant}M_0(\eps, d_j, R_j) {\leqslant}M_j^{\max}$; 3. $\Vert F^j_i \circ \Psi_j - P_j f_i \Vert_2 {\leqslant}\eps/2$; 4. $\Vert F^j_i \circ \Psi_j \Vert_4 {\leqslant}2$. We are now ready to describe the functions $F_1,\dots, F_r, g_1,\dots, g_r$ in the regularity lemma. Set $d' := d_j$, $\Psi' := \Psi_j$, $F_i:=F_i^j$ for all $i \in [r]$, and $$g_i := (P_{j+1} f_i -f_i)+ 1_{c^{-1}(i)} = P_{j+1} f_i + 1_{c^{-1}(i)} - 1_{\overline{c}^{-1}(i)}$$ for all $i \in [r]$. Note with this choice that the $g_i$s all map into $[-1,1]$ as required. We claim that these functions $F_i$ and $g_i$ do indeed verify (1) to (6) of the regularity lemma; we check these points in turn. *Point (1).* This is immediate from (ii) above since $$\Vert F_i\|_{{\operatorname{trig}}}=\Vert F_i^j \Vert_{{\operatorname{trig}}} {\leqslant}M_j^{\max} {\leqslant}M_J^{\max} =M_\Omega(r,d,\delta,\eps).$$ *Point (2).* This is immediate from since that tells us $$d' {\leqslant}d_j^{\max}{\leqslant}d_J^{\max} = D_\Omega(r,d,\delta,\eps).$$ *Point (3).* We know from point (iii) above, , and above that $$\begin{aligned} \Vert F_i\circ \Psi' - g_i\|_2 & = \|F_i^j\circ \Psi_j - g_i\|_2\\ & {\leqslant}\|F_i^j\circ \Psi_j -P_jf_i\|_2\\ & \qquad + \|P_jf_i - P_{j+1}f_i\|_2 + \|1_{c^{-1}(i)} - 1_{\overline{c}^{-1}(i)}\|_2\\ & {\leqslant}\frac{\eps}{2} + \sqrt{\frac{\eps^2}{100}} + \frac{\eps}{10}<\eps.\end{aligned}$$ *Point (4).* We have $$\Vert 1_{c^{-1}(i)} - g_i \Vert_{{\operatorname{QM}}} = \Vert f_i - P_{j+1} f_i \Vert_{{\operatorname{QM}}} {\leqslant}\delta_{j+1}.$$ By the choice of $\delta_{j+1}$, this is at most $1/\Omega(r,d,\delta,M_j^{\max}, d_j^{\max})$. Since $\|F_i\|_{{\operatorname{trig}}} {\leqslant}M_j^{\max}$ and $d' {\leqslant}d_j^{\max}$ and $\Omega$ is monotonic, it follows that this is at most $1/\Omega(r,d,\delta,\|F_i\|_{{\operatorname{trig}}},d')$ and (4) follows. *Point (5).* By design we have $\sum_{i=1}^r F_i \circ \Psi'(x) = \sum_{i = 1}^r F_i^j \circ \Psi_j(x)$. On the other hand, by (i) we have $F_i^j \circ \Psi_j {\geqslant}P_j f_i$ pointwise, and so summing over $i$ we have $$\begin{aligned} \sum_{i=1}^r F_i \circ \Psi'(x) & = \sum_{i = 1}^r F_i^j \circ \Psi_j(x)\\ & {\geqslant}\sum_{i = 1}^r P_jf_i(x)\\ & = P_j(f_1 + \dots + f_r)(x)\\ & = P_j(1_{\overline{c}^{-1}(1)} + \dots + 1_{\overline{c}^{-1}(r)})(x) = P_j(1_{B})(x)\end{aligned}$$ since $P_j$ is linear. However, since $R_j {\geqslant}2\delta^{-1}$ every atom of $\mathcal{B}_j$ which meets $B(\Psi,\delta/2)$ is entirely contained in $B$, and so $P_j 1_B {\geqslant}1$ pointwise on $B(\Psi,\frac{1}{2}\delta)$. *Point (6).* This follows immediately from (iv). The result is proved. Some results in Ramsey theory {#sec7} ============================= In this section we state and prove some auxiliary results of a Ramsey-theoretic nature, the main aim being to establish Proposition \[ramsey-prop\]. A model for the type of result we are interested in is the following: if $\N \times \N$ is finitely coloured, then there is a monochromatic triple of distinct elements $(t_1, u)$, $(t_2, u)$, $(t_3,t_2 - t_1)$. Such a result is certainly true, and can be easily proved as follows. Given an $r$-colouring of $[N]^2$, consider the colouring induced on $\{(1,1),\dots,(N,1)\}$ which is certainly at most an $r$-colouring. By van der Waerden’s theorem [@vdW] there is a monochromatic arithmetic progression $$\Delta:=\{(x,1),(x+d,1),\dots,(x+Md,1)\}$$ where $M \rightarrow \infty$ as $N \rightarrow \infty$ (for fixed $r$). If there is some $t_3$ such that $(t_3,jd)$ has the same colour as $\Delta$ for some $1 {\leqslant}j {\leqslant}M$ then we have a suitable monochromatic triple, namely $(x,1),(x+jd,1),(t_3,jd)$. Otherwise the box $[N] \times d[M]$ is $(r-1)$-coloured, and hence so is $(d,d)\cdot [M]^2$. This forms the basis for an induction on the number of colours, thereby giving the result. Of course this result is just the tip of the iceberg, and there is a vast generalisation available in recent work of Bergelson, Johnson, and Moreira [@bergelson-johnson-moreira]. Indeed, [@bergelson-johnson-moreira Corollary 3.7] contains the above as a special case by taking (in the language of that result) $G:=\N_0^2$, $m:=1$, $c:G \rightarrow G$ to be the identity, and letting $F_1$ be the set containing the two maps $$\N_0^2 \rightarrow \N_0^2; (x,y) \mapsto (0,0) \text{ and } \N_0^2 \rightarrow \N_0^2; (x,y) \mapsto (y,0).$$ (Verifying that this is the same theorem requires a little work: the set $D(m, \vec{F}, c; \mathbf{s})$ in [@bergelson-johnson-moreira Definition 3.1] then consists of the elements $(s_{00}, s_{01})$, $(s_{10}, s_{11})$ and $(s_{01} + s_{10}, s_{11})$, which are of the form $(t_1, u)$, $(t_2, u)$, $(t_3,t_2 - t_1)$ with $t_1 = s_{10}$, $t_2 = s_{01} + s_{10}$, $t_3 = s_{00}$, $u = s_{11}$.) While these extensions are clearly interesting we need to generalise our model in a different direction. In particular, we need not just one monochromatic triple but many triples. A result of this type extending Rado’s theorem was established in [@franklgrahamrodl Theorem 1], and extended to the case of torsion groups in [@serra-vena Theorem 1.3]. In particular it is worth noting that [@serra-vena Section 7.1] identifies some difficulties that emerge in the presence of torsion. We turn, now, to our arguments. Let us recall the statement we are aiming to prove. There is a monotonic function $\rho : \Z_{{\geqslant}0} \times \Z_{{\geqslant}0} \times (0,1] \rightarrow (0,1]$ with the following property. Suppose that $X$ and $Y$ are compact Abelian groups with Haar probability measures $\mu_X, \mu_Y$, that $\pi_X : X \rightarrow (\R/\Z)^{d}$ and $\pi_Y : Y \rightarrow (\R/\Z)^{d}$ are continuous homomorphisms, and that $F_1,\dots, F_r : X \times X \times Y \rightarrow \R_{{\geqslant}0}$ are continuous functions with $\sum_{i=1}^r F_i(x_1, x_2, y) {\geqslant}1$ whenever we have $| \pi_X(x_1)|,|\pi_X(x_2)|,|\pi_Y(y)| {\leqslant}\frac{1}{4}\delta$. Let $\mu = \mu_X \times \mu_X \times \mu_Y$. Then $${\int F_i(t,u,v) F_i(t + u', u, v') F_i(t', u',v)} d\mu(t,u,v) d\mu(t',u',v') {\geqslant}\rho(r,d,\delta)$$ for some $i \in [r]$. We shall prove this via the following result. \[prop.mainramseydriver\] Suppose that $G$ is a compact Abelian group, that $T \subset G$ is open, and that $A_1,\dots,A_r$ are the measurable colour classes of some $r$-colouring on $T \times (T-T)$. Then $$\sum_{i=1}^r \int{1_{A_i}(t_1,t_4-t_5)1_{A_i}(t_2,t_4-t_5)} 1_{A_i}(t_3,t_2-t_1)d\mu_T^{\otimes 5}(t_1,t_2,t_3,t_4,t_5) {\geqslant}r^{-O(r)}.$$ Here, $\mu_T := \mu_G(T)^{-1} \mu_G$, where $\mu_G$ is the normalised Haar measure on $G$ which makes sense since $T$ is open and so is measurable and of positive measure. We have written the bound here explicitly, first to show that it may be taken to be monotonically decreasing in $r$, and secondly because it is not too far from the right order. Indeed, suppose $G=\F_2^r$ and we have a $(2r+1)$-colouring of the pairs $(t,u) \in G\times (G-G)$ with colour classes defined by $$A_0 := \F_2^r \times \{0\} \text{ and }A_i = \{(t,u) : u_1 = \dots = u_{i-1} = 0, u_i = 1, t_i = 0\},$$ and $$A_{r + i} = \{(t,u) : u_1 = \dots = u_{i-1} = 0, u_i = 1, t_i = 1\},$$ where in both cases $i$ ranges over $\{1,\dots,r\}$. There are no triples $(t,u),(t',u),(t'',t+t')$ all lying in $A_i$ for any $i > 0$ since this would imply that $t_i=t_i'$ and so $(t+t')_i=0$, a contradiction. Hence all the monochromatic triples lie in $A_0$, which implies that their total measure is at most $4^{-r}$. Before turning to the proof of Proposition \[prop.mainramseydriver\], let us see how it implies Proposition \[ramsey-prop\]. Put $f_i:=\min\{F_i(x),1\}$ so that the $f_i$s take values in $[0,1]$, are continuous and have $\sum_i{f_i(x_1,x_2,y)} {\geqslant}1$ whenever $| \pi_X(x_1)|,|\pi_X(x_2)|,|\pi_Y(y)| {\leqslant}\delta/4$. We start by simplifying the dependence on $v$ and $v'$ by noting that since the $f_i$s all take values in $[0,1]$ we have, for each $i \in [r]$, $$\begin{aligned} \nonumber & \int f_i(t,u,v) f_i(t + u', u, v') f_i(t', u',v) d\mu(t,u,v) d\mu(t',u',v')\\ \nonumber & {\geqslant}\int{ f_i(t,u,v) f_i(t + u', u, v') f_i(t', u',v)}\\ \nonumber & \quad \times f_i(t,u,v') f_i(t + u', u, v) f_i(t', u',v') d\mu(t,u,v) d\mu(t',u',v')\\ \nonumber & = \int{\left(\int{f_i(t,u,v)f_i(t+u',u,v)f_i(t',u',v)d\mu_Y(v)}\right)^2d\mu_{X}^{\otimes 4}(t,u,t',u')}\\ \label{eqn.cssimp} & {\geqslant}\left(\int{f_i(t,u,v)f_i(t+u',u,v)f_i(t',u',v)d\mu_{X}^{\otimes 4}(t,u,t',u')d\mu_Y(v)}\right)^2,\end{aligned}$$ the last step here being a consequence of the Cauchy-Schwarz inequality. Let $B \subset X \times X$ be the set of pairs $(t,u)$ such that $|\pi_X(t)|, |\pi_X(u)| {\leqslant}\frac{1}{4}\delta$. For each $v \in Y$, define $$A_i^{(v)}:=\big\{(t,u)\in B: f_i(t,u,v) {\geqslant}1/r \text{ and } (t,u) \not \in \bigcup_{j=1}^{i-1}{A_j^{(v)}}\big\}.$$ By hypothesis we have $$\sum_i{f_i(t,u,v)} {\geqslant}1 \text{ whenever } | \pi_X(t)|,|\pi_X(u)|,|\pi_Y(v)| {\leqslant}\textstyle\frac{1}{4}\displaystyle\delta$$ and so by averaging $$\bigcup_{i=1}^r{A_i^{(v)}} = B \text{ whenever } |\pi_Y(v)| {\leqslant}\textstyle\frac{1}{4}\displaystyle\delta.$$ Moreover since the $f_i$s and $\pi_X$ are continuous the sets $A_i^{(v)}$ are measurable. We claim (and shall prove later) that for any choice of measurable sets $(A_i)_{i=1}^r$ with $\bigcup_i{A_i} = B$ we have $$\label{77claim} \sum_{i=1}^r{\int{1_{A_i}(t,u)1_{A_i}(t+u',u)1_{A_i}(t',u')d\mu_{X}^{\otimes 4}(t,u,t',u')}} {\geqslant}r^{-O(r)}(\delta/8)^{5d}.$$ Assuming this for now, by and the design of the sets $A_i^{(v)}$ we have $$\begin{aligned} \sup_i&\int f_i(t,u,v) f_i(t + u', u, v') f_i(t', u',v) d\mu(t,u,v) d\mu(t',u',v') \\ & {\geqslant}\bigg(\sup_i \int{f_i(t,u,v)f_i(t+u',u,v)f_i(t',u',v)}d\mu_{X}^{\otimes 4}(t,u,t',u')d\mu_Y(v)\bigg)^2\\ & {\geqslant}\bigg(\frac{1}{r^4}\int {\left(\sum_{i=1}^r{\int{1_{A_i^{(v)}}(t,u)1_{A_i^{(v)}}(t+u',u)} 1_{A_i^{(v)}}(t',u')}d\mu_{X}^{\otimes 4}(t,u,t',u')\right)d\mu_Y(v)}\bigg)^2.\end{aligned}$$ By and the comments before it, the inner integral is bounded below by $r^{-O(r)}(\delta/8)^{5d}$ uniformly for $v$ with $|\pi_Y(v)| {\leqslant}\frac{1}{4}\delta$. By Lemma \[pigeon-projection\] we have $$\int_{|\pi_Y(v)| {\leqslant}\delta/4} d\mu_Y(v) {\geqslant}(\delta/4)^d.$$ Putting these facts together we get that $$\begin{aligned} \sup_i&\int F_i(t,u,v) F_i(t + u', u, v') F_i(t', u',v) d\mu(t,u,v) d\mu(t',u',v')\\ & {\geqslant}\sup_i\int f_i(t,u,v) f_i(t + u', u, v') f_i(t', u',v) d\mu(t,u,v) d\mu(t',u',v') \\ & {\geqslant}\bigg(\sup_i \int{f_i(t,u,v)f_i(t+u',u,v)f_i(t',u',v)}d\mu_{X}^{\otimes 4}(t,u,t',u')d\mu_Y(v)\bigg)^2\\ & {\geqslant}\bigg(\frac{1}{r^4}\cdot r^{-O(r)}(\delta/8)^{5d}\cdot (\delta/4)^d\bigg)^2.\end{aligned}$$ This tells us that we may define $\rho$ monotonically in the appropriate way, and concludes the proof of Proposition \[ramsey-prop\] assuming Proposition \[prop.mainramseydriver\], except for the need to establish the claim . We turn now to the proof of this claim. First of all we introduce a dummy integration over $X$, so the claim to be established becomes $$\sum_{i = 1}^r \int 1_{A_i}(t,u) 1_{A_i}(t + u', u) 1_{A_i}(t', u') d\mu_X^{\otimes 5}(t,u,t,u',v) {\geqslant}r^{-O(r)}(\delta/8)^{5d}.$$ Now make the change of variables $t = t_1$, $u = t_4 - t_5$, $t' = t_3$, $u' = t_2 - t_1$, $v = t_5$. This change of variables is invertible (by $t_1 = t$, $t_2 = t + u'$, $t_3 = t'$, $t_4 = u + v$, $t_5 = v$) and preserves the Haar measure, so it is enough to show that $$\sum_{i = 1}^r \int 1_{A_i}(t_1, t_4 - t_5) 1_{A_i}(t_2, t_4 - t_5) 1_{A_i}(t_3, t_2 - t_1) d\mu_X^{\otimes 5}(t_1, t_2, t_3, t_4, t_5){\geqslant}r^{-O(r)}(\delta/8)^{5d}.$$ Now we are told that $\bigcup_i A_i = B = \{(x,x') : |\pi_X(x)|, |\pi_X(x')| {\leqslant}\frac{1}{4}\delta\}$. It follows that the $A_i$ restrict to give an $r$-colouring of $T \times (T - T)$, where $T := \{x \in X : |\pi_X(x)| {\leqslant}\frac{1}{8}\delta\}$. Since $\mu_X(T) {\geqslant}(\delta/8)^d$ by Lemma \[pigeon-projection\], the claim now follows from Proposition \[prop.mainramseydriver\]. The remaining task of the section, then, is to establish Proposition \[prop.mainramseydriver\]. A key ingredient of this is a “dependent random selection” result of the type pioneered by Gowers [@gowers-4ap]. It is given in almost this specific form as [@taovu Lemma 6.17] the proof of which is itself contained in [@sudszevu Lemma 4.2]. We need a weighted version of their result so we provide a self-contained proof, though the argument is more-or-less identical. \[lem.gowersweakregularity\] Suppose that $(X,\nu_X)$ and $(Y,\nu_Y)$ are probability spaces, $A \subset X \times Y$ is measurable with $(\nu_X \times \nu_Y)(A) = \alpha$, and let $\eta \in (0,1]$ be a parameter. Then there is a measurable set $X' \subset X$ with $\nu_X(X') {\geqslant}\frac{1}{2}\alpha$ such that the set $$E:=\big\{(x_1,x_2) \in X \times X: \nu_Y\left(\left\{ y \in Y : (x_1, y), (x_2, y) \in A\right\}\right) {\leqslant}\textstyle\frac{1}{2}\displaystyle\eta \alpha^2\big\}$$ which is visibly measurable has $\nu_X^{\otimes 2}(E \cap (X' \times X')) {\leqslant}\eta \nu_X(X')^2$. In words, a (weighted) proportion $1 - \eta$ of the “edges” in $X'$ have many common neighbours in $Y$. For $x \in X$, write $N_Y(x) := \{y \in Y: (x,y) \in A\}$ and for $y \in Y$ write $N_X(y) := \{x \in X : (x,y) \in A\}$. By Fubini’s theorem we have $$\int \nu_Y(N_Y(x_1) \cap N_Y(x_2)) d\nu_{X}^{\otimes 2}(x_1,x_2) = \int \nu_X(N_X(y))^2 d\nu_Y(y),$$ with both sides being equal to $$\int 1_{A}(x_1,y)1_A(x_2,y)d\nu_X^{\otimes 2}(x_1,x_2) d\nu_Y(y).$$ By the Cauchy-Schwarz inequality and the identity $$\int \nu_X(N_X(y)) d\nu_Y(y) = \iint 1_{A}(x,y) d\nu_X(x) d\nu_Y(y) = \alpha,$$ we see that $$\label{eq421} \int \nu_Y(N_Y(x_1) \cap N_Y(x_2)) d\nu_X^{\otimes 2}(x_1, x_2) {\geqslant}\alpha^2.$$ By definition of $E$ we have $$\iint 1_E(x_1,x_2) \nu_Y(N_Y(x_1) \cap N_Y(x_2)) d\nu_X^{\otimes 2}(x_1, x_2) {\leqslant}\textstyle\frac{1}{2}\displaystyle\eta \alpha^2.$$ Putting this together with , we see that $$\int \big(1 - \frac{1}{\eta} 1_E(x_1,x_2)\big) \nu_Y(N_Y(x_1) \cap N_Y(x_2)) d\nu_X^{\otimes 2}(x_1, x_2) {\geqslant}\textstyle\frac{1}{2}\displaystyle\alpha^2,$$ or in other words (by Fubini’s theorem) $$\iint \big(1 - \frac{1}{\eta} 1_E(x_1,x_2) \big) 1_{N_X(y)}(x_1)1_{N_X(y)}(x_2) d\nu_X^{\otimes 2}(x_1, x_2) d\nu_Y(y) {\geqslant}\textstyle\frac{1}{2}\displaystyle\alpha^2.$$ In particular, there is some specific choice of $y$ for which ($N_X(y)$ is measurable and) $$\int \big(1 - \frac{1}{\eta} 1_E(x_1,x_2) \big) 1_{N_X(y)}(x_1)1_{N_X(y)}(x_2) d\nu_X^{\otimes 2}(x_1, x_2) {\geqslant}\textstyle\frac{1}{2}\displaystyle\alpha^2.$$ For this $y$ set $X' := N_X(y)$, and we have both $$\int 1_{X'}(x_1)1_{X'}(x_2) d\nu_X^{\otimes 2}(x_1, x_2) {\geqslant}\textstyle\frac{1}{2}\displaystyle\alpha^2,$$ whence $\nu_X(X') {\geqslant}\frac{1}{2}\alpha$, and $$\int 1_{X'}(x_1)1_{X'}(x_2) d\nu_X^{\otimes 2}(x_1, x_2) {\geqslant}\frac{1}{\eta} \int 1_E(x_1,x_2) 1_{X'}(x_1)1_{X'}(x_2) d\nu_X^{\otimes 2}(x_1, x_2),$$ or in other words $\nu_X^{\otimes 2} (E \cap (X' \times X')) {\leqslant}\eta \nu_X^{\otimes 2}(X')^2$. The result is proved. In what follows we shall be working in products $T \times (T - T)$, where $T \subset G$. If $A \subset G \times G$ is measurable, we define $$\delta_T(A) := \int{1_{A}(t,t_1-t_2)d\mu_T(t)d\mu_T(t_1)d\mu_T(t_2)},$$ and $$\Lambda_T(A) := \int{1_A(t_1,t_4-t_5)1_A(t_2,t_4-t_5)1_A(t_3,t_2-t_1)}d\mu_T^{\otimes 5}(t_1,t_2,t_3,t_4,t_5).$$ In this notation, Proposition \[prop.mainramseydriver\] may be restated as follows: if $c : T \times (T - T) \rightarrow [r]$ is a colouring then there is some $i \in [r]$ for which $\Lambda_T(c^{-1}(i)) {\geqslant}r^{-O(r)}$. We shall establish this by induction on the number of colours (one of two places in our paper where we do this, the other being in the proof of Proposition \[main-prop-refined\]). To carry this out we need to prove a slightly stronger statement. Set $\eps_r := 2^{-7r+1} (r!)^{-3}$. Suppose that $G$ is a compact Abelian group, $T \subset G$ is an open set, $E \subset T \times (T-T)$ is measurable and $c : T \times (T - T) \rightarrow [r]$ is a measurable partial colouring, defined outside of a set $E$ with $\delta_T(E) {\leqslant}\eps_r$. Then there is $i \in [r]$ with $\Lambda_T(c^{-1}(i)) {\geqslant}\eps_r^2$. We proceed by induction on $r$, the result being vacuously true when $r = 0$ since $\eps_0 = \frac{1}{2}$ and there are no $0$-colourings of non-empty sets. Suppose we know the result for $r-1$, and that we have a partial $r$-colouring of $T \times (T - T)$ as described. Certainly $\eps_r < \frac{1}{2}$, so by the pigeonhole principle there is some $i$ for which $\delta_T(A_i) {\geqslant}1/2r$ where $A_1,\dots,A_r$ are the colour classes of $c$. We shall apply Lemma \[lem.gowersweakregularity\] with $X = T$, $Y = T - T$, $A = c^{-1}(i)$ and with $\eta = \frac{1}{4}\eps_{r-1}$. Let $\mu_X$ be the probability measure induced on $T$ by the Haar probability measure $\mu_G$ and let the measure on $Y$ be given by $\mu_Y:=\mu_T \ast \mu_{-T}$. The lemma outputs a measurable set $T'$ (that is, $X'$ in the lemma) with $\mu_T(T') {\geqslant}1/4r$ and and a measurable set $Z \subset T' \times T'$ consisting of a proportion at least $1 - \frac{1}{4}\eps_{r-1}$ of all pairs $(t_1, t_2)$ in $T' \times T'$ such that $$\label{882} \int{1_{A_i}(t_1,t_4-t_5)1_{A_i}(t_2,t_4-t_5)d\mu_T(t_4)d\mu_T(t_5)}{\geqslant}\frac{\eta}{8r^2}$$ whenever $(t_1, t_2) \in Z$. Suppose in the first instance that $\delta_{T'}(A_i) {\geqslant}\frac{1}{2}\eps_{r-1}$. Then $$\begin{aligned} \Lambda_T(A_i) &{\geqslant}\mu_T(T')^3\int{1_{A_i}(t_3,t_2-t_1)\int{1_{A_i}(t_1,t_4-t_5)}1_{A_i}(t_2,t_4-t_5)}d\mu_T(t_4)d\mu_T(t_5)d\mu_{T'}^{\otimes 3}(t_1,t_2,t_3)\\ & {\geqslant}\frac{\eta\mu_T(T')^3}{8r^2}\int{1_{A_i}(t_3,t_2-t_1)1_Z(t_1,t_2)}d\mu_{T'}^{\otimes 3}(t_1,t_2,t_3)\\ & {\geqslant}\frac{\eta\mu_T(T')^3}{8r^2}\bigg(\int{1_{A_i}(t_3,t_2-t_1)d\mu_{T'}^{\otimes 3}(t_1,t_2,t_3)} - \textstyle\frac{1}{4}\displaystyle\eps_{r-1} \bigg)\\ & {\geqslant}\frac{\eta\mu_T(T')^3}{8r^2}\cdot \frac{\epsilon_{r-1}}{4} {\geqslant}2^{-13} r^{-5} \eps^2_{r-1} > \eps_r^2.\end{aligned}$$ The alternative is that $\delta_{T'}(A_i) {\leqslant}\frac{1}{2}\eps_{r-1}$. Noting that $$\delta_{T'}(E) {\leqslant}\mu_T(T')^{-3} \delta_T(E) {\leqslant}64 r^3 \eps_r = \textstyle\frac{1}{2}\displaystyle\eps_{r-1},$$ it follows that $c$ restricts to give a partial colouring $c' : T' \times (T' - T') \rightarrow [r] \setminus \{i\}$, defined outside of a set $E' = (E \cup c^{-1}(i)) \cap (T' \times (T' - T'))$ with $\delta_{T'}(E') {\leqslant}\eps_{r-1}$. By our inductive hypothesis, there is $j$ such that $\Lambda_{T'}(A_j) {\geqslant}\eps_{r-1}^2$, which implies that $$\Lambda_T(A_j) {\geqslant}\mu_T(T')^5 \Lambda_{T'}(A_j) {\geqslant}(4r)^{-5}\eps_{r-1}^2 {\geqslant}\eps_r^2.$$ This concludes the proof. The baby counting lemma and the counting lemma {#sec8} ============================================== The objective of this section is to prove the baby counting lemma from §\[sec3\] and the counting lemma from §\[sec4\], the first of these acting as a kind of warm up to the second. The proofs require various lemmas on exponential sums with characters, and we begin by assembling these. There is a great deal of general theory on this topic; see, for example, [@iwaniec-kowalski Chapter 11]. We develop just what we need for our application, namely Proposition \[add-mult\] below, eschewing any temptation to seek the strongest available bounds. In fact, a bound of $o(1)$ times the trivial bound is all we need. Shkredov [@shkredov] makes use of the following result from Johnsen [@johnsen] which he (Shkredov) records as [@shkredov Theorem 2.4]. \[char-theorem\] Suppose that $\K$ is a finite field, $\chi_1,\dots,\chi_t$ are $t<|\K|$ multiplicative characters on $\K$ with $\chi_i$ non-principal for some $i$, and $h_1,\dots,h_t$ are distinct elements of $\K$. Then $$\big|\sum_{x \in \K}{\chi_1(x+h_1)\dots\chi_t(x+h_t)}\big| {\leqslant}(t-1)\sqrt{|\K|}$$ We shall also use this result but could equally take *e.g.* [@schmidt Theorem 2C’, p.43] or [@iwaniec-kowalski Theorem 11.23] instead. *Remark.* Note, of course, that *our* multiplicative characters are extended to the whole of $\F$ in a slightly different way to usual since they are $1$ at $0$. This can only add an error of size $t$ in the sum in Theorem \[char-theorem\] which will be of no consequence to us. With this in hand we are in a position to prove our key proposition. \[add-mult\] Suppose that $q : \F \rightarrow \F; x \mapsto ax^2 + bx$, $\chi, \chi' : \F \rightarrow \C^*$ are multiplicative characters, and $h \in \F^*$. Then we have $$\E_x e_p(q(x)) \chi(x) \chi'(x + h)=o_{p \rightarrow \infty}(1),$$ uniformly in $a,b,\chi,\chi'$, unless $a = b = 0$ and $\chi = \chi' = 1$. Suppose first that $\chi = \chi' = 1$. Then the sum reduces to $$\E_x e_p(ax^2 + bx),$$ and it follows immediately from the standard Gauss sum estimate that this is bounded by $O(p^{-1/2})$ unless $a = 0$; if $a=0$ then the sum is $0$ by orthogonality of characters unless $b = 0$. Suppose, then, that we do not have $\chi = \chi' = 1$. Writing $G(x):=e_p(-q(x))$ and $F(x):=\chi(x) \chi'(x + h)$ we have $$\big|\E_x e_p(q(x)) \chi(x) \chi'(x + h)\big| = \big|\E_{x,z_1,z_2,z_3}{F(x)\!\!\!\!\prod_{ \omega \in \{0,1\}^3 \setminus (0,0,0)}{\mathcal{C}^{|\omega|} G(x+\omega\cdot z)}}\big| {\leqslant}\|F\|_{U^3}$$ by the Gowers-Cauchy-Schwarz inequality (see, for example, [@taovu Equation (11.6)]; here $\Vert \cdot \Vert_{U^3}$ is the Gowers $U^3$-norm on $\F$, discussed in many places including [@taovu Chapter 11], and $\mathcal{C}$ is the complex conjugation operator). Thus it suffices to establish that $\Vert F \Vert_{U^3} = o(1)$, uniformly in $\chi, \chi'$ and $h$; we shall in fact establish the stronger bound $\|F\|_{U^3}=O(p^{-1/16})$. In other words, we shall prove that $$\begin{aligned} \label{to-prove-7} &\E_{x, z_1, z_2, z_3} \prod_{\omega \in \{0,1\}^3} \mathcal{C}^{|\omega|} \chi(x + \omega\cdot z) \chi'(x + h + \omega\cdot z) =O(p^{-1/2}), \end{aligned}$$ where $\omega\cdot z:=\omega_1z_1+\omega_2z_2+\omega_3z_3$. Since $h \neq 0$, there are $O(p^2)$ triples $z \in \F^3$ such that there are some $\omega,\omega' \in \{0,1\}^3$ with $h+\omega\cdot z=\omega'\cdot z$. For all other triples we may apply Theorem \[char-theorem\] to see that $$\begin{aligned} &\E_{x} \prod_{\omega \in \{0,1\}^3} \mathcal{C}^{|\omega|} \chi(x + \omega\cdot z) \chi'(x + h + \omega\cdot z) =O(p^{-1/2}). \end{aligned}$$ Equation (\[to-prove-7\]) follows immediately, and so does the proposition. This concludes our discussion of character sum estimates. We turn now to the proofs of the counting lemma and baby counting lemma. Before proceeding it may help the reader to recall some of the definitions from §\[sec3\], particularly the definition of the trig-norm, Definition \[deftrig\]. We begin by recalling some basic facts about duality. \[duality-fact\] Suppose that $\Lambda \subset \Z^d$ is a lattice of full rank. Write $G^+ := \{g \in (\R/\Z)^d: \xi \cdot g = 0 \; \mbox{for all $\xi \in \Lambda$}\}$. Suppose that $\lambda \in \Z^d$ satisfies $\lambda \cdot g = 0$ for all $g \in G^+$. Then $\lambda \in \Lambda$. Suppose that $\lambda \notin \Lambda$. Let $e_1,\dots, e_d$ be an integral basis for $\Lambda$. Suppose that $\lambda = \lambda_1 e_1 + \dots + \lambda_d e_d$ where, without loss of generality, $\lambda_1 \notin \Z$. By linear algebra there is some $x \in \R^d$ with $e_1 \cdot x = 1$ and $e_2 \cdot x = \dots = e_d \cdot x = 0$. In particular $\xi \cdot x \in \Z$ for $\xi \in \Lambda$, but $\lambda \cdot x = \lambda_1 \notin \Z$. Taking $g = \pi(x)$, where $\pi : \R^d \rightarrow (\R/\Z)^d$ is the natural projection, it follows that $g \in G^+$ but that $\lambda \cdot g \neq 0$ in $(\R/\Z)^d$. (In fact, the full rank hypothesis is unnecessary, but it is satisfied in our applications.) An easy consequence of this is the following. \[mult-cor\] Suppose that $\Lambda \subset \Z^d$ is a lattice of full rank. Write $G^{\times} := \{z \in (S^1)^d: z^{\xi} = 1 \; \mbox{for all $\xi \in \Lambda$}\}$. Suppose that $\lambda \in \Z^d$ satisfies $z^{\lambda} = 1$ for all $z \in G^{\times}$. Then $\lambda \in \Lambda$. This follows immediately from Lemma \[duality-fact\] using the isomorphism $\pi : (\R/\Z)^d \rightarrow (S^1)^d$ defined by $\pi(\theta_1,\dots, \theta_d) = (e(\theta_1),\dots, e(\theta_d))$. Using these facts, we may establish the following key orthogonality relations. We have the orthogonality relations $$\label{orth-plus} \int e(\xi \cdot t) d\mu_{G_{\Psi}^+}(t) = 1_{\Lambda_{\Psi}^+}$$ and $$\label{orth-times} \int v^{\xi} d\mu_{G_{\Psi}^{\times}}(v) = 1_{\Lambda_{\Psi}^{\times}}.$$ As noted in §\[sec3\], $\Lambda_{\Psi}^+, \Lambda_{\Psi}^{\times}$ both have full rank. We begin with . It is clear that if $\xi \in \Lambda_{\Psi}^+$ then the integral is $1$. Conversely, suppose that the integral is nonzero. Let $g \in G_{\Psi}^+$, and make the substitution $t = t' + g$, which preserves the Haar measure. We obtain $$\int e(\xi \cdot t) d\mu_{G_{\Psi}^+}(t) = e(\xi \cdot g) \int e(\xi \cdot t') d\mu_{G_{\Psi}^+}(t'),$$ and so $e(\xi \cdot g) = 1$, which implies that $\xi \cdot g = 0$ in $(\R/\Z)^d$. Since $g$ was arbitrary, it follows from Lemma \[duality-fact\] and the definition of $G_{\Psi}^+$ that $\xi \in \Lambda_{\Psi}^+$. The proof of is extremely similar, using Corollary \[mult-cor\] in place of Lemma \[duality-fact\]. Now we turn to the main business of the section, beginning with the baby counting lemma, the statement of which was as follows. The proof is quite straightforward. Let $\Psi$ be a ${\operatorname{QM}}$-system of dimension $d$, and let $F : \G^d \rightarrow \C$ be a trigonometric polynomial. Then we have $$\E_{x \in \F} F(\Psi(x)) = \int F d\mu_{H_{\Psi}} + o_{p \rightarrow \infty}(\Vert F \Vert_{{\operatorname{trig}}}).$$ Suppose that $\Vert F \Vert_{{\operatorname{trig}}} = M$. Expand $F$ as $$\label{F-fourier} F(\theta_1, \theta_2, z) = \sum_{\xi_1,\xi_2,\xi_3} \hat{F}(\xi_1, \xi_2,\xi_3) e(\xi_1 \cdot \theta_1 + \xi_2 \cdot \theta_2) z^{\xi_3},$$ where $\sum |\hat{F}(\xi_1, \xi_2, \xi_3)| {\leqslant}M$. Suppose that we write $\Psi(x) = ((a_i x^2/p, 2a_i x/p, \psi_i(x))_{i = 1}^d)$, and set $a := (a_1,\dots, a_d)$. Then we have $$\E_{x \in \F} F(\Psi(x)) = \sum_{\xi_1,\xi_2, \xi_3}\hat{F}(\xi_1, \xi_2,\xi_3) \E_{x \in \F} e_p(\xi_1 \cdot a x^2 + \xi_2 \cdot 2a x) \psi^{\xi_3}(x),$$ where $\psi^{\xi_3}(x):=\prod_i{\psi_i^{(\xi_3)_i}(x)}$ which is a multiplicative character. By Proposition \[add-mult\] (with $\chi' = 1$) the inner average is $o_{p \rightarrow \infty}(1)$ unless $\xi_1 \cdot a=0$, $\xi_2\cdot 2a=0$, and $\psi^{\xi_3}=1$. It follows from the definitions that $\xi_1, \xi_2 \in \Lambda_{\Psi}^+$ and $\xi_3 \in \Lambda_{\Psi}^{\times}$, and in that case the inner average equals 1. Since $\sum_{\xi_1,\xi_2,\xi_3} |\hat{F}(\xi_1, \xi_2,\xi_3)| {\leqslant}M$, the contribution from those $\xi_1,\xi_2,\xi_3$ not satisfying these conditions is $o_{p \rightarrow \infty}(M)$ and hence $$\E_{x \in \F} F(\Psi(x)) = \sum_{\xi_1, \xi_2 \in \Lambda_{\Psi}^+, \xi_3 \in \Lambda_{\Psi}^{\times}} \hat{F}(\xi_1,\xi_2,\xi_3) + o_{p \rightarrow \infty}(M).$$ We claim that $$\label{8-claim} \sum_{\xi_1, \xi_2 \in \Lambda_{\Psi}^+, \xi_3 \in \Lambda_{\Psi}^{\times}} \hat{F}(\xi_1,\xi_2,\xi_3) = \int F d\mu_{H_{\Psi}},$$ which is enough to conclude the proof. To prove the claim, replace $F$ by its Fourier expansion on the right hand side, and recall that $\mu_{H_{\Psi}} = \mu_{G_{\Psi}^+} \times \mu_{G_{\Psi}^+} \times \mu_{G_{\Psi}^{\times}}$. The right hand side is then $$\int \sum_{\xi_1,\xi_2 ,\xi_3} \hat{F}(\xi_1,\xi_2,\xi_3) e(\xi_1 \cdot \theta_1)e(\xi_2 \cdot \theta_2) z^{\xi_3} d\mu_{G_{\Psi}^+}(\theta_1) d\mu_{G_{\Psi}^+}(\theta_2) d\mu_{G_{\Psi}^{\times}}(z) .$$ By the orthogonality relations and , this is precisely the left-hand side of . *Remark.* We did not make any use of the fact that $\hat{F}$ was supported where $\|\xi_1\|_1,\|\xi_2\|_1,\|\xi_3\|_1 {\leqslant}M$ in this argument, but this will be important in the next argument. We turn now to the counting lemma itself, the proof of which is related to [@shkredov Theorem 1.2]. Suppose $\Psi$ is a $d$-dimensional ${\operatorname{QM}}$-system, $F : \G^d \rightarrow \C$ is a trigonometric polynomial, and that $S \subset B(\Psi,\eps)$. Then $$\begin{aligned} & T(F \circ \Psi, 1_S, F \circ \Psi, F \circ \Psi) \\ & \qquad = \mu_{\F}(S) \int F(t,u,v) F(t + u', u, v') F(t', u', v) d\mu_{H_{\Psi}}(t,u,v) d\mu_{H_{\Psi}}(t',u',v') \\ & \qquad \qquad + O(\epsilon \mu_{\F}(S)\Vert F \Vert_{{\operatorname{trig}}}^4) + o_{p \rightarrow \infty}(\Vert F \Vert_{{\operatorname{trig}}}^{9d}).\end{aligned}$$ As before, write $M = \Vert F \Vert_{{\operatorname{trig}}}$ and expand $F$ as $$F(\theta_1, \theta_2, z) = \sum_{\|\xi_1\|_1,\|\xi_2\|_1, \|\xi_3\|_1 {\leqslant}M} \hat{F}(\xi_1, \xi_2,\xi_3) e(\xi_1 \cdot \theta_1 + \xi_2 \cdot \theta_2) z^{\xi_3}$$ with $\sum |\hat{F}(\xi_1,\xi_2,\xi_3)| {\leqslant}M$. We define the function $$\begin{aligned} E(y) :&= \E_{x \in \F} F(\Psi(x)) F(\Psi(x + y)) F(\Psi(xy))\\ & = \sum_{\substack{\xi_1,\dots, \xi_9 \in \Z^d \\ \|\xi_i\|_1 {\leqslant}M}} \nonumber \hat{F}(\xi_1,\xi_2,\xi_3) \hat{F}(\xi_4,\xi_5,\xi_6) \hat{F}(\xi_7, \xi_8, \xi_9)\\ & \qquad \qquad \times \E_{x \in \F} \bigg( e_p\big(\xi_1 \cdot ax^2 + \xi_2 \cdot 2ax + \xi_4 \cdot a(x+y)^2\\ & \qquad \qquad \qquad \qquad + \xi_5 \cdot 2a(x + y) + \xi_7 \cdot a x^2 y^2 + \xi_8 \cdot 2ax y\big) \\ & \qquad \qquad \qquad \qquad \qquad \times \psi(x)^{\xi_3} \psi(x + y)^{\xi_6} \psi(xy)^{\xi_9}\bigg),\end{aligned}$$ where we write $\psi(x)^{\xi_i}=\prod_j{\psi_j(x)^{(\xi_i)_j}}$ etc. Now, $\psi_i(xy)=\psi_i(x)\psi_i(y)$ unless $x=0$ or $y=0$, so writing $$\begin{aligned} S(y) := \E_{x \in \F} & e_p\big((\xi_1 + \xi_4) \cdot a x^2 +\xi_7 \cdot ax^2y^2\\ & \qquad + (\xi_2 + \xi_5) \cdot 2a x + (\xi_4+\xi_8)\cdot 2axy\big)\\ & \qquad \qquad \times \psi(x)^{\xi_3+\xi_9} \psi(x + y)^{\xi_6}, \end{aligned}$$ if $y\neq 0$ we get $$\begin{aligned} \label{eqn.bg} E(y) & = \sum_{\substack{\xi_1,\dots, \xi_9 \in \Z^d \\ \|\xi_i\|_1 {\leqslant}M}} \nonumber \hat{F}(\xi_1,\xi_2,\xi_3) \hat{F}(\xi_4,\xi_5,\xi_6) \hat{F}(\xi_7, \xi_8, \xi_9) S(y)e_p(\xi_4 \cdot ay^2 + \xi_5 \cdot 2ay) \psi(y)^{\xi_9}\\ \nonumber & \qquad \qquad \qquad + O(p^{-1}M^3),\end{aligned}$$ since $$\sum_{\substack{\xi_1,\dots, \xi_9 \in \Z^d \\ \|\xi_i\|_1 {\leqslant}M}} |\hat{F}(\xi_1,\xi_2,\xi_3) \hat{F}(\xi_4,\xi_5,\xi_6) \hat{F}(\xi_7, \xi_8, \xi_9)| {\leqslant}M^3.$$ We are only interested in $E(y)$ when $y \in B(\Psi,\eps)$. In this case, since $\|\xi_4\|_1,\|\xi_5\|_1,\|\xi_9\|_1 {\leqslant}M$ we have $$\begin{aligned} |e_p (\xi_4 & \cdot ay^2 + \xi_5 \cdot 2ay) \psi(y)^{\xi_9}-1| \\ & {\leqslant}\|\xi_4\|_1\sup_i{|e_p(a_iy^2)-1|} + \|\xi_5\|_1\sup_i{|e_p(2a_iy)-1|} + \|\xi_9\|_1\sup_i{|\psi_i(y)-1|} = O(\eps M).\end{aligned}$$ It follows that for $y \in B(\Psi,\eps)\setminus \{0\}$ we have $$\label{e-s} E(y) = \sum_{\substack{\xi_1,\dots, \xi_9 \in \Z^d \\ \|\xi_i\|_1 {\leqslant}M}} \hat{F}(\xi_1,\xi_2,\xi_3) \hat{F}(\xi_4,\xi_5,\xi_6) \hat{F}(\xi_7, \xi_8, \xi_9) S(y) +O(\eps M^4) + O(p^{-1}M^3).$$ We may re-write $S$ as $$S(y) = \E_{x \in \F} e_p\bigg((\xi_1 + \xi_4 + \xi_7 \overline{y^2}) \cdot a x^2 + \big((\xi_2 + \xi_5) + (\xi_4 + \xi_8)\overline{y}\big)\cdot 2a x\bigg) \psi(x)^{\xi_3+\xi_9} \psi(x + y)^{\xi_6},$$ where $\overline{t}$ denotes the lift of $t \in \F$ to $\{0,\dots, p-1\}$. Now by Proposition \[add-mult\], since $y \in \F^*$, $S(y)$ is $o_{p \rightarrow \infty}(1)$ unless $$\label{add-const} \xi_1 + \xi_4 + \xi_7 \overline{y^2} , \xi_2 + \xi_5 + \overline{y}(\xi_4 + \xi_8) \in \Lambda_{\Psi}^+$$ and $$\label{mult-const} \xi_3 + \xi_9, \xi_6 \in \Lambda_{\Psi}^{\times}.$$ The reader may again wish to take a moment and recall the definitions of $\Lambda_{\Psi}^+$ and $\Lambda_{\Psi}^\times$, given at the start of §\[sec3\]. It follows from this and the bound $\Vert F \Vert_{{\operatorname{trig}}} {\leqslant}M$ that for $y \in B(\Psi,\eps)\setminus \{0\}$ we have $$\begin{aligned} \nonumber E(y) & = o_{p \rightarrow \infty}(M^3) + O(\eps M^4) +\\ & \qquad \qquad \sum_{\substack{\xi_1,\dots, \xi_9 : \|\xi_i\|_1 {\leqslant}M \\ \xi_1 + \xi_4 +\xi_7\overline{y^2}, \xi_2 + \xi_5+\overline{y}(\xi_4 + \xi_8) \in \Lambda_{\Psi}^+ \\ \xi_3 + \xi_9, \xi_6 \in \Lambda_{\Psi}^{\times}}}\!\!\!\!\!\! \hat{F}(\xi_1,\xi_2,\xi_3) \hat{F}(\xi_4,\xi_5,\xi_6) \hat{F}(\xi_7, \xi_8, \xi_9).\label{eq567}\end{aligned}$$ We say that $y$ is *exceptional* if, for some tuple $(\xi_1,\dots,\xi_9)$ with $\|\xi_i\|_1 {\leqslant}M$ for all $i$, one of the following is true: 1. there are at most $3$ values of $z$ for which $\xi_1 + \xi_4 + \xi_7 \overline{z^2} \in \Lambda_{\Psi}^+$, and $y$ is one of those values; or 2. there are at most $2$ values of $z$ for which $\xi_2 + \xi_5 + \overline{z}(\xi_4 + \xi_8) \in \Lambda_{\Psi}^+$, and $y$ is one of those values. The number of exceptional $y$ is clearly $O(M^{9d})$. Suppose that $y$ is not exceptional, and that $\xi_1,\dots,\xi_9$ lies in the support of the sum , thus and hold. Then there are at least 3 values of $z$ for which for which $\xi_1 + \xi_4 + \xi_7 \overline{z^2} \in \Lambda_{\Psi}^{+}$. Take two of these, $z_1$ and $z_2$, for which $\overline{z_1^2} \neq \overline{z_2^2}$. Then $\xi_7(\overline{z_1^2} - \overline{z_2^2}) \in \Lambda_{\Psi}^+$, which implies (from the definition of $\Lambda_{\Psi}^+$) that $\xi_7 \in \Lambda_{\Psi}^+$. Therefore, from , $\xi_1 + \xi_4 \in \Lambda_{\Psi}^+$ as well. Furthermore there are 2 values of $z$ for which $\xi_2 +\xi_5 + \overline{z}(\xi_4 + \xi_8) \in \Lambda_{\Psi}^+$. By much the same argument it follows that $\xi_2+ \xi_5, \xi_4 + \xi_8 \in \Lambda_{\Psi}^+$. It follows that if $y$ is not exceptional then may be replaced by the stronger conclusion that $$\label{add-const-stronger} \xi_1 + \xi_4 , \xi_7 , \xi_2 + \xi_5 , \xi_4 + \xi_8 \in \Lambda_{\Psi}^+.$$ Putting all this together, if $y$ is not exceptional (or $0$) then $$E(y) = o_{p \rightarrow \infty}(M^3) + O(\eps M^4) + \sum_{\substack{\xi_1,\dots, \xi_9 \\ \xi_1 + \xi_4, \xi_7,\xi_2 + \xi_5, \xi_4 + \xi_8 \in \Lambda_{\Psi}^+ \\ \xi_3 + \xi_9, \xi_6 \in \Lambda_{\Psi}^{\times}}} \hat{F}(\xi_1,\xi_2,\xi_3) \hat{F}(\xi_4,\xi_5,\xi_6) \hat{F}(\xi_7, \xi_8, \xi_9).$$ We claim that the sum on the right here is precisely $$I(F) := \int F(t,u,v) F(t + u', u, v') F(t', u', v) d\mu_{H_{\Psi}}(t,u,v) d\mu_{H_{\Psi}}(t', u', v') .$$ To see this, start with this latter expression and expand each copy of $F$ as a Fourier series. Recall also that $\mu_{H_{\Psi}} = \mu_{G_{\Psi}^+} \times \mu_{G_{\Psi}^+} \times \mu_{G_{\Psi}^{\times}}$. Then we get $$\begin{aligned} I(F)& =\sum_{\xi_1,\dots,\xi_9} \hat{F}(\xi_1, \xi_2,\xi_3) \hat{F}(\xi_4,\xi_5,\xi_6) \hat{F}(\xi_7,\xi_8,\xi_9)\\ & \qquad \times \int e(\xi_1 \cdot t + \xi_2 \cdot u + \xi_4 \cdot (t + u') + \xi_5 \cdot u + \xi_7 \cdot t' + \xi_8 \cdot u') d\mu_{G_{\Psi}^+}^{\otimes 4}(t,u,t',u')\\ & \qquad \times \int v^{\xi_3 + \xi_9} (v')^{\xi_6} d\mu_{G_{\Psi}^{\times}}^{\otimes 2}(v,v').\end{aligned}$$ The claim now follows from the orthogonality relations , . Thus if $y$ is not exceptional (or $0$) then $$E(y) = o_{p \rightarrow \infty}(M^3) + O(\eps M^4) + I(F).$$ On the other hand, whatever the value of $y$ we have $|E(y)| {\leqslant}1$ and the measure of the exceptional elements is at most $O(M^{9d}/p)$ since there are $O(M^{9d})$ of them. It follows that $$\begin{aligned} T(F \circ \Psi, 1_S , F \circ \Psi, F \circ \Psi) & = \E_y 1_S(y) E(y) \\ & = \mu(S) (I(F)+O(\epsilon M^4)) + o_{p \rightarrow \infty}(M^{9d})\end{aligned}$$ and the result is proved. 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Ben Green\ The Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG\ ben.greenmathsoxacuk\ Tom Sanders\ The Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG\ tom.sandersmathsoxacuk\ [^1]: Supported by ERC Starting Grant 279438 *Approximate algebraic structure and applications*, and by a Simons Investigator Grant.

--- author: - 'Xiao Fang$^*$, Shige Peng$^\dagger$, Qi-Man Shao$^*$, Yongsheng Song$^\ddagger$' date: '*The Chinese University of Hong Kong$^*$, Shandong University$^\dagger$, Chinese Academy of Sciences$^\ddagger$*' title: Limit theorems with rate of convergence under sublinear expectations --- [**Abstract:**]{} Under the sublinear expectation $\mathbbm{E}[\cdot]:=\sup_{\theta\in \Theta} E_\theta[\cdot]$ for a given set of linear expectations $\{E_\theta: \theta\in \Theta\}$, we establish a new law of large numbers and a new central limit theorem with rate of convergence. We present some interesting special cases and discuss a related statistical inference problem. We also give an approximation and a representation of the $G$-normal distribution, which was used as the limit in Peng (2007)’s central limit theorem, in a probability space. [**AMS 2010 subject classification:**]{} 60F05. [**Keywords and phrases:**]{} sublinear expectation, law of large numbers, central limit theorem, $G$-normal distribution, rate of convergence, Stein’s method. Introduction ============ Let $\{P_\theta: \theta\in \Theta\}$ be a set of probability measures on a measurable space $(\Omega, \mathcal{F})$. Let $E_\theta$ denote the expectation under $P_\theta$. For a random variable $X: \Omega \to \mathbb{R}$ such that $E_\theta[X]$ exists for all $\theta \in \Theta$, we define its [*sublinear*]{} expectation as It is clear that the sublinear expectation [00]{} satisfies the following: (i) [*monotonicity*]{} ($\E[X]{\geqslant}\E[Y]$ if $X{\geqslant}Y$), (ii) [*constant preservation*]{} ($\E[c]=c$ for $c\in \mathbb{R}$), (iii) [*sub-additivity*]{} ($\E[X+Y]{\leqslant}\E[X]+\E[Y]$), and (iv) [*positive homogeneity*]{} ($\E[\lambda X]=\lambda \E[X]$ for $\lambda{\geqslant}0$). From (iii), we have In the special case where $Y$ does not have the mean uncertainty, that is, From [103]{} and (i), we have Such a notion of sublinear expectation is often used in situations where it is difficult or impossible to find the “true” probablity $P_{\hat{\theta}}$ among a set of uncertain probability models $\{P_{\theta}\}_{\theta\in\Theta}$. To the best of our knowledge, the definition [00]{} of sublinear expectation first appeared in [@Hu81], who called it the upper expectation. It was also called the upper prevision in the theory of imprecise probabilities. See, for example, [@Wa91]. A type of nonlinear expectation adapted with a Brownian filtration, called $g$-expectation, was defined in [@p97]. The sublinear situation of $g$-expectation was applied in [@Chen-Epstein02] to describe the investors’ ambiguity aversions. The notion of coherent risk measures introduced in [@ArDe99] is also a type of sublinear expectation. See also [@FoSc11]. The motivation for these related notions is to use the set of probability measures $\{P_\theta: \theta\in \Theta\}$ to model the uncertainty of probabilities and distributions in real data, and use the sublinear expectation $\E$ as a robust method to measure the risk loss $X$. We also refer to [@DPR2010] and [@Pe10] for more information on sublinear expectations, dynamical risk measures and general nonlinear expectations. According to [@Pe07], we say two random variables $X$ and $Y$ are [*identically distributed*]{}, denoted by $X\overset{d}{=}Y$, if for all bounded continuous functions $\varphi$. $X\overset{d}{=}Y$ means that the distribution uncertainties of $X$ and $Y$ are the same. There are different notions of independence under sublinear or nonlinear expectations. See, for example, [@Wa91], [@Ma99] and [@MaMa05]. We adopt the notion introduced by [@Pe10] and say that a random vector $Y\in \mathbb{R}^n$ is [*independent*]{} of another random vector $X\in \mathbb{R}^m$ if for all bounded continuous functions $\varphi: \mathbb{R}^{m+n}\to \mathbb{R}$. This independence often occurs in many situations where the value of $X$ is realized before that of $Y$, but the distribution uncertainty of $Y$ does not change after this realization. A sequence of random variables $\{X_i\}_{i=1}^{\infty}$ is said to be [*i.i.d.*]{} if for each $i=1,2,\dots,$ $X_{i+1}$ is identically distributed as $X_1$ and independent of $(X_1,\dots, X_i)$. Under sublinear expectations, “$Y$ is independent of $X$" does not imply automatically that “$X$ is independent of $Y$". Example 3.13 of [@Pe10] provides such an example. In the special case that $\Theta$ is a singleton, [00]{} reduces to the usual definition of expectation, and the definition of i.i.d.  random variables reduces to that in the classical setting. [@Pe10] formulated a law of large numbers (LLN) under the sublinear expectation [00]{} as follows. Let $\{X_i\}_{i=1}^\infty$ be an i.i.d.  sequence of random variables with $\E[X_1]=\overline{\mu},$ $-\E[-X_1]=\underline{\mu}$, both being finite. By the definition of sublinear expectation [00]{}, we have $\underline{\mu}{\leqslant}\overline{\mu}$. Let $\overline{X}_n=(X_1+\dots+ X_n)/n$. Then, we have where $lip(\mathbb{R})$ denotes the class of Lipschitz functions. We refer to [001]{} as the weak convergence of $\overline{X}_n$ to the [*maximal distribution*]{} with parameters $\underline{\mu}$ and $\overline{\mu}$. By assuming further that $\underline{\mu}=\overline{\mu}=:\mu$ and [@Pe07] obtained a central limit theorem (CLT): where $\{u(t, x): (t,x)\in [0,\infty)\times \mathbb{R}\}$ is the unique viscosity solution to the following parabolic partial differential equation (PDE) defined on $[0,\infty)\times \mathbb{R}$: where $G=G_{\underline{\sigma}, \overline{\sigma}}(\alpha)$ is the following function parametrized by $\underline{\sigma}$ and $\overline{\sigma}$: Here we denote $\alpha^+:=\max\{0, \alpha\}$ and $\alpha^-:=(-\alpha)^+$. We refer to [002]{} as the weak convergence of $\sqrt{n}(\overline{X}_n-\mu)$ to the [*$G$-normal distribution*]{} with parameters $\underline{\sigma}^2$ and $\overline{\sigma}^2$. We will denote the right-hand side of [002]{} by $\mathcal{N}_G[\varphi]$ and suppress its dependence on $\underline{\sigma}^2$ and $\overline{\sigma}^2$ for the ease of notation. Recently, [@So17] obtained a convergence rate for Peng’s CLT [002]{}, which is of the order $O(1/n^{\alpha/2})$ with an unspecified $\alpha\in (0,1)$. In the special case that $\varphi$ is a convex function, we can verify by the Gaussian integration by parts formula and $G(\partial^2_{xx}u)=\frac{\overline{\sigma}^2}{2} \partial^2_{xx} u$ from the convexity of $\varphi$ that is the solution to the PDE [102]{}, where $Z$ is a standard Gaussian random variable. Therefore, the limit in [002]{} is a normal distribution. The same conclusion holds for concave $\varphi$, except that $\overline{\sigma}$ is replaced by $\underline{\sigma}$. The limit in [002]{} is also normal for any $\varphi\in lip(\mathbb{R})$ if $\overline{\sigma}=\underline{\sigma}$. Note that [001]{} and [002]{} reduce to classical LLN and CLT if $\Theta$ in [00]{} is a singleton. In this case $\E$ is a linear expectation. The goal of this paper is to obtain convergence rates for the above LLN and a new type of renormalized CLT with explicitly given bounds in the framework of sublinear expectations. For the LLN, we prove that where This upper bound provides us with a quantitative version of the fact that for large $n$, the sample mean is sufficiently concentrated inside the interval $[\underline{\mu}, \overline{\mu}]$. We deduce this upper bound from a new law of large numbers, which may be of independent interest. We will discuss a related statistical inference problem under sublinear expectations. We also discuss extensions to the multi-dimensional setting. With respect to the CLT in [002]{}, for the special case that $\varphi$ is a convex function, we prove that where $Z$ is a standard Gaussian random variable and $||\cdot||$ denotes the supremum norm of a function. A similar bound for $\varphi$ being a concave function is also obtained. For the general case where the mean of $X_1$ is uncertain (that is, $\underline{\mu}\ne \overline{\mu}$) and $\varphi$ may not be convex or concave, we formulate a new central limit theorem for where $\mu_i$ equals $\overline{\mu}$ or $\underline{\mu}$ depending on previous $\{X_j: j<i\}$ and the solution to the heat equation, and $\sigma_i$ depends furthermore on the set of the possible first two moments of $X_1$. Our main tool for proving the rate of convergence for the CLT is a combination of Lindeberg’s swapping argument and Stein’s method. This approach was used by [@Ro17] for proving a martingale CLT. The sublinear expectation [00]{} is defined through a class of probability measures, and in general, cannot be represented in a single probability space. However, for the $G$-normal distribution, which was used as the limit in Peng’s CLT [002]{}, we can give an approximation and a representation in a probability space. The rest of this paper is organized as follows. In Section 2, we present our results on the law of large numbers. Section 3 contains the results related to the CLT. A new representation of the $G$-normal distribution is derived in Section 4. Most of the proofs are deferred to Section 5. Law of large numbers ==================== In this section, we first provide a rate of convergence for Peng’s law of large numbers, then discuss its implication on the statistical inference for uncertain distributions, and finally, we present a new law of large numbers with rates that may be of independent interest. Rate of convergence ------------------- Let $\{X_i\}_{i=1}^\infty$ be an i.i.d.  sequence of random variables under a sublinear expectation $\E$ such that for a family of linear expectations $\{E_\theta: \theta\in \Theta\}$. Suppose both $\overline{\mu}=\E[X_1]\ \text{and} \ \underline{\mu}=- \E [-X_1]$ are finite. Define If $\overline{\sigma}^2$ is finite, then we can control the expected deviation of the sample mean $\overline{X}_n=\sum_{i=1}^n X_i/n$ from the interval $[\underline{\mu}, \overline{\mu}]$. \[t7\] Under the above setting, we have We can rewrite [1001]{} as where for $A\subset \mathbb{R}^d$ and $x\in \mathbb{R}^d$, $d_A(x):=\inf_{y\in A}|y-x|$. Clearly, for any interval $I$ larger than $[\underline{\mu}, \overline{\mu}]$, i.e., $[\underline{\mu}, \overline{\mu}]\subset I$, the conclusion of Theorem \[t7\] still holds for $d^2_{I}(\overline{X}_n)$. In fact, $[\underline{\mu}, \overline{\mu}]$ is the smallest interval satisfying Theorem \[t7\]. According to (\[001\]), if $[\underline{\nu},\overline{\nu}]\nsupseteq [\underline{\mu},\overline{\mu}]$, then $$\lim_{n\to \infty}\mathbb{E}\big[d_{[\underline{\nu},\overline{\nu}]}(\overline{X}_n)\big]=\sup_{x\in[\underline{\mu}, \overline{\mu}]}d_{[\underline{\nu},\overline{\nu}]}(x)>0.$$ [001]{} presents a law of large numbers under sublinear expectations where the convergence is in the distribution. In fact, if $\overline{\mu}>\underline{\mu}$, the convergence would not be in the strong sense: there does not exist a random variable $\eta$ such that Indeed, if (\[se1\]) holds, then by [001]{}, $\eta$ must be maximally distributed. Set $g(x)=\min\big\{\max\{x, \underline{\mu}\big\}, \overline{\mu}\}-\underline{\mu}$. On one hand, (\[se1\]) implies that $$\lim_{n\rightarrow\infty}\mathbb{E}\big[-g(\overline{X}_n)(\eta-\underline{\mu}+1)\big]=\mathbb{E}\big[-g(\eta)(\eta-\underline{\mu}+1)\big]=0.$$ On the other hand, as $\eta$ is independent of $g(S_n)$, we have $$\begin{aligned} \lim_{n\rightarrow\infty}\mathbb{E}\big[-g(\overline{X}_n)(\eta-\underline{\mu}+1)\big]&=&\lim_{n\rightarrow\infty}\mathbb{E}\big[g(\overline{X}_n)]\mathbb{E}[-(\eta-\underline{\mu}+1)\big]\\ &=&\mathbb{E}[g(\eta)]\mathbb{E}[-(\eta-\underline{\mu}+1)]=-(\overline{\mu}-\underline{\mu}).\end{aligned}$$ This is a contradiction. Theorem \[t7\] can be generalized to the multi-dimensional setting. \[t9\] Let $\{X_i\}_{i=1}^\infty$ be an i.i.d.  sequence of $d$-dimensional random vectors under a sublinear expectation $\E=\sup_{\theta\in \Theta} E_\theta$. Suppose that the convex hull of the closure of all the possible means $\{E_\theta [X_1]: \theta\in \Theta\}$ is a bounded convex polytope $\mathcal{P}$ with $m$ vertices. We have where $\overline{X}_n=\sum_{i=1}^n X_i/n$, $|\cdot|$ denotes the Euclidean norm, and $diam(\mathcal{P})$ denotes the diameter of the polytope. Theorem \[t9\] reduces to Theorem \[t7\] in the one-dimensional case by regarding $[\underline{\mu}, \overline{\mu}]$ as a polytope with $m=2$ vertices. Theorem \[t9\] follows from a new law of large numbers stated in Section 2.3, which may be of independent interest. \[r3\] Based on Theorem \[t9\], we can also give a convergence rate of $\E [d_{\mathcal{P}} (\overline{X}_n)]$ when $\mathcal{P}$ is a general convex set in $\mathbb{R}^d$ with a regular boundary. For example, if $\mathcal{P}$ is a disk of radius $R$ in a plane, we have (proof deferred to Section 5.1) $$\mathbb{E}[d_{\mathcal{P}}(\overline{X}_n)]\le \frac{\big(7\pi^2R+\sqrt{\overline{\sigma}^2+16R^2}\big)}{n^{\frac{2}{5}}}.$$ Statistical inference for uncertain distributions ------------------------------------------------- The upper bound in Theorem \[t7\] provides us with a quantitative version of the fact that for large $n$, the sample mean is sufficiently concentrated inside the interval $[\underline{\mu}, \overline{\mu}]$. This is related to the estimation of $\underline{\mu}$ and $\overline{\mu}$ described below. Given an i.i.d.  sequence of random variables $X_1, \dots, X_N$ under linear expectations, the usual estimator for their mean is Here, we consider a statistical estimation under sublinear expectations. Let $X_1,\dots, X_N$ be an i.i.d.  sequence of random variables under a sublinear expectation $\E$ such that for a family of linear expectations $\{E_\theta: \theta\in \Theta\}$. Suppose that $N=nk$ and the data are expressed as follows: [@JiPe16] proposed to estimate the lower mean $\underline{\mu}$ and the upper mean $\overline{\mu}$ of $X_1$ by and respectively. Applying Theorem \[t7\] and the union bound, we have the following result. \[p1\] Suppose $\E[X_1^2]<\infty$. We have where $C$ is a constant depending only on $\underline{\mu}, \overline{\mu}$ and $\overline{\sigma}^2$ in [120]{}. Define We have, by the union bound and Theorem \[t7\], The second inequality follows from the same argument. Proposition \[p1\] ensures that as $n\to \infty$ and $k=o(n)$, the estimators by [@JiPe16] are sufficiently concentrated inside $[\underline{\mu}, \overline{\mu}]$. A new law of large numbers -------------------------- We first formulate a new law of large numbers for the one-dimensional case. \[t2\] Let $\{X_i\}_{i=1}^\infty$ be an i.i.d.  sequence of random variables under a sublinear expectation $\E$ such that for a family of linear expectations $\{E_\theta: \theta\in \Theta\}$. Suppose that $\E[ X_1^2]<\infty$. Denote Then, for $\varphi$ differentiable such that $\varphi'\in lip(\mathbb{R})$, we have where and Theorem \[t2\] is a direct consequence of the following multivariate version, which will be proved in Section 5.1. \[t4\] Let $X_1, X_2, \dots$ be an i.i.d.  sequence of $d$-dimensional random vectors under a sublinear expectation $\E$ such that for a family of linear expectations $\{E_\theta: \theta\in \Theta\}$. Let be all possible means of $X_1$. Let $\mathcal{P}$ be the convex hull of the closure of $\mathcal{M}_1$. We have, for $\varphi: \mathbb{R}^d\to \mathbb{R}$ differentiable such that the gradient $D\varphi: \mathbb{R}^d\to \mathbb{R}^d$ is a Lipschitz function, where $\mu_i:=\operatorname*{argsup}_{\mu\in \mathcal{P}}\big\{\mu\cdot D\varphi\big[\frac{\sum_{j=1}^{i-1}(X_j-\mu_j)}{n}\big]\big\}$ (if the $\operatorname*{argsup}$ is not unique, choose any value), $\lambda_*$ is the supremum norm of the operator norm of the Hessian $D^2\varphi$, and $diam(\mathcal{P})$ denotes the diameter of $\mathcal{P}$. Central limit theorem with rate of convergence ============================================== As explained in the Introduction, in the special case where $\varphi$ is a convex or concave test function, the limit in Peng’s CLT in [002]{} is a usual normal distribution. We first provide a rate of convergence for this special case. Moreover, unlike in [002]{}, we do not need to impose the [*identically distributed*]{} assumption. \[t6\] Suppose $X_1,\dots, X_n$ are independent under a sublinear expectation $\E$ with Let For convex test functions $\varphi(\cdot)\in lip(\mathbb{R})$, we have where $Z$ is a standard Gaussian random variable. For concave functions $\varphi$, if we let then The proof of Theorem \[t6\] follows from a similar and simpler proof of Theorem \[t1\] below and is deferred to Section 5.2. Theorem \[t6\] has the following corollary if the $X_i$’s are assumed to be i.i.d.  \[t0\] Under the conditions of Theorem \[t6\], suppose further that $X_1,\dots, X_n$ are i.i.d., and denote Then, for a convex test function $\varphi\in lip(\mathbb{R})$, we have where $Z$ is a standard Gaussian random variable. If $\varphi$ is concave, then we have Corollary \[t0\] follows directly from Theorem \[t6\] by and the fact that $1+\dots+\frac{1}{n}{\leqslant}\log n+1.$ For the general case where the mean of $X_1$ is uncertain (that is, $\underline{\mu}\ne \overline{\mu}$) and $\varphi$ may not be convex or concave, we formulate a new CLT for where $\mu_i$ equals $\overline{\mu}$ or $\underline{\mu}$ depending on previous $\{X_j: j<i\}$ and the solution to the heat equation, and $\sigma_i$ depends furthermore on the set of the possible first two moments of $X_1$. As above, let $\{X_i\}_{i=1}^\infty$ be an i.i.d.  sequence of random variables under a sublinear expectation $\E$ such that for a family of linear expectations $\{E_\theta: \theta\in \Theta\}$. Suppose that $\E[ |X_1|^3]<\infty$. Define and for each possible mean $\mu$ of $X_1$, define We impose the following assumption: [**Assumption A**]{}. *Regarded as functions of $\mu$, $\overline{\sigma}_\mu^2$ and $\underline{\sigma}_\mu^2$ are continuous at, or can be continuously extended to, $\mu=\overline{\mu}$ and $\mu=\underline{\mu}$.* Denote There is no conflict of notation between [121]{} and [122]{} by Assumption A. We assume further that [**Assumption B**]{}. *All the four quantities in [122]{} are positive.* Let be the set of all possible pairs of mean and variance of $X_1$. Define and On the basis of Assumptions A and B, we have $\sigma_0^2>0$. We have the following theorem. \[t1\] Under the above setting, we have the following CLT: for each $\varphi\in lip(\mathbb{R})$, In [15]{}, $Z$ is a standard Gaussian random variable, with $\mu_i=\mu_i((X_j, \mu_j, \sigma_j): j<i)$ are defined as $\sigma_i=\sigma_i((X_j, \mu_j, \sigma_j): j<i, \mu_i)$ are defined as where $V_{i-1}:= V(t_{i-1}, W_{i-1})$ and $V(\cdot, \cdot)$ is the solution to the heat equation The proof of Theorem \[t1\] is deferred to Section 5.2. From the definition of $\mu_i$ in [13]{}, the first term of $f_{i-1, b}(\mu, \sigma^2)$ in [17]{} is ${\leqslant}0$ for $(\mu, \sigma^2)\in \mathcal{M}_2$ in [18]{}. It is straightforward to show, by checking the values of the supremum in [14]{} at the boundary points below and by the fact that $\sup_{(\mu, \sigma^2)\in \mathcal{M}_2} [f_{i-1, b}(\mu, \sigma^2) ]$ is continuous for $b$ in a compact set in $(0,\infty)$, that in [14]{}, if $\partial^2_{xx} V_{i-1}{\geqslant}0$, then and if $\partial^2_{xx} V_{i-1}< 0$, then Therefore, $\sigma_i^2$ is well-defined and is bounded below by $\sigma_0^2$ in [51]{}. In Theorem \[t1\], if we assume that $\overline{\mu}=\underline{\mu}=:\mu$, then it is easy to check that If we assume further that $\varphi$ is a convex (concave resp.) function and hence $V(t,\cdot)=E\varphi(\cdot+\sqrt{t}Z)$ is convex (concave resp.), then $\sigma_i$ is further reduced to $\overline{\sigma}_{\mu}$ ($\underline{\sigma}_{\mu} $ resp.). In this special case, Theorem \[t1\] reduces to Corollary \[t0\] except for the constant. Representation of $G$-normal distribution ========================================= Under the sublinear expectation, the $G$-normal distribution $\mathcal{N}_G$ plays the same role as the classical normal distribution does in a probability space (cf. [002]{}). However, since $\mathcal{N}_G$ is linked with a fully nonlinear PDE, which is called $G$-heat equation, generally we cannot give an explicit expression for $\mathcal{N}_G[\varphi]$ like the linear case. So it would be important to give a representation or approximation for $\mathcal{N}_G[\varphi] $ using random variables or processes in a probability space. Theorem \[t1\] shows that under a certain normalization, the partial sum of i.i.d random variables in a sublinear expectation space converges to the standard normal distribution. Motivated by this, in this section, we give an approximation of the $G$-normal distribution by using a suitably normalized partial sum of i.i.d. random variables in a probability space. Moreover, the continuous-time counterpart provides a representation of the $G$-normal distribution using (non-time-homogeneous) SDEs. This refines a result given in [@DHP11], Proposition 49, which implies that the $G$-normal distribution can be represented by Itô integrals with respect to a Brownian motion. Approximation of $G$-normal distribution ---------------------------------------- Let $X_1, X_2, \cdots$ be a sequence of i.i.d random variables with $E[X_1]=0$ and $E[X_1^2]=1$ in a probability space $(\Omega,\mathcal{F}, P)$. Suppose further that $E[|X_1|^3]<\infty$. Denote by $\Sigma^{\mathbb{N}}_G$ the collection of all the sequences of measurable functions $\{\sigma_i\}_{i=1}^{\infty}$ with $\sigma_i: \mathbb{R}\rightarrow [\underline{\sigma},\overline{\sigma}]$ for any $i\in\mathbb{N}$. Fix $n\in\mathbb{N}$. For a mapping $\sigma\in\Sigma^{\mathbb{N}}_G$, set $W^\sigma_{0,n}=0$, and, for $1\le i\le n$, set $$\begin{aligned} \label {D2-SDE} W^\sigma_{i,n}=W^\sigma_{i-1,n}+\sigma_{i}(W^\sigma_{i-1,n})\frac{X_i}{\sqrt{n}}.\end{aligned}$$ Thus, we have $W^\sigma_{i,n}=\frac{1}{\sqrt{n}}\sum\limits_{k=1}^i\Big(\sigma_{k}(W_{k-1,n}^\sigma) X_k\Big)=:\frac{1}{\sqrt{n}}\sum_{k=1}^i X^{\sigma}_{k,n}$. Write $W^\sigma_n=W^\sigma_{n,n}$ for simplicity. \[CLT\] For any $\varphi\in lip(\mathbb{R})$, we have $$\Big|\sup\limits_{\sigma\in\Sigma^{\mathbb{N}}_G}E[\varphi(W_n^\sigma)]-\mathcal{N}_G[\varphi]\Big|\le C_{\alpha,G}\bar{\sigma}^{2+\alpha}\|\varphi'\|\frac{E[|X_1|^{2+\alpha}]}{n^{\frac{\alpha}{2}}},$$ where $\alpha\in (0,1)$, and $C_{\alpha,G}>0$ are constants depending on $\underline{\sigma}$ and $\overline{\sigma}$. The proof of Theorem \[CLT\] is deferred to Section 5.3. We first express $\sup_{\sigma\in\Sigma^{\mathbb{N}}_G}E[\varphi(W_n^\sigma)]$ as a sublinear expectation of a sum of i.i.d. random variables. The theorem then follows from the error bound by [@So17] for [@Pe07]’s CLT. Representation of $G$-normal distribution ----------------------------------------- Roughly speaking, the continuous-time form of Eq. (\[D2-SDE\]) is $$\begin{aligned} \label {phi-SDE} dW^\sigma_t=\sigma(t,W^\sigma_t)dB_t, \ t\in(0,1],\end{aligned}$$ where $B$ is a standard Brownian motion in a filtered probability space $(\Omega,\mathcal{F}, \mathbb{F}, P)$. Denote as $\Sigma_G$ the collection of all smooth functions $\sigma: [0,1]\times\mathbb{R}\rightarrow [\underline{\sigma},\overline{\sigma}]$ with $$\sup\limits_{(t,x)\in[0,1]\times\mathbb{R}}|\partial_x\sigma(t,x)|<\infty.$$ For $\sigma\in\Sigma_G$, we consider the following stochastic differential equation SDE (\[phi-SDE\]) with the initial value $x$: $$\begin{aligned} \label {sigma-SDE} \begin{split} dW^{\sigma,x}_t&= \sigma(t,W^{\sigma,x}_t)dB_t‚‚,\ t\in(0,1],\\ W^{\sigma,x}_0&= x. \end{split}\end{aligned}$$ We write $W^{\sigma}$ for $W^{\sigma,0}$. Denote $\Theta_G:=\{P\circ (W_1^\sigma)^{-1}| \ \sigma\in\Sigma_G\}$. For a function $\sigma: [0,1]\times\mathbb{R}\rightarrow \mathbb{R}$, set $\widetilde{\sigma}(t,x)=\sigma(1-t,x)$. \[t10\] For any $\varphi\in lip(\mathbb{R})$, we have $$\mathcal{N}_G[\varphi]=\sup_{\mu\in\Theta_G}\mu[\varphi].$$ Note that in the above representation, we need to use non-time-homogeneous SDEs. If we only consider time-homogeneous SDEs, the representation will be strictly smaller than the $G$-normal distribution. Proofs ====== Proofs in Section 2 ------------------- In this subsection, we first prove Theorem \[t4\] and then use it to prove Theorem \[t9\]. Finally, we provide a simple explanation for Remark \[r3\]. Denote and denote For arbitrary random vectors $X$ and $Y$, denote We will prove the following claim. \[claim1\] For any $k=1,\dots, n$, we have Using telescoping sum and the independence assumption and applying Claim \[claim1\] recursively from $k=n$ to $k=1$, we have The lower bound is proved by changing ${\leqslant}$ to ${\geqslant}$ and changing $+$ to $-$ for the error terms. Therefore, we obtain Theorem \[t4\], subject to Claim \[claim1\]. To prove Claim \[claim1\], we first write By the property [101]{} of the sublinear expectation and the definition of $\lambda_*$, we have Note that Hence, By the definition of sublinear expectation, As $\mathcal{M}_1\subset \mathcal{P}$, it is clear that On the other hand, for $\lambda_1, \lambda_2{\geqslant}0$ such that $\lambda_1+\lambda_2=1$ and $\mu_1, \mu_2\in \overline{\mathcal{M}}_1$, the closure of $\mathcal{M}_1$, Therefore, and by the choice of $\mu_k$, we have This, together with [claim1-1]{}, proves Claim \[claim1\]. Here, $\mathcal{P}$ is a bounded convex polytope with $m$ vertices. Denote the set of vertices by $\mathcal{V}$. For each vertex $v\in \mathcal{V}$, define It is clear that $\mathcal{P}=\cap_{v\in \mathcal{V}} T_v$ where the intersection is over all the $m$ vertices. (Just to clarify the definitions, consider, for example, $d=1$ and $\mathcal{P}=[\underline{\mu},\overline{\mu}]$. It has two vertices $\mathcal{V}=\{\underline{\mu},\overline{\mu}\}$. Thus, we have $T_{\underline{\mu}}=[\underline{\mu},\infty)$, $T_{\overline{\mu}}=(-\infty, \overline{\mu}]$ and $\mathcal{P}=T_{\underline{\mu}} \cap T_{\overline{\mu}}$.) We will prove that and hence To prove [c3]{}, we take the function $\varphi$ in Theorem \[t4\] to be where $T_v-v=\{u-v: u\in T_v\}$. We will prove the following lemma. \[l4\] For this $\varphi$, we have that $\varphi$ is differentiable, $D\varphi: \mathbb{R}^d\to \mathbb{R}^d$ is a Lipschitz function, and On the basis of this lemma, we can take $\mu_i=v$ for all $i$ in Theorem \[t4\]. This implies the following: The left-hand side is precisely $d_{T_v}^2(\frac{\sum_{i=1}^n X_i}{n})$; hence, we obtain [c3]{}. We now prove Lemma \[l4\]. Without loss of generality, we assume that $v=0$; hence, $T_v-v=T_v=T_0$. For each $x$ such that $d(x,T_0)>0$, define Because of the convexity of $T_0$, $x_0$ is unique for each $x$, and moreover, $x_0$ as a function of $x$ is continuous. Based on this definition, Let $\mathcal{E}$ and $\mathcal{S}$ denote the set of “edges" and “surfaces" of $T_0$, respectively. The $d$-dimensional set $\mathcal{R}=\{x: d(x, T_0)>0\}$ can be divided into a finite number of disjoint parts as where and For each $x\in \mathcal{R}_s$, we change the coordinates such that $x_0$ is the origin and regard $\mathbb{R}^d$ as $s^\perp \bigotimes s$, where $s^\perp$ is the orthogonal space of $s$. Suppose that $s^\perp$ is $d_1$-dimensional. Then, under this new coordinate system and for $y\in \mathcal{R}_s$, we have Hence $D^2 \varphi(y)$ is a diagonal matrix with the first $d_1$ diagonal entries being $2$ and the rest being $0$, and $||D^2\varphi(y)||_{op}{\leqslant}2.$ Similar arguments and results apply to $x\in \mathcal{R}_e$ and to $x\in \mathcal{R}_0$. Recall that $y_0$ is a continuous function of $y$. We conclude that $D\varphi$ is continuous. Therefore, we have [42]{}. We now prove [41]{}. Recall that we assumed that $v=0$. On one hand, as $0\in \mathcal{P}$, $$\sup_{\mu\in \mathcal{P}}\mu \cdot D\varphi(x){\geqslant}0.$$ On the other hand, by considering $x\in \mathcal{R}_0, \mathcal{R}_e, \mathcal{R}_s$ separately as above, as $\mu\in \mathcal{P}$ points “inwards" and $D\varphi(x)=2(x-x_0)$ points “outwards", it is clear that which proves [41]{}. Let $\mathbf{B}_0(R)$ denote a disk of radius $R$ in a plane. For $m\in \mathbb{N}$, denote as $P_m$ a regular $m$-sided polygon with $\mathbf{B}_0(R)$ as the inscribed circle (see Figure 1 below). (0,0) circle (0.866)\[thick\]; iin [0,1,2,3,4,5]{} [ (i 60:1) edge\[thick\] ([(i+ 1) \* 60]{}:1); ]{} (120:1) edge \[left\] node [$r_m$]{} (0,0); (0,0.866) edge \[right\] node [R]{} (0,0); at (0, -1.3) [Figure 1: $\mathcal{P}$ & $P_m$]{}; Write $r_m$ as the radius of the regular $m$-sided polygon. Then, $r_m=\frac{R}{\cos\frac{\pi}{m}}$. We can easily check that $$r_m-R{\leqslant}\frac{7\pi^2R}{m^2} \ \emph{for} \ m\ge 3$$ and $$\lim_{m\rightarrow+\infty}m^2(r_m-R)= \frac{\pi^2R}{2}.$$ Now, we expand the set $\Theta$ as $\Theta_m$ such that $\{E_\theta [X_1]: \theta\in \Theta_m\}=P_m$ and $$\sup_{\theta\in \Theta_m} E_\theta[|X_1-E_\theta[X_1]|^2]=\sup_{\theta\in \Theta} E_\theta[|X_1-E_\theta[X_1]|^2]=:\bar{\sigma}^2.$$ Set $\mathbb{E}_m=\sup_{\theta\in \Theta_m}E_\theta$. Then, $$\begin{aligned} \mathbb{E}[d_{\mathcal{P}}(\overline{X}_n)]&\le& \mathbb{E}_m[d_{\mathcal{P}}(\overline{X}_n)]\\ &\le& \mathbb{E}_m[d_{P_m}(\overline{X}_n)]+r_m-R\\ &\le& r_m-R+\sqrt{\frac{m}{n}}\sqrt{\overline{\sigma}^2+4r^2_m}.\end{aligned}$$ By setting $m=n^{\frac{1}{5}}$, we have $$\mathbb{E}[d_{\mathcal{P}}(\overline{X}_n)]\le (7\pi^2R+\sqrt{\overline{\sigma}^2+16R^2}) n^{-\frac{2}{5}}$$ and $$\limsup_{n\rightarrow+\infty}\Big(n^{\frac{2}{5}}\mathbb{E}[d_{\mathcal{P}}(\overline{X}_n)]\Big)\le \frac{\pi^2R}{2}+\sqrt{\overline{\sigma}^2+4R^2}.$$ Proofs in Section 3 ------------------- In this subsection, we first introduce Stein’s method, which is our main tool for proving the results presented in Section 3. Then, we prove Theorem \[t1\]. Finally, we discuss the modification of the proof of Theorem \[t1\] for obtaining Theorem \[t6\]. ### Stein’s method for distributional approximations Stein’s method was introduced by [@St72] for distributional approximations. The book by [@ChGoSh10] contains an introduction to Stein’s method and many recent advances. Here, we will explain the basic ideas in the context of normal approximation. Let $W$ be a random variable with mean $x$ and variance $t>0$, and let $Z_{x, t}\sim N(x, t)$ be a Gaussian random variable. The Wasserstein distance between their distributions is defined as Inspired by the fact that $Y\sim N(x, t)$ if and only if for all absolutely continuous functions $f$ for which the above expectations exist, we consider the following Stein equation: A bounded solution to [1]{} is known to be Hereafter, we denote the standard Gaussian random variable $Z_{0,1}$ as $Z$. Setting $w=W$ and taking the expectation on both sides of [1]{}, we have The Wasserstein distance between the distribution of $W$ and $N(x,t)$ is then bounded by using the properties of $f_\varphi$ and by exploiting the dependence structure of $W$. We will need to use the following properties of $f_\varphi$. The first lemma provides an upper bound for $f''_\varphi$. \[l2\] For the solution [2]{} to Stein’s equation [1]{}, we have Define We have It is known that $g(s)$ is a bounded solution to and \[see, for example, (2.13) of [@ChGoSh10]\] This implies [25]{}. It is known that $V(t,x):=E\varphi(x+\sqrt{t}Z)$ is the solution to the heat equation The next lemma relates the solution to the Stein equation to the solution to the heat equation. \[l3\] Let $V(\cdot, \cdot)$ be the solution to the heat equation [24]{}. Let $f_\varphi$ be the solution [2]{} to Stein’s equation [1]{}. We have Define again We have and $g(s)$ is a bounded solution to As to prove [26]{}, we only need to show From (2.87) of [@ChGoSh10], we have where $\Phi(\cdot)$ denotes the standard normal distribution function. We have Let $\phi(u)$ be the standard normal density function. We have Therefore, This proves [28]{} and hence, the lemma. ### Proofs of Theorems \[t1\] and \[t6\] The proof is by Lindeberg’s swapping argument and Stein’s method. The approach was used by [@Ro17] for a martingale CLT. See also [@So17]. We note that in general $X_i$ is not independent of $\{X_j: j\ne i\}$. This fact prevents us from using some of the techniques in Stein’s method. Without loss of generality, we assume that $||\varphi'||=1.$ Denote and denote For arbitrary random vectors $X$ and $Y$, denote We will prove the following claim. \[claim2\] Let $\phi_{\sigma}(\cdot)$ be the density function of $N(0,\sigma^2)$ and let $*$ denote the convolution of functions. For any k=1,…, n, we have where $C_1$ is as in the statement of Theorem \[t1\]. Using telescoping sum and the independence assumption and applying Claim \[claim2\] recursively from $k=n$ to $k=1$ as in the argument below Claim \[claim1\], we have Therefore, we obtain Theorem \[t1\], subject Claim \[claim2\]. To prove Claim \[claim2\], let $\eta_1,\dots, \eta_n$ be an i.i.d.  sequence of random variables distributed as $N(0, \frac{1}{n})$ and be independent of $\{X_1, \dots, X_n\}$, and let We have where as in Section 5.2.1, $Z_{x,t}\sim N(x,t)$. Given $W_{k-1}$, let $f$ be the solution to (cf. [1]{}) Based on Lemma \[l2\] and $||\varphi'||= 1$, From [6]{}, we can rewrite [4]{} as Recall that $T_{n-k}\sim N(0, \frac{n-k}{n})$ and is independent of $\{X_1,\dots, X_n\}$. Using [21]{} with $Y=T_{n-k}$, $x=0, t=(n-k)/n$, we have and Therefore, from [32]{} and [33]{} where we regard $Y$ as the third and fourth terms on the right-hand side of [32]{}, we obtain Rewrite where and Based on [7]{} and the fact that $X_k$ is independent of $W_{k-1}$ and $\eta_k\sim N(0,\frac{1}{n})$ is independent of $\{X_1,\dots, X_n\}$, we have From [104]{}, [34]{} and the estimates above, we have where Note that and Therefore, [105]{} is further bounded by where $C_1$ is as in the statement of Theorem \[t1\]. We are left to show that $A$ in [106]{} equals 0. Since $\eta_k$ has mean 0 and is independent of $\{X_1,\dots, X_n\}$ and $T_{n-k}$, we have By the property [33]{} of sublinear expectation, we have Using Lemma \[l3\] and $t_i=\frac{n-i}{n}$ in the statement of the theorem, we have Moreover, by the definition of $\xi_k$ and $V_{i}$ below [17]{}, we have and by the definition of $\E$, Finally, by the choice of $\mu_k$ and $\sigma_k$ in [13]{} and [14]{}, we have $A=0$. Note that part of the reason for the particular expansion of [34]{} is to find connections to $V$. This, together with [105]{}, proves Claim \[claim2\]. The proof is similar to that of Theorem \[t1\]. We use a slightly different expansion (cf. [111]{}) and make use of the convexity (concavity) of $\varphi$ (cf. [113]{} and [114]{}). We only prove the case where $\varphi$ is convex. The concave case follows from a similar argument. Without loss of generality, we assume that $\mu=0$ and $||\varphi'|| = 1$. Denote and denote Define We will prove the following claim. \[claim3\] Let $\phi_{\sigma}(\cdot)$ be the density function of $N(0,\sigma^2)$ and let $*$ denote the convolution of functions. For any k=1,…, n, we have Using telescoping sum and the independence assumption and applying Claim \[claim3\] recursively from $k=n$ to $k=1$ as in the argument below Claim \[claim2\], we obtain the theorem. To prove Claim \[claim3\], let $\eta_1,\dots, \eta_n$ be an independent sequence of random variables distributed as $\eta_i\sim N(0, \frac{\overline{\sigma}_i^2}{\overline{B}_n^2})$ and be independent of $\{X_1, \dots, X_n\}$, and let As in [4]{}, we have Given $W_{k-1}$, let $f$ be the solution to Based on lemma \[l2\] and $||\varphi'||= 1$, we have By a similar argument leading to [104]{}, we have The appropriate change to [34]{} is as follows: where and Based on [7p]{} and the fact that $X_k$ is independent of $W_{k-1}$ and $\eta_k$ is independent of $\{X_1,\dots, X_n\}$, Therefore, we have where By the definition of $\xi_k$, we have Since we have assumed that $\E(X_k)=\E(-X_k)=0$, we have, using the property [33]{} of the sublinear expectation and also the fact that $T_{n-k}$ is independent of $\{X_1,\dots, X_n\}$, From Lemma \[l3\] and the fact that $T_{n-k}$ is independent of $\{X_1,\dots, X_n\}$, we have Since we have assumed that $\varphi$ is convex, the solution to the PDE [102]{} (cf. [107]{}) is also convex in the argument $x$, that is, $\partial^2_{xx} V{\geqslant}0$. Therefore, by the definition of sublinear expectation, and hence by [108]{}, This proves Claim \[claim3\]. Proofs in Section 4 ------------------- \[Proof of Theorem \[CLT\]\] Define $\xi=(\xi_1,\cdots, \xi_n):\mathbb{R}^n\rightarrow\mathbb{R}^n$ by $\xi_i(x)=x_i$, $i=1,\cdots, n$. Denote as $\mathcal{H}$ the collection of continuous real-valued functions $h$ on $\mathbb{R}^n$ with $|h(x)|\le C(1+|x|^3)$ for some constant $C>0$. For a function $h\in\mathcal{H}$, set $$\mathbb{E}[h(\xi)]:=\sup\limits_{\sigma\in\Sigma^{\mathbb{N}}_G}E[h(X^{\sigma}_{1,n},\cdots, X^{\sigma}_{n,n})].$$ Then, $\mathbb{E}[\xi_i]=\mathbb{E}[-\xi_i]=0,$ $ \mathbb{E}[\xi_i^2]=\overline{\sigma}^2$ and $-\mathbb{E}[-\xi_i^2]=\underline{\sigma}^2$, $i=1,2,\cdots,n.$ Moreover, for a function $\varphi\in lip(\mathbb{R})$, we have $$\mathbb{E}[\varphi(\xi_i)]=\sup\limits_{\lambda\in[\underline{\sigma},\overline{\sigma}]}E[\varphi(\lambda X_i)]=:\mathcal{N}[\varphi],$$ i.e., $\xi_1, \cdots, \xi_n$ are identically distributed under $\mathbb{E}$. Set $W_{i,n}=\frac{\xi_1+\cdots+\xi_i}{\sqrt{n}}$. We next prove that, for any function $\varphi\in lip(\mathbb{R})$, $$\begin{aligned} \label {se2} \mathbb{E}[\varphi(W_{i+1,n})]=\mathbb{E}[\mathbb{E}[\varphi(s+\frac{\xi_{i+1}}{\sqrt{n}})]\big|_{s=W_{i,n}}].\end{aligned}$$ On the one hand, we have, for any $\sigma\in \Sigma^{\mathbb{N}}_G$, $$E[\varphi(W^{\sigma}_{i+1,n})]=E[E[\varphi(s+\sigma_{i+1}(s)\frac{X_{i+1}}{\sqrt{n}})]\big|_{s=W^{\sigma}_{i,n}}]\le E[\mathbb{E}[\varphi(s+\frac{\xi_{i+1}}{\sqrt{n}})]\big|_{s=W^{\sigma}_{i,n}}].$$ Therefore we obtain $$\mathbb{E}[\varphi(W_{i+1,n})]\le\mathbb{E}[\mathbb{E}[\varphi(s+\frac{\xi_{i+1}}{\sqrt{n}})]\big|_{s=W_{i,n}}].$$ On the other hand, for each $s\in\mathbb{R}$, we choose $\lambda^{\varphi,n}(s)\in[\underline{\sigma},\overline{\sigma}]$ such that $$E[\varphi(s+\lambda^{\varphi,n}(s)\frac{X_{1}}{\sqrt{n}})]=\sup\limits_{\lambda\in [\underline{\sigma},\overline{\sigma}]}E[\varphi(s+\lambda\frac{X_{1}}{\sqrt{n}})]=\mathbb{E}[\varphi(s+\frac{\xi_{1}}{\sqrt{n}})].$$ Here, we are not sure about the measurability of the function $\lambda^{\varphi,n}(s)$. Therefore, we replace it by measurable approximations. Write $\Phi(s,t, X_1)=\varphi(s+\lambda^{\varphi,n}(t)\frac{X_{1}}{\sqrt{n}})$. For any two real numbers $s,t$, we have $$\begin{aligned} & &E[\Phi(s,s, X_1)]\\ &=&E[\Phi(t,s, X_1)] +\big(E[\Phi(s,s, X_1)]-E[\Phi(t,s, X_1)]\big)\\ &\le&E[\Phi(t,t, X_1)]+ L^\varphi|t-s|\\ &=&E[\Phi(s,t, X_1)] +\big(E[\Phi(t,t, X_1)]-E[\Phi(s,t, X_1)]\big)+L^\varphi|t-s|\\ &\le&E[\Phi(s,t, X_1)]+2L^\varphi|t-s|,\end{aligned}$$ where $L^\varphi$ is the Lipschitz constant of the function $\varphi$. For any $\epsilon>0$, set $\delta=\frac{\epsilon}{2L^\varphi}$ and $$\lambda^{\varphi,n}_\epsilon(s)=\sum_{k\in\mathbb{Z}}\lambda^{\varphi,n}(k\delta)1_{(k\delta,(k+1)\delta]}(s).$$ Then, for any $s\in \mathbb{R}$, $$E[\varphi(s+\lambda^{\varphi,n}_\epsilon(s)\frac{X_{1}}{\sqrt{n}})]\ge\mathbb{E}[\varphi(s+\frac{\xi_{1}}{\sqrt{n}})]-\epsilon.$$ For any $\sigma\in\Sigma^{\mathbb{N}}_G$ with $\sigma_{i+1}(s)=\lambda^{\varphi,n}_\epsilon(s)$, we have $$E[\varphi(W^{\sigma}_{i+1,n})]=E[E[\varphi(s+\sigma_{i+1}(s)\frac{X_{i+1}}{\sqrt{n}})]\big|_{s=W^{\sigma}_{i,n}}]\ge E[\mathbb{E}[\varphi(s+\frac{\xi_{i+1}}{\sqrt{n}})]\big|_{s=W^{\sigma}_{i,n}}]-\epsilon.$$ Therefore, $$\mathbb{E}[\varphi(W_{i+1,n})]\ge\mathbb{E}[\mathbb{E}[\varphi(s+\frac{\xi_{i+1}}{\sqrt{n}})]\big|_{s=W_{i,n}}].$$ Combining the above arguments, we prove equality (\[se2\]). Let $\tilde{\xi}_1,\cdots, \tilde{\xi}_n$ be i.i.d random variables under a sublinear expectation $\tilde{\mathbb{E}}$ with $\tilde{\xi}\sim \mathcal{N}$, the distribution of $\xi_1$. On the basis of (\[se2\]), we have, for any $\varphi\in lip(\mathbb{R})$, $$\mathbb{E}[\varphi(W_n)]=\tilde{\mathbb{E}}[\varphi(\frac{\tilde{\xi}_1+\cdots+\tilde{\xi}_n}{\sqrt{n}})].$$ Therefore, by using Theorem 4.5 of [@So17], we obtain the desired estimate. \[Proof of Theorem \[t10\]\] Without loss of generality, we shall only consider $\varphi$ that vanishes at infinity. Let $u$ be the solution to the $G$-heat equation with initial value $\varphi$. Set $\sigma_\varphi(t,x)=2G(\textmd{sgn}[\partial_{xx}^2u(1-t,x)])$, $(t,x)\in [0,1)\times\mathbb{R}$, where $$\textmd{sgn}[a]= \begin{cases} 1, & \mbox{if }a\ge0; \\ -1, & \mbox{if }a<0. \end{cases}$$ Then, $u$ satisfies $$\begin{aligned} \partial_t u-\frac{1}{2}\widetilde{\sigma}_\varphi^2\partial^2_{xx} u&=&0, \ (t,x)\in (0,1]\times\mathbb{R},\\ u(0,x)&=& \varphi (x).\end{aligned}$$ By the mollification procedure, we can find $\{\sigma_n\}\subset\Sigma_G$ such that $\|\sigma_n-\sigma_\varphi\|_{L^2([0,1]\times \mathbf{B}(R))}\rightarrow0$ as $n\rightarrow\infty$ for any $R<\infty$. Next, set $v_n(t,x):=E[\varphi(W^{\sigma_n,x}_t)]$. Then, $v_n$ is the solution to the following equation: $$\begin{aligned} \partial_t v_n-\frac{1}{2}\widetilde{\sigma}_n^2\partial^2_{xx} v_n&=&0, \ (t,x)\in (0,1]\times\mathbb{R},\\ v_n(0,x)&=& \varphi (x).\end{aligned}$$ As $\varphi$ vanishes at infinity, $$\mathbf{M}(R):=\mathop{\max_{|x|\ge R;}}_{1\ge t\ge 0}\big\{|u(t,x)|, |v_n(t,x)|: \ n\in\mathbb{N} \big\}$$ approaches zero as $R$ approaches $+\infty$. Also, we have $$\mathbf{m}(\epsilon):=\max_{(t,x)\in [0,\epsilon]\times\mathbb{R}}\big\{|u(t,x)-\varphi(x)|, |v_n(t,x)-\varphi(x)|: \ n\in\mathbb{N} \big\}$$ goes to zero as $\epsilon$ goes to $0$. Set $w_n=u-v_n$ and $\varepsilon_n=\widetilde{\sigma}_n^2-\widetilde{\sigma}_\varphi^2$. Then, $w_n$, which is nonnegative, satisfies $$\begin{aligned} \partial_t w_n-\frac{1}{2}\widetilde{\sigma}_n^2\partial^2_{xx} w_n&=&\frac{1}{2}\varepsilon_n\partial_{xx}^2u, \ (t,x)\in (0,1]\times\mathbb{R},\\ w_n(0,x)&=&0.\end{aligned}$$ According to the Aleksandrov-Bakel’man-Pucci-Krylov maximum principle (see, for instance, Theorem 7.1 of [@Lie]), $$\sup\limits_{(t,x)\in (\epsilon,1]\times\mathbf{B}(R)}w_n\le 2\mathbf{M}(R)+2\mathbf{m}(\epsilon)+c_0 (\frac{R}{\underline{\sigma}})^{1/2}\|\varepsilon_n\partial_{xx}^2u\|_{L^2([\epsilon,1]\times \mathbf{B}(R))},$$ where $c_0$ is a universal constant. Note that, following the interior regularity of $G$-heat equation, $$\|\varepsilon_n\partial_{xx}^2u\|_{L^2([\epsilon,1]\times \mathbf{B}(R))}\le 2\overline{\sigma}\|\partial_{xx}^2u\|_{\infty;[\epsilon,1]\times\mathbb{R}}\|\sigma_n-\sigma_\varphi\|_{L^2([0,1]\times \mathbf{B}(R))}\rightarrow0$$ as $n$ approaches $+\infty$. Thus, $$\mathbf{O}(R,\epsilon):=\limsup_{n\rightarrow\infty}\Big(\sup\limits_{(t,x)\in (\epsilon,1]\times\mathbf{B}(R)}w_n\Big)\le 2(\mathbf{M}(R)+\mathbf{m}(\epsilon))$$ and $$\mathbf{O}(R,\epsilon)\le\lim_{R\rightarrow\infty, \epsilon\rightarrow0}\mathbf{O}(R,\epsilon)\le\lim_{R\rightarrow\infty, \epsilon\rightarrow0} 2(\mathbf{M}(R)+\mathbf{m}(\epsilon))\le0.$$ In particular, we have $$\mathcal{N}_G[\varphi]=u(0,1)=\lim_{n\rightarrow\infty}v_n(0,1)=\lim_{n\rightarrow\infty}E[\varphi(W_1^{\sigma_n})].$$ Acknowledgements {#acknowledgements .unnumbered} ================ We thank the two anonymous referees for their detailed comments which led to many improvements. Fang X. was partially supported by a CUHK start-up grant. Peng S. was supported by NSF (No. 11626247). Shao Q. M. was partially supported by Hong Kong RGC GRF 14302515 and 14304917. Song Y. was supported by NCMIS, NSFC (No. 11688101) and Key Research Program of Frontier Sciences, CAS (No. QYZDB-SSW-SYS017). 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--- abstract: | The dynamics of a single impurity interacting with a many particle background is one of the central problems of condensed matter physics. Recent progress in ultracold atom experiments makes it possible to control this dynamics by coupling an artificial gauge field specifically to the impurity. In this paper, we consider a narrow toroidal trap in which a Fermi gas is interacting with a single atom. We show that an external magnetic field coupled to the impurity is a versatile tool to probe the impurity dynamics. Using Bethe Ansatz (BA) we calculate the eigenstates and corresponding energies exactly as a function of the flux through the trap. Adiabatic change of flux connects the ground state to excited states due to flux quantization. For repulsive interactions, the impurity disturbs the Fermi sea by dragging the fermions whose momentum matches the flux. This drag transfers momentum from the impurity to the background and increases the effective mass. The effective mass saturates to the total mass of the system for infinitely repulsive interactions. For attractive interactions, the drag again increases the effective mass which quickly saturates to twice the mass of a single particle as a dimer of the impurity and one fermion is formed. For excited states with momentum comparable to number of particles, effective mass shows a resonant behavior. We argue that standard tools in cold atom experiments can be used to test these predictions. PACS numbers : author: - 'F. Nur Ünal' - 'B. Hetényi' - 'M. Ö. Oktel' bibliography: - 'PaperPRAResubmit.bib' title: Impurity coupled to an artificial magnetic field in a Fermi gas in a ring trap --- Introduction ============ Ultracold atom systems are effectively used as a test bed for condensed matter models. They are preferred because of the high degree of control in experiments such as tunable interactions, impurities and direct measurements by optical techniques. Certain theoretical models of condensed matter such as resonant interactions [@resonance] or bosonic Mott transition [@mott] have been realized for the first time using cold atoms. Many models of one dimensional systems have been realized using two dimensional optical lattices to form narrow tubes [@1DTubeWeiss; @1DTubeParades; @1DTubeEsslinger]. One of the powerful theoretical tools to describe one dimensional systems is the Bethe Ansatz (BA). BA solution has been generalized to many integrable models, e.g. systems with multiple components, different statistics or spin [@Bethe; @LiebLiniger; @Yang; @McGuire; @GuanReview]. This exact solution method has been employed to explain experimental data on a number of instances [@1DTubeParades; @MeanfieldBreakdown]. However, as BA methods are restricted to one dimension, they have not been used to describe systems where an external artificial gauge field is present. In one dimension, such an external magnetic field can be disregarded by using a gauge transformation, unless the one dimensional system closes onto itself. Thus, if the particles are confined to a ring as opposed to a tube, the artificial magnetic field will significantly effect the physics. Such rings, in the form of toroidal traps, have been realized experimentally [@StamperKurnCircularWaveguide; @StamperKurn2015; @RingCJFoot; @RingSchmiedmayer; @RingChapman; @RingPrentiss; @RingRiis; @RingCampbell]. Although none of these experiments have included an artificial gauge field so far. In this work, we consider such a toroidal trap containing non-interacting fermions and describe the behavior of a single charged impurity interacting with background atoms. We argue that an artificial magnetic field coupling to the impurity is an efficient way to probe the polaron state forming due to the interactions. Artificial magnetic fields are created by coupling light to the internal states of the atoms [@spielman; @KetterleArtificial; @BlochHofstadter]. Hence, they are highly specific to the internal state making it possible to create effective magnetic fields coupling only to one type of atom. The charged particle is expected to drag the uncharged fermions along with itself around the ring. Because of the interactions between the impurity and the background atoms a collective excitation usually called a polaron is formed [@Polaron]. This excitation will couple to the external magnetic field with the charge of the impurity particle, however, its mass will critically depend on the interaction strength. The amount of angular momentum carried by the impurity and the uncharged fermions also depend on the total external flux through the ring. By changing the artificial magnetic field strength, it is possible to access excited states of the system adiabatically. We show that an artificial magnetic field coupling specifically to the impurity would be a very effective tool to probe polaron physics. We describe this system exactly using a Bethe Ansatz (BA) solution for contact interactions which are justified for cold atoms as the dominant scattering is s-wave. For strongly attractive interactions, the impurity forms a bound state with one of the background fermions and the physics reduces to the motion of a dimer with twice the mass of the particle. In the other limit of infinitely repulsive interaction, effective mass saturates to total particle number. We calculate the energy and momentum distributions, total transferred momentum and the effective mass for all interaction strengths. We believe these results can be experimentally checked with state of the art toroidal traps and techniques for artificial gauge field generation. The paper is organized as follows. In the next section, we define the model, introduce the notation and review earlier studies. In Section III, we solve the system for two particles and then generalize to any particle number using the BA. Sec. IV contains the analytical solution of the BA equations in certain limits and comparison with numerical solutions. We present our results for several quantities such as energy, angular momentum and effective mass of the charged particle. We give our conclusions along with a brief discussion of possible experiments in Sec. V. ![ (Color online) A simple illustration of the system. $N-1$ uncharged fermions (light gray) and a single charged impurity (dark gray) are trapped on a ring. The impurity is interacting with the fermions via Delta-function interaction. An artificial magnetic field couples exclusively to the impurity. The dynamics of the system depends on the interaction strength between particles and the total flux through the ring $\beta=\frac{qRA}{\hbar}$.[]{data-label="illusration"}](Fig1){width="15.00000%"} THE MODEL ========= The first quantized Hamiltonian for one charged particle among $N-1$ uncharged fermions under a magnetic field reads $${\cal H}=\frac{1}{2m}\Big(\frac{\hbar}{i}\frac{\partial}{\partial x_1}-qA\Big)^2- \frac{\hbar^2}{2m}\sum_{j=2}^N\frac{\partial^2}{\partial x_j^2}+2c\sum_{j=2}^N \delta(x_1-x_j).$$ All particles are assumed to be on a ring of radius R, $0\leq x_i \leq 2\pi R$. The position of the charged particle is $x_1$ and $A$ is the vector potential in the symmetric gauge. The Hamiltonian can be made dimensionless by using, $\tilde{x}_j=\frac{x_j}{R}$, $\tilde{E}=E\frac{2mR^2}{\hbar^2}$, $\tilde{c}=c\frac{2mR}{\hbar^2}$ and $\beta=\frac{qRA}{\hbar}$. $\beta$ is the total magnetic flux through the ring in units of flux quantum $q/h$. Dropping the tildes $${\cal H}=\Big(-i\frac{\partial}{\partial x_1}-\beta\Big)^2- \sum_{j=2}^N\frac{\partial^2}{\partial x_j^2}+2c\sum_{j=2}^N \delta(x_1-x_j).$$ The effect of the magnetic field can be shifted to the boundary conditions by a gauge transformation [@ShastrySutherland]. Namely, when the first particle makes a full circle around the ring the wave function gains a phase factor of $e^{i\beta 2\pi}$ where the periodic boundary conditions (PBCs) for the uncharged particles remain unaffected by the gauging process, $${\cal H}\rightarrow e^{-i\beta x_1}{\cal H}e^{i\beta x_1}.$$ Apart from the twisted BCs, the $\delta$-function interaction can be handled as a two-sided boundary condition (BC) between two different regions of N-particle space corresponding to different permutations of particles. The discontinuity relation at the boundary $x_1=x_j$ (which is obtained by passing to the center of mass and relative coordinates and then integrating the Hamiltonian) is $$\label{discont genel} (\partial_j-\partial_1)\psi\Big|_{x_1<x_j}-(\partial_j-\partial_1)\psi\Big|_{x_j<x_1}=2c\psi\Big|_{x_j=x_1} ,\quad j\neq1.$$ This one dimensional problem of two-component fermions has been studied by using the BA in the previous century. First, the one-spin deviate problem in a Fermi sea is solved by McGuire [@McGuire] and Flicker and Lieb [@FlickerLieb] solved the two-spin deviate problem. Yang [@Yang] elegantly derived the BA equations for the general $M$ down-spins among $N$ up-spins. Twisted BCs have been used throughout the BA literature as a way to probe ground state properties. However, with the possibility of optically inducing artificial magnetic fields, it is important to calculate the properties of the system at finite flux as opposed to infinitesimal values near zero. It is also necessary to consider cases where different components in the system experience different gauge fields. Our calculation takes both of these constraints into account and allows us to exactly study the dynamics resulting from the dragging effect of the charged particle on the uncharged particles. The resulting polaron physics has attracted great interest in the context of cold atoms over the last few years [@Polaron; @AnnaMinguzzi]. THE ANSATZ ========== As the interactions are reduced to BCs, the wave function for a given permutation of the particles is a superposition of plane waves. As the collisions of equal mass particles in one dimension conserve magnitudes of the incoming momenta, the interacting problem is integrable. Hence, in a given region only a finite number of plane waves are needed to construct the wave function. To make our notation clear, we first start with the case of one charged particle with one neutral particle. N=2 Particles ------------- For two particles we have $2!=2$ regions and the wave function in these regions is expressed as follows: $$\begin{aligned} \Psi_{12}(x_1,x_2)=(12)_{12}e^{i(k_1x_1+k_2x_2)}+(21)_{12}e^{i(k_2x_1+k_1x_2)},\nonumber\quad\\ \Psi_{21}(x_1,x_2)=(12)_{21}e^{i(k_1x_1+k_2x_2)}+(21)_{21}e^{i(k_2x_1+k_1x_2)},\quad\end{aligned}$$ where we use parenthesis with a subscript to indicate the coefficients of plane waves. In this notation numbers in the parenthesis indicate the order the wave vectors $k_1,k_2$ are distributed to the coordinates in the exponent and the subscript indices indicate the ordering of the particles on the ring, *i.e.* $\Psi_{12}$ means $x_1<x_2$. At $x_1=x_2$, the wave functions in the two regions should be equal whereas their derivative should obey Eq.\[discont genel\]. Equating the coefficients of each plane wave on both sides, we obtain: **BCs:** at $x_1=x_2$, $$(12)_{21}+(21)_{21}=(12)_{12}+(21)_{12},$$ $$(12)_{21}-(21)_{21}=(12)_{12}\Big(1+\frac{2}{s_{12}}\Big)+(21)_{12}\Big(-1+\frac{2}{s_{12}}\Big), \label{disc.}$$ where $s_{12}=i(k_1-k_2)/c$. Combined BCs give $$\label{bla} \begin{pmatrix} (12)\\\\(21) \end{pmatrix}_{21} = \begin{pmatrix} 1+\frac{1}{s_{12}} & \frac{1}{s_{12}} \\\\ \frac{-1}{s_{12}} & 1-\frac{1}{s_{12}} \end{pmatrix} \begin{pmatrix} (12)\\\\(21) \end{pmatrix}_{12}.$$ Allowed values for $k_1,k_2$ are found by applying the PBCs. PBC for one of the particles gives the BA equation. **BCs:** at $2\pi$ as $ x_2:0\rightarrow2\pi,\qquad \Psi_{21}(x_2=0)=\Psi_{12}(x_2=2\pi)$, $$\qquad(12)_{21}=(12)_{12}e^{ik_22\pi},\qquad (21)_{21}=(21)_{12}e^{ik_12\pi}. \label{pbc2}$$ as $ x_1:0\rightarrow2\pi,\qquad \Psi_{12}(x_1=0)=e^{i\beta2\pi}\Psi_{21}(x_1=2\pi)$, $$\quad(12)_{12}=(12)_{21}e^{i(k_1+\beta)2\pi},\quad (21)_{12}=(21)_{21}e^{i(k_2+\beta)2\pi}. \label{pbc1}$$ Combining the two BCs at $2\pi$, we obtain another constraint $k_1+k_2+\beta=n$, for $n\in\mathbb{Z}$. This is a reflection of the total angular momentum conservation in the system. Eqs.\[bla\] and Eq.\[pbc1\] have non-trivial solutions only when the determinant below vanishes, $$\begin{vmatrix} 1+\frac{1}{s_{12}}-e^{-i(k_1+\beta)2\pi} & \frac{1}{s_{12}} \\\\ \frac{-1}{s_{12}} & 1-\frac{1}{s_{12}}-e^{-i(k_2+\beta)2\pi} \end{vmatrix}=0.$$ ![ (Color online) Energy of the lowest three states vs. interaction strength for $N=2$ particles, for zero total angular momentum. Only scattering states are displayed. Energy is calculated by three different methods. Lines are from Eq.\[Tez Eq.\] direct analytical solution without employing BA, which is algebraically same with the two-particle BA calculation (Eq.\[BA for N=2\]). Diamonds are from the general $N$-particle BA calculation (Eq.\[BA eqs\]).[]{data-label="EvsC N2 fig"}](Fig2){width="47.00000%"} Solution of this determinant gives the BA equation, $$\alpha=\frac{c}{2}\cot\Big(\frac{\pi}{2}(\alpha+n-\beta)\Big)+\frac{c}{2}\cot\Big(\frac{\pi}{2}(\alpha-n+\beta)\Big), \label{BA for N=2}$$ where energy is $E=\frac{(n-\beta)^2+\alpha^2}{2}$ for $\alpha=k_2-k_1$. For the two-particle case, this problem can also be solved exactly without using the BA [@tez], $$c=\alpha\Big(\frac{\cos(\pi(n-\beta))}{\sin(\pi\alpha)}-\cot(\pi\alpha)\Big).\label{Tez Eq.}$$ These two equations analytically reproduce each other and the numerical results match perfectly (Fig.\[EvsC N2 fig\]). Extension of this method to N particles is straightforward if cumbersome. ![(Color online) Ground state energy vs. flux $\beta$ for $N=2$ particles with total angular momentum $n$. Eigenstates for flux $\beta$ with total angular momentum *n* are also eigenstates for flux $\beta+1$ with total angular momentum $n+1$. The system can be analyzed by considering flux values between $-1/2<\beta<1/2$ for all $n$. As the flux is increased by one adiabatically, the system evolves to a higher excited state which has one more unit of total angular momentum. As can be observed, the crossings between different eigenstates is not a problem for adiabatic evolution since only states with different total angular momentum *n* are degenerate.[]{data-label="E vs beta"}](Fig3){width="47.00000%"} N-1 Fermions, One Charged Particle ---------------------------------- The distinguishable charged particle is denoted again by $x_1$ and the wave function is defined in $N!$ regions corresponding to different permutations [@McGuire]. In each one of these regions the wave function consists of $N!$ plane waves in its most general form without imposing the antisymmetry between the fermions. As a total we have $N!\times N!$ coefficients: $$\begin{aligned} \Psi_{123\ldots}=(123\ldots)_{123\ldots}e^{i(k_1x_1+k_2x_2+k_3x_3+\ldots)}+(213\ldots)_{123\ldots}e^{i(k_2x_1+k_1x_2+k_3x_3+\ldots)}+\ldots\nonumber\\ \Psi_{213\ldots}=(123\ldots)_{213\ldots}e^{i(k_1x_1+k_2x_2+k_3x_3+\ldots)}+(213\ldots)_{213\ldots}e^{i(k_2x_1+k_1x_2+k_3x_3+\ldots)}+\ldots\nonumber\\ \Psi_{132\ldots}=(123\ldots)_{132\ldots}e^{i(k_1x_1+k_2x_2+k_3x_3+\ldots)}+(213\ldots)_{132\ldots}e^{i(k_2x_1+k_1x_2+k_3x_3+\ldots)}+\ldots\nonumber\\ \qquad\vdots\qquad\qquad\qquad\qquad\qquad\qquad\vdots\qquad\qquad\qquad\qquad\end{aligned}$$ where $k_1,k_2,\ldots k_N$ are distinct wavenumbers. BCs at $x_1=x_2$ are not effected by the addition of other fermions at the end of the sequence: $$\begin{pmatrix} (123\ldots)\\\\(213\ldots) \end{pmatrix}_{213\ldots} = \begin{pmatrix} 1+\frac{1}{s_{12}} & \frac{1}{s_{12}} \\\\ \frac{-1}{s_{12}} & 1-\frac{1}{s_{12}} \end{pmatrix} \begin{pmatrix} (123\ldots)\\\\(213\ldots) \end{pmatrix}_{123\ldots}.$$ BCs at $2\pi$ follow the same logic; as $ x_2:0\rightarrow2\pi,\; \Psi_{213\ldots N}(x_2=0)=\Psi_{13\ldots N2}(x_2=2\pi)$, yielding $$\begin{aligned} \;(123\ldots)_{213\ldots N}=(123\ldots)_{13\ldots N2}e^{ik_22\pi}, \nonumber\\ (213\ldots)_{213\ldots N}=(213\ldots)_{13\ldots N2}e^{ik_12\pi}.\end{aligned}$$ Number of independent coefficients decreases considerably by requiring antisymmetry upon exchange of fermions. Every coefficient of a plane wave in region $x_1<x_3<x_2<\ldots<x_N$ is identical with the coefficient of the same plane wave in region $x_1<x_2<x_3<\ldots<x_N$. This can be shown by noticing that at $x_2=x_3$ the wave functions must vanish requiring *e.g.* $(123\ldots)_{123\ldots N}=-(132\ldots)_{123\ldots N}$. Fermionic antisymmetry also relates the wave function in separate regions $\Psi_{123\ldots N}=-\Psi_{132\ldots N}$. As a result, the coefficients only depend on the position of the charged particle in the order. We can move indistinguishable fermions through one another at will and the $N!$ regions reduce to $N$ regions. After this simplification it is easy to combine the BC at a $\delta$-function with the overall PBC. $$\begin{aligned} \label{Nparticle det.} \begin{pmatrix} (123\ldots)\\\\(213\ldots) \end{pmatrix}_{213\ldots N} &=& \begin{pmatrix} 1+\frac{1}{s_{12}} & \frac{1}{s_{12}} \\\\ \frac{-1}{s_{12}} & 1-\frac{1}{s_{12}} \end{pmatrix} \begin{pmatrix} (123\ldots)\\\\(213\ldots) \end{pmatrix}_{123\ldots N} \nonumber\\ &=&\begin{pmatrix} e^{ik_22\pi}(123\ldots)_{123\ldots N}\\\\e^{ik_12\pi}(213\ldots)_{123\ldots N} \end{pmatrix}.\end{aligned}$$ The determinant can only vanish if $k_1$ and $k_2$ satisfy, $$k_1-\frac{c}{2}\cot{\pi k_1}=k_2-\frac{c}{2}\cot{\pi k_2}.$$ The same procedure can be applied to any pair of wavenumbers $k_i,k_j$. Thus, all the wavenumbers must satisfy $$k_1-\frac{c}{2}\cot{\pi k_1}=k_2-\frac{c}{2}\cot{\pi k_2}=k_3-\frac{c}{2}\cot{\pi k_3}=\ldots=\lambda,$$ where $\lambda$ is a real constant. This form is equivalent to the usual BA equations [@LiebLiniger]. Hence, the N wavenumbers which define an eigenstate must be chosen as N distinct roots of the equation: $$k-\lambda=\frac{c}{2}\cot{\pi k}.$$ However, there is another constraint. Applying PBCs sequentially on all particles restricts $\lambda$. $$\text{As } x_1:0\rightarrow2\pi,\; \Psi_{123\ldots N}(x_1=0)=e^{i\beta2\pi}\Psi_{23\ldots N1}(x_1=2\pi),$$ $$\;(123)_{123}=(123)_{231}e^{i(\beta+k_1)2\pi},$$ $$\begin{aligned} \text{as } x_2:0\rightarrow2\pi, \quad(123)_{231}=(123)_{312}e^{ik_22\pi},\\ \text{as } x_3:0\rightarrow2\pi, \quad(123)_{312}=(123)_{123}e^{ik_32\pi},\\ \vdots\qquad\qquad\qquad\qquad\vdots\qquad\qquad\qquad\qquad\end{aligned}$$ In combination: $$(123\ldots)_{123\ldots N}=e^{i(k_1+k_2+\ldots+k_N+\beta)2\pi}(123\ldots N)_{123\ldots N}$$ reflecting angular momentum conservation, sum of all the wavenumbers plus the flux must be integer on a ring. In short, the BA equation is solved by finding N roots of a simple equation subject to the angular momentum constraint: $$\label{BA eqs} k-\lambda=\frac{c}{2}\cot{\pi k}, \qquad\sum_j^Nk_j=n-\beta,\quad n\in\mathbb{Z}.$$ So far our treatment implicitly assumed repulsive interactions. In which case, all the wavevectors $k_i$ are real. The ansatz can easily be extended to attractive interactions yielding exactly the same equations Eq.\[BA eqs\] [@McGuireAttractive]. However, for negative $c$ two of the roots will be complex, as the $\delta$-potential in one dimension has only a single bound state. ![ Ground state energy vs. interaction strength for $N=1000$ particles and zero total angular momentum. Numerical solution of BA equation (dots) Eq.\[BA eqs\] are virtually indistinguishable from analytical solution (circles) $E=$(Fermi energy of $N-1$ fermions)$+\Delta E$. Error between the numerical and analytical solutions are too small to observe even in the regimes where the assumptions for analytical calculation fails. []{data-label="E vs c"}](Fig4){width="47.00000%"} SOLUTION OF THE BA EQUATION =========================== The $\cot{\pi k}$ term in Eq.\[BA eqs\] diverges at every integer k, thus, regardless of the value of $\beta\:(\text{or }\lambda)$ there is a root between every consecutive integer (Fig.\[cots\]). By changing the value of $\lambda$, all roots can be adjusted so that the total angular momentum constraint is satisfied. All the eigenstates in this problem can be labeled identically by choosing N distinct integers corresponding to the different branches of $\cot{\pi k}$ and the total angular momentum $n\in\mathbb{Z}$. The energy of an eigenstate is simply the sum of squares of all wavenumbers $$E=\sum_{i=1}^Nk_i^2.$$ ![(Color online) A representation of graphical solution to BA equation. The $\cot{\pi k}$ term in Eq.\[BA eqs\] diverges at every integer $k$ and there is a root between every consecutive integer independently from $\lambda(\beta)$. By changing the value of $\lambda$, all roots can be adjusted so that the total angular momentum constraint is satisfied. []{data-label="cots"}](Fig5){width="47.00000%"} For the simplest case of $\beta=0$, the ground state corresponds to $\lambda=0$ and the total angular momentum $n=0$. The roots $k$ are distributed symmetrically around zero for even N, hence, automatically satisfy the total angular momentum condition. The wavevectors for the ground state are in the N branches of *cot* from $-N/2$ to $N/2$. Excitations above this ground state can be generated by two procedures. First, by changing $\lambda$, N roots which are on the same branches of *cot* can be generated so as to create an eigenfunction with non-zero total angular momentum ($n\neq0$). Second, at least one of the roots can be chosen to reside on a branch that is not occupied for the ground state. For such a particle-hole excitation $\lambda$ must be adjusted to ensure the total angular momentum constraint. Inclusion of the magnetic field affects only the total angular momentum constraint. As that constraint is defined only up to an integer (*n*), the problems with values of $\beta$ differing by an integer are identical. Eigenstates for flux $\beta$ which have total angular momentum *n* are also eigenstates for flux $\beta+1$ which have total angular momentum $n+1$. This is a restatement of flux quantization. We can analyze the system by considering flux values between $-1/2<\beta<1/2$. However, in an experimental setting slowly increasing the value of the flux through the ring is a useful method to access excited states. As the flux is increased adiabatically from zero to one, the ground state evolves to an eigenstate which has its roots exactly in the same branches as the ground state, but, has a total angular momentum of minus one at zero flux (Fig.\[E vs beta\]). The crossings between different eigenstates do not pose a problem for adiabatic evolution as only states with different total angular momentum *n* can be degenerate in energy. The BA equation Eq.\[BA eqs\] can be very efficiently solved once the regions for the roots are determined. We used the Newton-Raphson algorithm to find a solution within a particular region. As all the roots depend monotonically on $\lambda$, another Newton-Raphson search is employed to satisfy the total angular momentum condition. We have found numerical solutions for systems of up to 10000 particles with high accuracy. Although numerically solving the BA equation is efficient and accurate, an analytic solution can provide more insight about the physics of the system. Analytic formulae for energy, angular momentum and effective mass also would be desirable to make correspondence with experimental observations. In the limit of strong interactions $1/c\ll1$ and large particle number $N/c\gg1$, such an analytic form can be obtained by approximating the roots of the BA equation. In this limit, because the *cot* diverges quickly near integers, most of the roots are close to integers. Apart from the few roots near $k\sim\lambda$, the deviation of the root $\Delta$ from an integer $s$ is small [@McGuire]. Solving for this small deviation we find that the roots occur at $$\begin{aligned} \label{k guess} k_s^+&\!\!\!=\!&s+\frac{1}{\pi}\text{acot}\frac{2}{c}(s-\lambda),\nonumber\\ k_s^-&\!\!=\!\!&-s-\frac{1}{\pi}\text{acot}\frac{2}{c}(s+\lambda),\quad s=0,1,\ldots,\frac{N}{2}-1,\quad\end{aligned}$$ where $acot$ is defined in the continuous region ($0,\pi$) for Eqs.\[k guess\] to be accurate guesses. Here we have restricted $s$ to analyze the ground state and excited states with roots on the same *cot* branches. Applying the total angular momentum condition we get, $$\begin{aligned} n-\beta&\!=\!\!\!&\sum_{s=0}^{N/2-1}k_s\nonumber\\ &\!\!\!\!=\!\!\!&\frac{1}{\pi}\sum_s^{N/2-1} \Big(\text{acot}\frac{2}{c}(s-\lambda)-\text{acot}\frac{2}{c}(s+\lambda) \Big)\nonumber\\ &\!\!\!\!=\!\!\!&\frac{c}{2\pi}\int_0^{x_F} {\mathrm{d}}x\Big(\text{acot}(x-b)-\text{acot}(x+b)\Big), \label{mom.rel.sum}\end{aligned}$$ with $b=2\lambda/c$ and $x_F=(N-1/2)/c$. Here the initial assumption of strong interactions and large particle numbers allow us to approximate the sum by an integral. For the ground state and the first few excited states $n-\beta$ is small compared to $N$ and the integral can be approximated as $$n-\beta=\frac{c}{2\pi}\int_{x_F-b}^{x_F+b} {\mathrm{d}}x \text{ atan}{x}\approx\frac{cb}{\pi} \text{atan}{x_F}.\label{mom.rel.}$$ Through this relation $b$, hence $\lambda$, is obtained for any flux value, allowing us to find expressions for all the roots in a self consistent way. Energy ------ Using these expressions for the roots, the total energy is $$\begin{aligned} E&\!=\!\!\!&\sum_{s=0}^{N/2-1}k_s^2\nonumber\\ &\!\!\!\!=\!\!\!&\sum_s^{N/2-1} \bigg\{ 2s^2+ \frac{2s}{\pi}\Big(\text{acot}\frac{2}{c}(s-\lambda)+\text{acot}\frac{2}{c}(s+\lambda)\Big) \nonumber\\ &&+\frac{1}{\pi^2}\Big((\text{acot}\frac{2}{c}(s-\lambda))^2+(\text{acot}\frac{2}{c}(s+\lambda))^2\Big) \bigg\}.\end{aligned}$$ The first term above is the total ground state energy of $N-1$ non-interacting fermions. Interactions result in the second and third terms which are first and second order corrections in our expansion. When $\Delta$’s are small, the third term is negligible. In this limit, the energy shift due to interactions is for repulsive interactions: $$\Delta E=cb(n-\beta)+\frac{c^2x_F^2}{4}-\frac{c^2}{4\pi}\bigg\{\Big((x_F+b)^2+1\Big)\text{atan}{(x_F+b)} +\Big((x_F-b)^2+1\Big)\text{atan}{(x_F-b)}-2x_F\bigg\}, \label{deltaE analy}$$ $$\text{with } \:b=\frac{\pi(n-\beta)}{c\text{ atan}(x_F)}.$$ This approximate form for energy successfully reproduces numeric results for particle numbers as small as 4 throughout all the interaction range. Ground state energy as a function of interaction strength is plotted for a typical case in Fig.\[E vs c\] for 1000 particles at $\beta=0.2$ flux. The deviation between numerical and analytical results are too small to observe in this plot. For attractive interactions, the $\delta$-function interaction supports one bound state in one dimension. Corresponding imaginary wavevectors appear as solutions of the BA equation. For $k=\alpha+i\sigma$ with $(\alpha,\sigma)\in\mathbb{R}$, the BA equation has only two roots with $\sigma\neq0$. The charged particle is bound with only one of the background fermions. When $1/|c|\ll1$, the complex roots are at $k=\lambda\pm i c/2$ while the rest of the roots preserve their form of Eqs.\[k guess\]. If the bound state is narrow, Pauli repulsion between the fermion in the bound pair and background fermions decreases the effective interaction. Within these approximations, we analytically calculate the total energy, for attractive interactions: $$\Delta E=-\frac{c^2(b^2+1)}{2}+cb(n-\beta)+\frac{c^2x_F^2}{4}+\frac{c^2}{4\pi}\bigg\{\Big((x_F+b)^2+1\Big)\text{atan}{(x_F+b)} +\Big((x_F-b)^2+1\Big)\text{atan}{(x_F-b)}-2x_F\bigg\}, \label{deltaE analy -c}$$ $$\text{with } \:b=\frac{(n-\beta)}{c\Big(1-\frac{1}{\pi}\text{atan}(x_F)\Big)}.$$ Angular Momentum ---------------- To understand the physics of the system and make correspondence to possible experiments, it is important to calculate other measurable quantities. In particular, for this system we are interested in how the dynamics of the impurity particle is affected by the fermion background. To this end, it is instructive to calculate angular momentum carried by the impurity $L_1$ and the related effective mass. As the impurity is interacting with the fermions, this effective mass is not only the mass of the impurity but also gets a contribution from the fermions dragged along with it. Such a compound object is generally called the polaron state or dimer state especially for attractive interactions. As stated above, one of the most interesting physical quantities in this system is the angular momentum carried by the charged particle, represented by the operator $\hat{P}_1=-i\frac{\partial}{\partial x_1}$. As this particle is coupled to the external magnetic field, $\hat{P}_1$ is the canonical momentum not the kinetic momentum. However, canonical momentum is the quantity that is generally measured by expansion imaging in artificial magnetic field experiments. The expectation value of $\langle\hat{P}_1\rangle=L_1$ is easily obtained by taking the derivative of the total energy with respect to flux, $$L_1=\frac{-1\:}{2}\,\frac{\partial\Delta E}{\partial\beta}.$$ ![ Angular momentum of the impurity vs. interaction strength for $N=100$ particles for (a) attractive, (b) repulsive interactions from the analytic calculation, numerical solutions produce the same results. In the non-interacting limit, the impurity carries all the angular momentum ($n-\beta$). $L_1$ saturates to almost zero for infinitely strong repulsive interactions as the total angular momentum is shared equally between all particles. The same behavior holds for excited states. For strongly attractive interactions, $L_1$ saturates to half the total value signifying dimer formation with one background particle. The insets in both figures focus on the weak interaction limit. []{data-label="L1 vs c1"}](Fig6){width="48.00000%"} Using the approximate form for the energy Eq.\[deltaE analy\], we obtain $$\label{L1eq} L_1=\frac{\pi(n-\beta)}{\text{atan}{x_F}}-\frac{c}{4\text{atan}{x_F}}\bigg\{(x_F+b)\text{atan}{(x_F+b)} -(x_F-b)\text{atan}{(x_F-b)} \bigg\}.$$ This form is valid for positive $c$ and easy to interpret. In the non-interacting limit, the canonical momentum of the charged particle is fixed by the external flux. Hence, all the angular momentum is carried by the charged particle. As interaction is turned on, the charged particle drags the background fermions and transfers some of its angular momentum to them. Stronger interactions increase the fraction of the transferred angular momentum and in the limit of infinitely repulsive interactions angular momentum is equally shared by N particles. On the other hand, for strong attractive interactions, $L_1$ saturates to half of the angular momentum in the system proving the formation of a dimer with one background fermion. ![(Color online) The angular momentum of the charged particle vs. flux at varying interaction strength for $N=100$ particles. As expected $L_1$ depends linearly on flux. The slope of the line decreases with increasing interaction strength indicating higher values of effective mass of the impurity.[]{data-label="L1 vs beta"}](Fig7){width="45.00000%"} The behavior of $L_1$ is displayed in Fig.\[L1 vs c1\] and Fig.\[L1 vs beta\] as a function of $c$ and $\beta$. Even for $N=100$ particles, the difference between numerical calculation of the derivative and the expression given above is negligible. As a function of interaction strength, the rapid decrease and eventual saturation of $L_1$ validates the scenario discussed above. The linear dependence on flux is expected, however, the slope of $L_1$ decreases as interaction gets stronger. This slope carries valuable information as it is related to the effective mass of the composite excitation formed by the impurity and background fermions. Effective Mass -------------- ![ (Color online) Effective mass of the impurity vs. interaction strength for $N=50$ particles and zero total angular momentum $n=0$. (a) For attractive interactions, $m^*$ saturates to twice the mass of the impurity due to the formation of a tightly bound pair. The inset shows the behavior around zero interaction in more detail. For attractive interaction, the effective mass (given by Eq.\[ddE\]) is almost insensitive to flux change. (b) For repulsive interactions, $m^*$ converges to $N$. As flux $\beta$ increases, this saturation gets faster. The dependence on the flux is more prominent for small particle numbers. []{data-label="m* vs c"}](Fig8){width="48.00000%"} We define the effective mass as $$\label{m* eq} m^*=\frac{2}{\frac{\partial^2\Delta E}{\partial\beta^2}}.$$ In the non-interacting limit, the effective mass is equal to $m$, however, its behavior is very different for attractive and repulsive interactions. As repulsive interactions are increased, it gets harder for the impurity to tunnel through the fermions and the dragged particles increase the effective mass. The increase in the effective mass saturates only when all the particles are moving together with the impurity. Thus, at large repulsive interaction, the effective mass reaches $Nm$. For weak attractive interactions, the first effect is once again the drag increasing the effective mass. However, the attractive $\delta$-function has a single bound state in one dimension. Thus, the impurity captures one of the background fermions and as the size of the bound state gets smaller, Pauli exclusion effectively repels the other fermions. The effective mass for attractive interactions increases and reaches $2m$ for infinitely attractive interaction where a dimer is formed from the impurity and one fermion. For attractive interactions, the analytical expression in the strongly interacting limit is useful to calculate the dimer mass, $$\begin{aligned} \label{ddE} \frac{\partial^2\Delta E}{\partial\beta^2}&=&\frac{2}{1-\frac{\text{atan}{(x_F)}}{\pi}}-\frac{1}{(1-\frac{\text{atan}{(x_F)}}{\pi})^2}\nonumber\\ &&+\frac{1}{2\pi(1-\frac{\text{atan}{(x_F)}}{\pi})^2} \bigg\{\text{atan}{(x_F+b)}+\text{atan}{(x_F-b)}+\frac{x_F+b}{1+(x_F+b)^2}+\frac{x_F-b}{1+(x_F-b)^2} \bigg\}.\end{aligned}$$ We calculated the effective mass numerically and analytically. For the ground state, the above scenario is validated by these calculations (see Fig.\[m\* vs c\]). The dependence of the effective mass on the external magnetic field is strongest for small particle number as this limit is the strongly interacting limit in one dimension. As the number of fermions increases, effective mass in the ground state has weak dependence on $\beta$. In this case, effective mass is essentially determined locally as the ability for the impurity particle to complete a full rotation is hampered. ![ Resonant behavior in $m^*$ for $\beta$ comparable to $N$. When the drag effect applied by the background particles overcomes the driving force of the magnetic field, $m^*$ can become negative. When the second derivative of the energy with respect to flux becomes zero, $m^*$ diverges. This divergence does not change the infinitely strong interaction limit. []{data-label="m* vs c resonance"}](Fig9){width="45.00000%"} The utility of an external magnetic field is the access it provides to excited states through adiabatic pumping. Excited states in this system are expected to be stable due to angular momentum conservation. It is thus reasonable to expect effective mass measurements to be carried out on such states in a cold atom setting. For the excited states, with angular momentum $|n|<N/4$ the main effect is faster saturation of effective mass as $c$ increases. However, for higher excited states, there is resonant behavior (Fig.\[m\* vs c resonance\]). Due to the nature of BA solution, a state for which all the roots are on the *cot* branches from $-N/2$ to $N/2$ can have at most $N/2$ units of angular momentum. When the total angular momentum of an excited state is comparable to particle number N, the sharing of this angular momentum between the impurity and the background is limited by this constraint. Thus, it is possible for this system to support negative effective mass if the external force acted by the magnetic field is overcome by the back reaction from the fermions. We numerically find this behavior for both low and high particle number (see Fig.\[m\* vs c resonance\]). Experimentally this effect should be more accessible for small number of particles as it is easier to pump angular momentum comparable to particle number. ![ Effective momentum density in k-space for different interaction strengths. In the strongly interacting limit, the distance between adjacent BA roots (wavevectors) is one. For small c, the roots are closer to each other around $(n-\beta)$. The impurity carrying $n-\beta$ units of angular momentum in the non-interacting limit first disturbs the fermions which are momentum matched to that value. []{data-label="k-dist beta10.2"}](Fig10){width="45.00000%"} Correlations ------------ Apart from the single particle properties related to the impurity, it is instructive to look at global properties to understand how the external particle disturbs the one dimensional Fermi liquid. A common way to visualize the disturbance in the Fermi sea is to plot the deviation of the distance between the BA roots (wavevectors) from one. For an undisturbed Fermi sea, this deviation is always one. For a weakly interacting impurity, the deviation is confined to a narrow region in k-space around $n-\beta$ (see Fig.\[k-dist beta10.2\]). This is expected as the impurity carrying $n-\beta$ units of angular momentum in the non-interacting limit first starts dragging fermions which are matched in momentum. As the strength of repulsion increases, so does the effected region in k-space, however, the deviation gets smaller. For infinitely repulsive interactions, the impurity becomes indistinguishable from the background fermions (Fig.\[k-dist beta10.2\]). For highly excited states where $n$ is comparable to N, particle-hole excitations complicate this picture similar to the effect we discussed for the effective mass. Another important physical property is the two-particle correlation function. Although for $\delta$-function interactions only the value of this function at zero determines the interaction energy, its general form is experimentally accessible through Hanbury-Brown-Twiss[@HanburyBrownTwiss] type measurements. This correlation also can be regarded as the real-space form of the bound state created by the impurity. To calculate the two-particle correlation function, we need to determine the coefficients of the plane waves in each region. Following McGuire [@McGuire] we choose the first coefficient in the first region $x_1<x_2<\ldots<x_N$, $$(123..N)_{123..N}=(1-e^{i2\pi k_1}).$$ Other coefficients in this region determined by BCs yield very similar expressions. The wavenumber associated with the distinguishable particle appears in the exponent and the sign of the permutation multiplies the coefficient: $$\begin{aligned} (213..N)_{123..N}&=&-(1-e^{i2\pi k_2}),\\ (312..N)_{123..N}&=&(1-e^{i2\pi k_3})\\ &\vdots& \nonumber\end{aligned}$$ The coefficients in other regions are related to the same coefficient in the first region with a phase factor determined by the PBCs. This phase factor is a full circle rotation around the ring of the particles that $x_1$ has to pass to be in the given region. Thus, the momenta belonging to the particles that $x_1$ has passed multiply the coefficient, e.g. $$(21354..N)_{2314..N}=(1-e^{i2\pi k_2})e^{i2\pi(k_1+k_3)}\ldots$$ In the simple form, the wave functions are not normalized, but we normalize the correlation function at the end. The two-particle correlation function in any state is given as $$g_{12}(x_1,x_2)=\int_0^{2\pi}{\mathrm{d}}x_3\cdots\int_0^{2\pi}{\mathrm{d}}x_N \Psi^*\Psi.$$ ![(Color online) Two-particle correlation function for $N=2$ particles. As expected, $g_{12}$ at zero separation decreases with increasing interaction strength. For weak interactions, the correlation function at zero decreases with increasing flux. However, for strong interactions, the correlation is almost insensitive to flux change due to the fermionization of the charged particle. []{data-label="g12N2 fig"}](Fig11){width="47.00000%"} For the $N=2$ particle case, the correlation function is simply the absolute square of the wave function. As could be expected, the correlation function is highly affected by the flux for the two-particle case. Using the numerically and analytically found wavenumbers in the expression, $$\begin{aligned} g_{12}(x_1,x_2)&=4-2\cos(2\pi k_1)-2\cos(2\pi k_2)\nonumber\\ &-8\sin(\pi k_1)\sin(\pi k_2)\cos(k_1-k_2)(x_2-x_1-\pi),\end{aligned}$$ we observe that inclusion of the flux generally decreases the two-particle correlation function (Fig.\[g12N2 fig\]). However, if the interactions are strong enough so that the particles are almost fermionized, this decrease is very small. It is also notable that although the flux breaks time-reversal symmetry and the wave functions choose a direction on the ring, correlation function is even with respect to $x_1-x_2$. This property holds for any particle number. ![(Color online) Two-particle correlation function for $N=50$ particles. (a) For weak interactions, Friedel oscillations occur as interference of two waves with wavelengths related to $k_F-\beta$ and $k_F+\beta$. (b) At strong interactions, the correlation becomes zero at zero separation since the impurity is effectively indistinguishable and $g_{12}$ and the frequency of Friedel oscillations are almost insensitive to flux change. []{data-label="g12N50 fig"}](Fig12){width="47.00000%"} For the general N particle case, we arrange the wave function in a better form to evaluate the integrals. We assign two wavenumbers to $x_1$ and $x_2$, and the rest of the particles are represented by a Slater determinant since they are indistinguishable fermions. For example, if we have $k_1,k_4$ associated with $x_1,x_2$ respectively, the Slater determinant is represented by $\mathbb{D}_{14}$ indicating the use of all wavenumbers except $k_1$ and $k_4$ in the exponents, $$\mathbb{D}_{14}= \begin{vmatrix} e^{ik_2x_3} & e^{ik_3x_3} & e^{ik_5x_3}\,\ldots \\\\ e^{ik_2x_4} & e^{ik_3x_4} & e^{ik_5x_4}\,\ldots \\\\ \vdots & & & \\\\ e^{ik_2x_N} & e^{ik_3x_N} & e^{ik_5x_N}\,\ldots \end{vmatrix}.$$ Hence, the wave function in the first region can be written as, $$\begin{aligned} \Psi=&\Big((12\ldots N)_{12\ldots N}+(21\ldots N)_{12\ldots N} \Big)\mathbb{D}_{12}\nonumber\\ &\:+\Big((13\ldots N)_{12\ldots N}+(31\ldots N)_{12\ldots N} \Big)\mathbb{D}_{13}+\ldots\end{aligned}$$ Integrating $\Psi^*\Psi$ over $x_3,\ldots,x_N$, the Slater determinants are orthogonal in large particle number limit, as the outer roots of *cot*’s are very close to integer values. The correlation function is then expressed as a sum over pairs of momenta associated with $x_1$ and $x_2$, $$\begin{aligned} g_{12}(x)=&\sum^N_{t<s}\sum^N|(ts\ldots N)|^2+|(st\ldots N)|^2\\ &+2Re\Big\{(ts\ldots N)^*(st\ldots N)e^{i(k_t-k_s)x} \Big\},\end{aligned}$$ where $x=x_2-x_1$. Here, $x>0$ but the symmetry of the correlation function for $x<0$ can be easily seen by using the region $x_2<x_3\ldots x_N<x_1$ instead of the first region. Finally, the correlation function is normalized to average density on the ring so that it saturates to one. The correlation function for $N=2$ and $N=50$ particles are given in Fig.\[g12N2 fig\] and Fig.\[g12N50 fig\] respectively, for different interaction strengths and flux. As expected, the correlation function at zero separation $g_{12}(0)$ decreases with increasing interaction strength until it saturates to zero. The other important feature of the correlation function is the Friedel oscillations [@Friedel] reflecting the sharpness of the Fermi surface in one dimension. The two-particle correlation function is a local quantity while the external flux changes the system properties globally. For any pair to feel the effect of the flux, one of the particles must go a full circle through the ring. Hence, as could be expected, the effect of the artificial magnetic field on the correlation function decreases as the number of particles increases or if they interact strongly. Even for the lowest lying excited states, we find the primary effect of the flux is on the Friedel oscillations while the shape of the correlation hole is unchanged. Finally, we calculate a thermodynamic quantity which is also related to $g_{12}(0)$. Derivative of the energy with respect to interaction strength $c$ gives us interaction potential, so, the kinetic and interaction contributions to the total energy can be separated. In Fig.\[Kin Pot\], one can notice that the interaction energy makes a peak and then decreases for increasing $c$. The initial increase is expected, however, as interactions become stronger, the tendency of the fermions to avoid the impurity dominates and the impurity is effectively fermionized. This is apparent in the $\delta$-function BC Eq.\[discont genel\]. Additionally, the interaction potential is equal to the correlation function at zero $g_{12}((x_2-x_1)=0)$ times the interaction strength which reproduces the results obtained by taking the derivative of the total energy. ![ (Color online) Kinetic energy of the particles vs. interaction strength for $\beta=0.2$ and $\beta=10.2$ for $N=50$ particles. The inset shows the interaction potential contribution to the total energy. The initial increase in interaction energy follows the increase in the interaction strength. However, beyond a certain strength, the tendency of the fermions to avoid the impurity is more dominant. These plots are obtained by taking the derivative of the total energy with respect to $c$. Alternatively, the interaction potential energy is also obtained by using the two-particle correlation function at zero separation. Both results are plotted in the inset showing remarkable agreement. []{data-label="Kin Pot"}](Fig13){width="48.00000%"} CONCLUSION ========== The problem of a single impurity interacting with a fermion background has attracted attention of the condensed matter society for years. In this paper, we argue that an artificial gauge field coupling exclusively to the impurity is an effective tool to probe the physics of this system at any interaction strength. We consider a Fermi gas in a narrow ring trap and an artificial magnetic field coupling only to a single impurity. We solve this system exactly by using the BA for contact interactions and calculate the dependence of measurable quantities on the external magnetic flux. We observe this dependence for total energy, angular momentum of the charged particle, effective mass and the two-particle correlation function. Using an artificial magnetic field in this system has two advantages. The usual measurement tools such as expansion imaging become probes of thermodynamic quantities by comparing measurements at different flux values. For example, the change of the momentum carried by the impurity caused by the magnetic field is a direct probe of the effective mass of the impurity. The second advantage is obtained by adiabatically increasing the flux value. Although the Hamiltonian of the system is periodic with flux, adiabatic evolution connects the ground state at zero flux to excited states at integer flux. In a cold atom experiment, such excited states can be expected to have long lifetimes due to total angular momentum conservation. Thus, we have calculated the physical properties for not only the ground state but also for excited states adiabatically connected to it. Our results show that the system can be described by a simple physical picture. The charged particle interacting with the background particles drags them along with itself around the ring. In the non-interacting limit, all of the angular momentum in the system is carried by the impurity. As interactions are turned on, fermions which are close to the impurity in momentum are disturbed more and start to gain momentum. At the limit of infinitely repulsive interactions, the charged particle is effectively indistinguishable from the background and the total momentum is shared equally between all particles. The pair correlation function also confirms this picture. The value of the correlation function near zero is mostly insensitive to the external flux while away from the correlation hole frequency of the Friedel oscillations sensitively depends on it. For strongly repulsive interactions, the effective mass saturates the total mass of the particles since it is dragging all of the background fermions along with itself around the ring. For attractive interactions, the impurity forms a bound pair with one of the fermions. The effect of dimer formation can be clearly seen in the angular momentum and the effective mass. For infinitely strong attractive interactions, angular momentum carried by the impurity saturates half the value of total angular momentum and the effective mass saturates twice the mass of the particle which confirm the presence of the dimer as a composite particle. The physical properties calculated in this paper are experimentally accessible through the standard tools of ultracold atom experiments. While artificial magnetic fields have been demonstrated in a variety of settings, they have not been used in combination with a toroidal trap to our knowledge. We believe our exact results would be relevant for such an experiment. F.N.Ü. is supported by Türkiye Bilimsel ve Teknolojik Arat[i]{}rma Kurumu (TÜBİTAK) Scholarship No. 2211. M.Ö.O. was supported by Türkiye Bilimsel ve Teknolojik Arat[i]{}rma Kurumu (TÜBİTAK) Grant No. 112T974. B.H. is supported by Türkiye Bilimsel ve Teknolojik Arat[i]{}rma Kurumu (TÜBİTAK) Grant No. 113F334.

--- abstract: | Motivated by applications in redistricting, we consider the uniform capacitated $k$-median and uniform capacitated $k$-means problems in bounded doubling metrics. We provide the first QPTAS for both problems and the first PTAS for the uniform capacitated $k$-median problem for points in ${\mathbb{R}}^2$. This is the first improvement over the bicriteria QPTAS for capacitated $k$-median in low-dimensional Euclidean space of Arora, Raghavan, Rao \[STOC 1998\] ($1+{\varepsilon}$-approximation, $1+{\varepsilon}$-capacity violation) and arguably the first polynomial-time approximation algorithm for a non-trivial metric. author: - | Vincent Cohen-Addad\ Sorbonne Université, CNRS, LIP6 bibliography: - 'facilitylocationptas.bib' title: Approximation Schemes for Capacitated Clustering in Doubling Metrics ---

--- abstract: 'The equilibrium configuration of an engineering structure, able to withstand a certain loading condition, is usually associated with a local minimum of the underlying potential energy. However, in the nonlinear context, there may be other equilibria present, and this brings with it the possibility of a transition to an alternative (remote) minimum. That is, given a sufficient disturbance, the structure might buckle, perhaps suddenly, to another shape. This paper considers the dynamic mechanisms under which such transitions (typically via saddle points) occur. A two-mode Hamiltonian is developed for a shallow arch/buckled beam. The resulting form of the potential energy—two stable wells connected by rank-1 saddle points—shows an analogy with resonance transitions in celestial mechanics or molecular reconfigurations in chemistry, whereas here the transition corresponds to switching between two stable structural configurations. Then, from Hamilton’s equations, the analytical equilibria are determined and linearization of the equations of motion about the saddle is obtained. After computing the eigenvalues and eigenvectors of the coefficient matrix associated with the linearization, a symplectic transformation is given which puts the Hamiltonian into normal form and simplifies the equations, allowing us to use the conceptual framework known as tube dynamics. The flow in the equilibrium region of phase space as well as the invariant manifold tubes in position space are discussed. Also, we account for the addition of damping in the tube dynamics framework, which leads to a richer set of behaviors in transition dynamics than previously explored.' author: - Jun Zhong - 'Lawrence N. Virgin' - 'Shane D. Ross' title: 'A Tube Dynamics Perspective Governing Stability Transitions: An Example Based on Snap-through Buckling' --- potential energy ,transients ,tube dynamics ,dynamic buckling ,invariant manifolds ,Hamiltonian Introduction ============ The nonlinear behavior of slender structures under loading is often dominated by a potential energy function that possesses a number of stationary points corresponding to various equilibrium configurations [@WiVi2016; @collins2012isomerization]. Some are stable (local minima, or ‘well’), some are unstable (local maxima or ‘hill-top’), and some correspond to saddle points, i.e., a shape with opposite curvature in different directions, but still unstable, having both stable and unstable directions. Interestingly, although difficult to observe experimentally, it is these saddle points that can have a profound organizing effect on global trajectories in a dynamics context. Thus, under a nominally fixed set of loads or a given configuration we may have the situation in which a system is at rest in a position of stable equilibrium, but, given sufficiently large perturbation (input of energy) may transition to a remote stable equilibrium [@virgin2017geometric], or even collapse completely [@das2009symmetry; @das2009pull]. The path taken during this transition is associated with the least energetic route, and this will typically correspond to a passage close to a saddle point: it is easier to take a path around a mountain than going directly over its peak. For a single mechanical degree of freedom the transition from one potential energy minimum to another is relatively unambiguous [@mann2009energy; @Thompson1984]. We can think of a twin-well oscillator and how it has no choice but to pass over an intermediate hilltop in transitioning to an adjacent minimum. For high-order systems trajectories have many more possible paths. But a system with two mechanical degrees of freedom (configuration space), and thus a 4 dimensional phase space, offers an intermediate situation: compelling conceptual clarity (i.e., the potential energy can be thought of as a surface or landscape), but still retaining a wider range of potential behavior over and above the aforementioned single oscillator (i.e., multiple ways of traversing and perhaps escaping from one potential well to another). For the two degree of freedom system, the analog of the hilltop is the saddle point of the potential energy surface. The linearized dynamics near such a point yield an oscillatory mode and an exponential mode, with both asymptotically stable and unstable directions. For energies slightly above the saddle point, there is a bottleneck to the energy surface [@NaRo2017; @KoLoMaRo2000]. Transitions from one side of the bottleneck can be understood in terms of sets of trajectories which are bounded by topological cylinders. The transition dynamics, which has in some contexts been known as tube dynamics [@Conley1968; @LlMaSi1985; @OzDeMeMa1990; @DeMeTo1991; @DeLeon1992; @Topper1997; @KoLoMaRo2000; @GaKoMaRo2005; @GaKoMaRoYa2006; @MaRo2006; @KoLoMaRo2011], has been developed for studying transitions between stable states (the potential wells) in a number of disparate contexts, and here it is applied to a structural mechanics situation in which snap-through buckling [@collins2012isomerization] is the key phenomenological transition. Conditions are determined whereby the initial energy imparted to the system is characterized in terms of subsequent escape from the initial potential well. The Paradigm: Snap-through of an Arch/Buckled Beam ================================================== A classic example of a saddle-node bifurcation in structural mechanics is the symmetric snap-through buckling of a shallow arch, in an essentially co-dimension 1 bifurcation [@Thompson1984]. However, if the arch (or equivalently a buckled beam) is [*not*]{} shallow then the typical mechanism of instability is an asymmetric snap-through, requiring two modes (symmetric and asymmetric) for characterization, and the instability corresponds to a subcritical pitchfork bifurcation. In both of these cases the transition is sudden and associated with a fast dynamic jump, since there is no longer any locally available stable equilibrium. This behavior is generic regardless of boundary conditions and is also exhibited by the laterally-loaded buckled beam [@Murphy1996; @Wiebe2013]. We shall focus on this latter example, for relative simplicity of introduction. The essential focus here is that the underlying potential energy of this system consists of two potential energy wells (the original unloaded equilibrium and the snapped-through equilibrium), an unstable hilltop (the intermediate, straight, unstable equilibrium) and two saddle-points. The symmetry of this system is broken by small geometric imperfections. The question is: [*how does the system escape its local potential energy well*]{} in a dynamical systems sense? Suppose we have a moderately buckled beam. If a central point load is applied then the beam deflects initially in a purely symmetric mode, as shown by the red line in Fig. \[fig:arch\](a), following the $\alpha$ loading path. ![ (a) A schematic load-deflection characteristic, (b) the two dominant degrees of freedom.[]{data-label="fig:arch"}](arch1_mod.png){width="100.00000%"} Upon a quasi-static increase in the load $P$, point $C$ is reached (a subcritical pitchfork bifurcation) and the arch quickly snaps-through (a thoroughly dynamic event) with a significant asymmetric component in the deflection and the system settles into its inverted position $D$ [@virgin2017geometric]. This behavior is captured by considering a two-mode analysis: sag $S$ (symmetric) and angle $A$ (asymmetric), or alternatively we can use the harmonic coordinates $X$ and $Y$, respectively, corresponding to the modes in part (b). In an approximate analysis they might be the lowest two buckling modes or free vibration modes from a standard eigen-analysis. Suppose we load the beam to a value slightly below the snap value at $P_C$, and fix it at that value. In this case there will be the five equilibria mentioned earlier: three equilibria purely in sag (two stable and an unstable one between them), and two saddles, with significant angular components but geometrically opposed [@WiVi2016]. Small geometric imperfections (in $A$ and/or $S$) will break the symmetry and influence which path is more likely to be followed. In this fixed configuration we can then think of the system in dynamic terms, and consider the range of initial conditions (including velocity, perhaps caused by an impact force) that might push the system from a point on path $\alpha$ to a point on path $\phi$. #### Governing equations In this analysis a slender buckled beam with thickness $d$, width $b$ and length $L$ is considered. A Cartesian coordinate system $o \textendash xyz$ is established on the mid-plane of the beam in which $o$ is the origin, $x,y$ the directions along the length and width directions and $z$ the downward direction normal to the mid-plane. Based on Euler-Bernoulli beam theory [@zhong2016analysis; @WiVi2016], the displacement field $(u_1,u_3)$ of the beam along $(x,z)$ directions can be written as $$\begin{split} u_1(x,z,t)&= u(x,t)-z \frac{\partial w(x,t)}{\partial x}\\ u_3(x,z,t)&= w(x,t) \label{disp_field} \end{split}$$ where $u(x,t)$ and $w(x,t)$ are the axial and transverse displacements of an arbitrary point on the mid-plane of the beam. Considering the von Kámán-type geometrical nonlinearity, the total axial strain can be obtained as $$\begin{split} \varepsilon^*_x= \frac{\partial u}{\partial x} - z \frac{\partial^2 w}{ \partial x^2}+ \frac{1}{2} \left(\frac{\partial w}{\partial x}\right)^2 \label{total_strain} \end{split}$$ For a moderately buckled-beam, we need to consider the initial strain $\varepsilon_0$ produced by initial deflection $w_0$ which is $$\begin{split} \varepsilon_0 = -z \frac{\partial^2 w_0}{ \partial x^2} + \frac{1}{2} \left(\frac{\partial w_0}{\partial x}\right)^2 \label{init_strain} \end{split}$$ Then the change in strain $\varepsilon_x$ can be expressed as $$\begin{split} \varepsilon_x = \varepsilon^*_x - \varepsilon_0 = \frac{\partial u}{\partial x} - z \left(\frac{\partial^2 w}{\partial x^2} - \frac{\partial^2 w_0}{\partial x^2}\right) + \frac{1}{2} \left[ \left(\frac{\partial w}{\partial x}\right)^2 - \left(\frac{\partial w_0}{\partial x}\right)^2\right] \label{strain_change} \end{split}$$ Here we just consider homogeneous isotropic materials with Young’s modulus $E$, and allow for the possibility of thermal loading. The axial stress $\sigma_x$ can be obtained according to the one dimensional constitutive equation, as $$\begin{split} \sigma_x= E \varepsilon_x - E \alpha_x \Delta T \label{constitutive} \end{split}$$ where $\alpha_x$ is the thermal expansion coefficient and $\Delta T $ is the temperature increment from the reference temperature at which the beam is in a stress free state. Thermal loading is introduced as a convenient way of controlling the initial equilibrium shapes (and hence the potential energy landscape) via axial loading. The strain energy $\mathcal{V}(x,z,t)$ is $$\begin{split} \mathcal{V}(x,z,t) &= \frac{b}{2} \int_0^L \int_{- \frac{d}{2}}^{ \frac{d}{2}} \sigma_x \varepsilon_x \mathrm{d}z \mathrm{d}x \label{strain_energy} \end{split}$$ Ignoring the axial inertia term, the kinetic energy $\mathcal{T}(x,z,t)$ of the buckled beam is $$\begin{split} \mathcal{T}(x,z,t)= \frac{b}{2} \int_0^L \int_{- \frac{d}{2}}^{ \frac{d}{2}} \rho \dot w^2 \mathrm{d}z \mathrm{d}x \label{kinetic_energy} \end{split}$$ where $\rho$ is the mass density. In addition, the dot over the quantity is the derivative with respective to time. The governing equations can be obtained by Hamilton’s principle which requires that $$\begin{split} \delta \int_{t_0}^t \left[ \mathcal{T}(x,z,t) - \mathcal{V}(x,z,t) \right] \mathrm{d} t \label{Hamilton_prin} + \int_{t_0}^{t} \delta W_{nc} \mathrm{d}t =0 \end{split}$$ where $\delta$ denotes the variational operator, $t_0$ and $t$ the initial and current time. $\delta W_{nc}$ is the variation of the virtual work done by non-conservative force (damping) which is expressed as $$\begin{split} \delta W_{nc} = - c_d \dot w \delta w \end{split}$$ where $c_d$ is the coefficient of (linear viscous) damping. In subsequent analysis, and related to typical practical situations, the damping will be small. After some manipulation, the governing equations in terms of axial force $N_x$ and bending moment $M_x$ can be obtained as [@zhong2016analysis] $$\begin{split} &\frac{\partial N_x}{\partial x}=0\\ &\frac{\partial^2 M_x}{\partial x^2}+ N_x \frac{\partial^2 w}{\partial x^2} = \rho A \ddot w + c_d \dot w \label{govern_eq_force} \end{split}$$ where $N_x$ and $M_x$ are defined as $$\begin{split} \left(N_x, M_x \right)=b \int_{- \frac{d}{2}}^{ \frac{d}{2}} \end{split} \sigma_x \left(1,z \right) \mathrm{d} z \label{force_definition}$$ By using , and , the force $N_x$ and moment $M_x$ in can be rewritten as $$\begin{split} N_x&=E A \left[\frac{\partial u}{\partial x} + \frac{1}{2} \left( \left(\frac{\partial w}{\partial x}\right)^2 - \left(\frac{\partial w_0}{\partial x}\right)^2\right) \right] -N_T\\ M_x&=- E I \left(\frac{\partial^2 w}{\partial x^2} - \frac{\partial^2 w_0}{\partial x^2}\right) \label{force_moment} \end{split}$$ where $A$ and $I$ denote the cross-sectional area and moment of inertia; $N_T=EA \alpha_x \Delta T$, the axial thermal loads. Thus, $EA$ and $EI$ are the axial stiffness and bending stiffness, respectively. Here we just consider a clamped-clamped beam with in-plane immovable ends. The boundary conditions are $$\begin{split} x=0,L: u=w=\frac{\partial w}{\partial x}=0 \label{boundary_condi} \end{split}$$ Note that from the first equation in , it is clear that the axial force $N_x$ is constant along the axial direction. In this case, integrating the axial force along the $x$ axis and using the boundary conditions $u(0,t)=u(L,t)=0$, one can obtain $$\begin{split} N_x= \frac{E A}{2L} \int_0^L \left[ \left(\frac{\partial w}{\partial x} \right)^2 - \left(\frac{\partial w_0}{\partial x} \right)^2 \right] \mathrm{d} x -N_T \label{axial_force} \end{split}$$ Using $M_x$ in and $N_x$ in , the second equation in can be described in terms of the transverse displacement $w$ as [@WiVi2016] $$\begin{split} \rho A \ddot{w} + c_d \dot w + EI \left(\frac{\partial^4 w}{\partial x^4} - \frac{\partial^4 w_0}{\partial x^4}\right) + \left[N_T - \frac{EA}{2L} \int_0^L \left( \left(\frac{\partial w}{\partial x} \right)^2 - \left(\frac{\partial w_0}{\partial x} \right)^2 \right) \mathrm{d} x \right] \frac{\partial^2 w}{\partial x^2} =0 \label{eq:PDE} \end{split}$$ where $w$ and $w_0$ are the current deflection and initial geometrical imperfection, respectively; $\rho$ is the mass density; $c_d$ is the damping coefficient; $A$ and $I$ are the area and the moment of inertia of the cross-section, respectively; $E$ is the Young’s modulus. Given the immovable ends it is natural to consider the effective externally applied axial force to be replaced by a thermal loading term: this is the primary destabilizing nonlinearity in the system. As mentioned earlier, clamped-clamped boundary conditions are considered. Thus we make use of the mode shapes $$\begin{split} &\phi_n = \alpha_n \left[\sinh \frac{\beta_n x}{L} - \sin \frac{\beta_n x}{L} + \delta_n \left(\cosh \frac{\beta_n x}{L} - \cos \frac{\beta_n x}{L} \right) \right],\\ &\delta_n = \frac{\sinh \beta_n - \sin \beta_n}{\cos \beta_n - \cosh \beta_n},\\ &\cos \beta_n \cosh \beta_n =1,\\ & \alpha_1 = -0.6186, \ \ \ \alpha_2 = -0.6631 \label{mode:Virgin} \end{split}$$ and describe the deflected shape in terms of a two-degree-of-freedom approximation $$\begin{split} w(x,t)&= X(t) \phi_1(x) +Y(t) \phi_2(x),\\ w_0(x)&= \gamma_1 \phi_1(x) + \gamma_2 \phi_2(x) \end{split}$$ where the initial imperfections are given by $w_0$. Substituting the assumed solution into the equation of motion  \[eq:PDE\] yields $$\begin{split} &\rho A \int_0^L \phi_i \ddot w \mathrm{d}x + c_d \int_0^L \phi_i \dot w \mathrm{d}x+ EI \int_0^L \frac{\partial^2 \phi_i}{\partial x^2} \left(\frac{\partial^2 w}{\partial x^2} - \frac{\partial^2 w_0}{\partial x^2} \right) \mathrm{d} x\\ & - \left[N_T - \frac{EA}{2L} \int_0^L \left(\left(\frac{\partial w}{\partial x} \right)^2 - \left(\frac{\partial w_0}{\partial x} \right)^2 \right) \mathrm{d}x \right] \int_0^L \frac{\partial \phi_i}{\partial x} \frac{\partial w}{\partial x} \mathrm{d}x =0 \label{virtual} \end{split}$$ Using the specific forms of $\phi_i$ in and noticing each mode shape is orthogonal, the nonlinear equations can be obtained $$\begin{split} & M_1 \ddot X + C_1 \dot X + K_1 \left(X - \gamma_1 \right) - N_T G_1 X - \frac{EA}{2L}G_1^2 \left(\gamma_1^2 X -X^3 \right) - \frac{EA}{2L} G_1 G_2 \left(\gamma_2^2 X -X Y^2 \right)=0\\ & M_2 \ddot Y + C_2 \dot Y + K_2 \left(Y - \gamma_2 \right) - N_T G_2 Y - \frac{EA}{2L}G_2^2 \left(\gamma_2^2 Y -Y^3 \right) - \frac{EA}{2L} G_1 G_2 \left(\gamma_1^2 Y -X^2 Y \right)=0 \label{odes} \end{split}$$ where $$\begin{split} \left(M_i, C_i \right) = \left(\rho A, c_d \right) \int_0^L \phi_i^2 \mathrm{d}x, \ \ K_i = EI \int_0^L \left(\frac{\partial^2 \phi_i}{\partial x^2} \right)^2 \mathrm{d}x, \ \ G_i = \int_0^L \left(\frac{\partial \phi_i}{\partial x} \right)^2 \mathrm{d}x \label{Galerkin-coefficient} \end{split}$$ The kinetic energy and potential energy, respectively, can be represented as $$\begin{split} \mathcal{T}(\dot X, \dot Y)=& \frac{1}{2} M_1 \dot X^2 + \frac{1}{2} M_2 \dot Y^2,\\ \mathcal{V}(X, Y)=& - K_1 \gamma_1 X - K_2 \gamma_2 Y + \frac{1}{2} K_1 X^2 + \frac{1}{2} K_2 Y^2 - \frac{1}{2} N_T\left( G_1 X^2 + G_2 Y^2 \right) \\ & - \frac{EA}{2L}G_1^2 \left(\frac{1}{2}\gamma_1^2 X^2 - \frac{1}{4}X^4 \right) - \frac{EA}{2L}G_2^2 \left(\frac{1}{2} \gamma_2^2 Y^2 - \frac{1}{4}Y^4 \right) \\ & - \frac{EA}{2L} \frac{G_1 G_2}{2} \left(\gamma_2^2 X^2 + \gamma_1^2 Y^2 -X^2 Y^2 \right).\\ % & + \frac{EA}{8L} \left(\gamma_1^2 G_1 + \gamma_2^2 G_2 \right)^2 + \frac{1}{2} K_1 \gamma_1^2 + \frac{1}{2} K_2 \gamma_2^2 \end{split}$$ For physically reasonable coefficients we have a number of equilibrium possibilities. For small values of $N_T$ we have an essentially linear system, dominated by the trivial (straight) equilibrium configuration, and thus an isolated center (minimum of the potential energy). This relates back to the situation in Figure \[fig:arch\] for a small value of $P$. But for larger values of $P$, for example a little below $P_c$, the system typically possesses a number of equilibria, some of which are stable and some of which are not. Some typical forms are shown in Figure \[fig:shape\](a) in which the five dots are the equilibrium points where W$_1$ and W$_2$ are within the two stable wells; S$_1$ and S$_2$ two unstable saddle points; H the unstable hilltop. Thus, we might have the system sitting (in equilibrium) at point W$_1$. If it is then subject to a disturbance [*with the right size and direction*]{} (in the dynamical context), we might expect the system to transition to the remote equilibrium at W$_2$. This might occur when the system is subject to a large impact force, for example [@Wiebe2013]. It is anticipated (and will later be shown) that the typically easiest transition will be associated with (an asymmetric) passage close to S$_1$ or S$_2$, and generally avoiding H. In Figure \[fig:shape\](b) is shown the same system but now with a small geometric imperfection in both modes (i.e., $\gamma_1 \ne 0$ and $\gamma_2 \ne 0$). In this case the symmetry of the system is broken, and given the relative energy associated with the saddle points it is anticipated (and will also be shown later) that optimal escape will tend to occur via S$_1$. ![ Contours of potential energy: (a) the symmetric system, $\gamma_1 = \gamma_2 = 0$, (b) with small initial imperfections in both modes, i.e., $\gamma_1$ and $\gamma_2 $ are nonzero.[]{data-label="fig:shape"}](shape_mod.png){width="100.00000%"} Note that eqs.  can also be obtained from Lagrange’s equations, $$\frac{d}{dt}\left( \frac{\partial \mathcal{L}}{ \partial \dot q_i}\right) -\frac{\partial \mathcal{L}}{ \partial q_i} = - C_i \dot q_i, \quad i =1, 2$$ when $q_1 = X$ and $q_2=Y$, and the Lagrangian is $$\mathcal{L}(X,Y,\dot X,\dot Y) = \mathcal{T}(\dot X, \dot Y) - \mathcal{V}(X, Y)$$ To transform this to a Hamiltonian system, one defines the generalized momenta, $$\begin{split} p_i = \frac{\partial \mathcal{L}}{ \partial \dot q_i} = M_i \dot q_i \end{split}$$ so $p_X = M_1 \dot X$ and $p_Y = M_2 \dot Y$, in which case, the kinetic energy is $$\mathcal{T}(X,Y,p_X,p_Y) = \frac{1}{2 M_1} p_X^2 + \frac{1}{2 M_2} p_Y^2$$ and the Hamiltonian is $$\mathcal{H}(X,Y,p_X,p_Y) = \mathcal{T} + \mathcal{V}$$ and Hamilton’s equations (with damping) [@Greenwood2003] are $$\begin{split} \dot X &= \frac{\partial \mathcal{H}}{\partial p_X}=\frac{p_X}{M_1} \\ \dot Y &= \frac{\partial \mathcal{H}}{\partial p_Y}=\frac{p_y}{M_2} \\ \dot p_X &= - \frac{\partial \mathcal{H}}{\partial X} - C_H p_X=- \frac{\partial \mathcal{V}}{\partial X} - C_H p_X \\ \dot p_Y &= - \frac{\partial \mathcal{H}}{\partial Y} - C_H p_Y=- \frac{\partial \mathcal{V}}{\partial Y} - C_H p_Y \\ \label{eq:eomHam} \end{split}$$ where $$\begin{split} \frac{\partial \mathcal{V}}{\partial X}=& K_1 \left(X - \gamma_1 \right) - N_T G_1 X - \frac{EA}{2L}G_1^2 \left(\gamma_1^2 X -X^3 \right) - \frac{EA}{2L} G_1 G_2 \left(\gamma_2^2 X -X Y^2 \right),\\ \frac{\partial \mathcal{V}}{\partial Y}=& K_2 \left(Y - \gamma_2 \right) - N_T G_2 Y - \frac{EA}{2L}G_2^2 \left(\gamma_2^2 Y -Y^3 \right) - \frac{EA}{2L} G_1 G_2 \left(\gamma_1^2 Y -X^2 Y \right) \end{split}$$ and $C_H=C_1/M_1=C_2/ M_2$ is the damping coefficient in the Hamiltonian system which can be easily found by comparing and , and using the relations of $M_i$ and $C_i$ in . We assume the lower saddle point S$_1$ has the smaller potential energy compared to S$_2$, thus the energy of S$_1$ is the critical energy for snap-though, and we initially focus attention on the dynamic behavior around the region of S$_1$. The linearized equations of about S$_1$ with position $(X_e,Y_e)$ can be written as $$\begin{split} \dot x&= \frac{p_x}{M_1}\\ \dot y&= \frac{p_y}{M_2}\\ \dot p_x&= A_{31} x + A_{32} y - C_H p_x\\ \dot p_y&= A_{32} x + A_{42} y - C_H p_y \label{linearization} \end{split}$$ where $(x,y,p_x,p_y)= (X,Y,p_X,p_Y) - (X_e,Y_e,0,0)$ and $$\begin{split} & A_{31}= -K_1 + N_T G_1 + \frac{E A G_1^2 \left(\gamma_1^2 - 3 X_e^2 \right)}{2L} + \frac{E A G_1 G_2 \left(\gamma_2^2 -Y_e^2 \right)}{2L},\\ & A_{32}= - \frac{E A G_1 G_2 X_e Y_e}{L},\\ & A_{42}= -K_2 + N_T G_2 + \frac{E A G_2^2 \left(\gamma_2^2 - 3 Y_e^2 \right)}{2L} + \frac{E A G_1 G_2 \left(\gamma_1^2 -X_e^2 \right)}{2L} \label{lin paras} \end{split}$$ If we replace the position of S$_1$ by the position of W$_1$, we can still use the linearized equations in to obtain the natural frequencies of the shallow arch near W$_1$ as $$\omega_{1,2}^{(d)} = w_{1,2}^{(c)} \sqrt{1-\xi_{1,2}^2}$$ where $\omega_{1,2}^{(c)}$ are the first two natural frequencies for the conservative system and $\xi_{1,2}$ are the viscous damping factors with the forms $$\omega_{1,2}^{(c)}=\frac{(b_{\omega} \mp \sqrt{b_{\omega}^2 - 4 c_{\omega}} )}{2}, \hspace{0.5in} \xi_{1,2} = \frac{C_H}{2 \omega_{1,2}^{(c)}}$$ and $$b_{\omega}=- \frac{A_{31}}{M_1} - \frac{A_{42}}{M_2}, \hspace{0.5in} c_{\omega}=\frac{A_{31} A_{42} - A_{32}^2}{M_1 M_2}$$ #### Non-dimensional equations of motion In order to reduce the parameters, some non-dimensional quantities are introduced, $$\begin{split} &\left( L_x, L_y \right)= L \left(1,\sqrt{ \frac{M_1}{M_2}} \right), \omega_0= \frac{ \sqrt{- A_{32} }}{ \left( M_1 M_2\right)^ \frac{1}{4}}, \tau= \omega_0 t, \left(\bar q_1 , \bar q_2 \right)= \left( \frac{x}{L_x}, \frac{y}{L_y} \right),\\ &\left(\bar p_1 , \bar p_2 \right)= \frac{1}{\omega_0} \left( \frac{p_x}{ L_x M_1},\frac{p_y}{ L_y M_2}\right), \left( c_x , c_y \right)= \frac{1}{ \omega_{0}^2} \left( \frac{A_{31}}{M_1}, \frac{A_{42}}{M_2} \right), c_1= \frac{C_H}{\omega_0} \label{dimless quan} \end{split}$$ Using the non-dimensional parameters in , the non-dimensional linearized equations are written as $$\begin{split} \dot {\bar q}_1 &= \bar p_1,\\ \dot {\bar q}_2 &= \bar p_2,\\ \dot {\bar p}_1 &= c_x \bar q_1 - \bar q_2 - c_1 \bar p_1,\\ \dot {\bar p}_2 &= - \bar q_1 + c_y \bar q_2 - c_1 \bar p_2 \label{nond eq} \end{split}$$ Written in matrix form, with column vector $\bar z=(\bar q_1 , \bar q_2 , \bar p_1 , \bar p_2)$, we have $$\dot {\bar z} = A \bar z + D \bar z$$ where $$A = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ c_x & -1 & 0 & 0 \\ -1 & c_y & 0 & 0 \end{pmatrix}, \hspace{0.5in} D = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & -c_1 & 0 \\ 0 & 0 & 0 & -c_1 \end{pmatrix} \label{A_and_D_matrix}$$ are the Hamiltonian part and damping part of the linear equations, respectively. Linearized Conservative Hamiltonian System {#linearization of conserve} ========================================== Solutions near the equilibria ----------------------------- #### Eigenvalues and eigenvectors In this section, we will discuss the linear dynamical behaviors of a buckled beam in the Hamiltonian system without taking account of any energy dissipation which makes $c_1=0$ (i.e., $C_H = 0$). Thus, the equations of motion are given as $$\dot {\bar z} = A \bar z \label{conservative eqns}$$ The system can be viewed as resulting from a quadratic Hamiltonian, $$\mathcal{H}_2= \tfrac{1}{2}\bar p_1 ^2 + \tfrac{1}{2}\bar p_2 ^2 - \tfrac{1}{2}c_x \bar q_1 ^2 - \tfrac{1}{2}c_y \bar q_2 ^2 + \bar q_1 \bar q_2 \label{H_2_bar}$$ which can be written in matrix form $$\mathcal{H}_2 = \frac{1}{2} \bar z ^T B \bar z$$ where $$\begin{split} B = J^{T}A =\begin{pmatrix} -c_x & 1 & 0 & 0 \\ 1 & -c_y & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \end{split}$$ and ${J}$ is the $4 \times 4$ canonical symplectic matrix $$\begin{aligned} \begin{split} {J} = \begin{pmatrix} {\bf 0} & {I}_2\\ -{I}_2 & {\bf 0}\\ \end{pmatrix} \end{split}\end{aligned}$$ where ${I}_2$ is the $2 \times 2$ identity matrix. The characteristic polynomial of is $$\begin{aligned} p( \beta ) = \beta^4 - ( c_x + c_y ) \beta^2 + c_x c_y - 1\end{aligned}$$ Let $\alpha= \beta^2$, then the roots of $ p(\alpha)=0 $ are as follows $$\begin{split} \alpha_1= \frac{c_x + c_y + \sqrt{ \left( c_x -c_y \right)^2 + 4}}{2},\\ \alpha_2= \frac{c_x + c_y - \sqrt{ \left( c_x - c_y \right)^2 + 4}}{2} \label{charoots} \end{split}$$ Generally, in $c_x > 0$ and $c_y <0$. In this case, $\alpha_1 > 0$ and $\alpha_2 < 0$. It follows that this equilibrium point is of the type saddle $\times$ center. Here we define $\lambda=\sqrt{\alpha_1}$ and $\omega_p=\sqrt{-\alpha_2}$. Thus, the eigenvectors are given by $$\begin{split} \left(1, c_x - \beta^2 , \beta , c_x \beta -\beta^3 \right), \label{gener egvec} \end{split}$$ where $\beta$ denotes one of the eigenvalues. After substituting $\beta = i \omega_p $ into and separating real and imaginary parts as $u_{\omega_p} + i v_{\omega_p}$, we obtain two corresponding eigenvectors $$\begin{split} u_ {\omega_p}&= \left( 1, c_x+ \omega_p^2 , 0 , 0 \right),\\ v_ {\omega_p}&= \left( 0, 0 , \omega_p ,c_x \omega_p+ \omega_p^3 \right), \label{eivect I} \end{split}$$ Moreover, the other two eigenvectors associated with the pair of real eigenvalues $\pm \lambda $ can be taken as $$\begin{split} u_{+ \lambda}&= \left( 1, c_x - \lambda^2 , \lambda , c_x \lambda - \lambda^3 \right),\\ u_{- \lambda}&=- \left( 1, c_x - \lambda^2 , - \lambda , \lambda^3 - c_x \lambda \right) \label{eivect R} \end{split}$$ #### Symplectic change of variables We consider the linear symplectic change of variables from $(\bar q_1, \bar q_2, \bar p_1, \bar p_2)$ to $(q_1 , q_2 , p_1 , p_2)$, $$\left( \begin{array}{c c c c} \bar q_1 \\ \bar q_2 \\ \bar p_1 \\ \bar p_2 \end{array} \right) = C \left( \begin{array}{c c c c} q_1 \\ q_2 \\ p_1 \\ p_2 \end{array} \right)$$ \[CT\] where the columns of the matrix $C$ are given by the eigenvectors, $$\begin{split} C=\left(u_{ + \lambda} , u_{ \omega_p} , u_{ - \lambda} , v_{\omega_p} \right) \label{change matrix} \end{split}$$ and where the vectors are written as column vectors. Then we find $$\begin{split} C^T J C=\begin{pmatrix} 0 & \bar D\\ -\bar D & 0 \end{pmatrix}, \hspace{0.5in} \bar D= \begin{pmatrix} d_\lambda & 0\\ 0 & d_{\omega_p} \end{pmatrix} \end{split}$$ where $$\begin{split} d_\lambda &= \lambda [4 - 2 (c_x -c_y) ( \lambda^2 -c_x)]\\ d_{\omega_p}&= \frac{\omega_p}{2} [4 +2 (c_x - c_y) ( \omega_p^2 + c_x)] \end{split}$$ In order to obtain a symplectic form which satisfies $C^T J C =J$, we need to rescale the columns of $C$. The scaling is given by factors $s_1 = \sqrt{ d_\lambda}$ and $s_2 = \sqrt{d_{\omega_p}}$. In this case, the final form of the symplectic matrix $C$ is given by $$\begin{split} C= \begin{pmatrix} \frac{1}{s_1} & \frac{1}{s_2} & - \frac{1}{s_1} & 0\\ \frac{c_x - \lambda^2}{s_1} & \frac{ \omega_p^2 + c_x}{s_2} & \frac{\lambda^2 - c_x}{s_1} & 0\\ \frac{\lambda}{s_1} & 0 & \frac{ \lambda}{s_1} & \frac{\omega_p}{s_2} \\ \frac{c_x \lambda - \lambda^3}{s_1} & 0 & \frac{c_x \lambda - \lambda^3}{s_1} & \frac{c_x \omega_p + \omega_p^3 }{s_2} \end{pmatrix} \label{sym matrix} \end{split}$$ The Hamiltonian can be rewritten in the simplified, normal form, $$\label{Enlin} \mathcal{H}_2= \lambda q_1 p_1 + \tfrac{1}{2} \omega_p(q_2 ^2 +p_2^2)$$ with corresponding linearized equations, $$\begin{split} \dot q_1&= ~~\lambda q_1, \\ \dot p_1&= - \lambda p_1,\\ \dot q_2&= ~~\omega_p p_2, \\ \dot p_2&= - \omega_p q_2 \label{new equa} \end{split}$$ Written in matrix form, with column vector $z=(q_1 , q_2 , p_1 , p_2)$, we have $$\dot z = \Lambda z$$ where $$\begin{split} \Lambda = C^{-1}AC = \left( \begin{array}{rrrr} \lambda & 0 & 0 & 0 \\ 0 & 0 & 0 & \omega_p \\ 0 & 0 & -\lambda & 0 \\ 0 & -\omega_p & 0 & 0 \end{array} \right) \label{Lambda_matrix} \end{split}$$ The solution of can be written as $$\begin{split} & q_1= q_1^0 e^{ \lambda t}, \ \ \ p_1= p_1^0 e^{ - \lambda t}\\ & q_2 + i p_2 = \left(q_2^0 + i p_2^0 \right) e^{-i \omega_p t} \end{split}$$ Note that the three functions $$f_1 = q_1 p_1, \quad f_2 = q_2^2 + p_2^2, \quad f_3 = \mathcal{H}_2$$ are constants of motion under the Hamiltonian system . Boundary of transit and non-transit orbits {#sec:separatrix} ------------------------------------------ #### The Linearized Phase Space For positive $h$ and $c$, the equilibrium or bottleneck region $\mathcal{R}$ (sometimes just called the neck region), which is determined by $$\mathcal{H}_2=h, \quad \mbox{and} \quad |p_1-q_1|\leq c,$$ is homeomorphic to the product of a 2-sphere and an interval $I$, $S^2\times I$; namely, for each fixed value of $p_1 -q_1 $ in the interval $I=[-c,c]$, we see that the equation $\mathcal{H}_2=h$ determines a 2-sphere $$\label{2-sphere} \tfrac{\lambda }{4}(q_1 +p_1 )^2 + \tfrac{1}{2}\omega_p (q_2^2+p_2^2) =h+\tfrac{\lambda }{4}(p_1 -q_1 )^2.$$ Suppose $a \in I$, then can be re-written as $$\label{2-sphere2} x_1^2 + q_2^2+p_2^2 = r^2,$$ where $x_1 = \sqrt{\tfrac{1 }{2}\tfrac{\lambda}{\omega_p}}(q_1 +p_1 )$ and $r^2=\tfrac{2}{\omega_p}(h+\tfrac{\lambda }{4}a^2)$, which defines a 2-sphere of radius $r$ in the three variables $x_1$, $q_2$, and $p_2$. The bounding 2-sphere of $\mathcal{R}$ for which $p_1 -q_1 = c$ will be called $n_1$ (the “left” bounding 2-sphere), and that where $p_1 -q_1 = -c$, $n_2$ (the “right” bounding 2-sphere). See Figure \[fig5\]. ![\[fig5\][ The flow in the equilibrium region has the form saddle $\times$ center. On the left is shown the projection onto the $(p_1,q_1)$ plane, the saddle projection. For the conservative dynamics, the Hamiltonian function $\mathcal{H}_2$ remains constant at $h>0$. Shown are the periodic orbit (black dot at the center), the asymptotic orbits (labeled A), two transit orbits (T) and two non-transit orbits (NT). ]{}](linear_projections_n_mod.png){width="\textwidth"} We call the set of points on each bounding 2-sphere where $q_1 + p_1 = 0$ the equator, and the sets where $q_1 + p_1 > 0$ or $q_1 + p_1 < 0$ will be called the northern and southern hemispheres, respectively. #### The Linear Flow in $\mathcal{R}$ To analyze the flow in $\mathcal{R}$, consider the projections on the ($q_1, p_1$)-plane and the $(q_2,p_2)$-plane, respectively. In the first case we see the standard picture of a saddle point in two dimensions, and in the second, of a center consisting of harmonic oscillator motion. Figure \[fig5\] schematically illustrates the flow. With regard to the first projection we see that $\mathcal{R}$ itself projects to a set bounded on two sides by the hyperbola $q_1p_1 = h/\lambda $ (corresponding to $q_2^2+p_2^2=0$, see ) and on two other sides by the line segments $p_1-q_1= \pm c$, which correspond to the bounding 2-spheres, $n_1$ and $n_2$, respectively. Since $q_1p_1$ is an integral of the equations in $\mathcal{R}$, the projections of orbits in the $(q_1,p_1)$-plane move on the branches of the corresponding hyperbolas $q_1p_1 =$ constant, except in the case $q_1p_1=0$, where $q_1 =0$ or $p_1 =0$. If $q_1p_1 >0$, the branches connect the bounding line segments $p_1 -q_1 =\pm c$ and if $q_1p_1 <0$, they have both end points on the same segment. A check of equation shows that the orbits move as indicated by the arrows in Figure \[fig5\]. To interpret Figure \[fig5\] as a flow in $\mathcal{R}$, notice that each point in the $(q_1,p_1)$-plane projection corresponds to a 1-sphere $S^1$ in $\mathcal{R}$ given by $$q_2^2+p_2^2 =\tfrac{2 }{\omega_p}(h-\lambda q_1p_1) .$$ Of course, for points on the bounding hyperbolic segments ($q_1p_1 =h/\lambda $), the 1-sphere collapses to a point. Thus, the segments of the lines $p_1-q_1 =\pm c$ in the projection correspond to the 2-spheres bounding $\mathcal{R}$. This is because each corresponds to a 1-sphere crossed with an interval where the two end 1-spheres are pinched to a point. We distinguish nine classes of orbits grouped into the following four categories: 1. The point $q_1 =p_1 =0$ corresponds to an invariant 1-sphere $S^1_h$, an unstable [**period orbit**]{} in $\mathcal{R}$. This 1-sphere is given by $$\label{3-sphere} q_2^2+p_2^2=\tfrac{2 }{\omega_p}h, \hspace{0.3in} q_1 =p_1 =0.$$ It is an example of a normally hyperbolic invariant manifold (NHIM) (see [@Wiggins1994]). Roughly, this means that the stretching and contraction rates under the linearized dynamics transverse to the 1-sphere dominate those tangent to the 1-sphere. This is clear for this example since the dynamics normal to the 1-sphere are described by the exponential contraction and expansion of the saddle point dynamics. Here the 1-sphere acts as a “big saddle point”. See the black dot at the center of the $(q_1,p_1)$-plane on the left side of Figure \[fig5\]. 2. The four half open segments on the axes, $q_1p_1 =0$, correspond to four cylinders of orbits asymptotic to this invariant 1-sphere $S^1_h$ either as time increases ($p_1 =0$) or as time decreases ($q_1 =0$). These are called [**asymptotic**]{} orbits and they form the stable and the unstable manifolds of $S^1_h$. The stable manifolds, $W^s_{\pm}(S^1_h)$, are given by $$\label{stable_manifold} q_2^2+p_2^2=\tfrac{2 }{\omega_p}h, \hspace{0.3in} q_1 =0, \hspace{0.3in} p_1 ~{\rm arbitrary}.$$ $W^s_+(S^1_h)$ (with $p_1>0$) is the branch going entering from $n_1$ and $W^s_-(S^1_h)$ (with $p_1<0$) is the branch going entering from $n_2$. The unstable manifolds, $W^u_{\pm}(S^1_h)$, are given by $$\label{unstable_manifold} q_2^2+p_2^2=\tfrac{2 }{\omega_p}h, \hspace{0.3in} p_1 =0, \hspace{0.3in} q_1 ~{\rm arbitrary}$$ $W^u_+(S^1_h)$ (with $q_1>0$) is the branch exiting from $n_2$ and $W^u_-(S^1_h)$ (with $q_1<0$) is the branch exiting from $n_1$. See the four orbits labeled A of Figure \[fig5\]. 3. The hyperbolic segments determined by $q_1p_1 ={\rm constant}>0$ correspond to two cylinders of orbits which cross $\mathcal{R}$ from one bounding 2-sphere to the other, meeting both in the same hemisphere; the northern hemisphere if they go from $p_1-q_1 =+c$ to $p_1-q_1 =-c$, and the southern hemisphere in the other case. Since these orbits transit from one realm to another, we call them [**transit**]{} orbits. See the two orbits labeled T of Figure \[fig5\]. 4. Finally the hyperbolic segments determined by $q_1p_1 = {\rm constant}<0$ correspond to two cylinders of orbits in $\mathcal{R}$ each of which runs from one hemisphere to the other hemisphere on the same bounding 2-sphere. Thus if $q_1 >0$, the 2-sphere is $n_1$ ($p_1 -q_1 =-c$) and orbits run from the southern hemisphere ($q_1 +p_1 <0$) to the northern hemisphere ($q_1 +p_1 >0$) while the converse holds if $q_1 <0$, where the 2-sphere is $n_2$. Since these orbits return to the same realm, we call them [**non-transit**]{} orbits. See the two orbits labeled NT of Figure \[fig5\]. Trajectories in the neck region ------------------------------- We now examine the appearance of the orbits in configuration space, that is, in $(\bar q_1,\bar q_2)$-plane. In configuration space, $\mathcal{R}$ appears as the neck region connecting two realms, so trajectories in $\mathcal{R}$ will be transformed back to the neck region. It should pointed out that at each moment in time, all trajectories must evolve within the energy boundaries which are zero velocity curves (corresponding to $ \bar p_1 = \bar p_2 = 0$) given by solving for $\bar q_2$ as a function of $\bar q_1$, $$\bar q_2( \bar q_1) = \frac{\bar q_1 \pm \sqrt{\bar q_1^2 - 2 c_y (h + \tfrac{c_x}{2} \bar q_1^2)}}{c_y}$$ Recall that in order to obtain the analytical solutions for $\bar z = (\bar q_1, \bar q_2, \bar p_1, \bar p_2)$, system $\bar z$ has been transformed into system $z = (q_1, q_2, p_1, p_2)$ by using the symplectic matrix $C$ consisting of generalized (re-scaled) eigenvectors $u_{+ \lambda}, u_{- \lambda}, u_{\omega_p}, v_{\omega_p}$ with corresponding eigenvalues $ \pm \lambda$ and $\pm i \omega_p$. Thus, the system $z$ should be transformed back to system $\bar z$ which generates the following general (real) solution with the form $$\begin{split} \bar z(t) = \left(\bar q_1, \bar q_2, \bar p_1, \bar p_2 \right)^T = q_1^0 e^{\lambda t} u_{+ \lambda} + p_1^0 e^{- \lambda t} u_{- \lambda} + \mathrm{Re} \left[\beta_0 e^{- i \omega_p t} \left(u_{\omega_p} - i v_{\omega_p} \right) \right] \label{whole-gener-sol} \end{split}$$ where $q_1^0, p_1^0$ are real and $\beta_0 = q_2^0 + i p_2^0$ is complex. Upon inspecting this general solution, we see that the solutions on the energy surface fall into different classes depending upon the limiting behaviors of $ \bar q_1, \bar q_2$ as $t$ tends to plus or minus infinity. Notice that $$\begin{split} \bar q_1(t) &= \frac{q_1^0}{s_1} e^{\lambda t} - \frac{p_1^0}{s_1} e^{-\lambda t} + \frac{1}{s_2} \left( q_2^0 \cos \omega_p t + p_2^0 \sin \omega_p t \right)\\ \bar q_2(t) &= \frac{c_x-\lambda^2}{s_1} q_1^0 e^{\lambda t} + \frac{\lambda^2 - c_x}{s_1} p_1^0 e^{-\lambda t} + \frac{\omega_p^2 +c_x}{s_2} \left( q_2^0 \cos \omega_p t + p_2^0 \sin \omega_p t \right)\\ \label{conser-sol} \end{split}$$ Thus, if $t\rightarrow + \infty$, then $\bar q_1 (t)$ is dominated by its $q_1^0$ term. Hence, $\bar q_1 (t)$ tends to minus infinity (staying on the left-hand side), is bounded (staying around the equilibrium point), or tends to plus infinity (staying on the right-hand side) according to $q_1^0 < 0 $, $q_1^0=0$ and $q_1^0>0$. See Figure \[fig:Conley\]. The same statement holds if $t\rightarrow - \infty$ and $-p_1^0$ replaces $q_1^0$. Different combinations of the signs of $q_1^0$ and $p_1^0$ will give us again the same nine classes of orbits which can be grouped into the same four categories. 1. If $q_1^0 = p_1^0 = 0$, we obtain a periodic solution. The periodic orbit projects onto the $(\bar q_1, \bar q_2)$ plane as a line with the following expression $$\begin{split} \bar q_1 & = \frac{1}{s_2} \left( q_2^0 \cos \omega_p t + p_2^0 \sin \omega_p t \right)\\ \bar q_2 & = \frac{ \omega_p^2 + c_x}{s_2} \left(q_2^0 \cos \omega_p t + p_2^0 \sin \omega_p t \right)\\ & =\left( \omega_p^2 + c_x \right) \bar q_1 \end{split}$$ Notice and $\mathcal{H}_2$ now can be rewritten as $\mathcal{H}_2 = \omega_p |\beta_0|^2 / 2$. Thus, since $\mathcal{H}_2=h$, the length of the periodic orbit is $\sqrt{ 2 h \left[(\omega_p^2 + c_x)^2+1 \right] / \left(\omega_p s_2^2 \right)}$. Note that the length of the line goes to zero with $h$. 2. Orbits with $q_1^0 p_1^0=0$ are asymptotic orbits. They are asymptotic to the periodic orbit. 1. When $q_1^0=0$ , the general solutions for $\bar q_1, \bar q_2$ are $$\begin{split} \bar q_1 &= - \frac{p_1}{s_1}+\frac{q_2}{s_2} \\ \bar q_2 &= \frac{ \lambda^2 - c_x}{s_1} p_1 + \frac{\omega_p^2 + c_x}{s_2} q_2 \\ & = \left(c_x - \lambda^2 \right) \bar q_1 + \frac{\lambda^2 + \omega_p^2}{s_2} \left(q_2^0 \cos \omega_p t + p_2^0 \sin \omega_p t \right) \end{split}$$ Thus, the orbits with $q_1^0=0$ project into a strip $S$ in the $(\bar q_1, \bar q_2)$-plane bounded by $$\begin{split} \bar q_2 =\left(c_x - \lambda^2 \right) \bar q_1 \pm \frac{\lambda^2 + \omega_p^2}{s_2} \sqrt{\frac{2 h}{\omega_p}} \label{strip boundary} \end{split}$$ 2. For $p_1^0=0$, following the same procedure as $q_1^0=0$, we have $$\begin{split} \bar q_1 &= \frac{q_1}{s_1} + \frac{q_2}{s_2}\\ \bar q_2 &= \left(c_x - \lambda^2\right) \bar q_1 + \frac{\lambda^2 + \omega_p^2}{s_2} \left(q_2^0 \cos \omega_p t + p_2^0 \sin \omega_p t \right) \end{split}$$ Notice that these two asymptotic orbits with $q_1^0=0$ and $p_1^0=0$ share the same strip $S$ and the same boundaries governed by . Also, since the slopes of the periodic orbit and the strip satisfies $\left(c_x - \lambda^2 \right) \left(c_x + \omega_p^2 \right)=-1$, the periodic orbit is perpendicular to the strip. In other words, the length of the periodic orbit is exactly the same as the width of the strip. 3. Orbits with $q_1^0 p_1^0 >0$ are transit orbits because they cross the equilibrium region $R$ from $- \infty$ (the left-hand side) to $+ \infty$ (the right-hand side) or vice versa. 4. Orbits with $q_1^0 p_1^0 <0$ are non-transit orbits ![ The flow of the conservative system in $\mathcal{R}$, the equilibrium region projected onto the $xy$ configuration space, for a fixed value of energy, $\mathcal{H}_2=h>0$. For any point on the bounding vertical lines $n_1$ or $n_2$ (dashed), there is a wedge of velocity directions inside of which the trajectories are transit orbits, and outside of which are non-transit orbits. The boundary of the wedge gives the orbits asymptotic to the single unstable periodic orbit in the neck for this energy. Shown are a typical asymptotic orbit; two transit orbits (dashed); and two non-transit orbits (dotted). []{data-label="fig:Conley"}](lin_conser_position_local_paper.png){width="67.00000%"} To study the flow in position space, Figure \[fig:Conley\] gives the four categories of orbits. From , we can see that for transit orbits and non-transit orbits, the signs of $q_1^0 p_1^0$ must satisfy $q_1^0 p_1^0>0$ and $q_1^0 p_1^0<0$,respectively. In Figure \[fig:Conley\], $S$ is the strip mentioned above. Outside of the strip, the signs of $q_1^0$ and $p_1^0$ are independent of the direction of the velocity. These signs can be determined in each of the components of the equilibrium region $\mathcal{R}$ complementary to the strip. For example, in the left two components, $q_1^0<0$ and $p_1^0>0$, while in the right two components $q_1^0>0$ and $p_1^0<0$. Therefore, $q_1^0 p_1^0<0$ in all components and only non-transit orbits project on to these four components. Inside the strip the situation is more complicated since in $S$ the signs of $q_1^0$ and $p_1^0$ depend on the direction of the velocity. At each position $(\bar q_1, \bar q_2)$ inside the strip there exists the so-called ‘wedge’ of velocities in which $q_1^0 p_1^0>0$ which was first found by Conley (1968) [@Conley1968] in the restricted three-body problem. See the shaded wedges in Figure \[fig:Conley\]. The existence and the angle of the wedge of velocity will be given in the next part. For simplicity we have indicated this dependence only on the two vertical bounding line segments in Figure \[fig:Conley\]. For example, consider the intersection of strip $S$ with left-most vertical line. On the subsegment so obtained there is at each point a wedge of velocity in which both $q_1^0$ and $p_1^0$ are positive, so that orbits with velocity interior to the wedge are transit orbits $(q_1^0 p_1^0>0)$. Of course, orbits with velocity on the boundary ot the wedge are asymptotic $(q_1^0 p_1^0 = 0)$, while orbits with velocity outside of the wedge are non-transit. The situation on the other subsegment is similar. #### The wedge of velocities To establish the wedge of velocity and obtain its angle, we need to use the following fact that all the inner products of one generalized eigenvector and another generalized eigenvector associated with $B$ are zero except for $$\begin{split} u_{+ \lambda}^T B u_{- \lambda} &=u_{- \lambda}^T B u_{+ \lambda} = \lambda \\ u_{\omega_p}^T B u_{\omega_p} &= v_{\omega_p}^T B v_{\omega_p} = \omega_p \\ \end{split}$$ Using this condition, we have the following relations, as $$\begin{split} \lambda &= u_{+ \lambda}^T B u_{- \lambda}\\ \Rightarrow \lambda q_1^0 &= q_1^0 u_{+ \lambda}^T B u_{- \lambda}\\ \Rightarrow \lambda q_1^0 &= e^{- \lambda t} \left(q_1^0 e^{\lambda t} u_{+ \lambda} \right)^T B u_{- \lambda}\\ \Rightarrow \lambda q_1^0 &= e^{- \lambda t} \bar z^T B u_{- \lambda}\\ \Rightarrow \lambda q_1^0 &= e^{- \lambda t} \left(\frac{\lambda^2}{s_1} \bar q_1 - \frac{ 1-c_xc_y + c_y \lambda^2}{s_1} \bar q_2 + \frac{\lambda}{s_1} \bar p_1 + \frac{c_x \lambda - \lambda^3}{s_1} \bar p_2 \right) \end{split}$$ Using similar arguments, we can also obtain $$\begin{split} \lambda p_1^0 &= e^{\lambda t} \left(- \frac{\lambda^2}{s_1} \bar q_1 + \frac{1- c_x c_y + c_y \lambda^2}{s_1} \bar q_2 + \frac{\lambda}{s_1} \bar p_1 + \frac{c_x \lambda - \lambda^3}{s_1} \bar p_2 \right) \end{split}$$ Thus, we obtain the following relations $$\begin{split} \lambda q_1^0 e^{\lambda t}&= \frac{\lambda^2}{s_1} \bar q_1 - \frac{1- c_x c_y + c_y \lambda^2}{s_1} \bar q_2 + \frac{\lambda}{s_1} \bar p_1 + \frac{c_x \lambda - \lambda^3}{s_1} \bar p_2\\ \lambda p_1^0 e^{- \lambda t} &=- \frac{\lambda^2}{s_1} \bar q_1 + \frac{1- c_x c_y + c_y \lambda^2}{s_1} \bar q_2 + \frac{\lambda}{s_1} \bar p_1 + \frac{c_x \lambda - \lambda^3}{s_1} \bar p_2 \label{chi prepare} \end{split}$$ Let $\chi$ be the angles determined by $$\begin{split} \cos \chi &= \frac{1}{\sqrt{ \left(\lambda^2 -c_x \right)^2 + 1}}, \ \ \ \ \ \ \sin \chi = \frac{\lambda^2 -c_x}{\sqrt{ \left(\lambda^2 -c_x \right)^2 + 1}}.\\ % \cos \chi_2 &= - \frac{1}{\sqrt{ \left(\lambda^2 -c_x \right)^2 + 1}}, \ \ \ \ \sin \chi_2 = - \frac{\lambda^2 -c_x}{\sqrt{ \left(\lambda^2 -c_x \right)^2 + 1}} \end{split}$$ Furthermore, let $$\begin{split} \bar p_1 = \rho \cos \theta, \ \ \ \bar p_2= \rho \sin \theta \end{split}$$ and $$\begin{split} \gamma &= \left( \frac{\lambda^2}{s_1} \bar q_1 - \frac{1- c_x c_y + c_y \lambda^2}{s_1} \bar q_2 \right) \left[\frac{\lambda^2}{s_1^2} \left(\bar p_1^2 + \bar p_2 ^2\right) \left(\left(\lambda^2 -c_x \right)^2 + 1 \right) \right]^{- \frac{1}{2}}\\ \label{gamma} \end{split}$$ Using , can be rewritten as $$\begin{split} \lambda q_1^0 e^{\lambda t} \left[\frac{\lambda^2}{s_1^2} \left(\bar p_1^2 + \bar p_2 ^2\right) \left(\left(\lambda^2 -c_x \right)^2 + 1 \right) \right]^{- \frac{1}{2}} &= \gamma + \cos(\theta - \chi)\\ \lambda p_1^0 e^{- \lambda t} \left[\frac{\lambda^2}{s_1^2} \left(\bar p_1^2 + \bar p_2 ^2\right) \left(\left(\lambda^2 -c_x \right)^2 + 1 \right) \right]^{- \frac{1}{2}} &= - \gamma + \cos(\theta - \chi) \label{angle of wedge} \end{split}$$ So far, the signs of $q_1^0$ and $p_1^0$ can be determined using Eq. . From Eq. , it can be concluded that $\gamma$ is only dependent on the position $(\bar q_1, \bar q_2)$, because $\bar p_1^2 + \bar p_2 ^2$ can be obtained from Eq. once the position is given. Outside the strip, we have $\mid \gamma \mid >1 $. In this case, the signs of $q_1^0$ and $p_1^0$ are independent of the direction of velocity and are always opposite, which makes $q_1^0 p_1^0<0$. Thus, only non-transit orbit exist in these regions. Inside the strip, we have $\mid \gamma \mid <1 $. This situation is quite different since the signs of $q_1^0$ and $p_1^0$ are dependent on the angle of velocity. Because for transit orbits, the sign of $q_1^0 p_1^0$ must be positive. Thus, we can vary $\theta$ (the direction of velocity) to satisfy this condition, and the wedge of velocity can be determined. It should be noted that the wedge of velocity can only exist inside the strip $S$: outside of $S$, no transit orbit exists. Linearized Dissipative Hamiltonian System {#linearization of dissipative} ========================================= Solutions near the equilibria ----------------------------- For the dissipative system, we still use the symplectic matrix $C$ as in to transform to the eigenbasis, i.e., transform $\bar z=(\bar q_1 , \bar q_2 , \bar p_1 , \bar p_2)$ to $z=(q_1 , q_2 , p_1 , p_2)$. The equations of motion now become $$\dot z = \Lambda z + \Delta z$$ where $\Lambda = {C}^{-1}{ {A}}{C}$ from before and the transformed damping matrix is, $$\begin{split} \Delta = { C}^{-1}{{D}}{C} = -c_1 \left( \begin{array}{rrrr} \tfrac{1}{2} & 0 & \tfrac{1}{2} & 0 \\ 0 & 0 & 0 & 0 \\ \tfrac{1}{2} & 0 & \tfrac{1}{2} & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \label{zeta_matrix} \end{split}$$ which results in $$\begin{aligned} &\begin{cases} \dot q_1 = \left(\lambda - \frac{ c_1}{2} \right) q_1 - \frac{ c_1}{2} p_1\\ \dot p_1 = - \frac{ c_1}{2} q_1 + \left(- \lambda -\frac{ c_1 }{2} \right) p_1 \label{Hd q1} \end{cases}\\ &\begin{cases} \dot q_2 = \omega_p p_2\\ \dot p_2 = - \omega_p q_2 - c_1 p_2 \label{Hd q2} \end{cases} \end{aligned}$$ Notice that the dynamics on the $(q_1,p_1)$ plane and $(q_2,p_2)$ plane are uncoupled. The fourth-order characteristic polynomial is thus decomposable into $p(\beta)= p_1(\beta) p_2(\beta)$, where the second-order characteristic polynomials for and are p\_1()= \^2 + c\_1 - \^2\ p\_2()= \^2 + c\_1 + \_p\^2 \[Hd poly\] Considering $c_1$ is positive and $c_1^2$ is smaller compared with $4 \omega_p^2$, the determinants for are \_1 = c\_1\^2 + 4 \^2 &gt; 0\ \_2 = c\_1\^2 - 4 \_p\^2 &lt;0 The corresponding eigenvalues are $$\begin{aligned} &\begin{cases} \beta_1= \frac{ - c_1 + \sqrt{ c_1^2 + 4 \lambda^2}}{2}\\ \beta_2= \frac{ - c_1 - \sqrt{ c_1^2 + 4 \lambda^2}}{2} \end{cases}\\ &\begin{cases} \beta_3= -\delta + i \omega_d \\ \beta_4= -\delta - i \omega_d \\ \end{cases} \end{aligned}$$ where $\delta = \frac{c_1}{2}, \omega_d =\omega_p \sqrt{ 1 - \xi_d^2}$ and $\xi_d=\frac{\delta}{\omega_p}$, with the corresponding eigenvectors $$\begin{split} u_{\beta_1}&= \left( \frac{c_1}{2}, \lambda - \frac{1}{2} \sqrt{c_1^2 + 4 \lambda^2} \right)\\ u_{\beta_2}&= \left( \frac{c_1}{2}, \lambda + \frac{1}{2} \sqrt{c_1^2 + 4 \lambda^2} \right)\\ u_{\beta_3}&= \left( \omega_p, - \delta + i \omega_d \right)\\ u_{\beta_4}&= \left( \omega_p, - \delta - i \omega_d \right) \label{Hd imagin} \end{split}$$ Thus, the general solutions for the $\left(q_1,p_1\right)$ and $\left(q_2,p_2\right)$ systems are $$\begin{aligned} &\begin{cases} q_1= k_1 e^{\beta_1 t} + k_2 e^{\beta_2 t}\\ p_1= k_3 e^{\beta_1 t} + k_4 e^{\beta_2 t} \end{cases}\\ &\begin{cases} q_2= k_5 e^{- \delta t} \cos{\omega_d t} + k_6 e^{- \delta t} \sin{\omega_d t}\\ p_2= \frac{k_5 }{\omega_p} e^{- \delta t} \left(-\delta \cos{\omega_d t} - \omega_d \sin{\omega_d t} \right) +\frac{k_6 }{\omega_p} e^{- \delta t} \left(\omega_d \cos{\omega_d t - \delta \sin{\omega_d t}} \right)\\ \end{cases} \end{aligned}$$ \[Hd gener solut\] where $$\begin{split} k_1 &= \frac{q_1^0 \left(2 \lambda + \sqrt{c_1^2 + 4 \lambda^2} \right)-c_1 p_1^0 }{2\sqrt{c_1^2 + 4 \lambda^2}}, \hspace{0.5in} k_2 =\frac{q_1^0 \left(-2 \lambda + \sqrt{c_1^2 + 4 \lambda^2} \right)+c_1 p_1^0 }{2\sqrt{c_1^2 + 4 \lambda^2}},\\ k_3 &= \frac{p_1^0 \left(-2 \lambda + \sqrt{c_1^2 + 4 \lambda^2} \right)-c_1 q_1^0 }{2\sqrt{c_1^2 + 4 \lambda^2}}, \hspace{0.5in} k_4 = \frac{p_1^0 \left(2 \lambda + \sqrt{c_1^2 + 4 \lambda^2} \right)+c_1 q_1^0 }{2\sqrt{c_1^2 + 4 \lambda^2}},\\ k_5&=q^0_2 , \hspace{0.5in} k_6=\frac{p^0_2 \omega_p + q^0_2 \delta}{\omega_d} \end{split}$$ Note that $k_1=q_1^0$, $k_4=p_1^0$, $k_2=k_3=0$, $k_5=q_2^0$ and $k_6=p_2^0 $ if $c_1=c_2=0$. Taking total derivative with respect to $t$ of the Hamiltonian along trajectories gives us $$\begin{split} \frac{d \mathcal{H}_2}{d t} = - \tfrac{1}{2} c_1 \lambda \left(q_1 + p_1 \right)^2 - c_1 \omega_p p_2^2 \le 0 \label{H2-rate of change-equal} \end{split}$$ which means the Hamiltonian is non-increasing, and will generally decrease due to damping. Boundary of transit and non-transit orbits {#boundary-of-transit-and-non-transit-orbits} ------------------------------------------ #### The Linear Flow in $\mathcal{R}$ Similar to the discussions for the conservative system, we still choose an equilibrium region $\mathcal{R}$ bounded by regions which project to the lines $n_1$ and $n_2$ in the $(q_1,p_1)$-plane (see Figure \[flow-damped\]). To analyze the flow in $\mathcal{R}$, we consider the projections on the $(q_1,p_1)$-plane and the $(q_2,p_2)$-plane, respectively. In the first case we see the standard picture of saddle point, now rotated compared to the conservative case, and in the second, of a stable focus which is a damped oscillator with frequency $\omega_d=\omega_p \sqrt{1- \xi_d^2}$, where $\xi_d=\frac{c_1}{2 \omega_p}$ - the viscous damping factor (damping ralative to critical damping). Notice that the frequency $\omega_d$ for the damped system decreases with increased damping, but only very slightly for lightly damped systems. ![\[flow-damped\][ The flow in the equilibrium region around S$_1$ for the dissipative system has the form saddle $\times$ focus. On the left is shown the projection onto the $(p_1,q_1)$ plane, the saddle projection. The asymptotic orbits (labeled A) on this projection are the saddle-type asymptotic orbits, and are rotated clockwise compared to the conservative system. They still form the separatrix between transit orbits (T) and two non-transit orbits (NT). The black dot at the center represents trajectories with only a focus projection, thus oscillatory dynamics decaying onto the point S$_1$. As the energy, the Hamiltonian function $\mathcal{H}_2$, is decreasing, the boundary is no longer equal to $q_1 p_1 = h/\lambda$, as it is for the conservative case, where $\mathcal{H}_2=h$ is the initial value of the energy for those trajectories entering through the left or right side bounding sphere (i.e., $n_1$ or $n_2$, respectively). These boundaries (the boundary of the shaded region) still correspond to the fastest trajectories through the neck region for a given $h$. ]{}](eigenspace_damp_both_paper.png){width="\textwidth"} We distinguish nine classes of orbits grouped into the following four categories: 1. The point $q_1=p_1=0$ corresponds to a [**focus-type asymptotic**]{} orbit with motion purely in the $(q_2,p_2)$-plane (see black dot at the origin of the $(q_1,p_1)$-plane in Figure \[flow-damped\]). Such orbits are asymptotic to the equilibrium point S$_1$ itself. Due to the effect of damping, the periodic orbit in the conservative system, which is an invariant 1-sphere $S_h^1$ mentioned in , does not exist. 2. The four half open segments on the lines governed by $q_1= c_1 p_1/(2 \lambda \pm \sqrt{c_1^2 + 4 \lambda^2}) $ correspond to [**saddle-type asymptotic**]{} orbits. See the four orbits labeled A in Figure \[flow-damped\]. These orbits have motion in both the $(q_1,p_1)$- and $(q_2,p_2)$-planes. 3. The segments which cross $\mathcal{R}$ from one boundary to the other, i.e., from $p_1 - q_1=+c$ to $p_1 - q_1=-c$ in the northern hemisphere, and vice versa in the southern hemisphere, correspond to [**transit**]{} orbits. See the two orbits labeled $T$ of Figure \[flow-damped\]. 4. Finally the segments which run from one hemisphere to the other hemisphere on the same boundary, namely which start from $p_1 - q_1 = \pm c$ and return to the same boundary, correspond to [**non-transit**]{} orbits. See the two orbits labeled NT of Figure \[flow-damped\]. Trajectories in the neck region ------------------------------- Following the same procedure of analysis as for conservative system, the general solution to the dissipative system can be obtained by $\bar z=C z$ which gives $$\begin{split} \bar q_1 &= \frac{k_1 - k_3}{s_1} e^{\beta_1 t} - \frac{k_4 - k_2}{s_1} e^{\beta_2 t} + \frac{q_2}{s_2}\\ \bar q_2 &= \frac{k_1 - k_3}{s_1} (c_x-\lambda^2) e^{\beta_1 t} - \frac{k_4 - k_2}{s_1} (c_x-\lambda^2) e^{\beta_2 t} + \frac{\omega_p^2 + c_x}{s_2} q_2 \label{diss sol} \end{split}$$ Similar to the situation in the conservative system, the solutions for the dissipative system on the energy surface fall into different classes depending upon the limiting behaviors. See Figure \[lin\_damp\_position\_paper\]. ![\[lin\_damp\_position\_paper\][ The flow of the dissipative system in $\mathcal{R}$, the equilibrium region projected onto the $xy$ configuration space, for trajectories starting at a fixed value of energy, $\mathcal{H}_2=h$, on either the right or left side vertical boundaries. As before, for any point on a bounding vertical line (dashed), there is a wedge of velocities inside of which the trajectories are transit orbits, and outside of which are non-transit orbits. For a given fixed energy, the wedge for the dissipative system is a subset of the wedge for the conservative system. The boundary of the wedge gives the orbits asymptotic (saddle-type) to the equilibrium point S$_1$. ]{}](lin_damp_position_paper.png){width="\textwidth"} From we know that the conditions $k_1-k_3>0$, $k_1 - k_3=0$ and $k_1-k_3<0$ make $\bar q_1$ tend to minus infinity, are bounded or tend to plus infinity if $t \rightarrow \infty$. See Figure \[flow-damped\]. The same statement holds if $t \rightarrow- \infty$ and $k_2 -k_4$ replaces $k_1- k_3$. Nine classes of orbits can be given according to different combinations of the sign of $k_1-k_3$ and $k_2 - k_4$ which can be classified into the following four categories: 1. Orbits with $k_1 - k_3 = k_4-k_2 = 0$ are [**focus-type asymptotic**]{} orbits $$\begin{split} \bar q_1 = q_2 / s_2, \hspace{0.5in} \bar q_2 = \left(\omega_p^2 + c_x \right) \bar q_1 \end{split}$$ Notice the presence of $q_2$ in reveals that the amplitude of the periodic orbit will gradually decease at the rate of $e^{-\delta t}$ with time. The larger the damping, the faster the rate will be. 2. Orbits with $ \left(k_1 - k_3\right) \left(k_4 - k_2 \right)= 0$ are [**saddle-type asymptotic**]{} orbits $$\bar q_2 =\left(c_x-\lambda^2 \right) \bar q_1 + \frac{\lambda^2+\omega_p^2}{s_2} q_2 \label{damped-sympt}$$ In similarity with the shrinking of the length of the periodic orbit, the amplitude of asymptotic orbits are also shrinking. 3. Orbits with $\left(k_1 - k_3\right) \left(k_4 - k_2 \right)>0$ are [**transit**]{} orbits 4. Orbits with $\left(k_1 - k_3\right) \left(k_4 - k_2 \right)<0$ are [**non-transit**]{} orbits #### Wedge of velocities We previously obtained the wedge of velocities for the conservative system. However, this method is no longer effective for the dissipative system. Thus, another approach will be pursued here. Based on the eigenvectors in , we can conclude that the directions of stable asymptotic orbits are along $u_{\beta_2}=\left(\frac{c_1}{2}, \lambda + \frac{1}{2} \sqrt{c_1^2 + 4 \lambda^2} \right)$. In this case, all asymptotic orbits in the transformed system must start on the line $$q_1=k_p p_1 \label{R:asymp-formula}$$ where $k_p=c_1 / (2 \lambda + \sqrt{c_1^2 + 4 \lambda^2})$. For a specific point $\left(\bar q_{10}, \bar q_{20} \right)$, the initial conditions in position space and transformed space are defined as $\left(\bar q_{10}, \bar q_{20}, \bar p_{10}, \bar p_{20} \right)$ and $\left(q_{10}, q_{20}, p_{10}, p_{20} \right)$, respectively. Using Eq. and the change of variables in , we can obtain $p_{10}$, $q_{20}$, $p_{20}$ and $\bar p_{20}$ in terms of $\bar q_{10}$, $\bar q_{20}$ and $\bar p_{10}$. With $p_{10}$, $q_{20}$, $p_{20}$ and $\bar p_{20}$ in hand, the normal form of the Hamiltonian can be rewritten as $$a_p \bar p_{10}^2 + b_p \bar p_{10}+c_p=0 \label{quadratic}$$ where $$\begin{split} & a_p=\frac{s_2^2}{2 \omega_p}, \hspace{0.5in} b_p=\frac{\lambda s_2^2 (1+k_p) \left[\bar q_2- \bar q_1 \left(c_x + \omega_p^2 \right) \right]}{\omega_p \left(k_p-1 \right) \left(\lambda^2 + \omega_p^2 \right)},\\ & c_p = \left(\sum\limits_{i=1}^{4} c_p^{(i)}\right)/ \left[2 \omega_p \left(k_p-1 \right)^2 \left(\lambda^2 + \omega_p^2 \right)^2 \right]- h,\\ & c_p^{(1)}=2 k_p s_1^2 \lambda \omega_p \left[\bar q_2- \bar q_1 \left(c_x + \omega_p^2 \right) \right]^2, \\ & c_p^{(2)}= 8 k_p s_2^2 \lambda^2 \omega_p^2 \bar q_1 \left(c_x \bar q_1 - \bar q_2 \right),\\ & c_p^{(3)}=s_2^2 \lambda^2 \left(1+k_p \right)^2 \left[\left(c_x \bar q_1- \bar q_2 \right)^2+ \bar q_1^2\omega_p^4 \right],\\ & c_p^{(4)}=s_2^2 \omega_p^2 \left(k_p -1 \right)^2 \left[\left(c_x \bar q_1 - \bar q_2) \right)^2+ \bar q_1^2 \lambda^4 \right] \end{split}$$ For the existence of real solutions, the determinant of quadratic equation should satisfy the condition $\vartriangle = b_p^2 - 4 a_p c_p \geq 0$: $\vartriangle=0$ is the critical condition for $p_{10}$ to have real solutions. Noticing $\left(c_x - \lambda^2 \right) \left(c_x + \omega_p^2 \right)=-1$, the critical condition gives an ellipse of the form $$\frac{\left(\bar q_{10} \cos \vartheta + \bar q_{20} \sin \vartheta \right)^2}{a_e^2} +\frac{\left(- \bar q_{10} \sin \vartheta + \bar q_{20} \cos \vartheta \right)^2}{b_e^2} = 1, \label{R:standard-ellipse}$$ where $$\begin{split} & a_e= \sqrt{\frac{2 h \left(\lambda^2 + \omega_p^2\right)^2 \left(c_x + \omega_p^2 \right)^2}{\omega_p s_2^2 \left[ \left(c_x + \omega_p^2 \right)^2 + 1\right]}}, \hspace{0.7in} b_e= \sqrt{\frac{h \left(k_p - 1\right)^2 \left(\lambda^2 + \omega_p^2 \right)^2}{\lambda k_p s_1^2 \left[ \left(c_x + \omega_p^2 \right)^2 + 1\right]}},\\ & \cos \vartheta = \frac{1}{\sqrt{\left(c_x + \omega_p^2 \right)^2 + 1 }}, \hspace{1.3in} \sin \vartheta= \frac{\left(c_x + \omega_p^2 \right)}{\sqrt{\left(c_x + \omega_p^2 \right)^2 + 1 } }\\ \end{split}$$ The ellipse is counterclockwise tilted by $\vartheta$ from a standard ellipse $\bar q_{10}^2/a_e^2 + \bar q_{20}^2 /b_e^2=1$. The ellipse governed by is the critical condition that $\bar p_{10}$ exists, so it is the boundary for asymptotic orbits. In other words, inside the ellipse, transit orbits exist, while outside the ellipse, transit orbits do not exist. As a result, we refer to the ellipse as the [**ellipse of transition**]{} (see Figure \[lin\_damp\_position\_paper\](b)). Note that on the boundary of the ellipse, there is only one asymptotic orbit (i.e., the wedge has collapsed into a single direction). The solutions to are given by $$\bar p_{10}=\frac{-b_p \pm \sqrt{b_p^2 - 4 a_p c_p}}{2 a_p}$$ and the expression for $\bar p_{20}$ is $$\begin{aligned} \bar p_{20}=\bar p_{10} \left(c_x + \omega_p^2 \right) + \frac{\lambda \left(1+ k_p \right) \left[\bar q_{20} -\bar q_{10} \left(c_x + \omega_p^2 \right) \right]}{k_p -1}\end{aligned}$$ Up to now, the initial conditions $(\bar q_{10}, \bar q_{20}, \bar p_{10}, \bar p_{20})$ for the asymptotic orbits at a specific position have been obtained. The interior angle determined by these two initial velocities defines the wedge of velocites: $\theta =\arctan \left(\bar p_{20}/\bar p_{10} \right)$. The boundary of this wedge correspond to the asymptotic orbits. In fact, the wedge for the conservative system can be obtained by this method by taking $c_1$ as zero. Figure \[lin\_damp\_position\_paper\] illustrates the projection on the configuration space in the equilibrium region. In the dissipative system, one important finding is the existence of the ellipse of transition given by . The length of the major and minor axes of the ellipse are $a_e$ and $b_e$, respectively. For small damping, the major axis is much larger than the minor axis so that it reaches far beyond the neck region. Thus, here we give the local flow near the neck region as shown in Figure \[lin\_damp\_position\_paper\](a). We show a zoomed-out view revealing the entire ellipse in Figure \[lin\_damp\_position\_paper\](b). The asymptotic orbits in the dissipative system are bounded by the ellipse (which is different from the asymptotic orbits in the conservative system, which are bounded by the strip). Moreover, in the conservative system, all asymptotic orbits can reach the boundary of the strip with the period of $ 2 \pi / \omega_p$, while the asymptotic orbits in the dissipative system can never reach the boundary of the ellipse after they start due to damping. Notice that $a_e$ goes to zero when $c_1$ is large enough. Outside the ellipse, $\vartriangle= b_p^2 - 4 a_p c_p<0$, only non-transit orbits project onto this region. Thus we can conclude that the signs of $k_1 - k_3$ and $k_4 - k_2$ are independent of the direction of the velocity and can be determined in each of the components of the equilibrium region $R$ complementary to the ellipse. For example, in the left-most component, $k_1 - k_3$ is negative and $k_4 - k_2$ is positive, while in the right-most components, $k_1 - k_3$ is positive and $k_4 - k_2$ is negative. Inside the ellipse the situation is more complex due to the existence of the wedge of velocity. For simplicity we still just show the wedges on the two vertical bounding line segments in Figure \[lin\_damp\_position\_paper\]. For example, consider the intersection of the strip with the left-most vertical line. At each position on the subsegment, one wedge of velocity exists in which $k_1 - k_2 $ is positive. The orbits with velocity interior to the wedge are transit orbits, and $k_4 - k_2$ is always positive. Orbits with velocity on the boundary of the wedge are asymptotic ($(k_1 - k_3) (k_4 - k_2)=0$), while orbits with velocity outside of the wedge are non-transit ($(k_1 - k_3) (k_4 - k_2)<0$). Notice that in Figure \[lin\_damp\_position\_paper\], the grey shaded wedges are the wedges for the dissipative system, while the blacked shaded wedges partially covered by the grey ones are for conservative system (hardly visible for the parameters shown in the figure). The shrinking of the wedges from the conservative system to the dissipative system is caused by damping. This confirms the expectation that an increase in damping decreases the proportion of the transit orbits. Transition Tubes ================ In this section, we go step by step through the numerical construction of the boundary between transit and non-transit orbits in the nonlinear system . We combine the geometric insight of the previous sections with numerical methods to demonstrate the existence of ‘transition tubes’ for both the conservative and damped systems. Particular attention is paid to the modification of phase space transport as damping is increased, as this has not been considered previously. #### Tube dynamics The dynamic snap-through of the shallow arch can be understood as trajectories escaping from a potential well with energy above a critical level: the energy of the saddle point S$_1$. However, even if the energy of the system is higher than critical, the snap-through may not occur. The dynamical boundary between snap-through and non-snap-through behavior can be systematically understood by [**tube dynamics**]{}. Tube dynamics [@Conley1968; @LlMaSi1985; @OzDeMeMa1990; @DeMeTo1991; @DeLeon1992; @Topper1997; @KoLoMaRo2000; @GaKoMaRo2005; @GaKoMaRoYa2006; @MaRo2006; @KoLoMaRo2011] supplies a large-scale picture of transport; transport between the largest features of the phase space—the potential wells. In the conservative system, the stable and unstable manifolds with a $S^1 \times \mathbb{R}$ geometry act as [**tubes**]{} emanating from the periodic orbits. While found above for the linearized system near S$_1$, these structures persist in the full nonlinear system The manifold tubes (usually called [**transition tubes**]{} in tube dynamics), formed by pieces of asymptotic orbits, separate two distinct types of orbits: transit orbits and non-transit orbits, corresponding to snap-through and non-snap-through in the present problem. The transit orbits, passing from one region to another through the bottleneck, are those inside the transition tubes. The non-transit orbits, bouncing back to their region of origin, are those outside the transition tubes. Thus, the transition tubes can mediate the global transport of states between snap-through and non-snap-through. In the dissipative system, similar transition tubes also exist. Even in systems where stochastic effects are present, the influence of these structures remains [@NaRo2017]. Algorithm for computing transition tubes ---------------------------------------- For the conservative system, Ref. [@KoLoMaRo2011] gives a general numerical method to obtain the transition tubes. The key steps are (1) to find the periodic orbits restricted to a specified energy using differential correction and numerical continuation based on the initial conditions obtained from the linearized system at first, then (2) to compute the manifold tubes of the periodic orbits in the nonlinear system (i.e., ‘globalizing’ the manifolds), and finally (3) to obtain the intersection of the Poincaré surface of section and global manifolds. See details in Ref. [@KoLoMaRo2011]. The method is effective in the conservative system, but not applicable to the dissipative system, since due to loss of conservation of energy, no periodic orbit exists. Thus, we provide another method as follows. [**Step 1: Select an appropriate energy.**]{} We first need to set the energy to an appropriate value such that the snap-though behavior exists. Once the energy is given, it remains constant in the conservative system. In our example, the critical energy for snap-through is the energy of S$_1$. Thus, we can choose an energy which is between that of S$_1$ and S$_2$. In this case, all transit orbits can just escape from W$_1$ to W$_2$ through S$_1$. Notice that the potential energy determines the width of the bottleneck and the size of the transition tubes which hence determines the relative fraction of transit orbits in the phase space. A representative energy case is shown in Figure \[non\_sections\_paper\], which also establishes our location for Poincaré sections $\Sigma_1$ and $\Sigma_2$ which are at $X=$constant lines passing through W$_1$ and W$_2$ respectively, and with $p_X>0$. ![\[non\_sections\_paper\][ For a representative energy above the saddle point S$_1$, we show the unstable periodic orbit in the neck region around S$_1$. It projects to a single line going between the upper and lower energy boundary curves, and arrows are shown for convenience. We show the Poincaré sections $\Sigma_1$ and $\Sigma_2$ which are defined by $X$ values equal to that of the two stable equilibria in the center of the left and right side wells, W$_1$ and W$_2$, respectively. The arrows on the vertical lines indicate that these Poincaré sections are also defined by positive $X$ momentum. ]{}](non_sections_paper.png){width="70.00000%"} [**Step 2: Compute the approximate transition tube and its intersection on a Poincaré section.**]{} We have analyzed the flow of linearized system in both phase space and position space which classifies orbits into four classes. It shows that in the conservative system the stable manifolds correspond to the boundary between transit orbits and non-transit orbits. Thus, we can choose this manifold as the starting point. We start by considering the approximation of transition tubes for the conservative system. [*Determine the initial condition.*]{} The stable manifold divides the transit orbits and non-transit orbits for all trajectories headed toward a bottleneck. Thus, we can use the stable manifold to obtain the initial condition. Considering the general solutions of the linearized equations , we can let $p_1^0=c$, $q_1^0=0$, $q_2^0=A_q$ and $p_2^0=A_p$. Notice that $$A_q^2+A_p^2=2 h/\omega_p \label{circle}$$ which forms a circle in the center projection, so in the next computational procedure we should pick up $N$ points on the circle with a constant arc length interval. Each $A_q$ and $A_p$ determined by these sampling points along with $p_1^0=c$ and $q_1^0=0$ can be used as initial conditions. When first transformed back to the position space and then transformed to dimensional quantities, this yields an initial condition $$\begin{pmatrix} X_0\\ Y_0\\p_{X0} \\ p_{Y0} \end{pmatrix} = \begin{pmatrix} x_e\\ y_e\\0\\ 0 \end{pmatrix} + \begin{pmatrix} L_x\\ L_y\\ \omega_0 L_x M_1\\ \omega_0 L_y M_2 \end{pmatrix}^T C \begin{pmatrix} c\\0\\A_q\\A_p \end{pmatrix} \label{Initial}$$ [*Integrate backward and obtaining Poincaré section.*]{} Using the $N$ initial conditions yielded by varying $A_q$ and $A_p$ governed by and integrating the nonlinear equations of motions in in the backward direction, we obtain a tube, formed by the $N$ trajectories, which is a linear approximation for the transition tube. Choosing the Poincaré surface-of-section $\Sigma_1$ is shown in Figure \[non\_sections\_paper\], corresponding to $X=X_{{\rm W}_1}$ and $p_X>0$. [**Step 3: Compute the real transition tube by the bisection method.**]{} We have obtained a Poincaré section which is the intersection of the approximate transition tube and the surface $\Sigma_1$. First pick a point (noted as $p_i$) which is almost the center of the closed curve. The line from $p_i$ to each of the $N$ points on the Poincaré map will form a ray. The point $p_i$ inside the curve in general is a transit orbit. Then choose another point on each radius which is a non-transit orbit, noted as $p_o$. With the approach described above, we can use the bisection method to obtain the boundary of the transition tube on a specific radius (cf. [@AnEaLo2017]). Picking the midpoint (marked by $p_m$) as the initial condition and carrying out forward integration for the nonlinear equation of motion in , we can estimate if this trajectory can transit or not. If it is a transit orbit, note it as $p_i$, or note it as $p_o$. Continuing this procedure again until the distance between $p_i$ and $p_o$ reaches a specified tolerance, the boundary of the tube on this ray is estimated. Thus, the real transition tube for the conservative system can be obtained if the same procedure is carried out for all angles. A related method is described in [@OnYoRo2017], adapting an approach of [@GaMaDuCa2009]. For the dissipative system, the size of the transition tubes for a given energy on $\Sigma_1$ will shrink. Using the bisection method and following the same procedure as for conservative system, the transition tube for the dissipative system will be obtained. Numerical results and discussion -------------------------------- To visualize the tube dynamics for the arch, several examples will be given. According to the steps mentioned above, we can obtain the transition tubes for both the conservative system and dissipative systems. For all results, the geometries of the arch are selected as $b=12.7$ mm $d=0.787$ mm, $L=228.6$ mm. The Young’s modulus and the mass density are $E=153.4$ GPa and $\rho=7567 \ \mathrm{kg \ m^{-3}}$. The selected thermal load corresponds to $184.1$ N, while the initial imperfections are $\gamma_1 = 0.082$ mm and $\gamma_2 = -0.077$ mm. These values match the parameters given in the experimental study [@WiVi2016]. For all the numerical results given in this section, the initial energy of the system is set at 3.68$\times 10^{-4}$ J - above the energy of saddle point S$_1$, so that the equilibrium point $W_1$ is inside the configuration space projection. This choice of initial energy will make it possible to compare the numerical results with the experimental results which are planned for future work. #### Transition tubes for conservative system For conservative system, the Hamiltonian is a constant of motion. In Figure \[non\_conservative\_tube\_all\_paper\], we show the configuration space projection of the transition tube and the Poincaré sections on $\Sigma_1$ and $\Sigma_2$ which are closed curves. In Figure \[non\_conservative\_tube\_all\_paper\] are shown all the trajectories which form the transition tube boundary starting from $\Sigma_1$ and ending up at $\Sigma_2$, flowing from left to right through the neck region. ![\[non\_conservative\_tube\_all\_paper\][ A transition tube from the left well to the right well, obtained using the method described in the text. The upper figure shows the configuration space projection. The lower left shows the tube boundary (closed curve) on Poincaré section $\Sigma_1$, which separates transit and non-transit trajectories. The lower right shows the corresponding tube boundary (closed curve) on Poincaré section $\Sigma_2$. ]{}](non_conservative_tube_all_paper.png){width="\textwidth"} Due to the the conservation of energy, the size of the transition tube is constant during evolution, which corresponds to the cross-sectional area of the transition tube. It should be noted that the areas of the tube Poincaré sections on $\Sigma_1$ and $\Sigma_2$ in Figure \[non\_conservative\_tube\_all\_paper\] are equal, due to the integral invariants of Poincaré for a system obeying Hamilton’s canonical equations (with no damping). Moreover, note that the size of the transition tube, the boundary of the transit orbits, is determined by the energy. For a lower energy, the size of the transition tube is smaller or vice versa. In other words, the area of the Poincaré sections on $\Sigma_1$ and $\Sigma_2$ is determined by the energy. In fact, the cross-sectional area of the transition tube is proportional to the energy above the saddle point S$_1$ [@MacKay1990]. As mentioned before, the transition tube separates the transit orbits and non-transit orbits, which correspond to snap-through and non-snap-through. The orbit inside the transition tube can transit, while the orbit outside the transition tube cannot transit. #### Transition tubes for dissipative system Unlike the conservation of energy in conservative system, the energy in the dissipative system is decreasing with time. ![\[non\_damp\_tube\_all\_paper\][ A transition tube from the left well to the right well, obtained using the method described in the text, for the case of damping. The upper figure shows the configuration space projection. The lower left shows the tube boundary (closed curve) on Poincaré section $\Sigma_1$ which separates transit and non-transit trajectories for initial conditions all with a given fixed initial energy. The lower right shows the corresponding image under the flow on Poincaré section $\Sigma_2$. Due to the damping, and a range of times spent in the neck region, spiraling is visible in this 2D projection since trajectories which spend longer in the neck will be at lower total energies. Compare with Figure \[non\_conservative\_tube\_all\_paper\]. ]{}](non_damp_tube_all_paper.png){width="\textwidth"} Figure \[non\_damp\_tube\_all\_paper\] shows the configuration space projection of the transition tube and the Poincaré sections on $\Sigma_1$ and $\Sigma_2$. In Figure \[non\_damp\_tube\_all\_paper\] the transition tube starts from $\Sigma_1$ and ends up with $\Sigma_2$ flowing from left to right through the neck region, as shown previously for the conservative system. From the figure, we can observe the distinct reduction in the size of the transition tube, especially near the neck region. To show this, the scale of the Poincaré section projections is the same as in Figure \[non\_conservative\_tube\_all\_paper\]. During the evolution, the energy of the system is decreasing due to damping. The trajectories spend a great amount of time crossing the neck region, resulting in the total energy decreasing dramatically (and influencing the size of the transition tubes to the right of the neck region). Thus, the transition tube is spiraling in the neck region so that Poincaré $\Sigma_2$ is not a closed curve, nor are the trajectories at a constant energy. The $\Sigma_2$ plot is merely a projection onto the $(Y,p_Y)$-plane to give an idea of the actual co-dimension 1 tube boundary in the 4-dimensional phase space. Note the clear differences between Figure \[non\_conservative\_tube\_all\_paper\] and Figure \[non\_damp\_tube\_all\_paper\]. The dramatic shrinking of tubes near the neck region is due almost entirely to the linearized dynamics near the saddle point. To confirm this, we present the linear transition tube obtained by the analytical solutions for the linearized dissipative system in Figure \[lin\_damp\_tube\_paper\]. ![\[lin\_damp\_tube\_paper\][ A transition tube from the left side boundary ($n_1$) to the right side boundary ($n_2$) of the equilibrium region around saddle point S$_1$, obtained for the linear damped system. Notice that the shrinking of the tube is observed as in the nonlinear system, Figure \[non\_damp\_tube\_all\_paper\], here seen in terms of the width of the projected strip onto configuration space. ]{}](lin_damp_tube_paper.png){width="70.00000%"} #### Effect of damping on the transition tubes In order to further quantify how damping affects the size of transition tubes, we present the tube Poincaré section on $\Sigma_1$ with different damping in Figure \[multi\_damping\_Poincare\_both\_paper\]. In Figure \[multi\_damping\_Poincare\_both\_paper\](a), we can see the canonical area ($\int_\mathcal{A} p_Y dY$) decreases with increasing damping. Thus, the proportion of transition trajectories will be fewer if the damping increases. Note that when the damping changes, different transition tubes almost share the same center which corresponds to the fastest trajectories. Figure \[multi\_damping\_Poincare\_both\_paper\](b) shows the relation between the damping and the projected canonical area ($\int_\mathcal{A} p_Y dY$), which is related to the relative number of transit compared to non-transit orbits. It shows that an increase in damping decreases the projected area. When the damping is small, the relation between the damping and the area is linear, while when the damping is large, the relation becomes slightly nonlinear. Note that generally in mechanical/structural experiments the non-dimensional damping factor $\xi_d$ is less than $5\%$ which corresponds to a damping coefficient $C_H$ less than $107.3 \ \mathrm{s^{-1}}$ (see the shaded region in Figure \[multi\_damping\_Poincare\_both\_paper\](b)). Furthermore, note that for the initial energy depicted in Figure \[multi\_damping\_Poincare\_both\_paper\], there are are no transit orbits starting on $\Sigma_1$ for $C_H$ greater than about $185 \ \mathrm{s^{-1}}$. ![\[multi\_damping\_Poincare\_both\_paper\][ The effect of the damping coefficient $C_H$ on the area of the transition tube on Poincaré section $\Sigma_1$ is shown. For a fixed initial energy above the saddle, the projection on the canonical plane $(Y,p_Y)$ is shown in (a) and the area is plotted in (b). In (b), the shaded region indicates the experimentally observed range of damping coefficients, which correspond to non-dimensional damping factor $\xi_d$ less than 5%. ]{}](multi_damping_Poincare_both_paper.png){width="\textwidth"} #### Demonstration of trajectories inside and outside the transition tube To illustrate the effectiveness of the transition tubes, we choose three points on $\Sigma_1$ (see A, B and C in Figure \[time-history-Poincare-all-mod-paper\](a)) as the initial conditions and integrate forward to see their evolution. ![\[time-history-Poincare-all-mod-paper\][ Several example trajectories are shown, starting from the stable well point W$_1$. The initial conditions from Poincaré section $\Sigma_1$ are shown in (a) for a fixed initial energy, along with the transition tube boundaries for the conservative case and a damped case. In (b), we show the trajectories for points A and B, for the conservative case where A is just outside the tube boundary and B is just inside. In (c), we show the trajectories for points B and C, for the damped case where B is just outside the tube boundary and C is just inside. In (d), we illustrate the effect of damping by starting the same initial condition, B, but showing the trajectory in the conservative case as trajectory B and the damped case as trajectory B$^{\prime}$. ]{}](time-history-Poincare-all-mod-paper.png){width="\textwidth"} Note that all the trajectories corresponding to these three points have the same initial energy and start from a configuration identical to the equilibrium point $W_1$, but with different initial velocity directions. Figure \[time-history-Poincare-all-mod-paper\](b) shows the trajectories A and B in the conservative system where A is outside the tube boundary and B is inside the tube boundary. In the figure, trajectory B transits through the neck region and trajectory A bounces back. Figure \[time-history-Poincare-all-mod-paper\](c) shows trajectories B and C in the dissipative system. Like the situation in the conservative system, trajectory C which is inside the tube can transit, while trajectory B which is outside the tube cannot. Figure \[time-history-Poincare-all-mod-paper\](d) shows the effect of damping on the transit condition for the trajectories B and B$^{\prime}$ with the same initial condition. Trajectory B is simulated using the conservative system and trajectory B$^{\prime}$ is simulated using the dissipative system. It shows that the damping changes the transit condition that a transit orbit B in the conservative system becomes non-transit orbit B$^{\prime}$ in the dissipative system, both starting from the same initial condition. From Figure \[time-history-Poincare-all-mod-paper\], we can conclude the transition tube can effectively estimate the snap-through transitions both in conservative systems and dissipative systems. Finally, we point out that the transition tubes are the boundary for transit orbits that transition [*the first time*]{}. For example, trajectory A in Figure \[time-history-Poincare-all-mod-paper\](b) stays outside of the transition tube so that it returns near the neck region at first, but, unless it happens to be on a KAM torus or a stable manifold of such a torus, it will ultimately transit as long as it does not form a periodic orbit near the potential well W$_1$, since the energy is above the critical energy for transition and is conservative. Conclusions =========== Tube dynamics is a conceptual dynamical systems framework initially used to study the isomerization reactions in chemistry [@OzDeMeMa1990; @DeMeTo1991; @DeLeon1992; @Topper1997; @JaFaUz1999] as well as other fields, like resonance transitions in celestial mechanics [@LlMaSi1985; @KoLoMaRo2000; @JaRoLoMaFaUz2002; @GaKoMaRoYa2006; @MaRo2006] and capsize in ship dynamics [@NaRo2017]. Here we extend the application of tube dynamics to structural mechanics: the snap-through of a shallow arch, or buckled-beam. In general, slender elastic structures are capable of exhibiting a variety of (co-existing) equilibrium shapes, and thus, given a disturbance, tube dynamics sheds light on how such a system might be caused to transition between available, stable equilibrium configurations. Moreover, it is the first time, to the best of our knowledge, that tube dynamics has been worked out for a dissipative system, which increases the generality of the approach. The snap-through transition of an arch was studied via a two-mode truncation of the governing partial differential equations based on Euler-Bernoulli beam theory. Via analysis of the linearized Hamiltonian equations around the saddle, the analytical solutions for both the conservative and dissipative systems were determined and the corresponding flows in the equilibrium region of eigenspace and configuration space were discussed. The results show that all transit orbits, corresponding to snap-through, must evolve from a wedge of velocities which are restricted to a strip in configuration space in the conservative system, and by an ellipse in the corresponding dissipative system when damping is included. Using the results from the linearization as an approximation, the transition tubes based on the full nonlinear equations for both the conservative and dissipative system were obtained by the bisection method. The orbits inside the transition tubes can transit, while the orbits outside the tubes cannot. Results also show that the damping makes the size of the transition tubes smaller, which corresponds to the degree, or amount, of orbits that transit. When the damping is small, it has a nearly linear effect on the size of the transition tubes. Further study of the dynamic behaviors of the arch can lead to more immediate application structural mechanics. For example, many structural systems possess multiple equilibria, and the manner in which the governing potential energy changes with a control parameter is, of course, the essence of bifurcation theory. However, under nominally fixed conditions, the present paper directly assesses the energy required to (dynamically) perturb a structural system beyond the confines of its immediate potential energy well. In future work, a three-mode truncation may be introduced to study such systems. High order approximations will present higher index saddles which will modify the tube dynamics framework presented here (cf. [@collins2011index] [@HallerUzer2011] [@Nagahata2013]). Furthermore, experiments will be carried out to show the effectiveness of the present approach to prescribe initial conditions which lead to dynamic buckling. Acknowledgements ================ This work was supported in part by the National Science Foundation under awards 1150456 (to SDR) and 1537349 (to SDR and LNV). 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--- author: - | JLQCD Collaboration: $^a$[^1], G. Cossu$^{a}$, S. Hashimoto$^{a,b}$, T. Kaneko$^{a,b}$, J. Noaki$^{a}$, M. Tomii$^{b}$\ High Energy Accelerator Research Organization (KEK), Ibaraki 305-0801, Japan Address\ Department of Particle and Nuclear Science, SOKENDAI (The Graduate University for Advanced Studies), Ibaraki 305-0801, Japan bibliography: - 'lattice\_2015\_fahy.bib' title: | (0,0)(0,0) (300,75)[(0,0)\[l\][[KEK-CP-340]{.nodecor} ]{}]{} Decay constants and spectroscopy of mesons in lattice QCD using domain-wall fermions --- Introduction ============ The JLQCD collaboration has recently produced ensembles of lattice configurations with fine lattice spacings and good chiral symmetry. Lattice simulations of 2+1-flavor QCD were performed using the Möbius domain-wall fermions [@mobius] with tree-level Symanzik gauge action. Table \[tab:lattices\] shows the $15$ gauge ensembles generated [@finelattice]. These lattices have lattice spacings of $1/a \approx 2.4$, $3.6$, and $4.5\text{ GeV}$ with pion masses between $230$ MeV and $500$ MeV. For each ensemble, $10,000$ molecular dynamics (MD) times were run after thermalization. Using domain-wall fermions the Ginsparg-Wilson relation is only approximate. The violation of the Ginsparg-Wilson relation is given by the residual mass. The Möbius representation as well as using stout link-smearing [@stout] make the residual mass small of $\mathcal{O}(1\text{ MeV})$ on the coarsest ($\beta=4.17$) lattices and $<0.2\text{ MeV}$ on the finer lattices [@residual]. Good chiral symmetry enables simpler renormalization such as $Z_V = Z_A$, and simplifies the calculation of the pseudo-scalar decay constants directly utilizing the PCAC relation. With chiral symmetry preserved, observables can be used to compare lattice QCD results to those of Chiral Perturbation Theory (ChPT). In this work we present results of masses and decay constants of light and charmed pseudo-scalar mesons. These measurements are used to determine the low energy constants (LEC) in ChPT. Additionally the fine lattice spacing enables exploration of charm physics with manageable cutoff effects. In this work the scale for these lattices was determined from Wilson flow using $t_0$. We use the value from [@scale] as an input. The determination was done using a linear extrapolation to the physical point in $m_\pi^2$ as well as an interpolation of the strange quark mass to reproduce the physical $(M^\text{phys}_{s\bar{s}})^2 = 2M_K^2 - M_\pi^2$. The fitted parameters to describe the mass dependence for the two smaller $\beta$ values were then used to determine the scale on our finest lattice. The results for $a^{-1}$ on each $\beta$ are listed in Table \[tab:lattices\]. Lattice Spacing $L^3\times T$ $L_5$ $a m_{ud}$ $a m_s$ $ m_\pi \text{ [MeV]} $ $ m_{\pi}L $ --------------------------------- ----------------------------------------- ------- ------------ --------- -------------------------- -------------- $\beta = 4.17,$ $ 32^3\times64$ $(L=2.6 \text{ fm})$ 12 0.0035 0.040 230 3.0 $a^{-1}=2.453(4)\text{ GeV}$ 0.0070 0.030 310 4.0 0.0070 0.040 310 4.0 0.0120 0.030 400 5.2 0.0120 0.040 400 5.2 0.0190 0.030 500 6.5 0.0190 0.040 500 6.5 $48^3\times96 $ $ (L=3.9 \text{ fm})$ 12 0.0035 0.040 230 4.4 $\beta= 4.35,$ $48^3\times 96 $ $(L=2.6 \text{ fm})$ 8 0.0042 0.018 300 3.9 $a^{-1}=3.610(9)\text{ GeV} $ 0.0042 0.025 300 3.9 0.0080 0.018 410 5.4 0.0080 0.025 410 5.4 0.0120 0.018 500 6.6 0.0120 0.025 500 6.6 $\beta = 4.47,$ $64^3\times128 $ $(L=2.7 \text{ fm}) $ 8 0.0030 0.015 280 4.0 $a^{-1} = 4.496(9) \text{ GeV}$ : Parameters of the JLQCD gauge ensembles used in this work. Pion masses are rounded to the nearest $10$ MeV. The ensemble with $m_\pi L \approx 3.0$ is excluded in all analysis below to avoid possible finite volume effects. \[tab:lattices\] Computation of observables ========================== Pseudo-scalar correlators were produced utilizing our QCD software package Iroiro++ [@iroiro]. These correlators were computed on $200$ gauge configurations separated by $50$ MD times and from two source locations, producing $400$ measurements of the light correlators and $300$ measurements of heavy correlators for each ensemble, except for the $\beta=4.17$ ensemble on the larger volume, which has $600$ light and $400$ heavy measurements. Correlators were produced with unsmeared point sources as well as smeared sources using Gaussian smearing, and the same point and smeared operators are used also for the sinks. Gaussian smearing is defined by the operator $(1-(\alpha/N) \Delta)^N$ where $\Delta$ as the Laplacian and in this work the parameters $\alpha= 20.0$ and $N=200$ were used. The amplitudes of the unsmeared local operators are required to compute the decay constants. Two-point correlation functions of the form ${\langle P^L(x){P^G}^\dagger(0)\rangle}$ were fit simultaneously with correlators ${\langle P^G(x){P^G}^\dagger(0)\rangle}$ where $L$ indicates an unsmeared local operator while $G$ denotes Gaussian smeared operators. The two-point correlation functions were fit to the functional form $$\begin{aligned} \label{eq:amp} C = \underbrace{\frac{1}{2m_\pi} {\langle 0\vert}P {\vert \pi\rangle}{\langle \pi\vert}P^\dagger{\vert 0\rangle}}_{A_{PP}} {\left( e^{-m_\pi t}+e^{-m_\pi (N_t-t)} \right)}\end{aligned}$$ for large $t$ to determine the masses and amplitudes where $P$ is either $P^L$ or $P^G$. The matrix element of ${\langle 0\vert}P{\vert \pi\rangle}$ of the unsmeared operator $P^L$ can be reconstructed from the simultaneous fit of ${\langle P^L(x)P^G(0)\rangle}$ and ${\langle P^G(x)P^G(0)\rangle}$. The decay constants are calculated by utilizing the axial Ward-Takahashi identity ${Z_A\partial_\mu A_\mu=(m_{q_1}+m_{q_2})P}$, where $A_\mu$ is the lattice axial current, and $m_q$’s are the quark masses of the pseudo-scalar meson of interest. This leads to the formula for $f_P$, $$\begin{aligned} \label{eq:fpi} f_P = (m_{q_1} + m_{q_2})\sqrt{\frac{2A_{PP}}{m_\pi^3}},\end{aligned}$$ which does not rely on the renormalization constant $Z_A$. The mass $m_q$ used is the bare quark mass plus the residual mass. We use the convention $F_\pi = f_\pi/\sqrt{2}$. Pion masses and decay constants =============================== Our measurements of the pion masses and decay constants for ensembles at different bare light quark masses allow us to investigate the consistency with $SU(2)$ ChPT. The quark mass dependence of $M_\pi$ and $F_\pi$ at next-to-next-to-leading order [@PhysRevD.90.114504] is $$\begin{aligned} \frac{M_\pi^2}{\bar{m}_q } &= 2B{\left[ 1-\frac{1}{2}x \ln\frac{\Lambda_3^2}{M^2} + \frac{17}{8} x^2{\left( \ln\frac{\Lambda_M^2}{M^2} \right)}^2 + k_M x^2 +\mathcal{O}{\left( x^3 \right)} \right]}, \label{mx-expand}\\ F_\pi &= F{\left[ 1 + x \ln\frac{\Lambda_4^2}{M^2}-\frac{5}{4} x^2{\left( \ln\frac{\Lambda_F^2}{M^2} \right)}^2 + k_F x^2 +\mathcal{O}{\left( x^3 \right)} \right]}. \label{fx-expand}\end{aligned}$$ These are expanded using the parameter $x = M^2/(4\pi F)^2$ where $M^2 = B (\bar{m}_q +\bar{m}_q ) = 2\bar{m}_q \Sigma/F^2$. $\bar{m}_q$ is the appropriately renormalized quark mass, where the renormalization factor is discussed in [@renormalization]. The parameters $\Lambda_3$ and $\Lambda_4$ are related to the effective coupling constants of ChPT through $\bar{\ell}_n = \ln {\Lambda_n^2/M_\pi^2}$. $\Lambda_M$ and $\Lambda_F$ are linear combinations of different $\Lambda_n$’s [@PhysRevD.90.114504]. The chiral expansions above are fit to the data for $F_\pi$ and $M_\pi^2/\bar{m}_q $ simultaneously at both NLO and NNLO. At NLO only terms up to $\mathcal{O}{{\left( x^2 \right)}}$ in (\[mx-expand\]) and (\[fx-expand\]) are included leaving the free parameters $F, B,\Lambda_3$, and $\Lambda_4$. For NNLO there are the additional free parameters $k_M$, and $k_F$, while the values of $\Lambda_{1}$ and $\Lambda_{2}$ were fixed to the phenomenological value from [@colangelo]. To account for the strange-quark mass dependence the fit function was corrected by a term proportional to $M_{s\bar{s}}^2 = 2M_K^2 - M_\pi^2$. Combining with a lattice spacing dependence, all fits were performed with a prefactor $(1+ \gamma_{a} a^2 + \gamma_{s}(M_{s\bar{s}} - M_{s\bar{s}}^{\text{phys}}) )$. At NLO the fits have $\chi^2$ less than $1.5$ if including only the ensembles with pion masses $M_\pi < 450 \text{ MeV}$, so the other ensembles were excluded for the NLO fits. For NNLO fits the ensembles of all pion masses were included. The results of the NLO and NNLO fits in the continuum and physical strange quark mass limits are shown in Figure \[fig:Fpi\_x\] by dashed lines. ![Plots of $M_\pi^2/\bar{m}_q $ (left panel) and $F_\pi$ (right panel), both vs. $x=2\bar{m}_q B/(4\pi F)^2$. Fit lines show the best NLO (blue) and NNLO (green) fits in the continuum and physical strange quark mass limits. The NLO fits only include the ensembles for $M_\pi < 450 \text{ MeV}$[]{data-label="fig:Fpi_x"}](combined_mpisqrbymq_x_NNLO_all_comb.eps "fig:"){width="49.00000%"} ![Plots of $M_\pi^2/\bar{m}_q $ (left panel) and $F_\pi$ (right panel), both vs. $x=2\bar{m}_q B/(4\pi F)^2$. Fit lines show the best NLO (blue) and NNLO (green) fits in the continuum and physical strange quark mass limits. The NLO fits only include the ensembles for $M_\pi < 450 \text{ MeV}$[]{data-label="fig:Fpi_x"}](combined_Fpi_NNLO_all_x.eps "fig:"){width="49.00000%"} ![Same as Figure \[fig:Fpi\_x\] except plotted vs. $\xi = M_\pi^2 / (4\pi F_\pi)^2$.[]{data-label="fig:Fpi_xi"}](mpisqrbymq_XI_NNLO_comb.eps "fig:"){width="49.00000%"} ![Same as Figure \[fig:Fpi\_x\] except plotted vs. $\xi = M_\pi^2 / (4\pi F_\pi)^2$.[]{data-label="fig:Fpi_xi"}](Fpi_NNLO_xi_inverse_comb.eps "fig:"){width="49.00000%"} Alternatively the ChPT expansions can be reorganized using the parameter $\xi = M_\pi^2 /(4\pi F_\pi)^2$. The expansions are $$\begin{aligned} \frac{M_\pi^2}{\bar{m}_q } &= 2B / {\left[ 1+\frac{1}{2} \xi \ln\frac{\Lambda_3}{M_\pi^2} - \frac{5}{8} \xi^2{\left( \ln\frac{\Omega_M^2}{M_\pi^2} \right)}^2 + c_M \xi^2 +\mathcal{O}{\left( \xi^3 \right)} \right]},\\ F_\pi &= F / {\left[ 1 - \xi \ln\frac{\Lambda_4^2}{M_\pi^2}-\frac{1}{4} \xi^2{\left( \ln\frac{\Omega_F^2}{M_\pi^2} \right)}^2 + c_F \xi^2 +\mathcal{O}{\left( \xi^3 \right)} \right]}, \label{xi-expand}\end{aligned}$$ where similarly the values $\Omega_M$ and $\Omega_F$ are combinations of other LEC’s [@PhysRevD.90.114504]. The pion masses and decay constants are plotted against $\xi$ in Figure \[fig:Fpi\_xi\]. The curves represent the fits of NLO and NNLO. Our preliminary results for the LEC’s from the NLO fits expanded in $x$ are $F = 83.2(6.3)\text{ MeV}$, $\Sigma^{1/3}[2\text{ MeV}] = 287.9(3.7)\text{ MeV}$, $\bar{\ell}_3 = 3.11(44)$ and $\bar{\ell}_3 = 4.37(22)$. The chiral condensate, $\Sigma$, is renormalized to the one in the $\overline{\text{MS}}$ scheme at $\mu = 2 \text{ GeV}$ using the renormalization factor calculated in [@renormalization]. The values obtained with the two expansion parameters as well as those from NLO and NNLO fits are all consistent within statistical error though the NNLO results have slightly larger uncertainty. $F$ is the decay constant in the chiral limit but at the physical pion mass value we obtain $F_\pi = 88.9(5.2)\text{ MeV}$. Charmed mesons ============== The lattice spacings of the JLQCD ensembles were chosen to treat heavy physics with minimal cutoff effects. We produced charmed correlators using domain-wall heavy quarks at three masses close to the charm mass. All results shown are first interpolated to the charm mass using the spin averaged $c\bar{c}$ masses. Charmonium correlators are also used in the analysis of their time-moments to determine the charm quark mass $m_c$ and strong coupling constant $\alpha_s$ [@charm]. Figure \[fig:charmedmasses\] shows the masses of the $D$ and $D_s$ mesons as well as linear fits in $M_\pi^2$ accounting for a dependence on the lattice spacing $a^2$ and interpolated in $m_s$ using $2M_K^2 - M_\pi^2$. The raw data for $M_{D_s}$ appear scattered because the data points with different input strange quark masses are plotted together. After interpolating in $2M_K^2 - M_\pi^2$, the data at different $\beta$ are more consistent with each other. The results after extrapolation are $M_D =1867.7(9.5)\text{ MeV}$ and $M_{D_s} = 1964.2(5.0)\text{ MeV}$. Their experimental values are $M_D^{\text{exp}} = 1864.8 \text{ MeV}$ and $M_{D_s}^{\text{exp}} = 1968.3 \text{ MeV}$. The dependence upon the lattice cutoff $a$ turned out to be minimal with a difference of $\mathcal{O}(1\%)$ between the fitted value at $\beta=4.17$ and the continuum limit. ![Masses of the $D$ meson (left) and $D_s$ meson (right) vs. $M_\pi^2$. These were fit linearly in $M_\pi^2$ accounting for a dependence on the lattice spacing $a^2$ and interpolated in the strange quark mass using $2M_K^2 - M_\pi^2$. The blue dashed line indicated the linear fit extrapolated to the continuum limit while the green dashed line shows the linear fit for the value of $a$ corresponding to our coarsest lattice $\beta=4.17$.[]{data-label="fig:charmedmasses"}](mD_s0.eps "fig:"){width="49.00000%"} ![Masses of the $D$ meson (left) and $D_s$ meson (right) vs. $M_\pi^2$. These were fit linearly in $M_\pi^2$ accounting for a dependence on the lattice spacing $a^2$ and interpolated in the strange quark mass using $2M_K^2 - M_\pi^2$. The blue dashed line indicated the linear fit extrapolated to the continuum limit while the green dashed line shows the linear fit for the value of $a$ corresponding to our coarsest lattice $\beta=4.17$.[]{data-label="fig:charmedmasses"}](mDs_s0.eps "fig:"){width="49.00000%"} The decay constants of the charmed mesons are also computed using the same process for the pion using the pseudo-scalar current and the appropriate quark masses. These results can be seen in Figure \[fig:charmeddecay\]. The fitted values after linear extrapolation in $M_\pi^2$ and $a^2$ are $f_D = 209.6(5.2)\text{ MeV}$ and $f_{D_s} = 244.4(4.1)\text{ MeV}$ with the dependence on the lattice spacing turning out to be negligible. For the decay constant of the $D$ meson we attempted to analyze with the ChPT fit at NLO [@Grinstein1992369] as well as a linear fit. It favors a smaller value for $f_D$ because of the chiral logarithm, but more precise data would be necessary to confirm, especially because the current fit is strongly influenced by the lightest data point, which has a relatively large error. ![Charmed meson decay constants $f_D$ (left panel) and $f_{D_s}$ (right panel) vs. $M_\pi^2$. On both plots the blue dashed line indicate a linear fit in $M_\pi^2$ accounting for a dependence on the lattice spacing $a^2$ and interpolated in strange quark mass using $2M_K^2 - M_\pi^2$. The left plot of $f_D$ includes a simple linear fit as well as the chiral NLO fit for the ensembles with $M_\pi < 450 \text{MeV}$.[]{data-label="fig:charmeddecay"}](fD_s0_both.eps "fig:"){width="49.00000%"} ![Charmed meson decay constants $f_D$ (left panel) and $f_{D_s}$ (right panel) vs. $M_\pi^2$. On both plots the blue dashed line indicate a linear fit in $M_\pi^2$ accounting for a dependence on the lattice spacing $a^2$ and interpolated in strange quark mass using $2M_K^2 - M_\pi^2$. The left plot of $f_D$ includes a simple linear fit as well as the chiral NLO fit for the ensembles with $M_\pi < 450 \text{MeV}$.[]{data-label="fig:charmeddecay"}](fDs_s0.eps "fig:"){width="49.00000%"} Summary {#sec:sum} ======= We have shown first results from the recently generated JLQCD lattices. The good chiral properties of these lattices enable successful fits of quantities to NLO and NNLO ChPT. Measurements are still in progress and the precision of the decay constants and LECs should be improved with better statistics and the use of stochastic noise sources. The fine lattice spacings allow us to compute charmed masses and decay constants with small dependence upon $a$. We plan to produce results with quarks heavier than the charm mass to investigate the lattice spacing dependence for heavy domain wall quarks. If the dependence continues to remain small it may be possible to extrapolate to $B$ physics. Numerical simulations are performed on the IBM System Blue Gene Solution at High Energy Accelerator Research Organization (KEK) under a support of its Large Scale Simulation Program (No. 13/14-04, 14/15-10). We thank P. Boyle for helping in the optimization of the code for BGQ. This work is supported in part by the Grant-in-Aid of the Japanese Ministry of Education (No. 26400259, 26247043, and 15K05065) and the SPIRE (Strategic Program for Innovative Research) Field5 project. [^1]: Email:bfahy@post.kek.jp

--- abstract: 'In this paper, we consider a two-dimensional heterogeneous cellular network scenario consisting of one base station (BS) and some mobile stations (MSs) whose locations follow a Poisson point process (PPP). The MSs are equipped with multiple radio access interfaces including a cellular access interface and at least one short-range communication interface. We propose a nearest-neighbor cooperation communication (NNCC) scheme by exploiting the short-range communication between a MS and its nearest neighbor to collaborate on their uplink transmissions. In the proposed cooperation scheme, a MS and its nearest neighbor first exchange data by the short-range communication. Upon successful decoding of the data from each other, they proceed to send their own data, as well as the data received from the other to the BS respectively in orthogonal time slots. The energy efficiency analysis for the proposed scheme is presented based on the characteristics of the PPP and the Rayleigh fading channel. Numerical results show that the NNCC scheme significantly improves the energy efficiency compared to the conventional non-cooperative uplink transmissions.[^1]' author: - bibliography: - 'manu\_icc2015-01.bib' title: 'On Energy Efficiency of the Nearest-Neighbor Cooperative Communication in Heterogeneous Networks' --- Cooperative communication; Poisson point process; heterogeneous cellular network Introduction ============ Nowadays many of mobile stations (MSs), e.g, smart cellular phones, tablets and PADs, are equipped with multiple radio access interfaces, e.g, cellular radio access, wireless local area network (WLAN), Bluetooth interfaces and etc.. As multi-mode MSs, they can constitute heterogeneous cellular networks (HCNs) and make it possible to improve the performance of cellular uplinks by serving as a relay to their neighboring MSs. By some of short-range communication methods provided by the multi-mode MSs, they can communicate with each other with significantly high efficiency and quite low cost. Then the MSs can exploit the short-range communication links among them along with the uplinks to the base station (BS) to form the cooperative communication, which can improve the performance of the HCNs with regard to rate, outage probability, coverage, energy efficiency and etc.. This paper focuses on the improvement of the energy efficiency of uplink cellular communications based on cooperation between neighboring MSs in HCNs. Cooperative diversity has already emerged as a new and effective technique to combat fading and to decrease energy consumption in wireless networks. The nearest neighbor relay scheme that relay is chosen to be the nearest-neighbor to the user towards the BS (access-point) always has been applied in [@SAK2010Relay12; @ALT2012Balance14]. [@SAK2010Relay12] proposes and analyzes the performance of two schemes: a distributed nearest-neighbor relay assignment in which users can act as relays, and an infrastructure-based relay assignment in which fixed relay nodes are deployed in the network to help the users forward their data. [@ALT2012Balance14] explores the balance between cooperation through relay nodes and aggregated interference generation in large decentralized wireless networks using decode-and-forward by the nearest neighbor relay scheme. [@ESM2010EnergyEfficient13] proposes an energy-efficient cooperative multicasting scheme by properly selecting relay agents (RAs) based on their location, channel condition and coverage. [@NB2013Contract15] studies the relay selection schemes to reduce energy consumption, and the optimal number of cooperative is also given. Besides, [@YU2010Channel18] is based on coded cooperation, which combines cooperation and channel coding together. To save bandwidth and improve the information transmission rate, network coding [@Li2003tit4] is often used after the MSs receive each others information successfully. But based on some criteria, [@ka2014Unicast17] finds more scenarios where network coding has no gain on throughput or energy saving. Further more, many existing works concentrate on the resource allocation in cooperative networks. [@Adeane2006spawc6] presents both a centralized and a distributed power allocation schemes to optimize the BER performance of cooperative networks. To maximize the overall throughput, [@Zhang2013globecom7] proposes an optimal power allocation. An adaptive coded cooperative protocol based on incremental redundancy by a ACK/NACK feedback is proposed in [@Alazem2008WiMob9]. Many of the above works have presented valuable theories, methods and technologies of cooperative communication. But there are still some improvements left to perform. Zou *et al*. in [@zou2013tcom] investigate user terminals cooperating with each other in transmitting their data packets to the BS, by exploiting the multiple network access interfaces, which is called inter-network cooperation. Given a target outage probability and data rate requirements, they analyze the energy consumption of conventional schemes as compared to the proposed inter-network cooperation. The results show that the inter-network cooperation can significantly improve the energy efficiency of the uplink cellular communications. However in practical view there are some limits of this scheme work. In [@zou2013tcom], it is required that the cooperative MSs have the same distances to the BS and know the instantaneous fading coefficients of both the short-range communication channel and each cellular channel to form an orthogonal matrix used in cooperative. In contrast, our paper intends to propose a more general and yet efficient cooperative scheme for HCNs, in which the cooperative MSs do not need to locate at the same distance away from the BS and are able to perform the cooperative communication without knowing the channel status. The main contributions of this paper are summarized as follows. At first, we present a nearest-neighbor cooperative communication (NNCC) scheme in a HCN consisting of different radio access networks, i.e., a short-range communication network and a cellular network. Then we compare the proposed NNCC scheme to conventional schemes without user cooperation under target outage probability and data rate requirements. Secondly, we derive the energy efficiency of NNCC scheme in a Rayleigh fading environment. Further more, given a target outage probability, data rate requirements and distances between MSs and the BS, we derive the cumulative distribution function (CDF) and the probability density function (PDF) of energy consumption by considering the MSs that follow a Poisson point process (PPP). The remainder of this paper is organized as follows. Section \[sec:Cellular-Uplink-Transmission\] presents the network model and the NNCC scheme. In Section \[sec:Energy-Consumption-Analysis\], we present the desired power consumption analysis, then we derive the CDF and the PDF of the system desired power consumption by considering stochastic spatial distribution of cooperative MSs. Section \[sec:Numerical-Results\] gives the numerical results. Finally, Section \[sec:Conclusions\] concludes the paper. System Model\[sec:Cellular-Uplink-Transmission\] ================================================ In this section, we first present a two-dimensional network model of a HCN environment. Then, we propose a NNCC scheme by exploiting the short-range network to assist cellular uplink transmissions. ![\[fig:System Model\]System Model](huitu2){width="0.75\columnwidth"} Network Model ------------- Consider a HCN consisting of a BS and some MSs whose locations follow a homogeneous Poisson point process (PPP) with density $\rho$. The MSs are assumed to equip with multiple radio access interfaces including at least a short-range communication interface and a cellular access interface. The MSs can communicate with their neighboring MSs by the short-range communication. The packet size of data is assumed to be the same across cellular communication link and all short-range communication links between MSs. Without loss of generality, we consider the BS is located at the origin of coordinate, and a specific MS U1 located at coordinate $\left(r_{0},0\right)$ intends to communicate with the BS. Our model is shown in Fig. \[fig:System Model\], U1 will choose its nearest neighboring MS, denoted by U2, to cooperatively communicate with the BS. According to the properties of the homogeneous PPP, the distance $r$ between U1 and U2 satisfies the following PDF [@stoyan1995stochastic] $$f_{r}(r)=2\pi\rho r\exp\left(-\pi\rho r^{2}\right),\label{eq:fr}$$ then the coordinate of U2 can be obtained as $\left(r_{0}+r\cos\theta,r\sin\theta\right)$, where $\theta$ follows a uniform distribution between $-\frac{\pi}{2}$ and $\frac{3\pi}{2}$. Denote the distance between U1 and the BS by $r_{1}$. The distance $r_{\mathrm{2}}$ between the MS U2 and BS can be obtained as $$r_{\mathrm{2}}^{2}=r{}^{2}+r_{1}^{2}+2r_{1}r\text{\ensuremath{\cos}}\left(-\theta\right)=r{}^{2}+r_{1}^{2}+2r_{1}r\cos\theta.\label{eq:d2b}$$ We consider a general channel model that incorporates the radio frequency, path loss and fading effects in characterizing wireless transmissions, i.e., $$\mathcal{P}_{{\rm \mathrm{R}}}=\mathcal{P}_{{\rm \mathrm{T}}}\left(\frac{\lambda}{4\pi d}\right)^{2}G_{{\rm \mathrm{T}}}G_{{\rm \mathrm{R}}}\left|h\right|^{2},\label{eq:pt-pr}$$ where $\mathcal{P}_{\mathrm{R}}$ is the received power, $\mathcal{P}_{\mathrm{T}}$ is the transmitted power, $\lambda$ is the carrier wavelength, $d$ is the transmission distance, $G_{\mathrm{T}}$ is the transmit antenna gain, $G_{\mathrm{R}}$ is the receive antenna gain, and $h$ is the channel fading coefficient. In this paper, we consider a Rayleigh fading model to characterize the channel fading, i.e., $|h|^{2}$ is modeled as an exponential random variable. The NNCC Scheme --------------- ![\[fig:NNCC\]In NNCC scheme, transmissions happen in short range network channel in time slot 1, and in cellular network channel in time slot 2 (and time slot 3).](schemes22){width="0.7\columnwidth"} In the NNCC scheme, when MS U1 and its nearest neighbor MS U2 intend to send data $D_{1}$ and $D_{2}$ to the BS, respectively, they cooperate with each other according to the following steps: 1. U1 and U2 exchange their data over the short-range communication network in time slot 1. 2. If both U1 and U2 succeed in decoding the data from each other, defined as the case $\delta=0$, both of them will send their own data as well as the data received from the other side to the BS in time slot 2 and time slot 3, respectively, i.e., they send both $D_{1}$ and $D_{2}$ to the BS over two orthogonal cellular uplink channels. Otherwise, defined as the case $\delta=1$, U1 and U2 will send only their own data to the BS separately in time slot 2, just like a conventional non-cooperation communication. Assuming that the short-range channels among MSs and the cellular channels to the BS are orthotropic and there is no interference at the BS among the MSs’ signals, we only consider the channel noise when analyzing the performance of the scheme. Considering that Ui transmits $D_{{\rm i}}$ to Uj with the signal power $\mathcal{P}_{ij}^{\mathrm{NC}}$, we can obtain the received signal-to-noise-ratio (SNR) between MSs by NNCC scheme as $$\gamma_{\mathrm{ij}}^{\mathrm{NC}}=\frac{\mathcal{P}_{\mathrm{ij}}^{\mathrm{NC}}}{N_{0}B_{\mathrm{s}}}\left(\frac{\lambda_{\mathrm{s}}}{4\pi r}\right)^{2}G_{\mathrm{U1}}G_{\mathrm{U2}}\left|h_{\mathrm{ij}}\right|^{2},\label{eq:r12nc}$$ where $\mathrm{i}=1\,\mathrm{or}\,2$, $\mathrm{j}=2\,\mathrm{or}\,1$, ${\rm i}\neq{\rm j},$ $\lambda_{s}$ is the carrier wavelength of the short-range communication, $G_{{\rm U}1}$ is the antenna gain at U1, $G_{\mathrm{U2}}$ is the antenna gain at U2, and $h_{{\rm ij}}$ is the fading coefficient of the channel from Ui to Uj. The noise is modeled as $N_{0}B_{s}$, where $N_{0}$ is the noise power spectral density and $B_{s}$ is the channel bandwidth. In step 2, MSs will transmit the data to the BS, and the received SNR at the BS from MS U1 or U2 over the cellular channel can be obtained as $$\gamma_{\mathrm{ib}}=\frac{\mathcal{P}_{\mathrm{ib}}}{N_{0}B_{\mathrm{c}}}\left(\frac{\lambda_{\mathrm{c}}}{4\pi r_{i}}\right)^{2}G_{\mathrm{Ui}}G_{\mathrm{BS}}\left|h_{\mathrm{ib}}\right|^{2},\label{eq:r1bnc1}$$ where $\mathrm{i}=1\,\mathrm{or}\,2$, $\lambda_{c}$ is the cellular carrier wavelength, $B_{c}$ is the cellular spectrum bandwidth, $G_{\mathrm{BS}}$ is the receive antenna gain at BS, and $h_{\mathrm{ib}}$ is the fading coefficient of channel from Ui to BS. Energy Efficiency Analysis of NNCC scheme\[sec:Energy-Consumption-Analysis\] ============================================================================ In this section, we analyze the energy efficiency of the proposed NNCC scheme compared to the conventional scheme without cooperation, under the requirements of target outage probability $P_{\mathrm{out}}$ and data rate $R$. Energy Consumption in the NNCC scheme ------------------------------------- \[thm:NNCC\_energy\_consumpution\]Under the situation that the BS succeeds in receiving the complete data from both MSs, the power consumption for the NNCC scheme can be obtained as $$\mathcal{P}^{\mathrm{NC}}=\mathcal{P}_{\mathrm{12}}^{\mathrm{NC}}+\mathcal{P}_{\mathrm{21}}^{\mathrm{NC}}+(1+(1-P_{{\rm out}})^{2})(\mathcal{P}_{\mathrm{1b}}^{\mathrm{NC}}+\mathcal{P}_{\mathrm{2b}}^{\mathrm{NC}}),\label{eq:pnc1}$$ where is the target outage probability,and to meet the target outage probability, where $\mathcal{P}_{12}^{\mathrm{NC}}$, $\mathcal{P}_{21}^{\mathrm{NC}}$, $\mathcal{P}_{1\mathrm{b}}^{\mathrm{NC}}$ and $\mathcal{P}_{2\mathrm{b}}^{\mathrm{NC}}$ are the desired transmission power from U1 to U2, U2 to U1, U1 to the BS and U2 to the BS, respectively, which are given by and . Due to the limited error correction capability in practical communication systems, both the short-range and cellular communications cannot achieve the Shannon capacity. Therefore, let $\Delta_{\mathrm{s}}>1$ and $\Delta_{\mathrm{c}}>1$ denote the performance gaps for the short-range communication and the cellular communication from their respective capacity limits, respectively. Using and considering the performance gap $\Delta_{\mathrm{s}}$ away from Shannon capacity, we obtain the maximum achievable rate from U1 to U2 of the short-range communication of the NNCC scheme as $$\begin{aligned} C_{\mathrm{12}}^{\mathrm{NC}} & =B_{\mathrm{s}}\log_{2}(1+\frac{\gamma_{\mathrm{12}}^{\mathrm{NC}}}{\Delta_{\mathrm{s}}})\nonumber \\ & =B_{\mathrm{s}}\log_{2}\left(1+\frac{\mathcal{P}_{\mathrm{12}}^{\mathrm{NC}}G_{\mathrm{U1}}G_{\mathrm{U2}}\left|h_{12}\right|^{2}}{\Delta_{\mathrm{s}}N_{0}B_{\mathrm{s}}}\left(\frac{\lambda_{\mathrm{s}}}{4\pi r}\right)^{2}\right).\label{eq:c12NC}\end{aligned}$$ In a Rayleigh fading channel, all random variables $\left|h_{12}\right|^{2}$, $\left|h_{21}\right|^{2}$, $\left|h_{\mathrm{1b}}\right|^{2}$ and $\left|h_{\mathrm{2b}}\right|^{2}$ follow independent exponential distributions with means $\sigma_{12}^{2}$, $\sigma_{21}^{2}$, $\sigma_{\mathrm{1b}}^{2}$ and $\sigma_{\mathrm{2b}}^{2}$, respectively. As we know, an outage event occurs when the channel capacity falls below the required data rate. Using and considering the performance gap $\Delta_{\mathrm{s}}$ away from Shannon capacity, we can obtain the outage probability of the short-range transmission from U1 to U2 as $$\begin{aligned} P_{\mathrm{out12}}^{\mathrm{NC}} & =\Pr\left(C_{\mathrm{12}}^{\mathrm{NC}}<R\right)\nonumber \\ & =1-\exp\left(-\frac{16\pi^{2}\triangle_{\mathrm{s}}N_{0}B_{\mathrm{s}}r^{2}\left(2^{\frac{R}{B_{\mathrm{s}}}}-1\right)}{\mathcal{P}_{\mathrm{12}}^{\mathrm{NC}}\sigma_{\mathrm{12}}^{2}G_{\mathrm{U1}}G_{\mathrm{U2}}\lambda_{\mathrm{s}}^{2}}\right).\label{eq:pout12nc}\end{aligned}$$ Assuming $P_{\mathrm{out12}}^{\mathrm{NC}}=P_{\mathrm{out}}$, we can obtain the desired power consumption of MSs for short-range communication $\mathcal{P}_{\mathrm{ij}}^{\mathrm{NC}}$ from as $$\mathcal{P}_{\mathrm{ij}}^{\mathrm{NC}}=-\frac{16\pi^{2}\triangle_{\mathrm{s}}N_{0}B_{\mathrm{s}}\left(2^{\frac{R}{B_{\mathrm{s}}}}-1\right)}{\sigma_{\mathrm{ij}}^{2}G_{\mathrm{U1}}G_{\mathrm{U2}}\lambda_{\mathrm{s}}^{2}\ln\left(1-P_{\mathrm{out}}\right)}r^{2}.\label{eq:p12nc}$$ Given $\sigma_{\mathrm{21}}^{2}=\sigma_{\mathrm{12}}^{2}$, we can obtain $\mathcal{P}_{\mathrm{12}}^{\mathrm{NC}}=\mathcal{P}_{\mathrm{21}}^{\mathrm{NC}}=\zeta r^{2}$, where $\zeta=-\frac{16\pi^{2}\triangle_{\mathrm{s}}N_{0}B_{\mathrm{s}}\left(2^{\frac{R}{B_{\mathrm{s}}}}-1\right)}{\sigma_{\mathrm{ij}}^{2}G_{\mathrm{U1}}G_{\mathrm{U2}}\lambda_{\mathrm{s}}^{2}\ln\left(1-P_{\mathrm{out}}\right)}$. As discussed before, case $\delta=0$ implies that both U1 and U2 succeed in decoding each others signals through short range communications, and $\delta=1$ means that either U1 or U2 (or both) fails to decode in the short-range transmissions. We can describe $\delta=0$ and $\delta=1$ as follows. $\delta=0$: $$B_{\mathrm{s}}\log_{2}\left(1+\frac{\gamma_{12}^{\mathrm{NC}}}{\Delta_{\mathrm{s}}}\right)\geqslant R\,{\rm and}\,B_{\mathrm{s}}\log_{2}\left(1+\frac{\gamma_{21}^{\mathrm{NC}}}{\Delta_{\mathrm{s}}}\right)\geqslant R.\label{eq:xita0}$$ $\delta=1:$ $$B_{\mathrm{s}}\log_{2}\left(1+\frac{\gamma_{12}^{\mathrm{NC}}}{\Delta_{\mathrm{s}}}\right)<R\,{\rm or}\,B_{\mathrm{s}}\log_{2}\left(1+\frac{\gamma_{21}^{\mathrm{NC}}}{\Delta_{\mathrm{s}}}\right)<R.\label{eq:xita1}$$ Denote the target outage probability for short-range communication between U1 and U2 by $P_{{\rm out}}$, given $P_{\mathrm{out}1\mathrm{2}}^{\mathrm{NC}}=P_{\mathrm{out21}}^{\mathrm{NC}}=P_{\mathrm{out}}$, we have $$\Pr\left(\delta=0\right)=\left(1-P_{{\rm out}}\right)^{2},$$ and $$\Pr\left(\delta=1\right)=1-\left(1-P_{{\rm out}}\right)^{2}.$$ Moreover, denote the target outage probability for cellular communication from U1, U2 to the BS by $P_{\mathrm{out}}^{{\rm NC}}$, and given $P_{\mathrm{out}1\mathrm{b}}^{\mathrm{NC}}=P_{\mathrm{out2b}}^{\mathrm{NC}}=P_{\mathrm{out}}^{{\rm NC}}$, we have $\Pr\left(C_{\mathrm{1b}}<R\right)=\Pr\left(C_{\mathrm{2b}}<R\right)$, then we obtain the outage probability of the NNCC scheme by from and as $P_{\mathrm{out}}=\left(1-P_{{\rm out}}\right)^{2}*\left(P_{\mathrm{out}}^{{\rm NC}}\right)^{2}+\left(1-\left(1-P_{{\rm out}}\right)^{2}\right)*\left(1-\left(1-P_{\mathrm{out}}^{{\rm NC}}\right)^{2}\right)$. Then, we obtain $$P_{\mathrm{out}}^{{\rm NC}}=\frac{\sqrt{\left(1-\epsilon\right)^{2}+P_{\mathrm{out}}\left(2\epsilon-1\right)}-\left(1-\epsilon\right)}{2\epsilon-1}\text{,}\label{eq:p11}$$ where $\epsilon=\left(1-P_{{\rm out}}\right)^{2}$. We can obtain the power consumption of Ui for cellular communication from as $$\mathcal{P}_{\mathrm{ib}}^{\mathrm{NC}}=-\frac{16\pi^{2}\triangle_{\mathrm{c}}N_{0}B_{\mathrm{c}}\left(2^{\frac{R}{B_{\mathrm{c}}}}-1\right)}{\sigma_{\mathrm{ib}}^{2}G_{\mathrm{Ui}}G_{\mathrm{BS}}\lambda_{\mathrm{c}}^{2}\ln\left(1-P_{\mathrm{out}}^{{\rm NC}}\right)}r_{i}^{2}.\label{eq:p1bnc1}$$ Given $\sigma_{\mathrm{1b}}^{2}=\sigma_{\mathrm{2b}}^{2}$, we can obtain $\mathcal{P}_{\mathrm{1b}}^{\mathrm{NC}}=\eta r_{1}^{2}$ and $\mathcal{P}_{\mathrm{2b}}^{\mathrm{NC}}=\eta r_{2}^{2}$, where $\eta=-\frac{16\pi^{2}\triangle_{\mathrm{c}}N_{0}B_{\mathrm{c}}\left(2^{\frac{R}{B_{\mathrm{c}}}}-1\right)}{\sigma_{\mathrm{ib}}^{2}G_{\mathrm{U2}}G_{\mathrm{BS}}\lambda_{\mathrm{c}}^{2}\ln\left(1-P_{\mathrm{out}}^{{\rm NC}}\right)}$. Notice that in case of $\delta=0$, cooperation communication is employed and there are energy consumption at both time slot 2 and time slot 3, resulting in that a total power consumption of $2(\mathcal{P}_{\mathrm{1b}}^{\mathrm{NC}}+\mathcal{P}_{\mathrm{2b}}^{\mathrm{NC}})$ is consumed by U1 and U2 in transmitting to BS. In case of $\delta=1$, U1 and U2 consume a total power consumption of $(\mathcal{P}_{\mathrm{1b}}^{\mathrm{NC}}+\mathcal{P}_{\mathrm{2b}}^{\mathrm{NC}})$ for transmitting to BS. Therefore, considering both the short-range communication and cellular transmissions, the total power consumption by the NNCC scheme is given by $$\begin{aligned} & \mathcal{P}^{\mathrm{NC}}\\ & =\mathcal{P}_{\mathrm{12}}^{\mathrm{NC}}+\mathcal{P}_{\mathrm{21}}^{\mathrm{NC}}+\left(2Pr\left(\delta=0\right)+Pr\left(\delta=1\right)\right)(\mathcal{P}_{\mathrm{1b}}^{\mathrm{NC}}+\mathcal{P}_{\mathrm{2b}}^{\mathrm{NC}})\\ & =\mathcal{P}_{\mathrm{12}}^{\mathrm{NC}}+\mathcal{P}_{\mathrm{21}}^{\mathrm{NC}}+(1+(1-P_{{\rm out}})^{2})(\mathcal{P}_{\mathrm{1b}}^{\mathrm{NC}}+\mathcal{P}_{\mathrm{2b}}^{\mathrm{NC}}).\end{aligned}$$ In order to compare our method and the traditional method, we give the same target interrupt probability definition about two schemes, and then we will derive the power consumption for the conventional scheme without user cooperation, in which both U1 and U2 succeed in transmitting their data to the BS separately. Similarly to the power consumption analysis of the NNCC scheme, assuming that $P_{\mathrm{out1b}}^{\mathrm{C}}=P_{\mathrm{out2b}}^{\mathrm{C}}=P_{{\rm out}}^{{\rm C}}$, the total power consumption of U1 and U2 under conventional non-cooperative communication can be obtained by $$\mathcal{P}^{\mathrm{C}}=\sum_{i=1}^{2}\mathcal{P}_{\mathrm{ib}}^{\mathrm{C}},\label{eq:pt}$$ where $$\mathcal{P}_{\mathrm{ib}}^{\mathrm{C}}=-\frac{16\pi^{2}\triangle_{\mathrm{c}}N_{0}B_{\mathrm{c}}r_{\mathrm{i}}^{2}\left(2^{\frac{R}{B_{\mathrm{c}}}}-1\right)}{\sigma_{\mathrm{ib}}^{2}G_{\mathrm{Ui}}G_{\mathrm{BS}}\lambda_{\mathrm{c}}^{2}\ln\left(1-P_{{\rm out2}}^{{\rm C}}\right)},\label{eq:p1bt}$$ and $$P_{{\rm out}}^{{\rm C}}=1-\sqrt{1-P_{\mathrm{out}}}.\label{eq:pOUT12C}$$ As and show, the NNCC scheme can save more energy than the conventional scheme because it can work under larger target outage probability. and give the relations between the desired transmission powers and the target outage probabilities as well as other impact factors, e.g., path loss, fading, and thermal noise, under the NNCC scheme and the conventional non-cooperative scheme, respectively. Based on them, some further performance analysis, such as the energy efficiency analysis in Section , can be presented. Energy Efficiency Analysis based on PPP\[sub:Energy-Efficiency-Analysis\] -------------------------------------------------------------------------- In order to get more performance analysis about the NNCC scheme, we put the NNCC scheme on a more general environment, we will consider MSs satisfy Poisson point process, which meet actual situation. The result is more practical and performance analysis is more accurate. Moreover, considering the stochastic spatial distribution of the MSs, we can derive the CDF and PDF of the desired transmission power which meet the target outage probability and rate requirement. \[thm:pdf\_cdf\_ee\] When the spatial distribution of the MSs follows a homogeneous PPP with density $\rho$, the PDF of is and the CDF of is where $$\varepsilon=1+(1-P_{{\rm out}})^{2}\text{,}\label{eq:I}$$ $$R_{2}=\frac{-\varepsilon\eta r_{1}\cos\theta+\varDelta_{R}}{2\zeta+\varepsilon\eta},\label{eq:r2}$$ and $$R_{1}=\frac{-\varepsilon\eta r_{1}\cos\theta-\varDelta_{R}}{2\zeta+\varepsilon\eta}.\label{eq:r1}$$ $Q_{1}$ stands for and $Q_{2}$ stands for Due to independence of random variables $r$ and $\theta$, considering $\theta$ follows the uniform distribution between $-\frac{\pi}{2}$ and $\frac{3\pi}{2}$, $r$ satisfies , the PDF of $r$ and $\theta$ can be obtained as $$f(r,\theta)=\rho r\exp\left(-\pi\rho r^{2}\right).\label{eq:frjiao-1-1-1}$$ Substituting into , we obtain $$\mathrm{\mathcal{P}}^{\mathrm{NC}}=\left(2\zeta+\varepsilon\eta\right)r^{2}+2\varepsilon\eta r_{1}\cos\theta r+2\varepsilon\eta r_{1}^{2}.\label{eq:grjiao}$$ So the the CDF of $\mathrm{\mathcal{P}}^{\mathrm{NC}}$ is $$\begin{aligned} & F_{\mathcal{P}^{\mathrm{NC}}}\left(p^{{\rm NC}}\right)=\Pr\left(P^{{\rm NC}}\leqslant p^{{\rm NC}}\right)\nonumber \\ & =\Pr\left(\left(2\zeta+\varepsilon\eta\right)r^{2}+2\varepsilon\eta r_{1}\cos\theta r+2\varepsilon\eta r_{1}^{2}\leqslant p^{{\rm NC}}\right).\label{eq:cdf_pnc}\end{aligned}$$ Based on and , we can acquire the CDF and PDF of desired power for NNCC scheme as expresses. From , we will know the PDF and CDF of $\mathrm{\mathcal{P}}^{\mathrm{NC}}$ and can easily obtain its expectation. It can help us to find the energy efficiency. Based on , it is easy to derive the relations between desired transmission power of the NNCC scheme and the target outage probability as well as other impact factors. Considering a successful transmission delivering both data of U1 and U2 by the total power consumption $\mathrm{\mathcal{P}}^{\mathrm{NC}}$, the energy efficiency of the NNCC scheme can be derived by $$EE^{\mathrm{NC}}=\frac{2R}{E\left(\mathrm{\mathrm{\mathcal{P}}^{\mathrm{NC}}}\right)}.\label{eq:ENERGY EFFICIENCY}$$ In , some numerical results about the desired transmission power and the energy efficiency of the NNCC scheme are given. Numerical Results\[sec:Numerical-Results\] ========================================== In this section, we present the analytical results of the proposed NNCC scheme and compare it to the existed schemes. We compare different schemes under the same definition of target outage probability. In our simulation, the frequency and bandwidth of the short-range communication are given as $f_{{\rm s}}=2.4{\rm \,GHz}$ and ${\rm B_{s}}=2{\rm \,MHz}$, respectively. In the cellular communication they are given as $f_{{\rm c}}=2100{\rm \,MHz}$ and ${\rm B}_{{\rm c}}=5\,{\rm MHz}$, respectively. The antenna gains of U1 and U2 are set as $G_{{\rm U1}}=G_{{\rm U2}}=0\,{\rm dB}$ and the BSs antenna gain is set as $G_{{\rm BS}}=5\,{\rm dB}$. The performance gaps $\Delta_{{\rm s}}$ and $\Delta_{{\rm c}}$ of short-range and cellular communication are given by $\Delta_{{\rm s}}=4\,{\rm dB}$ and $\Delta_{{\rm c}}=2\,{\rm dB}$. ![\[fig:power changes with distance\]Energy consumption by various transmission schemes with target outage probability $P_{\mathrm{out}}=10^{-3}$, effective rate $R=100000\,\mathrm{bits/s}$, and inter-user distance $r=20\,\mathrm{m}$. ](power_3_kinds){width="0.94\columnwidth"} Energy consumption is influenced by some parameters, such as distances between the MSs and the BS, distance between MSs, the density of MS and the target outage probability. In Fig. \[fig:power changes with distance\], we discuss the impact of distances between the MSs and the BS on energy consumption. We compare the desired transmission power of the NNCC scheme to the conventional non-cooperative scheme and the inter-network cooperative communication scheme by orthogonal matrix proposed in [@zou2013tcom]. The result shows that the power consumption of the NNCC scheme is significantly lower than the non-operative scheme and slightly higher than the inter-network cooperative scheme. It demonstrates the effectiveness of the NNCC scheme, while considering the simplicity and feasibility of the NNCC scheme, which does not need the knowledge of the instantaneous status of the channel and can be easily applied in the scenario of the MSs with stochastic spatial distribution. Similarly to Fig. \[fig:power changes with distance\], we present the comparison of the energy efficiency among the schemes in Fig. \[fig:Energy efficiency two way\]. ![\[fig:Energy efficiency two way\]Energy efficiency by various transmission schemes with target outage probability $P_{\mathrm{out}}=10^{-3}$, effective rate $R=100000\mathrm{\,bits/s}$, and inter-user distance $r=20\mathrm{\,m}$. ](ee_comp_3_kinds){width="0.94\columnwidth"} In Fig. \[fig:power and density \], we present the relations between the desired transmission power consumption and the density of the MSs under different distances between the MS U1 and the BS. It is shown that the power consumption decreases as the density of MSs increases, because the distance between the cooperative MSs tends to be smaller when the density of the MSs increases, and thus less power consumption is required in step 1. Fig. \[fig:power and outage probability\] demonstrates that the energy consumptions of the NNCC scheme decreases as the target outage probability increases with various densities of the MSs. ![\[fig:power and density \]Relation between the desired the transmission power and the density of MSs, with target outage probability $P_{\mathrm{out}}=10^{-3}$, effective rate $R=10000000\mathrm{\,bits/s}$, and U1-BS distance $r_{1}=2000\mathrm{\,m}$. ](rel_density_power){width="0.94\columnwidth"} ![\[fig:power and outage probability\]Desired transmission power with regard to the target outage probability with effective rate $R=1000000\,\mathrm{bits/s}$, and U1-BS distance $r_{1}=150\mathrm{\,m}$.](power_in_diff_out_prob){width="0.94\columnwidth"} Conclusions\[sec:Conclusions\] ============================== In this paper, we propose a cooperative communication scheme, namely the NNCC scheme, in which a MS in HCN and its nearest neighbor MS exploit a short-range communication network to assist the cellular transmissions. The energy efficiency of the NNCC scheme is analyzed and derived in closed-form expressions, which provide insights into the relations between energy efficiency and many important factors, e.g., MS density, the distance between the MSs and the BS, the target outage probability and etc.. Numerical results show that the NNCC scheme is simple, yet efficient compared to the existing schemes. Although this article studies collaboration between two MSs only, it’s obvious that there are vast energy saving comparing with the traditional scheme, no matter from the aspects of the distance between BS and MSs or expectation. In this paper, we have only considered the scenario of a single BS and allow only two MSs cooperate with each other. In the future, we will extend the work to the multi-cell scenario with multi-MS cooperative communications. [^1]: The corresponding author is Lijun Wang. The authors would like to acknowledge the support from the International Science and Technology Cooperation Program of China (Grant No. 2014DFA11640, 2012DFG12250 and 0903), the National Natural Science Foundation of China (NSFC) (Grant No. 61471180, 61271224 and 61301128), the NSFC Major International Joint Research Project (Grant No. 61210002), the Ministry of Science and Technology 863 program (Grant No. 2014AA01A707), the Hubei Provincial Science and Technology Department (Grant No. 2013CFB188), the Fundamental Research Funds for the Central Universities (Grant No. 2013ZZGH009 2013QN136, and 2014QN155), and Special Research Fund for the Doctoral Program of Higher Education (Grant No. 20130142120044), and EU FP7-PEOPLE-IRSES (Contract/Grant No. 247083, 318992 and 610524).

--- abstract: 'The $\gamma + p \rightarrow K^0 + \Sigma^+$ photoproduction reaction is investigated in the energy region from threshold to $E_\gamma = 2250$MeV. The differential cross section exhibits increasing forward-peaking with energy, but only up to the $K^*$ threshold. Beyond, it suddenly returns to a flat distribution with the forward cross section dropping by a factor of four. In the total cross section a pronounced structure is observed between the $K^*\Lambda$ and $K^*\Sigma$ thresholds. It is speculated whether this signals the turnover of the reaction mechanism from $t$-channel exchange below the $K^*$ production threshold to an $s$-channel mechanism associated with the formation of a dynamically generated $K^*$-hyperon intermediate state.' bibliography: - '&lt;your-bib-database&gt;.bib' title: 'Anomaly in the $K^0_S\,\Sigma^+$ photoproduction cross section off the proton at the $K^*$ threshold' --- meson photoproduction , strangeness ,cross section ,meson-baryon interaction ,dynamically generated resonance Introduction {#sec:Introduction} ============ The CBELSA/TAPS experiment at the Electron Stretcher Accelerator ELSA [@Hillert06] of Bonn University is investigating the structure of the nucleon at low energies. The excitation spectrum provides a fingerprint of the intra-baryonic interaction dynamics in the non-perturbative regime of QCD. Lattice calculations made impressive progress over recent years and provide ab initio calculations with almost realistic quark masses of the spectrum of ground state baryon masses [@Duerr08]. However, understanding excited states is still a challenge using lattice QCD, and so quark models are often used to describe the nucleon excitation spectrum. In a simple though extreme view the baryon structure is determined by a strong interaction potential which is mutually generated by the constituent quarks. While corresponding models [@HK83; @CR00; @LMP01] are successful in reproducing crucial parameters of low mass states, e.g. magnetic moments and electromagnetic couplings, important aspects of the excitation spectrum still remain dubious: (i) An essential fraction of higher mass states expected within the $SU(6) \times O(3)$ quark models has not yet been observed. It is an open issue whether this reflects deficits of the models or of incomplete experiments. (ii) Even basic features of low-lying states are difficult to understand in genuine 3-quark models, e.g. the parity ordering of the lowest lying nucleon excitations $N(1440)P_{11}$ (positive parity) and $N(1535)S_{11}$ (negative parity), which naturally would be reversed in any three dimensional static potential. This is a point of controversy also in lattice QCD calculations [@Mathur05; @Lin11]. Further prominent problems are the unusually large decay into the $\eta$ channel of the $N(1535)S_{11}$ compared to both, the nearby angular momentum partner $N(1520)D_{13}$ and the $N(1650)S_{11}$ with even the same quantum numbers, or the large 115MeV mass gap within the angular-momentum doublet $\Lambda(1405)S_{01}$ and $\Lambda(1520)D_{03}$. These facts were speculated to signal other than 3-quark dynamics to affect the observed resonance structure. Associated with the spontaneous breaking of chiral symmetry, some models indeed assign an essential role for baryon dynamics to meson fields [@GR96; @MG84] and meson-baryon interactions [@Dalitz; @SW88; @KWW97; @Inoue2001; @Hyodo2002; @G-RLN04; @LK04; @MRR04; @Borasoy07]. In particular in the vicinity of thresholds, the formation of (unstable) hadronic molecules in the sense of states which are dynamically generated by baryon-meson interaction is expected. They should show up, at least, as strong baryon-meson Fock components. For some low-lying states meson cloud effects seem to be indicated by meson electroproduction experiments at small momentum transfers [@Aznauryan09], but a direct experimental observation of molecular components is still missing. Meson photoproduction provides a sensitive tool to investigate these issues. In the experiment presented here the reaction $\gamma \, p \,\rightarrow\, K^0 \, \Sigma^+$ was studied from threshold ($E_\gamma = 1047.6$MeV) to a photon energy of $E_\gamma = 2250$MeV, i.e. across the $K^*$ production threshold at $E_\gamma = 1678.2$MeV for the $K^{*+}\Lambda$ final state, and at $E_\gamma = 1848.1$MeV for $K^{*0}\Sigma^+$. Compared to charged $K^+$ photoproduction, which has been extensively studied during recent years [@Bradford07], the neutral $K^0$ channel has tended to be sidelined. This appears entirely unjustified, though. To study $s$-channel resonance excitations (Figure\[fig:diagrams\](a)), the photoproduction of neutral kaons offers some advantages over charged ones, because the photons cannot directly couple to the (vanishing) charge of the meson. Hence, the $t$-channel diagram (c) in Figure \[fig:diagrams\] does not contribute to the production process. Since this may become dominant in charged kaon photoproduction, the neutral channel provides a cleaner probe for $s$-channel excitations. However, $t$-channel processes are not entirely suppressed in $K^0$ photoproduction. The photon coupling to the $K^0$-$K^{0*}$ vertex remains non-zero and renders a $t$-channel exchange of a $K^*$ meson possible as is visualised in Figure\[fig:diagrams\](e). This opens the opportunity to get a hand on explicit meson-baryon dynamics: Should $K^*$-hyperon dynamics play a significant role, then $K^*$ production via diagrams of the type Figure\[fig:diagrams\](f) may yield $K^0$ photoproduction markedly different above and below the $K^{0*}$ threshold, unmasked by the strong charge-dominated t-exchange in $K^+$ production. These considerations provided the motivation for our study of the $\gamma \, p \, \rightarrow K^0 \, \Sigma^+$ photoproduction reaction presented here. Previous data of Crystal Barrel [@Castelijns08] and SAPHIR [@Lawall05] agree rather well in general, however differences show up just in the energy region of the $K^*$ threshold, prohibiting any clear conclusions on $K^*$-hyperon dynamics. The goal of the present experiment was to improve this unsatisfactory situation. ![Diagrams contributing to charged and uncharged kaon photoproduction. The Born diagrams are shown in (a) - (d). Non-strange resonances may contribute as intermediate states in the $s$-channel (a) and $u$-channel (b). The $t$-channel meson exchange (c) and the seagull term (d) are proportional to the meson charge, hence contribute only to the charged kaon channel. Vector meson exchange (e) is allowed also for $K^0$ photoproduction. Diagram (f) visualizes subthreshold $K^*$ production with subsequent coupling into the kaon channel through $\pi^0$ rescattering, as explained in the text. []{data-label="fig:diagrams"}](figures/AllDiagrams.png){width="90.00000%"} Experiment {#sec:Experiment} ========== The experiment was performed with the combined Crystal Barrel [@Aker92] and TAPS [@Novotny91; @Gabler94] detector system at the tagged photon beam of ELSA using an electron beam of $E_0 = 3.2$ GeV. Bremsstrahlung was produced from a $500\,\mu$m thick diamond crystal. Coherent bremsstrahlung peaks which carry linear polarisation were subsequently set at 1305, 1515 and 1814 MeV. For the present analysis the azimuthal distributions were summed over, however, such that the effective photon linear polarisation was zero. The bremsstrahlung electrons were momentum analysed in a magnetic dipole (tagging-) spectrometer. The spectrometer’s electron detector covered a photon energy range of $E_\gamma = 0.18$–$0.92 E_0$. 14 slightly overlapping scintillator bars provided the tagger timing, whereas the energy was determined by a 480 channel scintillating fibre detector at low photon energies (i.e. high rates), complemented by a MWPC at high energies (low rates). An energy resolution between 10 and 25MeV was achieved, depending on the energy of the electron incident on the tagging detector. The tagging system was run at electron rates up to $10^7$Hz. The absolute photon flux was determined from the tagged electron spectrum in combination with a fast total absorbing PbWO$_4$ detector to measure the energy dependent tagging efficiency. The photon beam hit a $5.3$ cm long liquid hydrogen target with 80$\mu$m Kapton windows. A three layer scintillating fibre detector (ISFD) [@Suft05] surrounded the target within the polar angular range from 15 to 165 degrees. It determined a point-coordinate for charged particles. Both, charged particles and photons, were detected in the [Crystal Barrel]{} detector. The 1290 CsI(Tl) crystals of 16 radiation lengths were arranged cylindrically around the target in 23 rings, covering a polar angular range of 30 – 168 degrees, and read out by photo diodes attached to a wavelength shifter. For photons an energy resolution of $\sigma_{E}/E = 2.5\,\%/ \sqrt[4]{E/\textrm{GeV}}$ and an angular resolution of $\sigma_\textrm{angle} \simeq 1.1$degree was obtained. The $5.8$ – 30 degree forward cone was covered by the [TAPS]{} detector, set up in one hexagonally shaped wall of 528 BaF$_2$ modules at a distance of $118.7$cm from the target. For photons between 45 and 790 MeV an energy resolution of $\sigma_{E}/E = \left(0.59/\sqrt{E/\textrm{GeV}}+1.9\right)\%$ was achieved. The position of photon incidence could be resolved within 20mm. The [TAPS]{} detectors were individually equipped with photomultiplier readouts. Each [TAPS]{} module had a 5mm plastic scintillator in front of it to measure the energy loss signal of charged particles. The first level trigger was derived from [TAPS]{}. For this purpose the detector was subdivided into 8 sectors of crystals, each provided with two discriminator thresholds. A TAPS-high trigger corresponds to at least one group above the high threshold, whereas TAPS-low requires at least two sectors above the low threshold. The thresholds were chosen as low as possible while keeping the overall TAPS trigger rate at a tolerable level of about 100 kHz. A cluster finder algorithm (FAst Cluster Encoder – FACE) for the [Crystal Barrel]{} provided a second level trigger. Here a minimum of either one or two clusters could be demanded. The detector system was read out upon one of the two global trigger conditions: (1) TAPS-low without FACE requirement or (2) TAPS-high along with FACE. In both cases, the coincidence with the tagging spectrometer was additionally required. With the TAPS detector a 1-sigma width of 390ps was achieved in the time calibration for coincident photon hits in different detector modules and of 690ps relative to the photon tagger. Starting with the energy loss induced by cosmics to set the individual high voltages, the energy calibration for both the Crystal Barrel and TAPS detectors was performed iteratively by an offline gain adjustment to minimise the width of the $\pi^0$ peak in the 2-photon invariant mass distribution. In addition, every few hours the Crystal Barrel gains were monitored by means of a light-pulser system with calibrated attenuators. The quality of the resulting calorimeter calibrations were then cross-checked through the width of the $\eta$ peak in the 6-photon invariant mass spectrum. A 1-sigma width of 39MeV was obtained. The energy calibration of the tagging system was performed by deflecting low intensity electron beams of known energy directly into the tagging detector plane. An absolute accuracy of better than 10MeV was achieved. Event selection and data analysis {#sec:Analysis} ================================= The Crystal-Barrel/TAPS detector setup is optimised for multi-photon final states. Therefore, the $K^0 \Sigma^+$ reaction was studied in the neutral decay modes $K^0 \rightarrow \pi^0 \pi^0$ (B.R. $31.4\,\%$) and $\Sigma^+ \rightarrow p \pi^0$ (B.R. $51.6\,\%$), which in total yields 6 photons along with the proton. Hence, event topologies with 7 cluster hits in the calorimeters were selected. Charged particles were recognised through their signals in the ISFD or the TAPS $\Delta E$ plastic scintillators. Two charged particle events were discarded. For one charged particle in the final state the proton assignment was unique. 7 neutral cluster hits were also accepted. In this case, all combinatorial possibilities for the proton assignment were processed. The resulting combinatorial background was largely reduced through kinematic requirements in further analysis. Since the proton detection efficiency was limited, in particular at low proton momentum, 6 neutral cluster hits were also accepted. The tagged photon energy range extended below the $K^0\Sigma^+$ threshold. However, in the analysis a photon energy $E_\gamma > 1047.5$MeV was required to reduce random background. Furthermore, it was requested that 3 pairs of neutral (so-called $\gamma$) hits can be found with invariant masses in the range of the $\pi^0$, namely $110\,\textrm{MeV} \leq M_{\gamma\gamma} \leq 160\,\textrm{MeV}$. The $\eta p \rightarrow p\,6\gamma$ final state provided the most significant background. It was practically eliminated by requiring the invariant mass of the three identified $\pi^0$’s to [*not*]{} lie within the range $470\,\textrm{MeV} \leq M_{\pi^0\pi^0} \leq 620\,\textrm{MeV}$. This corresponds to a 4-sigma wide anti-cut around the $\eta$ mass. Finally, neutral background events from the electron beamdump (which was located below the floor in front of the Crystal Barrel calorimeter) were rejected through their special angular topology. Based on this preselection, the remaining events were subjected to a kinematic fit to the $\gamma \, p \rightarrow p\,3\pi^0$ reaction. The photon energy was defined by the tagging spectrometer. Energy and angle parameters for the fit were provided by the final state photon hits. The proton candidate did not enter the fit. In contrast, its momentum vector was determined through the kinematic conditions. With the four components of the proton momentum to be determined and seven conditions (overall energy-momentum and three $\pi^0$ masses) the fit still was threefold overdetermined. $\Sigma^+$ and $K^0$ masses were not used as conditions of the kinematic fit. After the kinematic fit an acceptable signal to background ratio was achieved. This is demonstrated in Figure\[fig:SigmaVsKaon\] where the $2\pi^0$ invariant mass is plotted against the $p\pi^0$ invariant mass distribution. A culmination of events is clearly visible around $M_{\pi^0\pi^0} = 490$ and $M_{p\pi^0} = 1190$MeV, corresponding to the masses of $K^0$ and $\Sigma^+$. ![ $\pi^0\pi^0$ against $p\pi^0$ invariant mass distribution after event preselection and kinematic cut, showing a concentration of events in the $K^0\Sigma^+$ final state. []{data-label="fig:SigmaVsKaon"}](figures/Fig-6-28-sigmavskaon-sw4.png){width="80.00000%"} As an example, Figure\[fig:pi0pi0InvMass\] shows the $\pi^0\pi^0$ invariant mass distribution for the bin 1350–1450MeV in photon energy and 0 – $0.33$ in $\cos\Theta^K_\textrm{cms}$, the kaon center-of-mass angle, after a cut on the $\Sigma^+$ mass region in the $p\pi^0$ mass distribution: $1170\,\textrm{MeV} \leq M_{p\pi^0} \leq 1210\,\textrm{MeV}$. The cut limits were obtained by minimising the related systematic error induced by background subtraction. In Monte Carlo simulations it turns out that the far dominating background is associated with uncorrelated photoproduction of three neutral pions. The simulated background distribution is shown as the hatched area in Figure\[fig:pi0pi0InvMass\]. The spectra agree very well with the experimental distributions in all bins. The simulated yield is scaled outside the area of the $K^0$ signal peak. ![ $\pi^0\pi^0$ invariant mass distribution for the bin 1350 – 1450MeV in photon energy and 0 – $0.33$ in $\cos\Theta^K_\textrm{cms}$ after a cut on the $\Sigma^+$ mass in Figure \[fig:SigmaVsKaon\]: $1170\,\textrm{MeV} \leq M_{p\pi^0} \leq 1210\,\textrm{MeV}$. The grey area represents the simulated background from uncorrelated 3$\pi^0$ photoproduction, scaled to the experimental yield outside the signal area (cf. text). []{data-label="fig:pi0pi0InvMass"}](figures/pinull_massen-1.png){width="60.00000%"} The photon energy range was divided into 12 bins of $\pm 50$MeV width, ranging from 1100 to 2200MeV. Monte Carlo simulations determined the experimental acceptance individually for each of the three selected event topologies. An important benefit of the almost $4\pi$ detection system is the practically flat acceptance. ![ Simulated acceptance for the $K^0\Sigma^+$ final state. The three event topologies are treated separately: 6 uncharged and 1 charged hits (grey), 7 uncharged hits (white), and 6 uncharged hits (light grey). The upper histogram (black) represents the total acceptance. []{data-label="fig:Accept_Winkel"}](figures/akzeptanz.pdf){width="70.00000%"} This is visualized in Figure\[fig:Accept\_Winkel\]. Results and Discussion {#sec:Results} ====================== Based on the absolute normalisation of the photon flux provided by the tagging system, differential cross sections were determined separately for the 6 and 7 hit topologies. Both agree very well and were then combined into the full squares which are shown in Figure\[fig:DiffWQ\_comparison\]. Associated with the data points are the total statistical errors. The combined systematic error is indicated by the grey bars on the abscissa. It has contributions from the photon flux ($\simeq 5\,\%$), the cuts applied in the analysis ($\simeq 5 - 6\,\%$), the kinematic fit ($\simeq 5\,\%$) and the simulated acceptance ($\simeq 5.6\,\%$). The error of the kinematic fit is induced by a cut on the confidence level, which was required to exceed $0.1$. All errors associated with cuts were estimated through the variation of the cuts over a wide range. Hence, they may be considered as upper error limits. The most probable errors would be significantly smaller. Also, the error in the photon flux affects the extracted absolute cross sections, but, within a given energy bin, it leaves the form of the angular distribution unchanged. In comparison to our new data, Figure\[fig:DiffWQ\_comparison\] also shows the results of previous measurements of Crystal Barrel [@Castelijns08] and SAPHIR [@Lawall05]. ![Measured differential cross sections for $K^0\Sigma^+$ photoproduction as a function of the kaon center-of-mass angle in $\pm 50$MeV wide bins of photon energy from 1100 to 2200 MeV. The present results (full squares) are compared to previous measurements of Crystal Barrel (open squares) [@Castelijns08] and SAPHIR (triangles) [@Lawall05]. The error bars are purely statistical. An estimate of the systematic uncertainty is given by the bars on the abscissa (cf. text). []{data-label="fig:DiffWQ_comparison"}](figures/diff_Wq.pdf){width="95.00000%"} While the data sets agree relatively well in general, there remain significant discrepancies in forward directions and, important for the present investigation, in the energy range of the $K^*$ threshold ($E_\gamma = 1750$ – 1850MeV bin). These discrepancies could be resolved by our new data. Below the $K^*$ threshold mostly the SAPHIR data is favored. However, at the $K^*$ threshold and higher energies the previous Crystal Barrel data is clearly supported. The differential cross sections of a CLAS measurement presented at conferences (not shown in Figure\[fig:DiffWQ\_comparison\]) [@Carnahan03], which detected the charged $K^0 \rightarrow \pi^+\pi^-$ decay, agrees well with our new data within the common detector acceptance. As can be seen from Figure\[fig:DiffWQ\_comparison\], directly above the $K^0\Sigma^+$ threshold a differential cross section of $\simeq 0.02\,\mu$barn/sr is obtained with flat angular dependence, typical for s-wave production. The cross section rises with increasing photon energy and also develops an increasing forward peaking, suggesting increasing $t$-channel contributions. This forward peaking is most pronounced in the $E_\gamma = 1700 \pm 50$MeV bin. In sharp contrast, the next energy bin exhibits an entirely flat distribution again and a drop of the cross section back to $\simeq 0.02\,\mu$barn/sr. This is at $E_\gamma = 1800$MeV, i.e. right between the thresholds of $K^*\Lambda$ and $K^*\Sigma$ photoproduction. The differential cross section then remains almost flat and practically constant up to the highest measured energies. This suggests that in the $K^*$ threshold region there is a sudden cross over from a $t$-channel mechanism back to $s$-channel production of $K^0\Sigma^+$ with increasing photon energy. As is shown in Figure\[fig:total\_x-sec\], this effect is strong enough to become clearly visible in the total cross section which, given the full $4\pi$ acceptance, is simply obtained by integration of the differential cross sections. Due to the forward peaking of the cross section below the $K^*$ thresholds, the effect is most pronounced in the most forward angular bin, however. Here, the cross section drops by a factor of four in the vicinity of the $K^*$ thresholds, cf. Figure\[fig:forward\_x-sec\]. It remains to be investigated whether a cusp-like structure develops, i.e. a discontinuity in the slope of the cross section. ![Total cross section for $K^0\Sigma^+$ photoproduction as a function of the centre-of-mass energy from the present experiment (full squares) in comparison to the previous Crystal Barrel (open squares) [@Castelijns08] and SAPHIR (triangles) [@Lawall05] data. The vertical lines indicate the $K^*\Lambda$ and $K^*\Sigma^+$ thresholds at $W = 2007.4$ and $2085.5$MeV, respectively. The SAID parameterisation [@SAID] is represented by the dashed-dotted curve. A K-MAID calculation with standard parameters yields the dashed curve. The full curve is obtained with the modifications described in the text. Above the $K^*$ threshold the grey circles represent the sum of the $K^0\Sigma^+$ cross section of the present experiment and the $K^{0*}\Sigma^+$ cross section of the work of Nanova et al. [@Nanova08]. The vertical bars on the abscissa again indicate the systematic error of the present experiment, the errors plotted with the data symbols are purely statistical. []{data-label="fig:total_x-sec"}](figures/tot_wq_K0Sigma+.pdf){width="80.00000%"} ![Cross section for $K^0\Sigma^+$ photoproduction as a function of the centre-of-mass energy from the present (full squares) and a previous (open squares) [@Castelijns08] Crystal Barrel experiment in the most forward angular bin of Figure\[fig:DiffWQ\_comparison\]. Plotted errors and curves represent the same as in Figure\[fig:total\_x-sec\], the vertical lines as well. []{data-label="fig:forward_x-sec"}](figures/vorwaertsbin-diff_wq-K0Sigma+W.pdf){width="80.00000%"} The structure seems related to a sudden change in the reaction mechanism. Below the $K^*$ threshold, the angular distributions suggest that t-exchange according to Figure\[fig:diagrams\](e) plays a major role, which is associated with $K^{0*}$ exchange. This hypothesis was tested with the K-MAID parameterisation [@K-MAID_webseite], where it is possible to manually change the $K^*$ exchange strength. With standard parameters both, K-MAID and SAID [@SAID], deliver an unsatisfactory description of the data, cf. the dashed and dashed-dotted curves in Figure\[fig:total\_x-sec\], respectively. Below the $K^*$ threshold this can be drastically improved by adjusting the couplings of the $S_{31}(1900)$ state to $G_1 = 0.3$ and $G_2 = 0.3$ [@K-MAID_webseite] and reduction of the Born-couplings from 1 to $0.7$. To check whether the sudden drop in the total cross section can be reproduced by changing the $K^{0*}$ exchange, this contribution was retained below the $K^*\Sigma^+$ threshold but manually set to zero above. As is demonstrated by the full curve in Figure\[fig:total\_x-sec\], with these modifications K-MAID yields a significantly improved description of the cross section, including the structure at the $K^*$ thresholds. In forward directions (cf. Figure\[fig:forward\_x-sec\]) the modified K-MAID still yields unsatisfactory results at smaller energies. The drop of the cross section at the $K^*$ thresholds is however rather well reproduced by switching off the $K^{0*}$ exchange contribution. In the vicinity of the threshold, a $K^*$ would be produced almost on mass shell. It then strongly couples to a $K^0$ and a pion. We speculate that, in this way, close-to-threshold $K^*$ production feeds the $K^0$ channel: The $K^0$ is ejected and observed in the final state while the pion is reabsorbed by the hyperon. Such a process resembles diagram (f) of Figure\[fig:diagrams\]. A vector-meson - hyperon interaction in the intermediate state is predominantly expected in a relative $s$-wave, which in turn would lead to the observed flat $K^0$ angular distribution beyond the $K^*$ thresholds. If, above the $K^*$ threshold, the vector meson is produced as a free particle, then diagram\[fig:diagrams\](f) no longer contributes to the $K^0\Sigma^+$ channel. The strength, which at the dip of the cross section is vanishing from the $K^0$ channel, is then expected to contribute to $K^{0*}\Sigma^+$. In order to test this idea, the measured total cross section of the reaction $\gamma p \rightarrow K^{0*}\Sigma^+$ [@Nanova08] was added to the observed $K^0\Sigma^+$ cross section above the $K^*$ threshold. The result is shown in Figure\[fig:total\_x-sec\] as open circles. Using the sum of the two cross sections, a smooth transition is obtained from below to above the $K^*$ thresholds and the dip structure vanishes. This is taken as further indication of a production mechanism as outlined above. This discussion suggests that the situation depicted in Figure\[fig:diagrams\](f) can be seen as the coupling of the initial photon to a dynamically generated $(K^*\Sigma)^+$ or $(K^*\Lambda)^+$ state in the vicinity of the $K^*$ threshold. The dip in the cross section occurs above the $K^*\Lambda$ but [*below*]{} the $K^*\Sigma$ threshold. This fact may indicate that indeed an intermediate state is formed with a strong $K^*\Sigma$ component. Such states are expected in chiral unitary approaches through the interaction of the nonet of vector mesons with the octet of baryons [@Oset-Ramos10]. If the vector meson and baryon couple in a relative s-wave, doublets of $J^P = (1/2)^-$ and $(3/2)^-$ are expected. In ref.[@Oset-Ramos10] a non-strange isospin $1/2$ doublet is indeed predicted at a mass of 1972MeV, i.e. close to the $K^*$ threshold. It remains to be seen whether, in a partial wave analysis (PWA), the reported structures can be reproduced by $(1/2)^-$ and $(3/2)^-$ partial waves. To make the PWA as unambiguous as possible, we are presently analysing polarisation observables in addition to the cross sections. In contrast to a $t$-channel dominated production mechanism, an $s$-channel intermediate state will provide a genuine spin filter. It is hence expected that, in addition to recoil polarisation and photon asymmetry, in particular the beam-target as well as the beam-recoil asymmetries will shed further light on the mechanism of $K^0$ photoproduction in the vicinity of the observed dip structure. The reported structure in the cross section is also close to the $\eta' p$ threshold. In Ref. [@OR11] a significant coupling of vectormeson-baryon to pseudoscalar-baryon channels with the same quantum numbers is expected. Consequently, one may speculate that the possible $K^*$-hyperon states may affect the $\eta' p$ cross sections at threshold as well, and thus help to solve the puzzle of $\eta' N$ interactions in both, hadronic and photoinduced reactions [@OR11]. Summary and outlook =================== Using the Crystal-Barrel/TAPS detector setup at the electron accelerator facility ELSA of Bonn University, the reaction $\gamma + p \rightarrow K^0 + \Sigma^+$ was investigated from threshold to $E_\gamma = 2250$MeV. We find an unexpected structure in the differential cross section between the $K^{0*}\Lambda$ and $K^{0*}\Sigma$ thresholds: The angular distribution exhibits a sudden transition from forward peaked to flat with increasing photon energy. In forward directions the cross section drops by a factor of four, generating a pronounced structure even in the total cross section. It is speculated that the effect may be due to close-to-threshold $K^{0*}$ production via the formation of a $K^*$-hyperon quasibound state, as expected in Reference[@Oset-Ramos10]. Above the $K^*$ threshold a real $K^*$ may be produced and the associated strength then vanishes from the $K^0$ channel. This is supported by the total cross section of $K^{0*}\Sigma^+$ photoproduction, which corresponds in size to the reduction of the $K^0\Sigma^+$ cross section in the dip region. To shed more light on the threshold structure it will be mandatory to exploit polarisation observables. In particular, the photon beam asymmetry should be sensitive to the parity character of the $t$-channel contributions, while recoil polarisation and beam-target asymmetry will strongly constrain the quantum numbers of an intermediate $s$-channel resonance. In the meanwhile polarisation observables have been measured by the collaboration and are presently under analysis. In addition, a partial wave analysis is under way where it will be interesting to see whether indications for $(1/2)^-$ and $(3/2)^-$ partial waves can be identified as they would be expected for a $K^*$-$\Lambda/\Sigma$ quasibound state. Acknowledgements {#acknowledgements .unnumbered} ================ Helpful discussions with M. Lutz, E. 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--- abstract: 'Our proximity and external vantage point make M31 an ideal testbed for understanding the structure of spiral galaxies. The Andromeda Optical and Infrared Disk Survey (ANDROIDS) has mapped M31’s bulge and disk out to R=40 kpc in $ugriJK_s$ bands with CFHT using a careful sky calibration. We use Bayesian modelling of the optical-infrared spectral energy distribution (SED) to estimate profiles of M31’s stellar populations and mass along the major axis. This analysis provides evidence for inside-out disk formation and a declining metallicity gradient. M31’s $i$-band mass-to-light ratio ($M/L_i^*$) decreases from 0.5 dex in the bulge to $\sim 0.2$ dex at 40 kpc. The best-constrained stellar population models use the full $ugriJK_s$ SED but are also consistent with optical-only fits. Therefore, while NIR data can be successfully modelled with modern stellar population synthesis, NIR data do not provide additional constraints in this application. Fits to the $gi$-SED alone yield $M/L_i^*$ that are systematically lower than the full SED fit by 0.1 dex. This is still smaller than the 0.3 dex scatter amongst different relations for $M/L_i$ via $g-i$ colour found in the literature. We advocate a stellar mass of $M_*(30~\mathrm{kpc}) =10.3^{+2.3}_{-1.7}\times 10^{10}~\mathrm{M}_\odot$ for the M31 bulge and disk.' author: - 'Jonathan Sick,$^1$ Stephane Courteau,$^1$ Jean-Charles Cuillandre,$^2$ Julianne Dalcanton,$^3$ Roelof de Jong,$^4$ Michael McDonald,$^5$ Dana Simard,$^1$' - 'R. Brent Tully$^6$' title: 'The Stellar Mass of M31 as inferred by the Andromeda Optical & Infrared Disk Survey' --- Introduction ============ The ANDROIDS programme has used the MegaCam and WIRCam cameras on the Canada-France-Hawaii Telescope (CFHT) to map M31’s bulge and disk homogeneously within $R=40$ kpc with $ugriJK_s$ bands and enable global studies of M31’s structure and stellar populations using both resolved stars and integrated spectral energy distributions (SEDs). In this contribution, we use ANDROIDS to estimate the stellar mass profile of the M31 disk with Bayesian modelling of the optical to near-IR (NIR) SED. This approach is more rigorous than the colour-$M/L^*$ prescriptions [e.g. @Zibetti:2009; @Taylor:2011; @Into:2013] often employed by pixel-by-pixel stellar mass estimation studies that use only a $g-i$ colour and marginalize over all likely star formation histories. By studying M31 in detail, an overall goal of ANDROIDS is to explore systematic uncertainties in studies of more distant and poorly resolved systems. M31 Surface Brightness Calibration ================================== Background subtraction is the most significant challenge for observational studies of M31’s structure since we cannot observe its disk and blank sky simultaneously. This is particularly acute in our NIR maps where skyglow is 3-dex brighter than the disk, while also having strong spatial and temporal variations. In [@Sick:2014], we describe our ANDROIDS/WIRCam sky-target nodding and background subtraction schemes and find that the NIR background cannot be known to better than 2% given the scale of sky-target nods required for M31. We overcome this uncertainty by solving for sky offsets that formally minimize surface brightness differences between overlapping pairs of images. Such sky offsets are $\sim1$% of the NIR brightness, but systematically uncertain up to a zeropoint normalization that is 0.16% of the sky level. In optical bands, the sky background is both more stable and somewhat dimmer, though we still employ sky-target nodding with the Elixir-LSB method for CFHT/MegaCam to build a real-time map of sky and scattered light backgrounds over one-hour sliding windows. With Elixir-LSB we easily identify low surface brightness features in M31’s outer disk, such as the Northern Spur, at levels below $\mu_g\sim26$ mag arcsec$^{-2}$ [@Sick:2013a]. The aforementioned sky offset zeropoint uncertainty requires that our surface brightness profiles be finely calibrated against external datasets. Resolved stellar catalogs transformed into surface brightness maps, such as our own WIRCam star catalog, and even Panchromatic Hubble Andromeda Treasury [PHAT; @Dalcanton:2012], provide a useful dataset up to the limit of completeness corrections. Extremely wide-field imaging is also useful as it enables a simultaneous mapping of the background and the disk light. We are currently using Dragonfly [@Abraham:2014] to image M31 as a replacement for the venerable wide-field plates of [@Walterbos:1987]. SED Stellar Mass Modelling ========================== ![Posterior stellar population profiles for different bandpass sets: $ugriJK_s$ (blue), $ugri$ (green), $gi$ (red). A declining metallicity gradient and inside-out disk formation (seen by an increase in the e-folding time, $\log \tau$, of the exponentially declining star formation history model) are clearly evident.[]{data-label="fig:pop_profile"}](figure1){width="0.7\columnwidth"} ![Posterior stellar $M/L_i^*$ (left) and stellar mass (right) major axis profiles. The $ugriJK_s$ (blue) and $ugri$ (green) fits are consistent, while fit of only $gi$ (red) are lower by 0.1 dex in $M/L_i^*$. Equivalent $gi$–$M/L_i^*$ relations in the literature can vary by 0.3 dex of $M/L_i^*$.[]{data-label="fig:mass_profile"}](figure2){width="0.7\columnwidth"} From the calibrated surface brightness profiles, we model the SED at each radial bin to estimate the stellar population, and hence the stellar mass-to-light ratio, $M/L_i^*$. Our modelling engine is the Flexible Stellar Population Synthesis (FSPS) software [@Conroy:2009; @Conroy:2010]. We chose FSPS for its reliable calibration and “lighter” AGB contribution than older SP models [e.g., @Bruzual:2003], and allowance for deep customization of the computed stellar populations.[^1] We use a Markov Chain Monte Carlo approach to modelling SEDs extracted along the northern major axis of the M31 disk implemented with the `emcee` python package [@Foreman-Mackey:2013]. We tested different star formation history parameterizations and found that a simple ‘$\tau$’ model, involving constant plus exponentially declining star formation rate components minimized residuals compared to more sophisticated ‘delayed $\tau$’ and late burst models. Of the dust attenuation treatments, the default power-law attenuation law with separate components for young and older stellar populations also minimized residuals compared to Milky Way or starburst attenuation models. We found that posterior SED residuals are minimized by fitting the entire $ugriJK_s$ SED. This contradicts [@Taylor:2011] and [@Zibetti:2009] who advocated against using NIR bands in mass estimation due to uncertain AGB treatments of the previous generation of stellar population synthesis models [e.g. @Bruzual:2003; @Maraston:2005]. Much like the NIR, the $griJK_s$-SED fit has little predictive power over the crucial $u$-band. This result should therefore encourage SED modellers to incorporate as many bandpasses as possible, including UV and IR, to obtain the best constraints on stellar populations and masses. We modelled SEDs extracted from a logarithmically-sized wedge [e.g. @Courteau:2011 their Fig. 2] to produce stellar population profiles (shown in Fig. \[fig:pop\_profile\]). Interestingly, the $ugri$-fit and $ugriJK_s$-fit SEDs produce statistically identical stellar population profiles, with the only exception being a slightly tighter posterior credible region from the full-SED fits. Although the consistency of optical and optical-NIR SED fits is reassuring from the perspective of NIR calibrations, it is also disappointing that the NIR data has not produced a remarkably improved posterior stellar population estimate. Clearly evident is that poorly sampled SEDs can bias results. Fitting only the $gi$ SED (that is, using an input information equivalent to those using colour-$M/L^*$ look-up-tables) clearly biases the posterior stellar population distribution, with significantly lower dust opacities and lower mass-to-light ratios. By comparison, we have also plotted mass-to-light ratios predicted by three colour-$M/L^*$ relations [@Zibetti:2009; @Taylor:2011; @Into:2013]. These fits systematically vary by 0.3 dex, far larger than the 0.1 dex of internal systematic uncertainty typically claimed by $g-i$ – $M/L^*$ fits [@Courteau:2013]. Compared to our full SED fits, modelling of the $gi$ SED is less biased than these other $M/L^*$ fits, which are based on other stellar population synthesis models. This serves as reminder that stellar mass estimates remain dominated by prior assumptions such as choices of IMF, dust, and details of AGB treatments, among other concerns. Discussion ========== We have used $ugriJK_s$ SEDs to map the stellar mass of M31’s disk and find a stellar mass, within $30$ kpc, of $M_{ugri}^{*} = 10.3^{+2.3}_{-1.7}\times 10^{10}~\mathrm{M}_\odot$. This result is consistent with the stellar bulge and disk masses quoted by [@Tamm:2012] ($10.1\times10^{10}~\mathrm{M}_\odot$). Future work will extend this analysis to a full 2D mapping of the M31 stellar mass distribution. We are matching these stellar mass maps with dynamical tracers of gas and stars to construct a mass model of M31’s stellar, gas, and dark matter components (Simard et al., in progress). The DiskFit code [@Spekkens:2007] allows us to correct the H<span style="font-variant:small-caps;">i</span> velocity fields of [@Saglia:2010] and [@Chemin:2009] for non-circular motions in M31’s central parts. The success of this mass model will be determined by the stability of a dynamical N-body realization. Acknowledgements {#acknowledgements .unnumbered} ================ J.S. and S.C. acknowledge support through respective Graduate Scholarship and Discovery grants from the Natural Sciences and Engineering Research Council of Canada. We thank the Canadian Advanced Network for Astronomical Research (CANFAR) for enabling the computing facilities needed for this work. [19]{} natexlab\#1[\#1]{} , R. G., & [van Dokkum]{}, P. G. 2014, [PASP]{}, 126, 55 , G., & [Charlot]{}, S. 2003, [MNRAS]{}, 344, 1000 , L., [Carignan]{}, C., & [Foster]{}, T. 2009, [ApJ]{}, 705, 1395 , C., [Gunn]{}, J. E., & [White]{}, M. 2009, [ApJ]{}, 699, 486 , C., [White]{}, M., & [Gunn]{}, J. E. 2010, [ApJ]{}, 708, 58 , S., [Widrow]{}, L. M., [McDonald]{}, M., [et al.]{} 2011, [ApJ]{}, 739, 20 , S., [Cappellari]{}, M., [de Jong]{}, R. S., [et al.]{} 2014, Rev. Mod. Phys., 86, 47 , J. J., [Williams]{}, B. F., [Lang]{}, D., [et al.]{} 2012, [ApJS]{}, 200, 18 , D., [Hogg]{}, D. W., [Lang]{}, D., & [Goodman]{}, J. 2013, [PASP]{}, 125, 306 , T., & [Portinari]{}, L. 2013, [MNRAS]{}, 430, 2715 , C. 2005, [MNRAS]{}, 362, 799 , R. P., [Fabricius]{}, M., [Bender]{}, R., [et al.]{} 2010, [A&A]{}, 509, A61 , J., [Courteau]{}, S., & [Cuillandre]{}, J.-C. 2013, 1310.4832 , J., [Courteau]{}, S., [Cuillandre]{}, J.-C., [et al.]{} 2014, [AJ]{}, 147, 109 , K., & [Sellwood]{}, J. A. 2007, [ApJ]{}, 664, 204 , A., [Tempel]{}, E., [Tenjes]{}, P., [Tihhonova]{}, O., & [Tuvikene]{}, T. 2012, [A&A]{}, 546, A4 , E. N., [Hopkins]{}, A. M., [Baldry]{}, I. K., [et al.]{} 2011, [MNRAS]{}, 418, 1587 , R. A. M., & [Kennicutt]{}, Jr., R. C. 1987, [A&AS]{}, 69, 311 , S., [Charlot]{}, S., & [Rix]{}, H. 2009, [MNRAS]{}, 400, 1181 [^1]: The lead author (J.S.) contributes to the maintenance of a Python-language wrapper for FSPS: <http://dan.iel.fm/python-fsps>

--- abstract: 'We consider a one dimensional elliptic distributed optimal control problem with pointwise constraints on the derivative of the state. By exploiting the variational inequality satisfied by the derivative of the optimal state, we obtain higher regularity for the optimal state under appropriate assumptions on the data. We also solve the optimal control problem as a fourth order variational inequality by a $C^1$ finite element method, and present the error analysis together with numerical results.' address: - 'Susanne C. Brenner, Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA' - 'Li-yeng Sung, Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA' - 'Winnifried Wollner, Department of Mathematics, Technische Universität Darmstadt, 64293 Darmstadt, Germany' author: - 'S.C. Brenner' - 'L.-Y. Sung' - 'W. Wollner' title: A One Dimensional Elliptic Distributed Optimal Control Problem with Pointwise Derivative Constraints --- Ø [^1] Introduction {#sec:Introduction} ============ Let $I$ be the interval $(-1,1)$ and the function $J:L_2(I)\times L_2(I)\longrightarrow \R$ be defined by $$\label{eq:JDef} J(y,u)=\frac12 \big(\|y-y_d\|_\LT^2+\beta\|u\|_\LT^2\big),$$ where $y_d\in L_2(I)$ and $\beta$ is a positive constant. The optimal control problem is to $$\label{eq:OCP} \text{find}\quad(\bar y,\bar u)=\argmin_{(y,u)\in \bbK}J(y,u),$$ where $(y,u)\in H^1_0(I)\times L_2(I)$ belongs to $\bbK$ if and only if $$\begin{aligned} {3} \int_I y'z'dx&=\int_I (u+f)z\,dx&\qquad&\forall\,z\in H^1_0(I), \label{eq:PDEConstraint}\\ y'&\leq \psi &\qquad&\text{a.e. on $I$}.\label{eq:DerivativeConstraint}\end{aligned}$$ We assume that $$\label{eq:DataRegularity} f\in H^1(I), \; \psi\in H^2(I)$$ and $$\label{eq:psiConstraint} \int_I \psi\,dx>0.$$ \[rem:Literature\] The optimal control problem defined by – is a one dimensional analog of the optimal control problems considered in [@CB:1988:Gradient; @CF:1993:Gradient; @DGH:2009:Gradient; @OW:2011:Gradient; @Wollner:2012:Gradient]. It was solved by a $C^1$ finite element method in [@BSW:2020:OneD] under the assumptions that $$\label{eq:OldData} f\in H^{\frac12-\epsilon}(I)\quad\text{and}\quad \psi\in H^{\frac32-\epsilon}(I).$$ Since the constraint implies $y\in H^2(I)$ by elliptic regularity, we can reformulate the optimization problem – as follows: $$\label{eq:ROCP} \text{Find}\quad \bar y=\argmin_{y\in K}\frac12\big(\|y-y_d\|_\LT^2 +\beta\|y''+f\|_\LT^2\big),$$ where $$\label{eq:KDef} K=\{y\in H^2(I)\cap H^1_0(I):\,y'\leq \psi \;\text{on}\;I\}.$$ According to the standard theory [@ET:1999:Convex; @KS:1980:VarInequalities], the minimization problem defined by - has a unique solution characterized by the fourth order variational inequality $$\beta\int_I (\bar y''+f)(y''-\bar y'')dx+ \int_I (\bar y-y_d) (y-\bar y)dx\geq 0 \qquad\forall\,y\in K,$$ which can also be written as $$\label{eq:VI} a(\bar y,y-\bar y)\geq \int_I y_d(y-\bar y)dx-\beta\int_I f(y''-\bar y'')dx \qquad\forall\,y\in K,$$ where $$\label{eq:aDef} a(y,z)=\beta\int_I y'z'dx+\int_I yz\,dx.$$ \[rem:VI\] The reformulation of state constraint optimal control problems as fourth order variational inequalities was discussed in [@PS:1996:OC], and a nonconforming finite element based on this idea was introduced in [@LGY:2009:Control]. Other finite element methods can be found in [@GY:2011:State; @BSZ:2013:OptimalControl; @BDS:2014:PUMOC; @BOPPSS:2016:OC3D; @BGPS:2018:Morley; @BGS:2018:Nonconvex; @BSZ:2019:Neumann]. \[rem:Nontrivial\]Note that implies $$\int_I\psi\,dx\geq \int_I y'\,dx=0 \qquad \forall\,y\in K$$ and hence $\int_I \psi\,dx\geq0$ is a necessary condition for $K$ to be nonempty. It is also a sufficient condition because the function $y$ defined by $$y(x)=\int_{-1}^x (\psi(t)-\bar\psi)dt$$ belongs to $K$, where $\bar\psi$ is the mean of $\psi$ over $I$. Furthermore, $$0=\int_I \psi\,dx=\int_I(\psi-y')dx$$ together with implies $\psi=y'$ identically on $I$ and hence $K=\{\psi\}$ is a singleton. Therefore we impose the condition to ensure that the optimization problem defined by – is nontrivial. Our goal is to show that $\bar y\in H^3(I)$ under the assumptions in and consequently / can be solved by a $C^1$ finite element method with $O(h)$ convergence in the energy norm. Note that previously $\bar y\in H^{\frac52-\epsilon}$ was the best regularity result in the literature for Dirichlet elliptic distributed optimal control problems on smooth/convex domains with pointwise constraints on the gradient of the state. The rest of the paper is organized as follows. The $H^3$ regularity of $\bar y$ is obtained in Section \[sec:DerivativeVI\] through a variational inequality for $\bar y'$ that can be interpreted as a Neumann obstacle problem for the Laplace operator. The $C^1$ finite element method for / is analyzed in Section \[sec:Discrete\], followed by numerical results in Section \[sec:Numerics\]. We end with some remarks on the extension to higher dimensions in Section \[sec:Conclusions\]. A Variational Inequality for $\bar y'$ {#sec:DerivativeVI} ====================================== Observe that the set $\{y':\,y\in K\}$ is the subset $\cK$ of $H^1(I)$ given by $$\begin{aligned} \label{eq:cKDef} \cK&=\{v\in H^1(I):\,\int_I v\,dx=0\quad\text{and}\quad v\leq\psi \;\text{on}\;I\}, %=\{v\in H^1(I)/\R:\,v\leq\psi \;\text{on}\;I\},\end{aligned}$$ and the variational inequality is equivalent to $$\begin{aligned} \label{eq:NeumannVI} &\int_I (\Phi-f')(q-p)dx+\int_I p'(q'-p')dx\\ &\hspace{40pt}+[f(1)(q(1)-p(1))-f(-1)(q(-1)-p(-1))]\geq0 \qquad \forall\,q\in \cK,\notag\end{aligned}$$ where $p=\bar y'$, $q=y'$, and $\Phi\in H^1(I)$ is determined by $$\begin{aligned} \beta\Phi'&=y_d-\bar y \label{eq:PhiDef1} \intertext{and} \int_I\Phi\,dx&=0.\label{eq:PhiDef2}\end{aligned}$$ Moreover is the variational inequality that characterizes the solution of the following minimization problem: $$\label{eq:NeumannObstacle} \text{Find}\quad p=\argmin_{q\in \cK}\Big[\frac12 \int_I (q')^2dx +\int_I(\Phi-f')q\,dx +f(1) q(1)- f(-1) q(-1)\Big].$$ A Neumann Obstacle Problem {#subsec:Neumann} -------------------------- The minimization problem , which is a Neumann obstacle problem, can be written more conveniently as $$\label{eq:ObstacleProblem} p=\argmin_{q\in \cK}\Big[\frac12 b(q,q)+(\phi,q)+ \tau q(1) -\sigma q(-1)\Big],$$ where $\sigma=f(-1)$, $\tau=f(1)$, $$\label{eq:Shorthands} b(q,r)=\int_I q'r'\,dx,\quad (\phi,q)=\int_I \phi\, q\,dx \quad\text{and}\quad \phi=\Phi-f'.$$ Note that we have a compatibility condition $$\label{eq:Compatibility} \int_I \phi\,dx+\tau-\sigma=0$$ that follows from , and the Fundamental Theorem of Calculus for absolutely continuous functions. Since $b(\cdot,\cdot)$ is coercive on $H^1(I)/\R$, the obstacle problem defined by and has a unique solution $p$ characterized by the variational inequality $$\label{eq:VI1} b(p,q-p)+(\phi,q-p)+\tau(q(1)-p(1))-\sigma(q(-1)-p(-1))\geq 0 \qquad\forall\,q\in \cK.$$ \[thm:pRegularity\] The solution $p=\bar y'\in \cK$ of / belongs to $H^2(I)$. We begin by observing that $$\label{eq:VI2} b(p,q-p)+(\phi,q-p)+\tau(q(1)-p(1))-\sigma(q(-1)-p(-1))\geq 0 \qquad\forall\,q\in\tilde K,$$ where $$\label{eq:tKDef} \tilde K=\{q\in H^1(I):\,q\leq\psi\;\;\text{in}\;I\;\; \text{and}\;\int_I q\,dx\geq0\}.$$ Indeed, $q\in \tilde K$ implies $q-\bar q\in K$, where $\bar q$ is the mean of $q$ over $I$, and hence, in view of , the definition of $b(\cdot,\cdot)$ in and , $$\begin{aligned} &b(p,q-p)+(\phi,q-p)+\tau(q(1)-p(1))-\sigma((q(-1)-p(-1))\\ &\hspace{10pt}=b(p,q-\bar q-p)+(\phi,q-\bar q-p)+ \tau(q(1)-\bar q-p(1))-\sigma(q(-1)-\bar q-p(-1))\\ &\hspace{10pt}\geq 0\end{aligned}$$ for all $q\in\tilde{K}$. Let $\fK\subset H^1(I)$ be defined by $$\label{eq:bKDef} \fK=\{q\in H^1(I):\;q\leq \;\psi\;\;\text{in}\;I\},$$ and $G:H^1(I)\longrightarrow [0,\infty)$ be defined by $$\label{eq:gDef} G(q)=\int_I q\,dx.$$ Then the function $\psi$ belongs to $\fK$ and $$\label{eq:Slater} G(\psi)>0$$ by . It follows from the Slater condition and the theory of Lagrange multipliers [@IK:2008:Lagrange Chapter 1, Theorem 1.6] that there exists a nonnegative number $\lambda$ such that $$\label{eq:VI3} b(p,q-p)+(\phi,q-p)+\tau((q(1)-p(1))-\sigma(q(-1)-p(-1))-\lambda\int_I (q-p)dx\geq 0$$ for all $q\in \fK$. Finally we observe that $$\label{eq:VI4} \tilde b(p,q-p)+(F,q-p)+\tau(q(1)-p(1))-\sigma(q(-1)-p(-1)) \geq 0\qquad\forall\,q\in\fK,$$ where $$\label{eq:taDef} \tilde b(q,r)=\int_I q'r'dx+\int_I qr\,dx$$ and $$\label{eq:FDef} F=\phi-\lambda-p.$$ The variational inequality defined by , and characterizes the solution of a coercive Neumann obstacle problem on $H^1(I)$. Since $F\in \LT$ and $\psi\in H^2(I)$, we can apply the result in [@Rodrigues:1987:Obstacle Chapter 5, Theorem 3.4] to conclude that $p\in H^2(I)$. We can deduce the regularity of $(\bar y,\bar u)$ from the relations $p=\bar y'$ and $\bar u=-(\bar y''+f)$. \[cor:Regularity\] Under the assumption on the data, the solution $(\bar y,\bar u)$ of the optimal control problem – belongs to $H^3(I)\times H^1(I)$. The result in [@Rodrigues:1987:Obstacle], which is for dimensions $\geq 2$, requires a compatibility condition between $\p\psi/\p n$ and the Neumann boundary condition so that the boundary trace of the normal derivative of the solution of the obstacle problem belongs to the correct Sobolev space. This is not needed in one dimension since the boundary values of the normal derivative are just numbers. The Karush-Kuhn-Tucker Conditions {#subsec:KKT} --------------------------------- It follows from , Theorem \[thm:pRegularity\] and integration by parts that $$\label{eq:ObstacleLM} b(p,q)+(\phi,q)+\tau q(1)-\sigma q(-1)-\lambda\int_I q\,dx +\int_I q\,d\nu=0 \qquad\forall\,q\in H^1(I),$$ where the regular Borel measure $\nu$ is given by $$\label{eq:ObstaclenuDef} d\nu=(p''-\phi+\lambda)dx+[p'(-1)+\sigma]d\delta_{-1}-[p'(1) +\tau]d\delta_{1},$$ and $\delta_{-1}$ (resp., $\delta_1$) is the Dirac point measure at $-1$ (resp., $1$). Let $\fA$ be the active set of the derivative constraint , i.e., $$\label{eq:ActiveSet} \fA=\{x\in [-1,1]:\,\bar y'(x)=\psi(x)\}=\{x\in[-1,1]:\,p(x)=\psi(x)\}.$$ By a standard argument, $p$ satisfies if and only if $$\label{eq:ObstaclenuProperties} \text{$\nu$ is nonnegative and supported on $\fA$.}$$ We can translate – into Karush-Kuhn-Tucker (KKT) conditions for the solution $\bar y'=p\in \cK$ of /, which is summarized in the following theorem. \[thm:KKT\] There exists a nonnegative number $\lambda$ such that $$\begin{aligned} &\int_I p'q'dx+\int_I(\Phi-f')q\,dx+f(1)q(1)-f(-1)q(-1)+ \int_I q\,d\nu\label{eq:KKT1}\\ &\hspace{100pt}=\lambda\int_I q\,dx&\qquad\forall\,q\in H^1(I),\notag\\ &\int_{[-1,1]}(p-\psi)d\nu=0,\label{eq:KKT2}\\[4pt] &d\nu=\rho\,dx+\gamma d\delta_{-1}+\zeta d\delta_1,\label{eq:KKT3}\end{aligned}$$ where $$\begin{aligned} &\text{$\rho=p''+f'-\Phi+\lambda\in L_2(I)$ is nonnegative a.e.},\label{eq:KKT4}\\ &\text{$\gamma=p'(-1)+f(-1)$ and $\zeta=-[p'(1)+f(1)]$ are nonnegative numbers},\label{eq:KKT5}\end{aligned}$$ and $\Phi\in H^1(I)$ satisfies –. \[rem:KKT\] The KKT conditions – are also sufficient conditions for . Indeed, they imply, for any $q\in\cK$, $$\begin{aligned} &\int_I p'(q'-p')dx+\int_I(\Phi-f')q\,dx+f(1)\big(q(1)-p(1)\big) -f(-1)\big(q(-1)-p(-1)\big)\\ &\hspace{50pt}= \lambda\int_I(q-p)dx-\int_I (q-p)d\nu\\ &\hspace{50pt}=-\int_I (q-\psi)d\nu\geq 0,\end{aligned}$$ which then also implies $\bar y(x)=\int_{-1}^x p(t)dt$ is the solution of . Finally we observe that Theorem \[thm:KKT\] implies $$\label{eq:KKT} \beta\int_I (\bar y''+f)z''dx+\int_I(\bar y-y_d)z\,dx+\int_{[-1,1]}z'd\mu=0 \qquad\forall\, z\in H^2(I)\cap H^1_0(I),$$ where $$\label{eq:muDef} \text{$\mu=\beta\nu$ is a nonnegative Borel measure},$$ and $$\label{eq:Complementarity} \int_{[-1,1]} (\bar y'-\psi)d\mu=0.$$ The Discrete Problem {#sec:Discrete} ==================== Let $V_h\subset H^2(I)\cap H^1_0(I)$ be the cubic Hermite finite element space (cf. [@Ciarlet:1978:FEM; @BScott:2008:FEM]) associated with a triangulation/partition $\cT_h$ of $I$ with mesh size $h$. The discrete problem is to find $$\label{eq:DiscreteProblem} \bar y_h=\argmin_{y_h\in K_h}\frac12\big(\|y_h-y_d\|_\LT^2 +\beta\|y_h''+f\|_\LT^2\big),$$ where $$\label{eq:KhDef} K_h=\{y_h\in V_h:\, P_hy_h'\leq P_h\psi\},$$ and $P_h$ is the interpolation operator associated with the $P_1$ finite element space associated with $\cT_h$, i.e., the constraint is only enforced at the grid points. The nodal interpolation operator from $C^1([-1,1])$ onto $V_h$ is denoted by $\Pi_h$. We will use the following standard estimates for $P_h$ and $\Pi_h$ (cf. [@Ciarlet:1978:FEM; @BScott:2008:FEM]) in the error analysis: $$\begin{aligned} {3} \|\zeta- P_h\zeta\|_\LT&\leq Ch|\zeta|_{H^1(I)} &\qquad&\forall\,\zeta\in H^1(I),\label{eq:PhError1}\\ \|\zeta-P_h\zeta\|_\LT&\leq Ch^2|\zeta|_{H^2(I)} &\qquad&\forall\,\zeta\in H^2(I),\label{eq:PhError2}\\ |\zeta-\Pi_h\zeta|_{H^1(I)}+h|\zeta-\Pi_h\zeta|_{H^2(I)}&\leq Ch^2|\zeta|_{H^3(I)} &\qquad&\forall\,\zeta\in H^3(I).\label{eq:PihEst}\end{aligned}$$ Here and below we use $C$ to denote a generic positive constant that is independent of the mesh size $h$. The unique solution $\bar y_h\in K_h$ of the minimization problem defined by and is characterized by the discrete variational inequality $$\beta\int_I (\bar y_h''+f)(y_h''-\bar y_h'')dx +\int_I (\bar y_h-y_d)(y_h-\bar y_h)dx\geq0 \qquad\forall\,y_h\in K_h,$$ which can also be written as $$\label{eq:DVI} a(\bar y_h,y_h-\bar y_h)\geq \int_I y_d(y_h-\bar y_h)dx -\beta\int_I f(y_h''-\bar y_h'')dx\qquad\forall\,y_h\in K_h,$$ where the bilinear form $a(\cdot,\cdot)$ is defined in . The error analysis of the finite element method is based on the approach in [@BSung:2017:State] for state constrained optimal control problems that was extended to one dimensional problems with constraints on the derivative of the state in [@BSW:2020:OneD]. We will use the energy norm $\|\cdot\|_a$ defined by $$\label{eq:EnergyNorm} \|v\|_a^2=a(v,v)=\|v\|_\LT^2+\beta|v|_{H^2(I)}^2.$$ Note that $$\label{eq:NormEquivalence} \|v\|_a\approx \|v\|_{H^2(I)} \qquad\forall\,v\in H^2(I)\cap H^1_0(I)$$ by a Poincaré-Friedrichs inequality [@Necas:2012:Direct]. An Abstract Error Estimate {#subsec:Abstract} -------------------------- In view of , and the Cauchy-Schwarz inequality, we have $$\begin{aligned} \label{eq:Preliminary} \|\bar y-\bar y_h\|_a^2&=a(\bar y-\bar y_h,\bar y-y_h)+ a(\bar y-\bar y_h,y_h-\bar y_h)\notag\\ &\leq \frac12\|\bar y-\bar y_h\|_a^2+\frac12\|\bar y-y_h\|_a^2 +a(\bar y,y_h-\bar y_h)\\ &\hspace{60pt} -\int_I y_d(y_h-\bar y_h)dx+ \beta\int_I f(y_h''-\bar y_h'')dx\qquad\forall\,y_h\in K_h.\notag\end{aligned}$$ It follows from , and that $$\begin{aligned} \label{eq:KKTRelation} &a(\bar y,y_h-\bar y_h)-\int_I y_d(y_h-\bar y_h)dx+ \beta\int_I f(y_h''-\bar y_h'')dx\notag\\ &\hspace{40pt}=\int_{[-1,1]} (\bar y_h'-y_h')d\mu\notag\\ &\hspace{40pt}=\int_{[-1,1]} (\bar y_h'-P_h\bar y_h')d\mu+ \int_{[-1,1]} (P_h\bar y_h'-P_h\psi)d\mu+\int_{[-1,1]} (P_h\psi-\psi)d\mu\\ &\hspace{90pt}+\int_{[-1,1]} (\psi-\bar y')d\mu+ \int_{[-1,1]} (\bar y'-y_h')d\mu\notag\\ &\hspace{40pt}\leq \int_{[-1,1]} (\bar y_h'-P_h\bar y_h')d\mu +\int_{[-1,1]} (P_h\psi-\psi)d\mu +\int_{[-1,1]} (\bar y'-y_h')d\mu \notag\end{aligned}$$ for all $y_h\in K_h$. Putting and together, we arrive at the abstract error estimate $$\begin{aligned} \label{eq:Abstract} \|\bar y-\bar y_h\|_a^2&\leq 2\Big(\int_{[-1,1]} (\bar y_h'-P_h\bar y_h')d\mu +\int_{[-1,1]}(P_h\psi-\psi)d\mu\Big)\\ &\hspace{50pt} +\inf_{y_h\in K_h}\Big(\|\bar y-y_h\|_a^2+ 2\int_{[-1,1]} (\bar y'-y_h')d\mu\Big).\notag\end{aligned}$$ Concrete Error Estimates {#subsec:Concrete} ------------------------ The three terms on the right-hand side of can be estimated as follows. First of all, we have $$\begin{aligned} \label{eq:Est1} \int_{[-1,1]}(\bar y_h'-P_h\bar y_h')d\mu& =\int_{[-1,1]} \big[(\bar y_h'-\bar y')-P_h(\bar y_h'-\bar y')\big]d\mu+ \int_{[-1,1]} (\bar y'-P_h\bar y')d\mu\notag\\ &=\beta \Big(\int_I \big[(\bar y_h'-\bar y')-P_h(\bar y_h'-\bar y')\big]\rho\,dx+ \int_I (\bar y'-P_h\bar y')\rho\,dx\Big)\\ &\leq C\big(h\|\bar y-\bar y_h\|_a+h^2|y|_{H^3(I)}\big),\notag\end{aligned}$$ by Corollary \[cor:Regularity\], , , , , and the fact that $\zeta-P_h\zeta$ vanishes at the points $\pm1$ for any $\zeta\in H^1(I)$. Similarly we can derive $$\label{eq:Est2} \int_{[-1,1]}(P_h\psi-\psi)d\mu=\beta\int_I (P_h\psi-\psi)\rho\,dx \leq Ch^2|\psi|_{H^2(I)}$$ by and . Finally we have $$\begin{aligned} \label{eq:Est3} &\inf_{y_h\in K_h}\Big(\|\bar y-y_h\|_a^2+ 2\int_{[-1,1]} (\bar y'-y_h')d\mu\Big)\notag\\ &\hspace{50pt}\leq \|\bar y-\Pi_h \bar y\|_a^2+2\int_{[-1,1]} \big[\bar y'-(\Pi_h\bar y)'\big]d\mu\\ &\hspace{50pt}= \|\bar y-\Pi_h \bar y\|_a^2+2\beta\int_I \big[\bar y'-(\Pi_h\bar y)'\big]\rho\,dx \leq Ch^2\big[|\bar y|_{H^3(I)}^2+|\bar y|_{H^3(I)}\big],\notag\end{aligned}$$ by Corollary \[cor:Regularity\], , , , and the fact that $\bar y'-(\Pi_h\bar y)'$ vanishes at $\pm1$. It follows from – and Young’s inequality that $$\label{eq:EnergyError} \|\bar y-\bar y_h\|_a\leq Ch,$$ which immediately implies the following result, where $\bar u_h=-(\bar y_h''+f)$ is the approximation for $\bar u=-(\bar y+f)$. \[thm:ErrorEstimates\] Under the assumptions on the data in , we have $$|\bar y-\bar y_h|_\HO+\|\bar u-\bar u_h\|_\LT\leq Ch.$$ \[rem:Sharp\] Numerical results in Section \[sec:Numerics\] indicate that the estimate for $\|\bar u-\bar u_h\|_\LT$ in Theorem \[thm:ErrorEstimates\] is sharp. \[rem:Comparison\] For comparison, the error estimate $$|\bar y-\bar y_h|_\HO+\|\bar u-\bar u_h\|_\LT\leq C_\epsilon h^{\frac12-\epsilon}$$ was obtained in [@BSW:2020:OneD] under the assumptions in . A Numerical Experiment {#sec:Numerics} ====================== We begin by constructing an example for the problem / with a known exact solution. An Example {#subsec:Example} ---------- Let $\beta=1$, $$\label{eq:psiExample} \psi(x)=\begin{cases} 1-\frac92 x^2&\qquad -1\leq x\leq 0\\[2pt] 1&\qquad \phantom{-}0\leq x\leq 1 \end{cases},$$ and $$\label{eq:ExactSolution} \bar y(x)=\int_{-1}^x p(t)dt,$$ where $$\label{eq:pDef} p(x)=\begin{cases} 1-\frac{81}{32}(x-\frac13)^2&\qquad -1\leq x\leq \frac13 \\[2pt] 1&\qquad\hspace{10pt} \frac13\leq x\leq 1 \end{cases}.$$ We have $\psi\in H^2(I)$, $$\label{eq:psiExampleConstraint} \int_I\psi\, dx =\frac12,$$ $p\in H^2(I)$, $$\label{eq:pSD} p''(x)=\begin{cases} -81/16& \qquad -1<x<\frac13\\[2pt] 0&\qquad\hspace{10pt} \frac13<x<1 \end{cases},$$ $p'(1)=0$, $p'(-1)=27/4$, $$\label{eq:pProperties} \int_I p\,dx=0, \quad p\leq\psi \quad\text{and}\quad \fA=\{-1\}\cup[1/3,1].$$ Let $f\in H^1(I)$ be defined by $$\label{eq:fDef} f(x)=\begin{cases}\d \frac{2}{9\pi}\sin (\pi(3x-1))&\qquad -1< x\leq \frac13 \\[8pt] -(x-\frac13)^2 &\qquad \hspace{10pt} \frac13\leq x<1 \end{cases}.$$ We have $f(-1)=0$, $f(1/3)=0$, $f_-'(1/3)=2/3$, $f'_+(1/3)=0$ and $f(1)=-4/9$. Therefore the function $$\label{eq:PhiExample} \Phi(x)=\begin{cases} f'(x)&\qquad -1<x< \frac13\\[2pt] f'(x)+\frac23&\qquad\hspace{10pt} \frac13< x<1 \end{cases}$$ belongs to $H^1(I)$ and $$\label{eq:PhiIntegralExample} \int_I\Phi\,dx=\int_I f'(x)+\int_\frac13^1\frac23\,dx =f(1)-f(-1)+\frac49=0.$$ Finally we take $\lambda=81/16$ and $y_d=\bar y+\Phi'$. Then the KKT conditions – are satisfied with $$\begin{aligned} d\nu&=[p''+f'-\Phi+\lambda]dx+[p'(-1)+f(-1)]d\delta_{-1} -[p'(1)+f(1)]d\delta_1\\ &=(211/48)\chi_{[1/3,1]} dx+(27/4)d\delta_{-1}+(4/9)d\delta_1,\end{aligned}$$ where $\chi_{[1/3,1]}$ is the characteristic function of the interval $[1/3,1]$. Numerical Results {#subsec:Results} ----------------- We solved the problem in Section \[subsec:Example\] by the finite element method in Section \[sec:Discrete\] on uniform meshes. The results are displayed in Table \[table:Results\]. [|l|c|c|c|c|]{} ------------------------------------------------------------------------ $2/h$ &$\|\bar y-\bar y_h\|_{L_2(I)}$&$\|\bar y-\bar y_h\|_{L_\infty(I)}$ &$|\bar y-\bar y_h|_{H^1(I)}$ &$|\bar y-\bar y_h|_{H^2(I)}$\ &&&&\ $1+2^0$ & 1.430334e-01& 1.625937e-01& 2.581989e-01& 8.660252e-01\ $1+2^1$ & 1.216070e-01& 1.385037e-01& 2.199480e-01& 7.486796e-01\ $1+2^2$ & 4.306657e-02 & 4.679253e-02 & 8.061916e-02 & 4.485156e-01\ $1+2^3$ & 1.613494e-02 & 1.850729e-02 & 2.919318e-02 & 2.573315e-01\ $1+2^4$ & 3.439341e-03 & 3.849954e-03 & 6.315816e-03 & 1.266029e-01\ $1+2^5$ & 9.590453e-04 & 1.087740e-03 & 1.741244e-03 & 6.470514e-02\ $1+2^6$ & 2.256478e-04& 2.542346e-04 &4.125212e-04 & 3.223430e-02\ $1+2^7$ & 5.874304e-05 & 6.639870e-05& 1.067193e-04&1.618687e-02\ $1+2^8$ & 1.425640e-05 &1.608790e-05 &2.549283e-05 & 8.086258e-03\ $1+2^9$ & 3.657433e-06 & 4.124680e-06 & 6.430499e-06 & 4.047165e-03\ We observe $O(h)$ convergence in the $H^2$ norm which agrees with Theorem \[thm:ErrorEstimates\]. On the other hand the convergence in the $H^1$ norm is $O(h^2)$, better than the $O(h)$ convergence predicted by Theorem \[thm:ErrorEstimates\]. The convergence in $L_2$ and $L_\infty$ is also $O(h^2)$. Concluding Remarks {#sec:Conclusions} ================== We have shown that higher regularity for the solutions of one dimensional Dirichlet elliptic distributed optimal control problems with pointwise constraints on the derivative of the state can be obtained through a variational inequality satisfied by the derivative of the optimal state. A similar result for one dimensional optimal control problems with mixed boundary conditions was obtained earlier in [@BSW:2020:OneD]. A natural question is: Can these results be extended to higher dimensions? For analogs of – on a smooth/convex domain $\O\in \R^d$ ($d=2,3$), where $f\in H^1(\O)$ and $\bm{\Psi}\in [H^2(\O)]^d$, one can also derive a variational inequality for the gradient of the optimal state. Observe that the space $\bG$ of the gradients of the states is characterized by (cf. [@GR:1986:NS Chapter I, Section 2.3]) $$\begin{aligned} \bG&=\{\nabla y:\,y\in H^2(\O)\cap H^1_0(\O)\}\notag\\ &=\{\bq\in[ H^1(\O)]^d:\,\Curl\bq=0 \;\text{on}\;\O\;\text{and} \;\bn\times\bq=0 \; \text{on}\;\p\O\},\end{aligned}$$ where $\bn$ is the unit outward normal along $\p\O$. Let $\bK$ be the subset of $\bG$ defined by $$\bK=\{\bq\in\bG:\,\bq\leq\bm{\Psi}\;\text{a.e. in $\O$}\}.$$ We assume that $\bK$ is nonempty, which is the case for example if $\bm{\Psi}\geq{\bf 0}$. The analog of is given by $$\label{eq:DirichletHigher} \int_\O (\bPhi-\nabla f)\cdot(\bq-\bp)dx +\int_\O \Div\bp\,\Div(\bq-\bp)\,dx +\int_{\p\O} f(\bq-\bp)\cdot\bn\,dS\geq 0$$ for all $\bq\in \bK$, where $\bp=\nabla\bar y\in\bK$, and $\bPhi\in \bG$ is defined by $ \beta\Div\bPhi=y_d-\bar y$, which is an analog of . The variational inequality is uniquely solvable because (cf. [@GR:1986:NS Chapter I, Sections 3.2 and 3.4]) $$\int_\O (\Div \bq)^2dx\geq C_\O|\bq|_{H^1(\O)}^2\qquad \forall\,\bq\in \bG.$$ We can also write as $$\begin{aligned} \label{eq:DirichletVector} &\int_\O (\bPhi-\nabla f)\cdot(\bq-\bp)dx+ \int_\O \big[\Div\bp\,\Div(\bq-\bp) +\Curl\bp\cdot\Curl(\bq-\bp)\big]\,dx\\ &\hspace{100pt}+\int_{\p\O} f(\bq-\bp)\cdot\bn\,dS \geq 0\qquad\forall\,\bq\in \bK,\notag\end{aligned}$$ which can be interpreted as an obstacle problem for the vector Laplacian operator with natural boundary conditions. In order to obtain higher regularity for the optimal state $\bar y$, one will need regularity results for /, which unfortunately are not available. Therefore the problem of extending the results in this paper to higher dimensions remains open. [10]{} S.C. Brenner, C.B. Davis, and L.-Y. Sung. . , 276:612–626, 2014. S.C. Brenner, J. Gedicke, and L.-Y. 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Optimal control of elliptic equations with pointwise constraints on the gradient of the state in nonsmooth polygonal domains. , 50:2117–2129, 2012. [^1]: The work of the first and second authors was supported in part by the National Science Foundation under Grant No. DMS-19-13035.

--- abstract: 'The validity of Hubble’s law defies the determination of the center of the big bang expansion, even if it exists. Every point in the expanding universe looks like the center from which the rest of the universe flies away. In this article, the author shows that the distribution of apparently circular galaxies is not uniform in the sky and that there exists a special direction in the universe in our neighborhood. The data is consistent with the assumption that the tidal force due to the mass distribution around the universe center causes the deformation of galactic shapes depending on its orientation and location relative to the center and our galaxy. Moreover, the cmb dipole data can also be associated with the center of the universe expansion, if the cmb dipole at the center of our supercluster is assumed to be due to Hubble flow. The location of the center is estimated from the cmb dipole data. The direction to the center from both sets of data is consistent and the distance to the center is computed from the cmb dipole data.' author: - Yukio Tomozawa title: The CMB Dipole and Circular Galaxy Distribution --- \[sec:level1\]Introduction ========================== Since the discovery of Hubble’s law in 1929 and the big bang interpretation of its data, the question lingers whether a center for the expansion exists and if so where. The Hubble law, **v** = H$_{\text{0}}$ **r**, yields the relationship, **v**$_{\text{2}}$ - **v**$_{\text{1}}$= H$_{\text{0}}$ ( **r**$_{\text{2}}$ - **r**$_{\text{1}}$) for any two galaxies with positions and velocities, **r**$_{\text{1}}$, **v**$_{\text{1}}$ and **r**$_{\text{2}}$ , **v**$_{\text{2}}$ respectively, where H$_{\text{0}}$ = 100 h km/s Mpc is the Hubble constant (with h = 0.5 0.85). For convenience of discussion, we assume the value of H$_{\text{0}}$ to be 70.0 km/s Mpc in this article. The last equation implies that every point appears to be the center of the expansion. Besides, the distribution of galaxies surrounding us seems to be isotropic on some large distance scale. In order to get information about the existence of a center for expansion of the universe, one has to search for observational data beyond the Hubble law. The author will show that such information can be obtained from data for the distribution of circular galixies and the cmb dipole in the temperature distribution. These independent data indicate that the direction to the center is consistent for both, and one set of data gives a determination of the distance. Distribution of Circular Galaxies ================================= Circular galaxies appear when the circular faces of spiral galaxies or elliptical galaxies are facing us; otherwise they are elliptical in shape. The fraction of circular galaxies in any direction depends on the probability of the orientation of the symmetry axis coinciding with the line of sight. If one assumes that such a probability is uniform in any direction, then one expects the distribution of circular galaxies to be isotropic. The author will show that that is not the case for the existing data. Fig. 1 (a) shows the distribution of 23,011 bright galaxies in galactic coordinates compiled in $\ $RC3[@rc3]. The galaxies in the galactic plane are missing for the obvious reason that they are prevented from observation due to our own Milky Way. Plots in equatorial coordinates and supergalactic coordinates show a similar uniform distribution. The number of galaxies on the two sides of the galactic plane is 12,323 in the north and 10,628 in the south with a ratio of 1.1651. The difference in these numbers may be partly due to a statistical fluctuation and partly due to a historical bias in past observation. Anyway, any study hereafter can be normalized with this ratio. Fig. 1 (b) shows the velocity distribution for the compiled galaxies. The number of galaxies for which the velocities, cz, are known is 10633 (5923 in the north and 4710 in the south). The approximate upper bound is $\sim$15,000 km/s and the average, $\sim$7,500 km/s, corresponds to a distance of 110 Mpc. In Fig. 2 (a), (b) and (c), we show the distribution of galaxies with circular shape in galactic, supergalactic and equatorial J2000.0 coordinates, respectively. These galaxies correspond to those with the value R25 = 0.00 (the error is in the range of $\sim$0.10) in RC3, where R25 stands for the logarithm of the ratio of the apparent major and minor axes. The number of such galaxies is 682 in the north, 416 in the south with 1098 for the total. Also shown in Fig. 2 (d) is the velocity distribution, which is similar to Fig 1 (b). The pertinent characteristics of Fig. 2 are the following: \(i) Apparently circular galaxies are not distributed uniformly in the sky. If the orientation of galaxies is distributed at random, Fig 2 (a)-(c) should show statistically uniform distributions. It definitely shows directionality. \(ii) The southern part is more compact compared to the northern part. Since a closed curve surrounding the pole is extended to all longitudes in Fig 1 (a), the views in other coordinates, supergalactic coordinates in Fig 2 (b) and equatorial coordinates J2000.0 in Fig 2 (c), provide intuitively clearer distribution plots. Fig 2 (b) shows similar shapes for both, northern and southern, distributions. The normalized ratio of the northern to the southern distribution is (682)/(416)/(1.1651) = 1.4071. The shapes in supergalactic coordinates in Fig 2 (b) match this ratio. Note that the left and right blobs in Fig 2 (b) approximately correspond to those in the north and the south in Fig 2 (a). Fig 3 (a)-(d) shows the distribution of galaxies with R25 = 0.01-0.10, 0.11-0.20, 0.21-0.40 and 0.41-1.00. They do not show a distinct directionality except for a higher density in the neighborhood of (120d, -30d). The total number of galaxies, the numbers in the north and south as well as the normalized north/south ratio are given for sliced values of R25 in Table 1. In summary, a distinct directionality exists only for R25 = 0.00, and not for the other values of R25. The meaning of this directionality in the distribution of apparently circular galaxies is an important question .The author suggests that the existence of a center for universe expansion is consistent with the presented data and suggests the direction to the center, if not the determination of the distance. \(I) Galaxies are subject to tidal forces due to the gravity of the masses which are contained in a sphere of radius equal to the distance from the center to the galaxies in question. Circular galaxies in the radial direction from the center through us stay circular in the presence of tidal forces due to axial symmetry, while those in directions perpendicular to the radial seen from us are deformed by the squeeze and stretch of tidal forces and therefore become elliptical. This explains the directionality in Fig. 2 (a)-(c). \(II) If the center lies at a large distance compared with the observed galaxies in the RC3 catalog, then the circular galaxy blobs in the both hemispheres should be approximately equal in size, since the defining cone for circular invariance constitutes parallel lines. The disparity in the south and the north suggests that the center is nearby and the direction to the center must be towards the north, since the tidal force near the center is small so that the distortation is minimal in the neighborhood. This explains the larger extension of circular galaxies in the northern hemisphere. The central value of the direction of the center is in the neighborhood of $$l_{c}=120{{}^\circ}+180{{}^\circ}=300{{}^\circ}\pm15{{}^\circ},\text{ \ \ \ }b_{c}=30{{}^\circ}\pm15{{}^\circ}. \label{center}$$ The distance to the center, $R_{c}$, is bounded by 10000km/s ==$>$143Mpc. \(III) If the matter density is constant, it produces a gravitational force that is linear in the radial direction, thus yielding a tidal force that is spherically symmetric and attractive. Obviously, such a tidal force does not explain the data of Fig.2 (a)-(c). Although the assumption of homogeneity and isotropy in the Friedman-Robertson-Walker metric yields a constant matter density at a fixed time, experimental evidence for those assumptions is yet to come. As a matter of fact, the analysis of this article finds that the matter density of the universe is not constant on this length scale. We are discussing a length scale that involves our local supercluster which is not uniform in density. In Fig 4(a), circular galaxy distribution (R25 = 0) from a newer compilation, the Hyperleda catalog (PGC 2003) with 12,008 items out of a total of 983,261 is shown. Three blobs in the equatorial slab are stars. Away from the equatorial slab, most of data points are galaxies. The data away from the equatorial slab has features similar to that of the data from the RC3 catalog, Fig 2(a). Fig 4(b) shows a spherical plot of Fig 4(a). The cmb Dipole in the Temperature Distribution ============================================== A dipole component was observed in the temperature distribution in the cmb (cosmic microwave background radiation) measurement. The blue shift dipole value for the solar system [@dipole] is v = 371$\pm$0.5 km/s, l = 264.4$\pm$0.3${{}^\circ}$, b = 48.4$\pm$0.5${{}^\circ}$. Or alternatively, the redshift dipole value is$$v=371\pm0.5\text{ }km/s,\text{ \ \ \ }l=84.4\pm0.3{{}^\circ},\text{ \ \ \ }b=-48.4\pm0.5{{}^\circ}.$$ By using the observation of a peculiar velocity for the solar system, one can compute the dipole component of the cluster (Virgo) center and that of the supercluster (Great Attractor) center. Then, one can associate the last dipole component with Hubble flow. In order to understand the relationship among Hubble flow, peculiar velocity and the cmb dipole, the author presents pertinent theorems. With the assumption of the existence of a center for expansion of the universe, Hubble flow creates a cmb dipole with red shift in the direction of the Hubble flow with the magnitude of the Hubble flow velocity. Let the velocities of the Hubble flow and the cmb emitter in the direction of the Hubble flow be $v_{H}$ and $v$, respectively. The relative velocity of the two is $v_{+}=(v-v_{H})/(1-vv_{H}/c^{2})$. The analogous velocities in the opposite direction are $-v_{H}$ and $v-2v_{H}$, and the relative velocity is $v_{-}=(v-2v_{H}+v_{H})/(1+(v-2v_{H})v_{H}/c^{2})$ in the opposite direction. The difference of these opposite relative velocities is then $$v_{+}-v_{-}=(v-v_{H})(2v-2v_{H})v_{H}/c^{2}(1-vv_{H}/c^{2})(1+(v-v_{H})v_{H}/c^{2})\approx2v_{H},\text{ \ \ \ \ \ \ \ }for\text{ \ }v\approx c\text{ and }v_{H}\ll c.$$ and$$v_{\pm}\approx v-v_{H}\pm v_{H}$$ This proves the statement. The cmb dipole vanishes at the center of the universe. The cmb dipoles for red shift at the center of a cluster and at a member galaxy with a peculiar velocity $\mathbf{v}_{p}$ are related by$$\mathbf{v}(dipole\text{ }at\text{ }the\text{ }clustercenter)=\mathbf{v}(dipole\text{ }at\text{ }the\text{ }galaxy)-\mathbf{v}_{p}+\mathbf{v}(clustercenter-galaxy),$$ where$$\mathbf{v}(A-B)=\mathbf{v}(A)-\mathbf{v}(B).$$ The peculiar velocity plays a role similar to Hubble flow, as far as the cmb dipole is concerned. This can be seen by considering the case where a peculiar velocity, $\mathbf{v}_{p}$, and a Hubble flow, $\mathbf{v}_{H}$, are parallel to each other. In this case, the galaxy is at rest with respect to a galaxy with a Hubble flow, $\mathbf{v}_{H}+\mathbf{v}_{p}$. Since both galaxies should have the same cmb dipole, a peculiar velocity yields a redshift cmb dipole with $\mathbf{v}_{p}$. That is the reason for the subtraction. The term, $\mathbf{v}(clustercenter-galaxy)=$ $\mathbf{v}(clustercenter)-\mathbf{v}(galaxy)$, is for adjustament of the Hubble law. The center of the universe should have an intrinsically vanishing value for the cmb dipole. Hereafter a cmb dipole implies a redshift dipole unless otherwise stated. The peculiar velocity of the sun relative to the Virgo center of the local cluster is estimated to be [@sciama] $$v=415\text{ }km/s,\text{ \ \ \ }l=335{{}^\circ},\text{ \ \ \ }b=7{{}^\circ} \label{sciama1}$$ or$$v=630\text{ }km/s,\text{ \ \ \ }l=330{{}^\circ},\text{ \ \ \ }b=45{{}^\circ} \label{sciama2}$$ We examine these two possibilities. The outcomes are listed in this order for each case. Using Theorem 2, one computes the cmb dipole at the Virgo center, which is located at$$v=1050\pm200\text{ }km/s,\text{ \ \ \ }l=287{{}^\circ},\text{ \ \ \ }b=72.3{{}^\circ}$$ corresponding to a distance of $15\pm3$ $Mpc$. Hereafter, all distances are expressed in the form of receding velocities due to the Hubble law, $\mathbf{v}=H_{0}\mathbf{r}$. The key formula is$$\mathbf{v}(dipole\text{ }at\text{ }Virgo)=\mathbf{v}(dipole\text{ }at\text{ }the\text{ }Sun)-\mathbf{v}_{p}(Sun/Virgo)+\mathbf{v}(Virgo). \label{dipole1}$$ For the cmb dipole at the Virgo center, one obtains $$v=728.3\pm148\text{ }km/s,\text{ \ \ \ }l=336.0\pm8.2{{}^\circ},\text{ \ \ \ }b=67.4\pm11.3{{}^\circ}$$$$v=418.8\pm47\text{ }km/s,\text{ \ \ \ }l=328.8\pm6.4{{}^\circ},\text{ \ \ \ }b=41.5\pm28.0{{}^\circ}$$ depending on the two choices of peculiar velocity. We note that the Cartesian coordinates for $(v,l,b)$ are expressed as $(v\cos(b)\cos l,v\cos(b)\sin l,v\sin(b).$ Further, the Virgo cluster is considered to be a part of a supercluster centered around the GA (great attractor). In order to compute the cmb dipole at the GA, one assumes the velocity of the Virgo center to be corresponding to a distance of $15\pm3$ $Mpc$. The position of the GA is taken to be [@ga]$$v=4200\text{ }km/s,\text{ \ \ \ }l=309{{}^\circ},\text{ \ \ \ }b=18{{}^\circ} \label{eqga1}$$ or$$v=3000\text{ }km/s,\text{ \ \ \ }l=305{{}^\circ},\text{ \ \ \ }b=18{{}^\circ} \label{eqga2}$$ and the infall velocity of the Virgo center to the GA to be $$v_{in}=1000\pm200km/s.$$ The direction of the infall is determined by$$\mathbf{v}(GA-V)=\mathbf{v}(GA)-\mathbf{v}(Virgo)$$ resulting in$$v(GA-V)=3712.3\pm75\text{ }km/s,\text{ \ \ \ }l(GA-V)=310.9\pm0.4{{}^\circ},\text{ \ \ \ }b(GA-V)=4.6\pm2.8{{}^\circ}\text{ \ \ \ } \label{infall1}$$ for Eq.(\[eqga1\]) and$$v(GA-V)=2552.5\pm59\text{ }km/s,\text{ \ \ \ }l(GA-V)=307.2\pm1.6{{}^\circ},\text{ \ \ \ }b(GA-V)=1.6\pm3.0{{}^\circ} \label{infall2}$$ for Eq. (\[eqga2\]). Using Theorem 2, one can compute the cmb dipole at the GA by$$\mathbf{v}(dipole\text{ }at\text{ }GA)=\mathbf{v}(dipole\text{ }at\text{ }Vigo)-\mathbf{v}(Virgo\text{ infall})+\mathbf{v}(GA-V), \label{dipole2}$$ where the direction of the infall is given by Eq. (\[infall1\]) or Eq. (\[infall2\]). Then, the cmb dipole at the GA is given by$$v=2609.3\pm120\text{ }km/s,\text{ \ \ \ }l=308.1\pm0.2{{}^\circ},\text{ \ \ \ }b=19.9\pm3.4{{}^\circ}$$$$v=2459.1\pm98\text{ }km/s,\text{ \ \ \ }l=308.6\pm0.6{{}^\circ},\text{ \ \ \ }b=11.6\pm3.9{{}^\circ}$$ for Eq. (\[eqga1\]), and$$v=1455.6\pm142\text{ }km/s,\text{ \ \ \ }l=301.3\pm0.6{{}^\circ},\text{ \ \ \ }b=25.5\pm5.3{{}^\circ}$$$$v=1286.6\pm104\text{ }km/s,\text{ \ \ \ }l=302.0\pm0.7{{}^\circ},\text{ \ \ \ }b=10.4\pm7.3{{}^\circ}$$ for Eq. (\[eqga2\]). Based on Theorem 1, one may assume that the cmb dipole at the GA calculated above is due to the Hubble flow of the the center of the GA. Then, one can compute the position of the center for expansion of the universe by$$\mathbf{v}(universe\text{ }center)=\mathbf{v}(GA\text{ }center)-\mathbf{v}(the\text{ }cmb\text{ }dipole\text{ }at\text{ }GA) \label{dipole3}$$ The position of the Universe center thus obtained is$$v_{c}=1595.3\pm196\text{ }km/s,\text{ \ \ \ }l_{c}=310.5\pm0.1{{}^\circ},\text{ \ \ \ }b_{c}=14.8\pm1.5{{}^\circ} \label{center1}$$$$v_{c}=1779.6\pm184km/s,\text{ \ \ \ }l_{c}=309.7\pm0.2{{}^\circ},\text{ \ \ \ }b_{c}=26.8\pm2.7{{}^\circ} \label{center2}$$ for Eq. (\[eqga1\]), and$$v_{c}=1585.4\pm196\text{ }km/s,\text{ \ \ \ }l_{c}=308.1\pm0.2{{}^\circ},\text{ \ \ \ }b_{c}=13.0\pm1.6{{}^\circ} \label{center3}$$$$v_{c}=1759.3\pm181\text{ }km/s,\text{ \ \ \ }l_{c}=307.4\pm0.0{{}^\circ},\text{ \ \ \ }b_{c}=25.3\pm2.9{{}^\circ} \label{center4}$$ for Eq. (\[eqga2\]). Conversion to the ordinary distance scale yields $22.8\pm2.8$ $Mpc$, $25.4\pm2.6$ $Mpc$ for Eq. (\[eqga1\]) and $22.6\pm2.8$ $Mpc$, $25.1\pm2.6Mpc$ for Eq. (\[eqga2\]). The angles obtained here for the direction of the universe center are quite consistent with those obtained in the previous section from the circular galaxy distribution, Eq. (\[center\]). Combining all the processes, Eq. (\[dipole1\]), Eq.(\[dipole2\]) and Eq. (\[dipole3\]), one gets$$\mathbf{v}(universe\text{ }center)=-(\mathbf{v}(dipole\text{ }at\text{ }the\text{ }Sun)-\mathbf{v}_{p}(total))$$ where$$\mathbf{v}_{p}(total)=\mathbf{v}_{p}(Sun/Virgo)+\mathbf{v}_{p}(Virgo/GA)$$ is the total peculiar velocity of the sun towards the GA. This equation gives identical results for the center of the universe, Eq. (\[center1\])-Eq. (\[center4\]). In other words, the author has arrived at the following theorem. The Hubble flow of the solar system is given by$$v_{H}(the\text{ }Sun)=\mathbf{v}(the\text{ }cmb\text{ }dipole\text{ }at\text{ }the\text{ }Sun)-\mathbf{v}_{p}(the\text{ }total\text{ peculiar }velocity\text{ }of\text{ }the\text{ }Sun)$$ The similarity of the final results for the location of the center of the universe, Eq. (\[center1\])-Eq. (\[center2\]) and Eq. (\[center3\])-Eq. (\[center4\]), is easily understood from this theorm, since the directions of the two solutions for the GA position, Eq. (\[eqga1\]) and Eq. (\[eqga2\]), are similar and therefore the peculiar velocities from Virgo towards the GA are similar, despite a large difference between the distances to the GA. Therefore, the difference comes from the peculiar velocity for the sun towaeds the Virgo center, Eq. (\[sciama1\]) or Eq. (\[sciama2\]). In order to settle on the distance to the universe center, one has to settle on one of the two choices for the peculiar velocity, Eq. (\[sciama1\]) or Eq. (\[sciama2\]). Summary and Discussion ====================== The data compiled in RC3 is an accumulation by various groups, so that the criterion for circularity may not be uniform across observational groups. Nevertheless, the consistency between the cmb dipole data and the circular galaxy distribution for the direction of the universe center suggests consistency among the many groups that contributed to the RC3 catalog in that respect. The recent measurement by SDSS increases the data on galaxies in number and quality. It is desirable, however, to enlarge the scope of sky coverage into the southern hemisphere to get improved data for the circular galaxy distribution. Finally, the author suggests that the deformation of galaxies by cosmic tidal forces discussed in this article be taken into account for the systematic study of the triaxiality of galaxies [@triax] and asymmetric spiral galaxies [@spiral]. From statistical analysis of these galaxy deformations, one may be able to reach a more accurate determination of the new cosmological parameter, the position of the center of the universe. As for the discussion of the cmb dipole, the observed cmd dipole cannot be explained merely by the presence of the peculiar velocity of the solar system. One should be involved with the Hubble flow even only from dimensional considerations. Theorem 3 suggests an important departure from normal thinking in cosmology. If further information on the peculiar velocity of the GA supercluster is found in the future, a similar procedure should be applied to the new situation. Then, the final result for the center of the universe expansion might be changed. However, the conclusion from the circular galaxy distribution gives a stringent constraint on the distance and the direction of the universe center. There is a proposal and discussion of a rotation of the universe based on data of polarization of radio waves[@birch]. In general, an intrinsic polarization of radio waves can be related to the existence of the center of the universe, since the deformation of a galaxy by tidal forces can cause an asymmetric magnetic field that creates polarization at the source which is called intrinsic. Then, some features of the observed polarization can be explained by the mere existence of the center of the universe, but it is not clear whether all the features can be explained from that or not. If not, one can ask the question whether there exists a unified description of all the data by the center and a rotation of the universe. Certainly, any rotation axis of the universe should pass through the center in a simple picture. In fact, the angles reported in the two works are close to each other: For the center of the universe, (l, b) = (310${{}^\circ}$, 30${{}^\circ}$) –$>$ (ra, dec) = (13h17m, -32d), and for the rotation axis, (ra, dec) = (14h35m,-35d) with an error bar of (30m, 30m). The proximity of these angles would make the construction of a unified picture easier. The author would like to thank the members of the Physics Department and the Astronomy Department of the University of Michigan for useful information. Correspondence should be addressed to the author at tomozawa@umich.edu. [9]{} de Vaucouleurs, G. et. al., Third Reference Catalogue of Bright Galaxies, Vol I-III (Spring-Verlag, New York, 1991). Melchiorri, B. et. al., New Astron. Rev. 46, 693 (2002); Fixsen, D.J. et. al., Apj. 473, 576 (1996). Sciama, D. W., Phys. Rev. Letters 18, 1065 (1967); Rees, M. and Sciama, D. W., Nature 213, 374 (1967); Stewart, J. M. and Sciama, D. W., Nature 216, 748 (1967). Faber, S. M. and Burstein, D., in Large-Scale Motions in the Universe (ed. Rubin, V. C. and Coyne, G. V., Princeton University Press 1988) p.115; Lynden-Bell, D. et.al., Apj. 326, 19 (1988). Binney, J. & Tremaine, S., Galactic Dynamics, (Princeton University Press, Princeton,1987). Schoenmakers, R. H. M., Franx, M. and de Zeeuw, P. T., Measuring non-axisymmetry in spiral galaxies. MNRAS 292, 349-364 (1997); Schoenmakers, R. H. M., Asymmetries in Spiral Galaxies, (University Press, Veenendaal, 2000) and references therein. Birch, P., Nature 298, 451 (1982); Okunov, Yu. N., Gen. Relat. Grav. 24, 121 (1992); Noland, B. and Ralston, J. P., Phys. Rev. Lett., 78, 3043 (1997); ibid. 79, 1958 (1997); and many others. Figure Caption. Fig.1 Distribution of bright galaxies compiled in RC3 [@rc3]. (a) Galaxy distribution in galactic coordinates. (b) Velocity distribution vs. galactic latitudes. Fig.2 The distribution of apparently circular galaxies which have the value R25= 0.00: (a) In galactic coordinates. (b) In supergalactic coordinates. (c) In equatorial J2000.0 coordinates. (d) Velocity distribution vs. galactic latitudes. Fig.3 Distribution of galaxies in various ranges of the value of R25: (a) 0.01-0.10, (b) 0.11-0.20, (c) 0.21-0.40 and (d) 0.41-1.00. Fig.4(a): Circular galaxy distribution (R25 = 0) from the Hyperleda catalog (PGC 2003) in galactic coordinates (12,008 items out of a total of 983,261). Fig.4(b): Spherical plot of Fig.4(a). Table Caption. Table1. The total number of galaxies, the numbers in the north and south and the normalized north/south ratio in various ranges of R25. \[c\][lllll]{}Range of R25 & Total & North & South & North/South Normalized Ratio\ 0.00-0.00 & 1098 & 682 & 416 & 1.4071\ 0.01-0.10 & 4573 & 2330 & 2243 & 0.8916\ 0.11-0.20 & 4448 & 2322 & 2126 & 0.9374\ 0.21-0.30 & 3286 & 1714 & 1572 & 0.9358\ 0.31-0.40 & 2246 & 1148 & 1098 & 0.8974\ 0.41-0.50 & 1638 & 861 & 787 & 0.9281\ 0.51-1.00 & 4050 & 2210 & 1835 & 1.0337

--- abstract: | As the number of possible predictors generated by high-throughput experiments continues to increase, methods are needed to quickly screen out unimportant covariates. Model-based screening methods have been proposed and theoretically justified, but only for a few specific models. Model-free screening methods have also recently been studied, but can have lower power to detect important covariates. In this paper we propose EEScreen, a screening procedure that can be used with any model that can be fit using estimating equations, and provide unified results on its finite-sample screening performance. EEScreen thus generalizes many recently proposed model-based and model-free screening procedures. We also propose iEEScreen, an iterative version of EEScreen, and show that it is closely related to a recently studied boosting method for estimating equations. We show via simulations for two different estimating equations that EEScreen and iEEScreen are useful and flexible screening procedures, and demonstrate our methods on data from a multiple myeloma study. Keywords: Estimating equations; Ultra-high-dimensional data; Sure independence screening; Variable selection author: - | Sihai Dave Zhao, Department of Biostatistics, Harvard School of Public Health\ Yi Li, Department of Biostatistics, University of Michigan bibliography: - 'refs.bib' title: 'Sure screening for estimating equations in ultra-high dimensions' --- Introduction {#sec:intro} ============ Modern high-throughput experiments are producing high-dimensional datasets with extremely large numbers of covariates. Traditional regression modeling strategies work poorly in such situations, leading to recent interest in regularized regression methods such as the lasso [@Tibshirani1996], the Dantzig selector [@CandesTao2007], and SCAD [@FanLi2001]. These procedures can perform well in estimation and prediction even when the number of covariates $p_n$ is larger than the sample size $n$, where here we are allowing $p_n$ to grow with $n$. However, when $p_n$ is extremely large compared to $n$, these methods can become inaccurate and computationally infeasible [@FanLv2008]. Thus there is a need for methods to quickly screen out unimportant covariates before using regularization methods. A number of screening strategies have so far been proposed, and choosing which one to use depends on what model we believe is most suitable for the data. Under the ordinary linear model, @FanLv2008 proposed a procedure with the sure screening property, where the covariates retained after screening will contain the truly important covariates with probability approaching one, even in the ultra-high-dimensional realm where $p_n$ grows exponentially with $n$. @FanSong2010 and @ZhaoLi2012 subsequently proposed procedures that maintain this property for generalized linear models and the Cox model, respectively. Screening methods have also been proposed for nonparametric additive models [@FanFengSong2011], linear transformation models [@Li2011], and single-index hazard models [@GorstRasmussenScheike2011]. In a recent development, @Zhu2011 proposed a screening method valid for any single-index model, a class so large that their screening procedure is nearly model-free. They used a new measure of dependence which can detect a wide variety of functional relationships between the covariates and the outcome, and proved that their method has the sure screening property for any single-index model. They also showed in simulations that it could significantly outperform model-based screening methods when the models were incorrectly specified. On the other hand, model-based screening can have greater power to detect important covariates, a consequence of the bias-variance tradeoff. However, there are often situations where we wish to use some model other than the ones mentioned above. For example, studies involving clustered observations, missing data, or censored outcomes are frequently encountered in genomic medicine, and are often analyzed with more complicated regression models for which no screening methods have yet been developed. In theory it is not difficult to propose a screening procedure for any given model: fit $p_n$ marginal regressions, one for each covariate, and retain those covariates with the largest marginal estimates, in absolute value. But fitting $p_n$ marginal regressions can still be time-consuming, especially if $p_n$ is very large and the fitting procedure is slow, and theoretical properties such as sure screening must still be studied on a case-by-case basis. In this paper we propose EEScreen, a unified approach to screening which can be used with any statistical model that can be fit using estimating equations. This is convenient because estimating equations are frequently used to analyze the previously mentioned correlated, missing, or censored data situations. EEScreen is also fundamentally different from most other screening procedures in that it only requires evaluating $p_n$ estimating equations at a fixed parameter value, rather than solving for $p_n$ marginal regressione estimates, making it exceedingly computationally convenient. We prove theoretical results about the screening properties of EEScreen that hold for any model that can be fit using U-statistic-based estimating equations. Furthermore, because we can design estimating equations to incorporate more or fewer modeling assumptions, we can use our EEScreen framework to span the range between model-based and model-free screening. In particular, we show that EEScreen can provide a screening method very similar to that of @Zhu2011 when used with a particular estimating equation. This estimating equation actually cannot be used for estimation in practice because it involves unknown parameters, but interestingly can still be used to derive a useful screening procedure. Finally, when covariates are highly correlated, @FanLv2008 suggested an iterative version of their screening procedure, which they found to outperform marginal screening in some cases. In this paper we provide an iterative version of EEScreen (iEEScreen), and we also demonstrate a novel connection between iEEScreen and EEBoost, a recently proposed boosting algorithm for estimation and variable selection in estimating equations [@Wolfson2011]. This connection may provide a means for a theoretical analysis of iterative screening methods, something which so far has been difficult to study. We introduce EEScreen in Section \[sec:eescreen\], where we also give some examples, establish its theoretical properties, and briefly discuss how to choose the number of covariates to retain after screening. We derive a new screening method similar to that of @Zhu2011 in Section \[sec:zhu2011\], and discuss iEEScreen in Section \[sec:ieescreen\]. We conduct a thorough simulation study in Section \[sec:sims\], using two different estimating equations, before applying our methods to analyze an issue in multiple myeloma in Section \[sec:data\]. We conclude with a discussion in Section \[sec:discussion\], and provide proofs in the Appendix. EEScreen: sure screening for estimating equations {#sec:eescreen} ================================================= Method {#sec:method} ------ Let $Y_i=(Y_{i1},\ldots,Y_{iK_i})^T$ be a $K_i\times1$ outcome vector and $\bX_i=(\bX_{i1},\ldots,\bX_{iK_i})^T$ be a $K_i\times p_n$ matrix of covariates for units $i=1,\ldots,n$. Then let $\bY=(\bY_1,\ldots,\bY_n)^T$ be a $\sum_iK_i\times1$ vector and $\bX=(\bX_1^T,\ldots,\bX_n^T)^T$ be a $\sum_iK_i\times p_n$ matrix. Assuming some regression model, we can construct a $p_n\times1$ estimating equation $\bU(\bbeta)$ that depends on $\bY_i$ and $\bX_i$ such that $\E\{\bU(\bbeta_0)\}=\b0$, where $\bbeta_0$ is the true $p_n\times1$ parameter vector. Let the set of true regression parameters $\mathcal{M}=\{j:\beta_{0j}\ne0\}$ have size $\vert\mathcal{M}\vert=s_n$, where $\beta_{0j}$ is the $j^{th}$ component of $\bbeta_0$. It is commonly assumed that $s_n$ is small and fixed or growing slowly. When $p_n<n$, $\bbeta_0$ is estimated by finding the $\hat{\bbeta}$ such that $\bU(\hat{\bbeta})=0$, but when $p_n>n$ there are an infinite number of solutions for $\hat{\bbeta}$, in which case regularized regression is used [@Fu2003; @JohnsonLinZeng2008; @Wolfson2011]. However, when $p_n$ is much greater than $n$, these methods can lose accuracy and be too computationally demanding, hence the need for screening methods to quickly reduce $p_n$. Most previously proposed screening methods proceed by fitting $p_n$ regression models, one covariate at a time, to get $p_n$ marginal estimates $\hat{\alpha}_j$. They then retain the covariates with $\vert\hat{\alpha}_j\vert$ above some threshold. This is akin to conducting $p_n$ Wald tests, though without standardizing the $\hat{\alpha}_j$ by their variances. However, in the case of estimating equations, even this procedure can be time-consuming if $p_n$ is large or $\bU$ is cumbersome to fit. Here, instead of marginal Wald tests, we construct marginal score tests for the $\beta_{0j}$ using $\bU$. To motivate our procedure, we first consider the case where the marginal model is correct for $\beta_{01}$. In other words, $\beta_{01}\ne0$ while $\beta_{0j}=0$ for all $j\ne1$. Then $\E[\bU\{(\beta_{01},0,\ldots,0)\}]=\b0$, so that each component of $\bU$ is a valid estimating equation for $\beta_{01}$. This implies that each component of $\bU(\b0)$ is the numerator of a score test for the null hypothesis $\beta_{01}=0$. If the marginal model is correct for $\beta_{01}$, then to achieve sure screening we must reject the score test. Therefore we use as our screening statistic the component of $\bU(\b0)$ that gives the most powerful test, which we denote $U_1(\b0)$. For each $j$, we can identify the component $U_j(\b0)$ of $\bU(\b0)$ that is most powerful for testing $\beta_{0j}=0$ under the marginal model that $\beta_{0j}$ is the only non-zero parameter. In many situations the first component of $\bU(\b0)$ will be associated with $\beta_{01}$, the second with $\beta_{02}$, and so on. When this is not the case, we can follow the construction above to relabel the components of $\bU(\b0)$ appropriately. We propose using the relabeled $U_j(\b0)$ as surrogate measures of association between the outcome and the $j^{th}$ covariate, after first standardizing the covariates to have equal variances. Instead of just taking the numerators of the score tests we could divide each $U_j(\b0)$ by an estimate of its standard deviation, but this would add computational complexity to our procedure, and even without doing so we will be able to achieve good results and prove finite-sample performance guarantees. One advantage to using score tests is that they do not require parameter estimation and so are more computationally convenient than performing $p_n$ marginal regressions. Furthermore, this framework will also allow us to give a unified treatment of the theoretical results for a large class of estimating equations. Specifically, we propose the following screening procedure: 1. Standardize the $p_n$ covariates to have variance 1. 2. For the $j^{th}$ parameter identify the marginal estimating equations $U_j$ as described above. 3. Set a threshold $\gamma_n$. 4. Retain the parameters $\{j:\vert U_j(\b0)\vert\geq\gamma_n\}$. We will denote the set of retained parameters by $\hat{\mathcal{M}}$. Note that this procedure only requires evaluating $p_n$ estimating equations at $\b0$, which can be computed very quickly. The convenience of score tests, however, comes at the price of ambiguity in the proper treatment of nuisance parameters, such as the intercept term in a regression model. Without loss of generality, let $\beta_{01}$ be the intercept term. We can first fit the intercept without any covariates in the model to get an estimate $\hat{\beta}_{01}$. This only needs to be done once, since $\hat{\beta}_{01}$ will remain the same for each $U_j$. We then screen by evaluating each $U_j$ at $\boldeta=(\hat{\beta}_{01},\b0)$ instead of at $\b0$. Our score test idea was motivated by the EEBoost algorithm [@Wolfson2011], a boosting procedure for estimating equations which uses components of the estimating equation $\bU$ as a surrogate measure of association. We therefore refer to our method as EEScreen, and we will draw more connections between EEScreen and EEBoost in Section \[sec:ieescreen\]. Examples {#sec:examples} -------- Here we provide some examples of EEScreen for various estimating equations, assuming throughout that $\E(\bX_i)=\b0$ and $\var(X_{ij})=1$. For the linear model with $K_i=1$, the usual linear regression score equation is $\bU(\bbeta)=\bX^T(\bY-\bX\bbeta)$, so $\bU(\b0)=\bX^T\bY$. Under the marginal model that $\beta_{0j'}$ is the only non-zero parameter, the $j^{th}$ component of $\E\{\bU(\b0)\}$ equals $\cor(X_{ij},X_{ij'})\beta_{0j'}$, where $X_{ij}$ is the $j^{th}$ component of the $i^{th}$ covariate vector. Clearly this is maximized when $j=j'$ for any value of $\beta_{0j'}$, so the component of $\bU(\b0)$ that gives the most powerful test is $U_{j'}(\b0)$. EEScreen then retains the parameters $\{j:\vert\sum_iX_{ij}Y_i\vert\geq\gamma_n\}$. Note that this is equivalent to the original screening procedure proposed by @FanLv2008. Under the Cox model, when $K_i=1$ with survival outcomes, let $T_i$ be the survival time, $C_i$ the censoring time, $Y_i=\min(T_i,C_i)$, and $\delta_i=I(T_i\leq C_i)$. The Cox model score equation is $$\bU(\bbeta)=\sum^n_{i=1}\int\left\{\bX_i-\frac{\sum^n_{i=1}\bX_i\tilde{Y}_i(x)\exp(\bX_i^T\bbeta)}{\sum^n_{i=1}\tilde{Y}_i(x)\exp(\bX_i^T\bbeta)}\right\}d\tilde{N}_i(x),$$ where $\tilde{N}_i(x)=I(T_i\leq x,\delta_i)$ is the observed failure process and $\tilde{Y}_i(x)=I(Y_i\geq x)$ is the at-risk process. Under the marginal model that $\beta_{0j'}$ is the only non-zero parameter, @GorstRasmussenScheike2011 show that the largest component of the limiting estimating equation evaluated at $\b0$ is found for the $j$ that maximizes $\int\cor\{X_{ij},F(t\mid X_{ij'})\}$, where $F(t\mid X_{ij'})$ is the distribution function of $T_i$, conditional on $X_{ij'}$. Thus the component of $\bU(\b0)$ that gives the most powerful test is again the $j'^{th}$ component. EEScreen then retains the parameters $$\left[j:\left\vert\sum^n_{i=1}\int\left\{X_{ij}-\frac{\sum^n_{i=1}X_{ij}\tilde{Y}_i(x)}{\sum^n_{i=1}\tilde{Y}_i(x)}\right\}d\tilde{N}_i(x)\right\vert\geq\gamma_n\right].$$ This is exactly the screening statistic of @GorstRasmussenScheike2011. This example illustrates the computational advantages that EEScreen can enjoy. @ZhaoLi2012 proposed screening for the Cox model based on fitting marginal Cox regressions, which requires $p_n$ applications of the Newton-Raphson algorithm. In contrast, @GorstRasmussenScheike2011 and EEScreen only require evaluating the $U_j(\b0)$. The ordinary linear model and the Cox model have already been studied in the screening literature, but EEScreen is most useful for models for which no screening procedures exist yet. In Sections \[sec:sims\] we study its performance on two such models: a $t$-year survival model [@Jung1996] and the accelerated failure time model [@Tsiatis1996; @Jin2003]. Theoretical properties {#sec:theory} ---------------------- One advantage of our EEScreen framework is that we can provide very general theoretical guarantees on its screening performance that hold for a large class of models, without needing to study each model on a case-by-case basis. We require three assumptions on the marginal estimating equations $U_j$ to prove that EEScreen has the sure screening property, where the probability that the retained parameters $\hat{\mathcal{M}}$ contains the true parameters $\mathcal{M}$ approaches 1. Let the expected full estimating equations be denoted $\bu(\bbeta)=\E\{\bU(\bbeta)\}$, so that the expected marginal estimating equations are $u_j(\bbeta)$. \[ass:ustat\] Let $\bX_{ij}$ be the $K_i\times1$ vector of the $j^{th}$ covariate for the $i^{th}$ unit. Each estimating equation $U_j$ has the form $$U_j(\bbeta)=\binom{n}{m}^{-1}\sum_{1\leq i_1<\ldots<i_m}h_j\{\bbeta;(\bY_{i_1},\bX_{i_1}),\ldots,(\bY_{i_m},\bX_{i_m})\}$$ for all $j$, where $n\geq m$ and $h_j$ is a real-valued kernel function that depends on $\beta$ and is symmetric in the $(\bY_{i_1},\bX_{i_1}),\ldots,(\bY_{i_m},\bX_{i_m})$. \[ass:bernstein\] There exist some constants $b>0$ and $\Sigma^2>0$ such that for all $j$, $\vert U_j(\b0)-u_j(\b0)\vert\leq b$ and $\var[h_j\{\b0;(\bY_{i_1},\bX_{i_1}),\ldots,(\bY_{i_m},\bX_{i_m})\}]\leq\Sigma^2$. Assumption \[ass:ustat\] requires that each $U_j$ be a U-statistic of order $m$, which encompasses a large number of important estimating equations. Assumption \[ass:bernstein\] amounts to conditions on the moments of the $U_j$, and they can often be satisfied by assuming bounded outcomes and covariates. These conditions are necessary for stating a Bernstein-type inequality for the $U_j$, which gives the probability bounds in Theorems \[thm:surescreening\] and \[thm:size\]. They can therefore be relaxed as long as there exists a similar probability inequality for $U_j$. For example, Bernstein-type inequalities exist for martingales [@vandeGeer1995], which would allow $U_j$ to be the Cox model score equations. \[ass:signal\] There exists some constant $c_1>0$ such that $\min_{j\in\mathcal{M}}\vert u_j(\b0)\vert\geq c_1[n/m]^{-\kappa}$ with $0<\kappa<1/2$, where $m$ is defined in Assumption \[ass:ustat\] and $[n/m]$ is the largest integer less than $n/m$. Assumption \[ass:signal\] is an assumption on the marginal signal strengths of the covariates in $\mathcal{M}$. In EEScreen these signals are quantified by the $u_j(\b0)$, and Assumption \[ass:signal\] requires them to be of at least a certain order so that they are detectable given a sample size $n$. An assumption of this type is always needed in a theoretical analysis of a screening procedure. For example, in the generalized linear model setting, our Assumption \[ass:signal\] is exactly equivalent to the assumption of @FanSong2010 that the magnitude of the covariance between $\E(\bY_i\mid\bX_i)$ and the $j^{th}$ covariate be of order $n^{-\kappa}$. Since EEScreen is similar to conducting $p_n$ score tests, Assumption \[ass:signal\] is similar to requiring that the expected value of the marginal score test statistic for $j\in\mathcal{M}$ be of a certain order. As previously mentioned, we could standardize the screening statistic $\vert u_j(\b0)\vert$ by its variance, in which case the score test analogy would be exact. It is very reasonable to use the marginal score test statistic as a proxy for the marginal association of the covariates. Under these assumptions, we can show that EEScreen possesses the sure screening property. \[thm:surescreening\] Under Assumptions \[ass:ustat\]–\[ass:signal\], if $\gamma_n=c_1[n/m]^{-\kappa}/2$ for $0<\kappa<1/2$, with $m$ defined in Assumption \[ass:ustat\], then $$\pr(\mathcal{M}\subseteq\hat{\mathcal{M}}) \geq 1-2s_n\exp\left\{-\frac{c_1^2[n/m]^{1-2\kappa}/4}{2\Sigma^2+bc_1[n/m]^{-\kappa}/3}\right\},$$ with $\Sigma^2$ and $b$ defined in Assumption \[ass:bernstein\]. Theorem \[thm:surescreening\] guarantees that all important covariates will be retained by EEScreen with high probability. Similar to previous work, we find that this probability bound depends only on $s_n$ and not on $p_n$. The bound also depends on $m$, the order of the U-statistic, so that EEScreen may not perform as well for larger $m$. Theorem \[thm:surescreening\] is almost an immediate consequence of properties of U-statistics, and the simplicity of the proof is due to the fact that EEScreen uses score tests instead of Wald tests. We therefore do not need to estimate any parameters, nor prove probability inequalities for those estimates, which is a major source of technical difficulty in previous work on screening. Theorem \[thm:surescreening\] is most useful if the size of the $\hat{\mathcal{M}}$ produced by EEScreen is small. In other words, we hope that $\hat{\mathcal{M}}$ does not contain too many false positives. With two more assumptions, we can provide a bound on $\vert\hat{\mathcal{M}}\vert$ that holds with high probability. \[ass:inf\] The expected full estimating equation $\bu(\bbeta)$ is differentiable with respect to $\bbeta$. Let the negative Jacobian $-\partial\bu/\partial\bbeta$ be denoted $\bi(\bbeta)$. \[ass:beta0\] There exists some constant $c_2>0$ such that $\Vert\bbeta_0\Vert_2\leq c_2$. Assumption \[ass:inf\] can hold even if the sample estimating equation $\bU$ is nondifferentiable. Assumption \[ass:beta0\] merely requires that there exist an upper bound on the size of the true $\bbeta_0$ that does not grow with $n$, which is a reasonable condition. \[thm:size\] Under Assumptions \[ass:ustat\]–\[ass:beta0\], if $\gamma_n=c_1[n/m]^{-\kappa}/2$ as in Theorem \[thm:surescreening\], then $$\pr\left[\vert\hat{\mathcal{M}}\vert\leq \frac{16c_2^2\sigma_{\max}^{*2}}{c_1^2[n/m]^{-2\kappa}}\right] \geq 1-2p_n\exp\left\{-\frac{c_1^2[n/m]^{1-2\kappa}/16}{2\Sigma^2+bc_1[n/m]^{-\kappa}/6}\right\},$$ where $\Sigma^2$ and $b$ are defined in Assumption \[ass:bernstein\] and $\sigma_{\max}^*=\sup_{0<t<1}\sigma_{\max}\{\bi(t\bbeta_0)\}$, where $\sigma_{\max}(\bA)$ denotes the largest singular value of the matrix $\bA$. Like Theorem \[thm:surescreening\], Theorem \[thm:size\] is also almost a simple consequence of properties of U-statistics. Theorem \[thm:size\] provides a finite-sample probability bound on $\vert\hat{\mathcal{M}}\vert$, but asymptotically we would need assumptions on $\bi(\bbeta^*)$ to guarantee that $\sigma_{\max}^*$ will not increase too quickly. In particular, if $\sigma_{\max}^*$ increased only polynomially in $n$, $\vert\hat{\mathcal{M}}\vert$ would increase polynomially. At the same time, the probability that the bound holds tends to one even if $\log p_n=o([n/m]^{1-2\kappa})$, so the false positive rate would decrease quickly to zero with probability approaching one even in ultra-high dimensions. A similar phenomenon was found by @FanFengSong2011. The presence of $\sigma_{\max}^*$ in Theorem \[thm:size\] reflects the dependence of $\vert\hat{\mathcal{M}}\vert$ on the degree of collinearity of our data. For general estimating equations, collinearity not only depends on the design matrix, but also varies across the parameter space. For example, @MackinnonPuterman1989 and @LesaffreMarx1993 showed that generalized linear models can be collinear even if their design matrices are not, and vice versa. In our situation, we are concerned with collinearity along the line segment between $\bbeta_0$ and $\b0$. Note that because $\sigma_{\max}^*$ depends only on $\bi$, $\bbeta_0$, and $\b0$, which are all nonrandom quantities, $\sigma_{\max}^*$ is nonrandom as well. Choosing $\gamma_n$ {#sec:gamma} ------------------- Theorems \[thm:surescreening\] and \[thm:size\] specify optimal rates for $\gamma_n$, and a number of methods have been proposed for choosing $\gamma_n$ in practice. @FanLv2008 suggested choosing $\gamma_n$ such that $\vert\hat{\mathcal{M}}\vert=n-1$ or $n/\log n$. Because these values are hard to interpret, @ZhaoLi2012 showed that $\gamma_n$ is related to the expected false positive rate of screening. @Zhu2011 also recently proposed a thresholding method based on adding artificial auxiliary variables, and provided a bound relating the number of added variables to the probability of including an unimportant covariate. These methods offer more interpretable ways of choosing how many covariates to retain with EEScreen. A related strategy is to set a desired false discovery rate. @BuneaWegkampAuguste2006 showed that FDR methods can achieve the sure screening property in the ordinary linear model, and @Sarkar2004 proposed an FDR method than can also control the false negative rate. It would be interesting to pursue this type of idea for EEScreen. In practice, however, we are often concerned with the prediction error of the estimator obtained by fitting a regularized regression method after EEScreen. If we used the methods above we would still need to choose a false positive rate or false discovery rate, but so far it is not clear what choices would give optimal prediction. In this case another option is to retain different numbers of covariates, fit the regularized regression for each screened model $\hat{\mathcal{M}}$, and select the $\hat{\mathcal{M}}$ that gives the lowest cross-validated estimate of prediction error. This is the approach we take in Section \[sec:data\], where we use EEScreen to analyze data from a multiple myeloma clinical trail. Model-free screening {#sec:zhu2011} ==================== @Zhu2011 recently proposed a screening statistic that can achieve sure screening for any single-index model. Specifically, for a completely observed response $\tilde{Y}_i$ and a $p$-dimensional covariate vector $\bX_i$, they assumed that $F(y\mid\bX_i)=F_0(y\mid\bX_i^T\bbeta_0)$, where $F(y\mid\bX_i)=\pr(\tilde{Y}_i<y\mid\bX_i)$ and $F_0$ is some distribution function that depends on $\bX_i$ only through the index $\bX_i^T\bbeta_0$, so that $j\in\mathcal{M}$ if and only if $\beta_{0j}\ne0$. This is a very mild assumption that holds for a large class of models, making the screening method of @Zhu2011 almost model-free. To simplify things, they assumed that $\E(\bX_i)=\b0$ and $\var(\bX_i)=\bI_{p_n}$, where $\bI_{p_n}$ is the $p_n\times p_n$ identity matrix. They quantified the marginal relationship between the covariates and an outcome $y$ by using the novel statistic $$\bOmega(y)=\E\{\bX_iF(y\mid\bX_i)\}=\cov\{\bX_i,F(y\mid\bX_i)\}=\cov\{\bX_i,I(\tilde{Y}_i<y)\}.$$ Intuitively, the covariance between $X_{ij}$ and $F(\tilde{Y}_i\mid\bX_i)$, where $X_{ij}$ is the $j^{th}$ component of $\bX_i$, should be large in magnitude if $j\in\mathcal{M}$. They therefore used $\omega_j=\E\{\Omega_j(\tilde{Y}_i)^2\}$ as a measure of marginal association, where $\Omega_j(y)$ is the $j^{th}$ component of $\bOmega(y)$, leading to the screening statistic $$\tilde{\omega}_j=n^{-1}\sum_{k=1}^n\left\{n^{-1}\sum_{i=1}^nX_{ij}I(\tilde{Y}_i<\tilde{Y}_k)\right\}^2.$$ This derivation of the screening procedure of @Zhu2011 makes no mention of estimation of $\bbeta_0$, making it seemingly irreconcilable with our EEScreen, which requires an estimating equation. However, we can actually show that EEScreen, combined with a particular estimating equation, leads to a very similar screening procedure. This further illustrates the flexibility and wide applicability of our proposed screening strategy. Note that conditional on $\bX_i$ and $\bX_k$, $F_0(\tilde{Y}_i\mid\bX_i^T\bbeta_0)$ and $F_0(\tilde{Y}_k\mid\bX_k^T\bbeta_0)$ are independent and identically distributed uniform random variables. Therefore, we know that $$\begin{aligned} &\pr\left\{F_0(\tilde{Y}_i\mid\bX_i^T\bbeta_0)<F_0(\tilde{Y}_k\mid\bX_k^T\bbeta_0)\right\}=\\ &\E\left[\pr\left\{F_0(\tilde{Y}_i\mid\bX_i^T\bbeta_0)<F_0(\tilde{Y}_k\mid\bX_k^T\bbeta_0)\mid\bX_i,\bX_k\right\}\right]= \frac{1}{2}.\end{aligned}$$ This fact can be used to construct the marginal estimating equations. Consider $$\label{eq:zhuesteq} \bU(\bbeta)=n^{-2}\sum_{k=1}^n\sum_{i=1}^n\bX_i\left[I\{F_0(\tilde{Y}_i\mid\bX_i^T\bbeta)<F_0(\tilde{Y}_k\mid\bX_k^T\bbeta)\}-\frac{1}{2}\right].$$ Since $\E\{\bU(\bbeta_0)\}=\b0$, (\[eq:zhuesteq\]) is an unbiased estimating equation for $\bbeta_0$. Furthermore, it is a U-statistic of order $m=2$, which is covered by the framework of Section \[sec:theory\]. It is important to note that (\[eq:zhuesteq\]) cannot be implemented in practice, because the functional form of $F_0(y\mid\bX^T\bbeta)$ is unknown, yet it is still useful for constructing a screening procedure. Recall that EEScreen uses the statistic $\bU(\b0)$, and for (\[eq:zhuesteq\]), $$\begin{aligned} \bU(\b0) &= n^{-2}\sum_{k=1}^n\sum_{i=1}^n\bX_i\left[I\{F_0(\tilde{Y}_i\mid\bX_i^T\b0)<F_0(\tilde{Y}_k\mid\bX_k^T\b0)\}-\frac{1}{2}\right]\\ &= n^{-2}\sum_{k=1}^n\sum_{i=1}^n\bX_i\left\{I(\tilde{Y}_i<\tilde{Y}_k)-\frac{1}{2}\right\},\end{aligned}$$ because $F_0(y\mid\bX_i^T\b0)=F_0(y\mid\bX_k^T\b0)=F_0(y\mid\b0)$, which is a monotonic function since $F_0$ is a distribution function. Under the marginal model that $\beta_{0j'}$ is the only non-zero parameter, the $j^{th}$ component of $\E\{\bU(\b0)\}$ is $\cor\{X_{ij},F(\tilde{Y}_i\mid X_{ij'})\}$. Thus the $j'^{th}$ component of $\bU(\b0)$ gives the most powerful score test, so EEScreen with (\[eq:zhuesteq\]) retains parameters $$\left[j:\left\vert n^{-2}\sum_{k=1}^n\sum_{i=1}^nX_{ij}\left\{I(\tilde{Y}_i<\tilde{Y}_k)-\frac{1}{2}\right\}\right\vert\geq\gamma_n\right],$$ or equivalently, $$\left\{j:\left\vert n^{-2}\sum_{k=1}^n\sum_{i=1}^nX_{ij}I(\tilde{Y}_i<\tilde{Y}_k)\right\vert\geq\gamma_n\right\},$$ because the $\bX_i$ are standardized to have mean $\b0$. In the notation of @Zhu2011, this is equivalent to using $\vert E\{\Omega_j(\tilde{Y}_i)\}\vert$ as the screening statistic for the $j^{th}$ covariate, rather than $\E\{\Omega_j(\tilde{Y}_i)^2\}$. The $\tilde{Y}_i$ may not be fully observed in the presence of censoring. If $C_i$ are the censoring times, let $Y_i=\min(\tilde{Y}_i,C_i)$ and $\delta_i=I(\tilde{Y}_i\leq C_i)$. Then if we assume that the $C_i$ are independent of the $\tilde{Y}_i$ and $\bX_i$, we can see that $$\begin{aligned} \E\left\{\frac{\delta_iI(Y_i<Y_k)}{S_C^2(Y_i)}\bigg\vert\bX_i,\bX_k\right\} &= \E\left[\E\left\{\frac{I(\tilde{Y}_i\leq C_i)I(\tilde{Y}_i\leq C_k)I(\tilde{Y}_i\leq\tilde{Y}_k)}{S_C^2(\tilde{Y}_i)}\bigg\vert\tilde{Y}_i,\bX_i,\bX_k\right\}\right]\\ &= \E\left[\E\left\{\frac{S_C^2(\tilde{Y}_k)I(\tilde{Y}_i\leq\tilde{Y}_k)}{S_C^2(\tilde{Y}_i)}\bigg\vert\tilde{Y}_i,\bX_i,\bX_k\right\}\right]\\ &= \E\{I(\tilde{Y}_k<\tilde{Y}_k)\mid\bX_i,\bX_k\},\end{aligned}$$ where $S_C$ is the survival function of the $C_i$. If the support of the $C_i$ is less than that of the $\tilde{Y}_i$, the $S_C(Y_i)$ term above could equal 0 for some $Y_i$. Thus this method of accommodating censoring could cause difficulty if it were used in the estimating equation (\[eq:zhuesteq\]) and could lead to inconsistent estimation of $\bbeta_0$ [@FineYingWei1998]. For simplicity, we will assume here that the support of $C_i$ is greater than or equal to that of $\tilde{Y}_i$. This then suggests that in the presence of censoring, the screening statistic of @Zhu2011 should become $$n^{-1}\sum_{k=1}^n\left\{n^{-1}\sum_{i=1}^nX_{ij}\frac{\delta_iI(Y_i<Y_k)}{\hat{S}_C^2(Y_i)}\right\}^2,$$ and the screening statistic derived using EEScreen should become $$\label{eq:zhueescreen} \left\vert n^{-2}\sum_{k=1}^n\sum_{i=1}^nX_{ij}\frac{\delta_iI(Y_i<Y_k)}{\hat{S}_C^2(Y_i)}\right\vert,$$ where $\hat{S}_C$ is the Kaplan-Meier estimate of $S_C$. This illustrates that the EEScreen framework is flexible enough to allow us to derive something similar to the approach of @Zhu2011, which was originally motivated by very different considerations. It also suggests that EEScreen can provide a sensible screening procedure for a particular model, such as the single-index model, even if the associated estimating equation (\[eq:zhuesteq\]) is not implementable in practice. iEEScreen {#sec:ieescreen} ========= Though the simplicity of EEScreen and related screening procedures is appealing, if the covariates are highly correlated, then in finite samples these univariate screening methods may not be able to achieve sure screening without incurring a large number of false positives. To address this issue, @FanLv2008 and @FanSamworthWu2009 proposed iterative screening, where the general idea is as follows. Below, $\mathcal{M}_l$ and $\mathcal{A}_l$ denote sets of covariate indices. In other words, $\mathcal{M}_l,\mathcal{A}_l\subseteq\{1,\ldots,p_n\}$. 1. Set $\mathcal{M}_0$ to be the empty set. 2. For $l=1:L$, 1. controlling for the variables in $\mathcal{M}_{l-1}$, screen the remaining covariates 2. select a set $\mathcal{A}_l$ of the most important of these covariates 3. use a multivariate variable selection method, such as lasso or SCAD, on the covariates in $\mathcal{M}_{l-1}\cup\mathcal{A}_l$ to get a reduced set $\mathcal{M}_l$ We can adapt these ideas to develop an iterative version of EEScreen, which we will call iEEScreen. However, to operationalize iEEScreen and iterative screening algorithms in general, we must first specify a number of parameters, such as how large $\vert\mathcal{A}_l\vert$ and $\vert\mathcal{M}_l\vert$ should be, what multivariate variable selection procedure to use, and how many iterations to run. @FanFengSong2011 recommended choosing the $\mathcal{A}_l$ using a permutation-based procedure, and the $\mathcal{M}_l$ using a SCAD-type variable selector [@FanLi2001] with cross-validation. Their iterations stop when either $\vert\mathcal{M}_l\vert>\vert\mathcal{A}_1\vert$, or $\mathcal{M}_l=\mathcal{M}_{l-1}$. These are sensible choices, but the many different layers of this procedure make it difficult to analyze. Instead, here we will show that the EEBoost method of @Wolfson2011, viewed as a variable selector rather than an estimation procedure, can actually be thought of as a version of iEEScreen. By linking iterative screening and boosting, we embed iEEScreen in the theoretical framework already developed for EEBoost and other boosting methods. In the future, this theoretical framework could in turn be applied to analyze the properties of iterative screening. We first briefly describe the EEBoost algorithm [@Wolfson2011]. For some small $\epsilon>0$ and the full estimating equation $\bU$, 1. Set $\bbeta^{(0)}=\b0$. 2. For $t=1:T$, 1. compute $\bDelta=\vert\bU(\bbeta^{(t-1)})\vert$ 2. identify $j_t=\argmax_j\Delta_j$, where $\Delta_j$ is the $j^{th}$ component of $\bDelta$ 3. set $\beta^{(t)}_{j_t}=\beta^{(t-1)}_{j_t}-\epsilon\cdot\sign(\Delta_{j_t})$, where $\beta^{(t)}_{j_t}$ is the $j_t^{th}$ component of $\bbeta^{(t)}$ Here, $T$ serves as the regularization parameter, and for a given $T$ only a certain number of $\bbeta^{(t)}_{j_t}$ will have been updated from their initial values of zero, effecting variable selection. @Wolfson2011 recommends choosing $\epsilon$ in the range \[0.001,0.05\], and $T$ can be chosen with some tuning procedure. To express EEBoost as an iterative version of EEScreen, note that at the beginning of EEBoost, $\Delta_j$ corresponds to the screening statistic $\vert U_j(0)\vert$ used in EEScreen. Evaluating $\bU$ at subsequent $\bbeta^{(t-1)}$ is a way of controlling for the variables that have already been selected into the model by EEBoost, which is step 2(a) of iterative screening. In particular, for $i=0,1,\ldots$ define $t_i$ such that $\Vert\bbeta^{(t_i)}\Vert_0\ne\Vert\bbeta^{(t_i+1)}\Vert_0$. In other words, $t_0$ is the first time that the number of nonzero components of $\bbeta^{(t)}$ changes, $t_1$ is the second time this happens, and so on. Then looking back at the iterative screening algorithm, for $l=1,\ldots,L$ we can identify $\mathcal{M}_{l-1}$ to be $\{j:\beta^{(t_{l-1})}_j\ne0\}$, $\mathcal{A}_l$ to be $\{j_{t_l}\}$, and $\mathcal{M}_l$ as being obtained by running EEBoost for $t_l$ iterations starting from the covariates in $\mathcal{M}_{t_l-1}\cup\mathcal{A}_l$. We can choose $L$ by tuning EEBoost with a generalized cross-validation-type criterion. We will thus implement iEEScreen using the EEBoost algorithm. In the remainder of this paper we study the effects of using EEScreen and iEEScreen as preprocessing steps before fitting regularized regression models. In particular, we will use EEBoost to fit the regressions, for two reasons. First, we would like to compare the effects of retaining different numbers of covariates after screening, from keeping only one or two covariates to keeping tens of thousands. Therefore we require a regularization method for estimating equations that can handle an arbitrarily large number of covariates. Second, in Section \[sec:aftsim\] we study a discrete estimating equation, so we require a regularization method which works well in that situation. To our knowledge, EEBoost is the only procedure that meets both of these criteria. However, this leads to a unique problem. We would naturally like to compare the effects of using EEScreen versus iEEScreen. But a careful inspection of the EEBoost algorithm reveals that running EEBoost twice, in other words first selecting covariates using EEBoost, and then using only those covariates in another instance of EEBoost, is actually identical to using EEBoost only once. This means that screening with the version of iEEScreen described in this section has no effect if EEBoost is then used for model-fitting. This behavior is different from, say, the lasso, where running two iterations of the lasso has been termed the relaxed lasso [@Meinshausen2007] and can give different results from the regular lasso. Therefore while we will be able to compare the variable selection properties of EEScreen and iEEScreen in simulations, where we will know the true model, we will not be able to compare EEScreen+EEBoost versus iEEScreen+EEBoost. We would like to address this issue in future work. Simulations {#sec:sims} =========== In our simulation studies, we evaluated the performances of EEScreen and iEEScreen with two different estimating equations, one for a $t$-year survival model and the other for an accelerated failure time model. We implemented iEEScreen by using EEBoost, as described in Section \[sec:ieescreen\], with $\epsilon=0.01$. We compared these to the naive approach of fitting $p_n$ marginal regressions, as well as to the method of @Zhu2011 and our EEScreen-derived method (\[eq:zhueescreen\]) from Section \[sec:zhu2011\]. We studied $p_n=20000$ covariates and set the true parameter vector $\bbeta_0$ to be such that $\beta_{0j}=1.5,j=1,\ldots,10$, $\beta_{0j}=-0.8,j=11,\ldots,20$, and $\beta_{0j}=0,j=21,\ldots,p_n$. We generated covariates $\bX_i$ from a $p_n$-dimensional zero-mean multivariate normal. To simulate an easy setting we used a covariance matrix that satisfied the partial orthogonality condition of @FanSong2010, where the important covariates were independent of the unimportant covariates. The covariance matrix consisted of 9 blocks of 10 covariates, 1 block of 910 covariates, and 19 blocks of 1000 covariates. Each block had a compound symmetry structure with the same correlation parameter $\rho$, which was equal to either 0.5 or 0.9, and the blocks were independent from each other. We matched the non-zero components of $\bbeta_0$ with two of the 10-dimensional blocks. To simulate a more difficult setting we let the entire covariance matrix have a compound symmetry structure with $\rho$ equal to either 0.3 or 0.5. The $t$-year survival model {#sec:tyearsim} --------------------------- We first considered a $t$-year survival model. Let $T_i$ and $\bX_i$ be the survival time and the covariate vector of the $i^{th}$ patient, respectively. We modeled the probability of surviving beyond some time $t_0$ conditional on covariates as $$\logit\{\pr(T_i\geq t_0\mid\bX_i)\}=\bX_i^T\bbeta_0.$$ This model is very useful in clinical investigations, and in fact we apply it to data from clinical trials of multiple myeloma therapies in Section \[sec:data\]. However, we cannot use the logistic regression because the $T_i$ are not directly observed. Let $C_i$ be the censoring time, such that we only observe $Y_i=\min(T_i,C_i)$ and $\delta_i=I(T_i\leq C_i)$. Without modeling the $C_i$, it is difficult to specify a full likelihood model for this data, so we instead turn to estimating equations. To account for the censored data, @Jung1996 assumed that the $C_i$ were independent of the $T_i$ and the $\bX_i$ and proposed using the estimating equation $$\label{eq:tyear} \bU(\bbeta) = n^{-1}\sum_{i=1}^n\frac{\bX_i\pi'(\bX_i^T\bbeta)}{\pi(\bX_i^T\bbeta)\{1-\pi(\bX_i^T\bbeta)\}}\left\{\frac{I(Y_i\geq t_0)}{\hat{S}_C(t_0)}-\pi(\bX_i^T\bbeta)\right\},$$ where $\pi(\eta)=\logit^{-1}(\eta)$, $\pi'(\eta)=\partial\pi/\partial\eta$, and $\hat{S}_C(t)$ is the Kaplan-Meier estimate of the survival function of the $C_i$. According to our procedure, after some simplification we see that EEScreen will retain the parameters $$\left[j:\lAbs\sum_{i=1}^nX_{ij}\frac{I(Y_i\geq t_0)}{\hat{S}_C(t_0)}\rAbs\geq\gamma_n\right]$$ Though $U_j$ does not satisfy Assumption \[ass:ustat\] because of the $\hat{S}_C(t)$ term, @Jung1996 showed that it can be written in the appropriate form, plus a negligible $o_P(1)$ term. To fit the $p_n$ regressions for the marginal screening method we used a simple Newton-Raphson procedure to solve $U_j$. Tuning EEBoost and iEEScreen was difficult because commonly used criteria such as AIC or BIC are not defined in the absence of a likelihood. We instead chose to minimize the GCV-type criterion $\widehat{BS}/(1-n^{-1}\Vert\hat{\bbeta}\Vert_0)^2$, where $\Vert\hat{\bbeta}\Vert_0$ is the number of nonzero components of $\hat{\bbeta}$, and $\widehat{BS}$ is the estimate of the Brier score at $t_0$. If $\hat{\pi}(t_0\mid\bX_i)$ is the predicted survival probability of patient $i$ at $t_0$, then $\widehat{BS}$ is defined by @Graf1999 as $$\widehat{BS}=n^{-1}\sum_i\left[\frac{\{0-\hat{\pi}(t_0\mid\bX_i)\}^2}{\hat{S}_C(X_i)}I(Y_i\leq t_0,\delta_i=1)+\frac{\{1-\hat{\pi}(t_0\mid\bX_i)\}^2}{\hat{S}_C(t_0)}I(Y_i\geq t_0)\right].$$ We generated survival times for $n=100$ subjects from $\log(T_i)=\bX_i^T\bbeta_0+\varepsilon_i$ with $\varepsilon_i$ having a logistic distribution with mean -0.5 and scale 1. Under this scheme the model of @Jung1996 is correctly specified. We generated $C_i$ from an exponential distribution to give approximately 50% censoring. We observed that the $20^{th}$ percentile of the simulated survival times was roughly $t_0=0.005$, so we used this $t_0$ when implementing the estimating equation. We simulated 200 such datasets. ------------------- ------------------- ----------------- ------------------ ----------------- $\rho=0.5$ $\rho=0.9$ $\rho=0.3$ $\rho=0.5$ EEScreen 2849 (6180) 22 (249.5) 19666.5 (610.5) 19676 (559.5) Marginal 2908 (6278) 22 (228.75) 19659 (611.5) 19696 (550.5) Zhu et al. (2011) 9614.5 (9497.75) 2043.5 (7687) 19647.5 (655.5) 19531.5 (737) Method (2) 7559.5 (11737.75) 944.5 (4121.25) 19614.5 (716.75) 19545.5 (726.5) ------------------- ------------------- ----------------- ------------------ ----------------- : \[tab:tyearmms\]Median minimum model size (interquartile range) for the $t$-year survival model ------------------- ---------------- ------------------- ------------------- ------------------ $\rho=0.5$ $\rho=0.9$ $\rho=0.3$ $\rho=0.5$ EEScreen 1.29 (0.09) 1.38 (0.47) 1.38 (0.36) 1.32 (0.16) Marginal 617.79 (61.99) 1023.79 (1405.58) 1608.09 (2594.27) 1054.86 (252.32) Zhu et al. (2011) 1.52 (0.08) 1.58 (0.45) 1.88 (4.47) 1.49 (0.2) Method (2) 1.54 (0.09) 1.58 (0.45) 2.13 (8.02) 1.48 (0.18) ------------------- ---------------- ------------------- ------------------- ------------------ : \[tab:tyeartiming\]Average runtime in seconds (standard deviation) for the $t$-year survival model Table \[tab:tyearmms\] reports the median sizes of the smallest models $\hat{\mathcal{M}}$ found by the different screening methods that still contained the true model $\mathcal{M}$. The performance is best under the partial orthogonality setting when $\rho=0.9$, which is not surprising because this setting leads to the greatest separation between the important and unimportant covariates. EEScreen and marginal screening show similar performances, while our method (\[eq:zhueescreen\]) appears to actually outperform the method of @Zhu2011 in the partial orthogonality setting. Though EEScreen and marginal screening produce similar results, Table \[tab:tyeartiming\] shows that marginal screening, at least for this $t$-year survival model, can take much longer. These simulations were run on the Orchestra cluster supported by the Harvard Medical School Research Information Technology Group, on machines with 3.6 GHz Intel Xeon processors with at least 12GB of memory, and marginal screening took at least 10 minutes. On the other hand, the EEScreen-type methods and the method of @Zhu2011 were completed in a few seconds, showing the EEScreen can be much more computationally efficient than standard screening methods. ![\[fig:tyearROCs\]Screening performances for the $t$-year survival model](tyearROCs.eps) To better understand the performances of these various screening methods, we studied in Figure \[fig:tyearROCs\] the average number of false positives corresponding to a given number of false negatives achieved by the screened model $\hat{\mathcal{M}}$. We again see that the methods perform best in the partial orthogonality setting when the correlation is high. Furthermore, given the same setting, EEScreen performs better than the model-free methods. This is most likely because the model used by EEScreen is correctly specified, and thus should be more powerful than the model-free methods. This type of phenomenon was also pointed out by @Zhu2011. As in Table \[tab:tyearmms\], our method (\[eq:zhueescreen\]) again appears to outperform that of @Zhu2011. Figure \[fig:tyearROCs\] also shows that in all cases, the variable selection performance of iEEScreen far outperforms the other methods, particularly in the compound symmetry setting. However, we found that iEEScreen is not able to include all of the important covariates. In the partial orthogonality setting, it can only include up to 17 or 18 of the important covariates, while in the compound symmetry setting it cannot achieve fewer than 15 false negatives. It turns out that the boosting procedure we use to implement iEEScreen saturates at some point in its fitting, perhaps due to the fact that there are more parameters than covariates, or perhaps because our choice for the boosting parameter $\epsilon=0.01$ might be too large. ![\[fig:tyearMSEs\]Mean squared errors for the $t$-year survival model](tyearMSEs.eps) Next we studied the effect on estimation accuracy of using screening before fitting a regularized regression model with EEBoost. Figure \[fig:tyearMSEs\] reports the average mean squared error of estimation (MSE) as a function of $\vert\hat{\mathcal{M}}\vert$, the number of variables kept after screening. Here we defined MSE as $\Vert\hat{\bbeta}-\bbeta_0\Vert^2_2$, where $\hat{\bbeta}$ is the estimate obtained by EEBoost after screening. It is clear that using EEScreen first can improve the estimation accuracy of EEBoost, especially in the compound symmetry setting. Screening with the model-free methods does not appear to reduce the MSE, perhaps because they need to retain a large number of covariates before they include the important variables (Table \[tab:tyearmms\]). ![\[fig:tyearPEs\]Out-of-sample AUCs for the $t$-year survival model](tyearPEs.eps) On the other hand, estimation error is not so meaningful in the absence of a correctly specified model. We therefore considered the out-of-sample predictive ability, as measured by the AUC statistic [@Uno2007] at time $t_0$, of the models fit by EEBoost after screening in Figure \[fig:tyearPEs\]. In the partial orthogonality settings, using EEScreen first does not appear to have much of an effect on the AUC, while in the compound symmetry setting it does improve the predictive ability of the subsequent fitted model. Our model-free method (\[eq:zhueescreen\]) does not seem to have much of an effect on AUC in either setting, but appears to perform slightly better than the method of @Zhu2011. The accelerated failure time model {#sec:aftsim} ---------------------------------- The $t$-year survival model is useful when we are interested in a fixed event time. To study the entire survival distribution, one useful approach is the accelerated failure time (AFT) model, which posits that $$\log(T_i)=\bX_i^T\bbeta_0+\varepsilon_i,$$ where the $\varepsilon_i$ are independent and identically distributed, and the $\varepsilon_i$ can have an arbitrary distribution. The $\bbeta$ can be estimated using the U-statistic-based estimating equation $$\label{eq:aft} \bU(\bbeta)=n^{-2}\sum_{i=1}^n\sum_{k=1}^n(\bX_k-\bX_i)I\{e_i(\bbeta)\leq e_k(\bbeta)\}\delta_i,$$ where $e_i(\bbeta)=\log(Y_i)-\bX_i^T\bbeta$ [@Tsiatis1996; @Jin2003; @CaiHuangTian2009]. Following our procedure, after some simplification we see that EEScreen will retain the parameters $$\left\{j:\lAbs\sum_{i=1}^n\sum_{k=1}^n(X_{kj}-X_{ij})I(Y_i\leq Y_k)\delta_i\rAbs\geq\gamma_n\right\}.$$ This is a U-statistic of order $m=2$ and therefore satisfies our assumptions in Section \[sec:theory\]. Despite being a discrete estimating equation, (\[eq:aft\]) poses no additional problems to EEScreen or iEEScreen. To fit the $p_n$ regressions for the marginal screening method we used the method of @Jin2003, available in the R package `lss`. To tune EEBoost and iEEScreen, consider the function $$\label{eq:aftobj} L(\bbeta)=n^{-2}\sum_{i=1}^n\sum_{j=1}^n\{e_j(\bbeta)-e_i(\bbeta)\}I\{e_i(\bbeta)\leq e_j(\bbeta)\}\delta_i.$$ @CaiHuangTian2009, in their work on regularized estimation for the AFT model, argued that $L(\bbeta)$ is an adequate measure of the accuracy of estimation. They and @Jin2003 also noted that $\bU(\bbeta)$ is the “quasiderivative” of $-L(\bbeta)$. For these reasons, we tuned EEBoost by minimizing the GCV-type criterion $$L(\hat{\bbeta})/(1-n^{-1}\Vert\hat{\bbeta}\Vert_0)^2,$$ where we used $L(\bbeta)$ in place of a negative log-likelihood. We generated $n=100$ survival times from $\log(T_i)=\bX_i^T\bbeta_0+\varepsilon_i$ with $\varepsilon_i$ having a standard normal distribution. We generated $C_i$ independently from an exponential distribution to give approximately 50% censoring, and we simulated 200 datasets. ----------------------------- ------------------- ---------------- ------------------ ------------------ $\rho=0.5$ $\rho=0.9$ $\rho=0.3$ $\rho=0.5$ EEScreen 997 (2968.75) 20 (2) 19829.5 (316.25) 19822.5 (401.25) Marginal 1750.5 (3742.25) 21 (144) 19835 (353) 19764 (436) Zhu et al. (2011) 10761.5 (9416) 747 (3804.5) 19482 (854) 19464.5 (922.5) Method (\[eq:zhueescreen\]) 7940.5 (11962.75) 282.5 (2230.5) 19501.5 (800.25) 19522 (785.75) ----------------------------- ------------------- ---------------- ------------------ ------------------ : \[tab:aftmms\]Median minimum model size (interquartile range) for the AFT model ----------------------------- ----------------- --------------- ------------------ ------------------ $\rho=0.5$ $\rho=0.9$ $\rho=0.3$ $\rho=0.5$ EEScreen 1.58 (0.15) 1.53 (0.1) 1.51 (0.1) 1.51 (0.11) Marginal 1024.71 (114.2) 971.85 (82.5) 1081.56 (149.64) 1203.19 (106.81) Zhu et al. (2011) 1.6 (0.16) 1.46 (0.11) 1.44 (0.1) 1.46 (0.11) Method (\[eq:zhueescreen\]) 1.6 (0.15) 1.46 (0.11) 1.44 (0.09) 1.45 (0.11) ----------------------------- ----------------- --------------- ------------------ ------------------ : \[tab:afttiming\]Average runtime in seconds (standard deviation) for the AFT model We report for the different screening methods the smallest $\hat{\mathcal{M}}$ that still contained $\mathcal{M}$ in Table \[tab:aftmms\]. As with the $t$-year survival model, the methods perform best in the partial orthogonality setting with $\rho=0.9$. We also again see that our method (\[eq:zhueescreen\]) outperforms the method of @Zhu2011. In addition, Table \[tab:afttiming\] shows that marginal screening is much more time-consuming than the EEScreen-based methods or the procedure of @Zhu2011. ![\[fig:aftROCs\]Screening performances for the AFT model](aftROCs.eps) Figure \[fig:aftROCs\] reports the average number of false positives contained in $\hat{\mathcal{M}}$ as the number of allowed false negatives is varied. As in the $t$-year survival model simulations, iEEScreen performs better than non-iterative EEScreen, though in the compound symmetry case it also saturates before it can select all of the important covariates. We also see that the EEScreen outperforms the model-free methods again, and that our method (\[eq:zhueescreen\]) somewhat outperforms the method of @Zhu2011. The plots in Figure \[fig:aftROCs\] for the model-free methods look very similar to the corresponding ones in Figure \[fig:tyearROCs\], and this is because the models used to generate both survival times were both AFT models, differing only in the distributions of the error terms. ![\[fig:aftMSEs\]Mean squared errors for the AFT model](aftMSEs.eps) The average mean square errors of the models fit after screening are plotted in Figure \[fig:aftMSEs\]. Similar to the results for the $t$-year survival model, we see that screening using model-free methods does not improve the estimation accuracy of the subsequent regularized regression fit. Interestingly, for the AFT model it appears that screening with EEScreen only barely decreases the MSE under partial orthogonality, and is actually detrimental to the MSE in the compound symmetry setting, in contrast to the results for the $t$-year survival model. ![\[fig:aftPEs\]Out-of-sample C-statistics for the AFT model](aftPEs.eps) We see something similar when we examine the out-of-sample predictive abilities of the models fit by EEBoost after screening. We calculated the C-statistics [@Uno2011] of the fitted models on independently generated datasets and report them in Figure \[fig:aftPEs\]. EEScreen does not have much of an effect on the C-statistic, while using the model-free methods tend to decrease the predictive ability of the fitted model. The results in Figures \[fig:aftMSEs\] and \[fig:aftPEs\] are in contrast to the corresponding $t$-year survival simulation results, which showed the EEScreen can indeed improve MSE and prediction. This may be due to the way these figures were generated: to plot these figures we varied the size of $\hat{\mathcal{M}}$ from between 400 to 20000 in increments of 400. However, the advantages of screening in the AFT setting perhaps may only be seen if fewer than 400 covariates are retained. Data example {#sec:data} ============ We illustrate our methods on data from a multiple myeloma clinical trial. Multiple myeloma is the second-most common hematological cancer, but despite recent advances in therapy the sickest patients have seen little improvement in their prognoses. It is of great interest to explore whether genomic data can be used to predict which patients will fall into this high-risk subgroup, so that they might be targeted for more aggressive or experimental therapies. ![\[fig:km\]Kaplan-Meier estimates from multiple myeloma clinical trials](km.eps) The MicroArray Quality Control Consortium II (MAQC-II) study posed exactly this question to 36 teams of analysts representing academic, government, and industrial institutions [@maqcii]. It used data from newly diagnosed multiple myeloma patients who were recruited into clinical trials UARK 98-026 and UARK 2003-33, which studied the treatment regimes total therapy II (TT2) and total therapy III (TT3), respectively [@Zhan2006; @Shaughnessy2007]. Teams were asked to predict the probability of surviving past $t_0=24$ months, which is roughly the median survival time of high-risk myeloma patients [@KyleRajkumar2008], using the TT2 arm as the training set and the TT3 arm as the testing set. There were 340 patients in TT2, with 126 events and an average follow-up time of 55.82 months, and 214 patients in TT3, with 43 events and an average follow-up of 37.03 months. The Kaplan-Meier estimates of the survival curves are given in Figure \[fig:km\]. Gene expression values for 54675 probesets were measured for each subject using Affymetrix U133Plus2.0 microarrays, and 13 clinical variables were also recorded, including age, gender, race, and serum $\beta_2$-microglobulin and albumin levels. Figure \[fig:km\] shows that there was a patient in TT2 censored before 24 months, so we cannot model these data using simple logistic regression. We therefore considered the $t$-year survival model with estimating equation (\[eq:tyear\]), from Section \[sec:tyearsim\]. Because we had a total of 54688 covariates and only 340 patients in TT2, we first implemented a screening step, where we considered EEScreen, our model-free method (\[eq:zhueescreen\]), and the method of @Zhu2011. We then fit the screened variables using EEBoost, with the generalized cross-validation criterion described in Section \[sec:tyearsim\]. To choose the size of $\hat{\mathcal{M}}$, we used 5-fold cross-validation and selected the value of $\vert\hat{\mathcal{M}}\vert$ that gave the best average AUC statistic. The values we considered were 10, 50, 100, 500, 1000, and the numbers from 5000 to 54688 in increments of 5000. Finally, we validated our model in the TT3 arm. Method Optimal $\vert\hat{\mathcal{M}}\vert$ 5-fold CV AUC (SD) AUC in TT3 ----------------------------- --------------------------------------- -------------------- ------------ EEScreen ($t$-year) 5000 0.61 (0.03) 0.61 Method (\[eq:zhueescreen\]) 10 0.63 (0.06) 0.58 Zhu et al. (2011) 100 0.67 (0.08) 0.59 EEScreen (AFT) 100 0.65 (0.08) 0.70 : \[tab:aucs\]AUCs for probability of surviving past $t_0=24$ months Table \[tab:aucs\] summarizes our results. We first focused on the AUCs estimated using five-fold cross-validation. Surprisingly, we found that EEScreen gave us the lowest AUC, and that the model-free methods required fewer covariates while giving better prediction. However, note that screening using the $t$-year survival estimating equation (\[eq:tyear\]) essentially dichotomizes the observed times to binary outcomes, because we are only modeling whether they are larger than $t_0$. In contrast, we can see from the forms of method (\[eq:zhueescreen\]) and the procedure of @Zhu2011 that they use continuous outcomes. We therefore hypothesized that the model-free methods had more power than EEScreen based on equation (\[eq:tyear\]) to detect covariate effects, even though they did not incorporate any modeling assumptions. To test this hypothesis we examined the performance of using EEScreen based on the AFT model estimating equation (\[eq:aft\]). This strategy does not dichotomize the survival outcomes and is also a more restrictive model than the $t$-year model because it makes a global assumption on the distribution of the survival times. After screening we still used the $t$-year survival model to fit the retained covariates. Indeed, Table \[tab:aucs\] shows that with this strategy, we needed to retain only 100 covariates to achieve a high AUC. Turning now to the validation AUCs calculated in the TT3 arm, we found that though the model-free methods gave higher AUCs in cross-validation, their validation AUCs were essentially comparable to that of EEScreen based on the $t$-year survival model. This might perhaps indicate that the model-free methods actually overfit to patients in the TT2 arm, and thus their results didn’t generalize well to patients treated with TT3. In contrast, the EEScreen method based on the AFT model gave a much higher validation AUC of 70%. The final fitted model contained 37 covariates, which in addition to various gene expression levels also included $\beta_2$-microglobulin, albumin, and lactate dehydrogenase levels. Thus our method was able to select important clinical predictors in addition to identifying potentially important genomic factors. Discussion {#sec:discussion} ========== In this paper we introduced EEScreen, a new computationally convenient screening method that can be used with any estimating equation-based regression method. We proved finite-sample performance guarantees that hold for any model that can be fit with U-statistic-based estimating equations, and in addition showed that our approach could be used to derive a model-free screening procedure very similar to one proposed by @Zhu2011. Finally, we have drawn a connection between screening and boosting methods, showing that the EEBoost algorithm of @Wolfson2011 can be viewed as a form of iterative screening. Our simulation results, conducted using a $t$-year survival model as well as the AFT model, support the use of EEScreen in practice. They suggest that EEScreen is capable of retaining most of the important covariates without also including too many false positives, unless the covariates are very highly correlated. In terms of estimation and prediction, when the working model is correctly specified, using EEScreen will usually not give worse results than not using screening at all, and at the very least will dramatically reduce the required computation time. This does not always appear to be true of the model-free methods. On the other hand, in our multiple myeloma example we saw that using different models for the screening step and the regression step can offer better performance than keeping to one model throughout. This illustrates the difficulty in choosing a default screening procedure that works well in all cases. However, our myeloma results suggest that one key consideration is the power of the screening step. The AFT model-based screening appeared to have greater power than the $t$-year model, and perhaps its modeling assumptions prevented it from overfitting to the TT2 arm, as the model-free methods seemed to do. This insight implies that different situations will require choosing different screening methods in order to achieve the greatest power. Estimating equations give us access to a wide range of models to choose from, with more parametric models offering lower variance but higher bias, and models with fewer assumptions offering the opposite tradeoff. Thus our EEScreen approach is perfectly suited to this screening strategy, offering quick computation and good theoretical properties for whichever model we decide to use. Acknowledgments {#acknowledgments .unnumbered} =============== We thank Professors Lee Dicker and Julian Wolfson for reading an earlier version of this manuscript. We also thank Professors Tianxi Cai, Tony Cai, Jianqing Fan, Hongzhe Li, and Xihong Lin for their many helpful comments and suggestions. Sihai Zhao is grateful for the support provided by NIH-NIGMS training grant T32-GM074897. Proof of Theorem \[thm:surescreening\] {#pf:surescreening} ====================================== The event $\{\mathcal{M}\subseteq\hat{\mathcal{M}}\}$ equals $\{\min_{j\in\mathcal{M}}\vert U_j(0)\vert\geq\gamma_n\}$, so it is easy to see that $$\pr(\mathcal{M}\subseteq\hat{\mathcal{M}}) \geq 1-\sum_{j\in\mathcal{M}}\pr(\vert U_j(0)\vert<\gamma_n).$$ By the triangle inequality, we know that for all $j$, $\vert u_j(0)\vert\leq\vert U_j(0)-u_j(0)\vert+\vert U_j(0)\vert$, and by Assumption \[ass:signal\] we see that $c_1[n/m]^{-\kappa}-\vert U_j(0)\vert\leq\vert U_j(0)-u_j(0)\vert$ for all $j\in\mathcal{M}$. Therefore, $\vert U_j(0)\vert<\gamma_n$ for $j\in\mathcal{M}$ implies $\vert U_j(0)-u_j(0)\vert\geq c_1[n/m]^{1-\kappa}/2$. We can conclude from Assumptions \[ass:ustat\] and \[ass:bernstein\] and Bernstein’s inequality for U-statistics [@Hoeffding1963] that $$\pr(\mathcal{M}\subseteq\hat{\mathcal{M}}) \geq 1-2s_n\exp\left\{-\frac{c_1^2[n/m]^{1-2\kappa}/4}{2\Sigma^2+bc_1[n/m]^{-\kappa}/3}\right\}$$ Proof of Theorem \[thm:size\] {#pf:size} ============================= For the marginal estimating equations $U_j$ and their expected values $u_j$, we know from Assumptions \[ass:ustat\] and \[ass:bernstein\] and Bernstein’s inequality for U-statistics [@Hoeffding1963] that $$\pr\{\max_j\vert U_j(0)-u_j(0)\vert\leq c_1[n/m]^{-\kappa}/4\} \geq 1-2p_n\exp\left\{-\frac{c_1^2[n/m]^{1-2\kappa}/16}{2\Sigma^2+bc_1[n/m]^{-\kappa}/6}\right\}.$$ Also, if $\max_j\vert U_j(0)-u_j(0)\vert\leq c_1[n/m]^{-\kappa}/4$, then $\vert U_j(0)\vert\geq\gamma_n$ implies that $\vert u_j(0)\vert\geq c_1[n/m]^{-\kappa}/4$. This means that $$\vert\hat{\mathcal{M}}\vert = \vert\{j:\vert U_j(0)\vert\geq\gamma_n\}\vert \leq \vert\{j:\vert u_j(0)\vert\geq c_1[n/m]^{-\kappa}/4\}\vert \leq \frac{16}{c_1^2[n/m]^{-2\kappa}}\sum_ju_j(\b0)^2.$$ From our EEScreen procedure described in Section \[sec:method\], we see that the $u_j(\b0)$ are the possibly relabeled components of the expected full estimating equation $\bu(\b0)$. Thus $\sum_ju_j(\b0)^2=\Vert\bu(\b0)\Vert_2^2$, and by the generalization of the mean value theorem to vector-valued functions [@HallNewell1979] and Assumptions \[ass:beta0\] and \[ass:inf\], $$\Vert\bu(\b0)\Vert_2 = \Vert\bu(\bbeta_0)-\bu(\b0)\Vert_2 \leq \sup_{0<t<1}\Vert\bi(t\bbeta_0)\Vert_2\Vert\bbeta_0\Vert_2 \leq c_2\sup_{0<t<1}\sigma_{\max}\{\bi(t\bbeta_0)\} = c_2\sigma_{\max}^*,$$ so that $$\begin{aligned} \pr\left[\vert\hat{\mathcal{M}}\vert\leq \frac{16c_2^2\sigma_{\max}^{*2}}{c_1^2[n/m]^{-2\kappa}}\right] &\geq \pr\{\max_j\Vert U_j(0)-u_j(0)\Vert_\infty\leq c_1[n/m]^{-\kappa}/4\}\\ &\geq 1-2p_n\exp\left\{-\frac{c_1^2[n/m]^{1-2\kappa}/16}{2\Sigma^2+bc_1[n/m]^{-\kappa}/6}\right\}.\end{aligned}$$

--- abstract: | An important ingredient for applications of nuclear physics to e.g. astrophysics or nuclear energy are the cross sections for reactions of neutrons with rare isotopes. Since direct measurements are often not possible, indirect methods like $(d,p)$ reactions must be used instead. Those $(d,p)$ reactions may be viewed as effective three-body reactions and described with Faddeev techniques. An additional challenge posed by $(d,p)$ reactions involving heavier nuclei is the treatment of the Coulomb force. To avoid numerical complications in dealing with the screening of the Coulomb force, recently a new approach using the Coulomb distorted basis in momentum space was suggested. In order to implement this suggestion separable representations of neutron- and proton-nucleus optical potentials, which are not only complex but also energy dependent, need to be introduced. Including excitations of the nucleus in the calculation requires a multichannel optical potential, and thus separable representations thereof.\ [**Keywords:**]{} [*Energy dependent separable representation of optical potentials, multi-channel optical potentials, nonlocal optical potentials, (d,p) Reactions*]{} bibliography: - 'coulomb.bib' --- \[1\] \[1\] \[1\] **** Energy Dependent Separable Optical Potentials for (d,p) Reactions \ [$^a$*Institute of Nuclear and Particle Physics, and Department of Physics and Astronomy,\ Ohio University, Athens, OH 45701, USA*]{}\ [$^b$*National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA* ]{} Introduction ============ Nuclear reactions are an important probe to learn about the structure of unstable nuclei. Due to the short lifetimes involved, direct measurements are usually not possible. Therefore indirect measurements using ($d,p$) reactions have been proposed (see e.g. Refs. [@RevModPhys.84.353; @jolie; @Kozub:2012ka]). Deuteron induced reactions are particularly attractive from an experimental perspective, since deuterated targets are readily available. From a theoretical perspective they are equally attractive because the scattering problem can be reduced to an effective three-body problem [@Nunes:2011cv]. Traditionally deuteron-induced single-neutron transfer ($d,p$) reactions have been used to study the shell structure in stable nuclei, nowadays experimental techniques are available to apply the same approaches to exotic beams (see e.g. [@Schmitt:2012bt]). Deuteron induced $(d,p)$ or $(d,n)$ reactions in inverse kinematics are also useful to extract neutron or proton capture rates on unstable nuclei of astrophysical relevance. Given the many ongoing experimental programs worldwide using these reactions, a reliable reaction theory for $(d,p)$ reactions is critical. One of the most challenging aspects of solving the three-body problem for nuclear reactions is the repulsive Coulomb interaction. While for very light nuclei, exact calculations of (d,p) reactions based on momentum-space Faddeev equations in the Alt-Grassberger-Sandhas (AGS) [@ags] formulation can be carried out [@Deltuva:2009fp] by using a screening and renormalization procedure  [@Deltuva:2005wx; @Deltuva:2005cc], this technique leads to increasing technical difficulties when moving to computing (d,p) reactions with heavier nuclei [@hites-proc]. Therefore, a new formulation of the Faddeev-AGS equations, which does not rely on a screening procedure, was presented in Ref. [@Mukhamedzhanov:2012qv]. Here the Faddeev-AGS equations are cast in a momentum-space Coulomb-distorted partial-wave representation instead of the plane-wave basis. Thus all operators, specifically the interactions in the two-body subsystems must be evaluated in the Coulomb basis, which is a nontrivial task (performed recently for the neutron-nucleus interaction [@upadhyay:2014]). The formulation of Ref. [@Mukhamedzhanov:2012qv] requires the interactions in the subsystems to be of separable form. Separable representations of the forces between constituents forming the subsystems in a Faddeev approach have a long tradition, specifically when considering the nucleon-nucleon (NN) interaction (see e.g. [@Haidenbauer:1982if; @Haidenbauer:1986zza; @Entem:2001it]) or meson-nucleon interactions [@Ueda:1994ur; @Gal:2011yp]. Here the underlying potentials are Hermitian, and a scheme for deriving separable representations suggested by Ernst-Shakin-Thaler [@Ernst:1973zzb] (EST) is well suited, specifically when working in momentum space. It has the nice property that the on-shell and half-off-shell transition matrix elements of the separable representation are exact at predetermined energies, the so-called EST support points. However, when dealing with neutron-nucleus (nA) or proton-nucleus (pA) phenomenological optical potentials, which are in general complex to account for absorptive channels that are not explicitly treated, as well as energy-dependent, extensions of the EST scheme have to be made. Separable Representation of Single Channel\ Energy Dependent Optical Potentials =========================================== The pioneering work by Ernst, Shakin and Thaler [@Ernst:1973zzb] constructed separable representations of Hermitian potentials. To apply this formalism to optical potentials, it needs to be extended to handle complex potentials [@Hlophe:2013xca]. We briefly recall the most important features, namely that a separable representation for a complex, energy-independent potential $U_l$ in a fixed partial wave of orbital angular momentum $l$ is given by [@Hlophe:2013xca] $$u_l = \sum\limits_{ij} U_l|\psi_{l,i}^+ \rangle\lambda_{ij}^{(l)}\langle\psi_{l,j}^-|U_l, \label{eq:1}$$ where $|\psi_{l,i}^+\rangle$ is a solution of the Hamiltonian $H=H_0+U_l$ with outgoing boundary conditions at energy $E_i$, and $|\psi_{l,i}^-\rangle$ is a solution of the Hamiltonian $H=H_0+U_l^*$ with incoming boundary conditions. The energies $E_i$ are referred to as EST support points. The free Hamiltonian $H_0$ has eigenstates $|k_i\rangle$ with $k_i^2=2\mu E_i$, $\mu$ being the reduced mass of the neutron-nucleus system. The EST scheme constrains the matrix $\lambda_{ij}^{(l)}$ with the conditions $$\begin{aligned} \delta_{kj}&=&\sum\limits_{i}\langle\psi_{l,k}^-|U_l|\psi_{l,i}^+\rangle\lambda_{ij}^{(l)} \cr \delta_{ik}&=&\sum\limits_{j}\lambda_{ij}^{(l)}\langle\psi_{l,j}^-|U_l|\psi_{l,k}^+\rangle, \label{eq:2}\end{aligned}$$ where the subscript $i=1 \dots N$ indicates the rank of the separable potential. Those two constraints of Eq. (\[eq:2\]) on $\lambda_{ij}^{(l)}$ are an essential feature of the EST scheme and ensure that at the EST support points $E_i$, both, the original $U$ and the separable potential $u$, have identical wavefunctions or half-shell $t$ matrices. The corresponding separable $t$ matrix takes the form $$t_l(E) = \sum\limits_{ij}U_l|\psi_{l,i}^+\rangle\tau^{(l)}_{ij}(E) \langle\psi_{l,j}^-|U_l \label{eq:3}$$ with $$\left(\tau^{(l)}_{ij}(E)\right)^{-1}= \langle\psi_{l,i}^-|U_l-U_lg_0(E)U_l|\psi_{l,j}^+\rangle. \label{eq:4}$$ Here $g_0(E) = (E -H_0 +i \varepsilon)^{-1}$ is the free propagator. The form factors are given as half-shell $t$-matrices $$T_l(E_i)|k_i\rangle\equiv U_l|\psi_{l,i}^+\rangle, \label{eq:5}$$ and are obtained through solving a momentum space Lippmann-Schwinger (LS) equation. However, when applying the same formulation to an energy-dependent complex potential $U(E)$, one obtains $$u_l = \sum\limits_{ij}U_l(E_i)|\psi_{l,i}^+\rangle\lambda^{(l)}_{ij}\langle \psi_{l,j}^-|U_l(E_j), \label{eq:form4}$$ with the constraints $$\begin{aligned} \delta_{kj}&=&\sum\limits_{i}\langle\psi_{l,k}^-|U_l(E_i)|\psi_{l,i}^+\rangle \lambda^{(l)}_{ij} \cr \delta_{ik}&=&\sum\limits_{j}\lambda^{(l)}_{ij}\langle\psi_{l,j}^-|U_l(E_j)| \psi_{l,k}^+\rangle. \label{eq:form4b} \end{aligned}$$ Omitting the partial wave index $l$ the two constraints on $\lambda$ can be written in matrix form as $$\mathcal{U}^t \; \lambda = {\bf 1} =\lambda \; \mathcal{U}, \label{form5}$$ with $$\mathcal{U}_{ij} = \langle\psi_i^-|U(E_i)|\psi_j^+\rangle. \label{form5b}$$ For a separable potential of rank $N>1$ it is obvious that the matrix $\mathcal{U}_{ij}$ is not symmetric in the indices $i$ and $j$. This leads to an asymmetric matrix $\lambda$ and thus a $t$ matrix which violates reciprocity. Therefore, a different approach must be taken in order to construct separable representations for energy-dependent potentials. Here we note that although the potential $u$ contains some of the energy dependence of $U(E)$ through the form factors calculated at the different fixed energy support points $E_i$, it has no explicit energy dependence. Thus, this separable construction needs to be considered as energy-independent EST representation. ![ The $s$-wave off-shell $t$-matrix elements for the $n$+$^{48}$Ca system calculated from the CH89 optical potential [@Varner:1991zz] as function of the off-shell momenta $k$ and $k'$ at 20 MeV incident neutron laboratory kinetic energy. Panels (a) and (c) depict the real and imaginary $t$-matrix elements corresponding to the CH89 global optical potential. The real and imaginary parts of the eEST separable representation of the off- shell $t$-matrix are shown in panels (b) and (d). The on-shell momentum is $k=~0.978$ fm$^{-1}$. []{data-label="fig1"}](fig1.eps){width="13.0cm"} A separable expansion for energy-dependent Hermitian potentials was suggested by Pearce [@Pearce:1987zz]. It is straightforward to apply this suggestion to complex potentials by using the insights previously gained in [@Hlophe:2013xca]. In analogy, we define the EST separable representation for complex, energy-dependent potentials (eEST) by allowing an explicit energy dependence of the coupling matrix elements $\lambda_{ij}$. $$\begin{aligned} u(E) = \sum\limits_{ij}U(E_i)|\psi_i^+\rangle\lambda_{ij}(E)\langle\psi_j^-|U(E_j), \label{eq:form9}\end{aligned}$$ where the partial wave index $l$ has been omitted for simplicity. In order to obtain a constraint on the matrix $\lambda(E)$, we require that the matrix elements of the potential $U(E)$ and its separable form $u(E)$ between the states $|\psi_i^+\rangle$ be the same at all energies $E$. This condition ensures that the potentials $U(E)$ and $u(E)$ lead to identical wavefunctions at the EST support points, just like in the energy-independent EST scheme. The constraints on $\lambda_{ij}(E)$ become $$\begin{aligned} \langle\psi_m^-|U(E)|\psi_n^+\rangle &=&\langle\psi_m^-|u(E)|\psi_n^+\rangle\cr &=&\sum\limits_{i}\langle\psi_m^-|U(E_i)|\psi_i^+\rangle\lambda_{ij}(E) \langle\psi_j^-|U(E_j)|\psi_n^+\rangle. \label{eq:form10}\end{aligned}$$ The corresponding separable $t$-matrix then takes the form $$\begin{aligned} t(E) = \sum\limits_{ij}U(E_i)|\psi_i^+\rangle\tau_{ij}(E)\langle\psi_j^-|U(E_j). \label{eq:form11}\end{aligned}$$ Substituting Eqs. (\[eq:form9\])$-$(\[eq:form11\]) into the LS equation leads to constraint for the matrix $\tau(E)$ such that $$R(E)\cdot\tau(E) \equiv \mathcal{M}(E), \label{eq:form12}$$ where $$R_{ij}(E) = \langle\psi_i^-|U(E_i)|\psi_j^+\rangle-\sum\limits_{n}\mathcal{M}_{in}(E) \langle\psi_n^-|U(E_n)\; g_0(E)\; U(E_j)|\psi_j^+\rangle, \label{eq:form12b}$$ with $$\begin{aligned} \mathcal{M}_{in}(E) \equiv[\mathcal{U}^e(E)\cdot \mathcal{U}^{-1}]_{in}. \label{eq:form12c}\end{aligned}$$ The matrix elements of $\mathcal{U}$ are defined in Eq. (\[form5b\]), and $$\mathcal{U}^e_{ij}(E) \equiv \langle\psi_i^-|U(E)|\psi_j^+\rangle. \label{eq:form13a}$$ For energy-independent potentials $\mathcal{U}^e(E)$ becomes $\mathcal{U}$ and the matrix $\mathcal{M}(E)$ is the unit matrix. The matrix element $\mathcal{U}^e_{ij}(E)$ is explicitly given as $$\begin{aligned} \mathcal{U}^e_{ij}(E) &=&U(k_i,k_j,E)+\int\limits_0^\infty dp p^2 \;T(p,k_i;E_i)\;g_0(E_i,p)\;U(p,k_j,E) \cr &+&\int\limits_0^\infty dp p^2\; U(k_i,p,E)\;g_0(E_j,p)\;T(p,k_j;E_j)\\ &+&\int\limits_0^\infty dp p^2 \int\limits_0^\infty dp' p'^2\; T(p,k_i;E_i)\;g_0(E_i,p)\; U(p,p',E) \;g_0(E_j,p')\;T(p',k_j;E_j)\nonumber, \label{eq:esep8d0}\end{aligned}$$ where $g_0(E,p)=[E-p^2/2\mu+i\varepsilon]^{-1}$. For the evaluation of $\mathcal{U}^e_{ij}(E)$ for all energies $E$ within the relevant energy regime, the form factors $T(p',k_j;E_j)$ are needed at the specified EST support points and the matrix elements of the potential $U(p',p,E)$ at all energies. The explicit derivation of the above expressions is given in Refs.[@Hlophe:2015rqn; @LindaHlophe2016], together with suggestions to simplify the calculation of $U(p',p,E)$. ![The unpolarized differential cross section for elastic scattering of protons from $^{48}$Ca (upper) and $^{208}$Pb (lower) as function of the c.m. angle. For $^{48}$Ca the cross section is calculated at a laboratory kinetic energy of 38 MeV and is scaled by a factor 4. The calculation for $^{208}$Pb is carried out at $E_{lab}$ = 45 MeV. The solid lines ($i$) depict the cross section calculated in momentum space based on the rank-5 separable representation of the CH89 [@Varner:1991zz] phenomenological optical potential, while the crosses ($ii$) represent the corresponding coordinate space calculations [@Nunes-private]. []{data-label="fig2"}](fig2.eps){width="9.0cm"} To apply the formulation to proton-nucleus scattering one first realizes that the proton-nucleus potential consists of the point Coulomb force, $V^c$, together with a short-ranged nuclear as well as a short-ranged Coulomb interaction representing the charge distribution of the nucleus, which we refer to as $U^s(E)$. While the point Coulomb potential has a simple analytical form, an optical potential is employed to model the short-range nuclear potential. The extension of the energy-independent EST separable representation to proton-nucleus optical potentials was carried out in Ref. [@Hlophe:2014soa]. In that work it was shown that the form factors of the separable representation are solutions of the LS equation in the Coulomb basis, and that they are obtained using methods introduced in Refs. [@Elster:1993dv; @Chinn:1991jb]. It was also demonstrated that the extension of the energy-independent EST separable representation scheme to proton-nucleus scattering involves two steps. First, the nuclear wavefunctions $|\psi_{l,i}^{(+)}\rangle$ are replaced by Coulomb-distorted nuclear wavefunctions $|\psi_{l,i}^{sc~(+)}\rangle$. Second, the free resolvent $g_0(E)$ is replaced by the Coulomb Green’s function, $g_c(E) =(E-H_0 -V^c +i\varepsilon)^{-1}$, and third, the energy-dependent scheme must be generalized. Upon suppressing the index $l$ we obtain a constraint similar to Eq. (\[eq:form12\]), $$\begin{aligned} R^{c}(E)\cdot\tau^{c}(E) =\mathcal{M}^{c}_{ij}(E), \label{pform2}\end{aligned}$$ with the matrix elements of $R^{c}(E)$ satisfying $$\begin{aligned} R_{ij}^{c}(E) &= & \langle\psi_{i}^{sc~(-)}|U^s(E_i)|\psi_{j}^{sc~(+)}\rangle \cr &-& \sum_i\mathcal{M}^{c}_{in}(E)\langle\psi_{n}^{sc~(-)}|U^s(E_n)g_c(E) U^s(E_j)|\psi_{j}^{sc~(+)} \rangle. \label{eq:pform3}\end{aligned}$$ The matrix $\mathcal{M}^{c}(E)$ is the Coulomb distorted counterpart of $\mathcal{M}(E)$ of Eq. (\[eq:form12c\]), and is defined as $$\mathcal{M}^c_{in}(E) = \left[\mathcal{U}^{e,sc}(E)\cdot ({\mathcal{U}^{sc}})^{-1}\right]_{in}, \label{eq:pform4}$$ with $$\begin{aligned} \mathcal{U}^{sc}_{ij}&\equiv& \langle\psi_{i}^{sc~(-)}|U^s(E_i)|\psi_{j}^{sc~(+)}\rangle,\cr \mathcal{U}^{e,sc}_{ij}(E)&\equiv&\langle \psi_{k_i}^{sc~(-)} | U^s(E) | \psi_{k_j}^{sc~(+)} \rangle. \label{eq:pform4b}\end{aligned}$$ If the potential is energy-independent the matrix $\mathcal{M}^c(E)$ becomes a unit matrix just like $\mathcal{M}(E)$. Further details for the explicit evaluation are given in Refs. [@Hlophe:2015rqn; @LindaHlophe2016]. In order to illustrate the quality of the separable representation of energy-dependent optical potentials for neutron as well as proton elastic scattering, the differential cross sections for proton scattering off $^{48}$Ca at laboratory kinetic energy 38 MeV and $^{208}$Pb at 45 MeV are shown in Fig. \[fig2\] and compared to the equivalent coordinate space calculations. We observe that the separable representation provides an excellent description on both cases. The power of a separable representation based on the EST scheme lies in the choice of the basis, namely here the half-shell t-matrices calculated at specific energies. This basis contains a lot of information about the system considered, and thus only a small number of basis states, represented by the rank of the separable potential, are needed to have this excellent representation. ![ The $p_{3/2}$ form factors $h_{0,i}$ for the $n+^{48}$Ca system obtained from the CH89 optical potential [@Varner:1991zz]. Panel (a) illustrates the form factors as function of momentum $p$ while panel (b) depicts its Fourier transform as function of the position coordinate $r$. The indices $i=$ 1, 2, and 3 correspond to the support points 5, 21, and 47 MeV. \[fig3\] ](fig3.eps){width="9.0cm"} ![ The $s$-wave form factors $h_{0,i}$ for the $n+^{48}$Pb system obtained from the CH89 optical potential [@Varner:1991zz]. Panel (a) illustrates the form factors as function of momentum $p$ while panel (b) depicts its Fourier transform as function of the position coordinate $r$. The indices $i=$ 1, 2, and 3 correspond to the support points 5, 21, and 47 MeV. []{data-label="fig4"}](fig4.eps){width="9.0cm"} Coordinate Space Separable Representation of\ Single Channel Optical Potentials ============================================= ![ The off-shell potential elements $u_l^{j_p}(r',r,E)$ of the separable representation of the CH89 optical potential [@Varner:1991zz] for the $n$+$^{48}$Ca system as function of the coordinates $r$ and $r'$ at $E=$ 20 MeV incident neutron laboratory kinetic energy. Panels (a) and (c) depict the real and imaginary potential matrix elements for the $s_{1/2}$ partial wave. The real and imaginary parts of the $p_{3/2}$ separable potential are shown in panels (b) and (d). \[fig5\] ](fig5.eps){width="13cm"} The formal scheme for deriving separable representations to Hermitian potentials was given by Ernst, Shakin, and Thaler in Ref. [@Ernst:1973zzb], and the application of the of the scheme to a two-body coordinate space potential representing an s-wave bound and scattering state in Ref. [@Ernst:1974up]. The authors chose to carry out their construction of the separable representation in coordinate space, which makes the procedure more cumbersome compared to the momentum space construction we employ, leading to a momentum space separable representation of either the transition matrix or the potential. Since coordinate space techniques have long tradition in nuclear physics, it can be useful to consider an EST based separable representation of potentials in coordinate space. Separable potentials are inherently nonlocal. Using the EST formulation leads to a well defined behavior of this non-locality. However, instead of implementing the EST construction in coordinate space, one can carry out the entire scheme in momentum space and then Fourier transform the momentum space result to coordinate space. This is quite simple, since it involves only a one-dimensional Fourier transform of the form factors. To illustrate a coordinate space realization of an EST separable representation, we show in Fig. \[fig3\] the form factors $h_{l,i}$ as function of the momentum $p$ for the $n+^{48}$Ca system in panel (a) together with their Fourier transformed counterparts in coordinate space in panel (b). The index $i$ refers to the EST support points used. The form factors are well behaved functions in momentum space as well as coordinate space. In Fig. \[fig4\] the s-wave form factors for the $n+^{208}$Pb system are shown, and we note that for the heavier nucleus $^{208}$Pb they extend to larger values of $r$ as should be expected considering the larger size of the heavier nucleus. The separable representation of the coordinate space potential in a given partial wave is obtained by summing over the rank of the potential according to Eq. (\[eq:1\]). The resulting nonlocal separable coordinate space representation of the CH89 optical potential is shown in Fig. \[fig5\] for the $n+^{48}$Ca system for the $s_{1/2}$ and $p_{3/2}$ channels. The non-locality is symmetric in $r$ and $r'$ as required by reciprocity and its extension in $r$ and $r'$ is given by the fall-off behavior of the form factors. It also shows a more intricate behavior than the often employed Perey-Buck Gaussian-type [@Perey:1962] non-locality construct. Employing the nonlocal separable representation in solving the integro-differential Schrödinger equation [@Titus:2016gvp] reveals that resulting coordinate space wavefunction exactly agree with the wavefunctions obtained from solving the Schrödinger equation with the local CH89 optical potential [@Ross-private]. Separable Representation of Multi-Channel\ Energy Dependent Optical Potentials ========================================== To generalize the energy-dependent EST (eEST) scheme to multichannel potentials, we proceed analogously to Ref. [@Pieper74] and replace the single-channel scattering wavefunctions with their multichannel counterparts, leading to a multichannel separable potential $$u(E)=\sum\limits_{\rho\sigma}\sum\limits_{ij} \left(\sum\limits_{\gamma JM} U(E_i)\big|\gamma JM\; \Psi_{\gamma\rho,i}^{J(+)}\big\rangle\right) \; \lambda_{ij} ^{\rho\sigma}(E)\; \left(\sum\limits_{\gamma JM}\big\langle \Psi_{\gamma\sigma,j}^{J(-)}\;\gamma JM\big|U(E_j)\right). \label{eq:mch1a-1}$$ The indices $i$ and $j$ stand for the EST support points. Using the definition of a multichannel half-shell $t$ matrix [@Gloecklebook], $$T(E_i)|\rho JM \; k_i^\rho \rangle=\sum_{\gamma} U(E_i)|\gamma\;JM\Psi_{{\gamma\rho}}^{J(+)}\rangle, \label{eq:nad3c}$$ Eq. (\[eq:mch1a-1\]) can be recast as $$u(E)=\sum\limits_{JM}\sum\limits_{J'M'}\sum\limits_{\rho\sigma}\sum\limits_{ij} T(E_i)\big|\rho JM\; k_i^\rho\big\rangle \;\lambda_{ij}^{\rho\sigma}(E)\; \big\langle k_j^\sigma\;\sigma J'M'\big|T(E_j). \label{eq:mch21a}$$ To determine the constraint on $u(E)$, we first generalize the matrices ${\cal U}^{e}(E)$ and ${\cal U}$ to multichannel potentials. This is accomplished by replacing the single-channel scattering states by the multichannel scattering states so that $$\begin{aligned} {\cal U}^{e,\alpha\beta}_{mn}(E)&\equiv&\sum\limits_{\gamma\nu}\big\langle \Psi_{\gamma\alpha,m}^{J(-)}\;\gamma JM | U(E)|\nu JM\;\Psi_{\nu\beta,n}^{J(+)}\rangle,\cr &=&\sum\limits_{\gamma\nu}\big\langle \Psi_{\gamma\alpha,m}^{J(-)}| U_{\gamma\nu}^J(E)|\Psi_{\nu\beta,n}^{J(+)}\rangle, \label{eq:mch21b0}\end{aligned}$$ and $${\cal U}^{\alpha\beta}_{mn}\equiv{\cal U}^{e,\alpha\beta}_{mn}(E_m) =\sum\limits_{\gamma\nu}\big\langle \Psi_{\gamma\alpha,m}^{J(-)}\big| U_{\gamma\nu}^J(E_m)\big|\Psi_{\nu\beta,n}^{J(+)}\big\rangle. \label{eq:mch21b1}$$ The $J$ dependence of matrix elements ${\cal U}^{e,\alpha\beta}_{mn}(E)$ and ${\cal U}^{\alpha\beta}_{mn}$ is omitted for simplicity. One one hand, Eq. (\[eq:mch21b1\]) shows that the matrix ${\cal U}$ depends only on the support energies $E_m$ and $E_n$. On other hand, we see from Eq. (\[eq:mch21b0\]) that ${\cal U}^{e}(E)$ depends on the projectile energy $E$ as well as the support energies. The constraint on the separable potential is obtained by substituting the multichannel matrices ${\cal U}^e$ and ${\cal U}$ into Eq. (\[eq:form10\]) leading to $$\begin{aligned} {\cal U}^{e,\alpha\beta}_{mn}(E) &=& \sum\limits_{\rho\sigma}\sum\limits_{ij}\big({\cal U}^t\big)^{\alpha\rho}_{mi}\; \lambda_{ ij}^{\rho\sigma}(E)\;{\cal U}^{\sigma\beta}_{jn},\cr &=&\left[{\cal U}^t\cdot\lambda(E)\cdot {\cal U} \right]_{mn}^{\alpha\beta}. \label{eq:mch21b2}\end{aligned}$$ To evaluate the separable multichannel $t$ matrix, we insert Eqs. (\[eq:mch21a\])-(\[eq:mch21b2\]) into a multi-channel LS equation and obtain $$\begin{aligned} t(E)&=&\sum\limits_{\rho\sigma}\sum\limits_{ij} \left(\sum\limits_{\gamma JM} U(E_i)\big|\gamma JM\; \Psi_{\gamma\rho,i}^{J(+)}\big\rangle\right) \;\tau_{ij} ^{\rho\sigma}(E)\; \left(\sum\limits_{\gamma JM}\big\langle \Psi_{\gamma\sigma,j}^{J(-)}\;\gamma JM\big|U(E_j)\right) \cr &=&\sum\limits_{JM}\sum\limits_{J'M'}\sum\limits_{\rho\sigma}\sum\limits_{ij} T(E_i)\big|\rho JM\; k_i^\rho\big\rangle \;\tau_{ij} ^{\rho\sigma}(E)\; \big\langle k_j^\sigma\;\sigma J'M'\big|T(E_j). \label{eq:mch21c}\end{aligned}$$ The coupling matrix elements $\tau_{ij}^{\rho\sigma}(E)$ fulfill $$\begin{aligned} R(E)\cdot\tau(E)=\mathcal{M}(E), \label{eq:mch24a}\end{aligned}$$ where $$\begin{aligned} R^{\rho\sigma}_{ij}(E)&=&\Big\langle k_i^{\rho}\Big|~T_{{\rho\sigma}}^J(E_i)~+~\sum\limits_{ {\beta}} T_{{\rho\beta}}^J(E_i)G_{{\beta}}(E_j)T_{ {\beta\sigma}}^J(E_j)\Big| k_j^{\sigma} \Big\rangle\cr\cr &-&~\sum\limits_{ {\beta\beta}'}\sum\limits_{n}{\cal M}_{in}^{ {\rho\beta}}\langle k_n^{\beta}\Big| T_{ {\beta\beta'}}^J(E_n)G_{ {\beta'}}(E)T_{ {\beta'\sigma}}^J(E_j) \Big| k_j^{\sigma}\Big\rangle, \label{eq:mch24b}\end{aligned}$$ and $${\cal M}^{\rho\sigma}_{ij}(E)=\left[{\cal U}^e(E)\cdot{\cal U}^{-1}\right]^{\rho\sigma}_{ij}. \label{eq:mch25c}$$ The expression for the matrix $R^{\rho\sigma}_{ij}(E)$ is analogous to the one for the single-channel case except for the extra channel indices. ![ The differential cross sections for scattering in the $n+^{12}$C system computed at different incident neutron energies with the eEST separable representation of the Olsson 89 DOMP [@Olsson:1989npa] (solid lines). The left hand panel shows the differential cross section for elastic scattering, while the right hand panel depicts the differential cross section for inelastic scattering to the $2^+$ state of $^{12}$C. The dashed lines indicate cross sections computed with the spherical Olsson 89 [@Olsson:1989npa] OMP. The filled diamonds represent the data taken from Ref. [@Olsson:1989npa]. The cross sections are scaled up by multiples of 10. The results at 21.6 MeV are multiplied by 10, those at 20.9 MeV are multiplied by 100, etc. \[fig:fig6\] ](fig6.eps){width="12cm"} To illustrate the implementation of the multichannel eEST separable representation scheme, we consider the scattering of neutrons from the nucleus $^{12}$C. The $^{12}$C nucleus possesses selected excited states, with the first and second levels having $I^\pi=2^+$ and $I^\pi=4^+$ and being located at 4.43 and 14.08 MeV above the $0^+$ ground state. The collective rotational model [@ThompsonNunes] is assumed to the coupling between the ground state and these excited states. We consider here elastic scattering and inelastic scattering to the $2^+$ rotational state. To test the multichannel eEST separable representation we use the deformed optical potential model (DOMP) derived by Olsson et [*al.*]{} [@Olsson:1989npa] and fitted to elastic and inelastic scattering data between 16 and 22 MeV laboratory kinetic energy. In Fig. \[fig:fig6\] the differential cross sections for elastic and inelastic scattering for the $n+^{12}$C system are shown at various incident neutron energies. The left hand panel shows the differential cross section for elastic scattering, and the right hand panel the differential cross section for inelastic scattering to the $2^+$ state of $^{12}$C. The support points are at $E_{lab}=$ 6 and 40 MeV. The separable representation describes both differential cross sections very well. In addition, it is in good agreement with the coupled-channel calculations shown in Fig. 1 of Ref. [@Olsson:1989npa]. The dashed lines indicate cross sections computed with the spherical Olsson 89 [@Olsson:1989npa] OMP. Summary and Outlook =================== In a series of steps we developed the input that will serve as a basis for Faddeev-AGS three-body calculations of $(d,p)$ reactions, which will not rely on the screening of the Coulomb force. To achieve this, Ref. [@Mukhamedzhanov:2012qv] formulated the Faddeev-AGS equations in the Coulomb basis using separable interactions in the two-body subsystems.We developed separable representations of phenomenological optical potentials of Woods-Saxon type for neutrons and protons. First we concentrated on neutron-nucleus optical potentials and generalized the Ernst-Shakin-Thaler (EST) scheme [@Ernst:1973zzb] so that it can be applied to complex and energy-dependent optical potentials [@Hlophe:2013xca; @Hlophe:2015rqn]. In order to consider proton-nucleus optical potentials, we further extended the EST scheme so that it can be applied to the scattering of charged particles with a repulsive Coulomb force [@Hlophe:2014soa]. Finally we extended the EST formulation to incorporate multi-channel optical potentials [@Hlophe:2016isw]. The results demonstrate, that separable representations based on a generalized EST scheme reproduce standard coordinate space calculations of neutron and proton scattering cross sections very well. We also showed that from momentum space separable representations corresponding coordinate space representations can be obtained using Fourier transforms of the form factors. From those solutions, observables for $(d,p)$ transfer reactions using a Faddeev-AGS formulation should be readily calculated. Work along these lines is in progress. [**Acknowledgments**]{} This material is based on work in part supported by the U. S. Department of Energy, Office of Science of Nuclear Physics under contract No. DE-FG02-93ER40756 with Ohio University. The authors thank F.M. Nunes and A. Ross for fruitful discussions.

--- abstract: | We calculate the electromagnetic form factors of a bound proton. The Chiral Quark-Soliton model provides the quark and antiquark substructure of the proton, which is embedded in nuclear matter. This procedure yields significant modifications of the form factors in the nuclear environment. The sea quarks are almost completely unaffected, and serve to mitigate the valence quark effect. In particular, the ratio of the isoscalar electric to the isovector magnetic form factor decreases by 20% at $Q^2=1 \text{ GeV}^2$ at nuclear density, and we do not see a strong enhancement of the magnetic moment. author: - 'Jason R. Smith' - 'Gerald A. Miller' title: 'Chiral solitons in nuclei: Electromagnetic form factors' --- Introduction {#sec:intro} ============ Recent polarization transfer experiments at TJNAF [@Strauch:2002wu] observed a difference in the electromagnetic form factors of a proton bound in a Helium nucleus compared to a free one. This, along with other effects, such as the nuclear EMC effect [@Aubert:1983xm], seems to suggest the modification of hadrons in the nuclear medium. There is extensive work on the medium modifications of electromagnetic properties of the nucleon in the literature (for example, see Refs. [@RuizArriola:ex; @Frank:1995pv; @Yakhshiev:2002sr; @Lu:1998tn] ). This includes effective Lagrangians as well as models that include the quark substructure of hadrons. While in principle these effects could be couched in terms of effective field theory operators, it is our thesis that such results may be more transparent, physically intuitive or straightforward to calculate when viewed as a change in the internal structure of the hadrons. We will use the Chiral-Quark Soliton model (CQS) [@Diakonov:2000pa; @Christov:1995vm], which has a direct connection to QCD via the Instanton Liquid model, to provide our subnuclear degrees of freedom. The primary motivation is that this model includes sea quarks which we have seen to be important in the nuclear EMC effect [@Smith:2003hu]. In that case, the large medium modification in the valence quark sector is reduced through the lack of such an effect in the sea (which can be seen directly in Drell-Yan experiments [@Alde:im]). The CQS is combined with the nuclear medium in a self-consistent quark-meson coupling calculation as in our previous work [@Smith:2003hu], and the electromagnetic form factors are extracted via the wave functions of the quarks using the results of Ref. [@Christov:1995hr]. The overall procedure is similar to the Quark-Meson Coupling model (QMC) [@Lu:1998tn], which uses the MIT bag model for the nucleon. The bag model does not include sea quarks. It is a confining model, whereas the CQS model is not. Additionally, the QMC model calculation, when coupled with a Relativistic Distorted Wave Impulse Approximation (RDWIA) calculation [@Udias:1999tm] or a Relativistic Multiple-Scattering Glauber Approximation (RMSGA) calculation [@Ryckebusch:2003fc; @Lava:2004mp], improves the agreement between theory the TJNAF data [@Strauch:2002wu]. With our study, we hope to reinforce the interpretation of the medium effect in terms of quark degrees of freedom, as well as provide an alternate model when the accuracy of the data is improved. We begin with a brief description of the CQS model in Section \[sec:model\]. In Section \[sec:medium\], we motivate and present our procedure to embed this model in nuclear matter. This description differs only slightly from that in our previous work [@Smith:2003hu]; it is repeated for completeness. Subsequently, we describe the numerical methods, and proceed to the results in Section \[sec:results\]. Chiral Quark-Soliton Model {#sec:model} ========================== The CQS model Lagrangian with (anti)quark fields $\overline{\psi},\psi$, and profile function $\Theta(r)$ is [@Diakonov:2000pa] $$\mathcal{L} = \overline{\psi} ( i \partial \!\!\!\!\!\:/\, - M e^{ i \gamma_{5}\bm{n}\cdot\bm{\tau} \Theta(r) } ) \psi, \label{eq:lagrangian}$$ where $\Theta(r\rightarrow\infty) = 0$ and $\Theta(0) = -\pi$ to produce a soliton with unit winding number. The quark spectrum consists of a single bound state and a filled negative energy Dirac continuum; the vacuum is the filled negative continuum with $\Theta = 0$. The wave functions in this spectrum provide the input for the electromagnetic form factors. We work to leading order in the number of colors ($N_{C}=3$), with $N_{f}=2$, and in the chiral limit. While the former characterizes the primary source of theoretical error, one could systematically expand in $N_{C}$ to calculate corrections. We take the constituent quark mass to be $M=420\text{ MeV}$, which reproduces, for example, the $N$-$\Delta$ mass splitting at higher order in the $N_{C}$ expansion, as well as many electromagnetic properties [@Christov:1995vm; @Christov:1995hr]. The theory contains divergences that must be regulated. We use a single Pauli-Villars subtraction. The Pauli-Villars mass is determined by reproducing the measured value of the pion decay constant, $f_{\pi} = 93 \text{ MeV}$, with the relevant divergent loop integral regularized using $M_{PV}\simeq 580 \text{ MeV}$. The nucleon mass is given by a sum of the energy of a single valence level ($E^{v}$), and the regulated energy of the soliton ($E_{\Theta}$, equal to the sum of energy levels, $E_{n}$, in the negative Dirac continuum with the sum of the energy levels in the vacuum, $E_{n}^{(0)}$, subtracted) \[eq:mn\] $$\begin{aligned} M_{N} & = & N_{C} E^{v}+ E_{\Theta}(M)-\frac{M^{2}}{M_{PV}^{2}}E_{\Theta}(M_{PV})\\ E_{\Theta}(M') & = & \sum_{E_{n},E_{n}^{(0)}\leq 0} E_{n} - E_{n}^{(0)}\Bigg|_{M=M'}.\end{aligned}$$ The field equation for the profile function, which follows from the Lagrangian (\[eq:lagrangian\]), is $$\Theta(r) = \arctan \frac{\rho_{ps}^{q}(r)}{\rho_{s}^{q}(r)},\label{eq:thetafe}$$ where $\rho_{s}^{q} \text{ and } \rho_{ps}^{q}$ are the quark scalar and pseudoscalar densities, respectively, and are given by sums of the wave functions of every occupied energy level. The electromagnetic form factors are also given in terms of the wave functions, and are derived in Ref. [@Christov:1995hr]. The formulae are reproduced here, with a Pauli-Villars regulator, for convenience. To leading order in $N_{C}$, we have only the isoscalar electric and isovector magnetic form factors ($G_{X}^{T=0,1} = G_{X}^{p} \pm G_{X}^{n}$) \[eq:GEM\] $$\begin{aligned} G_{E}^{T=0}(q^{2}) & \stackrel{N_{C}\rightarrow\infty}{=} & \frac{N_{C}}{3} \int d\bm{r}\: e^{i\bm{q}\cdot\bm{r}} \Bigg\{ \sum_{E_{n}\leq E^{v}} \psi_{n}^{\dag}(\bm{r})\psi_{n}(\bm{r}) -\sum_{E_{n}^{(0)}\leq 0} \psi_{n}^{(0)\dag}(\bm{r})\psi_{n}^{(0)}(\bm{r}) \Bigg\}\label{eq:GE}\\ G_{M}^{T=1}(q^{2}) & \stackrel{N_{C}\rightarrow\infty}{=} & \frac{N_{C} M_{N}}{3} \varepsilon^{jkl} \frac{iq^{j}}{|q^{2}|} \int d\bm{r}\: e^{i\bm{q}\cdot\bm{r}} \Bigg\{ \sum_{E_{n}\leq E^{v}} \psi_{n}^{\dag}(\bm{r}) \gamma^{0}\gamma^{k}\tau^{l} \psi_{n}(\bm{r})\nonumber\\ & & -\frac{M^{2}}{M_{PV}^{2}}\sum_{E_{n}^{(PV)}\leq 0} \psi_{n}^{(PV)\dag}(\bm{r}) \gamma^{0}\gamma^{k}\tau^{l} \psi_{n}^{(PV)}(\bm{r}) \Bigg\}\label{eq:GM}.\end{aligned}$$ The $\psi^{(PV)}_{n}(\bm{r})$ are the solutions of the Dirac equation with the replacement $M\rightarrow M_{PV}$. In the nuclear medium, Eqs. (\[eq:GEM\]) acquire a dependence on the Fermi momentum $G_{X}^{T=0,1}(q^{2})\rightarrow G_{X}^{T=0,1}(q^{2},k_{F})$ through the wave functions. This dependence is the subject of the next section. Nuclear Physics {#sec:medium} =============== We will begin with some motivation for our procedure to couple the quark substructure of the nucleon to the nuclear medium. Through the use of QCD sum rules, Ioffe [@Ioffe:kw] derived a relationship between the vacuum scalar condensate, $\langle \overline{\psi}\psi\rangle_{0}$, and the nucleon mass. One can re-derive this estimate in a constituent quark field theory such as we are using here. We begin with the scalar condensate $$\begin{aligned} \langle \overline{\psi}\psi\rangle_{0} & = & - \text{tr} \int^{\Lambda} \frac{d^{4}p}{(2 \pi)^{4}} \frac{1}{p\!\!\!\!\;/\, - M} \nonumber\\ & \sim & -\frac{N_{C}M \Lambda^{2}}{4 \pi^{2}}, \label{eq:qqvac}\end{aligned}$$ where the divergent integral is regulated by a momentum cutoff (playing the role of the Borel mass in the QCD sum rule approach). Using the fact that constituent quarks are essentially defined as having a mass $\sim M_{N}/N_{C}$, we can rewrite Eq. (\[eq:qqvac\]) as $$\begin{aligned} M_{N} & \sim & -\frac{4 \pi^{2}}{\Lambda^{2}} \langle \overline{\psi}\psi\rangle_{0}. \label{eq:ioffe}\end{aligned}$$ Although Eq. (\[eq:ioffe\]) is not a very accurate estimate, it does highlight the role of the condensate. It will be modified in the presence of other nucleons. The condensate at finite density can be written in terms of the nuclear scalar density $\rho_{s}^{N}$ and the nucleon sigma term $\sigma_{N}$ [@Cohen:1991nk] as $$\begin{aligned} \langle \overline{\psi}\psi\rangle_{\rho} & = & \langle\overline{\psi}\psi\rangle_{0} - \langle\overline{\psi}\psi\rangle_{0}\frac{\sigma_{N}}{m_{\pi}^{2}f_{\pi}^{2}} \rho_{s}^{N}\label{eq:qqmed}.\end{aligned}$$ We can then substitute Eq. (\[eq:qqmed\]) into Eq. (\[eq:ioffe\]) to obtain a schematic picture of the effect of the nuclear medium on the nucleon mass $$\begin{aligned} M_{N}(\rho) & \sim & -\frac{4 \pi^{2}}{\Lambda^{2}} \left[ \langle\overline{\psi}\psi\rangle_{0} - c_{s} \rho_{s}^{N} \right], \label{eq:ioffemed}\end{aligned}$$ where $c_{s}$ is the combination of the vacuum condensate, pion mass, decay constant and the the sigma term in Eq. (\[eq:qqmed\]). Using this dependence of the nucleon mass on the nuclear medium as a guide, we incorporate the medium dependence in the model by simply letting the quark scalar density in the field equation (\[eq:thetafe\]) contain a (constant) contribution arising from other nucleons present in symmetric nuclear matter. This models a scalar interaction via the exchange of multiple pairs of pions between nucleons. We take the scalar density to consist of three terms: 1) the constant condensate value $\langle \overline{\psi}\psi\rangle_{0}$ (in the vacuum or at large distances from a free nucleon), 2) the valence contribution $\rho_{s}^{v}$ and 3) the contribution from the medium which takes the form of the convolution of the nucleon $\rho_{s}^{N}$ and valence quark scalar densities as in the QMC model [@Lu:1998tn] \[eq:rhos\] $$\begin{aligned} \rho_{s}^{q}(\bm{r}) & \simeq & \langle \overline{\psi}\psi\rangle_{0} + \rho_{s}^{v}(\bm{r}) + \tilde{c}_{s} \int d\bm{r}'\rho_{s}^{N}(\bm{r}-\bm{r}')\rho_{s}^{v}(\bm{r}')\\ & = & \langle \overline{\psi}\psi\rangle_{0} + \rho_{s}^{v}(r) + \tilde{c}_{s} \rho_{s}^{N}S\\ & & S \equiv \int d\bm{r}' \rho_{s}^{v}(\bm{r}').\end{aligned}$$ We take the pseudoscalar density to have only the valence term $\rho_{ps}^{q} \simeq \rho_{ps}^{v}$; the two other contributions analogous to the first and third terms of Eq. (\[eq:rhos\]) vanish due to symmetries of the QCD vacuum and nuclear matter. These approximations to the densities neglect the precise form of the negative continuum wave functions in Eq. (\[eq:thetafe\]). The resulting free nucleon profile function has no discernible difference from a fully self-consistent treatment, demonstrating the excellence of this approximation. We take $\tilde{c}_{s} = c_{s}/S$ in Eqs. (\[eq:ioffemed\]) and (\[eq:rhos\]) to be a free parameter, which we vary to fit nuclear binding. This can be seen as either varying $\sigma_{N}$ in Eq. (\[eq:qqmed\]) or the vacuum value of the condensate in Eq. (\[eq:rhos\]) with $\rho_{\Gamma}^{q}\rightarrow\rho_{\Gamma}^{q}/\tilde{c}_{s}$, as was done in Ref. [@Smith:2003hu], since the overall normalization cancels in Eq. (\[eq:thetafe\]). The nucleon scalar density is determined by solving the nuclear self-consistency equation $$\rho_{s}^{N} = 4 \int^{k_{F}} \frac{d^{3}k}{(2\pi)^{3}} \frac{M_{N}(\rho_{s}^{N})}{\sqrt{k^{2}+M_{N}(\rho_{s}^{N})^{2}}}.\label{eq:nsc}$$ The dependence of the nucleon mass, and any other properties calculable in the model, on the Fermi momentum $k_{F}$ enters through Eq. (\[eq:nsc\]). Thus there are two coupled self-consistency equations: one for the profile, Eq. (\[eq:thetafe\]), and one for the density, Eq. (\[eq:nsc\]). These are iterated until the change in the nucleon mass Eq. (\[eq:mn\]) is as small as desired (in our case, $\Delta M_{N}\lesssim 0.1 \text{ MeV}$) for each value of the Fermi momentum. We use the Kahana-Ripka (KR) basis [@Kahana:be], with momentum cutoff $\Lambda$ and box size $L$ extrapolated to infinity (from a maximum value of $\Lambda L = 150$, comparable to that in Ref. [@Christov:1995hr]), to evaluate the energy eigenvalues and wave functions used as input for the densities, nucleon mass, and electromagnetic form factors. While the vacuum value of the condensate does not vary with the Fermi momentum by definition, the effective condensate, $\langle \overline{\psi}\psi\rangle_{0} + \tilde{c}_{s} \rho_{s}^{N}(k_{F}) S(k_{F})$, falls $\sim 30 \%$ at nuclear density, *q.v.* Eq. (\[eq:qqmed\]). This is consistent with the model independent result [@Cohen:1991nk] that predicts a value 25-50% below the vacuum value. A phenomenological vector meson (mass $m_{v}=770\text{ MeV}$) exchanged between nucleons (but not quarks in the same nucleon), is introduced as a substitute for uncalculated soliton-soliton interactions in order to obtain the necessary short distance repulsion which stabilizes the nucleus. This does not affect the form factors Eqs. (\[eq:GE\]) and (\[eq:GM\]). The resulting energy per nucleon is $$\frac{E}{A} = \frac{4}{\rho_{B}(k_{F})} \int^{k_{F}} \frac{d^{3}k}{(2\pi)^{3}} \sqrt{k^{2} + M_{N}(k_{F})^{2}} +\frac{1}{2}\frac{g_{v}^{2}}{m_{v}^{2}}\rho_{B}(k_{F}) \label{eq:epn}.$$ The mass of a free nucleon is computed to be $M_{N}(k_{F}=0)=1209\text{ MeV}$. The $\sim 30\%$ difference is as expected in the model at leading order in $N_{C}$. We evaluate the nucleon mass Eq. (\[eq:mn\]) and energy per nucleon Eq. (\[eq:epn\]) as a function of $k_F$. We choose our free parameters to fit $E/A - M_{N}(0) \equiv B = -15.75 \text{ MeV}$ at the minimum. We use the value $\tilde{c}_{s} = 1.27$ (corresponding to $\sigma_{N} = 41.4\text{ MeV}$), and vector coupling $g_{v}^{2}/4\pi = 10.55$, which gives a Fermi momentum of $k_{F} = 1.38\text{ fm}^{-1}$ in nuclear matter consistent with the known value $k_{F} = 1.35 \pm 0.05 \text{ fm}^{-1}$ [@Blaizot:tw]. We plot the binding energy per nucleon using Eq. (\[eq:epn\]) in Fig. \[fig:bepn\]. The compressibility is $K = 348.5\text{ MeV}$ which is above the experimental value $K = 210 \pm 30 \text{ MeV}$, but well below the Walecka model [@Walecka:qa] value of $560 \text{ MeV}$. The self-consistent calculation results in the profile functions for zero density, $0.5\rho_{0}$, $1.0\rho_{0}$ and $1.5\rho_{0}$ in Fig. \[fig:profile\] (where $\rho_{0}$ is nuclear density). ![Profile functions in nuclear matter. The solid line is the profile function for $1.5\rho_{0}$; the curves with progressively longer dashes correspond to $1.0\rho_{0}$, $0.5\rho_{0}$ and zero density respectively.[]{data-label="fig:profile"}](bepn4.eps) ![Profile functions in nuclear matter. The solid line is the profile function for $1.5\rho_{0}$; the curves with progressively longer dashes correspond to $1.0\rho_{0}$, $0.5\rho_{0}$ and zero density respectively.[]{data-label="fig:profile"}](prof4.eps) Results and Discussion {#sec:results} ====================== We use Eqs. (\[eq:GE\]) and (\[eq:GM\]) to calculate the form factors, which we present in Figs. \[fig:eff\] and \[fig:mff\]. ![The isovector magnetic form factor at nuclear density (solid) and at zero density (dashes).[]{data-label="fig:mff"}](eff4.eps) ![The isovector magnetic form factor at nuclear density (solid) and at zero density (dashes).[]{data-label="fig:mff"}](mff4.eps) We also present the results in terms of the ratios $$\frac{G_{E,M}^{T=0,1}(Q^{2},k_{F})}{G_{E,M}^{T=0,1}(Q^{2},0)} \equiv \frac{G_{X}^{*}(Q^{2})}{G_{X}(Q^{2})}\label{eq:ffr},$$ where $-q^{2} \equiv Q^{2}$, $X \text{ is } E (T=0) \text{ or } M (T=1)$, and the double ratio $$\frac{G_{E}^{*}(Q^{2})/G_{M}^{*}(Q^{2})}{G_{E}(Q^{2})/G_{M}(Q^{2})}\label{eq:ffdr}.$$ These ratios are plotted in Figs. \[fig:ffr\] and \[fig:ffdr\] for $0.5\rho_{0}$, $1.0\rho_{0}$ and $1.5\rho_{0}$. ![The double ratio Eq. (\[eq:ffdr\]) of the electric to magnetic form factors in nuclear matter and in the vacuum from the CQS model (heavy) and the QMC model [@Lu:1998tn] (light). Three densities are shown: $0.5\rho_{0}$ (long dashes), $1.0\rho_{0}$ (solid) and $1.5\rho_{0}$ (short dashes).[]{data-label="fig:ffdr"}](ffr.eps) ![The double ratio Eq. (\[eq:ffdr\]) of the electric to magnetic form factors in nuclear matter and in the vacuum from the CQS model (heavy) and the QMC model [@Lu:1998tn] (light). Three densities are shown: $0.5\rho_{0}$ (long dashes), $1.0\rho_{0}$ (solid) and $1.5\rho_{0}$ (short dashes).[]{data-label="fig:ffdr"}](ffdr.eps) The electric form factor is dominated by the valence contribution and shows a dramatic effect, while the magnetic form factor has equally important contributions from the valence and the sea. The latter shows almost no change in nuclear matter; it shows only a 1.3% enhancement of the magnetic moment at nuclear density, and a 2.3% enhancement at 1.5 times nuclear density. The effect in the electric form factor calculated here is comparable to that of the QMC model [@Lu:1998tn]; the main difference from that calculation lies in the lack of enhancement in the magnetic form factor, specifically the practically unchanged value of the magnetic moment. While both form factors use the same wave functions, the isovector magnetic form factor includes an extra weighting by a factor of the angular momentum of the state (relative to the electric form factor) due to the $\gamma^{k}$ in Eq. (\[eq:GM\]). This extra factor is not only responsible for making the regularization of Eq. (\[eq:GM\]) necessary, but for making the sea contribution as important as the valence. In the CQS model, the orbital angular momentum carried by the sea is comparable to the orbital angular momentum carried by the valence quarks [@Wakamatsu:1990ud] (the sum of which make up about 60% of the total angular momentum of the nucleon state, with the remainder belonging to the intrinsic spin of the constituent quarks). Conversely, the isoscalar electric form factor (which is finite, after the vacuum subtraction) does not have as large of a contribution from the sea. The valence level is the most important piece, even at $Q^{2}>0$, since the $Q^{2}$ dependence in the form factors arises from the wave functions [@Christov:1995hr]. The negative Dirac continuum wave functions largely cancel in the vacuum subtraction in Eq. (\[eq:GE\]). The magnetic form factors are sensitive to the tail of the quark wave functions, and the mere existence of a tail is due to the lack of confinement. This is one reason for the discrepancy between the current results and the QMC model [@Lu:1998tn], but the primary source is due to the resistance to change of the sea. The former accounts for only a few percent of the difference; it is the latter that is our most important result. We see that the role of antiquarks is again prevalent as in our previous work [@Smith:2003hu]. The double ratio obtained in Fig. \[fig:ffdr\] has the same trend as the QMC model [@Lu:1998tn], but differs in the details. Since we obtain a similar double ratio, we expect to have similar results if we compare these results with the polarization transfer data [@Strauch:2002wu]. This requires one to take the final state and relativistic effects into account through the use of the RDWIA [@Udias:1999tm] or the RMSGA [@Lava:2004mp], which accounts for a few percent of the discrepancy between the results for bound and free protons. A RMSGA calculation for the Helium reaction studied in Ref. [@Strauch:2002wu] has been done with these CQS model results [@Lava:private], and it delivers remarkably similar results to the same calculation done with the QMC model [@Lava:2004mp]. The CQS model predicts a smaller deviation than the QMC model from a Relativistic Plane Wave Impulse Approximation (RPWIA) calculation, which is taken as a baseline in Ref. [@Strauch:2002wu]. While it slightly worsens the agreement with the data at $Q^{2}\lesssim 1$, the differences are of the same order of magnitude as the current experimental error, and both models under predict the observed deviation from a RPWIA calculation. At higher $Q^{2}$, the two models produce nearly identical results for Helium. We ignore important corrections due to the rotation of the soliton that are suppressed by $1/N_{C}$. These corrections break the $N-\Delta$ degeneracy, and improve the agreement of the vacuum form factors with experiment [@Christov:1995hr]. More relevant to the calculation presented here, these corrections do not affect the $Q^2$ dependence, but instead affect the normalization of the form factors [@Christov:1995hr]. However, there is no reason at that level to continue to ignore quantum fluctuations of the the pion field (quark loops), and treat the profile function as a purely self-consistent mean field. We will save this difficult problem for the future. We have calculated the electric and magnetic form factors at leading order in $N_{C}$ at nuclear density using the CQS model. Our results help validate the apparent success of the QMC model in describing the polarization transfer experiment [@Strauch:2002wu; @Lu:1998tn], and provide a counterpoint to be distinguished when finer resolution becomes available in the data. In fact, the difference between the CQS model double ratio and the QMC model [@Lu:1998tn] is roughly the size as the current experimental error. 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--- author: - | Yuxuan Xing, Hulya Seferoglu\ [ECE Department, University of Illinois at Chicago]{}\ [yxing7@uic.edu, hulya@uic.edu]{} [^1] bibliography: - 'refs.bib' title: Predictive Edge Computing with Hard Deadlines --- [^1]: This work was supported in part by U.S. Department of Commerce, National Institute of Standards and Technology award 70NANB17H188, and ARL grant W911NF-17-1-0032.

--- abstract: 'An extended Keldysh formalism, well suited to properly take into account the initial correlations, is used in order to deal with the time-dependent current response of a resonant tunneling system. We use a *partition-free* approach by Cini in which the whole system is in equilibrium before an external bias is switched on. No fictitious partitions are used. Despite a more involved formulation, this partition-free approach has many appealing features being much closer to what is experimentally done. In particular, besides the steady-state responses one can also calculate physical dynamical responses. In the noninteracting case we clarify under what circumstances a steady-state current develops and compare our result with the one obtained in the partitioned scheme. We prove a Theorem of asymptotic Equivalence between the two schemes for arbitrary time-dependent disturbances. We also show that the steady-state current is independent of the history of the external perturbation (Memory Loss Theorem). In the so called wide-band limit an analytic result for the time-dependent current is obtained. In the interacting case we work out the lesser Green function in terms of the self energy and we recover a well known result in the long-time limit. In order to overcome the complications arising from a self energy which is nonlocal in time we propose an exact non-equilibrium Green function approach based on Time Dependent Density Functional Theory. The equations are no more difficult than an ordinary Mean Field treatment. We show how the scattering-state scheme by Lang follows from our formulation. An exact formula for the steady-state current of an arbitrary interacting resonant tunneling system is obtained. As an example the time-dependent current response is calculated in the Random Phase Approximation.' author: - 'Gianluca Stefanucci and Carl-Olof Almbladh' title: 'Time-Dependent Partition-Free Approach in Resonant Tunneling Systems' --- ł Ł ¶ § c Introduction {#intro} ============ A resonant tunneling system is essentially a mesoscopic region, typically a semiconductor heterostructure, coupled to two metallic leads, which play the role of charge reservoirs. In a real experiment the whole system is in thermodynamic equilibrium before the external disturbance is switched on and one can assign a unique temperature $\b^{-1}$ and chemical potential $\m$. Therefore, the initial density matrix is $\r\sim \exp[-\b(H-\m N)]$ where $H$ is the total Hamiltonian and $N$ is the total number of particles. By applying a bias to the leads at a given time, charged particles will start to flow through the central device from one lead to the other. As far as the leads are treated as *noninteracting*, it is not obvious that in the long-time limit a steady-state current can ever develop. The reason behind the uncertainty is that the bias represents a large perturbation and, in the absence of dissipative effects, *e.g.*, electron-electron or electron-phonon scatterings, the return of time-translational invariance is not granted. An alternative approach to this quantum transport problem has been suggested by Caroli *et al.*[@caroli1; @caroli2] who state: “It is usually considered that a description of the system as a whole does not permit the calculation of the current”.[@caroli1] Their approach is based on a fictitious *partition* where the left and right leads are treated as two isolated subsystems in the remote past. Then, one can fix a chemical potential $\m_{\a}$ and a temperature $\b^{-1}_{\a}$ for each lead, $\a=L,R$. In this picture the initial density matrix is given by $\r\sim \exp[-\b_{L}(H_{L}-\m_{L} N_{L})]\exp[-\b_{R}(H_{R}-\m_{R} N_{R})]$, where $H_{L,R}$ and $N_{L,R}$ now refer to the isolated $L,R$ lead. The current will flow through the system once the contacts between the device and the leads have been established. Hence, the time-dependent perturbation is a lead-device hopping rather than a local one-particle level-shift. Since the device is a mesoscopic object, it is reasonable to assume that the hopping perturbation does not alter the thermal equilibrium of the left and right charge reservoir and that a non-equilibrium steady state will eventually be reached. This argument is very strong and remains valid even for *noninteracting* leads. Actually, the partitioned approach by Caroli *et al.* was originally applied to a tight-binding model[@caroli1] describing a metal-insulator-metal tunneling junction and then extended to the case of free electrons subjected to an arbitrary one-body potential.[@caroli2] This extension was questioned by Feuchtwang;[@feuchtwang1; @feuchtwang2] the controversy was about the appropriate choice of boundary conditions for the uncontacted-system Green functions. In later years the non-equilibrium Green function techniques[@kb-book; @keldysh-jetp1965] in the partitioned approach framework were mainly applied to investigate steady-state situations. An important breakthrough in time-dependent non-equilibrium transport was achieved by Wingreen *et al.*[@wingreen; @jauho; @jauhocond; @haug] Still in the framework of the partitioned approach, they derive an expression for the fully nonlinear, time-dependent current in terms of the Green functions of the mesoscopic region (this embedding procedure holds only for noninteracting leads). Under the physical assumption that the initial correlations are washed out in the long-time limit, their formula is well suited to study the response to external time-dependent voltages and contacts. The limitations of the partitioned approach are essentially three. First, it is difficult to partition the electron-electron interactions between the leads and between the leads and the device. These interactions are important for establishing dipole layers and charge transfers which shape the potential landscape in the device region. Second, there is a crucial assumption of equivalence between the long-time behavior of the 1) initially partitioned and biased system once the *coupling* between the subsystems is established and 2) the whole partition-free system in thermal equilibrium once the *bias* is established. Third, the transient current has no direct physical interpretation since in a real experiment one switches on the bias and not the contacts; moreover, there is no well defined prescription which fixes the initial equilibrium distribution of the isolated central device. In this paper we use a partition-free scheme without the above limitations. This conceptually different time-dependent approach has been proposed by Cini.[@cini] He developed the general theory for the case of free electrons described in terms of a discrete set of states and a continuum set of states with focus on semiconductor junction devices. For a one-dimensional free-electron system subjected to a time-dependent perturbation of the form $U\Theta(t)\Theta(x)$, where $U$ is the applied bias and $(x,t)$ is the space-time variable, the Cini theory yields a current-voltage characteristics which agrees with the one obtained by Feuchtwang[@feuchtwang1; @feuchtwang2] in the partitioned approach. This result is particularly important since it shows that a steady state in a partition-free scheme develops even in the noninteracting case. Moreover, it demonstrates an equivalence which had previously been assumed. In the present work we extend the partition-free approach to noninteracting resonant tunneling systems and also to *interacting* such systems - in both cases using arbitrary time-dependent disturbances. We shall clarify under what circumstances a non-equilibrium steady state can develop and discuss the equivalence of the current-voltage characteristics obtained by Jauho *et al.*[@jauho] and that obtained by us. One of the advantages of the partition-free scheme over the traditional methods lies in the ability of the former to calculate transient physical (*i.e.* measurable) current responses. The plan of the paper is the following. In Section \[keldyshth\] we develop the general formalism which properly accounts for the initial correlations. We derive a solution of the Keldysh equations for the lesser and the greater Green functions in noninteracting and interacting systems. An exact and alternative treatment based on Time Dependent Density Functional Theory[@rg] (TDDFT) is proposed in order to calculate the total nonlinear time-dependent current. The current response of a noninteracting resonant tunneling system is discussed in Section \[model\]. We specify when the partitioned and the partition-free schemes yield the same asymptotic current (Theorem of Equivalence) and how this current may depend on history (Memory Loss Theorem). The general results are illustrated by model calculations. In Section \[interacting\] we consider an interacting resonant tunneling system with interacting leads. The TDDFT approach is compared with earlier works by Lang *et al.*[@Lang1; @Lang2] and Taylor *et al.*[@Taylor1; @Taylor2] Assuming that a steady state is reached we write down an exact formula for the nonlinear steady-state current. As a simple example we also study the current response in the Random Phase Approximation (RPA) of a capacitor-device-capacitor junction. Our main conclusions are summarized in Section \[conclusion\]. General Formulation {#keldyshth} =================== Noninteracting Systems in the Presence of an External Disturbance {#nonint} ----------------------------------------------------------------- Let us consider a system of noninteracting electrons described by an unperturbed Hamiltonian $$H_{0}=\sum_{mn}T_{m,n}c_{m}^{\dag}c_{n}, \quad\quad (\bT)_{m,n}=T_{m,n} \label{eq:h0}$$ and by a time-dependent disturbance of the form $$H_{U}(t)=\sum_{mn}U_{m,n}(t)c^{\dag}_{m}c_{n}, \quad [\bU(t)]_{m,n}=U_{m,n}(t), \label{eq:hu}$$ with $\bU(t)=0$ for any $t\leq t_{0}$. In Eqs. (\[eq:h0\])-(\[eq:hu\]), $c_m, c^\dagger_n$ are Fermi operators in some suitable basis, and we use boldface to indicate matrices in one-electron labels. Without loss of generality one can take $t_{0}=0$. The system is in equilibrium for negative times. ### Elementary Derivation We first obtain the Green function by elementary means without resorting to any Keldysh techniques. For a noninteracting system everything is known once we know how to propagate the one-electron orbitals in time and how they are populated before the system is perturbed. The time evolution is fully described by the retarded or advanced Green functions $\bG^{\rm R,A}$, and the initial population at zero time, *i.e.*, by $\bG^<(0;0)$. The real-time Green functions are defined by $$\bG^{\rm R,A}(t;t^\prime) = \pm \Theta(\pm t \mp t^\prime)\left[ \bG^{>}(t;t^\prime)-\bG^{<}(t;t^\prime) \right],$$ with $$\begin{aligned} G^>_{m,n}(t;t^\prime) &=& -i \langle c_m(t) c^\dagger_n(t^\prime) \rangle , \nonumber \\ G^<_{m,n}(t;t^\prime) &=& i \langle c^\dagger_n (t^\prime) c_m(t) \rangle, \nonumber \end{aligned}$$ where the operators are Heisenberg operators and where the averages are with respect to the equilibrium grand-canonical ensemble. Because there are no inter-particle interactions, the equation of motion for the electron operators simplifies to $$i \dot{c}_m(t) = \sum_n K_{m,n}(t)c_n(t)$$ where $\bK(t)\equiv \bT + \bU(t)$ is the full one-body Hamiltonian matrix. Consequently, the time evolution of $c_m$ is given by the one-electron evolution matrix $\bSS(t)$, $c_m(t) = \sum_n S_{m,n}(t) c_n(0)$, where $\bSS$ obeys $i\dot{\bSS}(t)=\bK(t)\bSS(t)$, with initial value $\bSS(0)$ = 1. We insert the time-evolved operators in the definitions of the $\bG$ matrices to obtain $$\bG^{\rm R,A}(t;t')=\mp i \Theta(\pm t \mp t^\prime) \bSS(t)\bSS^{\dag}(t'), \label{gra}$$ and $$\begin{aligned} \bG^{\lessgtr}(t; t') &=& \bSS(t) \bG^{\lessgtr}(0;0)\bSS^{\dag}(t') \nonumber \\ &=& \bG^{\rm R}(t;0) \bG^{\lessgtr}(0;0)\bG^{\rm A}(0;t'), \label{g<}\end{aligned}$$ where the last equality holds for any $t,t'>0$. We observe that the instantaneous current can be expressed in terms of $\bG^{<}(t; t)$, and thus the problem of finding the current is reduced to that of finding the retarded Green function and the equilibrium population of the one-electron levels. We note in passing that the initial populations can be expressed as $\bG^{<}(0;0)=i f(\bT)$, where $f$ is the Fermi function. Because $\bT$ is a matrix, so is $f(\bT)$. The above solution for the lesser/greater Green function was also derived by Cini[@cini] with an equation-of-motion approach. He also pointed out that they can be derived in the framework of the Keldysh formalism[@keldysh-jetp1965] as a finite-temperature extension of a treatment by Blandin *et al.*[@blandin] ### Derivation Based on the Keldysh Technique In this Subsection we give an alternative derivation of Eq. (\[g&lt;\]) using an extension of the Keldysh formalism. There are two reasons for giving another derivation. On one hand, we will use the Keldysh formalism taking due account of the prescribed integration along the imaginary axis. This will allow us to understand what kind of approximations are made in the partitioned approach. On the other hand, the derivation below clearly shows how the electron-electron interaction can be included. We introduce the Green function $$G_{m,n}(z;z')= -i\bra {\cal T} [c_{m}(z)\bar{c}_{n}(z')] \ket \label{gf}$$ which is path-ordered on the oriented contour $\g$ of Fig. \[transcontour\]. In Eq. (\[gf\]) $z=t+\t$ is the complex variable running on $\g$ with $t=\Re[z]$, $\t=i\,\Im[z]$ while $A$ and $B$ are the end-points of $\g$. Further, $c_m(z)$ and $\bar{c}_n(z)$ are Heisenberg operators defined by the non-unitary evolution operator for complex times $z$. They are in general not Hermitian conjugates of one another, but the usual equal-time anticommutation relations $\{ c_m(z), \bar{c}_n(z) \} = \delta_{m,n}$ are still obeyed. ![[Contour suited to include the effect of the initial correlations, see also Section \[intkel\].]{}[]{data-label="transcontour"}](fig1.eps) As before the average is the grand-canonical average. On the vertical track going from 0 to $-i\b$ we have $\bK(\t)=\bK(0)=\bT$ independent of $\t$. Therefore, the Green function satisfies the relations $$\begin{array}{l} \bG(A;z')=-{\rm e}^{\b\m}\bG(B;z')\\ \bG(z;A)=-{\rm e}^{-\b\m}\bG(z;B) \end{array}. \label{mrg<>}$$ Next, we write the total Hamiltonian $H(t)=H_{0}+H_{U}(t)$ as the sum of a diagonal term and an off-diagonal one $$H(t)=\sum_{m}\ve_{m}(t)c^{\dag}_{m}c_{m}+ \sum_{mn}V_{m,n}(t)c^{\dag}_{m}c_{n}.$$ The quantities $\ve_{m}(\t)=\ve_{m}$ and $V_{m,n}(\t)=V_{m,n}$ are constants on the vertical track. \[The decomposition above is completely general. In our model examples discussed later, the diagonal part $[\bcalE(z)]_{m,n}=\d_{m,n}\ve_{m}(z)$ will represent an uncontacted system and the off-diagonal $[\bV(z)]_{m,n}=V_{m,n}(z)$ the contacts.\] The Green function $\bG(z;z')$ is obtained by solving the equation of motion $$\left\{i\frac{d}{dz}-\bcalE(z)-\bV(z)\right\}\bG(z;z')=\d(z-z'), \label{eomg}$$ \[and its adjoint\] with boundary conditions (\[mrg&lt;&gt;\]). We define ${\bf g}(z;z')$ as the uncontacted Green function. The ${\bf g}$ satisfies Eq. (\[eomg\]) with $\bV=0$ and obeys the same boundary conditions of the contacted $\bG$. The unique ${\bf g}$ resulting from such a scheme belongs to the Keldysh space[@daniele] and can be written as $${\bf g}(z;z')=\Theta(z,z'){\bf g}^{>}(z;z')+\Theta(z',z){\bf g}^{<}(z;z'),$$ where $\Theta(z,z')=1$ if $z$ is later than $z'$ on $\g$ and zero otherwise. ${\bf g}^{>}(z;z')$ is analytic for any $z$ later than $z'$ while ${\bf g}^{<}(z;z')$ is analytic for any $z'$ later than $z$; they are given by $$\begin{aligned} {\bf g}^{<}(z;z')&=&if(\bcalE ){\rm e}^{-i\int_{z'}^{z}d\bar{z}\bcalE(\bar{z})}, \nonumber \\ {\bf g}^{>}(z;z')&=&i[f(\bcalE)-1]{\rm e}^{-i\int_{z'}^{z}d\bar{z}\bcalE(\bar{z})}, \label{lg<>}\end{aligned}$$ where $\bcalE\equiv\bcalE(0)$ and the integral appearing in the exponential function is a contour integral along $\g$ going from $z'$ to $z$. Choosing $z$ and $z'$ on the real axis ${\bf g}^{<}$ and ${\bf g}^{>}$ reduce to the real-time lesser and greater component. From Eqs. (\[lg&lt;&gt;\]) one can easily verify that the corresponding retarded and advanced component can be written as $${\bf g}^{\rm R,A}(t;t')=\mp i\Theta(\pm t\mp t') {\rm e}^{-i\int_{t'}^{t}d\bar{t}\bcalE(\bar{t})}. \label{lgra}$$ The uncontacted ${\bf g}$ allows to convert Eqs. (\[eomg\]) into an integral equation which preserves the relations (\[mrg&lt;&gt;\]): $$\bG(z;z')={\bf g}(z;z')+\int_{\g} d\bar{z}\;{\bf g}(z;\bar{z})\bV(\bar{z})\bG(\bar{z};z').$$ Using the Langreth theorem[@langreth] one finds $$\bG^{\lessgtr}=\left[\d+ \bG^{\rm R}\cdot \bV\right]\cdot {\bf g}^{\lessgtr} + \bG^{\lessgtr}\cdot \bV \cdot {\bf g}^{\rm A} +\bG^{\rceil}\star \bV \star {\bf g}^{\lceil}, \label{newder1}$$ where we have used the short hand notation $\cdot$ to denote integrals along the real axis, going from 0 to $\inf$, and $\star$ for integrals along the imaginary vertical track, going from 0 to $-i\b$. For the sake of clarity we have also introduced the symbols $\rceil$ and $\lceil$: any function with the superscript $\rceil$ is intended to have a real first argument and an imaginary second argument; the opposite is specified by $\lceil$. In Eq. (\[newder1\]), $\bV(z;z')\equiv \d(z-z')\bV(z)$; for $\bV$ we don’t need to say more since it is always foregone and followed by $\cdot$ or $\star$ so that no ambiguity arises. In particular we note that $\star \bV \star$ implies a simple matrix multiplication since along the vertical track $\bV$ is a constant matrix times the delta function. The equation for $\bG^{\lessgtr}$ contains $\bG(t;\t)$ with one real and one imaginary argument. This coupling does not allow to get a closed equation for $\bG(t;t')$ with two real arguments, unless $\bV=0$ on the vertical track. Conversely, $\bG^{\rm R}$ and $\bG^{\rm A}$ satisfy an integral equation without any coupling: $$\bG^{\rm R,A}={\bf g}^{\rm R,A}+ \bG^{\rm R,A}\cdot \bV\cdot {\bf g}^{\rm R,A}. \label{newder2}$$ Eq. (\[newder1\]) can be solved for $\bG^{\lessgtr}$ and one obtains $$\begin{aligned} \bG^{\lessgtr}&=&\left[\d+ \bG^{\rm R}\cdot \bV \right]\cdot {\bf g}^{\lessgtr}\cdot \left[ \d+\bV\cdot \bG^{\rm A} \right] \nonumber \\ &&+ \bG^{\rceil}\star \bV\star {\bf g}^{\lceil}\cdot \left[ \d+\bV\cdot \bG^{\rm A} \right]\;. \label{newder3}\end{aligned}$$ From Eq. (\[lg&lt;&gt;\]) and Eq. (\[lgra\]) we have ${\bf g}^{\lessgtr}(t;t')={\bf g}^{\rm R}(t;0) {\bf g}^{\lessgtr}(0;0){\bf g}^{\rm A}(0;t')$ and $ {\bf g}(\t;t)=-i{\bf g}(\t;0){\bf g}^{\rm A}(0;t)$, so that Eq. (\[newder3\]) can be rewritten as $$\begin{aligned} \bG^{\lessgtr}(t;t')&=&\bG^{\rm R}(t;0){\bf g}^{\lessgtr}(0;0) \bG^{\rm A}(0;t')\nonumber \\ && -i \left[\bG^{\rceil}\star \bV\star {\bf g}\right](t;0) \bG^{\rm A}(0;t'). \label{newder4}\end{aligned}$$ The above expression for $\bG^{\lessgtr}$ deserves a brief comment. Indeed, the first term on the r.h.s. is exactly what one got in the partitioned approach, where the hopping parameters $V_{m,n}$ vanish along the vertical track. It is usually argued that if $t,t'\ra\inf$ the second term vanishes. However, we point out that in the noninteracting case this is not true. If in the long-time limit some physical response functions, *e.g.*, the current, are correctly reproduced by using the partitioned $\bG^{\lessgtr}(t;t')= \bG^{\rm R}(t;0){\bf g}^{\lessgtr}(0;0)\bG^{\rm A}(0;t')$ other kind of argumentations should be invoked. We shall come to this point later on. To proceed further we need the Dyson equation for $\bG(t;\t)$. Exploiting the identity ${\bf g}(t;\t)=i{\bf g}^{\rm R}(t;0) {\bf g}(0;\t)$, we find $$\bG(t;\t)=i\bG^{\rm R}(t;0){\bf g}(0;\t)+ \left[\bG^{\rceil}\star \bV\star {\bf g}\right](t;\t). \label{newder6}$$ Eq. (\[newder6\]) can be solved for $\bG(t;\t)$. From the Dyson equation $\bG(\t;\t')={\bf g}(\t;\t')+ \left[\bG\star \bV\star {\bf g}\right](\t;\t')$, it follows that $\left[ \d-\bV\star {\bf g} \right]^{-1}(\bar{\t};\t)= \left[\d+\bV\star \bG\right](\bar{\t};\t)$ and hence $$\bG(t;\t)=i\bG^{\rm R}(t;0)\bG(0;\t). \label{newder9}$$ Substituting Eq. (\[newder9\]) into Eq. (\[newder4\]) one gets $$\bG^{\lessgtr}(t;t')=\bG^{\rm R}(t;0)\bG^{\lessgtr}(0;0) \bG^{\rm A}(0;t'). \label{newder11}$$ Eq. (\[newder11\]) coincides with Eq. (\[g&lt;\]), as it should. Interacting Systems in the Presence of an External Disturbance {#intkel} -------------------------------------------------------------- In the interacting case we keep track of the interactions by introducing a self-energy matrix. Then, Eq. (\[eomg\]) becomes $$\begin{aligned} \left\{i\frac{d}{dz}-\bcalE(z)-\bV(z)- \bS^{\d}(z)\right\}\bG(z;z') \nonumber \\ =\d(z-z') +\int_{\g}d{\bar z}\;\bS_{\rm c}(z;{\bar z})\bG({\bar z};z'). \label{ieom1td}\end{aligned}$$ Here $\bS^{\d}$ is the self-energy part which is local in time and it consists of a Hartree and an exchange term. The remaining part of the self energy $\bS_{\rm c}$ contains the contributions coming from the correlation and belongs to the Keldysh space:[@daniele] $$\bS_{\rm c}(z;z')=\Theta(z,z')\bS^{>}(z;z')+\Theta(z',z)\bS^{<}(z;z').$$ Like $\bG$, the self energy and its components are matrices in the one-electron labels. No simple expressions, like (\[gra\])-(\[g&lt;\]), can now be directly obtained from the equation of motion and the Keldysh formalism is unavoidable. A proper treatment of the initial correlations naturally leads to an extension of the Keldysh equations. The generalization was put forth by Wagner[@wagner] who obtained a minimal set of five independent integro-differential equations for the unknowns $\bG^{\rm R}$, $\bG^{\rm A}$, $\bG^{<}$ (or $\bG^{>}$) (or the Keldysh Green function $\bG^{\rm K}\equiv \bG^{>}+\bG^{<}$), $\bG^{\lceil}$ (or $\bG^{\rceil}$) and the thermal Green function $\bG$ with two imaginary arguments. In Appendix \[z\] we exploit the results of the previous Subsection to prove that the solution for $\bG^{\lessgtr}$ can be written as $$\bG^{\lessgtr}(t;t')=\bG^{\rm R}(t;0) \bG^{\lessgtr}(0;0)\bG^{\rm A}(0;t')+ \bDelta^{\lessgtr}(t;t'), \label{GGGG}$$ where $$\begin{aligned} \bDelta^{\lessgtr}(t;t')&=& i\bG^{\rm R}(t;0)\bG^{>}(0;t')- i\bG^{<}(t;0)\bG^{\rm A}(0;t') \nonumber \\ &&- \;\bG^{\rm R}(t;0) \bG^{\rm K}(0;0) \bG^{\rm A}(0;t') \nonumber \\ && + \left[\bG^{\rm R}\cdot\left[\bS^{\lessgtr}+ \bS^{\rceil}\star \bG\star \bS^{\lceil} \right]\cdot \bG^{\rm A}\right](t;t'). \quad\;\;\, \nonumber\end{aligned}$$ This result clearly reduces to Eq. (\[newder11\]) if the self energy vanishes since $\bG^{\rm R}(0;0)= [\bG^{\rm A}(0;0)]^{\dag}=-i$. We observe that if the Green functions vanish when the separation of their time arguments goes to infinity, Eq. (\[GGGG\]) yields a well known identity $$\lim_{t,t'\ra\inf}\bG^{\lessgtr}(t;t')= \left[\bG^{\rm R}\cdot\bS^{\lessgtr}\cdot \bG^{\rm A}\right](t;t'). \label{rsimp}$$ Eq. (\[rsimp\]) is well suited to study the *long-time* response of an interacting system subjected to an external time-dependent disturbance. On the other hand, if one is interested in the *short-time* response Eq. (\[GGGG\]) cannot be simplified. In some cases it might be simpler to use an alternative approach. Below we propose an exact non-equilibrium Green function treatment based on TDDFT and discuss the relations to ordinary Mean Field approximations. Mean Field Theory and Relations to TDDFT {#mftddft} ---------------------------------------- Any Mean Field Theory is a one-particle-like approximation in which each particle moves in an effective average potential independently of all other particles. The mean-field potential is local in time, meaning that $\bS_{\rm c}$ is discarded. Consequently, all the results of the Section \[nonint\] can be reused provided we substitute $\bK$ by $\bK+\bS^{\d}$. Thus, no extra complications arise if we treat an interacting system at the Hartree-Fock level. To be specific, let us focus on the Coulomb interaction and on paramagnetic systems (so that the self energy and the Green function are diagonal in the spin indices). Then, it is natural to choose the one-electron index as the coordinate ${\bf r}$ of the particle and to split the self energy $\S^{\d}_{{\bf r},{\bf r}'}(z)\equiv \S^{\d}({\bf r},{\bf r}',z)$ as a sum of the Hartree and the exchange term $$\S^{\d}({\bf r},{\bf r}',z)=V_{\rm H}({\bf r},z)\d({\bf r}-{\bf r}')+ \S_{\rm x}({\bf r},{\bf r}',z).$$ For extended systems, the Hartree potential $V_{\rm H}$ and the Coulomb potential from the nuclei $V_{\rm n}$ are separately infinite but with a finite sum. Together with the external field $U$ these terms form the classical electrostatic potential $U_{\rm C}=U+V_{\rm H}+V_{\rm n}$. The Green function $G_{{\bf r},{\bf r}'}(z;z')\equiv G({\bf r},z;{\bf r}',z')$ can be obtained from the self-consistent solution of the equation of motion and the lesser/greater component can be written as $$\begin{aligned} G^{\lessgtr}_{\rm MF}({\bf r},t;{\bf r}',t')&=& \int d{\bar {\bf r}}d{\bar {\bf r}}'\; G^{\rm R}_{\rm MF}({\bf r},t;{\bar {\bf r}},0) \nonumber \\ &&\times G^{\lessgtr}_{\rm MF}({\bar {\bf r}},0;{\bar {\bf r}}',0) G^{\rm A}_{\rm MF}({\bar {\bf r}}',0;{\bf r}',t'), \nonumber\end{aligned}$$ where the subscript MF has been used to stress that it is a Mean Field approximate result. In the ordinary Many Body Theory one has to abandon the one-particle picture in order to improve the approximation beyond the Hartree-Fock level. This leads to a self energy nonlocal in time and hence to the complicated solution (\[GGGG\]). In the case we only ask for the density $n({\bf r},t)= -i G^{<}({\bf r},t;{\bf r},t)$ the original Density Functional Theory[@hk; @ks] and its finite-temperature generalization[@mermin] has been extended to time-dependent phenomena.[@rg; @litong] The theory applies only to those cases where the external disturbance is local in space, *i.e.*, $U_{{\bf r},{\bf r}'}(t)=\d({\bf r}-{\bf r}')U({\bf r},t)$. For $t>0$ we switch on an external potential $U({\bf r},t)$ to obtain a density $n({\bf r},t)$. The Runge-Gross theorem states that if we instead had switched on a different $U'({\bf r},t)$ \[giving a different $n'({\bf r},t)$\], then $n({\bf r},t)=n'({\bf r},t)$ implies $U({\bf r},t)=U'({\bf r},t)$. Thus $U({\bf r},t)$ is a unique functional of $n({\bf r},t)$. Runge and Gross also show that one can compute $n({\bf r},t)$ in a one-particle manner using an effective potential $$U^{\rm eff}({\bf r},t)=U_{\rm C}({\bf r},t)+ v_{\rm xc}({\bf r},t).$$ Here, $v_{\rm xc}$ accounts for exchange and correlations and is obtained from an exchange-correlation action functional, $v_{\rm xc}({\bf r},t)= \d A_{\rm xc}[n]/\d n({\bf r},t)$. In our earlier language this corresponds to an effective self energy which is local in both space and time. The TDDFT one-particle scheme corresponds to a fictitious Green function ${\cal G}({\bf r},z;{\bf r}',z')$ which satisfies the equations of motion (\[eomg\]) with $[{\cal E}_{{\bf r},{\bf r}'}(z)+V_{{\bf r},{\bf r}'}(z)]$ replaced by $\d({\bf r}-{\bf r}')[-\nabla^{2}_{\bf r}/2+U^{\rm eff}({\bf r},z)]$. As a consequence we have $$\begin{aligned} {\cal G}^{\lessgtr}({\bf r},t;{\bf r}',t')&=& \int d{\bar {\bf r}}d{\bar {\bf r}}'\; {\cal G}^{\rm R}({\bf r},t;{\bar {\bf r}},0) \nonumber \\ &&\times {\cal G}^{\lessgtr}({\bar {\bf r}},0;{\bar {\bf r}}',0) {\cal G}^{\rm A}({\bar {\bf r}}',0;{\bf r}',t'). \nonumber \end{aligned}$$ The fictitious $\bcalG$ will not in general give correct one-particle properties. However by definition $\bcalG^{<}$ gives the correct density $$n({\bf r},t)=-2i {\cal G}^{<}({\bf r},t;{\bf r},t)$$ (where the factor of 2 comes from spin). Also total currents are correctly given by TDDFT. If for instance $J_{\a}$ is the total current from a particular region $\a$ we have $$J_{\a}(t)=-e\int_{\a}d {\bf r}\;\frac{d}{dt}n({\bf r},t) \label{cudft}$$ where the space integral extends over the region $\a$ ($e$ is the electron charge). The Density Functional Theory and the Runge-Gross extension refer specifically to the ${\bf r}$ basis. However, the arguments remain valid if we instead consider the diagonal density $n_{i}=\bra c^{\dag}_{i}c_{i}\ket$ in some other basis provided the interactions commute with the diagonal density operator. The latter condition is essential for the Runge-Gross theorem. Thus, for instance, if the one-electron indices refer to a particular lead one can still use Eq. (\[cudft\]) to calculate the corresponding total current (see Section \[interacting\]). For later references we now derive an expression for the lesser/greater Green function in the linear approximation. We consider the partition-free system described in the one-particle scheme of Mean Field Theory or TDDFT. Let $\d \bU^{\rm eff}(t)$ be the small time-dependent effective perturbation and $\d \bG^{\rm R,A}=\bG_{0}^{\rm R,A}\cdot \d\bU^{\rm eff}\cdot \bG_{0}^{\rm R,A}$ be the first order variation of the retarded and advanced Green functions with respect to their equilibrium counterparts $\bG_{0}^{\rm R,A}$. Then, from Eq. (\[newder11\]) we get $$\begin{aligned} \d \bG^{\lessgtr}(t;t') \quad\quad\quad\quad\quad \quad\quad\quad\quad\quad \quad\quad\quad\quad\quad \quad\quad\quad \label{gamdp2} \\ =\int d{\bar t}\; \bG_{0}^{\rm R}(t;{\bar t}) \d \bU^{\rm eff}({\bar t})\bG^{\lessgtr}(0;0)\bG_{0}^{\rm R}({\bar t};0) \bG_{0}^{\rm A}(0;t') \nonumber \\ \;\; + \int d{\bar t}\;\bG_{0}^{\rm R}(t;0) \bG_{0}^{\rm A}(0;{\bar t}) \bG^{\lessgtr}(0;0)\d \bU^{\rm eff}({\bar t})\bG_{0}^{\rm A}({\bar t};t'), \nonumber\end{aligned}$$ where we have taken into account that $\bG_{0}^{\rm R,A}$ commutes with $\bG^{\lessgtr}(0;0)$. The above expression takes an elegant form when $t'=t$. Indeed, for any $t>{\bar t}>0$ one has $\bG_{0}^{\rm R}({\bar t};0) \bG_{0}^{\rm A}(0;t)=-i\bG_{0}^{\rm A}({\bar t};t)$ and $\bG_{0}^{\rm R}(t;0)\bG_{0}^{\rm A}(0;{\bar t})= i\bG_{0}^{\rm R}(t;{\bar t})$. Since the integrands in Eq. (\[gamdp2\]) vanish for ${\bar t}>t$ due to the $\Theta$ function in $\bG_{0}^{\rm R}$ in the first term and in $\bG_{0}^{\rm A}$ in the second term, we conclude that for any positive time $t$ $$\d \bG^{\lessgtr}(t;t)=-i\int d{\bar t}\; \bG_{0}^{\rm R}(t;{\bar t})[\d \bU^{\rm eff}({\bar t}),\bG^{\lessgtr}(0;0)] \bG_{0}^{\rm A}({\bar t};t). \label{ga20df;}$$ We shall use this equation later on to calculate the linear current response in noninteracting and interacting resonant tunneling systems. Noninteracting Resonant Tunneling Systems {#model} ========================================= As a first application of the partition-free approach we study the time-dependent current response of a noninteracting resonant tunneling system. For the sake of simplicity the central device will be modeled by a single localized level. All the results of this Section can be generalized to the case of a multi-level noninteracting central device without any conceptual complications. There are many different geometries one can conceive beyond a one-level model, *e.g.*, a double quantum dots model,[@ziegler] a quantum wire coupled to a quantum dot,[@liang] a one-dimensional quantum-dot array[@shangguam] or a mesoscopic multi-terminal system.[@sun] However, the present paper is not intended to give a description of a series of applications. Rather, we prefer to illustrate how the partition-free approach works in a simple noninteracting model. We also emphasize that *all* the results of this Section remain valid in the interacting case if the bare external potential is replaced by the *exact* effective potential of TDDFT, see Section \[interacting\]. The whole system is described by a quadratic Hamiltonian $$\begin{aligned} H_{0}&=&\sum_{k\a}\ve_{k\a}c^{\dag}_{k\a}c_{k\a}+\ve_{0}c^{\dag}_{0}c_{0} \label{qham} \\ &&+ \sum_{k\a}V_{k\a}[c^{\dag}_{k\a}c_{0}+c^{\dag}_{0}c_{k\a}] \equiv \sum_{mn}T_{m,n}c^{\dag}_{m}c_{n}, \nonumber \end{aligned}$$ where $\a=L,R$ denotes the left, right lead and $m,n$ are collective indices for $k\a$ and 0. We assume the system in thermodynamic equilibrium at a given inverse temperature $\b$ and chemical potential $\m$ before the time-dependent perturbation $$\begin{aligned} H_{U}(t)&=&\sum_{k\a}U_{k\a}(t)c^{\dag}_{k\a}c_{k\a}+ U_{0}(t)c^{\dag}_{0}c_{0} \nonumber \\ &\equiv& \sum_{mn}U_{m,n}(t)c^{\dag}_{m}c_{n} \nonumber\end{aligned}$$ is switched on. In principle the time-dependent perturbation may have off-diagonal matrix elements. In order to model a uniform potential deep inside the electrodes such off-diagonal terms must be of lower order with respect to the system size. However their inclusion is trivial and it does not lead to any qualitative changes. The current from the $\a$ contact through the $\a$ barrier to the central region can be calculated from the time evolution of the occupation number operator $N_{\a}$ of the $\a$ contact. From the obvious generalization of Eq. (\[cudft\]) one readily finds $$\begin{aligned} J_{\a}(t)&=& 2e\sum_{k}\;\Re\left[G^{<}_{0,k\a}(t;t)\right]V_{k\a} \label{current1} \\ &=& 2e\sum_{k}\;\Re\left[\bG^{\rm R}(t;0)\bG^{<}(0;0) \bG^{\rm A}(0;t)\right]_{0,k\a}V_{k\a}. \nonumber\end{aligned}$$ The above expression is manifestly gauge-invariant. Indeed, if $U_{m,n}(t)\ra U_{m,n}(t)+\d_{m,n}\chi(t)$ then $\bG^{\rm R}(t;0)\ra {\rm e}^{-i\int_{0}^{t}\chi(\bar{t})d\bar{t}}\bG^{\rm R}(t;0)$ while $\bG^{\rm A}(0;t)\ra {\rm e}^{i\int_{0}^{t}\chi(\bar{t})d\bar{t}}\bG^{\rm A}(0;t)$ and the time-dependent shift $\chi(t)$ has no effect on the current response. In the same way it is invariant under a simultaneous shift of $\m$ and the initial potential. The matrix $\bG^{<}(0;0)$ can be written as[@lw] $$\bG^{<}(0;0)=\int_{\G}\frac{d\z}{2\p} \frac{f(\z){\rm e}^{\eta\z}}{\z-\bcalE-\bV},$$ where $\G$ is the contour surrounding all the Matzubara frequencies $\w_{n}=(2n+1)\p i/\b+\m$ clockwise \[see Fig. \[lwcontour\] in Appendix \[a2\]\] while $\eta$ is an infinitesimally small positive constant. It is therefore convenient to define the kernel $$Q_{\a}(\z;t)=\sum_{k}\left[ \bG^{\rm R}(t;0)\bG(\z)\bG^{\rm A}(0;t)\right]_{0,k\a}V_{k\a}, \label{qalfa}$$ with $\bG(\z)=[\z-\bcalE-\bV]^{-1}$, and to write the current in the form $$J_{\a}(t)= 2e\;\Re\left[\int_{\G}\frac{d\z}{2\p}f(\z){\rm e}^{\eta\z} Q_{\a}(\z;t)\right]. \label{current2}$$ It is worth noticing that the partitioned approach leads to Eq. (\[current1\]) with ${\bf g}^{<}(0;0)=if(\bcalE)$ in place of $\bG^{<}(0;0)=if(\bcalE+\bV)$. It is our intention to clarify under what circumstances, if any, the long-time behavior of the time-dependent current is not affected by this replacement. As a side remark we also observe that $J_{\a}(t=0)$ in Eq. (\[current1\]) correctly vanishes. Letting $|\l_{n}\ket$ and $\l_{n}$ be the eigenvectors and eigenvalues of $\bT=\bcalE+\bV$, we have $$J_{\a}(0)= -2e\sum_{k}\sum_{n}\Im\left[\bra 0|\l_{n}\ket f(\l_{n}) \bra\l_{n}|k\a\ket\right]V_{k\a}=0,$$ since $\bra 0|\l_{n}\ket$ and $\bra\l_{n}|k\a\ket$ can always be chosen as real quantities for systems with time reversal symmetry. Step-Like Modulation -------------------- The first exactly solvable model we wish to consider is a step-like modulation, *i.e.*, $U_{m,n}(t)=\Theta(t)U_{m,n}$. From Eq. (\[gra\]) it follows that for any $t>0$ $$\bG^{\rm R}(t;0)=-i{\rm e}^{-i[\bcalE+\bU+\bV]t} \equiv \int\frac{d\w}{2\p}{\rm e}^{-i\w t}\bG^{\rm R}(\w), \label{grt>0}$$ and $\bG^{\rm A}(0;t)=[\bG^{\rm R}(t;0)]^{\dag}$. The device component of $\bG^{\rm R,A}(\w)$ can be written as $$G^{\rm R,A}_{0,0}(\w)=\frac{1}{\w-\tilde{\ve}_{0}-\S^{\rm R,A}(\w)\pm i\eta}, \label{g00}$$ where $\tilde{\ve}_{0}=\ve_{0}+U_{0}$. Here, $\S^{\rm R,A}(\w)= \sum_{\a}\S^{\rm R,A}_{\a}(\w)$ is the retarded/advanced self energy induced by back and forth virtual hopping processes from the localized level to the leads and is given by $$\S^{\rm R,A}_{\a}(\w)= \sum_{k}\frac{V_{k\a}^{2}}{\w-\tilde{\ve}_{k\a}\pm i\eta}, \label{ser}$$ where we have used the short-hand notation $\tilde{\ve}_{k\a}= \ve_{k\a}+U_{k\a}$. ### $U_{k\a}=U_{\a}$: Steady-State Current If $U_{k\a}=U_{\a}$ the energy levels of the $\a$ lead are equally shifted. From Eq. (\[current1\]) it follows that we need to estimate the matrix elements $G^{\rm R}_{0,0}(t;0)$, $G^{\rm R}_{0,k\a}(t;0)$ of the retarded Green function and the two contractions $\sum_{k}G^{\rm A}_{0,k\a}(0;t)V_{k\a}$, $\sum_{k}G^{\rm A}_{k'\a',k\a}(0;t)V_{k\a}$ in the long-time limit. We assume that $\S_{\a}^{\rm R,A}(\w)$ is a smooth function for all real $\w$. Then, using the the Riemann-Lebesgue theorem one can prove (see Appendix \[a1\]) that the kernel $Q_{\a}(\z;t)$ has the following asymptotic behavior $$\begin{aligned} \lim_{t\ra\inf}Q_{\a}(\z;t)= \int\frac{d\ve}{2\p}\frac{\G_{\a}(\ve)}{\z-\ve+ U_{\a}} G^{\rm R}_{0,0}(\ve)\quad\quad\quad\quad \label{INTERM} \\ + \sum_{\a'}\int\frac{d\ve}{2\p} \frac{\G_{\a'}(\ve)}{\z-\ve+ U_{\a'}} \left|G^{\rm R}_{0,0}(\ve)\right|^{2} \S^{\rm A}_{\a}(\ve), \nonumber\end{aligned}$$ where $$\G_{\a}(\ve)=-2\Im[\S^{\rm R}_{\a}(\ve)]= 2\p\sum_{k}\d(\ve-\tilde{\ve}_{k\a})V^{2}_{k\a}. \label{gammalfa}$$ In Eq. (\[INTERM\]) the r.h.s. has a simple pole structure in the $\z$ variable and therefore the integration along the $\G$ contour can be easily performed. Using the identity $\int_{\G}(d\z/2\p)f(\z){\rm e}^{\eta\z} (\z-\ve)^{-1}=if(\ve)$ the stationary current $J^{(\rm S)}_{\a}\equiv \lim_{t\ra\inf}J_{\a}(t)$ has the following expression $$\begin{aligned} J^{(\rm S)}_{R}&=&-e\int \frac{d\ve}{2\p} \frac{\G_{R}(\ve)\G_{L}(\ve)}{[\ve-\tilde{\ve}_{0}-\L(\ve)]^{2}+ [\G(\ve)/2]^{2}} \nonumber \\ &&\quad\quad \times [f(\ve-U_{L})-f(\ve-U_{R})] =-J^{(\rm S)}_{L}, \label{js0}\end{aligned}$$ where $\L(\ve)=\Re[\S^{\rm R}(\ve)]$ is the Hilbert transform of $\G(\ve)=\sum_{\a}\G_{\a}(\ve)$: $$\L(\w)=P\int\frac{d\w'}{2\p}\frac{\G(\w')}{\w-\w'}. \label{lambdaalfa}$$ It is of interest to note that the dependence on the bias $U_{\a}$ appears not only in the distribution function $f$ but also in the quantities $\G$ and $\L$, see Eqs. (\[gammalfa\])-(\[lambdaalfa\]). The dependence of the self energy on the level-shifts is physical since when the particle visits the reservoirs experiences the applied potential. We also remark that Eq. (\[js0\]) is of the Landauer type.[@landauer] More generally the Landauer formula is valid for any mesoscopic device provided it is noninteracting. This result agrees with the one obtained in the partitioned approach by Jauho and coworkers.[@jauho; @haug] There the leads are decoupled from the central device and in thermal equilibrium at different chemical potential $\m_{L}$ and $\m_{R}$ and inverse temperature $\b_{L}$ and $\b_{R}$ in the remote past. In order to preserve charge neutrality each energy level $\ve_{k\a}$ must be shifted by $\m_{\a}-\m$ where $\m$ is the chemical potential of the two undisturbed leads. The stationary current is then obtained by switching on the contacts, *i.e.*, the hybridization part of the Hamiltonian. By tuning $\b_{R}=\b_{L}=\b$ and $\m_{R}-\m_{L}=U_{R}-U_{L}$ the current is given by Eq. (\[js0\]). To summarize we have found that for noninteracting leads a steady state develops in the long-time limit whenever 1) The one-body levels of the charge reservoirs form a continuum and 2) The self energy due to the hopping term is a smooth function. Under these hypotheses the time-translational invariance is restored by means of a *dephasing mechanism*. The comparison of our result with the one obtained in the partitioned scheme provides the criteria of equivalence: besides the tuning $\m_{L}-\m_{R}=U_{L}-U_{R}$ one needs to shift the levels of the $\a$ reservoir by $\m_{\a}-\m$. ### $U_{k\a}=U_{\a}$: Time-Dependent Current in the Wide-Band Limit The calculation of the stationary current is greatly simplified by the long-time behavior of the various terms coming from Eq. (\[current1\]). However, as far as we are interested in the current at any finite time we need to specify the structure of the retarded (advanced) self energy. Here, we consider the so called wide-band limit where the level-width functions $\G_{\a}(\w)\equiv 2\g_{\a}$ are assumed to be constant and hence, from Eq. (\[lambdaalfa\]), $\L_{\a}(\w)=0$. In this case $G^{\rm R}_{0,0}(\w)$ has a simple-pole structure and the calculations are slightly simplified. We emphasize that what follows is the first explicit result of a time-dependent current in a model system in the framework of a partition-free approach and therefore also a simple model could be of some interest. Without loss of generality we can always choose $\ve_{0}=0$; for the sake of simplicity we also consider $U_{0}=0$. We defer the reader to Appendix \[a2\] for the details. Here, we just write down the final result for $J_{\a}(t)$: $$\begin{aligned} &&J_{\a}(t)=J_{\a}^{(\rm S)}-4e\g_{\a}{\rm e}^{-\g t} \int\frac{d\w}{2\p}f(\w) \label{DJFNFCO} \\ &&\times\left[U_{\a} \;\Im\left\{\frac{{\rm e}^{i(\w+U_{\a})t}} {(\w+i\g)(\w+U_{\a}+i\g)}\right\}+ \sum_{\a'}\g_{\a'}U_{\a'}\; \quad \right.\nonumber \\ &&\left.\times \frac{U_{\a'}{\rm e}^{-\g t}+2\w\cos[(\w+U_{\a'})t]+2\g\sin[(\w+U_{\a'})t]} {[\w^{2}+\g^{2}][(\w+U_{\a'})^{2}+\g^{2}]}\right], \nonumber \end{aligned}$$ where $J_{\a}^{(\rm S)}$ is the stationary current of Eq. (\[js0\]) and $\g=\g_{R}+\g_{L}$. One can easily check that 1) For $t\ra\inf$ Eq. (\[DJFNFCO\]) yields the result in Eq. (\[js0\]), 2) For $t=0$ the current vanishes, that is $J_{\a}(0)=0$ and 3) For $U_{L}=U_{R}=0$ the current vanishes for any $t$. Eq. (\[DJFNFCO\]) can be rewritten in a more physical and compact way if we exploit the particle-number conservation. Denoting by $n_{0}$ the particle number operator in the central device we have $$J_{R}(t)+J_{L}(t)=e\frac{d}{dt}\bra n_{0}\ket,$$ so that $$\begin{aligned} J_{R}(t)=J_{R}^{(\rm S)}+e\frac{\g_{R}}{\g}\frac{d}{dt}\bra n_{0}\ket- 4e\frac{\g_{R}\g_{L}}{\g}{\rm e}^{-\g t} \int\frac{d\w}{2\p} \quad \nonumber \\\times \Im\left\{\frac{f(\w)}{\w+i\g}\left[ U_{R}\frac{{\rm e}^{i(\w+U_{R})t}}{\w+U_{R}+i\g}- U_{L}\frac{{\rm e}^{i(\w+U_{L})t}}{\w+U_{L}+i\g} \right]\right\}; \nonumber\end{aligned}$$ $J_{L}(t)$ is obtained by exchanging $R\leftrightarrow L$ in the r.h.s. of the above expression. Therefore, $J_{R}(t)\neq -J_{L}(t)$ for any finite time $t$, even in the symmetric case $\g_{R}=\g_{L}$; the time derivative of $\bra n_{0}\ket$ contributes to $J_{R}$ and $J_{L}$ in the same way. Our formula for the nonlinear transient current clearly differ from the one obtained by Jauho *et al.*[@jauho] in the partitioned scheme. Indeed, the prescribed integration along the imaginary axis gives extra terms (see Appendix \[a2\]) which are absent if the system is uncontacted for negative times. We have explicitly verified that by discarding these terms our formula reduces to the one obtained in the partitioned scheme. For long times, the extra terms vanish and our scheme reproduces the earlier steady-state results. If one of the two leads does not undergo any level shift, *e.g.* $U_{R}=0$, from Eq. (\[DJFNFCO\]) we get $$\begin{aligned} J_{R}(t)=J_{R}^{(\rm S)} -4e\g_{R}\g_{L}U_{L}{\rm e}^{-\g t}\int\frac{d\w}{2\p}f(\w) \quad\quad\quad\quad \label{tdcr} \\ \times \frac{U_{L}{\rm e}^{-\g t}+2\w\cos[(\w+U_{L})t]+2\g\sin[(\w+U_{L})t]} {[\w^{2}+\g^{2}][(\w+U_{L})^{2}+\g^{2}]}. \nonumber\end{aligned}$$ The transient behavior of the time-dependent quantity $J_{\a}(t)- J_{\a}^{(\rm S)}$ is not simply an exponential decay. In Fig. \[transiente1\] we have plotted $J_{R}(t)$ in Eq. (\[tdcr\]) versus $t$ for different values of the applied bias $U_{L}$ at zero temperature. The current strongly depends on $U_{L}$ for small $U_{L}$ while it is fairly independent of it in the strong bias regime; using the parameter specified in the caption, the time-dependent current has essentially the same shape for any $U_{L}\gtrsim 8$. ![[Time-dependent current $J_{R}(t)$ for different values of the applied bias $U_{L}=0.8,\;2.0,\;4.0$ and $6.0$. The numerical integration has been done with $\g_{R}=\g_{L}=0.2$, $\m=0$ and zero temperature.]{}[]{data-label="transiente1"}](fig3.eps) ![[Time-dependent current for $U_{L}=6$, $\m=0$ and zero temperature versus time for three different values of the line widths $\g_{R}=\g_{L}=0.2,\;0.5$ and $1.0$.]{}[]{data-label="transiente2"}](fig4.eps) In Fig. \[transiente2\] the current $J_{R}(t)$ is plotted for different values of the total line width $\g$ and for a fixed value $U_{L}=6$ of the applied bias. As expected, the larger is $\g$ the bigger is the slope of the current in $t=0$. ![[Stationary current versus the applied bias at zero temperature and chemical potential for three different values of the line widths $\g_{R}=\g_{L}=0.2,\;0.5$ and $1.0$.]{}[]{data-label="stationary"}](fig5.eps) Finally, in Fig. \[stationary\] we report the trend of the stationary current $J_{R}^{(\rm S)}$ as a function of the bias $U_{L}$ for three different choices of the level widths. As one can see the bigger is $\g$ and the wider is the range of validity of the Ohm law. Arbitrary Modulation: Theorem of Equivalence {#toe} -------------------------------------------- We have shown that the steady-state current induced by a step-like modulation does not change if one uses the $\bG^{<}$ of the partitioned approach, given by the first term on the r.h.s. of Eq. (\[newder4\]), in place of the one coming from the partition-free approach. This reasonable result is now *proved* and not simply *postulated*. The equivalence between the two expressions for the current is of special importance since it is much easier to work in the partitioned scheme. However, it has been proved only for step-like modulations with $U_{k\a}=U_{\a}$. Here, we prove that the above equivalence remains true under very general assumptions. To this end we consider the quantity $$\S^{\rm R,A}_{\a,{\cal V}}(t;t')= \sum_{k}\,\tg^{\rm R,A}_{k\a}(t;t') {\cal V}^{2}_{k\a} \label{fse}$$ where ${\cal V}$ is an arbitrary complex function of $k\a$. Then, the following theorem holds *: If $$\lim_{t\ra\inf}\S^{\rm R}_{\a,{\cal V}}(t;t')= \lim_{t\ra\inf}\S^{\rm A}_{\a,{\cal V}}(t';t)=0 \label{limco}$$ for any nonsingular ${\cal V}$, then $$\lim_{t\ra\inf} [Q_{\a}(\z;t)-q_{\a}(\z;t)]=0, \label{eoe}$$ where $q_{\a}(\z;t)\equiv\sum_{k} [\bG^{\rm R}(t;0){\bf g}(\z)\bG^{\rm A}(0;t)]_{0,k\a}V_{k\a}$, and ${\bf g}(\z)=[\z-\bcalE]^{-1}$ is the uncontacted Green function.* Eq. (\[eoe\]) says that if we apply the same time-dependent perturbation the same asymptotic current will emerge in the partitioned and partition-free approaches. ** : In terms of the self energy $\S^{\rm R}_{V}=\sum_{\a}\S^{\rm R}_{\a,V}$, the equation of motion for $G^{\rm R}_{0,0}$ takes the form $$\left\{i\frac{d}{dt}-\ve_{0}(t)\right\} G^{\rm R}_{0,0}(t;t')- [\S^{\rm R}_{V}\cdot G^{\rm R}_{0,0}](t;t')=\d(t-t'),$$ where the symbol $\cdot$ denotes the real-time convolution. We now consider the limit $t\ra\inf$. The hypothesis (\[limco\]) implies $$\lim_{t\ra\inf}G^{\rm R}_{0,0}(t;t')=0, \label{gr00t0}$$ which in turn implies that $[\S^{\rm R}_{V}\cdot G^{\rm R}_{0,0}](t;t') \stackrel{t\ra\inf}{\ra}0$. Furthermore, from the Dyson equation for $G^{\rm A}_{0,k\a}$ we find $$\sum_{k}G^{\rm A}_{0,k\a}(t';t)V_{k\a}= [G^{\rm A}_{0,0}\cdot\S^{\rm A}_{\a,V}](t';t) \stackrel{t\ra\inf}{\ra}0. \label{asycon}$$ We note that the above two asymptotic relations have been obtained for $t'=0$ in the special case of a step-like modulation, see Eq. (\[asymg00\]). Here, we have shown that they hold in a more general context. As a consequence of these two identities, the asymptotic difference $[Q_{\a}(\z;t)-q_{\a}(\z;t)]$ can be written as $$\left[G^{\rm R}_{0,0}\cdot\S^{\rm R}_{\cal V} \right](t;0)\times G_{0,0}(\z)\times\S^{\rm A}_{\cal V}\cdot \left[\d+G^{\rm A}_{0,0}\cdot\S^{\rm A}_{\a,V}\right](0;t).$$ Here, $\S^{\rm R}_{\cal V}=\sum_{\a}\S^{\rm R}_{\a,\cal V}$ is given by Eq. (\[fse\]) with ${\cal V}^{2}_{k\a}=V^{2}_{k\a}/(\z-\ve_{k\a})$. Since $\z\in\G$, $\Im[\z]\neq 0$ and hence ${\cal V}^{2}_{k\a}$ is nonsingular, meaning that Eq. (\[limco\]) holds. Eq. (\[limco\]) together with Eq. (\[asycon\]) imply the equation of equivalence (\[eoe\]). As a simple application of the Theorem of Equivalence one can calculate the stationary current for an arbitrary step-like modulation. The quantity $q_{\a}(\z;t)$ is simply given by the first two terms of Eq. (\[statcurint\]). Both have a simple pole structure in the $\z$ variable and we can perform the integration along the contour $\G$. Using the definition in Eq. (\[gammalfa\]), with $\tilde{\ve}_{k\a}=\ve_{k\a}+U_{\a}(\ve_{k\a})$, one obtains $$\begin{aligned} J^{(\rm S)}_{R}&=&-e\int\frac{d\ve}{2\p}f(\ve) \label{stacg} \\ &&\times\left\{\G^{(0)}_{L}(\ve)\G_{R}(\ve+U_{L}(\ve)) \left|G^{\rm R}_{0,0}(\ve+U_{L}(\ve)) \right|^{2}\right. \nonumber \\ &&\left.+ \G^{(0)}_{R}(\ve)\G_{L}(\ve+U_{R}(\ve)) \left|G^{\rm R}_{0,0}(\ve+U_{R}(\ve)) \right|^{2} \right\} \nonumber\end{aligned}$$ The quantity $\G^{(0)}_{\a}(\ve)\equiv 2\p\sum_{k}V_{k\a}^{2}\d(\ve-\ve_{k\a})$ is the equilibrium line width. Eq. (\[stacg\]) reduces to Eq. (\[js0\]) if $U_{\a}(\ve)=U_{\a}$ since in this case $\G^{(0)}_{\a}(\ve-U_{\a})=\G_{\a}(\ve)$. In the noninteracting case it is reasonable to assume that Eq. (\[stacg\]) yields the steady-state current even for an arbitrary time-dependent disturbance such that $\lim_{t\ra\inf}U_{k\a}(t)=U_{k\a}$ and $\lim_{t\ra\inf}U_{0}(t)=U_{0}$. In the next Section we shall prove that the asymptotic current has no memory and depends only on the asymptotic value of the external perturbation. Memory Loss Theorem ------------------- If the condition (\[limco\]) of the Theorem of Equivalence is fulfilled, the asymptotic value of the nonlinear time-dependent current in Eq. (\[current2\]) simplifies to $$\begin{aligned} J_{\a}(t)=&& 2e\;\Re\left[\int_{\G}\frac{d\z}{2\p}f(\z){\rm e}^{\eta\z} q_{\a}(\z;t)\right]\label{asymcur} \\ &&=2e\;\Re\left[ \sum_{k\b}G^{\rm R}_{0,k\b}(t;0)\tg^{<}_{k\b}(0;0) \right. \nonumber \\ && \quad\quad\quad\quad \quad\quad\left.\times \sum_{k'}G^{\rm A}_{k\b,k'\a}(0;t)V_{k'\a}\right]. \nonumber\end{aligned}$$ We note in passing that expressing $G^{\rm R}_{0,k\b}$ and $G^{\rm A}_{k\b,k'\a}$ in terms of $G^{\rm R}_{0,0}$ and $G^{\rm A}_{0,0}$ respectively, Eq. (\[asymcur\]) can be rewritten in terms of $\S^{<}_{\a,V}=\sum_{k}\tg^{<}_{k\a}V^{2}_{k\a}$ $$J_{\a}(t)=2e\;\Re\left\{ [G^{\rm R}_{0,0}\cdot\S^{<}_{\a,V}](t;t)+ [G^{<}_{0,0}\cdot\S^{\rm A}_{\a,V}](t;t) \right\},$$ where the asymptotic relation $G^{<}_{0,0}=\sum_{\a}G^{\rm R}_{0,0}\cdot \S^{<}_{\a,V}\cdot G^{\rm A}_{0,0}$ has been used \[see Eq. (\[rsimp\])\]. This agrees with the result obtained by Wingreen *et al.*[@wingreen; @jauho] in the partitioned approach, as it should. In general $J_{\a}(t\ra\inf)$ is not a constant unless the external perturbation tends to a constant in the distant future. In this case the following theorem holds *: If $$\lim_{t\ra\inf}U_{\a}(\ve,t)=U_{\a}(\ve), \quad \lim_{t\ra\inf}U_{0}(t)=U_{0}$$ the current $J_{\a}(t)$ tends to a constant, given by Eq. (\[stacg\]), in the long-time limit.* ** : Is convenient to denote with ${\bar G}$ and $\bar{\tg}$ the Green functions corresponding to the step-like modulation with coefficients $U_{\a}(\ve)$ and $U_{0}$. We have already shown that in the long-time limit Eq. (\[asymcur\]) yields Eq. (\[stacg\]) if $G^{\rm R,A}=\bar{G}^{\rm R,A}$. The Memory Loss Theorem is then proved if $$\lim_{t\ra\inf}\frac{\bar{G}^{\rm R}_{0,k\b}(t;0)} {G^{\rm R}_{0,k\b}(t;0)}= {\rm e}^{i\D_{k\b}}= \lim_{t\ra\inf}\frac{\sum_{k'}G^{\rm A}_{k\b,k'\a}(0;t)V_{k'\a}} {\sum_{k'}\bar{G}^{\rm A}_{k\b,k'\a}(0;t)V_{k'\a}} \label{wie3of3}$$ for some real constant $\D_{k\b}$ . According to Eq. (\[gr00t0\]), the device component of the retarded Green function $G^{\rm R}_{0,0}(t\ra\inf;t')$ vanishes for any finite $t'$. Since $\lim_{t,t'\ra\inf} [\tg_{0}^{\rm R}(t;t')/\bar{\tg}_{0}^{\rm R}(t;t')]=1$, from $G^{\rm R}_{0,0}=\tg_{0}^{\rm R}+\tg_{0}^{\rm R}\cdot\S^{\rm R}_{V}\cdot G^{\rm R}_{0,0}$ it follows that $\lim_{t,t'\ra\inf}[G^{\rm R}_{0,0}(t;t')/\bar{G}^{\rm R}_{0,0}(t;t')]= 1$. Let us now consider the Dyson equation $$G^{\rm R}_{0,k\a}(t;t')=\int d\bar{t}\; G^{\rm R}_{0,0}(t;\bar{t})V_{k\a} \tg^{\rm R}_{k\a}(\bar{t};t'), \label{gr0kacp}$$ with $t'=0$ and $t\ra\inf$. Since the integrand vanishes for any finite $\bar{t}$, we can substitute $G^{\rm R}_{0,0}$ with $\bar{G}^{\rm R}_{0,0}$. Furthermore, since the applied bias tends to a constant in the distant future, $\lim_{t\ra\inf}[\tg^{\rm R}_{k\a}(t;0)/\bar{\tg}^{\rm R}_{k\a}(t;0)]= {\rm e}^{-i\D_{k\a}}$ for some real quantity $\D_{k\a}$. The l.h.s. of Eq. (\[wie3of3\]) is then proved. A similar reasoning leads to the r.h.s. of Eq. (\[wie3of3\]). Linear Response in the Wide-Band Limit -------------------------------------- In the case of small time-dependent perturbations, one can use Eq. (\[ga20df;\]) to calculate the lesser Green function. In order to carry on the calculations analytically we consider the wide-band limit and we choose $\d U_{k\a}(t)=\d U_{\a}(t)$. For simplicity we omit the subscript 0 in the retarded and advanced equilibrium Green functions. By explicitly writing down the matrix product in Eq. (\[ga20df;\]) one readily realize that we have to calculate the functions $G_{0,0}^{\rm R}(t-{\bar t})$, $G_{0,k'\a'}^{\rm R}(t-{\bar t})$, $\sum_{k}V_{k\a}G_{0,k\a}^{\rm A}({\bar t}-t)$ and $\sum_{k}V_{k\a}G_{k'\a',k\a}^{\rm A}({\bar t}-t)$. They are easily obtained from Eqs. (\[az,‘/\]) by simply replacing $\tilde{\ve}\ra\ve$ and $t\ra t-{\bar t}$. The calculations are rather similar to those already performed to derive the expression (\[DJFNFCO\]) and they are left to the reader. Denoting by $\d J_{\a}(t)$ the time-dependent current in the linear regime one ends up with $$\begin{aligned} \d J_{\a}(t)=4e\g_{\a}\Re\left\{ \int_{0}^{t}d{\bar t} \; \int\frac{d\w}{2\p}f(\w) \frac{{\rm e}^{i(\w-\W_{0})(t-{\bar t})}}{\w-\W_{0}}\right. \nonumber\\ \left. \times\left[\d U_{0}({\bar t})- \d U_{\a}({\bar t}) +2i\sum_{\a'}\g_{\a'}\frac{\d U_{0}({\bar t})- \d U_{\a'}({\bar t})}{\w-\W_{0}^{\ast}}\right]\right\} \;\; \label{sfwkw}\end{aligned}$$ where $\W_{0}=\ve_{0}-i\g$. In the special case $\ve_{0}=0$, $\d U_{0}(t)=0$ and $\d U_{\a}(t)=\d U_{\a}=$ const, $\d J_{\a}(t)$ reduces to the time-dependent current in Eq. (\[DJFNFCO\]) to first order in $\d U_{\a}$, as it should. Eq. (\[sfwkw\]) takes a very simple form if $\ve_{0}= \d U_{0}=\d U_{R}=0$ and $\a=R$: $$\begin{aligned} \d J_{R}(t)&=&4e\g_{R}\g_{L}\int_{0}^{t}d{\bar t} \; \d U_{L}(t-{\bar t}) \nonumber \\ && \times\left\{ \frac{\Im[f(i\g)]}{\g}{\rm e}^{-2\g {\bar t}} -\frac{2}{\b}\Re\left[\sum_{n=0}^{\inf} \frac{{\rm e}^{(i\w_{n}-\g){\bar t}}}{\w_{n}^{2}+\g^{2}} \right]\right\}, \nonumber \end{aligned}$$ where $\w_{n}=(2n+1)\p i/\b+\m$ are the Matzubara frequencies and the identity $$\int_{-\inf}^{\inf}\frac{d\w}{2\p}f(\w) \frac{{\rm e}^{i\w {\bar t}}}{\w^{2}+\g^{2}}= f(i\g)\frac{{\rm e}^{-\g {\bar t}}}{2\g}- \frac{i}{\b}\sum_{n=0}^{\inf} \frac{{\rm e}^{i\w_{n}{\bar t}}}{\w_{n}^{2}+\g^{2}},$$ has been used. In the special case of a vanishing chemical potential, the Matzubara frequencies are imaginary numbers and $\d J_{R}(t)$ simplifies $$\begin{aligned} \d J_{R}(t)= -2e\frac{\g_{R}\g_{L}}{\g}\int_{0}^{t}d{\bar t} \; \d U_{L}(t-{\bar t}){\rm e}^{-2\g {\bar t}} \nonumber \\ \times \left\{\tan[\frac{\b\g}{2}]+ \frac{{\rm e}^{-[\frac{\p}{\b}-\g]{\bar t}}}{\p} F({\bar t}) \right\}, \label{djmfkf}\end{aligned}$$ where $$F({\bar t})=\sum_{m=0,1} (-)^{m} \F[{\rm e}^{-\frac{2\p{\bar t}}{\b}},1,\frac{\p+(-)^{m}\b\g}{2\p}]$$ is a linear combination of the Lerch transcendent functions $\F[z,s,a]=\sum_{n=0}^{\inf}z^{n}/(a+n)^{s}$. ![[Current versus time for two different external disturbance $\d U_{L}$. In both figures $\g_{R}=\g_{L}=0.5$; the current is plotted for two different inverse temperature $\b=2$ and $\b=100$. In (a) $\d U_{L}$ is a square bump-like modulation whose duration is 1 while in (b) the duration is 5.]{}[]{data-label="lcurrent_sb"}](fig6.eps) In Fig. \[lcurrent\_sb\] we show the trend of $\d J_{R}(t)$ for square bump-like modulations. On the top $\d U_{L}(t)=\Theta(t)\Theta(1-t)$ while on the bottom $\d U_{L}(t)=\Theta(t)\Theta(5-t)$; both disturbances are considered for two different inverse temperature $\b=2$ and $\b=100$. As one can see the effect of an increasing temperature consists in a sort of rescaling of the time-dependent current. The line widths $\g_{\a}$ have been taken equal and large enough to justify the linear approximation. Since the disturbance is of order 1, from Fig. \[stationary\] one can see that $\g_{R}=\g_{L}=0.5$ is a good choice. ![[Current versus time for an oscillating external disturbance $\d U_{L}$. In both figures $\g_{R}=\g_{L}=0.5$ and the inverse temperature is $\b=100$. $\d U_{L}(t)=\sin\w_{0}t$ with $\w_{0}=5,\;10,\;20$ in (a), (b) and (c) respectively.]{}[]{data-label="lcurrent_ac"}](fig7.eps) ![[Current versus time for three different periodic square bump-like modulations. In any figure $\g_{R}=\g_{L}=0.5$ and $\b=100$. The thin lines represent $\d U_{L}$ while the thick lines represent $\d J_{R}$.]{}[]{data-label="steppe"}](fig8.eps) The ac current in the linear approximation is plotted in Fig. \[lcurrent\_ac\] for $\b=100$ and $\g_{R}=\g_{L}=0.5$. The time-dependent disturbance is taken to be $\d U_{L}(t)=\sin\w_{0}t$ with $\w_{0}=5,\;10,\;20$ in (a), (b) and (c) respectively. Finally, in Fig. \[steppe\] we have considered the current response to a periodic square bump-like modulation for different values of the period. Interacting Systems {#interacting} =================== In earlier theoretical works on quantum transport one can distinguish at least two schools. In one school one tries to keep the full atomistic structure of the conductor and the leads, but all works so far are at the level of the Local Density Approximation (LDA) and only the steady state has been considered. The advantage of this approach is that the interaction in the leads and in the conductor are treated on the same footing via self-consistent calculations on the current-carrying system. It also allows for detailed studies of how the contacts influence the conductance properties. The other school is using simplified models which allows the analysis to be carried much further. Considerable progresses have been made in this respect for a localized level described by a Lundquist-like model[@kral; @lundin; @xin] and for the so called “Coulomb island”[@meir; @you] where $H_{0}$ in Eq. (\[qham\]) is replaced by the Anderson Hamiltonian. However, all these works treat the leads as non-interacting, which prohibits a realistic description of the contacts and of the long-range aspects of the Coulomb interaction[@Büttiker]. The model approach are based on a partitioned scheme which makes the time-dependent results difficult to interpret. We here want to show how the current LDA by Lang *et al.*[@Lang1; @Lang2] follows from the TDDFT scheme described in Section \[mftddft\]. We also present an exact result for the steady-state current of an interacting resonant tunneling system. Finally, the transient behavior of a capacitor-device-capacitor system is investigated on the level of Mean Field. Steady-State Limit of TDDFT --------------------------- In Section \[model\] we showed that under certain conditions a steady state is reached in the long-time limit, and that this limit is independent of history. We also showed that the partitioned and partition-free treatments give an equivalent description of the steady state. The mechanism for the loss of memory was pure dephasing, and it holds provided the leads are macroscopic while the device is finite. Another important ingredient is that the applied bias is uniform deep inside the leads. With these assumptions, our results can be generalized also to more general cases than the simplified model explicitly considered in Section \[model\]. In TDDFT, the full interacting problem is reduced to a fictitious noninteracting one and *all* the results of Section \[model\] can be recycled. In the case of Time Dependent Local Density Approximation (TDLDA), the exchange-correlation potential $v_{\rm xc}$ depends only on the instantaneous local density and has no memory at all. If the density tends to a constant, so does the effective potential $U^{\rm eff}$, which again implies that the density tends to a constant. Owing to the non-linearity of the problem there might still be more than one steady-state solution or none at all. If a steady state is reached in TDDFT, we can go directly to the long-time limit of the Dyson equation and work in the frequency space. We may with no restriction use a partitioned approach and split the fictitious one-electron Hamiltonian matrix in a non-conducting part $\bcalE$ and a correction $\bV$ involving one-body hopping terms between the two leads and the device. The lesser Green function of TDDFT fulfills $$\bcalG^{<}(\ve)= [ 1 + \bcalG^{\rm R}(\ve) \bV ]\; \bg^<(\ve) [ 1 + \bV \bcalG^{\rm A}(\ve) ] ,$$ where $\bg$ is the uncontacted TDDFT Green function \[*cf.* Eq. (\[newder3\])\]. In direct space, the uncontacted $\bg^<$ can be written $$g^<({\bf r}, {\bf r}', \ve) = 2 \pi i \sum_{m\alpha} f_\alpha(e_{m\alpha}) \phi_{m\alpha}({\bf r}) \phi_{m\alpha}^*({\bf r}') \d(\ve-e_{m\alpha})$$ in terms of diagonalizing orbitals $\phi_{m\alpha}$ with fictitious eigenvalues $e_{m\alpha}$ for the left and right leads ($\alpha = L,R$) and the device ($\alpha = D$) and Fermi functions $f_\alpha$ with chemical potential $\mu_\alpha$. The chemical potentials for the two leads differ, and the final result is independent of the chosen chemical potential for the device. When we apply $1 + \bcalG^{\rm R} \bV= \bcalG^{\rm R}[\bg^{\rm R}]^{-1}$ to an unperturbed orbital $\phi_{m\alpha}$, it is transformed to an interacting, *i.e.*, contacted eigenstate $\psi_{m\alpha}$. Above the conductance threshold, states originating from the left lead become right-going scattering states, and states from the right lead become left-going scattering states. In addition, fully reflected waves and discrete state may arise which contribute to the density but not to the current. Thus, $${\cal G}^<({\bf r}, {\bf r}', \ve) = 2 \pi i \sum_{m\alpha} f_\alpha(e_{m \alpha}) \psi_{m\alpha}({\bf r}) \psi_{m\alpha}^*({\bf r}') \d(\ve-e_{m\alpha}).$$ These results correspond closely to the general approach by Lang and coworkers.[@Lang1; @Lang2] In their approach, the continuum is split into left and right-going parts, which are populated according to two different chemical potentials. The density is then calculated self-consistently. Lang *et al.* further approximate exchange and correlation by the LDA and the leads by homogeneous jellia, but apart from these approximations it is clear that his method implements TDDFT, as described in Section \[mftddft\], in the steady state. It is also clear that the correctness of Lang’s approach relies on the Theorem of Equivalence between the partitioned and partition-free approaches and the Memory Loss Theorem derived here. The equivalence between the scattering state approach by Lang *et al.* and the partitioned non-equilibrium approach used by Taylor *et al.*[@Taylor1; @Taylor2] has also been shown by Brandbyge *et al.*[@brand] As shown above, the steady state of TDDFT can always be formulated in terms of orbitals which diagonalize the asymptotic one-particle Hamiltonian matrix. The current-carrying orbitals can always be grouped into a right-going class and a left-going class. As a consequence, the current can be expressed in a Landauer formula $$J_{R}^{(\rm S)} =- e \sum_m [ f_L(e_{mL}) {\cal T}_{mL} - f_R(e_{mR}) {\cal T}_{mR} ] \label{kefrv}$$ in terms of fictitious transmission coefficients ${\cal T}_{m\a}$ and energy eigenvalues $e_{m\a}$, $\a=L,R$. We also wish to emphasize that the steady-state current in Eq. (\[kefrv\]) comes out from a pure dephasing mechanism in the fictitious noninteracting problem. The memory-loss effects from scatterings is described by $A_{\rm xc}$ and $v_{\rm xc}$. One-Level Resonant Tunneling System ----------------------------------- In this Section we consider a resonant tunneling system described by the quadratic Hamiltonian of Eq. (\[qham\]) and an inter-particle interaction $$H_{W}=\frac{1}{2}\sum_{m\neq n}W_{m,n}n_{m}n_{n},$$ where $n_{m}=c^{\dag}_{m}c_{m}$ is the occupation number operator of the level $m$ and $W_{m,n}=W_{n,m}$ is a symmetric matrix. (If $H_{W}$ includes long-range terms, the regrouping of potential terms as discussed in Section \[mftddft\] must be done.) In the generalized TDDFT scheme (based on the $n_{m}$ occupations rather than on density) outlined in Section \[mftddft\] the fictitious Green function ${\cal G}_{m,n}$ is obtained by solving the Dyson equations with $\bK=\bT+\bU^{\rm eff}$, where $$U^{\rm eff}_{m,n}(t)=\d_{m,n}[U_{m}(t)+V_{{\rm H},m}(t)+v_{{\rm xc},m}(t)].$$ If $\bU^{\rm eff}$ satisfies the hypothesis of the Theorem of Equivalence and of the Memory Loss Theorem we can use Eq. (\[stacg\]) and write an exact formula for the steady-state current of an *interacting* resonant tunneling system: $$\begin{aligned} J^{(\rm S)}_{R}&=&-e\int\frac{d\ve}{2\p}f(\ve) \label{stacgdft} \\ &&\times\left\{\G^{(0)}_{L}(\ve)\G_{R}(\ve+U^{\rm eff}_{L}(\ve)) \left|{\cal G}^{\rm R}_{0,0}(\ve+U^{\rm eff}_{L}(\ve)) \right|^{2}\right. \nonumber \\ &&\left.\;\;+ \G^{(0)}_{R}(\ve)\G_{L}(\ve+U^{\rm eff}_{R}(\ve)) \left|{\cal G}^{\rm R}_{0,0}(\ve+U^{\rm eff}_{R}(\ve)) \right|^{2} \right\}. \nonumber\end{aligned}$$ For normal-metal electrodes we expect that the effective potential $U^{\rm eff}_{\a}(\ve,t)\ra U^{\rm eff}_{\a}(\ve)={\rm const}$ provided $U_{\a}(\ve,t)\ra U_{\a}(\ve)={\rm const}$ when $t\ra\inf$. The constant $U^{\rm eff}_{\a}(\ve)$ may depend on the history of $U_{\a}(\ve,t)$ while the steady-state current is independent of the history of $U^{\rm eff}_{\a}(\ve,t)$. ${\cal G}^{\rm R}_{0,0}(\w)$ is given by Eq.(\[g00\]) with $\tilde{\ve}_{0}=\ve_{0}+ \lim_{t\ra\inf}U^{\rm eff}_{0}(t)$ and with $\S^{\rm R}$ from Eq.(\[ser\]) with $\tilde{\ve}_{k\a}=\ve_{k\a}+\lim_{t\ra\inf} U^{\rm eff}_{k\a}(t)$. For the sake of clarity, Eq. (\[stacgdft\]) has been written for systems having a one-to-one correspondence between the one-body indices $k\a$ and the one-body energies $\ve_{k\a}+U^{\rm eff}_{k\a}(0)$. The generalization to systems with degenerate levels is straightforward and it is left to the reader. As a further example we study the RPA time-dependent current response in the partition-free approach. In the Hartree approximation the Green function $\bG^{\rm H}$ satisfies the equation of motion (\[ieom1td\]) with $\bS_{c}=0$ and $$\S^{\d}_{m,n}(z)\equiv \S^{\rm H}_{m,n}(z)= \d_{m,n}\sum_{l:\;l\neq n}W_{n,l}\;n^{\rm H}_{l}(z),$$ where $n^{\rm H}_{l}(z)=-iG^{\rm H,<}_{l,l}(z;z)$. According with the results obtained in Section \[keldyshth\], the lesser Green function $\bG^{\rm H,<}$ is given by Eq. (\[newder11\]) with $\bG\ra \bG^{\rm H}$. Therefore, in the linear approximation we have $$\begin{aligned} &&\d \bG^{<}(t;t)\label{dief} \\ && \quad =-i\int d{\bar t}\; \bG_{0}^{\rm H,R}(t;{\bar t})[\d \bU^{\rm eff}({\bar t}),\bG^{\rm H,<}(0;0)] \bG_{0}^{\rm H,A}({\bar t};t), \nonumber\end{aligned}$$ with $$\d \bU^{\rm eff}(t)=\d \bU(t)+\d\bS^{\rm H}(t). \label{efc;}$$ Eqs. (\[dief\])-(\[efc;\]) form a coupled system of integral equations for the unknowns $\d \bG^{<}(t;t)$ and $\d \bU^{\rm eff}(t)$. For a capacitor-device-capacitor system one can take $$W_{m,n}=\left\{ \begin{array}{ll} W_{\a\a'} & {\rm if}\quad m=k\a,\; n=k'\a' \\ 0 & {\rm otherwise} \end{array} \right..$$ Thus, putting an extra particle in the isolated $\a$ capacitor costs an energy $W_{\a\a}$ per particle. This means that the transfer of a finite number of particles from one capacitor to the other causes a finite change of the effective applied bias. We expect that the current vanishes in the long-time limit unless the applied bias continues to grow up. The coefficients $W_{RL}=W_{LR}$ mimic the repulsion energy between two particles in different capacitors. Actually, one can also consider the interaction between a particle in the central device and another in one of the two capacitors. No extra complications arise if $W_{0,k\a}=W_{\a}$, $\forall k$, and the results we are going to obtain can be easily extended. Switching a bias $\d U_{k\a}(t)=\d U_{\a}(t)$, from Eq. (\[efc;\]) one gets $\d U^{\rm eff}_{m,n}(t)=\d_{m,n}\d U^{\rm eff}_{n}(t)$ with $\d U^{\rm eff}_{k\a}(t)=\d U^{\rm eff}_{\a}(t)$, $\forall k$, and $$\d U^{\rm eff}_{\a}(t)=\d U_{\a}(t)- \frac{1}{e}\sum_{\b}\int_{0}^{t}d\bar{t}\;W_{\a\b}\d J_{\b}(\bar{t}), \label{dur}$$ where it has been taken into account that $\d N^{\rm H}_{\a}(t)\equiv \sum_{k} \d n^{\rm H}_{k\a}(t)=-\frac{1}{e} \int_{0}^{t}d\bar{t}\;\d J_{\a}(\bar{t})$. Since $\d \bU^{\rm eff}$ has the same matrix structure of the bare $\d \bU$, in the wide-band limit the linear time-dependent current $\d J_{\a}(t)$ is given by Eq. (\[sfwkw\]) with $\d \bU$ replaced by $\d \bU^{\rm eff}$. (It is worth noticing that the wide band limit still makes sense if the line width is approximately constant in a small interval around the chemical potential $\m$.) In this way the system of Eqs. (\[dief\])-(\[efc;\]) is reduced to a system of 4 coupled integral equations for the 4 scalar unknowns $\d U^{\rm eff}_{\a}$, $\d J_{\a}$ with $\a=L,R$. The symmetric case $\g_{R}=\g_{L}=\g/2$, $W_{RR}=W_{LL}$ allows a further simplification. Let us define $\d U^{\rm eff}_{\pm}= \d U^{\rm eff}_{R}\pm \d U^{\rm eff}_{L}$, $\d U_{\pm}= \d U_{R}\pm \d U_{L}$, $\d J_{\pm}=\d J_{R}\pm \d J_{L}$ and $W_{\pm}=W_{RR}\pm W_{RL}= W_{LL}\pm W_{LR}$. Then, from Eq. (\[dur\]) we find $$\d U^{\rm eff}_{\pm}(t)=\d U_{\pm}(t)- \frac{W_{\pm}}{e}\int_{0}^{t}d\bar{t}\;\d J_{\pm}(\bar{t}), \label{dutot}$$ while from Eq. (\[sfwkw\]) $$\d J_{+}(t)=2e\g\int_{0}^{t}d{\bar t} \; C_{+}(t-{\bar t})[2\d U_{0}({\bar t})-\d U^{\rm eff}_{+}({\bar t})], \label{sieklefl}$$ $$\d J_{-}(t)=-2e\g\int_{0}^{t}d{\bar t} \; C_{-}(t-{\bar t})\d U^{\rm eff}_{-}({\bar t}), \label{didke}$$ where $$C_{\pm}(t)=\Re\left\{\int\frac{d\w}{2\p}f(\w) \frac{{\rm e}^{i(\w-\ve_{0}+i\g)t}}{\w-\ve_{0}\mp i\g} \right\}$$ is the conductivity kernel. Once $\d J_{\pm}$ has been obtained, one can calculate $\d J_{R}=(\d J_{+}+\d J_{-})/2$ and $\d J_{L}=(\d J_{+}-\d J_{-})/2$. ![[Numerical solutions of Eqs. (\[dutot\])-(\[sieklefl\])-(\[didke\]) in the zero temperature limit with $\m=\ve_{0}=0$, $W_{-}=5$ and an external disturbance as described in the main text. The thick lines are the current in (a) and the effective potential in (b) for the step like modulation. The currents and the effective potentials for $t_{0}=1,\;2,\;4,\;6$ unstick from the thick line and start to oscillate and eventually vanish after a time $t\propto 1/W_{-}$. The vertical lines are the bare applied potentials.]{}[]{data-label="plasmonjdc"}](fig9.eps) In order to illustrate what is the time-dependent response of this model we have considered the zero temperature case with $\d U_{R}(t)=-\d U_{L}(t)=(1/2)\Theta(t)\Theta(t_{0}-t)$ and $\d U_{0}(t)=0$. Then, $\d U_{+}(t)=0$ and hence $\d J_{+}(t)=\d U^{\rm eff}_{+}(t)=0$. It follows that $\d J_{R}(t)=-\d J_{L}(t)=\d J_{-}(t)/2$ and $\d U^{\rm eff}_{R}(t)=-\d U^{\rm eff}_{L}(t)= \d U^{\rm eff}_{-}(t)/2$ for any time $t$. In Fig. \[plasmonjdc\](a) we display the time-dependent current for square bump-like modulations with $t_{0}=1,\;2,\;4,\;6$ and $W_{-}=5$. The thick line is the current for the step like modulation $\d U_{-}(t)=\Theta(t)$; depending on the value of $t_{0}$ the current unsticks itself from the thick line giving rise to different damped oscillating curves. In correspondence of each $t_{0}$ a vertical line has been drawn; it represents the bare applied potential $\d U_{-}(t)$. Fig. \[plasmonjdc\](b) shows the time-dependent effective potential $\d U^{\rm eff}_{-}(t)$. As the current response, it drops to zero in the long-time limit since the interactions completely screen the applied bias after a time $t\propto 1/W_{-}$. Summary and Concluding Remarks {#conclusion} ============================== In the present work we have used a partition-free scheme in order to treat the time-dependent current response of a mesoscopic system coupled to macroscopic leads. To this end, we have further developed the Keldysh formalism and we have formulated a formally exact theory which is more akin to the way the experiments are carried out. Among the advantages of the partition-free scheme we stress the possibility to calculate physical dynamical responses and to include the interactions between the leads and between the leads and the device in a quite natural way. In the noninteracting case we have shown that a perfect destructive interference takes place provided the energy levels of the leads form a continuum. The steady-state develops due to a *dephasing mechanism*. The comparison of our steady-state current with that obtained in the partitioned scheme shows that the two currents are equivalent if the energy levels are properly shifted in order to preserve charge neutrality. This kind of equivalence remains true for any time-dependent external potentials (Theorem of Equivalence). The Theorem of Equivalence has then been used in order to prove that the steady-state current depends only on the asymptotic value of the external perturbation (Memory Loss Theorem). For the sake of clarity, the Theorem of Equivalence and the Memory Loss Theorem have been proved for a single-level central device. The generalization to a multi-level central device is straightforward, as can be readily verified. In the wide band limit we have obtained an analytic result for the time-dependent current in the case of a step-like modulation and for arbitrary modulations in the linear regime. The interacting case represents a more difficult challenge and the expression for the lesser Green function at any finite time is more complicated than that commonly used to calculate steady-state response functions. As an alternative to a full many-body treatment we have proposed a formally exact one-particle scheme based on TDDFT. Then, *all* the results obtained in the noninteracting case can be recycled provided we substitute the external potential with the exact effective potential of TDDFT. Although it is difficult to prove any rigorous results for the effective TDDFT potential, we expect the interactions to reduce the memory effects even further compared to the noninteracting case. Thus, any nonlinear steady-state current can been expressed in a Landauer-like formula in terms of fictitious transmission coefficients and one-particle energy eigenvalues. The steady-state current depends on history only through the asymptotic shape of the effective TDDFT potential. This exact result may prompt for new approximations to the exchange-correlation action functional $A_{\rm xc}$. In the effective one-particle scheme of TDDFT the steady-state current comes out from a pure dephasing mechanism. The damping mechanism (due to the electron-electron scatterings) of the real problem is described by $A_{\rm xc}$. As an illustrative example we have also calculated the RPA time-dependent current of a capacitor-device-capacitor system and we have displayed the effect of the charge oscillations in the discharge process. We would like to acknowledge useful discussions with U. von Barth, P. Bokes, M. Cini, R. Godby, A.-P. Jahuo, B. I. Lundqvist, P. Hyldgaard, and B. Tobiyaszewska. This work was supported by the RTN program of the European Union NANOPHASE (contract HPRN-CT-2000-00167). Proof of Eq. (\[GGGG\]) {#z} ======================= It is convenient to define $\bG_{0}$ as the solution of Eqs. (\[ieom1td\]) with $\bS_{\rm c}=0$. $\bG_{0}$ satisfies all the relations we have derived for a noninteracting system in the presence of an external disturbance. By using the Langreth theorem, we get $$\begin{aligned} \bG^{\lessgtr}&=&\left[\d+ \bG^{\rm R}\cdot\bS^{\rm R}\right]\cdot \bG_{0}^{\lessgtr}+ \bG^{\lessgtr}\cdot\bS^{\rm A}\cdot \bG_{0}^{\rm A} \nonumber \\ &&+ \left[\bG^{\rm R}\cdot\bS^{\lessgtr}+ \bG^{\rceil}\star\bS^{\lceil}\right] \cdot \bG_{0}^{\rm A} \nonumber \\ &&+ \bG^{\rm R}\cdot \bS^{\rceil}\star \bG_{0}^{\lceil}+ \bG^{\rceil}\star\bS\star \bG_{0}^{\lceil} \nonumber\end{aligned}$$ and solving for $\bG^{\lessgtr}$ $$\begin{aligned} \bG^{\lessgtr}&=&[\d+\bG^{\rm R}\cdot\bS^{\rm R}]\cdot \bG_{0}^{\lessgtr} \cdot [\d+\bS^{\rm A}\cdot \bG_{0}^{\rm A}] \nonumber \\ &&+ \left[\bG^{\rm R}\cdot\bS^{\lessgtr}+ \bG^{\rceil}\star\bS^{\lceil}\right]\cdot \bG^{\rm A} \nonumber \\ &&+ [\bG^{\rm R}\cdot \bS^{\rceil}\star \bG_{0}^{\lceil} +\bG^{\rceil}\star\bS\star \bG_{0}^{\lceil}]\cdot [\d+\bS^{\rm A}\cdot \bG_{0}^{\rm A}]. \nonumber\end{aligned}$$ Next, we use $$\bG_{0}^{\lessgtr}(t;t')=\bG_{0}^{\rm R}(t;0)\bG_{0}^{\lessgtr}(0;0) \bG_{0}^{\rm A}(0;t')$$ and $$\bG_{0}(\t;t)=-i\bG_{0}(\t;0)\bG_{0}^{\rm A}(0;t),$$ so that $$\begin{aligned} &&\bG^{\lessgtr}(t;t')=\bG^{\rm R}(t;0)\bG_{0}^{\lessgtr}(0;0) \bG^{\rm A}(0;t') \label{tg<>}\\ &&\quad\quad +\left[\bG^{\rm R}\cdot\bS^{\lessgtr}\cdot \bG^{\rm A}\right](t;t')+ \left[\bG^{\rceil}\star\bS^{\lceil}\cdot \bG^{\rm A}\right](t;t') \nonumber \\ &&\quad\quad -i\left[ \bG^{\rm R}\cdot\bS^{\rceil}\star \bG_{0}+ \bG^{\rceil}\star \bS\star \bG_{0} \right](t;0)\bG^{\rm A}(0;t'). \nonumber\end{aligned}$$ As in the noninteracting case, we proceed by writing down the Dyson equation for $\bG(t;\t)$. Taking into account that $$\bG(\t;\t')=\bG_{0}(\t;\t')+\left[ \bG_{0}\star\bS\star \bG \right](\t;\t') \label{termg}$$ and that $$\bG_{0}(t;\t)=i\bG^{\rm R}_{0}(t;0)\bG_{0}(0;\t),$$ we have $$\bG(t;\t)=\left[ \bG^{\rm R}\cdot \bS^{\rceil}\star \bG \right](t;\t)+i\bG^{\rm R}(t;0)\bG(0;\t). \label{g-|}$$ Similarly, it is straightforward to show that $$\bG(\t;t)=\left[\bG\star \bS^{\lceil}\cdot \bG^{\rm A} \right](\t;t)-i\bG(\t;0)\bG^{\rm A}(0;t). \label{g|-}$$ Substituting Eq. (\[g-|\]) into Eq. (\[tg&lt;&gt;\]) and using Eq. (\[termg\]) one finds $$\begin{aligned} \bG^{\lessgtr}(t;t')&=&\bG^{\rm R}(t;0)\bG^{\lessgtr}(0;0) \bG^{\rm A}(0;t') \nonumber \\ && + \left[\bG^{\rm R}\cdot\left[\bS^{\lessgtr}+ \bS^{\rceil}\star \bG\star \bS^{\lceil} \right]\cdot \bG^{\rm A}\right](t;t') \nonumber \\ &&+i\bG^{\rm R}(t;0)\left[ \bG\star \bS^{\lceil}\cdot \bG^{\rm A} \right](0;t')\nonumber \\ && -i\left[ \bG^{\rm R}\cdot\bS^{\rceil}\star \bG \right](t;0)\bG^{\rm A}(0;t'). \label{tg<>3}\end{aligned}$$ Using Eqs. (\[g-|\])-(\[g|-\]) to express the last two terms as $$\begin{aligned} i\bG^{\rm R}(t;0)\left[ \bG\star \bS^{\lceil}\cdot \bG^{\rm A} \right](0;t') \quad\quad\quad\quad\quad \quad\quad\quad\quad \nonumber \\ = i\bG^{\rm R}(t;0)\bG^{>}(0;t')- \bG^{\rm R}(t;0)\bG^{>}(0;0)\bG^{\rm A}(0;t') \nonumber \end{aligned}$$ and $$\begin{aligned} -i\left[ \bG^{\rm R}\cdot\bS^{\rceil}\star \bG \right](t;0)\bG^{\rm A}(0;t') \quad\quad\quad\quad \quad\quad\quad\quad\quad \nonumber \\ =-i\bG^{<}(t;0)\bG^{\rm A}(0;t')- \bG^{\rm R}(t;0)\bG^{<}(0;0)\bG^{\rm A}(0;t'), \nonumber\end{aligned}$$ we end up with Eq. (\[GGGG\]). Proof of Eq. (\[INTERM\]) {#a1} ========================= Due to the smoothness of the self energy, in the long-time limit we can use the Riemann-Lebesgue theorem to obtain the following asymptotic behaviors $$\lim_{t\ra\inf}G^{\rm R}_{0,0}(t;0)= \lim_{t\ra\inf}\sum_{k}G^{\rm A}_{0,k\a}(0;t)V_{k\a}=0 \label{asymg00}$$ and $$\lim_{t\ra\inf}G^{\rm R}_{0,k\a}(t;0)= -iV_{k\a}{\rm e}^{-i\tilde{\ve}_{k\a}t}G^{\rm R}_{0,0}(\tilde{\ve}_{k\a}).$$ $$\begin{aligned} \lim_{t\ra\inf}\sum_{k}G^{\rm A}_{k'\a',k\a}(0;t)V_{k\a}= iV_{k'\a'}{\rm e}^{i\tilde{\ve}_{k'\a'}t} \quad\quad \nonumber \\ \times\left[ \d_{\a,\a'}+G^{\rm A}_{0,0}(\tilde{\ve}_{k'\a'}) \S^{\rm A}_{\a}(\tilde{\ve}_{k'\a'}) \right]. \nonumber\end{aligned}$$ From the above results and the definition (\[qalfa\]) one has $$\begin{aligned} \lim_{t\ra\inf}Q_{\a}(\z;t)= \sum_{k'}\frac{V^{2}_{k'\a}}{\z-\ve_{k'\a}} G^{\rm R}_{0,0}(\tilde{\ve}_{k'\a}) \quad\quad\quad\quad \label{statcurint} \\ \quad\quad +\sum_{k'\a'}\frac{V^{2}_{k'\a'}}{\z-\ve_{k'\a'}} G^{\rm R}_{0,0}(\tilde{\ve}_{k'\a'}) G^{\rm A}_{0,0}(\tilde{\ve}_{k'\a'}) \S^{\rm A}_{\a}(\tilde{\ve}_{k'\a'}) \nonumber \\ \quad\quad+ \lim_{t\ra\inf}G_{0,0}(\z)\sum_{k'\a'}\frac{V^{2}_{k'\a'}} {\z-\ve_{k'\a'}} G^{\rm R}_{0,0}(\tilde{\ve}_{k'\a'}){\rm e}^{-i\tilde{\ve}_{k'\a'}t} \nonumber \\ \quad\quad \quad\quad\quad\quad \quad\quad\quad\quad\quad\;\; \times\sum_{k''}\frac{V^{2}_{k''\a}}{\z-\ve_{k''\a}} {\rm e}^{i\tilde{\ve}_{k''\a}t} \nonumber \\ \quad\quad +\lim_{t\ra\inf}G_{0,0}(\z)\sum_{k'\a'}\frac{V^{2}_{k'\a'}} {\z-\ve_{k'\a'}}G^{\rm R}_{0,0}(\tilde{\ve}_{k'\a'}) {\rm e}^{-i\tilde{\ve}_{k'\a'}t} \nonumber \\ \quad\quad\times \sum_{k''\a''}\frac{V^{2}_{k''\a''}} {\z-\ve_{k''\a''}} G^{\rm A}_{0,0}(\tilde{\ve}_{k''\a''}) \S^{\rm A}_{\a}(\tilde{\ve}_{k''\a''}) {\rm e}^{i\tilde{\ve}_{k''\a''}t} \nonumber \\ \nonumber \\ = \int\frac{d\ve}{2\p}\frac{\G_{\a}(\ve)}{\z-\ve+ U_{\a}} G^{\rm R}_{0,0}(\ve) \quad\quad\quad \quad\quad\quad \quad\quad\quad \nonumber \\ \quad\quad+\sum_{\a'}\int\frac{d\ve}{2\p} \frac{\G_{\a'}(\ve)}{\z-\ve+ U_{\a'}} \left|G^{\rm R}_{0,0}(\ve)\right|^{2} \S^{\rm A}_{\a}(\ve) \nonumber \\ \quad\quad + \lim_{t\ra\inf}G_{0,0}(\z)\int\frac{d\ve}{2\p} \G_{\a}(\ve)\frac{{\rm e}^{i\ve t}}{\z-\ve+U_{\a}} \nonumber \\ \quad\quad\quad\quad\quad \times\sum_{\a'}\int\frac{d\ve'}{2\p}\G_{\a'}(\ve') \frac{{\rm e}^{-i\ve' t}}{\z-\ve'+U_{\a'}} G^{\rm R}_{0,0}(\ve') \nonumber \\ + \lim_{t\ra\inf}G_{0,0}(\z) \sum_{\a'}\int\frac{d\ve'}{2\p}\G_{\a'}(\ve') G^{\rm R}_{0,0}(\ve')\frac{{\rm e}^{-i\ve' t}}{\z-\ve'+U_{\a'}} \nonumber \\ \times \sum_{\a''}\int\frac{d\ve''}{2\p}\G_{\a''}(\ve'') G^{\rm A}_{0,0}(\ve'')\S^{\rm A}_{\a}(\ve'') \frac{{\rm e}^{i\ve'' t}}{\z-\ve''+U_{\a''}}, \nonumber\end{aligned}$$ where the relation $$G_{k\a,k'\a'}(\z)=\frac{\d_{k\a,k'\a'}}{\z-\ve_{k\a}} +\frac{V_{k\a}}{\z-\ve_{k\a}}\frac{V_{k'\a'}}{\z-\ve_{k'\a'}} G_{0,0}(\z),$$ has been explicitly used. Since $\z\in\G$ the quantity $[\z-\ve-U_{\a}]^{-1}$ is a smooth function of $\ve$ for any real $\ve$. In the limit $t\ra\inf$ the last two terms in Eq. (\[statcurint\]) vanish according with the Riemann-Lebesgue theorem and Eq. (\[INTERM\]) is recovered. Proof of Eq. (\[DJFNFCO\]) {#a2} ========================== The quantity $Q_{\a}(\z;t)$ involves the multiplication of three matrices and we can recognize four contributions, two containing $G^{\rm R}_{0,0}$ and other two containing $G^{\rm R}_{0,k'\a'}$. It is straightforward to verify that $$G_{0,0}(\z)= \left\{ \begin{array}{ll} \frac{1}{\z+i\g} & \Im[\z]>0 \\ & \\ \frac{1}{\z-i\g} & \Im[\z]<0 \end{array}\right. , \label{tgooz}$$ and that $G^{\rm R,A}_{0,0}(\w)=[\w\pm i\g]^{-1}$, where $\g=\g_{R}+\g_{L}$. Hence $$\begin{aligned} && G^{\rm R}_{0,0}(t;0)=-i{\rm e}^{-\g t}, \label{az,`/} \\ &&G^{\rm R}_{0,k\a}(t;0)= -iV_{k\a}\frac{{\rm e}^{-i\tilde{\ve}_{k\a}t}-{\rm e}^{-\g t}} {\tilde{\ve}_{k\a}+i\g} \nonumber \\ && \sum_{k}V_{k\a}G^{\rm A}_{0,k\a}(0;t)= -\g_{\a}{\rm e}^{-\g t}, \nonumber \\ && \sum_{k'}V_{k'\a'}G^{\rm A}_{k\a,k'\a'}(0;t)= i\d_{\a,\a'}V_{k\a}{\rm e}^{i\tilde{\ve}_{k\a}t} \nonumber \\ && \quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\;\; -\g_{\a'}V_{k\a}\frac{{\rm e}^{i\tilde{\ve}_{k\a}t}-{\rm e}^{-\g t}} {\tilde{\ve}_{k\a}-i\g}. \nonumber\end{aligned}$$ ![[Contour $\G$ of Eq. (\[current2\]). The black dots correspond to the position of the Matzubara frequencies in the complex $\z$ plane.]{}[]{data-label="lwcontour"}](fig2.eps) Eqs. (\[tgooz\])-(\[az,‘/\]) are all what we need in order to evaluate the quantity $Q_{\a}(\z;t)$ in Eq. (\[qalfa\]). The time-dependent current is then obtained integrating $Q_{\a}(\z;t)f(\z){\rm e}^{\eta\z}$ over $\z$ along the contour $\G$ of Fig. \[lwcontour\], according with Eq. (\[current2\]). Using Eqs. (\[az,‘/\]) and expressing $G_{k\a,0}(\z)=G_{0,k\a}(\z)$ and $G_{k\a,k'\a'}(\z)$ in terms of $G_{0,0}(\z)$ we obtain $$G^{\rm R}_{0,0}(t;0)G_{0,0}(\z)\sum_{k}V_{k\a}G^{\rm A}_{0,k\a}(0;t) =i\g_{\a}G_{0,0}(\z){\rm e}^{-2\g t}, \label{tdcr1}$$ $$\begin{aligned} \sum_{k'\a'}G^{\rm R}_{0,0}(t;0)G_{0,k'\a'}(\z) \sum_{k}V_{k\a}G^{\rm A}_{k'\a',k\a}(0;t)= \quad\quad\quad \label{tdcr2} \\ = 2\g_{\a}{\rm e}^{-\g t}G_{0,0}(\z) \int\frac{d\ve}{2\p}\frac{{\rm e}^{i\ve t}}{\z-\ve+U_{\a}} + \nonumber \\ 2i\g_{\a}{\rm e}^{-\g t}G_{0,0}(\z)\sum_{\a'}\g_{\a'} \int\frac{d\ve}{2\p} \frac{{\rm e}^{i\ve t}-{\rm e}^{-\g t}}{(\z-\ve+U_{\a'})(\ve-i\g)}. \nonumber \end{aligned}$$ We are left with the contributions containing $G^{\rm R}_{0,k'\a'}$. One of them is quite easy to evaluate and yields: $$\begin{aligned} \sum_{k'\a'}G^{\rm R}_{0,k'\a'}(t;0)G_{k'\a',0}(\z) \sum_{k}V_{k\a}G^{\rm A}_{0,k\a}(0;t)= \quad\quad\quad \label{tdcr3} \\ = 2i\g_{\a}G_{0,0}(\z)\sum_{\a'}\g_{\a'} \int\frac{d\ve}{2\p} \frac{{\rm e}^{-i\ve t-\g t}-{\rm e}^{-2\g t}}{(\z-\ve+U_{\a'})(\ve+i\g)}. \nonumber\end{aligned}$$ The other one is much more involved, but nothing more than standard algebra is needed to get the following expression $$\begin{aligned} \sum_{k'\a'}\sum_{k''\a''} G^{\rm R}_{0,k'\a'}(t;0)G_{k'\a',k''\a''}(\z) \times \quad\quad\quad\quad\quad \quad\quad \label{tdcr5} \\ \sum_{k}V_{k\a}G^{\rm A}_{k''\a'',k\a}(0;t)= \nonumber \\ =2\g_{\a}\int\frac{d\ve}{2\p} \frac{1-{\rm e}^{i\ve t}{\rm e}^{-\g t}} {(\ve+i\g)(\z-\ve+U_{\a})}+ \quad\quad\quad\quad\quad \quad\quad \nonumber \\ 2i\g_{\a}\sum_{\a'}\g_{\a'}\int\frac{d\ve}{2\p} \frac{1}{\z-\ve+U_{\a'}}\left| \frac{{\rm e}^{i\ve t}-{\rm e}^{-\g t}} {\ve+i\g}\right|^{2}- \nonumber \\ 2iG_{0,0}(\z)\sum_{\a'}\g_{\a'}\int\frac{d\ve}{2\p} \frac{{\rm e}^{-i\ve t}-{\rm e}^{-\g t}} {(\ve+i\g)(\z-\ve+U_{\a'})}\int\frac{d\ve'}{2\p} \times \nonumber \\ \left[ 2i\g_{\a} \frac{{\rm e}^{i\ve' t}}{\z-\ve' +U_{\a}}- \frac{{\rm e}^{i\ve' t}-{\rm e}^{-\g t}} {(\ve'-i\g)(\z-\ve'+U_{\a''})} \right]. \nonumber\end{aligned}$$ The r.h.s. of the above four equations must now be multiplied by $f(\z){\rm e}^{\eta\z}$ and integrated over $\z$ along the contour $\G$. Smearing the branches $\G_{+}$ and $\G_{-}$ on the real axis and taking into account Eq. (\[tgooz\]), the r.h.s. of Eq. (\[tdcr1\]) yields the following contribution to the current $$4e\g_{\a}{\rm e}^{-2\g t}\;\Im\left\{ \int\frac{d\w}{2\p}f(\w)\frac{1}{\w+i\g} \right\}, \label{ctdcn1}$$ where the integration over $\w$ has to be understood from $-\inf$ to $+\inf$. Another contribution comes from the first term on the r.h.s. of Eq. (\[tdcr2\]). By closing the contour of the $\ve$ integration on the complex upper half plane, it is non vanishing only if ${\rm Im}[\z]>0$. Therefore, only the upper branch $\G_{+}$ of $\G$ contributes. $\G_{+}$ can then be smeared on the real axis and one gets $$-4e\g_{\a}{\rm e}^{-\g t}\;\Im\left\{ \int\frac{d\w}{2\p}f(\w)\frac{{\rm e}^{i(\w+U_{\a})t}}{\w+i\g} \right\}. \label{ctdcn2}$$ A similar procedure can be adopted to evaluate the contribution coming from the second term on the r.h.s. of Eq. (\[tdcr2\]). One more time we can close the contour of the $\ve$ integration on the complex upper half plane. The pole in $\ve=i\g$ does not contribute since its residue is zero. The other pole is in $\ve=\z+U_{\a'}$ and hence one obtains $$-4e\g_{\a}\;\Re\left\{ \int\frac{d\w}{2\p}\frac{f(\w)}{\w+i\g}\sum_{\a'}\g_{\a'} \frac{{\rm e}^{i(\w+U_{\a'}) t-\g t}-{\rm e}^{-2\g t}}{\w+U_{\a'}-i\g}\right\}. \label{ctdcn3}$$ Next, we have to calculate the contribution coming from Eq. (\[tdcr3\]). By the same reasoning leading to Eq. (\[ctdcn3\]) it is readily verified that it yields the same result. Therefore we have to keep in mind that Eq. (\[ctdcn3\]) should be multiplied by 2 at the end. Let us now consider the contribution coming from the first two terms on the r.h.s. of Eq. (\[tdcr5\]). Since the discontinuous function $G_{0,0}(\z)$ does not appear in the integrand we can perform the contour integral over $\z$. We find $$\begin{aligned} &-&4e\g_{\a}\;\Im\left\{\int\frac{d\w}{2\p} f(\w-U_{\a})\frac{1-{\rm e}^{i\w t}{\rm e}^{-\g t}} {\w+i\g}\right\}- \label{ctdcn5} \\ &&4e\g_{\a}\;\Re\left\{ \sum_{\a'}\g_{\a'}\int\frac{d\w}{2\p} f(\w-U_{\a'}) \left| \frac{{\rm e}^{i\w t}-{\rm e}^{-\g t}} {\w+i\g}\right|^{2} \right\}. \nonumber\end{aligned}$$ The contribution coming from the last two terms on the r.h.s. of Eq. (\[tdcr5\]) vanishes. Indeed the integral over $\ve$ can be closed on the complex lower half plane. The pole in $\ve=-i\g$ does not contribute since its residue is zero. The other pole contributes only if $\Im[\z]<0$. At the same time we can also perform the integration over $\ve'$ by closing the contour in the complex upper half plane. The first term in the square brackets of Eq. (\[tdcr5\]) is non vanishing only if $\Im[\z]>0$. The same holds for the second term since the pole $\ve'=i\g$ has vanishing residue. By collecting all the results obtained one sees that they can be grouped into three broad categories: those which are time independent and that give rise to the stationary current, those which are proportional to ${\rm e}^{-\g t}$ and those which are proportional to ${\rm e}^{-2\g t}$. These last ones can be rewritten as $$-4e\g_{\a}{\rm e}^{-2\g t}\int\frac{d\w}{2\p}f(\w) \sum_{\a'}\frac{\g_{\a'}U^{2}_{\a'}}{[\w^{2}+\g^{2}][(\w+U_{\a'})^{2}+\g^{2}]}. \label{e2gt}$$ Let us now group the terms proportional to ${\rm e}^{-\g t}$. Two of them comes from Eq. (\[ctdcn2\]) and the first term of Eq. (\[ctdcn5\]); their sum can be written as $$-4e\g_{\a}U_{\a}{\rm e}^{-\g t}\int\frac{d\w}{2\p}f(\w) \;\Im\left\{\frac{{\rm e}^{i(\w+U_{\a})t}} {(\w+i\g)(\w+U_{\a}+i\g)}\right\}. \label{egt1}$$ The other two pieces come from Eq. (\[ctdcn3\]) (which we recall must be multiplied by 2) and the last term of Eq. (\[ctdcn5\]). By writing explicitly the real part, after some algebra one finds $$\begin{aligned} -8e\g_{\a}{\rm e}^{-\g t}\int\frac{d\w}{2\p}f(\w) \sum_{\a'}\g_{\a'}U_{\a'}\times \quad\quad\quad\quad \quad\quad \label{egt2} \\ \frac{\w\cos[(\w+U_{\a'})t]+\g\sin[(\w+U_{\a'})t]} {[\w^{2}+\g^{2}][(\w+U_{\a'})^{2}+\g^{2}]}. \nonumber\end{aligned}$$ The sum of Eqs. (\[e2gt\])-(\[egt1\])-(\[egt2\]) gives exactly the quantity $J_{\a}(t)-J_{\a}^{(\rm S)}$ of Eq. (\[DJFNFCO\]). [10]{} C. Caroli, R. Combescot, P. Nozìeres, and D. Saint-James, J. Phys. C [**4**]{}, 916 (1971). C. Caroli, R. Combescot, D. Lederer, P. Nozìeres, and D. Saint-James, J. Phys. C [**4**]{}, 2598 (1971). T. E. Feuchtwang, Phys. Rev. B [**10**]{}, 4121 (1974). T. E. Feuchtwang, Phys. Rev. B [**10**]{}, 4135 (1974). L. P. Kadanoff and G. Baym, [*Quantum Statistical Mechanics*]{} (W. A. Benjamin, Inc. New York, 1962). L. V. Keldysh, JETP [**20**]{}, 1018 (1965). Ned S. Wingreen, A.-P. Jauho, and Y. Meir, Phys. Rev. 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--- abstract: 'We show that the degree of spin polarization of photoelectrons from the surface states of topological insulators is 100 % if fully-polarized light is used as in typical photoemission measurements, and hence can be significantly [*higher*]{} than that of the initial state. Further, the spin orientation of these photoelectrons in general can also be very different from that of the initial surface state and is controlled by the photon polarization; a rich set of predicted phenomena have recently been confirmed by spin- and angle-resolved photoemission experiments.' author: - 'Cheol-Hwan Park' - 'Steven G. Louie' title: Spin Polarization of Photoelectrons from Topological Insulators --- Three-dimensional topological insulators (TIs) are strong spin-orbit interaction materials characterized by a bulk electronic gap and metallic topological surface state (TSS) bands with linear energy dispersions [@PhysRevLett.98.106803; @PhysRevB.75.121306; @PhysRevB.79.195322]. The predicted linear energy dispersion of the TSSs in TIs were first observed in angle-resolved photoemission spectroscopy (ARPES) measurements [@hsieh_nature2008]. TIs are considered to be a promising candidate for spintronic devices because of their spin-momentum locking [@zhang:natphys; @PhysRevLett.105.266806; @RevModPhys.82.3045]. Aspects of the spin distribution of the TSSs in TIs have been measured by spin-resolved ARPES experiments [@hsieh_science2009; @hsieh_nature2009; @PhysRevLett.106.216803; @arXiv:1101.3985; @PhysRevLett.106.257004; @PhysRevB.84.165113]. Because the spin orientation of the photoelectron in the specific conditions used in previous spin-resolved ARPES studies agreed with the expected picture of the spin distribution of the TSS electrons in TIs \[Fig. 1(a)\], photoemission matrix element effects were neglected in analyzing the spin polarization of the photoelectrons. (On the other hand, matrix element effects have been used in analyzing the circular dichroism of the TSS electrons in a TI [@PhysRevLett.107.077601; @PhysRevLett.107.207602; @PhysRevLett.108.046805; @arxiv:1108.1053].) Here, we find that the degree of spin polarization of photo-ejected electrons as defined in typical measurements with fully-polarized light is significantly higher (can in principle be 100%) than that of the TSSs ($\sim50$% for Bi$_2$Se$_3$ and Bi$_2$Te$_3$ according to first-principles calculations [@PhysRevLett.105.266806]), explaining the origin of high values of the measured degree of polarization ($75\%$ and $>85\%$ in Pan [*et al.*]{} [@PhysRevLett.106.257004] and Jozwiak [*et al.*]{} [@PhysRevB.84.165113], respectively, from experiments on Bi$_2$Se$_3$). Moreover, using the symmetries of the TI surface, we find that electron-photon interactions can completely alter the spin orientation of the photo-ejected electrons relative to that of the initial state and that the spin orientation of these photoelectrons can be controlled via light polarization tuning. For linearly polarized light, the detected spin orientation is significantly altered except for the specific case of the wavevector [**k**]{} of an initial state being parallel to the in-plane component of the light polarization $\hat{\epsilon}$. For [**k**]{} orthogonal to $\hat{\epsilon}$, the two spins are predicted to be antiparallel to each other. Moreover, for in-plane circularly polarized light, the spins of the photoelectrons are oriented either completely parallel or completely antiparallel to the surface normal depending on the handedness of the circular polarization. The measured degree of spin polarization is defined through the relation $$P_{\rm max}=\max_{\{\hat{t}\}}\frac{I_{\hat t}-I_{-{\hat t}}}{I_{\hat t}+I_{-{\hat t}}}\,, \label{eq:pol}$$ where $I_{\hat t}$ and $I_{-{\hat t}}$ are the intensities for the electron spin being aligned and anti-aligned with ${\hat t}$, respectively. The unit vector ${\hat t}={\hat t}_{\rm max}$ which maximizes Eq. (\[eq:pol\]) defines the direction of the electron spin polarization. Hsieh [*et al.*]{} [@hsieh_nature2009] reported in-plane $P_{\rm max}$ \[i.e., restricting $\hat{t}$ in Eq. (\[eq:pol\]) in the surface plane\] to be $20\%$ for photoelectrons from the TSSs of Bi$_2$Se$_3$, Souma [*et al.*]{} [@PhysRevLett.106.216803] and Xu [*et al.*]{} [@arXiv:1101.3985] reported $60\%$ for those of Bi$_2$Te$_3$, and Pan [*et al.*]{} [@PhysRevLett.106.257004] and Jozwiak [*et al.*]{} [@PhysRevB.84.165113] reported $75\%$ and $>85\%$, respectively, for those of Bi$_2$Se$_3$. On the other hand, the degree of in-plane spin polarization of the TSSs for both Bi$_2$Se$_3$ and Bi$_2$Te$_3$ obtained from first-principles calculations is $\sim50\%$ [@PhysRevLett.105.266806]. It is puzzling that the measured spin polarization of the photoelectrons [@PhysRevLett.106.257004; @PhysRevB.84.165113] can be significantly [*higher*]{} than that of the corresponding TSS obtained from theory [@PhysRevLett.105.266806]. Due to, e.g., spin-independent background signals, the measured degree of spin polarization is expected to always be [lower]{} than that from calculation. We find that the degree of spin polarization of photoelectrons from the TSSs of a TI is 100%. To show this, we start from the general expression for photocurrent from a detector (that selects a specific three-dimensional wavevector and hence a specific energy) with the spin quantization axis aligned with $\hat{t}$: $$I_{\hat t}\propto\left|\sum_{\{f| E_f=E_i+h\nu\}}\left<\hat{t},{\bf R}_D\right| \left.f\right>\left<f\right|H^{\rm int}\left|i\right>\right|^2\,, \label{eq:detected}$$ where $\left|i\right>$ is the initial TSS, $\left|f\right>$ the photo-excited state, $E_i$ and $E_f$ their respective energies, $h\nu$ the photon energy, and $H^{\rm int}$ the light-matter interaction Hamiltonian connecting the two states. State $\left|\hat{t},{\bf R}_D\right>$ is the detected state: (i) its spin part is the eigenstate of ${\bf s}\cdot\hat{t}$ with eigenvalue $+1$ (${\bf s}$ is the Pauli matrix vector for spin half) and (ii) its spatial part is localized at ${\bf R}_D$, where the detector is, far away from the sample surface. (Note that ${\bf R}_D$ is on the trajectory of the final-state wavepacket.) We dropped the obvious prefactor $\delta(E-E_i-h\nu)$ in front of the right hand side of Eq. (\[eq:detected\]) due to the energy-resolving detector which collects electrons with energy $E$. A formal derivation of Eq. (\[eq:detected\]) is reserved for interested readers [@suppl]. Here we discuss the physical meaning of Eq. (\[eq:detected\]). First, we note that the detector reads spin character of the photo-excited state at ${\bf R}_D$; hence, the near-surface part of the wavefunction affects the measured spin only indirectly through the matrix element $\left<f\right|H^{\rm int}\left|i\right>$. Second, a summation of the transition amplitude over degenerate photo-excited states $\left|f\right>$’s is necessary because, even in principle, we cannot tell which $\left|f\right>$ is involved in the detection [@feynman3]. If we denote $$\left|f'\right>=\sum_{\{f| E_f=E_i+h\nu\}}\left|f\right>\left<f\right|H^{\rm int}\left|i\right>\,, \label{eq:ip}$$ then we may rewrite Eq. (\[eq:detected\]) as $$I_{\hat t}\propto\left|\left<\hat{t},{\bf R}_D\right|\left.f'\right>\right|^2\,. \label{eq:detected2}$$ Since the measurement is performed at ${\bf R}_D$, far away from the sample where $\left|f\right>$’s are eigenstates of the free-electron Hamiltonian, the Bloch periodic part of the state $\left|f'\right>$ can be regarded as a position-independent two-element spinor in evaluating $I_{\hat{t}}$ by Eq. (\[eq:detected2\]). Then, from basic spin physics, we always can find a vector $\hat{t'}$ satisfying $\left|f'\right>$ being the eigenstate of ${\bf s}\cdot\hat{t'}$ with eigenvalue $+1$ and the eigenstate of ${\bf s}\cdot(-\hat{t'})$ with eigenvalue $-1$. Obviously $I_{{\hat t'}}\neq0$ and $I_{-{\hat t'}}=0$ for this particular orientation and we have $P_{\rm max}=100\%$ from Eq. (\[eq:pol\]), i.e., the degree of spin polarization of photo-ejected electrons is always 100% regardless of that of the initial electronic state. An important ingredient that led us to this result is that the initial TSS electronic state $\left|i\right>$ is not degenerate. If it is, a hole will be left in one of those [*different*]{} degenerate TSSs $\left|i\right>$’s after photodetection. Therefore, we can distinguish in principle which initial TSS is involved in the measurement; the detection probability amplitude corresponding to each $\left|i\right>$ should be first squared and then summed, and not the other way round [@feynman3], i.e., the photocurrent $I_{{\hat t}}$ will be the sum of contributions coming from all the degenerate initial TSSs $\left|i\right>$’s. If this happens, $P_{\rm max}\neq100\,$%. The simplest example is, for a normal spin-degenerate material, $I_{{\hat t}}$ and $I_{-{\hat t}}$ will always be the same regardless of the choice $\hat{t}$, making $P_{\rm max}=0$. Now we apply this general consideration to the case of the TSSs of a TI. According to recent [*ab initio*]{} calculations [@PhysRevLett.105.266806], the averaged degree of spin polarization of the TSSs, $$P^{\rm TSS}_{\rm ave}=\left| \left<\psi(n,{\bf k})\right|{\bf s}\left|\psi(n,{\bf k})\right> \right|\,, \label{eq:Pave}$$ where $\left|\psi(n,{\bf k})\right>$ is the two-component spinor wavefunction, is roughly 50%. Because $\left|\psi(n,{\bf k})\right>$ is not an eigenstate of a spin operator $\hat{t}\cdot{\bf s}$ for any $\hat{t}$, the degree of spin polarization for the TSSs had to be defined as an averaged quantity. On the contrary, the degree of spin polarization of the photoelectrons $P_{\rm max}$ \[Eq. (\[eq:pol\])\] is 100% and that is so even if we take into account the imaginary part of the self energies of the involved electronic states because the detector probes the spin character (of the photoexcited states) at ${\bf R}_D$ instead of taking its average over the entire space. If the imaginary part of the initial-state self energy is considered, the condition that “the initial electronic state is not degenerate” is naturally modified to “there are no other initial electronic states in the energy window within the width of the imaginary part of the initial-state self energy.” This result tells us that a direct comparison between $P_{\rm max}$ \[Eq. (\[eq:pol\])\] and $P^{\rm TSS}_{\rm ave}$ \[Eq. (\[eq:Pave\])\] is not meaningful, and solves the apparent puzzle that the former from experiments [@PhysRevLett.106.257004; @PhysRevB.84.165113] is higher than the latter from theory ($\sim50$%) [@PhysRevLett.105.266806]. Although the predicted degree of spin polarization of the photoelectrons from the TSSs is 100%, the measured value will always be lower than this due, e.g., to spin-unpolarized background signals and finite resolution of the apparatus. Equation (\[eq:detected\]) and our discussion based on it are not confined to topological insulators and can provide a guidance for the interpretation of any spin-resolved photoemission experiment using fully-polarized light. {width="1.8\columnwidth"} So far, we have discussed the magnitude of the spin polarization, without resorting to the details of the system. Now, by obtaining the state $\left|f'\right>$ in Eq. (\[eq:ip\]) using the specific symmetries of the TI surface, we determine the orientation of the spin polarization. We adopt the commonly used effective Hamiltonian $H^0_{\rm TI}({\bf k})$ for the Bloch periodic part $\left|\phi(n,{\bf k})\right>$ of the wavefunction of TSSs in a TI with (in-plane) Bloch wavevector ${\bf k}=k_x\hat{x}+k_y\hat{y}$ ($\hat{z}$ is along the surface normal) given by $$H^0_{\rm TI}({\bf k})=\hbar v\,k\,(\sin\,\theta_{\bf k}\,\sigma_x- \cos\,\theta_{\bf k}\,\sigma_y)\,, \label{eq:H_TI_0}$$ where $v$ is the band velocity, $\theta_{\bf k}$ the angle between [**k**]{} and the $+k_x$ direction, and $\sigma_x$ and $\sigma_y$ are the Pauli matrices acting on a two-component wavefunction, the so-called pseudospins. In constructing the effective Hamiltonian in Eq. (\[eq:H\_TI\_0\]), the basis states defining the $\left(\begin{array}{c}1\\0\end{array}\right)$ and $\left(\begin{array}{c}0\\1\end{array}\right)$ column vectors, corresponding to pseudospin up and down states, are constructed from the two degenerate states at ${\bf k}=0$. Here, we consider a class of materials having the symmetry of the surface of Bi$_2$Se$_3$ or Bi$_2$Te$_3$; however, the development can straightforwardly be extended to other classes. The two states at ${\bf k}=0$ which are used as basis, $\left|\phi_{1}\right>$ and $\left|\phi_{2}\right>$, can then be uniquely fixed by using the symmetry properties: $$\begin{aligned} \Theta\left|\phi_{1}\right>=-\left|\phi_{2}\right>\,&,&\,\,\, \Theta\left|\phi_{2}\right>=\left|\phi_{1}\right>\,,\nonumber\\ M\left|\phi_{1}\right>=i\left|\phi_{2}\right>\,&,&\,\,\, M\left|\phi_{2}\right>=i\left|\phi_{1}\right>\,,\nonumber\\ C_3\left|\phi_{1}\right>=e^{-i\pi/3}\left|\phi_{1}\right>\,&,&\,\,\, C_3\left|\phi_{2}\right>=e^{+i\pi/3}\left|\phi_{2}\right>\,, \label{eq:sym}\end{aligned}$$ where $\Theta$ is the time-reversal operator, $M$ the reflection operator, $x\to-x$ ($\hat x$ is along the $\Gamma$K direction), and $C_3$ the operator for ${2\pi}/{3}$ rotation around the $z$ axis. The eigenvalue and Bloch periodic eigenstate are $$E(n,{\bf k})=n\,\hbar\, v\,k\,, \label{eq:E_TI}$$ and $$\left|\phi(n,{\bf k})\right>= \frac{1}{\sqrt{2}}\left( \left|\phi_1\right> -n\,i\,e^{i\theta_{\bf k}}\left|\phi_2\right> \right)=\left( \begin{array}{c} 1\\ -n\,i\,e^{i\theta_{\bf k}} \end{array} \right)\,, \label{eq:wfn_TI}$$ respectively, where $n=\pm1$ is the band index. The pseudospin expectation value of the TSS \[Eq. (\[eq:wfn\_TI\])\] is thus given by $$\left<{\vec \sigma}\right>=n\left(\sin\theta_{\bf k}\,\hat{x} -\cos\theta_{\bf k}\,\hat{y}\right)\,. \label{eq:sigma0}$$ It is known that the actual spin expectation value $\left<{\bf s}\right>$ is aligned with the pseudospin expectation value [@RevModPhys.82.3045], i.e., $$\left<{\bf s}\right>\propto n\left(\sin\theta_{\bf k}\,\hat{x} -\cos\theta_{\bf k}\,\hat{y}\right)\,. \label{eq:s0}$$ The orientation of the spinor eigenstates in the upper band ($n=+1$) is shown in Fig. 1(a). In calculating photoemission matrix elements, we use the theoretical framework of Wang [*et al.*]{} [@PhysRevLett.107.207602]. At small [**k**]{}, it is assumed that the final photoemission states $\left|f\right>$’s are spin-degenerate because they are well within the spin-degenerate bulk band continuum. In matrix element calculations for small [**k**]{}, we will approximate the periodic part of the final state Bloch wavefunctions $\left|\phi^f_\uparrow({\bf k},k_\perp)\right>$ and $\left|\phi^f_\downarrow({\bf k},k_\perp)\right>$ by those with ${\bf k}=0$ and $k_\perp=\sqrt{ \frac{2m_e}{\hbar^2} (h\nu-E_{\rm D})}$, where $k_\perp$ is the surface normal component of the photoelectron wavevector, $m_e$ the electron mass and $E_{\rm D}$ the energy of two-fold degenerate TSSs at ${\bf k}=0$. We denote these two ([**k**]{}-independent) states by $\left|\phi^f_\uparrow\right>$ and $\left|\phi^f_\downarrow\right>$, respectively, and use the same symmetry relations as in Eq. (\[eq:sym\]) to define these states. Also, we will neglect the momentum dependence of the velocity operator ${\bf v}({\bf k})=e^{-i{\bf r}\cdot{\bf k}}\,{\bf v}\,e^{i{\bf r}\cdot{\bf k}}$ \[see Eq. (\[eq:v\])\]. that has to be used in calculating the optical transition matrix element between the periodic parts of the Bloch wavefunctions. This theoretical setup [@PhysRevLett.107.207602] was employed to find the energy- and momentum-dependent spin polarization of the TSS in Bi$_2$Se$_3$ from time-of-flight ARPES measurement with laser of energy 6.2 eV. The spin polarization of TSS electrons thus obtained is in excellent agreement with theory [@PhysRevLett.103.266801] and other spin-resolved ARPES experiments [@hsieh_science2009; @PhysRevLett.106.216803; @arXiv:1101.3985; @PhysRevLett.106.257004]. Therefore, our theoretical development and predictions below should be valid when low-energy light source is employed. An important point to note here is that $\left|\phi^f_\uparrow\right>$ and $\left|\phi^f_\downarrow\right>$ are [*the actual spin-up and spin-down states along $\hat{z}$ far away from the surface in vacuum where the measurement is performed*]{}, because we have imposed the symmetry constraints of the system in Eq. (\[eq:sym\]) [@zhang:natphys; @PhysRevLett.103.266801]. Even though the real spin character of those two basis states at the surface can be very complicated, we can regard the chosen degenerate doublet $\left|\phi^f_\uparrow\right>$ and $\left|\phi^f_\downarrow\right>$ the actual spin-up and spin-down states, respectively, from a measurement point of view. Next, the microscopic Hamiltonian $H_{\rm D}$ of an electron with spin-orbit coupling [@PhysRev.100.580] is given by $$H_{\rm D}=\frac{{\bf p}^2}{2m_e}+V+\frac{\hbar}{4m_e^2c^2} \left(\nabla V\times{\bf p}\right)\cdot{\bf s}\,, \label{eq:H_D}$$ where $V$ is the one-electron potential. Using Peierls substitution ${\bf p}\to{\bf p}-\frac{e}{c}{\bf A}$, where ${\bf A}$ is the vector potential of the electromagnetic wave, and the relation $H^{\rm int}_{\rm D}({\bf A}) = H_{\rm D}\left({\bf p}-\frac{e}{\hbar c}{\bf A}\right)-H_{\rm D}({\bf p})$, we obtain the electron-photon interaction Hamiltonian $$H^{\rm int}_{\rm D}({\bf A})=-\frac{e}{c}{\bf A}\cdot{\bf v}\,, \label{eq:Hint_D}$$ where $${\bf v}=\frac{\bf p}{m_e}+ \frac{\hbar}{4m_e^2c^2}\left({\bf s}\times \nabla V\right) \label{eq:v}$$ is the velocity operator [@PhysRev.100.580]. We first consider the matrix elements $${\bf v}_{s,i}=\left<\phi^f_s\right|{\bf v} \left|\phi_{i}\right>\,, \label{eq:matrix_elt2}$$ where $s=\uparrow$ or $\downarrow$ and $i=1$ or $2$ are the index of the photo-excite state and the pseudospin basis index of the TSS state, respectively. If we define $$v_\pm=v_x\pm i\,v_y\, \label{eq:vpm}$$ and use the symmetry of the system for both the initial and final states \[Eq. (\[eq:sym\])\], only the following four are non-zero among twelve possible combinations [@PhysRevLett.107.207602]: $$\begin{aligned} \left<\phi^f_\uparrow\right|v_+\left|\phi_2\right> &=& \left<\phi^f_\downarrow\right|v_-\left|\phi_1\right>^* = i\,\alpha\nonumber\\ \left<\phi^f_\uparrow\right|v_z\left|\phi_1\right>&=& \left<\phi^f_\downarrow\right|v_z\left|\phi_2\right> =i\,\beta\,, \label{eq:matrix_elt3}\end{aligned}$$ where $\alpha$ and $\beta$ are [*real*]{} constants which can be determined from, e.g., first-principles calculations. Plugging Eqs. (\[eq:matrix\_elt2\]) and (\[eq:matrix\_elt3\]) into Eq. (\[eq:Hint\_D\]), we obtain the interaction Hamiltonian matrix $H^{\rm int}_{\rm TI}({\bf A})$ connecting the two basis functions of the TSSs, $\left|\phi_1\right>$ and $\left|\phi_2\right>$, to the spin-up and spin-down photoexcited states, $\left|\phi^f_\uparrow\right>$ and $\left|\phi^f_\downarrow\right>$: $$H^{\rm int}_{\rm TI}({\bf A}) = \frac{\alpha}{2c}\,e\, \left[\left(A_y\,\sigma_x-A_x\,\sigma_y\right) +i\,\left(\frac{2\beta}{\alpha}\right)\,A_z\,I \right]\,, \label{eq:H_TI_int_general}$$ where $I$ is the $2\times2$ identity matrix. In this study, we neglect the last term that depends on the [*z*]{} component of the light polarization, i.e., we set $A_z=0$, in order to see the new physics clearly. Because this term is proportional to the identity matrix, it alone does not contribute to alteration of the spin direction of a photoemitted electron from that of the initial state. First we consider the case where the light is linearly polarized. Then, $$H^{\rm int}_{\rm TI}({\bf A}) = \frac{\alpha}{2c}\,eA\, (\sin\,\theta_{\bf A}\,\sigma_x-\cos\,\theta_{\bf A}\,\sigma_y)\,, \label{eq:H_TI_int}$$ where $\theta_{\bf A}$ is the angle between the vector potential [**A**]{} and the $+x$ direction. As we discussed before, the photocurrent is given by Eq. (\[eq:detected\]), where in our case $\left|i\right>=\left|\phi(n,{\bf k})\right>$ \[Eq. (\[eq:wfn\_TI\])\] and $\left|f\right>$’s are $\left|\phi^f_\uparrow\right>$ and $\left|\phi^f_\downarrow\right>$. Since the photocurrent $I_{\hat{t}}$ is nothing but the squared projection of the state $\left|f'\right>$ in Eq. (\[eq:ip\]) to the detector state $\left|\hat{t},{\bf R}_D\right>$, it is essential to know the spin orientation of $\left|f'\right>$. Using Eq. (\[eq:H\_TI\_int\]), we can write $\left|f'\right>$ \[Eq. (\[eq:ip\])\] in the basis of $\left|\phi^f_\uparrow\right>$ and $\left|\phi^f_\downarrow\right>$ as $$\left|\phi'(n,{\bf k})\right>= \frac{1}{\sqrt{2}} \left( \begin{array}{c} 1\\ -n\,i\,e^{i(2\theta_{\bf A}-\theta_{\bf k})} \end{array} \right)\,. \label{eq:wfn_TI2}$$ Comparing Eqs. (\[eq:wfn\_TI\]) with (\[eq:wfn\_TI2\]) and using Eqs. (\[eq:sigma0\]) and (\[eq:s0\]), we find that the net effect of photoexcitation on the detected electron spin polarization direction defined by the direction $\hat{t}$ maximizing $(I_{\hat{t}}-I_{-\hat{t}})/(I_{\hat{t}}+I_{-\hat{t}})$ in Eq. (\[eq:pol\]) is a rotation in direction through a change $$\theta_{\bf k}\to \theta_{\bf k}+2\Delta\theta_{{\bf A},{\bf k}}\,, \label{eq:trans}$$ where $\Delta\theta_{{\bf A},{\bf k}}\equiv\theta_{\bf A}-\theta_{\bf k}$ is the angle between the in-plane light polarization and the Bloch wavevector. The spin expectation value of the photoelectron arriving at the detector $\left<{\bf s}\right>_f$ in units of $\hbar/2$ is thus given by $$\left<{\bf s}\right>_f=n\left[ \sin\left(\theta_{\bf k}+2\Delta\theta_{{\bf A},{\bf k}}\right)\hat{x} -\cos\left(\theta_{\bf k}+2\Delta\theta_{{\bf A},{\bf k}}\right)\hat{y} \right]\,. \label{eq:s}$$ (Note that the magnitude of the spin polarization is 100% in agreement with the above, general discussion.) For the special case of $\theta_{\bf k}=\theta_{\bf A}$, the spin orientation of the photoelectron \[Eq. (\[eq:s\])\] is the same as that of the initial TSS electron \[Eq. (\[eq:s0\])\]. However, in general, the two are different. Especially, for states whose Bloch wavevector [**k**]{} is perpendicular to the in-plane component of the light polarization [**A**]{} \[i.e., $\left|\Delta\theta_{{\bf A},{\bf k}}\right|=\pi/2$ in Eq. (\[eq:trans\])\], the initial and final spins are [*antiparallel*]{} to each other. For in-plane circularly polarized light, the vector potential is given by ${\bf A}=A(\hat{x}\pm\,i\,\hat{y})/\sqrt{2}$ with the $+$ and $-$ signs denoting left-handed and right-handed circular polarizations as defined from the viewpoint of the light source, respectively. Then, within the effective Hamiltonian formalism \[Eq. (\[eq:H\_TI\_int\_general\])\], $$H^{\rm int}_{\rm TI}({\bf A}) = \frac{\alpha}{2\sqrt{2}c}\,eA\,(\pm\,i\,\sigma_x-\sigma_y)\,. \label{eq:H_TI_int_circ}$$ Applying the same argument as before, the spin polarization direction of the photoelectron measured by a spin-resolved detector is that of $$\left|\phi'(n,{\bf k})\right>_{\rm LHC}= \left( \begin{array}{c} 1\\ 0 \end{array} \right) =\left|\phi^f_\uparrow\right> \label{eq:wfn_TI2_cw}$$ and $$\left|\phi'(n,{\bf k})\right>_{\rm RHC}= \left( \begin{array}{c} 0\\ 1 \end{array} \right) =\left|\phi^f_\downarrow\right> \,, \label{eq:wfn_TI2_ccw}$$ for left-handed and right-handed circular polarizations, respectively. The spin polarization direction of the photoelectrons are thus pointed along the parallel ($+\hat{z}$) and antiparallel ($-\hat{z}$) directions to the surface normal, for left-handed \[Fig. 1(c)\] and right-handed \[Fig. 1(d)\] circular polarized light, respectively. The phenomenon of photo-induced spin rotation in a TI predicted here was not observed previously. The reason is that $\Delta\theta_{{\bf A},{\bf k}}$ in Eq. (\[eq:s\]) was held fixed or was allowed to change little in conventional spin-resolved ARPES measurements by linearly polarized light as different [**k**]{} states were probed. Recently, Lanzara and coworkers [@jozwiak] have tuned $\Delta\theta_{{\bf A},{\bf k}}$ and confirmed the photo-induced spin rotations, as predicted here for both linearly \[Eq. (\[eq:s\])\] and circularly \[Eqs. (\[eq:wfn\_TI2\_cw\]) and (\[eq:wfn\_TI2\_ccw\])\] polarized lights. In conclusion, we have shown that in spin-resolved photoemission experiments, the measured degree of spin polarization of photoelectrons with a specific energy is 100% under ideal condition regardless of that of the initial state if the initial state is not degenerate and fully-polarized light is used. We have used this general principle to explain why the degree of spin polarization of the photoelectrons from the surface states of a topological insulator can be higher than that of the initial states. Using the specific symmetries of the system, we have further shown that the spin polarization direction of photoexcited electrons from the topological surface states is in general very different from that of the initial states and is dictated by the light polarization. Our results provide a theoretical basis for manipulation of the spin polarization of the photoelectrons from the topological surface states [@jozwiak; @PhysRevLett.105.057401; @gedik:nnano]. 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Fermi’s golden rule ---------------------- Consider an electronic system described by a single-particle Hamiltonian $H_0$ with energy eigenvalues and corresponding eigenstates $E_n$ and $\left|n\right>$, respectively, where $n$ is the state index. Suppose now that we have a time-dependent perturbation of the form $H^{\rm int}(t)=H^{\rm int}({\bf A})\,e^{-2\pi i \nu t}$, where ${\bf A}$ is the vector potential and $h\nu$ the photon energy, in addition to the original Hamiltonian $H_0$. The total Hamiltonian is given by $$H(t)=H_0+H^{\rm int}({\bf A})\,e^{-2\pi i\nu t}\,. \label{eq:H}$$ We are interested in the evolution of an electron which is at $\left|i\right>$ before turning on the time-dependent perturbation. Assuming that the perturbation is turned on at $t=0$, the wavefunction satisfying normalization condition ($\left<\psi(t)|\psi(t)\right>=1$) to first order can be written as $$\left|\psi(t)\right>=e^{-iE_it/\hbar}\,\left|i\right>+\sum_f\,c_f(t)\,e^{-iE_ft/\hbar}\left|f\right>\,, \label{eq:c_f}$$ where $c_f(t)$ is the probability amplitude of finding an electron in state $\left|f\right>$ at a given time $t$. Obviously, $$c_f(0)=0\,. \label{eq:bc}$$ Using Eqs. (\[eq:H\]), (\[eq:c\_f\]), and (\[eq:bc\]), we can solve the time-dependent Schrödinger equation and arrive at $$c_f(t)=-i\,e^{i\,t\,\frac{\Delta E_f}{2\hbar}}\,\frac{\sin{t\,\frac{\Delta E_f}{2\hbar}}}{\frac{\Delta E_f}{2}}\, \left<f\right|\,H^{\rm int}\,\left|i\right>\,, \label{eq:c_ft}$$ where $$\Delta E_f\equiv (E_f-E_i)-h\nu\,. \label{eq:dE}$$ Equation (\[eq:c\_ft\]) leads to $$P_f(t)\equiv|c_f(t)|^2=\frac{t}{\hbar^2}\,\frac{\sin^2{t\,\frac{\Delta E_f}{2\hbar}}}{t\,\left(\frac{\Delta E_f}{2\hbar}\right)^2}\, \left|\left<f\right|\,H^{\rm int}\,\left|i\right>\right|^2\,. \label{eq:c2_ft}$$ Using the following relation $$\lim_{a\to\infty}\frac{1}{\pi}\frac{\sin^2(ax)}{ax^2}=\delta(x)\,, \label{eq:delta}$$ we finally obtain $$\begin{aligned} \Gamma_{f\leftarrow i}&=&\lim_{t\to\infty}P_f(t)/t\nonumber\\ &=&\frac{\pi}{\hbar^2}\delta\left(\frac{\Delta E_f}{2\hbar}\right) \left|\left<f\right|\,H^{\rm int}\,\left|i\right>\right|^2\nonumber\\ &=&\frac{2\pi}{\hbar}\delta(\Delta E_f) \left|\left<f\right|\,H^{\rm int}\,\left|i\right>\right|^2\,, \label{eq:goldenrule}\end{aligned}$$ which is Fermi’s golden rule. 2. Spin-resolved photoemission experiment ----------------------------------------- Now we find an expression for photocurrent $I_{\hat{t}}$ in spin-resolved photoemission experiment with a detector aligned such that it captures electrons in a state $\left|\hat{t},{\bf R}_D\right>$ of which (i) the spin part is the eigenvector of ${\bf s}\cdot\hat{t}$ with eigenvalue $+1$ and (ii) the spatial part is localized at the detector (${\bf r}={\bf R}_D$) as in the main manuscript. The photocurrent is then given by $$I_{\hat{t}}\propto\lim_{t\to\infty}\left|\left<\hat{t},{\bf R}_D|\psi(t)\right>\right|^2/t\,. \label{eq:Iph}$$ Note that in the previous section, we have used $\left|f\right>$ in the place of $\left|\hat{t},{\bf R}_D\right>$ to derive Fermi’s golden rule; replacement of the final state is the only difference. Using Eqs. (\[eq:c\_f\]) and (\[eq:c\_ft\]), we can write $\left<\hat{t},{\bf R}_D|\psi(t)\right>$ as $$\begin{aligned} \left<\hat{t},{\bf R}_D|\psi(t)\right>&=& \sum_{E}\sum_{\{f|E_f=E\}} -i\,e^{i\,t\,\frac{\Delta E_f}{2\hbar}}\,\frac{\sin{t\,\frac{\Delta E_f}{2\hbar}}}{\frac{\Delta E_f}{2}}\, \left<f\right|\,H^{\rm int}\,\left|i\right> \,e^{-iE_ft/\hbar}\,\left<\hat{t},{\bf R}_D|f\right>\nonumber\\ &=& \sum_{E} -i\,e^{i\,t\,\frac{\Delta E-2E}{2\hbar}}\,\frac{\sin{t\,\frac{\Delta E}{2\hbar}}}{\frac{\Delta E}{2}}\, \sum_{\{f|E_f=E\}} \,\left<\hat{t},{\bf R}_D|f\right> \left<f\right|\,H^{\rm int}\,\left|i\right>\,, \nonumber\\ \label{eq:step}\end{aligned}$$ where $\Delta E=(E-E_i)-h\nu$. An important knowledge we made use of in Eq. (\[eq:step\]) is that $\left<\hat{t},{\bf R}_D|i\right>=0$ because the initial state is localized at the crystal and does not extend to the detector. Obviously, a product of the terms in Eq. (\[eq:step\]) corresponding to different $E$ values does not contribute to the photocurrent calculated from Eq. (\[eq:Iph\]). Plugging Eq. (\[eq:step\]) into Eq. (\[eq:Iph\]) and using Eq. (\[eq:delta\]) again, we obtain the expression for the photocurrent: $$\begin{aligned} I_{\hat{t}}&\propto& \sum_{E} \delta(E-E_i-h\nu) \left|\sum_{\{f|E_f=E\}}\, \left<\hat{t},{\bf R}_D|f\right> \left<f\right|\,H^{\rm int}\,\left|i\right>\right|^2\nonumber\\ &\propto& \left|\sum_{\{f|E_f=E_i+h\nu\}}\, \left<\hat{t},{\bf R}_D|f\right> \left<f\right|\,H^{\rm int}\,\left|i\right>\right|^2\,, \label{eq:Iph2}\end{aligned}$$ which is Eq. (\[eq:detected\]) of the main manuscript. Equivalently, Eq. (\[eq:Iph2\]) can also be written as $$I_{\hat{t}}\propto \left|\sum_f\, \delta(E_f-E_i-h\nu)\, \left<\hat{t},{\bf R}_D|f\right> \left<f\right|\,H^{\rm int}\,\left|i\right>\right|^2\,. \label{eq:Iph3}$$

--- abstract: | It was proved that for any finite set of elements of a free product of residually finite groups such that no two of them belong to conjugate cyclic subgroups and each of them do not belong to a subgroup which is conjugate to a free factor there exists a homomorphism of the free product onto a finite group such that the order of the image of each fixed element is an arbitrary multiple of a constant number. *Key words:* free products, residual properties, omnipotence. *MSC:* 20E26, 20E06. --- **Vladimir V. Yedynak** *Faculty of Mechanics and Mathematics, Moscow State University* *Moscow 119992, Leninskie gory, MSU* *edynak\_vova@mail.ru* Introduction ============ Order separabilities are connected with the investigation of the correlation between the orders of elements’ images after a homomorphism of a group onto a finite group. For example in \[1\] it was proved that for each elements $u$ and $v$ of a free group $F$ such that $u$ is conjugate to neither $v$ nor $v^{-1}$ there exists a homomorphism of $F$ onto a finite group such that the images of $u$ and $v$ have different orders. In \[6\] it was proved that this property is inherited by free products. This paper is devoted to the proof of the theorem that strengthens the property of order separability for the class of free products of groups. **Theorem.** Consider the group $G=A\ast B$ where the subgroups $A$ and $B$ are residually finite. Consider the elements $u_1,\ldots, u_n$ such that $u_i\in G\setminus\{\,\cup_{g\in G}(g^{-1}Ag\cup g^{-1}Bg)\}, u_i, u_j$ belongs to conjugate cyclic subgroups whenever $i=j$. Then there exists the natural number $K$ such that for each ordered sequence $l_1,\ldots, l_n$ of natural numbers there exists a homomorphism $\varphi$ of $G$ onto a finite group such that the order of $\varphi(u_i)$ is equal to $Kl_i$ The property under study in this work is closely connected with omnipotence which was investigated in \[2\], \[3\] where it was shown that free groups and fundamental groups of compact hyperbolic surfaces are omnipotent. Besides all finite sets of independent elements whose orders are infinite in a Fuchsian group of the first type also satisfy the property of omnipotence \[4\]. Definition. The group $G$ is called omnipotent if for each elements $u_1,\ldots, u_n$ such that no two of them have conjugate nontrivial powers there exists a number $K$ such that for each ordered sequence of natural numbers $l_1,\ldots, l_n$ there exists a homomorphism $\varphi$ of $G$ onto a finite group such that the order of $\varphi(u_i)$ equals $Kl_i$. The familiar property was also investigated in \[7\] where some sufficient conditions were found for $n$-order separability of free products. The group $G$ is said to be $n$-order separable if for a set $S=\{\,s_1,\ldots, s_n\mid s_i\neq h^{-1}s_j^{\pm1}h, i\neq j\}$ of $n$ elements of $G$ there exists a homomorphism of $G$ onto a finite group mapping $S$ onto a set whose elements have pairwise different orders. Notice that the theorem of this paper will enable to investigate the residual properties of the fundamental group of graphs of groups whose vertex groups are residually finite free products and edge groups are cyclic not belonging to subgroups conjugate to free factors of vertex groups. Notations and Definitions ========================= We consider that for every graph there exists a mapping $\eta$ from the set of edges of this graph onto itself. For every edge $e$ this mapping corresponds an edge which is inverse to $e$. Besides the following conditions are true: $\eta(\eta(e))=e$ for each $e$, $\eta$ is a bijection, for every edge $e$ the beginning of $e$ coincides with the end of the edge $\eta(e)$. The graph is called oriented if from every pair of mutually inverse edges one of them is fixed. The fixed edge is called positively oriented and the inverse edge is called negatively oriented. Let $G$ be a free product of groups $A$ and $B$. There exists a correspondence such that for every action of $G$ on the set $X$ at which both $A$ and $B$ act freely there exists a graph $\Gamma$ satisfying the following properties: 1\) for each $c\in A\cup B$ and for each vertex $p$ of $\Gamma$ there exists exactly one edge labelled by $c$ going into $p$ and there exists exactly one edge labelled by the element $c$ which goes away from $p$. 2\) for every vertex $p$ of $\Gamma$ the maximal connected subgraph $A(p)$ of $\Gamma$ containing $p$ whose positively oriented edges are laballed by the elements of $A$ is the Cayley graph of the group $A$ with generators $\{\,A\}$; we define analogically the subgraph $B(p)$. 3\) we consider that for every edge $e$ from the first item there exists the edge inverse to $e$ which does not bear a label; two edges with labels are not mutually inverse; edges with labels are positively oriented. Definition 1. We say that a graph is the free action graph of the group $G=A\ast B$ if it satisfies the properties 1), 2), 3). Note that if $\varphi$ is the homomorphism of the group $G$ such that $\varphi_{A\cup B}$ is the bijection then the Cayley graph $Cay(\varphi(G); \{\,\varphi(A)\cup\varphi(B)\})$ of the group $\varphi(G)$ with respect to the set of generators $\{\,\varphi(A)\cup\varphi(B)\}$ is the free action graph of the group $G$. Remark. In what follows appending a new edge with label to a free action graph we shall consider that it is positively oriented and the inverse edge would have been appended. And if we delete an edge with label the inverse edge would have been deleted. If $e$ is the edge then $\alpha(e), \omega(e)$ are vertices which coincide with the beginning and the end of $e$ correspondingly. If we have the free action graph $\Gamma$ of the group $G$ then there exists the action of $G$ on the set of vertices of $\Gamma$ which is defined as follows. Let $p$ be an arbitrary vertex of $\Gamma$. Then according to the definition of the free action graph for each element $c$ from $A\cup B$ there exist edges $e$ and $f$ whose labels are equal to $c$ such that $\alpha(e)=p, \omega(f)=p$. In this case the action of $c$ on $p$ is defined as follows: $p\circ c = \omega(e), p\circ c^{-1} = \alpha(f)$. Remark also that if we change the property 2) in the definition of the free action graph supposing that $A(p)$ and $B(p)$ are the Cayley graphs of the homomorphic images of the groups $A$ and $B$ correspondingly we also obtain the graph such that there exists the action of the group $G$ on the set of its vertices. Such a graph will be referred to as an action graph of the group $G$. Since there exists the action of $G$ on the set of vertices of an action graph $\Gamma$ there exists a homomorphism of $G$ onto the group $S_n$, where $n$ is the cardinal number of the set of vertices of the graph $\Gamma$. Having a group $G$ and its action graph $\Gamma$ we shall denote this homomorphism as $\varphi_{\Gamma}$. If $e$ is the positively oriented edge of the action graph, then Lab$(e)$ is the label of $e$. Definition 2. Let $u$ be a cyclically reduced element of the group $G$ which belongs to neither $A$ nor $B$ and $\Gamma$ is the action graph of $G$. Fix a vertex $p$ of $\Gamma$. Then $u$-cycle in this action graph going from $p$ is the cycle $R=e_1\ldots e_n$ which satisfies the following properties: 1\) the path $P$ is a closed path such that its beginning $\alpha(P)=p$ 2\) consider $u=u_1\ldots u_k$ where $u_i\in A\cup B, u_i, u_{i+1}$ as well as $u_1, u_k$ do not belong to one free factor simultaneously; then $k$ divides $n$ and the edge $e_{ik+j}$ is positively oriented and has a label $u_j$, $1\leqslant j\leqslant k$ (indices are modulo $n$) 3\) the cycle $P$ is the minimal cycle which satisfies properties 1), 2). Definition 3. Suppose we have a path $S=e_1\cdots e_n$ in the action graph. Then the label of this path is the element of the group which is equal to $\prod_{i=1}^n$Lab$(e_i)'$, where Lab$(e_i)'$ equals either the label of $e_i$, if this edge is positively oriented, or Lab$(e_i)'= $ Lab$(\eta(e_i))^{-1}$ otherwise. We shall denote the label of the path $S$ as Lab$(S)$. Definition 4. Fix the graph $\Gamma$, $p$ and $q$ are vertices from $\Gamma$. Then we define the distance between $p$ and $q$ as $\rho(p, q)=\min_{S}\ l(S)$, where $S$ is an arbitrary path connecting $p$ and $q$, $l(S)$ is the number of edges in $S$. Notice that if a cycle $S$ does not have $l$-near vertices then each subpath of $S$ of length which less or equal than $l$ is geodesic. Definition 5. Fix an arbitrary graph and a cycle $S=e_1\cdots e_n$ in it. For every nonnegative integer number $l$ we shall say that $S$ does not have $l$-near vertices, if for every $i, j, i\neq j, 1\leqslant i, j\leqslant n$ the distance between the vertices $\alpha(e_i), \alpha(e_j)$ is greater or equal than $\min(l+1, |i-j|, n-|i-j|)$. Definition 6. Suppose we have the $u$-cycle $S$. It is obvious that its label equals the $k$-th power of $u$ for some $k$. Then we say that the length of the $u$-cycle $S$ is equal to $k$. Note that for the action graph $\Gamma$ and cyclically reduced element $u\in G\setminus\{\,A\cup B\}$ the order of $\mid\varphi_{\Gamma}(u)\mid$ coincides with the less common multiple of lengths of all $u$-cycles in the graph $\Gamma$. Hence if there exists a $u$-cycle in the action graph whose length equals $t$ then $\mid\varphi_{\Gamma}(u)\mid$ is a multiple of $t$. Suppose $u$ is an element of $A\ast B$ and $u=u_1\cdots u_n$ is the irreducible form of $u$. Then the length of $u$ is the number $l(u)=n$. The cyclic length $l'(u)$ of an element $u$ is the length of the cyclically reduced element which is conjugate to $u$. Auxiliary lemmas ================ **Lemma 1.** Consider the group $G=A\ast B$ where $A$ and $B$ are finite, $l$ and $n$ are natural numbers, $Q$ is a finite set of elements from $G$ which are cyclically reduced and whose lengths are greater than 1. Then for each $v\in Q$ there exists the homomorphism $\varphi$ of $F$ onto a finite group such that for each $u$ from $Q$ the $u$-cycles in the Cayley graph $Cay(\varphi(G); \{\,\varphi(A)\cup\varphi(B)\})$ of the group $\varphi(G)$ do not have $l$-near vertices, $\mid\varphi(v)\mid>n$, and $\varphi_{A\cup B\cup Q}$ — injection. *Proof.* For each $q\in Q$ define the set $L_q$ which consists of the elements from $G$ whose length is less or equal than $l+2l(q)+10$ and which do not belong to the subgroup generated by the element $q$. It is well known that free group are subgroup separable \[5\]. Besides if a group is virtually subgroup separable than it is subgroup separable (see \[2\] for example). Considering the above there exists the homomorphism $\varphi_q$ of the group $G$ onto a finite group such that $\varphi_q(L_q)\cap\<\varphi_q(q)\>$ is an empty set. There also exists the homomorphism $\varphi_v'$ of $G$ onto a finite group such that $\varphi_v'(v^i)\neq1$ where $i=1,\ldots, n$ and $\varphi_v'|_{A\cup B\cup Q}$ is the injection since virtually free groups are residually finite. The homomorphism $\varphi: G\rightarrow(\times_{h\in Q}\varphi_h(G))\times\varphi_v'(G), \varphi: f\mapsto\prod_{h\in Q}(\varphi_h(f))\varphi_v'(f)$ is as required. Lemma 1 is proved. The following statement was proved in \[2\] **Lemma 2.** Consider a group $G$ and its elements $g_1,\ldots, g_n$ possessing the property that for each $j, 1\leqslant j\leqslant n,$ there exist constants $K_{j, 1},\ldots, K_{j, n}$ such that for each natural $m$ there exists a homomorphism $\varphi_{j, m}$ of $G$ to a finite group with the condition that $|\varphi_{j, m}(g_k)|=K_{j, k}$ for all $k\neq j$ and $|\varphi_{j, m}(g_j)|=mK_{j, j}$. Then there exists the number $K$ such that for each ordered sequence of natural numbers $l_1,\ldots, l_n$ there exists the homomorphism $\psi$ of $G$ onto a finite group satisfying the property that $|\psi(g_i)|=Kl_i$. Proof of the theorem ==================== It follows from lemma 2 that that the theorem can be derived from the following proposition. **Proposition.** Let $G=A\ast B$ be a free product of residually finite groups $A$ and $B$, $u, v_1,\ldots, v_n \in G$. Elements $u$ and $v_i$ do not belong to conjugate cyclic subgroups. Besides $u$ does not belong to a subgroup which is conjugate to either $A$ or $B$. Then there exist natural numbers $L, K_1,\ldots, K_n$ such that for each natural $i$ there exists a homomorphism $\varphi$ of $G$ onto a finite group such that $\mid\varphi(u)\mid=Li, \mid\varphi(v_i)\mid=K_i, 1\leqslant i\leqslant n$. *Proof.* Since $A, B$ are residually finite we may consider that $A$ and $B$ are finite. Consider also that the elements $u, v_1,\ldots, v_n$ of $A\ast B$ are cyclically reduced. Let us to define the following notation. Consider the action graph $\Gamma$ of the group $K\ast L$. Let $S$ be the subset of $\Gamma$ (e. g. vertex, edge, path, subgraph etc). Then having a set $\Gamma_1,\ldots, \Gamma_n$ of copies of $\Gamma$ we consider that $S^i$ denotes the subset of $\Gamma_i$ corresponding to $S$ in $\Gamma$. Put $s=\max_{b\in\{\,u, v_1,\ldots, v_n\}}l(b)$, and let $k'$ be an arbitrary natural number such that $k'l(u)\geqslant10s$. Put $k=k'l(u)$. Denote by $P$ the set of all nonunit elements whose length is less or equal than $10k$. For $Q=\{\,u, v_1,\ldots, v_n\}\cup P$ according to lemma 1 there exists the homomorphism $\varphi$ of $G$ onto a finite group such that $\varphi_{A\cup B\cup Q}$ is the injection and for each $s\in S$ which is cyclically reduced and whose length is greater than 1 each $s$-cycle in the graph $\Gamma=Cay(\varphi(G); \{\,\varphi(A)\cup\varphi(B)$ has no $(k+4)$-near vertices and $\mid\varphi(u)\mid>10k$. Fix a natural number $m>2$ whose value we shall choose later. Consider $m$ copies of the graph $\Gamma$: $\Gamma_{i}, 1\leqslant i\leqslant m$. In the graph $\Gamma$ we fix a $u$-cycle $S=e_1\cdots e_r$. Without loss of generality we consider that Lab$(e_1)\in A$. Put $p_i=\alpha(e_i)$, Lab $(e_i)=u_i$ (see Figure 1). For each $i, 1\leqslant i\leqslant m,$ we delete edges incident to $p_2^i$ whose labels belong to $A$ and delete also edges labelled by the elements of $A$ whose begin or end points are $p_{k+2}^i$. For each $i$ we shall denote the obtained graph as $\Gamma_i'$.  Let $\psi$ be the bijection between the subgraphs $A(p_1)$ and $A(p_{k+1})$ which saves labels of edges and $\psi(p_1)=p_{k+1}$. Fix an arbitrary edge $e$ of $\Gamma$ from the subgraph $A(p_1)$ such that the corresponding edge $e^i$ of $\Gamma_i$ was deleted. Let $q=\alpha(e), r=\omega(e)$. For each $i, 1\leqslant i\leqslant m,$ if $q\neq p_2$ we connect the vertices $q^i$ and $\psi(r)^{i+1}$ by the new edge $f_i$. If $r\neq p_2$ we connect $r^i$ and $\psi(q)^{i+1}$ by the edge $f_i$. In both cases the label of $f_i$ coincides with Lab $(e)$, besides if $q\neq p_2$ then $f_i$ goes away from $q^i$ and if $r\neq p_2$ then $f_i$ goes into $r^i$. Now we need to complement the structure of obtained graph for to get the action graph of the group $A\ast B$. But it will not be the free action graph. For each $i, 1\leqslant i\leqslant m,$ let us to add one new vertex $n_i$ to the subgraph $\Gamma_i'$. Consider an arbitrary edge $e$ from $A(p_{k+2})$ such that the corresponding edge $e^i$ was deleted from $\Gamma_i$. Put $q=\alpha(e), r=\omega(e)$. If $q=p_{k+2}$ then connect the vertices $r^i$ and $n_i$ by the edge $g_i$. If $r=p_{k+2}$ then the new edge $g_i$ connects the vertices $q^i$ and $n_i$. Put Lab $(g_i)=$ Lab $(e)$. The begin point of $g_i$ coincides with either $n_i$ or $q^i$. For each $c\in A\cup B$ and for each vertex $p$ of the obtained graph which is not incident to an edge with label $c$ add a loop with label $c$ going from $p$. If we fix $j$ then the union of the graph $\Gamma_{j}'$ and $A(p_{k+1}^j), A(p_{k+2}^j), A(p_1^j)$ is denoted by $\Delta_j$. We constructed the new graph $\Delta$ which contains subgraphs $\Gamma_{j}'$ and $\Delta_j$ and is the action graph of the group $G$. In the graph $\Delta$ the $u$-cycle $S'$ going from the vertex $p_{1}^1$ has the length $(\mid\varphi(u)\mid - k')m$. From the properties of the homomorphism $\varphi$ it follows that $\mid\varphi(u)\mid>10k=10k'l(u)>k'$. Hence $\mid\varphi_{\Delta}(u)\mid\geqslant(\mid\varphi(u)\mid - k')m>m$. Let us to prove that for each $i$ and for each $v_i$-cycle $T$ in the graph $\Delta$ all vertices of $T$ belong to two subgraphs $\Delta_{j_1}, \Delta_{j_1+1}$ for some $j_1$. Suppose the contrary. That is we suppose that there exist pairwise different numbers $j_1, j_2, j_3$ such that the vertices of $T$ belong to all three subgraphs $\Delta_{j_1}, \Delta_{j_2}, \Delta_{j_3}$. Note that different subgraphs $\Delta_{k_1}, \Delta_{k_2}$ has the nonempty intersection if and only if $\mid k_1-k_2\mid=1$ and their intersection equals the subgraph $A(p_1^l)$ since $m>2$ where $l$ is equal to either $k_1$ or $k_2$. So if $\Delta_{j_1}, \Delta_{j_2}, \Delta_{j_3}$ contain vertices of $T$ there exists the number $j$ such that the subgraphs $\Delta_j, \Delta_{j+1}, \Delta_{j+2}$ contain the vertices of $T$ and there exists the path $R$ which is the part of $T$ and which belongs to $\Delta_j\cup\Delta_{j+1}\cup\Delta_{j+2}$, $R$ goes away from the vertex of $\Delta_j$ and goes into the vertex of $\Delta_{j+2}$ (indices are modulo $m$). From the properties of $R$ it follows that $R$ contains its first and the last edges $t_j, r_j$ correspondingly such that $t_j\in A(p_1^j), \omega(t_j)=p_{k+2}^{j+1}, r_j\in A(p_1^{j+1}), \omega(r_j)=p_{k+2}^{j+2}$, and the rest edges of $R$ are in $\Delta_{j+1}$. Because of our supposition that $T$ goes from $\Delta_{j+1}$ into $\Delta_{j+2}$ it is possible to deduce that $R$ contains the subpath $s_1\cdots s_l$ such that $s_2\cdots s_{l-1}$ belongs to $\Gamma_{j+1}'$ and edges $s_1, s_l$ satisfy the following properties: $\alpha(s_1)\in A(n_{j+1})\cup B(p_{k+2}^{j+1}), \omega(s_l)\in A(p_1^{j+1})\cup B(p_2^{j+1})$. Denote the path $e^{j+1}_1e^{j+1}_2\cdots e^{j+1}_{k+1}$ as $S_u$ and $s_2\cdots s_{l-1}$ as $S_{v_i}$ (see Figure 2). Note that $\rho(\alpha(S_u), \alpha(S_{v_i}))\leqslant1, \rho(\omega(S_u), \omega(S_{v_i}))\leqslant2$ (the function $\rho$ is taken with respect to $\Gamma_i$). Besides $S_{v_i}$ is a part of some $v_i$-cycle, $S_u$ is a part of the $u$-cycle $S'$. Since the elements $u, v_i$ of the group $A\ast B$ do not belong to conjugate cyclic subgroups and the length of the path $S_u$ is greater than $10s=10\max_{z\in\{\,u, v_1,\ldots, v_n\}}(l(z))$ the paths $S_{v_i}$ and $S_u$ are different. Suppose that the length of the path $S_{v_i}$ is less or equal than $k+4=l(S_u)+3$. The paths $S_{v_i}$ and $S_u$ and perhaps several edges whose number is less than 4 compose the loop. Let $g$ be the label of this loop. Then $g$ is an element of the group $G$ whose length is less or equal than $2l(S_u)+6=2k+8<10k$ and $\varphi(g)=1$. But this contradicts the condition on $\varphi$ and the set $Q$. Thus the length of the path $S_{v_i}$ is greater than $k+4=l(S_u)+3$. By the symmetry we may also assume that the length of the path $T\setminus S_{v_i}$ is greater than $k+4$: the structure of the part of $T$ in $\Delta_{j+2}$ is the same as in $\Delta_{j+1}$. But in this case $\rho(\alpha(S_{v_i}), \omega(S_{v_i}))\leqslant\min(l(S_u)+3, l(S_{v_i}), l(T\setminus S_{v_i}))=\min(k+4, l(S_{v_i}), l(T\setminus S_{v_i}))=k+4$, since $l(S_{v_i}), l(T'\setminus R')>k+4$. So the $v_i$-cycle $T$ containing $S_{v_i}$ has $(k+4)$-near vertices. This also contradicts the conditions on $\varphi$. Thus it is proved that for each $i$ and for each $v_i$-cycle $T$ in the graph $\Delta$ there exists $j, 1\leqslant j\leqslant m,$ such that all vertices of $T$ are contained in $\Delta_j\cup\Delta_{j+1}$ (indices are modulo $m$). We deduce also that each $u$-cycle of $\Delta$ which does not start at $p_1^1$ belongs to two subgraphs $\Delta_{k_1}, \Delta_{k_1+1}$. This can be established by the same way as it was shown that the analogical statement is true for $v_i$-cycles. Now we shall denote the obtained graph $\Delta$ for number $m$ as $\Delta_m'$. Consider the set of graphs $\Lambda_m=\Delta_{3m}', m=1, 2,..$. We shall show now that $|\varphi_{\Lambda_m}(v_i)|$ equals some constant number $K_i$ which does not depend on $m$. Let $R_{i, m}$ be the set of lengths of all $v_i$-cycles of $\Lambda_m$. The local structure of $\Lambda_m$ is the same: using the above notations and regarding that $\Lambda_m$ is the union of $\Delta_1,\ldots, \Delta_{3m}$ it is obvious that the subgraphs $\Delta_k\cup\Delta_{k+1}$ and $\Delta_l\cup\Delta_{l+1}$ are isomorphic and do not depend on $m$. Hence $R_{i, m}$ coincides with the set of lengths of $v_i$-cycles concentrated in $\Delta_1\cup\Delta_2$ and thereby $R_{i, m}=R_{i, t}$ for all $m, t$. The same reasonings are true for all $u$-cycles of $\Lambda_m$ except for the $u$-cycle whose length is the multiple of $3m$ so $|\varphi_{\Lambda_m}(u)|=mK$ for some constant $K$ which does not depend on $m$. Proposition is proved and therefore the theorem is also proved. I am grateful to Anton A. Klyachko, Ashot Minasyan, Denis Osin and Henry Wilton for valuable conversations and for information about omnipotence. 1\. Klyachko, A. A. Equations over groups, quasivarieties, and a residual property of a free group. *Journal of group theory* **2**: 319-327, 1999. 2\. Wise, Daniel T. Subgroup separability of graphs of free groups with cyclic edge groups. *Q. J. Math.* 51, No.1, 107-129 (2000). \[ISSN 0033-5606; ISSN 1464-3847\] 3\. Jitendra Bajpai. Omnipotence of surface groups. Masters Thesis, McGill University, 2007. 4\. Henry Wilton. Virtual retractions, conjugacy separability and omnipotence. J. Algebra 323 (2010), pp. 323-335. 5\. Marshall Hall, Jr. Coset representations in free groups. *Trans. Amer. Math. Soc.*, 67:421–432, 1948. 6\. Yedynak, V. V. Separability with respect to order. *Vestnik Mosk. Univ. Ser. I Mat. Mekh.* **3**: 56-58, 2006. 7\. Yedynak, V. V. Multielement order separability in free products of groups. *Communications in Algebra* 38(**3**): 3448 –- 3455, 2010.

--- abstract: 'Quantum simulators are engineered devices controllably designed to emulate complex and classically intractable quantum systems. A key challenge lies in certifying whether the simulator is truly mimicking the Hamiltonian of interest. However, neither classical simulations nor quantum tomography are practical to address this task because of their exponential scaling with system size. Therefore, developing novel certification techniques, suitable for large systems, is highly desirable. Here, in the context of fermionic spin-based simulators, we propose a global many-body spin to charge conversion scheme, which crucially does not require local addressability. A limited number of charge configuration measurements performed at different detuning potentials along a spin chain allow to discriminate the low-energy eigenstates of the simulator. This method, robust to charge decoherence, opens the way to certify large spin array simulators as the number of measurements is independent of system size and only scales linearly with the number of eigenstates to be certified.' author: - Abolfazl Bayat - Benoit Voisin - Gilles Buchs - Joe Salfi - Sven Rogge - Sougato Bose title: 'Certification of spin-based quantum simulators ' --- *Introduction.–* Quantum simulators [@georgescu2014quantum; @cirac2012goals; @buluta2009quantum] are devices designed to emulate the behavior of quantum systems whose complexity generically increases exponentially with size. Their importance is multifold as they: (i) can provide new insights into complex quantum phenomena, e.g. high temperature superconductivity [@anderson1987resonating; @lee2006doping], non-Abelian gauge theories [@banerjee2013atomic], scattering effects [@zhang2018experimental], quantum criticality [@PhysRevA.94.063626], and long-term many-body dynamics [@daley2012measuring; @trotzky2012probing; @lv2018quantum]; and (ii) realize models that do not exist naturally, e.g. Kitaev Hamiltonian for Toric code [@kitaev2003fault]. One of the main challenges in quantum technology is to certify that an engineered quantum simulator, non-tractable classically, truly emulates the system of interest [@wiebe2014hamiltonian; @gao2017quantum; @hangleiter2017direct; @aolita2015reliable]. A necessary first step consists in matching the simulator low energy eigenstates with their expected counterparts in the emulated model. This task is highly challenging as it usually requires full quantum state tomography, because eigenstates may differ only by their global entanglement structure. However, this requires local addressability along with a number of measurements which scales exponentially with system size [@PhysRevLett.105.150401]. Recently, quantum simulators have been implemented in various setups, including cold atoms [@bernien2017probing], ion traps [@zhang2017observation; @dutta2012nonequilibrium], superconducting devices [@barends2014superconducting; @o2016scalable; @roushan2017spectroscopic; @barends2015digital], semiconductor quantum dots [@watson2018programmable; @hensgens2017quantum; @li2018crossbar; @zajac2016scalable; @nakajima2017robust] and dopant arrays [@salfi2016quantum]. Semiconductors offer a scalable platform with the natural presence of Fermi statistics (as opposed to simulating fermions with bosonic qubits via non-local interactions) and of Coulomb, electron-phonon and spin-orbit interactions. However, state-of-the-art spin readout techniques for single [@elzerman2004single; @pla2012single] and adjacent pairs [@petta2005coherent; @shulman2012demonstration; @broome2017high; @gray2016unravelling; @banchi2016entanglement; @gray2018machine] of electrons requires challenging single site accessibility. Therefore it is crucial to develop new certification schemes applicable to large spin systems. ![ **Schematic of the system.** (a) A chain of interacting electrons confined in the sites of a regular lattice. (b) Potential gradient (tilt) applied across the chain to change the charge configuration. (c) Low energy spectrum and corresponding charge configuration in tilted system.[]{data-label="Fig_Schematic"}](Fig_Schematic){width="7cm" height="4cm"} Here, we propose a global spin to charge conversion readout scheme to discriminate between the low-energy entangled spin eigenstates of a spin chain. The basic principle is to measure the charge configuration of the simulator under different potential gradients (called tiltings) applied across the chain. Importantly the number of tilts is independent of the system size and only scales linearly with the number of eigenstates to discriminate. This readout scheme can be used to both certify (when the solution is known) and measure (when an unknown process is being simulated) the system evolution in the low energy regime. Our scheme can greatly facilitate the realization of a solid state spin-based quantum simulator as: (i) charge detections are easier to perform than direct spin measurements [@watson2018programmable; @hensgens2017quantum; @li2018crossbar; @zajac2016scalable; @nakajima2017robust; @salfi2016quantum]; (ii) a single capacitive detector is able to readout charge configurations of multiple sites [@nakajima2017robust]; (iii) global potential tilts are sufficient as opposed to local addressability; and, most importantly, (iv) the distinction of eigenstates sharing the same symmetries and the total spin (differing only in their entanglement structure) is possible without quantum tomography. *Model.–* The Heisenberg spin chain is a key model in condensed matter physics [@sachdev2011quantum; @amico2008entanglement], spintronics [@vzutic2004spintronics] and quantum technologies [@farooq2015adiabatic; @yang2010spin]. To simulate this model we consider $N$ interacting electrons hopping among $N$ sites (i.e. half filling) in a regular 1D lattice. The Hamiltonian is characterized by the Fermi-Hubbard model $$\begin{aligned} \label{Hubbard_Hamiltonian} H &=& t \sum_{k=1}^{N-1} \sum_{\sigma=\uparrow,\downarrow} \left( c_{k,\sigma}^\dagger c_{k+1,\sigma}+ c_{k+1,\sigma}^\dagger c_{k,\sigma} \right) + \sum_{k=1}^{N} \tilde{\epsilon}_k n_k \cr &+& V \sum_{k=1}^{N-1} n_k n_{k+1} + \frac{U}{2} \sum_{k=1}^{N} n_k (n_k-1),\end{aligned}$$ where $c_{k,\sigma}$ ($c_{k,\sigma}^\dagger$) is the annihilation (creation) fermionic operator for an electron at site $k$ with spin $\sigma$, number operator $n_k=\sum_{\sigma=\uparrow,\downarrow}c_{k,\sigma}^\dagger c_{k,\sigma}$ counts the number of electrons at site $k$, $t$ is the tunnel coupling between neighboring sites, $\tilde{\epsilon}_k$ is the local potential at site $k$, $V$ is the Coulomb interaction between adjacent sites and $U$ is the on-site energy. In the case of a homogeneous 1D array, i.e. $\tilde{\epsilon}_k{=}0$, the Hamiltonian (\[Hubbard\_Hamiltonian\]) is solvable [@Elliott1968]. Throughout this letter we consider a chain made of an even number of sites $N$, with on-site energy $U/t{=}40$, Coulomb interaction $V/t{=}10$ and local potential of the form $\tilde{\epsilon}_k=(k-1)\epsilon$ where $\epsilon$ is the potential difference between two adjacent sites. A schematic picture of the system is shown in Fig. \[Fig\_Schematic\](a). In a homogeneous lattice ($\tilde{\epsilon}_k{=}0$), whenever $U{\gg} t$, the low energy eigenstates take the charge configuration $(1,1,\cdots,1)$ and the system effectively becomes a Heisenberg spin chain with exchange coupling $J{\sim} t^2/U$ (with possible corrections due to $V$) [@pica2014exchange]. These eigenstates form a low energy manifold separated by units of $U$ from the eigenstates with double charge occupancies for which the map to the Heisenberg model fails. For even $N$ the ground state $|S_1\rangle$ is always a global singlet with total spin $S_{tot}{=}0$. The first two excited states $|T_1\rangle$ and $|T_2\rangle$ are triplets with the total spin $S_{tot}{=}1$. The fourth eigenstate is again another global singlet $|S_2\rangle$. In a chain of length $N=4$ these four eigenstates form the low energy manifold. *Charge configurations.–* Many-body spin eigenstate measurement is challenging. For example, $|S_1\rangle$ and $|S_2\rangle$ have the same total spin $S_{tot}{=}0$ and share various symmetries (e.g. SU(2) invariance) making them difficult to be distinguished locally. To achieve spin eigenstate readout, we apply a potential tilt across the chain, i.e. a finite $\epsilon$, to provide enough energy for electrons to overcome $U$, as shown in Fig. \[Fig\_Schematic\](b), the charge configuration is measured. Since the eigenstates are always orthogonal, their charge configuration, which are experimentally measurable, depend on their spin state. This is the core of our certification method. ![ **Singlet charge configurations.** Charge occupancies of a chain of length $N=4$ for the state: (a) $|S_1\rangle$; and (b) $|S_2\rangle$. (c) Energy spectrum of the first three singlet eigenstates. []{data-label="Fig_Charge_S_N4"}](Fig_Charge_S_N4){width="7.5cm" height="5.5cm"} We now develop the evolution of the charge configurations versus the tilt for a chain of $N{=}4$. Longer chains are discussed in the Supplementary Material (SM). The charge configuration of the two singlet eigenstates $|S_1\rangle$ and $|S_2\rangle$ as a function of $\epsilon/t$ are plotted in Figs. \[Fig\_Charge\_S\_N4\](a)-(b). The charge configuration changes for both eigenstates around $\epsilon/t {\sim} 13.4$ and one electron moves from either site $4$ (in the case of $|S_1\rangle$) or site $3$ (in the case of $|S_2\rangle$) to site $1$, creating two different charge configurations for $|S_1\rangle$ and $|S_2\rangle$. At around $\epsilon/t{\sim} 30$ in the eigenstate $|S_2\rangle$ an electron moves from site $4$ to site $2$ resulting in the charge configuration $(2,2,0,0)$. Finally, at $\epsilon/t {\sim} 50$ the charge configuration of $|S_2\rangle$ evolves to $(2,1,1,0)$ while $|S_1\rangle$ rearranges to $(2,2,0,0)$. All these charge configurations are summarized in Fig. \[Fig\_Schematic\](c). To understand this charge dynamics we plot the energies of the first three singlet eigenstates in Fig. \[Fig\_Charge\_S\_N4\](c). Any charge movement in the eigenstates corresponds to an anti-crossing between two eigenstates with the same $S_{tot}$. This is evident at $\epsilon/t {\sim} 13.4$, $\epsilon/t {\sim} 30$ and $\epsilon/t {\sim} 50$ where $E_{S_1}$ and $E_{S_2}$, $E_{S_2}$ and $E_{S_3}$ and $E_{S_1}$ and $E_{S_2}$ again, anti-cross. ![ **Triplet charge configurations.** Charge occupancies of a chain of length $N=4$ for the state: (a) $|T_1\rangle$; and (b) $|T_2\rangle$. (c) Energy spectrum of the first two triplet eigenstates. []{data-label="Fig_Charge_T_N4"}](Fig_Charge_T_N4){width="7.5cm" height="6cm"} A similar analysis can be performed for the triplet states. The charge configurations of the two triplets $|T_1\rangle$ and $|T_2\rangle$ are depicted in Figs. \[Fig\_Charge\_T\_N4\](a)-(b), respectively. The charge configuration of both eigenstates changes around $\epsilon/t {\sim} 13.4$ and one electron moves from either site $4$ (in the case of $|T_1\rangle$) or site $3$ (in the case of $|T_2\rangle$) to site $1$. In Fig. \[Fig\_Charge\_T\_N4\](c) we plot the energy eigenvalues of both $|T_1\rangle$ and $|T_2\rangle$ as functions of $\epsilon/t$ which show an anti-crossing at the charge transition point $\epsilon/t {\sim} 13.4$. For larger systems (see the SM), the final charge configurations are $(2,\cdots,2,0,\cdots,0)$ for $|S_1\rangle$ and $(2,\cdots,2,1,1,0,\cdots,0)$ for $|T_1\rangle$. This important feature will be used for certification later in the letter. *Adiabatic tilting.–* In order to readout the many-body spin eigenstate, we tilt the system, initially prepared in one of the low energy eigenstates, adiabatically such that it remains in the local eigenvector of the Hamiltonian at any time $\tau$. The eigenstates can be discriminated by measuring the charge configuration at different potentials $\epsilon$. The tilt potential varies as $$\label{epsilon_tau} \epsilon (\tau)= \left\{ {\begin{array}{c} \frac{\tau}{T_{max}} \epsilon_{max}, \quad \text{ for: } \tau \le T_{max} \\ \epsilon_{max}, \quad \qquad \text{for: } \tau > T_{max} \\ \end{array} } \right.$$ where $\epsilon_{max}$ is the maximum tilt potential considered here to be $\epsilon_{max}/t=70$. For any initial state $|\Psi(0)\rangle$ the system evolves to the state $|\Psi(\tau)\rangle$ according to the Schrödinger equation under the action of the time dependent Fermi-Hubbard Hamiltonian described in Eq. (\[Hubbard\_Hamiltonian\]). The choice of $T_{max}$ is important as it results in different system dynamics. Adiabaticity, which notably protects the evolution against Landau-Zener transitions while sweeping through anticrossings, is achieved for slow dynamics and large $T_{max}$. However, faster dynamics minimizes charge decoherence effects at these transitions. In Fig. \[Fig\_Adiabatic\_ST\_N4\](a) we plot the charge occupancies for the quantum state $|\Psi(\tau)\rangle$, taking $T_{max}=2\times 10^4/t$, as a function of time when the system is initially prepared in the state $|S_1\rangle$. The charge configurations are very similar to the real eigenstates displayed in Fig. \[Fig\_Charge\_S\_N4\](a), with the fidelity of the evolution $F=\left| \langle \Psi(\tau)|S_1(\tau)\rangle \right|^2$ remaining above 0.98 throughout the evolution, which demonstrates that the adiabatic condition is well satisfied. In Fig. \[Fig\_Adiabatic\_ST\_N4\](b) we depict the charge occupancies when the system is initialized in the state $|T_1\rangle$. Again the charge configurations are very similar to the ones for the real eigenstate shown in Fig. \[Fig\_Charge\_T\_N4\](a) with the fidelity above $0.97$ throughout the evolution. In Figs. \[Fig\_Adiabatic\_ST\_N4\](c) and (d) we plot the charge occupancies of the state $|\Psi(\tau)\rangle$ when the system is initially in the state $|T_2\rangle$ and $|S_2\rangle$, respectively. In these two cases, the evolution is very different from the charge configurations of the local eigenstates given in Fig. \[Fig\_Charge\_T\_N4\](b) and Fig. \[Fig\_Charge\_S\_N4\](b), respectively. Here $T_{max}$ is not large enough to keep an adiabatic evolution for these two eigenstates and their fidelity reaches levels as low as ${\sim} 0.2$. In the SM, we show that $T_{max}$ values in the order of $(10^7-10^8)/t$ would be required to ensure and adiabatic evolution of $|S_2\rangle$ and $|T_2\rangle$, due to smaller gaps between higher energy eigenstates. Nonetheless, as we will show below, only an adiabatic evolution of $|S_1\rangle$ and $|T_1\rangle$ is enough to distinguish all four eigenstates, enabling complete certification. ![ **Adiabatic evolution.** Charge occupancies in the evolution of a system of length $N=4$ when $T_{max}{=}2\times 10^4/t$ and the system is initialized in the state: (a) $|S_1\rangle$; (b) $|T_1\rangle$; (c) $|T_2\rangle$; and (d) $|S_2\rangle$. This choice of $T_{max}$ results in an adiabatic evolution only for $|S_1\rangle$ and $|T_1\rangle$. []{data-label="Fig_Adiabatic_ST_N4"}](Fig_Adiabatic_ST_N4){width="7.5cm" height="6cm"} *State discrimination.–* First we consider the ideal case in which the potential tilting is performed adiabatically for all eigenstates. The number of required tilts depends on the number of eigenstates to be discriminated. For instance, if one wants to distinguish between $|S_1\rangle$ and $|T_1\rangle$ then only one tilt, namely $\epsilon/t\simeq 50-60$, is enough as $|S_1\rangle$ takes the configuration $(2,2,0,0)$ and $|T_1\rangle$ goes to $(2,1,1,0)$. Only two tilts are then required to fully distinguish the four lowest eigenstates. For instance, by tilting to $\epsilon/t{=}35$ we can fully distinguish $|S_2\rangle$, with configuration $(2,2,0,0)$, and $|T_2\rangle$, with configuration $(2,1,0,1)$. However, both $|S_1\rangle$ and $|T_1\rangle$ share the same configuration $(2,1,1,0)$ and cannot be distinguished. Therefore, another charge configuration measurement must be performed at a larger detuning $\epsilon/t {\sim} 50-60$ when the charge configuration for $|S_1\rangle$ changes to $(2,2,0,0)$ while $|T_1\rangle$ remains in the $(2,1,1,0)$ configuration. The key feature of our proposal lies in its scalability: only two tilts are needed to fully distinguish the four lowest eigenstates, irrespective of the system size (see SM). In fact, for distinguishing $n$ low-energy eigenstates only $n/2$ tilts are required. Now, we consider an evolution which is only adiabatic for $|S_1\rangle$ and $|T_1\rangle$, like depicted in Fig. \[Fig\_Adiabatic\_ST\_N4\]. For $|S_2\rangle$, the outcome of the charge measurement will be time averaged over the charge occupancies due to rapid charge oscillations. Therefore, by using the same procedure, at $\epsilon/t{=}35$ the states $|S_2\rangle$ and $|T_2\rangle$ take the configurations $(1.5,1,1.5,0)$ and $(1.2,1.6,1,0.2)$, respectively, which are very distinct from each other as well as from the configuration of $|S_1\rangle$ and $|T_1\rangle$. Note that the partial charges mean that the quantum states are in a superposition of multiple charge states. This means that even when the evolution for $|S_2\rangle$ and $|T_2\rangle$ is non-adiabatic the proposed discrimination procedure still holds. ![ **Decoherence.** The time evolution in the presence of decoherence in a system of length $N=4$ when $T_{max}{=}2\times 10^4/t$. The charge occupancies are given for $\gamma/t=0.001$ when the system is initialized in the eigenstate: (a) $|S_1\rangle$; and (b) $|T_1\rangle$. (c) The distance between the charge probability distributions of $|S_1\rangle$ and $|T_1\rangle$ as a function of $\gamma$ when $\epsilon/t=70$. []{data-label="Fig_Decoherence_ST_N4"}](Fig_Decoherence_ST_N4){width="7.5cm" height="5.5cm"} *Decoherence.–* Interaction with the environment results in non-unitary dynamics and decoherence. For itinerant particles, the most common source of decoherence is charge fluctuations [@wardrop2014exchange; @nakajima2018coherent] which destroys the superposition of different charge configurations. Therefore, if $\{L_n\}$ represent the projection operators on $n$-th charge configuration then the non-unitary dynamics can be modeled using the Lindblad master equation $$\nonumber \frac{\partial \rho}{\partial \tau}=-i[H(\tau),\rho]+\gamma \sum_n \left( L_n \rho L_n^\dagger -\frac{1}{2} L_n^\dagger L_n \rho -\frac{1}{2} \rho L_n^\dagger L_n \right) $$ where $\gamma$ represents the decoherence strength, $\rho$ is the density matrix of the system and $L_n$’s are the Lindblad operators which depend on the decoherence source. In Fig. \[Fig\_Decoherence\_ST\_N4\](a) we plot the charge occupancies for the evolution of $|S_1\rangle$ in a chain of $N=4$ when $\gamma/t{=} 10^{-3}$ and $T_{max}=2\times 10^4/t$. Decoherence leads to partial charge transitions for the quantum states to become mixtures of charge configurations. The same evolution for the triplet state $|T_1\rangle$ is depicted in Fig. \[Fig\_Decoherence\_ST\_N4\](b). The evolution for $|T_1\rangle$ is less affected than $|S_1\rangle$ as there are less charge transitions. In the SM we discuss the fidelities and entropy production resulting from this evolution. As decoherence affects charge transitions, it is important to address its impact on on our protocol to distinguish quantum states. Each measurement outcome is associated to a charge projection operator $L_n$ with respective probability $p_n=\operatorname{Tr}{(\rho L_n)}$. Distinguishing the two eigenstates, e.g. $|S_1\rangle$ and $|T_1\rangle$, equivalent to distinguishing two probability distributions $\{p_n: p_n=\operatorname{Tr}{(\rho_{S_1} L_n)}\}$ and $\{q_n: q_n=\operatorname{Tr}{(\rho_{T_1} L_n)}\}$, where $\rho_{S_1}$ ($\rho_{T_1}$) is the solution of the above Lindblad master equation with the initial state $|S_1\rangle$ ($|T_1\rangle$). Experimentally the true probability distribution can be obtained by averaging over $M$ charge measurements at each tilt. The distance (or relative entropy) defined as $d(S_1,T_1){=}\sum_n p_n \log_2 \frac{p_n}{q_n}$ can be used to quantify the distinguishability between the two distributions. The error in discriminating the two probability distributions after $M$ samples scales as ${\sim} 2^{-Md}$ [@vedral1997statistical], for $M$ large. Therefore, by repeating the experiments at each tilt for $M{\sim} 10^2{-}10^3$ one can reconstruct the probability distributions and discriminate between the eigenstates when $d{>}1$. In Fig. \[Fig\_Decoherence\_ST\_N4\](c) we plot $d(S_1,T_1)$ versus $\gamma$ for a tilt set to $\epsilon/t{=}70$. The distance drops as $\gamma$ increases, however it remains above $10$ even for $\gamma/t{=}0.01$, and discrimination is still achievable. *Experimental realization.–* The most relevant platforms to realize our proposal are fermionic optical lattices [@schreiber2015observation] and dopant arrays [@fuechsle2012single; @salfi2016quantum]. We specifically consider the latter. The atomic precision of scanning tunneling microscopy lithography [@fuechsle2012single] provides the required versatility to fabricate 1D or 2D phosphorus donor-bound spin arrays in silicon with charge sensors in their proximity calibrated to accurately deduce charge configurations [@mahapatra2011charge]. The dopant charging energy is $U{\sim} 47$ meV for bulk donors and both $t$ and $V$ can be engineered via the physical separation between sites. For dopants placed $10$ nm apart, $t$ is about $1$ meV [@gamble2015multivalley] and $V$ around $10$ meV as considered in this letter. From these values, an adiabatic evolution is achieved for $T_{max} {\ge} 13$ ns. Experimental charge dephasing values can be converted to $\gamma {\sim} 0.02-1$ $\mu$eV [@pashkin2003quantum; @dupont2013coherent; @van2018microwave; @van2018time]. More precisely the ratio $\gamma/t$ is relevant, which is found to be ${\sim} 10^{-5}{-}10^{-3}$ as strong tunneling interactions are considered here. As shown in Fig. \[Fig\_Decoherence\_ST\_N4\](c), this results in $d{>}20$ (and fidelities above $0.8$, see SM), and hence precise certification to be achievable in dopant systems. The hyperfine interactions, coupling electron and nuclear spins, are another possible source of errors in dopant systems as they mix the singlet and triplet subspaces. For the hyperfine coupling of $A{\sim} 0.4$ $\mu$eV this mixing rate is ${\sim} A^2/(E_{T_1}-E_{S_1})$ due to the energy difference between $|S_1\rangle$ and $|T_1\rangle$. As the minimum $E_{T_1}{-}E_{S_1} {\sim} 100$ $\mu$eV is found for $N{=}4$ the role of hyperfine interaction can be neglected. However, as the energy gap scales as $1/N^2$, we predict that hyperfine interactions will be relevant for $N{>}20$ making the nuclear spin initialization essential. *Conclusion.–* We have proposed an efficient procedure for certifying the performance of spin-based quantum simulators via discriminating between the low energy eigenstates without using quantum tomography. This is a nontrivial task as the eigenstates cannot be distinguished locally because of: (i) being many-body entangled; and (ii) having the same symmetries and total spin. Given the fact that our scheme can be implemented without local addressability, it opens up the possibility to scale up the simulators to large sizes. The proposed mechanism can potentially be exploited to detect low energy phenomena such as quantum phase transitions, electronic thermometry and emergent Kondo screening clouds. After certification of the spin Hamiltonian in the low energy regime, the same simulator can be used to reveal classically inaccessible features such as high energy long-time dynamics and complex two-dimensional structures. Relevant platofrms to implement our certification method include dopant arrays [@fuechsle2012single; @salfi2016quantum] and fermionic optical lattices [@schreiber2015observation]. *Acknowledgment.–* The authors would like to thank Didier ST Medar for helpful discussions. We acknowledge support from the ARC Centre of Excellence for Quantum Computation and Communication Technology (CE170100012) and an ARC Discovery Project (DP180102620). JS acknowledges support from an ARC DECRA fellowship (DE160101490). AB thanks the National Key R&D Program of China, Grant No. 2018YFA0306703. SB acknowledges support from the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/R029075/1. 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Supplementary Material ====================== In the supplementary material we provide further investigations on some of the subjects which were not discussed in details in the main text of the paper. 1. Charge Configurations and State Discrimination for Large Chains ------------------------------------------------------------------ The proposed mechanism can also be applied to large chains with $N>4$. The charge configurations become more diverse as the size of the system increases. Using the same parameters as in the main text, namely $U/t=40$ and $V/t=10$, one can plot the charge occupancies of each site as a function of the tilting potential $\epsilon$. As a typical example we plot the data for $N=8$ in Fig. \[Fig\_Supp\_Charge\_ST\_N8\] for the first four eigenstates, namely $|S_{1,2}\rangle$ and $|T_{1,2}\rangle$. As the figure shows the overall picture is similar to the case of $N=4$ except that there are more charge movements. The eigenstate charge configurations as a function of $\epsilon/t$ for chains of length $N=6$ and $N=8$ are represented schematically in Figs. \[Fig\_Supp\_Schematic\](a)-(b). It can be shown that the final configuration of the eigenstate $|S_1\rangle$ is always $(2,\cdots,2,0,\cdots,0)$ and for the eigenstate $|T_1\rangle$ it is $(2,\cdots,2,1,1,0,\cdots,0)$. An important feature which arises in large chains is that the final charge configuration of $|T_2\rangle$ shows partial charge occupancies. This is due to a superposition of charges. Remarkably, independently of the system size we can discriminate between the four eigenstates using only two potential tilts. For instance, in the case of $N=6$, with $\epsilon/t{=}35$ we can fully discriminate the eigenstate $|S_2\rangle$ and $|T_2\rangle$ from the rest but we cannot distinguish $|S_1\rangle$ from $|T_1\rangle$. Note that, at this value of the potential tilt the charge measurement outcome for $|T_2\rangle$ is not unique as that eigenstate is a superposition of different charge configurations but due to orthogonality it does not share any charge configuration with $|T_1\rangle$ (which has the same charge configuration as $|S_1\rangle$) and $|S_2\rangle$. If the charge measurement shows the configuration $(2,2,1,1,0,0)$ this means that the quantum state is either $|S_1\rangle$ or $|T_1\rangle$ and to discriminate between them one has to tilt the system further to $\epsilon/t{=}70$ for which the two eigenstates take different charge configurations. The same argument is valid for $N=8$ and $N=10$ (data not shown) in which the two potential tilts should still be performed at $\epsilon/t{=}35$ and $\epsilon/t{=}70$ for full discrimination between the four eigenstates. ![ **Charge configuration for a chain of size $N=8$.** The charge configuration of a spin simulator with $N=8$, $U/t=40$ and $V/t=10$. The charge configrations are given for: (a) the ground state $|S_1\rangle$; (b) the first singlet excited state $|S_2\rangle$; (c) the first triplet excited state $|T_1\rangle$; and (d) the second triplet eigenstate $|T_2\rangle$. []{data-label="Fig_Supp_Charge_ST_N8"}](Fig_Supp_Charge_ST_N8){width="8cm" height="7cm"} 2. Adiabatic Evolution ---------------------- {width="12cm" height="6cm"} [ | m[1cm]{} | m[1cm]{}| m[1cm]{} | m[1cm]{}| m[1cm]{} | ]{} N & 4 & 6 & 8 & 10\ $\Delta E_{S}/t$ & 0.2231 & 0.1255 & 0.1011 & 0.0710\ $\Delta E_{T}/t$ & 0.0913 & 0.0757 & 0.0684 & 0.0575\ As mentioned in the main text, the time $T_{max}$ needed to keep the evolution of $|S_2\rangle$ and $|T_2\rangle$ adiabatic must lie in the range $10^7-10^8$. An example is given for $T_{max} {\sim} 8\times 10^7/t$. In Fig. \[Fig\_Supp\_Adiabatic\_ST\_N4\](a) we plot the charge occupancies as functions of time when the initial state is $|S_2\rangle$ for $N=4$. As the figure shows the charge occupancies are almost identical to the charge configuration shown in Fig. \[Fig\_Charge\_S\_N4\](b) for the local eigenstate. Similarly, in Fig. \[Fig\_Supp\_Adiabatic\_ST\_N4\](b) we plot the charge occupancies for the initial state $|T_2\rangle$ which is also close to the charge configuration of the local eigenstate given in Fig. \[Fig\_Charge\_T\_N4\](b). To asses the adiabatic evolution we plot the fidelities for $|S_2\rangle$ and $|T_2\rangle$ in Fig. \[Fig\_Supp\_Adiabatic\_ST\_N4\](c). The fidelity for $|S_2\rangle$ is always above $0.99$ and for $|T_2\rangle$ it always remains above $0.9$. The better fidelity for $|S_2\rangle$ is due to a larger energy gap for higher singlet eigenstates. It is worth emphasizing that, as discussed in the main text, a large value of $T_{max}$ to keep the evolution of $|S_2\rangle$ and $|T_2\rangle$ adiabatic is not needed for state discrimination between the first four eigenstates. A crucial issue for the adiabatic evolution is the estimation of $T_{max}$ needed to evolve larger systems. As we discussed above, it is important to keep the evolution for both $|S_1\rangle$ and $T_1\rangle$ adiabatic, even if the higher energy eigenstates do not follow an adiabatic evolution. The criteria for the validity of the adiabatic theorem has been the subject of research over many years [@comparat2009general; @marzlin2004inconsistency; @tong2007sufficiency]. The standard criteria implies that one has to satisfy $|\frac{\langle \dot{S}_2(\tau)|S_1(\tau)\rangle }{E_{S_2}(\tau)-E_{S_1}(\tau)}|\ll 1$ where $|\dot{S}_2(\tau)\rangle$ is the time derivative of the eigenstate $|S_2\rangle$ with respect to $\tau$. Similar criteria can be written for triplets as well. Using perturbation theory one can show that in a pessimistic estimation $\langle \dot{S}_2(\tau)|S_1(\tau)\rangle \sim T_{max}^{-1} [ E_{S_2}(\tau)-E_{S_1}(\tau)]^{-1}$. This implies that for the validity of the adiabatic evolution one has to keep $T_{max}>1/\Delta E^2$ where $\Delta E$ is the the energy gap. To see how the energy gap scales with system size we present the minimum energy gap during the adiabatic evolution for both singlets (i.e. $\Delta E_S=E_{S_2}-E_{S_1}$) and triplets (i.e. $\Delta E_T=E_{T_2}-E_{T_1}$) in TABLE \[Table\_Energy\_Gap\]. As the data show, the energy gap decreases fairly linearly as the system size increases. This means that for a chain of size $N=10$ the time $T_{max}$ is almost $10$ times larger than the one needed for $N=4$. ![ **Adiabatic evolution of $|S_2\rangle$ and $|T_2\rangle$.** The adiabatic evolution for the excited states $|S_2\rangle$ and $|T_2\rangle$ in a system with $N=4$ when the ramping time is chosen to be $T_{max}=8\times 10^7/t$. (a) The charge occupancies as functions of time when the system is initialized in the eigenstate $|S_2\rangle$. (b) The charge occupancies as functions of time when the system is initialized in the eigenstate $|T_2\rangle$. (c) The fidelity of the adiabatic evolution for both $|S_2\rangle$ and $|T_2\rangle$. []{data-label="Fig_Supp_Adiabatic_ST_N4"}](Fig_Supp_Adiabatic_ST_N4){width="8cm" height="7cm"} 3. Decoherence -------------- In order to understand the full effect of decoherence in the Lindblad master equation, we consider an adiabatic evolution of both $|S_1\rangle$ and $|T_1\rangle$ in a system of length $N=4$ with the total evolution time $T_{max}=2\times 10^4/t$. In Fig. \[Fig\_Supp\_Fidelity\_Entropy\_Decoherence\_ST\_N4\](a) we plot the fidelity of the evolution for the state $|S_1\rangle$ as a function of time $\tau$ for different values of noise strength $\gamma$. As the figure shows, by increasing $\gamma$ the fidelity decreases. To understand this, it is important to note that such dynamics is not unitary. This means that the quantum state of the system becomes mixed during the time evolution. To see this, one can compute the von Neumann entropy of the whole system which is defined as $$\label{von_Neumann_Entropy} S(\rho)=-\operatorname{Tr}\left( \rho\log_2 \rho \right).$$ In Fig. \[Fig\_Supp\_Fidelity\_Entropy\_Decoherence\_ST\_N4\](b) we plot the von Neumann entropy of the system when the quantum state is initially $|S_1\rangle$ as a function of time $\tau$ for different values of noise strength $\gamma$. As the figure shows the entropy increases monotonically and sharp rises happen during the charge movements when the charge state is delocalized. In Fig. \[Fig\_Supp\_Fidelity\_Entropy\_Decoherence\_ST\_N4\](c) we also plot the fidelity for the quantum state $|T_1\rangle$ keeping all the parameters the same as for the singlet $|S_1\rangle$. Finally, in Fig. \[Fig\_Supp\_Fidelity\_Entropy\_Decoherence\_ST\_N4\](d) we plot the von Neumann entropy of the evolution of the triplet state $|T_1\rangle$ as a function of time. Figs. \[Fig\_Supp\_Fidelity\_Entropy\_Decoherence\_ST\_N4\](c)-(d) show that the fidelity of the triplet is slightly higher and its von Neumann entropy is smaller in comparison with the singlet. This is due to less charge movements for triplets, or equivalently fewer energy anti-crossings between the eigenstates, which makes the triplet evolution less prone to decoherence. ![ **Fidelity and Entropy.** The evolution of a system with length $N{=}4$ in the presence of decoherence. (a) The fidelity $F$ versus time for $|S_1\rangle$. (b) The von Neumann entropy $S$ versus time for $|S_1\rangle$. (c) The fidelity $F$ versus time for $|T_1\rangle$. (d) The von Neumann entropy $S$ versus time for $|T_1\rangle$. []{data-label="Fig_Supp_Fidelity_Entropy_Decoherence_ST_N4"}](Fig_Supp_Fidelity_Entropy_Decoherence_ST_N4){width="8cm" height="7cm"}

--- abstract: 'Primary angle closure glaucoma (PACG) is the leading cause of irreversible blindness among Asian people. Early detection of PACG is essential, so as to provide timely treatment and minimize the vision loss. In the clinical practice, PACG is diagnosed by analyzing the angle between the cornea and iris with anterior segment optical coherence tomography (AS-OCT). The rapid development of deep learning technologies provides the feasibility of building a computer-aided system for the fast and accurate segmentation of cornea and iris tissues. However, the application of deep learning methods in the medical imaging field is still restricted by the lack of enough fully-annotated samples. In this paper, we propose a novel framework to segment the target tissues accurately for the AS-OCT images, by using the combination of weakly-annotated images (majority) and fully-annotated images (minority). The proposed framework consists of two models which provide reliable guidance for each other. In addition, uncertainty guided strategies are adopted to increase the accuracy and stability of the guidance. Detailed experiments on the publicly available AGE dataset demonstrate that the proposed framework outperforms the state-of-the-art semi-/weakly-supervised methods and has a comparable performance as the fully-supervised method. Therefore, the proposed method is demonstrated to be effective in exploiting information contained in the weakly-annotated images and has the capability to substantively relieve the annotation workload.' author: - Munan Ning - Cheng Bian - Donghuan Lu - 'Hong-Yu Zhou' - Shuang Yu - Chenglang Yuan - Yang Guo - Yaohua Wang - Kai Ma - Yefeng Zheng bibliography: - 'SemiSeg.bib' title: 'A Macro-Micro Weakly-supervised Framework for AS-OCT Tissue Segmentation' --- =5000 =1000 Introduction ============ Glaucoma is the leading cause of irreversible vision loss world-widely that is predicted to affect more than 100 million people by year 2040 [@tham2014global]. Primary angle closure glaucoma (PACG), as a major subtype of glaucoma, develops when the angle between the iris and cornea is closed or narrowed, resulting in the blockage of drainage canals and sudden rise in intraocular pressure [@sun2017primary]. In the clinical practice, the anterior segment optical coherence technology (AS-OCT) [@radhakrishnan2001real] is widely utilized to obtain both quantitative and qualitative information on the anatomical structures of cornea and iris for the PACG diagnosis [@fu2017segmentation; @li2012anterior; @niwas2016automated; @nolan2007detection]. However, manual analysis of each image is laborious and requires professional knowledge. Although the rapid development of deep learning technologies reveals the feasibility of fully automatic anatomical structure segmentation with high accuracy [@fu2018multi], it still requires a large quantity of images with pixel-wise annotations for the related structures, which is time-consuming and expertise-demanding. To alleviate the intensive annotation workload of clinicians, a lot of efforts have been made on semi-/weakly-supervised segmentation [@cui2019semi; @hung2018adversarial; @kervadec2019constrained; @perone2018deep; @tang2018normalized; @tang2018regularized; @yu2019uncertainty]. The semi-supervision based methods aim to extract information from a large amount of unlabeled images with the assistance of some fully-annotated images or samples. For example, Perone [*et al*. ]{} [@perone2018deep] proposed a semi-supervised teacher-student framework, which leveraged the supervised knowledge learned from the teacher model to improve the segmentation performance of the student model. Yu [*et al*. ]{} [@yu2019uncertainty] further adopted the uncertainty information to the teacher-student model to fully exploit the information of the unlabeled data by following the prediction consistencies under different perturbations. Hung [*et al*. ]{} [@hung2018adversarial] proposed an adversarial based strategy, which introduced a new discriminator to predict the confidence map for utilizing the information of unlabeled images. However, current semi-supervised methods still require a considerable quantity of fully-annotated images for a satisfactory performance. Another strategy is to improve the workload efficiency by adopting weak annotations for training. For example, Kervadec [*et al*. ]{} [@kervadec2019constrained] introduced a differentiable term into the proposed loss function to impose the soft size constraints extracted from the weak annotations on the target region. Tang [*et al*. ]{} [@tang2018normalized; @tang2018regularized] proposed to attain better performance by jointly optimizing the normalized cut with a deep learning model and CRFs for the weakly-supervised task. Although these weakly-supervised methods might relieve the annotation workload to some extent, their segmentation could be error-prone due to the lack of sufficient pixel-wise annotation information. In the clinical practice, apart from a large number of weakly-annotated samples, there is also a small number of full annotations, which might be combined together and employed to improve the model’s performance. To address the above issues of semi-/weakly-supervised learning, an intuitive solution is to integrate both the fully-annotated images and the weakly-labeled samples into the training process, so that the former images can provide accurate pixel-wise tutorial while the latter ones offer more high-level region proposals for segmentation. In this paper, we propose an uncertainty-aware macro-micro (UAMM) framework for the segmentation of the cornea and iris with a few fully-annotated data and a relatively large number of weakly-labeled samples. The network of the proposed UAMM approach consists of two main components with two flows: the macro model with the microscopic flow and the micro model with the macroscopic flow. Unlike the teacher-student framework in which only the teacher model provides guidance to the student model, the macro model and the micro model in the proposed framework offer information for each other to achieve better segmentation performance. Specifically, the macro model utilizes the weakly-labeled samples to learn segmentation proposals to induce the semantic clues for the optimization of the micro model (a.k.a, microscopic flow), while the micro model employs fully-annotated images to present pixel-wise tutorial to guide the learning process of the macro model (a.k.a, macroscopic flow). The main contributions of this study are four folds: - We propose a novel weakly-supervised methodology for the segmentation of cornea and iris in the AS-OCT images, which outperforms state-of-the-art semi-/weakly-supervised methods and achieves comparable performance as the fully-supervised network. - Besides the informative features distilled from the weakly-labeled samples, we propose to add the macroscopic flow from the micro model to provide pixel-wise guidance for the optimization of the macro model. - Other than pixel-wise annotation information learned from the fully-annotated images, the microscopic flow from the macro model is designed to offer more high-level semantic information for the training of the micro model. - We propose to introduce uncertainty guidance strategies into the microscopic flow and macroscopic flow for more accurate and stable guidance. Method ====== Fig. \[fig:framework\] displays the diagram of the proposed UAMM framework, which consists of the micro model and the macro model. Both models have the same network architecture, [*i*.*e*., ]{}DeepLabV3+ [@chen2018encoder], with different parameters. The proposed framework is optimized via a two-stage training strategy. In the first stage, the two models are trained individually using the fully-annotated images and weakly-labeled samples, i.e., the individual training stage, marked as and in Fig. \[fig:framework\]. In the second stage, the two models are trained jointly using only the weakly-labeled samples, [*i*.*e*., ]{}the joint training stage marked as in Fig. \[fig:framework\], which provide guidance (the macroscopic and microscopic flows, marked as and ) for each other to achieve better segmentation performance. To prevent potential misleading of the incorrect information, uncertainty guidance strategies are proposed to provide more accurate and stable guidance for the model training procedure. To clarify notations, $x \in \mathbb{R}^{H \times W \times 3}$ denotes the input image, where $H,W$ and $3$ represent the height, width and three channels of the input RGB image, respectively; $y^{s}, y^{w}\in\{0,1\}^{H \times W \times C}$ stand for the $C$-way full and weak annotations, respectively; $f$, $\theta$ and $m$ indicate the non-learning transformation of DeepLabV3+, the model parameters and model output, respectively. ![The framework of our uncertainty-aware micro-macro framework. We only use full annotations in stage , while weakly-labeled images in the other.[]{data-label="fig:framework"}](Framework){width="1.0\columnwidth"} Loss Functions for the Macro and Micro Model -------------------------------------------- In the first stage, the macro model and micro model are trained separately, [*i*.*e*., ]{}the macro model is optimized with the weakly-labeled samples, while the micro model is trained with the fully-annotated images. Specifically, suppose there are $N$ fully-annotated images denoted as $\mathcal{D}_{s}=\left\{\left(x_{i}, y^{s}_{i}\right)\right\}_{i=1}^{N}$, and $M$ weakly-labeled samples represented by $ \mathcal{D}_{w}=\left\{\left(x_{j}, y^{w}_{j}\right)\right\}_{j=1}^{M}$. The loss function for each model in the individual training stage is defined as: $$\label{eq:loss_for_micro} \mathcal{L}_{micro} (x_i)=-\frac{1}{K \times C} \sum_{k=1}^{K} \sum_{c=1}^{C} y_{i}^{s}(k,c) \log m^s_i(k,c)$$ $$\label{eq:loss_for_macro} \mathcal{L}_{macro} (x_j)=-\frac{1}{\sum_{k=1}^{K} s_{j}(k) \times C} \sum_{k=1}^{K} \sum_{c=1}^{C} s_{j}(k) \cdot y_{j}^{w}(k,c) \log m^w_j(k,c)$$ where $s_j \in\{0,1\}^{H \times W \times 1}$ is the binary indicator denoting the weakly-annotated pixels; $k$ iterates over all locations with $K$ = $H\times W $ and $c$ iterates over $C$ classes; $m^s_i=f(x_{i}; \theta^{s})$ and $m^w_j=f(x_{j};\theta^{w})$ represent the outputs of the micro model and macro model, respectively. Eq. \[eq:loss\_for\_micro\] represents the vanilla cross-entropy loss [@tang2018normalized] for the micro model, while Eq. \[eq:loss\_for\_macro\] denotes the partial-cross-entropy (pCE) loss [@tang2018normalized] for the macro-model. The pCE loss only considers the weak label proposals and the relevant regions during the training process, and thus can discourage the probability of mistakenly classifying the unlabeled pixels as the background. Uncertainty-aware KL Loss for the Macroscopic Flow -------------------------------------------------- Because pixel-wise labels are not available for the weakly-labeled images, the macro model trained on them can hardly deliver satisfactory segmentation performance. In the second stage, to further improve the accuracy, we utilize the output of the micro model to guide the optimization of the macro model. Specifically, we adopt the KL-divergence loss between the output of the two models to fine-tune the macro model. Despite the capability of KL-divergence to align the distributions of two models, the potential mistake of the micro model can result in inaccurate tutorials and mislead the optimization of the macro model. Therefore, we propose to use the uncertainty map to select the reliable pixels for guidance. By using the Monte Carlo dropout (MCD) method [@gal2016dropout], the uncertainty map can be easily inferred, which serves as an indicator of the reliability of the model’s prediction. Specifically, we modify the micro network with several dropout layers, and then repetitively perform the forward pass $T$ times to obtain $T$ Monte Carlo samples $\left\{p_{t}\right\} ^{T}_{t=1}$, where $p_{t}^{c} \in \mathbb{R}^{H \times W \times C}$ denotes the softmax probability map of the $c^{th}$ class at the $t^{th}$ forward pass. Because the variance of Monte Carlo samples can be treated as an approximation of the epistemic uncertainty [@smith2018understanding], the uncertainty map $U$ of the micro model can be formulated as: $$\label{eq:calculate_U} \mu_c=\frac{1}{T}\sum_{t=1}^{T}{p_{t}^{c}}\quad\quad {\rm and} \quad\quad U=\frac{1}{T \times C}\sum_{t=1}^{T}\sum_{c=1}^{C}(p_{t}^c-\mu_c)^2.$$ Furthermore, an empirical threshold $\tau$ is applied on the uncertainty map to obtain a binary indicator map, in which the positive values represent the reliable pixels. Then, the element-wise multiplication is performed between the KL-loss and the binary indicator map to select the reliable loss for back-propagation. Therefore, for the microscopic flow in the joint training stage, the macro model can be updated via the uncertainty guided KL loss, as defined below: $$\begin{aligned} \mathcal{L}_{UKL}&=\frac{\mathbb{I}(U<\tau) \cdot \mathcal{L}_{KL}\left(m^w_j||m^s_i\right)}{\sum\mathbb{I}(U<\tau)}\\ &=\frac{1}{\sum_{k=1}^{K} \mathbb{I}(U(k)<\tau) \times C} \sum_{k=1}^{K} \sum_{c=1}^{C} \mathbb{I}(U(k)<\tau) \cdot m^w_j(k,c) \log \left(\frac{m^w_j(k,c)}{m^s_i(k,c)}\right). \end{aligned} \label{eq:microscopic_flow}$$ where $U$ denotes the uncertainty map, $\mathbb{I}(\cdot)$ represents the binary map and the threshold $\tau$ is set to $0.5$ for all the experiments. Note that only weakly-labeled images are used in this step, because the micro model has extremely high confidence for the fully-annotated images, which has already been used to train the model in the first stage. Uncertainty-aware EMA as the Microscopic Flow --------------------------------------------- As previously stated, the micro model is first trained with the fully-annotated images. Despite the fact that the fully-annotated images contain informative pixel-wise annotation, optimization with a limited number of samples can easily result in overfitting and deteriorate the generalization capability of the model. Therefore, in the second stage, we use the segmentation proposals learned from the macro model to induce the semantic clues for the micro model. Unlike in the macroscopic flow where the output of the micro model can be directly used as the tutorial, the output of the macro model trained with weakly-labeled samples may not be accurate enough to be used for guidance. Yu [*et al*. ]{} [@yu2019uncertainty] proposed an asynchronous updating solution for two collaborative models, [*i*.*e*., ]{}the exponential moving average (EMA) mechanism, based on the idea that the weights of the model would contain implicit information of the inference evidence. In this work, the weights of the macro model contain critical information learnt from the weakly-labeled regions and could be useful for the training of the micro model. However, adopting the classic EMA strategy to partially update the micro model with the weights of the macro model requires a predefined updating rate, which may not be the optimal solution. Instead, we propose an uncertainty-aware exponential moving average (UEMA) mechanism for the microscopic flow. $\theta^{s}$ and $\theta^{w}$ are used to represent the weight parameters of the micro and macro model, respectively. The proposed UEMA in the joint training stage can be summarized as: $$\theta^{s}= \alpha\theta^{s}+(1-\alpha)\theta^{w} \quad\quad {\rm and} \quad\quad \alpha=\frac{\sum_{k=1}^{K} \mathbb{I}\left(U(k)<\tau\right)}{\sum_{k=1}^{K} \mathbbm{1}},$$ where the $\mathbbm{1} \in\{1\}^{H \times W}$ denotes the unit map with the same shape as $U$. Note that $U$ represents the uncertainty map the same as in Eq. \[eq:microscopic\_flow\]. The updating rate $\alpha$ is calculated by dividing the sum of uncertainty binary map $\mathbb{I}(U(k)<\tau)$ with the sum of $\mathbbm{1}$. It is used to control the updating rate of UEMA. The less certain the micro model is, the more its parameters are going to be affected by the macro model. Through this asynchronous updating strategy, the segmentation proposal learnt by the macro model can effectively guide the micro model towards better generalization ability with adaptive updating rates. Experiment ========== #### **Experimental setup** The proposed method is evaluated on a publicly available dataset: the Angle closure Glaucoma Evaluation (AGE) Challenge [@petb-fy10-19], which provides 3200 AS-OCT images with the dimension of $998\times 2130$ pixels. The original challenge dataset provides annotation for the angle closure classification label and location of the scleral spur. In order to further realize the quantitative analysis of iris and cornea, we have the two key tissues manually re-annotated by experienced ophthalmologists, and offered two types of annotations, [*i*.*e*., ]{}the full annotation and the weak annotation. Pixel-wise masks of iris and cornea are provided by the full annotation, meanwhile, for the weak annotation, line strokes inside the tissues are marked. It is worth mentioning that the original PACG classification problem is reformulated to the tissue segmentation problem, therefore we do not use the original annotation in this work. We randomly select $60\%$ of the images for training, $20\%$ for evaluation and $20\%$ for test (only full annotations are used for evaluation and test). All the images and the corresponding annotations are resized to 240 $\times$ 512 pixels, and the image intensities are normalized into the range of \[-1, 1\]. The framework is implemented with PyTorch on an NVIDIA Tesla P40 GPU. We utilize the SGD optimizer with $weight\ decay = 0.0005$ and $momentum = 0.9$ to update the network parameters. The batch size is set to 4 for both micro and macro models. Dice coefficient (Dice, represented with percentage) and average distance of boundaries (ADB, represented with millimeter) [@bian2018pyramid] are used as the evaluation criteria. Higher Dice and lower ADB imply better segmentation performance. For convenience, we denote Dice1/ADB1 and Dice2/ADB2 as the evaluation metrics of the cornea and the peripheral iris in this work. #### **Ablation study.** To demonstrate the effectiveness of the proposed modules, we conduct ablation studies as well as experiments with different annotation composition. As shown in Table \[table:ablation\], the performance has improved around 5.80% and 2.29% in average Dice by adding the macroscopic and microscopic flow, respectively. In order to evaluate the effect of the proposed uncertainty strategies, the results of flows without uncertainty are presented as well, [*i*.*e*., ]{}marked by the asterisk symbol. To be more specific, we use the conventional EMA for the macroscopic flow and the KL-loss for the microscopic flow directly. As expected, the result without uncertainty shows inferior performance (2.65% lower for macroscopic flow and 1.52% lower for microscopic flow, respectively), demonstrating that the proposed uncertainty strategies can improve the effectiveness of the tutorials. To evaluate the stability of the proposed method, we conducted additional experiments with different percentages of fully-annotated images. As expected, the more fully-annotated images we utilize, the better performance the method achieves, indicating that the proposed method can exploit the information from full annotations as well. ![Visualization of the segmentation results by different methods and ours.[]{data-label="fig:visualization"}](visualization){width="0.75\columnwidth"} #### **Comparison with State-of-the-art** As illustrated in Table \[table:sota\], the two columns within the annotation composition represent the percentages of fully-annotated and weakly-labeled images used for training. The results of state-of-the-art semi-/weakly-supervised methods, including WACT [@perone2018deep], UAMT [@yu2019uncertainty], AdvSemi [@hung2018adversarial], and CRF-rloss [@tang2018regularized], are presented for comparison. In the training set for the proposed UAMM method, only 1% images are fully-annotated while the rest 99% samples are weakly-labeled. For the semi-supervised methods, [*i*.*e*., ]{}WACT, UAMT and AdvSemi, generally weakly-annotated samples will not be utilized in their studies. Similarly, the full-annotated samples are not used in the weakly-supervised studies, [*i*.*e*., ]{}CRF-rloss, either. For a fair comparison, both of full and weakly-annotated samples will be integrated in the training procedure and provide two versions of results, so as to keep the model comparison under the same evaluation criteria. Oracle indicates using only the micro model, [*i*.*e*., ]{}a single DeepLabV3+ network [@chen2018encoder]. As the baseline method, Oracle has been applied on four training sets with different percentages of fully-annotated images and weakly-labeled samples, as denoted in Row 2. With the same training data setup, the proposed UAMM method has achieved the best performance among these methods, with 91.64% in average Dice score and 0.3 in ADB. Furthermore, the evaluation metrics of UAMM are close to the metrics of fully-annotated trained Oracle (only 2.01% lower on average Dice), demonstrating that the proposed method can exploit segmentation guidance from the weak annotations. The visualization of representative examples is displayed in Fig. \[fig:visualization\]. Conclusion ========== In this work, we proposed a macro-micro weakly-supervised framework to tackle the problem of cornea and iris segmentation for the AS-OCT images. Specifically, an uncertainty-aware KL loss is designed for the macroscopic flow to assist the training of the macro model by the prediction priors from the micro model. Then, the microscopic flow is obtained with an uncertainty-aware moving average mechanism, which updates the micro-model by gradually involving the weights of the macro model. Our approach outperformed state-of-the-art semi-/weakly-supervised methods on the cornea and iris segmentation task for AS-OCT images. In addition, it achieved comparable performance by using only 1% of fully-annotated data with that of DeepLabV3+ using all fully-annotated images. Acknowledgment ============== This work was funded by the Key Area Research and Development Program of Guangdong Province, China (No. 2018B010111001), National Key Research and Development Project (No. 2018YFC2000702) and Science and Technology Program of Shenzhen, China (No. ZDSYS201802021814180).

--- abstract: | Motivated by recent measurements of a relatively large $\theta^{}_{13}$ in the Daya Bay and RENO reactor neutrino experiments, we carry out a systematic analysis of the hybrid textures of Majorana neutrino mass matrix $M^{}_\nu$, which contain one texture zero and two equal nonzero matrix elements. We show that three neutrino masses $(m^{}_1, m^{}_2, m^{}_3)$ and three leptonic CP-violating phases $(\delta, \rho, \sigma)$ can fully be determined from two neutrino mass-squared differences $(\delta m^2, \Delta m^2)$ and three flavor mixing angles $(\theta^{}_{12}, \theta^{}_{23}, \theta^{}_{13})$. Out of sixty logically possible patterns of $M^{}_\nu$, thirty-nine are found to be compatible with current experimental data at the $3\sigma$ level. We demonstrate that the texture zero of $M^{}_\nu$ is stable against one-loop quantum corrections, while the equality between two independent elements not. Phenomenological implications of $M^{}_\nu$ for the neutrinoless double-beta decay and leptonic CP violation are discussed, and a realization of the texture zero and equality by means of discrete flavor symmetries is illustrated. --- = 16.9cm = 23.9cm = -25mm = -18mm [**Hybrid Textures of Majorana Neutrino Mass Matrix and\ Current Experimental Tests**]{} [**Ji-Yuan Liu**]{} [^1]\ [*College of Science, Tianjin University of Technology, Tianjin 300384, China*]{}\ [^2]\ [*Department of Theoretical Physics, School of Engineering Sciences,\ KTH Royal Institute of Technology, 106 91 Stockholm, Sweden*]{} PACS numbers: 14.60.Lm, 14.60.Pq Introduction ============ Recent years have seen great progress in neutrino physics [@XZ]. Thanks to a number of elegant solar, atmospheric, accelerator and reactor neutrino oscillation experiments [@PDG], two neutrino mixing angles are found to be quite large (i.e., $\theta^{}_{12} \approx 34^\circ$ and $\theta^{}_{23} \approx 40^\circ$), while two independent neutrino mass-squared differences $\delta m^2 \equiv m^2_2 - m^2_1$ and $\Delta m^2 \equiv m^2_3 - (m^2_1 + m^2_2)/2$ are measured with a good degree of accuracy (i.e., $\delta m^2 \approx 7.5\times 10^{-5}~{\rm eV}^2$ and $|\Delta m^2| \approx 2.5\times 10^{-3}~{\rm eV}^2$). The latest results from the Daya Bay [@Daya] and RENO [@Reno] experiments reveal that $\theta^{}_{13} \approx 9^\circ$ is relatively large, which is very crucial to determine the neutrino mass hierarchy (i.e., the sign of $\Delta m^2$) and to discover the leptonic CP violation (i.e., the Dirac CP-violating phase $\delta$) in the future long-baseline neutrino oscillation experiments. However, the absolute scale of neutrino masses and whether neutrinos are Dirac or Majorana particles are still unknown. On the theoretical side, a satisfactory description of tiny neutrino masses and leptonic mixing pattern is still lacking. Although the seesaw mechanisms can be responsible for the generation of light neutrino masses [@SS1; @SS2; @SS3], they leave the lepton flavor structure intact. In fact, it was shown one decade ago that the large leptonic mixing can be achieved by taking two independent elements of Majorana neutrino mass matrix $M^{}_\nu$ to be zero, in the flavor basis where the charged-lepton mass matrix $M^{}_l$ is diagonal [@FGM; @xing1; @xing2; @Guo]. Recently, several authors have demonstrated that these seven two-zero textures of $M^{}_\nu$ still survive the current neutrino oscillation data [@FXZ; @Ludl]. Furthermore, it has been pointed out that those texture zeros can be realized by implementing the $Z^{}_n$ flavor symmetry in the type-II seesaw model, where the Higgs triplets are introduced to account for tiny Majorana neutrino masses [@FXZ]. Apart from texture zeros in the neutrino mass matrix, possible correlations between two matrix elements of $M^{}_\nu$ have recently been investigated [@Grimus]. In the present paper, we perform a systematic study of $M^{}_\nu$ with one texture zero and two equal nonzero elements, which has been termed as “hybrid texture" in the literature [@Kaneko; @Dev; @hybrids]. The motivation for such an investigation is three-fold. First, from the phenomenological point of view, either one texture zero or an equality between two independent entries in $M^{}_\nu$ imposes one constraint condition and thus reduces the number of real free model parameters by two. Hence the hybrid textures are as predictive as the well-known two-zero textures, and deserve a detailed analysis. See, e.g., Refs. [@Kaneko] and [@Dev], for previous studies of hybrid textures. The textures with two equalities and other phenomenological assumptions have also been considered [@Dev2013; @Frigerio]. Second, it has been proved that a texture zero in any position in $M^{}_\nu$ can be realized by using Abelian flavor symmetries $Z^{}_n$ [@Grimus1; @Grimus2] or $U(1)$ [@Heeck]. However, the equality between two nonzero matrix elements should come from a non-Abelian flavor symmetry. Third, now that a good knowledge about three neutrino mixing angles and two neutrino mass-squared differences has been obtained, it is timely to reexamine the possible structure of $M^{}_\nu$ and explore the underlying symmetry in the lepton sector. Taking into account current neutrino oscillation data at the $3\sigma$ level, we have found that thirty-nine out of sixty logically possible hybrid textures of $M^{}_\nu$ are viable. If the $1\sigma$ ranges of neutrino mixing parameters are considered, only thirteen hybrid textures can survive. The remaining part of our paper is organized as follows. In section 2, we introduce the hybrid textures and present some useful analytical formulas. Then, the stability of texture zeros and equalities in $M^{}_\nu$ against quantum corrections is briefly discussed by using the one-loop renormalization group equation. Section 3 is devoted to the analytical analysis of six viable patterns, which serve as typical examples of hybrid textures. We show that three neutrino masses $(m^{}_1, m^{}_2, m^{}_3)$ and three CP-violating phases $(\delta, \rho, \sigma)$ can fully be determined from neutrino mixing angles $(\theta^{}_{12}, \theta^{}_{23}, \theta^{}_{13})$ and neutrino mass-squared differences $(\delta m^2, \Delta m^2)$. In section 4, a thorough numerical analysis has been performed, and the allowed parameter space of the chosen six viable patterns is given. To illustrate how to realize a hybrid texture, we give a concrete example in section 5, where a type-II seesaw model with an $S^{}_3\otimes Z^{}_3$ flavor symmetry is considered. Finally, we summarize our main conclusions in section 6. Hybrid Textures =============== Classification -------------- At low energies, lepton mass spectra and flavor mixing are determined by the charged-lepton mass matrix $M^{}_l$ and the effective neutrino mass matrix $M^{}_\nu$. We assume massive neutrinos to be Majorana particles, as in various seesaw models [@SS1; @SS2; @SS3], so $M^{}_\nu$ is in general a $3\times 3$ symmetric complex matrix. If one of six independent matrix elements of $M^{}_\nu$ is taken to be zero and two of the rest are equal, we finally arrive at ${\rm C}^1_6 \cdot {\rm C}^2_5 = 60$ logically possible textures. We enumerate all the 39 hybrid textures, which are compatible with current neutrino oscillation data at the $3\sigma$ level and can be classified into six categories: $$\begin{aligned} && {\bf A}^{}_1: \left(\matrix{ 0 & \times & \times\cr \times & \triangle & \triangle\cr \times & \triangle & \times} \right),~~~~ {\bf A}^{}_2: \left(\matrix{ 0 & \times & \times\cr \times & \triangle & \times\cr \times & \times & \triangle} \right),~~~~ {\bf A}^{}_3: \left(\matrix{ 0 & \times & \times\cr \times & \times & \triangle \cr \times & \triangle & \triangle} \right) ; $$ $$\begin{aligned} && {\bf B}^{}_1: \left(\matrix{ \triangle & 0 & \triangle \cr 0 & \times & \times \cr \triangle & \times & \times} \right),~~~~ {\bf B}^{}_2:\left(\matrix{ \times & 0 & \triangle \cr 0 & \triangle & \times \cr \triangle & \times & \times} \right), ~~~~ {\bf B}^{}_3:\left(\matrix{ \times & 0 & \triangle \cr 0 & \times & \times \cr \triangle & \times & \triangle} \right),\nonumber \\ && {\bf B}^{}_4:\left(\matrix{ \times & 0 & \times \cr 0 & \triangle & \triangle \cr \times & \triangle & \times} \right),~~~~ {\bf B}^{}_5:\left(\matrix{ \times & 0 & \times \cr 0 & \times & \triangle \cr \times & \triangle & \triangle} \right); $$ $$\begin{aligned} && {\bf C}^{}_1:\left(\matrix{ \triangle & \triangle & 0 \cr \triangle & \times & \times \cr 0 & \times & \times} \right),~~~~ {\bf C}^{}_2:\left(\matrix{ \times & \triangle & 0 \cr \triangle & \triangle & \times \cr 0 & \times & \times} \right), ~~~~ {\bf C}^{}_3:\left(\matrix{ \times & \triangle & 0 \cr \triangle & \times & \times \cr 0 & \times & \triangle} \right),\nonumber \\ && {\bf C}^{}_4:\left(\matrix{ \times & \times & 0 \cr \times & \triangle & \triangle \cr 0 & \triangle & \times} \right),~~~~ {\bf C}^{}_5:\left(\matrix{ \times & \times & 0 \cr \times & \times & \triangle \cr 0 & \triangle & \triangle} \right); $$ $$\begin{aligned} && {\bf D}^{}_1:\left(\matrix{ \triangle & \triangle & \times \cr \triangle & \times & 0 \cr \times & 0 & \times} \right),~~~~ {\bf D}^{}_2:\left(\matrix{ \triangle & \times & \triangle \cr \times & \times & 0 \cr \triangle & 0 & \times} \right),~~~~ {\bf D}^{}_3:\left(\matrix{ \times & \triangle & \times \cr \triangle & \triangle & 0 \cr \times & 0 & \times} \right), \nonumber \\ && {\bf D}^{}_4:\left(\matrix{ \times & \triangle & \times \cr \triangle & \times & 0 \cr \times & 0 & \triangle} \right),~~~~ {\bf D}^{}_5:\left(\matrix{ \times & \times & \triangle \cr \times & \triangle & 0 \cr \triangle & 0 & \times} \right),~~~~ {\bf D}^{}_6:\left(\matrix{ \times & \times & \triangle \cr \times & \times & 0 \cr \triangle & 0 & \triangle} \right); $$ $$\begin{aligned} && {\bf E}^{}_1:\left(\matrix{ \triangle & \triangle & \times \cr \triangle & 0 & \times \cr \times & \times & \times} \right),~ {\bf E}^{}_2:\left(\matrix{ \triangle & \times & \triangle \cr \times & 0 & \times \cr \triangle & \times & \times} \right),~ {\bf E}^{}_3:\left(\matrix{ \triangle & \times & \times \cr \times & 0 & \triangle \cr \times & \triangle & \times} \right),~ {\bf E}^{}_4:\left(\matrix{ \triangle & \times & \times \cr \times & 0 & \times \cr \times & \times & \triangle} \right),\nonumber \\ && {\bf E}^{}_5:\left(\matrix{ \times & \triangle & \triangle \cr \triangle & 0 & \times \cr \triangle & \times & \times} \right), ~ {\bf E}^{}_6:\left(\matrix{ \times & \triangle & \times \cr \triangle & 0 & \triangle \cr \times & \triangle & \times} \right),~ {\bf E}^{}_7:\left(\matrix{ \times & \triangle & \times \cr \triangle & 0 & \times \cr \times & \times & \triangle} \right),~ {\bf E}^{}_8:\left(\matrix{ \times & \times & \triangle \cr \times & 0 & \triangle \cr \triangle & \triangle & \times} \right), \nonumber \\ && {\bf E}^{}_9:\left(\matrix{ \times & \times & \triangle \cr \times & 0 & \times \cr \triangle & \times & \triangle} \right),~ {\bf E}^{}_{10}:\left(\matrix{ \times & \times & \times \cr \times & 0 & \triangle \cr \times & \triangle & \triangle} \right); $$ $$\begin{aligned} && {\bf F}^{}_1:\left(\matrix{ \triangle & \triangle & \times \cr \triangle & \times & \times \cr \times & \times & 0} \right),~ {\bf F}^{}_2:\left(\matrix{ \triangle & \times & \triangle \cr \times & \times & \times \cr \triangle & \times & 0} \right),~ {\bf F}^{}_3:\left(\matrix{ \triangle & \times & \times \cr \times & \triangle & \times \cr \times & \times & 0} \right),~ {\bf F}^{}_4:\left(\matrix{ \triangle & \times & \times \cr \times & \times & \triangle \cr \times & \triangle & 0} \right), \nonumber \\ && {\bf F}^{}_5:\left(\matrix{ \times & \triangle & \triangle \cr \triangle & \times & \times \cr \triangle & \times & 0} \right),~ {\bf F}^{}_6:\left(\matrix{ \times & \triangle & \times \cr \triangle & \triangle & \times \cr \times & \times & 0} \right),~ {\bf F}^{}_7:\left(\matrix{ \times & \triangle & \times \cr \triangle & \times & \triangle \cr \times & \triangle & 0} \right),~ {\bf F}^{}_8:\left(\matrix{ \times & \times & \triangle \cr \times & \triangle & \times \cr \triangle & \times & 0} \right), \nonumber \\ && {\bf F}^{}_9:\left(\matrix{ \times & \times & \triangle \cr \times & \times & \triangle \cr \triangle & \triangle & 0} \right),~ {\bf F}^{}_{10}:\left(\matrix{ \times & \times & \times \cr \times & \triangle & \triangle \cr \times & \triangle & 0} \right) \; , $$ where the triangles “$\triangle$" denote equal and nonzero elements, while the crosses “$\times$" stand for arbitrary and nonzero ones. If one more element of $M^{}_\nu$ is assumed to be zero (or two more elements are equal), the number of free parameters in $M^{}_\nu$ will be further reduced and the viable textures should be much less (see, for example, Ref. [@Grimus]). However, we have numerically checked that all those textures have already been excluded by current neutrino oscillation data at the $3\sigma$ level. Important Relations ------------------- In the basis where the charged-lepton mass matrix $M^{}_l$ is diagonal, the Majorana neutrino mass matrix $M^{}_\nu$ can be reconstructed from the leptonic mixing matrix $V$ and three neutrino masses: $$M^{}_\nu = V \left(\matrix{m^{}_1 & 0 & 0 \cr 0 & m^{}_2 & 0 \cr 0 & 0 & m^{}_3}\right) V^T \; . $$ The leptonic mixing matrix can be parametrized as $V = U \cdot P$, where the unitary matrix $U$ contains three mixing angles $(\theta^{}_{12}, \theta^{}_{23}, \theta^{}_{13})$ and one Dirac-type CP-violating phase $\delta$, namely, $$U=\left( \matrix{c^{}_{12} c^{}_{13} & s^{}_{12} c^{}_{13} & s^{}_{13} \cr -c^{}_{12} s^{}_{23} s^{}_{13} - s^{}_{12} c^{}_{23} e^{-i \delta } & -s^{}_{12} s^{}_{23} s^{}_{13} + c^{}_{12} c^{}_{23} e^{-i \delta } & s^{}_{23} c^{}_{13} \cr -c^{}_{12} c^{}_{23} s^{}_{13} + s^{}_{12} s^{}_{23} e^{-i \delta } & -s^{}_{12} c^{}_{23} s^{}_{13}- c^{}_{12} s^{}_{23} e^{-i \delta} & c^{}_{23} c^{}_{13}} \right) \; ; $$ and $P = {\rm Diag}\{e^{i\rho}, e^{i\sigma}, 1\}$ is a diagonal matrix with $\rho$ and $\sigma$ being two Majorana-type CP-violating phases. Here we have defined $s^{}_{ij} \equiv \sin \theta^{}_{ij}$ and $c^{}_{ij} \equiv \cos \theta^{}_{ij}$ for $ij = 12, 23, 13$. For later convenience, we rewrite $$M^{}_\nu = U \left(\matrix{\lambda^{}_1 & 0 & 0 \cr 0 & \lambda^{}_2 & 0 \cr 0 & 0 & \lambda^{}_3}\right) U^T \; , $$ where $\lambda^{}_1 \equiv m^{}_1 e^{2i\rho}$, $\lambda^{}_2 \equiv m^{}_2 e^{2i\sigma}$ and $\lambda^{}_3 \equiv m^{}_3$. If one matrix element is zero \[e.g., $(M^{}_\nu)^{}_{ab} = 0$\] and two other elements are equal \[e.g., $(M^{}_\nu)^{}_{\alpha \beta} = (M^{}_\nu)^{}_{cd}$\], where three different independent elements of $M^{}_\nu$ are considered, we obtain $$\begin{aligned} ~~~~~~~~~~~~~~~~ \sum_{i=1}^3 U^{}_{ai} U^{}_{bi} \lambda^{}_i = 0 ~~ {\rm and} ~~ \sum_{i=1}^3 (U^{}_{\alpha i} U^{}_{\beta i} - U^{}_{c i} U^{}_{d i}) \lambda^{}_i = 0 \; , $$ which lead to $$\begin{aligned} \frac{\lambda^{}_1}{\lambda^{}_3} &=& \frac{U^{}_{a3} U^{}_{b3} U^{}_{\alpha 2} U^{}_{\beta 2} - U^{}_{a2} U^{}_{b2} U^{}_{\alpha 3} U^{}_{\beta 3} + U^{}_{a2} U^{}_{b2} U^{}_{c3} U^{}_{d3} - U^{}_{a3} U^{}_{b3} U^{}_{c2} U^{}_{d2}}{U^{}_{a2} U^{}_{b2} U^{}_{\alpha 1} U^{}_{\beta 1} - U^{}_{a1} U^{}_{b1} U^{}_{\alpha 2} U^{}_{\beta 2} + U^{}_{a1} U^{}_{b1} U^{}_{c2} U^{}_{d2} - U^{}_{a2} U^{}_{b2} U^{}_{c1} U^{}_{d1}} \; , \nonumber \\ \frac{\lambda_2}{\lambda_3}&=&\frac{U^{}_{a1} U^{}_{b1} U^{}_{\alpha 3} U^{}_{\beta 3} - U^{}_{a3} U^{}_{b3} U^{}_{\alpha 1} U^{}_{\beta 1} + U^{}_{a3} U^{}_{b3} U^{}_{c1} U^{}_{d1} - U^{}_{a1} U^{}_{b1} U^{}_{c3} U^{}_{d3}}{U^{}_{a2} U^{}_{b2} U^{}_{\alpha 1} U^{}_{\beta 1} - U^{}_{a1} U^{}_{b1} U^{}_{\alpha 2} U^{}_{\beta 2} + U^{}_{a1} U^{}_{b1} U^{}_{c2} U^{}_{d2} - U^{}_{a2} U^{}_{b2} U^{}_{c1} U^{}_{d1}} \; . \label{eps.lambda} $$ With the help of Eq. (\[eps.lambda\]), one can figure out two neutrino mass ratios $\xi \equiv m^{}_1/m^{}_3 = |\lambda^{}_1/\lambda^{}_3|$ and $\zeta \equiv m^{}_2/m^{}_3 = |\lambda^{}_2/\lambda^{}_3|$, as well as two Majorana CP-violating phases $\rho = \arg[\lambda^{}_1/\lambda^{}_3]/2$ and $\sigma = \arg[\lambda^{}_2/\lambda^{}_3]/2$. As we shall show later, these important relations are quite useful in the determination of both neutrino mass spectrum and leptonic CP-violating phases from current experimental observations. If both $(\theta^{}_{12}, \theta^{}_{23}, \theta^{}_{13})$ and $\delta$ are precisely measured in neutrino oscillation experiments, the unitary matrix $U$ is fixed, thus both $(\xi, \zeta)$ and $(\rho, \sigma)$ can be determined from Eq. (11). The neutrino mass ratios $\xi$ and $\zeta$ are related to the ratio of two independent neutrino mass-squared differences as $$R^{}_\nu \equiv \frac{\delta m^2}{|\Delta m^2|} = \frac{2(\zeta^2 - \xi^2)}{|2-(\zeta^2+\xi^2)|} \; , $$ and to three neutrino masses as $$m^{}_3 = \sqrt{\frac{\delta m^2}{\zeta^2 - \xi^2}}\; , ~~~~ m^{}_2 = m^{}_3 \zeta \; ,~~~~ m^{}_1 = m^{}_3 \xi \; . $$ At present, the CP-violating phase $\delta$ is essentially unconstrained in neutrino oscillation experiments. If one of the hybrid textures is assumed, and three neutrino mixing angles and two neutrino mass-squared differences are given, then $\delta$ can be predicted from Eq. (12). For the normal neutrino mass hierarchy, the latest global-fit analysis yields at the $3 \sigma$ level [@Fogli] $$\begin{aligned} 0.259 \leq &\sin^2 \theta^{}_{12}& \leq 0.359 \; , \nonumber \\ 0.331 \leq &\sin^2 \theta^{}_{23}& \leq 0.637 \; , \nonumber \\ 0.017 \leq &\sin^2 \theta^{}_{13}& \leq 0.031 \; , $$ and $$\begin{aligned} 6.99\times 10^{-5}~{\rm eV}^2 \leq &\delta m^2& \leq 8.18 \times 10^{-5}~{\rm eV}^2 \; , \nonumber \\ 2.19\times 10^{-3}~{\rm eV}^2 \leq & \hspace{-0.25cm} \Delta m^2 \hspace{-0.25cm} & \leq 2.62 \times 10^{-3}~{\rm eV}^2 \; . $$ For the inverted neutrino mass hierarchy, the $3\sigma$ ranges of neutrino mixing angles and mass-squared differences are slightly different, so we shall use the same values as in the case of normal neutrino mass hierarchy. In Table 1, the best-fit values together with the $1\sigma$, $2\sigma$, and $3\sigma$ ranges are summarized [^3]. Parameter $\delta m^2~(10^{-5}~{\rm eV}^2)$ $\Delta m^2~(10^{-3}~{\rm eV}^2)$ $\theta^{}_{12}$ $\theta^{}_{23}$ $\theta^{}_{13}$ ----------------- ----------------------------------- ----------------------------------- ---------------------------- ---------------------------- --------------------------- Best fit $7.54$ $2.43$ $33.6^\circ$ $38.4^\circ$ $8.9^\circ$ $1\sigma$ range $[7.32, 7.80]$ $[2.33, 2.49]$ $[32.6^\circ, 34.8^\circ]$ $[37.2^\circ, 40.0^\circ]$ $[8.5^\circ, 9.4^\circ]$ $2\sigma$ range $[7.15, 8.00]$ $[2.27, 2.55]$ $[31.6^\circ, 35.8^\circ]$ $[36.2^\circ, 42.0^\circ]$ $[8.0^\circ, 9.8^\circ]$ $3\sigma$ range $[6.99, 8.18]$ $[2.19, 2.62]$ $[30.6^\circ, 36.8^\circ]$ $[35.1^\circ, 53.0^\circ]$ $[7.5^\circ, 10.2^\circ]$ : The latest global-fit results of three neutrino mixing angles $(\theta^{}_{12}, \theta^{}_{23}, \theta^{}_{13})$ and two neutrino mass-squared differences $\delta m^2 \equiv m^2_2 - m^2_1$ and $\Delta m^2 \equiv m^2_3 - (m^2_1 + m^2_2)/2$ in the case of normal neutrino mass hierarchy [@Fogli]. Quantum Corrections ------------------- The stability of texture zeros in $M^{}_\nu$ against radiative corrections has been extensively studied in the literature [@FXZ]. By using the one-loop renormalization group equation (RGE), one can demonstrate that the texture zeros in $M^{}_\nu$ at a high-energy scale remain there at low-energy scales. In this subsection, we examine the stability of texture zeros and equality between two matrix elements against quantum corrections. To be explicit, we consider the unique dimension-5 Weinberg operator of massive Majorana neutrinos in an effective field theory after the heavy degrees of freedom are integrated out [@Weinberg]: $$\frac{{\cal L}^{}_{\rm d =5}}{\Lambda} = \frac{1}{2} \kappa^{}_{\alpha \beta} \overline{\ell^{}_{\alpha \rm L}} \tilde{H} \tilde{H}^T \ell^c_{\beta \rm L} + {\rm h.c.} \; , $$ where $\Lambda$ is the cutoff scale, $\ell^{}_{\rm L}$ denotes the left-handed lepton doublet, $\tilde{H} \equiv i\sigma^{}_2 H^*$ with $H$ being the standard-model Higgs doublet, and $\kappa$ stands for the effective neutrino coupling matrix. After spontaneous gauge symmetry breaking, $\tilde{H}$ gains its vacuum expectation value $\langle \tilde{H} \rangle = v/\sqrt{2}$ with $v \approx 246$ GeV. We are then left with the effective Majorana mass matrix $M^{}_\nu = \kappa v^2/2$ for three light neutrinos from Eq. (16). If the dimension-5 Weinberg operator is obtained in the framework of the minimal supersymmetric standard model, one will be left with $M^{}_\nu = \kappa (v \sin\beta)^2/2$, where $\tan\beta$ denotes the ratio of the vacuum expectation values of two Higgs doublets. Eq. (16) or its supersymmetric counterpart can provide a simple but generic way of generating tiny neutrino masses. There are a number of interesting possibilities of building renormalizable gauge models to realize the effective Weinberg mass operator, such as the well-known seesaw mechanisms at a superhigh energy scale $\Lambda$ [@SS1; @SS2; @SS3]. The running of $M^{}_\nu$ from $\Lambda$ to the electroweak scale $\mu \simeq M^{}_Z$ (or vice versa) is described by the RGE’s [@RGE]. In the chosen flavor basis and at the one-loop level, $M^{}_\nu (M^{}_Z)$ and $M^{}_\nu (\Lambda)$ are related to each other via $$\begin{aligned} M^{}_\nu (M^{}_Z) = I^{}_0 \left(\matrix{ I^{}_e & 0 & 0 \cr 0 & I^{}_\mu & 0 \cr 0 & 0 & I^{}_\tau \cr} \right) M^{}_\nu (\Lambda) \left(\matrix{ I^{}_e & 0 & 0 \cr 0 & I^{}_\mu & 0 \cr 0 & 0 & I^{}_\tau \cr} \right) \; , $$ where the RGE evolution function $I^{}_0$ denotes the overall contribution from gauge and quark Yukawa couplings, and $I^{}_\alpha$ (for $\alpha = e, \mu, \tau$) stand for the contributions from charged-lepton Yukawa couplings [@Mei]. Because of $I^{}_e < I^{}_\mu < I^{}_\tau$ as a consequence of $m^{}_e \ll m^{}_\mu \ll m^{}_\tau$, they can modify the texture of $M^{}_\nu$. In comparison, $I^{}_0 \neq 1$ only affects the overall mass scale of $M^{}_\nu$. Note, however, that the texture zeros of $M^{}_\nu$ are stable against such quantum corrections induced by the one-loop RGE’s. Taking [**Pattern**]{} ${\bf A}^{}_1$ of $M^{}_\nu$ for example, we have $$\begin{aligned} M^{{\bf A}^{}_1}_\nu (\Lambda) = \left(\matrix{ 0 & a & b \cr a & d & d \cr b & d & c \cr} \right) $$ at $\Lambda$, and thus $$\begin{aligned} M^{{\bf A}^{}_1}_\nu (M^{}_Z) = I^{}_0 \left(\matrix{ 0 & a I^{}_e I^{}_\mu & b I^{}_e I^{}_\tau \cr a I^{}_e I^{}_\mu & d I^2_\mu & d I^{}_\mu I^{}_\tau \cr b I^{}_e I^{}_\tau & d I^{}_\mu I^{}_\tau & c I^2_\tau \cr} \right) $$ at $M^{}_Z$. Although the texture zero is stable, the equality $(M^{{\bf A}^{}_1}_\nu)^{}_{\mu \mu} = (M^{{\bf A}^{}_1}_\nu)^{}_{\mu \tau}$ at $\Lambda$ is spoiled at the weak scale $M^{}_Z$. Nevertheless, it can be shown that $I^{}_\alpha \approx 1$ (for $\alpha = e, \mu, \tau$) hold as an excellent approximation in the standard model. This interesting feature implies that the important relations obtained in Eq. (11) hold approximately both at $\Lambda$ and $M^{}_Z$. In other words, if a seesaw or flavor symmetry model predicts a hybrid texture of $M^{}_\nu$ at $\Lambda$, one may simply study its phenomenological consequences at $M^{}_Z$ by taking account of the same texture zero and equality. However, the absolute values of neutrino masses are indeed changed when running from a high-energy scale to low-energy scales. Analytical Approximations ========================= First of all, we point out that there exists a permutation symmetry, which relates one texture to another in Eqs. (1)–(6). More explicitly, the permutation between 2- and 3-rows of $M^{}_\nu$, and that between 2- and 3-columns at the same time, change the position of one zero and two equal elements, giving rise to another hybrid texture $\tilde{M}^{}_\nu$. If $M^{}_\nu$ can be diagonalized by a unitary matrix $U$ with mixing parameters $(\theta^{}_{12}, \theta^{}_{23}, \theta^{}_{13}, \delta)$, while $\tilde{M}^{}_\nu$ by a unitary matrix $\tilde{U}$ with mixing parameters $(\tilde{\theta}^{}_{12}, \tilde{\theta}^{}_{23}, \tilde{\theta}^{}_{13}, \tilde{\delta})$, it is straightforward to show that these two sets of mixing parameters are related as follows [@FXZ]: $$\tilde{\theta}^{}_{12} = \theta^{}_{12} \; , ~~~ \tilde{\theta}^{}_{13} = \theta^{}_{13} \; , ~~~ \tilde{\theta}^{}_{23} = \frac{\pi}{2} - \theta^{}_{23} \; , ~~~ \tilde{\delta} = \pi - \delta \; . $$ Moreover, $M^{}_\nu$ and $\tilde{M}^{}_\nu$ have the same eigenvalues $\lambda^{}_i$ (for $i = 1, 2, 3$). Among 39 viable patterns, one can immediately verify that such a permutation symmetry exists between $$\begin{aligned} && {\bf A}^{}_1 \leftrightarrow {\bf A}^{}_3 \; , ~~~ {\bf B}^{}_1 \leftrightarrow {\bf C}^{}_1 \; , ~~~ {\bf B}^{}_2 \leftrightarrow {\bf C}^{}_3 \; , ~~~ {\bf B}^{}_3 \leftrightarrow {\bf C}^{}_2 \; , ~~~ {\bf B}^{}_4 \leftrightarrow {\bf C}^{}_5 \; ,\nonumber \\ && {\bf B}^{}_5 \leftrightarrow {\bf C}^{}_4 \; , ~~~ {\bf D}^{}_1 \leftrightarrow {\bf D}^{}_2 \; , ~~~ {\bf D}^{}_3 \leftrightarrow {\bf D}^{}_6 \; , ~~~ {\bf D}^{}_4 \leftrightarrow {\bf D}^{}_5 \; , ~~~ {\bf E}^{}_1 \leftrightarrow {\bf F}^{}_2 \; , \nonumber \\ && {\bf E}^{}_2 \leftrightarrow {\bf F}^{}_1 \; , ~~~~ {\bf E}^{}_3 \leftrightarrow {\bf F}^{}_4 \; , ~~~~ {\bf E}^{}_4 \leftrightarrow {\bf F}^{}_3 \; , ~~~ {\bf E}^{}_5 \leftrightarrow {\bf F}^{}_5 \; , ~~~ {\bf E}^{}_6 \leftrightarrow {\bf F}^{}_9 \; , \nonumber \\ && {\bf E}^{}_7 \leftrightarrow {\bf F}^{}_8 \; , ~~~~ {\bf E}^{}_8 \leftrightarrow {\bf F}^{}_7 \; , ~~~~ {\bf E}^{}_9 \leftrightarrow {\bf F}^{}_6 \; , ~~~ {\bf E}^{}_{10} \leftrightarrow {\bf F}^{}_{10} \; , $$ so the analytical results in Eq. (11) for one hybrid texture can be obtained from those for the corresponding paired one. Hence we are left with only twenty independent patterns. It is worthwhile to mention that ${\bf Pattern}~{\bf A}^{}_2$ is invariant under the permutations of 2- and 3-rows and columns. In the following, we focus on the approximate analytical results for the six patterns ${\bf A}^{}_1$, ${\bf B}^{}_1$, ${\bf B}^{}_5$, ${\bf D}^{}_1$, ${\bf E}^{}_1$, and ${\bf E}^{}_8$ and explore their implications for neutrino mass spectrum and the leptonic CP-violating phases. The detailed numerical studies will be given in section 4. The analytical approximations for the other patterns can be discussed in a similar way, but they are more or less dependent on whether the mixing angle $\theta^{}_{23}$ is close to $\pi/4$ and whether the Dirac CP-violating phase $\delta$ is nearly $\pi/2$. Another important motivation to choose these six patterns for illustration is that they provide very concrete predictions either for the mixing angles, or for the neutrino mass hierarchy, or for the Dirac CP-violating phase, or for the neutrinoless double-beta decays, which make them phenomenologically more interesting and experimentally more testable than the other patterns. - [${\bf Pattern}~{\bf A}^{}_1$]{} with $(M^{}_\nu)^{}_{ee} = 0$ and $(M^{}_\nu)^{}_{\mu\mu} = (M^{}_\nu)^{}_{\mu\tau}$. With the help of Eq. (\[eps.lambda\]), in the leading order of $\sin \theta^{}_{13}$, we have $$\begin{aligned} \frac{\lambda^{}_1}{\lambda^{}_3} &\approx& - \frac{\sin^2 \theta^{}_{12} \left(1 - \tan\theta^{}_{23}\right)} {\cos 2\theta^{}_{12} (1 + \cot \theta^{}_{23})}~e^{2 i \delta} \; , \nonumber \\ \frac{\lambda^{}_2}{\lambda^{}_3} &\approx& + \frac{\cos^2 \theta^{}_{12} \left(1 - \tan\theta^{}_{23}\right)} {\cos 2\theta^{}_{12} (1 + \cot \theta^{}_{23})}~e^{2 i \delta} \; , $$ which lead us to the neutrino mass ratios $$\begin{aligned} \xi &\approx& \frac{\sin^2 \theta^{}_{12} |1-\tan\theta _{23}|} {\cos 2\theta^{}_{12} (1 + \cot \theta^{}_{23})} \; , \nonumber\\ \zeta &\approx& \frac{\cos^2 \theta^{}_{12} |1-\tan\theta _{23}|} {\cos 2\theta^{}_{12} (1 + \cot \theta^{}_{23})} \; , $$ and the relations between Majorana and Dirac CP-violating phases: $\rho \approx \delta - \pi/2$ and $\sigma \approx \delta$ (for $\theta^{}_{23} < 45^\circ$) or $\rho \approx \delta$ and $\sigma \approx \delta - \pi/2$ (for $\theta^{}_{23} > 45^\circ$). Taking the $3\sigma$ ranges of neutrino mixing angles, we obtain $0.59 \leq \tan \theta^{}_{12} \leq 0.75$ and $0.70 \leq \tan \theta^{}_{23} \leq 1.3$, which yield $\xi < \zeta <1$. Therefore, only the normal neutrino mass hierarchy $m^{}_1 < m^{}_2 < m^{}_3$ or equivalently $\Delta m^2 > 0$ is allowed. Furthermore, we get $$\begin{aligned} R^{}_\nu \approx \zeta^2 - \xi^2 \approx \sec 2\theta^{}_{12} \left(\frac{1 - \tan \theta^{}_{23}} {1 + \cot \theta^{}_{23}}\right)^2, $$ which is actually independent of $\delta$. In order to determine or constrain $\delta$, we have to work in the next-to-leading order approximation. More explicitly, we obtain $$\begin{aligned} \xi &\approx& \frac{\sin^2 \theta^{}_{12} |1 - \tan \theta^{}_{23}|}{\cos 2\theta^{}_{12} (1 + \cot \theta^{}_{23})} \left(1 + \frac{1-\cot 2\theta^{}_{23}}{1 + \cot \theta^{}_{23}} \tan 2\theta^{}_{12} \sin \theta^{}_{13} \cos \delta\right) \; , \nonumber \\ \zeta &\approx& \frac{\cos^2 \theta^{}_{12} |1 - \tan \theta^{}_{23}|}{\cos 2\theta^{}_{12} (1 + \cot \theta^{}_{23})} \left(1 + \frac{1 - \cot 2\theta^{}_{23}} {1 + \cot \theta^{}_{23}} \tan 2\theta^{}_{12} \sin \theta^{}_{13} \cos \delta \right) \; , $$ and thus $$\begin{aligned} R^{}_\nu \approx \sec 2\theta^{}_{12} \left(\frac{1 - \tan \theta^{}_{23}} {1+\cot\theta _{23}}\right)^2 \left[1 + 2 \tan 2\theta^{}_{12} \sin \theta^{}_{13} \cos \delta \left(\frac{1 - \cot 2\theta^{}_{23}} {1 + \cot \theta^{}_{23}}\right)\right]. $$ Now it is straightforward to solve Eq. (26) for the CP-violating phase, namely, $$\begin{aligned} \delta &\approx& \cos^{-1} \left\{\frac{\cot2 \theta_{12}(1+\cot\theta_{23})} {2\sin\theta^{}_{13} \left(1 - \cot 2\theta^{}_{23}\right)} \left[\frac{R^{}_\nu (1 + \cot \theta^{}_{23})^2}{\sec 2\theta^{}_{12} (1 - \tan \theta^{}_{23})^2} - 1\right]\right\}. $$ For the best-fit values of $(\theta^{}_{12}, \theta^{}_{23}, \theta^{}_{13})$ and $R^{}_\nu$ from Table 1, there is no solution to Eq. (27). Taking $\theta^{}_{12} = 35^\circ$ and setting the other parameters to their best-fit values (i.e., $\theta^{}_{23} = 38.4^\circ, \theta^{}_{13} = 8.9^\circ, \delta m^2 = 7.54 \times 10^{-5}~{\textrm{eV}}^2$ and $\Delta m^2 = 2.43 \times 10^{-3}~{\textrm{eV}}^2$), one can figure out the Dirac and Majorana CP-violating phases $$\begin{aligned} \delta \approx 23^\circ \; , ~~~ \rho \approx -67^\circ \; , ~~~ \sigma \approx 23^\circ \; , $$ as well as the neutrino mass spectrum $$\begin{aligned} m^{}_3 &\approx& \sqrt{\Delta m^2} \approx 4.9 \times 10^{-2}~{\textrm{eV}}\; , \nonumber \\ m^{}_2 &\approx& m^{}_3 \zeta \approx 8.9 \times 10^{-3}~{\textrm{eV}}\; , \nonumber\\ m^{}_1 &\approx& m^{}_3 \xi \approx 4.3 \times 10^{-3}~{\textrm{eV}}\; . $$ Since $(M^{}_\nu)^{}_{ee} = 0$ holds for [**Pattern**]{} ${\bf A}^{}_1$, the effective neutrino mass $\langle m \rangle^{}_{\rm ee}$ in the neutrinoless double-beta ($0\nu2\beta$) decays is vanishing. The future observation of $0\nu2\beta$ decays will definitely rule out this pattern. See, e.g., Ref. [@Rodejohann], for recent reviews on the theoretical and experimental status of the $0\nu2\beta$ decays. - [**Pattern**]{} ${\bf B}^{}_1$ with $(M^{}_\nu)^{}_{e\mu}=0$ and $(M^{}_\nu)^{}_{ee} = (M^{}_\nu)^{}_{e\tau}$. In the leading order of $\sin \theta^{}_{13}$, one can obtain from Eq. (11) that $$\begin{aligned} \frac{\lambda^{}_1}{\lambda^{}_3} &\approx& \frac{\sin \theta^{}_{13}}{\cos \theta^{}_{23}} \left(1 + \tan \theta^{}_{12} \sin \theta^{}_{23} e^{i \delta}\right) \; , \nonumber \\ \frac{\lambda^{}_2}{\lambda^{}_3} &\approx& \frac{\sin \theta^{}_{13}}{\cos \theta^{}_{23}} \left(1 - \cot \theta^{}_{12} \sin \theta^{}_{23} e^{i \delta}\right) \; , $$ leading to the neutrino mass ratios $$\begin{aligned} \xi &\approx& \frac{\sin \theta^{}_{13}}{\cos \theta^{}_{23}} \left(1 + \tan^2 \theta^{}_{12} \sin^2 \theta^{}_{23} + 2 \tan \theta^{}_{12} \sin \theta^{}_{23} \cos \delta\right)^{1/2} \; , \nonumber \\ \zeta &\approx& \frac{\sin \theta^{}_{13}}{\cos \theta^{}_{23}} \left(1 + \cot^2 \theta^{}_{12} \sin^2 \theta^{}_{23} - 2 \cot \theta^{}_{12} \sin \theta^{}_{23} \cos \delta\right)^{1/2} \; , $$ and the Majorana CP-violating phases $$\begin{aligned} \rho &\approx& \frac{1}{2} \arg \left(1 + \tan \theta^{}_{12} \sin \theta^{}_{23} e^{i \delta}\right) \; , \nonumber\\ \sigma &\approx& \frac{1}{2} \arg \left(1 - \cot \theta^{}_{12} \sin \theta^{}_{23} e^{i \delta}\right) \; . $$ Taking the values of three neutrino mixing angles $(\theta^{}_{12}, \theta^{}_{23}, \theta^{}_{13})$ within their $3\sigma$ ranges, one can verify $\xi < \zeta<1$, implying that only the normal neutrino mass hierarchy ($m^{}_1 < m^{}_2 < m^{}_3$) is allowed. Furthermore, we obtain $$\begin{aligned} R^{}_\nu \approx \zeta^2 - \xi^2 \approx 4\sin^2 \theta^{}_{13} \csc 2\theta^{}_{12} \tan^2 \theta^{}_{23}(\cot2\theta^{}_{12} - \csc \theta^{}_{23} \cos\delta), $$ from which one can determine the CP-violating phase $$\begin{aligned} \delta\approx\cos^{-1}\left[\sin\theta _{23}\left(\cot2\theta _{12} - \frac{R_\nu \sin 2\theta _{12}}{4\sin^2 \theta _{13} \tan^2 \theta_{23}}\right)\right]. $$ Since neutrino oscillation experiments indicate $m^{}_1 < m^{}_2$ (i.e., $R^{}_\nu > 0$), the condition $\cos \delta < \cot2\theta^{}_{12} \sin\theta^{}_{23}$ should be satisfied. With the best-fit values of three neutrino mixing angles and two neutrino mass-squared differences given in Table 1, we arrive at $$\begin{aligned} \delta \approx 92^\circ \; , ~~~~ \rho \approx 11^\circ \; , ~~~~ \sigma \approx -21^\circ, $$ while three neutrino masses are $$\begin{aligned} m^{}_3 &\approx& \sqrt{|\Delta m^2|} \approx 4.9\times10^{-2}~{\textrm{eV}}\; , \nonumber\\ m^{}_2 &\approx& m^{}_3 \zeta \approx 1.4\times10^{-2}~{\textrm{eV}}\; , \nonumber\\ m^{}_1 &\approx& m^{}_3 \xi \approx 1.0\times10^{-2}~{\textrm{eV}}\; . $$ In addition, the effective neutrino mass in the $0\nu2\beta$ decays can be estimated as $\langle m \rangle_{\rm ee} \approx m_3 \sin\theta _{13} \sec\theta_{23} \approx 9.8 \times 10^{-3}~{\textrm{eV}}$, which is quite challenging for the experimental searches even at the next-generation facilities [@Rodejohann]. - [**Pattern**]{} ${\bf B}^{}_5$ with $(M^{}_\nu)^{}_{e\mu}=0$ and $(M^{}_\nu)^{}_{\mu\tau} = (M^{}_\nu)^{}_{\tau\tau}$. With the help of Eq. (\[eps.lambda\]), in the leading order of $\sin\theta^{}_{13}$, we get $$\begin{aligned} \frac{\lambda^{}_1}{\lambda^{}_3} \approx \frac{\lambda^{}_2}{\lambda^{}_3} \approx \frac{e^{2 i \delta } \left(1 - \cot\theta^{}_{23}\right)}{1 + \tan\theta^{}_{23}} \; , $$ from which follows $$\begin{aligned} \xi \approx \zeta \approx \frac{|1-\cot\theta^{}_{23}|} {1 + \tan\theta^{}_{23}} \; , $$ and $\rho \approx \sigma \approx \delta -\pi/2$ (for $\theta^{}_{23} < 45^\circ$) or $\rho \approx \sigma \approx \delta$ (for $\theta^{}_{23} > 45^\circ$). Since $R^{}_\nu \approx 0$ in the leading-order approximation, we have to work at the next-to-leading order $$\begin{aligned} \frac{\lambda^{}_1}{\lambda^{}_3} &\approx& \frac{e^{2 i \delta } \left(1-\cot\theta^{}_{23}\right)}{1+\tan\theta^{}_{23}} \left[1 + \frac{\sin\theta^{}_{13} \cos\delta \left[1 - i \sqrt{2} \tan \delta \sin (2\theta^{}_{23} -\pi/4)\right]} {\tan\theta^{}_{12} \cos^2\theta^{}_{23}(1 - \cot\theta^{}_{23})}\right] \; , \nonumber \\ \frac{\lambda^{}_2}{\lambda^{}_3} &\approx& \frac{e^{2 i \delta } \left(1 - \cot\theta^{}_{23}\right)}{1 + \tan\theta^{}_{23}} \left[1 - \frac{\sin\theta^{}_{13} \cos \delta \left[1 - i \sqrt{2} \tan\delta \sin (2\theta^{}_{23} -\pi/4)\right]} {\cot \theta^{}_{12} \cos^2\theta^{}_{23}(1 - \cot\theta^{}_{23})}\right] \; . $$ Hence, to the first order of $\sin \theta^{}_{13}$, one obtains $$\begin{aligned} R^{}_\nu \approx \zeta^2 - \xi^2 \approx \frac{2(1 - \tan\theta^{}_{23}) \sin\theta^{}_{13} \cos\delta} {(1 - \cot2\theta^{}_{23}) \sin 2\theta^{}_{12}} \; , $$ leading to $$\begin{aligned} \delta &\approx& \cos^{-1} \left[ \frac{(1 - \cot2\theta _{23}) \sin2\theta^{}_{12} R^{}_\nu} {2(1 - \tan\theta^{}_{23}) \sin\theta^{}_{13}}\right] \; , $$ and the difference between two neutrino mass ratios $$\begin{aligned} \zeta - \xi \approx \frac{(1 - \tan^2 \theta^{}_{23}) \sin\theta^{}_{13} \cos\delta}{(1 - \cot2\theta^{}_{23})|1 - \cot\theta^{}_{23}| \sin2\theta^{}_{12}} \; . $$ Taking the values of $\theta^{}_{23}$ within the $3\sigma$ range, we have $\xi \approx \zeta < 1$, implying that only the normal neutrino mass hierarchy is allowed. To ensure $\xi < \zeta$ or equivalently $m^{}_1 < m^{}_2$, we have to require $\cos \delta > 0$ for $\theta^{}_{23} < 45^\circ$, and $\cos\delta < 0$ for $\theta^{}_{23} > 45^\circ$. Furthermore, using the best-fit values of three neutrino mixing angles and two neutrino mass-squared differences, we arrive at $$\begin{aligned} \delta \approx 70^\circ \; , ~~~~ \rho \approx 12^\circ \; , ~~~~ \sigma \approx -30^\circ \; , $$ and the neutrino mass spectrum $$\begin{aligned} m^{}_3 &\approx& (1 + \tan \theta^{}_{23}) \sqrt{\frac{\Delta m^2 \sin 2\theta^{}_{23}}{1 - \cot 2\theta^{}_{23}}} \approx 5.0 \times 10^{-2}~{\textrm{eV}}\; ,\nonumber\\ m^{}_1 &\approx& m^{}_2 ~~ \approx ~~ m^{}_3 \cdot \frac{|1 - \cot\theta^{}_{23}|}{1 + \tan\theta^{}_{23}} \approx 7.3 \times 10^{-3}~{\textrm{eV}}\; . $$ It is straightforward to calculate the effective neutrino mass $\langle m\rangle^{}_{\rm ee} \approx 7.3 \times 10^{-3}~{\textrm{eV}}$ in the $0\nu2\beta$ decays. As expected for the case of normal neutrino mass hierarchy, $\langle m \rangle^{}_{\rm ee}$ is too small to be measured in the near future. - [**Pattern**]{} ${\bf D}^{}_1$ with $(M^{}_\nu)^{}_{\mu\tau} = 0$ and $(M^{}_\nu)^{}_{ee} = (M^{}_\nu)^{}_{e\mu}$. From Eq. (11), in the leading order of $\sin\theta_{13}$, we derive $$\begin{aligned} \frac{\lambda^{}_1}{\lambda^{}_3} &\approx& e^{2 i \delta} \frac{\cos\theta^{}_{23} - \tan\theta^{}_{12} e^{i \delta}} {\cos\theta^{}_{23} + 2 \cot 2\theta^{}_{12} e^{i \delta}} \; , \nonumber\\ \frac{\lambda^{}_2}{\lambda^{}_3} &\approx& e^{2 i \delta} \frac{\cos\theta^{}_{23} + \cot\theta^{}_{12} e^{i \delta}}{\cos\theta^{}_{23} + 2 \cot 2\theta^{}_{12} e^{i \delta}} \; . $$ From Eq. (45), it is straightforward to extract the neutrino mass ratios $$\begin{aligned} \xi &\approx& \left[\frac{\cos^2\theta^{}_{23} + \tan^2\theta^{}_{12} - 2\cos\delta \cos\theta^{}_{23} \tan\theta^{}_{12}}{\cos^2\theta^{}_{23} + 4 \cot^2 2\theta^{}_{12} + 4\cos\delta \cot 2\theta^{}_{12} \cos\theta^{}_{23}}\right]^{1/2} \; , \nonumber\\ \zeta &\approx& \left[\frac{\cos^2 \theta^{}_{23} + \cot^2 \theta^{}_{12} + 2\cos\delta \cos\theta^{}_{23} \cot\theta^{}_{12}} {\cos^2\theta^{}_{23} + 4 \cot^2 2\theta^{}_{12} + 4\cos\delta \cot 2\theta^{}_{12} \cos\theta^{}_{23}}\right]^{1/2} \; , $$ and the Majorana CP-violating phases $$\begin{aligned} \rho &\approx& \delta + \frac{1}{2} \arg \left[\frac{\cos\theta^{}_{23} - \tan\theta^{}_{12} e^{i \delta}} {\cos\theta^{}_{23} + 2 \cot 2\theta^{}_{12} e^{i \delta}}\right] \; , \nonumber\\ \sigma &\approx & \delta + \frac{1}{2}\arg \left[\frac{\cos\theta^{}_{23} + \cot\theta^{}_{12} e^{i \delta}} {\cos\theta^{}_{23} + 2 \cot 2\theta^{}_{12} e^{i \delta}}\right] \; . $$ Moreover, it is easy to show that $\cos \delta > - \cot 2\theta^{}_{12} \sec \theta^{}_{23}$ must be satisfied in order to guarantee $\zeta > \xi$ or equivalently $m^{}_2 > m^{}_1$. Inputting the neutrino mixing angles in their $3\sigma$ ranges and $\delta \in [0, 2\pi)$, we find $\zeta > \xi > 1$, so only the inverted neutrino mass hierarchy is possible. Thus we get $$\begin{aligned} R^{}_\nu = \frac{2(\zeta^2-\xi^2)}{(\zeta^2+\xi^2)-2} \approx - \frac{8\left(\cos 2\theta^{}_{12} + \sin 2\theta^{}_{12} \cos\theta^{}_{23} \cos\delta\right)}{1 + 3\cos 4\theta^{}_{12} + 2 \sin 4\theta^{}_{12} \cos\theta^{}_{23} \cos\delta} \; , $$ which leads us to $$\begin{aligned} \delta \approx \cos^{-1} \left[ -\frac{8 \cos 2\theta^{}_{12} +\left(1 + 3 \cos 4\theta^{}_{12}\right) R^{}_\nu}{4\cos\theta^{}_{23} \sin 2\theta^{}_{12} \left(2 + \cos 2\theta^{}_{12} R^{}_\nu\right)}\right] \; . $$ With the best-fit values of three neutrino mixing angles and two neutrino mass-squared differences, we obtain three CP-violating phases $$\begin{aligned} \delta \approx 122^\circ \; , ~~~~ \rho \approx 76^\circ \; , ~~~~ \sigma \approx -45^\circ \; , $$ and three neutrino masses $$\begin{aligned} m^{}_3 &\approx& \sqrt{\frac{\delta m^2}{\zeta^2 - \xi^2}} \approx 3.91 \times 10^{-2}~{\textrm{eV}}\; , \nonumber\\ m^{}_2 &\approx& m^{}_3 \zeta \approx 6.32 \times 10^{-2}~{\textrm{eV}}\; , \nonumber\\ m^{}_1 &\approx& m^{}_3 \zeta \approx 6.26 \times 10^{-2}~{\textrm{eV}}\; . $$ Note that the neutrino mass spectrum is nearly degenerate, so the effective neutrino mass in the $0\nu2\beta$ decays $\langle m\rangle^{}_{\rm ee} \approx m^{}_3/|1 + 2 e^{i \delta} \cot 2\theta _{12} \sec\theta _{23}| \approx 3.87 \times 10^{-2}~{\textrm{eV}}$ turns out to be accessible in the next-generation experiments. - [**Pattern**]{} ${\bf E}^{}_1$ with $(M^{}_\nu)^{}_{\mu\mu} = 0$ and $(M^{}_\nu)^{}_{ee} = (M^{}_\nu)^{}_{e\mu}$. With the help of Eq. (\[eps.lambda\]), in the leading order of $\sin\theta _{13}$, we have $$\begin{aligned} \frac{\lambda^{}_1}{\lambda^{}_3} &\approx& -e^{2 i\delta} \tan^2 \theta^{}_{23} \frac{\cos \theta^{}_{23} - \tan \theta^{}_{12} e^{i \delta}}{\cos \theta^{}_{23} + 2 \cot 2\theta^{}_{12} e^{i \delta}} \; ,\nonumber\\ \frac{\lambda^{}_2}{\lambda^{}_3} &\approx& -e^{2 i\delta} \tan^2 \theta^{}_{23} \frac{\cos \theta^{}_{23} + \cot \theta^{}_{12} e^{i \delta}}{\cos \theta^{}_{23} + 2\cot 2\theta^{}_{12} e^{i\delta}} \;. $$ From Eq. (52), it is straightforward to extract the neutrino mass ratios $$\begin{aligned} \xi &\approx& \tan^2 \theta^{}_{23} \left[\frac{\cos^2\theta^{}_{23} + \tan^2\theta^{}_{12} - 2\cos\delta \cos\theta^{}_{23} \tan\theta^{}_{12}}{\cos^2\theta^{}_{23} + 4 \cot^2 2\theta^{}_{12} + 4\cos\delta \cot 2\theta^{}_{12} \cos\theta^{}_{23}}\right]^{1/2} \; , \nonumber\\ \zeta &\approx& \tan^2 \theta^{}_{23} \left[\frac{\cos^2 \theta^{}_{23} + \cot^2 \theta^{}_{12} + 2\cos\delta \cos\theta^{}_{23} \cot\theta^{}_{12}} {\cos^2\theta^{}_{23} + 4 \cot^2 2\theta^{}_{12} + 4\cos\delta \cot 2\theta^{}_{12} \cos\theta^{}_{23}}\right]^{1/2} \; , $$ and the Majorana CP-violating phases $$\begin{aligned} \rho &\approx& \delta + \frac{1}{2} \arg \left[\frac{-\cos\theta^{}_{23} + \tan\theta^{}_{12} e^{i \delta}} {\cos\theta^{}_{23} + 2 \cot 2\theta^{}_{12} e^{i \delta}}\right] \; , \nonumber\\ \sigma &\approx & \delta + \frac{1}{2}\arg \left[\frac{-\cos\theta^{}_{23} - \cot\theta^{}_{12} e^{i \delta}} {\cos\theta^{}_{23} + 2 \cot 2\theta^{}_{12} e^{i \delta}}\right] \;. $$ Combining Eq. (53) with Eq. (12), we can determine the Dirac CP-violating phase $$\begin{aligned} \delta \approx \cos^{-1} \left\{-\frac{\cot 2\theta^{}_{12}} {\cos \theta^{}_{23}} \left[1 + \frac{\tan^2 2\theta^{}_{12} \left( \cot^4 \theta^{}_{23} - \csc^2 \theta^{}_{23}\right) R^{}_\nu}{4\sec 2\theta^{}_{12} + 2\left( 2 \cot^4 \theta^{}_{23} - 1\right) R^{}_\nu}\right]\right\} \; . $$ Taking the best-fit values of three neutrino mixing angles and two neutrino mass-squared differences, we obtain $$\begin{aligned} \delta \approx 122^\circ \; , ~~~~ \rho \approx -13^\circ \; , ~~~~ \sigma \approx 45^\circ \; , $$ and the neutrino mass spectrum is as follows $$\begin{aligned} m^{}_3 &\approx& \sqrt{\frac{\delta m^2}{\zeta^2 - \xi^2}} \approx 9.24 \times 10^{-2}~{\textrm{eV}}\; , \nonumber\\ m^{}_1 &\approx& m^{}_2 ~~\approx~~9.41 \times 10^{-2}~{\textrm{eV}}\; . $$ Finally, it is straightforward to figure out $\langle m\rangle^{}_{\rm ee} \approx m^{}_3 \tan^2 \theta^{}_{23}/|1 + 2e^{i \delta} \cot 2\theta^{}_{12} \sec \theta^{}_{23}| \approx 5.8\times10^{-2}~{\textrm{eV}}$, which is quite encouraging for the upcoming $0\nu2\beta$ experiments. It is worthwhile to mention that the analytical formulas for [**Pattern**]{} ${\bf D}^{}_1$ are identical to those for [**Pattern**]{} ${\bf E}^{}_1$ if $\theta^{}_{23} = \pi/4$ is assumed. Therefore, the precision measurement of $\theta^{}_{23}$ is crucial to distinguish between these two patterns of $M^{}_\nu$. - [**Pattern**]{} ${\bf E}^{}_8$ with $(M^{}_\nu)^{}_{\mu\mu} = 0$ and $(M^{}_\nu)^{}_{e\tau} = (M^{}_\nu)^{}_{\mu\tau}$. From Eq. (11), in the leading order of $\sin\theta_{13}$, we obtain $$\begin{aligned} \frac{\lambda^{}_1}{\lambda^{}_3} &\approx& -e^{2i\delta} (\tan^2 \theta^{}_{23} - \cot \theta^{}_{12} \sec \theta^{}_{23} e^{-i\delta}) \; , \nonumber\\ \frac{\lambda^{}_2}{\lambda^{}_3} &\approx& -e^{2i\delta} (\tan^2 \theta^{}_{23} + \tan \theta^{}_{12} \sec \theta^{}_{23} e^{-i \delta}) \; . $$ Then, from Eq. (58), it is easy to extract the neutrino mass ratios $$\begin{aligned} \xi &\approx& \left[\tan^4 \theta^{}_{23} + \cot^2 \theta^{}_{12} \sec^2 \theta^{}_{23} - 2\cot \theta^{}_{12} \tan^2 \theta^{}_{23} \sec \theta^{}_{23} \cos\delta \right]^{1/2} \; , \nonumber\\ \zeta &\approx& \left[\tan^4 \theta^{}_{23} + \tan^2 \theta^{}_{12} \sec^2 \theta^{}_{23} + 2\tan \theta^{}_{12} \tan^2 \theta^{}_{23} \sec \theta^{}_{23} \cos\delta\right]^{1/2} \; , $$ and the Majorana CP-violating phases $$\begin{aligned} \rho & \approx & \delta + \frac{1}{2} \arg \left[-\tan^2 \theta^{}_{23} + \cot \theta^{}_{12} \sec\theta^{}_{23} e^{-i\delta}\right] \; ,\nonumber\\ \sigma &\approx & \delta + \frac{1}{2} \arg \left[-\tan^2 \theta^{}_{23} - \tan \theta^{}_{12} \sec\theta^{}_{23} e^{-i\delta}\right] \; . $$ One immediately observes from Eq. (59) that $$\begin{aligned} \zeta^2 - \xi^2 &\approx& \frac{4(\cos\delta \tan^2 \theta^{}_{23} - \cot 2\theta^{}_{12} \sec\theta^{}_{23})}{\sin 2\theta^{}_{12} \cos \theta^{}_{23}} \; . $$ In order to ensure $m^{}_1 < m^{}_2$ or equivalently $\zeta^2 - \xi^2 > 0$, we have to require $\cos \delta > \cot 2\theta^{}_{12} \sec \theta^{}_{23} \cot^2 \theta^{}_{23} > 0$, implying $\delta < \pi/2$ or $\delta > 3\pi/2$. Taking the $3\sigma$ ranges of the mixing parameters, we find that only the inverted neutrino mass hierarchy is allowed. Moreover, we get $$\begin{aligned} R^{}_\nu \approx \frac{4 \csc 2\theta^{}_{12} (\cos\delta \tan^2 \theta^{}_{23} - \cot 2\theta^{}_{12} \sec \theta^{}_{23})} {\left(2 \cot^2 2\theta^{}_{12} + \tan^2 \theta^{}_{23} \right) \sec \theta^{}_{23} - 2 \cos\delta \cot 2\theta^{}_{12} \tan^2 \theta^{}_{23}} \; , $$ from which one can determine the Dirac CP-violating phase $$\begin{aligned} \delta &\approx& \cos^{-1} \left[\frac{\cot 2\theta^{}_{12}} {\tan\theta^{}_{23} \sin\theta^{}_{23}} + \frac{\sin 2\theta^{}_{12} \sec\theta^{}_{23} R^{}_\nu}{4 + 2 \cos 2\theta^{}_{12} R^{}_\nu} \right] \; . $$ With the best-fit values of three neutrino mixing angles and two neutrino mass-squared differences, we arrive at $$\begin{aligned} \delta \approx 30^\circ \; , ~~~~ \rho \approx 9^\circ \; , ~~~~ \sigma \approx -68^\circ \; . $$ In addition, three neutrino masses are found to be $$\begin{aligned} m^{}_3 &\approx& \left[\frac{\delta m^2 \sin 2\theta^{}_{12} \cos \theta^{}_{23}} {4(\cos\delta \tan^2\theta^{}_{23} - \cot 2\theta^{}_{12} \sec \theta^{}_{23})}\right]^{1/2} \approx 3.61 \times 10^{-2}~{\textrm{eV}}\; , \nonumber\\ m^{}_2 &\approx& m^{}_3 \zeta ~~ \approx ~~ 5.15 \times 10^{-2}~{\textrm{eV}}\; , \nonumber\\ m^{}_1 &\approx& m^{}_3 \xi ~~ \approx ~~ 5.11 \times 10^{-2}~{\textrm{eV}}\;, $$ which are nearly degenerate. As a consequence, the effective neutrino mass in the $0\nu2\beta$ decays $\langle m\rangle^{}_{\rm ee} \approx m^{}_3 |\tan^2 \theta^{}_{23} e^{i\delta} - 2 \cot 2\theta^{}_{12} \sec \theta^{}_{23}| \approx 2.2 \times 10^{-2}~{\textrm{eV}}$ could be probed in the future $0\nu2\beta$ decay experiments. The above analytical analyses serve as an explicit example for how to determine the leptonic CP-violating phases and neutrino masses, when a hybrid texture of $M^{}_\nu$ is taken. The full parameter space of these patterns will be analyzed in the following section. Numerical Results ================= As mentioned before, we have performed a numerical study of all the sixty hybrid textures of $M^{}_\nu$. It turns out that thirty-nine of them are consistent with current neutrino oscillation data at the $3\sigma$ level, while only thirteen patterns can survive if the $1\sigma$ ranges of neutrino mixing parameters are considered. Our numerical analysis has been done in the following way: 1) For each pattern of $M^{}_\nu$, we generate a set of random numbers for two neutrino mass-squared differences $(\delta m^2, \Delta m^2)$ and three neutrino mixing angles $(\theta^{}_{12}, \theta^{}_{23}, \theta^{}_{13})$ in their $3\sigma$ ranges [@Fogli], which have been shown in Table 1. As we have shown in the previous section, it is then possible to determine the Dirac CP-violating phase $\delta$ from Eq. (12). Instead, we generate a random number of $\delta$ in the range of $[0, 2\pi)$, and test whether Eq. (12) is satisfied by the generated random numbers. 2) In practice, with the random numbers above, we first calculate the neutrino mass ratios $\zeta$ and $\xi$, and then three neutrino masses $(m^{}_1, m^{}_2, m^{}_3)$. The criteria for whether a specific pattern of $M^{}_\nu$ is consistent with current experimental data are set as follows: (a) With $\zeta$ and $\xi$, we can determine from Eq. (12) the value of $R^{}_\nu$, which is required to fall into the range $\left[(\delta m^2)^{3\sigma}_\textrm{{\footnotesize}min} / (\Delta m^2)^{3\sigma}_\textrm{{\footnotesize}max}, (\delta m^2)^{3\sigma}_\textrm{{\footnotesize}max} / (\Delta m^2)^{3\sigma}_\textrm{{\footnotesize}min} \right]$. (b) Since only two possible neutrino mass hierarchies $m^{}_1 < m^{}_2 < m^{}_3$ and $m^{}_3 < m^{}_1 < m^{}_2$ are allowed by neutrino oscillation experiments, we further demand $(\xi^2 - 1) (\zeta^2 - 1) > 0$, namely, $\xi^2 < \zeta^2 < 1$ corresponds to the normal mass hierarchy while $\zeta^2 > \xi^2 > 1$ to the inverted mass hierarchy. (c) The absolute scale of neutrino masses receives constraints from the beta-decay and $0\nu2\beta$-decay experiments, however, the most restrictive one comes from cosmological observations [@WMAP9]. Recently, the Planck Collaboration has released the first data on cosmic microwave background (CMB) [@Planck]. Combined with the WMAP-polarization, high-resolution CMB, and BAO data sets, the Planck data have placed an upper bound on the sum of three neutrino masses $\sum m^{}_i < 0.23~{\textrm{eV}}$ at the $95\%$ confidence level. We require that this upper bound should be satisfied. Once one pattern of $M^{}_\nu$ survives all the above constraints, we will calculate its predictions for the effective neutrino mass $\langle m\rangle^{}_\textrm{{\footnotesize}ee}$ in the $0\nu2\beta$ decays, the two Majorana CP-violating phases $(\rho, \sigma)$ and the Jarlskog invariant for leptonic CP violation $J^{}_\textrm{{\footnotesize}CP} = s^{}_{12} c^{}_{12} s^{}_{23} c^{}_{23} s^{}_{13} c^2_{13} \sin\delta$ [@Jarlskog]. In addition, the allowed parameter space of other mixing parameters can be determined. 3) To illustrate our results, we present a series of figures for each viable pattern in Figs. \[fig.A1\]–\[fig.E8\], where each figure consists of twelve plots in four rows. The first row shows the allowed ranges of three flavor mixing angles $\theta^{}_{12}$, $\theta_{23}$ and $\theta_{13}$, versus the Dirac CP-violating phase $\delta$. In the second row, the histograms of three mixing angles are given, indicating their distributions in the allowed parameter space. As the ongoing and forthcoming neutrino oscillation experiments will provide us with more precise measurements of three neutrino mixing angles and the Dirac CP-violating phase, our numerical illustrations make it easy to see whether a currently viable hybrid texture can be ruled out by future experimental data. In the third row, we present the allowed ranges of three neutrino mass eigenvalues $(m^{}_1, m^{}_2, m^{}_3)$ and the effective neutrino mass $\langle m \rangle^{}_{\rm ee}$ versus $\delta$. From these plots, one can immediately figure out which neutrino mass hierarchy is predicted, and whether the effective neutrino mass $\langle m \rangle^{}_{\rm ee}$ is accessible in future $0\nu2\beta$ experiments. Finally, the Majorana CP-violating phases $\rho$, $\sigma$ and the Jarlskog invariant $J^{}_{\rm CP}$ are depicted in the last row. The next-generation long-baseline neutrino oscillation experiments are promising to discover the leptonic CP violation if the Jarlskog invariant $J^{}_{\rm CP}$ is at the percent level [@Branco]. We have carried out a thorough numerical study of all the sixty patterns of $M^{}_\nu$, however, it will render our paper unreadable if all the figures of the viable thirty-nine patterns are presented. Therefore, we will focus only on the six patterns ${\bf A}^{}_1$, ${\bf B}^{}_1$, ${\bf B}^{}_5$, ${\bf D}^{}_1$, ${\bf E}^{}_1$, and ${\bf E}^{}_8$, for which the analytical results have been given in the previous section. Some comments and discussions on the numerical results in Figs. \[fig.A1\]–\[fig.E8\] are in order. - **Pattern A$_1$** – As shown in first row of Fig. \[fig.A1\], the Dirac CP-violating phase $\delta$ is essentially unconstrained and the whole range $[0, 2\pi)$ of $\delta$ is experimentally allowed. This can be easily understood from Eq. (24), where $R^{}_\nu$ is found to be independent of $\delta$ and $\theta^{}_{13}$ at the leading order. In contrast with the mixing angles $\theta^{}_{12}$ and $\theta^{}_{13}$, which are only mildly constrained, the allowed range of $\theta^{}_{23}$ splits into two distinct branches: one for $\theta^{}_{23} < 45^\circ$, and the other for $\theta^{}_{23} < 45^\circ$. Moreover, the histogram of $\theta^{}_{23}$ in the second row indicates that $\theta^{}_{23} < 45^\circ$ is preferred, in particular the values as small as $\theta_{23}=35.1^\circ$ at the lower border of the $3\sigma$ range. Consequently, the precision measurement of $\theta^{}_{23}$ can finally tell us whether this pattern is allowed or not. It is also interesting that the histogram of $\theta^{}_{13}$ peaks around $\theta^{}_{13} = 8^\circ$, quite close to the best-fit value. Three neutrino masses are given in the third row, where one can observe that only the normal mass hierarchy (i.e., $m^{}_1 < m^{}_2 < m^{}_3$) is possible. For [**Pattern**]{} ${\bf A}^{}_1$, the effective neutrino mass in $0\nu2\beta$ decays is exactly zero, implying some cancellation takes place among the contributions of three neutrino mass eigenstates. In the fourth row, we can see that an approximately linear correlation exists between the Majorana CP phases $(\rho, \sigma)$ and $\delta$. For a maximal CP-violating phases $\delta = \pi/2$ or $3\pi/2$, the Jarlskog invariant $|J^{}_{\rm CP}| \sim 3\%$ can be achieved. - **Pattern B$_1$** – We can see clearly from the first row of Fig. \[fig.B1\] that only two narrow ranges around $\delta = \pi/2$ and $ \delta = 3\pi/2$ are experimentally allowed. Although three neutrino mixing angles $\theta^{}_{12}$, $\theta^{}_{23}$ and $\theta^{}_{13}$ turn out to be arbitrary in their $3\sigma$ ranges, the distributions of $\theta^{}_{23}$ and $\theta^{}_{13}$ seem to peak around their best-fit values (i.e., $\theta^{}_{23} = 38.4^\circ$ and $\theta^{}_{13} = 8.9^\circ$), as shown in the second row. Similar to [**Pattern**]{} ${\bf A}^{}_1$, only the normal neutrino mass hierarchy (i.e., $m^{}_1 < m^{}_2 < m^{}_3$) is possible, so the effective neutrino mass $\langle m \rangle^{}_{\rm ee}$ is in general small, which renders it very challenging to observe the $0\nu2\beta$ decays. Since the constraint on $\delta$ is quite restrictive, only a small faction of the parameter space of $\rho$ and $\sigma$ is allowed. Interestingly, the Jarlskog invariant $J_\textrm{{\footnotesize}CP}$ is predicted to be close to its maximum, i.e., $|J^{}_{\rm CP}| \geq 3\%$, which should be accessible to the next-generation long-baseline neutrino oscillation experiments [@Branco]. - **Pattern B$_5$** – From the first row of Fig. \[fig.B6\], one can observe that the allowed ranges of $\delta$ contain two disjointed regions: one around $\delta = \pi$, and the other around $\delta = 0$ or $2\pi$. This feature becomes clear, if we recall the analytical discussions on [**Pattern**]{} ${\bf B}^{}_5$ in the previous section, where $\cos\delta > 0$ (i.e., $\delta < \pi/2$ or $\delta>3\pi/2$) for $\theta^{}_{23} < 45^\circ$, and $\cos\delta < 0$ (i.e., $\pi/2 < \delta < 3\pi/2$) for $\theta^{}_{23} > 45^\circ$, are required to guarantee $m^{}_2 > m^{}_1$. While both $\theta^{}_{12}$ and $\theta^{}_{13}$ are mildly constrained, $\theta^{}_{23}$ is restricted to smaller regions far from the maximal mixing. Moreover, there is a strong correlation between $\theta^{}_{23}$ and $\delta$. For instance, if $\delta \approx \pi$ is taken, then $\theta^{}_{23} \approx 51^\circ$ holds; if $\delta \approx 0$ or $2\pi$ is assumed, then we obtain $\theta^{}_{23} \approx 39^\circ$. As shown in the second row, the distribution of $\theta^{}_{23}$ peaks around its best-fit value, namely, $\theta^{}_{23} = 38.4^\circ$. The numerical results in the third row indicate that only the normal mass hierarchy ($m^{}_1 < m^{}_2 < m^{}_3$) is allowed, and the effective neutrino mass $\langle m \rangle^{}_{\rm ee}$ is quite small. The Majorana CP-violating phases $(\rho, \sigma)$ and the Jarlskog invariant $J^{}_{\rm CP}$ are shown in the last row, where one can observe that the parameter space is tightly constrained. - **Pattern D$_1$** – The Dirac CP-violating phase $\delta$ falls into $(\pi/2, 3\pi/2)$, as shown in the first row of Fig. \[fig.D1\]. It is evident from Eq. (49) that only $\cos \delta <0$ (i.e., $\pi/2 < \delta < 3\pi/2$) is possible, given the $3\sigma$ ranges of mixing parameters. Although the $3\sigma$ ranges of three mixing angles are still allowed, only smaller (larger) values of $\theta^{}_{12}$ ($\theta^{}_{23}$) can survive if $\delta \approx \pi$ is assumed. In the second row, the distributions of three mixing angles do not show significant preference in any specific ranges. Different from the previous patterns of $M^{}_\nu$, [**Pattern**]{} ${\bf D}^{}_1$ predicts the inverted neutrino mass hierarchy ($m^{}_3 < m^{}_1 < m^{}_2$), as indicated in the third row. Consequently, the effective neutrino mass $\langle m \rangle^{}_{\rm ee}$ is larger than $0.02~{\textrm{eV}}$, which is reachable in next-generation $0\nu2\beta$ decay experiments. While $\rho$ is restricted to a relatively small range around $\pi/2$ or $-\pi/2$, $\sigma$ can take any values in $[-\pi/4, \pi/4]$. In addition, the magnitude of Jarlskog invariant $J^{}_{\rm CP}$ could be at the percent level if $\delta$ is close to $\pi/2$ or $3\pi/2$. - **Pattern E$_1$** – As shown in the first row of Fig. \[fig.E1\], the Dirac CP-violating phase $\delta$ is limited to two small regions around $3\pi/4$ and $5\pi/4$. From the second row, we can observe that $\theta^{}_{12}$ and $\theta^{}_{13}$ show no significant preference in their $3\sigma$ ranges, while $\theta^{}_{23}$ has two peaks around $37^\circ$ and $51^\circ$. Note that the former peak dominates over the latter and is quite close to the best-fit value of $\theta^{}_{23}$. The neutrino mass spectrum is nearly degenerate, namely, $m^{}_1 \approx m^{}_2 \approx m^{}_3 \approx 0.1~{\textrm{eV}}$, as shown in the third row. Therefore, the effective neutrino mass $\langle m \rangle^{}_{\rm ee}$ can be as large as $0.05~{\textrm{eV}}$, which is much larger than those in all the previous patterns. In addition, the parameter space of two Majorana CP-violating phases $(\rho, \sigma)$ and the Jarlskog invariant $J^{}_{\rm CP}$ is strongly constrained. Since $\sin \delta$ is nonzero, the leptonic CP violation is expected. - **Pattern E$_8$** – From the first row of Fig. \[fig.E8\], one can see that the Dirac CP-violating phase $\delta$ is restricted to the ranges of $\delta < \pi/2$ and $\delta > 3\pi/2$, which are consistent with our analytical results in section 3. The distribution of $\theta^{}_{23}$ peaks around $37^\circ$, while those of $\theta^{}_{12}$ and $\theta^{}_{13}$ do not show significant preference, as shown in the second row. Only the inverted neutrino mass hierarchy ($m^{}_3 < m^{}_1 < m^{}_2$) is allowed, and the effective neutrino mass $\langle m \rangle^{}_{\rm ee}$ turns out to be in the range $[0.025~{\textrm{eV}}, 0.040~{\textrm{eV}}]$, which is accessible in future $0\nu2\beta$ decay experiments. In the last row, the two Majorana CP-violating phases $\rho$ and $\sigma$ are found to be linearly correlated with the Dirac CP phase $\delta$. It is worthwhile to stress that the precision measurements of three neutrino mixing angles, in particular the octant of $\theta^{}_{23}$, as well as the discovery of leptonic CP violation and the $0\nu2\beta$ decays, are crucially important to distinguish between different hybrid textures of Majorana neutrino mass matrix. For instance, we have also checked that only thirteen hybrid textures (i.e., ${\bf A}^{}_3$, ${\bf C}^{}_{2,5}$, ${\bf D}^{}_{1,5,6}$, ${\bf E}^{}_{1,4,6,7}$, and ${\bf F}^{}_{1,7,8}$) are viable, if the $1\sigma$ ranges of neutrino mixing parameters and the Dirac CP-violating phase are taken into account [@Fogli]. In addition, the cosmological bound on absolute neutrino masses becomes very relevant. If the upper limit $\sum m^{}_i < 0.44~{\textrm{eV}}$ from the nine-year WMAP observations [@WMAP9] is used instead of the latest Planck result, two extra patterns $$\begin{aligned} {\bf B}^{}_6: \left(\matrix{ \triangle & 0 & \times \cr 0 & \times & \triangle \cr \times & \triangle & \times} \right) \; , ~~~~ {\bf C}^{}_6:\left(\matrix{ \triangle & \times & 0 \cr \times & \times & \triangle \cr 0 & \triangle & \times} \right) \; , $$ can survive the oscillation data at the $3\sigma$ level. Flavor Symmetry =============== It has been proved that the texture zeros in the Majorana neutrino mass matrix $M^{}_\nu$ can be realized by implementing the $Z^{}_n$ flavor symmetry [@Grimus1; @Grimus2]. Taking the two-zero textures for example, one can demonstrate that the seven viable patterns can be derived from $Z^{}_n$ symmetries in the type-II seesaw model [@FXZ]. To illustrate how to realize the hybrid textures of $M^{}_\nu$, we will work in the type-II seesaw model, which extends the scalar sector of the standard model with one or more $SU(2)^{}_{\rm L}$ scalar triplets [@SS2]. For $N$ scalar triplets, the gauge-invariant Lagrangian relevant for neutrino masses reads $$-{\cal L}^{}_{\Delta} = \frac{1}{2} \sum_{j} \sum_{\alpha, \beta} \left(Y^{}_{\Delta^{}_j}\right)^{}_{\alpha \beta} \overline{\ell^{}_{\alpha \rm L}} \Delta^{}_j i\sigma^{}_2 \ell^c_{\beta \rm L} + {\rm h.c.} \; , $$ where $\alpha$ and $\beta$ run over $e$, $\mu$ and $\tau$, $\Delta^{}_j$ denotes the $j$-th triplet scalar field (for $j = 1, 2, \cdots, N$), and $Y^{}_{\Delta^{}_j}$ is the corresponding Yukawa coupling matrix. After the triplet scalar acquires its vacuum expectation value $\langle \Delta^{}_j\rangle \equiv v^{}_j$, the Majorana neutrino mass matrix is given by $$M^{}_\nu = \sum_j Y^{}_{\Delta^{}_j} v^{}_j \; , $$ where the smallness of $v^{}_j$ is attributed to the largeness of the triplet scalar mass scale [@SS2]. It is worth mentioning that an equality between two matrix elements cannot be achieved by imposing any Abelian symmetries on the Lagrangian in Eq. (67), since the Abelian symmetry group has only one-dimensional irreducible representations and the Yukawa couplings for different representations are not necessarily the same. In this work, we just take the pattern ${\bf A}^{}_2$ as a typical example, i.e., $$M^{{\bf A}^{}_2}_\nu = \left(\matrix{ 0 & a & b \cr a & d & c \cr b & c & d }\right) \;. $$ In order to ensure $(M^{}_\nu)^{}_{ee} = 0$, we have to implement a $Z^{}_3$ symmetry. Furthermore, to guarantee $(M^{}_\nu)^{}_{\mu \mu} = (M^{}_\nu)^{}_{\tau \tau}$, we can make use of an $S^{}_3$ symmetry, which is the simplest discrete non-Abelian group. In the framework of type-II seesaw model, we find that at least five scalar triplets $\Delta^{}_i$ (for $i=1,2,3,4,5$) are necessary. The assignments of the scalar triplets and lepton doublets under the $Z^{}_3$ symmetry are given as follows: $$\begin{aligned} \ell^{}_{e {\rm L}} \sim \omega \; , ~~~~~ \ell^{}_{\mu {\rm L}},~ \ell^{}_{\tau {\rm L}} \sim \omega^2 \; , ~~~~~ \Delta^{}_1,~ \Delta^{}_2 \sim 1 \; , ~~~~~ \Delta^{}_3, ~\Delta^{}_4, ~\Delta^{}_5 \sim \omega \; ; $$ while the assignments under the $S^{}_3$ symmetry are $$\begin{aligned} \ell^{}_{e {\rm L}} \sim {\bf 1} \; , ~~~~~ \left(\matrix{ \ell^{}_{\mu {\rm L}} \cr ~ \cr \ell^{}_{\tau {\rm L}}} \right) \sim {\bf 2} \; , ~~~~~ \left(\matrix{ \Delta^{}_1 \cr ~ \cr \Delta^{}_2} \right) \sim {\bf 2} \; , ~~~~~ \left(\matrix{ \Delta^{}_3 \cr ~ \cr \Delta^{}_4} \right) \sim {\bf 2} \; , ~~~~~ \Delta^{}_5 \sim {\bf 1} \; . $$ Hence the $S^{}_3 \otimes Z^{}_3$-invariant Lagrangian relevant for neutrino masses reads $$\begin{aligned} {\cal L}^{}_\nu &=& -\frac{1}{2} y^{}_1 \overline{\ell^{}_{e {\rm L}}} \left( \Delta^{}_1 \ell^c_{\mu {\rm L}} + \Delta^{}_2 \ell^c_{\tau {\rm L}} \right) -\frac{1}{2} y^{}_2 \left(\overline{\ell^{}_{\mu {\rm L}}} \Delta^{}_5 \ell^c_{\mu {\rm L}} + \overline{\ell^{}_{\tau {\rm L}}} \Delta^{}_5 \ell^c_{\tau {\rm L}}\right) \nonumber \\ &~& -\frac{1}{2} y^{}_3 \left[\left(\overline{\ell^{}_{\mu {\rm L}}} \Delta^{}_3 \ell^c_{\tau {\rm L}} + \overline{\ell^{}_{\tau {\rm L}}} \Delta^{}_3 \ell^c_{\mu {\rm L}}\right) + \left(\overline{\ell^{}_{\mu {\rm L}}} \Delta^{}_4 \ell^c_{\mu {\rm L}} - \overline{\ell^{}_{\tau {\rm L}}} \Delta^{}_4 \ell^c_{\tau {\rm L}}\right)\right] + {\rm h.c.} \; , $$ where we have used the tensor product ${\bf 2} \otimes {\bf 2} = {\bf 1} + {\bf 1}^\prime + {\bf 2}$ for the $S^{}_3$ symmetry group [@flavorsym]. After the triplet scalar acquires its vacuum expectation value $\langle \Delta^{}_i \rangle = v^{}_i$, the Majorana neutrino mass matrix is given by $$M^{}_\nu = \left(\matrix{ 0 & y^{}_1 v^{}_1 & y^{}_1 v^{}_2 \cr y^{}_1 v^{}_1 & y^{}_2 v^{}_5 + y^{}_3 v^{}_4 & 2y^{}_3 v^{}_3 \cr y^{}_1 v^{}_2 & 2y^{}_3 v^{}_3 & y^{}_2 v^{}_5 - y^{}_3 v^{}_4}\right) \; . $$ Once $v^{}_4 = 0$ is obtained by minimizing the scalar potential, we arrive at the pattern ${\bf A}^{}_2$ in Eq. (69), with $a = y^{}_1 v^{}_1$, $b = y^{}_1 v^{}_2$, $c = 2y^{}_3 v^{}_3$, and $d = y^{}_2 v^{}_5$. On the other hand, we have to ensure that the charged-lepton mass matrix $M^{}_l$ is diagonal. This can be achieved if one introduces two $SU(2)^{}_{\rm L}$ scalar doublets $\Phi^{}_1$ and $\Phi^{}_2$, in addition to the standard-model Higgs doublet $H$, which is a singlet under the flavor symmetry. These two extra scalar doublets and three right-handed charged-lepton fields are assigned as: $$e^{}_{\rm R} \sim \omega \; , ~~~ \mu^{}_{\rm R} \;, \tau^{}_{\rm R} \sim \omega^2 \; , ~~~ \Phi^{}_1, \Phi^{}_2 \sim 1 \; $$ under the $Z^{}_3$ symmetry, while $$e^{}_{\rm R}, \tau^{}_{\rm R} \sim {\bf 1} \; , ~~~ \mu^{}_{\rm R} \sim {\bf 1}^\prime \; , ~~~ \left(\matrix{ \Phi^{}_1 \cr ~ \cr \Phi^{}_2} \right) \sim {\bf 2} \; $$ under the $S^{}_3$ symmetry. Three lepton doublets transform in the same way as in Eqs. (70) and (71). Therefore, the $S^{}_3 \otimes Z^{}_3$-invariant Lagrangian relevant for the charged-lepton masses is $${\cal L}^{}_l = - y^{}_e \overline{\ell^{}_{e{\rm L}}} H e^{}_{\rm R} - y^{}_\mu \left(\overline{\ell^{}_{\mu {\rm L}}} \Phi^{}_2 - \overline{\ell^{}_{\tau {\rm L}}} \Phi^{}_1\right) \mu^{}_{\rm R} - y^{}_\tau \left(\overline{\ell^{}_{\mu {\rm L}}} \Phi^{}_1 + \overline{\ell^{}_{\tau {\rm L}}} \Phi^{}_2\right) \tau^{}_{\rm R} + {\rm h.c.} \; , $$ leading to a diagonal charged-lepton mass matrix $M^{}_l \equiv {\rm Diag}\{m^{}_e, m^{}_\mu, m^{}_\tau\}$ with three mass eigenvalues $m^{}_e = y^{}_e v/\sqrt{2}$, $m^{}_\mu = y^{}_\mu u/\sqrt{2}$, and $m^{}_\tau = y^{}_\tau u /\sqrt{2}$, where the vacuum expectation values of three scalar doublets $\langle H \rangle = v/\sqrt{2}$, $\langle \Phi^{}_2 \rangle = u/\sqrt{2}$ and $\langle \Phi^{}_1 \rangle = 0$ have been taken. Our numerical analysis has demonstrated that the hybrid texture in Eq. (69) is consistent with current neutrino oscillation data at the $3\sigma$ level. To complete the above $S^{}_3 \otimes Z^{}_3$ flavor model, we have to examine in detail the invariant scalar potential and check whether vacuum alignments $v^{}_1 \neq v^{}_2$, $v^{}_3 \neq v^{}_4$, and $\langle \Phi^{}_1 \rangle \neq \langle \Phi^{}_2 \rangle$ can be actually realized. For this purpose, one can work in the supersymmetric version of type-II seesaw model and follow the method of driving flavon fields, as proposed in Refs. [@Altarelli; @flavorsym1]. In addition, the symmetry realization of all the thirty-nine viable hybrid textures of $M^{}_\nu$ in a systematic way deserves further studies and will be discussed elsewhere. Summary ======= Motivated by the recent measurements of $\theta^{}_{13}$ in reactor neutrino experiments, we have performed a thorough study of the so-called hybrid textures of Majorana neutrino mass matrix $M^{}_\nu$, where one texture zero and one equality between two nonzero matrix elements are assumed. We have found that thirty-nine out of sixty possible patterns are compatible with current experimental data at the $3\sigma$ level. In the following, our main results are summarized: - If three neutrino mixing angles $(\theta^{}_{12}, \theta^{}_{23}, \theta^{}_{13})$ and two neutrino mass-squared differences $(\delta m^2, \Delta m^2)$ are given, the leptonic CP-violating phases $(\rho, \sigma, \delta)$ and three neutrino masses $(m^{}_1, m^{}_2, m^{}_3)$ can be fully determined. For illustration, we have chosen six hybrid textures ${\bf A}^{}_1$, ${\bf B}^{}_1$, ${\bf B}^{}_5$, ${\bf D}^{}_1$, ${\bf E}^{}_1$, and ${\bf E}^{}_8$, and derived the analytical approximate formulas for the CP-violating phases and neutrino masses. - In Figs. \[fig.A1\]–\[fig.E8\], we show the numerical results for the above six hybrid textures. Some interesting observations should be mentioned: (1) The allowed regions of $\theta^{}_{23}$ from the patterns ${\bf A}^{}_1$ and ${\bf B}^{}_5$ turn out to be well separated and deviate significantly from the maximal mixing. Therefore, if $\theta^{}_{23}$ is finally measured to be very close to $\pi/4$, then both ${\bf A}^{}_1$ and ${\bf B}^{}_5$ can be excluded. (2) Except the pattern ${\bf A}^{}_1$, all the hybrid textures under consideration have specific predictions for the CP-violating phase $\delta$. In addition, their predictions for the effective neutrino mass $\langle m \rangle^{}_{\rm ee}$ could also be quite different. (3) Now the cosmological bound on the absolute scale of neutrino masses becomes quite relevant for the study of lepton flavor structure. Moreover, we have considered the stability of texture zeros and equality against one-loop quantum corrections. In a type-II seesaw model, we illustrate how to realize the hybrid texture ${\bf A}^{}_2$ by implementing an $S^{}_3 \otimes Z^{}_3$ flavor symmetry. The ongoing and upcoming neutrino oscillation experiments are expected to precisely measure the neutrino mixing parameters, in particular the smallest mixing angle $\theta^{}_{13}$, the deviation of $\theta^{}_{23}$ from $\pi/4$ and the Dirac CP-violating phase $\delta$. The sensitivity of future cosmological observations to the sum of neutrino masses $\sum m^{}_i$ and the sensitivity of the neutrinoless double-beta decay experiments to the effective mass term $\langle m \rangle^{}_{\rm ee}$ will probably reach $\sim0.05~{\rm eV}$ in the near future. 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[**76**]{}, 056201 (2013). ![Pattern ${\bf A}^{}_1$ of $M^{}_\nu$: The allowed ranges of flavor mixing angles $(\theta^{}_{12}, \theta^{}_{23}, \theta^{}_{13})$ versus the Dirac CP-violating phase $\delta$ at the $3\sigma$ level, and the probability distribution of three angles, are given in the first and second rows, respectively. In the third and fourth rows, the predictions for three neutrino masses $(m^{}_1, m^{}_2, m^{}_3)$, the Majorana CP-violating phases $(\rho, \sigma)$ and the Jarlskog invariant $J^{}_{\rm CP}$ are shown with respect to the Dirac CP-violating phase $\delta$.[]{data-label="fig.A1"}](A1.eps){width="90.00000%"} ![Pattern ${\bf B}^{}_1$ of $M^{}_\nu$: The allowed ranges of flavor mixing angles $(\theta^{}_{12}, \theta^{}_{23}, \theta^{}_{13})$ versus the Dirac CP-violating phase $\delta$ at the $3\sigma$ level, and the probability distribution of three angles, are given in the first and second rows, respectively. In the third and fourth rows, the predictions for three neutrino masses $(m^{}_1, m^{}_2, m^{}_3)$ and the effective neutrino mass in neutrinoless double-beta decays $\langle m \rangle^{}_{\rm ee}$, the Majorana CP-violating phases $(\rho, \sigma)$ and the Jarlskog invariant $J^{}_{\rm CP}$, are shown with respect to the Dirac CP-violating phase $\delta$.[]{data-label="fig.B1"}](B1.eps){width="90.00000%"} ![Pattern ${\bf B}^{}_5$ of $M^{}_\nu$: The allowed ranges of flavor mixing angles $(\theta^{}_{12}, \theta^{}_{23}, \theta^{}_{13})$ versus the Dirac CP-violating phase $\delta$ at the $3\sigma$ level, and the probability distribution of three angles, are given in the first and second rows, respectively. In the third and fourth rows, the predictions for three neutrino masses $(m^{}_1, m^{}_2, m^{}_3)$ and the effective neutrino mass in the neutrinoless double-beta decays $\langle m \rangle^{}_{\rm ee}$, the Majorana CP-violating phases $(\rho, \sigma)$ and the Jarlskog invariant $J^{}_{\rm CP}$, are shown with respect to the Dirac CP-violating phase $\delta$.[]{data-label="fig.B6"}](B5.eps){width="90.00000%"} ![Pattern ${\bf D}^{}_1$ of $M^{}_\nu$: The allowed ranges of flavor mixing angles $(\theta^{}_{12}, \theta^{}_{23}, \theta^{}_{13})$ versus the Dirac CP-violating phase $\delta$ at the $3\sigma$ level, and the probability distribution of three angles, are given in the first and second rows, respectively. In the third and fourth rows, the predictions for three neutrino masses $(m^{}_1, m^{}_2, m^{}_3)$ and the effective neutrino mass in the neutrinoless double-beta decays $\langle m \rangle^{}_{\rm ee}$, the Majorana CP-violating phases $(\rho, \sigma)$ and the Jarlskog invariant $J^{}_{\rm CP}$, are shown with respect to the Dirac CP-violating phase $\delta$.[]{data-label="fig.D1"}](D1.eps){width="90.00000%"} ![Pattern ${\bf E}^{}_1$ of $M^{}_\nu$: The allowed ranges of flavor mixing angles $(\theta^{}_{12}, \theta^{}_{23}, \theta^{}_{13})$ versus the Dirac CP-violating phase $\delta$ at the $3\sigma$ level, and the probability distribution of three angles, are given in the first and second rows, respectively. In the third and fourth rows, the predictions for three neutrino masses $(m^{}_1, m^{}_2, m^{}_3)$ and the effective neutrino mass in the neutrinoless double-beta decays $\langle m \rangle^{}_{\rm ee}$, the Majorana CP-violating phases $(\rho, \sigma)$ and the Jarlskog invariant $J^{}_{\rm CP}$, are shown with respect to the Dirac CP-violating phase $\delta$.[]{data-label="fig.E1"}](E1.eps){width="90.00000%"} ![Pattern ${\bf E}^{}_8$ of $M^{}_\nu$: The allowed ranges of flavor mixing angles $(\theta^{}_{12}, \theta^{}_{23}, \theta^{}_{13})$ versus the Dirac CP-violating phase $\delta$ at the $3\sigma$ level, and the probability distribution of three angles, are given in the first and second rows, respectively. In the third and fourth rows, the predictions for three neutrino masses $(m^{}_1, m^{}_2, m^{}_3)$ and the effective neutrino mass in the neutrinoless double-beta decays $\langle m \rangle^{}_{\rm ee}$, the Majorana CP-violating phases $(\rho, \sigma)$ and the Jarlskog invariant $J^{}_{\rm CP}$, are shown with respect to the Dirac CP-violating phase $\delta$.[]{data-label="fig.E8"}](E8.eps){width="90.00000%"} [^1]: E-mail: liujy@tjut.edu.cn [^2]: E-mail: shunzhou@kth.se [^3]: The global-fit analysis of current neutrino oscillation experiments has also been performed by two other groups [@Schwetz; @Valle]. Although their best-fit results of three flavor mixing angles are slightly different from those obtained in Ref. [@Fogli], such differences become insignificant at the $3\sigma$ level.

--- abstract: 'Motivated by station-keeping applications in various unmanned settings, this paper introduces a steering control law for a pair of agents operating in the vicinity of a fixed beacon in a three-dimensional environment. This feedback law is a modification of the previously studied three-dimensional constant bearing (CB) pursuit law, in the sense that it incorporates an additional term to allocate attention to the beacon. We investigate the behavior of the closed-loop dynamics for a two-agent *mutual pursuit* system in which each agent employs the beacon-referenced CB pursuit law with regards to the other agent and a stationary beacon. Under certain assumptions on the associated control parameters, we demonstrate that this problem admits *circling equilibria* wherein the agents move on circular orbits with a common radius, in planes perpendicular to a common axis passing through the beacon. As the common radius and distances from the beacon are determined by choice of parameters in the feedback law, this approach provides a means to engineer desired formations in a three-dimensional setting.' author: - 'Kevin S. Galloway$^{\dagger}$ and Biswadip Dey$^{\ddagger}$[^1] [^2] [^3]' bibliography: - 'Draft\_Refs.bib' title: '**Beacon-referenced Mutual Pursuit in Three Dimensions** ' --- Cooperative control; Multi-agent systems; Pursuit problems; Autonomous mobile robots Introduction {#sec:Intro} ============ As pursuit and collective motion play significant roles in various contexts of robotics and engineering, it seems appealing to seek inspiration from nature, which abounds with many such examples. various possible ways to pursue and intercept a moving target, evidence of constant bearing (CB) pursuit strategy can be in a variety of animal species (e.g. dogs [@Shaffer_Dog_2004], flies [@Collett1975; @Osorio281], humans [@CHARDENON200213], raptors [@Tucker3745]). The CB pursuit strategy dictates that an agent should move towards its target in such a way that the angle between the baseline (alternatively known as the line-of-sight) connecting the two individuals and its own velocity remains constant. By prescribing a fixed offset between the baseline and the pursuer’s velocity, this strategy provides further generalization of the classical pursuit strategy (wherein the pursuer always moves directly towards the current location of its target). The work in [@Pavone_ASME_commonOffset] exploited this pursuit strategy as a building block for designing formations in an engineered setting, and demonstrated that, by applying a homogeneous[^4] CB pursuit strategy in a cyclic manner, a collective of agents eventually converge to a common rendezvous point, a circular formation or a logarithmic spiral pattern. Another line of work [@Kevin_2011_CDC; @Galloway_PRS_13; @Galloway_PRS_16] explored a more general setting wherein individual agents pursue each other using a heterogeneous CB pursuit strategy, and has shown existence and stability of a richer class of behaviors (circular motion, rectilinear motion, shape preserving spirals and periodic orbits). While a majority of the prior research has considered only planar settings to investigate CB pursuit strategy in a cyclic interaction, the work in [@CP_3D_Kevin_2010; @Galloway_PRS_16] considers the three-dimensional setting as well. While this line of research has demonstrated existence of circling equilibria in which agents moved on a common circular trajectory, both the location of the circumcenter of the formation (with respect to some inertial frame) and its size were determined by initial conditions rather than control parameters. To overcome this aspect and to broaden its scope from a design perspective, we introduced a modified version of the CB control law in our earlier work [@KSG_BD_ACC_2015; @KSG_BD_ACC_2016; @KSG_BD_arXiv_17]. In this new setting, the pursuer pays attention to a beacon (which can represent an attractive food source in a biological setting, or some target of interest for an unmanned vehicle), in addition to its neighbor. Another line of work [@Mallik2015ConsensusApplications; @Daingade2015AImplementation] has also investigated this beacon-referenced (or target-centric) cyclic pursuit framework, albeit their work uses a different control formulation. In the current work, we extend this beacon-referenced approach to the three dimensional-setting. We first introduce a beacon-referenced version of the CB pursuit law in three dimensions, and then the mutual pursuit scenario in which two agents apply this pursuit law to one another as well as a stationary beacon. Earlier work [@MISCHIATI20114483; @MISCHIATI2012894; @UH_BD_ICRA_15] has demonstrated that mutual pursuit can lead to a variety of interesting motion patterns, while providing better tractability from an nonlinear analysis perspective, and it can be viewed as a building block towards understanding the more general cyclic pursuit framework. This paper is organized as follows. We begin by describing the self-steering particle model for agents moving in the three dimensions. Then, in the later part of Section \[sec:Prob\_SetUp\], we introduce the beacon-referenced constant bearing pursuit law. As the underlying self-steering particle model has a 1-dimensional rotational invariance, we can describe the evolution of the mutual pursuit system by considering a reduced system evolving on $\mathds{R}^3 \times \mathds{R}^3 \times \mathcal{S}^2 \times \mathcal{S}^2$. In the rest of Section \[sec:Closed-Loop-Dyna\], we define the effective shape space, identify the geometric constraints associated with these variables, and derive the closed loop dynamics after making some simplifying assumptions about the control parameters. In Section \[sec:Rel\_Equilibrium\], we analyze the closed loop shape dynamics, and explore existence conditions and characterization of the associated relative equilibria. Finally we conclude in Section \[sec:Conclusion\]. Modeling Mutual Interactions {#sec:Prob_SetUp} ============================ Generative Model: Agents as Self-Steering Particles --------------------------------------------------- Similar to earlier works ([@Justh_PSK_3Dformation; @CP_3D_Kevin_2010]), we treat the agents as unit-mass self-steering particles moving along twice-differentiable paths in a three-dimensional environment. This allows us to describe the motion of an agent in terms of its natural Frenet frame [@Nat_Frenet_Bishop], defined by its position $\mathbf{r}_i$ (with respect to an inertial reference frame) and an orthonormal triad of vectors $[\mathbf{x}_i,\mathbf{y}_i,\mathbf{z}_i]$. Then, by constraining the agents to move at equal and nonvanishing speed, we can assume without loss of generality that the agents are moving at unit speed, and express the dynamics of a pair of agents as $$\begin{aligned} \dot{\mathbf{r}}_i &= \mathbf{x}_i \\ \dot{\mathbf{x}}_i &= u_i \mathbf{y}_i + v_i \mathbf{z}_i \\ \dot{\mathbf{y}}_i &= - u_i \mathbf{x}_i \\ \dot{\mathbf{z}}_i &= - u_i \mathbf{x}_i, \end{aligned} \label{Explicit_MODEL}$$ for $i=1,2$. Here, $u_i$ and $v_i$ are the natural curvatures viewed as gyroscopic steering controls. Moreover, we assume that the *beacon* is located at position $\mathbf{r}_b \in \mathds{R}^3$. Then it directly follows that $\mathcal{M}_c = SE(3) \times SE(3) \times \mathds{R}^3$ defines the underlying *configuration space* of dimension 15. However, as we are only interested in the agents’ motion relative to each other and to the beacon, we can formulate a reduction to the 9-dimensional *shape space*, defined as $\mathcal{M}_s = \mathcal{M}_c / SE(3)$. Similar to the scalar shape variables employed in the planar case [@KSG_BD_ACC_2015], we can define the following set of scalar variables (for $i=1,2$) $$\begin{aligned} & \bar{x}_{i} \triangleq \mathbf{x}_i \cdot \frac{\mathbf{r}_{i,i+1}}{|\mathbf{r}_{i,i+1}|}, && \bar{y}_{i} \triangleq \mathbf{y}_i \cdot \frac{\mathbf{r}_{i,i+1}}{|\mathbf{r}_{i,i+1}|}, && \bar{z}_{i} \triangleq \mathbf{z}_i \cdot \frac{\mathbf{r}_{i,i+1}}{|\mathbf{r}_{i,i+1}|}, \\ & \bar{x}_{ib} \triangleq \mathbf{x}_i \cdot \frac{\mathbf{r}_{ib}}{|\mathbf{r}_{ib}|}, && \bar{y}_{ib} \triangleq \mathbf{y}_i \cdot \frac{\mathbf{r}_{ib}}{|\mathbf{r}_{ib}|}, && \bar{z}_{ib} \triangleq \mathbf{z}_i \cdot \frac{\mathbf{r}_{ib}}{|\mathbf{r}_{ib}|}, \\ & \rho_{ib} \triangleq |\mathbf{r}_{ib}|, && \rho \triangleq |\mathbf{r}_{12}|, && \tilde{x} \triangleq \mathbf{x}_1 \cdot \mathbf{x}_2 \end{aligned}$$ to parametrize the shape space $\mathcal{M}_s$. (See Fig. \[Scalar\_Shapes\].) Here, $\mathbf{r}_{ij} = \mathbf{r}_i - \mathbf{r}_j$, $i,j \in \{1,2\}$ represents the position of agent $i$ relative to agent $j$, $\mathbf{r}_{1b}$ and $\mathbf{r}_{2b}$ represent the positions of agent 1 and 2 relative to the beacon located at $\mathbf{r}_b$, respectively, and addition in the index variables should be interpreted modulo $2$. (This convention will be employed throughout this work.) Clearly, these variables overparameterize the underlying shape space. However, this overparameterization can be taken into account by considering the appropriate constraints (*e.g.* $\bar{x}_i^2 + \bar{y}_i^2 + \bar{z}_i^2 = 1$, $i = 1,2$). In what follows, we will prescribe that the agents should not be collocated with each other or with the beacon, i.e. we assume $\rho>0$, $\rho_{1b}>0$, and $\rho_{2b}>0$. These assumptions are made to keep the pursuit laws well-defined, but are not necessarily enforced by the closed-loop system dynamics. {height="2in"} \[Scalar\_Shapes\] Beacon-referenced Constant Bearing Pursuit in Three Dimensions -------------------------------------------------------------- Similar to our previous work for the planar setting [@KSG_BD_ACC_2015], we construct this feedback law as a convex combination of two fundamental building blocks, expressed as $$\begin{aligned} u_i &= (1 - \lambda)u_i^{CB} + \lambda u_i^B \\ v_i &= (1 - \lambda)v_i^{CB} + \lambda v_i^B \end{aligned} \label{Steering_Feedback_top_level}$$ for $i = 1,2$, where $\lambda \in [0,1]$ maintains a balance between the influence of the beacon and that of the neighboring agent. In this feedback law , $u_i^{CB}$, $v_i^{CB}$ are governed by the original CB pursuit law [@CP_3D_Kevin_2010], and $u_i^B$, $v_i^B$ represent the deviation from a desired bearing toward the beacon, In particular, by letting $\mu_i > 0$ denote a positive control gain, we choose $$\begin{aligned} u_i^{CB} &= -\mu_i (\bar{x}_i - a_i)\bar{y}_i \nonumber \\ & \qquad - \frac{1}{\left|{\bf r}_{i,i+1}\right|}\left[{\bf z}_i \cdot \left(\dot{\bf r}_{i,i+1} \times {\frac{{\bf r}_{i,i+1}}{\left|{\bf r}_{i,i+1}\right|}}\right) \right] \label{u_CB_i} \\ v_i^{CB} &= -\mu_i (\bar{x}_i - a_i)\bar{z}_i \nonumber \\ & \qquad + \frac{1}{\left|{\bf r}_{i,i+1}\right|}\left[{\bf y}_i \cdot \left(\dot{\bf r}_{i,i+1} \times {\frac{{\bf r}_{i,i+1}}{\left|{\bf r}_{i,i+1}\right|}}\right) \right], \label{v_CB_i}\end{aligned}$$ where the parameter $a_i \in [-1,1]$ represents the desired offset between the heading of agent $i$ and its bearing toward agent $(i+1)$. We choose the beacon tracking component as $$\begin{aligned} u_i^{B} &= -\mu_i^b (\bar{x}_{ib} - a_{ib})\bar{y}_{ib} \label{u_Beacon_i} \\ v_i^{B} &= -\mu_i^b (\bar{x}_{ib} - a_{ib})\bar{z}_{ib}, \label{v_Beacon_i}\end{aligned}$$ where $\mu_i^b > 0$ is the corresponding control gain and the parameter $a_{ib} \in [-1,1]$ represents the desired offset between the heading of agent $i$ and its bearing toward the beacon. In general, the neighbor- tracking goal may conflict with the beacon-referencing goal, Closed Loop Shape Dynamics {#sec:Closed-Loop-Dyna} ========================== In [@Justh_PSK_3Dformation], the authors have demonstrated the importance of considering a reduced system evolving on $\mathds{R}^3 \times \mathcal{S}^2 \times \mathcal{S}^2$ for analyzing certain types of two-agent systems with their dynamics defined on $SE(3) \times SE(3)$. Before delving into further analysis, we investigate similar aspects for the system under consideration, and show existence of a corresponding *reduced space*. We begin by computing $$\begin{aligned} \dot{\mathbf{x}}_i &= (1 - \lambda)\Big[ u_i^{CB}\mathbf{y}_i + v_i^{CB}\mathbf{z}_i \Big] + \lambda \Big[ u_i^B\mathbf{y}_i + v_i^B\mathbf{z}_i\Big] \nonumber \\ &= -(1 - \lambda)\mu_i (\bar{x}_i - a_i)\Big[ \bar{y}_i\mathbf{y}_i + \bar{z}_i\mathbf{z}_i \Big] \nonumber \\ & \qquad -\lambda\mu_i^b (\bar{x}_{ib} - a_{ib}) \Big[ \bar{y}_{ib}\mathbf{y}_i + \bar{z}_{ib}\mathbf{z}_i\Big] \nonumber \\ & \qquad - \frac{(1 - \lambda)}{\left|{\bf r}_{i,i+1}\right|} \left[\left({\bf z}_i \cdot \left(\dot{\bf r}_{i,i+1} \times {\frac{{\bf r}_{i,i+1}}{\left|{\bf r}_{i,i+1}\right|}}\right) \right)\mathbf{y}_i \right. \nonumber \\ & \qquad \qquad - \left.\left({\bf y}_i \cdot \left(\dot{\bf r}_{i,i+1} \times {\frac{{\bf r}_{i,i+1}}{\left|{\bf r}_{i,i+1}\right|}}\right) \right)\mathbf{z}_i\right] \label{lat_Accln}\end{aligned}$$ for $i \in \{1,2\}$. Then by using the *BAC-CAB* identity of vector algebra, we can express as $$\begin{aligned} \dot{\mathbf{x}}_i &= -(1 - \lambda)\mu_i (\bar{x}_i - a_i) \left[ \frac{{\bf r}_{i,i+1}}{\left|{\bf r}_{i,i+1}\right|} - \bar{x}_i\mathbf{x}_i \right] \nonumber \\ & \qquad -\lambda\mu_i^b (\bar{x}_{ib} - a_{ib}) \left[ \frac{{\bf r}_{ib}}{\left|{\bf r}_{ib}\right|} - \bar{x}_{ib}\mathbf{x}_i \right] \nonumber \\ & \qquad + \frac{(1 - \lambda)}{\left|{\bf r}_{i,i+1}\right|} \left[ {\bf x}_i \times \left(\dot{\bf r}_{i,i+1} \times {\frac{{\bf r}_{i,i+1}}{\left|{\bf r}_{i,i+1}\right|}}\right) \right]. \label{lat_Accln_Self_Contained}\end{aligned}$$ {width="45.00000%"} \[Fig:Reduction\_of\_Spaces\] As $\dot{\mathbf{r}}_{ib} = \mathbf{x}_i$ and $\mathbf{r}_{i,i+1}$ can also be expressed as $({\bf r}_{i,b} - {\bf r}_{i+1,b})$, it directly follows from that the evolution of $(\mathbf{r}_{1b}, \mathbf{r}_{2b}, \mathbf{x}_1, \mathbf{x}_2)$ is governed by a self-contained dynamics on the *reduced space* $\mathcal{M}_r = \mathds{R}^3 \times \mathds{R}^3 \times \mathcal{S}^2 \times \mathcal{S}^2$ of dimension 10, . Then, after solving the evolution of this reduced dynamics, one can reconstruct the evolution of the complete frame $[\mathbf{x}_i, \mathbf{y}_i, \mathbf{z}_i]$ by using the rule of quadrature. With this observation, we focus on the reduced dynamics on $\mathcal{M}_r$, instead of the full dynamics defined on $\mathcal{M}_c$. Furthermore, the reduced dynamics on $\mathcal{M}_r$ is invariant to any rotation with respect to an inertial reference. This allows us to carry out further reduction, and focus our attention to a reduced system defined on the 7-dimensional *effective shape space* $\mathcal{M}_e$. As we will see in the later analysis, the following set of scalar variables provide an efficient parametrization of this effective shape space: $$\begin{aligned} & \bar{x}_{1} = \mathbf{x}_1 \cdot \frac{\mathbf{r}_{12}}{|\mathbf{r}_{12}|}, && \bar{x}_{2} = \mathbf{x}_2 \cdot \frac{\mathbf{r}_{21}}{|\mathbf{r}_{21}|}, && \tilde{x} = \mathbf{x}_1 \cdot \mathbf{x}_2 \\ & \bar{x}_{1b} = \mathbf{x}_1 \cdot \frac{\mathbf{r}_{1b}}{|\mathbf{r}_{1b}|}, && \bar{x}_{2b} = \mathbf{x}_2 \cdot \frac{\mathbf{r}_{2b}}{|\mathbf{r}_{2b}|} && \\ & \rho_{1b} = |\mathbf{r}_{1b}|, && \rho_{2b} = |\mathbf{r}_{2b}|, && \rho = |\mathbf{r}_{12}|. \end{aligned} \label{scalar_Variables_EffShape}$$ As we will see in the following subsection, these scalar variables are subject to appropriate constraints of codimension 1. Constraints on the Effective Shape Space Variables {#sec:Constraints} -------------------------------------------------- If the vectors $\mathbf{r}_{1b}$, $\mathbf{r}_{2b}$ and $\mathbf{r}_{12}$ are collinear, in addition to lying on the same plane (which directly follows from their definition), either of the following constraints shall hold true: $$\begin{aligned} \rho_{1b} + \rho_{2b} &= \rho \label{Constrain_1a} \\ \textrm{or,} \qquad |\rho_{1b} - \rho_{2b}| &= \rho. \label{Constrain_1b}\end{aligned}$$ However, even if they are not collinear, we can still exploit the fact that $\mathbf{r}_{1b} - \mathbf{r}_{2b} = \mathbf{r}_{12}$, and obtain the relationships $$\begin{aligned} \rho_{1b} \bar{x}_{1b} - \rho_{2b} \left( \mathbf{x}_1 \cdot \frac{\mathbf{r}_{2b}}{|\mathbf{r}_{2b}|} \right) &= \rho \bar{x}_1 \label{Constraints_2_1a} \\ \textrm{and,} \quad \rho_{1b} \left( \mathbf{x}_2 \cdot \frac{\mathbf{r}_{1b}}{|\mathbf{r}_{1b}|} \right) - \rho_{2b} \bar{x}_{2b} &= - \rho \bar{x}_2 \label{Constraints_2_1b}\end{aligned}$$ by taking their projections on the normalized velocities $\mathbf{x}_1$ and $\mathbf{x}_2$, respectively. As the dot-product of two unit vectors lies in the interval $[-1,1]$, - lead to the following inequality constraints: $$\begin{aligned} & - \rho_{2b} \leq \rho_{1b} \bar{x}_{1b} - \rho \bar{x}_1 \leq \rho_{2b} \label{Constraints_2a} \\ \textrm{and,} \quad & - \rho_{1b} \leq \rho_{2b} \bar{x}_{2b} - \rho \bar{x}_2 \leq \rho_{1b}. \label{Constraints_2b}\end{aligned}$$ In addition to these inequality constraints, we can also demonstrate that the underlying geometry leads to an additional constraint which poses restriction on the possible values of $\tilde{x}$ for some fixed values of $\bar{x}_{1}$, $\bar{x}_{2}$, $\bar{x}_{1b}$, $\bar{x}_{2b}$, $\rho$, $\rho_{1b}$ and $\rho_{2b}$, i.e. for the rest of the shape variables. As $\mathbf{r}_{1b}$, $\mathbf{r}_{2b}$ and $\mathbf{r}_{12}$ constitute a triangle, these three vectors lie on a plane. It readily follows that for a fixed value of $\bar{x}_{i}$, the normalized velocity vector $\mathbf{x}_{i}$ lies on a particular circle around $\mathbf{r}_{12}$ (or $\mathbf{r}_{21}$) which itself lies on the surface of a unit sphere. This circle is marked as $\mathcal{C}_i$ in Figure \[Fig:Constraints\]. In a similar way, a fixed value of $\bar{x}_{ib}$ forces $\mathbf{x}_{i}$ to lies on a particular circle around $\mathbf{r}_{ib}$ which itself lies on the surface of a unit sphere (shown as $\mathcal{C}_{ib}$ in Figure \[Fig:Constraints\]). Clearly, these two circles $\mathcal{C}_i$ and $\mathcal{C}_{ib}$ intersect two points $P_i'$ and $P_i''$. Furthermore, it can be shown that $P_i'$ and $P_i''$ are reflections of each other with respect to the plane containing $\mathbf{r}_{1b}$, $\mathbf{r}_{2b}$ and $\mathbf{r}_{12}$. As a consequence, $\tilde{x} = \mathbf{x}_1 \cdot \mathbf{x}_2$ can assume one out of only two possible values. In what follows, we will see that this constraint can be exploited in the analysis of the closed loop dynamics. {width="30.00000%"} \[Fig:Constraints\] Closed Loop Dynamics on the Effective Shape Space ------------------------------------------------- Before going into detailed analysis of the dynamics at hand, we introduce the following simplifying assumptions[^5]: - The controller gains ($\mu_i$ and $\mu_i^b$) are equal and common for both agents, i.e. $\mu_1 = \mu_2 = \mu_1^b = \mu_2^b = \mu$. - The bearing offset parameters with respect to the beacon are common for both agents, i.e. $a_{1b} = a_{2b} = a_0$. - The bearing offset parameters with respect to the other agent are the same for both agents, i.e. $a_1 = a_2 = a$. Under these three assumptions (A1)-(A3), the following set of self-contained equations describe the closed-loop shape dynamics : $$\begin{aligned} \text{\footnotesize{ $ \dot{\rho} $}} & \text{\footnotesize{ $ = \bar{x}_1 + \bar{x}_{2} $}} \label{eqn:2AgentShapeDynamics_rho} \\ \text{\footnotesize{ $ \dot{\rho}_{1b} $}} & \text{\footnotesize{ $ = \bar{x}_{1b} $}} \label{eqn:2AgentShapeDynamics_rho-1b} \\ \text{\footnotesize{ $ \dot{\rho}_{2b} $}} & \text{\footnotesize{ $ = \bar{x}_{2b} $}} \label{eqn:2AgentShapeDynamics_rho-2b} \\ \text{\footnotesize{ $ \dot{\bar{x}}_1 $}} & \text{\footnotesize{ $ = \frac{\lambda}{\rho} \Bigl(1 - \tilde{x} - \bar{x}_1^2 - \bar{x}_1 \bar{x}_{2} \Bigr) - (1-\lambda)\mu (\bar{x}_1 - a) \left(1 - \bar{x}_1^2 \right) $}} \nonumber \\ & \qquad \text{\footnotesize{ $ -\lambda\mu(\bar{x}_{1b}-a_0)\left(\frac{\rho_{1b}^2 +\rho^2 - \rho_{2b}^2 }{2\rho \rho_{1b}} - \bar{x}_{1b}\bar{x}_1 \right), $}} \label{eqn:2AgentShapeDynamics_x-1} \\ \text{\footnotesize{ $ \dot{\bar{x}}_2 $}} & \text{\footnotesize{ $ = \frac{\lambda}{\rho} \Bigl(1 - \tilde{x} - \bar{x}_2^2 - \bar{x}_1 \bar{x}_{2} \Bigr) - (1-\lambda)\mu (\bar{x}_2 - a) \left(1 - \bar{x}_2^2 \right) $}} \nonumber \\ & \qquad \text{\footnotesize{ $ -\lambda\mu(\bar{x}_{2b}-a_0)\left(\frac{\rho_{2b}^2 +\rho^2 - \rho_{1b}^2 }{2\rho \rho_{2b}} - \bar{x}_{2b}\bar{x}_2 \right), $}} \label{eqn:2AgentShapeDynamics_x-2} \\ \text{\footnotesize{ $ \dot{\bar{x}}_{1b} $}} & \text{\footnotesize{ $ = - (1-\lambda) \Bigl(\mu (\bar{x}_1 - a) + \frac{1 - \tilde{x}}{\rho}\Bigr) \left(\frac{\rho_{1b}^2 +\rho^2 - \rho_{2b}^2 }{2\rho \rho_{1b}} - \bar{x}_{1b}\bar{x}_1 \right) $}} \nonumber \\ & \qquad \text{\footnotesize{ $ - (1-\lambda) \frac{\bar{x}_1}{\rho} \Biggl(\left(\frac{\rho_{2b}}{\rho_{1b}}\right)\bar{x}_{2b} - \left(\frac{\rho}{\rho_{1b}}\right)\bar{x}_{2} - \bar{x}_{1b}\tilde{x} \Biggr) $}} \nonumber \\ & \qquad \text{\footnotesize{ $ -\left(\lambda\mu(\bar{x}_{1b}-a_0) - \frac{1}{\rho_{1b}}\right)\Bigl(1 - \bar{x}_{1b}^2 \Bigr), $}} \label{eqn:2AgentShapeDynamics_x-1b} \\ \text{\footnotesize{ $ \dot{\bar{x}}_{2b} $}} & \text{\footnotesize{ $ = - (1-\lambda)\Bigl(\mu (\bar{x}_2 - a) + \frac{1 - \tilde{x}}{\rho}\Bigr) \left(\frac{\rho_{2b}^2 +\rho^2 - \rho_{1b}^2 }{2\rho \rho_{2b}} - \bar{x}_{2b}\bar{x}_2 \right) $}} \nonumber \\ & \qquad \text{\footnotesize{ $ - (1-\lambda)\frac{\bar{x}_2}{\rho} \Biggl(\left(\frac{\rho_{1b}}{\rho_{2b}}\right)\bar{x}_{1b} - \left(\frac{\rho}{\rho_{2b}}\right)\bar{x}_{1} - \bar{x}_{2b}\tilde{x} \Biggr) $}} \nonumber \\ & \qquad \text{\footnotesize{ $ -\left(\lambda\mu(\bar{x}_{2b}-a_0) - \frac{1}{\rho_{2b}}\right)\Bigl(1 - \bar{x}_{2b}^2 \Bigr), $}} \label{eqn:2AgentShapeDynamics_x-2b} \\ \text{\footnotesize{ $ \dot{\tilde{x}} $}} & \text{\footnotesize{ $ = -\lambda\mu(\bar{x}_{2b}-a_0)\left(-\left(\frac{\rho}{\rho_{2b}}\right)\bar{x}_{1} + \left(\frac{\rho_{1b}}{\rho_{2b}}\right)\bar{x}_{1b} - \bar{x}_{2b}\tilde{x}\right) $}} \nonumber \\ & \qquad \text{\footnotesize{ $ -\lambda\mu(\bar{x}_{1b}-a_0)\left(\left(\frac{\rho_{2b}}{\rho_{1b}}\right)\bar{x}_{2b} + \left(\frac{\rho}{\rho_{1b}}\right)(-\bar{x}_{2}) - \bar{x}_{1b}\tilde{x}\right) $}} \nonumber \\ & \qquad \text{\footnotesize{ $ - (1-\lambda) \Biggl[ \left(\mu (\bar{x}_1 - a) + \frac{1 - \tilde{x}}{\rho} \right) \left(-\bar{x}_2 - \tilde{x}\bar{x}_1 \right) \Biggr. $}} \nonumber \\ & \qquad \; \text{\footnotesize{ $ \Biggl. + \bar{x}_1 \left(\frac{1-\tilde{x}^2}{\rho}\right) \Biggr] - (1-\lambda) \Biggl[ \bar{x}_2 \left(\frac{1-\tilde{x}^2}{\rho}\right) \Biggr. $}} \nonumber \\ & \qquad \; \text{\footnotesize{ $ \Biggl. + \left(\mu (\bar{x}_2 - a) + \frac{1 - \tilde{x}}{\rho} \right) \left(-\bar{x}_1 - \tilde{x}\bar{x}_2 \right) \Biggr] $}}. \label{eqn:2AgentShapeDynamics_x-tilde}\end{aligned}$$ Existence of Circling Equilibria {#sec:Rel_Equilibrium} ================================ The rest of this work is focused on determining conditions for existence of equilibria for the closed-loop dynamics -. These equilibria correspond to the agents moving on circular orbits with a common radius, in planes perpendicular to a common axis passing through the beacon, and therefore we will refer to them as *circling equilibria*. We proceed by setting $\dot{\rho} = \dot{\rho}_{1b} = \dot{\rho}_{2b} = 0$, which yields $$\begin{aligned} \bar{x}_2 = -\bar{x}_1, \quad \bar{x}_{1b} = 0 = \bar{x}_{2b}. \label{EquiliBr_x-ib_N_x-i}\end{aligned}$$ the circles $\mathcal{C}_{1b}$ and $\mathcal{C}_{2b}$ will be two great circles on the unit sphere, which intersect at two distinct antipodal points (see Figure \[Fig:Constraints\]). Moreover, the circles $\mathcal{C}_1$ and $\mathcal{C}_2$ coincide at every relative equilibrium. Substituting into - and simplifying, the closed loop dynamics on the nullclines $\dot{\rho} = \dot{\rho}_{1b} = \dot{\rho}_{2b} = 0$ can be expressed as $$\begin{aligned} \label{eqn:2AgentDynamicsAtEquilibriumCommonCBParameter} \dot{\bar{x}}_1 &= -(1-\lambda)\mu (\bar{x}_1 - a) \left(1 - \bar{x}_1^2 \right) \nonumber \\ & \qquad + \lambda\mu a_0\left(\frac{\rho_{1b}^2 +\rho^2 - \rho_{2b}^2 }{2\rho \rho_{1b}} \right) + \frac{\lambda}{\rho} \Bigl(1 - \tilde{x}\Bigr), \nonumber \\ \dot{\bar{x}}_2 &= -(1-\lambda)\mu (-\bar{x}_1 - a) \left(1 - \bar{x}_1^2 \right) \nonumber \\ & \qquad+ \lambda\mu a_0\left(\frac{-\rho_{1b}^2 +\rho^2 + \rho_{2b}^2 }{2\rho \rho_{2b}} \right) + \frac{\lambda}{\rho} \Bigl(1 - \tilde{x}\Bigr), \nonumber \\ \dot{\bar{x}}_{1b} &= -(1-\lambda)\Bigl(\mu (\bar{x}_1 - a) + \frac{1}{\rho}\left(1 - \tilde{x} \right)\Bigr)\left(\frac{\rho_{1b}^2 +\rho^2 - \rho_{2b}^2 }{2\rho \rho_{1b}} \right) \nonumber \\ & \qquad - \frac{(1-\lambda)\bar{x}_1^2}{\rho_{1b}} + \lambda\mu a_0 + \frac{1}{\rho_{1b}}, \nonumber \\ \dot{\bar{x}}_{2b} &= -(1-\lambda)\Bigl(\mu (-\bar{x}_1 - a) + \frac{1}{\rho}\left(1 - \tilde{x} \right)\Bigr)\left(\frac{\rho_{2b}^2 + \rho^2 - \rho_{1b}^2 }{2\rho \rho_{2b}} \right) \nonumber \\ & \qquad- \frac{(1-\lambda)\bar{x}_1^2}{\rho_{2b}} + \lambda\mu a_0 + \frac{1}{\rho_{2b}}, \nonumber \\ \dot{\tilde{x}} &= -\mu \bar{x}_1\Bigl(2(1-\lambda)\bar{x}_1 + \frac{\lambda a_0 \rho }{\rho_{1b}\rho_{2b}} \left(\rho_{2b}-\rho_{1b} \right)\Bigr). \end{aligned}$$ Then by narrowing our focus to the special case when $a_0 = 0$, we arrive at the following result. Consider a beacon-referenced mutual CB pursuit system with shape dynamics - parametrized by $\mu$, $\lambda$, and the CB . Then, a *circling equilibrium* exists if and only if $a<0$, and the corresponding equilibrium values $$\begin{aligned} \bullet \quad & \bar{x}_1 = \bar{x}_2 = 0, \quad \bar{x}_{1b} = \bar{x}_{2b} = 0, \quad \tilde{x} = -1, \\ \bullet \quad & \rho_{1b} = \rho_{2b}, \quad \rho = \frac{2\lambda }{(1-\lambda)\mu (-a)}. \end{aligned} \label{eqn:a0SameAlphaCircValues}$$ \[prop:existenceProp\_1\] In this case, it is clear that $\dot{\tilde{x}}=0$ if and only if $\bar{x}_1=0$, and therefore, from , we can conclude that the following conditions must hold true at an equilibrium $$\begin{aligned} (1-\lambda)\mu a + \frac{\lambda}{\rho} \Bigl(1 - \tilde{x}\Bigr) &= 0, \label{eqn:2AgentDynamicsAtEquilibriumCommonCBParameter_a0Zero__1} \\ (1-\lambda)\left( \frac{1 - \tilde{x}}{\rho} - \mu a \right)\left(\frac{\rho_{1b}^2 +\rho^2 - \rho_{2b}^2 }{2\rho \rho_{1b}} \right) &= \frac{1}{\rho_{1b}}, \label{eqn:2AgentDynamicsAtEquilibriumCommonCBParameter_a0Zero__2} \\ (1-\lambda)\left( \frac{1 - \tilde{x}}{\rho} -\mu a \right)\left(\frac{\rho_{2b}^2 +\rho^2 - \rho_{1b}^2 }{2\rho \rho_{2b}} \right) &= \frac{1}{\rho_{2b}}. \label{eqn:2AgentDynamicsAtEquilibriumCommonCBParameter_a0Zero__3}\end{aligned}$$ Now, if $\tilde{x} = 1$, the first condition holds true if and only if $a=0$. But with these choices for $\tilde{x}$ and $a$, the last two conditions - lead to $\frac{1}{\rho_{1b}} = \frac{1}{\rho_{2b}} = 0$, which cannot be true since both $\rho_{1b}$ and $\rho_{2b}$ are finite. Therefore we must have $\tilde{x} \neq 1$ at an equilibrium, and then the first condition yields the equilibrium value of $\rho$ as $$\rho = \frac{\lambda(1 - \tilde{x})}{(1-\lambda)\mu (-a)}. \label{eqn:rhoEquilibrium_a0zero}$$ As $\rho$ must be positive and finite, yields a meaningful solution if and only if $a < 0$. Substituting this solution for $\rho$ into -, we have $$\begin{aligned} \frac{1}{2 \rho_{1b}}\biggl[-\left(1 - \tilde{x} \right)\left(\frac{\rho_{1b}^2 - \rho_{2b}^2 }{\rho^2} \right) + 1 + \tilde{x} \biggr] &=0, \label{eqn:EQ_Final_Conditions_on_Nullcline__1} \\ \frac{1}{2 \rho_{2b}}\biggl[-\left(1 - \tilde{x} \right)\left(1 - \frac{\rho_{1b}^2 - \rho_{2b}^2 }{2\rho^2} \right) + 2 \biggr] &= 0. \label{eqn:EQ_Final_Conditions_on_Nullcline__2}\end{aligned}$$ Clearly holds true if and only if either of the following conditions hold:\ (I) $\displaystyle \rho_{1b} = \rho_{2b}$ and $\displaystyle \tilde{x} = -1$, or\ (II) $\displaystyle \rho_{1b} \neq \rho_{2b}$ and $\displaystyle \left(\frac{\rho_{1b}^2 - \rho_{2b}^2 }{\rho^2} \right) = \frac{1+\tilde{x}}{1-\tilde{x}}$, . Then it is straightforward to verify that the first set of conditions (I) satisfy . However, by substituting the second set of conditions (II) into , we have $$\begin{aligned} && -(1-\tilde{x}) \left(1 - \frac{1}{2}\left(\frac{1+\tilde{x}}{1-\tilde{x}}\right) \right) +2 &= 0 \nonumber \\ \Rightarrow && -(1-\tilde{x}) + \frac{1+\tilde{x}}{2} + 2 &= 0 \nonumber \\ \Rightarrow && \frac{3}{2}\left( \tilde{x} + 1 \right) &= 0,\end{aligned}$$ which is true if and only if $\tilde{x} = -1$. But, this contradicts the stated condition (II). Therefore this option is not viable, and (I) must hold true at an equilibrium. This concludes our proof. {width="45.00000%"} \[Fig:OffsetCircle\] We now shift our attention to the case where $a_0 \neq 0$, and show that circling equilibria exist in this scenario as well. Consider a beacon-referenced mutual CB pursuit system with shape dynamics - parametrized by $\mu$, $\lambda$, and the CB . The following statements are true.\ (a) Whenever $(1-\lambda) a + \lambda a_0 < 0$, a circling equilibrium exists, and the corresponding equilibrium values are given by $$\begin{aligned} \bullet \quad & \bar{x}_1 = \bar{x}_2 = 0, \quad \bar{x}_{1b} = \bar{x}_{2b} = 0, \quad \tilde{x} = -1, \\ \bullet \quad & \rho_{1b} = \rho_{2b} = \frac{\lambda}{-\mu \bigl((1-\lambda)a + \lambda a_0\bigr)}, \\ \bullet \quad & \rho = 2\rho_{1b} = \frac{2\lambda}{-\mu \bigl((1-\lambda)a + \lambda a_0\bigr)}. \end{aligned}$$\ (b) Whenever $a_0 < 0$, $a>0$, and $(1-\lambda) a + \lambda a_0 < 0$, a circling equilibrium exists, and the corresponding equilibrium values are given by $$\begin{aligned} \bullet \quad & \bar{x}_1 = \bar{x}_2 = 0, \quad \bar{x}_{1b} = \bar{x}_{2b} = 0, \quad \tilde{x} = 1, \\ \bullet \quad & \rho_{1b} = \rho_{2b} = \frac{\lambda a_0}{\mu\Bigl((1-\lambda)^2 a^2 - \lambda^2 a_0^2\Bigr)}, \\ \bullet \quad & \rho = \frac{-2(1-\lambda) a}{\mu\Bigl((1-\lambda)^2 a^2 - \lambda^2 a_0^2\Bigr)}. \end{aligned}$$ \[prop:existenceProp\_2\] It directly follows from that $\dot{\tilde{x}}=0$ if $\bar{x}_1 = 0$, and in that situation we can express the closed loop dynamics on the nullclines $\dot{\rho} = \dot{\rho}_{1b} = \dot{\rho}_{2b} = \dot{\tilde{x}}=0$ as $$\begin{aligned} \dot{\bar{x}}_1 &= (1-\lambda)\mu a + \lambda\mu a_0\left(\frac{\rho_{1b}^2 +\rho^2 - \rho_{2b}^2 }{2\rho \rho_{1b}} \right) + \frac{\lambda}{\rho} \Bigl(1 - \tilde{x}\Bigr), \nonumber \\ \dot{\bar{x}}_2 &= (1-\lambda)\mu a + \lambda\mu a_0\left(\frac{\rho_{2b}^2 +\rho^2 - \rho_{1b}^2 }{2\rho \rho_{2b}} \right) + \frac{\lambda}{\rho} \Bigl(1 - \tilde{x}\Bigr), \nonumber \\ \dot{\bar{x}}_{1b} &= (1-\lambda)\left( \mu a - \frac{1 - \tilde{x}}{\rho} \right) \left(\frac{\rho_{1b}^2 + \rho^2 - \rho_{2b}^2}{2\rho \rho_{1b}} \right) \nonumber \\ & \qquad + \lambda\mu a_0 + \frac{1}{\rho_{1b}}, \nonumber \\ \dot{\bar{x}}_{2b} &= (1-\lambda)\left( \mu a - \frac{1 - \tilde{x}}{\rho} \right) \left(\frac{\rho_{2b}^2 + \rho^2 - \rho_{1b}^2}{2\rho \rho_{2b}} \right) \nonumber \\ & \qquad + \lambda\mu a_0 + \frac{1}{\rho_{2b}}. \label{eqn:2AgentDynamicsAtEquilibriumCommonCBParameter_x1_zero}\end{aligned}$$ We note that taking the difference of $\dot{\bar{x}}_1-\dot{\bar{x}}_2$ yields $$\begin{aligned} \label{eqn:x1DotDiff} \dot{\bar{x}}_1-\dot{\bar{x}}_2 &= \lambda\mu a_0 \left( \frac{\rho_{1b}^2 +\rho^2 - \rho_{2b}^2 }{2\rho \rho_{1b}} - \frac{\rho_{2b}^2 +\rho^2 - \rho_{1b}^2 }{2\rho \rho_{2b}} \right) \nonumber \\ &= \frac{\lambda\mu a_0(\rho_{1b}-\rho_{2b}) }{2\rho\rho_{1b}\rho_{2b}} \Bigl(\left(\rho_{1b} + \rho_{2b} \right)^2 -\rho^2 \Bigr),\end{aligned}$$ and similar calculations lead to $$\begin{aligned} \label{eqn:x1bDotDiff} \dot{\bar{x}}_{1b}-\dot{\bar{x}}_{2b} &= (1-\lambda) \left( \mu a - \frac{1 - \tilde{x}}{\rho} \right) \left(\frac{\rho_{1b}-\rho_{2b}}{2\rho\rho_{1b}\rho_{2b}}\right) \nonumber \\ &\qquad \times \Bigl( (\rho_{1b} + \rho_{2b})^2 -\rho^2 \Bigr) + \frac{\rho_{2b}-\rho_{1b}}{\rho_{1b}\rho_{2b}}.\end{aligned}$$ Then by setting both and equal to zero, i.e. by setting the derivatives of $\bar{x}_2$ and $\bar{x}_{2b}$ identical to the derivatives of $\bar{x}_1$ and $\bar{x}_{1b}$ respectively, we can conclude that $\rho_{2b}$ must be equal to $\rho_{1b}$ at an equilibrium. Substituting this equivalence into , we can further conclude that the following conditions must hold true at an equilibrium $$\begin{aligned} (1-\lambda)\mu a + \lambda\mu a_0 \left( \frac{\rho}{2\rho_{1b}} \right) + \lambda \left(\frac{1 - \tilde{x}}{\rho} \right) &= 0, \label{eqn:equilibriumDynamicsTwoAgentA0NonZeroTerm_1} \\ (1-\lambda) \left(\mu a - \frac{1 - \tilde{x}}{\rho} \right) \left(\frac{\rho}{2 \rho_{1b}} \right) + \lambda\mu a_0 + \frac{1}{\rho_{1b}} &= 0. \label{eqn:equilibriumDynamicsTwoAgentA0NonZeroTerm_2}\end{aligned}$$ If $\tilde{x}=1$, then allows us to express $\rho$ as $$\rho = -2 \left(\frac{1-\lambda}{\lambda} \right)\left(\frac{a}{a_0} \right)\rho_{1b}. \label{Existence_4.2_Cond1}$$ As both $\rho$ and $\rho_{1b}$ must be positive, is meaningful if and only if $a/a_0 < 0$. Also, since $\rho_{1b} = \rho_{2b}$, substituting into constraint yields $$\begin{aligned} \left(\frac{1-\lambda}{\lambda}\right) \left(\frac{-a}{a_0}\right) < 1, \label{eqn:prop42partBconstraint}\end{aligned}$$ with the strict inequality resulting from the fact that we have assumed that the agents and beacon are not collinear. The combination of with $a/a_0 < 0$ yields two possibilities: - Case 1: $a_0>0$, $a<0$, $(1-\lambda) a + \lambda a_0 > 0$; - Case 2: $a_0<0$, $a>0$, $(1-\lambda) a + \lambda a_0 < 0$. Also, substitution of into leads to $$-\mu\left(\frac{(1-\lambda)^2}{\lambda} \right) \left(\frac{a^2}{a_0} \right) + \lambda\mu a_0 + \frac{1}{\rho_{1b}} = 0,$$ which in turn yields $$\begin{aligned} \rho_{1b} = \frac{1}{\mu\left(\frac{(1-\lambda)^2}{\lambda} \right) \left(\frac{a^2}{a_0} \right) - \lambda\mu a_0} \nonumber \\ = \frac{\lambda a_0}{\mu\Bigl((1-\lambda)^2 a^2 - \lambda^2 a_0^2\Bigr)}.\end{aligned}$$ This yields a meaningful solution if and only if $$a_0\Bigl((1-\lambda)^2 a^2 - \lambda^2 a_0^2\Bigr) > 0,$$ i.e. $$a_0\Bigl((1-\lambda) a - \lambda a_0\Bigr)\Bigl((1-\lambda) a + \lambda a_0\Bigr) > 0.$$ It is straightforward to verify that Case 2 (but not Case 1) satisfies this constraint, leading to the conditions of part (b) of the proposition. On the other hand, if $\tilde{x}=-1$, then - simplifies to $$\begin{aligned} \frac{1}{2\rho\rho_{1b}}\Bigl[2\rho_{1b}\bigl((1-\lambda)\mu a \rho + 2\lambda\bigr) + \lambda \mu a_0 \rho^2\Bigr] &= 0, \label{Existence_4.2_Cond2} \\ \frac{1}{2\rho_{1b}}\Bigl[(1-\lambda)\mu a \rho + 2\lambda + 2\lambda\mu a_0 \rho_{1b} \Bigr] &= 0. \label{Existence_4.2_Cond3}\end{aligned}$$ This completes the proof. {width="45.00000%"} \[Fig:CenteredCircle\] {width="45.00000%"} \[Fig:StackedCircle\] We note that Proposition \[prop:existenceProp\_2\] provides only sufficient (and not necessary) conditions for existence of circling equilibria in the case $a_0\neq 0$. This stems from the fact that there remains another possibility for $\dot{\tilde{x}}=0$ in , namely $\displaystyle \bar{x}_1 \neq 0$, $\displaystyle \rho_{1b} \neq \rho_{2b}$, and $\displaystyle \frac{\lambda a_0 \rho }{\rho_{1b}\rho_{2b}} \left(\rho_{2b}-\rho_{1b} \right) + 2(1-\lambda)\bar{x}_1 = 0$. In future work we will analyze this case to determine whether it presents a legitimate additional solution corresponding to circling equilibria. Conclusion {#sec:Conclusion} ========== In this paper we have presented a new control law which implements a beacon-referenced version of the CB control law. We have analyzed the 2-agent (with beacon) system and demonstrated the existence of circling equilibria for particular parameter choices. The circling equilibria obtained in this setting have a radius determined by control parameters rather than by initial conditions (as was the case for mutual CB pursuit without a beacon [@CP_3D_Kevin_2010]), which offers a method for designing circling trajectories with a desired diameter. Future work will focus on stability analysis for the special solutions presented here. Additional directions for research include exploration of the solution space for systems with $a_1 \neq a_2$, as well as analysis of the beacon-referenced cyclic pursuit system (i.e. $n>2$). [^1]: $^{\dagger}$Kevin S. Galloway is with the Electrical and Computer Engineering Department, United States Naval Academy, Annapolis, MD 21402 USA. [kgallowa@usna.edu]{} [^2]: $^{\ddagger}$Biswadip Dey is with the Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA. [biswadip@princeton.edu]{} [^3]: \*The second author’s research was supported in part by the Office of Naval Research under ONR grant N00014-14-1-0635. [^4]: *Homogeneous* in the sense that each individual attempts to maintain the same angular offset between its velocity and the baseline towards neighbor. [^5]:

--- abstract: 'We study thermoelectric devices where a single molecule is connected to metallic zigzag graphene nanoribbons (ZGNR) via highly transparent contacts that allow for injection of evanescent wave functions from ZGNRs into the molecular ring. Their overlap generates a peak in the electronic transmission, while ZGNRs additionally suppress hole-like contributions to the thermopower. Thus optimized thermopower, together with suppression of phonon transport through ZGNR-molecule-ZGNR structure, yield the thermoelectric figure of merit $ZT \sim 0.5$ at room temperature and $0.5 < ZT < 2.5$ below liquid nitrogen temperature. Using the nonequilibrium Green function formalism combined with density functional theory, recently extended to multiterminal devices, we show how the transmission resonance can also be manipulated by the voltage applied to a third ZGNR electrode, acting as the top gate covering molecular ring, to tune the value of $ZT$.' author: - 'Kamal K. Saha' - Troels Markussen - 'Kristian S. Thygesen' - 'Branislav K. Nikoli'' c' title: 'Multiterminal single-molecule–graphene-nanoribbon thermoelectric devices with gate-voltage tunable figure of merit $ZT$' --- =1 Thermoelectrics transform temperature gradients into electric voltage and vice versa. Although a plethora of thermoelectric-based energy conversion and cooling applications has been envisioned, their usage is presently limited by their small efficiency [@Vining2009]. Careful tradeoffs are needed to optimize the dimensionless figure of merit $ZT=S^2GT/\kappa$ quantifying the maximum efficiency of a thermoelectric conversion because $ZT$ contains unfavorable combination of the thermopower $S$, average temperature $T$, electrical conductance $G$ and the total thermal conductance $\kappa =\kappa_{\rm el} + \kappa_{\rm ph}$ (including contributions from electrons $\kappa_{\rm el}$ and phonons $\kappa_{\rm ph}$). The devices with $ZT > 1$ are regarded as good thermoelectrics, but $ZT > 3$ is required to compete with conventional generators [@Vining2009]. The major experimental efforts to increase $ZT$ have been focused on suppressing the phonon conductivity using either complex (through disorder in the unit cell) bulk materials [@Snyder2008] or bulk nanostructured materials [@Minnich2009]. A complementary approach engineers electronic density of states to obtain a sharp singularity [@Minnich2009] near the Fermi energy which can enhance the power factor $S^2G$. The very recent experiments [@Reddy2007] and theoretical studies [@Paulsson2003; @Ke2009; @Nozaki2010; @Entin-Wohlman2010; @Murphy2008] have ignited exploration of devices where a single molecule is attached to metallic [@Ke2009] or semiconducting [@Nozaki2010] electrodes, so that dimensionality reduction and possible strong electronic correlations [@Murphy2008] allow to increase $S$ concurrently with diminishing $\kappa_{\rm ph}$ while keeping the nanodevice disorder-free [@Markussen2009]. For example, creation of sharp transmission resonances near the Fermi energy $E_F$ by tuning the [*chemical properties*]{} of the molecule and molecule-electrode contact can substantially enhance the thermopower $S$ which depends on the derivative of the conductance near $E_F$. At the same time, the presence of a molecule in the electrode-molecule-electrode heterojunction severely disrupts phonon propagation when compared to homogenous clean electrode. ![(Color online) Schematic view of the proposed ZGNR$|$18-annulene$|$ZGNR three-terminal heterojunction. The contact between the source and drain 8-ZGNR (consisting of 8 zigzag chains) metallic electrodes and a ring-shaped molecule is made via 5-membered rings of carbon atoms (dark blue), while the electrodes are attached to atoms 1 and 10 of the molecule. The third electrode is coupled as an “air-bridge” top gate, made of ZGNR as well, covering only molecular ring at the distance . The two-terminal version of the device assumes that such top gate is absent. The hydrogen atoms (light yellow) are included to passivate the edge carbon atoms.[]{data-label="fig:setup"}](fig1) In this Letter, we exploit a transparent contact between metallic zigzag graphene nanoribbon (ZGNR) electrodes and a ring-shaped molecule to propose two-terminal (i.e., top gate absent in Fig. \[fig:setup\]) and three-terminal devices illustrated in Fig. \[fig:setup\]. Their thermoelectric properties are analyzed using the very recently developed nonequilibrium Green function combined with density functional theory formalism for multiterminal nanostructures (MT-NEGF-DFT) [@Saha2009a; @Saha2010]. The high contact transparency allows evanescent modes from the two ZGNR electrodes to tunnel into the molecular region and meet in the middle of it (when the molecule is short enough [@Ke2007]), which is a counterpart of the well-known metal induced gap states in metal-semiconductor Schottky junctions. This effect can induce a large peak (i.e., a resonance) in the electronic transmission function near $E_F$ \[Fig. \[fig:fig2\](a)\], despite the HOMO-LUMO energy gap of the isolated molecule. The enhancement of the thermopower \[Fig. \[fig:fig2\](b)\] due to transmission resonance around $E \ge E_F$ and suppression of hole-like contribution (i.e., negligible transmission around $E<E_F$) to $S$, together with several times smaller phonon thermal conductance (Fig. \[fig:fig3\]) when compared to infinite ZGNR, yields the maximum room-temperature $ZT \sim 0.5$ in the two-terminal device \[Fig. \[fig:fig2\](c),(d)\]. Furthermore, we discuss how a third top gate ZGNR electrode covering the molecule, while being separated by an air gap in Fig. \[fig:setup\], can tune the properties of the transmission resonance via the applied gate voltage thereby making possible further enhancement of $ZT$ (Fig. \[fig:fig4\]). ![(Color online) Physical quantities determining thermoelectric properties of the two-terminal device shown in Fig. \[fig:setup\] (without the top gate electrode): (a) zero-bias electronic transmission $\mathcal{T}_{\rm el}(E)$; (b) the corresponding thermopower $S$ at two different temperatures; (c) thermoelectric figure of merit $ZT$ vs. energy at two different temperatures; and (d) $ZT$ vs. temperature at three different energies.[]{data-label="fig:fig2"}](fig2) Among the recent theoretical studies of molecular thermoelectric devices via the NEGF-DFT framework [@Ke2009; @Nozaki2010], most have been focused [@Ke2009] on computing the thermopower $S$, with only a few [@Nozaki2010] utilizing DFT to obtain forces on displaced atoms and then compute $\kappa_{\rm ph}$. Moreover, due the lack of NEGF-DFT algorithms for [*multiterminal*]{} nanostructures, the possibility to tune thermoelectric properties of single-molecule devices via the usage of the third electrode has remained largely an unexplored realm [@Entin-Wohlman2010]. We note that the recent proposal [@Nozaki2010] for the two-terminal molecular thermoelectric devices with sophisticated combination of a local chemical tuning of the molecular states and usage of semiconducting electrodes has predicted much smaller $ZT \sim 0.1$ at room temperature. In addition, our $0.5<ZT<2.5$ at $E-E_F=-0.02$ eV (which can be set by the backgate electrode covering the whole device [@Zuev2009]) in the temperature range is much larger than the value achieved in conventional low-temperature bulk thermoelectric materials [@Snyder2008]. ![(Color online) (a) The phonon transmission function $\mathcal{T}_{\rm ph}(\omega)$ and (b) the corresponding thermal conductance $\kappa_{\rm ph}$ for an infinite 8-ZGNR and the two-terminal device shown in Fig. \[fig:setup\] (assuming absence of the third top gate electrode) whose source and drain electrodes are made of semi-infinite 8-ZGNR.[]{data-label="fig:fig3"}](fig3) The recent fabrication of GNRs with ultrasmooth edges [@Cai2010], where those with zigzag edges are insulating at very low temperatures due to edge magnetic ordering which is nevertheless easily destroyed above $\gtrsim 10$ K [@Yazyev2008], has opened new avenues for highly controllable molecular junctions with a well-defined molecule-electrode contact characterized by high transparency, strong directionality and reproducibility. This is due to the fact that strong molecule-GNR coupling makes possible formation of a continuous network across GNR and orbitals of conjugated organic molecules [@Ke2007]. Unlike the metallic carbon nanotubes (CNTs) employed experimentally [@Guo2006] to generate such networks [@Ke2007], GNRs have planar structure appropriate for aligning and patterning. The early experiments [@Guo2006] on CNT$|$molecule$|$CNT heterojunctions have measured surprisingly small conductances for a variety of sandwiched molecules. The first-principles analysis of different setups reveals that this is due to significant twisting forces when molecule is connected to CNT via, e.g., 6-membered rings [@Ke2007]. Therefore, to keep nearly parallel and in-plane configuration (hydrogen atoms of slightly deviate from the molecular plane) of our ZGNR$|$18-annulene$|$ZGNR junction, we use a 5-membered ring [@Ke2007] in Fig. \[fig:setup\] to chemically bond ZGNR to annulene. The carbon atoms of a ring-shaped molecule can be connected to ZGNR electrodes in configurations whose Feynman paths for electrons traveling around the ring generate either constructive or destructive quantum interference effects imprinted on the conductance [@Markussen2010]. For example, a $\pi$-electron at $E_F$ entering the molecule in setup (1,10) shown in Fig. \[fig:setup\] has the wavelength $k_F/2d$ ($d$ is the spacing between carbon atoms within the molecule), so that for the two simplest Feynman paths of length $9d$ (upper half of the ring) and $9d$ (lower half of the ring) the phase difference is 0. Note that the destructive quantum interference [@Markussen2010] would form an additional dip (i.e., antiresonance) within the main transmission peak in Fig. \[fig:fig2\](a) generated by the injection of overlapped evanescent modes [@Saha2010]. The effect of antiresonance on the thermopower $S$ for gold$|$18-annulene$|$gold junctions has been studied in Ref. [@Bergfield2009] as a possible sensitive tool to confirm the effects of quantum coherence on transport through single-molecule junctions. The computation of quantities entering $ZT$ for realistic single-molecule junctions requires quantum transport methods combined with [*first-principles*]{} input about atomistic and electronic structure to capture [*charge transfer*]{} in equilibrium (which is indispensable to obtain correct zero-bias transmission of, e.g., carbon-hydrogen systems [@Areshkin2010]), geometrically-optimized atomic positions of the molecular bridge including molecule-electrode separation in equilibrium, and forces on atoms when they are perturbed out of equilibrium. The state-of-the-art approach that can capture these effects, as long as the coupling between the molecule and the electrodes is strong enough to ensure transparent contact and diminish Coulomb blockade effects [@Murphy2008; @Cuniberti2005], is NEGF-DFT [@Cuniberti2005; @Areshkin2010]. The technical details of the construction of the nonequilibrium density matrix via NEGF-DFT for multiterminal devices are discussed in Ref. [@Saha2009a]. Our MT-NEGF-DFT code utilizes ultrasoft pseudopotentials and Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional. The localized basis set for DFT calculations is constructed from atom-centered orbitals (six per C atom and four per H atom) that are optimized variationally for the electrodes and the central molecule separately while their electronic structure is obtained concurrently. In the coherent transport regime, the NEGF post-processing of the result of the DFT self-consistent loop expresses the zero-bias electron transmission function between the left (L) and the right (R) electrodes as: $$\label{eq:telectron} \mathcal{T}_{\rm el}(E) = {\rm Tr} \left\{ {\bm \Gamma}_L (E) {\bf G}(E) {\bm \Gamma}_R (E) {\bf G}^\dagger(E) \right\}.$$ The matrices account for the level broadening due to the coupling to the electrodes, where ${\bm \Sigma}_{L,R}(E)$ are the self-energies introduced by the ZGNR electrodes [@Areshkin2010]. The retarded Green function matrix is given by ${\bf G}=[E{\bf S} - {\bf H} - {\bm \Sigma}_L - {\bm \Sigma}_R]^{-1}$, where in the local orbital basis $\{ \phi_i \}$ Hamiltonian matrix ${\bf H}$ is composed of elements $H_{ij} = \langle \phi_i |\hat{H}_{\rm KS}| \phi_{j} \rangle$ ($\hat{H}_{\rm KS}$ is the effective Kohn-Scham Hamiltonian obtained from the DFT self-consistent loop) and the overlap matrix ${\bf S}$ has elements $S_{ij} = \langle \phi_i | \phi_j \rangle$. The transmission function Eq. (\[eq:telectron\]) obtained within the NEGF-DFT framework allows us to compute the following integrals [@Esfarjani2006] $$\label{eq:kintegral} K_n(\mu) = \frac{2}{h} \int\limits_{-\infty}^{\infty} dE\, \mathcal{T}_{\rm el}(E) (E - \mu)^n \left(-\frac{\partial f(E,\mu)}{\partial E} \right),$$ where is the Fermi-Dirac distribution function at the chemical potential $\mu$. The knowledge of $K_n(\mu)$ finally yields all electronic quantities in the expression for $ZT$: $G=e^2K_0(\mu)$; $S=K_1(\mu)/[eTK_0(\mu)]$; and $\kappa_{\rm el} = \{K_2(\mu) - [K_1(\mu)]^2/K_0(\mu)\}/T$. The phonon thermal conductance is obtained from the phonon transmission function $\mathcal{T}_{\rm ph}(\omega)$ using the corresponding Landauer-type formula for the scattering region (molecule + portion of electrodes) attached to two semi-infinite electrodes: $$\label{eq:tphonon} \kappa_{\rm ph} = \frac{\hbar^2}{2\pi k_B T^2} \int\limits_{0}^{\infty} d\omega\, \omega^2 \mathcal{T}_{\rm ph}(\omega) \frac{ e^{\hbar\omega/k_BT}}{(e^{\hbar\omega/k_BT}-1)^2}.$$ The phonon transmission function $\mathcal{T}_{\rm ph} (\omega)$ can be calculated from the same Eq. (\[eq:telectron\]) with substitution ${\bf H} \rightarrow {\bf K}$ and ${\bf S} \rightarrow \omega^2 {\bf M}$, where ${\bf K}$ is the force constant matrix and ${\bf M}$ is a diagonal matrix with the atomic masses. We obtain the force constant matrix using GPAW, which is a real space electronic structure code based on the projector augmented wave method [@Enkovaara2010]. The electronic wave functions are expanded in atomic orbitals with a single-zeta polarized basis set, and PBE exchange-correlation functional is used. The whole scattering region, which includes 27 layers of ZGNR electrodes, is first relaxed to a maximum force of $0.01$ eV/Åper atom. Subsequently, we displace each atom $I$ by $Q_{I\alpha}$ in direction $\alpha=\{x,y,z\}$ to get the forces $F_{I\alpha,J\beta}$ on atom $J\neq I$ in direction $\beta$. The elements of ${\bf K}$ are then computed from finite differences, ${\bf K}_{I\alpha,J\beta} = [F_{J\beta}(Q_{I\alpha})-F_{J\beta}(-Q_{I\alpha})]/2Q_{I\alpha}$. The intra-atomic elements are calculated by imposing momentum conservation, such that $K_{I\alpha,I\beta} = -\Sigma_{J\neq I}K_{I\alpha,J\beta}$. ![(Color online) Physical quantities determining thermoelectric properties of the three-terminal device shown in Fig. \[fig:setup\] as a function of the applied gate voltage $V_g$: (a) zero-bias electronic transmission $\mathcal{T}_{\rm el}(E,V_g)$; (c) the corresponding thermopower $S$ at two different temperatures; (c) thermoelectric figure of merit $ZT$ vs. temperature at $V_g=0$; and (d) $ZT$ vs. $V_g$ at two different temperatures.[]{data-label="fig:fig4"}](fig4) Figure \[fig:fig2\](a) shows the zero-bias electronic transmission $\mathcal{T}_{\rm el}(E)$ for the two-terminal version of the device in Fig. \[fig:setup\], where the peak near $E_F$ is conspicuous. Additionally, the suppression of the hole-like transmission \[$\mathcal{T}_{\rm el}(E) \rightarrow 0$ around $E<E_F$\] evades unfavorable compensation [@Nozaki2010] of hole-like and electron-like contributions to the thermopower. This is due to the symmetry of the valence band propagating transverse mode in GNR semi-infinite electrode which changes sign at the two carbon atoms closest to the molecule for the geometry (molecule connected to the middle of the GNR edge) in Fig. \[fig:setup\]. We emphasize that these features of $\mathcal{T}_{\rm el}(E)$ are quite insensitive to the details of (short) conjugated molecules, and since they are governed by the ZGNR Bloch states, they are [*impervious*]{} to the usual poor estimates of the band gap size and molecular energy level position in DFT. The maximum value of the corresponding thermopower $S$ plotted in Fig. \[fig:fig2\](b) is slightly away from $E-E_F=0$ and it is an [*order of magnitude larger*]{} than the one measured on large-area graphene [@Zuev2009] or in molecular junctions with gold electrodes [@Reddy2007]. Moreover, the interruption of the infinite ZGNR by a molecule acts unfavorably to phonon transmission $\mathcal{T}_{\rm ph}(\omega)$, thereby generating three times smaller $\kappa_{\rm ph}$ at room temperature when compared in Fig. \[fig:fig3\] to the thermal conductance of an infinite 8-ZGNR. Figures \[fig:fig2\](c),(d) demonstrate that the interplay of large $S^2G$ and reduced $\kappa_{\rm ph}$ for the proposed two-terminal device yields the room-temperature $ZT \sim 0.5$ around $E-E_F=0$. The introduction of the narrow air-bridge top gate electrode in Fig. \[fig:setup\], which is positioned at the distance (ensuring negligible tunneling leakage current into such third ZGNR electrode) away from the two-terminal device underneath while covering only the molecular ring, makes possible tuning of the transmission resonance shown in Fig. \[fig:fig4\](a). Even in the absence of any applied gate voltage ($V_g=0$), $ZT$ vs. temperature plotted in Fig. \[fig:fig4\](c) is notably modified when compared to the corresponding functions in Fig. \[fig:fig2\](d) for the two-terminal device. This stems from slight hybridization of the top gate and molecular states. The narrowing of the transmission peak around $E_F$ due to the application of negative gate voltage enhances the thermopower in Fig. \[fig:fig4\](b), thereby increasing $ZT$ above its value at $V_g=0$, which can be substantial at low temperatures as shown in Fig. \[fig:fig4\](d). In conclusion, we predict that a single conjugated molecule attached to metallic GNR electrodes, where the transparent molecule-GNR contact allows evanescent modes to penetrate from the electrodes into the HOMO-LUMO molecular gap generating a transmission resonance, can act as an efficient thermoelectric device. Our first-principles simulations suggest that its figure of merit can reach $ZT \sim 0.5$ at room temperature or $0.5<ZT <2.5$ below liquid nitrogen temperature, which is much higher than $ZT$ found in other recent proposals for molecular thermoelectric devices [@Nozaki2010]. Moreover, introduction of the third air-bridge top gate covering the molecule can change the sharpness of the transmission resonance via the application of the gate voltage, thereby opening a path toward further optimization of $ZT$. 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--- abstract: 'Let $M$ be a $4k$-manifold whose Yamabe invariant $Y(M)$ is nonpositive. We show that $$Y(M\sharp\ l\ \Bbb HP^k\sharp \ m\ \overline{\Bbb HP^k})=Y(M),$$ where $l,m$ are nonnegative integers, and $\Bbb HP^k$ is a quaternionic projective space. When $k=4$, we also have $$Y(M\sharp\ l\ CaP^2\sharp \ m\ \overline{CaP^2})=Y(M),$$ where $CaP^2$ is a Cayley plane.' author: - | Chanyoung Sung[^1]\ National Institute for Mathematical Sciences\ 385-16 Doryong-dong Yuseong-gu Daejeon Korea title: | Connected sums with $\Bbb HP^{n}$ or $CaP^{2}$\ and the Yamabe invariant --- Introduction ============ The Yamabe invariant is an invariant of a smooth closed manifold defined using the scalar curvature. Let $M$ be a closed smooth $n$-manifold. By the well-known solution of the Yamabe problem, each conformal class of a smooth Riemannian metric on $M$ contains a so-called *Yamabe metric* which has constant scalar curvature. Moreover, letting $$[g]=\{\varphi g \mid \varphi:M \rightarrow \Bbb R^+ \ \textrm{is smooth} \}$$ be the conformal class of a Riemannian metric $g$, a Yamabe metric of $[g]$ actually realizes $$Y(M,[g]):=\inf_{\tilde{g} \in [g]} \frac{\int_M s_{\tilde{g}}\ dV_{\tilde{g}}}{(\int_M dV_{\tilde{g}})^{\frac{n-2}{n}}},$$ where $s_{\tilde{g}}$ and $dV_{\tilde{g}}$ respectively denote the scalar curvature and the volume element of $\tilde{g}$. The value $Y(M,[g])$, which is the value of the scalar curvature of a Yamabe metric with the total volume $1$ is the *Yamabe constant* of the conformal class. In a quest of a “best” Yamabe metric or more ambitiously a “canonical” metric on $M$, one naturally takes the supremum of the Yamabe constants over the set of all conformal classes on $M$. This is possible because by Aubin’s theorem [@aubin], the Yamabe constant of any conformal class on any $n$-manifold is always bounded by that of the unit $n$-sphere $S^n(1) \subset \Bbb R^{n+1}$, which is $n(n-1)(\textrm{Vol}(S^n(1)))^{2/n}$. The *Yamabe invariant* of $M$, $Y(M)$, is then defined as the supremum of the Yamabe constants over the set of all conformal classes on $M$. This supremum is not always attained, but if it is attained by a metric which is the unique Yamabe metric with total volume $1$ in its conformal class, then the metric has to be an Einstein metric.([@ander]) In general, one can hope a singular or degenerate Einstein metric leading to a kind of a “geometrization” from a maximizing sequence of Yamabe metrics. It is also noteworthy that the Yamabe invariant is a topological invariant of a closed manifold depending only on the smooth structure of the manifold. The Yamabe invariant of a compact orientable surfaces is just $4\pi \chi(M)$ where $\chi(M)$ denotes the Euler characteristic of $M$ by the Gauss-Bonnet theorem. In higher dimensions, it is not an easy task to compute the Yamabe invariant. Nevertheless recently there have been much progresses in dimension $3$ and $4$. In dimension $3$, the geometrization by the Ricci flow gives a lot of answers, and in dimension $4$, the Spin$^{c}$ structure and the Dirac operator are keys for computing the Yamabe invariant. In particular, LeBrun [@lb1; @lb3] showed that if $M$ is a compact Kähler surface whose Kodaira dimension is not equal to $- \infty$, then $$Y(M)=-4\sqrt{2}\pi\sqrt{(2\chi+3\sigma)(\tilde{M})},$$ where $\sigma$ denotes the signature and $\tilde{M}$ is the minimal model of $M$. Now based on this evidence, one can ask if the blowing-up does not change the Yamabe invariant of a closed orientable $4$-manifold with nonpostive Yamabe invariant, namely \[ques1\] Let $M$ be a closed orientable $4$-manifold with $Y(M)\leq 0$. Is there an orientation of $M$ such that $Y(M\sharp\ l\ \overline{\Bbb CP^{2}})=Y(M)$ for any integer $l>0$? What about in higher dimensions? Further one can also ask whether the analogous statement holds true for the “quaternionic blow-up”, i.e. a connected sum with a quaternionic projective space $\Bbb HP^{n}$, or even a connected sum with a Cayley plane $CaP^{2}$. The purpose of this paper is to prove an affirmative answer to this: \[th1\] Let $M$ be a closed $4k$-manifold with $Y(M)\leq 0$. Then $$Y(M\sharp\ l\ \Bbb HP^{k}\sharp \ m\ \overline{\Bbb HP^{k}})=Y(M),$$ where $l,m$ are nonnegative integers. When $k=4$, we also have $$Y(M\sharp\ l\ CaP^{2}\sharp \ m\ \overline{CaP^{2}})=Y(M).$$ Preliminaries ============= A computationally useful formula for the Yamabe constant is $$|Y(M,[g])|=\inf_{\tilde{g}\in [g]}(\int_{M} {|s_{\tilde{g}}|}^{\frac{n}{2}} d\mu_{\tilde{g}})^{\frac{2}{n}},$$ where the infimum is attained only by a Yamabe metric. (For a proof, see [@lb3; @sung3].) So when $Y(M,[g])\leq 0$, this implies that $$Y(M,[g]) = -\inf_{\tilde{g}\in [g]}(\int_{M} |s^{-}_{\tilde{g}}|^{\frac{n}{2}} d\mu_{\tilde{g}})^{\frac{2}{n}},$$ where $s_{g}^-$ is defined as $\min\{s_g,0\}$. Therefore when $Y(M)\leq 0$, $$\begin{aligned} \label{form1} Y(M)=-\inf_{g}(\int_{M} |s_{g}|^{\frac{n}{2}} d\mu_{g})^{\frac{2}{n}}=-\inf_{g}(\int_{M} |s_{g}^-|^{\frac{n}{2}} d\mu_{g})^{\frac{2}{n}}.\end{aligned}$$ Also essential is Kobayashi’s connected sum formula [@koba; @sung2]. $$Y(M_{1}\sharp M_{2}) \geq \left\{ \begin{array}{ll} -( |Y(M_1)|^{\frac{n}{2}}+ |Y(M_2)|^{\frac{n}{2}} )^{\frac{2}{n}} &\mbox{if } Y(M_i)\leq 0 \ \forall i\\ \min(Y(M_1),Y(M_2)) &\mbox{otherwise. } \end{array}\right.$$ We also need to know about the geometry and topology of $\Bbb HP^{k}$ and $CaP^{2}$. Both has the homogeneous Einstein metric of positive scalar curvature unique up to constant and can be viewed as the mapping cones of the (generalized) Hopf fibrations $\pi_{1}: S^{4k-1}\rightarrow \Bbb HP^{k-1}$ with $S^{3}$ fibers and $\pi_{2}: S^{15}\rightarrow S^{8}$ with $S^{7}$ fibers respectively. These fibrations have the associated geometries of Riemannian submersion with totally geodesic fibers. In case of $\pi_{1}$, $S^{4k-1}$ and $S^{3}$ are endowed with the round metric of constant curvature $1$, and $\Bbb HP^{k-1}$ is given the homogeneous Einstein metric with curvature ranging between $1$ and $4$. In case of $\pi_{2}$, the total space and the fibers have the round metric of curvature $1$, but the base has the round metric of curvature $4$. We will denote the round $n$-sphere with the metric of constant curvature $\frac{1}{a^{2}}$ by $S^{n}(a)$, i.e. the sphere of radius $a$ in the Euclidean $\Bbb R^{n+1}$. Proof of Theorem ================ It’s enough to prove for one connected sum. Let $M'$ be $M\sharp\ \Bbb HP^{k}$ or $M\sharp \ \overline{\Bbb HP^{k}}$, and set $n=4k$. First recall that $\Bbb HP^{k}$ admits a metric of positive scalar curvature meaning that $Y(\Bbb HP^k)>0$. Thus by the connected sum formula, $Y(M')\geq Y(M).$ The idea of the proof is to surger out an $\Bbb HP^{k-1}$ in $M'$ by performing the Gromov-Lawson surgery [@GL] to get back $M$ without decreasing the Yamabe constant much. To prove by contradiction, let’s assume $Y(M')>Y(M)+2c > Y(M)$ for a constant $c>0$ such that $c$ satisfies $c< \frac{|Y(M)|}{2}$ if $Y(M)<0$. Let $g$ be an unit-volume Yamabe metric on $M'$ such that $s_g\equiv Y(M',[g])= Y(M)+2c$. Let $W$ be an $\Bbb HP^{k-1}\subset \Bbb HP^{k}$ embedded in $M'$. Take a $\delta$-tubular neighborhood $N(\delta)=\{x\in M'|\ dist_{g}(x,W)< \delta\}$ of $W$ for $\delta>0$. We will take $\delta$ small enough so that $N(\delta)$ is diffeomorphic to $\Bbb HP^{k}- \{\textrm{a point}\}$ and the boundary of $N(\delta)$ is diffeomorphic to $S^{4k-1}$. We perform a Gromov-Lawson surgery described in [@sung1; @sung2] on $N(\delta)$ along $W$ keeping the scalar curvature bigger than $s_{g}-c$ to get a cylindrical end isometric to $(S^{4k-1}\times [0,1],\hat{g}+dt^2)$, where $(S^{4k-1},\hat{g})$ is a Riemannian submersion onto $(W,g_W=g|_{W})$ with totally geodesic fibers isometric to $S^{3}(\varepsilon)$, the round $3$-sphere of radius $\varepsilon\ll 1$. Here, the horizontal distribution is given by the connections on the normal bundle. By arranging $\varepsilon$ sufficiently small, $\hat{g}$ has positive scalar curvature. Moreover the volume of the deformed metric can be made arbitrarily small, say $\nu\ll 1$. (For a proof, one may refer to [@sung2]. Also a different method bypassing this is given in the remark below.) Now let’s take a homotopy $H_{b}(t)=\lambda(t)g_W+(1-\lambda(t))g_{std}$ of smooth metrics on $W$ from $g_W$ to the homogeneous Einstein metric $g_{std}$ of $\Bbb HP^{k-1}$ with curvature ranging from $1$ to $4$, where $\lambda:[0,1]\rightarrow [0,1]$ is a smooth decreasing function with the property that it is $1$ for $t$ near $0$ and $0$ near $1$. This induces a homotopy $H_{1}(t)$ of smooth metrics on $S^{4k-1}$ through a Riemannian submersion with totally geodesic fibers $S^3(\varepsilon)$. And then we homotope the horizontal distribution to that of the Hopf fibration through a Riemannian submersion with totally geodesic fibers $S^3(\varepsilon)$. Let’s denote this homotopy on $S^{4k-1}$ be $H_2(t)$ for $t\in [1,2]$. When $\varepsilon$ is sufficiently small, $H_{1}(t)+dt^2$ and $H_{2}(t)+dt^2$ will give a metric of positive scalar curvature on $S^{4k-1}\times [0,2]$, because it is a Riemannian submersion with totally geodesic fibers onto $\Bbb HP^{k-1}\times [0,2]$. We concatenate this part to the above one obtained from the Gromov-Lawson surgery to get a smooth metric with the boundary isometric to the squashed sphere $S^{4k-1}$ coming from the Hopf fibration. Let’s denote this metric on the boundary by $h_\varepsilon$ for a later purpose. We want to close it up by a $4k$-ball $B^{4k}$ equipped with a metric of positive scalar curvature. To construct such a metric we resort to the Gromov-Lawson surgery again. Take a sphere $S^{4k}$ with any metric of positive scalar curvature and let $p$ be any point on it. As before, we perform a Gromov-Lawson surgery in a sufficiently small neighborhood of $p$ to get a $4k$-ball with the positive scalar curvature and the cylindrical end isometric to $S^{4k-1}(\varepsilon')\times [0,1]$ for a $\varepsilon'>0$. And then we take a homothety of the whole thing by $\frac{1}{\varepsilon'}$ so that the boundary is isometric to the round sphere $(S^{4k-1}(1),h_{1})$. In order to glue this to the above obtained part, we have to homotope the metric on the boundary. We take a homotopy $H_{3}(t)=\lambda(t)h_{1}+(1-\lambda(t))h_{\varepsilon}$ for $t\in [0,1]$. The metric $H_{3}(t)$ on $S^{4k-1}$ has positive scalar curvature for every $t\in [0,1]$. Note that $h_{1}$ and $h_{\varepsilon}$ differ only by the size of the Hopf fiber. So for each $t$, $H_3(t)$ also has the same Riemannian submersion structure with the fiber isometric to the round $3$-sphere of radius $r(t):=\lambda(t)+(1-\lambda(t))\varepsilon$. By the O’Neill’s formula [@besse], $$\begin{aligned} s_{H_{3}(t)}&=&\frac{1}{r^{2}(t)}s_{f}+s_{b}\circ \pi-r^{2}(t)|A|^{2},\end{aligned}$$ where $s_{f}$, $s_{b}$, and $A$ denote the scalar curvature of the fiber and the base, and the integrability tensor for $t=0$ respectively. Thus $s_{H_{3}(t)}$ is constant for each $t$ and increases as $t$ increases. From the fact that $s_{H_{3}(0)}\equiv(4k-1)(4k-2)>0$, the result follows. Nevertheless the metric $H_{3}(t)+dt^2$ on $S^{4k-1}\times [0,1]$ may not have positive scalar curvature in general. But due to Gromov and Lawson’s lemma in [@GL], for a sufficiently large constant $L>0$, $H_{3}(\frac{t}{L})+dt^2$ on $S^{4k-1}\times [0,L]$ has positive scalar curvature. Now we have a desired $4k$-ball to be glued to the part made previously out of $M'$. After the gluing, what we get is just $M$ with a specially devised smooth metric which we denote by $\bar{g}$. Remember that the scalar curvature of $\bar{g}$ is bigger than $s_{g}-c$. Now we will derive a contradiction. In case that $Y(M)=0$, $$s_{\bar{g}}>s_{g} -c=Y(M)+c> Y(M)=0,$$ which is a contradiction. In case of $Y(M)<0$, we do the surgery so that $\nu^{\frac{2}{n}}< \frac{2c}{|Y(M)+c|}$. Then noting that $s_{g}<0$, $$\begin{aligned} -(\int_M |s^-_{\bar{g}}|^{\frac{n}{2}} d\mu_{\bar{g}})^{\frac{2}{n}} &>& -(\int_{M'-N(\delta)} |s_{g}|^{\frac{n}{2}} d\mu_{g}+ |s_{g}-c|^{\frac{n}{2}}\nu)^{\frac{2}{n}}\\ &>& -(\int_{M'} |s_{g}|^{\frac{n}{2}} d\mu_{g})^{\frac{2}{n}} + (s_{g}-c)\nu^{\frac{2}{n}}\\ &=& Y(M',[g]) + (Y(M)+c)\nu^{\frac{2}{n}} \\ &>& (Y(M)+2c)-2c\\ &=& Y(M).\end{aligned}$$ This gives a contradiction to the formula (\[form1\]), and completes a proof for the $\Bbb HP^{k}$ case. The case of $CaP^{2}$ can be proved in the same way using the fact that $CaP^{2}$ also admits a metric of positive scalar curvature, and is the mapping cone of the (generalized) Hopf fibration $\pi: S^{15}\rightarrow S^{8}$ with $S^{7}$ fibers as explained in the previous section. Since the smallness of $\nu$ was used only in the case of $Y(M)<0$, we will show a way of proof without using it when $Y(M)<0$. As done in LeBrun [@lb4], instead of doing surgery on $(N(\delta),g)$, we first take a conformal change $\varphi g$ of $(M',g)$ such that $\varphi \equiv 1$ outside $N(\delta)$ and the scalar curvature of $\varphi g$ is positive on a much smaller neighborhood $N(\delta')$ of $W$. Moreover one can arrange that it satisfies $$-(\int_{M'} |s^{-}_{\varphi g}|^{\frac{n}{2}} d\mu_{\varphi g})^{\frac{2}{n}} > -(\int_{M'} |s^{-}_{g}|^{\frac{n}{2}} d\mu_{g})^{\frac{2}{n}}-\epsilon$$ for any $\epsilon>0$. (This is possible because the codimension of $W$ is $\geq 3$.) Let’s just say $\epsilon < c$. Then we perform a Gromov-Lawson surgery on $(N(\delta'),\varphi g)$ keeping the scalar curvature positive. The rest is the same and finally we get $$\begin{aligned} -(\int_M |s^-_{\bar{g}}|^{\frac{n}{2}} d\mu_{\bar{g}})^{\frac{2}{n}} &=&-(\int_{M'} |s^{-}_{\varphi g}|^{\frac{n}{2}} d\mu_{\varphi g})^{\frac{2}{n}} \\ &>& (Y(M)+2c)-c\\ &>& Y(M).\end{aligned}$$ Example and Final remark ======================== Obviously the theorem is vacuous for the case of $\Bbb HP^{1}$ which is diffeomorphic to $S^{4}$. Let $H$ be a closed Hadarmard-Cartan manifold, i.e. one with a metric of nonpositive sectional curvature. By the well-known theorem of Gromov and Lawson [@GL2] on the enlargeable manifolds, $H$ cannot carry a metric with positive scalar curvature. Therefore $Y(H)\leq0.$ Applying our theorem to $H$, one has $$Y(H\sharp\ l\ \Bbb HP^{k}\sharp \ m\ \overline{\Bbb HP^{k}})=Y(H).$$ For a specific example, take $M=T^{n} \times H$, where $T^{n}$ is an $n$-dimensional torus and $H$ is as above, e.g. a product of closed real hyperbolic manifolds. Now since $M$ has an obvious $F$-structure, its Yamabe invariant is actually $0$ by collapsing the $T^{n}$-part. (Refer to Paternain and Petean [@pp].) Thus $$Y(M\sharp\ l\ \Bbb HP^{k}\sharp \ m\ \overline{\Bbb HP^{k}})=0.$$ Similar examples can also be constructed for $CaP^{2}$. Going back to the question \[ques1\] addressed in the introduction, our argument does not apply to the case of complex projective space $\Bbb CP^{k}$. We still have the fact that $\Bbb CP^{k}$ is the mapping cone of the Hopf fibration $\pi: S^{2k-1 }\rightarrow \Bbb CP^{k-1}$ with $S^{1}$ fibers. So the $\Bbb CP^{k-1}$ is embedded as a submanifold of codimension $2$ which is one less for the Gromov-Lawson surgery to work. Moreover the statement corresponding to the theorem \[th1\] can not be true at least in dimension $4$. This is because of Wall’s stabilization theorem [@wall]. Let $M$ be a simply-connected closed smooth $4$-manifold. Then there exists integers $l,m$ such that $$M \sharp\ l\ \Bbb CP^{2} \sharp\ m\ \overline{\Bbb CP^{2}}= a\ \Bbb CP^{2} \sharp\ b\ \overline{\Bbb CP^{2}},$$ where $a=l+\frac{1}{2}(b_{2}(M)+ \sigma(M))$ and $b=m+\frac{1}{2}(b_{2}(M)- \sigma(M)).$ But we know that $Y(a\ \Bbb CP^{2} \sharp\ b\ \overline{\Bbb CP^{2}})>0.$ Thus the Yamabe invariant changes drastically by taking connected sums with both $\Bbb CP^{2}$ and $\overline{\Bbb CP^{2}}$. We do not know whether the stabilization phenomenon of the Yamabe invariant is prevalent also in higher dimensions. But at least the question \[ques1\] is worth investigating in dimension both $4$ and higher. [99]{} Michael T. Anderson, [*On uniqueness and differentiability in the space of Yamabe metrics*]{}, Commun. Contemp. Math. [**7**]{} (2005), no. 3, 299–310. T. Aubin, [*Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire*]{}, J. Math. Pures Appl. [**55**]{} (1976), 269–296. A. 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Sung, [*Surgery, curvature, and minimal volume*]{}, Ann. Global Anal. Geom. [**26**]{} (2004), 209–229. C. Sung, [*Surgery and equivariant Yamabe invariant*]{}, Diff. Geom. Appl. [**24**]{} (2006), 271–287. C. Sung, [*Collapsing and monopole classes of $3$-manifolds*]{}, J. Geom. Phys. [**57**]{} (2007), 549–559. C. T. C. Wall, [*On simply-connected $4$-manifolds*]{}, J. London Math. [**39**]{} (1964), 141–149. [^1]: email address: cysung@nims.re.kr Key Words: Yamabe invariant, quaternionic projective space, Cayley plane, surgery  MS Classification(2000): 53C20,58E11,57R65  This work was supported by the Korea Research Council of Fundamental Science and Technology (KRCF), Grant No. C-RESEARCH-2006-11-NIMS.

--- abstract: 'We report direct-detection constraints on light dark matter particles interacting with electrons. The results are based on a method that exploits the extremely low levels of leakage current of the DAMIC detector at SNOLAB of  Acm$^{-2}$. We evaluate the charge distribution of pixels that collect $<10~\rm{e^-}$ for contributions beyond the leakage current that may be attributed to dark matter interactions. Constraints are placed on so-far unexplored parameter space for dark matter masses between 0.6 and 100 . We also present new constraints on hidden-photon dark matter with masses in the range $1.2$$30$ .' author: - 'A. Aguilar-Arevalo' - 'D. Amidei' - 'D. Baxter' - 'G. Cancelo' - 'B.A. Cervantes Vergara' - 'A.E. Chavarria' - 'E. Darragh-Ford' - 'J.R.T. de Mello Neto' - 'J.C. D’Olivo' - 'J. Estrada' - 'R. Gaïor' - 'Y. Guardincerri' - 'T.W. Hossbach' - 'B. Kilminster' - 'I. Lawson' - 'S.J. Lee' - 'A. Letessier-Selvon' - 'A. Matalon' - 'V.B.B. Mello' - 'P. Mitra' - 'J. Molina' - 'S. Paul' - 'A. Piers' - 'P. Privitera' - 'K. Ramanathan' - 'J. Da Rocha' - 'Y. Sarkis' - 'M. Settimo' - 'R. Smida' - 'R. Thomas' - 'J. Tiffenberg' - 'D. Torres Machado' - 'R. Vilar' - 'A.L. Virto' bibliography: - 'myrefs.bib' title: Constraints on Light Dark Matter Particles Interacting with Electrons from DAMIC at SNOLAB --- [^1] There is overwhelming astrophysical and cosmological evidence for Dark Matter (DM) as a major constituent of the universe. Still, its nature remains elusive. The compelling Weakly Interacting Massive Particle (WIMP) dark matter hypothesis [@kolb:1990vq]  implying DM is made of hitherto unknown particles with mass in the GeVTeV scale  has been intensely scrutinized during the last two decades by detectors up to the tonne-scale looking for nuclear recoils induced by coherent scattering of WIMPs. Despite the impressive improvements in sensitivity, notably by noble liquid experiments , WIMPs have so far escaped detection. Other viable candidates include DM particles from a hidden-sector [@battaglieri2017us], which couple weakly with ordinary matter through, for example, mixing of a hidden-photon with an ordinary photon . A phenomenological consequence is that hidden-sector DM particles also interact with electrons, with sufficiently large energy transfers to be detectable down to DM masses of $\approx$ MeV [@essig:2012]. Also, eV-mass hidden-photon DM particles can be probed through absorption by electrons in detection targets [@hochberg:2017]. The DAMIC (Dark Matter in CCDs) experiment [@Aguilar-Arevalo:2016] is well-suited for a sensitive search of this class of DM candidates. DAMIC detects ionization events induced in the bulk silicon of thick, fully depleted Charge-Coupled Devices (CCDs). By exploiting the charge resolution of the CCDs ($\approx 2$ ) and their extremely low leakage current ($\approx 4$ $\rm{mm^{-2}}\,\rm{d^{-1}}$), DAMIC has already placed constraints on hidden-photon DM with masses in the range $1.2$$30$  [@aguilar:2017] with data collected in a preliminary science run. In this Letter we apply a similar approach to explore DM- interactions with high-quality data from the DAMIC science run at the SNOLAB underground laboratory. We also present improved limits on hidden-photon DM particles. To model DM- interactions we follow Ref. [@essig:2016] where the bound nature of the electrons and crystalline band structure of the target are properly taken into account. The differential event rate in the detector for a DM mass , with transferred energy $E_e$, and momentum $q$ is parametrized as $$\label{eq:rate} \frac{dR}{dE_e} \propto \bar{\sigma_e} \int \frac{dq}{q^2}~\eta(m_\chi,q,E_e) |F_{DM}(q)|^2 |f_\textnormal{c}(q,E_e)|^2\,,$$ where  is a reference cross section for free electron scattering, $\eta$ includes properties of the incident flux of galactic DM particles, $F_{DM}$ is the dark matter form factor, and the crystal form factor  quantifies the atomic transitions of bound-state electrons. The DM form factor expresses the momentum-transfer dependence of the interaction, generically introduced as $F_{DM}=(\alpha m_e/q)^n$ {$n=0,1,2$}. The $n=0$ case corresponds to point-like interactions with heavy mediators or a magnetic dipole coupling, the $n=1$ case to an electric dipole coupling, and $n=2$ to massless or ultra-light mediators. The crystal form factor encodes target material properties and is calculated numerically from a density functional theory (DFT) approach, with results taken from Ref. [@essig:2016] for silicon. The DAMIC detector has taken data at SNOLAB since 2017 with seven CCDs (4k $\times$ 4k-pixel, 15 $\times$ 15 $^2$ pixel size, 675  thick for 6.0 g mass each). The devices are fully depleted and a drift field is maintained across the CCD thickness by the application of 70 V to a thin backside contact. The CCDs are operated at $\approx140$ K (stable to within $0.5$ K) inside a copper vacuum vessel kept at $\sim$ mbar. The CCD tower is shielded on all sides by at least 18 cm of lead, with the innermost 5 cm of ancient origin, and 42 cm of polyethylene to stop background radiation from environmental $\gamma$ rays and neutrons, respectively. Each CCD is read out serially by three-phase clocking, which first moves the charge in rows of pixels vertically ($y$-direction) into the serial register. Then, single pixels are shifted horizontally ($x$-direction) into the readout node, a charge to voltage amplifier located at a corner of the device. A second readout node at the opposite end of the serial register is also read out synchronously, providing a correlated noise-only measurement. An analog-to-digital converter (ADC) measures the readout node voltage, giving a pixel value $p$ in analog-to-digital converter units (ADU) linearly proportional to the number of charges in the pixel. The CCDs are individually calibrated in-situ by a red LED, with conversion constants $\Omega \approx 14.5$ ADU/. The standard mode of data taking consists of 30 ks ($\approx$ 8.3 hours) long exposures followed by readout. “Blank" images with a much shorter 30 s exposure are also taken immediately after each long exposure as a systematic check of the device operation. Details of device architecture, DAMIC infrastructure, calibration, and image processing are given in Refs. [@Aguilar-Arevalo:2016; @aguilar:2017]. The search reported here was performed on a special data set consisting of 38 exposures, each 100 ks ($\approx$ 1.16 days) long, collected in late 2017. This longer exposure time allows for a more precise determination of the leakage current. The data were acquired with 1$\times$100 binning, a readout mode where the charges of 100 consecutive pixels in a column are summed into the serial register before readout. The binned pixel size is thus 15$\times $1500 $^2$. Since readout noise is introduced each time the charge is measured, a better signal-to-noise ratio in the measurement of the charge collected over multiple pixels is achieved by binning. Hereafter, the term pixel will refer to a binned pixel. Each image contains 4272$\times$193 pixels, with a subset of 4116$\times$42 pixels corresponding to the active area of the CCD. The extra pixels, referred to as the $x$ and $y$ overscans, do not contain any charge since they are the result of clocking the CCD past the active region. ![Mean pixel ADU values, after the processing described in the text, as a function of row in the CCD. The first 42 rows correspond to the active region of the CCD, while rows $\ge 43$ correspond to the $y$ overscan. The offset observed in rows $\le42$ is due to charge accumulated in the pixels.[]{data-label="fig:ext212"}](Ext212.pdf){width="48.00000%"} Image processing begins with subtraction of the constant offset (“pedestal") present in each pixel introduced by the electronics chain. The pedestal is estimated on a per-row basis as the mean value of pixels in the $x$-overscan. To exclude an instrumental increase in transient noise at the boundaries of the CCDs, the analysis is restricted to 2500 columns in a central portion of the image. To remove correlated readout noise, we subtract from every pixel an appropriate linear combination of corresponding pixel values in the noise images acquired with the aforementioned second readout node. The subtraction coefficients are calculated to minimize the variance of the pixel noise. The resulting image noise is found to be $ \sigma_{pix} \approx 1.6$  as reported in Table \[tab:fits\]. Physical defects in the silicon lattice structure of the CCDs often result in localized regions of high dark current, generating hot pixels and columns recurring over multiple images. A mask obtained from a statistical analysis of 864 images of the full-science data set (see details of the methodology in Ref. [@aguilar:2017]) is applied, resulting in the removal of $\approx$0.25% of pixels. Clusters of pixels with signal larger than 8$\sigma_{pix}$, arising from ionization events by particles [@Aguilar-Arevalo:2016] that deposit more than $50$ eV, are also excluded as to limit the analysis to leakage current and signals from light dark matter. To mitigate the effect of charge trailing along rows from charge transfer inefficiency in the serial register, 200 pixels to the left of every cluster are masked along with 4 pixels to the right. Each pixel above and below these clusters is also masked to account for charge splitting across rows due to diffusion. This procedure removes $\approx$ 2.0% of pixels. After applying these image processing and pixel selection procedures, we calculate the mean value of pixels $\langle p \rangle$ in each row over the 38 images of the data set (Fig. \[fig:ext212\]). Rows 43 and higher correspond to the $y$ overscan, where $\langle p \rangle$ is consistent with zero. CCD data are contained in the first 42 rows of the image, where an offset is clearly present due to charge collected by the pixels. CCD numbers 2, 6 and 7 present a significantly higher leakage current that is non-uniform across the rows. This is likely due to external sources  e.g. optical or IR photons in the vessel  and inconsistent with DM which would produce charge uniformly distributed throughout the pixel array. Thus these CCDs are not considered any further in this analysis. For the four remaining CCDs, the analysis is restricted to rows 1-36 where $\langle p \rangle$ is found to be constant within uncertainty. The final selected region includes $\approx$  pixels for each of the four CCDs, with their corresponding pixel value distributions shown in Fig. \[fig:exthists\]. The total equivalent exposure of the search is 200 gd. The distribution of pixel values in a CCD is shown in Fig. \[fig:exthists\] and is modeled by the function $\Pi(p)$, which comes from the convolution of the pixel charge with the pixel readout noise. We take the pixel charge to be the sum of a Poisson-distributed leakage current $\lambda$ accumulated during the exposure and a DM signal $S$ derived from Eq. \[eq:rate\], where $S \equiv S(j \mid \bar{\sigma_e}, m_\chi)$ specifies the probability to produce $j$ charges in a pixel from specific DM interactions. The readout noise is parametrized from the pixel value distribution of blanks and overscans, and found to be well-described by the convolution of a Poisson with average $\lambda_{d}$ and a Gaussian of standard deviation $\sigma_{\rm pix}$. This parametrization reflects the presence of non-Gaussian features in the noise distribution. The pixel value distribution for a given CCD is then derived as: $$\begin{aligned} \label{eq:pixdist} \begin{split} \Pi(p) &= N \sum_{n_c=0}^{\infty} \sum_{n_l=0}^{\infty}\bigg( \bigg[ \sum_{j=0}^{n_c} S(j \mid \bar{\sigma_e}, m_\chi) \textnormal{Pois}(n_c-j \mid \lambda) \bigg] \textnormal{Pois}(n_l \mid \lambda_{d}) \textnormal{Gaus}(p \mid \Omega \big[(n_c + n_l) + \mu_0 \big], \Omega \sigma_{\rm pix})\, \bigg) \\ &= N \sum_{n_{\rm tot}=0}^{\infty} \bigg( \bigg[ \sum_{j=0}^{n_{\rm tot}} S(j \mid \bar{\sigma_e}, m_\chi) \textnormal{Pois}(n_{\rm tot}-j \mid \lambda_{\rm tot}) \bigg] \textnormal{Gaus}(p \mid \Omega \big[n_{\rm tot} + \mu_0 \big], \Omega \sigma_{\rm pix})\, \bigg)\,, \end{split}\end{aligned}$$ $$\label{eq:pixdist1} \rm{with}~~\textit{n}_{\rm{tot}} = \textit{n$_c$}+\textit{n$_l$} ~;~~ \lambda_{\rm tot} = \lambda_d+\lambda\,,$$ where $N$ is the number of pixels in the dataset, $n_{c}$ is the number of charges in a pixel from the DM signal and leakage current, $n_{l}$ is the number of charges in a pixel from readout shot noise, $\Omega$ is the to ADU calibration constant, and $\mu_0$ is an offset accounting for pedestal subtraction. The noise parameters $\sigma_{pix}$, $\lambda_d$ and $\mu_0$ reported in Table \[tab:fits\] are determined from a fit of the blanks and $y$-overscans. We then perform a maximum likelihood fit of the data to the leakage-only model (i.e. no contribution from DM- interactions, corresponding to $S(0)=1$ and $S(j\ge1)=0$) with $\sigma_{pix}$ and $\mu_0$ constrained with Gaussian penalty terms. The leakage current parameter $\lambda$ derived from the leakage-only best-fit value of $\lambda_{\rm tot}$ is reported in Table \[tab:fits\]; $\sigma_{pix}$ and $\mu_0$ from the constrained fit were found to be consistent with the blank and $y$-overscan values. Notice that $\lambda$ represents an upper limit to the leakage current, with $\lambda=1.0$ e$^-$mm$^{-2}$d$^{-1}$ ($\approx$  Acm$^{-2}$) for CCD 4, the lowest ever measured in a silicon device. -------- ---------------- ------------------------------ ----------- ------------------------------------- CCD n. $\sigma_{pix}$ $\lambda_{d}$ $\mu_{0}$ $\lambda = \lambda_{tot}-\lambda_d$ \[e$^-$\] \[e$^-$mm$^{-2}$img$^{-1}$\] \[e$^-$\] \[e$^-$mm$^{-2}$d$^{-1}$\] 1 1.628(1)  8.2(2) -0.185(3) 2.8(2) 3 1.572(1)  7.8(2) -0.160(4) 1.7(2) 4 1.594(1) 10.0(2) -0.219(4) 1.0(2) 5 1.621(1)  8.5(2) -0.183(4) 2.0(2) -------- ---------------- ------------------------------ ----------- ------------------------------------- : \[tab:fits\] Relevant parameters used in modeling the pixel value distribution, with statistical uncertainty in parentheses. The first three columns correspond to the fit of blanks and overscans, while the last column to the leakage-only fit to data. Where appropriate, units were converted from e$^-$pix$^{-1}$img$^{-1}$, as for Eq. \[eq:pixdist\], to e$^-$mm$^{-2}$d$^{-1}$. ![Distribution of pixel values (with aforementioned conversion constants $\Omega \approx 14.5$ ADU/) for the four CCDs selected for this analysis. An example of best fit result for the leakage-only model (no DM-) is given for CCD n. 1 (blue line); the dashed red line is the expectation for a DM- model with  = $1 \times 10^{-33}$ cm$^2$,  = 10 and $F_{DM}=1$. []{data-label="fig:exthists"}](exthists.pdf){width="48.00000%"} The DM signal is computed using Eq. \[eq:rate\]. We obtain the distribution of  from the binned output of the *QEDark* [@essig:2016; @priv_comm_sensei] module written for the *QuantumEspresso* [@quantumespresso] DFT code. To compute $\eta$ we assume halo parameters of dark matter density $\rho_{DM} = 0.3$  cm$^{-3}$, an isothermal Maxwellian velocity distribution with escape velocity v$_{\rm{esc}}$ = 544 kms$^{-1}$ and mean v$_0$ = 220 kms$^{-1}$, and periodic Earth motion with mean velocity v$_E$ = 232 kms$^{-1}$ [@lewin:1996]. The resulting ionization rate $dR/dE_e$ is then discretized into $dR/dn_e$, where $n_e$ is the number of ionization charges. For this purpose we use Monte-Carlo-derived probabilities $P(n_e \vert E_e)$ to produce electron-hole (e-h) pairs, informed from studies in Ref. , with the assumption that the initial energy deposit is split randomly between the e-h pair. Measurements of direct charge injection [@chang:1985; @scholze1998] validate the quantum yield of these prescriptions for deposits $<5$ eV; these prescriptions also match the Fano factor [@fano:1947] measured with similar CCDs in Ref. [@compton:2017]. The ionization rate is then obtained from $dR/dn_e = \int dE_e P(n_e \vert E_e)(dR/dE_e)$. Lastly, the effect of charge diffusion in the CCDs is included. In fact, a point-like charge deposit in the silicon bulk of the CCD may split over several pixels due to diffusion of the ionized charge as it drifts towards the pixel array. To derive the effective signal distribution, point-like charge deposits uniformly distributed across the depth $z$ of the CCD are simulated according to $dR/dn_e$. The charges are then distributed in the $x$-$y$ pixel array following the spatial variance $\sigma_{xy}^2(z)$ from a diffusion model derived from data [@Aguilar-Arevalo:2016], and a distribution of charges collected by a pixel is obtained. The procedure is repeated 1000 times to obtain the numerical distribution for the DM signal $S(j \mid \bar{\sigma_e}, m_{\chi})$. Examples of the DM model and leakage-only expectations are shown in Fig. \[fig:exthists\]. To constrain the DM signal, we implement a likelihood analysis in () space. For a fixed  and for every CCD $i$ we minimize the negative log-likelihood $\mathcal{LL}_i$ of $\Pi(p)$, leaving $\lambda_{\rm tot}$ as a free parameter while $\sigma_{pix}$ and $\mu_{0}$ are constrained to within their uncertainty (Table \[tab:fits\]), and report the total log-likelihood $\mathcal{LL}= \sum_{i=\textnormal{1}}^{4}\mathcal{LL}_i$. We find that non-zero values of  are preferred for DM masses above a few . This is mostly due to the presence of a few pixels with values $>6~\sigma_{pix}$ in the positive tail of the $p$ distribution (Fig. \[fig:exthists\]), consistent with the higher charge multiplicity expected for larger . However, the presence of a similar tail in the negative side of the $p$ distribution and of similar features in the blank images suggest a noise origin. In Table \[tab:pval\], we report the number of pixels found in the negative and positive tails of the $p$ distribution. The thresholds for the tails were chosen appropriately to obtain an expectation of two pixels from the fit with the leakage-only model. There is evidence for an overall excess with comparable numbers on both sides of the distribution and between blank and exposure images. We conclude that the preference for non-zero values of  in the fit is due to an imperfect modeling of the extreme tails of the noise distribution. Since we do not attempt to parametrize these tails further, more conservative limits are placed when the minimum of the total log-likelihood, $\mathcal{LL_{\rm min}}$, is found at a non-zero value of . For each  we obtain 90% C.L. constraints on  using the test statistic $\Lambda=2(\mathcal{LL}-\mathcal{LL_{\rm min})}$. ----------- ------- ------- ------- ------- CCD no. 1 3 4 5 Exposures 1 / 3 2 / 4 5 / 5 3 / 2 Blanks 3 / 5 4 / 1 2 / 1 2 / 3 ----------- ------- ------- ------- ------- : \[tab:pval\] Number of pixels in the negative and positive tails of the $p$ distribution, chosen such that there is an expectation of two pixels from the leakage-only fit. {width="1.\textwidth"} ![90% C.L. constraints upper limits on the hidden-photon DM kinetic mixing parameter $\kappa$ as a function of the hidden-photon mass $m_V$. Current best direct-detection limits from protoSENSEI at MINOS [@abramoff:2019], an analysis of the XENON10 data [@bloch:2017], a dish antenna [@suzuki:2015], and astrophysical solar limits [@bloch:2017; @An:2013yfc; @*Redondo:2013] are also shown for comparison.[]{data-label="fig:hplimit"}](HPlimit.pdf){width="48.00000%"} The 90% C.L. constraints on the DM- cross section from this analysis are compared in Fig. \[fig:limit\] to the current best direct-detection limits in Refs. [@abramoff:2019; @essig:2017xe; @agnese:2018; @emken:2019]. Complementary limits for heavier DM masses from noble liquid experiments can be found in Ref. . Note that for a high enough DM- cross section the DM flux at SNOLAB would be drastically reduced by interactions in the rock overburden [@emken:2019]. However, this region has already been excluded by experiments at shallower sites [@emken:2019]. Other constraints from analyses based on astrophysical modifications to the dark matter speed distribution can be found in Ref. . Several checks are performed to evaluate the robustness of the results. A $\pm$5% systematic uncertainty in the linearity of the calibration constant $\Omega$ changes the limits by $\mp$20% for  below few . We modify the ionization model by splitting the energy equally between the e-h or assigning it entirely to one of them, with limits changing by $<$10% for  below few . Lastly, we perform the analysis with different central portions of the CCD image, with limits changing by $<$10%. Our previous constraints on hidden-photon dark matter [@aguilar:2017] were obtained with a method analogous to the one presented in this Letter. The lower leakage current $\lambda$ and larger exposure of this data set result in more stringent constraints. The corresponding 90% C.L. upper limits on the hidden-photon kinetic mixing parameter $\kappa$ (also known as $\epsilon$ in literature) as a function of the hidden-photon mass $m_V$ are shown in Fig. \[fig:hplimit\]. In summary, we have established the best direct-detection limits on dark matter-electron scattering in the mass range of 0.6  to 6  by exploiting the excellent charge resolution and extremely low leakage current of DAMIC CCDs. We also place the best direct-detection constraints on hidden-photon dark matter in the mass range $1.2$$9$ . Further improvements with the SNOLAB apparatus will be explored by cooling the CCDs to 100 K and improving the light tightness of the cryostat, which may sensibly reduce the leakage current. Improvements of several orders of magnitude are expected with DAMIC-M, a kg-size detector with sub-electron resolution to be installed at the Laboratoire Souterrain de Modane in France [@settimo:2018]. We thank Rouven Essig, Tien-Tien Yu, and Tomer Volansky for assistance with the theoretical background of DM- scattering, and Chris Kouvaris for pointing to limit calculations for the overburden. We thank SNOLAB and its staff for support through underground space, logistical and technical services. SNOLAB operations are supported by the Canada Foundation for Innovation and the Province of Ontario Ministry of Research and Innovation, with underground access provided by Vale at the Creighton mine site. We acknowledge the financial support from the following agencies and organizations: Kavli Institute for Cosmological Physics at The University of Chicago through an endowment from the Kavli Foundation; National Science Foundation through Grant No. NSF PHY-1806974; Mexico’s Consejo Nacional de Ciencia y Tecnología (Grant No. 240666) and Dirección General de Asuntos del Personal Académico - Universidad Nacional Autónoma de México (Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica Grant No. IN108917). [^1]: Deceased January 2017

--- abstract: 'Hadron structure is considered in the frame of Strongly Correlated Quark Model (SCQM). It is shown that quark correlations result in fluctuations of hadronic matter distributions and single diffractive dissociation processes in hadronic collisions, hard or soft, are manifestation of these fluctuations inside colliding hadrons.' author: - 'G. Musulmanbekov' title: Quark Correlations Inside Hadrons And Single Diffraction --- Introduction ============ Elastic and inelastic $pp$ and $\overline{p}p$ $-$interactions play an important role in understanding of structure of hadrons. Variety of scattering processes, differentiation them into hard and soft ones, single and double diffraction and central hadron production tell us that this structure is rather complicated. What is the source of complexity? If hard processes are the area of application of current quark consideration then soft ones are described in constituent quark approach. Interplay between hard and soft interactions in high energy experiments strictly relates to interconnection between small and large size quark configurations inside a hadron. We think that this interplay is a manifestation of fluctuations of hadronic matter distributions inside hadrons and these fluctuations in turn are results of correlated motion of valence quarks inside hadrons. The question arises: is it possible to construct the dynamical system of quarks which can be observed at one instant of time as constituent (dressed) quarks and at the another one - as current (bare) quarks? Proposed by author the semiclassical model of strongly correlated quarks, SCQM, demonstrated how these configurations could be realized inside hadrons[@genis]. In this paper we give further elaboration of the model whichpossesses the features of both constituent and current quark models and allows one to extract confining potential and force(Section 2). In Section 3 we apply this model for description of diffractive processes and show that explicit manifestation of quark correlation inside hadrons is a single diffractive scattering. Strongly Correlated Quark Model =============================== Let us imagine the following hypothetical picture: single, colored quark imbedded into vacuum. Because of vacuum fluctuations one can observe two competing processes: first, polarization of $\overline{q}q$ sea by the color field of valence quark (VQ) and second, the tend of vacuum fluctuations to destroy this polarization. As a result, one can say about a vacuum pressure on a single, colored quark. This effect can be interpreted as the ”response” of the vacuum on the presence of, say, point defect or dislocation like in solid state physics. What happens if we place in vicinity of this quark the corresponding antiquark? By virtue of opposite color signs their polarization fields interfere in the overlapping region **destructively**. So the pressure of vacuum on quark (antiquark) from outside exceeds that one going from inner space between quark and antiquark. This results in an **attractive** force between quark (dislocation) and antiquark (antidislocation). The density of the remaining part of polarization field around quark (antiquark) is identified with hadronic matter distribution. At maximum displacement in $\overline{q}q-$ system, that corresponds to small overlapping of polarization fields, hadronic matter distributions have maximum values. So quark and antiquark located nearby start moving towards each other. The closer they to one another, the larger destructive interference effect and the smaller hadronic matter distributions are around VQs and the larger their kinetic energies. For such interacting $\overline{q}q-$ system the total Hamiltonian is $$H=\frac{m_{\overline{q}}}{(1-\beta^{2})^{1/2}}+\frac{m_{q}}{(1-\beta ^{2})^{1/2}}+V_{\overline{q}q}(2x),$$ were $m_{\overline{q}}$, $m_{q}-$ masses of valence antiquark and quark, $\beta=\beta(x)-$ their velocity depending on displacement $x$ and $V_{\overline{q}q}-$ quark–antiquark potential energy with separation $2x.$ It can be rewritten as $$H=\left[ \frac{m_{\overline{q}}}{(1-\beta^{2})^{1/2}}+U(x)\right] +\left[ \frac{m_{q}}{(1-\beta^{2})^{1/2}}+U(x)\right] =H_{\overline{q}}+H_{q},$$ were $U(x)=\frac{1}{2}V_{\overline{q}q}(2x)$ is potenial energy of quark or antiquark. Therefore, keeping in mind that quark and antiquark are strongly correlated we consider each of them separately as undergoing oscillatory motion in 1+1 dimension. Generalization to three–quark system in baryons is performed according to $SU(3)_{color}$ symmetry: an antiquark is replaced by two correspondingly colored quarks to get color singlet baryon and destructive interference takes place between color fields of three valence quarks. Putting aside the mass and charge differences of valence quarks we may say that inside baryon three quarks oscillate along the bisectors of equilateral triangle. Hereinafter we consider that axis $Z$ is perpendicular to the plane of oscillation $XY$. VQ with its polarized surroundings (hadronic matter distribution) form constituent quark. According to our approach potential energy of valence quark, $U(x),$ corresponds to the mass $M_{Q}$ of constituent quark: $$U(x)=const\int_{-\infty}^{\infty}dz^{\prime}\int_{-\infty}^{\infty}dy^{\prime }\int_{-x}^{\infty}dx^{\prime}\rho(x,\mathbf{r}^{\prime})\approx M_{Q}(x)$$ with $$\rho(x,\mathbf{r}^{\prime})=\left| \varphi(x,\mathbf{r}^{\prime})\right| ^{2}=\left| \varphi_{Q}(x^{\prime}+x,y^{\prime},z^{\prime})-\varphi_{\overline{Q}% }(x^{\prime}-x,y^{\prime},z^{\prime})\right| ^{2}.$$ The knowledge of the mechanism and structure of vacuum polarization around valence quark would give the information about the confining potential. We cannot say at the moment for sure what is the microscopical mechanism of interaction of valence quark with vacuum. It could be instanton induced interactions, excitation of fractal structure of space–time, etc. So we assume, as a first approximation, that the polarization field can be taken in gaussian form:$$\varphi_{Q}(\mathbf{r})=\varphi_{Q}(x,y,z)=\varphi_{Q}(x_{1},x_{2}% ,x_{3})=\frac{(\det\hat{A})^{1/2}}{(\pi)^{3/2}}\exp\left( -\mathbf{X}^{T}% \hat{A}\mathbf{X}\right) ,$$ where exponent is written in quadratic form. The same is$\ $for $\varphi _{\overline{Q}}(\mathbf{r}).$ We define the mass of constituent quark at maximum displacement $$M_{Q(\overline{Q})}(x_{\max})=\frac{1}{3}\left( \frac{m_{\Delta}+m_{N}}% {2}\right) \approx360\ MeV,$$ where $m_{\Delta}$ and $m_{N}$ are masses delta–isobar and nucleon correspondingly. Therefore, the parameters of the model are the masses of VQs,$\ m_{q(\overline{q})},$ which are chosen to be $5$ $MeV$, maximum displacement, $x_{\max},$ and parameters of gaussian function, $\sigma _{x,y,z}.$ As shown below, $x_{\max}$ and $\sigma_{x,y,z}$ are adjusted by comparison of calculated and experimental values of inelastic cross sections, $\sigma_{in}(s),$ for $pp$ and $\overline{p}p-$ collisions. {width="4.5in"} Using (3)–(5) we can calculate the confining potential $U(x)$ and force $F(x)=-\frac{dU}{dx}$ . They are shown in Fig. 1. As one can see the confining potential is essentially nonlinear. The behavior of potential evidently demonstrates the relationship between constituent and current quark states inside a hadron. At maximum displacement quark is nonrelativistic, constituent one (VQ surrounded by ”polarized sea”), since according to (3) the confining potential corresponds to the mass of constituent quark. At the origin of oscillation, $x=0,$ antiquark–quark in mesons and 3 quarks in baryons, being close to each other, have maximum kinetic energy and correspondingly minimum potential energy and mass: they are relativistic, current quarks (bare VQs). Intermediate region corresponds to increasing (decreasing) of quark’s mass by dressing (undressing) of quarks due to vacuum polarization. This mechanism agrees with local gauge invariance principle. Indeed, phase rotation of wave function of single quark in color space $\psi_c$ on angle $\theta$ depending on displacement $x$ of the quark in coordinate space $$\psi_{c}(x)\rightarrow e^{i\theta(x)}\psi_{c}(x)$$ results in it’s dressing (undressing) by quark and gluon condensate that corresponds to the transformation of gauge field $A_\mu =(\varphi,0,0,0)$ $$A_\mu(x)\rightarrow A_\mu(x)+\partial_\mu\theta(x).$$ Here we dropped color indices and took into account only scalar component, $\varphi$, of gauge field. Thus gauge transformation maps internal (isotopic) space of colored quark onto coordinate space. On the other hand this dynamical picture of VQ dressing (undressing) corresponds to chiral symmetry breaking (restoration). The behaviour of field $\varphi$ and hadronic matter distribution,$\rho$, for quark–antiquark system during their oscillations is shown in Fig. 2. ![Evolution of field $\varphi$, (Eq. (5)), and hadronic matter distribution $\rho$, (Eq. (4)), in quark–antiquark system during one–half of the period of oscillations; $d=2x$ – distance in [*fermi*]{} between quark and antiquark depicted as dots.](fig2.eps){width="4.7in"} Due to this mechanism of VQs oscillations nucleon runs over the states corresponding to the certain terms of the infinite series of Fock space $$\mid B\rangle=c_{1}\mid q_{1}q_{2}q_{3}\rangle+c_{2}\mid q_{1}q_{2}% q_{3}\overline{q}q\rangle+...$$ Confining force drastically differs from the one given by string models. When VQs are close each other it is very weak and fulfills the ”asymptotic freedom” behavior of quarks of QCD. At larger distances between VQs it starts growing rapidly, then reaching maximum value goes down, asymptotically approaching zero. Thus at large distances inside hadrons quarks being in a constituent state are almost free. Hence, it is clear why additive quark model, where quarks are treated as massive, almost unbound and extended objects, works well. We must emphasize that interaction between VQs is not direct but a result of polarization of surrounding vacuum combined with destructive interference. Attractive force between VQs in ground state hadrons does not appear as gluon string but goes from vacuum suppression that predominates from outside. Therefore, our approach reflects the features of bag models, as well. The model is in agreement with the experiments (Fig. 3) for description of VQ structure function inside a nucleon $$F_{2}^{ep}-F_{2}^{en}=\frac{x_{F}}{3}\left[ u_{v}(x_{F})-d_{v}(x_{F})\right] .$$ ![Valence quark structure functions in nucleons; data point are from papers[@emc; @bcdms].](fig3.eps){width="3.6495in"} Of cause, our description is classical and we must take into account quantum corrections. It will be the subject of forthcoming papers. Nevertheless, classical consideration of VQ oscillations is justified by E. Schrodinger’s paper[@schrod] where he, analyzing the motion of wave packet solution of time dependent Schrodinger equation for harmonic oscillator, demonstrated that this wave packet moves in exactly the same way as corresponding classical oscillator. In our model VQ with its surroundings can be treated as (nonlinear) wave packet. In forthcoming paper we’ll show that these wave packets possess soliton–like features. Application to Diffractive Processes ==================================== Different configurations of quark contents in colliding hadrons realized at the instant of collision result in different types of reactions. The probability of finding any quark configuration inside a hadron is defined by the probability of VQ’s displacement in proper frame of a hadron: $$P(x)dx=\frac{Adx}{\sqrt{1-m_{q}^{2}/(E-V)^{2}}}%$$ with $$\int P(x)dx=1.$$ Configurations with nonrelativistic constituent quarks ( $x\simeq x_{max}$) in both colliding hadrons lead to multiparticle production in central and fragmentation regions. Hard scattering with jet production and large angle elastic scattering take place when configurations with current VQs ($x\simeq 0$) in both colliding hadrons are realized. Intermediate configurations inside one (both) of colliding hadron are responsible for single (double) diffraction processes and semihard scattering. Using impact parameter representation, namely Inelastic Overlap Function (IOF), we can calculate total, inelastic, elastic and single diffractive cross sections for $pp$ and $\overline{p}p$ collisions. In impact parameter representation IOF can be specified via the unitarity equation $$2Imf(s,b)=|f(s,b)|^{2}+G_{in}(s,b),$$ where $f(s,b)-$elastic scattering amplitude and $G_{in}(s,b)$ is IOF. IOF is connected with inelastic differential cross sections in impact parameter space: $$\frac{1}{\pi}(d\sigma_{in}/db^{2})=G_{in}(s,b).$$ ![Inelastic overlap function for $pp$ and $\overline{p}p-$ collisions at $\sqrt{s}=53$ and $540$ $GeV$; triangles are Henzi and Valin approsimation[@henzi].](fig4.eps){width="2.2433in"} Then inelastic, elastic and total cross sections can be expressed via IOF as $$\sigma_{in}(s)=\int G_{in}(s,\mathbf{b})d^{2}\mathbf{b},$$$$\sigma_{el}(s)=\int\left[ 1-\sqrt{1-G_{in}(s,\mathbf{b})}\right] ^{2}% d^{2}\mathbf{b},$$$$\sigma_{tot}(s)=2\int\left[ 1-\sqrt{1-G_{in}(s,\mathbf{b})}\right] d^{2}\mathbf{b}.$$ ![Total cross section for $pp$ and $\overline{p}p-$ collisions; data points are compilations of experimental data taken from electronic data base HEPDATA[@hep].](fig5.eps){width="4.7in"} Since IOF relates to the probability of inelastic interaction at given impact parameter, (12), we carried out Monte Carlo simulation of inelastic nucleon–nucleon interactions. Inelastic interaction takes place at definite impact parameter $b$ if at least one pion is produced in the region where hadronic matter distributions of colliding protons overlap $$4M_{q_{i}}\gamma_{q_{i}}M_{p_{j}}\gamma_{p_{j}}\int\rho_{q_{i}}(\mathbf{r}% )\rho_{p_{j}}(\mathbf{r}-\mathbf{r}^{\prime})~d^{3}\mathbf{r}\geq m_{\pi_{\perp}}^{2},$$ where indices $q_{i}$ and $p_{j}$ refer to quarks from different nucleons and $i,j=1,2,3$, $M_{q_{i}}$, $M_{p_{j}}$ – masses of hadronic matter composed in constituent quarks $q_{i}$ and $p_{j}$, $\gamma_{q_{i}}$, $\gamma_{p_{j}}$ – their $\gamma$-factors; intergrand expression is convolution of hadronic matter density distributions of quarks $q_{i}$ and $p_{j}.$ This condition corresponds Hisenberg picture[@heisen] with modified right hand side: $m_{\pi}^{2}$ in the original Heisenberg inequality is replaced by $m_{\pi_{\perp}}^{2}.$ It is justified by the fact that the average transverse momentum of produced particles increases with energy. Specifying the quark configurations in each colliding nucleons by (9) and (3)–(5) we calculate $G_{in}(s,b)$ for certain values of impact parameter $b$ and then cross sections $\sigma_{in},\sigma_{el}$ and $\sigma_{tot}.$ The values of adjusted parameters of the model, $x_{\max}=0.64$ $fm,$ $\sigma_{x,y}=0.24$ $fm$ and $\sigma_{z}=0.12$ $fm,$ are chosen by comparison of calculated IOF with so called “BEL”–parametrization[@henzi], (Fig. 4), and calculated inelastic cross sections with experimental ones, $\sigma_{in}(s),$ in $pp$ and $\overline{p}p-$ collisions at $\sqrt{s}_{pp}=540$ $GeV.$ Fig. 5 shows the result of calculation for $pp$ total cross section at wide range of collision energies. One can see that the model with parameters fixed at one energy ($540$ $GeV)$ describes the energetic behavior of $\sigma_{tot}.$ The growth of cross sections with energy is due to the continuous tails of polarization fields around VQs not compensated by destructive interference with fields of two other VQs. With rising collision energy these tails result in the increasing effective size of hadronic matter distribution inside nucleons and correspondingly the increasing radius of interactions. The model gives for cross sections linear logarithmic energy dependence and the curve deviates from data points at very high energies. The reason of this is that in our geometrical approach, ((12) – (16)), we didn’t take into account the real part of scattering amplitude. At energies $\sqrt{s}<30$ $GeV$ calculated cross sections were corrected on contributions of Regge poles exchange by using Donnachie and Landshoff parametrization[@donlans]:$\Delta\sigma_{R}% ^{tot}=56.08s^{-0.4526}$ {width="4.0in"} An oscillatory motion of VQs appearing as interplay between constituent and bare (current) quark configurations results in fluctuations of hadronic matter distribution inside colliding nucleons. The manifestation of these fluctuations is a variety of scattering processes, hard and soft, in particular, the process of single diffraction (SD). We select SD–events among inelastic $pp\rightarrow pX$ events with the criterion $1-x_{F}<0.1$, where $x_{F}=\frac{2}{\sqrt{s}}(k_{1}+k_{2}+k_{3})$ (Fig. 6). Here $k_{1},$ $k_{2},$ $k_{3}$ are momenta of quarks forming the final state proton. As one can see, SD–events correspond to constituent quark configuration inside one colliding nucleon and (semi)bare quark configuration inside another one. Fig. 7 shows that calculated SD cross section slightly depends on energy; our calculation for LHC energy gives $\sigma_{SD}^{LHC}(pp)=8.30\pm0.15$ $mb.$ ![Single diffractive dissociation cross section for $pp$ and $\overline{p}p-$ collisions; data points are compilations of experimental data taken from electronic data base HEPDATA[@hep].](fig7.eps){width="4.7in"} Discussions and Summary ======================= Proposed dynamical model of hadron structure, SCQM, possesses some important features. Parameters of the model, maximal displacement of VQ, $x_{max}=0.64 fm$, and extension of quark and gluon condensate around VQs, $\sigma_{x,y}=0.24 fm$ and $\sigma_{z}=0.12 fm$ characterize extended sizes of hadron. Owing to noncompensated tails of vacuum condensates at outer sides of VQs and condition (18), the model describes the energy dependence of inelastic and total cross sections as a result of increasing effective sizes of colliding hadrons. On the other hand, there is no overlap of hadronic matter distributions in space between VQs of each interacting hadron even at very high energy collisions because destructive interference of their polarization fields reduces resulting hadronic matter at the center of quark system to zero value. So, the model keeps unitarity and does not need of inclusion of such a questionable effect as ”antishadowing” [@ashad]. According to our model diffraction dissociation is not shadowing effect of nondiffractive inelastic process – both compose inelastic process, they merely differ by configurations of quarks inside interacting hadrons. Because of plane oscillations of VQs and flattened (perpendicular to the plane of oscillation) form of hadronic matter distributions nucleons are deformed, non-spherical objects. In paper [@spin], assuming that spin of VQs are perpendicular to the plane of oscillations, we showed that such a deformation of proton could manifest itself in total cross section differences between longitudinal and transversal polarization states in proton–proton collisions. Apparently experimental evidence of such a deformation are quadrupole transition amplitudes in $N\rightarrow\Delta$ electroproduction which are sensitive to quadrupole deformation of the nucleon[@feassler]. Furthermore hadronic matter distribution inside hadrons and in turn the sizes of hadrons are fluctuating quantities. Fluctuations of extended sizes of nucleons allows one to understand why ’black disk” limit is not saturated in $pp$ and $\overline{p}p-$ collisions up to very high energies. We must mention the paper[@barshay] of S. Barshay with co-authors where they incorporated fluctuations in the eikonal into geometric picture for explaining of diffractive processes. There is a connection between eikonal and Inelastic Overlap Function used in our calculations of cross sections through the expression $$G_{in}(b,s)=1-e^{-2Im\chi(b,s)},$$ were $\chi(b,s)$ is eikonal. It is obvious that fluctuations of hadronic matter distributions caused by interplay between constituent and current quark configurations inside colliding hadrons lead to fluctuation of eikonal. In their approach the fluctuations of eikonal were controlled by phenomenological parameter, depending on energy. They predicted that single diffraction cross section increases reaching 14–15 mb at cms energies 20$\div$40 TeV then starts to decrease. Contrary to this prediction there is no apparent reason for any maximum, according to our approach, and this cross section increases steadily, very slightly up to asymptotic energies. Our unified geometrical explanation of diffractive and nondiffractive processes could give an answer on long standing question: what is pomeron? Historically the concept of “pomeron”, starting from simple Regge pole with intercept $a_0=1$, transformed to a rather complicated object with relatively arbitrary features and smooth meaning. To produce rising cross sections it must have intercept such that $a_0=1+\varepsilon$. The fact that the parameter $\varepsilon$ is universal, independent of particles being scattered in hadronic and DIS interactions, could say us that the nature of the cross section growth is the same for all processes. Our interpretation of pomeron is geometrical one. Both diffractive and nondiffractive particle production emerge from disturbance (excitation) of overlapped continuous vacuum polarization fields (gluon and $\overline{q}q$ condensate) around valence quarks of colliding hadrons followed by fragmentation process. The type of interaction depends on quark configurations inside both colliding hadrons occurring at the instant of interaction and the value of impact parameter. So, what we used to call “Pomeron” in $t-$ channel is solely continuum states in $s-$ channel and we claim that Pomeron is unique in elastic, inelastic (diffractive and nondiffractive) and DIS. This research was partly supported by the Russian Foundation of Basics Research, grants 99-07-90383 and 99-01-01103. [99]{} G. Musulmanbekov, *Proc. of XVII Int. Kazimierz Meeting on Particle Theory and Phenomenology, Iowa, 1995*, World Scientific, 1996, p. 347-353.; *Proc. of XXVI Int.Symp. on Multiparticle Dynamics, Sept. 1-5, Faro, 1996*, World Scientific, 1997, p. 357–363; Nucl. Phys. Suppl. **B71**(1999) 117. E. Schrodinger, *”Der stetige Ubergang von der Mikro-zur Makromechanik”,* Naturwissenschaften **14** (1926) 664. R. Henzi and P. Valin, Phys. Lett. **132B** (1983) 443; R. Henzi,* *Proc. of the 4th Topical Workshop on $\overline{p}p$ Collider Physics, Bern, 1984. W. Heisenberg, Z. Phys. **133** (1952) 65. J.J. Aubert et al., Nucl. Phys. **293B** (1987) 740. A.C. Benvenuti et al., Phys. Lett., **237B** (1990) 599. A.Donnachie and P.V. Landshoff, CERN–TH 6635/92. S.M. Troshin and N.E. Tyurin, hep-ph/9810495. G.Musulmanbekov, Proc. VII Workshop on High Energy Spin Phys. July 7–12 1997, Dubna, Dubna 1997, p. 165-170. A.Feassler, Prog. Part. Nucl. Phys. **44** (2000) 197. http://durpdg.durham.ac.uk/HEPDATA. S.Barshay, P. Heiliger, D. Rein, Z. Phys., **56C** (1992) 77.

--- author: - 'H. Stiele' - 'W. Pietsch' - 'F. Haberl' - 'D. Hatzidimitriou R. Barnard' - 'B. F. Williams' - 'A. K. H. Kong' - 'U. Kolb' bibliography: - 'papers2.bib' - '/Users/apple/work/papers/my1990.bib' - '/Users/apple/work/papers/my2000.bib' - '/Users/apple/work/papers/my2001.bib' - '/Users/apple/work/papers/catalog.bib' - '/Users/apple/work/papers/my2007.bib' - '/Users/apple/work/papers/my2008.bib' - '/Users/apple/work/papers/my2010.bib' date: 'Received / Accepted ' title: 'The deep XMM-Newton Survey of M 31 [^1] [^2] ' --- Introduction {#Sec:Intro} ============ Our nearest neighbouring large spiral galaxy, the Andromeda galaxy, also known as 31 or , is an ideal target for an X-ray source population study of a galaxy similar to the Milky Way. Its proximity [distance 780 kpc, @1998AJ....115.1916H; @1998ApJ...503L.131S] and the moderate Galactic foreground absorption [= 7[$\times 10^{20}$ cm$^{-2}$]{}, @1992ApJS...79...77S] allow a detailed study of source populations and individual sources. After early detections of 31 with X-ray detectors mounted on rockets [  @1974ApJ...190..285B] and the [*Uhuru*]{} satellite [@1974ApJS...27...37G], the imaging X-ray optics flown on the  X-ray observatory permitted the resolution of individual X-ray sources in 31 for the first time. In the entire set of  imaging observations of 31, @1991ApJ...382...82T [hereafter TF91] found 108 individual X-ray sources brighter than $\sim6.4$[$\times 10^{36}$ ]{}, of which 16 sources showed variability [@1979ApJ...234L..45V; @1990ApJ...356..119C]. In July 1990, the bulge region of 31 was observed with the  High Resolution Imager (HRI) for $\sim 48$ks. @1993ApJ...410..615P [hereafter PFJ93] reported 86 sources brighter than $\sim1.8$[$\times 10^{36}$ ]{} in this observation. Of the  HRI sources located within 75 of the nucleus, 18 sources were found to vary when compared to previous  observations and about three of the sources may be “transients". Two deep PSPC (Position Sensitive Proportional Counter) surveys of 31 were performed with , the first in July 1991 , the second in July/August 1992 . In total 560 X-ray sources were detected in the field of 31; of these, 491 sources were not detected in previous  observations. In addition, a comparison with the results of the  survey revealed long term variability in 18 sources, including 7 possible transients. Comparing the two  surveys, 34 long term variable sources and 8 transient candidates were detected. The derived luminosities of the detected 31 sources ranged from 5[$\times 10^{35}$ ]{} to 5[$\times 10^{38}$ ]{}. Another important result obtained with  was the establishment of supersoft sources (SSSs) as a new class of 31 X-ray sources and the identification of the first SSS with an optical nova in 31 . @2000ApJ...537L..23G reported on first observations of the nuclear region of 31 with . They found that the nuclear source has an unusual X-ray spectrum compared to the other point sources in 31. @2002ApJ...577..738K report on eight  ACIS-I observations taken between 1999 and 2001, which cover the central $\sim 17\arcmin\!\times\!17\arcmin$ region of 31. They detected 204 sources, of which $\sim$50% are variable on timescales of months and 13 sources were classified as transients. @2002ApJ...578..114K detected 142 point sources ($L_X=2\!\times\!10^{35}$ to 2[$\times 10^{38}$ ]{} in the 0.1–10keV band) in a 47ks /HRC observation of the central region of 31. A comparison with a  observation taken 11yr earlier, showed that 46$\pm$26% of the sources with $L_X>5$[$\times 10^{36}$ ]{} are variable. Three different 31 disc fields, consisting of different stellar population mixtures, were observed by . @2002ApJ...570..618D investigated bright X-ray binaries (XRBs) in these fields, while @2004ApJ...610..247D examined the populations of supersoft sources (SSSs) and quasisoft sources (QSSs), including observations of the central field. Using  HRC observations, @2004ApJ...609..735W measured the mean fluxes and long-term time variability of 166 sources detected in these data. used  data to examine the low mass X-ray binaries (LMXBs) in the bulge of 31. Good candidates for LMXBs are the so-called transient sources. Studies of transient sources in 31 are presented in numerous papers, e.g. @2006ApJ...643..356W, @2006ApJ...645..277T [ hereafter TPC06], @2005ApJ...632.1086W, @2006ApJ...637..479W [ hereafter WGM06], and . Using  and  data, @2004ApJ...616..821T detected 43 X-ray sources coincident with globular cluster candidates from various optical surveys. They studied their spectral properties, time variability and logN-logS relations. used  Performance Verification observations to study the variability of X-ray sources in the central region of 31. They found 116 sources brighter than a limiting luminosity of 6[$\times 10^{35}$ ]{} and examined the $\sim60$ brightest sources for periodic and non-periodic variability. At least 15% of these sources appear to be variable on a time scale of several months. used  to study the X-ray binary RX J0042.6+4115 and suggested it as a Z-source. @2006ApJ...643..844O studied the population of SSSs and QSSs with . Recently, @2008ApJ...676.1218T reported the discovery of 217s pulsations in the bright persistent SSS XMMU J004252.5+411540. presented the results of a complete spectral survey of the 335 X-ray point sources they detected in five  observations located along the major axis of 31. They obtained background subtracted spectra and lightcurves for each of the 335 X-ray sources. Sources with more than 50 source counts were individually spectrally fitted. In addition, they selected 18 HMXB candidates, based on a power law photon index of $0.8\!\le\!\Gamma\!\le\!1.2$. prepared a catalogue of 31point-like X-ray sources analysing all observations available at that time in the  archive which overlap at least in part with the optical ${\mathrm}{D}_{25}$ extent of the galaxy. In total, they detected 856 sources. The central part of the galaxy was covered four times with a separation of the observations of about half a year starting in June 2000. PFH2005 only gave source properties derived from an analysis of the combined observations of the central region. Source identification and classification were based on hardness ratios, and correlations with sources in other wavelength regimes. In follow-up work, (i) searched for X-ray burst sources in globular cluster (GlC) sources and candidates and identified two X-ray bursters and a few more candidates, while (ii) searched for correlations with optical novae. They identified 7 SSSs and 1 symbiotic star from the catalogue of PFH2005 with optical novae, and identified anadditional  source with an optical nova. This work was continued and extended on archival  HRC-I and ACIS-I observations by . presented a time variability analysis of all of the 31 central sources. They detected 39 sources not reported at all in PFH2005. 21 sources were detected in the July 2004 monitoring observations of the low mass X-ray binary RX J0042.6+4115 (PI Barnard), which became available in the meantime. Six sources, which were classified as “hard" sources by PFH2005, show distinct time variability and hence are classified as XRB candidates in SPH2008. The SNR classifications of three other sources from PFH2005 had to be rejected due to the distinct time variability found by SPH2008. reported on the first two SSSs ever discovered in the 31 globular cluster system, and discussed the very short supersoft X-ray state of the classical nova M31N 2007-11a. A comparative study of supersoft sources detected with ,  and , examining their long-term variability, was presented by @2010AN....331..212S. An investigation of the logN-logS relation of sources detected in the 2.0–10.0keV range will be presented in a forthcoming paper (Stiele et al. 2011 in prep.). In this work the contribution of background objects and the spatial dependence of the logN-logS relations for sources of 31 is studied. In this paper we report on the large  survey of 31, which covers the entire ${\mathrm}{D}_{25}$ ellipse of 31, for the first time, down to a limiting luminosity of $\sim$[$10^{35}$ erg s$^{-1}$]{} in the 0.2–4.5keV band. In Sect.\[sec:obsana\] information about the observations used is provided. The analysis of the data is presented in Sect.\[Sec:analys\]. Section \[Sec:coim\] presents the combined colour image of all observations used. The source catalogue of the deep  survey of 31 is described in Sect.\[Sec:srccat\]. The results of the temporal variability analysis are discussed in Sect.\[Sec:var\]. Cross-correlations with other 31 X-ray catalogues are discussed in Sect.\[Sec:CrossX-ray\], while Sect.\[SEC:CCow\] discusses cross-correlations with catalogues at other wavelengths. Our results related to foreground stars and background sources in the field of 31 are presented in Sect.\[Sec:fgback\]. Individual source classes belonging to M31 are discussed in Sect.\[Sec:Srcsm31\]. We draw our conclusions in Sect.\[Sec:Concl\]. [lcrcll]{} Paper & S$^{+}$ & \#ofSrc$^{*}$ & & field & comments\ & & & erg cm$^{-2}$ s$^{-1}$&\ & E & 108 & $6.4\!\times\!10^{36}$–$1.3\!\times\!10^{38}$ & entire set of  & 16 sources showed variability\ & & & (0.2–4keV) &imaging observations &\ & R (HRI) & 86 & $\ga1.8\!\times\!10^{36}$ & bulge region & 18 sources variable; $\sim$3 transients\ & & & (0.2–4keV) & &\ & R (PSPC) & 560 & $5\!\times\!10^{35}$–$5\!\times\!10^{38}$ & whole galaxy & two deep surveys\ (SPH97, SHL2001) & & & (0.1–2.4keV) & & 491 sources not detected with\ & & & & & 11 sources variable, 7 transients compared to\ & & & & & 34 sources variable, 8 transients between  surveys\ & X & 116 & $\ga6\!\times\!10^{35}$ & centre & examined the $\sim60$ brightest sources for variability\ & & & (0.3–12keV) & &\ & C (ACIS-I) & 204 & $\ga2\!\times\!10^{35}$ & central $\sim 17\arcmin\!\times\!17\arcmin$ & observations between 1999 and 2001\ & & & (0.3–7keV) & & $\sim$50% of the sources are variable, 13 transients\ & C (HRC) & 142 & $2\!\times\!10^{35}$–$2\!\times\!10^{38}$ & centre & one 47ks observation; 46$\pm$26% of the sources\ & & & (0.1–10keV) & & with $L_X>5$[$\times 10^{36}$ ]{} are variable\ & C (ACIS-I/S) & 28 & $5\!\times\!10^{35}$–$3\!\times\!10^{38}$ & 3 disc fields & bright X-ray binaries\ & & & (0.3–7keV) & &\ & C (ACIS-S S3) & 33 & & 3 disc fields + centre & supersoft sources and quasisoft sources\ & C (HRC) & 166 & $1.4\!\times\!10^{36}$–$5\!\times\!10^{38}$ & major axis + centre & $\ga$25% showed significant variability\ & & & (0.1–10keV) & &\ & C, X & 43 & $\sim10^{35}$–$\sim10^{39}$ & major axis + centre & globular cluster study\ & & & (0.3–10keV) & &\ & X & 856 & $4.4\!\times\!10^{34}$–$2.8\!\times\!10^{38}$ & major axis + centre & source catalogue\ & & & (0.2–4.5keV) & &\ & C, R, X & 21 & $\sim10^{35}$–$\sim10^{38}$ & centre & correlations with optical novae\ & & & (0.2–1keV) & &\ & C, X & 42 & $6\!\times\!10^{35}$–$\sim10^{39}$ & major axis + centre & supersoft sources and quasisoft sources\ & & & (0.2–2keV) & &\ & & & (0.3–10keV) & &\ & C, X & 46 & $\sim10^{35}$–$\sim10^{38}$ & centre & correlations with optical novae\ & & & (0.2–1keV) & &\ & C & 263 & $5\!\times\!10^{33}$–$1.5\!\times\!10^{38}$ & bulge region & low mass X-ray binary study\ & & & (0.5–8keV) & &\ & X & 39 & $7\!\times\!10^{34}$–$6\!\times\!10^{37}$ & centre & re-analysis of archival and new 2004 observations\ & & 300 & $4.4\!\times\!10^{34}$–$2.8\!\times\!10^{38}$ & & time variability analysis; 149 sources with a significance\ & & & (0.2–4.5keV) & & for variability $>$3; 6 new X-ray binary candidates,\ & & & & & 3 supernova remnant classifications were rejected\ & X & 335 & $\sim10^{34}$–$\sim10^{39}$ & 5 fields along & background subtracted spectra and lightcurves for\ & & & (0.3–10keV) & major axis & each source; 18 HMXB candidates, selected from their\ & & & & & power law photon index\ & X & 40 & & whole galaxy & supersoft sources; comparing ,  and\ & & & & &  catalogues\ \[Tab:VarSNRs1\] Notes:\ $^{ +~}$: X-ray satellite(s) on which the study is based: E for , R for , C for , and X for  (EPIC)\ $^{ *~}$: Number of sources\ $^{ \dagger~}$: observed luminosity range in the indicated energy band, assuming a distance of 780kpc to 31 Observations {#sec:obsana} ============ Figure \[fig:deepsurveyfields\] shows the layout of the individual  observations over the field of 31. The observations of the “Deep  Survey of 31” (PI Pietsch) mainly point at the outer parts of 31, while the area along the major axis is covered by archival  observations (PIs Watson, Mason, Di Stefano). To treat all data in the same way, we re-analysed all archival  observations of 31, which were used in . In addition we included an  target of opportunity (ToO) observation of source CXOM31 J004059.2+411551 and the four observations of source RX J0042.6+4115 (PI Barnard). All observations of the “Deep  Survey of 31” and the ToO observation were taken between June 2006 and February 2008. All other observations were available via the  Data Archive[^3] and were taken between June 2000 and July 2004. The journal of observations is given in Table \[tab:observations\]. It includes the 31 field name (Column 1), the identification number (2) and date (3) of the observation and the pointing direction (4, 5), while col. 6 contains the systematic offset (see Sect.\[SubSec:AstCorr\]). For each EPIC camera the filter used and the exposure time after screening for high background is given (see Sect.\[sec:Screening\]). [llllrrrlrlrlr]{} & & & & & & &\ & & & & & & & & & & &\ & & & & & & & & & & &\ Centre 1 & (c1) & 0112570401 & 2000-06-25 & 0:42:36.2 & 41:16:58 & $-1.9,+0.1$ & medium & 23.48(23.48) & medium & 29.64(29.64) &medium & 29.64(29.64)\ Centre 2 & (c2) & 0112570601 & 2000-12-28 & 0:42:49.8 & 41:14:37 & $-2.1,+0.2$ & medium & 5.82( 5.82) & medium & 6.42( 6.42) &medium & 6.42( 6.42)\ Centre 3 & (c3) & 0109270101 & 2001-06-29 & 0:42:36.3 & 41:16:54 & $-3.2,-1.7$ & medium & 21.71(21.71) & medium & 23.85(23.85) &medium & 23.86(23.86)\ N1 & (n1) & 0109270701 & 2002-01-05 & 0:44:08.2 & 41:34:56 & $-0.3,+0.7$ & medium & 48.31(48.31) & medium & 55.68(55.68) &medium & 55.67(55.67)\ Centre 4 & (c4) & 0112570101 & 2002-01-06/07 & 0:42:50.4 & 41:14:46 & $-1.0,-0.8$ & thin & 47.85(47.85) & thin & 52.87(52.87) &thin & 52.86(52.86)\ S1 & (s1) & 0112570201 & 2002-01-12/13 & 0:41:32.7 & 40:54:38 & $-2.1,-1.7$ & thin & 46.75(46.75) & thin & 51.83(51.83) &thin & 51.84(51.84)\ S2 & (s2) & 0112570301 & 2002-01-24/25 & 0:40:06.0 & 40:35:24 & $-1.1,-0.3$ & thin & 22.23(22.23) & thin & 24.23(24.23) &thin & 24.24(24.24)\ N2 & (n2) & 0109270301 & 2002-01-26/27 & 0:45:20.0 & 41:56:09 & $-0.3,-1.5$ & medium & 22.73(22.73) & medium & 25.22(25.22) &medium & 25.28(25.28)\ N3 & (n3) & 0109270401 & 2002-06-29/30 & 0:46:38.0 & 42:16:20 & $-2.3,-1.7$ & medium & 39.34(39.34) & medium & 43.50(43.50) &medium & 43.63(43.63)\ H4 & (h4) & 0151580401 & 2003-02-06 & 0:46:07.0 & 41:20:58 & $+0.3,+0.0$ & medium & 10.14(10.14) & medium & 12.76(12.76) &medium & 12.76(12.76)\ RX 1 & (b1)$^{\ddagger}$ & 0202230201 & 2004-07-16 & 0:42:38.6 & 41:16:04 & $-1.3,-1.2$ & medium & 16.32(16.32) & medium & 19.21(19.21) &medium & 19.21(19.21)\ RX 2 & (b2) & 0202230301 & 2004-07-17 & 0:42:38.6 & 41:16:04 & $-1.0,-0.9$ & medium & 0.0(0.0) & medium & 0.0(0.0) &medium & 0.0(0.0)\ RX 3 & (b3)$^{\ddagger}$ & 0202230401 & 2004-07-18 & 0:42:38.6 & 41:16:04 & $-1.7,-1.5$ & medium & 12.30(12.30) & medium & 17.64(17.64) &medium & 17.68(17.68)\ RX 4 & (b4)$^{\ddagger}$ & 0202230501 & 2004-07-19 & 0:42:38.6 & 41:16:04 & $-1.4,-1.8$ & medium & 7.94(7.94) & medium & 10.12(10.12) &medium & 10.13(10.13)\ S3 & (s3) & 0402560101 & 2006-06-28 & 0:38:52.8 & 40:15:00 & $-3.1,-3.0$ & thin & 4.99(4.99) & medium & 6.96(6.96) &medium & 6.97(6.97)\ SS1 & (ss1) & 0402560201 & 2006-06-30 & 0:43:28.8 & 40:55:12 & $-4.4,-3.7$ & thin & 14.07(9.57) & medium & 24.56(10.65) &medium & 24.58(10.66)\ SN1 & (sn1) & 0402560301 & 2006-07-01 & 0:40:43.2 & 41:17:60 & $-2.7,-1.5$ & thin & 41.23(35.42) & medium & 47.60(39.40) &medium & 47.64(39.44)\ SS2 & (ss2) & 0402560401 & 2006-07-08 & 0:42:16.8 & 40:37:12 & $-1.2,-1.3$ & thin & 21.64(9.92) & medium & 25.59(11.04) &medium & 25.64(11.05)\ SN2 & (sn2) & 0402560501 & 2006-07-20 & 0:39:40.8 & 40:58:48 & $-0.8,-0.7$ & thin & 48.79(21.45) & medium & 56.13(23.85) &medium & 56.17(23.86)\ SN3 & (sn3) & 0402560701 & 2006-07-23 & 0:39:02.4 & 40:37:48 & $-0.9,-2.0$ & thin & 23.80(15.43) & medium & 28.02(17.16) &medium & 28.04(17.17)\ SS3 & (ss3) & 0402560601 & 2006-07-28 & 0:40:45.6 & 40:21:00 & $-1.8,-1.7$ & thin & 27.77(20.22) & medium & 31.92(22.49) &medium & 31.94(22.5)\ S2 & (s21) & 0402560801 & 2006-12-25 & 0:40:06.0 & 40:35:24 & $-1.6,-0.7$ & thin & 39.12(39.12) & medium & 45.19(45.19) &medium & 45.21(45.21)\ NN1 & (nn1) & 0402560901 & 2006-12-26 & 0:41:52.8 & 41:36:36 & $-1.5,-1.5$ & thin & 37.9(37.9) & medium & 43.08(43.08) &medium & 43.1(43.1)\ NS1 & (ns1) & 0402561001 & 2006-12-30 & 0:44:38.4 & 41:12:00 & $-1.0,-1.3$ & thin & 45.11(45.11) & medium & 50.9(50.9) &medium & 50.93(50.93)\ NN2 & (nn2) & 0402561101 & 2007-01-01 & 0:43:09.6 & 41:55:12 & $-0.0,-1.2$ & thin & 41.73(41.73) & medium & 46.45(46.45) &medium & 46.47(46.47)\ NS2 & (ns2) & 0402561201 & 2007-01-02 & 0:45:43.2 & 41:31:48 & $-2.3,-1.7$ & thin & 34.96(34.96) & medium & 40.55(40.55) &medium & 40.58(40.58)\ NN3 & (nn3) & 0402561301 & 2007-01-03 & 0:44:45.6 & 42:09:36 & $-1.4,-0.7$ & thin & 31.04(31.04) & medium & 34.81(34.81) &medium & 34.81(34.81)\ NS3 & (ns3) & 0402561401 & 2007-01-04 & 0:46:38.4 & 41:53:60 & $-2.1,+0.3$ & thin & 39.41(39.41) & medium & 45.50(45.50) &medium & 45.52(45.52)\ N2 & (n21) & 0402561501 & 2007-01-05 & 0:45:20.0 & 41:56:09 & $-2.6,-1.3$ & thin & 37.18(37.18) & medium & 41.98(41.98) &medium & 42.03(42.03)\ SS1 & (ss11) & 0505760201 & 2007-07-22 & 0:43:28.8 & 40:55:12 & $-2.5,-2.6$ & thin & 30.07(23.90) & medium & 34.01(26.70) &medium & 34.02(26.72)\ S3 & (s31) & 0505760101 & 2007-07-24 & 0:38:52.8 & 40:15:00 & $-1.8,-1.0$ & thin & 21.86(15.74) & medium & 24.74(17.65) &medium & 24.74(17.65)\ CXOM31& (sn11)$^{\diamond}$ & 0410582001 & 2007-07-25 & 0:40:59.2 & 41:15:51 & $-1.2,-0.3$ & thin & 11.27(11.27) & medium & 14.01(14.01) &medium & 14.02(14.02)\ SS3 & (ss31) & 0505760401 & 2007-12-25 & 0:40:45.6 & 40:21:00 & $-1.0,+0.1$ & thin & 23.56(22.82) & medium & 28.18(25.8) &medium & 28.2(25.82)\ SS2 & (ss21) & 0505760301 & 2007-12-28 & 0:42:16.8 & 40:37:12 & $+1.3,-0.1$ & thin & 35.28(35.28) & medium & 40.00(40.00) &medium & 40.01(40.01)\ SN3 & (sn31) & 0505760501 & 2007-12-31 & 0:39:02.4 & 40:37:48 & $-1.6,-1.3$ & thin & 24.26(24.26) & medium & 28.77(28.77) &medium & 28.78(28.78)\ S3 & (s32) & 0511380101 & 2008-01-02 & 0:38:52.8 & 40:15:00 & $-1.7,-3.3$ & thin & 38.31(38.31) & medium & 44.92(44.92) &medium & 44.95(44.95)\ SS1 & (ss12) & 0511380201 & 2008-01-05 & 0:43:28.8 & 40:55:12 & $-0.9,-1.4$ & thin & 8.85( 8.85) & medium & 11.28(11.28) &medium & 11.29(11.29)\ SN2 & (sn21) & 0511380301 & 2008-01-06 & 0:39:40.8 & 40:58:48 & $-0.2,-0.4$ & thin & 24.79(24.79) & medium & 29.28(29.28) &medium & 29.29(29.29)\ SS1 & (ss13) & 0511380601 & 2008-02-09 & 0:43:28.8 & 40:55:12 & $-0.8,-1.8$ & thin & 13.35(13.35) & medium & 15.07(15.07) &medium & 15.08(15.08)\ Data analysis {#Sec:analys} ============= In this section, the basic concepts of the X-ray data reduction and source detection processes are described. Screening for high background {#sec:Screening} ----------------------------- The first step was to exclude times of increased background, due to soft proton flares. Most of these times are located at the start or end of an orbit window. We selected good time intervals (GTIs) – intervals where the intensity was lower than a certain threshold – using 7–15keV light curves constructed from source-free regions of each observation. The GTIs with PN and MOS data were determined from the higher statistic PN light curves. Outside the PN time coverage, GTIs were determined from the combined MOS light curves. For each observation, the limiting thresholds for the count rate were adjusted individually; this way we avoided cutting out short periods (up to a few hundred seconds) of marginally increased background. Short periods of low background, which were embedded within longer periods of high background, were omitted. For most observations, the PN count rate thresholds were 2–8ctsks$^{-1}$arcmin$^{-2}$. As many of the observations were affected by strong background flares, the net exposure which can be used for our analysis was strongly reduced. The GTIs of the various observations ranged over 6–56ks, apart from observation b2 (ObsID 0202230301) which had to be rejected, because it showed high background throughout the observation. The exposures for all three EPIC instruments are given in Cols. 8, 10 and 12 of Table \[tab:observations\]. The observations obtained during the summer visibility window of 31 were affected more strongly by background radiation than those taken during the winter window. The most affected observations of the deep survey were reobserved. After screening for times of enhanced particle background, the second step was to examine the influence of solar wind charge exchange. This was done by producing soft energy ($<\!2$keV) background light curves. These lightcurves varied only for 10 observations, for which additional screening was necessary. The screening of enhanced background due to solar wind charge exchange was applied to the observations only for the creation of colour images, in order to avoid that these observations will appear in the mosaic image with a tinge of red. The screening was not used for source detection. The third and last step includes the study of the background due to detector noise. The processing chains take into account all known bad or hot pixels and columns and flag the affected pixels in the event lists. We selected data with [(FLAG & 0xfa0000)=0]{}, excluded rows and columns near edges, and searched by eye for additional warm or hot pixels and columns in each observation. To avoid background variability over the PN images, we omitted the energy range from 7.2–9.2keV where strong fluorescence lines cause higher background in the outer detector area [@2004SPIE.5165..112F]. An additional background component can occur during the EPIC PN offset map calculation. If this period is affected by high particle background, the offset calculation will lead to a slight underestimate of the offset in some pixels which can then result in blocks of pixels ($\approx 4\!\times\!4$) with enhanced low energy signal.[^4] These blocks will be found by the [SAS]{} detection tools and appear as sources with extremely soft spectrum (so called supersoft sources). To reduce the number of false detections in this source class, we decided to include the task [epreject]{} in [epchain]{}, which locates the pixels with a slight underestimate of the offset and corrects this underestimate. To ensure that [epreject]{} produces reliable results, difference images of the event lists obtained with and without [epreject]{}, were created. Only events with energies above 200eV were used. We checked whether [epreject]{} removed all pixels with an enhanced low energy signal. Only in observation ns1 the difference image still shows a block of pixels with enhanced signal. As this block is also visible at higher energies (PHA$>30$) it cannot be corrected with [epreject]{}. Additionally, we ascertained that almost all pixels not affected during the offset map calculation have a value consistent with zero in the difference images, with two exceptions discussed in Sect.\[Sec:srccat\]. Images {#Sec:Images} ------ For each observation, the data were split into five energy bands: (0.2–0.5)keV, (0.5–1.0)keV, (1.0–2.0)keV, (2.0–4.5)keV, and (4.5–12)keV. For the PN data, we used only single-pixel events (PATTERN$=$0) in the first energy band, while for the other bands, single-pixel and double-pixel events were selected (PATTERN$\le$4). In the MOS cameras, single-pixel to quadruple-pixel events (PATTERN$\le$12) were used. We created images, background images and exposure maps (with and without vignetting correction) for PN, MOS1 and MOS2 in each of the five energy bands and masked them for the acceptable detector area. The image bin size is 2. The same procedure was applied in our previous 31 and M 33 studies . To create background images, the [SAS]{} task [eboxdetect]{} was run in local mode, in which it determines the background from the surrounding pixels of a sliding box, with box sizes of $5\times5$, $10\times10$ and $20\times20$ pixels (10$\times$10, 20$\times$20and 40$\times$40). The detection threshold is set to [likemin=15]{}, which is a good compromise between cutting out most of the sources and leaving sufficient area to derive the appropriate background. For the background calculation, a two dimensional spline is fitted to a rebinned and exposure corrected image (task [esplinemap]{}). The number of bins used for rebinning is controlled by the parameter [nsplinenodes]{}, which is set to 16 for all but the observations of the central region, where it was set to 20 (maximum value). For PN, the background maps contain the contribution from the “out of time (OoT)" events. Source detection {#Sec:SrcDet} ---------------- For each observation, source detection was performed simultaneously on 5 energy bands for each EPIC camera, using the XMM-[SAS]{} detection tasks [eboxdetect]{} and [emldetect]{}, as such fitting provides the most statistically robust measurements of the source positions by including all of the data. This method was also used to generate the 2XMM catalog . In the following we describe the detection procedure used. The source detection procedure consists of two consecutive detection steps. An initial source list is created with the task [eboxdetect]{} ( Sect.\[Sec:Images\]). To select source candidates down to a low statistical significance level, a low likelihood threshold of four was used at this stage. The background was estimated from the previously created background images (see Sect.\[Sec:Images\]). This list is the starting point for the XMM-[SAS]{} task [emldetect]{} (v. 4.60.1). The [emldetect]{} task performs a Maximum Likelihood fit of the distribution of source counts [based on Cash C-statistics approach; @1979ApJ...228..939C], using a point spread function model obtained from ray tracing calculations. If $P$ is the probability that a Poissonian fluctuation in the background is detected as a spurious source, the likelihood of the detection is then defined as $\mathcal{L}=-\ln{\left}( P {\right})$.[^5] The fit is performed simultaneously in all energy bands for all three cameras by summing the likelihood contribution of each band and each camera. Sources exceeding the detection likelihood threshold in the full band (combination of the 15 bands) are regarded as detections; the catalogue is thus full band selected. The detection threshold used is 7, as in PFH2005. Some other parameters differ from the values used in PFH2005, as in this work a parameter setting optimised for the detection of extended sources was used (G. Lamer; private communication). The parameters in question are the event cut-out ([ecut=30.0]{}) and the source selection radius ([scut=0.9]{}) for multi-source fitting, the maximum number of sources into which one input source can be split ([nmulsou=2]{}), and the maximum number of sources that can be fitted simultaneously ([nmaxfit=2]{}). Multi-PSF fitting was performed in a two stage process for objects with a detection likelihood larger than ten. All of the sources were also fitted with a convolution of a $\beta$-model cluster brightness profile with the  point spread function, in order to detect any possible extension in the detected signal. Sources which have a core radius significantly larger than the PSF are flagged as extended. The free parameters of the fit were the source location, the source extent and the source counts in each energy band of each telescope. To derive the X-ray flux of a source from its measured count rate, one uses the so-called energy conversion factors (ECF): $${\mathrm}{Flux}=\frac{{\mathrm}{Rate}}{{\mathrm}{ECF}}$$ These factors were calculated using the detector response, and depended on the used filter, the energy band in question, and the spectrum of the source. As we wanted to apply the conversion factors to all sources found in the survey, we assumed a power law model with photon index $\Gamma\!=\!1.7$ and the Galactic foreground absorption of $N_{{\mathrm}{H}}\!=\!7\times10^{20}$cm$^{-2}$ [@1992ApJS...79...77S see also PFH2005] to be the universal source spectrum for the ECF calculation. The ECFs (see Table \[tab:ECFvalues\]) were derived with [XSPEC]{}[^6](v 11.3.2) using response matrices (V.7.1) available from the  calibration homepage[^7]. As all necessary corrections of the source parameters ( vignetting corrections) were included in the image creation and source detection procedure[^8], the *on axis* ECF values were derived . The fluxes determined with the ECFs given in Table \[tab:ECFvalues\] are absorbed ( observed) fluxes and hence correspond to the observed count rates, which are derived in the [emldetect]{} task. During the mission lifetime, the MOS energy distribution behaviour has changed. Near the nominal boresight positions, where most of the detected photons hit the detectors, there has been a decrease in the low energy response of the MOS cameras [@2006ESASP.604..925R]. To take this effect into account, different response matrices for observations obtained before and after the year 2005 were used (see Table \[tab:ECFvalues\]). [lrrrrrr]{} & & & & & &\ & &\ EPIC PN & thin & $11.33$ & $8.44$ & $5.97$ & $1.94$ & $0.58$\ & medium & $10.05$ & $8.19$ & $5.79$ & $1.94$ & $0.58$\ EPIC MOS1 & thin & $2.25$ & $1.94$ & $2.06$ & $0.76$ & $0.14$\ & medium & $2.07$ & $1.90$ & $2.07$ & $0.75$ & $0.15$\ EPIC MOS2 & thin & $2.29$ & $1.98$ & $2.09$ & $0.78$ & $0.15$\ & medium & $2.06$ & $1.90$ & $2.04$ & $0.75$ & $0.15$\ EPIC MOS1 & thin & $2.59$ & $2.04$ & $2.12$ & $0.76$ & $0.15$\ OLD & medium & $2.33$ & $1.98$ & $2.09$ & $0.76$ & $0.15$\ EPIC MOS2 & thin & $2.58$ & $2.04$ & $2.13$ & $0.76$ & $0.15$\ OLD & medium & $2.38$ & $1.99$ & $2.09$ & $0.75$ & $0.16$\ \[tab:ECFvalues\] For most sources, band 5 just adds noise to the total count rate. If converted to flux, this noise often dominates the total flux due to the small ECF. To avoid this problem we calculated count rates and fluxes for detected sources in the “XID" (0.2–4.5)keV band (bands 1 to 4 combined). While for most sources this is a good solution, for extremely hard or soft sources there may still be bands just adding noise. This, then, may lead to rate and flux errors that seem to falsely indicate a lower source significance. A similar effect occurs in the combined rates and fluxes, if a source is detected primarily by one instrument ( soft sources in PN). Sources are entered in the  catalogue from the observation in which the highest source detection likelihood is obtained (either combined or single observations). For variable sources this means that the source properties given in the  catalogue (see Sect.\[Sec:srccat\] and Table 5) are those observed during their brightest state. We rejected spurious detections in the vicinity of bright sources. In regions with a highly structured background, the [SAS]{} detection task [emldetect]{} registered some extended sources. We also rejected these “sources" as spurious detections. In an additional step we checked whether an object had visible contours in at least one image out of the five energy bands. The point-like or extended nature, which was determined with [emldetect]{}, was taken into account. In this way, “sources" that are fluctuations in the background, but which were not fully modelled in the background images, were detected. In addition, objects located on hot pixels, or bright pixels at the rim or in the corners of the individual CCD chips (which were missed during the background screening) were recognised and excluded from the source catalogue, especially if they were detected with a likelihood larger than six in one detector only. To allow for a statistical analysis, the source catalogue only contains sources detected by the [SAS]{} tasks [eboxdetect]{} and [emldetect]{} as described above,  the few sources that were not detected by the analysis program, despite being visible on the X-ray images, have not been added by hand as it was done in previous studies (SPH2008; PFH2005). To classify the source spectra, we computed four hardness ratios. The hardness ratios and errors are defined as: $${\mathrm}{HR}i = \frac{B_{i+1} - B_{i}}{B_{i+1} + B_{i}}\; \mbox{and}\;\; {\mathrm}{EHR}i = 2 \frac{\sqrt{(B_{i+1} EB_{i})^2 + (B_{i} EB_{i+1})^2}}{(B_{i+1} + B_{i})^2}, \label{Eq:hardr}$$ for [*i*]{} = 1 to 4, where $B_{i}$ and $EB_{i}$ denote count rates and corresponding errors in energy band [*i*]{}. Astrometrical corrections {#SubSec:AstCorr} ------------------------- To obtain astrometrically-corrected positions for the sources of the five central fields we used the [SAS]{}-task [eposcorr]{} with  source lists [@2002ApJ...577..738K; @2002ApJ...578..114K; @2004ApJ...609..735W]. For the other fields we selected sources from the USNO-B1 [@2003AJ....125..984M], 2MASS [@2006AJ....131.1163S] and Local Group Galaxy Survey [LGGS; @2006AJ....131.2478M] catalogues[^9]. ### Astrometry of optical/infrared catalogues In a first step, we examined the agreement between the positions given by the various optical catalogues.[^10] A close examination of the shifts obtained, showed significant differences between the positions given in the individual catalogues. In summary, between the USNO-B1 and LGGS catalogues we found an offset of: $-$0197 in R.A. and 0067 in Dec[^11]; and between the USNO-B1 and 2MASS catalogues we found an offset of: $-$0108 in R.A. and 0204 in Dec. We chose the USNO-B1 catalogue as a reference, since it covers the entire field observed in the Deep  survey, and in addition it provides values for the proper motion of the optical sources. Since the optical catalogues, as well as the Deep  catalogue, are composed of individual observations of sub-fields of 31, we searched for systematic drifts in the positional zero points from region to region. However no systematic offsets were found. Finally, we applied the corrections found to the sources in the LGGS and 2MASS catalogues, to bring all catalogues to the USNO-B1 reference frame. The offsets found between the USNO-B1 and 2MASS catalogues can be explained by the independent determination of the astrometric solutions for these catalogues. Given that the positions provided in the LGGS catalogue are corrected with respect to the USNO-B1 catalogue [see @2006AJ....131.2478M], the offset found in right ascension was totally unexpected and cannot be explained. ### Corrections of the X-ray observations From the positionally corrected catalogues, we selected sources which either correlate with globular clusters from the Revised Bologna Catalogue or with foreground stars, characterised by their optical to X-ray flux ratio [@1988ApJ...326..680M] and their hardness ratio . For sources selected from the USNO-B1 catalogue, we used the proper motion corrected positions. We then used the [SAS]{}-task [eposcorr]{} to derive the offset of the X-ray aspect solution. Four observations did not have enough optical counterparts to apply this method. The lack of counterparts is due to the very short exposure times resulting after the screening for high background (obs. s3, ss12, ss13) and the location of the observation (obs. sn11). In these cases, we used bright persistent X-ray sources, which we correlated with another observation of the same field. We checked for any residual systematic uncertainty in the source positions and found it to be well characterised by a conservative $1\sigma$ value of 05. This uncertainty is due to positional errors of the optical sources as well as inaccuracy in the process of the determination of the offset between optical and X-ray sources, and is called systematic positional error. The appropriate offset, given in Col. 6 of Table \[tab:observations\], was applied to the event file of each pointing, and images and exposure maps were then reproduced with the corrected astrometry.\ Fields that were observed at least twice are treated in a special way, which is described in the following section. Multiple observations of selected fields ---------------------------------------- The fields that were observed more than once were the central field, the fields pointing on RX J0042.6+4115[^12], two fields located on the major axis of 31  (S2, N2) and all fields of the “Large Survey" located in the southern part of the galaxy (SS1, SS2, SS3, S3, SN3, SN2, SN1). To reach higher detection sensitivity we merged the images, background images and exposure maps of observations which have the same pointing direction and were obtained with the same filter setting. Subsequently, source detection, as described in Sect. \[Sec:SrcDet\], was repeated on the merged data. For the S2 field, there are two observations with different filter settings. In this case, source detection was performed simultaneously on all 15 bands of both observations,  on 30 bands simultaneously. The N2 field was treated in the same way. For the central field images, background images and exposure maps of observations c1, c2 and c3 were merged. These merged data were used together with the data of observation c4 to search for sources simultaneously; in this way it was possible to take into account the different ECFs for the different filters. One field was observed twice with slightly different pointing direction in observations sn1 and sn11; simultaneous source detection was used for these observations also. Variability calculation {#Sec:DefVar} ----------------------- To examine the time variability of each source listed in the total source catalogue, we determined the XID flux at the source position in each observation or at least an upper limit for the XID flux. We used the task [emldetect]{} with fixed source positions when calculating the total flux. To get fluxes and upper limits for all sources in the input list we set the detection likelihood threshold to 0. A starting list was created from the full source catalogue, which only contains the identification number and position of each source located in the field examined. To give correct results, the task [emldetect]{} has to process the sources from the brightest one to the faintest one. We, therefore, had to first order the sources in each observation by the detection likelihood. For sources not visible in the observation in question we set the detection likelihood to 0. This list was used as input for a first [emldetect]{} run. In this way we achieved an output list in which a detection likelihood was allocated to every source. For a final examination of the sources in order of detection likelihood, a second [emldetect]{} run was necessary. We only accepted XID fluxes for detections $\ge$ 3 $\sigma$; otherwise we used a 3 $\sigma$ upper limit. To compare the XID fluxes between the different observations, we calculated the significance of the difference $$S=\frac{F_{{\mathrm}{max}}- F_{{\mathrm}{min}}}{\sqrt{\sigma_{{\mathrm}{max}}^2+\sigma_{{\mathrm}{min}}^2}}$$ and the ratio of the XID fluxes $V=F_{{\mathrm}{max}}/F_{{\mathrm}{min}}$, where $F_{{\mathrm}{max}}$ and $F_{{\mathrm}{min}}$ are the maximum and minimum (or upper limit) source XID flux, and $\sigma_{{\mathrm}{max}}$ and $\sigma_{{\mathrm}{min}}$ are the errors of the maximum and minimum flux, respectively. This calculation was not performed whenever $F_{{\mathrm}{max}}$ was an upper limit. Finally, the largest XID flux of each source was derived, excluding upper limits. Spectral analysis ----------------- To extract the X-ray spectrum of individual sources, we selected an extraction region and a corresponding background region which was at least as large as the source region, was located on the same CCD at a similar off axis angle as the source, and did not contain any point sources or extended emission. For EPIC PN, we only accepted single-pixel events for the spectra of supersoft sources, while for all other spectra single and double-pixel events were used. For the EPIC-MOS detectors, single-pixel through to quadruple-pixel events were always used. Additionally, we only kept events with FLAG$=$0 for all three detectors. For each extraction region, we produced the corresponding response matrix files and ancillary response files. For each source, the spectral fit was obtained by fitting all three EPIC spectra simultaneously, using the tool [XSPEC]{}. For the absorption, we used the [TBabs]{} model, with abundances from @2000ApJ...542..914W and photoelectric absorption cross-sections from @1992ApJ...400..699B with a new He cross-section based on @1998ApJ...496.1044Y. Cross correlations {#Sec:CrossCorr_Tech} ------------------ Sources were regarded as correlating if their positions overlapped within their 3$\sigma$ (99.73%) positional errors, defined as : $$\Delta{\mathrm}{pos}\le3.44\times\sqrt{\sigma_{{\mathrm}{stat}}^2 + \sigma_{{\mathrm}{syst}}^2}+3\times\sigma_{{\mathrm}{ccat}} \label{Eq:Cor}$$ where $\sigma_{{\mathrm}{stat}}$ is the statistical and $\sigma_{{\mathrm}{syst}}$ the systematic error of the X-ray sources detected in the present study. The statistical error was derived by [emldetect]{}. The determination of the systematic error is described in Sect.\[SubSec:AstCorr\]. We use a value of 05, for all sources. The positional error of the sources in the catalogue used for cross-correlation is given by $\sigma_{{\mathrm}{ccat}}$. The values of $\sigma_{{\mathrm}{ccat}}$ (68% error) used for the different X-ray catalogues can be found in Table \[Tab:XrayRefCat\]. Exceptions to Eq. \[Eq:Cor\] are sources that are listed in more than one catalogue or that are resolved into multiple sources with . The first case is restricted to catalogues with comparable spatial resolution and hence positional uncertainty. To identify the X-ray sources in the field of 31 we searched for correlations with catalogues in other wavelength regimes. The  source catalogue was correlated with the following catalogues and public data bases: Globular Clusters: : Bologna Catalogue , @2009AJ....137...94C [$\sigma_{{\mathrm}{ccat}}=0.\arcsec2$], @2009AJ....138..770H [$\sigma_{{\mathrm}{ccat}}=0.\arcsec5$], @2008PASP..120....1K [$\sigma_{{\mathrm}{ccat}}=0.\arcsec2$], @2007PASP..119....7K [$\sigma_{{\mathrm}{ccat}}=0.\arcsec2$], @2005PASP..117.1236F, @1993PhDT........41M [$\sigma_{{\mathrm}{ccat}}=1\arcsec$] Novae: : Nova list of the 31 Nova Monitoring Project[^13] ($\sigma_{{\mathrm}{ccat}}$ is given for each individual source), PHS2007, @2010AN....331..187P Supernova Remnants: : , and , ; An X-ray source is considered as correlating with a SNR, if the X-ray source position (including 3$\sigma$ error) lies within the extent given for the SNR. Radio Catalogues: : @2005ApJS..159..242G [$\sigma_{{\mathrm}{ccat}}$ is given for each individual source], @2004ApJS..155...89G [$\sigma_{{\mathrm}{ccat}}$ is given for each individual source], @2008AJ....136..684K [$\sigma_{{\mathrm}{ccat}}=3\arcsec$], @1990ApJS...72..761B [$\sigma_{{\mathrm}{ccat}}$ is given for each individual source], NVSS [NRAO/VLA Sky Survey[^14]; @1998AJ....115.1693C $\sigma_{{\mathrm}{ccat}}$ is given for each individual source] H [II]{} Regions, H $\alpha$ Catalogue: : , @2007AJ....134.2474M [$\sigma_{{\mathrm}{ccat}}=0.\arcsec2$] Optical Catalogues: : USNO-B1 [@2003AJ....125..984M $\sigma_{{\mathrm}{ccat}}$ is given for each individual source], Local Group Survey [LSG; @2006AJ....131.2478M $\sigma_{{\mathrm}{ccat}}=0.\arcsec2$] Infrared catalogues: : 2MASS [@2006AJ....131.1163S $\sigma_{{\mathrm}{ccat}}$ is given for each individual source], @2008ApJ...687..230M [$\sigma_{{\mathrm}{ccat}}=0.\arcsec8$, for Table 2: $\sigma_{{\mathrm}{ccat}}=0.\arcsec5$] Data bases: : the SIMBAD catalogue[^15] (Centre de Données astronomiques de Strasbourg; hereafter SIMBAD) , the NASA Extragalactic Database[^16] (hereafter NED) [lrlr]{} & & &\ PFH2005 & $*$ & DKG2004 & 03\ SPH2008 & $*$ & WNG2006 & 03\ SHP97 & $*$ & VG2007 & 04\ SHL2001 & $*$ & OBT2001 & 3\ PFJ93 & $*$ & O2006 & 1\ TF91 & $*$ & SBK2009 & 3$^{+}$\ Ka2002 & 03 & D2002 & 05\ KGP2002 & $*$ & TP2004 & 1\ WGK2004 & 1$^{+}$ & ONB2010 & 1\ \[Tab:XrayRefCat\] Notes:\ $^{ {\dagger}~}$: $*$ indicates that the catalogue provides $\sigma_{{\mathrm}{ccat}}$ values for each source individually\ $^{ +~}$: value taken from indicated paper\ $^{ {\ddagger}~}$: TF91: @1991ApJ...382...82T, PFJ93: @1993ApJ...410..615P, SHP97: , SHL2001: , OBT2001: , D2002: @2002ApJ...570..618D, KGP2002: @2002ApJ...577..738K, Ka2002: @2002ApJ...578..114K, WGK2004: @2004ApJ...609..735W, DKG2004: @2004ApJ...610..247D, TP2004: @2004ApJ...616..821T, PFH2005: , O2006: @2006ApJ...643..844O, WNG2006: @2006ApJ...643..356W, VG2007: , SPH2008: , SBK2009: , ONB2010: @2010ApJ...717..739O Colour image {#Sec:coim} ============ Figure \[Fig:cimage\] shows the combined, exposure corrected EPIC PN, MOS1 and MOS2 RGB (red-green-blue) mosaic image of the Deep Survey and archival data. The colours represent the X-ray energies as follows: red: 0.2–1.0keV, green: 1.0–2.0keV and blue: 2.0–12keV. The optical extent of 31 is indicated by the $\mathrm{D_{25}}$ ellipse and the boundary of the observed field is given by the green contour. The image is smoothed with a 2D-Gaussian of 20 FWHM. In some observations, individual noisy MOS1 and MOS2 CCDs are omitted. The images have not been corrected for the background of the detector or for vignetting.\ The colour of the sources reflects their class. Supersoft sources appear in red. Thermal SNRs and foreground stars are orange to yellow. “Hard" sources (background objects, mainly AGN, and X-ray binaries or Crab-like SNRs) are blue to white. {width="12cm"} Logarithmically scaled  EPIC low background images made up of the combined images from the PN, MOS1 and MOS2 cameras in the (0.2–4.5) keV XID band for each 31 observation can be found in the Appendix. The images also show X-ray contours, and the sources from the  catalogue are marked with boxes. Source catalogue () {#Sec:srccat} =================== The source catalogue of the Deep  survey of 31 (hereafter  catalogue) contains 1897 X-ray sources. Of these sources 914 are detected for the first time in X-rays. The source parameters are summarised in Table 5, which gives the source number (Col. 1), detection field from which the source was entered into the catalogue (2), source position (3 to 9) with $3\sigma$ (99.73%) uncertainty radius (10), likelihood of existence (11), integrated PN, MOS1 and MOS2 count rate and error (12,13) and flux and error (14,15) in the (0.2–4.5) keV XID band, and hardness ratios and errors (16–23). Hardness ratios are calculated only for sources for which at least one of the two band count rates has a significance greater than $2\sigma$. Errors are the properly combined statistical errors in each band and can extend beyond the range of allowed values of hardness ratios as defined previously (–1.0 to 1.0; Eq. \[Eq:hardr\]). The “Val” parameter (Col 24) indicates whether the source is within the field of view (true or false, “T” or “F”) in the PN, MOS1 and MOS2 detectors respectively. Table 5 also gives the exposure time (25), source existence likelihood (26), the count rate and error (27, 28) and the flux and error (29, 30) in the (0.2–4.5)keV XID band, and hardness ratios and errors (31–38) for the EPIC PN. Columns 39 to 52 and 53 to 66 give the same information corresponding to Cols. 25 to 38, but for the EPIC MOS1 and MOS2 instruments. Hardness ratios for the individual instruments were again screened as described above. From the comparison between the hardness ratios derived from the integrated PN, MOS1 and MOS2 count rates (Cols. 16–23) and the hardness ratios from the individual instruments (Cols. 31–38, 45–52 and 59–66), it is clear that the combined count rates from all instruments yielded a significantly larger fraction of hardness ratios above the chosen significance threshold. Column 67 shows cross correlations with published 31 X-ray catalogues ( Sect.\[Sec:CrossCorr\_Tech\]). We discuss the results of the cross correlations in Sects.\[Sec:fgback\] and \[Sec:Srcsm31\]. In the remaining columns of Table 5, we give information extracted from the USNO-B1, 2MASS and LGGS catalogues ( Sect.\[Sec:CrossCorr\_Tech\]). The information from the USNO-B1 catalogue (name, number of objects within search area, distance, B2, R2 and I magnitude of the brightest[^17] object) is given in Cols. 68 to 73. The 2MASS source name, number of objects within search area, and the distance can be found in Cols. 74 to 76. Similar information from the LGGS catalogue is given in Cols. 77 to 82 (name, number of objects within search area, distance, V magnitude, V-R and B-V colours of the brightest[^18] object). To improve the reliability of source classifications we used the USNO-B1 B2 and R2 magnitudes to calculate $$\log{\left}(\frac{f_{{\mathrm}{x}}}{f_{{\mathrm}{opt}}}{\right}) = \log{\left}(f_{{\mathrm}{x}}{\right}) + \frac{m_{{\mathrm}{B2}} + m_{{\mathrm}{R2}}}{2\times2.5} + 5.37, \label{Eq:fxopt}$$ and the LGGS V magnitude to calculate $$\log{\left}(\frac{f_{{\mathrm}{x}}}{f_{{\mathrm}{opt}}}{\right}) = \log{\left}(f_{{\mathrm}{x}}{\right}) + \frac{m_{{\mathrm}{V}}}{2.5} + 5.37, \label{Eq:fxvopt}$$ following @1988ApJ...326..680M [ see Cols. 83–86]. The X-ray sources in the  catalogue are identified or classified based on properties in X-rays (HRs, variability, extent) and of the correlated objects in other wavelength regimes (Cols. 87 and 88 in Table 5). For classified sources the class name is given in angled brackets. Identification and classification criteria are summarised in Table \[Tab:class\], which provides, for each source class (Col.1), the classification criteria (2), and the numbers of identified (3) and classified (4) sources. The hardness ratio criteria are based on model spectra. Details on the definition of these criteria can be found in Sect.6 of PFH2005. As we have no clear hardness ratio criteria to discriminate between XRBs, Crab-like supernova remnants (SNRs) or AGN we introduced a $<$hard$>$ class for those sources. If such a source shows strong variability (i.e. V$\ge$10) on the examined time scales it is likely to be an XRB. Compared with SPH2008 the HR2 selection criterion for SNRs was tightened (from HR2$<\!-0.2$ to HR2$+$EHR2$<\!-0.2$) to exclude questionable SNR candidates from the class of SNRs. If we applied the former criterion to the survey data, $\sim$35 sources would be classified as SNRs in addition to those listed in Table \[Tab:class\]. Most of the 35 sources are located outside the D$_{25}$ ellipse, and none of them correlates with an optically identified SNR, a radio source, or an H[II]{} region. In addition, the errors in HR2 are of the same order as the HR2 values. It is therefore very likely that these sources do belong to other classes, since the strip between $-0.3\!<$HR2$<$0 is populated by foreground stars, XRBs, background objects, and candidates for these three classes. Outcomes of the identification and classification processes are discussed in detail in Sects.\[Sec:fgback\] and \[Sec:Srcsm31\]. The last column (89) of Table 5 contains the  source name as registered to the IAU Registry. Source names consist of the acronym XMMM31 and the source position as follows: XMMM31 Jhhmmss.s+ddmmss, where the right ascension is given in hours (hh), minutes (mm) and seconds (ss.s) truncated to decimal seconds and the declination is given in degrees (dd), arc minutes (mm) and arc seconds (ss) truncated to arc seconds, for equinox 2000. In the following, we refer to individual sources by their source number (Col.1 of Table 5), which is marked with a “" at the front of the number. Of the 1897 sources, 1247 can only be classified as $<$hard$>$ sources, while 123 sources remain without classification. Two of them ( 482,  768) are highly affected by optical loading; both “X-ray sources" coincide spatially with very bright optical foreground stars (USNO-B1 R2 magnitudes of 6.76 and 6.74 respectively). The spectrum of source  482 is dominated by optical loading. This becomes evident from the hardness ratios which indicate an SSS. For  768 the hardness ratios would allow a foreground star classification. The obtained count rates and fluxes of both sources are affected by the usage of [epreject]{}, which neutralises the corrections applied for optical loading. Therefore residuals are visible in the difference images created from event lists obtained with and without [epreject]{}. As we cannot exclude the possibility that some of the detected photons are true X-rays – especially for source  768 –, we decided to include them in the  catalogue, but without a classification. [llrr]{} & & &\ fg Star & ${\rm log}({{f}_{\rm x} \over {f}_{\rm opt}})\!<\!-1.0$ and HR2$-$EHR$2\!<\!0.3$ and HR3$-$EHR$3\!<\!-0.4$ or not defined & 40 & 223\ AGN & Radio source and not classification as SNR from HR2 or optical/radio & 11 & 49\ Gal & optical id with galaxy & 4 & 19\ GCl & X-ray extent and/or spectrum & 1 & 5\ SSS & HR$1\!<\!0.0$, HR2$-$EHR$2\!<\!-0.96$ or HR2 not defined, HR3, HR4 not defined & & 30\ SNR & HR$1\!>\!-0.1$ and HR2$+$EHR$2\!<\!-0.2$ and not a fg Star, or id with optical/radio SNR & 25 & 31\ GlC & optical id & 36 & 16\ XRB & optical id or X-ray variability & 10 & 26\ hard & HR2$-$EHR$2\!>\!-0.2$ or only HR3 and/or HR4 defined, and no other classification& & 1247\ \[Tab:class\] Flux distribution {#Sec:flux_dist} ----------------- The faintest source ( 526) has an XID band flux of 5.8[$\times 10^{-16}$ ]{}. The source with the highest XID Flux ( 966, XID band flux of 3.75[$\times 10^{-12}$ ]{}) is located in the centre of 31 and identified as a Z-source LMXB . This source has a mean absorbed XID luminosity of 2.74[$\times 10^{38}$ ]{}. Figure \[Fig:XIDfluxdist\] shows the distribution of the XID (0.2–4.5keV) source fluxes. Plotted are the number of sources in a certain flux bin. We see from the inlay that the number of sources starts to decrease in the bin from 2.4 to 2.6[$\times 10^{-15}$ ]{}. This XID flux roughly determines the completeness limit of the survey and corresponds to an absorbed 0.2–4.5keV limiting luminosity of $\sim\!2$[$\times 10^{35}$ ]{}. Previous X-ray studies [@2004ApJ...609..735W and references therein] noted a lack of bright sources ($L_{{\mathrm}{X}}\!\ga$[$10^{37}$ erg s$^{-1}$]{}; 0.1–10keV) in the northern half of the disc compared to the southern half. This finding is not supported in the present study. Excluding the pointings to the centre of 31, we found in the remaining observations 13 sources in each hemisphere that were brighter than $L_{{\mathrm}{X\,abs}}\!\ga$[$10^{37}$ erg s$^{-1}$]{}.[^19] The reason our survey does not support the old results is that we found several bright sources in the outer regions of the northern half of the disk, which have not been covered in @2004ApJ...609..735W [and references therein]. In the central field of 31, a total of 41 sources brighter than $L_{{\mathrm}{X}}\!\ga$[$10^{37}$ erg s$^{-1}$]{} (0.2–4.5keV) were found. Figure \[Fig:brightS\] shows the spatial distribution of the bright sources. Striking features are the two patches located north and south of the centre. The southern one seems to point roughly in the direction of M 32 ( 995), while the northern one ends in the globular cluster B116 ( 947). However there is no association to any known spatial structure of 31, like  the spiral arms. Exposure map {#Sec:ExpMap} ------------ Figure \[Fig:ExpMap\] shows the exposure map used to create the colour image of all  Large Survey and archival observations (Fig.\[Fig:cimage\]). The combined MOS exposure was weighted by a factor of 0.4, before being added to the PN exposure. However, this map does not quite represent the exposures used in source detection; overlapping regions were not combined during source detection. From Fig.\[Fig:ExpMap\] we see that the exposure for most of the surveyed area is rather homogeneous. Exceptions are the central area, overlapping regions and observation h4. Hardness ratio diagrams ----------------------- We plot X-ray colour/colour diagrams based on the HRs (see Fig.\[Fig:HR\_diagrams\]). Sources are plotted as dots if the error in both contributing HRs is below 0.2. Classified and identified sources are plotted as symbols in all cases. Symbols including a dot therefore mark the well-defined HRs of a class. From the HR1-HR2 diagram (upper panel in Fig.\[Fig:HR\_diagrams\]) we note that the class of SSSs is the only one that can be defined based on hardness ratios alone. In the part of the HR1-HR2 diagram that is populated by SNRs, most of the foreground stars and some background objects and XRBs are also found. Foreground star candidates can be selected from the HR2-HR3 diagram (middle panel in Fig.\[Fig:HR\_diagrams\]), where most of them are located in the lower left corner. The HR3-HR4 diagram (lower panel in Fig.\[Fig:HR\_diagrams\]) does not help to disentangle the different source classes. Thus, we need additional information from correlations with sources in other wavelengths or on the source variability or extent to be able to classify the sources. Extended sources {#Sec:ExtSrcs} ---------------- The  catalogue contains 12 sources which are fitted as extended sources with a likelihood of extension larger than 15. This value was chosen so as to minimise the number of spurious detections of extended sources (H. Brunner; private communication), as well as keeping all sources that can clearly be seen as extended sources in the X-ray images. A convolution of a $\beta$-model cluster brightness profile with the  point spread function was used to determine the extent of the sources ( Sect.\[Sec:SrcDet\]). This model describes the brightness profile of galaxy clusters, as $$f{\left}(x,y{\right})={\left}(1+\frac{{\left}(x-x_0{\right})^2+{\left}(y-y_0{\right})^2}{r_{\rm{c}}^2}{\right})^{-3/2},$$ where $r_{\rm{c}}$ denotes the core radius; this is also the extent parameter given by [emldetect]{}. Table \[Tab:ExtSrcs\] gives the source number (Col. 1), likelihood of detection (2), the extent found (3) and its associated error (4) in arcsec, the likelihood of extension (5), and the classification of the source (6, see Sect.\[SubSec:Gal\_GCl\_AGN\]) for each of the 12 extended sources. Additional comments taken from Table 5 are provided in the last column. [rrrrrrrcl]{} & & & & & & & &\ & & & & & & & &\ 141 & 65.08 & 11.22 & 1.29 & 23.68 & 1.45 & 0.20 & $<$GCl$>$ & GLG127(Gal), 37W 025A (IR, RadioS; NED)\ 199 & 275.16 & 17.33 & 1.05 & 174.73 & 4.31 & 0.29 & $<$hard$>$ &\ 252 & 222.05 & 14.64 & 1.12 & 81.60 & 4.40 & 0.49 & $<$GCl$>$ & 5 optical objects in error box\ 304 & 299.75 & 15.10 & 0.92 & 133.62 & 2.20 & 0.18 & $<$GCl$>$ & B242 \[CHM09\]; RBC3.5: $<$GlC$>$\ 442 & 33.76 & 11.60 & 1.71 & 15.44 & 1.62 & 0.28 & $<$hard$>$ &\ 618 & 271.08 & 6.20 & 0.73 & 42.86 & 3.15 & 0.21 & $<$hard$>$ &\ 718 & 77.75 & 7.18 & 1.23 & 21.47 & 0.58 & 0.07 & Gal & B052 \[CHM09\], RBC3.5\ 1130 & 168.31 & 10.80 & 0.97 & 44.23 & 3.27 & 0.31 & $<$hard$>$ &\ 1543 & 70.49 & 11.87 & 1.37 & 28.63 & 1.51 & 0.19 & $<$GCl$>$ & \[MLA93\] 1076 PN (SIM,NED)\ 1795 & 11416.36 & 18.79 & 0.29 & 4169.74 & 98.87 & 1.43 & GCl & GLG253 (Gal), \[B90\] 473, z=0.3 \[KTV2006\]\ 1859 & 107.09 & 13.73 & 1.40 & 43.89 & 1.23 & 0.19 & $<$hard$>$ &\ 1912 & 332.06 & 23.03 & 1.23 & 213.90 & 5.43 & 0.37 & $<$GCl$>$ & cluster of galaxies candidate\ \[Tab:ExtSrcs\] Notes:\ $^{ +~}$: Extent and error of extent in units of 1; 1 corresponds to 3.8pc at the assumed distance of 31\ $^{ *~}$: XID Flux and flux error in units of 1[$\times 10^{-14}$ ]{}\ $^{ \dagger~}$: Taken from Table 5 The extent parameter found for the sources ranges from 62 to 2303 (see Fig.\[Fig:extdist\]). The brightest source ( 1795), which has the highest likelihood of extension and the second largest extent, was identified from its X-ray properties as a galaxy cluster located behind 31 [@2006ApJ...641..756K]. The iron emission lines in the X-ray spectrum yield a cluster redshift of $z\!=\!0.29$. For further discussion see Sect.\[SubSec:Gal\_GCl\_AGN\]. Variability between *XMM-Newton* observations {#Sec:var} ============================================= To examine the long-term time variability of each source, we determined the XID flux at the source position in each observation or at least an upper limit for the XID flux. The XID fluxes were used to derive the variability factor and the significance of variability ( Sect.\[Sec:DefVar\]). The sources are taken from the  catalogue (Table 5). Table 8 contains all information necessary to examine time variability. Sources are only included in the table if they are observed at least twice. Column 1 gives the source number. Columns 2 and 3 contain the flux and the corresponding error in the (0.2–4.5) keV XID band. The hardness ratios and errors are given in columns 4 to 11. Column 12 gives the type of the source. All this information was taken from Table 5. The subsequent 140 columns provide information related to individual observations in which the position of the source was observed. Column 13 gives the name of one of these observations, which we will call observation 1. The EPIC instruments contributing to the source detection in observation 1, are indicated by three characters in the “obs1\_val" parameter (Col. 14, first character for PN, second MOS1, third MOS2), each one being either a “T" if the source is inside the FoV, or “F" if it lies outside the FoV. Then the count rate and error (15,16) and flux and error (17,18) in the (0.2–4.5) keV XID band, and hardness ratios and error (19–26) of observation 1 are given. Corresponding information is given for the remaining observations which cover the position of the source: obs. 2 (cols. 27–40), obs. 3 (41–54), obs. 4 (55–68), obs. 5 (69–82), obs. 6 (83–96), obs. 7 (97–110), obs. 8 (111–124), obs. 9 (125–138), obs. 10 (139–152). Whether the columns corresponding to obs. 3 – obs. 10 are filled in or not, depends on the number of observations in which the source has been covered in the combined EPIC FoV. This number is indicated in column 153. The maximum significance of variation and the maximum flux ratio (fvar\_max) are given in columns 154 and 155. As described in Sect.\[Sec:DefVar\], only detections with a significance greater than 3$\sigma$ were used, otherwise the 3$\sigma$ upper limit was adopted. Column 156 indicates the number of observations that provide only an upper limit. The maximum flux (fmax) and its error are given in columns 157 and 158. In a few cases a maximum flux value could not be derived, because each observation only yielded an upper limit. There can be two reasons for this: The first reason is that faint sources detected in merged observations may not be detected in the individual observations at the 3$\sigma$ limit. The second reason is that in cases where the significance of detection was not much above the 3$\sigma$ limit, it can become smaller than the 3$\sigma$ limit when the source position is fixed to the adopted final mean value from all observations. [llrrrcl]{} & & & & & &\ 966 & 1.63 & 49.01 & 46.73 & 0.59 & XRB & 1(sv,z), 2, 10(v), 12(v), 13, 14, 20, 22(v), 25(LMXB), 27, 28(1.56)\ 877 & 3.13 & 49.13 & 16.06 & 0.20 & $<$hard$>$ & 1(sv), 2, 10(v), 12(v), 13, 14, 20(v), 22(v), 27, 28(3.05)\ 745 & 2.43 & 26.89 & 12.65 & 0.18 & AGN & 13, 14\ 1157 & 1.32 & 11.10 & 9.87 & 0.25 & GlC & 1(sv), 2, 5, 10, 12, 13, 14, 20, 21, 22(v), 27, 28(1.37)\ 1060 & 2.13 & 30.00 & 9.04 & 0.14 & $<$XRB$>$ & 1(sv), 2, 10, 12, 13, 14, 20(v, NS-LMXB), 22(v), 27\ 1171 & 4.14 & 18.86 & 9.02 & 0.41 & GlC & 1(d,sv), 2(t, 53.4), 5, 10, 12, 13, 14, 16, 20, 22, 27, 28(2.47)\ 1116 & 3.76 & 51.98 & 8.16 & 0.10 & GlC & 1(sv), 2(t, 58.6), 3(t, 33), 5, 10, 12, 13, 14, 16, 20, 21, 22(v,t), 27\ \[Tab:varlist\_bright\] Notes:\ $^{ {\ddagger}~}$: maximum XID flux and error in units of 1[$\times 10^{-13}$ ]{} or maximum absorbed 0.2–4.5keV luminosity and error in units of 7.3[$\times 10^{36}$ ]{}\ $^{ {+}~}$: class according to Table \[Tab:class\]\ $^{{\dagger}~}$: for comment column see Table \[Tab:varlist\] Figure \[Fig:var\_fmax\] shows the variability factor plotted versus maximum detected XID flux. Apart from XRBs, or XRBs in GlCs, or candidates of these source classes, which were selected based on their variability, there are a few SSS candidates showing pronounced temporal variability. The sources classified or identified as AGN, background galaxies or galaxy clusters all show $F_{{\mathrm}{var}}\!<\!4$. Most of the foreground stars show $F_{{\mathrm}{var}}\!<\!4$. Out of the 1407 examined sources, we found 317 sources with a variability significance $>\!3.0$,  182 more than reported in SPH2008. For bright sources it is much easier to detect variability than for faint sources, because the difference between the maximum observed flux and the detection limit is larger. Therefore the significance of the variability declines with decreasing flux. This is illustrated by the distribution of the sources marked in green in Fig.\[Fig:var\_fmax\]. Table \[Tab:varlist\] lists all sources with a variability factor larger than five. There are 69 such sources (34 in addition to SPH2008). The sources are sorted in descending order with respect to their variability factors. Table \[Tab:varlist\] gives the source number (Col. 1), maxima of flux variability (2) and maxima of the significance parameter (3). The next columns (4, 5) indicate the maximum observed flux and its error. Column 6 contains the class of the source. Sources with $F_{{\mathrm}{var}}\!\ge\!10$ that were not already classified as SSS or foreground stars, were classified as XRB. Time variability can also be helpful to verify a SNR candidate classification. If there is significant variability, the SNR classification must be rejected, and if an optical counterpart is detected, the source has to be re-classified as foreground star candidate. Column 7 contains references to the individual sources in the literature. In some cases the reference provides information on the temporal behaviour and a more precise classification (see brackets). The numbers given in connection with and @2006ApJ...643..356W are the variability factors obtained in these papers from  data. From the 69 sources of Table \[Tab:varlist\], ten show a flux variability larger than 100. With a flux variability factor $>\!690$ source  523 is the most variable source in our sample. Source  57 has the largest significance of variability, with a value of $\approx 97$. The variability significance is below 10 for just 33 sources, 15 of which show significance values below 5. Thirty-five of the variable sources are classified as XRBs or XRB candidates, and eight of them are located in globular clusters. Nine of the variable sources are SSS candidates, while six variable sources are classified as foreground stars and foreground star candidates. Table \[Tab:varlist\_bright\] lists all “bright" sources with a maximum flux larger than 8[$\times 10^{-13}$ ]{} and a flux variability smaller than five (the description of the columns is the same as in Table \[Tab:varlist\]). All seven sources listed in Table \[Tab:varlist\_bright\] (three in addition to SPH2008) have a significance of variability $>\!10$. Apart from two sources, they are XRBs (three in globular clusters) or XRB candidates. The most luminous source in our sample is source  966 with an absorbed 0.2–4.5keV luminosity of $\approx 3.3$[$\times 10^{38}$ ]{} at maximum. Figure \[Fig:var\_hr\] shows the relationship between the variability factor and the hardness ratios HR1 and HR2, respectively. The hardness ratios are taken from Table 5. The HR1 plot shows that the sample of highly variable sources includes SSS and XRB candidates, which occupy two distinct regions in this plot . The SSSs marked by triangles, appear on the left hand side, while the XRBs or XRB candidates have much harder spectra, and appear on the right. It seems that foreground stars, SSSs and XRBs can be separated, on the HR2 diagram, although there is some overlap between foreground stars and XRBs. Individual sources are discussed in the Sects.\[Sec:fgback\] and \[Sec:Srcsm31\]. Cross-correlations with other 31 X-ray catalogues {#Sec:CrossX-ray} ================================================= Cross-correlations were determined by applying Eq.\[Eq:Cor\] to the sources of the  catalogue and to sources reported in earlier X-ray catalogues. The list of X-ray catalogues used is given in Table \[Tab:XrayRefCat\]. Previous *XMM-Newton* catalogues {#SubSec:prevXMM} -------------------------------- Previous source lists based on archival  observations were presented in , PFH2005, @2006ApJ...643..844O, SPH2008, and SBK2009. Of these four studies, PFH2005 covers the largest area of 31. Table \[Tab:CompXMM\] lists all sources from previous  studies that are not detected in the present investigation. [ll]{}\ \ 6 not detected, LH$>$100: & 327 ($<$SNR$>$,LH$=$2140.0), 384 (XRB,667.0), 332 ($<$SNR$>$,654.0),\ & 316 ($<$SNR$>$,259.0), 312 ($<$SNR$>$,241.0), 281($<$hard$>$,160.0)\ 10 not detected, 20$\le$LH$<$50: & 75 ($<$SSS$>$), 423 ($<$fg Star$>$), 120 ($<$hard$>$), 505 ($<$hard$>$),\ & 220 ($<$SNR$>$), 304 ($<$fg Star$>$), 819 ($<$hard$>$), 799 ($<$SSS$>$), 413 ($<$SNR$>$), 830 ($<$hard$>$)\ 14 not detected, 15$\le$LH$<$20: & 427($<$hard$>$), 734 ($<$hard$>$), 424 ($<$hard$>$), 518 ($<$SSS$>$),\ & 232 ($<$hard$>$), 339 ($<$hard$>$), 446 ($<$SSS$>$), 219 ($<$fg Star$>$), 567 ($<$hard$>$), 256 ($<$fg Star$>$),\ & 356 ($<$hard$>$), 248 ($<$hard$>$), 160 ($<$hard$>$), 399 ()\ 21 not detected, 10$\le$LH$<$15: & 375 ($<$hard$>$), 17 ($<$hard$>$), 195 ($<$hard$>$), 417 ($<$SNR$>$),\ & 783 ($<$hard$>$), 803 ($<$hard$>$), 829 ($<$hard$>$), 135 ($<$hard$>$), 151 ($<$hard$>$), 131 ($<$hard$>$),\ & 426 ($<$hard$>$), 593 ($<$fg Star$>$), 526 ($<$hard$>$), 250 ($<$hard$>$), 62 ($<$hard$>$), 67 ($<$hard$>$),\ & 188 ($<$hard$>$), 186 ($<$AGN$>$), 510 ($<$hard$>$), 529 ($<$hard$>$), 754 ($<$hard$>$)\ 52 not detected, LH$<$10: & 599 ($<$hard$>$), 439 ($<$hard$>$), 809 ($<$hard$>$), 14 ($<$SNR$>$), 743 ($<$hard$>$),\ & 433 ($<$hard$>$), 5 (), 210 ($<$hard$>$), 97 ($<$hard$>$), 708 ($<$hard$>$), 476 (), 534 ($<$hard$>$), 501 (),\ & 170 ($<$hard$>$), 146 (SNR), 769 (), 838 ($<$hard$>$), 571 ($<$hard$>$), 816 ($<$hard$>$, 554 (), 627 ($<$hard$>$),\ & 464 ($<$fg Star$>$), 811 ($<$hard$>$), 655 ($<$hard$>$), 184 ($<$hard$>$), 447 ($<$hard$>$), 380 ($<$hard$>$),\ & 566 ($<$hard$>$), 137 ($<$fg Star$>$), 63 (), 48 (), 152 ($<$fg Star$>$), 291 ($<$hard$>$), 559 ($<$hard$>$),\ & 102 ($<$hard$>$), 740 ($<$hard$>$), 540 ($<$fg Star$>$), 240 ($<$hard$>$), 485 (), 668 ($<$hard$>$), 44 (),\ & 560 ($<$hard$>$), 836 ($<$hard$>$), 436 ($<$hard$>$), 484 ($<$fg Star$>$), 216 ($<$hard$>$), 362 ($<$hard$>$), 527 ($<$$>$), 179 ($<$hard$>$),\ & 834 ($<$hard$>$), 86 ($<$hard$>$), 455 ()\ \ \ 3 not detected, 50$\le$LH$<$100: & 874 ($<$SNR$>$,LH$=$85.5), 895 ($<$hard$>$,75.9), 882 ( ,56.4)\ 6 not detected, 10$\le$LH$<$50: & 869 (), 885 ($<$SNR$>$), 863 ($<$hard$>$), 875 ($<$SSS$>$), 893 ($<$hard$>$), 866 ($<$hard$>$)\ 6 not detected, LH$<$10: & 870 ($<$SNR$>$), 891 ($<$hard$>$), 889 ($<$hard$>$), 872 ($<$SNR$>$), 867 ($<$hard$>$), 862 ($<$SNR$>$)\ \ \ & 4 ($<$hard$>$), 18 ($<$hard$>$), 29 ($<$hard$>$), 32 ($<$hard$>$), 34 ($<$hard$>$), 45 ($<$SSS$>$), 67 ($<$hard$>$),\ & 102 ($<$hard$>$), 106 ($<$hard$>$), 117 ($<$hard$>$), 149 ($<$hard$>$), 152 ($<$hard$>$), 179 ($<$hard$>$),\ & 183 ($<$hard$>$), 184 ($<$hard$>$), 188 ($<$hard$>$), 191 ($<$hard$>$), 192 ($<$AGN$>$), 202 ($<$hard$>$),\ & 204 ($<$fg star$>$), 217 ($<$hard$>$), 249 ($<$hard$>$), 250 ($<$hard$>$), 260 ($<$hard$>$), 274 ($<$hard$>$),\ & 279 ($<$hard$>$), 285 ($<$hard$>$), 295 ($<$hard$>$), 306 ($<$hard$>$), 325 ($<$hard$>$), 333 ($<$hard$>$)\ \[Tab:CompXMM\] In the ten observations covering the major axis, and a field in the halo of 31, PFH2005 detected 856 X-ray sources with a detection likelihood threshold of 7 ( Sect.\[Sec:SrcDet\]). Of these 856 sources, only 753 sources are also present in the  catalogue,  103 sources of PFH2005 were not detected. This can be due to: the search strategy; the parameter settings used in the [emldetect]{} run; the determination of the extent of a source for the  catalogue; the more severe screening for GTIs for the  catalogue, which led to shorter final exposure times; the use of the [epreject]{} task and last but not least due to the [SAS]{} versions and calibration files applied. The search strategy of PFH2005 was optimised to detect sources located close to each other in crowded fields. This point especially explains the non-detection of the bright PFH2005 sources \[PFH2005\] 281, 312, 316, 327, 332, 384 ($\mathcal{L}>50$) in the present study, as four of them (\[PFH2005\] 312, 316, 327, 332) are located in the innermost central region of 31 where source detection is complicated by the bright diffuse X-ray emission, while \[PFH2005\] 281 and 384 lie in the immediate vicinity of two bright sources (\[PFH2005\] 280 and 381 at distances of 7.7and 5.5, respectively). The changes in the [SAS]{} versions and the GTIs, in particular, affect sources with small detection likelihoods ($\mathcal{L}<10$). The improvements in the [SAS]{} detection tools and calibration files should reduce the number of spurious detections, which increase with decreasing detection likelihood. However, this does not necessarily imply that *all* undetected sources with $\mathcal{L}<10$ of PFH2005 are spurious detections. The changes in the [SAS]{} versions, calibration files and GTIs do not only affect the source detection tasks, but also can cause changes in the background images. These changes may increase the assumed background value at the position of a source, which would result in a lower detection likelihood. Going from [mlmin=7]{} to [mlmin=6]{}, but leaving everything else unchanged, we detected an additional nine sources of PFH2005. One of the previously undetected sources (\[PFH2005\] 75) was classified as $<$SSS$>$, but correlates with blocks of pixels with enhanced low energy signal in the PN offset map and was corrected by [epreject]{}. Another source classified as $<$SSS$>$ (\[PFH2005\] 799) is only detected in the MOS1 camera, but not in MOS2. From an examination by eye, it seems that source \[PFH2005\] 799 is the detection of some noisy pixels at the rim of the MOS1 CCD6 and not a real X-ray source. SPH2008 extended the source catalogue of PFH2005 by re-analysing the data of the central region of 31 and also including data of monitoring observations of LMXB RX J0042.6+4115. Of the 39 new sources presented in SPH2008, 24 are also listed in the  catalogue,  15 sources of SPH2008 were not detected. Differences between the two studies include the detection likelihood thresholds used for [eboxdetect]{} (SPH2008: [likemin]{}=5) and [emldetect]{} (SPH2008: [mlmin]{}=6), the lower limit for the likelihood of extention (SPH2008: [dmlextmin]{}=4; : 15), the screening for GTIs, the use of the [epreject]{} task and the [SAS]{} versions and calibration files used. Concerning the GTIs, images, background images and exposure maps SPH2008 followed the same procedures as in PFH2005. The arguments given above are therefore also valid here. From the 14 undetected sources, three sources were detected in SPH2008 with [mlmin]{} $<$ 7. One source (\[SPH2008\] 882) was added by hand to the final source list, as SPH2008 could not find any reason why [emldetect]{} did not automatically find it. The two extended sources (\[SPH2008\] 863, 869) detected with extent likelihoods between 4.7 and 5.1 in SPH2008, are neither detected as extended nor as pointlike sources in the present study, where the extent likelihood has to be larger than 15. SBK2009 re-analysed the  observations located along the major axis of 31, ignoring all observations pointing to the centre of the galaxy. They used a detection likelihood threshold of ten. Of the 335 sources detected by SBK2009, 304 sources are also contained in the  catalogue,  31 sources are not detected. Of the 304 re-detected sources, two (\[SBK2009\] 298, 233) are found with a detection likelihood below ten. Of the 31 undetected sources, 27 were also not detected in PFH2005. The remaining four sources correlate with PFH2005 sources, which were not detected in the present study. SBK2009 state that they find 34 sources not present in the source catalogue of PFH2005. A possible reason for this may be that SBK2009 used different energy bands for source detection. They also had five bands, but they combined bands 2 and 3 from PFH2005 into one band in the range 0.5–2keV, and on the other hand they split band 5 of PFH2005 into two bands from 4.5–7keV and from 7–12keV, respectively. This might also explain why most of the additional found sources were classified as $<$hard$>$. @2006ApJ...643..844O addressed the population of SSSs and QSSs based on the same archival observations as PFH2005. @2006ApJ...643..844O detected 15 SSSs, 18 QSSs and 10 SNRs of which one (\[O2006\] Table4, Src.3) is also listed as an SSS (\[O2006\] Table2, Src.13). Of these sources two SSSs, four QSSs and two SNRs (among them is the source \[O2006\] Table4, Src.3) are not contained in the  catalogue. These seven sources are also not present in the PFH2005 catalogue. The nine bright variable sources from were all detected. *Chandra* catalogues {#SubSec:Chcat} -------------------- The  catalogues used for cross-correlations were presented in Sect.\[Sec:Intro\] (see also Table \[Tab:XrayRefCat\]). Details of the comparison between the  catalogue and the different  catalogues can be found in Table \[Tab:CompChan\]. Here, we only give a few general remarks. A non-negligible number of  sources not reported in the  catalogue have already been classified as transient or variable sources. Thus, it is not surprising that these sources were not detected in the  observations . One  source (n1-66) lies outside the field of 31 covered by the  observations. For the innermost central region of 31, the point spread function of  causes source confusion and therefore only  observations are able to resolve the individual sources, especially if they are faint compared to the diffuse emission or nearby bright sources . This explains why a certain number of these sources are not detected in  observations. [ll]{}\ \ 5 transient: & r3-46,r3-43,r2-28,r1-23,r1-19\ 20 variable: & r3-53,r3-77,r3-106,r3-76,r2-52,r2-31,r2-23,r1-31,r2-20,r1-24,r1-28,r1-27,r1-33,r1-21,r1-20,r1-7,r2-15,r1-17,r1-16,r2-47\ 33 unclassified: & r3-102,r3-92,r3-51,r3-75,r3-91,r3-89,r3-101,r3-88,r2-44,r2-55,r2-54,r3-32,r2-53,r1-30,r3-99,r1-22,r1-26,r1-18,r3-26,\ &r2-41,r2-40,r3-71,r2-50,r2-49,r2-38,r3-97,r2-46,r3-12,r3-66,r3-104,r3-82,r3-5,r3-4\ \ \ 3 transient: & J004217.0+411508,J004243.8+411604,J004245.9+411619\ 7 variable: & J004232.7+411311,J004242.0+411532,J004243.1+411640,J004244.3+411605,J004245.2+411611,J004245.5+411608,\ &J004248.6+411624\ 16 unclassified: & J004207.3+410443,J004229.1+412857,J004239.5+411614,J004239.6+411700,J004242.5+411659,J004242.7+411503,\ &J004243.1+411604,J004244.2+411614,J004245.0+411523,J004246.1+411543,J004247.4+411507,J004249.1+411742,\ & J004251.2+411639,J004252.3+411734,J004252.5+411328,J004318.5+410950\ \ \ 12 transient: & s1-79,s1-80,s1-82,r3-46,r2-28,r1-23,r1-19,r2-69,r1-28,r1-35,r1-34,n1-85\ 7 variable: & r2-31,r1-31,r1-24,r1-20,r1-7,r1-17,r1-16\ 9 unclassified: & s1-81,r2-68,s1-85,r1-30,r1-22,r1-26,r1-18,n1-77,n1-84\ \ \ 11 transient: & 6,12,29,32,41,51,59,84,118,130,146\ 15 variable: & 3,5,8,9,18,22,24,27,44,63,92,96,99,149,169\ 78 unclassified: & 4,19,21,25,26,30,37,39,40,42,48,49,53,56,57,58,60,62,65,70,73,75,76,77,80,82,84,86,87,89,91,94,97,98,104,109,114,\ &115,117,119,122,124,129,133,138,141,143,144,145,150,152,158,162,164,167,171,173,182,183,188,189,191,193,194,\ &197,202,205,206,210,213,217,219,220,225,256,257,263\ \ \ 25 transient: & n1-26,n1-85,n1-86,n1-88,n1-89,r1-19,r1-23,r1-28,r1-34,r1-35,r2-28,r2-61,r2-62,r2-66,r2-69,r2-72,r3-43,r3-46,s1-18,\ & s1-27,s1-69,s1-79,s1-80,s1-82,s2-62\ \ \ 9 transient: & s2-62,s1-27,s1-69,s1-18,n1-26,r2-62,r1-35,r2-61,r2-66\ 5 unclassified: & s2-27,s2-10,n1-29,n1-46,r2-54\ 1 not in FoV: & n1-66\ \ \ 2 unclassified: & 17 ($\hat{=}$ r2-15), 28 ($\hat{=}$ r3-71)\ \[Tab:CompChan\] Notes:\ Variability information (transient, variable) is taken from the papers. “Unclassified" denotes sources which are not indicated as transient or variable sources in the papers. Of the 28 bright X-ray sources located in globular clusters [@2002ApJ...570..618D], two were not found in the  data (see Table \[Tab:CompChan\]). They are also not included in the source catalogue of PFH2005 and SPH2008. Hence, both objects are good candidates for being transient or at least highly variable sources ( Sect.\[SubSub:comp\_GlC\]). Another study of the globular cluster population of 31 is presented by @2004ApJ...616..821T. Their work is based on  and  data and contains 43 X-ray sources. Of these sources three were not found in the present study. One of them (\[TP2004\] 1) is located well outside the field of 31 covered by the Deep  Survey[^20]. The second source (\[TP2004\] 21) correlates with r3-71, which is discussed above (see @2002ApJ...570..618D in Table \[Tab:CompChan\]). The transient nature of the third source (\[TP2004\] 35), and the fact that it was not observed in any  observation taken before 2004 was already reported by @2004ApJ...616..821T. The source was first detected with  in the observation from 31 December 2006. *ROSAT* catalogues ------------------ Of the 86 sources detected with  HRI in the central $\sim$34 of 31 (PFJ93), all but eight sources (\[PFJ93\] 1,2,31,33,40,48,63,85) are detected in the  observations. Six of these eight sources (\[PFJ93\] 1,2,31,33,63,85) have already been discussed in PFH2005 and classified as transients. Sources \[PFJ93\] 40 and 48 correlate with \[PFH2005\] 312 and 332, respectively, which are discussed in Sect.\[SubSec:prevXMM\]. In addition to these eight sources, PFH2005 did not detect source \[PFJ93\] 51. This source was detected in the  observations centred on RX J0042.6+4115 and was thus classified as a recurrent transient (see SPH2008). In each of the two  PSPC surveys of 31, 396 individual X-ray sources were detected (SHP97 and SHL2001). From the SHP97 catalogue 130 sources were not detected. Of these sources 48 are located outside the FoV of our  31 survey. From the SHL2001 catalogue, 93 sources are not detected, 60 of which lie outside the  FoV. For information on individual sources see Table \[Tab:CompRos\]. [ll]{}\ \ 48 outside FoV: & 1,2,3,4,5,7,8,14,31,41,72,91,98,104,120,125,159,202,209,271,276,285,286,290,300,312,314,\ & 320,342,350,363,367,371,374,383,385,386,387,388,389,390,391,392,393,394,395,396\ 1 transient: & 69\ 21 not detected, LH$<$12: & 19,24,27,33,46,52,59,63,68,71,133,149,161,264,273,275,307,329,330,358,377\ 16 not detected, 12$\le$LH$<$15: & 12,15,49,82,93,113,114,128,196,230,262,283,334,364,372,376\ 44 not detected, LH$\ge$15: & 16(LH$=$26.6),32(30.2),43(18.2),45(51.2),60(20.1),66(36.2),67(4536.2),78(20.5),80(16.3),81(26.6),\ & 88(33.7),95(548.0),102(16.4),126(217.3),141(843.3),145(46.9),146(673.7),166(17.4),167(90.0),\ & 171(54.3),182(454.4),186(39.8),190(113.0),191(54.5),192(54.3),203(103.3),214(400.2),215(251.0),\ & 232(104.4),245(26.0),260(54.6),263(38.1),265(24.6),268(54.3),270(40.4),277(15.6),309(81.8),\ & 319(23.4),331(19.5),335(51.2),340(27.5),341(28.1),365(22.4),373(69.5)\ \ \ 60 outside FoV: & 1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,21,22,32,39,58,67,69,75,77,81,83,85,90,93,125,141,146,\ & 160,164,192,202,243,260,282,296,298,302,325,326,328,355,371,372,378,379,383,388,389,390,\ & 391,392,393,394,395,396\ 4 not detected, LH$<$12: & 62,96,238,269\ 2 not detected, 12$\le$LH$<$15: & 231,361\ 27 not detected, LH$\ge$15: & 51(LH$=$28.4),104(901.2),121(94.1),126(46.2),143(34.7),168(131.9),171(43.0),173(317.8),190(215.8),\ & 207(98.0),208(298.8),226(73.1),230(75.6),232(1165.6),240(218.4),246(39.9),248(219.6),256(60.0),\ & 267(22.2),271(52.8), 322(2703.3),324(147.7),344(40.7),356(15.3),365(19.0),380(17.4),384(15.8)\ \[Tab:CompRos\] Forty-four (out of 302) sources from SHP97 and 27 (out of 293) sources from SHL2001, have  detection likelihoods larger than 15, but are not listed in the  catalogue. These sources have to be regarded as transient or at least highly variable. *Einstein* catalogue -------------------- The list of  X-ray sources in the field of 31 reported by TF91 contains 108 sources, with 81 sources taken from the  HRI data with an assumed positional error of 3 [reported by @1984ApJ...284..663C], and 27 sources based on  IPC data with a 45 positional error. Applying the above mentioned correlation procedure to the  HRI sources, 64 of these sources are also detected in this work and listed in the  catalogue,  17 sources are not detected (\[TF91\] 29, 31, 35, 39, 40, 43, 46, 50, 53, 54, 65, 66, 72, 75, 78, 93, 96). For the  IPC sources only the 1 $\sigma$ positional error was used to search for counterparts among the  sources. Of the 27  IPC sources six remain without a counterpart in our catalogue (\[TF91\] 15, 99, 100, 106, 107, 108), of which \[TF91\] 15 and 108 are located outside the field of 31 covered by the  catalogue. Sources \[TF91\] 50 and 54 correlate with \[PFH2005\] 312 and 316, respectively. Both sources were already discussed in Sect.\[SubSec:prevXMM\]. Apart from \[TF91\] 106, which is suggested as a possible faint transient by SHL2001, the remaining 18 sources are also not detected by PFH2005. They classified those sources as transient. Cross-correlations with catalogues at other wavelengths {#SEC:CCow} ======================================================= The  catalogue was correlated with the catalogues and public data bases given in Sect.\[Sec:CrossCorr\_Tech\]. Two sources (from the  and from the reference catalogues) were be considered as correlating, if their positions matched within the uncertainty (see Eq. \[Eq:Cor\]). However, the correlation of an X-ray source with a source from the reference catalogue does not necessarily imply that the two sources are counterparts. To confirm this, additional information is needed, like corresponding temporal variability of both sources or corresponding spectral properties. We should also take into account the possibility that the counterpart of the examined X-ray source is not even listed in the reference catalogue used (due to faintness for example). The whole correlation process will get even more challenging if an X-ray source correlates with more than one source from the reference catalogue. In this case we need a method to decide which of the correlating sources is the most likely to correspond to the X-ray source in question. Therefore, the method used should indicate how likely the correlation is with each one of the sources from the reference catalogue. Based on these likelihoods one can define criteria to accept a source from the reference catalogue as being the most likely source to correspond to the X-ray source. The simplest method uses the spatial distance between the X-ray source and the reference sources to derive the likelihoods. In other words, the source from the reference catalogue that is located closest to the X-ray source is regarded as the most likely source corresponding to the X-ray source. An improved method is a “likelihood ratio" technique, were an additional source property ( an optical magnitude in deep field studies) is used to strengthen the correlation selection process. This technique was applied successfully to deep fields to find optical counterparts of X-ray sources [  @2007ApJS..172..353B]. A drawback of this method is that one a priori has to know the expected probability distribution of the optical magnitudes of the sources belonging to the studied object. In our case, this means that we have to know the distribution function for all optical sources of 31 that can have X-ray counterparts, *without* including foreground and background sources. Apart from the fact that such distribution functions are unknown, an additional challenge would be the time dependence of the magnitude of the optical sources ( of novae) and of the connection between optical and X-ray sources ( optical novae and SSSs). Therefore it is not possible to apply this “likelihood ratio" technique to the sources in the  survey. The whole correlation selection process becomes even more challenging if more than one reference catalogue is used. To be able to take all available information into account, we decided not to automate the selection process, but to select the class and most likely correlations for each source by hand (as it was done  in PFH2005). Therefore the source classification, and thus the correlation selection process, is based on the cross correlations between the different reference catalogues, on the X-ray properties (hardness ratios, extent and time variability), and on the criteria given in Table \[Tab:class\]. For reasons of completeness we give for each X-ray source the number of correlations found in the USNO-B1, 2MASS and LGGS catalogues in Table 5. The caveat of this method is that it cannot quantify the probability of the individual correlations. Foreground stars and background objects {#Sec:fgback} ======================================= Foreground stars {#Sec:fgStar} ---------------- X-ray emission has been detected from many late-type – spectral types F, G, K, and M – stars, as well as from hot OB stars [see review by @2000RvMA...13..115S]. Hence, X-ray observations of nearby galaxies also reveal a significant fraction of Galactic stars. With typical absorption-corrected luminosities of $L_{\mathrm{0.2-10\,keV}}\!<$[$10^{31}$ erg s$^{-1}$]{}, single stars in other galaxies are too faint to be detected with present instruments. However, concentrations of stars can be detected, but not resolved. Foreground stars (fg Stars) are a class of X-ray sources which are homogeneously distributed over the field of 31 (Fig.\[Fig:fgS\_spdist\]). The good positional accuracy of  and the available catalogues USNO-B1, 2MASS and LGGS allow us to efficiently select this type of source. The selection criteria are given in Table \[Tab:class\]. The optical follow-up observations of and have confirmed the foreground star nature of bright foreground star candidates selected in PFH2005, based on the same selection criteria as used in this paper. Somewhat different criteria were applied for very red foreground stars, with an LGGS colour ${\mathrm}{V}-{\mathrm}{R}\!>\!1$ or USNO-B1 colour ${\mathrm}{B2}-{\mathrm}{R2}\!>\!1$. These are classified as foreground star candidates, if $f_{{\mathrm}{x}}/f_{{\mathrm}{opt}}\!<\!-0.65$ and $f_{{\mathrm}{x}}/f_{{\mathrm}{opt,R}}\!<\!-1.0$. A misclassification of symbiotic systems in 31 as foreground objects by this criterion can be excluded, as symbiotic systems typically have X-ray luminosities below [$10^{33}$ erg s$^{-1}$]{}, which is more than a factor 100 below the detection limit of our survey. If the foreground star candidate lies within the field covered by the LGGS we checked its presence in the LGGS images (as the LGGS catalogue itself does not list bright stars, because of saturation problems). Otherwise DSS2 images were used. Correlations with bright optical sources from the USNO-B1 catalogue, with an $f_{{\mathrm}{x}}/f_{{\mathrm}{opt}}$ in the range expected for foreground stars, that were not visible in the optical images were rejected as spurious. We found 223 foreground star candidates. Fourty sources were identified as foreground stars, either because they are listed in the globular cluster catalogues as spectroscopically confirmed foreground stars or because they have a spectral type assigned to them in the literature . Two of the foreground star candidates close to the centre of 31 ( 826,  1110) have no entry in the USNO-B1 and LGGS catalogues, and one has no entry in the USNO-B1 R2 and B2 columns ( 976). However, they are clearly visible on LGGS images, they are 2MASS sources and they fulfil the X-ray hardness ratio selection criteria. Therefore, we also classify them as foreground stars. The following 19 sources were selected as very red foreground star candidates:  54,  118,  384,  391,  393,  585,  646,  651,  711,  1038,  1119,  1330,  1396,  1429,  1506,  1605,  1695,  1713 and  1747. A further 10 sources ( 210,  269,  278,  310,  484,  714,  978,  1591,  1908 and  1930) fulfil the hardness ratio criteria, but violate the $f_{{\mathrm}{x}}/f_{{\mathrm}{opt}}$ criteria and are therefore marked as “foreground star candidates" in the comment column of Table 5. \ \ \ \ Six sources ( 473,  780,  1551,  1585,  1676,  1742), classified as foreground star candidates, have X-ray light curves that in a binning of 1000s showed flares (see Fig.\[Fig:fgS\_flare\]). These observations strengthen the foreground star classification. A seventh source ( 714) is classified as a foreground star candidate, since its hardness ratios and its $f_{{\mathrm}{x}}/f_{{\mathrm}{opt}}$ ratio in the quiescent state fulfil the selection criteria of foreground star candidates. In addition, the source shows a flare throughout observation ss3. Hence, the $f_{{\mathrm}{x}}/f_{{\mathrm}{opt}}$ ratio for this observation, in which the source is brightest, is too high to be consistent with the range of values expected for foreground stars. [rrrrrr]{} & & & & &\ 473 & 12.984 & 12.681 & 12.558 & K0 & 0.4\ 714 & 14.310 & 13.618 & 13.458 & M0 & 0.2\ 780 & 14.251 & 13.595 & 13.351 & M3 & 0.1\ 1551 & 12.666 & 12.009 & 11.806 & M2 & 0.1\ 1585 & 13.488 & 12.899 & 12.650 & M2 & 0.1\ 1676 & 10.460 & 9.878 & 9.798 & K1 & 0.2\ 1742 & 13.722 & 13.138 & 12.896 & M1 & 0.2\ \[Tab:fgStar\_flare\] Notes:\ $^{ *~}$: spectral type\ $^{ +~}$: error (in subtypes) Table \[Tab:fgStar\_flare\] gives the J, H and K magnitudes taken from the 2MASS catalogue for each of the six flaring foreground stars. Using the standard calibration of spectral types for dwarf stars based on their near infrared colours (from the fourth edition of Allen’s astrophysical quantities, edt. A.Cox, p.151) we derived the spectral classification for the objects, using both H-K and J-K. The spectral types (and “error”) we give in Table \[Tab:fgStar\_flare\] are derived from averaging the two classes (derived from the two colours). The spectral types are entirely consistent with those expected for flare stars (usually K and M types). Figure \[Fig:fgS\_fldist\] shows the XID flux distribution for foreground stars and foreground star candidates, which ranges from 6.9[$\times 10^{-16}$ ]{} to 2.0[$\times 10^{-13}$ ]{}. Most of the foreground stars and candidates (257 sources) have fluxes below 5[$\times 10^{-14}$ ]{}. ### Comparing *XMM-Newton*, *Chandra* and *ROSAT* catalogues In the combined  PSPC survey (SHP97, SHL2001) 55 sources were classified as foreground stars. Of these, 14 sources remain without counterparts in the present  survey. Five of these 14 sources are located outside the field observed with . Forty-one  foreground star candidates have counterparts in the  catalogue. Of these counterparts, 16 were classified as foreground star candidates and four were identified as foreground stars . In addition 12 sources were listed as $<$hard$>$, two as AGN candidates and one as a globular cluster candidate in the  catalogue. The counterparts of three  sources remain without classification in the  catalogue. Another three  sources have more than one counterpart in the  data. Source \[SHP97\] 109 correlates with sources  597,  604,  606, and  645. The former three are classified as $<$hard$>$, while source  645 is classified as a foreground star candidate. However source  645 has the largest distance from the position of \[SHP97\] 109 compared to the other three  counterparts. Furthermore, this source had a flux below the  detection threshold (about a factor 2.6) in the  observations and is about a factor 3–34 fainter than the three other possible  counterparts. Thus it is very unlikely that \[SHP97\] 109 represents the X-ray emission of a foreground star. Source \[SHL2001\] 156 has two  counterparts and is discussed in Sect.\[Sec:SSS\_comp\]. The third source (\[SHL2001\] 374) correlates with sources  1922 and  1924. The two  sources are classified as $<$hard$>$ and as a foreground star candidate, respectively. In the source catalogue of SHL2001 source \[SHP97\] 369 is listed as the counterpart of \[SHL2001\] 374. The source in the first  survey has a smaller positional error and only correlates with source  1924. Although this seems to indicate that source  1924 is the counterpart of \[SHL2001\] 374, we cannot exclude the possibility that \[SHL2001\] 374 is a blend of both  sources, as these two sources have similar luminosities in the  observations. @2002ApJ...577..738K classified four sources as foreground stars. For two sources ( 960$\hat{=}$r2-42 and  976$\hat{=}$r3-33) the classification is confirmed by our study. The third source ( 1000$\hat{=}$r2-19) remained without classification in the  catalogue, as it is too soft to be classified as $<$hard$>$ and the optical counterpart found in the LGGS catalogue does not fulfil the $f_{{\mathrm}{x}}/f_{{\mathrm}{opt}}$ criteria. The fourth source (r2-46) was not detected in the  observations. The foreground star classification of three sources (s1-74, s1-45, n1-82) in @2004ApJ...609..735W is confirmed by the  study ( 289,   603,  1449). For source  289 the spectral type F0 was determined . The source list of DKG2004 contains six sources (s2-46, s2-29, s2-37, s1-45, s1-20, r3-122) that are classified as foreground stars. All six sources are confirmed as foreground star candidates by our  study ( Table 5). For source  696 ($\hat{=}$s1-20) obtained the spectral type G0. Of the four sources listed as foreground stars in only one source ( 936$\hat{=}$ \[VG2007\] 168) was confirmed as a foreground star, based on the entry in the RBCV3.5 and @2009AJ....137...94C. The second source ( 1118$\hat{=}$\[VG2007\] 180) is listed in the RBCV3.5 and @2009AJ....137...94C as a globular cluster. The third source ( 829$\hat{=}$\[VG2007\] 181) does not have a counterpart in the USNO-B1, 2MASS or LGGS catalogues, nor does it fulfil the hardness ratio criteria for foreground stars. Hence, the source is classified as $<$hard$>$. The fourth source (\[VG2007\] 81) is not spatially resolved from its neighbouring source \[VG2007\] 79 in our  observations (source  1078). Hence source  1078 is classified as $<$hard$>$. Galaxies, galaxy clusters and AGN {#SubSec:Gal_GCl_AGN} --------------------------------- The majority of background sources belong to the class of active galactic nuclei (AGN). This was shown by the recent deepest available surveys of the X-ray background . The class of AGN is divided into many sub-sets. The common factor in all the sub-sets is that their emission emanates from a small, spatially unresolved galactic core. The small size of the emitting region is implied by the X-ray flux variability observed in many AGN, which is on time scales as short as several minutes (to years). The observed X-ray luminosities range from [$10^{39}$]{} to [$10^{46}$ erg s$^{-1}$]{}, sometimes even exceeding [$10^{46}$ erg s$^{-1}$]{}. Although AGN show many different properties, like the amount of radio emission or the emission line strengths and widths, they are believed to be only different facets of one underlying basic phenomenon [  @1995PASP..107..803U]: the accretion of galactic matter onto a supermassive black hole ($\sim\!10^{6}\!-\!10^{9}$M) in the centre of the galaxy. It is difficult and, to some extent, arbitrary to distinguish between active and normal galaxies, since most galaxies are believed to host a black hole at the position of their kinetic centre [@2005ApJ...631..280B]. In normal galaxies the accretion rate to the central supermassive BH is so low, that only weak activity can be detected – if at all. The overall thermal emission of the nuclear region is due to bremsstrahlung from hot gas. The total X-ray luminosity of a normal galaxy can reach some [$10^{41}$ erg s$^{-1}$]{}, at most. It consists of diffuse emission and emission of unresolved individual sources. Galaxy clusters (GCls) are by far the largest and most massive virialised objects in the Universe. Their masses lie in the range of $10^{14}$–$10^{15}$M and they have sizes of a few megaparsecs (Mpc). A mass-to-light ratio of $M/L\!\simeq\!200\,$M/L indicates that galaxy clusters are clearly dominated by their dark matter content. Furthermore, galaxy clusters allow us to study the baryonic matter component, as they define the only large volumes in the Universe from which the majority of baryons emit detectable radiation. This baryonic gas, the [*hot intracluster medium*]{} (ICM), is extremely thin, with electron densities of $n_{\mathrm{e}}\!\simeq\!10^2$–10$^5$m$^{-3}$, and fills the entire cluster volume. Owing to the plasma temperatures of $k_{\mathrm{B}}\,T\!\simeq\!2$–10keV, the thermal ICM emission gives rise to X-ray luminosities of $L_{\mathrm{X}}\!\simeq\!10^{43}$–$3\!\times\!10^{45}$ergs$^{-1}$. Therefore galaxy clusters are the most X-ray luminous objects in the Universe next to AGN. We identified four sources as background galaxies and 11 as AGN, and classified 19 galaxy and 49 AGN candidates. The classification is based on SIMBAD and NED correlations and correlations with sources listed as background objects in the globular cluster catalogues [RBCV3.5 and @2009AJ....137...94C]. Sources are classified as AGN candidates, if they have a radio counterpart [NVSS; @1990ApJS...72..761B; @2004ApJS..155...89G] with the additional condition of being neither a SNR nor a SNR candidate from X-ray hardness ratios, as well as not being listed as a “normal" background galaxy in @2004ApJS..155...89G. Most AGN will be classified as $<$hard$>$ ((HR2$-$EHR2)$>-$0.2, see Table \[Tab:class\]) because of their intrinsic power law component. Additional absorption in the line of sight by the interstellar medium of 31 will lead to an even higher HR2. Only the few AGN with a dominant component in the measured flux below 1keV may lead to a classification $<$SNR$>$ or $<$fg Star$>$ in our adapted scheme. One ( 995) of the four identified galaxies is M 32. An overview of previous X-ray observations of this galaxy is given in PFH2005. They also discuss the fact that  resolved the X-ray emission of M 32 into several distinct point sources (maximum separation of the three central  sources 83). Although M 32 is located closer to the centre of the FoV in the observations of field SS1, than it was in the s1 observation used in PFH2005,  still detects only one source. The remaining three sources ( 88,  403,  718) are identified as galaxies, because they are listed as background galaxies in both the RBCV3.5 and @2009AJ....137...94C. For source  403 (B007) NED gives a redshift of $0.139692\pm0.000230$ [@2007AJ....134..706K]. Eleven X-ray sources are identified as AGN. The first one ( 363) correlates with a BL Lac object located behind 31 (NED, see also PFH2005). The second source ( 745) correlates with a Seyfert 1 galaxy (5C 3.100), which has a redshift of $\approx 0.07$ (SIMBAD). The third source ( 1559) correlates with a quasar (Sharov 21) that showed a single strong optical flare, during which its UV flux has increased by a factor of $\sim$20 . The remaining sources were spectroscopically confirmed (from our optical follow-up observations) to be AGN (D. Hatzidimitriou, private communication; and Hatzidimitriou et al. (2010) in prep.). [rcllc]{} & & & &\ 141 & $1.19^{+1.63}_{-0.88}$ & $2.17^{+2.30}_{-0.68}$ & $0.24^{+1.24}_{-0.11}$ & 78.5/53\ 252 & $0.61^{+1.16}_{-0.43}$ & $1.95^{+0.64}_{-0.29}$ & $0.22^{+0.15}_{-0.07}$ & 56.4/151\ 304 & $2.68^{+2.64}_{-1.85}$ & $0.95^{+3.32}_{-1.95}$ & $0.12^{+0.07}_{-0.05}$ & 50.9/57\ 1543 & $2.74^{+6.91}_{-1.76}$ & $2.08^{+2.31}_{-1.11}$ & $0.61^{+1.11}_{-0.26}$ & 32.9/34\ \[Tab:spfit\_ext\] In Sect.\[Sec:ExtSrcs\] the 12 extended sources in the  catalogue were presented. @2006ApJ...641..756K showed that the brightest of these sources ( 1795) is a galaxy cluster located at a redshift of $z\!=\!0.29$. For the remaining 11 sources, X-ray spectra were created and fitted with the [MEKAL]{} model in [XSPEC]{}. Unfortunately, for most of the examined sources the spectral parameters (foreground absorption, temperature and redshift) are not very well constrained. Nevertheless four sources ( 141,  252,  304,  1543) with temperatures in the range of $\sim\!1$–2keV and proposed redshifts between 0.1–0.6 were found (Table \[Tab:spfit\_ext\]). Inspection of optical images (DSS2 images and if available LGGS images) revealed an agglomeration of optical sources at the positions of these four extended X-ray sources. Thus they are classified as galaxy cluster candidates. Although, B242 (the optical counterpart of source  304) is listed as a globular cluster candidate in the RBC3.5 catalogue, @2009AJ....137...94C classified this source as a background object. Our findings from the X-rays favour the background object classification. Hence a globular cluster classification for this source seems to be excluded. Source  1912 was already classified as a galaxy cluster candidate in PFH2005. The spectrum confirms this classification. The best fit parameters are $=\!1.29^{+0.53}_{-0.41}$[$\times 10^{21}$ cm$^{-2}$]{}, $T\!=\!2.8^{+0.8}_{-0.5}$keV and redshift of $0.06^{+0.03}_{-0.04}$. A plot of the spatial distribution of the classified/identified background sources is given in Fig.\[Fig:BG\_spdist\], which shows that these sources are rather homogeneously distributed over the observed field. However, in the fields located along the major axis of 31 we mainly see AGN, which are bright enough to be visible through 31, while most of the galaxies and galaxy clusters are detected in the outer fields. ### Comparing *XMM-Newton*, *Chandra* and *ROSAT* catalogues Of the ten  PSPC survey sources classified as background galaxies one is located outside the field of the Deep  Survey. The remaining objects are confirmed to be background sources and are classified or identified as galaxies or AGN. The only case which is worth discussing in more detail is the source pair \[SHP97\] 246 and \[SHL2001\] 252. From the  observations it is evident that this source pair is not one source, as indicated in the combined  PSPC source catalogue (SHL2001), but consists of three individual sources ( 1269,  1279 and  1280). \[SHL2001\] 252 correlates spatially with all three  sources, while \[SHP97\] 246 correlates only with source  1269, which is identified as a foreground star of type K2 (SIMBAD). The two other  counterparts of \[SHL2001\] 252 are classified as a galaxy candidate and an AGN candidate, respectively. In summary, \[SHL2001\] 252 is most likely a blend of both background sources and maybe even a blend of all three  sources, while \[SHP97\] 246 seems to be the X-ray counterpart of the foreground star mentioned above. @2002ApJ...577..738K classified source r3-83 ( 1132) as an extragalactic object, as it is listed in SIMBAD and NED as an emission line object. Following PFH2005, we classified source  1132 as $<$hard$>$. The BL Lac object ( 363) was also detected in  observations [@2004ApJ...609..735W]. M 31 sources {#Sec:Srcsm31} ============ Supersoft sources ----------------- Supersoft source (SSS) classification is assigned to sources showing extremely soft spectra with equivalent blackbody temperatures of $\sim$15–80eV. The associated bolometric luminosities are in the range of [$10^{36}$]{}–[$10^{38}$ erg s$^{-1}$]{} . Because of the phenomenological definition, this class is likely to include objects of several types. The favoured model for these sources is that they are close binary systems with a white dwarf (WD) primary, burning hydrogen on the surface . Close binary SSSs include post-outburst, recurrent, and classical novae, the hottest symbiotic stars, and other LMXBs containing a WD (cataclysmic variables, CVs). Symbiotic systems, which contain a WD in a wide binary system, may also be observed as SSSs . Because matter that is burned can be retained by the WD, some SSS binaries may be progenitors of type-Ia supernovae . The  catalogue contains 30 SSS candidates that were selected on the basis of their hardness ratios (see Fig.\[Fig:HR\_diagrams\] and Table \[Tab:class\]). ### Spatial and flux distribution Figure \[Fig:SSS\_spdist\] shows the spatial distribution of the SSSs. Clearly visible is a concentration of sources in the central field. There are two explanations for that central enhancement. The first is that the central region was observed more often than the remaining fields and therefore there is a higher chance of catching a transient SSS in outburst. The second reason is that the major class of SSSs in the centre of 31 are optical novae (PFF2005, PHS2007). Optical novae are part of the old stellar population which is much denser in the centre of 31. Figure \[Fig:SSS\_fldist\] gives the distribution of 0.2–1.0keV source fluxes for all SSSs (black) and for those correlating with optical novae (blue). The unabsorbed fluxes were determined assuming a 50eV blackbody model (PFF2005). The two brightest SSSs ($F_{{\mathrm}{X}}>$[$10^{-12}$ erg cm$^{-2}$ s$^{-1}$]{}) consist of a persistent source with 217s pulsations [ 1061; @2008ApJ...676.1218T] and the nova M31N 2001-11a [ 1416; @2006IBVS.5737....1S]. A large fraction of SSSs are rather faint, with fluxes below 5[$\times 10^{-14}$ ]{}. Four sources have absorption-corrected luminosities below [$10^{36}$ erg s$^{-1}$]{} (0.2–1.0keV), which was indicated as the limiting luminosity for SSSs. That does not necessarily imply that these sources are not SSSs, since it is possible that the blackbody fit chosen does not represent well the properties of these sources. A higher absorption or a lower temperature would lead to increased unabsorbed luminosities. We also have to take into account that we might have observed the source during a phase of rising or decaying luminosity,  not at maximum luminosity. ### Correlations with optical novae {#SubSec:opt_novae} By cross-correlating with the nova catalogue[^21] indicated in Sect.\[Sec:CrossCorr\_Tech\], 14 of the 30 SSSs can be classified as X-ray counterparts of optical novae. Of these 14 novae, eight ( 748,  993,  1006,  1046,  1051,  1076,  1100, and  1236) are already discussed in PFF2005 and PHS2007. Nova M31N 2001-11a was first detected as a supersoft X-ray source. Motivated by that SSS detection, @2006IBVS.5737....1S found an optical nova at the position of the SSS in archival optical plates which had been overlooked in previous nova searches. Nova M31N 2007-06b has been discussed in . The remaining four novae are discussed individually in more detail below. As was shown in the / 31 nova monitoring project[^22], it is absolutely necessary to have a homogeneous and dense sample of deep optical and X-ray observations in order to study optical novae and their connections to supersoft X-ray sources. In the optical, the outer regions of 31 are regularly observed down to a limiting magnitude of $\sim$17 mag (Texas Supernova Search (TSS); @Quimby2006), while in X-rays only “snapshots" are available. Hence, the correlations of optical novae with detected SSSs have to be regarded as lucky coincidences. That also means that the identified nova counterparts are detected at a random stage of their SSS evolution which does not allow us to constrain the exact start or end point of the SSS phase, nor the maximum luminosity of the SSS. We also cannot exclude the possibility that some of the SSSs observed in the outer parts of 31 correspond to the supersoft phase of optical novae for which the optical outburst was missed. In the outer regions of 31, the samples of optical novae and X-ray SSSs are certainly incomplete, due to the rather high luminosity limit in the optical monitoring, and the lack of complete monitoring in X-rays, respectively. So one should be cautious in deriving properties of the disc nova population of 31 from the available data. #### Nova M31N 1997-10c was detected on 2 October 1997 at a B-band magnitude of $16.6$ [ShA 58; @1998AstL...24..641S]. An upper limit of 19 mag on 29 September 1997 was reported by the same authors. They classified this source as a very fast nova. In the  observation c1 (25 June 2000), an SSS ( 871), located within $\sim$19 of the optical nova, was detected. The source was fitted with an absorbed blackbody model. The formal best fit parameters of the  EPIC PN spectrum are: absorption $N_{{\mathrm}{H}}\approx3.45$[$\times 10^{21}$ cm$^{-2}$]{} and $k_{{\mathrm}{B}}T\approx41$eV. The unabsorbed luminosity in the 0.2–1keV band is $\approx5.9$[$\times 10^{37}$ ]{}. Confidence contours for absorption column density and blackbody temperature are shown in Fig.\[Fig:M31N1997-10c\_ccont\]. In the subsequent  observation of that region taken about half a year later (c2; 27 December 2000) the source is not detected. Although the source position is covered in observations c3 (29 June 2001), c4 (6/7 January 2002) and b (16–19 July 2004) the source was not re-detected. Using the count rates derived for the variability study (see Sect.\[Sec:var\]) and assuming the same spectrum for the source as in observation c1, upper limits of the source luminosity can be derived, which are given in Table \[Tab:M31N1997-10c\_uplim\]. [cc]{} &\ c2 & 10.8$^{+}$\ c3 & 1.9\ c4 & 1.0\ \[Tab:M31N1997-10c\_uplim\] Notes:\ $^{ +~}$: The count rate detected in observation c2 gives a luminosity of 2.4$\pm$2.8[$\times 10^{37}$ ]{}, which results in the upper limit given in the Table. The fact that this upper limit is higher than the luminosity detected in observation c1 is, at least in part, attributed to the very short effective observing time of less than 6000s. #### Nova M31N 2005-01b was discovered on 19 January 2005 at a white light magnitude of 16.3 by R. Quimby.[^23] An SSS ( 764) that correlates with the optical nova (distance: 43; 3$\sigma$ error: 55) was found in observation ss2 taken on 8 July 2006, which is 535 days after the discovery of the optical nova. Due to the severe background screening applied to observation ss2, there is not enough statistics to obtain a spectrum of the X-ray source. To get an estimate of the spectral properties of that source we created a spectrum in the 0.2–0.8keV range of the *unscreened* data. Although the spectrum was background corrected, we cannot totally exclude a contribution from background flares. The spectrum is best fitted by an absorbed blackbody model with an absorption of $N_{{\mathrm}{H}}\approx1.03$[$\times 10^{21}$ cm$^{-2}$]{} and a blackbody temperature of $k_{{\mathrm}{B}}T\approx45$eV. The unabsorbed 0.2–1keV luminosity is $L_{{\mathrm}{X}}\sim$1.0[$\times 10^{37}$ ]{}. In another  observation taken 1073 days after the optical outburst (ss21; 28 December 2007) the X-ray source is no longer visible. The 3$\sigma$ upper limit of the unabsorbed source luminosity is $\sim3.3$[$\times 10^{35}$ ]{} in the 0.2–4.5keV band, assuming the spectral model used for source detection. #### Nova M31N 2005-01c was discovered on 29 January 2005 at a white light magnitude of 16.1 by R. Quimby.[^24] In the  observation from 02 January 2007 (ns2, 703 days after optical outburst) an SSS was detected ( 1675) at a position consistent with that of the optical nova (distance: 09). The X-ray spectrum (Fig.\[Fig:M31N2005-01c\_spec\]) can be well fitted by an absorbed blackbody model with the following best fit parameters: absorption $N_{{\mathrm}{H}}=1.58^{+0.65}_{-0.45}$[$\times 10^{21}$ cm$^{-2}$]{} and $k_{{\mathrm}{B}}T=40\pm6$eV. The unabsorbed 0.2–1keV luminosity is $L_{{\mathrm}{X}}\sim$1.2[$\times 10^{38}$ ]{}. Confidence contours for absorption column density and blackbody temperature are shown in Fig.\[Fig:M31N2005-01c\_ccont\]. #### Nova M31N 2005-09b was discovered in optical images taken on 01 and 02 September 2005 at white light magnitudes of $\sim$18.0 and $\sim$16.5 respectively. From 31 August 2005, an upper limit of $\sim$18.7mag was reported [@2005ATel..600....1Q]. The nova was spectroscopically confirmed [@2006ATel..850....1P] and classified as a possible Fe[II]{} or hybrid nova[^25]. An X-ray counterpart ( 92) was detected in the  observation s3 (299 days after the optical outburst). Its position is consistent with that of the optical nova (distance: 057). As observation s3 was heavily affected by background flares, we only could estimate the spectral parameters from the *unscreened* data (see also paragraph about Nova M31N 2005-01b). A blackbody fit of the 0.2–0.8keV gives $N_{{\mathrm}{H}}\approx2.7$[$\times 10^{21}$ cm$^{-2}$]{}, k$T\approx35$eV, and an unabsorbed 0.2–1keV luminosity of $L_{{\mathrm}{X}}\sim$5.4[$\times 10^{38}$ ]{}. The X-ray source was no longer visible in observation s31, which was taken 391 days after observation s3. ### Comparing *XMM-Newton*, *Chandra* and *ROSAT* catalogues {#Sec:SSS_comp} The results and a detailed discussion of a study of the long-term variability of the SSS population of 31 are presented in @2010AN....331..212S. In summary our comparative study of SSS candidates in 31 detected with ,  and  demonstrated that strict selection criteria have to be applied to securely select SSSs. It also underlined the high variability of the sources in this class and the connection between SSSs and optical novae. Supernova remnants {#Sec:SNR_Diss} ------------------ After an supernova explosion the interaction between the ejected material and the ISM forms a supernova remnant (SNR). The SNR X-ray luminosities typically vary between $10^{35}$ and [$10^{37}$ erg s$^{-1}$]{} (0.2–10keV). SNRs can be divided into two categories, (i) sources where the thermal components dominate the X-ray spectrum below 2keV, and (ii) the so-called “plerions" or Crab-like SNRs with power law spectra. The former are located in areas of the X-ray colour/colour diagrams that overlap only with foreground star locii. If we assume that we have identified all foreground star candidates from the optical correlation and inspection of the optical images, the remaining sources can be classified as SNR candidates using the criteria given in Table \[Tab:class\]. Similar criteria were used to select supernova remnant candidates in  observations of M 33 . @2005AJ....130..539G and @2010ApJS..187..495L confirmed the supernova remnant nature of many of these candidates based on optical and radio follow-up observations. They also used a hardness ratio criterion to select supernova remnant candidates from  data. An X-ray source is classified as a SNR candidate if it either fulfils the hardness ratio criterion given in Table \[Tab:class\] (these are 25 such sources), or if it correlates with a known optical or radio SNR candidate (six sources). The sources assigned the classification of a SNR candidate based on the latter criterion alone, are marked in the comment column of Table 5 with the flag ‘*only correlation*’. As these six SNR candidates would be classified as $<$hard$>$ on the basis of their hardness ratios, they are good candidates for being “plerions". SNRs are taken as identified when they coincide with SNR candidates from the optical or radio and fulfil the hardness ratio criterion. For a discussion of detection of SNRs in different wavelength bands see @2010ApJS..187..495L. All together, we identified 25 SNRs and 31 SNR candidates in the  catalogue. This number is in the range expected from an extrapolation of the X-ray detected SNRs in the Milky Way as shown below. Assuming that our own Galaxy contains about 1440 X-ray sources which are brighter than $\sim$1[$\times 10^{35}$ ]{} , and that it contains $\sim$110 SNRs detected in X-rays [@2009BASI...37...45G], we would expect to detect $\sim$50 SNRs in the  catalogue ($0.4\times{\left}(1\,897 {\mathrm}{sources} - 263 {\mathrm}{fg Stars} {\right})$). This number is in good agreement with the number of identified and classified SNRs. The XID fluxes for SNRs range between 5.9[$\times 10^{-14}$ ]{} for source   1234 and 1.5[$\times 10^{-15}$ ]{} for source  419. These fluxes correspond to luminosities of 4.3[$\times 10^{36}$ ]{} to 1.1[$\times 10^{35}$ ]{}. A diagram of the flux distribution of the detected SNRs and candidates is shown in Fig.\[Fig:SNR\_fldist\]. Among the 25 identified SNRs, there are 20 SNRs from the PFH2005 catalogue. Source \[PFH2005\] 146, which correlates with the radio source \[B90\] 11 and the SNR candidate BA146, was not found in the present study. Source \[SPH2008\] 858, which coincides with a source reported as a ring-like extended object in  observations that was also detected in the optical and radio wavelength regimes and identified as a SNR [@2003ApJ...590L..21K], was re-detected ( 1050). Of the 31 SNR candidates ten have been reported by PFH2005. In the following, we first discuss in more detail the remaining four identified SNRs, that appear in the new catalogue but were not included in PFH2005: #### XMMM31 J003923.5+404419 ( 182) was classified as a SNR candidate from its \[S[II]{}\]:H$\alpha$ ratio. It appears as an *‘irregular ring with southerly projection’* and correlates with a radio source [@1969MNRAS.144..101P]. X-ray radiation of that source was first detected in the present study. #### XMMM31 J004413.5+411954 ( 1410) was classified as a SNR candidate from its \[S[II]{}\]:H$\alpha$ ratio . From Fig.\[Fig:src1410\_opt\] we can see that the source *‘appears as a bright knot’*, as was already reported by . The source has counterparts in the radio [@1990ApJS...72..761B] and X-ray (SHP97) range. It was reported as a SNR by SHP97. #### XMMM31 J004510.5+413251 and XMMM31 J004512.3+420029 ( 1587 and  1593, respectively) are new X-ray detections and correlate with the radio sources: \#354 and \#365 in the list of @1990ApJS...72..761B. Source  1587 also correlates with source 37W209 from the catalogue of . No optical counterparts were reported in the literature. In the following, we discuss two SNR candidates in more detail: #### XMMM31 J004434.8+412512 ( 1481) lies in the periphery of a super-shell with \[S[II]{}\]:H$\alpha\!>$0.5 . Located next to this source is a SNR candidate reported in , which has a radio counterpart from the NVSS catalogue.  1481 also correlates with  source \[SPH97\] 284, which was identified as a SNR in SPH97 due to its spatial correlation with source 3-086. Figure \[Fig:src1481\_opt\] shows the  error circle over-plotted on LGGS images. From the  source position it looks more likely that the X-rays are emitted from the HII region rather than from the SNR candidate visible in the optical and radio wavelengths. Nevertheless the  source detected is point-like and its hardness ratios lie in the range expected for SNRs. If the X-ray emission originated from the -region, it should have been detected as spatially extended emission. Thus,  1481 is classified as SNR candidate. A puzzling fact, however, is the pronounced variability between  and  observations of $F_{{\mathrm}{var}}\!=\!9.82$ with a significance of $S_{{\mathrm}{var}}\!\approx\!4$ (see Table \[Tab:VarSNRs1\]), which is not consistent with the long term behaviour of SNRs. There is still the possibility that the detected X-ray emission does not belong to either the -region or a SNR at all. #### XMMM31 J004239.8+404318 ( 969) was already observed with  (SHP97, SHL2001) and  [@2004ApJ...609..735W s1-84]. No optical counterpart is visible on the LGGS images. The X-ray spectrum, which is shown in Fig.\[Fig:src969\_sp\], is well fitted by an absorbed non-equilibrium ionisation model with the following best fit values: an absorption of $N_{{\mathrm}{H}}=1.76^{+0.46}_{-0.60}$[$\times 10^{21}$ cm$^{-2}$]{}, a temperature of $k_{{\mathrm}{B}}T=219^{+32}_{-19}$eV, and an ionisation timescale of $\tau=1.75^{+0.82}_{-1.75}\times10^8$scm$^{-3}$. The unabsorbed 0.2–5keV luminosity is $L_{{\mathrm}{X}}\sim6.5$[$\times 10^{37}$ ]{}. The soft spectrum with the temperature of $\sim$200eV is in good agreement with spectra of old SNRs  in the SMC . Although the unabsorbed luminosity is rather high for an old SNR, it is still in the range found for other SNRs [  @2002ApJ...580L.125K; @2007ApJ...663..234G]. Hence, XMMM31 J004239.9+404318 is classified as a SNR candidate. ### Comparing SNRs and candidates in *XMM-Newton*, *Chandra* and *ROSAT* catalogues The second  PSPC catalogue (SHL2001) contains 16 sources classified as SNRs. The counterparts of 12 of these sources are also classified as SNRs or SNR candidates in the  catalogue. [rrccrcrrl]{}  & & & & & & fvar & svar & reason why indicated vaiability is not reliable\ & & & &\ 474 & 5.27 $\pm$ 0.56 & 21.18 $\pm$ 4.46 & & & & 4.01 & 3.54\ 668 & 7.94 $\pm$ 1.36 & 26.30 $\pm$ 6.69 & & & & 3.31 & 2.69\ 883 & 2.83 $\pm$ 0.33 & & & 3.33 $\pm$ 0.83 & & 1.18 & 0.56\ 1040 & 7.12 $\pm$ 0.47 & & & 12.49 $\pm$ 1.67 & & 1.75 & 3.11\ 1050 & 8.25 $\pm$ 0.70 & & & 2.50 $\pm$ 0.83 & & 3.30 & 5.28\ 1066 & 28.35 $\pm$ 1.16 & & 256.16$\pm$ 16.19 & 39.13 $\pm$ 3.33 & 25.29 $\pm$ 5.32 & 10.13 & 14.06 &  source is a blend of two  sources\ 1234 & 59.12 $\pm$ 1.10 & 152.91 $\pm$13.82 & 268.98 $\pm$17.09 & 54.11 $\pm$ 3.33 & 109.13 $\pm$ 11.31 & 4.97 & 12.34 & embedded in diffuse emission in central area of 31\ 1275 & 23.88 $\pm$ 1.08 & 53.50 $\pm$ 8.47 & 79.39 $\pm$ 9.90 & & & 3.32 & 5.58\ 1328 & 9.25 $\pm$ 0.74 & & 26.99 & & & 1.00 & 0.00\ 1351 & 4.96 $\pm$ 0.68 & 24.96 $\pm$ 8.92 & 17.77 & & & 1.00 & 0.00\ 1372 & 2.12 $\pm$ 0.84 & & 29.91 & & & 1.00 & 0.00\ 1410 & 7.40 $\pm$ 0.94 & 29.87 $\pm$ 7.13 & & & & 4.04 & 3.12\ 1481 & 3.43 $\pm$ 0.97 & 33.66 $\pm$ 7.36 & & & & 9.82 & 4.07 & see Sect.\[Sec:SNR\_Diss\]\ 1535 & 14.73 $\pm$ 1.31 & 53.94 $\pm$ 9.14 & 34.41 $\pm$ 7.20 & & & 3.66 & 4.25\ 1599 & 16.08 $\pm$ 0.92 & 54.39 $\pm$10.03 & 33.51 $\pm$ 6.97 & & & 3.38 & 3.80\ 1637 & 12.72 $\pm$ 1.33 & & 27.21 & & & 1.00 & 0.00\ \[Tab:VarSNRs1\] Notes:\ $^{ +~}$: KGP2002: @2002ApJ...577..738K,WGK2004: @2004ApJ...609..735W\  and  count rates are converted to 0.2–4.5keV fluxes, using WebPIMMS and assuming a foreground absorption of $=\!6.6$[$\times 10^{20}$ cm$^{-2}$]{} and a photon index of $\Gamma\!=\!1.7$: ECF$_{{\mathrm}{SHP97}}\!=\!2.229\times$10$^{-14}$ergcm$^{-2}$cts$^{-1}$, ECF$_{{\mathrm}{SHL2001}}\!=\!2.249\times$10$^{-14}$ergcm$^{-2}$cts$^{-1}$, and ECF$_{{\mathrm}{KGP2002}}\!=\!8.325\times$10$^{-14}$ergcm$^{-2}$cts$^{-1}$. For WGK2004 the luminosity given in Table 2 of WGK2004 was converted to XID flux using $F_{{\mathrm}{XID}}$\[ergcm$^{-2}$s$^{-1}$\]$=\!6.654\times$10$^{-15}\!\times\!L_{{\mathrm}{WGK2004}}$\[10$^{36}$ergs$^{-1}$\]. Table \[Tab:VarSNRs1\] lists the , , and  fluxes of all SNRs and SNR candidates from the  catalogue that have counterparts classified as SNRs in  or  source lists. In addition, the maximum flux variability and the maximum significance of the variability (following the variability calculation of Sect.\[Sec:DefVar\]) are given. Three SNRs that have  counterparts show variability changing in flux by more than a factor of five. The most variable source ( 1066) is discussed below, the second source was discussed in Sect.\[Sec:SNR\_Diss\] (XMMM31 J004434.8+412512,  1481), and the third source ( 1234) is embedded in the diffuse emission of the central area of 31. In this environment the larger PSF of  results in an overestimate of the source flux, since the contribution of the diffuse emission could not be totally separated from the emission of the point source. The remaining four  sources classified as SNRs and their  counterparts are discussed in the following paragraph. [lrccrcrrr]{}  & & & & & & fvar & svar & remark$^{\ddagger}$\ & & & &\ & & & & & & & &\ 294 & $18.50 \pm 0.85$ & & &$53.27 \pm 6.69$ & $46.78 \pm 7.87$ & 2.88 & 5.16 &\ 472 & $ 3.15 \pm 0.69$ & & & & $26.09 \pm 6.07$ & 8.28 & 3.76 & 468 brt\ 969 & $53.51 \pm 1.35$ & $84.51\pm 15.97^{+}$ & & $34.55 \pm 6.91$ & $89.06 \pm 11.92$ & 2.58 & 3.96 &\ 1079 & $ 4.19 \pm 0.59$ & & & $20.06 \pm 6.24 $& & 4.79 & 2.53 & brt\ 1291 & $14.55 \pm 0.75$ & $16.04^{*} $ & $>$24.0 & $35.22 \pm 8.47$ & $40.93 \pm 7.87$ & 2.81 & 3.33 &\ 1741 & $ 4.12 \pm 0.65$ & $4.17^{\dagger} $ & & & & 1.01 & — & brt\ 1793 & $ 3.70 \pm 0.52$ & & & $26.08 \pm 6.46$ & & 7.06 & 3.46 & 1799 brt\ \[Tab:VarSNRs\] Notes:\ $^{ \ddagger~}$: Source number (from  catalogue) of another (brighter)  source which correlate with the same  source as the  source given in Col. 1; brt:  flux is below the  detection threshold (5.3[$\times 10^{-15}$ ]{}).\  and  count rates are converted to 0.2–4.5keV fluxes, using WebPIMMS and assuming a foreground absorption of $=\!6.6$[$\times 10^{20}$ cm$^{-2}$]{} and a photon index of $\Gamma\!=\!1.7$: ECF$_{{\mathrm}{SHP97}}\!=\!2.229\times$10$^{-14}$ergcm$^{-2}$cts$^{-1}$, ECF$_{{\mathrm}{SHL2001}}\!=\!2.249\times$10$^{-14}$ergcm$^{-2}$cts$^{-1}$, ECF$_{{\mathrm}{HRI}}\!=\!6.001\times$10$^{-14}$ergcm$^{-2}$cts$^{-1}$, $^{ \dagger~}$: ECF$_{{\mathrm}{DKG2004}}\!=\!5.56\times$10$^{-12}$ergcm$^{-2}$cts$^{-1}$. $^{ +~}$: For WGK2004 the luminosity given in Table 2 of WGK2004 was converted to XID flux using $F_{{\mathrm}{XID}}$\[ergcm$^{-2}$s$^{-1}$\]$=\!6.654\times$10$^{-15}\!\times\!L_{{\mathrm}{WGK2004}}$\[10$^{36}$ergs$^{-1}$\]. $^{ *~}$: For VG2007 the luminosity given in Table 2 of VG2007 was converted to XID flux using $F_{{\mathrm}{XID}}$\[ergcm$^{-2}$s$^{-1}$\]$=\!9.433\times$10$^{-15}\!\times\!L_{{\mathrm}{VG2007}}$\[10$^{36}$ergs$^{-1}$\]. SHP97 report that \[SHP97\] 203 and \[SHP97\] 211 ($\hat{=}$\[SHL2001\] 206) correlate with the same SNR (\[DDB80\] 1.13), have the same spectral properties and have luminosities within the range of SNRs. A correlation with the  HRI catalogue (PFJ93) reveals that the true X-ray counterpart of \[DDB80\] 1.13 is located between the two  PSPC sources. Furthermore, PFJ93 report that this SNR is located ‘*within 19of a brighter X-ray source*’ which matches positionally with \[SHP97\] 211. These findings are confirmed by  and  observations. The X-ray counterpart of \[DDB80\] 1.13 is source  1066 in the  catalogue (or \[PFH2005\] 354 or r3-69 in [@2002ApJ...577..738K]). The second source, which correlates with \[SHP97\] 211, is the  source  1077, which has a “hard" spectrum and is $\sim\!6.7$ times brighter than  1066. Hence, \[SHP97\] 211 is a blend of the two  sources  1066 and  1077. This also explains the pronounced variability between \[SHL2001\] 206 and  1066 given in Table \[Tab:VarSNRs1\]. Comparing the  detections of the SNR counterpart with the  flux gives a variability factor of $F_{{\mathrm}{var}}\!\approx\!1.12$. The distance between \[SPH97\] 203 and \[DDB80\] 1.13 is $\ga\!20$. \[SPH97\] 203 was reported only in the first  PSPC catalogue. It was not detected in the observations of the second  PSPC catalogue or in any  or  observation of that region. Thus it seems very likely that \[SPH97\] 203 was either a transient source or a false detection. In both cases \[SPH97\] 203 cannot be a SNR. As the field of \[DDB80\] 1.13 was observed many times with , and as  has detected weak SNRs in the central part of 31 [@2003ApJ...590L..21K and  ],  should have detected X-ray emission corresponding to the  source \[SPH97\] 203, if it really belonged to a SNR. The remaining two  SNRs correlate with  sources, which were not classified as SNRs or SNR candidates. Source \[SHP97\] 258 correlates with source  1337 and has a 3$\sigma$ positional error of 30. From the improved spatial resolution of  the total positional error reduces to 23. Hence, we can see that the X-ray source belongs to a foreground star candidate ( Table 5) and not to the very nearby SNR. Source \[SHL2001\] 129 correlates with sources  743 and  761, which are classified as a GlC and a GlC candidate, respectively. The SNR candidate listed as the counterpart of \[SHL2001\] 129 is located between these two  sources. In addition PFH2005 gives a third source which lies within the error circle of \[SHL2001\] 129 and which is classified as an AGN candidate. Thus it is very likely that \[SHL2001\] 129 is a blend of these three  sources and that the correlation with the SNR candidate has to be considered as a chance coincidence. From the sources listed as SNRs in the different  studies many are re-detected. Nevertheless two SNRs from  were not detected in the  observations. Source n1-85 has been reported as spatially correlated with an optical SNR by @2004ApJ...609..735W, but has also been classified as a repeating transient source in the same paper. An  counterpart to n1-85 was detected neither in the study of PFH2005 nor in the  catalogue. The transient nature of this source is at odds with the SNR classification. Source CXOM31 J004247.8+411556 [@2003ApJ...590L..21K], which correlates with the radio source \[B90\] 95, is located in the vicinity of two bright sources and close to the centre of 31. Due to ’s larger point spread function this source cannot be resolved by  in this environment. The larger PSF of  is also the reason why source  1050 has a significant variability in Table \[Tab:VarSNRs1\], since this source is located within the central diffuse emission of 31. Finally, we wanted to determine whether any of the  SNRs and SNR candidates were previously observed, but not classified as SNRs. In total there are seven such sources. One of them ( 1741) is classified as a SNR candidate based on its  hardness ratios, and correlates with the  source n1-48 (DKG2004). The fluxes obtained with  and  are in good agreement (see Table \[Tab:VarSNRs\]), but below the  detection threshold (5.3 [$\times 10^{-15}$ ]{}). For a further four sources, the corresponding sources were only detected previously with ROSAT. One of them ( 1793$\hat{=}$\[SHP97\] 347) also correlates with a radio source [source 472 of @1990ApJS...72..761B] and is therefore identified as a SNR. The rather high flux variability between the  and  observations (see Table \[Tab:VarSNRs\]) can be attributed to source  1799, which is located within 199 of  1793. This suggests that \[SHP97\] 347 is a combination of both  sources, but as \[SHP97\] 347 was not detected in SHL2001, we cannot exclude a transient source or false detection as an explanation for the  source. Source  472 ($\hat{=}$\[SHL2001\] 84), source  294 ($\hat{=}$\[SHP97\] 53$\hat{=}$\[SHL2001\] 56), and source  1079 ($\hat{=}$\[SHP97\] 212) are SNR candidates based on their hardness ratios. The pronounced flux variability of source  472 is due to source  468, which is located within 185 of  472 and is $\sim$8.6 times brighter than  472. The observed flux for source  1079 was below the ROSAT threshold. Furthermore, the ROSAT source \[SHP97\] 212 was classified as a SNR, but did not appear in the SHL2001 catalogue. Hence ROSAT may have detected an unrelated transient instead. Sources corresponding to the remaining two  sources were detected with  and . Source  969 was detected in both  PSPC surveys (\[SHP97\] 185$\hat{=}$\[SHL2001\] 186) and correlates with  source s1-84 [@2006ApJ...643..356W]. We classify it as a SNR candidate due to its hardness ratios and X-ray spectrum (see XMMM31 J004239.8+404318). Counterparts for source  1291 were reported in the literature as \[PFJ93\] 84, \[SHP97\] 251, \[SHL2001\] 255, \[VG2007\] 261 and source 4 in Table 5 of @2006ApJ...643..844O. Based on the  hardness ratios and the correlation with radio source \[B90\] 166 [@1990ApJS...72..761B], we identified the source as a SNR. For sources  294,  969, and  1291 the variability between different observations may not be real because of systematic cross-calibration uncertainties. Therefore, we keep the $<$SNR$>$ and SNR classifications for these sources. ### The spatial distribution To examine the spatial distribution of the  SNRs and SNR candidates, we determined projected distances from the centre of M 31. The distribution of SNRs and SNR candidates (normalised per deg$^{2}$) is shown in Fig.\[Fig:SNRdepro\_dist\]. It shows an enhancement of sources around $\sim$3kpc, which corresponds to the SNR population in the ’inner spiral arms’ of M 31. In addition, a second enhancement of sources around $\sim$10kpc is detected; this corresponds to the well known dust ring or star formation ring in the disc of 31 [@2006Natur.443..832B]. Only a few sources are located beyond this ring. Figure \[Fig:SNR\_dudist\] shows the spatial distribution of the SNRs and SNR candidates from the  catalogue plotted over the IRAS 60$\mu$m image [@1994STIN...9522539W]. We see that most of the SNRs and SNR candidates are located on features that are visible in the IRAS image. This again demonstrates that SNRs and SNR candidates are coincident with the dust ring at $\sim\!10$kpc. In addition, the locations of star forming regions obtained from *GALEX* data [@2009ApJ...703..614K and private communication] are indicated in Fig.\[Fig:SNR\_dudist\]. We see that many of the SNRs and SNR candidates are located within or next to star forming regions in M 31. X-ray binaries {#SubSec:XRB} -------------- X-ray binaries consist of a compact object plus a companion star. The compact object can either be a white dwarf (these systems are a subclass of CVs), a neutron star (NS), or a black hole (BH). A common feature of all these systems is that a large amount of the emitted X-rays is produced due to the conversion of gravitational energy from the accreted matter into radiation by a mass-exchange from the companion star onto the compact object. X-ray binaries containing an NS or a BH are divided into two main classes, depending on the mass of the companion star: - Low mass X-ray binaries (LMXBs) contain companion stars of low mass ($M\la$ 1M) and late type (type A or later), and have a typical lifetime of $\sim$[$10^{8-9}$]{} yr . LMXBs can be located in globular clusters. Mass transfer from the companion star into an accretion disc around the compact object occurs via Roche-lobe overflow. - High mass X-ray binaries (HMXBs) contain a massive O or B star companion [$M_{{\mathrm}{star}}\ga10$M, @Verbunt1994] and are short-lived with lifetimes of $\sim$[$10^{6-7}$]{}yr . One has to distinguish between two main groups of HMXBs: super-giant and the Be/X-ray binaries. In these systems wind-driven accretion onto the compact object powers the X-ray emission. Mass-accretion via Roche-lobe overflow is less frequent in HMXBs, but is still known to occur in several bright systems ( LMCX-4, SMCX-1, CenX-3). HMXBs are expected to be located in areas of relatively recent star formation, between 25–60Myr ago [@2010ApJ...716L.140A]. We should expect about 45 LMXBs in 31, following a similar estimation as the one presented in Sect.\[Sec:SNR\_Diss\]. Here the number of LMXBs in the Galaxy was estimated from . In the  catalogue 88 sources are identified/classified as XRBs. This is not surprising as we may expect 31 to have a higher fraction of XRBs than the Galaxy since it is an earlier type galaxy composed of a higher fraction of old stars. XRBs are the main contribution to the population of “hard" X-ray sources in 31. Despite some more or less reliable candidates, not a single, definitely detected HMXB is known in 31. The results of a new search for HMXB candidates are presented in Sect.\[SubSec:XRB\_HMXB\]. The LMXBs can be separated into two sub-classes: the field LMXBs (discussed in this section) and those located in globular clusters. Sources belonging to the latter sub-class are discussed in Sect.\[SubSec:GlC\]. The sources presented here are classified as XRBs, because they have HRs indicating a $<$hard$>$ source and are either transient or show a variability factor larger than ten (see Sect.\[Sec:var\]). In total 10 sources are identified and 26 are classified as XRBs by us, according to the classification criteria given in Table \[Tab:class\]. Apart from source  57 (XMMM31 J003833.2+402133, see below), the identified XRBs had been reported as X-ray binaries in the literature (see comment column of Table 5). Figure \[Fig:XRB\_fldist\] (red histogram) shows the flux distribution of XRBs. We see that this class contains only rather bright sources. This is not surprising as the classification criterion for XRBs is based on their variability, which is more easily detected for brighter sources (  Sect.\[Sec:var\]). The XID fluxes range from 1.4[$\times 10^{-14}$ ]{} ( 378) to 3.75[$\times 10^{-12}$ ]{} ( 966), which correspond to luminosities from 1.0[$\times 10^{36}$ ]{} to 2.7[$\times 10^{38}$ ]{}. It is clear from Fig.\[Fig:XRB\_spdist\], which shows the spatial distribution of the XRBs, that nearly all sources classified or identified as XRBs (yellow dots) are located in fields that were observed more than once (centre and southern part of the disc). This is partly a selection effect, caused by the fact that these particular fields were observed several times, thus allowing the determination of source variability. For sources located outside these fields, especially the northern part of the disc, the transient nature must have been reported in the literature to mark them as an XRBs. The source density of LMXBs, which follows the overall stellar density, is higher in the centre than in the disc of 31. One would not expect HMXBs in the central region which is dominated by the bulge (old stellar population). From Fig.\[Fig:XRB\_spdist\_IRAS\], which shows the spatial distribution of the XRBs over-plotted on an IRAS 60$\mu$m image [@1994STIN...9522539W], we see that only a few sources, classified or identified as XRBs, are located in the vicinity of star forming regions. References for the sources, selected from their temporal variability, are given in Table \[Tab:varlist\]. TPC06 report on four bright X-ray transients, which they detected in the observations of July 2004 and suggested to be XRB candidates. We also found these sources and classified source  705 and identified sources  985,  1153,  1177 as XRBs. One of the identified XRBs ( 1177) shows a very soft spectrum. @2005ApJ...632.1086W observed source  1153 with  and *HST*. From the location and X-ray spectrum they suggest it to be an LMXB. They propose that the optical counterpart of the X-ray source is a star within the X-ray error box , which shows an optical brightness change (in B) by $\simeq$1 mag. Source  985 was first detected in January 1979 by TF91 with the  observatory. WGM06 rediscovered it in  observations from 2004. Their coordinated *HST* ACS imaging does not reveal any variable optical counterpart. From the X-ray spectrum and the lack of a bright star, WGM06 suggest that this source is an LMXB with a black hole primary. In the following subsections we discuss three transient XRBs in more detail. #### XMMM31 J003833.2+402133 ( 57) was first detected in the  observation from 02 January 2008 (s32) at an unabsorbed 0.2–10keV luminosity of $\sim\!2$[$\times 10^{38}$ ]{}. From two observations, taken about 0.5yr (s31) and 1.5yr (s3) earlier, we derived upper limits for the fluxes, which were more than a factor of 100 below the values obtained in January 2008. The combined EPIC spectrum from observation s32 (Fig.\[SubFig:spec\_1\]) is best fitted with an absorbed disc blackbody plus power-law model, with $N_{{\mathrm}{H}}\!=\!1.68^{+0.42}_{-0.48}$[$\times 10^{21}$ cm$^{-2}$]{}, temperature at the inner edge of the disc $k_{{\mathrm}{B}}T_{{\mathrm}{in}}\!=\!0.462\pm0.013$keV and power-law index of $2.55^{+0.33}_{-1.05}$. The contribution of the disc blackbody luminosity to the total luminosity is $\sim 59\,\%$. Formally acceptable fits are also obtained from an absorbed disc blackbody and an absorbed bremsstrahlung model (see Table \[Tab:specprop\]). We did not find any significant feature in a fast Fourier transformation (FFT) periodicity search. The combined EPIC light curve during observation s32 was consistent with a constant value. To identify possible optical counterparts we examined the LGGS images and the images taken with the  optical monitor during the X-ray observation (UVW1 and UVW2 filters). The absence of optical/UV counterparts and of variability on short timescales, as well as the spectral properties suggest that this source is a black hole LMXB in the steep power-law state [@2006csxs.book..157M]. #### CXOM31 J004059.2+411551: @2007ATel.1147....1G reported on the detection of a previously unseen X-ray source in a 5ks  ACIS-S observation from 05 July 2007. In an  ToO observation [sn11, @2007ATel.1191....1S] taken about 20 days after the  detection, the source ( 523) was still bright. The position agrees with that found by . We detected the source at an unabsorbed 0.2–10keV luminosity of $\sim\!1.1$[$\times 10^{38}$ ]{}. The combined EPIC spectrum (Fig.\[SubFig:spec\_2\]) can be well fitted with an absorbed disc blackbody model with $N_{{\mathrm}{H}}\!=\!{\left}(2.00\pm{0.16}{\right})$[$\times 10^{21}$ cm$^{-2}$]{} and with a temperature at the inner edge of the disc of $k_{{\mathrm}{B}}T_{{\mathrm}{in}}\!=\!0.538\pm0.017$keV (Table \[Tab:specprop\]). The spectral parameters and luminosity did not change significantly compared to the  values of @2007ATel.1147....1G. We did not find any significant feature in an FFT periodicity search. The combined EPIC light curve was consistent with a constant value. The examination of LGGS images and of images taken with the  optical monitor (UVW1 and UVW2 filters) during the X-ray observation did not reveal any possible optical/UV counterparts. The lack of bright optical counterparts and the X-ray parameters (X-ray spectrum, lack of periodicity, transient nature, luminosity) are consistent with this source being a black hole X-ray transient, as already mentioned in @2007ATel.1147....1G. #### XMMU J004144.7+411110 ( 705) was detected by @2006ApJ...645..277T in  observations b1–b4 (July 2004) at an unabsorbed luminosity of 3.1–4.4[$\times 10^{37}$ ]{} in the 0.3–7keV band, using a [DISKBB]{} model. We detected the source in observation sn11 (25 July 2007) with an unabsorbed 0.2–10keV luminosity of $\sim$1.8[$\times 10^{37}$ ]{}, using also a [DISKBB]{} model. In observation sn11, the source was bright enough to allow spectral analysis. The spectra can be well fitted with an absorbed power-law, disc blackbody or bremsstrahlung model (Table \[Tab:specprop\]). The obtained spectral shapes (absorption and temperature as well as photon index) are in agreement with the values of @2006ApJ...645..277T. An FFT periodicity search did not reveal any significant periodicities in the 0.3s to 2000s range. No optical counterparts were evident in the images taken with the  optical monitor UVW1 and UVW2 during the sn11 observation, nor in the LGGS images. The lack of a bright optical counterpart and the X-ray parameters support that this source is a black hole X-ray transient, as classified by @2006ApJ...645..277T. \ [ccccccccc]{} & & & & & & & &\ & & & & & & & &\ & & & & &\ s32 &PL+DISCBB&$1.68^{+0.42}_{-0.48}$&$0.462\pm0.013$&$106^{+9}_{-10}$ &$2.55^{+0.33}_{-1.05}$ &173.89(145)&2.04&PN+M1+M2\ s32 &DISCBB&$1.06\pm0.06$&$0.511\pm0.009$&$95\pm4$ & &270.01(147)&1.46&PN+M1+M2\ s32 &BREMSS&$1.91\pm0.07$&$1.082^{+0.029}_{-0.030}$& & &208.65(147)&2.12&PN+M1+M2\ & & & & &\ sn11 &DISCBB&$2.00\pm0.16$&$0.538\pm0.017$&$75\pm6$ & &97.70(79)&1.12&PN+M1+M2\ sn11 &BREMSS&$3.13\pm0.19$&$1.097^{+0.060}_{-0.056}$& & &93.17(79)&1.72&PN+M1+M2\ & & & & &\ sn11 &DISCBB&$2.32^{+1.03}_{-0.87}$&$0.586^{+0.100}_{-0.087}$&$26^{+13}_{-8}$ & &29.74(23)&0.18&PN+M1+M2\ sn11 &BREMSS&$3.72^{+1.14}_{-1.00}$&$1.216^{+0.373}_{-0.269}$& & &29.48(23)&0.29&PN+M1+M2\ sn11 &PL&$6.17^{+1.72}_{-1.47}$&& &$3.23^{+0.46}_{-0.40}$& 31.57(23)&1.12&PN+M1+M2\ \[Tab:specprop\] Notes:\ $^{ *~}$: effective inner disc radius, where $i$ is the inclination angle of the disc\ $^{ {\dagger}~}$: unabsorbed luminosity in the $0.2$–$10.0$keV energy range in units of [$10^{38}$ erg s$^{-1}$]{}\ ### Sources from the XMM-LP total catalogue that were not detected by *ROSAT* To search for additional XRB candidates, we selected all sources from the  catalogue, that were classified as $<$hard$>$ and which did not correlate with a source listed in the  catalogues (PFJ93, SHP97 and SHL2001). The flux distribution of the selected sources is shown in Fig.\[Fig:noROSdist\], and Table \[Tab:noROSdist\] gives the number of sources brighter than the indicated flux limit. [rr]{} &\ &\ 5.5E-15 & 541\ 1E-14 & 242\ 5E-14 & 7\ 1E-13 & 1\ \[Tab:noROSdist\] Possible, new XRB candidates are sources that have an XID flux that lies at least a factor of ten above the  detection threshold (5.3[$\times 10^{-15}$ ]{}). These sources fulfil the variability criterion used to classify XRBs ( Sect.\[Sec:var\]). The  catalogue lists five sources without  counterparts that have XID fluxes above 5.3[$\times 10^{-14}$ ]{}. These are:  239,  365,  910,  1164, and  1553. Between the  and  observations more than ten years have elapsed. On this time scale AGN can also show strong variability. To estimate the number of AGN among the five sources listed above, we investigated how many sources of the identified and classified background objects from the  catalogue with an XID flux larger than 5.3[$\times 10^{-14}$ ]{} were not detected by . The result is that  detected all background sources with an XID flux larger than 5.3[$\times 10^{-14}$ ]{} that are listed in the  catalogue. Thus, the probability that any of the five sources listed above is a background object is very small, in particular if the source is located within the D$_{25}$ ellipse of 31. Therefore, the two sources located within the D$_{25}$ ellipse are listed in the  catalogue as XRB candidates, while the remaining three sources, which are located outside the D$_{25}$ ellipse, are classified as $<$hard$>$. All five sources are marked in the comment column of Table 5 with ‘XRB cand. from  corr.’. ### Detection of high mass X-ray binaries {#SubSec:XRB_HMXB} As already mentioned, until now not a single secure HMXB in 31 has been confirmed. The reason for this is that the detection of HMXBs in 31 is difficult. @2004ApJ...602..231C showed that the hardness ratio method is very inefficient in selecting HMXBs in spiral galaxies. The selection process is complicated by the fact, that the spectral properties of BH HMXBs, which have power-law spectra with indices of $\sim$1– $\sim$2 are similar to LMXBs and AGN. Therefore the region in the HR diagrams where BH HMXB are located is contaminated by other hard sources (LMXBs, AGN, and Crab like SNRs). For the NS HMXBs, which have power-law indices of $\sim$1, and thus should be easier to select, the uncertainties in the hardness ratios lead at best to an overlap – in the worst case to a fusion – with the area occupied by other hard sources [@2004ApJ...602..231C]. Based on the spectral analysis of individual sources of 31, SBK2009 identified 18 HMXB candidates with power-law indices between 0.8 and 1.2. One of these sources (\[SBK2009\] 123) correlates with a globular cluster, and hence it is rather an LMXB in a very hard state rather than an HMXB [  @2004ApJ...616..821T]. Four of their sources (\[SBK2009\] 34, 106, 149, and 295) do not have counterparts in the  catalogue. @Peter developed a selection algorithm for HMXBs in the SMC, which also uses properties of the optical companion. X-ray sources were selected as HMXB candidates if they had HR2$+$EHR2$>$0.1 as well as an optical counterpart within 25 of the X-ray source, with $-0.5\!<$B$-$V$<\!0.5$mag, $-1.5\!<$U$-$B$<\!-0.2$mag and V$<$17mag. We tried to transfer this SMC selection algorithm to 31 sources. In doing so, we encountered two problems: The first problem is that the region of the U-B/B-V diagram is also populated by globular clusters (LMXB candidates) in 31. The second problem is that due to the much larger distance to 31, the range of detected V magnitudes of HMXBs in the SMC of $\sim$13$<$V$<$17mag translates to a $\sim$19$<$V$<$23mag criterion for 31. Thus the V magnitude of optical counterparts of possible HMXB candidates lies in the same range as the optical counterparts of AGN. Therefore the V mag criterion, which provided most of the discriminatory power in the case of the SMC, fails totally in the case of 31. A few of the sources selected from the optical colour-colour diagram and HR diagrams are bright enough to allow the creation of X-ray spectra. That way two additional ( not given in SBK2009) HMXB candidates were found. In addition, we determined the reddening free Q parameter: $$Q = (\rm{U}-\rm{B})-0.72(\rm{B}-\rm{V})$$ [for definition see   @cox2001allen] which allowed us to keep only the intrinsically bluest stars, using Q $\le\!-0.4$ [O-type stars typically have Q$<\!-0.9$, while -0.4 corresponds to a B5 dwarf or giant or an A0 supergiant, @2007AJ....134.2474M]. U$-$B and B$-$V were taken from the LGGS catalogue. #### XMMM31 J004557.0+414830 ( 1716) has an USNO-B1 (R2$=$18.72mag), a 2MASS and an LGGS (V$=$20.02mag; Q$=\!-0.44$) counterpart. The EPIC spectrum is best fitted ($\chi^2_{red}\!=\!0.93$) by an absorbed power-law with $=\!7.4^{+6.0}_{-3.9}$[$\times 10^{21}$ cm$^{-2}$]{} and photon index $\Gamma\!=\!1.2\pm0.4$. The absorption corrected X-ray luminosity in the 0.2–10keV band is $\sim$7.1[$\times 10^{36}$ ]{}. #### XMMM31 J004506.4+420615 ( 1579) has an USNO-B1 (B2$=$20.87mag), a 2MASS and an LGGS (V$=$20.77mag; Q$=\!-1.04$) counterpart. The EPIC PN spectrum is best fitted ($\chi^2_{red}\!=\!1.6$) by an absorbed power-law with $=\!0.48^{+2.4}_{-1.0}$[$\times 10^{21}$ cm$^{-2}$]{} and photon index $\Gamma\!=\!1.0^{+0.7}_{-0.5}$. The absorption corrected X-ray luminosity in the 0.2–10keV band is $\sim$8.6[$\times 10^{36}$ ]{}.\ To strengthen these classifications spectroscopic optical follow-up observations of the optical counterparts are needed. An FFT periodicity search did not reveal any significant periodicities for either of the two sources and the light curves do not show eclipses. From the sources reported as HMXB candidates in SBK2009, three sources (\[SBK2009\] 21, 236, and 256) are located in the region of the U-B/B-V diagram, that we used. Another three sources (\[SBK2009\] 123, 172, and 226) are located outside that region. The remaining sources of SBK2009 have either no counterparts with a U-B colour entry in the LGGS catalogue (\[SBK2009\] 99, 234, 294, and 302) or have no optical counterpart from the LGGS catalogue at all (\[SBK2009\] 9, 160, 197, and 305). The reddening free Q parameter for the SBK2009 sources that have counterparts in the LGGS catalogue are given in Table \[Tab:SBK\_Q\]. [rrlr]{} & & &\ 312 & 21 & J004001.50+403248.0 & -0.34\ 1668 & 236 & J004538.23+421236.0 & -0.77\ 1724 & 256 & J004558.98+420426.5 & -0.81\ 1109 & 123 & J004301.51+413017.5 & +1.77\ 1436 & 172 & J004420.98+413546.7 & -0.65\ & & J004421.01+413544.3$^{*}$ & -0.29\ 1630 & 226 & J004526.68+415631.5 & -0.92\ & & J004526.58+415633.1$^{*}$ & -0.72\ \[Tab:SBK\_Q\] Notes:\ $^{ *~}$: counterparts listed in SBK2009 Globular cluster sources {#SubSec:GlC} ------------------------ A significant number of the luminous X-ray sources in the Galaxy and in 31 are found in globular clusters. X-ray sources corresponding to globular clusters are identified by cross-correlating with globular cluster catalogues (see Sect.\[Sec:CrossCorr\_Tech\]). Therefore changes between the  catalogue and the catalogue of PFH2005 in the classification of sources related to globular clusters are based on the availability of and modifications in recent globular cluster catalogues. In total 52 sources of the  catalogue correlate with (possible) globular clusters. Of these sources 36 are identified as GlCs because their optical counterparts are listed as globular clusters in the catalogues given in Sect.\[Sec:CrossCorr\_Tech\], while the remaining 16 sources are only listed as globular cluster candidates. The range of source XID fluxes goes from 3.1[$\times 10^{-15}$ ]{} ( 924) to 2.7[$\times 10^{-12}$ ]{} ( 1057), or in luminosity from 2.3[$\times 10^{35}$ ]{} to 2.0[$\times 10^{38}$ ]{} (Fig.\[Fig:XRB\_fldist\]; green histogram). Compared to the fluxes found for the XRBs discussed in Sect.\[SubSec:XRB\], 14 sources that correlate with GlCs have fluxes below the lowest flux found for field XRBs. The reason for this finding is that the classification of field XRBs is based on the variable or transient nature of the sources, which can only be to detected for brighter sources ( Sect.\[Sec:var\]) and not just by positional coincidence that is also possible for faint sources. Figure \[Fig:GlC\_spdist\] shows the spatial distribution of the GlC sources. X-ray sources correlating with GlCs follow the distribution of the optical GlCs, which are also concentrated towards the central region of 31. The three brightest globular cluster sources, which are located in the northen disc of 31, are  1057 (XMMM31 J004252.0+413109),  694 (XMMM31 J004143.1+413420), and  1692 (XMMM31 J004545.8+413941). They are all brighter than 8.4[$\times 10^{37}$ ]{}. Source  694 was classified as a black hole candidate, due to its variability observed at such high luminosities. A detailed discussion of the three sources is given in @2008ApJ...689.1215B. XMMM31 J004303.2+412121 ( 1118) was identified as a foreground star in PFH2005, based on the classification in the “Revised Bologna Catalogue" . took the classification from , which is based on the velocity dispersion of that source. Recent ‘*HST images unambiguously reveal that this* \[B147\] *is a well resolved star cluster, as recently pointed out also by @2007AJ....133.2764B*’ . That is why source  1118 is now identified as an XRB located in globular cluster B147. ### Integrated optical properties of the globular clusters in which the X ray sources are located [lcclllc]{} & & & & & &\ B005 & confirmed & old & 15.69 & 1.29 & 1.15 & old\ SK055B & candidate & - & 18.991 & 0.388 & 0.248 & –\ B024 & confirmed & old & 16.8 & 1.15 & 1.01 & old\ SK100C & candidate & na & 18.218 & 1.181 & 1.041 & old\ B045 & confirmed & old & 15.78 & 1.27 & 1.13 & old\ B050 & confirmed & old & 16.84 & 1.18 & 1.04 & old\ B055 & confirmed & old & 16.67 & 1.68 & 1.54 & old\ B058 & confirmed & old:: & 14.97 & 1.1 & 0.96 & old-inter\ MITA140 & confirmed & old & 17 & 9999 & - & –\ B078 & confirmed & old & 17.42 & 1.62 & 1.48 & old\ B082 & confirmed & old & 15.54 & 1.91 & 1.77 & old\ B086 & confirmed & old & 15.18 & 1.26 & 1.12 & old\ SK050A & confirmed & - & 18.04 & 1.079 & 0.939 & old-inter\ B094 & confirmed & old & 15.55 & 1.26 & 1.12 & old\ B096 & confirmed & old & 16.61 & 1.48 & 1.34 & old\ B098 & confirmed & old & 16.21 & 1.13 & 0.99 & old-inter\ B107 & confirmed & old & 15.94 & 1.28 & 1.14 & old\ B110 & confirmed & old & 15.28 & 1.28 & 1.14 & old\ B117 & confirmed & old:: & 16.34 & 1 & 0.86 & inter\ B116 & confirmed & old & 16.79 & 1.86 & 1.72 & old\ B123 & confirmed & old & 17.416 & 1.29 & 1.15 & old\ B124 & confirmed & old & 14.777 & 1.147 & 1.007 & old\ B128 & confirmed & old:: & 16.88 & 1.12 & 0.98 & old-inter\ B135 & confirmed & old & 16.04 & 1.22 & 1.08 & old\ B143 & confirmed & old & 16 & 1.22 & 1.08 & old\ B144 & confirmed & old:: & 15.88 & 0.59 & 0.45 & young\ B091D & confirmed & old & 15.44 & 9999 & - & –\ B146 & confirmed & old:: & 16.95 & 1.09 & 0.95 & interm\ B147 & confirmed & old & 15.8 & 1.27 & 1.13 & old\ B148 & confirmed & old & 16.05 & 1.17 & 1.03 & old\ B150 & confirmed & old & 16.8 & 1.28 & 1.14 & old\ B153 & confirmed & old & 16.24 & 1.3 & 1.16 & old\ B158 & confirmed & old & 14.7 & 1.15 & 1.01 & old\ B159 & confirmed & old & 17.2 & 1.41 & 1.27 & old\ B161 & confirmed & old & 16.33 & 1.1 & 0.96 & old-inter\ B182 & confirmed & old & 15.43 & 1.29 & 1.15 & old\ B185 & confirmed & old & 15.54 & 1.18 & 1.04 & old\ B193 & confirmed & old & 15.33 & 1.28 & 1.14 & old\ SK132C & candidate & - & 18.342 & 1.84 & 1.7 & old\ B204 & confirmed & old & 15.75 & 1.17 & 1.03 & old\ B225 & confirmed & old & 14.15 & 1.39 & 1.25 & old\ B375 & confirmed & old & 17.61:: & 1.02 & 0.88 & interm\ B386 & confirmed & old & 15.547 & 1.154 & 1.014 & old\ \[Tab:GlC\_optprop\] Notes:\ $^{ *~}$: classification as confirmed or otherwise comes from the revised Bologna catalogue (December 2009, Version 4) <http://www.bo.astro.it/M31/RBC_Phot07_V4.tab>\ $^{ +~}$: age comes from @2009AJ....137...94C\ V, and V$-$I are integrated colours that come from the revised Bologna catalogue (December 2009, Version 4) <http://www.bo.astro.it/M31/RBC_Phot07_V4.tab>\ (V$-$I)$_{\rm{o}}$ is the dereddened V-I integrated colour, assuming E(B-V)=0.10+-0.03, which is the average of the reddenings of all 31 clusters in @2005AJ....129.2670R. (this E(B$-$V) corresponds to E(V$-$I)$=\!0.14$)\ $^{ \dagger~}$: This dereddened colour (V$-$I)$_{\rm{o}}$ is used to estimate the age on the basis of the plots (V$-$I)$_{\rm{o}}$ versus logAge from @2007AJ....133..290S. For each X-ray source which correlates with a globular cluster or globular cluster candidate in the optical, we investigated its integrated V-I colour and derived age estimates. Table \[Tab:GlC\_optprop\] lists the name of the optical counterpart, its classification according to RBC V.4 , the age classification of @2009AJ....137...94C, the V magnitude and V-I colour given in RBC V.4, the dereddened V-I colour, and the age estimate derived by ourselves. The integrated V$-$I colours of the clusters can be found in RBC V.4 and can be used to provide estimates of the ages of the clusters, in conjunction with reddening values. We have adopted a reddening of E(B$-$V)$=\!0.10\pm0.03$, which is the average of the reddenings of all 31 clusters in @2005AJ....129.2670R. Using these values, we have derived (V$-$I)$_{\rm{o}}$ for our clusters. In most cases (V$-$I)$_{\rm{o}}\!>\!1.0$ suggesting clusters older than $\simeq$2 Gyr according to @2007AJ....133..290S. The histogram in Fig.\[Fig:GlC\_agedist\] shows the distribution of (V$-$I)$_{\rm{o}}$ for our clusters, with the approximate age-ranges marked. In general there is good agreement between the @2009AJ....137...94C and our age estimates. This result indicates that the great majority of the objects are indeed old globular clusters. Figure \[Fig:optGlC\_agedist\] shows the distribution of (V$-$I)$_{\rm{o}}$ for all confirmed and candidate globular clusters, listed in the RBC V.4, which are located in the  field, and which have V as well as I magnitudes given. A comparison with Fig.\[Fig:GlC\_agedist\] again reveals that mainly counterparts of old globular clusters (age $\ga$2Gyr) are detected in X-rays. ### Comparing GlC and candidates in *XMM-Newton*, *Chandra* and *ROSAT* catalogues {#SubSub:comp_GlC} The combined  PSPC catalogue (SHP97 and SHL2001) contains 33 sources classified as globular cluster counterparts. Of these sources one is located outside the field observed with . Another two sources do not have counterparts in the  catalogue. The first one is \[SHL2001\] 232, which is not visible in any  observation taken before December 2006 as was already reported in @2004ApJ...616..821T. The second source (\[SHL2001\] 231) correlates with B164 which is identified as a globular cluster in RBC V3.5. In addition \[SHL2001\] 231 is listed in PFH2005 as the counterpart of the source \[PFH2005\] 423. Due to the improved positional accuracy of the X-ray source in the  observations, PFH2005 rejected the correlation with B164 and instead classified \[PFH2005\] 423 as a foreground star candidate. Three  GlC candidates have more than one counterpart in the  catalogue. \[SHL2001\] 249 correlates with sources  1262 and  1267, where the latter is the X-ray counterpart of the globular cluster B185. \[SHL2001\] 254 correlates with sources  1289 and  1293, where the former is the X-ray counterpart of the globular cluster candidate mita311 [@1993PhDT........41M]. \[SHL2001\] 258 has a 1$\sigma$ positional error of 48 and thus correlates with sources  1297,  1305 and  1357.[^26] The brightest of these three sources ( 1305), which is actually located closest to the  position, correlates with the globular cluster candidate SK132C (RBC V3.5). Table \[Tab:ROSAT\_GlC\_tvar\] gives the variability factors (Cols. 6, 8) and significance of variability (7, 9) for sources classified as GlC candidates in the  PSPC surveys. For most sources only low variability is detected. The two sources with the highest variability factors found ( 1262,  1293) belong to  sources with more than one  counterpart. In these cases the  sources that correlate with the same  source and the optical globular cluster source show much weaker variability. Interestingly, a few sources show low, but very significant variability. Among these sources is the Z-source identified in and two of the sources discussed in @2008ApJ...689.1215B [ 1057,  1692]. {width="12cm"} [rrrrrrrrrccc]{} & & & & & & & & & & &\ & & & & & & & & & & &\ 383 & 73 & 68 & 1.45E-12 & 1.05E-14 & 1.26 & 11.58 & 1.23 & 9.04 & GlC & \* & \*\ 403 & 79 & 74 & 2.55E-14 & 2.36E-15 & 2.61 & 2.09 & 7.57 & 4.92 & Gal & &\ 422 & & 76 & 2.01E-14 & 1.43E-15 & & & & & $<$hard$>$ & &\ 694 & 122 & 113 & 1.52E-12 & 1.04E-14 & 1.38 & 10.56 & 1.26 & 7.46 & GlC & \* & \*\ 793 & 138 & 136 & 4.76E-14 & 2.33E-15 & 1.10 & 0.55 & 1.02 & 0.14 & $<$Gal$>$ & \* &\ 841 & 150 & 147 & 1.40E-12 & 1.61E-14 & 1.77 & 19.81 & 2.11 & 26.37 & GlC & \* & –\ 855 & 158 & 154 & 4.21E-13 & 3.70E-15 & 1.01 & 0.14 & 3.54 & 48.75 & GlC & \* &\ 885 & 168 & 163 & 1.56E-14 & 1.60E-15 & 1.70 & 1.54 & & & GlC & &\ 923 & 175 & 175 & 1.47E-13 & 2.07E-15 & 1.87 & 7.68 & 1.30 & 3.44 & GlC & &\ 933 & 178 & & 3.67E-14 & 1.64E-15 & 3.03 & 7.03 & & & GlC & &\ 947 & 180 & 179 & 3.24E-13 & 7.46E-15 & 2.43 & 12.78 & 2.29 & 12.16 & GlC & \* & –\ 966 & 184 & 184 & 3.51E-12 & 9.21E-15 & 1.00 & 0.20 & 2.31 & 151.58 & XRB & &\ 1057 & 205 & 199 & 2.67E-12 & 2.05E-14 & 1.72 & 26.26 & 1.79 & 28.48 & GlC & \* & \*\ 1102 & 217 & 211 & 3.23E-13 & 2.93E-15 & 1.06 & 1.04 & 5.51 & 60.80 & GlC & \* &\ 1109 & 218 & 212 & 3.25E-13 & 9.08E-15 & 1.91 & 9.70 & 2.10 & 10.72 & GlC & \* & \*\ 1118 & 222 & 216 & 1.16E-13 & 2.03E-15 & 1.46 & 3.63 & 1.89 & 7.46 & GlC & &\ 1122 & 223 & 217 & 2.48E-13 & 2.72E-15 & 2.08 & 12.02 & 7.03 & 73.27 & GlC & &\ 1157 & 228 & 223 & 7.59E-13 & 4.48E-15 & 1.06 & 1.68 & 1.08 & 2.75 & GlC & \* & \*\ 1171 & 229 & 227 & 4.68E-13 & 4.93E-15 & 1.82 & 12.92 & 1.75 & 12.19 & GlC & \* & \*\ 1262 & 247 & 249 & 2.94E-14 & 3.10E-15 & 14.11 & 18.40 & 14.64 & 20.24 & & &\ 1267 & 247 & 249 & 4.80E-13 & 4.57E-15 & 1.16 & 3.04 & 1.11 & 2.44 & GlC & \* & \*\ 1289 & 250 & 254 & 2.88E-14 & 1.91E-15 & 1.16 & 0.57 & 1.98 & 3.15 & $<$GlC$>$ & \* &\ 1293 & 250 & 254 & 6.70E-15 & 9.42E-16 & 3.73 & 2.80 & 8.53 & 5.72 & $<$AGN$>$ & &\ 1296 & 253 & 257 & 3.89E-14 & 1.58E-15 & 4.03 & 9.38 & 1.25 & 1.17 & GlC & &\ 1297 & 252 & 258 & 5.59E-15 & 9.61E-16 & 4.46 & 2.69 & & & $<$hard$>$& &\ 1305 & & 258 & 1.69E-14 & 9.87E-16 & & & & & $<$GlC$>$ & &\ 1340 & 261 & 266 & 6.07E-14 & 3.01E-15 & 1.77 & 4.20 & 1.30 & 1.64 & GlC & & \*\ 1357 & & 258 & 7.04E-15 & 1.18E-15 & & & & & $<$hard$>$& &\ 1449 & 281 & 289 & 2.34E-14 & 1.00E-15 & 3.10 & 5.36 & 1.79 & 2.39 & fg Star & &\ 1463 & 282 & 290 & 7.51E-13 & 8.38E-15 & 1.13 & 3.33 & 1.33 & 7.64 & GlC & \* & \*\ 1634 & 302 & 316 & 7.70E-14 & 2.91E-15 & 3.14 & 5.60 & 1.89 & 4.33 & $<$hard$>$& \* & \*\ 1692 & 318 & 336 & 1.15E-12 & 2.00E-14 & 2.86 & 45.59 & 2.62 & 38.63 & GlC & \* &\ 1803 & 349 & 354 & 8.72E-13 & 9.17E-15 & 1.32 & 7.87 & 1.03 & 0.93 & GlC & \* & \*\ \[Tab:ROSAT\_GlC\_tvar\] Notes:\ $^{ *~}$: SI: SHP97, SII: SHL2001\ $^{ +~}$: XID Flux and error in ergcm$^{-2}$s$^{-1}$\ $^{ {\dagger}~}$: Variability factor and significance of variability, respectively, for comparisons of  XID fluxes to  fluxes listed in SPH97 and SHL2001, respectively.\ $^{ {\ddagger}~}$: An asterisk indicates that the XID flux is larger than the corresponding  flux.  count rates are converted to 0.2–4.5keV fluxes, using WebPIMMS and assuming a foreground absorption of $=\!6.6$[$\times 10^{20}$ cm$^{-2}$]{} and a photon index of $\Gamma\!=\!1.7$: ECF$_{{\mathrm}{SHP97}}\!=\!2.229\times$10$^{-14}$ergcm$^{-2}$cts$^{-1}$ and ECF$_{{\mathrm}{SHL2001}}\!=\!2.249\times$10$^{-14}$ergcm$^{-2}$cts$^{-1}$ The 18 X-ray sources correlating with globular clusters which were found in the  HRI observations (PFJ93) were all re-detected in the  data. From the numerous studies of X-ray globular cluster counterparts in 31 based on  observations , only eight sources are undetected in the present study. One of them (\[TP2004\] 1) is located far outside the field of 31 covered by the Deep  Survey. The transient nature of \[TP2004\] 35, and the fact that it is not observed in any  observation taken before December 2006 was mentioned in Sect.\[SubSec:Chcat\]. The six remaining sources (r2-15, r3-51, r3-71, \[VG2007\] 58, \[VG2007\] 65, \[VG2007\] 82) are located in the central area of 31 and are also not reported in PFH2005. Figure \[Fig:posGlCtrans\_pos\] shows the position of these six sources (in red) and the sources of the  catalogue (in yellow). If the brightness of the six sources had not changed between the  and  observations, they would be in principle bright enough to be detected by  in the merged observations of the central field, which have in total an exposure $\ga$ 100ks. Two sources (r2-15 and \[VG2007\] 65) are located next to sources detected by . Source r2-15 is located within 13 of  1012 and within 17 of  1017 and has – in the  observation – a similar luminosity to both  sources. The distance between  1012 and  1017 is 17 and within 20 of  1012,  detected source  1006, which is about a factor 4.6 fainter than  1012. Therefore, when in a bright state, source r2-15 should be detectable with . Source \[VG2007\] 65 is located within 17of  1100, which is at least 3.5 times brighter than \[VG2007\] 65. This may complicate the detection of \[VG2007\] 65 with . The variability of \[VG2007\] 58, \[VG2007\] 65, and \[VG2007\] 82 is supported by the fact that these three sources were not detected in any  study, prior to . Hence, these six sources are likely to be at least highly variable or even transient. Several sources identified with globular clusters in previous studies have counterparts in the  catalogue but are not classified as GlC sources by us. Source  403 (\[SHL2001\] 74) correlates with B007, which is now identified as a background galaxy [@2009AJ....137...94C; @2007AJ....134..706K RBC V3.5]. Sources  793 (\[SHL2001\] 136, s1-12) and  796 (s1-11) are the X-ray counterparts of B042D and B044D, respectively, which are also suggested as background objects by @2009AJ....137...94C. Source  948 (s1-83) correlates with B063D, which is listed as a globular cluster candidate in RBC V3.5, but might be a foreground star [@2009AJ....137...94C]. Due to this ambiguity in classification we classified the source as $<$hard$>$. Source  966 correlates with \[SHL2001\] 184, which was classified as the counterpart of the globular cluster NB21 (RBC V3.5) in the  PSPC survey (SHL2001). In addition, source  966 also correlates with the  source r2-26 [@2002ApJ...577..738K]. Due to the much better spatial resolution of  compared to , @2002ApJ...577..738K showed that source r2-26 does not correlate with the globular cluster NB21. identified this source as the first Z-source in 31. The nature of source  1078 is unclear as RBC V3.5 reported that source to be a foreground star, while @2009AJ....137...94C classified it as an old globular cluster. Due to this ambiguity in the classification and due to the fact that source  1078 is resolved into two  sources (r2-9, r2-10), we decided to classify the source as $<$hard$>$. Due to the transient nature [@2002ApJ...577..738K; @2006ApJ...643..356W] and the ambiguous classifications reported by RBC V3.5 (GlC) and @2009AJ....137...94C [H[II]{} region], we adopt the classification of PFH2005 ($<$XRB$>$) for source  1152. SBK2009 classified the source correlating with source  1293 as a globular cluster candidate. We are not able to confirm this classification, as none of the globular cluster catalogues used, contains an entry at the position of source  1293. Instead we found a radio counterpart in the catalogues of @2005ApJS..159..242G, @1990ApJS...72..761B and NVSS. We therefore classified the source as an AGN candidate, as was also done in PFH2005. For source  1449 (\[SHL2001\] 289) the situation is more complicated. SHL2001 report \[MA94a\] 380 as the globular cluster correlating with this X-ray source. Based on the same reference, @2005PASP..117.1236F included the optical source in their statistical study of globular cluster candidates. However, the paper with the acronym \[MA94a\] is not available. An intensive literature search of the papers by Magnier did not reveal any work relating to globular clusters in 31, apart from @1993PhDT........41M which is cited in @2005PASP..117.1236F as “MIT". In addition the source is not included in any other globular cluster catalogues listed in Sect.\[Sec:CrossCorr\_Tech\]. In the X-ray studies of @2004ApJ...609..735W and PFH2005 and in the source is classified as a foreground star (candidate). Hence, we also classified source  1449 as a foreground star candidate, but suggest optical follow-up observations of the source to clarify its true nature. A similar case is source  422 (\[SHL2001\] 76), which is classified as a globular cluster by SHL2001, based on a correlation with \[MA94a\] 16. Here again the source is not listed in any of the globular cluster catalogues used. We found one correlation of source  422 with an object in the USNO-B1 catalogue, which has no B2 and R2 magnitude. Two faint sources (V$>\!22.5$mag) of the LGGS catalogue are located within the X-ray positional error circle. Thus source  422 is classified as $<$hard$>$. While RBC V3.5 classified the optical counterpart of source  1634 (\[SHL2001\] 316) as a globular cluster candidate, @2009AJ....137...94C regard SK182C as being a source of unknown nature. Therefore we decided to classify source  1634 as $<$hard$>$. Conclusions {#Sec:Concl} =========== This paper presents the analysis of a large and deep  survey of the bright Local Group SA(s)b galaxy 31. The survey observations were taken between June 2006 and February 2008. Together with re-analysed archival observations, they provide for the first time full coverage of the M31 ${\mathrm}{D}_{25}$ ellipse down to a 0.2–4.5keV luminosity of $\sim$[$10^{35}$ erg s$^{-1}$]{}. The analysis of combined and individual observations allowed the study of faint persistent sources as well as brighter variable sources. The source catalogue of the Large  Survey of 31 contains 1897 sources in total, of which 914 sources were detected for the first time in X-rays. The XID source luminosities range from $\sim$4.4[$\times 10^{34}$ ]{} to 2.7[$\times 10^{38}$ ]{}. The previously found differences in the spatial distribution of bright ($\ga$[$10^{37}$ erg s$^{-1}$]{}) sources between the northern and southern disc could not be confirmed. The identification and classification of the sources was based on properties in the X-ray wavelength regime: hardness ratios, extent and temporal variability. In addition, information obtained from cross correlations with 31 catalogues in the radio, infra-red, optical and X-ray wavelength regimes were used. The source catalogue contains 12 sources with spatial extent between 62 and 230. From spectral investigation and comparison with optical images, five sources were classified as galaxy cluster candidates. 317 out of 1407 examined sources showed long term variability with a significance $>$3$\sigma$ between the  observations. These include 173 sources in the disc that were not covered in the study of the central field (SPH2008). Three sources located in the outskirts of the central field could not have been detected as variable in the study presented in SPH2008, as they only showed variability with a significance $>$3$\sigma$ between the archival and the “Large Project" observations. For 69 sources the flux varied by more than a factor of five between XMM-Newton observations; ten of these varied by a factor $>$100. Discrepancies in source detection between the Large  Survey catalogue and previous  catalogues could be explained by different search strategies, and differences in the processing of the data, in the parameter settings of the detection runs and in the software versions used. Correlations with previous  studies showed that those sources not detected in this study are strongly time variable, transient, or unresolved. This is particularly true for sources located close to the centre of 31, where ’s higher spatial resolution resolves more sources. Some of the undetected sources from previous  studies were located outside the field covered with . However, there were several sources detected by  that had a  detection likelihood larger than 15. If these sources were still in a bright state they should have been detected with . Thus, the fact that these sources are not detected with  implies that they are transient or at least highly variable sources. On the other hand 242 $<$hard$>$  sources were found with XID fluxes larger than [$10^{-14}$ erg cm$^{-2}$ s$^{-1}$]{}, which were not detected with . To study the properties of the different source populations of 31, it was necessary to separate foreground stars (40 plus 223 candidates) and background sources (11 AGN and 49 candidates, 4 galaxies and 19 candidates, 1 galaxy cluster and 5 candidates) from the sources of 31. 1247 sources could only be classified as $<$hard$>$, while 123 sources remained without identification or classification. The majority (about two-thirds, see Stiele et al. 2011 in preparation) of sources classified as $<$hard$>$ are expected to be background objects, especially AGN. The catalogue of the Large  survey of 31 contains 30 SSS candidates, with unabsorbed 0.2–1.0keV luminosities between 2.4[$\times 10^{35}$ ]{} and 2.8[$\times 10^{37}$ ]{}. SSSs are concentrated to the centre of 31, which can be explained by their correlation with optical novae, and by the overall spatial distribution of 31 late type stars ( enhanced density towards the centre). Of the 14 identifications made of optical novae, four were presented in more detail. The 25 identified and 31 classified SNRs had XID luminosities between 1.1[$\times 10^{35}$ ]{} and 4.3$\times$10$^{36}$ ergs$^{-1}$. Three of the 25 identified SNRs were detected for the first time in X-rays. For one SNR the  classification can be confirmed. Six of the SNR candidates were selected from correlations with sources in SNR catalogues from the literature. As these six sources had rather “hard" hardness ratios they are good candidates for “plerions". An investigation of the spatial distribution showed that most SNRs and candidates are located in regions of enhanced star formation, especially along the 10kpc dust ring in 31. This connection between SNRs and star forming regions, implies that most of the remnants are from type II supernovae. Most of the SNR classifications from previous studies have been confirmed. However, in five cases these classifications are doubtful. The population of “hard" 31 sources mainly consists of XRBs. These rather bright sources (XID luminosity range: 1.0[$\times 10^{36}$ ]{} to 2.7[$\times 10^{38}$ ]{}) were selected from their transient nature or strong long term variability (variability factor $>$10; 10 identified, 26 classified sources). The spectral properties of three transient sources were presented in more detail. A sub-class of LMXBs is located in globular clusters. They were selected from correlations with optical sources included in globular cluster catalogues (36 identified, 16 classified sources). The XID luminosity of GlCs ranges from 2.3[$\times 10^{35}$ ]{} to 1.0[$\times 10^{38}$ ]{}. The spatial distribution of this source class also showed an enhanced concentration to the centre of 31. From optical and X-ray colour-colour diagrams possible HMXB candidates were selected. If the sources were bright enough, an absorbed power-law model was fitted to the source spectra. Two of the candidates had a photon index consistent with the photon index range of NS HMXBs. Hence these two sources were suggested as new HMXB candidates. Follow-up studies in the optical as well as in radio are in progress or are planned. They will allow us to increase the number of identified sources and help us to classify or identify sources which can up to now only be classified as $<$hard$>$ or are without any classification. This work focused on the overall properties of the source population of individual classes and gave us deeper insights into the long-term variability, spatial and flux distribution of the sources in the field of 31 and thus helped us to improve our understanding of the X-ray source population of 31. This publication makes use of the USNOFS Image and Catalogue Archive operated by the United States Naval Observatory, Flagstaff Station (http://www.nofs.navy.mil/data/fchpix/), of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation, of the SIMBAD database, operated at CDS, Strasbourg, France, and of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The XMM-Newton project is supported by the Bundesministerium für Wirtschaft und Technologie/Deutsches Zentrum für Luft- und Raumfahrt (BMWI/DLR, FKZ 50 OX 0001) and the Max-Planck Society. HS acknowledges support by the Bundesministerium für Wirtschaft und Technologie/Deutsches Zentrum für Luft- und Raumfahrt (BMWI/DLR, FKZ 50 OR 0405). \[App:XID\_Images\] [^1]: Based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. [^2]: Tables 5 and 8 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ [^3]: <http://xmm.esac.esa.int/xsa/> [^4]: See also <http://xmm2.esac.esa.int/docs/documents/CAL-TN-0050-1-0.ps.gz> [^5]: This is a simplified description as [emldetect]{} transforms the derived likelihoods to equivalent likelihoods, corresponding to the case of two free parameters. This allows comparison between detection runs with different numbers of free parameters. [^6]: <http://heasarc.gsfc.gov/docs/xanadu/xspec> [^7]: <http://xmm2.esac.esa.int/external/xmm_sw_cal/calib/epic_files.shtml> [^8]: especially in the [emldetect]{} task [^9]: For the remainder of the subsection we will call all three catalogues “optical catalogues" for easier readability, although the 2MASS catalogue is an infrared catalogue. [^10]: From the LGGS catalogue only sources brighter than 21mag were used in order to be comparable to the brightness limit of the USNO-B1 catalogue. [^11]: the offset in declination is negligible [^12]: The combination of observations b1, b3 and b4 is called b. [^13]: <http://www.mpe.mpg.de/~m31novae/opt/m31/M31_table.html> [^14]: <http://www.cv.nrao.edu/nvss/NVSSlist.shtml> [^15]: <http://simbad.u-strasbg.fr/simbad> [^16]: <http://nedwww.ipac.caltech.edu> [^17]: in B2 magnitude [^18]: in B magnitude [^19]: The luminosity is based on XID Fluxes. Using the total 0.2–12keV band the result does not change (23 in the northern half and 24 in the southern half). [^20]: The source was observed with  on 11 January 2001. Obs. id.: 0065770101 [^21]: <http://www.mpe.mpg.de/~m31novae/opt/m31/M31_table.html> [^22]: <http://www.mpe.mpg.de/~m31novae/xray/index.php> [^23]: <http://www.supernovae.net/sn2005/novae.html> [^24]: <http://www.supernovae.net/sn2005/novae.html> [^25]: <http://cfa-www.harvard.edu/iau/CBAT_M31.html> [^26]: In addition \[SHL2001\] 258 correlates with  1275,  1289 and  1293. However these sources have each an additional  counterpart.

--- author: - Sophie Morel and Junecue Suh bibliography: - 'sign.bib' title: 'The standard sign conjecture on algebraic cycles: the case of Shimura varieties' --- Introduction ============ First we recall the standard sign conjecture, its origin, statement and significance. Let $k$ be a field and fix a Weil cohomology theory ${\mathrm{H}}^*$ on the category of smooth projective varieties over $k$ with coefficients in a field $F$ of characteristic zero (*cf.* [@An] 3.3.1). Denote by $M_{hom}(k)_F$ the category of homological motives over $k$ associated with ${\mathrm{H}}^*$, and by $M_{num}(k)_F$ the category of numerical motives over $k$, both with coefficients in $F$ (*cf.* [@An] 4.1). The functor ${\mathrm{H}}^*$ defines a realization functor from $M_{hom}(k)_F$ to the category of graded $F$-vector spaces, that we will denote by ${\mathrm{H}}^*$ (*cf.* [@An] 4.2.5). The K[ü]{}nneth standard conjecture (cf. [@An] 5.1.1) states that, for every smooth projective variety $X/k$, the K[ü]{}nneth projectors $p^i_X$ onto the direct factor ${\mathrm{H}}^i(X)$ of ${\mathrm{H}}^{\ast}(X)$ are given by algebraic cycles. In the classical theory of motives (of Grothendieck), one uses it to modify the sign in the commutativity constraints in the $\otimes$-structure in order to get the Tannakian category of homological motives. For classical cohomology theories, the conjecture would be a consequence of the Hodge conjecture over the complex numbers and the Tate conjecture over finitely generated fields. Namely, as noted by Grothendieck (see [@Ta] p.99), the projectors (and any linear combination thereof) clearly are morphisms of Hodge structures and commute with the Galois action, and these cohomology classes make natural test cases for the Hodge and Tate conjectures. The strongest evidence for the Künneth conjecture is given by Katz and Messing ([@KM], Theorem 2): It is true when $k$ is algebraic over a finite field and ${\mathrm{H}}^{\ast}$ is either the $\ell$-adic cohomology for a prime $\ell\neq \mathrm{char}(k)$, or the crystalline cohomology. For the purpose of modifying the commutativity constraints and getting the Tannakian category, one needs somewhat less, and the necessary weakening is called the standard “sign” conjecture (terminology proposed by Jannsen; *cf.* [@An] 5.1.3 for the formulation and see [@An] 6.1.2.1 for obtaining the Tannakian category): For every $M\in\operatorname{\mathrm{Ob}}M_{hom}(k)_F$, there exists a decomposition $M=M^+\oplus M^-$ such that ${\mathrm{H}}^*(M^+)$ (resp. ${\mathrm{H}}^*(M^-)$) is concentrated in even (resp. odd) degrees. Equivalently, for every smooth projective $X/k$, the sum $p^{+}_X$ (resp. $p^{-}_X$) of the even (resp. odd) Künneth projectors on ${\mathrm{H}}^*(X)$ is given by an algebraic cycle. Note that such a decomposition is necessarily unique (up to unique isomorphism). This conjecture has, in addition to the consequences in terms of the category of homological motives and the algebraicity of (Hodge or Tate) cohomology classes, also the following interesting consequence, due to André and Kahn. Recall that the numerical and the homological equivalences on algebraic cycles on projective smooth varieties are conjectured to be the same. ([@An] 9.3.3.3) Let ${{\mathscr}{M}}$ be an additive $\otimes$-subcategory of $M_{hom}(k)_F$ and let ${{\mathscr}{M}}_{num}$ be its image in $M_{num}(k)_F$. If the sign conjecture is true for every object of ${{\mathscr}{M}}$, then the functor ${{\mathscr}{M}}{\longrightarrow}{{\mathscr}{M}}_{num}$ admits a section compatible with $\otimes$, unique up to $\otimes$-isomorphism. Next we turn to the main geometric objects of this paper, Shimura varieties. In this paper, we will take for $k$ a subfield of ${\mathbb{C}}$, and for ${\mathrm{H}}^*$ the cohomology theory that sends a smooth projective variety $X$ over $k$ to the Betti cohomology of $X({\mathbb{C}})$ with coefficients in number fields $F$. Let $({{\mathbf}G},{{\mathscr}{X}},h)$ be pure Shimura data (cf [@De-VS] 2.1.1 or [@P2] 3.1), $E\subset{\mathbb{C}}$ the reflex field and ${\mathrm{K}}$ a neat open compact subgroup of ${{\mathbf}G}({\mathbb{A}_f})$. Denote by $S^{\mathrm{K}}$ the Shimura variety at level ${\mathrm{K}}$ associated to $({{\mathbf}G},{{\mathscr}{X}},h)$; it is a smooth quasi-projective variety over $E$. Assume that $E\subset k$. If $S^{\mathrm{K}}$ is projective, denote by $M(S^{\mathrm{K}})$ the image of $S^{\mathrm{K}}$ in $M_{hom}(k)_{{\mathbb{Q}}}$. For a general connected reductive group ${{\mathbf}G}$ over ${\mathbb{Q}}$, we say that ${{\mathbf}G}$ satisfies condition (C), if - Arthur’s conjectures (cf section \[L2A\]) are known for ${{\mathbf}G}$, - the cohomological Arthur parameters for ${{\mathbf}G}$ satisfy a certain condition that will be spelled out at the end of section \[L2A\] (roughly, that what happens at the finite places determines the parameter) and - the classification of cohomological representations of ${{\mathbf}G}({\mathbb{R}})$ giv en by Adams and Johnson in [@AJ] agrees with the classification given by Arthur’s conjectures. Given the current state of knowledge of (C) (see below), we will also consider a weaker condition. We say that ${{\mathbf}G}$ satisfies condition (C'), if there exists a ${\mathbb{Q}}$-algebraic subgroup ${{\mathbf}G}'$ of ${{\mathbf}G}$ which contains the derived group ${{\mathbf}G}^{{\mathrm{der}}}$ and satisfies (C). The goal of this paper is to prove the following theorem : \[thA\] Let $({{\mathbf}G},{{\mathscr}{X}},h)$ be simple PEL Shimura data. Assume that ${{\mathbf}G}$ is anisotropic over ${\mathbb{Q}}$ modulo its center (so that $S^{\mathrm{K}}$ is projective and smooth) and that it satisfies condition (C'). Then $M(S^{\mathrm{K}})\in M_{hom}(k)_{{\mathbb{Q}}}$ satisfies the sign conjecture. If ${{\mathbf}G}$ is not anisotropic modulo its center, then $S^{\mathrm{K}}$ is not projective, so we can not talk about its homological motive. A possible generalization is the motive representing the intersection cohomology of the minimal compactification of $S^{\mathrm{K}}$. Such a motive is not known to exist for general varieties (though we certainly expect that it does), but in the case of minimal compactifications of Shimura varieties it has been constructed by Wildeshaus, even in the category of Chow motives over $E$ (cf [@Wil] Theorems 0.1 and 0.2). We then have the following generalization of the previous theorem : \[thB\] Let $({{\mathbf}G},{{\mathscr}{X}},h)$ be simple PEL Shimura data, and assume that ${{\mathbf}G}$ satisfies condition (C'). Denote by $IM(S^{\mathrm{K}})$ the “intersection motive” of the minimal compactification of $S^{\mathrm{K}}$. Then $IM(S^{\mathrm{K}})$ satisfies the sign conjecture. We actually have versions of these two theorems for motives with coefficients in smooth motives (whose Betti realizations are automorphic local systems), see Theorem \[th-sign-int-coeff\]. We will deduce Theorems \[thA\] and \[thB\] from another result that we need some more notation to state. Let ${{\mathbf}G}$ be a connected reductive group over ${\mathbb{Q}}$, ${\mathrm{A}}_{{\mathbf}G}$ the maximal ${\mathbb{Q}}$-split torus in the center of ${{\mathbf}G}$, ${\mathrm{K}}_\infty$ a maximal compact subgroup of ${{\mathbf}G}({\mathbb{R}})$, ${\mathrm{K}}_\infty'={\mathrm{A}}_G({\mathbb{R}})^\circ{\mathrm{K}}_\infty$, and ${{\mathscr}{X}}={{\mathbf}G}({\mathbb{R}})/{\mathrm{K}}'_\infty$. We assume that ${{\mathscr}{X}}$ is a Hermitian symmetric domain; this is satisfied by the group ${{\mathbf}G}$ in any Shimura data, and also by any subgroup thereof as in condition (C'). With the notation and under this assumption, the double coset space $$S^{\mathrm{K}}={{\mathbf}G}({\mathbb{Q}}){\setminus}({{\mathscr}{X}}\times{{\mathbf}G}({\mathbb{A}}_f)/{\mathrm{K}})$$ still makes sense for open compact subgroups ${\mathrm{K}}$ of ${{\mathbf}G}({\mathbb{A}}_f)$, and is a finite disjoint union of quotients of Hermitian symmetric domains by arithmetic subgroups of ${{\mathbf}G}({\mathbb{Q}})$. In particular, it is a disjoint union of locally symmetric Riemannian manifolds if ${\mathrm{K}}$ is neat. Moreover, by a theorem of Baily and Borel (cf [@BB] Theorem 10.11), $S^{\mathrm{K}}$ is a quasi-projective complex algebraic variety, smooth if ${\mathrm{K}}$ is neat. Let ${{\mathscr}{H}}_{\mathrm{K}}=C_c^\infty({\mathrm{K}}{\setminus}{{\mathbf}G}({\mathbb{A}_f})/{\mathrm{K}},{\mathbb{Q}})$ be the algebra of functions ${\mathrm{K}}{\setminus}{{\mathbf}G}({\mathbb{A}_f})/{\mathrm{K}}{\longrightarrow}{\mathbb{Q}}$ that are locally constant and have compact support, with multiplication given by the convolution product. Let $j:S^{\mathrm{K}}{\longrightarrow}\overline{S}^{\mathrm{K}}$ be the embedding of $S^{\mathrm{K}}$ in its minimal compactification. Let $W$ be an irreducible algebraic representation of ${{\mathbf}G}$ defined over a field $F$, and denote by ${{\mathscr}{F}}_W$ the associated $F$-local system on $S^{\mathrm{K}}$ (cf [@KR], p 113). Let $d$ be the dimension of $S^{\mathrm{K}}$ (as an algebraic variety). The *intersection complex* of $\overline{S}^{\mathrm{K}}$ with coefficients in ${{\mathscr}{F}}_W$ (or $W$) is the complex $${\mathrm{IC}}^{\mathrm{K}}_W:=(j_{!*}({{\mathscr}{F}}_W[d]))[-d].$$ The *intersection cohomology* of $\overline{S}^{\mathrm{K}}$ with coefficients in ${{\mathscr}{F}}_W$ (or $W$) is $${\mathrm{IH}}^*(S^{\mathrm{K}},W):={\mathrm{H}}^*(\overline{S}^K,{\mathrm{IC}}^{\mathrm{K}}_W).$$ It admits an $F$-linear action of ${{\mathscr}{H}}_{\mathrm{K}}\otimes_{{\mathbb{Q}}} F$ (cf [@KR] p 122-123). We write $$\label{eqn-Hecke-dec}\tag{1} {\mathrm{IH}}^i(S^{\mathrm{K}},W)\otimes_F {\mathbb{C}}=\bigoplus_{\pi_f}\pi_f^{{\mathrm{K}}}\otimes\sigma^i(\pi_f),$$ where the sum is over all irreducible admissible representations $\pi_f$ of ${{\mathbf}G}({\mathbb{A}_f})$, $\pi_f^{\mathrm{K}}$ is the space of ${\mathrm{K}}$-invariant vectors in $\pi_f$ (a representation of ${{\mathscr}{H}}_K\otimes{\mathbb{C}}$) and the $\sigma^i(\pi_f)$ are finite-dimensional ${\mathbb{C}}$-vector spaces. Then : \[thC\] Assume that ${{\mathbf}G}$ satisfies (C) and let $\pi_f$ be as above. Then, either $\sigma^i(\pi_f)=0$ for every $i$ even, or $\sigma^i(\pi_f)=0$ for every $i$ odd. In general, for $\pi_f$ fixed, there can be several degrees $i$ with $\sigma^i(\pi_f)\not=0$ (as is clear on the formula for ${\mathrm{IH}}^i$ in section \[proof\]). Hence the methods of this paper cannot be used to prove the full Künneth conjecture. This is also clear from the fact that the Lefschetz operator on ${\mathrm{IH}}^*$ commutes with the action of the Hecke operators. Note that the action of ${\mathbb{C}}^\times$ on the ${\mathrm{IH}}^i$ that gives the pure Hodge structure (whose existence follows from M. Saito’s theory of mixed Hodge modules) also commutes with the action of the Hecke operators (and with the Lefschetz operator). See page 8 of Arthur’s review paper [@A-L2bis] for a more precise version of these two statements. Here is the present state of knowledge about condition (C) : - Arthur’s conjectures (with substitute parameters) are known for split symplectic and quasi-split special orthogonal groups, by the book [@A-livre] of Arthur, modulo the stabilization of the twisted trace formula and a local theorem at the archimedean place (see the end of the introduction of [@A-livre]). They are also known for quasi-split unitary groups by work of Mok ([@Mok]) and for their inner forms by work of Kaletha-Minguez-Shin-White ([@KMSW]), modulo the same hypotheses. Finally, still assuming the same hypotheses, the conjectures are known for tempered representations of split general symplectic and quasi-split general orthogonal groups, by work of Bin Xu ([@Xu]).[^1] - This condition, in the cases where Arthur’s conjectures are (almost) known, follows easily from strong multiplicity one for the groups ${{\mathbf}{GL}}_n$. - The agreement of the classifications of Arthur and Adams-Johnson for cohomological representations of ${{\mathbf}G}({\mathbb{R}})$ is still open, though it should be accessible. In the case of even special orthogonal groups, Arthur’s methods don’t allow to distinguish between a representation and its conjugate under the orthogonal group. This doesn’t affect the methods of this paper and thus is not a problem for us, see the end of section \[proof\]. In the final section, we discuss the possibility of using finite correspondences to attack the standard Künneth (or sign) conjecture for general projective smooth varieties. #### Acknowledgments {#acknowledgments .unnumbered} We thank Pierre Deligne for helpful discussions, especially regarding the material on finite correspondences in the final section. We also thank Colette Moeglin for pointing out some misconceptions about Arthur’s conjectures in a previous version of this article. Reduction to Theorem \[thC\] ============================ First, we review the motivic constructions of coefficient systems and intersection motives, by Ancona and Wildeshaus. This will allow us to state Theorem \[th-sign-int-coeff\] with coefficients, which, together with Theorem \[thC\], implies both \[thA\] and \[thB\]. We then make certain reduction steps necessary for passing from PEL Shimura varieties to the associated connected Shimura varieties. Finally we prove Theorem \[th-sign-int-coeff\], modulo Theorem \[thC\]. Review of motivic constructions ------------------------------- Let $F$ be a number field and let $W$ be a finite dimensional algebraic representation of ${{\mathbf}G}_F$. One applies Théorème 4.7 and Remarque 4.8 of [@Anc] to get a Chow motive $\tilde{\mu}(W)$ over $S^{{\mathrm{K}}}$, whose Betti realization over $S^{{\mathrm{K}}}({\mathbb{C}})$ is the local system corresponding to $W$. By construction, it is a direct sum of Tate twists of direct summands in the motives $$\pi^{(r)}_{\ast} \mathbf{1}_{A^r}$$ where $r\ge 0$ and $\pi^{(r)}: A^r {\longrightarrow}S^{{\mathrm{K}}}$ is the $r$th power of the Kuga-Sato abelian scheme. Then one applies[^2] the main result of [@Wil] (Theorems 0.1 and 0.2 and Corollary 0.3) to obtain the intermediate extension $j_{!\ast} \tilde{\mu}(W)$ on the minimal compactification $\overline{S}^{{\mathrm{K}}}$, whose Betti realization is naturally isomorphic to the intersection complex. Finally, by taking the direct image of $j_{!\ast} \tilde{\mu}(W)$ under the structure morphism $m:\overline{S}^{{\mathrm{K}}} {\longrightarrow}\operatorname{Spec}k$, one gets the intersection motive $IM(S^K, W)$ whose Betti realization is canonically isomorphic to the intersection cohomology ${\mathrm{IH}}^{\ast}(S^K({\mathbb{C}}), W)$. [^3] Also constructed in [@Wil] (Theorem 0.5) is an endomorphism $KgK$ of $IM(S^{{\mathrm{K}}}, W)$, for each double coset $KgK \in {\mathrm{K}}{\setminus}{{\mathbf}G}({\mathbb{A}_f})/{\mathrm{K}}$, whose Betti realization coincides with the usual action of the Hecke operator for the coset on the intersection cohomology. The construction uses the compatibility of the motivic middle extension and the “change of level” maps $[h\cdot\;]_{\ast}$, see Theorem 0.4 of [@Wil]. To avoid possible confusion, we will use the notation $\widetilde{KgK}$ for the endomorphisms of $IM(S^{{\mathrm{K}}}, W)$.[^4] Sign conjecture with coefficients --------------------------------- Now we are ready to state a version with coefficients. Let $W$ be an irreducible algebraic representation of ${{\mathbf}G}$ over a number field $F$. We denote by ${\mathrm{IH}}^{+}$ (resp. ${\mathrm{IH}}^{-}$) the direct sum of ${\mathrm{IH}}^i(S^{{\mathrm{K}}}({\mathbb{C}}), W)$ for $i$ even (resp. odd), and denote by $p_W^{+} = p_{W, F}^{+}$ (resp. $p_W^{-} = p_{W, F}^{-}$) the corresponding projector on ${\mathrm{IH}}^{\ast}(S^{{\mathrm{K}}}({\mathbb{C}}), W)$. \[th-sign-int-coeff\] Assume that the group ${{\mathbf}G}$ in the simple PEL data satisfies condition (C'), so that there exists a subgroup ${{\mathbf}G}'$ which contains ${{\mathbf}G}^{{\mathrm{der}}}$ and satisfies condition (C). Then there exists an endomorphism $\tilde{p}_W^{+}$ (resp. $\tilde{p}_W^{-}$) of the image of $IM(S^{{\mathrm{K}}}, W)$ in $M_{hom}(k)_F$ whose Betti realization is $p_W^{+}$ (resp. $p_W^{-}$). It follows that the image of $IM(S^{{\mathrm{K}}}, W)$ in $M_{hom}(k)_F$ admits a decomposition $$IM(S^{{\mathrm{K}}}, W)_{hom} = IM(S^{{\mathrm{K}}}, W)_{hom}^{+} \oplus IM(S^{{\mathrm{K}}}, W)_{hom}^{-}$$ such that ${\mathrm{H}}^{\ast} (IM(S^{{\mathrm{K}}}, W)_{hom}^{+}) = {\mathrm{IH}}^{+}$ and ${\mathrm{H}}^{\ast} (IM(S^{{\mathrm{K}}}, W)_{hom}^{-}) = {\mathrm{IH}}^{-}$ are concentrated in even and odd degrees, respectively. Reduction steps {#sec-red-steps} --------------- *Change of base field*: First, we reduce to the case where $k={\mathbb{C}}$. For this, note that the vector spaces of algebraic correspondences modulo an adequate equivalence that is coarser than the algebraic equivalence (in particular the homological equivalence) are invariant under the change of ground field from $\bar{k}$ to ${\mathbb{C}}$. Then use the fact that the even and odd projectors are invariant under the action of $\operatorname{Gal}(\bar{k}/k)$. *Connected components*: As the minimal compactification $\overline{S}^{{\mathrm{K}}}$ is normal by construction, its connected components and irreducible components coincide. Thus the even projector for $S^{{\mathrm{K}}}$ is the sum of the even projectors for the connected components, and the sign conjecture is true for $S^{{\mathrm{K}}}$ and $W$ iff it is true for each of its connected components. *Raising the level*: \[ssec-level\] Finally, we may pass from a neat level subgroup ${\mathrm{K}}$ to any level subgroup ${\mathrm{K}}'\subset{\mathrm{K}}$. It suffices to show that $IM(S^{{\mathrm{K}}}, W)$ is a direct factor of $IM(S^{K'},W)$ in $M_{\hom}({\mathbb{C}})_F$. Take a connected component $\overline{X}$ of $\overline{S}^{{\mathrm{K}}}$, with $X = \overline{X} \cap S^{{\mathrm{K}}}$. The change of level map $f:=[1\cdot]: S^{{\mathrm{K}}'} {\longrightarrow}S^{{\mathrm{K}}}$ is a finite étale surjection that extends to a finite surjection $\overline{f}: \overline{S}^{{\mathrm{K}}'} {\longrightarrow}\overline{S}^{{\mathrm{K}}}$. Let $\overline{Y}$ be the inverse image $\overline{f}^{-1}(\overline{X})$ and $Y := \overline{Y} \cap S^{{\mathrm{K}}'} = f^{-1} (X)$. Over $X$ we have the adjunction map for direct image and the trace map: $${\mathrm{adj}}_f: {{\mathscr}{F}}_W {\longrightarrow}f_{\ast} {{\mathscr}{F}}_W \,\, \mbox{ and } \,\, {\mathrm{Tr}}_f: f_{\ast} {{\mathscr}{F}}_W {\longrightarrow}{{\mathscr}{F}}_W.$$ Since $\overline{f}$ is finite, these maps extend to $$\overline{{\mathrm{adj}}_f} : j_{!*} {{\mathscr}{F}}_W {\longrightarrow}\overline{f}_{\ast} (j_{!*} {{\mathscr}{F}}_W) \,\, \mbox{ and } \,\, \overline{{\mathrm{Tr}}_f}: \overline{f}_{\ast} ( j_{!*}{{\mathscr}{F}}_W ){\longrightarrow}j_{!*}{{\mathscr}{F}}_W$$ (here $j_{!*} {{\mathscr}{F}}_W$ means $j_{!*}({{\mathscr}{F}}_W [\dim X])[-\dim X]$). In Betti cohomology, these maps give rise to $$\xymatrix{ {\mathrm{IH}}^i (S^{{\mathrm{K}}}, W) \ar[r]^-{{\mathrm{adj}}} & {\mathrm{IH}}^i (S^{{\mathrm{K}}'}, W) \ar[r]^-{{\mathrm{Tr}}} & {\mathrm{IH}}^i (S^{{\mathrm{K}}}, W) }$$ The composite map is equal to multiplication by $\deg(f)$. We may replace $\overline{Y}$ (hence $Y$) with any connected component, denoting the restriction of $\overline{f}$ (also $f$) by the same letter. It suffices to show that $$\overline{{\mathrm{Tr}}_f} \circ \overline{{\mathrm{adj}}_f} = \deg(f) \;\; \mbox{ on } j_{!*}{{\mathscr}{F}}_W.$$ By construction ${{\mathscr}{F}}= {{\mathscr}{F}}_W$ is a semisimple local system, and we may replace it with a direct summand and assume it is irreducible. Then $j_{!*} {{\mathscr}{F}}$ is a simple perverse sheaf (*cf.* Théorème 4.3.1(ii) of [@BBD]), and it suffices to show the equality over the dense open subset $X$. This last follows from Théorème 2.9 (Var 4) (I), exposé XVIII, SGA4. As we have recalled, Wildeshaus constructs $[1\cdot]_{\ast}$ between intersection motives; see also the construction leading up to Corollary 8.8 in [@Wil]. Thus $IM(S^{{\mathrm{K}}}, {{\mathscr}{F}}_W)$ is a direct factor of $IM(S^{{\mathrm{K}}'}, {{\mathscr}{F}}_W)$, modulo homological equivalence. Proof of Theorem \[th-sign-int-coeff\] modulo Theorem \[thC\] ------------------------------------------------------------- By the previous reduction steps, we may pass to the connected Shimura varieties, see 2.1.2, 2.1.7 and 2.1.8 in [@De-VS]: The projective system of connected locally symmetric varieties depend only on the triple $({{\mathbf}G}^{{\mathrm{ad}}}, {{\mathbf}G}^{{\mathrm{der}}}, {{\mathscr}{X}}^{+})$. The motivic constructions of the coefficient systems and the intersection cohomology can be therefore transferred to the connected locally symmetric varieties attached to the subgroup ${{\mathbf}G}'$, for all small enough level subgroups. \[lem-hecke-elts\] If Theorem \[thC\] is true for $W$, then there exist elements $$h^{\pm}_{W, F} \in {{\mathscr}{H}}_{{\mathrm{K}}} \otimes_{{\mathbb{Q}}} F$$ which act as $p^{\pm}_W$ on ${\mathrm{IH}}^{\ast}(S^{{\mathrm{K}}}({\mathbb{C}}), W)$. [of Lemma]{} First we prove the statement over the field of coefficients $F'={\mathbb{C}}$. Let $\Sigma$ be the finite set consisting of the irreducible admissible representations $\pi_f$ that have nonzero contribution to the right hand side of (\[eqn-Hecke-dec\]) for some $i$. Then as representations of the Hecke algebra ${{\mathscr}{H}}_K\otimes_{{\mathbb{Q}}} {\mathbb{C}}$, $( \pi_f^K )_{\pi_f \in \Sigma}$ are irreducible and pairwise inequivalent. By Jacobson’s density theorem, for each $\pi_f \in \Sigma$ there exists an element $h_{\pi_f} \in {{\mathscr}{H}}_K \otimes_{{\mathbb{Q}}} {\mathbb{C}}$ that acts as $1$ on $\pi_f^K$ and as $0$ on $\pi'^K_f$ for every other $\pi'_f$ in $\Sigma$. By Theorem \[thC\], $\Sigma$ is the disjoint union of two subsets $\Sigma^{\pm}$, consisting of those $\pi_f \in \Sigma$ that have contribution in even or odd degrees, respectively. Therefore $$h^{\pm}_{W, {\mathbb{C}}} = \sum_{\pi_f \in \Sigma^{\pm}} h_{\pi_f} \in {{\mathscr}{H}}_{{\mathrm{K}}} \otimes_{{\mathbb{Q}}} {\mathbb{C}}$$ acts as $p^{\pm}_{W, {\mathbb{C}}} = p^{\pm}_{W, F} \otimes_F 1_{{\mathbb{C}}}$. To conclude the proof, use the fact: If $f: H {\longrightarrow}E$ is an $F$-linear map of $F$-vector spaces and $F'$ is an extension field of $F$, then an element $p\in E$ lies in the image of $f$ iff $p\otimes 1_{F'}$ is in the image of $f\otimes_F 1_{F'}$. Now Theorem \[th-sign-int-coeff\] follows easily from the lemma: Writing $$h^{\pm}_{W, F} = \sum_{g \in K {\setminus}{{\mathbf}G}({\mathbb{A}_f})/{\mathrm{K}}} c^{\pm}_g \; [1_{KgK}], \;\; c^{\pm}_g \in F$$ the endomorphism of $IM(S^{{\mathrm{K}}}, W)$ in $M_{rat}({\mathbb{C}})_F$ and also its image in $M_{hom}({\mathbb{C}})_F$ $$\tilde{p}^{\pm}_W := \sum c^{\pm}_g \; \widetilde{KgK}$$ has Betti realization $p^{\pm}_W$. In the case $W$ is the trivial representation defined over ${\mathbb{Q}}$, we get Theorems \[thA\] and \[thB\]. We have focused on PEL Shimura varieties, in order to apply the known constructions. However, the deduction via Lemma \[lem-hecke-elts\] of the sign conjecture from Theorem \[thC\] is valid for more general varieties. First, for the trivial coefficient system, the motivic construction of coefficient systems is unnecessary, and we do not need to restrict to PEL types. Then for compact Shimura varieties (that is, in case the group is anisotropic over ${\mathbb{Q}}$ modulo center), we do not need Wildeshaus’ construction of intersection motives, and the Shimura data do not need to satisfy his condition (+) on its central torus. Over the complex numbers (or even over ${\overline{{\mathbb{Q}}}}$, see [@Fal]), the sign conjecture can be verified for the locally symmetric varieties considered in Theorem \[thC\]. Through the work of Shimura, Deligne, Milne, and others we have a complete theory of canonical models of Shimura varieties over reflex fields, and the sign conjecture holds for these models. Finally, if we know the sign conjecture for the varieties attached to a ${\mathbb{Q}}$-anisotropic semisimple group ${{\mathbf}G}$, we also know it for the varieties attached to any isogenous quotient group of ${{\mathbf}G}$. For any variety of the latter kind admits a finite étale covering from a variety of the former kind, and we can apply an argument similar to the one in \[ssec-level\]. Of course, Theorem \[thC\] is essential in all these generalizations. Arthur’s conjectures {#L2A} ==================== We follow the presentation of Kottwitz in section 8 of [@K-SVLR]. As before, ${{\mathbf}G}$ is a connected reductive group over ${\mathbb{Q}}$. Let $\xi:{\mathrm{A}}_{{\mathbf}G}({\mathbb{R}})^\circ{\longrightarrow}{\mathbb{C}}^\times$ be a character of ${\mathrm{A}}_{{\mathbf}G}({\mathbb{R}})^\circ$. Let $L^2_{{\mathbf}G}$ be the space of functions $f:{{\mathbf}G}({\mathbb{Q}}){\setminus}{{\mathbf}G}({\mathbb{A}}){\longrightarrow}{\mathbb{C}}$ such that : - $f(zg)=\xi(z)f(g)$ $\forall z\in{\mathrm{A}}_{{\mathbf}G}({\mathbb{R}})^\circ,g\in{{\mathbf}G}({\mathbb{A}})$; - $f$ is square-integrable modulo ${\mathrm{A}}_{{\mathbf}G}({\mathbb{R}})^\circ$. (Cf. the beginning of section 2 of [@A-L2].) Then ${{\mathbf}G}({\mathbb{A}})$ acts on $L^2_{{\mathbf}G}$ by right multiplication on the argument of the function. We say that an irreducible representation $\pi$ of ${{\mathbf}G}({\mathbb{A}})$ is *discrete automorphic* if it appears as a direct summand in the representation $L^2_{{\mathbf}G}$. In that case, we write $m(\pi)$ for the multiplicity of $\pi$ in $L^2_{{\mathbf}G}$; it is known to be finite. We denote by $\Pi_{disc}({{\mathbf}G})$ the set of equivalence classes of discrete automorphic representations of ${{\mathbf}G}({\mathbb{A}})$ and by $L^2_{{{\mathbf}G},disc}$ the discrete part of $L^2_{{\mathbf}G}$ (ie the completed direct sum of the isotypical components of the $\pi\in\Pi_{disc}({{\mathbf}G})$). Arthur conjectured that $$L^2_{{{\mathbf}G},disc}\simeq\bigoplus_\psi\bigoplus_{\Pi_\psi}m(\psi,\pi)\pi,$$ where the $\psi$ are equivalence classes of global Arthur parameters, the $\Pi_\psi$ are sets of (isomorphism classes of) smooth admissible representations of ${{\mathbf}G}({\mathbb{A}})$ called Arthur packets and $m(\psi,\pi)$ are nonnegative integers that we will define later. Note that we are not saying that the representations $\pi$ are irreducible. (They are not in general.) The traditional statement of Arthur’s conjectures involves the conjectural Langlands group ${{\mathscr}{L}}_{\mathbb{Q}}$ of ${\mathbb{Q}}$, and Arthur parameters are morphisms ${{\mathscr}{L}}_{\mathbb{Q}}\times{{\mathbf}{SL}}_2({\mathbb{C}}){\longrightarrow}{}^L{{\mathbf}G}$, where ${}^L{{\mathbf}G}=\widehat{{{\mathbf}G}}\rtimes W_{\mathbb{Q}}$ is the Langlands dual group of ${{\mathbf}G}$. In some cases, it is possible to use instead substitute parameters defined in terms of cuspidal automorphic representations of general linear groups. This is the point of view that is taken in the proofs of Arthur’s conjectures by Arthur for symplectic and orthogonal groups (cf [@A-livre]) and by Mok in the case of quasi-split unitary groups (cf [@Mok]). In any case, a global Arthur parameter $\psi$ gives rise to : - a character $\xi_\psi:{\mathrm{A}}_{{\mathbf}G}({\mathbb{R}})^\circ{\longrightarrow}{\mathbb{C}}^\times$; - a reductive subgroup $S_\psi$ of $\widehat{{{\mathbf}G}}$ such that $S_\psi^\circ\subset Z(\widehat{{{\mathbf}G}})^\Gamma\subset S_\psi$, where $\Gamma=\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$; - a character $\varepsilon_\psi$ of the finite group ${\mathfrak{S}}_\psi:=S_\psi/Z(\widehat{{{\mathbf}G}})^\Gamma S_\psi^\circ$ with values in $\{\pm 1\}$. In the sum above, we only take the parameters $\psi$ such that $\xi_\psi=\xi$. Part of Arthur’s conjectures is that there should be a map $\pi^0{\longmapsto}\langle ., \pi^0\rangle$ from the set of isomorphism classes of irreducible constituents of elements of $\Pi_\psi$ to $\widehat{{\mathfrak{S}}}_\psi$, such that $\langle .,\pi^0\rangle= \langle .,\pi^1\rangle$ if $\pi^0$ and $\pi^1$ are two irreducible constituents of the same $\pi\in\Pi_\psi$, and that the multiplicity $m(\psi,\pi)$ is given by the following formula : $$m(\psi,\pi)=m(\psi,\pi^0):= |{\mathfrak{S}}_\psi|^{-1}\sum_{x\in{\mathfrak{S}}_\psi}\varepsilon_\psi(x) \langle x,\pi^0\rangle,$$ if $\pi^0$ is an irreducible constituent of $\pi$. We can now state part (ii) of condition (C). It says that, for every irreducible admissible representation $\pi_f$ of ${{\mathbf}G}({\mathbb{A}_f})$, there is at most one Arthur parameter $\psi$ such that $\pi_f$ is the finite part of an irreducible constituent of an element of $\Pi_\psi$. There are also local versions of Arthur’s conjectures involving local Arthur parameters and local Arthur packets. We will not give details here (see for example chapter I of Arthur’s book [@A-livre]). Proof of Theorem \[thC\] {#proof} ======================== We use the notation of Theorem \[thC\] and of section \[L2A\], and we take for $\xi:{\mathrm{A}}_{{\mathbf}G}({\mathbb{R}})^\circ{\longrightarrow}{\mathbb{C}}^\times$ the inverse of the character by which ${\mathrm{A}}_{{\mathbf}G}({\mathbb{R}})^\circ$ acts on $W({\mathbb{R}})$. If $\pi$ is an irreducible representation of ${{\mathbf}G}({\mathbb{A}})$, we can write $\pi=\pi_f\otimes \pi_\infty$, where $\pi_f$ (resp. $\pi_\infty$) is an irreducible representation of ${{\mathbf}G}({\mathbb{A}}_f)$ (resp. ${{\mathbf}G}({\mathbb{R}})$). Let ${\mathfrak{g}}$ be the complexified Lie algebra of ${{\mathbf}G}({\mathbb{R}})$. If $\pi_\infty$ is an irreducible representation of ${{\mathbf}G}({\mathbb{R}})$, we write ${\mathrm{H}}^*({\mathfrak{g}},{\mathrm{K}}'_\infty; \pi_\infty\otimes W)$ for the $({\mathfrak{g}},{\mathrm{K}}'_\infty)$-cohomology of the space of ${\mathrm{K}}'_\infty$-finite vectors in $\pi_\infty\otimes W$ (cf chapter I of [@BW]). It follows from Zucker’s conjecture (a theorem of Looijenga ([@Lo]), Saper-Stern ([@SS]) and Looijenga-Rapoport([@LR])) and from Matsushima’s formula (proved by Matsushima for $S^{\mathrm{K}}$ compact and by Borel and Casselman in the general case, cf Theorem 4.5 of [@BC]) that there is a ${{\mathscr}{H}}_{\mathrm{K}}\otimes{\mathbb{C}}$-equivariant isomorphism, for every $k\in{\mathbb{Z}}$, $${\mathrm{IH}}^k(S^{\mathrm{K}},W)\simeq\bigoplus_{\pi\in\Pi_{disc}({{\mathbf}G})}\pi_f^{{\mathrm{K}}}\otimes {\mathrm{H}}^k({\mathfrak{g}},{\mathrm{K}}'_\infty;\pi_\infty\otimes W)^{m(\pi)}$$ (see also (2.2) of Arthur’s article [@A-L2]). If $\pi_f$ is an irreducible representation of ${{\mathbf}G}({\mathbb{A}}_f)$, let $\Pi_\infty (\pi_f)$ be the set of equivalence classes of irreducible representations $\pi_\infty$ of ${{\mathbf}G}({\mathbb{R}})$ such that $\pi:=\pi_f\otimes\pi_\infty\in\Pi_{disc} ({{\mathbf}G})$. Then, for every irreducible admissible representation $\pi_f$ of ${{\mathbf}G}({\mathbb{A}}_f)$ and every $k\in{\mathbb{Z}}$, $$\dim\sigma^k(\pi_f)=\sum_{\pi_\infty\in\Pi_\infty(\pi_f)}m(\pi_f\otimes \pi_\infty)\dim{\mathrm{H}}^k({\mathfrak{g}},{\mathrm{K}}'_\infty;\pi_\infty\otimes W).$$ Vogan and Zuckerman have classified all the admissible representations $\pi_\infty$ of ${{\mathbf}G}({\mathbb{R}})$ such that ${\mathrm{H}}^*({\mathfrak{g}},{\mathrm{K}}'_\infty;\pi_\infty\otimes W) \not=0$ in [@VZ], and Adams and Johnson have constructed local Arthur packets for these representations in [@AJ]. (It is part of our assumptions that their construction is compatible with the local and global Arthur conjectures of section \[L2A\].) We will follow Kottwitz’s exposition of their results in section 9 of [@K-SVLR]. Let $\theta$ be the Cartan involution of ${{\mathbf}G}({\mathbb{R}})$ that is the identity on ${\mathrm{K}}_\infty$. For every real reductive group $H$, let $q(H)=\frac{1}{2}\dim(H/ K_H)$, where $K_H$ is a maximal compact-modulo-center subgroup of $H$. Fix $\pi_f$ such that $\Pi_\infty(\pi_f)\not=\varnothing$. By part (ii) of condition (C), $\pi_f$ determines a global Arthur parameter $\psi$, and we write $\psi_\infty$ for the local Arthur parameter of ${{\mathbf}G}_{\mathbb{R}}$ defined by $\psi$. The set $\Pi_\infty(\pi_f)$ is a subset of the local Arthur packet associated to $\psi_\infty$. If $\pi_\infty\in\Pi_\infty(\pi_f)$ and $\pi=\pi_f\otimes\pi_\infty$, then the character $\langle .,\pi\rangle$ of ${\mathfrak{S}}_\psi$ factors as $\langle .,\pi_f\rangle\langle .,\pi_\infty\rangle$, where both factors are characters of ${\mathfrak{S}}_\psi$, and the first (resp. second) factor depends only on $\pi_f$ (resp. $\pi_\infty$). By the multiplicity formula in section \[L2A\], the fact that $m(\pi)=m(\psi,\pi)\not=0$ means that the character $\langle .,\pi_\infty\rangle$ of ${\mathfrak{S}}_\psi$ is uniquely determined by $\pi_f$. [^5] Let $\pi_\infty\in\Pi_\infty(\pi_f)$. Then there is a relevant pair $(L,Q)$ such that $\pi_\infty$ comes by cohomological induction from a 1-dimensional representation of $L$, cf pages 194-195 of [@K-SVLR]. Here $Q$ is a parabolic subgroup of ${{\mathbf}G}_{\mathbb{C}}$ and $L$ is a Levi component of $Q$ that is defined over ${\mathbb{R}}$. By proposition 6.19 of [@VZ], $\pi_\infty\otimes W$ can only have $({\mathfrak{g}},{\mathrm{K}}'_\infty)$-cohomology in degrees belonging to $R+2{\mathbb{N}}$, with $R=\dim_{\mathbb{C}}(\mathfrak{u}\cap\mathfrak{p})$, where $\mathfrak{u}$ is the Lie algebra of the unipotent radical of $Q$ and $\mathfrak{p}$ is the $-1$-eigenspace for $\theta$ acting on ${\mathfrak{g}}$. Let $\mathfrak{l}$ be the (complex) Lie algebra of $L$. As $\mathfrak{l}$ and $\mathfrak{u}$ are invariant under $\theta$ (by construction of $L$ and $Q$), we see easily that $$\dim_{\mathbb{C}}(\mathfrak{p})=2R+\dim_{\mathbb{C}}(\mathfrak{l}\cap\mathfrak{p}),$$ hence $R=q({{\mathbf}G}_{\mathbb{R}})-q(L)$. So the parity of $R$ is determined by the parity of $q(L)$. Now lemma 9.1 of [@K-SVLR] says that $$(-1)^{q(L)}=\langle \lambda_{\pi_\infty},s_\psi\rangle,$$ where $s_\psi\in{\mathfrak{S}}_\psi$ is determined by the global parameter $\psi$ (if we see global parameters as morphisms $\psi:{{\mathscr}{L}}_{\mathbb{Q}}\times{{\mathbf}{SL}}_2({\mathbb{C}}){\longrightarrow}{}^L{{\mathbf}G}$, then $s_\psi$ is the image by $\psi$ of the nontrivial central element of ${{\mathbf}{SL}}_2({\mathbb{C}})$) and $\lambda_{\pi_\infty}$ is the character of ${\mathfrak{S}}_{\psi_\infty}\supset{\mathfrak{S}}_\psi$ defined on page 195 of [@K-SVLR]. But lemma 9.2 of [@K-SVLR] implies that the product $\lambda_{\pi_\infty}\langle .,\pi_\infty\rangle$ is independent of $\pi_\infty$ in the Arthur packet of $\psi_\infty$, so the restriction of $\lambda_{\pi_\infty}$ to ${\mathfrak{S}}_\psi$ depends only on $\pi_f$. This implies that the parity of $R$ depends only on $\pi_f$, which gives Theorem \[thC\]. We have to be a bit careful if ${{\mathbf}G}$ is a quasi-split even special orthogonal group, because in that case Arthur proved his conjectures only up to conjugacy by the quasi-split even orthogonal group ${{\mathbf}G}'\supset{{\mathbf}G}$. But, if $\pi_\infty$ is a representation of ${{\mathbf}G}({\mathbb{R}})$ with nonzero $({\mathfrak{g}},{\mathrm{K}}'_\infty)$-cohomology, then the integer $R$ associated to $\pi_\infty$ as above does not change if we replace $\pi_\infty$ by a ${{\mathbf}G}'({\mathbb{R}})$-conjugate (because the relevant pair $(L,Q)$ is just replaced by a ${{\mathbf}G}'({\mathbb{R}})$-conjugate). So the proof above still applies. Künneth conjecture and finite correspondences ============================================= From the proofs of the theorem of Katz and Messing and that of ours, one may wonder if the Künneth conjecture or the sign conjecture can be proved for more general projective smooth varieties, only using finite correspondences. More precisely, consider the ${\mathbb{Q}}$-subspace $$Z_{\mathrm{fin}, {\mathrm{H}}^{\ast}}^d \subseteq {\mathrm{H}}^{2d}(X\times_k X)(d)$$ spanned by the cohomology classes of all the cycles of codimension $d$ on $X\times_k X$, that are *finite* in both projections to $X$ (where $d=\dim X$). \[conj-fin-corr\] For every projective smooth variety $X/k$ and every $i\in {\mathbb{Z}}$, the Künneth projector $\pi^i_X$ (resp. the projector $\pi^{+}_X$) belongs to $Z^d_{\mathrm{fin}, {\mathrm{H}}^{\ast}}$. This is a priori stronger than the Künneth (resp. the sign) conjecture. It turns out that the apparent strength is only illusory, if either (a) $k$ is algebraically closed or (b) $k$ is perfect and $H^{\ast}$ is a classical Weil cohomology theory. The case (a) is a consequence of the following proposition. \[moving-lemma\] Suppose that $k$ is an algebraically closed field. Then the abelian group $$Z_{\mathrm{fin}, \sim_{\mathrm{rat}}}^d \subseteq Z_{\sim_{\mathrm{rat}}}^d$$ generated by the cycles mapping finitely to $X$ in both projections, in the group of codimension $d$ cycles on $X$ modulo rational equivalence, is in fact equal to the whole $Z^d_{\sim_{\mathrm{rat}}}$. It is enough to prove that any irreducible closed subscheme of codimension $d$ on $X\times_k X$ is rationally equivalent to a cycle that is finite in both projections. Because a proper quasi-finite map is finite, it is the matter of finding a cycle in the rational equivalence class, that meets all the closed fibres over $k$-rational points in both projections properly, that is, in dimension at most zero. This follows from the generalized moving lemma [@FL] of Friedlander and Lawson: In any fixed projective embedding, all the fibres of the first (resp. second) projection have the same degree, as they are all algebraically equivalent. Now, in the case (b), let $k$ be a perfect field, and suppose that $Z$ is an algebraic cycle of codimension $d$ on $X\times_k X$. If ${\mathrm{H}}^{\ast}$ is a classical Weil cohomology theory, then we have a corresponding cohomology theory ${\mathrm{H}}^{\ast}_{/k'}$ for every algebraic extension $k'$ of $k$, compatible with the cycle class maps in an obvious sense. Let $\overline{k}$ be an algebraic closure of $k$. By Proposition \[moving-lemma\], $Z\otimes_k \overline{k}$ is rationally — hence homologically — equivalent to a cycle $Z'$ which is finite over $X \otimes_k \overline{k}$ in both projections. Let $k'$ be a finite Galois extension of $k$ over which $Z'$ is defined. Taking the “average” of the $\operatorname{Gal}(k'/k)$-translates of $Z'$ (which requires ${\mathbb{Q}}$-coefficients), one gets a cycle $Z'_0$ on $X$, defined over $k$, that is finite in both projections and has the same cohomology class as $Z$. This means that, in the two cases, if the Künneth conjecture is true for $X/k$, then each $\pi^i_X$ is in fact a linear combination of the cohomology classes of finite correspondences over $X$. Finding enough such finite correspondences for general $X/k$ (which can be turned into the problem of finding certain finite extensions of the function field $k(X)$) seems to be an interesting open problem. [^1]: Note that we only need condition (C’) for theorem \[thB\], so Arthur’s results already allow us to get theorem \[thB\] for the Shimura varieties of split general symplectic groups. [^2]: Wildeshaus’ construction requires a condition, which he names (+) and is the same as (3.1.5) in [@P2], on the central torus in the Shimura data. It is satisfied by any PEL Shimura data: See for instance the analysis of the maximal torus quotient of ${{\mathbf}G}$ in §7 of [@K-PSSV]. [^3]: Strictly speaking, Wildeshaus’ construction works only for direct factors $N$ of $\pi^{(r)}_{\ast} \mathbf{1}_{A^r}$. Given a Tate twist $N(m)$, one takes the $m$th Tate twist of $IM(S^{{\mathrm{K}}}, N)$, which has Betti realization ${\mathrm{IH}}^{\ast}(S^K({\mathbb{C}}), N(m))$. [^4]: Wildeshaus does not construct an action of the Hecke algebra on $IM(S^{{\mathrm{K}}}, W)$. One expects, but does not know at the moment, that there is a canonical choice of $\widetilde{KgK}$. [^5]: $\langle ,.\pi_\infty\rangle$ is actually a character of the bigger group ${\mathfrak{S}}_{\psi_\infty}$, but its values on ${\mathfrak{S}}_{\psi_\infty}$ are not determined by $\pi_f$, otherwise $\Pi_{\infty}(\pi_f)$ would be a singleton, and this is not the case in general (cf case 3 on page 90 of Rogawski’s paper [@Ro] for a counterexample if ${{\mathbf}G}={{\mathbf}{GU}}(2,1)$).

--- abstract: 'About 2 million seconds of  observing time have been devoted to the Galactic center (GC), including large-scale surveys and deep pointings. These observations have led to the detection of about 4000 discrete X-ray sources and the mapping of diffuse X-ray emission in various energy bands. In this review, I first summarize general results from recent studies and then present close-up views of the three massive star clusters (Arches, Quintuplet, and GC) and their interplay with the Galactic nuclear environment.' address: '$^1$ Department of Astronomy, University of Massachusetts, Amherst, MA 01003' author: - 'Q. Daniel Wang$^1$' title: 'Chandra Observations of Galactic Center: High Energy Processes at Arcsecond Resolution' --- Introduction ============ While Sgr A\* itself is only weakly active at present, much of the high-energy activity in the GC is initiated apparently by the three young massive stellar clusters, located within 50 pc radius of the super-massive black hole Sgr A\* (Fig. \[f:global\]): Arches \[with an age of $(2-3) \times 10^6$ yrs\], Quintuplet \[$(3-6) \times 10^6$ yrs\], and GC \[$(3-7) \times 10^6$ yrs\] (e.g., Figer et al. 2004; Stolte et al. 2002, 2005; Genzel et al. 2003). Massive stars themselves can be moderately bright X-ray sources (e.g., colliding stellar wind binaries). Such stars also release large amounts of mechanical energy in form of fast stellar winds and supernovae, heating and shaping the surrounding interstellar medium (ISM) and affecting the accretion of the black hole. Furthermore, stellar end-products of massive stars (neutron stars and black holes) can also be strong X-ray sources. X-ray observations are thus a powerful tool for probing such high-energy phenomena and processes. The GC is a region where the spatial resolution of X-ray observations matters. This is why  has invested heavily on the GC, conducting repeated large-scale raster surveys \[existing 360 ks (Wang et al. 2002a) and upcoming 600 ks (PI: Muno)\] and numerous deep pointed observations: about 1 Ms on Sgr A (Baganoff et al. 2003), 100 ks each on Sgr B (Takagi et al. 2002), Sgr C (PI: Murakami), and Arches (Wang et al. 2006a), as well as 50 ks on Radio Arc (Yusef-Zadeh et al. 2002a). The main instrument used in these  observations is the ACIS-I, which covers a field of $17^\prime \times 17^\prime$ and an energy range of 0.5-10 keV. But below $\sim 2$ keV, X-rays from the GC are heavily absorbed by the ISM. The spatial resolution ranges from $\sim 1^{\prime\prime}$ on-axis to $\sim 10^{\prime\prime}$ at the outer boundaries of the field. While much of the data analysis is still ongoing, here I will first summarize some of the general results on detected discrete sources and diffuse X-ray emission and will then focus on the three massive star clusters, highlighting various high-energy phenomena and processes involved. General results on point-like X-ray sources =========================================== The  observations have led to the detection of about 4000 discrete sources; about 2400 of them are from the deep Sgr A observations (Wang et al. 2002a; Muno et al. 2003, 2006; Wang et al. 2006a). Most of these sources are located physically in the vicinity of the GC, judged from their X-ray spectral characteristics and number statistics. The number-flux relation (or the so-called log$N$-log$S$ relation) of the GC sources can be approximated as a power law with an index of $1.5\pm0.1$. But the relation is flatter (over the luminosity range of $5 \times 10^{31} {\rm~ergs~s^{-1}}\lesssim L_x \lesssim 10^{34} {\rm~ergs~s^{-1}}$) in the vicinity of the  and  clusters (Muno et al. 2006; Wang et al. 2006a), apparently due to the concentrations of massive stars and possibly their end-products. The GC X-ray sources represent a heterogeneous population of high-energy objects. Bright sources (1E 1740.7-2942 and 1E 1743.1-2843) with 2-10 keV luminosities $L_x \gtrsim 10^{36} {\rm~ergs~s^{-1}}$ are low-mass X-ray binaries (LMXBs). There is a clear dearth of X-ray sources with $10^{34} {\rm~ergs~s^{-1}}\lesssim L_x \lesssim 10^{36} {\rm~ergs~s^{-1}}$, although a few transients are detected in this intermediate luminosity range (e.g., Muno et al. 2005; Sakano et al. 2005; Wijnands et al. 2006; Wang et al. 2006a). Sources within the range of $10^{33} {\rm~ergs~s^{-1}}\lesssim L_x \lesssim 10^{34} {\rm~ergs~s^{-1}}$ tend to be colliding wind massive star binaries (Wang et al. 2006a) and possibly young pulsars (Wang et al. 2002b; Wang et al. 2006b). The nature of fainter sources ($10^{31} {\rm~ergs~s^{-1}}\lesssim L_x \lesssim 10^{33} {\rm~ergs~s^{-1}}$) are less certain; many of them are likely to be CVs, consistent with the lack of bright stellar counterparts in near-IR and radio surveys (e.g., Laycock et al. 2005; Bandyopadhyay 2005). A fraction of these sources have very hard X-ray spectra (e.g., with power law photon indices less than 0; Muno et al. 2004a). Such sources are most likely intermediate polars, in which X-ray-emitting regions are at least partially obscured and the observed X-rays represent the reflected emission (Ruiter, Belczynski, & Harrison 2006). General results on diffuse X-ray emission ========================================= Despite of their large number, the detected discrete X-ray sources typically account for only about 10% of the total observed X-ray emission from the GC. The spectra of the remaining “diffuse” emission and the accumulated spectrum of the sources are similar, in terms of both the continuum shape and the presence of prominent emission lines such as He-like S and Fe K$\alpha$ transitions, which indicate a broad thermal plasma temperature range of $\sim 1 - 10$ keV (Wang et al. 2002a; Muno et al. 2004b). The spatial distribution of the diffuse emission appears to follow closely the K-band stellar light (Muno et al. 2006). Therefore, a bulk of the emission likely originates in the old stellar population (coronally active binaries and CVs), as has been proposed for the Galactic ridge hard X-ray emission (Revnivtsev et al. 2005). There is no clear evidence for a significant presence of truly diffuse gas with $T \gtrsim 8$ keV, as was believed for many years. But a non-thermal diffuse hard X-ray component does seem to be important (Wang et al. 2002a; see also the presentation by B. Warwick). The strongest evidence for this component is the ubiquitous presence of the 6.4-keV line emission, globally correlated with the molecular gas in the GC region. The line emission arises from the filling of inner shell vacancies of neutral or weakly ionized irons, which can be effectively produced by ionizing radiation with energies $\gtrsim 7.1$ keV. Because of the high equivalent width ($\gtrsim 1$ keV) of the observed line, the source of the ionizing radiation must be currently obscured or absent. This reflection interpretation gives the most convincing interpretation for the 6.4-keV line emission associated with the giant molecular clouds, Sgr B2 and Sgr C. The required ionizing source is believed to be Sgr A\*, which might be substantially brighter ($\gtrsim 10^{39} {\rm~ergs~s^{-1}}$) several hundred years ago. Indeed, the continuum emission from Sgr B, presumably due to the reflection through electron Thompson scattering, is shown to have a hard X-ray spectrum (power law photon index $\Gamma \approx 1.8$) in the 2-200 keV range, consistent with what is expected from a typical AGN (Revnivtsev et al. 2004). However, the reflection interpretation is less successful in explaining the 6.4-keV line emission observed in fields closer to Sgr A\* (e.g., Wang et al. 2002a). While the line intensity is globally correlated with the emission from trace molecules, a peak-to-peak correlation is often absent. The lack of such a correlation is not expected, because molecular clouds should be optically thin to the X-ray radiation. But it is possible that their density peaks may be optically thick to the emission such as CS (J=2-1). An alternative mechanism to produce the K-shell vacancies is the collision of irons with low energy cosmic ray electrons (LECRe; Valinia et al. 2000). In this case, one also expect a hard X-ray emission from the bremsstrahlung process. The problem with this mechanism is the low efficiency of the emission (typically $10^{-4} - 10^{-5}$); most of the energy is consumed in the ionization loss. It is not clear how the required energy density of the LECRe could be maintained [*globally*]{} even in the GC region.  view of the massive star clusters in the GC region {#ss:aqc} =================================================== Fig. \[f:global\] presents the global perspective of the  and  clusters in the GC environment. This most active region also includes various prominent thermal and nonthermal radio filaments, dense molecular clouds, and strong diffuse X-ray emission, all of which are concentrated on the positive Galactic longitude side of Sgr A$^*$ (Wang et al. 2006a and references therein). This lopsided distribution of these features may partly be a chance coincidence. But some of the features are likely to be related, although they seem to have very different line-of-sight velocities. The thermal mid-IR-emitting filaments are clearly due to the radiative heating of the  and  clusters, while the strong 6.4-keV line emission is associated with the dense molecular gas (Wang et al. 2002a; Yusef-Zadeh et al. 2002b). Fig. \[f:ill\_cloud\] illustrates a scenario that provides a unified interpretation of various distinct interstellar features observed in the region. The densest cloud G0.13-0.13 in the region (Handa et al. 2006) is projected inside the west part of the distinct mid-IR cavity (Fig. \[f:global\]; Price et al. 2001) and is probably located at the far side of a tunnel dug out by the cloud (Fig. \[f:ill\_cloud\]). This tunnel in projection is seen as the cavity and is apparently filled with hot plasma, responsible for the enhanced diffuse X-ray emission (Fig. \[f:global\]; Wang et al. 2006a). This scenario is consistent with the lack of an enhanced near-IR extinction toward G0.13-0.13 and with the high excitation of molecular gas associated with the cloud, apparently due to shock-heating (Handa et al. 2006). The compression of the inter-cloud medium (the cavity wall in Fig. \[f:ill\_cloud\]), hence the attached magnetic field, may even explain the nonthermal radio filaments. The diffuse X-ray emission, both thermal and nonthermal, are likely products of the mechanical energy input from the massive star clusters. Much work is still required to clarify what is actually going on in this unusually dynamic region of the GC. ![Panoramic views of the GC environment of the three massive star clusters: MSX 24 $\mu$m intensity distributions (gray-scale; Price et al. 2001) and  ACIS-I 1-9 keV intensity contours. The region with strong nonthermal radio filaments are outlined by the dashed lines, while the cloud G0.13-0.13 is circled.[]{data-label="f:global"}](prefig/fig1.eps){width="36pc"} ![A plausible configuration of several major ISM components in the vicinity of the   clusters, as viewed from the Galactic north pole. The observed line-of-sight velocities of the components are marked.[]{data-label="f:ill_cloud"}](prefig/ill_cloud.eps){width="22pc"} The Arches and  Clusters ------------------------ Both the  and  clusters are associated with local enhancements of X-ray emission (e.g., Fig. \[f:im\_multi\_bw\]), which was discovered serendipitously at large off-axis angles in early [*Chandra*]{} observations (Yusef-Zadeh et al. 2002a; Wang et al. 2002a; Law & Yusef-Zadeh 2004). A recent 100 ks observation allows for an in-depth study of both discrete and diffuse components of the X-ray emission from these two clusters (Wang et al. 2006a). ![Adaptively smoothed ACIS-I 1-9 keV band images of the  (a) and  (b) clusters. The intensity contour levels are at 20, 23, 27, 33, 57, 57, 80, 114, 180, 314, 482, 682, 1351, and 3358 (above a local background of 13.4) for (a), and at 17, 29, 33, 42, 54, and 72 (above 17) for (b); all in units of $10^{-3} {\rm~counts~s^{-1}~arcmin^{-2}}$. The two large squares in (a) and (b) outline the fields covered by the [*HST*]{} NICMOS near-IR images of the  (c) and  (d), respectively (Figer et al. 2004). The contours are the same as in (a) and (b), except for excluding the first four levels in (c) for clarity. The detected sources are marked with [*crosses*]{} in (a) and (b). Several bright X-ray sources named previously are labeled (Yusef-Zadeh et al. 2002a; Law & Yusef-Zadeh 2004). []{data-label="f:im_multi_bw"}](prefig/im_multi_bw.eps){width="38pc"} The three bright X-ray sources in the  core region all have near-IR counterparts, classified as WN stars (Fig. \[f:im\_multi\_bw\]b). These sources have remarkably similar spectra, in terms of both the continuum shape and the strong presence of the 6.7-keV line (Fig. \[f:spec\_a\]). The spectrum can be characterized by an optically-thin thermal plasma with a temperature of $\sim 2$ keV and a metal (chiefly iron) abundance of 1.8$\times$ solar. The 0.3-8 keV luminosity of each source is $\sim 1 \times 10^{34} {\rm~ergs~s^{-1}}$. With these properties, the sources are most likely colliding wind massive star binaries, although their luminosities are somewhat higher than all known such objects. The X-ray metal abundance measurement, based primarily on the He-like Fe K$\alpha$ line, is also interesting, which is insensitive to the exact temperature of the plasma. The optically-thin thermal emission process is also quite simple, astrophysically. Furthermore, the iron abundance in the stellar winds should not be contaminated by the nuclear synthesis of these stars and thus reflect the value in the ISM of the GC. ![ ACIS-I spectrum of the diffuse X-ray emission southeast of the Arches cluster (Fig. \[f:cs\_dx\]) and the best-fit power law plus 6.4-keV Gaussian line model.[]{data-label="f:spec_aline"}](prefig/spec_a_total.ps){width="12pc"} ![ ACIS-I spectrum of the diffuse X-ray emission southeast of the Arches cluster (Fig. \[f:cs\_dx\]) and the best-fit power law plus 6.4-keV Gaussian line model.[]{data-label="f:spec_aline"}](prefig/line.ps){width="12pc"} The nature of the source-removed “diffuse” emission is more complicated. The overall spectrum of the emission shows both 6.4-keV and 6.7-keV emission lines. The 6.7-keV line arises predominately in the region close to the core of the  cluster. The region has an extent of $\sim 30^{\prime\prime}$ and appears to be elongated towards the east, morphologically matching an extinction deficit around the cluster (Stolte et al. 2002). The 6.4-keV line emission is more widely distributed, but is particularly enhanced in the southeast extension of the diffuse X-ray emission (Fig. \[f:im\_multi\_bw\]a and Fig. \[f:spec\_aline\]). All these can be interpreted as the collision between the cluster wind of the  cluster and a dense gas cloud (Fig. \[f:ill\]). The steep decline of the observed surface intensity of the diffuse X-ray emission with the off-cluster radius within $\sim 10^{\prime\prime}$ is consistent with the prediction of the emission from the expanding cluster wind (Fig. \[f:rbp\]a), whereas the flattening of the intensity distribution at larger radii (but $\lesssim 15^{\prime\prime}$) likely reflects the reverse shock heating and confinement of the wind. ![\[f:ill\] An illustration of the proposed cluster-cloud collision scenario. The shocked cloud gas is partly traced by the CS and 6.4-keV lines (Fig. \[f:cs\_dx\]), whereas the shocked cluster wind plasma near the cluster is by the 6.7-keV line. ](prefig/cs_dx.eps){width="18pc"} ![\[f:ill\] An illustration of the proposed cluster-cloud collision scenario. The shocked cloud gas is partly traced by the CS and 6.4-keV lines (Fig. \[f:cs\_dx\]), whereas the shocked cluster wind plasma near the cluster is by the 6.7-keV line. ](prefig/ill.eps){width="11pc"} ![\[f:rbp\] Radial ACIS-I 1-9 keV intensity profiles ([*crosses*]{} with $1\sigma$ error bars) around the  (a) and  (b) clusters, compared with the respective NICMOS F205W stellar light distributions (connected [*triangles*]{}). The cluster wind predictions are shown approximately as the solid line from 3-D simulations for the “standard” stellar wind mass-loss rates of the two clusters (Rockefeller et al. 2005)](prefig/rbp.eps){width="38pc"} . The presence of the cloud in the vicinity of the  cluster is known. In the CS map made by Serabyn & Gusten (1987) with the 30 [*IRAM*]{} telescope, the cloud is labeled as “Peak 2”. Fig. \[f:cs\_dx\] shows a recent interferometry map made with [*OVRO*]{} (Wang et al. 2006a). The cloud has a line-of-sight velocity of $-25 {\rm~km~s^{-1}}$ (i.e., moving toward us), which is similar to many other clouds or filaments in the region. In comparison, the cluster is moving at $\sim 95 {\rm~km~s^{-1}}$ away from us. Therefore, the relative velocity between the two is at least about $120 {\rm~km~s^{-1}}$. Because the filling factor of molecular clouds in the region is estimated to be about 0.3 (Serabyn & Gusten 1987), a chance collision between the cluster and such a cloud is not rare. The far-IR spectroscopy further shows the presence of a component of dusty gas at a velocity of $-70 {\rm~km~s^{-1}}$, unique at the location of the  cluster (Cotera et al. 2005). This component may represent shocked cloud gas, deflected toward us. The collision between the cluster and the cloud (Fig. \[f:ill\]) also provides a plausible explanation of the large-scale diffuse X-ray emission (Fig. \[f:cs\_dx\]). Indeed, the overall morphology of the diffuse X-ray emission resembles a bow shock. Presumably, the motion of the cloud relative to the cluster is roughly toward the east in the sky. The diffuse X-ray emission, except in the core region around the  cluster, is reasonably well correlated with the 6.4-keV line intensity, but shows little peak-to-peak correlation with the CS line intensity. The diffuse X-ray emission is enhanced particularly in the region southeast of the cluster, probably tracing the eastern edge of the cloud. But the northern part of the cloud, where the CS line emission is the strongest, shows no enhancement in the 6.4-keV line emission. The cloud does not seem to be dense enough to be optically thick to the CS line emission. Therefore, the radiation reflection interpretation does not work in this case. Certainly, the source of the radiation cannot be the  cluster itself. The luminosity of the cluster falls several orders of magnitude short of the required. The most probable origin of the diffuse X-ray X-ray emission is then LECRe that may be produced locally in and around the clusters (Wang et al. 2006a). The older and looser  cluster is weak in X-ray emission, both point-like and diffuse (Fig. \[f:im\_multi\_bw\]b and d, Fig. \[f:rbp\]b). The X-ray sources in the core of the cluster also show more diverse spectral characteristics, with substantially different intrinsic spectral shapes and absorptions. Some of these sources are probably colliding dusty winds in massive star binaries. The Pistol star, despite of its enormous bolometric luminosity, is not detected. But the 3$\sigma$ upper limit to the 0.3-8 keV luminosity, 3$\times10^{33}$ ${\rm~ergs~s^{-1}}$, is consistent with the nominal relation $L_{x}/L_{bol} \sim 10^{-7}$ for massive stars. The diffuse X-ray emission from the  cluster ($L_X \sim 2 \times 10^{33} {\rm~ergs~s^{-1}}$) is about a factor of 10 lower than that from the  cluster and can be naturally explained by the cluster wind and a limited number of low-mass pre-main sequence young stellar objects (YSOs). There appears to be a general deficiency of YSOs in the two clusters, relative to the prediction from the standard Miller & Scalo initial mass function (IMF). Compared with the X-ray emission from YSOs in the Orion nebula (Feigelson et al. 2005), the observed total diffuse X-ray luminosities from the Arches and Quintuplet clusters suggest that they contain no more than $2 \times 10^4$ and $3 \times 10^3$ YSOs. These numbers are a factor of 10 and 5 smaller than what would be expected from the IMF and the massive star populations observed in the cores of the two clusters. One possibility is that the IMF flattens at intermediate masses, as indicated in a near-IR study of the inner regions of the   cluster (Stolte et al. 2005). The top heavy IMF may be a result of the star formation in the extreme environment of the GC. In particular, collisions between dense molecular clouds such as G0.13-0.13 may be responsible for the formation of massive clusters. The GC cluster -------------- The deficiency of YSOs in the GC cluster has been proposed by Nayakshin et al. (2005), based on the observed diffuse X-ray intensity of the region (Fig. \[f:gc\_x\]; Baganoff et al. 2003). From the X-ray constraint on the total mass of the cluster, they further conclude that it cannot be a remnant of a massive star cluster that originated at several tens of parsecs away from Sgr A\* and then dynamically spiralled in. The GC cluster was thus most likely formed [*in situ*]{} in a self-gravitating circum-nuclear disk. Therefore, the top heavy IMF appears to be a general characteristic of star formation in the GC region. But the exact form of the IMF is yet to be determined for the clusters. ![ACIS-I spectra of  (a), Diffuse (b), IRA 13 (c), and Sgr A$^*$ (d). The relative deviations from the best-fit models (Wang et al. 2006b) are shown in the respective bottom panels. \[f:spec\]](prefig/ir_x_c.eps){width="14pc"} ![ACIS-I spectra of  (a), Diffuse (b), IRA 13 (c), and Sgr A$^*$ (d). The relative deviations from the best-fit models (Wang et al. 2006b) are shown in the respective bottom panels. \[f:spec\]](prefig/spec.eps){width="22pc"} The diffuse X-ray emission from the GC cluster region appears significantly different from those from the  and  clusters. The spectrum of the emission shows a Fe K$\alpha$ emission line at an intermediate energy $\sim 6.55$ keV (e.g., Fig. \[f:spec\]b; Baganoff et al. 2003), which is a clear signature for the non-equilibrium ionization (NEI) state of the X-ray-emitting plasma with an ionization timescale $n_e t \sim 3 \times 10^{10} {\rm~cm^{-3}~s}$ (Wang et al. 2006b). The source of the plasma is likely to be the stellar winds from massive stars in the GC cluster. We may assume that the pre-shock stellar winds are cold ($T \lesssim 10^4$ K) and generally have velocities of $\sim 1 \times 10^3 {\rm~km~s^{-1}}$). Fig. \[f:gc\_x\] shows that the distribution of the stars is not very centrally concentrated, except for the compact sub-cluster complex IRS 13. The density of the shocked wind plasma is typically not very high (order $\sim 10 {\rm~cm^{-3}}$). Thus, much of the shock-heated plasma may be out of the collisional ionization equilibrium, at least in individual wind-wind shock regions. Further constraints on the properties of the plasma may be obtained from studying its interaction with individual stars or sub-clusters. The X-ray study of IRS 13 provides such a possibility. The X-ray emission from this sub-cluster is clearly resolved and shows a non-NEI plasma spectrum (Fig. \[f:spec\]c). Therefore, the source most likely represents stellar wind-wind collisions. The X-ray emission may be enhanced due to a strong ram-pressure confinement by the overall GC cluster wind, plus a potential outflow from Sgr A\*, as predicted in the radiatively inefficient accretion flow models (Yuan, Quataert, & Narayan 2003). The confinement is indicated by the X-ray morphology that is slightly offset from the centroid position of IRS 13 and elongated toward to the west (Wang et al. 2006b). The properties of the plasma around Sgr A\* should also affect its accretion. The quiescent X-ray emission from Sgr A\* is resolved to have a size of about a couple of arc-seconds. This size corresponds to the Bondi accretion radius of the super-massive black hole. The accretion flow is believed to be responsible for the X-ray emission. Interestingly, the observed spectrum of the emission resembles that of the surrounding diffuse plasma and shows the Fe K$\alpha$ line at $\sim 6.6$ keV (Fig. \[f:spec\]d; Wang et al. 2006b; Xu et al. 2006). Therefore, the flow is also in an NEI state. A detailed modeling of the flow, confronted with the observed spatial and spectral distributions of the X-ray emission, will help to constrain the accretion dynamics. The X-ray emission from the GC region is complicated by the presence of a comet-like pulsar wind nebula (PWN), a few arc-seconds northwest of Sgr A\* (Fig. \[f:gc\_x\]; Wang et al. 2006b). The suspected pulsar, presumably moving at a high speed relative to the ambient medium, is located at the northern head of the nebula. This PWN interpretation naturally explains not only the morphology, but also the nonthermal spectrum (Fig. \[f:spec\]a) and the spectral steepening with the off-source distance due to the synchrotron cooling of ultra-relativistic electrons (and positrons). Furthermore, the inverse-Compton scattering of the intense ambient infrared photon field by the electrons provides a ready explanation for the TeV emission from the GC (Aharonian et al. 2004). The lack of a synchrotron-emitting radio or infrared counterpart of the PWN is at least partly due to the fast Compton-cooling (hence little accumulation) of relatively low-energy electrons, for which the Klein-Nishina suppression of the inverse-Compton scattering efficiency is not important. The presence of this young pulsar raises a number of questions: where does it originate? and where is the supernova remnant? The most logic origin of the pulsar is the GC cluster itself. But the remnant is more difficult to isolate, the properties of which depend sensitively on the density and temperature structure of the medium (e.g., Tang & Wang 2005). Considering the orbital motion around Sgr A\*, it is even possible that the pulsar may be produced together with the well-known SNR Sgr A West, although an alternative pulsar candidate has already been proposed for the remnant (see the contribution by S. Park). Clearly, the above is a very incomplete and subjective review of recent [*Chandra*]{} results on the GC. We will certainly learn a great deal more from the existing and upcoming [*Chandra*]{} observations before GC-2009 Workshop in China. Some of the ideas presented above are developed during or after many stimulating discussions with workshop participants, to whom I am grateful. I also thank my collaborators for their contributions to the various research projects mentioned above and the organizers of the workshop for the invitation and the hospitality. 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--- abstract: | Let $(U_{t},V_{t})$ be a bivariate Lévy process, where $V_{t}$ is a subordinator and $U_{t}$ is a Lévy process formed by randomly weighting each jump of $V_{t}$ by an independent random variable $X_{t}$ having cdf $F$. We investigate the asymptotic distribution of the self-normalized Lévy process $U_{t}/V_{t}$ at 0 and at $\infty$. We show that all subsequential limits of this ratio at 0 ($\infty$) are continuous for any nondegenerate $F$ with finite expectation if and only if $V_{t}$ belongs to the centered Feller class at 0 ($\infty$). We also characterize when $U_{t}/V_{t}$ has a non-degenerate limit distribution at 0 and $\infty$. *AMS Subject Classification:* MSC 60G51; MSC 60F05. *Keywords:* Lévy process; Feller class; self-normalization; stable distributions. author: - | Péter Kevei[^1]\ Analysis and Stochastics Research Group of the Hungarian Academy of Sciences\ Bolyai Institute, Aradi vértanúk tere 1, 6720 Szeged, Hungary and\ CIMAT, Callejón Jalisco S/N, Mineral de Valenciana, Guanajuato 36240, Mexico\ e-mail: `kevei@math.u-szeged.hu`\ David M. Mason[^2]\ Statistics Program, University of Delaware\ 213 Townsend Hall, Newark, DE 19716, USA\ e-mail: `davidm@udel.edu` title: 'Randomly Weighted Self-normalized Lévy Processes' --- Introduction and statements of two main results =============================================== We begin by defining the bivariate Lévy process $\left( U_{t},V_{t}\right) ,$ $t\geq0,$ that will be the object of our study. Let $F$ be a cumulative distribution function \[cdf\] satisfying $$\int_{-\infty}^{\infty}\left\vert x\right\vert F\left( \mathrm{d}x\right) <\infty\label{E}$$ and $\Lambda$ be a Lévy measure on $\mathbb{R}$ with support in $\left( 0,\infty\right) $ such that $$\int_{0}^{1}y\Lambda(\mathrm{d}y)<\infty.\label{VV}$$ We define the* Lévy function* $\overline{\Lambda}\left( x\right) =\Lambda\left( x,\infty\right) $ for $x\geq0$. Using Corollary 15.8 on page 291 of Kallenberg [@kallenberg] and assumptions (\[E\]) and (\[VV\]), we can define via $F$ and $\Lambda$ the bivariate Lévy process $\left( U_{t},V_{t}\right) ,$ $t\geq0$, having the joint characteristic function $$E\exp\left( \mathrm{i}\theta_{1}U_{t}+\mathrm{i}\theta_{2}V_{t}\right) =:\phi\left( t,\theta_{1},\theta_{2}\right) =\exp\left( t\int_{(0,\infty )}\int_{-\infty}^{\infty}\left( e^{\mathrm{i}(\theta_{1}u+\theta_{2}v)}-1\right) \Pi(\mathrm{d}u,\mathrm{d}v)\right) ,\label{holds}$$ with$$\Pi\left( \mathrm{d}u,\mathrm{d}v\right) =F\left( \mathrm{d}u/v\right) \Lambda\left( \mathrm{d}v\right) .\label{pi}$$ From the form of $\phi\left( t,\theta_{1},\theta_{2}\right) $ it is clear that $V_{t}$ is a driftless subordinator.  Throughout this paper $\left( U_{t},V_{t}\right) $, $t\geq0,$ denotes a Lévy process satisfying (\[E\]) and (\[VV\]) and having joint characteristic function (\[holds\]). Now let $\left\{ X_{s}\right\} _{s\geq0}$ be a class of i.i.d. $F$ random variables independent of the $V_{t}$ process. We shall soon see that for each $t\geq0$ the bivariate process$$\left( U_{t},V_{t}\right) \overset{\mathrm{D}}{=}\left( \sum_{0\leq s\leq t}X_{s}\Delta V_{s},\sum_{0\leq s\leq t}\Delta V_{s}\right) ,\label{WL}$$ where $\Delta V_{s}=V_{s}-V_{s-}$. Notice that in the representation (\[WL\]) each jump of $V_{t}$  is weighted by an independent $X_{t}$ so that $U_{t}$ can be viewed as a randomly weighted Lévy process. Here is a graphic way to picture this bivariate process. Consider $\Delta V_{s}$ as the intensity of a random shock to a system at time $s>0$ and $X_{s}\Delta V_{s}$ as the cost of repairing the damage that it causes. Then $V_{t}$, $U_{t}$ and $U_{t}/V_{t}$ represent, respectively, up to time $t$, the total intensity of the shocks, the total cost of repair and the average cost of repair with respect to shock intensity. For instance, $\Delta V_{s}$ can represent a measure of the intensity of a tornado that comes down in a Midwestern American state at time $s$ during tornado season and $X_{s}$ the cost of the repair of the damage per intensity that it causes. Note that $X_{s}$ is a random variable that depends on where the tornado hits the ground, say a large city, a medium size town, a village, an open field, etc. It is assumed that a tornado is equally likely to strike anywhere in the state. We shall be studying the asymptotic distributional behavior of the randomly weighted self-normalized Lévy process $U_{t}/V_{t}$ near 0 and infinity. Note that $\overline{\Lambda}(0+)=\infty$ implies that $V_{t}>0$ a.s. for any $t>0$. Whereas if $\overline{\Lambda}(0+)<\infty$, then, with probability 1, $V_{t}=0$ for all $t$ close enough to zero. For such $t>0$, $U_{t}/V_{t}$ $=0/0:=0$. Therefore to avoid this triviality, when we consider the asymptotic behavior of $U_{t}/V_{t}$ near 0 we shall always assume that $\overline {\Lambda}(0+)=\infty$.   Our study is motivated by the following results for weighted sums. Let $\left\{ Y,Y_{i}:i\geq1\right\} $ denote a sequence of i.i.d. random variables, where $Y$ is non-negative and nondegenerate with cdf $G$. Now let $\left\{ X,X_{i}:i\geq1\right\} $ be a sequence of i.i.d. random variables, independent of $\{Y,Y_{i}:i\geq1\}$. Assume that $X$ has cdf $F$ and is in the class $\mathcal{X}$ of nondegenerate random variables $X$ satisfying $E|X|<\infty.$ Consider the self-normalized sums $$T(n)=\frac{\sum_{i=1}^{n}X_{i}Y_{i}}{\sum_{i=1}^{n}Y_{i}}.$$ We define $0/0:=0$. Theorem 4 of Breiman [@Brei] says that $T(n)$ converges in distribution along the full sequence $\left\{ n\right\} $ for *every* $X\in\mathcal{X}$ with at least one limit law being nondegenerate if and only if $Y\in D(\beta)$, with $0\leq\beta<1$, which means that for some function $L$ slowly varying at infinity, $P\left\{ Y>y\right\} =y^{-\beta}L(y),y>0.$ In the case $0<\beta<1$ this is equivalent to $Y\geq0$ being in the domain of attraction of a positive stable law of index $\beta$. Breiman [@Brei] has shown in his Theorem 3 that in this case the limit has a distribution related to the arcsine law. At the end of his paper Breiman conjectured that $T(n)$ converges in distribution to a nondegenerate law for *some* $X\in\mathcal{X}$ if and only if $Y\in D(\beta),$ with $0\leq\beta<1.$ Mason and Zinn [@mz] partially verified his conjecture. They established the following: Whenever $X$ is nondegenerate and satisfies $E|X|^{p}<\infty$ for some $p>2, $ then $T(n)$ converges in distribution to a nondegenerate random variable if and only if $Y\in D(\beta)$, $0\leq\beta<1$. Recently, Kevei and Mason [@KM] investigated the subsequential limits of $T(n)$. To state their main result we need some definitions. A random variable $Y$ (not necessarily non-negative) is said to be in the *Feller class* if there exist sequences of centering and norming constants $\{a_{n}\}_{n\geq1}$ and $\{b_{n}\}_{n\geq1}$ such that if $Y_{1},Y_{2},\dots$ are i.i.d. $Y$ then for every subsequence of $\{n\}$ there exists a further subsequence $\{n^{\prime}\}$ such that $$\frac{1}{b_{n^{\prime}}}\left\{ \sum_{i=1}^{n^{\prime}}Y_{i}-a_{n^{\prime}}\right\} \overset{\mathrm{D}}{\longrightarrow}W,\text{ as }n^{\prime }\rightarrow\infty,$$ where $W$ is a nondegenerate random variable. We shall denote this by $Y\in\mathcal{F}$. Furthermore, $Y$ is in the *centered Feller class*, if $Y$ is in the *Feller class* and one can choose $a_{n}=0$, for all $n\geq1$. We shall denote this as $Y\in\mathcal{F}_{c}$. The main theorem in [@KM] connects $Y\in\mathcal{F}_{c}$ with the continuity of all of the subsequential limit laws of $T(n)$. It says that all of the subsequential distributional limits of $T(n)$ are continuous for any $X$ in the class $\mathcal{X}$, if and only if $Y\in\mathcal{F}_{c}$. The notions of Feller class and centered Feller class carry over to Lévy processes. In particular, a Lévy process $Y_{t}$ is said to be in the *Feller class* at infinity if there exists a norming function $B\left( t\right) $ and a centering function $A\left( t\right) $ such that for each sequence $t_{k}$ $\rightarrow\infty$ there exists a subsequence $t_{k}^{\prime}\rightarrow\infty$ $\ $such that $$\left( Y_{t_{k}^{\prime}}-A(t_{k}^{\prime})\right) /B(t_{k}^{\prime })\overset{\mathrm{D}}{\longrightarrow}W,\ \text{ as }k\rightarrow\infty,$$ where $W$ is a nondegenerate random variable. The Lévy process $Y_{t}$ belongs to the *centered Feller class* at infinity if it is in the Feller class at infinity and the centering function $A\left( t\right) $ can be chosen to be identically zero. For the definitions of *Feller class* at zero and *centered Feller class* at zero replace $t_{k}$ $\rightarrow\infty$ and $t_{k}^{\prime}$ $\rightarrow\infty$, by $t_{k}$ $\searrow0$ and $t_{k}^{\prime}$ $\searrow0$, respectively. See Maller and Mason [@MM2] and [@MM3] for more details. In this paper, we consider the continuous time analog of the results described above, i.e. we investigate the asymptotic properties of the self-normalized Lévy process $$T_{t}=U_{t}/V_{t},\label{ratio}$$ as $t\searrow0$ or $t\rightarrow\infty$. The expression *continuous time analog* is justified by Remark 2 in [@KM], where it is pointed out that under appropriate regularity conditions, norming sequence $\left\{ b_{n}\right\} _{n\geq1}$ and subsequences $\left\{ n^{\prime}\right\} $, $$\left( \frac{\sum_{1\leq i\leq n^{\prime}t}X_{i}Y_{i}}{b_{n^{\prime}}},\frac{\sum_{1\leq i\leq n^{\prime}t}Y_{i}}{b_{n^{\prime}}}\right) \overset{\mathrm{D}}{\longrightarrow}(a_{1}t+U_{t},a_{2}t+V_{t}),\text{as }n^{\prime}\rightarrow\infty.\label{fd}$$ In light of (\[fd\]) the results that we obtain in the case $t\rightarrow \infty$ are perhaps not too surprising given those just described for weighted sums. However, we find our results in the case $t\searrow0$ unexpected. Our main goal is to establish the following two theorems about the asymptotic distributional behavior of $U_{t}/V_{t}$. In the process we shall uncover a lot of information about its subsequential limit laws. First, assuming that $E|X|^{p}<\infty$, for some $p>2$, we obtain a partial solution to the continuous time version of the Breiman conjecture, i.e. the continuous time version of the result of Mason and Zinn [@mz]. \[Th2\] Assume that $X$ is nondegenerate and for some $p>2$, $E|X|^{p}<\infty$. Also assume that $\Lambda$ satisfies (\[VV\]) and, in the case $t\searrow0$, that $\overline{\Lambda}\left( 0+\right) =\infty$. The following are necessary and sufficient conditions for $U_{t}/V_{t}$ to converge in distribution as $t\searrow0$ (as $t\rightarrow\infty)$ to a random variable $T$, in which case it must happen that $\left( EX\right) ^{2}\leq ET^{2}\leq EX^{2}$. \(i) $U_{t}/V_{t}\overset{\mathrm{D}}{\rightarrow}T$ and $\left( EX\right) ^{2}<ET^{2}<EX^{2}$ if and only if $\overline{\Lambda}$ is regularly varying at zero (infinity) with index $-\beta\in(-1,0)$, in which case the random variable $T$ has cumulative distribution function $$P\left\{ T\leq x\right\} =\frac{1}{2}+\frac{1}{\pi\beta}\arctan\left[ \frac{\int|u-x|^{\beta}\mathrm{sgn}(x-u)F(\mathrm{d}u)}{\int|u-x|^{\beta }F(\mathrm{d}u)}\tan\frac{\pi\beta}{2}\right] ,\text{ }x\in\left( -\infty,\infty\right) ;\label{integ}$$ \(ii) $U_{t}/V_{t}\overset{\mathrm{D}}{\rightarrow}T$ and $ET^{2}=EX^{2}$ if and only if $\overline{\Lambda}$ is slowly varying at zero (infinity), in which case $T \overset{\mathrm{D}}{=}X;$ \(iii) $U_{t}/V_{t}\overset{\mathrm{D}}{\rightarrow}T$ and $ET^{2}=\left( EX\right) ^{2}$ if and only if $\overline{\Lambda}$ is regularly varying at zero (infinity) with index $-1$, in which case $T=EX.$ \[R11\]The assumption that $E|X|^{p}<\infty$ for some $p>2$ is only used in the proof of necessity in Theorem \[Th2\]. For the sufficiency parts of the theorem we only need to assume that $X$ is nondegenerate and $E\left\vert X\right\vert <\infty$. In line with the Breiman [@Brei] conjecture we suspect that $U_{t}/V_{t}\overset{\mathrm{D}}{\rightarrow}T$, as $t\searrow0$ (as $t\rightarrow\infty$), where $T$ is nondegenerate only if $\overline{\Lambda}$ satisfies the conclusion of parts (i) or (ii), and in the case that $T$ is degenerate only if $\overline{\Lambda}$ satisfies the conclusion of (iii). \[R1\]A special case of Theorem \[Th2\] shows that if $W_{t},$ $t>0,$ is standard Brownian motion, $V_{t}=\inf\left\{ s\geq0:W_{s}>t\right\} $ and each $X_{t}$ in (\[WL\]) is a zero/one random variable $X$ with $P\left\{ X=1\right\} =1/2$, then$\ U_{t}/V_{t}$ converges in distribution to the arcsine law as $t\searrow0$ or $t\rightarrow\infty.$ This is a consequence of the fact that $V_{t}$ is a stable process of index $1/2$, since in this case we can set $\beta=1/2$ and let $F$ be the cdf of $X$ in (\[integ\]), which yields after a little calculation that $T$ has the arcsine density $g_{T}\left( t\right) =\pi^{-1}\left( t(1-t)\right) ^{-1/2}$ for $0<t<1$. Moreover,$\ U_{t}/V_{t}\overset{\mathrm{D}}{=}U_{1}/V_{1}$, for all $t>0$, which can be seen by using the self-similar property of the $1/2$-stable process. \[R1b\]Theorem \[Th2\] has an interesting connection to some results of Barlow, Pitman and Yor [@BPY] and Watanabe [@Wat]. Suppose $V_{t}$ is a strictly stable process of index $0<\beta<1$ and each $X_{t}$ in (\[WL\]) is a zero/one random variable $X$ with $P\left\{ X=1\right\} =p$, with $0<p<1.$ Then $\ $Theorem \[Th2\] implies that $U_{t}/V_{t}$ converges in distribution as $t\searrow0$ or $t\rightarrow\infty$ to a random variable $Y_{\beta,p}$ with density defined for $0<x<1$, by $$g_{Y_{\beta,p}}\left( x\right) =\frac{\sin\left( \pi\beta\right) }{\pi }\frac{p\left( 1-p\right) x^{\beta-1}\left( 1-x\right) ^{\beta-1}}{p^{2}\left( 1-x\right) ^{2\beta}+\left( 1-p\right) ^{2}x^{2\beta }+2p\left( 1-p\right) x^{\beta}\left( 1-x\right) ^{\beta}\cos\left( \pi\beta\right) }.$$ Furthermore, since $V_{t}$ is self-similar, one sees that $U_{t}/V_{t}\overset{\mathrm{D}}{=}U_{1}/V_{1}$, for all $t>0$. Barlow, Pitman and Yor [@BPY] and Watanabe [@Wat] show that $g_{Y_{\beta,p}}$ is the density of the random variable $$p^{1/\beta}V_{1}/\left( p^{1/\beta}V_{1}+\left( 1-p\right) ^{1/\beta}V_{1}^{\prime}\right) ,$$ where $V_{1}\overset{\mathrm{D}}{=}V_{1}^{\prime}$ with $V_{1}$ and $V_{1}^{\prime}$ independent. Moreover, Theorem 2 of Watanabe [@Wat] says that if $A_{t}$ is the occupation time of $Z_{s}$, a $p-$skewed Bessel process of dimension $2-2\beta$, defined as $$A_{t}=\int_{0}^{t}\mathbf{1}\left\{ Z_{s}\geq0\right\} \mathrm{d}s,$$ then for all $t>0$, $A_{t}/t$ has a distribution with density $g_{Y_{\beta,p}}$. We point out that two additional representations can be given for $Y_{\beta,p}$ using Propositions \[prop1-repr\] and \[jump-repr\] in the next section. For more about the distribution of $Y_{\beta,p}$ as well as that of closely related random variables refer to James [@Jam]. \[R2\]Let $V_{t}$ be a subordinator and for each $x\geq0$ let $T\left( x\right) $ denote $\inf\left\{ t\geq0:V_{t}>x\right\} .$ Theorem \[Th2\] is analogous to Theorem 6, Chapter 3, of Bertoin [@bert], which says that $x^{-1}V_{T\left( x\right) -}$ converges in distribution as $x\searrow0$, (as $x\rightarrow\infty$) if and only if $V_{t}$ satisfies the necessary assumptions of Theorem \[Th2\] for some $-\beta\in\left[ -1,0\right] .$ The $\beta=0$ case corresponds to $\overline{\Lambda}$ being slowly varying at zero (infinity). When $-\beta\in\left( -1,0\right) $, the limiting distribution is the generalized arcsine law. Our most significant result about subsequential laws of $U_{t}/V_{t}$ is the following. Note that contrary to Theorem \[Th2\] we only assume finite expectation of $X$. \[subseq\] Assume $\left( U_{t},V_{t}\right) $, $t\geq0$, satisfies (\[E\]) and (\[VV\]) and has joint characteristic function (\[holds\]). All subsequential distributional limits of $U_{t}/V_{t}$, as $t\searrow0$, (as $t\rightarrow\infty$) are continuous for any cdf $F$ in the class $\mathcal{X} $, if and only if $V_{t}$ is in the centered Feller class at 0 $(\infty)$. \[R4\]The proof of Theorem \[subseq\] shows that if $F$ is in the class $\mathcal{X}$ and $V_{t}$ is in the centered Feller class at 0 $(\infty)$, all of the subsequential limit laws of $U_{t}/V_{t}$, as $t\searrow0$, (as $t\rightarrow\infty$) are not only continuous, but also have Lebesgue densities on $\mathbb{R}$. The rest of the paper is organized as follows. Section 2 contains two representations of the 2-dimensional Lévy process $(U_{t},V_{t})$. The first one plays a crucial role in the proof of Theorem \[Th2\], while the second one points out the connection between the continuous and discrete time versions of $V_{t}$. We provide a fairly exhaustive list of properties of the subsequential limit laws of $(U_{t},V_{t})$ in Section 3, and we prove our main results in Section 4. The Appendix contains some technical results needed in the proofs. Preliminaries ============= Representations for $\left( U_{t},V_{t}\right) $ -------------------------------------------------- Let $\left( U_{t},V_{t}\right) ,$ $t\geq0$, be a Lévy process satisfying (\[E\]) and (\[VV\]) with joint characteristic function (\[holds\]). We establish two representations for the bivariate Lévy process. Let $\varpi_{1},\varpi_{2},\dots$ be a sequence of i.i.d. exponential random variables with mean $1$, and for each integer $i\geq1$ set $S_{i}=\sum _{j=1}^{i}\varpi_{j}.$ Independent of $\varpi_{1},\varpi_{2},\dots$ let $X_{1},X_{2},\dots$ be a sequence of i.i.d. random variables with cdf $F$, which by (\[E\]) satisfies $E\left\vert X_{1}\right\vert <\infty.$ Consider the Poisson process $N(t)$ on $[0,\infty)$ with rate $1$, $$N\left( t\right) =\sum_{j=1}^{\infty}\mathbf{1}_{\{S_{j}\leq t\}}\mbox{, }t\geq0.\label{Poisson-proc}$$ Define for $s>0$, $$\varphi\left( s\right) =\sup\left\{ y:\overline{\Lambda}(y)>s\right\} ,\label{phi}$$ where the supremum of the empty set is taken as 0. It is easy to check that (\[VV\]) and Lemma \[int-trfo\] below imply that for all $\delta>0$, $$\int_{\delta}^{\infty}\varphi\left( s\right) \mathrm{d}s<\infty .\label{delta}$$ We have the following distributional representation of $\left( U_{t},V_{t}\right) $: \[prop1-repr\] For each fixed $t>0$, $$\left( U_{t},V_{t}\right) \overset{\mathrm{D}}{=}\left( \sum_{i=1}^{\infty }X_{i}\varphi\left( \frac{S_{i}}{t}\right) ,\sum_{i=1}^{\infty}\varphi\left( \frac{S_{i}}{t}\right) \right) .\label{ss}$$ It is important to note that this representation only holds for fixed $t>0$ and not for the process in $t$. As a first consequence of this representation we obtain that $E|U_{t}|/V_{t}\leq E|X|<\infty$, in particular, by Markov’s inequality, $U_{t}/V_{t}$ is stochastically bounded. Now let $\left\{ X_{s}\right\} _{s\geq0}$ be a class of i.i.d. $F$ random variables. Consider for each $t\geq0$ the process$$\left( \sum_{0\leq s\leq t}X_{s}\Delta V_{s},\sum_{0\leq s\leq t}\Delta V_{s}\right) ,$$ where $\Delta V_{s}=V_{s}-V_{s-}$. The following representation reveals the analogy between the continuous and discrete time self-normalization. \[jump-repr\] For each fixed $t\geq0$, $$\left( U_{t},V_{t}\right) \overset{\mathrm{D}}{=}\left( \sum_{0\leq s\leq t}X_{s}\Delta V_{s},\sum_{0\leq s\leq t}\Delta V_{s}\right) .\label{RW}$$ \[R5\]Notice that the process on the right hand side of (\[RW\]) is a stationary independent increment process. Since it has the same characteristic function as $\left( U_{t},V_{t}\right) $, the distributional representation in (\[RW\]) holds as a process in $t\geq0.$ Proofs of Propositions \[prop1-repr\] and \[jump-repr\] ------------------------------------------------------- In the proofs of Propositions \[prop1-repr\] and \[jump-repr\] we shall assume that $\Lambda\left( \left( 0,\infty\right) \right) =\infty$. The case $\Lambda\left( \left( 0,\infty\right) \right) <\infty$ follows by the same methods. First we state a useful lemma giving a well-known change of variables formula (see Revuz and Yor [@ry], Prop. 4.9, p.8, or Brémaud [@brem], p.301), where the integrals are understood to be Riemann–Stieltjes integrals. \[int-trfo\] Let $h$ be a measurable function defined on $(a,b]$, $0<a<b<\infty$, and $R$ a measure on $\left( 0,\infty\right) $ such that $$\overline{R}(x):=R\{(x,\infty)\},\ x>0,$$ is right continuous and $\overline{R}\left( \infty\right) =0$. Assume $\int_{0}^{\infty}\left\vert h\left( x\right) \right\vert R\left( \mathrm{d}x\right) <\infty$, and define for $s>0$ $$\varphi\left( s\right) = \sup\left\{ y:\overline{R}\left( y\right) >s\right\} ,$$ where the supremum of the empty set is defined to be $0$. Then we have $$\int_{0}^{\infty}h\left( x\right) R\left( \mathrm{d}x\right) =\int _{0}^{\infty}h\left( \varphi\left( s\right) \right) \mathrm{d}s.\label{chvar}$$ **Proof of Proposition \[prop1-repr\].** We only consider the process on $[0,1]$. Applying the Lévy–Itô integral representation of a Lévy process to our case we have that a.s. for each $t\geq0$ $$(U_{t},V_{t})=\int_{\mathbb{R}^{2}\backslash\{0\}}(u,v)N([0,t],\mathrm{d}u,\mathrm{d}v),\label{levy-ito}$$ where $N$ is a Poisson point process on $(0,1)\times\mathbb{R}\times \lbrack0,\infty)$, with intensity measure $\text{Leb}\times\Pi$, where $\Pi$ is the Lévy measure as in (\[pi\]). For the Poisson point process we have the representation $$N=\sum_{i=1}^{\infty}\delta_{(U_{i},X_{i}\varphi(S_{i}),\varphi(S_{i}))},\label{Pp-repr}$$ where $\{U_{i}\}$ are i.i.d. Uniform$(0,1)$ random variables, independent from $\{X_{i}\}$ and $\{\varpi_{i}\}$. (At this step we consider the Lévy process on $[0,1].$) To see this, let $$M=\sum_{i=1}^{\infty}\delta_{(U_{i},X_{i},S_{i})},$$ which is a marked Poisson point process on $[0,1]\times\mathbb{R}\times(0,\infty)$, with intensity measure $\nu=\text{Leb}\times F\times \text{Leb}$. Put $h(u,x,s)=(u,x\varphi(s),\varphi(s))$. Then $\nu\circ h^{-1}=\text{Leb}\times\Pi$. Thus Proposition 2.1 in Rosiński [@Rosinski] implies that the sequences $\{U_{i}\},\{X_{i}\},\{S_{i}\}$ can be defined on the same space as $N$ such that (\[Pp-repr\]) holds. Using (\[Pp-repr\]) for $N$, from (\[levy-ito\]) we obtain that a.s. for each $t \in[0,1]$ $$\label{proc-repr}(U_{t}, V_{t}) = \sum_{i=1}^{\infty}\left( X_{i} \varphi(S_{i}), \varphi(S_{i}) \right) \mathbf{1}_{ \{ U_{i} \leq t \} }.$$ To finish the proof note that if $\sum_{i=1}^{\infty}\delta_{x_{i}}$ is a Poisson point process and independently $\{\beta_{i}\}$ is an i.i.d. Bernoulli$(t)$ sequence, then $$\sum_{i=1}^{\infty}\delta_{x_{i}}\mathbf{1}_{\{\beta_{i}=1\}}\overset {\mathrm{D}}{=}\sum_{i=1}^{\infty}\delta_{x_{i}/t},$$ i.e. for a Poisson point process independent Bernoulli thinning and scaling are distributionally the same. Since the process representation (\[proc-repr\]) can be extended to any finite interval $[0,T]$ (see the final remark in [@Rosinski]), this completes the proof. $\,\hfill\Box$ We point out that Proposition \[prop1-repr\]** **can also be proved by the same way as Proposition 5.1 in Maller and Mason [@MM]. Next we turn to the proof of the second representation. **Proof of Proposition \[jump-repr\].** Let $\left\{ N_{n}\right\} _{n\geq1}$ be a sequence of independent Poisson processes on $\left[ 0,\infty\right) $ with rate $1$. Independent of $\left\{ N_{n}\right\} _{n\geq1}$ let $\left\{ \xi_{i,n}\right\} _{i\geq1,n\geq1}$ be an array of independent random variables such that for each $i\geq1,n\geq 1$, $\xi_{i,n}$ has distribution $P_{i,n}$ defined for each Borel subset of $A$ of $\mathbb{R}$ by $$P_{i,n}\left( A\right) =P\left\{ \xi_{i,n}\in A\right\} =\Lambda\left( A\cap\left[ a_{n},a_{n-1}\right) \right) /\mu_{n}\text{,}$$ where $a_{n}$ is a strictly decreasing sequence of positive numbers converging to zero such that $a_{0}=\infty$ and for all $n\geq1,$ $0<\mu_{n}=\Lambda\left( \left[ a_{n},a_{n-1}\right) \right) <\infty$. The process $V_{t},$ $t\geq0$, has the representation as the Poisson point process$$V_{t}=\sum_{n=1}^{\infty}\sum_{i\leq N_{n}\left( t\mu_{n}\right) }\xi _{i,n}=:\sum_{n=1}^{\infty}V_{t}^{\left( n\right) }.$$ See Bertoin [@bert], page 16. In this representation $$V_{t}^{\left( n\right) }=\sum_{0\leq s\leq t}\Delta V_{s}\mathbf{1}_{\{a_{n}\leq\Delta V_{s}<a_{n-1}\}}$$ and $$\Delta V_{s}\mathbf{1}_{\{a_{n}\leq\Delta V_{s}<a_{n-1}\}}=\sum_{i\leq N_{n}\left( s\mu_{n}\right) }\xi_{i,n}-\sum_{i\leq N_{n}\left( s\mu _{n}-\right) }\xi_{i,n}.$$ Moreover if $\Delta V_{s}>0$ there exists a unique pair $\left( i,n\right) $ such that $\Delta V_{s}=\xi_{i,n}$. Clearly$$\begin{split} & \bigg(\sum_{0\leq s\leq t}X_{s}\Delta V_{s}\mathbf{1}_{\{a_{n}\leq\Delta V_{s}<a_{n-1}\}},\sum_{0\leq s\leq t}\Delta V_{s}\mathbf{1}_{\{a_{n}\leq\Delta V_{s}<a_{n-1}\}}\bigg)\\ & \overset{\mathrm{D}}{=}\bigg(\sum_{i\leq N_{n}\left( t\mu_{n}\right) }X_{i,n}\,\xi_{i,n},\sum_{i\leq N_{n}\left( t\mu_{n}\right) }\xi _{i,n}\bigg)=:\left( U_{t}^{\left( n\right) },V_{t}^{\left( n\right) }\right) , \end{split} \label{comp}$$ where $\{X_{i,n}\}_{i\geq1,n\geq1}$ is an array of i.i.d. random variables with common distribution function $F$. Notice that the process $\left( U_{t}^{\left( n\right) },V_{t}^{\left( n\right) }\right) $ in (\[comp\]) is a compound Poisson process. Keeping this in mind, we see after a little calculation that $$E\exp\left( \mathrm{i}\left( \theta_{1}U_{t}^{\left( n\right) }+\theta _{2}V_{t}^{\left( n\right) }\right) \right) =\exp\left( t\int_{\left[ a_{n},a_{n-1}\right) }\int_{-\infty}^{\infty}\left( e^{\mathrm{i}(\theta _{1}u+\theta_{2}v)}-1\right) F\left( \mathrm{d}u/v\right) \Lambda\left( \mathrm{d}v\right) \right) .$$ Since the random variables $\left\{ \left( U_{t}^{\left( n\right) },V_{t}^{\left( n\right) }\right) \right\} _{n\geq1}$ are independent we readily conclude that (\[holds\]) holds. $\Box\medskip$ Additional asymptotic distribution results along subsequences ============================================================= Let $\mathrm{id}(a,b,\nu)$ denote an infinitely divisible distribution on $\mathbb{R}^{d}$ with characteristic exponent $$\mathrm{i}u^{\prime}b-\frac{1}{2}u^{\prime}au+\int\left( e^{\mathrm{i}u^{\prime}x}-1-\mathrm{i}u^{\prime}x \mathbf{1}_{\{ |x|\leq1 \}} \right) \nu(\mathrm{d}x),$$ where $b\in\mathbb{R}^{d}$, $a\in\mathbb{R}^{d\times d}$ is a positive semidefinite matrix, $\nu$ is a Lévy measure on $\mathbb{R}^{d}$ and $u^{\prime}$ stands for the transpose of $u$. In our case $d$ is $1$ or $2$. For any $h>0$ put $$a^{h}=a+\int_{|x|\leq h}xx^{\prime}\nu(\mathrm{d}x)\text{ and }b^{h}=b-\int_{h<|x|\leq1}x\nu(\mathrm{d}x).$$ When $d=1$, id$(b,\Lambda),$ with Lévy measure $\Lambda$ on $\left( 0,\infty\right) $, such that (\[VV\]) holds, and $b\geq0$, denotes a non-negative infinitely divisible distribution with Laplace transform $$\exp\left( -\theta b-\int_{0}^{\infty}\left( 1-e^{-\theta u}\right) \Lambda\left( \mathrm{d}u\right) \right) .$$ In this section it will be convenient to use the following representation for the joint characteristic function of the Lévy process $\left( U_{t},V_{t}\right) $, $t\geq0,$ satisfying (\[E\]) and (\[VV\]) and having joint characteristic function (\[holds\]): $$\begin{split} \phi\left( t,\theta_{1},\theta_{2}\right) & =\exp\left( \mathrm{i}t(\theta_{1}b_{1}+\theta_{2}b_{2})\right) \times\\ & \exp\left( t\int_{(0,\infty)}\int_{-\infty}^{\infty}\left( e^{\mathrm{i}(\theta_{1}u+\theta_{2}v)}-1-\left( \mathrm{i}\theta_{1}u+\mathrm{i}\theta_{2}v\right) \mathbf{1}_{\{u^{2}+v^{2}\leq1\}}\right) \Pi\left( \mathrm{d}u,\mathrm{d}v\right) \right) , \end{split} \label{cf-uv}$$ where $\Pi\left( \mathrm{d}u,\mathrm{d}v\right) $ is as in (\[pi\]) and $$\mathbf{b}=\left( \begin{array} [c]{c}b_{1}\\ b_{2}\end{array} \right) =\left( \begin{array} [c]{c}\int_{0<u^{2}+v^{2}\leq1}u\Pi\left( \mathrm{d}u,\mathrm{d}v\right) \\ \int_{0<u^{2}+v^{2}\leq1}v\Pi\left( \mathrm{d}u,\mathrm{d}v\right) \end{array} \right) .\label{b}$$ Note that assumptions (\[E\]) and (\[VV\]) insure that (\[b\]) is well defined. First we investigate the possible subsequential distributional limits of $\left( U_{t},V_{t}\right) $. The following theorem is an analog of Theorem 1 in [@KM]. \[2-dim-conv\] Consider the bivariate Lévy process $\left( U_{t},V_{t}\right) ,$ $t\geq0$, satisfying (\[E\]) and (\[VV\]) with joint characteristic function (\[cf-uv\]). Assume that for some deterministic sequences $t_{k}\searrow0$ ($t_{k}\rightarrow\infty)$ and $B_{k}$ the distributional convergence $$\frac{V_{t_{k}}}{B_{k}}\overset{\mathrm{D}}{\longrightarrow}V\label{v-conv}$$ holds, where $V$ has $\mathrm{id}(b,\Lambda_{0})$ distribution with Lévy measure $\Lambda_{0}$ on $\left( 0,\infty\right) $. Then $$\left( \frac{U_{t_{k}}}{B_{k}},\frac{V_{t_{k}}}{B_{k}}\right) \overset {\mathrm{D}}{\longrightarrow}(U,V),\label{uv-conv}$$ where $(U,V)$ has $\mathrm{id}(\mathbf{0},\mathbf{c},\Pi_{0})$ distribution with Lévy measure $\Pi_{0}\left( \mathrm{d}u,\mathrm{d}v\right) =F\left( \mathrm{d}u/v\right) \Lambda_{0}\left( \mathrm{d}v\right) $ on $\mathbb{R}\times\left( 0,\infty\right) $ and $$\mathbf{c}=\left( \begin{array} [c]{c}c_{1}\\ c_{2}\end{array} \right) =\left( \begin{array} [c]{c}bEX+\int_{0<u^{2}+v^{2}\leq1}u\Pi_{0}\left( \mathrm{d}u,\mathrm{d}v\right) \\ b+\int_{0<u^{2}+v^{2}\leq1}v\Pi_{0}\left( \mathrm{d}u,\mathrm{d}v\right) \end{array} \right) ,\label{c}$$ i.e. it has characteristic function $$\begin{split} \Psi(\theta_{1},\theta_{2}) & =Ee^{\mathrm{i}(\theta_{1}U+\theta_{2}V)}=\exp\bigg\{\mathrm{i}(\theta_{1}c_{1}+\theta_{2}c_{2})\\ & \phantom{=}+\int_{0}^{\infty}\int_{-\infty}^{\infty}\left( e^{\mathrm{i}(\theta_{1}u+\theta_{2}v)}-1-\left( \mathrm{i}\theta_{1}u+\mathrm{i}\theta_{2}v\right) \mathbf{1}_{\{u^{2}+v^{2}\leq1\}}\right) F\left( \mathrm{d}u/v\right) \Lambda_{0}\left( \mathrm{d}v\right) \bigg\}. \end{split} \label{limit-chfunc-proc}$$ Theorem \[2-dim-conv\] has some immediate consequences concerning the subsequential limits of $(U_{t},V_{t})$. The first part of the following corollary is deduced from Theorem \[2-dim-conv\] and classical theory, i.e. Theorem 15.14 in [@kallenberg]. The second part follows by Fourier inversion. \[cor5\] Let $(U_{t},V_{t}),$ $t\geq0$, be as in Theorem \[2-dim-conv\]. For deterministic constants $t_{k},B_{k}$ the vector $B_{k}^{-1}(U_{t_{k}},V_{t_{k}})$ converges in distribution to $\left( U,V\right) $ as $t_{k}\searrow0$ (as $t_{k}\rightarrow\infty)$ having characteristic function (\[limit-chfunc-proc\]) if, and only if $t_{k}\overline{\Lambda}(vB_{k})\rightarrow\overline{\Lambda}_{0}(v)$ for every continuity point of $\Lambda_{0}$, and $\int_{0}^{h}xt_{k}\Lambda(\mathrm{d}B_{k}x)\rightarrow \int_{0}^{h}x\Lambda_{0}(\mathrm{d}x)+b$ for some continuity point $h$ of $\Lambda_{0}$. Moreover, if $\overline{\Lambda}(0+)=\infty$, or $b>0$ then $V>0$ a.s., and so $U_{t_{k}}/V_{t_{k}}\overset{}{\overset{\mathrm{D}}{\longrightarrow}U/V},$ and with $\Psi$ as in (\[limit-chfunc-proc\]) for all $x$ $$P\left\{ U/V\leq x\right\} =\frac{1}{2}-\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{\infty}\frac{\Psi(u,-ux)}{u}\mathrm{d}u.$$ The remaining results in this section, though interesting in their own right, are crucial for the proof of Theorem \[subseq\]. The following proposition provides a sufficient condition for $\left( U,V\right) $ to have a $C^{\infty}$ 2-dimensional density. It also gives an alternative proof for Theorem 3 in [@KM]. We require the following notation: Put for $v>0$,$$V_{2}\left( v\right) =\int_{0<u\leq v}u^{2}\Lambda(\mathrm{d}u).\label{V2}$$ \[prop-fc0\] Assume that $(U,V)$ has joint characteristic function$$Ee^{\mathrm{i}(\theta_{1}U+\theta_{2}V)}=\exp\left\{ \int_{\left( 0,\infty\right) }\int_{\mathbb{R}}\left( e^{\mathrm{i}(\theta_{1}u+\theta_{2}v)}-1\right) F\left( \frac{\mathrm{d}u}{v}\right) \Lambda(\mathrm{d}v)\right\} ,$$ where $\int_{0}^{1}v\Lambda(\mathrm{d}v)<\infty$ and $F$ is in the class $\mathcal{X}$. Whenever $$\limsup_{v\searrow0}\frac{v^{2}\overline{\Lambda}\left( v\right) }{V_{2}\left( v\right) }<\infty\label{ss1}$$ holds, then $\left( U,V\right) $ has a $C^{\infty}$ density. As a consequence we obtain the following \[cor1\] Let $(U_{t},V_{t}),$ $t\geq0$, be as in Theorem \[2-dim-conv\]. Assume that $V_{t}$ is in the centered Feller class at zero (infinity) and $F$ is in the class $\mathcal{X}$. Then for a suitable norming function $B(t) $ any subsequential distributional limit of $$\left( \frac{U_{t_{k}}}{B(t_{k})},\frac{V_{t_{k}}}{B(t_{k})}\right)$$ along a subsequence $t_{k}\searrow0$ $(t_{k}\rightarrow\infty)$, say $\left( W_{1},W_{2}\right) $, has a $C^{\infty}$ Lebesgue density $f$ on $\mathbb{R}^{2}$, which implies that the asymptotic distribution of the corresponding ratio along the subsequence $\{t_{k}\}$ has a Lebesgue density $g_{T}$ on $\mathbb{R}$. The following corollary is an immediate consequence of Theorem \[2-dim-conv\]. Note that a Lévy process $Y_{t}$ that is in the Feller class at zero (infinity) but not in the centered Feller class at zero (infinity) has the required property. \[cor2\]Let $(U_{t},V_{t}),$ $t\geq0$, be as in Theorem \[2-dim-conv\]. Suppose along a subsequence $t_{k}\searrow0$ ($t_{k}\rightarrow\infty)$ $$\frac{V_{t_{k}}-A(t_{k})}{B(t_{k})}\overset{\mathrm{D}}{\longrightarrow}W,$$ where $W$ is nondegenerate and $A(t_{k})/B(t_{k})\rightarrow\infty,$ as $k\rightarrow\infty.$ Then $$\frac{U_{t_{k}}}{V_{t_{k}}}\overset{\mathrm{D}}{\longrightarrow}EX,\quad\text{as }k\rightarrow\infty.$$ For $t>0$ and $\varepsilon\in(0,1)$ put $$A_{t}(\varepsilon)=\left\{ \frac{\varphi(S_{1}/t)}{\sum_{i=1}^{\infty}\varphi(S_{i}/t)}>1-\varepsilon\right\} ,\label{AAa}$$ and $$\Delta_{t}=\left\vert \frac{\sum_{i=1}^{\infty}X_{i}\varphi(S_{i}/t)}{\sum_{i=1}^{\infty}\varphi(S_{i}/t)}-X_{1}\right\vert .$$ \[sums-th4\] Assume that for a subsequence $t_{k} \searrow0$ or $t_{k} \to\infty$ $$\label{A-assump}\lim_{\varepsilon\to0} \liminf_{k \to\infty} P \{ A_{t_{k}} ( \varepsilon) \} = \delta> 0,$$ then $$\lim_{\varepsilon\to0} \liminf_{k \to\infty} P \{ \Delta_{t_{k}} \leq\varepsilon\} \geq\delta.$$ Together with the stochastic boundedness of $U_{t} / V_{t}$ this implies the following. \[cor3\]Let $(U_{t},V_{t}),$ $t\geq0$, be as in Theorem \[2-dim-conv\]. Assume that (\[A-assump\]) holds for $V_{t}$, and $P\{X=x_{0}\}>0$ for some $x_{0}$. Then there exists a subsequence $t_{k}\searrow0$ $(t_{k}\rightarrow\infty)$ such that $U_{t_{k}}/V_{t_{k}}\overset{\mathrm{D}}{\longrightarrow}T,$ with $P\{T=x_{0}\}>0$. Put $$R_{t}=\frac{\sum_{i=1}^{\infty}\varphi^{2}\left( \frac{S_{i}}{t}\right) }{\left( \sum_{i=1}^{\infty}\varphi\left( \frac{S_{i}}{t}\right) \right) ^{2}}.\label{RT}$$ \[non-feller\] Assume that $R_{t}^{-1}\neq O_{P}(1)$ as $t\searrow0$ or $t\rightarrow\infty$, then there exists a subsequence $t_{k}\searrow0$ or $t_{k}\rightarrow\infty$ such that $U_{t_{k}}/V_{t_{k}}\overset{\mathrm{D}}{\longrightarrow}T,$ with $P\{T=EX\}>0$. The proofs of Propositions \[sums-th4\] and \[non-feller\] are adaptations of those of Theorems 4 and 5 in [@KM]. Therefore we only sketch the proof of the first one, and omit the proof of the second one. Proofs of results ================= Recall that throughout this paper $\left( U_{t},V_{t}\right) $, $t\geq0,$ denotes a Lévy process satisfying (\[E\]) and (\[VV\]) and having joint characteristic function (\[holds\]). We start with the proof of Theorem \[2-dim-conv\] since this result is crucial for both the proofs of Theorem \[Th2\] and \[subseq\]. Proof of Theorem \[2-dim-conv\] ------------------------------- Recall the notation at the beginning of Section 3. Since $V_{t}$ is a driftless subordinator, by Theorem 15.14 (ii) in [@kallenberg], (\[v-conv\]) is equivalent to the convergences $$t_{k}\overline{\Lambda}(vB_{k})\rightarrow\overline{\Lambda}_{0}(v),\quad\text{as }k\rightarrow\infty,\label{Levy-conv-V}$$ for any $v>0$ continuity point of $\overline{\Lambda}_{0}$, and $$\int_{0}^{v}xt_{k}\Lambda(\mathrm{d}B_{k}x)\rightarrow\int_{0}^{v}x\Lambda _{0}(\mathrm{d}x)+b,\quad\text{as }k\rightarrow\infty,\label{1stmoment-V}$$ where $v>0$ is a fixed continuity point of $\overline{\Lambda}_{0}$. Notice that using (\[cf-uv\]) we have that $$\begin{split} E & e^{\mathrm{i}\left( \theta_{1}\frac{U_{t_{k}}}{B_{k}}+\theta_{2}\frac{V_{t_{k}}}{B_{k}}\right) }=\exp\left\{ \mathrm{i}\frac{t_{k}}{B_{k}}(\theta_{1}b_{1}+\theta_{2}b_{2})\right\} \\ & \phantom{=}\times\exp\left\{ \int\left[ e^{\mathrm{i}(\theta_{1}u+\theta_{2}v)/B_{k}}-1-\frac{\mathrm{i}}{B_{k}}(\theta_{1}u+\theta _{2}v)\mathbf{1}_{\{0<u^{2}+v^{2}\leq1\}}\right] t_{k}\Pi(\mathrm{d}u,\mathrm{d}v)\right\} \\ & =\exp\left\{ \mathrm{i}\frac{t_{k}}{B_{k}}(\theta_{1}b_{1}+\theta_{2}b_{2})\right\} \\ & \phantom{=}\times\exp\left\{ \int\left[ e^{\mathrm{i}(\theta_{1}x+\theta_{2}y)}-1-\mathrm{i}(\theta_{1}x+\theta_{2}y)\mathbf{1}_{\{0<x^{2}+y^{2}\leq B_{k}^{-2}\}}\right] \Pi_{k}(\mathrm{d}x,\mathrm{d}y)\right\} , \end{split}$$ where $\Pi$ is the Lévy measure on $\left( 0,\infty\right) \times\mathbb{R}$ defined by (\[pi\]) and for each $k\geq1,$ $\Pi_{k}$ is the Lévy measure on $\left( 0,\infty\right) \times\mathbb{R}$ defined by$$\Pi_{k}(\mathrm{d}x,\mathrm{d}y)=t_{k}\Pi(B_{k}\mathrm{d}x,B_{k}\mathrm{d}y).$$ Further, for each $k\geq0$ and $h>0$ with $\Pi_{0}(\left\{ x:|x|=h\right\} )=0$, in accordance with the notation at the beginning of Section 3, let $$\begin{aligned} a_{k}^{h} & =\int_{x^{2}+y^{2}\leq h^{2}}\left( \begin{array} [c]{cc}x^{2} & xy\\ xy & y^{2}\end{array} \right) \Pi_{k}(\mathrm{d}x,\mathrm{d}y),\\ b_{k}^{h} & =\frac{t_{k}}{B_{k}}\mathbf{b}-\int_{1<x^{2}+y^{2}\leq B_{k}^{-2}}(x,y)\Pi_{k}(\mathrm{d}x,\mathrm{d}y)-\int_{h^{2}<x^{2}+y^{2}\leq 1}(x,y)\Pi_{k}(\mathrm{d}x,\mathrm{d}y)\text{ }\\ & =\int_{x^{2}+y^{2}\leq h^{2}}(x,y)\Pi_{k}(\mathrm{d}x,\mathrm{d}y),\end{aligned}$$ where we used (\[b\]). We set $a^{h}:=a_{0}^{h}$ and $b^{h}:=b_{0}^{h}.$ To show (\[uv-conv\]), by Theorem 15.14 (i) in [@kallenberg] we have to prove that as $k\rightarrow\infty$, $$\Pi_{k}\overset{v}{\rightarrow}\Pi_{0}\text{, on }\mathbb{R}^{2}-\left\{ \mathbf{0}\right\} \label{pi-conv-uv}$$ and for some (any) $h>0$ with $\Pi_{0}(\left\{ x:|x|=h\right\} )=0,$ as $k\rightarrow\infty$, $$\begin{aligned} a_{k}^{h} & \rightarrow a^{h},\label{a-conv-uv}\\ b_{k}^{h} & \rightarrow b^{h}.\label{b-conv-uv}$$ To establish (\[pi-conv-uv\]) it suffices to show that for each $\left( u,v\right) $ with $u\geq0$, $v>0$, and $\left( u,v\right) $, with $u>0$, $v=0 $, that when $\left( u,v\right) $ is a continuity point of $\overline{\Pi}_{0}$, $$t_{k}\overline{\Pi}(B_{k}u,B_{k}v)\rightarrow\overline{\Pi}_{0}(u,v),\text{ as }k\rightarrow\infty,\text{ }$$ and when $\left( -u,v\right) $ is a continuity point of $\Pi_{0}$, $$t_{k}\Pi(-B_{k}u,B_{k}v)\rightarrow\Pi_{0}(-u,v),\text{ as }k\rightarrow \infty;$$ where for $u\geq0,v>0$, $$t_{k}\overline{\Pi}(B_{k}u,B_{k}v)=\int_{v}^{\infty}\overline{F}(u/y)t_{k}\Lambda(\mathrm{d}B_{k}y),$$$$\overline{\Pi}_{0}(u,v)=\int_{v}^{\infty}\overline{F}(u/y)\Lambda _{0}(\mathrm{d}y),$$$$t_{k}\Pi(-B_{k}u,B_{k}v)=\int_{v}^{\infty}F(-u/y)t_{k}\Lambda(\mathrm{d}B_{k}y)$$ and $$\Pi_{0}(-u,v)=\int_{v}^{\infty}F(-u/y)\Lambda_{0}(\mathrm{d}y).$$ This follows with obvious changes of notation exactly as in the proof of Proposition 1 in [@KM]. The proofs that (\[a-conv-uv\]) and (\[b-conv-uv\]) hold follow exactly as in Propositions 2 and 3 in [@KM]. It turns out that $a^{h}$ converges to the zero matrix as $h\searrow0$ and by (\[1stmoment-V\]) $$b^{h}=\left( \begin{array} [c]{c}bEX+\int_{0}^{h}\psi(v)\Lambda_{0}(\mathrm{d}v)\\ b+\int_{0}^{h}\phi(v)v\Lambda_{0}(\mathrm{d}v) \end{array} \right) ,$$ where $\psi$ and $\phi$ are the following functions of $v\in(0,h]$: $$\phi\left( v\right) =\int_{\left[ -\sqrt{h^{2}-v^{2}},\sqrt{h^{2}-v^{2}}\right] }F\left( \frac{\mathrm{d}u}{v}\right) \text{ and }\psi\left( v\right) =\int_{\left[ -\sqrt{h^{2}-v^{2}},\sqrt{h^{2}-v^{2}}\right] }uF\left( \frac{\mathrm{d}u}{v}\right) .$$ (Refer to [@KM] for details.) Thus $$\lim_{h\rightarrow0}b^{h}=\left( \begin{array} [c]{c}bEX\\ b \end{array} \right) ,$$ and the theorem follows with the stated constants. $\,$ $\Box$ Proof of Theorem \[Th2\] ------------------------ The following three lemmas establish the in which case parts of (i), (ii) and (iii) of Theorem \[Th2\]. \[cor4\] If $\overline{\Lambda}$ is regularly varying at zero (infinity) with index $-\beta$ with $\beta\in\left( 0,1\right) $, then for an appropriate norming function $B_{t}$ the random variable $B_{t}^{-1}(U_{t},V_{t})$ converges in distribution as $t\searrow0$ (as $t\rightarrow \infty$) to $\left( U,V\right) $, having joint characteristic function $$\phi\left( \theta_{1},\theta_{2}\right) =\exp\left( \int_{(0,\infty)}\int_{-\infty}^{\infty}\left( e^{\mathrm{i}(\theta_{1}u+\theta_{2}v)}-1\right) F\left( \mathrm{d}u/v\right) \beta v^{-1-\beta}\mathrm{d}v\right) \label{stable}$$ and thus $$T_{t}=\frac{U_{t}}{V_{t}}\overset{\mathrm{D}}{\longrightarrow}\frac{U}{V}\text{, as }t\searrow0\ (\text{as }t\rightarrow\infty).\label{UV}$$ Moreover, the cdf of $U/V$ is given by (\[integ\]). **Proof.** We can find a function $B_{t}$ on $\left[ 0,\infty\right) $ such that $$B_{t}=L^{\ast}\left( t\right) t^{1/\beta},\text{ }t>0\text{,}$$ with $L^{\ast}$ defined on $\left[ 0,\infty\right) $ slowly varying at zero (infinity) satisfying for all $y>0$, $$\overline{\mu}_{t}\left( y\right) :=t\overline{\Lambda}\left( yB_{t}\right) \rightarrow\overline{\Lambda}_{0}\left( y\right) =y^{-\beta }\text{, as }t\searrow0\text{ (as }t\rightarrow\infty\text{).}$$ It is routine to show using well-known properties of regularly varying functions that for any $y>0$, as $t\searrow0$ (as $t\rightarrow\infty$) $$a_{t}^{h}:=\int_{0<y\leq h}y\mu_{t}\left( \mathrm{d}y\right) \rightarrow \frac{\beta h^{1-\beta}}{1-\beta}=\int_{0<y\leq h}y\Lambda_{0}\left( \mathrm{d}y\right) =:a^{h}.$$ Thus by applying Theorem 15.14 (ii) in [@kallenberg] we find that $B_{t}^{-1}V_{t}$ converges in distribution as $t\searrow0$ (as $t\rightarrow \infty$) to $V$, having characteristic function $\phi\left( 0,\theta _{2}\right) .$ This says that $V$ is a subordinator with an id$(0,\Lambda _{0})$ distribution. Theorem \[2-dim-conv\] completes the proof of (\[stable\]). Next, using Fubini’s theorem and the explicit formula for the $\beta$-stable characteristic function (Meerschaert and Scheffler [@meerschaert] p.266), we have for an appropriate constant $c>0$ $$\begin{aligned} & \int_{(0,\infty)}\int_{-\infty}^{\infty}\left( e^{\mathrm{i}(\theta _{1}u+\theta_{2}v)}-1\right) F\left( \mathrm{d}u/v\right) \beta v^{-1-\beta}\mathrm{d}v\\ & =\int_{-\infty}^{\infty} F(\mathrm{d} u)\int_{0}^{\infty}\left[ e^{\mathrm{i}(\theta_{1}u+\theta_{2})y}-1\right] \Lambda_{0}(\mathrm{d}y)\\ & =-c\int_{-\infty}^{\infty}|\theta_{1}u+\theta_{2}|^{\beta}\left( 1-\mathrm{i}\,\mathrm{sgn}(\theta_{1}u+\theta_{2})\tan\frac{\pi\beta}{2}\right) F(\mathrm{d}u).\end{aligned}$$ We see now that the characteristic function of $U-Vx$ is $$\label{f(t,x)}Ee^{\mathrm{i}t(U-Vx)} = \exp\left\{ -|t|^{\beta}c\int|u-x|^{\beta}F(\mathrm{d}u)\left[ 1-\mathrm{i}\,\mathrm{sgn}\left( t\right) \tan\frac{\pi\beta}{2}\frac{\int|u-x|^{\beta}\mathrm{sgn}(u-x)F(\mathrm{d}u)}{\int|u-x|^{\beta}F(\mathrm{d}u)}\right] \right\} .$$ Proposition 4 in [@Brei] now shows that $T$ has cdf (\[integ\]). $\Box\medskip$ \[sv\] If $\overline{\Lambda}$ is slowly varying at zero (at infinity), then $$T_{t}=\frac{U_{t}}{V_{t}}\overset{\mathrm{D}}{\longrightarrow}X\text{, as }t\searrow0\ (\text{as }t\rightarrow\infty),\label{AA}$$ where in the $t\searrow0$ case we also assume $\overline{\Lambda}(0+)=\infty$. **Proof.** The proof follows the lines of that of Lemma 5.3 in [@MM]. We shall only prove the $t\rightarrow\infty$ case. The $t\searrow0$ case is nearly identical. Now $\overline{\Lambda}$ slowly varying at infinity implies that $\varphi$ is non-increasing and rapidly varying at $0$ with index $-\infty$. (See the argument in Item 5 on p.22 of de Haan [@haan].) This means that for all $0<\lambda<1$ $$\varphi\left( x\lambda\right) /\varphi\left( x\right) \rightarrow \infty,\mbox{ as }x\searrow0.$$ By Theorem 1.2.1 on p. 15 of [@haan], rapidly varying at $0$ with index $-\infty$ implies that $$\frac{\int_{x}^{\overline{\Lambda}(0+)}\varphi\left( y\right) \mathrm{d}y}{x\varphi\left( x\right) }\rightarrow0,\mbox{ as }x\searrow0.\label{x}$$ By Lemma \[psi-lemma\] in the Appendix, we have $$\begin{aligned} E\left( \frac{\sum_{i=2}^{\infty}\left\vert X_{i}\right\vert \varphi\left( \frac{S_{i}}{t}\right) }{\varphi\left( \frac{S_{1}}{t}\right) }\Bigg|S_{1}\right) & =E\left\vert X\right\vert E\left( \frac{\sum _{i=2}^{\infty}\varphi\left( \frac{S_{i}}{t}\right) }{\varphi\left( \frac{S_{1}}{t}\right) }\Bigg|S_{1}\right) \\ & =E\left\vert X\right\vert S_{1}\frac{\int_{S_{1}/t}^{\overline{\Lambda }(0+)}\varphi\left( y\right) \mathrm{d}y}{\frac{S_{1}}{t}\varphi\left( \frac{S_{1}}{t}\right) },\end{aligned}$$ and by (\[x\]) $$E\left\vert X\right\vert S_{1}\frac{\int_{S_{1}/t}^{\overline{\Lambda}(0+)}\varphi\left( y\right) \mathrm{d}y}{\frac{S_{1}}{t}\varphi\left( \frac{S_{1}}{t}\right) }\overset{\mathrm{P}}{\rightarrow}0,\mbox{ as }t\rightarrow\infty.$$ From this we can readily conclude that $$\sum_{i=1}^{\infty}\varphi\left( \frac{S_{i}}{t}\right) =\varphi\left( \frac{S_{1}}{t}\right) \left( 1+o_{P}\left( 1\right) \right) ,\mbox{ as }t\rightarrow\infty,\label{lo1}$$ and $$\sum_{i=1}^{\infty}X_{i}\varphi\left( \frac{S_{i}}{t}\right) =X_{1}\varphi\left( \frac{S_{1}}{t}\right) \left( 1+o_{P}\left( 1\right) \right) ,\mbox{ as }t\rightarrow\infty.\label{lo2}$$ From the representation (\[ss\]), (\[lo1\]) and (\[lo2\]) we see that$$\frac{U_{t}}{V_{t}}\overset{D}{=}\frac{X_{1}\varphi\left( \frac{S_{1}}{t}\right) \left( 1+o_{P}\left( 1\right) \right) }{\varphi\left( \frac{S_{1}}{t}\right) \left( 1+o_{P}\left( 1\right) \right) }=X_{1}+o_{P}\left( 1\right) ,\mbox{ as }t\rightarrow\infty.$$ Obviously $T_{t}$ converges in distribution as $t\rightarrow\infty$ to $X$. $\Box$ \[beta11\] If $\overline{\Lambda}$ is regularly varying at zero (at infinity) with index $-1$, $$T_{t}=\frac{U_{t}}{V_{t}}\overset{\mathrm{D}}{\longrightarrow}EX\text{, as }t\searrow0\ (\text{as }t\rightarrow\infty).\label{EX}$$ **Proof.** Since $\overline{\Lambda}$ is regularly varying at zero (at infinity) with index $-1$, we can find norming and centering functions $b\left( t\right) $ and $a\left( t\right) $ such that $b(t)/a(t)\rightarrow0$ as $t\searrow0\ ($as $t\rightarrow\infty)$ and $$b\left( t\right) ^{-1}\left( V_{t}-a\left( t\right) \right) \overset{\mathrm{D}}{\longrightarrow}W,\text{ as }t\searrow0\ (\text{as }t\rightarrow\infty).$$ (Here we apply part (i) of Theorem 15.14 in [@kallenberg].) From this we see that $$V_{t}/a(t)\overset{\mathrm{P}}{\longrightarrow}1,\text{ as }t\searrow 0\ (\text{as }t\rightarrow\infty).$$ A straightforward application of Theorem \[2-dim-conv\] now shows that$$\left( \frac{U_{t}}{a(t)},\frac{V_{t}}{a(t)}\right) \overset{\mathrm{P}}{\longrightarrow}\left( EX,1\right) \text{, as }t\searrow0\ (\text{as }t\rightarrow\infty).$$ $\Box$ Next we turn to the necessary and sufficient parts of (i), (ii) and (iii). Assume that for some random variable $T$ $$T_{t}\overset{\mathrm{D}}{\longrightarrow}T,\text{ as }t\searrow0\ (\text{as }t\rightarrow\infty),\label{conT}$$ where in the case $t\searrow0$ we assume that $\overline{\Lambda}(0+)=\infty$. Our basic tool will be Proposition \[prop1-repr\], which says that $$T_{t}=\frac{U_{t}}{V_{t}}\overset{\mathrm{D}}{=}\frac{\sum_{i=1}^{\infty}X_{i}\varphi\left( \frac{S_{i}}{t}\right) }{\sum_{i=1}^{\infty}\varphi\left( \frac{S_{i}}{t}\right) }.\label{repr}$$ Since we assume that $$E\left\vert X\right\vert ^{p}<\infty\label{delta1}$$ for some $p>2$, we get by Jensen’s inequality that $$E\left\vert T_{t}\right\vert ^{p}\leq E\left\vert X\right\vert ^{p}<\infty.$$ (This is the only place in the proof that we use assumption (\[delta1\]).) Notice that (\[conT\]) and (\[delta1\]) imply that $$ET_{t}^{2}\rightarrow ET^{2},\text{ as }t\searrow0\ (\text{as }t\rightarrow \infty).\label{varconv}$$ Obviously $ET_{t}=EX$ and a little calculation gives that $$ET_{t}^{2}=(EX)^{2}+{Var}(X)ER_{t},$$ where $R_{t}$ is defined as in (\[RT\]). Clearly, $R_{t}\in\lbrack0,1]$ and whenever (\[varconv\]) holds, then for some $0\leq\beta\leq1$ $$ER_{t}\rightarrow1-\beta\text{, as }t\searrow0\text{ (as }t\rightarrow \infty),\label{rt}$$ which is equivalent to $$\left( EX\right) ^{2}\leq ET^{2}\leq EX^{2}.\label{vE}$$ It turns out that the value of $0\leq\beta\leq1$ determines the asymptotic distribution of $T_{t}$ as $t\searrow0$ (as $t\rightarrow\infty)$ and the behavior of the Lévy function $\overline{\Lambda}$ near zero (at infinity). For instance, when $\beta=1$, $Var\left( T_{t}\right) \rightarrow0 $, which implies that $$T_{t}\overset{\mathrm{P}}{\longrightarrow}EX\text{, as }t\searrow0\text{ (as }t\rightarrow\infty).\label{beta1}$$ In general we have the following result, which in combination with Lemmas \[cor4\], \[sv\] and \[beta11\] will complete the proof of Theorem \[Th2\]. \[prop-rt\] If (\[rt\]) holds for some $0\leq\beta\leq1$, then $\overline{\Lambda}$ is regularly varying at zero (infinity) with index $-\beta$. (In the case $t\searrow0$ we assume $\overline{\Lambda}(0+)=\infty$.) **Proof.** Recall the definition of $N(t)$ in (\[Poisson-proc\]) and notice that by (\[RT\]) for any $t>0$ we can write $$R_{t}=\frac{\int_{0}^{\infty}\varphi^{2}\left( s\right) N(\mathrm{d} ts)}{\left( \int_{0}^{\infty}\varphi\left( s\right) N(\mathrm{d} ts)\right) ^{2}}.$$ Define for $T>0$ its truncated version $$R_{t}(T)=\frac{\int_{0}^{T}\varphi^{2}\left( s\right) N( \mathrm{d} ts)}{\left( \int_{0}^{T}\varphi\left( s\right) N( \mathrm{d} ts)\right) ^{2}}.\label{RT1}$$ Given that $N(Tt)=n$ $$R_{t}(T)\overset{\mathrm{D}}{=}\frac{\sum_{i=1}^{n}\varphi^{2}(V_{i})}{\left( \sum_{i=1}^{n}\varphi(V_{i})\right) ^{2}},$$ where $V_{1},\ldots,V_{n}$ are i.i.d. Uniform$(0,T)$. The same computation as in Maller and Mason [@MM] gives$$ER_{t}(T)=t\int_{0}^{\infty}u\left( \int_{0}^{T}\varphi^{2}(s)e^{-u\varphi (s)}\mathrm{d}s\right) e^{-t\int_{0}^{T}(1-e^{-u\varphi(s)})\mathrm{d}s}\,\mathrm{d}u.$$ Clearly $R_{t}(T)\leq1$. Also $R_{t}(T)\overset{\mathrm{D}}{\rightarrow}R_{t} $ as $T\rightarrow\infty$ and thus $$ER_{t}(T)\rightarrow ER_{t},\text{ as }T\rightarrow\infty.\label{conv}$$ For each $T>0$ and $u>0$, set$$\Phi_{T}\left( u\right) =\int_{0}^{T}(1-e^{-u\varphi(s)})\mathrm{d}s\text{, }\Phi\left( u\right) =\int_{0}^{\infty}(1-e^{-u\varphi(s)})\mathrm{d}s\text{ and}$$$$f_{T,t}\left( u\right) =-tu\Phi_{T}^{\prime\prime}\left( u\right) e^{-t\Phi_{T}\left( u\right) }=tu\left( \int_{0}^{T}\varphi^{2}(s)e^{-u\varphi(s)}\mathrm{d}s\right) e^{-t\int_{0}^{T}(1-e^{-u\varphi (s)})\mathrm{d}s}.\label{fT}$$ Also for $u>0$, set $$f_{\left( t\right) }\left( u\right) =-tu\Phi^{\prime\prime}\left( u\right) e^{-t\Phi\left( u\right) }=tu\left( \int_{0}^{\infty}\varphi ^{2}(s)e^{-u\varphi(s)}\mathrm{d}s\right) e^{-t\int_{0}^{\infty }(1-e^{-u\varphi(s)})\mathrm{d}s}.\label{def-f(u)}$$ We have in this notation, $$ER_{t}(T)=\int_{0}^{\infty}f_{T,t}\left( u\right) \mathrm{d}u.\label{ERT}$$ *Case 1: $\beta\in\lbrack0,1)$*. In this case we must first show that as $T\rightarrow\infty$ $$ER_{t}(T)=\int_{0}^{\infty}f_{T,t}\left( u\right) \mathrm{d}u\rightarrow \int_{0}^{\infty}f_{\left( t\right) }\left( u\right) \mathrm{d}u,\label{PL}$$ which by (\[conv\]) implies $$\int_{0}^{\infty}f_{\left( t\right) }\left( u\right) \mathrm{d}u=ER_{t}.\label{int1}$$ It turns out to be surprisingly tricky to justify the passing-to-the-limit in (\[PL\]). Lemma \[lemma-rt-inf\] and Proposition \[P1\] in the Appendix handle this problem, and imply that expression (\[int1\]) is valid for $ER_{t}$. After this identity is established, the proof is completed by an easy modification of that of Proposition 5.2 in [@MM], which is based on Tauberian theorems. Therefore we omit it. *Case 2: $\beta=1$*. In this case, it is not necessary to verify (\[PL\]). Note that we have that by (\[rt\]) with $\beta=1$ $$ER_{t}\rightarrow0\text{, as }t\searrow0\text{ }(t\rightarrow\infty).$$ Therefore since $$ER_{t}(T)\rightarrow ER_{t}\geq\int_{0}^{\infty}f_{\left( t\right) }\left( u\right) \mathrm{d}u,$$ we can conclude that as $t\searrow0$ ($t\rightarrow\infty$), $$-t\int_{0}^{\infty}u\Phi^{\prime\prime}(u)e^{-t\Phi(u)}\mathrm{d}u=\int _{0}^{\infty}f_{\left( t\right) }\left( u\right) \mathrm{d}u\rightarrow 0,\label{z}$$ which is all we need for the following argument to work for $\beta=1.$ Applying Lemma \[int-trfo\], we get $$\Phi(u)=\int_{0}^{\infty}(1-e^{-ux})\Lambda\left( \mathrm{d}x\right) ,$$ which by integrating by parts and using (\[VV\]) is equal to $$\Phi(u)=u\int_{0}^{\infty}\overline{\Lambda}(y)e^{-uy}\mathrm{d}y.$$ Let $q(y)$ denote the inverse function of $\Phi$. From the expression for $f_{\left( t\right) }\left( u\right) $ in (\[def-f(u)\]) and (\[z\]) we obtain $$t^{-1}\int_{0}^{\infty}f_{\left( t\right) }\left( u\right) \mathrm{d}u=-\int_{0}^{\infty}e^{-ty}q(y)\Phi^{\prime\prime}(q(y))q(\mathrm{d}y)\sim o\left( t^{-1}\right) ,$$ as $t\rightarrow0$ ($t\rightarrow\infty$). Using Theorem 1.7.1 (Theorem 1.7.1’) in Bingham et al [@bgt] we obtain $$-\int_{0}^{x}q(y)\Phi^{\prime\prime}(q(y))q(\mathrm{d}y)\sim o\left( x\right) ,$$ as $x\rightarrow\infty$ ($x\rightarrow0$). Changing the variables and putting $x=\Phi(v)$ we have $$-\int_{0}^{v}u\Phi^{\prime\prime}(u)\mathrm{d}u=o\left( \Phi(v)\right) ,$$ as $v\rightarrow\infty$ ($v\rightarrow0$). Integrating by parts we get $$-\int_{0}^{v}u\Phi^{\prime\prime}(u)\mathrm{d}u=-v\Phi^{\prime}(v)+\Phi (v)=o\left( \Phi(v)\right) ,$$ which gives $$\frac{v\Phi^{\prime}(v)}{\Phi(v)}\rightarrow1,$$ as $v\rightarrow\infty$ ($v\rightarrow0$). This last limit readily implies that $$v^{-1}\Phi(v)=\int_{0}^{\infty}\overline{\Lambda}(y)e^{-vy}\mathrm{d}y$$ is slowly varying at infinity (zero). By Theorem 1.7.1’ (Theorem 1.7.1) in [@bgt] we obtain that $\int_{0}^{x}\overline{\Lambda}(y)\mathrm{d}y$ is slowly varying at zero (infinity), which by Theorem 1.7.2.b (Theorem 1.7.2) in [@bgt] implies that $\overline{\Lambda}$ is regularly varying at zero with index $-1$ (at infinity). $\Box$ Proof of Theorem \[subseq\] --------------------------- Before we proceed with the proofs it will be helpful to first cite some results from Maller and Mason [@MM2], [@MM3] and [@MM4]. Let $Y_{t}$ be a Lévy process with Lévy triplet $\ (\sigma^{2},\gamma,\nu)$, i.e. $Y_{1}$ has $\mathrm{id}(\sigma^{2},\gamma,\nu)$ distribution$.$ Theorem 1 in Maller and Mason [@MM2] states $Y_{t}$ belongs to the Feller class at infinity, if and only if $$\limsup_{x\rightarrow\infty}\frac{x^{2}\nu\{(-\infty,-x)\cup(x,\infty )\}}{\sigma^{2}+\int_{|y|\leq x}y^{2}\nu(\mathrm{d}y)}<\infty,\label{fcc}$$ and furthermore $Y_{t}$ belongs to the *centered Feller class* at infinity if and only if $$\limsup_{x\rightarrow\infty}\frac{x^{2}\nu\{(-\infty,-x)\cup(x,\infty )\}+x\left\vert \gamma+\int_{1<|y|\leq x}y\nu(\mathrm{d}y)\right\vert }{\sigma^{2}+\int_{|y|\leq x}y^{2}\nu(\mathrm{d}y)}<\infty.\label{cfc}$$ For the corresponding equivalences of *Feller class* at zero and *centered Feller class* at zero replace $x\rightarrow\infty$ by $x\searrow0$, respectively; see Theorems 2.1 and 2.3 in [@MM3]. It turns out by using the assumption that $V_{t}$ is a subordinator and by arguing as in the proof of Propositions \[prop1-repr\] or of Proposition 5.1 in [@MM] we get that $$\sqrt{R_{t}^{-1}}=\frac{\sum_{i=1}^{\infty}\varphi\left( \frac{S_{i}}{t}\right) }{\sqrt{\sum_{i=1}^{\infty}\varphi^{2}\left( \frac{S_{i}}{t}\right) }}\overset{\mathrm{D}}{=}\frac{V_{t}}{\sqrt{\sum_{0\leq s\leq t}\left( \Delta V_{t}\right) ^{2}}}.$$ From this distributional equality one can conclude that $\sqrt{R_{t}^{-1}}$ is stochastically bounded as $t\searrow0$ $\left( t\rightarrow\infty\right) $ if and only if $$\limsup_{t\searrow0\text{ }\left( t\rightarrow\infty\right) }\frac{t\int _{0}^{t}x\Lambda(\mathrm{d}x)}{\int_{0}^{t}x^{2}\Lambda(\mathrm{d}x)+t^{2}\overline{\Lambda}(t)}<\infty.\label{sc-condition}$$ by applying Theorem 3.1 in [@MM4] in the case $t\rightarrow\infty$, and Proposition 5.1 in [@MM3] (with $a(t)\equiv0$ there, and a small modification) when $t\searrow0$. The partial sum version of this result was proved by Griffin [@griff]. **Proof of Proposition \[prop-fc0\].** We first assume $X$ is nondegenerate and $EX=0$, which implies that there is an $a\geq1$ such that $$F\left( a\right) -F\left( 0\right) >0\text{ and }F\left( 0\right) -F\left( -a\right) >0.\label{aa}$$ We need the following lemma. Whenever (\[ss1\]) holds and $X$ is nondegenerate and $EX=0$, there exist $0<\kappa<1$ and $d>0$ such that with $a\geq1$ as in (\[aa\]), if $2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \geq1$, then $$\mathfrak{Re}\left\{ \int_{\left( 0,\infty\right) }\int_{\mathbb{R}}\left( e^{\mathrm{i}(\theta_{1}x+\theta_{2}v)}-1\right) F\left( \frac{\mathrm{d}x}{v}\right) \Lambda(\mathrm{d}v)\right\} \leq-d\left( \left\vert \theta_{1}\right\vert ^{\kappa}+\left\vert \theta_{2}\right\vert ^{\kappa }\right) .\label{ss3}$$ **Proof.** Notice that $$\mathfrak{Re}\int_{\left( 0,\infty\right) }\int_{\mathbb{R}}\left( e^{\mathrm{i}(\theta_{1}x+\theta_{2}v)}-1\right) F\left( \frac{\mathrm{d}x}{v}\right) \Lambda(\mathrm{d}v)=\int_{\left( 0,\infty\right) }\int_{\mathbb{R}}\left( \cos(\theta_{1}x+\theta_{2}v)-1\right) F\left( \frac{\mathrm{d}x}{v}\right) \Lambda(\mathrm{d}v)$$$$\leq\int_{0\leq v\leq1/\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) }\int_{\left\vert x\right\vert \leq va}\left( \cos(\theta_{1}x+\theta_{2}v)-1\right) F\left( \frac{\mathrm{d}x}{v}\right) \Lambda(\mathrm{d}v).$$ Observe that whenever $\left\vert x\right\vert \leq av$ with $a\geq1$ and $0\leq v\leq1/\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) $, $$\left\vert \theta_{1}x\right\vert +\left\vert \theta_{2}v\right\vert \leq\left( \left\vert a\theta_{1}\right\vert +\left\vert \theta _{2}\right\vert \right) v\leq1.$$ For some $c>0$, $$\sup_{0\leq\left\vert u\right\vert \leq1}\frac{\cos u-1}{u^{2}}\leq-c,$$ thus $$\int_{0\leq v\leq1/\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) }\int_{\left\vert x\right\vert \leq va}\left( \cos(\theta_{1}x+\theta_{2}v)-1\right) F\left( \frac{\mathrm{d}x}{v}\right) \Lambda(\mathrm{d}v)$$$$\leq-c\int_{0\leq v\leq1/\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) }\int_{\left\vert x\right\vert \leq av}\left( \theta_{1}x+\theta_{2}v\right) ^{2}F\left( \frac{\mathrm{d}x}{v}\right) \Lambda(\mathrm{d}v).$$ Now when $\theta_{1}\theta_{2}\geq0$ we have $\theta_{1}\theta_{2}\int_{0\leq x\leq va}xF\left( \frac{\mathrm{d}x}{v}\right) \geq0$, and we get that the last bound is $$\leq-c\int_{0\leq v\leq1/\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) }\int_{0\leq x\leq av}\left( \theta_{1}^{2}x^{2}+\theta_{2}^{2}v^{2}\right) F\left( \frac{\mathrm{d}x}{v}\right) \Lambda(\mathrm{d}v),$$ and when $\theta_{1}\theta_{2}<0$ we have $\theta_{1}\theta_{2}\int_{-va\leq x\leq0}xF\left( \frac{\mathrm{d}x}{v}\right) \geq0$, which gives$$\int_{0\leq v\leq1/\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) }\int_{\left\vert x\right\vert \leq va}\left( \cos(\theta_{1}x+\theta_{2}v)-1\right) F\left( \frac{\mathrm{d}x}{v}\right) \Lambda(\mathrm{d}v)$$$$\leq-c\int_{0\leq v\leq1/\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) }\int_{-av\leq x\leq 0}\left( \theta_{1}^{2}x^{2}+\theta_{2}^{2}v^{2}\right) F\left( \frac{\mathrm{d}x}{v}\right) \Lambda(\mathrm{d}v).$$ Notice that $$c\int_{0\leq v\leq1/\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) }\int_{0\leq x\leq av}\theta_{2}^{2}v^{2}F\left( \frac{\mathrm{d}x}{v}\right) \Lambda (\mathrm{d}v)$$$$=c\left( F\left( a\right) -F\left( 0\right) \right) \int_{0\leq v\leq1/\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) }\theta_{2}^{2}v^{2}\Lambda (\mathrm{d}v).$$ We get by arguing as on the top of page 968 in Pruitt [@pruitt] or in the remark after the proof of Proposition 6.1 in Buchmann, Maller and Mason [@BMM], that for some $c_{1}>0$ and $0<\kappa<1$, that whenever $2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \geq1$$$-c\left( F\left( a\right) -F\left( 0\right) \right) \theta_{2}^{2}\int_{0\leq v\leq1/\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) }v^{2}\Lambda (\mathrm{d}v)\leq-\frac{c_{1}\theta_{2}^{2}}{4a^{2}\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) ^{2}}\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta _{2}\right\vert \right) \right) ^{\kappa}.$$ Next, $$-c\int_{0\leq v\leq1/\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) }\int_{0\leq x\leq av}\theta_{1}^{2}x^{2}F\left( \frac{\mathrm{d}x}{v}\right) \Lambda (\mathrm{d}v)$$$$=-c\theta_{1}^{2}\int_{0\leq x\leq a}u^{2}F\left( \mathrm{d}u\right) \int_{0\leq v\leq1/\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) }v^{2}\Lambda (\mathrm{d}v),$$ which by the previous argument is for some $c_{2}>0$, for $2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \geq1$$$\leq-\frac{c_{2}\theta_{1}^{2}}{\left( 2a\left( \left\vert \theta _{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) ^{2}}\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta _{2}\right\vert \right) \right) ^{\kappa}.$$ Thus with $c_{3}=c_{1}\wedge c_{2}$, $$-c\int_{0\leq v\leq1/\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) }\int_{0\leq x\leq av}(\theta_{1}^{2}x^{2}+\theta_{2}^{2}v^{2})F\left( \frac{\mathrm{d}x}{v}\right) \Lambda(\mathrm{d}v)$$$$\leq-c_{3}\left( \frac{\theta_{1}^{2}}{4a^{2}\left( \left\vert \theta _{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) ^{2}}+\frac{\theta_{2}^{2}}{4a^{2}\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) ^{2}}\right) \left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) ^{\kappa}.$$ Notice that$$\frac{\theta_{1}^{2}}{4a^{2}\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) ^{2}}+\frac{\theta_{2}^{2}}{4a^{2}\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta _{2}\right\vert \right) ^{2}}\geq\frac{1}{4a^{2}}.$$ Hence when $\theta_{1}\theta_{2}>0$ and $2a\left( \left\vert \theta _{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \geq1$ for some $c_{4}>0$, $$-c\int_{0\leq v\leq1/\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) }\int_{0\leq x\leq av}(\theta_{1}^{2}x^{2}+\theta_{2}^{2}v^{2})F\left( \frac{\mathrm{d}x}{v}\right) \Lambda(\mathrm{d}v)\leq-c_{4}\left( \left\vert \theta _{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) ^{\kappa }.\label{e1}$$ The analogous inequality holds when $\theta_{1}\theta_{2}\leq0$ and $2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \geq1$, namely for some $c_{5}>0,$$$\int_{0\leq v\leq1/\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) }\int_{\left\vert x\right\vert \leq va}\left( \cos(\theta_{1}x+\theta_{2}v)-1\right) F\left( \frac{\mathrm{d}x}{v}\right) \Lambda(\mathrm{d}v)$$ $$\leq-c\int_{0\leq v\leq1/\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) }\int_{-av\leq x\leq 0}\left( \theta_{1}^{2}x^{2}+\theta_{2}^{2}v^{2}\right) F\left( \frac{\mathrm{d}x}{v}\right) \Lambda(\mathrm{d}v)\leq-c_{5}\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) ^{\kappa }.\label{e2}$$ Note that since $0<\kappa<1$ the function $\rho\left( u\right) =\left\vert u\right\vert ^{\kappa}$ is concave on $\left( 0,\infty\right) $, and thus $$\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) ^{\kappa}\geq\left\vert \frac{\left\vert \theta_{1}\right\vert +\left\vert \theta_{2}\right\vert }{2}\right\vert ^{\kappa}\geq\frac {\left\vert \theta_{1}\right\vert ^{\kappa}+\left\vert \theta_{2}\right\vert ^{\kappa}}{2},$$ which, in combination with (\[e1\]) and (\[e2\]), gives for some $d>0,$ whenever $2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta _{2}\right\vert \right) \geq1,$ $$\int_{0\leq v\leq1/\left( 2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \right) }\int_{\left\vert x\right\vert \leq va}\left( \cos(\theta_{1}x+\theta_{2}v)-1\right) F\left( \frac{\mathrm{d}x}{v}\right) \Lambda(\mathrm{d}v)\leq-d\left( \left\vert \theta_{1}\right\vert ^{\kappa}+\left\vert \theta_{2}\right\vert ^{\kappa }\right) .$$ $\Box$ The lemma implies that whenever $2a\left( \left\vert \theta_{1}\right\vert \vee\left\vert \theta_{2}\right\vert \right) \geq1$, then for some $d>0$ and $0<\kappa<1,$$$\left\vert Ee^{\mathrm{i}(\theta_{1}U+\theta_{2}V)}\right\vert \leq\exp\left( -d\left( \left\vert \theta_{1}\right\vert ^{\kappa}+\left\vert \theta _{2}\right\vert ^{\kappa}\right) \right) .$$ As in [@pruitt] this allows us to apply the inversion formula for densities and shows that it may be repeatedly differentiated, from which we readily infer that $\left( U,V\right) $ has a $C^{\infty}$ density when $EX=0.$ If $EX=\mu\neq0$, the same argument applied to $(U^{\prime},V)=(U-\mu V,V)$ shows that $(U^{\prime},V)$ has a $C^{\infty}$ density, which by a simple transformation implies that $\left( U,V\right) $ does too. $\Box$ **Proof of Corollary \[cor1\].** Note that each $V_{t_{k}}/B(t_{k})$ is an infinitely divisible random variable without a normal component with Lévy measure concentrated on $\left( 0,\infty\right) $ given by $t_{k}\Lambda\left( \cdot B(t_{k})\right) $ with characteristic function $$\Psi_{k}\left( \theta\right) =\exp\left\{ \mathrm{i}\theta b_{k}+\int _{0}^{\infty}\left( e^{\mathrm{i}\theta x}-1-\mathrm{i}\theta x\mathbf{1}_{\{0<x\leq1\}}\right) t_{k}\Lambda(B(t_{k})\mathrm{d}x)\right\} ,$$ where $$b_{k}=\int_{0}^{1}xt_{k}\Lambda(B(t_{k})\mathrm{d}x).$$ Since $V_{t_{k}}/B(t_{k})\overset{\mathrm{D}}{\rightarrow}W_{2},$ by Proposition 7.8 of Sato [@sato], $W_{2}$ is infinitely divisible. Since $W_{2}$ is necessarily non-negative, it does not have a normal component and has a Lévy measure $\Lambda_{0}$ concentrated on $\left( 0,\infty\right) $. Now by Theorem \[2-dim-conv\] and its proof, necessarily $\int_{0}^{1}x\Lambda_{0}(\mathrm{d}x)<\infty$ and $W_{2}$ has characteristic function $$\Psi_{0}\left( \theta\right) =\exp\left\{ \mathrm{i}\theta b+\int _{0}^{\infty}\left( e^{\mathrm{i}\theta x}-1\right) \Lambda_{0}(\mathrm{d}x)\right\} ,$$ where $b\geq0$. By (\[Levy-conv-V\]) and (\[1stmoment-V\]) in the proof of Theorem \[2-dim-conv\] for any continuity point $v>0$ of $\overline{\Lambda }_{0}$, $$t_{k}\overline{\Lambda}(vB(t_{k}))\rightarrow\overline{\Lambda}_{0}(v),\quad\text{as }k\rightarrow\infty,\label{v1}$$ and $$\int_{0}^{v}xt_{k}\Lambda(B(t_{k})\mathrm{d}x)\rightarrow\int_{0}^{v}x\Lambda_{0}(\mathrm{d}x)+b,\quad\text{as }k\rightarrow\infty.\label{v2}$$ From (\[v2\]) we easily get that for any continuity point $v>0$ of $\overline{\Lambda}_{0}$,$$\int_{0}^{v}x^{2}t_{k}\Lambda(B(t_{k})\mathrm{d}x)\rightarrow\int_{0}^{v}x^{2}\Lambda_{0}(\mathrm{d}x)=V_{0,2}\left( v\right) ,\quad\text{as }k\rightarrow\infty.\label{v3}$$ (Recall the notation (\[V2\]).) Now, since $V_{t}$ is in the centered Feller class, (\[cfc\]) implies that for some $K>0$ $$\limsup_{k\rightarrow\infty}\frac{v^{2}B^{2}(t_{k})\overline{\Lambda}\left( vB(t_{k})\right) }{V_{2}\left( vB(t_{k})\right) }\leq K\text{.}\label{K}$$ Note $$\frac{v^{2}B^{2}(t_{k})\overline{\Lambda}\left( vB(t_{k})\right) }{V_{2}\left( vB(t_{k})\right) }=\frac{v^{2}t_{k}\overline{\Lambda}(vB(t_{k}))}{\int_{0}^{v}x^{2}t_{k}\Lambda(B(t_{k})\mathrm{d}x)},$$ which by (\[v1\]) and (\[v3\]) converges to $v^{2}\overline{\Lambda}_{0}(v)/V_{0,2}\left( v\right) $ for each continuity point $v>0$ of $\overline{\Lambda}_{0}$. This with (\[K\]) implies that $$\sup_{v>0}\frac{v^{2}\overline{\Lambda}_{0}(v)}{\int_{0}^{v}x^{2}\Lambda _{0}(\mathrm{d}x)}\leq K,$$ so Proposition \[prop-fc0\] applies. $\Box$ **Proof of Proposition \[sums-th4\].** The proof is a simple adaptation of the proof of Theorem 4 in [@KM], so we only sketch it here. Putting $$B_{t}(k)=\left\{ \frac{\left\vert \sum_{i=2}^{\infty}X_{i}\varphi (S_{i}/t)\right\vert }{\sum_{i=1}^{\infty}\varphi(S_{i}/t)}\leq\frac {E|X|}{\sqrt{k}}\right\} ,$$ and recalling definition (\[AAa\]), the conditional version of Chebyshev’s inequality implies that $P\{B_{t}(k)|A_{t}(k^{-1})\}\geq1-1/\sqrt{k}$. Noticing that on the set $B_{t}(k)\cap A_{t}(k^{-1})$ $$\Delta_{t}\leq\frac{|X_{1}|}{k}+\frac{E|X|}{\sqrt{k}},$$ a tightness argument finishes the proof. $\Box$ Now we are ready to prove Theorem \[subseq\]. Choose any cdf $F$ in the class $\mathcal{X}$. Corollary \[cor1\] says whenever $V_{t}$ is in the centered Feller class at 0 $(\infty)$ then every subsequential limit law of $U_{t}/V_{t}$, as $t\searrow0$, (as $t\rightarrow \infty$) has a Lebesgue density on $\mathbb{R}$ and hence is continuous. Suppose $V_{t}$ is in the Feller class at 0 $(\infty),$ but not in the centered Feller class at 0 $(\infty)$. In this case Corollary \[cor2\] implies that one of the subsequential limits is the constant $EX$ and thus not continuous. Next Proposition 5.5 in [@MM3] in the case $t\searrow0$ and Proposition 3.2 in [@MM4] in the case $t\rightarrow\infty$ show that whenever $V_{t}$ is not in the Feller class at 0 ($\infty$), that is $$\limsup_{t\searrow0\text{ }\left( t\rightarrow\infty\right) }\frac {t^{2}\overline{\Lambda}(t)}{\int_{0}^{t}y^{2}\Lambda(\mathrm{d}y)}=\infty,$$ and (\[sc-condition\]) holds, then there exist a subsequence $t_{k}\searrow0$ ($t_{k}\rightarrow\infty$), such that (\[A-assump\]) holds, which by Corollary \[cor3\] for any $X$ such that $P\{X=x_{0}\}>0$ for some $x_{0}$, there exists a subsequence $t_{k}\searrow0$ $(t_{k}\rightarrow \infty)$ such that $U_{t_{k}}/V_{t_{k}}\overset{\mathrm{D}}{\longrightarrow}T$, with $P\{T=x_{0}\}>0$, that is, $T$ is not continuous. Finally, assume that (\[sc-condition\]) does not hold, then by Proposition \[non-feller\] there exists a subsequence $t_{k}\searrow0$ or $t_{k}\rightarrow\infty$ such that $U_{t_{k}}/V_{t_{k}}\overset{\mathrm{D}}{\longrightarrow}T,$ with $P\{T=EX\}>0$, and again $T$ is not continuous. This completes the proof of Theorem \[subseq\]. Appendix ======== To finish the proofs of Proposition \[prop-rt\] and thus Theorem \[Th2\] we shall require the following technical result. \[P1\] Assume that $$\liminf_{s\searrow0} \frac{s \overline\Lambda(s) }{ \int_{0}^{s} \overline\Lambda( x ) \mathrm{d}x } >0 \text{,}\label{inf}$$ then $$ER_{t}=\int_{0}^{\infty}f_{\left( t\right) }\left( u\right) \mathrm{d}u=-t\int_{0}^{\infty}u\Phi^{\prime\prime}\left( u\right) e^{-t\Phi\left( u\right) }\mathrm{d}u.\label{int}$$ **Proof.** Clearly for each $u>0$, $f_{T,t}\left( u\right) \rightarrow f_{\left( t\right) }\left( u\right) $, as $T\rightarrow\infty $. Therefore by Fatou’s lemma $$\int_{0}^{\infty}f_{\left( t\right) }\left( u\right) \mathrm{d}u\leq\liminf_{T\rightarrow\infty}\int_{0}^{\infty}f_{T,t}\left( u\right) \mathrm{d}u=\liminf_{T\rightarrow\infty}ER_{t}\left( T\right) \leq 1.\label{Tone}$$ Keeping in mind (\[conv\]) and (\[ERT\]), this implies that $$\int_{0}^{\infty}f_{\left( t\right) }\left( u\right) \mathrm{d}u\leq ER_{t}\leq1.$$ Therefore on account of (\[conv\]) to prove (\[int\]) it suffices to establish (\[PL\]), as $T\rightarrow\infty.$ One can readily check using (\[delta\]) that for some constants $C_{1}>0$ and $C_{2}>0$ and all $u>0$ $$0\leq-tu\Phi^{\prime\prime}(u)\leq t\left( C_{1}+u^{-1}C_{2}\right) .$$ To see this note that for each $u>0$$$\begin{aligned} -u\Phi^{\prime\prime}(u) & =u\int_{0}^{\infty}x^{2}e^{-ux}\Lambda (\mathrm{d}x)\\ & =\int_{0}^{1}x^{2}ue^{-ux}\Lambda(\mathrm{d}x)+u^{-1}\int_{1}^{\infty}u^{2}x^{2}e^{-ux}\Lambda(\mathrm{d}x),\\ & \leq\max_{0\leq y}ye^{-y}\int_{0}^{1}x\Lambda(\mathrm{d}x)+u^{-1}\overline{\Lambda}\left( 1\right) \max_{0\leq y}y^{2}e^{-y}=:C_{1}+u^{-1}C_{2}\text{.}$$ Thus since $$f_{T,t}\left( u\right) \leq-ut\Phi_{T}^{\prime\prime}(u)\leq-ut\Phi ^{\prime\prime}(u),$$ we get by the bounded convergence theorem that for all $D>\delta>0$$$\lim_{T\rightarrow\infty}\int_{\delta}^{D}f_{T,t}\left( u\right) \mathrm{d}u=\int_{\delta}^{D}f_{\left( t\right) }\left( u\right) \mathrm{d}u.$$ Notice that since $$\int_{0}^{\infty}f_{\left( t\right) }\left( u\right) \mathrm{d}u\leq1,$$ we have$$\lim_{\delta\rightarrow0}\int_{0}^{\delta}f_{\left( t\right) }\left( u\right) \mathrm{d}u=0\text{ and }\lim_{D\rightarrow\infty}\int_{D}^{\infty }f_{\left( t\right) }(u)\mathrm{d}u=0.$$ We see now that to complete the verification of (\[PL\]) we have to show that $$\lim_{\delta\rightarrow0}\limsup_{T\rightarrow\infty}\int_{0}^{\delta}f_{T,t}\left( u\right) \mathrm{d}u=0\label{delta0}$$ and $$\lim_{D\rightarrow\infty}\limsup_{T\rightarrow\infty}\int_{D}^{\infty}f_{T,t}(u)\mathrm{d}u=0.\label{Dinfty}$$ The first condition (\[delta0\]) is easy to show. Recalling (\[fT\]), notice that $$f_{T,t}(u)\leq tu\int_{0}^{\infty}\varphi^{2}(s)e^{-u\varphi(s)}\mathrm{d}s,$$ and so by Fubini $$\begin{aligned} \int_{0}^{\delta}f_{T,t}(u)\mathrm{d}u & \leq t\int_{0}^{\infty}\varphi ^{2}(s)\mathrm{d}s\int_{0}^{\delta}ue^{-u\varphi(s)}\mathrm{d}u\\ & =t\int_{0}^{\infty}\left[ -\varphi(s)\delta e^{-\delta\varphi (s)}+(1-e^{-\delta\varphi(s)})\right] \mathrm{d}s\\ & =t\left( \Phi(\delta)-\delta\Phi^{\prime}(\delta)\right) \leq t\Phi(\delta),\end{aligned}$$ which goes to $0$ as $\delta\rightarrow0$ and thus (\[delta0\]) holds. For the second condition (\[Dinfty\]), choose $D>0.$ We see that for all large enough $T>0$$$\int_{D}^{\infty}f_{T,t}(u)\mathrm{d}u=\int_{D}^{1/\varphi(T)}f_{T,t}(u)\mathrm{d}u+\int_{1/\varphi(T)}^{\infty}f_{T,t}(u)\mathrm{d}u.\label{qq}$$ Recall that $$f_{T,t}(u)=tu\int_{0}^{T}\varphi^{2}(s)e^{-u\varphi(s)}\mathrm{d}s\exp\left\{ -t\int_{0}^{T}\left( 1-e^{-u\varphi(s)}\right) \mathrm{d}s\right\} .\label{f2}$$ We shall first bound the second integral on the right side of (\[qq\]). For $u\varphi(T)\geq1$ and keeping mind that $\varphi(s)\geq\varphi(T)$ for $0<s\leq T$, we have $$\exp\left\{ -t\int_{0}^{T}\left( 1-e^{-u\varphi(s)}\right) \mathrm{d}s\right\} \leq e^{-t(1-e^{-1})T}$$ and so $$\int_{1/\varphi(T)}^{\infty}f_{T,t}(u)\mathrm{d}u\leq te^{-t(1-e^{-1})T}\int_{1/\varphi(T)}^{\infty}u\int_{0}^{T}\varphi^{2}(s)e^{-u\varphi (s)}\mathrm{d}s\mathrm{d}u.$$ Using Fubini’s theorem the last integral is easy to calculate. We get $$\begin{aligned} \int_{1/\varphi(T)}^{\infty}u\int_{0}^{T}\varphi^{2}(s)e^{-u\varphi (s)}\mathrm{d}s\mathrm{d}u & =\int_{0}^{T}\varphi^{2}(s)\mathrm{d}s\int_{1/\varphi(T)}^{\infty}ue^{-u\varphi(s)}\mathrm{d}u\\ & =\int_{0}^{T}\left( e^{-\varphi(s)/\varphi(T)}+\frac{\varphi(s)}{\varphi(T)}e^{-\varphi(s)/\varphi(T)}\right) \mathrm{d}s\\ & \leq T\left( 1+\max_{y\geq0}ye^{-y}\right) \leq2T.\end{aligned}$$ So we obtain $$\int_{1/\varphi(T)}^{\infty}f_{T,t}(u)\mathrm{d}u\leq2Tte^{-t(1-e^{-1})T},\label{D1}$$ which tends to $0$ as $T\rightarrow\infty$. Therefore to complete the verification that (\[Dinfty\]) holds and thus (\[PL\]) we must prove that $$\lim_{D\rightarrow\infty}\limsup_{T\rightarrow\infty}\int_{D}^{1/\varphi (T)}f_{T,t}(u)\mathrm{d}u=0.\label{D}$$ We shall bound $f_{T,t}\left( u\right) $ in the integral (\[D\]). Since $1/u\geq\varphi(T)$, and thus $\overline{\Lambda}\left( 1/u\right) \leq\overline{\Lambda}\left( \varphi(T)\right) \leq T$, we get that the second factor of $f_{T,t}\left( u\right) $ given in (\[f2\]) is $$\begin{split} \exp\left\{ -t\int_{0}^{T}\left( 1-e^{-u\varphi(s)}\right) \mathrm{d}s\right\} & \leq\exp\left\{ -t\int_{0}^{\overline{\Lambda}\left( 1/u\right) }\left( 1-e^{-u\varphi(s)}\right) \mathrm{d}s\right\} \\ & \leq e^{-t(1-e^{-1})\overline{\Lambda}(1/u)}. \end{split}$$ While for the first factor in $f_{T,t}(u)$ given in (\[f2\]) we use the simple bound $$tu\int_{0}^{T}\varphi^{2}(s)e^{-u\varphi(s)}\mathrm{d}s\leq tu\int_{0}^{\infty}\varphi^{2}(s)e^{-u\varphi(s)}\mathrm{d}s=:t\psi_{\Lambda}\left( u\right) .$$ We see that$$\begin{split} \int_{D}^{1/\varphi(T)}f_{T,t}(u)\mathrm{d}u & \leq t\int_{D}^{1/\varphi (T)}\psi_{\Lambda}\left( u\right) e^{-t(1-e^{-1})\overline{\Lambda}(1/u)}\mathrm{d}u\\ & \leq t\int_{D}^{\infty}\psi_{\Lambda}\left( u\right) e^{-t(1-e^{-1})\overline{\Lambda}(1/u)}\mathrm{d}u. \end{split}$$ Clearly (\[Dinfty\]) holds whenever for all $\gamma>0$, $$\int_{1}^{\infty}\psi_{\Lambda}\left( u\right) e^{-\gamma\overline{\Lambda }(1/u)}\mathrm{d}u<\infty.\label{levy-cond}$$ \[L1\] Whenever (\[inf\]) is satisfied, then for all $\gamma>0$, (\[levy-cond\]) holds. **Proof.** Recall the definition (\[def-f(u)\]). Notice that by (\[Tone\]) for all $t>0$$$\int_{0}^{\infty}f_{\left( t\right) }\left( u\right) \mathrm{d}u<\infty.\label{t}$$ Write$$\int_{0}^{\infty}(1-e^{-u\varphi(s)})\mathrm{d}s=\int_{0}^{1/u}(1-e^{-ux})\Lambda(\mathrm{d}x)+\int_{1/u}^{\infty}(1-e^{-ux})\Lambda\left( \mathrm{d}x\right) .$$ We see that$$\int_{1/u}^{\infty}(1-e^{-ux})\Lambda\left( \mathrm{d}x\right) \leq \overline{\Lambda}\left( 1/u\right)$$ and$$\begin{split} \int_{0}^{1/u}(1-e^{-ux})\Lambda\left( \mathrm{d}x\right) & =-\left( 1-e^{-1}\right) \overline{\Lambda}(1/u)+\int_{0}^{1/u}u\overline{\Lambda }\left( x\right) e^{-ux}\mathrm{d}x\\ & \leq\int_{0}^{1/u}u\overline{\Lambda}\left( x\right) e^{-ux}\mathrm{d}x\leq u\int_{0}^{1/u}\overline{\Lambda}\left( x\right) \mathrm{d}x. \end{split}$$ By assumption (\[inf\]) for some $\eta>0$ for all $u$ large$$u\int_{0}^{1/u}\overline{\Lambda}\left( x\right) \mathrm{d}x\leq \eta\overline{\Lambda}(1/u).\label{tail}$$ This implies that$$t\int_{0}^{\infty}(1-e^{-u\varphi(s)})\mathrm{d}s\leq\left( 1+\eta\right) t\overline{\Lambda}(1/u).$$ Thus for all large enough $D>0$ and all $t>0$ $$\int_{D}^{\infty}f_{\left( t\right) }\left( u\right) \mathrm{d}u\geq \int_{D}^{\infty}t\psi_{\Lambda}\left( u\right) \exp\left\{ -\left( 1+\eta\right) t\overline{\Lambda}(1/u)\right\} \mathrm{d}u,$$ and hence since (\[t\]) holds for all $t>0$, we get that for all $\gamma>0 $, (\[levy-cond\]) is satisfied. $\Box\medskip$ We see from Lemma \[L1\] that (\[levy-cond\]) holds whenever assumption (\[inf\]) is satisfied and thus by the arguments preceding the lemma the limit (\[PL\]) is valid.** **This completes the proof of Proposition \[P1\]. $\Box$ Return to the proofs of Proposition \[prop-rt\] and Theorem \[Th2\] ------------------------------------------------------------------- We shall now finish the proof of Proposition \[prop-rt\]. To do this we shall need three more lemmas. Let $X_{t}$ be a subordinator with canonical measure $\Lambda$. Assume that $X_{t}$ is without drift. Define $$I(x)=\int_{0}^{x}\overline{\Lambda}(y)\mathrm{d}y.$$ We give a criterion for subsequential relative stability of $X$ at $0$. \[th1\] Let $X$ be a driftless subordinator with $\overline{\Lambda }(0+)>0$. There are nonstochastic sequences $t_{k}\downarrow0$ and $B_{k}>0$, such that, as $k\rightarrow\infty$, $$\frac{X(t_{k})}{B_{k}}\overset{\mathrm{P}}{\longrightarrow}1\label{0.1}$$ if and only if $$\liminf_{x\downarrow0}\frac{x\overline{\Lambda}(x)}{I(x)}=0.\label{0.2}$$ **Proof.** From the convergence criteria for subordinators, e.g. part (ii) of Theorem 15.14 of [@kallenberg], p. 295, (\[0.1\])** **is equivalent to $$\lim_{t_{k}\rightarrow0}t_{k}\overline{\Lambda}(xB_{k})=0\text{ for every }x>0\text{ }\mathrm{and}\text{ }\lim_{t_{k}\rightarrow0}t_{k}\int_{0}^{1}x\Lambda(\mathrm{d}B_{k}x)=1.\label{kall}$$ Noting that $I(x)=\int_{0}^{x}y\Lambda(\mathrm{d}y)+x\overline{\Lambda}(x)$, we see that (\[kall\])** **implies $$t_{k}B_{k}^{-1}I(B_{k})=t_{k}B_{k}^{-1}\int_{0}^{B_{k}}x\Lambda(\mathrm{d}x)+t_{k}\overline{\Lambda}(B_{k})\rightarrow1\text{,}\label{0.3}$$ and clearly (\[0.3\]) and $t_{k}\overline{\Lambda}(B_{k})\rightarrow0$ imply (\[0.2\]). (Note that necessarily $B_{k}\rightarrow0$.)  Conversely, let (\[0.2\]) hold and choose a subsequence $c_{k}\rightarrow0$ as $k\rightarrow\infty$ such that $$\lim_{k\rightarrow\infty}\frac{c_{k}\overline{\Lambda}(c_{k})}{I(c_{k})}=0.$$ Define $$t_{k}:=\sqrt{\frac{c_{k}}{\overline{\Lambda}(c_{k})I(c_{k})}}.$$ Then $$\lim_{k\rightarrow\infty}t_{k}\overline{\Lambda}(c_{k})=\lim_{k\rightarrow \infty}\sqrt{\frac{c_{k}\overline{\Lambda}(c_{k})}{I(c_{k})}}=0,$$ and $$\lim_{k\rightarrow\infty}\frac{t_{k}I(c_{k})}{c_{k}}=\lim_{k\rightarrow\infty }\sqrt{\frac{I(c_{k})}{c_{k}\overline{\Lambda}(c_{k})}}=\infty.$$ Then set $B_{k}:=t_{k}I(c_{k})$, so $\lim_{k\rightarrow\infty}B_{k}=0$ and $\lim_{k\rightarrow\infty}B_{k}/c_{k}=\infty$. Given $x>0$ choose $k$ so large that $xB_{k}\geq c_{k}$. Then $$t_{k}\overline{\Lambda}(xB_{k})\leq t_{k}\overline{\Lambda}(c_{k})\rightarrow0.\label{0.4}$$ Furthermore, by writing $$\frac{t_{k}I(B_{k})}{B_{k}}=\frac{t_{k}I(c_{k})}{B_{k}}+\frac{t_{k}\left( I(B_{k})-I(c_{k})\right) }{B_{k}}=1+\frac{t_{k}\left( I(B_{k})-I(c_{k})\right) }{B_{k}}$$ and noting that for all large $k$ $$0\leq\frac{t_{k}\left( I(B_{k})-I(c_{k})\right) }{B_{k}}\leq\frac{B_{k}t_{k}\overline{\Lambda}(c_{k})}{B_{k}}\rightarrow0,$$ we also have $t_{k}B_{k}^{-1}I(B_{k})\rightarrow1$ and thus by (\[0.4\]) and the identity in (\[0.3\]) $$\lim_{t_{k}\rightarrow0}t_{k}\int_{0}^{1}x\Lambda(\mathrm{d}B_{k}x)=1$$ which in combination with (\[0.4\]) implies (\[0.1\]), by (\[kall\]). $\Box\medskip$ To continue we need the following lemma from [@MM]. \[psi-lemma\] Let $\Psi$ be a non-negative measurable real valued function defined on $(0,\infty)$ satisfying $$\int_{0}^{\infty}\Psi\left( y\right) \mathrm{d}y<\infty.$$ Then $$E \left( \sum_{i=1}^{\infty}\Psi\left( S_{i}\right) \right) =\int _{0}^{\infty}\Psi\left( y\right) \mathrm{d}y\label{ee1}$$ and $\lim_{n\rightarrow\infty}E\left( \sum_{i=n}^{\infty}\Psi\left( S_{i}\right) \right) =0.$ \[lemma-rt-inf\] (i) Assume that (\[rt\]) holds as $t \searrow0$ with $\beta< 1$. Then (\[inf\]) holds. \(ii) Assume that (\[rt\]) holds as $t \to\infty$ with $\beta< 1$. Then without loss of generality we can assume that (\[inf\]) holds. **Proof.** (i) We shall show that (\[rt\]) implies (\[inf\]). Assume on the contrary that (\[inf\]) does not hold. Then, since $V_{t}$ is a driftless subordinator by Lemma \[th1\] for some sequences $B_{k}>0$, $t_{k}\downarrow0$, $V_{t_{k}}/B_{k}\overset{\mathrm{P}}{\rightarrow}1.$ By Proposition \[prop1-repr\] the infinite sum $\sum _{i=1}^{\infty}\varphi\left( \frac{S_{i}}{t}\right) $ is equal in distribution to $V_{t}$ and $\sum_{i=1}^{\infty}\varphi^{2}\left( \frac {S_{i}}{t}\right) $ is equal in distribution to the subordinator $W_{t}$ with Lévy measure $\Lambda_{2}$ on $\left( 0,\infty\right) $ defined by $$\overline{\Lambda}_{2}\left( x\right) =\overline{\Lambda}\left( \sqrt {x}\right) .$$ From (\[kall\]) in the proof of Lemma \[th1\] above $$t_{k}\overline{\Lambda}\left( xB_{k}\right) \rightarrow0\text{ and }\int _{0}^{1}t_{k}\overline{\Lambda}\left( xB_{k}\right) \mathrm{d}x\rightarrow1,\label{twoL}$$ with $t_{k}\rightarrow0$ and $B_{k}\rightarrow0$. Thus we easily see that $$t_{k}\overline{\Lambda}_{2}\left( xB_{k}^{2}\right) =t_{k}\overline{\Lambda }\left( \sqrt{x}B_{k}\right) \rightarrow0$$ and $$\int_{0}^{1}t_{k}\overline{\Lambda}_{2}\left( xB_{k}^{2}\right) \mathrm{d}x=\int_{0}^{1}t_{k}\overline{\Lambda}\left( \sqrt{x}B_{k}\right) \mathrm{d}x=2\int_{0}^{1}yt_{k}\overline{\Lambda}\left( yB_{k}\right) \mathrm{d}y,$$ which for any $0<\delta<1$ is $$\leq2\delta\int_{0}^{1}t_{k}\overline{\Lambda}\left( xB_{k}\right) \mathrm{d}x+2\int_{\delta}^{1}t_{k}\overline{\Lambda}\left( xB_{k}\right) \mathrm{d}x.$$ Clearly by (\[twoL\]) $$\limsup_{k\rightarrow\infty}\left( 2\delta\int_{0}^{1}t_{k}\overline{\Lambda }\left( xB_{k}\right) \mathrm{d}x+2\int_{\delta}^{1}t_{k}\overline{\Lambda }\left( xB_{k}\right) \mathrm{d}x\right) =2\delta.$$ Thus since $0<\delta<1$ can be made arbitrarily small we get $$\lim_{k\rightarrow\infty}\int_{0}^{1}t_{k}\overline{\Lambda}_{2}\left( xB_{k}^{2}\right) \mathrm{d}x=0.$$ Hence applying Theorem 15.14 on page 295 of [@kallenberg], we get $W_{t_{k}}/B_{k}^{2}\overset{\mathrm{P}}{\rightarrow}0$ and thus $$R_{t_{k}}\overset{\mathrm{D}}{=}W_{t_{k}}/\left( V_{t_{k}}\right) ^{2}\overset{\mathrm{P}}{\rightarrow}0,$$ which since $R_{t_{k}}\leq1$ implies $ER_{t_{k}}\rightarrow0$, as $t_{k}\downarrow0$, which clearly contradicts to (\[rt\]). So we have (\[inf\]) in this case. \(ii) We shall first assume that $$\int_{0}^{\infty}\varphi\left( u\right) \mathrm{d}u=\infty,\label{infin}$$ which by (\[delta\]) implies $$\int_{0}^{1}\varphi\left( u\right) \mathrm{d}u=\infty.\label{phh2}$$ Set$$V\left( t\right) :=\sum_{i=1}^{\infty}\varphi\left( \frac{S_{i}}{t}\right) \mathbf{1}\left\{ \frac{S_{i}}{t}\leq1\right\} \text{ and }\overline {V}\left( t\right) :=\sum_{i=1}^{\infty}\varphi\left( \frac{S_{i}}{t}\right) \mathbf{1}\left\{ \frac{S_{i}}{t}>1\right\} .$$ We see that$$V\left( t\right) \geq\sum_{k=1}^{\infty}\varphi\left( 2^{-k+1}\right) \sum_{i=1}^{\infty}\mathbf{1}\left\{ 2^{-k}<\frac{S_{i}}{t}\leq 2^{-k+1}\right\} .$$ Now for each fixed $L\geq1,$ as $t\rightarrow\infty$, $$t^{-1}\sum_{k=2}^{L+1}\left( \varphi\left( 2^{-k+1}\right) \sum _{i=1}^{\infty}\mathbf{1}\left\{ 2^{-k}<\frac{S_{i}}{t}\leq2^{-k+1}\right\} \right) \overset{\mathrm{P}}{\rightarrow}\sum_{k=1}^{L}\varphi\left( 2^{-k}\right) 2^{-k-1}\geq2^{-1}\int_{2^{-L}}^{1}\varphi(u)\mathrm{d}u.$$ Thus since $L\geq1$ can be made arbitrarily large, on account of (\[phh2\]), $$t^{-1}V\left( t\right) \overset{\mathrm{P}}{\rightarrow}\infty,\text{ as }t\rightarrow\infty\text{.}\label{L11}$$ Next, using (\[ee1\]), we get $$t^{-1}E\overline{V}\left( t\right) =t^{-1}\int_{t}^{\infty}\varphi (y/t)\mathrm{d}y=\int_{1}^{\infty}\varphi(u)\mathrm{d}u<\infty\text{,}$$ which implies that $$t^{-1}\overline{V}\left( t\right) =O_{P}\left( 1\right) ,\text{ as }t\rightarrow\infty.\label{L22}$$ Hence by (\[L11\]) and (\[L22\]) $$\overline{V}\left( t\right) /V_{t}\overset{\mathrm{P}}{\rightarrow}0,\text{ as }t\rightarrow\infty.\label{L33}$$ We get then that $$V_{t}=\sum_{i=1}^{\infty}\varphi\left( \frac{S_{i}}{t}\right) =V\left( t\right) \left( 1+o\left( 1\right) \right) ,\text{ as }t\rightarrow \infty\text{.}\label{vt}$$ Now set$$W_{t}:=\sum_{i=1}^{\infty}\varphi^{2}\left( \frac{S_{i}}{t}\right) \text{, }W\left( t\right) :=\sum_{i=1}^{\infty}\varphi^{2}\left( \frac{S_{i}}{t}\right) \mathbf{1}\left\{ \frac{S_{i}}{t}\leq1\right\}$$$$\text{ and }\overline{W}\left( t\right) :=\sum_{i=1}^{\infty}\varphi ^{2}\left( \frac{S_{i}}{t}\right) \mathbf{1}\left\{ \frac{S_{i}}{t}>1\right\} .$$ Clearly $$t^{-1}E\overline{W}\left( t\right) =t^{-1}\int_{t}^{\infty}\varphi ^{2}(y/t)\mathrm{d}y=\int_{1}^{\infty}\varphi^{2}(u)\mathrm{d}u<\infty\text{,}$$ which says that $t^{-1}\overline{W}\left( t\right) =O_{P}\left( 1\right) $ as $t\rightarrow\infty$. Hence by (\[L33\]), $\overline{W}\left( t\right) /V_{t}\overset{\mathrm{P}}{\rightarrow}0$ as $t\rightarrow\infty$, which when combined with (\[vt\]) gives $$R_{t}=\frac{W_{t}}{V_{t}^{2}}=\frac{W\left( t\right) }{V^{2}\left( t\right) }+o_{P}(1)\text{, as }t\rightarrow\infty.\label{rp}$$ Notice that $V(t)$ is a Lévy process with canonical measure $\Lambda_{1} $ defined via $$\overline{\Lambda}_{1}\left( x\right) =\overline{\Lambda}\left( x\right) ,\text{ for }x\geq\varphi\left( 1\right) \text{, and }\overline{\Lambda}_{1}\left( x\right) =\overline{\Lambda}\left( \varphi\left( 1\right) \right) \text{ for }0<x<\varphi\left( 1\right) .$$ Set $\varphi_{1}(s)=\varphi(s)\mathbf{1}\{s<1\}$. Note that we have $$\varphi_{1}\left( s\right) =\sup\left\{ y:\overline{\Lambda}_{1}(y)>s\right\} ,\text{ }s>0,$$ where the supremum of the empty set is taken as 0. Let $R_{t}^{\left( 1\right) }$ be defined as $R_{t}$ with $\varphi$ replaced by $\varphi_{1}$, that is, $$R_{t}^{\left( 1\right) }=\frac{W\left( t\right) }{\left( V\left( t\right) \right) ^{2}}=\frac{\sum_{i=1}^{\infty}\varphi_{1}^{2}\left( \frac{S_{i}}{t}\right) }{\left( \sum_{i=1}^{\infty}\varphi_{1}\left( \frac{S_{i}}{t}\right) \right) ^{2}}.$$ Since $R_{t}\left( 1\right) =R_{t}^{\left( 1\right) }$, we see by formula (\[ERT\]) that $$ER_{t}^{\left( 1\right) }=\int_{0}^{\infty}f_{1,t}\left( u\right) \mathrm{d}u.\label{int2}$$ Next from (\[rp\]), we get $R_{t}^{\left( 1\right) }-R_{t}\overset {\mathrm{P}}{\rightarrow}0$, as $t\rightarrow\infty$, which implies that $$\lim_{t\rightarrow\infty}ER_{t}=\lim_{t\rightarrow\infty}ER_{t}^{\left( 1\right) }.$$ Clearly the tail behavior conclusions about $\Lambda_{1}(x)$, as $x\rightarrow\infty$, will be identical to those for $\Lambda(x)$, as $x\rightarrow\infty$. Moreover, since $\overline{\Lambda}_{1}(0+)$ is finite (\[inf\]) trivially holds for $\Lambda_{1}$. Therefore in our proof in the case $t\rightarrow\infty$ we can without loss of generality assume that (\[inf\]) is satisfied. The case $\mu:=$ $\int_{0}^{\infty}\varphi\left( u\right) \mathrm{d}u<\infty$ cannot occur when $\beta<1$ in (\[rt\]). In this case it is easily checked that $$t\overline{\Lambda}\left( x\mu t\right) \rightarrow0\text{ for all }x>0\text{ and }\int_{0}^{1}t\overline{\Lambda}\left( x\mu t\right) \mathrm{d}x\rightarrow1.$$ Therefore by proceeding exactly as above we get that $ER_{t}\rightarrow0$ as $t\rightarrow\infty$, which forces $\beta=1$. $\hspace*{10pt}$ $\Box$ Returning to the proof of Proposition \[prop-rt\], in the case $t\searrow0 $, Lemma \[lemma-rt-inf\] shows that the assumption of Proposition \[P1\] holds and, in the case $t\rightarrow\infty$, it says that we can assume without loss of generality that it is satisfied. 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--- abstract: 'For $\delta>0$ sufficiently small and $A\subset \mathbb{Z}^k$ with $|A+A|\le (2^k+\delta)|A|$, we show either $A$ is covered by $m_k(\delta)$ parallel hyperplanes, or satisfies $|{\widehat{\operatorname{co}}}(A)\setminus A|\le c_k\delta |A|$, where ${\widehat{\operatorname{co}}}(A)$ is the smallest convex progression (convex set intersected with a sublattice) containing $A$. This generalizes the Freiman-Bilu $2^k$ theorem, Freiman’s $3|A|-4$ theorem, and recent sharp stability results of the present authors for sumsets in $\mathbb{R}^k$ conjectured by Figalli and Jerison.' author: - 'Peter van Hintum, Hunter Spink, Marius Tiba' bibliography: - 'references.bib' title: 'Sets in $\mathbb{Z}^k$ with doubling $2^k+\delta$ are near convex progressions' --- Introduction ============ One of the central questions in additive combinatorics is the inverse sumset problem of characterizing the finite subsets $A$ of abelian groups with small *doubling constant* $|A+A|\cdot|A|^{-1} \le \lambda $ for fixed $\lambda>0$. In this paper, we will consider the inverse sumset problem in torsion-free abelian groups $\mathbb{Z}^k$, which has been studied from a variety of perspectives by Freiman [@Freiman], Green and Tao [@GreenTao], Chang [@Chang], and Sanders [@Revisited] among others. For $A \subset \mathbb{Z}^k$, define the convex progression ${\widehat{\operatorname{co}}}(A)$ to be the intersection of the real convex hull $\widetilde{{\operatorname{co}}}(A)$ with the affine sub-lattice $\Lambda_A$ spanned by $A$. An $r$-dimensional generalized arithmetic progression in $\mathbb{Z}^k$ is a subset of the form $B(n_1,\ldots,n_r;v_1,\ldots,v_r;b):=\left\{b+\sum_{i=1}^r\ell_iv_i: 0 \le \ell_i < n_i\right\}.$ Motivated by the fact that for $\widetilde{A}\subset \mathbb{R}^k$ the doubling constant (with respect to volume) is at least $2^k$, we define $d_k(A)=|A+A|-2^k|A|$ for $A\subset \mathbb{Z}^k$. Our main result describes the structure of $A\subset \mathbb{Z}^k$ with $d_k(A)\le \Delta_k|A|$, i.e. $|A+A|\le (2^k+\Delta_k)|A|$. \[mainthm\]\[quant\] a) For $k\geq 1$, there are constants $\Delta_k, m_k,\epsilon_k,$ such that for $A\subset \mathbb{Z}^k$ with $d_k(A)\leq \Delta_k|A|$, either $A$ is covered by $m_k$ parallel hyperplanes, or $A$ lies in some $B=B(n_1,\ldots,n_k;v_1,\ldots,v_k;b)$ with $v_1,\ldots,v_k$ linearly independent and $|A|\ge \epsilon_k |B|$. b) For $k\geq 1$, there are constants $c_k< (4k)^{5k}$, and constants $ \Delta_k(\epsilon_0), g_k(\epsilon_0)$ for all $\epsilon_0>0$, such that for $A\subset \mathbb{Z}^k$ with $d_k(A) \le \Delta_k(\epsilon_0)|A|$, if $A$ lies in some $B=B(n_1,\ldots,n_k;v_1,\ldots,v_k;b)$ with $v_1,\ldots,v_k$ linearly independent and $|A|\ge \epsilon_0 |B|$, then $$|{\widehat{\operatorname{co}}}(A)\setminus A|\le c_kd_k(A)+g_k(\epsilon_0)\min\{n_i\}^{-\frac{1}{1+\frac{1}{2}(k-1)\lfloor k/2 \rfloor}}|A|.$$ \[maincor\] There are constants $c_k< (4k)^{5k}$, $\Delta_k$, $m_k(\delta)$ such that for $\delta\in (0,\Delta_k]$ and $A\subset \mathbb{Z}^k$ with $d_k(A)\leq \delta |A|$, either $A$ is covered by $m_k(\delta)$ parallel hyperplanes, or $$|{\widehat{\operatorname{co}}}(A)\setminus A|\le c_k\delta |A|.$$ \[mainfigalli\] There are constants $c_k< (4k)^{5k}, \Delta_k$ such that for $\widetilde{A} \subset \mathbb{R}^k$ of positive measure with $|\widetilde{A} +\widetilde{A} | \le (2^k+\Delta_k)|\widetilde{A} |$, we have $ |{\widetilde{\operatorname{co}}}(\widetilde{A}) \setminus \widetilde{A}| \le c_k (|\widetilde{A} +\widetilde{A} | - 2^k|\widetilde{A} |).$ Here $|\cdot|$ denotes the outer Lebesgue measure. directly generalizes the Freiman-Bilu $2^k$ theorem [@Freiman; @Bilu] (as improved by Green and Tao [@GreenTao]), Freiman’s $3|A|-4$ theorem [@Freiman] for $A\subset \mathbb{Z}$, and the sharp stability of the Brunn-Minkowski inequality for equal sets in $\mathbb{R}^k$, conjectured by Figalli and Jerison [@Semisum], and recently proved by the authors of the present paper [@HomoBM]. As a consequence of a) and we have that for $A\subset \mathbb{Z}^k$ not contained in a bounded (in terms of $\delta$) number of parallel hyperplanes, the fact that $d_k(A)\le O(\delta)|A|$ is the same as the fact that $A$ satisfies $|{\widehat{\operatorname{co}}}(A)\setminus A|\le O(\delta)|A|$ and has density at least $\epsilon_k$ in a $k$-dimensional generalized arithmetic progression. We note that for a set $A$ symmetric about a lattice point, the discrete John’s theorem of Tao and Vu [@TAOVu] implies ${\widehat{\operatorname{co}}}(A)$ has positive density in a $k$-dimensional generalized arithmetic progression, so in this case the density condition is superfluous. The reverse implication in the previous paragraph follows from a weak converse to b), that if $A\subset B=B(n_1,\ldots,n_k;v_1,\ldots,v_k;b)$, $|A| \ge \epsilon_0 |B|$ and $|{\widehat{\operatorname{co}}}(A)\setminus A|\leq \delta' |A|$ then $d_k(A)\le (2^k\delta' + O(\min\{n_i\})^{-1})|A|$ (see and ), and $A$ is covered by $\min\{n_i\}$ parallel hyperplanes. , conjectured by Figalli and Jerison [@Semisum] and recently resolved by the authors of the present paper [@HomoBM] without any digression to the discrete setting, follows by standard approximation techniques from . This result similarly yields a characterization of positive measure $\widetilde{A}\subset \mathbb{R}^k$ with $d_k(\widetilde{A}) \le O(\delta)|\widetilde{A}|$ as equivalently having $|\widetilde{{\operatorname{co}}}(\widetilde{A})\setminus \widetilde{A}| \le O(\delta)|\widetilde{A}|$. First, we recall Green and Tao’s improvement [@GreenTao] to the classical Freiman-Bilu $2^k$-theorem. \[mainbilu\] Given $\delta>0$, there is a constant $m_k(\delta)$ such that if $A\subset \mathbb{Z}^k$ has $|A+A|\le (2^k-\delta)|A|$, then $A$ is covered by $m_k(\delta)$ parallel hyperplanes. The Freiman-Bilu theorem shows that the correct notion of degeneracy in $\mathbb{Z}^k$ is being covered by a bounded number of parallel hyperplanes, and non-degenerate sets $A$ have doubling constant bounded below by roughly $2^k$. This reflects the continuous analogue for measurable $\widetilde{A}\subset \mathbb{R}^k$. formally implies the Freiman-Bilu theorem, and extends the scope of the theorem beyond the $2^k$ threshold. Next, we recall Freiman’s $3|A|-4$ theorem, which marked the beginning of the study of inverse problems in additive combinatorics. \[3A-4\] Let $A\subset \mathbb{Z}$ be a subset of the integers with $d_1(A)\le |A|-4$. Then $|{\widehat{\operatorname{co}}}(A)\setminus A| \le d_1(A)+1$. This result is sharp, both in the linear bound on $d_1(A)$ in terms of $|A|$ (there are examples of sets $A$ with fixed $d_1(A)=|A|-3$ and $|{\widehat{\operatorname{co}}}(A)\setminus A|$ arbitrarily large in terms of $|A|$), and in the linear bound on $|{\widehat{\operatorname{co}}}(A)\setminus A|$ in terms of $d_1(A)$. is a direct generalization of Freiman’s $3|A|-4$ theorem to arbitrary dimension. The bound $c_k\delta$ is optimal up to the constant $c_k$. The bound $\Delta_k$ on $\delta$ is necessary as before. The additional condition of not being covered by $m_k(\delta)$ hyperplanes is also necessary, and has no analogue when $k=1$ (as subsets $A\subset \mathbb{Z}$ cannot exhibit lower dimensional degeneracies). Consider for example the set $A=(\{1,\ldots,n_0\}\times\{1,\ldots,2n\}^{k-1}) \cup \{(-1,1,1,\ldots,1)\}$, where $n_0$ is constant and $n\gg n_0$, which has $d_k(A)<0$ and $|{\operatorname{co}}(A) \setminus A| = \frac{|A|-1}{2^{k-1} n_0}$. Furthermore, is the first result for $A\subset \mathbb{Z}^k$ with doubling constant beyond the $2^k$ threshold besides the coarse characterizations given by Freiman’s general theorem on sets with small doubling [@Freiman] and subsequent optimizations (see the beginning of ). To our knowledge even a weaker result with $c_k\delta$ replaced with a function $\omega(\delta)$ with $\omega(\delta) \rightarrow 0$ as $\delta \to 0$ was not previously known. As it turns out, much of the work in proving the quantitative statement of is devoted to proving the following qualitative statement. \[quali\]\[qual\] Given $\epsilon_0,\delta>0$ and $k\geq 1$, there exist $n_k(\epsilon_0,\delta)$ and $\omega(\epsilon_0,\delta)$ where for fixed $\epsilon_0$ we have $\omega(\epsilon_0,\delta)\to 0$ as $\delta \to 0$ such that the following is true. If $A$ lies in some $B=B(n_1,\ldots,n_k;v_1,\ldots,v_k;b)$ with $v_1,\ldots,v_k$ linearly independent and $n_1,\ldots,n_k \ge n_k(\epsilon_0,\delta)$ ,such that $|A|\ge \epsilon_0 |B|$ and $d_{k}(A)\le \delta |A|$, then $|{\widehat{\operatorname{co}}}(A)\setminus A|\le \omega(\epsilon_0,\delta)|A|.$ A continuous analogue of proved by Christ [@Christ] and strengthened by Figalli and Jerison [@FigJerJems] was a key step in the study of stability results for the Brunn-Minkowski inequality. However, our methods are largely different from [@Christ] and [@FigJerJems], especially because of phenomena which occur in the discrete setting which have no continuous analogue. We mention a particularly nice intermediate result we show during the proof of . For a real-valued function $f$ on a convex progression $A={\widehat{\operatorname{co}}}(A)$, we define the *infimum-convolution* (see e.g. Strömberg’s extensive survey [@Infconvsurvey]) $f^{\square}:A+A\to \mathbb{R}$ by $$f^{\square}(z)=\min_{x+y=z}\{f(x)+f(y)\}.$$ \[squarepropintro\] There exist constants $c_{k+1}'$ and $g_{k+1}'(\epsilon_0)>0$ for $\epsilon_0>0$ such that the following is true. Let $A={\widehat{\operatorname{co}}}(A)\subset B(n_1,\ldots,n_k;v_1,\ldots,v_k;b)$ with $v_1,\ldots,v_k$ linearly independent, $|A|\ge \epsilon_0|B|$, and $f:A\to [0,1]$ a function. Then with $\widehat{f}:A\to [0,1]$ the lower convex hull function, $$\sum_{x\in A}(f-\widehat{f})(x)\le c'_{k+1}\left( 2^{k+1}\sum_{x\in A}f(x)-\sum_{x'\in A+A} f^{\square}(x')\right)+g_{k+1}'(\epsilon_0)\min\{n_i\}^{-\frac{1}{1+\frac{1}{2}k\lfloor(k+1)/2\rfloor}}|A|.$$ follows from our main results applied to the epigraph $A'_{f,N,M}=\{(a,x):a\in A,x \in [Nf(a),M]\}\subset A\times [0,M]$ for $1 \ll N \ll M$ large constants for fixed $A$ (note that the choice of $M$ allows us to avoid having the condition given by $\Delta_{k+1}$). (used in the proof of ) is essentially this statement for $A$ a simplex with a smaller error term $g_{k+1}'(\epsilon_0)\min\{n_i\}^{-1}|A|$ (and with $c_k'\mapsto c_k'^{-1}$). We remark that the constant $c_k$ in (as well as the constants from and ) can be taken to be less than $(4k)^{5k}$ (which was the constant found in [@HomoBM]), and the constant $c_k'$ can be taken to be at most $c_k$. We have the lower bound $c_k \ge \frac{2^k}{k}$, attained for example by the set $A$ with $ \frac{1}{n}A:=((\widetilde{T} \times [-2,0]) \cup (V(\widetilde{T}) \times \{1\})) \cap (\frac{1}{n}\mathbb{Z})^k$, where $\widetilde{T}\subset \mathbb{R}^{k-1}$ is a fixed simplex with vertices $V(\widetilde{T})\subset \mathbb{Z}^{k-1}$ and $n$ sufficiently large. We also have $c_k'\ge \frac{2^k}{k}$ by taking the functional version of this example, namely with $f:(n\widetilde{T}\cap \mathbb{Z}^k)\to [0,1]$ whose value is $0$ at the vertices and $1$ elsewhere. We believe that the optimal values of $c_k,c_k'$ lie closer to the lower bound $\frac{2^k}{k}$. What are the optimal values of $c_k$ and $c_k'$? Finally, we remark on the exponent $-\frac{1}{1+(k-1)\lfloor k/2\rfloor}$ in . Our proof of reduces the case that ${\widetilde{\operatorname{co}}}(A)$ is a simplex. For a general $A$ we first approximate ${\widetilde{\operatorname{co}}}(A)$ from within by a polytope $\widetilde{P}$, and then triangulate $\widetilde{P}$ into simplices via a triangulation of $\partial \widetilde{P}$. To approximate the volume of ${\widetilde{\operatorname{co}}}(A)$ by a polytope $\widetilde{P}$ with $|\widetilde{P}|\ge(1-\alpha)|{\widetilde{\operatorname{co}}}(A)|$ requires $\ell=O(\alpha^{-\frac{2}{k-1}})$ vertices by Gordon, Meyer, and Reisner [@BestApproximation], and the proof of the upper bound conjecture by Stanley [@Stanley] implies that a triangulation of $\partial \widetilde{P}$ has at most $O(\ell^{\lfloor k/2 \rfloor})$ simplicies. These two bounds end up giving the exponent in . Prior Work {#PriorWork} ---------- Freiman’s theorem [@Freiman] on sets with small doubling says there are constants $b_1(\lambda),b_2(\lambda),b_3(\lambda)$ such that any finite subset $A\subset \mathbb{Z}^k$ with doubling constant less than $\lambda$ can be covered by $b_1(\lambda)$ translates of a generalized arithmetic progression of size at most $b_2(\lambda)|A|$, and of dimension at most $b_3(\lambda)$. Freiman originally formulated his theorem in terms of convex progressions (images of sets of the form ${\widehat{\operatorname{co}}}(A')$ under affine linear maps) instead of generalized arithmetic progressions, and much of the literature focuses on this formulation. Generalizations of convex progressions were used implicitly by Bourgain [@Bourgain], and Green-Sanders [@GreenSanders] (see Sander’s extensive survey [@Revisited] for more information). We note that many authors require convex progressions to be symmetric, but in this paper we impose no such assumptions. Although we focus on subsets of $\mathbb{Z}^k$, we remark briefly that Freiman’s theorem has been generalized to arbitrary abelian groups by Green and Ruzsa [@GreenRusza], and the recent literature on approximate groups seeks to describe analogous characterizations in non-abelian groups (see for example the seminal work of Breuillard, Green and Tao [@ApproxGroup]). The constants $b_1(\lambda)$, $b_2(\lambda)$, and $\exp(b_3(\lambda))$ cannot all be brought down to polynomial as shown by Lovett and Regev [@counterexample], but the analogous question reformulated in terms of (symmetric) convex progressions is open (the *polynomial Freiman-Ruzsa conjecture*). Green and Ruzsa [@GreenRusza], Chang [@Chang], Bourgain [@Bourgain], and Green and Tao [@GreenTao2] showed the constants could be reduced to $b_1(\lambda)=b_2(\lambda)=\exp(b_3(\lambda))=\exp(O(\lambda^C))$ for some constant $C$, improved by Schoen [@Schoen] to $\exp(\exp(O(\sqrt{\log(\lambda)}))$, and finally improved by Sanders [@Revisited] to $\exp(O(\log^{3+o(1)}\lambda))$. For $\mathbb{Z}^k$, Green and Tao [@GreenTao] showed we can obtain optimal bounds for the dimension and sizes of these generalized arithmetic progressions, at the cost of the number of translates. In particular, for $\lambda\in [2^k,2^{k+1})$ we may take $b_2(\lambda)=1$ and $b_3(\lambda)=k$. Our a) shows that when $\lambda \le 2^k+\Delta_k$, then under the non-degeneracy hypothesis that $A$ is not covered by $m_k$ parallel hyperplanes, we can take $b_1(\lambda)=1,b_3(\lambda)=k$. The “thickness” of a subset $A\subset \mathbb{Z}^k$, is the minimum number of parallel hyperplanes required to cover $A$. The central result relating the doubling of a set $A$ to its thickness is the Freiman-Bilu $2^k$-theorem [@Bilu; @Freiman; @GreenTao], . There is a large literature of classifications of subsets $A\subset \mathbb{Z}^k$ with doubling at most $2^k-\delta$ (see e.g. Fishburn [@Fishburn], Freiman [@Freiman], Grynkiewicz and Serra [@Grynkiewicz], and Stanchescu [@Stanchescu1; @Stanchescu1.5; @Stanchescu2; @Stanchescu3]). For $k=1$ Freiman’s $3|A|-4$ theorem [@Freiman], , and subsequent improvements by Jin [@Jin] go beyond this threshold, but even for $k=2$ there do not appear to have been any such results beyond $2^k-\delta$. Without thickness assumptions, Gardner and Gronchi [@Gardner] proved for $A,C\subset \mathbb{Z}^{k}$ not lying in hyperplanes an optimal lower bound for $|A+C|$, but the bound is far worse than $(|A|^{\frac{1}{k}}+|C|^{\frac{1}{k}})^{k}$ predicted by the Brunn-Minkowski inequality for measurable sets in $\mathbb{R}^k$. Under thickness assumptions, the situation is better. For example, by a result of Green and Tao [@GreenTao] (following an approach of Bollobás and Leader [@Bollobas]), if $A,C \subset B(n_1,\ldots,n_k;v_1,\ldots,v_k;b)$ and $|A|,|C| \ge \epsilon |B|$ then $|A+C|\ge (2^k + O(\epsilon\min(n_i))^{-1})\min(|A|,|C|)$. Showing a general form of the Brunn-Minkowski inequality for thick sets is open (see [@Ruzsa Conjecture 3.10.12]), though progress in this direction has been made by Cifre, María, and Iglesias [@Cifre]. The sharp stability result for the Brunn-Minkowski inequality for equal sets, conjectured by Figalli and Jerison [@Semisum], was recently resolved by the authors of the present paper in [@HomoBM]. Our is a discrete analogue of this continuous stability result, and in fact implies it as mentioned earlier. A crucial component used in [@HomoBM] was having a “qualitative result” which shows that $|{\widetilde{\operatorname{co}}}(A)\setminus \widetilde{A}| |\widetilde{A}|^{-1}\to 0$ as $|\widetilde{A}+\widetilde{A}|\cdot |\widetilde{A}|^{-1}\to 2^k$. Such a result was first proved by Christ [@Christ] and later with explicit constants by Figalli and Jerison [@FigJerJems]. Proving discrete analogues of stability results for the Brunn-Minkowski inequality for thick subsets $A,C \subset \mathbb{Z}^k$, such as that of Christ [@Christ] and Figalli and Jerison [@FigJerAdv] for general sets, or sharp stability results such as Barchiesi and Julin [@Barchiesi] for one of the sets being convex and the present authors [@1911.11945] for arbitrary two-dimensional sets, would be extremely interesting, and we believe would be a worthwhile goal to pursue. Outline of the Paper {#outlineofpaper} -------------------- As mentioned before, most of the work is devoted to proving . The strategy is to construct in stages a highly structured set $A_{\star}$ from $A$ with the properties that $|A \Delta A_{\star}| =o_{\epsilon_0}(1) |A|$ and $d_k(A_{\star}) =o_{\epsilon_0}(1)|A_{\star}|=o_{\epsilon_0}(1)|A|$ (where for fixed $\epsilon_0$ we have $o_{\epsilon_0}(1)\to 0$ as $\delta \to 0$). The additional structure of $A_{\star}$ enables us to conclude that $|{\widehat{\operatorname{co}}}(A_{\star}) \setminus A_{\star}| =o_{\epsilon_0}(1)|A_{\star}|$, which finally implies that $|{\widehat{\operatorname{co}}}(A) \setminus A| =o_{\epsilon_0}(1)|A|$. At each stage we produce a new set $A_{\text{new}}$ from an existing set $A_{\text{old}}$ which satisfies $|A\Delta A_{old}|=o_{\epsilon_0}(1)|A|$ and $d_k(A_{old})=o_{\epsilon_0}(1)|A|$. With a single exception, this is done by throwing away rows $R_x$ in the first coordinate direction which are are in some sense unstructured. If we can show $|A_{old}\setminus A_{new}|=o_{\epsilon_0}(1)|A|$, then $|A\Delta A_{new}|=o_{\epsilon_0}(1)|A|$ and hence we have $d_k(A_{new})=o_{\epsilon_0}(1)|A|$. To bound $|A_{\text{old}} \setminus A_{\text{new}}|$ from above, we introduce an operation $(+)$ in order to create a “reference set” $A_{\text{old}}(+)A_{\text{old}}\subset A_{\text{old}}+A_{\text{old}}$, whose size we can guarantee to be approximately $2^k|A_{\text{old}}|$. For one dimensional sets $X, Y$ we define $X(+)Y:= (X+\min Y)\cup (Y+ \max X) \subset X+Y$; in general, we define $A_{\text{old}}(+)A_{\text{old}}:=\bigsqcup_{\vec{v}\in \{0,1\}^{k-1}}\bigsqcup_{x}R_x(+)R_{x+\vec{v}} \subset A_{\text{old}}+A_{\text{old}}$. In order to control the size of the unstructured rows $U:=\bigsqcup_{x\text{ - unstructured}}R_x$, we construct a set $D \subset U+A_{\text{old}} \subset A_{\text{old}}+A_{\text{old}} $ of comparable size to $U$, disjoint from $A_{\text{old}}(+)A_{\text{old}}$. Then, we will obtain $|A_{\text{old}} \setminus A_{\text{new}}|=|U| \approx |D| \le |A_{\text{old}}+A_{\text{old}}| - |A_{\text{old}}(+)A_{\text{old}}| \approx d_k(A_{\text{old}})=o_{\epsilon_0}(1)|A|$. For the last step in the proof of and for the proof of b), we use two versions of an argument inspired by the one used in [@HomoBM]. For the last part of , we prove that functions on convex domains with small infimum-convolution are close to their convex hulls (which is essentially ). For b), we proceed as follows. By choosing an appropriately small $\Delta_k(\epsilon_0),$ ensures $|{\widehat{\operatorname{co}}}(A)\setminus A|\cdot |A|^{-1}$ is as small as we like. This guarantees a large interior region of ${\widehat{\operatorname{co}}}(A+A)$ is contained in $A+A$. We control the size of ${\widehat{\operatorname{co}}}(A)\setminus A$ by inductively controlling the size of ${\widehat{\operatorname{co}}}(A)\setminus A$ restricted to certain homothetic copies of $\widetilde{{\operatorname{co}}}(A)$ used to cover a thickened boundary of $\widetilde{{\operatorname{co}}}(A)$. This will allow us to show that $|{\widehat{\operatorname{co}}}(A+A)\setminus (A+A)|\leq (2^k-c_k')|{\widehat{\operatorname{co}}}(A)\setminus A|+o_{\epsilon_0}(1)|B|$ for some constant $c_k'$, which allows us to conclude b). Finally, for a), we will start with a result of Green and Tao [@GreenTao] that $A$ is covered by a bounded number of generalized arithmetic progressions of dimension $k$ and size at most $|A|$, and show that we can reduce to a single generalized arithmetic progression of size $O(|A|)$. In Section 2, we make some initial definitions, conventions and observations, which will be used throughout the paper. In Sections 3 and 4, we prove and b) with a simultaneous induction on dimension. In Section 5, we prove a), whose proof is independent of Sections 3 and 4. Finally, for completeness, we include in Appendix A a proof of and from . Definitions, Conventions, and Observations ========================================== In this section, we introduce our definitions and conventions, as well as observations we will be using throughout the remaining sections. Definitions and Conventions --------------------------- For $A'\subset \mathbb{Z}^k$, we denote by - $\widetilde{{\operatorname{co}}}(A')\subset \mathbb{R}^k$ for the convex hull, - ${\operatorname{co}}(A')=\widetilde{{\operatorname{co}}}(A')\cap \mathbb{Z}^k$, - $\Lambda_{A'}=\langle A'-a\rangle+a\subset \mathbb{Z}^k$ for any $a\in A'$, the affine sublattice of $\mathbb{Z}^k$ spanned by $A'$, and - ${\widehat{\operatorname{co}}}(A')={\operatorname{co}}(A')\cap \Lambda_{A'}$, the smallest convex progression containing $A'$. We say that $A'$ is *reduced* if $\Lambda_{A'}=\mathbb{Z}^k$, or equivalently ${\operatorname{co}}(A')={\widehat{\operatorname{co}}}(A')$ and $A'$ is not contained in a hyperplane. We will typically denote regions of $\mathbb{R}^k$ with a tilde such as $\widetilde{A'}\subset \mathbb{R}^k$. By abuse of notation, we will use $|\cdot |$ to refer both to cardinality of sets, and for volumes of sets. It will be clear with the tilde notation whether we intend to use discrete or continuous volume, and from context what dimension we are considering. When we define a polytope or affine subspace $\widetilde{P}\subset \mathbb{R}^k$, we let $P=\widetilde{P}\cap \mathbb{Z}^k$. Given numbers $n_1,\ldots,n_k$, we define the discrete box $$B(n_1,\ldots,n_k)=\prod_{i=1}^k \{1,\ldots,n_i\}.$$ We write $B$ instead of $B(n_1,\ldots,n_k)$ when $n_1,\ldots,n_k$ are clear from context. In and b), we shall assume without loss of generality that the vectors $v_i$ coincide with basis vectors $e_i$ of $\mathbb{R}^k$ and $b=\vec{1}$, writing the generalized arithmetic progression $B=B(n_1,\ldots,n_k;v_1,\ldots,v_k;b)=B(n_1,\ldots,n_k)$. \[redconv\] In the proofs of and b), we shall assume $A$ is reduced, as allowed by below. We define the projection $$\pi:\mathbb{Z}^k=\mathbb{Z}\times\mathbb{Z}^{k-1}\to \{0\}\times\mathbb{Z}^{k-1}$$ to be the projection away from the first coordinate. For $\widetilde{A'}\subset \mathbb{R}^k$ a subset such that ${\widetilde{\operatorname{co}}}(\widetilde{A'})$ is a polytope with integral vertices, we define $V(\widetilde{A'})\subset \mathbb{Z}^{k}$ to be the vertices of ${\widetilde{\operatorname{co}}}(\widetilde{A}')$, and $V_\pi(\widetilde{A'}):=\pi(V(\widetilde{A}'))\subset \{0\}\times \mathbb{Z}^{k-1}$. A *row* of $A'\subset \mathbb{Z}^k$ is $R_x=\pi^{-1}(x)\cap A'$ for some $x\in \{0\}\times \mathbb{Z}^{k-1}$. When talking about the rows of a set $A'$, we will use the notation $R_x$ without further clarification. It will always be clear from context which set $A'$ is being referred to. For $X,Y\subset \mathbb{Z}$, define $X(+)Y:=(X+\min Y)\cup (Y+\max X)\subset X+Y$ if $X,Y$ are both nonempty, and $\emptyset$ otherwise. For $A'\subset \mathbb{Z}^k$, we define $$A'(+)A':=\sum_{\vec{v}\in \{0\}\times\{0,1\}^{k-1}}\sum_{x\in \{0\}\times \mathbb{Z}^{k-1}}R_x(+)R_{x+\vec{v}}\subset A'+A'.$$ In , by choosing $\omega(\epsilon_0,\delta)=1$ for large values of $\delta$, we may assume $\delta$ is sufficiently small and $n_k(\epsilon_0,\delta)$ is sufficiently large to make all statements true without actually specifying the exact bounds. \[nkconv\] We omit the sentence “for all $\delta$ sufficiently small and $n_k(\delta,\epsilon_0)$ sufficiently large” from the end of all of our statements, unless stated otherwise. The big and little o notations $O(1)$ and $o(1)$ are with respect to fixed $\epsilon_0$ as $\delta \to 0$ throughout. Finally, we introduce a small constant which we will use to absorb errors into exponents through the paper. \[cdef\] We let $c=10^{-10}$. Observations ------------ The first observation guarantees that hyperplanes $\widetilde{H}$ have small intersections with discrete boxes. In particular, large subsets of $B$ cannot be covered by few hyperplanes. \[hypboxsmall\] Given a hyperplane $\widetilde{H}$ and a box $B=B(n_1,\ldots,n_k)$, we have $$|\widetilde{H}\cap B|\le \min\{n_i\}^{-1}|B|.$$ In particular, a subset $A'\subset B$ with $|A'|>m\min\{n_1\}^{-1}|B|$ cannot be covered by $m$ hyperplanes. If coordinate basis vector $e_i$ is not parallel to $\widetilde{H}$, then the projection of $\widetilde{H}\cap B$ away from the $i$th coordinate injects into the same projection for $B$, and therefore $|\widetilde{H}\cap B|\le n_i^{-1}|B|$. The next observation is used to assume $A$ is reduced so holds. \[nonreduced\] For all $\epsilon_0>0$, the following holds. Given $n_i$ sufficiently large in terms of $\epsilon_0$, for a subset $A'\subset B=B(n_1,\ldots,n_k)$ with $|A'|\geq \epsilon_0|B|$, we can find a subset $A''\subset B(2^kn_1,\ldots,2^kn_k)$ such that $A''$ is reduced, $|A'|=|A''|$, $d_k(A')=d_k(A'')$, and $|{\widehat{\operatorname{co}}}(A')\setminus A'|=|{\operatorname{co}}(A'')\setminus A''|$. We first note that $A'$ is not contained inside a hyperplane by . Take some $a\in A'$, then $A'-a\in C:=\prod_{i=1}^k \{-n_i+1,\ldots,n_i-1\}$ and the affine sub-lattice $\Lambda_{A'-a}$ is actually a subgroup $\langle v_1, \hdots, v_k\rangle\subset \mathbb{Z}^k$ generated by linearly independent vectors $v_i=(v_{i,1}, \hdots, v_{i,k})$. Without loss of generality, suppose $n_1\le \ldots \le n_k$. By the Euclidean algorithm we may assume that $|v_{j, i}| = 0$ for all $j>i$, and $|v_{j,i}| \le |v_{i,i}|$ for all $j \le i$. Because $v_1,\ldots,v_i$ are linearly independent, we have $v_{i,i}\ne 0$. Consider a point $p=p_1v_1 + \hdots + p_kv_k \in C$. We will show by induction on $i$ that $|p_i|\le 2^{i-1}(n_i-1)$. Indeed, by considering the $i$th coordinate, we have that $$|p_1 v_{1,i} + p_2 v_{2,i}+ \hdots +p_i v_{i,i}| \le n_{i}-1$$ and hence $$|p_i| |v_{i,i}| \le |p_1| |v_{1,i}| + \ldots + |p_{i-1}| |v_{i-1,i}| + n_{i}-1 \le (|p_1|+ \ldots + |p_{i-1}| + n_{i}-1)|v_{i,i}|.$$ This shows that $A'-a\subset \{p_1 v_1 + \ldots p_kv_k \text{ : } |p_i| \le 2^{i-1}(n_{i}-1) \}$. If we let $$A''':=\{(p_1, \hdots p_k) \text{ : } p_1v_1 + \hdots p_kv_k \in A'-a \}\subset \prod_{i=1}^k \{-2^{i-1}(n_i-1),\ldots,2^{i-1}(n_i-1)\},$$ then $A'''$ is reduced in $\mathbb{Z}^k$, and is obtained from $A'$ by applying an element of $GL_n(\mathbb{Q})$ followed by a translation, so $|A'|=|A'''|$, $d_k(A''')=d_k(A')$, and $|{\operatorname{co}}(A''') \setminus A'''| =|{\widehat{\operatorname{co}}}(A')\setminus A'|.$ We conclude by taking $A''$ a suitable translation of $A'''$. We now prove an observation lower bounding $d_k$ for subsets of boxes, an easy corollary of a Lemma of Green and Tao [@GreenTao]. \[negdk\] For any subsets $X\subset B=B(n_1,\ldots,n_k)$ and $Y\subset \pi(B)$ we have $$d_{k}(X) \ge -2^{2k}\min\{n_i\}^{-1}|B|\text{, and }d_{k-1}(Y) \ge -2^{2(k-1)}\min\{n_i\}^{-1}n_1^{-1}|B|.$$ More generally, for $X_1,X_2\subset B$ and $Y_1,Y_2\subset \pi(B)$ we have $$\begin{aligned} |X_1+X_2|&\ge 2^{k}\min(|X_1|,|X_2|)- 2^{2k}\min\{n_i\}^{-1}|B|\text{, and}\\ |Y_1+Y_2|&\ge 2^{k-1}\min(|Y_1|,|Y_2|) -2^{2(k-1)}\min\{n_i\}^{-1}n_1^{-1}|B|.\end{aligned}$$ Because $B$ and $\pi(B)$ are downsets, the result follows from [@GreenTao Lemma 2.8], and the trivial estimates that the size of each coordinate projection of $B+B$ and $\pi(B)+\pi(B)$ have sizes at most $2^k\min\{n_i\}^{-1}|B|$ and $2^{k-1}\min\{n_i\}^{-1}n_1^{-1}|B|$, respectively. We frequently need the following observation when considering $A'(+)A'$ to show it has size roughly $2^k|A'|$ as described in . \[Surfobs\] Let $Y\subset \pi(B)$ with $B=B(n_1,\ldots,n_k)$, and let $0\ne \vec{v}\in \{0\}\times\{0,1\}^{k-1}$. Then $$\left|\{x \in \mathbb{Z}^{k-1} : |\{x,x+\vec{v}\}\cap {\operatorname{co}}(Y)|=1\}\right| \le 2(k-1) \min\{n_i\}^{-1}n_1^{-1}|B|.$$ Consider all lines in the direction $\vec{v}$ intersecting $\pi(B)$. On each such line there are at most 2 values of $x$ such that $|\{x,x+\vec{v}\}\cap {\operatorname{co}}(Y)|=1$, and there are at most $(k-1) \min\{n_i\}^{-1} |\pi(B)|$ such lines which intersect $\pi(B)$. The next observation relates $d_k$ between sets and subsets. In particular, it allows us to guarantee that all auxiliary sets we construct in the proof of are reduced and have similar $d_k$ solely because they are close in symmetric difference to the original set $A$. \[dkobs\] If $X\subset Y$, then $$\begin{aligned} \label{dk}d_k(X)\le d_k(Y)+2^k|Y\setminus X|.\end{aligned}$$ In particular, for reduced $A\subset B=B(n_1,\ldots,n_k)$ with $|A|\ge \epsilon_0|B|$, $d_k(A)\le \delta |B|$, $\delta$ sufficiently small in terms of $\epsilon_0$, and $n_1,\ldots,n_k$ sufficiently large in terms of $\epsilon_0,\delta$, if $A'\subset B$ has $$\begin{aligned} \label{redcond}|A\Delta A'|\le 2^{-(k+1)}\epsilon_0|B|\end{aligned}$$ then $A'$ is reduced. For , we have $$d_k(X)=|X+X|-2^k|X|\le |Y+Y|-2^k|Y|+2^k|Y\setminus X|=d_k(Y)+2^k|Y\setminus X|.$$ For , it suffices to show $A\cap A'$ is reduced, so we may assume $A'\subset A$. Assume for the sake of contradiction that $A'$ is not reduced. Then there is an $a\in A$, such that $a+A'$ is disjoint from $A'+A'$. Hence, we have $|A+A|\ge |A'+A'|+|A'|$, and in particular $$\delta |B| \ge d_k(A)\ge d_k(A')-(2^k+1)|A\setminus A'|+|A| \ge \left(-2^{2k}\min \{n_i\}^{-1}+\left(\frac{1}{2}-\frac{1}{2^{k+1}}\right)\epsilon_0\right)|B|,$$ a contradiction. We next have an observation which allows us to transition between convex sets and reduced convex progressions with a loss proportional to the surface area of a containing box. \[contdisc\] Let $B=B(n_1,\ldots,n_k)$, and suppose we have a convex polytope $\widetilde{P}\subset {\widetilde{\operatorname{co}}}(B)$. Then with $P=\widetilde{P}\cap \mathbb{Z}^k$, we have $\left||\widetilde{P}|-|P|\right|\leq 2k(k+1) \min\{n_i\}^{-1}|B|$. This is more generally true for any subset $\widetilde{P}\subset{\widetilde{\operatorname{co}}}(B)$ given as the intersection of finitely many open and closed half-spaces. By perturbing the defining half-spaces slightly, we may replace $\widetilde{P}$ with a polytope without changing $P$, so we assume $\widetilde{P}$ is a polytope from now on. Consider the set $X:=\left\{z\in \mathbb{Z}^k: (z+[0,1]^k)\cap \partial\widetilde{P}\neq \emptyset\right\}.$ We first show $|X|$ is small. $|X|\leq 2k(k+1)\min\{n_i\}^{-1}|B|$ For $1\le i \le k$, let $\pi_i:\mathbb{Z}^k\to\mathbb{Z}^{i-1}\times \{0\}\times \mathbb{Z}^{k-i}$ be the projection away from the $i$th coordinate. Let $f_i^+,f_i^{-}:\pi_i(X) \to \mathbb{Z}$ be defined by $$\begin{aligned} f_i^+:x \mapsto \max(\pi_i^{-1}(x)\cap X)\qquad f_i^{-}:x \mapsto \min(\pi_i^{-1}(x)\cap X),\end{aligned}$$ and for every $x=(x_1,\ldots,x_{i-1},0,x_{i+1},\ldots,x_k)\in \pi_i(X)$, let $$\begin{aligned} X^+_{i,x}&=\{(x_1,\dots,x_{i-1},j,x_{i+1},\dots,x_k):f^+_i(x)-k\leq j\leq f^+_i(x)\}\\ X^-_{i,x}&=\{(x_1,\dots,x_{i-1},j,x_{i+1},\dots,x_k):f^-_i(x)\leq j\leq f^-_i(x)+k\}\end{aligned}$$ be the $k+1$ elements of $\mathbb{Z}^k$ in the $x$-row in direction $i$ of $X$ below the maximum element and above the minimum element respectively. From these definitions it is immediate that $$\left|\bigcup_{i=1}^k X_{i,x}^+\cup X_{i,x}^-\right|\leq 2k(k+1)\min\{n_i\}^{-1}|B|,$$ so it suffices to show that $X\subset\bigcup X_{i,x}^+\cup X_{i,x}^-$. Suppose for the sake of contradiction that there is some $z\in X\setminus\left(\bigcup X_{i,x}^+\cup X_{i,x}^-\right)$. Then $$f^+_i(\pi_i(z))\geq z_i+k+1, \text{ and }f_i^-(\pi_i(z))\leq z_i-k-1$$ for all $i$, so there are $r_i^+,r_i^- \ge k+1$ such that $ z+r_i^+e_i+[0,1]^k$ and $z-r_i^-e_i+[0,1]^k$ intersect $\widetilde{P}$. As $z+[0,1]^k$ intersects $\widetilde{P}$, by convexity of $\widetilde{P}$ for all $i\in [k]$ there are points $$y^+_i\in (z+(k+1)e_i+[0,1]^k)\cap \widetilde{P},\qquad y^-_i\in (z-(k+1)e_i+[0,1]^k)\cap \widetilde{P}.$$ We claim that $$z+[0,1]^k\subset int(\widetilde{co}(\{y^+_1,\dots,y^+_k,y^-_1,\dots,y^-_k\}))\subset int (\widetilde{P}).$$ Write $y_i^+=z+(\frac{1}{2},\ldots,\frac{1}{2})+p_i^+$ and $ y_i^-=z+(\frac{1}{2},\ldots,\frac{1}{2})+p_i^-$ where $p_i^+= (k+1)e_i+\epsilon_i^+$ and $p_i^-= -(k+1)e_i+\epsilon_i^-$ with $\epsilon_i^{\pm}\in [-\frac{1}{2},\frac{1}{2}]^k$. Then this is equivalent to showing $$\left[-\frac{1}{2},\frac{1}{2}\right]^k\subset int(\widetilde{{\operatorname{co}}}(\{p_1^+,\ldots,p_k^+,p_1^-,\ldots,p_k^-\})).$$ We will show that ${\widetilde{\operatorname{co}}}(\{p_1^+,\ldots,p_k^+,p_1^-,\ldots,p_k^-\})$ has facets ${\widetilde{\operatorname{co}}}(p_1^{\pm},\ldots,p_k^{\pm})$ for the $2^k$ choices of $\pm$, and $\left[-\frac{1}{2},\frac{1}{2}\right]^k$ lies on the same side of these facets as ${\widetilde{\operatorname{co}}}(\{p_1^+,\ldots,p_k^+,p_1^-,\ldots,p_k^-\})$. To show this, let $\epsilon\in [-\frac{1}{2},\frac{1}{2}]^k$. We claim that it suffices to show that $\epsilon$ and $p_1^-$ lie on the same side of the hyperplane $\widetilde{H}$ through $p_1^+,\ldots,p_k^+$. Indeed, if this is the case, then by symmetry all vertices lie on the same side of $\widetilde{H}$, which implies that ${\widetilde{\operatorname{co}}}(\{p_1^+,\ldots,p_k^+\})$ is a facet, and ${\widetilde{\operatorname{co}}}(\{p_1^+,\ldots,p_k^+,p_1^-,\ldots,p_k^-\})$ lies on the same side of this facet as $\epsilon$. This is equivalent in turn to showing that for $w\in \{p_1^-,\epsilon\}$, the determinants of the matrices whose columns are $p_i^+-w$ for $1 \le i \le k$ have the same signs. We will in fact show that this sign is positive for both. For $w=\epsilon$, the matrix we are considering is $M+(k+1)I$ where $M$ has as its $i$th column $\epsilon_i^+-\epsilon$. Note that $M$ has entries of magnitude at most $1$, so the spectral radius of $M$ is at most $k$. But if $\det(M+(k+1)I)\le 0$, then as $\det(M+\lambda I) \to \infty$ as $\lambda \to \infty$ there must exist $\lambda\ge k+1$ with $\det(M+\lambda I)=0$, a contradiction as $|-\lambda|>k$. Hence $\det(M+(k+1)I)>0$ as desired. For $w=p_1^-$, note that we have already shown that $\epsilon_1^-$ lies on the positive side of $\widetilde{H}$, and $p_1^-\in \epsilon_1^-+\mathbb{R}_{\le 0}e_1$. Hence it suffices to show that the point $\epsilon_1^+-Ne_1$ lies on the positive side of $\widetilde{H}$ for all $N>0$ sufficiently large. This is equivalent to saying that the matrix $M_N$ whose $i$th column is $p_i^++Ne_1-\epsilon_1^-$ has positive determinant for all $N> 0$ sufficiently large. Subtracting the first column from all subsequent columns and then considering the coefficient of $N$ in $\det(M_N)$, this follows from an identical argument. Hence we have $z+[0,1]^k\subset int(\widetilde{P})$, contradicting $(z+[0,1]^k)\cap \partial \widetilde{P}\ne \emptyset$. Returning to the proof of , consider the translates of $[0,1]^k$ by $P$, i.e. $P+[0,1]^k$. Each of these translates is either contained in $\widetilde{P}$ or intersects $\partial{\widetilde{P}}$. Hence, $|P|\leq |\widetilde{P}|+|X|$. On the other hand, consider the set of all integer translates of $[0,1]^k$ intersecting $\widetilde{P}$. All of these translates intersect $\partial{\widetilde{P}}$ or are of the form $a+[0,1]^k$ with $a\in P$. As these clearly cover $\widetilde{P}$, we find $|\widetilde{P}|\leq |P|+|X|$. Finally, the following observation implies $A'$ being close to its convex hull implies $d_k(A')$ is small, which as mentioned in the introduction yields a weak converse to b). \[cvxsd\] Given a set $A'\subset B=B(n_1,\ldots,n_k)$, we have $$d_k(A')\leq 2^k|{\operatorname{co}}(A')\setminus A'|+2^{k+2}k(k+1)\min\{n_i\}^{-1}|B|$$ By , we only need to show $d_k({\operatorname{co}}(A'))\le 2^{k+2}k(k+1)\min\{n_i\}^{-1}|B|$. This follows as by we have $$\begin{aligned} d_k({\operatorname{co}}(A'))&= |{\operatorname{co}}(A')+{\operatorname{co}}(A')|-2^k |{\operatorname{co}}(A')|\\ &\leq |({\widetilde{\operatorname{co}}}(A')+{\widetilde{\operatorname{co}}}(A'))\cap \mathbb{Z}^k|-2^k |{\operatorname{co}}(A')|\\ &\leq |{\widetilde{\operatorname{co}}}(A')+{\widetilde{\operatorname{co}}}(A')|-2^k |{\operatorname{co}}(A')|+2^{k+1}k(k+1) \min\{n_i\}^{-1}|B|\\ &= 2^k|{\widetilde{\operatorname{co}}}(A')|-2^k |{\operatorname{co}}(A')|+2^{k+1}k(k+1) \min\{n_i\}^{-1}|B|\\ &\leq 2^{k+2}k(k+1) \min\{n_i\}^{-1}|B|.\end{aligned}$$ Proof of for $k$ given b) for $k-1$ {#1.6section} =================================== For $k=1$, and are implied by Freiman’s $3|A|-4$ theorem [@Freiman], , so we suppose from now on that $k\ge 2$. In this section, we prove for dimension $k$ given b) for dimension $k-1$. We recall by that we will assume that $A$ is reduced. Define $\epsilon\ge \epsilon_0$ to be the density of $A$ in $B$, so we have $$\begin{aligned} \label{Ainfo}|A|=\epsilon|B|\text{, and }d_k(A)\le \delta |B|.\end{aligned}$$ Outline of the proof -------------------- We will create sets $$A\supset A_1\supset A_2\supset A_3\supset A_4\supset A_5\subset A_+\supset A_\star$$ (note that $A_5\subset A_+$) such that $|A\Delta A_\star|$ is small, and $A_\star$ has a large number of properties which allow us to show that $A_\star$ is close to ${\operatorname{co}}(A_\star)$. From this we will be able to conclude that $A$ is close to ${\operatorname{co}}(A)$. In , we derive a general reduction to sets for which the projection under $\pi$ satisfies the induction hypothesis. In , we construct $A\supset A_1\supset A_2\supset A_3$ such that $A_3$ is reduced, has large rows $R_x$ close to ${\widehat{\operatorname{co}}}(R_x)$, and has $\pi(A_3)$ close to ${\operatorname{co}}(\pi(A_3))$. In , we construct $A_3\supset A_4\supset A_5$ such that $A_5$ has the same properties as $A_3$ and the arithmetic progressions ${\widehat{\operatorname{co}}}(R_x)$ have the same step size $d$. In , we show that $d=1$, i.e. ${\widehat{\operatorname{co}}}(R_x)={\operatorname{co}}(R_x)$ is an interval for all rows $R_x$ of $A_5$. In , we show that filling in the rows of $A_5$ to make a set $A_+\supset A_5$ preserves the properties that $A_5$ had (this is the only step we deviate from throwing away a subset of rows). In , we show that we can approximate $A_+$ with a subset $A_\star\subset A_+$ which has simultaneously 1. Few vertices on $\widetilde{{\operatorname{co}}}(A_\star)$ 2. $\pi(A_\star)$ close to ${\operatorname{co}}(\pi(A_\star))$ 3. The technical condition . Up to this point, we were able to show that $|A\Delta A_+| \le \delta^{O(1)}|A|$. However obtaining $A_\star$ involves a double recursion, and we are only able to show $|A\Delta A_\star|=o(1)|A|$ where $o(1)\to 0$ as $\delta \to 0$. In , we show that $A_\star$ is close to its convex hull. The key step is to convert the problem to one of bounding the size of the epigraph of a certain infimum-convolution of a function by the size of the epigraph of the original function. Finally, in we finish the proof of by showing that $A_\star$ being close to its convex hull implies $A$ is close to its convex hull. Exploiting the inductive hypothesis {#inductiveh} ----------------------------------- In this section we prove a result, relying on the inductive hypothesis, which we will frequently apply that allows us to remove a small number of rows from a set $A'$ to ensure that the projection $\pi(A')$ is close to ${\widehat{\operatorname{co}}}(\pi(A'))$. Recall we introduced in a small constant $c$. \[alphaprop’\] Let $\sigma=\sigma(\delta)$ be a function with $\sigma \to 0$ as $\delta \to 0$ and let $\lambda>\alpha>0$. Let $A'\subset B$ with $|A'|=\epsilon'|B|\ge \frac{\epsilon_0}{2}|B|$ and $d_k(A')\le \sigma^{\lambda}|B|$. Then there exists $A''\subset A'$ formed as a union of rows $R_x$ of $A'$ with $$\begin{aligned} |{\widehat{\operatorname{co}}}(\pi(A''))\setminus \pi(A'')| \le\sigma^{\alpha} |\pi(B)|,\quad |A'\setminus A''| \le \sigma^{\lambda-\alpha-c}|B|.\end{aligned}$$ Furthermore, if $A'$ is reduced then $A''$ is reduced and in particular ${\widehat{\operatorname{co}}}(\pi(A''))={\operatorname{co}}(\pi(A''))$. \[alphaproprmk\] For a fixed function $\sigma(\delta)$ and for fixed parameters $\lambda$ and $\alpha$, the proof of requires us to impose certain bounds on $\sigma$ and $n_k(\epsilon_0, \delta)$ so that for example $\sigma<1$. However, by we have an implicit bound on $n_k(\epsilon_0,\delta)$ and on $\delta$ depending on $\sigma$ (and hence in particular a bound on $\sigma(\delta)$). We shall not make remarks about in any subsequent statement. We can take $A''=\emptyset$ if $\lambda \le \alpha+c$, so suppose $\lambda > \alpha+c$. Let $$\begin{aligned} E_i=\{x\in \pi(A') : |\pi^{-1}(x)\cap A'| \ge i\},\qquad F_i=\{x \in \pi(A'+A') : |\pi^{-1}(x)\cap (A'+A')| \ge i\}.\end{aligned}$$ Note that $E_1\supset E_2 \supset \ldots$ and $F_1 \supset F_2 \supset \ldots$, and we have $$\begin{aligned} \label{EiFiequation} |A'|=\sum_{i=1}^{n_1} |E_i|,\text{ and }|A'+A'|=\sum_{i=1}^{2n_1-1} |F_i|.\end{aligned}$$ We note that $E_i+E_i \subset F_{2i-1},F_{2i-2}$, so we have $$\begin{aligned} |A'+A'| &\ge -2^{k-1}n_1^{-1}|B|+ 2\sum_{i=1}^{n_1} |E_i+E_i|\\&\ge -\sigma^{\lambda}|B|+2\sum_{i=1}^{n_1}|E_i+E_i|.\end{aligned}$$ Subtracting $2^k|A'|=2\sum_{i=1}^{n_1} 2^{k-1}|E_i|$, we obtain $d_k(A') \ge -\sigma^{\lambda}|B|+2\sum_{i=1}^{n_1} d_{k-1}(E_i),$ so by the hypothesis $\sigma^{\lambda}|B|\ge d_k(A')$ we see that $$\begin{aligned} \label{sigmalambdagreaterEi} \sigma^\lambda|B| \ge \sum_{i=1}^{n_1}d_{k-1}(E_i).\end{aligned}$$ Let $i_0$ be the first index with $d_{k-1}(E_{i_0})\le \sigma^{\alpha+c/2} |E_{i_0}|$, which exists as otherwise by ,, $$\sigma^{\lambda}|B|\ge\sigma^{\alpha+c/2}|A'|\ge \sigma^{\alpha+c/2}\frac{\epsilon_0}{2}|B|>\sigma^{\lambda}|B|.$$ Let $A'':=\pi^{-1}(E_{i_0})\cap A'\subset A'$ be the union of all rows of size at least $i_0$. By construction, $$\begin{aligned} \label{dk-1A''small} d_{k-1}(\pi(A''))=d_{k-1}(E_{i_0})\le \sigma^{\alpha+c/2}|E_{i_0}|= \sigma^{\alpha+c/2}|\pi(A'')|.\end{aligned}$$ Also as $|E_i|$ is decreasing in $i$, $\sum_{i=1}^{i_0-1} |E_i|\ge \frac{i_0-1}{n}|A'|$ by . Thus by and , $$\begin{aligned} \sigma^{\lambda}|B| \ge\sum_{i=1}^{n_1} d_{k-1}(E_i)\ge& \sigma^{\alpha+c/2}\frac{i_0-1}{n_1}|A'|-n_12^{2(k-1)}n_{k}(\epsilon_0,\delta)^{-1}n_1^{-1}|B|\\ \ge& \sigma^{\alpha+c/2}\frac{i_0-1}{n_1}\cdot\frac{\epsilon_0}{2}|B|-\sigma^{\lambda}|B|.\end{aligned}$$ Thus we obtain $$i_0-1 \le 4\sigma^{\lambda-\alpha-c/2} \epsilon_0^{-1}n_1\le \sigma^{\lambda-\alpha-c}n_1.$$ As the $|\pi(A'\setminus A'')|\le |\pi(B')|=n_1^{-1}|B|$ nonempty rows in $A'\setminus A''$ have size at most $\sigma^{\lambda-\alpha-c}n_1$, we have $$|A'\setminus A''| \le \sigma^{\lambda-\alpha-c}|B|.$$ We have $|A'\setminus A''|\le 2^{-(k+1)}\epsilon_0|B|$ so $A''$ is reduced by , and $|A''|\ge \frac{\epsilon_0}{4}|B|$. In particular, $\pi(A'')$ is reduced and $|\pi(A'')|\ge \frac{\epsilon_0}{4}|\pi(B)|$. The set $\pi(A'')$ has $d_{k-1}(\pi(A''))\le \sigma^{\alpha+c/2}|\pi(A'')|$ by , and has density at least $\frac{\epsilon_0}{4}$ in $\pi(B)$, which has side lengths at least $n_k(\epsilon_0,\delta)$. By , the number of parallel hyperplanes needed to cover $\pi(A'')$ is at least $\frac{\epsilon_0}{4}n_k(\epsilon_0,\delta)>m_{k-1}(\sigma^{\alpha+c/2})$, so by for dimension $k-1$ we deduce that $$|{\operatorname{co}}(\pi(A''))\setminus \pi(A'')|=|{\widehat{\operatorname{co}}}(\pi(A''))\setminus \pi(A'')|\le c_{k-1}\sigma^{\alpha+c/2}|\pi(B)|\le \sigma^{\alpha}|\pi(B)|.$$ \[alphaprop\] Let $\lambda> \alpha >0$. Let $A'\subset B$ with $|A'|=\epsilon'|B|\ge \frac{\epsilon_0}{2}|B|$ and $d_k(A')\le \delta^{\lambda}|B|$. Then there is an $A''\subset A'$ formed by a union of rows of $A'$ with $$\begin{aligned} |{\widehat{\operatorname{co}}}(\pi(A''))\setminus \pi(A'')|\le\delta^{\alpha} |\pi(B)|,\quad |A'\setminus A''| \le \delta^{\lambda-\alpha-c}|B|.\end{aligned}$$ Furthermore, if $A'$ is reduced then $A''$ is reduced and in particular ${\widehat{\operatorname{co}}}(\pi(A''))={\operatorname{co}}(\pi(A''))$. Reductions Part 1: All rows are dense in large APs {#prelimreductions} -------------------------------------------------- We start by constructing in a sequence of steps a set $A_3\subset A$ such that $|A\setminus A_3|$ is small, $\pi(A_3)$ is close to ${\operatorname{co}}(\pi(A_3))$ and the rows $R_x$ of $A_3$ are large and close to ${\widehat{\operatorname{co}}}(R_x)$. In the continuous setting, a similar preliminary reduction was carried out at the beginning of [@FigJerJems]. ### $A_1\subset A$ has $\pi(A_1)$ close to its convex progression: Construction Apply to $A$ with $\alpha=\frac{1}{2}$, $\lambda=1$ and $\epsilon'=\epsilon \ge \frac{\epsilon_0}{2}$ (by ) to obtain a reduced set $A_1\subset A$ with $$\begin{aligned} \label{A1info}|{\operatorname{co}}(\pi(A_1))\setminus \pi(A_1)| \le \delta^{\frac{1}{2}} |\pi(B)|,\quad|A\setminus A_1|\le \delta^{\frac{1}{2}-c}|B|.\end{aligned}$$ By , we have $$\begin{aligned} \label{A1dk}d_k(A_1)\le \delta |B|+2^k\delta^{\frac{1}{2}-c}|B|\le \delta^{\frac{1}{2}-2c}|B|.\end{aligned}$$ ### $A_2$ has large rows close to their convex progressions: Setup We show that assuming ${\operatorname{co}}(\pi(A'))\setminus \pi(A')$ is small, we can create a subset $A''\subset A'$ by deleting rows with big doubling or small size without changing the size of $A'$ too much. \[betaprop\] Let $\lambda>\alpha>\beta>0$, $\gamma>0$ and $A'\subset B$ with $$\begin{aligned} d_k(A') \le \delta^{\lambda} |B|,\qquad |{\operatorname{co}}(\pi(A'))\setminus \pi(A')|\le \delta^{\alpha}|\pi(B)|. \end{aligned}$$ If $A''\subset A'$ is the union all rows $R_x$ which satisfy $d_1(R_x)\le \delta^{\beta} n_1$ and $|R_x|\geq \delta^{\gamma} n_1$, then $$|A'\setminus A''|\le (\delta^{\alpha-\beta-c}+\delta^{\gamma})|B|.$$ Let $A'''$ be the union all rows $R_x$ of $A'$ which satisfy $d_1(R_x)\le \delta^{\beta} n_1$. For $0\ne\vec{v}\in \{0\}\times\{0,1\}^{k-1}$ and $x\in \{0\}\times \mathbb{Z}^{k-1}$, we have $$|R_x+R_{x+\vec{v}}|-|R_x|-|R_{x+\vec{v}}|\ge \begin{cases}0&|\{x,x+\vec{v}\}\cap \pi(A')|=0\\-n_1 & |\{x,x+\vec{v}\}\cap \pi(A')|=1\\-1 & |\{x,x+\vec{v}\}\cap \pi(A')|= 2\end{cases}$$ From $|{\operatorname{co}}(\pi(A'))\setminus \pi(A')|\le\delta^{\alpha}n_1^{-1}|B|$ and , we have $$\begin{aligned} \left|\left\{x\in\{0\}\times \mathbb{Z}^{k-1}: |\{x,x+\vec{v}\}\cap \pi(A')|=1\right\}\right|&\le 2\left|{\operatorname{co}}(\pi(A'))\setminus \pi(A')\right|+2(k-1)n_1^{-1}n_k(\epsilon_0,\delta)^{-1}|B|\\ &\le \delta^{\alpha-c/4}n_1^{-1}|B|\end{aligned}$$ and $$\begin{aligned} |\{x\in \{0\}\times\mathbb{Z}^{k-1}: |\{x,x+\vec{v}\}\cap \pi(A')|=2\}|\le |\pi(B)|\le n_k(\epsilon_0,\delta)^{-1}|B|\le \delta^{\alpha-c/4}|B|.\end{aligned}$$ Hence, as $\sum_{x\in\{0\}\times \mathbb{Z}^{k-1}}|R_x|=|A'|$, we have (taking $\vec{v}\in \{0\}\times \{0,1\}^{k-1}$ and $x\in \{0\}\times \mathbb{Z}^{k-1}$) $$\begin{aligned} |A'+A'| \ge& \sum_{\vec{v}}\sum_{x}|R_x+R_{x+\vec{v}}|\\ =&\left(\sum_{j=0}^2\sum_{0\ne\vec{v}}\sum_{|\{x,x+\vec{v}\}\cap \pi(A')|=j}|R_x+R_{x+\vec{v}}|\right)+\sum_{x\in \pi(A')}|R_x+R_{x}|\\ \ge& \left(\sum_{0\ne \vec{v}}\sum_{x}|R_{x}|+|R_{x+\vec{v}}|\right)-2(2^{k-1}-1)\delta^{\alpha-c/4}|B|+\sum_{x\in \pi(A')}|R_x+R_{x}|\\ \geq&(2^{k}-2)|A'|-\delta^{\alpha-c/2}|B|+\sum_{x\in \pi(A')}|R_x+R_x|.\end{aligned}$$ In particular, as $\sum_{x \in \pi(A')}|R_x|=|A'|$ and $d_1(R_x)\ge -1$ for all $x$, we have $$\begin{aligned} \delta^{\lambda}|B|\ge d_k(A')\ge& -\delta^{\alpha-c/2}|B|+\sum_{x\in \pi(A')}d_1(R_x)\\\ge&-\delta^{\alpha-c/2}|B|-n_k(\delta,\epsilon_0)^{-1}|B|+ \sum_{x\in \pi(A'\setminus A''')}d_1(R_x)\\ \ge&-\delta^{\alpha-3c/4}|B|+|\pi(A'\setminus A''')|\delta^{\beta} n_1,\end{aligned}$$ so $$|A'\setminus A'''|\le n_1|\pi(A'\setminus A''')|\le (\delta^{\alpha-\beta-3c/4}+\delta^{\lambda-\beta})|B|\le \delta^{\alpha-\beta-c}|B|.$$ Finally note that $A''\subset A'''$ satisfies $|A'''\setminus A''|\leq \delta^{\gamma}n_1|\pi(B)|\leq \delta^{\gamma}|B|$, from which the conclusion follows. ### $A_2$ has large rows close to their convex progression: Construction Let $A_2\subset A_1$ be the set obtained from applied to $A_1$ with $\lambda=\frac{1}{2}-2c$, $\alpha=\frac{1}{2}-3c$, $\beta=\frac{3}{10}$, and $\gamma=\frac{1}{5}$ (by ,). Then for all rows $R_x\subset A_2$, we have $$\begin{aligned} \label{A2betagamma}d_1(R_x)\le \delta^{\frac{3}{10}} n_1,\quad|R_x|\ge \delta^{\frac{1}{5}} n_1,\end{aligned}$$ and by we additionally have $$\begin{aligned} \label{AA2}|A\setminus A_2|\le |A\setminus A_1|+|A_1\setminus A_2|\le \left(\delta^{\frac{1}{2}-c}+\delta^{\frac{1}{5}-4c}+\delta^{\frac{1}{5}}\right)|B|\le \delta^{\frac{1}{5}-5c}|B|.\end{aligned}$$ By , we have $A_2$ is reduced and $$\begin{aligned} \label{A2dk} d_k(A_2)\le \left(\delta+2^k\delta^{\frac{1}{5}-5c}\right)|B|\le \delta^{\frac{1}{5}-6c}|B|.\end{aligned}$$ Freiman’s $3k-4$ theorem [@Freiman], , says that for any $R\subset \mathbb{Z}$, we have $$d_1(R)\ge \min(|R|-3,|{\widehat{\operatorname{co}}}(R)\setminus R|-1).$$ Therefore, because $\delta^{\frac{3}{10}} n_1 <\delta^{\frac{1}{5}} n_1-3$, we have by that every row $R_x$ of $A_2$ satisfies $$\begin{aligned} \label{coint}\text{ }|{\widehat{\operatorname{co}}}(R_x)\setminus R_x|\le \delta^{\frac{3}{10}}n_1+1\le \delta^{\frac{1}{10}}|R_x|+1\le 2\delta^{\frac{1}{10}}|R_x|.\end{aligned}$$ \[dxrmk\] In particular, this means that for each non-empty row $R_x$ we have $|R_x|>\left\lceil \frac{|{\widehat{\operatorname{co}}}(R_x)|}{2}\right\rceil$, so there exist two elements $z_1, z_2 \in R_x$ with $z_1-z_2=(d_x, 0, 0 \hdots, 0)$, where $d_x$ is the common difference in the arithmetic progression ${\widehat{\operatorname{co}}}(R_x)$. ### $A_3\subset A_2$ has $\pi(A_3)$ close to its convex progression: Construction Apply to $A_2$ with $\alpha=\frac{1}{10}$, $\epsilon'\ge\epsilon-\delta^{\frac{1}{5}-5c}\ge \frac{\epsilon_0}{2}$ and $\lambda=\frac{1}{5}-6c$ (by ,) to obtain a reduced set $A_3\subset A_2$ with $$\begin{aligned} \label{A3info}|{\operatorname{co}}(\pi(A_3))\setminus A_3|\le \delta^{\frac{1}{10}}|\pi(B)|, \qquad |A_2\setminus A_3|\le \delta^{\frac{1}{10}-7c}|B|.\end{aligned}$$ By and , we have $$\begin{aligned} \label{AA3}|A\setminus A_3|\le |A\setminus A_2|+|A_2\setminus A_3|\le \left(\delta^{\frac{1}{5}-5c}+\delta^{\frac{1}{10}-7c}\right)|B|\le \delta^{\frac{1}{10}-8c}|B|,\end{aligned}$$ and by , we have $$\begin{aligned} \label{A3dk}d_k(A_3)\le \left(\delta+2^k\delta^{\frac{1}{10}-8c}\right)|B|\le \delta^{\frac{1}{10}-9c}|B|.\end{aligned}$$ Finally, as the rows of $A_3$ are a subset of the rows of $A_2$, by and , we have $$\begin{aligned} \label{A3beta} |R_x|\ge \delta^{\frac{1}{5}}n_1,\qquad |{\widehat{\operatorname{co}}}(R_x)\setminus R_x|\le 2\delta^{\frac{1}{10}}|R_x|.\end{aligned}$$ Reductions Part 2: All rows are in APs of the same step size {#reductionssamesize} ------------------------------------------------------------ We now find a set $A_5\subset A_3$ which has the same properties as $A_3$, and furthermore has the property that for each row $R_x$, the arithmetic progressions ${\widehat{\operatorname{co}}}(R_x)$ have the same step sizes. To do this, we carefully analyze a discrete analogue of Voronoi cells. Let $d_x$ be the smallest consecutive difference between two consecutive elements in $R_x$, which as noted in is also the common difference of ${\widehat{\operatorname{co}}}(R_x)$, and let $d=\min d_x$. ### $A_4\subset A_3$ has all rows in same step size APs: Setup We now show that the rows with $d_x>d$ carry small weight. \[approp\] Let $\lambda>\alpha>0$ and $A'\subset B$ with $d_k(A')\le \delta^{\lambda}|B|$ and $|{\operatorname{co}}(\pi(A'))\setminus \pi(A')|\le \delta^{\alpha}|\pi(B)|$. Let $d_x$ be the smallest consecutive difference between two elements of row $R_x\subset A'$, and let $d=\min d_x$. If $A''\subset A'$ is the subset of rows $R_x$ with $d_x=d$, then $$|A'\setminus A''|\le \delta^{\alpha-c}|B|.$$ Before starting the proof of , we need to prove some claims. shows that $|R_x+R_y|$ is large if $d_x \ne d_y$, and creates a large set of disjoint row sums of this form. is used to prove , which shows that this set of disjoint row sums has small intersection with $A'(+)A'$. Finally, shows $A'(+)A'$ is large, and we can carry out the proof outline described in . \[dxdy\] Let $X,Y \subset \mathbb{Z}$ with $|X| \ge 2$ such that the smallest differences $d_X,d_Y$ between consecutive elements of $X$ and $Y$ respectively satisfy $d_X<d_Y$. Then $$|X+Y| \ge |X|+2|Y|-2.$$ Consider elements $x,x' \in X$ such that $x'-x=d_X$. Let $X_{<x}$ be the set of elements less than $x$ in $X$ and analogously $X_{>x'}$ those elements greater than $x'$. Then the following four sets $$X_{<x}+\min(Y), (Y+x), (Y+x'),X_{> x'}+\max(Y),$$ are disjoint subsets of $X+Y$. Now, we define $$f:\pi(A')\setminus \pi(A'')\to \pi(A'')$$ by letting $f(x)\in \pi(A'')$ be a closest point to $x$ in Euclidean distance (breaking ties arbitrarily). Fibers of $f$ should be thought of as a discrete analogue of Voronoi cells associated to $\pi(A'')$. \[xfnotyf\] We have $x+f(x)\ne y+f(y)\text{ for distinct }x,y\in \pi(A')\setminus \pi(A'')$. In particular, $$Z_1:=\bigsqcup_{x_1\in \pi(A')\setminus \pi(A'')}R_{x_1}+R_{f(x_1)}\subset A'+A'$$ is a disjoint union. Indeed, otherwise $x,y,f(x),f(y)$ form a parallelogram with distinct vertices with diagonals $xf(x)$ and $yf(y)$. However, in any parallelogram (even degenerated as long as the vertices are distinct), the longest diagonal is longer than all sides. Hence, if say $xf(x)$ is the longest diagonal, then $|x-f(x)|>|x-f(y)|$, a contradiction. Let $$Z:=A'(+)A'=\bigsqcup_{\vec{v}\in \{0\} \times \{0,1\}^{k-1}}\bigsqcup_{x_2\in \{0\} \times \mathbb{Z}^{k-1}}R_{x_2}(+)R_{x_2+\vec{v}}\subset A'+A'.$$ We now analyze when $R_{x_1}+R_{f(x_1)}$ and $R_{x_2}(+)R_{x_2+\vec{v}}$ can intersect. \[diagonal\] If $x_1\in \pi(A')\setminus \pi(A'')$, $x_2\in\{0\}\times \mathbb{Z}^{k-1}$, $\vec{v}\in\{0\}\times \{0,1\}^{k-1}$ are such that $R_{x_1}+R_{f(x_1)} \cap R_{x_2}(+)R_{x_2+\vec{v}} \not= \emptyset$, then either $\{x_1,f(x_1)\}=\{x_2,x_2+\vec{v}\}$ or $x_2,x_2+\vec{v}\in \pi(A') \setminus \pi(A'')$. The points $x_1, x_2, f(x_1), x_2+\vec{v}$ form a parallelogram with diagonals $x_1f(x_1)$ and $x_2(x_2+\vec{v})$. Assuming that $\{x_1,f(x_1)\}\not=\{x_2,x_2+\vec{v}\}$, this parallelogram has distinct vertices. The number of odd coordinates of $x_1-f(x_1)$ is the same as the number of odd coordinates of $x_1+f(x_1)=2x_2+\vec{v}$, which is the same as the number of non-zero coordinates of $v$. Hence, $|x_1-f(x_1)|\ge |\vec{v}|$, or equivalently $|x_1-f(x_1)|\ge |x_2-(x_2+\vec{v})|$. Therefore, $x_1f(x_1)$ is the longest diagonal of the above parallelogram. As in a parallelogram (even degenerated as long as the vertices are distinct) the largest diagonal is strictly longer than all sides, we deduce that the diagonal $x_1f(x_1)$ is strictly longer than $x_1x_2$ and $x_1(x_2+\vec{v})$. By definition of $f(x_1)$ this implies $x_2,x_2+\vec{v}\not\in \pi(A'')$. As $R_{x_2},R_{x_2+\vec{v}}$ are nonempty we also have $x_2,x_2+\vec{v}\in \pi(A')$ and the result follows. \[diagonal2\] For any $x_1\in \pi(A') \setminus \pi(A'')$ we have $$|(R_{x_1}+R_{f(x_1)})\setminus Z|\ge |R_{x_1}|-1.$$ We have $|(R_{x_1}+R_{f(x_1)})\setminus Z|=|(R_{x_1}+R_{f(x_1)})\setminus (R_{x_2}+R_{x_2+\vec{v}})|$ for the unique $x_2\in \{0\}\times \mathbb{Z}^{k-1}$ and $\vec{v}\in \{0\}\times \{0,1\}^{k-1}$ such that $x_1+f(x_1)=x_2+(x_2+\vec{v})$. Clearly $|(R_{x_1}+R_{f(x_1)})|\ge |R_{x_1}|-1$, so assume $(R_{x_1}+R_{f(x_1)}) \cap (R_{x_2}(+)R_{x_2+\vec{v}}) \not= \emptyset$. By we have that $\text{either }\{x_1,f(x_1)\}=\{x_2,x_2+\vec{v}\} \text{ or }x_2,x_2+\vec{v}\in \pi(A') \setminus \pi(A'')$. In the former case, by we have that $$|(R_{x_1}+R_{f(x_1)}) \setminus (R_{x_2}(+)R_{x_2+\vec{v}})|=|R_{x_1}+R_{f(x_1)}|-|R_{x_1}|-|R_{f(x_1)}|+1\ge |R_{x_1}|-1.$$ Assume now we are in the latter case. Let $z_1,z_2\in R_{f(x_1)}$ such that $z_1-z_2=(d,0,\ldots,0)$. As the smallest difference in $R_{x_2}(+)R_{x_2+\vec{v}}$ is strictly larger than $d$, for every element $z\in R_{x_1}$ either $z+z_1$ or $z+z_2$ is not in $R_{x_2}(+)R_{x_2+\vec{v}}$, and if there were $z,z'\in R_{x_1}$ with $z+z_1=z'+z_2$ then $z'-z=(d,0,\dots,0)$ contradicting $x_1\not\in \pi(A'')$. Hence $$|(R_{x_1}+R_{f(x_1)}) \setminus (R_{x_2}(+)R_{x_2+\vec{v}})|\ge |R_{x_1}|>|R_{x_1}|-1.$$ \[rs\] We have $|Z| \ge 2^k|A'| - 2^k\delta^{\alpha-c/2}|B|$. Note that for $x_2\in \{0\}\times \mathbb{Z}^{k-1}$, $0\ne\vec{v} \in \{0\}\times \{0,1\}^{k-1}$, we have $$|R_{x_2}(+)R_{{x_2}+\vec{v}}|-|R_{x_2}|-|R_{x_2+\vec{v}}|\ge\begin{cases}0 & |\{x_2,x_2+\vec{v}\}\cap \pi(A')|=0\\ -n_1 & |\{x_2,x_2+\vec{v}\}\cap \pi(A')|=1\\ -1& |\{x_2,x_2+\vec{v}\}\cap \pi(A')|=2.\end{cases}$$ From $|{\operatorname{co}}(\pi(A'))\setminus \pi(A')|\le\delta^{\alpha}n_1^{-1}|B|$ and , we have $$\begin{aligned} |\{x\in \{0\}\times\mathbb{Z}^{k-1}: |\{x,x+\vec{v}\}\cap \pi(A')|=1\}|&\le 2|{\operatorname{co}}(\pi(A'))\setminus \pi(A')|+2(k-1)n_1^{-1}n_k(\epsilon_0,\delta)^{-1}|B|\\ &\le \delta^{\alpha-c/4}n_1^{-1}|B|\end{aligned}$$ and $$\begin{aligned} |\{x\in \{0\}\times\mathbb{Z}^{k-1}: |\{x,x+\vec{v}\}\cap \pi(A')|=2\}|\le |\pi(B)|\le n_k(\epsilon_0,\delta)^{-1}|B|\le \delta^{\alpha-c/4}|B|.\end{aligned}$$ As $\sum_{x\in\{0\}\times \mathbb{Z}^{k-1}}|R_x|=|A'|$, we have (taking $\vec{v}\in \{0\}\times \{0,1\}^{k-1}$ and $x\in \{0\}\times \mathbb{Z}^{k-1}$) $$\begin{aligned} |Z| =& \sum_{\vec{v}}\sum_{x}|R_x(+)R_{x+\vec{v}}|\\ =&\left(\sum_{j=0}^2\sum_{0\ne\vec{v}}\sum_{|\{x,x+\vec{v}\}\cap \pi(A')|=j}|R_x(+)R_{x+\vec{v}}|\right)+\sum_{x\in \pi(A')}|R_x(+)R_x|\\ \ge& -2(2^{k-1}-1)\delta^{\alpha-c/4}|B|-n_k(\epsilon_0,\delta)^{-1}|B|+\sum_{\vec{v} }\sum_{x}|R_{x}|+|R_{x+\vec{v}}|\\ \geq&2^{k}|A'|-2^k\delta^{\alpha-c/2}|B|.\end{aligned}$$ Note that $$|\pi(A')\setminus \pi(A'')|\le |\pi(B)|\le n_k(\epsilon_0,\delta)^{-1}|B|\le \delta^{\alpha-c/2}|B|.$$ Thus, by , , and , we have $$\begin{aligned} |A'+A'|&\ge |Z\cup Z_1|\\ &=|Z|+\sum_{x_1\in \pi(A')\setminus \pi(A'')}|(R_{x_1}+R_{f(x_1)})\setminus Z|\\ &\ge2^k|A'| -2^k\delta^{\alpha-c/2}|B| +\sum_{x_1\in \pi(A')\setminus \pi(A'')}(|R_{x_1}|-1)\\ &\ge2^k|A'|-\delta^{\alpha-3c/4}|B|+|A'\setminus A''|.\end{aligned}$$ We conclude $$|A'\setminus A''| \le d_k(A')+\delta^{\alpha-3c/4}|B|\le \delta^{\lambda} |B|+\delta^{\alpha-3c/4}|B|\le \delta^{\alpha-c}|B|.$$ ### $A_4\subset A_3$ has all rows in same step size APs: Construction Let $A_4\subset A_3$ be the set given by applied to $A_3$ with $\lambda=\frac{1}{10}-9c$ $\alpha=\frac{1}{10}-10c$ (by ,). Then we have $$\begin{aligned} \label{A4info} |A_3\setminus A_4|\le \delta^{\frac{1}{10}-11c}|B|,\end{aligned}$$ and thus by and , $$\begin{aligned} \label{AA4} |A\setminus A_4|\le |A\setminus A_3|+|A_3\setminus A_4|\le (\delta^{\frac{1}{10}-8c}+\delta^{\frac{1}{10}-11c})|B|\le \delta^{\frac{1}{10}-12c}|B|.\end{aligned}$$ By , $A_4$ is reduced and we have $$\begin{aligned} \label{A4dk} d_k(A_4)\le (\delta+2^k\delta^{\frac{1}{10}-12c})|B|\le \delta^{\frac{1}{10}-13c}|B|.\end{aligned}$$ By construction and , ${\widehat{\operatorname{co}}}(R_x)$ has the same step size $d$ for all rows $R_x$ of $A_4$. Finally, by , we have $$\begin{aligned} \label{A4beta} |R_x|\ge \delta^{\frac{1}{5}}n_1,\qquad |{\widehat{\operatorname{co}}}(R_x)\setminus R_x|\le 2\delta^{\frac{1}{10}}|R_x|.\end{aligned}$$ ### $A_5\subset A_4$ with $\pi(A_5)$ close to its convex progression: Construction Apply to $A_4$ with $\lambda=\frac{1}{10}-13c$, $\alpha=\frac{1}{20}$ and $\epsilon'\ge \epsilon-\delta^{\frac{1}{10}-12c}\ge \frac{\epsilon_0}{2}$ (by ,) to obtain a reduced set $A_5\subset A_4$ with $$\begin{aligned} \label{A5daggerinfo} |{\operatorname{co}}(\pi(A_5))\setminus \pi(A_5)|\le \delta^{\frac{1}{20}}|\pi(B)|\qquad |A_4\setminus A_5|\le \delta^{\frac{1}{20}-14c}|B|.\end{aligned}$$ By and , we have $$\begin{aligned} \label{AA5dagger} |A\setminus A_5|\le |A\setminus A_4|+|A_4\setminus A_5|\le (\delta^{\frac{1}{10}-12c}+\delta^{\frac{1}{20}-14c})|B|\le \delta^{\frac{1}{20}-15c}|B|,\end{aligned}$$ and by , we have $$\begin{aligned} \label{A5dk} d_k(A_5)\le (\delta+2^k\delta^{\frac{1}{20}-15c})|B|\le \delta^{\frac{1}{20}-16c}|B|.\end{aligned}$$ Furthermore, all rows of $A_5$ are also rows of $A_4$, so have the same step size $d$ and satisfy , so $$\begin{aligned} \label{A5beta} |R_x|\ge \delta^{\frac{1}{5}}n_1,\qquad |{\widehat{\operatorname{co}}}(R_x)\setminus R_x|\le 2\delta^{\frac{1}{10}}|R_x|.\end{aligned}$$ Reductions Part 3: Showing the rows of $A_5$ are almost intervals {#almostintervals} ----------------------------------------------------------------- We now show that the arithmetic progressions ${\widehat{\operatorname{co}}}(R_x)$ containing the rows $R_x$ of $A_5$ are in fact intervals i.e. ${\widehat{\operatorname{co}}}(R_x)={\operatorname{co}}(R_x)$. We suppose by way of contradiction that the rows $R_x$ have convex progression ${\widehat{\operatorname{co}}}(R_x)$ with the same step size $d\ne 1$. Let $\pi':\mathbb{Z}^k\to \mathbb{Z}$ be the projection onto the second coordinate. For a set $A'\subset \mathbb{Z}^k$, we let a “hyperplane” $H_y$ be $\pi'^{-1}(y)\cap A'$. We shall make a series of temporary reductions in order to arrive at a contradiction, and we shall notate sets used in this proof by contradiction with the dagger symbol $\dagger$. The hyperplanes $H_y$ of a set $A'$ are unions of rows $R_x$. ### $A_6^{\dagger}\subset A_5$ has big hyperplanes with small doubling: Setup First, we show that assuming ${\operatorname{co}}(\pi(A'))\setminus \pi(A')$ is small, we can create a subset $A''\subset A'$ by deleting hyperplanes with big doubling or small size without changing the size of $A'$ too much. This is analogous to for rows. \[hypbetaprop\] Let $\lambda>\alpha>\beta >0, \gamma>0$ and let $A'\subset B$ with $$d_k(A')\le \delta^{\lambda}|B|,\qquad |{\operatorname{co}}(\pi(A'))\setminus \pi(A')|\le \delta^{\alpha}|\pi(B)|.$$ If $A''\subset A'$ is the union of all hyperplanes $H_y$ with $d_{k-1}(H_y)\le \delta^{\beta} n_2^{-1}|B|$ and $|H_y|\ge \delta^{\gamma}n_2^{-1}|B|$, then $$|A'\setminus A''|\le (\delta^{\alpha-\beta-c}+\delta^{\gamma})|B|.$$ Let $A'''\subset A'$ be the union of the hyperplanes $H_y$ with $d_{k-1}(H_y)\le \delta^{\beta}n_2^{-1}|B|$. For $0\ne \vec{v}\in \{0\}\times\{1\}\times \{0,1\}^{k-2}$ we have $$|R_x+R_{x+\vec{v}}|-|R_x|-|R_{x+\vec{v}}|\ge \begin{cases}0&|\{x,x+\vec{v}\}\cap \pi(A')|=0\\-n_1 & |\{x,x+\vec{v}\}\cap \pi(A')|=1\\-1 & |\{x,x+\vec{v}\}\cap \pi(A')|= 2\end{cases}$$ From $|{\operatorname{co}}(\pi(A'))\setminus \pi(A')|\le \delta^{\alpha}n_1^{-1}|B|$ and , we have $$\begin{aligned} |\{x\in\{0\}\times \mathbb{Z}^{k-1}: |\{x,x+\vec{v}\}\cap \pi(A')|=1\}|&\le 2|{\operatorname{co}}(\pi(A'))\setminus \pi(A')|+2(k-1)n_1^{-1}n_k(\epsilon_0,\delta)^{-1}|B|\\ &\le \delta^{\alpha-c/4}n_1^{-1}|B|\end{aligned}$$ and $$\begin{aligned} |\{x\in\{0\}\times \mathbb{Z}^{k-1}: |\{x,x+\vec{v}\}\cap \pi(A')|=2\}|\le |\pi(B)|\le n_k(\epsilon_0,\delta)^{-1}|B|\le \delta^{\alpha-c/4}|B|.\end{aligned}$$ Hence as $\sum_{x \in \{0\}\times \mathbb{Z}^{k-1}}|R_x|=|A'|$, we have (taking $\vec{v}\in \{0\}\times \{1\}\times \{0,1\}^{k-2}$ and $x\in \{0\}\times \mathbb{Z}^{k-1}$) $$\begin{aligned} |A'+A'|\ge& \left(\sum_{y\in \mathbb{Z}} |H_y+H_y|\right)+\sum_{j=0}^2\sum_{\vec{v}}\sum_{|\{x,x+\vec{v}\}\cap \pi(A')|=j}|R_x+R_{x+\vec{v}}|\\ \ge& \left(\sum_{y \in \mathbb{Z}}|H_y+H_y|\right)+\left(\sum_{\vec{v}}\sum_{x}|R_x|+|R_{x+\vec{v}}|\right) -2\cdot 2^{k-2}\delta^{\alpha-c/4}|B|\\ \ge&\left(\sum_{y\in \mathbb{Z}}|H_y+H_y|\right)+2^{k-1}|A'|-\delta^{\alpha-c/2}|B|.\end{aligned}$$ Now, as $\sum_{y \in \mathbb{Z}}|H_y|=|A'|$, by with $H_y$ and the box $\pi'^{-1}(y)\cap B$, we have $$\begin{aligned} \delta^{\lambda}|B|\ge d_k(A')\ge& \left(\sum_{y\in \pi'(A')}d_{k-1}(H_y)\right)-\delta^{\alpha-c/2}|B|\\ \ge& \left(\sum_{y\in \pi'(A')\setminus \pi'(A''')}d_{k-1}(H_y)\right)-n_2\cdot 2^{2(k-1)}n_k(\epsilon_0,\delta)^{-1}n_2^{-1}|B|-\delta^{\alpha-c/2}|B|\\ \ge& |\pi'(A'\setminus A''')|\cdot \delta^{\beta} n_2^{-1}|B|-\delta^{\alpha-3c/4}|B|\\ \ge&\delta^{\beta} |A'\setminus A'''|-\delta^{\alpha-3c/4}|B|.\end{aligned}$$ Therefore, $$|A'\setminus A'''| \le (\delta^{\lambda-\beta}+\delta^{\alpha-\beta-3c/4})|B|\le \delta^{\alpha-\beta-c}|B|.$$ Finally, note that $A''\subset A'''$ satisfies $|A'''\setminus A''|\le n_2\delta^{\gamma}n_2^{-1}|B|=\delta^{\gamma}|B|$, from which the conclusion follows. ### $A_6^{\dagger}\subset A_5$ has big hyperplanes with small doubling: Construction Applying to $A_5$ with $\lambda=\frac{1}{20}-16c$, $\alpha=\frac{1}{40}$, $\beta=\frac{1}{80}$, and $\gamma=\frac{1}{160}-c$, we obtain $$\begin{aligned} \label{A6daggerinfo}|A_5\setminus A_6^{\dagger}|\le (\delta^{\frac{1}{80}-c}+\delta^{\frac{1}{160}-c})|B|\le \delta^{\frac{1}{160}-2c} |B|\end{aligned}$$ and for every hyperplane $H_y\subset A_6^{\dagger}$ we have $$\begin{aligned} \label{hypdoub}d_{k-1}(H_y)\le \delta^{\frac{1}{80}}n_2^{-1}|B|, \qquad |H_y|\ge \delta^{\frac{1}{160}-c} n_2^{-1}|B|.\end{aligned}$$ We also have by and that $$\begin{aligned} \label{AA6dagger} |A\setminus A_6^{\dagger}|\le |A\setminus A_5|+|A_5\setminus A_6^{\dagger}|\le (\delta^{\frac{1}{20}-15c}+\delta^{\frac{1}{160}-2c})|B|\le \delta^{\frac{1}{160}-3c}|B|,\end{aligned}$$ and by we have $A_6^{\dagger}$ is reduced and $$\begin{aligned} \label{A6dk} d_k(A_6^{\dagger})\le (\delta+2^k\delta^{\frac{1}{160}-3c})|B|\le \delta^{\frac{1}{160}-4c}|B|.\end{aligned}$$ Consider the set $H_y$ contained inside a box $B_y:=\pi'^{-1}(y)\cap B$ with sides at least $n_k(\epsilon_0,\delta)$. We have $|H_y| \ge \delta^{\frac{1}{160}-c} |B_y|$ and $d_{k-1}(H_y) \le \delta^{\frac{1}{160}+c}|H_y|$. By , the number of parallel hyperplanes needed to cover $H_y$ is at least $n_k(\epsilon_0,\delta)\delta^{\frac{1}{160}-c}>m_{k-1}(\delta^{\frac{1}{160}+c})$. Hence by for dimension $k-1$, we deduce that $$\begin{aligned} \label{convhyp}|{\widehat{\operatorname{co}}}(H_y)\setminus H_y|\le c_{k-1}\delta^{\frac{1}{160}+c}|B_y|\le\delta^{\frac{1}{160}}|B_y|.\end{aligned}$$ \[stepsize\] For a hyperplane $H_y\subset A_6^{\dagger}$, the smallest affine sublattice $\Lambda_{H_y}\subset \pi'^{-1}(y)=\mathbb{Z}\times \{y\}\times \mathbb{Z}^{k-2}$ containing $H_y$ has the property that the nonempty rows of $\Lambda_{H_y}$ have step size $d$. For each row $R_x$ contained in a hyperplane $H_y$, the arithmetic progression ${\widehat{\operatorname{co}}}(R_x)$ has step size $d$. Let $d'$ be the uniform step size of the nonempty rows of $\Lambda_{H_y}$ (which exists by Lagrange’s theorem), and hence of the nonempty rows of ${\widehat{\operatorname{co}}}(H_y)$. Assume for the sake of contradiction $d'\not=d$. As $d'$ divides $d$, for every row $R_x$ of $H_y$ and corresponding row $R_x'$ of ${\widehat{\operatorname{co}}}(H_y)$, we have $|R_x'\setminus R_x|\ge \frac{1}{2}|R_x|$ (as each row $R_x$ has at least $2$ elements by ). Adding this over all rows $R_x$ of $H_y$, we obtain from $$|{\widehat{\operatorname{co}}}(H_y)\setminus H_y| \ge \frac{1}{2}|H_y|\ge \frac{1}{2}\delta^{\frac{1}{160}-c} |B_y|$$ contradicting that $\delta^{\frac{1}{160}}|B_y|\ge |{\widehat{\operatorname{co}}}(H_y)\setminus H_y|.$ ### $A_7^{\dagger}\subset A_6^{\dagger}$ with $\pi(A_7^{\dagger})$ close to its convex progression: Construction Apply to $A_{6}^{\dagger}$ with $\lambda=\frac{1}{160}-4c$, $\alpha=\frac{1}{320}$, $\epsilon'\ge\epsilon-\delta^{\frac{1}{160}-3c}\ge \frac{\epsilon_0}{2}$ (by ,) to obtain a reduced set $A_7^{\dagger}\subset A_6^{\dagger}$ with $$\begin{aligned} \label{A7daggerinfo} |A_6^{\dagger}\setminus A_7^{\dagger}|\le \delta^{\frac{1}{320}-5c}|B|,\qquad |{\operatorname{co}}(\pi(A_7^{\dagger}))\setminus \pi(A_7^{\dagger})|\le \delta^{\frac{1}{320}}|\pi(B)|.\end{aligned}$$ By and , we have $$\begin{aligned} \label{AA7dagger} |A\setminus A_7^{\dagger}|\le |A\setminus A_6^{\dagger}|+|A_6^{\dagger}\setminus A_7^{\dagger}| \le (\delta^{\frac{1}{160}-3c}+\delta^{\frac{1}{320}-5c})|B|\le \delta^{\frac{1}{320}-6c}|B|,\end{aligned}$$ and by we have $$\begin{aligned} \label{A7daggerdk} d_k(A_7^{\dagger})\le (\delta+2^k\delta^{\frac{1}{320}-6c})|B|\le \delta^{\frac{1}{320}-7c}|B|.\end{aligned}$$ As we pass from $A_6^{\dagger}$ to $A_7^{\dagger}$, the affine sub-lattice $\Lambda_{H_y}$ shrinks, so by the nonempty rows of $\Lambda_{H_y}$ have step size at least $d$. As the nonempty rows of $A_7^{\dagger}$ have step size $d$, this forces the nonempty rows of $\Lambda_{H_y}$ to have step size exactly $d$. We note that we do not know that the hyperplanes of $A_7^{\dagger}$ have big size or small doubling. ### $A_8^{\dagger}\subset A_7^{\dagger}$ has $\pi(H)$ reduced for all hyperplanes $H$: Construction Let $A_8^{\dagger}\subset A_7^{\dagger}$ be the union of all hyperplanes $H_y$ such that $\pi(H_y)$ is reduced in $\{0\}\times \{y\}\times \mathbb{Z}^{k-2}$. Recall we let $B_y=\pi'^{-1}(y)\cap B$, so $H_y\subset B_y$. If $\pi(H_y)$ is not reduced inside $\{0\}\times\{y\}\times \mathbb{Z}^{k-2}$, then there is a direction $e_j$ with $j\in \{3,\hdots, k\}$ such that $\pi(H_y)\cap(\pi(H_y)+e_j)=\emptyset$. Hence, letting $\pi_j:\mathbb{Z}^k\to \mathbb{Z}^{k-1}$ be the projection away from the $j$th coordinate and $S_x=\pi_j^{-1}(x)\cap \pi(H_y)$, we have $|{\operatorname{co}}(S_x)\setminus S_x| \ge \frac{1}{2}(|S_x|-1)$. Summing the above inequality over all $x\in \pi_j(\pi(H_y))\subset \pi_j(\pi(B_y))$, we deduce $$|{\operatorname{co}}(\pi(H_y))\setminus \pi(H_y)|\ge \sum_{x\in \pi_j(\pi(H_y))}|{\operatorname{co}}(S_x)\setminus S_x| \ge \frac{1}{2}(|\pi(H_y)|-n_k(\epsilon_0,\delta)^{-1}|\pi(B_y)|).$$ Adding this over all $y$ with $\pi(H_y)$ not reduced, we obtain by that $$\begin{aligned} \nonumber|\pi(A_7^{\dagger}\setminus A_8^{\dagger})|&\le n_k(\epsilon_0,\delta)^{-1}|\pi(B)|+2\sum_{\pi(H_y)\text{ not reduced}}|{\operatorname{co}}(\pi(H_y))\setminus \pi(H_y)|\\\label{piA7A8}&\le n_k(\epsilon_0,\delta)^{-1}|\pi(B)|+2|{\operatorname{co}}(\pi(A_7^{\dagger}))\setminus \pi(A_7^{\dagger})|\le 3\delta^{\frac{1}{320}}|\pi(B)|\le \delta^{\frac{1}{320}-c}|\pi(B)|,\end{aligned}$$ so in particular we have $$|A_7^{\dagger}\setminus A_8^{\dagger}|\le \delta^{\frac{1}{320}-c}|B|.$$ Hence by we have $$\begin{aligned} \label{AA8dagger} |A\setminus A_8^{\dagger}|\le |A\setminus A_7^{\dagger}|+|A_7^{\dagger}\setminus A_8^{\dagger}|\le \left(\delta^{\frac{1}{320}-6c}+\delta^{\frac{1}{320}-c}\right)|B|\le \delta^{\frac{1}{320}-7c}|B|,\end{aligned}$$ and by we have $A_8^{\dagger}$ is reduced and $$\begin{aligned} \label{A8dk}d_k(A_8^{\dagger})\le \left(\delta+2^k\delta^{\frac{1}{320}-7c}\right)|B|\le \delta^{\frac{1}{320}-8c}|B|.\end{aligned}$$ Note that $|{\operatorname{co}}(\pi(A_8^{\dagger}))\setminus \pi(A_8^{\dagger})|\le |{\operatorname{co}}(\pi(A_7^{\dagger}))\setminus \pi(A_7^{\dagger})|+|\pi(A_7^{\dagger}\setminus A_8^{\dagger})|$, so we have by , that $$\begin{aligned} \label{copiA8dagger}|{\operatorname{co}}(\pi(A_8^{\dagger}))\setminus \pi(A_8^{\dagger})|\le \left(\delta^{\frac{1}{320}}+\delta^{\frac{1}{320}-c}\right)|\pi(B)|\le \delta^{\frac{1}{320}-2c}|\pi(B)|.\end{aligned}$$ As we pass from $A_7^{\dagger}$ to $A_8^{\dagger}$, the affine sub-lattice $\Lambda_{H_y}$ shrinks, so the nonempty rows of $\Lambda_{H_y}$ have step size at least $d$. As the nonempty rows of $A_8^{\dagger}$ have step size $d$, this forces the nonempty rows of $\Lambda_{H_y}$ to have step size exactly $d$. Furthermore, the reducedness of $\pi(H_y)$ implies $\pi(\Lambda_{H_y})=\{0\}\times \{y\}\times\mathbb{Z}^{k-2}$. ### Contradiction We now derive a contradiction. Let $H_{y_1},\ldots,H_{y_\ell}$ be the nonempty hyperplanes of $A_8^{\dagger}$ with $y_1<\ldots<y_\ell$, and for notational convenience set $H_{i}:=H_{y_i}$ and $\Lambda_i:=\Lambda_{H_{y_i}}$. Let $$\Phi_i:\pi(\Lambda_{i})=\{0\}\times\{y_i\}\times \mathbb{Z}^{k-2}\to \mathbb{Z}/d\mathbb{Z}$$ be the affine-linear function defined by taking $\Phi_i(0,y_i,z)=z'\mod d$ where $(z'+d\mathbb{Z},y_i,z)$ is a row in $\Lambda_{i}$. We create $r$ subintervals $I_i\subset \{1,\ldots,\ell\}$ satisfying the following properties. - $1\in I_1$ - $\bigsqcup_{j\in I_i}H_j$ is not reduced in $\mathbb{Z}^k$, and $H_{1+\max I_i}\sqcup\bigsqcup_{j \in I_i}H_j$ is reduced, for $1\leq i\le r-1$. - $\min I_{i+1}=\max I_i+\begin{cases} 0 & H_{\max I_i}\sqcup H_{1+\max I_{i}}\text{ is not reduced.}\\1 & H_{\max I_i}\sqcup H_{1+\max I_i}\text{ is reduced.} \end{cases}$ These conditions uniquely determine intervals $I_1,\ldots,I_r$ which cover $\{1,\ldots,\ell\}$, and as $A_8^{\dagger}$ is reduced we have $r\ge 2$. \[minmaxrmk\] If $\max I_i=\min I_{i+1}$ then $|I_i|\ge 2$. If instead $\max I_i+1=\min I_{i+1}$, then with $j=\max I_i$ we have $y_{j}+1=y_{j+1}$, and $\Phi_j(w)-\Phi_{j+1}(w+e_2): \{0\}\times \{y_{j}\}\times \mathbb{Z}^{k-2}\to \mathbb{Z}$ is non-constant. For $1\le i \le r-1$, let $z_i\in H_{1+\max I_i}$ be a point not in the affine sub-lattice containing $\bigsqcup_{j \in I_i} H_{j}$. Let $$f:\bigsqcup_{j\in [1,\min I_r-1]}H_j\to A_8^{\dagger}$$ be defined by setting $$f\left(\bigsqcup_{j \in [\min I_i,\min I_{i+1}-1]}H_j\right)=z_i.$$ If $z',z''\in \bigsqcup_{j\in[1,\min I_r-1]}H_j$ are distinct, then we have $$z'+f(z')\ne z''+f(z'').$$ Indeed, if they were equal then $$\pi'(z')+\pi'(f(z'))=\pi'(z'')+\pi'(f(z'')),$$ and if without loss of generality $\pi'(z')\le \pi'(z'')$, then $\pi'(f(z'))\le \pi'(f(z''))$, so we must have $\pi'(z')=\pi'(z'')$. Therefore $f(z')=f(z')$, so $z'=z''$, a contradiction. Hence the set $$Z_1=\bigsqcup_{i=1}^{r-1}\bigsqcup_{j\in [\min I_i,\max I_{i}-1]}(H_j+z_i)\subset A_8^{\dagger}+A_8^{\dagger}$$ is a disjoint union as $[\min I_{i},\max I_i-1]\subset [\min I_i,\min I_{i+1}-1]$. For $ \vec{v}\in \{0\}\times\{0,1\}^{k-1}$ and $x\in \{0\}\times \mathbb{Z}^{k-1}$, and $z\in H_t$ with $t\in [\min I_i, \max I_{i}-1]$ for some $1\le i \le r-1$, we have $$z+f(z)\not \in R_x + R_{x+\vec{v}}.$$ Assume to the contrary that $z+f(z)\in R_x+R_{x+\vec{v}}$. First note that $\pi'(x)\ge y_{\min I_i}$ since $$2y_{\min I_i}\le \pi'(z)+\pi'(f(z))=\pi'(x)+\pi'(x+\vec{v})\le 2\pi'(x)+1.$$ Next, we note that $\pi'(x+\vec{v})\le y_{\max I_i}$. Indeed, if $\pi'(x+\vec{v})>y_{\max I_i}$, then as $R_x$ and $R_{x+\vec{v}}$ are non-empty, we have $\pi'(x+\vec{v})\geq y_{1+\max I_i}$ and $\pi'(x)\geq y_{\max I_i}$. However, then $$\pi'(x+\vec{v})+\pi'(x)\geq y_{1+\max I_i}+y_{\max I_i}>\pi'(f(z))+\pi'(z),$$ a contradiction. Hence, $z,R_x,R_{x+\vec{v}}$ are all contained in the affine sub-lattice containing $\sqcup_{j \in I_i}H_j$, and $f(z)$ is not in this affine sub-lattice by construction, contradicting $z+f(z)\in R_x+R_{x+\vec{v}}$. Hence the sets $$Z:=A_8^{\dagger}(+)A_8^{\dagger}= \bigsqcup_{\vec{v}\in \{0\}\times \{0,1\}^{k-1}}\bigsqcup_{x\in \{0\}\times\mathbb{Z}^{k-1}}R_x(+)R_{x+\vec{v}}\subset A_8^{\dagger}+A_8^{\dagger}$$ and $Z_1$ are disjoint. The set of indices $\mathcal{I}=[1,\min I_{r}-1]\setminus \bigcup_{i=1}^{r-1} [\min I_i,\max I_{i}-1]$ whose corresponding hyperplanes $H_j$ were not accounted for by $Z_1$ are precisely those indices $j$ such that there exists $1 \le i\le r-1$ with $j=\max I_i=\min I_{i+1}-1$. We will now find a third set $Z_2$ disjoint from $Z$, and $Z_1$ which accounts for the hyperplanes with indices in $\mathcal{I}$. Consider two consecutive hyperplanes $H_j$, and $H_{j+1}$ with $j\in \mathcal{I}$, and let $i$ be such that $j=\max I_i$ and $j+1=\min I_{i+1}$. Note that by , we have $y_j+1=y_{j+1}$ and the affine-linear function $\Phi_j(w)-\Phi_{j+1}(w+e_2):\{0\}\times \{y_j\}\times \mathbb{Z}^{k-2}\to \mathbb{Z}/d\mathbb{Z}$ is non-constant. In particular, there is an index $s_j\in \{3,\ldots,k\}$ so that the standard basis vector $e_{s_j}$ satisfies $\Phi_j(w+e_{s_j})-\Phi_{j+1}(w+e_{s_j}+e_2)\ne \Phi_j(w)-\Phi_{j+1}(w+e_2)$ for all $w$. Rearranging, $$\Phi_j(w+e_{s_j})+\Phi_{j+1}(w+e_2)\ne \Phi_j(w)+\Phi_{j+1}(w+e_{s_j}+e_2)$$ for all $w$. Hence we have $$\left(R_{w+e_2}+R_{w+e_{s_j}}\right) \cap \left(R_{w}+R_{w+e_2+e_{s_j}}\right) = \emptyset$$ for all $w$ since they lie in different translated $d\mathbb{Z}$-progressions. For $w\in \{0\}\times \{y_j\}\times \mathbb{Z}^{k-2}$, we have $$(R_{w+e_2}+R_{w+e_{s_j}})\cap Z=\emptyset.$$ If $(w+e_2)+(w+e_{s_j})=x+(x+\vec{v})$ for some $x\in \{0\}\times \mathbb{Z}^{k-1}$ and $\vec{v}\in \{0\}\times \{0,1\}^{k-1}$, then by looking at the odd coordinates we see that $\vec{v}=e_2+e_{s_j}$ and hence $x=w$. But then $$(R_{w+e_2}+R_{w+e_{s_j}})\cap Z=(R_{w+e_2}+R_{w+e_{s_j}})\cap (R_{w}(+)R_{w+e_2+e_{s_j}})=\emptyset.$$ Hence the disjoint union $$Z_2:=\bigsqcup_{j\in \mathcal{I}}\bigsqcup_{w \in \{0\}\times \{y_j\}\times \mathbb{Z}^{k-2}}R_{w+e_2}(+)R_{w+e_{s_j}}$$ is disjoint from $Z$. Finally, we prove a claim which implies $Z_1$ is disjoint from $Z_2$. For any $1 \le s \le r-1$ and for any $z\in H_t$ with $t \in [\min I_s, \max I_s-1]$ and for any $w\in \{0\}\times \{y_j\}\times \mathbb{Z}^{k-2}$ with $j \in \mathcal{I}$, we have $$z+f(z)\not \in R_{w+e_2}+R_{w+e_{s_j}}.$$ Assume for the sake of contradiction that $z+f(z)\in R_{w+e_2}+R_{w+e_{s_j}}$. Let $i$ be such that $j=\max I_i=\min I_{i+1}-1$. First, suppose that $\pi'(z)=y_t\le y_j-1$. Then $s \le i$ as if $i<s$ then $j=\min I_{i+1}-1< \min I_s\leq t$, and therefore $y_j<y_t$, a contradiction. Therefore $\pi'(f(z))= y_{\max I_s +1}\le y_{\max I_i +1}= y_{j+1}=y_j+1$ by , so we obtain the contradiction $$\pi'(z+f(z))\le y_j-1+(y_j+1)<2y_j+1=\pi'(w+e_2)+\pi'(w+e_{s_j}).$$ Next, suppose $\pi'(z)=y_t=y_j$. Then $j=t$, contradicting that $\mathcal{I}$ is disjoint from $[\min I_s,\max I_s-1]$ by construction of $\mathcal{I}$. Finally, suppose that $\pi'(z)=y_t\ge y_j+1$. Then as $\pi'(f(z))> \pi'(z)$, we have the contradiction $$\pi'(z+f(z))> 2y_j+2>2y_j+1=\pi'(w+e_2)+\pi'(w+e_{s_j}).$$ Hence $Z,Z_1$ and $Z_2$ are disjoint subsets of $A_8^\dagger+A_8^\dagger$. Note that for $x_1\ne x_2$ we have $$|R_{x_1}(+)R_{x_2}|-|R_{x_1}|-|R_{x_2}|\ge \begin{cases}0&|\{x_1,x_2\}\cap \pi(A')|=0\\-n_1 & |\{x_1,x_2\}\cap \pi(A')|=1\\-1 & |\{x_1,x_2\}\cap \pi(A')|= 2\end{cases}$$ For $0\ne\vec{v}\in \{-1,0,1\}^k$ we have from and , $$\begin{aligned} |\{x\in \mathbb{Z}^{k-1}: |\{x,x+\vec{v}\}\cap \pi(A^\dagger_8)|=1\}|&\le 2|{\operatorname{co}}(\pi(A^\dagger_8))\setminus \pi(A^\dagger_8)|+2(k-1)n_1^{-1}n_k(\epsilon_0,\delta)^{-1}|B|\\ &\le \delta^{\frac{1}{320}-3c}n_1^{-1}|B|\end{aligned}$$ and $$\begin{aligned} |\{x\in \mathbb{Z}^{k-1}: |\{x,x+\vec{v}\}\cap \pi(A^\dagger_8)|=2\}|\le |\pi(B)|\le n_k(\epsilon_0,\delta)^{-1}|B|\le \delta^{\frac{1}{320}-3c}|B|.\end{aligned}$$ Therefore, we have (taking $\vec{v}\in \{0\}\times \{0,1\}^{k-1}$ and $x\in \{0\}\times \mathbb{Z}^{k-1}$) $$\begin{aligned} |A_8^{\dagger}+A_8^{\dagger}| \ge& |Z| + |Z_1| + |Z_2|\\ =&\left(\sum_{\vec{v}}\sum_{x} |R_x(+)R_{x+\vec{v}}|\right)+\sum_{i=1}^{r-1}\sum_{j\in [\min I_i,\max I_{i+1}-1]}|H_j|\\&+\sum_{j \in \mathcal{I}}\sum_{w\in \{0\}\times \{y_j\}\times \mathbb{Z}^{k-2}}|R_{w+e_2}(+)R_{w+e_{s_j}}|\\ \ge&\left(\sum_{\vec{v}}\sum_{x} |R_x|+|R_{x+\vec{v}}|\right)+\sum_{i=1}^{r-1}\sum_{j\in [\min I_i,\max I_{i+1}-1]}|H_j|\\&+\left(\sum_{j \in \mathcal{I}}\sum_{w\in \{0\}\times \{y_j\}\times \mathbb{Z}^{k-2}}|R_{w+e_{s_j}}|\right)-\delta^{\frac{1}{320}-4c}|B|\\ =&2^k|A_8^{\dagger}|+\left(\sum_{j\in [1,\min I_{r}-1]}|H_j|\right)-\delta^{\frac{1}{320}-4c}|B|.\end{aligned}$$ If we consider the same process ran in reverse, we produce another collection of intervals $I'_1, \hdots I'_{r'}\subset \{1,\ldots \ell\}$ with $\ell\in I_1'$ such that $$|A_8^{\dagger}+A_8^{\dagger}| \ge 2^k|A_8^{\dagger}|+\left(\sum_{j\in [\max I'_{r'}+1,l]}|H_j|\right)-\delta^{\frac{1}{320}-4c}|B|.$$ As $A_8^{\dagger}$ is reduced, we have that $r'\ge 2$. As $\sqcup_{i \in I_r} H_{i}$ is not reduced, we have that $I_r \subset I_1'\subset [\max I_r'+1,\ell]$. Averaging the two inequalities we get by that $$d_k(A_8^{\dagger})\ge \frac{1}{2}|A_8^{\dagger}|-\delta^{\frac{1}{320}-4c}|B|\ge \frac{\epsilon_0}{4}|B|,$$ contradicting that $d_k(A_8^{\dagger})\le \delta^{\frac{1}{320}-8c}|B|$. The conclusion follows. Reductions Part 4: Filling in the rows to create $A_+\supset A_5$ {#reductionAplus} ----------------------------------------------------------------- We recall that we have just shown that all rows $R_x$ of $A_5$ satisfy ${\operatorname{co}}(R_x)={\widehat{\operatorname{co}}}(R_x)$. We now show that filling in all of the rows of $A_5$ does not change the size of $A_5$ or $d_k(A_5)$ too much. ### $A_+ \supset A_5$ with all rows filled in Let $A_+ \supset A_5$ be obtained by replacing each row $R_x$ of $A_5$ with ${\operatorname{co}}(R_x)$. We have $A_+$ is reduced as $A_5$ is reduced. Also by and we have $$\begin{aligned} \label{AA+}|A \Delta A_+|\le |A\setminus A_5|+|A_+\setminus A_5|\le \delta^{\frac{1}{20}-15c}|B|+2\delta^{\frac{1}{10}}|A_5|\le \delta^{\frac{1}{20}-16c}|B|.\end{aligned}$$ Furthermore, $\pi(A_+)=\pi(A_5)$, so by we have $$|{\operatorname{co}}(\pi(A_+))\setminus \pi(A_+)|=|{\operatorname{co}}(\pi(A_5))\setminus \pi(A_5)|\le \delta^{\frac{1}{20}}|\pi(B)|.$$ $d_k(A_+)\leq \delta^{\frac{1}{20}-17c}|B|.$ We begin the proof with the following general claim. \[disjointunion\] Given a finite family of finite subsets $Z_i\subset \mathbb{Z}$ and a parameter $\rho$ such that $|{\operatorname{co}}(Z_i)| \le (1+\rho)|Z_i|$, we have $$\left|\bigcup {\operatorname{co}}(Z_i)\right| \le (1+2\rho)\left|\bigcup Z_i\right|.$$ Let $\mathcal{F}, \mathcal{G}$ be families of indices such that $\bigcup {\operatorname{co}}(Z_i)=(\bigsqcup_{i \in \mathcal{F}} {\operatorname{co}}(Z_i))\cup (\bigsqcup_{i \in \mathcal{G}}{\operatorname{co}}(Z_i))$, where both are disjoint unions. Then $$\left|\bigcup {\operatorname{co}}(Z_i)\setminus \bigcup Z_i\right| \le \sum_{i\in \mathcal{F}\cup \mathcal{G}} |{\operatorname{co}}(Z_i)\setminus Z_i| \le \sum_{i\in \mathcal{F}\cup \mathcal{G}} \rho|Z_i|\le 2\rho\left|\bigcup Z_i\right|.$$ Now, for nonempty rows $R_{x_{1}},R_{x_{2}}$ of $A_5$, we have by that $|{\operatorname{co}}(R_{x_{1}})|\le (1+2\delta^{\frac{1}{10}})|R_{x_{1}}|$ and $|{\operatorname{co}}(R_{x_{2}})| \le (1+2\delta^{\frac{1}{10}})|R_{x_{2}}|$, so $$|{\operatorname{co}}(R_{x_{1}}+R_{x_{2}})|=|{\operatorname{co}}(R_{x_{1}})|+|{\operatorname{co}}(R_{x_{2}})|-1 \le (1+2\delta^{\frac{1}{10}})(|R_{x_{1}}|+|R_{x_{2}}|)-1\le (1+4\delta^{\frac{1}{10}})|R_{x_{1}}+R_{x_{2}}|.$$ Taking the sets $Z_i$ to be the pairwise row sums $R_{x_{1}}+R_{x_{2}}$ with $x_1+x_2=x$ fixed, and summing the inequality in over all $x\in\{0\}\times\mathbb{Z}^{k-1}$, we obtain $$|A_++A_+| \le (1+8\delta^{\frac{1}{10}})|A_5+A_5|.$$ Hence by , we thus have $$\begin{aligned} d_k(A_+) &\le (1+8\delta^{\frac{1}{10}})d_k(A_5)+2^k\cdot8\delta^{\frac{1}{10}}|A_5|\\ &\le \delta^{\frac{1}{20}-17c}|B|.\end{aligned}$$ Reductions Part 5: Approximating $A_+$ by $A_\star\subset A_+$ with few vertices in ${\operatorname{co}}(A_\star)$ and an extra technical condition {#sectionAstarapprox} --------------------------------------------------------------------------------------------------------------------------------------------------- We now construct a set $A_\star\subset A_+$ with $|A_+\setminus A_\star|=o(1)|B|$, which has simultaneously 1. $|V(A_\star)|$, which we recall is the number of vertices of ${\widetilde{\operatorname{co}}}(A_\star)$, is bounded by a function of $\delta$ 2. $|{\operatorname{co}}(\pi(A_\star))\setminus \pi(A_\star)|=o(1)|\pi(B)|$ 3. The technical condition holds. We show this using a double recursion, and the bounds we obtain will no longer be powers of $\delta$. In , we prove , which shows that we can ensure that 1 holds. This is accomplished by showing an analogous approximation result for (continuous) polytopes, and then transitioning to the discrete setting using . In , we prove , which shows that we can ensure that both 1 and 2 hold. by itself shows that 2 holds, so we alternate applications of and , and show that at some point both 1 and 2 hold simultaneously. In , we prove , which shows that we can ensure that all of 1,2,3 hold. To do this, we show , which shows that we can ensure 1 and 3 hold. Similarly to the proof of , we alternate applications of and , and show that at some point 1,2,3 hold simultaneously. Finally, in , we apply to $A_+$ to construct $A_\star$. ### $A_\star\subset A_+$ with $|V(A_\star)|$ small: Setup Part 1 {#AstarPart1} \[polyapprox\] For any $\alpha\ge 16 \epsilon_0^{-1} k(k+1) \min\{n_i\}^{-1}$, there exists an $\ell=\ell(\alpha)$ such that the following is true. For any set of points $C\subset B$ with $|{\operatorname{co}}(C)|\ge \frac{\epsilon_0}{2}|B|$, there exists $Q={\operatorname{co}}(Q)\cap C\subset C$ with $|V(Q)|\le \ell$ and $V(Q)\subset V(C)$, such that $|{\operatorname{co}}(Q)|\ge (1-\alpha)|{\operatorname{co}}(C)|$, and if $C$ has all rows intervals then $Q$ has all rows intervals. There exists a constant $\tau_k$ such that for $n_i$ sufficiently large and $\alpha=\min\{n_i\}^{-\frac{2}{(k-1)\lfloor k/2\rfloor+2}}$, we can take $\ell=\tau_k\min\{n_i\}^{\frac{k-1}{(k-1)\lfloor k/2\rfloor+2}}$. To do this, we first consider a continuous analogue, which was proved constructively by Gordon, Meyer, and Reisner [@BestApproximation]. \[contpolyapprox\] For any $\alpha>0$, there exists $\ell'=\ell'(\alpha)$ such that the following is true. For any polytope $\widetilde{C}$, there is a polytope $\widetilde{Q}$ which is the convex hull of at most $\ell'$ vertices of $\widetilde{C}$ with $|\widetilde{Q}|\ge (1-\alpha)|\widetilde{C}|$. There is an absolute constant $\tau$ independent of $k$ such that for $\alpha$ sufficiently small in terms of $k$ we can take $\ell'=(\frac{\tau}{k}\alpha)^{-\frac{k-1}{2}}$. It is enough to find such a polytope $\widetilde{Q}$ with vertices contained inside $\widetilde{C}$. Indeed, a simple convexity argument shows that as we vary the vertices of $\widetilde{Q}$ the maximum volume is attained when all vertices of $\widetilde{Q}$ are among the vertices of $\widetilde{C}$. The result then follows from [@BestApproximation Theorem 3]. Let $\widetilde{C}={\widetilde{\operatorname{co}}}(C)$ be the continuous convex hull of $C$. By , there exists a polytope $\widetilde{P}$ with $|V(\widetilde{P})|\le \ell'(\frac{\alpha}{2})$, $V(\widetilde{P})\subset V(\widetilde{C})=V(C)$, and $|\widetilde{P}|\ge (1-\frac{\alpha}{2})|\widetilde{C}|$. Let $Q=\widetilde{P}\cap C$, and note that ${\widetilde{\operatorname{co}}}(Q)=\widetilde{P}$. We thus have by that $$\begin{aligned} |{\operatorname{co}}(Q)|&\ge |\widetilde{P}|-2k(k+1)\min\{n_i\}^{-1}|B|\\ &\ge \left(1-\frac{\alpha}{2}\right)|\widetilde{C}|-2k(k+1)\min\{n_i\}^{-1}|B|\\ &\ge \left(1-\frac{\alpha}{2}\right)|{\operatorname{co}}(C)|-4k(k+1)\min\{n_i\}^{-1}|B|\\ &\ge (1-\alpha)|{\operatorname{co}}(C)|.\end{aligned}$$ For $\alpha=\min\{n_i\}^{-\frac{2}{(k-1)\lfloor k/2\rfloor+2}}$ (sufficiently small in terms of $k$ as $n_i$ is sufficiently large) we see that $\ell=\ell'(\frac{\alpha}{2})$, yielding $\ell=\tau_k\min\{n_i\}^{\frac{k-1}{(k-1)\lfloor k/2\rfloor+2}}$. ### $A_\star\subset A_+$ with $|V(A_\star)|$ and $|{\operatorname{co}}(\pi(A_\star))\setminus \pi(A_\star)|$ small: Setup part 2 {#AstarPart2} At this point in the proof, we will lose polynomial control over the doubling constant, so for convenience we will work with purely qualitative statements from now on. The following proposition is the qualitative analogue of and . Crucially, this qualitative version holds for all $\delta>0$, rather than for $\delta$ sufficiently small. \[qualalphaprop\] There are functions $h_{1}(t),h_{2}(t)$ with $h_{1},h_{2}\to 0$ as $t \to 0$ such that for any function $f=f(\delta)$ with $f\to 0$ as $\delta \to 0$ the following is true. For every $\delta>0$, if $A'\subset B$ and $|A'|\ge \frac{\epsilon_0}{2}$ with $d_k(A')\le f|B|$, then there is a subset of the rows $A''\subset A'$ such that $$|A'\setminus A''|\le h_{1}(f)|B|,\qquad|{\operatorname{co}}(\pi(A''))\setminus \pi(A'')|\le h_{2}(f)|\pi(B)|.$$ According to , the first sentence of should be read as follows. Given $\epsilon_0$ there are increasing functions $h_{1}(t),h_{2}(t)$ with $h_{1},h_{2}\to 0$ as $t \to 0$ such that given a function $f(\delta)$ with $f\to 0$ as $\delta \to 0$, there exist a function $n_k(\delta, \epsilon_0)$ such that following is true. We shall omit this type clarification in the rest of the document. Let $\Delta'(\epsilon_0)$ denote the implicit bound on $f(\delta)$ required to apply with $\sigma=f$, $\lambda=1$ and $\alpha=\frac12$. Note under , yields a bound on $\delta$ which may depend on $\sigma$ rather than a bound on $\sigma(\delta)$, but in we note that the proof works with a bound on $\sigma(\delta)$ instead. For $t\geq \Delta'(\epsilon_0)$, set $h_1(t)=h_2(t)=1$ and for $f<\Delta'(\epsilon_0)$ gives the result with $h_1(t)=t^{\frac{1}{2}-c}$ and $h_2(t)=t^{\frac{1}{2}}$. \[firstit\] There are functions $h_3(t),h_4(t)\to 0$ as $t \to 0$ such that for any function $f=f(\delta)\to 0$ as $\delta \to 0$ the following is true. For every $\delta>0$, if $A'\subset B$ has all rows intervals, $|A'|\ge \frac{2\epsilon_0}{3}|B|$ and $d_k(A')\le f|B|$, then there exists a subset $A''\subset A'$ with each row an interval, such that 1. $|{\operatorname{co}}(\pi(A''))\setminus \pi(A'')|\le h_3(f)|\pi(B)|$ 2. $|A'\setminus A''|\le h_4(f)|B|$ 3. $V(A'')\le \ell(f)$ with $\ell$ as in . We start by giving a high level outline of how the proof will work. Starting with $A'=A_0'$, we will create a nested sequence of sets $A_0'\supset A_0''\supset A_1'\supset A_1''\supset \ldots \supset A_{\gamma(f)}'$, where $\gamma$ is some function, $A_i''\subset A_i'$ is obtained through an application of , and $A_{i+1}'\subset A_i''$ is obtained through an application of . All $A_{i}',A_{i}''$ automatically satisfy the second point, and all of the $A_i'$ automatically satisfy the third point, so it suffices to show there is an $A_i'$ which satisfies the first point. But $|{\operatorname{co}}(\pi(A_{i-1}''))\setminus \pi(A_{i-1}'')|$ is always extremely small by construction, so if $|{\operatorname{co}}(\pi(A_i'))\setminus \pi(A_i'))|>h_3(f)|\pi(B)|$, then $|\pi(A_{i-1}'')\setminus \pi(A_i')|$ must be at least roughly $h_3(f)|\pi(B)|$. This inequality can only happen approximately $h_3(f)^{-1}$ times however before the $A_i'$ become empty, which is smaller than $\gamma(f)$. Let $h_1,h_2$ be as in . We recursively define functions with $g_0(t)=t$ and $$g_{i}(t)=g_{i-1}(t)+2^k h_1(g_{i-1}(t))+2^kt.$$ For fixed $i$ we have $g_i(t)\to 0$ as $t \to 0$. We define a decreasing sequence of real numbers $r_1,r_2,\ldots \to 0$ with the properties &\_[t(0,r\_i\]]{}+\_[j=0]{}\^[i-1]{}h\_2(g\_j(t))0,i ,&\[h1h2cond0\]\ \[h1h2cond1\]&\_[t(0,r\_i\]]{}it+\_[j=0]{}\^[i-1]{}h\_1(g\_j(t))0,i\ \[h1h2cond2\]&\_[t(0,r\_i\]]{} i t+\_[j=0]{}\^[i-1]{}h\_1(g\_j(t)). For $t\le r_1$ define $\gamma(t)=\max\{i:t\leq r_i\}$, and note $t \in (0,r_{\gamma(t)}]$, and $\gamma(t) \rightarrow \infty$ as $ t \rightarrow 0$. Let $$h_3(t):=\begin{cases}\frac{2}{\gamma(t)}+\sum_{j=0}^{\gamma(t)-1}h_2(g_{j}(t)) & \text{if } t \le r_1 \\1& \text{if } t > r_1\end{cases},\quad h_4(t):=\begin{cases}\gamma(t)t+\sum_{j=0}^{\gamma(t)-1}h_1(g_j(t)) & \text{if } t \le r_1 \\ 1 & \text{if } t>r_1\end{cases},$$ and note that $h_3, h_4 \rightarrow 0$ as $t\rightarrow 0$ by ,. If $f>r_1$, then the conclusion trivially holds with $A''=\emptyset$. Otherwise, if $f\le r_1$, we will see that we can iterate the below construction $\gamma(f)$ times while always satisfying the conditions in and that the corresponding set under consideration has size at least $\frac{\epsilon_0}{2}|B|$. We will now recursively construct sets $A'=A'_0\supset A'_1\supset \ldots \supset A_{\gamma(f)}'$ with all rows intervals such that $$\begin{aligned} \label{RecursiveAi}d_k(A'_{i})\le g_{i}(f)|B|,\qquad |A'\setminus A'_{i}|\leq \left[i f+\sum_{j=0}^{i-1}h_1(g_j(f))\right]|B|.\end{aligned}$$ Suppose that for some $i\leq \gamma(f)$, we have constructed $A'_{i-1}$, we will now construct $A'_i$. By for $i-1$, for $i=\gamma(f)$, and the fact that $f\le r_{\gamma(f)}$, we have $$\begin{aligned} |A'\setminus A_{i-1}'|\le \left[(i-1)f+\sum_{j=0}^{i-2}h_1(g_j(f))\right]|B| \le \left[\gamma(f)f+\sum_{j=0}^{\gamma(f)-1}h_1(g_j(f))\right] |B| \le \frac{\epsilon_0}{6}|B|,\end{aligned}$$ so $|A_{i-1}'|\ge (\frac{2}{3}-\frac{1}{6})\epsilon_0|B|=\frac{\epsilon_0}{2}|B|$. Applying to $A'_{i-1}$ we obtain $A''_i\subset A'_{i-1}$ with all rows intervals such that $$\begin{aligned} \label{Ai-1'Ai''1}|A'_{i-1}\setminus A''_i|\le h_1(g_{i-1}(f))|B|,\qquad |{\operatorname{co}}(\pi(A''_i))\setminus \pi(A''_i)|\le h_2(g_{i-1}(f))|\pi(B)|.\end{aligned}$$ An identical proof shows $|A''_{i}|\geq (\frac23-\frac16) \epsilon_0 |B| \ge \frac{\epsilon_0}{2} |B|$. We now apply with $\alpha=f$, to find a subset $A'_i\subset A''_i$ with all rows intervals such that $|V(A'_i)|\le \ell(f)$ and $A_i'={\operatorname{co}}(A_i')\cap A_i''$, with the property that $$\begin{aligned} \label{Ai''Ai'1}|A_i''\setminus A_i'|=|A_i''\setminus {\operatorname{co}}(A_i')|\le \left|{\operatorname{co}}(A''_i)\setminus {\operatorname{co}}(A'_i)\right|\le f|{\operatorname{co}}(A'_i)|\le f|B|,\end{aligned}$$ so together with for $i-1$ and , we deduce $$\begin{aligned} |A'\setminus A'_i|\leq \left[(i-1) f+\sum_{j=0}^{i-2}h_1(g_j(f))+f+h_1(g_{i-1}(f))\right]|B|=\left[if+\sum_{j=0}^{i-1}h_1(g_j(f))\right]|B|.\end{aligned}$$ By for $i-1$, ,, and , $$d_k(A_i')\le d_k(A_{i-1}')+2^k(h_1(g_{i-1}(f))+f)|B|\le (g_{i-1}(f)+2^k h_1(g_{i-1}(f))+2^kf)|B|=g_i(f)|B|,$$ verifying for $A_i'$. Writing $\gamma$ for $\gamma(f)$, we have $|A'\setminus A_{\gamma}'|\le (\gamma f+\sum_{j=0}^{\gamma-1}h_1(g_j(f)))|B|=h_4(f)|B|$. If all of $A_1',\ldots,A_{\gamma}'$ have the property that $|{\operatorname{co}}(\pi(A_i'))\setminus \pi(A_i')| > h_3(f)|\pi(B)|$, then noting that because $\pi(A_i')\subset \pi(A_i'')$ we have $|{\operatorname{co}}(\pi(A_i'))\setminus \pi(A_i')|-|{\operatorname{co}}(\pi(A_i''))\setminus \pi(A_i'')|=|\pi(A_i'')\setminus \pi(A_i')|-|{\operatorname{co}}(\pi(A_i'))\setminus {\operatorname{co}}(\pi(A_i''))|$, we deduce $$\begin{aligned} |{\operatorname{co}}(\pi(A_1'))\setminus \pi(A'_\gamma)|&\ge |{\operatorname{co}}(\pi(A_1'))\setminus \pi(A_1')|+\sum_{i=2}^\gamma |\pi(A_i'')\setminus \pi(A_i')|\\&\geq \sum_{i=1}^\gamma |{\operatorname{co}}(\pi(A'_i))\setminus\pi(A'_i)|-|{\operatorname{co}}(\pi(A''_i))\setminus\pi(A''_i)|\\ &\geq \sum_{i=1}^{\gamma}(h_3(f)-h_2(g_{i-1}(f)))|\pi(B)|\\ &=2|\pi(B)|,\end{aligned}$$a contradiction. Let $1\le i_0\leq \gamma$ be an index such that $|{\operatorname{co}}(\pi(A'_{i_0}))\setminus \pi(A'_{i_0})|\leq h_3(f)|\pi(B)|$. Then, as $i_0\leq \gamma$, we have by and the definition of $h_4$ that $|A'\setminus A'_{i_0}|\leq |A'\setminus A'_\gamma|\leq h_4(f)|B|.$ Moreover, $|V(A'_{i_0})|\le \ell(f)$ vertices. We thus conclude by setting $A''=A'_{i_0}$. ### $A_\star\subset A_+$ with $|V(A_\star)|$ and $|{\operatorname{co}}(\pi(A_\star))\setminus \pi(A_\star)|$ small and one further technical condition: Setup part 3 {#AstarPart3} For every $A'\subset B$, let $\mathcal{T}^+(A')$ (resp. $\mathcal{T}^-(A')$) be a triangulation of the upper (resp. lower) convex hull of ${\widetilde{\operatorname{co}}}(A')$ with respect to the $e_1$ direction, projected under $\pi$ to $\{0\}\times\mathbb{R}^{k-1}$, so in particular every $\widetilde{T}\in\mathcal{T}^+(A')$ has $\widetilde{T}\subset \{0\}\times\mathbb{R}^{k-1}$. We ensure that if ${\widetilde{\operatorname{co}}}(A'_1)={\widetilde{\operatorname{co}}}(A'_2)$, then $\mathcal{T}^+(A'_1)=\mathcal{T}^+(A'_2)$ and $\mathcal{T}^-(A'_1)=\mathcal{T}^-(A'_2)$. For a simplex $\widetilde{T}\subset \{0\}\times \mathbb{R}^{k-1}$, we will write $$T:=\widetilde{T}\cap \{0\}\times \mathbb{Z}^{k-1},\text{ and }T^o:=\widetilde{T}^o\cap (\{0\}\times \mathbb{Z}^{k-1})$$ where $\widetilde{T}^o$ is the interior of $\widetilde{T}$. \[YofT\] Given $\widetilde{T} \subset \{0\}\times \mathbb{R}^{k-1}$ with integral vertices, and a set $\mathcal{W} \subset \{0\}\times \mathbb{Z}^{k-1}$, for every $x\in T$ we define the set $$Y_{\mathcal{W}}(x):=((x+\mathcal{W})\cap T)\cup V(T)\subset T.$$ \[overpulling\] Let $\nu>0$, $A'\subset B$ with $|A'|\ge \frac{2\epsilon_0}{3}|B|$, $d_k(A')\le f_1(\delta)|B|$, $|{\operatorname{co}}(\pi(A'))\setminus \pi(A')|\le f_2(\delta)|\pi(B)|$. Suppose we have sets $\mathcal{W}_T\subset \{0\}\times \mathbb{Z}^{k-1}$ with $|\mathcal{W}_T| \le \nu$ for every $\widetilde{T} \in\mathcal{T}^+(A')\cup\mathcal{T}^{-}(A')$. Then there exists a subset of rows $A''\subset A'$ such that $$|A'\setminus A''|\le (2\nu+2)(f_1(\delta)+2^{k+1}f_2(\delta))|B|$$ which satisfies the following additional properties. 1. ${\operatorname{co}}(A'')={\operatorname{co}}(A')$. 2. For every $\widetilde{T}\in \mathcal{T}^+(A')\cup \mathcal{T}^-(A')=\mathcal{T}^+(A'') \cup \mathcal{T}^-(A'')$, if $x\in T^o \setminus V_\pi(A'')$, $y\in Y_{\mathcal{W}_T}(x)$, $z \in \pi(B)$ and $\vec{v}\in\{0\}\times\{0,1\}^{k-1}$ with $x+y=z+z+\vec{v}$ and $R_{x}+R_{y}, R_{z}+R_{z+\vec{v}}$ nonempty, then $(R_{x}+R_{y})\cap (R_{z}+R_{z+\vec{v}})\ne \emptyset$. For every $\widetilde{T} $, write $\mathcal{W}_T:=\{\vec{w}_{T,i} : 1\le i \le \nu \}$. Let $X_{T,i}\subset T^o\setminus V_\pi(A')$ be those $x$ such $x+\vec{w}_{T,i}\in Y_{\mathcal{W}_T}(x)$, and writing $x+x+\vec{w}_{T,i}=z+z+\vec{v}$ with $\vec{v}\in\{0\}\times\{0,1\}^{k-1}$, we have $$R_{x}+R_{x+\vec{w}_{T,i}},R_{z}+R_{z+\vec{v}}\text{ nonempty and }(R_{x}+R_{x+\vec{w}_{T,i}}) \cap (R_{z}+R_{z+\vec{v}}) = \emptyset.$$ For $\star\in \{+,-\}$, let $X^\star_i:=\bigsqcup_{\widetilde{T}\in \mathcal{T}^\star(A')} X_{T,i}$, let $y_i^\star$ be defined on $X_i^\star$ by setting $y^\star_i(x)=x+\vec{w}_{T,i}$ for $x\in X_{T,i}$, and set the disjoint union $$Z^\star_i:=\bigsqcup_{\widetilde{T}\in \mathcal{T}^\star(A') }\bigsqcup_{x\in X_{T,i}} R_x+R_{y^\star_i(x)}\subset A'+A'.$$ Here we note the union is disjoint as $\frac{1}{2}(x+y_i^\star(x))\in \widetilde{T}^{\circ}$, which are disjoint for distinct $\widetilde{T}$, and for a given $\widetilde{T}$ we have $\{2x+\vec{w}_{T,i}\}_{x\in T}$ are distinct. For $\star \in \{+,-\}$ let $X^\star_0\subset \pi(B)\setminus V_\pi(A')$ be those $x$ such that there exists $\widetilde{T}\in\mathcal{T}^\star(A')$ with $x\in T^o$ and $y^\star_0(x)\in V(T)$, such that writing $x+y^\star_0(x)=z+z+\vec{v}$ with $\vec{v}\in\{0\}\times\{0,1\}^{k-1}$, $$R_{x},R_{z}+R_{z+\vec{v}}\text{ nonempty, and }(R_{x}+R_{y^\star_0(x)}) \cap (R_{z}+R_{z+\vec{v}})= \emptyset.$$ Set the disjoint union $$Z^{\star}_0:=\bigsqcup_{x\in X^{\star}_0} R_x+R_{y^\star_0(x)}\subset A'+A'.$$ Here, the union is disjoint because $\frac{1}{2}(x+y_0^\star(x))\in \widetilde{T}^{\circ}$, which are disjoint for distinct $\widetilde{T}$, and for a given $\widetilde{T}$ we have $\{\frac{1}{2}(\widetilde{T}^{\circ}+y)\}_{y\in V(T)}$ are disjoint. Finally, set $X:=\bigcup_{(\star,i) \in \{+,-\}\times \{0,\ldots,\nu\}}X^\star_i$, and let $A''=A'\setminus \bigsqcup_{x\in X} R_x$, so that $|A'\setminus A''|=\sum_{x \in X}|R_x|$. By construction $A''$ satisfies the properties 1 and 2, so it suffices to show $|A'\setminus A''|\le (2\nu+2)(f_1(\delta)+2^{k+1}f_2(\delta))|B|$. Set the disjoint union $$Z:=A'(+)A'=\bigsqcup_{\vec{v}\in \{0\}\times \{0,1\}^{k-1}}\bigsqcup_{x\in \{0\}\times\mathbb{Z}^{k-1}}R_x(+)R_{x+\vec{v}}\subset A'+A',$$ and note that by construction $Z\cap Z^\star_i=\emptyset$ for all $\star,i$. Choose $(\star,i)\in \{+,-\}\cup \{0,\ldots,\nu\}$ so that $$|A'\setminus A''|=\sum_{x \in X}|R_x| \le (2\nu+2) \sum_{x \in X^{\star}_i}|R_x|.$$ Note that for $0\ne \vec{v}\in \{0\}\times \{0,1\}^{k-1}$ and $x\in \{0\}\times \mathbb{Z}^{k-1}$ we have $$|R_{x}(+)R_{x+\vec{v}}|-|R_{x}|-|R_{x+\vec{v}}|\ge \begin{cases}0&|\{x_1,x_2\}\cap \pi(A')|=0\\-n_1 & |\{x_1,x_2\}\cap \pi(A')|=1\\-1 & |\{x_1,x_2\}\cap \pi(A')|= 2\end{cases}$$ By , we have for $0\ne \vec{v}\in \{0\}\times\{0,1\}^{k-1}$ that $$\begin{aligned} |\{x\in \{0\}\times\mathbb{Z}^{k-1}: |\{x,x+\vec{v}\}\cap \pi(A')|=1\}|&\le 2|{\operatorname{co}}(\pi(A'))\setminus \pi(A')|+2(k-1)n_1^{-1}n_k(\epsilon_0,\delta)^{-1}|B|\\ &\le 3f_2(\delta)n_1^{-1}|B|\end{aligned}$$ and $$\begin{aligned} |\{x\in \{0\}\times\mathbb{Z}^{k-1}: |\{x,x+\vec{v}\}\cap \pi(A')|=2\}|\le |\pi(B)|\le n_k(\epsilon_0,\delta)^{-1}|B|\le f_2(\delta)|B|.\end{aligned}$$ We therefore have (taking $\vec{v}\in \{0\}\times \{0,1\}^{k-1}$ and $x\in \{0\}\times \mathbb{Z}^{k-1}$) $$\begin{aligned} |A'+A'| \ge& |Z| + |Z^\star_i| \\ =&\left(\sum_{\vec{v}}\sum_{x} |R_x(+)R_{x+\vec{v}}|\right)+\sum_{x\in X^\star_i}|R_x+R_{y^\star_i(x)}|\\ \ge&2^k|A'|-(2^{k-1}-1)(3f_2(\delta)+f_2(\delta))|B|-|\pi(B)|+\sum_{x\in X^\star_i}|R_x|\\\ge& 2^k|A'|-2^{k+1}f_2(\delta)|B|+\frac{1}{2\nu+2}|A'\setminus A''|.\end{aligned}$$ Hence, $$|A'\setminus A''|\leq (2\nu+2)( d_k(A')+2^{k+1}f_2(\delta)|B|)\leq (2\nu+2)(f_1(\delta)+2^{k+1}f_2(\delta))|B|.$$ \[Astarreduction\] There exist functions $h_5(t), h_6(t), H_7(t)$ such that $h_5, h_6,\rightarrow 0$ as $t \rightarrow 0$ and $H_7(t)\to \infty$ as $t \to 0$ such that for any function $e:=e(\delta)$ such that $e \rightarrow 0$ as $\delta \rightarrow 0$ the following is true. For every $\delta$, if $A'\subset B$ has all rows intervals and $|A'|\ge \frac{3\epsilon_0}{4}|B|$, $d_k(A')\le e|B|$, and if for every simplex $\widetilde{T}\subset \{0\}\times \mathbb{R}^{k-1}$ with integral vertices we have a set $\mathcal{W}_T\subset \{0\}\times \mathbb{Z}^{k-1}$ with $|\mathcal{W}_T| \le \nu$ for a constant $\nu=\nu(k)$, then there exists $A''\subset A'$ with all rows intervals and $$|A'\setminus A''|\le h_5(e)|B|,\qquad|{\operatorname{co}}(\pi(A''))\setminus \pi(A'')|\le h_6(e)|\pi(B)|$$ which satisfies the following additional properties. 1. $V(A'')\le H_7(e)$ 2. We have for every $\widetilde{T}\in\mathcal{T}^+(A'')\cup\mathcal{T}^{-}(A'')$, if $x\in T^o \setminus V_\pi(A'')$, $y\in Y_{\mathcal{W}_T}(x)$, $z \in \pi(B)$ and $\vec{v}\in\{0\}\times\{0,1\}^{k-1}$ with $x+y=z+z+\vec{v}$ and $R_{x}+R_{y}, R_{z}+R_{z+\vec{v}}$ nonempty, that $(R_{x}+R_{y}) \cap (R_{z}+R_{z+\vec{v}}) \ne \emptyset$. The high level overview of this proof is identical to that given at the start of the proof of , except with $h_3$ replaced with $h_6$, replaced with and replaced with . We recursively define functions $g_0(t)=t$ and $$g_i(t)=(2\nu+2)\left[\left(2^k+\frac{1}{2\nu+2}\right)g_{i-1}(t)+\left(2^{2k}+\frac{2^k}{2\nu+2}\right)h_4(g_{i-1}(t))+2^{2k+1}h_3(g_{i-1}(t))\right]$$ For fixed $i$ we have $g_i(t)\to 0$ as $t \to 0$. We define a decreasing sequence of real numbers $r_1,r_2,\ldots \to 0$ with the following properties: \[h3h4cond0\]&\_[t(0,r\_i\]]{}+\_[j=0]{}\^[i-1]{}h\_3(g\_[j]{}(t))0i&\ \[h3h4cond1\]&\_[t(0,r\_i\]]{}(2+2)0i\ \[h3h4cond2\]&\_[t(0,r\_i\]]{}(2+2)\_0. For $t\le r_1$ set $\gamma(t)=\max\{i: t\leq r_i\}$. Note that $t \in (0,r_{\gamma(t)}]$ and that $\gamma(t) \rightarrow \infty$ as $ t \rightarrow 0$. Set $$h_5(t):=\begin{cases}(2\nu+2)\left[\sum_{j=0}^{\gamma(t)-1} g_{j}(t)+\left(2^k+\frac{1}{2\nu+2}\right)h_4(g_{j}(t))+2^{k+1}h_3(g_{j}(t))\right] & \text{if } t \le r_1 \\ 1 & \text{if } t>r_1 \end{cases},$$ $$h_6(t):=\begin{cases}\frac{2}{\gamma(t)}+\sum_{j=0}^{\gamma(t)-1}h_3(g_j(t)) & \text{if } t \le r_1 \\ 1 & \text{if } t>r_1 \end{cases},$$ and with $\ell$ as in $$H_7(t):=\max_{0 \le j \le \gamma(t)-1}(\ell(g_{j}(t))$$ and note that $h_5, h_6 \rightarrow 0$ as $t\rightarrow 0$ by ,. If $e>r_1$, then the conclusion trivially holds with $A''=\emptyset$. Otherwise, if $e\le r_1$, we will see that we can iterate the below construction $\gamma(e)$ times while always satisfying the conditions in and that the corresponding set under consideration has size at least $\frac{2\epsilon_0}{3}|B|$. We will now recursively construct sets $A'=A_0'\supset A_1'\supset \ldots\supset A'_{\gamma(e)}$ with all rows intervals, such that $$\begin{aligned} \label{dkAi-1'}d_k(A'_{i})&\leq g_{i}(e)|B|,\\\label{A'Ai-1'}|A'\setminus A'_{i}|&\leq (2\nu+2)\left[\sum_{j=0}^{i-1} g_{j}(e)+\left(2^k+\frac{1}{2\nu+2}\right)h_4(g_{j}(e))+2^{k+1}h_3(g_{j}(e))\right]|B|.\end{aligned}$$ Suppose that for some $i\le \gamma(e)$, we have constructed $A'_{i-1}$, we will now construct $A_i'$. By for $i-1$, for $i=\gamma(e)$, and the fact that $e\le r_{\gamma(e)}$, we have $|A'_{i-1}|\geq (\frac34-\frac{1}{12})\epsilon_0 |B|=\frac23 \epsilon_0|B|$. Applying to $A'_{i-1}$, we find $A''_{i}$ with all rows intervals, $|V(A_i'')|\le \ell(g_{i-1}(e))$, and $$\begin{aligned} \label{A'i-1A''i2}|A'_{i-1}\setminus A''_i|\leq h_4(g_{i-1}(e)) |B|,\qquad |{\operatorname{co}}(\pi(A''_i))\setminus \pi(A''_i)|\leq h_3(g_{i-1}(e))|\pi(B)|.\end{aligned}$$ By for $i-1$,, and , we have $$\begin{aligned} \label{dkA''i}d_k(A''_i)\leq \left[g_{i-1}(e)+2^kh_4(g_{i-1}(e))\right]|B|.\end{aligned}$$ An identical proof shows $|A''_i|\geq(\frac34-\frac{1}{12})\epsilon_0 |B|=\frac23 \epsilon_0|B|$. Now by and , applying to $A''_i$ gives a set $A'_i\subset A''_i$ with all rows intervals, ${\operatorname{co}}(A'_i)={\operatorname{co}}(A''_i)$, property 2 with $A'=A_i'$, and $$\begin{aligned} \label{Ai'Ai''2}|A''_{i}\setminus A'_i|\leq (2\nu+2)\left[g_{i-1}(e)+2^kh_4(g_{i-1}(e))+2^{k+1}h_3(g_{i-1}(e))\right] |B|,\end{aligned}$$ so together with for $i-1$ and , we deduce $$\begin{aligned} |A'\setminus A'_i|\leq (2\nu+2)\left[\sum_{j=0}^{i-1} g_{j}(e)+\left(2^k+\frac{1}{2\nu+2}\right)h_4(g_{j}(e))+2^{k+1}h_3(g_{j}(e))\right] |B|.\end{aligned}$$ By , and , we have $$\begin{aligned} d_k(A'_i)&\leq (2\nu+2)\left[\left(2^k+\frac{1}{2\nu+2}\right)g_{i-1}(e)+\left(2^{2k}+\frac{2^k}{2\nu+2}\right)h_4(g_{i-1}(e))+2^{2k+1}h_3(g_{i-1}(e))\right]|B|\\&=g_i(e)|B|.\end{aligned}$$ Write $\gamma$ for $\gamma(e)$. If for $1 \le i \le \gamma$ we have $|{\operatorname{co}}(\pi(A'_i))\setminus \pi(A'_i)|>h_6(e)|\pi(B)|$, then we have removed at least $$\begin{aligned} |{\operatorname{co}}(\pi(A_1'))\setminus \pi(A'_\gamma)|&\ge |{\operatorname{co}}(\pi(A_1'))\setminus \pi(A_1')|+\sum_{j=2}^{\gamma}|\pi(A_j'')\setminus \pi(A_j')|\\&\geq \sum_{j=1}^\gamma |{\operatorname{co}}(\pi(A'_j))\setminus\pi(A'_j)|-|{\operatorname{co}}(\pi(A''_j))\setminus\pi(A''_j)|\\ &\geq \sum_{j=1}^{\gamma}(h_6(e)-h_3(g_{j-1}(e)))|\pi(B)|\\ &=2|\pi(B)|\end{aligned}$$rows in $\pi(B)$, a contradiction. Let $1 \le i_0\leq \gamma$ be an index such that $|{\operatorname{co}}(\pi(A'_{i_0}))\setminus \pi(A'_{i_0})|\leq h_6(e)|\pi(B)|$. As $i_0\leq \gamma$, we have by and the definition of $h_5$ that $$\begin{aligned} |A'\setminus A'_{i_0}|\leq |A'\setminus A'_\gamma|\le h_5(e)|B|.\end{aligned}$$ Moreover, $V(A'_{i_0})=V(A''_{i_0})\le \ell(g_{i_0-1}(e))\leq H_7(e)$, and as remarked earlier, $A'_{i_0}$ satisfies property 2. We thus conclude by setting $A''=A'_{i_0}$. ### $A_\star\subset A_+$ with $|V(A_\star)|$ and $|{\operatorname{co}}(\pi(A_\star))\setminus \pi(A_\star)|$ small and one further technical condition: Construction {#AstarConstruction} Before we proceed we need to introduce the following definition. \[sij\] Given a simplex $\widetilde{T}\subset \mathbb{R}^k$ with vertices $x_0, \hdots, x_k$ construct inductively a dense family of translates of $\frac{1}{2^i}\widetilde{T}$ inside $\widetilde{T}$ as follows. Set $$\begin{aligned} \mathcal{S}_{i,0}(\widetilde{T})&:= \left\{\left(1-\frac{1}{2^i}\right)x_r+ \frac{1}{2^i}\widetilde{T} : 0\le r \le k \right\}\\ \mathcal{S}_{i,j+1}(\widetilde{T})&:= \left\{ \frac{\widetilde{S}_1+ \widetilde{S}_2}{2}: \widetilde{S}_1, \widetilde{S}_2 \in \mathcal{S}_{i,j} \right\}\end{aligned}$$ Given a simplex $\widetilde{T}\subset \mathbb{R}^k$ we define $$\begin{aligned} \mathcal{U}_{i,j}({\widetilde{T}}):=\{\vec{u}:\exists \widetilde{S}_1,\widetilde{S}_2\in \mathcal{S}_{i,j}\text{ with }\widetilde{S}_1+\vec{u}=\widetilde{S}_2\}.\end{aligned}$$ Before preceeding, we remark that we will now need a future result, , to define certain constants $\mu_1$ and $\mu_2$ depending only on $k$. The proof is entirely self-contained, and while we could include the result and its proof at this point, we feel it is better to defer them. \[VofT\] We define constants $\mu_1=\mu_1(k), \mu_2=\mu_2(k)$ as those produced by . Given a simplex $\widetilde{T}$ we set $$\mathcal{W}_T:= \bigcup_{\vec{u}\in \mathcal{U}_{\mu_1, \mu_2}(\widetilde{T})} \mathcal{R}(\vec{u})\cup (-\mathcal{R}(\vec{u})),$$ where $\mathcal{R}(\vec{u})=\lfloor \vec{u} \rfloor + \{0\} \times \{0,1\}^{k-1} $. This satisfies $|\mathcal{W}_T| \le \nu$ for $\nu=\nu(k)$ the constant $2^k|\mathcal{U}_{\mu_1,\mu_2}|$, which is independent of the simplex $\widetilde{T}$. Note that $0\in\mathcal{U}_{\mu_1, \mu_2}(\widetilde{T})$ so $\{0\}\times \{0,1\}^{k-1}\subset \mathcal{W}_T$, and $\mathcal{W}_T=-\mathcal{W}_T$. With this construction we apply , obtaining a subset $A_\star\subset A_+$ with all rows intervals satisfying the following properties. By , we have $$\begin{aligned} \label{AAstar}|A\Delta A_\star|\le|A\Delta A_{+}|+|A_+\setminus A_\star|\le (\delta^{\frac{1}{20}-16c}+h_5(\delta^{\frac{1}{20}-17c}))|B|=:h_8(\delta)|B|,\end{aligned}$$ and by , we have $A_\star$ is reduced and $$\begin{aligned} \label{dkAstar}d_k(A_{\star}) \le (\delta^{\frac{1}{20}-17c} + 2^kh_8(\delta) )|B| = :h_9(\delta)|B|.\end{aligned}$$ Also, $$\begin{aligned} \label{copiAstar}|{\operatorname{co}}(\pi(A_{\star})) \setminus \pi(A_{\star})|&\le h_6(\delta^{\frac{1}{20}-17c})|\pi(B)|,\text{ and }\\ \label{Astarfewvert}V(A_\star)&\le H_7(\delta^{\frac{1}{20}-17c}).\end{aligned}$$ Here, $h_6,h_8,h_9 \rightarrow 0$ as $\delta \rightarrow 0$. \[pullover\] Finally, we have for every $\widetilde{T}\in\mathcal{T}^+(A_\star)\cup\mathcal{T}^{-}(A_\star)$, if $x\in T^o \setminus V_\pi(A_\star)$, $y\in Y_{\mathcal{W}_T}(x)$, $z \in \pi(B)$ and $\vec{v}\in\{0\}\times\{0,1\}^{k-1}$ with $x+y=z+z+\vec{v}$ and $R_{x}+R_{y}, R_{z}+R_{z+\vec{v}}$ nonempty, then $(R_{x}+R_{y}) \cap (R_{z}+R_{z+\vec{v}}) \ne \emptyset$. $A_\star$ is close to ${\operatorname{co}}(A_\star)$ {#Astarclosetoco} ---------------------------------------------------- In this section, we show that $|{\operatorname{co}}(A_\star)\setminus A_\star|= o(1)|B|$. It is easy to show that $2^k|{\operatorname{co}}(A_\star)\setminus A_\star|\le |{\operatorname{co}}(A_\star+A_\star)\setminus (A_\star+A_\star)|+o(1)|B|$, so it will suffice to show that $$\begin{aligned} \label{vaguedescription2kck}|{\operatorname{co}}(A_\star+A_\star)\setminus (A_\star+A_\star)|\le (2^k-c_k')|{\operatorname{co}}(A_\star)\setminus A_\star|+o(1)|B|\end{aligned}$$ for some constant $c_k'$. We now give a motivating outline. For $\widetilde{T}\in \mathcal{T}^+(A_\star)\cup \mathcal{T}^-(A_\star)$, we will define functions $g_{T}^+,g_T^-:T\to [0,n_1]$ (actually we will need $[0,2n_1]$ for technical reasons), which encode the distances from the nonempty rows of $A_\star$ in $T$ to the upper and lower convex hulls of $A_\star$ respectively. Then we can estimate $$\begin{aligned} \label{vaguedesc1}|{\operatorname{co}}(A_\star)\setminus A_\star|=o(1)|B|+\sum_{\ast \in \{+,-\}}\sum_{\widetilde{T}\in \mathcal{T}^\ast} \sum_{x\in T}g_T^\ast(x).\end{aligned}$$ Moreover, we will define functions $g^{+\square}_T,g^{-\square}_T$ as certain restricted infimum convolutions of $g^+_T$ and $g^-_T$ with themselves. These will encode the distance between the rows of a certain subset of $A_\star+A_\star$ (which we will guarantee to be intervals by ) to the upper and lower convex hulls of $A_\star+A_\star$ respectively. This subset accounts for almost all rows. Then we can similarly estimate $$\begin{aligned} \label{vaguedesc2}|{\operatorname{co}}(A_\star+A_\star)\setminus (A_\star+A_\star)| \le o(1)|B|+\sum_{\ast \in \{+,-\}}\sum_{\widetilde{T}\in \mathcal{T}^\ast}\sum_{x'\in 2\widetilde{T}\cap \{0\}\times \mathbb{Z}^{k-1}} g^{\ast\square}_{T}(x').\end{aligned}$$ To prove the inequality, it will therefore suffice to show for every $\ast \in \{+,-\}$ and $\widetilde{T}\in \mathcal{T}^\ast$, given a function $g:T\to [0,2n_1]$ which is $0$ at the vertices of $T$, that $$\begin{aligned} \label{vaguedesc3}\sum_{x\in T}g(x) \le o(1)|B|+ (2^k-c_k')\sum_{x'\in 2\widetilde{T}\cap \{0\}\times \mathbb{Z}^{k-1}}g^{\square}(x').\end{aligned}$$ In , we properly define the functions $g^\ast_T(x)$ and $g^{\ast \square}_T(x)$ and show in and in , thus reducing the problem to showing . In , we prove in . Finally, in , we combine these results and conclude in . ### Transitioning from $A_\star$ to functions and their infimum convolutions {#Transitioningsubsection} We focus on the gaps in the $e_1$-direction between $A_\star$ and the convex hull of $A_\star$ via functions on $\pi(A_\star)$. We denote by $V_\pi=V_\pi(A_\star)$ the projection of the vertices of ${\widetilde{\operatorname{co}}}(A_{\star})$ under $\pi$ to $\{0\}\times \mathbb{Z}^{k-1}$. We denote the empty rows by $E:={\operatorname{co}}(\pi(A_\star))\setminus \pi(A_\star).$ Finally, we write $\mathcal{T}^+:=\mathcal{T}^+(A_\star)$ and $\mathcal{T}^-:=\mathcal{T}^-(A_\star)$. Recall that for a simplex $\widetilde{T}\subset {\widetilde{\operatorname{co}}}(\pi(B))$, we denote $T=\widetilde{T}\cap \mathbb{Z}^{k}=\widetilde{T}\cap (\{0\}\times \mathbb{Z}^{k-1})$. \[gplusgminusdefn\] Denoting $\Psi_{A_\star}^+,\Psi_{A_\star}^-:{\widetilde{\operatorname{co}}}(\pi(A_\star))\to \mathbb{R}$ the upper and lower convex hull function of $A_\star$ in the $e_1$-direction respectively, we define for $\widetilde{T}^+\in \mathcal{T}^+$ and $\widetilde{T}^-\in \mathcal{T}^-$ the functions $$g^+_{T^+}:T^+\to [0,2n_1],\qquad g^-_{T^-}:T^-\to [0,2n_1]$$ according to the formulas $$g^+_{T^+}(x)=\begin{cases}\Psi_{A^\star}^+(x)-\max R_x & \text{if }x\in V(T^+)\text{ or }x\not\in V_\pi\cup E\\ 2n_1 & \text{otherwise}.\end{cases},$$ and $$g^-_{T^-}(x)=\begin{cases}\min R_x-\Psi_{A^\star}^-(x) & \text{if }x\in V(T^-)\text{ or }x\not\in V_\pi\cup E\\ 2n_1 & \text{otherwise.}\end{cases}.$$ \[gplusgminusrmk\] For $x\in V(T^\ast)$ or $x\not\in V_{\pi}\cup E$, $g^\ast_{T^\ast}(x)\in [0,n_1-1]$ is the distance in the $e_1$-direction from the row $R_x$ to the upper convex hull of $A_\star$ for $\ast=+$ and lower convex hull of $A_\star$ for $\ast=-$, and we always have $g^\ast_{T^\ast}(x)\le 2n_1$. In particular, for $x\in V(T^\ast)$ we note that $g^\ast_{T^\ast}(x)=0$. \[gsimple\] We have the following estimate for some function $h_{10}(\delta)\to 0$ as $\delta\to 0$ that $$\begin{aligned} \left(\sum_{\ast \in \{+,-\}}\sum_{\widetilde{T}^\ast\in\mathcal{T}^\ast}\sum_{x\in T^\ast}g^\ast_{T^\ast}(x)\right)-|{\operatorname{co}}(A_\star)\setminus A_\star|\le h_{10}(\delta)|B|.\end{aligned}$$ Writing $|{\operatorname{co}}(A_\star)\setminus A_\star|=\sum_{x\in {\operatorname{co}}(\pi(A_\star))}|({\operatorname{co}}(A_\star)\setminus A_\star)\cap \pi^{-1}(x)|$, we upper bound the contribution separately on the left hand side for each $x\in {\operatorname{co}}(\pi(A_\star))$. - We estimate the contribution of $x\in \partial\widetilde{T}^+$ for some $\widetilde{T}^+\in \mathcal{T}^+$ or with $x\in\partial\widetilde{T}^-$ for some $\widetilde{T}^-\in \mathcal{T}^-$. There are at most $\binom{|V_\pi|}{k}$ simplices, each with $k$ facets, with each facet having at most $n_k(\delta,\epsilon_0)^{-1}|\pi(B)|$ integral points by , each of which can in turn contribute $2n_1+2n_1$ to the left hand side. - We estimate the contribution of $x \in V_\pi\cup E$ that lie in the interior of at most one simplex in $\mathcal{T}^+$ and at most one simplex in $\mathcal{T}^-$. There are at most $|E|+|V_\pi|$ of these points, each of which can in turn contribute $2n_1+2n_1$ to the left hand side. - Finally, each remaining $x$ lies in a unique simplex of $\widetilde{T}^+\in\mathcal{T}^+$ and $\widetilde{T}^-\in\mathcal{T}^-$, and $x\not \in V_\pi\cup E$. For such $x$, we have $|({\operatorname{co}}(A_\star)\setminus A_\star)\cap R_x|=\lfloor g^+_{T^+}(x)\rfloor + \lfloor g^-_{T^-}(x) \rfloor$. This discrepancy with $g^+_{T^+}(x)+g^-_{T^-}(x)$ is crudely bounded by $2|\pi(B)|$ for each of the at most $\binom{|V_\pi|}{k}$ simplices. Combining these errors, and noting that $|E|$ and $|V_\pi|$ are bounded by and , we conclude by choosing $h_{10}(\delta)$ so that $$4k\binom{|V_\pi|}{k}n_k(\delta,\epsilon_0)^{-1}|B|+(|E|+|V_\pi|)4n_1+2\binom{|V_\pi|}{k}|\pi(B)|\le h_{10}(\delta)|B|.$$ Recall in , we introduced for a simplex $\widetilde{T}\subset \{0\}\times \mathbb{R}^{k-1}$ with integral vertices and a subset $\mathcal{W}\subset \{0\}\times \mathbb{Z}^{k-1}$ the notation $Y_{\mathcal{W}}(x)=((x+\mathcal{W})\cap T)\cup V(T)$. We now define a restricted infimum convolution with respect to $\mathcal{W}$. \[infconv\] Given a simplex $\widetilde{T}\subset \{0\}\times \mathbb{R}^{k-1}$ with integral vertices, a subset $\mathcal{W}\subset \{0\}\times \mathbb{Z}^{k-1}$, and a function $g : T \rightarrow \mathbb{R}_{\ge 0}$, we define the restricted infimum convolution $g^{\square}_{\mathcal{W}}:T+T\to \mathbb{R}_{\ge 0}$ by $$g^{\square}_{\mathcal{W}}(x)=\begin{cases}\min\{g(x_1)+g(x_2)\}& \text{over all }x_1+x_2=x \text{ with } x_1\in T^{\circ}\text{ and }x_2 \in Y_{\mathcal{W}}(x_1)\\ 0& \text{no such $x_1,x_2$ exist.} \end{cases}$$ For $g=g^\ast_{T^\ast}$ for some $\ast\in \{+,-\}$, we will always take $\mathcal{W}=\mathcal{W}_T$ as defined in in the infimum convolution $g^{\ast\square}_{T^\ast,\mathcal{W}}$. We will always omit the subscript $\mathcal{W}$, writing $g^{\ast\square}_{T^\ast}$ instead. \[gsquare\] We have the following estimate for some function $h_{11}(\delta)\to 0$ as $\delta\to 0$ that $$\begin{aligned} |{\operatorname{co}}(A_\star+A_\star)\setminus (A_\star+A_\star)|-\sum_{\ast\in \{+,-\}} \sum_{\widetilde{T}^\ast\in\mathcal{T}^\ast}\sum_{x'\in (T^\ast+T^\ast)}g_{T^\ast}^{\ast\square}(x')\le h_{11}(\delta)|B|.\end{aligned}$$ Writing $|{\operatorname{co}}(A_\star+A_\star)\setminus (A_\star+A_\star)|=\sum_{x'\in {\operatorname{co}}(\pi(A_\star+A_\star))}|({\operatorname{co}}(A_\star+A_\star)\setminus (A_\star+A_\star))\cap \pi^{-1}(x')|$, we upper bound the contribution for each $x'\in {\operatorname{co}}(\pi(A_\star+A_\star))$, which we express uniquely as $x'=x+x+\vec{v}$ with $\vec{v}\in\{0\}\times\{0,1\}^{k-1}, x \in\{0\}\times\mathbb{Z}^{k-1}$. Fix $x'=x+x+\vec{v}$ as above. Suppose for each $\ast \in \{+,-\}$ there exists $\widetilde{T}^\ast\in \mathcal{T}^\ast$ with $x,x+\vec{v}\in T^{\ast o}\setminus (V_\pi\cup E)$. Then $$|{\operatorname{co}}(A_\star+A_\star)\cap \pi^{-1}(x')|\le g_{T^+}^{+\square}(x')+g_{T^-}^{-\square}(x').$$ Note that we can write $x'=x+(x+\vec{v})$, with $x\in T^{\circ}$ and $x+\vec{v}\in Y_{\mathcal{W}_{T^\ast}}(x)$ since $\vec{v}\in \mathcal{W}_{T^{\ast}}$. Hence by there exists $x_1^*\in T^{\ast o}$ and $x_2^*\in Y_{\mathcal{W}_{T^\ast}}(x_1^\ast)$ so that $x_1^\ast+x_2^\ast=x'$ and $g^{\ast\square}_{T^\ast}(x')=g_{T^\ast}^\ast(x_1^\ast)+g_{T^\ast}^\ast(x_2^\ast)$. If either $x_1^\ast$ or $x_2^\ast$ are in $(V_\pi\cup E)\setminus V(T^\ast)$, the corresponding term on the right hand side is at least $2n_1$ and the inequality is trivially true. Hence, we may assume $x_1^*,x_2^*\not \in (V_\pi\cup E)\setminus V(T^\ast)$. In particular, we have $x_1^*\in T^{\ast\circ}\setminus V_\pi$ and both $x_1^\ast,x_2^\ast\not\in E$. Let $\Psi_{A_\star+A_\star}^+,\Psi_{A_\star+A_\star}^-:{\widetilde{\operatorname{co}}}(\pi(A_\star+A_\star))\to \mathbb{R}$ be the upper and lower convex hull function on $A_\star+A_\star$ in the $e_1$-direction respectively. We have $$\begin{aligned} \Psi^+_{A_\star+A_\star}(x')-\max (R_{x_1^+}+R_{x_2^+})&=\Psi_{A_\star}(x_1^+)+\Psi_{A_\star}(x_2^+)-\max R_{x_1^+}-\max R_{x_2^+}\\ &=g_{T^+}^+(x_1^+)+g_{T^+}^+(x_2^+)=g^{+\square}_{T^+}(x')\end{aligned}$$ and $$\begin{aligned} \min (R_{x_1^-}+R_{x_2^-})-\Psi^-_{A_\star+A_\star}(x')&=\min R_{x_1^-}+\min R_{x_2^-}-\Psi_{A_\star}(x_1^-)-\Psi_{A_\star}(x_2^-)\\ &=g_{T^-}^-(x_1^-)+g_{T^-}^-(x_2^-)=g^{-\square}_{T^-}(x').\end{aligned}$$ By , as $x_1^*\in T^\circ\setminus V_\pi$, $x_2^\ast\in Y_{\mathcal{W}_{T^\ast}}(x_1^\ast)$, and $x_1^\ast,x_2^\ast,x,x+\vec{v}\not\in E$, the intervals $R_{x_1^+}+R_{x_2^+}$ and $R_{x_1^-}+R_{x_2^-}$ both overlap $R_x+R_{x+\vec{v}}$, so $$I_{x'}:=(R_{x_1^-}+R_{x_2^-})\cup (R_{x}+R_{x+\vec{v}})\cup (R_{x_1^+}+R_{x_2^+})$$ is an interval. Therefore $$\begin{aligned} &|({\operatorname{co}}(A_\star+A_\star)\setminus (A_\star+A_\star))\cap \pi^{-1}(x')|\\\le &|({\operatorname{co}}(A_\star+A_\star)\cap \pi^{-1}(x'))\setminus I_{x'}|\\ \le&\lfloor \Psi^+_{A_\star+A_\star}(x')\rfloor-\max (R_{x_1^+}+R_{x_2^+})+\min (R_{x_1^-}+R_{x_2^-})-\lceil \Psi^-_{A_\star+A_\star}(x')\rceil\\ \le& g^{+\square}_{T^+}(x')+g^{-\square}_{T^-}(x').\end{aligned}$$ Returning to the proof of , we have the following estimates. Recall that we uniquely write $x'=x+x+\vec{v}$ with $x'\in {\operatorname{co}}(\pi(A_\star+A_\star))$, $x\in \{0\}\times \mathbb{Z}^{k-1}$ and $\vec{v}\in \{0\}\times \{0,1\}^{k-1}$. - We estimate the contribution of $x,\vec{v}$ such that for some $\ast \in \{+,-\}$, there is no $\widetilde{T}^\ast\in \mathcal{T}^\ast$ with $x,x+\vec{v}\in T^{\ast\circ}$. For the simplex $\widetilde{T}^\ast$ containing $\frac{1}{2}(x+(x+\vec{v}))=\frac{1}{2}x'\in \pi({\widetilde{\operatorname{co}}}(A_\star))$, there is a hyperplane $\widetilde{H}$ containing a facet of $\widetilde{T}^\ast$ that separates (or contains one of) $x$ and $x+\vec{v}$. For each $\vec{v}$, there are at most $\binom{|V_\pi|}{k}$ simplices, each with $k$ facets, and each facet separating (or containing) at most $2$ such $x,x+\vec{v}$ pairs on each of the at most $(k-1)n_k(\epsilon_0,\delta)^{-1}|\pi(B)|$ $\vec{v}$-fibers of $\pi(B)$, and each such $x,\vec{v}$ contributes at most $2n_1$ to the left hand side. - We now estimate the contribution for those $x,\vec{v}$ such that one of $x,x+\vec{v}$ lies in $V_\pi\cup E$. There are $2^{k-1}$ choices of $v$, and for each of these choices there are at most $2(|V_\pi|+|E|)$ such values of $x$, each of which contributes at most $2n_1$ to the right hand side. - The remaining $x,\vec{v}$ have $x,x+\vec{v}\in T^{+o}\setminus (V_\pi\cup E), T^{-o}\setminus (V_\pi\cup E)$ for unique simplices $\widetilde{T}^+\in \mathcal{T}^+$ and $\widetilde{T}^-\in \mathcal{T}^-$. But the above claim shows that the contribution of such $x'$ is non-positive. Combining these errors, and noting that $|E|$ and $|V_\pi|$ are bounded by and , we conclude by taking $h_{11}(\delta)$ so that $$\begin{aligned} 2^{k+1}(k-1)\binom{|V_\pi|}{k} n_k(\epsilon_0,\delta)^{-1}|B|+2^{k+1}(|V_\pi|+|E|)n_1\le h_{11}(\delta)|B|.\end{aligned}$$ ### Infimum convolution of functions {#infconvolutionsubsection} In this section we prove a general result about the infimum convolution (see ) of functions, related to the fact that small doubling implies being close to the convex hull. \[squareprop\] There exist constants $c_k',c_k''>0$ such that the following is true. Let $T\subset \pi(B)$ be a discrete simplex with integral vertices $x_0,\ldots,x_{k-1}\in \{0\}\times\mathbb{Z}^{k-1}$, and let $g:T\to [0,2n_1]$ with $g(x_i)=0$ for all $i$. Then $$\sum_{x' \in T+T}g_{\mathcal{W}_T}^{\square}(x')\le (2^k-c_k')\sum_{x \in T}g(x)+c_k''\min\{n_i\}^{-1}|B|$$ We omit the subscript $\mathcal{W}_T$ from now on. Throughout the entire proof we shall consider the sets $\mathcal{S}_{i,j}=\mathcal{S}_{i,j}(\widetilde{T})$ in with parameters $i,j$ bounded above by $\mu_1,\mu_2$, respectively. For a subset $\widetilde{P} \subset \widetilde{T}$, we define $g(\widetilde{P})= \sum_{x \in \widetilde{P}\cap T} g(x)$, and for a subset $\widetilde{Q}\subset \widetilde{T}+\widetilde{T}$, define $g^{\square}(\widetilde{Q})=\sum_{x\in \widetilde{Q}\cap (T+T)}g^{\square}(\widetilde{Q})$. For a subset $\widetilde{P} \subset \widetilde{T}$, we define $d'_k(\widetilde{P})=-g^{\square}(\widetilde{P}+\widetilde{P})+2^kg(\widetilde{P})$. \[dk’obs\] For a polytope $\widetilde{P} \subset \widetilde{T}$, we have $d'_k(\widetilde{P}) \ge -2^{2k+5}\min\{n_i\}^{-1}|B|$. In particular, with $P=\widetilde{P}\cap (\{0\}\times\mathbb{Z}^{k-1})$ we have $d'_k(P) \ge -2^{2k+5}\min\{n_i\}^{-1}|B|$. More generally, the same conclusion holds for any region $\widetilde{P}\subset \widetilde{T}$ defined as the intersection of open and closed half-spaces. Note that if $\widetilde{P}\subset \widetilde{T}$ is defined as the intersection of open and closed half-spaces, then we can perturb the open half-spaces to closed ones without changing the lattice points in $\widetilde{P}$ or $\widetilde{P}+\widetilde{P}$, so we may assume that $\widetilde{P}$ is a polytope. Recall that $\{0\}\times \{0,1\}^{k-1} \subset \mathcal{W}_T$. Note that if we write $z\in (\widetilde{P}+\widetilde{P})\cap T=2\widetilde{P}\cap T$ as $z=x+(x+\vec{v})$ with $\vec{v}\in \{0\}\times \{0,1\}^{k-1}$, then as $\widetilde{P}$ is convex, either $x,x+\vec{v}\in P^o$ or the segment $[x,x+\vec{v}]$ intersects $\partial\widetilde{P}$. Therefore with $\vec{v}$ ranging over $\{0\}\times\{0,1\}^{k-1}$ we have $$\begin{aligned} d'_k(\widetilde{P})&= 2^kg(\widetilde{P})-g^{\square}(\widetilde{P}+\widetilde{P}) \nonumber\\ &\ge 2^k\sum_{x\in P} g(x) - \sum_{\vec{v}}\sum_{x,x+\vec{v}\in P^o}g^{\square}(x+x+\vec{v}) - \sum_{\vec{v}}\sum_{[x,x+\vec{v}]\cap\partial \widetilde{P}\ne \emptyset}g^{\square}(x+x+\vec{v})\nonumber\\\label{3.44line3} &\ge 2^k\sum_{x\in P} g(x) - \sum_{\vec{v}}\sum_{x,x+\vec{v} \in P^o }g(x)+g(x+\vec{v})- \sum_{\vec{v}}\sum_{[x,x+\vec{v}]\cap\partial \widetilde{P}\ne \emptyset}4n_1\\ &\ge-\sum_{\vec{v}}\sum_{[x,x+\vec{v}]\cap\partial \widetilde{P}\ne \emptyset}4n_1 \nonumber\\ &\ge -8n_1\sum_{\vec{v}}|(\widetilde{P} \Delta (\widetilde{P}-\vec{v})) \cap \{0\}\times\mathbb{Z}^{k-1}| \nonumber\\ &\ge -8n_1\sum_{\vec{v}}|(\widetilde{P} \Delta (\widetilde{P}-\vec{v}))| -8n_12^{k-1}\cdot 3\cdot 2(k-1)k \min\{n_i\}^{-1} |\pi(B)| \label{3.44line6}\\ &\ge -8n_1\sum_{\vec{v}}|\vec{v}||\partial\widetilde{P}| -3\cdot 2^{k+3}(k-1)k \min\{n_i\}^{-1} |B| \nonumber\\ &\ge -8n_12^{k-1}\sqrt{k-1}\cdot2(k-1)\min\{n_i\}^{-1} |\pi(B)| -3\cdot 2^{k+3}(k-1)k \min\{n_i\}^{-1} |B| \label{3.44line8}\\ &\ge -2^{2k+5}\min\{n_i\}^{-1}|B|.\nonumber\end{aligned}$$ In we used the fact that $\vec{v}\in \mathcal{W}_T$ so $x+\vec{v}\in Y_{\mathcal{W}_T}(x)$, and $g^{\square}\le 2\max g \le 4n_1$. In we have used to upper bound $$|(\widetilde{P} \Delta (\widetilde{P}-\vec{v})) \cap \{0\}\times\mathbb{Z}^{k-1}|=|\widetilde{P}\cap \{0\}\times\mathbb{Z}^{k-1}|+|(\widetilde{P}-\vec{v})\cap \{0\}\times\mathbb{Z}^{k-1}|-|(\widetilde{P}\cap(\widetilde{P}-\vec{v}))\cap \{0\}\times\mathbb{Z}^{k-1}|,$$ and in the fact that $|\partial \widetilde{P}|\le |\partial \widetilde{\pi(B)}|\le 2(k-1)\min\{n_i\}^{-1}|\pi(B)|$. For a vertex $x\in V(\widetilde{T})$ and simplices $\widetilde{S}=(1-2^{-i})x+2^{-i}\widetilde{T} \in \mathcal{S}_{i,0}$ and $\widetilde{S}'=\frac{1}{2}(x+\widetilde{S}) \in \mathcal{S}_{i+1,0}$, we have $$g(S')\leq 2^{-k} g(S)+2^{-k}d_k'(T) +2^{k+6}\min\{n_i\}^{-1}|B|$$ By we have (letting $\widetilde{S}'^c$ be the complement of $\widetilde{S}'$ inside $\widetilde{T}$) that $$\begin{aligned} \nonumber d_k'(T)\ge d_k'(\widetilde{T})&=2^k g(\widetilde{T})-g^\square(\widetilde{T}+\widetilde{T})\\\nonumber &\ge 2^k g(\widetilde{S}')-g^\square(\widetilde{S}'+\widetilde{S}') + 2^k g(\widetilde{S}'^c)-g^\square(\widetilde{S}'^c+\widetilde{S}'^c)\\\nonumber &= 2^k g(\widetilde{S}')-g^\square(\widetilde{S}'+\widetilde{S}') + d_k'(\widetilde{S}'^c)\\\nonumber &\ge 2^k g(\widetilde{S}')-g^\square(\widetilde{S}'+\widetilde{S}')-2^{2k+5}\min\{n_i\}^{-1}|B|\\\nonumber &\ge 2^k g(\widetilde{S}')-g^\square(\widetilde{S}^{\circ}+x)-g^{\square}(\partial\widetilde{S}+x)-2^{2k+5}\min\{n_i\}^{-1}|B|\\\label{3.49line6} &\ge 2^k g(\widetilde{S}')-g(\widetilde{S}^{\circ})-k(4n_1)\min\{n_i\}^{-1}|\pi(B)|-2^{2k+5}\min\{n_i\}^{-1}|B|\\\nonumber &\ge2^k g(S')-g(S)-2^{2k+6}\min\{n_i\}^{-1}|B|.\end{aligned}$$ where in we used that $x\in Y_{\mathcal{W}_T}(y)$ for all $y\in \widetilde{T}^{\circ}$, to estimate the number of lattice points on each facet of $\widetilde{S}$, and the fact that $\max g^{\square}\le 4n_1$. For $\widetilde{S}'\in\mathcal{S}_{i,0}$, we have $$\begin{aligned} g(S'')&\leq 2^{-ik} g(T)+\max\left(0,\frac{1}{2^k-1} d_k'(T)\right)+ \frac{2^k}{2^k-1}2^{k+6}\min\{n_i\}^{-1}|B|\\&\le 2^{-ik} g(T)+\frac{1}{2^k-1} d_k'(T)+2^{2k+6}\min\{n_i\}^{-1}|B|\end{aligned}$$ \[averageg\] Let $i\leq \mu_1$ and $j\leq \mu_2-1$. For $\widetilde{S}'_1,\widetilde{S}'_2\in\mathcal{S}_{i,j}$ and $\widetilde{S}''=\frac12(\widetilde{S}'_1+\widetilde{S}'_2)\in\mathcal{S}_{i,j+1}$, we have $$g(S'')\leq \frac12( g(S'_1)+g(S'_2))+2^{-k}d_k'(T)+ 2^{5k}\min\{n_i\}^{-1}|B|$$ Let $\vec{u}$ be the vector such that $\widetilde{S}'_1+\vec{u}=\widetilde{S}'_2$. Then by , we have $\mathcal{R}(\vec{u})=\lfloor \vec{u} \rfloor +\{0\}\times\{0,1\}^{k-1}\subset \mathcal{W}_{T}$. Let $$\widetilde{S}''^{c}=\widetilde{P}_1\sqcup\dots\sqcup \widetilde{P}_{k}$$ be a partition into convex regions, the intersection of open and closed half-spaces. Indeed this can be obtained by taking the defining equations $x \cdot \vec{c}_i\le \vec{b}_i$ for $1 \le i \le k$ of $\widetilde{S}''$, and defining $\widetilde{P}_j$ by setting $x \cdot \vec{c}_i\le\vec{b}_i$ for $1 \le i <j$ and $x \cdot \vec{w}_j > \vec{b}_j$ inside $\widetilde{T}$. Hence, by we find $$\begin{aligned} d_k'(T)\ge d_k'(\widetilde{T})=2^k g(\widetilde{T})-g^\square(\widetilde{T}+\widetilde{T}) &=2^kg(\widetilde{S}'')-g^\square(\widetilde{S}''+\widetilde{S}'')+ \sum_j 2^kg( \widetilde{P}_{j})-g^\square( \widetilde{P}_{j}+\widetilde{P}_{j})\\ &=2^kg(\widetilde{S}'')-g^\square(\widetilde{S}_1'+\widetilde{S}_2')+ \sum_jd_k'(\widetilde{P}_{j})\\ &\ge 2^kg(\widetilde{S}'')-g^\square(\widetilde{S}_1'+\widetilde{S}_2') -k2^{2k+5}\min\{n_i\}^{-1}|B|.\end{aligned}$$ Note that every point $x' \in (\widetilde{S}_1'+\widetilde{S}_2') \cap \{0\}\times\mathbb{Z}^{k-1}$ we can write uniquely as $x'=x+x+\vec{w}$ for some $\vec{w}\in \mathcal{R}(\vec{u})$ (this is true in fact for every $x' \in \{0\}\times \mathbb{Z}^{k-1}$), and for this $\vec{w}$ (in fact for any $\vec{w}\in \mathcal{R}(u)$) we have $\vec{w}-\vec{u}\in\{0\}\times (-1,1]^{k-1}$. We have $\vec{u}\in \{0\}\times \prod_{i=2}^k[-n_i+1,n_i-1]$ and $x'\in (\pi(\widetilde{B})+\pi(\widetilde{B}))\cap \{0\}\times \mathbb{Z}^{k-1}=\{0\}\times \prod_{i=2}^k \{2,\ldots,2n_i\}$, so $$x=\left\lfloor \frac{x'-\lfloor\vec{u}\rfloor}{2}\right\rfloor\in B':=\{0\}\times\prod_{i=2}^k \left\{\left\lfloor\frac{3-n_{i}}{2}\right\rfloor,\ldots,\left\lfloor\frac{3n_i-1}{2}\right\rfloor\right\},$$ where $B'$ is a translate of $B(2n_2-1,\ldots,2n_k-1)$. Also, as the midpoint $\frac{1}{2}(x+(x+(\vec{w}-\vec{u})))$ lies in $\widetilde{S}'_1$, either $x\in \widetilde{S}_1'$ and $x+(\vec{w}-\vec{u}) \in \widetilde{S}_1'$ (equivalently $x+\vec{w}\in \widetilde{S}_2'$), or $x$ and $x+(\vec{w}-\vec{u})$ are separated by some hyperplane $\widetilde{H}$ containing one of the $k$ facets of $\widetilde{S}_1'$. Given $\vec{w}\in \mathcal{R}(\vec{u})$ and a hyperplane $\widetilde{H}$, there are at most $2^{2k}\min\{n_i\}^{-1}|\pi(B)|$ many choices of $x \in B'$ with $x,x+(\vec{w}-\vec{u})$ separated by $\widetilde{H}$. Indeed, set $\widetilde{G}_{\vec{w},\widetilde{H}}$ to be the convex region of the box $\widetilde{B'}$ between the hyperplanes $\widetilde{H}$ and $\widetilde{H}-(\vec{w}-\vec{u})$. Note that $$|\widetilde{G}_{\vec{w},\widetilde{H}}|\le |\vec{w}-\vec{u}|\cdot|\partial \widetilde{B'}|\le (k-1)^{\frac{1}{2}} 2(k-1)2^{k-2}\min\{n_i\}^{-1}|\pi(B)|\le 2^{2k-1}\min\{n_i\}^{-1}|\pi(B)|.$$ By applied to $B'$, $$\begin{aligned} \label{discGsize}|\widetilde{G}_{\vec{w},\widetilde{H}}\cap (\{0\}\times \mathbb{Z}^{k-1})|\le (2^{2k-1} + 2(k-1)k2^{k-2})\min\{n_i\}^{-1}|\pi(B)| \le 2^{2k}\min\{n_i\}^{-1}|\pi(B)|.\end{aligned}$$ From the above discussion, if $x+x+\vec{w}\in \widetilde{S}_1'+\widetilde{S}_2'$, and either $x\not \in \widetilde{S}_1'$ or $x+\vec{w}\not\in \widetilde{S}_2'$, then $x\in \widetilde{G}_{\vec{w},\widetilde{H}}$ for some $\widetilde{H}$ containing a facet of $\widetilde{S_1}'$. Hence from (taking $\vec{w}\in \mathcal{R}(\vec{u})$ and $x\in \{0\}\times\mathbb{Z}^{k-1}$) we deduce $$\sum_{\vec{w}}\sum_{\substack{x+x+\vec{w} \in \widetilde{S}_1'+\widetilde{S}_2' \\ x \not\in \widetilde{S}_1' \text{ or }x+\vec{w} \not\in \widetilde{S}_2' }} g^{\square}(x+x+\vec{w}) \le 2^{k-1}k2^{2k}\min\{n_i\}^{-1}|\pi(B)|\max g^{\square} \le k2^{3k+1}\min\{n_i\}^{-1}|B|.$$ Also, as $\mathcal{R}(u) \subset \mathcal{W}_T$ and $\max g^{\square}\le 4n_1$, we have $$\begin{aligned} \sum_{\vec{w}}\sum_{\substack{x+x+\vec{w} \in \widetilde{S}_1'+\widetilde{S}_2' \\ x \in \widetilde{S}_1' \text{ and }x+\vec{w} \in \widetilde{S}_2' }} g^{\square}(x+x+\vec{w}) &\le \sum_{\vec{w}}\sum_{\substack{x+x+\vec{w} \in \widetilde{S}_1'+\widetilde{S}_2'\nonumber \\ x \in (\widetilde{S}_1')^{\circ} \text{ and }x+\vec{w} \in \widetilde{S}_2' }} (g(x)+g(x+\vec{w}))+\sum_{x\in \partial \widetilde{S}_1'}4n_1\\ &\le 2^{k-1}(g(\widetilde{S}_1')+ g(\widetilde{S}_2'))+4k\min\{n_i\}^{-1}|B|\label{3.51},\end{aligned}$$ where in we used on each of the facets of $\widetilde{S}_1'$. Putting it all together, $$\begin{aligned} d_k'(T)\ge d_k'(\widetilde{T})\ge& 2^kg(\widetilde{S}'')-k2^{2k+5}\min\{n_i\}^{-1}|B|\\&-\sum_{\substack{x+x+\vec{w} \in \widetilde{S}_1'+\widetilde{S}_2' \\ x \in \widetilde{S}_1' \text{ and }x+\vec{w} \in \widetilde{S}_2' }} g^{\square}(x+x+\vec{w})- \sum_{\substack{x+x+\vec{w} \in \widetilde{S}_1'+\widetilde{S}_2' \\ x \not\in \widetilde{S}_1' \text{ or }x+\vec{w} \not\in \widetilde{S}_2' }} g^{\square}(x+x+\vec{w})\\ \ge& 2^{k}g(\widetilde{S}'')-2^{k-1}(g(\widetilde{S}_1')+ g(\widetilde{S}_2'))-(4k+k2^{2k+5}+k2^{3k+1})\min\{n_i\}^{-1}|B|\\ \ge& 2^{k}g(S'')-2^{k-1}(g(S_1')+ g(S_2'))-2^{6k}\min\{n_i\}^{-1}|B|.\end{aligned}$$ \[mu1mu2\] For $\widetilde{S}'\in\mathcal{S}_{\mu_1,\mu_2}$ we have $$g(S')\leq 2^{-\mu_1 k} g(T)+\left(\frac{1}{2^k-1}+ \mu_22^{-k}\right)d_k'(T)+(2^{2k+6}+\mu_22^{5k})\min\{n_i\}^{-1}|B|$$ Finally, before we prove , we prove the following result which as mentioned before constructs the constants $\mu_1,\mu_2$. \[coveringfamily\] There exist $\mu_1=\mu_1(k)$ and $\mu_2=\mu_2(k)$ and a family $\mathcal{F}\subset \mathcal{S}_{\mu_1,\mu_2}(\widetilde{T})$ such that $\widetilde{T}\subset \bigcup_{\widetilde{S}\in \mathcal{F}}\widetilde{S}$ and $\sum_{\widetilde{S}\in \mathcal{F}} |\widetilde{S}|\leq 2^{\mu_1-1}|\widetilde{T}|$, i.e. $|\mathcal{F}|\leq 2^{\mu_1k-1}$. Without loss of generality assume $\widetilde{T}$ is regular of volume $1$ centered at the origin. Extend a finite covering of $[0,1]^{k-1}$ with $q_k$ translates of $\widetilde{T}$ to a periodic covering $\mathcal{C}$ of $\mathbb{R}^{k-1}$ with average density $q_k$, and let $\mu_1(k):= \lceil\log_2(q_k)\rceil+2k-1$. We have that $2^{-\mu_1-1} \mathcal{C}$ is a periodic covering of $\mathbb{R}^{k-1}$ by translates of $2^{-\mu_1-1}\widetilde{T}$ with average density $q_k$, so for any polytope $\widetilde{P}$ there exists a $\vec{u}$ with $\sum_{\widetilde{S}\in \vec{u}+2^{-\mu_1+1}\mathcal{C}}|\widetilde{S}\cap \widetilde{P}|\le q_k|\widetilde{P}|$. Take $\widetilde{P}=2\widetilde{T}$, and let $\mathcal{C}'\subset \vec{u}+2^{-\mu_1-1}\mathcal{C}$ be the set of simplices which intersect $\widetilde{T}$, so that $\widetilde{T}\subset \bigcup_{\widetilde{S}\in\mathcal{C}'} \widetilde{S}$. Each $\widetilde{S}\subset \mathcal{C}'$ is contained in $\widetilde{T}+2^{-\mu_1-1}\widetilde{T}-2^{-\mu_1-1}\widetilde{T}\subset \widetilde{T}+2^{-\mu_1-1}\widetilde{T}+2^{-\mu_1-1}(k-1)\widetilde{T}\subset 2\widetilde{T}$, so $$\sum_{\widetilde{S}\in\mathcal{C}'}|\widetilde{S}|=\sum_{\widetilde{S}\in \mathcal{C}'}|\widetilde{S}\cap 2\widetilde{T}|\leq q_k|2\widetilde{T}|=2^{k-1}q_k.$$ For each $\widetilde{S} \in \mathcal{C}'$, there exists a translate $f(\widetilde{S})$ of $2^{-\mu_1-1}\widetilde{T}$ such that $\widetilde{S} \cap \widetilde{T} \subset f(\widetilde{S}) \subset \widetilde{T}$, and we construct $\mathcal{C}'':=\{f(S) : \widetilde{S} \in \mathcal{C}'\}$. Then $\sum_{\widetilde{S}'\in\mathcal{C}''}|\widetilde{S}'|\leq 2^{k-1}q_k$, all simplices in $\mathcal{C}''$ are contained in $\widetilde{T}$, and $\widetilde{T}\subset \bigcup _{\widetilde{S}'\in\mathcal{C}''}\widetilde{S}'$. The collection $\cup_{j\ge 0}\mathcal{S}_{\mu_1,j}(\widetilde{T})$ is a dense collection of translates of $2^{-\mu_1}\widetilde{T}$ contained inside $\widetilde{T}$, and in fact for every (possibly lower dimensional) face $\widetilde{F}$ of $\widetilde{T}$, the sub-collection of simplices in $\cup_{j\ge 0}\mathcal{S}_{\mu_1,j}(\widetilde{T})$ intersecting $\widetilde{F}$ is dense among all translates of $2^{-\mu_1-1}\widetilde{T}$ contained in $\widetilde{T}$ which intersect $\widetilde{F}$. Therefore for each element $\widetilde{S} \in \mathcal{C''}$, there exist a translate $h(\widetilde{S})\in\cup_{j\ge 0}\mathcal{S}_{\mu_1,j}(\widetilde{T}')$ which contains $\widetilde{S}$. Finally, we can construct the family $\mathcal{F} := \{h(\widetilde{S}) : \widetilde{S} \in \mathcal{C}'' \}$. As $\mathcal{C''}$ is a fixed finite set, there exist $\mu_2=\mu_2(k)$ such that $\mathcal{F} \subset \mathcal{S}_{\mu_1,\mu_2}(\widetilde{T}')$. Hence, $\sum_{\widetilde{S}\in\mathcal{F}} |\widetilde{S}|\leq 2^{2k-2}q_k\leq 2^{\mu_1-1}$ as desired. Recall by we find a family $\mathcal{F}\subset \mathcal{S}_{\mu_1,\mu_2}$ such that $\widetilde{T}\subset \bigcup_{\widetilde{S}\in \mathcal{F}}\widetilde{S}$ and $\sum_{\widetilde{S}\in \mathcal{F}} |\widetilde{S}|\leq 2^{\mu_1-1}|\widetilde{T}|$, i.e. $|\mathcal{F}|\leq 2^{\mu_1k-1}$. By , we conclude that $$\begin{aligned} g(T)&\leq \sum_{\widetilde{S}\in\mathcal{F}} g(S)\\ &\leq\sum_{\widetilde{S}\in\mathcal{F}} \left[2^{-\mu_1 k} g(T)+\left(\frac{1}{2^k-1}+ \mu_22^{-k}\right)d_k'(T)+(2^{2k+6}+\mu_22^{5k})\min\{n_i\}^{-1}|B|\right]\\ &\leq 2^{\mu_1k-1}\left[2^{-\mu_1 k} g(T)+\left(\frac{1}{2^k-1}+ \mu_22^{-k}\right)d_k'(T)+(2^{2k+6}+\mu_22^{5k})\min\{n_i\}^{-1}|B|\right]\end{aligned}$$ Hence as $d_k'(T)=-g^{\square}(T+T)+2^kg(T)$, we have $$g^\square(T+T)\leq \left(2^{k}-\frac{2^{-\mu_1k}}{\frac{1}{2^k-1} +\mu_22^{-k}}\right)g(T)+ \frac{2^{2k+6}+\mu_22^{5k}}{\frac{1}{2^k-1}+ \mu_22^{-k}}\min\{n_i\}^{-1}|B|.$$ ### $A_\star$ is close to ${\operatorname{co}}(A_\star)$: Construction {#AstarclosetocoAstarsubsection} In this section we prove that $|{\operatorname{co}}(A_\star)\setminus A_\star||B|^{-1}\to 0$ as $\delta \to 0$. \[AstarclosetocoAstarprop\] We have for some function $h_{\star}(\delta) \rightarrow 0$ as $\delta \rightarrow 0$ that $|{\operatorname{co}}(A_\star) \setminus A_\star| \le h_{\star}(\delta) |B|$. By ,,, and by , , and , we have that $$\begin{aligned} 2^k|{\operatorname{co}}(A_\star) \setminus A_\star|\le& d_k(A_\star) -d_k({\operatorname{co}}(A_\star))+ |{\operatorname{co}}(A_\star+A_\star) \setminus (A_\star+A_\star)| \\ \le& h_9(\delta)|B| +2^{2k}n_k(\epsilon_0,\delta)^{-1}|B|+h_{11}(\delta)|B|\\ &+\sum_{T^+}\sum_{x\in (T^++T^+)}g_{T^+}^{+\square}(x) +\sum_{T^-}\sum_{x\in (T^-+T^-)}g_{T^-}^{-\square}(x)\\ \le& h_9(\delta)|B| + 2^{2k}n_k(\epsilon_0,\delta)^{-1}|B|+h_{11}(\delta)|B|\\ &+\binom{H_7(\delta^{\frac{1}{20}-17c})}{k}c_k''n_k(\delta,\epsilon_0)^{-1}|B|\\&+(2^k-c_k')\left(\sum_{T^+}\sum_{x\in T^+}g^+_{T^+}(x)+\sum_{T^-}\sum_{x \in T^-}g^-_{T^-}(x)\right)\\ \le& h_9(\delta)|B| + 2^{2k}n_k(\epsilon_0,\delta)^{-1}|B|+h_{11}(\delta)|B|\\ &+\binom{H_7(\delta^{\frac{1}{20}-17c})}{k}c_k''n_k(\delta,\epsilon_0)^{-1}|B|+ (2^k-c_k') |{\operatorname{co}}(A_\star) \setminus A_\star| \\&+ (2^k-c_k') h_{10}(\delta)|B| \end{aligned}$$ We conclude that $|{\operatorname{co}}(A_\star) \setminus A_\star| \le h_{\star}(\delta) |B|$, for a function $h_{\star} \rightarrow 0$ as $\delta \rightarrow 0$. $A$ is close to ${\operatorname{co}}(A)$ {#AclosetocoA} ---------------------------------------- Recall that we have $d_k(A)\le \delta|B|, |A|\ge \epsilon_0|B|$, and for some functions $h_8,h_9,h_\star\to 0$ as $\delta\to 0$ that $$|{\operatorname{co}}(A_\star)\setminus A_\star|\le h_\star(\delta)|B|\text{, }\quad |A\Delta A_\star|\le h_8(\delta)|B|\text{, and}\quad d_k(A_\star)\le h_9(\delta)|B|.$$ We will now show that $|{\operatorname{co}}(A)\setminus A|\le h(\delta)|B|$ for some function $h\to 0$ as $\delta \to 0$. \[closestarlem\] Given a polytope $\widetilde{Q}$ and $\lambda>0$, let $o$ be the barycenter of the largest volume simplex $\widetilde{T}\subset\widetilde{Q}$. If $p\not \in (1+\lambda)\widetilde{Q}$, then there is a subpolytope $\widetilde{P}\subset \widetilde{Q}$ with $|\widetilde{P}|= \left(\frac{\lambda}{2k^2}\right)^k|\widetilde{Q}|$ such that $$\frac{\widetilde{P}+p}{2}\cap\widetilde{Q}=\emptyset.$$ We may assume $p\in \partial (1+\lambda)\widetilde{Q}$, and that $\widetilde{T}$ is regular with inradius $1$, with $o$ at the origin. Then $$\widetilde{T}\subset \widetilde{Q}\subset -k\widetilde{T},$$ and we estimate the diameter of $\widetilde{Q}$ is strictly less than $2k^2$, as this is an upper bound for the side length of $-k\widetilde{T}$ (as the distance from a vertex to $o$ is $k^2$). Also, note that the distance $s$ from $o$ to $\partial \widetilde{Q}$ is at least $1$, the inradius of $\widetilde{T}$. Let $q=op\cap \partial \widetilde{Q}$, let $H$ be the homothety with center $q$ and ratio $\frac{\lambda}{2k^2}$, and let $\widetilde{P}=H(\widetilde{Q})$. Clearly $\widetilde{P}$ has the desired volume, and has diameter strictly less than $\lambda$. Let $H'$ be the homothety with center $q$ and ratio $-\frac{\lambda}{2}$. Then $H'(\widetilde{Q})$ and $\widetilde{Q}$ are separated by the supporting hyperplane at the point $q$. It is enough to show that $\frac{\widetilde{P}+p}{2}$ is contained in the interior of $H'(\widetilde{Q})$. Note that the distance from $H'(o)$ to $\partial H'(\widetilde{Q})$ is at least $\frac{\lambda}{2}$. As $\frac{\widetilde{P}+p}{2}$ is a set of diameter strictly less than $\frac{\lambda}{2}$ containing $H'(o)=\frac{p+q}{2}$, it is contained in the interior of $H'(\widetilde{Q})$ as desired. Note that $|{\widetilde{\operatorname{co}}}(A_\star)|\ge \frac{\epsilon_0}{2}|B|$ by . Let $\widetilde{T}\subset {\widetilde{\operatorname{co}}}(A_\star)$ be the largest volume simplex, and let $o$ be its barycenter. Consider the homothety $H$ with center $o$ and ratio $1+\lambda(\delta)$ where $\lambda(\delta)$ is a function which goes to $0$ as $\delta \to 0$ which will be chosen later. Let $\widetilde{R}=H({\widetilde{\operatorname{co}}}(A_{\star}))$. We will show now that $A\subset \widetilde{R}$. Indeed, suppose not, let $x \in A\setminus \widetilde{R}$. Then by , there is a subset $\widetilde{P}\subset {\widetilde{\operatorname{co}}}(A_\star)$ with volume $(\frac{\lambda}{2k^2})^k|{\widetilde{\operatorname{co}}}(A_\star)|$ such that $\widetilde{P}+x$ is disjoint from $2{\widetilde{\operatorname{co}}}(A_\star)$. Then, by and , $$\begin{aligned} |A+A| &\ge |x+(\widetilde{P}\cap A_\star)|+|A_\star+A_\star|\\&\ge|x+(\widetilde{P}\cap {\operatorname{co}}(A_\star))|-h_{\star}(\delta)|B|+2^k|A_\star|-d_k(A_\star)\\ &\ge|\widetilde{P}\cap \mathbb{Z}^k|-h_{\star}(\delta)|B|+ 2^k(|A|-h_8(\delta)|B|)-h_{9}(\delta)|B|\\ &\geq|\widetilde{P}|+2^k|A|-h_{12}(\delta)|B|\\ &\ge \left(\frac{\lambda}{2k^2}\right)^k|{\widetilde{\operatorname{co}}}(A_\star)|+2^k|A|-h_{12}(\delta)|B|\\ &\ge 2^k|A|+\left(\frac{\epsilon_0}{2}\left(\frac{\lambda}{2k^2}\right)^k-h_{12}(\delta)\right)|B|.\end{aligned}$$ where $h_{12}(\delta)$ is a function with $h_{12}(\delta)\ge 2^kh_8(\delta)+h_\star(\delta)+ 2k(k+1)n_k(\epsilon_0,\delta)^{-1}$. Hence, $$\delta |B|\ge d_k(A)\ge \left(\frac{\epsilon_0}{2}\left(\frac{\lambda}{2k^2}\right)^k-h_{12}(\delta)\right)|B|.$$ Choosing $\lambda(\delta)$ such that $\frac{\epsilon_0}{2}(\frac{\lambda}{2k^2})^k-h_{12}(\delta)>\delta$ for all sufficiently small $\delta$ yields the desired contradiction. Therefore $A\subset (1+\lambda){\widetilde{\operatorname{co}}}(A_\star)$, so ${\widetilde{\operatorname{co}}}(A)\subset (1+\lambda){\widetilde{\operatorname{co}}}(A_\star)$. Hence by , $$\begin{aligned} |{\operatorname{co}}(A)|&\le (1+\lambda)^k|{\operatorname{co}}(A_\star)|+(1+(1+\lambda)^k)2k(k+1)n_k(\epsilon_0,\delta)^{-1}|B|\\ &\le (1+2\lambda)^k|{\operatorname{co}}(A_\star)|\\ &\le |{\operatorname{co}}(A_\star)|+((1+2\lambda)^k-1)|B|\\ &\le |A_\star|+((1+2\lambda)^k-1+h_\star(\delta))|B|\\ &\le |A|+(h_8(\delta)+(1+2\lambda)^k-1+h_\star(\delta))|B|.\end{aligned}$$ Defining $\omega(\epsilon_0,\delta)=h_8(\delta)+(1+2\lambda)^k-1+h_\star(\delta)$, we have $$|{\operatorname{co}}(A)\setminus A|\le \omega(\epsilon_0,\delta)|B|$$ with $\omega(\epsilon_0,\delta)\to 0$ as $\delta \to 0$ for any fixed $\epsilon_0>0$. Proof of b) for $k$ given for $k$ {#quantfromqualsection} ================================= To prove b), we first prove the following closely related proposition. \[quantprop\] There are constants $c_k$ (we can take $c_k=(4k)^{5k}$), $f_k$ and $\rho_k(\epsilon_0),n_k(\epsilon_0)$ for all $\epsilon_0>0$ such that the following is true. For every box $B=B(n_1,\ldots,n_k)$ with $n_1,\ldots,n_k \ge n_k(\epsilon_0)$, and for $A'\subset B$ a reduced set with $|A'|\ge \epsilon_0|B|$, $|{\operatorname{co}}(A')\setminus A'|\le \rho_k(\epsilon_0)|A'|$, and a triangulation $\mathcal{T}$ of $\partial{\widetilde{\operatorname{co}}}(A')$, we have that $$|{\operatorname{co}}(A')\setminus A'|\le c_{k}d_k(A')+f_k|\mathcal{T}|\min\{n_i\}^{-1}|B|.$$ We will see that this result follows from the following result. \[magicprop\] There are constants $c^1_k,c^2_k,\eta_k>0$ (we can take $c^1_k\leq 2^{2k}(2k)^{5k}$) such that the following is true. For every box $B=B(n_1,\ldots,n_k)$ and for $\widetilde{T}\subset \widetilde{B}$ a simplex with vertices $o=x_0,x_1,\ldots,x_k$, and $A\subset T=\widetilde{T}\cap \mathbb{Z}^k$ with $\{o,x_1,\ldots,x_k\}\subset A$ we have $$|(T\setminus (1-\eta_k)\widetilde{T})\setminus A|\le \frac{1}{2}|T\setminus A|+c_k^1d_k(A)+c^2_k\min\{n_i\}^{-1}|B|.$$ We shall write $A_S:=A\cap S$. Recall in , which recursively constructs a family of simplices $\mathcal{S}_{i,j}(\widetilde{T})$ such that $\mathcal{S}_{0,0}=\{\widetilde{T}\}$, $\mathcal{S}_{i,0}$ are the averages of a vertex in $V(T)$ with a simplex in $\mathcal{S}_{i-1,0}$, and $\mathcal{S}_{i,j}$ are the averages of two simplices in $\mathcal{S}_{i,j-1}$. For $x$ a vertex of $\widetilde{T}$ and $\widetilde{S}=(1-2^{-i})x+2^{-i}\widetilde{T} \in \mathcal{S}_{i,0}$ and $\widetilde{S}'=\frac{1}{2}(x+\widetilde{S}) \in \mathcal{S}_{i+1,0}$, we have $$|A_{S'}| \ge 2^{-k}|A_{ S}|-2^{-k}d_k(A)-2^k\min\{n_i\}^{-1}|B|.$$ By , with $\widetilde{S}'^c$ the complement of $\widetilde{S}'$ in $\widetilde{T}$, $$\begin{aligned} d_k(A)=|A+A|-2^k|A|&= |(A+A)\cap 2\widetilde{S'}|-2^k|A_{S'}|+|(A+A)\cap 2\widetilde{S'}^c|-2^k|A_{S'^c}|\\ &\ge |(A+A)\cap 2\widetilde{S'}|-2^k|A_{S'}|+|A_{S'^c}+A_{S'^c}|-2^k|A_{S'^c}|\\ &\ge |x+A_{S}|-2^k|A_{S'}|-2^{2k}\min\{n_i\}^{-1}|B|\\ &=|A_{S}|-2^k|A_{S'}|-2^{2k}\min\{n_i\}^{-1}|B|.\end{aligned}$$ For $\widetilde{S}\in \mathcal{S}_{i,0}$ we have $$|A_{S}| \ge 2^{-ik}|A|-\frac{1-2^{-ik}}{2^k-1}d_k(A)-2^{k+1}\min\{n_i\}^{-1}|B|.$$ For $\widetilde{S}_1, \widetilde{S}_2\in \mathcal{S}_{i,j}$, and $\widetilde{S}'=\frac{1}{2}(\widetilde{S}_1+\widetilde{S}_2)\in \mathcal{S}_{i,j+1}$, we have $$|A_{S'}| \ge \min(|A_{S_1}|,|A_{S_2}|)-2^{-k}d_k(A)-(k+2)2^{k}\min\{n_i\}^{-1}|B|.$$ Let $\widetilde{P}_1,\ldots,\widetilde{P}_{k+1}$ be a partition of $\widetilde{S}'^c$ into convex sets as in the proof of . Then by , we have $$\begin{aligned} d_k(A)&=|A+A|-2^k|A|\\ &= |(A+A)\cap 2\widetilde{S'}|-2^k|A_{S'}|+|(A+A)\cap 2\widetilde{S'}^c|-2^k|A_{S'^c}|\\ &\ge|A_{S_1}+A_{S_2}|-2^k|A_{S'}|+\sum_{i=1}^{k+1}|A_{\widetilde{P}_i}+A_{\widetilde{P}_i}|-2^k|A_{\widetilde{P}_i}|\\ &\ge 2^k(\min(|A_{S_1}|,|A_{S_2}|)-2^k|A_{S'}|-(k+2)2^{2k}\min\{n_i\}^{-1}|B|.\end{aligned}$$ \[FinalAScor\] For $\widetilde{S}\in \mathcal{S}_{i,j}$ we have $$\begin{aligned} |A_{S}|&\ge 2^{-ik}|A|-\left(\frac{1-2^{-ik}}{2^k-1}+j2^{-k}\right)d_k(A)-\left(2^{k+1}+j(k+2)2^{k}\right)\min\{n_i\}^{-1}|B|.\end{aligned}$$ In particular, by applied to $S$ and $T$, we have $$|S\setminus A_{S}|\leq 2^{-ik}|T\setminus A|+c^{1}_{i,j}d_k(A)+c^{2}_{i,j}\min\{n_i\}^{-1}|B|,$$ with $c^{1}_{i,j}=\frac{1-2^{-ik}}{2^k-1}+j2^{-k}$ and $c^{2}_{i,j}=(1+2^{-ik})2k(k+1)+2^{k+1}+j(k+2)2^{k}$. Let $i=\left\lceil\log_{\frac12}\left(\frac{k^{1/k}}{(2k)^5}\right)\right\rceil$ and $j=16k\log(2k)$. Let $c_k^1=(2k)^{5k}c^1_{i,j}\leq 2^{2k}(2k)^{5k}$ and $c_k^2=(2k)^{5k}c_{i,j}^2$ where $c^1_{i,j}$ and $c^2_{i,j}$ are as in . By [@HomoBM Claim 4.2], there exists a constant $\eta_k$ and a family of simplices $\mathcal{F}\subset \mathcal{S}_{i,j}$ with $|\mathcal{F}|\leq (2k)^{5k}$ with $$\sum_{\widetilde{S}\in \mathcal{F}}|\widetilde{S}|\le \frac{1}{2}|\widetilde{T}|,$$ and $$\widetilde{T}\setminus (1-\eta_k)\widetilde{T}\subset \bigcup \mathcal{F}.$$ We prove with parameters $c_k^1,c_k^2, \eta_k$ as above. Noting that $2^{-ik}=\frac{|\widetilde{S}|}{|\widetilde{T}|}$, we have $$\begin{aligned} |(T\setminus A)\setminus (1-\eta_k)\widetilde{T}|&\le \sum_{\widetilde{S}\in \mathcal{F}}|S\setminus A_{S}|\\ &\le \sum_{\widetilde{S}\in \mathcal{F}}\left(2^{-ik}|T\setminus A|+c^1_{i,j}d_k(A)+c^2_{i,j}\min\{n_i\}^{-1}|B|\right)\\ &\le \frac{1}{2}|T\setminus A|+c_{k}^1d_k(A)+c_{k}^2\min\{n_i\}^{-1}|B|.\end{aligned}$$ We fix $\eta_k$ as in . We need one final lemma to prove . \[containedheart\] For every $\epsilon_0>0$, there exists a constant $\rho_k(\epsilon_0)>0$ such that if $A'\subset B$ with $|A'|\ge \epsilon_0|B|$, $|{\operatorname{co}}(A')\setminus A'|\le \rho_k(\epsilon_0)|B|$, then there exists $o\in\mathbb{Z}^k$, such that (scaling with respect to $o$ and recalling we take $\eta_k$ as in ) we have $$(1-\eta_k){\widetilde{\operatorname{co}}}(A'+A')\cap \mathbb{Z}^k\subset A'+A'.$$ By , $|{\widetilde{\operatorname{co}}}(A')|\geq \left(\epsilon_0-2k(k+1) \min\{n_i\}^{-1}\right)|B|\ge \frac{\epsilon_0}{2}|B|$, so by John’s Lemma [@John], there exists an ellipsoid $\widetilde{F'}\subset {\widetilde{\operatorname{co}}}(A')$ with $|\widetilde{F'}|\geq k^{-k}\frac{\epsilon_0}{2}|B|$. Let $o'$ be the centre of this ellipsoid $\widetilde{F'}$. Let $o\in\mathbb{Z}^k$ be a point closest to $o'$ and let $p\in\partial \widetilde{F'}$ be the intersection of the ray $o'o$ with $\partial\widetilde{F'}$. Let $H'$ be the homothety centred at $p$ with ratio $\frac{|op|}{|o'p|}\geq 1-\frac{\sqrt{k}}{|o'p|}$, so that $H'(o')=o$. If $|o'p|\le 2\sqrt{k}$, then as the cross-sectional area of $\widetilde{F}'$ perpendicular to $o'p$ is at most $|\partial {\widetilde{\operatorname{co}}}(B)|$, we see that $|\widetilde{F}'|\le 2\sqrt{k}|\partial{\widetilde{\operatorname{co}}}(B)|<k^{-k}\frac{\epsilon_0}{2}|B|$, a contradiction. Hence $\widetilde{F}=H'(\widetilde{F'})\subset {\widetilde{\operatorname{co}}}(A')$ is an ellipse with center $o$ and $|\widetilde{F}|\ge(2k)^{-k}\frac{\epsilon_0}{2}|B|$. Taking a point $x'\in(1-\eta_k){\widetilde{\operatorname{co}}}(A'+A')\cap \mathbb{Z}^k$, our goal is to show that $x'\in A'+A'$. Let $x=\frac{1}{2}(x'+o)\in \mathbb{R}^k$, and let $y$ be the intersection of the ray $ox$ with $\partial {\widetilde{\operatorname{co}}}(A')$. Note that the ratio $r=|xy|/|oy| \ge \eta_k$. Let $H$ be the homothety with center $y$ and ratio $r$. This homothety sends $o$ to $x$ and ${\widetilde{\operatorname{co}}}(A')$ to $H({\widetilde{\operatorname{co}}}(A'))$. Note that $H({\widetilde{\operatorname{co}}}(A')) \subset {\widetilde{\operatorname{co}}}(A')$ because ${\widetilde{\operatorname{co}}}(A')$ is convex. Note that $H(\widetilde{F})$ is symmetric around $o$ and satisfies $|H(\widetilde{F})|= r^k |\widetilde{F}|$. By , $$|H(\widetilde{F})\cap \mathbb{Z}^k|\geq r^k |\widetilde{F}|-2k(k+1) \min\{n_i\}^{-1}|B|\ge \eta_k^k(2k)^{-k}\frac{\epsilon_0}{4}|B|> 2\rho_k(\epsilon_0).$$ for $\rho_k(\epsilon_0)$ sufficiently small. In particular, as $H(\widetilde{F})\subset{\widetilde{\operatorname{co}}}(A')$, $$\begin{aligned} |H(\widetilde{F})\cap A'|&\geq |H(\widetilde{F})\cap \mathbb{Z}^k|-|{\operatorname{co}}(A')\setminus A'|> \frac{1}{2}|H(\widetilde{F})\cap \mathbb{Z}^k|.\end{aligned}$$ By the symmetry of $H(\widetilde{F})$ around $x$, we have that $z\in H(\widetilde{F})\cap \mathbb{Z}^k$ implies that also $x'-z \in H(\widetilde{F})\cap \mathbb{Z}^k$. Hence, as $H(\widetilde{F})\cap A'$ contains more than half the elements in $H(\widetilde{F})\cap \mathbb{Z}^k$, we can find $z,z'\in H(\widetilde{F})\cap A'$, such that $x'=z+z'$ and thus $x'\in A'+A'$. Let $c_k=2c_k^1+2^{1-k}\leq (4k)^{5k}$ and $f_k=(2(k+1)(\frac{1}{2}+2^kc_k^1)+8k(k+1)+2^{k+1}+2c_k^2+(2+2^{1-k})2k(k+1)$. Let $o$ be the point supplied by . Note that as $o+A'\subset A'+A'$, we find $d_k(A'\cup\{o\})\leq d_k(A')$ and $|{\operatorname{co}}(A'\cup\{o\})\setminus (A'\cup\{o\})|\geq |{\operatorname{co}}(A')\setminus A'|-1 $, so we may assume $o\in A'$. Let $\mathcal{T}'$ be a triangulation of ${\widetilde{\operatorname{co}}}(A)$ obtained by coning off the simplices in $\mathcal{T}$ at $o$, so in particular $|\mathcal{T}|=|\mathcal{T}'|$. For each $\widetilde{T}\in \mathcal{T}'$ all vertices are in $A$. By , we have $$\begin{aligned} |{\operatorname{co}}(A+A)\setminus (A+A)| &\le \sum_{\widetilde{T}\in \mathcal{T}'}|{\operatorname{co}}(T+T)\setminus(A+A)|\\ &= \sum_{\widetilde{T}\in\mathcal{T}'}|({\operatorname{co}}(T+T)\setminus (A+A))\setminus (1-\eta_k)2\widetilde{T}|\\ &\le \sum_{\widetilde{T}\in\mathcal{T}'}|(2\widetilde{T}\setminus (1-\eta_k)2\widetilde{T})\cap \mathbb{Z}^k)\setminus (A_{\widetilde{T}\setminus (1-\eta_k)\widetilde{T}}+A_{\widetilde{T}\setminus (1-\eta_k)\widetilde{T}})|\\ &\le\sum_{\widetilde{T}\in \mathcal{T}'}|(2\widetilde{T}\setminus (1-\eta_k)2\widetilde{T})\cap \mathbb{Z}^k|-|A_{\widetilde{T}\setminus (1-\eta_k)\widetilde{T}}+A_{\widetilde{T}\setminus (1-\eta_k)\widetilde{T}}|.\end{aligned}$$ By , , and , this is $$\begin{aligned} \le& \sum_{\widetilde{T}\in\mathcal{T}'}\left(|2\widetilde{T}\setminus(1-\eta_k)2\widetilde{T}|-2^k|A_{\widetilde{T}\setminus (1-\eta_k)\widetilde{T}}|+(2^{k}2k(k+1)+2^{2k})\min\{n_i\}^{-1}|B|\right)\\ =&2^k\sum_{\widetilde{T}\in \mathcal{T}'}\left(|\widetilde{T}\setminus(1-\eta_k)\widetilde{T}|-|A_{\widetilde{T}\setminus (1-\eta_k)\widetilde{T}}|\right)+|\mathcal{T}|(2^{k+1}k(k+1)+2^{2k})\min\{n_i\}^{-1}|B|\\ \le& 2^k\sum_{\widetilde{T}\in \mathcal{T}'}\left(|(T\setminus A)\setminus (1-\eta_k)\widetilde{T}|\right)+|\mathcal{T}|(2^{k+2}k(k+1)+2^{2k})\min\{n_i\}^{-1}|B|.\\ \le& 2^{k}\sum_{\widetilde{T}\in \mathcal{T}'}\left(\frac{1}{2}|T\setminus A| + c_k^1d_k(A\cap T)\right)+|\mathcal{T}|(2^{k+2}k(k+1)+2^{2k}+2^kc_k^2)\min\{n_i\}^{-1}|B|\\ \le& 2^{k-1}|{\operatorname{co}}(A)\setminus A|+2^kc_k^1d_k(A)\\&+|\mathcal{T}|\left(2^k(k+1)\left(\frac{1}{2}+2^kc_k^1\right)+2^{k+2}k(k+1)+2^{2k}+2^kc_k^2\right)\min\{n_i\}^{-1}|B|.\end{aligned}$$ Hence as $d_k(A)=|A+A|-2^k|A|$ and $2^k| {\operatorname{co}}(A)|-|{\operatorname{co}}(A+A)|\le (2^k+1)2k(k+1)\min\{n_i\}^{-1}|B|$ by , we conclude $$\begin{aligned} |{\operatorname{co}}(A)\setminus A|\le& (2c_k^1+2^{1-k})d_k(A)\\ &+|\mathcal{T}|\left(2(k+1)\left(\frac{1}{2}+2^kc_k^1\right)+8k(k+1)+2^{k+1}+2c_k^2+(2+2^{1-k})2k(k+1)\right)\\&\quad\min\{n_i\}^{-1}|B|\\ =&c_kd_k(A)+f_k|\mathcal{T}|\min\{n_i\}^{-1}|B|\end{aligned}$$ Let $\alpha=\min\{n_i\}^{-\gamma}$ with $\gamma=\frac{1}{1+\frac{1}{2}(k-1)\lfloor k/2 \rfloor}$ and $\ell=\tau_{k}\alpha^{-\frac{k-1}{2}}$ with $\tau_{k}$ as in . Note that this $\gamma$ satisfies $-\gamma=\frac{k-1}{2}\lfloor k/2 \rfloor \gamma-1$. Take $\Delta_k(\epsilon_0)$ sufficiently small so that by , we have $|{\operatorname{co}}(A)\setminus A|\le \rho_k(\frac{\epsilon_0}{2})|B|$ where $\rho_k$ is the constant from . By , we find a subset $A'\subset A$ such that $|{\operatorname{co}}(A)\setminus {\operatorname{co}}(A')|\le \alpha |B|$ and $A'=A\cap {\operatorname{co}}(A')$ such that ${\operatorname{co}}(A')$ has at most $\ell$ vertices. In particular, we have $|A\setminus A'|\le \alpha |B|$. By , we have $d_k(A')\le 2^k\alpha |B|+d_k(A)$. Now, by Stanley’s resolution of the upper bound conjecture [@Stanley], as $\partial{\widetilde{\operatorname{co}}}(A')$ is a combinatorial sphere with $\ell$ vertices, if we take a triangulation $\mathcal{T}$ of $\partial {\widetilde{\operatorname{co}}}(A')$ we have $|\mathcal{T}|\le f_k'\ell^{\lfloor k/2 \rfloor}=f_k'\tau_{k-1}\alpha^{-\frac{k-1}{2}\lfloor k/2\rfloor}$ for some constant $f_k'$. By applied to $A'={\operatorname{co}}(A')\cap A\subset A$, we thus have $$|{\operatorname{co}}(A')\setminus A'|\le c_k(2^k\alpha |B|+d_k(A))+\alpha^{-\frac{k-1}{2}\lfloor k/2\rfloor}f_k'\tau_{k}f_k\min\{n_i\}^{-1}|B|.$$ Thus, dropping , provided that $\min\{n_i\}$ is bounded from below by a function of $k,\epsilon_0$ given by and , we get as $-\gamma=\frac{k-1}{2}\lfloor k/2 \rfloor \gamma-1$ that $$\begin{aligned} |{\operatorname{co}}(A)\setminus A|&\le c_k(2^k+1)\alpha |B|+c_kd_k(A)+\alpha^{-\frac{k-1}{2}\lfloor k/2 \rfloor}f_k'\min\{n_i\}^{-1}|B|\\ &\leq c_kd_k(A)+g_k\min\{n_i\}^{-\frac{1}{1+\frac{1}{2}(k-1)\lfloor k/2 \rfloor}}|A|.\end{aligned}$$ By taking $g_k=g_k(\epsilon_0)$ sufficiently large in terms of $\epsilon_0$ we can guarantee that the above inequality actually holds for any choice of $\{n_i\}$, which concludes the proof. Proving a) {#1.1asection} ========== We will now prove a). To do this, we will need the following special case of the main result of Green and Tao [@GreenTao]. \[GTspecialcase\] There exist constants $w_k$ such that for any $A\subset \mathbb{Z}^k$ with $d_k(A)\le |A|$, there exists a generalized arithmetic progression $P$ of dimension at most $k$ and size at most $|A|$, along with vectors $x_1,\ldots,x_{w_k}$ such that $$A\subset \bigcup_{i=1}^{w_k}P+x_i.$$ We apply , obtaining a generalized arithmetic progression $P$ and $x_1,\ldots,x_{w_k}\in \mathbb{Z}^k$ such that $A\subset \bigcup_{i=1}^{w_k}P+x_i$. Take $n_k^0$ to be a large threshold, chosen later. If $P$ is contained inside a hyperplane, then $A$ is covered by $w_k$ parallel hyperplanes and we are done. Similarly, if one of the side lengths of $P$ is at most $n_k^0$, then we can cover $P$ by $n_k^0$ parallel hyperplanes, so $A$ can be covered by $w_kn_k^0$ parallel hyperplanes. Therefore, we may assume that $$P=B(n_1,\ldots,n_k;v_1,\ldots,v_k;0)$$ is non-degenerate and $n_i\ge n_k^0$ for all $i$. By applying a linear transformation from $GL_k(\mathbb{Q})$ taking $v_i$ to the standard basis vectors $e_i$ and then scaling up to clear denominators, we may assume that $v_i=be_i$, where $b\in \mathbb{N}$. \[clm5.2\] There exist a factor $b'$ of $b$ such that $b'\ge w_k!^{-w_k} b$ and the following holds. If we consider the decomposition $$A=A_1\sqcup \ldots \sqcup A_r$$ associated to the cosets $y_1,\ldots,y_r\in ( \mathbb{Z}/b'\mathbb{Z})^k$, then after possibly relabeling we have $|A_1|\ge |A_j|$ for all $j$, and for every $p\ne 1$ there exists $j_p \ne 1$ such that $$y_p+y_{j_p}\ne y_1+y_k$$ for $k\in \{1,\ldots,r\}$. Set $b_{0}'=b$, and $y_{1,0},\ldots,y_{r_0,0}\in (\mathbb{Z}/b_{0}'\mathbb{Z})^k=( \mathbb{Z}/b\mathbb{Z})^k$ the distinct representatives of $A$, or equivalently the distinct representatives of $x_1,\ldots,x_{w_k}$. We note that in particular, this implies that $r_0\le w_k$. We recursively construct factors $b_{j+1}'$ of $b_{j}'$ with $b_{j+1}'\ge w_k!^{-1} b_{j}'$ such that if $y_{1,j},\ldots,y_{r_j,j}\in ( \mathbb{Z}/(b'_{j}\mathbb{Z})^k$ are the distinct representatives of $A$, then the following is true. If we consider the associated coset decomposition $$A=A_{1,j}\sqcup \ldots \sqcup A_{r_j,j},$$ possibly relabeling so that $|A_{1,j}|\ge |A_{p,j}|$ for $1 \le p \le r_j$, then either for every $p\ne 1$ there exists a $\lambda(p,j)\ne 1$ such that $$y_{p,j}+y_{\lambda(p,j),j}\ne y_{1,j}+y_{\ell,j}$$ for $1 \le \ell \le r_j$, or else we have $r_{j+1}<r_j$. Suppose that $y_{p,j}$ does not have the property that there exists $\lambda(p,j)$ such that $y_{p,j}+y_{\lambda(p,j),j}\ne y_{1,j}+y_{\ell,j}$ for all $\ell$. This is equivalent to saying that $y_{p,j}$ has the property that for all $\lambda$, there is an $\ell$ such that $(y_{p,j} - y_{1,j})+ (y_{\lambda,j} -y_{1,j})= y_{\ell,j}-y_{1,j}$. Then the cyclic group generated by $y_{p,j}-y_{1,j}$ lies entirely inside $\{0,y_{2,j}-y_{1,j},\ldots,y_{r_j,j}-y_{1,j}\}$, so has order at most $r_j \le w_k$. Setting $b_{j+1}=b_{j}/\gcd(b_{j},w_k!)$, we obtain that $y_{p,j}-y_{1,j}=0$ in $( \mathbb{Z}/b_{j+1}\mathbb{Z})^k$, so $r_{j+1}<r_j$. As $r_j$ can decrease at most $w_k$ times from $r_0\le w_k$, there exists a $j\le w_k$ for which $r_j=r_{j+1}$. Taking $j_p=\lambda(p,j)$, $b'=b_{j}$ and $y_p=y_{p,j}$ the distinct representatives of $A$ in $(\mathbb{Z}/b_{j}\mathbb{Z})^k=(\mathbb{Z}/b'\mathbb{Z})^k$, we obtain the desired result. Returning to the proof of a), let $P'=B(n_1,\ldots,n_k;b'e_1,\ldots,b'e_k;0)$ where $b'$ is furnished by . Let $w_k'=w_k \cdot w_k!^{k\cdot w_k}$, and $x_1',\ldots,x_{w_k'}'$ be translation vectors such that $$A\subset \bigcup_{i=1}^{w_k'}P'+x_i'.$$ Also note that $|P'|\le |A|$. \[clm5.3\] There exists an $x$ such that $A\subset x+(b'\mathbb{Z})^k$. Before we begin the proof of the claim we need the following lemma. \[error\] For $X,Y\subset A$, then $$|X+Y|\ge 2^k\min(|X|,|Y|)-2^{2k}(w_k'n_k^0)^{-1}w_k'^k|A|.$$ Let $C(X),C(Y)$ be obtained by compressing $X,Y$ in each of the coordinate directions. Then $C(X),C(Y)$ are contained in the down-set $C(A)\subset C(\bigcup P'+x_i')$, which in turn is contained inside a box of side lengths $w_k'n_1,\ldots,w_k'n_k\ge w_k'n_k^0$, which has volume at most $w_k'^k\prod n_i \le w_k'^k|A|$. Therefore by [@GreenTao], we obtain $$|X+Y|\ge |C(X)+C(Y)|\ge 2^k\min(|X|,|Y|)-2^{2k}(w_k'n_k^0)^{-1}w_k'^k|A|.$$ Let $$A=A_1\sqcup \ldots \sqcup A_r$$ be the coset decomposition as in the claim with $|A_1|$ maximal, and $r \le w_k'$. We want to show that $r=1$. We have for any $p\ne 1$ that $$\begin{aligned} |A+A|&\ge |A_p+A_{j_p}|+\sum_i|A_1+A_i|\\ &\ge |A_p|+\sum_{i=1}^r (2^k|A_i|-2^{2k}(w_k'n_k^0)^{-1}w_k'^k|A|)\\ &\ge |A_p|+2^k|A|-2^{2k}(n_k^0)^{-1}w_k'^k|A|\end{aligned}$$ so averaging over all $p\ne 1$ we obtain $$\begin{aligned} \label{5.3first}d_k(A)\ge \frac{1}{w_k'-1}|A\setminus A_1|-2^{2k}(n_k^0)^{-1}w_k'^k|A|.\end{aligned}$$ On the other hand, assuming $r \ge 2$ we have $$|A+A|\ge |A_1+A_2|+|A_1+A_1| \ge |A_1|+2^k|A_1|-2^{2k}(w_k'n_k^0)^{-1}w_k'^{k}|A|$$ so $$\begin{aligned} \label{5.3second}d_k(A)\ge |A|-(2^k+1)|A\setminus A_1|-2^{2k}(w_k'n_k^0)^{-1}w_k'^k|A|.\end{aligned}$$ Adding $(w_k'-1)(2^k+1)$ of to , and using $\Delta_k|A| \ge d_k(A)$, we obtain $$((w_k'-1)(2^k+1)+1)\Delta_k|A| \ge |A| - ((w_k'-1)(2^k+1)+w_k'^{-1})2^{2k}(n_k^0)^{-1}w_k'^k|A|,$$ which gives the desired contradiction provided $\Delta_k$ is sufficiently small and $n_k^0$ is sufficiently large. Returning to the proof of a), after translating $A$ we can assume that $A\subset (b'\mathbb{Z})^k$, so we may scale down and assume that $b'=1$. We now show that the boxes are in some sense “near” each other. There exists a universal constant $f_k$ so that for $P''=B(f_kn_1,\dots,f_kn_k;e_1,\dots,e_k;0)$ we have that $A\subset P''+x$ for some $x$. Recall that $x_1',\ldots,x_{w_k'}'$ are the translation vectors for $P'=B(n_1,\dots,n_k;e_1,\dots,e_k;0)$ which cover $A$. Suppose that $|A\cap (P'+x_1)|$ is maximal, so $|A\cap (P'+x_1)|\ge \frac{1}{w_k'}|A|$. Let $A_1=A\cap (P'+x_1)$. We first show that the width in the $j$-direction is bounded by a fixed multiple of $n_j$ for each $j$. So fix a $j$ and let $\pi_j$ be the projection in the $j$th coordinate. For a subset $\mathcal{B}\subset \{1,\ldots,w_k'\}$, let $h_j(\mathcal{B})$ be the difference between the largest and smallest values in $\{\pi_j(x_i')\}_{i\in \mathcal{B}}$. Suppose we have a set $\mathcal{B}\subset \{1,\ldots,w_k'\}$ containing $1$. If $\mathcal{B}$ is not the whole set and there is no $i \not \in \mathcal{B}$ such that $h_j(\mathcal{B}\cup \{i\})\le 100h_j(\mathcal{B})+100n_j$, then taking $\mathcal{B}'$ to be either $\{x_\ell': \pi_j(x_\ell')\ge \min \{\pi_j(x_i')\}_{i \in \mathcal{B}}\}$ or $\{x_\ell': \pi_j(x_\ell')\le \max \{\pi_j(x_i')\}_{i \in \mathcal{B}}\}$ (whichever has $\mathcal{B}'^c$ non-empty), then the sets $$Z_1=\bigcup_{i \in \mathcal{B}'} P'+x_i',\quad Z_2=\bigcup_{i\in (\mathcal{B}')^c}P+x_i'$$ have the following property. Let $z$ be the closest point of $Z_2\cap A$ in the $e_j$-direction to $Z_1$. Then by considering the projections under $\pi_j$, the sets $$(Z_1\cap A)+(Z_1\cap A),(Z_2\cap A)+(Z_2\cap A),z+A_1$$ are disjoint. By applied to these sets we obtain $$\begin{aligned} |A+A|&\ge |(Z_1\cap A)+(Z_1\cap A)|+|(Z_2\cap A)+(Z_2\cap A)|+|z+A_1|\\ &\ge 2^k|Z_1\cap A|+2^k|Z_2\cap A|+|A_1|-2^{2k+1}(w_k'n_k^0)^{-1} w_k'^k|A|\\ &\ge 2^k|A|+\frac{1}{w_k'}|A|-2^{2k+1}(w_k'n_k^0)^{-1} w_k'^k|A|,\end{aligned}$$ a contradiction provided $\Delta_k$ is sufficiently small and $n^0_k$ sufficiently large. Hence, if $\mathcal{B}$ is not the whole set $\{1,\dots,w_k'\}$, then there is an $i\not\in \mathcal{B}$ such that $h_j(\mathcal{B}\cup\{i\})\leq 100h_j(\mathcal{B})+100n_j$. Start with $\mathcal{B}=\{1\}$ and $h_j(\{1\})=0$. Repeatedly applying this, we see that $h_j(\{1,\ldots,w_k'\})\le f_k n_j-1$ for some universal integer constant $f_k\leq 101^{w_k'}$, independent of $j$. We deduce there is a translate $x$ such that $P'+x_i'\subset f_kP'+x$ for all $i$. Returning to the proof of a), taking $\epsilon_k\leq f_k^{-k}$, then as $|P'|\le |A|$ we find $A$ is a set of density at least $\epsilon_k$ inside the generalised arithmetic progression $f_kP'+x$. Proof of and {#appendixproofs} ============= Decorate constants from with a dash to distinguish them from constants with the same name in . Let $\Delta_k=\min\{\Delta_k',\Delta'_k(\epsilon_k')\}$ as given by . Let $c_k=c_k'+\frac12 ((4k)^{5k}-c_k')$, and $m_k(\delta)=\max\left\{m_k',\left(\frac{2g'_k(\epsilon'_k)\delta^{-1}}{(4k)^5k-c'_k}\right)^{1+\frac12(k-1)\lfloor k/2\rfloor}=\left(\frac{g'_k(\epsilon'_k)\delta^{-1}}{c_k-c_k'}\right)^{1+\frac12(k-1)\lfloor k/2\rfloor}\right\}$. As $d_k(A)\leq \delta |A|\leq \Delta_k' |A|$, by a), either $A$ is covered by $m_k'\leq m_k(\delta)$ parallel hyperplanes, or there is some generalized arithmetic progression $B=B(n_1,\dots,n_k;v_1,\dots,v_k;b)$ with the $v_i$ linearly independent, $A\subset B$ and $|A|\geq \epsilon_k'|B|$. If $n_i\leq m_k(\delta)$ for some $i$, then $B$ (and thus also $A$) is covered by $n_i\leq m_k(\delta)$ parallel hyperplanes. Hence, we may assume $\min\{n_i\}\geq m_k(\delta)$. As $d_k(A)\leq \Delta'_k(\epsilon'_k)$, $A$ and $B$ satisfy the conditions of b), so that $$\begin{aligned} |{\widehat{\operatorname{co}}}(A)\setminus A|&\leq c_k'd_k(A)+g'_k(\epsilon'_k)\min\{n_i\}^{-\frac{1}{1+\frac{1}{2}(k-1)\lfloor k/2 \rfloor}}|A|\\ &\leq c'_k \delta |A|+g'_k(\epsilon'_k)m_k(\delta)^{-\frac{1}{1+\frac{1}{2}(k-1)\lfloor k/2 \rfloor}}|A|\\ &= c_k \delta |A|.\end{aligned}$$ That concludes the proof. Let $m_k(\delta),\Delta_k',c_k'$ be the constants from . Let $(4k)^{5k}>c_k>c_k'$ be any constant and $\Delta_k\le \frac{c_k'}{c_k}\Delta_k'$. By standard approximations (see e.g. [@Semisum p.3 footnote 2]) we may assume that $\widetilde{A}$ is a finite union of positive measure convex polytopes, and $\widetilde{A}$ satisfies the condition $|\widetilde{A}+\widetilde{A}| \le (2^k+\Delta_k)|\widetilde{A}|$ from . Define $\delta=\frac{|\widetilde{A}+\widetilde{A}|-2^k|\widetilde{A}|}{|\widetilde{A}|}$. Let $N\in \mathbb{N}$, let $A_N=(\frac{1}{N}\mathbb{Z})^k\cap \widetilde{A}$, and let $(A+A)_N=(\frac{1}{N}\mathbb{Z})^k\cap (\widetilde{A}+\widetilde{A})$. Note that for each $r\in \mathbb{N}$, $A_N$ contains a translate of the combinatorial box $B(r,\ldots,r)$ for $N$ sufficiently large. In particular, this implies $A_N$ is reduced. As $N \to \infty$ we have $\frac{1}{N^k}|A_N| \to |\widetilde{A}|$, $\frac{1}{N^k}|{\operatorname{co}}(A_N)|\to |{\widetilde{\operatorname{co}}}(\widetilde{A})|$, and $\frac{1}{N^k}|(A+A)_N|\to |\widetilde{A}+\widetilde{A}|$. Thus for $N$ sufficiently large $\frac{1}{N^k}d_k(A_N) \le \frac{1}{N^k}(|(A+A)_N|-2^k|A_N|)\le\frac{1}{N^k}\frac{c_k}{c_k'}\delta |A_N|$, so $$d_k(A_N)\le \frac{c_k}{c_k'}\delta |A_N|\le \frac{c_k}{c_k'}\Delta_k|A_N|\le\Delta_k'|A_N|.$$ By , the number of hyperplanes needed to cover $A_N$ is larger than $m_k(\frac{c_k}{c_k'}\delta)$ for $N$ sufficiently large. Therefore by , we have $|{\operatorname{co}}(A_N)\setminus A_N| \le c_k' (\frac{c_k}{c_k'}\delta)|A_N|=c_k\delta|A_N|$ for $N$ sufficiently large. Dividing by $N^k$ and taking the limit as $N\to \infty$ of both sides yields $$|{\operatorname{co}}(\widetilde{A})\setminus \widetilde{A}| \le c_k |\widetilde{A}|.$$

--- abstract: 'In this paper we give an elementary treatment of the dynamics of skew tent maps. We divide the two-parameter space into six regions. Two of these regions are further subdivided into infinitely many regions. All of the regions are given explicitly. We find the attractor in each subregion, determine whether the attractor is a periodic orbit or is chaotic, and also determine the asymptotic fate of every point. We find that when the attractor is chaotic, it is either a single interval or the disjoint union of a finite number of intervals; when it is a periodic orbit, all periods are possible. Sometimes, besides the attractor, there exists an invariant chaotic Cantor set.' author: - | [**Kaijen Cheng**]{}\ [*College of Mathematics and Computer Science,*]{}\ [*Quanzhou Normal University*]{}\ [*Quanzhou 362000, Fujian, P.R.China*]{}\ [*e-mail: kaijen@cycu.org.tw* ]{}\ [**Kenneth J. Palmer**]{}\ [*Department of Mathematics,*]{}\ [*National Taiwan University*]{}\ [*Taipei - Taiwan*]{}\ [*e-mail: palmer@math.ntu.edu.tw*]{}\ title: On the dynamics of skew tent maps --- Introduction ============ In this paper we give an elementary treatment of the dynamics of skew tent maps following on from [@CP] where a special case was studied. Most of the results given here were already proved in [@ITN1; @ITN] but using non-elementary methods. The papers [@II; @MV; @MV1] also concern skew tent maps but confine themselves to the discussion of sophisticated concepts such as kneading sequences, entropy and renormalization. Here our modest aim is to find the attractor, determine whether the dynamics on the attractor is chaotic or not (in the sense of Devaney [@D]) and determine the asymptotic fate of all points and to do these things using elementary methods and with complete proofs. Two other elementary approaches have been given by Bassein [@B] and Lindstr" om and Thunberg [@LT]. Also recently a description of most of the results, but without many of the proofs, has been given by Sushko et al. in [@SAG]. We divide the parameter space into six regions. Bassein [@B] considers only four of these regions. Lindstr" om and Thunberg [@LT] also omit a couple of these regions. Here we study all six regions and also obtain more detail about the dynamics than Bassein [@B] or Lindstr" om and Thunberg [@LT] in the two regions (Sections 4 and 5), where the more complicated behaviour occurs. In fact, the main results here are in Sections 4 and 5 which form the bulk of the paper. As in Sushko et al. [@SAG], we find the attractor in each subregion but we also determine the asymptotic fate of every point. In Section 4 we show that apart from the points whose orbits go to infinity, all other points except those which are preimages of a finite set of unstable periodic orbits go to the attractor, which is a disjoint union of a finite number of closed intervals (a so-called band attractor) on which the dynamics is chaotic. In Section 5 we show that apart from the points whose orbits go to infinity, all other points, except a chaotic Cantor set and its preimages or a certain periodic orbit and its preimages, go to the attractor, which can be a periodic orbit or a band attractor on which the dynamics is chaotic. In remarks 3.2, 3.4, 3.6, 3.8, 4.5 and 5.12, we give more information about relation between our results and those in [@B; @ITN1; @ITN; @LT; @SAG]. More detail about the results in the paper are given at the end of the next section after we have introduced the requisite notation. Preliminaries ============= A general (continuous) tent map can be defined as follows: we take 2 non-horizontal straight lines, not parallel, which intersect at a point $(x_0,y_0)$, one with slope $r$ and the other with slope $-k$. Then we define our tent map to be $$\label{0} f(x)=\begin{cases} s+rx & (x\le x_0)\\ t-kx & (x\ge x_0),\end{cases}$$ where $s+rx_0=t-kx_0=y_0$. If $rk<0$, the map is a homeomorphism and the dynamics is trivial. Also if $r<0$, $k<0$, the map is conjugate via the transformation $h(x)=-x$ to $$(h^{-1}fh)(x)=\begin{cases} -t-kx & (x\le -x_0)\\ -s+rx & (x\ge -x_0).\end{cases}$$ So we may assume that $r>0$, $k>0$ in . Then if we define $H(x)=x+x_0$, we find that $$g(x)=(H^{-1}fH)(x) =\begin{cases} \gamma+rx & (x\le 0)\\ \gamma-kx & (x\ge 0),\end{cases}$$ where $\gamma=y_0-x_0$. Now if $\gamma\le 0$, we see that $g(x)\le 0$ for all $x$. So we can restrict to $x\le 0$ where $g$ is strictly increasing so that the dynamics is trivial. So the only interesting case is $\gamma>0$. Then if we take $\ell(x)=\gamma x$, we find that $$(\ell^{-1}g\ell)(x) =\begin{cases} 1+rx & (x\le 0)\\ 1-kx & (x\ge 0),\end{cases}.$$ So, without loss of generality, we can consider maps $$\label{1} f(x)=\begin{cases} 1+rx & (x\le 0)\\ 1-kx& (x\ge 0).\end{cases}$$ where $r>0$, $k>0$. Note that $f(x)\le 1$ for all $x$. This is essentially the same map studied in (3) in [@SAG]. Their $a_{{\cal L}}=r$ and $a_{{\cal R}}=-k$. First we give some preliminary results about the map in . \[lem1\] Let $f$ be as in . When $r>1$, set $I=[\alpha,\beta]$, where $$\alpha=-\frac{1}{r-1},\quad \beta=\frac{r}{k(r-1)}$$ and when $0<r\le 1$, set $I=(-\infty,\infty)$. Then [(i)]{} when $r>1$, $f(\alpha)=\alpha$, $f(\beta)=\alpha$ and $x\notin I\Longrightarrow f^{n}(x)\to -\infty\quad{\rm as}\quad n\to\infty$; [(ii)]{} for all $x\in{\rm int}(I)$, there exists $n\ge 0$ such that $f^{n}(x)\in [1-k,1]$; [(iii)]{} when $k\le 1$, or $k>1$ and $r\le k/(k-1)$, then $[1-k,1]\subset I$, $f(I)\subset I$ and $f([1-k,1])\subset [1-k,1]$; [(iv)]{} when $k>1$ and $r\le k/(k-1)$, if we define $h:[0,1]\to [1-k,1]$ by $h(x)=1-k+kx$, then $h$ is bijective and $g=h^{-1}\circ f\circ h$ maps $[0,1]$ onto itself and is given by $$\label{2} g(x)=\begin{cases} b+rx & (0\le x\le a)\\ k(1-x) & (a\le x\le 1),\end{cases}$$ where $a=1-1/k$, $b=1-ra$. \(i) Clearly $f(\alpha)=\alpha$, $f(\beta)=\alpha$. If $x<\alpha$, $$f(x)-\alpha=r(x-\alpha)<0.$$ It follows that for $n\ge 0$, $f^n(x)<\alpha$ and $f^{n}(x)-\alpha=r^{n}(x-\alpha)$. Hence $f^{n}(x)-\alpha\to -\infty$ as $n\to\infty$. If $x>\beta$, then $$f(x)=1-kx<1-k\beta=\alpha$$ and it follows from the previous part that $f^{n}(x)-\alpha\to -\infty$ as $n\to\infty$. \(ii) If $\alpha<x<\beta$, then $\alpha<f(x)\le 1$. So we may assume $\alpha<x\le 1$ where $\alpha=-\infty$ if $r\le 1$. Then we show there exists $n\ge 0$ such that $f^{n}(x)\in [0,1]$. Suppose $\alpha< f^{n}(x)<0$ for $n\ge 0$. If $r>1$, by induction on $n$, it follows that $f^{n}(x)-\alpha=r^{n}(x-\alpha)$ which $\to \infty$ as $n\to\infty$. If $r=1$, $f^{n}(x)=n+x$ which again $\to \infty$ as $n\to\infty$. On the other hand, if $r<1$, $$f^n(x)= 1+r+\cdots+r^{n-1}+r^nx\to 1/(1-r)>0$$ as $n\to\infty$. So in all cases, there exists $n\ge 0$ such that $f^{n}(x)\ge 0$ or $f^{n}(x)\le\alpha$. Let $n$ be the first such $n$. If $n=0$, then $x\ge 0$ since $\alpha<x\le 1$. If $n\ge 1$, then $\alpha<f^{n-1}(x)<0$ implies that $\alpha=f(\alpha) < f^{n}(x)<1$. Hence $f^{n}(x)\in [0,1]$. Then $f^{n+1}(x)=1-kf^n(x)\in [1-k,1]$. \(iii) First we show $[1-k,1]\subset I$. We need only consider $r>1$. Then if $k\le 1$ or $k>1$, $r\le k/(k-1)$, it is clear that $\alpha\le 1-k$ and $\beta\ge 1$. It follows that $[1-k,1]\subset I$. Next we show $f(I)\subset I$. If $r\le 1$, this is trivial. If $r>1$, $$\alpha\le x\le 0\Longrightarrow 1+r\alpha\le 1+rx\le 1 \Longrightarrow \alpha\le f(x)\le 1.$$ $$0\le x\le \beta\Longrightarrow 1 \ge 1-kx\ge 1-k\beta\Longrightarrow 1\ge f(x)\ge \alpha.$$ However $k\le 1$, or $k>1$, $r\le k/(k-1)$ implies $\beta \ge 1$. Hence $f(I)\subset I$. Next we show $f([1-k,1])\subset [1-k,1]$. When $k>1$, $$f([1-k,1])=[1+r(1-k),1]\cup [1-k,1]\subset [1-k,1]$$ if $1+r(1-k)\ge 1-k$, that is, $r\le k/(k-1)$. When $0<k\le 1$, $$f([1-k,1])=[1-k,1-k(1-k)]\subset [1-k,1].$$ \(iv) This is simple algebra. Note that $g$ in is the map studied in [@B] and [@ITN]. These authors restrict themselves to the parameter range $k>1$, $0<r<k/(k-1)$, thereby excluding the cases in Propositions \[prop1\] and \[prop4\] below. Now we summarize the results in the rest of the paper. In Section 3, we study the parameter ranges $k<1$, where there is a stable fixed point; $k>1$, $r<1/k$ where there is a stable $2-$cycle; $k>1$, $\max\{1,k/(k^2-1)\}<r<k/(k-1)$ or $k/(k^2-1)<r<1/(k-1)$, where the map is chaotic on the whole of $[1-k,1]$, and $k>1$, $r>k/(k-1)$, where the orbits of all points go to infinity except those lying on a chaotic Cantor set. In Section 4, we study the parameter range $k>1$, $1/k<r<k/(k^2-1)$. This is divided into subranges (which are explicitly described as in [@SAG]), where the attractor consists of a finite number of disjoint closed intervals and, apart from the points whose orbits go to infinity, all other points except a finite set of unstable periodic orbits and their preimages, go to the attractor on which the dynamics is chaotic. In Section 5, we study the parameter range $k>1$, $1/(k-1)<r<1$. Corresponding to each integer $m\ge 2$, there is a subrange. Inside each subrange there are 4 further subranges, each with a different kind of attractor: (i) a stable periodic orbit with period $m+1$, (ii) the union of $m+1$ disjoint closed intervals on which the dynamics is chaotic, (iii) the union of $2(m+1)$ disjoint closed intervals on which the dynamics is chaotic or (iv) $[1-k,1]$ itself is chaotic. As in [@SAG], the different subranges are described explicitly. Moreover, we show that in the cases (i)-(ii), the orbits of all points, except those which lie on a chaotic Cantor set or are preimages of this set, go to the attractor, in case (iii) the orbits of all points, except those which lie on a chaotic Cantor set or on a certain periodic orbit or are preimages of the set or the periodic orbit, go to the attractor, whereas in case (iv) the orbits of all points go to the attractor. ![Regions in $(k,r)-$parameter space: attracting period 2 cycle (green), chaos on $[1-k,1]$ (blue), chaotic band attractors (red), attracting periodic orbits and chaotic band attractors (yellow). ](figure1.pdf){width="80.00000%"} The simple cases ================= The attractor is a fixed point ------------------------------ \[prop1\] Let $f$ be as in and $I$ as in Lemma \[lem1\]. When $k<1$ and $x$ is in int$(I)$, then $f^n(x)$ converges to the fixed point $x^*=\frac{1}{k+1}$. From Lemma \[lem1\] (ii), (iii), we can assume $f^{n}(x)\in [1-k,1]$ for all $n\ge 0$. Then $\vert f(x)-x^*\vert=|1-kx-(1-kx^*)|=k\vert x-x^*\vert$ and by induction $\vert f^{n}(x)-x^*\vert=k^{n}\vert x-x^*\vert$ for $n\geq 0$. So $f^{n}(x)\rightarrow x^*$ as $n\rightarrow \infty.$ The proposition follows. This region is not considered in [@B] or [@ITN]. The same result is shown in [@SAG]; see the first two bullet points on page 587 and the first part of Section 3.1. See also Theorem 4.1 (c), (d) in [@LT]. The attractor is a stable period 2 orbit ---------------------------------------- \[prop2\] Let $f$ be as in . Suppose $k>1$, $r<1/k$. Then $f$ has a stable $2-$cycle and all orbits are attracted to this $2-$cycle except $x^*=1/(k+1)$ and its preimages. First from Lemma \[lem1\], since $1/k<k/(k-1)$ when $k>1$, the interval $[1-k,1]$ is invariant under $f$ and for all real $x$, there exist $n\ge 0$ such that $f^n(x)\in [1-k,1]$. So we may restrict $f$ to $[1-k,1]$ and study $g=h^{-1}fh:[0,1]\to [0,1]$ as defined in . Then $$b-(1-a+a^2)=\frac{(k-1)(1-rk)}{k^2}>0$$ and it follows from the discussion in Section 2 of [@B] that $g$ has an attracting $2-$cycle and all orbits in $[0,1]$ are attracted to this $2-$cycle except the fixed point $c=1/(2-a)$ and its preimages. So $f=hgh^{-1}$ has an attracting $2-$cycle and all orbits in $[1-k,1]$ are attracted to this $2-$cycle except $x^{*}=h(c)=\frac{1}{k+1}$ and its preimages. Our proof comes from [@B]. The same result is shown in [@SAG]; see page 590–591. See also Theorem 4.1 (f) in [@LT]. This case is also mentioned in (1) on the first page of [@ITN]. The attractor is a chaotic interval $[1-k,1]$ --------------------------------------------- \[prop3\] Let $f$ be as in and $I$ as in Lemma \[lem1\]. Suppose $k>1$ and either $\max\{1,k/(k^2-1)\}<r<k/(k-1)$ or $k/(k^2-1)<r<1/(k-1)$. Then $[1-k,1]$ is a chaotic attractor for $f$ in [int]{}$(I)$. In Lemma \[lem1\], we saw the interval $[1-k,1]$ is invariant under $f$. Moreover for all $x\in$ int$(I)$, there exists $n\ge 0$ such that $f^n(x)\in [1-k,1]$. So we may restrict $f$ to $[1-k,1]$ and study $g=h^{-1}fh:[0,1]\to [0,1]$ as defined in . We see that $$b-a=\frac{(k-1)(1/(k-1)-r)}{k}>0$$ when $r<1/(k-1)$, and when $r>1$ $$\frac{1-b}{a}=r>1$$ so that $b<1-a$. Also $$b-c=b-\frac{1}{2-a}=-\frac{rk^2-r-k}{k(k+1)}<0.$$ Then it follows from Propositions 2 and 4 in [@B] that $g$ is chaotic on $[0,1]$ and hence $f$ on $[1-k,1]$. Our proof comes from [@B]. Similar results are shown in [@SAG], although we did not find them easily. They are included in the results referring to ${\cal A}_{1}$. The first region here is called $D^*$ in [@ITN] and the second region is their $D_1$. These two regions are not separately considered in [@LT]. Also there are no results about chaos in [@LT], only results concerning positive Lyapunov exponent. All points escape except those on a chaotic Cantor set ------------------------------------------------------ \[prop4\] Let $f$ be as in and $\alpha$, $\beta$ as in Lemma \[lem1\]. Suppose $k>1$, $r>k/(k-1)$. Then if $x$ is real, either $f^n(x)\to-\infty$ as $n\to\infty$ or $x\in \Lambda\subset [\alpha,\beta]$, where $$\Lambda=\{x\in [\alpha,\beta]: f^n(x)\in [\alpha,\beta]\;{\rm for}\; n\ge 0\}$$ is an invariant Cantor set on which $f$ is chaotic. If we define $h:(-\infty,\infty)\to(-\infty,\infty)$ by $$h(x)=\frac{r+k}{k(r-1)}x-\frac{1}{r-1},$$ then $h([0,1])=[\alpha,\beta]$ and $g=h^{-1}\circ f\circ h$ is given by $$g(x)=\begin{cases} rx & (x\le a)\\ k(1-x) & (x\ge a),\end{cases}$$ where $a=k/(r+k)$ satisfies $g(a)=ra>1$. Then we define $$I_0=[0,1/r],\quad I_1=[1-1/k,1]$$ and $$\tilde\Lambda=\{x\in [0,1]: g^{n}(x)\in [0,1]\;{\rm for}\; n\ge 0\}.$$ As for the logistic map $\mu x(1-x)$ for $\mu>4$ (see pages 34–38 in [@D] or pages 98–101 and 131 in [@E]), we show $\tilde\Lambda$ is an invariant Cantor set on which $g$ is chaotic and such that if $x\notin \Lambda$, $g^n(x)\to -\infty$ as $n\to\infty$. The proposition follows with $\Lambda=h(\tilde\Lambda)$. Compare equation (14) on page 592 in [@SAG]. This region is not considered in [@B] or [@ITN]. In Theorem 4.1 (j), it is only shown that almost all orbits tend to $-\infty$. The existence of the Cantor set is not shown. Chaotic band attractors ======================= Now when $k>1$, $k/(k^2-1)<r$ (equivalently $rk^2-k-r>0$) and $r<1/(k-1)$, we see from Proposition \[prop3\] that $f$ is chaotic on $[1-k,1]$. In this section we suppose that $$\label{s0eq} (k,r)\in S_{1}=\{(k,r): k>1, 1/k<r\le k/(k^{2}-1)\}.$$ Now we state the main theorem. \[prop5\] Let $f$ be as in with $(k,r)\in S_{1}$ and let $I$ be as in Lemma \[lem1\]. Then there exists a decreasing sequence $\{S_{p}\}^{\infty}_{p=1}$ of subsets of $S_{1}$ such that when $p\ge 1$ and $(k,r)\in {\rm int}(S_p)$, $$\tilde\Lambda(k,r,p)=\bigcup_{i=0}^{2^{p}-1}J_{p,i}=\bigcup_{i=0}^{2^{p}-1}f^{i}([f^{2^{p}}(1),1]).$$ is invariant under $f$, all points in int$(I)$, except those on a finite set of periodic orbits or preimages of these orbits, are attracted to it, and when $k\to 1$, $\tilde\Lambda(k,r,p)\to \{1-k,1\}$. More precisely, for $p\ge 1$: [(i)]{} When $(k,r)\in {\rm int}(S_p)$, $J_{p,i}\bigcap J_{p,j}$ is empty for $0\le i,j\le 2^p-1$ and $i\neq j$; also if $(k,r)\in {\rm int}(S_{p+1})$ and $0\le i\le 2^{p}-1$, $J_{p+1,i}$ and $J_{p+1,2^{p}+i}$ are subintervals of $J_{p,i}$ obtained by deleting a central open interval. [(ii)]{} When $(k,r)\in {\rm int}(S_p)$, the orbits of all points in int$(I)$, except those on the orbits of certain unstable periodic points $\tilde C_m$, $0\le m\le p-1$ or preimaqes of these orbits, land in $\tilde\Lambda(k,r,p)$. [(iii)]{} if $(k,r)\in {\rm int}(S_p)\setminus S_{p+1}$, $f$ is chaotic on $\tilde\Lambda(k,r,p)$. Proof of Theorem \[prop5\] for $g$ ---------------------------------- First we prove this result for the map $g$ in . The lemma below is the key to the proof (and it is an important element in the next section). In this lemma, we show that $[0,1]$ splits into three intervals $I_0\cup J\cup I_1 =[0,g(b)]\cup(g(b),b)\cup[b,1]$ such that $g(I_1)=I_0$ and $g(I_0)=I_1$. The middle open interval $J$ contains an unstable fixed point and the orbits of all other points in $J$ eventually land in $I_0\cup I_1$. This means that $g$ can no longer be chaotic on $[0,1]$ but, if an additional condition is satisfied, we find that $g^2$ is chaotic on $I_1$ and hence that $g$ is chaotic on $I_0\cup I_1$. Actually this lemma is essentially the same as Lemma 1.1 in [@ITN]. \[lem2\] Consider the map $g$ in with $a=1-1/k$, $b=1-ra$ and suppose $(k,r)\in S_{1}=\{(k,r): k>1, 1/k<r\le k/(k^{2}-1)\}$. Then [(i)]{} $C_0=k/(k+1)$ is an unstable fixed point of $g$; [(ii)]{} $0<a<g(b)\le \frac{k}{k+1}\le b$, with strict inequality when $rk^2-k-r<0$ (that is, $r<k/(k^2-1)$) and $g([b,1])=[0,g(b)]=[0,k(1-b)]$; [(iii)]{} $g^2([b,1])=[b,1]$ and $$g^2(x)=\begin{cases} B+R(x-b) & (b\le x\le A)\\ b+K(1-x) & (A\le x\le 1),\end{cases}$$ where $A=1-a/k$, $B=g^{4}(1)=k(1-k)+k^{2}b$, $R=k^2$, $K=rk$; [(iv)]{} For each $x\in [0,1]$, $x\neq C_0=k/(k+1)$, $g^{n}(x)\in [0,g(b)]\cup[b,1]$ for sufficiently large $n>0$; [(v)]{} If $r^2k^3-k-r>0$, then $g^2$ is chaotic on $[b,1]$ and $g$ is chaotic on $[0,g(b)]\cup[b,1]$. \(i) Clearly $C_0=k/(k+1)$ is a fixed point of $g$ and it is unstable since $g'(C_0)=-k<-1$. \(ii) Clearly $a>0$. Also $b>a$ so that $g(b)=k(1-b)$ and $g([b,1])=[0,g(b)]$. Then $$g(b)-a=(kr-1)a>0,\quad \frac{k}{k+1}-g(b)=(k-1)\left(\frac{k}{k^2-1}-r\right)\ge 0$$ and $$b-\frac{k}{k+1}= \frac{k-1}{k}\left(\frac{k}{k^2-1}-r\right)\ge 0$$ with strict inequality if $rk^2-k-r<0$. \(iii) Note that on $[a,1]$, $$g^2(x)=\begin{cases} k(1-k)+k^2b+k^2(x-b) & (a\le x\le A=1-a/k)\\ b+rk(1-x) & (A\le x\le 1).\end{cases}$$ Now $$A-b=\frac{(rk-1)(k-1)}{k^2}>0.$$ So $a<b<A$. The range of $g^2$ on $[b,A]$ is $[g^2(b),1]$ and on $[A,1]$ the range is $[b,1]$. So $g^2([b,1])=[b,1]$ if and only if $g^2(b)\ge b$. However, using (ii), $$g^{2}(b)-b=k(1-k)+k^2b-b=(k^{2}-1)\left(b-\frac{k}{k+1}\right)\ge 0.$$ Lastly note that $k(1-k)+k^{2}b=g^{2}(b)=g^{4}(1)$. \(iv) Let $x\in (g(b),b)$, $x\neq C_0$. Then since $g(b)>a$, $g(x)=k(1-x)$. So $g(x)-C_0=k(1-x)-k(1-C_0)=k(C_0-x)$. Then if $g^n(x)\in (g(b),b)$ for all $n\ge 0$, we would have $|g^n(x)-C_0|=k^n|x-C_0|\to\infty$ as $n\to\infty$. \(v) If $h(x)=b+(1-b)x$, $$(h^{-1}g^{2}h)(x)=\begin{cases} \tilde B+Rx &(0\le x\le \tilde A)\\ K(1-x) &(\tilde A\le x\le 1),\end{cases}$$ where $\tilde A=h^{-1}(A)=1-1/K$, $\tilde B=1-R\tilde A$. Then $$\frac{1-\tilde B}{\tilde A}=R>1,\quad \tilde B-\frac{1}{2-\tilde A} =-\frac{RK^{2}-K-R}{K(K+1)}.$$ Since also $RK^{2}-K-R=k(r^2k^3-k-r)>0$. it follows from Propositions 4 and 2 in [@B] that $h^{-1}g^{2}h$ is chaotic on $[0,1]$ so that $g^2$ is chaotic on $[b,1]$. Then $g$ is chaotic on $[b,1]\cup g([b,1])=[0,g(b)]\cup[b,1]$. In the previous lemma, we considered the behaviour of $g$ on $[0,1]$ and $g^2$ on $[b,1]$, where $b=g^2(1)$. In the lemma below, for $p\ge 1$, we consider the behaviour of $g^{2^p}$ on $[B_p,1]$ and $g^{2^{p+1}}$ on $[B_{p+1},1]$, where $B_p=g^{2^p}(1)$. We prove it by repeatedly applying the previous lemma. To state the new lemma, we need to define the sets $S_{p}$ mentioned in Theorem \[prop5\]. Let $r_p$, $k_p$ be defined by the recurrence relations $$r_{p+1}=k^2_p,\quad k_{p+1}=r_pk_p,\quad p\ge 0$$ where $r_0=r$, $k_0=k$. For $p\ge 1$, write $$S_{p}=\{(k,r): k>1, r>1/k, k_{p-1}^2r_{p-1}-k_{p-1}-r_{p-1}\le 0\}.$$ When $k>1$ and $kr>1$, it is easy to see that $k_p>1$, $r_p>1$ for $p\ge 1$ and hence that $k_pr_p>1$ for $p\ge 0$. Then $$r_{p+1}k^{2}_{p+1}-k_{p+1}-r_{p+1}=k_{p}(r^2_{p}k^3_{p}-k_{p}-r_{p}) >k_{p}(r_{p}k^2_{p}-k_{p}-r_{p}).$$ So $r_{p+1}k^{2}_{p+1}-k_{p+1}-r_{p+1}\le 0$ implies that $r_{p}k^{2}_{p}-k_{p}-r_{p}<0$. Since it is easy to see by induction that $r_p$ and $k_p$ are continuous functions of $(k,r)$ so that $${\rm int}(S_{p})=\{(k,r): k>1, r>1/k, k_{p-1}^2r_{p-1}-k_{p-1}-r_{p-1}< 0\},$$ it follows that $S_{p+1}\subset$int$(S_p)$ for $p\ge 1$. \[lem3\] Consider the map $g$ in with $a=1-1/k$, $b=1-ra$. For $p\ge 0$, define $$\label{ABC} B_p=g^{2^p}(1),\quad A_p=1-(1-B_p)/k_p,\quad C_p=\frac{B_{p}+k_{p}}{k_{p}+1}.$$ Then for $p\ge 0$, if $(k,r)\in S_{p+1}$, [(i)]{} $C_{p}$ is an unstable fixed point of $g^{2^{p}}$; [(ii)]{} $$B_{p}<A_{p}<g^{2^{p}}(B_{p+1})\le C_{p}\le B_{p+1}$$ with the last two inequalities strict when $(k,r)\in{\rm int}(S_{p+1})$ and $$g^{2^{p}}([B_{p+1},1])=[B_{p},g^{2^{p}}(B_{p+1})];$$ [(iii)]{} $g^{2^{p+1}}([B_{p+1},1])=[B_{p+1},1]$ and $$g^{2^{p+1}}(x)=\begin{cases} B_{p+2}+r_{p+1}(x-B_{p+1}) &(B_{p+1}\le x\le A_{p+1})\\ B_{p+1}+k_{p+1}(1-x) &(A_{p+1}\le x\le 1).\end{cases}$$ [(iv)]{} For each $x\in [B_{p},1]$, $x\neq C_{p}$, $g^{n2^{p}}(x)\in [B_{p},g^{2^{p}}(B_{p+1})]\cup[B_{p+1},1]$ for sufficiently large $n>0$. [(v)]{} When $(k,r)\in{\rm int}(S_{p+1})$, the intervals $g^i([B_{p+1},1])$, $i=0,\ldots, 2^{p+1}-1$, are disjoint and for $i=0,\ldots,2^p-1$, $g^i([B_{p+1},1])$ and $g^{2^p+i}([B_{p+1},1])$ are contained in $g^i([B_{p},1])$ and are obtained by removing a central open interval, which contains $g^i(C_p)$, from $g^i([B_{p},1])$. [(vi)]{} When $(k,r)\in{\rm int}(S_{p+1})$, the orbits of all points in $[0,1]$, except for the orbits of $C_q$, $0\le q\le p$, and their preimages, eventually land in the union $\Lambda(k,r,p+1)$ of the disjoint intervals $g^i([B_{p+1},1])$, $i=0,\ldots, 2^{p+1}-1$. [(vii)]{} If $(k,r)\in{\rm int}(S_{p+1})\setminus S_{p+2}$, then $g^{2^{p+1}}$ is chaotic on $[B_{p+1},1]$ and therefore $g$ is chaotic on $\Lambda(k,r,p+1)$. For $p\ge 0$, if $r_{p}k^{2}_{p}-k_{p}-r_{p}\le 0$, we prove (i)-(vii) hold by induction on $p$. For $p=0$, the statements (i)-(vii) follow from Lemma \[lem2\]. Assuming the statements (i)-(vii) are true for $p\ge 0$, we prove them for $p+1$. So we are assuming $r_{p+1}k^{2}_{p+1}-k_{p+1}-r_{p+1}\le 0$, which implies that $r_{p}k^{2}_{p}-k_{p}-r_{p}< 0$. So by the induction hypothesis, $$g^{2^{p+1}}([B_{p+1},1])=[B_{p+1},1]$$ and $$g^{2^{p+1}}(x)=\begin{cases} B_{p+2}+r_{p+1}(x-B_{p+1}) &(B_{p+1}\le x\le A_{p+1})\\ B_{p+1}+k_{p+1}(1-x) &(A_{p+1}\le x\le 1).\end{cases}$$ Observe since $$g^{2^{p+1}}(A_{p+1})=B_{p+2}+r_{p+1}(A_{p+1}-B_{p+1}) =B_{p+1}+k_{p+1}(1-A_{p+1})=1,$$ it follows on elimination of $A_{p+1}$ that $$\label{speqn}(1-B_{p+2})=r_{p+1}(1-1/k_{p+1})(1-B_{p+1}).$$ Now if $h(x)=B_{p+1}+(1-B_{p+1})x$, $$G(x)=(h^{-1}g^{2^{p+1}}h)(x)=\begin{cases} b_{p+1}+r_{p+1}x &(0\le x\le a_{p+1})\\ k_{p+1}(1-x) &(a_{p+1}\le x\le 1),\end{cases}$$ where $$\label{bp} a_{p+1}=h^{-1}(A_{p+1}),\quad b_{p+1}=h^{-1}(B_{p+2})$$ but also $$a_{p+1}=1-1/k_{p+1},\quad b_{p+1}=1-r_{p+1}a_{p+1}$$ since $$G(a_{p+1})=(h^{-1}g^{2^{p+1}}h)(a_{p+1})=h^{-1}(g^{2^{p+1}}(A_{p+1}))=h^{-1}(1)=1.$$ $G$ satisfies the conditions of Lemma \[lem2\] since $k_{p+1}>1$, $k_{p+1}r_{p+1}>1$ and $r_{p+1}k^{2}_{p+1}-k_{p+1}-r_{p+1}\le 0$. Now we prove (i) for $p+1$. We know from Lemma \[lem2\] (i) applied to $G$ that $k_{p+1}/(k_{p+1}+1)$ is an unstable fixed point of $G$. It follows that $$\label{cpeq} h(k_{p+1}/(k_{p+1}+1)) =(B_{p+1}+k_{p+1})/(k_{p+1}+1)=C_{p+1}$$ is an unstable fixed point of $hGh^{-1}=g^{2^{p+1}}$. So (i) is proved for $p+1$. Now we prove (ii) for $p+1$. We know from Lemma \[lem2\] (ii) applied to $G=h^{-1}g^{2^{p+1}}h$ that $G([b_{p+1},1])=[0,G(b_{p+1})]$ and so, using , $$\begin{array}{rl} g^{2^{p+1}}([B_{p+2},1])&=(hGh^{-1})([B_{p+2},1])=(hG)([b_{p+1},1])\\ \\ &=h([0,G(b_{p+1})])=[B_{p+1},g^{2^{p+1}}(h(b_{p+1}))]\\ \\ &=[B_{p+1},g^{2^{p+1}}(B_{p+2})].\end{array}$$ Also, again from Lemma \[lem2\] (ii), $$0<a_{p+1}<G(b_{p+1})\le \frac{k_{p+1}}{k_{p+1}+1}\le b_{p+1},$$ with strict inequality when $r_{p+1}k^{2}_{p+1}-k_{p+1}-r_{p+1}<0$, so that, applying $h$, $$B_{p+1}<A_{p+1}<g^{2^{p+1}}(B_{p+2})\le \frac{B_{p+1}+k_{p+1}}{k_{p+1}+1} \le B_{p+2}$$ with strict inequality when $r_{p+1}k^{2}_{p+1}-k_{p+1}-r_{p+1}<0$. This proves (ii) for $p+1$. Next we prove (iii) for $p+1$. By Lemma \[lem2\] (iii) applied to $G$, $$\label{Geq}G^2([b_{p+1},1])=[b_{p+1},1]$$ and $$G^2(x)=\begin{cases} B+R(x-b_{p+1}) & (b_{p+1}\le x\le A)\\ b_{p+1}+K(1-x) & (A\le x\le 1),\end{cases}$$ where $A=1-a_{p+1}/k_{p+1}$, $$\label{newone} B=G^2(b_{p+1})=(h^{-1}g^{2^{p+2}}h)(b_{p+1})=h^{-1}(g^{2^{p+2}}(B_{p+2}))=h^{-1}(B_{p+3}),$$ $R=k_{p+1}^2=r_{p+2}$, $K=r_{p+1}k_{p+1}=k_{p+2}$. Then implies $$g^{2^{p+2}}([B_{p+2},1])=(hG^2)([b_{p+1},1])=h([b_{p+1},1])=[B_{p+2},1],$$ and using and , $$g^{2^{p+2}}(x)=(hG^2h^{-1})(x) =\begin{cases} B_{p+3}+r_{p+2}(x-B_{p+2}) & (B_{p+2}\le x\le h(A))\\ B_{p+2}+k_{p+2}(1-x) & (h(A)\le x\le 1),\end{cases}$$ where, using , $h(A)=B_{p+1}+(1-B_{p+1})A=1-(1-B_{p+1})a_{p+1}/k_{p+1} =1-(1-B_{p+2})/(r_{p+1}k_{p+1})=1-(1-B_{p+2})/k_{p+2}=A_{p+2}$. This completes the proof of (iii) for $p+1$. We also know from Lemma \[lem2\] (iv) applied to $G=h^{-1}g^{2^{p+1}}h$ that if $x\in [0,1]$, $x\neq k_{p+1}/(k_{p+1}+1)$, then $G^n(x)\in [0,G(b_{p+1})]\cup [b_{p+1},1]$ for some $n\ge 0$. This means that if $x\in [B_{p+1},1]$ and $x\neq h(k_{p+1}/(k_{p+1}+1))=C_{p+1}$ (see ), then $g^{n2^{p+1}}(x)\in [B_{p+1},g^{2^{p+1}}(B_{p+2})]\cup [B_{p+2},1]$ for some $n\ge 0$. This proves (iv) for $p+1$. Now we show (v) for $p+1$. By the induction hypothesis, the intervals $g^i([B_{p+1},1])$, $i=0,\ldots, 2^{p+1}-1$ are disjoint. Then for $i=0,\ldots, 2^{p+1}-1$, $$\begin{array}{l} g^i([B_{p+2},1])\subset g^i([B_{p+1},1]),\\ \\ g^{2^{p+1}+i}([B_{p+2},1])=g^i([B_{p+1},g^{2^{p+1}}(B_{p+2})])\subset g^i([B_{p+1},1]). \end{array}$$ Since $g^{2^{p+1}}([B_{p+1},1])=[B_{p+1},1]$, we have $a\in g^{2^{p+1}-1}([B_{p+1},1])$. Thus $g$ is one to one on $g^i([B_{p+1},1])$ for $0\le i<2^{p+1}-1$ and hence $g^i$ is one to one on $$[B_{p+1},1]=[B_{p+1},g^{2^{p+1}}(B_{p+2})]\cup(g^{2^{p+1}}(B_{p+2}),B_{p+2})\cup[B_{p+2},1]$$ for the same $i$. It follows that $g^i([B_{p+2},1])$ and $$g^{2^{p+1}+i}([B_{p+2},1])=g^i([B_{p+1},g^{2^{p+1}}(B_{p+2})])$$ are contained in $g^i([B_{p+1},1])$, are disjoint and each contains an endpoint of $g^i([B_{p+1},1])$. Hence $g^i([B_{p+2},1])$ and $g^{2^{p+1}+i}([B_{p+2},1])$ are what remains after removing a central open interval from $g^i([B_{p+1},1])$. Also since $g^{2^{p+1}}(B_{p+2})< C_{p+1}<B_{p+2}$ when $r_{p+1}k^{2}_{p+1}-k_{p+1}-r_{p+1}<0$ and $g^i$ is one to one on $[B_{p+1},1]$ for $0\le i <2^{p+1}-1$, it follows that $g^i(C_{p+1})$ is in the central open interval for $0\le i \le 2^{p+1}-1$. Then (v) follows for $p+1$. Now we prove (vi) for $p+1$. Suppose $x\in [0,1]$ does not lie on the orbit of $C_m$ for $0\le m\le p+1$. Then by the induction hypothesis, there is some $n\ge 0$ such that $g^n(x)\in \Lambda(k,r,p+1)$. Hence there exists $i$, $0\le i<2^{p+1}$, such that $g^n(x)\in g^i([B_{p+1},1])$ so that $g^{n+2^{p+1}-i}(x)\in g^{2^{p+1}}([B_{p+1},1])=[B_{p+1},1]$. Since $g^{n+2^{p+1}-i}(x)\neq C_{p+1}$, it follows from (iv) for $p+1$ as we have just proved that there exists $m>0$ such that $$g^{m2^{p+1}+n+2^{p+1}-i}(x)\in [B_{p+1},g^{2^{p+1}}(B_{p+2})]\cup[B_{p+2},1] =g^{2^{p+1}}([B_{p+2},1])\cup[B_{p+2},1]$$ which is contained in $\Lambda(k,r,p+2)$. This proves (vi) for $p+1$. Finally we show (vii) for $p+1$. We know from Lemma \[lem2\] (v) applied to $G^2$ that if $RK^2-K-R>0$, then $G^2$ is chaotic on $[b_{p+1},1]$. So if $r_{p+1}k^{2}_{p+1}-k_{p+1}-r_{p+1}>0$, $g^{2^{p+2}}$ is chaotic on $h([b_{p+1},1])=[B_{p+2},1]$. It follows that $g$ is chaotic on $\Lambda(k,r,p+2)$. This completes the induction proof that (i)-(vii) hold for $p\ge 1$. \[rem1\] It follows from (v) and (vi) in Lemma \[lem3\] that if $p\ge 0$ and $(k,r)\in{\rm int}(S_{p+1})$ then $\Lambda(k,r,p+1)\subset \Lambda(k,r,p)$. So for $p\ge 1$ and $(k,r)\in{\rm int}(S_{p})$, $\Lambda(k,r,p)\subset \Lambda(k,r,1)=I_{1,0}\cup I_{1,1}=[0,k(1-b)]\cup[b,1]=[0,rk-r]\cup[1-r+r/k,1]$ so that $\Lambda(k,r,p)\to \{0,1\}$ as $k\to 1$. Proof of Theorem \[prop5\] -------------------------- First it follows from Lemma \[lem1\] that if $x\in$ int$(I)$, that there exists $m\ge 0$ such that $f^{n}(x)\in [1-k,1]$ for $n\ge m$. Also $f$ maps $[1-k,1]$ into itself and on $[1-k,1]$, $f=hgh^{-1}$, where $g:[0,1]\to [0,1]$ is as in and $h(x)=1-k+kx$. Now suppose $p\ge 1$ and $(k,r)\in {\rm int}(S_{p})$. Then $$h(g^{i}([B_{p},1]))=f^{i}(h([B_{p},1])),$$ where, using , $$\label{beq} B_{p}=g^{2^{p}}(1)=h^{-1}(f^{2^{p}}(h(1)))=h^{-1}(f^{2^{p}}(1))$$ so that $$h([B_{p},1])=[h(B_{p}),1]=[f^{2^{p}}(1),1].$$ Thus $$h(g^{i}([B_{p},1]))=f^i(h([B_{p},1]))=f^{i}([f^{2^{p}}(1),1])=J_{p,i}.$$ Then (i) follows from (v) in Lemma \[lem3\]. Now we see that $$\tilde\Lambda(k,r,p)=\bigcup_{i=0}^{2^{p}-1}J_{p,i}=\bigcup_{i=0}^{2^{p}-1}h(g^i([B_p,1])) = h(\Lambda(k,r,p)),$$ where $\Lambda(k,r,p)$ is as in Lemma \[lem3\]. Since $\Lambda(k,r,p)$ is invariant under $g$, $\tilde\Lambda(k,r,p)$ is invariant under $f$, and since from Remark \[rem1\], $\Lambda(k,r,p)\to \{0,1\}$ as $k\to 1$, it follows that $\tilde\Lambda(k,r,p)\to h(\{0,1\})=\{1-k,1\}$ as $k\to 1$. Since the $g-$orbits of all points in $[0,1]$ except those on the orbits of the unstable periodic points $C_m$, $0\le m\le p-1$, or their preimages, land in $\Lambda(k,r,p)$, it follows that the $f-$orbits of all points in $[1-k,1]$ except those on the orbits of the unstable periodic points $\tilde C_m=h(C_{m})$, $0\le m\le p-1$, or their preimages, land in $\tilde\Lambda(k,r,p)$. Note, using and , $$h(C_{m})=1-k+kC_{m}=1-k+k\frac{B_{m}+k_{m}}{k_{m}+1}=\frac{f^{2^{m}}(1)+k_{m}}{k_{m}+1}.$$ Thus (ii) is proved. Finally (iii) follows directly from (vii) in Lemma \[lem3\]. In [@SAG], Proposition 4.6 on page 612, the band attractors are described but it is not shown as we do here that all points in int$(I)$, except those which lie on a finite set of periodic orbits or are preimages of these orbits, are attracted to the attractor. In Section 5 in [@B], where this parameter range is considered, it is only proved that some power of the map exhibits chaos on some interval and that there are no attracting periodic orbits. The region is not divided into infinitely many subregions each with a chaotic band attractor. The same comment applies to [@LT]. In fact, in [@LT] this region is not separately considered. The region int$(S_p)\setminus S_{p+1}$ corresponds to $D^{(p+1)}_0$ in [@ITN]. Geometry of the regions $S_{p}$ ------------------------------- Now we describe the geometry of the regions $S_p$. First we derive formulae for $r_p$ and $k_p$. \[lem5\] The recurrence relations $$r_{p+1}=k^2_p,\quad k_{p+1}=r_pk_p,$$ where $r_0=r$, $k_0=k$, have the solution $$k_p=r^{\chi(p)/2}k^{\chi(p)+(-1)^p},\quad r_p=r^{\chi(p)/2+(-1)^p}k^{\chi(p)},$$ with $\chi(p)=(2^{p+1}+2(-1)^{p+1})/3$. Also for $p\ge 0$, $$r_pk^2_p-k_p-r_p =\begin{cases} r^{\chi_(p)/2-1}k^{\chi(p)-1}[r^{\chi(p)}k^{2\chi(p)-1}-k-r]& (p\; {\rm odd})\\ \\ r^{\chi_(p)/2}k^{\chi(p)}[r^{\chi(p)+1}k^{2\chi(p)+2}-k-r]& (p\; {\rm even}).\end{cases}$$ If we define $x_p=r_p/k_p$, we see that $x_{p+1}=x^{-1}_p$ and so $x_p=x_0^{(-1)^p}=(r/k)^{(-1)^p}$. Then $$k_{p+1}=x_pk^2_p=\left(\frac{r}{k}\right)^{(-1)^p}k^2_p,$$ which we solve as $$k_p=r^{\chi(p)/2}k^{\chi(p)+(-1)^p}.$$ Then $$r_p=x_pk_p=r^{\chi(p)/2+(-1)^p}k^{\chi(p)}.$$ Next $$\begin{array}{rl} &r_pk^2_p-k_p-r_p\\ \\ &=k_{p}(r_{p}k_{p}-x_{p}-1)\\ \\ &=r^{\chi_(p)/2}k^{\chi(p)+(-1)^p}[r^{\chi(p)+(-1)^p}k^{2\chi(p)+(-1)^p} -r^{(-1)^p}k^{(-1)^{p+1}}-1]\\ \\ &=\begin{cases} r^{\chi(p)/2}k^{\chi(p)-1}[r^{\chi(p)-1}k^{2\chi(p)-1} -r^{-1}k-1]& (p\; {\rm odd})\\ r^{\chi(p)/2}k^{\chi(p)+1}[r^{\chi(p)+1}k^{2\chi(p)+1} -rk^{-1}-1]& (p\; {\rm even})\end{cases}\\ \\ &=\begin{cases} r^{\chi(p)/2-1}k^{\chi(p)-1}[r^{\chi(p)}k^{2\chi(p)-1} -k-r]& (p\; {\rm odd})\\ r^{\chi(p)/2}k^{\chi(p)}[r^{\chi(p)+1}k^{2\chi(p)+2} -k-r]& (p\; {\rm even}).\end{cases} \end{array}$$ It follows from this lemma that $r_pk^2_p-k_p-r_p$ has the same sign as the polynomial $$\label{tpeq} t_{p}(k,r)=\begin{cases} r^{\chi(p)}k^{2\chi(p)-1}-k-r& (p\; {\rm odd})\\ r^{\chi(p)+1}k^{2\chi(p)+2}-k-r& (p\; {\rm even}).\end{cases}$$ Next for $p\ge 0$, we determine the sign of the polynomials $t_p(k,r)$, and hence the sign of $r_pk^2_p-k_p-r_p$ in the region $S_1$. Thus we determine the $S_{p}$. \[lem6\] Let $t_p(k,r)$ be as in with $p\ge 0$. [(i)]{} for $p\ge 0$, there is a strictly decreasing function $\rho_{p}(k)>0$, defined for $k>1$ if $p=0$ and for $k\ge 1$ if $p\ge 1$, such that $t_{p}(k,r)$ has the same sign as $r-\rho_{p}(k)$ in $k>1$, $r>0$, where $$\rho_0(k)=\frac{k}{k^2-1}>\frac{1}{k},\quad \rho_1(k)=\frac{1+\sqrt{1+4k^4}}{2k^3}>\frac{1}{k},\quad k>1;$$ [(ii)]{} there is a decreasing sequence $K_{p}$, $p\ge 2$, with $1<K_p<2$ and tending to $1$ as $p\to\infty$ such that $$\rho_{p}(k)\begin{cases} >1/k & (1\le k<K_{p}),\\ =1/k & (k=K_{p}),\\ <1/k & (k>K_{p}).\end{cases}$$ [(iii)]{} $\rho_{p+1}(k)<\rho_{p}(k)$ for $1<k\le K_{p+1}$, $p\ge 0$ with $\le K_1$ interpreted as $<\infty$. [(iv)]{} for $p\ge 1$, $$S_{p}=\{(k,r): 1<k<K_{p-1}, 1/k<r\le \rho_{p-1}(k)\},$$ where $K_0=K_1=\infty$. \(i) For $p=0$, since $t_0(k,r)=rk^2-k-r$, the statement is true. For $p=1$, $$t_{1}(k,r)=k^{3}r^2-r-k=k^3(r-(1+\sqrt{1+4k^4})/(2k^3))(r+(\sqrt{1+4k^4}-1)/(2k^3))$$ and hence $t_1(k,r)$, has the same sign as $r-\rho_{1}(k)$. It is easy to see that $\rho_0(k)>1/k$ and $\rho_1(k)>1/k$. For $p\ge 2$, note that for fixed $k>1$, $t_p(k,r)$ is a polynomial in $r$ which is strictly convex in $r$ in $r\ge 0$ and tends to $\infty$ as $r\to\infty$. Moreover when $r=0$, it is negative. It follows that for $p\ge 2$ and fixed $k>1$, there is a well-defined function $\rho_{p}(k)>0$ such that $t_{p}(k,r)=0$ in $r\ge 0$ if and only if $r=\rho_{p}(k)$. Next when $p\ge 2$ is even and $r=\rho_{p}(k)$, $$\frac{\partial}{\partial r}t_p(k,r)=\chi(p)+(\chi(p)+1)k/r>0,$$ and $$\frac{\partial}{\partial k}t_p(k,r)=2\chi(p)+1+(2\chi(p)+2)r/k>0.$$ When $p\ge 3$ is odd and $r=\rho_{p}(k)$, $$\frac{\partial}{\partial r}t_p(k,r)=\chi(p)(1+k/r)-1>0,$$ and $$\frac{\partial}{\partial k}t_p(k,r)=2\chi(p)-2+(2\chi(p)-1)r/k>0.$$ From the implicit function theorem, it follows that $\rho'_{p}(k)$ exists and is negative so that $\rho_{p}(k)$ is strictly decreasing. Also since both derivatives with respect to $r$ are positive at $r=\rho_{p}(k)$, $t_{p}(k,r)$ has the same sign as $r-\rho_{p}(k)$. \(ii) Now we have $p\ge 2$. Since $t_{p}(1,1)=-1$, $t_{p}(2,1/2)>0$ and $\frac{d}{dk}t_p(k,1/k)>0$ for $k\ge 1$, $t_p(k,1/k)$ has a unique zero $K_p$ in $k\ge 1$ which lies in $(1,2)$. Then $t_p(k,1/k)$ has the same sign as $k-K_{p}$. Since $t_{p+1}(k,1/k)>t_{p}(k,1/k)$ in $k\ge 1$, it follows that $K_{p+1}<K_{p}$. Now if $k<K_{p}$ (resp. $=, >$), then $t_p(k,1/k)<0$ (resp. $=, >$) which implies that $1/k<\rho_{p}(k)$ (resp. $=, >$). \(iii) With $r=\rho_{p+1}(k)$ where $1<k\le K_{p+1}$ with $\le K_1$ interpreted as $<\infty$, $t_{p+1}(k,r)=0$ implies that $r_{p+1}k^2_{p+1}-k_{p+1}-r_{p+1}=0$, which by the remarks before Lemma \[lem3\], implies that $r_pk^2_p-k_p-r_p<0$. By Lemma \[lem5\], $r_pk^2_p-k_p-r_p$ has the same sign as $t_p(k,r)$ and hence the same sign as $r-\rho_p(k)$ which must therefore be negative. That is, $\rho_{p+1}(k)-\rho_p(k)<0$. \(iv) For $p=1$, we have the formula in where $\rho_0(k)=k/(k^2-1)$. For $p\ge 2$, $$\begin{array}{rl} S_{p} &=\{(k,r): k>1,\; r>1/k,\; k_{p-1}^2r_{p-1}-k_{p-1}-r_{p-1}\le 0\}\\ \\ &=\{(k,r): k>1,\; r>1/k,\; t_{p-1}(k,r)\le 0\}\\ \\ &=\{(k,r): k>1,\; r>1/k,\; 0< r\le \rho_{p-1}(k)\}\\ \\ &=\{(k,r): k>1,\; 1/k< r\le \rho_{p-1}(k)\}\\ \\ &=\{(k,r): 1<k<K_{p-1},\; 1/k< r\le \rho_{p-1}(k)\}, \end{array}$$ where $K_1=\infty$. In fact we can solve the cubic $k^6r^3-k-r=0$ to get $$\rho_2(k)=\left(\frac{1+\sqrt{1-\frac{4}{27k^{8}}}}{2k^5}\right)^{1/3} +\left(\frac{1-\sqrt{1-\frac{4}{27k^{8}}}}{2k^5}\right)^{1/3}.$$ {width="80.00000%"} Redwood City Attracting periodic orbits, chaotic band attractors and chaotic Cantor sets =========================================================================== Suppose $f$ is as in with $$\label{reg} (k,r)\in \{(k,r): 0<r<1,\; k>1+1/r\}.$$ The region in the parameter space corresponds to the region defined in equation (13) in [@SAG]. However equation (13) appears to be wrong as it implies that both slopes have absolute values $>1$, inconsistent with the existence of stable periodic orbits. In fact, the region in (13) should be $$\{(a_{{\cal L}},a_{{\cal R}}): 0<a_{{\cal L}}<1, a_{{\cal R}}<-(a_{{\cal L}}+1)/a_{{\cal L}}\}$$ which in our notation is . Note that the $n$ used in [@SAG] equals our $m+1$, where $m$ is defined below. This region is studied in Section 6 in [@B]. Corresponding to each integer $m\ge 2$, we define $$\label{altkm} K_m(r)=1+1/r+1/r^{2}+\cdots+1/r^{m-1} =\frac{1-r^m}{r^{m-1}(1-r)}.$$ Then the sets $$\{(k,r): 0<r<1, K_m(r)<k\le K_{m+1}(r)\},\quad m\ge 2$$ partition the space $\{(k,r):0<r<1, k>1+1/r\}$. We define $$T_m=\{(k,r): 0<r<1, K_m(r)<k<K_{m+1}(r)\},\quad m\ge 2.$$ (Note that $T_m$ is called $D_m$ in [@ITN].) We study separately the behaviour of the map in these subranges. Now we state the main theorem proved in this section. \[thm2\] Let $f$ be as in and let $m\ge 2$. Then $T_{m}$ can be divided into four subranges $$\begin{array}{l} R_{m1}=\{(k,r)\in T_m: kr^m<1\},\\ \\ R_{m2}=\{(k,r)\in T_m: kr^m>1,\; r^mk^2-k-r<0,\; r^{2m}k^3-k-r>0\},\\ \\ R_{m3}=\{(k,r)\in T_m: kr^m>1,\; r^mk^2-k-r<0,\; r^{2m}k^3-k-r<0\},\\ \\ R_{m4}=\{(k,r)\in T_m: kr^m>1,\;r^mk^2-k-r>0\},\end{array}$$ in each of which $f$ has an attractor. In $R_{m1}$ and $R_{m2}$, the orbits of all points not lying in a certain chaotic Cantor set or preimages of this Cantor set eventually go to the attractor; in $R_{m3}$ the orbits of all points not lying in a certain chaotic Cantor set or on a certain unstable periodic orbit or preimages of this Cantor set or periodic orbit eventually go to the attractor; in $R_{m4}$ the orbits of all points eventually lie in the attractor. In $R_{m1}$ the attractor is a periodic orbit with period $m+1$, in $R_{m4}$ it is the interval $[1-k,1]$ on which the dynamics is chaotic and in $R_{m2}$ and $R_{m3}$, the attractor is also chaotic and consists respectively of $m+1$ and $2m+2$ disjoint closed intervals. From Lemma \[lem1\], we know that for all real $x$, there exists $m\ge 0$ such that $f^{n}(x)\in [1-k,1]$ for $n\ge m$. Then if we restrict $f$ to $[1-k,1]$, the map $g=h^{-1}fh:[0,1]\to[0,1]$, where $h(x)=1-k+kx$, has the form with $$a=1-1/k,\quad b=1-ra.$$ Note that when $0<r<1$, $k>1+1/r$, $$b<a.$$ In subsections 5.1 to 5.5, we prove the results in Theorem \[thm2\] for the map $g$. Theorem \[thm2\] is proved in Section 5.6. The geometry of the $R_{mi}$ is described in Section 5.7. Properties of the iterates of $g$ in when $(k,r)\in T_m$ -------------------------------------------------------- In this subsection we derive a formula for $g^{m+1}$. First we introduceRedwood City the important quantity $x_m$. If $0<r<1$ and $K_m(r)<k <K_{m+1}(r)$, then $$r^m<\frac{1}{r+k(1-r)}< r^{m-1}$$ and hence $$\frac{1-r^{m-1}}{1-r}b< a<\frac{1-r^{m}}{1-r}b$$ and therefore $$(1+r+\cdots+r^{m-2})b<a<(1+r+\cdots+r^{m-1})b.$$ Then we see that for $1\le i\le m-1$, $$\label{yeq1} g^{i}(0)=(1+r+\cdots+r^{i-1})b < a$$ but $$\label{yeq2} g^{m}(0)=(1+r+\cdots+r^{m-1})b > a.$$ Note by and , there exists $x_m$ with $0<x_m<b<a$ such that $$\label{rel3} g^{m-1}(x_{m})=(1+r+\cdots +r^{m-2})b+r^{m-1}x_m=a.$$ Then since $b=1-ra$, $$\label{rel4} r^{m}x_{m}=1-(1+r+\cdots +r^{m-1})b$$ and since $a=1-1/k$, it further follows that $$\label{rel5} x_{m}=1-\frac{1-r^{m}}{kr^{m-1}(1-r)}.$$ $x_m$ plays an important role in the sequel. Next we derive a formula for the first three segments of $g^{m+1}(x)$ when $(k,r)\in T_m$. \[lem7\] Let $(k,r)\in T_m$. Then $$g^{m+1}(x)=\begin{cases} -kr^m(x-x_m) & (0\le x\le x_m)\\ \\ k^2r^{m-1}(x-x_m) & (x_m\le x\le x_m+{1\over k^2r^{m-1}})\\ \\ b+r-kr^m(x-x_m) & (x_m+{1\over k^2r^{m-1}}\le x \le b+rx_m).\end{cases}$$ Note that if $0\le x\le x_m$, using , $$(1+r+\cdots +r^{m-2})b+r^{m-1}x\le (1+r+\cdots +r^{m-2})b+r^{m-1}x_m=a.$$ So $(1+r+\cdots +r^{i-1})b+r^{i}x\le a$ if $1\le i\le m-1$ and hence $$\label{rel1} g^i(x)=(1+r+\cdots +r^{i-1})b+r^{i}x\quad{\rm if}\quad 1\le i\le m,\; 0\le x\le x_m.$$ Similarly if $0\le x\le b+rx_m$, $$(1+r+\cdots +r^{m-3})b+r^{m-2}x\le (1+r+\cdots +r^{m-2})b+r^{m-1}x_m=a.$$ So $(1+r+\cdots +r^{i-1})b+r^{i}x\le a$ if $1\le i\le m-2$ and hence $$\label{rel2} g^i(x)=(1+r+\cdots +r^{i-1})b+r^{i}x\quad{\rm if}\quad 1\le i\le m-1,\; 0\le x\le b+rx_m.$$ Next if $x_m\le x\le b+rx_m$, $$g^{m-1}(x)=(1+r+\cdots +r^{m-2})b+r^{m-1}x\ge (1+r+\cdots +r^{m-2})b+r^{m-1}x_m=a$$ so that $g^m(x)=k(1-g^{m-1}(x))$. Using this and , , we conclude that $$g^m(x) = \begin{cases}(1+r+\cdots +r^{m-1})b+r^{m}x & (0\le x\le x_m)\\ \\ k(1-(1+r+\cdots +r^{m-2})b-r^{m-1}x) & (x_m\le x\le b+rx_m).\end{cases}$$ Using and , we deduce that $$\label{gmform} g^m(x)= \begin{cases}1-r^m(x_m-x) & (0\le x\le x_m)\\ \\ 1-kr^{m-1}(x-x_m)& (x_m\le x\le b+rx_m).\end{cases}$$ Now if $0\le x\le x_m$, we have $$a<g^{m}(0)=1-r^mx_m\le g^m(x).$$ So if $0\le x\le x_m$, $$g^{m+1}(x)=k(1-(1-r^m(x_m-x)))=-kr^m(x-x_m).$$ Next in $[x_m,b+rx_m]$, $g^m(x)=1-kr^{m-1}(x-x_m)$ decreases from $1$ to $g^{m}(b+rx_{m})=g^m(g(x_m))=g^{m+1}(x_m)=g^2(g^{m-1}(x_m))=g^{2}(a)=0$ and $$1-kr^{m-1}(x-x_m)=a \quad{\rm iff}\quad x=x_m+{1\over k^2r^{m-1}}.$$ So if $x_m\le x\le x_m+{1\over k^2r^{m-1}}$, $$g^{m+1}(x)=k(1-(1-kr^{m-1}(x-x_m)))=k^2r^{m-1}(x-x_m)$$ and if $x_m+{1\over k^2r^{m-1}}\le x\le b+rx_m$, $$g^{m+1}(x)=b+r(1-kr^{m-1}(x-x_m))=b+r-kr^{m}(x-x_m).$$ This completes the proof of the lemma. Two periodic orbits, a forward invariant open set and a chaotic Cantor set -------------------------------------------------------------------------- For $(k,r)\in T_m$, we first show the existence of two periodic orbits, one of which is always unstable. \[prop8\] Suppose $(k,r)\in T_m$. Then [(i)]{} $g$ in has a periodic point $$a_1=kr^mx_m/(kr^m+1)\in (0,x_m)$$ with minimal period $m+1$ and if we define $a_{i+1}=g^i(a_1)$, then $$g^{i-1}(0)<a_{i}<g^{i-1}(x_m)<g^{i}(0),\quad i=1,\ldots,m$$ and $$a<g^{m}(0)<a_{m+1}<1.$$ [(ii)]{} Next $$b_1=k^2r^{m-1}x_m/(k^2r^{m-1}-1)\in (x_m,x_m+1/(k^2r^{m-1}))$$ is an unstable periodic point of $g$ with minimal period $m+1$. Moreover, if we define $b_{i+1}=g^i(b_1)$ for $i=0,\ldots,m$, then $$\label{gi} g^{i-1}(x_m)<b_i<g^i(x_m)\; (\le a),\quad i=1,\ldots,m-1$$ and $$\label{gm}a<b_m<\frac{k}{1+k}< b_{m+1}.$$ \(i) From Lemma \[lem7\], we see that $g^{m+1}(x)=x$ has the solution $a_1$ in $(0,x_m)$. Since $0<a_1<x_m<b=g(0)$, we have using that $g^i(0)<g^i(a_1)<g^i(x_m)<g^{i+1}(0)$ if $0\le i\le m-2$, that $g^{m-1}(0)<g^{m-1}(a_1)<g^{m-1}(x_m)=a<g^{m}(0)$ and using that $a<g^m(0)<g^m(a_1)<1$. In particular, $a_i$ is an increasing sequence for $i=1,\ldots,m+1$ and hence is a periodic orbit for $g$ with minimal period $m+1$. \(ii) Again from the formula for $g^{m+1}$ in Lemma \[lem7\], we see that $g^{m+1}(x_m)=0<x_m$ and $g^{m+1}(x_m+1/(k^2r^{m-1}))=1>x_m+1/(k^2r^{m-1})$. So there is a unique $x\in (x_m,x_m+1/(k^2r^{m-1}))$ such that $g^{m+1}(x)=x$. In fact, by solving the equation $g^{m+1}(x)=x$, we see that $x=b_1$. Since $|(g^{m+1})'(b_1)|=k^2r^{m-1}>1$, it follows that $b_1$ is unstable. Next since $x_m<b_1<b+rx_m=g(x_m)$, it follows from that $g^{i-1}(x_m)<b_i<g^i(x_m)\le a$ for $i=1,\ldots,m-1$. So is proved. Then applying $g$ again, $g^{m-1}(x_m)=a<b_m<g^m(x_m)=1$. Next note that since $x_m<b_1<x_m+1/(k^2r^{m-1})<b+rx_m$ and by $g^m$ is decreasing in $(x_m,b+rx_m)$, we have $g^m(x_m)>g^m(b_1)>g^m(x_m+1/(k^2r^{m-1}))$ and so $$1>b_{m+1}>a.$$ Next suppose $k/(1+k)\le b_m$. Then $a<k/(1+k)\le b_m$, since $$\frac{k}{1+k}-a=\frac{k}{1+k}-\left(1-\frac{1}{k}\right)=\frac{1}{k(k+1)}>0.$$ Applying $g(x)=k(1-x)$ once, we get $1>k/(1+Redwood Cityk)\ge b_{m+1}$ and again, we get $0<k/(1+k)\le b_1$, implying the absurdity that $k/(1+k)<a$. Hence $b_m<k/(1+k)$. Then $$b_{m+1}-\frac{k}{1+k}=g(b_m)-\frac{k}{1+k}=k(1-b_m)-\frac{k}{1+k} =k\left(\frac{k}{1+k}-b_m\right)>0.$$ Hence $k/(1+k)<b_{m+1}$ and the proof of is completed. Finally we see that $b_i$ strictly increases for $i=1,\ldots, m+1$ and so $m+1$ is the minimal period. Next, under an additional condition which will play an important role, we give more information about the relative positions of the two periodic orbits and show the existence of a forward invariant open set associated with the unstable periodic orbit. \[prop9\] Suppose $(k,r)\in T_m$ and $r^mk^2-k-r<0$. Then $$kr^mx_m<b_1<b,$$ $$\label{rel6} g^{i-1}(0)<a_{i}<g^{i-1}(x_m)<b_i<g^{i}(0)<g^i(x_m)\; (\le a), \quad i=1,\ldots,m-1$$ and $$\label{rel7} g^{m-1}(0)<a_m<a<b_m<b_{m+1}<g^m(0)<a_{m+1}.$$ Next there exists a unique sequence $\hat b_i$, $i=2,\ldots,m+1$ such that $\hat b_{m+1}=b_{m+1}$ and $$\label{hatin} g(\hat b_i)=\hat b_{i+1},\quad b_{i-1}<\hat b_i<g^{i-1}(0), \quad i=2,\ldots,m.$$Redwood City Moreover the intervals $[0,b_1)$, $(\hat b_2,b_2)$,...., $(\hat b_m,b_m)$, $(b_{m+1},1]$ are disjoint and $$U=[0,b_1)\cup \bigcup^{m}_{i=2}(\hat b_i,b_i)\cup (b_{m+1},1]$$ is an open set such that $g(U)\subset U$ and for all $x\in U$, there exists $n\ge 0$ such that $g^n(x)\in [0,x_m]$. First we observe that $$\label{kb} kr^mx_m-b_1 =kr^{m-1}x_m\frac{r^mk^2-k-r}{k^2r^{m-1}-1}=\frac{r^mk^2-k-r}{k}b_1<0$$ since $r^mk^2-k-r<0$. Then if $b_{m+1}\ge g^m(0)$, when we apply $g$ we get $b_1\le g^{m+1}(0)=kr^mx_m$. So we must have $$\label{rel8}b_{m+1}<g^m(0).$$ Then using and , we have $$\begin{array}{rl} b-b_1 &=b-kr^mx_m+kr^mx_m-b_1\\ \\ &= \displaystyle 1-r(1-1/k)-kr^m+\frac{r(1-r^m)}{1-r}+\frac{r^mk^2-k-r}{k}b_1\\ \\ &=\displaystyle -\frac{1}{k}[r^mk^2-k-r]+\frac{r^2(1-r^{m-1})}{1-r}+\frac{r^mk^2-k-r}{k}b_1 \end{array}$$ so that $$b-b_1=-\frac{1}{k}[r^mk^2-k-r](1-b_1)+\frac{r^2(1-r^{m-1})}{1-r}>0.$$ So we have proved $kr^mx_m<b_1<b$. Next since $$0<a_1<x_m<b_1<b<b+rx_m=g(x_m),$$ equation follows using . Applying $g$ to with $i=m-1$, we get $$g^{m-1}(0)<a_{m}<a<b_{m}<g^{m}(0).$$ The rest of follows from Proposition \[prop8\] and . We prove by backwards induction on $i$. The range of $g$ on $(0,a)$ is $(b,1)$. From Proposition \[prop8\], we know that $b<a<b_{m+1}<1$. So there exists $x\in (0,a)$ such that $g(x)=b_{m+1}$ and this $x$ is unique since $g$ is strictly increasing on $(0,a)$. Define $\hat b_m=x$. Then since $b_m<b_{m+1}<g^m(0)$ and $g$ is strictly increasing on $[b_{m-1},g^{m-1}(0)]$, we have $b_{m-1}<\hat b_m<g^{m-1}(0)$. Thus holds for $i=m$. Now we assume holds for some $i$ with $3\le i\le m$ and prove it for $i-1$. So we know that $$g(\hat b_i)=\hat b_{i+1},\quad b_{i-1}<\hat b_i<g^{i-1}(0).$$ The range of $g$ on $(b_{i-2},g^{i-2}(0))$ is $(b_{i-1},g^{i-1}(0))$. So there exists a point $x$ in $(b_{i-2},g^{i-2}(0))$ such that $g(x)=\hat b_{i}$ and this $x$ is unique since $g$ is strictly increasing on $(0,a)$. Define $\hat b_{i-1}=x$. Then holds for $i-1$ and the induction proof is complete. Note that $g([0,b_1))=[b,b_2)\subset (\hat b_2,b_2)$, $g((\hat b_i,b_i))= (\hat b_{i+1},b_{i+1})$ for $i=2,\ldots m-1$, $g((\hat b_m,b_m))=(b_{m+1},1]$ and $g((b_{m+1},1])=[0,b_1)$. It follows that $g(U)\subset U$ and if $x\in U$, there exists $n\ge 0$ such that $y=g^n(x)\in [0,b_1)$. If $x_m<y<b_1$, then we know from the graph of $g^{m+1}$ in Lemma \[lem7\] that $g^{Redwood Citym+1}(y)<y$. Since there are no fixed points of $g^{m+1}$ in $(x_m,b_1)$, it follows that there exists $p>0$ such that $g^{p(m+1)}(y)<x_m$. Hence if $x\in U$, either there exists $n\ge 0$ such that $g^n(x)\in [0,x_m]$ or there exists $p>0$ such that $g^{p(m+1)+n}(x)\in [0,x_m)$. Next we show that under the conditions of Proposition \[prop9\], the orbits of all points in $[0,1]$ either eventually land in $U$ or stay on an invariant Cantor set on which the dynamics is chaotic. \[prop10\] Suppose $(k,r)\in T_m$ and $r^mk^2-k-r<0$. Let $U$ be as in Proposition \[prop9\]. Then there exists a Cantor set $S$ in $[0,1]\setminus U$ such that $g(S)=S$ and such that if $x\in [0,1]\setminus U$, then either $x\in S$ or there exists $n>0$ such that $g^n(x)\in U$. Moreover the dynamics on $S$ is chaotic. Define $$S=\{x\in [0,1]: g^n(x)\in [0,1]\setminus U,\; n\ge 0\},$$ where $$[0,1]\setminus U=I_1\cup I_2\cup\cdots\cup I_{m-1}\cup I_m =[b_1,\hat b_2]\cup [b_2,\hat b_3]\cup\cdots\cup [b_{m-1},\hat b_m]\cup [b_m,b_{m+1}].$$ $S=\bigcap^{\infty}_{n=0}(g^n)^{-1}([0,1]\setminus U)$ is closed. Clearly $g(S)\subset S$. If $x\in S$ there exists $y\in [0,1]$ such that $g(y)=x$. Then $y\in [0,1]\setminus U$, since $y\in U$ implies $x\in U$, contradicting $x\in S$. So $y\in S$. Thus $S\subset g(S)$ and $g(S)=S$. We note that $g(I_j)=I_{j+1}$ for $1\le j\le m-1$ but $g(I_m)=[b_1,b_{m+1}]$, which contains $I_1\cup I_2\cup\cdots\cup I_{m-1}\cup I_m$. Let $\Sigma_m$ be the set of sequences $\{a_k\}^{\infty}_{k=0}$ such that $a_k\in\{1,\ldots,m\}$ and if $a_k<m$, then $a_{k+1}=a_k+1$. $\Sigma_m$ is invariant under the shift $\sigma$ and the dynamics of $\sigma$ on $\Sigma_m$ is chaotic. If $x\in S$, we define its itinerary to be the sequence $a\in\Sigma_m$ such that $g^k(x)\in I_{a_k}$. This defines a mapping $\phi:S\to \Sigma_m$ such that $\phi\circ g=\sigma\circ \phi$. By standard arguments (see, for example, pages 94-99 in [@D] where the case $m=2$ is considered), we show that $\phi$ is continuous and surjective and, furthermore, we may conclude that $\phi$ is a conjugacy and $S$ is a Cantor set, provided we can show that $S$ is a hyperbolic set. First note that if $x\in I_m$, then $|g'(x)|=k$ but if $x\in I_j$ with $j<m$, then $|g'(x)|=r$. Now suppose $x\in S$. If $x\in I_m$, then $|g'(x)|=k>1$. Suppose $x\in I_j$, where $1\le j\le m-1$. Then $g^{i}(x)\in I_{i+j}$ for $0\le i\le m-j$. In particular, $g^{m-j}(x)\in I_{m}$. So $g^{m-j+1}(x)\in I_{\ell}$ for some $\ell$, $1\le \ell\le m$. Suppose (a) $g^{m-j+1}(x)\in I_m$ also. Then $|(g^{m-j+2})'(x)|=r^{m-j}k^2$ which is $>1$, since $$k^2r^{m-j}\ge k^2r^{m-1}>k^2r^{2m-2}>K^2_m(r)r^{2m-2}>1.$$ Otherwise (b) $g^{m-j+1}(x)\in I_{\ell}$, where $1\le \ell\le m-1$. Then $g^{m-j+1+p}(x)\in I_{\ell+p}$ for $0\le p\le m-\ell$. It followsRedwood City that $$|(g^{m-j+1+m-\ell+1})'(x)|=r^{m-j}kr^{m-\ell}k=k^2r^{2m-j-\ell} \ge k^2r^{2m-2}>1.$$ Hence, by Lemma 4 in [@K], $S$ is hyperbolic and the proof is complete. Attracting periodic orbit in $R_{m1}$ ------------------------------------- \[prop6\] If $(k,r)\in R_{m1}=\{(k,r)\in T_m, k<1/r^m\}$, $a_1$ from Proposition \[prop8\] is an attracting periodic point for $g$ in . Moreover the open set $U$ from Proposition \[prop9\] is contained in its domain of attraction and all points in $[0,1]$ are attracted to the periodic orbit except those in the Cantor set $S$. Since $|(g^{m+1})'(a_1)|=kr^m$ and $kr^m<1$, $a_1$ is an attracting fixed point of $g^{m+1}$. Since $kr^m<1$, we have $r^mk^2-k-r<0$. So by Proposition \[prop9\], for each $x\in U$ there exists $n\ge 0$ such that $g^n(x)\in [0,x_m]$. Then we see from the graph of $g^{m+1}$ in Lemma \[lem7\] that $g^{\ell(m+1)}(g^n(x))\to a_1$ as $\ell\to\infty$. So $x$ is in the domain of attraction of the orbit of $a_1$. It also follows from Proposition \[prop10\] that the only points not attracted to the periodic orbit are those in the Cantor set $S$. Chaotic band attractors in $R_{m2}$ and $R_{m3}$ ------------------------------------------------ ### A band attractor for $g$ in $R_{m2}$ and $R_{m3}$ First we show the existence of a band attractor for $g$ as in , when $kr^m>1$ and $r^mk^2-k-r<0$. \[prop7\] If $(k,r)\in T_m$, $kr^m>1$ and $r^mk^2-k-r<0$, [(i)]{} the inequalities $$\label{lotin} x_m< kr^mx_m<b_1<b$$ hold and $$g^{m+1}([0,kr^mx_m])=[0,kr^mx_m],$$ where $x_m$ is as defined in and $b_1$ is as in Proposition \[prop8\]. [(ii)]{} With $p=kr^mx_m$, the intervals $g^i([0,p])$ are disjoint for $i=0,\ldots,m$ and if we define $$\Lambda= \bigcup^{m}_{i=0}g^i([0,p]),$$ then $g(\Lambda)=\Lambda$ and the orbits of all points in $[0,1]$ eventually land in $\Lambda$ except those in the Cantor set $S$ from Proposition \[prop10\]. \(i) $k>1/r^m$ implies that $x_m<kr^mx_m$. Then we see that the rest of follows from Proposition \[prop9\]. Next we consider $$\begin{array}{rl} g^{m+1}(kr^mx_m)-kr^mx_m &=k^2r^{m-1}(kr^mx_m-x_m)-kr^mx_m\\ \\ & =kr^{m-1}[r^mk^2-k-r]x_m\\ \\ &<0,\end{array}$$ where we have used Lemma \[lem7\]. Thus $$\label{gmk}g^{m+1}(kr^mx_m)<kr^mx_m.$$ Then since $x_m<kr^mx_m<b_1<x_m+\frac{1}{k^2r^{m-1}}$ and looking at the graph of $g^{m+1}$ in Lemma \[lem7\], we see that $g^{m+1}([0,kr^mx_m])=[0,kr^mx_m]$ follows at once from . \(ii) Since from , we have $0<p<b_1<b=g(0)<a$, it follows using that $$\label{yeq3}g^{i-1}(0)<g^{i-1}(p)<b_i<g^i(0)$$ for $i=1,\ldots,m$. This shows the disjointness of $g^i([0,p])$ for $i=0,\ldots,m-1$ and $g^i([0,p])=[g^i(0),g^i(p)]$ for the same $i$. Then we have $$g^{m}([0,p])=g(g^{m-1}([0,p])=g([g^{m-1}(0), g^{m-1}(p)])= [\min\{g^{m}(0), g^{m}(p)\},1],$$ since $$\label{gm1} g^{m-1}(0)<a=g^{m-1}(x_m)<g^{m-1}(p).$$ Using the formula for $g^{m}$, we have $$g^{m}(0)-g^{m}(p)=1-r^{m}x_{m}-(1-kr^{m-1}(p-x_{m}))=r^{m-1}(r^{m}k^{2}-k-r)x_{m}<0.$$ Hence $$\label{gmp}g^{m}([0,p])=[g^{m}(0),1].$$ Next since by , $g^{m-1}$ is increasing on $[0,b+rx_m]$ and $0<p<b$, it follows that $$g^{m-1}(p) < g^{m-1}(b)=g^m(0).$$ Thus $g^{m}([0,p])$ lies strictly to the right of all the intervals $g^{i}([0,p])$, $i=0,\ldots m-1$. Hence the intervals $g^i([0,p])$ are disjoint for $i=0,\ldots,m$. Next $$\begin{array}{rl} g(\Lambda) &=\displaystyle\bigcup^{m+1}_{i=1}g^i([0,p])=\bigcup^{m}_{i=1}g^i([0,p])\cup g^{m+1}([0,p]) =\bigcup^{m}_{i=1}g^i([0,p])\cup [0,p]\\ \\ &=\displaystyle\bigcup^{m}_{i=0}g^i([0,p])\\ \\ &=\Lambda.\end{array}$$ Since by and , $\hat b_{i+1}<g^i(0)$ for $i=1,\ldots,m-1$ and $g^i(p)<b_{i+1}$ for $i=0,\ldots,m-1$, it follows that $[0,p]\subset [0,b_1)$, $g^i([0,p])=[g^i(0),g^i(p)]\subset (\hat b_{i+1},b_{i+1})$ for $i=1,\ldots,m-1$. Also, using and , $g^m([0,p])=[g^m(0),1]\subset (b_{m+1},1]$. Hence $\Lambda\subset U$. Now suppose $x\in U$. Then by Proposition \[prop9\], there exists $n\ge 0$ such that $g^n(x)\in [0,x_m]\subset \Lambda$. Finally by Proposition \[prop10\], it follows that if $x\in [0,1]$, then either $x\in S$ or its orbit eventually lands in $\Lambda$. ### Dynamics on the attractor in $R_{m2}$ and $R_{m3}$ Finally we determine the dynamics of $g$ in on $\Lambda$, the invariant set from Proposition \[prop7\]. \[prop11\] [(i)]{} If $(k,r)$ is in $$R_{m2}=\{(k,r)\in T_m: kr^m>1,\; r^mk^2-k-r<0,\;r^{2m}k^3-k-r>0\},$$ $g$ is chaotic on $\Lambda=\bigcup^m_{i=0}g^i([0,kr^mx_m])$; [(ii)]{} if $(k,r)$ is in $$R_{m3}=\{(k,r)\in T_m: kr^m>1,\; r^mk^2-k-r<0,\;r^{2m}k^3-k-r<0\},$$ $g$ is chaotic on the union of the disjoint intervals $$\Lambda_1=\bigcup^{2m+1}_{i=0}g^i([0,k^2r^{m-1}(kr^m-1)x_m])\subset \Lambda$$ and if $x\in \Lambda\setminus\Lambda_1$, there exists $n\ge 0$ such that $g^n(x)\in \Lambda_1$ except for those $x$ on the orbit of the periodic point $a_1=kr^mx_m/(kr^m+1)$. $\Lambda_1$ is obtained from $\Lambda$ by removing an interval from the middle of each interval in $\Lambda$. Note in both (i) and (ii) we have $r^mk^2-k-r<0$. Then by Proposition \[prop7\], $g^{m+1}([0,p])=[0,p]$ with $p=kr^mx_m$. If we define $H:[0,1]\to [0,p]$ by $H(x)=kr^mx_m(1-x)$, then using Lemma \[lem7\] noting that $0\le H(x)\le kr^mx_m$, where by $x_m<kr^mx_m<b_1<x_m+1/(k^2r^{m-1})$, we find that $G=H^{-1}g^{m+1}H:[0,1]\to[0,1]$ is given byRedwood City $$\label{defG} G(x)=\begin{cases} B+Rx & (0\le x\le A)\\ K(1-x) & (A\le x\le 1),\end{cases}$$ where $$\label{yeq4} R=k^2r^{m-1},\quad K=kr^m,\quad A=1-1/K,\quad B=1-RA.$$ We see that $$K=kr^{m}>1,\quad R=k^{2}r^{m-1}>K>1,\quad \frac{K}{K-1}-R=-\frac{K(r^{m}k^{2}-k-r)}{r(K-1)}>0$$ What we have just done holds for both (i) and (ii). Now we prove (i). Then $$\label{yeq5} R-\frac{K}{K^{2}-1}=\frac{K(r^{2m}k^{3}-k-r)}{r(K^{2}-1)}>0.$$ This means that $K>1$ and $\max\{1,K/(K^{2}-1)\}<R<K/(K-1)$ so that it follows from what we have proved for $g$ in the proof of Proposition \[prop3\] that $G$ is chaotic on $[0,1]$ and hence that $g^{m+1}$ is chaotic on $[0,p]$. Then it follows that $g$ is chaotic on the union of the disjoint intervals $g^i([0,p])$ for $0\le i\le m$. Thus (i) is proved. Now we prove (ii). Then $$k^6r^{4m-1}-k-r>\frac{k^2}{r}-k-r =r\left[\left(\frac{k}{r}\right)^2-\left(\frac{k}{r}\right)-1\right]>0$$ if $k>r(1+\sqrt{5})/2$. However $k>K_m(r)\ge 1+1/r>2>2r>r(1+\sqrt{5})/2$. Hence if $(k,r)\in R_{m3}$, $$\label{yeq6} k^6r^{4m-1}-k-r>0.$$ Next since $k>1/r^m$, for $K$ and $R$ in , we have $K=kr^m>1$ and $R=k^2r^{m-1}>1$ and since $k^3r^{2m}-k-r<0$, using we have $R< K/(K^2-1)$. Then by Lemma \[lem2\] applied to $G$ defined in , $$\label{yeq7} 0<G(B)=K(1-B)<\frac{K}{K+1}<B<1,$$ $$\label{yeq8}G([B,1])=[0,K(1-B)],\quad G([0,K(1-B)])=[B,1],\quad G^2([B,1])=[B,1],$$ and $$G^2(x)=\begin{cases} B_2+R_1(x-B) & (B\le x\le A_1)\\ B+K_1(1-x) & (A_1\le x\le 1),\end{cases}$$ where $G^2(A_1)=1$, $B_2=G^{4}(1)$, $R_1=K^2$, $K_1=RK$. Also if $x$ is in $[0,1]$, $x\neq K/(K+1)$, we have $G^n(x)\in [0,G(B)]\cup[B,1]$ for sufficiently large $n>0$. Next since by $$R_1K^2_1-K_1-R_1=k^2r^{2m-1}(k^6r^{4m-1}-k-r)>0,$$ $G^2$ is chaotic on $[B,1]$. Now we see what these conclusions about $G$ meRedwood Cityan for $g^{m+1}=HGH^{-1}$. First if $$p=kr^mx_m,\quad p_1=H(B)=kr^mx_m(1-B)=k^2r^{m-1}(kr^m-1)x_m$$ and $$p_2=H(G(B))=kr^mx_m(1-K(1-B))=p-kr^mp_1,$$ we have, using , $0<p_1<p_2<p$ and, using , $$g^{m+1}([0,p_1])=[p_2,p],\quad g^{m+1}([p_2,p])=[0,p_1],\quad g^{2m+2}([0,p_1])=[0,p_1].$$ Next if $x\in [0,p]$, $x\neq H(K/(K+1))=a_1$ (see Proposition \[prop8\]), then $g^{n(m+1)}(x)\in [0,H(B)]\cup[H(G(B)),kr^mx_m]=[0,p_1]\cup[p_2,p]\subset \Lambda_1$ for sufficiently large $n>0$. Also $g^{2m+2}$ is chaotic on $H([B,1])=[0,kr^mx_m(1-B)]=[0,p_1]$. Suppose now that $x\in\Lambda$ and is not on the orbit of $a_1$. Then we have $x=g^i(y)$ for some $i$, $0\le i\le m$, and $y\in [0,p]$, $y\neq a_1$. Then $g^{n(m+1)}(y)\in\Lambda_1$ and hence $g^{n(m+1)+i}(x)\in\Lambda_1$ for sufficiently large $n>0$. Now we show that the intervals $g^i([0,p_1])$ are disjoint for $i=0,\ldots,2m+1$. Since $g^i([0,p_1])\subset g^i([0,p])$ for $i=0,\ldots,m$ and $g^{m+1+i}([0,p_1])=g^i([p_2,p])\subset g^i([0,p])$ for $i=0,\ldots,m$, we need only show that $g^{m+1+i}([0,p_1])$ and $g^i([0,p_1])$ are disjoint for $i=0,\ldots,m$. But if $g^{m+1+i}([0,p_1])\cap g^i([0,p_1])\neq\emptyset$, applying $g^{m+1-i}$, we have $[0,p_1]\cap [p_2,1]\neq\emptyset$, which is absurd. Hence the intervals $g^i([0,p_1])$ are disjoint for $i=0,\ldots,2m+1$. Note that for $i=1,\ldots, m-1$, $g^{i}$ is strictly increasing on $[0,p]$ because of and because $p<b<b+rx_{m}$ by . Hence since $0<p_1<p_2<p$, we have $$g^{i}(0)<g^{i}(p_{1})<g^{i}(p_{2})<g^{i}(p),\quad i=0,\ldots,m-1.$$ It follows that for $i=0,\ldots,m-1$, the intervals $g^{i}([0,p_1])=[g^{i}(0),g^{i}(p_{1})]$ and $g^{m+1+i}([0,p_1]) =g^i([p_2,p])=[g^{i}(p_{2}),g^{i}(p)]$ are obtained from $g^i([0,p])=[g^{i}(0),g^{i}(p)]$ by removing a middle interval. Next note that $g^m([0,p_1])$ and $g^m([p_2,p])$ are disjoint because their respective images under $g$ are $[p_2,p]$ and $[0,p_1]$. Also $g^m([0,p_1])$ contains $g^m(0)$, which is the left endpoint of $g^m([0,p])=[g^m(0),1]$ (see Eq. ), and $g^m([p_2,p])$ contains $1$ since $g^{m+1}([p_2,p])=[0,p_1]$ contains $0$. Hence $g^m([0,p_1])$ and $g^{2m+1}([0,p_1])=g^m([p_2,p])$ are obtained from $g^m([0,p])$ by removing a middle interval. Finally since $g^{2m+2}$ is chaotic on $[0,p_1]$, it follows that $g$ is chaotic on the union $\Lambda_1$ of the intervals $g^i([0,p_1])$ for $i=0,\ldots,2m+1$. Chaos in $R_{m4}$ ----------------- \[prop12\] When $$(k,r)\in R_{m4}=\{(k,r)\in T_m: r^mk^2-k-r>0\},$$ the map $g$ in is chaotic on $[0,1]$. In view of Propositions 2 and 4 in [@B], we need only show that if $J$ is a nontrivial interval, then $g^n(J)=[0,1]$ for some $n>0$. In the following, $a_i$ and $b_i$ are the periodic orbits from Proposition \[prop8\], where we note that $$a_1<a_2<\cdots<a_m<a<a_{m+1}.$$ First suppose $a_1\in J$. Then $a_1\in g^{n(m+1)}(J)$ for all $n\ge 0$. Then we cannot have $g^{n(m+1)}(J)\subset [0,x_m)$ for all $n\ge 0$ for otherwise, looking at the graph of $g^{m+1}$ in Lemma \[lem7\], the length of $g^{n(m+1)}(J)$ would be $(kr^m)^n$ times the length of $J$ which $\to\infty$ as $n\to\infty$. So there exists $n>0$ such that $[a_1,x_m]\subset g^{n(m+1)}(J)$. Then $[0,a_1]=g^{m+1}([a_1,x_m])\subset g^{(n+1)(m+1)}(J)$. So we can assume $J=[0,a_1]$. Looking at the graph of $g^{m+1}$ as described in Lemma \[lem7\], we see that $g^{m+1}(J)=[a_1,kr^mx_m]$. Since $r^mk^2-k-r>0$, $b_1<kr^mx_m$ (see Eq. \[kb\]). So we can assume $J=[a_1,\alpha]$, where $\alpha>b_1$. Then $g^{n(m+1)}(J)$ contains $[a_1,g^{n(m+1)}(\alpha)]$ for all $n\ge 0$. Suppose $g^{n(m+1)}(\alpha)\le x_m+1/(k^2r^{m-1})$ for all $n\ge 0$. Then we see from the graph of $g^{m+1}$ that $g^{n(m+1)}(\alpha)$ is an increasing sequence whose limit would be a fixed point of $g^{m+1}$ in $(b_1,x_m+1/(k^2r^{m-1})]$. However there is no such fixed point. Hence there exists $n\ge 0$ such that $g^{n(m+1)}(J)$ contains $[a_1,x_m+1/(k^2r^{m-1})]$ and so $g^{(n+1)(m+1)}(J)=[0,1]$. Thus we have proved that if $a_1\in J$, then $g^n(J)=[0,1]$ for large $n$. What remains is to show that the situation that $a_1, a_2,\ldots, a_{m+1}\notin g^n(J)$, where $a_{i+1}=g^i(a_1)$, for all $n\ge 0$ is not possible. Then we must have that for all $n\ge 0$, $g^n(J)$ is a subset of one of the intervals $[0,a_1)$, $(a_i,a_{i+1})$ for $i=1,\ldots,m$ and $(a_{m+1},1]$. If $J\subset [0,a_1)$, then $|g^{m+1}(J)|=kr^m|J|$, where $|\cdot|$ denotes length here. Suppose $J\subset (a_1,a_2)$. Then from Proposition \[prop8\], $$a_1<x_m<b=g(0)<a_2<g(x_m)=b+rx_m.$$ Then if $[x_m,x_m+1/(k^2r^{m-1})]\subset J$, $g^{m+1}(J)=[0,1]$, a possibility which can be excluded. Then either (a) $J\subset (a_1,x_m+1/(k^2r^{m-1}))$ or (b) $J\subset (x_m,a_2)$. If (a) holds, then either $x_m\notin J$, in which case $|g^{m+1}(J)|\ge kr^m|J|$ or $|g^{m+1}(J)|\ge k^2r^{m-1}|J|$, or $x_m\in J$ and $|g^{m+1}(J)|\ge (k^2r^m/(k+r))|J|$ since if we write $J=(\alpha,\beta)$ so that $x_m=\theta\alpha+(1-\theta)\beta$, then $$\begin{array}{rl} |g^{m+1}(J)| &=\max\{kr^{m}(x_m-\alpha),k^2r^{m-1}(\beta-x_m)\}\\ \\ &\ge kr^{m-1}|J|\min_{0\le \theta\le 1}\max\{r(1-\theta),k\theta\}\\ \\ &=\displaystyle\frac{k^2r^m}{k+r}|J|;\end{array}$$ if (b) holds, then either $x_m+1/(k^2r^{m-1})\notin J$, in which case $|g^{m+1}(J)|\ge k^2r^{m-1}|J|$ or $|g^{m+1}(J)|\ge kr^{m}|J|$, or $x_m+1/(k^2r^{m-1}) \in J$ and $|g^{m+1}(J)|\ge (k^2r^m/(k+r))|J|$ since if we write $J=(\alpha,\beta)$, $x_m+1/(k^2r^{m-1})=\theta\alpha+(1-\theta)\beta$, $$|g^{m+1}(J)|\ge kr^{m-1}|J|\min_{0\le \theta\le 1}\max\{k(1-\theta),r\theta\}=\frac{k^2r^m}{k+r}|J|.$$ Hence if $J\subset (a_1,a_2)$, $|g^{m+1}(J)|\ge L|J|$, where $$L=\min\{kr^m, k^2r^{m-1}, k^2r^m/(k+r)\}=k^2r^m/(k+r)>1$$ since $r^mk^2-k-r>0$. If $J\subset (a_i,a_{i+1})$ with $2\le i\le m$, then $J=g^{i-1}(\tilde J)$, where $\tilde J\subset (a_1,a_2)$ and here $g(x)=b+rx$ since $a_i<a$ for $i=1,\ldots,m$. Then $|g^{m+1}(\tilde J)|\ge L|\tilde J|$ and $|J|=r^{i-1}|\tilde J|$. Hence $$|g^{m+1-i+1}(J)|=|g^{m+1}(\tilde J)|\ge L|\tilde J|=(L/r^{i-1})|J|\ge L|J|.$$ If $J\subset (a_{m+1},1]$, then $|g(J)|\ge k|J|$. Since the length of the interval $J$ is expanded by some iterate of $g$ with coefficient of expansion at least $L>1$, it would follow that $|g^n(J)|$ is unbounded as $n\to\infty$, which is not possible. The proof is finished. The condition $(k,r)\in R_{m4}$ is the same as Bassein’s $((1-b)/a)^m>(1-b+ab)(1-a)/a$ on page 129 of [@B]. She does not give the details on how to prove the chaos. Proof of Theorem \[thm2\] ------------------------- When $(k,r)\in R_{m1}$, it follows from Proposition \[prop6\], that $h(a_1)$ is an attracting periodic point for $f$ with period $m+1$, where $h(x)=1-k+kx$ as in Lemma \[lem1\] (iv). Also the orbits of all points in $[1-k,1]=h([0,1])$, except those in the Cantor set $h(S)$, on which according to Proposition \[prop10\] the dynamics is chaotic, are attracted to the periodic orbit. Moreover, using Lemma \[lem1\], the orbits of all other points on the real line except those on the Cantor set $h(S)$ or preimages of this set are attracted to the periodic orbit. When $(k,r)\in R_{m2}$, it follows from Propositions \[prop7\] and \[prop11\] (i) that $h(\Lambda)\subset [1-k,1]$ is an invariant set for $f$ consisting of $m+1$ disjoint closed intervals on which the dynamics is chaotic. Moreover, using also Lemma \[lem1\] (ii), the orbits of all points on the real line except those on the Cantor set $h(S)$ or preimages of this set, are attracted to $h(\Lambda)$. Again, according to Proposition \[prop10\], the dynamics on $h(S)$ is chaotic. When $(k,r)\in R_{m3}$, it follows from Propositions \[prop7\] and \[prop11\] (ii) that $h(\Lambda_1)\subset [1-k,1]$ is an invariant set for $f$ consisting of $2m+2$ disjoint closed intervals on which the dynamics is chaotic. Moreover, using also Lemma \[lem1\] (ii), the orbits of all points on the real line except those on the Cantor set $h(S)$ or on the orbit of the periodic point $h(a_1)$ or preimages of the set or periodic orbit, eventually lie in $h(\Lambda_1)$. Again, according to Proposition \[prop10\], the dynamics on $h(S)$ is chaotic. When $(k,r)\in R_{m4}$, it follows from Proposition \[prop12\] that $h([0,1])=[1-k,1]$ is an invariant set for $f$ on which the dynamics is chaotic. Moreover, using Lemma \[lem1\] (ii), the orbits of all points on the real line eventually lie in $[1-k,1]$. In [@SAG], $R_{m1}$ corresponds to Proposition 3.1 on page 595, $R_{m2}$ to Proposition 4.1 on page 603, $R_{m3}$ to Proposition 4.2 on page 604 and $R_{m4}$ to ${\cal A}_1$ on page 604. However these authors do not describe the asymptotic fate of all points as we have. For the map $g$ in , this parameter region is studied in Section 6 in [@B]. As here, she defines a subrange corresponding to each integer $m\ge 2$, which coincides with our $T_m$. Inside each subrange she shows that the attractor is am $m+1-$periodic orbit (corresponding to our $R_{m1}$), or the interval $I$ on which the dynamics is chaotic (our $R_{m4}$); otherwise she shows that the $m+1-$th iterate of the map is chaotic on some subinterval (our $R_{m2}$ and $R_{m3}$). However she does not describe the attractor in $R_{m2}$ and $R_{m3}$ as we have. She does not show the existence of the invariant Cantor set in $R_{m1}$, $R_{m2}$ and $R_{m3}$. In Theorem 4.1 (g) in [@LT], the region $R_{m1}$ is studied and the existence of the attracting periodic orbit is shown. However they do not show the existence of the invariant Cantor set. A detailed analysis of the dynamics in $R_{m2}$, $R_{m3}$ and $R_{m4}$ is not given. In [@ITN], our $R_{m1}$ is $D^{(1)}_m$, our $R_{m2}\cup R_{m3}$ is $D^{(2)}_m$ and our $R_{m4}$ is $D^*_m$. Geometry of the four regions ---------------------------- In Theorem \[thm2\], we have divided $T_m$ into four regions $R_{m1}$, $R_{m2}$, $R_{m3}$ and $R_{m4}$. Now we give some information about the geometry of these regions. First note when $r>0$ and $k>0$ that $r^mk^2-k-r$ has the same sign as $k-L_m(r)$, where $$L_m(r)=\frac{1+\sqrt{1+4r^{m+1}}}{2r^m}.$$ Next note that using , $$\begin{array}{rl} K_{m+1}(r)-L_m(r) &=\displaystyle \frac{2(1+r+\cdots+r^m)}{2r^m}-\frac{1+\sqrt{1+4r^{m+1}}}{2r^m}\\ \\ &>\displaystyle \frac{1+2r-\sqrt{1+4r^{m+1}}}{2r^m}\\ \\ &>0\quad{\rm since}\quad (1+2r)^2>1+4r^{m+1}\;{\rm if}\; 0<r<1. \end{array}$$ Next if $p(k)=r^{2m}k^{3}-k-r$, we see that $p'(k)=3(kr^{m})^{2}-1>0$ if $k>1/r^{m}$. Also $p(1/r^{m})=-r<0$ and $p(k)\to\infty$ as $k\to\infty$. Hence $p(k)$ has a unique zero in $(1/r^{m},\infty)$, which we denote as $N_{m}(r)$. Thus $r^{2m}k^{3}-k-r$ has the same sign as $k-N_m(r)$ when $k>1/r^m$. Then, since $r^{2m}k^{3}-k-r>r^mk^2-k-r$ when $k>1/r^m$. it follows that $N_{m}(r)<L_{m}(r)$. [*Conclusion:*]{} $$\frac{1}{r^m}<N_m(r)<L_m(r)<K_{m+1}(r)$$ and $R_{m2}$ is defined by $N_m(r)<k<L_m(r)$, $R_{m3}$ by $1/r^m<k<N_m(r)$ and $R_{m4}$ by $k>L_m(r)$. The remaining problem is how $K_m(r)$ relates to $1/r^m$, $N_m(r)$ and $L_m(r)$. First note that $$\frac{1}{r^m}-K_m(r)=\frac{1}{r^m}-\frac{1-r^m}{r^{m-1}(1-r)} =\frac{1-2r+r^{m+1}}{r^m(1-r)}.$$ $p_{\alpha}(r)=1-2r+r^{m+1}$ has the properties: $p_{\alpha}(0)=1$, $p_{\alpha}(1)=0$ and $p_{\alpha}$ strictly decreases to a negative minimum at $(2/(m+1))^{1/m}$ and then strictly increases to $0$. So there is a number $\alpha_m$ where $0<\alpha_m<(2/(m+1))^{1/m}$ such that $p_{\alpha}(r)>0$ if $0<r<\alpha_m$, $p_{\alpha}(\alpha_m)=0$ and $p_{\alpha}(r)<0$ if $\alpha_m<r<1$. Since $p_{\alpha}(0.5)>0$ it follows that $\alpha_m>0.5$ and since $1-2r+r^{m+2}<1-2r+r^{m+1}$, it follows that $\alpha_{m+1}<\alpha_m$ so that $\alpha_m$ is a decreasing sequence. [*Conclusion:*]{} $K_m(r)-1/r^m$ has the same sign as $r-\alpha_m$, where $0.5<\alpha_m<1$. To compare $L_m(r)$ with $K_m(r)$, we look at $$\begin{array}{rl} &r^mK_m(r)^2-K_m(r)-r\\ \\ &=\displaystyle r^m\left(1+\frac{1}{r}+\cdots+\frac{1}{r^{m-1}}\right)^2-\left(1+\frac{1}{r}+\cdots+\frac{1}{r^{m-1}}\right)-r\\ \\ &=\displaystyle\frac{P(r)}{r^{m-1}},\end{array}$$ where $$P(r)=r(1+r+\cdots+r^{m-1})^2-(1+r+\cdots+r^m)=-1+r^2+\sum^{2m-1}_{\ell=3}c_{\ell}r^{\ell},$$ where $c_{\ell}>0$. Hence for $0<r<1$, $$P'(r)=2r+\sum^{2m-1}_{\ell=3}c_{\ell}\ell r^{\ell-1}>0.$$ Note also that $P(0)=-1$ and $P(1)=\sum^{2m-1}_{\ell=3}c_{\ell}>0$. It follows that $P(r)$ has a unique zero $\beta_m$ in $(0,1)$. Also $P(r)<0$ if $0<r<\beta_m$ and $>0$ if $\beta_m<r<1$. Note that since $P(r)$ is increasing in $m$, $\beta_m$ is a decreasing sequence. Also since when $K_m(r)=1/r^m$, $$r^mK_m(r)^2-K_m(r)-r=-r<0$$ so that $P(r)<0$, it follows that $\alpha_m<\beta_m$. [*Conclusion:*]{} $K_m(r)-L_m(r)$ has the same sign as $r-\beta_m$, where $\alpha_m<\beta_m<1$. To compare $K_m(r)$ with $N_m(r)$, we look at $$\begin{array}{rl} &r^{2m}K_m(r)^3-K_m(r)-r\\ \\ &=\displaystyle r^{2m}\left(1+\frac{1}{r}+\cdots+\frac{1}{r^{m-1}}\right)^3-\left(1+\frac{1}{r}+\cdots+\frac{1}{r^{m-1}}\right)-r\\ \\ &=\displaystyle\frac{Q(r)}{r^{m-1}},\end{array}$$ where $$Q(r)=r^2(1+r+\cdots+r^{m-1})^3-(1+r+\cdots+r^m)=-1-r+\sum^{3m-1}_{\ell=3}c_{\ell}r^{\ell},$$ where $c_{\ell}>0$. Hence for $0<r<1$, $$Q''(r)=6c_3r+\sum^{3m-1}_{\ell=4}c_{\ell}\ell(\ell-1)r^{\ell-2}>0.$$ Note also that $Q(0)=-1$ and $Q(1)=m^3-m-1>0$. From the convexity of $Q$, the existence of a unique root $\gamma_m$ of $Q$ in $(0,1)$ follows. Also since when $K_m(r)=1/r^m$ so that $r=\alpha_m$, $$r^{2m}K_m(r)^3-K_m(r)-r=-r<0,$$ it follows that $\alpha_m<\gamma_m$. On the other hand, when $r=\beta_m$, $$r(1+r+\cdots+r^{m-1})^2=(1+r+\cdots+r^m)$$ so that $$\begin{array}{rl} Q(r)&= r(1+r+\cdots+r^m)(1+r+\cdots+r^{m-1})-(1+r+\cdots+r^m)\\ \\ &= -(1+r+\cdots+r^m)(1-2r+r^{m+1})/(1-r)\\ \\ &=-(1+r+\cdots+r^m)p_{\alpha}(r)/(1-r)\\ \\ &>0\end{array}$$ since $\beta_m>\alpha_m$. Hence $\beta_m>\gamma_m$. [*Conclusion:*]{} $K_m(r)-N_m(r)$ has the same sign as $r-\gamma_m$, where $\alpha_m<\gamma_m<\beta_m$. It follows from the above conclusions that $$\begin{array}{rl} R_{m1}&=\{(k,r)\in T_m: k<1/r^m\}=\{(k,r):0<r<\alpha_m,\; K_m(r)<k<1/r^m\}\\ \\ R_{m2}&=\{(k,r)\in T_m: N_m(r)<k<L_m(r)\}\\ \\ &=\{(k,r): 0<r<\beta_m,\; \max\{K_m(r),N_m(r)\}<k<L_m(r)\}\\ \\ R_{m3}&=\{(k,r)\in T_m: 1/r^m<k<N_m(r)\}\\ \\ &=\{(k,r):0<r<\gamma_m,\; \max\{1/r^m,K_m(r)\}<k<N_m(r)\} \\ \\ R_{m4} &=\{(k,r)\in T_m: k>L_m(r)\}\\ \\ &=\{(k,r): 0<r<1,\; \max\{K_m(r),L_m(r)\}<k<K_{m+1}(r)\}. \end{array}$$ ![Regions $R_{21}$ (yellow), $R_{22}$ (blue), $R_{23}$ (red), $R_{24}$ (green) in $(r,k)-$parameter space. In $R_{21}$ there is an attracting periodic orbit with period 3, in $R_{22}$ there is a chaotic band attractor consisting of 3 intervals, in $R_{23}$ there is a chaotic band attractor consisting of 6 intervals and in $R_{24}$, the interval $[1-k,1]$ is a chaotic attractor.](figure3.pdf){width="80.00000%"} Acknowledgement {#acknowledgement .unnumbered} =============== The authors wish to thank Professor Laura Gardini for helpful remarks. [3]{} Bassein S. The dynamics of a family of one-dimensional maps. [*Amer. Math. Monthly*]{} [**105**]{} (1998), 118–130. Cheng K, Palmer K. Chaos in a model for masting. [*Discr. Cont. Dyn. Sys. B*]{} [**20**]{} (2015), 1917–1932. Devaney RL. An Introduction to Chaotic Dynamical Systems, 2nd Ed.. Redwood City, Addison-Wesley, 1989. Elaydi S. Discrete Chaos. Boca Raton, Chapman and Hall/CRC, 1999. Ichimura K, Ito M. Dynamics of skew tent maps. [*RIMS Kokyuroku*]{} [**1042**]{} (1998), 92–98. Ito S, Tanaka S, Nakada H. On unimodal transformations and chaos I. [*Tokyo J. Math*]{} [**2**]{} (1979), 221–239. Ito S, Tanaka S, Nakada H. On unimodal transformations and chaos II. [*Tokyo J. Math*]{} [**2**]{} (1979), 241–259. Kraft R. Chaos, Cantor sets and hyperbolicity for the logistic maps. [*Amer. Math. Monthly*]{} [**106**]{} (1999), 400–408. Lindstr" om, T., Thunberg, H. An elementary approach to dynamics and bifurcations of skew tent maps. [*J. Difference Eqns. Appl.*]{} [**14**]{} (2008), 819–833. Marcuard J, Visinescu E. Monotonicity properties of some skew tent maps. [*Ann. Inst. Henri Poincar' e, Probab. Stat.*]{} [**28**]{} (1992), 1–29. Misiurewicz M, Visinescu E. Kneading sequences of skew tent maps. [*Ann. Inst. Henri Poincar' e, Probab. Stat.*]{} [**27**]{} (1991), 125–140. Sushko I, Avrutin V, Gardini, L. Bifurcation structure in (the skew tent map and its application as a border collision normal form. [*J. Difference Eqns. Appl.*]{} [**22**]{} (2016), 1040–1087.

--- abstract: 'To determine the sharp constants for the one dimensional Lieb–Thirring inequalities with exponent $\gamma \in (1/2,3/2)$ is still an open problem. According to a conjecture by Lieb and Thirring the sharp constant for these exponents should be attained by potentials having only one bound state. Here we exhibit a connection between the Lieb–Thirring conjecture for $\gamma=1$ and an isporimetric inequality for ovals in the plane.' address: - 'Department of Physics, P. Universidad Católica de Chile, Casilla 306, Santiago 22, Chile' - 'School of Mathematics, Georgia Institute of Technology' author: - 'Rafael D. Benguria' - Michael Loss title: 'Connection between the Lieb–Thirring conjecture for Schrödinger operators and an isoperimetric problem for ovals on the plane' --- [^1] [^2] Introduction ============ The Lieb–Thirring inequalities are one of the main tools in the proof of the stability of matter [@LiTh75] (see also the review article [@Li76] or [@Li00]). Let $H=-\Delta + V$ be the Schrödinger operator acting on $L^2({\mathbb{R}}^n)$, $n \ge 1$ and denote by $e_1 \le e_2 \le \dots < 0$ the negative eigenvalues of $H$. The Lieb–Thirring inequalities are given by $$\sum_{j \ge 1} {\vert e_j \vert}^{\gamma} \le L_{\gamma,n} \int_{{\mathbb{R}}^n} V_{-}(x)^{\gamma+n/2} \, dx, \label{eq:1.1}$$ where $V_{-}(x) \equiv \max(-V(x),0)$ is the negative part of the potential. The above inequalities hold for $\gamma \ge 1/2$ when $n=1$, for $\gamma >0$ when $n=2$, and for $\gamma \ge 0$ for $n \ge 3$. The case $\gamma=1/2$, $n=1$ was established by T. Weidl [@We96]. The case $\gamma=0$, $n \ge 3$ was established independently by M. Cwikel, E.H. Lieb and G.V. Rosenbljum. One can show in general that $L_{\gamma,n} \ge L_{\gamma,n}^c$, where $$L_{\gamma,n}^c = 2^{-n} \pi^{-n/2} \frac{\Gamma(\gamma+1)} {\Gamma(\gamma+1+n/2)} \label{eq:1.2}$$ are the semiclassical constants. Define $R_{\gamma,n} \equiv L_{\gamma,n}/ L_{\gamma,n}^c \ge 1$. Aizenman and Lieb proved that $R_{\gamma,n}$ decreases as $\gamma$ increases [@AiLi78]. In [@LiTh76] it is proven that $L_{3/2,1} = L_{3/2,1}^c$ and thus, $L_{\gamma,1} = L_{\gamma,1}^c$, for all $\gamma \ge 3/2$. For $n>1$, Laptev and Weidl [@LaWe00a] proved $L_{3/2,n} = L_{3/2,n}^c$ hence, $L_{\gamma,n} = L_{\gamma,n}^c$, for all $\gamma \ge 3/2$. The sharp constant for $\gamma=1/2$ and $n=1$, $L_{1/2,1}=1/2$ was proved in [@HuLiTh98]. For best constants up to date see [@HuLaWe00]. For $n=1$, the sharp constants $L_{\gamma,1}$ are not known for values of $\gamma$ in the interval $(1/2,3/2)$. However, in 1976 Lieb and Thirring [@LiTh76] conjectured that the sharp constants are attained for potentials that have only one bound state, and therefore $$L_{\gamma,1} \equiv L_{\gamma , 1}^1 = \frac{1}{\sqrt{\pi}} \frac{1}{\gamma-1/2} \frac{\Gamma(\gamma+1)}{\Gamma(\gamma+1/2)} \left(\frac{\gamma-1/2}{\gamma+1/2} \right)^{\gamma+1/2}. \label{eq:1.3}$$ In this manuscript we establish a connection between the Lieb–Thirring conjecture for $\gamma=1$ and $n=1$ and an isoperimetric inequality for closed curves in the plane which are smooth, have positive curvature and length $2 \pi$. The rest of the article is organized as follows. In Section 2, we provide a new and direct method for maximizing the lowest eigenvalue of one dimensional Schrödinger operators. In Section 3 we establish the aforementioned connection with a problem for closed curves in the plane. We should emphasize that the isoperimetric problem that we allude to is also still open. Maximizing the first eigenvalue =============================== The problem of maximizing the lowest eigenvalue of the one-dimensional Schrödinger operator on the line subject to a constraint on integrals of powers of the potential was first considered by Joseph Keller in 1961 [@Ke61]. See also [@LiTh76; @AsHa87; @BeVe92; @Ve02]. Consider the Schrödinger operator, $$H=-\frac{d^2}{dx^2} + V \label{eq:2.1}$$ defined on $L^2({\mathbb{R}})$, and let $-\lambda_1$ be the lowest eigenvalue. Then, $$\lambda_1^{\gamma} \le L_{\gamma , 1}^1 \int_{-\infty}^{\infty} V_{-}(x)^{\gamma+1/2} \, dx, \label{eq:2.2}$$ for all $\gamma>1/2$, where the sharp constants $L_{\gamma,1}^1$ are given by, $$L_{\gamma,1}^1 = \frac{1}{\sqrt{\pi}} \frac{1}{\gamma-1/2} \frac{\Gamma(\gamma+1)}{\Gamma(\gamma+1/2)} \left(\frac{\gamma-1/2}{\gamma+1/2} \right)^{\gamma+1/2}. \label{eq:2.3}$$ Keller’s proof uses the Direct Calculus of Variations. When the exponent $\gamma=1$ there is a very simple argument to compute the best constant. We give the full argument in the sequel, because it is important in our later derivation of the connection between the Lieb–Thirring conjecture and an isoperimetric inequality for ovals in ${\mathbb{R}}^2$. Let $u_1$ and $-\lambda_1$ be the normalized ground state and the lowest eigenvalue of the Schrödinger operator $H=-d^2/dx^2 - V$ on $L^2({\mathbb{R}})$, where $V \ge 0$. Thus, $$-u_1''-V \, u_1 = - \lambda_1 u_1, \qquad \mbox{in ${\mathbb{R}}$}. \label{eq:2.4}$$ Multiplying (\[eq:2.4\]) by $u_1$ and integrating in ${\mathbb{R}}$, we get $$\lambda_1 = \int_{{\mathbb{R}}} V \, u_1^2 \, dx - \int_{{\mathbb{R}}} (u_1')^2 \, dx. \label{eq:2.5}$$ Since, $$V \, u_1^2 \le K \, V^{3/2} + \frac{4}{27 \, K^2} u_1^6,$$ for all $K >0$, from (\[eq:2.5\]) we get, $$\lambda_1 \le K \int_{{\mathbb{R}}} V^{3/2} \, dx + \frac{4}{27 \, K^2} \int_{{\mathbb{R}}} u_1^6 \, dx - \int_{{\mathbb{R}}} (u_1')^2 \, dx. \label{eq:2.6}$$ However, if $\int_{{\mathbb{R}}} u_1^2 \, dx=1$, $$\int_{{\mathbb{R}}} (u_1')^2 \, dx \ge \frac{\pi^2}{4} \int_{{\mathbb{R}}} u_1^6 \, dx, \label{eq:2.7}$$ so, choosing $K=4/(3 \sqrt{3} \pi)$, we finally get $$\lambda_1 \le \frac{4}{3 \sqrt{3} \pi} \int_{{\mathbb{R}}} V^{3/2} \, dx = L_{1,1}^1 \int_{{\mathbb{R}}} V^{3/2} \, dx \label{eq:2.8}$$ which is Keller’s result for $\gamma=1$. For completeness we give an elementary proof of (\[eq:2.7\]). First make the change of variables $x \to s$ given by $$s = \int_{-\infty}^{x} u_1^2 \, dy. \label{eq:2.9}$$ Here, $s: 0 \to 1$, and $ds/dx=u_1^2$. With this change of variables we have, $$\int_{-\infty}^{\infty} u_1^6 \, dx = \int_0^1 u_1^4 \, ds, \label{eq:2.10}$$ and $$\int_{-\infty}^{\infty} (u_1')^2 \, dx = \int_0^1 (\dot u_1)^2 u_1^2 \, ds, \label{eq:2.11}$$ where $\dot u_1 \equiv d u_1/ds$. Since $u_1$ goes to zero at $x=\pm \infty$ we have $u_1(s=0)=u_1(s=1)=0$. Finally, if we call $w \equiv u_1^2$, $\int_{-\infty}^{\infty} u_1^6 = \int_0^1 w^2 \, ds$ and $\int_{-\infty}^{\infty} (u_1')^2 \, dx =(1/4) \int_0^1 {\dot w}^2 \, ds$. In terms of $w(s)$, (\[eq:2.7\]) is given by $$\int_0^1 {\dot w}^2 \, ds \ge \pi^2 \int_0^1 w^2 \, ds, \label{eq:2.12}$$ which follows from the fact that the first Dirichlet eigenvalue of the interval $(0,1)$ is $\pi^2$. One can obtain the cases with $\gamma \neq 1$ in (\[eq:2.2\]) in a similar way (see the Appendix). Maximizing the sum of the first two eigenvalues and the connection with a geometric problem in ${\mathbb{R}}^2$. ================================================================================================================ Consider the Schrödinger operator $$H=-\frac{d^2}{dx^2} - V,$$ on $L^2({\mathbb{R}})$ with $V \ge 0$ such that $\int V^{3/2} \, dx < \infty$. Assume $H$ has at least two negative eigenvalues, and denote by $-\lambda_1$ and $-\lambda_2$ the lowest two eigenvalues and $u_1$, $u_2$ the corresponding normalized eigenfunctions. As before, we have $$\lambda_1 = \int_{{\mathbb{R}}} V \, u_1^2 \, dx - \int_{{\mathbb{R}}} (u_1')^2 \, dx, \label{eq:3.1}$$ and $$\lambda_2 = \int_{{\mathbb{R}}} V \, u_2^2 \, dx - \int_{{\mathbb{R}}} (u_2')^2 \, dx. \label{eq:3.2}$$ Adding these two equations and using the pointwise bound, $$V (u_1^2+u_2^2) \le K \, V^{3/2} + \frac{4}{27 \, K^2} (u_1^2+u_2^2)^3,$$ we get $$\lambda_1+\lambda_2 \le K \int_{{\mathbb{R}}} V^{3/2} \, dx + \frac{4}{27 \, K^2} \int_{{\mathbb{R}}} (u_1^2+u_2^2)^3 \, dx - \int_{{\mathbb{R}}} \left((u_1')^2+(u_2')^2 \right)\, dx. \label{eq:3.3}$$ In order to prove the Lieb–Thirring conjecture for $\gamma=1$ in the special case of potentials having only two eigenvalues, it would be enough to prove $$\int_{{\mathbb{R}}} \left((u_1')^2+(u_2')^2 \right)\, dx \ge \frac{\pi^2}{4} \int_{{\mathbb{R}}} (u_1^2+u_2^2)^3 \, dx, \label{eq:3.4}$$ for any pair of functions $u_1$, $u_2$ such that $\int_{{\mathbb{R}}} u_1^2 \, dx= \int_{{\mathbb{R}}} u_2^2 \, dx=1$, and $\int_{{\mathbb{R}}} u_1 \, u_2 \, dx=0$ (i.e., for any pair of mutually orthogonal, normalized functions). For then, it would follow from (\[eq:3.3\]) and (\[eq:3.4\]) that $$\lambda_1+\lambda_2 \le L_{1,1}^1 \int_{{\mathbb{R}}} V^{3/2} \, dx. \label{eq:3.5}$$ To prove (\[eq:3.4\]) is still an open problem. Here we will show that (\[eq:3.4\]) is equivalent to an (open) isoperimetric inequality for ovals on the plane. To establish this connection, we perform a change of variables similar to the one used in the previous section to prove Keller’s result on the lowest eigenvalue. First we change the independent variable $$x \to s \equiv \pi \int_{-\infty}^{x} \left(u_1^2+u_2^2 \right) \, dy. \label{eq:3.6}$$ Since $u_1$ and $u_2$ are both normalized, it follows that $s$ runs from $0$ to $2 \pi$. From (\[eq:3.6\]) we have $$\frac{ds}{dx} = \pi \left(u_1^2+u_2^2 \right).$$ Moreover, set $$u_1=\rho \cos \theta, \qquad \mbox{and} \qquad u_2=\rho \sin \theta, \label{eq:3.7}$$ so that $$u_1^2+u_2^2= \rho^2, \qquad \mbox{and} \qquad {u_1'}^2+{u_2'}^2= {\rho'}^2 + \rho^2 {\theta'}^2. \label{eq:3.8}$$ With this change of variables we can write $$\int_{{\mathbb{R}}}\left({u_1'}^2+{u_2'}^2 \right) \, dx = \pi \int_{0}^{2 \pi} \left(\rho^2 {\dot \rho}^2 + \rho^4 {\dot \theta}^2 \right) \, ds, \label{eq:3.9}$$ and $$\int_{{\mathbb{R}}}\left(u_1^2+u_2^2 \right)^3 \, dx = \frac{1}{\pi} \int_{0}^{2 \pi} \rho^4 \, ds. \label{eq:3.10}$$ Furthermore, set $$R=\rho^2,$$ and $$\varphi= 2\theta.$$ In these new variables, the desired inequality (\[eq:3.4\]) is equivalent to $$\frac {\int_0^{2\pi} \left({\dot R}^2+R^2{\dot \varphi}^2 \right) \, ds} {\int_0^{2\pi} R^2\, ds} \ge 1, \label{eq:3.11}$$ subject to the fact that $u_1$ and $u_2$ are orthonormal, fact that we ought to express in terms of the new variables. In the new variables, $$0=\int_{{\mathbb{R}}} u_1 \, u_2 \, dx = \frac{1}{2 \pi}\int_0^{2 \pi} \sin \varphi (s) \, ds.$$ Concerning the other side constraints (i.e., the fact that $u_1$ and $u_2$ are normalized), given the definition of $s$ and the fact that $s$ runs from $0$ to $2\pi$, it is enough to consider the combination $$0=\int_{{\mathbb{R}}} (u_1^2-u_2^2) \, dx = \frac{1}{\pi}\int_0^{2 \pi} \cos \varphi (s) \, ds.$$ Thus, in the new variables, the fact that $u_1$ and $u_2$ are orthonormal imply $$\int_0^{2 \pi} \sin \varphi (s) \, ds= \int_0^{2 \pi} \cos \varphi (s) \, ds=0. \label{eq:3.12}$$ These latter conditions can be given a simple geometrical interpretation. If one considers a closed curve in ${\mathbb{R}}^2$ and denote by $\cos \varphi (s)$ and $\sin \varphi (s)$ the components of the unit tangent, with respect to a fixed frame, as a function of arc–length, (\[eq:3.12\]) just says that the curve in question is closed. Moreover, the curvature of the curve is given by $$\kappa(s)=\frac{d \varphi}{ds}. \label{eq:3.13}$$ Let’s denote by $C$ a closed curve in the plane, of length $2 \pi$, with positive curvature, and let $$H(C) \equiv -\frac{d^2}{ds^2} + \kappa^2 \label{eq:3.14}$$ acting on $L^2(C)$ with periodic boundary conditions. Then, (\[eq:3.4\]), and for that matter (\[eq:3.11\]), is equivalent to saying that the lowest eigenvalue of $H(C)$, $\lambda_1(C)$ say, is larger or equal to $1$, for any closed curve on the plane of length $2 \pi$. It is a simple fact to see that if $C$ is a circle of length $2 \pi$, the lowest eigenvalue of $H(C)$ is precisely $1$. Unfortunately we are far from proving the desired bound for general curves. It is relatively simple to show that the lowest eigenvalue of the Hamiltonian $H(C)$ is bounded below by $1/2$. To see this one first notes that the corresponding eigenfunction can be chosen to be positive. The quadratic form $$(f, H(C) f) = \int_0^{2 \pi} |f'(s)|^2 ds + \int _0^{2 \pi} \kappa^2(s) f(s)^2 ds$$ can be written as $$\int_0^{2 \pi} | \frac{d}{ds} (e^{i \varphi(s)} f(s))|^2 ds \ ,$$ which we have to minimize over non negative functions $f$ satisfying $\int_0^{2 \pi} f(s)^2 ds =1$. Expanding the function $e^{i\varphi(s)} f(s)$ into a Fourier series $$e^{i\varphi(s)} f(s) = \sum_{n=-\infty}^\infty c_n \frac{ e^{ins}}{\sqrt{2 \pi}} \ ,$$ we find that since $f(s) \geq 0$ $$|c_0|^2 \leq \frac{1}{2 \pi} (\int_0^{2 \pi} f(s) ds)^2 \ .$$ Moreover, since the functions $1/\sqrt{2 \pi}$ and $e^{i \varphi(s)}/\sqrt{2 \pi}$ are orthogonal in the innerproduct of $L^2([0,2 \pi])$ we find that $$|c_0|^2 + \frac{1}{2 \pi} (\int_0^{2 \pi}f(s) ds)^2 \leq \int_0^{2\pi} f(s)^2 ds =1 \ .$$ Thus, $$|c_0|^2 \leq \min\{\frac{1}{2 \pi} (\int_0^{2 \pi} f(s) ds)^2, 1-\frac{1}{2 \pi} (\int_0^{2 \pi} f(s) ds)^2\} \leq 1/2 \ .$$ Since $\sum_n |c_n|^2 =1$ we learn that $$\sum_{n \not= 0} |c_n|^2 \geq \frac{1}{2} \ .$$ Clearly, $$(f, H(C) f) = \sum_{n=-\infty}^{\infty} n^2 |c_n|^2 \geq \sum_{n \not= 0} |c_n|^2 \geq \frac{1}{2} \ ,$$ hence $\lambda_1 (C) \ge 1/2$. [*Remarks:*]{} i\) A word of warning should be made at this point. In principle, the function $R$ defined from the eigenfunctions $u_1$ and $u_2$, via $\rho$ through equation (\[eq:3.8\]) above, must vanish at $s=0$ and $s=2 \pi$. For the [*curve problem*]{}, however, we drop this boundary condition. Thus, a priori the conjecture for the [*curve problem*]{} is stronger than the Lieb–Thirring conjecture for the two bound states, although we believe it amounts to the same. ii\) The best bound to date on $L_{1,1}$ is the bound of Eden and Foias [@EdFo91] who proved, $$L_{1,1} \le \frac{2}{9} \sqrt{3} \approx 0.3849 \dots \label{eq:ed-fo-bound}$$ Our bound $\lambda_1(C) \ge 1/2$ yields the bound $$L_{1,1} \le \frac{4}{9 \pi} \sqrt{6} \approx 0.3465 \dots,$$ which although better than (\[eq:ed-fo-bound\]), only applies to Schrödinger operators with two bound states. Just for comparison, the conjectured sharp value for $L_{1,1}$ is $4\sqrt{3}/(9 \pi) \approx 0,2450$. iii\) In recent years several authors have obtained isoperimetric inequalities for the lowest eigenvalues of a variant of $H(C)$, and we give a short summary of the main results in the sequel. Consider the Schrödinger operator $$H_g(C) \equiv -\frac{d^2}{ds^2} + g \kappa^2 \label{eq:3.15}$$ defined on $L^2(C)$ with periodic boundary conditions. As before, $C$ denotes a closed curve in ${\mathbb{R}}^2$ with positive curvature $\kappa$, and length $2 \pi$. Here, $s$ denotes arclength. If $g<0$, the lowest eigenvalue of $H_g(C)$, say $\lambda_1(g,C)$ is uniquely maximized when $C$ is a circle [@DuEx95]. When $g=-1$, the second eigenvalue, $\lambda_2(-1,C)$ is uniquely maximized when $C$ is a circle [@HaLo98]. If $0,g \le 1/4$, $\lambda_1(g,C)$ is uniquely minimized when $C$ is a circle [@ExHaLo99]. It is an open problem to determine the curve $C$ that minimizes $\lambda_1(g,C)$ in the cases, $1/4 < g \le 1$, and $g<0, g \neq -1$. If $g > 1$ the circle is not a minimizer for $\lambda_1(g,C)$ (see, e.g., [@ExHaLo99; @Ha02] for more details on the subject). To conclude this section we give an alternative interpretation of the minimization principle (\[eq:3.11\]) subject to the side constraints (\[eq:3.12\]). Interpret now $s$ as time (instead of arclength) and, given $R(s)$ and $\varphi(s)$ as before, define $$x(s) = R(s) \cos \varphi (s),$$ and $$y(s) = R(s) \sin \varphi (s).$$ Then, the minimization problem (\[eq:3.11\]), (\[eq:3.12\]) is equivalent to the following, $$\frac{\int_0^{2 \pi} \left( {\dot x}^2 + {\dot y}^2 \right) \, ds} {\int_0^{2 \pi} \left( x^2 + y^2 \right) \, ds} \ge 1, \label{eq:3.16}$$ where $x(s)$ and $y(s)$ are periodic, of period $2 \pi$ and satisfy the side constraints, $$\int_0^{2 \pi} \frac{x(s)}{\sqrt{x(s)^2+y(s)^2}} \, ds= \int_0^{2 \pi} \frac{y(s)}{\sqrt{x(s)^2+y(s)^2}} \, ds= 0. \label{eq:3.17}$$ Notice that (\[eq:3.16\]) certainly holds if one replaces the side constraints (\[eq:3.17\]) by $\int_0^{2 \pi} x(s) \, ds =0$ and $\int_0^{2 \pi} y(s) \, ds =0$, for then both functions $x(s)$ and $y(s)$ would be orthogonal to the constants and one would have $\int_0^{2 \pi} {\dot x}^2 \, ds \ge \int_0^{2 \pi} x(s)^2 \, ds$ and $\int_0^{2 \pi} {\dot y}^2 \, ds \ge \int_0^{2 \pi} y(s)^2 \, ds$, independently. Appendix ======== To obtain inequality (\[eq:2.2\]) for $\gamma \neq 1$ we start from equation (\[eq:2.5\]) as before. Using Hölder’s inequality we get $$\lambda_1 \le \left( \int_{-\infty}^{\infty} V^{\gamma+(1/2)} \, dx \right)^{2/2\gamma+1} \left(\int_{-\infty}^{\infty}u_1^{2 (2\gamma+1)/(2 \gamma-1)} \, dx \right)^{(2\gamma-1)/(2\gamma+1)} - \int_{-\infty}^{\infty} (u_1')^2 \, dx. \label{eq:a0}$$ We claim that if $\int_{-\infty}^{\infty} u_1^2 \, dx=1$, $$\int_{-\infty}^{\infty} {u_1'}^2 \, dx \ge c(\gamma) \left( \int_{-\infty}^{\infty} u_1^{2(2\gamma+1)/(2\gamma-1)} \, dx \right)^{2 \gamma-1}, \label{eq:a1}$$ where $$c(\gamma)= \left[\sqrt{\frac{\pi}{2}} \frac{\gamma^{\gamma} \Gamma(\gamma+1/2)} {\Gamma(\gamma+1)(\gamma-1/2)^{\gamma-1/2}} \right]^2.$$ Using the claim and denoting $$A \equiv \left( \int_{-\infty}^{\infty} V^{\gamma + (1/2)}\, dx \right)^{2/2\gamma+1},$$ and $$Y \equiv \left( \int_{-\infty}^{\infty} u_1^{2(2\gamma+1)/(2\gamma-1)} \, dx \right)^{2 \gamma-1},$$ we get $$\lambda_1 \le A Y^{1/(2\gamma+1)} - c(\gamma) Y. \label{eq:a2}$$ Maximizing the left side of (\[eq:a2\]) over $Y$ (for $\gamma> 1/2$), we get $$\lambda_1 \le \tilde c(\gamma) \left( \int_{-\infty}^{\infty} V^{\gamma + (1/2)}\, dx \right)^{1/\gamma}, \label{eq:a3}$$ where $$\tilde c (\gamma) = \frac{2 \gamma}{c(\gamma)^{1/(2 \gamma)} (2\gamma+1)^{(2\gamma+1)/(2 \gamma)}}.$$ Hence, $$\lambda_1^{\gamma} \le L_{\gamma,1}^1 \int_{-\infty}^{\infty} V^{\gamma + (1/2)}\, dx. \label{eq:a4}$$ To conclude we need only to prove the claim (\[eq:a1\]) whenever $\int_{-\infty}^{\infty} u_1^2 \, dx=1$. Introducing the same change of variables as in Section 2, i.e., $$x \to s = \int_{-\infty}^{x} u_1^2 \, dy,$$ and $$w\equiv u_1^2,$$ the claim reduces to proving $$\frac{1}{4} \int_0^1 {\dot w}^2 \, ds \ge c(\gamma) \left( \int_0^1 w^{2/2\gamma-1} \, ds \right)^{2 \gamma -1}, \label{eq:a5}$$ which follows from Sobolev’s inequality in one dimension. Acknowledgements ================ It is a pleasure to thank the organizers of the Pan-American Advanced Studies Institute (PASI) on Partial Differential Equations, Inverse Problems and Non-Linear Analysis for their kind invitation to present these results. We thank Mark S. Ashbaugh, Evans Harrell and Elliott Lieb for useful discussions. [A]{} M. Aizenman, and E. H. Lieb, *On Semi–Classical Bounds for Eigenvalues of Schrödinger Operators.* Physics Letters A **66** (1978), 427–429. M.S. Ashbaugh, and E.M. Harrell, *Maximal and minimal eigenvalues and their associated nonlinear equations.* J. Math. Phys. **28** (1987), 1770–1786. R. 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Lieb, *Lieb–Thirring Inequalities* in Encyclopaedia of Mathematics, Suppl. II, Kluwer, Dordrecht 2000, pp. 311–312. E.J.M. Veling, *Lower bounds for the infimum of the spectrum of the Schrödinger operator in $\mathbb R\sp N$ and the Sobolev inequalities.* J. Inequal. Pure Appl. Math. **3** (2002), Article 63, 22 pp. T. Weidl, *On the Lieb–Thirring constants $L_{\gamma,1}$ for $\gamma \ge 1/2$.* Commun. Math. Phys., **178** (1996), 135–146. [^1]: This work was supported in part by Fondecyt (Chile), Projects \# 102–0844 and \# 702–0844 [^2]: Work partially supported by NSF Grant DMS 03–00349

--- abstract: 'For $n\ge5$, it is well known that the moduli space $\mathfrak{M_{0,\:n}}$ of unordered $n$ points on the Riemann sphere is a quotient space of the Zariski open set $K_n$ of $\mathbb C^{n-3}$ by an $S_n$ action. The stabilizers of this $S_n$ action at certain points of this Zariski open set $K_n$ correspond to the groups fixing the sets of $n$ points on the Riemann sphere. Let $\alpha$ be a subset of $n$ distinct points on the Riemann sphere. We call the group of all linear fractional transformations leaving $\alpha$ invariant the stabilizer of $\alpha$, which is finite by observation. For each non-trivial finite subgroup $G$ of the group ${\rm PSL}(2,{\Bbb C})$ of linear fractional transformations, we give the necessary and sufficient condition for finite subsets of the Riemann sphere under which the stabilizers of them are conjugate to $G$. We also prove that there does exist some finite subset of the Riemann sphere whose stabilizer coincides with $G$. Next we obtain the irreducible decompositions of the representations of the stabilizers on the tangent spaces at the singularities of $\mathfrak{M_{0,\:n}}$. At last, on $\mathfrak{M_{0,\:5}}$ and $\mathfrak{M_{0,\:6}}$, we work out explicitly the singularities and the representations of their stabilizers on the tangent spaces at them.' author: - 'Yue Wu$^\dagger$ and Bin Xu$^\ddagger$' title: 'Moduli Spaces of Unordered $n\ge5$ Points on the Riemann Sphere and Their Singularities' --- [^1] Introduction ============ This manuscript is originated from the bachelor’s thesis of the first author. We study in great detail some algebraic properties of orbifold singularities of the moduli space $\mathfrak{M_{0,\:n}}$ of $n\geq 5$ unordered points on the Riemann sphere, such as the stabilizers of them and the corresponding linear representations over the tangent spaces at them. A lot of interesting results are observed and proved by hand computation, at least some of which, we do admit, may be well known to experts. This elementary and systematic exposition on $\mathfrak{M_{0,\:n}}$ might be still of some value to be open access to the community through arXiv. We should mention some references for the widely known classical cases of $n=4,\,5$ and $6$ about the stabilzers. In particular, see [@Wiman] and [@Dol Chapter 8 ] for the case of $n=4$ and $5$, and [@Bolza] for the case of $n=6$. The Moduli Space $\mathfrak{M_{0,\:n}}$ --------------------------------------- Let $C$ and $C'$ be two compact Riemann surfaces of genus $g$, and $$\{p_1,\:p_2,\:\cdots,\:p_n\}\subseteq C,\: \{p'_1,\:p'_2,\:\cdots,\:p'_n\}\subseteq C'.$$ $(C,\:\{p_1,\:p_2,\:\cdots,\:p_n\})$ and $(C',\:\{p'_1,\:p'_2,\:\cdots,\:p'_n\})$ are isomorphic if there exists some biholomorphic map $f:\:C\to C'$ such that $$f(\{p_1,\:p_2,\:\cdots,\:p_n\})=\{p'_1,\:p'_2,\:\cdots,\:p'_n\}.$$ The moduli space $\mathfrak{M_{g,\:n}}$ is the set of isomorphism classes of compact Riemann surfaces of genus $g$ with $n$ unordered marked points. It is well-known that $\mathfrak{M_{g,\:n}}$ is both a complex orbifold and an irreducible quasiprojective variety of dimension $3g-3+n$, where $n\ge3$ if $g=0$, $n\ge1$ if $g=1$ and $n\ge0$ if $g\ge2$. In Section \[orbif\] using elementary methods we shall prove that $\mathfrak{M_{0,\:n}}=K_n/G$ for $n\ge4$, where $K_n$ is the Zariski open set $$K_n=\{\boldsymbol{\lambda}=(\lambda_1,\:\lambda_2,\:\cdots,\:\lambda_{n-3})\in\mathbb C^{n-3}|\lambda_i\ne0,\:1,\:\lambda_i\ne\lambda_j,\:\forall i,\:j=1,\:2,\:\cdots,\:n-3,\:i\ne j\}$$ and $G$ is a finite group of birational transformations of $\mathbb C^{n-3}$, whose restrictions to $K_n$ are isomorphisms on $K_n$. $\mathfrak{M_{0,\:n}}=K_n/G$ is a complex orbifold of dimension $n-3$ for $n\ge4$. $\mathfrak{M_{0,\:n}}$ is a single point if $n\le3$. Each point in $\mathfrak{M_{0,\:4}}$ can be viewed as an elliptic curve, and vice versa. Hence $\mathfrak{M_{0,\:4}}$ is isomorphic to the complex plane. In Section \[grou\] we shall find that $G$ is isomorphic to $S_n$ when $n\ge5$. The moduli space $\mathfrak{M_{0,\:n}}$ is the quotient space of a Zariski open set $K_n$ of $\mathbb C^{n-3}$ by an $S_n$ action when $n\ge5$. For $\boldsymbol {\lambda}\in K_n$, set $$[\boldsymbol {\lambda}]=\{0,\:1,\:\infty,\:\lambda_1,\:\lambda_2,\:\cdots,\:\lambda_{n-3}\}$$ and $G_{\boldsymbol \lambda}$ the stabilizer of $\boldsymbol {\lambda}$ $$G_{\boldsymbol \lambda}=\{g_{\sigma}\in G|g_\sigma(\boldsymbol \lambda)=\boldsymbol \lambda\}.$$ For $n\ge5$, we call $\overline{[\boldsymbol {\lambda}]}\in \mathfrak{M_{0,\:n}}$ an oribfold singularity of $\mathfrak{M_{0,\:n}}$ if $G_{\boldsymbol \lambda}$ is non-trivial. Given a finite subset $\alpha$ of the Riemann sphere, let $\mathcal{A}_{\alpha}$ denote the group of linear fractional transformations that fix $\alpha$. In Section \[sing\] we shall prove that For $n\ge5$, $G_{\boldsymbol \lambda}$ is isomorphic to $\mathcal{A}_{[\boldsymbol {\lambda}]}$. Thus the study of the orbifold singularities of $\mathfrak{M_{0,\:n}}$ for $n\geq 5$ can be reduced to that of $\mathcal{A_\alpha}$ for subsets $\alpha$ with $n$ elements on the Riemann sphere. Next we shall find a way to determine $\mathcal{A_\alpha}$ for any finite subset $\alpha$ of the Riemann sphere when $|\alpha|\ge4$. The Subset Fixed by a Specific Group ------------------------------------ Given a finite subset $\alpha$ of the Riemann sphere, $|\alpha|=n$, $n\ge4$. It is easy to see that $\mathcal{A}_{\alpha}$ has at most $n(n-1)(n-2)$ elements. Thus $\mathcal{A}_{\alpha}$ is finite. There are exactly five kinds of non-trivial finite linear fractional transformation groups: the icosahedral group $I$, the octahedral group $O$, the tetrahedral group $T$, the dihedral group $D_n$ and the finite cyclic group $\mathbb Z_n$, $n\ge2$. Let $G$ be a finite group of linear fractional transformations. In Section \[orbit\], by discussing the orbits of the action of $G$ on $S^2$ and $\widehat{\mathbb{C}}$, we shall find all finite subsets $\alpha$ of $\widehat{\mathbb{C}}$ such that $\mathcal{A}_{\alpha}\simeq G$. Thus for any given finite $\alpha\subseteq \widehat{\mathbb{C}}$, $|\alpha|\ge4$, we can find $\mathcal{A}_{\alpha}$. The following five theorems constitute the whole result. Notice that we shall not distinguish a point on $S^2$ from its image on the extended complex plane under the stereographic projection. Suppose that $G$ is the icosahedral group $I$ that fixes a regular dodecahedron whose center is the origin. Let $V_I,\:F_I,\:E_I$ denote the vertices, the projections of the central points of the faces on $S^2$ and the projections of the middle points of the edges on $S^2$ respectively, with the origin being the central of the projection. For any $X\in S^2 \backslash (V_I\cup F_I\cup E_I)$, define $B_I(X)$ as the orbit of $X$. For any finite subset $\alpha$ of $S^2$, $\mathcal{A}_{\alpha}=G(\:\simeq A_5)$ if and only if $\alpha$ is a union of certain elements in $\{V_I,\:F_I,\:E_I\}\cup\{B_I(X)|X\in S^2 \backslash (V_I\cup F_I\cup E_I)\}$. Now suppose $G$ is the octahedral group $O$ that fixes a cube whose center is the origin. Let $V_O,\:F_O,\:E_O$ denote the vertices, the projections of the central points of the faces on $S^2$ and the projections of the middle points of the edges on $S^2$ respectively, with the origin being the center of the projection. For any $X\in S^2 \backslash (V_O\cup F_O\cup E_O)$, define $B_O(X)$ as the orbit of $X$. For any finite subset $\alpha$ of $S^2$, $\mathcal{A}_{\alpha}=G(\:\simeq S_4)$ if and only if $\alpha$ is a union of certain elements in $\{V_O,\:F_O,\:E_O\}\cup\{B_O(X)|X\in S^2 \backslash (V_O\cup F_O\cup E_O)\}$. Now suppose $G$ is the tetrahedral group $T$ that fixes a regular tetrahedron whose center is the origin. Let $V_T,\:F_T,\:E_T$ denote the vertices, the projections of the central points of the faces on $S^2$ and the projections of the middle points of the edges on $S^2$ respectively, with the origin being the center of the projection. For any $X\in S^2 \backslash (V_T\cup F_T\cup E_T)$, define $B_T(X)$ as the orbit of $X$. Thus we have (see Figure \[iA\_5, A\_4\]) $$V_I=V_T\cup F_T\cup B_T(B);\: F_I=B_T(N);\: E_I=E_T\cup B_T(Q)\cup B_T(R);$$$$B_I(X)=B_T(X)\cup B_T(g(X))\cup B_T(g^2(X))\cup B_T(g^3(X))\cup B_T(g^4(X)),$$ for $X\in S^2 \backslash (V_I\cup F_I\cup E_I)$. ![The dodecahedron fixed by $\mathcal{A}_{\alpha}$ and the tetrahedron fixed by $G$.[]{data-label="iA_5, A_4"}](a_5a_4.png){width="3in"} And (see Figure \[iS\_4, A\_4\]) $$V_O=V_T\cup F_T; F_O=E_T;\: E_O=B_T(P);\:$$$$B_O(X)=B_T(X)\cup B_T(g(X))$$ for $X\in S^2 \backslash (V_O\cup F_O\cup E_O)$. ![The dodecahedron fixed by $\mathcal{A}_{\alpha}$ and the tetrahedron fixed by $G$.[]{data-label="iS_4, A_4"}](s_4a_4.png){width="2.5in"} For any finite subset $\alpha$ of $S^2$, $\mathcal{A}_{\alpha}=G(\:\simeq A_4)$ if and only if all of the three claims are true: 1. $\alpha$ is a union of certain elements in $$\{V_T,\:F_T,\:E_T\}\cup\{B_T(X)|X\in S^2 \backslash (V_T\cup F_T\cup E_T)\};$$ 2. $\alpha$ is NOT a union of certain elements in $$\{V_I,\:F_I,\:E_I\}\cup\{B_I(X)|X\in S^2 \backslash (V_I\cup F_I\cup E_I)\};$$ 3. $\alpha$ is NOT a union of certain elements in $$\{V_O,\:F_O,\:E_O\}\cup\{B_O(X)|X\in S^2 \backslash (V_O\cup F_O\cup E_O)\}.$$ [Theorem 11 does not claim the existence of $\alpha$ though it does give a necessary and sufficient condition for $\alpha$ under which $\mathcal{A}_\alpha$ coincides with $A_4$. Theorems 12 and 13 have the same flavor as Theorem 11. We overcome this shortcoming by giving a unified existence result for $\alpha$ in Theorem 24. ]{} Now we go back to $\widehat{\mathbb C}$. Assume that $$f(z)=e^{\frac{1}{n}2\pi i}z,\:g(z)=\frac{1}{z},\:n\ge2$$ and $$G=\langle f, g\rangle\simeq D_n.$$ Set $$V=\{0, \infty\},$$ $$A_n=\{e^{\frac{k}{n}2\pi i}|k\in\mathbb{Z}\},$$ $$B_n=\{e^{\frac{2k+1}{2n}2\pi i}|k\in\mathbb{Z}\}$$ and $$C_n(z)=\{ze^{\frac{k}{n}2\pi i}|k\in\mathbb{Z}\}\cup\{z^{-1}e^{\frac{k}{n}2\pi i}|k\in\mathbb{Z}\}$$ for $z\in{\mathbb C}^*\backslash\{e^{\frac{l}{2n}2\pi i}|l\in\mathbb{Z}\}$. For any finite subset $\alpha$ of $S^2$, $|\alpha|\ge4$, $\mathcal{A}_{\alpha}=G(\simeq D_n)$ if and only if all of the four claims are true: 1. $\alpha$ is a union of certain elements in $$\{V,\:A_n,\:B_n\}\cup\{C_n(z)|z\in{\mathbb C}^*\backslash\{e^{\frac{l}{2n}2\pi i}|l\in\mathbb{Z}\}\};$$ 2. $\alpha$ is NOT a union of certain elements in $$\{V,\:A_{pn},\:B_{pn}\}\cup\{C_{pn}(z)|z\in{\mathbb C}^*\backslash\{e^{\frac{l}{2n}2\pi i}|l\in\mathbb{Z}\}\},\:p\ge2;$$ 3. $\alpha$ is NOT in the icosahedral, the octahedral or the tetrahedral case; 4. when $n=2$, $\alpha$ is NOT a union of certain elements in $$\{A_2,\:\phi^{-1}(A_{2p}),\:\phi^{-1}(B_{2p})\}\cup\{\phi^{-1}(C_{2p}(z))|z\in\mathbb C^*\backslash\{e^{\frac{l}{4p}2\pi i}|l\in\mathbb{Z}\}\},\:p\ge2,$$ where $$\phi(z)=\frac{1+z}{1-z}.$$ Assume that $$G=\langle z \mapsto e^{\frac{2\pi i}{n}}z\rangle\simeq\mathbb{Z}_n,\:n\ge 2.$$ Set $$C_n(z)=\{e^{\frac{k}{n}2\pi i}z|k\in\mathbb{Z}\}$$ for $z\in\mathbb C^*$. For any finite subset $\alpha$ of $S^2$, $|\alpha|\ge4$, $\mathcal{A}_{\alpha}=G(\simeq \mathbb Z_n), n\ge2$ if and only if all of the three claims are true: 1. $\alpha$ is a union of certain elements in $$\{\{\infty\},\:\{0\}\}\cup\{C_n(z)|z\in{\mathbb C}^*\};$$ 2. $\alpha$ is NOT a union of certain elements in $$\{\{\infty\},\:\{0\}\}\cup\{C_{pn}(z)|z\in{\mathbb C}^*\},\:p\ge2;$$ 3. $\alpha$ is NOT in the icosahedral, the octahedral the tetrahedral or the dihedral case. Representations of the Stabilizers of Singularities --------------------------------------------------- For any $\boldsymbol {\lambda}\in K_n$, ${n\ge5}$ such that $\overline{[\boldsymbol {\lambda}]}$ is a singularity of $\mathfrak{M_{0,\:n}}$, each $g_\sigma \in G_{\boldsymbol {\lambda}}$ introduces a tangential mapping $g_\sigma^*$ of $\text{T}_{\boldsymbol {\lambda}}K_n$ such that $$g^*_\sigma(\frac{\partial}{\partial\lambda^i}\bigg|_{\boldsymbol{\lambda}})(\lambda^j)=\frac{\partial}{\partial\lambda^i}\bigg|_{\boldsymbol{\lambda}}(\lambda^j\circ g_\sigma)=\frac{\partial}{\partial\lambda^i}\bigg|_{\boldsymbol{\lambda}}(g^{(j)}_\sigma)=\frac{\partial g^{(j)}_\sigma}{\partial\lambda^i}\bigg|_{\boldsymbol{\lambda}}$$ for $\:i,\:j=1,\:2,\:\cdots,\:n-3.$ Let $\boldsymbol{J_\sigma}$ denote the Jacobian matrix of $g_\sigma$. Define a representation of $G_{\boldsymbol{\lambda}}$ $$X_{\boldsymbol{\lambda}}:\:G_{\boldsymbol{\lambda}}\to\text{GL}_{n-3},\:g_\sigma\mapsto \boldsymbol{J_\sigma}\bigg|_{\boldsymbol{\lambda}}.$$ It is obvious that $X_{\boldsymbol{\lambda}}$ is a representation of $G_{\boldsymbol {\lambda}}$ of degree $n-3$. Let $K^{(1)}_{\boldsymbol{\lambda}},\: K^{(2)}_{\boldsymbol{\lambda}},\:\cdots,\:K^{(k)}_{\boldsymbol{\lambda}}$ denote the conjugate classes of $G_{\boldsymbol{\lambda}}$, and $X^{(1)}_{\boldsymbol{\lambda}},\: X^{(2)}_{\boldsymbol{\lambda}},\:\cdots,\:X^{(k)}_{\boldsymbol{\lambda}}$ its irreducible representations (for the exact definition, please refer to Section \[rep\]). For any $\boldsymbol {\lambda}\in K_n$, ${n\ge5}$ such that $\overline{[\boldsymbol {\lambda}]}$ is a singularity of $\mathfrak{M_{0,\:n}}$, assume that $X_{\boldsymbol{\lambda}}=p_1X^{(1)}_{\boldsymbol{\lambda}}\oplus\cdots\oplus p_kX^{(k)}_{\boldsymbol{\lambda}}$. Then call $(p_1,\:p_2,\:\cdots,\:p_k)$ the **multiplicity vector** of $\boldsymbol{\lambda}$. From the above discussion we know that if $G_{\boldsymbol{\lambda}}$ is isomorphic to the icosahedral group $I$, $[\boldsymbol{\lambda}]$ is equivalent to one of the following sets (which are categorized into eight types): 1. **F+mB**: $F_I\cup B_I(a_1)\cup B_I(a_2)\cup\cdots\cup B_I(a_m)$, where $B_I(a_1),\:B_I(a_2),\:\cdots,\:B_I(a_m)$ are different orbits of order 60, $m\in\mathbb {N}$; 2. **V+mB**: $V_I\cup B_I(a_1)\cup B_I(a_2)\cup\cdots\cup B_I(a_m)$, where $B_I(a_1),\:B_I(a_2),\:\cdots,\:B_I(a_m)$ are different orbits of order 60, $m\in\mathbb {N}$; 3. **E+mB**: $E_I\cup B_I(a_1)\cup B_I(a_2)\cup\cdots\cup B_I(a_m)$, where $B_I(a_1),\:B_I(a_2),\:\cdots,\:B_I(a_m)$ are different orbits of order 60, $m\in\mathbb {N}$; 4. **FV+mB**: $F_I\cup V_I\cup B_I(a_1)\cup B_I(a_2)\cup\cdots\cup B_I(a_m)$, where $B_I(a_1),\:B_I(a_2),\:\cdots,\:B_I(a_m)$ are different orbits of order 60, $m\in\mathbb {N}$; 5. **VE+mB**: $V_I\cup E_I\cup B_I(a_1)\cup B_I(a_2)\cup\cdots\cup B_I(a)_m)$, where $B_I(a_1),\:B_I(a_2),\:\cdots,\:B_I(a_m)$ are different orbits of order 60, $m\in\mathbb {N}$; 6. **EF+mB**: $E_I\cup F_I\cup B_I(a_1)\cup B_I(a_2)\cup\cdots\cup B_I(a_m)$, where $B_I(a_1),\:B_I(a_2),\:\cdots,\:B_I(a_m)$ are different orbits of order 60, $m\in\mathbb {N}$; 7. **FVE+mB**: $F_I\cup V_I\cup E_I\cup B_I(a_1)\cup B_I(a_2)\cup\cdots\cup B_I(a_m)$, where $B_I(a_1),\:B_I(a_2),\:\cdots,\:B_I(a_m)$ are different orbits of order 60, $m\in\mathbb {N}$; 8. **(1+m)B**: $B_I(a_0)\cup B_I(a_1)\cup\cdots\cup B_I(a_m)$, where $B_I(a_0),\:B_I(a_1),\:\cdots,\:B_I(a_m)$ are different orbits of order 60, $m\in\mathbb {N}$. For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the icosahedral group $I$, we have found its multiplicity vector 1. **F+mB**: $(m,\:1+4m,\:1+5m,\:3m,\:3m)$; 2. **V+mB**: $(m,\:1+4m,\:2+5m,\:3m,\:1+3m)$; 3. **E+mB**: $(m,\:2+4m,\:2+5m,\:1+3m,\:2+3m)$; 4. **FV+mB**: $(m,\:2+4m,\:3+5m,\:1+3m,\:1+3m)$; 5. **VE+mB**: $(m,\:3+4m,\:4+5m,\:2+3m,\:3+3m)$; 6. **EF+mB**: $(m,\:3+4m,\:3+5m,\:2+3m,\:2+3m)$; 7. **FVE+mB**: $(m,\:4+4m,\:5+5m,\:3+3m,\:3+3m)$; 8. **(1+m)B**: $(1+m,\:4+4m,\:5+5m,\:2+3m,\:3+3m)$. If $G_{\boldsymbol{\lambda}}$ is isomorphic to the octahedral group $O$, $[\boldsymbol{\lambda}]$ is equivalent to one of the following sets (which are categorized into eight types): 1. **F+mB**: $F_O\cup B_O(a_1)\cup B_O(a_2)\cup\cdots\cup B_O(a_m)$, where $B_O(a_1),\:B_O(a_2),\:\cdots,\:B_O(a_m)$ are different orbits of order 24, $m\in\mathbb {N}$; 2. **V+mB**: $V_O\cup B_O(a_1)\cup B_O(a_2)\cup\cdots\cup B_O(a_m)$, where $B_O(a_1),\:B_O(a_2),\:\cdots,\:B_O(a_m)$ are different orbits of order 24, $m\in\mathbb {N}$; 3. **E+mB**: $E_O\cup B_O(a_1)\cup B_O(a_2)\cup\cdots\cup B_O(a_m)$, where $B_O(a_1),\:B_O(a_2),\:\cdots,\:B_O(a_m)$ are different orbits of order 24, $m\in\mathbb {N}$; 4. **FV+mB**: $F_O\cup V_O\cup B_O(a_1)\cup B_O(a_2)\cup\cdots\cup B_O(a_m)$, where $B_O(a_1),\:B_O(a_2),\:\cdots,\:B_O(a_m)$ are different orbits of order 24, $m\in\mathbb {N}$; 5. **VE+mB**: $V_O\cup E_O\cup B_O(a_1)\cup B_O(a_2)\cup\cdots\cup B_O(a_m)$, where $B_O(a_1),\:B_O(a_2),\:\cdots,\:B_O(a_m)$ are different orbits of order 24, $m\in\mathbb {N}$; 6. **EF+mB**: $E_O\cup F_O\cup B_O(a_1)\cup B_O(a_2)\cup\cdots\cup B_O(a_m)$, where $B_O(a_1),\:B_O(a_2),\:\cdots,\:B_O(a_m)$ are different orbits of order 24, $m\in\mathbb {N}$; 7. **FVE+mB**: $F_O\cup V_O\cup E_O\cup B_O(a_1)\cup B_O(a_2)\cup\cdots\cup B_O(a_m)$, where $B_O(a_1),\:B_O(a_2),\:\cdots,\:B_O(a_m)$ are different orbits of order 24, $m\in\mathbb {N}$; 8. **(1+m)B**: $B_O(a_0)\cup B_O(a_1)\cup B_O(a_2)\cup\cdots\cup B_O(a_m)$, where $B_O(a_1),\:B_O(a_2),\:\cdots,\:B_O(a_m)$ are different orbits of order 24, $m\in\mathbb {N}$. For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the octahedral group $O$, we have found its multiplicity vector 1. **F+mB**: $(m,\:m,\:1+3m,\:3m,\:2m)$; 2. **V+mB**: $(m,\:m,\:1+3m,\:3m,\:1+2m)$; 3. **E+mB**: $(m,\:1+m,\:1+3m,\:1+3m,\:1+2m)$; 4. **FV+mB**: $(m,\:m,\:2+3m,\:1+3m,\:1+2m)$; 5. **VE+mB**: $(m,\:1+m,\:2+3m,\:2+3m,\:2+2m)$; 6. **EF+mB**: $(m,\:1+m,\:2+3m,\:2+3m,\:1+2m)$; 7. **FVE+mB**: $(m,\:1+m,\:3+3m,\:3+3m,\:2+2m)$; 8. **(1+m)B**: $(1+m,\:1+m,\:3+3m,\:2+3m,\:2+2m)$. If $G_{\boldsymbol{\lambda}}$ is isomorphic to the Tetrahedral Group $T$, then $[\boldsymbol{\lambda}]$ is equivalent to one of the following sets (which are categorized into six types): 1. **F+mB**: $F_T\cup B_T(a_1)\cup B_T(a_2)\cup\cdots\cup B_T(a_m)$, where $B_T(a_1),\:B_T(a_2),\:\cdots,\:B_T(a_m)$ are different orbits of order 12, $m\in\mathbb {N^*}$; 2. **E+mB**: $E_T\cup B_T(a_1)\cup B_T(a_2)\cup\cdots\cup B_T(a_m)$, where $B_T(a_1),\:B_T(a_2),\:\cdots,\:B_T(a_m)$ are different orbits of order 12, $m\in\mathbb {N^*}$; 3. **FV+mB**: $F_T\cup V_T\cup B_T(a_1)\cup B_T(a_2)\cup\cdots\cup B_T(a_m)$, where $B_T(a_1),\:B_T(a_2),\:\cdots,\:B_T(a_m)$ are different orbits of order 12, $m\in\mathbb {N^*}$; 4. **FE+mB**: $F_T\cup E_T\cup B_T(a_1)\cup B_T(a_2)\cup\cdots\cup B_T(a_m)$, where $B_T(a_1),\:B_T(a_2),\:\cdots,\:B_T(a_m)$ are different orbits of order 12, $m\in\mathbb {N}$; 5. **FVE+mB**: $F_T\cup V_T\cup E_T\cup B_I(a_1)\cup B_I(a_2)\cup\cdots\cup B_I(a_m)$, where $B_I(a_1),\:B_I(a_2),\:\cdots,\:B_I(a_m)$ are different orbits of order 12, $m\in\mathbb {N^*}$; 6. **(1+m)B**: $B_T(a_1)\cup B_T(a_2)\cup\cdots\cup B_T(a_m)$, where $B_T(a_1),\:B_T(a_2),\:\cdots,\:B_T(a_m)$ are different orbits of order 12, $m\in\mathbb {N}$. For each singularity $[\boldsymbol{\lambda}]$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the tetrahedral group $T$, we have found its multiplicity vector 1. **F+mB**: $(m,\:1+m,\:m,\:3m)$; 2. **E+mB**: $(m,\:m,\:m,\:1+3m)$; 3. **FV+mB**: $(m,\:1+m,\:1+m,\:1+3m)$; 4. **FE+mB**: $(m,\:1+m,\:m,\:2+3m)$; 5. **FVE+mB**: $(m,\:1+m,\:1+m,\:3+3m)$; 6. **(1+m)B**: $(m,\:1+m,\:1+m,\:2+3m)$. For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the dihedral group $D_p$, $[\boldsymbol{\lambda}]$ is equivalent to one of the following sets (which are categorized into six types): 1. **mC**: $C_p(a_1)\cup C_p(a_2)\cup\cdots\cup C_p(a_m)$, where $C_p(a_1),\:C_p(a_2),\:\cdots,\:C_p(a_m)$ are different orbits of order $2p$, $m\in\mathbb {N^*}$; 2. **A+mC**: $A_p\cup C_p(a_1)\cup C_p(a_2)\cup\cdots\cup C_p(a_m)$, where $C_p(a_1),\:C_p(a_2),\:\cdots,\:C_p(a_m)$ are different orbits of order $2p$, $m\in\mathbb {N}$; 3. **AB+mC**: $A_p\cup B_p\cup C_p(a_1)\cup C_p(a_2)\cup\cdots\cup C_p(a_m)$, where $C_p(a_1),\:C_p(a_2),\:\cdots,\:C_p(a_m)$ are different orbits of order $2p$, $m\in\mathbb {N^*}$; 4. **2+mC**: $\{0,\:\infty\}\cup C_p(a_1)\cup C_p(a_2)\cup\cdots\cup C_p(a_m)$, where $C_p(a_1),\:C_p(a_2),\:\cdots,\:C_p(a_m)$ are different orbits of order $2p$, $m\in\mathbb {N^*}$; 5. **A+2+mC**: $A_p\cup\{0,\:\infty\}\cup C_p(a_1)\cup C_p(a_2)\cup\cdots\cup C_p(a_m)$, where $C_p(a_1),\:C_p(a_2),\:\cdots,\:C_p(a_m)$ are different orbits of order $2p$, $m\in\mathbb {N}$; 6. **AB+2+mC**: $A_p\cup B_p\cup\{0,\:\infty\}\cup C_p(a_1)\cup C_p(a_2)\cup\cdots\cup C_p(a_m)$, where $C_p(a_1),\:C_p(a_2),\:\cdots,\:C_p(a_m)$ are different orbits of order $2p$, $m\in\mathbb {N^*}$. For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the dihedral group $D_p$ ($p$ is odd), we have found its multiplicity vector 1. **mC**: - $(2m-1,\:m,\:m-1)$ for $p=3$; - $(2m-1,\:2m,\:\cdots,\:2m,\:m,\:m-1)$ for $p\ge5$; 2. **A+mC**: - $(2m,\:m,\:m)$ for $p=3$; - $(2m,\:2m+1,\:\cdots,\:2m+1,\:m,\:m)$ for $p\ge5$; 3. **AB+mC**: - $(2m+1,\:m,\:m+1)$ for $p=3$; - $(2m+1,\:2m+2,\:\cdots,\:2m+2,\:m,\:m+1)$ for $p\ge5$; 4. **2+mC**: $(2m,\:\cdots,\:2m,\:m,\:m-1)$; 5. **A+2+mC**: $(2m+1,\:\cdots,\:2m+1,\:m,\:m)$; 6. **AB+2+mC**: $(2m+2,\:\cdots,\:2m+2,\:m,\:m+1)$. For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the dihedral group $D_p$ ($p$ is even), we have found its multiplicity vector 1. **mC**: - $(m,\:m-1,\:m-1,\:m-1)$ for $p=2$; - $(2m-1,\:m,\:m-1,\:m,\:m)$ for $p=4$; - $(2m-1,\:2m,\:\cdots,\:2m,\:m,\:m-1,\:m,\:m)$ for $p\ge6$; 2. **A+mC**: - $(m,\:m,\:m-1,\:m)$ for $p=2$; - $(2m,\:m,\:m,\:m,\:m+1)$ for $p=4$; - $(2m,\:2m+1,\:\cdots,\:2m+1,\:m,\:m,\:m,\:m+1)$ for $p\ge6$; 3. **AB+mC**: - $(m,\:m+1,\:m,\:m)$ for $p=2$; - $(2m+1,\:m,\:m+1,\:m+1,\:m+1)$ for $p=4$; - $(2m+1,\:2m+2,\:\cdots,\:2m+2,\:m,\:m+1,\:m+1,\:m+1)$ for $p\ge6$; 4. **2+mC**: - $(m,\:m-1,\:m,\:m)$ for $p=2$; - $(2m,\:\cdots,\:2m,\:m,\:m-1,\:m,\:m)$ for $p\ge4$; 5. **A+2+mC**: - $(m,\:m,\:m,\:m+1)$ for $p=2$; - $(2m+1,\:\cdots,\:2m+1,\:m,\:m,\:m,\:m+1)$ for $p\ge4$; 6. **AB+2+mC**: - $(m,\:m+1,\:m+1,\:m+1)$ for $p=2$; - $(2m+2,\:\cdots,\:2m+2,\:m,\:m+1,\:m+1,\:m+1)$ for $p\ge4$. For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the cyclic group $\mathbb{Z}_p$, $[\boldsymbol{\lambda}]$ is equivalent to one of the following sets (which are categorized into three types): 1. **mC**: $C_p(a_1)\cup C_p(a_2)\cup\cdots\cup C_p(a_m)$, where $C_p(a_1),\:C_p(a_2),\:\cdots,\:C_p(a_m)$ are different orbits of order $p$, $m\in\mathbb {N^*}$; 2. **1+mC**: $\{0\}\cup C_p(a_1)\cup C_p(a_2)\cup\cdots\cup C_p(a_m)$, where $C_p(a_1),\:C_p(a_2),\:\cdots,\:C_p(a_m)$ are different orbits of order $p$, $m\in\mathbb {N^*}$; 3. **2+mC**: $\{0,\:\infty\}\cup C_p(a_1)\cup C_p(a_2)\cup\cdots\cup C_p(a_m)$, where $C_p(a_1),\:C_p(a_2),\:\cdots,\:C_p(a_m)$ are different orbits of order $p$, $m\in\mathbb {N^*}$. For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the cyclic group $\mathbb{Z}_p$, we have found its multiplicity vector 1. **mC**: - $(m-2,\:m-1)$ for $p=2$; - $(m-1,\:m-1,\:m-1)$ for $p=3$; - $(m-1,\:m,\:\cdots,\:m,\:m-1,\:m-1)$ for $p\ge4$; 2. **1+mC**: - $(m-1,\:m-1)$ for $p=2$; - $(m,\:\cdots,\:m,\:m-1,\:m-1)$ for $p\ge3$. 3. **2+mC**: $(m,\:\cdots,\:m,\:m-1)$. The Group that Fixes Five of Six Points --------------------------------------- In Section \[n=5\] and \[n=6\] we will investigate $\mathcal A_\alpha$ when $|\alpha|=5$ or $6$, and find their classifications and multiplicity vectors. From the above discussion we shall drive the following conclusion Set $\alpha=\{z_1,\: z_2,\: z_3,\: z_4,\: z_5\}\subseteq\widehat {\mathbb{C}}$. 1. If there exists some linear fractional transformation $\psi$ such that $$\psi(\alpha)=\{1,\: w,\: w^2,\: w^3\:, w^4\},\:w=e^{\frac{2\pi}{5}i},$$ then $$\mathcal{A_\alpha}={\psi}^{-1}\langle z \mapsto e^{\frac{2\pi}{5}i}z,\:z \mapsto \frac{1}{z}\rangle {\psi}\simeq D_5,$$ and its multiplicity vector is $(0,\:1,\:0,\:0)$; 2. if there exists some linear fractional transformation $\psi$ such that $$\psi(\alpha) =\{0,\: 1,\: i,\: -1,\: -i\},$$ then $$\mathcal{A_\alpha}={\psi}^{-1}\langle z \mapsto iz\rangle {\psi}\simeq \mathbb Z _4,$$ and its multiplicity vector is $(1,\:1,\:0,\:0)$; 3. if there exists some linear fractional transformation $\psi$ such that $$\psi(\alpha)=\{0,\: \infty,\: 1, \:w,\: w^2\},\:w=e^{\frac{2\pi}{3}i},$$ then $$\mathcal{A_\alpha}={\psi}^{-1}\langle z \mapsto e^{\frac{2\pi}{3}i}z,\:z \mapsto \frac{1}{z}\rangle\psi \simeq D_3,$$ and its multiplicity vector is $(1,\:0,\:0)$; 4. if there exists some linear fractional transformation $\psi$ such that $$\psi(\alpha)=\{0,\: 1, \:-1,\: a,\: -a\},\:a\ne0,\:\pm 1,$$ then 1. if $a=\pm(\sqrt{5}+2)\text{ or }\pm(\sqrt{5}-2)$, then there exists some linear fractional transformation $\phi$ such that $$\phi(\alpha)=\{1,\: w,\: w^2,\: w^3\:, w^4\},\:w=e^{\frac{2\pi}{5}i}$$ and this is case 1; 2. if $a=\pm i$, then $$\alpha =\{0,\: 1,\: i,\: -1,\: -i\}$$ and this is case 2; 3. if $a=\pm\sqrt{3}i\text{ or }\pm\frac{1}{\sqrt{3}}i$, then there exists some linear fractional transformation $\phi$ such that $$\phi(\alpha)=\{0,\: \infty,\: 1, \:w,\: w^2\},\:w=e^{\frac{2\pi}{3}i}$$ and this is case 3; 4. otherwise, $$\mathcal{A_\alpha}={\psi}^{-1}\langle z \mapsto -z\rangle {\psi}\simeq \mathbb Z _2,$$ and its multiplicity vector is $(1,\:1)$; 5. otherwise, $$\mathcal{A_\alpha}=\{\mathrm{Id}\}.$$ Set $\alpha=\{z_1,\: z_2,\: z_3,\: z_4,\: z_5,\: z_6\}\subseteq \widehat{\mathbb{C}}$. 1. If there exists some linear fractional transformation $\psi$ such that $$\psi(\alpha)=\{1,\: w,\: w^2,\: w^3,\: w^4, \:w^5\},\:w=e^{\frac{\pi}{3}i},$$ then $$\mathcal{A_\alpha}={\psi}^{-1}\langle z \mapsto e^{\frac{\pi}{3}i}z,\:z \mapsto \frac{1}{z}\rangle {\psi}\simeq D_6,$$ and its multiplicity vector is $(0,\:1,\:0,\:0,\:0,\:1)$; 2. if there exists some linear fractional transformation $\psi$ such that $$\psi(\alpha) =\{0,\: 1,\: w, \:w^2, \:w^3,\: w^4\},\:w=e^{\frac{2\pi}{5}i},$$ then $$\mathcal{A_\alpha}={\psi}^{-1}\langle z \mapsto e^{\frac{2\pi}{5}i}z\rangle {\psi}\simeq \mathbb Z _5,$$ and its multiplicity vector is $(1,\:1,\:1,\:0,\:0)$; 3. if there exists some linear fractional transformation $\psi$ such that $$\psi(\alpha) =\{0,\: \infty, \:1, \:i, \:-1,\: -i\},$$ then $$\mathcal{A_\alpha}={\psi}^{-1}\langle z\mapsto iz,\: z\mapsto \frac{iz+1}{z+i}\rangle {\psi}\simeq S_4,$$ and its multiplicity vector is $(0,\:0,\:1,\:0,\:0)$; 4. if there exists some linear fractional transformation $\psi$ such that $$\psi(\alpha)=\{1,\:w,\:w^2,\:a,\:aw,\:aw^2\},\:|a|\ge1,\:a\ne1,\:w,\:w^2,\:w=e^{\frac{2\pi}{3}i},$$ 1. if $a=\sqrt w,\:w\sqrt w\text{ or }w^2\sqrt w$, $$\alpha=\{1,\:\sqrt w,\: w,\:w\sqrt w,\: w^2,\:w^2\sqrt w\}$$ and this is case 1 ; 2. if $a=-(2+\sqrt3),\:-(2+\sqrt3)w\text{ or }-(2+\sqrt3)w^2$, then there exists some linear fractional transformation $\phi$ such that $$\phi(\alpha) =\{0,\: \infty, \:1, \:i, \:-1,\: -i\}$$ and this is case 3; 3. otherwise $$\mathcal{A_\alpha}={\psi}^{-1}\langle z \mapsto e^{\frac{2\pi}{3}i}z,\:z \mapsto \frac{a}{z}\rangle\psi \simeq D_3,$$ and its multiplicity vector is $(1,\:1,\:0)$; \[a\] 5. if there exists some linear fractional transformation $\psi$ such that $$\psi(\alpha)=\{1,\: -1, \:a,\: -a,\: b, \:-b\},\:a\ne\pm b, \:a,\:b \in \mathbb{C} \backslash \{ 0, \pm 1 \},$$ 1. if $\{\pm1,\:\pm a,\: \pm b\}=$ $$\{\pm(7-4\sqrt3), \:\pm(2-\sqrt3),\:\pm 1\},\:\{\pm(2-\sqrt3),\:\pm 1,\:\pm(2+\sqrt3)\},\:\{\pm 1,\: \pm(2+\sqrt3),\:\pm(7+4\sqrt3)\}$$ then there exists some linear fractional transformation $\phi$ such that $$\phi(\alpha)=\{1,\: w,\: w^2,\: w^3,\: w^4, \:w^5\},\:w=e^{\frac{\pi}{3}i}$$ and this is case 1; 2. if $\{\pm1,\:\pm a,\: \pm b\}=$ $$\{\pm (\sqrt2-1)i,\:\pm(\sqrt2+1)i,\:\pm 1\},\:\{\pm(3-2\sqrt2), \:\pm 1,\pm(\sqrt2-1)i\},\:\{\pm 1, \:\pm(3+2\sqrt2),\:\pm(\sqrt2+1)i\},$$ then there exists some linear fractional transformation $\phi$ such that $$\phi(\alpha) =\{0,\: \infty, \:1, \:i, \:-1,\: -i\}$$ and this is case 3; 3. if $a^2b+ab^2+a^2-6ab+b^2+a+b=0,$ then there exists some linear fractional transformation $\phi$ such that $$\phi(\alpha)=\{1,\:w,\:w^2,\:c,\:cw,\:cw^2\},\:c\ne0,\:1,\:w,\:w^2,\:w=e^{\frac{2\pi}{3}i}$$ and this is case 4c; 4. otherwise, 1. if $ab=\pm1$, $$\mathcal{A_\alpha}={\psi}^{-1}\langle z \mapsto -z,\:z \mapsto \frac{1}{z}\rangle\psi \simeq K_4,$$ and its multiplicity vector is $(1,\:0,\:1,\:1)$ (viewed as type 2+mC); 2. if $ab\ne\pm1$, $$\mathcal{A_\alpha}={\psi}^{-1}\langle z \mapsto -z\rangle\psi \simeq \mathbb{Z}_2,$$ and its multiplicity vector is $(1,\:2)$; 6. otherwise, $$\mathcal{A_\alpha}=\{\mathrm{Id}\}.$$ Finally we will come to the conclusion in Section \[all\] that every non-trivial finite group of linear fractional transformations is a stabilizer of certain finite subset of $\widehat{\mathbb{C}}$. For any finite non-trivial group $G$ of linear fractional transformations, there exists a subset $\alpha=\{z_1, z_2, \cdots, z_n\}\subseteq \widehat{\mathbb{C}}$ such that $\mathcal{A}_{\alpha}\simeq G$. The Moduli Space $\mathfrak{M_{0,\:n}}$ {#orbifold} ======================================= It is well-known that The moduli space $\mathfrak{M_{g,\:n}}$ is both a complex orbifold and an irreducible quasiprojective variety of dimension $3g-3+n$, where $n\ge3$ if $g=0$, $n\ge1$ if $g=1$ and $n\ge0$ if $g\ge2$.[@De-Mu] In this section using elementary methods we shall prove that $\mathfrak{M_{0,\:n}}$ is a complex orbifold of dimension $n-3$ when $n\ge4$. Moreover, we shall show that it is the quotient space of a Zariski open set $K_n$ of $\mathbb C^{n-3}$ by an $S_n$ action when $n\ge5$. The stabilizers of this $S_n$ action at points of this Zariski open set correspond to the groups fixing the sets of $n$ points on the Riemann sphere. The Orbifold Structure of Moduli Space $\mathfrak{M_{0,\:n}}$ {#orbif} ------------------------------------------------------------- For $n\in\mathbb Z_{\ge4}$, set $$K_n=\{\boldsymbol{\lambda}=(\lambda^1,\:\lambda^2,\:\cdots,\:\lambda^{n-3})\in\mathbb C^{n-3}|\lambda^i\ne0,\:1,\:\lambda^i\ne\lambda^j,\:\forall i,\:j=1,\:2,\:\cdots,\:n-3,\:i\ne j\}.$$ For any $\boldsymbol{\lambda}\in K_n$, set $$z^{\boldsymbol{\lambda}}_1=0,\:z^{\boldsymbol{\lambda}}_2=1,\:z^{\boldsymbol{\lambda}}_3=\infty,\:z^{\boldsymbol{\lambda}}_{i+3}=\lambda^{i},\:i=1,\:2,\:\cdots,\:n-3.$$ For any $\boldsymbol{\lambda}\in K_n$ and $\sigma\in S_n$, define $f^{\boldsymbol {\lambda}}_\sigma$ as the unique linear fractional transformation such that $$f^{\boldsymbol {\lambda}}_\sigma (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(1)})=0,\:f^{\boldsymbol {\lambda}}_\sigma (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(2)})=1,\:f^{\boldsymbol {\lambda}}_\sigma (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(3)})=\infty.$$ For $i=1,\:2,\:\cdots,\:n-3$ and $\sigma\in S_n$ define a mapping $g^{(i)}_\sigma$ $$g^{(i)}_\sigma:\:K_n\to\mathbb{C},\:\boldsymbol{\lambda}\mapsto f^{\boldsymbol {\lambda}}_\sigma (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(i+3)}),$$ and $g_\sigma$ $$g_\sigma :\:K_n\to K_n,\:\boldsymbol{\lambda}\mapsto(f^{\boldsymbol {\lambda}}_\sigma (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(4)}),\:f^{\boldsymbol {\lambda}}_\sigma (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(5)}),\:\cdots,\:f^{\boldsymbol {\lambda}}_\sigma (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(n)})).$$ By definition $g_\sigma$ is a rational map from $\mathbb C^{n-3}$ to $\mathbb C^{n-3}$ which restricts to an isomorphism of $K_n$. Here is a useful observation: For any $\boldsymbol{\lambda}\in K_n$ and $\sigma\in S_n$, we have $$f^{\boldsymbol {\lambda}}_\sigma (z^{\boldsymbol{\lambda}}_k)=z^{g_\sigma(\boldsymbol{\lambda})}_{\sigma(k)}$$ holds for $k=1,\:2,\:\cdots,\:n$. Define $G$ $$G=\{g_\sigma|\sigma\in S_n\}.$$ Notice that $|G|\le n!$ and $G$ is a finite group acting on $K_n$. To prove the theorem first notice that $$f^{\boldsymbol {\lambda}}_\sigma (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(k)})=z^{g_\sigma(\boldsymbol{\lambda})}_k,\:k=1,\:2,\:\cdots,\:n.$$ So we have $$f^{g_\sigma(\boldsymbol {\lambda})}_\pi\circ f^{\boldsymbol {\lambda}}_\sigma (z^{\boldsymbol{\lambda}}_{(\pi\cdot\sigma)^{-1}(k)})=f^{g_\sigma(\boldsymbol {\lambda})}_\pi(z^{g_\sigma(\boldsymbol{\lambda})}_{\pi^{-1}(k)})=z^{g_\pi\circ g_\sigma(\boldsymbol {\lambda})}_k,\:k=1,\:2,\:\cdots,\:n.$$ Thus we conclude that $$f^{g_\sigma(\boldsymbol {\lambda})}_\pi\circ f^{\boldsymbol {\lambda}}_\sigma =f^{\boldsymbol {\lambda}}_{\pi\cdot\sigma},\:z^{g_\pi\circ g_\sigma(\boldsymbol {\lambda})}_k=z_k^{g_{\pi\cdot\sigma}(\boldsymbol {\lambda})},\:k=1,\:2,\:\cdots,\:n.$$ So $g_\pi\circ g_\sigma=g_{\pi\cdot\sigma}\in G$. Thus $G$ is a group. As $G$ is a finite group acting on $K_n$, we have another conclusion $\mathfrak{M_{0,\:n}}=K_n/G.$ Given two elements $$\overline{\{0,\:1,\:\infty,\:\lambda^1,\:\lambda^2,\:\cdots,\:\lambda^{n-3}\}},\:\overline{\{0,\:1,\:\infty,\:\mu^1,\:\mu^2,\:\cdots,\:\mu^{n-3}\}}\in \mathfrak{M_{0,\:n}},$$ set $$\boldsymbol{\lambda}=(\lambda^1,\:\lambda^2,\:\cdots,\:\lambda^{n-3}),\:\boldsymbol{\mu}=(\mu^1,\:\mu^2,\:\cdots,\:\mu^{n-3}).$$ If $\boldsymbol{\mu}\in G(\boldsymbol{\lambda})$, then there exists some $\sigma\in S_n$ s.t. $\boldsymbol{\mu}=g_\sigma(\boldsymbol{\lambda}).$ So we have $$f^{\boldsymbol {\lambda}}_\sigma (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(k)})=z^{g_\sigma(\boldsymbol{\lambda})}_k=z^{\boldsymbol{\mu}}_k,\:k=1,\:2,\:\cdots,\:n,$$ which means $$f^{\boldsymbol {\lambda}}_\sigma(\{0,\:1,\:\infty,\:\lambda^1,\:\lambda^2,\:\cdots,\:\lambda^{n-3}\})=\{0,\:1,\:\infty,\:\mu^1,\:\mu^2,\:\cdots,\:\mu^{n-3}\}.$$ So $$\overline{\{0,\:1,\:\infty,\:\lambda^1,\:\lambda^2,\:\cdots,\:\lambda^{n-3}\}}=\overline{\{0,\:1,\:\infty,\:\mu^1,\:\mu^2,\:\cdots,\:\mu^{n-3}\}}.$$ On the other hand, if $$\overline{\{0,\:1,\:\infty,\:\lambda^1,\:\lambda^2,\:\cdots,\:\lambda^{n-3}\}}=\overline{\{0,\:1,\:\infty,\:\mu^1,\:\mu^2,\:\cdots,\:\mu^{n-3}\}},$$ then there exists some linear fractional transformation $h$ s.t. $$h(\{0,\:1,\:\infty,\:\lambda^1,\:\lambda^2,\:\cdots,\:\lambda^{n-3}\})=\{0,\:1,\:\infty,\:\mu^1,\:\mu^2,\:\cdots,\:\mu^{n-3}\}$$ and some $\sigma\in S_n$ s.t.  $$h(z^{\boldsymbol{\lambda}}_{\sigma^{-1}(k)})=z^{\boldsymbol{\mu}}_k,\:k=1,\:2,\:\cdots,\:n.$$ So $h=f^{\boldsymbol {\lambda}}_\sigma$ and $\boldsymbol{\mu}=g_\sigma(\boldsymbol{\lambda})$. Thus we conclude that $$\overline{\{0,\:1,\:\infty,\:\lambda^1,\:\lambda^2,\:\cdots,\:\lambda^{n-3}\}}=\overline{\{0,\:1,\:\infty,\:\mu^1,\:\mu^2,\:\cdots,\:\mu^{n-3}\}}$$ if and only if $$\boldsymbol{\mu}\in G(\boldsymbol{\lambda}),$$ and therefore $$\mathfrak{M_{0,\:n}}=K_n/G.$$ $\mathfrak{M_{0,\:n}}$ is a complex orbifold of dimension $n-3$. The Explicit Structure of the Group $G$ {#grou} --------------------------------------- Now let us explore $G$. Define subgroup $V$ of $S_n$ $$V=S_{\{1,\:2,\:3\}}\times S_{\{4,\:5,\:\cdots,\:n\}}$$ and subset $S$ of $S_n$ $$\begin{split} S=\{&e,\:(p,\:1),\:(p,\:2),\:(p,\:3),\:(p,\:1)(q,\:2),\:(p,\:2)(q,\:3),\:(p,\:3)(q,\:1),\:(p,\:1)(q,\:2)(r,\:3)|\\ &p,\:q,\:r=4,\:5,\:\cdots,\:n,\:p\ne q,\:q\ne r,\:r\ne p\}. \end{split}$$ Thus $S$ forms a complete list of left coset representatives of $V\subset S_n$. Define group $H$ of linear fractional transformations $$H=\{\mathrm{Id},\:\lambda\mapsto 1-\lambda,\: \lambda\mapsto \frac{1}{\lambda},\:\lambda\mapsto\frac{\lambda}{\lambda-1} ,\:\lambda\mapsto \frac{\lambda-1}{\lambda},\:\lambda\mapsto \frac{-1}{\lambda-1}\},$$ then we have $$\{g_\sigma|\sigma\in V\}=\{\boldsymbol{\lambda}\mapsto(h(\lambda^{\tau(1)}),\:h(\lambda^{\tau(2)}),\:\cdots,\:h(\lambda^{\tau(n-3)}))|h\in H,\:\tau\in S_{n-3}\}.$$ For $p=4,\:5,\:\cdots,\:n$, set $\sigma=(1,\:p)$ and we have $$f^{\boldsymbol {\lambda}}_{(1,\:p)}(\lambda^{p-3})= f^{\boldsymbol {\lambda}}_{(1,\:p)} (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(1)})=0,$$ $$f^{\boldsymbol {\lambda}}_{(1,\:p)}(1)= f^{\boldsymbol {\lambda}}_{(1,\:p)}(z^{\boldsymbol{\lambda}}_{\sigma^{-1}(2)})=1,$$ $$f^{\boldsymbol {\lambda}}_{(1,\:p)}(\infty)= f^{\boldsymbol {\lambda}}_{(1,\:p)}(z^{\boldsymbol{\lambda}}_{\sigma^{-1}(3)})=\infty.$$ So we have $$f^{\boldsymbol {\lambda}}_{(1,\:p)}(z)=\frac{z-\lambda^{p-3}}{1-\lambda^{p-3}},$$ and $$g_{(1,\:p)}^{(k)}(\boldsymbol {\lambda})= \begin{dcases} \frac{\lambda^k-\lambda^{p-3}}{1-\lambda^{p-3}}, &k\ne p-3; \\ \frac{-\lambda^{p-3}}{1-\lambda^{p-3}}, &k=p-3. \end{dcases}$$ Set $\sigma=(2,\:p)$ and we have $$f^{\boldsymbol {\lambda}}_{(2,\:p)}(0) =f^{\boldsymbol {\lambda}}_{(2,\:p)}(z^{\boldsymbol{\lambda}}_{\sigma^{-1}(1)})=0,$$ $$f^{\boldsymbol {\lambda}}_{(2,\:p)}(\lambda^{p-3}) =f^{\boldsymbol {\lambda}}_{(2,\:p)} (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(2)})=1,$$ $$f^{\boldsymbol {\lambda}}_{(2,\:p)}(\infty) =f^{\boldsymbol {\lambda}}_{(2,\:p)}(z^{\boldsymbol{\lambda}}_{\sigma^{-1}(3)})=\infty.$$ So we have $$f^{\boldsymbol {\lambda}}_{(2,\:p)}(z)=\frac{z}{\lambda^{p-3}},$$ and $$g_{(2,\:p)}^{(k)}(\boldsymbol {\lambda})= \begin{dcases} \frac{\lambda^k}{\lambda^{p-3}}, &k\ne p-3; \\ \frac{1}{\lambda^{p-3}}, &k=p-3. \end{dcases}$$ Set $\sigma=(3,\:p)$ and we have $$f^{\boldsymbol {\lambda}}_{(3,\:p)}(0)= f^{\boldsymbol {\lambda}}_{(3,\:p)} (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(1)})=0,$$ $$f^{\boldsymbol {\lambda}}_{(3,\:p)}(1)= f^{\boldsymbol {\lambda}}_{(3,\:p)}(z^{\boldsymbol{\lambda}}_{\sigma^{-1}(2)})=1,$$ $$f^{\boldsymbol {\lambda}}_{(3,\:p)}(\lambda^{p-3})= f^{\boldsymbol {\lambda}}_{(3,\:p)}(z^{\boldsymbol{\lambda}}_{\sigma^{-1}(3)})=\infty.$$ So we have $$f^{\boldsymbol {\lambda}}_{(3,\:p)}(z)=\frac{z(1-\lambda^{p-3})}{z-\lambda^{p-3}},$$ and $$g_{(3,\:p)}^{(k)}(\boldsymbol {\lambda})= \begin{dcases} \frac{\lambda^k(1-\lambda^{p-3})}{\lambda^k-\lambda^{p-3}}, &k\ne p-3; \\ \frac{\lambda^{p-3}-1}{\lambda^{p-3}}, &k=p-3. \end{dcases}$$ For $p,\:q=4,\:5,\:\cdots,\:n$ and $p\ne q$ set $\sigma=(1,\:p)(2\:,q)$ and we have $$f^{\boldsymbol {\lambda}}_{(1,\:p)(2\:,q)}(\lambda^{p-3})= f^{\boldsymbol {\lambda}}_{(1,\:p)(2\:,q)} (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(1)})=0,$$ $$f^{\boldsymbol {\lambda}}_{(1,\:p)(2\:,q)}(\lambda^{q-3})= f^{\boldsymbol {\lambda}}_{(1,\:p)(2\:,q)}(z^{\boldsymbol{\lambda}}_{\sigma^{-1}(2)})=1,$$ $$f^{\boldsymbol {\lambda}}_{(1,\:p)(2\:,q)}(\infty)= f^{\boldsymbol {\lambda}}_{(1,\:p)(2\:,q)}(z^{\boldsymbol{\lambda}}_{\sigma^{-1}(3)})=\infty.$$ So we have $$f^{\boldsymbol {\lambda}}_{(1,\:p)(2\:,q)}(z)=\frac{z-\lambda^{p-3}}{\lambda^{q-3}-\lambda^{p-3}},$$ and $$g_{(1,\:p)(2\:,q)}^{(k)}(\boldsymbol {\lambda})= \begin{dcases} \frac{\lambda^k-\lambda^{p-3}}{\lambda^{q-3}-\lambda^{p-3}}, &k\ne p-3,\:q-3;\\ \frac{-\lambda^{p-3}}{\lambda^{q-3}-\lambda^{p-3}}, &k=p-3;\\ \frac{1-\lambda^{p-3}}{\lambda^{q-3}-\lambda^{p-3}}, &k=q-3. \end{dcases}$$ Set $\sigma=(2,\:p)(3,\:q)$ and we have $$f^{\boldsymbol {\lambda}}_{(2,\:p)(3,\:q)}(0) =f^{\boldsymbol {\lambda}}_{(2,\:p)(3,\:q)}(z^{\boldsymbol{\lambda}}_{\sigma^{-1}(1)})=0,$$ $$f^{\boldsymbol {\lambda}}_{(2,\:p)(3,\:q)}(\lambda^{p-3}) =f^{\boldsymbol {\lambda}}_{(2,\:p)(3,\:q)} (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(2)})=1,$$ $$f^{\boldsymbol {\lambda}}_{(2,\:p)(3,\:q)}(\lambda^{q-3}) =f^{\boldsymbol {\lambda}}_{(2,\:p)(3,\:q)}(z^{\boldsymbol{\lambda}}_{\sigma^{-1}(3)})=\infty.$$ So we have $$f^{\boldsymbol {\lambda}}_{(2,\:p)(3,\:q)}(z)=\frac{(\lambda^{p-3}-\lambda^{q-3})z}{\lambda^{p-3}(z-\lambda^{q-3})},$$ and $$g_{(2,\:p)(3\:,q)}^{(k)}(\boldsymbol {\lambda})= \begin{dcases} \frac{(\lambda^{p-3}-\lambda^{q-3})\lambda^k}{\lambda^{p-3}(\lambda^k-\lambda^{q-3})},&k\ne p-3,\:q-3;\\ \frac{\lambda^{p-3}-\lambda^{q-3}}{\lambda^{p-3}(1-\lambda^{q-3})},&k=p-3;\\ \frac{\lambda^{p-3}-\lambda^{q-3}}{\lambda^{p-3}},&k=q-3. \end{dcases}$$ Set $\sigma=(3,\:p)(1,\:q)$ and we have $$f^{\boldsymbol {\lambda}}_{(3,\:p)(1,\:q)}(\lambda^{q-3})= f^{\boldsymbol {\lambda}}_{(3,\:p)(1,\:q)} (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(1)})=0,$$ $$f^{\boldsymbol {\lambda}}_{(3,\:p)(1,\:q)}(1)= f^{\boldsymbol {\lambda}}_{(3,\:p)(1,\:q)}(z^{\boldsymbol{\lambda}}_{\sigma^{-1}(2)})=1,$$ $$f^{\boldsymbol {\lambda}}_{(3,\:p)(1,\:q)}(\lambda^{p-3})= f^{\boldsymbol {\lambda}}_{(3,\:p)(1,\:q)}(z^{\boldsymbol{\lambda}}_{\sigma^{-1}(3)})=\infty.$$ So we have $$f^{\boldsymbol {\lambda}}_{(3,\:p)(1,\:q)}(z)=\frac{(\lambda^{p-3}-1)(z-\lambda^{q-3})}{(\lambda^{q-3}-1)(z-\lambda^{p-3})},$$ and $$g_{(3,\:p)(1\:,q)}^{(k)}(\boldsymbol {\lambda})= \begin{dcases} \frac{(\lambda^{p-3}-1)(\lambda^k-\lambda^{q-3})}{(\lambda^{q-3}-1)(\lambda^k-\lambda^{p-3})},&k\ne p-3,\:q-3;\\ \frac{\lambda^{p-3}-1}{\lambda^{q-3}-1},&k=p-3;\\ \frac{(\lambda^{p-3}-1)\lambda^{q-3}}{(\lambda^{q-3}-1)\lambda^{p-3}},&k=q-3. \end{dcases}$$ For $p,\:q,\:r=4,\:5,\:\cdots,\:n$, and $p\ne q,\:q\ne r,\:r\ne p$, set $\sigma=(1,\:p)(2\:,q)(3,\:r)$ and we have $$f^{\boldsymbol {\lambda}}_{(1,\:p)(2\:,q)(3,\:r)}(\lambda^{p-3})= f^{\boldsymbol {\lambda}}_{(1,\:p)(2\:,q)(3,\:r)} (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(1)})=0,$$ $$f^{\boldsymbol {\lambda}}_{(1,\:p)(2\:,q)(3,\:r)}(\lambda^{q-3})= f^{\boldsymbol {\lambda}}_{(1,\:p)(2\:,q)(3,\:r)}(z^{\boldsymbol{\lambda}}_{\sigma^{-1}(2)})=1,$$ $$f^{\boldsymbol {\lambda}}_{(1,\:p)(2\:,q)(3,\:r)}(\lambda^{r-3})= f^{\boldsymbol {\lambda}}_{(1,\:p)(2\:,q)(3,\:r)}(z^{\boldsymbol{\lambda}}_{\sigma^{-1}(3)})=\infty.$$ So we have $$f^{\boldsymbol {\lambda}}_{(1,\:p)(2\:,q)(3,\:r)}(z)=\frac{(\lambda^{q-3}-\lambda^{r-3})(z-\lambda^{p-3})}{(\lambda^{q-3}-\lambda^{p-3})(z-\lambda^{r-3})},$$ and $$g_{(1,\:p)(2\:,q)(3,\:r)}^{(k)}(\boldsymbol {\lambda})= \begin{dcases} \frac{(\lambda^{q-3}-\lambda^{r-3})(\lambda^k-\lambda^{p-3})}{(\lambda^{q-3}-\lambda^{p-3})(\lambda^k-\lambda^{r-3})}, &k\ne p-3,\:q-3,\:r-3;\\ \frac{(\lambda^{q-3}-\lambda^{r-3})\lambda^{p-3}}{(\lambda^{q-3}-\lambda^{p-3})\lambda^{r-3}},&k=p-3;\\ \frac{(\lambda^{q-3}-\lambda^{r-3})(1-\lambda^{p-3})}{(\lambda-{q-3}-\lambda^{p-3})(1-\lambda^{r-3})},&k=q-3;\\ \frac{\lambda^{q-3}-\lambda^{r-3}}{\lambda^{q-3}-\lambda^{p-3}},&k=r-3. \end{dcases}$$ So we have the conclusions $G=\tilde S\tilde V$, where $$\tilde V=\{\boldsymbol{\lambda}\mapsto(h(\lambda^{\tau(1)}),\:h(\lambda^{\tau(2)}),\:\cdots,\:h(\lambda^{\tau(n-3)}))|h\in H,\:\tau\in S_{n-3}\};$$ $$H=\{\mathrm{Id},\:\lambda\mapsto 1-\lambda,\: \lambda\mapsto \frac{1}{\lambda},\:\lambda\mapsto\frac{\lambda}{\lambda-1} ,\:\lambda\mapsto \frac{\lambda-1}{\lambda},\:\lambda\mapsto \frac{-1}{\lambda-1}\};$$ $$\begin{split} \tilde S=\{g_\sigma|&\sigma=e,\:(p,\:1),\:(p,\:2),\:(p,\:3),\:(p,\:1)(q,\:2),\:(p,\:2)(q,\:3),\:(p,\:3)(q,\:1),\:(p,\:1)(q,\:2)(r,\:3)\\ &p,\:q,\:r=4,\:5,\:\cdots,\:n,\:p\ne q,\:q\ne r,\:r\ne p\}; \end{split}$$ and $$g_{(1,\:p)}^{(k)}(\boldsymbol {\lambda})= \begin{dcases} \frac{\lambda^k-\lambda^{p-3}}{1-\lambda^{p-3}}, &k\ne p-3; \\ \frac{-\lambda^{p-3}}{1-\lambda^{p-3}}, &k=p-3; \end{dcases}$$ $$g_{(2,\:p)}^{(k)}(\boldsymbol {\lambda})= \begin{dcases} \frac{\lambda^k}{\lambda^{p-3}}, &k\ne p-3; \\ \frac{1}{\lambda^{p-3}}, &k=p-3; \end{dcases}$$ $$g_{(3,\:p)}^{(k)}(\boldsymbol {\lambda})= \begin{dcases} \frac{\lambda^k(1-\lambda^{p-3})}{\lambda^k-\lambda^{p-3}}, &k\ne p-3; \\ \frac{\lambda^{p-3}-1}{\lambda^{p-3}}, &k=p-3; \end{dcases}$$ $$g_{(1,\:p)(2\:,q)}^{(k)}(\boldsymbol {\lambda})= \begin{dcases} \frac{\lambda^k-\lambda^{p-3}}{\lambda^{q-3}-\lambda^{p-3}}, &k\ne p-3,\:q-3;\\ \frac{-\lambda^{p-3}}{\lambda^{q-3}-\lambda^{p-3}}, &k=p-3;\\ \frac{1-\lambda^{p-3}}{\lambda^{q-3}-\lambda^{p-3}}, &k=q-3; \end{dcases}$$ $$g_{(2,\:p)(3\:,q)}^{(k)}(\boldsymbol {\lambda})= \begin{dcases} \frac{(\lambda^{p-3}-\lambda^{q-3})\lambda^k}{\lambda^{p-3}(\lambda^k-\lambda^{q-3})},&k\ne p-3,\:q-3;\\ \frac{\lambda^{p-3}-\lambda^{q-3}}{\lambda^{p-3}(1-\lambda^{q-3})},&k=p-3;\\ \frac{\lambda^{p-3}-\lambda^{q-3}}{\lambda^{p-3}},&k=q-3; \end{dcases}$$ $$g_{(3,\:p)(1\:,q)}^{(k)}(\boldsymbol {\lambda})= \begin{dcases} \frac{(\lambda^{p-3}-1)(\lambda^k-\lambda^{q-3})}{(\lambda^{q-3}-1)(\lambda^k-\lambda^{p-3})},&k\ne p-3,\:q-3;\\ \frac{\lambda^{p-3}-1}{\lambda^{q-3}-1},&k=p-3;\\ \frac{(\lambda^{p-3}-1)\lambda^{q-3}}{(\lambda^{q-3}-1)\lambda^{p-3}},&k=q-3; \end{dcases}$$ $$g_{(1,\:p)(2\:,q)(3,\:r)}^{(k)}(\boldsymbol {\lambda})= \begin{dcases} \frac{(\lambda^{q-3}-\lambda^{r-3})(\lambda^k-\lambda^{p-3})}{(\lambda^{q-3}-\lambda^{p-3})(\lambda^k-\lambda^{r-3})}, &k\ne p-3,\:q-3,\:r-3;\\ \frac{(\lambda^{q-3}-\lambda^{r-3})\lambda^{p-3}}{(\lambda^{q-3}-\lambda^{p-3})\lambda^{r-3}},&k=p-3;\\ \frac{(\lambda^{q-3}-\lambda^{r-3})(1-\lambda^{p-3})}{(\lambda^{q-3}-\lambda^{p-3})(1-\lambda^{r-3})},&k=q-3;\\ \frac{\lambda^{q-3}-\lambda^{r-3}}{\lambda^{q-3}-\lambda^{p-3}},&k=r-3. \end{dcases}$$ $G$ is isomorphic to $S_n$ when $n\ge5$. The Singularities of Moduli Space $\mathfrak{M_{0,\:n}}$ {#sing} --------------------------------------------------------- For $\boldsymbol {\lambda}\in K_n$, $n\ge5$, set $$[\boldsymbol {\lambda}]=\{0,\:1,\:\infty,\:\lambda^1,\:\lambda^2,\:\cdots,\:\lambda^{n-3}\}=\{z^{\boldsymbol {\lambda}}_k|k=1,\:2,\:\cdots,\:n\}$$ and $G_{\boldsymbol \lambda}$ the stabilizer of $\boldsymbol {\lambda}$ $$G_{\boldsymbol \lambda}=\{g_{\sigma}\in G|g_\sigma(\boldsymbol \lambda)=\boldsymbol \lambda\}.$$ \[def\] When $n\ge5$, call $\overline{[\boldsymbol {\lambda}]}\in \mathfrak{M_{0,\:n}}$ an oribfold singularity of $\mathfrak{M_{0,\:n}}$ if $G_{\boldsymbol \lambda}$ is non-trivial. Set $$\alpha=\{z_1, z_2, \cdots, z_n\}\subseteq \widehat{\mathbb{C}},$$ and $\mathcal{A}_{\alpha}$ the group of linear fractional transformations fixing $\alpha$. For any $\boldsymbol {\lambda}\in K_{n}$, $n\ge5$, define $\Phi_{\boldsymbol \lambda}$ as the mapping $$\Phi_{\boldsymbol \lambda}:\:G_{\boldsymbol \lambda}\to\mathcal{A}_{[\boldsymbol {\lambda}]},\: g_\sigma\mapsto f^{\boldsymbol {\lambda}}_\sigma.$$ For each $g_\sigma\in G_{\boldsymbol \lambda}$, we have $$f^{\boldsymbol {\lambda}}_\sigma (z^{\boldsymbol{\lambda}}_{\sigma^{-1} (k)})=z^{g_\sigma(\boldsymbol{\lambda})}_k=z^{\boldsymbol{\lambda}}_k,\:k=1,\:2,\:\cdots,\:n.$$ Thus $f^{\boldsymbol {\lambda}}_\sigma\in\mathcal{A}_{[\boldsymbol {\lambda}]}$, and $\Phi_{\boldsymbol \lambda}$ is well-defined. The map $\Phi_{\boldsymbol \lambda}$ is a group isomorphism between $G_{\boldsymbol \lambda}$ and $\mathcal{A}_{[\boldsymbol {\lambda}]}$. First notice that for $g_\pi,\:g_\sigma\in G_{\boldsymbol \lambda}$ $$f^{\boldsymbol {\lambda}}_\pi\circ f^{\boldsymbol {\lambda}}_\sigma (z^{\boldsymbol{\lambda}}_{(\pi\cdot\sigma)^{-1}(k)}) =f^{\boldsymbol {\lambda}}_\pi(z^{\boldsymbol{\lambda}}_{\pi^{-1}(k)})=z^{\boldsymbol {\lambda}}_k,\:k=1,\:2,\:\cdots,\:n.$$ So $f^{\boldsymbol {\lambda}}_\pi\circ f^{\boldsymbol {\lambda}}_\sigma =f^{\boldsymbol {\lambda}}_{\pi\cdot\sigma}$, which means that $\Phi_{\boldsymbol \lambda}$ is a group homomorphism. For any $h\in\mathcal A_{[\boldsymbol {\lambda}]}$, there exists a unique $\sigma\in S_n$ such that $$h(z^{\boldsymbol {\lambda}}_{\sigma^{-1}(k)})=z^{\boldsymbol {\lambda}}_k,\:k=1,\:2,\:\cdots,\:n.$$ Thus we have $h=f^{\boldsymbol \lambda}_{\sigma}$ and $g_\sigma(\boldsymbol \lambda)=\boldsymbol \lambda$, and $$\Phi_{\boldsymbol \lambda}(g_\sigma)=f^{\boldsymbol \lambda}_{\sigma}=h.$$ So $\Phi_{\boldsymbol \lambda}$ is a group epimorphism. For any $g_\sigma\in G_{\boldsymbol \lambda}$ such that $$\Phi_{\boldsymbol \lambda}(g_\sigma)=f^{\boldsymbol {\lambda}}_\sigma=\mathrm{Id},$$ we have $$z^{\boldsymbol{\lambda}}_k =f^{\boldsymbol {\lambda}}_\sigma (z^{\boldsymbol{\lambda}}_k) =z^{\boldsymbol{\lambda}}_{\sigma(k)}$$ holds for $k=1,\:2,\:\cdots,\:n$. Thus $\sigma$ is the identity element. Hence $\Phi_{\boldsymbol \lambda}$ is a group isomorphism. Now the study of the orbifold singularities of $\mathfrak{M_{0,\:n}}$ for $n\geq 5$ can be reduced to that of $\mathcal{A_\alpha}$ for subset $\alpha$ with $n$ elements on the Riemann sphere. In Section \[orbit\] I shall develope a method to find $\mathcal{A_\alpha}$ for an arbitrary finite subset $\alpha$ of the Riemann sphere when $|\alpha|\ge4$. The Subset Fixed by a Specific Group {#orbit} ==================================== Given a finite subset $\alpha=\{z_1, z_2, \cdots, z_n\}\subseteq \widehat{\mathbb{C}}$, $n\ge4$, let $\mathcal{A}_{\alpha}$ be the group of linear fractional transformations that fix $\alpha$. For any $f$ in $\mathcal{A}_{\alpha}$, since $f(z_1),\:f(z_2),\:f(z_3)$ are all in $\alpha$, there are at most $n(n-1)(n-2)$ ways to choose the images of $z_1, z_2, z_3$. Thus $\arrowvert \mathcal{A}_{\alpha} \arrowvert \le n(n-1)(n-2)$. Let $G$ be a finite group of linear fractional transformations. There are only five kinds of non-trivial finite linear fractional transformation groups: the icosahedral group $I$, the octahedral group $O$, the tetrahedral group $T$, the dihedral group $D_n$ and the finite cyclic group $\mathbb Z_n$, $n\ge2$. In the rest of this section, by discussing the orbits of the action of $G$ on $S^2$ and $\widehat{\mathbb{C}}$, we shall find all finite subsets $\alpha$ of $\widehat{\mathbb{C}}$ such that $\mathcal{A}_{\alpha}\simeq G$. Thus given a finite $\alpha\subseteq \widehat{\mathbb{C}}$, $|\alpha|\ge4$, we can find $\mathcal{A}_{\alpha}$. Notice that we shall not distinguish a point on $S^2$ from its image on the extended complex plane under the stereographic projection. $G$ is the Icosahedral Group $I$ {#A_5} -------------------------------- Suppose that $G$ is the icosahedral group $I$ that fixes a regular dodecahedron whose center is the origin. {width="3in"} There are $10$ axes joining opposite vertices of the dodecahedron. Rotations about each axis of angles $2\pi/3$ and $4\pi/3$ carry the dodecahedron into itself. There are, thus, $2\times10=20$ such rotations in $G$. There are $6$ axes joining central points of opposite faces of the dodecahedron. Rotations about each axis of angles $2\pi/5$, $4\pi/5$, $6\pi/5$ and $8\pi/5$ carry the dodecahedron into itself. There are, thus, $4\times6=24$ such rotations in $G$. There are $15$ axes joining middle points of opposite edges of the dodecahedron. Rotations about each axis of angle $\pi$ carry the dodecahedron into itself. There are, thus, $1\times15=15$ such rotations in $G$. We have already got $20+24+15=59$ rotations. To these add the identity transformation, and we get the whole group. There are four kinds of orbits of $G$ on $S^2$. Let $V_I,\:F_I,\:E_I$ denote the vertices, the projections of the central points of the faces on $S^2$ and the projections of the middle points of the edges on $S^2$ respectively, with the origin being the center of the projection. Then $V_I,\:F_I,\:E_I$ are three different orbits of $G$ and $$|V_I|=20,\:|F_I|=12,\:|E_I|=30.$$ For any $X\in S^2 \backslash (V_I\cup F_I\cup E_I)$, define $B_I(X)$ as the orbit of $X$. As $X$ is not fixed by any non-trivial element in $G$, it is obvious that $$\arrowvert B_I(X) \arrowvert =\arrowvert G \arrowvert=60.$$ For any finite subset $\alpha$ of $S^2$ such that $\mathcal{A}_{\alpha} = G$, $\alpha$ is a finite union of the orbits of $G$. On the other hand, if $\alpha$ is a finite union of the orbits of $G$, we have $G\subseteq\mathcal{A}_{\alpha}$. Since none of $S_4,\:A_4,\:D_n,\:Z_n,\:n\ge2$ has a subgroup isomorphic to $A_5$, we conclude that $G=\mathcal{A}_{\alpha}$. Thus we conclude that \[tA\_5\] For any finite subset $\alpha$ of $S^2$, $\mathcal{A}_{\alpha}=G\simeq A_5$ if and only if $\alpha$ is a union of certain elements in $\{V_I,\:F_I,\:E_I\}\cup\{B_I(X)|X\in S^2 \backslash (V_I\cup F_I\cup E_I)\}$. $G$ is the Octahedral Group $O$ {#S_4} ------------------------------- Now suppose $G$ is the octahedral group $O$ that fixes a cube whose center is the origin. There are $4$ axes joining opposite vertices of the cube. Rotations about each axis of angles $2\pi/3$ and $4\pi/3$ carry the cube into itself. There are, thus, $2\times4=8$ such rotations in $G$. There are $3$ axes joining central points of opposite faces of the cube. Rotations about each axis of angles $\pi/2$, $\pi$ and $3\pi/2$ carry the cube into itself. There are, thus, $3\times3=9$ such rotations in $G$. There are $6$ axes joining middle points of opposite edges of the cube. Rotations about each axis of angle $\pi$ carry the cube into itself. There are, thus, $1\times6=6$ such rotations in $G$. We have already got $8+9+6=23$ rotations. To these add the identity transformation, and we get the whole group. There are four kinds of orbits on $S^2$. Let $V_O,\:F_O,\:E_O$ denote the vertices, the projections of the central points of the faces on $S^2$ and the projections of the middle points of the edges on $S^2$ respectively, with the origin being the center of the projection. Then $V_O,\:F_O,\:E_O$ are three different orbits of $G$ and $$|V_O|=8,\:|F_O|=6,\:|E_O|=12.$$ For any $X\in S^2 \backslash (V_O\cup F_O\cup E_O)$, define $B_O(X)$ as the orbit of $X$. As $X$ is not fixed by any non-trivial element in $G$, it is obvious that $$|B_O(X)|=|G|=24.$$ For any finite subset $\alpha$ of $S^2$ such that $\mathcal{A}_{\alpha} = G$, $\alpha$ is a finite union of the orbits of $G$. On the other hand, if $\alpha$ is a finite union of the orbits of $G$, we have $G\subseteq\mathcal{A}_{\alpha}$. Since none of $A_5,\:A_4,\:D_n,\:Z_n,\:n\ge2$ has a subgroup isomorphic to $S_4$, we conclude that $G=\mathcal{A}_{\alpha}$. Thus we conclude that \[tS\_4\] For any finite subset $\alpha$ of $S^2$, $\mathcal{A}_{\alpha}=G\simeq S_4$ if and only if $\alpha$ is a union of certain elements in $\{V_O,\:F_O,\:E_O\}\cup\{B_O(X)|X\in S^2 \backslash (V_O\cup F_O\cup E_O)\}$. $G$ is the Tetrahedral Group $T$ {#A_4} -------------------------------- ### The Orbits of $T$ Now suppose $G$ is the tetrahedral group $T$ that fixes a regular tetrahedron whose center is the origin. There are $4$ axes, each joining a central point of a face to the opposite vertex. Rotations about each axis of angles $2\pi/3$ and $4\pi/3$ carry the tetrahedral into itself. There are, thus, $2\times4=8$ such rotations in $G$. There are $3$ axes joining middle points of opposite edges of the tetrahedral. Rotations about each axis of angle $\pi$ carry the tetrahedral into itself. There are, thus, $1\times3=3$ such rotations in $G$. We have already got $8+3=11$ rotations. To these add the identity transformation, and we get the whole group. There are three kinds of orbits on $S^2$, too. Let $V_T,\:F_T,\:E_T$ denote the vertices, the projections of the central points of the faces on $S^2$ and the projections of the middle points of the edges on $S^2$ respectively, with the origin being the center of the projection. Then $V_T,\:F_T,\:E_T$ are three different orbits of $G$ and $$|V_T|=|F_T|=4,\:|E_T|=6.$$ For any $X\in S^2 \backslash (V_T\cup F_T\cup E_T)$, define $B_T(X)$ as the orbit of $X$. As $D$ is not fixed by any non-trivial element in $G$, it is obvious that $$|B_T(X)|=|G|=12.$$ For any finite subset $\alpha$ of $S^2$ such that $\mathcal{A}_{\alpha} = G$, $\alpha$ is a finite union of the orbits of $G$. On the other hand, if $\alpha$ is a finite union of the orbits of $G$, we have $G\subseteq\mathcal{A}_{\alpha}$. Since neither of $D_n, Z_n, n\ge2$ has a subgroup isomorphic to $S_4$, we conclude that $\mathcal{A}_{\alpha}$ is isomorphic to $A_5, S_4$ or $A_4$. In the following sections we shall investigate the two cases when $\mathcal{A}_{\alpha}$ is isomorphic to $A_5$ and $S_4$. ### Case 1: $\mathcal{A}_{\alpha}$ is Isomorphic to $A_5$ If $\mathcal{A}_{\alpha}$ is isomorphic to $A_5$, we know that $\mathcal{A}_{\alpha}$ fixes some regular dodecahedron whose center is the origin. For every vertex of the tetrahedron, there exists some element $f\in G\subseteq\mathcal{A}_{\alpha}$ of order three that fixes it. But for any element $g\in\mathcal{A}_{\alpha}$, if $g$ is of order three, both of the fixed points of $g$ are vretices of the dodecahedron. Thus we conclude that every vertex of the tetrahedron is also a vertex of the regular dodecahedron. Their relative positions are shown in Figure \[A\_5, A\_4\]. ![The dodecahedron fixed by $\mathcal{A}_{\alpha}$ and the tetrahedron fixed by $G$.[]{data-label="A_5, A_4"}](a_5a_4.png){width="5in"} It is obvious that $$V_T\subseteq V_I.$$ Connect the two points $G$ and $G'$. As they are opposite points of the dodecahedron, the segment $GG'$ goes through the origin $O$. Assume that $GG'$ intersects the triangle $AD'H$ at $G''$, then $G''$ must be the center of $AD'H$, and thus $G\in F_T$. As the four points in $F_T$ are congruent, we have $$F_T\subseteq V_I.$$ Let $P$ be the midpoint of $BC$. Connect the origin $O$ and $P$. Let $f$ be the rotation of $\pi$ with $OP$ as its axis. By observation we have $f\in\mathcal {A}_\alpha$ and $$f(A)=H,\:f(H)=A.$$ So $OP$ intersects the segment $AH$ at its midpoint $P''$. As the six points in $E_T$ are congruent, we have $$E_T\subseteq E_I.$$ From the above discussion we see that $$B\notin V_T\cup F_T\cup E_T.$$ So we have $$|B_T(B)|=12.$$ Since $$V_T\subseteq V_I,\:F_T\subseteq V_I,\:B_T(B)\subseteq V_I,$$ and $$|V_T|=|F_T|=4,\:|B_T(B)|=12,\:|V_I|=20,$$ we have $$V_I=V_T\cup F_T\cup B_T(B).$$ Let $N$ denote the center of the pentagon $ABCDE$. It is obvious that $$N\notin V_T\cup F_T\cup E_T.$$ So we have $$|B_T(N)|=12.$$ Since $$B_T(N)\subseteq F_I,$$ and $$|B_T(N)|=|F_I|=12,$$ we have $$F_I=B_T(N).$$ Let $Q$, $R$ denote the midpoints of $AB$, $DE$. It is obvious that $$Q, R\notin V_T\cup F_T\cup E_T.$$ Note that for any $f\in G$, $f(Q)\ne R$. So we have $$B_T(Q)\ne B_T(R).$$ Since $$E_T\subseteq E_I,\:B_T(Q)\subseteq E_I,\:B_T(R)\subseteq E_I,$$ and $$|E_T|=6,\:|B_T(Q)|=|B_T(R)|=12,\:|E_I|=30,$$ we have $$E_I=E_T\cup B_T(Q)\cup B_T(R).$$ For any $X\in S^2 \backslash (V_I\cup F_I\cup E_I)$, $f(X)\notin V_I\cup F_I\cup E_I$, $\forall f\in \mathcal A_\alpha$. Now let $g$ denote the rotation which fixes the pentagon $ABCDE$ and transforms $A$ to $B$. We have $$X,\:g(X),\:g^2(X),\:g^3(X),\:g^4(X)\in S^2 \backslash (V_T\cup F_T\cup E_T).$$ Next we shall prove that $$B_T(X),\:B_T(g(X)),\:B_T(g^2(X)),\:B_T(g^3(X)),\:B_T(g^4(X))$$ are five different orbits of $G$. Suppose that $$B_T(g^i(X))=B_T(g^j(X))$$ for some $i,j=0,1,2,3,4,\:i\ne j$. Then there exists some $h\in G\simeq A_4$ such that $$g^i(X)=h(g^j(X))=h(g^{j-i}(g^i(X))).$$ Since $g^i(X)\notin V_I\cup F_I\cup E_I$, it can not be fixed by any non-trivial element in $\mathcal A_\alpha$. Thus we have $$h\circ g^{j-i}=\mathrm{I},$$ which contradicts the assumption that $$h\in G\simeq A_4.$$ Now we see that $$B_T(X),\:B_T(g(X)),\:B_T(g^2(X)),\:B_T(g^3(X)),\:B_T(g^4(X))$$ are five different orbits of $G$. As $$B_T(X),\:B_T(g(X)),\:B_T(g^2(X)),\:B_T(g^3(X)),\:B_T(g^4(X))\subseteq B_I(X)$$ and $$|B_T(X)|=|B_T(g(X))|=|B_T(g^2(X))|=|B_T(g^3(X))|=|B_T(g^4(X))|=12,$$ $$|B_I(X)|=60,$$ we have $$B_I(X)=B_T(X)\cup B_T(g(X))\cup B_T(g^2(X))\cup B_T(g^3(X))\cup B_T(g^4(X)).$$ In summary, $$V_I=V_T\cup F_T\cup B_T(B),$$ $$F_I=B_T(N),$$ $$E_I=E_T\cup B_T(Q)\cup B_T(R),$$ $$B_I(X)=B_T(X)\cup B_T(g(X))\cup B_T(g^2(X))\cup B_T(g^3(X))\cup B_T(g^4(X)),$$ $X\in S^2 \backslash (V_I\cup F_I\cup E_I)$. As $\mathcal{A}_{\alpha}$ is isomorphic to $A_5$, $\alpha$ is a finite union of these above orbits. ### Case 2: $\mathcal{A}_{\alpha}$ Is isomorphic to $S_4$ If $\mathcal{A}_{\alpha}$ is isomorphic to $S_4$, we know that $\mathcal{A}_{\alpha}$ fixes some cube whose center is the origin. For every vertex of the tetrahedron, there exists some element $f\in G\subseteq \mathcal{A}_{\alpha}$ of order three that fixes it. But for any element $g\in\mathcal{A}_{\alpha}$, if $g$ is of order three, both of the fixed points are vretices of the cube. Thus we conclude that every vertex of the tetrahedron is also a vertex of the cube. Their relative positions are shown in Figure \[S\_4, A\_4\]. ![The cube fixed by $\mathcal{A}_{\alpha}$ and the tetrahedron fixed by $G$.[]{data-label="S_4, A_4"}](s_4a_4.png){width="3in"} It is obvious that $$V_T\subseteq V_O.$$ Connect the two points $A$ and $A'$. As they are opposite points of the cube, the segment $AA'$ goes through the origin $O$. Assume that $AA'$ intersects the triangle $B'CD'$ at $A''$, then $A''$ must be the center of $B'CD'$, and $A'\in F_T$. As the four points in $F_T$ are congruent, we have $$F_T\subseteq V_I.$$ From the above discussion we see that $$V_T\subseteq V_O,\:F_T\subseteq V_O,$$ and $$|V_T|=|F_T|=4,\:|V_O|=8,$$ so we have $$V_O=V_T\cup F_T.$$ It is obvious that $$F_O=E_T.$$ Let $P$ denote the midpoint of $AB$. It is obvious that $$P\notin V_T\cup F_T\cup E_T.$$ Since $$B_T(P)\subseteq E_O,$$ and $$|B_T(P)|=|E_O|=12,$$ we have $$E_O=B_T(P).$$ For any $X\in S^2 \backslash (V_O\cup F_O\cup E_O)$, $f(X)\notin V_O\cup F_O\cup E_O$, $\forall f\in \mathcal A_\alpha$. Now let $g$ denote the rotation which fixes the square $ABCD$ and transforms $A$ to $B$. We have $$X,\:g(X)\in S^2 \backslash (V_T\cup F_T\cup E_T).$$ Next we shall prove that $$B_T(X),\:B_T(g(X))$$ are two different orbits of $A_4$. Suppose that $$B_T(X)=B_T(g(X)).$$ Then there exists some $h\in G\simeq A_4$ such that $$X=h(g(X)).$$ Since $X\notin V_O\cup F_O\cup E_O$, it can not be fixed by any non-trivial element in $\mathcal A_\alpha$. Thus we have $$h\circ g=\mathrm{I},$$ which contradicts the assumption that $$h\in G\simeq A_4.$$ Now we see that $$B_T(X),\:B_T(g(X))$$ are two different orbits of $A_4$. As $$B_T(X),\:B_T(g(X))\subseteq B_O(X)$$ and $$|B_T(X)|=|B_T(g(X))|=12,\:|B_O(X)|=24,$$ we have $$B_O(X)=B_T(X)\cup B_T(g(X)).$$ In summary, $$V_O=V_T\cup F_T,$$ $$F_O=E_T,$$ $$E_O=B_T(P),$$ $$B_O(X)=B_T(X)\cup B_T(g(X)),$$ $X\in S^2 \backslash (V_O\cup F_O\cup E_O)$. As $\mathcal{A}_{\alpha}$ is $S_4$, $\alpha$ is a finite union of these above orbits. ### Conclusion From the discussion above we conclude that \[tA\_4\] For any finite subset $\alpha$ of $S^2$, $\mathcal{A}_{\alpha}=G\simeq A_4$ if and only if all of the three claims are true: 1. $\alpha$ is a union of certain elements in $$\{V_T,\:F_T,\:E_T\}\cup\{B_T(X)|X\in S^2 \backslash (V_T\cup F_T\cup E_T)\};$$ 2. $\alpha$ is NOT a union of certain elements in $$\{V_I,\:F_I,\:E_I\}\cup\{B_I(X)|X\in S^2 \backslash (V_I\cup F_I\cup E_I)\};$$ 3. $\alpha$ is NOT a union of certain elements in $$\{V_O,\:F_O,\:E_O\}\cup\{B_O(X)|X\in S^2 \backslash (V_O\cup F_O\cup E_O)\};$$ where (Figure \[A\_5, A\_4\]) $$V_I=V_T\cup F_T\cup B_T(B);$$ $$F_I=B_T(N);$$ $$E_I=E_T\cup B_T(Q)\cup B_T(R);$$ $$B_I(X)=B_T(X)\cup B_T(g(X))\cup B_T(g^2(X))\cup B_T(g^3(X))\cup B_T(g^4(X))$$ and (Figure \[S\_4, A\_4\]) $$V_O=V_T\cup F_T;$$ $$F_O=E_T;$$ $$E_O=B_T(P);$$ $$B_O(X)=B_T(X)\cup B_T(g(X)).$$ $G$ is the Dihedral Group $D_n$ {#D_n} ------------------------------- In this section, we go back to $\widehat{\mathbb C}$. Assume that $$f(z)=e^{\frac{1}{n}2\pi i}z,\:g(z)=\frac{1}{z},\:n\ge2$$ and $$G=\langle f, g\rangle\simeq D_n.$$ There are three kinds of orbits. Set $$V=\{0, \infty\},$$ $$A_n=\{e^{\frac{k}{n}2\pi i}|k\in\mathbb{Z}\},$$ $$B_n=\{e^{\frac{2k+1}{2n}2\pi i}|k\in\mathbb{Z}\}$$ and $$C_n(z)=\{ze^{\frac{k}{n}2\pi i}|k\in\mathbb{Z}\}\cup\{z^{-1}e^{\frac{k}{n}2\pi i}|k\in\mathbb{Z}\}$$ if $z\in{\mathbb C}^*\backslash\{e^{\frac{l}{2n}2\pi i}|l\in\mathbb{Z}\}$. If $\mathcal A_\alpha=G$, $\alpha$ is a finite union of the above orbits. If $\alpha$ is a finite union of the above orbits, we have $G\subseteq\mathcal{A}_{\alpha}$. So we conclude that $\mathcal{A}_{\alpha}$ is isomorphic to $A_5,\:S_4,\:A_4$ or $D_{pn}$ for $p\in\mathbb Z^+$. The $A_5,\:S_4,\:A_4$ case have already been explored. Now we shall discuss the case that $\mathcal{A}_{\alpha}$ is isomorphic to $D_{pn}$, $p\in\mathbb Z,\:p\ge2$. Set that $$\mathcal{A}_{\alpha}=\langle \rho,\pi|(\rho)^{pn}=(\pi)^2=(\rho\pi)^2=e\rangle\simeq D_{pn}.$$ *Case 1: $f=\rho^q$ for some $q\in\mathbb Z$.* As $f=\rho^q$, $\rho$ fixes $0$ and $\infty$ as $f$. As $\rho$ is of order $pn$, we see that $$\rho(z)=e^{\frac{m}{pn}2\pi i}z$$ for some $m$ prime to $pn$. Thus $$\mathcal{A}_{\alpha}=\langle z\mapsto e^{\frac{1}{pn}2\pi i}z,\:z\mapsto\frac{1}{z} \rangle.$$ If $p$ is odd, we have $$A_{pn}=A_n\cup C_n({e^{\frac{1}{pn}2\pi i}})\cup C_n({e^{\frac{2}{pn}2\pi i}})\cup\cdots\cup C_n({e^{\frac{p-1}{2pn}2\pi i}}),$$ $$B_{pn}=B_n\cup C_n({e^{\frac{1}{2pn}2\pi i}})\cup C_n({e^{\frac{3}{2pn}2\pi i}})\cup\cdots\cup C_n({e^{\frac{p-2}{2pn}2\pi i}}),$$ $$C_{pn}(z)=C_n(z)\cup C_n({e^{\frac{1}{pn}2\pi i}}z)\cup C_n({e^{\frac{2}{pn}2\pi i}}z)\cup\cdots\cup C_n({e^{\frac{p-1}{pn}2\pi i}}z).$$ If $p$ is even, we have $$A_{pn}=A_n\cup B_n\cup C_n({e^{\frac{1}{pn}2\pi i}})\cup C_n({e^{\frac{2}{pn}2\pi i}})\cup\cdots\cup C_n({e^{\frac{p-2}{2pn}2\pi i}}),$$ $$B_{pn}=C_n({e^{\frac{1}{2pn}2\pi i}})\cup C_n({e^{\frac{3}{2pn}2\pi i}})\cup\cdots\cup C_n({e^{\frac{p-1}{2pn}2\pi i}}),$$ $$C_{pn}(z)=C_n(z)\cup C_n({e^{\frac{1}{pn}2\pi i}}z)\cup C_n({e^{\frac{2}{pn}2\pi i}}z)\cup\cdots\cup C_n({e^{\frac{p-1}{pn}2\pi i}}z).$$ *Case 2: $f=\pi$.* Thus we see that $n=2$ and $g=\rho^p$. Define the linear fractional transformation $\phi$ $$\phi(z)=\frac{1+z}{1-z}.$$ Now we have $$\phi^{-1}\circ\rho^p\circ\phi(z)=-z,\:\phi^{-1}\circ\pi\circ\phi(z)=\frac{1}{z},$$ and thus $$\phi^{-1}\circ\rho\circ\phi(z)=e^{\frac{m}{2p}2\pi i}z,$$ for some $m$ prime to $2p$. So we have $$\phi^{-1}\mathcal A_\alpha\phi=\langle z\mapsto e^{\frac{1}{2p}2\pi i}z,\:z\mapsto \frac{1}{z}\rangle\simeq D_{2p}.$$ So the three kinds of orbits of $\mathcal A_\alpha$ are $\phi^{-1}(V)$, $\phi^{-1}(A_{2p})$, $\phi^{-1}(B_{2p})$ and $\phi^{-1}(C_{2p}(z))$ for $z\in\mathbb C^*\backslash\{e^{\frac{l}{4p}2\pi i}|l\in\mathbb{Z}\}$. If $n$ is odd, we have $$\phi^{-1}(V)=A_2,$$ $$\phi^{-1}(A_{2p})=V\cup C_2(i\tan(\frac{1}{4p}2\pi))\cup C_2(i\tan(\frac{2}{4p}2\pi))\cup\cdots\cup C_2(i\tan(\frac{\frac{p-1}{2}}{4p}2\pi)),$$ $$\phi^{-1}(B_{2p})=B_2\cup C_2(i\tan(\frac{1}{8p}2\pi))\cup C_2(i\tan(\frac{3}{8p}2\pi))\cup\cdots\cup C_2(i\tan(\frac{p-2}{8p}2\pi)),$$ $$\phi^{-1}(C_{2p}(z))=C_2(\phi^{-1}(z))\cup C_2(\phi^{-1}(e^{\frac{1}{2p}2\pi i}z))\cup \cdots\cup C_2(\phi^{-1}(e^{\frac{p-1}{2p}2\pi i}z))$$ for $z\in\mathbb C^*\backslash\{e^{\frac{l}{4p}2\pi i}|l\in\mathbb{Z}\}$. If $n$ is even, we have $$\phi^{-1}(V)=A_2,$$ $$\phi^{-1}(A_{2p})=V\cup B_2\cup C_2(i\tan(\frac{1}{4p}2\pi))\cup C_2(i\tan(\frac{2}{4p}2\pi))\cup\cdots\cup C_2(i\tan(\frac{\frac{p-2}{2}}{4p}2\pi)),$$ $$\phi^{-1}(B_{2p})=C_2(i\tan(\frac{1}{8p}2\pi))\cup C_2(i\tan(\frac{3}{8p}2\pi))\cup\cdots\cup C_2(i\tan(\frac{p-1}{8p}2\pi)),$$ $$\phi^{-1}(C_{2p}(z))=C_2(\phi^{-1}(z))\cup C_2(\phi^{-1}(e^{\frac{1}{2p}2\pi i}z))\cup \cdots\cup C_2(\phi^{-1}(e^{\frac{p-1}{2p}2\pi i}z))$$ for $z\in\mathbb C^*\backslash\{e^{\frac{l}{4p}2\pi i}|l\in\mathbb{Z}\}$. In conclusion, we have \[tD\_n\] For any finite subset $\alpha$ of $S^2$, $|\alpha|\ge4$, $\mathcal{A}_{\alpha}=G\simeq D_n,\: n\ge2$ if and only if all of the four claims are true: 1. $\alpha$ is a union of certain elements in $$\{V,\:A_n,\:B_n\}\cup\{C_n(z)|z\in{\mathbb C}^*\backslash\{e^{\frac{l}{2n}2\pi i}|l\in\mathbb{Z}\}\};$$ 2. $\alpha$ is NOT a union of certain elements in $$\{V,\:A_{pn},\:B_{pn}\}\cup\{C_{pn}(z)|z\in{\mathbb C}^*\backslash\{e^{\frac{l}{2n}2\pi i}|l\in\mathbb{Z}\}\},\:p\ge2;$$ 3. when $n=2$, $\alpha$ is NOT a union of certain elements in $$\{A_2,\:\phi^{-1}(A_{2p}),\:\phi^{-1}(B_{2p})\}\cup\{\phi^{-1}(C_{2p}(z))|z\in\mathbb C^*\backslash\{e^{\frac{l}{4p}2\pi i}|l\in\mathbb{Z}\}\},\:p\ge2;$$ 4. $\alpha$ is NOT in the icosahedral, the octahedral or the tetrahedral case, where $$A_{pn}=A_n\cup C_n({e^{\frac{1}{pn}2\pi i}})\cup C_n({e^{\frac{2}{pn}2\pi i}})\cup\cdots\cup C_n({e^{\frac{p-1}{2pn}2\pi i}}),$$ $$B_{pn}=B_n\cup C_n({e^{\frac{1}{2pn}2\pi i}})\cup C_n({e^{\frac{3}{2pn}2\pi i}})\cup\cdots\cup C_n({e^{\frac{p-2}{2pn}2\pi i}}),$$ $$C_{pn}(z)=C_n(z)\cup C_n({e^{\frac{1}{pn}2\pi i}}z)\cup C_n({e^{\frac{2}{pn}2\pi i}}z)\cup\cdots\cup C_n({e^{\frac{p-1}{pn}2\pi i}}z),$$ $$\phi^{-1}(V)=A_2,$$ $$\phi^{-1}(A_{2p})=V\cup C_2(i\tan(\frac{1}{4p}2\pi))\cup C_2(i\tan(\frac{2}{4p}2\pi))\cup\cdots\cup C_2(i\tan(\frac{\frac{p-1}{2}}{4p}2\pi)),$$ $$\phi^{-1}(B_{2p})=B_2\cup C_2(i\tan(\frac{1}{8p}2\pi))\cup C_2(i\tan(\frac{3}{8p}2\pi))\cup\cdots\cup C_2(i\tan(\frac{p-2}{8p}2\pi)),$$ $$\phi^{-1}(C_{2p}(z))=C_2(\phi^{-1}(z))\cup C_2(\phi^{-1}(e^{\frac{1}{2p}2\pi i}z))\cup \cdots\cup C_2(\phi^{-1}(e^{\frac{p-1}{2p}2\pi i}z))$$ for $z\in\mathbb C^*\backslash\{e^{\frac{l}{4p}2\pi i}|l\in\mathbb{Z}\}$ if $p$ is odd, and $$A_{pn}=A_n\cup B_n\cup C_n({e^{\frac{1}{pn}2\pi i}})\cup C_n({e^{\frac{2}{pn}2\pi i}})\cup\cdots\cup C_n({e^{\frac{p-2}{2pn}2\pi i}}),$$ $$B_{pn}=C_n({e^{\frac{1}{2pn}2\pi i}})\cup C_n({e^{\frac{3}{2pn}2\pi i}})\cup\cdots\cup C_n({e^{\frac{p-1}{2pn}2\pi i}}),$$ $$C_{pn}(z)=C_n(z)\cup C_n({e^{\frac{1}{pn}2\pi i}}z)\cup C_n({e^{\frac{2}{pn}2\pi i}}z)\cup\cdots\cup C_n({e^{\frac{p-1}{pn}2\pi i}}z),$$ $$\phi^{-1}(V)=A_2,$$ $$\phi^{-1}(A_{2p})=V\cup B_2\cup C_2(i\tan(\frac{1}{4p}2\pi))\cup C_2(i\tan(\frac{2}{4p}2\pi))\cup\cdots\cup C_2(i\tan(\frac{\frac{p-2}{2}}{4p}2\pi)),$$ $$\phi^{-1}(B_{2p})=C_2(i\tan(\frac{1}{8p}2\pi))\cup C_2(i\tan(\frac{3}{8p}2\pi))\cup\cdots\cup C_2(i\tan(\frac{p-1}{8p}2\pi)),$$ $$\phi^{-1}(C_{2p}(z))=C_2(\phi^{-1}(z))\cup C_2(\phi^{-1}(e^{\frac{1}{2p}2\pi i}z))\cup \cdots\cup C_2(\phi^{-1}(e^{\frac{p-1}{2p}2\pi i}z))$$ for $z\in\mathbb C^*\backslash\{e^{\frac{l}{4p}2\pi i}|l\in\mathbb{Z}\}$ if $p$ is even. We also have two corollaries \[D\_n1\] For $n\ge5$, set $\alpha=\{e^{\frac{k}{n}2\pi i}|k\in\mathbb Z\}$, we have $$\mathcal A_\alpha=\langle z\mapsto e^{\frac{1}{n}2\pi i}z,\:z\mapsto \frac{1}{z} \rangle\simeq D_n.$$ \[D\_n2\] For $n\ge5$, set $\alpha=\{0,\:\infty\}\cup\{e^{\frac{k}{n}2\pi i}|k\in\mathbb Z\}$, we have $$\mathcal A_\alpha=\langle z\mapsto e^{\frac{1}{n}2\pi i}z,\:z\mapsto \frac{1}{z} \rangle\simeq D_{n}.$$ $G$ is the Cyclic Group $\mathbb{Z}_n$ {#Z_n} -------------------------------------- Things are the same as in the section above. We still work in $\widehat{\mathbb C}$. We may assume that $$G=\langle z \mapsto e^{\frac{2\pi i}{n}}z\rangle\simeq\mathbb{Z}_n,\:n\ge 2.$$ There are three kinds of orbits. Let $N$ be the orbit $\{\infty\}$, $S$ the orbit $\{0\}$ and $C_n(z)$ the orbit $\{e^{\frac{k}{n}2\pi i}z|k\in\mathbb{Z}\}$ for $z\in\mathbb C^*$. If $\mathcal A_\alpha=G$, $\alpha$ is a finite union of the above orbits. If $\alpha$ is a finite union of the above orbits, we have $G\subseteq\mathcal{A}_{\alpha}$. So we conclude that $\mathcal{A}_{\alpha}$ is isomorphic to $A_5,\:S_4,\:A_4,\:D_{pn}$ or $\mathbb{Z}_{pn}$ for $p\in{\mathbb Z}^+$. The $A_5,\:S_4,\:A_4,\:D_{pn}$ case have already been explored. Now we shall discuss the case that $\mathcal{A}_{\alpha}$ is isomorphic to ${\mathbb Z}_{pn}$, $p\in\mathbb Z,\:p\ge2$. We have $$C_{pn}(z)=C_n(z)\cup C_n(e^{\frac{1}{pn}2\pi i}z)\cup C_n(e^{\frac{2}{pn}2\pi i}z)\cup\cdots\cup C_n(e^{\frac{p-1}{pn}2\pi i}z)$$ for $z\in\mathbb C^*$. In conclusion, we have \[tZ\_n\] For any finite subset $\alpha$ of $S^2$, $|\alpha|\ge4$, $\mathcal{A}_{\alpha}=G\simeq \mathbb Z_n, n\ge2$ if and only if all of the three claims are true: 1. $\alpha$ is a union of certain elements in $$\{N,\:S\}\cup\{C_n(z)|z\in{\mathbb C}^*\};$$ 2. $\alpha$ is NOT a union of certain elements in $$\{N,\:S\}\cup\{C_{pn}(z)|z\in{\mathbb C}^*\},\:p\ge2;$$ 3. $\alpha$ is NOT in the icosahedral, the octahedral the tetrahedral or the dihedral case, where $$C_{pn}(z)=C_n(z)\cup C_n(e^{\frac{1}{pn}2\pi i}z)\cup C_n(e^{\frac{2}{pn}2\pi i}z)\cup\cdots\cup C_n(e^{\frac{p-1}{pn}2\pi i}z)$$ for $z\in\mathbb C^*$. We also have one corollary \[cZ\_n\] For $n\ge4$, set $\alpha=\{0\}\cup\{e^{\frac{k}{n}2\pi i}|k\in\mathbb Z\}$, we have $$\mathcal A_\alpha=\langle z\mapsto e^{\frac{1}{n}2\pi i}z \rangle\simeq \mathbb Z_n.$$ Representations of the Stabilizers of Singularities {#rep} =================================================== Definition and Some Prior Work ------------------------------ For any $\boldsymbol {\lambda}\in K_n$ such that $\overline{[\boldsymbol {\lambda}]}$ is a singularity of $\mathfrak{M_{0,\:n}}$, each $g_\sigma \in G_{\boldsymbol {\lambda}}$ introduces a tangential mapping $g_\sigma^*$ of $\text{T}_{\boldsymbol {\lambda}}K_n$ such that $$g^*_\sigma(\frac{\partial}{\partial\lambda^i}\bigg|_{\boldsymbol{\lambda}})(\lambda^j)=\frac{\partial}{\partial\lambda^i}\bigg|_{\boldsymbol{\lambda}}(\lambda^j\circ g_\sigma)=\frac{\partial}{\partial\lambda^i}\bigg|_{\boldsymbol{\lambda}}(g^{(j)}_\sigma)=\frac{\partial g^{(j)}_\sigma}{\partial\lambda^i}\bigg|_{\boldsymbol{\lambda}}$$ for $\:i,\:j=1,\:2,\:\cdots,\:n-3.$ Let $\boldsymbol{J_\sigma}$ denote the Jacobian matrix of $g_\sigma$. Define a representation of $G_{\boldsymbol{\lambda}}$ $$X_{\boldsymbol{\lambda}}:\:G_{\boldsymbol{\lambda}}\to\text{GL}_{n-3},\:g_\sigma\mapsto \boldsymbol{J_\sigma}\bigg|_{\boldsymbol{\lambda}}.$$ It is obvious that $X_{\boldsymbol{\lambda}}$ is an representation of $G_{\boldsymbol {\lambda}}$ of degree $n-3$. In the rest of this section by calculating its character $\chi_{\boldsymbol{\lambda}}$, we shall explore the representation $X_{\boldsymbol{\lambda}}$ for each $\boldsymbol {\lambda}\in K_n$ such that $\overline{[\boldsymbol {\lambda}]}$ is a singularity of $\mathfrak{M_{0,\:n}}$. To make the calculation easier, we shall introduce two lemmas first. \[tec\] For any $\sigma\in S_n$, let $\boldsymbol{J_\sigma}$ denote the Jacobian matrix of $g_\sigma$. Then $$\mathbf{tr}\boldsymbol{J_\sigma}=\sum_{\substack{\sigma(k+3)\\=1,\:2,\:3\:\mathrm{or}\:k+3}}\frac{\partial g^{(k)}_\sigma(\boldsymbol {\lambda})}{\partial \lambda^{k}}.$$ Set $$\Psi_\sigma:\:\:K_n\times\widehat{\mathbb C}\to\widehat{\mathbb C},\:\:(\boldsymbol {\lambda},\:z)\mapsto f^{\boldsymbol {\lambda}}_\sigma(z).$$ By definition $f^{\boldsymbol {\lambda}}_\sigma$ is the linear fractional transformation such that $$f^{\boldsymbol {\lambda}}_\sigma (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(1)})=0,\:f^{\boldsymbol {\lambda}}_\sigma (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(2)})=1,\:f^{\boldsymbol {\lambda}}_\sigma (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(3)})=\infty.$$ Thus $\Psi_\sigma(\boldsymbol {\lambda},\:z)$ is a fraction of $z$ and $\lambda^{j}$, where $j$ satisfies the condition $\sigma(j+3)=1,\:2$ or $3$. So we have $$\frac{\partial \Psi_\sigma(\boldsymbol {\lambda},\:z)}{\partial \lambda^{l}}=0$$ if $\sigma(l+3)\ne1,\:2$ or $3$. As $$g^{(k)}_\sigma(\boldsymbol {\lambda}) =f^{\boldsymbol {\lambda}}_\sigma (z^{\boldsymbol{\lambda}}_{\sigma^{-1}(k+3)}) =\Psi_\sigma(\boldsymbol {\lambda},\:z^{\boldsymbol{\lambda}}_{\sigma^{-1}(k+3)})$$ we have $$\frac{\partial g^{(k)}_\sigma(\boldsymbol {\lambda})}{\partial \lambda^{l}}=0$$ if $\sigma(l+3)\ne1,\:2,\:3$ or $k+3$. Thus $$\mathbf{tr}\boldsymbol{J_\sigma} =\sum_{k=1}^{n-3}\frac{\partial g^{(k)}_\sigma(\boldsymbol {\lambda})}{\partial \lambda^{k}} =\sum_{\substack{\sigma(k+3)\\=1,\:2,\:3\:\mathrm{or}\:k+3}}\frac{\partial g^{(k)}_\sigma(\boldsymbol {\lambda})}{\partial \lambda^{k}}.$$ \[relation\] If $\boldsymbol {\lambda},\:\boldsymbol {\mu}\in K_n$, and $[\boldsymbol {\lambda}]$, $[\boldsymbol {\mu}]$ are equivalent, then for any linear fractional transformation $\varphi$ such that $$\varphi ([\boldsymbol {\lambda}])=[\boldsymbol {\mu}],$$ there exists a unique element $\pi\in S_n$ satisfying $$\varphi=f^{\boldsymbol {\lambda}}_\pi,\: \boldsymbol {\mu}=g_\pi(\boldsymbol{\lambda}),\: G_{\boldsymbol{\mu}}=g_\pi G_{\boldsymbol{\lambda}}g_{\pi^{-1}}$$ and $$\Phi_{\boldsymbol\mu}(g_{\pi\cdot\sigma\cdot\pi^{-1}}) =f^{\boldsymbol {\lambda}}_\pi\circ\Phi_{\boldsymbol\lambda}(g_\sigma)\circ({f^{\boldsymbol {\lambda}}_\pi})^{-1}, \:\:\chi_{\boldsymbol{\mu}}(g_{\pi\cdot\sigma\cdot\pi^{-1}})=\chi_{\boldsymbol{\lambda}}(g_{\sigma})$$ for each $g_\sigma\in G_{\boldsymbol{\lambda}}$, with $\Phi_{\boldsymbol\lambda}$, $\Phi_{\boldsymbol\mu}$ defined in \[def\] and $\chi_{\boldsymbol{\lambda}}$, $\chi_{\boldsymbol{\mu}}$ denoting the characters for $X_{\boldsymbol{\lambda}}$ and $X_{\boldsymbol{\mu}}$ respectively. For any linear fractional transformation $\varphi$ such that $$\varphi ([\boldsymbol {\lambda}])=[\boldsymbol {\mu}],$$ there exists a unique element $\pi\in S_n$ such that $$\varphi(z^{\boldsymbol{\lambda}}_{\pi^{-1}(k)})=z^{\boldsymbol{\mu}}_k,\:k=1,\:2,\:\cdots,\:n.$$ Thus we have $$\varphi=f^{\boldsymbol {\lambda}}_\pi,\:\boldsymbol {\mu}=g_\pi(\boldsymbol{\lambda}).$$ For any $g_\sigma\in G_{\boldsymbol{\lambda}}$, $$f^{\boldsymbol {\lambda}}_\pi\circ f^{\boldsymbol {\lambda}}_\sigma\circ (f^{\boldsymbol {\lambda}}_\pi)^{-1}(z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}\cdot\pi^{-1}(k)}) =f^{\boldsymbol {\lambda}}_\pi\circ f^{\boldsymbol {\lambda}}_\sigma(z^{\boldsymbol{\lambda}}_{\sigma^{-1}\cdot\pi^{-1}(k)}) =f^{\boldsymbol {\lambda}}_\pi(z^{\boldsymbol{\lambda}}_{\pi^{-1}(k)}) =z^{\boldsymbol{\mu}}_k$$ for $k=1,\:2,\:\cdots,\:n$. Thus $$f^{\boldsymbol {\lambda}}_\pi\circ f^{\boldsymbol {\lambda}}_\sigma\circ(f^{\boldsymbol {\lambda}}_\pi)^{-1}=f^{\boldsymbol {\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}},\: g_{\pi\cdot\sigma\cdot\pi^{-1}}\in G_{\boldsymbol{\mu}}.$$ We conclude that $$g_\pi G_{\boldsymbol{\lambda}}g_{\pi^{-1}}=G_{\boldsymbol{\mu}}$$ and $$\Phi_{\boldsymbol\mu}(g_{\pi\cdot\sigma\cdot\pi^{-1}}) =f^{\boldsymbol {\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}} =f^{\boldsymbol {\lambda}}_\pi\circ f^{\boldsymbol {\lambda}}_\sigma\circ(f^{\boldsymbol {\lambda}}_\pi)^{-1} =f^{\boldsymbol {\lambda}}_\pi\circ \Phi_{\boldsymbol\lambda}(g_\sigma)\circ(f^{\boldsymbol {\lambda}}_\pi)^{-1}.$$ Notice that $$X_{\boldsymbol{\mu}}(g_{\pi\cdot\sigma\cdot\pi^{-1}}) =\boldsymbol{J_{\pi\cdot\sigma\cdot\pi^{-1}}}\bigg|_{\boldsymbol{\mu}} =\boldsymbol{J_\pi}\bigg|_{g_\sigma(g_{\pi^{-1}}({\boldsymbol{\mu}}))} \boldsymbol{J_\sigma}\bigg|_{g_{\pi^{-1}}({\boldsymbol{\mu}})} \boldsymbol{J_{\pi^{-1}}}\bigg|_{\boldsymbol{\mu}} =\boldsymbol{J_\pi}\bigg|_{\boldsymbol{\lambda}} \boldsymbol{J_\sigma}\bigg|_{\boldsymbol{\lambda}} \boldsymbol{J_{\pi^{-1}}}\bigg|_{\boldsymbol{\mu}}.$$ As $g_\pi\circ g_{\pi^{-1}}=g_\varepsilon$, we have $$\mathrm{I}_{n-3} =X_{\boldsymbol{\mu}}(g_\varepsilon) =\boldsymbol{J_{\varepsilon}}\bigg|_{\boldsymbol{\mu}} %=\boldsymbol{J_{\pi\cdot\pi^{-1}}}\bigg|_{\boldsymbol{\mu}} =\boldsymbol{J_\pi}\bigg|_{g_{\pi^{-1}}({\boldsymbol{\mu}})} \boldsymbol{J_{\pi^{-1}}}\bigg|_{\boldsymbol{\mu}} =\boldsymbol{J_\pi}\bigg|_{\boldsymbol{\lambda}} \boldsymbol{J_{\pi^{-1}}}\bigg|_{\boldsymbol{\mu}}.$$ Thus $$X_{\boldsymbol{\mu}}(g_{\pi\cdot\sigma\cdot\pi^{-1}}) =\boldsymbol{J_\pi}\bigg|_{\boldsymbol{\lambda}} X_{\boldsymbol{\lambda}}(g_\sigma)(\boldsymbol{J_{\pi}}\bigg|_{\boldsymbol{\lambda}})^{-1}$$ and $$\chi_{\boldsymbol{\mu}}(g_{\pi\cdot\sigma\cdot\pi^{-1}})=\chi_{\boldsymbol{\lambda}}(g_{\sigma}).$$ Calculation of the Characters {#cha} ----------------------------- Before the calculation, notice that For each $g_\sigma\in G_{\boldsymbol{\lambda}}$, $$f^{\boldsymbol{\lambda}}_\sigma(z^{\boldsymbol{\lambda}}_k)=z^{\boldsymbol{\lambda}}_{\sigma(k)}$$ for $k=1,\:2,\:\cdots,\:n$. So if $\sigma$ is of order $m$, $f^{\boldsymbol{\lambda}}_\sigma$ is of order $m$, too. Also note that $\sigma$ has at most two fixed points if $g_\sigma$ is not the identity element. ### Characters of Elements with Two Fixed Points For any $\boldsymbol{\lambda_0}\in K_n$ and $g_\sigma\in G_{\boldsymbol{\lambda_0}}$ ($g_\sigma$ is not the identity element), if there are two fixed points of $\sigma$, then $\chi_{\boldsymbol{\lambda_0}}(g_{\sigma})=-1$. Suppose that $$\sigma(a)=a,\:\sigma(d)=b,\:\sigma(c)=c$$ where $a,\:b,\:c$ and $d$ are four different integers. Let $\varphi$ be the linear fractional transformation such that $$\varphi(z^{\boldsymbol{\lambda_0}}_a)=0,\:\varphi(z^{\boldsymbol{\lambda_0}}_b)=1,\:\varphi(z^{\boldsymbol{\lambda_0}}_c)=\infty$$ and $\boldsymbol{\mu_0}$ an element of $K_n$ such that $$[\boldsymbol{\mu_0}]=\varphi([\boldsymbol{\lambda_0}]).$$ From Lemma \[relation\] we know that there exists a unique element $\pi\in S_n$ such that $\varphi=f^{\boldsymbol {\lambda_0}}_\pi$ and $$G_{\boldsymbol{\mu_0}}=g_\pi G_{\boldsymbol{\lambda_0}}g_{\pi^{-1}}, \:\:\chi_{\boldsymbol{\mu_0}}(g_{\pi\cdot\sigma\cdot\pi^{-1}})=\chi_{\boldsymbol{\lambda_0}}(g_{\sigma}).$$ Note that $$\pi(a)=1,\:\pi(b)=2,\:\pi(c)=3,\:\pi(d)>3.$$ Thus $$\pi\cdot\sigma^{-1}\cdot\pi^{-1}(1)=1,\: \pi\cdot\sigma^{-1}\cdot\pi^{-1}(2)=\pi(d),\: \pi\cdot\sigma^{-1}\cdot\pi^{-1}(3)=3$$ and $$\{k\in\mathbb{N}^*\big|\pi\cdot\sigma\cdot\pi^{-1}(k+3)=1,\:2,\:3\:\:\mathrm{or}\:\:k+3\}=\{\pi(d)-3\}.$$ For any $\boldsymbol{\mu}\in K_n$, by the definition of $f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}$ we have $$0=f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}\cdot\pi^{-1}(1)}) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_1) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(0),$$ $$1=f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}\cdot\pi^{-1}(2)}) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi(d)}) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(\mu^{\pi(d)-3}),$$ $$\infty=f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}\cdot\pi^{-1}(3)}) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_3) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(\infty).$$ Thus $$f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z) =\frac{z}{\mu^{\pi(d)-3}}.$$ Particularly $$f^{\boldsymbol{\mu_0}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z) =\frac{z}{\mu_0^{\pi(d)-3}}.$$ Suppose $\sigma$ is of order $m$ ($m\ge2$). As $g_{\pi\cdot\sigma\cdot\pi^{-1}}\in G_{\boldsymbol{\mu_0}}$, $f^{\boldsymbol{\mu_0}}_{\pi\cdot\sigma\cdot\pi^{-1}}$ is of order $m$, too. Thus $\mu_0^{\pi(d)-3}$ is the primitive $m$th root of unity. As $$\mu_0^{\pi(d)-3} =z^{\boldsymbol{\mu_0}}_{\pi(d)} =f^{\boldsymbol{\mu_0}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu_0}}_{\pi\cdot\sigma^{-1}(d)}) =\frac{z^{\boldsymbol{\mu_0}}_{\pi\cdot\sigma^{-1}(d)}}{\mu_0^{\pi(d)-3}}$$ we have $$z^{\boldsymbol{\mu_0}}_{\pi\cdot\sigma^{-1}(d)}=(\mu_0^{\pi(d)-3})^2.$$ $$\begin{aligned} \chi_{\boldsymbol{\lambda_0}}(g_\sigma) =&\chi_{\boldsymbol{\mu_0}}(g_{\pi\cdot\sigma\cdot\pi^{-1}}) =\mathbf{tr}\boldsymbol{J_{\pi\cdot\sigma\cdot\pi^{-1}}}\bigg|_{\boldsymbol{\mu_0}}\\ =&\sum_{\substack{\pi\cdot\sigma\cdot\pi^{-1}(k+3)\\=1,\:2,\:3\:\mathrm{or}\:k+3}} \frac{\partial g^{(k)}_{\pi\cdot\sigma\cdot\pi^{-1}}({\boldsymbol{\mu}})}{\partial \mu^{k}} \bigg|_{\boldsymbol{\mu_0}} =\frac{\partial g^{(\pi(d)-3)}_{\pi\cdot\sigma\cdot\pi^{-1}}({\boldsymbol{\mu}})} {\partial \mu^{\pi(d)-3}}\bigg|_{\boldsymbol{\mu_0}}\\ =&\frac{\partial f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}\cdot\pi^{-1}(\pi(d))})} {\partial \mu^{\pi(d)-3}}\bigg|_{\boldsymbol{\mu_0}} =\frac{\partial f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}(d)})} {\partial \mu^{\pi(d)-3}}\bigg|_{\boldsymbol{\mu_0}}\\ =&\bigg(\frac{\partial}{\partial \mu^{\pi(d)-3}} \bigg(\frac{z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}(d)}}{\mu^{\pi(d)-3}}\bigg)\bigg)\bigg|_{\boldsymbol{\mu_0}}.\end{aligned}$$ As $\pi\cdot\sigma^{-1}(d)-3\ne\pi(d)-3$, we know that $$\frac{\partial z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}(d)}}{\partial \mu^{\pi(d)-3}}=0.$$ Thus $$\chi_{\boldsymbol{\lambda_0}}(g_\sigma) =\bigg(\frac{\partial}{\partial \mu^{\pi(d)-3}} \big(\frac{z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}(d)}}{\mu^{\pi(d)-3}}\big)\bigg)\bigg|_{\boldsymbol{\mu_0}} =\frac{-z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}(d)}}{(\mu^{\pi(d)-3})^2}\bigg|_{\boldsymbol{\mu_0}} =\frac{-z^{\boldsymbol{\mu_0}}_{\pi\cdot\sigma^{-1}(d)}}{(\mu_0^{\pi(d)-3})^2} =-1.$$ ### Characters of Elements with One Fixed Point For any $\boldsymbol{\lambda_0}\in K_n$ and $g_\sigma\in G_{\boldsymbol{\lambda_0}}$, if $\sigma$ has only one fixed point and is of order two, then $\chi_{\boldsymbol{\lambda_0}}(g_{\sigma})=0$. Suppose that $$\sigma(a)=a,\:\sigma(b)=c,\:\sigma(c)=b$$ where $a,\:b$ and $c$ are three different integers. Let $\varphi$ be the linear fractional transformation such that $$\varphi(z^{\boldsymbol{\lambda_0}}_a)=0,\:\varphi(z^{\boldsymbol{\lambda_0}}_b)=1,\:\varphi(z^{\boldsymbol{\lambda_0}}_c)=\infty$$ and $\boldsymbol{\mu_0}$ an element of $K_n$ such that $$[\boldsymbol{\mu_0}]=\varphi([\boldsymbol{\lambda_0}]).$$ From Lemma \[relation\] we know that there exists a unique element $\pi\in S_n$ such that $\varphi=f^{\boldsymbol {\lambda_0}}_\pi$ and $$\chi_{\boldsymbol{\mu_0}}(g_{\pi\cdot\sigma\cdot\pi^{-1}})=\chi_{\boldsymbol{\lambda_0}}(g_{\sigma}).$$ Note that $$\pi(a)=1,\:\pi(b)=2,\:\pi(c)=3$$ and $$\pi\cdot\sigma^{-1}\cdot\pi^{-1}(1)=1,\: \pi\cdot\sigma^{-1}\cdot\pi^{-1}(2)=3,\: \pi\cdot\sigma^{-1}\cdot\pi^{-1}(3)=2.$$ Thus $$\{k\in\mathbb{N}^*\big|\pi\cdot\sigma\cdot\pi^{-1}(k+3)=1,\:2,\:3\:\:\mathrm{or}\:\:k+3\}=\emptyset$$ and $$\chi_{\boldsymbol{\lambda_0}}(g_\sigma) =\chi_{\boldsymbol{\mu_0}}(g_{\pi\cdot\sigma\cdot\pi^{-1}}) =\mathbf{tr}\boldsymbol{J_{\pi\cdot\sigma\cdot\pi^{-1}}}\bigg|_{\boldsymbol{\mu_0}} =\sum_{\substack{\pi\cdot\sigma\cdot\pi^{-1}(k+3)\\=1,\:2,\:3\:\mathrm{or}\:k+3}} \frac{\partial g^{(k)}_{\pi\cdot\sigma\cdot\pi^{-1}}({\boldsymbol{\mu}})}{\partial \mu^{k}} \bigg|_{\boldsymbol{\mu_0}} =0.$$ For any $\boldsymbol{\lambda_0}\in K_n$ and $g_\sigma\in G_{\boldsymbol{\lambda_0}}$, if $\sigma$ has only one fixed point $a_1$, and there exists some linear fractional transformation $\psi$ such that $$\psi\circ f^{\boldsymbol{\lambda_0}}_\sigma\circ\psi^{-1}(z) =e^{\frac{2q\pi}{p}i}z$$ where $p,\:q$ are co-prime positive integers, and $$\psi(z^{\boldsymbol{\lambda_0}}_{a_1})=0,$$ then $$\chi_{\boldsymbol{\lambda_0}}(g_{\sigma})=-1-e^{\frac{-2q\pi}{p}i}.$$ Notice that $\sigma$ is of order $p$ as $f^{\boldsymbol{\lambda_0}}_\sigma$ is of order $p$. Thus from the previous lemma it is obvious that the result holds when $p=2$. Next assume that $p\ge3$. Suppose that $$\sigma(a_2)=a_3,\:\sigma(a_3)=a_4,\:\cdots,\:\sigma(a_{p+1})=a_2$$ where $a_1,\:a_2,\:\cdots,\:a_{p+1}$ are $p+1$ different integers. Let $\varphi$ be the linear fractional transformation such that $$\varphi\circ\psi(z^{\boldsymbol{\lambda_0}}_{a_1})=0,\:\varphi\circ\psi(z^{\boldsymbol{\lambda_0}}_{a_2})=1,\:\varphi\circ\psi(z^{\boldsymbol{\lambda_0}}_{a_3})=\infty$$ and $\boldsymbol{\mu_0}$ an element of $K_n$ such that $$[\boldsymbol{\mu_0}]=\varphi\circ\psi([\boldsymbol{\lambda_0}]).$$ From Lemma \[relation\] we know that there exists a unique element $\pi\in S_n$ such that $\varphi\circ\psi=f^{\boldsymbol {\lambda_0}}_\pi$ and $$\boldsymbol {\mu_0}=g_\pi(\boldsymbol{\lambda_0}), \:\:G_{\boldsymbol{\mu_0}}=g_\pi G_{\boldsymbol{\lambda_0}}g_{\pi^{-1}}, \:\:\chi_{\boldsymbol{\mu_0}}(g_{\pi\cdot\sigma\cdot\pi^{-1}})=\chi_{\boldsymbol{\lambda_0}}(g_{\sigma}).$$ Note that $$\pi(a_1)=1,\:\pi(a_2)=2,\:\pi(a_3)=3,\:\pi(a_{p+1})>3.$$ Thus $$\pi\cdot\sigma^{-1}\cdot\pi^{-1}(1)=1,\: \pi\cdot\sigma^{-1}\cdot\pi^{-1}(2)=\pi(a_{p+1}),\: \pi\cdot\sigma^{-1}\cdot\pi^{-1}(3)=2$$ and $$\{k\in\mathbb{N}^*\big|\pi\cdot\sigma\cdot\pi^{-1}(k+3)=1,\:2,\:3\:\:\mathrm{or}\:\:k+3\}=\{\pi(a_{p+1})-3\}.$$ For any $\boldsymbol{\mu}\in K_n$, by the definition of $f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}$ we have $$0=f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}\cdot\pi^{-1}(1)}) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_1) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(0),$$ $$1=f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}\cdot\pi^{-1}(2)}) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi(a_{p+1})}) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(\mu^{\pi(a_{p+1})-3}),$$ $$\infty=f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}\cdot\pi^{-1}(3)}) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_2) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(1).$$ Thus $$f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z) =\frac{(\mu^{\pi(a_{p+1})-3}-1)z}{\mu^{\pi(a_{p+1})-3}(z-1)}.$$ Now we have $$\begin{aligned} \chi_{\boldsymbol{\lambda_0}}(g_\sigma) =&\chi_{\boldsymbol{\mu_0}}(g_{\pi\cdot\sigma\cdot\pi^{-1}}) =\mathbf{tr}\boldsymbol{J_{\pi\cdot\sigma\cdot\pi^{-1}}}\bigg|_{\boldsymbol{\mu_0}}\\ =&\sum_{\substack{\pi\cdot\sigma\cdot\pi^{-1}(k+3)\\=1,\:2,\:3\:\mathrm{or}\:k+3}} \frac{\partial g^{(k)}_{\pi\cdot\sigma\cdot\pi^{-1}}({\boldsymbol{\mu}})}{\partial \mu^{k}} \bigg|_{\boldsymbol{\mu_0}} =\frac{\partial g^{(\pi(a_{p+1})-3)}_{\pi\cdot\sigma\cdot\pi^{-1}}({\boldsymbol{\mu}})} {\partial \mu^{\pi(a_{p+1})-3}}\bigg|_{\boldsymbol{\mu_0}}\\ =&\frac{\partial f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}\cdot\pi^{-1}(\pi(a_{p+1}))})} {\partial \mu^{\pi(a_{p+1})-3}}\bigg|_{\boldsymbol{\mu_0}} =\frac{\partial f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi(a_p)})} {\partial \mu^{\pi(a_{p+1})-3}}\bigg|_{\boldsymbol{\mu_0}}.\end{aligned}$$ Set $w=e^{\frac{2\pi}{p}i}$. Thus $\psi\circ f^{\boldsymbol{\lambda_0}}_\sigma\circ\psi^{-1}(z)=w^qz$. 1. If $p=3$, we have $$\psi(z^{\boldsymbol{\lambda_0}}_{a_1})=0,$$ $$\psi(z^{\boldsymbol{\lambda_0}}_{a_3}) =\psi\circ f^{\boldsymbol{\lambda_0}}_\sigma\circ\psi^{-1} (\psi(z^{\boldsymbol{\lambda_0}}_{a_2})) =w^q\psi(z^{\boldsymbol{\lambda_0}}_{a_2}),$$ $$\psi(z^{\boldsymbol{\lambda_0}}_{a_4}) =\psi\circ f^{\boldsymbol{\lambda_0}}_\sigma\circ\psi^{-1} (\psi(z^{\boldsymbol{\lambda_0}}_{a_3})) =w^{2q}\psi(z^{\boldsymbol{\lambda_0}}_{a_2}).$$ Thus $$\begin{aligned} \mu_0^{\pi(a_4)-3} =&[0,\:1,\:\mu_0^{\pi(a_4)-3},\:\infty]\\ =&[\varphi\circ\psi(z^{\boldsymbol{\lambda_0}}_{a_1}),\:\varphi\circ\psi(z^{\boldsymbol{\lambda_0}}_{a_2}),\:\varphi\circ\psi(z^{\boldsymbol{\lambda_0}}_{a_4}),\:\varphi\circ\psi(z^{\boldsymbol{\lambda_0}}_{a_3})]\\ =&[\psi(z^{\boldsymbol{\lambda_0}}_{a_1}),\:\psi(z^{\boldsymbol{\lambda_0}}_{a_2}),\:\psi(z^{\boldsymbol{\lambda_0}}_{a_4}),\:\psi(z^{\boldsymbol{\lambda_0}}_{a_3})]\\ =&[0,\:1,\:w^{2q},\:w^{q}]\\ =&-w^q.\end{aligned}$$ $$\begin{aligned} \chi_{\boldsymbol{\lambda_0}}(g_\sigma) =&\frac{\partial f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi(a_3)})} {\partial \mu^{\pi(a_4)-3}}\bigg|_{\boldsymbol{\mu_0}} =\frac{\partial f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_3)} {\partial \mu^{\pi(a_4)-3}}\bigg|_{\boldsymbol{\mu_0}}\\ =&\bigg(\frac{\partial}{\partial \mu^{\pi(a_4)-3}}\bigg( \frac{\mu^{\pi(a_4)-3}-1}{\mu_0^{\pi(a_4)-3}} \bigg)\bigg)\bigg|_{\boldsymbol{\mu_0}} =\frac{1}{(\mu^{\pi(a_4)-3})^2}\bigg|_{\boldsymbol{\mu_0}} =w^{-2q}.\end{aligned}$$ As $p=3$, we have $$w^{-2q}=-1-w^{-q}$$ for $q=1$ and $2$. 2. If $p\ge4$, we have $$\psi(z^{\boldsymbol{\lambda_0}}_{a_1})=0,$$ $$\psi(z^{\boldsymbol{\lambda_0}}_{a_{p+1}}) =\psi\circ f^{\boldsymbol{\lambda_0}}_\sigma\circ\psi^{-1} (\psi(z^{\boldsymbol{\lambda_0}}_{a_p})) =w^q\psi(z^{\boldsymbol{\lambda_0}}_{a_p}),$$ $$\psi(z^{\boldsymbol{\lambda_0}}_{a_2}) =\psi\circ f^{\boldsymbol{\lambda_0}}_\sigma\circ\psi^{-1} (\psi(z^{\boldsymbol{\lambda_0}}_{a_{p+1}})) =w^{2q}\psi(z^{\boldsymbol{\lambda_0}}_{a_p}),$$ $$\psi(z^{\boldsymbol{\lambda_0}}_{a_3}) =\psi\circ f^{\boldsymbol{\lambda_0}}_\sigma\circ\psi^{-1} (\psi(z^{\boldsymbol{\lambda_0}}_{a_2})) =w^{3q}\psi(z^{\boldsymbol{\lambda_0}}_{a_p}).$$ Thus $$\begin{aligned} \mu_0^{\pi(a_{p+1})-3} =&[0,\:1,\:\mu_0^{\pi(a_{p+1})-3},\:\infty]\\ =&[\varphi\circ\psi(z^{\boldsymbol{\lambda_0}}_{a_1}),\:\varphi\circ\psi(z^{\boldsymbol{\lambda_0}}_{a_2}),\:\varphi\circ\psi(z^{\boldsymbol{\lambda_0}}_{a_{p+1}}),\:\varphi\circ\psi(z^{\boldsymbol{\lambda_0}}_{a_3})]\\ =&[\psi(z^{\boldsymbol{\lambda_0}}_{a_1}),\:\psi(z^{\boldsymbol{\lambda_0}}_{a_2}),\:\psi(z^{\boldsymbol{\lambda_0}}_{a_{p+1}}),\:\psi(z^{\boldsymbol{\lambda_0}}_{a_3})]\\ =&[0,\:w^{2q},\:w^q,\:w^{3q}]\\ =&\frac{1}{1+w^q}\end{aligned}$$ and $$\begin{aligned} \mu_0^{\pi(a_p)-3} =&[0,\:1,\:\mu_0^{\pi(a_p)-3},\:\infty]\\ =&[\varphi\circ\psi(z^{\boldsymbol{\lambda_0}}_{a_1}),\:\varphi\circ\psi(z^{\boldsymbol{\lambda_0}}_{a_2}),\:\varphi\circ\psi(z^{\boldsymbol{\lambda_0}}_{a_p}),\:\varphi\circ\psi(z^{\boldsymbol{\lambda_0}}_{a_3})]\\ =&[\psi(z^{\boldsymbol{\lambda_0}}_{a_1}),\:\psi(z^{\boldsymbol{\lambda_0}}_{a_2}),\:\psi(z^{\boldsymbol{\lambda_0}}_{a_p}),\:\psi(z^{\boldsymbol{\lambda_0}}_{a_3})]\\ =&[0,\:w^{2q},\:1,\:w^{3q}]\\ =&\frac{1}{1+w^q+w^{2q}}.\end{aligned}$$ $$\begin{aligned} \chi_{\boldsymbol{\lambda_0}}(g_\sigma) =&\frac{\partial f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi(a_p)})} {\partial \mu^{\pi(a_{p+1})-3}}\bigg|_{\boldsymbol{\mu_0}} =\frac{\partial f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(\mu^{\pi(a_p)-3})} {\partial \mu^{\pi(a_{p+1})-3}}\bigg|_{\boldsymbol{\mu_0}}\\ =&\bigg(\frac{\partial}{\partial \mu^{\pi(a_{p+1})-3}}\bigg( \frac{(\mu^{\pi(a_{p+1})-3}-1)\mu^{\pi(a_p)-3}} {\mu^{\pi(a_{p+1})-3}(\mu^{\pi(a_p)-3}-1)} \bigg)\bigg)\bigg|_{\boldsymbol{\mu_0}}\\ =&\frac{\mu^{\pi(a_p)-3}} {(\mu^{\pi(a_{p+1})-3})^2(\mu^{\pi(a_p)-3}-1)} \bigg|_{\boldsymbol{\mu_0}} =-1-w^{-q}.\end{aligned}$$ ### Characters of Elements with No Fixed Point For any $\boldsymbol{\lambda_0}\in K_n$ and $g_\sigma\in G_{\boldsymbol{\lambda_0}}$, if $\sigma$ has no fixed point and is of order two, then $\chi_{\boldsymbol{\lambda_0}}(g_{\sigma})=1$. Suppose that $$\sigma(a)=c,\:\sigma(c)=a,\:\sigma(d)=b,\:\sigma(b)=d$$ where $a,\:b,\:c$ and $d$ are four different integers. Let $\varphi$ be the linear fractional transformation such that $$\varphi(z^{\boldsymbol{\lambda_0}}_a)=0,\:\varphi(z^{\boldsymbol{\lambda_0}}_b)=1,\:\varphi(z^{\boldsymbol{\lambda_0}}_c)=\infty$$ and $\boldsymbol{\mu_0}$ an element of $K_n$ such that $$[\boldsymbol{\mu_0}]=\varphi([\boldsymbol{\lambda_0}]).$$ From Lemma \[relation\] we know that there exists a unique element $\pi\in S_n$ such that $\varphi=f^{\boldsymbol {\lambda_0}}_\pi$ and $$G_{\boldsymbol{\mu_0}}=g_\pi G_{\boldsymbol{\lambda_0}}g_{\pi^{-1}}, \:\:\chi_{\boldsymbol{\mu_0}}(g_{\pi\cdot\sigma\cdot\pi^{-1}})=\chi_{\boldsymbol{\lambda_0}}(g_{\sigma}).$$ Note that $$\pi(a)=1,\:\pi(b)=2,\:\pi(c)=3,\:\pi(d)>3.$$ Thus $$\pi\cdot\sigma^{-1}\cdot\pi^{-1}(1)=3,\: \pi\cdot\sigma^{-1}\cdot\pi^{-1}(2)=\pi(d),\: \pi\cdot\sigma^{-1}\cdot\pi^{-1}(3)=1$$ and $$\{k\in\mathbb{N}^*\big|\pi\cdot\sigma\cdot\pi^{-1}(k+3)=1,\:2,\:3\:\:\mathrm{or}\:\:k+3\}=\{\pi(d)-3\}.$$ For any $\boldsymbol{\mu}\in K_n$, by the definition of $f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}$ we have $$0=f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}\cdot\pi^{-1}(1)}) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_3) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(\infty),$$ $$1=f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}\cdot\pi^{-1}(2)}) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi(d)}) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(\mu^{\pi(d)-3}),$$ $$\infty=f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}\cdot\pi^{-1}(3)}) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_1) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(0).$$ Thus $$f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z) =\frac{\mu^{\pi(d)-3}}{z}.$$ $$\begin{aligned} \chi_{\boldsymbol{\lambda_0}}(g_\sigma) =&\chi_{\boldsymbol{\mu_0}}(g_{\pi\cdot\sigma\cdot\pi^{-1}}) =\mathbf{tr}\boldsymbol{J_{\pi\cdot\sigma\cdot\pi^{-1}}}\bigg|_{\boldsymbol{\mu_0}}\\ =&\sum_{\substack{\pi\cdot\sigma\cdot\pi^{-1}(k+3)\\=1,\:2,\:3\:\mathrm{or}\:k+3}} \frac{\partial g^{(k)}_{\pi\cdot\sigma\cdot\pi^{-1}}({\boldsymbol{\mu}})}{\partial \mu^{k}} \bigg|_{\boldsymbol{\mu_0}} =\frac{\partial g^{(\pi(d)-3)}_{\pi\cdot\sigma\cdot\pi^{-1}}({\boldsymbol{\mu}})} {\partial \mu^{\pi(d)-3}}\bigg|_{\boldsymbol{\mu_0}}\\ =&\frac{\partial f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}\cdot\pi^{-1}(\pi(d))})} {\partial \mu^{\pi(d)-3}}\bigg|_{\boldsymbol{\mu_0}} =\frac{\partial f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_2)} {\partial \mu^{\pi(d)-3}}\bigg|_{\boldsymbol{\mu_0}}\\ =&\frac{\partial \mu^{\pi(d)-3}}{\partial \mu^{\pi(d)-3}}\bigg|_{\boldsymbol{\mu_0}} =1.\end{aligned}$$ For any $\boldsymbol{\lambda_0}\in K_n$ and $g_\sigma\in G_{\boldsymbol{\lambda_0}}$, if $\sigma$ has no fixed point and is of order three, then $\chi_{\boldsymbol{\lambda_0}}(g_{\sigma})=0$. Suppose that $$\sigma(a)=b,\:\sigma(b)=c,\:\sigma(c)=a$$ where $a,\:b$ and $c$ are three different integers. Let $\varphi$ be the linear fractional transformation such that $$\varphi(z^{\boldsymbol{\lambda_0}}_a)=0,\:\varphi(z^{\boldsymbol{\lambda_0}}_b)=1,\:\varphi(z^{\boldsymbol{\lambda_0}}_c)=\infty$$ and $\boldsymbol{\mu_0}$ an element of $K_n$ such that $$[\boldsymbol{\mu_0}]=\varphi([\boldsymbol{\lambda_0}]).$$ From Lemma \[relation\] we know that there exists a unique element $\pi\in S_n$ such that $\varphi=f^{\boldsymbol {\lambda_0}}_\pi$ and $$\chi_{\boldsymbol{\mu_0}}(g_{\pi\cdot\sigma\cdot\pi^{-1}})=\chi_{\boldsymbol{\lambda_0}}(g_{\sigma}).$$ Note that $$\pi(a)=1,\:\pi(b)=2,\:\pi(c)=3$$ and $$\pi\cdot\sigma^{-1}\cdot\pi^{-1}(1)=3,\: \pi\cdot\sigma^{-1}\cdot\pi^{-1}(2)=1,\: \pi\cdot\sigma^{-1}\cdot\pi^{-1}(3)=2.$$ Thus $$\{k\in\mathbb{N}^*\big|\pi\cdot\sigma\cdot\pi^{-1}(k+3)=1,\:2,\:3\:\:\mathrm{or}\:\:k+3\}=\emptyset$$ and $$\chi_{\boldsymbol{\lambda_0}}(g_\sigma) =\chi_{\boldsymbol{\mu_0}}(g_{\pi\cdot\sigma\cdot\pi^{-1}}) =\mathbf{tr}\boldsymbol{J_{\pi\cdot\sigma\cdot\pi^{-1}}}\bigg|_{\boldsymbol{\mu_0}} =\sum_{\substack{\pi\cdot\sigma\cdot\pi^{-1}(k+3)\\=1,\:2,\:3\:\mathrm{or}\:k+3}} \frac{\partial g^{(k)}_{\pi\cdot\sigma\cdot\pi^{-1}}({\boldsymbol{\mu}})}{\partial \mu^{k}} \bigg|_{\boldsymbol{\mu_0}} =0.$$ For any $\boldsymbol{\lambda_0}\in K_n$ and $g_\sigma\in G_{\boldsymbol{\lambda_0}}$, if $\sigma$ has no fixed point and $f^{\boldsymbol{\lambda_0}}_\sigma$ is conjugate to a rotation of $\frac{2q\pi}{p}$ ($p,\:q$ are co-prime positive integers), then $$\chi_{\boldsymbol{\lambda_0}}(g_{\sigma})=-1-2\cos{\frac{2q\pi}{p}}.$$ Notice that $\sigma$ is of order $p$ as $f^{\boldsymbol{\lambda_0}}_\sigma$ is of order $p$. Thus from the previous lemmas it is obvious that the result holds when $p=2$ and $3$. Next assume that $p\ge4$. Suppose that $$\sigma(a_1)=a_2,\:\sigma(a_2)=a_3,\:\cdots,\:\sigma(a_p)=a_1$$ where $a_1,\:a_2,\:\cdots,\:a_p$ are $p$ different integers. Let $\varphi$ be the linear fractional transformation such that $$\varphi(z^{\boldsymbol{\lambda_0}}_{a_1})=0,\:\varphi(z^{\boldsymbol{\lambda_0}}_{a_2})=1,\:\varphi(z^{\boldsymbol{\lambda_0}}_{a_3})=\infty$$ and $\boldsymbol{\mu_0}$ an element of $K_n$ such that $$[\boldsymbol{\mu_0}]=\varphi([\boldsymbol{\lambda_0}].$$ From Lemma \[relation\] we know that there exists a unique element $\pi\in S_n$ such that $\varphi=f^{\boldsymbol {\lambda_0}}_\pi$ and $$\boldsymbol {\mu_0}=g_\pi(\boldsymbol{\lambda_0}), \:\:G_{\boldsymbol{\mu_0}}=g_\pi G_{\boldsymbol{\lambda_0}}g_{\pi^{-1}}, \:\:\chi_{\boldsymbol{\mu_0}}(g_{\pi\cdot\sigma\cdot\pi^{-1}})=\chi_{\boldsymbol{\lambda_0}}(g_{\sigma}).$$ Note that $$\pi(a_1)=1,\:\pi(a_2)=2,\:\pi(a_3)=3,\:\pi(a_p)>3.$$ Thus $$\pi\cdot\sigma^{-1}\cdot\pi^{-1}(1)=\pi(a_p),\: \pi\cdot\sigma^{-1}\cdot\pi^{-1}(2)=1,\: \pi\cdot\sigma^{-1}\cdot\pi^{-1}(3)=2$$ and $$\{k\in\mathbb{N}^*\big|\pi\cdot\sigma\cdot\pi^{-1}(k+3)=1,\:2,\:3\:\:\mathrm{or}\:\:k+3\}=\{\pi(a_p)-3\}.$$ For any $\boldsymbol{\mu}\in K_n$, by the definition of $f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}$ we have $$0=f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}\cdot\pi^{-1}(1)}) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi(a_p)}) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(\mu^{\pi(a_p)-3}),$$ $$1=f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}\cdot\pi^{-1}(2)}) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_1) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(0),$$ $$\infty=f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}\cdot\pi^{-1}(3)}) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_2) =f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(1).$$ Thus $$f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z) =\frac{z-\mu^{\pi(a_p)-3}}{\mu^{\pi(a_p)-3}(z-1)}.$$ $$\begin{aligned} \chi_{\boldsymbol{\lambda_0}}(g_\sigma) =&\chi_{\boldsymbol{\mu_0}}(g_{\pi\cdot\sigma\cdot\pi^{-1}}) =\mathbf{tr}\boldsymbol{J_{\pi\cdot\sigma\cdot\pi^{-1}}}\bigg|_{\boldsymbol{\mu_0}}\\ =&\sum_{\substack{\pi\cdot\sigma\cdot\pi^{-1}(k+3)\\=1,\:2,\:3\:\mathrm{or}\:k+3}} \frac{\partial g^{(k)}_{\pi\cdot\sigma\cdot\pi^{-1}}({\boldsymbol{\mu}})}{\partial \mu^{k}} \bigg|_{\boldsymbol{\mu_0}} =\frac{\partial g^{(\pi(a_p)-3)}_{\pi\cdot\sigma\cdot\pi^{-1}}({\boldsymbol{\mu}})} {\partial \mu^{\pi(a_p)-3}}\bigg|_{\boldsymbol{\mu_0}}\\ =&\frac{\partial f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi\cdot\sigma^{-1}\cdot\pi^{-1}(\pi(a_p))})} {\partial \mu^{\pi(a_p)-3}}\bigg|_{\boldsymbol{\mu_0}} =\frac{\partial f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi(a_{p-1})})} {\partial \mu^{\pi(a_p)-3}}\bigg|_{\boldsymbol{\mu_0}}.\end{aligned}$$ Set $$w=e^{\frac{2\pi}{p}i}.$$ As $f^{\boldsymbol{\lambda_0}}_\sigma$ is conjugate to a rotation of $\frac{2q\pi}{p}$, there exists a linear fractional transformation $\psi$ such that $$\psi(z^{\boldsymbol{\lambda_0}}_{a_k})=w^{kq}$$ for $k=1,\:2,\:\cdots,\:p$. 1. If $p=4$, we have $$\begin{aligned} \mu_0^{\pi(a_4)-3} =&[0,\:1,\:\mu_0^{\pi(a_4)-3},\:\infty]\\ =&[\varphi(z^{\boldsymbol{\lambda_0}}_{a_1}),\:\varphi(z^{\boldsymbol{\lambda_0}}_{a_2}),\:\varphi(z^{\boldsymbol{\lambda_0}}_{a_4}),\:\varphi(z^{\boldsymbol{\lambda_0}}_{a_3})]\\ =&[\psi(z^{\boldsymbol{\lambda_0}}_{a_1}),\:\psi(z^{\boldsymbol{\lambda_0}}_{a_2}),\:\psi(z^{\boldsymbol{\lambda_0}}_{a_4}),\:\psi(z^{\boldsymbol{\lambda_0}}_{a_3})]\\ =&[1,\:w^q,\:w^{-q},\:w^{2q}]\\ =&\frac{-w^q}{1+w^q+w^{2q}}\\ =&-1.\end{aligned}$$ $$\begin{aligned} \chi_{\boldsymbol{\lambda_0}}(g_\sigma) =&\frac{\partial f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi(a_3)})} {\partial \mu^{\pi(a_4)-3}}\bigg|_{\boldsymbol{\mu_0}} =\frac{\partial f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_3)} {\partial \mu^{\pi(a_4)-3}}\bigg|_{\boldsymbol{\mu_0}}\\ =&\bigg(\frac{\partial}{\partial \mu^{\pi(a_4)-3}}\bigg( \frac{1}{\mu_0^{\pi(a_4)-3}} \bigg)\bigg)\bigg|_{\boldsymbol{\mu_0}} =\frac{-1}{(\mu_0^{\pi(a_4)-3})^2}\bigg|_{\boldsymbol{\mu_0}}=-1.\end{aligned}$$ As $p=4$, we have $$-1=-1-2\cos{\frac{2q\pi}{p}}$$ when $p,\:q$ are co-prime positive integers. 2. If $p\ge4$, we have $$\begin{aligned} \mu_0^{\pi(a_p)-3} =&[0,\:1,\:\mu_0^{\pi(a_p)-3},\:\infty]\\ =&[\varphi(z^{\boldsymbol{\lambda_0}}_{a_1}),\:\varphi(z^{\boldsymbol{\lambda_0}}_{a_2}),\:\varphi(z^{\boldsymbol{\lambda_0}}_{a_p}),\:\varphi(z^{\boldsymbol{\lambda_0}}_{a_3})]\\ =&[\psi(z^{\boldsymbol{\lambda_0}}_{a_1}),\:\psi(z^{\boldsymbol{\lambda_0}}_{a_2}),\:\psi(z^{\boldsymbol{\lambda_0}}_{a_p}),\:\psi(z^{\boldsymbol{\lambda_0}}_{a_3})]\\ =&[1,\:w^q,\:w^{-q},\:w^{2q}]\\ =&\frac{-w^q}{1+w^q+w^{2q}}.\end{aligned}$$ and $$\begin{aligned} \mu_0^{\pi(a_{p-1})-3} =&[0,\:1,\:\mu_0^{\pi(a_{p-1})-3},\:\infty]\\ =&[\varphi(z^{\boldsymbol{\lambda_0}}_{a_1}),\:\varphi(z^{\boldsymbol{\lambda_0}}_{a_2}),\:\varphi(z^{\boldsymbol{\lambda_0}}_{a_{p-1}}),\:\varphi(z^{\boldsymbol{\lambda_0}}_{a_3})]\\ =&[\psi(z^{\boldsymbol{\lambda_0}}_{a_1}),\:\psi(z^{\boldsymbol{\lambda_0}}_{a_2}),\:\psi(z^{\boldsymbol{\lambda_0}}_{a_{p-1}}),\:\psi(z^{\boldsymbol{\lambda_0}}_{a_3})]\\ =&[1,\:w^q,\:w^{-2q},\:w^{2q}]\\ =&\frac{-w^q}{1+w^{2q}}.\end{aligned}$$ $$\begin{aligned} \chi_{\boldsymbol{\lambda_0}}(g_\sigma) =&\frac{\partial f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(z^{\boldsymbol{\mu}}_{\pi(a_{p-1})})} {\partial \mu^{\pi(a_p)-3}}\bigg|_{\boldsymbol{\mu_0}} =\frac{\partial f^{\boldsymbol{\mu}}_{\pi\cdot\sigma\cdot\pi^{-1}}(\mu^{\pi(a_{p-1})-3})} {\partial \mu^{\pi(a_p)-3}}\bigg|_{\boldsymbol{\mu_0}}\\ =&\bigg(\frac{\partial}{\partial \mu^{\pi(a_p)-3}}\bigg( \frac{\mu^{\pi(a_{p-1})-3}-\mu^{\pi(a_p)-3}}{\mu^{\pi(a_p)-3}(\mu^{\pi(a_{p-1})-3}-1)} \bigg)\bigg)\bigg|_{\boldsymbol{\mu_0}}\\ =&\frac{-\mu^{\pi(a_{p-1})-3}}{(\mu^{\pi(a_p)-3})^2(\mu^{\pi(a_{p-1})-3}-1)} \bigg|_{\boldsymbol{\mu_0}}\\ =&-(1+2\cos{\frac{2q\pi}{p}}).\end{aligned}$$ Representations of the Icosahedral Group $I$ {#A_5prep} -------------------------------------------- For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the icosahedral group $I$, recall from Section \[sing\] that $$\Phi_{\boldsymbol{\lambda}}:\:G_{\boldsymbol{\lambda}}\to\mathcal{A}_{[\boldsymbol{\lambda}]},\:g_\sigma\mapsto f^{\boldsymbol{\lambda}}_\sigma$$ is a group isomorphism. There are five conjugacy classes of $G_{\boldsymbol{\lambda}}$: - $K^{(1)}_{\boldsymbol{\lambda}}$: the identity element; - $K^{(2)}_{\boldsymbol{\lambda}}$: elements whose images under $\Phi_{\boldsymbol{\lambda}}$ are conjugate to a rotation of $\frac{2\pi}{5}$; - $K^{(3)}_{\boldsymbol{\lambda}}$: elements whose images under $\Phi_{\boldsymbol{\lambda}}$ are conjugate to a rotation of $\frac{4\pi}{5}$; - $K^{(4)}_{\boldsymbol{\lambda}}$: elements whose images under $\Phi_{\boldsymbol{\lambda}}$ are conjugate to a rotation of $\frac{2\pi}{3}$; - $K^{(5)}_{\boldsymbol{\lambda}}$: elements whose images under $\Phi_{\boldsymbol{\lambda}}$ are conjugate to a rotation of $\pi$. Table \[chaA\_5\] is the character table of $G_{\boldsymbol{\lambda}}$, with $X^{(1)}_{\boldsymbol{\lambda}},\: X^{(2)}_{\boldsymbol{\lambda}},\:\cdots,\:X^{(5)}_{\boldsymbol{\lambda}}$ representing the five different irreducible representations of $G_{\boldsymbol{\lambda}}$ and $\chi^{(1)}_{\boldsymbol{\lambda}},\: \chi^{(2)}_{\boldsymbol{\lambda}},\:\cdots,\:\chi^{(5)}_{\boldsymbol{\lambda}}$ their characters. $$\begin{array}{c|ccccc} & K^{(1)}_{\boldsymbol{\lambda}} & K^{(2)}_{\boldsymbol{\lambda}} & K^{(3)}_{\boldsymbol{\lambda}} & K^{(4)}_{\boldsymbol{\lambda}} & K^{(5)}_{\boldsymbol{\lambda}} \\ \hline \chi^{(1)}_{\boldsymbol{\lambda}}& 1 & 1 & 1 & 1 & 1 \\ \chi^{(2)}_{\boldsymbol{\lambda}}& 4 & -1 & -1 & 1 & 0 \\\chi^{(3)}_{\boldsymbol{\lambda}}& 5 & 0 & 0 & -1 & 1 \\ \chi^{(4)}_{\boldsymbol{\lambda}}& 3 & \frac{1+\sqrt5}{2} & \frac{1-\sqrt5}{2} & 0 & -1 \\ \chi^{(5)}_{\boldsymbol{\lambda}}& 3 & \frac{1-\sqrt5}{2} & \frac{1+\sqrt5}{2} & 0 & -1 \\ \end{array}$$ For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the icosahedral group $I$, assume that $X_{\boldsymbol{\lambda}}=p_1X^{(1)}_{\boldsymbol{\lambda}}\oplus\cdots\oplus p_5X^{(5)}_{\boldsymbol{\lambda}}$. Then call $(p_1,\:p_2,\:\cdots,\:p_5)$ the **multiplicity vector** of $\boldsymbol{\lambda}$. Now all we have to do is to find the multiplicity vector for any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the icosahedral group $I$. From Section \[A\_5\] we know that $[\boldsymbol{\lambda}]$ is equivalent to one of the following sets (which are categorized into eight types): 1. **F+mB**: $F_I\cup B_I(a_1)\cup B_I(a_2)\cup\cdots\cup B_I(a_m)$, where $B_I(a_1),\:B_I(a_2),\:\cdots,\:B_I(a_m)$ are different orbits of order 60, $m\in\mathbb {N}$; 2. **V+mB**: $V_I\cup B_I(a_1)\cup B_I(a_2)\cup\cdots\cup B_I(a_m)$, where $B_I(a_1),\:B_I(a_2),\:\cdots,\:B_I(a_m)$ are different orbits of order 60, $m\in\mathbb {N}$; 3. **E+mB**: $E_I\cup B_I(a_1)\cup B_I(a_2)\cup\cdots\cup B_I(a_m)$, where $B_I(a_1),\:B_I(a_2),\:\cdots,\:B_I(a_m)$ are different orbits of order 60, $m\in\mathbb {N}$; 4. **FV+mB**: $F_I\cup V_I\cup B_I(a_1)\cup B_I(a_2)\cup\cdots\cup B_I(a_m)$, where $B_I(a_1),\:B_I(a_2),\:\cdots,\:B_I(a_m)$ are different orbits of order 60, $m\in\mathbb {N}$; 5. **VE+mB**: $V_I\cup E_I\cup B_I(a_1)\cup B_I(a_2)\cup\cdots\cup B_I(a)_m)$, where $B_I(a_1),\:B_I(a_2),\:\cdots,\:B_I(a_m)$ are different orbits of order 60, $m\in\mathbb {N}$; 6. **EF+mB**: $E_I\cup F_I\cup B_I(a_1)\cup B_I(a_2)\cup\cdots\cup B_I(a_m)$, where $B_I(a_1),\:B_I(a_2),\:\cdots,\:B_I(a_m)$ are different orbits of order 60, $m\in\mathbb {N}$; 7. **FVE+mB**: $F_I\cup V_I\cup E_I\cup B_I(a_1)\cup B_I(a_2)\cup\cdots\cup B_I(a_m)$, where $B_I(a_1),\:B_I(a_2),\:\cdots,\:B_I(a_m)$ are different orbits of order 60, $m\in\mathbb {N}$; 8. **(1+m)B**: $B_I(a_0)\cup B_I(a_1)\cup\cdots\cup B_I(a_m)$, where $B_I(a_0),\:B_I(a_1),\:\cdots,\:B_I(a_m)$ are different orbits of order 60, $m\in\mathbb {N}$. From the theorems in Section \[cha\] we have already known the character of $X_{\boldsymbol{\lambda}}$. The results are shown in Table \[chaA5\]. $$\begin{array}{c|ccccc} & K^{(1)}_{\boldsymbol{\lambda}} & K^{(2)}_{\boldsymbol{\lambda}} & K^{(3)}_{\boldsymbol{\lambda}} & K^{(4)}_{\boldsymbol{\lambda}} & K^{(5)}_{\boldsymbol{\lambda}} \\ \hline F+mB & 9+60m & -1 & -1 & 0 & 1 \\ V+mB & 17+60m & \frac{-\sqrt5-1}{2} & \frac{\sqrt5-1}{2} & -1 & 1 \\ E+mB & 27+60m & \frac{-\sqrt5-1}{2} & \frac{\sqrt5-1}{2} & 0 & -1 \\ FV+mB & 29+60m & -1 & -1 & -1 & 1 \\ VE+mB & 47+60m & \frac{-\sqrt5-1}{2} & \frac{\sqrt5-1}{2} & -1 & -1 \\ EF+mB & 39+60m & -1 & -1 & 0 & -1 \\ FVE+mB & 59+60m & -1 & -1 & -1 & -1 \\ (1+m)B & 57+60m & \frac{-\sqrt5-1}{2} & \frac{\sqrt5-1}{2} & 0 & 1 \\ \end{array}$$ Thus we have the conclusion For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the icosahedral group $I$, we have found its multiplicity vector 1. **F+mB**: $(m,\:1+4m,\:1+5m,\:3m,\:3m)$; 2. **V+mB**: $(m,\:1+4m,\:2+5m,\:3m,\:1+3m)$; 3. **E+mB**: $(m,\:2+4m,\:2+5m,\:1+3m,\:2+3m)$; 4. **FV+mB**: $(m,\:2+4m,\:3+5m,\:1+3m,\:1+3m)$; 5. **VE+mB**: $(m,\:3+4m,\:4+5m,\:2+3m,\:3+3m)$; 6. **EF+mB**: $(m,\:3+4m,\:3+5m,\:2+3m,\:2+3m)$; 7. **FVE+mB**: $(m,\:4+4m,\:5+5m,\:3+3m,\:3+3m)$; 8. **(1+m)B**: $(1+m,\:4+4m,\:5+5m,\:2+3m,\:3+3m)$. There is an interesting pattern of the result. For $\boldsymbol {\lambda_1}\in K_{n_1},\:\boldsymbol {\lambda_2}\in K_{n_2}$ with $(a_1,\:a_2,\:\cdots,\:a_5)$ and $(b_1,\:b_2,\:\cdots,\:b_5)$ their multiplicity vectors respectively, if there exist some linear fractional transformations $\phi_1$ and $\phi_2$ such that $$\mathcal{A}_{\phi_1([\boldsymbol {\lambda_1}])}=\mathcal{A}_{\phi_2([\boldsymbol {\lambda_2}])}\simeq A_5$$ with $\phi_1([\boldsymbol{\lambda_1}])\ne\phi_2([\boldsymbol {\lambda_2}])$, then choose $\boldsymbol {\mu}\in K_{n_1+n_2+3}$ such that there exists some linear fractional transformation $\phi$ satisfying $$\phi([\boldsymbol {\mu}])=\phi_1([\boldsymbol{\lambda_1}])\cup\phi_2([\boldsymbol {\lambda_2}]),$$ then $G_{\boldsymbol {\mu}}\simeq A_5$ and the multiplicity vector of $\boldsymbol {\mu}$ is $$(a_1+b_1,\:a_2+b_2,\:a_3+b_3,\:a_4+b_4+1,\:a_5+b_5).$$ Representations of the Octahedral Group $O$ ------------------------------------------- For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the octahedral group $O$, let $\Psi_{\boldsymbol{\lambda}}$ be the isomorphism which maps $G_{\boldsymbol{\lambda}}$ to the symmetric group $S_4$. There are five conjugacy classes of $G_{\boldsymbol{\lambda}}$: - $K^{(1)}_{\boldsymbol{\lambda}}$: the identity element; - $K^{(2)}_{\boldsymbol{\lambda}}$: elements whose images under $\Psi_{\boldsymbol{\lambda}}$ are conjugate to $(1234)$; - $K^{(3)}_{\boldsymbol{\lambda}}$: elements whose images under $\Psi_{\boldsymbol{\lambda}}$ are conjugate to $(123)$; - $K^{(4)}_{\boldsymbol{\lambda}}$: elements whose images under $\Psi_{\boldsymbol{\lambda}}$ are conjugate to $(12)(34)$; - $K^{(5)}_{\boldsymbol{\lambda}}$: elements whose images under $\Psi_{\boldsymbol{\lambda}}$ are conjugate to $(12)$. Table \[chaS\_4\] is the character table of $G_{\boldsymbol{\lambda}}$, with $X^{(1)}_{\boldsymbol{\lambda}},\: X^{(2)}_{\boldsymbol{\lambda}},\:\cdots,\:X^{(5)}_{\boldsymbol{\lambda}}$ representing the five different irreducible representations of $G_{\boldsymbol{\lambda}}$ and $\chi^{(1)}_{\boldsymbol{\lambda}},\: \chi^{(2)}_{\boldsymbol{\lambda}},\:\cdots,\:\chi^{(5)}_{\boldsymbol{\lambda}}$ their characters. $$\begin{array}{c|ccccc} & K^{(1)}_{\boldsymbol{\lambda}} & K^{(2)}_{\boldsymbol{\lambda}} & K^{(3)}_{\boldsymbol{\lambda}} & K^{(4)}_{\boldsymbol{\lambda}} & K^{(5)}_{\boldsymbol{\lambda}} \\ \hline \chi^{(1)}_{\boldsymbol{\lambda}}& 1 & 1 & 1 & 1 & 1 \\ \chi^{(2)}_{\boldsymbol{\lambda}}& 1 & -1 & 1 & 1 & -1 \\\chi^{(3)}_{\boldsymbol{\lambda}}& 3 & -1 & 0 & -1 & 1 \\ \chi^{(4)}_{\boldsymbol{\lambda}}& 3 & 1 & 0 & -1 & -1 \\ \chi^{(5)}_{\boldsymbol{\lambda}}& 2 & 0 & -1 & 2 & 0 \\ \end{array}$$ For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the octahedral group $O$, assume that $X_{\boldsymbol{\lambda}}=p_1X^{(1)}_{\boldsymbol{\lambda}}\oplus\cdots\oplus p_5X^{(5)}_{\boldsymbol{\lambda}}$. Then call $(p_1,\:p_2,\:\cdots,\:p_5)$ the **multiplicity vector** of $\boldsymbol{\lambda}$. Now all we have to do is to find the multiplicity vector for any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the octahedral group $O$. From Section \[S\_4\] we know that $[\boldsymbol{\lambda}]$ is equivalent to one of the following sets (which are categorized into eight types): 1. **F+mB**: $F_O\cup B_O(a_1)\cup B_O(a_2)\cup\cdots\cup B_O(a_m)$, where $B_O(a_1),\:B_O(a_2),\:\cdots,\:B_O(a_m)$ are different orbits of order 24, $m\in\mathbb {N}$; 2. **V+mB**: $V_O\cup B_O(a_1)\cup B_O(a_2)\cup\cdots\cup B_O(a_m)$, where $B_O(a_1),\:B_O(a_2),\:\cdots,\:B_O(a_m)$ are different orbits of order 24, $m\in\mathbb {N}$; 3. **E+mB**: $E_O\cup B_O(a_1)\cup B_O(a_2)\cup\cdots\cup B_O(a_m)$, where $B_O(a_1),\:B_O(a_2),\:\cdots,\:B_O(a_m)$ are different orbits of order 24, $m\in\mathbb {N}$; 4. **FV+mB**: $F_O\cup V_O\cup B_O(a_1)\cup B_O(a_2)\cup\cdots\cup B_O(a_m)$, where $B_O(a_1),\:B_O(a_2),\:\cdots,\:B_O(a_m)$ are different orbits of order 24, $m\in\mathbb {N}$; 5. **VE+mB**: $V_O\cup E_O\cup B_O(a_1)\cup B_O(a_2)\cup\cdots\cup B_O(a_m)$, where $B_O(a_1),\:B_O(a_2),\:\cdots,\:B_O(a_m)$ are different orbits of order 24, $m\in\mathbb {N}$; 6. **EF+mB**: $E_O\cup F_O\cup B_O(a_1)\cup B_O(a_2)\cup\cdots\cup B_O(a_m)$, where $B_O(a_1),\:B_O(a_2),\:\cdots,\:B_O(a_m)$ are different orbits of order 24, $m\in\mathbb {N}$; 7. **FVE+mB**: $F_O\cup V_O\cup E_O\cup B_O(a_1)\cup B_O(a_2)\cup\cdots\cup B_O(a_m)$, where $B_O(a_1),\:B_O(a_2),\:\cdots,\:B_O(a_m)$ are different orbits of order 24, $m\in\mathbb {N}$; 8. **(1+m)B**: $B_O(a_0)\cup B_O(a_1)\cup B_O(a_2)\cup\cdots\cup B_O(a_m)$, where $B_O(a_1),\:B_O(a_2),\:\cdots,\:B_O(a_m)$ are different orbits of order 24, $m\in\mathbb {N}$. From the theorems in Section \[cha\], we have already found the character of $X_{\boldsymbol{\lambda}}$. The results are shown in Table \[chaS4\]. $$\begin{array}{c|ccccc} & K^{(1)}_{\boldsymbol{\lambda}} & K^{(2)}_{\boldsymbol{\lambda}} & K^{(3)}_{\boldsymbol{\lambda}} & K^{(4)}_{\boldsymbol{\lambda}} & K^{(5)}_{\boldsymbol{\lambda}} \\ \hline F+mB & 3+24m & -1 & 0 & -1 & 1 \\ V+mB & 5+24m & -1 & -1 & 1 & 1 \\ E+mB & 9+24m & -1 & 0 & 1 & -1 \\ FV+mB & 11+24m & -1 & -1 & -1 & 1 \\ VE+mB & 17+24m & -1 & -1 & 1 & -1 \\ EF+mB & 15+24m & -1 & 0 & -1 & -1 \\ FVE+mB & 23+24m & -1 & -1 & -1 & -1 \\ (1+m)B & 21+24m & -1 & 0 & 1 & 1 \\ \end{array}$$ For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the octahedral group $O$, we have found its multiplicity vector 1. **F+mB**: $(m,\:m,\:1+3m,\:3m,\:2m)$; 2. **V+mB**: $(m,\:m,\:1+3m,\:3m,\:1+2m)$; 3. **E+mB**: $(m,\:1+m,\:1+3m,\:1+3m,\:1+2m)$; 4. **FV+mB**: $(m,\:m,\:2+3m,\:1+3m,\:1+2m)$; 5. **VE+mB**: $(m,\:1+m,\:2+3m,\:2+3m,\:2+2m)$; 6. **EF+mB**: $(m,\:1+m,\:2+3m,\:2+3m,\:1+2m)$; 7. **FVE+mB**: $(m,\:1+m,\:3+3m,\:3+3m,\:2+2m)$; 8. **(1+m)B**: $(1+m,\:1+m,\:3+3m,\:2+3m,\:2+2m)$. There is an interesting pattern of the result. For $\boldsymbol {\lambda_1}\in K_{n_1},\:\boldsymbol {\lambda_2}\in K_{n_2}$ with $(a_1,\:a_2,\:\cdots,\:a_5)$ and $(b_1,\:b_2,\:\cdots,\:b_5)$ their multiplicity vectors respectively, if there exist some linear fractional transformations $\phi_1$ and $\phi_2$ such that $$\mathcal{A}_{\phi_1([\boldsymbol {\lambda_1}])}=\mathcal{A}_{\phi_2([\boldsymbol {\lambda_2}])}\simeq S_4$$ with $\phi_1([\boldsymbol{\lambda_1}])\ne\phi_2([\boldsymbol {\lambda_2}])$, then choose $\boldsymbol {\mu}\in K_{n_1+n_2+3}$ such that there exists some linear fractional transformation $\phi$ satisfying $$\phi([\boldsymbol {\mu}])=\phi_1([\boldsymbol{\lambda_1}])\cup\phi_2([\boldsymbol {\lambda_2}]),$$ then $G_{\boldsymbol {\mu}}\simeq S_4$ and the multiplicity vector of $\boldsymbol {\mu}$ is $$(a_1+b_1,\:a_2+b_2,\:a_3+b_3,\:a_4+b_4+1,\:a_5+b_5).$$ Representations of the Tetrahedral Group $T$ -------------------------------------------- Things are a little complicated in the tetrahedral case, as all rotations of order three are conjugate in the liner fractional transformation group. Let $F_T$ denote the projections of the four central points of the faces of the tetrahedron on the Riemann sphere. That means $$F_T=\{A,\:B,\:C,\:D\} =\{0,\:\sqrt2\:,\sqrt2e^{\frac{2\pi}{3}i},\:\sqrt2e^{\frac{4\pi}{3}i}\}.$$ Let $V_T$, $E_T$ denote the four vertices and the projections of the six middle points of the edges of the tetrahedron on the Riemann sphere. That means $$V_T=\{E,\:F,\:G,\:H\} =\{\infty,\:\frac{-1}{\sqrt2}\:,\frac{-1}{\sqrt2}e^{\frac{2\pi}{3}i},\:\frac{-1}{\sqrt2}e^{\frac{4\pi}{3}i}\},$$ $$E_T=\{I,\:J,\:K,\:L,\:M,\:N\} =\{\frac{\sqrt2}{1-\sqrt3},\:\frac{\sqrt2}{1-\sqrt3}e^{\frac{2\pi}{3}i},\:\frac{\sqrt2}{1-\sqrt3}e^{\frac{4\pi}{3}i},\:\frac{\sqrt2}{1+\sqrt3},\:\frac{\sqrt2}{1+\sqrt3}e^{\frac{2\pi}{3}i},\:\frac{\sqrt2}{1+\sqrt3}e^{\frac{4\pi}{3}i}\}.$$ Set $$G_0=\langle z\mapsto e^{\frac{2\pi}{3}i}z,\: z\mapsto \frac{\sqrt2-z}{\sqrt2z+1}\rangle\simeq A_4.$$ Thus $G_0$ fixes $F_T$, $V_T$ and $E_T$. ![Center points on the faces, vertices and middle points on the edges of a tetrahedron.[]{data-label="A4"}](a41.png "fig:"){width="2.3in"} ![Center points on the faces, vertices and middle points on the edges of a tetrahedron.[]{data-label="A4"}](a42.png "fig:"){width="2.1in"} From Section \[A\_4\] we know that if $G_{\boldsymbol{\lambda}}$ is isomorphic to the Tetrahedral Group $T$, then $[\boldsymbol{\lambda}]$ is equivalent to one of the following sets (which are categorized into six types): 1. **F+mB**: $F_T\cup B_T(a_1)\cup B_T(a_2)\cup\cdots\cup B_T(a_m)$, where $B_T(a_1),\:B_T(a_2),\:\cdots,\:B_T(a_m)$ are different orbits of order 12, $m\in\mathbb {N^*}$; 2. **E+mB**: $E_T\cup B_T(a_1)\cup B_T(a_2)\cup\cdots\cup B_T(a_m)$, where $B_T(a_1),\:B_T(a_2),\:\cdots,\:B_T(a_m)$ are different orbits of order 12, $m\in\mathbb {N^*}$; 3. **FV+mB**: $F_T\cup V_T\cup B_T(a_1)\cup B_T(a_2)\cup\cdots\cup B_T(a_m)$, where $B_T(a_1),\:B_T(a_2),\:\cdots,\:B_T(a_m)$ are different orbits of order 12, $m\in\mathbb {N^*}$; 4. **FE+mB**: $F_T\cup E_T\cup B_T(a_1)\cup B_T(a_2)\cup\cdots\cup B_T(a_m)$, where $B_T(a_1),\:B_T(a_2),\:\cdots,\:B_T(a_m)$ are different orbits of order 12, $m\in\mathbb {N}$; 5. **FVE+mB**: $F_T\cup V_T\cup E_T\cup B_I(a_1)\cup B_I(a_2)\cup\cdots\cup B_I(a_m)$, where $B_I(a_1),\:B_I(a_2),\:\cdots,\:B_I(a_m)$ are different orbits of order 12, $m\in\mathbb {N^*}$; 6. **(1+m)B**: $B_T(a_1)\cup B_T(a_2)\cup\cdots\cup B_T(a_m)$, where $B_T(a_1),\:B_T(a_2),\:\cdots,\:B_T(a_m)$ are different orbits of order 12, $m\in\mathbb {N}$. Set $$w=e^{\frac{2\pi}{3}i}.$$ ### Representations of $T$ when $\boldsymbol{\lambda}$ is of type E+mB, FV+mB, EFV+mB or (1+m)B There are four conjugacy classes of $G_{\boldsymbol{\lambda}}$: - $K^{(1)}_{\boldsymbol{\lambda}}$: the identity element; - $K^{(2)}_{\boldsymbol{\lambda}}$: elements of order two; - $K^{(3)}_{\boldsymbol{\lambda}}$: one of the two left classes; - $K^{(4)}_{\boldsymbol{\lambda}}$: the other of the two left classes. Now fix $K^{(1)}_{\boldsymbol{\lambda}}$, $K^{(2)}_{\boldsymbol{\lambda}}$, $K^{(3)}_{\boldsymbol{\lambda}}$, $K^{(4)}_{\boldsymbol{\lambda}}$. Table \[chaA\_41\] is the character table of $G_{\boldsymbol{\lambda}}$, with $X^{(1)}_{\boldsymbol{\lambda}},\:X^{(2)}_{\boldsymbol{\lambda}},\:X^{(3)}_{\boldsymbol{\lambda}}$ and $X^{(4)}_{\boldsymbol{\lambda}}$ representing the four different irreducible representations of $G_{\boldsymbol{\lambda}}$ and $\chi^{(1)}_{\boldsymbol{\lambda}},\: \chi^{(2)}_{\boldsymbol{\lambda}},\:\chi^{(3)}_{\boldsymbol{\lambda}}$ and $\chi^{(4)}_{\boldsymbol{\lambda}}$ their characters. $$\begin{array}{c|cccc} & K^{(1)}_{\boldsymbol{\lambda}} & K^{(2)}_{\boldsymbol{\lambda}} & K^{(3)}_{\boldsymbol{\lambda}} & K^{(4)}_{\boldsymbol{\lambda}}\\ \hline \chi^{(1)}_{\boldsymbol{\lambda}}& 1 & 1 & 1 & 1 \\ \chi^{(2)}_{\boldsymbol{\lambda}}& 1 & 1 & w & w^2 \\ \chi^{(3)}_{\boldsymbol{\lambda}}& 1 & 1 & w^2 & w \\ \chi^{(4)}_{\boldsymbol{\lambda}}& 3 &-1 & 0 & 0 \\ \end{array}$$ For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the tetrahedral group $T$ and $\boldsymbol{\lambda}$ is of type E+mB, FV+mB, EFV+mB or (1+m)B, assume that $X_{\boldsymbol{\lambda}}=p_1X^{(1)}_{\boldsymbol{\lambda}}\oplus\cdots\oplus p_4X^{(4)}_{\boldsymbol{\lambda}}$. Then call $(p_1,\:p_2,\:p_3,\:p_4)$ the **multiplicity vector** of $\boldsymbol{\lambda}$. The definition seems ambiguous as $K^{(3)}_{\boldsymbol{\lambda}}$ and $K^{(4)}_{\boldsymbol{\lambda}}$ are chosen randomly. But we shall prove that $(p_1,\:p_2,\:p_3,\:p_4)$ remains the same however we choose $K^{(3)}_{\boldsymbol{\lambda}}$ and $K^{(4)}_{\boldsymbol{\lambda}}$, which makes the notion multiplicity vector well-defined. From the theorems in Section \[cha\], we have already found the character of $X_{\boldsymbol{\lambda}}$. The results are shown in Table \[chaA41\]. $$\begin{array}{c|cccc} & K^{(1)}_{\boldsymbol{\lambda}} & K^{(2)}_{\boldsymbol{\lambda}} & K^{(3)}_{\boldsymbol{\lambda}} & K^{(4)}_{\boldsymbol{\lambda}} \\ \hline E+mB & 3+12m &-1 & 0 & 0 \\ FV+mB & 5+12m & 1 & -1 & -1 \\ FVE+mB & 11+12m&-1 & -1 & -1 \\ (1+m)B & 9+12m & 1 & 0 & 0 \\ \end{array}$$ Thus we have the conclusion that the multiplicity vector of $\boldsymbol{\lambda}$ is 1. **E+mB**: $(m,\:m,\:m,\:1+3m)$; 2. **FV+mB**: $(m,\:1+m,\:1+m,\:1+3m)$; 3. **FVE+mB**: $(m,\:1+m,\:1+m,\:3+3m)$; 4. **(1+m)B**: $(m,\:1+m,\:1+m,\:2+3m)$. ### Representations of $T$ when $\boldsymbol{\lambda}$ is of type F+mB or FE+mB If $\boldsymbol{\lambda}$ is of type F+mB or FE+mB, let $\psi$ be a linear fractional transformation such that $$\psi([\boldsymbol{\lambda}])=F_T\cup B_T(a_1)\cup B_T(a_2)\cup\cdots\cup B_T(a_m)$$ or $$\psi([\boldsymbol{\lambda}])=F_T\cup E_T\cup B_T(a_1)\cup B_T(a_2)\cup\cdots\cup B_T(a_m).$$ Let $\rho$ denote the element in $G_{\boldsymbol{\lambda}}$ such that $$\psi\circ\Phi_{\boldsymbol{\lambda}}(\rho)\circ\psi^{-1}(z)=wz.$$ There are four conjugacy classes of $G_{\boldsymbol{\lambda}}$: - $K^{(1)}_{\boldsymbol{\lambda}}$: the identity element; - $K^{(2)}_{\boldsymbol{\lambda}}$: elements of order two; - $K^{(3)}_{\boldsymbol{\lambda}}$: the conjugate class of $\rho$; - $K^{(4)}_{\boldsymbol{\lambda}}$: the conjugate class of $\rho^2$. Now fix $K^{(1)}_{\boldsymbol{\lambda}}$, $K^{(2)}_{\boldsymbol{\lambda}}$, $K^{(3)}_{\boldsymbol{\lambda}}$, $K^{(4)}_{\boldsymbol{\lambda}}$. Table \[chaA\_42\] is the character table of $G_{\boldsymbol{\lambda}}$, with $X^{(1)}_{\boldsymbol{\lambda}},\:X^{(2)}_{\boldsymbol{\lambda}},\:X^{(3)}_{\boldsymbol{\lambda}}$ and $X^{(4)}_{\boldsymbol{\lambda}}$ representing the four different irreducible representations of $G_{\boldsymbol{\lambda}}$ and $\chi^{(1)}_{\boldsymbol{\lambda}},\: \chi^{(2)}_{\boldsymbol{\lambda}},\:\chi^{(3)}_{\boldsymbol{\lambda}}$ and $\chi^{(4)}_{\boldsymbol{\lambda}}$ their characters. $$\begin{array}{c|cccc} & K^{(1)}_{\boldsymbol{\lambda}} & K^{(2)}_{\boldsymbol{\lambda}} & K^{(3)}_{\boldsymbol{\lambda}} & K^{(4)}_{\boldsymbol{\lambda}}\\ \hline \chi^{(1)}_{\boldsymbol{\lambda}}& 1 & 1 & 1 & 1 \\ \chi^{(2)}_{\boldsymbol{\lambda}}& 1 & 1 & w & w^2 \\ \chi^{(3)}_{\boldsymbol{\lambda}}& 1 & 1 & w^2 & w \\ \chi^{(4)}_{\boldsymbol{\lambda}}& 3 &-1 & 0 & 0 \\ \end{array}$$ For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the tetrahedral group $T$ and $\boldsymbol{\lambda}$ is of type F+mB or FE+mB, assume that $X_{\boldsymbol{\lambda}}=p_1X^{(1)}_{\boldsymbol{\lambda}}\oplus\cdots\oplus p_4X^{(4)}_{\boldsymbol{\lambda}}$. Then call $(p_1,\:p_2,\:p_3,\:p_4)$ the **multiplicity vector** of $\boldsymbol{\lambda}$. The definition seems ambiguous as $\psi$ is chosen randomly. But we shall prove that $(p_1,\:p_2,\:p_3,\:p_4)$ remains the same however we choose $\psi$, which makes the notion multiplicity vector well-defined. From the theorems in Section \[cha\], we have already found the character of $X_{\boldsymbol{\lambda}}$. The results are shown in Table \[chaA42\]. $$\begin{array}{c|cccc} & K^{(1)}_{\boldsymbol{\lambda}} & K^{(2)}_{\boldsymbol{\lambda}} & K^{(3)}_{\boldsymbol{\lambda}} & K^{(4)}_{\boldsymbol{\lambda}} \\ \hline F+mB & 1+12m & 1 & w & w^2 \\ FE+mB & 7+12m &-1 & w & w^2 \\ \end{array}$$ Thus we have the conclusion that the multiplicity vector of $\boldsymbol{\lambda}$ is 1. **F+mB**: $(m,\:1+m,\:m,\:3m)$; 2. **FE+mB**: $(m,\:1+m,\:m,\:2+3m)$. For each singularity $[\boldsymbol{\lambda}]$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the tetrahedral group $T$, we have found its multiplicity vector 1. **F+mB**: $(m,\:1+m,\:m,\:3m)$; 2. **E+mB**: $(m,\:m,\:m,\:1+3m)$; 3. **FV+mB**: $(m,\:1+m,\:1+m,\:1+3m)$; 4. **FE+mB**: $(m,\:1+m,\:m,\:2+3m)$; 5. **FVE+mB**: $(m,\:1+m,\:1+m,\:3+3m)$; 6. **(1+m)B**: $(m,\:1+m,\:1+m,\:2+3m)$. Representations of the Dihedral Group $D_p$ {#D_pprep} ------------------------------------------- For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the dihedral group $D_p$, recall from Section \[sing\] that $$\Phi_{\boldsymbol{\lambda}}:\:G_{\boldsymbol{\lambda}}\to\mathcal{A}_{[\boldsymbol{\lambda}]},\:g_\sigma\mapsto f^{\boldsymbol{\lambda}}_\sigma$$ is a group isomorphism. From Section \[D\_n\] we know that $[\boldsymbol{\lambda}]$ is equivalent to one of the following sets (which are categorized into six types): 1. **mC**: $C_p(a_1)\cup C_p(a_2)\cup\cdots\cup C_p(a_m)$, where $C_p(a_1),\:C_p(a_2),\:\cdots,\:C_p(a_m)$ are different orbits of order $2p$, $m\in\mathbb {N^*}$; 2. **A+mC**: $A_p\cup C_p(a_1)\cup C_p(a_2)\cup\cdots\cup C_p(a_m)$, where $C_p(a_1),\:C_p(a_2),\:\cdots,\:C_p(a_m)$ are different orbits of order $2p$, $m\in\mathbb {N}$; 3. **AB+mC**: $A_p\cup B_p\cup C_p(a_1)\cup C_p(a_2)\cup\cdots\cup C_p(a_m)$, where $C_p(a_1),\:C_p(a_2),\:\cdots,\:C_p(a_m)$ are different orbits of order $2p$, $m\in\mathbb {N^*}$; 4. **2+mC**: $\{0,\:\infty\}\cup C_p(a_1)\cup C_p(a_2)\cup\cdots\cup C_p(a_m)$, where $C_p(a_1),\:C_p(a_2),\:\cdots,\:C_p(a_m)$ are different orbits of order $2p$, $m\in\mathbb {N^*}$; 5. **A+2+mC**: $A_p\cup\{0,\:\infty\}\cup C_p(a_1)\cup C_p(a_2)\cup\cdots\cup C_p(a_m)$, where $C_p(a_1),\:C_p(a_2),\:\cdots,\:C_p(a_m)$ are different orbits of order $2p$, $m\in\mathbb {N}$; 6. **AB+2+mC**: $A_p\cup B_p\cup\{0,\:\infty\}\cup C_p(a_1)\cup C_p(a_2)\cup\cdots\cup C_p(a_m)$, where $C_p(a_1),\:C_p(a_2),\:\cdots,\:C_p(a_m)$ are different orbits of order $2p$, $m\in\mathbb {N^*}$. ### Representations of $D_p$ when $p$ is odd {#D_poddprep} When $G_{\boldsymbol{\lambda}}$ is isomorphic to the dihedral group $D_p$ ($p$ is odd), there are $\frac{p+3}{2}$ conjugacy classes of $G_{\boldsymbol{\lambda}}$: - $K^{(l)}_{\boldsymbol{\lambda}}$: elements whose images under $\Phi_{\boldsymbol{\lambda}}$ are conjugate to a rotation of $\frac{2l\pi}{p}$, where $l=1,\:2,:\cdots,\:\frac{p-1}{2}$; - $K^{(\frac{p+1}{2})}_{\boldsymbol{\lambda}}$: the identity element; - $K^{(\frac{p+3}{2})}_{\boldsymbol{\lambda}}$: elements of order two. Table \[chaD\_podd\] is the character table of $G_{\boldsymbol{\lambda}}$, with $X^{(1)}_{\boldsymbol{\lambda}},\: X^{(2)}_{\boldsymbol{\lambda}},\:\cdots,\:X^{(\frac{p+3}{2})}_{\boldsymbol{\lambda}}$ representing the $\frac{p+3}{2}$ different irreducible representations of $G_{\boldsymbol{\lambda}}$ and $\chi^{(1)}_{\boldsymbol{\lambda}},\: \chi^{(2)}_{\boldsymbol{\lambda}},\:\cdots,\:\chi^{(\frac{p+3}{2})}_{\boldsymbol{\lambda}}$ their characters. $$\begin{array}{c|ccccccc} & K^{(1)}_{\boldsymbol{\lambda}} & \cdots & K^{(k)}_{\boldsymbol{\lambda}} & \cdots & K^{(\frac{p-1}{2})}_{\boldsymbol{\lambda}} & K^{(\frac{p+1}{2})}_{\boldsymbol{\lambda}} & K^{(\frac{p+3}{2})}_{\boldsymbol{\lambda}} \\ \hline \chi^{(1)}_{\boldsymbol{\lambda}} & 2\cos{\frac{2\pi}{p}} & \cdots & 2\cos{\frac{2k\pi}{p}} & \cdots & 2\cos{\frac{(p-1)\pi}{p}} & 2 & 0 \\ \vdots & \vdots & & \vdots & & \vdots & \vdots & \vdots \\ \chi^{(l)}_{\boldsymbol{\lambda}} & 2\cos{\frac{2l\pi}{p}} & \cdots & 2\cos{\frac{2kl\pi}{p}} & \cdots & 2\cos{\frac{(p-1)l\pi}{p}} & 2 & 0 \\ \vdots & \vdots & & \vdots & & \vdots & \vdots & \vdots \\ \chi^{(\frac{p-1}{2})}_{\boldsymbol{\lambda}} & 2\cos{\frac{(p-1)\pi}{p}} & \cdots & 2\cos{\frac{(p-1)k\pi}{p}} & \cdots & 2\cos{\frac{(p-1)^2\pi}{2p}} & 2 & 0 \\ \chi^{(\frac{p+1}{2})}_{\boldsymbol{\lambda}} & 1 & \cdots & 1 & \cdots & 1 & 1 & 1 \\ \chi^{(\frac{p+3}{2})}_{\boldsymbol{\lambda}} & 1 & \cdots & 1 & \cdots & 1 & 1 & -1 \\ \end{array}$$ For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the dihedral group $D_p$ ($p$ is odd), assume that $X_{\boldsymbol{\lambda}}=a_1X^{(1)}_{\boldsymbol{\lambda}}\oplus\cdots\oplus a_{\frac{p+3}{2}}X^{(\frac{p+3}{2})}_{\boldsymbol{\lambda}}$. Then call $(a_1,\:a_2,\:\cdots,\:a_{\frac{p+3}{2}})$ the **multiplicity vector** of $\boldsymbol{\lambda}$. From the theorems in Section \[cha\] we have already known the character of $X_{\boldsymbol{\lambda}}$. The results are shown in Table \[chaDpodd\]. $$\begin{array}{c|ccccccc} & K^{(1)}_{\boldsymbol{\lambda}} & \cdots & K^{(k)}_{\boldsymbol{\lambda}} & \cdots & K^{(\frac{p-1}{2})}_{\boldsymbol{\lambda}} & K^{(\frac{p+1}{2})}_{\boldsymbol{\lambda}} & K^{(\frac{p+3}{2})}_{\boldsymbol{\lambda}} \\\hline mC & -1-2\cos{\frac{2\pi}{p}} & \cdots & -1-2\cos{\frac{2k\pi}{p}} & \cdots & -1-2\cos{\frac{(p-1)\pi}{p}} & 2mp-3 & 1 \\ A+mC & -1-2\cos{\frac{2\pi}{p}} & \cdots & -1-2\cos{\frac{2k\pi}{p}} & \cdots & -1-2\cos{\frac{(p-1)\pi}{p}} & (2m+1)p-3 & 0 \\ AB+mC & -1-2\cos{\frac{2\pi}{p}} & \cdots & -1-2\cos{\frac{2k\pi}{p}} & \cdots & -1-2\cos{\frac{(p-1)\pi}{p}} & (2m+2)p-3 & -1 \\ 2+mC & -1 & \cdots & -1 & \cdots & -1 & 2mp-1 & 1 \\ A+2+mC & -1 & \cdots & -1 & \cdots & -1 & (2m+1)p-1 & 0 \\ AB+2+mC & -1 & \cdots & -1 & \cdots & -1 & (2m+2)p-1 & -1 \\ \end{array}$$ Thus we have the conclusion For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the dihedral group $D_p$ ($p$ is odd), we have found its multiplicity vector 1. **mC**: - $(2m-1,\:m,\:m-1)$ for $p=3$; - $(2m-1,\:2m,\:\cdots,\:2m,\:m,\:m-1)$ for $p\ge5$; 2. **A+mC**: - $(2m,\:m,\:m)$ for $p=3$; - $(2m,\:2m+1,\:\cdots,\:2m+1,\:m,\:m)$ for $p\ge5$; 3. **AB+mC**: - $(2m+1,\:m,\:m+1)$ for $p=3$; - $(2m+1,\:2m+2,\:\cdots,\:2m+2,\:m,\:m+1)$ for $p\ge5$; 4. **2+mC**: $(2m,\:\cdots,\:2m,\:m,\:m-1)$; 5. **A+2+mC**: $(2m+1,\:\cdots,\:2m+1,\:m,\:m)$; 6. **AB+2+mC**: $(2m+2,\:\cdots,\:2m+2,\:m,\:m+1)$. ### Representations of $D_p$ when $p$ is even {#D_pevenprep} When $G_{\boldsymbol{\lambda}}$ is isomorphic to the dihedral group $D_p$ ($p$ is even), there are $\frac{p+6}{2}$ conjugacy classes of $G_{\boldsymbol{\lambda}}$: - $K^{(l)}_{\boldsymbol{\lambda}}$: elements whose images under $\Phi_{\boldsymbol{\lambda}}$ are conjugate to a rotation of $\frac{2l\pi}{p}$, where $l=1,\:2,:\cdots,\:\frac{p}{2}$; - $K^{(\frac{p+2}{2})}_{\boldsymbol{\lambda}}$: the identity element. There are only two conjugacy classes left. If $\boldsymbol{\lambda}$ is of type A+mC or A+2+mC, set - $K^{(\frac{p+4}{2})}_{\boldsymbol{\lambda}}$: elements which have two fixed points in $[\boldsymbol{\lambda}]$ and are not in $K^{(l)}_{\boldsymbol{\lambda}}$, $l=1,\:2,:\cdots,\:\frac{p+2}{2}$; - $K^{(\frac{p+6}{2})}_{\boldsymbol{\lambda}}$: elements which have no fixed point in $[\boldsymbol{\lambda}]$ and are not in $K^{(l)}_{\boldsymbol{\lambda}}$, $l=1,\:2,:\cdots,\:\frac{p+2}{2}$. If $\boldsymbol{\lambda}$ is of type mC, AB+mC, 2+mC or AB+2+mC, just set $K^{(\frac{p+4}{2})}_{\boldsymbol{\lambda}}$ to be any of the left two conjugacy classes and $K^{(\frac{p+6}{2})}_{\boldsymbol{\lambda}}$ the other one. Now fix $K^{(1)}_{\boldsymbol{\lambda}},\:K^{(2)}_{\boldsymbol{\lambda}},\cdots, K^{(\frac{p+6}{2})}_{\boldsymbol{\lambda}}$. Table \[chaD\_peven\] is the character table of $G_{\boldsymbol{\lambda}}$, with $X^{(1)}_{\boldsymbol{\lambda}},\: X^{(2)}_{\boldsymbol{\lambda}},\:\cdots,\:X^{(\frac{p+6}{2})}_{\boldsymbol{\lambda}}$ representing the $\frac{p+6}{2}$ different irreducible representations of $G_{\boldsymbol{\lambda}}$ and $\chi^{(1)}_{\boldsymbol{\lambda}},\: \chi^{(2)}_{\boldsymbol{\lambda}},\:\cdots,\:\chi^{(\frac{p+6}{2})}_{\boldsymbol{\lambda}}$ their characters. $$\begin{array}{c|cccccccc} & K^{(1)}_{\boldsymbol{\lambda}} & \cdots & K^{(k)}_{\boldsymbol{\lambda}} & \cdots & K^{(\frac{p}{2})}_{\boldsymbol{\lambda}} & K^{(\frac{p+2}{2})}_{\boldsymbol{\lambda}} & K^{(\frac{p+4}{2})}_{\boldsymbol{\lambda}} & K^{(\frac{p+6}{2})}_{\boldsymbol{\lambda}} \\ \hline \chi^{(1)}_{\boldsymbol{\lambda}} & 2\cos{\frac{2\pi}{p}} & \cdots & 2\cos{\frac{2k\pi}{p}} & \cdots & -2 & 2 & 0 & 0 \\ \vdots & \vdots & & \vdots & & \vdots & \vdots & \vdots &\vdots\\ \chi^{(l)}_{\boldsymbol{\lambda}} & 2\cos{\frac{2l\pi}{p}} & \cdots & 2\cos{\frac{2kl\pi}{p}} & \cdots & 2(-1)^l & 2 & 0 & 0 \\ \vdots & \vdots & & \vdots & & \vdots & \vdots & \vdots &\vdots\\ \chi^{(\frac{p-2}{2})}_{\boldsymbol{\lambda}} & 2\cos{\frac{(p-2)\pi}{p}} & \cdots & 2\cos{\frac{(p-2)k\pi}{p}} & \cdots & 2(-1)^{\frac{p-2}{2}} & 2 & 0 & 0\\ \chi^{(\frac{p}{2})}_{\boldsymbol{\lambda}} & 1 & \cdots & 1 & \cdots & 1 & 1 & 1 & 1 \\ \chi^{(\frac{p+2}{2})}_{\boldsymbol{\lambda}} & 1 & \cdots & 1 & \cdots & 1 & 1 & -1 & -1\\ \chi^{(\frac{p+4}{2})}_{\boldsymbol{\lambda}} & -1 & \cdots & (-1)^k & \cdots & (-1)^{\frac{p}{2}} & 1 & 1&-1\\ \chi^{(\frac{p+6}{2})}_{\boldsymbol{\lambda}} & -1 & \cdots & (-1)^k & \cdots & (-1)^{\frac{p}{2}} & 1 & -1&1\\ \end{array}$$ For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the dihedral group $D_p$ ($p$ is even), assume that $X_{\boldsymbol{\lambda}}=a_1X^{(1)}_{\boldsymbol{\lambda}}\oplus\cdots\oplus a_{\frac{p+6}{2}}X^{(\frac{p+6}{2})}_{\boldsymbol{\lambda}}$. Then call $(a_1,\:a_2,\:\cdots,\:a_{\frac{p+6}{2}})$ the **multiplicity vector** of $\boldsymbol{\lambda}$. The definition seems ambiguous when $\boldsymbol{\lambda}$ is of type mC, AB+mC, 2+mC or AB+2+mC as $K^{(\frac{p+4}{2})}_{\boldsymbol{\lambda}}$ and $K^{(\frac{p+6}{2})}_{\boldsymbol{\lambda}}$ are chosen randomly. But we shall prove that $(a_1,\:a_2,\:\cdots,\:a_{\frac{p+6}{2}})$ remains the same however we choose $K^{(\frac{p+4}{2})}_{\boldsymbol{\lambda}}$ and $K^{(\frac{p+6}{2})}_{\boldsymbol{\lambda}}$, which makes the notion multiplicity vector well-defined. From the theorems in Section \[cha\] we have already known the character of $X_{\boldsymbol{\lambda}}$. The results are shown in Table \[chaDpeven\]. $$\begin{array}{c|cccccccc} \centering & K^{(1)}_{\boldsymbol{\lambda}} & \cdots & K^{(k)}_{\boldsymbol{\lambda}} & \cdots & K^{(\frac{p}{2})}_{\boldsymbol{\lambda}} & K^{(\frac{p+2}{2})}_{\boldsymbol{\lambda}} & K^{(\frac{p+4}{2})}_{\boldsymbol{\lambda}} & K^{(\frac{p+6}{2})}_{\boldsymbol{\lambda}} \\ \\\hline \mathrm{mC} & -1-2\cos{\frac{2\pi}{p}} & \cdots & -1-2\cos{\frac{2k\pi}{p}} & \cdots & -1-2\cos{\frac{(p-1)\pi}{p}} & 2mp-3 & 1 & 1 \\ \mathrm{A+mC} & -1-2\cos{\frac{2\pi}{p}} & \cdots & -1-2\cos{\frac{2k\pi}{p}} & \cdots & -1-2\cos{\frac{(p-1)\pi}{p}} & (2m+1)p-3 & -1 & 1 \\ \mathrm{AB+mC} & -1-2\cos{\frac{2\pi}{p}} & \cdots & -1-2\cos{\frac{2k\pi}{p}} & \cdots & -1-2\cos{\frac{(p-1)\pi}{p}} & (2m+2)p-3 & -1 & -1 \\ \mathrm{2+mC} & -1 & \cdots & -1 & \cdots & -1 & 2mp-1 & 1 & 1 \\ \mathrm{A+2+mC} & -1 & \cdots & -1 & \cdots & -1 & (2m+1)p-1 & -1 & 1 \\ \mathrm{AB+2+mC} & -1 & \cdots & -1 & \cdots & -1 & (2m+2)p-1 & -1 & -1 \\ \end{array}$$ Thus we have the conclusion For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the dihedral group $D_p$ ($p$ is even), we have found its multiplicity vector 1. **mC**: - $(m,\:m-1,\:m-1,\:m-1)$ for $p=2$; - $(2m-1,\:m,\:m-1,\:m,\:m)$ for $p=4$; - $(2m-1,\:2m,\:\cdots,\:2m,\:m,\:m-1,\:m,\:m)$ for $p\ge6$; 2. **A+mC**: - $(m,\:m,\:m-1,\:m)$ for $p=2$; - $(2m,\:m,\:m,\:m,\:m+1)$ for $p=4$; - $(2m,\:2m+1,\:\cdots,\:2m+1,\:m,\:m,\:m,\:m+1)$ for $p\ge6$; 3. **AB+mC**: - $(m,\:m+1,\:m,\:m)$ for $p=2$; - $(2m+1,\:m,\:m+1,\:m+1,\:m+1)$ for $p=4$; - $(2m+1,\:2m+2,\:\cdots,\:2m+2,\:m,\:m+1,\:m+1,\:m+1)$ for $p\ge6$; 4. **2+mC**: - $(m,\:m-1,\:m,\:m)$ for $p=2$; - $(2m,\:\cdots,\:2m,\:m,\:m-1,\:m,\:m)$ for $p\ge4$; 5. **A+2+mC**: - $(m,\:m,\:m,\:m+1)$ for $p=2$; - $(2m+1,\:\cdots,\:2m+1,\:m,\:m,\:m,\:m+1)$ for $p\ge4$; 6. **AB+2+mC**: - $(m,\:m+1,\:m+1,\:m+1)$ for $p=2$; - $(2m+2,\:\cdots,\:2m+2,\:m,\:m+1,\:m+1,\:m+1)$ for $p\ge4$. When $G_{\boldsymbol{\lambda}}$ is isomorphic to $D_2$, $\boldsymbol{\lambda}$ is of type 2+mC if and only if $\boldsymbol{\lambda}$ is of type A+mC, and $\boldsymbol{\lambda}$ is of type A+2+mC if and only if $\boldsymbol{\lambda}$ is of type AB+mC. Notice that the multiplicity vectors are different when $\boldsymbol{\lambda}$ is viewed as an element of different types. Representations of the Cyclic Group $\mathbb{Z}_p$ {#Z_pprep} -------------------------------------------------- For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the cyclic group $\mathbb{Z}_p$, recall from Section \[sing\] that $$\Phi_{\boldsymbol{\lambda}}:\:G_{\boldsymbol{\lambda}}\to\mathcal{A}_{[\boldsymbol{\lambda}]},\:g_\sigma\mapsto f^{\boldsymbol{\lambda}}_\sigma$$ is a group isomorphism. From Section \[Z\_n\] we know that $[\boldsymbol{\lambda}]$ is equivalent to one of the following sets (which are categorized into three types): 1. **mC**: $C_p(a_1)\cup C_p(a_2)\cup\cdots\cup C_p(a_m)$, where $C_p(a_1),\:C_p(a_2),\:\cdots,\:C_p(a_m)$ are different orbits of order $p$, $m\in\mathbb {N^*}$; 2. **1+mC**: $\{0\}\cup C_p(a_1)\cup C_p(a_2)\cup\cdots\cup C_p(a_m)$, where $C_p(a_1),\:C_p(a_2),\:\cdots,\:C_p(a_m)$ are different orbits of order $p$, $m\in\mathbb {N^*}$; 3. **2+mC**: $\{0,\:\infty\}\cup C_p(a_1)\cup C_p(a_2)\cup\cdots\cup C_p(a_m)$, where $C_p(a_1),\:C_p(a_2),\:\cdots,\:C_p(a_m)$ are different orbits of order $p$, $m\in\mathbb {N^*}$. Set $$w=e^{\frac{2\pi}{p}i}.$$ ### Representations of $\mathbb{Z}_p$ when $\boldsymbol{\lambda}$ is of type mC or 2+mC When $\boldsymbol{\lambda}$ is of type mC or 2+mC, let $\rho$ denote an element in $G_{\boldsymbol{\lambda}}$ such that $\Phi_{\boldsymbol{\lambda}}(\rho)$ is a rotation of $\frac{2\pi}{p}$. Set $$K^{(k)}_{\boldsymbol{\lambda}}=\{\rho^k\}$$ for $k=1,\:2,\:\cdots,\:p$, and we have got the $p$ conjugate classes of $G_{\boldsymbol{\lambda}}$. Now fix $K^{(1)}_{\boldsymbol{\lambda}},\:K^{(2)}_{\boldsymbol{\lambda}},\cdots, K^{(p)}_{\boldsymbol{\lambda}}$. Table \[chaZ\_p1\] is the character table of $G_{\boldsymbol{\lambda}}$, with $X^{(1)}_{\boldsymbol{\lambda}},\: X^{(2)}_{\boldsymbol{\lambda}},\:\cdots,\:X^{(p)}_{\boldsymbol{\lambda}}$ representing the $p$ different irreducible representations of $G_{\boldsymbol{\lambda}}$ and $\chi^{(1)}_{\boldsymbol{\lambda}},\: \chi^{(2)}_{\boldsymbol{\lambda}},\:\cdots,\:\chi^{(p)}_{\boldsymbol{\lambda}}$ their characters. $$\begin{array}{c|ccccccc} & K^{(1)}_{\boldsymbol{\lambda}} & K^{(2)}_{\boldsymbol{\lambda}} & \cdots & K^{(k)}_{\boldsymbol{\lambda}} & \cdots & K^{(p)}_{\boldsymbol{\lambda}} \\ \hline \chi^{(1)}_{\boldsymbol{\lambda}} & w & w^2 & \cdots & w^k & \cdots & 1 \\ \chi^{(2)}_{\boldsymbol{\lambda}} & w^2 & w^4 & \cdots & w^{2k} & \cdots & 1 \\ \vdots & \vdots & \vdots & & \vdots & & \vdots \\ \chi^{(l)}_{\boldsymbol{\lambda}} & w^l & w^{2l} & \cdots & w^{kl} & \cdots & 1 \\ \vdots & \vdots & \vdots & & \vdots & & \vdots \\ \chi^{(p)}_{\boldsymbol{\lambda}} & 1 & 1 & \cdots & 1 & \cdots & 1 \\ \end{array}$$ For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the cyclic group $\mathbb{Z}_p$ and $\boldsymbol{\lambda}$ is of type mC or 2+mC, assume that $X_{\boldsymbol{\lambda}}=a_1X^{(1)}_{\boldsymbol{\lambda}}\oplus\cdots\oplus a_{p}X^{(p)}_{\boldsymbol{\lambda}}$. Then call $(a_1,\:a_2,\:\cdots,\:a_p)$ the **multiplicity vector** of $\boldsymbol{\lambda}$. The definition seems ambiguous as $\rho$ is chosen randomly. But we shall prove that $(a_1,\:a_2,\:\cdots,\:a_p)$ remains the same however we choose $\rho$, which makes the notion multiplicity vector well-defined. From the theorems in Section \[cha\] we have already known the character of $X_{\boldsymbol{\lambda}}$. The results are shown in Table \[chaZp1\]. $$\begin{array}{c|ccccccc} & K^{(1)}_{\boldsymbol{\lambda}} & K^{(2)}_{\boldsymbol{\lambda}} & \cdots & K^{(k)}_{\boldsymbol{\lambda}} & \cdots & K^{(p-1)}_{\boldsymbol{\lambda}} & K^{(p)}_{\boldsymbol{\lambda}} \\ \hline mC & -1-2\cos{\frac{2\pi}{p}} & -1-2\cos{\frac{4\pi}{p}} & \cdots & -1-2\cos{\frac{2k\pi}{p}} & \cdots & -1-2\cos{\frac{2(p-1)\pi}{p}} & mp-3 \\ 2+mC & -1 & -1 & \cdots & -1 & \cdots & -1 & mp-1 \\ \end{array}$$ Thus we have the conclusion that the multiplicity vector of $\boldsymbol{\lambda}$ is 1. **mC**: - $(m-2,\:m-1)$ for $p=2$; - $(m-1,\:m-1,\:m-1)$ for $p=3$; - $(m-1,\:m,\:\cdots,\:m,\:m-1,\:m-1)$ for $p\ge4$; 2. **2+mC**: $(m,\:\cdots,\:m,\:m-1)$. ### Representations of $\mathbb{Z}_p$ when $\boldsymbol{\lambda}$ is of type 1+mC When $\boldsymbol{\lambda}$ is of type 1+mC, suppose that $\sigma(a)=a$ for each $g_\sigma\in G_{\boldsymbol{\lambda}}$. Let $\psi$ be any linear fractional transformation such that $$\psi(z^{\boldsymbol{\lambda}}_a)=0$$ and $$\psi\circ\Phi_{\boldsymbol{\lambda}}(G_{\boldsymbol{\lambda}})\circ\psi^{-1} =\langle z\mapsto wz\rangle\simeq \mathbb{Z}_p.$$ Let $\rho$ denote the element in $G_{\boldsymbol{\lambda}}$ such that $$\psi\circ \Phi_{\boldsymbol{\lambda}}(\rho)\circ\psi^{-1}(z)=wz.$$ Set $$K^{(k)}_{\boldsymbol{\lambda}}=\{\rho^k\}$$ for $k=1,\:2,\:\cdots,\:p$, and we have got the $p$ conjugate classes of $G_{\boldsymbol{\lambda}}$. Now fix $K^{(1)}_{\boldsymbol{\lambda}},\:K^{(2)}_{\boldsymbol{\lambda}},\cdots, K^{(p)}_{\boldsymbol{\lambda}}$. Table \[chaZ\_p2\] is the character table of $G_{\boldsymbol{\lambda}}$, with $X^{(1)}_{\boldsymbol{\lambda}},\: X^{(2)}_{\boldsymbol{\lambda}},\:\cdots,\:X^{(p)}_{\boldsymbol{\lambda}}$ representing the $p$ different irreducible representations of $G_{\boldsymbol{\lambda}}$ and $\chi^{(1)}_{\boldsymbol{\lambda}},\: \chi^{(2)}_{\boldsymbol{\lambda}},\:\cdots,\:\chi^{(p)}_{\boldsymbol{\lambda}}$ their characters. $$\begin{array}{c|ccccccc} & K^{(1)}_{\boldsymbol{\lambda}} & K^{(2)}_{\boldsymbol{\lambda}} & \cdots & K^{(k)}_{\boldsymbol{\lambda}} & \cdots & K^{(p)}_{\boldsymbol{\lambda}} \\ \hline \chi^{(1)}_{\boldsymbol{\lambda}} & w & w^2 & \cdots & w^k & \cdots & 1 \\ \chi^{(2)}_{\boldsymbol{\lambda}} & w^2 & w^4 & \cdots & w^{2k} & \cdots & 1 \\ \vdots & \vdots & \vdots & & \vdots & & \vdots \\ \chi^{(l)}_{\boldsymbol{\lambda}} & w^l & w^{2l} & \cdots & w^{kl} & \cdots & 1 \\ \vdots & \vdots & \vdots & & \vdots & & \vdots \\ \chi^{(p)}_{\boldsymbol{\lambda}} & 1 & 1 & \cdots & 1 & \cdots & 1 \\ \end{array}$$ For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the cyclic group $\mathbb{Z}_p$ and $\boldsymbol{\lambda}$ is of type 1+mC, assume that $X_{\boldsymbol{\lambda}}=a_1X^{(1)}_{\boldsymbol{\lambda}}\oplus\cdots\oplus a_{p}X^{(p)}_{\boldsymbol{\lambda}}$. Then call $(a_1,\:a_2,\:\cdots,\:a_p)$ the **multiplicity vector** of $\boldsymbol{\lambda}$. The definition seems ambiguous as $\psi$ is chosen randomly. But we shall prove that $(a_1,\:a_2,\:\cdots,\:a_p)$ remains the same however we choose $\psi$, which makes the notion multiplicity vector well-defined. From the theorems in Section \[cha\] we have already known the character of $X_{\boldsymbol{\lambda}}$. The results are shown in Table \[chaZp2\]. $$\begin{array}{c|ccccccc} & K^{(1)}_{\boldsymbol{\lambda}} & K^{(2)}_{\boldsymbol{\lambda}} & \cdots & K^{(k)}_{\boldsymbol{\lambda}} & \cdots & K^{(p-1)}_{\boldsymbol{\lambda}} & K^{(p)}_{\boldsymbol{\lambda}} \\ \hline 1+mC & -1-w^{-1} & -1-w^{-2} & \cdots & -1-w^{-k} & \cdots & -1-w^{-(p-1)} & mp-2 \\ \end{array}$$ Thus we have the conclusion that the multiplicity vector of $\boldsymbol{\lambda}$ is 1. **1+mC**: - $(m-1,\:m-1)$ for $p=2$; - $(m,\:\cdots,\:m,\:m-1,\:m-1)$ for $p\ge3$. For any $\boldsymbol{\lambda}\in K_n$ such that $G_{\boldsymbol{\lambda}}$ is isomorphic to the cyclic group $\mathbb{Z}_p$, we have found its multiplicity vector 1. **mC**: - $(m-2,\:m-1)$ for $p=2$; - $(m-1,\:m-1,\:m-1)$ for $p=3$; - $(m-1,\:m,\:\cdots,\:m,\:m-1,\:m-1)$ for $p\ge4$; 2. **1+mC**: - $(m-1,\:m-1)$ for $p=2$; - $(m,\:\cdots,\:m,\:m-1,\:m-1)$ for $p\ge3$. 3. **2+mC**: $(m,\:\cdots,\:m,\:m-1)$. The Group that Fixes Four Points {#n=4} ================================ Let $\alpha=\{z_1, z_2, z_3, z_4\} \subseteq \widehat {\mathbb C}$, we define $\lambda$ to be the cross-ratio $[{z}_1, {z}_2, {z}_3, {z}_4]$. Define a function $\pi: {S}_4 \to \mathbb C \;\; \sigma \mapsto [{z}_{\sigma(1)}, {z}_{\sigma(2)}, {z}_{\sigma(3)}, {z}_{\sigma(4)}$\]. Then we have $$\pi((1,2))=[{z}_2, {z}_1, {z}_3, {z}_4]=\frac{({z}_2-{z}_3)({z}_1-{z}_4)}{({z}_2-{z}_1)({z}_3-{z}_4)}=1-\lambda;$$ $$\pi((1,3))=[{z}_3, {z}_2, {z}_1, {z}_4]=\frac{({z}_3-{z}_1)({z}_2-{z}_4)}{({z}_3-{z}_2)({z}_1-{z}_4)}=\frac{\lambda}{\lambda-1};$$ $$\pi((1,4))=[{z}_4, {z}_2, {z}_3, {z}_1]=\frac{({z}_4-{z}_3)({z}_2-{z}_1)}{({z}_4-{z}_2)({z}_3-{z}_1)}=\frac{1}{\lambda}.$$ By calculation we have $$\pi((1,2))=\pi((3,4))=\pi((1,3,2,4))=\pi((1,4,2,3))=1-\lambda;$$ $$\pi((1,3))=\pi((2,4))=\pi((1,2,3,4))=\pi((1,4,3,2))=\frac{\lambda}{\lambda-1};$$ $$\pi((1,4))=\pi((2,3))=\pi((1,2,4,3))=\pi((1,3,4,2))=\frac{1}{\lambda};$$ $$\pi((1,2,3))=\pi((2,4,3))=\pi((3,4,1))=\pi((4,2,1))=\frac{\lambda-1}{\lambda};$$ $$\pi((1,3,2))=\pi((2,3,4))=\pi((3,1,4))=\pi((4,1,2))=\frac{1}{1-\lambda};$$ $$\pi((1,2)(3,4))=\pi((1,3)(2,4))=\pi((1,4)(3,2))=\lambda.$$ On one hand, for any $f\in\mathcal{A_\alpha}$, we have $$\pi(\sigma_f)=[z_{\sigma_f(1)}, z_{\sigma_f(2)}, z_{\sigma_f(3)}, z_{\sigma_f(4)}]=[f(z_1), f(z_2), f(z_3), f(z_4)]=[z_1, z_2, z_3, z_4]=\lambda.$$ On the other hand, if there exists some $\sigma \in {S}_4$ s.t.  $$\pi(\sigma)=[z_{\sigma(1)}, z_{\sigma(2)}, z_{\sigma(3)}, z_{\sigma(4)}]=[z_1, z_2, z_3, z_4]=\lambda,$$ there must be some $f\in\mathcal{A_\alpha}$ s.t. $f(z_i)=z_{{\sigma}(i)}, i=1, 2, 3, 4$. Now, all we have to do is to find $\sigma \in {S}_4 \textrm{ s.t. } \lambda = \pi (\sigma)$. If $\lambda=1/2$, we have $1-\lambda=\lambda, \frac{\lambda}{\lambda-1}\ne\lambda, \frac{1}{\lambda}\ne\lambda, \frac{\lambda-1}{\lambda}\ne\lambda, \frac{1}{1-\lambda}\ne\lambda$. $$\{\sigma_f\in S_4|f\in\mathcal{A_\alpha}\}=\{ e,(1,2),(3,4),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3),(1,3,2,4),(1,4,2,3)\}\cong{D}_4.$$ If $\lambda=2$, we have $\frac{\lambda}{\lambda-1}=\lambda, 1-\lambda\ne\lambda, \frac{1}{\lambda}\ne\lambda, \frac{\lambda-1}{\lambda}\ne\lambda, \frac{1}{1-\lambda}\ne\lambda$. $$\{\sigma_f\in S_4|f\in\mathcal{A_\alpha}\}=\{e,(1,3),(2,4),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3),(1,2,3,4),(1,4,3,2)\}\cong {D}_4$$ If $\lambda=-1$, we have $\frac{1}{\lambda}=\lambda, 1-\lambda\ne\lambda, \frac{\lambda}{\lambda-1}\ne\lambda, \frac{\lambda-1}{\lambda}\ne\lambda, \frac{1}{1-\lambda}\ne\lambda$. $$\{\sigma_f\in S_4|f\in\mathcal{A_\alpha}\}=\{e,(1,4),(2,3),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3),(1,2,4,3),(1,3,4,2)\}\cong {D}_4$$ If $\lambda = \frac{1\pm \sqrt 3 i}{2}$, we have $ \frac{\lambda-1}{\lambda}=\frac{1}{1-\lambda}=\lambda, 1-\lambda\ne\lambda, \frac{\lambda}{\lambda-1}\ne\lambda, \frac{1}{\lambda}\ne\lambda$. $$\begin{aligned} \{\sigma_f\in S_4|f\in\mathcal{A_\alpha}\} &=& \{e,(1,2)(3,4),(1,3)(2,4),(1,4)(2,3),(1,2,3),(1,3,2),(2,3,4),(2,4,3),\\ && \:\:\:(3,4,1),(3,1,4),(4,1,2),(4,2,1)\}={A}_4.\end{aligned}$$ If $\lambda \ne 1/2, 2, -1, \frac{1\pm \sqrt 3 i}{2}$, now $1-\lambda\ne\lambda, \frac{\lambda}{\lambda-1}\ne\lambda, \frac{1}{\lambda}\ne\lambda, \frac{\lambda-1}{\lambda}\ne\lambda, \frac{1}{1-\lambda}\ne\lambda$. $$\{\sigma_f\in S_4|f\in\mathcal{A_\alpha}\}=\{e,(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}\cong K_4.$$ The Group that Fixes Five Points {#n=5} ================================ Let ${ z }_{ 1 },{ z }_{ 2 },{ z }_{ 3 },{ z }_{ 4 }$ be $0, 1, \lambda, \infty $, respectively, $\lambda \in \mathbb{C} \backslash \{0, 1\}$. For $\sigma \in S_4$, let $f_\sigma$ denote the linear fractional transformation which maps $z_i$ to $z_{\sigma (i)}, i=1, 2, 3, 4$ (if $f_\sigma$ exists). From the previous section we know that ${ f }_{ (1,2)(3,4) },{ f }_{ (1,3)(2,4) },{ f }_{ (1,4)(2,3) }$ always exist regardless of the choice of $\lambda$. By calculation we know that $$f_{(1,2)(3,4)}(z)=\frac{\lambda z - \lambda}{z-\lambda},{ f }_{ (1,3)(2,4) }(z)=\frac{z-\lambda}{z-1},{ f }_{ (1,4)(2,3)}(z)=\frac{\lambda}{z},$$ which fix $\lambda \pm \sqrt { { \lambda }^{ 2 }-\lambda }, 1\pm \sqrt { 1-\lambda } ,\pm \sqrt { \lambda } $ respectively. When $n=5$, $\alpha=\{z_1, z_2, z_3, z_4, z_5\}\subseteq\widehat{\mathbb{C}}$. Since any non-trivial $f\in\mathcal{A_\alpha}$ is elliptic, $\sigma_f$ must be of Type $(5),(4,1),(3,1,1)$ or $(2,2,1)$. There are five possibilities: 1. There exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type $(5)$; 2. For any $f\in\mathcal{A_\alpha}, \sigma_f$ is not of Type $(5)$, but there exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type $(4, 1)$; 3. For any $f\in\mathcal{A_\alpha}, \sigma_f$ is not of Type $(5)$ or $(4,1)$, but there exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type $(3,1,1)$; 4. For any $f\in\mathcal{A_\alpha}, \sigma_f$ is not of Type $(5),(4,1)$ or $(3,1,1)$, but there exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type $(2,2,1)$; 5. $\mathcal{A_\alpha}$ is the trivial group. In the rest of this section we shall discuss the five possibilities one by one, and prove the following theorem For $\alpha=\{z_1,\: z_2, \:z_3,\: z_4, \:z_5\}\subseteq\widehat{\mathbb{C}}$, $\mathcal{A_\alpha}$ is isomorphic to $D_5$, $\mathbb{Z}_4$, $D_3$, $\mathbb{Z}_2$ or the trivial group $\{\mathrm{Id}\}$. $\mathcal{A_\alpha}$ is Isomorphic to $D_5$ {#n=5, D_5} ------------------------------------------- In the first case assume that there exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type $(5)$. This assumption amounts to the existence of some linear fractional transformation $\psi$ s.t. $$\psi(\alpha)=\{1, w, w^2, w^3, w^4\},\: w=e^{\frac{2\pi}{5}i}.$$ Without lose of generality we assume that $$\alpha=\left\{ 1,w,{ w }^{ 2 },{ w }^{ 3 },{ w }^{ 4 } \right\},\: f(z)=e^{\frac{2\pi}{5}i}z.$$ Define another linear fractional transformation $g$ $$g(z)=\frac{1}{z}.$$ It is obvious that $f,g \in \mathcal{A_\alpha}$. Note that $\langle f,g\rangle\simeq D_5$ acts transitively on $\alpha$. For any $k \in \mathcal{A_\alpha}$, there exists some $l\in \langle f,g\rangle$ s.t. $h=l\circ k$ fixes $w^4$. We have $$h(\{1, w, w^2, w^3\})=\{1, w, w^2, w^3\},\: h(w^4)=w^4.$$ Let $\phi$ be the linear fractional transformation which maps $1, w, w^3$ to $0, 1, \infty$ respectively. Define $\lambda$ to be the image of $w^2$, so we have $$\lambda=\phi(w^2)=[0,1,\phi(w^2),\infty]=[1, w, w^2, w^3]=w^4+w+2,$$ and $$\phi(w^4)=[0,1,\phi(w^4),\infty]=[1, w, w^4, w^3]=w^3+w^2.$$ Thus $$\phi \circ h \circ \phi^{-1}(\{0,1,\lambda,\infty\})=\{0,1,\lambda,\infty\},\: \phi \circ h \circ \phi^{-1}(w^3+w^2)=w^3+w^2.$$ As a result, $\phi \circ h \circ \phi^{-1} \in \mathcal{A}_{\{0, 1, \lambda, \infty\}}$. As $[0, 1, \lambda, \infty]=\lambda=w^4+w+2=\frac{\sqrt 5 +3}{2} \ne 2, \frac{1}{2}, -1, \frac{1\pm \sqrt3 i}{2}$, we know from \[n=4\] that $\mathcal{A}_{\{0, 1, \lambda, \infty\}}=\{I, { f }_{ (1,2)(3,4) },{ f }_{ (1,3)(2,4) },{ f }_{ (1,4)(2,3) }\}.$ However $f_{(1,2)(3,4)}, f_{(1,3)(2,4)}, f_{(1,4)(2,3)}$ fix $\lambda \pm \sqrt { { \lambda }^{ 2 }-\lambda }, 1\pm \sqrt { 1-\lambda } ,\pm \sqrt { \lambda }$ respectively, and $$w^3+w^2 \ne \lambda \pm \sqrt { { \lambda }^{ 2 }-\lambda }, 1\pm \sqrt { 1-\lambda } ,\pm \sqrt { \lambda }.$$ So we have $\phi \circ h \circ \phi^{-1} = I$, and thus $k = l^{-1}\in \langle f,g\rangle$. In conclusion we have $$\mathcal{A_\alpha} = \langle z \mapsto e^{\frac{2\pi}{5}i}z,z \mapsto \frac{1}{z}\rangle \simeq D_5.$$ $\mathcal{A_\alpha}$ is Isomorphic to $\mathbb{Z}_4$ ---------------------------------------------------- In this case assume that for any $h\in\mathcal{A_\alpha}$, $\sigma_h$ is not of Type $(5)$, but there exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type $(4, 1)$. Under this assumption, there exists some linear fractional transformation $\psi$ s.t. $$\psi(\alpha) =\{0, 1, i, -1, -i\}.$$ On the other hand, for any $\alpha=\{z_1, z_2, z_3, z_4, z_5\}\subseteq \widehat{\mathbb C}$, if there exists some linear fractional transformation $\psi$ s.t. $\psi(\alpha)=\{0, 1, i, -1, -i\}$, we see that ${\psi}^{-1} \circ f \circ \psi$ fixes $\alpha$ and is of Type $(4,1)$, where $f(z)=iz$. However, since the five points in $\alpha$ are not concyclic, then for any $h\in\mathcal{A_\alpha}$, $\sigma_h$ is not of Type $(5)$. Thus we see that the assumption amounts to the existence of some linear fractional transformation $\psi$ s.t. $$\alpha=\{0, 1, i, -1, -i\},\: f(z)=iz.$$ It is obvious that $f \in\mathcal{A_\alpha}$. For any $g \in \mathcal{A_\alpha}$, it is easy to see that $$|g(\{\pm 1, \pm i\})\cap\{\pm 1, \pm i\}|\ge3.$$ So $g$ fixes the unit circle, and thus $g(0)=0$. The linear fractional transformation $$\phi(z)=\frac{(1-i)z+i-1}{z+i}$$ takes $1, i, -i$ and $0$ to $0, 1, \infty$ and $1+i$ respectively. Define $\lambda$ to be the image of $-1$, so we have $$\lambda=\phi(-1)=[0,1,\phi(-1),\infty]=[1, i, -1, -i]=2.$$ As $g$ fixes $0$, we see that $$\phi \circ g \circ \phi^{-1}(\{0,1,\lambda,\infty\})=\{0,1,\lambda,\infty\}, \phi \circ g \circ \phi^{-1}(1+i)=1+i.$$ As $[0, 1, \lambda, \infty]=\lambda=2$, from \[n=4\] we know that $$\mathcal{A}_{\{0, 1, \lambda, \infty\}}=\{I, f_{(1,2,3,4)}, f_{(1,3)(2,4)}, f_{(1,4,3,2)}, f_{(1,3)}, f_{(2,4)}, f_{(1,2)(3,4)}, f_{(1,4)(2,3)}\}.$$ But $f_{(1,2)(3,4)}, f_{(1,4)(2,3)}$ fix $\lambda\pm\sqrt{{\lambda}^2-\lambda}=2\pm\sqrt2$ and $ \pm\sqrt{\lambda}=\pm\sqrt2$ respectively, and $f_{(1,3)}, f_{(2,4)}$ fixes $\pm i$ and $\pm 1$ respectively. So none of them fixes $1+i$. Thus we have $$\phi \circ g \circ \phi^{-1}=f_{(1,2,3,4)},$$ or $$\phi \circ g \circ \phi^{-1}=f_{(1,3)(2,4)},$$ or $$\phi \circ g \circ \phi^{-1}=f_{(1,4,3,2)},$$ or $$\phi \circ g \circ \phi^{-1}=I.$$ However $$\phi^{-1} \circ f_{(1,2,3,4)}\circ\phi=f,2: \phi^{-1} \circ f_{(1,3)(2,4)}\circ\phi=f^2,\: \phi^{-1} \circ f_{(1,4,3,2)}\circ\phi=f^3,\: \phi^{-1} \circ I\circ\phi=I.$$ So we have $g\in \langle f \rangle$. In conclusion we have $$\mathcal{A_\alpha} = \langle z \mapsto iz\rangle \simeq \mathbb Z _4.$$ $\mathcal{A_\alpha}$ is Isomorphic to $D_3$ ------------------------------------------- In this case assume that for any $h\in\mathcal{A_\alpha}$, $\sigma_h$ is not of Type $(5)$ or $(4, 1)$, but there exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type $(3, 1, 1)$. Under this assumption, there exists some linear fractional transformation $\psi$ s.t. $$\psi(\alpha)=\{0, \infty, 1, w, w^2\},\: w=e^{\frac{2\pi}{3}i}.$$ On the other hand, for any $\alpha=\{z_1, z_2, z_3, z_4, z_5\}\subseteq \widehat{\mathbb C}$, if there exists some linear fractional transformation $\psi$ s.t. $\psi(\alpha)=\{0, \infty, 1, w, w^2\}$, we see that ${\psi}^{-1} \circ f \circ \psi$ fixes $\alpha$ and is of Type $(3, 1, 1)$, where $f(z)=wz$. However, since no four points are concyclic, then for any $h\in\mathcal{A_\alpha}$, $\sigma_h$ is not of Type $(5)$ or $(4, 1)$. Thus we see that the assumption amounts to the existence of some linear fractional transformation $\psi$ s.t. $$\psi(\alpha)=\{0, \infty, 1, w, w^2\},\: w=e^{\frac{2\pi}{3}i}.$$ Without lose of generality assume that $$\alpha=\{0, \infty, 1, w, w^2\},\: f(z)=wz.$$ Define another linear fractional transformation $g$ $$g(z)=\frac{1}{z}.$$ It is obvious that $f,g \in \mathcal{A_\alpha}$. Note that there are two orbits of $\langle f,g\rangle$: $\{0, \infty\}$, and $\{1, w, w^2\}$. For any $k\in\mathcal{A_\alpha}$, it is easy to see that $$|k(\{1, w, w^2\})\cap\{1, w, w^2\}|\ge1.$$ So there exists some $i, j\in \mathbb{Z}$ s.t. $k(w^i)=w^j$. Let $h=f^{1-j}\circ k\circ f^{i-1}$, we see that $h(w)=w$. Thus $$h (\{0,1,w^2,\infty\})=\{0,1,w^2,\infty\},\: h(w)=w.$$ As a result, $h \in \mathcal{A}_{\{0, 1, w^2, \infty\}}$. Set $\lambda=w^2$. As $[0, 1, \lambda, \infty]=\lambda=w^2\ne 2, \frac{1}{2}, -1, \frac{1\pm \sqrt3 i}{2}$, we know from \[n=4\] that $\mathcal{A}_{\{0, 1, \lambda, \infty\}}=\{I, { f }_{ (1,2)(3,4) },{ f }_{ (1,3)(2,4) },{ f }_{ (1,4)(2,3) }\}.$ But $f_{(1,2)(3,4)}, f_{(1,3)(2,4)}$ fix $\lambda \pm \sqrt { { \lambda }^{ 2 }-\lambda }, 1\pm \sqrt{1-\lambda}$ respectively, and $$w \ne \lambda \pm \sqrt { { \lambda }^{ 2 }-\lambda },\: 1\pm \sqrt { 1-\lambda }.$$ So we have $h=f_{(1,4)(2,3)}$ or $h=I$. However $$f_{(1,4)(2,3)}=f\circ g\in\langle f,g\rangle,\: \textrm{I}\in\langle f,g\rangle,$$ thus we conclude that $k=f^{1-i}\circ h\circ f^{j-1}\in\langle f,g\rangle$ In conclusion we have $$\mathcal{A_\alpha} = \langle z \mapsto e^{\frac{2\pi}{3}i}z,z \mapsto \frac{1}{z}\rangle \simeq D_3.$$ $\mathcal{A_\alpha}$ is Isomorphic to $\mathbb{Z}_2$ ---------------------------------------------------- In this case assume that for any $h\in\mathcal{A_\alpha}$, $\sigma_h$ is not of Type $(5),(4,1)$ or $(3,1,1)$, but there exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type $(2, 2, 1)$. Under this assumption, there exists some linear fractional transformation $\psi$ s.t. $$\psi(\alpha)=\{0, 1, -1, a, -a\},\: a\ne0, \pm 1.$$ On the other hand, for any $\{0, 1, -1, a, -a\}, a \in \mathbb{C} \backslash \{ 0, \pm 1 \}$, the linear fractional transformation $z\mapsto -z$ is in $\mathcal{A}_{\{0, 1, -1, a, -a\}}$ and of Type $(2, 2, 1)$. Now we aim to find out the specific value $a$ takes when for each element $h\in \mathcal{A}_{\{0, 1, -1, a, -a\}}$, $\sigma_h$ is not of Type $(5), (4, 1)$ or $(3, 1, 1)$. *Case 1*: There exists some $h\in\mathcal{A}_{\{0, 1, -1, a, -a\}}$ s.t. $\sigma_h$ is of Type (5). This amounts to the existence of some linear fractional transformation $\psi$ s.t. $$\psi(\{0, 1, -1, a, -a\})=\{1, w, w^2, w^3, w^4\}, \:w=e^{\frac{2\pi}{5}i}.$$ We conclude that $0, 1, -1, a, -a$ lie on the same circle: the real axis, and $a\in\mathbb{R}$. Without lose of generality assume that $r=a>0$. If $r>1$, we may assume that $ \psi(-r)=w^4, \psi(-1)=1, \psi(0)=w, \psi(1)=w^2, \psi(r) =w^3$. We have $$[1,w,w^2,w^3]=[-1,0,1,r],$$ and thus $$r=2w^4+2w+3=\sqrt{5}+2.$$ If $0<r<1$, we assume that $\psi(-1)=w^4, \psi(-r)=1, \psi(0)=w, \psi(r)=w^2, \psi(1)=w^3$. By exactly the same means we conclude that $$r=2w^4+2w-1=\sqrt{5}-2.$$ On the other hand, the linear fractional transformation $$\psi_1(z)=\frac{(w^2-w)z+w^2+w}{(1-w)z+w+1}$$ maps $-\sqrt 5 -2, -1, 0, 1$ and $\sqrt 5+2$ to $w^4, 1, w, w^2$ and $w^3$ respectively, and $$\psi_2(z)=\frac{(w^2+3w+1)z+w^4-w^3}{(w^3+w^2-2)z+w^3-w^2}$$ maps $-1,-\sqrt 5 +2, 0, \sqrt 5-2$ and $1$ to $w^4, 1, w, w^2$ and $w^3$ respectively. And we conclude that there exists some $h\in\mathcal{A}_{\{0, 1, -1, a, -a\}}$ s.t. $\sigma_h$ is of Type (6) if and only if $z=\pm\sqrt{5}\pm2$. *Case 2*: There exists some $h\in\mathcal{A}_{\{0, 1, -1, a, -a\}}$ s.t. $\sigma_h$ is of Type (4, 1). This amounts to the existence of some linear fractional transformation $\psi$ s.t. $$\psi(\{0, 1, -1, a, -a\})=\{0, 1, i, -1, -i\}.$$ It is obvious to see that such a $\psi$ exists if and only if $a=\pm i$. *Case 3*: There exists some $h\in\mathcal{A}_{\{0, 1, -1, a, -a\}}$ s.t. $\sigma_h$ is of Type (3, 1, 1). This amounts to the existence of some linear fractional transformation $\psi$ s.t. $$\psi(\{0, 1, -1, a, -a\})=\{0, \infty, 1, w, w^2\}, \:w=e^{\frac{2\pi}{3}i}.$$ The situation now is a little complex. We can see that the linear fractional transformation $f(z)=-z$ leaves $\{0, 1, -1, a, -a\}$ invariant, is of order 2, and fixes the point $0$. However, from the above section we know that $\mathcal{A}_{\psi(\{0, 1, -1, a, -a\})}=\langle z\mapsto wz, z\mapsto\frac{1}{z}\rangle \simeq D_3 $. The only three linear fractional transformations of order 2 in $\mathcal{A}_{\psi(\{0, 1, -1, a, -a\})}$ are $$z\mapsto\frac{1}{z}, z\mapsto\frac{w}{z}, z\mapsto\frac{w^2}{z},$$ which fixes $1, w^2, w$ respectively. So $\psi(0)\ne0,\infty$. Without lose of generality we assume that $\psi(0)=1$. The only element of order 2 fixing $1$ in $\mathcal{A}_{\psi(\{0, 1, -1, a, -a\})}$ is $h(z)=\frac{1}{z}$. However, the element $f(z)=-z$ is of order 2, fixes $0$, and is in $\mathcal{A}_{\{0, 1, -1, a, -a\}}$. So we have $h=\psi \circ f \circ {\psi}^{-1}$. As $h$ fixes $\{0, \infty \}$ and $\{1, -1\}$, and $f$ fixes $\{1,-1\}$ and $\{a, -a\}$, we see that there are only two possibilities. The first is $\psi(\{1, -1\})=\{0, \infty \}$, $\psi(\{a, -a\})=\{w, w^2\}$, and the second is $\psi(\{1, -1\})=\{w, w^2\}$, $\psi(\{a, -a\})=\{0, \infty \}$. In the former situation, assume that $\psi(0)=1, \psi(1)=\infty, \psi(-1)=0, \psi(a)=w, \psi(-a)=w^2$, and we have$$[0, 1, w, \infty]=[-1, 0, a, 1]$$ and $$a=\sqrt{3}i.$$ In the latter situation assume that $\psi(0)=1, \psi(1)=w, \psi(-1)=w^2, \psi(a)=0, \psi(-a)=\infty$, and we have $$[0, 1, w, \infty]=[a, 0, 1, -a]$$ and $$a=\frac{i}{\sqrt{3}}.$$ On the other hand, the linear fractional transformation $$\psi_1(z)=\frac{1+z}{1-z}$$ maps $0, 1, -1, \sqrt3 i, -\sqrt3 i$ to $1, \infty, 0, w, w^2$ respectively, and $$\psi_2(z)=\frac{i-\sqrt{3}z}{i+\sqrt{3}z}$$ maps $0, 1, -1, \frac{1}{\sqrt{3}}i, -\frac{1}{\sqrt{3}}i$ to $1, w, w^2, \infty, 0$ respectively. And we conclude that there exists some $h\in\mathcal{A}_{\{0, 1, -1, a, -a\}}$ s.t. $\sigma_h$ is of Type (3, 1, 1) if and only if $a=\pm\sqrt{3}i$ or $\pm\frac{1}{\sqrt{3}}i$. From the above discussion we see that the assumption amounts to the existence of some linear fractional transformation $\psi$ s.t. $$\psi(\alpha)=\{0, 1, -1, a, -a\},\: a \ne0, \pm 1, \pm\sqrt{5}\pm2, \pm i, \pm\sqrt{3}i\mathrm{~or}\pm\frac{i}{\sqrt{3}}.$$ Without lose of generality assume $$\alpha=\{0, 1, -1, a, -a\},\: a \ne0, \pm 1, \pm\sqrt{5}\pm2, \pm i, \pm\sqrt{3}i\mathrm{~or}\pm\frac{i}{\sqrt3},\: f(z)=-z.$$ So $\sigma_f =(1)(2,3)(4,5)$. If $\exists g\in\mathcal{A_\alpha}$ s.t. $\sigma_g(1)\ne1$, without lose of generality assume that $\sigma_g(1)=2$. We have $$\sigma_g=(1,2)(3,4)(5)=:\pi_1$$ or $$\sigma_g=(1,2)(4,5)(3)=:\pi_2$$ or $$\sigma_g=(1,2)(5,3)(4)=:\pi_3.$$ But $$\pi_1\sigma_f=(1,2,4,5,3)$$ and $$\pi_2\sigma_f=(1,2,3)$$ and $$\pi_3\sigma_f=(1,2,5,4,3).$$ thus such a $g$ does not exist, and every element in $\mathcal{A_\alpha}$ fixes $0$. If $\exists g \in {\mathcal{A_\alpha}}$ s.t. $g \ne f, g \ne I$, assume that $g(1)=a, g(-1)=-a, g(a)=1, g(-a)=-1$. So $$g(z)=\frac{a}{z}.$$ However, $g(0)\ne0$. So such a $g$ does not exist, too. In conclusion we have $$\mathcal{A_\alpha} = \langle z \mapsto -z\rangle \simeq \mathbb Z _2.$$ Conclusion {#n=5, con} ---------- From the above discussion we shall drive the following conclusion Set $\alpha=\{z_1,\: z_2,\: z_3,\: z_4,\: z_5\}\subseteq\widehat {\mathbb{C}}$. 1. If there exists some linear fractional transformation $\psi$ such that $$\psi(\alpha)=\{1,\: w,\: w^2,\: w^3\:, w^4\},\:w=e^{\frac{2\pi}{5}i},$$ then $$\mathcal{A_\alpha}={\psi}^{-1}\langle z \mapsto e^{\frac{2\pi}{5}i}z,\:z \mapsto \frac{1}{z}\rangle {\psi}\simeq D_5,$$ and its multiplicity vector is $(0,\:1,\:0,\:0)$; 2. if there exists some linear fractional transformation $\psi$ such that $$\psi(\alpha) =\{0,\: 1,\: i,\: -1,\: -i\},$$ then $$\mathcal{A_\alpha}={\psi}^{-1}\langle z \mapsto iz\rangle {\psi}\simeq \mathbb Z _4,$$ and its multiplicity vector is $(1,\:1,\:0,\:0)$; 3. if there exists some linear fractional transformation $\psi$ such that $$\psi(\alpha)=\{0,\: \infty,\: 1, \:w,\: w^2\},\:w=e^{\frac{2\pi}{3}i},$$ then $$\mathcal{A_\alpha}={\psi}^{-1}\langle z \mapsto e^{\frac{2\pi}{3}i}z,\:z \mapsto \frac{1}{z}\rangle\psi \simeq D_3,$$ and its multiplicity vector is $(1,\:0,\:0)$; 4. if there exists some linear fractional transformation $\psi$ such that $$\psi(\alpha)=\{0,\: 1, \:-1,\: a,\: -a\},\:a\ne0,\:\pm 1,$$ then 1. if $a=\pm(\sqrt{5}+2)\text{ or }\pm(\sqrt{5}-2)$, then there exists some linear fractional transformation $\phi$ such that $$\phi(\alpha)=\{1,\: w,\: w^2,\: w^3\:, w^4\},\:w=e^{\frac{2\pi}{5}i}$$ and this is case 1; 2. if $a=\pm i$, then $$\alpha =\{0,\: 1,\: i,\: -1,\: -i\}$$ and this is case 2; 3. if $a=\pm\sqrt{3}i\text{ or }\pm\frac{1}{\sqrt{3}}i$, then there exists some linear fractional transformation $\phi$ such that $$\phi(\alpha)=\{0,\: \infty,\: 1, \:w,\: w^2\},\:w=e^{\frac{2\pi}{3}i}$$ and this is case 3; 4. otherwise, $$\mathcal{A_\alpha}={\psi}^{-1}\langle z \mapsto -z\rangle {\psi}\simeq \mathbb Z _2,$$ and its multiplicity vector is $(1,\:1)$; 5. otherwise, $$\mathcal{A_\alpha}=\{\mathrm{Id}\}.$$ The Group that Fixes Six Points {#n=6} =============================== There is no much difference here from the previous section. When $n=6$, $\alpha=\{z_1, z_2, z_3, z_4, z_5, z_6\}\subseteq \widehat{\mathbb{C}}$. Since any non-trivial $f\in\mathcal{A}_\alpha$ is a rotation of finite order, $\sigma_f$ must be of Type (6), (5, 1), (4, 1, 1), (3, 3), (2, 2, 2), (2, 2, 1, 1) or identity. There are seven possibilities: 1. There exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type (6); 2. For any $f\in\mathcal{A_\alpha}$, $\sigma_f$ is not of Type (6), but there exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type (5, 1); 3. For any $f\in\mathcal{A_\alpha}$, $\sigma_f$ is not of Type (6) or (5, 1), but there exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type (4, 1, 1); 4. For any $f\in\mathcal{A_\alpha}$, $\sigma_f$ is not of Type (6), (5, 1) or (4, 1, 1), but there exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type (3, 3); 5. For any $f\in\mathcal{A_\alpha}$, $\sigma_f$ is not of Type (6), (5, 1), (4, 1, 1) or (3, 3), but there exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type (2, 2, 2); 6. For any $f\in\mathcal{A_\alpha}$, $\sigma_f$ is not of Type (6), (5, 1), (4, 1, 1), (3, 3) or (2, 2, 2), but there exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type (2, 2, 1, 1); 7. $\mathcal{A_\alpha}$ is the trivial group. In the rest of this section we shall discuss the five possibilities one by one, and prove the following theorem For $\alpha=\{z_1, z_2, z_3, z_4, z_5, z_6\}\subseteq\widehat{\mathbb{C}}$, $\mathcal{A_\alpha}$ is isomorphic to $D_5$, $\mathbb{Z}_5$, $S_4$, $D_3$, $K_4$, $\mathbb{Z}_2$ or the trivial group $\{\mathrm{Id}\}$. $\mathcal{A_\alpha}$ is Isomorphic to $D_6$ {#n=6, D_6} ------------------------------------------- In the first case assume that there exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type (6). This amounts to the existence of some linear fractional transformation $\psi$ s.t. $$\psi(\alpha)=\{1, w, w^2, w^3, w^4, w^5\}, w=e^{\frac{\pi}{3}i}.$$ Without lose of generality assume that $$\alpha=\{1, w, w^2, w^3, w^4, w^5\},\: f(z)=wz.$$ Define another linear fractional transformation $g$ $$g(z)=\frac{1}{z}.$$ It is obvious that $f,g \in \mathcal{A_\alpha}$. Note that $\langle f,g\rangle\simeq D_6$ acts transitively on $\alpha$. For any $k \in \mathcal{A_\alpha}$, there exists some $l\in \langle f,g\rangle$ s.t. $h=l\circ k$ fixes $-1$. We have $$h(\{w^4, w^5, 1, w, w^2\})=\{w^4, w^5, 1, w, w^2\},\: h(-1)=-1.$$ Thus $h\in\mathcal{A}_{\{w^4, w^5, 1, w, w^2\}}$. Define another linear fractional transformation $\phi$ $$\phi(z)=\frac{(-w^2-w)z+w^2+w}{z+1}.$$ Notice that $$\phi(w^4)=-3,\: \phi(w^5)=-1,\: \phi(1)=0,\: \phi(w)=1, \: \phi(w^2)=3,\: \phi(-1)=\infty.$$ Thus we have $$\phi\circ h\circ\phi^{-1}\in\mathcal{A}_{\{0, \pm 1, \pm 3\}}, \:\phi\circ h\circ\phi^{-1}(\infty)=\infty.$$ From Section \[n=5, con\] we know that $$\phi\circ h\circ\phi^{-1}(z)=-z \mathrm{~or~} \phi\circ h\circ\phi^{-1}=\mathrm{I},$$ which amounts to $$h=g \mathrm{~or~} h=\mathrm{I}.$$ So $k=l^{-1}\circ h\in\langle f,g\rangle$. In conclusion we have $$\mathcal{A_\alpha} = \langle z \mapsto e^{\frac{\pi}{3}i}z,z \mapsto \frac{1}{z}\rangle \simeq D_6.$$ $\mathcal{A_\alpha}$ is Isomorphic to $\mathbb{Z}_5$ {#n=6, Z_5} ---------------------------------------------------- In this case assume that for any $h\in\mathcal{A_\alpha}$, $\sigma_h$ is not of Type (6), but there exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type (5, 1). Under this assumption, there exists some linear fractional transformation $\psi$ s.t. $$\psi(\alpha) =\{0, 1, w, w^2, w^3, w^4\},\: w=e^{\frac{2\pi}{5}i}.$$ On the other hand, for any $\alpha=\{z_1, z_2, z_3, z_4, z_5, z_6\}\subseteq \widehat{\mathbb C}$, if there exists some linear fractional transformation $\psi$ s.t. $\psi(\alpha)=\{0, 1, w, w^2, w^3, w^4\}$, we see that ${\psi}^{-1} \circ f \circ \psi$ fixes $\alpha$ and is of Type (5, 1), where $f(z)=wz$. However, since the six points in $\alpha$ are not concyclic, for any $h\in\mathcal{A_\alpha}$, $\sigma_h$ is not of Type (6). Thus we see that the assumption amounts to the existence of the linear fractional transformation $\psi$ s.t. $$\psi(\alpha)=\{0, 1, w, w^2, w^3, w^4\}.$$ Without lose of generality we assume that $$\alpha=\{0, 1, w, w^2, w^3, w^4\},\: f(z)=wz.$$ It is obvious that $f \in\mathcal{A_\alpha}$. For any $g \in \mathcal{A_\alpha}$, it is easy to see that $$|g(\{1, w, w^2, w^3, w^4\})\cap\{1, w, w^2, w^3, w^4\}|\ge4.$$ So $g$ fixes the unit circle, and thus $g(0)=0$. As $g(0)=0$, $g\in\mathcal{A}_{\{1, w, w^2, w^3, w^4\}}$. From Section \[n=5, D\_5\] we know that $$\mathcal{A}_{\{1, w, w^2, w^3, w^4\}}=\langle z \mapsto e^{\frac{2\pi}{5}i}z,z \mapsto \frac{1}{z}\rangle.$$ As $g$ fixes $0$, we see that $g\in\langle f\rangle$. In conclusion we have $$\mathcal{A_\alpha} = \langle z \mapsto e^{\frac{2\pi}{5}i}\rangle \simeq \mathbb{Z}_5.$$ $\mathcal{A_\alpha}$ is Isomorphic to $S_4$ {#n=6, S_4} ------------------------------------------- In this case assume that for any $h\in\mathcal{A_\alpha}$, $\sigma_h$ is not of Type (6) or (5, 1), but there exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type (4, 1, 1). Under this assumption, there exists some linear fractional transformation $\psi$ s.t. $$\psi(\alpha) =\{0,\infty, 1, i, -1, -i\}.$$ On the other hand, for any $\alpha=\{z_1, z_2, z_3, z_4, z_5, z_6\}\subseteq \widehat{\mathbb C}$, if there exists some linear fractional transformation $\psi$ s.t. $\psi(\alpha)=\{0, \infty, 1, i, -1, -i\}$, we see that ${\psi}^{-1} \circ f \circ \psi$ fixes $\alpha$ and is of Type (4, 1, 1), where $f(z)=iz$. However, since no five points in $\alpha$ are concyclic, we see that for any $h\in\mathcal{A_\alpha}$, $\sigma_h$ is not of Type (6) or (5, 1). Thus the assumption amounts to the existence of the linear fractional transformation $\psi$ s.t. $$\psi(\alpha)=\{0, \infty, 1, i, -1, -i\}.$$ Without lose of generality we assume that $$\alpha=\{0, \infty, 1, i, -1, -i\}.$$ Let $\varphi$ be the stereographic projection from $\widehat{\mathbb{C}}$ to $S^2$. We have $$\varphi(\alpha)=\{(1,0,0),(-1,0,0),(0,1,0),(0,-1,0),(0,0,1),(0,0,-1)\}.$$ So $\varphi$ maps $\alpha$ to the six vertices of a regular octahedron with its center at the origin. In Section \[S\_4\] we shall prove that $$\mathcal{A_\alpha}=\langle z\mapsto iz, z\mapsto \frac{iz+1}{z+i} \rangle \simeq S_4.$$ $\mathcal{A_\alpha}$ is Isomorphic to $D_3$ {#n=6, D_3} ------------------------------------------- In this case assume that for any $h\in\mathcal{A_\alpha}$, $\sigma_h$ is not of Type (6), (5, 1) or (4, 1, 1), but there exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type (3, 3). Under this assumption, there exists some linear fractional transformation $\psi$ s.t. $$\psi(\alpha)=\{1, w, w^2, a, aw, aw^2\},\: |a|\ge 1,\: a\ne 1, w, w^2,\: w=e^{\frac{2\pi}{3}i}.$$ On the other hand, for any $\{1, w, w^2, a, aw, aw^2\}, |a|\ge 1, a\ne 1, w, w^2$, the linear fractional transformation $z\mapsto wz$ is in $\mathcal{A}_{\{1, w, w^2, a, aw, aw^2\}}$ and of Type (3, 3). Now we aim to find out the specific value $a$ takes when for each element $h\in \mathcal{A}_{\{1, w, w^2, a, aw, aw^2\}}$, $\sigma_h$ is not of Type (6), (5, 1) or (4, 1, 1). *Case 1: $|a|=1$.* Suppose $h\in \mathcal{A}_{\{1, w, w^2, a, aw, aw^2\}}$, and $\sigma_h$ is of Type (6), (5, 1) or (4, 1, 1). As $|a|=1$, the six points in $\{1, w, w^2, a, aw, aw^2\}$ are concyclic. Thus no element in $\mathcal{A}_{\{1, w, w^2, a, aw, aw^2\}}$ is of Type (5, 1) or (4, 1, 1), and $\sigma_h$ has to be of Type (6). From Section \[n=6, D\_6\] we know that there exists some linear fractional transformation $\psi$ s.t. $$\psi(\{1, w, w^2, a, aw, aw^2\})=\{1, \sqrt w, w, w\sqrt w, w^2, w^2\sqrt w\},\: \sqrt w=e^{\frac{\pi}{3}i}.$$ It is obvious that this amounts to $a=\sqrt w, w\sqrt w$ or $w^2\sqrt w$. *Case 2: $|a|>1$.* Suppose $h\in \mathcal{A}_{\{1, w, w^2, a, aw, aw^2\}}$, and $\sigma_h$ is of Type (6), (5, 1) or (4, 1, 1). As $|a|\ge 1$, no five points in $\{1, w, w^2, a, aw, aw^2\}$ are concyclic. Thus no element in $\mathcal{A}_{\{1, w, w^2, a, aw, aw^2\}}$ is of Type $(6)$ or $(5, 1)$, and $\sigma_h$ has to be of Type $(4, 1, 1)$. From Section \[n=6, S\_4\] we know that there exists some linear fractional transformation $\psi$ s.t. $$\psi(\{1, w, w^2, a, aw, aw^2\})=\{0, \infty, \pm 1, \pm i\}.$$ Thus there exists a subset $\Sigma \subseteq \alpha$, $|\Sigma|=4$ s.t. all four points in $\Sigma$ lie on the same circle $C$, and the two points in $\alpha\backslash\Sigma$ lie on different sides of $C$. As $|a|>1$, it is easy to see that $$|\Sigma\cap\{1, w, w^2\}|=|\Sigma\cap\{a, aw, aw^2\}|=2.$$ Without lose of generality we assume that $$\Sigma=\{w, aw, w^2, aw^2\}.$$ Thus we have $$\frac{-3a}{(1-a)^2}=[w, aw, w^2, aw^2]\in \mathbb{R}.$$ As $|a|>1$, the above condition equals to $a\in\mathbb{R}$. {width="2.3in"} {width="2.3in"} If $a>1$, $1$ and $a$ lie on the same side of $C$. It con not be the case. As $a<-1$, we have $$[w, w^2, aw, aw^2]=2,$$ which equals to that $$a=-2-\sqrt3.$$ Define the linear fractional transformation $\varphi$ $$\varphi(z)=\frac{-(1+\sqrt3)(1+i)}{2}\frac{z-1}{z+2+\sqrt3},$$ then we have $$\varphi(1)=0, \varphi(-2-\sqrt3)=\infty, \varphi(w)=1, \varphi(w^2)=i, \varphi((-2-\sqrt3)w)=-1, \varphi((-2-\sqrt3)w^2)=-i.$$ From the above discussion we see that the assumption amounts to the existence of some linear fractional transformation $\psi$ s.t. $$\psi(\alpha)=\{1, w, w^2, a, aw, aw^2\},$$ where $$w=e^{\frac{2\pi}{3}i},\: |a|\ge 1,\: a\ne(\sqrt w)^j \mathrm{~or~} (-2-\sqrt3)w^j \mathrm{~for~any~} j \in \mathbb{Z}.$$ Without lose of generality assume that $$\alpha=\{1, w, w^2, a, aw, aw^2\}, \: f(z)=wz, \: g(z)=\frac{a}{z}.$$ It is obvious that $f,g \in \mathcal{A_\alpha}$. Note that $\langle f,g\rangle\simeq D_3$ acts transitively on $\alpha$. For any $k \in \mathcal{A_\alpha}$, there exists some $l\in \langle f,g\rangle$ s.t. $h=l\circ k$ fixes $1$. We have $$h(\{w, w^2, a, aw, aw^2\})=\{w, w^2, a, aw, aw^2\},\: h(1)=1.$$ Thus $h\in\mathcal{A}_{\{w, w^2, a, aw, aw^2\}}$. Since no element in $\mathcal{A_\alpha}$ is of Type (6), (5, 1) or (4, 1, 1), and $h(1)=1$, $\sigma_h$ is of Type (2, 2, 1, 1) or identity. Next we shall prove that $h$ has to be the identity. Suppose that $\sigma_h$ is of Type (2, 2, 1, 1). Thus there exists some linear fractional transformation $\varphi$ s.t. $$\varphi(\alpha)=\{0, \infty, 1, -1, z_0, -z_0\},\: |z_0|\ge1.$$ Since $h$ fixes $1$, assume $$\varphi(1)=0.$$ As $0, \infty, 1, -1$ are concyclic, there exists some subset $\Sigma\subseteq\alpha, |\Sigma|=4, 1\in\Sigma$ s.t. all elements in $\Sigma$ lie on the same circle $C$. *Case 1: $|a|=1$.* Without lose of generality assume that $\arg a \in (0,\pi/3)\cup(\pi/3, 2\pi/3)$. {width="3in"} By observation we have $$\varphi(1)=0,\: \varphi(aw)=\infty,\: \varphi(a)=1, \:\varphi(aw^2)=-1,$$ or $$\varphi(1)=0, \:\varphi(aw)=\infty, \:\varphi(a)=-1, \:\varphi(aw^2)=1.$$ So we have $$[1,aw,a,aw^2]=[0,\infty,1,-1],$$ or $$[1,aw,a,aw^2]=[0,\infty,-1,1],$$ which equals to $$a=\sqrt w.$$ This contradicts the assumption that $a\ne \sqrt w$, so $h$ can not be of Type (2, 2, 1, 1). *Case 2: $|a|>1$.* As $|a|>1$, it is easy to see that $$|\Sigma\cap\{1,w,w^2\}|=|\Sigma\cap\{a,aw,aw^2\}|=2.$$ Without lose of generality assume that $$\Sigma=\{1,w,a,aw\}.$$ Then we have $$\frac{-3a}{(1-a)^2}=[1,w,a,aw]\in\mathbb R,$$ which amounts to $$a\in\mathbb R.$$ {width="2.3in"} {width="2.3in"} If $a>1$, $w^2$ and $aw^2$ lie on the same side of $C$, which contradicts the assumption that $\varphi(\alpha)=\{0, \infty, 1, -1, z_0, -z_0\}$. If $a<-1$, by observation we have $$\varphi(1)=0,\:\varphi(a)=\infty,\:\varphi(w)=1,\:\varphi(aw)=-1,$$ or $$\varphi(1)=0,\:\varphi(a)=\infty,\:\varphi(w)=-1,\:\varphi(aw)=1.$$ So we have $$[1, a, w, aw]=[0, \infty, 1, -1],$$ or $$[1, a, w, aw]=[0, \infty, -1, 1],$$ which amounts to $$a=-2-\sqrt3.$$ This also contradicts our assumption that $a\ne-2-\sqrt3$. So $h$ has to be the identity, and $k=l^{-1}\in\langle f,g\rangle$. In conclusion $$\mathcal A_\alpha=\langle z\mapsto wz, z\mapsto \frac{a}{z} \rangle \simeq D_3.$$ $\mathcal{A_\alpha}$ is Isomorphic to $K_4$ or $\mathbb{Z}_2$ ------------------------------------------------------------- In this case assume that for any $h\in\mathcal{A_\alpha}$, $\sigma_h$ is not of Type (6), (5, 1), (4, 1, 1) or (3, 3), but there exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type (2, 2, 2). Under this assumption, there exists some linear fractional transformation $\psi$ s.t. $$\psi(\alpha)=\{1, -1, a, -a, b, -b\},\: a\ne\pm b, \:a,b \in \mathbb{C} \backslash \{ 0, \pm 1 \}.$$ On the other hand, for any $\{1, -1, a, -a, b, -b\},a\ne\pm b, a,b \in \mathbb{C} \backslash \{ 0, \pm 1 \}$, the linear fractional transformation $z\mapsto -z$ is in $\mathcal{A}_{\{\pm 1, \pm a, \pm b\}}$ and of Type (2, 2, 2). Now we aim to find out the specific value $a$ and $b$ takes when for each element $h\in \mathcal{A}_{\{\pm 1, \pm a, \pm b\}}$, $\sigma_h$ is not of Type Type (6), (5, 1), (4, 1, 1) or (3, 3). *Case 1*: There exists some $h\in\mathcal{A}_{\{\pm 1, \pm a, \pm b\}}$ s.t. $\sigma_h$ is of Type (6). This amounts to the existence of some linear fractional transformation $\psi$ s.t. $$\psi(\{\pm 1, \pm a, \pm b\})=\{1, w, w^2, w^3, w^4, w^5\}, \:w=e^{\frac{\pi}{3}i}.$$ We conclude that $\pm 1, \pm a, \pm b$ lie on the same circle: the real axis or the unit circle. If $\pm 1, \pm a, \pm b$ lie on the unit circle, we have $$\{\pm 1, \pm a, \pm b\}=\{1, w, w^2, w^3, w^4, w^5\}.$$ If $\pm 1, \pm a, \pm b$ lie on the real axis, without lose of generality assume that $$0<a<b.$$ If $a<b<1$, we have $$[a,1,-1,-a]=[1,w^2,w^3,w^5],\:[b,1,-1,-b]=[1,w,w^2,w^3],$$ which amounts to $$a=7-4\sqrt3, \:b=2-\sqrt3.$$ If $a<1<b$, we have $$[a,1,-1,-a]=[1,w,w^4,w^5],\:[b,1,-1,-b]=[1,w^5,w^2,w],$$ which amounts to $$a=2-\sqrt3, \:b=2+\sqrt3.$$ If $1<a<b$, we have $$[a,1,-1,-a]=[1,w^5,w^4,w^3],\:[b,1,-1,-b]=[1,w^4,w^3,w],$$ which amounts to $$a=2+\sqrt3, \:b=7+4\sqrt3.$$ On the other hand, the linear fractional transformation $$f_1(z)=\frac{\sqrt3 z-(2-\sqrt3)}{(2+\sqrt3)z+\sqrt3}$$ is of Type (6) and in $\mathcal{A}_{\{\pm(7-4\sqrt3), \pm(2-\sqrt3),\pm 1\}}$ and $$f_2(z)=\frac{\sqrt3 z-1}{z+\sqrt3}$$ is of Type (6) and in $\mathcal{A}_{\{\pm(2-\sqrt3),\pm 1,\pm(2+\sqrt3)\}}$ and $$f_1(z)=\frac{\sqrt3 z-(2+\sqrt3)}{(2-\sqrt3)z+\sqrt3}$$ is of Type (6) and in $\mathcal{A}_{\{\pm 1, \pm(2+\sqrt3),\pm(7+4\sqrt3)\}}$. Thus there exists some $h\in\mathcal{A}_{\{\pm 1, \pm a, \pm b\}}$ s.t. $\sigma_h$ is of Type (6) if and only if $$\{\pm 1, \pm a, \pm b\}=\{\pm(7-4\sqrt3), \pm(2-\sqrt3),\pm 1\},$$ or $$\{\pm 1, \pm a, \pm b\}=\{\pm(2-\sqrt3),\pm 1,\pm(2+\sqrt3)\},$$ or $$\{\pm 1, \pm a, \pm b\}=\{\pm 1, \pm(2+\sqrt3),\pm(7+4\sqrt3)\}.$$ *Case 2*: There exists some $h\in\mathcal{A}_{\{\pm 1, \pm a, \pm b\}}$ s.t. $\sigma_h$ is of Type (5, 1). From \[n=6, Z\_5\] we know that $\mathcal{A}_{\{\pm 1, \pm a, \pm b\}}\simeq\mathbb{Z}_5$, and does not contain elements of order two. This contradicts the assumption that $\sigma_h$ is of Type (2, 2, 2). *Case 3*: There exists some $h\in\mathcal{A}_{\{\pm 1, \pm a, \pm b\}}$ s.t. $\sigma_h$ is of Type (4, 1, 1). This amounts to the existence of some linear fractional transformation $\psi$ s.t. $$\psi(\{\pm 1, \pm a, \pm b\})=\{0, \infty, \pm 1, \pm i\}.$$ Set $\Sigma=\psi^{-1}(\{1, i, -1, -i\})$. So all four points in $\Sigma$ lie on the circle $C$, and the two points in $\{\pm 1, \pm a, \pm b\}\backslash\Sigma$ are inverse points with respect to $C$. If $|\Sigma\cap\{1,-1\}|=0$, we have $\Sigma=\{\pm a, \pm b\})$, and thus $C$ is a Euclidean circle with its center at the origin, or a line through the origin. As $\pm 1$ are inverse points with respect to $C$, $C$ has to be the imaginary line. Assume that $$a=si,\:b=ti.$$ Without lose of generality assume that $$0<s<t.$$ So we have $$[ti,si,-si,-ti]=[1,ti,-si,-1]=[0,1,\infty,-1]=2,$$ which amounts to $$s=\sqrt2-1,\: t=\sqrt2+1.$$ If $|\Sigma\cap\{1,-1\}|=1$, without lose of generality assume that $$\Sigma=\{1, \pm a, b\}.$$ So we have $$[-1,1,-b,b]=[-1,a,-b,-a]=[0,1,\infty,-1]=2,$$ or $$[-1,1,-b,-a]=[-1,b,-b,a]=[0,1,\infty,-1]=2,$$ or $$[-1,1,-b,a]=[-1,b,-b,-a]=[0,1,\infty,-1]=2,$$ which amounts to $$a=\pm(\sqrt2+1)i,\:b=-3-2\sqrt2\mathrm{~or~}a=\pm(\sqrt2-1)i, \:b=-3+2\sqrt2,$$ or $$a=2+\sqrt3,\: b=-7-4\sqrt3\mathrm{~or~}a=2-\sqrt3,\: b=-7+4\sqrt3\$$ or $$a=-2-\sqrt3,\: b=-7-4\sqrt3\mathrm{~or~}a=-2+\sqrt3,\: b=-7+4\sqrt3\$$ respectively. As the six points in $\{\pm 1, \pm a, \pm b\}$ are not concyclic, $a, b$ can not both be real. If $|\Sigma\cap\{1,-1\}|=2$, let $\{\pm 1, \pm a, \pm b\}\backslash\Sigma=\{x, y\}$ and there are two possibilities. The first is that $x+y\ne0$, and the second is that $x+y=0$. When $x+y\ne0$, without lose of generality assume that $$\Sigma=\{\pm 1, a, b\}.$$ As the six points in $\{\pm 1, \pm a, \pm b\}$ are not concyclic, it is easy to see that $$a, b\notin\mathbb{R}.$$ As $-a, -b$ are inverse points with respect to $C$, we see that $$\mathrm{Im}(a)\mathrm{Im}(b)<0.$$ {width="3in"} So we have $$[-a,1,-b,-1]=[-a,a,-b,b]=[0,1,\infty,-1]=2$$ which amounts to $$a=(\pm\sqrt2\pm1)i,\:b=\frac{1}{a}.$$ When $x+y=0$, without lose of generality assume that $$\Sigma=\{\pm 1, \pm a\}.$$ So $C$ is a Euclidean circle with its center at the origin or is th real axis. As $\pm b$ are inverse points with respect to $C$, $C$ has to be the real axis. Without lose of generality assume that $$a>0.$$ So we have $$[b,1,-b,-a]=[1,a,-a,-1]=[0,1,\infty,-1]=2$$ which amounts to $$a=3+2\sqrt2,\:b=\pm(\sqrt2+1)i\mathrm{~or~}a=3-2\sqrt2,\:b=\pm(\sqrt2-1)i.$$ On the other hand, the linear fractional transformation $$f_1(z)=\frac{iz+1}{z+i}$$ is of Type (4, 1, 1) and in $\mathcal{A}_{\{\pm (\sqrt2-1)i,\pm(\sqrt2+1)i,\pm 1\}}$ and $$f_2(z)=\frac{z+\sqrt2-1}{-(\sqrt2+1)z+1}$$ is of Type (4, 1, 1) and in $\mathcal{A}_{\{\pm(3-2\sqrt2), \pm 1,\pm(\sqrt2-1)i\}}$ and $$f_3(z)=\frac{z+\sqrt2+1}{-(\sqrt2-1)z+1}$$ is of Type (4, 1, 1) and in $\mathcal{A}_{\{\pm 1, \pm(3+2\sqrt2),\pm(\sqrt2+1)i\}}$. Thus there exists some $h\in\mathcal{A}_{\{\pm 1, \pm a, \pm b\}}$ s.t. $\sigma_h$ is of Type (4, 1, 1) if and only if $$\{\pm 1, \pm a, \pm b\}=\{\pm (\sqrt2-1)i,\pm(\sqrt2+1)i,\pm 1\},$$ or $$\{\pm 1, \pm a, \pm b\}=\{\pm(3-2\sqrt2), \pm 1,\pm(\sqrt2-1)i\},$$ or $$\{\pm 1, \pm a, \pm b\}=\{\pm 1, \pm(3+2\sqrt2),\pm(\sqrt2+1)i\}.$$ *Case 4*: There exists some $h\in\mathcal{A}_{\{\pm 1, \pm a, \pm b\}}$ s.t. $\sigma_h$ is of Type (3, 3). This amounts to the existence of some linear fractional transformation $\psi$ s.t. $$\psi(\{\pm 1, \pm a, \pm b\})=\{1, w, w^2, z_0, z_0w, z_0w^2\},$$ where $$z_0\ne(\sqrt w)^j \mathrm{~or~} (-2-\sqrt3)w^j \mathrm{~for~any~} j \in \mathbb{Z}, \: w=e^{\frac{2\pi}{3}i}.$$ From Section \[n=6, D\_3\] we know that $$\mathcal A_{\psi(\{\pm 1, \pm a, \pm b\})}=\langle z\mapsto wz, z\mapsto \frac{z_0}{z} \rangle \simeq D_3.$$ The only three elements of order two in $\mathcal A_{\psi(\{\pm 1, \pm a, \pm b\})}$ are $$f_1(z)=\frac{z_0}{z},\:f_2(z)=\frac{wz_0}{z},\:f_3(z)=\frac{w^2z_0}{z}.$$ So we have $$f_i(\{1, w, w^2\})=\{z_0, z_0w, z_0w^2\},\:f_i(\{z_0, z_0w, z_0w^2\})=\{1, w, w^2\},\:i=1,2,3.$$ Define the linear fractional transformation $g$ $$g(z)=-z.$$ Then $g\in\mathcal{A}_{\{\pm 1, \pm a, \pm b\}}$ and is of order two. The we have $$g=\psi^{-1}\circ f_i \circ \psi$$ for some $i=1, 2,$ or $3$. So we have $$|\{1, w, w^2\}\cap\psi(\{\pm1\})|=|\{1, w, w^2\}\cap\psi(\{\pm a\})|=|\{1, w, w^2\}\cap\psi(\{\pm b\})|=1.$$ Without lose of generality assume that $$\psi(\{1, a, b\})=\{1, w, w^2\}$$ and $$\psi(1)=1,\:\psi(a)=w,\:\psi(b)=w^2.$$ Now there are two possibilities. The first is $$\psi(-a)=w\psi(-1),\:\psi(-b)=w\psi(-a).$$ and the second is $$\psi(-1)=w\psi(-a),\:\psi(-a)=w\psi(-b).$$ Define the linear fractional transformation $\varphi$ $$\varphi(z)=\frac{((a-2b+1)w+2a-b-1)z+(ab-2a+b)w-ab-a+2b}{((-2a+b+1)w+-a+2b-1)z+(ab+a-2b)w-ab+2a-b}$$ satisfies $$\varphi(1)=1,\:\varphi(a)=w,\:\varphi(b)=w^2.$$ The equations $$\left\{\begin{array}{rl} \varphi(-a)=w\varphi(-1) \\ \varphi(-b)=w\varphi(-a) \\ \end{array} \right.$$ reduces to $$\left\{\begin{array}{rl} a^2b^2+a^3-2a^2b-2ab+b^2+a=0 \\ a^2b^2+b^3-2ab^2-2ab+a^2+b=0 \\ \end{array} \right.$$ which amounts to $$a=w, b=w^2\mathrm{~or~}a=w^2, b=w$$ which can not be the case. The equations $$\left\{\begin{array}{rl} \varphi(-1)=w\varphi(-a) \\ \varphi(-a)=w\varphi(-b) \\ \end{array} \right.$$ reduces to $$\left\{\begin{array}{rl} a^3b+ a^2b^2+a^3-5a^2b+2ab^2+2a^2-5ab+b^2+a+b=0 \\ ab^3+ a^2b^2+b^3-5ab^2+2a^2b+2b^2-5ab+a^2+a+b=0. \\ \end{array} \right.$$ As $$a^3b+ a^2b^2+a^3-5a^2b+2ab^2+2a^2-5ab+b^2+a+b=(a+1)(a^2b+ab^2+a^2-6ab+b^2+a+b)$$ and $$ab^3+ a^2b^2+b^3-5ab^2+2a^2b+2b^2-5ab+a^2+a+b=(b+1)(a^2b+ab^2+a^2-6ab+b^2+a+b),$$ the above equations amounts to $$a^2b+ab^2+a^2-6ab+b^2+a+b=0.$$ Thus there exists some $h\in\mathcal{A}_{\{\pm 1, \pm a, \pm b\}}$ s.t. $\sigma_h$ is of Type (3, 3) if and only if $$a^2b+ab^2+a^2-6ab+b^2+a+b=0.$$ From the above discussion we see that the assumption amounts to the existence of some linear fractional transformation $\psi$ s.t. $$\psi(\alpha)=\{1, -1, a, -a, b, -b\},\: a\ne\pm b, \:a,b \in \mathbb{C} \backslash \{ 0, \pm 1 \},$$ where $\{\pm1,\pm a,\pm b\}$ is not the sets we discussed above. Without lose of generality assume $$\alpha=\{\pm1,\pm a,\pm b\},\: f(z)=-z,$$ where $\{\pm1,\pm a,\pm b\}$ is not the sets we discussed above. So $f\in\mathcal A_\alpha$ and $\sigma_f =(1,2)(3,4)(5,6)$. If $\exists g\in\mathcal{A_\alpha}$ s.t. $g\ne f, g\ne \mathrm{I}$, without lose of generality assume that $\sigma_g(1)\ne\sigma_f(1)=2$. *Case 1:* $\sigma_g(1)=1$. We see that $g$ is of Type (2, 2, 1, 1), and it has another fixed point besides $1$. There are two possibilities. If $\sigma_g(2)=2$, it could be that $$\sigma_g=(3,4)(5,6)=:\pi_1$$ or $$\sigma_g=(3,5)(4,6)=:\pi_2$$ or $$\sigma_g=(3,6)(4,5)=:\pi_3.$$ If $\sigma_g(2)\ne2$, as $3,4,5,6$ are congruent, we may assume without lose of generality that $$\sigma_g(3)=3.$$ So it could be that $$\sigma_g=(2,4)(5,6)=:\pi_4$$ or $$\sigma_g=(2,5)(4,6)=:\pi_5$$ or $$\sigma_g=(2,6)(4,5)=:\pi_6.$$ However $$\pi_1\sigma_f=(1,2),\:\pi_2\sigma_f=(1,2)(3,6)(4,5),\:\pi_3\sigma_f=(1,2)(3,5)(4,6),$$ $$\pi_4\sigma_f=(1,4,3,2),\:\pi_1\sigma_f=(1,5,4,3,6,2),\:\pi_1\sigma_f=(1,6,4,3,5,2).$$ As a result $$\sigma_g=\pi_2\mathrm{~or~}\pi_3.$$ *Case 2:* $\sigma_g(1)\ne1$. As $3,4,5,6$ are congruent, we may assume without lose of generality that $$\sigma_g(1)=3.$$ So it could be that $$\sigma_g=(1,3)(2,4)(5,6)=:\rho_1$$ or $$\sigma_g=(1,3)(2,5)(4,6)=:\rho_2$$ or $$\sigma_g=(1,3)(2,6)(4,5)=:\rho_3$$ or $$\sigma_g=(1,3)(2,4)=:\rho_4$$ or $$\sigma_g=(1,3)(2,5)=:\rho_5$$ or $$\sigma_g=(1,3)(2,6)=:\rho_6$$ So it could be that $$\sigma_g=(1,3)(4,5)=:\rho_7$$ or $$\sigma_g=(1,3)(4,6)=:\rho_8$$ or $$\sigma_g=(1,3)(5,6)=:\rho_9.$$ However $$\rho_1\sigma_f=(1,4)(2,3),\:\rho_2\sigma_f=(1,5,4)(2,3,6),\:\rho_3\sigma_f=(1,6,4)(2,3,5)\:$$ $$\rho_4\sigma_f=(1,4)(2,3)(5,6),\:\rho_5\sigma_f=(1,5,6,2,3,4),\:\rho_6\sigma_f=(1,6,5,2,3,4),\:$$ $$\rho_7\sigma_f=(1,2,3,5,6,4),\:\rho_8\sigma_f=(1,2,3,6,5,4),\:\rho_9\sigma_f=(1,2,3,4).$$ As a result $$\sigma_g=\rho_1\mathrm{~or~}\rho_4.$$ Notice that $$\langle \sigma_f, \pi_2\rangle=\{\mathrm{I},(1,2)(3,4)(5,6),(3,5)(4,6),(1,2)(3,6)(4,5)\},$$ $$\langle \sigma_f, \pi_3\rangle=\{\mathrm{I},(1,2)(3,4)(5,6),(3,6)(4,5),(1,2)(3,5)(4,6)\},$$ $$\langle \sigma_f, \rho_1\rangle=\{\mathrm{I},(1,2)(3,4)(5,6),(1,4)(2,3),(1,3)(2,4)(5,6)\},$$ $$\langle \sigma_f, \rho_4\rangle=\{\mathrm{I},(1,2)(3,4)(5,6),(1,3)(2,4),(1,4)(2,3)(5,6)\}.$$ If $\mathcal A_\alpha\ne\{\mathrm{I}, f\}$, then there exists some $h\in\mathcal A_\alpha$ s.t. $h$ is of Type (2, 2, 1, 1), and the two fixed points of $h$ are on the same orbit of $f$. Without lose of generality assume that $h$ fixes $\pm1$. Thus $$h(z)=\frac{pz+q}{qz+p},\: p,q \in\mathbb C,\:a^2\ne b^2,\:b\ne0.$$ We have $$h(a)=b,\:h(-a)=-b$$ or $$h(a)=-b,\:h(-a)=b.$$ Anyway $$h(a)+h(-a)=0$$ which means that $$p=0.$$ So we have $$ab=1 (\mathrm{or~}-1).$$ In conclusion, if $ab=\pm1$ $$\mathcal{A_\alpha} = \langle z \mapsto -z, z \mapsto \frac{1}{z} \rangle \simeq K_4.$$ Otherwise $$\mathcal{A_\alpha} = \langle z \mapsto -z\rangle \simeq \mathbb Z _2.$$ Another Possibility ------------------- In this case assume that for any $h\in\mathcal{A_\alpha}$, $\sigma_h$ is not of Type (6), (5, 1), (4, 1, 1), (3, 3) or (2, 2, 2) but there exists some $f\in\mathcal{A_\alpha}$ s.t. $\sigma_f$ is of Type (2, 2, 1, 1). Under this assumption, there exists some linear fractional transformation $\psi$ s.t. $$\psi(\alpha)=\{0, \infty, \pm1, \pm a\},\: a\ne 0, \pm1.$$ Define the linear fractional transformation $g$ $$g(z)=\frac{a}{z}.$$ We see that $g\in \mathcal{A_\alpha}$ and is of Type (2, 2, 2), which contradicts our assumption. This case is not possible. Conclusion {#n=6, con} ---------- From the above discussion we shall drive the following theorem Set $\alpha=\{z_1,\: z_2,\: z_3,\: z_4,\: z_5,\: z_6\}\subseteq \widehat{\mathbb{C}}$. 1. If there exists some linear fractional transformation $\psi$ such that $$\psi(\alpha)=\{1,\: w,\: w^2,\: w^3,\: w^4, \:w^5\},\:w=e^{\frac{\pi}{3}i},$$ then $$\mathcal{A_\alpha}={\psi}^{-1}\langle z \mapsto e^{\frac{\pi}{3}i}z,\:z \mapsto \frac{1}{z}\rangle {\psi}\simeq D_6,$$ and its multiplicity vector is $(0,\:1,\:0,\:0,\:0,\:1)$; 2. if there exists some linear fractional transformation $\psi$ such that $$\psi(\alpha) =\{0,\: 1,\: w, \:w^2, \:w^3,\: w^4\},\:w=e^{\frac{2\pi}{5}i},$$ then $$\mathcal{A_\alpha}={\psi}^{-1}\langle z \mapsto e^{\frac{2\pi}{5}i}z\rangle {\psi}\simeq \mathbb Z _5,$$ and its multiplicity vector is $(1,\:1,\:1,\:0,\:0)$; 3. if there exists some linear fractional transformation $\psi$ such that $$\psi(\alpha) =\{0,\: \infty, \:1, \:i, \:-1,\: -i\},$$ then $$\mathcal{A_\alpha}={\psi}^{-1}\langle z\mapsto iz,\: z\mapsto \frac{iz+1}{z+i}\rangle {\psi}\simeq S_4,$$ and its multiplicity vector is $(0,\:0,\:1,\:0,\:0)$; 4. if there exists some linear fractional transformation $\psi$ such that $$\psi(\alpha)=\{1,\:w,\:w^2,\:a,\:aw,\:aw^2\},\:|a|\ge1,\:a\ne1,\:w,\:w^2,\:w=e^{\frac{2\pi}{3}i},$$ 1. if $a=\sqrt w,\:w\sqrt w\text{ or }w^2\sqrt w$, $$\alpha=\{1,\:\sqrt w,\: w,\:w\sqrt w,\: w^2,\:w^2\sqrt w\}$$ and this is case 1 ; 2. if $a=-(2+\sqrt3),\:-(2+\sqrt3)w\text{ or }-(2+\sqrt3)w^2$, then there exists some linear fractional transformation $\phi$ such that $$\phi(\alpha) =\{0,\: \infty, \:1, \:i, \:-1,\: -i\}$$ and this is case 3; 3. otherwise $$\mathcal{A_\alpha}={\psi}^{-1}\langle z \mapsto e^{\frac{2\pi}{3}i}z,\:z \mapsto \frac{a}{z}\rangle\psi \simeq D_3,$$ and its multiplicity vector is $(1,\:1,\:0)$; \[a\] 5. if there exists some linear fractional transformation $\psi$ such that $$\psi(\alpha)=\{1,\: -1, \:a,\: -a,\: b, \:-b\},\:a\ne\pm b, \:a,\:b \in \mathbb{C} \backslash \{ 0, \pm 1 \},$$ 1. if $\{\pm1,\:\pm a,\: \pm b\}=$ $$\{\pm(7-4\sqrt3), \:\pm(2-\sqrt3),\:\pm 1\},\:\{\pm(2-\sqrt3),\:\pm 1,\:\pm(2+\sqrt3)\},\:\{\pm 1,\: \pm(2+\sqrt3),\:\pm(7+4\sqrt3)\}$$ then there exists some linear fractional transformation $\phi$ such that $$\phi(\alpha)=\{1,\: w,\: w^2,\: w^3,\: w^4, \:w^5\},\:w=e^{\frac{\pi}{3}i}$$ and this is case 1; 2. if $\{\pm1,\:\pm a,\: \pm b\}=$ $$\{\pm (\sqrt2-1)i,\:\pm(\sqrt2+1)i,\:\pm 1\},\:\{\pm(3-2\sqrt2), \:\pm 1,\pm(\sqrt2-1)i\},\:\{\pm 1, \:\pm(3+2\sqrt2),\:\pm(\sqrt2+1)i\},$$ then there exists some linear fractional transformation $\phi$ such that $$\phi(\alpha) =\{0,\: \infty, \:1, \:i, \:-1,\: -i\}$$ and this is case 3; 3. if $a^2b+ab^2+a^2-6ab+b^2+a+b=0,$ then there exists some linear fractional transformation $\phi$ such that $$\phi(\alpha)=\{1,\:w,\:w^2,\:c,\:cw,\:cw^2\},\:c\ne0,\:1,\:w,\:w^2,\:w=e^{\frac{2\pi}{3}i}$$ and this is case 4c; 4. otherwise, 1. if $ab=\pm1$, $$\mathcal{A_\alpha}={\psi}^{-1}\langle z \mapsto -z,\:z \mapsto \frac{1}{z}\rangle\psi \simeq K_4,$$ and its multiplicity vector is $(1,\:0,\:1,\:1)$ (viewed as type 2+mC); 2. if $ab\ne\pm1$, $$\mathcal{A_\alpha}={\psi}^{-1}\langle z \mapsto -z\rangle\psi \simeq \mathbb{Z}_2,$$ and its multiplicity vector is $(1,\:2)$; 6. otherwise, $$\mathcal{A_\alpha}=\{\mathrm{Id}\}.$$ Each Finite Subgroup of $\text{PSL}(2,\:\mathbb C)$ is a Stabilizer of Certain Finite Subset of $\widehat{\mathbb C}$ {#all} ===================================================================================================================== It is already known that for any finite subset $\alpha=\{z_1, z_2, \cdots, z_n\}\subseteq \widehat{\mathbb{C}}$, $n\ge4$, $\mathcal{A}_{\alpha}$ is finite. In this section we aim to prove that For any finite non-trivial group $G$ of linear fractional transformations, there exists a finite subset $\alpha=\{z_1, z_2, \cdots, z_n\}\subseteq \widehat{\mathbb{C}}$ such that $\mathcal{A}_{\alpha}\simeq G$. There are only five kinds of finite non-trivial linear fractional transformation groups: the icosahedral group $A_5$, the octahedral group $S_4$, the tetrahedral group $A_4$, the dihedral group $D_k$ and the finite cyclic group $\mathbb Z_k$, $k\ge2$. In Section \[A\_5\], \[S\_4\] and \[A\_4\] we can find $\alpha$ such that $\mathcal{A}_{\alpha}\simeq A_5,\:S_4$ and $A_4$. In Corollary \[D\_n1\], \[D\_n2\] and \[cZ\_n\] we can find $\alpha$ such that $\mathcal{A}_{\alpha}\simeq D_m$ and $\mathbb Z_n$, $m\ge5$, $n\ge4$. In Section \[n=4\] we can find $\alpha$ such that $\mathcal{A}_{\alpha}\simeq D_4$ and $D_2$. And in Section \[n=5, con\] we can find $\alpha$ such that $\mathcal{A}_{\alpha}\simeq D_3$ and $\mathbb Z_2$. So all we need to do is to find an $\alpha$ such that $\mathcal{A}_{\alpha}\simeq \mathbb Z_3$. Set $$\alpha=\{0,\:1,\:w,\:w^2,\:2,\:2w,\:2w^2\},\:w=e^{\frac{2\pi}{3}i},$$ and $$G=\langle z\mapsto wz\rangle\simeq \mathbb Z_3.$$ We have $G\subseteq\mathcal{A}_{\alpha}$. As $|\alpha|=7$, from Theorem \[tA\_5\], \[tS\_4\] and \[tA\_4\], we see that $\mathcal{A}_{\alpha}$ is not $A_5$, $S_4$ or $A_4$. Thus $G$ is isomorphic to $D_k$ or $\mathbb Z_{k}$. From Theorem \[tD\_n\] we see that if $\mathcal A_\alpha\simeq D_k$, $|\alpha|=nk$ or $nk+2$, $n\in\mathbb N^*$. As $|\alpha|=7$, $\mathcal A_\alpha\simeq D_7$ or $D_5$. From Theorem \[tZ\_n\] we see that if $\mathcal A_\alpha\simeq \mathbb Z_k$, $|\alpha|=nk$, $nk+1$ or $nk+2$, $n\in\mathbb N^*$. As $|\alpha|=7$, $\mathcal A_\alpha\simeq \mathbb Z_7$, $\mathbb Z_6$, $\mathbb Z_5$, $\mathbb Z_3$ or $\mathbb Z_2$. It is easy to see that no five points in $\alpha$ are concyclic. Thus $\mathcal{A}_{\alpha}$ is not $D_7$, $D_5$, $\mathbb Z_7$, $\mathbb Z_6$ or $\mathbb Z_5$. As $G\subseteq\mathcal{A}_{\alpha}$ and $G\simeq\mathbb Z_3$, $\mathcal{A}_{\alpha}$ is not $\mathbb Z_2$. Thus $\mathcal{A}_{\alpha}$ is isomorphic to $\mathbb Z_3$. [1]{} E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, , Volume I, Springer-Verlag New York Inc. 1985. O. Bolza, Math. Ann. [**30**]{} (1887) 546-552. I. V. Dolgachev, Cambridge University Press, Cambridge, 2012. M. Mulase and M. Penkava, , Unpublished lecture notes, https://www.math.ucdavis.edu/mulase/texfiles/1997moduli.pdf A. Wiman, Math. Ann. [**48**]{} (1896) 195-240 \ [^1]: $\dagger$ The first author is supported in part by the National Natural Science Foundation of China (Grant No. 11671371).\ $^\ddagger$ The second author is supported in part by the National Natural Science Foundation of China (Grant Nos. 11571330 and 11271343) and the Fundamental Research Funds for the Central Universities (Grant No. WK3470000003).

--- author: - Tai Li - 'and Yu-Feng Zhou' title: '**Strongly first order phase transition in the singlet fermionic dark matter model after LUX**' --- Introduction ============ The possibility of baryogenesis through electroweak phase transition (EWPhT) has been studied extensively (for reviews see e.g. Refs. [@Cohen:1993nk; @Rubakov:1996vz; @Trodden:1998ym; @Quiros:1999jp]). If the EWPhT is strongly first order, it can fulfill the condition of departure from thermal equilibrium which is one of the three conditions necessary for the generation of baryon number asymmetry in the Universe [@Sakharov:1967dj; @Kuzmin:1985mm]. In order to avoid the washout of the baryon number asymmetry, the baryon number violating interactions induced by electroweak sphalerons must be suppressed at the temperature when the bubbles enveloping the broken phase start to nucleate [@DeSimone:2011ek]. A commonly adopted assumption is that the sphaleronic interactions are suppressed immediately after the EWPhT, which leads to a requirement that $\varphi_c$ the vacuum expectation value (VEV) of the Higgs field in the broken phase is larger than the critical temperature, namely [@Shaposhnikov:1987tw; @Shaposhnikov:1986jp] $$\label{eq1} \frac{\varphi_c}{T_c} \gtrsim \mathcal{E}.$$ where $\mathcal{E}\approx 1$ is a constant. In the standard model (SM), the condition in Eq. (\[eq1\]) is satisfied only when the Higgs boson is very light, i.e., $m_h \lesssim 30 {\mbox{ GeV}}$ for $\mathcal{E} = 1$ [@Carrington:1991hz; @Anderson:1991zb; @Arnold:1992fb; @Arnold:1992rz; @Dine:1992wr], which is ruled out by the current experiments, especially after the discovery of a $125{\mbox{ GeV}}$ Higgs boson at the LHC [@Chatrchyan:2012ufa; @Aad:2012tfa]. Thus new physics beyond the SM must be introduced for a successful electroweak baryogenesis. Another clear indication of new physics is the existence of dark matter (DM), which has been well established by astrophysical and cosmological observations as well as N-body simulations. According to the latest analysis reported by the Planck Collaboration, the measured energy density of DM in the Universe is [@Ade:2013zuv] $$\label{omegah2} \Omega h^2 = 0.1187\pm0.0017.$$ Although the SM has been very successful in phenomenology, it can provide neither a strongly first order EWPhT for baryogenesis nor a valid candidate of DM. One of the simplest models with DM candidates is the extension of the SM with a gauge singlet scalar field [@Silveira:1985rk; @McDonald:1993ex; @Burgess:2000yq; @Davoudiasl:2004be; @He:2009yd; @Gonderinger:2009jp; @Bandyopadhyay:2010cc; @Guo:2010hq; @Mambrini:2011ik]. The stability of the scalar can be protected by an $ad\ hoc$ $Z_2$ symmetry. The $Z_2$ symmetry may be a residual symmetry from a global or local $U(1)$. In the extension of the left-right symmetric models with a gauge singlet scalar, the $Z_2$ symmetry may originate from the parity and CP symmetries [@Guo:2008si; @Guo:2010vy; @Guo:2010sy; @Guo:2011zze; @Liu:2012bm]. However, if EWPhT is also required, it was shown that the singlet scalar could contribute only up to $3\%$ of the DM energy density [@Cline:2012hg; @Cline:2013gha]. In the inert doublet model, an additional $SU(2)$ doublet is added to the SM. This model can provide a valid DM candidate and also trigger strongly first order EWPhT, due to the contributions from other charged and neutral scalars in the additional doublet [@Chowdhury:2011ga]. When taking into account the data of the LHC and DM direct detection experiments, the parameter space of this model is highly constrained [@Borah:2012pu; @Arhrib:2013ela]. The DM particle can also be a gauge singlet fermion which interacts with the SM sector through a gauge singlet real scalar. The phenomenology of this type of DM model has been explored in Refs. [@Kim:2008pp; @Lee:2008xy; @Kim:2009ke; @Baek:2011aa; @Baek:2012uj]. Light subGeV-scale singlet scalars exchanged by the fermionic DM particles can serve as a force-carrier in the mechanism of the Sommerfeld enhancement which has been considered to explain the large boost factors suggested by the data of various DM indirect detection experiments (see e.g. Refs. [@Sommerfeld31; @Hisano:2002fk; @Hisano:2003ec; @Cirelli:2007xd; @ArkaniHamed:2008qn; @Pospelov:2008jd; @MarchRussell:2008tu; @Cassel:2009wt; @Liu:2013vha; @Chen:2013bi]), such as PAMELA [@Adriani:2008zr], Fermi-LAT [@Abdo:2009zk; @1109.0521] and AMS-02 [@PhysRevLett.110.141102] (for a recent analysis see e.g. [@Jin:2013nta]). It is of interest to investigate whether the strongly first order EWPhT can also be realized in the singlet fermionic DM model. This question was addressed in Ref [@Fairbairn:2013uta] in which the discussion was limited to the case of tree-level barrier only. However, without the $Z_2$ symmetry, the strongly first order EWPhT can be achieved from the singlet scalar contributions via both tree- and loop-level effects due to the linear and cubic terms in the singlet scalar and Higgs potential, which is similar with the case of the SM plus a gauge singlet real scalar [@Espinosa:2011ax; @Chung:2012vg; @Choi:1993cv; @Ham:2004cf; @Ahriche:2007jp; @Profumo:2007wc; @Cline:2009sn; @Huang:2012wn]. In this work, we aim at an extensive and up-to-date analysis of the EWPhT in this model. In comparison with the previous analysis, we make the following improvements: - We go beyond the tree-level analysis by including the loop-level barrier induced from the thermal corrections to the effective potential. We show that when taking into account both the tree- and loop-level barriers the allowed parameter space is significantly enlarged. For instance, the upper limit on the mass of the second Higgs particle is about $100{\mbox{ GeV}}$ higher at $\sin\alpha=0.001$. At the same time the critical temperature after including the cubic terms from one-loop corrections is about $10\%$ higher. We show that in this case the allowed mass of the second Higgs particle can reach $\sim 600{\mbox{ GeV}}$. - We adopt an improved analytical approximation of the finite temperature effective potential which well matches both the usual high- and low-temperature approximations. This approximation makes our analysis valid for large values of $\varphi_c/T_c$, which is of crucial importance as the value of $\varphi_c/T_c$ can reach up to $10$ in this model. - We consider the contribution from the sphaleron magnetic moment to the sphaleron energy. We find that in this model the contribution from the sphaleron magnetic moment is weakened compared with the case of the SM, due to the extra scalar field. The sphaleron magnetic moment energy can lead to a difference between the values of $\varphi_c/T_c$ and $E_{\text{sph}}(T_c)/35T_c$ within $10\%$. - We include the latest upper limits on DM-neucleon scattering cross section from the LUX experiment [@Akerib:2013tjd] which is about one order of magnitude stronger than the previous one reported by XENON100 [@Aprile:2012nq]. As a consequence the mixing angle between the SM-like and the second Higgs particles is stringently constrained. - We focus on the constraints on the phenomenologically interesting physical parameters such as the mass of the second Higgs particle, the mixing angle and the DM particle mass. A numerical scan of the parameter space of this model is performed using a Markov Chain Monte Carlo (MCMC) approach. Taking into account the latest data from the LHC and the LUX experiments, and combining the constrains from the LEP experiment and the electroweak precision test, we find that the mass of the second Higgs particle is in the range $\sim 30 - 350{\mbox{ GeV}}$ and the mixing angle is constrained to $\alpha \lesssim 28^{\circ}$. We also find that the DM particle mass is predicted to be in the range $\sim 15-350{\mbox{ GeV}}$. This paper is organized as follows. We first give a brief overview of the singlet fermionic DM model in section \[smodel\]. In section \[sewpt\], we discuss the effective potential at finite temperature at the tree- and loop-level. A numerical analysis of parameter space is performed and the allowed parameter space is given in section \[spara\]. In section \[sec:sphaleron\] we discuss the correction of the sphaleron energy from the magnetic dipole and its effect on the parameter space allowed by the requirement of a strongly enough first order EWPhT. We then investigate the constraints from DM thermal relic density (section \[srelic\]), DM direct detection (section \[sdire\]), LHC data on Higgs signal strength (section \[sdiph\]), LEP data and electroweak precision test (section \[slep\]). The combined result is present in section \[sresu\]. Finally, conclusions and some discussions are given in section \[sconc\]. Singlet fermionic dark matter model {#smodel} =================================== We consider an extension of the SM with a gauge singlet Dirac fermion $\psi$ which interacts with SM particles through a gauge singlet scalar $S$. The tree-level Higgs potential of this model is given by $$\label{eq:Ls} V\left(\Phi,S\right)= -\mu_\phi^2 \Phi^\dagger \Phi + \lambda_\phi \left(\Phi^\dagger\Phi\right)^2 - \mu_1^3 S - \frac{1}{2}\mu_{s}^{2}S^{2} - \frac{1}{3}\mu_{3}S^{3} + \frac{1}{4}\lambda_{s}S^{4}+\mu \Phi^{\dagger}\Phi\,S + \frac{1}{2} \lambda \Phi^{\dagger}\Phi\,S^2,$$ where $\Phi$ is the SM Higgs doublet $$\Phi=\left(\begin{array}{c} G^{+}\\ \frac{1}{\sqrt{2}}\left(\phi^0-iG^0\right) \end{array}\right),$$ where $G^\pm$, $G^0$ are the would-be Goldstone bosons. The coefficient $\mu_1$ in Eq. (\[eq:Ls\]) can be eleminated by a shift of the field $S$, $S\rightarrow S+\sigma$, which only causes a redefinition of parameters. In general both $\phi^0$ and $S$ can develop non-zero VEVs at zero temperature which are defined as $\varphi_0 \equiv \langle \phi^0 \rangle\mid_{T=0}$ and $s_0 \equiv \langle S \rangle\mid_{T=0}$. The last two terms in Eq. (\[eq:Ls\]) lead to off-diagonal terms in the squared mass matrix of singlet scalar and the SM Higgs boson, which introduces a mixing between $\phi^0$ and $S$. The squared mass matrix of $\phi^0$ and $S$ is given by $$\label{massmatrix} \mathcal{M}^{2} = \left(\begin{array}{cc} \mathcal{M}_{11}^2 & \mathcal{M}_{12}^2 \\ \mathcal{M}_{21}^2 & \mathcal{M}_{22}^2 \end{array}\right),$$ where $$\begin{aligned} \label{mij} \mathcal{M}_{11}^2 &=& -\mu_{\phi}^{2}+3\lambda_{\phi}\varphi_0^{2}+ \frac{1}{2} \lambda s_0^2 + \mu s_0 , \notag \\ \mathcal{M}_{22}^2 &=& -\mu_{s}^{2}-2\mu_{3}s_0+3\lambda_{s}s_0^{2}+ \frac{1}{2} \lambda \varphi_0^{2} , \\ \mathcal{M}_{12}^2 &=& \mathcal{M}_{21}^2 = \mu \varphi_0+\lambda \varphi_0s_0 . \notag\end{aligned}$$ The squared mass matrix in Eq. (\[massmatrix\]) can be diagonalized by rotating $\phi^0$ and $S$ into mass eigenstates ($h$, $H$) $$\left(\begin{array}{c} h\\ H \end{array}\right)=\left(\begin{array}{cc} \cos\alpha & -\sin\alpha\\ \sin\alpha & \cos\alpha \end{array}\right)\left(\begin{array}{c} \phi^0\\ S \end{array}\right),$$ where the mixing angle $\alpha$ is $$\label{tan2a} \tan2\alpha = \frac{2 m_{12}^2}{\left(m_{22}^2-m_{11}^2\right)}.$$ The value of $\alpha$ is defined in the range $0^\circ-45^\circ$, such that $h$ plays the role of the SM-like Higgs particle while $H$ is singlet dominant. The interaction involving the singlet fermionic DM particle $\psi$ is given by the Lagrangian $$\mathcal{L}_{\psi}=i\,\bar{\psi} \partial\!\!\!/ \psi - y_{\psi}\bar{\psi}\psi S.$$ In general $S$ can develope a non-zero VEV, which contributes to the mass of the fermionic DM particle $\psi$. In this work we consider the case where $\psi$ only obtains mass from the VEV of $S$, namely $m_\psi=y_\psi s_0$, which makes the model more predictive. Effective potential and EWPhT {#sewpt} ============================= The tree-level potential for $\varphi = \langle\phi^0\rangle$ and $s = \langle S\rangle$ can be written as $$\label{v0} V_0\left(\varphi,s\right) = -\frac{1}{2} \mu_{\phi}^2 \varphi^2 - \frac{1}{2} \mu_s^2 s^2 - \frac{1}{3} \mu_3 s^3 + \frac{1}{2} \mu \, s \,\varphi^2 + \frac{1}{4} \lambda_{\phi} \varphi^4 + \frac{1}{4} \lambda_s s^4 + \frac{1}{4}\lambda \, s^2 \varphi^2 .$$ The coefficients $\mu_{\phi}$ and $\mu_s$ can be rewritten in terms of the VEVs $\varphi_0$ and $s_0$ according to the minimization conditions of the tree-level potential. However, the minimization conditions can not guarantee that $\left(\varphi_0, s_0\right)$ is the global minimum. Thus a check on whether there exists a deeper minimum is needed. In order to guarantee the stability of $\left(\varphi_0, s_0\right)$ as the global vacuum, it is also required that the potential is bounded-from-below. The parameters $\lambda_{\phi}$, $\mu$ and $\lambda$ can be rewritten in terms of three physical parameters, i.e. the masses of the two Higgs particles $m_h$, $m_H$ and the mixing angle $\alpha$, as follows $$\begin{aligned} \lambda_{\phi} & =& \frac{1}{2\varphi_0^{2}}\left(m_{h}^{2}\cos^{2}\alpha+m_{H}^{2}\sin^{2}\alpha\right) , \notag \\ \mu & =&-2\frac{s_0}{\varphi_0^{2}} \left(m_{h}^{2}\sin^{2}\alpha+m_{H}^{2}\cos^{2}\alpha+\mu_{3}s_0-2\lambda_{s}s_0^{2}\right) , \\ \lambda & =& \frac{1}{\varphi_0s_0}\left[\left(m_{H}^{2}-m_{h}^{2}\right)\sin\alpha\cos\alpha-\mu \varphi_0\right]. \notag\end{aligned}$$ We include one-loop Coleman-Weinberg correction of the potential at zero temperature [@Coleman:1973jx] $$\label{v1} V_1\left(\varphi,s\right) = \frac{1}{64\pi^2} \sum_{i} N_{i} m_{i}^4 \left(\varphi,s\right) \left[\log \frac{m_{i}^2 \left(\varphi,s\right)}{Q^2} - C_{i}\right],$$ where $i$ runs over all the particles in the loop, and $N_{i}$ is the degrees of freedom of the particle $i$, $C_i$ is a constant ($C_i = 6/5$ for gauge bosons, $C_i = 3/2$ for scalars and fermions), $Q$ is the renormalization scale which we fix at the mass of the top quark. The counter terms $V_{\text{CT}}\left(\varphi,s\right)$ needed to renormalize the potential are given in Appendix \[appA\]. The one-loop effective potential at finite temperature $T$ can be written as $$\label{veff} V_{\text{eff}}\left(\varphi,s;T\right)=V_{0}\left(\varphi,s\right) + V_{1}\left(\varphi,s\right) + V_{\text{CT}}\left(\varphi,s\right) + V_{1}\left(\varphi,s;T \right) ,$$ where $V_1\left(\varphi,s;T\right)$ is the one-loop thermal corrections $$V_{1}\left(\varphi,s;T\right) = \frac{T^4}{2\pi^2} \left[ \underset{i}{\sum} n_i I_{\text{B}}\left(a_i\right) + \underset{j}{\sum} n_j I_{\text{F}}\left(a_j\right) \right],$$ where $a=m^2\left(\varphi,s\right)/T^2$, $i$ ($j$) runs over all the bosons (fermions), $n_{i(j)}$ denotes the degrees of freedom of bosons (fermions), and $I_{{\text{B}}({\text{F}})} \left(a\right)$ is defined as $$\label{Ibf} I_{{\text{B}}({\text{F}})}\left(a\right) = \int_0^{\infty} dx \, x^2 \ln \left(1 \mp e^{-\sqrt{x^2 + a}} \right),$$ where the sign $-$ ($+$) is for bosons (fermions). Since the evaluation of the integration in Eq. (\[Ibf\]) is computationally expensive, it is necessary to have an analytical approximation. In the high temperature limit, i.e. $m\left(\varphi,s\right)/T \ll 1$, $I_{{\text{B}}({\text{F}})}\left(a\right)$ can be expanded as [@Dolan:1973qd] $$\begin{aligned} \label{IhighT} I_{{\text{B}}}^{(1)} \left(a\right) &=& -\frac{\pi^4}{45} + \frac{\pi^2}{12} a - \frac{\pi}{6} a^{\frac{3}{2}} - \frac{1}{32} a^2 \left[\log\left(a\right) - \gamma_{\text{B}}\right] , \label{highTb} \\ I_{{\text{F}}}^{(1)} \left(a\right) &=& -\frac{7\pi^4}{360} + \frac{\pi^2}{24} a + \frac{1}{32} a^2 \left[\log\left(a\right) - \gamma_{\text{F}}\right] , \label{highTf}\end{aligned}$$ where $\gamma_{\text{B}} = 5.40762$ and $\gamma_{\text{F}} = 2.63503$. The term cubic in $m/T$ in Eq. (\[highTb\]) gives rise to the barrier in the potential which makes the phase transition first order. In the low temperature limit, $I_{{\text{B}}({\text{F}})}\left(a\right)$ can be expanded as [@Anderson:1991zb] $$\label{lowT} I_{\text{B}}^{(2)}\left(a;n\right) = I_{\text{F}}^{(2)}\left(a;n\right) = - \sqrt{\frac{\pi}{2}} \, a^{\frac{3}{4}} \, e^{-a^{1/2}} \left( 1+\frac{15}{8}a^{\frac{1}{2}}+\frac{105}{128}a \right).$$ The high- and low-temperature approximations are shown in Fig. \[fI\]. It can be seen that the high temperature approximation starts to fail when $a \gtrsim 3$. By matching the high- and low-temperature approximations, we obtain a reasonable approximation to the integral $$\label{Iapp} I_{{\text{B}}({\text{F}})}^{(3)}\left(a\right) = t_{{\text{B}}({\text{F}})}\left(a\right) \, I_{{\text{B}}({\text{F}})}^{(1)} \left(a\right) + \left(1-t_{{\text{B}}({\text{F}})}\left(a\right) \right) \, I_{{\text{B}}({\text{F}})}^{(2)} \left(a;2\right),$$ where $t_{\text{B}}(a) = e^{-\left(a/6.3\right)^4}$ and $t_{\text{F}}(a) = e^{-\left(a/3.25\right)^4}$ are obtained by numerically fitting to the exact value of the integral. A comparison of different approximations of $I_{{\text{B}}({\text{F}})}(a)$ is shown in Fig. \[fI\]. For the approximation $I^{(3)}_{{\text{B}}({\text{F}})}(a)$ in Eq. (\[Iapp\]), the deviation to the exact value of $I_{{\text{B}}({\text{F}})}$ is less than $5\%$ in the region $0\leqslant a\leqslant 20$. The calculation of effective potential can be further improved by including thermal corrections to the boson masses which come from high order ring diagrams. After including the ring diagrams, the field-dependent squared mass matrix for the two Higgs particles is given by $$\mathcal{M}^2\left(\varphi, s; T\right) = \left(\begin{array}{cc} \mathcal{M}_{11}^2 & \mathcal{M}_{12}^2 \\ \mathcal{M}_{21}^2 & \mathcal{M}_{22}^2 \end{array}\right) + \left( \begin{array}{cc} c_\phi & 0 \\ 0 & c_s \end{array} \right) T^2,$$ where the matrix elements $\mathcal{M}_{ij}$ are defined analogously as in Eq. (\[mij\]) with replacements $\varphi_0\rightarrow\varphi$, $s_0 \rightarrow s$, $c_\phi$ and $c_s$ are defined as $$\begin{aligned} c_\phi &=& \frac{1}{48}\left(9g^2 + 3g^{\prime 2} + 12 y_t^2 + 24 \lambda_{\phi} + 2 \lambda\right), \\ c_s &=& \frac{1}{12}\left(2\lambda + 3 \lambda_s + 2 y_{\psi}^2\right),\end{aligned}$$ where $y_t$ is the top Yukawa coupling, $g$ and $g^{\prime}$ are the $SU(2)_L$ and $U(1)_Y$ gauge couplings, respectively. The thermal masses of the Goldstone bosons are given by $$m_{G^0,G^{\pm}}^2\left(\varphi, s; T\right) = -\mu_{\phi}^2 + \lambda_{\phi} \varphi^2 + \mu s + \frac{1}{2} \lambda s^2 + c_\phi T^2.$$ In order to trigger first order EWPhT, the thermal effective potential must have two degenerate minima separated by a barrier at the critical temperature. Due to the existence of the extra scalar field, there can exist two kinds of barriers in this model - [**Tree-level barrier.**]{} This kind of barrier arises from the terms linear and cubic in $s$ which are already present in the effective potential at tree-level. In the scenario with tree-level barrier only, one important implication is that a first order EWPhT is always related to a change of the VEV of the singlet scalar field at the critical temperature. If the VEV of the singlet scalar field is constant during the EWPhT, the tree-level potential would have the same structure as that in the SM case which has no barrier. - [**Loop-level barrier.**]{} This kind of barrier arises from the term cubic in $m/T$ which comes from the thermal one-loop corrections of the bosonic fields to the effective potential. It also exists in the SM case, which is however not enough to trigger a strongly first order EWPhT. In this model, the extra singlet scalar field can contribute to this kind of barrier and make it possible to trigger a strongly first order EWPhT. For the investigation of the tree-level barrier, it is enough to keep only the leading order terms which are quadratic in $m/T$ of the high-temperature approximation $$\label{V1lo} V_1^{\text{lo}}\left(\varphi, s; T\right) = \left(\frac{1}{2} \kappa_\phi \varphi^2 + \frac{1}{2} \kappa_s s^2 + \kappa_3 s \right) T^2,$$ where $$\begin{aligned} \kappa_\phi &=& \frac{1}{48} \left(9g^2 + 3g^{\prime 2} + 12 y_t^2 + 24 \lambda_{\phi} + 2\lambda \right), \notag \\ \kappa_s &=& \frac{1}{12} \left(2\lambda + 3 \lambda_s + 2 y_{\psi}^2 \right), \\ \kappa_3 &=& \frac{1}{12} \left(-\mu_3 + 2 \mu\right). \notag\end{aligned}$$ For an illustration of the tree-level barrier, we use $V_0\left(\varphi,s\right)+ V_1^{\text{lo}}\left(\varphi,s;T \right)$ as an approximation of the effective potential. The stationary points of this effective potential are located at the intersections of the curves determined by $\partial V_{\text{eff}}\left(\varphi, s;T\right) / \partial \varphi = 0$ and $\partial V_{\text{eff}}\left(\varphi, s;T\right) /\partial s = 0$ which lead to $$\label{path1} \varphi = 0 \quad \text{or} \quad \varphi^2 = f_h\left(s\right) = -\frac{\kappa_\phi T^2 - \mu_\phi^2 + \mu s + \frac{1}{2}\lambda s^2}{\lambda_\phi},$$ and $$\label{path2} \varphi^2 = f_s\left(s\right) = -2\cdot\frac{\kappa_3 T^2 + \left(\kappa_s T^2-\mu_s^2\right) s - \mu_3 s^2 +\lambda_s s^3}{\mu+\lambda s}.$$ ![Thermal evolution of the effective potential in the senario with tree-level barrier only. The parameters are fixed at $s_0=300{\mbox{ GeV}}$, $\lambda_\phi = 1$, $\mu_3 = 250{\mbox{ GeV}}$, $\lambda_s = 1$, $\mu = -250{\mbox{ GeV}}$, $\lambda = 0.1$ and $y_{\psi} = 0.5$. Left) The effective potentials at $T>T_c$, $T=T_c$, and $T<T_c$ from top to bottom, respectively. The global minima of the effective potentials are indicated by red dots. In (c) the path with lowest barrier between the two local minima is indicated by the red line. Right) Curves corresponding to $\partial V_{\text{eff}}\left(\varphi, s;T\right) / \partial \varphi = 0$ (solid line) and $\partial V_{\text{eff}}\left(\varphi, s;T\right) /\partial s = 0$ (dashed line). The global minima are located at the intersections of the two curves as indicated by the black dots.[]{data-label="fefftems"}](TgtTc.eps){width="50pt"} ![Thermal evolution of the effective potential in the senario with tree-level barrier only. The parameters are fixed at $s_0=300{\mbox{ GeV}}$, $\lambda_\phi = 1$, $\mu_3 = 250{\mbox{ GeV}}$, $\lambda_s = 1$, $\mu = -250{\mbox{ GeV}}$, $\lambda = 0.1$ and $y_{\psi} = 0.5$. Left) The effective potentials at $T>T_c$, $T=T_c$, and $T<T_c$ from top to bottom, respectively. The global minima of the effective potentials are indicated by red dots. In (c) the path with lowest barrier between the two local minima is indicated by the red line. Right) Curves corresponding to $\partial V_{\text{eff}}\left(\varphi, s;T\right) / \partial \varphi = 0$ (solid line) and $\partial V_{\text{eff}}\left(\varphi, s;T\right) /\partial s = 0$ (dashed line). The global minima are located at the intersections of the two curves as indicated by the black dots.[]{data-label="fefftems"}](TeqTc.eps){width="50pt"} ![Thermal evolution of the effective potential in the senario with tree-level barrier only. The parameters are fixed at $s_0=300{\mbox{ GeV}}$, $\lambda_\phi = 1$, $\mu_3 = 250{\mbox{ GeV}}$, $\lambda_s = 1$, $\mu = -250{\mbox{ GeV}}$, $\lambda = 0.1$ and $y_{\psi} = 0.5$. Left) The effective potentials at $T>T_c$, $T=T_c$, and $T<T_c$ from top to bottom, respectively. The global minima of the effective potentials are indicated by red dots. In (c) the path with lowest barrier between the two local minima is indicated by the red line. Right) Curves corresponding to $\partial V_{\text{eff}}\left(\varphi, s;T\right) / \partial \varphi = 0$ (solid line) and $\partial V_{\text{eff}}\left(\varphi, s;T\right) /\partial s = 0$ (dashed line). The global minima are located at the intersections of the two curves as indicated by the black dots.[]{data-label="fefftems"}](TltTc.eps){width="50pt"} We show the evolution of this effective potential with temperature in Fig. \[fefftems\]. Since at sufficiently high temperature the effective potential is dominated by the contributions from the thermal corrections in Eq. (\[V1lo\]), there is only one minimum at $\varphi = 0$, as shown in Fig. \[fhightem\]. As the temperature decreases, local minimum with $\varphi\neq0$ appears, but the original minimum at $\varphi = 0$ is still the global one. At the critical temperature $T_c$, the minimum at $\varphi = \varphi_c$ becomes degenerate with the minimum at $\varphi = 0$, as shown in Fig. \[fcriticaltem\]. The minimum at $\varphi = 0$ becomes meta-stable and the phase transition of $\varphi$ occurs. It can be seen that there is a barrier which separates the two degenerate minima and leads to first order EWPhT. After the phase transition of $\varphi$, the local minimum at $\varphi\neq0$ becomes the global one, as shown in Fig. \[flowtem\]. Parameter space for EWPhT {#spara} ========================= To check whether a EWPhT is strongly first order, we should first find the critical temperature which is defined as when there appear two degenerate minima. We search for $T_c$ in the range from $T_{\text{min}}=1{\mbox{ GeV}}$ to $T_{\text{max}}=1{\mbox{ TeV}}$. We start from $T_{\text{min}}$, then increase the temperature and check the minima of the potential. The critical temperature is obtained when the local minimum at $\varphi\neq0$ becomes degenerate with the one at $\varphi=0$. If the global minimum at $T_{\text{max}}$ is at $\varphi\neq0$, EWPhT will not occur. When the EWPhT occurs, there is a path connecting the two degenerate local minima which has the lowest barrier (see Fig. \[fcriticaltem\]). If there is no barrier along this path, the EWPhT is of the second order. In this case the local minimum corresponds to a flat direction of the potential. To identify this case we follow the method in Ref. [@Cline:2009sn] to check whether a putative minimum is a real minimum. We minimize the potential on small circles surrounding the putative local minimum. If the minima on the circles are greater than the putative minimum, it is indeed a true local minimum. We explore the full parameter space of the singlet fermionic DM model which includes: $m_H$, $\alpha$, $s_0$, $\mu_3$, $\lambda_s$, and $m_{\psi}$. We scan these parameters in the ranges $$\begin{aligned} 10{\mbox{ GeV}}\leqslant m_H \leqslant 1{\mbox{ TeV}}, \quad & 0 \leqslant \alpha \leqslant 45^{\circ}, \quad & -1{\mbox{ TeV}}< s_0 \leqslant 1{\mbox{ TeV}}, \notag \\ -1{\mbox{ TeV}}\leqslant \mu_3 \leqslant 1{\mbox{ TeV}}, \quad & 0 \leqslant \lambda_s \leqslant 3, \quad & -3 \leqslant y_{\psi} \leqslant 3.\end{aligned}$$ The mass of the SM-like Higgs particle is fixed at $m_h = 125{\mbox{ GeV}}$. We use an improved random walk sampling algorithm to scan the parameter space based on a MCMC method with the Metropolis algorithm. The likelihood of a given parameter set $\boldsymbol{x}$ is defined as $$\label{lik} \mathscr{L}\left(\boldsymbol{x}\right) = \min\{\varphi_c/T_c, 1\}.$$ We run multi-chain samplers with initial values uniformly distributed in the 6-dimensional parameter space and obtain a sample set containing about $5\times10^6$ sample points satisfying $\varphi_c/T_c>1$. ![The relative frequency distribution of the order parameter $\varphi_c/T_c$ of the samples satisfying $\varphi_c/T_c>1$ which are obtained using the likelihood function in Eq. (\[lik\]).[]{data-label="fvctc"}](vcTcdist.eps){width="200pt"} The relative frequency distribution of the order parameter $\varphi_c/T_c$ is shown in Fig. \[fvctc\]. Strongly first order EWPhTs are found with $\varphi_c/T_c$ up to $10$ in this model. The frequency distributions of the $6$ free parameters are shown in Fig. \[fparamdist\]. It can be seen that, for $\mathcal{E} = 1$, there exists an upper limit on the mass of the second Higgs particle around $600{\mbox{ GeV}}$, and $s_0$ is constrained to $\left|s_0\right|\lesssim600{\mbox{ GeV}}$. Heavier particles cannot trigger a strongly enough first order EWPhT, as the contributions of heavy particles suffer from exponential suppression as shown in Eq. (\[lowT\]). In this model, the extra scalar field leads to a tree-level barrier at the critical temperature. Both of the tree- and the loop-level barriers can trigger strongly first order EWPhT. A comparison between the tree- and loop-level barriers is shown in Fig. \[fparamspace2\] in which we plot the allowed regions for the case with tree-level barrier only and the case with both tree- and loop-level barriers. As shown by the figure, the allowed region with both the tree- and loop-level barriers is larger than that in tree-level only case. For instance, the upper limits of $m_H$ is about $100{\mbox{ GeV}}$ higher at $\sin\alpha=0.001$ for $\mathcal{E}=1$. The loop-level cubic terms also raise the critical temperature. As shown in Fig. \[fparamspace2:c\], the critical temperature has an upper limit around $150{\mbox{ GeV}}$, which is about $10\%$ lower in the case where only the tree-level barrier is considered. The effect of the sphaleron magnetic moment {#sec:sphaleron} =========================================== The condition for the sphaleronic interactions to be sufficiently suppressed to preserve the baryon asymmetry generated during the EWPhT is given by [@Shaposhnikov:1987tw; @Shaposhnikov:1986jp] $$\frac{E_{sph}(T_c, B)}{T_c} \gtrsim 35.$$ In the SM, the sphaleron energy relates to $\varphi$ the VEV of the Higgs field by $$E_{\text{sph}} = \frac{8 \pi m_W(\varphi)}{g^2} \mathcal{C},$$ where $m_W(\varphi)$ is the $W$-Boson mass, and $\mathcal{C} \sim 2$ is a constant determined by the sphaleron solution. Thus, this condition can be translated into Eq. (\[eq1\]) with $\mathcal{E}=1$. This conclusion can be modified if there exists a primordial magnetic field in the early universe [@DeSimone:2011ek; @Comelli:1999gt]. The magnetic field can be generated before or during the EWPhT through various mechanisms (for a review see Ref. [@Enqvist:1998fw]). Meanwhile, the electroweak sphaleron may develope a $U(1)_Y$ magnetic dipole moment. The interaction between the magnetic dipole and the background magnetic field can give negative contribution to the sphaleron energy. Consequently, the preservation of the baryon asymmetry requires a larger value of $\mathcal{E}$. In the presence of a background hypermagnetic field $B$, the sphaleron magnetic moment $\mu$ can lower the sphaleron energy $$E_{\text{sph}} \left(T, B\right) = E_{\text{sph}}(T) - \mu(T) B.$$ In this work, we parametrize the external hypermagnetic field as [@DeSimone:2011ek] $$B = b T^2,$$ where $b$ is a dimensionless parameter which is usually taken to be $b \lesssim 0.4$. To estimate the effect of sphaleron dipole moment in this model, we follow the approach adopted in Refs. [@DeSimone:2011ek; @Comelli:1999gt]. The formulas which give the sphaleron solution and the sphaleron magnetic moment are summarized in the Appendix \[sphaleron\]. In this model, the relation between $E_{\text{sph}}/(35T_c)$ and $\varphi_c/T_c$ is complicated and can only be calculated numerically. In Table \[sphaleron\_table\] we show the values of $E_{\text{sph}}/(35T_c)$ and $\varphi_c/T_c$ for several typical parameter sets. It can be seen that the presence of the sphaleron magnetic moment can lower the sphaleron energy, which makes the value of $E_{\text{sph}}/(35T_c)$ lower than the value of $\varphi_c/T_c$. However, the difference between them are within $10\%$. As can be seen in Table \[sphaleron\_table\], in the listed parameter sets, the values of $\varphi_c/T_c$ varies from 1.2 to 2.12, and all of the parameter sets can provide a strongly enough first order EWPhT. This is different from the conclusion in the case of the SM where the inclusion of the magnetic moment generally requires $\varphi_c/T_c \gtrsim 1.3$ [@Comelli:1999gt]. The reason is that the extra scalar field $S$ in this model raises the sphaleron energy but gives no contribution to the sphaleron magnetic moment, which weakens the contribution from the sphaleron magnetic moment. In Fig. \[fparamspace2:a\] and Fig. \[smallangle\], we show the boundary of the allowed parameter space for $\mathcal{E}=1.2$. It can be seen in Fig. \[smallangle\] that, after considering all the constrains from observables, the difference between the upper bound on the mass of the second Higgs particle for $\mathcal{E}=1.2$ and that for $\mathcal{E}=1$ is within $10\hbox{ GeV}$. $m_2$ $\sin\alpha$ $s_0$ $\mu_3$ $\lambda_s$ $E_{\text{sph}}(T_c)$ $E_{\text{dipole}}(T_c, B)$ $\frac{\varphi_c}{T_c}$ $\frac{E_{\text{sph}}(T_c, B)}{35T_c}$ ------- -------------- ------- --------- ------------- ----------------------- ----------------------------- ------------------------- ---------------------------------------- 256.8 0.05 42.7 125.8 1.05 1.16 0.08 1.20 1.14 120.0 0.074 136.6 267.6 0.75 1.39 0.05 2.12 2.08 97.2 0.002 212.8 235.5 0.70 0.90 0.06 1.21 1.16 197.1 0.14 100.2 464.1 0.92 1.33 0.04 1.65 1.62 127.2 0.02 118.8 61.8 0.80 1.18 0.05 1.38 1.34 : Sphaleron and magnetic dipole energies for several typical parameter sets. The sphaleron energy and magnetic dipole energy are in units of $4\pi\sqrt{\varphi_c^2+s_c^2}/g$. Other parameters $m_2$, $s_0$ and $u_3$ are in unit of GeV. The SM Higgs mass is set to $m_1 = 125 \hbox{ GeV}$. The magnetic field is fixed at $B = 0.4 T^2$. \[sphaleron\_table\] DM thermal relic density {#srelic} ======================== The fermionic DM particle $\psi$ can annihilate into final states $\bar{f}f$, $W^+W^-$, $ZZ$, $hh$, $HH$ or $hH$ via $s$-channel Higgs particle exchanges. For annihilation with final states $hh$, $HH$ or $hH$, the $t$- and $u$-channels are also possible. The Feynman diagrams for these processes are shown in Fig. \[fd\_anni\]. The cross sections for these processes are given in Appendix \[appB\]. ![Feynman diagrams for the annihilation of fermionic DM particle.[]{data-label="fd_anni"}](fd_anni.eps){width="420pt"} The thermal average of the cross section multiplied by the DM relative velocity $v_{\text{rel}}$ at a temperature $T$ is given by $$\left< \sigma v_{\text{rel}} \right> = \frac{1}{8m_{\psi}^4 T K_2^2 \left(m_{\psi}/T\right)} \int_{4m_{\psi}^2}^{\infty} d\mathfrak{s} \, \sigma \left(\mathfrak{s}\right) \left(\mathfrak{s}-4m_{\psi}^2\right) \sqrt{\mathfrak{s}} \, K_1\left(\frac{\sqrt{\mathfrak{s}}}{T}\right),$$ where $K_{1}$ ($K_2$) is the modified Bessel function of the first (second) kind, $\sqrt{\mathfrak{s}}$ denotes the center-of-mass energy. The temperature evolution of the abundance $Y$ which is defined as the number density devided by the entropy density of the DM particle is governed by the Boltzmann equation [@Gondolo:1990dk] $$\label{be} \frac{dY}{dT} = \sqrt{\frac{\pi g_{*}\left(T\right)}{45}} M_{\text{pl}} \langle\sigma v_{\text{rel}}\rangle \left[Y\left(T\right)^2 - Y_{\text{eq}}\left(T\right)^2 \right],$$ where $M_{\text{pl}}=1.22 \times 10^{19} {\mbox{ GeV}}$ is the Planck mass scale, $g_{*}$1 is the effective number of relativistic degrees of freedom, and $Y_{\text{eq}}$ is the abundance at equilibrium. The relic density is related to the present-day abundance $Y\left(T_0\right)$ by $$\Omega \, h^2 = 2.472 \times 10^8 {\mbox{ GeV}}^{-1} m_\psi Y\left(T_0\right),$$ where $T_0$ is the temperature of the microwave background. In this work we adopt the freeze-out approximation, and use micrOMEGAs3.3 for numerical calculation of the relic density [@Belanger:2001fz; @Belanger:2006qa]. The freeze-out temperature $T_f$ can be defined from the relation $Y\left(T_f\right) = \left(1+\delta\right)Y_{\text{eq}} \left(T_f\right)$ with $\delta$ being a constant and can be determined by solving $$\left.\frac{d \ln Y_{\text{eq}}}{d T}\right|_{T=T_f} = \delta\left(\delta+2\right) \sqrt{\frac{\pi g_{*}\left(T_f\right)}{45}} M_{\text{pl}} \langle \sigma v_{\text{rel}} \rangle Y_{\text{eq}}\left(T_f\right) ,$$ with $\delta = 1.5$ [@Belanger:2001fz]. Below the freeze-out temperature, $Y_{\text{eq}} \ll Y$, Eq. (\[be\]) can be integrated $$\frac{1}{Y\left(T_0\right)} = \frac{1}{Y\left(T_f\right)} + \sqrt{\frac{\pi}{45}} M_{\text{pl}} \int_{T_0}^{T_f} \sqrt{g_{*}\left(T\right)} \langle\sigma v_{\text{rel}}\rangle dT.$$ The deviation of this approximation from the exact solution of the Boltzmann equation Eq. (\[be\]) is within $2\%$ [@Belanger:2001fz]. ![DM thermal relic density as a function of the DM particle mass with $m_H=250{\mbox{ GeV}}$ for different values of $\alpha=2^\circ$, $20^\circ$ and $45^\circ$, respectively. Other parameters are fixed at $s_0=300{\mbox{ GeV}}$, $\mu_3=300{\mbox{ GeV}}$ and $\lambda_s=1$. The horizontal solid line indicates $\Omega h^2=0.1187$ [@Ade:2013zuv].[]{data-label="fRelicDensity"}](RelicDensity.eps){width="350pt"} Fig. \[fRelicDensity\] shows the thermal relic density as a function of the DM particle mass. Since the measurement on the DM relic density from the Planck experiment is very precise, the value of $m_\psi$ can actually be solved from the DM relic density up to a five-fold ambiguity. The ambiguity arises from the two resonant annihilations when $m_\psi \approx m_{h,H}/2$. Direct detection of DM {#sdire} ====================== For a Dirac DM particle the spin-independent DM-proton elastic scattering cross section is given by $$\sigma_{\text{SI}} \approx \frac{m_{r}^{2}}{\pi}\lambda_{p}^{2},$$ where $m_r$ is the DM-proton reduced mass $m_r = m_{\psi} m_p / \left(m_{\psi} + m_p\right)$ with $m_p$ the proton mass. The coupling $\lambda_p$ is given by $$\frac{\lambda_{p}}{m_{p}} = \sum_{q=u,d,s}f_{T_q}^{(p)}\frac{\lambda_{q}}{m_{q}} +\frac{2}{27}f_{T_{g}}^{(p)}\sum_{q=c,b,t}\frac{\lambda_{q}}{m_{q}}.$$ The coupling $\lambda_q$ at quark level in this model is $$\label{ddcs} \frac{\lambda_{q}}{m_{q}} = \frac{y_{\psi}\sin\alpha\cos\alpha}{\varphi_0}\left(\frac{1}{m_h^{2}}-\frac{1}{m_H^{2}}\right).$$ The parameters $f_{T_q}^{(p)}$ are defined from the nucleon matrix elements $m_p\, f_{T_q}^{(p)} \equiv \left<p\left|m_q \bar{q} q\right|p\right>$ for $q = u,d,s$ and $f_{T_{g}}^{(p)} = 1 - \sum_{q=u,d,s}f_{T_{q}}^{(p)}$. In numerical calculations we take the values $f_{T_{u}}^{(p)} = 0.0153$, $f_{T_{d}}^{(p)} = 0.0191$ and $f_{T_{s}}^{(p)} = 0.0447$ [@Belanger:2013oya]. For some of the recent studies of these parameters we refer to the Refs. [@Cheng:2012qr; @Crivellin:2013ipa; @Cirigliano:2013zta]. Currently the strongest upper limits on $\sigma_{\text{SI}}$ are given by the LUX experiment [@Akerib:2013tjd]. The allowed region in the $m_H-\sin\alpha$ plane is shown in Fig. \[smallangle\]. It can be seen that the mixing angle is severely constrained by the LUX data, for instance $\sin\alpha\lesssim0.1$ leading to $\alpha \lesssim 5.7^\circ$ at $m_H=350{\mbox{ GeV}}$. In the region where $\left|m_H-m_h\right| \lesssim 20{\mbox{ GeV}}$, the constraint from LUX data is significantly relaxed due to the destructive interference between the contributions from the two Higgs particles, as shown in Eq. (\[ddcs\]). In Fig. \[smallangle\] we also show the upper bound on the mixing angle corresponding to the data of the XENON100 experiment. It can be seen that the XENON100 constraint on the mixing angle is much weaker than the LUX constraint, for instance $\sin\alpha \lesssim 0.4$ leading to $\alpha\lesssim23^\circ$ at $m_H=350{\mbox{ GeV}}$. The next generation of DM direct detection experiments can push the upper bound on $\sigma_{\text{SI}}$ down to $\sim 10^{-47} \text{cm}^2$ [@Aprile:2012zx]. This upper bound can further constrain the mixing angle $\alpha$. In Fig. \[smallangle\], we show the upper bound on the mixing angle which corresponds to the projected exclusion limit of the future XENON1T experiment. It can be seen that $\sin\alpha$ can be further constrained to one order of magnitude lower than the upper bound from the LUX data in the regions off resonance, for instance $\sin\alpha \lesssim 0.01$ leading to $\alpha\lesssim0.57^\circ$ at $m_H=350{\mbox{ GeV}}$. Higgs signal strength at the LHC {#sdiph} ================================ The LHC experiment has reported the discovery of a SM-like Higgs boson [@Aad:2012tfa; @Chatrchyan:2012ufa]. Throughout our work we take the SM-like Higgs particle mass fixed at $m_h=125{\mbox{ GeV}}$. The Higgs signal strengths in different channels such as $\bar{b}b$, $\tau^+ \tau^-$, $\gamma \gamma$, $WW^*$ and $ZZ^*$ have been measured by the ATLAS, CMS and CDF experiments. The combined result on the Higgs signal strength with respect to the SM value shows no significant deviation from the SM prediction [@Ellis:2013lra] $$\label{eq:HiggsStrength} r_h = 1.02 ^{+0.11} _{-0.12},$$ with $r_h$ defined as the signal strength of the SM-like Higgs particle in new physics models relative to that in the SM. We consider $r_h$ in the range $0.78-1.24$ which corresponds to the approximately $95\%$ confidence level (CL) allowed range. The signal strength of the SM-like Higgs particle in this model with respect to the SM value is given by $$r_h = \frac{\sigma_{gg\rightarrow h} B_{h\rightarrow XX}}{\sigma_{gg\rightarrow h}^{\text{SM}} B^{\text{SM}}_{h\rightarrow XX}} = \frac{\sigma_{gg\rightarrow h}}{\sigma_{gg\rightarrow h}^{\text{SM}}} \times \frac{\Gamma_{h \rightarrow XX}}{\Gamma^{\text{SM}}_{h \rightarrow XX}} \times \frac{\Gamma^{\text{SM}}_{h}}{\Gamma_{h}},$$ where $X$ stands for any final state particle, $\sigma_{gg\rightarrow h}$ is the production cross section through gluon-gluon fusion of the SM-like Higgs particle, $\Gamma_{h \rightarrow XX}$ is the width of the SM-like Higgs particle decaying to $X$, $\Gamma_h$ is the total decay width of the SM-like Higgs particle, and $\sigma^{\text{SM}}_{gg\rightarrow h}$, $\Gamma^{\text{SM}}_{h \rightarrow XX}$ and $\Gamma_h^{\text{SM}}$ are the corresponding values in the SM. The mixing between the two Higgs particles leads to a universal $\cos\alpha$ suppression of all the couplings between the SM-like Higgs particle and the SM fermions and gauge bosons, which leads to $$\frac{\sigma_{gg\rightarrow h}}{\sigma_{gg\rightarrow h}^{\text{SM}}} = \frac{\Gamma_{h\rightarrow XX}}{\Gamma_{h\rightarrow XX}^{\text{SM}}} = \cos^2 \alpha.$$ Additionally, the signal strength of the SM-like Higgs particle is also suppressed by two possible new invisible decay channels which are $h \rightarrow \bar{\psi}\psi$ and $h \rightarrow H H$. The total decay width of the SM-like Higgs particle in this model can be written as $$\Gamma_{h} = \Gamma_{h}^{\text{SM}} \cos^2\alpha + \Gamma_{h\rightarrow \bar{\psi}\psi} + \Gamma_{h \rightarrow H H},$$ where $\Gamma_{h\rightarrow \bar{\psi}\psi}$ and $\Gamma_{h \rightarrow H H}$ are the decay widths of the SM-like Higgs particle via the two new channels $$\begin{aligned} \Gamma_{h \rightarrow \bar{\psi}\psi} &=& \frac{y_{\psi}^2 m_h}{8 \pi} \beta_{\psi}^3 \cdot \sin^2\alpha, \\ \Gamma_{h \rightarrow H H} &=& \frac{\lambda_{hHH}^2}{8\pi m_h} \beta_{H},\end{aligned}$$ where $\beta_{\psi(H)} = \sqrt{1-4m_{\psi(H)}^2/m_h^2}$ and $\lambda_{hHH}$ is the coupling of $h H H$ defined in Eq. (\[lamhHH\]) in Appendix \[appB\]. Thus, the signal strength of the SM-like Higgs particle can be written as $$\label{r1} r_h = \frac{\Gamma^{\text{SM}}_h \, \cos^4\alpha}{\Gamma^{\text{SM}}_h\cos^2\alpha + \Gamma_{h\rightarrow\bar{\psi}\psi} + \Gamma_{h\rightarrow H H}}.$$ Note that the signal strength $r_{h}$ is suppressed by $\cos^2{\alpha}$ even if the two new invisible decay channels are kinematically forbidden. In the parameter region where $m_H < m_h/2$, $\Gamma_{h\rightarrow HH}$ is still considerably large even if the mixing angle is very small. In the limit without mixing between the two Higgs particles, it is given by $$\Gamma_{h\rightarrow HH} = \frac{\lambda^2 \varphi_0^2}{16\pi m_h} \beta_H,$$ which results in a constraint on the parameter $\lambda$ $$\lambda^2 \lesssim 14.2 \frac{m_h \Gamma_h^{\text{SM}}}{\varphi_0^2 \beta_H}.$$ In the parameter region where $m_H\lesssim 30{\mbox{ GeV}}$, this constraint is strong enough to exclude all the sample points, as shown in Fig. \[smallangle\]. Analogously, the signal strength of the second Higgs particle is given by $$\label{r2} r_H = \frac{\Gamma^{\text{SM}}_H \, \sin^4\alpha}{\Gamma^{\text{SM}}_H\sin^2\alpha + \Gamma_{H\rightarrow\bar{\psi}\psi} + \Gamma_{H\rightarrow h h}}.$$ The signal strength of the second Higgs particle is proportional to $\sin^2\alpha$, which comes from the coupling between the second Higgs particle and the SM fermions and gauge bosons, and it is also suppressed by the decay channels $H\rightarrow\bar{\psi}\psi$ and $H\rightarrow hh$. The allowed region in the $m_H-\sin\alpha$ plane under this constraint is plotted in Fig. \[smallangle\]. It can be seen that the result on the signal strength of the SM-like Higgs particle imposes an upper bound on the mixing angle, due to the suppression factor $\cos^2\alpha$ in the signal strength in Eq. ([\[r1\]]{}). When the invisible decay of the SM-like Higgs particle through the channel $h\rightarrow HH$ is kinematically forbidden, i.e. $m_H > m_h/2$, the upper limit on the mixing angle is directly given by $\sin^2 \alpha \lesssim 0.22$, leading to $\alpha \lesssim 28^{\circ}$. When the channel $h\rightarrow HH$ is opened, i.e. $m_H < m_h/2$, the mixing angle is further constrained, for instance $\sin\alpha \lesssim 0.01$ leading to $\alpha\lesssim0.57^\circ$ at $m_H=50{\mbox{ GeV}}$. Besides the constraint on the signal strength of the SM-like Higgs particle, the current LHC data also set an upper bound on a Higgs particle with a mass larger than $145{\mbox{ GeV}}$ [@Chatrchyan:2013yoa], which can be translated into an upper bound on $r_H$ in this model. However, this constraint is much weaker than the constraint on $r_h$ as the invisible decay of the second Higgs particle can be very large. LEP constraint and the electroweak precision test {#slep} ================================================= The LEP data impose constraints on the ratio of Higgs-$Z$-$Z$ coupling strength with respect of the SM value $\xi^2_\mathcal{H} = \left(g_{\mathcal{H}ZZ}/g_{\mathcal{H}ZZ}^{SM}\right)^2$ with $\mathcal{H}=h,H$, as shown in Fig. 10(a) in Ref. [@Barate:2003sz]. In this model, the Higgs-$Z$-$Z$ coupling strength is suppressed by the mixing between the two Higgs particles $$\xi^2_h = \cos^2 \alpha, \qquad \xi^2_H = \sin^2 \alpha.$$ The allowed region in the $m_H-\sin\alpha$ plane under the constraint from LEP data at $95\%$ CL is shown in Fig. \[smallangle\]. This constraint sets an upper bound on the mixing angle in the region with $m_H < 114{\mbox{ GeV}}$, which is however much weaker compared with that from the LHC and the LUX experiments, as can be seen in the figure. The second Higgs particle in this model gives extra contributions to the gauge boson self-energy diagrams compared with the SM case, which can affect the oblique parameters $S$, $T$ and $U$ [@Peskin:1990zt; @Maksymyk:1993zm]. The shifts of the oblique parameters from the SM values $\Delta X \equiv X - X^{SM}$ are given by [@Baek:2011aa; @Barger:2007im] $$\begin{aligned} \Delta T &=& \frac{3}{16\pi s_W^2} \left[ \cos^2 \alpha \left\{ f_T\left(\frac{m_h^2}{m_W^2} \right) - \frac{1}{c_W^2} f_T\left(\frac{m_h^2}{m_Z^2}\right) \right\} \right. \notag \\ & &+ \sin^2 \alpha \left\{ f_T\left(\frac{m_H^2}{m_W^2}\right) - \frac{1}{c_W^2} f_T\left(\frac{m_H^2}{m_Z^2}\right)\right\} \notag \\ & & \left. - \left\{ f_T\left(\frac{m_h^2}{m_W^2}\right) - \frac{1}{c_W^2} f_T\left(\frac{m_h^2}{m_Z^2}\right) \right\} \right] , \\ \Delta S &=& \frac{1}{2\pi} \left[ \cos^2 \alpha f_S \left(\frac{m_h^2}{m_Z^2}\right) + \sin^2 \alpha f_S \left(\frac{m_H^2}{m_Z^2}\right) - f_S \left(\frac{m_h^2}{m_Z^2} \right) \right] , \\ \Delta U &=& \frac{1}{2\pi} \left[ \cos^2 \alpha f_S \left(\frac{m_h^2}{m_W^2}\right) + \sin^2 \alpha f_S \left(\frac{m_H^2}{m_W^2}\right) - f_S \left(\frac{m_h^2}{m_W^2} \right) \right] - \Delta S ,\end{aligned}$$ where $m_{W(Z)}$ is the masses of the $W$ ($Z$) gauge boson, $c_W^2=m_W^2/m_Z^2$ and $s_W^2=1-c_W^2$. The functions $f_T(x)$ and $f_S(x)$ are defined as $$\begin{aligned} f_T\left(x\right) &=& \frac{x \log x}{x-1} , \\ f_S\left(x\right) &=& \left\{ \begin{array}{ll} \frac{1}{12} \left\{ -2x^2+9x + \left[x^2-(6x-18)/x-1+18\right]x\log x \right. \\ \left. +2\sqrt{\left(x-4\right)} \left(x^2-4x+12\right) \right. \\ \left. \times \left[ \tanh^{-1}\sqrt{x}/\sqrt{x-4} - \tanh^{-1} (x-2)/\sqrt{(x-4)x} \right] \right\},&\text{for } 0<x<4 \vspace{10pt} \\ \frac{1}{12} \left\{ -2x^2+9x + \left[x^2-6x-18/(x-1)+18\right]x\log x \right. \\ \left. +\sqrt{\left(x-4\right)} \left(x^2-4x+12\right) \log \frac{1}{2} \left(x-\sqrt{(x-4)x}-2\right) \right\},&\text{for } x>4. \end{array} \right.\end{aligned}$$ The constraints from the oblique parameters given in Ref. [@Barger:2007im; @Baak:2011ze] can be translated into constraints on the mass of the second Higgs particle and the mixing angle. We show the $95\%$ CL allowed region in the $m_H-\sin\alpha$ plane in Fig. \[smallangle\]. It can be seen that this constraint is weaker compared with the LHC constraint and the LUX constraint. Combined Results {#sresu} ================ We combine the constraints from all the above mentioned observables such as the DM relic density, the DM-nucleon scattering cross section, the signal strength of the SM-like Higgs particle, the Higgs-$Z$-$Z$ coupling strength and the oblique parameters on the parameter space satisfying $\varphi_c/T_c > 1$. About $2\times10^5$ sample points surviving all the constraints are obtained. The frequency distributions of the 6 free parameters after considering the phenomenological constraints are shown in Fig. \[fparamdist\]. ![Allowed region in the $m_H-\sin\alpha$ plane satisfying $\varphi_c/T_c>1$ and all the constraints from the electroweak precision test (EWPT) at $95\%$ CL, the LEP data at $95\%$ CL, the Higgs search results at LHC, and the upper bound on DM-nucleon scattering cross section from the LUX experiment. The red dot-dashed line is the upper bound on the mixing angle from the $90\%$ CL XENON100 constraint and the red dashed line is that from the projected exclusion limit of the future XENON1T experiment. The dots are the sample points satisfying $\varphi_c/T_c>\mathcal{E}$ and all the constraints with $\mathcal{E} = 1.2$ (dark gray) and $\mathcal{E} = 1$ (light gray), respectively.[]{data-label="smallangle"}](ConstrainsLog.eps){width="400pt"} The allowed region in the $m_H-\sin\alpha$ plane is shown in Fig. \[smallangle\]. As shown in the figure, the most stringent constraints come from the data of the LHC and the LUX experiments. It can be seen that in the region where the mass of the second Higgs particle is nearly degenerate with that of the SM-like Higgs particle, the LUX constraint is significantly relaxed due to the destructive interference between the contributions from the two Higgs particles. Consequently, in this region the upper bound on the mixing angle is set by the LHC data which leads to $\alpha \lesssim 28^\circ$. In the region where $m_H < m_h/2$, the mixing angle is further constrained, as the invisible decay of the SM-like Higgs particle is opened. In other regions the upper limit on the mixing angle is determined by the LUX data, for instance, $\alpha \lesssim 5.7^\circ$ at $m_H=350{\mbox{ GeV}}$. As shown by the dots, the requirement of a strongly first order EWPhT sets an upper bound on the mass of the second Higgs particle around $350{\mbox{ GeV}}$ for $\mathcal{E} = 1$, which is expected as the contributions of very heavy particles to effective potential is suppressed exponentially. As shown by the dark gray dots, when considering $\mathcal{E} = 1.2$ the upper bound on the mass of the second Higgs particle becomes lower. But the difference between the upper bound for $\mathcal{E}=1.2$ and that for $\mathcal{E}=1$ is within $10 \hbox{ GeV}$. A lower bound on the mass of the second Higgs particle around $30{\mbox{ GeV}}$ is also imposed due to the constraint on $\lambda$ from the LHC data. The future XENON1T experiment can push the upper bound on $\sigma_{\text{SI}}$ down to $\sim 10^{-47} \text{cm}^2$ [@Aprile:2012zx]. The constraint from the projected exclusion limit of the future XENON1T experiment is also shown in Fig. \[smallangle\]. It can be seen that a significant proportion of the parameter space can be ruled out by the future XENON1T experiment. The mixing angle can be further constrained to one order of magnitude lower compared with the result of the LUX experiment, for instance $\alpha \lesssim 0.57^\circ$ at $m_H=350{\mbox{ GeV}}$. ![Allowed values of $m_H$ and $m_\psi$ from the sample points satisfying $\varphi_c/T_c>1$ and all the phenomenological constraints (see text for detailed explanation).[]{data-label="fmhmp"}](mhmp.eps){width="300pt"} The allowed values of $m_H$ and $m_\psi$ from the sample points are shown in Fig. \[fmhmp\]. The DM particle mass is solved from the DM thermal relic density which leads to a five-fold ambiguity. As shown in the figure, there are three branches which correspond to the two resonant annihilations when $m_\psi\approx m_{h,H}/2$ and the threshold of DM annihilation into Higgs particles. It can be seen that the DM particle mass is predicted to be in the range $\sim 15-350{\mbox{ GeV}}$. The distribution of $y_\psi$ is also significantly changed by the constraint from DM thermal relic density, as shown in Fig. \[fparamdist\]. Conclusion {#sconc} ========== In summary, we have systematically explored the parameter space of the singlet fermionic DM model which can lead to strongly enough first order EWPhT as required by electroweak baryogenesis. We have taken into account the loop-level barrier by including the high temperature approximation up to the terms quartic in $m/T$, and an analytical approximation of the effective potential which well matches both the high- and low-temperature approximations has been introduced, which allows for reliable calculations in low temperature region. It has been shown that the mixing angle is constrained to $\alpha\lesssim 28^\circ$ and the mass of the second Higgs particle is in the range $\sim 30-350{\mbox{ GeV}}$. The DM particle mass is predicted to be in the range $\sim 15-350{\mbox{ GeV}}$. The future XENON1T detector can rule out a large proportion of the parameter space. The constraint can be relaxed when the mass of the SM-like Higgs particle is degenerate with that of the second Higgs particle. In other regions the mixing angle can be further constrained to one order of magnitude lower compared with the result using the LUX data, for instance $\alpha \lesssim 0.57^\circ$ at $m_H=350{\mbox{ GeV}}$. Acknowledgement {#acknowledgement .unnumbered} =============== This work is supported in part by the National Basic Research Program of China (973 Program) under Grants No. 2010CB833000; the National Nature Science Foundation of China (NSFC) under Grants No. 10975170, No. 10821504, No. 10905084 and No. 11335012; and the Project of Knowledge Innovation Program (PKIP) of the Chinese Academy of Science. Appendices {#appendices .unnumbered} ========== Renormalization of the Higgs potential {#appA} ====================================== The counter-terms to renormalize the potential at zero temperature are given by $$\label{vct} V_{\text{CT}}\left(\varphi,s\right) = -\frac{\delta \mu _{\phi}^2}{2} \varphi^2 + \frac{\delta \lambda _{\phi}}{4} \varphi^4 -\frac{\delta \mu _s^2}{2} s^2-\frac{\delta \mu _3}{3} s^3+ \frac{\delta \lambda _s}{4} s^4+\frac{\delta \mu}{2} \varphi^2 s + \frac{\delta \lambda}{4} \varphi^2 s^2 .$$ We use the following renormalization conditions $$\label{renorm1} \left. \left(\frac{\partial}{\partial \varphi}, \frac{\partial}{\partial s}, \frac{\partial^2}{\partial \varphi^2}, \frac{\partial^2}{\partial s^2}, \frac{\partial^2}{\partial s \partial \varphi} \right) \left(V_1\left(\varphi,s\right)+V_{\text{CT}}\left(\varphi,s\right)\right) \right|_{(\varphi,s)=(\varphi_0,s_0)} = 0 ,$$ and $$\label{renorm2} \left(\left.\frac{\partial }{\partial s} \right|_{(\varphi,s)=(0,s_{\varphi})}, \left. \frac{\partial}{\partial v} \right|_{(\varphi,s)=(\varphi_{s},0)} \right) \left(V_1\left(\varphi,s\right)+V_{\text{CT}}\left(\varphi,s\right)\right) = 0,$$ where $s_{\varphi}$ ($\varphi_{s}$) is the location of the minimum on the $s$ ($\varphi$) directions. The conditions in Eq. (\[renorm1\]) keep the locations of tree-level VEVs and the mass of the two Higgs particles unchanged, and that in Eq. (\[renorm2\]) keep the locations of the minima on the $s$ and $\varphi$ direction unchanged. The solutions of the renormalization conditions Eq. (\[renorm1\]) and (\[renorm2\]) are $$\begin{aligned} \delta\mu_{\phi}^2 &=& \frac{\varphi_s^3 V_1^{(1,0)}\left(\varphi_0,s_0\right) + 2\varphi_0^3 V_1^{(1,0)}\left(\varphi_s,0\right) - \varphi_0 \varphi_s^3 V_1^{(2,0)}\left(\varphi_0,s_0\right)}{2 \varphi_0^3 \varphi_s} ,\\ \delta\lambda_{\phi} &=& \frac{V_1^{(1,0)}\left(\varphi_0,s_0\right)-\varphi_0 V_1^{(2,0)}\left(\varphi_0,s_0\right)}{2\varphi_0^3} , \\ \delta\mu_s^2 &=& \frac{1}{2s_0^2\left(s_0-s_\varphi\right)^2s_\varphi} \left\{s_0 \left[2s_0^3 V_1^{(0,1)}\left(0,s_\varphi\right)+s_\varphi^2\left(-6s_0+4s_\varphi\right)V_1^{(0,1)}\left(\varphi_0,s_0\right) \right. \right. \notag \\ & &\left.+ 2s_\varphi^2s_0\left(s_0-s_\varphi\right)V_1^{(0,2)}\left(\varphi_0,s_0\right) + s_\varphi^2 \varphi_0\left(2s_0-s_\varphi\right)V_1^{(1,1)}\left(\varphi_0,s_0\right)\right] \notag \\ & & \left. - \varphi_0^2 s_0^2 s_\varphi^2 \left(s_0-s_\varphi\right) \delta\lambda \right\} ,\end{aligned}$$ $$\begin{aligned} \delta\mu_3 &=& \frac{1}{2s_0^3\left(s_0-s_\varphi\right)^2 s_\varphi} \left\{ -2s_0 \left[2s_0^3 V_1^{(0,1)}\left(0,s_\varphi\right)+s_\varphi\left(-3s_0^2+s_\varphi^2\right)V_1^{(0,1)}\left(\varphi_0,s_0\right) \right. \right. \notag \\ & &\left.+ s_\varphi s_0\left(s_0^2-s_\varphi^2\right)V_1^{(0,2)}\left(\varphi_0,s_0\right) + s_\varphi \varphi_0 s_0^2 V_1^{(1,1)}\left(\varphi_0,s_0\right)\right] \notag \\ & & \left. - \varphi_0^2 s_0^2 s_\varphi \left(s_0^2-s_\varphi^2\right) \delta\lambda \right\} ,\end{aligned}$$ $$\begin{aligned} \delta\lambda_s &=& \frac{1}{2s_0^3\left(s_0-s_\varphi\right)^2 s_\varphi} \left\{ -s_0 \left[2s_0^2 V_1^{(0,1)}\left(0,s_\varphi\right)+s_\varphi\left(-4s_0+2s_\varphi\right)V_1^{(0,1)}\left(\varphi_0,s_0\right) \right. \right. \notag \\ & &\left.+ 2s_\varphi s_0\left(s_0-s_\varphi\right)V_1^{(0,2)}\left(\varphi_0,s_0\right) + s_\varphi \varphi_0 s_0 V_1^{(1,1)}\left(\varphi_0,s_0\right)\right] \notag \\ & & \left. - \varphi_0^2 s_0^2 s_\varphi \left(s_0-s_\varphi\right) \delta\lambda \right\} ,\\ \delta\mu &=& -\delta\lambda s_0 ,\\ \delta\lambda &=& \frac{1}{\varphi_0^3 s_0 \varphi_s} \left[\left(3\varphi_0^2 \varphi_s-\varphi_s^3\right)V_1^{(1,0)}\left(\varphi_0,s_0\right) - 2\varphi_0^3 V_1^{(1,0)}\left(\varphi_s,0\right) \right. \notag \\ & & \left.- \varphi_0^2 s_0 \varphi_s V_1^{(1,1)}\left(\varphi_0,s_0\right) - \varphi_0 \varphi_s\left(\varphi_0^2-\varphi_s^2\right)V_1^{(2,0)}\left(\varphi_0,s_0\right)\right],\end{aligned}$$ where $$V^{(m,n)}\left(\varphi,s\right) = \frac{\partial^{(m+n)}V\left(\varphi,s\right)}{\partial h^m \partial s^m}.$$ Cross sections for DM annihilation {#appB} ================================== The cross sections for DM particles annihilating into the SM fermions and gauge bosons are given by [@Kim:2008pp] $$\begin{gathered} \label{relic} \sigma v_{\text{rel}}\left(\bar{\psi}\psi\rightarrow \bar{f}f,W^+W^-,ZZ\right) = \frac{\left(y_\psi \sin\alpha \cos\alpha\right)^2}{16\pi} \left(1 - \frac{4m_{\psi}^2}{\mathfrak{s}}\right) \\ \times \left| \frac{1}{\mathfrak{s}-m_{h}^2 + i m_{h}\Gamma_{h}} + \frac{1}{\mathfrak{s}-m_{H}^2 + i m_{H}\Gamma_{H}}\right|^2 \cdot A_{f,W,Z},\end{gathered}$$ where $\Gamma_h$ ($\Gamma_H$) is the total decay width of the SM-like Higgs particle (the second Higgs particle), $\sqrt{\mathfrak{s}}$ denotes the center-of-mass energy, and $A_{f,W,Z}$ stands for the contributions from channels with final states $\bar{f}f$, $W^+W^-$ and $ZZ$ $$\begin{aligned} A_f &=& 6\, \mathfrak{s} \left(\frac{m_f}{\varphi_0}\right)^2 \times \left(1-\frac{4m_f^2}{\mathfrak{s}}\right)^{3/2}, \\ A_{W} &=& 4 \, \left(\frac{m_{W}^2}{\varphi_0}\right)^2 \times \left(2+\frac{(\mathfrak{s}-2m_{W}^2)^2}{4m_{W}^4}\right) \times \sqrt{1-\frac{4m_{W,Z}^2}{\mathfrak{s}}} .\end{aligned}$$ $A_{Z}$ is defined analogously with $A_{W}$ and there is an additional factor of $1/2$ for $A_Z$. The cross sections for DM particles annihilating into two identical Higgs particles through $s$-channele are given by [@Qin:2011za] $$\sigma v_{\text{rel}}^{(\text{s})}\left(\bar{\psi}\psi\rightarrow\mathcal{H}\mathcal{H}\right) = \frac{1}{2} \kappa_{\mathcal{H}} \left(\mathfrak{s} - 4m_\psi^2\right) \left| \frac{y_{h} \lambda_{h\mathcal{H}\mathcal{H}}}{\mathfrak{s}-m_{h}^2 + i m_{h}\Gamma_{h}} + \frac{y_{H} \lambda_{H\mathcal{H}\mathcal{H}}}{\mathfrak{s}-m_{H}^2 + i m_{H}\Gamma_{H}}\right|^2,$$ where $\mathcal{H}$ stands for $H$ or $h$, and $\kappa_{\mathcal{H}}$ is defined as $$\kappa_{\mathcal{H}} = \frac{1}{16\pi\,\mathfrak{s}^2} \sqrt{\mathfrak{s}^2 -4\mathfrak{s} m_{\mathcal{H}}^2},$$ The cross sections for DM particles annihilating into two identical Higgs particles through $t$- and $u$-channel are given by $$\begin{aligned} \sigma v_{\text{rel}}^{(\text{t+u})}\left(\bar{\psi}\psi\rightarrow\mathcal{H}\mathcal{H}\right) &=& \kappa_{\mathcal{H}} \, y_{\mathcal{H}}^4 \left\{ \frac{\left(4 m_\psi^2 - m_{\mathcal{H}}^2\right)^2}{D^2-A^2} - \log\left|\frac{A+D}{A-D}\right| \left[ \frac{\left(\mathfrak{s}+8m_\psi^2-2m_{\mathcal{H}}^2\right)}{2D} \right. \right. \nonumber \\ & & \left. \left.+\frac{\left(16m_\psi^4 -4m_\psi^2 \mathfrak{s}-m_{\mathcal{H}}^4\right)}{AD}\right]-2 \right\},\end{aligned}$$ where $A$ and $D$ are defined as $$A = \frac{1}{2}\left(2 m_{\mathcal{H}}^2 - \mathfrak{s}\right), \qquad D=\frac{\mathfrak{s}}{2}\beta_\psi \beta_{\mathcal{H}},$$ with $\beta_{\psi} = \sqrt{1-4m_\psi^2 / \mathfrak{s}}$ and $\beta_{\mathcal{H}} = \sqrt{1-4m_{\mathcal{H}}^2/\mathfrak{s}}$. The interference terms between the $s$- and $u$-, $t$-channels are given by $$\begin{aligned} \sigma v_{\text{rel}}^{(\text{int})}\left(\bar{\psi}\psi\rightarrow\mathcal{H}\mathcal{H}\right) &=& 2\kappa_{\mathcal{H}}\, y_{\mathcal{H}}^2\, m_{\psi} \left[ \frac{y_{h} \lambda_{h\mathcal{H}\mathcal{H}} \left(\mathfrak{s}-m_{h}^2\right)}{ \left(\mathfrak{s}-m_{h}^2\right)^2 + m_{h}^2 \Gamma_{h}^2} + \frac{y_{H} \lambda_{H\mathcal{H}\mathcal{H}} \left(\mathfrak{s}-m_{H}^2\right)}{ \left(\mathfrak{s}-m_{H}^2\right)^2 + m_{H}^2 \Gamma_{H}^2}\right] \nonumber \\ & & \times \log\left|\frac{A+D}{A-D}\right|\left( \frac{A}{D}+\frac{1}{2}\frac{\beta_\psi}{\beta_{\mathcal{H}}}-2\right).\end{aligned}$$ The cross sections for DM particles annihilating into $h$ and $H$ through $s$-channel are given by $$\sigma v_{\text{rel}}^{(\text{s})}\left(\bar{\psi}\psi\rightarrow hH\right) = \kappa_{hH} \left(\mathfrak{s} - 4m_\psi^2\right) \left| \frac{y_{h} \lambda_{Hhh}}{\mathfrak{s}-m_{h}^2 + i m_{h}\Gamma_{h}} + \frac{y_{H} \lambda_{hHH}}{\mathfrak{s}-m_{H}^2 + i m_{H}\Gamma_{H}}\right|^2,$$ where $\kappa_{hH}$ is defined as $$\kappa_{hH} = \frac{1}{16\pi\,\mathfrak{s}^2} \sqrt{\mathfrak{s}^2 -2\mathfrak{s}\left(m_{h}^2+m_{H}^2\right) + \left(m_{h}^2-m_{H}^2\right)^2}.$$ The cross sections for DM particle annihilation into $h$ and $H$ through $t$- and $u$-channel are given by $$\begin{aligned} \sigma v_{\text{rel}}^{(\text{t+u})}\left(\bar{\psi}\psi\rightarrow hH\right) &=& 2 \kappa_{hH}\, y_{h}^2 \, y_{H}^2 \left\{ \frac{\left(4 m_\psi^2 - m_{h}^2\right)\left(4 m_\psi^2 - m_{H}^2\right)}{D^2-A^2} \right. \nonumber \\ & & \left. - \log\left|\frac{A+D}{A-D}\right| \left[ \frac{\left(\mathfrak{s}+8m_\psi^2-m_h^2-m_H^2\right)}{2D} \right. \right. \nonumber \\ & & \left. \left.+\frac{\left(16m_\psi^4 -4m_\psi^2 \mathfrak{s}-m_h^2 m_H^2\right)}{AD}\right]-2 \right\},\end{aligned}$$ where $A$ and $D$ are defined as $$A = \frac{1}{2}\left( m_h^2 + m_H^2 - \mathfrak{s}\right), \qquad D=\frac{\mathfrak{s}}{2}\beta_\psi \beta_{hH},$$ with $$\beta_{hH} = \sqrt{1-\frac{\left(m_h+m_H\right)^2}{\mathfrak{s}}}\sqrt{1-\frac{\left(m_h-m_H\right)^2}{\mathfrak{s}}}.$$ The interference terms between the $s$- and $u$-, $t$-channels are given by $$\begin{aligned} \sigma v_{\text{rel}}^{(\text{int})}\left(\bar{\psi}\psi\rightarrow hH\right) &=& 4\kappa_{hH}\, y_h\, y_H\, m_{\psi} \left[ \frac{y_{h} \lambda_{Hhh} \left(\mathfrak{s}-m_{h}^2\right)}{ \left(\mathfrak{s}-m_{h}^2\right)^2 + m_{h}^2 \Gamma_{h}^2} + \frac{y_{H} \lambda_{hHH} \left(\mathfrak{s}-m_{H}^2\right)}{ \left(\mathfrak{s}-m_{H}^2\right)^2 + m_{H}^2 \Gamma_{H}^2}\right] \nonumber \\ & & \times \log\left|\frac{A+D}{A-D}\right|\left( \frac{A}{D}+\frac{1}{2}\frac{\beta_\psi}{\beta_{hH}}-2\right).\end{aligned}$$ The physical couplings in this model are given by $$\begin{aligned} y_{\mathcal{H}} &=& \left\{ \begin{array}{ll} y_\psi\sin\alpha, & \hbox{if $\mathcal{H}=h$;} \\ y_\psi\cos\alpha, & \hbox{if $\mathcal{H}=H$.} \end{array} \right. \nonumber \\ \lambda_{hhh} &=& c_\alpha^3 \lambda_\phi \varphi_0 - \frac{1}{2} c_\alpha^2 s_\alpha \lambda s_0 - \frac{1}{2} c_\alpha^2 s_\alpha \mu + \frac{1}{2} c_\alpha s_\alpha^2 \lambda \varphi_0 - s_\alpha^3 \lambda_s s_0 + \frac{1}{3} s_\alpha^3 \mu_3, \nonumber \\ \lambda_{hHH} &=& c^2_{\alpha} s_{\alpha}\lambda s_0-\frac{1}{2}s^3_{\alpha} \lambda s_0+\frac{1}{2}c^3_{\alpha} \lambda \varphi_0- c_{\alpha} s^2_{\alpha} \lambda \varphi_0-3 c^2_{\alpha} s_{\alpha} \lambda_s s_0 \notag \\ & & +3 c_{\alpha} s^2_{\alpha} \lambda_{\phi} \varphi_0 + c^2_{\alpha} s_{\alpha} \mu-\frac{1}{2}s^3_{\alpha} \mu+ c^2_{\alpha} s_{\alpha} \mu_{3} , \label{lamhHH} \\ \lambda_{Hhh} &=& \frac{1}{2} c_\alpha^3\lambda s_0 + \frac{1}{2} c_\alpha^3 \mu - c_\alpha^2 s_\alpha \lambda \varphi_0 + 3 c_\alpha^2 s_\alpha \lambda_\phi \varphi_0 - c_\alpha s_\alpha^2 \lambda s_0 \nonumber \\ & & + 3 c_\alpha s_\alpha^2 \lambda_s s_0 - c_\alpha s_\alpha^2 \mu - c_\alpha s_\alpha^2 \mu_3 + \frac{1}{2} s_\alpha^3 \lambda \varphi_0, \nonumber \\ \lambda_{HHH} &=& c_\alpha^3 \lambda_s s_0 - \frac{1}{3} c_\alpha^3 \mu_3 + \frac{1}{2} c_\alpha^2 s_\alpha \lambda \varphi_0 + \frac{1}{2} c_\alpha s_\alpha^2 \lambda s_0 + \frac{1}{2} c_\alpha s_\alpha^2 + s_\alpha ^3 \lambda_\phi \varphi_0. \nonumber\end{aligned}$$ where $c_\alpha$ and $s_\alpha$ stand for $\cos\alpha$ and $\sin\alpha$, respectively. Sphaleron solution with magnetic moment {#sphaleron} ======================================= The Lagrangian of the gauge and Higgs sectors of the singlet fermionic DM model is given by $$\mathcal{L} = - \frac{1}{4} F^a_{\mu\nu} F^{a, \mu\nu} - \frac{1}{4} f_{\mu\nu} f^{\mu\nu} + \left(D_\mu \Phi\right)^{\dagger} \left(D^{\mu}\Phi\right) + \frac{1}{2} \partial_\mu S \partial^{\mu} S - V\left(\Phi, S, T\right),$$ where $$\begin{aligned} F_{\mu\nu}^a &=& \partial_\mu W_\nu^a - \partial_\nu W_\mu^a + g \epsilon^{abc} W_\mu^b W_\nu^c, \nonumber \\ f_{\mu\nu} &=& \partial_\mu a_\nu - \partial_\nu a_\mu, \nonumber \\ D_\mu &=& = \partial_\mu - \frac{i}{2}g \sigma^a W_\mu^a - \frac{i}{2}g^{\prime} a_\mu H, \nonumber\end{aligned}$$ where $W^a_\mu (a = 1,2,3)$ and $a_\mu$ are the $SU(2)_L$ and $U(1)_Y$ gauge fields, respectively. The Higgs potential $V\left(\Phi, S, T\right)$ is the effective potential at temperature $T$. The corresponding energy functional is given by $$E = \int d^3 x \left[ \frac{1}{4} F_{ij}^a F_{ij}^a + \left(D_i\phi\right)^\dagger \left(D_i\phi\right) + \frac{1}{2} \partial_i S \partial_i S + V\left(\Phi, S, T\right) \right].$$ In the limit of vanishing weak mixing angle, $\theta_w \approx 0$, the $U(1)_Y$ gauge field decouples, and the sphaleron solution is spherically symmetric. We adopt the ansatz for the fields from Refs. [@Klinkhamer:1984di; @Klinkhamer:1990fi; @Enqvist:1992kd; @Choi:1994mf] $$\begin{aligned} g W^a_i \sigma^a d x^i &=& \left(1-f(\xi)\right)F_a \sigma^a, \\ \Phi &=& \frac{\varphi}{\sqrt{2}} \left( \begin{array}{c} 0 \\ h(\xi) \\ \end{array} \right), \\ S &=& s \, p(\xi),\end{aligned}$$ where $\xi \equiv gvr$ is the dimensionless distance, and the functions $F_a$ are defined as [@Klinkhamer:1990fi] $$\begin{aligned} F_1 &=& -2 \sin\phi d\theta - \sin 2\theta \cos\phi d\phi, \\ F_2 &=& -2 \cos\phi d\theta + \sin 2\theta \sin\phi d\phi, \\ F_3 &=& 2\sin^2\theta d\phi.\end{aligned}$$ The sphaleron energy can be minimized by the solving the variational field equations $$\begin{aligned} f^{\prime\prime} &=& \frac{2}{\xi^2}f(f-1)(1-2f) + \frac{1}{4} h^2 (f-1), \label{eq:0_sph_f} \\ h^{\prime\prime} + \frac{2}{\xi} h^{\prime} &=& \frac{2}{\xi^2} h (1-f)^2 + \frac{1}{g^2 \varphi^4} \frac{\partial V(h, p, T)}{\partial h}, \label{eq:0_sph_h} \\ p^{\prime\prime} + \frac{2}{\xi} p^{\prime} &=& \frac{1}{g^2 \varphi^2 s^2} \frac{\partial V(h,p,T)}{\partial p}, \label{eq:0_sph_p}\end{aligned}$$ where the prime denotes the derivative with respect to $\xi$. To ensure the smoothness at the origin and the asymptotic behavior at $\xi \rightarrow \infty$, the boundary conditions for $f(\xi)$, $h(\xi)$ and $p(\xi)$ are given by $$f(0) = h(0) = 0,$$ and $$f(\infty) = h(\infty) = p(\infty) = 1.$$ Note that the value of $S$ at the origin is not constrained by any condition. The boundary condition for $p(\xi)$ can be obtained from the Taylor expansion of the equations around $\xi=0$, which leads to $p^{\prime}(0) = 0$. For non-vanishing weak mixing angle, $\theta_W\neq 0$, the $U(1)_Y$ gauge field must be taken into account because its source term is nonzero. The source term of the $U(1)_Y$ gauge field $a_i$ is given by the current $$\partial_{ij} f_{ij} = J_i = - \frac{i}{2} g^{\prime} \left[\Phi^\dagger D_i \Phi - \left(D_i\Phi\right)^\dagger \Phi \right].$$ At the leading order in $\theta_W$, $a_i$ in the current can be neglected, which leads to $$J_i = -\frac{1}{2} g^\prime \varphi^2 \frac{1}{r^2} h^2(\xi) \left[1-f(\xi)\right] \epsilon_{3ij} x_j.$$ Thus, in the presence of a constant background magnetic field $B$ along the $z$-axis, the energy of the $U(1)_Y$ field is given by $$\label{edipole} E = - \int d^3 x a_i^{\text{bg}} J_i,$$ where $a_i^{\text{bg}} = -(B/2)\epsilon_{3ij} x_j$ is the vector potential of the background magnetic field. The sphaleron energy in Eq. (\[edipole\]) can be rewritten in the form of a magnetic moment $\mu$ along the $z$-axis in the background magnetic field $$E = E_{\text{dipole}} = - \mu B,$$ where the magnetic moment $\mu$ is defined as $$\mu = \frac{2\pi}{3} \frac{g^{\prime}}{g^3 \varphi(T)} \int_0^\infty d\xi \xi^2 h^2(\xi) [1-f(\xi)].$$ Thus, the non-vanishing weak mixing angle gives rise to a sphaleron magnetic moment [@Klinkhamer:1984di], and the sphaleron solution becomes axially symmetric [@Manton:1983nd]. In this case, the ansatz for the fields can be chosen as [@Klinkhamer:1990fi] $$\begin{aligned} g^{\prime} a_i dx^i &=& \left[1-f_0\left(\xi\right)\right] F_3, \\ g W_i^a \sigma^a dx^i &=& \left[1-f\left(\xi\right)\right] \left(F_1\sigma^1 + F_2\sigma^2\right) + \left[1-f_3\left(\xi\right)\right] F_3 \sigma^2, \\ \Phi &=& \frac{\varphi}{\sqrt{2}} \left( \begin{array}{c} 0 \\ h(\xi) \\ \end{array} \right), \\ S &=& s \, p(\xi),\end{aligned}$$ with $i = 1,2,3$. The energy functional is $$\begin{aligned} E &=& \frac{4 \pi \varphi}{g} \int_0^\infty d\xi \left\{ \frac{8}{3} f^{\prime 2} + \frac{4}{3} f_3^{\prime 2} + \frac{8}{\xi^2} \left[ \frac{2}{3} f_3^2\left(1-f\right)^2 + \frac{1}{3} \left(f(1-f)+f-f_3\right)^2\right] \right. \nonumber \\ & & \left. + \frac{4g^2}{3g^{\prime 2}} \left[f_0^{\prime 2} + \frac{2}{\xi^2} \left(1-f_0\right)^2 \right] + \frac{1}{2} \xi^2 h^{\prime 2} + h^2 \left[\frac{1}{3} \left(f_0-f_3\right)^2 + \frac{2}{3}\left(1-f\right)^2 \right] \right. \nonumber \\ & & \left. + \frac{s^2}{2\varphi^2} \xi^2 p^{\prime 2} + \frac{\xi^2}{g^2\varphi^4} V\left(h, p, T\right) \right\}.\end{aligned}$$ The energy functional can be minimized by solving the variational equations $$\begin{aligned} f^{\prime\prime} &=& \frac{2}{\xi^2}(f-1)[f(f-2) + f_3(1+f_3)] + \frac{1}{4} h^2 (f-1), \\ f_3^{\prime\prime} &=& \frac{2}{\xi^2}[3f_3+f(f-2)(1+2f_3)]+\frac{1}{4}h^2(f_3-f_0), \\ f_0^{\prime\prime} &=& \frac{2}{\xi^2}(f_0-1) + \frac{g^{\prime 2}}{4 g^2} h^2 (f_0-f_3), \\ h^{\prime\prime} + \frac{2}{\xi} h^{\prime} &=& \frac{2}{3\xi^2} h [2(1-f)^2 + (f_0-f_3)^2] + \frac{1}{g^2 \varphi^4} \frac{\partial V(h, p, T)}{\partial h}, \\ p^{\prime\prime} + \frac{2}{\xi} p^{\prime} &=& \frac{1}{g^2 \varphi^2 s^2} \frac{\partial V(h,p,T)}{\partial p},\end{aligned}$$ with boundary conditions given by $$f(0) = f_3(0) = h(0) = 0, \quad f_0(0) = 1, \quad p^{\prime}(0) = 0,$$ and $$f(\infty) = f_3(\infty) = f_0(\infty) = h(\infty) = p(\infty) = 1.$$ [10]{} A. G. 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--- abstract: 'In this paper we invastigate the notion of generalized $(\ci,\cj)$ - Luzin set. This notion generalize the standard notion of Luzin set and Sierpi[ń]{}ski set. We find set theoretical conditions which imply the existence of generalized $(\ci, \cj)$ - Luzin set. We show how to construct large family of pairwise non-equivalent $(\ci,\cj)$ - Luzin sets. We find a class of forcings which preserves the property of being $(\ci,\cj)$ - Luzin set.' address: ' Institute of Mathematics and Computer Science, Wroc[ł]{}aw University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wroc[ł]{}aw, Poland.' author: - 'Robert Ra[ł]{}owski and Szymon Żeberski' title: Generalized Luzin sets --- Notation and Terminology ======================== We will use standard set-theoretic notation following [@jech]. In particular for any set $X$ and any cardinal $\kappa$, $[X]^{<\kappa}$ denotes the set of all subsets of $X$ with size less than $\kappa.$ Similarly, $[X]^\kappa$ denotes the family of subsets of $X$ of size $\kappa.$ By $\cp(X)$ we denote the power set of $X$. If $A\subseteq X\times Y$ then for $x\in X$ and $y\in Y$ we put $$A_x=\{y\in Y:\; (x,y)\in A\},$$ $$A^y=\{x\in X:\; (x,y)\in A\}.$$ By $A\vartriangle B$ we denote the symmetric difference of sets $A$ and $B,$ i.e. $$A\vartriangle B= (A\setminus B)\cup (B\setminus A).$$ In this paper $\cx$ denotes uncountable Polish space. By $\Open(\cx)$ we denote the topology of $\cx.$ By $\Borel(\cx)$ we denote the $\sigma$-field of all Borel sets. Let us recall that each Borel set can be coded by a function from $\omega^\omega.$ Precise definition of such coding can be found in [@kechris]. If $x\in\omega^\omega$ is a Borel code then by $\#x$ we denote the Borel set coded by $x.$ $\ci$, $\cj$ are $\sigma $-ideals on $\cx,$ i.e. $\ci, \cj\subseteq\cp(\cx)$ are closed under countable unions and subsets. Additionally we assume that $[\cx]^\omega\subseteq\ci, \cj.$ Moreover $\ci, \cj$ have Borel base i.e each set from the ideal can be covered by a Borel set from the ideal. Standard examples of such ideals are the ideal $\bbl$ of Lebesgue measure zero sets and the ideal $\bbk$ of meager sets of Polish space. Let $M\subseteq N$ be standard transitive models of ZF.\ Coding Borel sets from the ideal $I$ is absolute iff $$(\forall x\in M\cap \omega^\omega)( M\models \#x\in I\iff N\models \#x\in I).$$ We say that $\ci$ satisfies $\kappa$ chain condition ($\kappa$-c.c.) if every family $\ca$ of Borel subsets of $\cx$ satisfying the following conditions: 1. $(\forall A\in\ca)(A\notin\ci)$ 2. $(\forall A,B\in\ca)(A\neq B\rightarrow A\cap B\in\ci)$ has size smaller than $\kappa.$ If $\ci$ is $\omega_1$-c.c. then we say that $\ci$ is c.c.c. Let us recall that a function $f:\cx\rightarrow\cx $ is $\ci$-measurable if the preimage of every open subset of $\cx$ is $\ci$-measurable i.e belongs to the $\sigma $-field generated by Borel sets and the ideal $\ci.$ In other words $f$ is $\ci$-measurable iff $$(\forall U\in \Open(\cx))(\exists B\in \Borel(\cx))(\exists I\in\ci)(f^{-1}[U]=B\vartriangle I).$$ Let us recall the following cardinal coefficients: $$\begin{array}{l@{\ =\,{} }l} non(\ci) & \min\{ |A|:\; A\subseteq \cx\land A\notin \ci\}\\ add(\ci) & \min\{ |\ca|:\;\ca\subseteq \ci\land \bigcup \ca\notin \ci \}\\ cov(\ci) & \min\{ |\ca|:\;\ca\subseteq \ci\land \bigcup \ca=\cx \}\\ cov_h(\ci) & \min\{ |\ca|:\;\ca\subseteq \ci\land (\exists B\in \Borel(\cx)\setminus\ci)( B\subseteq \bigcup \ca) \}\\ cof(\ci) & \min\{ |\ca|:\; \ca\subseteq \ci\land \ca\text{ is a base of }\ci\} \end{array}$$ where $\ca\text{ is a base of }\ci$ iff $\ca\subseteq\ci\land (\forall I\in\ci)(\exists A\in\ca)( I\subseteq A).$ Let us remark that above coefficients can be defined for larger class of families (not only ideals). We say that $L\subseteq \cx$ is a $(\ci,\cj)$ - Luzin set if - $L\notin \ci,$ - $(\forall B\in \ci)( B\cap L\in \cj).$ Assume that $\kappa $ is a cardinal number. We say that $L\subseteq \cx$ is a $(\kappa,\ci,\cj)$ - Luzin set iff $L$ is a $(\ci,\cj)$ - Luzin set and $|L|=\kappa$. The above definition generalizes the standard notion of Luzin and Sierpi[ń]{}ski sets. Namely, $L$ is Luzin set iff $L$ is generalized $(\bbl,[\bbr]^{\le\omega})$ - Luzin set and $S$ is Sierpi[ń]{}ski set iff $S$ is generalized $(\bbk,[\bbr]^{\le\omega})$ - Luzin set. The above notion generalizes also notions from [@C]. We say that ideals $\ci$ and $\cj$ are orthogonal if $$\exists A\in\cp(\cx)\; A\in I\land A^c\in \cj.$$ In such case we write $\ci\perp \cj$. Let $\cf\subseteq \cx^\cx$ be a family of functions. We say that $A,B\subseteq \cx$ are equivalent with respect to $\cf$ if $$(\exists f\in \cf)\; (B=f[A]\vee A=f[B])$$ We say that $A,B\subseteq \cx$ are Borel equivalent if $A, B$ are equivalent with respect to the family of all Borel functions. We say that $\ci$ has Fubini property iff for every Borel set $A\subseteq\cx\times\cx$ $$\{x\in\cx : A_x\notin\ci\}\in\ci\Longrightarrow\{y\in\cx : A^y\notin\ci\}\in\ci$$ Natural examples of ideals fulfilling Fubini property are the ideal of null sets $\bbl$ (by Fubini theorem) and the ideal of meager sets $\bbk$ (by Kuratowski-Ulam theorem). By definition we can obtain the following properties: Assume that $\ci\perp \cj.$ 1. There exist a $(\ci,\cj)$ - Luzin set. 2. If $L$ is a $(\ci,\cj)$ - Luzin set then $L$ is not $(\cj,\ci)$ - Luzin set. (Part 1) By the definition of $\ci\perp\cj$ we can find two sets $I\in\ci$ and $J\in\cj$ such that $I\cup J=\cx.$ We will show that $J$ is $(\ci,\cj)$ - Luzin set. $J$ is not in $\ci.$ Let us fix any set $A\in\ci.$ We have that $A\cap J\subseteq J\in\cj.$ (Part 2) By the definition of $\ci\perp\cj$ we can find two sets $I\in\ci$ and $J\in\cj$ such that $I\cup J=\cx.$ Assume that $L$ is $(\ci,\cj)$ - Luzin set and $(\cj,\ci)$ - Luzin set. We have that $$L\cap J\subseteq J\in \cj \text{ and } L\cap I\subset I\in \ci$$ By the property of being $(\cj,\ci)$ - Luzin set $$L\cap J\in\ci.$$ So $L= (L\cap J)\cup (L\cap I)\in \ci.$ what is a contradiction with being $(\ci, \cj)$ - Luzin set. We will try to find a wide class of forcings which preserves the property of being $(\ci, \cj)$ - Luzin set. We are mainly interested in so called definable forcings (see [@zapletal]). Let us recall that $\bbp$ is definable forcing if $\bbp$ is of the form $\Borel(\cx)\setminus\ci,$ where $\cx$ and $\ci$ have absolute definition for standard transitive models of ZF of the same hight. Existence of Luzin sets ======================== Let us start with a theorem which under suitable assumptions guarantees existence of uncountably many pairwise different $(\ci,\cj)$ - Luzin sets. Assume that $\kappa=cov(\ci)=cof(\ci)\le non(\cj).$ Let $\cf $ be a family of functions from $\cx$ to $\cx.$ Assume that $|\cf|\le\kappa.$ Then we can find a sequence $(L_\alpha)_{\alpha<\kappa}$ such that 1. $L_\alpha$ is $(\kappa,\ci,\cj)$ - Luzin set, 2. for $\alpha\neq\beta,\ L_\alpha$ is not equivalent to $L_\beta$ with respect to the family $\cf.$ Let us enumerate the family $\cf$: $$\cf=\{ f_\alpha:\alpha<\kappa\}.$$ Now, let us enumerate Borel base of ideal $\ci$: $$\cb_\ci=\{ B_\alpha:\;\alpha<\kappa\}.$$ Now without loss of generality we can assume that $$(\forall f\in\cf)(\forall \lambda<\kappa) ( \kappa\le |f[(\bigcup\nolimits_{\xi<\lambda}B_\xi)^c]| )$$ Indeed, since $cov(\ci)=\kappa$ a set $(\bigcup\nolimits_{\xi<\lambda}B_\xi)^c$ is not in the ideal $\ci.$ If the function $f$ does not have the above property and $L$ is a $(\ci,\cj)$-Luzin set then $$f[L]=f[L\cap \bigcup\nolimits_{\xi<\lambda}B_\xi]\cup f[L\cap (\bigcup\nolimits_{\xi<\lambda}B_\xi)^c]$$ and both sets has cardinality less than $\kappa.$ So $f[L]$ is not $(\ci,\cj)$-Luzin set. By induction we will construct the family $\{x^\eta_{\alpha,\zeta}: \eta,\zeta,\alpha<\kappa\}$ and $\{d^\eta_{\alpha,\zeta}: \eta,\zeta,\alpha<\kappa\}$ such that $$d^\eta_{\alpha,\zeta}=f_\zeta(x^\eta_{\alpha,\zeta})$$ and for any different $\eta,\eta'<\kappa$ $$\{x^\eta_{\alpha,\zeta}: \zeta,\alpha<\kappa\} \cap \{d^{\eta'}_{\alpha,\zeta}: \zeta,\alpha<\kappa\}=\emptyset$$ and $$x_{\alpha,\zeta}^\eta\in\cx\setminus \left(\{d^\eta_{\xi,\zeta}:\;\eta,\xi,\zeta<\alpha\}\cup \{x^\eta_{\xi,\zeta}:\;\eta,\xi,\zeta<\alpha\}\cup\bigcup_{\xi<\alpha }B_\xi\right)$$ for every $\eta,\zeta<\alpha.$ Assume that we are in $\alpha$-th step of construction. Fix $\eta,\zeta<\alpha$. It means that we have constructed the following set $$Old=\{ x^\lambda_{\beta,\xi},d^\lambda_{\beta,\xi}:\; \beta,\xi,\lambda<\alpha\}\cup \{ x^\lambda_{\alpha,\xi},d^\lambda_{\alpha,\xi}:\; \lambda<\eta\lor (\lambda=\eta\land \xi<\zeta)\}.$$ Since $|f_\zeta[(\bigcup_{\xi<\alpha} B_\xi)^c]|\ge\kappa$ and $|Old|<\kappa$ we get that $$|f_\zeta[(\bigcup_{\xi<\alpha} B_\xi\cup Old)^c]|\ge\kappa.$$ That’s why we can find $$d^\eta_{\alpha,\zeta}\in f_\zeta[(\bigcup_{\xi<\alpha} B_\xi\cup Old)^c]\setminus Old.$$ Let $x^\eta_{\alpha,\zeta}$ be such that $d^\eta_{\alpha,\zeta}=f_\zeta(x^\eta_{\alpha,\zeta})$. In this way we can finish the $\alpha$-th step of construction. Now, let us define $L_\alpha=\{x^\alpha_{\xi,\zeta} : \xi,\zeta<\kappa\}.$ Let us check that $L_\alpha$ is $(\ci,\cj)$ - Luzin set. Indeed, if $A\in \ci$ then there exists $\beta<\kappa$ s.t. $A\subset B_\beta$. Then we have $$A\cap L_\alpha\subset B_\beta\cap L_\alpha= B_\beta\cap \{x^\alpha_{\xi,\zeta}:\;\xi,\zeta<\beta\} \subseteq \{x^\alpha_{\xi,\zeta}:\;\xi,\zeta<\beta\}\in \cj$$ because $|\{x^\alpha_{\xi,\zeta}:\;\xi,\zeta<\beta\}|\le|\beta|<\kappa\le non(\cj).$ What is more, for every function $f=f_\alpha\in\cf$ and every $\beta\neq\gamma$ we have that $$\kappa\le |f[L_\gamma]\setminus L_\beta|$$ because $\{ d^{\gamma}_{\xi,\alpha}:\alpha<\xi<\kappa\}\subseteq f[L_\gamma]\setminus L_\beta$. So $L_\beta\neq f[L_\gamma].$ In fact we haved proved a little stronger result. \[stronger\] Assume that $\kappa=cov(\ci)=cof(\ci)\le non(\cj).$ Let $\cf $ be a family of functions from $\cx$ to $\cx.$ Assume that $|\cf|\le\kappa.$ Then we can find a sequence $(L_\alpha)_{\alpha<\kappa}$ such that 1. $L_\alpha$ is $(\kappa,\ci,\cj)$ - Luzin set, 2. for $\alpha\neq\beta$ and $f\in\cf$ we have that $\kappa\le |f[L_\alpha]\vartriangle L_\beta|.$ Let us notice that for every ideal $\ci$ we have the inequality $cov(\ci)\le cof(\ci).$ This gives the following corollary. \[conti\] If $2^\omega=cov(\ci)=non(\cj)$ then there exists continuum many different $(\ci,\cj)$ - Luzin sets which aren’t Borel equivalent. In particular, if CH holds then there exists continuum many different $(\omega_1,\ci,\cj)$ - Luzin sets which aren’t Borel equivalent. We can extend above corollary to a wilder class of functions - namely, $\ci$-measurable functions. \[conti2\] If $2^\omega=cov(\ci)=non(\cj)$ then there exists continuum many different $(\ci,\cj)$ - Luzin sets which aren’t equivalent with respect to all $\ci$-measurable functions. In particular, if CH holds then there exists continuum many different $(\omega_1,\ci,\cj)$ - Luzin sets which aren’t equivalent with respect to all $\ci$-measurable functions. First, let us notice that if a function $f$ is $\ci$-measurable then there exists a set $I\in\ci\cap\Borel(\cx)$ such that $f\upharpoonright (\cx\setminus I)$ is Borel. Indeed, it is enough to consider a countable base $\{U_n\}_{n\in\omega}$ of topology of $\cx.$ Then $f^{-1}[U_n]=B_n\vartriangle I_n,$ where $B_n$ is Borel and $I_n$ is from the ideal $\ci.$ Now, put $I=\bigcup_{n\in\omega}I_n.$ So we can consider a family of partial Borel functions which domain is Borel set with complement in the ideal $\ci.$ This family is naturally of size continuum. So we can use Corollary \[conti\] and Remark \[stronger\] to finish the proof. Now, let us concentrate on ideal of null and meager sets. 1. Assume that $cov(\bbl)=2^\omega.$ There exists continuum many different $(2^\omega,\bbl,\bbk)$ - Luzin sets which aren’t equivalent with respect to the family of Lebesgue - measurable functions. 2. Assume that $cov(\bbk)=2^\omega.$ There exists continuum many different $(2^\omega,\bbk,\bbl)$ - Luzin sets which aren’t equivalent with respect to the family of Baire - measurable functions. Let us notice that the equality $cov(\bbl)=2^\omega$ implies that $2^\omega=cov(\bbl)=cof(\bbl)=non(\bbk).$ Similarly, the equality $cov(\bbk)=2^\omega$ implies that $2^\omega=cov(\bbk)=cof(\bbk)=non(\bbl)$ (see [@cichondiagram]). Corollary \[conti2\] finishes the proof. Luzin sets and forcing ====================== Now, let us focus on the class of forcings which preserves being $(\ci,\cj)$-Luzin set. Let us start with a technical observation. \[jacek\] Assume that $\ci$ has Fubini property. Suppose that $\bbp_\ci=\Borel(\cx)\setminus \ci$ is a proper definable forcing. Let $B\in \ci$ be a set in $V^{\bbp_\ci}[G].$ Then $B\cap \cx^V\in \ci$. Let $\dot{B}$ – name for $B$, $\dot{r}$ – canonical name for generic real, $C\subseteq \cx\times\cx$ - Borel set from the ideal $\ci.$ $C$ is coded in ground model $V$ and $B=C_{\dot{r}_G}.$ Now by Fubini property: $$\{ x:\;\; C^x\notin \ci\}\in \ci.$$ Let $x\in B\cap\cx^V$ then $V[G]\models x\in B$ $$0<\lbv x\in \dot{B}\rbv =\lbv x\in C_{\dot{r}}\rbv =\lbv (\dot{r},x)\in C\rbv=\lbv\dot{r}\in C^x\rbv =[C^x]_\ci$$ Then we have: $$B\cap\cx^V\subseteq\{ x: C^x\notin \ci\}\in \ci.$$ But the last set is coded in ground model because the set $C$ was coded in $V$. Assume that $\omega<\kappa$ and $\ci, \cj$ are c.c.c. and have Fubini property. Suppose that $\bbp_\ci=\Borel(\cx)\setminus \ci$ and $\bbp_\cj=\Borel(\cx)\setminus \cj$ are definable forcings. Then $\bbp_\cj$ preserves $(\kappa,\ci,\cj)$ - Luzin set property. Let $L$ be a $(\kappa,\ci,\cj)$ - Luzin set in $V$. In $V[G]$ take any $B\in \ci$ then $L\cap B\cap V=L\cap B$ but $L\cap B\in \ci$ in $V$ so $L\cap B\in \cj$ in $V$ by definition of $L.$ Finally, by Lemma \[jacek\] $$L\cap B=L\cap B\cap V\in \cj\text{ in } V[G].$$ Let $(\bbp,\le)$ be a forcing notion such that $$\{ B : B\in\ci\cap \Borel(\cx), B \text{ is coded in } V\}$$ is a base for $\ci$ in $V^\bbp[G].$ Assume that Borel codes for sets from ideals $\ci,\cj$ are absolute. Then $(\bbp,\le)$ preserve being $(\ci,\cj)$ - Luzin sets. Let $L$ be a $(\ci,\cj)$ - Luzin set in ground model $V.$ We will show that $V^\bbp[G]\models L\text{ is }(\ci,\cj)\text{ - Luzin set}.$ Let us work in $V^\bbp[G].$ Fix $I\in\ci.$ $\ci$ has Borel base consisting of sets coded in $V$. So, there exists $b\in\omega^\omega\cap V$ such that $I\subseteq\#b\in\ci.$ By absoluteness of Borel codes from $\ci$ we have that $V\models\#b\in\ci.$ $L$ is a $(\ci,\cj)$ - Luzin set in the model $V.$ So, there is $c\in \omega^\omega\cap V$ which codes Borel set from the ideal $\cj$ such that $V\models L\cap\#b\subseteq\#c.$ By absoluteness of Borel codes from $\cj$ we get that $$V^\bbp[G]\models L\cap B\subseteq L\cap\#b\subseteq\#c\in\cj,$$ what proves that $L$ is a $(\ci,\cj)$ - Luzin set in generic extension. The above theorem gives us a series of corollaries. \[same-reals\] Let $(\bbp,\le)$ be any forcing notion which does not change the reals i. e. $(\omega^\omega)^V=(\omega^\omega)^{V^\bbp[G]}.$ Assume that Borel codes for sets from ideals $\ci,\cj$ are absolute. Then $(\bbp,\le)$ preserve being $(\ci,\cj)$ - Luzin sets. \[wniosek1\] Assume that $(\bbp,\le)$ is a $\sigma$-closed forcing and Borel codes for sets from ideals $\ci,\cj$ are absolute. Then $(\bbp,\le)$ preserve $(\ci,\cj)$ - Luzin sets. \[iteracja\] Let $\lambda\in On$ be an ordinal number. Let $\bbp_\lambda=\< (P_\alpha,\dot{Q}_\alpha):\;\;\alpha<\lambda\>$ be iterated forcing with countable support. Spouse that 1. for any $\alpha<\lambda$ $P_\alpha\force \dot{Q}_\alpha - \sigma\text{ closed }$, 2. Borel codes for sets from ideals $\ci, \cj$ are absolute, then $\bbp_\lambda$ preserve $(\ci, \cj)$ - Luzin sets. Our forcing $\bbp_\lambda$ is $\sigma$-closed because it is countable support iteration of $\sigma$ -closed forcings. So, we can apply Corollary \[wniosek1\] to finish the proof. Now, let us consider some properties of countable support iteration connected with preservation of some relation. We will follow notation given by Goldstern (see [@goldi]). First, let us consider measure case. Let $\Omega$ is a family of clopen sets of Cantor space $2^\omega$ and $$C^{random}=\{ f\in \Omega^\omega:(\forall n\in\omega) \mu(f(n))<2^{-n}\}$$ with discrite topology. If $f\in C^{random}$ then let us define the following set $A_f=\bigcap_{n\in\omega}\bigcup_{k\ge n} f(k)$. Now, we are ready to define the following relation $\sqsubseteq=\bigcup_{n\in\omega} \sqsubseteq_n$ where $$(\forall f\in C^{random})(\forall g\in 2^\omega) (f\sqsubseteq_n g\iff (\forall k\ge n)\; g\notin f(k)).$$ Definition of the notion of preservation of relation $\sqsubseteq^{random}$ by forcing notion $(\bbp,\le)$ can be found in paper [@goldi]. Let us focus on the following consequence of that definition. \[outer\] If $(\bbp,\le)$ preserves $\sqsubseteq^{random}$ then $\bbp\force \mu^*(2^\omega\cap V)=1$. Now, we say that forcing notion $\bbp$ preserves outer measure iff $\bbp$ preserves $\sqsubseteq^{random}$. It is well known that Laver forcing preserves some stronger property than $\sqsubseteq^{random}$ (see [@judah]). So, Laver forcing preserves outer measure. In [@goldi] we can find the following theorem: \[goldi\_iteration\] Let $\bbp_\lambda=((P_\alpha,Q_\alpha):\;\alpha<\gamma)$ be any countable support iteration such that $$(\forall \alpha<\gamma)\; P_\alpha\force Q_\alpha \text{ preserves }\sqsubseteq^{random}$$ then $\bbp_\gamma$ preserves the relation $\sqsubseteq^{random}$. \[random\_luzin\] Assume that $\bbp$ is a forcing notion which preserves $\sqsubseteq^{random}$. Then $\bbp$ preserves being $(\bbl, \bbk)$-Luzin set. Assume that $V\models L \mbox{ is } (\bbl,\bbk)\mbox{-Luzin set.}$ Let us work in $V^\bbp[G].$ Take any null set $A\in\bbl.$ Then there is a null set $B$ in ground model such that $A\cap V\subseteq B.$ Indeed, let us assume that there is no such $B\in V.$ Then without loss of generality $(2^\omega\setminus A)\cap V\in\bbl.$ But $A\in\bbl$ then we have that $2^\omega\cap V\subset A\cup ((2^\omega\setminus A)\cap V)$ which is a null set. But by Fact \[outer\] $\mu^*(2^\omega\cap V)=1.$ So we have a contradiction. Then intersection $A\cap L\subseteq B\cap L\in\bbk$ is a meager set in ground model. Then by absolutnes of borel codes of meager sets the set $A\cap L$ is a meager set what finishes the proof. In constructible universe $L$ let us consider the countable forcing iteration $P_{\omega_2}=((P_\alpha,Q_\alpha):\alpha<\omega_2)$ of the length $\omega_2$ as follows, for any $\alpha<\omega_2$ - if $\alpha$ is even then $P_\alpha\force ''Q_\alpha \text{ is random forcing}''$, - in odd case $P_\alpha\force ``Q_\alpha\text{ is Laver forcing}''$. Previously we noticed that both random and Laver forcing, preserves $\sqsubseteq^{random}$ and then by Theorem \[goldi\_iteration\] $P_{\omega_2}$ preserves relation $\sqsubseteq^{random}.$ By Theorem \[random\_luzin\] the $(\bbl,\bbk)$-Luzin sets are preserved by our iteration $P_{\omega_2}$. Moreover, in generic extension we have $cov(\bbl)=\omega_2$ and $2^\omega=\omega_2$ (for details see [@goldi]). Asuume that in the ground model $A$ is $(\bbl,\bbk)$-Luzin set with outer measure equal to one. Then in generic extension it has outer measure one and $|A|=\omega_1.$ So, it does not contain any Lebesgue positive Borel set. Thus $A$ is completely $\bbl$-nonmeasurable set. The analogous machinery can be used for ideal of meager sets $\bbk$. Let us recall the necessary definitions (see [@goldi]). Let $C^{Cohen}$ be set of all functions from $\omega^{<\omega}$ into itself. Then $\sqsubseteq^{Cohen}=\bigcup_{n\in\omega} \sqsubseteq_n^{Cohen}$ and for any $n\in\omega$ let $$(\forall f\in C^{Cohen})(\forall g\in \omega^\omega)( f\sqsubseteq_n^{Cohen} g\text{ iff } (\forall k<n)( g\upharpoonright k^{\frown} f(g\upharpoonright k)\subseteq g)).$$ Then finally we have the following theorem: \[cohen\_luzin\] Assume that $\bbp$ is a forcing notion which preserves $\sqsubseteq^{Cohen}$. Then $\bbp$ preserves being $(\bbk, \bbl)$-Luzin set. The another preservation theorem which is due to Shelah (see [@shelah] and also [@schliri]) is as follows \[shelah\_iteration\] Let $\bbp_\lambda=((P_\alpha,\dot{Q}_\alpha):\;\alpha<\lambda)$ be any countable support iteration such that $(\forall \alpha<\gamma)$ $P_\alpha\force Q_\alpha \text{is proper } $ and $$P_\alpha\force Q_\alpha\force \text{every new open dense set contains old open dense set}$$ then $\bbp_\lambda \force \text{ every new open dense contains old open dense set}$. We can easily derive Let $\bbp_\lambda=((P_\alpha,\dot{Q}_\alpha):\;\alpha<\lambda)$ be any countable support iteration such that $(\forall \alpha<\lambda)$ $P_\alpha\force Q_\alpha \text{is proper } $ and $$P_\alpha\force Q_\alpha\force \text{every new open dense set contains old open dense set}$$ Then $\bbp_\lambda$ preserves being $(\bbk,\bbl)$-Luzin set. [123]{} T. Bartoszynski, H. Judah, S. Shelah, [*The Cichon Diagram,*]{} J. Symbolic Logic vol. 58 (2) (1993), pp.401-423, J. Cicho[ń]{}, [*On two-cardinal properties of ideals*]{}, Trans. Am. Math. Soc. [**vol 314**]{}, no. 2 (1989), pp 693-708, J. Cicho[ń]{}, J. Pawlikowski, [*On ideals of subsets of the plane and on Cohen reals*]{}, J. Symbolic Logic [**vol. 51**]{}, no. 1 (1986), pp 560-569 M. Goldstern, [*Tools for your forcing construction,*]{} Israel Mathematical Conference Proceedings, vol. 06 (1992), pp.307-362, H. Judah, S. Shelah, [*The Kunnen Miller chart,*]{} Journal of Symbolic Logic, (1990). A. Kanamori, [*The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings*]{} Springer-Verlag (2009), Kechris, [*Classical Descriptive Set Theory*]{} (Graduate Texts in Mathematics) (v. 156) Springer (1995), T. Jech, [*Set Theory*]{} millenium edition, (2003), S. Shelah, [*Proper and improper forcing*]{}, (1998). Ch. Schlindwein, [*Understanding preservation theorems, II*]{}, to appear in Math. Log. Quart. [*arXiv:1001.0922v1*]{} J. Zapletal, [*Forcing Idealized*]{} (Cambridge Tracts in Mathematics) (2008).

--- abstract: 'Exclusive semileptonic $B$ decays into excited charmed mesons are investigated at order $\Lambda_{\rm QCD}/m_Q$ in the heavy quark effective theory. Differential decay rates for each helicity state of the four lightest excited $D$ mesons ($D_1$, $D_2^*$, $D_0^*$, and $D_1^*$) are examined. At zero recoil, $\Lambda_{\rm QCD}/m_Q$ corrections to the matrix elements of the weak currents can be written in terms of the leading Isgur-Wise functions for the corresponding transition and meson mass splittings. A model independent prediction is found for the slope parameter of the decay rate into helicity zero $D_1$ at zero recoil. The differential decay rates are predicted, including $\Lambda_{\rm QCD}/m_Q$ corrections with some model dependence away from zero recoil and including order $\alpha_s$ corrections. Ratios of various exclusive branching ratios are computed. Matrix elements of the weak currents between $B$ mesons and other excited charmed mesons are discussed at zero recoil to order $\Lambda_{\rm QCD}/m_Q$. These amplitudes vanish at leading order, and can be written at order $\Lambda_{\rm QCD}/m_Q$ in terms of local matrix elements. Applications to $B$ decay sum rules and factorization are presented.' address: 'California Institute of Technology, Pasadena, CA 91125' author: - 'Adam K. Leibovich, Zoltan Ligeti, Iain W. Stewart, Mark B. Wise' title: Semileptonic $B$ decays to excited charmed mesons --- Introduction ============ Heavy quark symmetry [@HQS] implies that in the $m_Q\to\infty$ limit matrix elements of the weak currents between a $B$ meson and an excited charmed meson vanish at zero recoil (where in the rest frame of the $B$ the final state charmed meson is also at rest). However, in some cases at order $\Lambda_{\rm QCD}/m_Q$ these matrix elements are not zero [@llsw]. Since most of the phase space for semileptonic $B$ decay to excited charmed mesons is near zero recoil, $\Lambda_{\rm QCD}/m_Q$ corrections can be very important. This paper is concerned with rates for $B$ semileptonic decay to excited charmed mesons, including the effects of $\Lambda_{\rm QCD}/m_Q$ corrections. The use of heavy quark symmetry resulted in a dramatic improvement in our understanding of the spectroscopy and weak decays of hadrons containing a single heavy quark, $Q$. In the limit where the heavy quark mass goes to infinity, $m_Q\to\infty$, such hadrons are classified not only by their total spin $J$, but also by the spin of their light degrees of freedom (i.e., light quarks and gluons), $s_\ell$ [@IWprl]. In this limit hadrons containing a single heavy quark come in degenerate doublets with total spin, $J_\pm = s_\ell \pm \frac12$, coming from combining the spin of the light degrees of freedom with the spin of the heavy quark, $s_Q = \frac12$. (An exception occurs for baryons with $s_\ell = 0$, where there is only a single state with $J = \frac12$.) The ground state mesons with $Q\,\bar q$ flavor quantum numbers contain light degrees of freedom with spin-parity $s_\ell^{\pi_\ell}=\frac12^-$, giving a doublet containing a spin zero and spin one meson. For $Q=c$ these mesons are the $D$ and $D^*$, while $Q=b$ gives the $B$ and $B^*$ mesons. Excited charmed mesons with $s_\ell^{\pi_\ell} = \frac32^+$ have been observed. These are the $D_1$ and $D_2^*$ mesons with spin one and two, respectively. (There is also evidence for the analogous $Q=b$ heavy meson doublet.) For $q=u,d$, the $D_1$ and $D_2^*$ mesons have been observed to decay to $D^{(*)}\,\pi$ and are narrow with widths around $20\,$MeV. (The $D_{s1}$ and $D_{s2}^*$ strange mesons decay to $D^{(*)}K$.) In the nonrelativistic constituent quark model these states correspond to $L=1$ orbital excitations. Combining the unit of orbital angular momentum with the spin of the light antiquark leads to states with $s_\ell^{\pi_\ell}=\frac12^+$ and $\frac32^+$. The $\frac12^+$ doublet, $(D_0^*,D_1^*)$, has not been observed. Presumably this is because these states are much broader than those with $s_\ell^{\pi_\ell}=\frac32^+$. A vast discrepancy in widths is expected since the members of the $\frac12^+$ doublet of charmed mesons decay to $D^{(*)}\,\pi$ in an $S$-wave while the members of the $\frac32^+$ doublet of charmed mesons decay to $D^{(*)}\,\pi$ in a $D$-wave. (An $S$-wave $D_1\to D^*\,\pi$ amplitude is allowed by total angular momentum conservation, but forbidden in the $m_Q\to\infty$ limit by heavy quark spin symmetry [@IWprl].) The heavy quark effective theory (HQET) is the limit of QCD where the heavy quark mass goes to infinity with its four velocity, $v$, fixed. The heavy quark field in QCD, $Q$, is related to its counterpart in HQET, $h_v^{(Q)}$, by $$\label{HQETfield} Q(x) = e^{-im_Qv\cdot x}\left[1 + \frac{iD\!\!\!\!\slash}{2m_Q} + \ldots \right] h_v^{(Q)} \,,$$ where $\vslash h_v^{(Q)} = h_v^{(Q)}$ and the ellipses denote terms suppressed by further powers of $\Lambda_{\rm QCD}/m_Q$. Putting Eq. (\[HQETfield\]) into the part of the QCD Lagrangian involving the heavy quark field, ${\mathcal L}=\bar Q\,(iD\!\!\!\!\slash - m_Q)\,Q$, gives $$\label{fulllag} {\mathcal L} = {\mathcal L}_{\mathrm HQET} + \delta {\mathcal L} + \ldots \,.$$ The HQET Lagrangian [@eft] $$\label{HQETlag} {\mathcal L}_{\mathrm HQET} = \bar h_v^{(Q)}\, iv\cdot D\, h_v^{(Q)}$$ is independent of the mass of the heavy quark and its spin, and so for $N_Q$ heavy quarks with the same four velocity $v$ there is a $U(2N_Q)$ spin-flavor symmetry. This symmetry is broken by the order $\Lambda_{\rm QCD}/m_Q$ terms [@eft/m] in $\delta{\mathcal L}$, $$\label{lag} \delta{\cal L} = \frac1{2 m_Q}\, \Big[ O_{{\rm kin},v}^{(Q)} + O_{{\rm mag},v}^{(Q)} \Big] \,,$$ where $$O_{{\rm kin},v}^{(Q)} = \bar h_v^{(Q)}\, (iD)^2\, h_v^{(Q)} \,, \qquad O_{{\rm mag},v}^{(Q)} = \bar h_v^{(Q)}\, \frac{g_s}2\, \sigma_{\alpha\beta} G^{\alpha\beta}\, h_v^{(Q)} \,.$$ The first term in Eq. (\[lag\]) is the heavy quark kinetic energy. It breaks the flavor symmetry but leaves the spin symmetry intact. The second is the chromomagnetic term, which breaks both the spin and flavor symmetries. (In the rest frame, it is of the form $\vec{\mu}_Q\cdot\vec B_{\mathrm color}$, where $\vec{\mu}_Q$ is the heavy quark color magnetic moment.) The hadron masses give important information on some HQET matrix elements. The mass formula for a spin symmetry doublet of hadrons $H_\pm$ with total spin $J_\pm = s_\ell \pm \frac12$ is $$\label{mass} m_{H_\pm} = m_Q + \bar\Lambda^H - {\lambda_1^H \over 2 m_Q} \pm {n_\mp\, \lambda_2^H \over 2m_Q} + \ldots \,,$$ where the ellipsis denote terms suppressed by more powers of $\Lambda_{\rm QCD}/m_Q$ and $n_\pm = 2J_\pm+1$ is the number of spin states in the hadron $H_\pm$. The parameter $\bar\Lambda$ is the energy of the light degrees of freedom in the $m_Q\to\infty$ limit, $\lambda_1$ determines the heavy quark kinetic energy[^1] $$\label{lambda1} \lambda_1^H = \frac1{2v^0\,m_{H_\pm}}\, \langle H_\pm(v)|\, \bar h_v^{(Q)}\, (iD)^2\, h_v^{(Q)}\, |H_\pm(v)\rangle \,,$$ and $\lambda_2$ determines the chromomagnetic energy $$\label{lambda2} \lambda_2^H = {\mp1\over 2v^0\,m_{H_\pm}n_\mp}\, \langle H_\pm(v)|\, \bar h_v^{(Q)}\, \frac{g_s}2\, \sigma_{\alpha\beta} G^{\alpha\beta}\, h_v^{(Q)}\, |H_\pm(v)\rangle \,.$$ $\bar\Lambda$ and $\lambda_1$ are independent of the heavy quark mass, while $\lambda_2$ has a weak logarithmic dependence on $m_Q$. Of course they depend on the particular spin symmetry doublet to which $H_\pm$ belong. In this paper, we consider heavy mesons in the ground state $s_\ell^{\pi_\ell} = \frac12^-$ doublet and the excited $s_\ell^{\pi_\ell} = \frac32^+$ and $\frac12^+$ doublets. We reserve the notation $\bar\Lambda$, $\lambda_1$, $\lambda_2$ for the ground state multiplet and use $\bar\Lambda'$, $\lambda_1'$, $\lambda_2'$ and $\bar\Lambda^*$, $\lambda_1^*$, $\lambda_2^*$ for the excited $s_\ell^{\pi_\ell} = \frac32^+$ and $\frac12^+$ doublets, respectively. The average mass $\overline{m}_H$, weighted by the number of helicity states $$\label{avemass} \overline{m}_H = \frac{n_-m_{H_-} + n_+m_{H_+}}{n_+ + n_-} \,,$$ is independent of $\lambda_2$. The spin average masses for the lowest lying charmed mesons is given in Table \[tab:charm\]. Identifying the $B^{(*)}\pi$ resonances observed at LEP with the bottom $s_\ell^{\pi_\ell}=\frac32^+$ meson doublet we can use their average mass, $\overline{m}_B'=5.73\,$GeV [@talks], to determine the differences $\bar\Lambda'-\bar\Lambda$ and $\lambda_1'-\lambda_1$: $$\begin{aligned} \label{HQET_diff} \bar\Lambda' - \bar\Lambda &=& {m_b\,(\overline{m}_B'-\overline{m}_B) - m_c\,(\overline{m}_D'-\overline{m}_D) \over m_b-m_c} \simeq 0.39\, {\rm GeV} \,, {\nonumber}\\* \lambda_1' - \lambda_1 &=& {2 m_c m_b\, [(\overline{m}_B'-\overline{m}_B) - (\overline{m}_D'-\overline{m}_D)]\over m_b-m_c} \simeq -0.23\, {\rm GeV}^2 \,. \end{aligned}$$ The numerical values in Eq. (\[HQET\_diff\]) follow from the choices $m_b=4.8\,$GeV and $m_c=1.4\,$GeV. To the order we are working, $m_b$ and $m_c$ in Eq. (\[HQET\_diff\]) can be replaced by $\overline{m}_B$ and $\overline{m}_D$. This changes the value of $\bar\Lambda'-\bar\Lambda$ only slightly, but has a significant impact on the value of $\lambda_1'-\lambda_1$. The value of $\bar\Lambda'-\bar\Lambda$ given in Eq. (\[HQET\_diff\]) has considerable uncertainty because the experimental error on $\overline{m}_B'$ is large, and because it is not clear that the peak of the $B^{(*)}\pi$ mass distribution corresponds to the narrow $\frac32^+$ doublet.[^2] $s_l^{\pi_l}$ Particles $J^P$ $\overline{m}$ (GeV) -- --------------- ------------------ -------------- ---------------------- -- $\frac12^-$ $D$, $D^*$ $0^-$, $1^-$ $1.971$ $\frac12^+$ $D_0^*$, $D_1^*$ $0^+$, $1^+$ $\sim2.40$ $\frac32^+$ $D_1$, $D_2^*$ $1^+$, $2^+$ $2.445$ : Charmed meson spin multiplets ($q=u,d$).[]{data-label="tab:charm"} At the present time, $\bar\Lambda$ and $\lambda_1$ are not well determined. A fit to the electron energy spectrum in semileptonic $B$ decay gives [@gklw] $\bar\Lambda \simeq 0.4\,$GeV and $\lambda_1 \simeq -0.2\,{\mathrm GeV}^2$, but the uncertainties are quite large [@AM]. (A linear combination of $\bar\Lambda$ and $\lambda_1$ is better determined than the individual values.) The measured $D^*-D$ mass difference ($142\,$MeV) and the measured $D_2^*-D_1$ mass difference ($37\,$MeV) fix $\lambda_2 = 0.10\,{\mathrm GeV}^2$ and $\lambda_2' = 0.013\,{\mathrm GeV}^2$. Note that the matrix element of the chromomagnetic operator is substantially smaller in the excited $s_\ell^{\pi_\ell} = \frac32^+$ multiplet than in the ground state multiplet. This is consistent with expectations based on the nonrelativistic constituent quark model. In this phenomenological model, the splitting between members of a $Q\,\bar q$ meson spin symmetry doublet arises mostly from matrix elements of the operator $\vec s_Q\cdot\vec s_{\bar q}\,\delta^3(\vec r\,)$, and these vanish for $Q\,\bar q$ mesons with orbital angular momentum. Semileptonic $B$ meson decays have been studied extensively. The semileptonic decays $B\to D\,e\,\bar\nu_e$ and $B\to D^*\,e\,\bar\nu_e$ have branching ratios of $(1.8\pm0.4)\%$ and $(4.6\pm0.3)\%$, respectively [@PDG], and comprise about 60% of the semileptonic decays. The differential decay rates for these decays are determined by matrix elements of the weak $b\to c$ axial-vector and vector currents between the $B$ meson and the recoiling $D^{(*)}$ meson. These matrix elements are usually parameterized by a set of Lorentz scalar form factors and the differential decay rate is expressed in terms of these form factors. For comparison with the predictions of HQET, it is convenient to write the form factors as functions of the dot-product, $w=v\cdot v'$, of the four-velocity of the $B$ meson, $v$, and that of the recoiling $D^{(*)}$ meson, $v'$. In the $m_Q\to\infty$ limit, heavy quark spin symmetry implies that the six form factors that parameterize the $B\to D$ and $B\to D^*$ matrix elements of the $b\to c$ axial-vector and vector currents can be written in terms of a single function of $w$ [@HQS]. Furthermore, heavy quark flavor symmetry implies that this function is normalized to unity at zero recoil, $w = 1$, where the $D^{(*)}$ is at rest in the rest frame of the $B$ [@NuWe; @VoSi; @HQS]. The functions of $w$ that occur in predictions for weak decay form factors based on HQET are usually called Isgur-Wise functions. There are perturbative $\alpha_s(m_Q)$ and nonperturbative $\Lambda_{\mathrm QCD}/m_Q$ corrections to the predictions of the $m_Q\to\infty$ limit for the $B\to D^{(*)}\,e\,\bar\nu_e$ semileptonic decay form factors. The perturbative QCD corrections do not cause any loss of predictive power. They involve the same Isgur-Wise function that occurs in the $m_Q\to\infty$ limit. At order $\Lambda_{\mathrm QCD}/m_Q$ several new Isgur-Wise functions occur; however, at zero recoil, there are no $\Lambda_{\mathrm QCD}/m_Q$ corrections [@Luke]. Expectations for the $B\to D^{(*)}\,e\,\bar\nu_e$ differential decay rate based on HQET are in agreement with experiment [@rich]. Recently, semileptonic $B$ decay to an excited heavy meson has been observed [@OPAL; @ALEPH; @CLEO]. With some assumptions, CLEO [@CLEO] and ALEPH [@ALEPH] find respectively the branching ratios ${\mathcal B}(B\to D_1\,e\,\bar\nu_e) = (0.49 \pm 0.14)\%$ and ${\mathcal B}(B\to D_1\,e\,\bar\nu_e) = (0.74 \pm 0.16)\%$, as well as the limits ${\mathcal B}(B\to D_2^*\,e\,\bar\nu_e)<1\%$ and ${\mathcal B}(B\to D_2^*\,e\,\bar\nu_e)<0.2\%$. In the future it should be possible to get detailed experimental information on the $B\to D_1\,e\,\bar\nu_e$ and $B\to D_2^*\,e\,\bar\nu_e$ differential decay rates. In this paper we study the predictions of HQET for $B$ semileptonic decay to excited charmed mesons. This paper elaborates on the work in Ref. [@llsw] and contains some new results. In the infinite mass limit the matrix elements of the weak axial-vector and vector current between the $B$ meson and any excited charmed meson vanish at zero recoil by heavy quark symmetry. Corrections to the infinite mass limit of order $\Lambda_{\mathrm QCD}/m_Q$ and order $\alpha_s(m_Q)$ are discussed. The corrections of order $\Lambda_{\mathrm QCD}/m_Q$ are very important, particularly near zero recoil. Section II discusses the differential decay rate ${\rm d}^2\Gamma/{\rm d}w\,{\rm d}\!\cos\theta$ for $B\to(D_1,D_2^*)\,e\,\bar\nu_e$, where $\theta$ is the angle between the the charged lepton and the charmed meson in the rest frame of the virtual $W$ boson. Corrections of order $\Lambda_{\mathrm QCD}/m_Q$ are included. At order $\Lambda_{\mathrm QCD}/m_Q$ the $B\to D_1$ zero recoil matrix element does not vanish and is expressible in terms of the leading $m_Q\to\infty$ Isgur-Wise function, $\tau$, and $\bar\Lambda'-\bar\Lambda$ (which is known in terms of hadron mass splittings from Eq. (\[HQET\_diff\])). Away from zero recoil new Isgur-Wise functions occur, which are unknown. These introduce a significant uncertainty. The $\Lambda_{\mathrm QCD}/m_Q$ corrections enhance considerably the $B$ semileptonic decay rate to the $D_1$ state, and for zero helicity the slope of ${\rm d}\Gamma(B\to D_1\,e\,\bar\nu_e)/{\rm d}w$ at $w=1$ is predicted. These corrections also reduce the ratio $R ={\cal B}(B\to D_2^*\,e\,\bar\nu_e)/{\cal B}(B\to D_1\,e\,\bar\nu_e)$ compared to its value in the $m_Q\to\infty$ limit. The value of $\tau$ at zero recoil is not fixed by heavy quark symmetry, and must be determined from experiment. The measured $B\to D_1\,e\,\bar\nu_e$ branching ratio is used to determine (with some model dependent assumptions) $|\tau(1)|=0.71$. The effects of perturbative QCD corrections are also discussed, with further details given in Appendix A. It is interesting to understand the composition of the inclusive $B$ semileptonic decay rate in terms of exclusive final states. In Section III, the HQET predictions for the differential decay rates for $B\to D_0^*\,e\,\bar\nu_e$ and $B\to D_1^*\,e\,\bar\nu_e$ are investigated. The situation for the excited $s_\ell^{\pi_\ell}=\frac12^+$ multiplet is similar to the $s_\ell^{\pi_\ell}=\frac32^+$ multiplet discussed in Section II. Using a quark model relation between the leading $m_Q\to\infty$ Isgur-Wise functions for $B$ decays to the $s_\ell^{\pi_\ell}=\frac32^+$ and $s_\ell^{\pi_\ell}=\frac12^+$ charmed mesons (and some other model dependent assumptions), the rates for $B\to D_0^*\,e\,\bar\nu_e$ and $B\to D_1^*\,e\,\bar\nu_e$ are predicted. Section IV discusses the contribution of other excited charmed mesons to the matrix elements of the vector and axial-vector current at zero recoil. Only excited charmed hadrons with $s_\ell^{\pi_\ell}=\frac12^-$, $\frac32^-$ and $\frac12^+$, $\frac32^+$ can contribute. The $\frac32^+$ and $\frac12^+$ doublets are discussed in Sections II and III. This section deals with the $\frac12^-$ and $\frac32^-$ cases, where the $\Lambda_{\mathrm QCD}/m_Q$ corrections to the states from $\delta{\mathcal L}$ give rise to non-vanishing zero recoil matrix elements. Section V examines other applications of our results. Using factorization, predictions are made for nonleptonic $B$ decay widths to $D_2^*\,\pi$, $D_1\,\pi$ and to $D_1^*\,\pi$, $D_0^*\,\pi$. The importance of our results for $B$ decay sum rules is discussed. Including the excited states dramatically strengthens the Bjorken lower bound on the slope of the $B\to D^{(*)}\,e\,\bar\nu_e$ Isgur-Wise function. Concluding remarks and a summary of our most significant predictions are given in Section VI. $B\to D_1\,\lowercase{e}\,\bar\nu_{\lowercase{e}}$ and $B\to D_2^*\,\lowercase{e}\,\bar\nu_{\lowercase{e}}$ decays ================================================================================================================== The matrix elements of the vector and axial-vector currents ($V^\mu=\bar c\,\gamma^\mu\,b$ and $A^\mu=\bar c\,\gamma^\mu\gamma_5\,b$) between $B$ mesons and $D_1$ or $D_2^*$ mesons can be parameterized as $$\begin{aligned} \label{formf1} {\langle D_1(v',\epsilon)|\, V^\mu\, |B(v)\rangle \over \sqrt{m_{D_1}\,m_B}} &=& f_{V_1}\, \epsilon^{*\mu} + (f_{V_2} v^\mu + f_{V_3} v'^\mu)\, (\epsilon^*\cdot v) \,, {\nonumber}\\* {\langle D_1(v',\epsilon)|\, A^\mu\, |B(v)\rangle \over \sqrt{m_{D_1}\,m_B}} &=& i\, f_A\, \varepsilon^{\mu\alpha\beta\gamma} \epsilon^*_\alpha v_\beta v'_\gamma \,, {\nonumber}\\* {\langle D^*_2(v',\epsilon)|\, A^\mu\, |B(v)\rangle \over\sqrt{m_{D_2^*}\,m_B}} &=& k_{A_1}\, \epsilon^{*\mu\alpha} v_\alpha + (k_{A_2} v^\mu + k_{A_3} v'^\mu)\, \epsilon^*_{\alpha\beta}\, v^\alpha v^\beta \,, {\nonumber}\\* {\langle D^*_2(v',\epsilon)|\, V^\mu\, |B(v)\rangle \over\sqrt{m_{D_2^*}\,m_B}} &=& i\,k_V\, \varepsilon^{\mu\alpha\beta\gamma} \epsilon^*_{\alpha\sigma} v^\sigma v_\beta v'_\gamma \,, \label{ff_nrwD}\end{aligned}$$ where the form factors $f_i$ and $k_i$ are dimensionless functions of $w$. At zero recoil ($v=v'$) only the $f_{V_1}$ form factor can contribute, since $v'$ dotted into the polarization ($\epsilon^{*\mu}$ or $\epsilon^{*\mu\alpha}$) vanishes. The differential decay rates can be written in terms of the form factors in Eq. (\[ff\_nrwD\]). It is useful to separate the contributions to the different helicities of the $D_1$ and $D_2^*$ mesons, since the $\Lambda_{\rm QCD}/m_Q$ corrections effect these differently, and the decay rates into different helicity states will probably be measurable. We define $\theta$ as the angle between the charged lepton and the charmed meson in the rest frame of the virtual $W$ boson, i.e., in the center of momentum frame of the lepton pair. The different helicity amplitudes yield different distributions in $\theta$. In terms of $w=v\cdot v'$ and $\theta$, the double differential decay rates are $$\begin{aligned} \label{rate1} {{\rm d}^2\Gamma_{D_1}\over {\rm d}w\,{\rm d}\!\cos\theta} &=& 3\Gamma_0\, r_1^3\, \sqrt{w^2-1}\, \bigg\{ \sin^2\theta\, \Big[ (w-r_1) f_{V_1}+(w^2-1) (f_{V_3}+r_1 f_{V_2}) \Big]^2 \\* && + (1-2r_1w+r_1^2)\, \Big[ (1+\cos^2\theta)\, [f_{V_1}^2 + (w^2-1) f_A^2] - 4\cos\theta\, \sqrt{w^2-1}\, f_{V_1}\, f_A \Big] \bigg\} \,,\nonumber\\* {{\rm d}^2\Gamma_{D_2^*}\over {\rm d}w\,{\rm d}\!\cos\theta} &=& \frac32\,\Gamma_0\, r_2^3\, (w^2-1)^{3/2}\, \bigg\{ \frac43\,\sin^2\theta\, \Big[ (w-r_2) k_{A_1}+(w^2-1) (k_{A_3}+r_2 k_{A_2}) \Big]^2 {\nonumber}\\* && + (1-2r_2w+r_2^2)\, \Big[ (1+\cos^2\theta)\, [k_{A_1}^2 + (w^2-1) k_V^2] - 4\cos\theta\, \sqrt{w^2-1}\, k_{A_1}\, k_V \Big] \bigg\} \,,\nonumber\end{aligned}$$ where $\Gamma_0 = {G_F^2\,|V_{cb}|^2\,m_B^5 /(192\pi^3)}$, $r_1=m_{D_1}/m_B$, $r_2=m_{D_2^*}/m_B$. The semileptonic $B$ decay rate into any $J\neq1$ state involves an extra factor of $w^2-1$. The $\sin^2\theta$ term is the helicity zero rate, while the $1+\cos^2\theta$ and $\cos\theta$ terms determine the helicity $\lambda=\pm1$ rates. Since the weak current is $V-A$ in the standard model, $B$ mesons can only decay into the helicity $|\lambda|=0,1$ components of any excited charmed mesons. The decay rate for $|\lambda|=1$ vanishes at maximal recoil, $w_{\rm max}=(1+r^2)/(2r)$, as implied by the $1-2rw+r^2$ factors above ($r=r_1$ or $r_2$). From Eq. (\[rate1\]) it is straightforward to obtain the double differential rate ${\rm d}^2\Gamma/{\rm d}w\,{\rm d}y$ using the relation $$\label{yct} y = 1 - rw - r\sqrt{w^2-1}\, \cos\theta \,,$$ where $y=2E_e/m_B$ is the rescaled lepton energy. The form factors $f_i$ and $k_i$ can be parameterized by a set of Isgur-Wise functions at each order in $\Lambda_{\rm QCD}/m_Q$. It is simplest to calculate the matrix elements in Eq. (\[formf1\]) using the trace formalism [@falk; @trace]. The fields $P_v$ and $P_v^{*\mu}$ that destroy members of the $s_l^{\pi_l}=\frac12^-$ doublet with four-velocity $v$ are in the $4\times4$ matrix $$H_v = \frac{1+\vslash}2\, \Big[ P_v^{*\mu} \gamma_\mu - P_v\, \gamma_5 \Big] \,. \label{Hdef}$$ while for $s_l^{\pi_l}=\frac32^+$ the fields $P_v^\nu$ and $P_v^{*\mu\nu}$ are in $$\label{Fdef} F_v^\mu = \frac{1+\vslash}2\, \bigg\{ P_v^{*\mu\nu} \gamma_\nu - \sqrt{\frac32}\, P_v^\nu \gamma_5 \bigg[ g^\mu_\nu - \frac13 \gamma_\nu (\gamma^\mu-v^\mu) \bigg] \bigg\} \,.$$ The matrices $H$ and $F$ satisfy the properties $\vslash H_v=H_v=-H_v\vslash$, , and $F_v^\mu\,\gamma_\mu=F_v^\mu\,v_\mu=0$. To leading order in $\Lambda_{\rm QCD}/m_Q$ and $\alpha_s$, matrix elements of the $b\to c$ flavor changing current between the states destroyed by the fields in $H_v$ and $F_{v'}^\sigma$ are $$\label{lo} \bar c\, \Gamma\, b = \bar h^{(c)}_{v'}\, \Gamma\, h^{(b)}_v = \tau(w)\, {\rm Tr}\, \Big\{ v_\sigma \bar F^\sigma_{v'}\, \Gamma\, H_v \Big\} \,.$$ Here $\tau(w)$ is a dimensionless function, and $h_v^{(Q)}$ is the heavy quark field in the effective theory ($\tau$ is $\sqrt3$ times the function $\tau_{3/2}$ of Ref. [@IWsr]). This matrix element vanishes at zero recoil for any Dirac structure $\Gamma$ and for any value of $\tau(1)$, since the $B$ meson and the $(D_1,D_2^*)$ mesons are in different heavy quark spin symmetry multiplets, and the current at zero recoil is related to the conserved charges of heavy quark spin-flavor symmetry. Eq. (\[lo\]) leads to the $m_Q\to\infty$ predictions for the form factors $f_i$ and $k_i$ given in Ref. [@IWsr]. At order $\Lambda_{\rm QCD}/m_Q$, there are corrections originating from the matching of the $b\to c$ flavor changing current onto the effective theory, and from order $\Lambda_{\rm QCD}/m_Q$ corrections to the effective Lagrangian. The current corrections modify the first equality in Eq. (\[lo\]) to $$\label{1/mcurrent} \bar c\, \Gamma\, b = \bar h_{v'}^{(c)}\, \bigg( \Gamma - \frac i{2m_c} \overleftarrow D\!\!\!\!\!\slash\, \Gamma + \frac i{2m_b}\, \Gamma \overrightarrow D\!\!\!\!\!\slash\ \bigg)\, h_v^{(b)} \,.$$ For matrix elements between the states destroyed by the fields in $F_{v'}^\sigma$ and $H_v$, the new order $\Lambda_{\rm QCD}/m_Q$ operators in Eq. (\[1/mcurrent\]) are $$\begin{aligned} \label{1/mc1} \bar h^{(c)}_{v'}\, i\overleftarrow D_{\!\lambda}\, \Gamma\, h^{(b)}_v &=& {\rm Tr}\, \Big\{ {\cal S}^{(c)}_{\sigma\lambda}\, \bar F^\sigma_{v'}\, \Gamma\, H_v \Big\} \,, \nonumber\\* \bar h^{(c)}_{v'}\, \Gamma\, i\overrightarrow D_{\!\lambda}\, h^{(b)}_v &=& {\rm Tr}\, \Big\{ {\cal S}^{(b)}_{\sigma\lambda}\, \bar F^\sigma_{v'}\, \Gamma\, H_v \Big\} \,. \label{curr}\end{aligned}$$ The most general form for these quantities is $$\label{1/mc2} {\cal S}^{(Q)}_{\sigma\lambda} = v_\sigma \Big[ \tau_1^{(Q)}\, v_\lambda + \tau_2^{(Q)}\, v'_\lambda + \tau_3^{(Q)}\, \gamma_\lambda \Big] + \tau_4^{(Q)}\, g_{\sigma\lambda} \,. \label{Sdef}$$ The functions $\tau_i$ depend on $w$, and have mass dimension one.[^3] They are not all independent. The equation of motion for the heavy quarks, $(v\cdot D)\,h_v^{(Q)}=0$, implies $$\begin{aligned} \label{const1} w\,\tau_1^{(c)} + \tau_2^{(c)} - \tau_3^{(c)} &=& 0 \,, \nonumber\\* \tau_1^{(b)} + w\,\tau_2^{(b)} - \tau_3^{(b)} + \tau_4^{(b)} &=& 0 \,.\end{aligned}$$ Four more relations can be derived using $$\label{momcons} i\partial_\nu\,(\bar h_{v'}^{(c)}\,\Gamma\,h_v^{(b)}) = (\bar\Lambda v_\nu-\bar\Lambda'v'_\nu)\, \bar h_{v'}^{(c)}\,\Gamma\,h_v^{(b)} \,,$$ which is valid between the states destroyed by the fields in $F_{v'}^\sigma$ and $H_v$. This relation follows from translation invariance and the definition of the heavy quark fields $h_v^{(Q)}$. It implies that $$\label{const2a} {\cal S}^{(c)}_{\sigma\lambda} + {\cal S}^{(b)}_{\sigma\lambda} = (\bar\Lambda v_\lambda-\bar\Lambda'v'_\lambda)\, v_\sigma\, \tau \,.$$ Eq. (\[const2a\]) gives the following relations[^4] $$\begin{aligned} \label{const2} \tau_1^{(c)} + \tau_1^{(b)} &=& \bar\Lambda\, \tau \,, \nonumber\\* \tau_2^{(c)} + \tau_2^{(b)} &=& -\bar\Lambda'\, \tau \,, \nonumber\\ \tau_3^{(c)} + \tau_3^{(b)} &=& 0 \,, \nonumber\\* \tau_4^{(c)} + \tau_4^{(b)} &=& 0 \,.\end{aligned}$$ These relations express the $\tau_j^{(b)}$’s in terms of the $\tau_j^{(c)}$’s. Furthermore, combining Eqs. (\[const1\]) with (\[const2\]) yields $$\begin{aligned} \label{const2b} \tau_3^{(c)} &=& w\, \tau_1^{(c)} + \tau_2^{(c)} \,, \nonumber\\* \tau_4^{(c)} &=& (w-1)\, (\tau_1^{(c)} - \tau_2^{(c)}) - (w \bar\Lambda' - \bar\Lambda)\, \tau \,.\end{aligned}$$ All order $\Lambda_{\rm QCD}/m_Q$ corrections to the form factors coming from the matching of the QCD currents onto those in the effective theory are expressible in terms of $\bar\Lambda\,\tau$ and $\bar\Lambda'\,\tau$ and two functions, which we take to be $\tau_1^{(c)}$ and $\tau_2^{(c)}$. From Eqs. (\[curr\]) and (\[Sdef\]) it is evident that only $\tau_4^{(Q)}$ can contribute at zero recoil. Eq. (\[const2b\]) determines this contribution in terms of $\tau(1)$ and measurable mass splittings given in Eq. (\[HQET\_diff\]), $$\label{tau41} \tau^{(b)}_4(1) = -\tau^{(c)}_4(1) = (\bar\Lambda'-\bar\Lambda)\,\tau(1) \,.$$ Note that with our methods Eq. (\[tau41\]) cannot be derived working exclusively at zero recoil. At that kinematic point, matrix elements of the operator $\bar h_v^{(c)}\,\Gamma\,h_v^{(b)}$ vanish between a $B$ meson and an excited charmed meson, and so Eq. (\[momcons\]) only implies that $\tau_4^{(c)}+\tau_4^{(b)}=0$. Eq. (\[tau41\]) relies on the assumption that the $\tau_j^{(Q)}(w)$ are continuous at $w=1$. Next consider the terms originating from order $\Lambda_{\rm QCD}/m_Q$ corrections to the HQET Lagrangian, $\delta {\cal L}$ in Eq. (\[lag\]). These corrections modify the heavy meson states compared to their infinite heavy quark mass limit. For example, they cause the mixing of the $D_1$ with the $J^P=1^+$ member of the $s_l^{\pi_l}=\frac12^+$ doublet. (This is a very small effect, since the $D_1$ is not any broader than the $D_2^*$.) For matrix elements between the states destroyed by the fields in $F_{v'}^\sigma$ and $H_v$, the time ordered products of the kinetic energy term in $\delta{\cal L}$ with the leading order currents are $$\begin{aligned} \label{kinetic} i \int {\rm d}^4x\, T\,\Big\{ O_{{\rm kin},v'}^{(c)}(x)\, \Big[ \bar h_{v'}^{(c)}\, \Gamma\, h_{v}^{(b)} \Big](0)\, \Big\} &=& \eta^{(c)}_{\rm ke}\, {\rm Tr}\, \Big\{ v_\sigma \bar F^\sigma_{v'}\, \Gamma\, H_v \Big\} \,, {\nonumber}\\* i \int {\rm d}^4x\, T\,\Big\{ O_{{\rm kin},v}^{(b)}(x)\, \Big[ \bar h_{v'}^{(c)}\, \Gamma\, h_{v}^{(b)} \Big](0)\, \Big\} &=& \eta^{(b)}_{\rm ke}\, {\rm Tr}\, \Big\{ v_\sigma \bar F^\sigma_{v'}\, \Gamma\, H_v \Big\} \,.\end{aligned}$$ These corrections do not violate spin symmetry, so their contributions enter the same way as the $m_Q\to\infty$ Isgur-Wise function, $\tau$. For matrix elements between the states destroyed by the fields in $F_{v'}^\sigma$ and $H_v$, the time ordered products of the chromomagnetic term in $\delta{\cal L}$ with the leading order currents are $$\begin{aligned} \label{magnetic} i \int {\rm d}^4x\, T\,\Big\{ O_{{\rm mag},v'}^{(c)}(x)\, \Big[ \bar h_{v'}^{(c)}\, \Gamma\, h_{v}^{(b)} \Big](0)\, \Big\} &=& {\rm Tr}\, \bigg\{ {\cal R}_{\sigma\alpha\beta}^{(c)}\, \bar F_{v'}^\sigma\, i\sigma^{\alpha\beta}\, \frac{1+\vslash'}2\, \Gamma\, H_v \bigg\} \,, \nonumber\\* i \int {\rm d}^4x\, T\,\Big\{ O_{{\rm mag},v}^{(b)}(x)\, \Big[ \bar h_{v'}^{(c)}\, \Gamma\, h_{v}^{(b)} \Big](0)\, \Big\} &=& {\rm Tr}\, \bigg\{ {\cal R}_{\sigma\alpha\beta}^{(b)}\, \bar F_{v'}^\sigma\, \Gamma\, \frac{1+\vslash}2\, i\sigma^{\alpha\beta} H_v \bigg\} \,. \end{aligned}$$ The most general parameterizations of ${\cal R}^{(Q)}$ are $$\begin{aligned} \label{Rdef} {\cal R}_{\sigma\alpha\beta}^{(c)} &=& \eta_1^{(c)}\, v_\sigma \gamma_\alpha \gamma_\beta + \eta_2^{(c)}\, v_\sigma v_\alpha \gamma_\beta + \eta_3^{(c)}\, g_{\sigma\alpha} v_\beta \,, \nonumber\\* {\cal R}_{\sigma\alpha\beta}^{(b)} &=& \eta_1^{(b)}\, v_\sigma \gamma_\alpha \gamma_\beta + \eta_2^{(b)}\, v_\sigma v'_\alpha \gamma_\beta + \eta_3^{(b)}\, g_{\sigma\alpha} v'_\beta \,.\end{aligned}$$ Only the part of ${\cal R}_{\sigma\alpha\beta}^{(Q)}$ antisymmetric in $\alpha$ and $\beta$ contributes when inserted into Eq. (\[magnetic\]). The functions $\eta_i$ depend on $w$, and have mass dimension one. Note that $g_{\sigma\alpha}\gamma_\beta$ is dependent on the tensor structures included in Eq. (\[Rdef\]) for matrix elements between these states. For example, for the $\Lambda_{\rm QCD}/m_c$ corrections the following trace identity holds $$\label{magicTr} {\rm Tr}\, \bigg\{ \Big[ v_\sigma \gamma_\alpha\gamma_\beta + 2g_{\sigma\alpha} v_\beta + 2(1+w)\, g_{\sigma\alpha} \gamma_\beta \Big] \bar F_{v'}^\sigma\, \sigma^{\alpha\beta}\, \frac{1+\vslash'}2\, \Gamma H_v \bigg\} = 0 \,.$$ All contributions arising from the time ordered products in Eq. (\[magnetic\]) vanish at zero recoil, since $v_\sigma\bar F_v^\sigma=0$ and . Thus we find that at zero recoil the only $\Lambda_{\rm QCD}/m_Q$ corrections that contribute are determined by measured meson mass splittings and the value of the leading order Isgur-Wise function at zero recoil. The form factors in Eq. (\[formf1\]) depend on $\eta_i^{(b)}$ only through the linear combination $\eta_b=\eta_{\rm ke}^{(b)}+6\,\eta_1^{(b)}-2(w-1)\,\eta_2^{(b)}+\eta_3^{(b)}$. Denoting $\varepsilon_Q=1/(2m_Q)$ and dropping the superscript on $\tau_i^{(c)}$ and $\eta_i^{(c)}$, the $B\to D_1\,e\,\bar\nu_e$ form factors are [@llsw] $$\begin{aligned} \label{expf} \sqrt6\, f_A &=& - (w+1)\tau - \varepsilon_b \{ (w-1) [(\bar\Lambda'+\bar\Lambda)\tau - (2w+1)\tau_1-\tau_2] + (w+1)\eta_b \} \nonumber\\* && - \varepsilon_c [ 4(w\bar\Lambda'-\bar\Lambda)\tau - 3(w-1) (\tau_1-\tau_2) + (w+1) (\eta_{\rm ke}-2\eta_1-3\eta_3) ] \,,\nonumber\\* \sqrt6\, f_{V_1} &=& (1-w^2)\tau - \varepsilon_b (w^2-1) [(\bar\Lambda'+\bar\Lambda)\tau - (2w+1)\tau_1-\tau_2 + \eta_b] \nonumber\\* && - \varepsilon_c [ 4(w+1)(w\bar\Lambda'-\bar\Lambda)\tau - (w^2-1)(3\tau_1-3\tau_2-\eta_{\rm ke}+2\eta_1+3\eta_3) ] \,, \nonumber\\ \sqrt6\, f_{V_2} &=& -3\tau - 3\varepsilon_b [(\bar\Lambda'+\bar\Lambda)\tau - (2w+1)\tau_1-\tau_2 + \eta_b] \nonumber\\* && - \varepsilon_c [ (4w-1)\tau_1+5\tau_2 +3\eta_{\rm ke} +10\eta_1 + 4(w-1)\eta_2-5\eta_3 ] \,, \nonumber\\* \sqrt6\, f_{V_3} &=& (w-2)\tau + \varepsilon_b \{ (2+w) [(\bar\Lambda'+\bar\Lambda)\tau - (2w+1)\tau_1-\tau_2] - (2-w)\eta_b \} \nonumber\\* && + \varepsilon_c [ 4(w\bar\Lambda'-\bar\Lambda)\tau + (2+w)\tau_1 + (2+3w)\tau_2 \nonumber\\* && \phantom{\varepsilon_c [} + (w-2)\eta_{\rm ke} - 2(6+w)\eta_1 - 4(w-1)\eta_2 - (3w-2)\eta_3 ] \,. \end{aligned}$$ The analogous formulae for $B\to D_2^*\,e\,\bar\nu_e$ are $$\begin{aligned} \label{expk} k_V &=& - \tau - \varepsilon_b [(\bar\Lambda'+\bar\Lambda)\tau - (2w+1)\tau_1-\tau_2 + \eta_b] - \varepsilon_c (\tau_1-\tau_2+\eta_{\rm ke}-2\eta_1+\eta_3) , \nonumber\\* k_{A_1} &=& - (1+w)\tau - \varepsilon_b \{ (w-1) [(\bar\Lambda'+\bar\Lambda)\tau - (2w+1)\tau_1-\tau_2] + (1+w)\eta_b \} \nonumber\\* && - \varepsilon_c [ (w-1)(\tau_1-\tau_2) + (w+1)(\eta_{\rm ke}-2\eta_1+\eta_3) ] , \nonumber\\ k_{A_2} &=& - 2\varepsilon_c (\tau_1+\eta_2) , \\* k_{A_3} &=& \tau + \varepsilon_b [(\bar\Lambda'+\bar\Lambda)\tau - (2w+1)\tau_1-\tau_2 + \eta_b] - \varepsilon_c (\tau_1+\tau_2-\eta_{\rm ke}+2\eta_1-2\eta_2-\eta_3) . \nonumber\end{aligned}$$ Recall that $f_{V_1}$ determines the zero recoil matrix elements of the weak currents. From Eqs. (\[expf\]) it follows that $$\sqrt6\, f_{V_1}(1) = -8\, \varepsilon_c\, (\bar\Lambda'-\bar\Lambda)\, \tau(1)\,.$$ The allowed kinematic range for $B\to D_1\,e\,\bar\nu_e$ decay is $1<w<1.32$, while for $B\to D_2^*\,e\,\bar\nu_e$ decay it is $1<w<1.31$. Since these ranges are fairly small, and at zero recoil there are some constraints on the $\Lambda_{\rm QCD}/m_Q$ corrections, it is useful to consider the decay rates given in Eq. (\[rate1\]) expanded in powers of $w-1$. The general structure of the expansion of ${\rm d}\Gamma/{\rm d}w$ is elucidated schematically below, $$\begin{aligned} \label{schematic} {{\rm d}\Gamma_{D_1}^{(\lambda=0)}\over {\rm d}w} &\sim& \sqrt{w^2-1}\, \Big[ (w-1)^0\, \Big( 0 + 0\,\varepsilon + \varepsilon^2 + \varepsilon^3 + \ldots \Big) \nonumber\\* && \phantom{\sqrt{w^2-1}} + (w-1)^1\, \Big(0 + \varepsilon + \varepsilon^2 + \ldots\Big) + (w-1)^2\, \Big(1 + \varepsilon + \ldots\Big) + \ldots \Big] \,, \nonumber\\* {{\rm d}\Gamma_{D_1}^{(|\lambda|=1)}\over {\rm d}w} &\sim& \sqrt{w^2-1}\, \Big[ (w-1)^0\, \Big( 0 + 0\,\varepsilon + \varepsilon^2 + \varepsilon^3 + \ldots \Big) \nonumber\\* && \phantom{\sqrt{w^2-1}} + (w-1)^1\, \Big(1 + \varepsilon + \ldots\Big) + (w-1)^2\, \Big(1 + \varepsilon + \ldots\Big) + \ldots \Big] \,, \nonumber\\* {{\rm d}\Gamma_{D_2}^{(|\lambda|=0,1)}\over {\rm d}w} &\sim& (w^2-1)^{3/2}\, \Big[ (w-1)^0\, \Big( 1 + \varepsilon + \ldots \Big) + (w-1)^1\, \Big(1 + \varepsilon + \ldots\Big) + \ldots \Big] \,. \end{aligned}$$ Here $\varepsilon^n$ denotes a term of order $(\Lambda_{\rm QCD}/m_Q)^n$. The zeros in Eq. (\[schematic\]) are consequences of heavy quark symmetry, as the leading contribution to the matrix elements of the weak currents at zero recoil is of order $\Lambda_{\rm QCD}/m_Q$. Thus, the $D_1$ decay rate at $w=1$ starts out at order $\Lambda_{\rm QCD}^2/m_Q^2$. Similarly, from Eq. (\[rate1\]) it is evident that the vanishing of $f_{V_1}(1)$ in the $m_Q\to\infty$ limit implies that at order $w-1$ the $D_1^{(\lambda=0)}$ rate starts out at order $\Lambda_{\rm QCD}/m_Q$. The $D_2^*$ decay rate is suppressed by an additional power of $w^2-1$, so there is no further restriction on its structure. In this paper we present predictions using two different approximations to the decay rates. In approximation A we treat $w-1$ as order $\Lambda_{\rm QCD}/m_Q$ and expand the decay rates in these parameters. In approximation B the known order $\Lambda_{\rm QCD}/m_Q$ contributions to the form factors are kept, as well as the full $w$-dependence of the decay rates. Expanding the terms in the square brackets in Eq. (\[rate1\]) in powers of $w-1$ gives $$\begin{aligned} \label{rt_expn1} {{\rm d}^2\Gamma_{D_1}\over {\rm d}w\,{\rm d}\!\cos\theta} &=& \Gamma_0\, \tau^2(1)\, r_1^3\, \sqrt{w^2-1}\, \sum_n\, (w-1)^n\, \bigg\{ \sin^2\theta\, s_1^{(n)} \\* && + (1-2r_1w+r_1^2)\, \Big[ (1+\cos^2\theta)\, t_1^{(n)} - 4\cos\theta\, \sqrt{w^2-1}\, u_1^{(n)} \Big] \bigg\} \,,\nonumber\\* {{\rm d}^2\Gamma_{D_2^*}\over {\rm d}w\,{\rm d}\!\cos\theta} &=& \frac32\,\Gamma_0\, \tau^2(1)\, r_2^3\, (w^2-1)^{3/2}\, \sum_n\, (w-1)^n\, \bigg\{ \frac43\,\sin^2\theta\, s_2^{(n)} {\nonumber}\\* && + (1-2r_2w+r_2^2)\, \Big[ (1+\cos^2\theta)\, t_2^{(n)} - 4\cos\theta\, \sqrt{w^2-1}\, u_2^{(n)} \Big] \bigg\} \,.\nonumber\end{aligned}$$ (We do not expand the factors of $\sqrt{w^2-1}$ that multiply $\cos\theta$). The subscripts of the coefficients $s,t,u$ denote the spin of the excited $D$ meson, while the superscripts refer to the order in the $w-1$ expansion. The $u_i^{(n)}$ terms proportional to $\cos\theta$ only affect the lepton spectrum, since they vanish when integrated over $\theta$. Eqs. (\[rate1\]), (\[expf\]), and (\[expk\]) yield the following expressions for the coefficients in the $D_1$ decay rate in Eq. (\[rt\_expn1\]), $$\begin{aligned} \label{stuD1} s_1^{(0)} &=& 32\varepsilon_c^2\, (1-r_1)^2\, (\bar\Lambda'-\bar\Lambda)^2 + \ldots \,, {\nonumber}\\* s_1^{(1)} &=& 32\varepsilon_c\, (1-r_1^2)\, (\bar\Lambda'-\bar\Lambda) + \ldots \,, {\nonumber}\\* s_1^{(2)} &=& 8\, (1+r_1)^2 + \ldots \,, \nonumber\\ t_1^{(0)} &=& 32\varepsilon_c^2\, (\bar\Lambda'-\bar\Lambda)^2 + \ldots \,, {\nonumber}\\* t_1^{(1)} &=& 4 + 8\varepsilon_c\, \Big[ 4(\bar\Lambda'-\bar\Lambda) + \hat\eta_{\rm ke} - 2\hat\eta_1 - 3\hat\eta_3 \Big] + 8\varepsilon_b\, \hat\eta_b + \ldots \,, {\nonumber}\\* t_1^{(2)} &=& 8\,(1+\hat\tau') + \ldots \,, {\nonumber}\\ u_1^{(0)} &=& 8\varepsilon_c\, (\bar\Lambda'-\bar\Lambda) + \ldots \,, {\nonumber}\\* u_1^{(1)} &=& 2 + \ldots \,. \end{aligned}$$ For the decay rate into $D_2^*$ the first two terms in the $w-1$ expansion are $$\begin{aligned} \label{stuD2} s_2^{(0)} &=& 4\, (1-r_2)^2\, \Big[1 + 2\varepsilon_b\,\hat\eta_b + 2\varepsilon_c\,(\hat\eta_{\rm ke}-2\hat\eta_1+\hat\eta_3) \Big] + \ldots \,,{\nonumber}\\* s_2^{(1)} &=& 4\,(1-r_2)^2\, (1+2\hat\tau') + \ldots \,, {\nonumber}\\ t_2^{(0)} &=& 4 + 8\varepsilon_b\,\hat\eta_b + 8\varepsilon_c\, (\hat\eta_{\rm ke}-2\hat\eta_1+\hat\eta_3) + \ldots \,, {\nonumber}\\* t_2^{(1)} &=& 2(3+4\hat\tau') + \ldots \,, {\nonumber}\\ u_2^{(0)} &=& 2 + \ldots \,. \end{aligned}$$ In Eqs. (\[stuD1\]) and (\[stuD2\]) the functions $\tau$, $\tau'={\rm d}\tau/{\rm d}w$, and $\eta_i$ are all evaluated at $w=1$, and the functions with a hat are normalized to $\tau(1)$ (e.g., $\hat\eta_i=\eta_i/\tau(1)$, $\hat\tau'=\tau'/\tau(1)$, etc.). The ellipses denote higher order terms in the $\Lambda_{\rm QCD}/m_Q$ expansion. The $u_i^{(n)}$ terms are suppressed by $\sqrt{w^2-1}$ compared to $s_i^{(n)}$ and $t_i^{(n)}$, therefore we displayed the $u$’s to one lower order than the $s$ and $t$ coefficients. (Note that $u_1^{(0)}$ also starts out at order $\Lambda_{\rm QCD}/m_Q$ as a consequence of the vanishing of $f_{V_1}(1)$ in the $m_Q\to\infty$ limit, as it was shown for $s_1^{(1)}$ after Eq. (\[schematic\]).) The order $\Lambda_{\rm QCD}/m_Q$ terms proportional to $\bar\Lambda'-\bar\Lambda$ are very significant for the $D_1$ decay rate. The decay rate into $D_2^*$ does not receive a similarly large enhancement from order $\Lambda_{\rm QCD}/m_Q$ terms proportional to $\bar\Lambda'-\bar\Lambda$. The coefficients $s_2^{(n)}$ and $t_2^{(n)}$ are independent of $\bar\Lambda'$ and $\bar\Lambda$ to the order displayed in Eq. (\[stuD2\]). The values of $s_1^{(0)}$ and $t_1^{(0)}$ are known to order $\Lambda_{\rm QCD}^2/m_Q^2$, and $s_1^{(1)}$ and $u_1^{(0)}$ are known to order $\Lambda_{\rm QCD}/m_Q$. At order $\Lambda_{\rm QCD}/m_Q$, the only unknowns in $t_1^{(1)}$, $s_2^{(0)}$, $t_2^{(0)}$ are the $\hat\eta_i$ functions that parameterize corrections to the HQET Lagrangian. The remaining coefficients in Eqs. (\[stuD1\]) and (\[stuD2\]) (i.e., $s_1^{(2)}$, $t_1^{(2)}$, $u_1^{(1)}$, $s_2^{(1)}$, $t_2^{(1)}$, $u_2^{(0)}$) are known in the infinite mass limit in terms of $\hat\tau'(1)$, the slope of the $m_Q\to\infty$ Isgur-Wise function at zero recoil. At order $\Lambda_{\rm QCD}/m_Q$, these six coefficients depend on the unknown subleading $\tau_i$ and $\eta_i$ functions. The values of $\tau'$, $\eta_i^{(Q)}$ and $\tau_{1,2}$ that occur in Eqs. (\[stuD1\]) and (\[stuD2\]) are not known ($\tau_i$ only appears in the terms replaced by ellipses). $\eta_{1,2,3}^{(Q)}$, which parameterize time ordered products of the chromomagnetic operator, are expected to be small (compared to $\Lambda_{\rm QCD}$), and we neglect them hereafter. This is supported by the very small $D_2^*-D_1$ mass splitting, and the fact that model calculations indicate that the analogous functions parameterizing time ordered products of the chromomagnetic operator for $B\to D^{(*)}\,e\,\bar\nu_e$ decays are small [@qcdsr]. On the other hand, there is no reason to expect $\tau_{1,2}$ and $\eta_{\rm ke}^{(Q)}$ to be much smaller than about $500\,$MeV. Note that the large value for $\lambda_1'$ is probably a consequence of the $D_1$ and $D_2^*$ being $P$-waves in the quark model, and does not necessarily imply that $O_{\rm kin}^{(Q)}$ significantly distorts the overlap of wave functions that yield $\eta_{\rm ke}^{(Q)}$. Even though $\varepsilon_c(\bar\Lambda'-\bar\Lambda)\simeq0.14$ is quite small, the order $\Lambda_{\rm QCD}/m_Q$ correction to $t_1^{(1)}$ proportional to $\varepsilon_c(\bar\Lambda'-\bar\Lambda)$ is as large as the leading $m_Q\to\infty$ contribution. This occurs because it has an anomalously large coefficient and does not necessarily mean that the $\Lambda_{\rm QCD}/m_Q$ expansion has broken down. For example, the part of the $\Lambda_{\rm QCD}^2/m_c^2$ corrections that involve $\bar\Lambda'$, $\bar\Lambda$, and $\tau'(1)$ affect $s_1^{(1)}$ by $(21+10\hat\tau')\%$, and $t_1^{(1)}$ by $(44+15\hat\tau')\%$ (using $\bar\Lambda=0.4\,$GeV [@gklw]). These corrections follow from Eq. (\[expf\]), but they are neglected in Eq. (\[stuD1\]) (i.e., approximation A), because there are other order $\Lambda_{\rm QCD}^2/m_Q^2$ effects we have not calculated. As the kinetic energy operator does not violate spin symmetry, effects of $\eta_{\rm ke}^{(Q)}$ can be absorbed into $\tau$ by the replacement of $\tau$ by $\widetilde\tau=\tau+\varepsilon_c\,\eta_{\rm ke}^{(c)}+\varepsilon_b\,\eta_{\rm ke}^{(b)}$. This replacement introduces an error of order $\Lambda_{\rm QCD}^2/m_Q^2$, in $t_1^{(1)}$, etc. But due to the presence of large $\Lambda_{\rm QCD}/m_Q$ corrections, the resulting $\Lambda_{\rm QCD}^2/m_Q^2$ error is also sizable, and is expected to be more like an order $\Lambda_{\rm QCD}/m_Q$ correction. Hereafter, unless explicitly stated otherwise, it is understood that the replacement $\tau\to\widetilde\tau$ is made. But we shall examine the sensitivity of our results to $\eta_{\rm ke}$ (assuming it has the same shape as $\tau$). In approximation A we treat $w-1$ as order $\Lambda_{\rm QCD}/m_Q$ [@llsw], and keep terms up to order $(\Lambda_{\rm QCD}/m_Q)^{2-n}$ in $s_1^{(n)}$ and $t_1^{(n)}$ ($n=0,1,2$) in Eq. (\[stuD1\]), and up to order $(\Lambda_{\rm QCD}/m_Q)^{1-n}$ in $s_2^{(n)}$ and $t_2^{(n)}$ ($n=0,1$) in Eq. (\[stuD2\]). Since the $u_i^{(n)}$ are suppressed by $\sqrt{w^2-1}$ compared to $s_i^{(n)}$ and $t_i^{(n)}$, we keep $u_i^{(n)}$ to one lower order than the $s$ and $t$ coefficients, i.e., to order $(\Lambda_{\rm QCD}/m_Q)^{1-n}$ ($n=0,1$) for $B\to D_1$ decay and order $(\Lambda_{\rm QCD}/m_Q)^{n}$ ($n=0$) for $B\to D_2^*$ decay. The terms included in approximation A are precisely the ones explicitly shown in Eqs. (\[stuD1\]) and (\[stuD2\]). This power counting has the advantage that the unknown functions, $\tau_1$ and $\tau_2$, do not enter the predictions.[^5] Neglecting higher order terms in the $w-1$ expansion in approximation A gives rise to a sizable error for the $B\to D_1\,e\,\bar\nu_e$ decay[^6]. The order $(w-1)^3$ term is important for the decay into helicity zero $D_1$ in the $m_Q\to\infty$ limit, since the helicity zero rate (which, as we shall see, dominates over the helicity one rate) starts out at order $(w-1)^2$ as shown in Eq. (\[schematic\]). In approximation B we do not expand the decay rates in powers of $w-1$. We keep the $\Lambda_{\rm QCD}/m_Q$ corrections to the form factors that involve $\bar\Lambda'$ and $\bar\Lambda$ and examine the sensitivity of our results to the corrections involving $\tau_1$ and $\tau_2$ (assuming that they have the same shape as $\tau$, which is not a strong assumption). This approximation retains some order $\Lambda_{\rm QCD}^2/m_Q^2$ terms away from zero recoil in the differential decay rates. Furthermore, a linear form for the Isgur-Wise function is assumed, $\tau(w)=\tau(1)\,[1+\hat\tau'(w-1)]$. The uncertainty in the $\Lambda_{\rm QCD}/m_Q$ corrections is parameterized by the functions $\tau_{1,2}(w)$. A different choice of $\tau_{1,2}(w)$ changes what is retained by terms involving $\bar\Lambda/m_Q$ and $\bar\Lambda'/m_Q$. In an approximation, which we shall refer to as B$_1$, we set $\tau_1=\tau_2=0$ in Eqs. (\[expf\]) and (\[expk\]). (This is identical to saturating the first two relations in Eq. (\[const2\]) by $\tau_{1,2}^{(b)}$, i.e., setting $\tau_1^{(b)}=\bar\Lambda\,\tau$ and $\tau_2^{(b)}=-\bar\Lambda'\,\tau$.) An equally reasonable approximation, which we refer to as B$_2$, is given by setting $\tau_1=\bar\Lambda\,\tau$ and $\tau_2=-\bar\Lambda'\,\tau$ in Eqs. (\[expf\]) and (\[expk\]). (This is identical to setting $\tau_{1,2}^{(b)}=0$.) If the first two relations in Eq. (\[const2\]) are taken as hints to the signs of $\tau_1$ and $\tau_2$, then the difference between approximations B$_1$ and B$_2$ gives a rough estimate of the uncertainty related to the unknown $\Lambda_{\rm QCD}/m_Q$ corrections. When our predictions are sensitive to $\tau_1$ and $\tau_2$, we shall vary these in a range larger than that spanned by approximations B$_1$ and B$_2$. Note that the infinite mass limits of B$_1$ and B$_2$ coincide. Predictions of approximation A are within the spread of the approximation B results, except for those that depend on the helicity zero $D_1$ rate. In that case, including the order $(w-1)^3$ term in the infinite mass limit alone, $s_1^{(3)}=8\,(1+r_1)^2\,(1+2\hat\tau')$, would bring the approximation A results close to approximation B. Eqs. (\[stuD1\]) and (\[stuD2\]) show that the heavy quark expansion for $B$ decays into excited charmed mesons is controlled by the excitation energies of the hadrons, $\bar\Lambda'$ and $\bar\Lambda$. For highly excited mesons that have $\bar\Lambda'$ comparable to $m_c$, the $1/m_Q$ expansion is not useful. For the $s_\ell^{\pi_\ell}=\frac32^+$ doublet $\varepsilon_c\,\bar\Lambda'\sim0.3$. However, near zero recoil only $\varepsilon_c\,(\bar\Lambda'-\bar\Lambda)\sim0.14$ occurs at order $\Lambda_{\rm QCD}/m_Q$. The expressions for the decay rates in terms of form factors in Eq. (\[rate1\]) imply that one form factor dominates each decay rate near zero recoil, independent of the helicity of the $D_1$ or $D_2^*$ ($f_{V_1}$ for $D_1$ and $k_{A_1}$ for $D_2^*$). Thus, to all orders in the $\Lambda_{\rm QCD}/m_Q$ expansion, $s_1^{(0)}/t_1^{(0)}=(1-r_1)^2$, and $s_2^{(0)}/t_2^{(0)}=(1-r_2)^2$. This implies that for $B\to D_1$ decay $\lim_{w\to1}\Big[({\rm d}\Gamma_{D_1}^{(\lambda=0)}/{\rm d}w)\Big/({\rm d}\Gamma_{D_1}^{(|\lambda|=1)}/{\rm d}w)\Big]=1/2$, and for $B\to D_2^*$ decay $\lim_{w\to1}\Big[({\rm d}\Gamma_{D_2^*}^{(\lambda=0)}/{\rm d}w)\Big/({\rm d}\Gamma_{D_2^*}^{(|\lambda|=1)}/{\rm d}w)\Big]=2/3$. Note that the first of these ratios would vanish if the rates were calculated in the $m_Q\to\infty$ limit. In that case $f_{V_1}(1)=0$, so the ratio of helicity zero and helicity one $B\to D_1$ rates is determined by the other form factors at zero recoil. Predictions ----------- The relationship between $s_1^{(0)}$ and $s_1^{(1)}$ implies a model independent prediction for the slope parameter of semileptonic $B$ decay into helicity zero $D_1$. This holds independent of the subleading Isgur-Wise functions that arise at order $\Lambda_{\rm QCD}/m_Q$. The semileptonic decay rate to a helicity zero $D_1$ meson is $$\label{mir1} {{\rm d}\Gamma_{D_1}^{(\lambda=0)} \over {\rm d}w} = \frac{128}3\, \Gamma_0\, r_1^3\, (1-r_1)^2\, \sqrt{w^2-1}\, \tau^2(1)\, \varepsilon_c^2\, (\bar\Lambda'-\bar\Lambda)^2\, \Big[ 1 - \rho_{D_1}^2\, (w-1) + \ldots \Big] \,,$$ where the slope parameter $\rho_{D_1}^2$ for helicity zero $D_1$ has the value $$\rho_{D_1}^2 = - {1+r_1\over1-r_1}\, {2m_c\over\bar\Lambda'-\bar\Lambda} + {\cal O}(1) \,.$$ Since the decay rate at zero recoil is suppressed, $\rho_{D_1}^2$ is of order $m_Q/\Lambda_{\rm QCD}$. Note that this slope parameter is negative. Recently the ALEPH [@ALEPH] and CLEO [@CLEO] Collaborations measured, with some assumptions, the $B\to D_1\,e\,\bar\nu_e$ branching ratio. The average of their results is $$\label{data} {\cal B}(B\to D_1\,e\,\bar\nu_e) = (6.0\pm1.1) \times 10^{-3} \,.$$ The $B\to D_2^*\,e\,\bar\nu_e$ branching ratio has not yet been measured; CLEO set the limit ${\cal B}(B\to~D_2^*\,e\,\bar\nu_e)<1\%$ [@CLEO], while ALEPH found ${\cal B}(B\to D_2^*\,e\,\bar\nu_e)<0.2\%$ [@ALEPH]. Predictions for various quantities of experimental interest are made in Table \[tab:sec2res\] using $\bar\Lambda'-\bar\Lambda=0.39\,$GeV, $\bar\Lambda=0.4\,$GeV, $\tau_B=1.6\,$ps, $|V_{cb}|=0.04$, $m_c=1.4\,$GeV, $m_b=4.8\,$GeV. Keeping $m_b-m_c$ fixed and varying $m_c$ by $\pm0.1\,$GeV only affects our results at the few percent level. These predictions depend on the shape of the Isgur-Wise function. In our approximations this enters through the slope parameter, $\hat\tau'=\tau'(1)/\tau(1)$, which is expected to be of order $-1$. We shall quote results for the “central value" $\hat\tau'=-1.5$, motivated by model predictions [@ISGW; @Cola; @VeOl; @More], and discuss the sensitivity to this assumption. For $B\to D_1\,e\,\bar\nu_e$ decay we use $r_1=0.459$ and $1<w<1.319$, whereas for $B\to D_2^*\,e\,\bar\nu_e$ decay $r_2=0.466$ and $1<w<1.306$. -------------------- ------------------------------------- ----------------------------------------------- --------------------------------------------------- --------------------------------------------------------- Approximation $R=\Gamma_{D_2^*}\Big/\Gamma_{D_1}$ $\Gamma_{D_1}^{(\lambda=0)}\Big/\Gamma_{D_1}$ $\Gamma_{D_2^*}^{(\lambda=0)}\Big/\Gamma_{D_2^*}$ $\tau(1)\, \bigg[\displaystyle {6.0\times10^{-3} \over {\cal B}(B\to D_1\,e\,\bar\nu_e)} \bigg]^{1/2}$ \[6pt\] A$_\infty$ $0.93$ $0.88$ $0.64$ $0.92$ B$_\infty$ $1.65$ $0.80$ $0.66$ $1.24$ A $0.40$ $0.81$ $0.64$ $0.60$ B$_1$ $0.52$ $0.72$ $0.63$ $0.71$ B$_2$ $0.67$ $0.77$ $0.64$ $0.75$ -------------------- ------------------------------------- ----------------------------------------------- --------------------------------------------------- --------------------------------------------------------- The order $\Lambda_{\rm QCD}/m_Q$ corrections are important for predicting $$R \equiv {{\cal B}(B\to D_2^*\,e\,\bar\nu_e) \over {\cal B}(B\to D_1\,e\,\bar\nu_e) } \,.$$ In the $m_Q\to\infty$ limit $R\simeq1.65$ for $\hat\tau'=-1.5$ (this is the B$_\infty$ result in Table \[tab:sec2res\]). The sizable difference between approximations A and B is mainly due to the order $(w-1)^3$ contribution to the helicity zero $D_1$ rate. For $\hat\tau'=-1.5$ this term by itself would shift the approximation A result for $R$ from 0.40 to 0.49 and the A$_\infty$ prediction from 0.93 to 1.65. The $\Lambda_{\rm QCD}/m_Q$ correction to the form factors yield a large suppression of $R$ as shown in Table \[tab:sec2res\] and Fig. \[fig:Rtau\]a. Fig. \[fig:Rtau\]a also shows that $R$ is fairly insensitive to $\hat\tau'$. The difference of the B$_1$ and B$_2$ results in Table \[tab:sec2res\] and Fig. \[fig:Rtau\]a shows that $R$ is sensitive to the unknown $\Lambda_{\rm QCD}/m_Q$ corrections, $\tau_1$ and $\tau_2$. In Fig. \[fig:Rtau\]b we plot $R$ in approximation B as a function of $\hat\tau_1$ setting $\hat\tau_2=0$ (solid curve), and as a function of $\hat\tau_2$ setting $\hat\tau_1=0$ (dashed curve). Fig. \[fig:Rtau\]b shows that $R$ is fairly insensitive to $\tau_2$, whereas it depends sensitively on $\tau_1$. In the range $-0.75\,{\rm GeV}<\hat\tau_1<0.75\,{\rm GeV}$, $R$ goes over $0.27<R<1.03$. This suppression of $R$ compared to the infinite mass limit is supported by the experimental data. (It is possible that part of the reason for the strong ALEPH bound ${\cal B}(B\to D_2^*\,e\,\bar\nu_e\,X)\times{\cal B}(D_2^*\to D^{(*)}\pi)\lesssim(1.5-2.0)\times10^{-3}$ [@ALEPH] is a suppression of ${\cal B}(D_2^*\to D^{(*)}\pi)$ compared to ${\cal B}(D_1\to D^*\pi)$.) =8truecm =8truecm The prediction for the fraction of helicity zero $D_1$’s in semileptonic $B\to D_1$ decay, $\Gamma_{D_1}^{(\lambda=0)}/\Gamma_{D_1}$, is surprisingly stable in the different approximations (see Table \[tab:sec2res\]). The weak dependence of this ratio on $\hat\tau'$ is well described in approximation B for $|1.5+\hat\tau'|<1$ by adding $0.05(1.5+\hat\tau')$. The dependence on $\tau_1$ is at the 0.01 level, while the $\tau_2$-dependence is $-0.07\,\hat\tau_2/{\rm GeV}$. This is why the B$_2$ result for this quantity is 0.05 larger than the B$_1$ prediction. A linear dependence of $({\rm d}\Gamma_{D_1}^{(\lambda=0)}/{\rm d}w)\Big/({\rm d}\Gamma_{D_1}/{\rm d}w)$ on $w$ between $\lim_{w\to1}\Big[({\rm d}\Gamma_{D_1}^{(\lambda=0)}/{\rm d}w)\Big/({\rm d}\Gamma_{D_1}/{\rm d}w)\Big]=1/3$ and $\Big[({\rm d}\Gamma_{D_1}^{(\lambda=0)}/{\rm d}w)\Big/({\rm d}\Gamma_{D_1}/{\rm d}w)\Big]=1$ at $w=w_{\rm max}$ is consistent with our result. A similar prediction exists for the fraction of helicity zero $D_2^*$’s in semileptonic $B\to D_2^*$ decay. As can be seen from Table \[tab:sec2res\], it is again quite stable. The dependence on $\hat\tau'$ in approximation B is given by adding $0.04(1.5+\hat\tau')$. However, $\Gamma_{D_2^*}^{(\lambda=0)}/\Gamma_{D_2^*}$ is sensitive to both $\tau_1$ and $\tau_2$ at the $(10-20)\%$ level, and the small difference between the B$_1$ and B$_2$ predictions for this quantity in Table \[tab:sec2res\] is due to an accidental cancellation. The prediction for the $w$ dependence of $({\rm d}\Gamma_{D_2^*}^{(\lambda=0)}/{\rm d}w)\Big/({\rm d}\Gamma_{D_2^*}/{\rm d}w)$ between $\lim_{w\to1}\Big[({\rm d}\Gamma_{D_2^*}^{(\lambda=0)}/{\rm d}w)\Big/({\rm d }\Gamma_{D_2^*}/{\rm d}w)\Big]=2/5$ and $\Big[({\rm d}\Gamma_{D_2^*}^{(\lambda=0)}/{\rm d}w)\Big/({\rm d}\Gamma_{D_2^*}/{\rm d}w)\Big]=1$ at $w=w_{\rm max}$ in this case is not linear. The predictions considered so far do not depend on the value of $\tau(1)$, but $\tau(1)$ affects some results that we discuss later. $\tau(1)$ can be determined from the measured $B\to D_1\,e\,\bar\nu_e$ branching ratio using the expressions in Eq. (\[rt\_expn1\]) and (\[stuD1\]). Using approximation B$_1$ and $\hat\tau'=-1.5$, we obtain $$\label{tau1} \tau(1)\, \bigg[ {6.0\times10^{-3} \over {\cal B}(B\to D_1\,e\,\bar\nu_e)} \bigg]^{1/2} = 0.71 \,.$$ The extracted value of $\tau(1)$ is plotted in Fig. \[fig:tau1\]a in approximations B$_\infty$, B$_1$, and B$_2$ as functions of $\hat\tau'$. The suppression of $\tau(1)$ compared to the infinite mass limit indicates that the order $\Lambda_{\rm QCD}/m_Q$ corrections enhance the semileptonic $B\to D_1$ width by about a factor of three. In approximation B the value of $\tau(1)$ changes by less than 0.01 as $\tau_2$ is varied in the range $-0.75\,{\rm GeV}<\hat\tau_2<0.75\,{\rm GeV}$, but $\tau(1)$ is sensitive to $\tau_1$ at the 15% level. In Fig. \[fig:tau1\]b we plot $\tau(1)$ as a function of $\hat\tau_1$ for $\hat\tau'=-1$ (dashed curve), $\hat\tau'=-1.5$ (solid curve), and $\hat\tau'=-2$ (dash-dotted curve). For $\tau_1>0$ (such as approximation B$_2$) $\tau(1)$ is enhanced compared to the B$_1$ value of 0.71. =8truecm =8truecm The value of $\tau(1)$ in approximation B is larger than that in approximation A. Most of the difference arises from the inclusion of the order $(w-1)^3$ term, $s_1^{(3)}$, which reduces the theoretical expression for the helicity zero $B\to D_1\,e\,\bar\nu_e$ rate (for $\hat\tau'<-0.5$), resulting in an increase in the value of $\tau(1)$ needed to accommodate the measured rate. For $\hat\tau'=-1.5$ this term by itself would shift the approximation A result from 0.60 to 0.66, and the A$_\infty$ prediction from 0.92 to 1.22. The ISGW nonrelativistic constituent quark model predicts $\tau(1)=0.54$, in rough agreement with Eq. (\[tau1\]) [@ISGW; @IWsr]. (For some other quark model predictions, see, e.g., Ref. [@VeOl; @More]. QCD sum rules can also be used to estimate $\tau$, see, e.g., Ref [@Cola].) The ALEPH and CLEO analyses that yield Eq. (\[data\]) assume that $B\to D_1\,e\,\bar\nu_e\,X$ is dominated by $B\to D_1\,e\,\bar\nu_e$, and that $D_1$ decays only into $D^*\,\pi$. If the first assumption turns out to be false then $\tau(1)$ will decrease, if the second assumption is false then $\tau(1)$ will increase compared to Eq. (\[tau1\]). The predictions discussed above would change if we had not absorbed into $\tau$ the time ordered product involving the kinetic energy operator. As discussed earlier (in the paragraph preceding the description of approximation A), the replacement of $\tau$ by $\widetilde\tau=\tau+\varepsilon_c\,\eta_{\rm ke}^{(c)}+\varepsilon_b\,\eta_{\rm ke}^{(b)}$ introduces an error, which is formally of order $\Lambda_{\rm QCD}^2/m_Q^2$. Absorbing $\eta_{\rm ke}$ into $\tau$ almost fully eliminates the $\eta_{\rm ke}$ dependence of the $D_2^*$ rate. For the $D_1$ rate, however, absorbing $\eta_{\rm ke}$ into $\tau$ generates at order $\Lambda_{\rm QCD}^2/m_Q^2$ a formally suppressed but numerically sizable $\eta_{\rm ke}$ dependence. This $\eta_{\rm ke}$ dependence is more like a typical $\Lambda_{\rm QCD}/m_Q$ correction, since the $\Lambda_{\rm QCD}/m_Q$ current corrections are as important as the infinite mass limit for the $D_1$ rate. Keeping $\hat\eta_{\rm ke}^{(Q)}=\eta_{\rm ke}^{(Q)}/\tau$ explicit in the results, the total $B\to D_1$ semileptonic rate in units of $\Gamma_0\,\tau^2(1)$ is $0.033\,(1+1.1\,\varepsilon_c\,\hat\eta_{\rm ke}^{(c)}+\ldots)$, while the $B\to D_2^*$ rate is $0.017\,(1+2.0\,\varepsilon_c\,\hat\eta_{\rm ke}^{(c)}+\ldots)$. From these expressions it is evident that, for $-0.75\,{\rm GeV}<\hat\eta_{\rm ke}<0.75\,{\rm GeV}$, $\tau(1)$ changes only by $\pm15\%$, while $R$ has a larger variation. In the future this uncertainty will be reduced if differential spectra can also be measured besides total rates in $B\to D_1,D_2^*$ decays. Note that $\eta_{\rm ke}$ does not enter into predictions for the $B\to D_1\,e\,\bar\nu_e$ decay rate near zero recoil. Order $\alpha_s$ corrections to the results of this section can be calculated in a straightforward way, using well-known methods. Details of this calculation are given in Appendix A. The order $\alpha_s$ corrections to the results shown in Table \[tab:sec2res\] are given in Table \[tab:alphas\]. These are smaller than the uncertainty in our results from higher order terms in the $\Lambda_{\rm QCD}/m_Q$ expansion that have been neglected. The corrections are most significant for $R=\Gamma_{D_2^*}\Big/\Gamma_{D_1}$ and $\tau(1)$ in approximation B; the central values of these quantities are reduced by about $9\%$ and $4\%$, respectively. Some of these $\alpha_s$ corrections depend sensitively on $\hat\tau'$, but they remain small for $0>\hat\tau'>-2$. For the remainder of this paper, we neglect the small $\alpha_s$ corrections. -------------------- --------------------------------------------------- --------------------------------------------------------------- ------------------------------------------------------------------- --------------------------------------------------------------- Approximation $\delta\Big(\Gamma_{D_2^*}\Big/\Gamma_{D_1}\Big)$ $\delta\Big(\Gamma_{D_1}^{(\lambda=0)}\Big/\Gamma_{D_1}\Big)$ $\delta\Big(\Gamma_{D_2^*}^{(\lambda=0)}\Big/\Gamma_{D_2^*}\Big)$ $\delta\tau(1)\, \bigg[\displaystyle {6.0\times10^{-3} \over {\cal B}(B\to D_1\,e\,\bar\nu_e)} \bigg]^{1/2}$ \[6pt\] A$_\infty$ $-0.68$ $0.10$ $0.02$ $-0.26$ B$_\infty$ $-1.63$ $0.19$ $-0.003$ $-0.32$ A $-0.22$ $0.04$ $0.05$ $-0.24$ B$_1$ $-0.55$ $0.06$ $-0.02$ $-0.32$ B$_2$ $-0.68$ $0.07$ $-0.05$ $-0.33$ -------------------- --------------------------------------------------- --------------------------------------------------------------- ------------------------------------------------------------------- --------------------------------------------------------------- Our predictions for the single differential $B\to(D_1,D_2^*)\,e\,\bar\nu_e$ spectra follow from Eqs. (\[rt\_expn1\]), (\[stuD1\]), and (\[stuD2\]). ${\rm d}\Gamma/{\rm d}w$ is given by integrating Eqs. (\[rt\_expn1\]) over ${\rm d}\!\cos\theta$. This amounts to the replacements $\sin^2\theta\to4/3$, $(1+\cos^2\theta)\to8/3$, and $\cos\theta\to0$. Thus ${\rm d}\Gamma/{\rm d}w$ is trivial to obtain using either approximations A or B. The electron energy spectra are obtained by expressing $\cos\theta$ in terms of $y$ (where $y=2E_e/m_B$ is the rescaled electron energy) using Eq. (\[yct\]), and integrating $w$ over $[(1-y)^2+r^2]/[(2r(1-y)]<w<(1+r^2)/(2r)$. They depend on the coefficients $u_i^{(n)}$ which did not enter our results so far. =8truecm =8truecm =8truecm =8truecm In Fig. \[fig:D1spect\] the electron spectrum for $B\to D_1\,e\,\bar\nu_e$ is plotted in units of $\Gamma_0\,\tau^2(1)$. Figs. \[fig:D1spect\]a and \[fig:D1spect\]b are the spectra for helicity zero and helicity one $D_1$, respectively. In these plots $\hat\tau'=-1.5$. The dotted curve shows the $m_Q\to\infty$ limit (B$_\infty$), the solid curve is approximation B$_1$, the dashed curve is B$_2$. Note that the kinematic range for $y$ is $0<y<1-r^2$. Near $y=0$ and $y=1-r^2$ the spectrum is dominated by contributions from $w$ near $w_{\rm max}$. In this case, we expect sizable uncertainties in our results, for example, from unknown terms that occur in the $u_i^{(n)}$ terms in Eq. (\[stuD1\]) at a lower order than in the $s$ and $t$ coefficients. Fig. \[fig:D1spect\] shows the large enhancement of the $D_1$ rate due to order $\Lambda_{\rm QCD}/m_Q$ corrections, and that the difference between approximations B$_1$ and B$_2$ is small compared to this enhancement. In Figs. \[fig:D2spect\]a and \[fig:D2spect\]b we plot the electron spectrum for $B\to D_2^*\,e\,\bar\nu_e$ for helicity zero and helicity one $D_2^*$, respectively. In this case the $\Lambda_{\rm QCD}/m_Q$ corrections are less important. $B\to D_0^*\,\lowercase{e}\,\bar\nu_{\lowercase{e}}$ and $B\to D_1^*\,\lowercase{e}\,\bar\nu_{\lowercase{e}}$ decays ==================================================================================================================== The other low lying states above the $D^{(*)}$ ground states occur in a doublet with $s_l^{\pi_l}=\frac12^+$. These states are expected to be broad since they can decay into $D^{(*)}\,\pi$ in an $S$-wave, unlike the $D_1$ and $D_2^*$ which can only decay in a $D$-wave. (An $S$-wave decay amplitude for the $D_1$ is forbidden by heavy quark spin symmetry [@IWprl].) This section repeats the analysis of the previous section for these states. Since the notation, methods, and results are similar to those used in Sec. II, the discussion here will be briefer. The matrix elements of the vector and axial currents between $B$ mesons and $D_0^*$ or $D_1^*$ mesons can be parameterized by $$\begin{aligned} \label{formf2} \langle D_0^*(v')|\, V^\mu\, |B(v)\rangle &=& 0, {\nonumber}\\* {\langle D_0^*(v')|\, A^\mu\, |B(v)\rangle \over\sqrt{m_{D_0^*}\,m_B}} &=& g_+\, (v^\mu+v'^\mu) + g_-\, (v^\mu-v'^\mu) \,, {\nonumber}\\* {\langle D_1^*(v',\epsilon)|\, V^\mu\, |B(v)\rangle \over\sqrt{m_{D_1^*}\,m_B}} &=& g_{V_1}\, \epsilon^{*\, \mu} + (g_{V_2} v^\mu + g_{V_3} v'^\mu)\, (\epsilon^*\cdot v) \,, {\nonumber}\\* {\langle D_1^*(v',\epsilon)|\, A^\mu\, |B(v)\rangle \over\sqrt{m_{D_1^*}\,m_B}} &=& i\, g_A\, \varepsilon^{\mu\alpha\beta\gamma}\, \epsilon^*_\alpha v_\beta\, v'_\gamma \,, \end{aligned}$$ where $g_i$ are functions of $w$. At zero recoil the matrix elements are determined by $g_+(1)$ and $g_{V_1}(1)$. In terms of these form factors the double differential decay rates for $B\to D_0^*\,e\,\bar\nu_e$ and $B\to D_1^*\,e\,\bar\nu_e$ decays are $$\begin{aligned} \label{rate2} {{\rm d}^2\Gamma_{D_0^*}\over {\rm d}w\,{\rm d}\!\cos\theta} &=& 3\Gamma_0\, r_0^{*3}\, (w^2-1)^{3/2}\, \sin^2\theta\, \Big[ (1+r_0^*)\,g_+ - (1-r_0^*)\, g_- \Big]^2 \,,\\* {{\rm d}^2\Gamma_{D_1^*}\over {\rm d}w\,{\rm d}\!\cos\theta} &=& 3\Gamma_0\, r_1^{*3}\, \sqrt{w^2-1}\, \bigg\{ \sin^2\theta\, \Big[ (w-r_1^*) g_{V_1}+(w^2-1) (g_{V_3}+r_1^* g_{V_2}) \Big]^2 \nonumber\\* && + (1-2r_1^*w+r_1^{*2})\, \Big[(1+\cos^2\theta)\, [g_{V_1}^2 + (w^2-1) g_A^2] - 4\cos\theta\, \sqrt{w^2-1}\, g_{V_1}\, g_A \Big] \bigg\} \,.\nonumber\end{aligned}$$ where $\Gamma_0 = {G_F^2\,|V_{cb}|^2\,m_B^5 /(192\pi^3)}$, $r_0^*=m_{D_0^*}/m_B$ and $r_1^*=m_{D_1^*}/m_B$. We follow the previous section to obtain expressions for the form factors $g_i$ in terms of Isgur-Wise functions to order $\Lambda_{\rm QCD}/m_Q$. The fields $P_v$ and $P_v^{*\mu}$ that destroy members of the $s_l^{\pi_l}=\frac12^+$ doublet with four-velocity $v$ are in the $4\times4$ matrix $$K_v = \frac{1+\vslash}2\, \Big[ P_v^{*\mu}\, \gamma_5\gamma_\mu + P_{v}\, \Big] \,.$$ This matrix $K$ satisfies $\vslash K_v=K_v=K_v\vslash$. In the infinite mass limit matrix elements of the leading order current operator are [@IWsr] $$\bar h^{(c)}_{v'}\, \Gamma\, h^{(b)}_v = \zeta(w)\, {\rm Tr}\, \Big\{ \bar K_{v'}\, \Gamma\, H_v \Big\}$$ Here $\zeta(w)$ is the leading order Isgur-Wise function ($\zeta$ is twice the function $\tau_{1/2}$ of Ref. [@IWsr]). Since the $(D_0^*,D_1^*)$ states are in a different spin multiplet than the ground state, $g_+(1)=g_{V_1}(1)=0$ in the infinite mass limit, independent of $\zeta(1)$. The order $\Lambda_{\rm QCD}/m_Q$ corrections to the current can be parameterized as $$\begin{aligned} \label{1/mc21} \bar h^{(c)}_{v'}\, i\overleftarrow D_{\!\lambda}\, \Gamma\, h^{(b)}_v &=& {\rm Tr}\, \Big\{ {\cal S}^{(c)}_{\lambda}\, \bar K_{v'}\, \Gamma\, H_v \Big\} \,, \nonumber\\* \bar h^{(c)}_{v'}\, \Gamma\, i\overrightarrow D_{\!\lambda}\, h^{(b)}_v &=& {\rm Tr}\, \Big\{ {\cal S}^{(b)}_{\lambda}\, \bar K_{v'}\, \Gamma\, H_v \Big\} \,.\end{aligned}$$ This is the analogue of Eq. (\[1/mc1\]), except that in the present case $$\label{1/mc22} {\cal S}^{(Q)}_{\lambda} = \zeta_1^{(Q)} v_\lambda + \zeta_2^{(Q)} v'_\lambda + \zeta_3^{(Q)} \gamma_\lambda \,.$$ The functions $\zeta_i^{(Q)}(w)$ have mass dimension one. The heavy quark equation of motion yield $$\begin{aligned} \label{constbr1} w\, \zeta_1^{(c)} + \zeta_2^{(c)} + \zeta_3^{(c)} &=& 0 \,,{\nonumber}\\* \zeta_1^{(b)} + w\, \zeta_2^{(b)} - \zeta_3^{(b)} &=& 0 \,.\end{aligned}$$ Eq. (\[momcons\]) implies ${\cal S}^{(c)}_\lambda+{\cal S}^{(b)}_\lambda = (\bar\Lambda v_\lambda-\bar\Lambda^*v'_\lambda)\, \zeta$, which gives three more relations $$\begin{aligned} \label{constbr2} \zeta_1^{(c)} + \zeta_1^{(b)} &=& \bar\Lambda\, \zeta \,, \nonumber\\* \zeta_2^{(c)} + \zeta_2^{(b)} &=& -\bar\Lambda^*\, \zeta \,, \nonumber\\ \zeta_3^{(c)} + \zeta_3^{(b)} &=& 0 \,.\end{aligned}$$ These relations express the $\zeta_j^{(b)}$’s in terms of the $\zeta_j^{(c)}$’s. Combining Eqs. (\[constbr1\]) with (\[constbr2\]) yields $$\begin{aligned} \label{constbr3} \zeta_2^{(c)} &=& -{w\bar\Lambda^*-\bar\Lambda\over w+1}\, \zeta - \zeta_1^{(c)} \,, \nonumber\\* \zeta_3^{(c)} &=& {w\bar\Lambda^*-\bar\Lambda\over w+1}\, \zeta - (w-1)\, \zeta_1^{(c)} \,.\end{aligned}$$ At zero recoil, only $\zeta_3^{(Q)}$ can give a non-vanishing contribution to the matrix elements of the weak currents in Eq. (\[formf2\]). It is determined in terms of $\bar\Lambda^*-\bar\Lambda$ and $\zeta(1)$, since Eqs. (\[constbr2\]) and (\[constbr3\]) imply that $$\zeta_3^{(c)}(1) = -\zeta_3^{(b)}(1) = {\bar\Lambda^*-\bar\Lambda\over2}\, \zeta(1) \,.$$ We use Eq. (\[constbr3\]) to eliminate $\zeta_2^{(c)}$ and $\zeta_3^{(c)}$ in favor of $\zeta_1^{(c)}$ and $\zeta$. There are also order $\Lambda_{\rm QCD}/m_Q$ corrections to the effective Lagrangian, given in Eq. (\[lag\]). Time ordered products involving $O_{\rm kin}$ can be parameterized as $$\begin{aligned} i \int {\rm d}^4x\, T\,\Big\{ O_{{\rm kin},v'}^{(c)}(x)\, \Big[ \bar h_{v'}^{(c)}\, \Gamma\, h_{v}^{(b)} \Big](0)\, \Big\} &=& \chi_{\rm ke}^{(c)}\, {\rm Tr}\, \Big\{ \bar K_{v'}\, \Gamma\, H_v \Big\} \,, {\nonumber}\\* i \int {\rm d}^4x\, T\,\Big\{ O_{{\rm kin},v}^{(b)}(x)\, \Big[ \bar h_{v'}^{(c)}\, \Gamma\, h_{v}^{(b)} \Big](0)\, \Big\} &=& \chi_{\rm ke}^{(b)}\, {\rm Tr}\, \Big\{ \bar K_{v'}\, \Gamma\, H_v \Big\} \,.\end{aligned}$$ These corrections do not contribute at zero recoil. The chromomagnetic corrections have the form $$\begin{aligned} \label{timeo2} i \int {\rm d}^4x\, T\,\Big\{ O_{{\rm mag},v'}^{(c)}(x)\, \Big[ \bar h_{v'}^{(c)}\, \Gamma\, h_{v}^{(b)} \Big](0)\, \Big\} &=& {\rm Tr}\, \bigg\{ {\cal R}_{\alpha\beta}^{(c)}\, \bar K_{v'}\, i\sigma^{\alpha\beta}\, \frac{1+\vslash'}2\, \Gamma\, H_v \bigg\} \,, \nonumber\\* i \int {\rm d}^4x\, T\,\Big\{ O_{{\rm mag},v}^{(b)}(x)\, \Big[ \bar h_{v'}^{(c)}\, \Gamma\, h_{v}^{(b)} \Big](0)\, \Big\} &=& {\rm Tr}\, \bigg\{ {\cal R}_{\alpha\beta}^{(b)}\, \bar K_{v'}\, \Gamma\, \frac{1+\vslash}2\, i\sigma^{\alpha\beta} H_v \bigg\} , \end{aligned}$$ In this case the most general form of ${\cal R}_{\alpha\beta}^{(Q)}$ is $$\label{Rdef2} {\cal R}_{\alpha\beta}^{(c)} = \chi_1^{(c)} \gamma_\alpha \gamma_\beta + \chi_2^{(c)} v_\alpha \gamma_\beta \,, \qquad {\cal R}_{\alpha\beta}^{(b)} = \chi_1^{(b)} \gamma_\alpha \gamma_\beta + \chi_2^{(b)} v'_\alpha \gamma_\beta \,.$$ At zero recoil the contribution of $\chi_2^{(Q)}$ vanish because $v_\alpha(1+\vslash)\sigma^{\alpha\beta}(1+\vslash)=0$, while that of $\chi_1^{(Q)}$ vanish because $(1-\vslash)\gamma_\alpha\gamma_\beta(1+\vslash)=(1-\vslash)(\gamma_\alpha v_\beta-\gamma_\beta v_\alpha)(1+\vslash)$. Using Eqs. (\[1/mc21\])–(\[timeo2\]), it is straightforward to express the form factors $g_i$ parameterizing $B\to D_0^*\,e\,\bar\nu_e$ and $B\to D_1^*\,e\,\bar\nu_e$ semileptonic decays in terms of Isgur-Wise functions. The order $\Lambda_{\rm QCD}/m_b$ Lagrangian corrections arise only in the combination $\chi_b=\chi_{\rm ke}^{(b)}+6\chi_1^{(b)}-2(w+1)\chi_2^{(b)}$. Dropping the $c$ superscript from $\zeta_1^{(c)}$ and $\chi_i^{(c)}$, we obtain $$\begin{aligned} \label{expg0} g_+ &=& \varepsilon_c\, \bigg[ 2(w-1)\zeta_1 - 3\zeta\, {w\bar\Lambda^*-\bar\Lambda\over w+1} \bigg] - \varepsilon_b\, \bigg[ {\bar\Lambda^*(2w+1)-\bar\Lambda(w+2)\over w+1}\, \zeta - 2(w-1)\,\zeta_1 \bigg] \,, \nonumber\\* g_- &=& \zeta + \varepsilon_c\, \Big[ \chi_{\rm ke}+6\chi_1-2(w+1)\chi_2 \Big] + \varepsilon_b\, \chi_b \,. \end{aligned}$$ The analogous formulae for $B\to D_1^*\,e\,\bar\nu_e$ are $$\begin{aligned} \label{expg1} g_A &=& \zeta + \varepsilon_c\, \bigg[ {w\bar\Lambda^*-\bar\Lambda \over w+1} \zeta +\chi_{\rm ke}-2\chi_1 \bigg] - \varepsilon_b\, \bigg[ {\bar\Lambda^*(2w+1)-\bar\Lambda(w+2)\over w+1}\, \zeta - 2(w-1)\,\zeta_1 - \chi_b \bigg] \,,\nonumber\\* g_{V_1} &=& (w-1)\,\zeta + \varepsilon_c\, \Big[(w\bar\Lambda^*-\bar\Lambda)\zeta + (w-1)(\chi_{\rm ke}-2\chi_1) \Big] \nonumber\\* && - \varepsilon_b\, \Big\{ [\bar\Lambda^*(2w+1)-\bar\Lambda(w+2)]\, \zeta - 2(w^2-1)\,\zeta_1 - (w-1)\chi_b \Big\} \,, {\nonumber}\\* g_{V_2} &=& 2\varepsilon_c\, (\zeta_1-\chi_2) \,, {\nonumber}\\* g_{V_3} &=& - \zeta - \varepsilon_c\, \bigg[ {w\bar\Lambda^*-\bar\Lambda \over w+1}\zeta + 2\zeta_1 + \chi_{\rm ke} - 2\chi_1 +2\chi_2 \bigg] \nonumber\\* && + \varepsilon_b\, \bigg[ {\bar\Lambda^*(2w+1)-\bar\Lambda(w+2)\over w+1}\, \zeta - 2(w-1)\,\zeta_1 - \chi_b \bigg] \,. \end{aligned}$$ These equations show that at zero recoil the leading contributions to $g_{V_1}$ and $g_+$ of order $\Lambda_{\rm QCD}/m_Q$ are determined in terms of $\bar\Lambda^*-\bar\Lambda$ and $\zeta(1)$. Explicitly, $$\begin{aligned} g_+(1) &=& -\frac32\, (\varepsilon_c+\varepsilon_b)\, (\bar\Lambda^*-\bar\Lambda)\, \zeta(1) \,, \nonumber\\* g_{V_1}(1) &=& (\varepsilon_c-3\,\varepsilon_b)\, (\bar\Lambda^*-\bar\Lambda)\, \zeta(1) \,.\end{aligned}$$ For approximation A we shall again expand the double differential decay rates in Eq. (\[rate2\]) in powers of $w-1$, $$\begin{aligned} {{\rm d}^2\Gamma_{D_0^*}\over {\rm d}w\,{\rm d}\!\cos\theta} &=& 3\Gamma_0\, \zeta^2(1)\, r_0^{*3}\, (w^2-1)^{3/2}\, \sin^2\theta\, \sum_n\, (w-1)^n\, s_0^{(n)} \,, \\* {{\rm d}^2\Gamma_{D_1^*}\over {\rm d}w\,{\rm d}\!\cos\theta} &=& 3\Gamma_0\, \zeta^2(1)\, r_1^{*3}\, \sqrt{w^2-1}\, \sum_n\, (w-1)^n\, \bigg\{ \sin^2\theta\, s_1^{(n)}\, \nonumber\\* && + (1-2r_1^*w+r_1^{*2})\, \Big[(1+\cos^2\theta)\, t_1^{(n)} - 4\cos\theta\, \sqrt{w^2-1}\, u_1^{(n)} \Big] \bigg\} \,.\nonumber\end{aligned}$$ The coefficients for the decay rate into $D_0^*$ are $$\begin{aligned} s_0^{(0)} &=& (1-r_0)^{*2}\, \Big[ 1 + 2 \varepsilon_c (\hat\chi_{\rm ke} + 6\hat\chi_1 - 4\hat\chi_2) + 4 \varepsilon_b \hat\chi_b \Big] + 3(\varepsilon_c+\varepsilon_b)\, (1-r_0^{*2})\, (\bar\Lambda^*-\bar\Lambda) + \ldots \,,{\nonumber}\\* s_0^{(1)} &=& 2(1-r_0^*)^2\, \hat\zeta' + \ldots \,.\end{aligned}$$ For the decay into $D_1^*$ the coefficients are $$\begin{aligned} s_1^{(0)} &=& (\varepsilon_c-3\varepsilon_b)^2\, (1-r_1^*)^2\, (\bar\Lambda^*-\bar\Lambda)^2 + \ldots \,, {\nonumber}\\* s_1^{(1)} &=& - 2(\varepsilon_c-3\varepsilon_b)\, (1-r_1^{*2})\, (\bar\Lambda^*-\bar\Lambda) + \ldots \,, {\nonumber}\\* s_1^{(2)} &=& (1+r_1^*)^2 + \ldots \,,{\nonumber}\\ t_1^{(0)} &=& (\varepsilon_c-3\varepsilon_b)^2\, (\bar\Lambda^*-\bar\Lambda)^2 + \ldots \,, {\nonumber}\\* t_1^{(1)} &=& 2 + 4(\varepsilon_c-3\varepsilon_b)\, (\bar\Lambda^*-\bar\Lambda) + 4\varepsilon_c (\hat\chi_{\rm ke}-2\hat\chi_1) + 4\varepsilon_b \hat\chi_b + \ldots \,, {\nonumber}\\* t_1^{(2)} &=& 2(1 + 2\hat\zeta') + \ldots \,,{\nonumber}\\ u_1^{(0)} &=& (\varepsilon_c-3\varepsilon_b)\, (\bar\Lambda^*-\bar\Lambda) + \ldots \,, \nonumber \\* u_1^{(1)} &=& 1 + \ldots \,. \end{aligned}$$ Note that at zero recoil and at order $w-1$ the contributions to $D_1^*$ decay proportional to $\bar\Lambda^*-\bar\Lambda$ depend on the anomalously small combination $\varepsilon_c-3\varepsilon_b\sim0.05{\rm GeV}^{-1}$. Thus $\Lambda_{\rm QCD}/m_Q$ corrections enhance $B\to D_1^*$ by a much smaller amount than they enhance $B\to D_1$ decay. On the other hand, the $B\to D_0^*$ decay rate receives a large enhancement from $\Lambda_{\rm QCD}/m_Q$ corrections, similar to $B\to D_1$. In approximation A, $B\to D_1^*$ is treated the same way as $B\to D_1$ in Sec. II. $B\to D_0^*$ is treated as $B\to D_2^*$ in Sec. II, since these rates contain an additional factor of $w^2-1$. Approximation B is also very similar to that in Sec. II, except that in the present case there is only one unknown $\Lambda_{\rm QCD}/m_Q$ Isgur-Wise function, $\zeta_1$ (once time ordered products involving the chromomagnetic operator are neglected, and the matrix elements of the time ordered products involving the kinetic energy operator are absorbed into the $m_Q\to\infty$ Isgur-Wise function, $\zeta$). In approximation B$_1$ we set $\zeta_1=0$ in Eqs. (\[expg0\]) and (\[expg1\]). This is identical to saturating the first relation in Eq. (\[constbr2\]) by $\zeta_1^{(b)}$, i.e., setting $\zeta_1^{(b)}=\bar\Lambda\,\zeta$. In approximation B$_2$ we set $\zeta_1=\bar\Lambda\,\zeta$ in Eqs. (\[expg0\]) and (\[expg1\]), which is identical to setting $\zeta_1^{(b)}=0$. To the extent the first relation in Eq. (\[constbr2\]) can be taken as a hint to the sign of $\zeta_1$, the difference between approximations B$_1$ and B$_2$ gives a crude estimate of the uncertainty related to the unknown $\Lambda_{\rm QCD}/m_Q$ corrections. As in the previous section, the expression for the decay rate in terms of form factors in Eq. (\[rate2\]) implies that $s_1^{(0)}/t_1^{(0)}=(1-r_1^*)^2$ to all orders in the $\Lambda_{\rm QCD}/m_Q$ expansion. Thus the ratio of helicity zero and helicity one $B\to D_1^*$ decay rates at zero recoil is $\lim_{w\to1}\Big[({\rm d}\Gamma_{D_1^*}^{(\lambda=0)}/{\rm d}w)\Big/({\rm d}\Gamma_{D_1^*}^{(|\lambda|=1)}/{\rm d}w)\Big]=1/2$. Predictions ----------- A model independent prediction similar to that in Sec. II can be made for the slope parameter of semileptonic $B$ decay into the helicity zero $D_1^*$. We write the semileptonic decay rate into the helicity zero $D_1^*$ as $$\label{mir2} {{\rm d}\Gamma_{D_1^*}^{(\lambda=0)} \over {\rm d}w} = 4 \Gamma_0\, r_1^{*3}\, (1-r_1^*)^2\, \sqrt{w^2-1}\, \zeta^2(1)\, (\varepsilon_c-3\varepsilon_b)^2\, (\bar\Lambda^*-\bar\Lambda)^2\, \Big[ 1 - \rho_{D_1^*}^2\, (w-1) + \ldots \Big] \,.$$ The relationship between $s_1^{(0)}$ and $s_1^{(1)}$ implies that the slope parameter $\rho_{D_1^*}^2$ for helicity zero $D_1^*$ is $$\rho_{D_1^*}^2 = {1+r_1^*\over1-r_1^*}\, {2\over(\varepsilon_c-3\varepsilon_b)\, (\bar\Lambda^*-\bar\Lambda)} + {\cal O}(1) \,.$$ As in Sec. II, this slope parameter is of order $m_Q/\Lambda_{\rm QCD}$. It would be very hard experimentally to test this model independent prediction, since the $D_1^*$ is expected to be of order $100\,$MeV broad, and also because $\varepsilon_c-3\varepsilon_b$ is so small. Predictions for the $B\to D_0^*\,e\,\bar\nu_e$ and $B\to D_1^*\,e\,\bar\nu_e$ rates are shown in the first two columns of Table \[tab:sec3res\], normalized to $\zeta^2(1)$ times the measured $B\to D_1\,e\,\bar\nu_e$ rate. These results are obtained using $\hat\zeta'=-1$, and $\bar\Lambda^*-\bar\Lambda\simeq0.35\,$GeV corresponding to $1<w<1.33$. This value of $\bar\Lambda^*-\bar\Lambda$ has at least a $50\,$MeV uncertainty at present, as it follows from model predictions for the masses of the $s_\ell^{\pi_\ell}=\frac12^+$ charmed mesons, $\overline{m}_D^{\,*}\simeq2.40\,$GeV [@GI], and from the fact that $\lambda_1^*=\lambda_1'$ in nonrelativistic quark models with spin-orbit independent potentials. Although the $D_1^*$ state is expected to be somewhat heavier than the $D_0^*$, we use the kinematic range $1<w<1.33$ for both decays. The results in the first two columns of Table \[tab:sec3res\] are quite sensitive to the value of $\hat\zeta'$ and $\zeta_1$. In approximation B$_1$, for example, ${\cal B}(B\to D_0^*\,e\,\bar\nu_e)/[\zeta^2(1)\times0.006]$ changes from 1.92 at $\hat\zeta'=0$ to 0.54 at $\hat\zeta'=-2$. In the same range of $\hat\zeta'$, ${\cal B}(B\to D_1^*\,e\,\bar\nu_e)/[\zeta^2(1)\times0.006]$ changes from 0.72 to 0.24. The effect of $\zeta_1$ is also important; in the range $-0.75\,{\rm GeV}<\hat\zeta_1<0.75\,{\rm GeV}$, the $D_0^*$ and $D_1^*$ branching ratios change from 1.68 to 0.66 and 0.30 to 0.63, respectively. Therefore, even if $\zeta$ were known from models or lattice calculations, there would still be a factor of two uncertainty in the theoretical predictions for the semileptonic $B\to D_0^*$ and $D_1^*$ rates; but the uncertainty in the sum of these two rates is smaller. -------------------- -------------------------------------------------------- -------------------------------------------------------- ----------------------------------------- Approximation ${\displaystyle{\cal B}(B\to D_0^*\,e\,\bar\nu_e)\over ${\displaystyle{\cal B}(B\to D_1^*\,e\,\bar\nu_e)\over $\Gamma_{D_0^*+D_1^*}\Big/\Gamma_{D_1}$ \displaystyle\zeta^2(1)\times0.006}$ \displaystyle\zeta^2(1)\times0.006}$ \[6pt\] A$_\infty$ $0.30$ $0.66$ $1.07$ B$_\infty$ $0.33$ $0.46$ $1.61$ A $1.03$ $0.65$ $0.80$ B$_1$ $1.11$ $0.44$ $1.03$ B$_2$ $0.85$ $0.53$ $1.05$ -------------------- -------------------------------------------------------- -------------------------------------------------------- ----------------------------------------- To obtain even a crude absolute prediction for the $B\to D_1^*,D_0^*$ rates, a relation between the $s_\ell^{\pi_\ell}=\frac12^+$ and $\frac32^+$ Isgur-Wise functions is needed. In any nonrelativistic constituent quark model with spin-orbit independent potential, $\zeta$ and $\tau$ are related by [@VeOl; @IWsr] $$\label{ztau} \zeta(w) = {w+1\over\sqrt3}\, \tau(w) \,,$$ since both of these spin symmetry doublets correspond to $L=1$ orbital excitations. This implies $$\label{zetarel} \zeta(1) = {2\over\sqrt3}\, \tau(1) \,, \qquad \hat\zeta' = \frac12 + \hat\tau' \,.$$ In the same approximation, $\hat\eta_{\rm ke}=\hat\chi_{\rm ke}$.[^7] Predictions for the $B$ semileptonic decay rate into the states in the $s_\ell^{\pi_\ell}=\frac12^+$ doublet that follow from Eq. (\[zetarel\]) are shown in the last column of Table \[tab:sec3res\]. (For this quantity, approximations B$_i$ ($i=1,2$) contain a somewhat ad hoc input of combining the B$_i$ prediction in Sec. II with the B$_i$ prediction for $B\to D_0^*,D_1^*$.) For $\hat\tau'=-1.5$, the $\frac12^+$ doublet contributes about $1.0\times{\cal B}(B\to D_1\,e\,\bar\nu_e)\sim0.6\%$ to the total $B$ decay rate. Varying $\tau_{1,2}$ and $\zeta_1$ in approximation B results in the range $(0.6-1.7)\times{\cal B}(B\to D_1\,e\,\bar\nu_e)$ for the sum of the $D_0^*$ and $D_1^*$ rates. This combined with our results for $R=\Gamma_{D_2^*}/\Gamma_{D_1}$ in Sec. II is consistent with the ALEPH measurement [@ALEPH] of the branching ratio for the sum of all semileptonic decays containing a $D^{(*)}\,\pi$ in the final state to be $(2.26\pm0.44)\%$. The semileptonic decay rate into $D$ and $D^*$ is about $6.6\%$ of the total $B$ decay rate [@PDG]. Our results then suggest that the six lightest charmed mesons contribute about 8.2% of the $B$ decay rate. Therefore, semileptonic decays into higher excited states and non-resonant multi-body channels should be at least 2% of the $B$ decay rate, and probably around 3% if the semileptonic $B$ branching ratio is closer to the LEP result of about 11.5%. Such a sizable contribution to the semileptonic rate from higher mass excited charmed mesons and non-resonant modes would soften the lepton spectrum, and may make the agreement with data on the inclusive lepton spectrum worse. Of course, the decay rates to the broad $\frac12^+$ states would change substantially if the nonrelativistic quark model prediction in Eq. (\[ztau\]) is wrong. Semileptonic $B$ decay rate to the six lightest charmed mesons could add up to close to $10\%$ if $\zeta$ were enhanced by a factor of two compared to the prediction of Eq. (\[ztau\]). However, model calculations [@More] seem to obtain a suppression rather than an enhancement of $\zeta$ compared to Eq. (\[ztau\]). Thus, taking the measurements for the $B\to D$, $D^*$, and $D_1$ semileptonic branching ratios on face value, a decomposition of the semileptonic rate as a sum of exclusive channels seems problematic both in light of our results and the above ALEPH measurement for the sum of all semileptonic decays containing a $D^{(*)}\,\pi$ in the final state. Other excited charmed mesons at zero recoil =========================================== In the previous two sections matrix elements of the weak vector current and axial-vector current between a $B$ meson and an excited charmed mesons with $s_\ell^{\pi_\ell}=\frac32^+$ and $\frac12^+$ quantum numbers were considered. Here we consider such matrix elements at zero recoil for excited charmed mesons with other $s_\ell^{\pi_\ell}$ quantum numbers. Only charmed mesons with spin zero or spin one can contribute at this kinematic point. The polarization tensor of a spin $n$ state is rank $n$, traceless and symmetric in its indices, and vanishes if it is contracted with the 4-velocity of the state. For matrix elements of the axial-vector or vector current, at least $n-1$ indices of the charmed meson polarization tensor are contracted with $v^\mu$, the four velocity of the $B$ meson. Consequently, for $n>1$ these matrix elements vanish at zero recoil, where $v=v'$. In this section we work in the rest frame, $v=v'=(1,\vec0\,)$, and four-velocity labels on the fields and states are suppressed. For spin zero and spin one excited charmed mesons, the possible spin parities for the light degrees of freedom are $s_\ell^{\pi_\ell}=\frac12^+$, $\frac32^+$, which we have already considered in the previous sections, and $s_\ell^{\pi_\ell}=\frac12^-$, $\frac32^-$. In the nonrelativistic constituent quark model, the $\frac12^-$ states are interpreted as radial excitations of the ground state $(D,D^*)$ doublet and the $\frac32^-$ states are $L=2$ orbital excitations. In the quark model, these states are typically expected to be broad. The mass of the lightest $s_l^{\pi_l}=\frac32^-$ doublet is expected around $2.8\,$GeV, while the lightest excited states with $s_l^{\pi_l}=\frac12^-$ are expected around $2.6\,$GeV [@GI].[^8] ($B$ decays into radial excitations of the $s_l^{\pi_l}\neq\frac12^-$ states have similar properties as the decay into the lightest state with the same quantum numbers.) In the $m_Q\to\infty$ limit, the zero recoil matrix elements vanish by heavy quark symmetry. For the excited $s_\ell^{\pi_\ell}=\frac12^-$ states, the $m_Q\to\infty$ Isgur-Wise functions vanish at zero recoil due to the orthogonality of the states. The matrix elements for the $s_\ell^{\pi_\ell}\neq\frac12^-$ states vanish at zero recoil due to spin symmetry alone, and therefore the corresponding $m_Q\to\infty$ Isgur-Wise functions need not vanish at zero recoil. Using the same methods as in Sections II and III, it is straightforward to show that $\Lambda_{\rm QCD}/m_Q$ corrections to the current do not contribute at zero recoil. For the $s_\ell^{\pi_\ell}=\frac12^-$ states, this follows from the heavy quark equation of motion. For the $s_\ell^{\pi_\ell}=\frac32^-$ states, the $\Lambda_{\rm QCD}/m_Q$ corrections to the current can be parameterized similar to Eqs. (\[1/mc1\]) and (\[1/mc2\]). In this case the analogue of $F_v^\mu$ in Eq. (\[Fdef\]) satisfies $\vslash F_v^\mu=F_v^\mu=F_v^\mu\,\vslash$. Recall that the $\tau_4^{(Q)}g_{\sigma\lambda}$ in Eq. (\[1/mc2\]) was the only term whose contribution at zero recoil did not vanish due to the $v_\mu F_v^\mu=0$ property of the Rarita-Schwinger spinors. Here, the analogous term is placed between $1-\vslash$ and $1+\vslash'$, and therefore also disappears at $v=v'$. It remains to consider the $\Lambda_{\rm QCD}/m_Q$ contributions to the $\frac12^-$ and $\frac32^-$ matrix elements coming from corrections to the Lagrangian in Eq. (\[lag\]). These are written as time ordered products of $O_{\rm kin}^{(Q)}(x)$ and $O_{\rm mag}^{(Q)}(x)$ with the leading order $m_Q\to\infty$ currents (e.g., Eq (\[kinetic\])). At zero recoil it is useful to insert a complete set of states between these operators. Since the zero recoil weak currents are charge densities of heavy quark spin-flavor symmetry, only one state from this sum contributes. For the $s_\ell^{\pi_\ell}=\frac12^-$ multiplet this procedure gives $$\begin{aligned} \label{css1} \frac{\langle D^{*(n)}(\varepsilon)|\, \vec A\, |B\rangle} {\sqrt{m_{D^{*(n)}}\,m_B}} &=& \frac{-\vec\varepsilon}{(\bar\Lambda^{(n)}-\bar\Lambda)}\, \Bigg\{ \left(\frac1{2m_c} + \frac3{2m_b}\right) \frac{\langle D^{*(n)}(\varepsilon)|\, O_{\rm mag}^{(c)}(0)\, |D^*(\varepsilon)\rangle}{\sqrt{m_{D^{*(n)}}\,m_{D^*}}} \nonumber\\* && \phantom{\frac{-\vec\varepsilon}{(\bar\Lambda^{(n)}-\bar\Lambda)}} + \left(\frac1{2m_c} - \frac1{2m_b}\right) \frac{\langle D^{*(n)}(\varepsilon)|\, O_{\rm kin}^{(c)}(0)\, |D^*(\varepsilon)\rangle}{\sqrt{m_{D^{*(n)}}\,m_{D^*}}} \Bigg\} \,.\end{aligned}$$ and $$\label{css2} \frac{\langle D^{(n)}|\, V^0\, |B\rangle} {\sqrt{m_{D^{(n)}}\,m_B}} = \frac{1}{(\bar\Lambda^{(n)}-\bar\Lambda)} \left(-\frac1{2m_c} + \frac1{2m_b}\right) \frac{\langle D^{(n)}|\, O_{\rm mag}^{(c)}(0)+O_{\rm kin}^{(c)}(0)\, |D\rangle}{\sqrt{m_{D^{(n)}}\,m_D}} \,.$$ Here we have denoted spin zero and spin one members of the excited $s_\ell^{\pi_\ell}=\frac12^-$ multiplet by $D^{(n)}$ and $D^{*(n)}$ respectively, and the analogues of $\bar\Lambda$ by $\bar\Lambda^{(n)}$. Heavy quark spin-flavor symmetry was used to write the effects of $O_{\rm kin}^{(b)}$ and $O_{\rm mag}^{(b)}$ in terms of matrix elements of $O_{\rm kin}^{(c)}$ and $O_{\rm mag}^{(c)}$. This neglects the weak logarithmic dependence on the heavy quark mass in the matrix elements of $O_{\rm mag}$. For the spin one member of the $s_\ell^{\pi_\ell} = \frac32^-$ multiplet, which we denote by $D_1^{**}$, $$\label{css3} \frac{\langle D_1^{**}(\varepsilon)|\, \vec A\, |B\rangle} {\sqrt{m_{D_1^{**}}\,m_B}} = \frac{-\vec\varepsilon}{(\bar\Lambda^{**}-\bar\Lambda)} \left(\frac1{2m_c}\right) \frac{\langle D_1^{**}(\varepsilon)|\, O_{\rm mag}^{(c)}(0)\, |D^*(\varepsilon)\rangle}{\sqrt{m_{D_1^{**}}\,m_D}}\,.$$ For the $s_\ell^{\pi_\ell}=\frac12^-$ and $\frac32^-$ excited charmed mesons, the correction to the Lagrangian, $\delta{\cal L}$ in Eq. (\[lag\]), gives rise to an order $\Lambda_{\rm QCD}/m_c$ contribution to the matrix elements of the weak currents at zero recoil. Formulae similar to those in Eqs. (\[css1\])–(\[css3\]) hold in the $s_\ell^{\pi_\ell}=\frac12^+$, $\frac32^+$ cases, but the corresponding matrix elements vanish due to the parity invariance of the strong interaction. Applications ============ Factorization ------------- Factorization should be a good approximation for $B$ decay into charmed mesons and a charged pion. Contributions that violate factorization are suppressed by $\Lambda_{\rm QCD}$ divided by the energy of the pion in the $B$ rest frame [@GrDu] or by $\alpha_s(m_Q)$. Furthermore for these decays, factorization also holds in the limit of large number of colors. Neglecting the pion mass, the two-body decay rate, $\Gamma_\pi$, is related to the differential decay rate ${\rm d}\Gamma_{\rm sl}/{\rm d}w$ at maximal recoil for the analogous semileptonic decay (with the $\pi$ replaced by the $e\,\bar\nu_e$ pair). This relation is independent of the identity of the charmed meson in the final state, $$\label{factor} \Gamma_\pi = {3\pi^2\, |V_{ud}|^2\, C^2\, f_\pi^2 \over m_B^2\, r} \times \left( {{\rm d} \Gamma_{\rm sl}\over {\rm d}w} \right)_{w_{\rm max}} .$$ Here $r$ is the mass of the charmed meson divided by $m_B$, $w_{\rm max}=(1+r^2)/(2r)$, and $f_\pi\simeq132\,$MeV is the pion decay constant. $C$ is a combination of Wilson coefficients of four-quark operators, and numerically $C\,|V_{ud}|$ is very close to unity. These nonleptonic decay rates can therefore be predicted from a measurement of ${\rm d}\Gamma_{\rm sl}/{\rm d}w$ at maximal recoil. The semileptonic decay rate near maximal recoil is only measured for $B\to D^{(*)}\,e\,\bar\nu_e$ at present. The measured $B\to D^{(*)}\,\pi$ rate is consistent with Eq. (\[factor\]) at the level of the 10% experimental uncertainties. In the absence of a measurement of the $B\to(D_1,D_2^*)\,e\,\bar\nu_e$ differential decay rates, we can use our results for the shape of ${\rm d}\Gamma_{\rm sl}/{\rm d}w$ to predict the $B\to D_1\,\pi$ and $B\to D_2^*\,\pi$ decay rates. These predictions depend on the semileptonic differential decay rates at $w_{\rm max}$, where we are the least confident that $\Lambda_{\rm QCD}/m_Q$ terms involving $\bar\Lambda$ and $\bar\Lambda'$ are the most important. With this caveat in mind, we find the results shown in Table \[tab:factor\]. -------------------- --------------------------------------------------- ----------------------------------------------- -- Approximation ${\displaystyle{\cal B}(B\to D_1\,\pi)\over ${\displaystyle{\cal B}(B\to D_2^*\,\pi)\over \displaystyle{\cal B}(B\to D_1\,e\,\bar\nu_e)}$ \displaystyle{\cal B}(B\to D_1\,\pi)}$ \[6pt\] A$_\infty$ $0.39$ $0.36$ B$_\infty$ $0.26$ $1.00$ A $0.29$ $0.21$ B$_1$ $0.19$ $0.41$ B$_2$ $0.20$ $0.56$ -------------------- --------------------------------------------------- ----------------------------------------------- -- At present there are only crude measurements of the ${\cal B}(B\to D_1\,\pi)$ and ${\cal B}(B\to D_2^*\,\pi)$ branching ratios. Assuming ${\cal B}(D_1(2420)^0\to D^{*+}\,\pi^-)=2/3$ and ${\cal B}(D_2^*(2460)^0\to D^{*+}\,\pi^-)=0.2$, the measured rates are [@CLEOfact] $$\begin{aligned} \label{factordata} {\cal B}(B^-\to D_1(2420)^0\,\pi^-) &=& (1.17\pm0.29) \times 10^{-3} \,, \nonumber\\* {\cal B}(B^-\to D_2^*(2460)^0\,\pi^-) &=& (2.1\pm0.9) \times 10^{-3} \,.\end{aligned}$$ A reduction of the experimental uncertainty in ${\cal B}(B\to D_2^*\,\pi)$ is needed to test the prediction in the second column of Table \[tab:factor\]. The prediction for ${\cal B}(B\to D_1\,\pi)/{\cal B}(B\to D_1\,e\,\bar\nu_e)$ in approximation B is fairly independent of $\tau_{1,2}$, but more sensitive to $\hat\tau'$. The latter dependence is plotted in Fig. \[fig:factor1\] for $0>\hat\tau'>-2$. Not absorbing $\eta_{\rm ke}$ into $\tau$ results in the following weak dependence: ${\cal B}(B\to D_1\,\pi)/{\cal B}(B\to D_1\,e\,\bar\nu_e)\propto1+0.27\,\varepsilon_c\,\hat\eta_{\rm ke}+\ldots$. Assuming that factorization works at the 10% level, a precise measurement of the ${\cal B}(B\to D_1\,\pi)$ rate may provide a determination of $\hat\tau'$. The present experimental data, ${\cal B}(B\to D_1\,\pi)/{\cal B}(B\to D_1\,e\,\bar\nu_e)\simeq0.2$, does in fact support $\hat\tau'\sim-1.5$, which we took as the “central value" in this paper, motivated by model calculations. =9truecm The prediction for ${\cal B}(B\to D_2^*\,\pi)/{\cal B}(B\to D_1\,\pi)$, on the other hand, only weakly depends on $\hat\tau'$, but it is more sensitive to $\tau_{1,2}$. Varying $\tau_{1,2}$ in the range $-0.75\,{\rm GeV}<\hat\tau_{1,2}<0.75\,{\rm GeV}$, we can accommodate almost any value of ${\cal B}(B\to D_2^*\,\pi)/{\cal B}(B\to D_1\,\pi)$ between 0 and 1.5. This quantity depends more sensitively on $\tau_1$ than on $\tau_2$. In Fig. \[fig:factor2\] we plot ${\cal B}(B\to D_2^*\,\pi)/{\cal B}(B\to D_1\,\pi)$ in approximation B as a function of $\hat\tau_1$ setting $\hat\tau_2=0$ (solid curve), and as a function of $\hat\tau_2$ setting $\hat\tau_1=0$ (dashed curve). Not absorbing $\eta_{\rm ke}$ into $\tau$ results in the following dependence: ${\cal B}(B\to D_2^*\,\pi)/{\cal B}(B\to D_1\,\pi)\propto1+0.75\,\varepsilon_c\,\hat\eta_{\rm ke}+\ldots$. This ratio and $R$ depend on $\hat\eta_{\rm ke}$ and $\hat\tau_1$. In the future experimental data on these ratios may lead to a determination of $\hat\eta_{\rm ke}$ and $\hat\tau_1$. If the experimental central value on ${\cal B}(B\to D_2^*\,\pi)$ does not decrease compared to Eq. (\[factordata\]), then it would suggest a huge value for $\hat\tau_1$, leading to a violation of the ALEPH bound on $R$ (see Fig. \[fig:Rtau\]). The approximation B results in Tables \[tab:sec2res\] and \[tab:factor\] can be combined to give ${\cal B}(B\to D_2^*\,\pi)/{\cal B}(B\to D_2^*\,e\,\bar\nu_e) = 0.15$. Varying $\hat\tau_i$, $\hat\eta_{\rm ke}$ and $\hat\tau'$ does not bring this quantity close to the current experimental limit. Therefore, if the branching ratio for $B\to D_2^*\,e\,\bar\nu_e$ is below the ALEPH bound, then ${\cal B}(B\to D_2^*\,\pi)$ should be smaller than the central value in Eq. (\[factordata\]). =9truecm Sum Rules --------- Our results are important for sum rules that relate inclusive $B\to X_c\,e\,\bar\nu_e$ decays to the sum of exclusive channels. The Bjorken sum rule bounds the slope of the $B\to D^{(*)}\,e\,\bar\nu_e$ Isgur-Wise function, defined by the expansion $\xi(w)=1-\rho^2\,(w-1)+\ldots\,$. Knowing $\rho^2$ would reduce the uncertainty in the determination of $|V_{cb}|$ from the extrapolation of the $B\to D^{(*)}\,e\,\bar\nu_e$ spectrum to zero recoil. The Bjorken sum rule [@Bjsr; @IWsr] is $$\label{bjorken} \rho^2 = \frac14 + \sum_m\, {|\zeta^{(m)}(1)|^2 \over 4} + 2 \sum_p\, {|\tau^{(p)}(1)|^2 \over 3} + \ldots \,.$$ Throughout this section the ellipses denote contributions from non-resonant channels. $\zeta^{(m)}$ and $\tau^{(p)}$ are the Isgur-Wise functions for the exited $s_\ell^{\pi_\ell}=\frac12^+$ and $\frac32^+$ states, respectively (for $m=p=0$ these are the orbitally excited states discussed in Sec. II and III, and $m,p\geq1$ are radial excitations of these).[^9] Since all terms in the sums, as well as the contributions replaced by ellipses, are non-negative, a lower bound on $\rho^2$ can be obtained by keeping only the first few terms on the right-hand-side of Eq. (\[bjorken\]). Using Eqs. (\[tau1\]) and (\[ztau\]), we find that the contribution of the lowest lying $s_\ell^{\pi_\ell}=\frac12^+$ and $\frac32^+$ states implies the bound $$\label{bjnew} \rho^2 > \frac14 + {|\zeta(1)|^2\over4} + 2\, {|\tau(1)|^2\over3} \simeq 0.75 \,.$$ The contribution of the $\frac12^+$ states through $\zeta(1)$ to this bound, which relies on the quark model result in Eq. (\[ztau\]), is only 0.17. An upper bound on $\rho^2$ follows from an upper bound on the excited states contribution to the right-hand-side of Eq. (\[bjorken\]). This sum rule was first derived by Voloshin [@Volo] $$\label{voloshin} \frac12\, \bar\Lambda = \sum_m\, (\bar\Lambda^{*\,(m)}-\bar\Lambda)\, {|\zeta^{(m)}(1)|^2\over4} + 2 \sum_p\, (\bar\Lambda'^{\,(p)}-\bar\Lambda)\, {|\tau^{(p)}(1)|^2\over3} + \ldots \,.$$ Here $\bar\Lambda^{*\,(m)}$ and $\bar\Lambda'^{\,(p)}$ are the analogues of $\bar\Lambda^*$ and $\bar\Lambda'$ for the exited $s_\ell^{\pi_\ell}=\frac12^+$ and $\frac32^+$ states, respectively. Eq. (\[voloshin\]) combined with Eq. (\[bjorken\]) implies that $\rho^2<1/4+\bar\Lambda/(2\varepsilon_1)$, where $\varepsilon_1$ is the excitation energy of the lightest excited charmed meson state. However, knowing $\zeta(1)$ and $\tau(1)$ does not strengthen this bound on $\rho^2$ significantly. On the other hand, Eq. (\[voloshin\]) implies the bound $\bar\Lambda>0.38\,$GeV (neglecting perturbative QCD corrections). The model dependent contribution of the $\frac12^+$ states to this bound is only $0.12\,$GeV; while the bound $\bar\Lambda>0.26\,{\rm GeV}$ from only the $\frac32^+$ states is fairly model independent. A class of zero recoil sum rules were considered in Ref. [@VVsr]. The axial sum rule, which bounds the $B\to D^*$ form factor (that is used to determine $|V_{cb}|$) only receives contributions from $s_\ell^{\pi_\ell}=\frac12^-$ and $\frac32^-$ states, which were discussed in Sec. IV. It has the form $$\label{aasr} |F_{B\to D^*}(1)|^2 + \sum_{X_c} {|\langle X_c(\varepsilon)|\, \vec A\, |B\rangle|^2 \over 4m_{X_c}\,m_B } = \eta_A^2 - {\lambda_2\over m_c^2} + {\lambda_1+3\lambda_2\over4}\, \bigg( {1\over m_c^2} + {1\over m_b^2} + {2\over 3m_c\,m_b} \bigg) \,,$$ where $\eta_A$ is the perturbative matching coefficient of the full QCD axial-vector current onto the HQET current, $X_c$ denotes spin one states (continuum or resonant) with $s_\ell^{\pi_\ell}=\frac12^-$ and $\frac32^-$, and $F_{B\to D^*}(1)$ is defined by $${\langle D^*(\varepsilon)|\, \vec A\, |B\rangle\over 2\sqrt{m_{D^*}\,m_B} } = F_{B\to D^*}(1)\, \vec\varepsilon \,.$$ Neglecting the contributions of the excited states $X_c$ to the left-hand-side, gives an upper bound on $|F_{B\to D^*}(1)|^2$. Using the nonrelativistic constituent quark model, we estimate using Eq. (\[css1\]) that the contribution of the first radial excitation of the $D^*$ to the sum over $X_c$ in Eq. (\[aasr\]) is about 0.1. For this estimate we took $\bar\Lambda^{(1)}-\bar\Lambda=450\,$MeV, $O_{\rm mag}^{(c)}=C\,\delta^3(r)\,\vec s_c\cdot\vec s_{\bar q}$ (fixing the constant $C$ by the measured $D^*-D$ mass splitting), $O_{\rm kin}^{(c)}=\vec\nabla^2$, and used the harmonic oscillator quark model wave functions of Ref. [@ISGW]. A 0.1 correction would significantly strengthen the upper bound on $F_{B\to D^*}(1)$ and have important consequences for the extraction of the magnitude of $V_{cb}$ from exclusive $B\to D^*e\bar\nu_e$ decay. Note that $s_\ell^{\pi_\ell}=\frac32^-$ states do not contribute to the zero recoil axial sum rule in the quark model, because their spatial wave functions vanish at the origin. The $J^P=1^+$ members of the $s_\ell^{\pi_\ell}=\frac12^+$ and $s_\ell^{\pi_\ell}=\frac32^+$ doublets contribute to the vector sum rule, which is used to bound $\lambda_1$. This sum rule reads [@VVsr; @llsw] $$\begin{aligned} \label{vvsr} && {(m_b-3m_c)^2 \over 4m_b^2\,m_c^2}\, \sum_m\, (\bar\Lambda^{*\,(m)}-\bar\Lambda)^2\, {|\zeta^{(m)}(1)|^2\over4} + {2\over m_c^2}\, \sum_p\, (\bar\Lambda'^{\,(p)}-\bar\Lambda)^2\, {|\tau^{(p)}(1)|^2\over3} + \ldots \nonumber\\* && \qquad\qquad = {\lambda_2\over m_c^2} - {\lambda_1+3\lambda_2\over4}\, \bigg( {1\over m_c^2} + {1\over m_b^2} - {2\over3m_c\,m_b} \bigg) \,.\end{aligned}$$ This relation can be simplified by setting $m_b/m_c$ to different values. Taking $m_b=m_c$ yields $$\label{l1sr} \lambda_1 = -3\, \sum_m\, (\bar\Lambda^{*\,(m)}-\bar\Lambda)^2\, {|\zeta^{(m)}(1)|^2\over4} - 6 \sum_p\, (\bar\Lambda'^{\,(p)}-\bar\Lambda)^2\, {|\tau^{(p)}(1)|^2\over3} + \ldots \,,$$ whereas $m_c\gg m_b\gg\Lambda_{\rm QCD}$ gives [@llsw] $$\label{vvnew} \lambda_1 + 3\lambda_2 = - 9 \sum_m\, (\bar\Lambda^{*\,(m)}-\bar\Lambda)^2\, {|\zeta^{(m)}(1)|^2\over4} + \ldots \,.$$ These relations can be combined to obtain a sum rule for $\lambda_2$, $$\label{l2sr} \lambda_2 = -2\, \sum_m\, (\bar\Lambda^{*\,(m)}-\bar\Lambda)^2\, {|\zeta^{(m)}(1)|^2\over4} + 2 \sum_p\, (\bar\Lambda'^{\,(p)}-\bar\Lambda)^2\, {|\tau^{(p)}(1)|^2\over3} + \ldots \,.$$ Eqs. (\[l1sr\]) and (\[l2sr\]) were previously obtained in Ref. [@srreview] using different methods. The strongest constraint on $\lambda_1$ is given by Eq. (\[vvnew\]) (the sum rule in Eq. (\[l1sr\]) only implies $-\lambda_1>(0.06+0.15)\,{\rm GeV}^2$). Including the contribution of the lightest $s_\ell^{\pi_\ell}=\frac12^+$ doublet to Eq. (\[vvnew\]) yields $$\label{l1new} \lambda_1 < - 3\lambda_2 - 9\, (\bar\Lambda^*-\bar\Lambda)^2\, {|\zeta(1)|^2\over4} \simeq - 3\lambda_2 - 0.18\,{\rm GeV}^2 \,,$$ neglecting perturbative QCD corrections. Note that only the broad $D_1^*$ state (and its radial excitations) contribute to this sum rule, so the result in Eq (\[l1new\]) is sensitive to the relation between $\tau(1)$ and $\zeta(1)$ in Eq. (\[ztau\]). Perturbative corrections to the sum rules in this section can be found in Ref. [@pert]. Summary and conclusions ======================= The branching ratios for $B\to D\,e\,\bar\nu_e$ and $B\to D^*\,e\,\bar\nu_e$ are $(1.8\pm0.4)\%$ and $(4.6\pm0.3)\%$, respectively [@PDG]. This implies that about $40\%$ of semileptonic $B$ decays are to excited charmed mesons and non-resonant final states. An excited charmed meson doublet $(D_1(2420),D_2^*(2460))$ with $s_\ell^{\pi_\ell}={3\over2}^+$ has been observed. These states are narrow and have widths around $20\,$MeV. With some assumptions, the CLEO and ALEPH collaborations have measured about a $(0.6\pm0.1)\%$ branching ratio for $B\to D_1\,e\,\bar\nu_e$. The decay $B\to D_2^*\,e\,\bar\nu_e$ has not been observed, and CLEO and ALEPH respectively report limits of $1\%$ and $0.2\%$ on its branching ratio. A detailed experimental study of semileptonic $B$ decays to these states should be possible in the future. The semileptonic $B$ decay rate to an excited charmed meson is determined by the corresponding matrix elements of the weak axial-vector and vector currents. At zero recoil (where the final excited charmed meson is at rest in the rest frame of the initial $B$ meson), these currents correspond to charges of the heavy quark spin-flavor symmetry. Consequently, in the $m_Q\to\infty$ limit, the zero recoil matrix elements of the weak currents between a $B$ meson and any excited charmed meson vanish. However, at order $\Lambda_{\rm QCD}/m_Q$ these matrix elements are not necessarily zero. Since for $B$ semileptonic decay to excited charmed mesons most of the available phase space is near zero recoil, the $\Lambda_{\rm QCD}/m_Q$ corrections can play a very important role. In this paper we studied the predictions of HQET for the $B\to D_1\,e\,\bar\nu_e$ and $B\to D_2\,e\,\bar\nu_e$ differential decay rates including the effects of $\Lambda_{\rm QCD}/m_Q$ corrections to the matrix elements of the weak currents. Since the matrix elements of the weak currents between a $B$ meson and any excited charmed meson can only be nonzero for spin zero or spin one charmed mesons at zero recoil, the $\Lambda_{\rm QCD}/m_Q$ corrections are more important for the spin one member of the $s_\ell^{\pi_\ell}={3\over2}^+$ doublet. The $\Lambda_{\rm QCD}/m_Q$ corrections to the matrix elements of the weak axial-vector and vector currents can be divided into two classes; corrections to the currents themselves and corrections to the states. For $B$ semileptonic decays to the $D_1$, parity invariance of the strong interactions forces the corrections to the states to vanish at zero recoil. Furthermore, the corrections to the current give a contribution which at zero recoil is expressible in terms of the leading, $m_Q\to\infty$, Isgur-Wise function and known meson mass splittings. This correction leads to an enhancement of the $B$ semileptonic decay rate to the $D_1$ over that to the $D_2$. With some model dependent assumptions, we made predictions for the differential decay rates for $B\to D_1\,e\,\bar\nu_e$ and $B\to D_2^*\,e\,\bar\nu_e$ and determined the zero recoil value of the leading $m_Q \to\infty$ Isgur-Wise function from the measured $B$ to $D_1$ semileptonic decay rate. The influence of perturbative QCD corrections on these decay rates were also considered but these are quite small. Factorization was used to predict the rates for the nonleptonic decays $B\to D_1\,\pi$ and $B\to D_2^*\,\pi$. The ALEPH limit on the semileptonic decay rate to $D_2^*$ implies a small branching ratio for $B\to D_2^*\,\pi$. The ratio ${\cal B}(B\to D_1\,\pi)/{\cal B}(B\to D_1\,e\,\bar\nu_e)$ can be used to determine $\hat\tau'$. The present experimental value for this quantity favors $\hat\tau'$ near $-1.5$. The most significant uncertainty at order $\Lambda_{\rm QCD}/m_Q$ arises from $\hat\tau_1$ and $\hat\eta_{\rm ke}$. It may be possible to determine these quantities from measurements of $R=\Gamma_{D_2^*}/\Gamma_{D_1}$ and ${\cal B}(B\to D_2^*\,\pi)/{\cal B}(B\to D_1\,\pi)$. The $w$-dependence of the semileptonic decay rates can provide important similar information. A broad multiplet of excited charmed mesons with masses near those of the $D_1$ and $D_2^*$ is expected. It has spin of the light degrees of freedom $s_\ell^{\pi_\ell}={1\over2}^+$, giving spin zero and spin one states that are usually denoted by $D_0^*$ and $D_1^*$. We studied the predictions of HQET for the $B\to D_0^*\,e\,\bar\nu_e$ and $B\to D_1^*\,e\,\bar\nu_e$ differential decay rates including the effects of $\Lambda_{\rm QCD}/m_Q$ corrections to the matrix elements of the weak current. The situation here is similar to that in the case of the $s_\ell^{\pi_\ell}={3\over2}^+$ doublet. Using a relation between the leading, $m_Q\to\infty$, Isgur-Wise functions for these two excited charmed meson doublets that is valid in the nonrelativistic constituent quark model with any spin-orbit independent potential (and a few other assumptions), we determined the rates for $B$ semileptonic decays to these excited charmed mesons. We find that branching ratio for $B$ semileptonic decays into the four states in the $s_\ell^{\pi_{\ell}}={1\over2}^+$ and ${3\over2}^+$ doublets is about $1.6\%$. Combining this with the measured rates to the ground state $D$ and $D^*$ implies that more than $2\%$ of the $B$ meson decays must be semileptonic decays to higher mass excited charmed states or nonresonant modes. Some of the more important results in Tables \[tab:sec2res\] and \[tab:sec3res\] are summarized in Table \[tab:concl\]. -------------------- ------------------------------------- --------------------------------------------------------- --------------------------------------------------- Approximation $R=\Gamma_{D_2^*}\Big/\Gamma_{D_1}$ $\tau(1)\, \bigg[\displaystyle {6.0\times10^{-3} \over $\Gamma_{D_1+D_2^*+D_1^*+D_0^*}\Big/\Gamma_{D_1}$ {\cal B}(B\to D_1\,e\,\bar\nu_e)} \bigg]^{1/2}$ \[6pt\] B$_\infty$ $1.65$ $1.24$ $4.26$ B$_1$ $0.52$ $0.71$ $2.55$ B$_2$ $0.67$ $0.75$ $2.71$ -------------------- ------------------------------------- --------------------------------------------------------- --------------------------------------------------- We considered the zero recoil matrix elements of the weak currents between a $B$ meson and other excited charmed mesons at order $\Lambda_{\rm QCD}/m_Q$. Only the corrections to the states contribute and these were expressed in terms of matrix elements of local operators. Our results have implications for $B$ decay sum rules, where including the contributions of the excited charmed meson states strengthens the bounds on $\rho^2$ (the slope of the Isgur-Wise function for $B\to D^{(*)}\,e\,\bar\nu_e$), on $\lambda_1$, and on the zero recoil matrix element of the axial-vector current between $B$ and $D^*$ mesons. The latter bound has implications for the extraction of $|V_{cb}|$ from exclusive $B\to D^*\,e\,\bar\nu_e$ decay. We thank A. Le Yaouanc, M. Neubert, A. Vainshtein for useful discussions, and A. Falk for keeping us from being disingenuous. This work was supported in part by the Department of Energy under grant no. DE-FG03-92-ER 40701. Perturbative order $\alpha_{\lowercase{s}}$ corrections ======================================================= In this Appendix we compute order $\alpha_s$ and order $\alpha_s\,\Lambda_{\rm QCD}/m_Q$ corrections to the $B\to(D_1,D_2^*)\,e\,\bar\nu_e$ form factors. At this order both the current in Eq. (\[1/mcurrent\]) and the order $\Lambda_{\rm QCD}/m_Q$ corrections to the Lagrangian in Eq. (\[lag\]) receive corrections. Matrix elements of the kinetic energy operator, $\eta_{\rm ke}^{(Q)}$, enter proportional to $\tau$ to all orders in $\alpha_s$ due to reparameterization invariance [@LuMa]. The matrix elements involving the chromomagnetic operator are probably very small and have been neglected. Order $\alpha_s$ corrections to the $b\to c$ flavor changing current in the effective theory introduce a set of new operators at each order in $\Lambda_{\rm QCD}/m_Q$, with the appropriate dimensions and quantum numbers. The Wilson coefficients for these operators are known $w$-dependent functions [@falk; @qcdcorr], which we take from [@physrep]. The vector and axial-vector currents can be written at order $\alpha_s$ as $$\begin{aligned} \label{A1} V^\mu &=& \bar h^{(c)}_{v'}\, \Bigg[ \gamma^\mu-{i\overleftarrow D\!\!\!\!\!\slash\, \gamma^\mu \over 2\,m_c} + {i\gamma^\mu \overrightarrow D\!\!\!\!\!\slash \over 2\,m_b}\, \Bigg]\, h^{(b)}_v + {\alpha_s\over\pi}\, \Big[ V^{\mu\,(1)} + V^{\mu\,(2)} \Big] +\ldots \,, \nonumber\\* A^\mu &=& \bar h^{(c)}_{v'}\, \Bigg[ \gamma^\mu\gamma_5-{i\overleftarrow D\!\!\!\!\!\slash\, \gamma^\mu\gamma_5 \over 2\,m_c} + {i\gamma^\mu\gamma_5 \overrightarrow D\!\!\!\!\!\slash \over 2\,m_b}\, \Bigg]\, h^{(b)}_v + {\alpha_s\over\pi}\, \Big[ A^{\mu\,(1)} + A^{\mu\,(2)} \Big] +\ldots \,,\end{aligned}$$ where the ellipses denote terms higher order in $\alpha_s$ and $\Lambda_{\rm QCD}/m_Q$. Superscripts $(1)$ denote corrections proportional to $\alpha_s$, $$\begin{aligned} \label{A2} V^{\mu\,(1)} &=& \bar h^{(c)}_{v'}\, \Big[ c_{V1}\gamma^\mu+c_{V2}v^\mu+c_{V3}v'^\mu \Big]\, h^{(b)}_v \,, {\nonumber}\\* A^{\mu\,(1)} &=& \bar h^{(c)}_{v'}\, \Big[ c_{A1}\gamma^\mu+c_{A2}v^\mu+c_{A3}v'^\mu \Big]\,\gamma_5\, h^{(b)}_v \,.\end{aligned}$$ The terms with superscript $(2)$ in Eq. (\[A1\]) denote corrections proportional to $\alpha_s\,\Lambda_{\rm QCD}/m_Q$, $$\begin{aligned} \label{A3} V^{\mu\,(2)} = \bar h^{(c)}_{v'}\, &\Bigg\{& {i\overrightarrow D_{\!\lambda}\over 2\,m_b}\, \bigg[ \Big(c_{V1}\gamma^\mu+c_{V2}v^\mu+c_{V3}v'^\mu\Big) \bigg(\gamma^\lambda + 2 v'^\lambda\,{\frac{\overleftarrow\partial} {\partial w}}\bigg) + 2 c_{V2} g^{\mu\,\lambda} \bigg] \\* &-& {i\overleftarrow D_{\!\lambda}\over 2\,m_c}\, \bigg[ 2 c_{V3} g^{\mu\,\lambda} + \bigg(\gamma^\lambda + 2v^\lambda\, {\frac{\overrightarrow\partial}{\partial w}}\bigg) \Big(c_{V1}\gamma^\mu+c_{V2}v^\mu+c_{V3}v'^\mu\Big) \bigg] \Bigg\}\, h^{(b)}_v, {\nonumber}\\* A^{\mu\,(2)} = \bar h^{(c)}_{v'}\, &\Bigg\{& {i\overrightarrow D_{\!\lambda}\over 2\,m_b}\, \bigg[ \Big(c_{A1}\gamma^\mu+c_{A2}v^\mu + c_{A3}v'^\mu\Big)\,\gamma_5\, \bigg(\gamma^\lambda + 2v'^\lambda\, {\frac{\overleftarrow\partial}{\partial w}}\bigg) + 2 c_{A2} g^{\mu\,\lambda}\,\gamma_5\, \bigg] {\nonumber}\\* &-& {i\overleftarrow D_{\!\lambda}\over 2\,m_c}\, \bigg[ 2 c_{A3} g^{\mu\,\lambda} + \bigg(\gamma^\lambda + 2v^\lambda\, {\frac{\overrightarrow\partial }{\partial w}}\bigg) \Big(c_{A1}\gamma^\mu+c_{A2}v^\mu+c_{A3}v'^\mu\Big) \bigg]\, \gamma_5\, \Bigg\}\, h^{(b)}_v \,.{\nonumber}\end{aligned}$$ In these expressions the covariant derivatives, $D_\lambda$, act on the fields $h_v^{(b)}$ or $h_{v'}^{(c)}$, and partial derivatives with respect to $w$, $\partial/\partial w$, act on the coefficient functions $c_{Vi}(w)$ and $c_{Ai}(w)$. Using Eqs. (\[A2\]) and (\[A3\]) it is straightforward to include the order $\alpha_s$ and $\alpha_s\,\Lambda_{\rm QCD}/m_Q$ corrections using trace formalism presented in Sec. II. The corrections with superscript (1) simply change the form of $\Gamma$ in Eq. (\[lo\]), while those with superscript (2) change $\Gamma$ in Eq. (\[1/mc1\]). The $B\to D_1\,e\,\bar\nu_e$ form factors were defined in Eq. (\[formf1\]), and their expansions in terms of Isgur-Wise functions at leading order in $\alpha_s$ were given in Eq. (\[expf\]). The order $\alpha_s$ and order $\alpha_s\,\Lambda_{\rm QCD}/m_Q$ corrections modify the results for $f_i$ in Eq. (\[expf\]) to $f_i+(\alpha_s/\pi)\,\delta f_i$. The functions $\delta f_i$ are given by $$\begin{aligned} \sqrt6\,\delta f_A &=& -(w+1)c_{A1}\tau -2\varepsilon_c\,(w\bar\Lambda'-\bar\Lambda)\Big[ 2c_{A1} +(w+1)c_{A1}' +c_{A3} \Big] \tau {\nonumber}\\* &+& \varepsilon_c\,(w-1)\, \Big\{ [ 3c_{A1}-2(w-1)c_{A3} ] \tau_1- (3c_{A1}+4c_{A3})\tau_2 \Big\} {\nonumber}\\* &-& \varepsilon_b\,\Big[(\bar\Lambda'+\bar\Lambda)(w-1)c_{A1}- 2(\bar\Lambda'-w\bar\Lambda)(w+1)c_{A1}'+2\,(w\bar\Lambda'-\bar\Lambda) c_{A2}\Big] \tau {\nonumber}\\* &+& \varepsilon_b\,(w-1)\, \Big\{ [(2w+1)c_{A1}-2(w-1)c_{A2} ] \tau_1+ (c_{A1}-4c_{A2})\tau_2 \Big\} , \end{aligned}$$ $$\begin{aligned} \sqrt6\,\delta f_{V_1} &=& (1-w^2)c_{V1}\tau - 2\varepsilon_c\, (w\bar\Lambda'-\bar\Lambda)(w+1)\Big[ 2c_{V1}+(w-1)c_{V1}'+2c_{V3} \Big] \tau {\nonumber}\\* &+& \varepsilon_c\, (w^2-1) \Big\{ [ 3c_{V1}+2(w+2)c_{V3} ] \tau_1 - (3c_{V1}+2c_{V3})\tau_2 \Big\} {\nonumber}\\* &-& \varepsilon_b\, (w+1) \Big[(\bar\Lambda'+\bar\Lambda)(w-1)c_{V1} - 2(\bar\Lambda'-w\bar\Lambda)(w-1)c_{V1}'+4(w\bar\Lambda'-\bar\Lambda)c_{V2} \Big] \tau {\nonumber}\\* &+& \varepsilon_b\,(w^2-1) \Big\{ [ (2w+1)c_{V1}+2(w+2)c_{V2} ] \tau_1 + (c_{V1}-2c_{V2})\tau_2 \Big\} , \end{aligned}$$ $$\begin{aligned} \sqrt6\, \delta f_{V_2} &=& - [3c_{V1}+2(w+1)c_{V2}] \tau - 2\varepsilon_c\, (w\bar\Lambda'-\bar\Lambda) \Big[ 3c_{V1}'+2c_{V2}+2(w+1)c_{V2}' \Big] \tau {\nonumber}\\* &-& \varepsilon_c \Big\{ [ (4w-1)c_{V1}-2(2w+1)(w-1)c_{V2}-2(w+2)c_{V3} ] \tau_1 {\nonumber}\\* &&\phantom{\varepsilon_c} + [5c_{V1}+2(1-w)c_{V2}+2c_{V3}]\tau_2 \Big\} {\nonumber}\\* &-& \varepsilon_b\, \Big\{ 3 (\bar\Lambda'+\bar\Lambda)c_{V1}- 6(\bar\Lambda'-w\bar\Lambda)c_{V1}'+2[(w-1)\bar\Lambda'+(3w+1)\bar\Lambda] c_{V2}{\nonumber}\\* &&\phantom{\varepsilon_b} - 4(\bar\Lambda'-w\bar\Lambda)(w+1)c_{V2}' \Big\}\tau {\nonumber}\\* &+& \varepsilon_b \Big\{ [ 3(2w+1)c_{V1}+2(2w^2+1)c_{V2} ] \tau_1 + [ 3c_{V1}+ 2(w-2)c_{V2} ] \tau_2 \Big\} ,\end{aligned}$$ $$\begin{aligned} \sqrt6\, \delta f_{V_3} &=& [ (w-2)c_{V1}-2(w+1)c_{V3} ] \tau {\nonumber}\\* &+& 2\varepsilon_c\, (w\bar\Lambda'-\bar\Lambda) \Big\{ 2c_{V1} +(w-2)c_{V1}'- 2 [ c_{V3}+(w+1)c_{V3}' ] \Big\} \tau {\nonumber}\\* &+& \varepsilon_c \Big\{ [ (2+w)c_{V1}+2(w^2-3w-1)c_{V3} ] \tau_1 + [ (3w+2)c_{V1}+(4w-2)c_{V3} ] \tau_2 \Big\} {\nonumber}\\* &+& \varepsilon_b\, \Big[ (\bar\Lambda'+\bar\Lambda)(w+2)c_{V1}+ 2(\bar\Lambda'-w\bar\Lambda) (2-w)c_{V1}'+4\bar\Lambda'(w+1)c_{V2}{\nonumber}\\* &&\phantom{\varepsilon_b} - 2(\bar\Lambda'+\bar\Lambda)(w-1)c_{V3} +4(\bar\Lambda'-w\bar\Lambda)(w+1)c_{V3}' \Big] \tau {\nonumber}\\* &-& \varepsilon_b\, \Big\{ [ (2w^2+5w+2)c_{V1}+2w(2+w)c_{V2}+2(1+w-2w^2)c_{V3} ] \tau_1 {\nonumber}\\* &&\phantom{\varepsilon_b} + [(2+w)c_{V1}-2wc_{V2}-2(w-1)c_{V3}]\tau_2 \Big\} .\end{aligned}$$ Here $c_{Vi}$ and $c_{Ai}$ are functions of $w$, and prime denotes a derivative with respect to $w$. Note that at zero recoil $\delta f_{V_1}$ is known in terms of $\bar\Lambda'-\bar\Lambda$ and $\tau(1)$, as expected from our results in Sec. II. For $B\to D_2^*\,e\,\bar\nu_e$ decay, the $\alpha_s$ and order $\alpha_s\,\Lambda_{\rm QCD}/m_Q$ corrections modify the leading order form factors in Eq. (\[expk\]) to $k_i\to k_i+(\alpha_s/\pi)\,\delta k_i$. The functions $\delta k_i$ are $$\begin{aligned} \delta k_V &=& -c_{V1}\tau -\varepsilon_c\,\Big[ 2c_{V1}'(w\bar\Lambda'-\bar\Lambda) \tau + (c_{V1}-2wc_{V3})\tau_1-(c_{V1}+2c_{V3})\tau_2 \Big] \\* &-& \varepsilon_b\,\Big\{ [(\bar\Lambda'+\bar\Lambda)c_{V1}-2 (\bar\Lambda'-w\bar\Lambda) c_{V1}']\tau - [(2w+1)c_{V1}+2wc_{V2}] \tau_1 - (c_{V1}+2c_{V2})\tau_2 \Big\}, {\nonumber}\end{aligned}$$ $$\begin{aligned} \delta k_{A_1} &=& -(w+1)c_{A1}\tau -\varepsilon_c\,\Big[ 2 (c_{A1}'+wc_{A1}'-c_{A3}) (w\bar\Lambda'-\bar\Lambda) \tau + (w-1)c_{A1} (\tau_1-\tau_2) {\nonumber}\\* && +2(w^2-1)c_{A3}\tau_1 \Big] - \varepsilon_b\, \Big\{ [(\bar\Lambda'+\bar\Lambda)(w-1)c_{A1}- 2(\bar\Lambda'-w\bar\Lambda)(w+1)c_{A1}'{\nonumber}\\* && -2(w\bar\Lambda'-\bar\Lambda)c_{A2}]\tau - (w-1)[c_{A1}(\tau_1+\tau_2) +2(wc_{A1}-wc_{A2} -c_{A2})\tau_1] \Big\},\end{aligned}$$ $$\begin{aligned} \delta k_{A_2} &=& c_{A2}\tau + \varepsilon_c\,\Big\{ 2c_{A2}' (w\bar\Lambda'-\bar\Lambda) \tau - [ 2c_{A1}-(2w+1)c_{A2}+2c_{A3} ] \tau_1 + c_{A2}\tau_2 \Big\} {\nonumber}\\* &+& \varepsilon_b\,\Big\{ [(\bar\Lambda'+3\bar\Lambda)c_{A2} - 2(\bar\Lambda'-w\bar\Lambda)c_{A2}' ]\tau - (2w+3)c_{A2}\tau_1 - c_{A2}\tau_2 \Big\}, \end{aligned}$$ $$\begin{aligned} \delta k_{A_3} &=& (c_{A1}+c_{A3})\tau +\varepsilon_c\,\Big[ 2(c_{A1}'+c_{A3}') (w\bar\Lambda'-\bar\Lambda) \tau -(c_{A1}-c_{A3})(\tau_1+\tau_2)+4wc_{A3}\tau_1\Big] {\nonumber}\\* &+& \varepsilon_b\,\Big\{ [ (\bar\Lambda'+\bar\Lambda)(c_{A1}+c_{A3}) -2 \bar\Lambda'c_{A2} - 2 (\bar\Lambda'-w\bar\Lambda) (c_{A1}'+c_{A3}')]\tau {\nonumber}\\* &&\phantom{\varepsilon_b} -(c_{A1}+c_{A3})(\tau_1+\tau_2) - 2w(c_{A1}-c_{A2}+c_{A3})\tau_1 \Big\} .\end{aligned}$$ To compute the corrections to the results obtained in Sec. II, it is sufficient to expand the Wilson coefficients $c_{Vi}$ and $c_{Ai}$ to linear order in $w$. We take $c_{Vi}$ and $c_{Ai}$ and their first derivatives at zero recoil from Ref. [@physrep]. To evaluate these, we choose to integrate out the $c$ and $b$ quarks at a common scale $\mu=\sqrt{m_c\,m_b}$, giving for $c_{Vi}$ and $c_{Ai}$ $$\begin{aligned} c_{V1}(1) &=& -\frac43 - {1+z\over1-z}\, \ln z \simeq 0.91 \,, {\nonumber}\\* c_{V2}(1) &=& - {2\,(1-z+z \ln z) \over 3 (1-z)^2} \simeq -0.46 \,, {\nonumber}\\* c_{V3}(1) &=& {2z\, (1-z+\ln z) \over 3 (1-z)^2} \simeq -0.20 \,, {\nonumber}\\ c_{A1}(1) &=& -\frac83 - {1+z\over1-z}\, \ln z \simeq -0.42 \,, {\nonumber}\\* c_{A2}(1) &=& - {2\,[3-2z-z^2+(5-z)z\ln z] \over 3(1-z)^3} \simeq -1.20 \,, {\nonumber}\\* c_{A3}(1) &=& {2z\, [1+2z-3z^2+(5z-1)\ln z] \over 3 (1-z)^3} \simeq 0.42 \,.\end{aligned}$$ The derivatives $c_{Vi}'$ and $c_{Ai}'$ at zero recoil are $$\begin{aligned} c_{V1}'(1) &=& - {2 [ 13-9z+9z^2-13z^3+3(2+3z+3z^2+2z^3)\,\ln z] \over 27\,(1-z)^3 } \simeq 0.20 \,, {\nonumber}\\* c_{V2}'(1) &=& {2 [ 2+3z-6z^2+z^3+6z\,\ln z ] \over 9\,(1-z)^4 } \simeq 0.21 \,, {\nonumber}\\* c_{V3}'(1) &=& {2z [1-6z+3z^2+2z^3-6z^2\,\ln z] \over 9\,(1-z)^4 } \simeq 0.05 \,, {\nonumber}\\ c_{A1}'(1) &=& - {2 [7+9z-9z^2-7z^3+3(2+3z+3z^2+2z^3)\,\ln z ]\over 27\,(1-z)^3} \simeq 0.64 \,, {\nonumber}\\* c_{A2}'(1) &=& {2 [2-33z+9z^2+25z^3-3z^4-6z(1+7z)\,\ln z]\over 9\,(1-z)^5 } \simeq 0.37 \,, {\nonumber}\\* c_{A3}'(1) &=& - {2z [3-25z-9z^2+33z^3-2z^4-6z^2(7+z)\,\ln z]\over 9\,(1-z)^5} \simeq -0.12 \,. \end{aligned}$$ Here $z=m_c/m_b$, and the numbers quoted are for $z=1.4/4.8$. Using these values and the $\alpha_s$ corrections for the form factors above, we find the corrections given in Table \[tab:alphas\] to the leading order results summarized in Table \[tab:sec2res\]. N. Isgur and M.B. Wise, Phys. Lett. B232 (1989) 113; Phys. Lett. B237 (1990) 527. A.K. Leibovich [*et al.*]{}, Phys. Rev. Lett. 78 (1997) 3995. N. Isgur and M.B. Wise, Phys. Rev. Lett. 66 (1991) 1130. E. Eichten and B. Hill, Phys. Lett. B234 (1990) 511;\ H. Georgi, Phys. Lett. B240 (1990) 447. E. Eichten and B. Hill, Phys. Lett. B243 (1990) 427;\ A.F. Falk [*et al.*]{}, Nucl. Phys. B357 (1991) 185. M. Feindt, Talk given at the Second International Conference on $B$ physics and $CP$ violation, Honolulu, March 1997;\ G. Eigen, Talk given at the Seventh International Symposium on Heavy Flavor Physics, Santa Barbara, July 1997. M. Gremm [*et al.*]{}, Phys. Rev. Lett. 77 (1996) 20;\ M. Gremm and I. Stewart, Phys. Rev. D55 (1997) 1226. M. Gremm and A. Kapustin, Phys. Rev. D55 (1997) 6924. 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Kapustin [*et al.*]{}, Phys. Lett. B375 (1996) 327;\ C.G. Boyd [*et al.*]{}, Phys. Rev. D55 (1997) 3027;\ A. Czarnecki [*et al.*]{}, TPI-MINN-97-19 \[hep-ph/9706311\]. M. Luke and A.V. Manohar, Phys. Lett. B286 (1992) 348. A.F. Falk and B. Grinstein, Phys. Lett. B247 (1990) 406;\ M. Neubert, Phys. Lett. B306 (1993) 357; Nucl. Phys. B416 (1994) 786. M. Neubert, Phys. Rept. 245 (1994) 259; and references therein. [^1]: Hadron states labeled by their four-velocity, $v=p_H/m_H$, satisfy the standard covariant normalization $\langle H(p_H')\,|\,H(p_H)\rangle = (2\pi)^3\,2E_H\,\delta^3(\vec{p}_H^{\,\prime} - \vec{p}_H)$. [^2]: The $B_{s1}$ and $B_{s2}^*$ masses could also be used to determine $\bar\Lambda'-\bar\Lambda$ from the relation $$\begin{aligned} \bar\Lambda' - \bar\Lambda = \bar\Lambda_s' - \bar\Lambda + (\overline{m}_D' - \overline{m}_{D_s}') + {\cal O}(\Lambda_{\rm QCD}\,m_s/m_c) \,, \nonumber\end{aligned}$$ with the analog of Eq. (\[HQET\_diff\]) used to fix $\bar\Lambda_s'-\bar\Lambda$, and $\overline{m}_D'-\overline{m}_{D_s}'=-114\,$MeV. The $B_s^*$ has not been observed, but its mass can be determined from $(m_{B_s^*}-m_{B_s})-(m_{B^*}-m_B) = (m_c/m_b)\,[(m_{D_s^*}-m_{D_s})-(m_{D^*}-m_D)]$. However, because of uncertainties in the $B_{s1}$ and $B_{s2}^*$ masses and the unknown order $(\Lambda_{\rm QCD}\,m_s/m_c)$ term, this relation does not give a more reliable determination of $\bar\Lambda'-\bar\Lambda$ than Eq. (\[HQET\_diff\]). [^3]: Order $\Lambda_{\rm QCD}/m_c$ corrections were also analyzed in Ref. [@Mannel]. We find that $\tau_4$ (denoted $\xi_4$ in [@Mannel]) does contribute in Eq. (\[curr\]) for $\Gamma=\gamma_\lambda\widetilde\Gamma$, and corrections to the Lagrangian are parameterized by more functions than in [@Mannel]. [^4]: In Ref. [@llsw] two out of these four relations were obtained (only those two were needed to get Eq. (\[tau41\])). We thank M. Neubert for pointing out that there are two additional constraints. [^5]: Approximation A differs from our discussion in Ref. [@llsw] only in the separation of the different helicity states of the excited charmed mesons, and keeping the $1-2rw+r^2$ factors for the helicity one states as well as the $(w^2-1)^{3/2}$ terms for the $D_2^*$ rates unexpanded. [^6]: We thank A. Le Yaouanc for pointing out the importance of these terms. [^7]: A relation between $\tau_{1,2}$ and $\zeta_1$ may also hold in this model. [^8]: The lightest $\frac12^-$ states may be narrow since decays to the $s_\ell^{\pi_\ell}=\frac12^-$ and $\frac32^-$ multiplets are suppressed by the available phase space, and decays to $D^{(*)}\,\pi$ in an $S$-wave are forbidden by parity. [^9]: In Ref. [@IWsr] $|\zeta^{(m)}(1)|^2/4$ was denoted by $|\tau_{1/2}^{(m)}(1)|^2$, and $|\tau^{(p)}(1)|^2/3$ was denoted by $|\tau_{3/2}^{(p)}(1)|^2$.

--- abstract: 'Using a density functional based electronic structure method, we study the effect of perturbations on the surface state Dirac cone of a strong topological insulator Bi$_2$Se$_3$ from both the intrinsic and extrinsic sources. We consider atomic relaxations, and film thickness as intrinsic and interfacial thin dielectric films as an extrinsic source of perturbation to the surface states. We find that atomic relaxations has no effect on the degeneracy of the Dirac cone whereas film thickness has considerable effect on the surface states inducing a gap which increases monotonically with decrease in film thickness. We consider two insulating substrates BN and quartz as dielectric films and show that surface terminations of quartz with or without passivation plays critical role in preserving Dirac cone degeneracy whereas BN is more inert to the TI surface states. The relative orbital contribution with respect to bulk is mapped out using a simple algorithm, and with the help of it we demonstrate the bulk band inversion when spin-orbit coupling is switched on. The layer projected charge density distributions of the surface states shows that these states are not strictly confined to the surface. The spatial confinement of these states extends upto two to three quintuple layers, a quintuple layer consists of five atomic layers of Bi and Se.' author: - Jiwon Chang - Priyamvada Jadaun - 'Leonard F. Register' - 'Sanjay K. Banerjee' - Bhagawan Sahu title: 'Intrinsic and extrinsic perturbations on the topological insulator Bi$_2$Se$_3$ surface states' --- Introduction ============ Three dimensional (3D) topological band insulators (TI), Bi$_2$X$_3$, X=Se, Te and Sb$_2$Te$_3$ have attracted considerable attention from the condensed matter physics community because of the relatively simple crystal structure that hosts its novel surface states[@zhang1]. Many more 3D TI materials have been predicted[@new3d] by now and quest for studying their novel surface state properties has increased dramatically. The surface state of 3D TI are time-reversal symmetric (TRS) in some regions of momentum space and are therefore protected against perturbations which cannot break TRS. These considerations also suggest the possibility of dissipationless transport within these surface states where it would be relatively insensitive to non-magnetic disorder or any perturbations that protect the TRS properties of these states[@zahid]. Such properties have caused excitement in the electron device community. The advances in undertanding of structural, electronic, magnetic and transport properties of 3D TI, made possible by both experimental and theoretical studies[@zhang2], have allowed it to gain attention in the scientific community[@star1]. It is, however, not clear whether perturbations, both intrinsic and extrinsic, can affect the surface state Dirac cone, by breaking the TRS. Therefore, it is necessary to study 3D TI and the novel surfaces states, using [*ab-initio*]{} methods. To understand the effects of these variables on the surface states, we report on the density functional based electronic structure studies of one such strong 3D TI Bi$_2$Se$_3$. Our study suggests a critical thickness of six quintuple layers (QLs) needed to maintain the degeneracy of the electron-hole bands at the Dirac point and the surface states have spatial extension to within 2$\sim$3QLs. Atomic relaxations have no effect on the linear spectrum but thin dielectric films (crystalline BN and SiO$_2$) are found to play a critical role in breaking the surface state degeneracy, depending upon the nature of surface terminations and passivations. Our studies will be useful for designing electronic devices using a 3D TI and will have implications in interpreting experiments. We begin by describing the bulk crystal structure of Bi$_2$Se$_3$ and the computational method used for this study in section II. In section III, we present the electronic structure of bulk TI and a simple algorithm to map out the surface states from the bulk band structure, and we use this method to explain band inversion when spin-orbit coupling is switched on. We then discuss the thin-film structure and the thickness dependent surface band structure of Bi$_2$Se$_3$ in section IV. The spatial extension of these surface states are then determined using charge densities projected onto each atomic layer and each atom. The effect of atomic relaxations and thin dielectric films on the surface electronic structure will be discussed in section V. Finally we present our summary and conclusions. Bulk Structure and Computational Method ======================================= The bulk crystal structure of Bi$_2$Se$_3$ is trigonal (or rhombohedral) with lattice constant of [*a*]{}= 0.984 nm and $\alpha$$\sim$25$^o$. The primitive cell consists of five atoms (two Bi and three Se atoms) arranged in a order Se2-Se1-Bi-Bi-Se1 where Se1 and Se2 are two inequivalent sites of Se atoms in the cell. Figure 1(a) shows a cartoon of bulk rhombohedral structure along with three primitive lattice vectors that span it (the tracing inside the large cell). The three-fold rotation axis is taken along the [*z*]{}-axis. Alternatively, one can construct a hexagonal cell from the trigonal cell so that the number of atoms is tripled with the lattice parameters [*a*]{}=0.4138 nm and [*c*]{}=2.8633 nm (Fig. 1(a), the larger cell). The building block of the hexagonal bulk Bi$_2$Se$_3$ crystal consists of five atomic layers, each layer containing only one atom. One such building block is referred to as quintuple layer (QL) (square shaded region in Fig. 1(a)). The hexagonal unit cell contains three such QLs, i.e., 15 atomic layers stacked along the [*z*]{}-direction. The atomic planes are arranged in a sequence Se1-Bi-Se2-Bi-Se1, which is different from the trigonal case, with the following interlayer distances: [*d*]{}(Se1-Bi)= 0.156344 nm, [*d*]{}(Bi-Se2)=0.190665 nm, [*d*]{}(Se2-Bi)=0.190665 nm, [*d*]{}(Bi-Se1)=0.156353 nm and [*d*]{}(Se1-Se1’)=0.259304 nm where Se1’ is the Se1 site in the next unit cell. Because of the relatively larger distance between two QLs, their anchoring along the [*z*]{}-direction is considered to be rather weak. Indeed, mechanical exfoliation of Bi$_2$Se$_3$ demonstrated recently[@balan1] hints at van der Waal’s type of bonding between two neighboring unit cells, much like a graphene exfoliation. We use a DFT-based electronic structure method implemented with projector-augmented wave basis and pseudopotentials[@vasp1]. Since the topological features of the surface states arise from spin-orbit coupled states of the bulk, we invoke its spin-orbit coupling (SoC) feature[@vasp2] and we choose exchange-correlation as Perdew-Burke-Ernzerof (PBE) type[@perdew]. The choice of the PBE over local-density approximation is guided by its use in previous structural studies on bulk Bi$_2$Se$_3$ which reproduced the photoemission spectra very well[@zhang2]. These studies find the optimized lattice constant of Bi$_2$Se$_3$ close to the experimental value, therefore we did not optimize lattice constants of the bulk structure in our calculations. A kinetic energy cut-off of 25 Ry and a [**k**]{}-point mesh of 10 $\times$ 10 $\times$ 10 in the first Brillouin Zone (BZ) (Fig. 1(b)) are used for self-consistent steps as well as generating the bulk band structures. Larger energy cut-offs and [**k**]{}-mesh size were tested for total energy convergence. The first BZ of the bulk structure contains four time-reversal invariant (TRI) points, namely, $\bf \Gamma$, [**L**]{}, [**Z**]{} and [**F**]{} (Fig.1(b)). Bulk band structure =================== In this section, we address the band structure of bulk Bi$_2$Se$_3$, with and without SoC. We demonstrate the band inversion with switching on of SoC with the help of an algorithm which maps out orbital contribution to the electronic states from bulk bands. Figure 2(a) and (b), respectively, shows energy dispersion curves for the bulk Bi$_2$Se$_3$ without and with SoC, along high-symmetry directions connecting the four TRI points (Fig. 1(b). The direct band gap, in both cases, forms at the $\Gamma$-point and is 200 meV(285 meV) without(with) SoC. These gaps compare well with those obtained in other DFT simulations[@zhang2]. It was suggested theoretically, using a model Hamiltonian based on atomic orbitals and without SoC, that the valence band maximum (VBM) and conduction band minimum (CBM) mainly consists of Se and Bi $p_z$ orbitals[@zhang3], respectively (Fig. 2(a)). Switching on the SoC inverts the bands at the $\bf \Gamma$-point (Fig. 2(b)) and consequently the contributions from Bi and Se ions associated with these bands. \ To understand the band inversion at the $\Gamma$-point visually, we adopt a simple algorithm to map out the most relevant orbital contribution to VBM and CBM[@park]. Since $p_z$ orbitals mainly contribute to these bands near the Fermi level, the algorithm projects the crystal wave-functions on to the angular momentum dependent spherical harmonics at a particular [**k**]{}-point with the associated band energies for each atom. The primitive unit cell of bulk Bi$_2$Se$_3$ contains three Se and two Bi atoms and using the above algorithm, we obtain [*s*]{} and three [*p*]{} orbitals contribution for each of the atom type at each [*k-point*]{} and band energy. We take the ratio of Se and Bi $p_z$ orbitals contributions compared to the total orbital contributions resulting from all the orbitals of five atoms at a particular [**k**]{}-point and the band energy. If this ratio is 40$\%$ (60$\%$) or greater for Bi (Se), we take the $p_z$ orbital contribution to that band from Bi or Se at that [**k**]{}-point with the associated band energy; otherwise, we neglect it. The choice of the percentage cut-off is guided by the number of each type of atoms in the unit cell: two for Bi versus three for Se. We find that changing the cut-off or critical percentage ratio makes the contributions to each band at each [**k**]{}-point different but the overall contribution to the VBM and CBM at the $\Gamma$ point remains the same. In this sense, the choice of critical percentage is arbitrary. We use the above algorithm both without and with SoC calculations, and as a result we could see the band inversion as seen in Fig. 2(b). In the literature, this inversion process is termed as phase transition from a trivial insulator (Bi$_2$Se$_3$ without SoC) to a topological insulator. \ Thin film band structures and surface state localization ======================================================== The most interesting feature emerges when thin-film structures are considered. A gapless linear spectrum with continuum in the bulk is evident at the TRI point $\Gamma$ (for example Figure 4(e) for a thin film consisting of 6QLs). These states are contributed by both the surfaces of the thin film and are spin-polarized. The thin film structure is constructed by stacking up the quintuple layers along the [*z*]{}-direction with a vacuum region that forms the supercell in the DFT calculation (Fig. 3(a)). We consider several thicknesses of the thin film structure: 2QLs, 3QLs, 4QLs, 5QLs and 6QLs to study the robustness of the linear energy dispersion and for each thickness we used a vacuum size of about 3 nm. We did not relax the atomic positions. The relaxations and their effect will be considered later. Kinetic energy cut-off of 25 Ry and 10 $\times$ 10 $\times$ 1 [**k**]{}-point mesh in the surface BZ of the hexagonal Bi$_2$Se$_3$ (Fig. 3(b)) is used. On decreasing the thickness from 6QLs to 5QLs, a gap seems to be induced at the degenerate point of the linear spectrum at $\Gamma$ (Fig. 4(d)). The magnitude of the induced gap increases monotonically going from 5QLs to 4QLs to 3QLs and finally to 2QLs (Figs. 4(a-c)). These gaps are tabulated in Table I. This suggests a critical thickness of at least 6 QLs in experiments in order to access the novel surface states of Bi$_2$Se$_3$. We note that thickness dependent gaps in Bi$_2$Te$_3$ were realized experimentally[@Li] and predicted theoretically[@park]. In Bi$_2$Se$_3$, we are aware of only one theoretical work which suggests thickness dependent electronic structure[@liu]. To understand the origin of the induced gap in the linear spectrum, we first map out all the orbital contributions to the bulk band structure from all the atoms in first few layers of each thickness, from the top and bottom region of the thin film (Figs. 4(a)-(e)). For thickness of 3QLs-5QLs, the contributions from the top or bottom one quintuple layer is chosen whereas for 2QLs system, we consider the top or bottom two atomic layers for estimating the contributions. The contributions are calculated using the same procedure as explained in Section III with two exceptions: 1) The sum of contributions from all the orbitals and atom types are considered (not just from the $p_z$ orbitals) and 2) the critical percentages, for accepting it as a contribution, are different for different thicknesses. The critical percentage for 2QLs is fixed at 30 $\%$ whereas for 3QLs-6QLs, percentages are respectivley, 60$\%$, 55$\%$, 50$\%$ and 45$\%$. The choice of these critical percentages are arbitrary. We find that choosing different percentages do not change the overall features in the band structures. We note that thinner structures have larger critical percentage for accepting the contributions with reduction in percentages for thicker structures. This is motivated by the fact that thicker structures have more QLs than the thinner ones and we fixed the contributions to only 1 QL for all thicknesses, except for 2 QL structures where two atomic layers are considered to be a cut-off limit. So keeping the same critical percentage for all thicknesses will offset the number of quintuple layers to be considered for acceptance as a contribution and different thickness will need different number of quintuple layers. Alternatively, one can fix the critical percentage for all the thicknessess to the same value and let the number of quintuple layers that contribute to the bulk band structure differ from one thickness to another. Either consideration leads to the same conclusion, namely the contribution to the Dirac cone near $\Gamma$ originates from the first few layers of the thin film. To locate the critical number of quintuple layers upto which surface states contributing to the Dirac cone is spread in real space, we use a method in which atomic layer dependent charge density from the surface wave functions is calculated. We choose an energy window of 50 meV around the Fermi level in 4QLs-6QLs bandstructures (Figs 4(c)-(e)) with the states in the neighborhood of the $\Gamma$-point and plot the layer projected relative charge density (Figs. 5(a)-(c)). The relative charge density is defined to be the ratio of maximum charge density in a given layer to the maximum among the maximal charge densities of all the layers in that structure. Our calculations hint at 2-3 QLs spatial extension of the surface states in 4QLs-6QLs. The charge density distribution for 2QLs and 3QLs is not shown because within the energy window of 50 meV considered for this calculations, bulk as well surface states both contribute and it is not a pure surface state contribution (Figs. 4(a) and (b)). The surface state localization length to within 2QLs-3QLs in real space hint at possible interactions due to their overlap in thinner structures. This surface state interactions result in opening of the band gap. However, the time-reversal symmetry is not broken at the $\Gamma$-point; the four-fold degenerate band splits into two-fold degenerate bands. The overlap in thinner structures results in finite charge density in a region where surface states for thicker structures had no densities (Figs. 5(a) and (b)). As the degree of overlap increases with decreasing thickness, the induced gap increases monotonically, as seen from Table I. The surface state localization deep inside the bulk region has implications for experiments where measured charge mobility can only be assigned to both suface and bulk states and it will be a challenge in experiments to separate these two contributions. Thickness (QL$\sim$1nm) 2 3 4 5 6 ------------------------- -------- -------- -------- -------- -------- Band gap 0.1281 0.0430 0.0155 0.0051 0.0000 at $\bar{\Gamma}$ : Induced band gaps (in eV) for thin-films with various thicknesses at the time-reversal invariant point $\bar{\Gamma}$. \ \ Effects of Atomic Relaxations and the Dielectric films ====================================================== As the Bi$_2$Se$_3$ thin film is contructed from the bulk crystal, atomic relaxations are inevitable. As a result, both Dirac cone degeneracy and surface state localization length may be affected. Since the 6QLs electronic spectrum has no gap, it is natural to consider this structure for the relaxation study. We considered the same computational parameter as the unrelaxed case but allowed only the [*z*]{}-component of the atom to relax. Since a vacuum along [*z*]{}-direction is present, atomic motions are more likely near the vacuum region than within the rigid planar structure. The total energy is assumed to have converged when the [*z*]{}-component of the Hellman-Feynman force is smaller than 0.025 eV/Å. We find that the Dirac cone degneracy is not perturbed with atomic relaxations (Fig. 6(a)) and overall band structure features are retained except for small rearrangements of bands throughout the spectrum. This result suggests the robustness of the topologically protected surface states with respect to atomic relaxations. The extraction of surface states for each band and each [**k**]{}-point is performed as explained in Section III, and the results are denoted by circles (Fig. 6(b)). The charge density distribution for each layer is determined in a similar way to that in section IV. Figure 6(c) shows the relative charge density of the relaxed structure compared to the non-relaxed case. The overall trends exhibited for the non-relaxed case is maintained, but the charge density seems to be enhanced in the second QL from each side of the surface. This hints at strong spread of the surface states in the second QL as compared to the non-relaxed case. We considered thin dielectric films of crystalline boron nitride (BN) and quartz (or SiO$_2$) to study their effect on surface states. We note here that amorphous SiO$_2$ is used in practical devices so our results for quartz are at best qualitative. We first discuss the interface structure of BN and Bi$_2$Se$_3$ and its effect on surface band structure of Bi$_2$Se$_3$. BN crystallizes in hexagonal structure with [*a*]{} = 0.2494 nm and [*c*]{}= 0.333 nm with d(B-N) =0.144 nm[@CrystalBN]. The bulk direct band gap is 5.97 eV[@kanda]. We use GGA for Bi$_2$Se$_3$ studies. However, use of GGA to optimize the bulk positions and lattice parameters for BN results in overestimated value of lattice constants and it underestimates the bandgap by 33$\%$ [@brink]. Therefore, for the interface studies, we used experimental values of bulk lattice parameters and positions. With these experimental values, our calculated bulk band gap value is same as fully optimized DFT calculations suggest[@openmx]. Our analysis suggests that to keep our computational burden in DFT-based calculations minimal, 3 $\times$ d(B-N) lattice structure can be matched in-plane with 1 $\times$ 1 Bi$_2$Se$_3$ cell or any multiple of this combination that can maintains 3:1 lattice size ratio resulting always in 4$\%$ natural strain. Any other combinations result in either difficulty in forming the periodic structure or large supercell-size with increase of number of atoms by orders of magnitude to allow strain less than 4$\%$. Along the stacking direction, 6 QLs of Bi$_2$Se$_3$ ($\sim$ 6 nm), which has no band gap for surface states, is put on six layers of BN ($\sim$ 1.7 nm). The choice of 6 BN layer is somewhat arbitrary and guided by the fact the size of vacuum and BN layers should be thick enough to avoid interactions of periodically repeated Bi$_2$Se$_3$ surface layers. Since the atomic displacements in Bi$_2$Se$_3$ play no role in breaking the surface state degeneracy, we strained the BN lattice, instead of the TI lattice, to fit with Bi$_2$Se$_3$ surface resulting in compressive strain. We did not relax the interfacial atomic positions to check whether the strained BN lattice can affect the Dirac cone at the $\Gamma$ point. We considered four configurations of Se positions with respect to boron and nitrogen positions in BN: Se on the top of B, on the top of N, on the hexagonal hole and in the interstitial region between B and N. We note here that BN layers cannot be put on both sides of Bi$_2$Se$_3$ film for Se on the top of B configuration because the same orientations are not maintained between the positions of Se and B (Fig. 7). This occurs because of need to satisfy simultaneously the symmetry of Bi$_2$Se$_3$ layers and BN layers in forming the supercell structure. The same holds true for Se on the top of N atom. However, for Se on the top of the hexagonal hole or at the interstitial region between B and N, the orientations are same on both sides (Fig. 8). For the sake of consistency, we calculated the band structures of the interface structures with BN only on one side of Bi$_2$Se$_3$ for all configurations. For the interface simulations, we used 7 $\times$ 7 $\times$ 1 [**k**]{}-point mesh and 2 nm thick vacuum layer. We adjusted the separation between the Se layer and the BN layer. We found the optimal separation of 0.3 nm, and all the studied configurations of Se with respect to BN layer are energetically close and give quite similar band structures. We chose one of these configurations namely Se on B for further discussions. Figure 9 shows the band structure of Bi$_2$Se$_3$ on BN at the optimal interfacial separation of 0.3 nm. It is evident that BN has negligible effect on the surface state dispersion of Bi$_2$Se$_3$ and the Dirac cone degeneracy at the $\Gamma$-point is preserved with no change in the band gap of BN layers. However, the bulk band gap value of Bi$_2$Se$_3$ in the slab structure changes slightly. We now discuss our results for surface state dispersion in presence of crystalline quartz (or $\alpha$-SiO$_2$). We put quartz on both sides of the TI film. The hexagonal crystal structure of quartz contains four-fold coordinated oxygens, forming a layered structure with Si. The lattice parameters are [*a*]{}=0.4914 nm and [*c*]{}=0.5408 nm[@quartz]. Our DFT calculations of optimized lattice paramters with PBE potentials are found to be close to the experimental values, accurate to within 0.1$\%$. Therefore, in this section we take experimental lattice parameters for building our interface structure of TI and quartz. The bulk amorphous quartz has direct band gap of 8.9 eV[@sio2gap] and our DFT calculation on crystalline SiO$_2$ results in a bandgap value of 6 eV. The interface supercell structure consists of six quintuple layers of Bi$_2$Se$_3$ and two unit cells of SiO$_2$ stacked along the [*z*]{}-axis. The size of the SiO$_2$ and the Bi$_2$Se$_3$ cell along [*x*]{}-[*y*]{} direction chosen is, respectively, 1 $\times$ 1 and 2 $\times$ 2 to keep the computational burden minimal. This produces about 2.75$\%$ compressive strain on both sides of the TI surface. Consideration of larger sizes results in lower strain but the total number of atoms in the cell increases atleast an order of magnitude (250 versus 2500). We did not relax the interfacial atomic positions to check whether the resulting strain can affect the TI Dirac cone. We use 7 $\times$ 7 $\times$ 1 [**k**]{}-point mesh for the BZ integration. Periodicity is assumed along [*x*]{}- [*y*]{} direction. We performed electronic structure calculations at various interplaner distances between the quartz and Bi$_2$Se$_3$ to fix the optimal distance at which the dielectric oxide effects can be assessed. We consider two surface terminations for quartz: Si- and O-termination each with and without hydrogen passivation of dangling states (Fig. 10). Without hydrogen saturation in O-terminated quartz, the Dirac cone of TI is destroyed whereas in Si-terminated quartz, it is preserved. The interplaner separation at the interface region is found to be 0.3 nm for Si-terimated quartz without passivation. Hydrogen saturation reduces the optimal separation to 0.25 nm in O-termination but the optimal distance is insenstitive to the hydrogen saturation in Si-termination. Figures 11 show band structures of the TI in presence of oxygen terminated quartz with hydrogen passivation. Similar band structure is obtained with Si-termination (Figure not shown). The Dirac cone is seen to be insensitive to both the terminations with hydrogen saturation. The band gaps of thin films of quartz with hydrogen saturated oxygen and with Si terminations are calculated to be 9.23 and 8 eV, respectively. In presence of TI, gaps of each of these terminations reduce but the bulk gaps of TI remain unchanged. Summary and Conclusions ======================= We use a density functional based electronic structure method to study the effect of intrinsic and extrinsic perturbations on the linear spectrum of a strong topological insulator Bi$_2$Se$_3$. Narrow Bi$_2$Se$_3$ film thickness, atomic relaxation, and applied dielectric films of BN and quartz are considered as perturbations. The thickness of the Bi$_2$Se$_3$ film has a considerable effect on the surface state Dirac cone, inducing a gap due to overlap of surface states orginating from two sides of the film. We estimated surface state localization length, by determining the layer projected valence charge density from the states with energies spanning few tens of meV around the Fermi level. The localization length is found to be within 2-3QLs from each side of the film. With increasing overlap between the surface states in thinner structures, the induced gap increases monotonically. We map out the atom and orbital projected bands from the crystal wave-functions. We then extract the surface state contributions to the thin-film band structure and explain the band inversion in bulk TI when spin-orbit interaction is included. 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--- abstract: 'We demonstrate gate-control of the electronic $g$-tensor in single and double quantum dots formed along a bend in a carbon nanotube. From the dependence of the single-dot excitation spectrum on magnetic field magnitude and direction, we extract spin-orbit coupling, valley coupling, spin and orbital magnetic moments. Gate control of the $g$-tensor is measured using the splitting of the Kondo peak in conductance as a sensitive probe of Zeeman energy. In the double-quantum-dot regime, the magnetic field dependence of the position of cotunneling lines in the two-dimensional charge stability diagram is used to infer the real-space positions of the two dots along the nanotube.' author: - 'R. A. Lai' - 'H. O. H. Churchill' - 'C. M. Marcus' bibliography: - 'Refs.bib' title: '$g$-Tensor Control in Bent Carbon Nanotube Quantum Dots' --- [^1] [^2] Carbon nanotubes have several attractive properties that make them favorable candidates for spin qubits and spintronics applications, with many recent experimental advances particularly in the area of few-electron quantum devices [@mason2004local; @grove2008triple; @jorgensen2008singlet; @buitelaar2008pauli; @steele2009tunable; @hughc132009]. In addition to a low concentration of nuclear spins and large confinement energies, nanotubes exhibit a unique circumferential spin-orbit coupling, which has been described theoretically [@izumida2009] and characterized experimentally [@ferdi2008; @hugh2009; @jespersen2011]. This spin-orbit coupling generates an effective magnetic field parallel to the axis of the nanotube. Bends in nanotubes couple position and spin by creating a spatial dependence of the direction and magnitude of this effective magnetic field [@flensberg2010]. Recently, this effect has been used to facilitate electron dipole spin resonance and qubit manipulation [@pei2012valley; @laird2012]. Here, we demonstrate another key feature of nanotubes with bends: control of the $g$-tensor via electrostatic gates [@flensberg2010]. In addition, we demonstrate the use of $g$-tensor anisotropy to extract the positions of gate-defined single and double quantum dots along a curved nanotube, taking advantage of the $g$-to-position mapping made possible by a bend. The device was based on a single-walled carbon nanotube grown by methane chemical vapor deposition. During growth, van der Waals forces bind large numbers of nanotubes to the silicon substrate in random orientations, including bent configurations. Though not used here, deliberate bending of nanotubes has also been demonstrated [@falvo1997nanotubes; @poncharal1999electrostatic; @biercuk2004; @geblinger2008self]. A bent tube was identified by scanning electron microscopy, and Ti/Pd contacts (5/50 nm thick) and a local side-gate (SG) proximal to the left arm of the bend were patterned by electron beam lithography \[Fig. 1(a)\]. The degenerately doped silicon substrate formed a global back-gate (BG) insulated by $0.5\,\mu$m of thermal oxide. The radius of curvature was $\sim\,0.5\, \mu$m, with the two arms of the bend at angles $130^{\circ}$ and $0^{\circ}$ relative to the x-axis. The device was measured in a dilution refrigerator with an electron temperature of 0.1 K using direct current and lock-in techniques with a vector magnetic field $\vec{B}$, which we label in polar and cartesian coordinates in the x-y plane of the tube as $\vec{B}=(B,{\varphi})=({B_{\rm{x}}},{B_{\rm{y}}})$. \[figure1\] {width="3.4in"} Coulomb diamonds with fourfold shell structure were observed in differential conductivity, $dI/dV$, as a function of source-drain bias, ${V_{\rm{SD}}}$, and back-gate voltage, ${V_{\rm{BG}}}$, with the side-gate grounded, as shown in Fig. 1(b) [@jarillo2005orbital; @liang2002]. A group of four resonances closely spaced in ${V_{\rm{BG}}}$, corresponding to consecutive filling of the spin-valley levels of a longitudinal orbital shell, can be seen in Fig. 1(b). A repeating pattern of three-two-three conductance ridges for ${V_{\rm{SD}}}<$ 0.5 mV are seen at occupancies of one, two, and three electrons, respectively, above a full shell, which we attribute to the Kondo features in even and odd occupied Coulomb blockade valleys. The single-dot spectrum as a function of ${B_{\rm{x}}}$ was extracted from conductance data in the Kondo regime at ${V_{\rm{BG}}}=0.616$ V and ${B_{\rm{y}}}= 0$, using the inflection point of conductance as a function of ${V_{\rm{SD}}}$ to measure level position [@rosch2003nonequilibrium], as shown in Fig 1(c). The dependence of the spectrum on ${B_{\rm{x}}}$ can be understood as resulting from a combination of spin-orbit coupling and/or valley mixing that breaks the degeneracy of the four spin-valley states, consistent with theory [@fang2008] and previous experiment [@jespersen2011]. Fitting the spectrum at negative bias to a model of a locally straight nanotube [@jespersen2011] yields spin (43 $\mu$eV/T) and valley (290 $\mu$eV/T) magnetic moments, spin-orbit (220 $\mu$eV) and valley (70 $\mu$eV) couplings, and magnetic field angle relative to the nanotube (15$^{\circ}$) \[Fig. 1(c)\]. The model also includes a threshold magnetic field of $0.15$ T of Kondo peak splitting [@kogan2004measurements; @costi2000kondo]. We observed that the cotunneling spectrum is asymmetric in bias, as seen previously [@Amasha2005Kondo]. Device parameters are consistent with previous measurements [@ferdi2008; @hugh2009; @jespersen2011], except for the spin $g$-factor (1.5), which is somewhat smaller than previously reported ($g=2$). \[figure2\] {width="3.4in"} Differential conductance, $dI/dV$, as a function of ${V_{\rm{SD}}}$ and ${V_{\rm{SG}}}$ with ${V_{\rm{BG}}}=0.72$ V showed Coulomb diamonds with Kondo ridges at zero bias for every other charge state \[Fig. 2(a)\], similar to those in Fig. 1(b). In this regime, the Kondo peak shows a larger splitting for ${B_{\rm{x}}}$ than for ${B_{\rm{y}}}$ \[Fig. 2(b)\], indicating that $g$ is anisotropic. Anisotropy of the $g$-tensor can also be seen in a plot of $d^2I/dV^2$ as function of bias and magnetic field angle in the plane \[Fig. 2(c)\]. Here, the two inflection points of conductance appear as maxima and minima at positive and negative bias, respectively, with a maximal splitting near ${\varphi}\sim 0^{\circ} (\rm{x})$ and a minimal splitting near ${\varphi}\sim 90^{\circ} (\rm{y})$. Control of the $g$-tensor is achieved by moving the position of the dot along the nanotube bend without changing its occupancy using two gates acting in opposition. For a many-electron quantum dot ($N\sim$ 70), we can introduce a single voltage axis, ${V^*}$, parametrized by ${V_{\rm{SG}}}$, that tracks a Coulomb resonance as both gate voltages (SG and BG) are swept, as shown in Fig. 2(d). At all values of $V^{*}$, we observed a sinusoidal dependence of the Kondo splitting on magnetic field angle in the plane \[Fig. 2(c)\], with the amplitude and phase of the sinusoid depending on $V^{*}$. As an example, Kondo splittings along with sinusoidal fits are shown in a polar plot in Fig. 2(e) for ${V^*}$=4.65 V (red), corresponding to ${V_{\rm{SG}}}=4.65$ V and ${V_{\rm{BG}}}=0.84$ V, and ${V^*}$=4.85 V (purple), corresponding to ${V_{\rm{SG}}}=4.85$ V and ${V_{\rm{BG}}}=0.76$ V. The angles of maximal splitting, ${\varphi_{\rm{max}}}$, are clearly different for the two cases. Figure 2(f) shows values for ${\varphi_{\rm{max}}}$ for five values of ${V^*}$, corresponding to the points marked in Fig. 2(a). The angle where the maximum splitting occurred was found to change monotonically with a rough rate of $\sim8^{\circ}$ for a change in ${V^*}$ of 0.2 V. Based on the increase in ${\varphi_{\rm{max}}}$ for increasing ${V^*}$, we note that in this instance the dot moved toward the right side of the bend as ${V^*}$ became more positive (while the global back gate became more negative). Given the position of the side gate, one might have expected the opposite direction of motion. Due to disorder, however, the dot can move either way as a function of ${V^*}$. Fabricating multigate devices and reducing disorder will both serve to increase control of dot position. Additional features associated with a bend are evident when the device is tuned to form a double quantum dot. This regime is characterized by the familiar honeycomb charge stability diagram, as seen in Fig. 3(a). To allow energy level shifts as a function of magnetic field to be examined for the two dots separately, we define two axes, $V1$ and $V2$, \[dotted lines in Fig. 3(a)\] midway between adjacent triple-point resonances. Changes in the position of points L and R, where axes $V1$ and $V2$ cross the cotunneling lines that define the charge stability boundaries, were used to track the evolution of energy levels of the left and right dots. (The left dot is the one closest to the side gate.) Accurate positions for points L and R were found by fitting cotunneling peaks along $V1$ and $V2$ to the form $I(V)=I_0\cosh^{-2}\left[(V-V_0)/2W\right]$ \[Fig. 3(a) inset\], where $W$ is the peak width due to both temperature and tunneling, expected to be valid for ${V_{\rm{SD}}}\ll W$ [@beenakker1991]. The observed behavior is well described by a capacitance model of a double quantum dot [@vanderwiel] and gives the tunnel coupling $t=0.59$ meV and the mutual charging energy $U_{\rm{Cm}}=1.2$ meV. From the size of the bias triangles at the triple-point between the points L and R in finite-bias measurements, we obtain the gate lever-arms that convert gate voltage and energy [@alphas]. We note that the same lever arm values were found at ${B_{\rm{x}}}=0.5$ T and $1.5$ T. \[figure3\] {width="3.3in"} Cotunneling current along $V1$ and $V2$ as a function of $B$ and ${\varphi}$ are shown in Figs. 3(b-e). The dependence of points L and R on ${\varphi}$ reflects anisotropy in ${g}$ for the left and right dots, respectively. In particular, larger shifts from the zero field values occur when the field is parallel to the section of the nanotube containing the dot; smaller shifts occur when the field is perpendicular to the nanotube segment. We note that the positions of the extrema in this dependence for the left and right dots do not occur at the same angles. From these dependences, we conclude that the left and right dots reside in segments of the tube that make different angles with respect to the applied field. That is, the double dot is on a bend. Two-dimensional plots showing the movement of points R and L from their zero-field values, defined as $\Delta V1$ and $\Delta V2$, as a function of both field angle and magnitude are shown in Figs. 4(a,b). Field direction dependences of $\Delta V1$ and $\Delta V2$ follow similar evolution, with an offset in phase. We also observe that there does not exist a field angle about which the level shifts are symmetric, especially evident for $B<0.5$ T. We interpret the lack of symmetry, which gives Figs. 4(a,b) their overall canted appearance, as indicating that each dot extends along a segment of bent tube. The bend breaks the symmetry that would be present if each dot were within straight segment of tube. To model the angle dependence of $\Delta V1$ and $\Delta V2$, we assume the energy levels of each dot respond to the applied field as a function of the difference of the field angle, ${\varphi}$, and the nanotube angle of a completely localized dot, ${\varphi_{\rm{max}}}$. Thus, the energy levels for the right and left dots in gate voltage respond periodically with magnetic field angle, with a maximum corresponding to parallel field when ${\varphi}={\varphi_{\rm{max}}}$ and a minimum corresponding to perpendicular field, $\Delta V1(2) \propto \left| \cos \frac{\pi}{180}\left({\varphi}-{\varphi_{\rm{max}}}^{R(L)}\right) \right|$. We expect the model to be valid at higher fields, when the orbital magnetic energy is larger than the spin Zeeman energy, valley scattering, spin-orbit coupling, and changes in the charging energy. Figure 4(c) shows good agreement between data at $B=1$ T, fit using the $\cosh^{-2}$ form given above, and the model of phase evolution. Deviations between data and model may reflect the finite extent of the electron distribution or shifts in that distribution with magnetic field angle. We interpret ${\varphi_{\rm{max}}}$ as the tangent angle of the nanotube at the mean position of each quantum dot. With this interpretation, we can convert ${\varphi_{\rm{max}}}$ to a position in real space using the angle distribution of the nanotube \[Fig. 4(d) inset\] and the micrograph of the device \[Fig. 4(d)\]. The positions of the two dots represented by ${\varphi_{\rm{max}}}$ are shown as a red circle (${\varphi_{\rm{max}}}^L=143^{\circ}$) and a purple circle (${\varphi_{\rm{max}}}^R=167^{\circ}$) in Fig. 4(d). We conclude that by using a bend in a carbon nanotube, both the magnitude and angular orientation of the $g$-tensor can be controlled using electrostatic gates. Future experiments could use these effects to produce rapid spin manipulation beyond EDSR, for instance, by initializing a spin in one dot and moving non-adiabatically to the other, resulting in spin rotation at the Larmor frequency. \[figure4\] {width="3in"} Support from the National Science Foundation through the Materials World Network and NRI, through the INDEX Center is acknowledged. We also thank IBM, Harvard University, and the Danish National Research Foundation for support. [^1]: Present address: Department of Physics, Stanford University, Stanford, California 94305, USA. [^2]: Present address: Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02138, USA.

--- abstract: 'The paper discusses an adaptive strategy to effectively control nonlinear manipulation motions of a dual arm robot (DAR) under system uncertainties including parameter variations, actuator nonlinearities and external disturbances. It is proposed that the control scheme is first derived from the dynamic surface control (DSC) method, which allows the robot’s end-effectors to robustly track the desired trajectories. Moreover, since exactly determining the DAR system’s dynamics is impractical due to the system uncertainties, the uncertain system parameters are then proposed to be adaptively estimated by the use of the radial basis function network (RBFN). The adaptation mechanism is derived from the Lyapunov theory, which theoretically guarantees stability of the closed-loop control system. The effectiveness of the proposed RBFN-DSC approach is demonstrated by implementing the algorithm in a synthetic environment with realistic parameters, where the obtained results are highly promising.' author: - Dung Tien Pham - Thai Van Nguyen - Hai Xuan Le - Linh Nguyen - Nguyen Huu Thai - Tuan Anh Phan - Hai Tuan Pham - Anh Hoai Duong bibliography: - 'References.bib' date: 'Received: date / Accepted: date' title: Adaptive neural network based dynamic surface control for uncertain dual arm robots --- Introduction ============ Robots have been increasingly moving into human based environments to replace or assist human workers. More specifically, anthropomorphic or dual arm robots (DAR) have more and more played a vital role in many industrial, health care or household environments [@RN51; @RN52; @RN53; @RN54; @RN55; @RN78]. For instance, dual arm manipulators have been effectively employed in a diversity of tasks including assembling a car, grasping and transporting an object or nursing the elderly [@RN73]. In those scenarios, the DAR have been expected to behave like a human, which is they should be able to manipulate an object similarly to what a person does [@RN53]. As compared to a single arm robot, the DAR have significant advantages such as more flexible movements, higher precision and greater dexterity for handling large objects [@RN56; @RN57]. Nevertheless, since the kinematic and dynamic models of the DAR system are much more complicated than those of a single arm robot, it has more challenges to effectively and efficiently control the DAR, where synchronously coordinating the robot arms are highly expected. In order to accurately and stabily track the robot arms along desired trajectories, a number of the control strategies have been proposed. For instance, the traditional methods such as nonlinear feedback control [@RN58] or hybrid force/position control relied on the kinematics and statics [@RN59; @RN60] have been proposed to simultaneously control both of the arms. In the works [@RN61; @RN62; @RN63], the authors have proposed to utilize the impedance control by considering the dynamic interaction between the robot and its surrounding environment while guaranteeing the desired movements. More importantly, robustness of the control performance is also highly prioritized in consideration of designing a controller for a highly uncertain and nonlinear DAR system. In literature of the modern control theory, sliding mode control (SMC) demonstrates a diverse ability to robustly control any system. Since the pioneer work [@RN64], the variable structure SMC has enjoyed widespread use and attention in many applications [@RN10; @RN11; @RN12; @RN65; @RN67; @RN68; @RN77; @Le2019b; @Le2017; @VA2018; @Le2019; @Le2019a]. Nonetheless, due to presence of discontinuities, the SMC law may cause undesirable oscillations, which is also called the chatterring phenomenon. Park *et al.* [@RN13] proposed a saturation function to replace the discontinuous signum function in the control signal to reduce effects of the chattering. Recently, the chattering phenomenon can be eliminated by designing the controller without a discontinuous term [@Le2017; @VA2018]. For robustly controlling nonlinear systems with unmatched uncertainties, the SMC control law is usually designed in conjunction with the backstepping method, where the sliding surface is aggregated in the last step [@RN14; @RN17]. For instance, Chen *et al.* [@RN75] developed a backstepping sliding mode controller to enhance the global ultimate asymptotic stability and invariability to uncertainties in a nonholonomic wheeled mobile manipulator. To address the problem of explosion of terms associated with the integrator backstepping technique, Swaroop *et al.* [@RN19] proposed the dynamic surface control (DSC) method by using a first-order low-pass filter in the synthetic input. Nonetheless, the aforesaid traditional control techniques are not really practical when they require to accurately model all the nonlinear dynamics of the DAR system, where its unknown parameters are highly uncertain and not easily estimated. It is noted that uncertainties of the DAR system can practically lead to degradation of its control performance. Furthermore, a number of unexpected disturbances and obstacles in the working environments can cause the DAR system to be unstable. To address the issues of accurately modelling all the nonlinear dynamics and estimating the unknown and uncertain parameters, some modern control approaches based on fuzzy logic or artificial neural network have been proposed in the past decades. For instance, by the use of the adaptive learning and function approximations, Lee and Choi [@RN70] introduced a radial basis function network (RBFN) for approximating the nonlinear dynamics of a SCARA-type robot manipulator. Similarly, Wang *et al.* [@RN72] employed the approximation of a neural network to deal with the nonlinearities and uncertainties of a single robot manipulator, where errors caused by the neural network approximation can be estimated by a proposed control robust term. In addition, the authors in [@RN73] designed an adaptive control system for a humanoid robot by using the RBFN to develop a scheme to adaptively estimate unknown and uncertain dynamics of the robot. Based on a multi-input multi-output fuzzy logic unit, Jiang *et al.* [@RN74] proposed an algorithm to adaptively estimate the dynamics of the DAM, given its nonsysmmetric deadzone nonlinearity. In the context of adaptive DSC, it was proposed to employ fuzzy techniques and neural networks to adaptively estimate parameters for the control laws in uncertain nonlinear systems [@Luo2009] and nonlinear systems with uncertain time delays [@Wu2016], respectively. In this paper, we propose an adaptive control strategy based on the DSC method and the RBFN to effectively and efficiently control the DAR system. The proposed approach provides the DAR system not only adaptive estimation of the nonlinear dynamics but also robustness to system uncertainties including the system parameter variations, actuator nonlinearities and external disturbances. In other words, the aggregated control scheme based on the DSC technique enables the manipulators to be capable of efficiently tracking the desired trajectories given large variation of the system information such as the undetermined volume and mass of the payload and significantly reducing chattering influences. The RBFN allows the proposed controller to be able to adaptively estimate the nonlinear and uncertain parameters of the DAR system. More importantly, the adaptation mechanism is designed based on Lyapunov method, which mathematically guarantees the stability of the closed-loop control system. The proposed algorithm was extensively validated in the synthetic environments, where the obtained results are highly promising. The rest of the paper is arranged as follows. We first introduce a model of the DAR system in Section \[sec\_2\]. We then present how to construct a RBFN-DSC controller for the DAR system based on the DSC and RBFN in Section \[sec\_3\]. Section \[sec\_4\] discusses validation of the proposed approach in a simulation environment before conclusions are drawn in Section \[sec\_5\]. Dual Arm Robot Model {#sec_2} ==================== Lets consider a dual two degree of free (DoF) arm robot that cooperatively manipulates an object with mass of $m$ as pictorically shown in Fig. \[fig1\]. It is assumed that both the manipulators rigidly attach to the load so that there is no slip between the grasping points and the grasped load. Let $m_i, I_i, l_i$ denote the mass, mass moment of inertia and length of the corresponding link in the model, respectively. We also define $d_1$ and $d_2$ as the length of the object and distance between the two arms at the robot’s base. The distance from the mass centre of a link to a joint is denoted as $k_i$ while the joint angle between a link and the base or its preceding link is denoted as $\theta_i$. ![Dual arm robot modelling[]{data-label="fig1"}](model.png){width="0.8\linewidth"} ![Operational motions of dual arm robot[]{data-label="fig2"}](motion.png){width="0.8\linewidth"} Operationally, in this work we consider that the robot manipulators make motions on the horizontal $xy$ plane. In other words, the robot arms first move towards the object. After the manipulators are firmly attached to the load, the robot then picks the object up and transports it to a new position by adjusting the motions to robustly follow the given trajectory, demonstrated in Fig. \[fig2\], where ($x_i$,$y_i$) and ($x_f$,$y_f$) are the initial and final locations of the payload, respectively. We let $x_m$ and $y_m$ denote the mass center of the payload on the $xy$ plane, the trajectory of the object can be specified by $$\begin{array}{r@{}l@{\qquad}l} {{x}_{m}}&{}=\frac{{{d}_{2}}}{2}+{{l}_{1}}\cos {{\theta}_{1}}+{{l}_{2}}\cos ({{\theta}_{1}}+{{\theta}_{2}})-\frac{{{d}_{1}}}{2} \\ &{}=-\frac{{{d}_{2}}}{2}+{{l}_{3}}\cos {{\theta}_{3}}+{{l}_{4}}\cos ({{\theta}_{3}}+{{\theta}_{4}})+\frac{{{d}_{1}}}{2}, \\ \nonumber {{y}_{m}}&{}={{l}_{1}}\sin {{\theta}_{1}}+{{l}_{2}}\sin ({{\theta}_{1}}+{{\theta}_{2}}) \\ \nonumber &{}={{l}_{3}}\sin {{\theta}_{3}}+{{l}_{4}}\sin ({{\theta}_{3}}+{{\theta}_{4}}). \end{array}$$ In order to transport the object to a new position, the robot manipulators apply forces $F_1$ and $F_2$ to the payload as illustrated in Fig. \[fig3\]. On the other hands, to rigidly hold the load up, friction forces $F_{s1}$ and $F_{s2}$ are needed. Let $F_{siy}$ and $F_{siz}$ denote the components of the friction forces in $y$ and $z$ directions, respectively. To prevent the load from rotating around $y$ and $z$ axes, it is supposed that $F_{s1y}=F_{s2y}$ and $F_{s1z}=F_{s2z}$. Then the dynamic equations of the object are as follows, $$\label{eq3} \begin{array}{r@{}l@{\qquad}l} &m{{\ddot{x}}_{m}}={{F}_{2}}-{{F}_{1}}, \\ %\label{eq4} &m{{\ddot{y}}_{m}}=2{{F}_{s1y}}=2{{F}_{s2y}}, \\ %\label{eq5} &mg=2{{F}_{s1z}}=2{{F}_{s2z}}, \end{array}$$ where $$\begin{aligned} \begin{array}{r@{}l@{\qquad}l} {{\ddot{x}}_{m(t)}}=&-{{L}_{1}}\left( {{{\dot{\theta }}}_{1}}\cos {{\theta }_{1}}+{{{\ddot{\theta }}}_{1}}\sin {{\theta }_{1}} \right)\\&-{{L}_{2}}\left[ {{\left( {{{\dot{\theta }}}_{1}}+{{{\dot{\theta }}}_{2}} \right)}^{2}}\cos \left( {{\theta }_{1}}+{{\theta }_{2}} \right)\right]\\ &-{{L}_{2}}\left[{{\left( {{{\ddot{\theta }}}_{1}}+{{{\ddot{\theta }}}_{2}} \right)}^{2}}\sin \left( {{\theta }_{1}}+{{\theta }_{2}} \right) \right], \end{array} \end{aligned}$$ $$\begin{aligned} \begin{array}{r@{}l@{\qquad}l} {{\ddot{y}}_{m(t)}}=&{{L}_{3}}\left( {{{\ddot{\theta }}}_{3}}\cos {{\theta }_{3}}+{{{\dot{\theta }}}_{3}}^{2}\sin {{\theta }_{3}} \right)\\&-{{L}_{4}}\left[ \left( {{{\ddot{\theta }}}_{3}}+{{{\ddot{\theta }}}_{4}} \right)\cos \left( {{\theta }_{3}}+{{\theta }_{4}} \right)\right]\\ &+{{L}_{4}}\left[{{\left( {{{\dot{\theta }}}_{3}}+{{{\dot{\theta }}}_{4}} \right)}^{2}}\sin \left( {{\theta }_{3}}+{{\theta }_{4}} \right) \right], \end{array} \end{aligned}$$ and $g=9.8 m/s^2$. And the relationship between the applied forces and the friction forces is presented by $$\label{eq6} \begin{array}{r@{}l@{\qquad}l} {{F}_{s1y}}^{2}+{{(\frac{mg}{2})}^{2}}<{{(\mu {{F}_{1}})}^{2}}, \\ %\label{eq7} {{F}_{s2y}}^{2}+{{(\frac{mg}{2})}^{2}}<{{(\mu {{F}_{2}})}^{2}}, \end{array}$$ where $\mu$ is the friction coefficient in dry condition. If ${{\ddot{x}}_{m(t)}}\ge 0$, both the applied forces $F_1$ and $F_2$ can be computed by $$\label{eq8} \begin{array}{r@{}l@{\qquad}l} {{F}_{1}}&=\frac{1}{\mu }\sqrt{{{\left( \frac{m{{{\ddot{y}}}_{m}}}{2} \right)}^{2}}+{{\left( \frac{mg}{2} \right)}^{2}}}, \\ %\label{eq9} {{F}_{2}}&=\frac{1}{\mu }\sqrt{{{\left( \frac{m{{{\ddot{y}}}_{m}}}{2} \right)}^{2}}+{{\left( \frac{mg}{2} \right)}^{2}}}+m{{\ddot{x}}_{m}}. \end{array}$$ Nonetheless, if ${{\ddot{x}}_{m(t)}}<0$, those forces can be obtained by $$\label{eq10} \begin{array}{r@{}l@{\qquad}l} {{F}_{1}}&=\frac{1}{\mu }\sqrt{{{\left( \frac{m{{{\ddot{y}}}_{m}}}{2} \right)}^{2}}+{{\left( \frac{mg}{2} \right)}^{2}}}-m{{\ddot{x}}_{m}}, \\ %\label{eq11} {{F}_{2}}&=\frac{1}{\mu }\sqrt{{{\left( \frac{m{{{\ddot{y}}}_{m}}}{2} \right)}^{2}}+{{\left( \frac{mg}{2} \right)}^{2}}}. \end{array}$$ ![Physical model of the robot arms[]{data-label="fig3"}](force.png){width="0.8\linewidth"} By the use of Lagrange multipliers, the dynamic model of the dual arm robot manipulating the payload can be summarized as follows, $$\label{eq12} M(\theta)\ddot{\theta}+C(\theta,\dot{\theta})\dot{\theta}={u}+{{J}^{T}}(\theta)F(\theta,\dot{\theta},\ddot{\theta})-{{T}_{d}}-\beta, \nonumber$$ where $u$ is a $4\times1$ control torque input vector, ${{T}_{d}}$ is a $4\times1$ vector presenting the noise effects on the robot arms and $\beta$ denotes the viscous friction forces on all the joints, which are specified as follows, $$\begin{array}{r@{}l@{\qquad}l} \theta&={{\left[ \begin{matrix} {{\theta}_{1}}\; {{\theta}_{2}}\; {{\theta}_{3}}\; {{\theta}_{4}} \nonumber \end{matrix} \right]}^{T}}, \\ \nonumber u&={{\left[ \begin{matrix} {{u}_{1}} \; {{u}_{2}} \; {{u}_{3}} \; {{u}_{4}} \nonumber \end{matrix} \right]}^{T}}, \\ \nonumber F&={{\left[ \begin{matrix} {{F}_{1}} \; {{F}_{s1y}} \; {{F}_{2}} \; {{F}_{s2y}} \nonumber \end{matrix} \right]}^{T}}, \\ \nonumber {{T}_{d}}&={{\left[ \begin{matrix} {{T}_{d1}} \;{{T}_{d2}} \; {{T}_{d3}}\; {{T}_{d4}} \nonumber \end{matrix} \right]}^{T}}, \\ \nonumber \beta &={{\left[ \begin{matrix} {{b}_{1}}{{{\dot{\theta}}}_{1}} \; {{b}_{2}}{{{\dot{\theta}}}_{2}} \; {{b}_{3}}{{{\dot{\theta}}}_{3}}\; {{b}_{4}}{{{\dot{\theta}}}_{4}} \end{matrix} \right]}^{T}}, \nonumber \end{array}$$ where $b_i$ is the viscous friction at the $i^{th}$ joint. $M(\theta)$ is a $4\times4$ matrix of the mass moment of inertia, whose components are specified by $$\begin{array}{r@{}l@{\qquad}l} & {{m}_{11}}={{A}_{1}}+{{A}_{2}}+2{{A}_{3}}\cos {{\theta}_{2}}, \\ \nonumber & {{m}_{12}}={{m}_{21}}={{A}_{2}}+{{A}_{3}}\cos {{\theta}_{2}}, \\ \nonumber & {{m}_{22}}={{A}_{2}}, \\ \nonumber & {{m}_{13}}={{m}_{14}}={{m}_{23}}={{m}_{24}}=0, \\ \nonumber & {{m}_{33}}={{A}_{4}}+{{A}_{5}}+2{{A}_{6}}\cos {{\theta}_{4}}, \\ \nonumber & {{m}_{34}}={{m}_{43}}={{A}_{5}}+{{A}_{6}}\cos {{\theta}_{4}},\\ \nonumber & {{m}_{44}}={{A}_{5}}, \\ \nonumber & {{m}_{31}}={{m}_{32}}={{m}_{41}}={{m}_{42}}=0 \nonumber \end{array}$$ with $$\begin{array}{r@{}l@{\qquad}l} & {{A}_{1}}={{m}_{1}}k_{1}^{2}+{{m}_{2}}l_{1}^{2}+{{I}_{1}}, \\ & {{A}_{2}}={{m}_{2}}k_{2}^{2}+{{I}_{2}}, \\ & {{A}_{3}}={{m}_{2}}{{l}_{1}}{{k}_{2}}, \\ & {{A}_{4}}={{m}_{3}}k_{3}^{2}+{{m}_{4}}l_{3}^{2}+{{I}_{3}}, \\ & {{A}_{5}}={{m}_{4}}{{k}_{4}}^{2}+{{I}_{4}}, \\ & {{A}_{6}}={{m}_{4}}{{l}_{3}}{{k}_{4}}. \end{array}$$ $C(\theta,\dot{\theta})$ is a $4\times1$ Coriolis-centripetal vector, whose elements are computed by $$\begin{array}{r@{}l@{\qquad}l} & {{c}_{11}}=-{{A}_{3}}\sin {{\theta}_{2}}(\dot{\theta}_{2}^{2}+{{{\dot{\theta}}}_{1}}{{{\dot{\theta}}}_{2}})+{{b}_{1}}{{{\dot{\theta}}}_{1}}, \\ & {{c}_{21}}={{A}_{3}}\dot{\theta}_{1}^{2}\sin {{\theta}_{2}}+{{b}_{2}}{{{\dot{\theta}}}_{2}}, \\ & {{c}_{31}}=-{{A}_{6}}\sin {{\theta}_{4}}(\dot{\theta}_{4}^{2}+{{{\dot{\theta}}}_{3}}{{{\dot{\theta}}}_{4}})+{{b}_{3}}{{{\dot{\theta}}}_{3}}, \\ & {{c}_{41}}={{A}_{6}}\dot{\theta}_{3}^{2}\sin {{\theta}_{4}}+{{b}_{2}}{{{\dot{\theta}}}_{4}}. \end{array}$$ Furthermore, $J$ is a $4\times4$ Jacobian matrix with the elements obtained by $$\begin{array}{r@{}l@{\qquad}l} & {{J}_{11}}=-{{L}_{1}}\sin {{\theta}_{1}}-{{L}_{2}}\sin ({{\theta}_{1}}+{{\theta}_{2}}), \\ & {{J}_{12}}=-{{L}_{1}}\cos {{\theta}_{1}}-{{L}_{2}}\cos ({{\theta}_{1}}+{{\theta}_{2}}), \\ & {{J}_{13}}={{J}_{14}}=0, \\ & {{J}_{21}}=-{{L}_{2}}\sin ({{\theta}_{1}}+{{\theta}_{2}}), \\ & {{J}_{22}}=-{{L}_{2}}\cos ({{\theta}_{1}}+{{\theta}_{2}}), \\ & {{J}_{23}}={{J}_{24}}=0, \\ & {{J}_{31}}={{J}_{32}}=0, \\ & {{J}_{33}}={{L}_{3}}\sin {{\theta}_{3}}+{{L}_{4}}\sin ({{\theta}_{3}}+{{\theta}_{4}}), \\ & {{J}_{34}}=-{{L}_{3}}\cos {{\theta}_{3}}-{{L}_{4}}\cos ({{\theta}_{3}}+{{\theta}_{4}}), \\ & {{J}_{41}}={{J}_{42}}=0, \\ & {{J}_{43}}={{L}_{4}}\sin ({{\theta}_{3}}+{{\theta}_{4}}), \\ & {{J}_{44}}=-{{L}_{4}}\cos ({{\theta}_{3}}+{{\theta}_{4}}). \end{array}$$ Control Approach {#sec_3} ================ In order to design a control law to efficiently and automatically adjust the robot manipulators, we first discuss a control scheme based on the dynamic surface control method. It is noted that due to the system uncertainties including parameter variations, actuator nonlinearities and external disturbances, system parameters in the designed controller are practically uncertain and unknown; then we introduce a radial basis function network based technique to adaptively estimate those uncertain and unknown dynamics. Generally speaking, the dynamic model of the dual arm robot (DAR) (\[eq12\]) can be represented as follows, $$\label{eq14} \begin{array}{r@{}l@{\qquad}l} & {{{{\dot{x}}}}_{1}}={{{{x}}}_{2}} \\ & {{{{\dot{x}}}}_{2}}={{M}^{-1}}(\theta )u+{{M}^{-1}}(\theta )[{{J}^{T}}\left( \theta \right)F(\theta ,\dot{\theta },\ddot{\theta })-T_d-\beta-C(\theta ,\dot{\theta })], \end{array}$$ where ${{{x}}_{1}}={{({{\theta }_{1}},{{\theta }_{2}},{{\theta }_{3}},{{\theta }_{4}})}^{T}}$ and ${{{x}}_{2}}={{({{\dot{\theta }}_{1}},{{\dot{\theta }}_{2}},{{\dot{\theta }}_{3}},{{\dot{\theta }}_{4}})}^{T}}$. Let $K(\theta,\dot{\theta},\ddot{\theta})={{J}^{T}}(\theta)F(\theta,\dot{\theta},\ddot{\theta})-C(\theta,\dot{\theta})-G(\theta)-\beta -{{T}_{d}}$, the (\[eq14\]) can be simplified by $$\label{eq15} \begin{array}{r@{}l@{\qquad}l} & {{{{\dot{x}}}}_{1}}={{{{x}}}_{2}} \\ & {{{{\dot{x}}}}_{2}}={{M}^{-1}}(\theta ){u}+{{M}^{-1}}(\theta ).K(\theta ,\dot{\theta },\ddot{\theta }) \\ \end{array}.$$ It can be clearly seen that the system uncertainties are now incorporated into $K(\theta,\dot{\theta},\ddot{\theta})$ that presents the complex nonlinear dynamic of the robot. For the purpose of simplicity, $K(\theta,\dot{\theta},\ddot{\theta})$ and $K$ will be used interchangeably. Dynamic surface control for certain DAR systems {#sec_3a} ----------------------------------------------- The aim of controlling a dual arm robot is to guarantee that ${{{x}}_{1}}$ tracks the reference ${{{x}}_{1r}}$. Therefore, we propose to design a control law using the dynamic surface control (DSC) structure as sequentially expressed by the following steps. **Step 1**: Let $$\label{eq16} {{{z}}_{1}}={{{x}}_{1}}-{{{x}}_{1r}}$$ denote the vector of tracking errors, and consider the first Lyapunov function candidate as follows, $$\label{eq17} {{V}_{1}}=\frac{1}{2}{{{z}}_{1}}^{T}{{{z}}_{1}}.$$ If differentiating ${{V}_{1}}$ with respect to time, one obtains $$\label{eq18} \begin{array}{r@{}l@{\qquad}l} {{\dot{V}}_{1}}&={{{z}}_{1}}^{T}{{\dot{{z}}}_{1}}={{{z}}_{1}}^{T}\left( {{{{\dot{x}}}}_{1}}-{{{{\dot{x}}}}_{1r}} \right)={{{z}}_{1}}^{T}({{{x}}_{2}}-{{{\dot{x}}}_{1r}})\\ &=-{{c}_{1}}{{{z}}_{1}}^{T}{{{z}}_{1}}+{{{z}}_{1}}^{T}\left( {{{{x}}}_{2}}-{{{{\dot{x}}}}_{1r}}+{{c}_{1}}{{{{z}}}_{1}} \right), \end{array}$$ where $c_1$ is a positive definite diagonal matrix. **Step 2**: Let $$\label{eq19} {{{z}}_{2}}={{{x}}_{2}}-{{{\alpha }}_{2f}}$$ define the error between the input ${{{x}}_{2}}$ and the virtual control ${{{\alpha }}_{2f}}$, which is also an output of the first-order filter when putting ${\alpha }$ through. ${\alpha}$ is assumed a virtual control, given by $$\label{eq23} {\alpha }={{\dot{x}}}_{1r}-{{c}_{1}}{{{z}}_{1}}.$$ If the first-order low-pass filter [@RN19] is presented by $$\label{eq20} \tau {{{\dot{\alpha }}}_{2f}}+{{{\alpha }}_{2f}}={\alpha },$$ where ${{\alpha }_{2f}}(0)=\alpha (0)$ and $\tau >0$ is the filter constant. Then, the sliding surface is defined as follow, $$\label{eq21} {s}={{\lambda }}{{{z}}_{1}}+{{{z}}_{2}},$$ where $\lambda $ is a positive definite diagonal matrix. The derivative of the sliding surface can be easily obtained by [@RN14] $$\label{eq22} \begin{array}{r@{}l@{\qquad}l} \dot{s}&={{\lambda }}{{{\dot{{z}}}}_{1}}+{{{{\dot{z}}}}_{2}} \\ &={{\lambda }}{{{\dot{{z}}}}_{1}}+({{{\dot{{x}}}_{2}}}-{{{{\dot{\alpha }}}}_{2f}}) \\ &={{\lambda }}{{{\dot{{z}}}}_{1}}+{{M}^{-1}}K+{{M}^{-1}}{u}-{{{{\dot{\alpha }}}}_{2f}} \\ &={{\lambda }}{{{\dot{{z}}}}_{1}}+{{M}^{-1}}\left( K+{u}-M{{{{\dot{\alpha }}}}_{2f}} \right). \end{array}$$ Now, let’s consider the second Lyapunov function candicate as follows, $$\label{eq24} {{V}_{2}}={{V}_{1}}+\frac{1}{2}{{{s}}^{T}}{s},$$ Differentiating ${{V}_{2}}$ with respect to time, one obtains $$\label{eq25} {{\dot{V}}_{2}}={{\dot{V}}_{1}}+{{{s}}^{T}}{\dot{s}}$$ In order to guarantee the sliding surface to ultimately converge to zero, the control scheme should include two sub-laws. The first is the switching law, which is employed to drive the system states towards a particular sliding surface. This switching control signal is given by $$\label{eq27} {{{u}}_{sw}}=-M\left( {{c}_{2}}sign\left( {{s}} \right)+{{c}_{3}}{s} \right),$$ $c_2$ and $c_3$ are the positive definite diagonal matrices. The second is the equivalent control law that is utilized to keep those states lying on the sliding surface. The equivalent control signal is formulated by $$\label{eq26} {{{u}}_{eq}}=M{{{{\dot{\alpha }}}}_{2f}}-K-M{{\lambda }}{{{\dot{{z}}}}_{1}}=-M\left( {{\lambda }}{{{\dot{{z}}}}_{1}}+{{M}^{-1}}K-{{{{\dot{\alpha }}}}_{2f}} \right)$$ Therefore, the total control signal can be formed by $$\label{eq28} \begin{array}{r@{}l@{\qquad}l} {u}=&{{{u}}_{sw}}+{{{u}}_{eq}} \\ =&-M\left( {{c}_{2}}sign\left( {{s}} \right)+{{c}_{3}}{s} \right)-M\left( {{\lambda }}{{{\dot{{z}}}}_{1}}+{{M}^{-1}}K-{{{{\dot{\alpha }}}}_{2f}} \right) . \end{array}$$ Stability of the proposed control scheme in (\[eq28\]) is analysed in the following theorem. **Theorem 1.** The proposed control law (\[eq28\]) guarantees the closed-loop system (\[eq14\]) to be asymptotically stable. From (\[eq25\]), the derivative of the second Lyapunov function candidate, ${{\dot{V}}_{2}}$, can be rewritten by $$\label{eq29} \begin{array}{r@{}l@{\qquad}l} {{{\dot{V}}}_{2}}=&-{{{{z}}}_{1}}^{T}{{c}_{1}}{{{{z}}}_{1}}+{{{{s}}}^{T}}\left( {{\lambda }}{{{\dot{{z}}}}_{1}}+{{M}^{-1}}\left( K+{u}-M{{{{\dot{\alpha }}}}_{2f}} \right) \right) \\ =&-{{{{z}}}_{1}}^{T}{{c}_{1}}{{{{z}}}_{1}}-{{s}}^{T}{{c}_{2}}sign({s})-{{s}}^{T}{{c}_{3}}{s} \\ &+{{{s}}^{T}}\left( {{c}_{2}}sign\left( s \right)+{{\lambda }}{{{\dot{{z}}}}_{1}}+{{M}^{-1}}\left( K+{u}-M{{{{\dot{\alpha }}}}_{2f}} \right)+{{c}_{3}}{s} \right). \end{array}$$ Substituting the control input in (\[eq28\]) into (\[eq29\]) leads to $$\label{30} {{\dot{V}}_{2}}=-{{{z}}_{1}}^{T}{{c}_{1}}{{{z}}_{1}}-{{s}}^{T}{{c}_{2}}sign\left( {{s}} \right)-{{s}^{T}}{{c}_{3}}{s}<0.$$ Therefore, based on the Lyapunov stability theory the sliding surface $s$ is asymptotically stable. Adaptive dynamic surface control for uncertain DAR systems ---------------------------------------------------------- The deterministic control law (\[eq28\]) can be effectively employed provided that the system parameters are certain. Nevertheless, in practice, the DAR system operates under system uncertainties such as parameter variations and nonlinearities or external disturbances. In other words, it is impractical to accurately determine the system parameters in $K(\theta ,\dot{\theta },\ddot{\theta })$. To deal with these challenges, it is proposed to utilize the radial basis function neural network (RBFN) to approximately estimate the dynamic model $K(\theta ,\dot{\theta },\ddot{\theta })$, given the system uncertainties. Overall, the structure of the RBFN [@RN27] as shown in Fig. \[fig4\] comprises the inputs $(x_1,x_2)$, the outputs $(\delta_1,\delta_2,\delta_3,\delta_4)$ and a number of neurons in the hidden layers. If ${r}={{\left( {{{{x}}}_{1}}^{T},{{{{x}}}_{2}}^{T} \right)}^{T}}$ and $\delta=(\delta_1,\delta_2,\delta_3,\delta_4)^T$, then the output of the RBFN can be presented by $$\delta\left( r \right)={{W}^{T}}h\left( r \right),$$ where $W$ is the weight matrix, $h\left( r \right)={{({{h}_{1}}\left( r \right),{{h}_{2}}\left( r \right),...,{{h}_{l}}\left( r \right) })^{T}}$, where ${{h}_{i}}\left( r \right)$ is an activation function. The widely used activation function, which is also employed in this work, is Gaussian, $$\label{eq33} {{h}_{i}}(r)=\frac{\exp \left( \frac{{{\left\| {{x}_{1}}-{{\rho }_{1i}} \right\|}^{2}}+{{\left\| {{x}_{2}}-{{\rho}_{2i}} \right\|}^{2}}}{{{b}_{i}}^{2}} \right)}{\sum\limits_{j=1}^{n}{\exp \left( -\frac{{{\left\| {{x}_{1}}-{{\rho}_{1j}} \right\|}^{2}}+{{\left\| {{x}_{2}}-{{\rho}_{2j}} \right\|}^{2}}}{{{b}_{j}}^{2}} \right)}}, \;\; i=1,2,...,n,$$ where $n$ is the number of neurons in the hidden layer, $\rho$ is the matrix of means and $b$ is the vector of variances. If $\hat{W}$ denotes estimation of the weight matrix $W$, which is updated by the adaptation mechanism as follows, $$\label{eq34} \dot{\hat{W}}=\Gamma \left( {h}{{{{s}}}^{T}}-\varsigma \left\| {{s}} \right\|\hat{W} \right)$$ where $\varsigma$ is positive and $\Gamma$ is the positive definite diagonal matrix of the adaptation constants, then the output of the RBFN $\delta(r)$ is approximated by $${\hat{\delta }}\left( r \right)={{\hat{W}}^{T}}{h}.$$ In this work, we employ the RBFN to adaptively estimate the uncertain dynamic $K$; therefore, the control input in (\[eq28\]) can be approximated by $$\label{eq35} {u}=-M\left( {{c}_{2}}sign\left( {{s}} \right)+{{c}_{3}}{s} \right)-M\left( {{\lambda }}{{{\dot{{z}}}}_{1}}+{{M}^{-1}}{{\hat{W}}^{T}}{h}-{{{{\dot{\alpha }}}}_{2f}} \right) .$$ ![Schematic diagram of RBF neural network.[]{data-label="fig4"}](noron.png){width="0.8\linewidth"} It is noticed that the estimation of the weight matrix in the RBFN is derived from the Lyapunov theory, which guarantees stability of the closed-loop system as presenting in the following theorem. **Theorem 2.** Given the adaptation mechanism (\[eq34\]), the proposed control scheme (\[eq35\]) can guarantee the closed-loop DAR system (\[eq15\]) to be input-to-state stable [@RN28] with the attractor $$\label{eq36} D=\left\{ {s}\in {{R}^{4}}|\left\| {s} \right\|> \frac{{{\varepsilon }_{N}}+\varsigma \frac{{{\left\| W \right\|}_{F}}^{2}}{4}}{{{c}_{3\min }}} \right\},$$ where ${c}_{3\min }$ is the minimum value of $c_3$, and ${\varepsilon }_{N}$ is a small positive number so that the approximation error $\varepsilon=\delta-\hat{\delta}$ satisfies $\left\| {\varepsilon} \right\|<{\varepsilon }_{N}$. Let $$\tilde{W}=W-\hat{W}$$ define the error between the ideal weight $W$ and the estimated weight $\hat{W}$ Considering the Lyapunov function candidate $$\label{eq37} {{V}_{2}}={{V}_{1}}+\frac{1}{2}{{{s}}^{T}}{s}+tr\left( {{{\tilde{W}}}^{T}}{{\Gamma }^{-1}}\tilde{W} \right)$$ and differentiating it with respect to time, one obtains $$\label{eq38} \begin{array}{r@{}l@{\qquad}l} {{{\dot{V}}}_{2}}=&{{{\dot{V}}}_{1}}+{{{{s}}}^{T}}\dot{{s}}+tr\left( {{{\tilde{W}}}^{T}}{{\Gamma }^{-1}}\dot{\tilde{W}} \right) \\ =&-{{{{z}}}_{1}}^{T}{{c}_{1}}{{{{z}}}_{1}}-{{{{s}}}^{T}}{{c}_{2}}sign\left( {{s}} \right)-{{{{s}}}^{T}}{{c}_{3}}{s}+{{{{s}}}^{T}}\left( {\delta }-{\hat{\delta }} \right) \\ &-tr\left( {{{\tilde{W}}}^{T}}{{\Gamma }^{-1}}\dot{\hat{W}} \right) \\ =&-{{{{z}}}_{1}}^{T}{{c}_{1}}{{{{z}}}_{1}}-{{{{s}}}^{T}}{{c}_{2}}sign\left( {{s}} \right)-{{{{s}}}^{T}}{{c}_{3}}{s}+{{{{s}}}^{T}}{{W}^{T}}{h}\\ &-{{{{s}}}^{T}}{{{\hat{W}}}^{T}}{h}-tr\left( {{{\tilde{W}}}^{T}}{{\Gamma }^{-1}}\dot{\hat{W}} \right) +{{{{s}}}^{T}}{\varepsilon} \\ =&-{{{{z}}}_{1}}^{T}{{c}_{1}}{{{{z}}}_{1}}-{{{{s}}}^{T}}{{c}_{2}}sign\left( {{s}} \right)-{{{{s}}}^{T}}{{c}_{3}}{s}+{{{{s}}}^{T}}{\varepsilon}\\ &+tr\left( {{{\tilde{W}}}^{T}}\left( {{{h}{{{s}}}^{T}-\Gamma }^{-1}}\dot{\hat{W}}\right) \right). \end{array}$$ If we substitute (\[eq35\]) into the derivative of the Lyapunov function (\[eq38\]), it yields $$\begin{aligned} \label{eq39} {{\dot{V}}_{2}}=&-{{{z}}_{1}}^{T}{{c}_{1}}{{{z}}_{1}}-{{{s}}^{T}}{{c}_{2}}sign\left( {s} \right)-{{{s}}^{T}}{{c}_{3}}{s}+{{{s}}^{T}}{\varepsilon}\\ \nonumber &+\varsigma \left\| {{s}} \right\|tr\left( {{{\tilde{W}}}^{T}}\left( W-\tilde{W} \right) \right).\end{aligned}$$ By the use of Cauchy-Schwarz inequality $$\label{eq40} tr\left[ {{{\tilde{W}}}^{T}}\left( W-\tilde{W} \right) \right]\le {{\left\| {\tilde{W}} \right\|}_{F}}{{\left\| W \right\|}_{F}}-{{\left\| {\tilde{W}} \right\|}_{F}}^{2},$$ we can easily compute the inequality of the derivative of the Lyapunov function as follows, $$\begin{aligned} \label{eq41} {{\dot{V}}_{2}}\le &-{{{z}}_{1}}^{T}{{c}_{1}}{{{z}}_{1}}-{{{s}}^{T}}{{c}_{2}}sign\left( {s} \right)-{{{s}}^{T}}{{c}_{3}}{s}+{{{s}}^{T}}{\varepsilon}\\ \nonumber &+\varsigma \left\| {{s}} \right\|\left( {{\left\| {\tilde{W}} \right\|}_{F}}{{\left\| W \right\|}_{F}}-{{\left\| {\tilde{W}} \right\|}_{F}}^{2} \right).\end{aligned}$$ Rearranging (\[eq41\]) by utilizing the attractor (\[eq36\]), it yields $$\label{eq42} \begin{array}{r@{}l@{\qquad}l} {{\dot{V}}_{2}}\le &-{{{s}}^{T}}{{c}_{2}}sign\left( {{s}} \right)-\varsigma \left\| {{s}} \right\|{{\left( {{\left\| {\tilde{W}} \right\|}_{F}}-\frac{1}{2}{{\left\| W \right\|}_{F}} \right)}^{2}}\\ &-\left\| {{s}} \right\|{{c}_{3\min }}\left\| {{s}} \right\|+{{\varepsilon }_{N}}\left\| {{s}} \right\|+\left\| {{s}} \right\|\varsigma \left( \frac{{{\left\| W \right\|}_{F}}^{2}}{4} \right) \end{array}$$ In other words, if the sliding surface is outside the attractor, which is $$\left\| {s} \right\|>\frac{{{\varepsilon }_{N}}+\varsigma \frac{{{\left\| W \right\|}_{F}}^{2}}{4}}{{{c}_{3\min }}},$$ we then have ${{\dot{V}}_{2}}\le 0$. Therefore, the sliding surface ${s}$ is input-to-state stable. Simulation Results {#sec_4} ================== To demonstrate effectiveness of the proposed control law, we conducted experiments in simulation environment. To simulate the DAR protocol, the two robot arms were first to track the desired trajectories to reach the object. The reference trajectories in the first 2 seconds are mathematically specified by $$\label{eq45} \begin{array}{r@{}l@{\qquad}l} {{x}_{a1}}(t)&={{x}_{f1}}+({{x}_{i1}}-{{x}_{f1}}){{e}^{-10{{t}^{2}}}}, \\ {{y}_{a1}}(t)&={{y}_{f1}}+({{y}_{i1}}-{{y}_{f1}}){{e}^{-10{{t}^{2}}}}, \\ {{x}_{a2}}(t)&={{x}_{f2}}+({{x}_{i2}}-{{x}_{f2}}){{e}^{-10{{t}^{2}}}}, \\ {{y}_{a2}}(t)&={{y}_{f2}}+({{y}_{i2}}-{{y}_{f2}}){{e}^{-10{{t}^{2}}}}, \end{array}$$ where $x_{a1},y_{a1},x_{a2},y_{a2}$ are the trajectories of the robot arms. $\left( {{x}_{i1}},{{y}_{i1}},{{x}_{i2}},{{y}_{i2}} \right)$ and $\left( {{x}_{f1}},{{y}_{f1}},{{x}_{f2}},{{y}_{f2}} \right)$ are the initial and final positions of the manipulators, respectively. After firmly holding the payload, the robot transports the object along the half of a circle so that it can avoid collision with an obstacle. The center of the object is expected to travel on a curve as follows, $$\begin{array}{r@{}l@{\qquad}l} {{x}_{mr}}(t)&={{x}_{0}}+{{r}_{m}}\cos(\phi t), \\ {{y}_{mr}}(t)&={{y}_{0}}+{{r}_{m}}sin(\phi t), \end{array}$$ where $\left( {{x}_{0}},{{y}_{0}} \right)$ is the position of the obstacle, which is also the center of the circle on which the center of the object travels. ${{r}_{m}}$ is the radius of the circle, while $\phi $ is a polar angle that varies from $-\pi$ to $0$. It is noted that the joint angles between the link and the base or its preceding link at the beginning $t=0$ were known, ${{q}_{1}}(0)=\frac{\pi }{6},\,\,{{q}_{2}}(0)=\frac{\pi }{2},\,\,{{q}_{3}}(0)=\pi$ and ${{q}_{4}}(0)=\frac{-2\pi }{3}$. In the synthetic experiments, the physical model parameters of the DAR system were given. Furthermore, the parameters of the DSC controller were known. Those information are summarized in Table \[tab1\]. It was supposed that there is no prior knowledge of the robot dynamics, then the weight matrix $W$ of the RBFN were initialized by zeros. Moreover, an unexpected disturbance as shown in Fig. \[fig5\], which exerts the applied forces, was taken into consideration to illustrate robustness of the proposed approach. ![External disturbance[]{data-label="fig5"}](disturbance.png){width="0.8\linewidth"} \[tab1\] Dynamic model parameters -------------------------------------------------------------------------------------------------------------------------------------------- ${{m}_{1}}={{m}_{2}}={{m}_{3}}={{m}_{4}}=1.5\,(kg)$; ${{I}_{1}}={{I}_{2}}={{I}_{3}}={{I}_{4}}=0.18\,(kg{{m}^{2}})$; ${{l}_{1}}={{l}_{2}}={{l}_{3}}={{l}_{4}}=1.2\,(m)$; ${{k}_{1}}={{k}_{2}}={{k}_{3}}={{k}_{4}}=0.48\,(m)$; ${{b}_{1}}={{b}_{2}}={{b}_{3}}={{b}_{4}}=110\,(Nm/s)$; ${{d}_{1}}=0.25\,(m)$; ${{d}_{2}}=1.2\,(m)$; $\mu =0.35$; $m=1.5\,(kg)$ Reference trajectory parameters $\left( {{x}_{i1}},{{y}_{i1}},{{x}_{i2}},{{y}_{i2}} \right)=\left( 0.76,\,0.6,\,-0.76,\,0.6 \right)$; $\left( {{x}_{f1}},{{y}_{f1}},{{x}_{f2}},{{y}_{f2}} \right)=\left( -0.275,\,1.4,\,-0.525,\,1.4 \right)$; $\left( {{x}_{0}},{{y}_{0}} \right)=\left( 0,\,1.4 \right)$; ${{r}_{m}}=0.4$; ${{\theta}_{1}}(0)=\frac{\pi }{6};\,\,{{\theta}_{2}}(0)=\frac{\pi }{2};\,\,{{\theta}_{3}}(0)=\pi ;\,\,{{\theta}_{4}}(0)=\frac{-2\pi }{3}$; ${{\dot{\theta}}_{1}}(0)={{\dot{\theta}}_{2}}(0)={{\dot{\theta}}_{3}}(0)={{\dot{\theta}}_{4}}(0)=0$ Controller parameters $\lambda =diag\left( 15,15,15,15 \right)$; ${{c}_{1}}=diag\left( 122,122,122,122 \right)$; ${{c}_{2}}=diag\left( 122,122,122,122 \right)$; ${{c}_{3}}=diag\left( 152,152,152,152 \right)$; $\hat{W}\left( 0 \right)=0$; $\Gamma =diag\left( 30,30,30,30 \right)$ : Parameters of the dual arm robot system Before examining the motions of the robot arms, let’s investigate the motions of the four links of the DAR system by considering the joint angles between the link and the base or its preceding link on $xy-plane$ as illustrated in Fig. \[fig5\]. For the purposes of comparisons, in this experimental example we implemented both the algorithms of the conventional DSC scheme as presented in Section \[sec\_3a\], where the system parameters were assumed to be determined and certain, and the proposed RBFN based DSC (RBFN-DSC) approach. It can be clearly seen in Fig. \[fig6\] that the results obtained by the two implemented algorithms were expected to approach the references, which are early obtained from the equations in (\[eq45\]), all the time. While the deterministic DSC control law quickly tracked the references in all the links, the proposed algorithm performed well in the lower links as shown in Figures \[fig6a\] and \[fig6c\] and insignificantly degraded in the upper links as shown in Figures \[fig6b\] and \[fig6d\], approximate 0.05 s behind the DSC method. This is understandable since the parameters of the DSC control law were given, while the RBFN-DSC technique needed time to adaptively estimate those. Nonetheless, in fact, given the system uncertainties including the system parameter variations, actuator nonlinearities and external disturbances, exactly determining the system parameters for the DSC algorithm is not really practical, while the proposed approach can estimate those parameters by the use of the RBFN. Errors of the joint angles at the links as illustrated in Fig. \[fig7\] consolidate efficacy of the proposed control scheme. \ \ \ \ More importantly, as can be seen in Fig. \[fig8\], the motion trajectories of the two end-effectors show that the proposed RBFN-DSC is really efficiently and effectively practical. Given the aim of transporting the payload along a half of a circle to avoid collision with an obstacle, the movements of both the left and right manipulators of the robot under the control of the deterministic DSC scheme in Fig. \[fig8b\] and the proposed RBFN-DSC law in Fig. \[fig8c\] were expected to track the ideal trajectories as shown in Fig. \[fig8a\]. It can be clearly seen that given the system parameters, the DSC method controlled the end-effectors to travel quite smoothly, including before approaching the payload and during transporting it, as compared with the desired references. Nevertheless, though having to estimate the system parameters under their uncertainties and nonlinearities, the trajectories obtained by the proposed RBFN-DSC control scheme in the whole protocol are highly comparable to not only those obtained by the DSC method but also the expectation. That is, the proposed algorithm guarantees that the DAR system to be able to adaptively learn its nonlinear parameters while safely transport the payload to the destination. The RBFN-DSC control law is highly applicable for the uncertain DAR systems. \ \ Conclusions {#sec_5} =========== The paper has introduced a new robust adaptive control approach for an uncertain DAR system, where the manipulators are expected to grasp and transport an object to a destination on the desired trajectories. To guarantee motions of the robot’s end-effectors to be robustly tracked on the references, the control scheme is designed based on the DSC technique. Nonetheless, due to the system uncertainties and nonlinearities, the DAR system dynamics are not practically determined, which leads to impracticality of the DSC algorithm. Hence, it has been proposed to adaptively learn the uncertain system parameters by employing the RBFN, where the adaptation mechanism has been derived from the Lyapunov function to guarantee the stability of the closed-loop control system. The results obtained by a synthetic implementation have verified the proposed control law. It is noted that the proposed algorithm will be implemented in the realistic DAR system in the future works.

--- abstract: | The quantum phase diagram of disordered electron systems as function of the concentration of magnetic impurities $n_m$ and the local exchange coupling $J$ is studied in the dilute limit. We take into account the Anderson localisation of the electrons by a nonperturbative numerical treatment of the disorder potential. The competition between RKKY interaction $J_{\rm RKKY}$ and the Kondo effect, as governed by the temperature scale $T_K$, is known to gives rise to a rich magnetic quantum phase diagram, the Doniach diagram. Our numerical calculations show that in a disordered system both the Kondo temperature $T_K$ and $J_{\rm RKKY}$ are widely distributed. Accordingly, also their ratio, $J_{\rm RKKY}/T_K$ is widely distributed as shown in Fig.\[fig:ratio\_kondo\_rkky\_square\](a). However, we find a sharp cutoff of that distribution, which allows us to define a critical density of magnetic impurities $n_c$ below which Kondo screening wins at all sites of the system above a critical coupling $J_c$, forming the Kondo phase\[see Fig.\[fig:ratio\_kondo\_rkky\_square\](b)\]. As disorder is increased, $J_c$ increases and a spin coupled phase is found to grow at the expense of the Kondo phase. From these distribution functions we derive the magnetic susceptibility which show anomalous power law behavior. In the Kondo phase that power is determined by the wide distribution of the Kondo temperature, while in the spin coupled phase it is governed by the distribution of $J_{\rm RKKY}$. At low densities and small $J< J_c$ we identify a paramagnetic phase. We also report results on a honeycomb lattice, graphene, where we find that the spin coupled phase is more stable against Kondo screening, but is more easily destroyed by disorder into a PM phase. author: - Hyun Yong Lee - Stefan Kettemann bibliography: - 'reference.bib' title: Magnetic Quantum Phase Diagram of Magnetic Impurities in 2 Dimensional Disordered Electron Systems --- Introduction ============ Phenomena which emerge from the interplay of strong correlations and disorder remain a challenge for condensed matter theory. Spin correlations and disorder effects are however relevant for a wide range of materials, including doped semiconductors like Si:P close to metal-insulator transitions,[@Loehneysen] and heavy Fermion systems, materials with 4f or 5f atoms, notably Ce, Yb, or U.[@Vojta] Many of these materials show a remarkable magnetic quantum phase transition which can be understood by the competition between indirect exchange interaction, the Ruderman-Kittel-Kasuya-Yoshida(RKKY) interaction between localised magnetic moments[@Kittel; @Kasuya; @Yoshida] and their Kondo screening. \ Thereby, one finds a suppression of long range magnetic order when exchange coupling $J$ is increased and Kondo screening succeeds. This results in a typical quantum phase diagram with a quantum critical point where the $T_c$ of the magnetic phase is vanishing, the Doniach diagram.[@Doniach] Recently, controlled studies of magnetic adatoms on the surface of metals,[@Zhou] on graphene,[@Chen] and on the conducting surface of topological insulators[@Hsieh; @gap; @rader; @reinert] with surface sensitive experimental methods like spin resolved STM and ARPES became possible. This demands a theoretical study of the Doniach diagram for magnetically doped disordered electron systems (DES), in particular 2D systems. In any material there is some degree of disorder. In doped semiconductors it arises from the random positioning of the dopants themselves, in heavy Fermion metals and in 2D metals it may arise from structural defects or impurities. Disorder is known to cause Anderson localisation, which therefore has to be taken into account when deriving the Doniach diagram of disordered electrons systems. Moreover, as noted already early,[@Anderson] the physics of random systems is fully described by probability distributions, not just averages. This must be particularly true for systems with random local magnetic impurities (MIs),[@Mott] since the magnetic impurities are exposed to the local density of states of the conduction electrons, which is widely distributed itself. In fact, it has been noticed that a wide distribution of the Kondo temperature $T_K$ of MIs in disordered host metals gives rise to non-Fermi liquid behavior, such as the low temperature power-law divergence of the magnetic susceptibility.[@Mott; @Vladimir1; @Vladimir2; @Bhatt; @Wolfle; @jones; @Grempel; @Kettemann1; @kskim2] Nonmagnetic disorder quenches the Kondo screening of MIs due to Anderson-localisation and the formation of local pseudogaps at the Fermi energy,[@Kettemann1; @Kettemann2; @Kettemann3; @Kettemann5] resulting in bimodal distributions of $T_K$ and a finite concentration of free, paramagnetic moments (PMs). However, in these studies the RKKY interaction $J_{\rm RKKY}$ between different MIs has not yet been taken into account. $J_{\rm RKKY}$ is mediated by the conduction electrons, and aligns the spins of the MIs ferromagnetically or antiferromagnetically, depending on their distance $R$. This is a long-ranged interaction, with a power law decay $J_{\rm RKKY}\sim 1/R^d$, where d is the dimension, and its typical value is not changed by weak disorder.[@Lerner; @Bulaevskii; @Bergmann; @HYLEE1] However, its amplitude has a wide log-normal distribution in disordered metals.[@Lerner; @HYLEE2] In this article we therefore intend to study the competition between RKKY interaction and the Kondo effect in disordered electron systems. In the next section we introduce the model, and provide the equations for the Kondo temperature and the RKKY coupling. In section III, we derive numerically the distribution function of $J_{\rm RKKY}$, and compare it with an analytical result, based on a perturbative expansion of the nonlinear sigma model. We derive next numerically the distribution function of $T_K$ finding excellent agreement with approximate analytical results which were obtained, taking into account the multifracatlity and power law correlations of wave functions. In section IV we present the main results, the distribution function of the Ratio of TK and RKKY Interaction, for various distances between magnetic impurities R. From that we show how to derive the zero temperature magnetic quantum phase diagram as function of magnetic impurity density and exchange coupling, for 2D disordered electronic systems. At low densities and small $J < J_c$ we identify a paramagnetic phase. For graphene we find that the spin coupled phase is more stable against Kondo screening, but is more easily destroyed by disorder into a PM phase. In section V we derive from the distribution functions the magnetic susceptibility as function of temperature, which show anomalous power law behavior. In the Kondo phase that power is found to be determined by the wide distribution of the Kondo temperature, while at small exchange coupling there we identify spin coupled phase where the magnetic susceptibility is governed by the distribution of $J_{\rm RKKY}$. In the final section we conclude and discuss the relevance and limitations of our results. Model ===== In order to obtain the Doniach diagram of random electron systems we extend the approach of Doniach[@Doniach] by calculating the distribution functions of $T_K$ and $J_{\rm RKKY}$ and their ratio. Thus, in our approach we try to draw conclusions on the quantum phase diagram of an electron system with a finite density of magnetic impurities, by considering the Kondo temperature of single impurities and the RKKY coupling of pairs of magnetic moments. We start from a microscopic description of the MIs, the Anderson impurity model coupled to a non-interacting disordered electronic Hamiltonian with on-site disorder. Then, we map it with the Schrieffer-Wolff transformation on a model of Kondo impurity spins coupled to the disordered host electron spins by the local coupling $J$.[@Kettemann5] We consider the single-impurity $T_K$ and the coupling $J_{\rm RKKY}$ between a pair of spins. For the numerical calculations we employ the single-band Anderson tight-binding model on a square lattice of size$L$ and lattice spacing$a$, $$\centering H = -t\sum_{{\langle}i,j {\rangle}} c_{i}^{\dagger} c_{j} + \sum_{i} (w_i - \tilde{E}_F)~c_{i}^{\dagger} c_{i}, \label{eq:Hamiltonian}$$ where $t$ is the hopping energy between nearest neighbours ${\langle}i,j {\rangle}$, $w_i$ is the on-site disorder potential distributed in the interval $[-W/2, W/2]$. $\tilde{E}_F = E_F + {\varepsilon}_{\rm edge}$, where $E_F$ is the Fermi energy measured from the band edge, in 2D ${\varepsilon}_{\rm edge} = -4t$. We use periodic boundary conditions. In the dilute limit, one can calculate the $T_K$ of a single magnetic impurity at position $\bm{R}_i$ from the Nagaoka-Suhl one-loop equation,[@Nagaoka; @Suhl] $$1 = \frac{J}{2} \int_0^{D} d{\varepsilon}\, \frac{\tanh[({\varepsilon}- E_F)/2 T_K]}{{\varepsilon}- E_F}\,\rho_{ii}({\varepsilon}), \label{eq:nagaoka_kpm}$$ with band width $D$. $\rho_{ii}({\varepsilon}) = \braket{i| \delta({\varepsilon}- H) |i}$ is the local density of states(LDOS). The RKKY coupling $ J_{{\rm RKKY}_{ij}}$ between two MIs located at positions $\bm{R}_i$, $\bm{R}_j$ is in the zero temperature limit ($T=0$) given by[@Didier; @HYLEE1] $$J_{{\rm RKKY}_{ij}} = -J^2\frac{S(S+1)}{2S^2} \int_{{\varepsilon}<E_F} d{\varepsilon}\int_{{\varepsilon}'>E_F} d{\varepsilon}' \frac{F({\varepsilon},{\varepsilon}')_{ij}}{{\varepsilon}-{\varepsilon}'}, \label{eq:J-KPM}$$ where $F({\varepsilon},{\varepsilon}')_{ij} = {\rm Re}[\rho_{ij}({\varepsilon})\rho_{ji}({\varepsilon}')]$, and $S$ is the magnitude of the MI spin. Distribution Functions ====================== Using the Kernel Polynomial method (KPM),[@Didier; @Weisse] one can evaluate the matrix elements of the density matrix $\rho_{ij}({\varepsilon}) = \braket{i|\delta ({\varepsilon}-\hat{H})|j}$[@HYLEE1; @Weisse; @Mucciolo] with a polynomial expansion of order $M$. Here, we increase the cutoff degree $M$ linearly with the linear system size $L$ based on our analysis for the convergence of RKKY interaction with respect $M$ in Ref.. It has been also carefully discussed in Ref. that the choice of $M \propto L$, not $M \propto L^2$, gives proper DOS and LDOS results avoiding finite size effect. Eq. yields in a clean 2D system $$\begin{aligned} J^{0}_{\rm 2D} = - \frac{m^*}{8 \pi} \sin (2k_F R) /(k_F R)^2 {\nonumber}\end{aligned}$$ in the asymptotic limit$k_F R \gg1$ with effective electron mass $m^*=1/(2a^2t)$ , and Fermi wave vector $k_F$. [@Kittel] Its geometrical average is close to the clean limit for distances $R$ smaller than localisation length $\xi$, and decays exponentially at larger distances,[@Sobota; @HYLEE1] $ e^{{{\langle}\frac{1}{2} \ln J_{\rm RKKY}}^2 {\rangle}} \sim e^{-R/\xi}. $ As shown in Figs.\[fig:rkky\_disordered\]a,b, the distribution of the absolute value of $J_{\rm RKKY}$ is well fitted by a log-normal, $$\begin{aligned} N(x) = \frac{N}{\sqrt{2\pi \sigma}} \exp\left[ -\frac{(x-x_0)^2}{2\sigma^2} \right], {\nonumber}\end{aligned}$$ where $x=\ln |J_{\rm RKKY}|$ and the fitting gives for $R= 5a$ and $W=2t,4t$, $x_0=5,6$ and width $\sigma = 5.3 + .85 W/t$ increasing with the disorder strength $W$. This is qualitatively consistent with analytical results obtained at weak disorder,[@Lerner] while analytical calculations at strong disorder have not been performed yet. This distribution width hardly depends on the distance $R$. We used $N=30\,000$ disorder configurations. \ The distribution of $T_K$ is shown in Fig.\[fig:rkky\_disordered\]c, as obtained from the numerical solution of Eq. for $L=40a$, $j=J/D=0.25$. Since for every sample only one single site is taken to avoid a distortion of the distribution due to intersite correlations, we had to use a huge number of $N=30\,000$ different random disorder configurations to get sufficient statistics. It has a strongly bimodal shape where the low $T_K$- peak becomes more distinctive with larger disorder amplitude $W$.[@Grempel; @Kettemann1; @Kettemann4] In Fig.\[fig:rkky\_disordered\]d we show these results for fixed disorder strength $W=5t$ for various exchange couplings $j$. Recently, an analytical derivation of the low $T_K$-tail of $P(T_K)$ was done, using the multifractal distribution and correlations of intensities.[@Kettemann5] These correlations are in 2D logarithmic with an amplitude of order $1/g$, where $g= E_F \tau$. For weak disorder, $g \gg 1$, it corresponds to a power law correlation with power $ \eta_{2D} = 2/ \pi g. $ The correlation energy is of the order of the elastic scattering rate $E_{c} \sim 1/\tau$. Thus, for $T_K \ll {\rm Max}\{\Delta_{\xi} = D/\xi^2, \Delta = D/L^2\}$,[@Kettemann5] $$\label{ptktailj} P(T_K) = \left( 1- p_{FM} \right)\left(\frac{E_c}{T_K} \right)^{1-j} ({\rm Min} \{ \xi, L \} )^{-\frac{d^2 j^2}{2\eta_{2D}}},$$ where $p_{FM} = n_{FM}(0)/n = ({\rm Min} \{ \xi, L \} )^{-\frac{d^2 j^2}{2\eta_{2D}}}$, the ratio of free PMs. Eq. has a power law tail with power $\beta_j=1-j$ in good agreement with the numerical results, Fig.\[fig:rkky\_disordered\]d, for $T_K/T_K^0 < .03$. For $T_K^0 > T_K > {\rm Max}\{\Delta_{\xi} = D/\xi^2, \Delta = D/L^2\}$ one finds[@Kettemann5] $$\begin{aligned} \label{ptktail} \frac{P(T_K)}{ 1- p_{FM}}= (\frac{E_c}{T_K})^{1-\frac{\eta_{2D}}{2 d}} \exp [ - \frac{(\frac{T_K}{E_c})^{\frac{\eta_{2D}}{d}}}{2 c_1} \ln^2 \left(\frac{T_K}{T_K^0} \right) ], \end{aligned}$$ where $c_1 = 7.51$. This expression is in agreement with the numerical results, see Fig.\[fig:rkky\_disordered\]d, using $\xi = g \exp (\pi g)$, and $1/\tau = \pi W^2/6 D$, fitting only $E_c \approx .73 t$ and the prefactor. Thus, we confirm that the power law tail is governed by the multifractal correlation with power $\eta_{2D}$. The quantum phase transition between the free paramagnetic moment phase (PM) and a Kondo screened phase can be studied by calculating the critical exchange coupling $J_c$ above which there is no more than one free magnetic moment in the sample volume $L^d$.[@Kettemann2] From the multifractality of the eigenfunction intensities it is found to be related to the power $\eta_{2D}$ of the power law correlations in the 2D DES as $J_{c} = \sqrt{ \eta_{2D}} D$ and thus to increase in 2D linearly with disorder strength$W$ as,[@Kettemann5] $$J_c = \sqrt{D/(3E_F)} W. \label{eq:jc}$$ In Fig.\[fig:jc\_wrt\_fermi\]a, Eq. is plotted together with numerical results as function of disorder strength$W$. We find good agreement. There are only deviations at large disorder, $g <1$, where the $1/g$ expansion breaks down. We plot $J_{c}$ as function of $E_F$ in Fig.\[fig:jc\_wrt\_fermi\]b, together with the density of states (DOS). We find that $J_{c}$ is increasing towards the band edge as $1/\sqrt{E_F}$ in agreement with Eq.. Far outside of ${\varepsilon}_{\rm edge}$ of the clean system it increases as $J_c/D = 1/\ln | {\varepsilon}_{\rm edge}- E_F|$ due to the gap in the DOS. Magnetic Phase Diagram at $T=0$ =============================== In clean systems the critical density $n_c =1/R_c^d$ above which the MIs are coupled with each other can be obtained from the condition that $|J^{0}_{\rm RKKY} (R_c) | = T_K$.[@Doniach] Thus, in 2D with $|J^{0}_{\rm RKKY}|_{k_F R \gg 1} = J^2 \frac{m}{8 \pi^2 k_F^2 R^2}$ and $T_K = c E_F \exp (- D/J)$, $c \approx 1.14$, one finds $ n_c = 16 \pi^2 c \frac{E_F^2}{J^2} \exp (-\frac{ D}{J})$. In disordered systems, $T_K$ of an MI at a given site competes with the RKKY coupling to another MI at distance $R$. Thus, the distribution function $N(x_{JK})$ of the ratio of these two energy scales $x_{JK}=|J_{\rm RKKY}(R)| / T_K$ for a given disordered sample with density of MIs $ n= 1/ R^2$, where $R$ is the average distance between the MIs, is crucial to determine its magnetic state. The distribution of $x_{JK}$ for $W=3t$ and $J/D=0.2$ is shown for several distances $R$ in Fig.\[fig:ratio\_kondo\_rkky\_square\](a) ($N=10\,000$, $L=100a$, $E_F= t$ and $M=300$). Likewise $N(T_K)$ and $N(J_{\rm RKKY})$, the distribution of $x_{JK}$ has an exponentially wide width characterized by a small-$x_{JK}$ tails and a sharp upper cutoff in $x_{JK}$ as shown in Fig.\[fig:ratio\_kondo\_rkky\_square\](a). As increasing the distance $R$ between the magnetic impurities the distribution $N(x_{JK})$ is shifted to the left(smaller $x_{JK}$), since the RKKY interaction decreases with $R$. The sharp upper cutoff in $x_{JK}$ allows us to define a critical density $n_{c}(J)=1/R_c^2$ below which the Kondo effect dominates in the competition with RKKY interaction at all sites. $n_{c}(J)$ is plotted in Fig.\[fig:ratio\_kondo\_rkky\_square\](b) for various values of disorder strength $W$. When the MI density $n$ exceeds $n_c$, magnetic clusters start to form at some sites and the MIs may be coupled by $J_{\rm RKKY}$. We see that this coupled moment phase (CM) expands at the expense of the Kondo phase with increasing $W$. When $R$ is larger than localisation length $\xi$ the coupling $J_{\rm RKKY}$ is exponentially small and there is a [*paramagnetic phase*]{} (PM) below $n_{\xi} = 1/\xi(g)^2,$ where MIs remain free up to exponentially small temperatures. In graphene the pseudogap at the Dirac point quenches the Kondo effect below $J_c =D/2$, independently on disorder amplitude$W$. Thus, in graphene there is a larger parameter space where the MIs are coupled (CM) than in a normal 2DES, see Fig.\[fig:graphene\_critical\_n\]. However, short range disorder localises the electrons, cutting off the RKKY-interaction and for $n<n_{\xi}$ there is a PM phase. Thus, the magnetic phase in graphene is more stable against Kondo screening but is more easily destroyed by disorder. ![(Color online) Critical MI density $n_c$ as function of $J/D$ for graphene($\varepsilon_F=3t$, Dirac point) $L$, $M$, $N$ as in Fig.\[fig:ratio\_kondo\_rkky\_square\]. []{data-label="fig:graphene_critical_n"}](figure4.eps){width="40.00000%"} Doniach Phase Diagram of disordered 2DES and Graphene. ====================================================== We find, that the Kondo phase splits at finite temperature into a [*Kondo Fermi-liquid (FL) phase*]{}, where all MIs are screened, and a [*Kondo Non-Fermi-liquid (NFL) phase*]{}, at $T > T^*(n)$, where some MIs remain unscreened and contribute to the magnetic susceptibility with an anomalous temperature dependence, given by,[@Kettemann5] $$\begin{aligned} \chi (T) \sim \frac{n}{E_{c}} \frac{2d}{\eta_{2D}} \left(\frac{T}{E_{c}} \right)^{\frac{\eta_{2d}}{2d}-1} {\rm for\,} T > T^*(n) > \frac{D}{\xi^2}. $$ The temperature $T^*(n)$, plotted schematically in Fig.\[fig:critical\_n\] (blue line), is given by the position of the low $T_K$ peak in the distribution $P(T_K)$, see Fig.\[fig:rkky\_disordered\]c. We note that $J$ may be distributed itself and may add a nonuniversal, material dependent contribution to the distribution of $T_K$[@Wolfle] and $J_{\rm RKKY}$. For $n>n_c$ there is a succession of phases, starting with the [ *RKKY phase*]{} where clusters are formed locally due to the widely distributed RKKY coupling. Anomalous power laws are observed when clusters are broken up successively as temperature is raised. From the log-normal distribution $N(|J_{\rm RKKY}|)$ one obtains for the magnetic susceptibility, $$\begin{aligned} \chi(T) T &=& n_{FM} (T) = \int_0^T d|J_{\rm RKKY}| N(|J_{\rm RKKY}|) {\nonumber}\\ &\sim& n \exp \Big[ - \ln^2(T/|J_{\rm RKKY}^0|)/(2 \sigma(W))^2 \Big],\end{aligned}$$ where width $\sigma(W)$ increases with disorder strength $W$. Accordingly, the excess specific heat is $$C(T) = T \frac{d n_{FM}}{d T}\sim \exp\Big[-\ln^2(T/|J_{\rm RKKY}^0|)/(2 \sigma(W)^2 \Big].$$ The detailed analysis of the quantum phase diagram at higher concentrations $n$ requires to go beyond our present analysis. One expects that at $n > n_{SG}$ a [*spin-glass phase*]{} appears, where the magnetic susceptibility shows a peak at spin glass temperature $T_{SG}$ as studied in Refs.. Above a critical density $n_{F}$ a phase with long range order may form below a critical temperature $T_c(n,J)$.[@Coqblin; @Varma; @Magalhaes; @Bouzerar] ![(Color online) Schematic Doniach diagram: temperature $T$ divided by $J^2$ versus $J/D$. Vertical dotted line: critical point $J_c(n)$ separating RKKY phase from Kondo phase. Blue line: $T^*(n)$ separating Kondo FL phase from Kondo NFL phase. For $n<n_{\xi}$ and $J<J_c$ a paramagnetic phase (PM) appears. []{data-label="fig:critical_n"}](figure5.eps){width="50.00000%"} Conclusions and Discussion ========================== We conclude that it is the full distribution function $N(x_{JK})$ of the ratio of the RKKY coupling and the Kondo temperature which determines the magnetic phase diagram of magnetic moments in disordered electron systems, especially at low concentrations. We identified a critical density of magnetic impurities $n_c$ below which Kondo wins at all positions in a disordered sample above a critical coupling $J_c$, which increases with the disorder amplitude. As a result, the Kondo phase is diminished as the disorder is increased, favoring a phase where the MI spins are coupled. The magnetic susceptibility obeys an anomalous power law behavior, which crosses over as function of $J$ from the Kondo regime where that power is determined by the wide distribution of the Kondo temperature $T_K$, to a spin coupled phase where it is governed by the log-normal distribution of $J_{\rm RKKY}$. At low densities and small $J< J_c$, we identify a paramagnetic phase. The distribution function of $|J_{\rm RKKY}|/T_K$ is expected to determine also the magnetic phase diagram of magnetically doped graphene and the surface of topological insulators with magnetic adatoms, see Fig. \[fig:graphene\_critical\_n\]. This distribution function may also be crucial to explain the anomalous magnetic properties of doped semiconductors in the vicinity of metal-insulator transition,[@Loehneysen] where we expect that $\eta$ is replaced by the universal value $\eta = 2(\alpha_0-d)$, $d=3$ with the universal multifractality parameter $\alpha_0$. In this work we considered the distribution function of the Kondo temperature of single impurities and the RKKY coupling of pairs of magnetic moments and extracted information on the quantum phase diagram of systems with finite concentrations of MIs. While this approach has its limitations, for example at finite concentration the RKKY coupling can reduce $T_K$ as has been already found by Tsay and Klein in the 70s.[@Tsay1; @Tsay2] However, they concluded that this reduction is minor. More importantly, later work revealed that the Kondo lattice of a finite density of magnetic moments, which is coupled to the conduction electrons, has a coherent low temperature heavy fermion phase, and a Kondo insulator phase at half filling of the magnetic moment levels. More recently, the Kondo lattice in 1 dimension was studied more rigorously (see the review by Tsunetsugu et. al[@sigrist]), and it was shown that, at least in 1D, the groundstate of this system can not be understood by the mere extension of the single and two-magnetic impurity problem, where the physics is governed by the competition between these two energy scales, the Kondo temperature and the RKKY coupling. However, the higher temperature behavior was found to be still governed by the competition between these two energy scales. Therefore, we expect that the consideration of the reduced problem of two impurity spins, will give important information on the physics of disordered electron systems at finite concentration of magnetic moments, which becomes more meaningful the lower the density and the higher the temperature is. Going beyond the limitations of this approach, one will have to study the disordered Kondo lattice where a finite density of magnetic moments is coupled to the conduction electrons. For a clean Kondo lattice it is known that a coherent low temperature heavy fermion phase, and a Kondo insulator phase at half filling of the magnetic moment levels appears.[@coleman; @read] It remains to see how these low temperature phases are modified by the presence of nonmagnetic disorder. We gratefully acknowledge useful discussions with Georges Bouzerar, Ki-Seok Kim, Eduardo Mucciolo and Keith Slevin, as well as the support by the BK21 Plus funded by the Ministry of Education, Korea (10Z20130000023).

--- author: - 'Junyi Feng, Songyuan Li, Xi Li, Fei Wu, Qi Tian, Ming-Hsuan Yang, and Haibin Ling [^1]' bibliography: - 'bibli.bib' title: 'TapLab: A Fast Framework for Semantic Video Segmentation Tapping into Compressed-Domain Knowledge' --- Acknowledgments {#acknowledgments .unnumbered} =============== Acknowledgment {#acknowledgment .unnumbered} ============== Junyi Feng and Songyuan Li contributed equally to this work. [^1]: (Correspongding author: Xi Li.)

--- abstract: 'We settle the question of tight thresholds for offline cuckoo hashing. The problem can be stated as follows: we have $n$ keys to be hashed into $m$ buckets each capable of holding a single key. Each key has $k \geq 3 $ (distinct) associated buckets chosen uniformly at random and independently of the choices of other keys. A hash table can be constructed successfully if each key can be placed into one of its buckets. We seek thresholds $c_k$ such that, as $n$ goes to infinity, if $n/m \leq c$ for some $ c < c_k$ then a hash table can be constructed successfully with high probability, and if $n/m \geq c$ for some $c > c_k$ a hash table cannot be constructed successfully with high probability. Here we are considering the offline version of the problem, where all keys and hash values are given, so the problem is equivalent to previous models of multiple-choice hashing. We find the thresholds for all values of $k > 2$ by showing that they are in fact the same as the previously known thresholds for the random $k$-XORSAT problem. We then extend these results to the setting where keys can have differing number of choices, and provide evidence in the form of an algorithm for a conjecture extending this result to cuckoo hash tables that store multiple keys in a bucket.' author: - 'Martin Dietzfelbinger[^1]' - 'Andreas Goerdt[^2]' - 'Michael Mitzenmacher[^3]' - | \ Andrea Montanari[^4] - 'Rasmus Pagh[^5]' - 'Michael Rink${}^\star$' title: Tight Thresholds for Cuckoo Hashing via XORSAT --- =1 =1 =1 Introduction ============ Consider a hashing scheme with $n$ keys to be hashed into $m$ buckets each capable of holding a single key. Each key has $k \geq 3$ (distinct) associated buckets chosen uniformly at random and independently of the choices of other keys. A hash table can be constructed successfully if each key can be placed into one of its buckets. This setting describes the offline load balancing problem corresponding to multiple choice hashing [@ABKU] and cuckoo hashing [@FPSS:2005; @cuckoo1] with $k \geq 3$ choices. An open question in the literature (see, for example, the discussion in [@MMsurvey]) is to determine a tight threshold $c_k$ such that if $n/m \leq c$ for some $ c < c_k$ then a hash table can be constructed successfully with high probability, and if $n/m \geq c$ for some $c > c_k$ a hash table cannot be constructed successfully with high probability. In this paper, we provide these thresholds. We note that, in parallel with this work, two other papers have similarly provided means for determining the thresholds [@Fountoulakis:2009; @Frieze:2009]. Our work differs from these works in substantial ways. Perhaps the most substantial is our argument that, somewhat surprisingly, the thresholds we seek were actually essentially already known. We show that tight thresholds follow from known results in the literature, and in fact correspond exactly to the known thresholds for the random $k$-XORSAT problem. We describe the $k$-XORSAT problem and the means for computing its thresholds in more detail in the following sections. Our argument is somewhat indirect, although all of the arguments appear to rely intrinsically on the analysis of corresponding random hypergraphs, and hence the alternative arguments of [@Fountoulakis:2009; @Frieze:2009] provide additional insight that may prove useful in further explorations. With this starting point, we extend our study of the cuckoo hashing problem in two ways. First, we consider [*irregular cuckoo hashing*]{}, where the number of choices corresponding to a key is not a fixed constant $k$ but itself a random variable depending on the key. Our motivations for studying this variant include past work on irregular low-density parity-check codes [@ldpc] and recent work on alternative hashing schemes that have been said to behave like cuckoo hashing with “3.5 choices” [@cuckoo35]. Beyond finding thresholds, we show how to optimize irregular cuckoo hashing schemes with a specified average number of choices per key; for example, with an average of 3.5 choices per key, the optimal scheme is the natural one where half of the keys obtain 3 choices, and the other half obtain 4. Second, we consider the generalization to the setting where a bucket can hold more than one key. We provide a conjecture regarding the appropriate threshold behavior for this setting, and provide a simple algorithm that, experimentally, appears to perform remarkably close to the thresholds predicted by our conjecture. =1 Paper overview -------------- Section \[sect:facts:cores\] presents an exposition of known results on cores of random hypergraphs. Readers familiar with this material may want to skip directly to Section \[sec:main\], which provides our proof that the thresholds for $k$-XORSAT and $k$-ary cuckoo hashing are identical. In Section \[sec:non-integer\] we extend the discussion of thresholds to the case where $k$ is any real number greater than 2. Finally, Section \[subsec:selfless:algorithm\] presents our simple algorithm to construct hash tables and presents experimental evidence that it is able to achieve load factors close to the thresholds. =1 Further details appear in the appendices. Technical background on cores {#sect:facts:cores} ============================= The key to our analysis will be the behavior of cores in random hypergraphs. We therefore begin by providing a review of this subject. To be clear, the results of this section are not new; the reader is encouraged to see [@MezMontBook:2009 Ch. 18], as well as references [@Coo:2004; @DubMan:2002; @Molloy:RSA:2005] for more background. We consider the set of all $k$-uniform hypergraphs with $m$ nodes and $n$ hyperedges ${\mathcal{G}^k_{m,n}}$. More precisely, each hypergraph $G$ from ${\mathcal{G}^k_{m,n}}$ consists of $n$ (labeled) hyperedges of a fixed size $k\ge2$, chosen independently at random, with repetition, from the $\binom{m}{k}$ subsets of $\{1,\ldots,m\}$ of size $k$. This model will be regarded as a probability space. We always assume $k$ is fixed, $m$ is sufficiently large, and $n=c m$ for a constant $c$. For $\ell\ge2$, the $\ell$[*-core*]{} of a hypergraph $G$ is defined as the largest induced sub-hypergraph that has minimum degree $\ell$ or larger. It is well known that the $\ell$-core can be obtained by the following iterative “peeling process”: While there are nodes with degree smaller than $\ell$, delete them and their incident hyperedges. By pursuing this process backwards one sees that the $\ell$-core, conditioned on the number of nodes and hyperedges it contains, is a uniform random hypergraph that satisfies the degree constraint. The fate of a fixed node $a$ after a fixed number of $h$ iterations of the peeling procedure is determined by the $h$-neighborhood of $a$, where the $h$-neighborhood of $a$ is the sub-hypergraph induced on the nodes at distance at most $h$ from $a$. For example, the $1$-neighborhood contains all hyperedges containing $a$. In our setting where $n$ is linear in $m$ the $h$-neighborhood of node $a$ is a hypertree of low degree (at most $\log \log m$) with high probability. We assume this in the discussion to come. We can see whether a node $a$ is removed from the hypergraph in the course of $h$ iterations of the peeling process in the following way. Consider the hypertree rooted from $a$ (so the children are nodes that share a hyperedge with $a$, and similarly the children of a node share a hyperedge with that node down the tree). First, consider the nodes at distance $h-1$ from $a$ and delete them if they have at most $\ell-2$ child hyperedges; that is, their degree is at most $\ell-1.$ Second, treat the nodes at distance $h-2 $ in the same way, and so on, down to distance $1$, the children of $a$. Finally, $a$ is deleted if its degree is at most $\ell-1.$ The analysis of such random processes on trees has been well-studied in the literature. (See, for example, [@BroderFriezeUpfal; @AndOrTrees] for similar analyses.) We wish to determine the probability $q_h$ that node $a$ is deleted after $h$ rounds of the peeling process. For $j < h$ let $p_j$ be the probability that a node at distance $h-j$ from $a$ is deleted after $j$ rounds of the peeling process. The discussion becomes easier for the binomial random hypergraph with an expected number of $cm$ hyperedges: Each hyperedge is present with probability $k! \cdot c /m^{k-1}$ independently. It is well known that ${\mathcal{G}^k_{m,n}}$ and the binomial hypergraph are equivalent as far as asymptotic behavior of cores are concerned when $c$ is a constant. Let ${\mathrm{Bin}}(N,p)$ denote a random variable with a binomial distribution, and ${\mathrm{Po}}(\beta)$ a random variable with a Poisson distribution. Below we make use of the Poisson approximation of the binomial distribution and the fact that the number of child hyperedges of a node in the hypertree asymptotically follows the binomial distribution. This results in additive terms that tend to zero as $m$ goes to infinity. We have $p_0 = 0$, $$\begin{aligned} p_1 & =& \Pr\bigg[{\mathrm{Bin}}\left( \binom{m-1}{k-1} \,\,,\,\,k! \cdot \frac{c}{m^{k-1}} \right) \le \ell-2 \bigg]\, \\ & & \\ & = & \Pr[{\mathrm{Po}}(k c ) \le \ell-2] \pm o(1), \\ \mbox{ } & \\ p_{j+1} &= & \Pr\bigg[{\mathrm{Bin}}\left(\binom{m-1}{k-1} \,\, , \,\, k! \cdot \frac{c}{m^{k-1}} \cdot (1- p_j)^{k-1}\right)\, \le \ell-2\bigg]\, \\ & & \\ &=& \Pr[ {\mathrm{Po}}(kc(1-p_j)^{k-1}) \le \ell-2] \pm o(1), \mbox{ for } j=1,\dots,h-2.\end{aligned}$$ The probability $q_h$ that $a$ itself is deleted is given by the following different formula: $$\label{del} q_h = \Pr[{\mathrm{Po}}(kc(1-p_{h-1})^{k-1} )\le \ell-1] \pm o(1).$$ The $p_j$ are monotonically increasing and $0 \le p_j \le 1$, so $p= \lim p_j$ is well-defined. The probability that $a$ is deleted approaches $p$ from below as $h$ grows. Continuity of the functions involved implies that $p$ is the smallest non-negative solution of $$\begin{aligned} p & = & \Pr[ {\mathrm{Po}}(kc(1-p)^{k-1}) \le \ell-2].\end{aligned}$$ Observe that $1$ is always a solution. Equivalently, applying the monotone function $t\mapsto kc(1-t)^{k-1}$ to both sides of the equation, $p$ is the smallest solution of $$\label{eq:def:p} kc(1-p)^{k-1} = kc \left( 1 - \Pr [ {\mathrm{Po}}(kc(1-p)^{k-1}) \le \ell-2] \right)^{k-1}.$$ Let $\beta = kc (1-p)^{k-1}$. It is helpful to think of $\beta$ with the following interpretation: Given a node in the hypertree, the number of child hyperedges (before deletion) follows the distribution ${\mathrm{Po}}(kc)$. Asymptotically, a given child hyperedge is not deleted with probability $(1-p)^{k-1}$, independently for all children. Hence the number of child hyperedges after deletion follows the distribution ${\mathrm{Po}}(kc(1-p)^{k-1}).$ And $\beta$ is the key parameter for the node giving the expected number of hyperedges containing it that could contribute to keeping it in the core. Note that (\[eq:def:p\]) is equivalent to $$c = \frac{1}{k}\cdot\frac{\beta}{\left (\Pr[{\mathrm{Po}}(\beta ) \,\ge \ell-1]\right)^{k-1}}.$$ This motivates considering the function $$g_{k,\ell}(\beta) =\frac{1}{k}\cdot\frac{\beta}{(\Pr[{\mathrm{Po}}(\beta) \ge \ell-1])^{k-1}}, \label{eq:definition:g}$$ which has the following properties in the range $(0,\infty)$: It tends to infinity for $\beta\to0$, as well as for $\beta\to\infty$. Since it is convex there is exactly one global minimum. Let $\beta^*_{k,\ell}=\arg\min_{\beta}g_{k,\ell}(\beta)$ and $c^*_{k,\ell}=\min g_{k,\ell}(\beta)$. For $\beta > \beta^*_{k,\ell}$ the function $g_{k,\ell}$ is monotonically increasing. For each $c>c^*_{k,\ell}$ let $\beta(c) \,= \, \beta_{k,\ell}(c)$ denote the unique $\beta>\beta^*_{k,\ell}$ such that $g_{k,\ell}(\beta)=c$. Coming back to the fate of $a$ under the peeling process, Equation (\[del\]) shows that $a$ is deleted with probability approaching $\Pr[{\mathrm{Po}}(\beta(c))\le \ell-1].$ This probability is smaller than $1$ if and only if $c > c^*_{k, \ell}$, which implies that the expected number of nodes that are [*not*]{} deleted is linear in $n$. As the $h$-neighborhoods of two nodes $a$ and $b$ are disjoint with high probability, by making use of the second moment we can show that in this case a linear number of nodes survive with high probability. (The sophisticated reader would use Azuma’s inequality to obtain concentration bounds.) Following this line of reasoning, we obtain the following results, the full proof of which is in [@Molloy:RSA:2005]. (See also the related argument of [@MezMontBook:2009 Ch. 18].) Note the restriction to the case $k + \ell > 4$, which means that the result does not apply to $2$-cores in standard graphs; since the analysis of standard cuckoo hashing is simple, using direct arguments, this case is ignored in the analysis henceforth. \[prop:one\] Let $k+\ell >4$ and $G$ be a random hypergraph from ${\mathcal{G}^k_{m,n}}$. Then $c^*_{k,\ell}$ is the threshold for the *appearance* of an $\ell$-core in $G$. That is, for constant $c$ and $m\to \infty$, - if $n/m = c < c^*_{k,\ell}$, then $G$ has an empty $\ell$-core with probability $1-o(1)$. - if $n/m = c > c^*_{k,\ell}$, then $G$ has an $\ell$-core of linear size with probability $1-o(1)$. In the following we assume $c > c^*_{k,\ell}$. Therefore $\beta(c)>\beta^*_{k,\ell}$ exists. Let $\hat m$ be the number of nodes in the $\ell$-core and $\hat n$ be the number of hyperedges in the $\ell$-core. We will find it useful in what follows to consider the [*edge density*]{} of the $\ell$-core, which is simply the ratio of the number of hyperedges to the number of nodes. =1 \[prop:two\] Let $c>c^*_{k,\ell}$ and $n/m = c\, (1\pm o(1))$. Then with high probability in ${\mathcal{G}^k_{m,n}}$ $$\hat m = \Pr[{\mathrm{Po}}({\beta(c))}\ge \ell]\cdot m \pm o(m)$$ and $$\hat n = (\Pr[{\mathrm{Po}}({\beta(c))}\ge \ell-1])^k \cdot n \pm o(m).$$ =1 \[prop:two\] Let $c>c^*_{k,\ell}$ and $n/m = c\, (1\pm o(1))$. Then with high probability in ${\mathcal{G}^k_{m,n}}$ $$\hat m = \Pr[{\mathrm{Po}}({\beta(c))}\ge \ell]\cdot m \pm o(m) \mbox{ and } \hat n = (\Pr[{\mathrm{Po}}({\beta(c))}\ge \ell-1])^k \cdot n \pm o(m).$$ The bound for $\hat m$ follows from the concentration of the expected number of nodes surviving when we plug in the limit $p$ for $p_h$ in equation (\[del\]). The result for $\hat n$ follows similar lines: Consider a fixed hyperedge $e$ that we assume is present in the random hypergraph. For each node of this hyperedge we consider its $h$-neighborhood modified in that $e$ itself does not belong to this $h$-neighborhood. We have $k$ disjoint trees with high probability. Therefore each of the $k$ nodes of $e$ survives $h$ iterations of the peeling procedure independently with probability $\Pr[{\mathrm{Po}}(\beta(c)) \ge \ell-1]$. Note that we use $\ell-1$ here (instead of $\ell$) because the nodes belong to $e.$ Then $e$ itself survives with $ (\Pr[{\mathrm{Po}}(\beta(c)) \ge \ell-1])^k.$ Concentration of the number of surviving hyperedges again follows from second moment calculations or Azuma’s inequality. With this we have the information needed regarding the edge density of the $\ell$-core. \[prop:three\] If $c>c^*_{k,\ell}$ and $n/m = c\, (1\pm o(1))$ then with high probability the edge density of the $\ell$-core of a random hypergraph from ${\mathcal{G}^k_{m,n}}$ is $$\frac{\beta(c)\cdot \Pr[{\mathrm{Po}}(\beta(c))\ge \ell-1]}{k\cdot\Pr[{\mathrm{Po}}(\beta(c))\ge \ell]}\pm o(1).$$ This follows directly from Proposition \[prop:two\], where we have also used equation (\[eq:definition:g\]) to simplify the expression for $\hat n$. We define $c_{k,\ell}$ as the unique $c$ that satisfies $$\label{ckll+1} \frac{\beta(c)\cdot \Pr[{\mathrm{Po}}(\beta(c))\ge \ell-1]} {k\cdot\Pr[{\mathrm{Po}}(\beta(c))\ge \ell]}\,= \, \ell-1.$$ The values $c_{k,\ell}$ will prove important in the work to come; in particular, we next show that $c_{k, 2}$ is the threshold for $k$-ary cuckoo hashing for $k > 2$. We also conjecture that $c_{k,\ell+1}$ is the threshold for $k$-ary cuckoo hashing when a bucket can hold $\ell$ keys instead of a single key. The following table contains numerical values of $c_{k,\ell}$ for $\ell=2, \ldots, 7$ and $k=2,\ldots,7$ (rounded to 10 decimal places). Some of these numbers are found or referred to in other works, such as [@Coo:2004 Sect. 5], [@MezRicZec:2003 Sect.4.4], [@MezMontBook:2009 p.423], [@FerRam:2007], and [@CaiSanWor:2007]. $\ell \backslash k$ 2 3 4 5 6 7 --------------------- -------------- -------------- -------------- -------------- -------------- -------------- 2 $-$ 0.9179352767 0.9767701649 0.9924383913 0.9973795528 0.9990637588 3 1.7940237365 1.9764028279 1.9964829679 1.9994487201 1.9999137473 1.9999866878 4 2.8774628058 2.9918572178 2.9993854302 2.9999554360 2.9999969384 2.9999997987 5 3.9214790971 3.9970126256 3.9998882644 3.9999962949 3.9999998884 3.9999999969 6 4.9477568093 4.9988732941 4.9999793407 4.9999996871 4.9999999959 5.0000000000 7 5.9644362395 5.9995688805 5.9999961417 5.9999999733 5.9999999998 6.0000000000 =1 Equality of thresholds for random $k$-XORSAT and $k$-ary cuckoo hashing {#sec:main} ======================================================================= We now recall the random $k$-XORSAT problem and describe its relationship to cores of random hypergraphs and cuckoo hashing. The $k$-XORSAT problem is a variant of the satisfiability problem in which every clause has $k$ literals and the clause is satisfied if the XOR of values of the literals is 1. Equivalently, since XORs correspond to addition modulo 2, and the negation of $X_i$ is just $1$ XOR $X_i$, an instance of the $k$-XORSAT problem corresponds to a system of linear equations modulo 2, with each equation having $k$ variables (none of which is negated), and randomly chosen right hand sides. (In what follows we simply use the addition operator where it is understood we are working modulo 2 from context.) For a random $k$-XORSAT problem, let ${{\Phi}^k_{m,n}}$ be the set of all sequences of $n$ linear equations over $m$ variables $x_1,\ldots,x_m,$ where an equation is $$x_{j_1}+\cdots+x_{j_k}=b_j,$$ where $b_j\in\{0,1\}$ and $\{j_1,\ldots,j_k\}$ is a subset of $\{1,\ldots,m\}$ with $k$ elements. We consider ${{\Phi}^k_{m,n}}$ as a probability space with the uniform distribution. Given a $k$-XORSAT formula $F$, it is clear that $F$ is satisfiable if and only if the formula obtained from $F$ by repeatedly deleting variables that occur only once (and equations containing them) is satisfiable. Now consider the $k$-XORSAT formula as a hypergraph, with nodes representing variables and hyperedges representing equations. (The values $b_j$ of the equations are not represented.) The process of repeatedly deleting all variables that occur only once, and the corresponding equations, is exactly equivalent to the peeling process on the hypergraph. Hence, after the peeling process, we obtain the 2-core of the hypergraph. This motivates the following definition. Let $\Psi^k_{m,n}$ be the set of all sequences of $n$ equations such that each variable appears at least twice. We consider $\Psi^k_{m,n}$ as a probability space with the uniform distribution. Recall that if we start with a uniformly chosen random $k$-XORSAT formula, and perform the peeling process, then conditioned on the remaining number of equations and variables ($\hat n$ and $\hat m$), we are in fact left with a uniform random formula from $\Psi^k_{\hat m,\hat n}$. Hence, the imperative question is when a random formula from $\Psi^k_{\hat m,\hat n}$ will be satisfiable. In [@DubMan:2002], it was shown that this depends entirely on the edge density of the corresponding hypergraph. If the edge density is smaller than 1, so that there are more variables than equations, the formula is likely to be satisfiable, and naturally, if there are more equations than variables, the formula is likely to be unsatisfiable. Specifically, we have the following theorem from [@DubMan:2002]. \[thresh\] \[DM:Theorem\] Let $k > 2$ be fixed. For $n/m = \gamma$ and $m\to\infty$, - if $\gamma>1$ then a random formula from $\Psi^k_{m,n}$ is unsatisfiable with high probability. - if $\gamma<1$ then a random formula from $\Psi^k_{m,n}$ is satisfiable with high probability. The proof of Theorem \[thresh\] in Section 3 of [@DubMan:2002] uses a first moment method argument for the simple direction (part (a)). Part (b) is significantly more complicated, and is based on the second moment method. Essentially the same problem has also arisen in coding theoretic settings; analysis and techniques can be found in for example [@MMU]. =1 It has been suggested by various readers of earlier drafts of this paper that previous proofs of Theorem \[thresh\] have been insufficiently complete, particularly for $k > 3$. We therefore provide a detailed proof in Appendix C for completeness. We have shown that the edge density is concentrated around a specific value depending on the initial ratio $c$ of hyperedges (equations) to nodes (variables). Let $c_{k,2}$ be the value of $c$ such that the resulting edge density is concentrated around 1. Then Proposition \[prop:three\] and Theorem \[thresh\] together with the preceding consideration implies: \[corthresh\] Let $k > 2$ and consider ${{\Phi}^k_{m,n}}.$ The satisfiability threshold with respect to the edge density $c=n/m$ is $c_{k, 2}$. Again, up to this point, everything we have stated was known from previous work. We now provide the connection to cuckoo hashing, to show that we obtain the same threshold values for the success of cuckoo hashing. That is, we argue the following: \[thm:equivalence:variant\] For $k > 2$, $c_{k,2}$ is the threshold for $k$-ary cuckoo hashing to work. That is, and with $n$ keys to be stored and $m$ buckets, with $c=n/m$ fixed and $m\to\infty$, - if $c > c_{k,2}$, then $k$-ary cuckoo hashing does not work with high probability. - if $c < c_{k,2}$, then $k$-ary cuckoo hashing works with high probability. Assume a set of $n$ keys $S$ is given, and for each $x \in S$ a random set $A_x\subseteq\{1,\ldots,m \}$ of size $k$ of possible buckets is chosen. To prove part (a), note that the sets $A_x$ for $x\in S$ can be represented by a random hypergraph from ${\mathcal{G}^k_{m,n}}$. If $n/m = c > c_{k,2}$ and $m\to\infty$, then with high probability the edge density in the 2-core is greater than 1. The hyperedges in the 2-core correspond to a set of keys, and the nodes in the 2-core to the buckets available for these keys. Obviously, then, cuckoo hashing does not work. To prove part (b), consider the case where $n/m = c < c_{k,2}$ and $m \to\infty$. Picking for each $x$ a random $b_x \in \{0, 1\}$, the sets $A_x$, $x\in S$, induce a random system of equations from ${{\Phi}^k_{m,n}}.$ Specifically, $A_x = \{ j_1 , \dots, j_k\}$ induces the equation $x_{j_1} + \dots + x_{j_k} = b_x.$ By Corollary \[corthresh\] a random system of equations from ${{\Phi}^k_{m,n}}$ is satisfiable with high probability. This implies that the the matrix $M$ made up from the left-hand sides of these equations consists of linearly independent rows with high probability. This is because a given set of left-hand sides with dependent rows is only satisfiable with probability at most $1/2$ when we pick the $b_x$ at random. Therefore we have an $n \times n$-submatrix in $M$ with a nonzero determinant. The expansion of the determinant of this submatrix as a sum of products by the Leibniz formula must contain a product with all factors being variables $x_{i_j}$ (as opposed to 0). This product term corresponds to a permutation mapping keys to buckets, showing that cuckoo hashing is indeed possible. We make some additional remarks. We note that the idea of using the rank of the key-bucket matrix to obtain lower bounds on the cuckoo hashing threshold is not new either; it appears in [@DP:2008]. There the authors use a result bounding the rank by Calkin [@Calkin] to obtain a lower bound on the threshold, but this bound is not tight in this context. More details can be found by reviewing [@Calkin Theorem 1.2] and [@MezMontBook:2009 Exercise 18.6]. Also, Batu et al. [@BBC] note that 2-core thresholds provide an upper bound on the threshold for cuckoo hashing, but fail to note the connection to work on the $k$-XORSAT problems. Non-integer choices {#sec:non-integer} =================== The analysis of $k$-cores in Section \[sec:main\] and the correspondence to $k$-XORSAT problems extends nicely to the setting where the number of choices for a key is not necessarily a fixed number $k$. This can be naturally accomplished in the following way: when a key $x$ is to be inserted in the cuckoo hash table, the number of choices of location for the key is itself determined by some hash function; then the appropriate number of choices for each key $x$ can also be found when performing a lookup. Hence, it is possible to ask about for example cuckoo hashing with 3.5 choices, by which we would mean an average of 3.5 choices. Similarly, even if we decide to have an average of $k$ choices per key, for an integer $k$, it is not immediately obvious whether the success probability in $k$-ary cuckoo hashing could be improved if we do not fix the number of possible positions for a key but rather choose it at random from a cleverly selected distribution. Let us consider a more general setting where for each $x\in U$ the set $A_x$ is chosen uniformly at random from the set of all ${\ensuremath{k}}_x$-element subsets of $[{\ensuremath{m}}]$, where ${\ensuremath{k}}_x$ follows some probability mass function ${\ensuremath{{\rho}}}_x$ on $\{2,\ldots,{\ensuremath{m}}\}$.[^6] Let ${\ensuremath{\kappa}}_x=E({\ensuremath{k}}_x)$ and ${\ensuremath{\kappa}}^*=\frac{1}{{\ensuremath{n}}}\sum_{x\in S}{\ensuremath{\kappa}}_x$. Note that ${\ensuremath{\kappa}}^*$ is the average (over all $x \in S$) worst case lookup time for successful searches. We keep ${\ensuremath{\kappa}}^*$ fixed and study which sequence $({\ensuremath{{\rho}}}_x)_{x\in S}$ maximizes the probability that cuckoo hashing is successful. We fix the sequence of the expected number of choices per key $({\ensuremath{\kappa}}_x)_{x\in S}$ and therefore ${\ensuremath{\kappa}}^*$. Furthermore we assume ${\ensuremath{\kappa}}_x\leq n-2$, for all $x \in S$; obviously this does not exclude interesting cases. For compactness reasons, there is a system of probability mass functions ${\ensuremath{{\rho}}}_x$ that maximizes the success probability. We will show the following: \[prop:constant\_expected\_degree\] Let $({\ensuremath{{\rho}}}_x)_{x \in S}$ be an optimal sequence. Then we have, for all $x \in S$: $${\ensuremath{{\rho}}}_x( \lfloor {\ensuremath{\kappa}}_x \rfloor )=1-({\ensuremath{\kappa}}_x-\lfloor {\ensuremath{\kappa}}_x\rfloor), \text{ and }{\ensuremath{{\rho}}}_x( \lfloor {\ensuremath{\kappa}}_x \rfloor+1 )={\ensuremath{\kappa}}_x-\lfloor {\ensuremath{\kappa}}_x\rfloor.$$ That is, the success probability is maximized if for each $x\in S$ the number of choices ${\ensuremath{k}}_x$ is concentrated on $\lfloor {\ensuremath{\kappa}}_x\rfloor$ and $\lfloor {\ensuremath{\kappa}}_x\rfloor +1$ (when the number of choices is non-integral). Further, in the natural case where all keys $x$ have the same expected number ${\ensuremath{\kappa}}^*$ of choices, the optimal assignment is concentrated on $\lfloor {\ensuremath{\kappa}}^* \rfloor$ and $\lfloor {\ensuremath{\kappa}}^*\rfloor +1$. Also, if ${\ensuremath{\kappa}}_x$ is an integer, then a fixed degree ${\ensuremath{k}}_x={\ensuremath{\kappa}}_x$ is optimal. This is very different from other similar scenarios, such as erasure- and error-correcting codes, where irregular distributions have proven beneficial [@ldpc]. =1 The proof is given in Appendix A. =1 We consider a random bipartite graph $G_S$ with left node set $S$, right node set $[{\ensuremath{m}}]$ and an edge between two nodes $x\in S$ and $a\in [{\ensuremath{m}}]$ if and only if $a\in A_x$. Let the sequence $({\ensuremath{\kappa}}_x)_{x\in S}$ be fixed. For each $x \in S$ we want to obtain a distribution ${\ensuremath{{\rho}}}_x$ for the degree ${\ensuremath{k}}_x$ (or, equivalently, the cardinality of $A_x$), such that we have $E({\ensuremath{k}}_x)={\ensuremath{\kappa}}_x$ and the following quantity is maximized: $$\label{eq:success_prob} \Pr(\text{``success''}):=\Pr( (A_x)_{x\in S} \text{ admits a left-perfect matching}\footnote{In the following ``matching'' and ``left-perfect matching'' are used synonymously.}\text{ in $G_S$}){\text{ }}.$$ We study the sequence $({\ensuremath{{\rho}}}_x)_{x\in S}$ that realizes the maximum. Let $z$ be an arbitrary but fixed element of $S$ with probability mass function ${\ensuremath{{\rho}}}_z$. To prove Proposition \[prop:constant\_expected\_degree\] it is sufficient to show that if there exist two numbers ${\ensuremath{l}}$ and ${\ensuremath{k}}$ with ${\ensuremath{l}}<{\ensuremath{\kappa}}_z < {\ensuremath{k}}$ and ${\ensuremath{k}}-{\ensuremath{l}}\geq2$ as well as ${\ensuremath{{\rho}}}_z({\ensuremath{l}})>0$ and ${\ensuremath{{\rho}}}_z({\ensuremath{k}})>0$ then cannot be maximal. We start by fixing ${\ensuremath{k}}_x$ and $A_x$ for each $x\in S-\{z\}$ and consider the corresponding bipartite graph $G_{S-\{z\}}$. Let $B\subseteq [{\ensuremath{m}}]$ be the set of right nodes in $G_{S-\{z\}}$ that are matched in every matching. Then there is a matching for the whole key set $S$ in $G_S$ if and only if $A_z \not \subseteq B$. Note that $0\leq |B|<{\ensuremath{m}}$, i.e., there must be at least one right node that is not matched. Let $p=\min\{ {\ensuremath{{\rho}}}_z({\ensuremath{l}}), {\ensuremath{{\rho}}}_z({\ensuremath{k}}) \}>0$ and $|B|=b$. We will show that changing ${\ensuremath{{\rho}}}_z$ to $$\begin{aligned} {\ensuremath{{\rho}}}_z'({\ensuremath{l}})&:={\ensuremath{{\rho}}}_z({\ensuremath{l}})-p &{\ensuremath{{\rho}}}_z'({\ensuremath{k}})&:={\ensuremath{{\rho}}}_z({\ensuremath{k}})-p \\ {\ensuremath{{\rho}}}_z'({\ensuremath{l}}+1)&:={\ensuremath{{\rho}}}_z({\ensuremath{l}}+1)+p &{\ensuremath{{\rho}}}_z'({\ensuremath{k}}-1)&:={\ensuremath{{\rho}}}_z({\ensuremath{k}}-1)+p,\end{aligned}$$ with ${\ensuremath{{\rho}}}_z'(j)={\ensuremath{{\rho}}}_z(j)$ for $j\notin\{{\ensuremath{l}},{\ensuremath{k}}\}$, increases , while leaving ${\ensuremath{\kappa}}_z$ unchanged. This is the case if and only if $$\label{eq:first_ineq} p\cdot \frac{\binom{b}{{\ensuremath{l}}}}{\binom{{\ensuremath{m}}}{{\ensuremath{l}}}} + p\cdot \frac{\binom{b}{{\ensuremath{k}}}}{\binom{{\ensuremath{m}}}{{\ensuremath{k}}}} \geq p\cdot \frac{\binom{b}{{\ensuremath{l}}+1}}{\binom{{\ensuremath{m}}}{{\ensuremath{l}}+1}} + p\cdot \frac{\binom{b}{{\ensuremath{k}}-1}}{\binom{{\ensuremath{m}}}{{\ensuremath{k}}-1}}, $$ and the strict inequality holds for at least one value $b$ that occurs with positive probability. The left sum of is the $2\cdot p$ fraction of the failure probability (by ${\ensuremath{{\rho}}}_z({\ensuremath{l}})$ and ${\ensuremath{{\rho}}}_z({\ensuremath{k}})$) before the change of ${\ensuremath{{\rho}}}_z$ under the condition that $B$ has cardinality $b$; the right sum is the corresponding fraction of the failure probability after the change. Depending on $b$ we have to distinguish several cases. $b={\ensuremath{m}}-1$. In this case both sides of are equal, i.e., the modification we do to ${\ensuremath{{\rho}}}_z$ will not change the success probability. ${\ensuremath{k}}\leq b < {\ensuremath{m}}-1$. Canceling $p$ and subtracting ${\binom{b}{{\ensuremath{k}}}}/{\binom{{\ensuremath{m}}}{{\ensuremath{k}}}}$ and ${\binom{b}{{\ensuremath{l}}+1}}/{\binom{{\ensuremath{m}}}{{\ensuremath{l}}+1}}$ from both sides of shows that the strict inequality holds if and only if $$\label{eq:intermediate} \begin{split} \frac{b\cdots(b-{\ensuremath{l}}+1)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{l}}+1)}-\frac{b\cdots(b-{\ensuremath{l}})}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{l}})}>\frac{b\cdots(b-{\ensuremath{k}}+2)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+2)}-\frac{b\cdots(b-{\ensuremath{k}}+1)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+1)} {\text{ }}. \end{split}$$ Factoring out $\frac{b\cdots(b-{\ensuremath{l}}+1)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{l}}+1)}$ on the left side and $\frac{b\cdots(b-{\ensuremath{k}}+2)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+2)}$ on the right side gives $$\frac{b\cdots (b-{\ensuremath{l}}+1)}{{\ensuremath{m}}\cdots ({\ensuremath{m}}-{\ensuremath{l}})}>\frac{b\cdots (b-{\ensuremath{k}}+2)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+1)} \Leftrightarrow \frac{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+1)}{{\ensuremath{m}}\cdots ({\ensuremath{m}}-{\ensuremath{l}})}>\frac{b\cdots (b-{\ensuremath{k}}+2)}{b\cdots (b-{\ensuremath{l}}+1)}{\text{ }}.$$ Since ${\ensuremath{l}}\leq {\ensuremath{k}}-2$, this is equivalent to $$({\ensuremath{m}}-{\ensuremath{l}}+1)\cdot({\ensuremath{m}}-{\ensuremath{l}})\cdots({\ensuremath{m}}-{\ensuremath{k}}+1)>(b-{\ensuremath{l}})\cdot(b-{\ensuremath{l}}-1)\cdots(b-{\ensuremath{k}}+2){\text{ }},$$ which is true for $m-1>b$. ${\ensuremath{l}}\leq b<{\ensuremath{k}}$. Calculations along the lines of case 2 show that the strict inequality of also holds in this case. Note that $\binom{b}{k}$, $\binom{b}{k-1}$ and $\binom{b}{l+1}$ can be zero. $0\leq b<{\ensuremath{l}}$. In this case both sides of are zero, i.e., the modifications we do to ${\ensuremath{{\rho}}}_z$ will not change the success probability. Since in cases 1 and 4 above there was no change in the success probability, to show that cannot be maximal when ${\ensuremath{k}}-{\ensuremath{l}}\geq2$ as we are considering, it remains to show that at least one of the Cases 2 and 3 occurs with positive probability. We construct a situation in which one of these cases applies, and which occurs with positive probability. Choose degrees ${\ensuremath{k}}_x$ for all elements $x \in S-\{z\}$ such that ${\ensuremath{k}}_x\leq {\ensuremath{\kappa}}_x$ and ${\ensuremath{{\rho}}}_x({\ensuremath{k}}_x)>0$. Consider a permutation of the elements $x \in S-\{z\}$ such that these degrees are ordered, i.e., ${\ensuremath{k}}_{x_1}\leq {\ensuremath{k}}_{x_2}\leq \ldots \leq {\ensuremath{k}}_{x_{{\ensuremath{n}}-1}}$. Choose the first element $x_i$ with $i \geq {\ensuremath{l}}$ and ${\ensuremath{k}}_{x_i}\leq i$. Such an element must exist, since we assume ${\ensuremath{k}}_x\leq {\ensuremath{\kappa}}_x\leq {\ensuremath{n}}-2$, in particular we have $l<{\ensuremath{n}}-2$. Arrange that $A_{x_j}\subseteq [i],1\leq j \leq i,$ such that there is a matching in $G_{\{x_1,\ldots,x_i\}}$. This implies $b\geq {\ensuremath{l}}$. Then arrange that $|A_{x_j}-\bigcup_{1\leq {j'}<j} A_{x_{j'}}|=1$, for all $i<j\leq n-2$, as well as $|A_{x_{n-1}}-\bigcup_{1\leq {j'}<n-1} A_{x_{j'}}|=2$, which implies $b<m-1$. This finishes the proof of Proposition \[prop:constant\_expected\_degree\]. Thresholds for non-integral degree distributions ------------------------------------------------ We now describe how to extend our previous analysis to derive thresholds for the case of a non-integral number of choices per key; equivalently, we are making use of thresholds for XORSAT problems with an irregular number of literals per clause. Following notation that is frequently used in the coding literature, we let $\Lambda_k$ be the probability that a key obtains $k$ choices, and define $\Lambda(x) = \sum_k \Lambda_k x^k$. Clearly, then, $\Lambda'(x) = \sum_k \Lambda_k k x^{k-1}$, and $\Lambda'(1)=\kappa^*$. (We assume henceforth that $\Lambda_0 = \Lambda_1 = 0$ and $\Lambda_k = 0$ for all $k$ sufficiently large for technical convenience.) We now follow our previous analysis from Section \[sect:facts:cores\]; to see if a node $a$ is deleted after $h$ rounds of the peeling process, we let $p_j$ be the probability that a node at distance $h-j$ from $a$ is deleted after $j$ rounds. We must now account for the differing degrees of hyperedges. Here, the appropriate asymptotics is given by a mixture of binomial hypergraphs, with each hyperedge of degree $k$ present with probability $k! \cdot c \Lambda_k /m^{k-1}$ independently. The corresponding equations are then given by $p_0 = 0$, $$\begin{aligned} p_1 & =& \Pr\bigg[\sum_k {\mathrm{Bin}}\left( \binom{m-1}{k-1} \,\,,\,\,k! \cdot \frac{c\Lambda_k}{m^{k-1}} \right) \le \ell-2 \bigg]\, \\ & = & \Pr\bigg[\sum_k {\mathrm{Po}}(k c \Lambda_k) \le \ell-2\bigg] \pm o(1), \\ & = & \Pr[{\mathrm{Po}}(c \Lambda'(1)) \le \ell-2] \pm o(1), \\ \mbox{ } & \\ p_{j+1} &= & \Pr\bigg[\sum_k {\mathrm{Bin}}\left(\binom{m-1}{k-1} \,\, , \,\, k! \cdot \frac{c \Lambda_k}{m^{k-1}} \cdot (1- p_j)^{k-1}\right)\, \le \ell-2\bigg]\, \\ &=& \Pr\bigg[ \sum_k {\mathrm{Po}}(kc \Lambda_k (1-p_j)^{k-1}) \le \ell-2\bigg] \pm o(1), \mbox{ for } j=1,\dots,h-2, \\ &=& \Pr[ {\mathrm{Po}}(c \Lambda'(1-p_j)) \le \ell-2] \pm o(1), \mbox{ for } j=1,\dots,h-2.\end{aligned}$$ Note that we have used the standard fact that the sum of Poisson random variables is itself Poisson, which allows us to conveniently express everything in terms of the generating function $\Lambda(x)$ and its derivative. As before we find $p= \lim p_j$, which is now given by the smallest non-negative solution of $$\begin{aligned} p & = & \Pr[ {\mathrm{Po}}(c\Lambda'(1-p)) \le \ell-2].\end{aligned}$$ When given a degree distribution $(\Lambda_k)_k$, we can proceed as before to find the threshold load that allows that the edge density of the 2-core remains greater than 1; using =1 a second moment argument, =1 the approach of Appendix C, this can again be shown to be the required property for the corresponding XORSAT problem to have a solution, and hence for there to be a permutation successfully mapping keys to buckets. Notice that this argument works for all degree distributions (subject to the restrictions given above), but in particular we have already shown that the optimal thresholds are to be found by the simple degree distributions that have all weight on two values, $\lfloor{\ensuremath{\kappa}}^*\rfloor$ and $\lfloor{\ensuremath{\kappa}}^*\rfloor + 1$. Abusing notation slightly, let $c_{{\ensuremath{\kappa}}^*,2}$ be the unique $c$ such that the edge density of the 2-core of the corresponding mixture is equal to 1, following the same form as in Proposition \[prop:three\] and equation (\[ckll+1\]). The corresponding extension to Theorem \[thm:equivalence:variant\] is the following: \[thm:equivalence:variant2\] For ${\ensuremath{\kappa}}^* > 2$, $c_{{\ensuremath{\kappa}}^*,2}$ is the threshold for cuckoo hashing with an average of ${\ensuremath{\kappa}}^*$ choices per key to work. That is, with $n$ keys to be stored and $m$ buckets, with $c=n/m$ fixed and $m\to\infty$, - if $c > c_{{\ensuremath{\kappa}}^*,2}$, for any distribution on the number of choices per key with mean ${\ensuremath{\kappa}}^*$, cuckoo hashing does not work with high probability. - if $c < c_{{\ensuremath{\kappa}}^*,2}$, then cuckoo hashing works with high probability when the distribution on the number of choices per key is given by ${\ensuremath{{\rho}}}_x( \lfloor {\ensuremath{\kappa}}_x \rfloor )=1-({\ensuremath{\kappa}}_x-\lfloor {\ensuremath{\kappa}}_x\rfloor)$ and ${\ensuremath{{\rho}}}_x( \lfloor {\ensuremath{\kappa}}_x \rfloor+1 )={\ensuremath{\kappa}}_x-\lfloor {\ensuremath{\kappa}}_x\rfloor$, for all $x \in S$. ![Thresholds for non-integral ${\ensuremath{\kappa}}^*$-ary cuckoo hashing, with optimal degree distribution. The values in the tables are rounded to the nearest multiple of $10^{-10}$.[]{data-label="fig:non-integral"}](\figPath/threshold-nonintegral.pdf){width="\linewidth"} ${\ensuremath{\kappa}}^*$ $c_{{\ensuremath{\kappa}}^*,2}$ --------------------------- --------------------------------- 2.25 0.6666666667 2.50 0.8103423635 2.75 0.8788457372 3.00 0.9179352767 3.25 0.9408047937 3.50 0.9570796377 3.75 0.9685811888 4.00 0.9767701649 ${\ensuremath{\kappa}}^*$ $c_{{\ensuremath{\kappa}}^*,2}$ --------------------------- --------------------------------- 4.25 0.9825693463 4.50 0.9868637629 4.75 0.9900548807 5.00 0.9924383913 5.25 0.9942189481 5.50 0.9955692011 5.75 0.9965961383 6.00 0.9973795528 We have determined the thresholds numerically for a range of values of ${\ensuremath{\kappa}}^*$. The results are shown in Figure \[fig:non-integral\]. One somewhat surprising finding is that the threshold for ${\ensuremath{\kappa}}^*\leq 2.25$ appears to simply be given by $c = 0.5/(3-{\ensuremath{\kappa}}^*)$. Consequently, in place of using 2 hash functions per key, simply by using a mix of 2 or 3 hash functions for a key, we can increase the space utilization by adding 33% more keys with the same (asymptotic) amount of memory. Algorithm for computing a placement {#subsec:selfless:algorithm} =================================== In this section, we describe an algorithm for finding a placement for the keys using $k$-ary cuckoo hashing when the set $S$ of keys is given an advance. The algorithm is an adaptation of the “selfless algorithm” proposed by Sanders [@Sanders:SOFSEM:2004], for the case $k=2$, and analyzed in [@CaiSanWor:2007], for orienting standard undirected random graphs so that all edges are directed and the maximum indegree of all nodes is at most $\ell$, for some fixed $\ell \geq 2$. We generalize this algorithm to hypergraphs, including hypergraphs where hyperedges can have varying degrees. Of course, maximum matching algorithms can solve this problem perfectly. However, there are multiple motivations for considering our algorithms. First, it seems in preliminary experiments that the running times of standard matching algorithms like the Hopcroft-Karp algorithm [@HK:1973] will tend to increase significantly as the edge density approaches the threshold (the details of this effect are not yet understood), while our algorithm has linear running time which does not change in the neighborhood of the threshold. This proves useful in our experimental evaluation of thresholds. Second, we believe that algorithms of this form may prove easier to analyze for some variations of the problem. We first describe the generalized selfless algorithm for bucket size $\ell=1$. A description in pseudocode is given as Algorithm \[algo:GeneralizedSelfless\]. The algorithm can deal with arbitrary hypergraphs, uniform or not. The aim is to “orient” the hyperedges of the hypergraph $G$, i.e., associate a node $v\in e$ to each hyperedge $e$ so that at most one hyperedge is directed towards any one node $v$. Initially, all hyperedges are unoriented. Nodes that have an hyperedge directed towards them are saturated and are not considered further, and similarly hyperedges once oriented are fixed. At each step, if there is a node $v$ that is incident to only one undirected hyperedge $e$, we direct $v$ to $e$, breaking ties arbitrarily. (In the pseudocode, this is realized by giving such nodes the highest *priority*, which is 0. Note that this rule entails that the algorithm starts by carrying out the peeling process for the $2$-core. But the rule is also applied when hyperedges from the 2-core have already been treated.) If there are no such nodes, every unoriented hyperedge is assigned as its *weight* the number of unsaturated nodes it contains. (Intuitively, a smaller weight means a higher need to direct the hyperedge.) The priority of a node $v$ then is the sum of the inverses of the weights of the hyperedges that contain $v$. This corresponds to the expected number of hyperedges $v$ would have directed toward it if all its unoriented hyperedges were directed to one of their nodes at random. Now a vertex $v$ of smallest (highest) priority is chosen, again breaking ties at random. If this priority is larger than 1, then the algorithm stops and reports “failure”. This is because the sum of all priorities is the number of undirected hyperedges, so if the smallest priority is bigger than 1, the number of undirected hyperedges is larger than the number of unsaturated nodes, and it is impossible to complete the process of directing the hyperedges. Otherwise the algorithm directs the minimum weight incident hyperedge of $v$ toward $v$, breaking ties randomly. (Intuitively, this means that the algorithm tries to continue the peeling process “on average”.) This step is repeated until all hyperedges have been oriented or failure occurs. =1 We ran the generalized selfless algorithm for hypergraphs with $10^5$ and $10^6$ nodes and tabulated the failure rate around the theoretical threshold values $c_{k,2}$ for $k=3,4,5$. Results demonstrate that the generalized selfless algorithm achieves results quite near the threshold; more details and figures are given in Appendix B. =1 We ran the generalized selfless algorithm for hypergraphs with $10^5$ and $10^6$ nodes and tabulated the failure rate around the theoretical threshold values $c_{k,2}$ for $k=3,4,5$. For each pair $({\ensuremath{m}},k)$ we considered $81$ edge densities $c=\frac{{\ensuremath{n}}}{{\ensuremath{m}}}$, spaced apart by $0.0001$, thus covering an interval of length $0.008$, which encloses the theoretical threshold value for the particular parameter pair $({\ensuremath{m}},k)$. The hyperedges of the hypergraphs were randomly chosen via pseudo random number generator MT19937 “Mersenne Twister” of the GNU Scientific Library [@GNU_Scientific]. We measured the average failure rate of the algorithm over $100$ random hypergraphs for each combination $({\ensuremath{m}},{\ensuremath{n}},k)$ within the parameter space. To get an estimation of the threshold, i.e., the rate $c$ where the algorithm switches from success to failure, we fit the sigmoid function $$\label{eq:fit_function} \sigma(c;a,b)=\frac{1}{1+e^{-(c-a)/b}}$$ to the measured failure rate (via gnuplot[^7]), using the method of least squares. We determined the parameters $a,b$ that lead to a (local) minimum of the sum of squares of the $81$ residuals, denoted by $\sum_{res}$. The parameter $a$ is the inflection point of and therefore the approximation of the threshold of the generalized selfless algorithm. Figures \[fig:gen\_selfless\_edge\_size\_3\], \[fig:gen\_selfless\_edge\_size\_4\] and \[fig:gen\_selfless\_edge\_size\_5\] show the results of the experiments. One observes that this simple algorithm is able to construct the placements for edge densities quite close to the calculated thresholds $c_{k,2}$. The slope of the sigmoid curve increases and $\sum_{res}$ decreases with growing ${\ensuremath{m}}$ and $k$, leading to a sharp transition from total success to total failure. Clearly the algorithm can fail on hypergraphs that admit a matching. Experimental comparisons with a perfect matching algorithm [@HK:1973] showed that this is very unlikely for random hypergraphs. An example is given in Figure \[fig:selfless\_vs\_perfect\_matching\], which shows the failure rate of perfect matching in comparison to the generalized selfless algorithm. Note that the plot shows an interval of size $0.004$, i.e., 41 data points instead of $81$. The differences in the failure rates of the algorithms become very small as ${\ensuremath{m}}$ grows. A conjecture, with evidence from a generalized selfless algorithm {#subsec:conjecture} ------------------------------------------------------------------ Now consider a situation in which buckets have a capacity of $\ell>1$ keys. There is as yet no rigorous analysis of the appropriate thresholds for cuckoo hashing for the cases $k > 2$ and $\ell>1$. However, our results of Section \[sect:facts:cores\] suggest a natural conjecture: \[conj:threshold\] For $k$-ary cuckoo hashing with bucket size $\ell$, it is *conjectured* that cuckoo hashing works with high probability if $n/m = c > c_{k,\ell+1}$, and does not work if $n/m = c < c_{k,\ell+1}$, i.e., that the threshold is at the point where the $(\ell+1)$-core of the cuckoo hypergraph starts having edge density larger than $\ell$. In order to provide evidence for this conjecture, we generalize our algorithm further so that it can deal with bucket size $\ell>1$. The pseudocode is given as Algorithm \[algo:GeneralizedSelfless2\]. In hypergraph language, we are now looking for an orientation of the hyperedges of $G$ so that every node has at most $\ell$ hyperedges directed toward it. Now a node is saturated if it has $\ell$ edges pointing to it. As long as there are nodes $v$ such that the number of hyperedges directed toward $v$ and the number of undirected hyperedges containing $v$ taken together does not exceed $\ell$, one such node is chosen and its undirected edges are directed toward it. Again, the effect of this rule is that the algorithm starts by carrying out the peeling process that finds the $(\ell+1)$-core. Otherwise, the algorithm assigned weights and priorities as before, and if all priorities exceed $\ell$, the algorithm stops and reports failure. If the smallest (highest) priority is at most $\ell$, a vertex of smallest priority is chosen and one of the incident undirected hyperedges of minimum weight is directed toward it. The process is carried out until all hyperedges have been directed or failure occurs. Experiments with Algorithm \[algo:GeneralizedSelfless2\] corroborate Conjecture \[conj:threshold\], in that they show that the failure rate of the algorithm changes from 0 to 1 very close to the possible threshold values suggested in Conjecture \[conj:threshold\]. =1 Again, numerical results are given in Appendix B. =1 (For an example see Figure \[fig:gen\_selfless\_edge\_size\_3\_bucket\_size=2\].) Conclusion ========== We have found tight thresholds for cuckoo hashing with 1 key per bucket, by showing that the thresholds are in fact the same for the previous studied $k$-XORSAT problem. 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Specifically, we show that if $({\ensuremath{{\rho}}}_x)_{x \in S}$ is an optimal sequence, then for all $x \in S$: $${\ensuremath{{\rho}}}_x( \lfloor {\ensuremath{\kappa}}_x \rfloor )=1-({\ensuremath{\kappa}}_x-\lfloor {\ensuremath{\kappa}}_x\rfloor), \text{ and }{\ensuremath{{\rho}}}_x( \lfloor {\ensuremath{\kappa}}_x \rfloor+1 )={\ensuremath{\kappa}}_x-\lfloor {\ensuremath{\kappa}}_x\rfloor.$$ We consider a random bipartite graph $G_S$ with left node set $S$, right node set $[{\ensuremath{m}}]$ and an edge between two nodes $x\in S$ and $a\in [{\ensuremath{m}}]$ if and only if $a\in A_x$. Let the sequence $({\ensuremath{\kappa}}_x)_{x\in S}$ be fixed. For each $x \in S$ we want to obtain a distribution ${\ensuremath{{\rho}}}_x$ for the degree ${\ensuremath{k}}_x$ (or, equivalently, the cardinality of $A_x$), such that we have $E({\ensuremath{k}}_x)={\ensuremath{\kappa}}_x$ and the following quantity is maximized: $$\label{eq:success_prob} \Pr(\text{``success''}):=\Pr( (A_x)_{x\in S} \text{ admits a left-perfect matching}\footnote{In the following ``matching'' and ``left-perfect matching'' are used synonymously.}\text{ in $G_S$}){\text{ }}.$$ We study the sequence $({\ensuremath{{\rho}}}_x)_{x\in S}$ that realizes the maximum. Let $z$ be an arbitrary but fixed element of $S$ with probability mass function ${\ensuremath{{\rho}}}_z$. To prove Proposition \[prop:constant\_expected\_degree\] it is sufficient to show that if there exist two numbers ${\ensuremath{l}}$ and ${\ensuremath{k}}$ with ${\ensuremath{l}}<{\ensuremath{\kappa}}_z < {\ensuremath{k}}$ and ${\ensuremath{k}}-{\ensuremath{l}}\geq2$ as well as ${\ensuremath{{\rho}}}_z({\ensuremath{l}})>0$ and ${\ensuremath{{\rho}}}_z({\ensuremath{k}})>0$ then cannot be maximal. We start by fixing ${\ensuremath{k}}_x$ and $A_x$ for each $x\in S-\{z\}$ and consider the corresponding bipartite graph $G_{S-\{z\}}$. Let $B\subseteq [{\ensuremath{m}}]$ be the set of right nodes in $G_{S-\{z\}}$ that are matched in every matching. Then there is a matching for the whole key set $S$ in $G_S$ if and only if $A_z \not \subseteq B$. Note that $0\leq |B|<{\ensuremath{m}}$, i.e., there must be at least one right node that is not matched. Let $p=\min\{ {\ensuremath{{\rho}}}_z({\ensuremath{l}}), {\ensuremath{{\rho}}}_z({\ensuremath{k}}) \}>0$ and $|B|=b$. We will show that changing ${\ensuremath{{\rho}}}_z$ to $$\begin{aligned} {\ensuremath{{\rho}}}_z'({\ensuremath{l}})&:={\ensuremath{{\rho}}}_z({\ensuremath{l}})-p &{\ensuremath{{\rho}}}_z'({\ensuremath{k}})&:={\ensuremath{{\rho}}}_z({\ensuremath{k}})-p \\ {\ensuremath{{\rho}}}_z'({\ensuremath{l}}+1)&:={\ensuremath{{\rho}}}_z({\ensuremath{l}}+1)+p &{\ensuremath{{\rho}}}_z'({\ensuremath{k}}-1)&:={\ensuremath{{\rho}}}_z({\ensuremath{k}}-1)+p,\end{aligned}$$ with ${\ensuremath{{\rho}}}_z'(j)={\ensuremath{{\rho}}}_z(j)$ for $j\notin\{{\ensuremath{l}},{\ensuremath{k}}\}$, increases , while leaving ${\ensuremath{\kappa}}_z$ unchanged. This is the case if and only if $$\label{eq:first_ineq} p\cdot \frac{\binom{b}{{\ensuremath{l}}}}{\binom{{\ensuremath{m}}}{{\ensuremath{l}}}} + p\cdot \frac{\binom{b}{{\ensuremath{k}}}}{\binom{{\ensuremath{m}}}{{\ensuremath{k}}}} \geq p\cdot \frac{\binom{b}{{\ensuremath{l}}+1}}{\binom{{\ensuremath{m}}}{{\ensuremath{l}}+1}} + p\cdot \frac{\binom{b}{{\ensuremath{k}}-1}}{\binom{{\ensuremath{m}}}{{\ensuremath{k}}-1}}, $$ and the strict inequality holds for at least one value $b$ that occurs with positive probability. The left sum of is the $2\cdot p$ fraction of the failure probability (by ${\ensuremath{{\rho}}}_z({\ensuremath{l}})$ and ${\ensuremath{{\rho}}}_z({\ensuremath{k}})$) before the change of ${\ensuremath{{\rho}}}_z$ under the condition that $B$ has cardinality $b$; the right sum is the corresponding fraction of the failure probability after the change. Depending on $b$ we have to distinguish several cases. $b={\ensuremath{m}}-1$. In this case both sides of are equal, i.e., the modification we do to ${\ensuremath{{\rho}}}_z$ will not change the success probability. ${\ensuremath{k}}\leq b < {\ensuremath{m}}-1$. Canceling $p$ and subtracting ${\binom{b}{{\ensuremath{k}}}}/{\binom{{\ensuremath{m}}}{{\ensuremath{k}}}}$ and ${\binom{b}{{\ensuremath{l}}+1}}/{\binom{{\ensuremath{m}}}{{\ensuremath{l}}+1}}$ from both sides of shows that the strict inequality holds if and only if $$\label{eq:intermediate} \begin{split} \frac{b\cdots(b-{\ensuremath{l}}+1)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{l}}+1)}-\frac{b\cdots(b-{\ensuremath{l}})}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{l}})}>\frac{b\cdots(b-{\ensuremath{k}}+2)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+2)}-\frac{b\cdots(b-{\ensuremath{k}}+1)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+1)} {\text{ }}. \end{split}$$ Factoring out $\frac{b\cdots(b-{\ensuremath{l}}+1)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{l}}+1)}$ on the left side and $\frac{b\cdots(b-{\ensuremath{k}}+2)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+2)}$ on the right side gives $$\frac{b\cdots (b-{\ensuremath{l}}+1)}{{\ensuremath{m}}\cdots ({\ensuremath{m}}-{\ensuremath{l}})}>\frac{b\cdots (b-{\ensuremath{k}}+2)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+1)} \Leftrightarrow \frac{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+1)}{{\ensuremath{m}}\cdots ({\ensuremath{m}}-{\ensuremath{l}})}>\frac{b\cdots (b-{\ensuremath{k}}+2)}{b\cdots (b-{\ensuremath{l}}+1)}{\text{ }}.$$ Since ${\ensuremath{l}}\leq {\ensuremath{k}}-2$, this is equivalent to $$({\ensuremath{m}}-{\ensuremath{l}}+1)\cdot({\ensuremath{m}}-{\ensuremath{l}})\cdots({\ensuremath{m}}-{\ensuremath{k}}+1)>(b-{\ensuremath{l}})\cdot(b-{\ensuremath{l}}-1)\cdots(b-{\ensuremath{k}}+2){\text{ }},$$ which is true for $m-1>b$. ${\ensuremath{l}}\leq b<{\ensuremath{k}}$. Calculations along the lines of case 2 show that the strict inequality of also holds in this case. Note that $\binom{b}{k}$, $\binom{b}{k-1}$ and $\binom{b}{l+1}$ can be zero. $0\leq b<{\ensuremath{l}}$. In this case both sides of are zero, i.e., the modifications we do to ${\ensuremath{{\rho}}}_z$ will not change the success probability. Since in cases 1 and 4 above there was no change in the success probability, to show that cannot be maximal when ${\ensuremath{k}}-{\ensuremath{l}}\geq2$ as we are considering, it remains to show that at least one of the Cases 2 and 3 occurs with positive probability. We construct a situation in which one of these cases applies, and which occurs with positive probability. Choose degrees ${\ensuremath{k}}_x$ for all elements $x \in S-\{z\}$ such that ${\ensuremath{k}}_x\leq {\ensuremath{\kappa}}_x$ and ${\ensuremath{{\rho}}}_x({\ensuremath{k}}_x)>0$. Consider a permutation of the elements $x \in S-\{z\}$ such that these degrees are ordered, i.e., ${\ensuremath{k}}_{x_1}\leq {\ensuremath{k}}_{x_2}\leq \ldots \leq {\ensuremath{k}}_{x_{{\ensuremath{n}}-1}}$. Choose the first element $x_i$ with $i \geq {\ensuremath{l}}$ and ${\ensuremath{k}}_{x_i}\leq i$. Such an element must exist, since we assume ${\ensuremath{k}}_x\leq {\ensuremath{\kappa}}_x\leq {\ensuremath{n}}-2$, in particular we have $l<{\ensuremath{n}}-2$. Arrange that $A_{x_j}\subseteq [i],1\leq j \leq i,$ such that there is a matching in $G_{\{x_1,\ldots,x_i\}}$. This implies $b\geq {\ensuremath{l}}$. Then arrange that $|A_{x_j}-\bigcup_{1\leq {j'}<j} A_{x_{j'}}|=1$, for all $i<j\leq n-2$, as well as $|A_{x_{n-1}}-\bigcup_{1\leq {j'}<n-1} A_{x_{j'}}|=2$, which implies $b<m-1$. This finishes the proof of Proposition \[prop:constant\_expected\_degree\]. Performance results for the generalized selfless algorithm ========================================================== We present some performance results for the generalized selfless algorithm We ran the generalized selfless algorithm for hypergraphs with $10^5$ and $10^6$ nodes and tabulated the failure rate around the theoretical threshold values $c_{k,2}$ for $k=3,4,5$. For each pair $({\ensuremath{m}},k)$ we considered $81$ edge densities $c=\frac{{\ensuremath{n}}}{{\ensuremath{m}}}$, spaced apart by $0.0001$, thus covering an interval of length $0.008$, which encloses the theoretical threshold value for the particular parameter pair $({\ensuremath{m}},k)$. The hyperedges of the hypergraphs were randomly chosen via pseudo random number generator MT19937 “Mersenne Twister” of the GNU Scientific Library [@GNU_Scientific]. We measured the average failure rate of the algorithm over $100$ random hypergraphs for each combination $({\ensuremath{m}},{\ensuremath{n}},k)$ within the parameter space. To get an estimation of the threshold, i.e., the rate $c$ where the algorithm switches from success to failure, we fit the sigmoid function $$\label{eq:fit_function} \sigma(c;a,b)=\frac{1}{1+e^{-(c-a)/b}}$$ to the measured failure rate (via gnuplot[^8]), using the method of least squares. We determined the parameters $a,b$ that lead to a (local) minimum of the sum of squares of the $81$ residuals, denoted by $\sum_{res}$. The parameter $a$ is the inflection point of and therefore the approximation of the threshold of the generalized selfless algorithm. Figures \[fig:gen\_selfless\_edge\_size\_3\], \[fig:gen\_selfless\_edge\_size\_4\] and \[fig:gen\_selfless\_edge\_size\_5\] show the results of the experiments. One observes that this simple algorithm is able to construct the placements for edge densities quite close to the calculated thresholds $c_{k,2}$. The slope of the sigmoid curve increases and $\sum_{res}$ decreases with growing ${\ensuremath{m}}$ and $k$, leading to a sharp transition from total success to total failure. Clearly the algorithm can fail on hypergraphs that admit a matching. Experimental comparisons with a perfect matching algorithm [@HK:1973] showed that this is very unlikely for random hypergraphs. An example is given in Figure \[fig:selfless\_vs\_perfect\_matching\], which shows the failure rate of perfect matching in comparison to the generalized selfless algorithm. Note that the plot shows an interval of size $0.004$, i.e., 41 data points instead of $81$. The differences in the failure rates of the algorithms become very small as ${\ensuremath{m}}$ grows. Similarly, we find our generalized algorithm for the case where the bucket size $\ell$ is greater than 1 has similar behavior. For an example see Figure \[fig:gen\_selfless\_edge\_size\_3\_bucket\_size=2\]. Proof of the threshold for $k$-XORSAT ===================================== In this section we give a full proof of the threshold for $k$-XORSAT (Corollary \[corthresh\]). The proof employs the notation and facts developed in Sections \[sect:facts:cores\] and \[sec:main\], especially Propositions \[prop:two\] and \[prop:three\], and the following fact (known as “Friedgut’s Theorem” for $k$-XORSAT [@CD03; @CD09]). \[fact:Friedgut\] For every $k\geq 3$ there exists a function $c_k(m)\leq 1$ such that, for every ${\ensuremath{\varepsilon}}>0$ and a random formula $F$ (system $Ax=b$ of equations) from $\Phi^k_{m,n}$ we have the following: $$\lim_{m\to\infty} \Pr[\text{$F$ is satisfiable}]= \begin{cases} 1, \text{ if } n=c_k(m)(1-{\ensuremath{\varepsilon}})m\\ 0, \text{ if } n=c_k(m)(1+{\ensuremath{\varepsilon}})m {\text{ }}.\\ \end{cases}$$ Recall from Section \[sec:main\] that $\Phi^k_{m,n}$ can be regarded as a probability space whose elements are pairs $(A,b)$ where $A$ is an $n\times m$ matrix with entries in $\{0,1\}$, each row containing $k$ 1’s, and $b\in\{0,1\}^n$. Alternatively, $A$ can be regarded as a node-edge incidence matrix $A_G$ of a $k$-uniform hypergraph $G\in {\mathcal{G}^k_{m,n}}$. Via the obvious correspondence we identify ${\mathcal{G}^k_{m,n}}$ with the set of bipartite graphs $G$ with $n$ left nodes (“check nodes”) and $m$ right nodes (“variable nodes”) and degree $k$ at each left node. Similarly, $\Psi^k_{\hat{m},\hat{n}}$ is the probability space whose elements are pairs $(\hat{A},\hat{b})$, $\hat{b}\in\{0,1\}^{\hat{n}}$, where $\hat{A}$ is either the incidence matrix of a $k$-uniform hypergraph $H$ with $\hat{m}$ nodes, $\hat{n}$ edges, and minimum degree 2 or the adjacency matrix $\hat{A}_H$ of a bipartite graph $H$ with $\hat{n}$ left nodes and $\hat{m}$ right nodes, with degree $k$ at each left node and minimum degree 2 at each right node. We use the same notation for both and let ${\mathcal{H}^k_{\hat{m},\hat{n}}}$ be the set of all these graphs.[^9] The following lemma is central. \[lemma:Z2\] For any $\delta>0$ there exists ${\ensuremath{\varepsilon}}= {\ensuremath{\varepsilon}}(\delta)>0$ such that the following happens. Let $H\in{\mathcal{H}^k_{\hat{m},\hat{n}}}$ be uniformly random with $\hat{n}<\hat{m}(1-\delta)$ and denote by $Z_{H}$ the number of solutions of the linear system $\hat{A}_Hx = 0$ (over $\mathrm{GF}[2]$). Then $$\begin{aligned} \Pr[Z_{H} = 2^{\hat{m}-\hat{n}}]\ge {\ensuremath{\varepsilon}}(\delta)>0 \, . $$ We note that a full proof of this lemma for the special case $k=3$, with $\Pr[Z_{H} = 2^{\hat{m}-\hat{n}}]=1-o(1)$, was given in [@DubMan:2002]. Consider the following two cases. $c^*_{k,2}<c<c_{k,2}$. Let a system $A_Gx=b$ be chosen at random from $\Phi^k_{m,n}$. Reducing $G$ to its 2-core $H$ leads to a system $\hat{A}_Hx=\hat{b}$ with $\hat{m}$ variables, $\hat{n}$ equations, and $\operatorname{rank}(\hat{A}_H)=\operatorname{rank}(A_G)-(m-\hat{m})$. The graph $H$ is random in ${\mathcal{H}^k_{\hat{m},\hat{n}}}$. By Propositions \[prop:two\] and \[prop:three\], with high probability $\hat{n}\leq (1-\delta)\hat{m}$ for some $\delta=\delta(c)$, and $\hat{m}=\Theta(m)$. By Lemma \[lemma:Z2\], for $m$ large enough, we get $\Pr[Z_{H} = 2^{\hat{m}-\hat{n}}]\geq {\ensuremath{\varepsilon}}(\delta)>0$. This implies $\Pr[A_Gx=b\text{ is satisfiable}] \geq \Pr[A_G \text{ has full row rank}]=\Pr[Z_G=2^{m-n}]\geq {\ensuremath{\varepsilon}}(\delta)$. $c>c_{k,2}$. Let $A_Gx=b$ and its reduced version $\hat{A}_Hx=\hat{b}$ be as in (i). By Propositions \[prop:two\] and \[prop:three\], with high probability $\hat{n}\geq (1+\delta)\hat{m}$ for some $\delta=\delta(c)$, and $\hat{m}=\Theta(m)$. We have $\operatorname{rank}(\hat{A})\leq \hat{m}$, and by the randomness of $\hat{b}$ we have $\Pr[\hat{A}_Hx=\hat{b}\text{ is satisfiable}]\leq2^{\hat{m}-\hat{n}}\leq2^{-\delta\hat{m}}$. Combining parts (i) and (ii) with Friedgut’s Theorem (Fact \[fact:Friedgut\]) shows that $\lim_{m\to\infty} c_k(m)=c_{k,2}$, which implies Corollary \[corthresh\]. We now move to the proof of Lemma \[lemma:Z2\], which focuses on the 2-core $H$ of the graph $G$, and we condition on its number of nodes. With a slight abuse of notation we will drop the “hat” from our notations. In other words, we now let $H$ be a uniformly random graph from ${\mathcal{H}^k_{m,n}}$ and let $\gamma = n / m$ (see Theorem \[thresh\]). It is convenient to introduce some additional notation. Given a formal series $p(z)$, $\operatorname{coeff}[p(z),z^r]$ denotes the coefficient of $z^r$ in $p(z)$. We further introduce the notations $$\begin{aligned} &q(z) = (e^{z}-1-z)\, ,\;\;\;\;\;\;\;\;\;\;\; Q(z) = \frac{z q'(z)}{q(z)}\, ,\\ &p_k(z) = \frac{1}{2}(1+z)^k+ \frac{1}{2}(1-z)^k\, ,\;\;\;\;\;\;\; P_k(z) = \frac{z p_k'(z)}{p_k(z)}\, . $$ It is easy to see that $z\mapsto Q(z)$ is a strictly increasing function with $\lim_{z\to 0}Q(z)=2$, and $\lim_{z\to\infty}Q(z) = \infty$. Further $z\mapsto P_k(z)$ is strictly increasing with $\lim_{z\to 0}P_k(z)=0$, and $\lim_{z\to\infty}P_k(z) = k$ for $k$ even, and $k-1$ otherwise. Further we define the domain sets $$\begin{aligned} {\mathcal{D}_{m,n}}&= &\big\{(w,l)\in{\mathbb{Z}}^2\,:\, 0\le w\le m\, , 2w\le l\le kn-2(m-w)\, , l\mbox{ even }\big\}\, , \\ {\mathcal{D}_{\gamma}}({\ensuremath{\varepsilon}})&= &\big\{(\omega,\lambda)\in{\mathbb{R}}^2\,:\, {\ensuremath{\varepsilon}}\le \omega\le 1-{\ensuremath{\varepsilon}}\, , \tfrac{2\omega}{k\gamma}+{\ensuremath{\varepsilon}}\le \lambda\le 1-\tfrac{2(1-\omega)}{k\gamma}-{\ensuremath{\varepsilon}}\big\}\, ,\\ {\mathcal{D}_{m,n}}({\ensuremath{\varepsilon}})&= &\big\{(w,l)\in{\mathcal{D}_{m,n}}\,:\, (\tfrac{w}{m},\tfrac{l}{kn})\in {\mathcal{D}_{\gamma}}({\ensuremath{\varepsilon}})\big\}\, ,\\ {\overline{\mathcal{D}}_{m,n}}({\ensuremath{\varepsilon}})&= & {\mathcal{D}_{m,n}}\setminus {\mathcal{D}_{m,n}}({\ensuremath{\varepsilon}})\, . $$ The assertion of Lemma \[lemma:Z2\] now follows from the following sequence of lemmas, to be proven in the subsections below. \[lemma:Comb\] Let $Z_{H}$ be the number of solutions of the linear system $A_Hx=0$. Then $$\begin{aligned} \operatorname{E}[Z_H] & = \frac{1}{N_0}\sum_{(w,l)\in{\mathcal{D}_{m,n}}} N(w,l)\, , $$ where we define $$\begin{aligned} N_0 & = (kn)!\,\operatorname{coeff}[(e^z-1-z)^m,z^{kn}]\, ,\\ N(w,l) & = \binom{m}{w}\, l!(kn-l)!\, \operatorname{coeff}[(e^z-1-z)^w,z^l]\,\operatorname{coeff}[(e^z-1-z)^{m-w},z^{kn-l}]\, \operatorname{coeff}[p_k(z)^n,z^l]\, . $$ \[lemma:Boundary\] For any $\delta>0$ there exists ${\ensuremath{\varepsilon}}>0$ such that, if $n\le m(1-\delta)$, then $$\begin{aligned} \frac{1}{N_0}\cdot\sum_{(w,l)\in{\overline{\mathcal{D}}_{m,n}}({\ensuremath{\varepsilon}})}\! \! N(w,l)\le 2^{m\delta}\, . $$ \[lemma:Exp\] For any $\delta>0, {\ensuremath{\varepsilon}}>0$ there exists $C= C(\delta,{\ensuremath{\varepsilon}})$ such that, if $m\delta\le n\le m(1-\delta)$ and $(w,l)\in{\mathcal{D}_{m,n}}({\ensuremath{\varepsilon}})$, then $$\begin{aligned} \frac{N(w,l)}{N_0}\le \frac{C}{m}\, \exp\Big(m\,\psi\big(\tfrac{w}{m},\tfrac{l}{kn}\big) \Big)\, , $$ where, letting $h(z) = -z\log z-(1-z)\log(1-z)$,[^10] we define $$\begin{aligned} \psi(\omega,\lambda) &= h(\omega)-k\gamma\, h(\lambda)- \log q(s)+ k\gamma\log s\label{eq:Phi}\\ &+\omega \log q(a)-k\gamma\lambda \log a+(1-\omega)\log q(b)-k\gamma(1-\lambda) \log b\nonumber\\ & +\gamma\log p_k(c)-k\gamma\lambda \log c\, .\nonumber $$ Finally, $a=a(\omega,\lambda)$, $b = b(\omega,\lambda)$, $c = c(\omega,\lambda)$, and $s$ are the unique non-negative solutions of $$\begin{aligned} Q(s) = k\gamma\, , \;\;\;\;\; Q(a) = \frac{k\gamma\lambda}{\omega}\, , \;\;\;\;\;Q(b) = \frac{k\gamma(1-\lambda)}{(1-\omega)}\, , \label{Qeq}\\ P_k(c) = k\lambda\, .\label{Peq} $$ \[lemma:Calculus\] For any $\gamma<1$, the function $\psi: {\mathcal{D}_{\gamma}}(0)\to{\mathbb{R}}$ achieves its unique global maximum at $(\omega,\lambda) =(1/2,1/2)$, with $\psi(1/2,1/2) = (1-\gamma)\log 2$. Further, there exists $\xi>0$ such that $-\operatorname{Hess}_{\psi}(1/2,1/2)\succeq\, \xi\, I_2$.[^11] Finally, let us recall a well known fact about lattice sums (see for instance [@BR]). \[lemma:Sum\] Let $D$ be an open domain in ${\mathbb{R}}^d$, and $F: D\to{\mathbb{R}}$ be continuously differentiable, achieving its unique maximum in $z_*\in D$, with $\operatorname{Hess}_F(z_*)\succeq \xi I_{d}$ for some $\xi>0$. Then there exists $C>0$ such that, for any $\delta\ge 0$ $$\begin{aligned} \sum_{x\in {\mathbb{Z}}^d\, :\; x\,\delta \in D} \exp\Big( \tfrac{1}{\delta}F(\delta x)\Big) \le \frac{C}{\delta^{d/2}}\, \exp\Big(\tfrac{1}{\delta}F(z_*)\Big)\, . $$ The proof is simply obtained by putting together Lemmas \[lemma:Comb\], \[lemma:Boundary\], \[lemma:Exp\], \[lemma:Calculus\] and using Lemma \[lemma:Sum\] (with $F(x_1,x_2)= \psi(x_1,2x_2/(k\gamma))$, $d=2$ and $\delta = 1/n$) to bound the sum. Proof of Lemma \[lemma:Comb\] ----------------------------- Clearly $N_0$ is the number of graphs in ${\mathcal{H}^k_{m,n}}$. Indeed it is the number of way of putting $nk$ distinct balls in $m$ bins in such a way that each bin contains at least $2$ balls. The claim follows by proving that, for each $(w,l)\in{\mathcal{D}_{m,n}}$, $N(w,l)$ is the number of couples $(H,x)$ where $H\in {\mathcal{H}^k_{m,n}}$ and $x\in\{0,1\}^m$ with $A_Hx = 0 \mod 2$, such that $x$ has $w$ ones and $H$ has $l$ edges incident on variable (right) nodes $i$ such that $x_i=1$. Indeed, $\binom{m}{w}$ gives the number of ways of choosing the ones. Paint by red the $l$ edges incident on these nodes, and by blue the other $(kn-l)$ edges. The coefficient factors give the number of ways of attributing red/blue edges to nodes on the two sides. The factorials give the number of ways of matching edges of the same color on the two sides. [$\square$]{} Proof of Lemma \[lemma:Exp\] ---------------------------- Let us start by proving a lower bound on $N_0$. For any $s>0$, we have $$\begin{aligned} N_0 = (kn)! \frac{q(s)^m}{s^{kn}}\, \Pr_{s}\Big[\sum_{i=1}^mX_i =kn\Big] \, , $$ where $X_1,\dots, X_m$ are i.i.d. Poisson random variable (with parameter $s$) conditioned to $X_i\ge 2$, i.e., for any $q\ge 2$, $$\begin{aligned} \Pr_s[X_i = q] = \frac{1}{e^s-1-s}\, \frac{s^q}{q!}\, . $$ By assumption $s$ is chosen such that $\operatorname{E}_s[X_i]=Q(s)=k\gamma\in (k\delta,k(1-\delta))$. By the local central limit theorem for lattice random variables of [@BR Corollary 22.3], we have $\Pr_{s}\big[\sum_{i=1}^mX_i =kn\big] \ge C'/\sqrt{m}$ for sone constant $C'(\delta)$, whence, using Stirling’s formula $$\begin{aligned} N_0 \ge C_1(\delta) \left(\frac{kn}{e}\right)^{kn} \frac{q(s)^m}{s^{kn}}\, .\label{eq:BoundN0} $$ Consider now $N(w,l)$. By the central limit theorem for the sum of Bernoulli random variables, for any $m\delta\le w\le m(1-\delta)$, we have $$\begin{aligned} \binom{m}{w}\le \frac{C_2(\delta)}{\sqrt{m}}\, e^{mh(w/m)}\, . $$ Treating the coefficient terms as above, and using Stirling’s formula for $l,(kn-l)=\Theta(m)$, we get $$\begin{aligned} N(w,l) \le \frac{C_3(\delta)}{m}\,e^{mh(w/m)}\, \left(\frac{l}{e}\right)^l \left(\frac{knl}{e}\right)^{(kn-l)}\frac{q(a)^w}{a^l}\, \frac{q(b)^{m-w}}{b^{kn-l}}\, \frac{p_k(c)^n}{c^l}\, .\label{eq:BoundNwl} $$ The claim is proved by taking the ratio of the bounds (\[eq:BoundNwl\]) and (\[eq:BoundN0\]). [$\square$]{} Proof of Lemma \[lemma:Calculus\], outline ------------------------------------------ We now present an outline of the proof of Lemma \[lemma:Calculus\]. Appendix \[app:fullproof5\] contains the additional details for a complete proof. For $(\omega,\lambda)=(1/2,1/2)$, Eqs. (\[Qeq\]), (\[Peq\]) admit the unique solution $a=b=s$ and $c=1$. A straightforward calculation yields $\psi(1/2,1/2) = (1-\gamma)\log 2$. Call $\Psi(\omega,\lambda;a,b,c)$ the right hand side of Eq. (\[eq:Phi\]). Notice that the derivatives of $\Psi$ with respect to $a,b,c$ vanish by Eqs. (\[Qeq\]), (\[Peq\]). Therefore it is easy to compute the partial derivatives $$\begin{aligned} \frac{\partial \psi}{\partial \omega} & = & \log \frac{1-\omega}{\omega} +\log\frac{q(a)}{q(b)}\, ,\\ \frac{\partial \psi}{\partial \lambda} & = & -k\gamma \log \frac{1-\lambda}{\lambda}-k\gamma\log\frac{a}{b}-k\gamma\log c\, . $$ Using the fact that $a=b=s$ and $c=1$ at $(\omega,\lambda)=(1/2,1/2)$, we get that the gradient of $\psi$ vanishes at $(1/2,1/2)$, and again, $\psi(1/2,1/2) = (1-\gamma)\log 2$. By a somewhat longer calculation, we obtain the following second derivatives $$\begin{aligned} \left.\frac{\partial^2 \psi}{\partial \omega^2} \right|_{1/2,1/2} & = & -4\,\Big(1+\frac{(k\gamma)^2}{s^2C}\Big)\, ,\\ \left. \frac{\partial^2 \psi}{\partial \lambda\partial\omega}\right|_{1/2,1/2} & = & 4\, \frac{(k\gamma)^2}{s^2C}\, , \\ \left.\frac{\partial^2 \psi}{\partial \omega^2} \right|_{1/2,1/2} & = & -4\, \frac{(k\gamma)^2}{s^2C}\, , $$ with $$\begin{aligned} C= \frac{q''(s)}{q(s)}-\frac{q'(s)^2}{q(s)^2}+\frac{k\gamma}{s^2}>0\, . $$ It is easy to deduce that $-\operatorname{Hess}_\psi(1/2,1/2)$ is positive definite. The function $\psi:{\mathcal{D}_{\gamma}}(0)\to{\mathbb{R}}$ is continuous in ${\mathcal{D}_{\gamma}}(0)$ and differentiable in its interior. Further, we have the following asymptotic behaviors (first two at fixed $\lambda$, second two at fixed $\omega$): $$\begin{aligned} \lim_{\omega\to 0}&\frac{\partial\psi}{\partial\omega}=+ \infty\, , \;\;\;\;\;\;\; \;\;\;\;\;\;\; \;\;\;\;\;\;\; \;\;\;\;\;\; \lim_{\omega\to 1}\frac{\partial\psi}{\partial\omega}=- \infty\, ,\\ \lim_{\lambda\to 2\omega/(k\gamma)} &\frac{\partial\psi}{\partial\lambda}=+ \infty\, , \;\;\;\;\;\;\;\;\;\; \lim_{\lambda\to 1-2(1-\omega)/(k\gamma)} \frac{\partial\psi}{\partial\lambda}=- \infty\, . $$ Therefore any global maximum of $\psi$ must be a stationary point in the interior of ${\mathcal{D}_{\gamma}}(0)$. We next will prove that $(1/2,1/2)$ is the only such point. Notice that $\Psi(\omega,\lambda;a,b,c)$ is convex with respect to $a,b,c$. As a consequence $$\begin{aligned} \psi(\omega,\lambda)=\min_{a,b,c}\Psi(\omega,\lambda;a,b,c)\, . \label{eq:Variational} $$ We will construct an upper bound on $\psi$ by choosing $a,b,c$ appropriately. The first remark is that $$\begin{aligned} \Psi(1-\omega,1-\lambda;b,a,1/c) = \Psi(\omega,\lambda;a,b,c) -\gamma\, \log \frac{p_k(c)}{c^{k}p_k(1/c)} \, . $$ Since, for $c\in[0,1]$ (which is guaranteed by Eq. (\[Peq\]) for $\lambda\in [0,1/2]$) we have $p_k(c)\ge c^{k}p_k(1/c)$, we can restrict without loss of generality to $\lambda\le 1/2$ (whence $c\in [0,1]$). Next notice that, maximizing $\Psi$ over $\omega$, we get $\Psi(\omega,\lambda;a,b,c)\le \Psi_1(\lambda;a,b,c)$, where $$\begin{aligned} \Psi_1(\lambda;a,b,c) = & \log\big(q(a)+q(b)\big)-k\gamma\, h(\lambda)- \log q(s)+ k\gamma\log s\\ &-k\gamma\lambda \log a-k\gamma(1-\lambda) \log b +\gamma\log p_k(c)-k\gamma\lambda \log c\, .\nonumber $$ Next fix $c = c(\lambda) = b\lambda/(a-a\lambda)$. Since this transformation is invertible, we can as well keep $c$ as a free parameter, and let $\lambda =ac/(ac+b)$. If we let $\Psi_2(a,b,c) = \Psi_1(ac/(ac+b);a,b,c)$, we get $$\begin{aligned} \Psi_2(a,b,c) = & \log\big(q(a)+q(b)\big)- \log q(s)+\gamma\log p_k(c)\\ &-k\gamma\log(ac+b)+k\gamma\lambda \log s\, .\nonumber\end{aligned}$$ Also, without loss of generality, we can rescale $a$ by a factor $s$, and set $b=s$, therefore defining $\Psi_3(a,c)=\Psi_2(sa,s,c)$. If we introduce the notation $$\begin{aligned} \Lambda_s(x)=\frac{q(sx)}{q(s)} = \frac{e^{sx}-1-sx}{e^{s}-1-s}\,, $$ we get the expression $$\begin{aligned} \Psi_3(a,c) = -k\gamma \log(1+ac) +\log\big(1+\Lambda_s(a)\big)+\gamma\log p_k(c)\, . $$ By the above derivation we have the following relation with $\psi(\omega,\lambda)$: $$\begin{aligned} &\psi(\omega,\lambda)\le \left.\Psi_4(c)\right|_{c=\lambda/a_*(c)(1-\lambda)} \, ,\\ &\Psi_4(c) = \Psi_3(a_*(c),c)\, ,\;\;\;\;\;\;\; a_*(c) = \arg\min_{a\ge 0}\Psi_3(a,c)\, . $$ A direct calculation shows that $a_*(1) = 1$ and $\Psi_3(1,1) = (1-\gamma)\log 2$. This point corresponds to $(\omega,\lambda)=(1/2,1/2)$ through the above derivation. We will show that $c=1$ is indeed the global maximum of $\Psi_4(c)$ for $c\in [0,1]$, which implies the assertion. Maximizing $\Psi_3(a,c)$ with respect to $c$ implies $a_*(c)$ to be the unique non-negative solution of the stationarity condition $$\begin{aligned} a = \frac{(1+c)^{k-1}-(1-c)^{k-1}}{(1+c)^{k-1}+(1-c)^{k-1}}\, . \label{Beq} $$ On the other hand, the stationarity condition with respect to $a$ yields $$\begin{aligned} c = \frac{\lambda_s(a)}{1+\Lambda_s(a)-a\lambda_s(a)}\, . \label{Ceq} $$ where we used the fact that $\Lambda_s'(1) = k\gamma$ and defined $\lambda_s(x) = \Lambda_s'(x)/\Lambda_s'(1)$. Equations (\[Beq\]) and (\[Ceq\]) admit the solutions $a=c=0$ and $a=c=1$, and is easy to check that these are both local maxima of $\Psi_4$. We will show that they admit only one more solution with $c\in(0,1)$, that necessarily is a local minimum of $\Psi_4$. Indeed, if we let $a=\tanh x$, $c= \tanh y$, Eq. (\[Beq\]) becomes $$\begin{aligned} x = (k-1) y\, . $$ Our claim is therefore implied by Lemma \[lemma:Elementary\] below. [$\square$]{} \[lemma:Elementary\] For $s> 0$, let $$\begin{aligned} \Lambda_s(t) = \frac{e^{st}-1-st}{e^{s}-1-s}\, , \;\;\;\;\;\;\; \lambda_s(t) = \frac{e^{st}-1}{e^{s}-1}\, . $$ Define $F_s:{\mathbb{R}}\to{\mathbb{R}}$ by $$\begin{aligned} F_s(x) = \operatorname{atanh}\big(f(\tanh x)\big)\, ,\;\;\;\;\;\;\;\;\; f_s(t)= \frac{\lambda_s(t)}{1+\Lambda_s(t)-t\lambda_s(t)}\, . $$ Then $F_s$ is convex on $[0,\infty)$. This can be seen simply by graphing $F_s(x)$, or by some calculus which we omit. Proof of Lemma \[lemma:Boundary\] --------------------------------- The proof is analogous to the one of Lemma \[lemma:Exp\]. We have just to be careful to the values of $w,l$ near the boundary of the domain ${\mathcal{D}_{m,n}}$. Luckily we only need a loose upper bound. Equation (\[eq:BoundN0\]) remains true in the present case (as it only hinges on $n=\Theta(m)$). On the other hand using $\binom{m}{w}\le \exp(mh(w/m))$, $\operatorname{coeff}[f(x)^k,x^l]\le f(a)^k/a^l$ and $m!\le \sqrt{2\pi}\, (m/e)^{m+1/2}$, we get $$\begin{aligned} N(w,l) \le 2\pi\, e^{-kn-1} e^{mh(w/m)}\, l^{l+1/2} (kn-l)^{(kn-l+1/2)}\frac{q(a)^w}{a^l}\, \frac{q(b)^{m-w}}{b^{kn-l}}\, \frac{p_k(c)^n}{c^l}\, . $$ for any $a,b,c>0$. Taking the ratio, and bounding polynomial factors $\sqrt{l(kn-l)}\le C m$ we get $$\begin{aligned} \frac{N(w,l)}{N_0}\le C\, m\, \exp\big(m\psi(w/m,\lambda/kn)\big)\, , $$ whence $$\begin{aligned} \frac{1}{N_0}\sum_{(w,l)\in{\overline{\mathcal{D}}_{m,n}}({\ensuremath{\varepsilon}})} N(w,l)\le Cm^3\, \exp\big(m\sup\{ \psi(\omega,\lambda):\, (\omega,\lambda)\in {\mathcal{D}_{\gamma}}(0)\setminus {\mathcal{D}_{\gamma}}({\ensuremath{\varepsilon}})\} \big) $$ with $\psi(\omega,\lambda)$ defined as in Eq. (\[eq:Phi\]). Notice that $\psi:{\mathcal{D}_{\gamma}}(\delta)\to {\mathbb{R}}$ is a continuous function. It is therefore sufficient to show that it is strictly smaller than $(1-\gamma)\log 2$ on the boundaries of its domain. This indeed follows from Lemma \[lemma:Calculus\]. [$\square$]{} [^1]: Fakultät für Informatik und Automatisierung, Technische Universität Ilmenau. Research supported by DFG grant DI 412/10-1. [{martin.dietzfelbinger,michael.rink}@tu-ilmenau.de]{} [^2]: Fakultät für Informatik, Technische Universität Chemnitz. [goerdt@informatik.tu-chemnitz.de]{} [^3]: Harvard University, School of Engineering and Applied Sciences. Part of this work was done while visiting Microsoft Research New England. [michaelm@eecs.harvard.edu]{} [^4]: Department of Electrical Engineering and Department of Statistics, Stanford University. Part of this work was done while visiting Microsoft Research New England. [montanar@stanford.edu]{} [^5]: Efficient Computation group, IT University of Copenhagen. [pagh@itu.dk]{} [^6]: We could in principle also consider the possibility of keys having only a single choice. However, this is generally not very interesting since even a small number of keys with a single choice would make an assignment impossible whp., by the birthday paradox. Hence, we restrict our attention to at least two choices. [^7]: gnuplot, an interactive plotting program, version 4.2, <http://www.gnuplot.info> [^8]: gnuplot, an interactive plotting program, version 4.2, <http://www.gnuplot.info> [^9]: For simplicity we assume that for each left node a sequence of $k$ right nodes is chosen at random, allowing and ignoring repetitions. The difference from $k$-uniform hypergraphs is negligible. [^10]: $\log$ means logarithm to the base $e$ [^11]: $\operatorname{Hess}_\psi$ denotes the Hessian matrix of $\psi$ and $I_2$ the $2\times 2$ unit matrix

--- abstract: 'A numerically efficient inverse method for parametric model uncertainty identification using maximum likelihood estimation is presented. The goal is to identify a probability model for a fixed number of model parameters based on a set of experiments. To perform maximum likelihood estimation, the output probability density function is required. Forward propagation of input uncertainty is established combining Polynomial Chaos and moment matching. High-order moments of the output distribution are estimated using the generalized Polynomial Chaos framework. Next, a maximum entropy parametric distribution is matched with the estimated moments. This method is numerically very attractive due to reduced forward sampling and deterministic nature of the propagation strategy. The methodology is applied on a wet clutch system for which certain model variables are considered as stochastic. The number of required model simulations to achieve the same accuracy as the brute force methodologies is decreased by one order of magnitude. The probability model identified with the high order estimates resulted into a true log-likelihood increase of about 4% since the accuracy of the estimated output probability density function could be improved up to 47%.' author: - | Wannes De Groote\*[^1]^1,2^, Tom Lefebvre\*^1,2^, Georges Tod^3^, Nele De Geeter^1,2^, Bruno Depraetere^3^,\ Suzanne Van Poppel^4^, Guillaume Crevecoeur^1,2^\ ^1^ EEMMeCS, Ghent University, Belgium\ ^2^ EEDT Decision & Control, Flanders Make, Belgium\ ^3^ DecisionS, Flanders Make, Belgium\ ^4^ CodesignS, Flanders Make, Belgium bibliography: - 'references.bib' title: 'Inverse Parametric Uncertainty Identification using Polynomial Chaos and high-order Moment Matching benchmarked on a Wet Friction Clutch' --- uncertainty identification ,polynomial chaos ,method of moments ,wet clutch ,shifting time Introduction ============ The increasing performance demands in design and control of mechatronic applications lead to an upward trend in the need for accurate and robust models [@wilamowski2018control]. These mathematical relations allow to make predictions on the behavior and performance of the system in a virtual computational environment. It enables efficient design processes, leading to faster successive optimization iterations, so that more reliable products can be made at lower production cost. In this research, we consider the modeling of a wet clutch system. Wet friction clutches are hydraulic-mechanical devices used to transmit torque from an input shaft to an output shaft by means of friction. These systems are used in various types of automatic transmissions to selectively engage gear elements. During engagement, multiple plates make contact which enables power transfer from motor to load. Contrary to dry clutches, the plates are bathed in oil to assure better heat conduction of the friction losses. Wet friction clutches thus result in higher torques which make them suitable and primordial in off-road vehicles and agricultural machines [@Widanage2011]. Models of clutch systems have been widely used for fault diagnosis purposes [@foulard2015] and numerical optimization of both design and control [@della2018]. Clutch models identified for the wet clutch considered in this paper have been previously implemented in condition monitoring [@agusmian2013] and control [@bruno2011]. Additionally, model based control techniques applied on the wet clutch setup, have proven shorter convergence time and can avoid excessive control inputs that lead to unsafe operations [@dekeyser2014]. ![Proposed identification method determines optimal ${\boldsymbol{\mathrm{\alpha}}}$, associated to $\mathcal{P}_X({\boldsymbol{\mathrm{\alpha}}})$ of the uncertain parameters within the model, via MLE. The input distribution $\mathcal{P}_X({\boldsymbol{\mathrm{\alpha}}})$ is propagated through the model towards a corresponding output distribution $\mathcal{P}_{Y_l}({\boldsymbol{\mathrm{\alpha}}})$ for each performed experiment $l \in \mathcal{N}$. The novelty lies in the sequential use of gPC and PDF construction based on moment matching of gPC based high-order moments.[]{data-label="fig:approach"}](approach.pdf){width=".9\columnwidth"} Although the value of having accurate and robust models for mechatronic applications, as illustrated with the wet friction clutch, is straightforward, the construction of accurate models remains a cumbersome process. Since these systems are plagued by their intrinsic complexity and nonlinear behavior [@Widanage2011], variations in the physical phenomena over time due to e.g. wear of plates or degradation and centrifugal effects of the oil in the wet friction clutches; model discrepancies are inevitable [@watson2005]. Due to these uncertaintes, deterministic models become less useful. Alternatively, one can consider a system model of stochastic nature that predicts an output distribution for given control input. Models with ingrained parametric uncertainty have proven increased robustness in the field of fault diagnosis [@touati2012] and control [@marconi2008; @abdeetedal2018]. In this research we adopt the available system model and focus on identifying a probabilistic model to several lumped model parameters. Mostly (mechatronic) models are derived from first principles. Lumped parameter values are then determined empirically by fitting the model to the experiments. Considering that this strategy generates a model that can explain the experiment set, it is no direct proof of the validity of the physical model. It may occur that the parameter value fitted on this experiment set compensates for an inherent model shortcoming. Rather than increasing the model complexity, we propose to couple a parameterized input probability distribution to some of the lumped model parameters [@hajiloo2012]. We assume that the uncertainty structure (i.e. the parameterized input probability distribution) is fixed, and only its parameters, such as mean and standard deviation (e.g. for a normal distribution), need to be identified based on what parameters best explain a number of experiments. Stochastic system identification can be performed by means of prevailing methods as (quasi) Monte Carlo (MC) sampling techniques for assessing the propagation of the uncertainty parameter to the output variables (response) of the system model [@stein1987large; @caflisch1998monte]. These techniques are however curtailed by the computational inconvenience that comes with the numerous forward simulations required to achieve an acceptable degree of accuracy. Furthermore, in an inverse uncertainty identification setting, the uncertain parameters need to be optimized with respect to the correspondence of model responses to measurements, leading to even higher computational costs. More recent work has revived the exploitation of generalized Polynomial Chaos (gPC) expansions that can lead to significant gains in terms of computational feasibility [@xiu2002wiener; @xiu2003modeling; @xiu2005high; @xiu2007efficient; @blatman2011adaptive]. In the gPC framework, the propagation of input uncertainty to output variables is realized by developing the stochastic subspace through a polynomial series expansion. We propose in this paper to choose the polynomial basis so that it is orthogonal with respect to the joint probability density function of the random input variables. Hence, an efficient mathematical context emerges that allows to express the statistical moments as a function of the polynomial coefficients. Once these moments are available, the conditional Probability Density Function (PDF) of the outcome of the experiment can be retrieved through the well-known method of moments [@bowman2004estimation; @munkhammar2017polynomial]. To our knowledge, application of the gPC framework to estimate probabilistic moments of a stochastic output model has remained limited to mean and variance estimates. In this work we devised an efficient algorithm to calculate high-order moments as well, so to increase the information content that can be passed to the moment matching algorithm. The major computational bottleneck is thereby isolated and can be executed as an offline step. Figure \[fig:approach\] illustrates the incorporation of the identified output distributions for each experiment within a Maximum Likelihood Estimation (MLE) framework [@bickel2015]. This methodology enables to find sufficient parametric uncertainty that a model needs to include so that different experiments can be explained [@Berx2014]. A Genetic Algorithm (GA) is used here to optimize the parameterized distributions of the model parameters, because of its exploratory characteristics [@chan2007]. First, we elaborate our inverse uncertainty identification method to define different methods and concepts that are included within our theoretical framework. Next we address the wet clutch system and discuss in detail our practical set-up. Lastly, the developed methods are applied on the clutch application to identify the input probability model characterized by high-order moments.\ Inverse uncertainty identification method {#sec:IUIM} ========================================= Introduction ------------ We consider physics based nonlinear forward models, $\mathcal{Y}:\mathcal{N}\times\mathbb{R}^n \rightarrow\mathbb{R}$, used to generate predictions, $Y_l=\mathcal{Y}(l;{\boldsymbol{\mathrm{x}}})$, for the univariate outcome of experiments, $y_l$, that are characterized by some index $l\in\mathcal{N}$. The index $l$ is assumed to contain sufficient information to render the experiments deterministic from the perspective of the model, i.e. the information contains the operational settings that can be adjusted by an experimenter to the mechatronic system. The variable ${\boldsymbol{\mathrm{x}}}\in\mathbb{R}^n$ contains any (lumped) circumstantial parameters that, once identified, are assumed to remain constant for all experiments. We consider a number ($N$) of experiments within the index set $\mathcal{N}=\{1,\cdots,N\}$. When we assume that the measurements are deterministic, the mismatch stems either from a lack of circumstantial knowledge or suggests a genuine shortcoming of the model. In the former case, a latent variable may be present while in the latter case, the model does not, or not correctly, take into account all underlying physical phenomena.\ In recent literature issues are addressed by associating a random variable, ${\boldsymbol{\mathrm{X}}}\in\mathcal{X}\sim\mathcal{P}_X$, to the model parameter set ${\boldsymbol{\mathrm{x}}}$ [@hajiloo2012]. This approach renders the forward model stochastic and rather than exact predictions the model generates a distribution of possible outcomes. Although this strategy may deny an underlying physical reality, it grants the model practical use as it hands the user a measure for how far off a prediction may be. With respect to this concept, our contribution is twofold.\ First, we propose an efficient numerical method to obtain an approximation of the nonlinear transformed output distribution of the random variable $Y_l = \mathcal{Y}(l;{\boldsymbol{\mathrm{X}}})$, by moment matching with a parametric distribution. The more moments are taken into account, the better the approximation will be. To facilitate cheap yet reliable moment estimates we engage the gPC framework and extend it so that high-order moments can be extracted rigorously. Such benefits both the MLE, as nonlinear effects can be taken into account, as well as the post identification usage of the stochastic model.\ Secondly, we propose a fast computational method to associate a probability model to the variable, ${\boldsymbol{\mathrm{X}}}$, based on a number of experimental observations, $\{y_l\}_{l\in\mathcal{N}}$. To that end we put forward a parametric model, $\mathcal{P}_X({\boldsymbol{\mathrm{\alpha}}})$, for the probability of the random variable, ${\boldsymbol{\mathrm{X}}}$, parameterized by the variable ${\boldsymbol{\mathrm{\alpha}}}$, and we identify an optimal value for ${\boldsymbol{\mathrm{\alpha}}}$ by means of Maximum Likelihood Estimation (MLE). Hereinafter, for notational convenience we make no distinction between model $\mathcal{P}_X({\boldsymbol{\mathrm{\alpha}}})$ and parameter ${\boldsymbol{\mathrm{\alpha}}}$, and refer to it as the (input) probability model. Maximum Log-Likelihood Estimate ------------------------------- The proposed identification procedure is formulated as an inverse problem (\[eq:MLE\]). That is, we want to find the probability model, ${\boldsymbol{\mathrm{\alpha}}}^*$, that maximizes the natural $\log$ of the likelihood, $ L({\boldsymbol{\mathrm{\alpha}}})$. The likelihood of a given probability model, ${\boldsymbol{\mathrm{\alpha}}}$, is calculated as the product (or summation of the natural logarithm) of the probability density functions, $f_{Y_l}(y_l|{\boldsymbol{\mathrm{\alpha}}})$ $l\in \mathcal{N}$. Probability $f_{Y_l}(y_l|{\boldsymbol{\mathrm{\alpha}}})$ is the conditional relative likelihood that the value of the stochastic model output, $Y_l$, of experiments $l$, as obtained through simulation given probability model ${\boldsymbol{\mathrm{\alpha}}}$, would equal the experimentally observed value, $y_l$. The higher the value of $L$, the more likely the forward model. $$\begin{aligned} {\boldsymbol{\mathrm{\alpha}}}^* &= \max_{{\boldsymbol{\mathrm{\alpha}}}} \log L\left({\boldsymbol{\mathrm{\alpha}}}\right)=\max_{{\boldsymbol{\mathrm{\alpha}}}} \sum_{l\in\mathcal{N}} \log L_l \\ &= \max_{{\boldsymbol{\mathrm{\alpha}}}} \sum_{l\in\mathcal{N}} \log f_{Y_l}\left(y_l|{\boldsymbol{\mathrm{\alpha}}}\right) \end{aligned} \label{eq:MLE}$$ With regard to the construction of the MLE cost function, $\log L$, it is clear that the computational bottleneck is predominated by the numerical evaluation of the conditional probabilities, $f_{Y_l}(y_l|{\boldsymbol{\mathrm{\alpha}}}),l\in\mathcal{N}$. Hence, we will require an efficient numerical procedure to quantify the probabilities, $f_{Y_l}$. A novel procedure of such kind is described here next. Moment Estimation {#sec:UP-PCE} ----------------- To evaluate the conditional output probability density functions, $f_{Y_l}\left(y_l|{\boldsymbol{\mathrm{\alpha}}}\right)$ we will engage the generalized Polynomial Chaos (gPC) expansion framework. The gPC framework accounts for a number of advantageous mathematical conditions to propagate uncertainty. Such are accomplished by approximating the forward nonlinear model with a polynomial expansion. By subsequently choosing the polynomial basis so that it satisfies orthogonality conditions with respect to the probability density function of the random input variables, an efficient mathematical context emerges that allows to express the statistical moments in function of the coefficients associated to the expansion. Conventional application is limited to the mean and variance. In this work we propose a novel method to incorporate high-order moments as well. ### Generalized polynomial chaos According to the polynomial approximation theorem, any smooth function, $y:\mathbb{R}^n\rightarrow\mathbb{R}$, is equivalent to an infinite polynomial series [@ghanem1991stochastic] $$y({\boldsymbol{\mathrm{x}}}) = \sum\nolimits_{i=1}^{\infty} c_i \psi_i({\boldsymbol{\mathrm{x}}})$$ From a computational perspective our interest is however reserved to the $d$-th order approximation. That is, let $\mathcal{P}^d_n$ be the $n$-variate polynomial space of at most degree $d$ and let ${\boldsymbol{\mathrm{\psi}}} = \{\psi_i\}_{i=1}^p$ serve as basis for $\mathcal{P}^d_n$ with $p = \frac{(n+d)!}{n!d!}$. The $d$-th order approximation is then given by (\[eq:poly-D\]) for given polynomial coefficients $\{c_i\}_{i=1}^p$. An $n$-variate basis ${\boldsymbol{\mathrm{\psi}}}$ can be constructed from $n$ univariate bases ${\boldsymbol{\mathrm{\phi}}}^{(k)} = \{\phi^{(k)}_j\}_{j=0}^d, k\in\{1,\dots,n\}$. Consider the basis vector elements, $\psi_{|\uline{i}|\leq d} = \prod_{k=1}^{n} \phi^{(k)}_{\uline{i}(k)}$, with multi-index $\uline{i} = (i_1,\dots,i_n)$ and where $|\uline{i}|$ is defined as $\sum_{k=1}^{n}i_k$. For notational convenience we exploit the bijection between index $\uline{i}$ and $i$ taking values in $\mathcal{I}=\{1,\dots,p\}$. $$\label{eq:poly-D} y^{(d)}({\boldsymbol{\mathrm{x}}}) = \sum\nolimits_{i\in\mathcal{I}} c_i \psi_i({\boldsymbol{\mathrm{x}}})$$ Within the context of uncertainty propagation this representation allows to establish advantageous computational conditions by a distinct choice of basis, ${\boldsymbol{\mathrm{\psi}}}$. Assume that ${\boldsymbol{\mathrm{x}}}$ is composed of $n$ independently distributed random variables, $X_k$, with known supports and PDF, $f_{X_k}:\mathcal{X}_k\subseteq\mathbb{R}\rightarrow\mathbb{R}_{\geq0}$. The joint support, $\mathcal{X}$, and PDF, $f_{{\boldsymbol{\mathrm{X}}}}$, are given by $\bigotimes_{k}\mathcal{X}_k$ and $\prod_{k}f_{X_k}$, respectively. Now recall that our goal is to propagate the input uncertainty on ${\boldsymbol{\mathrm{x}}}$ to output $y$. By choosing the univariate bases, ${\boldsymbol{\mathrm{\phi}}}^{(k)}$, so that they satisfy an orthogonality condition w.r.t. the PDFs, $f_{X_k}$, associated to the respective variables, $X_k$, the statistical moments can be calculated in function of the polynomial coefficients. Orthogonality of a basis is established in function of an inner product definition. We take interest in the inner product defined in (\[eq:improduct\]). A polynomial basis is orthogonal w.r.t. $f_{X}$ if it satisfies the orthogonality condition $\langle \psi_i,\psi_j\rangle = \delta_{ij}\langle \psi_i^2\rangle$. $$\label{eq:improduct} \left\langle \psi_i,\psi_j\right\rangle \equiv {\mathrm{E}\left\{\psi_i\psi_j\right\}} = \int_{\mathcal{X}} \psi_i({\boldsymbol{\mathrm{x}}}) \psi_j({\boldsymbol{\mathrm{x}}}) f_{{\boldsymbol{\mathrm{X}}}}({\boldsymbol{\mathrm{x}}}) \text{d}{\boldsymbol{\mathrm{x}}}$$ Note that if we construct the multivariate basis as described above and so that the generating univariate bases, ${\boldsymbol{\mathrm{\phi}}}^{(k)}$, satisfy the orthogonality condition w.r.t. to the PDFs, $f_{X_k}$, also the multivariate basis, ${\boldsymbol{\mathrm{\psi}}}$, will satisfy the orthogonality condition w.r.t. the joint PDF, $f_{{\boldsymbol{\mathrm{X}}}}$, considering that $$\begin{aligned} \left\langle \psi_i,\psi_j\right\rangle &= \prod_{k=1}^{n}\int_{\mathcal{X}_k} \phi^{(k)}_{i_k}(x_k) \phi^{(k)}_{j_k}(x_k) f_{X_k}(x_k) \text{d}x_k \\ &= \prod_{k=1}^{n}\left\langle \phi_{i_k}^{(k)},\phi_{j_k}^{(k)}\right\rangle = \delta_{ij} \end{aligned}$$ ### Stochastic relation Now recall that we desire to characterize the stochastic properties of the output by quantifying its stochastic moments, $\mu_m$ $$\label{eq:momentdef} \mu_m = {\mathrm{E}\left\{Y^m\right\}} = {\mathrm{E}\left\{y({\boldsymbol{\mathrm{X}}})^m\right\}} = \int_\mathcal{X} y({\boldsymbol{\mathrm{x}}})^m f_{{\boldsymbol{\mathrm{X}}}}({\boldsymbol{\mathrm{x}}}) \text{d}{\boldsymbol{\mathrm{x}}}$$ When we substitute the $d$-th order polynomial approximation of the output model in (\[eq:momentdef\]), we retrieve an approximate expression for the $m$-th moment in function of the polynomial expansion coefficients. $$\label{eq:moments} \begin{aligned} \mu^{(d)}_m &= {\mathrm{E}\left\{y^{(d)}({\boldsymbol{\mathrm{X}}})^m\right\}} = \int_{\mathcal{X}} \left(\sum_{i\in\mathcal{I}} c_i \psi_i({\boldsymbol{\mathrm{x}}})\right)^m f_{{\boldsymbol{\mathrm{X}}}}({\boldsymbol{\mathrm{x}}}) \text{d}{\boldsymbol{\mathrm{x}}} \\ &= \sum_{i_1\in \mathcal{I}} \cdots \sum_{i_m\in \mathcal{I}} c_{i_1} \cdots c_{i_m} \left\langle\psi_{i_1}\cdots \psi_{i_m}\right\rangle \end{aligned}$$ From this expression one may easily verify that the first two moments can be estimated as shown below. Efficient estimation of the high-order moments is discussed in \[sec:algo\]. $$\begin{aligned} \mu_1^{(d)} &= c_1 \langle\psi_1^2\rangle\\ \mu_2^{(d)} &= {\mathop{\mathsmaller{\sum}}\nolimits}_{i\in\mathcal{I}} c^2_i \langle\psi_i^2\rangle \end{aligned}$$ We emphasize that this result only holds when the series is expanded over a basis that satisfies the orthogonality condition w.r.t. $f_{{\boldsymbol{\mathrm{X}}}}$. Several standard probability distributions are associated to known polynomial families by the Wiener-Askey scheme [@xiu2002wiener]. An overview is presented in Table \[tab:wiener\]. If the input stochasticity does not correspond with a standard distribution, a variable transformation can be used. ### Variable transformation A set of models with parameterized uncertainty parameters ${\boldsymbol{\mathrm{X}}}$ can now be characterized. To that end we introduce the parametric transformation, ${\boldsymbol{\mathrm{x}}}(\cdot|{\boldsymbol{\mathrm{\alpha}}})$, that maps standard random variable, ${\boldsymbol{\mathrm{\Theta}}}\in\vartheta$, with entries that are distributed according to one of the distributions in Table \[tab:wiener\], to the random variable, ${\boldsymbol{\mathrm{X}}}$. The multivariate expansion basis, ${\boldsymbol{\mathrm{\psi}}}$, can then be generated from the corresponding standard polynomials. We emphasize that the forward model becomes a function of the variable, ${\boldsymbol{\mathrm{\Theta}}}$. Consequently, the conditional PDF of the outcome is now fully determined by parameter ${\boldsymbol{\mathrm{\alpha}}}$, and the distribution of ${\boldsymbol{\mathrm{\Theta}}}$. $$\label{eq:mapping} {\boldsymbol{\mathrm{X}}} = {\boldsymbol{\mathrm{x}}}({\boldsymbol{\mathrm{\Theta}}}|{\boldsymbol{\mathrm{\alpha}}}) \rightarrow Y = y\left({\boldsymbol{\mathrm{x}}}\left({\boldsymbol{\mathrm{\Theta}}}|{\boldsymbol{\mathrm{\alpha}}}\right)\right) \equiv \eta\left({\boldsymbol{\mathrm{\Theta}}}|{\boldsymbol{\mathrm{\alpha}}}\right)$$ distribution, $f_{X}$ polynomials, $\phi_i$ support, $\mathcal{X}$ ----------------------- ----------------------- ------------------------------- Gaussian Hermite $\left[-\infty,\infty\right]$ Gamma Laguerre $\left[0,\infty\right]$ Beta Jacobi $\left[-1,1\right]$ Uniform Legendre $\left[-1,1\right]$ : \[tab:wiener\]Wiener-Askey polynomial chaos. ### Coefficient determination The polynomial coefficients from (\[eq:poly-D\]) can be quantified numerically [@xiu2007efficient] by projection of the forward model on the polynomial space exploiting the properties of the inner product. All terms but one will vanish resulting in $$\label{eq:galerkin} c_{i}(l|{\boldsymbol{\mathrm{\alpha}}}) = \left\langle \eta(l,{\boldsymbol{\mathrm{\Theta}}}|{\boldsymbol{\mathrm{\alpha}}}),\psi_i\right\rangle = \int_{\vartheta} \eta(l,{\boldsymbol{\mathrm{\theta}}}|{\boldsymbol{\mathrm{\alpha}}})\psi_i({\boldsymbol{\mathrm{\theta}}})f_{\Theta}({\boldsymbol{\mathrm{\theta}}})\text{d}{\boldsymbol{\mathrm{\theta}}}$$ for all $i\in \mathcal{I}$. Since the scope of this work is on general nonlinear forward models, we can not evaluate the associated integral explicitly. We therefore approximate the integrals using Gaussian-quadrature (\[eq:quadrature\]). For details we refer to [@xiu2005high]. An univariate Gaussian-quadrature of order $q$ is defined as a signature of the same size. A signature is defined as a set $\{(w_j,{\boldsymbol{\mathrm{\theta}}}_j)\}_{j\in\mathcal{Q}},\mathcal{Q}=\{1,\dots,q\}$, where ${\boldsymbol{\mathrm{\theta}}}_j$ and $w_j$ are associated to the position and the weight attributed to each element, respectively. The signature will depend on the weighting function, $f_{\Theta}$, and is as such related to the polynomial basis. A quadrature of order $q$ is exact for polynomials up to degree $ 2q-1$. The order hence affects the number of coefficients that can be retrieved correctly and therefore the accuracy of the polynomial approximation. In the multivariate case, a full tensor product of univariate quadrature rules can be considered. We note that the number of collocation points will thus scale exponentially with the number of dimensions. $$\label{eq:quadrature} c_{i}(l|{\boldsymbol{\mathrm{\alpha}}}) \approx \sum_{j\in\mathcal{Q}} \eta\left(l,{\boldsymbol{\mathrm{\theta}}}_j|{\boldsymbol{\mathrm{\alpha}}}\right)\psi_i\left({\boldsymbol{\mathrm{\theta}}}_j\right)w_j$$ Note that, similar to MC techniques, the gPC approach requires a number of evaluations of the forward model, $\eta(l,{\boldsymbol{\mathrm{\theta}}}_j|{\boldsymbol{\mathrm{\alpha}}})$. The difference is that the estimation of the statistical moments is realized through a mathematical detour. The focus of approximation in the gPC framework is on modeling the polynomial response function whilst that of the MC approach is on the direct estimation of the output distribution. The accuracy of gPC depends on the capacity of the basis to capture the nonlinearity of the forward model rather than on the capacity of the sampling method to properly represent the input uncertainty by the spatial distribution of the sample points. It is the prevailing consensus that an equivalent level of accuracy can be achieved with only a fraction of the input points of any MC approach. PDF fitting with the method of moments {#sec:PDF-MM} -------------------------------------- Once the approximate stochastic moments, $\mu_{l,m}^{(d)}({\boldsymbol{\mathrm{\alpha}}})$, of the $l$-the experiment for given input stochastic model, ${\boldsymbol{\mathrm{\alpha}}}$, are obtained a model for the conditional PDF of the random outcome, $Y_l$, can be extracted by matching these moments to a known parametric PDF, the so called methods of moments. The complexity of the parametric PDF should match the stochasticity of the random input, the expected symmetry of the stochasticity of the outcome and the nonlinearity of the forward model. Two possibilities are discussed for extracting the conditional PDF. - [*Gaussian distribution*:]{} The most straightforward and prevailing approach is to fit a normal distribution to the output PDF based on estimates of the mean and variance. The conditional PDF is then approximated by $$f_{Y}(y|{\boldsymbol{\mathrm{\alpha}}}) = \varphi\left(\frac{y - \mu^{(d)}({\boldsymbol{\mathrm{\alpha}}})}{\sigma^{(d)}({\boldsymbol{\mathrm{\alpha}}})}\right)$$ - [*Maximum entropy distribution*:]{} To cope with an increased number of moments, one can fit a maximum entropy distribution. Here, we seek a distribution $f_{Y}$ that maximizes the entropy ${\mathcal{W}} =- \int_{\mathcal{Y}} f_Y(y) \log f_Y (y) \text{d}y$ subject to $M+1$ moment matching constraints. The solution to this optimization problem is given by the following parameterized PDF (see Appendix \[appendix:med\] for derivation) $$f_{Y}(y|{\boldsymbol{\mathrm{\lambda}}}) = \exp \left(-\mathsmaller{\sum}_{m=0}^M\lambda_{m} y^m\right)$$ Substituting the latter into the original constrained optimization problem, yields the following dual unconstrained optimization problem that can be solved to find values for the parameters $\lambda_m$ $$\max_{{\boldsymbol{\mathrm{\lambda}}}} \log \int_{\mathcal{X}} \exp \left(-\mathsmaller{\sum}_{m=1}^M \lambda_m x^m\right) \text{d}x + \mathsmaller{\sum}_{k=1}^M \lambda_m \mu^{(d)}_{m}(\alpha)$$ A solution exists for any $M$. However, in order to capture fourth order asymmetric effects (i.e. bimodal densities), the number of moments should $\geq 4$ so that the polynomial in the exponent can have two local maxima. Worth mentioning is the common conception in the literature [@lee2009comparative; @rajabi2015polynomial] to obtain the moments, or even directly the PDF, by applying MC techniques on the polynomial expansion (\[eq:poly-D\]) which is relatively cheap to evaluate. In this approach the expansion is used as a response function and the benefit of gPC is only exploited through the supposed superior convergence rate of the expansion for polynomials corresponding the input distribution (\[eq:poly-D\]). When compared to its peers, gPC is usually applied in this sense [@xiu2005high; @lee2009comparative; @najm2009uncertainty]. Estimation of high-order moments {#sec:algo} -------------------------------- The expression presented in (\[eq:moments\]) becomes inefficient for $m>2$. Therefore we expand the power of the sum by applying the multinomial theorem to obtain an expression which is numerically far more efficient. Remark that there is only one summation and each term in the series is rendered unique. Moreover, we decompose the high-order multivariate inner products into a product of univariate inner products. As a result we can compute the high-order inner products numerically using a univariate instead of a multivariate quadrature. The multi-index set $\mathcal{I}_{m,p}$ is defined as $\{\uline{i}\in\mathcal{I}_{m,p}:|\uline{i}|=m\}$. $$\label{eq:moments2.0} \begin{aligned} \mu_m^{(d)} = \sum_{\uline{i} \in\mathcal{I}_{mp} } \binom{m}{i_1 \cdots i_p} \cdot \prod_{j=1}^n \left\langle\prod_{k=1}^p {\phi_{\uline{k}(j)}^{(j)}}^{i_k}\right\rangle \cdot \prod_{k=1}^p c_k^{i_k} \end{aligned}$$ Details on deriving equation (\[eq:moments2.0\]) can be found in Appendix \[appendix:mathderiv\]. The algorithmic execution time still demands an upper-limit for $m$. That is because $\dim(\mathcal{I}_{m,p})=\binom{m+p-1}{m}$ where $p$ itself is a combinatorial number. Note that the orthogonality property extends to the high-order inner products, as they occur in (\[eq:moments\]) or (\[eq:moments2.0\]) for $m>2$, however in a less predictable fashion. Hence in practice we only require the multinomial powers $\uline{i}$ whose corresponding inner product is nonzero. Such are stored in the set $\mathcal{I}^*_{m,p}$. The corresponding multinomial coefficient and inner product are compressed into coefficients $a_{\uline{i}}^{n,d,m}$ and stored for later use. One may verify that a significant fraction of $\mathcal{I}_{m,p}$ result into zero valued inner products, see Fig. \[fig:sets\]. $$\label{eq:moments3.0} \begin{aligned} \mu_m^{(d)} = \sum_{\uline{i} \in \mathcal{I}^*_{m,p}} a_{\uline{i}}^{n,d,m} \prod_{k=1}^p c_k^{i_k} \end{aligned}$$ To compute the set $\mathcal{I}_{m,p}^*$, we devised an algorithm that calculates the next multi-index, satisfying $|\uline{i}|=m$, from the previous multi-index. Because our algorithm generates the admissible multi-indices one by one, it admits further increase of the computational efficiency. The process of generating the multi-indices and the corresponding inner product can be executed in parallel avoiding storage of the entire set $\mathcal{I}_{m,p}$. The general idea is presented in Fig. \[fig:sets\]. The multinomial power problem can be addressed as reorganizing a stack of $m$ items over $p$ possible locations. We use two logical operators, push ($\mathcal{P}$) and fork ($\mathcal{F}$). A row is ‘pushed’ if the last location is nonzero. Otherwise the trailing nonempty location is ‘forked’ which involves pushing $1$ item from this location to the next and placing the remaining items back to the last location. The algorithm is initiated with all items in the last position. The calculation time of the parallel offline step is shown in Fig. \[fig:zeroprod\] and could be limited, e.g. $(n,d,m)=(4,3,5)$, to $18\si{\second}$ nonetheless that $575,757$ inner products are validated. The generation of the multi-indices itself took only about $1.2 \si{\second}$. Calculations were performed on a 2.1GHz, 4 GB RAM laptop. Above procedure allowed us to compute up to the 5^th^ moment quite efficiently. As discussed in section \[sec:PDF-MM\], this allows us to fit bimodal output distributions rigorously. [.20]{} [.2368]{} [.375]{} ![*Left*: Illustration of the sets $\mathcal{I}_{5,10}$ (*left*) and $\mathcal{I}^*_{5,10}$ (*right*) for $(n,d)=(2,3)\rightarrow p = 10$. *Right*: Illustration of the calculation mechanism of $\mathcal{I}_{m,p}$ for $(m,p)=(5,4)$.[]{data-label="fig:sets"}](algorithm "fig:"){width="\columnwidth"} ![Calculation time of the offline step w.r.t. $(n,d,m)$.[]{data-label="fig:zeroprod"}](algorithmspeed){width=".5\columnwidth"} Forward uncertainty propagation ------------------------------- In order to illustrate the forward propagation of uncertainty concatenating the techniques described in the sections \[sec:UP-PCE\] to \[sec:algo\], we apply the methodology on an illustrative univariate nonlinear forward model, $y(x)$. $$\begin{aligned} y(x) &= \tan\left(\tfrac{1}{4}x\right) + \exp\left(\tfrac{1}{3}x-1\right) + \tanh(x)\end{aligned}$$ For convenience we further assume that the random variable, $X$, is distributed according to the standard normal distribution, $\mathcal{N}(0,1)$. The forward propagation of this random variable passed through the function $y(x)$, results into a *camelback* output distribution as is depicted in Fig. \[fig:PDFtransform\]. The lower right plot shows the density of the stochastic input, $X$. The density of the output random variable, $Y$, is plotted in the upper left graph. Additionally, an MC sample set of $N=50$ is visualized by the black dots with corresponding $10$ bin histogram in gray. Figure \[fig:coeff\] depicts the polynomial coefficients for varying quadrature order, $q$, and using Hermite chaos. One may observe that the higher order coefficients are poorly approximated with low order quadratures. Recall that a quadrature of order $q$ is exact for polynomials up to degree $ 2q-1$. Now, assuming that model $y(x)$ contains polynomials up to degree $d$ and that we desire to approximate the $d$-th coefficient, then the integrand in (\[eq:galerkin\]) will contain a polynomial of degree $2d$. In order to be exact, the quadrature should therefore have order $d+1$ at least. Remark that the coefficient estimate becomes increasingly precise for $q$ surpassing $d+1$, as can be seen for e.g. $i=3$. That is because, $y(x)$ is truly a polynomial of degree $\gg d$. In Fig. \[fig:moments\] we compare the stochastic moments numerically obtained from the true PDF[^2] with those obtained from gPC approximation (\[eq:poly-D\]) for varying $d$ and $q$. Note that we have chosen the quadrature order so that $q\geq d+1$. As a result, the poorly approximated coefficients from Fig. \[fig:coeff\] are never considered in the moment computation. This explains why even for low polynomial order, the higher order moment approximations remain very precise. In conclusion, the moments are employed to fit parametric distributions to the output distribution, $f_Y$. Results are depicted in Fig. \[fig:matching\] when using the Gaussian and maximum entropy distribution. The numerical correspondence between two PDFs is quantified by the Earth Mover’s Distance (EMD) (Appendix \[appendix:emd\]). Each fit is compared to the actual PDF. The PDF fit obtained with the exact moments serves as an upper bound for the amount of information that is contained within the moments. [.9]{} \[fig:coeff\] [.9]{} \[fig:moments\] [.47]{} [.47]{} [.475]{} [.475]{} Wet clutch shifting time {#sec:application} ======================== System description ------------------ The proposed methodology is applied to identify the parametric uncertainty of a wet clutch system. A clutch is a mechanical device that connects a motoring unit to a load that transforms power into useful work. The objective is to ensure smooth and on demand coupling of the driving shaft to the driven shaft. A wet clutch system realizes this by means of contact friction between two series of friction plates, see Fig. \[clutch\] for a schematic cross section [@bruno2011]. By construction these friction plates are fully submerged in oil so that slip induced heat can be transferred adequately. This avoids thermal overheat and increases the clutch’s life time. A wet clutch uses the same hydraulic system to activate the engagement process. By pressurizing the hydraulic chamber, a freewheeling piston is pressed against the series of interlocking friction plates realizing a power transfer. The pressure in the hydraulic chamber is controlled via a current signal that activates a servo-valve linking the hydraulic chamber to a pressurized reservoir. The pressure difference over the piston determines its position and therewith the clutch engagement. ![Schematic cross section of the wet clutch system.[]{data-label="clutch"}](Clutchschematic){width="0.6\columnwidth"} Since the wet-clutch set-up should be low cost, additional sensors are avoided and a feedforward control strategy using the parameterized current profile shown in Fig. \[fig:feedforward\] is applied. The signal is characterized by three parameters. An initial maximum current pulse, parameterized by its duration $\Delta t$, gets the piston as close as possible to the friction plates without making contact. During this phase, the hydraulic chamber fills up and initial torque transfer commences due to Couette flow phenomena. The pressure drop invoked by reducing the current to $u_0$ avoids a hard collision. Once the plates are in contact a second pulse, defined by $\Delta u$, is generated to have a proper pressure build-up for synchronizing the load. At synchronous speed the torque transfer caused by slip speeds drops away. Therefore, hazardous jerks should be avoided by having slow synchronization. This explains the downward flank of the current pulse. After syncing the pressure is increased to guarantee the plates cannot release again by accident and start slipping. The shifting time, defined as the time interval between initialization of the feedforward control signal and the moment that both shafts obtain the same angular speed, is therefore considered as key characteristic of the engagement process. ![Feedforward current signal.[]{data-label="fig:feedforward"}](feedforward){width="0.45\columnwidth"} Shifting time model {#sec:model&exp} ------------------- The shifting time is derived from a dynamic simulation of the clutch engagement. Prior research was on the building of a dynamic model of the engagement process based on first principle physics. Model parameter values were then identified via experimental system identification (from isolated subsystem testing....). The exact model identification is not in the scope of this paper and a general overview is given. We refer to [@Georges; @iqbal2015; @agusmian2013; @Widanage2011] for further reading. Figure \[DynamicScheme\] illustrates the simplified dynamical scheme of the wet clutch system. ![Dynamic torque equilibrium scheme.[]{data-label="DynamicScheme"}](dynScheme){width="0.8\columnwidth"} The motor is controlled towards a constant speed $\omega_m$ delivering a torque $T_m$ to the system. The motor is connected with a torque converter that drives the input shaft of the clutch with torque $T_1$ at rotational velocity $\omega_1$. The system can be modeled using a nonlinear torque ratio function $\frac{T_1}{T_m} = f_{t}(\frac{\omega_1}{\omega_m})$ and a capacity factor function $\frac{\omega_{m}}{\sqrt{T_m}}=f_{c}(\frac{\omega_1}{\omega_{m}})$ provided by the manufacturer. Consequently, the driving torque $T_1$ of the input shaft can be written as $$T_1(\omega_m,\omega_1) = \omega_m^2\frac{f_{t}(\frac{\omega_1}{\omega_m})}{f_{c}(\frac{\omega_1}{\omega_m})^2}. \\$$ The clutch is connected to a ratio selector with gear ratio $R=\frac{\omega_{in}}{\omega_{out}}$. Thereafter, the output shaft operates at $\omega_2$ as result of the driving torque $T_2$. Furthermore, the rotational movement of the output shaft and flywheel, characterized by inertia $J_2$, are counteracted by an additional brake torque $T_b$. $$T_b(\omega_2)=T_{b0} + b\omega_o$$ The system is fully engaged when both shafts have an equivalent speed $\omega_1=R \omega_2$. The dynamics of this engagement process can be described by a succession of two distinct phases.\ ### Asynchronous phase The first phase is characterized by a speed difference $\omega_1 > R\omega_2 $ of the clutch shafts. During this phase the hydraulic piston chamber starts filling up. The resulting overpressure $p_{hc}$ is measured and depicted in Figure \[fig:pressure\]. The dynamics of the pressure build-up can be captured by defining a state ${\boldsymbol{\mathrm{s}}}=\begin{bmatrix} p_{hc}&\dot{p}_{hc}&\tilde{z} \end{bmatrix}^T$. The state variable $\tilde{z}$ can easily be scaled and truncated within a feasible area to obtain the piston position $z\in[0, z_M]$, for which $z_M$ is the position that assures full contact between the friction plates. The dynamics are modeled by an affine system with parameters identified to optimally match the pressure $p_{hc}$. Mark that the control input $u$ is the feedforward current signal of the valve as was illustrated in Fig. \[fig:feedforward\]. $$\begin{aligned} \label{eq: pressureDyn} \dot{{\boldsymbol{\mathrm{s}}}}&={\mathrm{A}} {\boldsymbol{\mathrm{s}}}+ {\mathrm{B}} \begin{bmatrix} u \\ \dot{u} \end{bmatrix}+{\boldsymbol{\mathrm{c}}} \\ {\mathrm{A}}&=\begin{bmatrix} 0 & 1 & 0 \\ a_1 & a_2 & 0 \\ a_3 & a_4 & a_5\\ \end{bmatrix}; \ {\boldsymbol{\mathrm{B}}}=\begin{bmatrix} 0 & 0 \\ b_1 & b_2 \\ 0 & 0 \\ \end{bmatrix}; \ {\boldsymbol{\mathrm{C}}}=\begin{bmatrix} 0 \\ c_1 \\ c_2 \end{bmatrix} \end{aligned}$$ The detailed pressure dynamics are out of scope and we refer to prior research for more information [@Widanage2011; @Georges]. During the filling phase the plates do not make contact. However, the oil between the friction plates will behave as a planar Couette flow. The resulting shear stresses within the fluid will cause an initial torque transmission and acceleration of the load speed $\omega_2$. This phenomena is characterized by the constant $\gamma $, capturing the geometry of the plates and the fluid viscosity. A second influence on the torque transmission is the pressure on the plates $p$. This pressure is a fraction of the total hydraulic overpressure $p_{hc}$ as is illustrated in Fig. \[fig:pressure\]. There is assumed that the pressure build-up on the plates initiates when the piston reaches a constant threshold $z_p$. $$\label{eq: zp} p=\max(0,\frac{z-z_p}{z_M-z_p}) p_{hc}$$ The torque transfer due to plate pressure is modeled with parameter $\alpha$, based on the geometry of the plates and fluid characteristics. A naive approach to model the transferred clutch torque $T_c$ would be to switch between Couette flow and plate friction once $z$ equals $z_p$ in a discrete way. However, the sudden torque shift would cause unrealistic behavior. Therefore, a transient function $\delta_t \in [0,1]$ is defined (Fig. \[fig:pos\]) to have a smooth transition between the aforementioned sources of torque transfer. $$T_c(\omega_1,\omega_2,p,z) = \delta_t(z) \cdot \alpha p+(1-\delta_t(z) )\cdot\frac{\gamma}{z_{M}-z}\cdot(\omega_1-R \omega_2)$$ The expression for the counteracting clutch torque $T_c$ on the input shaft holds when $\omega_1> R\omega_2$. The driving torque of the output shaft equals $T_2=R T_c$ and is illustrated in Fig. $\ref{fig:torque}$. Hence, one can formulate the nonlinear system dynamics during the asynchronous phase by considering the torque equilibrium of both shafts. $$\begin{aligned} J_1\dot{\omega}_1&=T_1(\omega_m,\omega_1) -T_c(\omega_1,\omega_2,p,z) \\ J_2\dot{\omega}_2&=R T_c(\omega_1,\omega_2,p,z)- T_b(\omega_2) \end{aligned}$$. [.45]{} \[fig:pressure\] [.45]{} \[fig:pos\] ### Synchronous phase The system shifting process arrives in the synchronous phase once an equivalent speed for both shafts is obtained ($\omega_1=R \omega_2$). Henceforth, the shafts can be considered as fully engaged and the system becomes independent of the control signal $u$. The driving torque of the output shaft equals $T_2=R T_1$ since perfect torque transmission is assumed (Fig. $\ref{fig:torque}$). $$(J_1+ \frac{J_2}{R^2})\dot{\omega}_1=T_1(\omega_m,\omega_1) - \frac{1}{R}T_b\left(\frac{\omega_1}{R}\right) \\$$ ### Shifting time The purpose of this dynamical model is to determine the shifting time, which is key for various applications of the wet clutch. This is determined by measuring the time interval between initialization of the feedforward control signal and the moment that both shafts obtain the same angular velocity (note that the mutual speed does not need to equal the initial input velocity). Fig. \[fig:omega\] illustrates the acceleration of the output shaft $\omega_2$ towards a synchronous speed with the input shaft. The figure clearly illustrates the discrepancy between the measured shifting time $y_l$ and corresponding simulation $Y_l$. [.45]{} \[fig:torque\] [.45]{} \[fig:omega\] Practical set-up ---------------- The test set-up considered for this paper is depicted in Fig. \[testbench\]. An AC electric motor ($30$ ) is controlled to a constant speed via a high bandwidth motor drive. The motor is connected to a controlled transmission via a torque converter. Within this transmission, the clutch can be controlled to engage with the proper gear. Furthermore, the output shaft of the clutch is connected to a load transmission. The load consists of a flywheel ($2,5$ ) and a brake. This system was previously used to study smooth gearbox controllers [@bruno2011]. ![Wet clutch test bench[]{data-label="testbench"}](test_bench){width="0.6\columnwidth"} Design of experiments --------------------- We experimentally tested the influence of several control and operating conditions on the shifting time $y_l$ on the wet clutch test set-up. We performed an up-shift from neutral to first gear for various control parameters and conditions, spanning a test scenario space. More specifically, 18 distinct feedforward control signals were tested. We compared 3 different values for $\Delta t$ and $u_0$ and 2 values for $\Delta u$. Further we verified 3 different motor speeds $\omega_m$ (1200, 1350 and 1500 ) and 2 different friction load conditions $T_b$ (low, high). For these, a full factorial design of $l\in\{1,\cdots,108\}$ experiments has been tested. As an illustration of the modeling deficiency, we compare the actual measurements, $y_l$, with the simulated shifting times, $Y_l$, in Fig. \[fig:experiments\]. The global trend is captured by the deterministic model, however a distinct discrepancy between measurements and simulation is present. Parametric uncertainty model {#sec:parametric_UM} ---------------------------- The objective of this research is to associate and identify a probabilistic model for several of the lumped model parameters. As mentioned in section \[sec:model&exp\], several of these parameters were identified empirically based on isolated subsystem measurements. The inertia of the shafts are derived from materials properties and geometrical considerations. Experiments in steady state regime are used to determine values for the viscous and Coulomb friction coefficients. On the available set-up no measurements of the oil temperature are possible and the temperature effect on the dynamics is therefore eliminated by performing all experiments after warm-up of the machine. However, our model contains various other variables that could not be determined or measured directly. Initially, we determined the parameter values using a least-squared procedure on the test scenario set. The comparison in Fig. \[fig:experiments\] already illustrated the disadvantages of this strategy. Using a sensitivity analysis, we could identify two key parameters: $c_2$, the oil pressure bias in piston position computation (\[eq: pressureDyn\]) and the initial piston position $z_p$ (\[eq: zp\]). Therefore, we define the variables $x_1$ and $x_2$ that serve as scaling factor for respectively $c_2$ and $z_p$. We aim at identifying a probabilistic model for the random variable ${\boldsymbol{\mathrm{X}}}=(X_1;X_2)$. We propose a normal distribution, $\mathcal{P}_{{\boldsymbol{\mathrm{X}}}}({\boldsymbol{\mathrm{\alpha}}})=\mathcal{N}({\boldsymbol{\mathrm{\mu}}},{\mathrm{\Sigma}})$, with mean ${\boldsymbol{\mathrm{\mu}}}=(\mu_1;\mu_2)$ and covariance ${\mathrm{\Sigma}}$, corresponding the variable transformation, ${\boldsymbol{\mathrm{X}}} = {\boldsymbol{\mathrm{\mu}}} + {\mathrm{\Sigma}}{\boldsymbol{\mathrm{\Theta}}},{\boldsymbol{\mathrm{\Theta}}}\sim\mathcal{N}(0,{\mathrm{I}})$. Further we assume that both variables are independent so that ${\mathrm{\Sigma}} = \mathrm{diag}(\sigma_1,\sigma_2)$. Therefore, the probabilistic input model can be defined by ${\boldsymbol{\mathrm{\alpha}}}=(\mu_1,\mu_2, \sigma_1,\sigma_2)$. Note that a normal transformation implicitly assumes that forward model, $\mathcal{Y}$, can be evaluated for any value ${\boldsymbol{\mathrm{x}}}\in\mathbb{R}^2$. This can be avoided by clipping the distribution, constricting the stochastic domain of ${\boldsymbol{\mathrm{X}}}$ to a feasible area ${\boldsymbol{\mathrm{x}}} \in [{\boldsymbol{\mathrm{x}}}^-, {\boldsymbol{\mathrm{x}}}^+]$, i.e. we are only interested in the part of the normal distribution in between these boundaries. This is facilitated by the transformation, $\mathrm{clip}({\boldsymbol{\mathrm{\Theta}}})$. The random variables $\tilde{{\boldsymbol{\mathrm{\Theta}}}}_i = \mathrm{clip}({\boldsymbol{\mathrm{\Theta}}}_i)$ should be distributed according to $\frac{1}{\Phi(\theta_i^+)-\Phi(\theta_i^-)}\varphi(\tilde{\theta}_i),\tilde{\theta}_i\in[\theta_i^-,\theta_i^+]$ where $\theta^-_i = \frac{1}{\sigma_i}(x^-_i - \mu_i)$ and $\theta^+_i = \frac{1}{\sigma_i}(x^+_i - \mu_i)$[^3]. First we map ${\boldsymbol{\mathrm{\Theta}}}_i$ to a uniform distribution in between the values $\Phi(\theta^-_i)$ and $\Phi(\theta^+_i)$. Then we use the inverse transform sampling method to map the uniform variable back onto the standard normal distribution but now limited to the interval $[\theta_i^-,\theta_i^+]$. In conclusion the parametric probability model is defined through the transformations $$\begin{aligned} u_i &= \Phi(\theta^-_i) + \Phi(\theta_i) \left(\Phi(\theta^+_i) - \Phi(\theta^-_i)\right)\\ \tilde{{\theta}}_i &= \Phi^{-1}(u_i) \\ {\boldsymbol{\mathrm{x}}} &= {\boldsymbol{\mathrm{\mu}}} + {\mathrm{\Sigma}} \tilde{{\boldsymbol{\mathrm{\theta}}}} = {\boldsymbol{\mathrm{\mu}}} + {\mathrm{\Sigma}}\cdot \mathrm{clip}({\boldsymbol{\mathrm{\Theta}}}), ~{\boldsymbol{\mathrm{\Theta}}} \sim \mathcal{N}(0,{\mathrm{I}})\end{aligned}$$ [.4]{} \[fig:modeloutput\] [.4]{} {width="\columnwidth"} \[fig:distrclip\] [.9]{} \[fig:distribution\_experiments\] The clipping boundaries, ${\boldsymbol{\mathrm{x}}}^-$ and ${\boldsymbol{\mathrm{x}}}^+$, of the stochastic variables are chosen so that the space for which a feasible solution of the shifting time can be obtained for all 108 experiments, is maximized. Figures \[fig:modeloutput\] and \[fig:distrclip\] illustrate respectively the parametric model output and associated clipped normal input distribution. Figure \[fig:distribution\_experiments\] demonstrates the corresponding output distribution for each experiment. Visual comparison with Fig. \[fig:experiments\] demonstrates that the least-squared approach treats the problem in a non appropriate manner[^4]. Results and discussion ====================== In this section we apply the methodologies described in section \[sec:IUIM\] on the wet clutch system as was described in section \[sec:application\]. First we demonstrate the forward propagation of uncertainty by concatenating the techniques described in the sections \[sec:UP-PCE\] and \[sec:PDF-MM\], simply to determine the corresponding output distribution by moment matching for a given input distribution. Secondly we use the MLE method to identify the optimal parameter ${\boldsymbol{\mathrm{\alpha}}}^*$ that best describes observed experiments. We compare results obtained with both the Gaussian and max entropy parametric PDFs. To obtain a benchmark PDF we estimated the output distributions empirically using Monte Carlo (MC) uncertainty propagation of 200 000 simulation samples and fitted a spline curve on the associated histograms. The associated $\log L$ values are referred to as the true values. PDF fitting by the method of moments ------------------------------------ First we demonstrate the PDF estimation strategy combining high-order gPC moments with the method of moments. Corresponding the variable transformation and associated parametric probability model that were motivated in section \[sec:parametric\_UM\], and in concordance with Table \[tab:wiener\], we make use of Hermite polynomials. We used polynomial order $d=4$ and corresponding univariate quadrature order $q=5$, i.e. $25$ function simulations were required to estimate the output distribution for a single experiment. Due to numerical restrictions (see section \[sec:algo\]), we are limited to the first $m = 4$ moments. Estimates for the output distribution of the 16^th^ experiment are visualized in Fig. \[fig:fits\_nonlin\]. We compare the MC reference distribution with Gaussian and max entropy fits. We also included PDF estimates that are based on the stochastic moments extracted from the MC sample set. Based on these results we can compare how much information could be extracted by the gPC approach with respect to the amount of information that is contained within the first four moments about the true distribution. The minor difference between the gPC PDF estimates and the MC equivalent PDF estimates suggests that gPC is a viable approach with sparse function evaluations. For a fair comparison besides the visual verification, we calculate the Earth Mover’s Distance (EMD) (see Appendix \[appendix:emd\]) for every estimate w.r.t. the estimated true output distribution. Different values are presented in Fig. \[fig:emd\_nonlin\]. These results confirm that high-order estimates obtained through the extended gPC framework are capable of extracting statistical information from significantly fewer function evaluations when compared to the brute force methodologies such as MC. [.4]{} \[fig:fits\_nonlin\] [.4]{} \[fig:emd\_nonlin\] Maximum log-likelihood estimation --------------------------------- We demonstrated that the high-order gPC moment matching strategy provides cheap yet accurate output PDF estimates for given input uncertainty. Consequently it can be engaged as an efficient alternative to evaluate the $\log$-Likelihood of a number of experiments at a mere fraction of the computational cost associated to brute force strategies such as MC. Next we apply the method to identify the optimal parameter set ${\boldsymbol{\mathrm{\alpha}}}^*$ associated to the probability model described in section \[sec:parametric\_UM\]. To solve problem (\[eq:MLE\]), we used a genetic algorithm (GA) with population size 50 using the [$\mathrm{Matlab}$]{} optimization toolbox. We compared optimization results for both Gaussian and max entropy PDF abstractions. The performance of the optimal parameter sets are verified using MC estimates of the $\log$-Likelihood corresponding the identified probability models. ### Gaussian output distribution Using the Gaussian output distribution and gPC moment estimates, the following optimal parameter set was identified, ${\boldsymbol{\mathrm{\alpha}}}^*_{G}=(0.91,0.55,0.077,0.014)$. Figure \[fig:gauss\] compares measurement with the corresponding output distributions. The solution has an estimated $\log$-Likelihood of $\log\hat{L}^{G}({\boldsymbol{\mathrm{\alpha}}}^*_{G})=188.5$ using the moment matching approach. When we evaluate the $\log$-Likelihood using the MC approach, we obtain a smaller value $\log L({\boldsymbol{\mathrm{\alpha}}}^*_{G})=176.7$. ### Maximum entropy output distribution Using the max entropy distribution we obtain the following optimal parameter, ${\boldsymbol{\mathrm{\alpha}}}^*_{ME}=(0.91,0.55,0.081,0.046)$. Note that we retrieve the same mean as with the Gaussian output distributions, ${\boldsymbol{\mathrm{\mu}}}$, but that both covariance values have increased. Figure \[fig:ME\] visualizes the corresponding output distributions and measurements. Using the max entropy moment matching approach, the $\log$-likelihood is estimated on $\log \hat{L}^{ME}({\boldsymbol{\mathrm{\alpha}}}^*_{ME})=189$. Verification using MC delivers a slightly smaller $\log$-Likelihood $\log L({\boldsymbol{\mathrm{\alpha}}}^*_{ME})=183.7$. Note that the difference is significantly smaller compared to the $\log$-Likelihood mismatch obtained with the Gaussian PDF estimates. [.9]{} \[fig:gauss1\] [.9]{} \[fig:gauss2\] [.9]{} \[fig:ME1\] [.9]{} \[fig:ME2\] Discussion ---------- With respect to the global $\log L$ estimates, we remark the following. In both the max entropy and the Gaussian case the $\log L$ is overestimated using the proposed gPC moment matching method. However, for the max entropy distribution, the mismatch between true $\log L$ and estimated $\log L$ is significantly smaller (approximately 55%). Moreover the true $\log L$ obtained with ${\boldsymbol{\mathrm{\alpha}}}^*_{ME}$ exceeds that obtained with ${\boldsymbol{\mathrm{\alpha}}}^*_{G}$ by 4%. Both of these observations confirm that the use of high-order stochastic moments benefits the identification of an optimal input probability model and that the gPC framework can be used to obtain accurate estimates of these moments. Figure \[fig:LLHcomp\] compares the estimates $\log \hat{L}_l^{G}({\boldsymbol{\mathrm{\alpha}}}^*_{G})$ and $\log \hat{L}_l^{ME}({\boldsymbol{\mathrm{\alpha}}}^*_{ME})$, for each individual experiment $l \in \{1,\cdots, 108\}$, w.r.t. to their true value. One may note that for the majority of the experiments the $\log L$ is estimated with great precision for either output distribution. This indicates that for the majority of the experiments, the true output distribution is about Gaussian and no additional information is contained within the high-order moments. However for a small subset of experiments, the Gaussian overestimates the true value significantly due to its incapacity of modeling asymmetric distributions. As a result, measurements that are highly improbable and would strongly affect the global $\log L$ are penalized less by the Gaussian distribution. This effect was already visualized in Fig. \[fig:nonlinearfit\]. Following the argument made above, we compare the EMD metric averaged over all experiments in Fig. \[fig:rescomp\], as it proofs to be valuable to quantify the estimated output distribution’s accuracy. We found that the use of the maximum entropy distribution with 4 moments could reduce the EMD by approximately 42% for $\alpha_G^*$ and 47% for $\alpha_{ME}^*$ when compared with the averaged EMD value obtained with Gaussian output distributions. The minor difference compared to the EMD of the PDF estimates based on the stochastic moments directly extracted from the MC sample set indicates that the gPC can correctly estimate the statistical moments. The improved accuracy can therefore be fully attributed to the higher order moments gPC framework. Figure \[fig:evalComp\] illustrates the mean absolute error (MAE) of the $\log$-Likelihood estimates of all individual experiments $l \in \{1,\cdots, 108\}$ compared to the MC reference obtained by 200 000 simulation samples. The accuracy of the brute force propagation techniques increases with respect to the number of samples used for uncertainty propagation. The quasi Monte Carlo (qMC) technique considers an equally distributed grid of $S$ samples. The gPC results require a magnitude less expensive function evaluations to achieve the same accuracy as the brute force techniques. The interpolated values $S_{MQ}$ and $S_{qMQ}$ are depicted in Table \[table:overview\]. A comparison between the gPC techniques indicates that the MAE could be reduced by 15.3% for ${\boldsymbol{\mathrm{\alpha}}}_G^*$ and 24.8% for ${\boldsymbol{\mathrm{\alpha}}}_{ME}^*$ due to the incorporation of higher order estimates in the novel gPC framework. [0.49]{} [0.49]{} $\log L$ PDF est. MAE $S_{gPC}$ $S_{MC}$ $S_{qMC}$ -- ---------- ---------- ------- ----------- ---------- ----------- Gauss 0.183 25 458 577 ME 0.155 25 629 1171 Gauss 0.117 25 265 370 ME 0.088 25 383 1009 : Summary of log-Likelihood estimation.[]{data-label="table:overview"} Conclusion ========== In this paper we proposed and discussed an efficient numerical method that is tailored to parametric model uncertainty identification using maximum likelihood estimation. Here the objective is to find a parametric input distribution that best explains a series of observations. A novel method is proposed to determine the output distribution for given input probability model making use of the generalized Polynomial Chaos (gPC) expansion framework. We extended the gPC framework so that high-order stochastic moments can be obtained efficiently. These high-order estimates can then be fed to a Gaussian or maximum entropy distribution. This strategy enables a reduction in the required number of function evaluations to obtain an adequate estimation of the statistical moments of the output distribution. The method is applied to calibrate a model for the shifting time for wet clutch engagement based on a series of measurements verified in the lab. It is shown that output distribution estimations via high-order moment matching excels traditional identification methods that are based on mean and variance estimates whilst the computational cost remains the same due to efficient techniques that were developed. The high-order moment matching resulted into a $\log$-likelihood increase of about 4% since the accuracy of the estimated output probability density function could be improved up to 47% compared to Gaussian distributions. This methodology reveals high potential when scaling up the number of uncertain parameters that need to be identified through inverse uncertainty identification. In addition to the inclusion of parametric uncertainty, future work will investigate the incorporation of model structure uncertainty. Acknowledgments {#acknowledgments .unnumbered} =============== Wannes De Groote holds a doctoral grant strategic basis research (3S07219) of the Fund for Scientific research Flanders (FWO). Some of the computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation - Flanders (FWO) and the Flemish Government – department EWI. This study was performed in the frameworks of the EVIT ICON project. This research was supported by Flanders Make, the strategic research center for the manufacturing industry. Maximum entropy distribution {#appendix:med} ============================ The maximum entropy distribution, $\hat{p}$ provides a solution to the classical problem where an estimate of univariate density $p$ is sought from knowledge of its moments [@mead1984maximum]. The extent to which a density may be determined from its moments is a topic of discussion in mathematical literature. In practice only a finite number of moments, say $K+1$, are usually available so that there exists an infinite variety of functions whose first $K+1$ moments coincide and a unique reconstruction of $p$ is simply impossible. To remedy this issue, $\hat{p}$ is sought to maximize ${\mathcal{W}}[p] = -\int_{\mathcal{X}} p(x) \log p(x)\text{d}x$ under the condition that the first $K+1$ moments are equal to the true moments $\mu_k$. The Lagrangian $L$ corresponding to this problem is $$\label{eq:originalL} {\mathcal{L}}[p,{\boldsymbol{\mathrm{\lambda}}}] = {\mathcal{W}}[p] + \mathsmaller{\sum}_{k=0}^K \mathsmaller{\int}_{\mathcal{X}} \lambda_k(x^k - \mu_k) p(x) \text{d}x$$ Nullifying the functional variation to $p$ yields $$\delta_p {\mathcal{L}} = 0 \Rightarrow \hat{p}(x|{\boldsymbol{\mathrm{\lambda}}}) = \exp \left(-\mathsmaller{\sum}_{k=0}^K \lambda_k x^k\right)$$ Without loss of generality we may further assume that $\hat{p}$ is normalized so that $\mu_0 = 1$. Accordingly, we can express $\lambda_0$ in function of the remaining Lagrangian multipliers $$\begin{gathered} \mathsmaller{\int}_{\mathcal{X}} \exp \left(-\mathsmaller{\sum}_{k=0}^K \lambda_k x^k\right) \text{d}x = 1 \\ \Rightarrow \lambda_0 = \log \mathsmaller{\int}_{\mathcal{X}} \exp \left(-\mathsmaller{\sum}_{k=1}^K \lambda_k x^k\right) \text{d}x\end{gathered}$$ One may substitute this solution into the original Lagrangian (\[eq:originalL\]) so to obtain the dual unconstrained problem which can be solved accordingly. $$\min_{{\boldsymbol{\mathrm{\lambda}}}}\Gamma({\boldsymbol{\mathrm{\lambda}}}) = \lambda_0 + \mathsmaller{\sum}_{k=1}^K \lambda_k \mu_k$$ Earth mover’s distance {#appendix:emd} ====================== The Earth Mover’s distance (EMD) is a metric for the dissimilarity between two signatures that are each a compact representation of a distribution. In formal mathematics the metric is known as the Wasserstein distance between two probability distributions. The metric was coined in computer science due to an apparent analogy to its definition that can be modeled as the solution to a transportation problem [@rubner2000earth]. Given are two signatures $P=\{(u_i,{\boldsymbol{\mathrm{x}}}_i)\}_{i=1}^n$ and $Q=\{(v_j,{\boldsymbol{\mathrm{y}}}_j)\}_{j=1}^m$. An insightful interpretation is that $u_i$ and $v_j$ represent the amount of dirt at the respective positions ${\boldsymbol{\mathrm{x}}}_i$ and ${\boldsymbol{\mathrm{y}}}_j$. The EMD between the signatures $P$ and $Q$ is defined as the minimal (normalized) work required to reconfigure $P$ into $Q$ moving around the dirt. Formally that is $$\begin{aligned} \mathrm{EMD}(P,Q)=~ &\min_{F=\{f_{ij}\}} &&\frac{{\sum\nolimits}_{ij} f_{ij}d({\boldsymbol{\mathrm{x}}}_i,{\boldsymbol{\mathrm{y}}}_j)}{{\sum\nolimits}_{ij} f_{ij}} \\ &\text{subject to}&& \mathsmaller{{\sum\nolimits}}_i f_{ij} \leq v_j \\ &&&\mathsmaller{{\sum\nolimits}}_j f_{ij} \leq u_i \\ &&&\mathsmaller{{\sum\nolimits}}_{ij} f_{ij} = \min\left\{\mathsmaller{{\sum\nolimits}}_i u_i ,\mathsmaller{{\sum\nolimits}}_j v_j \right\} \\ &&&f_{ij} \geq 0 \end{aligned}$$ where $f_{ij}$ represents the dirt transferred from position ${\boldsymbol{\mathrm{x}}}_i$ to ${\boldsymbol{\mathrm{y}}}_j$, and, $d({\boldsymbol{\mathrm{x}}}_i,{\boldsymbol{\mathrm{y}}}_j)$ the distance to be covered for that transport. The transportation problem above is a special linear programming problem. Derivation of equation (\[eq:moments2.0\]) {#appendix:mathderiv} ========================================== Details are given about the derivation of formula (\[eq:moments2.0\]). We initiate the derivation from equation (\[eq:moments\]). First we expand the power of the sum using the multinomial theorem. Then we regroup the terms so that the high-order inner product is isolated from the coefficients. Finally we substitute expression $\prod_{j=1}^{n} \phi^{(j)}_{\uline{k}(j)}$ for $\psi_{\uline{k}}$ so to obtain equation (\[eq:moments2.0\]). $$\begin{aligned} \mu^{(d)}_m &= \int_{\mathcal{X}} \left(\sum_{i\in\mathcal{I}} c_i \psi_i\right)^m f_{{\boldsymbol{\mathrm{X}}}} \text{d}{\boldsymbol{\mathrm{x}}} \\ &= \int_{\mathcal{X}} \sum_{\uline{i}\in\mathcal{I}(m,p)} \binom{m}{\uline{i}} \prod_{k=1}^{p} c_k^{i_k} \psi_k^{i_k} f_{{\boldsymbol{\mathrm{X}}}} \text{d}{\boldsymbol{\mathrm{x}}} \\ &= \sum_{\uline{i}\in\mathcal{I}(m,p)} \binom{m}{\uline{i}} \cdot \left\langle \prod_{k=1}^{p} \psi_k^{i_k} \right\rangle \cdot \prod_{k=1}^{p}c_k^{i_k} \\ &= \sum_{\uline{i}\in\mathcal{I}(m,p)} \binom{m}{\uline{i}} \cdot \prod_{j=1}^{n} \left\langle \prod_{k=1}^{p} {\phi^{(j)}_{\uline{k}(j)}}^{i_k} \right\rangle \cdot \prod_{k=1}^{p}c_k^{i_k} \end{aligned}$$ [^1]: \*Both authors contributed equally to this work. [^2]: For a monotonic function, $y(x)$, with inverse, $x(y)$, this is equal to, $f_Y(y) = \lvert x'(y) \rvert \cdot f_X(x(y)) = \lvert y'(x(y)) \rvert^{-1} \cdot f_X(x(y)) $. [^3]: Here $\varphi(\cdot)$ and $\Phi(\cdot)$ are standard notation for respectively the PDF and the cumulative density function (CDF) of the standard normal distribution. [^4]: We may note that the least-squares approach maximizes the objective $- \sum \frac{1}{2} (y_l - \mathcal{Y}_l)^2$. Comparison with the $\log$-Likelihood approach reveals that this is equivalent with assuming that each experiment has an output probability density $\sim \exp(-\frac{1}{2}(y_l-\mathcal{Y}_l)^2)$. In other words the least-squares implicitly assumes that each experiment is normally distributed with variance $1$.

--- abstract: 'Criteria suitable for measuring entanglement between two different potential wells in a Bose-Einstein condensation (BEC) are evaluated. We show how to generate the required entanglement, utilizing either an adiabatic two-mode or dynamic four-mode interaction strategy, with techniques that take advantage of s-wave scattering interactions to provide the nonlinear coupling. The dynamic entanglement method results in an entanglement signature with spatially separated detectors, as in the Einstein-Podolsky-Rosen (EPR) paradox.' author: - 'Q. Y. He$^{1}$, M. D. Reid$^{1,2}$, C. Gross$^{2}$, M. Oberthaler$^{2}$, and P. D. Drummond$^{1,2}$' title: 'EPR entanglement strategies in two-well BEC ' --- One of the most important questions in modern physics is the problem of macroscopic spatial entanglement, which directly impinges on the nature of reality. Here we analyse how rapid advances in Bose-Einstein condensation (BEC) in ultra-cold atoms can help to resolve this issue. Recently, the observation of spin-squeezing has shown that measurement beyond the standard quantum limit is achievable [@Esteve2008; @Gross2010; @Riedel2010]. Spin squeezing is known to demonstrate entanglement between atoms [@Sorenson], but not which subsystems have been entangled. An important step forward beyond this would be to realise quantum entanglement in the Einstein-Podolsky-Rosen (EPR) sense; that is, having two spatially separated condensates entangled with each other [@Hines]. This is an important milestone towards future experiments involving entanglement of macroscopic mass distributions, thereby demonstrating quantum Schroedinger cat type superpositions of distinct mass distributions. In this Letter, we analyse some achievable entangled quantum states using a two-well BEC, and the measurable criteria that can be used to signify entanglement. The types of quantum state considered include number anti-correlated states prepared using adiabatic passage, as well as dynamically prepared spin-squeezed states. In particular, we focus on spin-entanglement, as a particularly useful route for achieving measurable EPR entanglement, without requiring atomic local oscillators. We note that spin orientation is easily coupled to magnetic forces to allow superpositions of different mass distributions, once spin entanglement is present. We consider different types of spin entanglement criteria, and analyze which quantum states these are sensitive to. We show that existing experimental techniques appear capable of generating spatial entanglement, with relatively minor changes. There are several possible routes available. Our most significant conclusion is that the criterion used to measure entanglement must be chosen carefully. Not all measures of entanglement are equivalent, and there is an important question as to what one regards as the fundamental subsystems, ie, particles or modes. The appropriate choice of measure depends on the entangled state, how it is prepared, and what type of detection is technologically feasible. To demonstrate and analyse this need to adapt the criterion to the state, we choose here to analyse two and four mode models of a BEC, indicated schematically in Fig. \[fig:Double-well\], where $a_{1}$, $a_{2}$ are operators for two internal states at $A$ and $b_{1},\ b_{2}$ are operators for two internal states at $B$. ![(a) Two internal modes $a_{,\ }b$ with spatial entanglement; (b) two pairs of modes $a_{1,\ }a_{2}$ and $b_{1,\ }b_{2}$ are entangled. \[fig:Double-well\]](Fig1){width="0.85\columnwidth"} In the limit of tight confinement and small numbers of atoms, this type of system can be treated using a simple coupled mode effective Hamiltonian, of form:$$\hat{H}/\hbar=\kappa\sum_{i}\hat{a}_{i}^{\dagger}\hat{b}_{i}+\frac{1}{2}\left[\sum_{ij}g_{ij}\hat{a}_{i}^{\dagger}\hat{a}_{j}^{\dagger}\hat{a}_{j}\hat{a}_{i}\right]+\left\{ \hat{a}_{i}\leftrightarrow\hat{b}_{i}\right\} \ .\label{eq:Hamiltonian}$$ Here $\kappa$ is the inter-well tunneling rate between wells, while $g_{ij}$ is the intra-well interaction matrix between the different spin components. #### Adiabatic preparation: We first consider two-mode states having a single spin orientation, with number correlations established using adiabatic passage in the ground state. This makes them practical to prepare following earlier experimental approaches [@Esteve2008; @Ketterle_Interferometer], as shown in Fig. \[fig:Double-well\](a). A recent multi-mode analysis shows that effects of other spatial modes may be relatively small [@Ferris]. In a two-mode analysis, we assume that $a_{1}$ and $b_{1}$ have been prepared in the many-body ground state of Eq (\[eq:Hamiltonian\]) with a fixed number of atoms $N$, while the second pair of spin states $a_{2}$ and $b_{2}$ remain in the vacuum state, so that we can write $a\equiv a_{1}$ and $b\equiv b_{1}$. In these cases there is only one nuclear spin orientation, and there is existing experimental data on phase coherence and number correlations [@Esteve2008; @Ketterle_Interferometer], with $10dB$ relative number squeezing being maximally indicated. A number of previous analyses have used entropic measures specific to pure states to study entanglement. These signatures cannot be readily measured, and are not applicable to realistic mixed states that are typically created in the laboratory. However, one generally demonstrate spatial entanglement between the two wells $a$ and $b$ using the non-Hermitian operator product criterion of Hillery and Zubairy (HZ) [@hillzub]. This is also related to a recently developed continuous-variable Bell inequality criterion [@cvbell2]. A sufficient entanglement criterion between $A$ and $B$ is the operator product measure: $$|\langle a^{\dagger}b\rangle|^{2}>\langle a^{\dagger}ab^{\dagger}b\rangle\,.\label{eq:HZ}$$ Interwell spin operators have already been measured in this environment. These are defined as: $\hat{J}_{AB}^{X}=\left(\hat{a}^{\dagger}\hat{b}+\hat{a}\hat{b}^{\dagger}\right)/2;\,\,\hat{J}_{AB}^{Y}=\left(\hat{a}^{\dagger}\hat{b}-\hat{a}\hat{b}^{\dagger}\right)/(2i);\,\,\hat{J}_{AB}^{Z}=\left(\hat{a}^{\dagger}\hat{a}-\hat{b}^{\dagger}\hat{b}\right)/2;\,\hat{J}_{AB}^{\pm}=\hat{J}_{AB}^{X}\pm i\hat{J}_{AB}^{Y};\,\hat{N}_{AB}=\hat{N}_{A}+\hat{N}_{B}=\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b}$. In spin language, the HZ criterion shows that spatial entanglement is proved for any state when$$\begin{aligned} E_{HZ} & = & \frac{\langle\Delta\hat{J}_{AB}^{+}\Delta\hat{J}_{AB}^{-}\rangle}{\langle\hat{N_{A}}\rangle}\nonumber \\ & = & \frac{\frac{1}{4}\langle\left[N_{A}+N_{B}\right]^{2}\rangle-\langle\left[\hat{J}_{AB}^{Z}\right]^{2}\rangle}{|\langle J_{AB}^{X}\rangle|^{2}+|\langle J_{AB}^{Y}\rangle|^{2}}<1\,.\label{eq:HZ-entanglement}\end{aligned}$$ This has similarities to the spin squeezing criterion [@sorenson] which has now been measured experimentally  [@Esteve2008; @Gross2010]. However, a crucial difference is that the spatial entanglement criterion (\[eq:HZ-entanglement\]) involves an increased relative number fluctuation, rather than the reduced relative number fluctuations found with the spin-squeezing criterion. Theoretically, we find that two-well entanglement exists in the ground state with the HZ criterion, although suppressed for increasingly strong repulsive interactions. This behaviour is also known from previous studies using an entropic $\varepsilon(\rho)$ entanglement measure [@Hines; @XieHai]. The strongest theoretical entropic entanglement is found when all atom numbers are equally represented in the superposition. We find that the closest state to this ‘super-entangled’ limit is obtained at a critical value of $Ng/\kappa\simeq-2$. This attractive interaction regime (as found in $^{41}K$ and $^{7}Li$ isotopes) gives rise to a maximal spread in the distribution of numbers in each well. Maximum entanglement results for this model have also been found [@XieHai] for entropic entanglement measures. In our calculations, we account for effects of finite temperatures by assuming a canonical ensemble of $\hat{\rho}=\exp\left[-\hat{H}/k_{B}T\right]$, with an interwell coupling of $\hbar\kappa/k_{B}=50nK$. Our result for the Hillery-Zubairy operator product signature is graphed below. This shows that two-well spatial entanglement is maximized for an attractive inter-atomic coupling, and the effect is relatively robust against thermal excitations: ![Adiabatic entanglement with interactions in a two-well potential. Dashed and dotted lines: HZ entanglement signature ($E_{HZ}<1$) at $T=0K,\,50nK,\,80nK$ - lowest lines at lowest temperature; solid line: entropic entanglement ($E_{entropic}=1-\varepsilon(\rho)/\epsilon_{max}<1$) at $T=0K$. \[fig:with adiabatic preparation\]](Fig2){width="0.9\columnwidth"} #### Dynamic preparation: To proceed further, EPR entanglement as we define it requires using measurements $O_{A}$ and $O_{B}$ that are individually defined either at well $A$ or well $B$. Thus, entanglement is shown by performing a set of simultaneous measurements on the spatially separated systems: typically by measuring correlations $<O_{A}O_{B}>$ or $P(O_{A},O_{B})$. This is necessary to justify Einstein’s no action at a distance assumption, that making one measurement at $A$ cannot affect the outcome of another measurement at $B$. One could achieve EPR entanglement with this criterion by making quadrature amplitude measurements. That is, by expanding $a=X_{a}+iP_{a}$ and $b=X_{b}+iP_{b}$, where $X_{a}$ etc are quadrature amplitudes, so that the moment $<ab^{\dagger}>$ is measured as four separate real correlations. Proposed methods for measuring entanglement in BEC experiments include time-reversed dynamics [@ZhangHelmersonSpinEnt], and interference with side-modes of a BEC moving through an optical lattice [@FerrisOlsen]. This shows that, in principle, such a quadrature-based entanglement measurement is not impossible. However, while feasible optically, this type of measurement is nontrivial with ultra-cold atoms owing to interaction induced phase fluctuations, and we propose a different strategy. To get good EPR measurements we consider instead the intra-well spins $J^{X},\ J^{Y},$ and $J^{Z}$ at site $A$ and $B$.****$\ $This means having at least four modes in total. To prove EPR entanglement using these measurements, one can define the spin measurements at $A$ to be in terms of $a_{1}$ and $a_{2}$: $\hat{J}_{A}^{X}=\left(\hat{a_{1}}^{\dagger}\hat{a_{2}}+\hat{a_{2}}^{\dagger}\hat{a_{1}}\right)/2,$ $\hat{J}_{A}^{Y}=\left(\hat{a_{1}}^{\dagger}\hat{a_{2}}-\hat{a_{2}}^{\dagger}\hat{a_{1}}\right)/(2i),$ $\hat{J}_{A}^{Z}=\left(\hat{a_{1}}^{\dagger}\hat{a_{1}}-\hat{a_{2}}^{\dagger}\hat{a_{2}}\right)/2,$ $\hat{N}_{A}=\hat{a_{1}}^{\dagger}\hat{a_{1}}+\hat{a_{2}}^{\dagger}\hat{a_{2}}$; also define raising and lowering operators as: $\hat{J}_{A}^{\pm}=\hat{J}_{A}^{X}\pm i\hat{J}_{A}^{Y}$, and similar definition for site B. These are measurable locally using Rabi rotations and number measurements, without local oscillators being required. The spin orientation measured at each site can be selected independently to optimise the criterion for the state used. One can then show EPR entanglement via spin measurements using the spin version of the Heisenberg-product entanglement criterion [@Bowen; @polarization; @ent-1] $$E_{product}=\frac{2\sqrt{\Delta^{2}\hat{J}_{AB}^{\theta\pm}\cdot\Delta^{2}\hat{J}_{AB}^{(\theta+\frac{\pi}{2})\pm}}}{|\langle J_{A}^{Y}\rangle|+|\langle J_{B}^{Y}\rangle|}<1\ ,\label{eq:product form}$$ or the sum criterion by Duan et al and Simon [@inseplur; @FiberEntangle] $$E_{sum}=\frac{\Delta^{2}\hat{J}_{AB}^{\theta\pm}+\Delta^{2}\hat{J}_{AB}^{(\theta+\frac{\pi}{2})\pm}}{|\langle J_{A}^{Y}\rangle|+|\langle J_{B}^{Y}\rangle|}<1\ ,\label{eq:SUM}$$ with general sum and difference spins $\hat{J}_{AB}^{\theta\pm}=\hat{J}_{A}^{\theta}\pm\hat{J}_{B}^{\theta}$, and $J^{\theta}=cos(\theta)J^{Z}+sin(\theta)J^{X}$. Here the conjugate Schwinger spin operators $J^{\theta}$ and $J^{\theta+\pi/2}$ obey the uncertainty relation $\Delta^{2}J^{\theta}\Delta^{2}J^{\theta+\pi/2}\geq\frac{1}{4}|\langle J^{Y}\rangle|$. In order to obtain ultra-cold atomic systems with four-mode entanglement, we consider a dynamical approach to EPR entanglement which utilizes phase as well as number correlations. This requires the BEC’s to evolve in time, in a similar way to successful EPR experiments in optical fibres [@FiberEntangle; @CRSD1987; @fiber; @experiment]. This is very different to the previous scheme, as the atom-atom interaction appears explicitly as part of the time-evolution. The best entanglement is obtained when the interaction between atoms of different spin is different to the interaction between the atoms of the same spin. In Rubidium, this either requires using a Feshbach resonance to break the symmetry, or else separating the two spin components spatially as in the successful fibre experiments [@fiber; @experiment] or in spin-squeezing atom-chip experiments [@Riedel2010]. At a Feshbach resonance, for alkali metals like Rubidium-87, the interactions between the different spin orientations can be reduced compared to the self-interactions. This allows an avenue for this type of entanglement with both the spin orientations remaining *in situ* in the same trap potential. To start with, we consider the conditions required to obtain the best squeezing of Schwinger spin operators by optimizing the phase choice $\theta$: $tg(2\theta)=2\langle J^{Z},\ J^{X}\rangle/(\Delta^{2}J^{Z}-\Delta^{2}J^{X})$. Entanglement can be generated by the interference of two squeezed states on a $50:50$ beam-splitter with a relative optical phase of $\varphi=\pi/2$. This has also been achieved in optical experiments [@FiberEntangle], although not yet in BEC. ![(a) Squeezing of Schwinger spin operators $S_{dB}$: $S_{+}=10log_{10}\left[\Delta^{2}(J_{A}^{\theta}-J_{B}^{\theta})/n_{0}\right]$ (solid), $S_{-}=10log_{10}\left[\Delta^{2}(J_{A}^{\theta+\pi/2}+J_{B}^{\theta+\pi/2})\right]/n_{0}$ (dashed), and $n_{0}=\frac{1}{2}(|\langle J_{A}^{X}\rangle|+|\langle J_{B}^{Y}\rangle|)$ is shot noise (dotted). (b) Entanglement ($E_{product}$) based on the criterion (\[eq:product form\]) by the solid curve and $E_{sum}$ in sum criterion (\[eq:SUM\]) by the dashed curve. Here the parameters correspond to $Rb$ atoms at magnetic field $B=9.131G$, with scattering lengths $a_{11}=100.4a_{0}$, $a_{22}=95.5a_{0},$ and $a_{12}=80.8a_{0}$. $a_{0}=53pm$. The coupling constant $g_{ij}\propto2\lyxmathsym{\textgreek{w}}_{\perp}a_{ij}$. The number of $Rb$ atoms is $N_{A}=200$. $\tau=g_{11}N_{A}t$.\[fig:with cross term\]](Fig3){width="0.9\columnwidth"} Here we again take a four modes approach, explicitly assuming$a_{1},\ b_{1}$ and $a_{2},\ b_{2}$ that are initially in coherent states for simplicity, i.e., assuming we have coherence between the wells. For simplicity, we suppose that the initial state is then prepared in an overall four-mode coherent state using a Rabi rotation. It is also possible to choose a constrained total particle number, but we have used the simplest model of coherence between the wells:$$|\psi>=|\alpha>_{a_{1}}|\alpha>_{b_{1}}|\alpha>_{a_{2}}|\alpha>_{b_{2}}$$ Next, we assume that the inter-well potential is increased so that each well evolves independently. Finally, we decrease the inter-well potential for a short time, so that it acts as a controllable, non-adiabatic beam-splitter [@olsen], to allow interference between the wells, followed by independent spin measurements in each well. For dynamics, we assume a simple two-spin evolution per well, which is exactly soluble. We can treat this using either Schroedinger or Heisenberg equations of motion. In the Heisenberg case, since the number of particles is conserved in each mode, this has the solution:$$\begin{aligned} \hat{a}_{i}\left(t\right) & = & \exp\left[-i\sum g_{ij}\hat{N}_{j}t\right]\hat{a}_{i}\left(0\right)\ ,\end{aligned}$$ where the couplings $g_{ij}$ are obtained from the known $Rb$ scattering lengths at a Feshbach resonance. ![Same as Fig. \[fig:with cross term\] but assuming NO cross-couplings, i.e., $g_{12}=0$. \[fig:without cross term\]](Fig4){width="0.9\columnwidth"} After dynamical evolution from an initial coherent state, we find spin-squeezing in each well, prior to using the beam-splitter as shown in Fig. \[fig:with cross term\](a). After using the beam-splitter, entanglement can be detected in principle as $E<1$, as shown in Fig. \[fig:with cross term\](b). Note, Fig. \[fig:without cross term\] shows that assuming NO cross-couplings, i.e., $g_{12}=0$ gives much better results than using the cross-couplings obtained in a $Rb$ Feshbach resonance. As discussed in the adiabatic approach, this would require spatially separated wells for the different spin-orientations, or a different type of Feshbach resonance, in order to eliminate cross-couplings. In summary, we have shown two feasible techniques for measuring EPR-type spatial BEC entanglement, using currently available double-well BEC approaches combined with available atomic detection methods. The simplest method employs an attractive ground-state adiabatic method, with a single spin orientation. This requires an essentially nonlocal detection strategy, in which the two BEC’s are expanded and interfere with each other. 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--- author: - Mengshu Liu - 'Leland W. Harriger' - Huiqian Luo - Meng Wang - 'R. A. Ewings' - 'T. Guidi' - Hyowon Park - Kristjan Haule - Gabriel Kotliar - 'S. M. Hayden' - Pengcheng Dai title: 'Nature of magnetic excitations in superconducting BaFe$_{1.9}$Ni$_{0.1}$As$_{2}$' --- Our BaFe$_{2-x}$Ni$_{x}$As$_{2}$ electron-doped samples were grown using self flux method as described before [@yanchao]. Since the electronic and superconducting properties of Co and Ni-doping of BaFe$_2$As$_2$ near optimal superconducting transition temperatures are almost identical [@budko], we chose to study spin excitations in optimally doped BaFe$_{1.9}$Ni$_{0.1}$As$_{2}$ with $T_c=20$ K. To further compare spin excitations in BaFe$_2$As$_2$ and BaFe$_{1.9}$Ni$_{0.1}$As$_{2}$, we show in Figure 1 spin excitations of these two materials at different energies. Consistent with data shown in Figs. 2-4 of the main text, we see that the effect of electron-doping is to mostly modify spin excitations below 80 meV.  Our theoretical method for computing the magnetic excitation spectrum employs abinitio full potential DFT+DMFT method, as implemented in Ref. [@Haule], which is based on the commercial DFT code of Wien2k [@Wien2K]. The DMFT method requires solution of the generalized quantum impurity problem, which is here solved by the numerically exact continuous-time quantum Monte Carlo method (CTQMC) [@Haule:07; @Werner:06]. The Coulomb interaction matrix for electrons on iron atom was determined by the self-consistent GW method in Ref. [@Kutepov:10], giving $U=5\,$eV and $J=0.8\,$eV for the local basis functions within the all electron approach employed in our DFT+DMFT method. ![ The Feynman diagrams for the Bethe-Salpeter equation. It relates the two-particle Green’s function ($\chi$) with the polarization ($\chi^{0}$) and the local irreducible vertex function ($\Gamma_{loc}^{irr}$). The non-local two-particle Green’s function is obtained by replacing the local propagator by the non-local propagator. \[fig:BSE\]](diagram) The dynamical magnetic susceptibility $\chi(\textbf{q},\omega)$ is computed from the *ab initio* perspective by extracting the two-particle vertex functions of DFT+DMFT solution $\Gamma_{loc}^{irr}$. The polarization bubble $\chi^{0}$ is computed from the fully interacting one particle Greens function. The full susceptibility is computed from $\chi^{0}$ and the two-particle irreducible vertex function $\Gamma_{loc}^{irr}$, which is assumed to be local in the same basis in which the DMFT self-energy is local, implemented here by the projector to the muffin-tin sphere [@Haule]. In order to extract $\Gamma_{loc}^{irr}$, we employ the Bethe-Salpeter equation (see Fig. \[fig:BSE\]) which relates the local two-particle Green’s function ($\chi_{loc}$), sampled by CTQMC, with both the local polarization function ($\chi_{loc}^{0}$) and $\Gamma_{loc}^{irr}$: $$\Gamma_{loc{\alpha_{1}\sigma_{1},\alpha_{2}\sigma_{2}\atop \alpha_{3}\sigma_{3},\alpha_{4}\sigma_{4}}}^{irr}(i\nu,i\nu^{\prime})_{i\omega}=\frac{1}{T}[(\chi_{loc}^{0})_{i\omega}^{-1}-\chi_{loc}^{-1}].$$ $\Gamma_{loc}^{irr}$ depends on three Matsubara frequencies ($i\nu$, $i\nu^{\prime}$; $i\omega$), and both the spin ($\sigma_{1-4}$) and the orbital ($\alpha_{1-4}$) indices, which run over $3d$ states on the iron atom. $T$ is the temperature. Once the irreducible vertex $\Gamma_{loc}^{irr}$ is obtained, the momentum dependent two-particle Green’s function is constructed again using the Bethe-Salpeter equation (Fig. \[fig:BSE\]) by replacing the local polarization function $\chi_{loc}^{0}$ by the non-local one $\chi_{\textbf{q},i\omega}^{0}$: $$\chi_{{\alpha_{1}\sigma_{1},\alpha_{2}\sigma_{2}\atop \alpha_{3}\sigma_{3},\alpha_{4}\sigma_{4}}}(i\nu,i\nu^{\prime})_{\textbf{q},i\omega}=[(\chi^{0})_{\textbf{q},i\omega}^{-1}-T\cdot\Gamma_{loc}^{irr}]^{-1}. \label{eq:BSE_imag}$$ Finally, the dynamic magnetic susceptibility $\chi(\textbf{q},i\omega)$ is obtained by closing the two particle green’s function with the magnetic moment $\mu=\mu_B(\textbf{L}+2\textbf{S})$ vertex, and summing over all internal degrees of freedom, i.e., orbitals ($\alpha_{1-4}$), spins ($\sigma_{1-4}$) and frequencies ($i\nu$,$i\nu^{\prime}$), on the four external legs $$\chi(\textbf{q},i\omega)=T\sum_{i\nu,i\nu^{\prime}}\sum_{{\alpha_{1}\alpha_{2}\atop\alpha_{3}\alpha_{4}}}\sum_{{\sigma_{1}\sigma_{2}\atop \sigma_{3}\sigma_{4}}}\mu_{{\alpha_{1}\sigma_{1}\atop\alpha_{3}\sigma_{3}}}^{z} \mu_{{\alpha_{2}\sigma_{2}\atop\alpha_{4}\sigma_{4}}}^{z} \;\chi_{{\alpha_{1}\sigma_{1},\alpha_{2}\sigma_{2}\atop \alpha_{3}\sigma_{3},\alpha_{4}\sigma_{4}}}(i\nu,i\nu^{\prime})_{\textbf{q},i\omega} \label{eq:chi}$$ We note that the same abinitio methodology which is here used to compute the magnetic excitation spectra, was previously shown to describe the photoemission, the optical spectra and the magnetic moments of this material [@Yin:11] in excellent agreement with experiment. Chen, Y. C., Lu, X. Y., Wang, M., Luo, H. Q., & Li, S. L., Systematic growth of BaFe$_{2-x}$Ni$_x$As$_2$ large crystals. Supercond. Sci. Technol. **24**, 065004 (2011). Bud’ko, S. L., Ni, N., & Canfield, P. C., Jump in specific heat at the superconducting transition temperature in Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ and Ba(Fe$_{1-x}$Ni$_x$)$_2$As$_2$ single crystals. Phys. Rev. B [**79**]{}, 220516(R) (2009). Haule, K., Yee, C.-H., Kim, K., Dynamical mean-field theory within the full-potential methods: Electronic structure of CeIrIn$_{5}$ , CeCoIn$_{5}$ , and CeRhIn$_{5}$. Phys. Rev. B [**81**]{}, 195107 (2010). Blaha, P., Schwarz, K., Madsen, G. K. H., Kvasnicka, K., & Luitz, J., Wien2K, Karlheinz Schwarz, Technische Universitat Wien, Austria (2001). Haule, K., Quantum Monte Carlo impurity solver for cluster dynamical mean-field theory and electronic structure calculations with adjustable cluster base. Phys. Rev. B [**75**]{}, 155113 (2007). Werner, P., Comanac, A., de’ Medici, L., Troyer, M., & Millis, A. J., Continuous-Time Solver for Quantum Impurity Models. Phys. Rev. Lett. [**97**]{}, 076405 (2006). Kutepov, A., Haule, K., Savrasov, S. Y., & Kotliar, G., Self-consistent $GW$ determination of the interaction strength: Application to the iron arsenide superconductors. Phys. Rev. B [**82**]{}, 045105 (2010). Yin, Z. P., Haule, K., & Kotliar, G., Magnetism and charge dynamics in iron pnictides. Nature Physics [**7**]{}, 294-297 (2011).

--- abstract: | Some known results regarding the Euler and Navier-Stokes equations were obtained by different authors. Existence and smoothness of the Navier-Stokes solutions in two dimensions have been known for a long time. Leray $\cite{jL34}$ showed that the Navier-Stokes equations in three space dimensions have a weak solution. Scheffer $\cite{vS76}, \cite{vS93}$ and Shnirelman $\cite{aS97}$ obtained weak solution of the Euler equations with compact support in spacetime. Caffarelli-Kohn-Nirenberg $\cite{CKN82}$ improved Scheffer’s results , and F.-H. Lin $\cite{fL98}$ simplified the proof of the results of J. Leray. Many problems and conjectures about the behavior of solutions of the Euler and Navier-Stokes equations are described in the book of Bertozzi and Majda $\cite{BM02}$ or Constantin $\cite{pC01}$. Solutions of the Navier-Stokes and Euler equations with initial conditions (Cauchy problem) for two and three dimensions are obtained in the convergence series form by the iterative method using the Fourier and Laplace transforms in this paper. For several combinations of problem parameters numerical results were obtained and presented as graphs. author: - 'A. Tsionskiy, M. Tsionskiy [^1]' title: 'Solution of the Cauchy problem for the Navier - Stokes and Euler equations' --- The mathematical setup ======================   The Navier-Stokes equations describe the motion of a fluid in $R^{N} ( N=2\;\rm{or}\; 3 ) $. We look for a viscous incompressible fluid filling all of $R^{N}$ here. The Navier-Stokes equations are then given by $$\label{eqn1} \frac{\partial u_{k}}{\partial t} \; + \; \sum_{n=1}^{N} u_{n}\frac{\partial u_{k}}{\partial x_{n}}\; =\;\nu\Delta u_{k}\; - \; \frac{\partial p}{\partial x_{k}} \; + \;f_{k}(x,t)\;\;\;\;\;(x\in R^{N},\;\;t\geq 0,\;\;{1\leq k \leq N})$$ $$\label{eqn2} \emph{div}\,\vec{u}\;=\; \sum_{n=1}^{N} \frac{\partial u_{n}}{\partial x_{n}}\; =\;0\;\;\;\;\;\;\;\;\;\; (x\in R^{N},t\geq 0)$$ with initial conditions $$\label{eqn3} \vec{u}(x,0)\; = \; \vec{u}^{0}(x)\;\;\;\;\;\;\;\;\;\; (x\in R^{N})$$ Here $\vec{u}(x,t)=(u_{k}(x,t)) \in R^{N},\;\; ({1\leq k \leq N}) \;-\; $is an unknown velocity vector $( N = 2\; \rm{or}\; 3 ),\; p\,(x,t)\;-\;$ is an unknown pressure, $\vec{u}^{0}(x)\;$ is a given, $C^{\infty}$ divergence-free vector field $,\; f_{k}(x,t)\;$are components of a given, externally applied force $\vec{f}(x,t)$, $\nu$ is a positive coefficient of the viscosity (if $\nu = 0$ then (1.1) - (1.3) are the Euler equations), and $ \Delta\;=\; \sum_{n=1}^{N} \frac{\partial^{2}}{\partial x_{n}^{2}}\;$ is the Laplacian in the space variables. Equation $(\ref{eqn1})$ is Newton’s law for a fluid element subject. Equation $(\ref{eqn2})$ says that the fluid is incompressible. For physically reasonable solutions, we accept $$\label{eqn4} u_{k}(x,t) \rightarrow 0\;\;, \;\; \frac{\partial u_{k}}{\partial x_{n}} \;\rightarrow\;0\;\; \rm {as} \;\;\mid x \mid \;\rightarrow\; \infty\;\;\;( {1\leq k \leq N} ,\;\; {1\leq n \leq N}) \;\;\;$$ Hence, we will restrict attention to initial conditions $\vec{u}^{0}$ and force $\vec{f}$ that satisfy $$\label{eqn5} \mid\partial_{x}^{\alpha}\vec{u}^{0}(x)\mid\;\leq\;C_{\alpha K}(1+\mid x \mid)^{-K} \quad \rm{on }\;R^{N}\;\rm{ for\; any }\;\alpha \;\rm{ and }\;K>0.$$ and $$\label{eqn6} \mid\partial_{x}^{\alpha}\partial_{t}^{\beta}\vec{f}(x,t)\mid\;\leq\;C_{\alpha \beta K}(1+\mid x \mid +t)^{-K} \quad \rm{on }\;R^{N}\times[0,\infty)\; \rm{ for \;any }\;\alpha,\beta \;\rm{ and }\;K>0.$$ We add ($ - \sum_{n=1}^{N} u_{n}\frac{\partial u_{k}}{\partial x_{n}}\;$) to both sides of the equations (\[eqn1\]). Then we have: $$\label{eqn7} \frac{\partial u_{k}}{\partial t}\;=\;\nu\,\Delta\,u_{k}\;-\;\frac{\partial p}{\partial x_{k}}\;+\;f_{k}(x,t)- \; \sum_{n=1}^{N} u_{n}\frac{\partial u_{k}}{\partial x_{n}}\;\;\;\;\;\;\;\;\;\;\;\; (x\in R^{N},\;\;t\geq 0,\;\;{1\leq k \leq N})$$ $$\label{eqn8} \emph{div}\,\vec{u}\;=\; \sum_{n=1}^{N} \frac{\partial u_{n}}{\partial x_{n}}\; =\;0\;\;\;\;\;\;\;\;\;\; (x\in R^{N},t\geq 0)$$ $$\label{eqn9} \vec{u}(x,0)\; = \; \vec{u}^{0}(x)\;\;\;\;\;\;\;\;\;\; (x\in R^{N})$$ $$\label{eqn10} u_{k}(x,t) \rightarrow 0\;\;, \;\;\frac{\partial u_{k}}{\partial x_{n}}\;\rightarrow\;0\;\; \rm{as} \;\;\mid x \mid \;\rightarrow\; \infty\;\;\;( {1\leq k \leq N} ,\;\; {1\leq n \leq N}) \;\;\;$$ $$\label{eqn11} \mid\partial_{x}^{\alpha}\vec{u}^{0}(x)\mid\;\leq\;C_{\alpha K}(1+\mid x \mid)^{-K} \quad\rm{on }\;R^{N}\;\rm{ for \; any }\;\alpha\;\rm{ and }\;K>0.$$ $$\label{eqn12} \mid\partial_{x}^{\alpha}\partial_{t}^{\beta}\vec{f}(x,t)\mid\;\leq\;C_{\alpha \beta K}(1+\mid x \mid +t)^{-K} \quad\rm{on }\;R^{N}\times[0,\infty)\;\rm{ for \; any }\;\alpha,\beta\;\rm{ and }\;K>0.$$ We shall solve the system of equations (\[eqn7\]) - (\[eqn12\]) by the iterative method. To do so we write this system of equations in the following form: $$\label{eqn13} \frac{\partial u_{jk}}{\partial t}\;=\;\nu\,\Delta\,u_{jk}\;-\;\frac{\partial p_{j}}{\partial x_{k}}\;+\;f_{jk}(x,t) \;\;\;\;\;\;\;\;\;\;\;\; (x\in R^{N},\;\;t\geq 0,\;\;{1\leq k \leq N})$$ $$\label{eqn14} \emph{div}\,\vec{u}_{j}\;=\; \sum_{n=1}^{N} \frac{\partial u_{jn}}{\partial x_{n}}\; =\;0\;\;\;\;\;\;\;\;\;\; (x\in R^{N},t\geq 0)$$ $$\label{eqn15} \vec{u}_{j}(x,0)\; = \; \vec{u}^{0}(x)\;\;\;\;\;\;\;\;\;\; (x\in R^{N})$$ $$\label{eqn16} u_{j k}(x,t) \rightarrow 0\;\;, \;\;\frac{\partial u_{j k}}{\partial x_{n}}\;\rightarrow\;0\;\; \rm{as} \;\;\mid x \mid \;\rightarrow\; \infty\;\;\;( {1\leq k \leq N} ,\;\; {1\leq n \leq N}) \;\;\;$$ $$\label{eqn17} \mid\partial_{x}^{\alpha}\vec{u}^{0}(x)\mid\;\leq\;C_{\alpha K}(1+\mid x \mid)^{-K} \quad\rm{on }\;R^{N}\;\rm{ for \; any }\;\alpha\;,\;K>0\;\;\;\rm{ and } \;\;C_{\alpha K} > 0.$$ $$\label{eqn18} \mid\partial_{x}^{\alpha}\partial_{t}^{\beta}\vec{f}(x,t)\mid\;\leq\;C_{\alpha \beta K}(1+\mid x \mid +t)^{-K} \quad\rm{on }\;R^{N}\times[0,\infty)\;\rm{ for \; any }\;\alpha,\beta\;,\;K>0\;\;\rm{ and } \;\;C_{\alpha \beta K} > 0.$$ Here j is the number of the iterative process step (j = 1,2,3,...). $$\label{eqn19} f_{jk}(x,t)\; = \; f_{k}(x,t) \; - \; \sum_{n=1}^{N} u_{j-1,n}\frac{\partial u_{j-1,k}}{\partial x_{n}}\;\;\;\;\;\;({1\leq k \leq N})$$ or the vector form $$\label{eqn20} \vec{f}_{j}(x,t)\; = \; \vec{f}(x,t) \; - \;(\;\vec{u}_{j-1}\;\cdot\;\nabla\;)\;\vec{u}_{j-1}\;$$ \ For the first step of the iterative process (j = 1) we have: $$\\(\vec{u}_{0}\;\cdot\;\nabla\;)\;\vec{u}_{0}\;=\;0$$ and $$\\\vec{f}_{1}(x,t)\; = \; \vec{f}(x,t)$$ Solution. Case N = 2 ====================   We use Fourier transform (\[A2\]) for equations $(\ref{eqn13})\; - \;(\ref{eqn20})$ and get: $$\\U_{jk}( \gamma_{1},\gamma_{2},t)\;=\;F [u_{jk} ( x_{1}, x_{2},t)]$$ $$\\ F {\mbox{\Large [ \normalsize}} \frac{\partial^{2}u_{jk}( x_{1}, x_{2},t)}{\partial x^{2}_{s}}\mbox{\Large ]\normalsize}\;=\;-\gamma^{2}_{s}U_{jk}( \gamma_{1},\gamma_{2},t) \;\;\;\;\; \rm{[use (\ref{eqn16})]}$$ $$\\U_{k}^{0}( \gamma_{1},\gamma_{2})\;=\;F[u_{k}^{0} ( x_{1}, x_{2})]$$ $$\\P_{j}( \gamma_{1},\gamma_{2},t)\;=\;F[p_{j}\, ( x_{1}, x_{2},t)]$$ $$\\F_{jk}( \gamma_{1},\gamma_{2},t)\;=\;F[f_{jk} ( x_{1}, x_{2},t)]$$ $$\\ k,s\;=\;1,2$$ and then: $$\label{eqn21} \frac{\partial U_{j1}( \gamma_{1},\gamma_{2},t )}{ \partial t} \;=\;-\nu ( \gamma_{1}^{2}+\gamma_{2}^{2}) U_{j1}( \gamma_{1},\gamma_{2},t )\;+\;i\gamma_{1} P_{j}( \gamma_{1},\gamma_{2},t )\;+\; F_{j1}( \gamma_{1},\gamma_{2},t )$$ $$\label{eqn22} \frac{\partial U_{j2}( \gamma_{1},\gamma_{2},t )}{ \partial t} \;=\;-\nu ( \gamma_{1}^{2}+\gamma_{2}^{2}) U_{j2}( \gamma_{1},\gamma_{2},t )\;+\;i\gamma_{2} P_{j}( \gamma_{1},\gamma_{2},t )\;+\; F_{j2}( \gamma_{1},\gamma_{2},t )$$ $$\label{eqn23} \gamma_{1} U_{j1}( \gamma_{1},\gamma_{2},t )\;+\; \gamma_{2} U_{j2}( \gamma_{1},\gamma_{2},t )\;=\;0$$ $$\label{eqn24} U_{j1}(\gamma_{1},\gamma_{2},0)\;=\; U_{1}^{0}(\gamma_{1},\gamma_{2})$$ $$\label{eqn25} U_{j2}(\gamma_{1},\gamma_{2},0)\;=\; U_{2}^{0}(\gamma_{1},\gamma_{2})$$ Hence eliminate $P_{j}(\gamma_{1},\gamma_{2},t)$ from equations $(\ref{eqn21})$, $(\ref{eqn22})$ and find: $$\begin{aligned} \label{eqn26} \frac{\partial }{\partial t} {\mbox{\large [ \normalsize}} U_{j2}( \gamma_{1},\gamma_{2},t )\; -\;\frac{\gamma_{2}}{\gamma_{1}} U_{j1}( \gamma_{1},\gamma_{2},t) {\mbox{\large ] \normalsize}} \;= \quad\quad\quad\quad\quad\quad \quad\quad \quad \nonumber \\ \nonumber \\ -\nu( \gamma_{1}^{2}+\gamma_{2}^{2}) {\mbox{\large [ \normalsize}} U_{j2}( \gamma_{1},\gamma_{2},t )\; -\;\frac{\gamma_{2}}{\gamma_{1}} U_{j1}( \gamma_{1},\gamma_{2},t) {\mbox{\large ] \normalsize}} \;+ \;{\mbox{\large [ \normalsize}} F_{j2}( \gamma_{1},\gamma_{2},t )\; -\;\frac{\gamma_{2}}{\gamma_{1}} F_{j1}( \gamma_{1},\gamma_{2},t) {\mbox{\large ] \normalsize}} \end{aligned}$$ $$\label{eqn27} \gamma_{1} U_{j1}( \gamma_{1},\gamma_{2},t )\;+\; \gamma_{2} U_{j2}( \gamma_{1},\gamma_{2},t )\;=\;0$$ $$\label{eqn28} U_{j1}(\gamma_{1},\gamma_{2},0)\;=\; U_{1}^{0}(\gamma_{1},\gamma_{2})$$ $$\label{eqn29} U_{j2}(\gamma_{1},\gamma_{2},0)\;=\; U_{2}^{0}(\gamma_{1},\gamma_{2})$$ We use Laplace transform ($\ref{A4}$), ($\ref{A5}$) for equations ($\ref{eqn26}$), ($\ref{eqn27}$) and have: $$\\U_{jk}^{\otimes} (\gamma_{1},\gamma_{2},\eta) \;=\;L[\,U_{jk}(\gamma_{1},\gamma_{2},t)\,] \;\;\;\;\;\;\; \rm{k=1,2}$$ $$\begin{aligned} \label{eqn30} \eta \,{\mbox{\large [ \normalsize}} U_{j2}^{\otimes}( \gamma_{1},\gamma_{2},\eta )\; -\;\frac{\gamma_{2}}{\gamma_{1}}\, U_{j1}^{\otimes}( \gamma_{1},\gamma_{2},\eta) {\mbox{\large ] \normalsize}} \;-\; {\mbox{\large [ \normalsize}} U_{j2}( \gamma_{1},\gamma_{2},0 )\; -\;\frac{\gamma_{2}}{\gamma_{1}}\, U_{j1}( \gamma_{1},\gamma_{2},0) {\mbox{\large ] \normalsize}} \;= \nonumber\\ \nonumber\\ -\nu\,( \gamma_{1}^{2}+\gamma_{2}^{2}){\mbox{\large [ \normalsize}} U_{j2}^{\otimes} ( \gamma_{1},\gamma_{2},\eta )\; -\;\frac{\gamma_{2}}{\gamma_{1}}\, U_{j1}^{\otimes}( \gamma_{1},\gamma_{2},\eta) {\mbox{\large ] \normalsize}} \;+\; {\mbox{\large [ \normalsize}} F_{j2}^{\otimes}( \gamma_{1},\gamma_{2},\eta )\; -\;\frac{\gamma_{2}}{\gamma_{1}} \,F_{j1}^{\otimes}( \gamma_{1},\gamma_{2},\eta) {\mbox{\large ] \normalsize}} \end{aligned}$$ $$\label{eqn31} \gamma_{1} U_{j1}^{\otimes}( \gamma_{1},\gamma_{2},\eta )\;+\; \gamma_{2} U_{j2}^{\otimes} ( \gamma_{1},\gamma_{2},\eta )\;=\;0$$ $$\label{eqn32} U_{j1}(\gamma_{1},\gamma_{2},0)\;=\; U_{1}^{0}(\gamma_{1},\gamma_{2})$$ $$\label{eqn33} U_{j2}(\gamma_{1},\gamma_{2},0)\;=\; U_{2}^{0}(\gamma_{1},\gamma_{2})$$ The solution of the system of equations $(\ref{eqn30})\;-\; (\ref{eqn33})\;$ is: $$\label{eqn34} U_{j1}^{\otimes}( \gamma_{1},\gamma_{2},\eta )\;=\;\frac{[\gamma_{2}^{2} F_{j1}^{\otimes}( \gamma_{1},\gamma_{2},\eta)- \gamma_{1}\gamma_{2} F_{j2}^{\otimes}( \gamma_{1},\gamma_{2},\eta)+\gamma_{2}^{2} U_{1}^{0}(\gamma_{1},\gamma_{2}) -\gamma_{1}\gamma_{2} U_{2}^{0}(\gamma_{1},\gamma_{2})]}{ (\gamma_{1}^{2}+\gamma_{2}^{2}) [\eta+\nu (\gamma_{1}^{2}+\gamma_{2}^{2})]}$$ $$\label{eqn35} U_{j2}^{\otimes}( \gamma_{1},\gamma_{2},\eta )\;=\;\frac{[\gamma_{1}^{2} F_{j2}^{\otimes}( \gamma_{1},\gamma_{2},\eta)- \gamma_{1}\gamma_{2} F_{j1}^{\otimes}( \gamma_{1},\gamma_{2},\eta)+\gamma_{1}^{2} U_{2}^{0}(\gamma_{1},\gamma_{2}) -\gamma_{1}\gamma_{2} U_{1}^{0}(\gamma_{1},\gamma_{2})]}{ (\gamma_{1}^{2}+\gamma_{2}^{2}) [\eta+\nu (\gamma_{1}^{2}+\gamma_{2}^{2})]}$$ Then we use the convolution formula $(\ref{A6})$ and integral $(\ref{A7})$ for $(\ref{eqn34}) \;,\; (\ref{eqn35})\;$ and obtain: $$\begin{aligned} \label{eqn36} U_{j1}(\gamma_{1},\gamma_{2},t)\;=\; \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2}+\gamma_{2}^{2}) (t-\tau)} \frac{[\gamma_{2}^{2} F_{j1}( \gamma_{1}, \gamma_{2},\tau)-\gamma_{1}\gamma_{2} F_{j2}( \gamma_{1}, \gamma_{2},\tau )]}{ (\gamma_{1}^{2}+\gamma_{2}^{2}) }\,d\tau \;+ \nonumber\\ \nonumber\\ +\;{\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2}+\gamma_{2}^{2}) t} \;U_{1}^{0}(\gamma_{1},\gamma_{2}) \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \end{aligned}$$ $$\begin{aligned} \label{eqn37} U_{j2}(\gamma_{1},\gamma_{2},t)\;=\;\int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2}+\gamma_{2}^{2}) (t-\tau)} \frac{ [\gamma_{1}^{2} F_{j2}( \gamma_{1}, \gamma_{2},\tau)-\gamma_{1}\gamma_{2} F_{j1}( \gamma_{1}, \gamma_{2},\tau )] }{ (\gamma_{1}^{2}+\gamma_{2}^{2}) }\,d\tau\;+ \nonumber\\ \nonumber\\ +\; {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2}+\gamma_{2}^{2}) t} \;U_{2}^{0}(\gamma_{1},\gamma_{2}) \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \end{aligned}$$ \ $ P_{j}(\gamma_{1},\gamma_{2},t) $ is obtained from $(\ref{eqn21})\;$or$\; (\ref{eqn22})\;$: $$\label{eqn38} P_{j}(\gamma_{1},\gamma_{2},t) \;=\;i\frac{[\gamma_{1} F_{j1}( \gamma_{1}, \gamma_{2},t)+\gamma_{2} F_{j2}( \gamma_{1}, \gamma_{2},t )]}{ (\gamma_{1}^{2}+\gamma_{2}^{2}) }$$\ Use of the Fourier inversion formula $(\ref{A2})$ and find: $$\begin{aligned} \label{eqn39} u_{j1}(x_{1},x_{2},t)\;=\;\frac{1}{2\pi} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2}+\gamma_{2}^{2}) (t-\tau)} \frac{[\gamma_{2}^{2} F_{j1}( \gamma_{1}, \gamma_{2},\tau)-\gamma_{1}\gamma_{2} F_{j2}( \gamma_{1}, \gamma_{2},\tau )]}{ (\gamma_{1}^{2}+\gamma_{2}^{2}) } \,d\tau\;+ \nonumber\\ \nonumber\\ \nonumber\\ +\; {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2}+\gamma_{2}^{2}) t} \;U_{1}^{0}(\gamma_{1},\gamma_{2})\biggr] \;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2}\;= \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \nonumber\\ \nonumber\\ \nonumber\\ = \;\frac{1}{4\pi^2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\gamma_{2}^{2} }{(\gamma_{1}^{2}+\gamma_{2}^{2}) } \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2}+\gamma_{2}^{2}) (t-\tau)} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2})} f_{j1}(\tilde x_{1},\tilde x_{2},\tau)\,d\tilde x_{1}d\tilde x_{2}d\tau\cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2}\;- \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \nonumber\\ \nonumber\\ \nonumber\\ - \;\frac{1}{4\pi^2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\gamma_{1} \gamma_{2} }{(\gamma_{1}^{2}+\gamma_{2}^{2}) } \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2}+\gamma_{2}^{2}) (t-\tau)} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2})} f_{j2}(\tilde x_{1},\tilde x_{2},\tau)\,d\tilde x_{1}d\tilde x_{2}d\tau\cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2}\;+ \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \nonumber\\ \nonumber\\ \nonumber\\ + \;\frac{1}{4\pi^2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2}+\gamma_{2}^{2}) t} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2})} u^{0}_{1}(\tilde x_{1},\tilde x_{2})\,d\tilde x_{1}d\tilde x_{2}{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2}\;= \nonumber\\ \nonumber\\ \nonumber\\ = \;S_{11}(f_{j1})\;+\; S_{12}(f_{j2})\;+\;B(u^{0}_{1}) \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \end{aligned}$$ $$\begin{aligned} \label{eqn40} u_{j2} (x_{1},x_{2},t)\;=\;\frac{1}{2\pi} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2}+\gamma_{2}^{2}) (t-\tau)} \frac{[\gamma_{1}^{2} F_{j2}( \gamma_{1}, \gamma_{2},\tau)-\gamma_{1}\gamma_{2} F_{j1}( \gamma_{1}, \gamma_{2},\tau )]}{ (\gamma_{1}^{2}+\gamma_{2}^{2}) }\,d\tau\;+ \nonumber\\ \nonumber\\ \nonumber\\ +\; {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2}+\gamma_{2}^{2}) t} \;U_{2}^{0}(\gamma_{1},\gamma_{2})\biggr] \;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2}\;= \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \nonumber\\ \nonumber\\ \nonumber\\ = \;-\;\frac{1}{4\pi^2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\gamma_{1} \gamma_{2} }{(\gamma_{1}^{2}+\gamma_{2}^{2}) } \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2}+\gamma_{2}^{2}) (t-\tau)} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2})} f_{j1}(\tilde x_{1},\tilde x_{2},\tau)\,d\tilde x_{1}d\tilde x_{2}d\tau\cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2}\;+ \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \nonumber\\ \nonumber\\ \nonumber\\ + \;\frac{1}{4\pi^2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\gamma_{1}^{2} }{(\gamma_{1}^{2}+\gamma_{2}^{2}) } \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2}+\gamma_{2}^{2}) (t-\tau)} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2})} f_{j2}(\tilde x_{1},\tilde x_{2},\tau)\,d\tilde x_{1}d\tilde x_{2}d\tau\cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2}\;+ \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \nonumber\\ \nonumber\\ \nonumber\\ + \;\frac{1}{4\pi^2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2}+\gamma_{2}^{2}) t} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2})} u^{0}_{2}(\tilde x_{1},\tilde x_{2})\,d\tilde x_{1}d\tilde x_{2}{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2}\;= \nonumber\\ \nonumber\\ \nonumber\\ = \;S_{21}(f_{j1})\;+\; S_{22}(f_{j2})\;+\;B(u^{0}_{2}) \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \end{aligned}$$ $$\begin{aligned} \label{eqn41} p_{j}\,(x_{1},x_{2},t)\;=\; \frac{i}{2\pi} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{[\gamma_{1} F_{j1}( \gamma_{1}, \gamma_{2},t)+ \gamma_{2} F_{j2}( \gamma_{1}, \gamma_{2},t )]}{ (\gamma_{1}^{2}+\gamma_{2}^{2}) }\;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2}\;=\; \nonumber\\ \nonumber\\ \nonumber\\ = \;-\;\frac{i}{4\pi^2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\gamma_{1}}{(\gamma_{1}^{2}+\gamma_{2}^{2}) } \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2})} f_{j1}(\tilde x_{1},\tilde x_{2},t)\,d\tilde x_{1}d\tilde x_{2}\; {\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2}\;+ \nonumber\\ \nonumber\\ \nonumber\\ + \;\frac{i}{4\pi^2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\gamma_{2}}{(\gamma_{1}^{2}+\gamma_{2}^{2}) } \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2})} f_{j2}(\tilde x_{1},\tilde x_{2},t)\,d\tilde x_{1}d\tilde x_{2}\; {\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2}\;= \nonumber\\ \nonumber\\ =\;\tilde S_{1}(f_{j1})\;+\; \tilde S_{2}(f_{j2}) \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad\quad\quad\quad\quad\quad\quad \quad\quad \end{aligned}$$ So, the integrals $(\ref{eqn39})\;-\; (\ref{eqn41})\;$exist by the restrictions $(\ref{eqn17})\;, (\ref{eqn18})\;$. Here $S_{11}(), S_{12}(), S_{21}(), S_{22}(), B(), \tilde S_{1}(), \tilde S_{2}()$ are the integral - operators. $$S_{12}()\;= \;S_{21}()$$ We have for the vector $\vec{u}_{j}$ from the equations $(\ref{eqn39})\;-\; (\ref{eqn40})\;$: $$\label{eqn42} \vec{u}_{j}\;=\;\bar{\bar{S}}\;\cdot\;\vec{f}_{j}\;+\;B(\vec{u}^{0})\;,$$ where $\;\bar{\bar{S}} \; $ is the matrix - operator: $$\left( \begin{array}{ccc} S_{11} & S_{12} \\ S_{21} & S_{22} \end{array} \right)$$ We put $\vec{f}_{j}$ from equation $(\ref{eqn20})$ into equation $(\ref{eqn42})$ and have: $$\begin{aligned} \label{eqn44} \vec{u}_{j} = \bar{\bar{S}}\cdot(\;\vec{f}\;-\;(\;\vec{u}_{j-1}\cdot\nabla)\vec{u}_{j-1})\;+\;B(\vec{u}^{0})\;= \nonumber\\ \nonumber\\ =\;\bar{\bar{S}}\cdot\vec{f} \;-\;\bar{\bar{S}}\cdot(\vec{u}_{j-1}\;\cdot\;\nabla\;)\;\vec{u}_{j-1}\;+\;B(\vec{u}^{0})\; = \nonumber\\ \nonumber\\ =\;\vec{u}_{1}\;-\;\bar{\bar{S}}\cdot(\vec{u}_{j-1}\;\cdot\;\nabla)\;\vec{u}_{j-1} \quad\quad\quad\quad\quad\quad \quad\quad \end{aligned}$$ Here $\vec{u}_{1}\;$ is the solution of the system of equations $(\ref{eqn13})\; - \;(\ref{eqn20})$ with condition: $$\sum_{n=1}^{2} u_{n}\frac{\partial u_{k}}{\partial x_{n}}\;=\;0\;\;\;\;\;\;\; \rm{k=1,2}\;\;\;$$ For j = 1 formula $(\ref{eqn42})$ can be written as follows: $$\label{eqn47} \vec{u}_{1}\;=\;\bar{\bar{S}}\;\cdot\;\vec{f}_{1}\;+\;B(\vec{u}^{0})\;,\;\;\;\;\;\\\vec{f}_{1}(x,t)\; = \; \vec{f}(x,t)$$ If t $\rightarrow$ 0 then $\vec{u}_{1} \rightarrow \vec{u}^{0}$ (look at integral-operators $\bar{\bar{S}}, B()\;\;$ - integrals $\;(\ref{eqn39})\; , \;(\ref{eqn40})$). For j = 2 we define from equation $(\ref{eqn20})$: $$\vec{f}_{2}(x,t)\; = \; \vec{f}_{1}(x,t) \; - \;(\;\vec{u}_{1}\;\cdot\;\nabla\;)\;\vec{u}_{1}\;$$ We denote: $$\label{eqn48} \vec{f}_{2}^{*}\;=\;(\vec{u}_{1}\;\cdot\;\nabla)\;\vec{u}_{1}$$ and then we have: $$\vec{f}_{2}(x,t)\; = \; \vec{f}_{1}(x,t) \; - \vec{f}_{2}^{*}$$ Then we get $\vec{u}_{2}$ from $(\ref{eqn42}),(\ref{eqn47})$: $$\label{eqn50} \vec{u}_{2}\;=\;\bar{\bar{S}}\;\cdot\;\vec{f}_{2} \;+\;B(\vec{u}^{0})\;=\;\bar{\bar{S}}\;\cdot\;(\vec{f}_{1}\;-\;\vec{f}_{2}^{*}) \;+\;B(\vec{u}^{0})\;=\;\vec{u}_{1}\;-\;\vec{u}_{2}^{*}$$ Here we have: $$\label{eqn49} \vec{u}_{2}^{*}\;=\;\bar{\bar{S}}\;\cdot\;\vec{f}_{2}^{*}$$ If t $\rightarrow$ 0 then $\vec{u}_{2}^{*} \rightarrow$ 0 (look at integral-operator $\bar{\bar{S}}\;\;$- integrals $\;(\ref{eqn39})\; , \;(\ref{eqn40})$). Continue for j = 3. We define from equation $(\ref{eqn20})$: $$\vec{f}_{3}(x,t)\; = \; \vec{f}_{1}(x,t) \; - \;(\;\vec{u}_{2}\;\cdot\;\nabla\;)\;\vec{u}_{2}\;$$ Here we have: $$\label{eqn51} (\vec{u}_{2}\;\cdot\;\nabla)\;\vec{u}_{2}\;= \;((\vec{u}_{1}\;-\;\vec{u}_{2}^{*})\;\cdot\;\nabla\;) \;(\vec{u}_{1}\;-\;\vec{u}_{2}^{*})\;= \;\vec{f}_{2}^{*}\;+\;\vec{f}_{3}^{*}$$ We denote in $(\ref{eqn51})$: $$\vec{f}_{3}^{*}\;=\;-\; (\vec{u}_{1}\;\cdot\;\nabla)\;\vec{u}_{2}^{*}\; -\; (\vec{u}_{2}^{*}\;\cdot\;\nabla)\;\vec{u}_{1}\;+\; (\vec{u}_{2}^{*}\;\cdot\;\nabla)\;\vec{u}_{2}^{*}$$ and then we have: $$\vec{f}_{3}(x,t)\; = \; \vec{f}_{1}(x,t) \; - \vec{f}_{2}^{*}\; - \vec{f}_{3}^{*}$$ Then we get $\vec{u}_{3}$ from $(\ref{eqn42}) , (\ref{eqn47}) , (\ref{eqn49})$: $$\label{eqn53} \vec{u}_{3}\;=\;\bar{\bar{S}}\;\cdot\;\vec{f}_{3} \;+\;B(\vec{u}^{0})\;=\;\bar{\bar{S}}\;\cdot\;(\vec{f}_{1} \;-\;\vec{f}_{2}^{*}\;-\;\vec{f}_{3}^{*})\;+\;B(\vec{u}^{0})\;=\;\vec{u}_{1} \;-\;\vec{u}_{2}^{*}\;-\;\vec{u}_{3}^{*}$$ Here we denote: $$\label{eqn52} \vec{u}_{3}^{*}\;=\;\bar{\bar{S}}\;\cdot\;\vec{f}_{3}^{*}$$ If t $\rightarrow$ 0 then $\vec{u}_{3}^{*} \rightarrow$ 0 (look at integral-operator $\bar{\bar{S}}\;\;$- integrals $\;(\ref{eqn39})\; , \;(\ref{eqn40})$). For j = 4. We define from equation $(\ref{eqn20})$: $$\vec{f}_{4}(x,t)\; = \; \vec{f}_{1}(x,t) \; - \;(\;\vec{u}_{3}\;\cdot\;\nabla\;)\;\vec{u}_{3}\;$$ Here we have: $$\label{eqn54} (\vec{u}_{3}\;\cdot\;\nabla)\;\vec{u}_{3}\;= \;((\vec{u}_{2}\;-\;\vec{u}_{3}^{*})\;\cdot\;\nabla\;) \;(\vec{u}_{2}\;-\;\vec{u}_{3}^{*})\;= \;\vec{f}_{2}^{*}\;+\;\vec{f}_{3}^{*}\;+\;\vec{f}_{4}^{*}$$ \ We denote in $(\ref{eqn54})$: $$\vec{f}_{4}^{*}\;=\;-\; (\vec{u}_{2}\;\cdot\;\nabla)\;\vec{u}_{3}^{*}\; -\; (\vec{u}_{3}^{*}\;\cdot\;\nabla)\;\vec{u}_{2}\;+\; (\vec{u}_{3}^{*}\;\cdot\;\nabla)\;\vec{u}_{3}^{*}$$ and then we have: $$\vec{f}_{4}(x,t)\; = \; \vec{f}_{1}(x,t) \; - \vec{f}_{2}^{*}\; - \vec{f}_{3}^{*}\; - \vec{f}_{4}^{*}$$ Then we get $\vec{u}_{4}$ from $(\ref{eqn42}) , (\ref{eqn47}) , (\ref{eqn49}) , (\ref{eqn52})$: $$\label{eqn56} \vec{u}_{4}\;=\;\bar{\bar{S}}\;\cdot\;\vec{f}_{4}\;+\;B(\vec{u}^{0})\;=\;\bar{\bar{S}}\;\cdot\;(\vec{f}_{1} \; -\;\vec{f}_{2}^{*}\; -\;\vec{f}_{3}^{*}\;-\;\vec{f}_{4}^{*})\;+\;B(\vec{u}^{0})\;=\;\vec{u}_{1} \;-\;\vec{u}_{2}^{*}\;-\;\vec{u}_{3}^{*}\;-\;\vec{u}_{4}^{*}$$ Here we denote: $$\label{eqn55} \vec{u}_{4}^{*}\;=\;\bar{\bar{S}}\;\cdot\;\vec{f}_{4}^{*}$$ If t $\rightarrow$ 0 then $\vec{u}_{4}^{*} \rightarrow 0$ (look at integral-operator $\bar{\bar{S}}\;\;$- integrals $\;(\ref{eqn39})\; , \;(\ref{eqn40})$). For arbitrary number j $(j \geq 2)$. We define from equation $(\ref{eqn20})$: $$\vec{f}_{j}(x,t)\; = \; \vec{f}_{1}(x,t) \; - \;(\;\vec{u}_{j-1}\;\cdot\;\nabla\;)\;\vec{u}_{j-1}\;$$ Here we have: $$\label{eqn60} (\vec{u}_{j-1}\;\cdot\;\nabla)\;\vec{u}_{j-1}\;=\;\sum_{l=2}^{j} \vec{f}_{l}^{*}$$ and it follows: $$\label{eqn60a} \vec{f}_{j}\;=\;\vec{f}_{1}\;-\; \sum_{l=2}^{j} \vec{f}_{l}^{*}$$ Then we get $\vec{u}_{j}$ from $(\ref{eqn42}) , (\ref{eqn47}) $: $$\label{eqn57} \vec{u}_{j}\;=\;\bar{\bar{S}}\;\cdot\; \vec{f}_{j}\;+ \;B(\vec{u}^{0})\;=\;\bar{\bar{S}}\;\cdot\;( \vec{f}_{1}\;-\; \sum_{l=2}^{j} \vec{f}_{l}^{*})\;+ \;B(\vec{u}^{0})\;=\;\vec{u}_{1}\;-\;\sum_{l=2}^{j} \vec{u}_{l}^{*}$$ Here we denote: $$\label{eqn59} \vec{u}_{l}^{*}\;=\;\bar{\bar{S}}\;\cdot\;\vec{f}_{l}^{*}\;\;\;\;\;\;\;\;\;\;\;\;\;\;(2\;\leq\;l\;\leq\;j)$$ If t $\rightarrow$ 0 then $\vec{u}_{l}^{*} \rightarrow$ 0 (look at integral-operator $\bar{\bar{S}}\;\;$- integrals $\;(\ref{eqn39})\; , \;(\ref{eqn40})$). We consider the equations $(\ref{eqn47})$ - $(\ref{eqn59})$ and see that the series $(\ref{eqn57})$ converge for $j \rightarrow \infty$ with the conditions for the first step (j = 1) of the iterative process: $$\;\;\;\sum_{n=1}^{2} u_{0n}\frac{\partial u_{0k}}{\partial x_{n}} = 0\;\;\;\;\;\;\;\; \rm{k=1,2}$$ and conditions $$\label{eqn61} C_{\alpha K}\leq\;\frac{1}{2}\;\;,\;\;C_{\alpha \beta K}\leq\;\frac{1}{2}$$ Here $\;\;C_{\alpha K}\;\;$and$\;\;C_{\alpha \beta K}\;$ are received from $(\ref{eqn17})$ , $(\ref{eqn18}).$ Hence, we receive from equation $(\ref{eqn44})\;$ when $j \rightarrow \infty$: $$\label{eqn45} \vec{u}_{\infty}=\;\vec{u}_{1}\;-\;\bar{\bar{S}}\cdot(\vec{u}_{\infty}\;\cdot\;\nabla)\;\vec{u}_{\infty}$$ Equation $(\ref{eqn45})$ describes the converging iterative process. Then we have from formula $(\ref{eqn41})\;$: $$\label{eqn46a} p_{\infty}\,\;=\; \;\tilde S_{1}(f_{\infty 1})\;+\; \tilde S_{2}(f_{\infty 2})$$ Here $\vec{f}_{\infty}$ = ($f_{\infty 1} , f_{\infty 2}$) is received from formula $(\ref{eqn60a})\;$. On the other hand we can transform the original system of differential equations $(\ref{eqn7})\; - \;(\ref{eqn9})$ to the equivalent system of integral equations by the scheme of iterative process $(\ref{eqn42})\;, \;(\ref{eqn44})$ for vector $\vec{u}$: $$\label{eqn46} \vec{u}\;=\;\vec{u}_{1}\;-\;\bar{\bar{S}}\cdot(\vec{u}\;\cdot\;\nabla)\;\vec{u},$$ where $\vec{u}_{1}$ is from formula $(\ref{eqn47})$. We compare the equations $(\ref{eqn45})$ and $(\ref{eqn46})$ and see that the iterative process $(\ref{eqn45})$ converge to the solution of the system $(\ref{eqn46})$ and hence to the solution of the differential equations $(\ref{eqn7})\; - \;(\ref{eqn9})$ with conditions $(\ref{eqn61})$. **In other words there exist smooth functions** $\mathbf{p_{\infty}(x, t)}$, $\mathbf{u_{\infty i}(x, t)}$ **(i = 1, 2) on** $\mathbf{R^{2} \times [0,\infty)}$ **that satisfy** $\mathbf{(\ref{eqn1}), (\ref{eqn2}), (\ref{eqn3})}$ **and** $$\label{eqn46b} \mathbf{p_{\infty}, \;u_{\infty i} \in C^{\infty}(R^{2} \times [0,\infty)),} \nonumber\\ \nonumber\\$$ $$\label{eqn4aa} \mathbf{\int_{R^{2}}|\vec{u}_{\infty}(x, t)|^{2}dx < C }$$ **for all t** $\mathbf{\geq 0}$. Solution. Case N = 3 ====================   We use Fourier transform $(\ref{A3})$ for equations $(\ref{eqn13})\; - \;(\ref{eqn20})$ and get: $$\\U_{jk}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t)\;=\;F[u_{jk} ( x_{1}, x_{2}, x_{3}, t)]$$ $$F{\mbox{\Large [ \normalsize}} \frac{\partial^{2}u_{jk}( x_{1}, x_{2}, x_{3}, t)}{\partial x^{2}_{s}}{\mbox{\Large ] \normalsize}} \;=\;-\gamma^{2}_{s}U_{jk}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t) \;\;\;\;\; \rm{[use (\ref{eqn16})]}$$ $$\\U_{k}^{0}( \gamma_{1} ,\gamma_{2} ,\gamma_{3})\;=\;F[u_{k}^{0} ( x_{1}, x_{2}, x_{3})]$$ $$\\P_{j}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t)\;=\;F[p_{j}\, ( x_{1}, x_{2}, x_{3}, t)]$$ $$\\F_{jk}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t)\;=\;F[f_{jk} ( x_{1}, x_{2}, x_{3}, t)]$$ $$\\ k,s\;=\;1,2,3$$ and then: $$\label{eqn134} \frac{d U_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )}{d t} \;=\;-\nu ( \gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) U_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )\;+\;i\gamma_{1} P_{j}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )\;+\; F_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3},t )$$ $$\label{eqn135} \frac{d U_{j2}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )}{d t} \;=\;-\nu ( \gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) U_{j2}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )\;+\;i\gamma_{2} P_{j}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )\;+\; F_{j2}( \gamma_{1}, \gamma_{2}, \gamma_{3},t )$$ $$\label{eqn136} \frac{d U_{j3}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )}{d t} \;=\;-\nu ( \gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) U_{j3}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )\;+\;i\gamma_{3} P_{j}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )\;+\; F_{j3}( \gamma_{1}, \gamma_{2}, \gamma_{3},t )$$ $$\label{eqn137} \gamma_{1} U_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t ) \;+\; \gamma_{2}\, U_{j2}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t ) \;+\; \gamma_{3}\, U_{j3}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t ) \;=\;0$$ $$\label{eqn138} U_{j1}(\gamma_{1}, \gamma_{2}, \gamma_{3}, 0)\;=\; U_{1}^{0}(\gamma_{1} ,\gamma_{2} ,\gamma_{3})$$ $$\label{eqn139} U_{j2}(\gamma_{1}, \gamma_{2}, \gamma_{3}, 0)\;=\; U_{2}^{0}(\gamma_{1} ,\gamma_{2} ,\gamma_{3})$$ $$\label{eqn140} U_{j3}(\gamma_{1}, \gamma_{2}, \gamma_{3}, 0)\;=\; U_{3}^{0}(\gamma_{1} ,\gamma_{2} ,\gamma_{3})$$ Hence eliminate $P_{j}(\gamma_{1}, \gamma_{2}, \gamma_{3}, t)$ from equations $(\ref{eqn134})\;-\; (\ref{eqn136})\;$ and find: $$\begin{aligned} \label{eqn141} \frac{d}{dt} {\mbox{\large [ \normalsize}} U_{j2}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )\; -\;\frac{\gamma_{2}}{\gamma_{1}} \,U_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t) {\mbox{\large ] \normalsize}} \;= -\nu( \gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) {\mbox{\large [ \normalsize}} U_{j2}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )\; -\; \nonumber \\ \nonumber\\ -\;\frac{\gamma_{2}}{\gamma_{1}}\, U_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t) {\mbox{\large ] \normalsize}} + \; {\mbox{\large [ \normalsize}} F_{j2}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )\; -\;\frac{\gamma_{2}}{\gamma_{1}} \, F_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t) {\mbox{\large ] \normalsize}} \quad\quad\quad\quad\quad\quad \end{aligned}$$ $$\begin{aligned} \label{eqn142} \frac{d}{dt} {\mbox{\large [ \normalsize}} U_{j3}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )\; -\;\frac{\gamma_{3}}{\gamma_{1}}\, U_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t) {\mbox{\large ] \normalsize}} \;= -\nu( \gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) {\mbox{\large [ \normalsize}} U_{j3}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )\; -\; \nonumber \\ \nonumber\\ -\;\frac{\gamma_{3}}{\gamma_{1}}\, U_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t) {\mbox{\large ] \normalsize}} + \;{\mbox{\large [ \normalsize}} F_{j3}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )\; -\;\frac{\gamma_{3}}{\gamma_{1}}\, F_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t) {\mbox{\large ] \normalsize}} \quad\quad\quad\quad\quad\quad \end{aligned}$$ $$\label{eqn143} \gamma_{1} U_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t ) \;+\; \gamma_{2}\, U_{j2}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t ) \;+\; \gamma_{3}\, U_{j3}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t ) \;=\;0$$ $$\label{eqn144} U_{j1}(\gamma_{1}, \gamma_{2}, \gamma_{3}, 0)\;=\; U_{1}^{0}(\gamma_{1} ,\gamma_{2} ,\gamma_{3})$$ $$\label{eqn145} U_{j2}(\gamma_{1}, \gamma_{2}, \gamma_{3}, 0)\;=\; U_{2}^{0}(\gamma_{1} ,\gamma_{2} ,\gamma_{3})$$ $$\label{eqn146} U_{j3}(\gamma_{1}, \gamma_{2}, \gamma_{3}, 0)\;=\; U_{3}^{0}(\gamma_{1} ,\gamma_{2} ,\gamma_{3})$$ We use Laplace transform $(\ref{A4}), (\ref{A5})$ for equations $(\ref{eqn141})\;-\; (\ref{eqn143})\;$ and have: $$U_{jk}^{\otimes} (\gamma_{1}, \gamma_{2}, \gamma_{3}, \eta) \;=\;L[U_{jk}(\gamma_{1}, \gamma_{2}, \gamma_{3}, t)] \;\;\;\;\;\;\; \rm{k=1,2,3}$$ $$\begin{aligned} \label{eqn147} \eta {\mbox{\large [ \normalsize}} U_{j2}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta )\; -\;\frac{\gamma_{2}}{\gamma_{1}} U_{j1}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta) {\mbox{\large ] \normalsize}} \;-\; {\mbox{\large [ \normalsize}} U_{j2}( \gamma_{1}, \gamma_{2}, \gamma_{3}, 0 )\; -\;\frac{\gamma_{2}}{\gamma_{1}} U_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, 0) {\mbox{\large ] \normalsize}} \;= \nonumber \\ \nonumber\\ -\nu( \gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}){\mbox{\large [ \normalsize}} U_{j2}^{\otimes} ( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta )\; -\;\frac{\gamma_{2}}{\gamma_{1}} U_{j1}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta) {\mbox{\large ] \normalsize}} \;+ \quad\quad\quad\quad\quad\quad \nonumber \\ \nonumber\\ +\; {\mbox{\large [ \normalsize}} F_{j2}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta )\; -\;\frac{\gamma_{2}}{\gamma_{1}} F_{j1}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta) {\mbox{\large ] \normalsize}} \quad\quad\quad\quad\quad\quad \quad\quad\quad \quad\quad\quad \end{aligned}$$ $$\begin{aligned} \label{eqn148} \eta {\mbox{\large [ \normalsize}} U_{j3}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta )\; -\;\frac{\gamma_{3}}{\gamma_{1}} U_{j1}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta) {\mbox{\large ] \normalsize}} \;-\; {\mbox{\large [ \normalsize}} U_{j3}( \gamma_{1}, \gamma_{2}, \gamma_{3}, 0 )\; -\;\frac{\gamma_{3}}{\gamma_{1}} U_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, 0) {\mbox{\large ] \normalsize}} \;= \nonumber \\ \nonumber\\ -\nu( \gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}){\mbox{\large [ \normalsize}} U_{j3}^{\otimes} ( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta )\; -\;\frac{\gamma_{3}}{\gamma_{1}} U_{j1}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta) {\mbox{\large ] \normalsize}} \;+ \quad\quad\quad\quad\quad\quad \nonumber \\ \nonumber\\ +\; {\mbox{\large [ \normalsize}} F_{j3}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta )\; -\;\frac{\gamma_{3}}{\gamma_{1}} F_{j1}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta) {\mbox{\large ] \normalsize}} \quad\quad\quad\quad\quad\quad \quad\quad\quad \quad\quad\quad \end{aligned}$$ $$\label{eqn149} \gamma_{1} U_{j1}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta ) \;+\; \gamma_{2}\, U_{j2}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta ) \;+\; \gamma_{3}\, U_{j3}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta ) \;=\;0$$ $$\label{eqn150} U_{j1}(\gamma_{1}, \gamma_{2}, \gamma_{3}, 0)\;=\; U_{1}^{0}(\gamma_{1} ,\gamma_{2} ,\gamma_{3})$$ $$\label{eqn151} U_{j2}(\gamma_{1}, \gamma_{2}, \gamma_{3}, 0)\;=\; U_{2}^{0}(\gamma_{1} ,\gamma_{2} ,\gamma_{3})$$ $$\label{eqn152} U_{j3}(\gamma_{1}, \gamma_{2}, \gamma_{3}, 0)\;=\; U_{3}^{0}(\gamma_{1} ,\gamma_{2} ,\gamma_{3})$$ In the usual way the solution of the system of equations $(\ref{eqn147})\;-\; (\ref{eqn149})\;$with formulas $(\ref{eqn150})\;-\; (\ref{eqn152})\;$can be rewritten in the following form: $$\begin{aligned} \label{eqn153} U_{j1}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta )\;=\;\frac{[( \gamma_{2}^{2} +\gamma_{3}^{2}) F_{j1}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta) - \gamma_{1}\gamma_{2} F_{j2}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta) - \gamma_{1}\gamma_{3} F_{j3}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta)]}{ (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) [\eta+\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2})] }\;+\nonumber \\ \nonumber\\ +\; \frac{ U_{1}^{0}(\gamma_{1} , \gamma_{2} , \gamma_{3})}{[\eta+\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2})] } \quad\quad\quad\quad\quad\quad \quad\quad\quad \quad\quad\quad \quad\quad\quad\quad\quad\quad \end{aligned}$$ $$\begin{aligned} \label{eqn154} U_{j2}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta )\;=\;\frac{[( \gamma_{3}^{2} +\gamma_{1}^{2}) F_{j2}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta) - \gamma_{2}\gamma_{3} F_{j3}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta) - \gamma_{2}\gamma_{1} F_{j1}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta)]}{ (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) [\eta+\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2})] }\;+\nonumber \\ \nonumber\\ +\; \frac{ U_{2}^{0}(\gamma_{1} , \gamma_{2} , \gamma_{3})}{[\eta+\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2})] } \quad\quad\quad\quad\quad\quad \quad\quad\quad \quad\quad\quad \quad\quad\quad\quad\quad\quad \end{aligned}$$ $$\begin{aligned} \label{eqn155} U_{j3}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta )\;=\;\frac{[( \gamma_{1}^{2} +\gamma_{2}^{2}) F_{j3}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta) - \gamma_{3}\gamma_{1} F_{j1}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta) - \gamma_{3}\gamma_{2} F_{j2}^{\otimes}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \eta)]}{ (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) [\eta+\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2})] }\;+ \nonumber \\ \nonumber\\ +\; \frac{ U_{3}^{0}(\gamma_{1} , \gamma_{2} , \gamma_{3})}{[\eta+\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2})] } \quad\quad\quad\quad\quad\quad \quad\quad\quad \quad\quad\quad \quad\quad\quad\quad\quad\quad \end{aligned}$$ Then we use the convolution formula (\[A6\]) and integral (\[A7\]) for $(\ref{eqn153})\;-\; (\ref{eqn155})\;$and obtain: $$\begin{aligned} \label{eqn156} U_{j1}(\gamma_{1}, \gamma_{2}, \gamma_{3}, t)\;=\; \quad\quad\quad\quad\quad\quad \quad\quad\quad \quad\quad\quad \quad\quad\quad\quad\quad\quad \nonumber \\ \nonumber\\ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)} \frac{[( \gamma_{2}^{2} +\gamma_{3}^{2}) F_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \tau) -\gamma_{1}\gamma_{2} F_{j2}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \tau ) -\gamma_{1}\gamma_{3} F_{j3}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \tau ) ]}{ (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) }\,d\tau\;+ \nonumber \\ \nonumber\\ +\;{\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) t} \;U_{1}^{0}(\gamma_{1} ,\gamma_{2} ,\gamma_{3}) \quad\quad\quad\quad\quad\quad \quad\quad\quad \quad\quad\quad \quad\quad\quad\quad \end{aligned}$$ $$\begin{aligned} \label{eqn157} U_{j2}(\gamma_{1}, \gamma_{2}, \gamma_{3}, t)\;=\; \quad\quad\quad\quad\quad\quad \quad\quad\quad \quad\quad\quad \quad\quad\quad\quad\quad\quad \nonumber \\ \nonumber\\ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)} \frac{[( \gamma_{3}^{2} +\gamma_{1}^{2}) F_{j2}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \tau) -\gamma_{2}\gamma_{3} F_{j3}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \tau ) -\gamma_{2}\gamma_{1} F_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \tau ) ]}{ (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) }\,d\tau\;+ \nonumber \\ \nonumber\\ +\;{\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) t} \;U_{2}^{0}(\gamma_{1} ,\gamma_{2} ,\gamma_{3}) \quad\quad\quad\quad\quad\quad \quad\quad\quad \quad\quad\quad \quad\quad\quad\quad \end{aligned}$$ $$\begin{aligned} \label{eqn158} U_{j3}(\gamma_{1}, \gamma_{2}, \gamma_{3}, t)\;=\; \quad\quad\quad\quad\quad\quad \quad\quad\quad \quad\quad\quad \quad\quad\quad\quad\quad\quad \nonumber \\ \nonumber\\ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)} \frac{[( \gamma_{1}^{2} +\gamma_{2}^{2}) F_{j3}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \tau) -\gamma_{3}\gamma_{1} F_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \tau ) -\gamma_{3}\gamma_{2} F_{j2}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \tau ) ]}{ (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) }\,d\tau\;+ \nonumber \\ \nonumber\\ +\;{\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) t} \;U_{3}^{0}(\gamma_{1} ,\gamma_{2} ,\gamma_{3}) \quad\quad\quad\quad\quad\quad \quad\quad\quad \quad\quad\quad \quad\quad\quad\quad \end{aligned}$$ $ P_{j}(\gamma_{1}, \gamma_{2}, \gamma_{3}, t) $ is obtained from $(\ref{eqn134})\;[(\ref{eqn135})\;$or$\; (\ref{eqn136})]\;$: $$\label{eqn159} P_{j}(\gamma_{1}, \gamma_{2}, \gamma_{3}, t) \;=\;i\frac{[\gamma_{1} F_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t) +\gamma_{2} F_{j2}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t ) +\gamma_{3} F_{j3}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )]}{ (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) }$$ \ Use of the Fourier inversion formula $(\ref{A3})$ and find: $$\begin{aligned} \label{eqn160} u_{j1}(x_{1}, x_{2}, x_{3}, t)\;=\; \frac{1}{(2\pi)^{3/2}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)} \frac{[ ( \gamma_{2}^{2} +\gamma_{3}^{2}) F_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \tau)]} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) } \,d\tau \;- \nonumber \\ \nonumber\\ -\; \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)}\frac{ [\gamma_{1}\gamma_{2} F_{j2}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \tau ) + \gamma_{1}\gamma_{3} F_{j3}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \tau )]} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) }\,d\tau\; + \quad \nonumber\\ \nonumber\\ +\;{\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) t} \;U_{1}^{0}(\gamma_{1} ,\gamma_{2} ,\gamma_{3})\biggr] \;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3}\;= \nonumber\\ \nonumber\\ =\;\frac{1}{8\pi^{3}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{( \gamma_{2}^{2} +\gamma_{3}^{2})} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) } \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})} \cdot \nonumber\\ \nonumber\\ \cdot f_{j1}(\tilde x_{1},\tilde x_{2}, \tilde x_{3},\tau)\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}d\tau\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3}\;- \nonumber\\ \nonumber\\ -\;\frac{1}{8\pi^{3}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{ \gamma_{1}\gamma_{2}} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) } \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})} \cdot \nonumber\\ \nonumber\\ \cdot f_{j2}(\tilde x_{1},\tilde x_{2}, \tilde x_{3},\tau)\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}d\tau\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3}\;- \nonumber\\ \nonumber\\ -\;\frac{1}{8\pi^{3}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{ \gamma_{1}\gamma_{3}} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) } \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})} \cdot \nonumber\\ \nonumber\\ \cdot f_{j3}(\tilde x_{1},\tilde x_{2}, \tilde x_{3},\tau)\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}d\tau\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3}\;+ \nonumber\\ \nonumber\\ +\;\frac{1}{8\pi^{3}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) t}\biggl[ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})} \cdot \quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \cdot \; u_{1}^{0}(\tilde x_{1},\tilde x_{2}, \tilde x_{3})\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3} \;= \nonumber\\ \nonumber\\ =\; S_{11}(f_{j1})\;+\; S_{12}(f_{j2})\;+\; S_{13}(f_{j3})\;+\;B(u_{1}^0) \quad\quad\quad\quad\quad\quad \quad\quad\quad \quad\quad\quad \end{aligned}$$ $$\begin{aligned} \label{eqn161} u_{j2}(x_{1}, x_{2}, x_{3}, t)\;=\; \frac{1}{(2\pi)^{3/2}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)} \frac{[( \gamma_{3}^{2} +\gamma_{1}^{2}) F_{j2}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \tau)]} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) } \,d\tau \;- \nonumber\\ \nonumber\\ -\; \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)}\frac{[\gamma_{2}\gamma_{3} F_{j3}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \tau ) +\gamma_{2}\gamma_{1} F_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \tau )]} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) }\,d\tau\; + \quad \nonumber\\ \nonumber\\ +\;{\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) t} \;U_{2}^{0}(\gamma_{1} ,\gamma_{2} ,\gamma_{3})\biggr] \;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3}\;= \nonumber\\ \nonumber\\ =\;-\;\frac{1}{8\pi^{3}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{ \gamma_{2}\gamma_{1}} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) } \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})} \cdot \nonumber\\ \nonumber\\ \cdot f_{j1}(\tilde x_{1},\tilde x_{2}, \tilde x_{3},\tau)\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}d\tau\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3}\;+ \nonumber\\ \nonumber\\ +\;\frac{1}{8\pi^{3}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{ (\gamma_{3}^2 + \gamma_{1}^2)} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) } \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})} \cdot \nonumber\\ \nonumber\\ \cdot f_{j2}(\tilde x_{1},\tilde x_{2}, \tilde x_{3},\tau)\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}d\tau\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3}\;- \nonumber\\ \nonumber\\ -\;\frac{1}{8\pi^{3}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{ \gamma_{2}\gamma_{3}} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) } \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})} \cdot \nonumber\\ \nonumber\\ \cdot f_{j3}(\tilde x_{1},\tilde x_{2}, \tilde x_{3},\tau)\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}d\tau\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3}\;+ \nonumber\\ \nonumber\\ +\;\frac{1}{8\pi^{3}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) t}\biggl[ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})} \cdot \quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \cdot \; u_{2}^{0}(\tilde x_{1},\tilde x_{2}, \tilde x_{3})\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3} \;= \nonumber\\ \nonumber\\ =\; S_{21}(f_{j1})\;+\; S_{22}(f_{j2})\;+\; S_{23}(f_{j3})\;+\;B(u_{2}^0) \quad\quad\quad\quad\quad\quad \quad\quad\quad \quad\quad\quad \end{aligned}$$ $$\begin{aligned} \label{eqn162} u_{j3}(x_{1}, x_{2}, x_{3}, t)\;=\; \frac{1}{(2\pi)^{3/2}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)} \frac{[( \gamma_{1}^{2} +\gamma_{2}^{2}) F_{j3}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \tau)]} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) } \,d\tau \;- \nonumber\\ \nonumber\\ -\;\int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)}\frac{[\gamma_{3}\gamma_{1} F_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \tau ) +\gamma_{3}\gamma_{2} F_{j2}( \gamma_{1}, \gamma_{2}, \gamma_{3}, \tau )]} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) }\,d\tau\; + \quad \nonumber\\ \nonumber\\ +\; {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) t} \;U_{3}^{0}(\gamma_{1} ,\gamma_{2} ,\gamma_{3})\biggr] \;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3}\;= \nonumber\\ \nonumber\\ =\;-\;\frac{1}{8\pi^{3}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{ \gamma_{3}\gamma_{1}} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) } \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})} \cdot \nonumber\\ \nonumber\\ \cdot f_{j1}(\tilde x_{1},\tilde x_{2}, \tilde x_{3},\tau)\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}d\tau\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3}\;- \nonumber\\ \nonumber\\ -\;\frac{1}{8\pi^{3}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{ \gamma_{3}\gamma_{2}} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) } \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})} \cdot \nonumber\\ \nonumber\\ \cdot f_{j2}(\tilde x_{1},\tilde x_{2}, \tilde x_{3},\tau)\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}d\tau\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3}\;+ \nonumber\\ \nonumber\\ +\;\frac{1}{8\pi^{3}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{ (\gamma_{1}^2 + \gamma_{2}^2)} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) } \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) (t-\tau)} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})} \cdot \nonumber\\ \nonumber\\ \cdot f_{j3}(\tilde x_{1},\tilde x_{2}, \tilde x_{3},\tau)\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}d\tau\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3}\;+ \nonumber\\ \nonumber\\ +\;\frac{1}{8\pi^{3}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) t}\biggl[ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})} \cdot \quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \cdot \; u_{3}^{0}(\tilde x_{1},\tilde x_{2}, \tilde x_{3})\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3} \;= \nonumber\\ \nonumber\\ =\; S_{31}(f_{j1})\;+\; S_{32}(f_{j2})\;+\; S_{33}(f_{j3})\;+\;B(u_{3}^0) \quad\quad\quad\quad\quad\quad \quad\quad\quad \quad\quad\quad \end{aligned}$$ $$\begin{aligned} \label{eqn163} p_{j}\,(x_{1}, x_{2}, x_{3}, t)\;=\; \frac{i}{(2\pi)^{3/2}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\biggl[ \;\frac{[\gamma_{1} F_{j1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t) + \gamma_{2} F_{j2}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )]} { (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) }\;+ \nonumber\\ \nonumber\\ +\;\frac{ \gamma_{3} F_{j3}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )}{ (\gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) }\; \biggr] \;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1} +x_{2}\gamma_{2} +x_{3}\gamma_{3})}\,d\gamma_{1} d\gamma_{2} d\gamma_{3}\; = \; \nonumber\\ \nonumber\\ =\frac{i}{8\pi^{3}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{ \gamma_{1}} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) } \biggl[ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})} \cdot \nonumber\\ \nonumber\\ \cdot f_{j1}(\tilde x_{1},\tilde x_{2}, \tilde x_{3},t)\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3}\;+ \nonumber\\ \nonumber\\ +\;\frac{i}{8\pi^{3}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{ \gamma_{2}} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) } \biggl [ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})} \cdot \nonumber\\ \nonumber\\ \cdot f_{j2}(\tilde x_{1},\tilde x_{2}, \tilde x_{3},t)\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3}\;+ \nonumber\\ \nonumber\\ +\;\frac{i}{8\pi^{3}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{ \gamma_{3}} { (\gamma_{1}^{2} +\gamma_{2}^{2}+\gamma_{3}^{2} ) } \biggl[ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2}+\tilde x_{3}\gamma_{3})} \cdot \nonumber\\ \nonumber\\ \cdot f_{j3}(\tilde x_{1},\tilde x_{2}, \tilde x_{3},t)\,d\tilde x_{1}d\tilde x_{2} d\tilde x_{3}\biggr]{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{3}\gamma_{3})}\,d\gamma_{1}d\gamma_{2}d\gamma_{3}\;= \nonumber\\ \nonumber\\ =\;\tilde S_{1}(f_{j1})\;+\; \tilde S_{2}(f_{j2})\;+\; \tilde S_{3}(f_{j3}) \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad\quad\quad\quad\quad\quad\quad \quad\quad \end{aligned}$$ So, the integrals $(\ref{eqn160})\;-\; (\ref{eqn163})\;$exist by the restrictions $(\ref{eqn17})\;, (\ref{eqn18})\;$. Here $S_{11}(), S_{12}(), S_{13}(), S_{21}(), S_{22}(), S_{23}(), S_{31}(), S_{32}(), S_{33}(), B(), \tilde S_{1}(), \tilde S_{2}(), \tilde S_{3}()$ are the integral - operators. $$S_{12}()\;= \;S_{21}()$$$$S_{13}()\;= \;S_{31}()$$$$S_{23}()\;= \;S_{32}()$$ We have for the vector $\vec{u}_{j}$ from the equations $(\ref{eqn160})\;-\; (\ref{eqn162})\;$: $$\label{eqn164} \vec{u}_{j}\;=\;\bar{\bar{S}}\;\cdot\;\vec{f}_{j}\;+\;B(\vec{u}^{0})\;,$$ where $\;\bar{\bar{S}} \; $ is the matrix - operator: $$\left( \begin{array}{ccc} S_{11} & S_{12} & S_{13} \\ S_{21} & S_{22} & S_{23} \\ S_{31} & S_{32} & S_{33} \end{array} \right)$$ We put $\vec{f}_{j}$ from equation $(\ref{eqn20})$ into equation $(\ref{eqn164})$ and have: $$\begin{aligned} \label{eqn165} \vec{u}_{j} = \bar{\bar{S}}\cdot(\;\vec{f}\;-\;(\;\vec{u}_{j-1}\cdot\nabla)\vec{u}_{j-1})\;+\;B(\vec{u}^{0})\;= \nonumber\\ \nonumber\\ =\;\bar{\bar{S}}\cdot\vec{f} \;-\;\bar{\bar{S}}\cdot(\vec{u}_{j-1}\;\cdot\;\nabla\;)\;\vec{u}_{j-1}\;+\;B(\vec{u}^{0})\; = \nonumber\\ \nonumber\\ =\;\vec{u}_{1}\;-\;\bar{\bar{S}}\cdot(\vec{u}_{j-1}\;\cdot\;\nabla)\;\vec{u}_{j-1} \quad\quad\quad\quad\quad\quad \quad\quad \end{aligned}$$ Here $\vec{u}_{1}\;$ is the solution of the system of equations $(\ref{eqn13})\; - \;(\ref{eqn20})$ with condition: $$\sum_{n=1}^{3} u_{n}\frac{\partial u_{k}}{\partial x_{n}}\;=\;0\;\;\;\;\;\;\; \rm{k=1,2,3}\;\;$$ For j = 1 formula $(\ref{eqn164})$ can be written as follows: $$\label{eqn168} \vec{u}_{1}\;=\;\bar{\bar{S}}\;\cdot\;\vec{f}_{1}\;+\;B(\vec{u}^{0})\;,\;\;\;\;\;\\\vec{f}_{1}(x,t)\; = \; \vec{f}(x,t)$$ If t $\rightarrow$ 0 then $\vec{u}_{1} \rightarrow \vec{u}^{0}$ (look at integral-operators $\bar{\bar{S}}, B()\;\;$- integrals $\;(\ref{eqn160})\; - \;(\ref{eqn162})$). For j = 2 we define from equation $(\ref{eqn20})$: $$\vec{f}_{2}(x,t)\; = \; \vec{f}_{1}(x,t) \; - \;(\;\vec{u}_{1}\;\cdot\;\nabla\;)\;\vec{u}_{1}\;$$ We denote: $$\label{eqn169} \vec{f}_{2}^{*}\;=\;(\vec{u}_{1}\;\cdot\;\nabla)\;\vec{u}_{1}$$ and then we have: $$\vec{f}_{2}(x,t)\; = \; \vec{f}_{1}(x,t) \; - \vec{f}_{2}^{*}$$ Then we get $\vec{u}_{2}$ from $(\ref{eqn164}),(\ref{eqn168})$: $$\label{eqn171} \vec{u}_{2}\;=\;\bar{\bar{S}}\;\cdot\;\vec{f}_{2} \;+\;B(\vec{u}^{0})\;=\;\bar{\bar{S}}\;\cdot\;(\vec{f}_{1}\;-\;\vec{f}_{2}^{*}) \;+\;B(\vec{u}^{0})\;=\;\vec{u}_{1}\;-\;\vec{u}_{2}^{*}$$ Here we have: $$\label{eqn170} \vec{u}_{2}^{*}\;=\;\bar{\bar{S}}\;\cdot\;\vec{f}_{2}^{*}$$ If t $\rightarrow$ 0 then $\vec{u}_{2}^{*} \rightarrow$ 0 (look at integral-operator $\bar{\bar{S}}\;\;$- integrals $\;(\ref{eqn160})\; - \;(\ref{eqn162})$). Continue for j = 3. We define from equation $(\ref{eqn20})$: $$\vec{f}_{3}(x,t)\; = \; \vec{f}_{1}(x,t) \; - \;(\;\vec{u}_{2}\;\cdot\;\nabla\;)\;\vec{u}_{2}\;$$ Here we have: $$\label{eqn172} (\vec{u}_{2}\;\cdot\;\nabla)\;\vec{u}_{2}\;= \;((\vec{u}_{1}\;-\;\vec{u}_{2}^{*})\;\cdot\;\nabla\;) \;(\vec{u}_{1}\;-\;\vec{u}_{2}^{*})\;= \;\vec{f}_{2}^{*}\;+\;\vec{f}_{3}^{*}$$ We denote in $(\ref{eqn172})$: $$\vec{f}_{3}^{*}\;=\;-\; (\vec{u}_{1}\;\cdot\;\nabla)\;\vec{u}_{2}^{*}\; -\; (\vec{u}_{2}^{*}\;\cdot\;\nabla)\;\vec{u}_{1}\;+\; (\vec{u}_{2}^{*}\;\cdot\;\nabla)\;\vec{u}_{2}^{*}$$ and then we have: $$\vec{f}_{3}(x,t)\; = \; \vec{f}_{1}(x,t) \; - \vec{f}_{2}^{*}\; - \vec{f}_{3}^{*}$$ Then we get $\vec{u}_{3}$ from $(\ref{eqn164}) , (\ref{eqn168}) ,(\ref{eqn170})$: $$\label{eqn174} \vec{u}_{3}\;=\;\bar{\bar{S}}\;\cdot\;\vec{f}_{3} \;+\;B(\vec{u}^{0})\;=\;\bar{\bar{S}}\;\cdot\;(\vec{f}_{1} \;-\;\vec{f}_{2}^{*}\;-\;\vec{f}_{3}^{*})\;+\;B(\vec{u}^{0})\;=\;\vec{u}_{1} \;-\;\vec{u}_{2}^{*}\;-\;\vec{u}_{3}^{*}$$ Here we denote: $$\label{eqn173} \vec{u}_{3}^{*}\;=\;\bar{\bar{S}}\;\cdot\;\vec{f}_{3}^{*}$$ If t $\rightarrow$ 0 then $\vec{u}_{3}^{*} \rightarrow$ 0 (look at integral-operator $\bar{\bar{S}}\;\;$- integrals $\;(\ref{eqn160})\; - \;(\ref{eqn162})$). For j = 4. We define from equation $(\ref{eqn20})$: $$\vec{f}_{4}(x,t)\; = \; \vec{f}_{1}(x,t) \; - \;(\;\vec{u}_{3}\;\cdot\;\nabla\;)\;\vec{u}_{3}\;$$ Here we have: $$\label{eqn175} (\vec{u}_{3}\;\cdot\;\nabla)\;\vec{u}_{3}\;= \;((\vec{u}_{2}\;-\;\vec{u}_{3}^{*})\;\cdot\;\nabla\;) \;(\vec{u}_{2}\;-\;\vec{u}_{3}^{*})\;= \;\vec{f}_{2}^{*}\;+\;\vec{f}_{3}^{*}\;+\;\vec{f}_{4}^{*}$$ We denote in $(\ref{eqn175})$: $$\vec{f}_{4}^{*}\;=\;-\; (\vec{u}_{2}\;\cdot\;\nabla)\;\vec{u}_{3}^{*}\; -\; (\vec{u}_{3}^{*}\;\cdot\;\nabla)\;\vec{u}_{2}\;+\; (\vec{u}_{3}^{*}\;\cdot\;\nabla)\;\vec{u}_{3}^{*}$$ and then we have: $$\vec{f}_{4}(x,t)\; = \; \vec{f}_{1}(x,t) \; - \vec{f}_{2}^{*}\; - \vec{f}_{3}^{*}\; - \vec{f}_{4}^{*}$$ Then we get $\vec{u}_{4}$ from $(\ref{eqn164}) , (\ref{eqn168}) ,(\ref{eqn170}) ,(\ref{eqn173})$: $$\label{eqn177} \vec{u}_{4}\;=\;\bar{\bar{S}}\;\cdot\;(\vec{f}_{1} \; -\;\vec{f}_{2}^{*}\; -\;\vec{f}_{3}^{*}\;-\;\vec{f}_{4}^{*})\;+\;B(\vec{u}^{0})\;=\;\vec{u}_{1} \;-\;\vec{u}_{2}^{*}\;-\;\vec{u}_{3}^{*}\;-\;\vec{u}_{4}^{*}$$ Here we denote: $$\label{eqn176} \vec{u}_{4}^{*}\;=\;\bar{\bar{S}}\;\cdot\;\vec{f}_{4}^{*}$$ If t $\rightarrow$ 0 then $\vec{u}_{4}^{*} \rightarrow$ 0 (look at integral-operator $\bar{\bar{S}}\;\;$- integrals $\;(\ref{eqn160})\; - \;(\ref{eqn162})$). For arbitrary number j $(j \geq 2)$. We define from equation $(\ref{eqn20})$: $$\vec{f}_{j}(x,t)\; = \; \vec{f}_{1}(x,t) \; - \;(\;\vec{u}_{j-1}\;\cdot\;\nabla\;)\;\vec{u}_{j-1}\;$$ Here we have: $$\label{eqn181} (\vec{u}_{j-1}\;\cdot\;\nabla)\;\vec{u}_{j-1}\;=\;\sum_{l=2}^{j} \vec{f}_{l}^{*}$$ and it follows: $$\label{eqn181a} \vec{f}_{j}\;=\;\vec{f}_{1}\;-\; \sum_{l=2}^{j} \vec{f}_{l}^{*}$$ Then we get $\vec{u}_{j}$ from $(\ref{eqn164}) , (\ref{eqn168}) $ $$\label{eqn182} \vec{u}_{j}\;=\;\bar{\bar{S}}\;\cdot\; \vec{f}_{j}\;+ \;B(\vec{u}^{0})\;=\;\bar{\bar{S}}\;\cdot\;( \vec{f}_{1}\;-\; \sum_{l=2}^{j} \vec{f}_{l}^{*})\;+ \;B(\vec{u}^{0})\;=\;\vec{u}_{1}\;-\;\sum_{l=2}^{j} \vec{u}_{l}^{*}$$ Here we denote: $$\label{eqn183} \vec{u}_{l}^{*}\;=\;\bar{\bar{S}}\;\cdot\;\vec{f}_{l}^{*}\;\;\;\;\;\;\;\;\;\;\;\;\;\;(2\;\leq\;l\;\leq\;j)$$ If t $\rightarrow$ 0 then $\vec{u}_{l}^{*} \rightarrow$ 0 (look at integral-operator $\bar{\bar{S}}\;\;$- integrals $\;(\ref{eqn160})\; , \;(\ref{eqn162})$). We consider the equations $(\ref{eqn168})$ - $(\ref{eqn183})$ and see that the series $(\ref{eqn182})$ converge for $j \rightarrow \infty$ with the conditions for the first step (j = 1) of the iterative process: $$\;\;\;\sum_{n=1}^{3} u_{0n}\frac{\partial u_{0k}}{\partial x_{n}} = 0\;\;\;\;\;\;\;\; \rm{k=1,2,3}$$ and conditions $$\label{eqn184} C_{\alpha K}\leq\;\frac{1}{2}\;\;,\;\;C_{\alpha \beta K}\leq\;\frac{1}{2}.$$ Here $\;\;C_{\alpha K}\;\;$and$\;\;C_{\alpha \beta K}\;$ are received from $(\ref{eqn17})$ , $(\ref{eqn18}).$ Hence, we receive from equation $(\ref{eqn165})\;$ when $j \rightarrow \infty$: $$\label{eqn185} \vec{u}_{\infty}=\;\vec{u}_{1}\;-\;\bar{\bar{S}}\cdot(\vec{u}_{\infty}\;\cdot\;\nabla)\;\vec{u}_{\infty}$$ Equation $(\ref{eqn185})$ describes the converging iterative process. Then we have from formula $(\ref{eqn163})\;$: $$\label{eqn185a} p_{\infty}\,\;=\; \;\tilde S_{1}(f_{\infty 1})\;+\; \tilde S_{2}(f_{\infty 2})\;+\; \tilde S_{3}(f_{\infty 3})$$ Here $\vec{f}_{\infty}$ = ($f_{\infty 1} , f_{\infty 2}, f_{\infty 3}$) is received from formula $(\ref{eqn181a})\;$. On the other hand we can transform the original system of differential equations $(\ref{eqn7})\; - \;(\ref{eqn9})$ to the equivalent system of integral equations by the scheme of iterative process $(\ref{eqn164})\;, \;(\ref{eqn165})$ for vector $\vec{u}$: $$\label{eqn186} \vec{u}\;=\;\vec{u}_{1}\;-\;\bar{\bar{S}}\cdot(\vec{u}\;\cdot\;\nabla)\;\vec{u},$$ where $\vec{u}_{1}$ is from formula $(\ref{eqn168})$. We compare the equations $(\ref{eqn185})$ and $(\ref{eqn186})$ and see that the iterative process $(\ref{eqn185})$ converge to the solution of the system $(\ref{eqn186})$ and hence to the solution of the differential equations $(\ref{eqn7})\; - \;(\ref{eqn9})$ with conditions $(\ref{eqn184})$. **In other words there exist smooth functions** $\mathbf{p_{\infty}(x, t)}$, $\mathbf{u_{\infty i}(x, t)}$ **(i = 1, 2, 3) on** $\mathbf{R^{3} \times [0,\infty)}$ **that satisfy** $\mathbf{(\ref{eqn1}), (\ref{eqn2}), (\ref{eqn3})}$ **and** $$\label{eqn186b} \mathbf{p_{\infty}, \;u_{\infty i} \in C^{\infty}(R^{3} \times [0,\infty)),} \nonumber\\ \nonumber\\$$ $$\label{eqn186c} \mathbf{\int_{R^{3}}|\vec{u}_{\infty}(x, t)|^{2}dx < C }$$ **for all t** $\mathbf{\geq 0}$.\ In the following chapters 4 and 5 we describe in further details examples of the solutions for the Navier-Stocks and Euler problems with various applied forces and different values of the viscosity coefficient ${\nu}$.\ Example of the solution of the Cauchy problem for the Euler equations by the described iterative method with a particular applied force (N = 2) =============================================================================================================================================== We will consider an example of the solution of the Cauchy problem for the Euler equations ${(\nu = 0)}$ for N = 2 and with initial conditions: $$\label{eqn400} \vec{u}(x,0)\; = \; \vec{u}^{0}(x)\; = \;0\;\;\;\;\;\;\;\; (x\in R^{2})$$ Hence, and from formulas $(\ref{eqn39}), (\ref{eqn40})$ for arbitrary step j of the iterative process, it follows: $$\begin{aligned} \label{eqn401} u_{j1}(x_{1},x_{2},t)\;=\;\frac{1}{4\pi^2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\gamma_{2}^{2} }{(\gamma_{1}^{2}+\gamma_{2}^{2}) } \int_{0}^{t} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2})} f_{j1}(\tilde x_{1},\tilde x_{2},\tau)\,d \tilde x_{1}d \tilde x_{2}d\tau\cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2}\;- \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \nonumber\\ \nonumber\\ \nonumber\\ - \;\frac{1}{4\pi^2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\gamma_{1} \gamma_{2} }{(\gamma_{1}^{2}+\gamma_{2}^{2}) } \int_{0}^{t} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2})} f_{j2}(\tilde x_{1},\tilde x_{2},\tau)\,d \tilde x_{1}d \tilde x_{2}d\tau\cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2} \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \nonumber\\ \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \end{aligned}$$ $$\begin{aligned} \label{eqn402} u_{j2}(x_{1},x_{2},t)\;=\;-\frac{1}{4\pi^2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\gamma_{1} \gamma_{2} }{(\gamma_{1}^{2}+\gamma_{2}^{2}) } \int_{0}^{t} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2})} f_{j1}(\tilde x_{1},\tilde x_{2},\tau)\,d \tilde x_{1}d \tilde x_{2}d\tau\cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2}\;+ \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \nonumber\\ \nonumber\\ \nonumber\\ + \;\frac{1}{4\pi^2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\gamma_{1}^{2} }{(\gamma_{1}^{2}+\gamma_{2}^{2}) } \int_{0}^{t} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2})} f_{j2}(\tilde x_{1},\tilde x_{2},\tau)\,d \tilde x_{1}d \tilde x_{2}d\tau \cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2} \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \nonumber\\ \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \end{aligned}$$ We convert the Cartesian coordinates to the polar coordinates by formulas: $x_{1}$ = $r\;\cdot\;$cos$\varphi\;$;$\;\;x_{2}$ = $r\;\cdot\;$sin$\varphi\;$;$\;\;\gamma_{1}$ = $\rho\;\cdot\;$cos$\psi\;$;$\;\;\gamma_{2}$ = $\rho\;\cdot\;$sin$\psi\;$;$\;\;\tilde x_{1}$ = $\tilde r\;\cdot\;$cos$\tilde \varphi\;$;$\;\;\tilde x_{2}$ = $\tilde r\;\cdot\;$sin$\tilde \varphi$; and obtain from formulas $(\ref{eqn401}), (\ref{eqn402})$: $$\begin{aligned} \label{eqn403} u_{j1}(r,\varphi,t)\;=\;\frac{1}{4\pi^2} \int_{0}^{\infty}\int_{0}^{2\pi}{sin^{2}\psi } \int_{0}^{t} \int_{0}^{\infty}\int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)} f_{j1}(\tilde r,\tilde \varphi,\tau)\,\tilde r d \tilde r d \tilde \varphi d\tau\cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi\;- \quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ - \;\frac{1}{4\pi^2} \int_{0}^{\infty}\int_{0}^{2\pi}{sin\psi cos\psi } \int_{0}^{t} \int_{0}^{\infty}\int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)} f_{j2}(\tilde r,\tilde \varphi,\tau)\,\tilde r d \tilde r d \tilde \varphi d\tau\cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\; {\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi \quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ $$\begin{aligned} \label{eqn404} u_{j2}(r,\varphi,t)\;=\;-\frac{1}{4\pi^2} \int_{0}^{\infty}\int_{0}^{2\pi}{sin\psi cos\psi } \int_{0}^{t} \int_{0}^{\infty}\int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)} f_{j1}(\tilde r,\tilde \varphi,\tau)\,\tilde r d \tilde r d \tilde \varphi d\tau\cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\;\;\;\cdot\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi\;+ \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \nonumber\\ \nonumber\\ \nonumber\\ + \;\frac{1}{4\pi^2} \int_{0}^{\infty}\int_{0}^{2\pi}cos^{2}\psi \int_{0}^{t} \int_{0}^{\infty}\int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)} f_{j2}(\tilde r,\tilde \varphi,\tau)\,\tilde r d \tilde r d \tilde \varphi d\tau\cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\;\;\;\cdot\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \nonumber\\ \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \end{aligned}$$ We have the applied force $\vec{f}_{j}$ for arbitrary step j of the iterative process: $$\label{eqn405} f_{j\tilde r}(\tilde r,\tilde \varphi,\tau) = f_{j\tilde r}(\tilde r){\mbox{\Large e \normalsize}}^{in_{j}\tilde \varphi}f_{j\tau}(\tau)\;\;\;,\;\;\;f_{j\tilde \varphi}(\tilde r,\tilde \varphi,\tau) \equiv 0$$ or $$\label{eqn406} f_{j\tilde r}(\tilde r,\tilde \varphi,\tau)\equiv 0\;\;\;,\;\;\;f_{j\tilde \varphi}(\tilde r,\tilde \varphi,\tau) = f_{j\tilde \varphi}(\tilde r){\mbox{\Large e \normalsize}}^{in_{j}\tilde \varphi}f_{j\tau}(\tau)$$ where $\;\;f_{j\tilde r}(\tilde r,\tilde \varphi,\tau)\;\;\;,\;\;\;f_{j\tilde \varphi}(\tilde r,\tilde \varphi,\tau)\;\;\; - \;\;\;$ radial and tangential components of the applied force. $n_{j}$ - separate circumferential mode, $n_{j}$ = 0,1,2,3,... We take the radial and tangential components of the applied force $(\ref{eqn405}), (\ref{eqn406})$ with condition $(\ref{eqn18})\;$. For the radial component of the applied force we use De Moivre’s formulas (\[A8\]) and have: $$\begin{aligned} \label{eqn407} f_{j1}(\tilde r,\tilde \varphi,\tau) = f_{j\tilde r}(\tilde r){\mbox{\Large e \normalsize}}^{in_{j}\tilde \varphi}cos \tilde \varphi f_{j\tau}(\tau) = \frac{1}{2}f_{j\tilde r}(\tilde r)\bigl({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} + {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi}\bigr)f_{j\tau}(\tau) \nonumber\\ \nonumber\\ f_{j2}(\tilde r,\tilde \varphi,\tau) = f_{j\tilde r}(\tilde r){\mbox{\Large e \normalsize}}^{in_{j}\tilde \varphi}sin \tilde \varphi f_{j\tau}(\tau) = \frac{i}{2}f_{j\tilde r}(\tilde r)\bigl({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} - {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi}\bigr)f_{j\tau}(\tau)\end{aligned}$$ We put the applied force components $(\ref{eqn407})$ in formulas $(\ref{eqn403}), (\ref{eqn404})$ , change the order of integration and find: $$\begin{aligned} \label{eqn408} u_{jr1}(r,\varphi,t)\;=\;\frac{1}{8\pi^2} \biggl[ \int_{0}^{\infty}\int_{0}^{2\pi}{sin^{2}\psi } \int_{0}^{\infty} f_{j\tilde r}(\tilde r) \int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)} ({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} + {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi})d \tilde \varphi \tilde r d \tilde r \cdot \nonumber\\ \nonumber\\ \nonumber\\ \cdot\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi\;- \quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ -\;\; i \int_{0}^{\infty}\int_{0}^{2\pi}{sin\psi cos\psi } \int_{0}^{\infty} f_{j\tilde r}(\tilde r) \int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)}({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} - {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi})d \tilde \varphi \tilde r d \tilde r \cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi \biggr] \int_{0}^{t}f_{j\tau}(\tau) d\tau \quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ $$\begin{aligned} \label{eqn409} u_{jr2}(r,\varphi,t)\;=\;\frac{1}{8\pi^2} \biggl[ - \int_{0}^{\infty}\int_{0}^{2\pi}{sin\psi cos\psi} \int_{0}^{\infty} f_{j\tilde r}(\tilde r) \int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)} ({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} + {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi})d \tilde \varphi \tilde r d \tilde r \cdot \nonumber\\ \nonumber\\ \nonumber\\ \cdot\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi\;+ \quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ +\;\; i \int_{0}^{\infty}\int_{0}^{2\pi}{ cos^{2}\psi } \int_{0}^{\infty} f_{j\tilde r}(\tilde r) \int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)}({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} - {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi})d \tilde \varphi \tilde r d \tilde r \cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\; {\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi \biggr] \int_{0}^{t}f_{j\tau}(\tau) d\tau \quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ We denote: $$\label{eqn410} u_{jt}(t) = \int_{0}^{t}f_{j\tau}(\tau) d\tau$$ and from formulas $(\ref{eqn408}), (\ref{eqn409})$ it follows: $$\begin{aligned} \label{eqn411} u_{jr1}(r,\varphi,t)\;=\;u_{jr1}(r,\varphi)u_{jt}(t) \nonumber\\ \nonumber\\ u_{jr2}(r,\varphi,t)\;=\;u_{jr2}(r,\varphi)u_{jt}(t)\end{aligned}$$ where $$\begin{aligned} \label{eqn412} u_{jr1}(r,\varphi)\;=\;\frac{1}{8\pi^2} \biggl[ \int_{0}^{\infty}\int_{0}^{2\pi}{sin^{2}\psi } \int_{0}^{\infty} f_{j\tilde r}(\tilde r) \int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)} ({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} + {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi})d \tilde \varphi \tilde r d \tilde r \cdot \nonumber\\ \nonumber\\ \nonumber\\ \cdot\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi\;- \quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ -\;\; i \int_{0}^{\infty}\int_{0}^{2\pi}{sin\psi cos\psi } \int_{0}^{\infty} f_{j\tilde r}(\tilde r) \int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)}({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} - {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi})d \tilde \varphi \tilde r d \tilde r \cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\; {\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi \biggr] \quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ $$\begin{aligned} \label{eqn413} u_{jr2}(r,\varphi)\;=\;\frac{1}{8\pi^2} \biggl[ - \int_{0}^{\infty}\int_{0}^{2\pi}{sin\psi cos\psi} \int_{0}^{\infty} f_{j\tilde r}(\tilde r) \int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)} ({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} + {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi})d \tilde \varphi \tilde r d \tilde r \cdot \nonumber\\ \nonumber\\ \nonumber\\ \cdot\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi\;+ \quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ +\;\; i \int_{0}^{\infty}\int_{0}^{2\pi}{ cos^{2}\psi } \int_{0}^{\infty} f_{j\tilde r}(\tilde r) \int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)}({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} - {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi})d \tilde \varphi \tilde r d \tilde r \cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\; {\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi \biggr] \quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ Let us denote internal integrals in $(\ref{eqn412}), (\ref{eqn413})$ as $\;\;I_{\underline{+}}(\tilde r,\rho,\psi)\;\;$: $$\label{eqn414} I_{\underline{+}}(\tilde r,\rho,\psi) = \int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)} ({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi\;\;} \underline{+}\;\; {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi})d \tilde \varphi$$ We have two integrals here. Plus (+) is for the first part of each integral $(\ref{eqn412}), (\ref{eqn413})$ and minus (-) is for the second part. We substitute $\tilde \theta$ for $\tilde \varphi$:$\;\;\;\tilde \theta\;$ = $\;\tilde \varphi\;$ - $\psi\;\;$ , d$\tilde \theta\;$ = d$\tilde \varphi\;$ and receive: $$\label{eqn415} I_{\underline{+}}(\tilde r,\rho,\psi) = {\mbox{\Large e \normalsize}}^{i(n_{j}-1)\psi}\int_{-\psi}^{2\pi-\psi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos\tilde \theta + i(n_{j}-1)\tilde \theta} d \tilde \theta \;\;\underline{+}\;\;{\mbox{\Large e \normalsize}}^{i(n_{j}+1)\psi}\int_{-\psi}^{2\pi-\psi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos\tilde \theta + i(n_{j}+1)\tilde \theta} d \tilde \theta$$ Then we use the Bessel function’s integral representation $(\ref{A9})$ and have: $$\label{eqn416} I_{\underline{+}}(\tilde r,\rho,\psi) = 2\pi i^{(n_{j}-1)}{\mbox{\Large e \normalsize}}^{i(n_{j}-1)\psi}J_{n_{j}-1}(\tilde{r}\rho)\;\;\underline{+}\;\;2\pi i^{(n_{j}+1)}{\mbox{\Large e \normalsize}}^{i(n_{j}+1)\psi}J_{n_{j}+1}(\tilde{r}\rho)$$ Put $\;\;I_{\underline{+}}(\tilde r,\rho,\psi)\;\;$ from $(\ref{eqn416})$ in formulas $(\ref{eqn412}), (\ref{eqn413})\;$ , change order of integration and obtain: $$\label{eqn417} u_{jr1}(r,\varphi)\;=\;\frac{1}{8\pi^2}\int_{0}^{\infty}\int_{0}^{\infty}f_{j\tilde r}(\tilde r)\int_{0}^{2\pi}\biggl[{sin^{2} \psi} I_{+}(\tilde r,\rho,\psi)-i\;{sin\psi cos\psi}I_{-}(\tilde r,\rho,\psi)\biggr]\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}d\psi \tilde r d \tilde r \rho d\rho \nonumber\\ \nonumber\\ \quad\quad$$ $$\label{eqn418} u_{jr2}(r,\varphi)\;=\;\frac{1}{8\pi^2}\int_{0}^{\infty}\int_{0}^{\infty}f_{j\tilde r}(\tilde r)\int_{0}^{2\pi}\biggl[-{sin\psi}{cos\psi} I_{+}(\tilde r,\rho,\psi)+i\;{cos^{2}\psi}I_{-}(\tilde r,\rho,\psi)\biggr]\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}d\psi \tilde r d \tilde r \rho d\rho \nonumber\\ \nonumber\\ \quad\quad$$ Then we group parts in brackets of formulas $(\ref{eqn417}), (\ref{eqn418})\;$ , use De Moivre’s formulas $(\ref{A8})$ and the Bessel function’s properties. And we get: $$\label{eqn419} u_{jr1}(r,\varphi)\;=\;-\;\frac{n_{j}\;i^{n_{j}}}{2\pi}\int_{0}^{\infty}\int_{0}^{\infty}f_{j\tilde r}(\tilde r)\int_{0}^{2\pi}{sin\psi}\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)+i n_{j} \psi}\;d\psi \;J_{n_{j}}(\tilde r \rho)\;d \tilde r d\rho \quad\quad$$ $$\label{eqn420} u_{jr2}(r,\varphi)\;=\;\frac{n_{j}\;i^{n_{j}}}{2\pi}\int_{0}^{\infty}\int_{0}^{\infty}f_{j\tilde r}(\tilde r)\int_{0}^{2\pi}{cos\psi}\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)+i n_{j} \psi}\;d\psi \;J_{n_{j}}(\tilde r \rho)\;d \tilde r d\rho \quad\quad$$ We substitute $\theta$ for $\psi$:$\;\;\;\theta\;$ = $\;\psi\;$ - $\varphi\;\;$ , d$\theta\;$ = d$\psi\;$ in the internal integrals of formulas $(\ref{eqn419}), (\ref{eqn420})\;$, use De Moivre’s formulas $(\ref{A8})$ and the Bessel function’s integral representation $(\ref{A9})$ and have from formulas $(\ref{eqn419}), (\ref{eqn420})$: $$\label{eqn421} u_{jr1}(r,\varphi)\;=\;\frac{n_{j}}{2}\;{\mbox{\Large e \normalsize}} ^{i n_{j} \varphi}\int_{0}^{\infty}\biggl[{\mbox{\Large e \normalsize}} ^{i \varphi}J_{n_{j+1}}(r \rho)\;+\;{\mbox{\Large e \normalsize}} ^{-i \varphi}J_{n_{j-1}}(r \rho)\biggr]\int_{0}^{\infty}f_{j\tilde r}(\tilde r)\;J_{n_{j}}(\tilde r \rho)\;d \tilde r d\rho \quad\quad$$ $$\label{eqn422} u_{jr2}(r,\varphi)\;=\;\frac{i\; n_{j}}{2}\;{\mbox{\Large e \normalsize}} ^{i n_{j} \varphi}\int_{0}^{\infty}\biggl[{\mbox{\Large e \normalsize}} ^{i \varphi}J_{n_{j+1}}(r \rho)\;-\;{\mbox{\Large e \normalsize}} ^{-i \varphi}J_{n_{j-1}}(r \rho)\biggr]\int_{0}^{\infty}f_{j\tilde r}(\tilde r)\;J_{n_{j}}(\tilde r \rho)\;d \tilde r d\rho \quad\quad$$ Let us denote: $$\label{eqn423} R_{j,n_{j}-1,r}(r)\;=\;\int_{0}^{\infty}\int_{0}^{\infty}f_{j\tilde r}(\tilde r)\;J_{n_{j}}(\tilde r \rho)J_{n_{j}-1}(r \rho)\;d \tilde r d\rho \quad\quad$$ $$\label{eqn424} R_{j,n_{j}+1,r}(r)\;=\;\int_{0}^{\infty}\int_{0}^{\infty}f_{j\tilde r}(\tilde r)\;J_{n_{j}}(\tilde r \rho)J_{n_{j}+1}(r \rho)\;d \tilde r d\rho \quad\quad$$ Then we have from formulas $(\ref{eqn421}), (\ref{eqn422})$: $$\label{eqn425} u_{jr1}(r,\varphi)\;=\;\frac{n_{j}}{2}\bigl[R_{j,n_{j}-1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (n_{j}-1) \varphi}\;+\;R_{j,n_{j}+1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (n_{j}+1) \varphi}\bigr]$$ $$\label{eqn426} u_{jr2}(r,\varphi)\;=\;\frac{i\;n_{j}}{2}\bigl[R_{j,n_{j}-1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (n_{j}-1) \varphi}\;-\;R_{j,n_{j}+1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (n_{j}+1) \varphi}\bigr]$$ Then if $n_{j}\;=\;0$ it follows from $(\ref{eqn423}), (\ref{eqn424}) , (\ref{eqn425}), (\ref{eqn426})$ that $u_{jr1}(r,\varphi)\;=\;u_{jr2}(r,\varphi)\;=\;0\;$ and hence $u_{1}\;\;=\;\;u_{2}\;\;=\;\;0$. In the equations bellow we will consider $n_{j}\;\geq\;1$. We change the order of integration in formulas $(\ref{eqn423}), (\ref{eqn424}) $ and obtain: $$\label{eqn427} R_{j,n_{j}-1,r}(r)\;=\;\int_{0}^{\infty}f_{j\tilde r}(\tilde r)\int_{0}^{\infty}J_{n_{j}}(\tilde r \rho)J_{n_{j}-1}(r \rho)\;d\rho\; d \tilde r \quad\quad$$ $$\label{eqn428} R_{j,n_{j}+1,r}(r)\;=\;\int_{0}^{\infty}f_{j\tilde r}(\tilde r)\int_{0}^{\infty}J_{n_{j}}(\tilde r \rho)J_{n_{j}+1}(r \rho)\;d\rho\; d \tilde r \quad\quad$$ Internal integrals in formulas $(\ref{eqn427}), (\ref{eqn428})$ are established by the discontinuous integral of Weber and Schafheitlin $(\ref{A17})\; \cite{gW44}$. Then we have from $(\ref{eqn427}), (\ref{eqn428})$: $$\label{eqn429} R_{j,n_{j}-1,r}(r)\;=\;r^{n_{j}-1} \int_{r}^{\infty}\frac{f_{j\tilde r}(\tilde r)}{\tilde r^{n_{j}}} d \tilde r \quad\quad$$ $$\label{eqn430} R_{j,n_{j}+1,r}(r)\;=\;\frac{1}{r^{n_{j}+1}} \int_{0}^{r}\tilde r^{n_{j}} f_{j\tilde r}(\tilde r) d \tilde r \quad\quad$$ Now we integrate the solution $(\ref{eqn403}), (\ref{eqn404})$ by the tangential component of the applied force $(\ref{eqn406})$ for ($n_{j}\;\geq\;1$). Then we use De Moivre’s formulas $(\ref{A8})$ and have: $$\begin{aligned} \label{eqn431} f_{j1}(\tilde r,\tilde \varphi,\tau) = - f_{j\tilde \varphi}(\tilde r){\mbox{\Large e \normalsize}}^{in_{j}\tilde \varphi}sin \tilde \varphi f_{j\tau}(\tau) = - \frac{i}{2}f_{j\tilde \varphi}(\tilde r) \bigl( {\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} - {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi} \bigr) f_{j\tau}(\tau) \nonumber\\ \nonumber\\ f_{j2}(\tilde r,\tilde \varphi,\tau) \;\;=\;\; f_{j\tilde \varphi}(\tilde r){\mbox{\Large e \normalsize}}^{in_{j}\tilde \varphi}cos \tilde \varphi f_{j\tau}(\tau) \;\;= \; \frac{1}{2} f_{j\tilde \varphi}(\tilde r)\bigl({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} + {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi}\bigr)f_{j\tau}(\tau) \nonumber\\ \nonumber\\\end{aligned}$$ Hence formulas $(\ref{eqn431})$ are the components $f_{j1}$ and $f_{j2}$ from the tangential component of the applied force $(\ref{eqn406})$, while formulas $(\ref{eqn407})$ are the components $f_{j1}$ and $f_{j2}$ from the radial component of the applied force $(\ref{eqn405})$. Let us put $(\ref{eqn431})$ in formulas $(\ref{eqn403}), (\ref{eqn404})$ and do the operations as we did in $(\ref{eqn408}) - (\ref{eqn426})\; (\;n_{j}\;\geq\;1)$. We consider that $f_{j\tilde \varphi}(\tilde r)\cdot f_{j\tau}(\tau)$ is restricted by condition $(\ref{eqn18})$ and get: $$\label{eqn432} R_{j,n_{j}-1,\varphi}(r)\;=\;-\;\int_{0}^{\infty}\int_{0}^{\infty}\bigl(f_{j\tilde \varphi}(\tilde r)\;\tilde r\bigr)^{'}_{\tilde r}\;J_{n_{j}}(\tilde r \rho)J_{n_{j}-1}(r \rho)\;d \tilde r d\rho \quad\quad$$ $$\label{eqn433} R_{j,n_{j}+1,\varphi}(r)\;=\;-\;\int_{0}^{\infty}\int_{0}^{\infty}\bigl(f_{j\tilde \varphi}(\tilde r)\;\tilde r\bigr)^{'}_{\tilde r}\;J_{n_{j}}(\tilde r \rho)J_{n_{j}+1}(r \rho)\;d \tilde r d\rho \;, \quad\quad$$ Here $\bigl(\bigr)^{'}_{\tilde r} \equiv \frac{\partial}{\partial \tilde r}$. Hence we have: $$\label{eqn434} u_{j\varphi1}(r,\varphi)\;=\;-\;\frac{i}{2}\bigl[R_{j,n_{j}-1,\varphi}(r)\;{\mbox{\Large e \normalsize}} ^{i (n_{j}-1) \varphi}\;+\;R_{j,n_{j}+1,\varphi}(r)\;{\mbox{\Large e \normalsize}} ^{i (n_{j}+1) \varphi}\bigr]$$ $$\label{eqn435} u_{j\varphi2}(r,\varphi)\;=\;\frac{1}{2}\bigl[R_{j,n_{j}-1,\varphi}(r)\;{\mbox{\Large e \normalsize}} ^{i (n_{j}-1) \varphi}\;-\;R_{j,n_{j}+1,\varphi}(r)\;{\mbox{\Large e \normalsize}} ^{i (n_{j}+1) \varphi}\bigr]$$ We change the order of integration in formulas $(\ref{eqn432}), (\ref{eqn433}) $ and obtain: $$\label{eqn436} R_{j,n_{j}-1,\varphi}(r)\;=\;-\;\int_{0}^{\infty}\bigl(f_{j\tilde \varphi}(\tilde r)\;\tilde r\bigr)^{'}_{\tilde r}\int_{0}^{\infty}\;J_{n_{j}}(\tilde r \rho)J_{n_{j}-1}(r \rho)\;d\rho d \tilde r \quad\quad$$ $$\label{eqn437} R_{j,n_{j}+1,\varphi}(r)\;=\;-\;\int_{0}^{\infty}\bigl(f_{j\tilde \varphi}(\tilde r)\;\tilde r\bigr)^{'}_{\tilde r}\int_{0}^{\infty}\;J_{n_{j}}(\tilde r \rho)J_{n_{j}+1}(r \rho)\; d\rho d \tilde r \quad\quad$$ Internal integrals in formulas $(\ref{eqn436}), (\ref{eqn437})$ are established by the discontinuous integral of Weber and Schafheitlin $(\ref{A17})\cite{gW44}$. Then we have from $(\ref{eqn436}), (\ref{eqn437})$: $$\label{eqn438} R_{j,n_{j}-1,\varphi}(r)\;=\; -\; r^{n_{j}-1} \int_{r}^{\infty}\bigl(f_{j\tilde \varphi}(\tilde r)\;\tilde r\bigr)^{'}_{\tilde r}\;\frac{d \tilde r}{\tilde r^{n_{j}}} \quad\quad$$ $$\label{eqn439} R_{j,n_{j}+1,\varphi}(r)\;=\; -\; \frac{1}{r^{n_{j}+1}} \int_{0}^{r}\bigl(f_{j\tilde \varphi}(\tilde r)\;\tilde r\bigr)^{'}_{\tilde r}\;\tilde r^{n_{j}} d \tilde r \quad\quad$$ We have obtain formulas $(\ref{eqn407})\;$ - $\;(\ref{eqn439})$ for an arbitrary step j of the iterative process and the applied forces $(\ref{eqn405})$ or $(\ref{eqn406})$. Now we investigate the first step (j = 1 , $n_{1} = n = 1,2,3,...$) of the iterative process with the particular radial component of the applied force ${f}_{1}(x,t)\;\;$\[ look at $\;(\ref{eqn47})$\]: $$\begin{aligned} \label{eqn440} f_{1\tilde r}(\tilde r,\tilde \varphi,\tau) = f_{1\tilde r}(\tilde r){\mbox{\Large e \normalsize}}^{in\tilde \varphi}f_{1\tau}(\tau)\;\;\;,\;\;\;f_{1\tilde \varphi}(\tilde r,\tilde \varphi,\tau) \equiv 0 \nonumber\\ \nonumber\\ f_{1\tilde r}(\tilde r) = F_{n}\tilde r^{n+1}{\mbox{\Large e \normalsize}}^{-\mu_{n}\tilde r}\;\;,\;\;f_{1\tau}(\tau) = {\mbox{\Large e \normalsize}}^{-\sigma_{n}\tau} \quad\quad\end{aligned}$$ $F_{n}, \mu_{n}, \sigma_{n}$ - constants. $0 < F_{n}< \infty$ , $1 < \mu_{n}< \infty$ , $1 < \sigma_{n}< \infty$. Let us put the applied force $(\ref{eqn440})$ in formulas $(\ref{eqn429}), (\ref{eqn430})$, integrate and then we have: $$\begin{aligned} \label{eqn441} R_{1,n-1,r}(r)\;=\;r^{n-1} \int_{r}^{\infty}\frac{F_{n}\tilde r^{n+1}{\mbox{\Large e \normalsize}}^{-\mu_{n}\tilde r}}{\tilde r^{n}} d \tilde r\;=\;F_{n} r^{n-1} \int_{r}^{\infty}{\mbox{\Large e \normalsize}}^{-\mu_{n}\tilde r} \tilde r d \tilde r\;=\;\frac{F_{n} r^{n-1}}{\mu_{n}^2}\Gamma (2, \mu_{n}r) \nonumber\\ \nonumber\\ \quad\quad\end{aligned}$$ $$\begin{aligned} \label{eqn442} R_{1,n+1,r}(r)\;=\;\frac{1}{r^{n+1}} \int_{0}^{r} F_{n}\tilde r^{2n+1}{\mbox{\Large e \normalsize}}^{-\mu_{n}\tilde r} d \tilde r\;=\;\;\frac{F_{n}}{r^{n+1}\mu_{n}^{2n+2}}\gamma(2n+2, \mu_{n}r) \nonumber\\ \nonumber\\ \quad\quad\end{aligned}$$ $\Gamma (\alpha, x) , \gamma(\alpha, x)$ are the incomplete gamma functions $\cite{BE253}$. Hence, and from formulas $(\ref{eqn425}), (\ref{eqn426})$ it follows by j = 1: $$\label{eqn443} u_{1r1}(r,\varphi)\;=\;\frac{n}{2}\bigl[R_{1,n-1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;+\;R_{1,n+1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]$$ $$\label{eqn444} u_{1r2}(r,\varphi)\;=\;\frac{i\;n}{2}\bigl[R_{1,n-1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;-\;R_{1,n+1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]$$ and from formula $(\ref{eqn410})$: $$\label{eqn445} u_{1t}(t) = \int_{0}^{t}f_{1\tau}(\tau) d\tau = \int_{0}^{t}{\mbox{\Large e \normalsize}}^{-\sigma_{n}\tau}d\tau = \frac{1}{\sigma_{n}}\gamma(1,\sigma_{n}t)$$ And we have the velocity $\vec u_{1}$ \[look at $\;(\ref{eqn47})$\] from formulas $(\ref{eqn411})$: $$\begin{aligned} \label{eqn446} u_{1r1}(r,\varphi,t)\;=\;u_{1r1}(r,\varphi)\;u_{1t}(t) \nonumber\\ \nonumber\\ u_{1r2}(r,\varphi,t)\;=\;u_{1r2}(r,\varphi)\;u_{1t}(t)\end{aligned}$$ We obtain the following equations by performing appropriate transformations: $$\label{eqn447} u_{1 r}(r,\varphi, t)\;=\;\frac{n}{2}\bigl[R_{1,n-1,r}(r)\;+\;R_{1,n+1,r}(r)\bigr]\;{\mbox{\Large e \normalsize}} ^{i n \varphi}\;u_{1t}(t)$$ $$\label{eqn448} u_{1\varphi}(r,\varphi, t)\;=\;\frac{i\;n}{2}\bigl[R_{1,n-1,r}(r)\;-\;R_{1,n+1,r}(r)\bigr]\;{\mbox{\Large e \normalsize}}^{i n \varphi}\;u_{1t}(t)$$ $u_{1 r}(r,\varphi, t) ,\;\; u_{1\varphi}(r,\varphi, t)$ are the radial and tangential components of the velocity $\vec u_{1}$. We have from formula $(\ref{eqn445})$: $$\begin{array}{ll} lim\;\;u_{1t}(t)= 0 \\ t \rightarrow 0 \end{array}$$ Hence, and from formulas $(\ref{eqn446})-(\ref{eqn448})$: $$\begin{array}{ll} lim \;\;u_{1r1}(r,\varphi,t)= 0;\;\;\; lim \;\;u_{1r2}(r,\varphi,t)= 0;\\ t \rightarrow 0; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; t \rightarrow 0;\\ lim \;\;u_{1r}(r,\varphi,t)= 0;\;\;\; lim \;\;u_{1\varphi}(r,\varphi,t)= 0;\\ t \rightarrow 0; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; t \rightarrow 0;\\ \end{array}$$ In other words the velocity $\vec u_{1}$ satisfies the initial conditions $(\ref{eqn400})$. We use the asymptotic properties of the incomplete gamma functions $\Gamma (\alpha, x) ,\;\; \gamma(\alpha, x)$ and from formulas $(\ref{eqn441}) - (\ref{eqn444})\;,$ $(\ref{eqn447}),\;\; (\ref{eqn448})$ we have the velocity $\vec u_{1}$ satisfies conditions $(\ref{eqn16})$ (for $r \;\rightarrow\; \infty )$. Let us continue investigation for the second step (j = 2) of the iterative process. Find $\vec{f}_{2}^{*}(r,\varphi, t) = \{f_{21}^{*}, f_{22}^{*}\}$ - the first correction of the particular radial applied force ${f}_{1}(x,t)$ $(\ref{eqn440})$. We have for $\vec{f}_{2}^{*}$ from formula $(\ref{eqn48})$: $$\label{eqn449} f_{21}^{*} = u_{1r1}\;\frac{\partial u_{1r1}}{\partial x_{1}}\;+\;u_{1r2}\;\frac{\partial u_{1r1}}{\partial x_{2}}$$ $$\label{eqn450} f_{22}^{*} = u_{1r1}\;\frac{\partial u_{1r2}}{\partial x_{1}}\;+\;u_{1r2}\;\frac{\partial u_{1r2}}{\partial x_{2}}$$ where $u_{1r1},\;\;u_{1r2}$ are the components of $\vec{u_{1}}$ and were taken from formulas $(\ref{eqn446})$. We have here: $$\begin{aligned} \label{eqn451} \frac{\partial u_{1r1}(r,\varphi, t)}{\partial x_{1}} \; = \; \frac{\partial u_{1r1}(r,\varphi, t)}{\partial r}\; \frac{\partial r}{\partial x_{1}} \; + \; \frac{\partial u_{1r1}(r,\varphi, t)}{\partial \varphi}\; \frac{\partial \varphi}{\partial x_{1}} \nonumber\\ \nonumber\\ \frac{\partial u_{1r1}(r,\varphi, t)}{\partial x_{2}}\; = \; \frac{\partial u_{1r1}(r,\varphi, t)}{\partial r}\; \frac{\partial r}{\partial x_{2}} \; + \; \frac{\partial u_{1r1}(r,\varphi, t)}{\partial \varphi}\; \frac{\partial \varphi}{\partial x_{2}} \nonumber\\ \nonumber\\ \frac{\partial u_{1r2}(r,\varphi, t)}{\partial x_{1}}\; = \; \frac{\partial u_{1r2}(r,\varphi, t)}{\partial r}\; \frac{\partial r}{\partial x_{1}} \; + \; \frac{\partial u_{1r2}(r,\varphi, t)}{\partial \varphi}\; \frac{\partial \varphi}{\partial x_{1}} \nonumber\\ \nonumber\\ \frac{\partial u_{1r2}(r,\varphi, t)}{\partial x_{2}}\; = \; \frac{\partial u_{1r2}(r,\varphi, t)}{\partial r}\; \frac{\partial r}{\partial x_{2}} \; + \; \frac{\partial u_{1r2}(r,\varphi, t)}{\partial \varphi}\; \frac{\partial \varphi}{\partial x_{2}} \nonumber\\ \nonumber\\ \frac{\partial r}{\partial x_{1}} = cos \varphi \; , \; \frac{\partial \varphi}{\partial x_{1}} = \;-\; \frac{sin \varphi}{r}\;,\; \frac{\partial r}{\partial x_{2}} = sin \varphi \; , \;\frac{\partial \varphi}{\partial x_{2}} = \frac{cos \varphi}{r} \nonumber\\ \nonumber\\\end{aligned}$$ Hence, we use formulas $(\ref{eqn443}) - (\ref{eqn446})$ for $u_{1r1}(r,\varphi, t),\;\; u_{1r2}(r,\varphi, t)$ and have from $(\ref{eqn451})$: $$\begin{aligned} \label{eqn452} \frac{\partial u_{1r1}(r,\varphi, t)}{\partial x_{1}} \; = \; \frac{n}{2}\biggl\{\bigl[R_{1,n-1,r}^{'}(r)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;+\;R_{1,n+1,r}^{'}(r)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]\;cos \varphi\;+ \nonumber\\ +\;i\;\bigl[(n-1)R_{1,n-1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;+\;(n+1)R_{1,n+1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]\bigl(-\frac{sin \varphi}{r} \bigr ) \biggr \}\;u_{1t}(t) \nonumber\\ \nonumber\\ \frac{\partial u_{1r1}(r,\varphi, t)}{\partial x_{2}}\; = \; \frac{n}{2}\biggl\{\bigl[R_{1,n-1,r}^{'}(r)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;+\;R_{1,n+1,r}^{'}(r)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]\;sin \varphi\;+ \nonumber\\ +\;i\;\bigl[(n-1)R_{1,n-1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;+\;(n+1)R_{1,n+1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]\frac{cos \varphi}{r} \biggr \}\;u_{1t}(t) \nonumber\\ \nonumber\\ \frac{\partial u_{1r2}(r,\varphi, t)}{\partial x_{1}}\; = \; \frac{i n}{2}\biggl\{\bigl[R_{1,n-1,r}^{'}(r)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;-\;R_{1,n+1,r}^{'}(r)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]\;cos \varphi\;+ \nonumber\\ +\;i\;\bigl[(n-1)R_{1,n-1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;-\;(n+1)R_{1,n+1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]\bigl(-\frac{sin \varphi}{r} \bigr ) \biggr \}\;u_{1t}(t) \nonumber\\ \nonumber\\ \frac{\partial u_{1r2}(r,\varphi, t)}{\partial x_{2}}\; = \; \frac{i n}{2}\biggl\{\bigl[R_{1,n-1,r}^{'}(r)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;-\;R_{1,n+1,r}^{'}(r)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]\;sin \varphi\;+ \nonumber\\ +\;i\;\bigl[(n-1)R_{1,n-1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;-\;(n+1)R_{1,n+1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]\frac{cos \varphi}{r} \biggr \}\;u_{1t}(t) \nonumber\\ \nonumber\\\end{aligned}$$ where $$\begin{aligned} \label{eqn453} R_{1,n-1,r}^{'}(r)\;=\;\frac{d R_{1,n-1,r}(r)}{d r}\;=\;F_{n}\bigl [ \frac{(n-1) r^{n-2}}{\mu_{n}^2}\; \Gamma (2, \mu_{n}r) - r^{n} {\mbox{\Large e \normalsize}} ^{- \mu_{n}r}\bigr ] \quad\quad \nonumber\\ \nonumber\\ R_{1,n+1,r}^{'}(r)\;=\;\frac{d R_{1,n+1,r}(r)}{d r}\;=\;F_{n}\bigl [- \frac{ (n+1) }{r^{n+2} \mu_{n}^{2n+2}}\; \gamma (2n+2, \mu_{n}r) + r^{n} {\mbox{\Large e \normalsize}} ^{- \mu_{n}r}\bigr ] \nonumber\\ \nonumber\\ \quad\quad\end{aligned}$$ Let us put $u_{1r1},\; u_{1r2},\; \frac{\partial u_{1r1}}{\partial x_{1}},\; \frac{\partial u_{1r1}}{\partial x_{2}},\; \frac{\partial u_{1r2}}{\partial x_{1}},\; \frac{\partial u_{1r2}}{\partial x_{2}}\;$ from formulas $(\ref{eqn446}), (\ref{eqn452})$ in formulas $(\ref{eqn449}), (\ref{eqn450})$ for $f_{21}^{*} \;,\; f_{22}^{*}\;$.\ \ After completing appropriate operations we have: $$\label{eqn454} f_{21}^{*} (r,\varphi, t) = \frac{n^{2}}{2^{2}}\bigl [ T_{2,2n-1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (2n-1) \varphi} + T_{2,2n+1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (2n+1) \varphi} \bigr ] T_{2n}(t)$$ $$\label{eqn455} f_{22}^{*} (r,\varphi, t) = \frac{i n^{2}}{2^{2}}\bigl [ T_{2,2n-1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (2n-1) \varphi} - T_{2,2n+1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (2n+1) \varphi} \bigr ] T_{2n}(t)$$ where $$\begin{aligned} \label{eqn456} T_{2,2n-1,r}(r) = \bigl [ R_{1,n-1,r}(r) + R_{1,n+1,r}(r) \bigr ] R_{1,n-1,r}^{'}(r) - \frac{(n-1)R_{1,n-1,r}(r)}{r} \bigl [ R_{1,n-1,r}(r) - R_{1,n+1,r}(r) \bigr ] \quad\quad \nonumber\\ \nonumber\\ T_{2,2n+1,r}(r) = \bigl [ R_{1,n-1,r}(r) + R_{1,n+1,r}(r) \bigr ] R_{1,n+1,r}^{'}(r) - \frac{(n+1)R_{1,n+1,r}(r)}{r} \bigl [ R_{1,n-1,r}(r) - R_{1,n+1,r}(r) \bigr ] \quad\quad \nonumber\\ \nonumber\\ \quad\quad\end{aligned}$$ $$\label{eqn457} T_{2n} (t) = u_{1t}^{2} (t)$$ We use formulas $(\ref{eqn441}) , (\ref{eqn442})$ for $R_{1,n-1,r}(r)\;,\;R_{1,n+1,r}(r)$ and $(\ref{eqn453})$ for $R_{1,n-1,r}^{'}(r)\;,\;R_{1,n+1,r}^{'}(r)$ then do appropriate operations for $T_{2,2n-1,r}(r)\;,\;T_{2,2n+1,r}(r)$ and get: $$\begin{aligned} \label{eqn458} T_{2,2n-1,r}(r) = F_{n}^{2}\;\bigl [-\; \frac{ r^{2n-1}}{\mu_{n}^2}\;{\mbox{\Large e \normalsize}} ^{- \mu_{n}r}\; \Gamma (2, \mu_{n}r)\;+\;\frac{2(n-1)}{r^{3}\mu_{n}^{2n+4}}\;\gamma (2n+2, \mu_{n}r)\;\Gamma (2, \mu_{n}r)\;- \quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ -\;\frac{1}{r\mu_{n}^{2n+2}}\;{\mbox{\Large e \normalsize}} ^{- \mu_{n}r}\;\gamma (2n+2, \mu_{n}r) \bigr ] \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ T_{2,2n+1,r}(r) = F_{n}^{2}\;\bigl [ \frac{ r^{2n-1}}{\mu_{n}^2}\;{\mbox{\Large e \normalsize}} ^{- \mu_{n}r}\; \Gamma (2, \mu_{n}r)\;-\;\frac{2(n+1)}{r^{3}\mu_{n}^{2n+4}}\;\gamma (2n+2, \mu_{n}r)\;\Gamma (2, \mu_{n}r)\;+ \quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ +\;\frac{1}{r\mu_{n}^{2n+2}}\;{\mbox{\Large e \normalsize}} ^{- \mu_{n}r}\;\gamma (2n+2, \mu_{n}r) \bigr ] \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ For radial $f_{2r}^{*}$ and tangential $f_{2\varphi}^{*}$ components of the first correction $\vec{f_{2}^{*}}(r, \varphi, t)$ of the particular radial applied force we have: $$\label{eqn459} f_{2r}^{*} (r,\varphi, t) = \frac{n^{2}}{2^{2}}\bigl [ T_{2,2n-1,r}(r) + T_{2,2n+1,r}(r) \bigr ]\; {\mbox{\Large e \normalsize}} ^{i\; 2n \varphi}\;T_{2n}(t)\;=\;\frac{n^{2}}{2^{2}}T_{2,2n,r}(r)\; {\mbox{\Large e \normalsize}} ^{i\; 2n \varphi}\;T_{2n}(t)$$ $$\label{eqn460} f_{2\varphi}^{*} (r,\varphi, t) = \frac{i n^{2}}{2^{2}}\bigl [ T_{2,2n-1,r}(r) - T_{2,2n+1,r}(r) \bigr ]\; {\mbox{\Large e \normalsize}} ^{i\; 2n \varphi}\; T_{2n}(t)\;=\;\frac{i n^{2}}{2^{2}}T_{2,2n,\varphi}(r)\; {\mbox{\Large e \normalsize}} ^{i\; 2n \varphi}\;T_{2n}(t)$$ Here $$\label{eqn461} T_{2,2n,r}(r) = -\;F_{n}^{2}\;\frac{4}{r^{3}\mu_{n}^{2n+4}}\;\gamma (2n+2, \mu_{n}r)\;\Gamma (2, \mu_{n}r)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\$$ $$\begin{aligned} \label{eqn462} T_{2,2n,\varphi}(r) = -\;F_{n}^{2}\;\bigl [ \frac{ 2 r^{2n-1}}{\mu_{n}^2}\;{\mbox{\Large e \normalsize}} ^{- \mu_{n}r}\; \Gamma (2, \mu_{n}r)\;-\;\frac{4 n}{r^{3}\mu_{n}^{2n+4}}\;\gamma (2n+2, \mu_{n}r)\;\Gamma (2, \mu_{n}r)\;+ \nonumber\\ +\;\frac{2}{r\mu_{n}^{2n+2}}\;{\mbox{\Large e \normalsize}} ^{- \mu_{n}r}\;\gamma (2n+2, \mu_{n}r) \bigr ] \quad\quad \nonumber\\ \nonumber\\\end{aligned}$$ We compare the particular radial applied force $\vec{f_{1}}\;$ from $\;(\ref{eqn440})$ with the first correction $\vec{f_{2}^{*}}\;$ from $\;((\ref{eqn459})- (\ref{eqn462}))$ of this particular radial applied force, and we have: $$\label{eqn463} \mid\vec{f_{2}^{*}}\mid\; <<\; \mid\vec{f_{1}}\mid$$ with condition $$\label{eqn464} F_{n}\;\leq\;\frac{1}{n}$$ After the first step of the iterative process (j = 1) we obtained the velocity $\vec{u_{1}}\;$ \[ see $\; (\ref{eqn446})$\].\ Now we will calculate $\vec{u_{2}^{*}}\;$ - the first correction of the velocity $\vec{u_{1}} $. Solution of this problem has two stages. On the first stage we find the part of the first correction $\vec{u_{2r}^{*}}$, corresponding to the radial component of the first correction of applied force $f_{2r}^{*}\;$ from $\;(\ref{eqn459})$: $$\label{eqn465} f_{2r}^{*}(r,\varphi,t) = \;\frac{n^{2}}{2^{2}}T_{2,2n,r}(r)\; {\mbox{\Large e \normalsize}} ^{i\; 2n \varphi}\;T_{2n}(t)\;\;\;,\;\;\;f_{2\varphi}^{*}(r,\varphi,t) \equiv 0$$ On the second stage we calculate the other part of the first correction $\vec{u_{2\varphi}^{*}}\;$, corresponding to the tangential component of the first correction of applied force $f_{2\varphi}^{*}\;$ from $\;(\ref{eqn460})$: $$\label{eqn466} f_{2r}^{*}(r,\varphi,t) \equiv 0\;\;\;,\;\;\; f_{2\varphi}^{*}(r,\varphi,t) = \;\frac{i n^{2}}{2^{2}}T_{2,2n,\varphi}(r)\; {\mbox{\Large e \normalsize}} ^{i\; 2n\varphi}\;T_{2n}(t)$$ In other words $$\begin{aligned} \label{eqn467} \vec{u_{2}^{*}} = \vec{u_{2r}^{*}} + \vec{u_{2\varphi}^{*}}, \;\;\;\vec{u_{2r}^{*}} = \{u_{2r1}^{*}, u_{2r2}^{*}\}, \;\;\;\vec{u_{2\varphi}^{*}} = \{u_{2\varphi1}^{*}, u_{2\varphi2}^{*}\}.\end{aligned}$$ First stage: we use formulas $(\ref{eqn425})\;,\;(\ref{eqn426})\;$ for components $\;u_{jr1}(r, \varphi),\; u_{jr2}(r, \varphi)$ and formulas $(\ref{eqn429})\;,\;(\ref{eqn430})$ for $R_{j,n_{j}-1,r}(r),\; R_{j,n_{j}+1,r}(r)$ for j = 2 and then formulas $(\ref{eqn459})\;,\;(\ref{eqn461})$ for $f_{2r}^{*} (r,\varphi, t)\;,\;T_{2,2n,r}(r)$. We do appropriate operations and have: $$\label{eqn468} u_{2r1}^{*}(r,\varphi)\;=\;n\bigl[R_{2,2n-1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (2n-1) \varphi}\;+\;R_{2,2n+1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (2n+1) \varphi}\bigr]$$ $$\label{eqn469} u_{2r2}^{*}(r,\varphi)\;=\;i\;n\bigl[R_{2,2n-1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (2n-1) \varphi}\;-\;R_{2,2n+1,r}(r)\;{\mbox{\Large e \normalsize}} ^{i (2n+1) \varphi}\bigr]$$ Here $$\label{eqn470} R_{2,2n-1,r}(r)\;=\;\frac{n^{2}r^{2n-1}}{2^{2}} \int_{r}^{\infty}\frac{T_{2,2n,r}(\tilde r)}{\tilde r^{2n}} d \tilde r \quad\quad$$ $$\label{eqn471} R_{2,2n+1,r}(r)\;=\;\frac{n^{2}}{2^{2} r^{2n+1}} \int_{0}^{r}\tilde r^{2n} T_{2,2n,r}(\tilde r) d \tilde r \quad\quad$$ Second stage: we use formulas $(\ref{eqn434})\;,\;(\ref{eqn435})$ for components $u_{j\varphi1}(r, \varphi),\; u_{j\varphi2}(r, \varphi)$ and formulas $(\ref{eqn438})\;,\;(\ref{eqn439})$ for $R_{j,n_{j}-1,\varphi}(r),\; R_{j,n_{j}+1,\varphi}(r)$ for j = 2 and then formulas $(\ref{eqn460})\;,\;(\ref{eqn462})$ for $f_{2\varphi}^{*} (r,\varphi, t)\;,\;T_{2,2n,\varphi}(r)$. We do appropriate operations and have: $$\label{eqn472} u_{2\varphi1}^{*}(r,\varphi)\;=\;- \;\frac{i}{2}\bigl[R_{2,2n-1,\varphi}(r)\;{\mbox{\Large e \normalsize}} ^{i (2n-1) \varphi}\;+\;R_{2,2n+1,\varphi}(r)\;{\mbox{\Large e \normalsize}} ^{i (2n+1) \varphi}\bigr]$$ $$\label{eqn473} u_{2\varphi2}^{*}(r,\varphi)\;=\;\frac{1}{2}\bigl[R_{2,2n-1,\varphi}(r)\;{\mbox{\Large e \normalsize}} ^{i (2n-1) \varphi}\;-\;R_{2,2n+1,\varphi}(r)\;{\mbox{\Large e \normalsize}} ^{i (2n+1) \varphi}\bigr]$$ Here: $$\label{eqn474} R_{2,2n-1,\varphi}(r)\;=\;-\;\frac{i n^{2}r^{2n-1}}{2^{2}} \int_{r}^{\infty}\frac{(T_{2,2n,\varphi}(\tilde r)\cdot \tilde r)_{\tilde r}^{'}}{\tilde r^{2n}} d \tilde r \quad\quad$$ $$\label{eqn475} R_{2,2n+1,\varphi}(r)\;=\;-\;\frac{i n^{2}}{2^{2} r^{2n+1}} \int_{0}^{r}\tilde r^{2n} (T_{2,2n,\varphi}(\tilde r)\cdot \tilde r)_{\tilde r}^{'} d \tilde r \quad\quad$$ Then we have: $$\begin{aligned} \label{eqn476} u_{21}^{*}(r,\varphi) = u_{2r1}^{*}(r,\varphi) + u_{2\varphi1}^{*}(r,\varphi)\;= \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\ \nonumber\\ \nonumber\\ =\;\bigl [ n R_{2,2n-1,r}(r) - \frac{i}{2} R_{2,2n-1,\varphi}(r)\bigr ]\;{\mbox{\Large e \normalsize}} ^{i (2n-1) \varphi} + \bigl [ n R_{2,2n+1,r}(r) - \frac{i}{2} R_{2,2n+1,\varphi}(r)\bigr ]\;{\mbox{\Large e \normalsize}} ^{i (2n+1) \varphi} \nonumber\\ \nonumber\\\end{aligned}$$ $$\begin{aligned} \label{eqn477} u_{22}^{*}(r,\varphi) = u_{2r2}^{*}(r,\varphi) + u_{2\varphi2}^{*}(r,\varphi)\;= \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ =\;i\;\bigl [ n R_{2,2n-1,r}(r) - \frac{i}{2} R_{2,2n-1,\varphi}(r)\bigr ]\;{\mbox{\Large e \normalsize}} ^{i (2n-1) \varphi} - i\;\bigl [ n R_{2,2n+1,r}(r) - \frac{i}{2} R_{2,2n+1,\varphi}(r)\bigr ]\;{\mbox{\Large e \normalsize}} ^{i (2n+1) \varphi} \nonumber\\ \nonumber\\\end{aligned}$$ From formula $(\ref{eqn410})$ for j = 2 and formula $(\ref{eqn457})$ we have: $$\label{eqn478} u_{2t}(t) = \int_{0}^{t}T_{2n}(\tau) d\tau = \frac{1}{\sigma_{n}^{2}} \bigl [ t - \frac{2}{\sigma_{n}} \gamma (1, \sigma_{n}t) + \frac{1}{2 \sigma_{n}} \gamma (1,2 \sigma_{n}t) \bigr ]$$ Hence, and from equation $(\ref{eqn411})$ it follows: $$\begin{aligned} \label{eqn479} u_{21}^{*}(r,\varphi,t)\;=\;u_{21}^{*}(r,\varphi)\;u_{2t}(t) \nonumber\\ \nonumber\\ u_{22}^{*}(r,\varphi,t)\;=\;u_{22}^{*}(r,\varphi)\;u_{2t}(t)\end{aligned}$$ After completing appropriate operations we have: $$\begin{aligned} \label{eqn480} u_{2 r}^{*}(r,\varphi, t)\;=\;\bigl \{\bigl [ n R_{2,2n-1,r}(r) - \frac{i}{2} R_{2,2n-1,\varphi}(r)\bigr ] + \bigl [ n R_{2,2n+1,r}(r) - \frac{i}{2} R_{2,2n+1,\varphi}(r)\bigr ]\bigr \}\;{\mbox{\Large e \normalsize}} ^{i 2n \varphi}\;u_{2t}(t) \nonumber\\ \nonumber\\ \quad\quad\quad\quad\end{aligned}$$ $$\begin{aligned} \label{eqn481} u_{2\varphi}^{*}(r,\varphi, t)\;=\;i\;\bigl \{\bigl [ n R_{2,2n-1,r}(r) - \frac{i}{2} R_{2,2n-1,\varphi}(r)\bigr ] - \bigl [ n R_{2,2n+1,r}(r) - \frac{i}{2} R_{2,2n+1,\varphi}(r)\bigr ]\bigr \}\;{\mbox{\Large e \normalsize}} ^{i 2n \varphi}\;u_{2t}(t) \nonumber\\ \nonumber\\ \quad\quad\quad\quad\end{aligned}$$ Here $u_{2 r}^{*}(r,\varphi, t) ,\; u_{2\varphi}^{*}(r,\varphi, t)$ are the radial and tangential components of the first correction $\vec u_{2}^{*}$ of the velocity $\vec u_{1}$ and $$\begin{aligned} \label{eqn482} \bigl [ n R_{2,2n-1,r}(r) - \frac{i}{2} R_{2,2n-1,\varphi}(r)\bigr ]\;=\;\frac{F_{n}^{2}n^{2}r^{2n-1}}{2^{2}}\biggl [-\; \frac{\Gamma(1, 2\mu_{n}r)}{2\mu_{n}^{2}} -\;\frac{\Gamma(2, 2\mu_{n}r)}{2^{2}\mu_{n}^{2}} + \nonumber\\ \nonumber\\ +\; n\sum_{l=0}^{\infty}\frac{\Gamma(l+2, 2\mu_{n}r)}{(2n+2)_{l+1}2^{l+1}\mu_{n}^{2}} - \sum_{l=0}^{\infty}\frac{\Gamma(l+3, 2\mu_{n}r)}{(2n+2)_{l+1}2^{l+3}\mu_{n}^{2}} \biggr ] \nonumber\\ \quad\quad\quad\quad\end{aligned}$$ $$\begin{aligned} \label{eqn483} \bigl [ n R_{2,2n+1,r}(r) - \frac{i}{2} R_{2,2n+1,\varphi}(r)\bigr ]\;=\;\frac{F_{n}^{2}n^{2}}{2^{2}r^{2n+1}}\biggl [-\; \frac{n (2n+1)}{2^{4n-2}\mu_{n}^{4n+2}}\;\gamma(4n, 2\mu_{n}r) + \nonumber\\ \nonumber\\ +\;\frac{2n (2n+1)}{\mu_{n}^{2n+2}}\;\gamma(2n, \mu_{n}r)\;r^{2n}\; {\mbox{\Large e \normalsize}} ^{- \mu_{n}r}\biggr ] \nonumber\\ \quad\quad\quad\quad\end{aligned}$$ From formulas $(\ref{eqn479})$ or $(\ref{eqn480}), (\ref{eqn481})$ with properties of $u_{2t}(t)\;$ - $\;(\ref{eqn478})$ it follows: $$\begin{aligned} \begin{array}{ll} lim \;\;u_{21}^{*}(r,\varphi,t)= 0;\;\;\; lim \;\;u_{22}^{*}(r,\varphi,t)= 0;\\ t \rightarrow 0; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; t \rightarrow 0; \end{array}\end{aligned}$$ $$\begin{aligned} \begin{array}{ll} lim \;\;u_{2r}^{*}(r,\varphi,t)= 0;\;\;\; lim \;\;u_{2\varphi}^{*}(r,\varphi,t)= 0;\\ t \rightarrow 0; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; t \rightarrow 0; \end{array}\end{aligned}$$ and we have the velocity $\vec u_{2} = \vec u_{1} - \vec u_{2}^{*}$ \[ look at $\;(\ref{eqn50})$\] satisfying the initial conditions $(\ref{eqn400})$. We use the asymptotic properties of the incomplete gamma functions $\Gamma (\alpha, x) ,\; \gamma(\alpha, x)$ and from formulas $(\ref{eqn482}), (\ref{eqn483})\;$ we have: the first correction $\vec u_{2}^{*}$ and therefore the velocity $\vec u_{2}$ satisfies conditions $(\ref{eqn16})$ (for $r \;\rightarrow\; \infty $). Let us compare the solution $(\ref{eqn446})$ or $(\ref{eqn447}), (\ref{eqn448})$ for $\vec u_{1}$ of the first step of iterative process with the first correction $(\ref{eqn479})$ or $(\ref{eqn480}), (\ref{eqn481})$ for $\vec u_{2}^{*}$ , which is received on the second step of iterative process. We see that $$\label{eqn484} \mid\vec{u_{2}^{*}}\mid\; <<\; \mid\vec{u_{1}}\mid$$ with conditions $$\begin{aligned} \label{eqn485} F_{n}\;\leq\;\frac{1}{n} \nonumber\\ \nonumber\\ t\;\leq\;\sigma_{n}\end{aligned}$$ By continuing this iterative process we can obtain next parts $\;\vec{u_{3}^{*}}\;, \vec{u_{4}^{*}}\;,...$ of the converging series for $\vec{u}$. For arbitrary step j of the iterative process we have by using formula $(\ref{eqn57})$: $$\label{eqn486} \vec{u}_{j}\;=\;\vec{u}_{1}\;-\;\sum_{l=2}^{j} \vec{u}_{l}^{*}$$ and then: $$\label{eqn487} \begin{array}{ll} lim \;\;\vec{u_{j}}= \vec{u}\\ j \rightarrow \infty \end{array}$$ where $\vec{u}$ is the solution of the problem $(\ref{eqn1}) - (\ref{eqn6})$ for $\nu$ = 0.\ \ Below we provide numerical analysis of these results for the following values of problem’s parameters: Circumferential modes n = 1, 2, 3, 4, 5. $\sigma_{n}$ = 10. ${0\leq t \leq 10}$. Results were obtained for functions $\vec u_{1} - (\ref{eqn446})$ or $(\ref{eqn447}), (\ref{eqn448})$ ; $\vec u_{2}^{*} - (\ref{eqn479})$ or $(\ref{eqn480}), (\ref{eqn481})$ with calculations of the incomplete gamma functions $\cite{BE253}$. $\vec u_{2} = \vec u_{1} - \vec u_{2}^{*}$ and is shown in FIG. 4.1 - 4.5. The vector field $\vec u_{2}$ at distances r = 1, 2, 3, 5, 7 is represented by the dotted curves in left diagrams. The comparison of $\mid\vec u_{1}\mid$ (dashed plots) and $\mid\vec u_{2}^{*}\mid$ (solid plots) in plane $\varphi$ = \[0, $\pi$\], at distances 0 $\leq$ r $\leq$ 50 is represented in right diagrams. This comparison shows $\mid\vec{u_{2}^{*}}\mid\; <<\; \mid\vec{u_{1}}\mid$ and is corresponding to the conclusion $(\ref{eqn484})$. $\nonumber\\$ {height="60mm"}\ FIG.4.1. n = 1, $F_1$ = 1, $\mu_1$ = 1 {height="60mm"}\ FIG.4.2. n = 2, $F_2$ = 0.5, $\mu_2$ = 1 {height="60mm"}\ FIG.4.3. n = 3, $F_3$ = 0.33, $\mu_3$ = 1.3 {height="60mm"}\ FIG.4.4. n = 4, $F_4$ = 0.25, $\mu_4$ = 1.5 {height="60mm"}\ FIG.4.5. n = 5, $F_5$ = 0.2, $\mu_5$ = 1.7 $\nonumber\\ \nonumber\\$ Example of the solution of the Cauchy problem for the Navier - Stokes equations by the described iterative method with a particular applied force (N = 2) ========================================================================================================================================================= We will consider an example of the solution of the Cauchy problem for the Navier - Stokes equations for N = 2 and with initial conditions: $$\label{eqn200} \vec{u}(x,0)\; = \; \vec{u}^{0}(x)\; = \;0\;\;\;\;\;\;\;\; (x\in R^{2})$$ Hence, and from formulas $(\ref{eqn39}), (\ref{eqn40})$ for arbitrary step j of the iterative process, it follows: $$\begin{aligned} \label{eqn201} u_{j1}(x_{1},x_{2},t)\;=\;\frac{1}{4\pi^2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\gamma_{2}^{2} }{(\gamma_{1}^{2}+\gamma_{2}^{2}) } \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2}+\gamma_{2}^{2}) (t-\tau)}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2})} f_{j1}(\tilde x_{1},\tilde x_{2},\tau)\,d \tilde x_{1}d \tilde x_{2}d\tau\cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2}\;- \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \nonumber\\ \nonumber\\ \nonumber\\ - \;\frac{1}{4\pi^2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\gamma_{1} \gamma_{2} }{(\gamma_{1}^{2}+\gamma_{2}^{2}) } \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2}+\gamma_{2}^{2}) (t-\tau)}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2})} f_{j2}(\tilde x_{1},\tilde x_{2},\tau)\,d \tilde x_{1}d \tilde x_{2}d\tau\cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2} \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \nonumber\\ \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \end{aligned}$$ $$\begin{aligned} \label{eqn202} u_{j2}(x_{1},x_{2},t)\;=\;-\frac{1}{4\pi^2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\gamma_{1} \gamma_{2} }{(\gamma_{1}^{2}+\gamma_{2}^{2}) } \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2}+\gamma_{2}^{2}) (t-\tau)}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2})} f_{j1}(\tilde x_{1},\tilde x_{2},\tau)\,d \tilde x_{1}d \tilde x_{2}d\tau\cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2}\;+ \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \nonumber\\ \nonumber\\ \nonumber\\ + \;\frac{1}{4\pi^2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\gamma_{1}^{2} }{(\gamma_{1}^{2}+\gamma_{2}^{2}) } \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu (\gamma_{1}^{2}+\gamma_{2}^{2}) (t-\tau)}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mbox{\Large e \normalsize}} ^{i(\tilde x_{1}\gamma_{1}+\tilde x_{2}\gamma_{2})} f_{j2}(\tilde x_{1},\tilde x_{2},\tau)\,d \tilde x_{1}d \tilde x_{2}d\tau \cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\;{\mbox{\Large e \normalsize}} ^{-i(x_{1}\gamma_{1}+x_{2}\gamma_{2})}\,d\gamma_{1}d\gamma_{2} \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \nonumber\\ \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \end{aligned}$$ We convert the Cartesian coordinates to the polar coordinates by formulas: $x_{1}$ = $r\;\cdot\;$cos$\varphi\;$;$\;\;x_{2}$ = $r\;\cdot\;$sin$\varphi\;$;$\;\;\gamma_{1}$ = $\rho\;\cdot\;$cos$\psi\;$;$\;\;\gamma_{2}$ = $\rho\;\cdot\;$sin$\psi\;$;$\;\;\tilde x_{1}$ = $\tilde r\;\cdot\;$cos$\tilde \varphi\;$;$\;\;\tilde x_{2}$ = $\tilde r\;\cdot\;$sin$\tilde \varphi$; and obtain from formulas $(\ref{eqn201}), (\ref{eqn202})$: $$\begin{aligned} \label{eqn203} u_{j1}(r,\varphi,t)\;=\;\frac{1}{4\pi^2} \int_{0}^{\infty}\int_{0}^{2\pi}{sin^{2}\psi } \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)}\int_{0}^{\infty}\int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)} f_{j1}(\tilde r,\tilde \varphi,\tau)\,\tilde r d \tilde r d \tilde \varphi d\tau\cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi\;- \quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ - \;\frac{1}{4\pi^2} \int_{0}^{\infty}\int_{0}^{2\pi}{sin\psi cos\psi } \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)} \int_{0}^{\infty}\int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)} f_{j2}(\tilde r,\tilde \varphi,\tau)\,\tilde r d \tilde r d \tilde \varphi d\tau\cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\; {\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi \quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ $$\begin{aligned} \label{eqn204} u_{j2}(r,\varphi,t)\;=\;-\frac{1}{4\pi^2} \int_{0}^{\infty}\int_{0}^{2\pi}{sin\psi cos\psi } \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)} \int_{0}^{\infty}\int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)} f_{j1}(\tilde r,\tilde \varphi,\tau)\,\tilde r d \tilde r d \tilde \varphi d\tau\cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\;\;\;\cdot\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi\;+ \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \nonumber\\ \nonumber\\ \nonumber\\ + \;\frac{1}{4\pi^2} \int_{0}^{\infty}\int_{0}^{2\pi}cos^{2}\psi \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)}\int_{0}^{\infty}\int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)} f_{j2}(\tilde r,\tilde \varphi,\tau)\,\tilde r d \tilde r d \tilde \varphi d\tau\cdot \nonumber\\ \nonumber\\ \nonumber\\ \;\;\;\;\cdot\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \nonumber\\ \quad\quad\quad\quad\quad\quad \quad\quad \quad\quad \end{aligned}$$ We have the applied force $\vec{f}_{j}$ for arbitrary step j of the iterative process: $$\label{eqn205} f_{j\tilde r}(\tilde r,\tilde \varphi,\tau) = f_{j\tilde r}(\tilde r,\tau){\mbox{\Large e \normalsize}}^{in_{j}\tilde \varphi}\;\;\;,\;\;\;f_{j\tilde \varphi}(\tilde r,\tilde \varphi,\tau) \equiv 0$$ or $$\label{eqn206} f_{j\tilde r}(\tilde r,\tilde \varphi,\tau)\equiv 0\;\;\;,\;\;\;f_{j\tilde \varphi}(\tilde r,\tilde \varphi,\tau) = f_{j\tilde \varphi}(\tilde r,\tau){\mbox{\Large e \normalsize}}^{in_{j}\tilde \varphi}$$ where $\;\;f_{j\tilde r}(\tilde r,\tilde \varphi,\tau)\;\;\;,\;\;\;f_{j\tilde \varphi}(\tilde r,\tilde \varphi,\tau)\;\;\; - \;\;\;$ radial and tangential components of the applied force. $n_{j}$ - separate circumferential mode, $n_{j}$ = 0,1,2,3,... We take the radial and tangential components of the applied force $(\ref{eqn205}), (\ref{eqn206})$ with condition $(\ref{eqn18})\;$. For the radial component of the applied force we use De Moivre’s formulas (\[A8\]) and have: $$\begin{aligned} \label{eqn207} f_{j1}(\tilde r,\tilde \varphi,\tau) = f_{j\tilde r}(\tilde r,\tau){\mbox{\Large e \normalsize}}^{in_{j}\tilde \varphi}cos \tilde \varphi = \frac{1}{2}f_{j\tilde r}(\tilde r,\tau)\bigl({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} + {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi}\bigr) \nonumber\\ \nonumber\\ f_{j2}(\tilde r,\tilde \varphi,\tau) = f_{j\tilde r}(\tilde r,\tau){\mbox{\Large e \normalsize}}^{in_{j}\tilde \varphi}sin \tilde \varphi = \frac{i}{2}f_{j\tilde r}(\tilde r,\tau)\bigl({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} - {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi}\bigr)\end{aligned}$$ We put the applied force components $(\ref{eqn207})$ in formulas $(\ref{eqn203}), (\ref{eqn204})$ and find: $$\begin{aligned} \label{eqn208} u_{jr1}(r,\varphi,t)\;=\;\frac{1}{8\pi^2} \int_{0}^{\infty}\int_{0}^{2\pi}{sin^{2}\psi } \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)}\int_{0}^{\infty} f_{j\tilde r}(\tilde r, \tau) \cdot \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ \cdot\;\int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)} ({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} + {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi})d \tilde \varphi \tilde r d \tilde r d\tau\biggr] {\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi\;- \quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ -\;\; \frac{i}{8\pi^2} \int_{0}^{\infty}\int_{0}^{2\pi}{sin\psi cos\psi } \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)} \int_{0}^{\infty} f_{j\tilde r}(\tilde r, \tau) \cdot \quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\;\int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)}({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} - {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi}) d \tilde \varphi \tilde r d \tilde r d\tau \biggr] {\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi \quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ $$\begin{aligned} \label{eqn209} u_{jr2}(r,\varphi,t)\;=\;- \frac{1}{8\pi^2} \int_{0}^{\infty}\int_{0}^{2\pi}{sin\psi cos\psi} \biggl[ \int_{0}^{t}{\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)}\int_{0}^{\infty} f_{j\tilde r}(\tilde r, \tau) \cdot \quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ \cdot\;\int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)} ({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} + {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi})d \tilde \varphi \tilde r d \tilde r d\tau \biggr] {\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi\;+ \quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ +\;\; \frac{i}{8\pi^2} \int_{0}^{\infty}\int_{0}^{2\pi}{ cos^{2}\psi } \biggl[ \int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)}\int_{0}^{\infty} f_{j\tilde r}(\tilde r, \tau) \cdot \quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ \;\cdot\;\int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)}({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} - {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi}) d \tilde \varphi \tilde r d \tilde r d\tau \biggr] {\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}\,\rho d\rho d\psi \quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \nonumber\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ Let us denote internal integrals in $(\ref{eqn208}), (\ref{eqn209})$ as $\;\;I_{\underline{+}}(\tilde r,\rho,\psi)\;\;$: $$\label{eqn210} I_{\underline{+}}(\tilde r,\rho,\psi) = \int_{0}^{2\pi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos(\tilde \varphi-\psi)} ({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi\;\;} \underline{+}\;\; {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi})d \tilde \varphi$$ We have two integrals here. Plus (+) is for the first part of each integral $(\ref{eqn208}), (\ref{eqn209})$ and minus (-) is for the second part. We substitute $\tilde \theta$ for $\tilde \varphi$:$\;\;\;\tilde \theta\;$ = $\;\tilde \varphi\;$ - $\psi\;\;$ , d$\tilde \theta\;$ = d$\tilde \varphi\;$ and receive: $$\label{eqn211} I_{\underline{+}}(\tilde r,\rho,\psi) = {\mbox{\Large e \normalsize}}^{i(n_{j}-1)\psi}\int_{-\psi}^{2\pi-\psi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos\tilde \theta + i(n_{j}-1)\tilde \theta} d \tilde \theta \;\;\underline{+}\;\;{\mbox{\Large e \normalsize}}^{i(n_{j}+1)\psi}\int_{-\psi}^{2\pi-\psi}{\mbox{\Large e \normalsize}} ^{i\tilde r\rho cos\tilde \theta + i(n_{j}+1)\tilde \theta} d \tilde \theta$$ Then we use the Bessel function’s integral representation $(\ref{A9})$ and have: $$\label{eqn212} I_{\underline{+}}(\tilde r,\rho,\psi) = 2\pi i^{(n_{j}-1)}{\mbox{\Large e \normalsize}}^{i(n_{j}-1)\psi}J_{n_{j}-1}(\tilde{r}\rho)\;\;\underline{+}\;\;2\pi i^{(n_{j}+1)}{\mbox{\Large e \normalsize}}^{i(n_{j}+1)\psi}J_{n_{j}+1}(\tilde{r}\rho)$$ Let us put $\;\;I_{\underline{+}}(\tilde r,\rho,\psi)\;\;$ from $(\ref{eqn212})$ in formulas $(\ref{eqn208}), (\ref{eqn209})\;$ , change order of integration and obtain: $$\begin{aligned} \label{eqn213} u_{jr1}(r,\varphi, t)\;=\;\frac{1}{8\pi^2}\int_{0}^{\infty}\int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)} \int_{0}^{\infty}f_{j\tilde r}(\tilde r, \tau) \cdot \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \cdot \int_{0}^{2\pi}\biggl[{sin^{2} \psi} I_{+}(\tilde r,\rho,\psi)-i\;{sin\psi cos\psi}I_{-}(\tilde r,\rho,\psi)\biggr]\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}d\psi \tilde r d \tilde r d\tau \rho d\rho \nonumber\\ \nonumber\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ $$\begin{aligned} \label{eqn214} u_{jr2}(r,\varphi, t)\;=\;\frac{1}{8\pi^2}\int_{0}^{\infty}\int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)} \int_{0}^{\infty}f_{j\tilde r}(\tilde r, \tau) \cdot \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \cdot \int_{0}^{2\pi}\biggl[-{sin\psi}{cos\psi} I_{+}(\tilde r,\rho,\psi)+i\;{cos^{2}\psi}I_{-}(\tilde r,\rho,\psi)\biggr]\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)}d\psi \tilde r d \tilde r d\tau \rho d\rho \nonumber\\ \nonumber\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ Then we group parts in brackets of formulas $(\ref{eqn213}), (\ref{eqn214})\;$ , use De Moivre’s formulas $(\ref{A8})$ and the Bessel function’s properties. And we get: $$\label{eqn215} u_{jr1}(r,\varphi, t)\;=\;-\;\frac{n_{j}\;i^{n_{j}}}{2\pi}\int_{0}^{\infty}\int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)} \int_{0}^{\infty}f_{j\tilde r}(\tilde r, \tau)\int_{0}^{2\pi}{sin\psi}\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)+i n_{j} \psi}\;d\psi \;J_{n_{j}}(\tilde r \rho)\;d \tilde r d\tau d\rho \quad\quad$$ $$\label{eqn216} u_{jr2}(r,\varphi, t)\;=\;\frac{n_{j}\;i^{n_{j}}}{2\pi}\int_{0}^{\infty}\int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)} \int_{0}^{\infty}f_{j\tilde r}(\tilde r, \tau)\int_{0}^{2\pi}{cos\psi}\;{\mbox{\Large e \normalsize}} ^{-ir\rho cos(\psi-\varphi)+i n_{j} \psi}\;d\psi \;J_{n_{j}}(\tilde r \rho)\;d \tilde r d\tau d\rho \quad\quad$$ \ We substitute $\theta$ for $\psi$:$\;\;\;\theta\;$ = $\;\psi\;$ - $\varphi\;\;$ , d$\theta\;$ = d$\psi\;$ in the internal integrals of formulas $(\ref{eqn215}), (\ref{eqn216})\;$, use De Moivre’s formulas $(\ref{A8})$ and the Bessel function’s integral representation $(\ref{A9})$ and have from formulas $(\ref{eqn215}), (\ref{eqn216})$: $$\label{eqn217} u_{jr1}(r,\varphi, t)\;=\;\frac{n_{j}}{2}\;{\mbox{\Large e \normalsize}} ^{i n_{j} \varphi}\int_{0}^{\infty}\int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)} \biggl[{\mbox{\Large e \normalsize}} ^{i \varphi}J_{n_{j+1}}(r \rho)\;+\;{\mbox{\Large e \normalsize}} ^{-i \varphi}J_{n_{j-1}}(r \rho)\biggr]\int_{0}^{\infty}f_{j\tilde r}(\tilde r, \tau)\;J_{n_{j}}(\tilde r \rho)\;d \tilde r d\tau d\rho \quad\quad$$ $$\label{eqn218} u_{jr2}(r,\varphi, t)\;=\;\frac{i\; n_{j}}{2}\;{\mbox{\Large e \normalsize}} ^{i n_{j} \varphi}\int_{0}^{\infty}\int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)} \biggl[{\mbox{\Large e \normalsize}} ^{i \varphi}J_{n_{j+1}}(r \rho)\;-\;{\mbox{\Large e \normalsize}} ^{-i \varphi}J_{n_{j-1}}(r \rho)\biggr]\int_{0}^{\infty}f_{j\tilde r}(\tilde r, \tau)\;J_{n_{j}}(\tilde r \rho)\;d \tilde r d\tau d\rho \quad\quad$$ \ Let us denote: $$\label{eqn219} R_{j,n_{j}-1,r}(r, t)\;=\;\int_{0}^{\infty}\int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)} J_{n_{j}-1}(r \rho)\int_{0}^{\infty}f_{j\tilde r}(\tilde r, \tau)\;J_{n_{j}}(\tilde r \rho)\;d \tilde r d\tau d\rho \quad\quad$$ $$\label{eqn220} R_{j,n_{j}+1,r}(r, t)\;=\;\int_{0}^{\infty}\int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)} J_{n_{j}+1}(r \rho)\int_{0}^{\infty}f_{j\tilde r}(\tilde r, \tau)\;J_{n_{j}}(\tilde r \rho)\;d \tilde r d\tau d\rho \quad\quad$$ Then we have from formulas $(\ref{eqn217}), (\ref{eqn218})$: $$\label{eqn221} u_{jr1}(r,\varphi, t)\;=\;\frac{n_{j}}{2}\bigl[R_{j,n_{j}-1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n_{j}-1) \varphi}\;+\;R_{j,n_{j}+1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n_{j}+1) \varphi}\bigr]$$ $$\label{eqn222} u_{jr2}(r,\varphi, t)\;=\;\frac{i\;n_{j}}{2}\bigl[R_{j,n_{j}-1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n_{j}-1) \varphi}\;-\;R_{j,n_{j}+1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n_{j}+1) \varphi}\bigr]$$ Then if $n_{j}\;=\;0$ it follows from $(\ref{eqn219}), (\ref{eqn220}) , (\ref{eqn221}), (\ref{eqn222})$ that $u_{jr1}(r,\varphi, t)\;=\;u_{jr2}(r,\varphi, t)\;=\;0\;$ and hence $u_{1}\;\;=\;\;u_{2}\;\;=\;\;0$. In the equations bellow we will consider $n_{j}\;\geq\;1$. Now we integrate the solution $(\ref{eqn203}), (\ref{eqn204})$ by the tangential component of the applied force $(\ref{eqn206})$ ( for $n_{j}\;\geq\;1$). Then we use De Moivre’s formulas $(\ref{A8})$ and have: $$\begin{aligned} \label{eqn223} f_{j1}(\tilde r,\tilde \varphi,\tau) = - f_{j\tilde \varphi}(\tilde r, \tau){\mbox{\Large e \normalsize}}^{in_{j}\tilde \varphi}sin \tilde \varphi = - \frac{i}{2}f_{j\tilde \varphi}(\tilde r, \tau) \bigl( {\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} - {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi} \bigr) \nonumber\\ \nonumber\\ f_{j2}(\tilde r,\tilde \varphi,\tau) \;\;= \;\; f_{j\tilde \varphi}(\tilde r, \tau){\mbox{\Large e \normalsize}}^{in_{j}\tilde \varphi}cos \tilde \varphi \;= \; \frac{1}{2}f_{j\tilde \varphi}(\tilde r, \tau)\bigl({\mbox{\Large e \normalsize}}^{i(n_{j}-1)\tilde \varphi} + {\mbox{\Large e \normalsize}}^{i(n_{j}+1)\tilde \varphi}\bigr) \nonumber\\ \nonumber\\\end{aligned}$$ Hence formulas $(\ref{eqn223})$ are the components $f_{j1}$ and $f_{j2}$ from the tangential applied force $(\ref{eqn206})$, while formulas $(\ref{eqn207})$ are the components $f_{j1}$ and $f_{j2}$ from the radial applied force $(\ref{eqn205})$. Let us put $(\ref{eqn223})$ in formulas $(\ref{eqn203}), (\ref{eqn204})$ and do the operations as we did in $(\ref{eqn208}) - (\ref{eqn222})\; (\;n_{j}\;\geq\;1)$. We consider that $f_{j\tilde \varphi}(\tilde r, \tau)$ is restricted by condition $(\ref{eqn18})$ and get: $$\label{eqn224} R_{j,n_{j}-1,\varphi}(r, t)\;=\;-\;\int_{0}^{\infty}\int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)}J_{n_{j}-1}(r \rho)\int_{0}^{\infty}\bigl(f_{j\tilde \varphi}(\tilde r, \tau)\cdot\tilde r\bigr)^{'}_{\tilde r}\;J_{n_{j}}(\tilde r \rho)\;d \tilde r d\tau d\rho \quad\quad$$ $$\label{eqn225} R_{j,n_{j}+1,\varphi}(r, t)\;=\;-\;\int_{0}^{\infty}\int_{0}^{t} {\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)}J_{n_{j}+1}(r \rho)\int_{0}^{\infty}\bigl(f_{j\tilde \varphi}(\tilde r, \tau)\cdot\tilde r\bigr)^{'}_{\tilde r}\;J_{n_{j}}(\tilde r \rho)\;d \tilde r d\tau d\rho \;, \quad\quad$$ Here $\bigl(\bigr)^{'}_{\tilde r} \equiv \frac{\partial}{\partial \tilde r}$. Hence we have: $$\label{eqn226} u_{j\varphi1}(r,\varphi, t)\;=\;-\;\frac{i}{2}\bigl[R_{j,n_{j}-1,\varphi}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n_{j}-1) \varphi}\;+\;R_{j,n_{j}+1,\varphi}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n_{j}+1) \varphi}\bigr]$$ $$\label{eqn227} u_{j\varphi2}(r,\varphi, t)\;=\;\frac{1}{2}\bigl[R_{j,n_{j}-1,\varphi}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n_{j}-1) \varphi}\;-\;R_{j,n_{j}+1,\varphi}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n_{j}+1) \varphi}\bigr]$$ We have obtain formulas $(\ref{eqn207})\;$ - $\;(\ref{eqn227})$ for an arbitrary step j of the iterative process and the applied forces $(\ref{eqn205})$ or $(\ref{eqn206})$. Now we investigate the first step (j = 1 , $n_{1} = n = 1,2,3,...$) of the iterative process with the particular radial applied force ${f}_{1}(x,t)\;\;$\[look at $\;(\ref{eqn47})$\]: $$\begin{aligned} \label{eqn228} f_{1\tilde r}(\tilde r,\tilde \varphi,\tau) = f_{1\tilde r}(\tilde r, \tau){\mbox{\Large e \normalsize}}^{in\tilde \varphi}\;\;\;,\;\;\;f_{1\tilde \varphi}(\tilde r,\tilde \varphi,\tau) \equiv 0 \nonumber\\ \nonumber\\ f_{1\tilde r}(\tilde r, \tau) = F_{n}\tilde r^{n+1}{\mbox{\Large e \normalsize}}^{-\mu_{n}^2\tilde r^2}f_{1\tau}(\tau) \quad\quad\end{aligned}$$ $F_{n}, \mu_{n}$ - constants. , $0 < F_{n}< \infty$ , $1 < \mu_{n}< \infty$. Let us put the particular radial applied force $(\ref{eqn228})$ in formulas $(\ref{eqn219})$ , $(\ref{eqn220})$ and for the internal integral we have by using formula $ (\ref{A10}) , \cite{BE253}$: $$\label{eqn229} I(\rho, \tau) = \int_{0}^{\infty}f_{1\tilde r}(\tilde r, \tau)\;J_{n}(\tilde r \rho)\;d \tilde r = F_{n} f_{1\tau}(\tau) \int_{0}^{\infty} \tilde r^{n+1}{\mbox{\Large e \normalsize}}^{-\mu_{n}^2\tilde r^2}J_{n}(\tilde r \rho)\;d \tilde r = \frac{F_{n} f_{1\tau}(\tau)\rho^n}{(2\mu_{n}^2)^{n+1}}{\mbox{\Large e \normalsize}}^{-\frac{\rho^2}{4\mu_{n}^2}}$$ Now we put $I(\rho, \tau)$ from formula $(\ref{eqn229})$ in formulas $(\ref{eqn219})$ , $(\ref{eqn220})$ , change the order of integration and have by using formula $ (\ref{A11}) , \cite{BE253}$: $$\begin{aligned} \label{eqn230} R_{1,n-1,r}(r, t)\;=\;F_{n} \int_{0}^{t}f_{1\tau}(\tau) \int_{0}^{\infty} {\mbox{\Large e \normalsize}} ^{-\bigl [\nu (t-\tau)+\frac{1}{4\mu_{n}^2}\bigr ]\rho^2}\frac{\rho^n}{(2\mu_{n}^2)^{n+1}} J_{n-1}(r \rho) d\rho d\tau \;= \nonumber\\ \nonumber\\ =\;\frac{F_{n} r^{n-1}}{2\mu_{n}^2}\int_{0}^{t}\frac{f_{1\tau}(\tau)\Phi\bigl(n+2, n+2;\frac{-\mu_{n}^2 r^2}{\bigl[4\mu_{n}^2\nu (t-\tau)+ 1\bigr]}\bigr)}{[4\mu_{n}^2\nu (t-\tau)+ 1]^n}d\tau \quad\quad\quad\quad\quad\quad\end{aligned}$$ $$\begin{aligned} \label{eqn231} R_{1,n+1,r}(r, t)\;=\;F_{n} \int_{0}^{t}f_{1\tau}(\tau) \int_{0}^{\infty} {\mbox{\Large e \normalsize}} ^{-\bigl [\nu (t-\tau)+\frac{1}{4\mu_{n}^2}\bigr ]\rho^2}\frac{\rho^n}{(2\mu_{n}^2)^{n+1}} J_{n+1}(r \rho) d\rho d\tau\;= \nonumber\\ \nonumber\\ =\;\frac{F_{n} r^{n+1}}{2(n+1)}\int_{0}^{t}\frac{f_{1\tau}(\tau)\Phi\bigl(n+1, n+2;\frac{-\mu_{n}^2 r^2}{\bigl[4\mu_{n}^2\nu (t-\tau)+ 1\bigr]}\bigr)}{[4\mu_{n}^2\nu (t-\tau)+ 1]^{n+1}}d\tau \quad\quad\quad\quad\quad\quad\end{aligned}$$ Here $\Phi(a, c; x)$ is a confluent hypergeometric function $\cite{BE153}$. We substitute y for $\tau$: y = $\frac{1}{[4\mu_{n}^2\nu (t-\tau)+ 1]}$, dy = $\frac{4\mu_{n}^2\nu}{[4\mu_{n}^2\nu (t-\tau)+ 1]^2} d\tau$ and receive: $$\begin{aligned} \label{eqn232} R_{1,n-1,r}(r, t)\;=\;\frac{F_{n} r^{n-1}}{8\mu_{n}^4\nu}\int_{\frac{1}{[4\mu_{n}^2\nu t+1]}}^{1}f_{1\tau}(y)\cdot y^{n-2}\cdot \Phi\bigl(n+2, n+2;-\mu_{n}^2 r^2y\bigr)dy \quad\quad\quad\quad\quad\quad\end{aligned}$$ $$\begin{aligned} \label{eqn233} R_{1,n+1,r}(r, t)\;=\frac{F_{n} r^{n+1}}{8\mu_{n}^2\nu(n+1)}\int_{\frac{1}{[4\mu_{n}^2\nu t+1]}}^{1}f_{1\tau}(y)\cdot y^{n-1}\cdot\Phi\bigl(n+1, n+2;-\mu_{n}^2 r^2y\bigr)dy \quad\quad\quad\quad\quad\quad\end{aligned}$$ Let us denote $f_{1\tau}(y) = y^2$ and get: $$\begin{aligned} \label{eqn234} R_{1,n-1,r}(r, t)\;=\;\frac{F_{n} r^{n-1}}{8\mu_{n}^4\nu}\int_{\frac{1}{[4\mu_{n}^2\nu t+1]}}^{1}y^{n}\cdot \Phi\bigl(n+2, n+2;-\mu_{n}^2 r^2y\bigr)dy \quad\quad\quad\quad\quad\quad\end{aligned}$$ $$\begin{aligned} \label{eqn235} R_{1,n+1,r}(r, t)\;=\frac{F_{n} r^{n+1}}{8\mu_{n}^2\nu(n+1)}\int_{\frac{1}{[4\mu_{n}^2\nu t+1]}}^{1} y^{n+1}\cdot\Phi\bigl(n+1, n+2;-\mu_{n}^2 r^2y\bigr)dy \quad\quad\quad\quad\quad\quad\end{aligned}$$ We use formula $(\ref{A12})$ for integrand in the integral $(\ref{eqn234})$ and formula $(\ref{A13})$ for integrand in the integral $(\ref{eqn235})$ $\cite{BE153}$, integrate and then we have: $$\begin{aligned} \label{eqn236} R_{1,n-1,r}(r, t)\;=\;\frac{F_{n} r^{n-1}}{8\mu_{n}^4\nu(n+1)}\biggl[\Phi\bigl(n+1, n+2;-\mu_{n}^2 r^2\bigr) - \frac{\Phi\bigl(n+1, n+2;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)}{(4\mu_{n}^2\nu t+1)^{n+1}}\biggr] \nonumber\\ \nonumber\\ \quad\quad\quad\quad\quad\quad\end{aligned}$$ $$\begin{aligned} \label{eqn237} R_{1,n+1,r}(r, t)\;=\;\frac{F_{n} r^{n+1}}{8\mu_{n}^2\nu(n+1)(n+2)}\biggl[\Phi\bigl(n+1, n+3;-\mu_{n}^2 r^2\bigr) - \frac{\Phi\bigl(n+1, n+3;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)}{(4\mu_{n}^2\nu t+1)^{n+2}}\biggr] \nonumber\\ \nonumber\\ \quad\quad\quad\quad\quad\quad\end{aligned}$$ Hence, and from formulas $(\ref{eqn221}), (\ref{eqn222})$ it follows for j = 1: $$\label{eqn238} u_{1r1}(r,\varphi, t)\;=\;\frac{n}{2}\bigl[R_{1,n-1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;+\;R_{1,n+1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]$$ $$\label{eqn239} u_{1r2}(r,\varphi, t)\;=\;\frac{i\;n}{2}\bigl[R_{1,n-1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;-\;R_{1,n+1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]$$ We continue operations and get: $$\label{eqn240} u_{1 r}(r,\varphi, t)\;=\;\frac{n}{2}\bigl[R_{1,n-1,r}(r, t)\;+\;R_{1,n+1,r}(r, t)\bigr]\;{\mbox{\Large e \normalsize}} ^{i n \varphi}\;$$ $$\label{eqn241} u_{1\varphi}(r,\varphi, t)\;=\;\frac{i\;n}{2}\bigl[R_{1,n-1,r}(r, t)\;-\;R_{1,n+1,r}(r, t)\bigr]\;{\mbox{\Large e \normalsize}}^{i n \varphi}\;$$ $u_{1 r}(r,\varphi, t) , \;u_{1\varphi}(r,\varphi, t)$ are the radial and tangential components of the velocity $\vec u_{1}$. We use the properties of the confluent hypergeometric function $\Phi(a, c; x)$ and have from formulas $(\ref{eqn236})-(\ref{eqn241})$: $$\begin{array}{ll} lim \;\;u_{1r1}(r,\varphi,t)= 0;\;\;\; lim \;\;u_{1r2}(r,\varphi,t)= 0;\\ t \rightarrow 0; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; t \rightarrow 0;\\ lim \;\;u_{1r}(r,\varphi,t)= 0;\;\;\; lim \;\;u_{1\varphi}(r,\varphi,t)= 0;\\ t \rightarrow 0; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; t \rightarrow 0;\\ \end{array}$$ Hence we have velocity $\vec u_{1}$ satisfies the initial conditions $(\ref{eqn200})$. Then we use the asymptotic properties of the confluent hypergeometric function $\Phi(a, c; x)\; \cite{BE153}$ and from formulas $(\ref{eqn236})-(\ref{eqn241})$ we have velocity $\vec u_{1}$ satisfies conditions $(\ref{eqn16})$ ( for $r \;\rightarrow\; \infty )$. Let us continue investigation for the second step (j = 2) of the iterative process. Find $\vec{f}_{2}^{*}(r,\varphi, t) = \{f_{21}^{*}, f_{22}^{*}\}$ - the first correction of the particular radial applied force ${f}_{1}(x,t)$ $(\ref{eqn228})$. We have for $\vec{f}_{2}^{*}$ from formula $(\ref{eqn48})$: $$\label{eqn242} f_{21}^{*} = u_{1r1}\;\frac{\partial u_{1r1}}{\partial x_{1}}\;+\;u_{1r2}\;\frac{\partial u_{1r1}}{\partial x_{2}}$$ $$\label{eqn243} f_{22}^{*} = u_{1r1}\;\frac{\partial u_{1r2}}{\partial x_{1}}\;+\;u_{1r2}\;\frac{\partial u_{1r2}}{\partial x_{2}}$$ where $u_{1r1},\; u_{1r2}$ are the components of $\vec{u_{1}}$ and were taken from formulas $(\ref{eqn238}) , (\ref{eqn239})$. We have here: $$\begin{aligned} \label{eqn244} \frac{\partial u_{1r1}(r,\varphi, t)}{\partial x_{1}} \; = \; \frac{\partial u_{1r1}(r,\varphi, t)}{\partial r}\; \frac{\partial r}{\partial x_{1}} \; + \; \frac{\partial u_{1r1}(r,\varphi, t)}{\partial \varphi}\; \frac{\partial \varphi}{\partial x_{1}} \nonumber\\ \nonumber\\ \frac{\partial u_{1r1}(r,\varphi, t)}{\partial x_{2}}\; = \; \frac{\partial u_{1r1}(r,\varphi, t)}{\partial r}\; \frac{\partial r}{\partial x_{2}} \; + \; \frac{\partial u_{1r1}(r,\varphi, t)}{\partial \varphi}\; \frac{\partial \varphi}{\partial x_{2}} \nonumber\\ \nonumber\\ \frac{\partial u_{1r2}(r,\varphi, t)}{\partial x_{1}}\; = \; \frac{\partial u_{1r2}(r,\varphi, t)}{\partial r}\; \frac{\partial r}{\partial x_{1}} \; + \; \frac{\partial u_{1r2}(r,\varphi, t)}{\partial \varphi}\; \frac{\partial \varphi}{\partial x_{1}} \nonumber\\ \nonumber\\ \frac{\partial u_{1r2}(r,\varphi, t)}{\partial x_{2}}\; = \; \frac{\partial u_{1r2}(r,\varphi, t)}{\partial r}\; \frac{\partial r}{\partial x_{2}} \; + \; \frac{\partial u_{1r2}(r,\varphi, t)}{\partial \varphi}\; \frac{\partial \varphi}{\partial x_{2}} \nonumber\\ \nonumber\\ \frac{\partial r}{\partial x_{1}} = cos \varphi \; , \; \frac{\partial \varphi}{\partial x_{1}} = \;-\; \frac{sin \varphi}{r}\;,\; \frac{\partial r}{\partial x_{2}} = sin \varphi \; , \;\frac{\partial \varphi}{\partial x_{2}} = \frac{cos \varphi}{r} \nonumber\\ \nonumber\\\end{aligned}$$ Then we use formulas $(\ref{eqn238}) , (\ref{eqn239})$ for $u_{1r1}(r,\varphi, t),\; u_{1r2}(r,\varphi, t)$ and have from $(\ref{eqn244})$: $$\begin{aligned} \label{eqn245} \frac{\partial u_{1r1}(r,\varphi, t)}{\partial x_{1}} \; = \; \frac{n}{2}\biggl\{\bigl[R_{1,n-1,r}^{'}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;+\;R_{1,n+1,r}^{'}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]\;cos \varphi\;+ \nonumber\\ +\;i\;\bigl[(n-1)R_{1,n-1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;+\;(n+1)R_{1,n+1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]\bigl(-\frac{sin \varphi}{r} \bigr ) \biggr \} \nonumber\\ \nonumber\\ \frac{\partial u_{1r1}(r,\varphi, t)}{\partial x_{2}}\; = \; \frac{n}{2}\biggl\{\bigl[R_{1,n-1,r}^{'}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;+\;R_{1,n+1,r}^{'}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]\;sin \varphi\;+ \nonumber\\ +\;i\;\bigl[(n-1)R_{1,n-1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;+\;(n+1)R_{1,n+1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]\frac{cos \varphi}{r} \biggr \} \nonumber\\ \nonumber\\ \frac{\partial u_{1r2}(r,\varphi, t)}{\partial x_{1}}\; = \; \frac{i n}{2}\biggl\{\bigl[R_{1,n-1,r}^{'}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;-\;R_{1,n+1,r}^{'}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]\;cos \varphi\;+ \nonumber\\ +\;i\;\bigl[(n-1)R_{1,n-1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;-\;(n+1)R_{1,n+1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]\bigl(-\frac{sin \varphi}{r} \bigr ) \biggr \} \nonumber\\ \nonumber\\ \frac{\partial u_{1r2}(r,\varphi, t)}{\partial x_{2}}\; = \; \frac{i n}{2}\biggl\{\bigl[R_{1,n-1,r}^{'}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;-\;R_{1,n+1,r}^{'}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]\;sin \varphi\;+ \nonumber\\ +\;i\;\bigl[(n-1)R_{1,n-1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n-1) \varphi}\;-\;(n+1)R_{1,n+1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (n+1) \varphi}\bigr]\frac{cos \varphi}{r} \biggr \} \nonumber\\ \nonumber\\\end{aligned}$$ where $$\begin{aligned} \label{eqn246} R_{1,n-1,r}^{'}(r, t)\;=\;\frac{\partial R_{1,n-1,r}(r, t)}{\partial r}\;=\;\frac{F_{n}(n-1) r^{n-2}}{8\mu_{n}^4\nu(n+1)}\biggl[\Phi\bigl(n+1, n+2;-\mu_{n}^2 r^2\bigr) - \frac{\Phi\bigl(n+1, n+2;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)}{(4\mu_{n}^2\nu t+1)^{n+1}}\biggr] - \nonumber\\ \nonumber\\ \quad\quad\quad\quad\quad - \frac{F_{n} r^{n}}{4\mu_{n}^2\nu(n+2)}\biggl[\Phi\bigl(n+2, n+3;-\mu_{n}^2 r^2\bigr) - \frac{\Phi\bigl(n+2, n+3;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)}{(4\mu_{n}^2\nu t+1)^{n+2}}\biggr] \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\\end{aligned}$$ $$\begin{aligned} \label{eqn247} R_{1,n+1,r}^{'}(r, t)\;=\;\frac{\partial R_{1,n+1,r}(r, t)}{\partial r}\;=\;\frac{F_{n} r^{n}}{8\mu_{n}^2\nu(n+2)}\biggl[\Phi\bigl(n+1, n+3;-\mu_{n}^2 r^2\bigr) - \frac{\Phi\bigl(n+1, n+3;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)}{(4\mu_{n}^2\nu t+1)^{n+2}}\biggr] - \nonumber\\ \nonumber\\ - \frac{F_{n} r^{n+2}}{4\nu(n+2)(n+3)}\biggl[\Phi\bigl(n+2, n+4;-\mu_{n}^2 r^2\bigr) - \frac{\Phi\bigl(n+2, n+4;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)}{(4\mu_{n}^2\nu t+1)^{n+3}}\biggr] \quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\\end{aligned}$$ Let us put $u_{1r1},\; u_{1r2}, \;\frac{\partial u_{1r1}}{\partial x_{1}}, \;\frac{\partial u_{1r1}}{\partial x_{2}}, \;\frac{\partial u_{1r2}}{\partial x_{1}},\; \frac{\partial u_{1r2}}{\partial x_{2}}$ from formulas $(\ref{eqn238}) , (\ref{eqn239}) , (\ref{eqn245})$ in formulas $(\ref{eqn242}), (\ref{eqn243})$ for $f_{21}^{*} \;,\; f_{22}^{*}$.\ \ After compliting appropriate operations we have: $$\label{eqn248} f_{21}^{*} (r,\varphi, t) = \frac{n^{2}}{2^{2}}\bigl [ T_{2,2n-1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (2n-1) \varphi} + T_{2,2n+1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (2n+1) \varphi} \bigr ]$$ $$\label{eqn249} f_{22}^{*} (r,\varphi, t) = \frac{i n^{2}}{2^{2}}\bigl [ T_{2,2n-1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (2n-1) \varphi} - T_{2,2n+1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (2n+1) \varphi} \bigr ]$$ where $$\begin{aligned} \label{eqn250} T_{2,2n-1,r}(r, t) = \bigl [ R_{1,n-1,r}(r, t) + R_{1,n+1,r}(r, t) \bigr ] R_{1,n-1,r}^{'}(r, t) - \nonumber\\ \nonumber\\ - \frac{(n-1)}{r} R_{1,n-1,r}(r, t)\bigl [ R_{1,n-1,r}(r, t) - R_{1,n+1,r}(r, t) \bigr ] \nonumber\\ \nonumber\\ T_{2,2n+1,r}(r, t) = \bigl [ R_{1,n-1,r}(r, t) + R_{1,n+1,r}(r, t) \bigr ] R_{1,n+1,r}^{'}(r, t) - \nonumber\\ \nonumber\\ - \frac{(n+1)}{r} R_{1,n+1,r}(r, t)\bigl [ R_{1,n-1,r}(r, t) - R_{1,n+1,r}(r, t) \bigr ] \nonumber\\ \nonumber\\ \quad\quad\end{aligned}$$ For radial $f_{2r}^{*}$ and tangential $f_{2\varphi}^{*}$ components of the first correction $\vec{f_{2}^{*}}(r, \varphi, t)$ of the particular radial applied force we have: $$\label{eqn251} f_{2r}^{*} (r,\varphi, t) = \frac{n^{2}}{2^{2}}\bigl [ T_{2,2n-1,r}(r, t) + T_{2,2n+1,r}(r, t) \bigr ]\; {\mbox{\Large e \normalsize}} ^{i\; 2n \varphi}\;=\;\frac{n^{2}}{2^{2}}T_{2,2n,r}(r, t)\; {\mbox{\Large e \normalsize}} ^{i\; 2n \varphi}\;$$ $$\label{eqn252} f_{2\varphi}^{*} (r,\varphi, t) = \frac{i n^{2}}{2^{2}}\bigl [ T_{2,2n-1,r}(r, t) - T_{2,2n+1,r}(r, t) \bigr ]\; {\mbox{\Large e \normalsize}} ^{i\; 2n \varphi}\;=\;\frac{i n^{2}}{2^{2}}T_{2,2n,\varphi}(r, t)\; {\mbox{\Large e \normalsize}} ^{i\; 2n \varphi}\;$$ where ( see $(\ref{eqn250})$) $$\begin{aligned} \label{eqn253} T_{2,2n,r}(r, t) = \bigl [ R_{1,n-1,r}(r, t) + R_{1,n+1,r}(r, t) \bigr ] \bigl [R_{1,n-1,r}^{'}(r, t) + R_{1,n+1,r}^{'}(r, t)\bigr ] - \nonumber\\ \nonumber\\ - \frac{1}{r}\bigl [(n-1)R_{1,n-1,r}(r, t) + (n+1)R_{1,n+1,r}(r, t) \bigr ]\bigl [ R_{1,n-1,r}(r, t) - R_{1,n+1,r}(r, t) \bigr ] \nonumber\\ \nonumber\\ T_{2,2n,\varphi}(r, t) = \bigl [ R_{1,n-1,r}(r, t) + R_{1,n+1,r}(r, t) \bigr ] \bigl [R_{1,n-1,r}^{'}(r, t) - R_{1,n+1,r}^{'}(r, t)\bigr ] - \nonumber\\ \nonumber\\ - \frac{1}{r}\bigl [(n-1)R_{1,n-1,r}(r, t) - (n+1)R_{1,n+1,r}(r, t) \bigr ]\bigl [ R_{1,n-1,r}(r, t) - R_{1,n+1,r}(r, t) \bigr ] \nonumber\\ \nonumber\\ \quad\quad\end{aligned}$$ We use formulas $(\ref{eqn236}) , (\ref{eqn237})$ for $R_{1,n-1,r}(r, t)\;,\;R_{1,n+1,r}(r, t)$ and $(\ref{eqn246}) , (\ref{eqn247})$ for $R_{1,n-1,r}^{'}(r, t)\;$ , $\;R_{1,n+1,r}^{'}(r, t)$ then do the appropriate operations for $T_{2,2n-1,r}(r, t)\;,\;T_{2,2n+1,r}(r, t)$, using formula $(\ref{A14})$, and get: $$\begin{aligned} \label{eqn254} T_{2,2n,r}(r, t) = \frac{- F_{n}^2\cdot r^{2n-1}}{16\mu_{n}^6\nu^2(n+1)^2(n+2)}\biggl[\Phi\bigl(n+1, n+2;-\mu_{n}^2 r^2\bigr) - \frac{\Phi\bigl(n+1, n+2;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)}{(4\mu_{n}^2\nu t+1)^{n+1}}\biggr]\cdot \nonumber\\ \nonumber\\ \cdot\biggl[\Phi\bigl(n+1, n+3;-\mu_{n}^2 r^2\bigr) - \frac{\Phi\bigl(n+1, n+3;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)}{(4\mu_{n}^2\nu t+1)^{n+2}}\biggr] \quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ $$\begin{aligned} \label{eqn255} T_{2,2n,\varphi}(r, t) = \frac{- F_{n}^2\cdot r^{2n-1}}{16\mu_{n}^6\nu^2(n+1)(n+2)}\biggl[\Phi\bigl(n, n+2;-\mu_{n}^2 r^2\bigr) - \frac{\Phi\bigl(n, n+2;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)}{(4\mu_{n}^2\nu t+1)^{n+1}}\biggr]\cdot \nonumber\\ \nonumber\\ \cdot\biggl[\Phi\bigl(n+2, n+3;-\mu_{n}^2 r^2\bigr) - \frac{\Phi\bigl(n+2, n+3;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)}{(4\mu_{n}^2\nu t+1)^{n+2}}\biggr] + \quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ + \frac{F_{n}^2 \cdot n \cdot r^{2n-1}}{16\mu_{n}^6\nu^2(n+1)^2(n+2)}\biggl[\Phi\bigl(n+1, n+2;-\mu_{n}^2 r^2\bigr) - \frac{\Phi\bigl(n+1, n+2;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)}{(4\mu_{n}^2\nu t+1)^{n+1}}\biggr]\cdot \nonumber\\ \nonumber\\ \cdot\biggl[\Phi\bigl(n+1, n+3;-\mu_{n}^2 r^2\bigr) - \frac{\Phi\bigl(n+1, n+3;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)}{(4\mu_{n}^2\nu t+1)^{n+2}}\biggr] \quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ \quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ By comparing particular radial applied force $\vec{f_{1}}\;$ from $\;(\ref{eqn228})$ with the first correction $\vec{f_{2}^{*}}\;$ from $\;((\ref{eqn251})- (\ref{eqn255}))$ of this particular radial applied force we have: $$\label{eqn256} \mid\vec{f_{2}^{*}}\mid\; <<\; \mid\vec{f_{1}}\mid$$ with condition $$\label{eqn257} F_{n}\;\leq\;\frac{1}{n}$$ After the first step of the iterative process (j = 1) we had the velocity $\vec{u_{1}}\;$ - formulas $\; (\ref{eqn240})\;,\;(\ref{eqn241})$.\ Now we will calculate $\vec{u_{2}^{*}}\;$ - the first correction of the velocity $\vec{u_{1}} $. Solution of this problem has two stages. On the first stage we find the part of the first correction $\vec{u_{2r}^{*}}$, corresponding to the first correction $f_{2r}^{*}\;$ from formula $\;(\ref{eqn251})$ of the applied force: $$\label{eqn258} f_{2r}^{*}(r,\varphi,t) = \;\frac{n^{2}}{2^{2}}T_{2,2n,r}(r, t)\; {\mbox{\Large e \normalsize}} ^{i\; 2n \varphi}\;\;\;,\;\;\;f_{2\varphi}^{*}(r,\varphi,t) \equiv 0$$ On the second stage we calculate the other part of the first correction $\vec{u_{2\varphi}^{*}}\;$, corresponding to the first correction $f_{2\varphi}^{*}\;$ from formula $\;(\ref{eqn252})$ of the applied force: $$\label{eqn259} f_{2r}^{*}(r,\varphi,t) \equiv 0\;\;\;,\;\;\; f_{2\varphi}^{*}(r,\varphi,t) = \;\frac{i n^{2}}{2^{2}}T_{2,2n,\varphi}(r, t)\; {\mbox{\Large e \normalsize}} ^{i\; 2n\varphi}$$ In other words $$\begin{aligned} \label{eqn260} \vec{u_{2}^{*}} = \vec{u_{2r}^{*}} + \vec{u_{2\varphi}^{*}}, \;\;\;\vec{u_{2r}^{*}} = \{u_{2r1}^{*}, u_{2r2}^{*}\}, \;\;\;\vec{u_{2\varphi}^{*}} = \{u_{2\varphi1}^{*}, u_{2\varphi2}^{*}\}.\end{aligned}$$ First stage: After completing appropriate operations we have from formula $(\ref{eqn254})$: $$\begin{aligned} \label{eqn261} T_{2,2n,r}(r, t) = \frac{- F_{n}^2\cdot r^{2n-1}}{16\mu_{n}^6\nu^2(n+1)^2(n+2)}\biggl[\Phi\bigl(n+1, n+2;-\mu_{n}^2 r^2\bigr)\cdot \Phi\bigl(n+1, n+3;-\mu_{n}^2 r^2\bigr)- \nonumber\\ \nonumber\\ - \frac{1}{(4\mu_{n}^2\nu t+1)^{n+2}}\cdot\Phi\bigl(n+1, n+2;-\mu_{n}^2 r^2\bigr)\cdot\Phi\bigl(n+1, n+3;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr) - \quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ - \frac{1}{(4\mu_{n}^2\nu t+1)^{n+1}}\cdot\Phi\bigl(n+1, n+2;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)\cdot\Phi\bigl(n+1, n+3;-\mu_{n}^2 r^2\bigr) + \quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ + \frac{1}{(4\mu_{n}^2\nu t+1)^{2n+3}}\cdot \Phi\bigl(n+1, n+2;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)\cdot\Phi\bigl(n+1, n+3;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)\biggr] \quad\quad \nonumber\\ \nonumber\\ \quad\quad\quad\quad\quad\end{aligned}$$ We take formulas $(\ref{eqn221})\; ,\; (\ref{eqn222})$ for components $u_{jr1}(r, \varphi, t),\; u_{jr2}(r, \varphi, t)$ and formulas $(\ref{eqn219}) , (\ref{eqn220})$ for $R_{j,n_{j}-1,r}(r, t),\; R_{j,n_{j}+1,r}(r, t)$ for j = 2 and then formulas $(\ref{eqn258}) , (\ref{eqn261})$ for $f_{2r}^{*} (r,\varphi, t)\;,\;T_{2,2n,r}(r, t)$ . We do appropriate operations and have: $$\label{eqn264} u_{2r1}^{*}(r,\varphi, t)\;=\;n\bigl[R_{2,2n-1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (2n-1) \varphi}\;+\;R_{2,2n+1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (2n+1) \varphi}\bigr]$$ $$\label{eqn265} u_{2r2}^{*}(r,\varphi, t)\;=\;i\;n\bigl[R_{2,2n-1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (2n-1) \varphi}\;-\;R_{2,2n+1,r}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (2n+1) \varphi}\bigr]$$ After changing the order of integration we receive: $$\label{eqn266} R_{2,2n-1,r}(r, t)\;=\;\frac{n^{2}}{2^{2}}\int_{0}^{t}\int_{0}^{\infty} T_{2,2n,r}(\tilde r, \tau)\int_{0}^{\infty} {\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)} J_{2n-1}(r \rho)\;J_{2n}(\tilde r \rho)\;d\rho d \tilde r d\tau \quad\quad$$ $$\label{eqn267} R_{2,2n+1,r}(r, t)\;=\;\frac{n^{2}}{2^{2}}\int_{0}^{t}\int_{0}^{\infty} T_{2,2n,r}(\tilde r, \tau) \int_{0}^{\infty}{\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)} J_{2n+1}(r \rho)\;J_{2n}(\tilde r \rho)\;d\rho d \tilde r d\tau \quad\quad$$ Second stage: After operations with $T_{2,2n,\varphi}(r, t)$ from formula $(\ref{eqn255})$ and using formula $(\ref{A14})$, we obtain: $$\begin{aligned} \label{eqn278} \bigl(T_{2,2n,\varphi}( r, t)\cdot r\bigr)^{'}_{r} = \frac{\partial \bigl(T_{2,2n,\varphi}( r, t)\cdot r\bigr)}{\partial r} = T_{2,2n,\varphi, \varphi}(r, t) + T_{2,2n,\varphi, r}(r, t) \nonumber\\ \nonumber\\\end{aligned}$$ We denote here $$\begin{aligned} \label{eqn279} T_{2,2n,\varphi, \varphi}(r, t) = \frac{- F_{n}^2\cdot n \cdot r^{2n+1}}{8\mu_{n}^4\nu^2(n+1)(n+2)^2}\biggl[\Phi\bigl(n+1, n+3;-\mu_{n}^2 r^2\bigr) - \frac{\Phi\bigl(n+1, n+3;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)}{(4\mu_{n}^2\nu t+1)^{n+2}}\biggr]\cdot \nonumber\\ \nonumber\\ \cdot\biggl[\Phi\bigl(n+2, n+3;-\mu_{n}^2 r^2\bigr) - \frac{\Phi\bigl(n+2, n+3;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)}{(4\mu_{n}^2\nu t+1)^{n+2}}\biggr] + \quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ +\frac{F_{n}^2\cdot r^{2n+1}}{8\mu_{n}^4\nu^2(n+1)(n+3)}\biggl[\Phi\bigl(n, n+2;-\mu_{n}^2 r^2\bigr) - \frac{\Phi\bigl(n, n+2;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)}{(4\mu_{n}^2\nu t+1)^{n+1}}\biggr]\cdot \quad\quad\quad\quad \nonumber\\ \nonumber\\ \cdot\biggl[\Phi\bigl(n+3, n+4;-\mu_{n}^2 r^2\bigr) - \frac{\Phi\bigl(n+3, n+4;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)}{(4\mu_{n}^2\nu t+1)^{n+3}}\biggr] \quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\\end{aligned}$$ and $$\begin{aligned} \label{eqn280} T_{2,2n,\varphi, r}(r, t) = \frac{- F_{n}^2 \cdot n \cdot r^{2n-1}}{8\mu_{n}^6\nu^2(n+1)^2(n+2)}\biggl[\Phi\bigl(n+1, n+2;-\mu_{n}^2 r^2\bigr) - \frac{\Phi\bigl(n+1, n+2;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)}{(4\mu_{n}^2\nu t+1)^{n+1}}\biggr]\cdot \quad \nonumber\\ \nonumber\\ \cdot\biggl[\Phi\bigl(n+1, n+3;-\mu_{n}^2 r^2\bigr) - \frac{\Phi\bigl(n+1, n+3;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)}{(4\mu_{n}^2\nu t+1)^{n+2}}\biggr] = 2 n \cdot T_{2,2n, r}(r, t) \quad\quad\quad\quad \nonumber\\ \nonumber\\\end{aligned}$$ $T_{2,2n, r}(r, t)$ is taken from formulas $(\ref{eqn254})$ and $(\ref{eqn261})$. Then we do several operations and have from formula $(\ref{eqn279})$: $$\begin{aligned} \label{eqn281} T_{2,2n,\varphi, \varphi}(r, t) = \frac{- F_{n}^2\cdot n \cdot r^{2n+1}}{8\mu_{n}^4\nu^2(n+1)(n+2)^2}\biggl[\Phi\bigl(n+1, n+3;-\mu_{n}^2 r^2\bigr) \cdot\Phi\bigl(n+2, n+3;-\mu_{n}^2 r^2\bigr) - \nonumber\\ \nonumber\\ - \frac{1}{(4\mu_{n}^2\nu t+1)^{n+2}}\cdot\Phi\bigl(n+1, n+3;-\mu_{n}^2 r^2\bigr)\cdot\Phi\bigl(n+2, n+3;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr) - \quad\quad \nonumber\\ \nonumber\\ - \frac{1}{(4\mu_{n}^2\nu t+1)^{n+2}}\cdot\Phi\bigl(n+1, n+3;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)\cdot \Phi\bigl(n+2, n+3;-\mu_{n}^2 r^2\bigr) + \quad\quad \nonumber\\ \nonumber\\ + \frac{1}{(4\mu_{n}^2\nu t+1)^{2n+4}}\cdot\Phi\bigl(n+1, n+3;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)\cdot\Phi\bigl(n+2, n+3;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)\biggr ] + \nonumber\\ \nonumber\\ + \frac{F_{n}^2\cdot r^{2n+1}}{8\mu_{n}^4\nu^2(n+1)(n+3)}\biggl[\Phi\bigl(n, n+2;-\mu_{n}^2 r^2\bigr) \cdot\Phi\bigl(n+3, n+4;-\mu_{n}^2 r^2\bigr) - \quad\quad\quad\quad \nonumber\\ \nonumber\\ - \frac{1}{(4\mu_{n}^2\nu t+1)^{n+3}}\cdot\Phi\bigl(n, n+2;-\mu_{n}^2 r^2\bigr)\cdot\Phi\bigl(n+3, n+4;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr) - \quad\quad \quad\quad \nonumber\\ \nonumber\\ - \frac{1}{(4\mu_{n}^2\nu t+1)^{n+1}}\cdot\Phi\bigl(n, n+2;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)\cdot \Phi\bigl(n+3, n+4;-\mu_{n}^2 r^2\bigr) + \quad\quad\quad\quad \nonumber\\ \nonumber\\ + \frac{1}{(4\mu_{n}^2\nu t+1)^{2n+4}}\cdot\Phi\bigl(n, n+2;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)\cdot\Phi\bigl(n+3, n+4;-\frac{\mu_{n}^2 r^2}{(4\mu_{n}^2\nu t+1)}\bigr)\biggr ] \quad\quad \nonumber\\ \nonumber\\\end{aligned}$$ We transform formulas $(\ref{eqn226}), (\ref{eqn227})$ for components $u_{j\varphi1}(r, \varphi),\; u_{j\varphi2}(r, \varphi)$ and formulas $(\ref{eqn224}), (\ref{eqn225})\;\;$ for $R_{j,n_{j}-1,\varphi}(r),\; R_{j,n_{j}+1,\varphi}(r)$ for j = 2 and then use formula $(\ref{eqn259})$ for $f_{2\varphi}^{*} (r,\varphi, t)$. We do several operations and have: $$\label{eqn284} u_{2\varphi1}^{*}(r,\varphi, t)\;=\;- \;\frac{i}{2}\bigl[R_{2,2n-1,\varphi}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (2n-1) \varphi}\;+\;R_{2,2n+1,\varphi}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (2n+1) \varphi}\bigr]$$ $$\label{eqn285} u_{2\varphi2}^{*}(r,\varphi, t)\;=\;\frac{1}{2}\bigl[R_{2,2n-1,\varphi}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (2n-1) \varphi}\;-\;R_{2,2n+1,\varphi}(r, t)\;{\mbox{\Large e \normalsize}} ^{i (2n+1) \varphi}\bigr]$$ After changing the order of integration we obtain: $$\label{eqn286} R_{2,2n-1,\varphi}(r, t) = -\;\frac{in^{2}}{2^{2}}\int_{0}^{t}\int_{0}^{\infty} \bigl(T_{2,2n,\varphi}(\tilde r, \tau)\cdot\tilde r\bigr)^{'}_{\tilde r} \int_{0}^{\infty}{\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)}J_{2n-1}(r \rho)\;J_{2n}(\tilde r \rho)\;d\rho d \tilde r d\tau$$ $$\label{eqn287} R_{2,2n+1,\varphi}(r, t) = -\;\frac{in^{2}}{2^{2}}\int_{0}^{t}\int_{0}^{\infty} \bigl(T_{2,2n,\varphi}(\tilde r, \tau)\cdot\tilde r\bigr)^{'}_{\tilde r} \int_{0}^{\infty}{\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)}J_{2n+1}(r \rho)\;J_{2n}(\tilde r \rho)\;d\rho d \tilde r d\tau$$ Then we take $\bigl(T_{2,2n,\varphi}(\tilde r, \tau)\cdot\tilde r\bigr)^{'}_{\tilde r}$ from formula $(\ref{eqn278})$ and with use of formula $(\ref{eqn280})$ put it in formulas $(\ref{eqn286}), (\ref{eqn287})$ and have: $$\begin{aligned} \label{eqn288} R_{2,2n-1,\varphi}(r, t) = -\;\frac{in^{2}}{2^{2}}\int_{0}^{t}\int_{0}^{\infty} T_{2,2n,\varphi, \varphi}(\tilde r, \tau) \int_{0}^{\infty}{\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)}J_{2n-1}(r \rho)\cdot J_{2n}(\tilde r \rho)\;d\rho d \tilde r d\tau - \quad\quad\quad \nonumber\\ \nonumber\\ -\;\frac{in^{3}}{2}\int_{0}^{t}\int_{0}^{\infty} T_{2,2n,r}(\tilde r, \tau) \int_{0}^{\infty}{\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)}J_{2n-1}(r \rho)\cdot J_{2n}(\tilde r \rho)\;d\rho d \tilde r d\tau = R_{2,2n-1,\varphi, \varphi}(r, t) - 2niR_{2,2n-1,r}(r, t) \nonumber\\ \nonumber\\\end{aligned}$$ $$\begin{aligned} \label{eqn289} R_{2,2n+1,\varphi}(r, t) = -\;\frac{in^{2}}{2^{2}}\int_{0}^{t}\int_{0}^{\infty} T_{2,2n,\varphi, \varphi}(\tilde r, \tau) \int_{0}^{\infty}{\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)}J_{2n+1}(r \rho)\cdot J_{2n}(\tilde r \rho)\;d\rho d \tilde r d\tau - \quad\quad\quad \nonumber\\ \nonumber\\ -\;\frac{in^{3}}{2}\int_{0}^{t}\int_{0}^{\infty} T_{2,2n,r}(\tilde r, \tau) \int_{0}^{\infty}{\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)}J_{2n+1}(r \rho)\cdot J_{2n}(\tilde r \rho)\;d\rho d \tilde r d\tau = R_{2,2n+1,\varphi, \varphi}(r, t) - 2niR_{2,2n+1,r}(r, t) \nonumber\\ \nonumber\\\end{aligned}$$ where $$\begin{aligned} \label{eqn290} R_{2,2n-1,\varphi,\varphi}(r, t) = -\;\frac{in^{2}}{2^{2}}\int_{0}^{t}\int_{0}^{\infty} T_{2,2n,\varphi, \varphi}(\tilde r, \tau) \int_{0}^{\infty}{\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)}J_{2n-1}(r \rho)\cdot J_{2n}(\tilde r \rho)\;d\rho d \tilde r d\tau \end{aligned}$$ $$\begin{aligned} \label{eqn291} R_{2,2n+1,\varphi,\varphi}(r, t) = -\;\frac{in^{2}}{2^{2}}\int_{0}^{t}\int_{0}^{\infty} T_{2,2n,\varphi, \varphi}(\tilde r, \tau) \int_{0}^{\infty}{\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)}J_{2n+1}(r \rho)\cdot J_{2n}(\tilde r \rho)\;d\rho d \tilde r d\tau \end{aligned}$$ and $R_{2,2n-1,r}(r, t)\;,\; R_{2,2n+1,r}(r, t)$ we take from formulas $(\ref{eqn266}), (\ref{eqn267})$. Then we use formulas $(\ref{eqn260}), (\ref{eqn264}), (\ref{eqn265}), (\ref{eqn284}), (\ref{eqn285})$ and have: $$\begin{aligned} \label{eqn302} u_{21}^{*}(r,\varphi, t) = u_{2r1}^{*}(r,\varphi, t) + u_{2\varphi1}^{*}(r,\varphi, t)\;= \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ =\;\bigl [ n R_{2,2n-1,r}(r, t) - \frac{i}{2} R_{2,2n-1,\varphi}(r, t)\bigr ]\;{\mbox{\Large e \normalsize}} ^{i (2n-1) \varphi} + \bigl [ n R_{2,2n+1,r}(r, t) - \frac{i}{2} R_{2,2n+1,\varphi}(r, t)\bigr ]\;{\mbox{\Large e \normalsize}} ^{i (2n+1) \varphi} \nonumber\\ \nonumber\\\end{aligned}$$ $$\begin{aligned} \label{eqn303} u_{22}^{*}(r,\varphi, t) = u_{2r2}^{*}(r,\varphi, t) + u_{2\varphi2}^{*}(r,\varphi, t)\;= \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \nonumber\\ =\;i\;\bigl [ n R_{2,2n-1,r}(r, t) - \frac{i}{2} R_{2,2n-1,\varphi}(r, t)\bigr ]\;{\mbox{\Large e \normalsize}} ^{i (2n-1) \varphi} - i\;\bigl [ n R_{2,2n+1,r}(r, t) - \frac{i}{2} R_{2,2n+1,\varphi}(r, t)\bigr ]\;{\mbox{\Large e \normalsize}} ^{i (2n+1) \varphi} \nonumber\\ \nonumber\\\end{aligned}$$ We obtain by performing appropriate transformations: $$\begin{aligned} \label{eqn304} u_{2 r}^{*}(r,\varphi, t)\;=\;\bigl \{\bigl [ n R_{2,2n-1,r}(r, t) - \frac{i}{2} R_{2,2n-1,\varphi}(r, t)\bigr ] + \bigl [ n R_{2,2n+1,r}(r, t) - \frac{i}{2} R_{2,2n+1,\varphi}(r, t)\bigr ]\bigr \}\;{\mbox{\Large e \normalsize}} ^{i 2n \varphi} \nonumber\\ \nonumber\\ \quad\quad\quad\quad\end{aligned}$$ $$\begin{aligned} \label{eqn305} u_{2\varphi}^{*}(r,\varphi, t)\;=\;i\;\bigl \{\bigl [ n R_{2,2n-1,r}(r, t) - \frac{i}{2} R_{2,2n-1,\varphi}(r, t)\bigr ] - \bigl [ n R_{2,2n+1,r}(r, t) - \frac{i}{2} R_{2,2n+1,\varphi}(r, t)\bigr ]\bigr \}\;{\mbox{\Large e \normalsize}} ^{i 2n \varphi} \nonumber\\ \nonumber\\ \quad\quad\quad\quad\end{aligned}$$ Here $u_{2 r}^{*}(r,\varphi, t) ,\; u_{2\varphi}^{*}(r,\varphi, t)$ are the radial and tangential components of the first correction $\vec u_{2}^{*}$ of the velocity $\vec u_{1}$. $R_{2,2n-1,r}(r, t),\; R_{2,2n+1,r}(r, t)$ are taken from formulas $(\ref{eqn266}), (\ref{eqn267})$. $$\begin{aligned} \label{eqn306} R_{2,2n-1,\varphi}(r, t)\;=\;R_{2,2n-1,\varphi, \varphi}(r, t) - 2niR_{2,2n-1,r}(r, t) \quad\quad\quad\quad \nonumber\\ \nonumber\\ R_{2,2n+1,\varphi}(r, t)\;=\;R_{2,2n+1,\varphi, \varphi}(r, t) - 2niR_{2,2n+1,r}(r, t) \quad\quad\quad\quad \nonumber\\ \nonumber\\\end{aligned}$$ and $R_{2,2n-1,\varphi, \varphi}(r, t),\; R_{2,2n+1, \varphi, \varphi}(r, t)$ are taken from formulas $(\ref{eqn290}), (\ref{eqn291})$. Then we do appropriate operations and have from formulas $(\ref{eqn304}), (\ref{eqn305})$: $$\begin{aligned} \label{eqn304a} u_{2 r}^{*}(r,\varphi, t)\;=\;- \frac{i}{2}\bigl [ R_{2,2n-1,\varphi,\varphi}(r, t) + R_{2,2n+1,\varphi,\varphi}(r, t)\bigr ]\;{\mbox{\Large e \normalsize}} ^{i 2n \varphi} \nonumber\\ \quad\quad\quad\quad\end{aligned}$$ $$\begin{aligned} \label{eqn305a} u_{2\varphi}^{*}(r,\varphi, t)\;=\;\frac{1}{2}\bigl [ R_{2,2n-1,\varphi,\varphi}(r, t) - R_{2,2n+1,\varphi,\varphi}(r, t)\bigr ]\;{\mbox{\Large e \normalsize}} ^{i 2n \varphi} \nonumber\\ \quad\quad\quad\quad\end{aligned}$$ From formulas $(\ref{eqn302}), (\ref{eqn303})$ with properties of $R_{2,2n-1,r}(r, t),\; R_{2,2n+1,r}(r, t),\; R_{2,2n-1,\varphi}(r, t),\; R_{2,2n+1,\varphi}(r, t)$ it follows: $$\begin{aligned} \label{eqn307} \begin{array}{ll} lim \;\;u_{21}^{*}(r,\varphi,t)= 0;\;\;\; lim \;\;u_{22}^{*}(r,\varphi,t)= 0;\\ t \rightarrow 0; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; t \rightarrow 0; \end{array}\end{aligned}$$ $$\begin{aligned} \label{eqn308} \begin{array}{ll} lim \;\;u_{2r}^{*}(r,\varphi,t)= 0;\;\;\; lim \;\;u_{2\varphi}^{*}(r,\varphi,t)= 0;\\ t \rightarrow 0; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; t \rightarrow 0; \end{array}\end{aligned}$$ In other words the velocity $\vec u_{2} = \vec u_{1} - \vec u_{2}^{*}$ \[look at $(\ref{eqn50})$\] satisfies the initial conditions $(\ref{eqn200})$. We use the asymptotic properties of the confluent hypergeometric function $\Phi(a, c; x)$ and have from formulas $(\ref{eqn266}),(\ref{eqn267}), (\ref{eqn290}),$ $(\ref{eqn291})$: the first correction $\vec u_{2}^{*}$ and therefore the velocity $\vec u_{2}$ satisfies conditions $\;(\ref{eqn16})\;$ ( for $r \;\rightarrow\; \infty $). By comparing the solution $\vec u_{1}$ from $(\ref{eqn238}), (\ref{eqn239})$ or $(\ref{eqn240}), (\ref{eqn241})$ of the first step of iterative process with the first correction $\vec u_{2}^{*}$ from $(\ref{eqn302}), (\ref{eqn303})$ or $(\ref{eqn304a}), (\ref{eqn305a})$, which is received on the second step of iterative process, we see that $$\label{eqn309} \mid\vec{u_{2}^{*}}\mid\; <<\; \mid\vec{u_{1}}\mid$$ with conditions $$\begin{aligned} \label{eqn310} F_{n}\;\leq\;\frac{1}{n} \nonumber\\ \nonumber\\\end{aligned}$$ By continuing this iterative process we can obtain next parts $\;\vec{u_{3}^{*}}\;, \vec{u_{4}^{*}}...$, of the converging series for $\vec{u}$. For arbitrary step j of the iterative process we have by using formula $(\ref{eqn57})$: $$\label{eqn311} \vec{u}_{j}\;=\;\vec{u}_{1}\;-\;\sum_{l=2}^{j} \vec{u}_{l}^{*}$$ and then: $$\label{eqn312} \begin{array}{ll} lim \;\;\vec{u_{j}}= \vec{u}\\ j \rightarrow \infty \end{array}$$ where $\vec{u}$ is the solution of the problem $(\ref{eqn1}) - (\ref{eqn6})$.\ \ Below we provide numerical analysis of these results for the following values of problem’s parameters: Circumferential modes n = 1, 2, 3, 4, 5. $\mu_n$ = 1 (n = 1, 2, 3, 4, 5). Results were obtained for functions $\vec u_{1} - (\ref{eqn238}), (\ref{eqn239})$ or $(\ref{eqn240}), (\ref{eqn241})$ with calculations of the confluent hypergeometric functions $\cite{BE153}$; $\vec u_{2}^{*} - (\ref{eqn302}), (\ref{eqn303})$ or $(\ref{eqn304a}), (\ref{eqn305a})$ by using numerical integration of the triple integrals $(\ref{eqn290}), (\ref{eqn291})$. Each of those integrals is computed as an iterated integral. Let us consider first the calculation of the inner integrals from $(\ref{eqn290}), (\ref{eqn291})$: $$\begin{aligned} \label{eqn290a} I_{\underline{+}}(r, \tilde r, t, \tau) = \int_{0}^{\infty}{\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)}J_{2n_{\underline{+}}1}(r \rho)\cdot J_{2n}(\tilde r \rho)\;d\rho \end{aligned}$$ For condition t $< \tau$ the integrand is diminishing fast enough. It is easy to find upper limit of integration, so we can substitute integral $(\ref{eqn290a})$ for $$\begin{aligned} \label{eqn290b} I_{\underline{+}}(r, \tilde r, t, \tau) = \int_{0}^{A_1}{\mbox{\Large e \normalsize}} ^{-\nu \rho^2 (t-\tau)}J_{2n_{\underline{+}}1}(r \rho)\cdot J_{2n}(\tilde r \rho)\;d\rho, \end{aligned}$$ where ($0 < A_1 = 200 < \infty$), and hence we are integrating over the finite interval. For additional check let us increase $A_1$ in 1.5 times and change the number of steps of integration $n_1($ from 4001 to 6001 $)$. We see that the difference in result values of integral $(\ref{eqn290b})$ is within the range of required precision $\epsilon_1(10^{-14})$. For condition t $= \tau$ the integral $(\ref{eqn290a})$ is in fact an integral of Weber and Schafheitlin and it is possible to calculate it analytically $(\ref{A17})\; \cite{gW44}$. Let us now consider the calculation of middle integrals $$\begin{aligned} \label{eqn290c} \tilde I_{\underline{+}}(r, t, \tau) = \int_{0}^{\infty} T_{2,2n,\varphi, \varphi}(\tilde r, \tau) I_{\underline{+}}(r, \tilde r, t, \tau) d \tilde r \end{aligned}$$ We use asymptotical properties of confluent hypergeometric functions $\Phi(a, c; x)$ $ \;\;\cite{BE153}$ and we have: $$\begin{array}{ll} T_{2,2n,\varphi, \varphi}(\tilde r, \tau)\rightarrow (1/ \tilde r^{2n + 5})\\ \tilde r \rightarrow \infty \end{array}$$ Hence, we substitute integral $(\ref{eqn290c})$ for $$\begin{aligned} \label{eqn290d} \tilde I_{\underline{+}}(r, t, \tau) = \int_{0}^{A_2} T_{2,2n,\varphi, \varphi}(\tilde r, \tau) I_{\underline{+}}(r, \tilde r, t, \tau) d \tilde r \end{aligned}$$ where ($0 < A_2 = 20 < \infty$) and integration is really over the finite interval. For additional check let us increase value $A_2$ in 1.5 times and change the number of integration steps $n_2$(from 201 to 301). We see that the difference in result values of integral $(\ref{eqn290d})$ is within the range of required precision $\epsilon_2(10^{-11})$. Confluent hypergeometric functions $\Phi(a, c; x)$ were computed with precision $\epsilon(10^{-15})$. The outer integrals in $(\ref{eqn290}), (\ref{eqn291})$ are the integrals over finite interval (0, t = 10). These integrals are computed with precision $\epsilon_3(10^{-5})$ and the number of steps of integration $n_3$= 101. For additional check let us change the number of integration steps $n_3$(from 101 to 201), and we see that the difference in result integral values is within the required precision $\epsilon_3(10^{-5})$. All integrals were computed by Simpson’s method and $\epsilon_1(10^{-14}) < \epsilon_2(10^{-11}) < \epsilon_3(10^{-5})$. $\vec u_{2} = \vec u_{1} - \vec u_{2}^{*}$ and is shown in FIG. 5.1.1 - 5.1.15. The vector field $\vec u_{2}$ at distances r = 1, 2, 3, 5, 7 is represented by the dotted curves in top diagrams. The comparison of $\mid\vec u_{1}\mid$ (dashed plots) and $\mid\vec u_{2}^{*}\mid$ (solid plots) in plane $\varphi$ = \[0, $\pi$\], at distances $\;\;\;$ 0 $\leq$ r $\leq$ 50 is represented in bottom diagrams. This comparison shows $\mid\vec{u_{2}^{*}}\mid\; <<\; \mid\vec{u_{1}}\mid$ and is corresponding to the conclusion $(\ref{eqn484})$. $\nonumber\\$ $\nonumber\\$ {height="80mm"}\ FIG.5.1.1. n = 1, $F_1$ = 1, $\nu$ = 1.5 {height="80mm"}\ FIG.5.1.2. n = 1, $F_1$ = 1, $\nu$ = 1 {height="80mm"}\ FIG.5.1.3. n = 1, $F_1$ = 1, $\nu$ = 0.75 {height="80mm"}\ FIG.5.1.4. n = 2, $F_2$ = 0.5, $\nu$ = 1.5 {height="80mm"}\ FIG.5.1.5. n = 2, $F_2$ = 0.5, $\nu$ = 1 {height="80mm"}\ FIG.5.1.6. n = 2, $F_2$ = 0.5, $\nu$ = 0.75 {height="80mm"}\ FIG.5.1.7. n = 3, $F_3$ = 0.33, $\nu$ = 1.5 {height="80mm"}\ FIG.5.1.8. n = 3, $F_3$ = 0.33, $\nu$ = 1 {height="80mm"}\ FIG.5.1.9. n = 3, $F_3$ = 0.33, $\nu$ = 0.75 {height="80mm"}\ FIG.5.1.10. n = 4, $F_4$ = 0.25, $\nu$ = 1.5 {height="80mm"}\ FIG.5.1.11. n = 4, $F_4$ = 0.25, $\nu$ = 1 {height="80mm"}\ FIG.5.1.12. n = 4, $F_4$ = 0.25, $\nu$ = 0.75 {height="80mm"}\ FIG.5.1.13. n = 5, $F_5$ = 0.2, $\nu$ = 1.5 {height="80mm"}\ FIG.5.1.14. n = 5, $F_5$ = 0.2, $\nu$ = 1 {height="80mm"}\ FIG.5.1.15. n = 5, $F_5$ = 0.2, $\nu$ = 0.75 $\nonumber\\ \nonumber\\$ $\nonumber\\ \nonumber\\$ $\nonumber\\ \nonumber\\$ $\nonumber\\ \nonumber\\$ {#section .unnumbered} The Fourier integral can be stated in the forms: $\\N=1$ $$\label{A1} U(\gamma)=F[u(x)]= \frac{1}{(2\pi)^{1/2}} \int_{-\infty}^{\infty}u(x)\, {\mbox{\Large e \normalsize}} ^{i\gamma x}dx \;\;\;\;\;\;\;\;\;\; u(x)= \frac{1}{(2\pi)^{1/2}}\int_{-\infty}^{\infty}U(\gamma) {\mbox{\Large e \normalsize}} ^{-i\gamma x}d\gamma$$ $\\\\\\N=2$ $$\begin{aligned} \label{A2} U( \gamma_{1} , \gamma_{2})=F[ u(x_{1} , x_{2})]= \frac{1}{2\pi} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} u( x_{1} , x_{2})\,{\mbox{\Large e \normalsize}} ^{ i( \gamma_{1} x_{1} + \gamma_{2} x_{2}) } dx_{1} dx_{2} \nonumber\\ \nonumber\\ u( x_{1} , x_{2})= \frac{1}{2\pi} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} U( \gamma_{1} , \gamma_{2})\,{\mbox{\Large e \normalsize}} ^{- i( \gamma_{1} x_{1} + \gamma_{2} x_{2}) } d\gamma_{1} d\gamma_{2}\end{aligned}$$ $\\N=3$ $$\begin{aligned} \label{A3} U( \gamma_{1} , \gamma_{2} , \gamma_{3})=F[\, u(x_{1} , x_{2} , x_{3})]= \frac{1}{(2\pi)^{3/2}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} u( x_{1} , x_{2} , x_{3})\,{\mbox{\Large e \normalsize}} ^{ i( \gamma_{1} x_{1} + \gamma_{2} x_{2} + \gamma_{3} x_{3}) } dx_{1} dx_{2} dx_{3} \nonumber\\ \nonumber\\ u( x_{1} , x_{2} , x_{3})= \frac{1}{(2\pi)^{3/2}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} U( \gamma_{1} , \gamma_{2},\gamma_{3} )\, {\mbox{\Large e \normalsize}} ^{- i( \gamma_{1} x_{1} + \gamma_{2} x_{2} + \gamma_{3} x_{3}) } d\gamma_{1} d\gamma_{2} d\gamma_{3} \end{aligned}$$ $\\\\$ The Laplace integral is usually stated in the following form: $$\label{A4} U^{\otimes}(\eta)=L[\,u(t)\,]= \int_{0}^{\infty}u(t)\, {\mbox{\Large e \normalsize}} ^{-\eta t}dt \;\;\;\;\; u(t)=\frac{1}{2\pi i}\int_{c- i \infty }^{c + i \infty} U^{\otimes}(\eta) \,{\mbox{\Large e \normalsize}} ^{\eta t}d\eta \;\;\;\;\; c > c_{0}$$ $$\label{A5} L[\,u^{'}(t)\,]=\eta \,U^{\otimes}(\eta)-u(0)$$ \[section\] $\\\\$ If integrals $$U_{1}^{\otimes}(\eta)= \int_{0}^{\infty}u_{1}(t)\, {\mbox{\Large e \normalsize}} ^{-\eta t}d\,t \;\;\;\;\;\;\;\;\;\; U_{2}^{\otimes}(\eta)= \int_{0}^{\infty}u_{2}(t)\, {\mbox{\Large e \normalsize}} ^{-\eta t}d\,t$$ absolutely converge by $Re\, \eta > \sigma_{d}$, then $U^{\otimes}(\eta)\,= \,U_{1}^{\otimes}(\eta)\, U_{2}^{\otimes}(\eta)$ is Laplace transform of $$\label{A6} u(t)=\int_{0}^{t}u_{1}(t-\tau)\,u_{2}(\tau)\,d\,\tau \\$$ Useful *Laplace integral*: $$\label{A7} L[\,{\mbox{\Large e \normalsize}} ^{\eta_{k}t}\,]\,=\,\int_{0}^{\infty}{\mbox{\Large e \normalsize}} ^{-(\eta-\eta_{k})\,t}d\,t \;=\; \frac{1}{(\eta-\eta_{k})}\;\;\;\;\;\;\;\;\;(Re\,\eta\,>\,\eta_{k})$$ De Moivre’s formulas: $$\label{A8} cos \varphi = \frac{1}{2}({\mbox{\Large e \normalsize}}^{i\varphi} + {\mbox{\Large e \normalsize}}^{-i\varphi})\;\;,\;\;sin \varphi = \frac{1}{2i}({\mbox{\Large e \normalsize}}^{i\varphi} - {\mbox{\Large e \normalsize}}^{-i\varphi})$$ Bessel function’s integral representation: $$\label{A9} J_{n}(z) = \frac{i^{-n}}{2\pi}\oint_C{\mbox{\Large e \normalsize}} ^{iz cos\theta + in\theta} d \theta\;, \;\;\emph{C is the unit circle around the origin.}$$ The discontinuous integral of Weber and Schafheitlin: $$\label{A17} \int_{0}^{\infty}J_{\mu}(at)\cdot J_{\mu-1}(bt)\;dt\;\;=\;\;\Biggl\{ \begin{array}{lll} b^{\mu-1}/a^{\mu} & (b < a)\\ 1/2b & (b = a)\\ 0 & (b > a) \end{array}$$ $$\label{A10} \int_{0}^{\infty}J_{\mu}(\alpha t) {\mbox{\Large e \normalsize}}^{-\gamma^2 t^2}t^{\mu+1}\;d t = \frac{ \alpha^\mu}{(2\gamma^2)^{\mu+1}}{\mbox{\Large e \normalsize}}^{-\frac{\alpha^2}{4\gamma^2}},\;\;\;\;\;\;\;\;\; Re \;\mu > -1, \; Re \;\gamma^2 > 0.$$ $$\begin{aligned} \label{A11} \int_{0}^{\infty}J_{\mu}(\alpha t) {\mbox{\Large e \normalsize}}^{-\gamma^2 t^2}t^{\rho-1}\;d t = \frac{\gamma^{-\rho}}{2\cdot\Gamma(\mu+1)} \cdot \Gamma\biggl(\frac{\mu+\rho}{2} \biggr)\cdot \biggl(\frac{\alpha}{2\gamma} \biggr)^{\mu}\cdot \Phi \biggl(\frac{\mu+\rho}{2},\mu+1;-\frac{\alpha^2}{4\gamma^2}\biggr), \nonumber\\ \nonumber\\ Re \;\gamma^2 > 0, Re (\mu+\rho) > 0.\end{aligned}$$ $$\label{A12} \frac{d}{dy}[y^a \cdot \Phi (a,c;-\beta y)] = a \cdot y^{a-1} \cdot \Phi (a+1,c;-\beta y)$$ $$\label{A13} \frac{d}{dy}[y^{c-1} \cdot \Phi (a,c;-\beta y)] = (c - 1) \cdot y^{c-2} \cdot \Phi (a,c-1;-\beta y)$$ Formula describing connection between the contiguous confluent hypergeometric functions: $$\label{A14} c \cdot\Phi - c \cdot\Phi(a-) - x\cdot c \cdot\Phi(c+) = 0$$ \ \ \ \ [11]{} , *Vorticity and Incompressible Flows,* Cambridge U. Press, Cambridge, 2002. , *Some open problems and research directions in the mathematical study of fluid dynamics,* in Mathematics Unlimited-2001 and Beyond, Springer Verlag, Berlin, 2001, 353-360. , *The Mathematical Theory of Viscous Incompressible Flows,* (2nd edition), Gordon and Breach, 1969. , *Sur le Mouvement d’un Liquide Visquex Emplissent l’Espace.* Acta Math. J. 63 (1934), 193-248. , *Turbulence and Hausdorff dimension, in Turbulence and the Navier-Stokes Equations.* Lecture Notes in Math. No. 565, Springer Verlag, 1976, pp. 94-112. , *An inviscid flow with compact support in spacetime.* J. Geom. Analysis 3 No. 4 (1993), 343-401. , *On the nonuniqueness of weak solutions of the Euler equation.* Communications on Pure and Applied Math. 50 (1997), 1260-1286. , *Partial regularity of suitable weak solutions of the Navier-Stokes equations.* Communications on Pure and Applied Math. 35 (1982), 771-831. , *A new proof of the Caffarelli-Kohn-Nirenberg theorem.* Communications on Pure and Applied Math. 51 (1998), 241-257. , *Higher Transcendental functions.* Volume 1. New York, Toronto, London. Mc Graw-Hill Book Company, Inc. 1953. , *Higher Transcendental functions.* Volume 2. New York, Toronto, London. Mc Graw-Hill Book Company, Inc. 1953. , *A treatise on the theory of Bessel functions.* (2nd edition), Cambridge Univercity Press, 1944. [^1]: 2000 Mathematics Subject Classification. Primary 35Q30, Secondary 76D05.

--- abstract: '[It has been shown that]{} the symmetry-protected topological (SPT) phases with finite Abelian symmetries can be described by Chern-Simons field theory. We propose a topological response theory to uniquely identify the SPT orders, which allows us to obtain [a systematic scheme to classify]{} bosonic SPT phases with any finite Abelian symmetry group. We point out that even for finite Abelian symmetry, there exist bosonic SPT phases beyond the current Chern-Simons theory framework. We also apply the theory to fermionic SPT phases with $\mathbb{Z}_m$ symmetry and find the classification of SPT phases depends on the parity of $m$: for even $m$ there are $2m$ classes, $m$ out of which is intrinsically fermionic SPT phases and can not be realized in any bosonic system. Finally we propose a classification scheme of fermionic SPT phases for any finite, Abelian symmetry.' author: - Meng Cheng - 'Zheng-Cheng Gu' bibliography: - 'spt.bib' title: 'Topological Response Theory of Abelian Symmetry-Protected Topological Phases in Two Dimensions' --- [*Introduction.*]{} In recent years, the research on topological matter has revealed a new class of gapped quantum phases, namely symmetry-protected topological(SPT) phases [@Gu_PRB2009]. They are topologically distinct from trivial atomic insulators if (and only if) certain symmetries are not broken. A notable example is the electronic topological insulators in two and in three dimensions [@Kane2005b; @Bernevig2006; @Fu_PRL07; @Roy_PRB2009; @Moore_PRB07; @Kitaev2009], [protected by time reversal and charge conservation symmetries]{}. Although featureless in the bulk, SPT phases do support gapless boundary excitations protected by symmetry. This fact clearly distinguishes them from [a trivial product state or an atomic insulating state]{}. Recent theoretical discoveries, initiated by the group-cohomological construction [@Chen_arxiv2011; @Chen_science], have vastly extended our knowledge on the classification of SPT phases. Systematic constructions of bosonic SPT phases, with arbitrary symmetry groups and in any spatial dimensions, have been proposed [@Chen_PRB2011a; @Fidkowski_PRB2011; @Turner_PRB2011; @Schuch_PRB2011; @Chen_arxiv2011; @Chen_science; @Chen_PRB2011b]. Later on the construction is also generalized to fermionic systems, using the so-called group super-cohomology approach [@Gu_arxiv2012]. In addition to the general classification, representative ground state wavefunctions, as well as exact solvable parent lattice Hamiltonians are also naturally derived. However, the wavefunctions and parent Hamiltonians are often quite complicated and it is not easy to access the low-energy universal properties of [these]{} phases, e.g., the edge properties. Quite recently, field-theoretical approach have been taken to understand the physics of SPT phases in two dimensions protected by Abelian symmetries [@Levin_arxiv2012; @Lu_arxiv2012; @senthil_arxiv; @senthil_3D; @Liu_PRL2013]. There the SPT phases are described effectively by a multi-component $\mathbb{U}(1)$ Chern-Simons theory and the classification is derived from the equivalence classes of ${{\mathbf K}}$ matrix and the (perturbative) stability of edge theory. However, a (non-perturbative) bulk argument is missing and the equivalent classes of ${{\mathbf K}}$ matrix is also quite difficult to compute for arbitrary Abelian symmetry group. On the other hand, the underlying connection with (super)cohomology class is unclear as well. In this work we present a unified treatment of both bosonic and fermionic SPT phases in two dimensions, with finite, unitary Abelian symmetries. Building upon the work of Levin and Gu [@LevinGu_arxiv2012], we formulate a topological response theory to probe the bulk properties of SPT phases. This allows us to characterize bosonic SPT phases with finite Abelian symmetry group that can be described in the Chern-Simons theory framework. We also study fermionic SPT phases with $\mathbb{Z}_m$ symmetry and interestingly find an even-odd effect: the classification is $\mathbb{Z}_m$ for odd $m$ and $\mathbb{Z}_{2m}$ for even $m$. Finally we obtain a minimal [set]{} of fermionic SPT phases for any finite, Abelian symmetry. [*Topological Response Theory of SPT Phases.* ]{} [We start with a brief overview of the underlying strategy for our classification scheme. Because SPT phases are protected by a global symmetry, we can consider coupling the SPT phase to an external gauge field taking value in that symmetry group. Following the scheme proposed by Levin and Gu [@LevinGu_arxiv2012], [we use the braiding statistics of the gauge fluxes as a physical response to identify these SPT phases. ]{}A classification can be achieved once the following two problems are resolved: (a) How to compute the braiding statistics of a gauge flux. (b) How to identify the equivalence classes of the flux statistics. Heuristically, when inserting a flux into a SPT phase, the statistics of the flux depends on the charged particles bind to the flux due to unknown local energetics. Therefore, each SPT phase should be associated with a whole family of flux statistics obtained from attachment of gauge charges, i.e. the whole flux sector.]{} Now we describe the first part of our proposal, i.e. coupling the SPT phase to an external gauge field. We review the Chern-Simons field theory of SPT phases [@Lu_arxiv2012; @Levin_arxiv2012]. They are described by a multi-component Abelian Chern-Simons theory, which in its most general form is given by the following Lagrangian: $$\mathcal{L}_\text{CS}=\frac{1}{4\pi}\varepsilon^{\mu\nu\lambda}a_\mu^I K_{IJ}\partial_\nu a_\lambda^J. \label{}$$ Here ${{\mathbf K}}$ is a $N\times N$ integer symmetric matrix. Notice that for bosonic systems, all diagonal entries of ${{\mathbf K}}$ must be even while at least one of the diagonal entries is odd for fermionic systems. Quantization of this gauge theory gives $|\det {{\mathbf K}}|$ ground states on torus. Since a SPT phase can not have any intrinsic topological order, we require $\det {{\mathbf K}}=\pm 1$ so there is no topological degeneracy on torus. And ${{\mathbf K}}$ should have equal number of positive and negative eigenvalues to avoid net chirality [^1] We will now assume that the system under consideration has a global, on-site symmetry group $G$. We consider unitary, finite Abelian group, which generally can be written as $G=\mathbb{Z}_{m_1}\times\mathbb{Z}_{m_2}\times\cdots\mathbb{Z}_{m_k}$ where $m_k>1$. We assume that the $\mathbb{Z}_{m_j}$ subgroup is generated by $g_j$ with $ g_j^{m_j}=1$. The matter fields carry irreducible representations of the symmetry group. Since the symmetry is Abelian and finite, it amounts to assign $\mathbb{Z}_{m_1}\times\mathbb{Z}_{m_2}\times\cdots\mathbb{Z}_{m_k}$ charges $q_\alpha^I$ to the $I$-th matter field. Here $q^I_\alpha$are valued in $\{0,1,\dots, m_\alpha-1\}$ and the subscript $\alpha$ refers to symmetry subgroups. Different assignment of the charges (or equivalently, symmetry transformation properties) can lead to distinct symmetry-protected phases. [Next we]{} couple the SPT phase to a gauge field taking value in the gauge group $G$. This is achieved by writing down minimal coupling for the matter fields in the SPT phases, uniquely determined by the charges carried by the matter fields. To facilitate the continuum field theory formulation, we view the group $G$ as a discrete subgroup of $\mathbb{U}(1)^k$. This perspective allows us to introduce $k$ external $\mathbb{U}(1)$ gauge fields $A^\alpha_\mu,\alpha=1,\dots,k$ that minimally couple to the matter fields in the SPT phases and then by introducing $k$ Higgs fields $\varphi_\alpha$ with charge $m_\alpha$ we can obtain a continuum version of discrete gauge theory as the symmetry broken phase of the $\mathbb{U}(1)^k$ Higgs theory. The presence of Higgs condensation results in quantization of gauge fluxes. After we gauge the [global]{} symmetry, the minimal coupling between the matter fields in the bulk and the external gauge fields reads: $$\mathcal{L}_\text{coupling}=\sum_\alpha{{\mathbf q}}_{\alpha}^T{{\mathbf j}}{A}^\alpha \label{}$$ The full theory is given by $\mathcal{L}=\mathcal{L}_\text{CS}+\mathcal{L}_\text{coupling}$. We then substitute ${{\mathbf j}}^\mu=\frac{1}{2\pi}\varepsilon^{\mu\nu\lambda}\partial_\nu {{\mathbf a}}_\lambda$ and integrate out internal gauge fields ${{\mathbf a}}$, yielding an effective theory for the external fields $A^\alpha$: $$\mathcal{L}_\text{eff}=\frac{1}{4\pi}\varepsilon^{\mu\nu\lambda}{A}^\alpha_\mu{{\mathbf q}}_\alpha^T{{\mathbf K}}^{-1}{{\mathbf q}}_\beta\partial_\nu {A}_\lambda^\beta+\mathcal{L}_\text{Higgs}[\varphi_\alpha, A_\alpha]. \label{eqn:gaugedL}$$ We denote $\tilde{{{\mathbf K}}}_{\alpha\beta}={{\mathbf q}}_\alpha^T{{\mathbf K}}^{-1}{{\mathbf q}}_\beta$ in the following discussion. We have therefore derived an Abelian Chern-Simons-Higgs theory as the effective “response theory” for the gauged SPT phases. Physically, as long as the symmetry $G$ remains unbroken, we can always gauge the SPT phases. The braiding statistics of the fluxes are fully captured by the effective theory . Let us now identify the equivalence classes of the flux statistics, as advertised in the beginning of our presentation. We start from the simplest case $G=\mathbb{Z}_m$. Then we just have a single Chern-Simons term at the level $\tilde{K}={{\mathbf q}}^T{{\mathbf K}}^{-1}{{\mathbf q}}$. An “elementary” vortex in the Higgs field encloses $\frac{2\pi}{m}$ gauge flux, as a result of flux quantization. The exchange statistics of the vortices can be derived from a charge-vortex duality transformation : $\theta=-\frac{\pi \tilde{K}}{m^2}$. To claim $\theta$ (or equivalently, $\tilde{K}$) as a unique “topological invariant” of the $\mathbb{Z}_m$ SPT phases, we must understand when two seemingly different values of $\theta$ actually describe the same topological class. To this end, notice that $\theta$ is the statistical angle of a pure Higgs vortex. Depending on local energetics, there may be $\mathbb{Z}_m$ charges bound to it. Assuming $q$ elementary $\mathbb{Z}_m$ charges are attached to the vortex, the composite object has a statistical angle $\theta+\frac{2\pi q}{m}$. This implies that the exchange statistics of a $\mathbb{Z}_m$ flux as a “topological invariant” for SPT phases is defined module $\frac{2\pi}{m}$, or $\tilde{K}$ is defined module $2m$. Two SPT phases whose corresponding statistical angles differ by an integer multiple of $\frac{2\pi}{m}$ should be considered as being equivalent, since the statistics can be changed by just adjusting the local energetics without affecting the bulk. Therefore the equivalance classes of $\tilde{K}$ is at most $\mathbb{Z}_{2m}$. However, this is not the end of the story. The statistics of the underlying particles, bosons or fermions, make a big difference. We first consider the simpler case of bosonic systems. One can easily see that $\tilde{K}$ for bosons must be even. Thus the classification is reduced to $\mathbb{Z}_m$. They can be realized by the following $\{{{\mathbf K}},{{\mathbf q}}_g\}$: [$${{\mathbf K}}= \begin{pmatrix} 0 & 1\\ 1 & -2n \end{pmatrix}, {{\mathbf q}}_g= \begin{pmatrix} 1\\ 0 \end{pmatrix}, n=0,1, \dots, m-1. \label{eqn:znspt}$$ It corresponds to $\theta=\frac{2\pi n}{m^2}$.]{} We notice that if the external gauge field is treated dynamically, the intrinsic topological order described by is essentially a twisted [gauge theory]{} with gauge group $G$  [@Propitius_thesis; @twistedQD] (see the Supplementary Material [@suppl] for the derivation of this fact). [However, it is important to emphasize that the classification of SPT using the braiding statistics of the gauge fluxes is [*not*]{} equivalent to classifying the intrinsic topological orders described by the corresponding Chern-Simons-Higgs theory Eq. (also see the Supplementary Material [@suppl] for more discussion).]{} [ ]{} Having understood $G=\mathbb{Z}_m$, we move to the next level of complexity $G=\mathbb{Z}_m\times\mathbb{Z}_n$. In this case, we have to consider an $\mathbb{U}(1)\times \mathbb{U}(1)$ external gauge fields and the response theory is fully characterized by a $2\times 2$ matrix $\tilde{{{\mathbf K}}}$. Repeating our argument above, the two diagonal elements determine the exchange statistics of the two types of gauge fluxes corresponding to the two subgroups and give $\mathbb{Z}_m\times\mathbb{Z}_n$ classification. The off-diagonal element introduces a new ingredient, the braiding statistics between the two gauge fluxes, denoted by $\theta_{12}$: $\theta_{12}=\frac{2\pi \tilde{K}_{12}}{mn}$. To determine the equivalence classes of $\theta_{12}$, we notice that we can attach $\mathbb{Z}_m$ charge to $\mathbb{Z}_n$ flux or vice versa and the mutual statistical angles are changed by integer multiples of $\frac{2\pi}{m}$ and $\frac{2\pi}{n}$, respectively. Therefore we identify the equivalence relation as $$\theta_{12}\equiv \theta_{12}+\frac{2\pi k_1}{m}+\frac{2\pi k_2}{n}, k_1,k_2\in\mathbb{Z}. \label{}$$ In terms of $\tilde{K}_{12}$, $$\tilde{K}_{12}\equiv \tilde{K}_{12}+k_1n+k_2m. \label{}$$ [The set of integers generated by $k_1 n+ k_2 m, k_1, k_2\in\mathbb{Z}$ are simply the integer multiples of $(m,n)$]{}. Therefore we conclude that that $\tilde{K}_{12}$ is defined module $(m,n)$, which gives additional $\mathbb{Z}_{(m,n)}$ classes. The complete classification is thus $\mathbb{Z}_m\times\mathbb{Z}_n\times\mathbb{Z}_{(m,n)}$. What is the physical meaning of the mutual braiding statistics in the SPT phases? To give a concrete example, consider a SPT phase characterized by $\tilde{K}_{11}=\tilde{K}_{22}=0, \tilde{K}_{12}=l$. One can easily write down the following Chern-Simons theory: $${{\mathbf K}}= \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}, {{\mathbf q}}_{1}= \begin{pmatrix} 0\\ 1 \end{pmatrix}, {{\mathbf q}}_{2}= \begin{pmatrix} l\\ 0 \end{pmatrix} \label{eqn:znzn}$$ So the two “dual” degrees of freedoms in the SPT phases carry the two subgroup symmetries respectively. Other classes can be understood in a similar way. In general, all bosonic SPT phases with $\mathbb{Z}_m\times \mathbb{Z}_n$ symmetry are realized by [@note1] $${{\mathbf K}}=\sigma_x\otimes\mathbf{1}_{2\times 2}, {{\mathbf q}}_1= \begin{pmatrix} 0\\ 1\\ 1\\ p \end{pmatrix}, {{\mathbf q}}_2= \begin{pmatrix} l \\ 0 \\ 1\\ q \end{pmatrix}. \label{}$$ It corresponds to $\tilde{{{\mathbf K}}}=\begin{pmatrix} 2p & l+p+q\\ l+p+q & 2q\end{pmatrix}$. We are now well prepared to generalize the above picture to arbitrary finite Abelian group $G$ with $k\geq 2$ generators. The $\tilde{{{\mathbf K}}}$ matrix of the response theory has $k$ diagonal elements, yielding $\mathbb{Z}_{m_1}\times\mathbb{Z}_{m_2}\times\cdots\mathbb{Z}_{m_k}$ classification. The $\frac{k(k-1)}{2}$ off-diagonal elements determine the braiding statistics between the gauge fluxes and the classification is $\prod_{i<j}\mathbb{Z}_{(m_i,m_j)}$. Therefore, we conclude that for a given symmetry group $G=\prod_{i=1}^k \mathbb{Z}_{m_i}$, the Abelian Chern-Simons theory construction can give $\prod_{i=1}^k \mathbb{Z}_{m_i}\times\prod_{ i<j} \mathbb{Z}_{(m_i,m_j)}$ classification of possible SPT phases. In fact, what we have obtained turns out to be a subset of the cohomological classification [@Chen_arxiv2011], which gives additional $\prod_{i<j<k}\mathbb{Z}_{(m_i, m_j, m_k)}$ classes [^2]. It is quite clear what is the limitation of the present approach: the gauged theory is bound to be Abelian. However, even with Abelian symmetry group SPT phases can become non-Abelian after the symmetry group is gauged. Simplest such example is $G=\mathbb{Z}_n\times\mathbb{Z}_n\times\mathbb{Z}_n$. In fact, for any $(m_i, m_j, m_k)>1$, there exists “non-Abelian” SPTs. We leave the study of these phases for future publications. [*Fermionic SPT Phases.*]{} We now study fermionic SPT phases protected by a symmetry group $G=\mathbb{Z}_m$. The conservation of total fermion parity is considered as a physical constraint rather than a symmetry. We will show quite interestingly, the classification exhibits an even-odd effect: for odd $m$ the classification is $\mathbb{Z}_m$ while for even $m$, it is $\mathbb{Z}_{2m}$. The $\mathbb{Z}_m$ classes for odd $m$ are nothing but the $\mathbb{Z}_m$ classes for bosonic SPT phases. The main difference between fermionic systems and bosonic ones is that the “identity” particle is a fermion which itself has a statistical angle $\pi$. This implies that when we consider the equivalence classes of the braiding statistics of fluxes in the gauged theory, we are free to attach fermions to them and the statistical angle can be changed by $\pi$ [@Gu_2010ftop; @Freedman_AP2004]. We already see there is a difference between even and odd $m$, since $\mathbb{Z}_m$ gauge theory with even $m$ contains a dyonic excitation with fermionic statistics while for odd $m$ this is not the case. We first consider $m$ odd. Attachment of (bosonic) $\mathbb{Z}_m$ charges allows one to change the statistical angle of an elementary gauge flux by an amount $\frac{2\pi p}{m}, p=0,1,\dots, m-1$. By attaching $\mathbb{Z}_m$-charge-neutral fermions, the set of phases is [fractionalized]{} to $ \frac{\pi p}{m}, p=0,1,\dots, m-1$. One might wonder what if all fundamental fermions $\mathbb{Z}_m$ are charged. In this case, we can attach a composite of $m$ such fermions. The composite is $\mathbb{Z}_m$ neutral and still has fermionic statistics since $m$ is odd. As a result, given two SPT phases characterized by $\tilde{K}_1$ and $\tilde{K}_2$, the equivalence relation can be derived: $$\frac{\pi\tilde{K}_1}{m^2}-\frac{\pi\tilde{K}_2}{m^2}=\frac{\pi p}{m}, \label{}$$ which implies that $\tilde{K}\equiv \tilde{K}+{mp}$, in constrat to $\tilde{K}\equiv \tilde{K}+{2mp}$ for bosonic SPT phases. We then easily see that there are only $m$ different classes ([We allow both even and odd $\tilde{K}$ in the fermionic case]{}). Therefore, all fermionic SPT phases with $\mathbb{Z}_m$ symmetry can be identified with the bosonic ones. [A physical interpretation is the following: bosonic SPT phases can be realized in any fermionic system when strong on-site interactions completely suppress charge fluctuations and effectively we have a spin system.]{} This fact can also be derived directly (see the Supplementary Material for the derivation [@suppl]). Now we turn to the case when $m$ is even. As we have elaborated, attachment of $\mathbb{Z}_{m}$ charges allows the statistical angle of fluxes to be changed by $\frac{2\pi p}{m}$. Attaching fermions does not do much in this case since for $p=m/2$ we already have a fermionic quasiparticle. The equivalence relation is exactly the same as the one for bosonic SPT phases, but now we allow both even and odd $\tilde{K}$. When $\tilde{K}$ is odd, the corresponding SPT phases are intrinsically fermionic and [can not be realized in any bosonic system]{}. We are then led to the conclusion that the classification for $m$ even is $\mathbb{Z}_{2m}$. We notice that the $m=2$ case receives a lot of attention recently [@Qi_arxiv2012; @Ryu_arxiv2012; @Yao_arxiv2012; @Gu_2013] since it serves as a nice example of “collapse” of classification of non-interacting fermions (which is $\mathbb{Z}$ in the present case) when interactions are taken into account ($\mathbb{Z}_8$). The Chern-Simons field theory approach [gives rise to a]{} $\mathbb{Z}_4$ classification. The missing four classes are those with unpaired Majorana edge modes which are beyond the Chern-Simons field theory description. Having worked out the classification for $G=\mathbb{Z}_m$, it is straightforward to generalize to arbitrary finite Abelian group $G=\prod_\alpha \mathbb{Z}_{m_\alpha}$. We notice that the underlying fermionic statistics does not effect mutual braiding properties, so the additional classes due to nontrivial mutual braiding statistics between gauge fluxes in different conjugate classes are still classified by $\prod_{i<j}\mathbb{Z}_{(m_i,m_j)}$. Thus we obtained a classification of fermionic SPT based on Chern-Simons theory as $\prod_i \mathbb{Z}_{m_i^*}\prod_{i<j}\mathbb{Z}_{(m_i, m_j)}$ where $m^*$ is defined as $m^*=m$ for odd $m$ and $2m$ for even $m$. [*Edge theory for fermionic SPT phase.*]{} Having established the classification of fermionic SPT phases with $\mathbb{Z}_m$ symmetry, we move on to study the edge theory and its stability in more detail. Let us consider the simplest one of $\mathbb{Z}_m$(for even $m$) fermionic SPT phases described by the following ${{\mathbf K}}$ matrix and ${{\mathbf q}}$ vector: $${{\mathbf K}}= \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}, {{\mathbf q}}= \begin{pmatrix} 1\\ 0 \end{pmatrix}, \label{eqn:fspt}$$ It is well known that the Chern-Simons theory implies existence of gapless edge states, whose effective Lagrangian can be derived from gauge invariance principle [@Wen_AdvPhys1995]: $$\mathcal{L}_\text{edge}=\frac{1}{4\pi}(\partial_t\phi_I K_{IJ}\partial_x \phi_J-\partial_x\phi_I V_{IJ}\partial_x \phi_J). \label{}$$ with symmetry transformation: $\bm{\phi} \rightarrow \bm{\phi}+\frac{2\pi}{m}{{\mathbf q}}$. To include interaction effect, it is convinient to switch to a non-chiral basis $\phi_{1}= \varphi-\theta, \phi_{2}=\varphi+\theta$. We then have a generic Luttinger liquid model of the gapless edge: $$H=\int{\mathrm{d}}x\,\frac{u}{2\pi}\left[ K(\partial_x\theta)^2+K^{-1}(\partial_x\varphi)^2 \right]. \label{}$$ Here $u$ is the charge velocity and $K$ the Luttinger parameter. One can see that the $\mathbb{Z}_m$ transformation acts on the non-chiral bosonic fields as $\varphi\rightarrow\varphi+\frac{\pi}{m}, \theta\rightarrow\theta+\frac{\pi}{m}$. To understand the stability of the edge theory, we add the leading perturbations $\cos 2m\varphi$ and $\cos 2m\theta$ allowed by $\mathbb{Z}_m$ symmetry. They have scaling dimensions $\frac{m^2K}{2}$ and $\frac{m^2}{2K}$respectively. So demanding that these two perturbations are irrelevant, we find the stable region is $\frac{2}{m^2}<K<\frac{m^2}{2}$. Interestingly, if we create domain walls on the edge such that the two sides have distinct $\mathbb{Z}_m$-breaking mass gaps (i.e. $\varphi$ condensed or $\theta$ condensed), a localized Majorana zero mode has to appear on the domain wall in those intrinsically fermionic SPT phases, which can serve as an experimental signature. [*Conclusion and Discussion.*]{} In conclusion, we systematically investigate SPT phases with an Abelian finite group symmetry within the framework of Chern-Simons field theory. We develop a topological response theory to classify the SPT phases by gauging the symmetry group. A careful examination of the equivalence classes of the braiding statistics of gauge fluxes enables us to characterize all possible bosonic SPT phases that can be realized as Abelian Chern-Simons theories. We also compare our approach with the results of the group cohomology theory and discuss the limitation of $K$-matrix construction. Indeed, the topological response theory describes a non-perturbative effect in the bulk of SPT phases and provides us a unique way to identify different SPT phases. Finally, we extend the classification scheme to fermionic SPT phases. For the simplest symmetry group $G=\mathbb{Z}_m$, we find the classification of fermionic SPT phases has an intriguing even-odd dependence on $m$. We then generalize the classification to arbitrary finite Abelian groups. We also discuss the edge stability of those intrinsic fermionic SPT phases. For future studies, it would be very interesting to develop a topological response theory to describe those SPT phases with non-Abelian flux statistics. On the other hand, the concept of topological response theory is also very useful for the classification of symmetry enriched topological(SET) order. [*Acknowledgement.*]{} MC thanks Lukasz Fidkowski, Chetan Nayak, Zhenghan Wang and Juven Wang for insightful discussions. ZCG is supported in part by Frontiers Center with support from the Gordon and Betty Moore Foundation. [*Note added.*]{} By the completion of this work, we became aware of recent preprints [@Lu_arxiv2013; @Wen2013] which have some overlap with our results. [**Supplementary material**]{} Vortex Duality in Abelian Chern-Simons-Higgs Theory =================================================== In this section we review the Abelian Chern-Simons-Higgs theory using charge-vortex duality transformation. The Lagrangian density of the theory is given by $$\mathcal{L}_{\text{CSH}}=\frac{1}{4\pi}\varepsilon^{\mu\nu\lambda}A^\alpha_\mu \tilde{K}_{\alpha\beta}\partial_\nu A^\beta_\lambda-\sum_{\alpha}\Big[\frac{1}{2} \big|(\partial_\mu-im_{\alpha}A_\alpha) \varphi_\alpha)\big|^2-V(\varphi_\alpha)\Big]. \label{}$$ The amplitudes of the Higgs fields are fixed by the potential energy. So we write $\varphi_\alpha=v_\alpha e^{i\theta_\alpha}$, $$\mathcal{L}_\text{eff}=\frac{1}{4\pi}\varepsilon^{\mu\nu\lambda}A^\alpha_\mu \tilde{K}_{\alpha\beta}\partial_\nu A^\beta_\lambda-\sum_{\alpha}\frac{v_\alpha^2}{2}\Big(m_{\alpha} A_\mu^\alpha-{\partial_\mu\theta_\alpha}\Big)^2. \label{}$$ First we perform the Hubbard-Stratonovich transformation and write $$\mathcal{L}_\text{eff}=\frac{1}{4\pi}\varepsilon^{\mu\nu\lambda}A^\alpha_\mu \tilde{K}_{\alpha\beta}\partial_\nu A^\beta_\lambda-\sum_\alpha\Big[\frac{1}{v_\alpha^2}\xi_\alpha^2-\xi_\alpha^\mu\Big(m_{\alpha} A_\mu^\alpha-{\partial_\mu\theta_\alpha}\Big)\Big]. \label{}$$ Then decompose the phase field as $\theta_\alpha=\eta_\alpha+\zeta_\alpha$ where $\eta_\alpha$ is the smooth part of the phase fluctuation and $\zeta_\alpha$ is the singular(vortex) part. Integrate out the smooth part of the phase fields $\eta_\alpha$, we obtain the constraint $\partial_\mu\xi_\alpha^\mu=0$, which can be resolved as $\xi_\alpha^\mu=\frac{1}{2\pi}\varepsilon^{\mu\nu\lambda}\partial_\nu b_{\alpha\lambda}$. The dual representation $$\mathcal{L}=\frac{1}{4\pi}\varepsilon^{\mu\nu\lambda}A^\alpha_\mu \tilde{K}_{\alpha\beta}\partial_\nu A^\beta_\lambda+\frac{m_{\alpha}}{2\pi}\varepsilon^{\mu\nu\lambda}A^\alpha_{\mu}\partial_\nu b^\alpha_{\lambda}+\dots. \label{}$$ We then integrate out the $A$ fields to finish the duality transformation: $$\mathcal{L}_\text{dual}=\frac{1}{4\pi}\varepsilon^{\mu\nu\lambda}{{\mathbf b}}_{\mu}^T {{\mathbf m}} \tilde{{{\mathbf K}}}^{-1}{{\mathbf m}}^T \partial_\nu {{\mathbf b}}_\lambda. \label{}$$ From the dual action we can directly read off the self and mutual statistics of fluxes. If we treat the $A$ fields as fully dynamical, the above action can be written as a doubled Chern-Simons theory: $$\mathcal{L}=\frac{1}{4\pi}\varepsilon^{\mu\nu\lambda}({{\mathbf b}}_\mu^T, {{\mathbf A}}_\mu^T) \begin{pmatrix} \mathbf{0} & \mathbf{M}\\ \mathbf{M} & \tilde{{{\mathbf K}}} \end{pmatrix} \partial_\nu\begin{pmatrix} {{\mathbf b}}_\lambda\\ {{\mathbf A}}_\lambda \end{pmatrix} . \label{}$$ Here ${{\mathbf M}}$ is the diagonal matrix $\text{diag}(m_1, m_2, \dots )$. This Chern-Simons field theory represents a twisted $\prod_{\alpha}\mathbb{Z}_{m_\alpha}$ gauge theory. We therefore find the correspondence between the SPT phase and the Abelian topological phases which are essentially gauged SPTs. Intrinsic Topological Order and Classification of Topological Responses ======================================================================= We elaborate on the nonequivalence between the classification of the gauged SPT phases as a “topological response” theory and the classification of the intrinsic topological order. As emphasized in the main text, the classification of the response theory is not completely equivalent to the classification of the intrinsic topological order defined by the Chern-Simons-Higgs theory. In classifying the intrinsic topological orders, all the gauge fluxes are regarded as dynamical deconfined objects. Two (Abelian) topological phases are equivalent as long as they have the same quasiparticle braiding matrices (so-called $T$ and $S$ matrices), regardless of how the quasiparticles are labeled. This kind of equivalence relation is nothing but the $\mathbb{GL}(N,\mathbb{Z})$ equivalence for the ${{\mathbf K}}$ matrix. To illustrate the difference, first we start from $G=\mathbb{Z}_n$. As disucussed in the main text, the $n$ different SPTs after being gauged result in the following $n$ gauge theories: $${{\mathbf K}}= \begin{pmatrix} 0 & n\\ n & 2p \end{pmatrix}, p=0,1,\dots, n-1. \label{}$$ One might wonder these gauge theories are all distinct. However, this is not generally true. In fact, for every odd $n\geq 5$, we have the following $\mathbb{GL}(2,\mathbb{Z})$ equivalence between $p=2$ and $p=\frac{n+1}{2}$: $$W^T\begin{pmatrix} 0 & n\\ n & 4 \end{pmatrix}W= \begin{pmatrix} 0 & n\\ n & n+1 \end{pmatrix}, W= \begin{pmatrix} -2 & -1\\ n & \frac{n+1}{2} \end{pmatrix}. \label{}$$ This is just one example and there could be more “collapse” for general $n$. Our second example is the gauge group $G=\mathbb{Z}_2\times\mathbb{Z}_2$. The dynamical gauged theory is given by the following $\mathbf{K}$ matrices: $$\mathbf{K}= \begin{pmatrix} \mathbf{0}_{2\times 2} & \mathbf{2}_{2\times 2}\\ \mathbf{2}_{2\times 2} & \tilde{K} \end{pmatrix}, \tilde{K}= \begin{pmatrix} k & l\\ l & p \end{pmatrix}, k,p\in\{0,2\}, l\in \{0,1\}. \label{eqn:z2z2}$$ The possible choices of $k, p$ and $l$ yields all the $H^3(\mathbb{Z}_2\times\mathbb{Z}_2, \mathbb{U}(1))=\mathbb{Z}_2^3$ cohomology classes. In fact, the $8$ classes listed in reduce to only $4$ under generic $\mathbb{SL}(4,\mathbb{Z})$ equivalence. In terms of the $\tilde{K}$ matrices, there are only the following four different intrinsic topological orders: $$\begin{gathered} \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 2 & 1\\ 1 & 2 \end{pmatrix} \\ \begin{pmatrix} 2 & 0\\ 0 & 0 \end{pmatrix}\sim\begin{pmatrix} 0 & 0\\ 0 & 2 \end{pmatrix} \sim\begin{pmatrix} 2 & 0\\ 0 & 2 \end{pmatrix},\\ \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}\sim \begin{pmatrix} 2 & 1\\ 1 & 0 \end{pmatrix}\sim\begin{pmatrix} 0 & 1\\ 1 & 2 \end{pmatrix}. \end{gathered} \label{}$$ Here $\sim$ denotes the equivalence of the derived intrinsic topological orders. In the response theory, we are not allowed to permute the gauge fluxes from different subgroups of the symmetry group since they correspond to different physical symmetries. Equivalence between fermionic and bosonic SPT phases with $\mathbb{Z}_{m}$ symmetry ==================================================================================== We demonstrate directly that when $m$ is odd all $\mathbb{Z}_m$ fermionic SPT phase are equivalent to bosonic ones. Let us consider the bulk Chern-Simons theory of $\mathbb{Z}_m$ fermionic SPT phase $$\mathbf{K}= \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}, {{\mathbf q}}_g= \begin{pmatrix} q_1\\ q_2 \end{pmatrix}. \label{}$$ Here $g$ denotes the generator of the $\mathbb{Z}_m$ symmetry. We then add a trivial phase given by ${{\mathbf K}}=\sigma_z$ with trivial symmetry transformation on the edge bosons ${{\mathbf q}}'_g=\begin{pmatrix} p \\ p\end{pmatrix}$ where $p\in \mathbb{Z}$. We pack the whole system into a $4\times 4$ ${{\mathbf K}}$ given by ${{\mathbf K}}=\sigma_z\otimes \mathbf{1}_{2\times 2}$. We then perform the following $\mathbb{SL}(4, \mathbb{Z})$ transformation $${{\mathbf W}}= \begin{pmatrix} 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 0\\ -1 & 0 & 0 & 1\\ 1 & 0 & 1 & -1 \end{pmatrix}. \label{}$$ Under ${{\mathbf W}}$ the ${{\mathbf K}}$ matrix becomes ${{\mathbf K}}=\begin{pmatrix} \sigma_z & 0\\ 0 & \sigma_x\end{pmatrix}$. So the first two components are describing fermionic systems and the last two bosonic ones. We denote the symmetry vector for the collectively as $\tilde{{{\mathbf q}}}_g=(q_1,q_2, p, p)^T$. The edge modes in the new basis are denoted by $$\tilde{\bm{\phi}}= \begin{pmatrix} \phi_1^f\\ \phi_2^f\\ \phi_1^b\\ \phi_2^b \end{pmatrix} = \begin{pmatrix} \phi_1-\phi_1'-\phi_2'\\ \phi_2\\ \phi_1'+\phi_2'\\ \phi_1-\phi_2' \end{pmatrix} \label{}$$ Under the $\mathbb{SL}(4, \mathbb{Z})$ transformation ${{\mathbf W}}$, the vector $\tilde{{{\mathbf q}}}_g \rightarrow \mathbf{W}^{-1}\tilde{{{\mathbf q}}}_g=(q_1-2p, q_2, 2p, q_1-p)^T$. If $q_1-q_2$ is even, we let $p=\frac{q_1-q_2}{2}$ and then $\tilde{{{\mathbf q}}}_g=(q_2, q_2, q_1-q_2, \frac{q_1+q_2}{2})^T$. Thus the fermionic SPT is equivalent to a bosonic one with $\mathbf{K}=\sigma_x, {{{\mathbf q}}}^b_g=(q_1-q_2,\frac{q_1+q_2}{2})^T$. If $q_1-q_2$ is odd, it seems that $p=\frac{q_1-q_2}{2}$, being a half integer, is not physical. Here the fermionic nature plays a crucial. This is most easily understood from the edge modes. The edge modes $\bm{\phi}'=(\phi_1', \phi_2')^T$ transforms under the $\mathbb{Z}_m$ symmetry as $$U_g\bm{\phi}'U_g^\dag= \bm{\phi}'+\frac{2\pi p}{m} \begin{pmatrix} 1\\ 1 \end{pmatrix}. \label{}$$ When $p$ is a half integer, we have $$U_g^m\bm{\phi}'(U_g^\dag)^m=\bm{\phi}'+2\pi p\begin{pmatrix} 1\\ 1 \end{pmatrix} \equiv \bm{\phi}'+\pi \begin{pmatrix} 1\\ 1 \end{pmatrix} , \label{}$$ which is projectively the identity in a fermionic system. We now turn to the corresponding bosonic SPT. We notice that $$U_g^m \bm{\phi}^b(U_g^\dag)^m=\bm{\phi}^b+\pi \begin{pmatrix} 0\\ q_1+q_2 \end{pmatrix} \label{}$$ which is not consistent with $U_g^m=1$ in the bosonic case. Again the identity transformation in a fermionic system can be realized projectively: $$\begin{gathered} \phi_1\rightarrow \phi_1+\pi\\ \phi_2\rightarrow\phi_2+\pi, \end{gathered} \label{}$$ which means that $$\phi^b_2=\phi_1-\phi_2'\rightarrow \phi^b_2+\pi \label{}$$ is also an identity transformation. So we can freely add $\begin{pmatrix} 0\\ \pi\end{pmatrix}$ to $\bm{\phi}^b$ and as a result $U_g^m$ can differ by $\begin{pmatrix} 0 \\ m\pi\end{pmatrix}$, which makes the derived relation legitimate for odd $m$, but not for even $m$. We therefore prove that the fermionic SPT phases with $\mathbb{Z}_m$ are all equivalent to bosonic ones when $m$ is odd. [^1]: If the $K$ matrix has a nonzero signature (i.e. the numbers of positive and negative eigenvalues are not equal), the state supports chiral edge modes which can not be completely gapped out even without any symmetry. Thus such a state is not a SPT, since by definition SPT should be adiabatically connected to a trivial state when the symmetry is broken. [^2]: For ${G}=\prod_{i=1}^k \mathbb{Z}_{m_i}$, one finds [@Propitius_thesis] $H^3(G, \mathbb{U}(1))=\prod_{i} \mathbb{Z}_{m_i}\prod_{i<j} \mathbb{Z}_{(m_i,m_j)}\prod_{i<j<k}\mathbb{Z}_{(m_i, m_j, m_k)}$. Clearly, when $k\geq 3$ the Abelian Chern-Simons theory approach fails to capture the classes of SPT phases associated with $\mathbb{Z}_{(m_i, m_j, m_k)}$. Physically, this is because the gauge fluxes in the topological response theory carry non-Abelian statistics [@Propitius_thesis].

**Set Theory and the Analyst** by **N. H. Bingham and A. J. Ostaszewski** [*Then to the rolling heaven itself I cried*,]{}\ [*Asking what lamp had destiny to guide* ]{}\ [*Her little children stumbling in the dark*.]{}\ [*And ‘A blind understanding’ heaven replied.*\ – The Rubaiyat of Omar Khayyam ]{} **Abstract.** This survey is motivated by specific questions arising in the similarities and contrasts between (Baire) category and (Lebesgue) measure – category-measure duality and non-duality, as it were. The bulk of the text is devoted to a summary, intended for the working analyst, of the extensive background in set theory and logic needed to discuss such matters: to quote from the Preface of Kelley \[Kel\]: “what every young analyst should know”. **Table of Contents** 1\. Introduction2. Early history3. Gödel Tarski and their legacy4. Ramsey, Erdős and their legacy: infinite combinatorics4a. Ramsey and Erdős4b. Partition calculus and large cardinals4c. Partitions from large cardinals4d. Large cardinals continued 5\. Beyond the constructible hierarchy $L$ – I5a. Expansions via ultrapowers5b. Ehrenfeucht-Mostowski models: expansion via indiscernibles6. Beyond the constructible hierarchy $L\ $– II6a. Forcing and generic extensions6b. Forcing Axioms 7\. Suslin, Luzin, Sierpiński and their legacy: infinite games and large cardinals7a. Analytic sets.7b. Banach-Mazur games and the Luzin hierarchy8. Shadows9. The syntax of Analysis: Category/measure regularity versus practicality10. Category-Measure dualityCoda **1. Introduction** An analyst, as Hardy said, is a mathematician habitually seen in the company of the real or complex number systems. For simplicity, we restrict ourselves to the reals here, as the complex case is obtainable from this by a cartesian product. In mathematics, one’s underlying assumption is generally Zermelo-Fraenkel set theory (ZF), augmented by the Axiom of Choice (AC) when needed, giving ZFC. One should always be as economical as possible about one’s assumptions; it is often possible to proceed without the full strength of AC. Our object here is to survey, with the working analyst in mind, a range of recent work, and of situations in analysis, where one can usefully assume less than ZFC. This survey is rather in the spirit of that by Wright \[Wri\] forty years ago and Mathias’s ‘Surrealist landscape with figures’ survey \[Mat\]; a great deal has happened since in the area, and we feel that the time is ripe for a further survey along such lines. As classical background for what follows, we refer to the book of Oxtoby \[Oxt\] on (Lebesgue) measure and (Baire) category. This book explores the duality between them, focussing on their remarkable similarity; for Oxtoby, it is the measure case that is primary. Our viewpoint is rather different: for us, it is the *category* case that is primary, and we focus on their differences. Our motivation was that a number of results in which category and measure behave interchangeably *disaggregate* on closer examination. One obtains results in which what can be said depends explicitly on what axioms one assumes. One thus needs to adopt a flexible, or pluralist, approach in order to be able to handle apparently quite traditional problems within classical analysis. We come to this survey with the experience of a decade of work on problems with wide-ranging contexts, from the real line to topological groups, in which ‘the category method’ has been key. The connection with the Baire Category Theorem, viewed as an equivalent of the Axiom of Dependent Choices (DC: for every non-empty set $X$ and $R\subseteq X^{2}$ satisfying $\forall x\exists y[(x,y)\in R]$ there is $h\in X^{\mathbb{N}}$ with $(h(n),h(n+1))\in R$ for all $n\in \mathbb{N})$, has sensitized us to a reliance on this as a very weak version of the Axiom of Choice, AC, but one that is often adequate for analysis. This sensitivity has been further strengthened by settings of general theorems on Banach spaces reducible to the separable case (e.g. via Blumberg’s Dichotomy \[BinO6, Th. B\]; compare the separable approximations in group theory \[MonZ, Ch. II §2.6\]) where DC suffices. On occasion, it has been possible to remove dependence on the Hahn-Banach theorem, a close relative of AC. For the relative strengths of the usual Hahn-Banach Theorem HB and the Axiom of Choice AC, see \[Pin1,2\]; \[PinS\] provide a model of set theory in which the Axiom of Dependent Choices DC holds but HB fails. HB is derivable from the Prime Ideal Theorem PI, an axiom weaker than AC : for literature see again \[Pin1,2\]; for the relation of the Axiom of Countable Choice, ACC (below), to DC here, see \[HowR\]. Note that HB for separable normed spaces is not provable from DC \[DodM, Cor. 4\], unless the space is complete — see \[BinO10\]. When category methods fail, e.g. on account of ‘character degradation’, as when the *limsup* operation is applied to well-behaved functions (see §9), the obstacles may be removed by appeal to supplementary set-theoretic axioms, so leading either beyond, or sometimes away from, a classical setting. This calls for analysts to acquire an understanding of their interplay and their standing in relation to ‘classical intuition’ as developed through the historical narrative. Our aim here is to describe this hinterland in a language that analysts may appreciate. We list some sources that we have found useful, though we have tried to make the text reasonably self-contained. From logic and foundations of mathematics, we need AC and its variants, for which we refer to Jech \[Jec1\]. For set theory, our general needs are served by \[Jec2\]; see also Ciesielski \[Cie\], Shoenfield \[Sho\], Kunen \[Kun3\]. For descriptive set theory, see Kechris \[Kec2\] or \[MarK\]; for analytic sets see Rogers et al. \[Rog\]. For large cardinals, see e.g. Drake \[Dra\], Kanamori \[Kan\], Woodin \[Woo1\]. The paper is organised as follows. After a review of the early history of the axiomatic approach to set theory (including also a brief review of some formalities) we discuss the contributions of Gödel and Tarski and their legacy, then of Ramsey and Erdős and their legacy. We follow this with a discussion of the role of infinite combinatorics (partition calculus) and of the ‘large cardinals’. We then sketch the various ‘pre-Cohen’ expansions of $\ $Gödel’s universe of constructible sets $L$ (via the ultrapowers of Łoś, or the indiscernibles of Ehrenfeucht-Mostowski models, and the insights they bring to our understanding of $L$). This is followed by an introduction to the ‘forcing method’ and the generic extensions which it enables. We describe classical descriptive set theory: the early ‘definability theme’ pursued by Suslin, Luzin, Sierpiński and their legacy, and the completion of their programme more recently through a recognition of the unifying role of Banach-Mazur games; these require large cardinals for an analysis of their ‘consistency strength’, and are seen in some of the most recent literature as casting ‘shadows’ (§8) on the real line. We conclude in §9 with a discussion of the ‘syntax of analysis’ in order to draw on the ‘definability theme’, and in that light we turn finally in §10 to the additional axioms which permit a satisfying category-measure duality for the working analyst. *The canonical status of the reals* The English speak of ‘the elephant in the room’, meaning something that hangs in the air all around us, but is not (or little) spoken of in polite company. The canonical status of the reals is one such, and the *sets* of reals available is another such. We speak of ‘the reals’ (or ‘the real line’), $\mathbb{R}$ and ‘the rationals’, $\mathbb{Q}$ – as in the Hardy saying above. The definite article suggests canonical status, in some sense – what sense? The rationals are indeed canonical. We may think of them as a ‘tie-rack’, on which irrationals are ‘hung’. But how, and how many? In §6 we will review the forcing method for selecting from ‘outside the wardrobe’ an initial lot of almost any size that one may wish for, together with their ‘Skolem hull’ – further ones required by the operation of the axioms (see the Skolem functions of §2 below). In brief: the reals are *canonical modulo cardinality, but not otherwise*. This is no surprise, in view of the Continuum Hypothesis (CH), which directly addresses the cardinality of the real line/continuum, and which we know from P. J. Cohen’s work of 1963/64 \[Coh1,2\] will always be just that, a hypothesis. The canonical status of the reals rests on (at least) four things: \(i) (geometrical; ancient Greeks): lines in Euclidean geometry: any line can be made into a cartesian axis; \(ii) (analytical; 1872): Dedekind cuts; \(iii) (analytical; 1872): Cantor, equivalence classes of Cauchy sequences of rationals (subsequently extended topologically: completion of any metric space); \(iv) (algebraic; modern): any *complete* archimedean ordered field is isomorphic to $\mathbb{R}$ (see e.g. \[Cohn, § 6.6\]: our italics; here too Cauchy sequences are used to define completeness). None of these is concerned with cardinality; CH is. On the other hand, completeness depends on which $\omega $-sequences are available. Accordingly, the problems that confront the working analyst split, into two types. Some (usually the ‘less detailed’) do not hinge on cardinality, and for these the reals retain their traditional canonical status. By contrast, some do hinge on cardinality; these are the ones that lead the analyst into set-theoretic underpinnings involving an element of *choice*. Such choices emphasise the need for a *plural* approach, to axiomatic assumptions, and hence to the status of the reals. This is inevitable: as Solovay \[Sol1\] puts it, ‘it (the cardinality of the reals) can be anything it ought to be’. We turn now to the second of the ‘elephants’ above: which sets of reals are available. The spectrum of axiom possibilities which we review in §10 extends from the ‘prodigal’ (below – see §2) AC at one end (which yields for example non-measurable Vitali sets) to the restrictive DC with additional components of LM (‘all sets of reals are measurable’) and/or PB (‘all sets of reals have the Baire property’) at the other, and include intermediate positions for the additional component such as PD (‘all projective sets of reals are determined’), where the sets of reals with these so-called ‘regularity properties’ are qualified (see §§7 and 9). Underlying an analysis of these axioms is repeated appeal to simplification of contexts - a mathematical *ex oriente lux* – typified by passage to a ‘large’ homogeneous/monochromatic subset, as in Ramsey’s Theorem on $\mathbb{N}$ (§4a). This has generalizations to large cardinals $\kappa $, in particular ones that support a $\{0,1\}$-valued measure (equivalently, a ‘suitably complete’ ultrafilter – see below). On the one hand, the latter permits an extension of Suslin’s classical tree-like representation of an analytic set (§7) to sets of far greater logical complexity by witnessing membership of a set by means of infinite branches in a corresponding tree that pass through ‘large’ sets of nodes at each height/level (see §7). On the other hand, in the context of the ‘line’ of ordinals, one meets other forms of isomorphic behaviour on ‘large’ sets: on closed unbounded subsets of ordinals and on the related stationary sets (§5b, 6b). *Notes.* 1. This survey arose out of our decade-long probing of questions in regular variation \[BinGT\]. In \[BinO3\] we needed to disaggregate a classical theorem of Delange (see \[BinGT, Th. 2.01\]); the category and measure aspects need different set-theoretic assumptions. We regard the category case as primary, as one can obtain the measure case from it by working bitopologically (passing from the Euclidean to the *density topology*; see \[BinO2,7,8\]); also, measure theory needs stronger set-theoretic assumptions than category theory (§10.2 and §10.3 below). If one replaces the limits in regular variation by *limsups*, the Baire property or measurability may be lost; the resulting character degradation is studied in detail in \[BinO3 §3, 5§11\]. 2\. We close by a brief mention of ‘yet another elephant in the room’. One can never prove consistency (of sets of rich enough axioms), merely relative consistency. This is related to Gödel’s incompleteness theorems (§3). Thus we do not know that ZF or ZFC itself is consistent; this is something we have to live with; it is no reason to despair, or give up mathematics; quite the contrary, if anything. In what follows, ‘consistency’ means ‘consistency relative to ZF’. **2. Early history** A little historical background may not come amiss here. The essence of analysis – and the reason behind the Hardy quotation that we began with – is its concern with infinite or limiting processes – most notably, as in calculus, our most powerful single technique in mathematics (and indeed, in science generally). Life being only finitely long, the infinite – actual or potential – takes us beyond direct human experience, even in principle. This underlies the unease the ancient Greeks had with the irrationals (or reals), and why they missed calculus (at least in its differential form, despite their success with areas and volumes under the heading of the ‘method of exhaustion’). One can see, for example in the ordering of the material in the thirteen books of Euclid’s *Elements,* that they were at ease with rationals, and with geometrical objects such as line-segments etc., but not with reals. Traces of this unease survive in Newton’s handling of the material in his *Principia,* where he was at pains to use established geometrical arguments rather than his own ‘method of fluxions’. That there was unfinished business here shows, e.g., in the title of a work of one of the founding fathers of analysis, Bolzano, with his *Paradoxien des Unendlichen* (1852, posthumous). The bridge between the real line and the complex plane (the ‘Argand diagram’ – Argand, 1806, Wessel, 1799, Gauss, 1831) pre-dated this. The construction of the reals came independently in two different ways in 1872: Dedekind cuts (or sections), which still dominate settings where one has an *order,* and Cantor’s construction via (equivalence classes of) Cauchy sequences (of rationals) – still ubiquitous, as the completion procedure for metric spaces. *Cantor.* Cantor’s work, in the 1870s to 1890s, established set theory (*Mengenlehre)* as the basis on which to do mathematics, and analysis in particular. Here we find, for example, the countability of the rationals, and of the algebraic numbers (Cantor, 1874) and the uncountability of the reals (Cantor, 1895), established via the familiar Cantor diagonalisation argument. But note what is implicit here: Cantor diagonalisation (as used, say, to prove the countability of the rationals) is an *effective* argument. But to move from this to saying that ‘the union of countably many countable sets is countable’ (Cantor, 1885) needs the Axiom of Countable Choice (ACC), below. *Hilbert.* Moving to the 20th century: Hilbert famously said (in defence of Cantor against Kronecker): ‘No one shall expel us from the paradise that Cantor has created for us’. Hilbert addressed himself to the programme of re-working the mathematical canon of its time to (then) modern standards of rigour, witness his books on the foundations of geometry \[Hil1,2,3\] (1899) and of mathematics \[HilB\] (1934, 1939, with Bernays), cf. the Hilbert problems of 1900. As we shall see, Hilbert was a man of his time here, and his views on foundational questions were too naive. Meanwhile, Lebesgue introduced measure theory in 1902, Fréchet metric spaces in 1906, and Hausdorff general topology in 1905-1914 (three very different editions of his classic book *Grundzuge der Mengenlehre* appeared in 1914, 1927 and 1935). Hilbert space emerged c. 1916 (work of Hilbert and Schmidt; named by F. Riesz in 1926). Banach’s book \[Ban\] appeared in 1932, effectively launching the field of functional analysis; this magisterial work is still worth reading. But, Banach was a man of his time; he worked sequentially, rather than using the language of weak topologies, presumably because he felt it to be not yet in final form. However, the language and viewpoint of general topology was already available, and already a speciality of the new Polish school of mathematics, of which Banach himself was the supreme ornament. For a scholarly and sympathetic account of these matters, see Rudin \[Rud, Appendix B\]. The need for care in set theory had been dramatically shown by the Russell Paradox of 1902, and its role in showing the limitations of Frege’s programme in logic and foundations, especially his *Grundgesetze der Arithmetik* (vol. 2 of 1903). The Paradox, far from being a programme wrecker, was pregnant with consequences \[GabW\], just as with Gödel’s work later (below), and that too was ultimately based on a Paradox (the ‘Liar paradox’). See \[Hall2\] for a discussion. Foundational questions had been addressed in 1889 by Peano. Zermelo began his axiomatisation, and gave the Axiom of Choice (AC) in 1902. Fraenkel, Skolem and others continued and revised this work; what is known nowadays as Zermelo-Fraenkel set theory (ZF), together with ZF+AC, or ZFC, emerged by 1930 or so. AC is most often used in the (equivalent) form of Zorn’s Lemma of 1935 (a misnomer, as the result is due to Kuratowski in 1922, but the usage is now established). It will be helpful for later passages to note that the axioms include the operations of comprehension (the forming of a subset determined by a property), union and power set (denoted here by $\wp $), as well as foundation/regularity, asserting the well-foundedness of the relation of membership $\in $ (no descending $\in $-chains). In this context AC is a generator of sets par excellence, with effects of both positive and negative aspects: allowing the construction both to ‘satisfy intuition’ (as in the construction of ‘invariant means’) and to astound it (as in the Banach-Tarski paradox): see the comments in \[TomW, Ch. 15\]. The tension between ‘too many’ sets or ‘too few’ pervades the history of set theory through the lens of logic, all the way back to Cantor: see \[Hall1\]. For a discussion of approaches to axiomatization see \[Sco2\]. *Brouwer.* The interplay between analysis (specifically, topology) and foundations in this era is well exemplified by the work of Brouwer. Brouwer is best remembered for two contributions: his fixed-point theorem (of 1911, \[Bro1\]), and Intuitionism (1920, cf. \[Bro2\]). The first is beloved of economists, as it provides existence proofs of economic equilibria – the ‘invisible hand’ of Adam Smith, and his later ‘disciples’. But, his proof of the fixed-point theorem was a non-constructive existence proof, and Brouwer lost faith in these for foundational reasons. He reacted by seeking to re-formulate mathematics ‘intuitively’, on new foundations – differing from those in use then and now by, for instance, outlawing proof by contradiction. This led to serious conflict, for instance the *Annalenstreit* (Annals struggle) of 1928, where Hilbert, as Editor-in-Chief of the *Mathematische Annalen,* ejected Brouwer from the Editorial Board. *Von Neumann.* Von Neumann contributed to foundational questions, e.g. by formalising the (or a) construction of the natural numbers $\mathbb{N}$: $0:=\{\emptyset \}$, $1:=\{0\}$, $2:=\{0,1\}$, $3:=\{0,1,2\}$, etc.: $n+1:=n\cup \{n\},$ see \[Hal, §11\] (\[Neu1\], \[Neu2\], 1928), and work on amenable groups, with applications to the ‘Banach-Tarski paradox’ (as above) (\[TomW\]; \[Bin\]). The sets $x$ in Von Neumann’s definition are ordered by $\in $ and are *transitive*: if $z\in y\in x$, then $z\in x.$ Indeed the ordinals, which form the class $On$ (not a set), are initially introduced as transitive well-ordered *structures* $\langle x,\in _{x}\rangle $ with $\in _{x}$ the restriction to $x$ of the membership relation. Once ordinals $\alpha $ are established (this uses the axiom of regularity), the cumulative hierarchy $V_{\alpha }$ may be introduced inductively so that $V_{\alpha +1}=\wp (V_{\alpha }),$ with $\wp $ the power set operation, and $V_{\lambda }=\bigcup \{V_{\alpha }:\alpha <\lambda \}$ for $\lambda $ a limit ordinal. The class of sets is then $V=\bigcup \{V_{\alpha }:\alpha \in On\},$ and each set $x$ has a well-defined *rank*: the least $\alpha $ with $x\in V_{\alpha }.$ The formal language of set theory $LST$ builds formulas from a defined sequence of free variables (e.g. $v_{0},v_{1},...),$ the atomic ones taking the form $x\in y$ and $x=y,$ with $x$ and $y$ standing for free variables; the syntactically more complex ones then arise from the usual logical connectives and quantifiers ($\forall x$ and $\exists y$ – creating bound variables from the free variables $x,y).$ The idea is that the free and bound variables are restricted to range only over the elements in the universe of discourse (thus yielding a ‘first-order’ language). This language is a necessary ingredient of the axiomatic method, its first purpose being to give meaning to the notion of ‘property’ (so that e.g. $\{x\in y:\varphi (x)\}$ is recognized as a set when $\varphi $ is a formula with one free variable $x$). The language $LST$ is minimal as compared to the language of, say, group theory, whose type (officially: ‘signature’) involves more items (a designated constant $1$, functions like $y\circ z$ , relations, etc). Each such language is interpreted in a mathematical structure; for instance, at its simplest a group structure has the form $\mathcal{G}:=\langle G,1_{G},\circ _{G},\cdot ^{-1}\rangle $ and so lists its domain, designated elements and operations. Below structures are assumed to be *sets* unless otherwise qualified; it is sometimes convenient (despite formal complications) to allow a class as a domain, e.g. $\langle V,\in \rangle .$ The (metamathematical – i.e. ‘external’ to the discourse in the language) semantic relation $\models $ of satisfaction/truth (below), due to Tarski (see \[Tar2\], cf. \[BelS, Ch. 3 §2\]), is read as ‘models’, or informally as ‘thinks’ (adopting a common enough anthropomorphic stance). A formula $\varphi $ of $LST$ with free variables $x,y,...,z$ may be interpreted in the structure $\mathcal{M}:=\langle M,\in _{M}\rangle $ (with $\in _{M}$ now a binary set relation on the set $M$) for a given assignment $a,b,...,c$ in $M$ for these free variables, and one writes$$\mathcal{M}\models \varphi (x,y,...,z)[a,b,...,c],\text{ or by abbreviation }M\models \varphi \lbrack a,b,...,c]$$if the property holds; this requires an induction on the syntactic complexity of the formula starting with the atomic formulas (for instance, the atomic case $x\in y$ is interpreted under the assignment $a,b$ as holding iff $a\in _{M}b$ ). Compare the reduction of complexity in the forcing relation of §6 below. This apparatus enables definition of ‘suitably qualified’ forms of ‘definability’; by contrast, unrestricted ‘definability’ leads to such difficulties as the ‘least ordinal that is not definable’, so is to be avoided (compare §3 below with Tarski’s undefinability of truth). A simple example is that of an element $w\in M$ being definable over $M$ from a parameter $v\in M,$ in which case for some formula $\varphi (x,y)$ with two free variables:$$w\text{ is the unique }u\in M\text{ with }M\models \varphi (u,v).$$Thus Gödel introduced the constructible hierarchy $L_{\alpha }$ by analogy with $V_{\alpha }$: however, $L_{\alpha +1}$ comprises only sets definable over $L_{\alpha }$ from a parameter in $L_{\alpha };$ here $L_{\lambda }=\bigcup \{L_{\alpha }:\alpha <\lambda \}$ for $\lambda $ a limit ordinal, a matter we return to later, yielding the class $L=\bigcup \{L_{\alpha }:\alpha \in On\}$. Certain formulas, like $\varphi (x,y)$ above (which can be explicitly, and so effectively, enumerated, as $\varphi _{m}$ say), may give rise via the substitution of a parameter $v$ for $y$ to a family of not necessarily unique elements $u\in M$ satisfying $\varphi (u,v).$ An appeal, in general, to AC but in the ‘metamathematical’ setting (i.e. the context of the mathematics studying relations between the language and the structures), selects a witness $w$ of the relation $\varphi (x,v)$ holding in $M$: the function $v\mapsto w$ is called a *Skolem function* (for* *$M$ and $\varphi $); we will see a striking application presently – for background on this key notion see e.g. \[Hod\]. Evidently, a structure like $\mathcal{M}:=\langle L_{\alpha },\in _{L_{\alpha }}\rangle $ contains enough well-orderings of its initial parts $L_{\beta }$ for $\beta <\alpha $ (induced by the enumeration $\varphi _{m}$ and well-ordering of the ordinal parameters) that reference to AC here becomes unnecessary. (Incidentally, this is why AC holds in the class structure $\langle L,\in \rangle $.) We will refer to some other definability classes below in §6, so as an introduction we mention two classical ones. The class $OD$ of ordinally definable sets comprises those that are definable from ordinal parameters over $\langle V_{\alpha },\in _{V_{\alpha }}\rangle $ for some $\alpha .$ An element of a set in OD need not itself be in OD; the class HOD is the smaller class of those elements $x$ whose transitive closure consists entirely of sets in OD, so HOD is a transitive class; see \[MyhS\] for a discussion. In view of the finitary character of formulas, the Löwenheim-Skolem-Tarski theorem (see e.g. \[Hod\], or \[BelS, Ch. 4.3\]), as applied to the language of set theory $LST$, asserts that if a set $\Sigma $ of sentences is modelled in a structure $\mathcal{M}$, then there exist structures $\mathcal{N}$ of any infinite cardinality satisfying $\Sigma ,$ including *countable* ones. The latter ones are generated by induction by iterative application of all the Skolem functions; so this needs only the Axiom of Dependent Choices. A familiar example is the countable subring with domain $\mathbb{Q}$ of the ordered ring structure $\langle \mathbb{R},0,1,+,\times ,<\rangle $. Passing to above-continuum cardinalities yields models of non-standard analysis with infinitesimals and infinite integers (see below); but here AC is needed to construct Skolem functions with which to generate the much larger structure. The axioms of set theory include a finite set and an *axiom schema* corresponding to the Axiom of Replacement (which asserts that the image of a set under a functional relation $\varphi (x,y)$ expressed in $LST$ is again a set). In order to model these axioms in structures like $\langle M,\in _{M}\rangle $ with $M$ a set, it is necessary to restrict attention to the use of a finite number of instances of the axiom schema – causing no practical loss of generality, since any amount of mathematical argument will necessarily do just that (for instance, a deduction of an inconsistency). Thus, assuming the consistency of the axioms of set theory, any finite subset of the axioms has a model $\mathcal{M}$ (by the Gödel-Henkin Completeness Theorem; see e.g. \[BelS, Th. 4.2\]) and so also a countable model $\mathcal{N}$. This is conventionally and systematically rephrased as saying that the axioms of set theory have a countable model; compare \[Kun2, Ch. 7 §9\]. By its very nature the countable model $\mathcal{N}$ will contain far fewer bijections than exist in Cantor’s world $V.$ If transitive, the domain $N$ of $\mathcal{N}$ will have an initial segment of the ordinals in $V;$ however, there will be countable ordinals which $\mathcal{N}$ ‘thinks’ are uncountable, owing to missing bijections. The rule to observe is that *ordinals are absolute* whereas *cardinality is relative*. This is exploited in arranging the failure of the Continuum Hypothesis, CH, by the model extension process of forcing (see below for details and references). In the context of a transitive model of set theory $\mathcal{M}$ we will write e.g. $\omega _{1}^{M}$ for the ordinal which in $\mathcal{M}$ is its first uncountable. In the absence of a superscript the implied context is $V.$ Provided the Regularity axiom is included, the structure $\mathcal{N}=\langle N,\in _{N}\rangle ,$ being then well-founded, is isomorphic to a transitive structure; the isomorphism $\pi $ is given inductively by:$$\pi (x):=\{\pi (y):y\in _{N}x\},$$and is known as the *Mostowski collapse*. Thus, for example, $\pi (\emptyset ^{M})=\emptyset .$ **3. Gödel, Tarski and their legacy** The use of formal language brought greater clarity to the axiomatic method: thus Skolem helpfully clarified one of Zermelo’s axioms by replacing the latter’s use of the informal notion of ‘definite property’ with a formal rendering (i.e. by reference to formulas in a formal language). This was soon to be followed by the discovery of the limitations of formal language: the publication in 1931 of Gödel’s two incompleteness theorems, preceded by the results of his 1930 thesis on the completeness of first-order logic (that every universally valid sentence is provable – \[BelS, Th. 12.1.3\]) and on compactness (a corollary). The latter was to bear fruit at the hands of Tarski much later (1958 on). We note that the Compactness Theorem for predicate calculus (that a set of sentences has a model iff each finite subset has a model \[BelS, Ch. 5 §4\]), Tychonov’s Theorem in topology and AC are deeply connected; see \[Jec1\]. See also \[BelS, Ch. 5 esp. §5\] for the status of variants and the connection with the ultraproducts of §5 below. The two incompleteness theorems concerning any axiomatic system rich enough to encompass arithmetic (firstly, the existence in the formal language of the axioms of sentences that can be neither proved nor disproved, and secondly, the impossibility of such a system to provide a proof for its own consistency), rather than just wreck Hilbert’s programme, produced untold benefits to the richness of mathematics: the plurality of the possible interpretations of a set of axioms (as in Skolem’s non-standard arithmetic), and the accompanying search for choosing the ways to reduce incompleteness, on the one hand, and to test or justify any belief in consistency, on the other: especially in the case of the axioms of set theory. See \[Ste\]. Gödel’s enduring insight was the embedding by arithmetic coding (hence the need for the ‘rich enough’ presence of arithmetic) of (aspects of) a ‘metalanguage’ – the informal language of discourse needed to examine a formal language as a mathematical entity – back into the formal language, specifically the concepts of proof and provability – see below. Addressing the incompleteness of set theory, Gödel’s second legacy relates to ‘relative consistency’: proof in 1938 (published in 1940) of the consistency relative to ZF of both AC – a matter of supreme importance, given the Banach-Tarski paradox (dating back to 1924) – and of GCH. The key idea in the proof was the introduction (see §2 above) of the cumulative hierarchy $L_{\alpha }$ of constructible sets whose totality comprising the class $L$ is an *inner model* (i.e. a subuniverse of the universe $V\ $of von Neumann, specifically a transitive class containing $On$). This was to be the foundation stone for the advances of the ‘next one hundred years’ in two ways. The first was to invite extensions of $L$ by appropriate choice of sets outside $L.$ The second, more technical, derives from Skolem’s method (1912) of constructing countable sub-models, enshrined in a *condensation principle,* that if $M\ $is a countable ‘submodel’ of $L$ (more accurately an ‘elementary substructure’), then it is isomorphic to a set $L_{\alpha }.$ Contemporaneously with Gödel’s earliest contributions, and blending and intertwining with them, there occurs a ‘volcanic eruption’ of ideas and results from the fertile mind of Tarski: bursting forth in 1924 with the Banach-Tarski paradox (mentioned above) and evidenced by the working seminars of 1927-1929, laying the foundations of Tarski’s remarkable legacy, both that published in its time and that published later. This included work on the definability or otherwise (definable if ‘external’, not if ‘internal’) of the concept of truth, a result closely allied to Gödel’s incompleteness result and of similar vintage. Suffice it to point to the role of ‘elementary substructure’ (term due to Tarski) in the condensation principle above. Deficiencies in Hilbert’s approach to geometry (e.g., its tacit assumption of set theory) led Tarski to re-examine the axiomatic basis of geometry. In 1930 Tarski was able to prove the decidability of ‘elementary geometry’, via a reduction to ‘elementary algebra’ where he was able to generalize Sturm’s algorithm for counting zeros of polynomials — see \[Vau\] for references and \[SolAH\] for recent developments in this area. **4. Ramsey, Erdős and their legacy: infinite combinatorics; partition calculus and large cardinals** **4a. Ramsey and Erdős** Pursuing a special case of Hilbert’s *Entscheidungsproblem* of 1928 – proposing the task of finding an effective algorithm to decide the validity of a formula in first-order logic – Ramsey was led to results in both finite and *infinite combinatorics* (obtained late that same year, and published in 1930, \[Ram\]), the finite version of which yielded the desired algorithm for the special (though common) universal type of formula. In general no computable algorithm exists, as was shown by Church (using Gödel’s coding) in 1935, and independently by Turing in 1936 (via Turing machines). The *Infinite Ramsey Theorem* (which acted as a paradigm for its finite variants) asserts in its simplest form that if the distinct unordered pairs (doubletons) of natural numbers are partitioned into two (disjoint) classes, then there exists an infinite subset $\mathbb{M}\subseteq \mathbb{N}$ all doubletons from which fall in the same class; thus $\mathbb{M}$, which may be said to be a *homogeneous* (monochromatic) subset for the partition, is large – see \[Dra, Ch. 2.8.1, Ch. 7.2 which both use DC\]. (Homogeneity is a constantly recurring theme in what follows.) Thus, as a corollary, a Cauchy sequence in $\mathbb{R}$ contains either an increasing or a decreasing subsequence. The combinatorial result extends from doubletons to (unordered) $n$-tuples (called by Ramsey ‘combinations’) and from *dichotomous* partitions to ones allowing any finite number $k$ of partitioning classes. Further analogues and generalizations form the discipline of *partition calculus*, the founding fathers of which were Paul Erdős and Richard Rado: see \[ErdR\]. Given its origins, it is not altogether surprising that Ramsey’s theorem and its generalizations continue to play a key role in the logical foundations of set theory. **4b. Partitions from large cardinals** We are particularly concerned below with the partition property that follows. As usual we regard any ordinal (including any cardinal) as the set of its predecessors. The partition property (partition relation) of concern is$$\kappa \rightarrow (\alpha )_{2}^{<\omega },$$by which is meant that if $[\kappa ]^{<\omega }$ (the finite subsets of $\kappa $) is partitioned into two classes, then there is a homogeneous subset of $\kappa $ of order type $\alpha .$ (Ramsey’s result as stated above is recorded in this notation as $\omega \rightarrow (\omega )_{2}^{2},$ and its immediate generalization to $n$-tuples and $k$ classes as $\omega \rightarrow (\omega )_{k}^{n}.$) For any $\alpha \geq \omega $ the least cardinal $\kappa $ for which $\kappa \rightarrow (\alpha )_{2}^{<\omega }$ holds, denoted $\kappa (\alpha ),$ is called the $\alpha $-th *Erdős cardinal* (or *partition cardinal*); but do such cardinals exist? One may show in ZFC that$\ \kappa (\alpha ),$ if it exists, is regular (below), and when $\alpha $ is a limit ordinal, that $\kappa =\kappa (\alpha )$ is strongly inaccessible (below) \[Dra, Ch. 10\], and so $V_{\kappa }$ is a model of ZFC, written $V_{\kappa }\models $ZFC. Hence, by Gödel’s incompleteness theorem, we cannot deduce its existence in ZFC. Of particular importance are cardinals $\kappa $, in particular $\kappa =\kappa (\omega _{1})$, for which $\kappa \rightarrow (\omega _{1})_{2}^{<\omega }$ holds: see the next section. So if, as we do, we need them, then we must add their existence to our axiom system. To gauge the consistency strength of this assumption we refer to one of the earliest notions of a ‘large cardinal’: a *measurable cardinal* $\kappa .$ Such a cardinal was defined by Ulam \[Ula\] in 1930 by the condition that it supports a $\{0,1\}$-valued $\kappa $-additive (i.e. additive over families of cardinality $\lambda $, for all $\lambda <\kappa )$ non-trivial measure on the power set $\mathcal{\wp }(\kappa )$. This may be reformulated as asserting the existence of a $\kappa $-complete *ultrafilter* on $\kappa $ (\[Car2\], \[ComN\], \[Jec2\], \[GarP\]). It turns out that for $\kappa $ measurable, the stronger relation $\kappa \longrightarrow (\kappa )_{2}^{<\omega }$ holds. The latter is taken as the defining property of a *Ramsey cardinal*, through its similarity with $\omega \rightarrow (\omega )_{2}^{2}.$ We stop to notice that the relation $\kappa \rightarrow (\kappa )_{2}^{2}$ (taken to be the definition of a *weakly compact cardinal* \[Dra, Ch.10.2\]) holds iff $\kappa $ is strongly inaccessible and $\kappa $ has the tree property: every tree of cardinality $\kappa $ having less than cardinality $\kappa $ nodes at each level has a path, i.e. a branch of full length $\kappa .$ It is interesting that, as with the Cauchy sequences in $\mathbb{R}$ above, if $\kappa \rightarrow (\kappa )_{2}^{2},$ then every linearly ordered set of cardinality $\kappa $ has a subset of cardinality $\kappa $ which is either well-ordered or reversely well-ordered by the linear ordering. **4c. Large cardinals continued** The first notion of a large cardinal is motivated by the conceptual leap from the finite to the infinite, as exemplified by the set of natural numbers viewed as $\mathbb{N}$, or, better for this context, as the first infinite ordinal $\omega .$ The arithmetic operations of summation and multiplication/exponentiation (equivalently, the power set operation $\wp )$ applied to members of $\omega $ lead to members below $\omega .$ This observation can be copied by a direct reference to the two corresponding operations that generate a union of a given family and the power set of a given set, each operation being guaranteed by the corresponding axiom. Thus a cardinal is said to be *weakly inaccessible* if it is a limit cardinal above $\omega $ which is *regular* (a regular limit cardinal), meaning, firstly, that it is the *limit*, i.e. supremum (union), of all the preceding ordinals, and, secondly, that nonetheless it is not the union (supremum) of a smaller family of ordinals. A cardinal $\kappa $ is *strongly inaccessible*, or just (plain) *inaccessible*, if it is a regular strong limit cardinal, i.e. additionally $2^{\lambda }<\kappa $ for all $\lambda <\kappa . $ (Here $2^{\lambda }$ is the cardinality of $\wp (\lambda )$.) Further such notions (of hyper-inaccessibility), which we omit here, have been introduced by reference to the idea of a ‘large limit’ (limit over a large set) of ‘large cardinals’. The axioms ZFC, assumed consistent, cannot imply the existence of an inaccessible $\kappa $, as then $V_{\kappa }$, being a model for ZFC, provides proof within ZFC of the consistency of ZFC, a contradiction to Gödel’s incompleteness theorem. A second source of largeness is motivated by the study of infinitary languages, the idea being to overcome some of the limitations of first-order languages. For example, in the language $\mathcal{L}_{\kappa \kappa }$ one admits $\kappa $ many free variables and permits infinite conjunctions/disjunctions of a family of formulas of cardinality below $\kappa .$ This leads to the desirability of these languages having a compactness property analogous to Gödel’s compactness property of the ordinary language $\mathcal{L}_{\omega \omega }$ (see above). Examples of the failure of compactness abound; so it emerges that the desired $\kappa ,$ if it exists, needs to be large. Thus a cardinal $\kappa $ is called *strongly compact* \[Dra, Ch. 10.3\] if the language $\mathcal{L}_{\kappa \kappa }$ is $(\lambda ,\kappa )$-*compact* for each $\lambda \geq \kappa $, that is: for each $\lambda \geq \kappa $ and any set $\Sigma $ of sentences in that language with $|\Sigma |\leq \lambda ,$ if each subset $\Sigma ^{\prime }$ with $|\Sigma ^{\prime }|<\kappa $ has a model, then $\Sigma $ has a model. (So the cardinality of $\Sigma \ $here is not constrained.) The property may be characterized without reference to the language more simply as saying that every $\kappa $-complete filter can be extended to a $\kappa $-complete ultrafilter. Analogously, a cardinal $\kappa $ is *weakly compact* \[Dra, Ch. 10.3\] if the language $\mathcal{L}_{\kappa \kappa }$ is $(\kappa ,\kappa )$-compact: if any set of sentences $\Sigma $ with $|\Sigma |\leq \kappa $ such that each of its subsets of cardinality $<\kappa $ has a model, then $\Sigma $ has a model. A third, more promising, source is more in keeping with the first (‘operational’) viewpoint. It is motivated by the ‘substructures’ analysis initiated in Gödel’s proof that GCH holds in the universe of constructible sets. Attention focusses now on the properties that the operation of elementary embedding could or should have. We recall that the range of such an embedding is an elementary substructure. Suppose that $j:N\rightarrow M$ is an elementary embedding, where $N$ and $\ M$ are transitive classes and $j$ is definable in $N$ by a formula of set theory with parameters from $N$. Then $j$ must take ordinals to ordinals and $j$ must be strictly increasing. Also $j(\omega )=\omega $ and $j(\alpha )\geq \alpha ,$ so there is a least $\delta $ with $j(\delta )>\delta .$ This is the *critical point* of $j.$ Then$$\mathcal{U}:=\{X\subseteq \delta :\delta \in j(X)\}$$is a non-principal $\delta $-complete ultrafilter on $\delta ,$ i.e. $\delta $ is a measurable cardinal. In fact, the converse is also true – see \[SolRK, Th. 1.2\]. Interestingly, here a non-principal ultrafilter is defined by membership of a single point, albeit via images. The significance of this characterization lies in the ‘operations’ the function $j$ encodes which, on the one hand, pass the test of ‘elementarity’ and, on the other, introduce an upward jump at the critical point (roughly speaking, an ‘inaccessibility from below by elementarity’). We mention some further canonical large-cardinal notions obtained from variations on this elementary embedding theme; these will be useful not only presently for the establishement of a *reference scale* of consistency strength, but also later in relation to the regularity properties of subsets of $\mathbb{R}$ (such as Lebesgue measurability etc., considered in §7 and 10). A cardinal $\kappa $ is *supercompact* if it is $\lambda $*-supercompact* for all $\lambda \geq \kappa $; here $\kappa $ is $\lambda $-supercompact if there is a (necessarily non-trivial) elementary embedding $j=j_{\lambda }:V\rightarrow M$ with $M$ a transitive class, such that $j$ has critical point $\kappa $, and $M^{\lambda }\subseteq M$, i.e. $M$ is closed under arbitrary sequences of length $\lambda $. Under AC, w.l.o.g. $j(\kappa )>\lambda $. For $\kappa $ a cardinal and $\lambda >\kappa $ an ordinal, $\kappa $ is said to be $\lambda $*-strong* if for some transitive inner model (§3), $M$ say, there exists an elementary embedding $j_{\lambda }:V\rightarrow M$ with critical point $\kappa ,$ $j_{\lambda }(\kappa )\geq \lambda ,$ and$$V_{\lambda }\subseteq M.$$Furthermore, $\kappa $ is said to be a *strong cardinal* if it is $\lambda $-strong for all ordinals $\lambda >\kappa $. This notion may be relativized to subsets $S$ to yield the concept of $\lambda $-$S$*-strong* by requiring in place of the inclusion above only that$$j(S)\cap V_{\lambda }=S\cap V_{\lambda }.$$(One says that $j$ preserves $S\ $up to $\lambda $.) This provides passage to our last definition. The cardinal $\delta $ is a *Woodin cardinal* if $\delta $ is strongly inaccessible, and for each $S\subseteq V_{\delta }$ there exists a cardinal $\theta <\delta $ which is $\lambda $-$S$-strong for every $\lambda <$ $\theta .$ The *consistency strength* of various extensions of the standard axioms ZFC, by the addition of further axioms, may then be compared (perhaps even assessed on a well-ordered scale) by determining which canonical large-cardinal hypothesis will suffice to create a model for the proposed extension. Thus, for $\kappa $ supercompact, $V_{\kappa }\models \exists \mu \lbrack $$\mu $ is strong$], $ which places supercompact above strong. Likewise, for $\kappa $ strong, $V_{\kappa }\models \exists \mu \lbrack $$\mu $ is measurable$]$, placing measurability below strong. (And below that is the existence of a Ramsey cardinal, recalling earlier comments.) The consistency of Woodin cardinals is thus between strong and supercompact: diagramatically,$$\text{supercompact \TEXTsymbol{>} Woodin \TEXTsymbol{>} strong \TEXTsymbol{>} measurable \TEXTsymbol{>} Ramsey.}$$ **5. Beyond the constructible hierarchy** $L$ – **I** We have mentioned the Löwenheim-Skolem-Tarski theorem. How else may one construct structures that will contain a given one as an elementary embedding? In topology one naturally reaches for powers and products (as with Tychonov’s theorem), and also their various substructures such as function spaces. For example, Hewitt \[Hew\] in 1948 constructed hyper-real fields by using a quotient operation on the space of continuous functions via a maximal ideal; cf. \[DalW2\]. **5a. Expansions via ultrapowers and intimations of indiscernibles** Jerzy Łoś \[Łoś\] in 1955, though foreshadowed by Skolem’s construction \[Sko\] of non-standard arithmetic in 1934, and even Gödel 1930, introduced a natural algebraic way of constructing new structures. Łoś relied on the concept, introduced in 1937 by Cartan \[Car1,2\], of *ultrafilter*: a maximal filter in the power set of $I$, say. (The assumption of the existence of these – see PI in §1– is in general weaker than AC.) For a family of structures $\langle \mathcal{A}_{i}:i\in I\rangle ,$ all of identical type/signature, i.e. each having the same distinguished operations and relations on its domain $A_{i}$ (and possibly distinguished elements), one first defines the direct product as a structure (again of the same type) with domain the set $\tprod\nolimits_{i\in I}A_{i}$ (the product’s existence in general implicitly invoking AC, of course) by defining the operations and relations pointwise; thus any distinguished element $e$, say, if interpreted in $A_{i}$ as $e_{i}$, say, is interpreted in the product by the function $e:i\mapsto e_{i}$. Next, for $\mathcal{U}$ an ultrafilter on $I$, define $\mathcal{U}$-equivalence: $f\sim g$ according as $\{i\in I:f(i)=g(i)\}\in \mathcal{U}$, i.e. $f$ and $g$ are pointwise $\mathcal{U}$-almost equal. Then denote by $\tprod\nolimits_{i\in I}\mathcal{A}_{i}/\mathcal{U}$ the equivalence classes $[f]_{\mathcal{U}}$ and equip these with the requisite operations and relations suitably interpreted as relations that hold pointwise $\mathcal{U}$-almost always. By induction from the construction of these ‘atomic’ cases of relations, Łoś’s Theorem (ŁT below) asserts satisfaction in the ultraproduct of general properties/formulas $\varphi ,$ say for simplicity with one free variable $v,$ via $$\dprod\nolimits_{i\in I}\mathcal{A}_{i}/\mathcal{U}\models \varphi (v)[f]_{\mathcal{U}}\text{ iff }\{i\in I:\mathcal{A}_{i}\models \varphi (f(i))\}\in \mathcal{U},$$for $\varphi $ any first-order formula (in the language needed to describe a structure of that type – ‘signature’ above). If the $\mathcal{A}_{i}=\mathcal{A}$ are all equal (with domain $A),$ then $\mathcal{A}$ embeds elementarily into the *ultrapower* $\mathcal{A}^{I}/\mathcal{U}$, when $a\in A$ is identified with the constant map $f_{a}:i\mapsto a$. Consider $\mathcal{A}:=\langle \mathbb{R},+,\cdot ,\leq ,0,1\rangle ,$ $I=\mathbb{N}$ and $\mathcal{U}$ an ultrafilter extending the filter of co-finite subsets of $\mathbb{N}$ (again invoking, say, AC). Then, $\mathbb{R}$ embeds in $\mathbb{R}^{I}/\mathcal{U}$ , with any real number $a$ represented by the constant function $f_{a}:$ $n\mapsto a.$ Let us call the function $\mathrm{id}(i):=i$ for $i\in I$ a *dominating function* since it plays an important role and *dominates* any constant function $f_{m}$ for $m\in \mathbb{N}$; indeed, $[f_{m}]_{\mathcal{U}}\leq \lbrack \mathrm{id}]_{\mathcal{U}}$, since $\{n:m\leq n\}\in \mathcal{U}$, and so $\mathrm{id}$ is an element following all of $\mathbb{N}$, and so follows all of $\mathbb{R}$ in $\mathbb{R}^{I}/\mathcal{U}$. That is, $\mathrm{id}$ identifies an *infinite number*; likewise $1/\mathrm{id}$ identifies a positive (non-zero) element that may be interpreted as an* infinitesimal*. (This observation allowed Abraham Robinson \[Rob1,2\] to develop a *non-standard analysis* within which to interpret and interrogate rigorously Leibniz’s intuitive texts on infinitesimals; see \[Kei\] for an undergraduate rigorous development of calculus in this setting.) The argument just given may be repeated with $\mathcal{A}:=\langle A,\in _{A}\rangle $ for $A$ a transitive set and $\in _{A}$ the relation of membership in $A.$ If $\mathcal{A}$ is a countable model of ZF, then, provided $\mathcal{U}$ is countably complete (see e.g. \[Kan, Prop. 5.3\]), $\mathcal{A}^{I}/\mathcal{U}$ is well-founded under its ‘interpretation of the membership relation’, so will contain elements that form an interval of ordinals following the ordinals in $A.$ However, there are no means within $\mathcal{A}$ itself of ‘seeing’ the existence of this extra layer of ordinals: speaking informally (but see below), they are ‘indiscernible’. (Strictly speaking, $\mathcal{A}^{I}/\mathcal{U}$ needs to be replaced by an isomorphic structure which is a transitive set, known as the *Mostowski collapse*$,$ defined inductively by the collapsing function $\pi $:$$\pi ([f]_{\mathcal{U}})=\{\pi ([g]_{\mathcal{U}}):[g]_{\mathcal{U}}\in _{\mathcal{U}}[f]_{\mathcal{U}}\}$$(cf. §2); then interpretations of ordinals collapse to actual ordinals.) When $I=\kappa $ with $\kappa $ the least measurable cardinal and $\mathcal{U}$ the ($\kappa $-complete) corresponding ultrafilter, Dana Scott considered the extension of $L$ to $L[\mathcal{U}]$ (the Lévy class of sets ‘constructible relative to’ $\mathcal{U}$ – obtained by allowing definability over the ordinals to refer also to $\mathcal{U}$ – so a class closed under the intersection with $\mathcal{U}$; see \[Kan, Ch. 1 §3\], \[Dra, 5.6.2\]), and investigated the ultrapower $L[\mathcal{U}]^{I}/\mathcal{U}$ to conclude the non-existence of a measurable cardinal in $L.$ This is easiest to understand through the lens of the theorem that existence of a measurable cardinal contradicts $V=L$ \[Sco1\], \[Dra, 6.2.10\], \[BelS, Ch. 14 §6\] (so there is no measurable cardinal in $L$). This is done again by referring to the *dominating function* $\mathrm{id}(i)=i,$ which vies with $\kappa $ for the place of smallest measurable cardinal (in the Mostowski collapse). A proper proof needs to avoid doubtful manipulations of $\mathcal{U}$-equivalence classes of subclasses of $L[\mathcal{U}]$. (To achieve this, one represents any function $f$ by one of least rank $\mathcal{U}$-equivalent to it – the ‘Scott trick’; under these circumstances well-foundedness of the resulting model needs to be verified, using $\sigma $-additivity of $\mathcal{U}$.) The gist of the proof is to recreate the following contradictions stemming from ŁT. As before, $\mathrm{id}(i):=i$ for $i\in I,$ and $f_{\lambda }:i\mapsto \lambda $ is the constant function on $I$ embedding $\lambda $ into the ultrapower. By ŁT, the map $\lambda \mapsto f_{\lambda }$ is injective for $\lambda <\kappa $ (since $\{i:f_{\lambda }(i)=\lambda <\mu =f_{\mu }(i)\}=\kappa \in \mathcal{U},$ for $\lambda <\mu <\kappa ).$ By ŁT again, $f_{\kappa }$ is the smallest measurable cardinal in $\mathcal{A}$ (since $f_{\kappa }(i)=\kappa $ is such a cardinal, for all $i),$ hence $f_{\kappa }=\kappa $ (up to equivalence, really). Now $\mathrm{id}<f_{\kappa }$ (since $\{i\in \kappa :i=\mathrm{id}(i)<f_{\kappa }(i)=\kappa \}=\kappa \in \mathcal{U}$), so $\mathrm{id}<f_{\kappa }=\kappa .$ But, for each $\lambda <\kappa ,$ we have $f_{\lambda }<\mathrm{id}$ (since $\{i:f_{\lambda }(i)<\mathrm{id}(i)\}=\{i\in I:\lambda <i\}=\kappa \backslash (\lambda +1)\in \mathcal{U}$, as $\mathcal{U}$ is $\kappa $-complete). But $\{f_{\lambda }:\lambda <\kappa \}$ has cardinality $\kappa ,$ and so $\kappa \leq \mathrm{id},$ contradicting the earlier deduction that $\mathrm{id}<\kappa .$ Actually, these observations just demonstrate that the embedding $j=j_{\mathcal{U}}$ obtained by composing $\lambda \mapsto f_{\lambda }$ with the Mostowski collapse satisfies $j(\lambda )=\lambda $ for $\lambda <\kappa ,$ and $j(\kappa ),$ being the collapsed version of $[\mathrm{id}],$ lies strictly above $\kappa ;$ thus the ordinal $\kappa $ is the *critical point* of $j.$ This argument was further investigated by Haim Gaifman, from the point of view of iterating the ultrapower construction, and perfected by Kunen \[Kun1\]. **5b. Ehrenfeucht-Mostowski models: expansion via indiscernibles** At about the same time as Łoś introduced ultraproducts into model-theory, Ehrenfeucht and Mostowski \[EhrM\] in 1956 introduced a construction that expands a structure $\mathcal{A}$ by importing a linearly ordered set of elements in such a way that, speaking anthropomorphically, $\mathcal{A}$ is incapable of distinguishing between these imports and a certain infinite subset of its own domain. Less than a decade later, first Morley in 1962 (see e.g. \[Mor\]) and then Silver in his thesis in 1966 (see \[Sil\]) put these features to decisive use, by enabling the imported elements to generate various kinds of information about $\mathcal{A}$ consistent with that generated by $\mathcal{A}$ on its own. The original construction provided an elementary embedding of any infinite structure $\mathcal{A}$ into another ‘larger’ one – larger in possessing many non-trivial automorphisms, securing in particular a non-trivial elementary embedding. A (copy of a) linearly ordered set $X$ is adjoined to $A$ of elements $x$ which are to be ‘indiscernible’ from the viewpoint of $\mathcal{A}$ (except only in name – as the formal language must adjoin formal names $c_{x}$ to speak about them) in the sense that:$$(\mathcal{A},(c_{x})_{x\in X})\models \varphi (x_{1},...,x_{n})\Leftrightarrow \varphi (x_{1}^{\prime },....,x_{n}^{\prime }),$$for all formulas $\varphi $ having $n$ free variables, for all $n,$ and all $x_{1}<...<x_{n}<x_{1}^{\prime }<...<x_{n}^{\prime }$ in $X.$ That this is possible in general relies on the Compactness Theorem (and so on AC): the idea here being that if one takes the sentences true in $\mathcal{A}$ together with sentences $\varphi (c_{x_{1}},...,c_{x_{n}})\Leftrightarrow \varphi (c_{x_{1}^{\prime }},...,c_{x_{n}^{\prime }})$ (also the inequalities $c_{x}\neq c_{y}$), then one may satisfy a finite set $F$ of these by interpreting the finite number $m$ of $c_{x}$s in play in $F$, $c_{x_{1}},...,c_{x_{m}}$ say, with suitably chosen elements of $A,$ as follows. To effect the choice, partition all $m$-tuples of $A$ according as to whether or not $\mathcal{A}$ can distinguish between them on the basis of the properties defined by the finite number of formulas $\varphi (v_{1},...,v_{m})$ obtained from the $\varphi $ in $F.$ (That is: the free variables $v_{i}$ replace the constants $c_{x_{i}}$.) Then an infinite homogenous set for this partition yields a model for $F.$ In particular, for limit ordinal $\delta ,$ the structure $\mathcal{A}=\langle L_{\delta },\in \rangle $ (by abuse of notation $\in $ here and below denotes membership $\in $ restricted to $L_{\delta }$) can be expanded to a structure with a sequence of indiscernibles whose formal language names are $c_{n}$. Call that $\mathcal{A}_{0}.$ (Here AC may be avoided, as $L_{\delta }$ is well-ordered.) In turn, for any ordinal $\alpha ,$ that expanded structure $\mathcal{A}_{0}$ may be further extended to a structure $\mathcal{M}_{\alpha }(\mathcal{A})$ with a set of indiscernibles $X$ of order type $\alpha $ and with the following additional property: for any formula in the language of $\langle L_{\delta },\in \rangle ,$ $\varphi (v_{1},...,v_{n})$ say, $$(\mathcal{A},(c_{n})_{n\in \omega })\models \varphi (c_{1},...,c_{n})\text{ iff }\mathcal{M}_{\alpha }\models \varphi (\mathbf{x})\text{ for some }\mathbf{x}=(x_{1},...,x_{n})\in X^{n}.$$So, in particular, the indiscernibles $X$ can generate all the true sentences about $\mathcal{A}$. But are the structures $\mathcal{M}_{\alpha }(\mathcal{A})$ well-founded for *all* $\alpha $? That depends on whether the structures $\mathcal{M}_{\alpha }(\mathcal{A})$ for just $\alpha <\omega _{1}$ are all well-founded (the reduction here is possible, since any descending sequence occurring in the models with larger $\alpha $ can be captured by a countable submodel). This will be so when $\mathcal{A}=\langle L_{\kappa },\in \rangle $ and $\kappa $ satisfies the partition relation$$\kappa \rightarrow (\omega _{1})_{2}^{<\omega }.$$(With $\alpha <\omega _{1}$ as above, the argument is similar to but easier than that in the Ehrenfeucht-Mostowski result. Appealing to the partition relation above in place of Ramsey’s theorem, partition $(\xi _{1},...,\xi _{n})\in \lbrack \kappa ]^{<\omega }$ dichotomously according as to whether $\mathcal{M}_{\alpha }\models \varphi (\xi _{1},...,\xi _{n})$ holds or not; extract an $\omega _{1}$ homogeneous subset of $\kappa $ and use its first $\alpha $ members as the required indiscernibles. Their Skolem hull in $L_{\kappa }$, a well-founded set, is isomorphic to $\mathcal{M}_{\alpha }(\mathcal{A})$.) A first corollary (by appeal to indiscernibility, use of only the first $\omega $ indiscernibles, and then the countability of the formal language): only a countable number of subsets of $\omega $ are constructible in $L,$ even though from the viewpoint of $L$ there are uncountably many of them in $L$; but then, an embellishment of the analysis yields that $\omega _{1}^{L},$ the ordinal intepreted by $L$ as the first uncountable, is also countable. Silver deduced deeper results about $L$ along these lines. Some of these were then bettered by Kunen \[Kun1\], who devised a way for iterating the ultrapower construction of a structure $\mathcal{M}$ in a setting where the ultrafilter $\mathcal{U}$ need not be a member of $\mathcal{M}$. A most remarkable contribution from Silver was the introduction of the set now called $0^{\#}$ (*zero-sharp*) following Solovay (originally designated a ‘remarkable’ set); this is the set of Gödel codes $\lceil \varphi \rceil $ for all the true sentences $\varphi $ about $L$ generated by the $\omega $-sequence of indiscernibles $\{\omega _{1},\omega _{2},...,\omega _{n},...\},$ namely: $$0^{\#}:=\{\lceil \varphi \rceil :L\models \varphi (x_{1},...,x_{n})\text{ for }(x_{1},...,x_{n})\in \{\omega _{1},\omega _{2},...,\omega _{n},...\}\}.$$(The notation tacitly assumes that $n=n(\varphi )$ is the number of free variables in $\varphi .)$ This set’s very existence of course depends on suitable large-cardinal assumptions, such as $\kappa \rightarrow (\omega _{1})_{2}^{<\omega }$ holding for some $\kappa .$ The ‘existence of $0^{\#}$’ can be used as a large-cardinal assumption in its own right, lying below the existence of the Erdős cardinal. Indeed, in §7 we discuss the classical theory of analytic sets and thereafter the determinacy of infinite positional games with a target set $T,$ say; the assumption that sets with co-analytic target set are determined ($\mathbf{\Pi }_{1}^{1}$-determinacy) implies that 0$^{\#}$ exists, a result due to Harrington \[Har\]. We return to the indiscernibles for the structures $\mathcal{A}=\langle L_{\delta },\in \rangle $, *assuming the partition relation* just mentioned, which had been studied initially by Gaifman and by Rowbottom. Silver’s great contribution was to describe the structure, indeed the ‘very good behaviour’ (below), of a (proper) class $X$ of ordinal indiscernibles: closed (under limits – i.e. under suprema), unbounded in any cardinal $\lambda $ (with $X\cap \lambda $ of cardinality $\lambda )$; with $L_{\alpha }\prec L_{\beta }$ for $\alpha <\beta $ both in $X$ (indeed, stretching the notation to class structures, with $L_{\alpha }\prec L$); having the property that every set in $L$ is definable from parameters in $X.\ $Among the significant consequeness is the, already mentioned, countability of those sets in $L\ $that are definable over $L$ without any parameters (implying immediately that $V\neq L),$ and more importantly the definability of truth in $L.$ For details see e.g. \[Dra, Th. 4.8\]. We stress these results are subject to the partition assumption. The point (above) about good behaviour concerns particularly the ‘closed unbounded’ nature of $X$ above. Sets of ordinals with this property should be regarded as ‘large’, since they enable the very important ‘stationary sets’ of the next section to be thought of as non-negligible. The two concepts play a leading role in combinatorial principles (holding in $L)$ isolated by Jensen (see e.g. \[Dev1\]) from the fine structure of $L$. These include Jensen’s $\Diamond $ (diamond), used in constructing a ‘Suslin continuum’ as a counterexample to Suslin’s hypothesis (see below); $\square $ (square); derived ones like $\clubsuit $ (club), introduced by Ostaszewski \[Ost1\] (in ‘counterexample’ constructions for general topology); and generalizations $\clubsuit _{\text{NS}}$ studied by Woodin \[Woo1, Ch. 8\]. Compare the use of NT (for No Trump) in \[BinO1, 4\]. **6. Beyond the constructible hierarchy** $L$ – **II** **6a. Forcing and generic extensions** The undisputed game-changer for set theory was Cohen’s ‘method of forcing’, devised as a means of importing into a countable structure $\mathcal{M}=\langle M,\in _{M}\rangle $ additional sets from $V\backslash M$ ($V$ contains the reals; $M,$ being countable, does not), without disturbing the fact that $\mathcal{M}$ may be a model of ZF. Speaking anthropomorphically, the imported set may have the intention of introducing new information – say, the existence of a transfinite sequence of real numbers viewed by $\mathcal{M}$ as an $\omega _{2}^{M}$ sequence (reference here to the interpretation in $\mathcal{M}$ of the second uncountable cardinal), albeit viewed by $V$ as a countable sequence – without nevertheless encoding such catastrophic information as that $M$ itself is countable. Cohen described his method \[Coh3\] as ultimately analogous to the construction of a field extension: introduce a name for the algebraically absent element, and then describe its properties via polynomials in that element. In truth the extension method shares a family resemblance with non-constructive existence proofs, either via the Baire category method (the desired item has generic features), or the Erdős probabilistic method (measure-theoretic: the desired item has ‘random’ features). Indeed the two canonical instances of forcing to adjoin real numbers, Cohen’s and Solovay’s, are categorical (Cohen reals) or measure-theoretic (‘random reals’, or – perhaps better – ‘Solovay reals’). Indeed, following an idea of Ryll-Nardzewski and of Takeuti, Mostowski \[Most\] shows how to guide the selection of an imported set by reference to the points of a Baire topological space (one in which Baire’s theorem holds); avoiding a specified meagre set ensures that the extension of $\mathcal{M}$ will be a model of ZF. The two canonical cases then correspond to two topological spaces. For an alternative unification see \[Kun3\]. One views the forcing method as acting ‘over’ a structure $\mathcal{M}$ by providing a set $P$ in $M$ of partial descriptions of a generic object $G$ yet to be determined. $P$ is thus rendered as a partially ordered set, and under its ordering relation $q\leq p$ is understood as saying that $q$ contains more information about the object to be constructed than does $p.$ There is a syntactic relation $p\Vdash \varphi $ for $p\in P$ and $\varphi $ a sentence, read as ‘$p$ forces $\varphi $’, which may be ‘explained’ by an induction reminiscent of the Tarski inductive definition of truth ($\models , $ in §2), but with significant differences (below). Before embarking on the details, it is helpful to use an analogy with probability or statistical inference. Indeed $p\in P$ is usually called a ‘condition’; forcing is inspired by the language of ‘conditioning’; its inferences are concerned with information about $G$ given the information in $p.$ Thus the forcing relation must allow for further information which may become available ‘later’, so to speak. As a first pass, here is a brief glimpse of the character of the forcing relation: as this is a syntactical relation, we refer to a language whose terms are built from functions from $P$ to $M,$ and so we have (see \[Kun2, Cor. 3.7\]): $$\begin{aligned} p &\Vdash &\varphi \wedge \psi \text{ iff }p\Vdash \varphi \text{ and }p\Vdash \psi , \\ p &\Vdash &\lnot \varphi \text{ iff not}(\exists q\leq p)\text{ [}q\Vdash \varphi \text{],} \\ p &\Vdash &(\exists v)[v\in \sigma \wedge \varphi (v)]\text{ then }(\exists q\leq p)(\exists x\in \text{dom}(\sigma ))[\text{ }q\Vdash \varphi (x)].\end{aligned}$$ A clearer picture will emerge shortly. Whilst a variant of the forcing relation above was Cohen’s starting point, this is now a derived concept, the usual starting point being a set $G$ that is a filter on $P$ with the property that whenever $D\ $is a *dense* subset of $P$ (i.e. for each $p$ there is $q\leq p$ with $q\in D)$ and $D\in M,$ then$$G\cap D\neq \emptyset .$$Then $G$ is said to be $P$-*generic* over $\mathcal{M}$, or just *generic* over $\mathcal{M}$, when $P$ is understood. For $M$ countable, the dense subsets of $P$ lying in $M$ may be enumerated as a sequence $D_{n},$ and we may choose $p_{n}\in P$ starting with an arbitrary $p_{0}\in D_{0}$ and inductively $p_{n+1}\leq p_{n}$ with $$p_{n+1}\in D_{n+1}.$$The choice is possible precisely because $D_{n+1}$ is dense. Then $G:=\{q:(\exists n)$ $q\leq p_{n}\}$ meets each $D_{n},$ and so is generic over $\mathcal{M}$. This construction is sometimes called the *Cohen diagonalization argument,* since, in particular, $G$ decides every sentence $\varphi $. Indeed, the following set is dense:$$D_{\varphi }:=\{p:p\Vdash \lnot \varphi \text{ or }p\Vdash \varphi \}$$(as $p\notin D_{\varphi }$ implies not($p\Vdash \lnot \varphi )$ and so $(\exists q\leq p)$ \[$q\Vdash \varphi $\]). The idea is that the dense sets provide a structured way of hinting at the properties of $G$ and about the various ways that $G$ might be selected, but conditional on some given state of knowledge $p.$ The sequence $p_{n}$ above runs through all possible dense sets in an arbitrary order, and brings into existence a particular realization of $G.$ Before $G$ is created there are only names for $G$ and for all the possible objects in the intended extension, given simply by the *functions* in $M^{P}.$ (This corresponds to the use of *polynomials* in field extension.) But, once a generic $G$ is given, one may proceed inductively to give an interpretation $\tau ^{G}$ to the ‘names’ $\tau \in M^{P}$ of objects, inductively so that$$\tau ^{G}:=\{\sigma ^{G}:(\exists p\in G)[\sigma =\tau (p)]\}$$(mirroring the Mostowski collapse above), and so construct the extension $\mathcal{M}[G]$ as the set of $G$-interpretations. One then defines forcing (relative to $P$ and $\mathcal{M}$) by:$$p\Vdash \varphi \text{ iff (}\forall G\text{ generic over }\mathcal{M}\text{)[}p\in G\rightarrow \mathcal{M}[G]\models \varphi \text{].}$$This should clarify the three properties of the forcing relation introduced earlier. It emerges that if $\mathcal{M}\models $ ZFC, then $\mathcal{M}[G]\models $ ZFC. Furthermore, if $P$ satisfies the so-called *countable chain condition* (*‘ccc’*)* *(which actually calls for antichains of $P$ in $M\ $to be countable), then all ordinals that are cardinals from the viewpoint of $\mathcal{M}$ continue to be cardinals from the viewpoint of $\mathcal{M}[G],$ and their cofinalities \[Jec2\] remain the same. To secure the failure of CH, Cohen used as his conditions finite sets $p$ with elements of the form:$$\langle n,\alpha ,i\rangle \text{ for }n\in \omega ,\text{ }\alpha <\omega _{2},\text{ }i\in \{0,1\},$$which act as coded messages about objects, named as $c_{\alpha },$ to be imported from outside $M$ asserting that $n\notin c_{\alpha }$ if $i=0$ and $n\in c_{\alpha }$ if $i=1.$ As with the ‘dog that did not bark’, that which $p$ will never say allows us to infer that $c_{\alpha }$ will be a subset of $\omega :$ this is *forced* to be the case, since no extension of the coded message $p$ can say otherwise. Thus $p$ ‘hints at information’ by the absence of information. Formally, the corresponding $P,$ called $Add(\omega ,\omega _{2})$ since it adds $\omega _{2}$ many subsets of $\omega $, may be defined in $M$ to comprise ‘partial functions’ $p$ with finite domain contained in $\omega \times \omega _{2}^{M}$ and range in $\{0,1\}$, and with the ordering of ‘increasing informativeness’ that $q\leq p$ if $p\subseteq q,$ that is, $q$ contains at least all of the information in $p.$ The filter $G$ in $P\ $has the property that $\tbigcup G=\{\langle n,\alpha ,i(n,\alpha )\rangle :$ $n\in \omega ,\alpha \in M\cap \omega _{2}^{M}\}$ for some $i:(n,\alpha )\mapsto \{0,1\}.$ Indeed, for $n,\alpha $ as above, each of the sets$$D_{n,\alpha }:=\{p:\langle n,\alpha ,i\rangle \in p\text{ for some }i\in \{0,1\}\}$$is dense, as may be readily checked. (Hint: Given $p\notin D_{n,\alpha }$ choose $q$ to contain both $p$ and $\langle n,\alpha ,1\rangle .)$ So $G\ $must meet $D_{\alpha ,n}$ for each $\alpha \in M$ (as $\omega \subseteq M,$ since $\mathcal{M}\models $ ZFC$).$ For $\alpha \in M\cap \omega _{2}^{M},$ put$$G_{\alpha }:=\{n\in \omega :\langle n,\alpha ,i(n,\alpha )\rangle \in G,i(\alpha ,n)=1\}\subseteq \omega .$$Moreover, for distinct $\alpha ,\beta <\omega _{2}$, put$$\Delta _{\alpha ,\beta }:=\{p:\langle n,\alpha ,i\rangle ,\langle n,\beta ,1-i\rangle \in p\text{ for some }n\in \omega \text{ and some }i\in \{0,1\}\},$$which is dense. (Given $p\notin \Delta _{\alpha ,\beta }$ choose $q$ to contain both $p$ and $\langle m,\alpha ,1\rangle ,\langle m,\beta ,0\rangle $ for large enough $m.)$ So for distinct $\alpha ,\beta \in M\cap \omega _{2}^{M},$ $G$ contains $\langle n,\alpha ,i\rangle ,\langle n,\beta ,1-i\rangle $ for some $n$ and $i,$ with $i=1,$ say (w.l.o.g.). Then $n\in G_{\alpha }\backslash G_{\beta }$. Thus in $\mathcal{M}[G]$ there are $\omega _{2}^{M}$ distinct subsets of $\omega ,$ and so from the viewpoint of $\mathcal{M}[G]$ the continuum is at least $\omega _{2}$ (since $\omega _{2}^{M}$ is still the interpretation of $\omega _{2}$ in $\mathcal{M}[G]$ by the ccc, which is satisfied by $P$ here). We have just given an example of importing a set in order to increase the cardinality of the continuum. (Note that this construction may be repeated with $\omega _{2}^{M}$ replaced by $\omega _{\tau }^{M}$ for $\tau $ with any cofinality other than $\omega $, that being the only restriction on the cofinality of the continuum.) An important ingredient in Solovay’s result \[Sol3\] on LM (in constructing a model of ZF+DC in which all sets of reals are Lebesgue measurable – cf. \[Kan, Ch.13 §11\] – to which we refer in §10.2) uses a further partial order $P^{\kappa }=Coll(\omega \times \kappa ,\kappa )$, introduced by Lévy, whose function is to alter/collapse a (strongly) inaccessible cardinal $\kappa $ so that in the extension $\mathcal{N}=\mathcal{M}[G^{\kappa }]$ ($G^{\kappa }$ being $P^{\kappa }$-generic over $\mathcal{M}$) it is the ordinal $\kappa $ that appears as the first uncountable cardinal $\omega _{1}^{N}$. Consequently the ordinals below $\kappa $ are made to be countable by the importation of appropriate enumerations. Interest focuses on the substructure $\mathcal{N}_{1}$ with domain the sets that are hereditarily definable over $\mathcal{N}$, from a parameter in $\mathcal{N\cap }On^{\omega }$ (i.e. from a sequence of ordinals in $\mathcal{N}$), much as defined earlier. $\mathcal{N}_{1}$ satisfies the axioms ZF (see \[MyhS\]), and, significantly here, shares the same sequences of ordinals, in particular *the same reals*. (Here the reals are identified via binary expansion with characteristic functions of subsets of $\omega $.) The Lévy conditions (elements of $P^{\kappa })$ this time are partial functions with finite domain $\omega \times \kappa $ and range in $\kappa .$ Since there are no bounds placed on the range values of the partial function in this $P$, it follows that for $\alpha <\kappa $ the functions $$G_{\alpha }:=\{\langle n,\lambda (\alpha ,n)\rangle :\langle \alpha ,n,\lambda (\alpha ,n)\rangle \in G^{\kappa }\}$$will collectively witness (by enumeration) that each $\lambda <\kappa $ is countable. This ensures that $\kappa $ viewed from $\mathcal{M}[G^{\kappa }]$ is $\omega _{1}.$ Solovay ’s purpose is to turn any transfinite sequence of ordinals below an inaccessible $\kappa $ into an $\omega $-sequence. This helps him turn an arbitrary set of reals $A$ that lies in $\mathcal{N}_{1},$ initially definable in $\mathcal{N}$ via ordinal parameters, into one that is definable via a real $a$. (This also carries the advantage that, since $\kappa $ retains its inaccessibility in the extension $\mathcal{M}[a]$, one may w.l.o.g. argue as though $\mathcal{M}[a]$ is $\mathcal{M}.)$ As both $\mathcal{N}$ and $\mathcal{N}_{1}$ have the same reals, they also have the same Borel sets and the same *null* $\mathcal{G}_{\delta }$-sets. Solovay’s surprising innovation was to force over $\mathcal{M}[a]$ using the non-null Borel sets $\mathcal{B}_{+}$ ordered by inclusion (smaller sets yielding more information as to location). The key idea here is to introduce the notion of a *random real*, namely a real that cannot be covered by any null $\mathcal{G}_{\delta }$-set coded canonically by a real $c$ of the model $\mathcal{M}[a]$. (Solovay thought of these as ‘random’; we have already mentioned that Cohen reals are categorical, while random (‘Solovay’) reals are measure-theoretic; the term generic was already in use, so unavailable. Compare our earlier use of the language of probability and statistical inference above. One might also mention the term pseudo-random number in computer simulation.) But, $\mathcal{M}[a]$ being countable, there are only countably many such codes, so in $V$ the set of non-random reals is null. For a set $A\subseteq \omega ^{\omega }$ that is definable from an $\omega $-sequence of ordinals (i.e., by a sequence from $On^{\omega }$), suppose that with $a$ as above, for some formula $\varphi _{A}$ say, $A=\{x:\mathcal{M}[a][x]\models \varphi _{A}[a,x]\}$. Now one may choose a formula $\psi _{A}$ such that, for $G_{+}$ a $\mathcal{B}_{+}$-generic filter and any $x\in \mathcal{M}[G_{+}]\cap On^{\omega },$ $$\mathcal{M}[G_{+}]\models \varphi _{A}[a,x]\text{ iff }\mathcal{M}[x]\models \psi _{A}[a,x].$$In $\mathcal{B}_{+}$ choose a maximal (necessarily countable, by positivity of measure here) antichain of Borel sets $\mathcal{C}$ whose elements ‘decide’ the formula $\psi _{A}[\check{a},\dot{r}]$ (i.e. force the formula or its negation), where $\check{a}$ is a name for the set $a$ given above, and $\dot{r}$ is a name for a random real (cf. the use of $\dot{q}$ in §6b below). Then for $x$ random:$$x\in A\Longleftrightarrow x\in \tbigcup \{F_{c}:F_{c}^{\mathcal{M}[G]}\in \mathcal{C}\text{ and }F_{c}^{\mathcal{M}[G]}\Vdash \varphi _{A}[\check{a},\dot{r}]\}.$$Here $F_{c}$ is a non-null closed set canonically coded by $c.$ So modulo the null set of non-random reals, $A$ is an $\mathcal{F}_{\sigma }.$ **6b. Forcing Axioms** Solovay’s argument makes heavy use in various ways of ‘two-step extensions’ like $\mathcal{M[}G][H]$ with $G$ an $\mathcal{M}$-generic filter and $H$ an $\mathcal{M[}G]$-generic filter. By implication, $G$ is associated with a partial order $P$ in $\mathcal{M}$ and $H$ with a partial order $Q$ in $\mathcal{M[}G].$ This can be turned into a one-step extension $\mathcal{M[}K] $, but in a perspicuous way (more general than cartesian products), so that a generic extension of a generic extension is again a generic extension. Since the model $\mathcal{M[}G]$ is created by interpreting ‘names’ (using $G $ as in $\tau ^{G}$ above), the partial order for the equivalent single step needs to be built out of $P$ and out of a name $\dot{Q}$ for $Q,$ and must refer to pairs $(p,\dot{q})$ with $p\in P$ and $\dot{q}$ a name for something that is $P$-forced to lie in $\dot{Q};$ likewise, the order on the resulting composition of the two partial orders, denoted $P_{1}\ast \dot{Q}$, must make use of how the $P$-conditions $P$-force the extension property $\dot{q}\leq \dot{q}^{\prime }$ between names for elements of $\mathcal{M[}G][H].$ Thus a kind of syntactical analysis in $\mathcal{M}$ underlies this ‘iterated forcing’. More generally, any ordinal $\alpha $ of $\mathcal{M}$ can provide the basis for $\alpha $-step iterations, and, as with the topologies on products so too here, various kinds of $\alpha $-iterations may be constructed by appropriate constraints on the supports (e.g. finite or countable). We omit the details, except to mention that it was by use of such an iteration that Solovay and Tennenbaum \[SolT\] showed that it is consistent that no Suslin continuum exists (so otherwise than in $L,$ where such exists); this led to the more general observation, proved by Martin and Solovay: the consistency of Martin’s Axiom, MA (\[MarS\], cf. \[Fre1\]), namely the statement that for all cardinals $\kappa $ below the continuum ($\kappa <\mathfrak{c)}$ the following holds: $\mathrm{MA}(\kappa ):$ *for every partial order* $P$*satisfying the countable chain condition (ccc), and any family* $\mathcal{F}$ *with* $|\mathcal{F}|\leq \kappa $ *of dense subsets of* $P,$* there is a filter* $G$* in* $P$* which meets each* $D\in \mathcal{F}$. The reader will notice the similarity between the property of $G$ here and that of a filter $P$-generic over $\mathcal{M}$; indeed Martin (and independently Rowbottom) proposed this axiom as a combinatorial principle that is forcing-free – so, in particular, with the potential for immediate applicability without expertise in logic. That potential was so quickly realized both in theorem-proving and counterexample-manufacture – look no further than \[Fre1\] – that it became the ‘tool of first choice’ when abstaining from CH whilst harbouring CH-like intuitions, because, like Zorn’s Lemma, it encapsulates a ‘construction without (transfinite) induction’, replacing the latter with a side-condition swept away into $\mathcal{F}$, the family of dense sets. Of course, the ‘implied’ induction was performed, off-line so to speak, in the Martin-Solovay paper \[MarS\], aptly titled ‘Internal Cohen extensions’, reflecting the view that MA asserts that the universe of sets is closed under a large class of generic extensions. In regard to MA’s huge significance as an alternative to the continuum hypothesis: we cite after Martin and Solovay \[MarS\] the statistic that at least 71 of 82 consequences of CH, as given in Sierpiński’s monograph \[Sie\], are decided by MA or $[$MA & $2^{\aleph _{0}}>\aleph _{1}].$ Amongst these are that MA implies: \(1) $2^{\aleph _{0}}$ is not a real-valued measurable cardinal;(2) the union of less than $2^{\aleph _{0}}$ (Lebesgue) null /meagre sets of reals is null/meagre;(3) Lebesgue measure is $2^{\aleph _{0}}$-additive;and that $[$MA & $2^{\aleph _{0}}>\aleph _{1}]$ implies: (1) Suslin’s hypothesis, that every complete, dense, linear order without first and last elements in which every family of disjoint intervals is at most countable (the Suslin condition) is order-isomorphic to $\mathbb{R}$;(2) every $\mathbf{\Sigma }_{2}^{1}$ set of reals (for the $\mathbf{\Sigma }$ and $\mathbf{\Pi }$ notation of the projective hierarchy see §9) is Lebesgue measurable and has the Baire property;(3) every set of reals of cardinality $\aleph _{1}$ is $\mathbf{\Pi }_{1}^{1}$ (co-analytic) iff every $\aleph _{1}$ union of Borel sets is $\mathbf{\Sigma }_{2}^{1}$. It is worth remarking that an equivalent of MA is the topological statement that, in a compact Hausdorff space whose open sets satisfy the countable chain condition, the union of less than $2^{\aleph _{0}}$ meagre sets is meagre \[Wei\], \[Fre1\]. This identifies MA as a variant of Baire’s Theorem, and gives it a special role in the investigation of the *additivity properties* etc. of classical ideals such as the null and meagre sets, for which see \[BartJ\]. Given its particular usefulness and origin, MA, termed a *Forcing Axiom,* inspired the search for further, more powerful, forcing axioms. The first to occupy centre-stage is the Proper Forcing Axiom, PFA. This is an extension of $MA(\aleph _{1})$, which draws in more model theory. At the price of replacing *all* the cardinals $\kappa <\mathfrak{c}$ by allowing just $\kappa =\aleph _{1}$, PFA relaxes the ‘ccc’ restriction. (In fact, Todorčević and Veličković (\[Tod\], \[Vel\]) showed that PFA implies that $\mathfrak{c}=\aleph _{2},$ so allowing back in all the, rather few, cardinals $\kappa <\mathfrak{c.)}$ The relaxation widens access to the class of *proper* partial orders (below), and so asserts: PFA: *for every partial order* $P$* that is proper and any family* $\mathcal{F}$ *with* $|\mathcal{F}|\leq \aleph _{1}$ *of dense subsets of* $P,$* there is a filter* $G$* in* $P$* which meets each* $D\in \mathcal{F}$. The definition of properness refers to the interplay between the whole of the partial order $P$ and those fragments of $P$ that appear in ‘suitably rich’ countable structures, as follows. A partial order $P$ is *proper* if, for any regular uncountable cardinal $\kappa $ and countable model $\mathcal{M\prec }$ $H\mathcal{(\kappa )}$ (the family of sets hereditarily of cardinal less than $\kappa $ \[Dra, Ch. 3 §7\]) with $P\in M: $ *for each* $p\in P\cap M$* and each* $q\leq p,$*every antichain* $A\in M$* contains an element* $r$*compatible with* $p.$ (This formulation obviates the need to refer to ‘maximal antichains’.) The class of proper partial orders includes both those satisfying ccc (which preserves cardinality, and cofinality) and those with countable closure (i.e. guaranteeing a lower bound for any decreasing $\omega $-sequence). A consistency proof for PFA needs use of a supercompact cardinal. See \[Bau\] for applications and discussion (especially remarks after his Th. 3.1 concerning the need for a supercompact and its ‘reflection properties’), and also \[Dev2\], and the more recent \[Moo\]. A wider variant still is SPFA, based on $\aleph _{1}$-semiproper forcing. The maximal version, known as Martin’s Maximum, MM, was introduced by Foreman, Magidor and Shelah \[ForMS\], and like PFA needs a supercompact cardinal for a proof of its consistency. Here the role of $\omega _{1}$ as $\aleph _{1}$ (in merely prescribing a cardinality bound) changes in order to create an $\omega _{2}$-chain condition, as we shall see presently. Prominence is given now to the *stationary* subsets of $\omega _{1}$ (defined below), cf. §5b; these are the ‘non-negligible’ subsets in relation to coding, and their definition draws on some associated ‘large’ sets: the subsets that are closed and unbounded (cofinal) in $\omega _{1}$, with which we begin. A set $C\subseteq \omega _{1}$ is *closed* if it contains all its limit points (i.e. $\sup (C\cap \alpha )\in C\ $for limit $\alpha $ whenever $C\cap \alpha $ is cofinal in $\alpha );$ such sets form a filter, as any two unbounded closed sets meet. A subset $S\subseteq \omega _{1}$ is *stationary* if $S$ meets every closed unbounded set. In MM, the partial orders $P$ are required to preserve stationarity. This condition is motivated by a question about the *non-stationary ideal,* the ideal of non-stationary sets (denoted $\ell _{\text{NS}}$ or NS$_{\omega _{1}}$): whether it is $\omega _{2}$-saturated, i.e. whether every $\omega _{2}$-sequence of stationary sets contains at least two members intersecting again in a stationary set. If so, then the Boolean algebra $\wp (\omega _{1})/\ell _{\text{NS}}$ is complete and satisfies the $\omega _{2}$-chain condition. MM implies this. Woodin \[Woo1,2\] has forcefully argued for a canonical model where CH fails (cf. Coda); it is a forcing extension of $L(\mathbb{R}),$ i.e. of the Hajnal ‘constructible closure’ of $\mathbb{R}$ (the class of sets constructible from some real in $V\ $– \[Dra, Ch. 5 §6.1\], cf. \[Kan, Ch. 1, §3\]; this is not to be confused with the Lévy class of sets ‘constructible relative to a given set’ \[Dra, Ch. 5 §6.2\], which occurs in §5 in the shape of $L[\mathcal{U}]$ with distinct notation). **7. Suslin, Luzin, Sierpiński and their legacy: infinite games and large cardinals** After the (necessarily) extensive excursion into logic and model theory, we now re-anchor all this to analytic practice. Henceforth, we intertwine these two aspects. For the Analysts’s point of view of set theory, we can do no better at this point than to cite C. A. (Ambrose) Rogers, a modern-day analyst par excellence (with a pedigree of: Geometry of Numbers, Discrete geometry, Convexity, Hausdorff measures, Topological descriptive set theory). In his last phase (post 1960), Rogers famously ‘would often give talks entitled Which sets do we need?, his answer being: analytic sets’ (cited from \[Ost7\]). To these we now turn. For background here, see \[Rog\]. **7a. Analytic sets** Analytic subsets of $\mathbb{R}$ are precisely the sets that arise as projections of planar Borel sets. Their initial (‘classical’) study, principally by Suslin, Luzin and Sierpiński, was prompted by Lebesgue’s erroneous assertion, in the course of his research on functions that are ‘analytically representable’, that these projections were Borel. But they need not be, as was first observed by Suslin in 1916. Indeed, an analytic set is Borel iff its complement is also Borel \[Sou\]. Until that moment the typical sets considered by analysts were Borel. Fortunately for Lebesgue’s research goals, analytic sets are extremely well-behaved: in the first place projections of analytic sets are inevitably analytic, and furthermore they have the following three regularity properties (‘the classical regularity properties’ below): they are measurable \[Lus\], they have the property of Baire \[Nik\], and likewise the perfect-set property \[Ale\] (they are either countable or contain a perfect set), and in certain circumstances are well approximable from within by compact subsets (they are ‘capacitable’ – a property discovered independently by R. O. Davies \[Dav\] in 1952 and in a general topological context by G. Choquet in 1952 \[Cho1-4\]). The newly discovered sets emerged as the first-level sets of the *projective hierarchy* (also called the *Luzin hierarchy*) generated from the Borel sets by alternately applying the operation of projection and complementation (a fact later recognized also through the analysis of their logical complexity: counting how many *alternations* of existential and universal quantifiers over the reals are needed to define them, and identifying which the *preliminary* quantifier is: existential or universal). However, the very successful classical study of analytic sets struggled to promote much of the ‘good behaviour’ up the hierarchy. At the margins, of particular interest, was Kôndo’s uniformization theorem of 1939 (that a co-analytic planar set has a co-analytic *uniformization*, i.e. contains a co-analytic graph selecting one point from each vertical section). The message from set theory in Gödel’s inner universe of sets $L$ was particularly depressing: Kôndo’s theorem implied the existence in $L$ of an analytic sets whose complement failed to have the perfect-set property (the culprit was the well-ordering of $L$, which relative to $L$ lies at the second projective level). Further progress seemed doomed. But an unlikely development, in the shape of a game-theoretic rival to AC, unblocked the log-jam. However, it was left to a later generation to pore over the classical achievements to extract the necessary inspiration from the classicists by drawing in a further theme: the Banach-Mazur games. To explain this development we need to explore some analytic-set theory. Suslin’s characterization \[Sou\] in 1917 of analytic sets $S\subseteq \mathbb{R}$ asserts they may be represented in the form$$S=\dbigcup\nolimits_{\mathbf{i}\in \mathbb{N}^{\mathbb{N}}}F(\mathbf{i})=\dbigcup\nolimits_{\mathbf{i}\in \mathbb{N}^{\mathbb{N}}}\dbigcap\nolimits_{n=1}^{\infty }F(\mathbf{i}|n),$$where each of the *determining sets* $F(\mathbf{i}|n)$ is closed and of diameter at most $2^{-n}$ – so that $F(\mathbf{i})$ has at most one member; here$$\mathbf{i}|n:=(i_{0},...,i_{n-1}).$$(For this reason, the operation taking a determining system to the set $S$ above is now usually called the Suslin operation, though it is sometimes called the $A$-operation as in \[Kur\], apparently named for P. S. Alexandrov, who had devised it to construct perfect subsets of uncountable Borel sets \[Ale\].) Implicit in the formula is an operation on the determining system of sets $\langle F(\mathbf{i}|n):\mathbf{i}|n\in \mathbb{N}^{<\mathbb{N}}\rangle ,$ which includes countable intersection and countable union (and preserves analyticity if the determining system comprises analytic sets \[Rog, Part 1, §2.3\]). This goes beyond countable union seemingly towards a continuum union, but one that is constrained by the upper (h)semi-continuity of the map $\mathbf{i\mapsto }F(\mathbf{i})$. Under this ‘continuous union’ lie hidden the countable ordinals, by virtue of the countable tree $T$ of all finite sequences $\mathbf{i}|n$ (ordered by sequence extension). For any $x$ the associated subtree $$T_{x}:=\{\mathbf{i}|n:x\in F(\mathbf{i}|n)\}$$is well-founded iff $x\notin S,$ as then $T_{x}$ has no paths (infinite branches); indeed $x\notin F(\mathbf{i})$ for all $\mathbf{i}$. (This tree idea, with the $\mathbf{i}|n$ replaced by rationals, goes back, albeit under the name ‘sieve’, to Lebesgue’s construction of a measurable set that is not Borel.) The overall complexity of the subtree may then be measured by a countable ordinal, known as the *Luzin-Sierpiński index* of the tree $T_{x}$ (or of the point $x$) – \[LusS\]. This is obtained rather as the Cantor-Bendixon index of a scattered set is obtained by the repeated (inductive) removal of isolated points, except that here one removes at each stage the terminal nodes of a tree. (A moment’s reflection shows this corresponds to a linear ordering of the finite sequences, akin to lexicographic but adjusted to allow shorter sequences to preceed their longer extensions, such that the tree is well-ordered iff it is well-founded: this is the *Kleene-Brouwer order*.) When the determining system of $S$ (i.e. the family of sets $F(\mathbf{i}|n)$ above) consists of closed sets, it readily follows, via its countable transfinite definition, that the set of points $x$ in the complement of $S$ with index bounded by $\alpha <\omega _{1}$ is Borel. It is also immediate that the complement of an analytic set is a union of $\omega _{1}$ Borel sets, since the index is bounded by $\omega _{1}$. The important *boundedness property* of the index (that it remains bounded over any analytic set $S^{\prime }$ in the complement of $S$ by a corresponding countable ordinal, a matter that hinges on the ‘continuous union’ aspect) leads to a proof of the *First* *Separation Theorem*: disjoint analytic sets may be covered by disjoint Borel sets. From here, as an immediate corollary, an analytic set with analytic complement is Borel. **7b. Banach-Mazur games and the Luzin hierarchy** We recall that a Banach-Mazur game with target set $S\subseteq \mathbb{R}$ is an infinite positional game which may be viewed as played by two players ‘alternately picking ad infinitum’ the digits of a decimal expansion of a real number – but this needs the interpretation that each player selects a function (a strategy) determining that player’s choice of next digit, given the current position – with the first player declared the winner iff the real number generated from the play of the two strategies falls in $S,$ and otherwise the second. The target set $S\ $is said to be *determined* if one or other of the players has a winning strategy. Mazur proposed the game (this is Problem 43 in the Scottish Book, \[Mau\]), and Banach responded in 1935 by characterizing determinacy by the property of Baire. See \[GalMS\] for an alternative infinite game which offers a measure-theoretic result as a contrast to Banach’s category result. It is clear from its description that the game offers a natural interpretation for a sequence of choices in a manner related to the countable axiom of choice. In 1962 Mycielski and Steinhaus \[MycS\] proposed the Axiom of Determinacy, AD, as an alternative to AC – in essence setting the task of ascertaining its consistency relative to ZF. See \[Myc\] for an account of the consequences of AD current in 1964, making the case that, in a hoped-for subuniverse of sets in which AD holds, the well-known ‘paradoxes’ (Hausdorff, Banach-Tarski, ...) flowing from AC would be ruled out, while at the same time preserving standard analysis in $\mathbb{R}$ (since ‘countable choice’ for a countable family with union at most a continuum of members follows from AD – so, in view of the continuum restriction, it is usual to work with AD+DC). We may pass now to a generalization of Suslin’s representation for analytic sets, which enabled higher-level analogues of the classical regularity properties. Interpreting $\mathbb{N}^{\mathbb{N}}$ as the set of irrationals (via continued fraction expansion), we may w.l.o.g. assume that $S\subseteq \mathbb{N}^{\mathbb{N}}.$ This carries the simplifying advantage that, ignoring a countable set of lines, we may easily identify planar sets, regarded as lying in $\mathbb{N}^{\mathbb{N}}\times \mathbb{N}^{\mathbb{N}},$ with subsets of $\mathbb{N}^{\mathbb{N}}$ (merging a pair $(x,y)$ into a single sequence $\langle x,y\rangle $) and so regard projection as an operation from $\mathbb{N}^{\mathbb{N}}$ to $\mathbb{N}^{\mathbb{N}}.$ Replacing $F(\mathbf{i}|n)$ by its $2^{-n}$ open swelling $S(\mathbf{i}|n)$ yields that $s\in S$ iff for some $\mathbf{i}\in \mathbb{N}^{\mathbb{N}}$$$s|n\in S(\mathbf{i}|n)\qquad (n\in \mathbb{N)};$$here we interpret $s|n$ as a (rational) point of $\mathbb{R}$ (and implicitly refer to the metric of first difference: $d(x,y)=2^{-n},$ when $x,y$ differ first in their $n^{\text{th}}$ term). We can tidy up further while working in $\mathbb{R},$ by assuming compact $F(\mathbf{i}|n)$ and replacing $S(\mathbf{i}|n)$ with a union of a finite number of rational-ended closed intervals. Coding such finite unions by $\mathbb{N}$, we arrive at a reformulation of Suslin’s characterization: for $T\ $a tree of finite (pairs $(u,v)$ of ) sequences, define the projection of $T$ into $\mathbb{N}^{\mathbb{N}}$ by $$p(T):=\{x\in \mathbb{N}^{\mathbb{N}}:(\forall n)[(x|n,\mathbf{i}|n)\in T]\};$$then $S\ $is analytic iff $S=p(T)$ for some appropriate tree $T$ of finite sequences of elements of $\mathbb{N}\times \mathbb{N}$. The generalization to a $\gamma $*-Suslin* set for ordinals $\gamma $ is obtained by taking trees $T$ of finite sequences of elements from $\mathbb{N}\times \gamma ,$ and provides the context allowing the regularity properties of category and measure to be lifted up the projective hierarchy. A $\gamma $-Suslin set is said to be a *homogeneously Suslin set* if there is an $\omega _{1}$-complete ultrafilter $\mathcal{U}_{x|n}$ for each $x|n$ such that for all $n$$$\{\mathbf{i}|n\in \gamma ^{n}:(x|n,\mathbf{i}|n)\in T\}\in \mathcal{U}_{x|n}$$(membership witnessed via a ‘large’ set of nodes), and $$p(T)=\{x\in \mathbb{N}^{\mathbb{N}}:(\forall A_{n}\in \mathcal{U}_{x|n})\text{ }\exists \mathbf{i}\text{ }\forall n\text{ }[\mathbf{i}|n\in A_{n}]\}$$(projection equivalent to passage through a ‘large’ sets of nodes at each height/level; the sequence $\langle \mathcal{U}_{x|n}:n\in \mathbb{N}\rangle $ is then said to be countably complete). In using the index set $\gamma ^{<\omega }$ these generalizations sound muted echoes of the non-separable theory of analytic sets (pioneered by A. H. Stone and R. W. Hansell – see \[Sto\] and \[Ost3\]). Martin, generalizing \[Mar\], shows in \[MarSt, Th. 2.3\] that homogeneously $\gamma $-Suslin sets are determined (as well as having the classical regularity properties), and that if Ramsey cardinals exist, then co-analytic sets are homogenously Souslin. This last result is a re-interpretation of Martin’s earlier theorem \[Mar\] that if there is a Ramsey cardinal (e.g. if there is a measurable cardinal), then analytic games are determined. Two features of the analysis of a co-analytic set $C$ via the Luzin-Sierpiński index are of great significance to the study of projective sets. First, the index maps to the ordinals, i.e. into a well-ordered set, and so the index induces a prewellordering, rather than a well-ordering on the set $C$ (as distinct points of $C$ may be mapped to the same ordinal). Secondly, denoting the index by $\rho ,$ the relation$$R^{+}(x,y):=x\in C\text{ and }\rho (x)\leq \rho (y),$$and its negation $R^{-}(x,y)$ are both Borel, and so both co-analytic. Taking an abstract viewpoint, a class $\Gamma $ of sets in $\mathbb{N}^{\mathbb{N}}$ may be said to have the *prewellordering property* if for every set $C\in \Gamma $ there is a map $\rho :C\rightarrow On$ such that both of $R^{\pm }(x,y)$ are in $\Gamma .$ (The map is then called a $\Gamma $-*norm*.) Suppose that the complementary class $\check{\Gamma} $ (i.e. of sets with complement in $\Gamma $) is, like the analytic sets, closed under projection; then the class of sets $\exists ^{1}\Gamma $ obtained as the projections of sets in $\Gamma $ also has the prewellordering property. This would have been clear to Luzin and Sierpiński; but, with the introduction of determinacy, a new feature arises: *The First Periodicity Theorem* (\[Mar\], \[Mos2\]): For a class of sets $\Gamma $ for which the sets in the* ambiguous class* $\Delta _{\Gamma }:=\Gamma \cap \check{\Gamma}$ are determined: for every $C\in \Gamma ,$ if $C\ $admits a $\Gamma $-norm, then $\{y:\forall x[\langle x,y\rangle \in C]\}$ admits a norm in the class of sets $\forall ^{1}\exists ^{1}\Gamma ,$ i.e. in the class of sets of the form $\forall x\exists y[\langle x,y\rangle \in C^{\prime }]$ for some $C^{\prime }$ in $\Gamma .$ Thus, in particular: inductively, if the $\mathbf{\Sigma }_{2n}^{1}$-class (for the $\mathbf{\Sigma }$ and $\mathbf{\Pi }$ notation of the projective hierarchy, again see §9) has the prewellordering property, then so does the $\mathbf{\Pi }_{2n+1}^{1}$-class, assuming determinacy of the ambiguous class $\mathbf{\Delta }_{2n}^{1}$. The $\mathbf{\Pi }_{2n+1}^{1}$-class yields quite directly a prewellordering for the class $\mathbf{\Sigma }_{2n+2}^{1}:$ if $A(x)\equiv (\exists y)[\langle x,y\rangle \in C]$ for $C$ in $\mathbf{\Pi }_{2n+1}^{1}$ with norm $\rho _{C},$ then a norm (of the corresponding class) for $A$ may be defined by $$\rho _{A}(x):=\min \{\rho _{C}(x,y):\langle x,y\rangle \in C\}.$$Thus the prewellordering property ‘zig-zags’ between the $\mathbf{\Pi }$ and $\mathbf{\Sigma }$ classes. Part of the motivation to take a game-theoretic approach to the projective sets was the appearance in 1967 of a new proof of the earlier mentioned Suslin separation theorem for analytic sets given by David Blackwell \[Blac\] on the basis of the Gale-Stewart proof of the determinacy of open sets \[GalS\] of 1953. The wealth of insights thereafter is history: witness the very title of Mathias’s ‘Surrealist landscape with figures’ survey \[Mat\], capturing the spirit of the time. It was a careful reading of Kôndo’s proof of the uniformization of $\mathbf{\Pi }_{1}^{1}$-sets by a $\mathbf{\Pi }_{1}^{1}$ graph that initially led Moschovakis to isolate a more general kind of $\Gamma $-norm: that of a $\Gamma $-scale which refers to an $\omega $-sequence of $\Gamma $-norms $\rho _{m}$ defined on a set $C$ of $\Gamma $ with associated relations $R^{\pm }(m,x,y)$ in $\Gamma $ (as with the single $\Gamma $-norm above), but with an additional ‘convergence-guiding’ property: for any sequence $c_{n}\in C$ with $c_{n}\rightarrow c_{0}$, if for each $m$ $$\langle \rho _{m}(c_{n}):n\in \omega \rangle \text{ is eventually a constant, }\lambda _{m}\text{ say,}$$then $c_{0}\in C$ and $\rho _{m}(c_{0})\leq \lambda _{m}$ for all $m.$ (See e.g. \[MarK, §8.2\].) Mutatis mutandis, the Moschovakis *Second Periodicity Theorem* \[Mos2\] has the same form as the First but with $\Gamma $-scale replacing $\Gamma $-norm throughout. Analogously, the Second Theorem implies that the Kôndo uniformization property likewise zigzags between the $\mathbf{\Pi }$ and $\mathbf{\Sigma }$ classes – see \[Mos1\]. Guided by the original $\mathbf{\Pi }_{1}^{1}$-norm (the Luzin-Sierpiński index), having range in $\omega _{1}$ (less, if the $\mathbf{\Pi }_{1}^{1}$ set in question is Borel), one defines the *projective ordinal* of level $n$ by reference to the sets in the ambiguous class $\mathbf{\Delta }_{n}^{1}$$$\mathbf{\delta }_{n}^{1}:=\text{supremum of the lengths of prewellorderings in }\mathbf{\Delta }_{n}^{1}.$$Martin showed that $\mathbf{\delta }_{2}^{1}\leq \omega _{2},$ with equality implied under AD by the Moschovakis result that $\mathbf{\delta }_{n}^{1}$ for $n\geq 1$ is a cardinal and that, under PD, $\mathbf{\delta }_{2n}^{1}<\mathbf{\delta }_{2n+2}^{1}.$ Under AD+DC $\mathbf{\delta }_{2n}^{1}=(\mathbf{\delta }_{2n-1}^{1})^{+}$ (i.e. the even-indexed ordinal is the successor of the preceding odd-indexed one); furthermore, Jackson’s theorem \[Jac1,2\] asserts that under AD+DC$$\mathbf{\delta }_{2n-1}^{1}=\aleph _{w(2n-1)+1},$$where $w$ is defined via iterated exponentiation: inductively by $w(m+1)=\omega ^{w(m)}$ with $w(1)=\omega .$ A concerted effort to assess the consistency strength of the determinacy assumption for $\mathbf{\Delta }_{n}^{1}$ ultimately led to the result that this is equiconsistent with the existence of $n$ Woodin cardinals below a measurable cardinal. **8. Shadows** Here we wrap up our survey of the set-theoretical domain. We have seen how combinatorial properties, some ‘high up’, in Cantor’s world affect properties of the real line down below. When powerful axioms extend familiar properties in desirable ways one is led to ask whether one can get away with less and get if not the same outcome, then ‘almost’ the same (in some sense). To this end Mycielski and Tomkowicz \[MycT\] speak in very suggestive language of *shadows* of AC in their chosen setting of $L(\mathbb{R}), $ a model of set theory that resolves some of the hardest set-theory problems. Their quest is theorems of ZFC that have corollaries that are theorems of ZF+AD – see \[MycT\]. In $L(\mathbb{R})$ AD implies DC \[Kec1\], and the present authors have come to view DC as a natural ally for analysis. We give our favourite example of this, and then, after a brief review of syntactical teminology in §9, we survey in §10 results which give further succour, if one is willing in the interests of plurality to conduct mathematics in an appropriate helpful (indeed playful, to borrow the term from \[Mos1\], when games are enlisted) subuniverse. *An example with the Principle of D*ependent Choice DC in mind.** We begin with an example concerned with real-valued sublinear functions on $\mathbb{R}$ which ‘almost’ follow Banach’s enduring paradigmatic definition. They are subadditive, i.e. satisfying $f(x+y)\leq f(x)+f(y),$ but in one variant they are $\mathbb{N}$-homogeneous in the sense that $f(nx)=nf(x)$ for $n=0,1,2,...$, so $\mathbb{Q}_{+}$-homogeneous, and for all $x.$ In other variants the quantification over $x$ may also be thinned – see \[BinO6\]. In electing to study sublinear functions as possible realizations of norms, Berz (\[Berz\], \[BinO6\]) showed, for measurable $f,$ that the graph of $f$ is conical – comprises two half lines through the origin; however, his argument relied on AC, in the usual form of *Zorn’s Lemma*, which he used in the context of $\mathbb{R}$ over the field of scalars $\mathbb{Q}$. In spirit he follows Hamel’s construction of a discontinuous additive function, and so ultimately this rests on transfinite induction of *continuum* length requiring *continuum* many selections. Our own proof \[BinO6\] (cf. \[BinO7, 10\]) of Berz’s theorem, taken in a wider context including Banach spaces, depends in effect on the Baire Category Theorem BC, or the completeness of $\mathbb{R}$ (in either of the distinct roles of ‘Cauchy-sequential’ and ‘Cauchy-filter’ completeness, the latter stronger in the absence of AC, see \[FosM, §3\] and also \[DodM, §7, §2\]): we rely on generalizations of the *Kestelman-Borwein-Ditor Theorem*, KBD, asserting that for any (category/measure theoretic) non-negligible set $T\ $and any null sequence $z_{n}\rightarrow 0,$ for quasi all $t\in T$ the $t$-translate of some subsequence $z_{n(m)}$ (dependent on $t$) embeds in $T,$ i.e. $t+z_{n(m)}\in T.$ See \[MilO\] for a discussion of this ‘shift-compactness’ notion. KBD is a variant of BC. So the proof ultimately rests on elementary induction via the *Axiom (Principle) of Dependent Choice(s)* DC (thus named in 1948 by Tarski \[Tar2, p. 96\] and studied in \[Most\], but anticipated in 1942 by Bernays \[Ber, Axiom IV\*, p. 86\] – see \[Jec1, §8.1\], \[Jec2, Ch. 5\]); DC in turn is equivalent to BC by a result of Blair \[Bla\]. (For further results in this direction see also \[Pin3,4\], \[Gol\], \[HerK\], \[Wol\], and the textbook \[Her\].) The relevance of KBD in the setting of a Polish group comes from its various corollaries which include the Steinhaus-Weil Interior points Theorem \[BinO9\], the Open Mapping Theorem and its generalization to group actions: the Effros Theorem – see \[vMil\], \[Ost4,5,6\]. For a target set $T$ that is a dense $G_{\delta }$, embeddings which are performed simultaneously in any neighbourhood by a perfect subset of $T$ of a fixed set $Z$ (not necessarily a null sequence) into $T$ characterize those sets $Z$ that are strong measure zero – see\[GalMS\]. We note that DC is equivalent to a statement about trees: a pruned tree has an infinite branch (for which see \[Kec2, 20.B\]) and so by its very nature is an ingredient in set-theory axiom systems which consider the extent to which Banach-Mazur-type games (with underlying tree structure) are determined. The latter in turn have been viewed as generalizations of Baire’s Theorem ever since Choquet \[Cho5\] – cf. \[Kec2, 8C,D,E\]. Inevitably, determinacy and the study of the relationship between category and measure go hand in hand. **9. The syntax of Analysis: Category/measure regularity versus practicality** The Baire/measurable property discussed at various points above is usually satisfied in mathematical practice. Indeed, any analytic subset of $\mathbb{R}$ possesses these properties (\[Rog, Part 1 §2.9\], \[Kec2, 29.5\]), hence so do all the sets in the $\sigma $-algebra that they generate (the $C$-sets, \[Kec2, §29.D\], $C$ for *criblé* – see \[Bur1, 2\], cf. \[BinO4\]). There is a broader class still. Recall first that an analytic set may be viewed as a projection of a planar Borel set $P,$ so is definable as $\{x:\Phi (x)\}$ via the $\mathbf{\Sigma }_{1}^{1}$ formula $\Phi (x):=(\exists y\in \mathbb{R})[(x,y)\in P];$ here the notation $\mathbf{\Sigma }_{1}^{1}$ indicates one quantifier block (the subscripted value) of existential quantification, ranging over reals (type 1 objects – the superscripted value). Use of the *bold-face version* of the symbol indicates the need to refer to *arbitrary* coding (by reals not necessarily in an *effective* manner, for which see \[Gao, §1.5\]) of the various open sets needed to construct $P.$ (An open set $U$ is *coded* by the sequence of rational intervals contained in $U.)$ Effective variants are rendered in light-face. Consider a set $A$ such that both $A$ and $\mathbb{R}\backslash A$ may be defined by a $\mathbf{\Sigma }_{2}^{1}$ formula, say respectively as $\{x:\Phi (x)\}$ and $\{x:\Psi (x)\}$, where $\Phi (x):=(\exists y\in \mathbb{R})(\forall z\in \mathbb{R})(x,y,z)\in P\}$ now, and similarly $\Psi .$ This means that $A$ is both $\mathbf{\Sigma }_{2}^{1}$ and $\mathbf{\Pi }_{2}^{1}$ (with $\mathbf{\Pi }$ indicating a leading universal quantifier block), and so is in the ambiguous class $\mathbf{\Delta }_{2}^{1}.$ If in addition the equivalence$$\Phi (x)\Longleftrightarrow \lnot \Psi (x)$$is provable in $ZF,$ i.e. *without reference* to $AC,$ then $A$ is said to be *provably* $\mathbf{\Delta }_{2}^{1}.$ It turns out that such sets have the Baire/measurable property – see \[FenN\], where these are generalized to the *universally (=absolutely) measurable* sets (cf. \[BinO6, §2\]); the idea is ascribed to Solovay in \[Kan, Ch. 3 Ex. 14.4\]. How much further this may go depends on what axioms of set theory are admitted, a matter to which we now turn. Our interest in such matters derives from the *Character Theorems* of regular variation, noted in \[BinO3, §3\] (revisited in \[BinO5, §11\]), which identify the logical complexity of the function $$h^{\ast }(x):=\lim \sup_{t\rightarrow \infty }h(t+x)-h(t),$$which is $\mathbf{\Delta }_{2}^{1}$ if the function $h$ is Borel (and is $\mathbf{\Pi }_{2}^{1}$ if $h$ is analytic, and $\mathbf{\Pi }_{3}^{1}$ if $h$ is co-analytic). We argued in \[BinO3, §5\] that $\mathbf{\Delta }_{2}^{1}$ is a natural setting in which to study regular variation. **10. Category-Measure duality** *1. Practical axiomatic alternatives: LM, PB, AD, PD.* While ZF is common ground in mathematics, AC is not, and alternatives to it are widely used, in which for example all sets are Lebesgue-measurable (usually abbreviated to LM) and all sets have the Baire property, sometimes abbreviated to PB (as distinct from BP to indicate individual ‘possession of the Baire property’). One such is DC above. As Solovay \[Sol3, p. 25\] points out, this axiom is sufficient for the establishment of Lebesgue measure, i.e. including its translation invariance and countable additivity (“...positive results ... of measure theory...”), and may be assumed together with LM. Another is the *Axiom of Determinacy* AD mentioned above and introduced by Mycielski and Steinhaus \[MycS\]; this implies LM, for which see \[MycSw\], and PB, the latter a result, mentioned in §7, due to Banach – see \[Kec2, 38.B\]. Its introduction inspired remarkable and still current developments in set theory concerned with determinacy of ‘definable’ sets of reals (see \[ForK\] and particularly \[Nee\]) and consequent combinatorial properties (such as partition relations) of the alephs (see \[Kle\]); again see §7. Others include the (weaker) *Axiom of Projective Determinacy* PD \[Kec2, § 38.B\], cf. §7, restricting the operation of AD to the smaller class of projective sets. (The independence and consistency of DC versus AD was established respectively in Solovay \[Sol4\] and Kechris \[Kec2\] – see also \[KecS\]; cf. \[DalW1\], \[Ost2\].) *2. LM versus PB.* In 1983 Raissonier and Stern \[RaiS, Th. 2\] (cf. \[Bart1,2\]), inspired by then current work of Shelah (circulating in manuscript since 1980) and earlier work of Solovay, showed that *if every* $\Sigma _{2}^{1}$* set is Lebesgue measurable, then every* $\Sigma _{2}^{1}$* set has BP*, whereas the converse fails – for the latter see \[Ster\] – cf. \[BartJ, §9.3\] and \[Paw\]. This demonstrates that measurability is in fact the stronger notion – see \[JudSh, §1\] for a discussion of the consistency of analogues at level 3 and beyond – which is one reason why we regard category rather than measure as *primary*. For we have seen above how the category version of Berz’s theorem implies its measure version; see also \[BinO6, 10\]. Note that the assumption of Gödel’s *Axiom of Constructibility* $V=L,$ a strengthening of AC, yields $\mathbf{\Delta }_{2}^{1}$ non-measurable subsets, so that the Fenstad-Normann result on the narrower class of provably $\mathbf{\Delta }_{2}^{1}$ sets mentioned in §9 above marks the limit of such results in a purely ZF framework (at level 2). *3. Consistency and the role of large cardinals.* While LMand PB are inconsistent with AC, such axioms can be consistent with DC. Justification with scant exception involves some form of large-cardinal assumption, which in turn, as in §4, calibrates relative consistency strengths – see \[Kan\] and \[KoeW\] (cf. \[Lar\] and \[KanM\]). Thus Solovay \[Sol3\] in 1970 was the first to show the equiconsistency of ZF+DC+LM+PB with that of ZFC+‘*there exists an inaccessible cardinal*’. The appearance of the inaccessible in this result is not altogether incongruous, given its emergence in results (from 1930 onwards) due to Banach \[Ban\] (under GCH), Ulam \[Ula\] (under AC), and Tarski \[Tar1\], concerning the cardinalities of sets supporting a countably additive/finitely additive \[0,1\]-valued/$\{0,1\}$-valued measure (cf. \[Bog, 1.12(x)\], \[Fre2\]). Later in 1984 Shelah \[She1, 5.1\] showed in ZF+DC that already the measurability of all $\mathbf{\Sigma }_{1}^{3}$ sets implies that $\aleph _{1}^{L}$ is inaccessible (the symbol $\aleph _{1}^{L}$ refers to the substructure/ subuniverse of constructible sets and denotes the first uncountable ordinal therein – cf. §2). As a consequence, Shelah \[She1,5.1A\] showed that ZF+DC+LM is equiconsistent with ZF+‘*there exists an inaccessible*’, whereas \[She1, 7.17\] ZF+DC+PBis equiconsistent with just ZFC (i.e. without reference to inaccessible cardinals), so driving another wedge between classical measure-category symmetries (see \[JudSh\] for further, related ‘wedges’). The latter consistency theorem relies on the result \[She1, 7.16\] that any model of ZFC+ CH has a generic (forcing) extension satisfying ZF+ ‘*every set of reals (first-order) defined using a real and an ordinal parameter has BP*’. (Here ‘first-order’ restricts the range of any quantifiers.) For a topological proof see Stern \[Ster\]. *4. LM versus PB continued.* Raisonnier \[Rai, Th. 5\] (cf. \[She1, 5.1B\]) has shown that in ZF+DC one can prove that if there is an uncountable well-ordered set of reals (in particular a subset of cardinality $\aleph _{1}$), then there is a non-measurable set of reals. (This motivates Judah and Spinas \[JudSp\] to consider generalizations including the consistency of the $\omega _{1}$-variant of DC.) See also Judah and Rosłanowski \[JudR\] for a model (due to Shelah) in which ZF+DC+LM+$\lnot $PBholds, and also \[She2\] where an inaccessible cardinal is used to show consistency of ZF+LM+$\lnot $PB+‘*there is an uncountable set without a perfect subset*’. For a textbook treatment of much of this material see again \[BartJ\]. Raisonnier \[Rai, Th. 3\] notes the result, due to Shelah and Stern, that there is a model for ZF+DC+PB+$\aleph _{1}=\aleph _{1}^{L}$+ ‘*the ordinally definable subsets of reals are measurable*’. So, in particular by Raisonnier’s result, there is a non-measurable set in this model. Shelah’s result indicates that the non-measurable is either $\Sigma _{3}^{1}$ (light-face symbol: all open sets coded effectively) or $\mathbf{\Sigma }_{2}^{1}$ (bold-face). Thus here PB+$\lnot $LM holds. *5. Regularity of reasonably definable sets.* From the existence of suitably large cardinals flows a most remarkable result due to Shelah and Woodin \[SheW\] justifying the opening practical remark about BP, which is that every ‘reasonably definable’ set of reals is Lebesgue measurable: compare the commentary in \[BecK\] following their Th 5.3.2. This is a latter-day sweeping generalization of a theorem due to Solovay (cf. \[Sol2\]) that, subject to large-cardinal assumptions, $\mathbf{\Sigma }_{2}^{1}$ *sets are measurable* (and so also *have BP* by \[RaiS\]). **Coda** To return to the algebraic characterization of the reals as ‘the’ complete archimedean ordered field: it is the ‘complete’ which hides the ‘modulo cardinality’ and ‘modulo which sets are available’ aspects. It is always good to look at familiar mathematics, and ask oneself the analogous question in that context. As working analysts ourselves, we feel for those of our colleagues new to these matters, who may look fondly back to an age of ‘bygone innocence’, when ‘one didn’t need to worry about such things’. We prefer instead to marvel at the unfathomable richness of mathematics. As usual, Shakespeare puts his finger on it somewhere: [*There are more things in heaven and earth, Horatio,*]{}\ [*Than are dreamt of in our philosophy.*]{} So we have only mathematical ‘gut-feeling and belief’, as with Mickiewicz: [*Czucie i wiara silniej mówi do mnie*,]{}\ [*Niż mdrca szkiełko i oko.*]{}\ – ‘Feeling and faith more forcefully persuade, Than the lens and the eye of a sage’. Thus it is that we close with two ‘high-profile’ attitudes towards Solovay’s dictum that the continuum ‘can be anything it ought to be’, to both of which Woodin has contributed. 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Model details ============= For all datasets, we use [spaCy](https://spacy.io/) for tokenization. We map out of vocabulary words to a special $\texttt{<unk>}$ token and map all words with numeric characters to ‘qqq’. Each word in the vocabulary was initialized to pretrained embeddings. For general domain corpora we used either (i) [FastText Embeddings](https://s3-us-west-1.amazonaws.com/fasttext-vectors/wiki.simple.vec) (SST, IMDB, 20News, and CNN) trained on Simple English Wikipedia, or, (ii) [GloVe 840B](http://nlp.stanford.edu/data/glove.840B.300d.zip) embeddings (AGNews and SNLI). For the MIMIC dataset, we learned word embeddings using [Gensim](https://radimrehurek.com/gensim/) over all discharge summaries in the corpus. We initialize words not present in the vocabulary using samples from a standard Gaussian $\mathcal{N}$($\mu=0$, $\sigma^2=1$). BiLSTM ------ We use an embedding size of 300 and hidden size of 128 for all datasets except bAbI (for which we use 50 and 30, respectively). All models were regularized using $\ell_2$ regularization ($\lambda = 10^{-5}$) applied to all parameters. We use a sigmoid activation functions for binary classification tasks, and a softmax for all other outputs. We trained the model using maximum likelihood loss using the Adam Optimizer with default parameters in PyTorch. CNN --- We use an embedding size of 300 and 4 kernels of sizes \[1, 3, 5, 7\], each with 64 filters, giving a final hidden size of 256 (for bAbI we use 50 and 8 respectively with same kernel sizes). We use ReLU activation function on the output of the filters. All other configurations remain same as BiLSTM. Average ------- We use the embedding size of 300 and a projection size of 256 with ReLU activation on the output of the projection matrix. All other configurations remain same as BiLSTM. Graphs ====== To provide easy navigation of our (large set of) graphs depicting attention weights on various datasets/tasks under various model configuration we have created an interactive interface to browse these results, accessible at: <https://successar.github.io/AttentionExplanation/docs/>. Adversarial Heatmaps ==================== **SST** **Original**: **Adversarial**: **$\Delta\hat{y}$**: *0.005* **IMDB** **Original**: **Adversarial**: **$\Delta\hat{y}$**: *0.004* **20 News Group - Sports** **Original**: **Adversarial**: **$\Delta\hat{y}$**: *0.001* **ADR** **Original**: **Adversarial**: **$\Delta\hat{y}$**: *0.002* **AG News** **Original**: **Adversarial**: **$\Delta\hat{y}$**: *0.006* **SNLI** **Hypothesis**:a man is running on foot **Original Premise Attention**: **Adversarial Premise Attention**: **$\Delta\hat{y}$**: *0.002* **Babi Task 1** **Question**: Where is Sandra ? **Original Attention**: **Adversarial Attention**: **$\Delta\hat{y}$**: *0.003* **CNN-QA** **Question**:federal education minister @placeholder visited a @entity15 store in @entity17 , saw cameras **Original**: **Adversarial**: **$\Delta\hat{y}$**: *0.005*

--- abstract: 'With the advance of technology, entities can be observed in multiple views. Multiple views containing different types of features can be used for clustering. Although multi-view clustering has been successfully applied in many applications, the previous methods usually assume the complete instance mapping between different views. In many real-world applications, information can be gathered from multiple sources, while each source can contain multiple views, which are more cohesive for learning. The views under the same source are usually fully mapped, but they can be very heterogeneous. Moreover, the mappings between different sources are usually incomplete and partially observed, which makes it more difficult to integrate all the views across different sources. In this paper, we propose MMC (**M**ulti-source **M**ulti-view **C**lustering), which is a framework based on collective spectral clustering with a discrepancy penalty across sources, to tackle these challenges. MMC has several advantages compared with other existing methods. First, MMC can deal with incomplete mapping between sources. Second, it considers the disagreements between sources while treating views in the same source as a cohesive set. Third, MMC also tries to infer the instance similarities across sources to enhance the clustering performance. Extensive experiments conducted on real-world data demonstrate the effectiveness of the proposed approach.' author: - title: 'Multi-Source Multi-View Clustering via Discrepancy Penalty' --- Introduction ============ With the advance of technology, most of entities can be observed in multiple views. Multiple views containing different types of features can be used for clustering. Multiple view clustering [@MVC; @MVC_CCA; @MVC_co_reg] aims to enhance clustering performance by integrating different views. Moreover, as the information explodes, we can get information from multiple sources. Each source can contain multiple views that are fully aligned and available for clustering. Combining data from multiple sources, multiple views may help us get better clustering performance. However, several difficulties prevent us from combining different sources and views. First, the views may be very heterogeneous. Different views may have different feature spaces and distributions. Second, the instances mapping between different sources may be incomplete and partially observed. Different sources may have different instance sets, which means the instance mapping between different sources is not fully mapped. Also in real-world problems, the instance mapping is often partially observed. We may only get part of the instance mapping between different sources. Third, the views in one source are generally more cohesive than the views across different sources. This is very different from the traditional multi-view clustering problem. A good example is the social networks shown in Fig. \[sns\]. People usually use several social network services simultaneously, *e.g.*, those provided by Twitter and Foursquare. Each social network is an independent source containing several views that describe different aspects of the social network. We can use the profile information of users for both Twitter and Foursquare (view 3 in Fig. \[sns\]), the social connections of users (view 2 in Fig. \[sns\]), location check-in history, etc. Views in a single source describe the characters of the same set of users and focus on different aspects. The views in Twitter focus on the social activity aspects, while the views in Foursquare focus more on the location based aspects. Since not all people use both Twitter and Foursquare, the user mapping between Twitter and Foursquare is incomplete and not one-to-one. Furthermore, not all the shared users link their Twitter accounts with their Foursquare accounts. We can only observe part of the mapping between Twitter and Foursquare. Multi-view clustering [@MVC; @LongYZ08; @MVC_co_training] aims to utilize the multiple representations of instances in different features spaces to get better clustering performance. However, most of the previous methods are based on the assumption that all the views are fully mapped/aligned. They cannot deal with the multi-source multi-view scenario with incomplete/partial mapping across sources. Although there are some previous studies on dealing with multiple incomplete view clustering [@ShaoMVC; @PVC; @flexibleMVC; @PAKDDShao], none of them are suitable for multi-source multi-view scenario. They either cannot extend to more than two views or they do not treat the views in one source as a cohesive set. All of the previous methods only use the known mapping information. Furthermore, none of them try to extend the known mapping information to help improve clustering. ![An Example of Social Networks. Twitter and Foursquare are two sources. Twitter contains user connection view, user interaction view and user profile view, while Foursquare contains user connection view, user check-in location view, user profile view. The user mapping between Twitter and Foursquare is incomplete and partially observed. []{data-label="sns"}](sns.png){width="0.9\columnwidth"} These challenges and emerging applications call for novel clustering methods which can deal with multiple sources multiple views situations. In this paper, we propose MMC (**M**ulti-source **M**ulti-view **C**lustering) to integrate multiple sources for better clustering results. The main contributions of this paper are summarized as follows: 1. This paper is the first one to investigate the multi-source multi-view clustering problem, where multiple sources containing multiple views are available for clustering and the instance mappings between sources are incomplete and partially known. 2. We develop a method MMC, based on collective spectral clustering with discrepancy penalty within and between sources. MMC appreciates the cohesiveness of all the views in each source by pushing the latent feature matrice of the views in one source to a consensus for each source. MMC also considers the cross-source discrepancy by minimize the difference among the consensus latent feature matrices of different sources. 3. The proposed MMC not only generates clusters from multiple sources but also tries to infer the unknown instance similarity mapping between sources to improve the clustering performance. By using the learned consensus latent feature matrices of different sources and the incomplete/partial mappings across sources, MMC generates the instance similarities across different sources, which will in turn help enhance the clustering performance. 4. The proposed MMC is not limited to the multiple sources multiple views problem. In real-world applications, we may have multiple partial aligned views, *i.e.,* views with different numbers of instances. We can group the views into groups where each group has the same set of instances. Thus each group can be viewed as a source and our method can be applied. The experiment results on three groups of real data show the effectiveness of the proposed method by comparing it with other state-of-art methods. Problem Formulation =================== In this section, we will first define the problem of multi-source multi-view clustering. Then we will start from the single source problem to develop the objective function for multi-source multi-view clustering problem. Problem Definition ------------------ Before we define the problem of multi-source multi-view clustering, we summarize the notations in Table \[table\_symbol\]. Let $\mathcal{S} = \{S^k\}_{k=1}^K$ denote the set of the $K$ available sources. For each $S^k$, we have $S^k = \{X_i^k\}_{i=1}^{v_k}$, where $X_i^k$ denotes the $i$-th view in source $k$ and $v_k$ denotes the number of views in source $k$. We assume that source k $(1\le k\le K)$ contains $n_k$ instances. Let $ \mathcal{M} = \{M^{(i,j)}\}_{1\le i,j\le K}$ be the known instance mappings between sources, where $M^{(i,j)} \in \mathbb{R}_{+}^{n_i\times n_j}$ denotes the instance mapping between sources i and j, and its element is defined by $$M_{a,b}^{(i,j)} = \begin{cases} 1&\text{instance $ a $ in source $ i $ is mapped to } \\ & \text{instance $ b $ in source $ j $}.\\ 0&\text{otherwise.} \end{cases}$$ [max width=0.95]{} [|c|c|]{} Symbols & Description\ K & Number of sources.\ $v_k$ & Number of views for source $k$.\ $n_k$ & Number of instances in source $k$.\ $c_k$ & Number of clusters for source $k$.\ $X^{k}_i$ & i-th view in source $ k $\ $K^k_i$ & Kernel matrix for the $i$-th view in source $ k $\ $K_{U}$ & Kernel matrix of the matrix $ U $\ $L^k_i=(D^k_i)^{-1/2}K^k_i(D^k_i)^{-1/2}$ & ----------------------------------------------------------- Normalized graph Laplacian for the i-th view in source $k$, where $D^k_i$ is a diagonal matrix consisting of the row sums of $K_i^k$. ----------------------------------------------------------- : Summary of symbols and their meanings \ $M^{(i,j)}\in \mathbb{R}^{n_i\times n_j}$ & Instance mapping between source $ i $ and $ j $.\ $W^{(i,j)}\in \mathbb{R}^{n_i\times n_j}$ & Indicator matrix between source $ i $ and $ j $\ $\alpha^k_i$ & Importance of view $ i $ in source $ k $\ $\beta^{(i,j)}$ & Importance of penalty between source $ i $ and $ j $\ $U^k_i\in \mathbb{R}^{n_k\times c_k}$& Latent feature matrix for view $ i $ in source $ k $\ $U^{k*}\in \mathbb{R}^{n_k\times c_k}$ & Consensus latent feature matrix for source $ k $\ \[table\_symbol\] Our goal is to cluster the instances into $c_k$ clusters for each source k, while considering the other sources by using the cross-source mapping in $\mathcal{M}$. Single Source Multiple Views Clustering --------------------------------------- Clustering with multiple views within a single source can be seen as traditional multi-view clustering problem. However, in order to incorporate the cross-source disagreement, we would like to get a consensus clustering solution for each source. Let $K^k_i$ be the positive semi-definite similarity matrix or kernel matrix for view $ i $ in source $k$. The corresponding normalized graph Laplacian will be $L^k_i = {D^k_i}^{-1/2}K^k_i{D^k_i}^{-1/2}$, where $D^k_i$ is a diagonal matrix with the diagonal elements be the row sums of $K^k_i$. To perform spectral clustering for a single view $ i $ in source $ k $, as shown in [@MVC_co_reg; @SC_tutorial; @Andrew_sc], we only need to solve the following optimization problem for the normalized graph Laplacian $L^k_i$: $$\max_{U^k_i\in \mathbb{R}^{n_k\times c_k}}~ tr((U^k_i)^TL^k_iU^k_i), ~~s.t. (U^k_i)^TU^k_i = \mathit{I}, \label{single_view}$$ where $tr$ denotes the matrix trace, and $ U^k_i $ can be seen as a latent feature matrix of view $i$ in source $ k $, which can be given to the k-means algorithm to obtain cluster memberships. For clustering multiple views in the same source, we have to consider the disagreement between different views. We enforce the learned latent feature matrix for each view to look similar by regularizing them towards a common consensus. Similar to the regularization in [@MVC_co_reg], we define the discrepancy/dissimilarity between two latent feature matrices as: $$D(U^k_i, U^k_j) = \|K_{U^k_i}-K_{U^k_j}\|_F^2, \label{eq._in_source_disagree}$$ where $K_{U^k_i}$ and $K_{U^k_j}$ are the similarity/kernel matrices for $U^k_i$ and $U^k_j$, and $\|.\|_F$ denotes the Frobenius norm of the matrix. We use linear kernel as the similarity measure in Eq. (\[eq.\_in\_source\_disagree\]). Thus, we get $K_{U^k_i} = U^k_i(U^k_i)^T$ and $K_{U^k_j} = U^k_j(U^k_j)^T$. Using the properties of trace and the fact that $(U^k_i)^TU^k_i=\mathit{I}$, $(U^k_j)^TU^k_j=\mathit{I}$, Eq. (\[eq.\_in\_source\_disagree\]) can be rewritten as follows: $$\begin{adjustbox}{max width=1\columnwidth} $\displaystyle \begin{split} &D(U^k_i, U^k_j) = \|U^k_i(U^k_i)^T-U^k_j(U^k_j)^T\|_F^2 \\ &=tr\left(U^k_i(U^k_i)^T + U^k_j(U^k_j)^T - 2U^k_i(U^k_i)^TU^k_j(U^k_j)^T\right)\\ &=2c_k - 2tr(U^k_i(U^k_i)^TU^k_j(U^k_j)^T). \end{split} $ \end{adjustbox}$$ Ignoring the constant terms, we can get the discrepancy between two different latent feature matrices: $$D(U^k_i, U^k_j) = -tr(U^k_i(U^k_i)^TU^k_j(U^k_j)^T).$$ Then $ D(U^k_i, U^{k*}) = -tr(U^k_i(U^k_i)^TU^{k*}(U^{k*})^T)$ is the discrepancy between the $ i $-th view and the consensus. Considering the discrepancy/dissimilarity between each view and the consensus in the same source, the objective function for clustering all the views within the $ k $-th source will be as follows: $$\small \centering \begin{adjustbox}{max width=0.9\columnwidth} $\displaystyle \begin{split} \max_{\{U^k_{i}\},{U^{k*}}} \mathcal{J}_k &= \sum_{i=1}^{v_k}\left( tr({U^k_{i}}^TL^k_{i}U^k_{i}) + \alpha^k_{i}tr(U^k_{i}{U^k_{i}}^TU^{k*}{U^{k*}}^T)\right) ,\\ &\textit{s.t.}~~ {U^k_{i}}^TU^k_{i} = \mathit{I}, \forall 1\le i\le v_k, {U^{k*}}^TU^{k*} = \mathit{I} \end{split} $ \end{adjustbox} \label{eq:multi}$$ where $\mathcal{J}_k$ is the objective function for source $ k $, $U^k_i$ is the latent feature matrix for the $ i $-th view in source $ k $, $ U^{k*} $ is the consensus latent feature matrix for source $ k $, and $\alpha^k_i$ is the relative importance of view $ i $ in source $ k $. Multiple Sources Multiple Views Clustering ------------------------------------------ In this section, we will derive the objective function for Multi-source Multi-view Clustering problem. We model it as a joint matrix optimization problem. In order to incorporate multiple sources, we need to add penalty between sources to the single source multiple views clustering objective function. To appreciate the cohesiveness of the views within one source, we learn the consensus latent feature matrix for each source and only penalize the discrepancy between those consensus latent feature matrices. Similar to the discrepancy function across views within one source, the penalty function across sources should consider the discrepancy between the consensus clustering results for the sources. Since the mappings between sources are incomplete and partially known, we cannot directly apply the same penalty function as in the single source multiple views clustering objective function. However, by using the mapping matrices, we can project the learned latent feature matrix from one source to other sources, ${M^{(i,j)}}^TU^{i*}$ can be seen as projection of the instance in source $ i $ to the instances in source $ j $. The penalty function for discrepancy between source $ i $ and source $ j $ is as follow: $$\begin{adjustbox}{max width=0.85\columnwidth} $\displaystyle \widetilde{D}(U^{i*}, U^{j*}) = \|{M^{(i,j)}}^TU^{i*}({M^{(i,j)}}^TU^{i*})^T-U^{j*}{U^{j*}}^T\|_F^2. $ \end{adjustbox} \label{eq:penalty}$$ Observing that the known mapping between two sources is one-to-one and it is reasonable to assume that the unknown part is one-to-at-most-one, which means one instance in one source can be mapped to at most one instance in the other source. So we can approximately assume that $ M^{(i,j)}M^{(i,j)T} = I $ to help simplify the penalty function. The Eq. [\[eq:penalty\]]{} can be expressed as: $$\begin{adjustbox}{max width=0.872\columnwidth} $\displaystyle tr\left(\left({M^{(i,j)}}^TU^{i*}U^{i*T}M^{(i,j)} -U^{j*}U^{j*T}\right) \left({M^{(i,j)}}^TU^{i*}U^{i*T}M^{(i,j)} -U^{j*}U^{j*T}\right)\right)$ \end{adjustbox}$$ Using the fact that $ U^{i*T}U^{i*} = I $, $ tr(U^{i*T}U^{i*}) = c_i $, and $ M^{(i,j)}M^{(i,j)T} = I $, and ignoring the constant terms, the penalty function for discrepancy is $$\small \widetilde{D}(U^{i*}, U^{j*}) = -tr\left(U^{j*}U^{j*T}{M^{(i,j)}}^TU^{i*}U^{i*T}M^{(i,j)}\right) $$ Adding the penalty function between sources to the single source multiple views clustering objective function for all the sources, we get @size[7.5]{}@mathfonts $$\begin{split} \label{final} \max_{U^k_{i},U^{k*}(1\le k\le K, 1\le i\le v_k)} \mathcal{O} = \sum_{k=1}^{K} \mathcal{J}_k - \sum_{i\ne j, 1\le i,j\le K}\beta^{(i,j)}\widetilde{D}(U^{i*}, U^{j*}), \end{split} \notag$$ where $ \beta^{(i,j)} \ge 0$ is a parameter controlling the balance between the objective function for individual source and the inconsistency across sources. Optimization and MMC framework ============================== The proposed MMC framework simultaneously optimizes the latent featue matrices in multiple sources and infers the cross-source instance similarity mappings to help enhance the clustering performance. To optimize the objective function in Eq. (\[final\]), we employ an alternating scheme, that is, we optimize the objective function with respect to one variable while fixing others. Basically, we optimize the objective function using two stages. First, maximizing $\mathcal{O}$ over $U^k_{i}$s with fixed $U^{k*}$s. Second, maximizing $\mathcal{O}$ over $U^{k*}$s with fixed $U^k_{i}$s. We repeat these two steps, until it converges. Solving the optimization, we can iteratively learn the latent feature matrices for each source. However, when only a small portion of instances mapping between sources are observed, the clustering performance is affected by the incompleteness of the instance mapping across sources. Inferring the exact instance mapping is really challenging and usually additional information is required to infer such anchor link [@anchorlink; @Zhang:2014:MBM:2623330.2623645]. However, instead of inferring the exact instance mapping, we can try to infer the similarity mapping. This idea is based on the **Principle of Transitivity on Similarity**: Instances similar to the same instance in different sources should be similar. Fig. \[fig\_similar\] illustrates the principle. ![Principle of Transitivity on Similarity.[]{data-label="fig_similar"}](instance_similar.png){width="0.85\columnwidth"} In Fig. \[fig\_similar\], B and C are both similar to A in different sources. Given the known mapping between A in the two sources, we can infer that B and C are similar. MMC tries to use the similarity transitivity principle to help infer the cross-source instance similarity and to help improve the clustering performance. Next, we will talk about each step in the MMC framework in detail. Initialization -------------- Since the efficiency of the iterative optimization is affected by the initialization step, in this paper, we learn the initial value of $ U_i^k $ and $ U^{k*} $ rather than random initialization. For each $ U_i^k $, we solve Eq. (\[single\_view\]) to get an initial value. As we described in the previous section, Eq. (\[single\_view\]) is just the objective function for single view clustering, without considering the relation among views and sources. The solution $ U^k_i $ is given by the top-$ c_k $ eigenvectors of the Laplacian $ L_i^k $. For each $ U^{k*} $, we just solve Eq. (\[eq:multi\]) to get the initialization value with the initial $ U^k_i $. The objective function can be written as: $$\small \centering \begin{split} \max_{U^{k*}} ~& tr\left({U^{k*}}^T \left(\sum_{i=1}^{v_k}\alpha^k_{i} U^k_{i}{U^k_{i}}^T \right)U^{k*} \right),\\ &\textit{s.t.}~~ {U^{k*}}^TU^{k*} = \mathit{I} \end{split} \label{eq:init}$$ The solution is given by the top-$ c_k $ eigenvectors of the modified Laplacian $ \sum_{i=1}^{v_k}\alpha^k_{i} U^k_{i}{U^k_{i}}^T$. In the previous section, we assume that the mapping matrix between two sources is semi-orthogonal, *i.e.*, $ M^{(i,j)}M^{(i,j)T} = I $. However, we can only get part of the mapping information due to the incompleteness and partial known property of real-world problem. The mapping matrix between two sources can be expressed in two parts $ M_0^{(i,j)} $ and $ M_1^{i,j} $: $$M^{(i,j)} = \begin{bmatrix} M_0^{(i,j)} & 0 \\ 0 & M_1^{(i,j)} \end{bmatrix},$$ where $ M_0^{(i,j)} $ represents the known mapping between source $ i $ and $ j $ and $ M_1^{(i,j)}$ represents the unknown part. It is easy to find that $ M_0^{(i,j)}M_0^{(i,j)T} = I$. We only need to initialize the unknown part to make it semi-orthogonal. One natural way to estimate the unknown mapping is to use the instance similarity among the instances between two sources. Using the similarity transitivity principle, the instance similarity between two sources $ i $ and $ j $ can be estimated by: $\small K_{U^{i*}}\begin{bmatrix} M_0^{(i,j)} & 0 \\ 0 & 0 \end{bmatrix} K_{U^{j*}}, $ where $ K_{U^{i*}} $ and $ K_{U^{j*}} $ are the two kernel matrices for latent feature $ U^{i*} $ and $ U^{j*} $. It is worth to note that the kernel matrices $K_{U^{i*}}$ and $K_{U^{j*}}$ can be seen as the similarity matrices of the latent features for sources $ i $ and $ j $. The similarity transitivity principle allows us to use the known instance mapping as a bridge to connects unmapped instances in two sources. Using Fig. \[fig\_similar\] as an example, $\left(K_{U^{i*}}\right)_{b,a}$ provides the similarity between instances B and A in source $ i $, while $\left(K_{U^{j*}}\right)_{a,c}$ provides the similarity between instances A and C in source $ j $. If both instances A in both sources $ i $ and $j$ are mapped through $ M^{(i, j)} $, $\left(K_{U^{i*}}\begin{bmatrix} M_0^{(i,j)} & 0 \\ 0 & 0 \end{bmatrix}K_{U^{j*}}\right)_{b,c}$, denoted by $ (\widetilde{M}^{(i,j)})_{b,c} $, will provide the estimated similarity between instance B in source $ i $ and instance C in source $ j $. So for the unmapped instances, we can have an estimated similarity mapping $ \widetilde{M}_1^{(i,j)} $. Then we can orthogonalize it using SVD or other orthogonalization methods. Maximizing $\mathcal{O}$ over $U^k_{i}$s with fixed $U^{k*}$s ------------------------------------------------------------- With fixed $U^{k*}$s, for each $U^k_{i}$ we only need to maximize part of $\mathcal{J}_k$. $$\small \begin{split} \max_{U^k_{i}}~ \mathcal{L} &=tr({U^k_{i}}^TL^k_{i}{U^k_{i}})+\alpha^k_{i}tr({U^k_{i}}{U^k_{i}}^TU^{k*}{U^{k*}}^T)\\ &=tr({U^k_{i}}^T(L^k_{i}+\alpha^k_{i}U^{k*}{U^{k*}}^T){U^k_{i}})\\ &\textit{s.t.}~~ {U^k_{i}}^TU^k_{i} = \mathit{I} \end{split} \label{eq:op1}$$ This is a standard spectral clustering objective on source $k$ view $i$ with modified graph Laplacian $L^k_{i}+\alpha^k_{i}U^{k*}{U^{k*}}^T$. According to [@Andrew_sc], the solution $U^k_{i}$ is given by the top-$c_k$ eigenvectors of this modified Laplacian. With fixed $U^{k*}$s, we can calculate each $U^k_{i}$ to maximize the objective function.\ Maximizing $\mathcal{O}$ over $U^{k*}$s with fixed $U^k_{i}$s ------------------------------------------------------------- With fixed $U^k_{i}$s, for each $U^{k*}$, we only need to maximize: $$\resizebox{\hsize}{!}{$\displaystyle \begin{split} \max_{U^{k*}} \mathcal{Q} &= \sum_{i=1}^{v_k}\alpha^k_{i}tr(U^k_{i}{U^k_{i}}^TU^{k*}{U^{k*}}^T) - \sum_{1\le j\le K,j\ne k} \beta^{(k,j)}\widetilde{D}(U^{k*}, U^{j*})\\ &=\sum_{i=1}^{v_k}\alpha^k_{i}tr(U^k_{i}{U^k_{i}}^TU^{k*}{U^{k*}}^T) \\ &+ \sum_{j\ne k} \beta^{(k,j)} tr\left(U^{j*}U^{j*T}{M^{(k,j)}}^TU^{k*}U^{k*T}M^{(k,j)}\right)\\ &= tr\left( U^{k*T}L^{k*}U^{k*}\right)\\ &\textit{s.t.}~~ {U^{k*}}^TU^{k*} = \mathit{I}.\\ \end{split}$ } \label{eq:update_star}$$ where $$L^{k*} = \sum_{i=1}^{v_k}\alpha^k_{i}U^k_{i}{U^k_{i}}^T+\sum_{j\ne k}\beta^{(k,j)}M^{(k,j)} U^{j*}U^{j*T}{M^{(k,j)}}^T$$ The solution $U^{k*}$ is given by the top-$c_k$ eigenvectors of this modified Laplacian $L^{k*}$. Thus, with fixed $U^{k}_i$s, we can calculate the consensus $U^{k*}$ for each of the sources to maximize the objective function. Infer the Similarity Mapping between Sources -------------------------------------------- Using above optimization method, we can iteratively learn the latent feature matrices for each source. However, when the number of partially observed mapping is limited, *i.e.*, when only a small number of instances mapping between sources are observed, the estimated initial similarity mapping between two sources may not be accurate. Hence the improvement of clustering performance is limited. Based on the similarity transitivity principle, MMC proposes to use the learned latent feature matrices for multiple sources to help infer the similarity mapping across sources. Similar to the initialization, we use the instance mapping between two sources as a bridge to help transfer the similarity. The new estimated instance similarities between two sources can be written as: $$\widetilde{M}^{(i,j)} = K_{U^{i*}}M^{(i,j)}K_{U^{j*}},$$ where $ K_{U^{i*}} $ and $ K_{U^{j*}} $ are the two kernel matrices for latent feature matrices $ U^{i*} $ and $ U^{j*} $. In our experiment, we use linear kernel for the latent feature matrices. This estimated similarity mapping includes every instance across two sources. However, we want to preserve the already known instance mapping and only update the instance similarity mapping for instances whose mappings are unknown. We introduce the indicator matrix $W^{i,j}$, which has the same dimension as $M^{(i,j)}$ and was initialized with only 0 and 1. $W^{(i,j)}_{ab}$ equals to 1 if the mapping between $ a $-th instance from source $ i $ and the $ b $-th instance from source $ j $ is known, and 0 if unknown. The similarity mapping between source $ i $ and source $ j $ and is updated as follows: $$\begin{aligned} \begin{adjustbox}{max width=0.85\columnwidth} $\displaystyle M^{(i,j)} \leftarrow W^{(i,j)}\circ M^{(i,j)} + ( \mathbf{1} -W^{(i,j)})\circ \widetilde{M}^{(i,j)}, $ \end{adjustbox} \label{update_mapping}\end{aligned}$$ where $\circ$ indicates the element-wise multiplication, $\widetilde{M}^{(i,j)} = U^{i*}{U^{i*}}^TM^{(i,j)}U^{j*}{U^{j*}}^T$ and $ \mathbf{1} $ is an all-one matrix. By using the indicator matrix $W^{(i,j)}$ and element-wise multiplication, we can only update the unknown part of the mapping, and preserve the known part. Once we have a better mapping across sources, it will help learn better latent feature matrices. The better latent feature matrices will in-turn help infer the similarity mapping. This iteration continues until it converges. MMC framework ------------- The algorithm for the MMC framework is shown as Algorithm \[algorithm\_MMC\]. We first calculate the kernel matrices and the corresponding normalized graph Laplacian matrices for every view. In all the experiments throughout the paper, we use Gaussian kernel for computing the similarities unless mentioned otherwise. The standard deviation of the kernel is set equal to the median of the pair-wise Euclidean distances between the data points. We then initialize the latent feature matrices $ \{U_i^k\} $, $ \{U^{k*} \} $ and the instance mappings $ M^{(i,j)} $. Then we iteratively update $U_i^k$s, $U^{k*}$s and $M^{(i,j)}$s until they all converge. Data matrices for every view from each source $\{X^k_i\}$. Instance mappings between sources $\{M^{(i,j)}\}$. Indicator matrices $\{W^{(i,j)}\}$. The number of clusters for each source $\{c_k\}$. Parameters $ \{\alpha^k_i\} $ and $\{\beta^{(i,j)}\}$. Clustering results for each source. Calculate $ K^k_i $ and $L^k_i$ for every $ k $ and $ i $. Initialize $U_i^k$ for every $ k $ and $ i $. Initialize $U^{k*}$ by solving Eq.(\[eq:init\]). Update each $U_i^k$ by solving Eq. (\[eq:op1\]). Update each $U^{k*}$ by solving Eq. (\[eq:update\_star\]). Update the similarity mappings using Eq. (\[update\_mapping\]). Apply k-means on $U^{k*}$ for every source $ k $. Experiments and Results ======================= In this section, we compare MMC framework with a number of baselines on three real-world data sets. Comparison Methods ------------------ We compare the proposed MMC method with several state-of-art methods. Since no previous methods can be directly applied to the multi-source multi-view situation, in order to compare with the previous methods, we make some changes. The details of comparison methods is as follows: - **MMC:** MMC is the clustering framework proposed in this paper, which applies collective spectral clustering with discrepancy penalty across sources. The parameter $\alpha$ is set to 0.1 and $\beta$ is set to 1 for all the views and sources throughout the experiment. - **Concat:** Feature concatenation is one way to integrate all the views. We concatenate views within each source, so each source is a concatenated view. Since the instances between sources are not fully aligned, we extend each source by adding pseudo instances (average features). Thus, sources are fully aligned after extension. We then apply PCA and k-means to get the clustering results. - **Sym-NMF** Symmetric non-negative matrix factorization is proposed in [@sym_nmf] as a general framework for clustering. It factorizes a symmetric matrix containing pairwise similarity values. To apply Sym-NMF to multi-source multi-view situation, we apply Sym-NMF to every view from each source to get the latent feature matrices. Then we combine all the latent feature matrices in the same source to produce the final clustering results. - **MultiNMF:** MultiNMF is one of the state-of-art multi-view clustering methods based on joint nonnegative matrix factorization [@sdm2013_liu]. MultiNMF formulates a joint matrix factorization process with the constraint that pushes clustering solution of each view towards a common consensus instead of fixing it directly. Throughout the experiment, the parameter $\lambda_v$ is set to 0.01 as in the original paper. - **CoReg:** CoReg is the centroid based multi-view clustering method proposed in [@MVC_co_reg]. It aims to get clusters that are consistent across views by co-regularizing the clustering hypotheses. Throughout the experiment, the parameter $\lambda_v$ is set to 0.01 as suggested in the original paper. - **CGC:** CGC [@flexibleMVC] is the most recent work that deals with many-to-many instance relationship, which is similar to incomplete instance mapping. In order to run the CGC algorithm, we generated the relations between views within one source, which is complete one-to-one mapping. We also generate the relations between views across sources, which is incomplete and partially known. We run CGC on all the views across sources and report the best performance for each source. In the experiment, the parameter $\lambda$ is set to 1 as suggested in the original paper. It is worth to note that the two multi-view clustering methods MultiNMF and Co-Reg only work with views that are fully aligned. In our experiments, only views from the same source are fully mapped/aligned. Views across different sources are partially mapped. We apply MultiNMF and Co-Reg in two ways. ![Source alignment[]{data-label="fig:pseudo"}](pseudo.png){width="0.8\columnwidth"} The first way is to apply them to every single source, denoted as **MultiNMF-S** and **CoReg-S**. Thus, both MultiNMF-S and Co-Reg-S only co-regularize the views within a source without considering the discrepancy between sources. The second way is to apply MultiNMF and Co-Reg to multiple sources, denoted as **MultiNMF-M** and **CoReg-M**. However, the views across sources are not fully mapped/aligned. In order to apply MultiNMF and Co-Reg, we align the sources by adding average pseudo instances to every source. As shown in Fig. \[fig:pseudo\], for the unmapped instances in one source, we created the corresponding pseudo instances in other sources. Thus, the instances from different sources are fully mapped. MultiNMF and Co-Reg are then applied to all the aligned views, and performance is reported for every source. In our experiments, we use Gaussian kernel for computing the similarities unless mentioned otherwise. The standard deviation of the kernel is set equal to the median of the pair-wise Euclidean distances between the data points. K-means is used to get clustering results for all the methods. For each setting, we run k-means 20 times and report the average performance. Dataset ------- In this paper, three groups of real-world data sets are used to evaluate the proposed MMC method. The important statistics of them are summarized in Table \[tab:data\_stat\]. [max width=0.9]{} **data** **size** **\# view** **\# cluster** ----------------- ---------- ------------- ---------------- **Dutch** 2000 6 10 **USPS** 2000 2 10 **English** 1200 2 6 **Translation** 1200 2 6 **BBC** 352 2 6 **Reuters** 294 2 6 **Guardian** 302 2 6 : Statistics of the datasets[]{data-label="tab:data_stat"} - **Dutch-USPS** This data set comes from two sources, UCI Handwritten Dutch digit numbers and USPS digit data. The first source, **Dutch**[^1], consists of 2000 examples of handwritten numbers ’0’-’9’ (200 examples per class) extracted from a collection of Dutch utility maps. All the examples have been digitized in binary images. Each example is represented in the following six views: (1) 76 Fourier coefficients of the character shapes, (2) 216 profile correlations, (3) 64 Karhunen-Love coefficients, (4) 240 pixel averages in $2\times 3$ windows, (5) 47 Zernike moments, and (6) 6 morphological features. The second source, **USPS**[^2], consists of digit images with size $ 16\times 16$ for numbers ‘0’-‘9’. We randomly select 2000 examples corresponding to the examples in first source. From USPS data, we extract two views, the original pixel feature with dimension of 256 and the Gaussian similarity matrix between examples with dimension of 2000. - **English-Translations:** This data contains two sources, the original Reuters news documents written in English, and the machine translations in other four languages (French, German, Spanish and Italian) in 6 topics [@Amini09learningfrom]. From the first source, **English**, we use the document-term matrix and the cosine similarity matrix of the documents as two views. From the second source, **Translation**, we extract the document-term matrices from French and German as two views. We randomly sample 1200 documents from the first source in a balanced manner, with each category having 200 documents. We then select the corresponding 1200 documents from the second source. - **News Text data[^3]:** This news data has three sources: BBC, Reuters, and The Guardian. In total there are 948 news articles covering 416 distinct news stories from the period February to April 2009. Thus, the articles from these three sources are naturally partially mapped. Of these distinct stories, 169 were reported in all three sources, 194 in two sources, and 53 appeared in a single news source. Each story was annotated with one of the six topical labels: business, entertainment, health, politics, sport, technology. From each source, we extract two views, the document-term matrix and the cosine similarity matrix. From the three sources, we create three sets of data, **BBC**-**Reuters** (239 mapped instances), **BBC**-**Guardian** (250 mapped instances) and **Reuters**-**Guadian** (212 mapped instances). It is worth noting that both Dutch-USPS and English-Translation data are one-to-one fully mapped. In our experiments, we randomly delete part of the mappings across different sources. Results ------- The results for Dutch-USPS data and English-Translation data are shown in Table \[tab:two\_set\]. The results are obtained under 60% known mappings . We report the NMI (Normalized Mutual Information) for each source in Table \[tab:two\_set\]. From Table \[tab:two\_set\], we can observe that the proposed MMC framework outperforms all the other comparison methods on both Dutch-USPS and English-Translation data. For the Dutch-USPS data, although CoReg-M and CGC are close to MMC (less than 4%) on Dutch, MMC outperforms the other methods on USPS by at least 10 %. We can also observe that MultiNMF-M and CoReg-M perform better than MultiNMF-S and CoReg-S on Dutch-USPS. However, the performance of the multi-source methods is worse than single-source method on English-Translation. This suggests that combining multiple sources only using the incomplete instance mappings may even hurt the performance. The proposed MMC methods, however, iteratively discovers the similarity among unmapped instances and uses the similarity to help learning. We also reported the results on three sets of the news text data (BBC-Reuters, BBC-Guardian and Reuters-Guardian) in Table \[tab:result\_3source\]. From Table \[tab:result\_3source\], we can see that MMC outperforms other comparison methods by a large margin in most cases. On Reuters-Guardian, although MultiNMF-M is slightly better than MMC on Guardian, MMC is still better than all the other baselines on Reuters. From Table \[tab:two\_set\] and Table \[tab:result\_3source\], we can observe that MMC outperforms other comparison methods in most cases for all the three groups of data. We can also conclude that MMC reduces the impact of negative transfer by iteratively learns the similarity among unmapped instances and takes advantage of it. The other reason why MMC can have a better performance for all the sources is that MMC treats the views within each source as a cohesive unit for clustering while considering discrepancy/disagreements between sources. [max width=0.95]{} Method **Dutch** **USPS** **English** **Translation** ------------ ------------ ------------ ------------- ----------------- Concat 0.5734 0.3916 0.1914 0.1621 Sym-NMF 0.7778 0.3005 0.2783 0.1527 MultiNMF-S 0.5382 0.4010 0.3413 0.2708 MultiNMF-M 0.7585 0.4700 0.342 0.2164 CoReg-S 0.7503 0.4044 0.3381 0.2874 CoReg-M 0.7886 0.5257 0.2187 0.2198 CGC 0.7851 0.2780 0.2636 0.2536 MMC **0.8248** **0.6348** **0.3528** **0.3073** : NMI for **Dutch**-**USPS** and **English**-**Translation** at 60% cross source mapping known[]{data-label="tab:two_set"} [max width=0.95]{} ------------ ------------ ------------ ------------ ------------ ------------ ------------ BBC Reuters BBC Guardian Reuters Guardian Concat 0.3003 0.3118 0.3344 0.3603 0.2994 0.3073 Sym-NMF 0.4414 0.443 0.4261 0.4444 0.4332 0.4325 MultiNMF-S 0.4799 0.4158 0.4637 0.4677 0.4275 0.4656 MultiNMF-M 0.5453 0.5127 0.4539 0.5243 0.5372 **0.5465** CoReg-S 0.5488 0.5273 0.5532 0.5393 0.5273 0.536 CoReg-M 0.4311 0.4615 0.4927 0.5103 0.4644 0.454 CGC 0.4378 0.4159 0.4354 0.4171 0.4338 0.3921 MMC **0.5714** **0.5874** **0.5632** **0.5903** **0.5639** 0.5377 ------------ ------------ ------------ ------------ ------------ ------------ ------------ : NMI for News Text Data[]{data-label="tab:result_3source"} [max width=0.95]{} Method $c_1 = 2$ $c_1 = 4$ $c_1 = 6$ $c_1 = 8$ $c_1 = 10$ --------------- ------------ ------------ ------------ ------------ ------------ Concat(D) 0.7806 0.5847 0.5499 0.5737 0.5734 Sym-NMF(D) 0.9652 0.8907 0.8393 0.7912 0.7778 MultiNMF-S(D) 0.9041 0.6996 0.5687 0.5917 0.5382 MultiNMF-M(D) 0.9652 0.8717 0.8086 0.7460 0.7585 CoReg-S(D) 0.9387 0.8879 0.8322 0.7807 0.7503 CoReg-M(D) 0.9652 0.9035 0.8205 0.8295 0.7886 CGC(D) 0.9652 0.9108 0.8345 0.7976 0.7851 MMC(D) **0.9652** **0.9433** **0.8897** **0.8311** **0.8248** Concat(U) 0.4877 0.3470 0.3338 0.3730 0.3916 Sym-NMF(U) 0.5708 0.4412 0.3465 0.2875 0.3005 MultiNMF-S(U) 0.6261 0.6070 0.4783 0.4306 0.4010 MultiNMF-M(U) 0.7711 0.5006 0.6147 0.5585 0.4700 CoReg-S(U) 0.5375 0.5116 0.4207 0.3778 0.4044 CoReg-M(U) 0.5676 **0.7604** 0.5023 0.5146 0.5257 CGC(U) 0.5708 0.4513 0.2866 0.2642 0.2780 MMC(U) **0.7753** 0.7435 **0.7109** **0.6856** **0.6348** : NMI on Dutch-USPS with different clusters number[]{data-label="tab:clust_1"} Parameter Study --------------- There are two sets of parameters in the proposed MMC method: $\{\alpha_i^k\} $, the relative importance of view $i$ in source $k$, and $\{\beta^{(i,j)}\}$, the weight of the discrepancy penalty between source $i$ and $j$. Here we explore the influence of the view importance weights and the discrepancy penalty weights. We first fix $\{\beta^{(i,j)}\}$ to be 1, and run the proposed MMC method with various $\{\alpha_i^k\}$ values ($10^{-3}$ to $10^3$). We then fix $\{\alpha_i^{k}\}$ to be 0.1, and run the proposed MMC method with various $\{\beta^{(i,j)}\}$ values ($10^{-3}$ to $10^3$). Due to the limit of space, we only report the results on Dutch-USPS data with 60% known mapping in Fig. \[fig:alpha\] and Fig. \[fig:beta\]. From Fig. \[fig:alpha\], we can see that the performance is stable with $ \alpha_i^k $ smaller than 100, and the best performance is achieved when $ \alpha_i^k $ is around 0.1. In Fig. \[fig:beta\], The performance is stable with $ \beta^{(i,j)} $ between 0.1 and 100. The best performance is achieved when $ \beta^{(i,j)} $ is near 1. ![NMI v.s. $ \beta^{(i,j)} $ on Dutch-USPS.[]{data-label="fig:beta"}](alpha){width="\textwidth"} \ ![NMI v.s. $ \beta^{(i,j)} $ on Dutch-USPS.[]{data-label="fig:beta"}](beta){width="\textwidth"} Discussion ========== In this section, we aim at analyzing MMC more in detail in order to answer the following four questions:\ (1) How does the difficulty of the clustering problem affect the performance of these methods?\ (2) How does percentage of known mappings between sources affect the performance of MMC?\ (3) How good is the inferred similarity mapping?\ (4) How fast does MMC converge? To show the performance for clustering problem with different difficulties, we apply MMC to Dutch-USPS data but with different number of clusters (2 to 10). The difficulty of the clustering problem increases as the number of clusters increases. The results are shown in Table \[tab:clust\_1\]. The percentage of known mappings is also set to 60%. The NMIs for both sources are reported in separate rows (MMC(D) for Dutch and MMC(U) for USPS). From Table \[tab:clust\_1\], we can observe that as the number of clusters increases (the difficulty of the problem increases), the performance for all of the methods decrease. The proposed MMC outperforms other comparison methods in almost all cases with one exception. CoReg-M outperforms MMC on USPS when the cluster number is 4. However, the proposed MMC achieved the second best performance in that case. To answer the second question, we apply MMC on both Dutch-USPS data and English-Translation data with various percentages of known mapping between 30% to 100 % (10% interval). The results are shown in Tables \[tab:v1\] and \[tab:v2\]. It is worth noting that Sym-NMF, CoReg-S and MultiNMF-S do not utilize the instance mapping across sources. Thus, the performance of these three methods remain the same for different percentages. In Table \[tab:v1\] and Table \[tab:v2\], the proposed MMC outperforms the other comparison methods for both sources in almost all of the different parameter settings. It is important to notice that even with 100% mapping available, the proposed MMC is still better than other multi-view clustering methods. This is because MMC will treat views within one source as a cohesive set while other multi-view clustering algorithms treat the views from different sources equally. Method 30% known 40% known 50% known 60% known 70% known 80% known 90% known 100% known --------------- ------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------ Concat(D) 0.5515 0.5616 0.5631 0.5734 0.5770 0.5887 0.6064 0.6128 Sym-NMF(D) 0.7778 0.7778 0.7778 0.7778 0.7778 0.7778 0.7778 0.7778 MultiNMF-S(D) 0.5382 0.5382 0.5382 0.5382 0.5382 0.5382 0.5382 0.5382 MultiNMF-M(D) 0.6081 0.6676 0.7031 0.7585 0.8040 0.8130 0.7799 0.8356 CoReg-S(D) 0.7503 0.7503 0.7503 0.7503 0.7503 0.7503 0.7503 0.7503 CoReg-M(D) 0.7827 0.7861 0.7825 0.7886 0.8038 0.8343 0.8492 0.8596 CGC(D) **0.7947** 0.7891 0.8019 0.7851 0.7929 0.7840 0.7878 0.8003 MMC(D) 0.7931 **0.7903** **0.8177** **0.8248** **0.8371** **0.8537** **0.8611** **0.8746** Concat(U) 0.3067 0.3231 0.3623 0.3916 0.4354 0.4789 0.5350 0.6128 Sym-NMF(U) 0.3005 0.3005 0.3005 0.3005 0.3005 0.3005 0.3005 0.3005 MultiNMF-S(U) 0.401 0.4010 0.4010 0.4010 0.4010 0.4010 0.4010 0.4010 MultiNMF-M(U) 0.3468 0.3611 0.5029 0.4700 0.6007 0.6083 0.7011 0.7816 CoReg-S(U) 0.4044 0.4044 0.4044 0.4044 0.4044 0.4044 0.4044 0.4044 CoReg-M(U) 0.3527 0.4094 0.4607 0.5257 0.5808 0.6642 0.7541 0.8564 CGC(U) 0.2968 0.2902 0.2795 0.2780 0.2958 0.2882 0.2758 0.2898 MMC(U) **0.496** **0.5463** **0.6039** **0.6348** **0.6862** **0.7587** **0.8262** **0.8684** Method 30% known 40% known 50% known 60% known 70% known 80% known 90% known 100% known --------------- ------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------ Concat(E) 0.1144 0.1334 0.1617 0.1914 0.2037 0.2488 0.2495 0.2498 Sym-NMF(E) 0.2783 0.2783 0.2783 0.2783 0.2783 0.2783 0.2783 0.2783 MultiNMF-S(E) 0.3413 0.3413 0.3413 0.3413 0.3413 0.3413 0.3413 0.3413 MultiNMF-M(E) 0.3253 0.3289 0.3339 0.3420 **0.3689** 0.3531 0.3523 0.3508 CoReg-S(E) 0.3381 0.3381 0.3381 0.3381 0.3381 0.3381 0.3381 0.3381 CoReg-M(E) 0.2060 0.2088 0.2098 0.2187 0.2194 0.2193 0.2184 0.2182 CGC(E) 0.2388 0.2550 0.2656 0.2636 0.2707 0.2680 0.2737 0.2740 MMC(E) **0.3436** **0.3485** **0.3571** **0.3528** 0.3576 **0.3595** **0.3558** **0.3637** Concat(T) 0.1044 0.1124 0.1397 0.1621 0.1685 0.2098 0.2072 0.2198 Sym-NMF(T) 0.1527 0.1527 0.1527 0.1527 0.1527 0.1527 0.1527 0.1527 MultiNMF-S(T) 0.2708 0.2708 0.2708 0.2708 0.2708 0.2708 0.2708 0.2708 MultiNMF-M(T) 0.1945 0.2063 0.2146 0.2164 0.2146 0.2223 0.2541 0.2581 CoReg-S(T) 0.2874 0.2874 0.2874 0.2874 0.2874 0.2874 0.2874 0.2874 CoReg-M(T) 0.2160 0.2146 0.2178 0.2198 0.2171 0.2213 0.2210 0.2299 CGC(T) 0.2288 0.2450 0.2556 0.2536 0.2607 0.2580 0.2637 0.2640 MMC(T) **0.3028** **0.3075** **0.3072** **0.3073** **0.3090** **0.3144** **0.3196** **0.3230** From the results, we can conclude that MMC works for various percentages of known mapping across sources. The reason why MMC performs better is not only because it appreciates the cohesiveness of the views, but also for every iteration, MMC tries to infer the instance similarity mapping between different sources. Although the instance similarity mapping is not as the same as the instance mapping, it provides extra information about the partially known instance mapping. Thus the inferred instance similarity mappings will help improve clustering in the next iteration. To show how good the inference of similarity mapping is, we perform another set of experiments to measure the quality of the inferred similarity mappings among those not-aligned instances. For each not-aligned instance, we select the most similar instance mapped by the similarity mapping. Then we check if the two instances are in the same class. We test the accuracy for different number of clusters on Dutch-USPS data. Here the percentage of known instance mapping is set to 60%. We reported the number of instances that are not aligned by the known mapping, and the number of class matches by the inference of similarity mappings in Table \[tab:inference\]. [max width=0.8]{} $ c = 2 $ $ c = 4 $ $ c = 6 $ $ c = 8 $ $ c = 10 $ ----------- ----------- ----------- ----------- ----------- ------------ instances 160 320 480 640 800 matches 113 259 349 429 522 accuracy 0.7063 0.8094 0.7271 0.6703 0.6525 : The inference accuarcy under differnet cluster numbers on Dutch-USPS[]{data-label="tab:inference"} [max width=0.95]{} % Known $ 30\% $ $ 40\% $ $ 50\% $ $ 60\% $ $ 70\% $ $ 80\% $ $ 90\% $ --------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- \# iter 14 14 15 18 19 21 17 : The number of iterations until converge[]{data-label="tab:converge"} From Table \[tab:inference\], we can clearly observe that when the number of clusters is 4, the inference of similarity mapping can get an accuracy as high as 0.8094. As the number of clusters increases, the accuracy of the inference drops. However, even with the number of clusters being as high as 10, we still get an accuracy of 0.6525 for the inference. To have a better understanding of the inferred similarity mapping, we plot the similarity mapping among the unmapped instances from the Dutch-USPS data with four clusters and 60% known mapping in Fig. \[fig:inference\]. The instances are sorted by the class label for both sources (the first 80 instances belongs to class 1, the second 80 instances belong to class 2, etc,.) The X axis indicates the instances in Dutch, while the Y axis indicates the instances in USPS. From the figure, we can clearly see that there are four dark squares of width 80 on the diagonal line, which indicates that the same class of instances are more likely to be mapped together. ![Scatter plot of the inferred similarity mapping for Dutch-USPS data with four clusters.[]{data-label="fig:inference"}](test.png){width="0.85\columnwidth"} To show how fast MMC converges, we report the number of outer iterations until convergence for Dutch-USPS data in Table \[tab:converge\]. From the table we can see that the method converges fast (less than 20 iterations). Related Works ============= Multi-view learning [@Blum_co_training; @MVC_co_reg; @Nigam_co_training; @LongYZ08], is proposed to learn from instances which have multiple representations in different feature space. For example, [@MVC] developed and studied partitioning and agglomerative, hierarchical multi-view clustering algorithms for text data. [@MVC_co_reg; @MVC_co_training] are among the first works proposed to solve clustering problem via spectral projection. [@sdm2013_liu] proposed to solve multi-view clustering by joint non-negative matrix factorization. [@trivedi2010multiview; @ShaoMVC; @PVC; @DBLP:conf/pkdd/ShaoHY15] are among the first works to solve the multi-view clustering with partial/incomplete views. However, none of the previous multi-view clustering methods can deal with incomplete and partial known mapping between sources/views. Further more, All the previous methods fail to treat the views within one source as a cohesive unit. Consensus clustering [@consensus_clustering; @li08consensus] is also related to the proposed MMC framework. It deals with the situation in which a number of different clustering results have been obtained for a particular dataset and it is desired to find a single consensus clustering which is a better fit in some sense than the existing ones. [@goder08consensus] gives a report about consensus clustering algorithms comparison and refinement. [@LockD13] proposes a bayesian consensus clustering method. However, consensus clustering aims to find a single consensus clustering from fully mapped clustering solutions. None of the previous methods works for the incomplete and partially unknown mappings between the instances. Conclusion ========== This paper is the first to investigate the problem of clustering with multiple sources and multiple views. The proposed MMC framework treats views in the same source as a cohesive group for clustering by learning consensus latent feature matrices from the views within one source. It also incorporates multiple sources by using cross-source discrepancy penalty to enhance the clustering performance. MMC also uses the learned latent features to infer the cross-source unknown similarity mapping, which in turn will help improve the performance of clustering. 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--- abstract: 'We theoretically investigate the optical response of a quantum dot, embedded in a microcavity and incoherently excited by pulsed pumping. The exciton and biexciton transition are off-resonantly coupled with the left- and right-polarized mode of the cavity, while the two-photon resonance condition is fulfilled. Rich behaviours are shown to occur in the time dependence of the second-order correlation functions which refer to counter-polarized photons. The corresponding time-averaged quantities, which are accessible to experiments, confirm that such a dot-cavity system behaves as a good emitter of single, polarization-correlated photon pairs.' author: - 'J.I. Perea' - 'F. Troiani' - 'C. Tejedor' bibliography: - 'pulses.bib' title: | Correlated photon-pair emission from\ pumped-pulsed quantum dots embedded in a microcavity --- INTRODUCTION ============ Semiconductor quantum dots (QDs) were recently demonstrated to emit highly nonclassical light, under proper exciting conditions [@moreau; @santori]. Besides being interesting in its own right, such phenomena can be exploited for the implementation of solid-state quantum-information devices [@nielsen]. Among other things, these require high collection efficiencies and photon emission rates larger than the relevant decoherence ones: two properties which are typically not present in individual QDs, but are significantly approached when the QD is embedded in a microcavity (MC). In fact, cavity photons are mostly emitted with a high degree of directionality, thus allowing for large collection efficiencies [@solomon]. Moreover, the reduction of the exciton recombination time (Purcell effect) minimizes the decoherence due to exciton-phonon scattering  [@gerard; @solomon; @unitt; @varoutsis]. The experimental results obtained in the low-excitation limit (i.e., at most one exciton at a time in the QD) are relatively well understood [@moreau01; @gerard; @santori01; @zwiller02; @kiraz02; @michler; @pelton; @fattal04]. The QD optical selection rules and its discrete energy spectrum can however be exploited also for emitting polarization-correlated photon pairs. The main goal of the present work is that of investigating the time-dependence and average values of the photon correlations, in a condition where the pair-emission probability is enhanced by suitably configuring the coupling to the MC. The paper is organized as follows: in Sec. II we describe the system and the model we use; in Sec. III, we present the results for the time-dependent photon-coherence functions, as well as time-averaged quantities, which may directly be compared with experimental results. Finally, we drow our conclusions in Sec. IV. THE SYSTEM AND THE MODEL ========================= The discrete nature of the QD energy spectra significantly reduces the portion of the dot Hilbert space which is required for the dot description, at least in the low-excitation regime. Throughout the paper, we shall accordingly restrict ourselves to the following four dot eigenstates: the ground (or vacuum) state $ |G\rangle $; the two optically active single-exciton states $ | X_R \rangle $ and $ | X_L \rangle $, with $z$ components of the overall angular momentum $+1$ and $-1$, respectively; the lowest biexciton state $ | B \rangle $. The Coulomb interaction between the carriers, enhanced by their 3D confinement, results in an energy renormalization of the exciton transitions (the so called biexciton binding energy) of the order of a few meV. These, together with the polarization-dependent selection rules, allow a selective addressing of the different transitions, already at the ps timescale (see Fig. \[lscheme\]). In most self-assembled QDs, the spin degeneracy of the bright exciton states is removed by the anisotropic electron-hole exchange interaction, the actual eigenstates thus being the linearly polarized $ | X_x \rangle $ and $ | X_y \rangle $. This geometric effect can however be compensated (e.g., by an external magnetic field), thus recovering the circularly polarized exciton basis [@bayer]. Since our goal is the efficient emission of photon pairs, we consider the QD to be embedded in a cavity having the fundamental mode with the following essential property: [*its frequency $\omega _C$ is different from the transition energies between QD states, but as close as possible to one half the energy difference between $ | G \rangle $ and $ | B \rangle $*]{} ($ \Delta _1 = - \Delta _2 $ in Fig. \[lscheme\]). This [*two-photon resonance*]{} condition seems promising for enhancing the emission of photon pairs with respect to that produced by QD’s without cavity [@santori02; @stevenson; @ulrich03; @ulrich]. The cavity mode also presents a degeneracy with respect to the right and left polarizations. Although we do not impose any constriction on the number of photons inside the cavity, their number is limited to only a few units with the adopted range of physical parameters. In order to account for the open nature of the cavity-dot system, we simulate its dynamics by means of a density-matrix description. In particular, the evolution of the density operator $\rho$ is given by the following master equation in the Lindblad form [@scully] ($\hbar = 1$): $$\begin{aligned} \frac{d}{dt} \rho & = & i \left[ \rho, H_S \right] + \sum _{J=R,L} \frac{\kappa}{2} \left( 2 a_J \rho a^{\dagger}_J - a^{\dagger}_J a_J \rho - \rho a^{\dagger}_J a_J \right) + \frac{\gamma}{2} \sum_{i=1}^4 \sum_{j=1}^4 \left( 2 \sigma_i \rho \sigma^{\dagger}_j - \sigma^{\dagger}_i \sigma_j \rho - \rho \sigma^{\dagger}_i \sigma_j \right) \nonumber \\ & & \times (\delta_{i,j} + \delta_{|i-j|,2}) + \frac{P}{2} \sum_{n=1}^{N_P} \theta [t-(n-1)T+t_P] \ \theta [(n-1)T-t] \sum_{i=1}^4 \sum_{j=1}^4 \left( 2 \sigma^{\dagger}_i \rho \sigma_j \right. \nonumber \\ & & \left. - \sigma_i \sigma^{\dagger}_j\rho - \rho \sigma_i \sigma^{\dagger}_j \right) (\delta_{i,j} + \delta_{|i-j|,2}), \label{masterequation}\end{aligned}$$ where $a_{J}$ ($a_{J}^\dagger$) is the annihilation (creation) operator for a photon with polarization $J=R,L$, while $\sigma_{i=1,4} = | G \rangle \langle X_R |, | G \rangle \langle X_L |,| X_L \rangle \langle B |, | X_R \rangle \langle B |$ are the ladder operators. The second and third terms on the right-hand side of the equation account for the radiative relaxation of the cavity and of the dot, respectively; the last one corresponds to its pulsed, incoherent pumping (i.e., the electron-hole pairs being photogenerated in the wetting layer, before relaxing non-radiatively in the dot). The Hamiltonian of the coupled QD-MC system is $$\begin{aligned} H_{S} & = & (\omega _C + \Delta _1 ) \left[ \mid X_R \rangle \langle X_R \mid + \mid X_L \rangle \langle X_L \mid \right] + (2 \omega _C + \Delta _1 + \Delta _2) \mid B \rangle \langle B \mid \nonumber \\ & & + \sum _{J=R,L} \left[ \omega _C \left( a^\dagger_J a_J + 1/2 \right) + \sum _{i} q \left( \sigma _i a^\dagger _J + a_J \sigma ^\dagger _i \right) \right].\end{aligned}$$ In the following, we shall consider the case of degenerate excitons, while their transition energies are assumed to be detuned with respect to the cavity-mode frequency: $ E_{X_{R,L}} - E_G = \omega_C + \Delta_1 $ and $ E_B - E_{X_{R,L}} = \omega_C + \Delta_2 $. As already mentioned, we will focus on the two-photon resonance case, $ \Delta_1 = - \Delta_2 $. Besides, the light emission through the leaky modes is considered inefficient as compared to that from the MC ($ \gamma \ll \kappa $). Finally, the system is pumped by means of rectangular pulses, of intensity $ P $, duration $t_P$, and repetition rate $ 1/T $. The correlation properties of the emitted radiation are described by the first- and second-order coherence functions. Photon correlations outside the cavity can be considered proportional to those inside [@walls; @stace]. Therefore, the polarization-resolved second-order coherence functions we shall refer to in the following are: $$g_{J,J'}^{(2)}(t,t+\tau) = \frac{G_{J,J'}^{(2)}(t,t+\tau)}{\langle a_{J}^\dagger(t)a_{J}(t)\rangle \langle a_{J'}^\dagger(t+\tau)a_{J'}(t+\tau)\rangle} \, , \label{g2}$$ where $$G_{J,J'}^{(2)}(t,t+\tau) = \langle a_{J}^\dagger(t) a_{J'}^\dagger(t+\tau) a_{J'}(t+\tau)a_{J}(t) \rangle \, , \label{G2}$$ $J,J' = R,L$ being the photon polarizations. The dependence of the $G^{(2)}_{J,J'} (t,t+\tau)$ on the delay $ \tau $ is derived, given the system state $\rho (t)$, by means of the quantum regression theorem [@scully]. Further details on the method can be found in Ref. [@perea04]. RESULTS ======== Second order coherence functions -------------------------------- We start by illustrating the overall time-dependence of the second-order coherence functions, for typical values of the relevant physical parameters (specified in the figure captions), and a single excitation pulse ($N_P=1$). Figures \[g2ll\] and \[g2rl\] show $ g_{L,L}^{(2)} (t,\tau) $ and $ g_{R,L}^{(2)} (t,\tau) $, respectively ($ g_{R,R}^{(2)} = g_{L,L}^{(2)} $ and $ g_{L,R}^{(2)} = g_{R,L}^{(2)} $ due to the system’s symmetry with respect to light and exciton polarizations). The emitted light clearly exhibits non-classical features. The function $ g_{L,L}^{(2)} $ quite generally shows a strongly sub-poissonian statistics \[$ g^{(2)} (t,0) < 1 $\], and a clear anti-bunching behavior \[$ g^{(2)} (t,\tau) > g^{(2)} (t,0) $\]. In fact, even though the present excitation regime is high enough for the system to be multiply excited, the polarization properties of the biexciton states and the optical selection rules suppress the probability for more than one photon to be emitted with the same polarization. Besides, due to the pulsed nature of the system’s excitation, the memory of the first photon being collected is never lost: correspondingly, the asymptotic value the $ g^{(2)} (t,\tau) $ tends to for infinite delay is always well below 1, though larger than $ g^{(2)} (t,0) $. The value of 1, which characterizes the continuous-pumping case [@perea05], is recovered for increasing duration $t_P$ of the pumping pulse (not shown here). As may be expected, the features emerging from the analysis of the $ g_{R,L}^{(2)} (t,\tau) $ function are quite different. In fact, the emissions of the $R$ and $L$ photons are positively correlated, and a strongly super-poissonian statistics emerges. The asymptotic values $g_{R,L}^{(2)} (t,\tau \rightarrow \infty)$, depend on the initial time $t$, whereas the overall dependence on $\tau$ includes both bunching and anti-bunching features. In Fig. \[todoenuno\] we plot the functions $ g_{J,J'}^{(2)} (0,\tau) $, in order to better appreciate some of the details, and to compare the two-photon resonance case $\Delta _1=-\Delta_2=0.5$ meV with one where such condition is not fulfilled ($\Delta _1=\Delta _2=0.1$ meV). In the two-photon resonance case, $(J,J')=(R,L)$ exhibits the above-mentioned rich behavior (upper panel): it is close to 1 (Poissonian statistics) at zero delay; it then oscillates with a pseudo period of the about 50 ps, while remaining below its initial value (photon bunching); finally, it reaches an asymptotic value of about $1.3$ (red curve). On the contrary, for $ \Delta_1 \neq - \Delta_2 $ (black curve), any correlation between the light emission in the two polarizations is suppressed ($g^{(2)}_{R,L} (0,\tau )=1$). The case $ ( J , J' ) = ( L , L ) $ is shown in the lower panel: here, the two-photon resonance doesn’t play any role, because it doesn’t apply to two photons with the same polarization. The deep at zero delay demonstrates that the multiple excitation of each cavity mode is completely negligible, whereas that of the dot-cavity system is not (red curve): an $L$ excitation might still be transferred from the QD to the MC after an $L$ photon has been emitted at $t=0$. In this case, the differences with respect to the off-resonance case (black curve) is only due to the difference between the values of $\Delta_{1,2}$. Two-photon coincidences ----------------------- The behaviors sofar discussed are not directly accessible in experiments, for they occur on timescales which are shorter than those presently achievable, e.g., within a Hanbury-Brown-Twiss setup [@scully]. In fact, the single photon detectors currently used in this kind of experiments have detection times $t_{det}$ of the order of hundreds of ps, whereas the typical timescales $G_{J,J'}^{(2)}(t,t+\tau) $ evolves on are of tens of ps. Therefore, correlation functions have to be integrated on time intervals of the order of $t_{det}$, or larger, and normalized by analogous integrals of the populations $ \langle a_J^{\dagger} (t) a_J (t) \rangle $. Experimentally, the second-order correlation functions can be normalized by repeatedly exciting the dot by means of identical pulses, separated by time intervals $T$ of the order of $10$ ns; such a value is supposed to be much larger than any correlation time in the system. Therefore, when the start and the stop detections correspond to different pulses ($\tau > T$), no correlation at all is expected. By normalizing the second order correlation $G_{J,J'}^{(2)} (t,t+\tau) $ with such *coincidences at large delay*, the value of 1 is obtained in the case of a pulsed laser [@santori]. As a first step in the investigation of this aspect, we perform the calculation of $G_{J,J'}^{(2)}(t,t+\tau)$, for $ \tau > T $. This function is shown in Fig. \[2pulses\] for $t=0$, $T=10$ ns, and all the other parameters as in Figs. \[g2ll\] and \[g2rl\]; similar results are obtained for any other time $t$ after the first pulse. In the following we use the above results in order to quantify the two-photon correlations. The ideal normalization of $ G_{J,J'}^{(2)} $ is given by the time integral of the uncorrelated second-order correlation function, which is given by $ \langle n_J (t) \rangle \, \langle n_{J'} (t+\tau ) \rangle $: $$\overline{g}_{J,J'} = \frac{\int_{0}^{T} dt \int_{0}^{T} d\tau G_{J,J'}^{(2)}(t,t+\tau) } { \int_{0}^{T} dt \langle a_{J}^\dagger(t) a_{J}(t) \rangle \int_{0}^{T} dt \langle a_{J'}^\dagger(t) a_{J'}(t)\rangle } . \label{G2nor3}$$ In practice, if the dot is periodically excited by sequences of identical laser pulses, and if the time interval $T$ separating two consecutive such pulses is larger than the system’s memory, then $ G_{J,J'}^{(2)}(t,t+\tau) \simeq \langle n_J (t) \rangle \, \langle n_{J'} (t+\tau ) \rangle $ for $ \tau > T $. Correspondingly, $ \overline{g}_{J,J'} $ can be approximated by $$\overline{g}_{J,J'}' = \frac{\int_{0}^{T} dt \int_{0}^{T} d\tau \; G_{J,J'}^{(2)}(t,t+\tau) } { \int_{t}^{T} dt \int_{T-t}^{2T-t} d\tau \; G_{J,J'}^{(2)} (t,t+\tau)} . \label{G2nor2}$$ In other words, two-photon correlation is approximated by the ratio between the areas of the first and the second peaks in Fig. \[2pulses\]. Our simulations of the system’s evolution under the effect of two identical squared pulses ($N_P =2$) aim at understanding to which extent such an approximation actually holds. Being the overall integration time $2T$ orders of magnitude larger than the characteristic timescales of the dot-cavity dynamics and, thus, of the time-step of the numerical calculations, such simulations are extremely time-consuming. In Fig. \[inter\_G2\] we compare the values of $\overline{g}_{J,J'}$ and $\overline{g}_{J,J'}'$, for different system parameters, while the time delay between consecutive laser pulses is kept constant ($T=10$ ns). Our results show that, for small values of the dot-cavity coupling $q$, the approximation of $\overline{g}_{J,J'}'$ with $\overline{g}_{J,J'}$ is not completely adequate: although the occupations of all the excited states in the system decay to negligible values on timescales of the order of $T$, the coherence function $G_{J,J'}^{(2)}(t,t+\tau)$ still cannot be factorized, for its decay with $\tau$ is slower than that of the density matrix with $t$. Aside from the normalization issue, a strong dependence of the second-order correlations on the dot-cavity coupling constant emerges. The value of $ \overline{g}_{L,L} = \overline{g}_{R,R} $ monotonically increases with $q$, while it remains well below 1 in all the range of considered parameters. This confirms the system’s general tendency to emit not more than one photon of each polarization. In the limit of weak dot-cavity coupling, and thus of slow excitation transfer from the QD to the MC, the probability for the system to be further excited after the first photon emission has taken place is practically suppressed. In order to identify the weak and strong coupling regimes, one can make the following simple argument: the QD requires a time $ \sim 1 / P $ to be excited, and $ \sim 1 / q $ to transfer its excitation to the MC. Therefore, if $ 1 / q \ll t_P, 1 / P \sim 4$ ps, there is no time for the dot to relax by emitting a photon (in the cavity) and being subsequently re-excited by the laser pulse. In this respect, the region where $ q < 0.2 $ meV can be identified with the weak-coupling regime. On the other hand, $ \overline{g}_{R,L} $ decreases with increasing $q$. In particular, a strong correlation is observed at the weak coupling regime, whereas the probability of an $L$ photon being emitted becomes nearly independent from the previous observation of $R$ ones ($ \overline{g}_{R,L} \sim 1$) for $q \gtrsim 0.2$ meV. One could be tempted to assign some coherence properties to the regime $ \overline{g}_{R,L} \sim 1$, being this the experimental value characteristic of a pulsed laser [@santori]. However, as shown in the inset of Fig. \[inter\_G2\], this is not the case for $g_{R,L}^{(2)}(t,t+\tau)$. In fact, the second-order coherence function shows strong and fast oscillations, indicating no coherence at all. Therefore, $\overline{g}_{R,L}=1$ cannot be identified with coherence. CONCLUSIONS =========== We have studied the photon emission of a QD, embedded in a MC and incoherently excited by pulsed pumping. We have shown that in the two-photon resonance condition, strongly positive correlations between $R$ and $L$ radiation can be achieved, while keeping negligible the probability of emitting multiple, equally polarized photons. Under proper excitation conditions, thus, the QD behaves as an efficient emitter of counter-polarized photon pairs. The detailed analysis of the second-order coherence functions $g_{J,J'}^{(2)}$ shows a rich behavior, including strong oscillations on a $10$ ps timescale; the asymptotic values ($\tau \rightarrow \infty$) differ from 1, the value expected in the continuous-pumping case. Time-averaged correlations $ \overline{g}_{J,J'} $ have also been computed, in order to allow a more direct comparison with experimental results. The above-mentioned features are clearly reflected also in their values, specially for small values of the dot-cavity coupling constant $q$. ACKNOWLEDGEMENTS ================ This work has been partly supported by the Spanish MCyT under contract No. MAT2002-00139, CAM under Contract No. 07N/0042/2002, and the European Union within the Research Training Network COLLECT. ![Scheme showing the cavity-mode frequency and the QD levels. The arrows show the two types of circularly polarized optical transitions connecting the QD levels.[]{data-label="lscheme"}](4niveles.EPS){height="8cm" width="8.cm"} ![(Color online) Second-order coherence function $g_{L,L}^{(2)} ( t , \tau ) $ as a function of the two time arguments (in ps). The time origin $ t = 0 $ is fixed at the end of the first laser pulse. The values of the physical parameters are: $ q = 0.1 $ meV, $ \kappa = 0.1 $ meV, $\Delta_1 = - \Delta_2 = 0.5 $ meV, $\gamma = 0.01 $ meV, $ P = 1$ meV, and $ t_P = 3 $ ps.[]{data-label="g2ll"}](g2ll.eps){height="8cm" width="8.cm"} ![(Color on line) Second-order coherence function $ g_{R,L}^{(2)} (t,\tau)$ as a function of the two time arguments (in ps). The values of the physical parameters are the ones reported in the caption of Fig. \[g2ll\].[]{data-label="g2rl"}](g2rl.eps){height="8cm" width="8.cm"}

--- abstract: 'This paper deals with the isotropic realizability of a given regular divergence free field $j$ in $\RR^3$ as a current field, namely to know when $j$ can be written as $\si\nabla u$ for some isotropic conductivity $\si>0$, and some gradient field $\nabla u$. The local isotropic realizability in $\RR^3$ is obtained by Frobenius’ theorem provided that $j$ and $\curl j$ are orthogonal in $\RR^3$. A counter-example shows that Frobenius’ condition is not sufficient to derive the global isotropic realizability in $\RR^3$. However, assuming that $(j,\curl j,j\times\curl j)$ is an orthogonal basis of $\RR^3$, an admissible conductivity $\si$ is constructed from a combination of the three dynamical flows along the directions $j/|j|$, $\curl j/|\curl j|$ and $(j/|j|^2)\times\curl j$. When the field $j$ is periodic, the isotropic realizability in the torus needs in addition a boundedness assumption satisfied by the flow along the third direction $(j/|j|^2)\times\curl j$. Several examples illustrate the sharpness of the realizability conditions.' author: - 'M. Briane[^1]' - 'G.W. Milton[^2]' title: 'Isotropic realizability of current fields in $\RR^3$' --- [**Keywords:**]{} current field, isotropic conductivity, Frobenius’ condition, dynamical flow [**Mathematics Subject Classification:**]{} 35B27, 78A30, 37C10 Introduction ============ In the theory of composite conductors (see, [*e.g.*]{}, [@Mil]), we are naturally led to study periodic composites. The effective properties of a periodic composite are obtained by passing from a local Ohm’s law \[Ohm\] j=e, between a periodic divergence free current field $j$ and a periodic electric field $e$, to an effective Ohm’s law j=\^\*e,, where $\si(y)$ is the local conductivity which is isotropic, and $\si^*$ is the (constant) effective conductivity of the composite which is in general anisotropic. In this context, it is natural to characterize the periodic current fields arising in the solution of these equations among all the divergence free fields. More precisely, the paper deals with the following question: Given a periodic regular divergence free field $j$ from $\RR^3$ into $\RR^3$, under which conditions $j$ is an isotropically realizable current field, namely there exists an isotropic conductivity $\si>0$ and a gradient field such that $j=\si\nabla u$? An additional motivation comes from the success of transformation optics (see, [*e.g.*]{}, [@PSS1; @PSS2]) where the objective is to choose moduli (in our case the conductivity $\sigma$) to achieve desired fields (in our case the prescribed current field $j$). In [@BMT] we have studied the isotropic realizability of a given regular electric field $e=\nabla u$ in $\RR^d$, for any $d\geq 2$. The key ingredient of our approach was the associated gradient system \[X\] { X’(t,x)=u(X(t,x)), & t,\ X(0,x)=x, & . which allowed us to prove the following isotropic realizability result for a gradient field in the whole space and in the torus: \[thm.elerea\] Let $u$ be a function in $C^3(\RR^d)$ satisfying the non-vanishing condition \[Du�0\] \_[\^d]{}|u|&gt;0. Then, there exists a unique function $\tau\in C^1(\RR^d)$ such that for any $x\in\RR^d$, the trajectory $t\mapsto X(t,x)$ meets the equipotential $\{u=0\}$ at the times $\tau(x)$, namely \[tau(x)\] u(X((x),x))=0. Moreover, the positive function $\si$ defined in $\RR^d$ by \[sitau\] (x):=(\_0\^[(x)]{}u(X(s,x))ds),x\^d, satisfies the conductivity equation $\div(\si\nabla u)=0$ in $\RR^d$. On the other hand, when $\nabla u$ is periodic, the conductivity $\si$ can be chosen periodic if and only if there exists a constant $C>0$ such that \[bdDeu\] |\_0\^[(x)]{}u(X(t,x))dt|C,x\^d. In the case of a gradient field, the local isotropic realizability which follows from the non-vanishing condition thanks to the rectification theorem (see [@BMT], Theorem 2.2 $i)$), is thus equivalent to the global realizability given by the previous theorem. The case of a regular divergence free field $j$ in $\RR^3$ is much more intricate. First of all, a necessary condition for the isotropic realizability is the orthogonality of $j$ and $\curl j$ in $\RR^3$. Conversely, if $j$ is non-zero and orthogonal to $\curl j$ in $\RR^3$, then Frobenius’ theorem implies that $j$ is isotropically realizable locally in $\RR^3$ (see Proposition \[pro.locrea\]). However, contrary to the case of a gradient field, these two conditions are not sufficient to ensure the global realizability (see Section \[ss.cexF\] for a counter-example). This strictly local nature of Frobenius’ theorem is strongly connected to cohomology which is outside the scope of this paper. On the other hand, we cannot use for a current field the properties of a gradient system which permits us in particular to define the time $\tau(x)$ satisfying . Our approach concerning the isotropic realizability of a current field is still based on dynamical systems. But now, the procedure to construct an admissible conductivity associated with a given regular divergence free field $j$, uses a combination of three dynamical flows which are not of gradient type. To this end, we need that the three fields $j$, $\curl j$ and $j\times\curl j$ make an orthogonal basis of $\RR^3$, including in this way Frobenius’ condition. Then, the method consists in flowing from a fixed point $x_0\in\RR^3$, first with the flow $X_1$ along the direction $j/|j|$ during a time $t_1$, next with the flow $X_2$ along the direction $\curl j/|\curl j|$ during a time $t_2$, finally with the flow $X_3$ along the direction $j/|j|^2\times\curl j$ during a time $t_3$. So, we obtain the triple time dynamical flow \[Xti\] (t\_1,t\_2,t\_3)X\_[32]{}(t\_3,t\_2,t\_1)=X\_3(t\_3,X\_2(t\_2,X\_1(t\_1))),X\_[32]{}(0,0,0)=x\_0, which is assumed to be a $C^1$-diffeomorphism onto $\RR^3$. Under these assumptions we prove (see Theorem \[thm.glorea\]) that the field $j$ is isotropically realizable with the conductivity $\si$ defined by \[siX\] (X\_[32]{}(t\_1,t\_2,t\_3)):=(\_0\^[t\_3]{}[|j|\^2|j|\^2]{}(X\_[32]{}(s,t\_2,t\_1))ds),(t\_1,t\_2,t\_3)\^3, This result can be regarded as a global Frobenius’ theorem, and is illustrated by the very simple current field $j$ of Section \[ss.jsh\], which yields an infinite set of (not obvious) admissible conductivities. Unfortunately, Section \[ss.jplan\] shows that the approach with the triple time dynamical flow fails for a periodic regular field $j$ of a particular form which is everywhere perpendicular to a constant vector, since $\curl j$ does vanish in $\RR^3$ (see Remark \[rem.curlj=0\]). However, in this case the divergence free field can be written as an orthogonal gradient, which allows us to apply Theorem \[thm.elerea\] for a two-dimensional electric field. When the field $j$ is periodic, the isotropic realizability in the torus needs an extra assumption as in the case of a periodic electric field [@BMT] (Theorem 2.17). Under the former conditions which ensure the isotropic realizability of $j$ in the whole space $\RR^3$, we prove (see Corollary \[cor.perrea\]) that the field $j$ is isotropically realizable in the torus, namely reads as $\si\nabla u$ with both $\si$ and $\nabla u$ periodic, if and only if \[bdX\] \_[(t\_1,t\_2)\^2]{}(\_[-]{}\^[|j|\^2|j|\^2]{}(X\_[32]{}(s,t\_2,t\_1))ds)&lt;, which is equivalent to the boundedness from below and above of the conductivity  in $\RR^3$. The sharpness of condition is illustrated by Proposition \[pro.fgh\] and Example \[rem.f\] below. The paper is divided in two parts. In Section \[s.rea\] we study the validity of the isotropic realizability of a regular divergence free field first in the whole space $\RR^3$, then in the torus when the current field is assumed to be periodic. Section \[s.exa\] is devoted to examples and counter-examples which illustrate the theoretical results of Section \[s.rea\]. ### Notations {#notations .unnumbered} - $Y:=[0,1]^3$ and $Y':=[0,1]^2$. - $\langle\cdot\rangle$ denotes the average over $Y$. - $C^k_\sharp(Y)$ denotes the space of $k$-continuously differentiable $Y$-periodic functions on $\RR^d$. - $L^2_\sharp(Y)$ denotes the space of $Y$-periodic functions in $L^2_{\rm }(\RR^d)$, and $H^1_\sharp(Y)$ denotes the space of functions $\ph\in L^2_\sharp(Y)$ such that $\nabla\ph\in L^2_\sharp(Y)^d$. - For any open set $\Om$ of $\RR^d$, $C^\infty_c(\Om)$ denotes the space of smooth functions with compact support in $\Om$, and $\D'(\Om)$ the space of distributions on $\Om$. Results of isotropic realizability {#s.rea} ================================== Realizability in the whole space -------------------------------- Let us start by the following definition: Let $j$ be a divergence free field in $L^\infty(\RR^3)^3$ – $j$ will be taken regular in the sequel – The field $j$ is said to be [*isotropically realizable in*]{} $\RR^3$ as a current field, if there exist an isotropic conductivity $\si>0$ with $\si,\si^{-1}\in L^\infty(\RR^3)$, and a potential $u\in W^{1,\infty}(\RR^3)^3$, such that $j=\si\nabla u$. Moreover, when $j$ is $Y$-periodic, $j$ is said to be [*isotropically realizable in the torus*]{} if $\si$ and $\nabla u$ can be chosen $Y$-periodic. First of all we have the following result which provides a criterion for the local isotropic realizability of a regular current field: \[pro.locrea\] Let $j$ be a vector-valued function such that \[divj=0\] jC\^2(\^3)\^3,j0\^3,j=0\^3. Then, a necessary and sufficient condition for the current field $j$ to be locally isotropically realizable in $\RR^3$ with some positive $C^1$ conductivity $\si$ is that \[j.curlj=0\] jj=0\^3. If $j$ is isotropically realizable with some conductivity $\si\in C^1(\RR^3)$, then $j=\si\nabla u$ with $\si\in C^1(\RR^3)$, and \[curlj.j=0\] j=(u)=u+(u)=j\^3, which yields immediately . Conversely, if and are both satisfied, then by Frobenius’ theorem (see, [*e.g.*]{}, [@Car], Theorem 6.6.2 and example p. 279) there exists locally a non-zero $C^1$ function $\si$ and a $C^1$ function $u$, such that $j=\si\nabla u$. The function $\si$ can be chosen positive by a continuity argument, which shows the isotropic realizability of $j$ locally in $\RR^3$. Actually, the divergence free condition is not necessary to obtain the local realizability. Frobenius’ condition implies the local isotropic realizability for a current field $j$ satisfying condition . However, contrary to the case [@BMT] of an electric field for which the local realizability and the global realizability turn out to be equivalent, these two conditions are not sufficient to ensure the global isotropic realizability of $j$, as shown by the counter-example of Section \[ss.cexF\]. To overcome this difficulty we will use an alternative approach based on the flows along the three orthogonal directions $j$, $\curl j$ and $j\times\curl j$ under suitable assumptions which are detailed below: Let $j$ be a current field satisfying conditions . Beyond condition we assume that \[ortbasj\] (j,j,jj)\^3. Then, for a fixed $x_0\in\RR^3$ and for any $(t_1,t_2,t_3)\in\RR^3$, consider the flows $X_1(t,x)$, $X_2(t,x)$, $X_3(t,x)$ along the orthogonal directions $j/|j|$, $\curl j/|\curl j|$, $j/|j|^2\times\curl j$ respectively, that is \[Xi\] { (t,x)=[j|j|]{}(X\_1(t,x)), & X\_1(0)=x,\ (t,x)=[j|j|]{}(X\_2(t,x)), & X\_2(0)=x,\ (t,x)=[jj|j|\^2]{}(X\_3(t,x)), & X\_3(0,x)=x. . (t,x)\^3. Note that the flows $X_1$ and $X_2$ are well defined in the whole set $\RR\times\RR^3$, since by ${j/|j|}$ and ${\curl j/|\curl j|}$ belong to $C^1(\RR^3)$ and are bounded in $\RR^3$ (see, [*e.g.*]{}, [@Arn], Chap. 2.6). In the sequel, we will assume that the flow $X_3$ is also defined in the whole set $\RR\times\RR^3$. That is the case if for example $j\in C^2_\sharp(Y)^3$. \[rem.X3\] In view of the normalization of the flows $X_1$, $X_2$, and to avoid the latter assumption, it seems [*a priori*]{} more logical to renormalize the flow $X_3$ with $|j||\curl j|$ rather than $|j|^2$. The derivation of the isotropic realizability is quite similar in both cases (see Theorem \[thm.glorea\] and Remark \[rem.tX3\] just below). However, the normalization by $|j|^2$ arises naturally in the orthogonal decomposition which is a key-ingredient for the construction of an admissible conductivity associated with the isotropic realizability of $j$. Moreover, it gives a necessary condition for the isotropic realizability in the torus without the need to assume that $\curl j$ does not vanish in $\RR^3$ (see the first part of Corollary \[cor.perrea\] below). Actually, there are lots of examples where $\curl j$ vanishes somewhere (see Section \[ss.jplan\] below), but the normalization of the flow $X_3$ by $|j|^2$ may be relevant in some cases (see Proposition \[pro.jz=0\] and Remark \[rem.curlj=0\]). Next, denote for a fixed point $x_0\in\RR^3$, \[X1X23\] { X\_1(t\_1):=X\_1(t\_1,x\_0)\ X\_[23]{}(s\_2,s\_3,t\_1):=X\_2(s\_2,X\_3(s\_3,X\_1(t\_1,x\_0)))\ X\_[32]{}(t\_3,t\_2,t\_1):=X\_3(t\_3,X\_2(t\_2,X\_1(t\_1,x\_0))), . (s\_1,s\_2,s\_3,t\_1,t\_2,t\_3)\^6. So, the dynamical flow $X_{32}$ is obtained by flowing from the point $x_0$ along the direction $j/|j|$ during the time $t_1$, then from the point $X_1(t_1)$ along the direction $\curl j/|\curl j|$ during the time $t_2$, finally from the point $X_2(t_2,X_1(t_1))$ along the direction $(j/|j|^2)\times\curl j$ during the time $t_3$. The end point is thus $X_{32}(t_3,t_2,t_1)$. A similar construction holds for $X_{23}(t_2,t_3,t_1)$ by commuting the flows $X_2$ and $X_3$. Now, the main assumption is that any point $x$ in $\RR^3$ can be attained by the composition of the three flows, so that $x$ can be represented in a unique way by the system of coordinates $(t_1,t_2,t_3)$, that is \[diffX32\] (t\_1,t\_2,t\_3)X\_[32]{}(t\_3,t\_2,t\_1) Then, we have the following sufficient condition for the global isotropic realizability in $\RR^3$: \[thm.glorea\] Let $j$ be a field in $\RR^3$ satisfying and . Also assume that condition holds true. Then, the field $j$ is isotropically realizable in $\RR^3$ with the conductivity $e^w\in C^1(\RR^3)$, where the function $w$ is defined by \[w(x)\] w(x)=w(X\_[32]{}(t\_3,t\_2,t\_1)):=\_0\^[t\_3]{}[|j|\^2|j|\^2]{}(X\_[32]{}(s,t\_2,t\_1))ds,(t\_1,t\_2,t\_3)\^3. \[rem.tX3\] In view of Remark \[rem.X3\], if we renormalize the flow $X_3$ by $|j||\curl j|$, it is replaced by the flow $\tilde{X}_3$ defined by \[tX3\] [\_3t]{}(t,x)=[jj|j||j|]{}(\_3(t,x)),\_3(0,x)=x, which by condition is defined in $\RR\times\RR^3$. Then, similarly to , assuming that the triple flow \[tX32\] \_[32]{}:(t\_1,t\_2,t\_3)\_3(t\_3,X\_2(t\_2,X\_1(t\_1,x\_0)) is a $C^1$-diffeomorphism onto $\RR^3$, we obtain that the field $j$ is isotropically realizable in $\RR^3$ with the conductivity $e^{\tilde{w}}\in C^1(\RR^3)$, where \[tw(x)\] (x)=w(\_[32]{}(t\_3,t\_2,t\_1)):=\_0\^[t\_3]{}[|j||j|]{}(\_[32]{}(s,t\_2,t\_1))ds,(t\_1,t\_2,t\_3)\^3. The proof is quite similar to the proof of Theorem \[thm.glorea\] replacing formula by \[Dtwjxcurlj\] ([jj|j||j|]{})=[|j||j|]{} =([jj|j|\^2]{})([jj|j||j|]{})\^3. \[rem.homX32\] Alternatively, we can replace the diffeomorphism condition by \[homX32\] (t\_1,t\_2,t\_3)X\_[32]{}(t\_3,t\_2,t\_1) so that the Jacobian of $X_{32}$ may vanish somewhere. In compensation we have to assume that the function $w$ of belongs to $C^1(\RR^3)$. See the application to the example of Section \[ss.jsh\]. Condition is not sharp to ensure the isotropic realizability of the current field. Indeed, the planar example of Proposition \[pro.jz=0\] below shows that the isotropic realizability can be satisfied while condition is violated. See also Example \[rem.f\] below. [**Proof of Theorem \[thm.glorea\].**]{} ß[*First step:* ]{} Construction of an admissible conductivity. ßLet $j$ be a field satisfying . Assume that $j$ is isotropically realizable in $\RR^3$, namely there exists $u,w\in C^1(\RR^3)$ such that \[juw\] j=e\^wu\^3. It seems that we can choose the potential $u$ arbitrarily along the trajectory $X_1(t)$, provided $\nabla u$ does not vanish along this trajectory. So, define \[ut1\] u(X\_1(t)):=\_0\^[t]{}|j(X\_1(s))|ds,t. Taking the derivative with respect to $t$ of and using , , we get that =|j(X\_1(t))|=u(X\_1(t))X\_1(t) =e\^[-w(X\_1(t))]{}[jj|j|]{}(X\_1(t)), which implies that \[wXt1\] w(X\_1(t))=0,t. Next, taking the curl of we get that $\curl j=\nabla w\times j$, hence \[jxcurljDw\] [jj|j|\^2]{}=w-\_j(w)\^3, where $\Pi_j$ is the orthogonal projection on the subspace $\RR j$. Hence, integrating along the trajectory $X_3$, we have $$\ba{l} \dis w\big(X_{32}(t_3,t_2,t_1)\big)-w\big(X_{32}(0,t_2,t_1)\big)=w\big(X_{32}(t_3,t_2,t_1)\big)-w\big(X_2(t_2,X_1(t_1))\big) \\ \ecart \dis =\int_0^{t_3}\nabla w\big(X_{32}(s,t_2,t_1)\big)\cdot{\partial X_{32}\over\partial s}(s,t_2,t_1)\,ds =\int_0^{t_3}\left|{j\times\curl j\over|j|^2}\right|^2\big(X_{32}(s,t_2,t_1)\big)\,ds \\ \ecart \dis =\int_0^{t_3}{|\curl j|^2\over|j|^2}\big(X_{32}(s,t_2,t_1)\big)\,ds. \ea$$ Then, integrating along the trajectory $X_2$ and using , we get that $$\ba{l} \dis w\big(X_2(t_2,X_1(t_1))\big)-w\big(X_2(0,X_1(t_1))\big)=w\big(X_2(t_2,X_1(t_1))\big)-w\big(X_1(t_1)\big) \\ \ecart \dis =\int_0^{t_2}\nabla w\big(X_2(s,t_1)\big)\cdot{\partial X_2\over\partial s}(s,t_1)\,ds =\int_0^{t_2}\nabla w\big(X_2(s,t_1)\big)\cdot{\curl j\over|\curl j|}\big(X_2(s,t_1)\big)\,ds=0. \ea$$ The two previous equalities combined with yield the desired expression . [*Second step:* ]{} Construction of a grid on the surface $\{t_1=c_1\}$, generated by the flows $X_2,X_3$. ßLet us prove that for any $(c_1,t_2',t_2,t_3)\in\RR^4$, the flows $X_2$ and $X_3$ generate on the regular surface $\{t_1\!=\!c_1\}$, a thin grid whose: - step is of small enough size $\nu>0$, - horizontal lines are trajectories along the flow $X_2$, - vertical lines are trajectories along the flow $X_3$, - two opposite vertices are the points \[x’x\] x’:=X\_2(t\_2’,X\_1(c\_1))x:=X\_3(t\_3,X\_2(t\_2,x’))=X\_[32]{}(t\_3,t\_2’+t\_2,c\_1). First, we divide the flows in small time steps, as shown the diagram \[x’hvx\] & -.2cm x\ & -.2cm\ & -.2cm\ & -.2cm\ & -.2cm\ & -.2cm\ & -.2cm\ & -.2cm\ x’ & -.2cm where the horizontal arrows represent the flow $X_2$, and the vertical ones the flow $X_3$. Then, we commute step by step the flows $X_2$ and $X_3$, while remaining on the surface $\{t_1\!=\!c_1\}$, as shown the commutation diagram \[compq\] & & q & & & & & & q &\ & & & -.3cm X\_3 & & X\_3 -.3cm & & & & -.3cm X\_3\ p & \_[X\_2]{} & & & & & p & \_[X\_2]{} & & to finally obtain the desired grid \[Gx’xy’y\] y’ & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm x\ & & -.2cm & & -.2cm & & -.2cm & & -.2cm & & -.2cm\ & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm\ & & -.2cm & & -.2cm & & -.2cm & & -.2cm & & -.2cm\ & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm\ & & -.2cm & & -.2cm & & -.2cm & & -.2cm & & -.2cm\ & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm\ & & -.2cm & & -.2cm & & -.2cm & & -.2cm & & -.2cm\ x’ & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm y where by there exists $(s_2,s_3)\in\RR^2$ such that \[x’y’yx\] x’:=X\_2(t\_2’,X\_1(c\_1)),y’=X\_3(s\_3,x’),y:=X\_2(t\_2,x’),x=X\_3(t\_3,y)=X\_2(s\_2,y’). The vertices $x',y',y,x$ of the grid satisfy the commutative diagram \[X23=X32\] y’=X\_3(s\_3,x’) & & X\_2(s\_2,y’)= & -.3cm X\_3(t\_3,y)=x\ X\_3 & & & -.2cm X\_3\ x’=X\_2(t\_2’,X\_1(c\_1)) & \_[X\_2]{} & & -.3cm X\_2(t\_2,x’)=y. Note that the grid is schematic. A more realistic grid is represented in figure \[fig1\] below. Frobenius’ theorem will allow us to make the local switching thanks to a potential $u$ satisfying $j=\si\nabla u$. To this end, we proceed by induction on the number $n$ of switchings, for an appropriate time step $\nu>0$ which will be chosen later. The induction hypothesis, for a given $n\in\NN$, consists in the existence of a partial grid $G_n(\nu)$, with $n$ switchings, represented by the diagram \[x’phqx\] & & & & & & & & & & -.2cm x\ & & & & & & & & & & -.2cm\ & & & & & & & & & & -.2cm\ & & & & & & & & & & -.2cm\ & & & & & -.2cm & -.2cm q & -.2cm & -.2cm & -.2cm & -.2cm\ & & & & -.2cm & & -.2cm & & -.2cm & & -.2cm\ & -.2cm & -.2cm & -.2cm & -.2cm p & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm\ -.2cm & & -.2cm & & -.2cm & & -.2cm & & -.2cm & & -.2cm\ x’ & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm which lies on the surface $\{t_1\!=\!c_1\}$. First, the result holds for $n=0$. Indeed, the points $x'$ and $x$ defined by clearly belong to the surface $\{t_1\!=\!c_1\}$, so does the initial diagram for any time step $\tau$. Next, assume that after a number $n$ of switchings, we are led to the grid $G_n(\nu)$ which, by the induction hypothesis, lies on the surface $\{t_1\!=\!c_1\}$. By virtue of Frobenius’ theorem there exist an open neighborhood $V$ of $p$, and a potential $u\in C^1(V)$ such that $j=\si\nabla u$ in $V$. Then, we may chose $\tau>0$ small enough so that \[S32S23V\] q-.2cm & S\_[32]{}(p,):={X\_3(\_3,X\_2(\_2,p)):|\_2|+|\_3|&lt;}V\ & S\_[23]{}(p,):={X\_2(\_2,X\_3(\_3,p)):|\_2|+|\_3|&lt;}V, independently of the point $p$ in a given compact set of $\RR^3$. Since the potential $u$ is constant along the flows $X_2$ and $X_3$, we have for any point $r\in S_{32}(p,\tau)$, $$u(r)=u\big(X_3(\de_3,X_2(\de_2,p))\big)=u\big(X_3(0,X_2(\de_2,p))\big)=u\big(X_2(\de_2,p)\big)=u\big(X_2(0,p)\big)=u(p).$$ The same equality holds for any point $r\in S_{23}(p,\tau)$, hence \[S32S23u=up\] qS\_[32]{}(p,){u=u(p)}VS\_[23]{}(p,){u=u(p)}V. Moreover, since the trajectory along the flow $X_2$, passing through the point $p$ in diagram lies on the surface $\{t_1\!=\!c_1\}$ (by the induction hypothesis), thanks to the semi-group property satisfied by the flow $X_3$, we have that for any $|\de_2|+|\de_3|<t$, there exists $(t_2,t_3)\in\RR^2$ such that X\_3(\_3,X\_2(\_2,p))=X\_3(\_3,X\_[32]{}(t\_3,t\_2,c\_1))=X\_[32]{}(\_3+t\_3,t\_2,c\_1){t\_1=c\_1}. Hence, by the definition of $S_{32}(p,\tau)$, we get that \[S32t1=c1\] t&gt;0,S\_[32]{}(p,t){t\_1=c\_1}. However, thanks to the condition which yields $\nabla u\neq 0$ in $V$, the sets $S_{32}(p,\tau)$, $S_{23}(p,\tau)$ and $\{u=u(p)\}\cap V$ are regular open surfaces in $V$. Hence, from the inclusions we deduce that the surfaces $S_{32}(p,\tau)$ and $S_{23}(p,\tau)$ actually agree with the equipotential $\{u=u(p)\}$ in some neighborhood of $p$ containing $q$, but independent of $p$ in a given compact set $K$ of $\RR^3$. This combined with the condition (which does not depend on time), implies that the time step $\nu<\tau$ may be chosen small enough, but independently of the point $p$ in $K$, so that qS\_[23]{}(p,){t\_1=c\_1}. Therefore, the following grid $G_{n+1}(\nu)$ which completes \[xpvqx’\] & & & & & & & & & & -.2cm x\ & & & & & & & & & & -.2cm\ & & & & & & & & & & -.2cm\ & & & & & & & & & & -.2cm\ & & & & -.2cm & -.2cm & -.2cm q & -.2cm & -.2cm & -.2cm & -.2cm\ & & & & -.2cm & & -.2cm & & -.2cm & & -.2cm\ & -.2cm & -.2cm & -.2cm & -.2cm p & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm\ -.2cm & & -.2cm & & -.2cm & & -.2cm & & -.2cm & & -.2cm\ x’ & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm & -.2cm also lies on the surface $\{t_1=c_1\}$. The induction proof is thus done, which establishes the existence of the grid . [*Third step:* ]{} Proof of the isotropic realizability with the conductivity $e^w$. ßLet us prove that the function $w$ defined by combined with condition satisfies the equality . This yields the global isotropic realizability of $j$, since by , and we have \[curlewj\] (e\^[-w]{}j)=e\^[-w]{}(j-wj)=e\^[-w]{}(j-[(jj)j|j|\^2]{})=0. On the one hand, taking the derivative of with respect to $t_3$, we have w([jj|j|\^2]{})(X\_[32]{}(t\_3,t\_2,t\_1))=[|j|\^2|j|\^2]{}(X\_[32]{}(t\_3,t\_2,t\_1)),(t\_1,t\_2,t\_3)\^3, which together with condition implies that \[Dwjxcurlj\] w([jj|j|\^2]{})=[|j|\^2|j|\^2]{}=|[jj|j|\^2]{}|\^2\^3. Moreover, taking the derivative of with respect to $t_2$ for $t_3=0$, we get that $$\ba{ll} \dis 0=\nabla w\big(X_{32}(0,t_2,t_1)\big)\cdot{\partial X_{32}\over\partial t_2}(0,t_2,t_1) & \dis =\nabla w\big(X_2(t_2,X_1(t_1))\big)\cdot {\partial X_2\over\partial s}(t_2,X_1(t_1)) \\ \ecart & \dis =\nabla w\big(X_2(t_2,X_1(t_1))\big)\cdot{\curl j\over|\curl j|}\big(X_2(t_2,X_1(t_1))\big), \ea$$ which yields \[Dwcurljt3=0\] wj=0{X\_2(t\_2,X\_1(t\_1)):(t\_1,t\_2)\^2}. ![*The square $Q$ split in small squares $Q_k$ on the surface $\{t_1=c_1\}$*[]{data-label="fig1"}](Surface_t1.eps) On the other hand, for a constant $c_1\in\RR$, consider on the surface $\{t_1=c_1\}$ the curvilinear integral over $\partial Q$ defined by \[intQdom\] \_ :=\_[0]{}\^[t\_3]{}[|j|\^2|j|\^2]{}(X\_3(s,y))ds -\_[0]{}\^[s\_3]{}[|j|\^2|j|\^2]{}(X\_3(s,x’))ds, where $Q=Q_{x',y',y,x}$, with $x',y',y,x$ defined by , is the “square" lying on the surface $\{t_1=c_1\}$, whose: - horizontal sides are trajectories associated with the flow $X_2$ along $\curl j/|\curl j|$, - vertical sides are trajectories associated with the flow $X_3$ along $j/|j|^2\times\curl j$, - vertices $x',y',y,x$ make the loop . Now, consider the thin grid on $Q$, associated with the grid , whose lines are parallel to the trajectories along $\curl j/|\curl j|$ and $(j/|j|^2)\times\curl j$, and which splits $Q$ in small squares $Q_k$, as shown in .. According to the second step of the proof, we may choose the time step of the grid so that Frobenius’ theorem applies in some open neighborhood $V_k$ of $\RR^3$, containing the closed square $\overline{Q_k}$. Namely, there exists $w_k\in C^1(V_k)$ such that $$\curl(e^{-w_k}\,j)=0\quad\mbox{in }V_k,$$ which implies formulas and for $w_k$. Using the loops induced by the boundaries $\partial Q_k$, the contributions of the interior vertical lines (along $j/|j|^2\times\curl j$) two by two cancel (see .), which leads to \[intomQQk\] \_ =\_[k]{}\_ . However, since $$\nabla w_k\cdot\curl j=0\quad\mbox{in }V_k\supset\overline{Q_k},$$ the curvilinear integrals of $dw_k$ on the two horizontal lines of $\partial Q_k$ (along $\curl j/|\curl j|$) are equal to $0$, while by putting in the curvilinear integrals of $dw_k$ on the two vertical lines of $\partial Q_k$ (along $j/|j|^2\times\curl j$) agree with the curvilinear integral of ${\om}$ over $\partial Q_k$, which yields \[intomQk=0\] \_ =\_ dw\_k=0. The last equality is due to the fact that $dw_k$ is an exact differential on the closed loop $\partial Q_k$. Hence, from and we deduce that \[intQom=0\] \_ =0. Next, consider the function $w$ defined by . Taking the loop in the anti-clockwise direction, the contributions of the vertical lines (along $(j/|j|^2)\times\curl j$) of the curvilinear integral of $dw$ over $\partial Q$ in the anti-clockwise direction, read as, in view of and , \[verlin\] \_[0]{}\^[t\_3]{}w([jj|j|\^2]{})(X\_3(s,y))ds -\_[0]{}\^[s\_3]{}w([jj|j|\^2]{})(X\_3(s,x’))ds =\_ . Moreover, again by the contributions of the horizontal lines (along $\curl j/|\curl j|$) of the curvilinear integral of $dw$ over $\partial Q$ in the anti-clockwise direction, read as \[horlin\] \_[0]{}\^[t\_2]{}w(X\_2(t\_2,x’))ds -\_[0]{}\^[s\_2]{}w(X\_2(s,y’))ds. From and we deduce that $$\ba{ll} \dis 0 & \dis =\int_{\stackrel{\curvearrowleft}{\partial Q}} dw \\ \ecart & \dis =\int_{\stackrel{\curvearrowleft}{\partial Q}} {\om} +\int_{0}^{t_2}\nabla w\cdot{\curl j\over|\curl j|}\big(X_2(t_2,x')\big)\,ds -\int_{0}^{s_2}\nabla w\cdot{\curl j\over|\curl j|}\big(X_2(s,y')\big)\,ds, \ea$$ which by implies that \[intDwcurlj\] \_[0]{}\^[s\_2]{}w(X\_2(s,y’))ds =\_[0]{}\^[t\_2]{}w(X\_2(t\_2,x’))ds. Equality combined with and yields \[intDwcurlj2\] [1s\_2]{}\_[0]{}\^[s\_2]{}w(X\_2(s,X\_3(s\_3,x’)))ds=0,x’=X\_2(t\_2’,X\_1(c\_1)). Then, making $t_2\to 0$ – which, by the continuity of the flows in the diagram , implies that $s_2\to 0$ and $X_3(s_3,x')\to X_3(t_3,x')$ – and using the continuity of the integrand in the left-hand side of , we get that wj(X\_[32]{}(t\_3,t\_2’,c\_1))=wj(X\_3(t\_3,x’))=0,(c\_1,t\_2’,t\_3)\^3. This combined with the diffeomorphism condition leads to \[Dw.curlj=0\] wj=0\^3. Finally, combining the equalities and with , we derive the formula . Therefore, thanks to we may conclude that the current field $j$ is isotropically realizable with the conductivity $e^w$ defined by , which concludes the proof of Theorem \[thm.glorea\]. Isotropic realizability in the torus ------------------------------------ As in the case of an electric field [@BMT] the isotropic realizability of a periodic current field in $\RR^3$ does not imply in general the isotropic realizability in the torus, as shown in Example \[rem.f\] below. First of all, we have the following criterion of isotropic realizability in the torus: \[pro.perrea\] Let $j$ be a $Y$-periodic divergence free field in $L^\infty_\sharp(Y)^3$. Then, the field $j$ is isotropically realizable in the torus with a conductivity $\si>0$ satisfying $\si,\si^{-1}\in L^\infty_\sharp(Y)$, if and only if there exists $w\in L^\infty(\RR)$ such that \[curlwj\] (e\^[-w]{}j)=0\^3. Assume that $j$ is isotropically realizable in the torus with a conductivity $\si>0$ satisfying $\si,\si^{-1}\in L^\infty_\sharp(Y)$. Then, defining $w:=\ln\si\in L^\infty(\RR^3)$, the function $e^{-w}j$ is a gradient, which implies . Conversely, assume that holds with $w\in L^\infty(\RR)$. Then, there exists $u\in W^{1,\infty}(\RR^3)$ such that $e^{-w}j=\nabla u$, or equivalently $j=e^w\,\nabla u$ in $\RR^3$. It is not clear that $e^w$ is periodic. However, we can construct a suitable periodic conductivity by adapting the average argument of [@BMT] (proof of Theorem 2.17). To this end, define the sequence $(w_n)_{n\in\NN\setminus\{0\}}$, by $$w_n(x):={1\over (2n+1)^3}\kern -.2cm\sum_{k\in\ZZ^3,\,|k|_\infty\leq n}\kern -.4cm w(x+k), \quad\mbox{for }x\in\RR^3,\quad\mbox{where }|k|_\infty:=\max\left(|k_1|,|k_2|,|k_3|\right).$$ Since $w$ is in $L^\infty(\RR^3)$, the sequence $w_n$ is bounded in $L^\infty(\RR)$, and thus converges weakly-$*$ to some function $w_\sharp$ in $L^\infty(\RR^3)$. It is easy to check that $w_\sharp$ is $Y$-periodic. Moreover, by the periodicity of $j$ we have $$\curl(e^{-w_n}j)={1\over (2n+1)^3}\kern -.2cm\sum_{k\in\ZZ^3,\,|k|_\infty\leq n}\kern -.4cm\curl\big(e^{-w(\cdot+k)}j(\cdot+k)\big) ={1\over (2n+1)^3}\kern -.2cm\sum_{k\in\ZZ^3,\,|k|_\infty\leq n}\kern -.4cm\curl\big(\nabla u(\cdot+k)\big)=0.$$ As $e^{-w_n}j$ converges weakly-$*$ to $e^{-w_\sharp}j$ in $L^\infty(\RR^3)^3$, the previous equality leads to $$\curl(e^{-w_\sharp}j)=0\quad\mbox{in }\D'(\RR^3)^3.$$ Therefore, $e^{-w_\sharp}j$ is a periodic gradient and $j$ is isotropically realizable in the torus with the conductivity $e^{w_\sharp}$. As a consequence of Proposition \[pro.perrea\] and Theorem \[thm.glorea\], we have the following result: \[cor.perrea\] Let $j$ be a $Y$-periodic current field satisfying conditions and . - Assume that the field $j$ is isotropically realizable in the torus with a positive conductivity $\si\in C^1_\sharp(Y)$. Then, there exists a constant $C>0$ such that \[L2X3’\] \_[-]{}\^[|j|\^2|j|\^2]{}(X\_3(s,x))dsC,x\^3, where the flow $X_3$ is defined by . - Alternatively, assume that the conditions and are satisfied. Then, the field $j$ is isotropically realizable in the torus if and only if estimate holds true. Corollary \[cor.perrea\] with the boundedness condition  is illustrated in Proposition \[pro.fgh\] below. [**Proof of Corollary \[cor.perrea\].**]{} ß[*Proof of $i)$.*]{} Denoting $\si=e^w\in C^1(\RR^3)$, we have $j=e^w\nabla u$, with $\nabla u\in C^1(\RR^3)^3$. Then, from we deduce that $${\partial\over\partial t}\left[w\big(X_3(t,x)\big)\right]={|\curl j|^2\over |j|^2}\big(X_3(t,x)\big),\quad\forall\,(t,x)\in\RR\times\RR^3,$$ which yields \[wdX\] w(X\_3(t,x))-w(x)=\_0\^[t]{}[|j|\^2|j|\^2]{}(X\_3(s,x))ds,(t,x)\^3. Therefore, due to the boundedness of $w\in C^1_\sharp(Y)$, the estimate holds. [*Proof of $ii)$.*]{} By Theorem \[thm.glorea\] we already know that $j$ is isotropically realizable with the conductivity $\si=e^w\in C^1(\RR^3)$ defined by . If the field $j$ is isotropically realizable in the torus, then by $i)$ the estimate holds. Conversely, in view of the estimate combined with the definition of $X_{32}$, the function $w$ defined by clearly belongs to $L^\infty(\RR^3)$. Therefore, applying Proposition \[pro.perrea\] with the conductivity $e^w$, we get that $j$ is isotropically realizable in the torus. Examples and counter-examples {#s.exa} ============================= In this section a point of $\RR^3$ is denoted by the coordinates $(x,y,z)$, and $(e_x,e_y,e_z)$ denotes the canonical basis of $\RR^3$. Example of global realizability in the space {#ss.jsh} -------------------------------------------- We will illustrate the construction of Theorem \[thm.glorea\] with the current field \[jsh\] j(x,y,z):= 1\ x\ 0 ,(x,y,z)\^3, which is clearly non-zero and divergence free in $\RR^3$. We have j(x,y,z):= 0\ 0\ x jj(x,y,z)= xx\ -x\ 0 , so that the field $j$ satisfies and Frobenius’ condition . Noting that $|j|=|\curl j|=|\cosh x|$, and using the solutions to one-dimensional first-order odes, the flows $X_1$, $X_2$, $X_3$ of associated with the field $j$ are given by X\_1(t)= [argsinh]{}(t+(x\_1(0)))\ +y\_1(0)-(x\_1(0))\ z\_1(0) ,X\_2(t)= x\_2(0)\ y\_2(0)\ t+z\_2(0) ,\ X\_3(t)= [argsinh]{}(e\^t(x\_3(0)))\ -\_0\^[t]{}[ds]{}+y\_3(0)\ z\_3(0) . Hence, starting from the point $x_0=(0,1,0)$, the composed flows $X_{32}$, $X_{23}$ of are given by \[X23sh\] X\_[32]{}(t\_3,t\_2,t\_1)= [argsinh]{}(t\_1e\^[t\_3]{})\ -\_0\^[t\_3]{}[ds]{}+\ t\_2 , (t\_1,t\_2,t\_3)\^3. Making classical changes of variables in the integral term, formula can be written \[X23shf\] X\_[32]{}(t\_3,t\_2,t\_1)={ 0\ 1-t\_3\ t\_2 & t\_1=0\ \ [argsinh]{}(t\_1e\^[t\_3]{})\ ([1]{})+f(t\_1)\ t\_2 & t\_10, . where \[shf\] f(t):=-[argtanh]{}([1]{})+,t0. The function $f$ satisfies \[shff’\] f(t)=|t|+1-2+o(1)f’(t)=[t]{}=[1t]{}+o(1), and is a $C^1$-diffeomorphism from $(0,\infty)$ onto $\RR$, and from $(-\infty,0)$ onto $\RR$. Hence, we have \[fdiff\] (x,y){0}, !t\_1,xt\_1&gt;0y-[argtanh]{}([1x]{})=f(t\_1), which together with implies that for this $t_1$, \[xyzX32\] (x,y,z)=X\_[32]{}(t\_3,t\_2,t\_1){ { x=0\ y=1-t\_3\ z=t\_2 . & t\_1=0\ { x=t\_1e\^[t\_3]{}\ y-[argtanh]{}([1x]{})=f(t\_1)\ z=t\_2. . & t\_10. . As a consequence of , , the first equality of , and , the mapping $X_{32}$ define a homeomorphism onto $\RR^3$, which is of class $C^1$ by . Unhappily, we can check that the Jacobian of $X_{23}$ vanishes (exclusively) on the line $\{t_1=0\}$. However, taking into account the equality $\sinh x=t_1\,e^{t_3}$, $X_{23}$ does establish a $C^1$-diffeomorphism from the half-spaces $\{\pm\,t_1>0\}$ onto the half-spaces $\{\pm\,x>0\}$. Therefore, condition holds true restricting ourselves on these half-spaces. On the other hand, since $|j|=|\curl j|$, the function $w$ defined by reads as \[wsh\] w(x,y,z)=t\_3={ 1-y & x=0 &\ ([xt]{}) & x0, & xt&gt;0, y-[argtanh]{}([1x]{})=f(t). . It is easy to check that the asymptotic expansions satisfied by $f$ imply that $w\in C^1(\RR^3)$. Hence, the conditions of Theorem \[thm.glorea\] are fulfilled in the two half-spaces $\{\pm\,x>0\}$. Therefore, the field $j$ defined by is isotropically realizable in the half-spaces $\{\pm\,x>0\}$, with the conductivity $\si\in C^1(\RR^3)$ given by \[sish\] (x,y,z):={ e\^[1-y]{} & x=0 &\ & x0, & xt&gt;0, y-[argtanh]{}([1x]{})=f(t). . Finally, the $C^1$-regularity of $j$ and $\si$ ensure the isotropic realizability in the whole space $\RR^3$. Since $X_{32}$ is a homeomorphism onto $\RR^3$ of class $C^1$ and the function $w$ of is in $C^1(\RR^3)$, we can also conclude thanks to Remark \[rem.homX32\]. \[rem.jsh\] The previous example allows us to show that there exists an infinite set of admissible conductivities which cannot be derived from a multiple of the conductivity defined with the function . To this end, consider any function $f:\RR\setminus\{0\}\to\RR$ which is a $C^1$-diffeomorphism from $(0,\infty)$ onto $\RR$ and from $(-\infty,0)$ onto $\RR$, which satisfies the following asymptotic expansions around $0$: \[expff’\] f(t)=|t|+c+o(1)f’(t)=[1t]{}+c’+o(1),c,c’. Then, define the conductivity $\si_f$ by \[sif\] \_f(x,y,z):={ 2e\^[c-y]{} & x=0 &\ & x0, & xt&gt;0, y-[argtanh]{}([1x]{})=f(t). . Thanks to , and the function $\si_f$ belongs to $C^1(\RR)$. Moreover, we have for any $(x,y)\in\RR\setminus\{0\}\times\RR$, (\_f\^[-1]{}j)= \ t\ 0 = 0\ 0\ -[1x]{}[ty]{} . This combined with -[x]{}=[1x]{}=[tx]{}f’(t) 1=[ty]{}f’(t), yields that $\curl(\si_f^{-1}j)=0$ in $\RR\setminus\{0\}\times\RR$. Since $\si_f$ is in $C^1(\RR)$, the equality holds in $\RR^2$. Therefore, the field $j$ defined by is isotropically realizable in $\RR^3$ with any conductivity $\si_f$ defined by and . Example of non-global realizability under Frobenius’ condition {#ss.cexF} -------------------------------------------------------------- The following example shows that condition is not sufficient to derive a global realizability result in accordance with the local character of Frobenius’ theorem. This example is an extension to a divergence free non-vanishing field in $\RR^3$ of the counterexample of [@Boy] obtained for a non-vanishing field in $\RR^2$. Define the function $j$ in $\RR^3$ by \[jcex\] j(x,y,z):= f(x)\ f’(x)\ -zf’(x) ,f(x):=8x\^3-6x\^4-1, which satisfies condition . We have $$\curl j(x,y,z)=\begin{pmatrix} 0 \\ z\,f''(x) \\ f''(x) \end{pmatrix},$$ so that Frobenius’ condition holds, but not since $f''(\sqrt{2/3})=0$. Now, assume that $j$ can be written $\sigma\nabla u$ for some positive continuous function $\sigma$ in the closed strip $\{0\leq x\leq 1\}$. We have for any $x\in(0,1)$, $f'(x)\neq 0$ and \[fu\] { f(x)=[ux]{}\ f’(x)=[uy]{}, . =[f(x)f’(x)]{}[uy]{}. Then, the method of characteristics implies that for a fixed $z\in\RR$, there exists a function $g_z$ in $C^1(\RR)$ such that \[ugz\] u(x,y,z)=g\_z(y+F(x)),(x,y)(0,1), where $F$ is the primitive of $f/f'$ on $(0,1)$ defined by \[F\] F(x):=[x\^28]{}-[x12]{}-[(1-x)24]{}-[x24]{}+[124x]{},x(0,1). As $x\to 0$, $x>0$, we have by $$t:=y+F(x)\,\mathop{\sim}_{x\to 0}{1\over 24\,x}.$$ This combined with , , yields $$g'_z(t)={1\over\sigma(x,y,z)}\,f'(x)\,\mathop{\sim}_{t\to\infty}\,{1\over 24\,\sigma(0,y,z)}\,{1\over t^2}.$$ As $x\to 1$, $x>1$, we have by $$t=y+F(x)=y+{1\over 12}-{\ln(1-x)\over 24}+o(1),\quad\mbox{hence}\quad 1-x \,\mathop{\sim}_{t\to\infty}\,e^{24y+2}\,e^{-24t},$$ which implies that $$g'_z(t)={1\over\sigma(x,y,z)}\,f'(x)\,\mathop{\sim}_{t\to\infty}\,{24\,e^{24y+2}\over\sigma(1,y,z)}\,e^{-24t}.$$ Therefore, the two asymptotic expansions of $g_z(t)$ as $t\to\infty$, lead to a contradiction. The case where the current field lies in a fixed plane {#ss.jplan} ------------------------------------------------------ Consider a periodic field $j$ in $C^2_\sharp(Y)^3$ satisfying and which remains perpendicular to a fixed direction. By an orthogonal change of variables we may assume that $j$ lies in the plane $\{z=0\}$, namely $j=(j_x,j_y,0)$. From now on, any vector of $\RR^2$ will be identified to a vector of $\RR^3$ with zero $z$-coordinate. Hence, we deduce that \[j.curlj=02d\] jj=0\_z jj\^3. Then, using the representation of a divergence-free field by an orthogonal gradient in $\RR^2$, the most general expression for a divergence free field $j$ satisfying is \[jF2d\] j(x,y,z)=(x,y,z)\^v(x,y)=(x,y,z) -\_y v(x,y)\ \_x v(x,y)\ 0 & (x,y,z)\^3,\ \_x\_y v-\_y\_x v=0 & \^3, where $\al\in C^2_\sharp(Y')$, $\al>0$, and $v\in C^3(\RR^2)^2$, with $\nabla v$ $Y'$-periodic. For the sake of simplicity, we assume from now on that the function $\al$ only depends on the coordinate $z$. Therefore, we are led to \[jz=0\] j(x,y,z)=(z)\^v(x,y),(x,y,z)\^3, where $\al\in C^2_\sharp([0,1])$, $\al>0$, and $v\in C^3(\RR^2)^2$, with $\nabla v$ $Y'$-periodic and $\nabla v\neq 0$ in $\RR^2$ by . Following the isotropic realizability procedure for a gradient field [@BMT], consider the gradient system \[Z\] { Z’(t,x,y)=v(Z(t,x,y)), & t,\ Z(0,x,y)=(x,y)\^2. & . Then, by virtue of Theorem \[thm.elerea\] there exists a unique function $\tau_v\in C^1(\RR^2)$ such that \[tauv\] v(Z(\_v(x,y),x,y))=0,(x,y)\^2, and the function $w_v$ defined by \[wv\] w\_v(x,y):=-\_0\^[\_v(x,y)]{}v(Z(s,x,y))ds,(x,y)\^2, satisfies the conductivity equation $\div(e^{-w_v}\nabla v)=0$ in $\RR^2$. We have the following isotropic realizability result: \[pro.jz=0\] If the function $w_v$ of is in $L^\infty(\RR^2)$, then the field $j$ defined by is isotropically realizable in the torus. Conversely, if $j$ is isotropic realizable in the torus with a positive conductivity $\si\in C^1_\sharp(Y)$, then the function $w_v$ is in $L^\infty(\RR^2)$. On the other hand, if $w_v$ belongs to $L^\infty(\RR^2)$, then condition holds. \[rem.curlj=0\] The criterion for the isotropic realizability of $j$ in the torus given in Proposition \[pro.jz=0\], that is the boundedness of $w_v$, implies condition . The converse is not clear. The reason is that the field $j$ defined by does not satisfy condition . More precisely, the curl of $j$ \[curljjz=0\] j(x,y,z)=-’(z)v(x,y)+(z)v(x,y)e\_z,(x,y,z)\^3, does vanish in $\RR^3$. Indeed, due to the periodicity of $\al$ and $\nabla v$ we have $$\int_0^1\al'(z)\,dz=\int_{Y'}\De v(x,y)\,dx dy=0.$$ This combined with the continuity of $\al'$ and $\De v$ implies the existence of a point $(x,y,z)\in\RR^3$ such that $\al'(z)=\De v(x,y)=0$. Therefore, we cannot use Theorem \[thm.glorea\]. [**Proof of Proposition \[pro.jz=0\].**]{} Assume that the function $w_v$ of is in $L^\infty(\RR^2)$. Then, by Theorem \[thm.elerea\] the gradient field $\nabla v$ is isotropically realizable in the torus, namely there exists a periodic positive conductivity $\si$, with $\si,\si^{-1}\in L^\infty_\sharp(Y')$, such that $\div(\si\nabla v)=0$ in $\RR^2$. Hence, there exists a stream function $u$, with $\nabla u\in L^2_\sharp(Y')^2$, such that $\si\nabla v=-\nabla^\perp u$ in $\RR^2$. Therefore, by $j=\al\,\si^{-1}\nabla u$ is isotropically realizable in the torus. Conversely, assume that $j$ is isotropically realizable in the torus with a positive conductivity $\si\in C^1_\sharp(Y)$. Then, since $j_z=0$, we have $j=\si\nabla u$ in $\RR^3$, with $\nabla u\in C^1_\sharp(Y')^2$. Equating this equality with at $z=0$, we get that $${\al(0)\over\si(x,y,0)}\,\nabla v(x,y)=-\nabla^\perp u(x,y),\quad\mbox{for }(x,y)\in\RR^2.$$ Thus, $\nabla v$ is isotropically realizable in the torus with the conductivity $\al(0)\,\si^{-1}(\cdot,0)\in C^1_\sharp(Y')$, which by virtue of Theorem \[thm.elerea\] implies that $w_v$ belongs to $L^\infty(\RR^2)$. Now, assume that $w_v$ is in $L^\infty(\RR^2)$. By , and we have $${\partial X_3\over\partial t}={j\times\curl j\over |j|^2}(X_3)={\De v(X_3)\over|\nabla v(X_3)|^2}\,\nabla v(X_3)+{\al'(X_3\cdot e_z)\over\al(X_3\cdot e_z)}\,e_z.$$ Moreover, by we have $\div(e^{w_v}\nabla v)=0$ in $\RR^2$, or equivalently $\De v=-\nabla w_w\cdot\nabla v$ in $\RR^2$, hence \[Xjv\] [X\_3t]{}=-\_[v]{}(w\_v)(X\_3)+[’(X\_3e\_z)(X\_3e\_z)]{}e\_z, where $\Pi_{\nabla v}$ denotes the orthogonal projection on $\RR\nabla v$. However, since $\nabla w_v-\Pi_{\nabla v}(\nabla w_v)$ is parallel to $j$ by and thus orthogonal to $\partial_t X_3$ by , we get that $$\ba{ll} \dis \left|{\partial X_3\over\partial t}\right|^2 & \dis =-\Pi_{\nabla v}(\nabla w_v)(X_3)\cdot {\partial X_3\over\partial t}+{\al'(X_3\cdot e_z)\over\al(X_3\cdot e_z)}\,{\partial X_3\over\partial t}\cdot e_z \\ \ecart & \dis =-\nabla w_v(X_3)\cdot {\partial X_3\over\partial t}+{\al'(X_3\cdot e_z)\over\al(X_3\cdot e_z)}\,{\partial X_3\over\partial t}\cdot e_z ={\partial\over\partial t}\left[-\,w(X_3)+\ln\big(\al(X_3\cdot e_z)\big)\right] \ea$$ It follows that for any $(t,x,y,z)\in\RR^4$, \[lenXjz=0\] \_0\^t|[X\_3s]{}(s,x,y,z)|\^2ds=w\_v(x,y,z)-w\_v(X\_3(t,x,y,z)) +. The function $\ln\al$ is periodic and continuous in $\RR$, hence it is bounded. Therefore, equality shows that the boundedness of $w_v$ implies condition . \[rem.jsh2\] Note that the field current $j$ defined by has a zero $z$-coordinate, and j=\^vv(x,y):=x-y, so that we could [*a priori*]{} use the above method. However, in this case the solution of the gradient system is given by Z(t,x,y)= 2[argtanh]{}(e\^t(x/2))\ -t+y , which is clearly not a global solution. Therefore, the present two-dimensional approach does not work for the very simple current field . A particular class of current fields ------------------------------------ Let $f,g,h\in C^2(\RR)$ be three $1$-periodic functions such that $f$ has only isolated roots in $\RR$ which are not roots of $f'$, while $g,h$ do not vanish in $\RR$. Then, the following isotropic realizability result holds: \[pro.fgh\] The $Y$-periodic field $j$ defined by \[jfgh\] j(x,y,z):= g(y)h(z)\ f(x)h(z)\ f(x)h(z) ,(x,y,z)\^3, satisfies the conditions and . On the other hand, consider the following assertions: - the function $f$ does not vanish in $\RR^3$, - the field $j$ is isotropically realizable in the torus with a positive conductivity $\si\in C^1_\sharp(Y)$, - condition holds. Then, $(i)$ and $(ii)$ are equivalent conditions and they imply assertion $(iii)$. Moreover, when $g=h=1$ all three assertions $(i)$, $(ii)$ and $(iii)$ are equivalent. Condition clearly holds. We have \[curljfgh\] j= f(x)(g’(y)-h’(z))\ g(y)(h’(z)-f’(x))\ h(z)(f’(x)-g’(y)) ,(x,y,z)\^3, hence condition is also satisfied. However, note that condition does not hold. $(i)\Rightarrow(ii)$. If $f$ does not vanish in $\RR$, we can write \[ufgh\] j(x,y,z)=f(x)g(y)h(z)u(x,y,z),u(x,y,z):=\_0\^x[dtf(t)]{}+\_0\^y[dtg(t)]{}+\_0\^z[dth(t)]{}. Therefore, $j$ is isotropically realizable in the torus with the conductivity $$\si(x,y,z):=\big|f(x)\,g(y)\,h(z)\big|>0,$$ which belongs to $C^1_\sharp(Y)$. $(ii)\Rightarrow(i)$. More precisely, we will prove that if $f$ vanishes in $\RR$, then $j$ is not isotropically realizable with any positive continuous conductivity $\si(x,y,z)$ which is $Y'$-periodic with respect to $(y,z)$. To this end, assume by contradiction that both $f$ vanishes in $\RR$ and $j$ is isotropically realizable with a positive continuous conductivity $\si(x,y,z)$ which is $Y'$-periodic with respect to $(y,z)$, $Y'=[0,1]^2$. Let $a<b$ be such that $f(a)=0$ and $f(b)\neq 0$. Starting from the equality $\curl(\si^{-1} j)=0$, integrating by parts over the cube $[a,b]\times Y'$, and denoting by $n$ the outside normal, we have \[int=0\] 0 & =\_[\[a,b\]Y’]{}(\^[-1]{} j)e\_ydxdydz=\_[(\[a,b\]Y’)]{}(n\^[-1]{}j)e\_yds\ & =\_[{a,b}Y’]{}(n\^[-1]{}j)e\_yds+\_[\[a,b\]Y’]{}(n\^[-1]{}j)e\_yds. The integral over $[a,b]\times\partial Y'$ in is equal to zero due to the $Y'$-periodicity of $\si^{-1}j$ with respect to $(y,z)$. Hence, using that $f(a)=0$ and $(e_x\times j)\cdot e_y=-j_z=-f(x)\,g(y)$, it follows that the integral over $\{a,b\}\times Y'$ in satisfies 0=-\_[Y’]{}f(b)g(y)\^[-1]{}(b,y,z)dydz. This leads to a contradiction, since the function $(y,z)\mapsto f(b)\,g(y)\,\si^{-1}(b,y,z)$ is continuous and has a constant sign in $Y'$. $(ii)\Rightarrow(iii)$. This is a straightforward consequence of Corollary \[cor.perrea\] $i)$. [*$(iii)\Rightarrow(i)$, when $g=h=1$.*]{} Assume by contradiction that $f$ vanishes at some point $a\in\RR$. Since $f'(a)\neq 0$, we may assume that, for instance, there exists a real number $b>a$ such that $f>0$ in $(a,b]$ and $f'>0$ in $[a,b]$. By and we have $$j\times \curl j=\begin{pmatrix} 2f(x)f'(x) \\ -f'(x) \\ -f'(x) \end{pmatrix},\quad\mbox{for }x\in\RR,$$ hence the flow $X_3$ of reads as \[Xf\] { x’(t)=[2f(x(t))f’(x(t))2f\^2(x(t))+1]{}, & x(0)=x\ y’(t)=-[f’(x(t))2f\^2(x(t))+1]{}, & y(0)=y\ z’(t)=-[f’(x(t))2f\^2(x(t))+1]{}, & z(0)=z, . (x,y,z)\^3. Define the function $F$ in $(a,b]$ by $$F(x):=\int_b^x{2f^2(s)+1\over 2f(s)f'(s)}\,ds,\quad\mbox{for }x\in(a,b].$$ The function $F$ is an increasing bijection from $(a,b]$ onto $(-\infty,0]$, and the solution of the first equation of is given by $$x(t)=F^{-1}\big(t+F(x)\big),\quad\mbox{for }t\leq 0,$$ where $F^{-1}$ denotes the reciprocal of $F$. Making the change of variables $r=x(s)$, we have for any $t\leq 0$, $$\int_0^t \big(y'(s)\big)^2\,ds=\int_x^{x(t)}\left({f'(r)\over 2f^2(r)+1}\right)^2{2f^2(r)+1\over 2f(r)f'(r)}\,dr =\int_x^{x(t)}{f'(r)\over 2f(r)\,\big(2f^2(r)+1\big)}\,dr.$$ Then, since $x(t)$ tends to $a$ as $t\to-\infty$ and $\dis f(r)\mathop{\sim}_{a}f'(a)\,(r-a)$, we get that \_[-]{}\^0|[X\_3s]{}(s,x,y,z)|\^2ds-\_[t-]{}\_0\^t (y’(s))\^2ds=-\_x\^[a]{}[f’(r)2f(r)(2f\^2(r)+1)]{}dr=. Therefore, the $L^2(\RR)$-norm of $t\mapsto\partial_t X_3(t,x,y,z)$ is infinite for any $x\in(a,b]$. This proves the implication $(iii)\Rightarrow (i)$, when $g=h=1$. \[rem.f\] Consider the particular case of where $g=h=1$. When the function $f$ vanishes in $\RR$, the field $j\in C^2_\sharp(Y)^3$ still satisfies conditions and . Since condition does not hold, Theorem \[thm.glorea\] does not apply. However, the field $j$ is actually isotropically realizable in the whole space $\RR^3$, but not in the torus due to Proposition \[pro.fgh\]. It is not obvious how to derive an explicit conductivity associated with $j$, but we now proceed to do so. To this end, we may assume that $f\in C^2(\RR)$ is a $1$-periodic function satisfying $f(0)=0$, $f'(0)\neq 0$, and $f>0$ in $(0,1)$. Then, $j$ lies in the plane $\{j_y=j_z\}$, so that we can apply the procedure of Section \[ss.jplan\] based on the representation of two-dimensional divergence-free functions by orthogonal gradients. This combined with the approach of [@BMT] (Proposition 2.11) allows us to construct a conductivity $\si\in C^1(\RR^3)$ such that $\curl(\si^{-1}j)=0$ in $\RR^3$, as follows: Let $F$ be the function defined in $(0,1)$ by \[Ff\] F(x):=\_[[12]{}]{}\^x[dsf(s)]{},x(0,1). The function $F$ is a $C^1$-diffeomorphism from $(0,1)$ onto $\RR$. Then, denoting by $F^{-1}$ its reciprocal, an admissible conductivity $\si$ is given by \[sifF\] (x,y,z):={ & x(n,n+1)\ e\^[-f’(0)(y+z)]{} & x=n, . n, which is $1$-periodic with respect to $x$. Let us prove that $\si\in C^1(\RR^3)$, and $\curl(\si^{-1}j)=0$ in $\RR^3$. For $x\in(0,1)$ and $(y,z)\in\RR^2$, set $t:=F^{-1}(y+z+F(x))\in(0,1)$. Since $f'(0)\neq 0$ and $f\in C^2(\RR)$, we have for $n=0,1$, |y+z-[1f’(0)]{}([t-nx-n]{})| & =|F(t)-F(x)-[1f’(0)]{}([t-nx-n]{})|\ & =|\_x\^t([1f(s)]{}-[1f’(0)(s-n)]{})ds|C|t-x|, which implies that \[limtx\] \_[xn]{}[f(t)f(x)]{}=\_[xn]{}[t-nx-n]{}=e\^[f’(0)(y+z)]{},n=0,1. This shows the continuity of the function $\si$. Moreover, we have for any $x\in(0,1)$, \[Dsif\] [x]{}=[f’(x)-f’(t)f(t)]{}=[z]{}=-[f(x)f’(t)f(t)]{}, which together with implies that $\nabla\si$ has finite limits as $x\to 0$ or $1$. Therefore, the function $\si$ belongs to $C^1(\RR^3)$. Set $w:=\ln\si$. By and we have for any $x\in(0,1)$, $${\partial w\over\partial x}={1\over\si}\,{\partial\si\over\partial x}={f'(x)-f'(t)\over f(x)}\quad\mbox{and}\quad {\partial w\over\partial y}={\partial w\over\partial z}={1\over\si}\,{\partial\si\over\partial y}=-f'(t),$$ which implies that $w$ solves the equations \[eqwf\] f(x)[wx]{}-f’(x)=[wy]{}=[wz]{}\^3. By and with $g=h\equiv 1$, equations lead to the equation $$\curl j-\nabla w\times j=0\quad\mbox{in }\RR^3,$$ or equivalently, $\curl(e^{-w}j)=0$ in $\RR^3$. Therefore, the current field $j=\big(1,f(x),f(x)\big)$ is isotropically realizable with the conductivity $\si\in C^1(\RR^3)$ defined by and . Note that the function $\si$ is $1$-periodic with respect to $x$, but is not periodic with respect to $(y,z)$ in accordance with Proposition \[pro.fgh\]. Finally, the isotropic realizability of $j$ in $\RR^3$ can be written $$\begin{pmatrix} 1 \\ f(x) \\ f(x) \end{pmatrix}= \sigma(x,y,z)\,\nabla\left(x+\int_0^{y+z}f\big(F^{-1}(s+F(x-n))\big)\,ds\right),\quad\mbox{for }x\in(n,n+1),\ n\in\ZZ,$$ where the conductivity $\si$ is defined by . [**Acknowledgments.**]{} GWM thanks the [*National Science Foundation*]{} for support through grant DMS-1211359. Also the authors thank Andrejs Treibergs for his comments. [10]{} : [*Ordinary differential equations*]{}, translated from the third Russian edition by R. Cooke, Springer Textbook, Springer-Verlag, Berlin 1992, pp. 334. : “Carathéodory’s principle and the global existence of an integrating factor", [*Commun. Math. Phys.*]{}, [**10**]{} (1968), 52-68. : “Which electric fields are realizable in conducting materials?", [*ESAIM: Math. Model. Numer. Anal.*]{}, [**48**]{} (2) (2014), 307-323. : [*Calcul Différentiel*]{}, (French) Hermann, Paris 1967, 178 pp. : [*The Theory of Composites*]{}, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge 2002, pp. 719. : “Controlling Electromagnetic Fields", [*Science*]{} , [**312**]{} (5781) (2006), 1780-1782. : “Calculation of material properties and ray tracing in transformation media", [*Optic Express*]{}, [**14**]{} (21) (2006), 9794-9804. [^1]: INSA de Rennes, IRMAR (CNRS, UMR 6625), FRANCE – mbriane@insa-rennes.fr [^2]: Department of Mathematics, University of Utah, USA – milton@math.utah.edu

--- abstract: 'One of the factors limiting electron mobility in supported graphene is remote phonon scattering. We formulate the theory of the coupling between graphene plasmon and substrate surface polar phonon (SPP) modes, and find that it leads to the formation of interfacial plasmon-phonon (IPP) modes, from which the phenomena of dynamic anti-screening and screening of remote phonons emerge. The remote phonon-limited mobilities for SiO$_{2}$, HfO$_{2}$, h-BN and Al$_{2}$O$_{3}$ substrates are computed using our theory. We find that h-BN yields the highest peak mobility, but in the practically useful high-density range the mobility in HfO$_{2}$-supported graphene is high, despite the fact that HfO$_{2}$ is a high-$\kappa$ dielectric with low-frequency modes. Our theory predicts that the strong temperature dependence of the total mobility effectively vanishes at very high carrier concentrations. The effects of polycrystallinity on IPP scattering are also discussed.' author: - 'Zhun-Yong Ong' - 'Massimo V. Fischetti' bibliography: - 'SubstrateRemotePhonons\_v23July2012.bib' title: 'Theory of Interfacial Plasmon-Phonon Scattering in Supported Graphene' --- introduction ============ Graphene, a single-layer of hexagonally arranged carbon atoms [@KSNovoselov:Nature05], has been long considered a promising candidate material for post-Si CMOS technology and other nano-electronic applications on account of its excellent electrical [@EHHwang:PRL07] and thermal transport [@AABalandin:NL08] properties. In suspended single-layer graphene (SLG), the electron mobility has been demonstrated to be as high as 200,000 cm$^{2}$V$^{-1}$s$^{-1}$ [@KIBolotin:SSC08]. However, in real applications such as a graphene field-effect transistor (GFET), the graphene is physically supported by an insulating dielectric substrate such as SiO$_{2}$, and the carrier mobility in such supported-graphene structures is about one order of magnitude lower [@KIBolotin:SSC08]. This reduction in carrier mobility is further exacerbated in top-gated structures in which a thin layer of a high-$\kappa$ dielectric, such as HfO$_{2}$ or Al$_{2}$O$_{3}$, is deposited or grown on the graphene sheet [@MCLemme:SSE08; @JSMoon:EDL10; @JPezoldt:PSS10]. The degradation of the electrical transport properties is a result of exposure to environmental perturbations such as scattering by charge traps, surface roughness, and remote optical phonons which are a kind of surface excitation. Such environmental effects are encountered in metal-oxide-semiconductor (MOS) structures [@MVFischetti:JAP01]. Hess and Vogl first suggested that remote phonons [\[]{}sometimes also known as Fuchs-Kliewer (FK) [@RFuchs:PR65] surface optical (SO) phonons[\]]{} can have a substantial effect on the mobility of Si inversion layer carriers [@KHess:SSC79]. Fischetti and co-workers later studied the effects of remote phonon scattering in MOS structures and found that high-$\kappa$ oxide layers have a significant effect on carrier mobility in Si [@MVFischetti:JAP01] and Ge [@TORegan:JCompElec07]. This method was later applied by Xiu to study remote phonon scattering in Si nanowires [@KXiu:SISPAD11]. In Refs. [@MVFischetti:JAP01] and [@KXiu:SISPAD11] it was found that the plasmons in the channel material (Si) hybridized with the surface polar phonons (SPP) in the nearby dielectric material to form interfacial plasmon-phonon (IPP) modes. This hybridization occurrence naturally leads to the screening/anti-screening of the SPP from the dielectric material. Scattering with these IPP modes results in a further reduced channel electron mobility in 2D Si and Si nanowires. Likewise in supported graphene, remote phonon scattering is one of the mechanisms believed to reduce the mobility of supported graphene, with the form of the scattering mechanism varying with the material properties of the dielectric substrate. Experimentally, hexagonal boron nitride (h-BN) has been found to be a promising dielectric material for graphene, and it is commonly believed that this is at least partially due to the fact that remote phonon scattering is weak with a h-BN substrate [@CRDean:NatureNanotech10]. On other substrates such as SiO$_{2}$ [@JHChen:NatureNanotech08; @VEDorgan:APL10] and SiC [@JARobinson:NL09; @PSutter:NatureMat09], the mobility of supported graphene is lower. Thus, it is important to develop an accurate understanding of remote phonon scattering in order to find an optimal choice of substrate that will minimize the degradation of carrier mobility in supported graphene. Although the subject of remote phonon scattering in graphene [@SFratini:PRB08; @SVRotkin:NL09; @VPerebeinos:NL08; @AKonar:PRB10; @JKViljas:PRB10] and carbon nanotubes [@VPerebeinos:NL08; @SVRotkin:NL09]) has been broached in the recent past, the basic approach used in the aforementioned works does not deal adequately with the *dynamic* screening of the SPP modes. In graphene, dynamic screening of SPP modes has its origin in SPP-plasmon coupling, and the two time-dependent phenomena have to be treated within the same framework. Typically, the coupling phenomenon is ignored, and screening of the SPP modes is approximated with a Thomas-Fermi (TF) type of static screening [@AKonar:PRB10; @XLi:APL10; @SFratini:PRB08], which is adequate for the case of impurity scattering [@SAdam:SSC09; @SAdam:PNAS07] but can lead to a miscalculation of the scattering rates since the use of static screening underestimates the electron-phonon coupling strength [@SFratini:PRB08], especially for higher-frequency modes. The failure to incorporate correctly SPP-plasmon coupling into the approach means that the dispersion relation of the SPP (or, more accurately, of the IPP) modes is incorrect and that the dynamic screening of the remote phonons is not accounted for in a natural manner. To understand the screening phenomenon in our situation, let us first give a bird’s eye view of the physical picture. This picture is somewhat different from what is found in the more familiar semiconductor-inversion-layer/high-$\kappa$-dielectric geometry, since the absence of a gap in bulk graphene renders its dielectric response stronger and qualitatively different – almost metal-like, as testified by the presence of Kohn anomalies in the phonon spectra [@MLazzeri:PRL06; @SPisana:NatureMaterials07] – than the response of a two-dimensional electron gas. Graphene plasmons interact with the SPP modes through the time-dependent electric field generated by the latter, and the former are forced to oscillate at the frequency of the latter ($\omega$). When $\omega$ is less than the natural frequency of the plasmon ($\omega_{p}$), *i.e.* $\omega<\omega_{p}$ , the electrons can respond to the SPP mode and screen its electric field. On the other hand, when $\omega>\omega_{p}$, the motion of the plasmons lags that of the SPP mode, resulting in poor or no screening, or even in anti-screening, which can actually augment the scattering field [@BKRidley:Book99]. In bulk SiO$_{2}$, the main TO-phonon frequencies are around 56 and 138 meV. At long wavelengths ($\lambda>10^{-8}$m), the plasmon frequencies for a carrier density of $10^{12}\mathrm{cm^{-2}}$ are comparable or smaller than the TO-phonon frequencies. Thus, a TF-type approximation is inadequate especially for describing the screening (or more accurately, the *anti-screening*) of the 138 meV TO phonon modes. Our calculations suggest that, contrary to what is found in the semiconductor/high-$\kappa$ case [@MVFischetti:JAP01] and to the claims made in Ref. [@SFratini:PRB08], the higher-frequency SPP modes cannot be ignored despite their reduced Bose-Einstein occupation factors at room temperature. It is our intention in this paper to provide a systematic description of the coupling between the substrate SPP and the graphene plasmon modes, and relate this coupling to the dynamic screening phenomenon. Our theory can be generalized to graphene heterostructure such as double-gated graphene although this falls outside the scope of our paper and will be the subject of a future work. We begin by deriving our model of the IPP system. Its dispersion is then calculated from the model. The pure SO phonon and graphene plasmon branches are compared with the IPP branches. Also, we compute the electron-IPP and the electron-SPP coupling coefficients for different substrates (SiO$_{2}$, h-BN, HfO$_{2}$ and Al$_{2}$O$_{3}$). We show that the IPP modes can be interpreted as dynamically screened SPP modes. Scattering rates are then calculated and used to compute the remote phonon-limited mobility$\mu_{RP}$ for different substrates at room temperature (300 K) with varying carrier density. The temperature dependence of $\mu_{RP}$ at low and high carrier densities is compared. Using the$\mu_{RP}$ results, we analyze the suitability of the various dielectric materials for use as substrates or gate insulators in nanoelectronics applications. We also discuss the effects of polycrystallinity on remote phonon scattering. Model ===== Coupling between substrate polar phonons and graphene plasmons -------------------------------------------------------------- Our approach to constructing the theoretical model of the coupled plasmon-phonon systems follows closely that of Fischetti, Neumayer and Cartier [@MVFischetti:JAP01] although some modifications are needed to describe the plasmon-phonon coupling. One of the primary difficulties in describing the coupled system is the anisotropy in the dielectric response of graphene: graphene is polarizable in the plane but its out-of-plane response is presumably negligible. If the graphene sheet is modeled as a slab of finite thickness with a dynamic dielectric response in the in-plane direction [\[]{}$\epsilon_{gr}^{\parallel}(\omega)=\epsilon_{gr}(1-\omega_{p}^{2}/\omega^{2})$ where $\omega_{p}$ is the plasma frequency[\]]{} and none in the out-of-plane direction [\[]{}$\epsilon_{gr}^{\perp}(\omega)=\mathrm{constant}$[\]]{}, the dispersion of the SPPs remains unchanged, indicating that the SPP and plasmon modes are uncoupled. This absence of coupling is implicitly assumed in much of the current literature on SPP scattering in graphene [@SFratini:PRB08; @XLi:APL10; @AKonar:PRB10; @JKViljas:PRB10; @VPerebeinos:PRB10] although it has already been shown to be untrue in 2D Si [@MVFischetti:JAP01] and Si nanowires [@KXiu:SISPAD11]. Furthermore, there is considerable experimental support for the coupling of graphene plasmons to the SPPs [@YLiu:PRB08; @YLiu:PRB10; @ZFei:NL11; @RJKoch:PRB10]. As we will show later, accounting for this coupling results in the formation of IPP modes which are screened/anti-screened and scatter charge carriers in graphene more weakly/strongly than the unhybridized SPP modes. This ‘uncoupling problem’ persists even when one inserts a vacuum region between the graphene slab and the substrate. Ultimately, this alleged lack of coupling can be traced back to the continuity of the electric displacement field $\mathbf{D}$ at the interface between the graphene slab and the substrate/vacuum. Given that the dynamic response of the graphene is only in the in-plane directions and that the coupling should be with the *p*-polarized waves of the substrate, the slab approach is not likely to be correct. To overcome this difficulty, we find it is necessary to treat the graphene as a polarizable charge sheet (as shown in Fig. \[Fig:GrapheneSubstrateSetup\]) rather than as a finite slab with a particular in-plane dielectric function. This polarization charge then generates a discontinuity in the electric displacement along the surface of the graphene. It is this discontinuity that couples the dielectric response of the substrate to that of the graphene sheet. The basic setup is shown in Fig. \[Fig:GrapheneSubstrateSetup\]. The graphene is an infinitely thin sheet co-planar with the $x$-$y$ plane and floating at a height $d$ above the substrate which occupies the semi-infinite region $z<0$. Notation-wise, we try to follow Ref. [@MVFischetti:JAP01]. In the direction perpendicular to the interface, the (ionic) dielectric response of the substrate is assumed to be due to two optical phonon modes, an approximation used in Ref. [@MVFischetti:JAP01], that is: $$\epsilon_{ox}(\omega)=\epsilon_{ox}^{\infty}+(\epsilon_{ox}^{i}-\epsilon_{ox}^{\infty})\frac{\omega_{TO2}^{2}}{\omega_{TO2}^{2}-\omega^{2}}+(\epsilon_{ox}^{0}-\epsilon_{ox}^{i})\frac{\omega_{TO1}^{2}}{\omega_{TO1}^{2}-\omega^{2}}\label{Eq:DielectricEquation}$$ where $\omega_{TO1}$ and $\omega_{TO2}$ are the first and second transverse optical (TO) angular frequencies (with $\omega_{TO1}<\omega_{TO2}$), and $\epsilon_{ox}^{\infty}$, $\epsilon_{ox}^{i}$ and $\epsilon_{ox}^{0}$ are the optical, intermediate and static permittivities. We can also express $\epsilon_{ox}(\omega)$ in the generalized Lyddane-Sachs-Teller form: $$\epsilon_{ox}(\omega)=\epsilon_{ox}^{\infty}\frac{(\omega_{LO2}^{2}-\omega^{2})(\omega_{LO1}^{2}-\omega^{2})}{(\omega_{TO2}^{2}-\omega^{2})(\omega_{TO1}^{2}-\omega^{2})}$$ where $\omega_{LO1}$ and $\omega_{LO2}$ are the first and second longitudinal optical angular frequencies. The variables $\mathbf{Q}$ and $\mathbf{R}$ represent the two-dimensional wave and coordinate vector in the $(x,y)$ plane of the interface, respectively. ![Schematic of set up of graphene-substrate system. The SLG is modeled as a infinitely thin (in the $z$-direction) layer of polarizable charge. A gap of $d$ separates the graphene charge sheet and the substrate surface.[]{data-label="Fig:GrapheneSubstrateSetup"}](Fig_GrapheneSubstrateSetup){width="4in"} As in Ref. [@MVFischetti:JAP01], we try to derive the longitudinal electric eigenmodes of the system since the transverse modes (given by poles of the total electric response) correspond to a vanishing electric field and so to a vanishing coupling with the graphene carriers. In effect, the longitudinal modes are the transverse-magnetic (TM) solutions of Maxwell’s equations. It was also shown in Ref. [@MVFischetti:JAP01] that one may ignore the effects of retardation. Therefore, we need only to employ simpler electrostatics instead of the full Maxwell’s equations. We begin our derivation by writing down the Poisson equation for the *bare* scalar potential $\Phi$, $$-\nabla^{2}\Phi(\mathbf{R},z)=\frac{1}{\epsilon_{0}}\rho_{ox}(\mathbf{R},z,t),\label{Eq:BarePoisson}$$ where $\rho_{ox}$ is the (periodic) polarization charge distribution at the surface of the substrate that is the source of scattering, and $\epsilon_{0}$ is the permittivity of vacuum. Equation (\[Eq:BarePoisson\]) describes the electrostatic potential within the graphene. However, the *effective* scalar potential felt by the graphene carriers is different and should include the collective screening effect of the induced electrons/holes, which changes the RHS of Eq. (\[Eq:BarePoisson\]). Hence, we modify Eq. (\[Eq:BarePoisson\]) by adding a screening charge term on its RHS, and we obtain the Poisson equation for the *screened* scalar potential $\Phi_{scr}$, $$-\nabla^{2}\Phi_{scr}(\mathbf{R},z,t)=\frac{1}{\epsilon_{0}}\left[\rho_{ox}(\mathbf{R},z,t)+\rho_{scr}(\mathbf{R},z,t)\right],\label{Eq:EffectivePoisson}$$ where $\rho_{scr}$ is the screening charge term. The integral form of Eq. (\[Eq:EffectivePoisson\]) is: $$\Phi_{scr}(\mathbf{R},z,t)=\Phi(\mathbf{R},z,t)+\int d\mathbf{R}'dz'G(\mathbf{R}z,\mathbf{R}'z')\rho_{scr}(\mathbf{R}'z',t)\label{Eq:EffectivePoissonIntegral}$$ where $G(\mathbf{R}z,\mathbf{R}'z')$ is the Green function that satisfies the boundary conditions [\[]{}see Eqs. (\[Eq:GreenFunctionInterfaceBC\])[\]]{}, and the equation: $$-\nabla^{2}\left[\epsilon(\mathbf{R},z)G(\mathbf{R}z,\mathbf{R}'z')\right]=\delta(\mathbf{R}-\mathbf{R}',z-z')\ .\label{Eq:RealSpaceGreenFn}$$ The bare potential $\Phi(\mathbf{R},z,t)$ is defined as $\Phi(\mathbf{R},z,t)=\int d\mathbf{R}'dz'G(\mathbf{R}z,\mathbf{R}'z')\rho_{ox}(\mathbf{R}'z',t)$. The second term on the RHS of Eq. (\[Eq:EffectivePoissonIntegral\]) represents the screening charge distribution. The bare and screened potentials can be written as sums of their Fourier components: $$\Phi(\mathbf{R},z,t)=\sum_{\mathbf{Q}}\phi_{Q,\omega}(z)e^{-i\mathbf{Q}\cdot\mathbf{R}}e^{i\omega t}\ ,$$ $$\Phi_{scr}(\mathbf{R},z,t)=\sum_{\mathbf{Q}}\phi_{Q,\omega}^{scr}(z)e^{-i\mathbf{Q}\cdot\mathbf{R}}e^{i\omega t}$$ \[Eq:FourierComponents\] where it must be understood that only the real part of Eq. (\[Eq:FourierComponents\]) is to be taken here and in the following sections. Given the cylindrical symmetry of the problem, the Fourier components $\phi_{Q,\omega}$ and $\phi_{Q,\omega}^{scr}$ depend only on the magnitude of the wave vector $\mathbf{Q}$. From Eq. (\[Eq:EffectivePoissonIntegral\]) we obtain the following expression for the $z$-dependent part of the Fourier-transformed screened potential: $$\phi_{Q,\omega}^{scr}(z)e^{i\omega t}=\phi_{Q,\omega}(z)e^{i\omega t}+\int dz'G_{Q}(z,z')\rho_{Q,\omega}^{scr}(z',t)\ .\label{Eq:EffectivePoissonIntegralFourier}$$ Equation (\[Eq:EffectivePoissonIntegralFourier\]) is solvable if the polarization charge $\rho_{Q,\omega}^{scr}$ is expressed as a function of the screened scalar potential. Here, we assume that $\rho_{Q,\omega}^{scr}$ responds linearly to $\phi_{Q,\omega}^{scr}$, and write the screening charge term as: $$\rho_{Q,\omega}^{scr}(z,t)=e^{2}\Pi(Q,\omega)f(z)\phi_{Q,\omega}^{scr}(z)e^{i\omega t}\label{Eq:ScreeningCharge}$$ where $\Pi(Q,\omega)$ is the in-plane 2D polarization charge term, and $f(z)$ governs the polarization charge distribution in the out-of-plane direction. For convenience, we model the graphene as an infinitely thin sheet of polarized charge and set $f(z)=\delta(z-d)$. Combining Eqs. (\[Eq:EffectivePoissonIntegralFourier\]) and (\[Eq:ScreeningCharge\]), we obtain the expression: $$\phi_{Q,\omega}^{scr}(z)=\phi_{Q,\omega}(z)+e^{2}\int dz'G_{Q}(z,z')\Pi(Q,\omega)f(z')\phi_{Q,\omega}^{scr}(z')\ .\label{Eq:EffectivePoissonIntegralFourierFinal}$$ The expression in Eq. (\[Eq:EffectivePoissonIntegralFourierFinal\]) becomes: $$\begin{aligned} \phi_{Q,\omega}^{scr}(z) & =\phi_{Q,\omega}(z)+e^{2}G_{Q}(z,d)\Pi(Q,\omega)\phi_{Q,\omega}^{scr}(d)\\ & =\phi_{Q,\omega}(z)+e^{2}G_{Q}(z,d)\Pi(Q,\omega)\phi_{Q,\omega}(d)+e^{4}G_{Q}(z,d)\Pi(Q,\omega)G_{Q}(d,d)\Pi(Q,\omega)\phi_{Q,\omega}^{scr}(d)\\ & =\ldots\\ & =\phi_{Q,\omega}(z)+\frac{e^{2}G_{Q}(z,d)\Pi(Q,\omega)}{1-e^{2}G_{Q}(d,d)\Pi(Q,\omega)}\phi_{Q,\omega}(d)\end{aligned}$$ and the corresponding component of the electric field perpendicular to the interface at $z=0$ is: $$\hat{\mathbf{z}}\cdot\mathbf{E}_{Q,\omega}\Big|_{z=0}=-\frac{\partial}{\partial z}\phi_{Q,\omega}^{scr}(z)\Bigg|_{z=0}=-\frac{\partial\phi_{Q,\omega}(z,t)}{\partial z}-\frac{\partial G_{Q}(z,d)}{\partial z}\frac{e^{2}\Pi(Q,\omega)}{1-e^{2}G_{Q}(d,d)\Pi(Q,\omega)}\phi_{Q,\omega}(d)\Bigg|_{z=0}\ .$$ For notational simplicity, we write: $$\phi_{Q,\omega}^{scr}(z)=\phi_{Q,\omega}(z)+G_{Q}(z,d)\mathcal{P}_{Q,\omega}\phi_{Q,\omega}(d)\label{Eq:ScreenedScalar}$$ $$\hat{\mathbf{z}}\cdot\mathbf{E}_{Q,\omega}=-\frac{\partial\phi_{Q,\omega}(z,t)}{\partial z}-\frac{\partial G_{Q}(z,d)}{\partial z}\mathcal{P}_{Q,\omega}\phi_{Q,\omega}(d)\ ,$$ where $$\mathcal{P}_{Q,\omega}=\frac{e^{2}\Pi(Q,\omega)}{1-e^{2}G_{Q}(d,d)\Pi(Q,\omega)}\ .\label{Eq:Polarization}$$ Here, we emphasize that Eq. (\[Eq:ScreenedScalar\]) is the key to determining the dispersion relation as we shall show later. The Green function $G_{Q}(z,z')$ in Eq. (\[Eq:EffectivePoissonIntegralFourier\]) obeys the relation: $$-\left(\frac{\partial^{2}}{\partial z^{2}}-Q^{2}\right)G_{Q}(z,z')=\frac{1}{\epsilon_{0}}\delta(z-z')\ .\label{Eq:GreenFnPoissonEqn}$$ We require the Green function to satisfy the following conditions at and away from the interface ($z=0$). $$\epsilon_{0}\frac{dG_{Q}(z=0^{+},z')}{dz}=\epsilon_{ox}^{\infty}\frac{dG_{Q}(z=0^{-},z')}{dz}\label{Eq:GreenFnContinuity-1}$$ $$G_{Q}(z=0^{+},z')=G_{Q}(z=0^{-},z')\label{Eq:GreenFnContinuity-2}$$ $$G_{Q}(z<0,d)=G_{Q}(0,d)e^{+Qz}$$ $$G_{Q}(z>d,d)=G_{Q}(d,d)e^{-Q(z-d)}$$ \[Eq:GreenFunctionInterfaceBC\] The solution to Eq. (\[Eq:GreenFnPoissonEqn\]) is [@JDJackson:Book99]: $$G_{Q}(z,z')=\left\{ \begin{array}{ll} \frac{1}{2\epsilon_{0}Q}\left(e^{-Q|z-z'|}-\lambda e^{-Q|z+z'|}\right) & ,z>0\\ \frac{1}{2\epsilon_{0}Q}\left(1-\lambda\right)e^{-Q|z-z'|} & ,z\leq0 \end{array}\right.\label{Eq:FourierGreenFunction}$$ where $$\lambda=\frac{\epsilon_{ox}^{\infty}-\epsilon_{0}}{\epsilon_{ox}^{\infty}+\epsilon_{0}}\ .$$ The bare potential in Eq. (\[Eq:ScreenedScalar\]) can be written as: $$\phi_{Q,\omega}(z)=\left\{ \begin{array}{ll} A_{1}e^{-Qz} & ,\ z>0\\ A_{2}e^{+Qz} & ,\ z\leq0 \end{array}\right.$$ where $A_{1}$ and $A_{2}$ are the amplitudes of the bare potential for $z>0$ and $z\leq0$ respectively. Thus, the expression for the screened potential in Eq. (\[Eq:ScreenedScalar\]) is: $$\phi_{Q,\omega}^{scr}(z)=\left\{ \begin{array}{ll} A_{1}e^{-Qz}+G_{Q}(z,d)\mathcal{P}_{Q,\omega}A_{1}e^{-Qd} & ,z>0\\ A_{2}e^{+Qz}+G_{Q}(z,d)\mathcal{P}_{Q,\omega}A_{1}e^{-Qd} & ,z\leq0 \end{array}\right.\ .\label{Eq:ScreenedScalarField}$$ At the interface $z=0$, the continuity of the component of the electric field parallel to the interface requires the continuity of $\phi_{Q,\omega}^{scr}$, *i.e.* $\phi_{Q,\omega}^{scr}(z=0^{+})=\phi_{Q,\omega}^{scr}(z=0^{-})$, giving us: $$A_{1}+A_{1}G_{Q}(z=0^{+},d)\mathcal{P}_{Q,\omega}e^{-Qd}=A_{2}+A_{1}G_{Q}(z=0^{-},d)\mathcal{P}_{Q,\omega}e^{-Qd}\ .\label{Eq:ContinuityEField}$$ Similarly, the continuity of the perpendicular component of the electric displacement, *i.e.*, $\epsilon_{0}\frac{d\phi_{Q,\omega}^{scr}(z=0^{+})}{dz}=\epsilon_{ox}(\omega)\frac{d\phi_{Q,\omega}^{scr}(z=0^{-})}{dz}$ leads to: $$\epsilon_{0}\bigg[A_{1}-A_{1}\frac{1}{Q}\frac{dG_{Q}(z=0^{+},d)}{dz}\mathcal{P}_{Q,\omega}e^{-Qd}\bigg]=\epsilon_{ox}(\omega)\bigg[-A_{2}-A_{1}\frac{1}{Q}\frac{dG_{Q}(z=0^{-},d)}{dz}\mathcal{P}_{Q,\omega}e^{-Qd}\bigg]\ .\label{Eq:ContinuityEDisplacement}$$ Substituting Eqs. (\[Eq:GreenFnContinuity-1\]) and (\[Eq:GreenFnContinuity-2\]) into Eqs. (\[Eq:ContinuityEField\]) and (\[Eq:ContinuityEDisplacement\]), we obtain the following relations: $$A_{1}=A_{2}$$ $$\epsilon_{0}\left[1-\frac{1}{Q}\frac{\partial G_{Q}(z=0^{+},d)}{\partial z}\mathcal{P}_{Q,\omega}e^{-Qd}\right]=\epsilon_{ox}(\omega)\left[-1-\frac{1}{Q}\frac{\partial G_{Q}(z=0^{-},d)}{\partial z}\mathcal{P}_{Q,\omega}e^{-Qd}\right]\label{Eq:PreSecularEquation}$$ Rearranging the terms in Eq. (\[Eq:PreSecularEquation\]) we obtain: $$\epsilon_{0}+\epsilon_{ox}(\omega)+\left[\epsilon_{ox}(\omega)-\epsilon_{ox}^{\infty}\right]G_{Q}(0,d)\mathcal{P}_{Q,\omega}e^{-Qd}=0\ ,\label{Eq:SecularEquation}$$ which can be rewritten as: $$\left[\epsilon_{0}+\epsilon_{ox}(\omega)\right]\left\{ 1-\left[G_{Q}(d,d)-G_{Q}(0,d)e^{-Qd}\right]e^{2}\Pi(Q,\omega)\right\} -\left(\epsilon_{0}+\epsilon_{ox}^{\infty}\right)G_{Q}(0,d)e^{2}\Pi(Q,\omega)e^{-Qd}=0$$ or more explicitly: $$\left[\epsilon_{ox}(\omega)+\epsilon_{0}\right]\left[1-\left(1-e^{-2Qd}\right)\frac{e^{2}\Pi(Q,\omega)}{2\epsilon_{0}Q}\right]-\frac{e^{2}\Pi(Q,\omega)}{Q}e^{-2Qd}=0\ .\label{Eq:SecularEquation-1}$$ Equation (\[Eq:SecularEquation\]) gives us the dispersion of the coupled plasmon-phonon modes, and is sometimes called the secular equation [@MVFischetti:JAP01]. Physically, we expect three branches (two phonon and one plasmon). We write the coupled plasmon-phonon modes as $\omega_{Q}^{(1)}$, $\omega_{Q}^{(2)}$ and $\omega_{Q}^{(3)}$ for each $\mathbf{Q}$-point. In the limit $d\rightarrow\infty$, Eq. (\[Eq:SecularEquation-1\]) becomes: $$\left[\epsilon_{ox}(\omega)+\epsilon_{0}\right]\left[1-\frac{e^{2}\Pi(Q,\omega)}{2\epsilon_{0}Q}\right]=0$$ which gives us as expected the dispersion for the two uncoupled SPP branches and the single plasmon branch in isolated graphene. Plasmon and phonon content -------------------------- The solutions of Eq. (\[Eq:SecularEquation-1\]) represent excitations of the IPP modes. However, the effective scattering amplitude of a particular mode may not be substantial if it is plasmon-like. Scattering with a plasmon-like excitation does not necessarily lead to loss of momentum since the momentum is simply transferred to the constituent carriers of the plasmon excitation and there is no change in the total momentum of all the carriers. On the other hand, scattering with a phonon-like excitation does lead to a loss of momentum since phonons belong to a different set of degrees of freedom. Therefore, as in Ref. [@MVFischetti:JAP01], it is necessary to define the *phonon content* [@MEKim:PRB78] of each IPP mode. The phonon content quantifies the modal fraction that is phonon-like and modulates its scattering strength. Likewise, we can also define the plasmon content of the mode. To find the plasmon content, we first consider the two solutions $\omega_{Q}^{(-g,\alpha)}$ ($\alpha=1,2$) obtained from the secular equation Eq. (\[Eq:SecularEquation-1\]) by ignoring the polarization response [\[]{}setting $\Pi(Q,\omega)=0$[\]]{}. Following Ref. [@MVFischetti:JAP01], the plasmon content of the IPP mode $\omega_{Q}^{(i)}$ is defined here as: $$\Phi^{(g)}(\omega_{Q}^{(i)})=\left|\frac{(\omega_{Q}^{(i)2}-\omega_{Q}^{(-g,1)2})(\omega_{Q}^{(i)2}-\omega_{Q}^{(-g,2)2})}{(\omega_{Q}^{(i)2}-\omega_{Q}^{(j)2})(\omega_{Q}^{(i)2}-\omega_{Q}^{(k)2})}\right|\label{Eq:PlasmonContent}$$ where the indices $(i,j,k)$ are cyclical. Note that the expected ‘sum rule’ [@MVFischetti:JAP01] $$\sum_{i=1}^{3}\Phi^{(g)}(\omega_{Q}^{(i)})=1\ ,\label{Eq:PlasmonContentSumRule}$$ holds. Equation (\[Eq:PlasmonContentSumRule\]) implies that the total plasmon weight of the three solutions is equal to one (as it would be without hybridization). The (non-plasmon) phonon content is then defined as $1-\Phi^{(g)}(\omega_{Q}^{(i)})$. In order to distinguish the phonon-1 and phonon-2 parts of the non-plasmon content, we need to define the relative individual phonon content. For phonon-1, this is accomplished by ignoring its response and replacing $\epsilon_{ox}(\omega)$ in Eq. (\[Eq:SecularEquation-1\]) with $\epsilon_{ox}^{\infty}(\omega_{LO2}^{2}-\omega^{2})/(\omega_{TO2}^{2}-\omega^{2})$. From the solutions of the modified secular equation ($\omega_{Q}^{(-TO1,1)}$ and $\omega_{Q}^{(-TO1,2)}$), the relative phonon-1 content of mode $i$ will be: $$R^{(TO1)}(\omega_{Q}^{(i)})=\left|\frac{(\omega_{Q}^{(i)2}-\omega_{Q}^{(-TO1,1)2})(\omega_{Q}^{(i)2}-\omega_{Q}^{(-TO1,2)2})}{(\omega_{Q}^{(i)2}-\omega_{Q}^{(j)2})(\omega_{Q}^{(i)2}-\omega_{Q}^{(k)2})}\right|\label{Eq:Phonon1RelativeContent}$$ where, as before, $i$, $j$ and $k$ are cyclical. The relative phonon-2 content can be similarly defined by replacing the superscript $(-TO1,\alpha)$ with $(-TO2,\alpha)$. Hence, the TO-phonon-1 content will be: $$\Phi^{(TO1)}(\omega_{Q}^{(i)})=\frac{R^{(TO1)}(\omega_{Q}^{(i)})}{R^{(TO1)}(\omega_{Q}^{(i)})+R^{(TO2)}(\omega_{Q}^{(i)})}\left[1-\Phi^{(g)}(\omega_{Q}^{(i)})\right]\ .\label{Eq:Phonon1Content}$$ The TO-phonon-2 content $\Phi^{(TO2)}(\omega_{Q}^{(i)})$ can be similarly defined. Given Eqs. (\[Eq:PlasmonContent\]) and (\[Eq:Phonon1Content\]), the following sum rules have been numerically verified: $$\sum_{i=1}^{3}\Phi^{(TO1)}(\omega_{Q}^{(i)})=\sum_{i=1}^{3}\Phi^{(TO2)}(\omega_{Q}^{(i)})=1$$ $$\Phi^{(g)}(\omega_{Q}^{(i)})+\Phi^{(TO1)}(\omega_{Q}^{(i)})+\Phi^{(TO2)}(\omega_{Q}^{(i)})=1$$ \[Eq:PhononContentSumRules\] for each mode $\omega_{Q}^{(i)}$. Scattering strength ------------------- As we have seen earlier, the IPP modes that result from the SPP-plasmon coupling have a different dispersion from that of the uncoupled SPP and plasmon modes. The electric field generated by the IPP modes is also different from that of the uncoupled SPP and plasmon modes. Since the remote phonon-electron coupling is derived from the quantization of the energy density of the electric field [@MVFischetti:JAP01], we expect this difference in the electric field to be reflected in the scattering strength of the IPP modes. To find the scattering strength of an IPP mode, we have to determine the amplitude of its electric field. In Eq. (\[Eq:ScreenedScalarField\]) there are three unknowns ($A_{1}$, $A_{2}$ and $\omega$), two of which ($A_{1}$ and $\omega$) can only be eliminated through Eqs. (\[Eq:ContinuityEField\]) and (\[Eq:ContinuityEDisplacement\]). To find $A_{1}$, we follow the semi-classical approach in Ref. [@MVFischetti:JAP01], where the time-averaged total energy of the scattering field is set equal to the zero-point energy. In the following discussion, we set $A_{1}=A_{Q}.$ We first compute the time-averaged electrostatic energy $\langle\mathcal{U}_{Q,\omega}^{scr}\rangle$ associated with the screened field: $$\left\langle \mathcal{U}_{Q,\omega}^{scr}\right\rangle =\left\langle \frac{1}{2}\int dzd\mathbf{R}\,\epsilon(\omega_{Q}^{(i)})\left|\nabla\left[\phi_{Q,\omega}^{scr}(z)e^{i\mathbf{Q}\cdot\mathbf{R}-i\omega_{Q}^{(i)}t}\right]\right|^{2}\right\rangle \ .\label{Eq:ScreenedZeroPoint}$$ The angle brackets $\langle\ldots\rangle$ denote time average. The volume integral in Eq. (\[Eq:ScreenedZeroPoint\]) is the result of three contributions: one from the substrate ($z\leq0$), one from the graphene-substrate gap ($0<z\leq d$), and one from the region above the graphene ($z>d$). Each term can be converted into a surface integral. Adopting a ‘piecewise approach’ to evaluate the integral in Eq. (\[Eq:ScreenedZeroPoint\]), we must evaluate three surface integrals. To do so, we need the explicit expressions for $G_{Q}(z,d)$: $$G_{Q}(z,d)=\left\{ \begin{array}{ll} \frac{1}{2\epsilon_{0}Q}\left(1-\lambda\right)e^{+Q(z-d)} & ,\: z\leq0\\ \frac{1}{2\epsilon_{0}Q}\left[e^{+Q(z-d)}-\lambda e^{-Q(z+d)}\right] & ,\:0<z\leq d\\ \frac{1}{2\epsilon_{0}Q}\left[e^{-Q(z-d)}-\lambda e^{-Q(z+d)}\right] & ,\: z>d \end{array}\right.$$ and for $-\frac{\partial}{\partial z}G_{Q}(z,d)$: $$-\frac{\partial}{\partial z}G_{Q}(z,d)=\left\{ \begin{array}{ll} -\frac{1}{2\epsilon_{0}}\left(1-\lambda\right)e^{+Q(z-d)} & ,\: z\leq0\\ \frac{1}{2\epsilon_{0}}\left[-e^{+Q(z-d)}-\lambda e^{-Q(z+d)}\right] & ,\:0<z\leq d\\ \frac{1}{2\epsilon_{0}}\left[e^{-Q(z-d)}-\lambda e^{-Q(z+d)}\right] & ,\: z>d \end{array}\right.$$ We can now evaluate the electrostatic energy in the regions $z\leq0$, $0<z\leq d$ and $z>d$. $$\left\langle \mathcal{U}_{Q,\omega}^{scr}\right\rangle =\left\langle \mathcal{U}_{Q,\omega}^{scr}(z\leq0)\right\rangle +\left\langle \mathcal{U}_{Q,\omega}^{scr}(0<z\leq d)\right\rangle +\left\langle \mathcal{U}_{Q,\omega}^{scr}(z>d)\right\rangle \label{Eq:AllEnergyTerms}$$ As mentioned earlier, the volume integrals in Eq. (\[Eq:ScreenedZeroPoint\]) can be recast as surface integrals. Thus, $$\left\langle \mathcal{U}_{Q,\omega}^{scr}(z\leq0)\right\rangle =\frac{\epsilon_{0}\Omega A_{Q}^{2}Q}{2}\bigg[1-\frac{1}{Q}\frac{\partial G_{Q}(z=0^{+},d)}{\partial z}\mathcal{P}_{Q,\omega}e^{-Qd}\bigg]\bigg[1+G_{Q}(z=0^{+},d)\mathcal{P}_{Q,\omega}e^{-Qd}\bigg]\label{Eq:ZeroPointBottom}$$ $$\begin{split}\left\langle \mathcal{U}_{Q,\omega}^{scr}(0<z\leq d)\right\rangle =\frac{\overline{\epsilon}_{ox}(\omega_{Q}^{(i)})\Omega A_{Q}^{2}Q}{2}\bigg[1+\frac{1}{Q}\frac{\partial G_{Q}(z=0^{-},d)}{\partial z}\mathcal{P}_{Q,\omega}e^{-Qd}\bigg]\bigg[1+G_{Q}(z=0^{-},d)\mathcal{P}_{Q,\omega}e^{-Qd}\bigg]\\ +\frac{\epsilon_{0}\Omega A_{Q}^{2}Q}{2}\bigg[e^{-Qd}-\frac{1}{Q}\frac{\partial G_{Q}(z=d^{+},d)}{\partial z}\mathcal{P}_{Q,\omega}e^{-Qd}\bigg]\bigg[e^{-Qd}+G_{Q}(z=d^{+},d)\mathcal{P}_{Q,\omega}e^{-Qd}\bigg] \end{split} \label{Eq:ZeroPointMiddle}$$ $$\left\langle \mathcal{U}_{Q,\omega}^{scr}(z>d)\right\rangle =-\frac{\epsilon_{0}\Omega A_{Q}^{2}Q}{2}\bigg[e^{-Qd}-\frac{1}{Q}\frac{\partial G_{Q}(z=d^{-},d)}{\partial z}\mathcal{P}_{Q,\omega}e^{-Qd}\bigg]\bigg[e^{-Qd}+G_{Q}(z=d^{-},d)\mathcal{P}_{Q,\omega}e^{-Qd}\bigg]\ .\label{Eq:ZeroPointTop}$$ \[Eq:ZeroPointAll\] In Eq. (\[Eq:ZeroPointMiddle\]), $\bar{\epsilon}_{ox}(\omega)$ is the dielectric function of the substrate, which we distinguish with the overhead bar, and distinct from $\epsilon_{ox}(\omega)$. As we shall see later, as in Ref. [@MVFischetti:JAP01], the function $\bar{\epsilon}_{ox}(\omega)$ is chosen in a way consistent with the particular excitation that we want. Let us regroup the terms in Eqs. (\[Eq:ZeroPointAll\]) into those on the substrate surface at $z=0$ and those on the graphene at $z=d$. At $z=0$, we have $$\begin{split}\left\langle \mathcal{U}_{Q,\omega}^{scr}(z=0)\right\rangle =\frac{\epsilon_{0}\Omega A_{Q}^{2}Q}{2}\bigg[1-\frac{1}{Q}\frac{\partial G_{Q}(z=0^{+},d)}{\partial z}\mathcal{P}_{Q,\omega}e^{-Qd}\bigg]\bigg[1+G_{Q}(z=0^{+},d)\mathcal{P}_{Q,\omega}e^{-Qd}\bigg]\\ +\frac{\overline{\epsilon}_{ox}(\omega_{Q}^{(i)})\Omega A_{Q}^{2}Q}{2}\bigg[1+\frac{1}{Q}\frac{\partial G_{Q}(z=0^{-},d)}{\partial z}\mathcal{P}_{Q,\omega}e^{-Qd}\bigg]\bigg[1+G_{Q}(z=0^{-},d)\mathcal{P}_{Q,\omega}e^{-Qd}\bigg] \end{split} \label{Eq:SubstrateSurfaceZeroPoint}$$ while at $z=d$, we have: $$\begin{split}\left\langle \mathcal{U}_{Q,\omega}^{scr}(z=d)\right\rangle =\frac{\epsilon_{0}\Omega A_{Q}^{2}Q}{2}\bigg[e^{-Qd}-\frac{1}{Q}\frac{\partial G_{Q}(z=d^{+},d)}{\partial z}\mathcal{P}_{Q,\omega}e^{-Qd}\bigg]\bigg[e^{-Qd}+G_{Q}(z=d^{+},d)\mathcal{P}_{Q,\omega}e^{-Qd}\bigg]\\ -\frac{\epsilon_{0}\Omega A_{Q}^{2}Q}{2}\bigg[e^{-Qd}+\frac{1}{Q}\frac{\partial G_{Q}(z=d^{-},d)}{\partial z}\mathcal{P}_{Q,\omega}e^{-Qd}\bigg]\bigg[e^{-Qd}+G_{Q}(z=d^{-},d)\mathcal{P}_{Q,\omega}e^{-Qd}\bigg] \end{split} \label{Eq:GrapheneSurfaceZeroPoint}$$ We have to be careful in computing $\left\langle \mathcal{U}_{Q,\omega}^{scr}\right\rangle $. Mathematically, it may seem that we ought to set $\left\langle \mathcal{U}_{Q,\omega}^{scr}\right\rangle =\left\langle \mathcal{U}_{Q,\omega}^{scr}(z=0)\right\rangle +\left\langle \mathcal{U}_{Q,\omega}^{scr}(z=d)\right\rangle $. However, note that the term $\left\langle \mathcal{U}_{Q,\omega}^{scr}\right\rangle $ accounts for the various excitation effects, ionic and electronic, but the term $\left\langle \mathcal{U}_{Q,\omega}^{scr}(z=d)\right\rangle $ corresponds to the charge singularity in the zero-thickness graphene sheet. It is a ‘self-interaction’ of the charge distribution in the graphene which has no dependence on $\omega$, unlike $\left\langle \mathcal{U}_{Q,\omega}^{scr}(z=0)\right\rangle $, so that we should not expect it to contribute physically to the scattering of the graphene carriers. To see this more clearly, we rewrite Eq. (\[Eq:ScreenedZeroPoint\]) as: $$\left\langle \mathcal{U}_{Q,\omega}^{scr}\right\rangle =\left\langle \frac{1}{2}\int dzd\mathbf{R}\,\mathbf{D}\cdot\mathbf{E}\right\rangle =\left\langle \frac{1}{2}\int dzd\mathbf{R}\,(\epsilon_{0}\mathbf{E}+\mathbf{P}_{L}+\mathbf{P}_{e})\cdot\mathbf{E}\right\rangle$$ where $\mathbf{P}_{L}$ and $\mathbf{P}_{e}$ are the polarization fields of the lattice (substrate) and the graphene electronic excitation respectively. The following identification can be made: $$\begin{aligned} \left\langle \mathcal{U}_{Q,\omega}^{scr}(z=0)\right\rangle & =\left\langle \frac{1}{2}\int dzd\mathbf{R}\,(\epsilon_{0}\mathbf{E}+\mathbf{P}_{L})\cdot\mathbf{E}\right\rangle \\ \left\langle \mathcal{U}_{Q,\omega}^{scr}(z=d)\right\rangle & =\left\langle \frac{1}{2}\int dzd\mathbf{R}\,\mathbf{P}_{e}\cdot\mathbf{E}\right\rangle \end{aligned}$$ and since we are only interested in the interaction of the *lattice* polarization field with the graphene carriers, we set $\left\langle \mathcal{U}_{Q,\omega}^{scr}\right\rangle =\left\langle \mathcal{U}_{Q,\omega}^{scr}(z=0)\right\rangle $, *i.e.*: $$\begin{aligned} \left\langle \mathcal{U}_{Q,\omega}^{scr}\right\rangle & = & \frac{\Omega A_{Q}^{2}Q}{2}\left[\epsilon_{0}+\bar{\epsilon}_{ox}(\omega_{Q}^{(i)})+\left(\bar{\epsilon}_{ox}(\omega_{Q}^{(i)})-\epsilon_{ox}^{\infty}\right)G_{Q}(z=0,d)\mathcal{P}_{Q,\omega}e^{-Qd}\right]\:\nonumber \\ & & \times\left[1+G_{Q}(z=0,d)\mathcal{P}_{Q,\omega}e^{-Qd}\right]\label{Eq:ScreenedElectostaticEnergy}\end{aligned}$$ where we have used the relationship $\frac{1}{Q}\frac{\partial}{\partial z}G_{Q}(z=0^{-},d)=G_{Q}(z=0,d)$. In Eq. (\[Eq:ScreenedElectostaticEnergy\]), the first factor $[\epsilon_{0}+\bar{\epsilon}_{ox}(\omega)-\ldots]$ resembles the secular equation, Eq. (\[Eq:SecularEquation\]). Indeed, if we replace $\overline{\epsilon}_{ox}(\omega)$ with $\epsilon_{ox}(\omega)$, then $\left\langle \mathcal{U}_{Q,\omega}^{scr}\right\rangle =0$ as expected. This is no coincidence since $\mathcal{U}_{Q,\omega}^{scr}$ represents the energy of the charge distribution present at the substrate-vacuum interface and the secular equation in Eq. (\[Eq:SecularEquation\]) is a statement about the *absence* of charges at the substrate-vacuum interface. This also confirms our earlier choice of excluding the contribution from the surface charges at $z=d$, since that contribution does not disappear when we replace $\overline{\epsilon}_{ox}(\omega)$ with $\epsilon_{ox}(\omega)$. By regrouping the terms according to the position of their charge distribution in Eq. (\[Eq:AllEnergyTerms\]), we make the relationship to the secular equation Eq. (\[Eq:SecularEquation\]) manifest. Using Eq. (\[Eq:Polarization\]), the expression for the screened electrostatic energy can be rewritten as: $$\left\langle \mathcal{U}_{Q,\omega}^{scr}\right\rangle =\frac{\Omega A_{Q}^{2}Q}{2}\left[\bar{\epsilon}_{ox}(\omega_{Q}^{(i)})-\epsilon_{ox}(\omega_{Q}^{(i)})\right]\left(\frac{1-G_{Q}(d,d)e^{2}\Pi(Q,\omega_{Q}^{(i)})+G_{Q}(0,d)e^{2}\Pi(Q,\omega_{Q}^{(i)})e^{-Qd}}{1-G_{Q}(d,d)e^{2}\Pi(Q,\omega_{Q}^{(i)})}\right)^{2}\ .\label{Eq:ScreenedElectrostaticEnergyNeat}$$ We use the relationship $\langle\mathcal{W}_{Q,\omega}^{scr}\rangle=2\langle\mathcal{U}_{Q,\omega}^{scr}\rangle$ to obtain the time-averaged total energy, and set it equal to the zero-point energy, [*i.e.*]{}, $\frac{1}{2}\hbar\omega_{Q}^{(i)}=\langle\mathcal{W}_{Q,\omega}^{scr}\rangle$, so that: $$\frac{1}{2}\hbar\omega_{Q}^{(i)}=\Omega A_{Q}^{2}Q\left[\bar{\epsilon}_{ox}(\omega_{Q}^{(i)})-\epsilon_{ox}(\omega_{Q}^{(i)})\right]\left(\frac{1-G_{Q}(d,d)e^{2}\Pi(Q,\omega_{Q}^{(i)})+G_{Q}(0,d)e^{2}\Pi(Q,\omega_{Q}^{(i)})e^{-Qd}}{1-G_{Q}(d,d)e^{2}\Pi(Q,\omega_{Q}^{(i)})}\right)^{2}\ .$$ Therefore, the squared amplitude of the field is: $$A_{Q}^{2}=\frac{\hbar\omega_{Q}^{(i)}}{2\Omega Q\left[\bar{\epsilon}_{ox}(\omega_{Q}^{(i)})-\epsilon_{ox}(\omega_{Q}^{(i)})\right]}\left(\frac{1-G_{Q}(d,d)e^{2}\Pi(Q,\omega_{Q}^{(i)})+G_{Q}(0,d)e^{2}\Pi(Q,\omega_{Q}^{(i)})e^{-Qd}}{1-G_{Q}(d,d)e^{2}\Pi(Q,\omega_{Q}^{(i)})}\right)^{-2}\ .$$ To determine the strength of the scattering field, say for the TO1 phonon, we take the difference between (i) the squared amplitude of the field with the TO1 mode frozen and (ii) that of the field with the mode in full response. In (i), we set: $$\bar{\epsilon}_{ox}^{TO1,\infty}(\omega)=\epsilon_{ox}^{\infty}\left(\frac{\omega_{LO2}^{2}-\omega^{2}}{\omega_{TO2}^{2}-\omega^{2}}\right)$$ and $$\bar{\epsilon}_{ox}^{TO1,0}(\omega)=\epsilon_{ox}^{\infty}\left(\frac{\omega_{LO2}^{2}-\omega^{2}}{\omega_{TO2}^{2}-\omega^{2}}\right)\frac{\omega_{LO1}^{2}}{\omega_{TO1}^{2}}\ .$$ The squared amplitude of the TO1 scattering field for $\omega=\omega_{Q}^{(i)}$ is: $$\begin{aligned} A_{TO1}(Q,\omega_{Q}^{(i)})^{2} & = & \frac{\hbar\omega_{Q}^{(i)}}{2\Omega Q}\left(\frac{1}{\bar{\epsilon}_{ox}^{TO1,\infty}(\omega_{Q}^{(i)})-\epsilon_{ox}(\omega_{Q}^{(i)})}-\frac{1}{\bar{\epsilon}_{ox}^{TO1,0}(\omega_{Q}^{(i)})-\epsilon_{ox}(\omega_{Q}^{(i)})}\right)\Phi^{(TO1)}(\omega_{Q}^{(i)})\nonumber \\ & & \times\left(\frac{1-G_{Q}(d,d)e^{2}\Pi(Q,\omega_{Q}^{(i)})+G_{Q}(0,d)e^{2}\Pi(Q,\omega_{Q}^{(i)})e^{-Qd}}{1-G_{Q}(d,d)e^{2}\Pi(Q,\omega_{Q}^{(i)})}\right)^{-2}\ .\label{Eq:SqAmpScattField}\end{aligned}$$ The expression for the TO2 scattering field can be similarly obtained. Therefore, the TO1 *effective* scattering field can be written as: $$\phi_{Q,\omega}^{scr}(d)=A_{TO1}(Q,\omega_{Q}^{(i)})\left[e^{-Qd}+G_{Q}(z,d)\frac{e^{2}\Pi(Q,\omega_{Q}^{(i)})}{1-e^{2}G_{Q}(d,d)\Pi(Q,\omega_{Q}^{(i)})}e^{-Qd}\right]\ .\label{Eq:TO1EffScattField}$$ The scattering potential is: $$V(\mathbf{R},z)=\sum_{l=1}^{3}\sum_{\mu=1}^{2}eA_{TO1}(Q,\omega_{Q}^{(l)})\left[e^{-Qz}+G_{Q}(z,d)\mathcal{P}_{Q,\omega_{Q}^{(l)}}e^{-Qd}\right]e^{i\mathbf{Q}\cdot\mathbf{R}-i\omega_{Q}^{(l)}t}\left(a_{\mathbf{Q}}^{(l)}+a_{\mathbf{-Q}}^{(l)\dagger}\right)\label{Eq:ScatteringPotential}$$ where $a_{\mathbf{Q}}^{(l)}$ ($a_{\mathbf{Q}}^{(l)\dagger}$) is the annihilation (creation) operator for the mode corresponding to $\mathbf{Q}$ and $\omega_{Q}^{(l)}$. Generally, the graphene field operator can be written in the spinorial form as: $$\Psi(\mathbf{R},z)=\frac{1}{\sqrt{2\Omega}}\sum_{s=\pm1}\sum_{\mathbf{K}}\left[\left(\begin{array}{c} 1\\ se^{i\theta_{\mathbf{K}}} \end{array}\right)c_{\mathbf{K}}^{s\mathcal{K}}+\left(\begin{array}{c} e^{i\theta_{\mathbf{K}}}\\ s \end{array}\right)c_{\mathbf{K}}^{s\mathcal{K}'}\right]e^{i\mathbf{K}\cdot\mathbf{R}}\sqrt{\delta(z-d)}\label{Eq:GrapheneFieldOperator}$$ where $\mathcal{K}(\mathcal{K}')$ denotes the $\mathcal{K}(\mathcal{K}')$ valley, and the $+(-)$ sign corresponds to the $\pi(\pi^{*})$ band; $c_{\mathbf{K}}^{s\mathcal{K}}$($c_{\mathbf{K}}^{s\mathcal{K}\dagger}$) is the annihilation (creation) operator of the $s$-band $\mathbf{K}$ electron state at the $\mathcal{K}$ valley. Therefore, the interaction term is $$H_{int}=\int dz\int d\mathbf{R}\Psi^{\dagger}(\mathbf{R},z)V(\mathbf{R},z)\Psi(\mathbf{R},z)$$ and, if we neglect the inter-valley terms, simplifies to: $$H_{int}\approx\sum_{l=1}^{3}\sum_{s_{1},s_{2}}\sum_{\mathbf{K},\mathbf{Q}}M_{Q}^{(l)}\alpha_{s_{1}\mathbf{K+Q},s_{2}\mathbf{K}}\left(c_{\mathbf{K}+\mathbf{Q}}^{s_{1}\mathcal{K}\dagger}c_{\mathbf{K}}^{s_{2}\mathcal{K}}+s_{1}s_{2}c_{\mathbf{K}+\mathbf{Q}}^{s_{1}\mathcal{K}'\dagger}c_{\mathbf{K}}^{s_{2}\mathcal{K}'}\right)\left(a_{\mathbf{Q}}^{(l)}+a_{\mathbf{-Q}}^{(l)\dagger}\right)\label{Eq:InteractionTerm}$$ where $$\alpha_{s_{1}\mathbf{K_{1}},s_{2}\mathbf{K_{2}}}=\frac{1+s_{1}s_{2}e^{-i(\theta_{\mathbf{K}_{1}}-\theta_{\mathbf{K_{2}}})}}{2}$$ is the overlap integral that comes from the inner product of the spinors, and $$\begin{aligned} M_{Q}^{(l)} & = & \sum_{\mu=1}^{2}\left[\frac{e^{2}\hbar\omega_{Q}^{(l)}}{2\Omega Q}\left(\frac{1}{\bar{\epsilon}_{ox}^{TO\mu,\infty}(\omega_{Q}^{(l)})-\epsilon_{ox}(\omega_{Q}^{(l)})}-\frac{1}{\bar{\epsilon}_{ox}^{TO\mu,0}(\omega_{Q}^{(l)})-\epsilon_{ox}(\omega_{Q}^{(l)})}\right)\Phi^{(TO\mu)}(\omega_{Q}^{(l)})\right]^{1/2}\nonumber \\ & & \times\left|1-G_{Q}(d,d)e^{2}\Pi(Q,\omega_{Q}^{(l)})+G_{Q}(0,d)e^{2}\Pi(Q,\omega_{Q}^{(l)})e^{-Qd}\right|^{-1}\label{Eq:ElectronPhononMQ}\end{aligned}$$ is the electron-phonon coupling coefficient corresponding to the $\omega_{Q}^{(l)}$ mode. Landau damping -------------- At sufficiently short wavelengths, plasmons cease to be proper quasi-particle excitations because of Landau damping [@BKRidley:Book99]. To model this phenomenon, albeit approximately, we take that to be the case when the pure graphene plasmon excitation, whose dispersion $\omega=\omega_{p}(Q)$ is determined by the expression $1-e^{2}G_{Q}(d,d)\Pi(Q,\omega)=0$, enters the intra-band single-particle excitation (SPE) continuum [@EHHwang:PRB07]. This happens when the plasmon branch crosses the electron dispersion curve, *i.e.* when $\omega_{p}=\hbar v_{F}Q$, and the wave vector at which this happens is $Q_{c}$. When $Q<Q_{c}$, the electron-phonon coupling coefficient in Eq. (\[Eq:InteractionTerm\]) is that of Eq. (\[Eq:ElectronPhononMQ\]). Although the lower-frequency IPP branches may undergo Landau damping from intra-band SPE as $\omega_{Q}^{(l)}<v_{F}Q$, we still retain them because the sum rules in Eqs. (\[Eq:PlasmonContentSumRule\]) and (\[Eq:PhononContentSumRules\]) require us to maintain charge conservation [@BKRidley:Book99]. On the other hand, when $Q>Q_{c}$, Landau damping is assumed to dominate all the IPP modes and the coupling between the substrate SPP modes and the graphene plasmons can be ignored. Instead of scattering with three IPP modes for each given wave vector, we revert to using only two SPP modes. This allows us to satisfy the sum rules in Eq. (\[Eq:PhononContentSumRules\]). In this case, the electron-phonon coupling coefficient in Eq. (\[Eq:ElectronPhononMQ\]) can be rewritten as: $$M_{Q}^{(l)}=\sum_{\mu=1}^{2}\left[\frac{e^{2}\hbar\omega_{Q}^{(l)}}{2\Omega Q}\left(\frac{1}{\bar{\epsilon}_{ox}^{TO\mu,\infty}(\omega_{Q}^{(l)})+\epsilon_{0}}-\frac{1}{\bar{\epsilon}_{ox}^{TO\mu,0}(\omega_{Q}^{(l)})+\epsilon_{0}}\right)\Phi^{(TO\mu)}(\omega_{Q}^{(l)})\right]^{1/2}$$ where $l=\mathrm{SO_{1}},\mathrm{SO_{2}}$ indexes the SPP branch. Results and discussion ====================== Numerical evaluation -------------------- Having set up the theoretical framework for electron-IPP interaction, we compute the dispersion of the coupled interfacial plasmon-phonon modes and study the electrical transport properties. ### Interfacial plasmon-phonon dispersion In this section we compute the scattering rates from the remote phonons by employing the dispersion relation ($\omega_{Q}^{(l)}$) and the electron-phonon coupling coefficient ($M_{Q}^{(l)}$), which can be determined by solving Eqs. (\[Eq:SecularEquation\]) and (\[Eq:InteractionTerm\]), respectively. For simplicity, to solve the latter equations we use the zero-temperature, long-wavelength approximation for $\Pi(Q,\omega)$ [@BWunsch:NJP06; @EHHwang:PRB07]: $$\Pi(Q,\omega)=\frac{Q^{2}E_{F}}{\pi\hbar^{2}\omega^{2}}\label{Eq:LongWavelengthApprox}$$ where $E_{F}$ is the Fermi level which can be determined from the carrier density $n$ via the relation $n=E_{F}^{2}/(\pi\hbar^{2}v_{F}^{2})$. In Fig. \[Fig:Dispersion1E12\] we show the dispersion relation for an SiO$_{2}$ substrate with $n=10^{12}\mathrm{cm^{-2}}$. The three coupled IPP branches are drawn with solid lines and labeled I, II and III while the dispersion of the uncoupled modes is drawn in dashed lines in the figure. The branches labeled ‘$\mathrm{SO_{1}}$’ (61 meV) and ‘$\mathrm{SO_{2}}$’ (149 meV) have a flat dispersion and are determined from the quantity: $$\epsilon_{0}+\epsilon_{ox}(\omega)=0$$ while the branch labeled ‘Pure plasmon’ is determined from the zeros of the equation: $$1-G_{Q}(d,d)e^{2}\Pi(Q,\omega)\label{Eq:PurePlasmonDispersion}$$ which gives the dispersion of the pure graphene plasmons when the frequency dependence of the substrate dielectric function is neglected and only the effect of the substrate image charges is taken into account. We observe that in the long wavelength limit ($Q\rightarrow0$), branches I, II and III converge asymptotically to the ‘pure’ plasmon, $\mathrm{SO_{1}}$, and $\mathrm{SO_{2}}$ branches respectively. On the other end, as $Q\rightarrow\infty$, branches I, II and III converge asymptotically to the pure $\mathrm{SO_{1}}$, $\mathrm{SO_{2}}$ and plasmon branches respectively. At intermediate values of $Q$ the IPP branches are a mixture of the pure branches. The coupling between pure SO phonons and graphene plasmons has often been ignored in transport studies based on the dispersionless unscreened, decoupled SO modes [@AKonar:PRB10; @JKViljas:PRB10; @KZou:PRL10; @SFratini:PRB08; @SVRotkin:NL09; @VPerebeinos:PRB10; @XLi:APL10] On the other hand, using many-body techniques, Hwang, Sensarma and Das Sarma [@EHHwang:PRB10] have studied the remote phonon-plasmon coupling in supported graphene and were able to reproduce the coupled plasmon-phonon dispersion observed by Liu and Willis [@YLiu:PRB08; @YLiu:PRB10] in their angle-resolved electron-energy-loss spectroscopy experiments on epitaxial graphene grown on SiC. Similar results of strongly coupled plasmon-phonon modes were reported by Koch, Seyller and Schaefer [@RJKoch:PRB10]. Fei and co-workers also found evidence of this plasmon-phonon coupling in the graphene-SiO$_{2}$ system in their infrared nanoscopy experiments [@ZFei:NL11]. Given the increasing experimental support for the hybridization of the SPPs with the graphene plasmons, it is interesting to investigate the effect of these coupled modes on carrier transport in graphene. ![Dispersion relation of coupled interfacial plasmon-phonon system with $n=10^{12}\mathrm{cm^{-2}}$. The three hybrid IPP branches are labeled I (blue), II (green) and III (red). The uncoupled pure SO phonon (1 and 2) and plasmon branches are drawn with dashed lines. In the limits $Q\rightarrow0$ and $Q\rightarrow\infty$, the IPP branches converge to the pure phonon and plasmon branches. In between, they are a mix of the pure branches.[]{data-label="Fig:Dispersion1E12"}](Figure_IPP_Dispersion){width="4in"} ### Electron-phonon coupling Here, the electron-phonon coupling coefficients $M_{Q}^{(l)}$ of the IPP and the SPP modes are compared. Recall that IPP modes are formed through the hybridization of the SPP and graphene plasmon modes, and their coupling to the graphene electrons are different to that of the SPP modes. It is sometimes assumed [@AKonar:PRB10; @SFratini:PRB08] that the SPP modes are screened by the plasmons, and the IPP-electron coupling is weaker than the SPP-electron coupling. As we have discussed above, this assumption does not hold when the frequency of the IPP mode is higher than the plasmon frequency. We plot the $M_{Q}^{(l)}Q$ for the SPP and IPP modes in Fig. \[Fig:CouplingCoeffSiO2\].We first notice that at small $Q$, the coupling terms for branches I and II are actually larger than those for $\mathrm{SO_{1}}$ and $\mathrm{SO_{2}}$, even though I and II are phonon-like. This is because at long wavelengths, $\omega_{p}<\omega_{Q}^{(l)}$ for $l=1,2$, resulting in anti-screening, effect which enhances the SPP electric field. For the plasmon-like branch III, $M_{Q}^{(III)}$ is actually much larger than the those for $\mathrm{SO_{1}}$ and $\mathrm{SO_{2}}$ over the entire range of $Q$ values. When we take Landau damping into account, we use the coupling coefficients (shaded in gray in Fig. \[Fig:CouplingCoeffSiO2\]) of branches I, II and II for $Q<Q_{c}$ and of $\mathrm{SO_{1}}$ and $\mathrm{SO_{2}}$ for $Q\geq Q_{c}$. ![Plot of $M_{Q}^{(l)}Q$ for $n=10^{12}\mathrm{cm^{-2}}$ in SiO$_{2}$. The IPP branches are labeled I (blue), II (green) and III (red), and the SPP branches are labeled $\mathrm{SO_{1}}$ and $\mathrm{SO_{2}}$. The cutoff wave vector $Q_{c}$ is drawn in gray dashed lines. When $Q<Q_{c}$, we use the part of the IPP branches shaded in gray, and when $Q\geq Q_{c}$, we use the part of the SPP branches shaded in gray.[]{data-label="Fig:CouplingCoeffSiO2"}](Figure_ElectronPhononCoupling){width="4in"} Substrate-limited mobility -------------------------- The momentum relaxation rate for an electron in band $s$ with wave vector $\mathbf{K}$ can be written as: $$\begin{aligned} \Gamma_{RP}(s,\mathbf{K}) & =\frac{2\pi}{\hbar}\sum_{l}\sum_{s'}\sum_{\mathbf{Q}}\left|M_{Q}^{(l)}\alpha_{s\mathbf{K+Q},s'\mathbf{K}}\right|^{2}\left[1-ss'\cos(\theta_{\mathbf{K+Q}}-\theta_{\mathbf{K}})\right]\nonumber \\ & \ \times\Bigg\{\left[1+N_{B}(\omega_{Q}^{(l)})\right]\left[1-f(E_{s'\mathbf{K+Q}})\right]\delta(E_{s\mathbf{K}}-E_{s'\mathbf{K+Q}}-\hbar\omega_{Q}^{(l)})\nonumber \\ & \ +N_{B}(\omega_{Q}^{(l)})\left[1-f(E_{s'\mathbf{K+Q}})\right]\delta(E_{s\mathbf{K}}-E_{s'\mathbf{K+Q}}+\hbar\omega_{Q}^{(l)})\Bigg\}\label{Eq:MomentumRelaxationRate}\end{aligned}$$ where $N_{B}(\omega)=(e^{\hbar\omega/k_{B}T}-1)^{-1}$, $f(E)=[e^{(E-E_{F})/k_{B}T}+1]^{-1}$ and $E_{s\mathbf{K}}=s\hbar|\mathbf{K|}$. In assuming the latter expression, we use the Dirac-conical approximation. Equation (\[Eq:MomentumRelaxationRate\]) automatically includes the Fermi-Dirac distribution of the final states and remains applicable when the doping level is high. The individual scattering rates for the screened (I, II and III) and unscreened ($\mathrm{SO_{1}}$ and $\mathrm{SO_{2}}$) branches at the carrier concentration of $n=10^{12}\mathrm{cm^{-2}}$ in SiO$_{2}$ and HfO$_{2}$ are plotted in Fig. \[Fig:ScatteringRateSiO2\]. Landau damping is taken into account by setting the coupling coefficient of the IPP (SPP) modes to zero when $Q<Q_{c}$ ($Q\geq Q_{c}$). We observe that at low energies, the IPP scattering rates are much higher than the SPP ones. At higher energies, the SPP scattering rates increase rapidly. The dominant scattering mechanism around the Fermi level appears to be due to the plasmon-like branch III in SiO$_{2}$ and HfO$_{2}$. In addition, at the Fermi level in $\mathrm{HfO_{2}}$, the SPP branches have scattering rates comparable to those of branch III. This explains why the low density mobility of HfO$_{2}$ is less than that of SiO$_{2}$. ![Plot of scattering rates at $n=10^{12}\mathrm{\ cm^{-2}}$ for different substrates: (left) $\mathrm{SiO_{2}}$ and (right) HfO$_{2}$. In SiO$_{2}$, the plasmon-like branch III dominates the scattering rate at $E=E_{F}$. In HfO$_{2}$, branches III, $\mathrm{SO_{1}}$ and $\mathrm{SO_{2}}$ dominate the scattering rate at $E=E_{F}=117\mathrm{\ meV}$. The SPP branches ($\mathrm{SO_{1}}$ and $\mathrm{SO_{2}}$) do not contribute much to the Fermi-level scattering rate in SiO$_{2}$ because of their higher frequencies and smaller occupation factors.[]{data-label="Fig:ScatteringRateSiO2"}](Fig_ScattRate_SiO2_1E12_300K "fig:"){width="3in"} ![Plot of scattering rates at $n=10^{12}\mathrm{\ cm^{-2}}$ for different substrates: (left) $\mathrm{SiO_{2}}$ and (right) HfO$_{2}$. In SiO$_{2}$, the plasmon-like branch III dominates the scattering rate at $E=E_{F}$. In HfO$_{2}$, branches III, $\mathrm{SO_{1}}$ and $\mathrm{SO_{2}}$ dominate the scattering rate at $E=E_{F}=117\mathrm{\ meV}$. The SPP branches ($\mathrm{SO_{1}}$ and $\mathrm{SO_{2}}$) do not contribute much to the Fermi-level scattering rate in SiO$_{2}$ because of their higher frequencies and smaller occupation factors.[]{data-label="Fig:ScatteringRateSiO2"}](Fig_ScattRate_HfO2_1E12_300K "fig:"){width="3in"} The expression for the IPP/SPP-limited part of the electrical conductivity is: $$\sigma_{RP}=\frac{g_{s}g_{v}e^{2}}{4\pi\hbar^{2}k_{B}T}\int_{0}^{\infty}f(E-E_{F})[1-f(E-E_{F})]\Gamma_{tr}(E)^{-1}EdE\ .\label{Eq:ConductivityFormula}$$ where $g_{s}=2$ and $g_{v}=2$ are the spin and valley degeneracies respectively. Only the contribution from the conduction band is included in Eq. (\[Eq:ConductivityFormula\]). We use Eqs. (\[Eq:MomentumRelaxationRate\]) and (\[Eq:ConductivityFormula\]) to compute the IPP/SPP-limited electrical conductivity by setting: $$\Gamma_{tr}(E)=\Gamma_{RP}(s,\mathbf{K})\ .\label{Eq:SubLtdGamma}$$ In making this approximation, we ignore the other effects (ripples, charged impurity, acoustic phonons, optical phonons, etc). The scattering rates from the acoustic and optical phonons tend to be significantly smaller and are not the limiting factor in electrical transport in supported graphene [@RSShishir:JPhys09]. Impurity scattering tends to be the dominant limiting factor, but its effects can be reduced by varying fabrication conditions. Thus, the conductivity using Eq. (\[Eq:SubLtdGamma\]) gives us its upper bound. We calculate the remote phonon-limited mobility as: $$\mu_{RP}=\frac{\sigma_{RP}}{en}\label{Eq:SubLtdMobility}$$ where $n=\frac{g_{s}g_{v}}{2\pi\hbar^{2}v_{F}^{2}}\int_{0}^{\infty}f(E-E_{F})EdE$ is the carrier density. For $n=10^{12}\mathrm{cm^{-2}}$ in SiO$_{2}$, we obtain $\mu_{RP}\approx40,000\ \mathrm{cm^{2}V^{-1}s^{-1}}$. This is more than the corresponding values reported in the literature ($\sim1000-20,000\ \mathrm{cm^{2}V^{-1}s^{-1}}$) [@YWTan:PRL07; @KSNovoselov:Nature05] although we have to bear in mind that it is an upper limit. Nonetheless, it suggests that IPP/SPP scattering imposes a bound on the electron mobility. Mobility results ---------------- Although suspended graphene has an intrinsic mobility limit of $200,000\ \mathrm{cm^{2}V^{-1}s^{-1}}$ at room temperature [@KIBolotin:SSC08], typical numbers for graphene on SiO$_{2}$ tend to fall in the range 1000-20,000 $\mathrm{cm^{2}V^{-1}s^{-1}}$ [@JHChen:NatureNanotech08]. One significant reason for this drastic reduction in mobility is believed to be the presence of charged impurities in the substrate which causes long-range Coulombic scattering [@SAdam:PNAS07; @SAdam:SSC09; @CJang:PRL08] and much effort has been directed towards the amelioration of the effects of these charged impurities. For example, it has been suggested that modifying the dielectric environment of the graphene, either through immersion in a high-$\kappa$ liquid or an overlayer of high-$\kappa$ dielectric material, can lead to a weakening of the Coulombic interaction and an increase in electron mobility [@CJang:PRL08]. On the other hand, actual experimental evidence in favor of this theory is ambiguous. Electrical conductivity data from Jang and co-workers [@CJang:PRL08] as well as Ponomarenko and co-workers [@LAPonomarenko:PRL09] indicate a smaller-than-expected increase in mobility when a liquid overlayer is used. This suggests that mechanisms other than long and short-range impurity scattering are at play here. Here, we turn to the problem of scattering by IPP modes. ### Comparing different substrates Having set up the theoretical framework in the earlier sections, we now apply it to the study of the remote phonon-limited mobility of four commonly-used substrates: SiO$_{2}$, HfO$_{2}$, h-BN and Al$_{2}$O$_{3}$. Silicon dioxide is the most common substrate material while HfO$_{2}$ and Al$_{2}$O$_{3}$ are high-$\kappa$ dielectrics commonly used as top gate oxides [@KZou:PRL10; @NYGarces:JAP11]. Hexagonal boron nitride shows much promise as both a substrate and a top gate dielectric material [@CRDean:NatureNanotech10]. The study of the remote phonon-limited mobility in these substrates allows us to understand how electronic transport in supported graphene depends on the frequencies and relative permittivities of the substrate phonons. From Eq. (\[Eq:ConductivityFormula\]) with the effects of Landau damping taken into account, we compute the remote phonon-limited mobility numerically, using the well-known Gilat-Raubenheimer method [@GGilat:PR66] to discretize the sum in Eq. (\[Eq:ConductivityFormula\]). We plot $\mu_{RP}$ as a function of carrier density ($n=0.3\times10^{12}\mathrm{\ cm^{-2}}$ to $5.2\times10^{12}\mathrm{\ cm^{-2}}$) at 300 K in Fig. \[Fig:ScreenedResults\]. Note that the mobility values of the high-$\kappa$ substrates (HfO$_{2}$ and Al$_{2}$O$_{3}$) are substantially lower compared to SiO$_{2}$ and h-BN in the carrier density range $n<2.0\times10^{12}\mathrm{cm^{-2}}$. Similar results have been found in MOS systems [@MVFischetti:JAP01]. Hexagonal BN has the highest mobility at low carrier densities because of its high phonon frequencies, which corresponds to low Bose-Einstein occupancy, as well as its weak dipole coupling to graphene. In general, $\mu_{RP}$ for all four substrates increases with $n$ because the dynamic screening effect becomes stronger at higher carrier densities. At low carrier densities, the mobility is low for all the substrates because there is a large proportion of plasmons modes whose frequencies are lower than the SPP mode frequencies. Thus, their coupling to the SPP modes results in the formation of *anti-screened* IPP modes that couple *more* strongly to the carriers, a phenomenon that has been studied for polar semiconductors [@BKRidley:Book99]. However, as $n$ increases, the mobility for all four substrates rises because the plasmon frequency scales as $\omega_{p}\propto n^{1/4}$, resulting in higher-frequency plasmon modes. Thus, the plasmon-phonon coupling forms screened IPP modes that are weakly coupled to the carriers. Furthermore, at higher carrier densities, Landau damping becomes less important as a result of the increasing magnitude of the plasmon wave vector $Q_{c}$. Contrary to expectation, we find that the mobility for HfO$_{2}$ exceeds those of other substrates at larger densities ($n=5\times10^{12}\mathrm{\ cm^{-2}}$). At $n=5\times10^{12}\mathrm{\ cm^{-2}}$, HfO$_{2}$ has the highest remote-phonon mobility followed by h-BN, SiO$_{2}$ and Al$_{2}$O$_{3}$. This is because the proportion of screened IPP modes increases with increasing carrier density. Given the small values of $\omega_{TO1}$ and $\omega_{TO2}$ for HfO$_{2}$, its coupling coefficients are smaller as a result of stronger dynamic screening. This weaker coupling compensates in part the higher occupation factors. In contrast, the larger values of $\omega_{TO1}$ and $\omega_{TO2}$ for h-BN imply that screening does not play a significant role at low carrier densities. Hence, its coupling to the graphene carriers does not diminish as rapidly as carrier density increases. The computed $\mu_{RP}$ values for $\mathrm{HfO_{2}}$ and h-BN highlight the role of low-frequency excitations in carrier scattering. The low-frequency modes are highly occupied at room temperature and induce carrier significant scattering at low $n$. At higher $n$ when dynamic screening becomes important, the low-frequency modes are more strongly screened and their coupling to the carriers becomes diminished more rapidly than that of high-frequency modes. ### Dynamic screening effects To compute the mobility for the case without any screening or anti-screening effects, the Landau damping cutoff wave vector is decreased, *i.e.*, $Q_{c}\rightarrow0$, resulting in the replacement of all the IPP modes with SPP modes. We plot the SPP-limited mobility as a function of carrier density in Fig. \[Fig:ScreenedResults\] (solid symbols), and compare these results for the IPP-limited mobility. The SPP-limited mobility for different substrates spans a range of values varying over nearly two orders of magnitude. In the absence of dynamic screening or anti-screening, the SPP-limited mobility for HfO$_{2}$ is only around $\mathrm{1000\ cm^{2}V^{-1}s^{-1}}$ at $n=10^{12}\ \mathrm{cm}^{-2}$, more than an order of magnitude smaller than the corresponding IPP-limited mobility, because of its low phonon frequencies. This result is also clearly inconsistent with experimental observations, since significantly higher mobility values have been reported for HfO$_{2}$-covered graphene [@BFallahazad:APL10; @KZou:PRL10]. The drastic reduction of the computed mobility suggests that screening is very important for the determination of scattering rates in a coupled plasmon-phonon system with low frequency modes. In contrast, h-BN gives an SPP-limited mobility of $\sim110,000\ \mathrm{cm^{2}V^{-1}s^{-1}}$ at $n=0.3\times10^{12}\ \mathrm{cm}^{-2}$, which is still close to the IPP-limited mobility, indicating that its high frequency modes are relatively unaffected by screening. The maximum SPP-limited mobility for Al$_{2}$O$_{3}$ is around $\mathrm{8,400\ cm^{2}V^{-1}s^{-1}}$at $n=0.3\times10^{12}\ \mathrm{cm}^{-2}$, which is much smaller than the $\mathrm{19000\ cm^{2}V^{-1}s^{-1}}$ extracted by Jandhyala and co-workers [@SJandhyala:ACSNano12] who used Al$_{2}$O$_{3}$ for their top gate dielectric. This disagreement reinforces the necessity of including dynamic screening effects. Furthermore, the carrier density dependence of SPP-limited mobility is different from that of IPP-limited mobility. The IPP-limited $\mu_{RP}$ increases rapidly with carrier density because dynamic screening becomes stronger at higher $n$, an effect that is not found in SPP-limited mobility. In contrast, SPP-limited $\mu_{RP}$ decreases monotonically with increasing $n$. Our results suggest that HfO$_{2}$ remains a promising candidate material for integration with graphene since its high static permittivity can reduce the effect of charged impurities [@AKonar:PRB10] while its IPP scattering rates are relatively low when the carrier density is high. Although its surface excitations are low-frequency, which results in high Bose-Einstein occupancy, this is offset by its relatively strong dynamic screening at higher carrier densities. Thus, IPP scattering does not represent a problem for its integration with graphene field-effect transistors. As expected, h-BN is also a good dielectric material since its high phonon frequencies imply a low Bose-Einstein occupation factor. Furthermore, its smooth interface results in a smaller interface charge density and is less likely to induce mobility-limiting ripples in graphene. $\mathrm{SiO_{2}}$ h-BN $\mathrm{HfO_{2}}$ $\mathrm{Al_{2}O_{3}}$ ------------------------------------------- -------------------- -------- -------------------- ------------------------ $\epsilon_{ox}^{0}$ ($\epsilon_{0}$) 3.90 5.09 22.00 12.35 $\epsilon_{ox}^{i}$ ($\epsilon_{0}$) 3.05 4.57 6.58 7.27 $\epsilon_{ox}^{\infty}$ ($\epsilon_{0}$) 2.50 4.10 5.03 3.20 $\omega_{TO1}$ (meV) 55.60 97.40 12.40 48.18 $\omega_{TO2}$ (meV) 138.10 187.90 48.35 71.41 : Parameters [\[]{}see Eq. (\[Eq:DielectricEquation\])[\]]{} used in computing dispersion relation and scattering rates for SiO$_{2}$, h-BN, HfO$_{2}$ and Al$_{2}$O$_{3}$. They are taken from Refs. [@MVFischetti:JAP01] and [@VPerebeinos:PRB10].[]{data-label="Tab:SubstrateDielectricParameters"} ![Calculated conductivity remote phonon-limited mobility for different values of carrier density and different substrates (SiO$_{2}$, HfO$_{2}$, h-BN and Al$_{2}$O$_{3}$) at room temperature (300 K). The IPP-limited mobility values are plotted using solid lines with unfilled symbols while the SPP-limited mobility values are plotted using dotted lines with solid symbols.[]{data-label="Fig:ScreenedResults"}](Fig_Mobility_AllSubstrates_300K){width="4in"} ### Temperature dependence Remote phonon scattering exhibits a strong temperature dependence – stronger than for ionized impurity scattering – because the Bose-Einstein occupation of the remote phonons decreases with lower temperature. This change in the distribution of the remote phonons (IPP or SPP) necessarily implies that the electronic transport character of the SLG must change with temperature. At lower temperatures, scattering with the remote phonons decreases, resulting in a higher remote phonon-limited electrical mobility. The dependence of the change in mobility with temperature is related to the dispersion of the remote phonons and their coupling to the graphene electrons. By measuring the dependence of the mobility or conductivity with respect to temperature, it is possible to determine the dominant scattering mechanisms in the supported graphene. Given that our model of electron-IPP scattering differs from the more common electron-SPP scattering model, comparing the temperature dependence of the substrate-limited mobility can enable us to distinguish between the two models. The mobility of supported graphene over the temperature range of 100 to 500 K for the different substrates is computed at carrier densities of $n=10^{12}\ \text{cm}^{-2}$ and $n=10^{13}\ \text{cm}^{-2}$. For the purpose of comparison, we perform the calculation for the case with screening (IPP) and without screening (SPP). The results ($1/\mu_{RP}$ vs. $T$) are shown in Fig. \[Fig:InverseMobility\]a and b. In Fig. \[Fig:InverseMobility\]a, we plot the IPP- and SPP-limited inverse mobility at $n=10^{12}\ \text{cm}^{-2}$. As expected, the substrate-limited mobility decreases with rising temperatures for both the screened and unscreened cases. From the plots, we observe that there exists an ‘activation’ temperature for each substrate at which the inverse mobility increases precipitously. For SiO$_{2}$, that temperature is around 200 K in the screened case and around 120 K in the unscreened case. This difference is striking and may be used to distinguish the IPP model from the SPP model at low carrier densities. In all four substrates, the slope of $1/\mu_{RP}$ with respect to $T$ is also steeper in the IPP-limited case than in the SPP-limited case. In Fig. \[Fig:InverseMobility\]b, we plot again the IPP- and SPP-limited inverse mobility but at a much higher carrier density of $n=10^{13}\ \text{cm}^{-2}$. The IPP-limited $1/\mu_{RP}$ is about three orders of magnitude smaller than the SPP-limited $1/\mu_{RP}$ from 100 to 500 K. At high carrier densities, IPP scattering is insignificant and any changes in total mobility with respect to temperature cannot attribute to IPP scattering. The results in Fig. \[Fig:InverseMobility\] suggest that if IPP modes are the surface excitations that limit carrier transport in SiO$_{2}$-supported graphene at room temperature, then the mobility would have a significant increase at around 200 K for $n=10^{12}\ \text{cm}^{-2}$. However, this IPP temperature dependence disappears at much higher carrier densities ($\sim n=10^{13}\ \text{cm}^{-2}$) because the IPP coupling to electrons becomes so weak that it no longer contributes significantly to carrier scattering. The results in Fig. \[Fig:InverseMobility\] also shows that the $\mu_{RP}$ increases monotonically with $n$. This should be contrasted with the result of Fratini and Guinea [@SFratini:PRB08] who found that $\mu_{RP}$ *decreases* as $\sim1/\sqrt{n}$ at room temperature. This is because the coupling coefficient $\lim_{Q\rightarrow\infty}M_{\mathbf{Q}}$, which is proportional to the matrix element, scales as $1/\sqrt{Q}$ in the SPP model with static screening. In Fig. \[Fig:CouplingCoeffSiO2\], $\lim_{Q\rightarrow\infty}M_{\mathbf{Q}}Q$ decreases with $Q$, implying that $\lim_{Q\rightarrow\infty}M_{\mathbf{Q}}$ scales as $Q^{\alpha}$ where $\alpha<-1$. In other words, the coupling coefficient vanishes more rapidly with $Q$ in the IPP model than the SPP model. Our $\mu_{RP}$ results parallel those in Ref. [@ZRen:IEDM03] in which the remote phonon-limited mobility increases with the carrier mobility in a 2-dimensional electron gas system in the Si inversion layer with high-$\kappa$ insulators. In supported SLG, the carrier mobility is limited by three scattering mechanisms: long-range charged impurity, short-range defect and remote phonon scattering [@WZhu:PRB09]. The intrinsic phonon scattering processes in graphene can be effectively neglected. Of the three scattering mechanisms, only remote phonon scattering is strongly temperature dependent. The IPP model suggests that remote phonon scattering diminishes with increasing carrier density. Thus, the experimental consequence is that the temperature dependence of the mobility in supported-SLG should weaken at higher carrier densities. On the other hand, the SPP model predicts that the temperature dependence of the mobility should increase at higher carrier densities [@SFratini:PRB08]. This difference in the temperature dependence of the total mobility between the two models should be easily discriminable in experiments. ![Inverse SPP and IPP-limited mobility versus temperature at (a) $n=10^{12}\ \text{cm}^{-2}$ and (b) $n=10^{13}\ \text{cm}^{-2}$ for SiO$_{2}$, HfO$_{2}$, h-BN and Al$_{2}$O$_{3}$. $1/\mu_{RP}$ is strongly temperature dependent at low carrier densities only.[]{data-label="Fig:InverseMobility"}](Fig_Mobility_AllSubstrates_1E12_Temp_loglog "fig:"){width="4in"} ![Inverse SPP and IPP-limited mobility versus temperature at (a) $n=10^{12}\ \text{cm}^{-2}$ and (b) $n=10^{13}\ \text{cm}^{-2}$ for SiO$_{2}$, HfO$_{2}$, h-BN and Al$_{2}$O$_{3}$. $1/\mu_{RP}$ is strongly temperature dependent at low carrier densities only.[]{data-label="Fig:InverseMobility"}](Fig_Mobility_AllSubstrates_1E13_Temp_loglog "fig:"){width="4in"} ### Disordered graphene We discuss qualitatively the interfacial plasmon-phonon phenomenon in disordered graphene. It is well-known that graphene grown by chemical vapor deposition (CVD) [@XLi:Science09] is generally polycrystalline and contains a high density of defects. In supported graphene, charged impurities from the substrate and other defects scatter graphene carriers. These defects can affect the dynamics of plasmons in graphene which may in turn affect the hybridization between the plasmon and the SPP modes. At short wavelengths, the plasmon lifetime rapidly decreases as a result of Landau damping which results in the decay of the plasmons into single-particle excitations. At long wavelengths, the plasmon lifetime can be affected by defects in the graphene. As far as we know, there is no theory of graphene plasmon damping from defects. However, it has been pointed out that long-wavelength plasmons in polycrystalline metal undergo anomalously large damping due to scattering with structural defects [@VKrishan:PRL70]. If this is also true in polycrystalline or defective graphene, then it implies that the long-wavelength surface excitation in supported graphene are SPP, not IPP, modes. To model phenomenologically this damping of long-wavelength plasmon modes in polycrystalline graphene with defects, we set another cutoff wave vector $Q_{d}$ below which the surface excitations are SPP and not IPP modes. $Q_{d}$ is possibly related to the length scale $\lambda$ of the inhomogeneities or defects in graphene. As a guess, we choose $\lambda$= 6 nm, which is a typical autocorrelation length of ‘puddles’ in neutral supported graphene[@SAdam:PRB11], and set $Q_{d}=1/\lambda$. Hence, in our model, for $Q\geq Q_{c}$ and $Q\leq Q_{d}$, the surface excitations are SPP modes while for $Q_{d}<Q<Q_{c}$, they are IPP modes. We compute the remote phonon-limited mobility at 300 K and plot the results in Fig. \[Fig:DefectGrapheneMobility\]. We find that the long-wavelength SPP dramatically alters the carrier dependence of $\mu_{RP}$ in SiO$_{2}$, HfO$_{2}$ and Al$_{2}$O$_{3}$. In perfect monocrystalline graphene, $\mu_{RP}$ reaches $\mathrm{\sim2\times10^{6}\ cm^{2}V^{-1}s^{-1}}$ in HfO$_{2}$ and SiO$_{2}$ at $n=5\times10^{12}\ \text{cm}^{-2}$ . On the other hand, in polycrystalline graphene with defects, it drops to the range of $10^{4}$ to $\mathrm{10^{5}\ cm^{2}V^{-1}s^{-1}}$. For h-BN, $\mu_{RP}$ is quite relatively unaffected by the long-wavelength SPP modes except at low carrier densities ($n<0.5\times10^{6}\ cm^{-2}$). This change in remote phonon-limited mobility highlights the possible role of defects in the surface excitations of supported graphene. We emphasize that our treatment is purely phenomenological and a more rigorous treatment of plasmon damping is needed in order to obtain a more quantitatively accurate model. Nevertheless, it emphasizes the relationship between dynamic screening and plasmons. In highly defective graphene, the surface excitations may be unscreened SPPs rather than IPPs because of plasmon damping. This should be taken into account when interpreting electronic transport experimental data of exfoliated and CVD-grown graphene. ![Remote phonon-limited mobility in perfect monocrystalline (clear symbols) and defective polycrystalline graphene (solid symbols) for SiO$_{2}$, HfO$_{2}$, h-BN and Al$_{2}$O$_{3}$. As carrier density increases, $\mu_{RP}$ also increases. The use of long-wavelength SPP modes leads to a significant decrease in $\mu_{RP}$ in polycrystalline graphene.[]{data-label="Fig:DefectGrapheneMobility"}](Fig_Mobility_AllSubstrates_Polycrystal_300K){width="4in"} Conclusion ========== We have studied coupled interfacial plasmon-phonon excitations in supported graphene. The coupling between the pure graphene plasmon and the surface polar phonon modes of the substrates results in the formation of the IPP modes, and this coupling is responsible for the screening and anti-screening of the IPP modes. Accounting for these modes, we calculate the room temperature scattering rates and substate-limited mobility for $\mathrm{SiO_{2}}$, $\mathrm{HfO_{2}}$, h-BN and $\mathrm{Al_{2}O_{3}}$ at different carrier densities. The results suggest that, despite being a high-$\kappa$ oxide with low frequency modes, $\mathrm{HfO_{2}}$ exhibits a substrate-limited mobility comparable to that of h-BN at high carrier densities. We attribute this to the dynamic screening of the $\mathrm{HfO_{2}}$ low-frequency modes. The disadvantage of the higher Bose-Einstein occupation of these low-frequency modes is offset by the stronger dynamic screening which suppresses the electron-IPP coupling. Our study also indicates that the contribution to scattering by high-frequency substrate phonon modes cannot be neglected because of they are less weakly screened by the graphene plasmons. The temperature dependence of the remote phonon-limited mobility is also calculated within out theory. Its change with temperature is different at low and high carrier densities. We find that in the IPP model, the temperature dependence of the mobility diminishes with increasing carrier density only, in direct contrast to the predictions of the more commonly used SPP models. The implications of the damping of long-wavelength plasmons have also been studied. We find that the it leads to a substantial reduction in the remote phonon-limited mobility in SiO$_{2}$, HfO$_{2}$ and Al$_{2}$O$_{3}$. We gratefully acknowledge the support provided by Texas Instruments, the Semiconductor Research Corporation (SRC), the Microelectronics Advanced Research Corporation (MARCO), the Focus Center Research Project (FCRP) for Materials, Structures and Devices (MSD), and Samsung Electronics Ltd. We also like to thank David K. Ferry (Arizona State University), Eric Pop (University of Illinois), and Andrey Serov (University of Illinois) for engaging in valuable technical discussions.

--- abstract: 'Given $n$ points in the plane, a *covering path* is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least $n/2$ segments, and $n-1$ straight line segments obviously suffice even if the covering path is required to be noncrossing. We show that every set of $n$ points in the plane admits a (possibly self-crossing) covering path consisting of $n/2 +O(n/\log{n})$ straight line segments. If the path is required to be noncrossing, we prove that $(1-{\varepsilon})n$ straight line segments suffice for a small constant ${\varepsilon}>0$, and we exhibit $n$-element point sets that require at least $5n/9 -O(1)$ segments in every such path. Further, the analogous question for noncrossing *covering trees* is considered and similar bounds are obtained. Finally, it is shown that computing a noncrossing covering path for $n$ points in the plane requires $\Omega(n \log{n})$ time in the worst case.' author: - 'Adrian Dumitrescu[^1]' - 'Dániel Gerbner[^2]' - 'Balázs Keszegh[^3]' - 'Csaba D. Tóth[^4]' title: Covering Paths for Planar Point Sets --- Introduction {#sec:intro} ============ In this paper we study polygonal paths visiting a finite set of points in the plane. A *spanning path* is a directed Hamiltonian path drawn with straight line edges. Each edge in the path connects two of the points, so a spanning path can only turn at one of the given points. Every spanning path of a set of $n$ points consists of $n-1$ segments. A *covering path* is a directed polygonal path in the plane that visits all the points. A covering path can make a turn at any point, [[i.e.]{}]{}, either at one of the given points or at a (chosen) Steiner point. Obviously, a spanning path for a point set $S$ is also a covering path for $S$. If no three points in $S$ are collinear, every covering path consists of at least $\lceil n/2\rceil$ segments. A *minimum-link* covering path for $S$ is one with the smallest number of segments (links). A point set is said to be in *general position* if no three points are collinear. We study the following two questions concerning covering paths posed by Morić [@Mo10; @Mo11] as a generalization of the well-known puzzle of linking $9$ dots in a $3 \times 3$ grid with a polygonal path having only $4$ segments [@Lo1914]. Another problem which leads to these questions is separating red from blue points [@FKMU10]. 1. What is the minimum number, $f(n)$, such that every set of $n$ points in the plane can be covered by a (possibly self-intersecting) polygonal path with $f(n)$ segments? 2. What is the minimum number, $g(n)$, such that every set of $n$ points in the plane can be covered by a *noncrossing* polygonal path with $g(n)$ segments? If no three points are collinear, then each segment of a covering path contains at most two points, thus $\lceil n/2\rceil$ is a trivial lower bound for both $f(n)$ and $g(n)$. Morić conjectured that the answer to the first problem is $n(1/2+o(1))$ while the answer to the second is $n(1-o(1))$. We confirm his first conjecture (Theorem \[T1\]) but refute the second (Theorem \[paththm\])[^5]. A consideration of these questions in retrospect appears in [@DO11]. \[T1\] Every set of $n$ points in the plane admits a (possibly self-crossing) covering path consisting of $n/2 +O(n/\log{n})$ line segments. Consequently, $\lceil n/2\rceil \leq f(n) \leq n/2 +O(n/\log{n})$. A covering path with $n/2 +O(n/\log{n})$ segments can be computed in $O(n^{1+{\varepsilon}})$ time, for every ${\varepsilon}>0$. As expected, the noncrossing property is much harder to deal with. Every set of $n$ points in the plane trivially admits a noncrossing path consisting of $n-1$ straight line segments that visits all the points, [[e.g.]{}]{}, by sorting the points along some direction, and then connecting them in this order. On the other hand, again trivially, any such covering path requires at least $\lceil n/2 \rceil$ segments, if no three points are collinear. We provide the first nontrivial upper and lower bounds for $g(n)$, in particular disproving the conjectured relation $g(n)=n(1-o(1))$. \[paththm\] Every set of $n$ points in the plane admits a noncrossing covering path with at most $\lceil (1-1/601080391)n\rceil-1$ segments. Consequently, $g(n) \leq \lceil (1-1/601080391)n\rceil-1$. A noncrossing covering path with at most this many segments can be computed in $O(n\log n)$ time. \[thm:path-lower\] There exist $n$-element point sets that require at least $(5n-4)/9$ segments in any noncrossing covering path. Consequently, $g(n) \geq (5n-4)/9$. In the proof of Theorem \[paththm\], we construct a noncrossing covering path that can easily be extended to a noncrossing covering cycle by adding one Steiner point (and two segments). \[cor:cycle\] Every set of $n\geq 2$ points in the plane admits a noncrossing covering cycle with at most $\lceil (1-1/601080391)n\rceil+1$ segments. A noncrossing covering cycle of at most this many segments can be computed in $O(n\log n)$ time. #### Covering trees. For covering a finite point set in the plane, certain types of geometric graphs other than paths may also be practical. A noncrossing path or tree, for example, are equally useful for separating a red and blue set of points [@FKMU10], which is one of the motivating problems. A *covering tree* for a planar point set $S$ is a tree drawn in the plane with straight-line edges such that every point in $S$ lies at a vertex or on an edge of the tree. The lower and upper bounds $\lceil n/2\rceil \leq f(n) \leq n/2 +O(n/\log{n})$ of Theorem \[T1\] trivially carry over for the number of edges of covering trees (with possible edge crossings). Let $t(n)$ be the minimum integer such that every set of $n$ points in the plane admits a *noncrossing covering tree* with $t(n)$ straight-line edges. Since every path is a tree, we have $t(n)\leq g(n)\leq \lceil (1-1/601080391)\rceil$ from Theorem \[paththm\]. However, a noncrossing covering tree is significantly easier to obtain than a noncrossing covering path. By simplifying the proof of Theorem \[paththm\], we derive a stronger upper bound for covering trees. \[thm:trees-upper\] Every set of $n$ points in the plane admits a noncrossing covering tree with at most $\lfloor 5n/6\rfloor$ edges. Consequently, $t(n) \leq \lfloor 5n/6\rfloor$. A covering tree with at most $\lfloor 5n/6\rfloor$ edges can be computed in $O(n\log n)$ time. By modifying the lower bound analysis in the proof of Theorem \[thm:path-lower\], we show that the same point set used there yields a slightly weaker lower bound for noncrossing covering trees. \[thm:tree-lower\] There exist $n$-element point sets in the plane that require at least $(9n-4)/17$ edges in any noncrossing covering tree. Consequently, $t(n) \geq (9n-4)/17$. Instead of minimizing the number of edges in a covering tree, one can try to minimize the number of line segments, where each segment is either a single edge or a chain of several collinear edges of the tree. Let $s(n)$ be the minimum integer such that every set of $n$ points in the plane admits a *noncrossing covering tree* with $s(n)$ line segments. By definition, we trivially have $s(n) \leq t(n)$. In addition, we determine an exact formula for $s(n)$: \[prop:tree:s(n)\] We have $$s(n)= \begin{cases} n-1 \text{~~~~~~if } n=2,3,4 \\ \lceil n/2 \rceil \text{~~~~~~if } n \geq 5. \end{cases}$$ #### Bicolored variants. Let $S$ be a bicolored set of $n$ points, with $S=B\cup R$, where $B$ and $R$ are the set of blue and red points, respectively. Two covering paths, $\pi_R$ and $\pi_B$, one for the red and one for the blue points, are *mutually noncrossing* if each of $\pi_R$ and $\pi_B$ is noncrossing, and moreover, $\pi_R$ and $\pi_B$ do not cross (intersect) each other. A natural extension of the monochromatic noncrossing covering path problem is: What is the minimum number $j(n)$ such that every bicolored set of $n$ points in the plane can be covered by two monochromatic mutually noncrossing polygonal paths with $j(n)$ segments in total? Using the construction in the proof of Theorem \[thm:path-lower\] we obtain the following corollary. \[C1\] Given a bicolored set of $n$ points, there are two mutually noncrossing covering paths with a total of at most $3n/2+O(1)$ segments. Such a pair of paths can be computed in $O(n \log{n})$ time. On the other hand, there exist bicolored sets that require at least $5n/9 -O(1)$ segments in any pair of mutually noncrossing covering paths. Consequently, $5n/9 -O(1) \leq j(n) \leq 3n/2+O(1)$. Similarly, two covering trees $\tau_R$ and $\tau_B$, one for the red and one for the blue points, are *mutually noncrossing* if each is noncrossing and $\tau_R$ and $\tau_B$ do not cross each other. The analogous question is in this case: What is the minimum number $k(n)$ such that every bicolored set of $n$ points in the plane can be covered by two monochromatic mutually noncrossing polygonal trees with $k(n)$ edges in total? The construction in the proof of Theorem \[thm:tree-lower\] yields the following corollary. \[C2\] Given a bicolored set of $n$ points, there are two mutually noncrossing covering trees with a total of at most $n$ edges. Such a pair of trees can be computed in $O(n \log{n})$ time. On the other hand, there exist bicolored sets that require at least $9n/17 -O(1)$ edges in any pair of mutually noncrossing covering trees. Consequently, $9n/17 -O(1) \leq k(n) \leq n$. #### Computational complexity. We establish an $\Omega(n \log{n})$ lower bound for computing a noncrossing covering path for a set of $n$ points in the plane. \[T3\] The sorting problem for $n$ numbers is linear-time reducible to the problem of computing a noncrossing covering path for $n$ points in the plane. Therefore, computing a noncrossing covering path for a set of $n$ points in the plane requires $\Omega(n \log{n})$ time in the worst case in the algebraic decision tree model of computation. On the other hand, a noncrossing covering tree for $n$ points can be easily computed in $O(n)$ time; see also Section \[sec:conclusion\]. #### Related previous results. Given a set of $n$ points in the plane, the [minimum-link covering path]{} problem asks for a covering path with the smallest number of segments (links). Arkin [[et al.]{}]{} [@AMP03] proved that (the decision version of) this problem is NP-complete. Stein and Wagner [@SW01] gave a $O(\log z)$-approximation where $z$ is the maximum number of collinear points. Various upper and lower bounds on the minimum number of links needed in an axis-aligned path traversing an $n$-element point set in ${\mathbb{R}}^d$ have been obtained in [@BBD+08; @CM98; @C04; @KKM94]. Approximation algorithms with constant ratio (depending on the dimension $d$) for this problem are developed in [@BBD+08], while some NP-hardness results have been claimed in [@EHS10], and further revised in [@J12]. Other variants of Euclidean TSP can be found in a survey article by Mitchell [@Mi00]. Covering Paths with Possible Self-Crossings {#sec:cover1} =========================================== A set $X$ of $k$ points in general position in the plane, no two on a vertical line, is a [*$k$-cap*]{} ([*$k$-cup*]{}, respectively) if $X$ is in (strictly) convex position and all points of $X$ lie above (below, respectively) the line connecting the leftmost and the rightmost point of $X$. Similarly, caps and cups can be defined for arbitrary points (with allowed collinearities), with $X$ being in *weakly* convex position. By slightly abusing notation, we use the same terminology when referring to them, and distinguish them based on the underlying point sets. According to a classical result of Erdős and Szekeres [@ES35], every set of at least ${2k-4 \choose k-2}+1$ points in general position in the plane, no two on a vertical line, contains a $k$-cap or a $k$-cup. In particular, every such set contains $k$ points in convex position; see also [@ES60; @Ma02]. They also showed that this bound is the best possible, [[i.e.]{}]{}, there exist sets of ${2k-4 \choose k-2}$ points containing no $k$-cup or $k$-cap. More generally, there exist sets of ${k+l-4 \choose k-2}$ points containing neither $k$-cups nor $l$-caps. While Erdős and Szekeres originally proved the above results for points in general position, their arguments go though verbatim for arbitrary point sets (with allowed collinearities), and the same quantitative bounds hold for the resulting caps or cups (now in weakly convex position). Following the terminology coined by Welzl [@We11], a set $S$ of $n$ points in the plane is called [*perfect*]{} if it can be covered by a (possibly self-crossing) polygonal path consisting of at most $\lceil n/2 \rceil$ segments. It is easy to see that a cup or a cap is perfect: indeed, a suitable covering path can be obtained by extending the odd numbered edges of the $x$-monotone polygonal chain connecting the points (since no two points lie on a vertical line, any consecutive pair of these edges properly intersect). #### Proof of Theorem \[T1\]. Let $S$ be a set of $n$ points in the plane, no three of which are collinear. Choose an orthogonal coordinate system such that no two points have the same $x$-coordinate. By the result of Erdős and Szekeres [@ES35], every $m$-element subset of $S$ contains a $k$-cup or a $k$-cap for some $k=\Omega(\log{m})$. Since every such subset is perfect, it can be covered by a path of $\lceil k/2 \rceil$ segments. To construct a covering path, we partition $S$ into caps and cups of size $\Omega(\log n)$ each, and a set of less than $n/\log n$ “leftover” points. Set $T=S$. While $|T|\geq n/\log n$, repeatedly find a maximum-size cup or cap in $T$ and delete those elements from $T$. Note that $\log(n/\log{n}) = \Omega(\log{n})$, and we have found a $k$-cup or $k$-cap for some $k=\Omega(\log n)$ in each step. Therefore, we have found $O(n/\log n)$ pairwise disjoint caps and cups in $S$, and we are left with a set $T$ of less than $n/\log n$ points. For each $k$-cup (or $k$-cap), construct a covering sub-path with $\lceil k/2\rceil$ segments. Link these paths arbitrarily into one path, that is, append them one after another in any order. Finally append to this path an arbitrary spanning path of the remaining less than $n/\log{n}$ points in $T$, with one point per turn. A covering path for $S$ is obtained in this way. The total number of segments in this path is $n/2 + O(n/\log{n})$, as required. Chvátal and Klincsek [@CK80] showed that a maximum-size cap (and cup) in a set of $n$ points in the plane, no 3 of which are collinear, can be found in $O(n^3)$ time. With $O(n/\log n)$ calls to their algorithm, a covering path with $n/2 +O(n/\log n)$ segments can be constructed in $O(n^4/\log n)$ time in the RAM model of computation. Now if the problem can be solved in time $O(n^4/\log n)$, it can also be solved in time $O(n^{1+{\varepsilon}})$ for any ${\varepsilon}>0$: arbitrarily partition the points into $n^{1 -{\varepsilon}/3}$ subsets of $n^{{\varepsilon}/3}$ points each, solve each subset separately, move the leftover points to the next subset, then link the paths together with one extra segment per path. Non-Crossing Covering Paths: Upper Bound {#sec:noncrossing-up} ======================================== It is easy to see that for every set of $n$ points in the plane, there is a noncrossing covering path with at most $n-1$ segments. For example, an $x$-monotone spanning path for $n$ points has $n-1$ segments, no two crossing edges, and no Steiner points either. In this section, we prove Theorem \[paththm\] and show that $(1-{\varepsilon})n$ segments suffice for some small constant ${\varepsilon}>0$. In the proof of Theorem \[paththm\], however, we still use the trivial upper bound $n-1$ for several subsets of points with the additional constraint that the two endpoints of the path are two given points on the boundary of a convex region containing the points in its interior (Lemma \[trivi\]). \[trivi\] Let $X$ be a set of $n$ points in the interior of a convex region $C$, and let $a,b$ be two points on the boundary $\partial C$ of $C$. Then $X\cup \{a,b\}$ admits a noncrossing covering path with $|X|+1$ segments such that its two endpoints are $a$ and $b$, and its relative interior lies in the interior of $C$. Such a covering path can be constructed in $O(n\log n)$ time. We include the easy proof for completeness (a similar lemma was also an essential tool in [@FKMU10]). Let $\ell_a$ and $\ell_b$ be tangent lines at $C$ incident to $a$ and $b$, respectively. If $\ell_a$ and $\ell_b$ are not parallel, then let $O=\ell_a\cap \ell_b$; otherwise let $O$ be a point at infinity corresponding to the direction of the two parallel lines $\ell_a$ and $\ell_b$. Sort the points in $X$ in the order in which they are encountered by a rotating sweep line from $\ell_a$ to $\ell_b$ around $O$ (with ties broken arbitrarily). Let $\gamma$ be the polygonal path that starts at $a$, visits the points in $X$ in the above sweep order, and ends at $b$. The edges of $\gamma$ are pairwise noncrossing, since they lie in interior-disjoint wedges centered at $O$. By construction, $\gamma$ lies in the convex hull of $X\cup \{a,b\}\subseteq {\rm int}(C)\cup \{a,b\}$, hence the relative interior of $\gamma$ lies in the interior of $C$, as required. Before proving Theorem \[paththm\], we show how to reduce the trivial bound $n-1$ on the size of noncrossing covering paths by an arbitrarily large constant, provided that the number of points $n$ is sufficiently large. \[lenyeg\] Let $S$ be a set of $n$ points in the plane that contains a cap or cup of even size $k$. Then $S$ admits a noncrossing covering path $\gamma$ with at most $n+1-\lfloor k/6\rfloor$ segments. Furthermore, if $S$ lies in the interior of a vertical strip $H$ bounded by two vertical lines, $h_1$ and $h_2$, then we may require that the two endpoints of $\gamma$ lie on $h_1$ and $h_2$, respectively, and the relative interior of $\gamma$ lie in the interior of $H$. Let $H$, $S\subset H$, be a vertical strip bounded by two vertical lines, $h_1$ and $h_2$, from the left and right, respectively. Assume that $S$ contains a *cap* of size $k$ (the case of a *cup* is analogous). We construct a noncrossing covering path $\gamma$ with at most $n+1-\lfloor k/6\rfloor$ segments. We may assume that $k$ is a multiple of 6 (by decreasing $k$, if necessary, without changing the value of $\lfloor k/6\rfloor$). Let $P=\{p_1,p_2,\ldots , p_k\}\subset S$ be a cap of size of $k$, labeled in increasing order of $x$-coordinates. Note that $P$ admits a covering path $\gamma_0=(q_1,q_2,\ldots,q_{k/2+1})$ of $k/2$ segments, where each segment of $\gamma_0$ contains two consecutive points of the cap. We may assume (by extending or shortening $\gamma_0$ if necessary) that the endpoints of $\gamma_0$ are on the boundary of the vertical strip $H$, that is, $q_1\in h_1$ and $q_{k/2+1}\in h_2$. Let $q_0\in h_1$ and $q_{k/2+2}\in h_2$ be arbitrary points vertically below the endpoints of $\gamma_0$. With this notation, the polygonal path $(q_0q_1)\cup \gamma_0\cup (q_{k/2+1}q_{k/2+2})$ is a convex arc. Let $s_{k/2+1}\in h_2$ be an arbitrary Steiner point above the right endpoint of $\gamma_0$. The two endpoints of our final covering path for $S$ will be $q_0 \in h_1$ and $s_{k/2+1} \in h_2$. In the remainder of the proof, we first construct a noncrossing covering path for $S$ with $n+1$ segments from $q_0$ to $s_{k/2+1}$ and then modify this construction to “save” $\lfloor k/6\rfloor$ segments. ![A set with $n=36$ points. A covering path $\gamma_0$ for a cap of $k=18$ points (bold). The region above $\gamma_0$ is decomposed into 9 convex regions by vertical rays $r_i$, $i=2,\ldots,9$, each passing through a Steiner point $s_i$. We obtain a covering path with $n+1=37$ segments for all points by concatenating three covering paths.[]{data-label="fig:eps1"}](eps11){width="80.00000%"} #### Preliminary approach. The path $\gamma_0$ decomposes the vertical strip $H$ into two regions: a convex region below and a nonconvex region above, with $k/2-1$ reflex vertices at $q_2,\ldots , q_{k/2}$. Decompose the nonconvex region above $\gamma_0$ into $k/2$ convex regions by upward vertical rays $r_i$ emitted by $q_i$, for $i=2,\ldots,q_{k/2}$; and place an arbitrary Steiner point $s_i$ on the ray $r_i$. We construct a noncrossing covering path for $S$ as a concatenation of the following three paths: The first path is a covering path for the points in $S$ lying strictly below $\gamma_0$, from the Steiner point $q_0\in h_1$ to $p_k \in \gamma_0$, obtained by Lemma \[trivi\]. The second path is the part of $\gamma_0$ from $p_k$ to $p_1$. The third path consists of $k/2$ covering subpaths for the points in $S$ lying above $\gamma_0$; the first of these subpaths runs from $p_1$ to $s_2$, and the others run from $s_i$ to $s_{i+1}$, for $i=1,\ldots,k/2$, as obtained by Lemma \[trivi\]. The resulting path visits all points of $S$, since the second part visits all points along $\gamma_0$, and the convex regions jointly contain all points below or above $\gamma_0$. The $k/2+2$ parts of the covering path are pairwise noncrossing, since they lie either on $\gamma_0$ or in pairwise interior-disjoint regions whose interiors are also disjoint from $\gamma_0$. The second part of the covering path (the part along $\gamma_0$) covers $k$ points with $k/2$ segments, and the remaining $k/2+1$ parts each require one more segment than the number of points covered. Hence, the total number of segments is $n+1-(k/2)+(k/2+1)=n+1$. #### Modified construction. We now modify the above construction and “save” $\lfloor k/6\rfloor$ segments. The savings come from the following two ideas. (1) It is not necessary to decompose the entire region above $\gamma_0$ into convex pieces. If a region above $\gamma_0$ contains all points in $S$ above $\gamma_0$, and has fewer than $k/2$ reflex vertices, then we can decompose this region into fewer than $k/2$ convex pieces, using fewer than $k/2$ Steiner points and thus reducing the size of the resulting covering path. (2) If a ray emitted by a reflex vertex passes through a point of $S$ and decomposes the reflex angle into two convex angles, then a Steiner point can be replaced by a point of $S$, which saves one segment in the resulting covering path. ![Local modifications for $\gamma_0$. Case 1: $S\cap Q_9\neq \emptyset$, we choose a ray $r_9$ emitted by $q_9$ that passes through a point in $S\cap Q_9$. Case 2: $S\cap Q_3=\emptyset$, we construct ${\rm conv}(X_3)$, and replace the arc $(q_2,q_3,q_4)\subset \gamma_0$ by $(q_2,q_2',q_4',q_4)\subset \gamma_1$. Similarly, $S\cap Q_6=\emptyset$, we construct ${\rm conv}(X_6)$, and replace the arc $(q_5,q_6,q_7)\subset \gamma_0$ by $(q_5,q_5',q_7',q_7)\subset \gamma_1$. Note that the triangles $\Delta(c_3,q_2',q_4')$ and $\Delta(c_6,q_5',q_7')$ contain no points from $S$. []{data-label="fig:eps2"}](eps22){width="80.00000%"} We show that one of the two ideas is always applicable locally. Specifically, we modify $\gamma_0$ by replacing some of the arcs $(q_{i-1},q_i,q_{i+1})\subset \gamma_0$, where $i$ is a multiple of 3, by different arcs and obtain a new convex polygonal arc $\gamma_1$. The path $\gamma_1$ retains the property that every segment contains two points from $S$, however it may consist of fewer segments than $\gamma_0$. We keep the modifications “local” in the sense that a modification in the neighborhood of a vertex $q_i$, where $i>0$ is a multiple of 3, is carried out independently of modifications at all other vertices $q_j$, where $j>0$ is a multiple of 3. Even though $\gamma_0$ may be modified in the neighborhoods of vertices $q_i$ and $q_{i+3}$, where $i>0$ is a multiple of 3, the intermediate edge $q_{i+1}q_{i+2}$ of $\gamma_0$ will preserved: it will be either an edge of $\gamma_1$ or contained in a longer edge of $\gamma_1$. For every $i$, where $i$ is a positive multiple of 3, let $Q_i$ be the region above *both* lines $q_{i-1}q_i$ and $q_iq_{i+1}$, and between the vertical rays $r_{i-1}$ and $r_{i+1}$. We distinguish two cases (refer to Fig. \[fig:eps2\]): #### Case 1: $S\cap Q_i\neq \emptyset$. In this case, the polygonal arc $(q_{i-1},q_i,q_{i+1})$ of $\gamma_0$ is not modified but the ray $r_i$ is defined differently. In addition, the Steiner points $s_{i-1}\in r_{i-1}$ and $s_{i+1}\in r_{i+1}$ are defined differently. Pick an arbitrary point $s_i\in S\cap Q_i$, and let $r_i$ be the ray emitted by $q_i$ and passing through $s_i$. Let $r_{i-1}$ and $r_{i+1}$ be vertical rays emitted by $q_{i-1}$ and $q_{i+1}$, respectively, like before. Decompose the region above $(q_{i-1},q_i,q_{i+1})$ by the two vertical rays $r_{i-1}$ and $r_{i+1}$, and then by the (possibly nonvertical) ray $r_i$. Choose Steiner points $s_{i-1}\in r_{i-1}$ and $s_{i+1}\in r_{i+1}$, such that they each lie on the common boundary of two consecutive regions of the decomposition. #### Case 2: $S\cap Q_i= \emptyset$. In this case we modify the polygonal arc $(q_{i-1},q_i,q_{i+1})$ of $\gamma_0$. Let $c_i$ be the intersection point of lines $q_{i-2}q_{i-1}$ and $q_{i+1}q_{i+2}$ (possibly $q_{i-2}=q_0$ or $q_{i+2}=q_{k/2+2}$). Let $S_i$ denote the set of points of $S$ in the interior of the triangle $\Delta(q_{i-1}c_iq_{i+1})$; and let $X_i=S_i\cup \{q_{i-1},q_{i+1}\}$. Since $Q_i$ is empty and each of the segments $q_{i-1}q_i$ and $q_iq_{i+1}$ contains two points of $S$, the convex hull of $X_i$ has at least four vertices, [[i.e.]{}]{}, ${{\rm conv}}(X_i)$ is not a triangle. Let $a_ib_i$ be an arbitrary edge of ${{\rm conv}}(X_i)$, that is incident to neither $q_{i-1}$ nor $q_{i+1}$. The line $a_ib_i$ intersects the sides $q_{i-1}c_i$ and $q_{i+1}c_i$ of the triangle $\Delta(q_{i-1}c_iq_{i+1})$. Denote the intersection points by $q_{i-1}'\in q_{i-1}c_i$ and $q_{i+1}'\in q_{i+1}c_i$. Replace $(q_{i-1},q_i,q_{i+1})$ by $(q_{i-1},q_{i-1}',q_{i+1}',q_{i+1})$ to obtain $\gamma_1$. The segment $q_{i-1}'q_{i+1}'$ contains points $a_i,b_i\in S$. Notice that $q_{i-1}q_{i-1}'$ is collinear with $q_{i-2}q_{i-1}$; and similarly $q_{i+1}'q_{i+1}$ is collinear with $q_{i+1}q_{i+2}$. Therefore, $q_{i-1}$ and $q_{i+1}$ are not vertices of $\gamma_1$. Let $r_i$ be a vertical ray emitted by $c_i$, and pick an arbitrary Steiner point $s_i\in r_i$. Decompose the region above $(q_{i-1},q_{i-1}',q_{i+1}',q_{i+1})\subset \gamma_1$ by the polygonal arc $(q_{i-1},c_i, q_{i+1})$ and the vertical ray $r_i$. Since the triangle $\Delta(c_i,q_{i-1}',q_{i+1}')$ contains no point from $S$, the two convex regions adjacent to $r_i$ (only one region in the extremal case $i=k/2$) contain all the points of $S$ lying above $(q_{i-1},q_{i-1}',q_{i+1}',q_{i+1})$. ![The set of $n=36$ points from Fig. \[fig:eps1\]. A covering path $\gamma_1$ for a cap of $14$ points (bold). A dashed noncrossing path below $\gamma_1$, and a dashed noncrossing path above $\gamma_1$ cover all remaining points in $S$. The resulting covering path has $n-2=34$ segments.[]{data-label="fig:eps3"}](eps33){width="80.00000%"} After the local modifications, we proceed analogously to our initial approach. Construct a noncrossing covering path for $S$ as a concatenation of the following three paths: A trivial path for the points strictly below $\gamma_1$ from the Steiner point $q_0\in h_1$ to the rightmost vertex on $\gamma_1$ (Lemma \[trivi\]); followed by part of $\gamma_1$ from the rightmost to the leftmost point on $\gamma_1$; the third path visits all points in the convex regions above $\gamma_1$, passing though the points $s_i\in r_i$ between consecutive regions. The resulting path visits all points of $S$ lying below, on, and above $\gamma_1$ (in this order). We use $\lfloor k/6\rfloor$ fewer segments than in our initial construction, since each local modification saves one segment: In Case 1, we use a point $s_i\in S$ instead of a Steiner point. In Case 2, we decrease the number of segments along $\gamma_1$ by one, and decrease the number of relevant convex regions above $\gamma_1$ by two. This concludes the proof of Lemma \[lenyeg\]. Let $S$ be a set of $n+1$ points in the plane, no three of which are collinear. Assume, by rotating the point set if necessary, that no two points have the same $x$-coordinate. Lay out a raster of vertical lines in the plane such that there are exactly $m={32\choose 16}+1=601,080,391$ points between consecutive lines; no points on the lines or to the left of the leftmost line; and less than $m$ points to the right of the rightmost line. By the result of Erdős and Szekeres [@ES35], there is a cap or cup of 18 points between any two raster lines. By Lemma \[lenyeg\] (for $n=m$ and $k=18$), the $m$ points between consecutive raster lines admit a noncrossing covering path with $m+1-3=m-2$ segments such that the two endpoints of the path are Steiner points on the two raster lines, and the relative interior of the path lies strictly between the raster lines. The $n_0<m$ points to the right of the rightmost raster line can be covered by an $x$-monotone path with $n_0$ segments starting at the (unique) Steiner point on that line assigned by the previous group of points. The noncrossing covering paths between consecutive raster lines can be joined into a single noncrossing covering path for $S$ by adding one vertical segment on each raster line except for the first and the last one, as depicted in Fig. \[pathjoin\]. The total number of segments is $n-\lfloor n/m\rfloor-1\leq \lceil (1-1/601080391)n\rceil-1$, as claimed. ![Joining noncrossing covering paths in the last step.[]{data-label="pathjoin"}](pathjoin){width="60.00000%"} In the proof of Theorem \[paththm\], we constructed a noncrossing covering path $\gamma$ for $S$ such that the two endpoints of $\gamma$ are leftmost and the rightmost vertices of $\gamma$. Hence $\gamma$ can be augmented to a noncrossing covering *cycle* by adding a new vertex of sufficiently large $y$-coordinate, and thus proving Corollary \[cor:cycle\]. Noncrossing Covering Paths: Lower Bound {#sec:cover2} ======================================= #### Proof of Theorem \[thm:path-lower\] (outline). For every $k\in \mathbb{N}$, we construct a set $S$ of $n=2k$ points in the plane in general position, where all points are very close to the parabola $x\rightarrow x^2$. We then show that every noncrossing covering path $\gamma$ consists of at least $(5n-4)/9$ segments. The lower bound is based on a charging scheme: we distinguish *perfect* and *imperfect* segments in $\gamma$, containing 2 and fewer than 2 points of $S$, respectively. We charge every perfect segment to a “nearby” endpoint of an imperfect segment or an endpoint of $\gamma$, such that each of these endpoints is charged at most twice. This implies that at most about $\frac{4}{5}$ of the segments are perfect, and the lower bound of $(5n-4)/9$ follows. We continue with the details. #### A technical lemma. We start with a simple lemma, showing that certain segments in a noncrossing covering path are almost parallel. We say that a line segment $s$ *traverses* a circular disk $D$ if $s$ intersects the boundary of $D$ twice. =0.75 \[lem:par\] Let $\varphi \in (0,\frac{\pi}{2})$ be an angle. For every ${\varepsilon}>0$, there exists $\delta\in (0,{\varepsilon})$ such that if two noncrossing line segments $ab$ and $cd$ both traverse two concentric disks of radii ${\varepsilon}$ and $\delta$, then the supporting lines of the segments $ab$ and $cd$ meet at an angle at most $\varphi$. Let $ab$ and $cd$ be two noncrossing line segments that both traverse two concentric disks of radii ${\varepsilon}>\delta>0$. Refer to Fig. \[fig:parallel\]. By translating the segments, if necessary, we may assume that both are tangent to the disk of radius $\delta$. Clip the segments in the disk of radius ${\varepsilon}$ to obtain two noncrossing chords. The angle between two noncrossing chords is maximal if they have a common endpoint. In this case, they meet at an angle $2\arcsin(\delta/{\varepsilon})$. For every ${\varepsilon}>0$, $\exists \delta\in (0,{\varepsilon})$ such that $2\arcsin(\delta/{\varepsilon})<\varphi$. #### Construction. For every $k\in \mathbb{N}$, we define a set $S=\{a_1,\ldots , a_k, b_1,\ldots , b_k\}$ of $n=2k$ points. Initially, let $A=\{a_1,\ldots , a_k\}$ be a set of $k$ points on the first-quadrant part of the parabola $\alpha: x\rightarrow x^2$ such that no two lines determined by $A$ are parallel. (To achieve strong general position, we shall slightly perturb the points in $S$ in the last step of the construction.) Label the points in $A$ by $a_1,\ldots , a_k$ in increasing order of $x$-coordinates. Each point $b_i$ will be in a small $\delta$-neighborhood of $a_i$, for a suitable $\delta>0$ and $i=1,\ldots,k$. The pairs $\{a_i,b_i\}$ are called *twin*s. The value of $\delta>0$ is specified in the next paragraph. See Fig. \[58\] for a sketch of the construction. =0.45 For every $r>0$, let $D_i(r)$ denote the disk of radius $r$ centered at $a_i\in A$. Since the points in $A$ are in strictly convex position, points in $A$ determine ${k\choose 2}$ distinct lines. Let $(2\varphi)\in (0,\frac{\pi}{2})$ be the minimum angle between two lines determined by $A$ (recall that no two such lines are parallel). Let ${\varepsilon}>0$ be sufficiently small such if a line intersects two disks in $\{D_1({\varepsilon}),\ldots ,D_k({\varepsilon})\}$, then it meets the line passing through the centers of the two disks at an angle less than $\varphi/2$. It follows that any line intersects at most two disks $D_1({\varepsilon}),\ldots , D_k({\varepsilon})$ ([[i.e.]{}]{}, the ${\varepsilon}$-neighborhoods of at most two points in $A$). By Lemma \[lem:par\], there exists $\delta_0>0$ such that if two noncrossing segments traverse both $D_i({\varepsilon})$ and $D_i(\delta_0)$, then their supporting lines meet at an angle less than $\varphi$. For $i=1,\ldots , k-1$, let $\delta_i>0$ be the maximum distance between the supporting line of $a_ia_{i+1}$ and points on the arc of the parabola $\alpha$ between $a_i$ and $a_{i+1}$. We are ready to define $\delta>0$; let $\delta=\min\{\delta_i:i=0,1,\ldots, k-1\}$. We now choose points $b_i\in D_i(\delta)$, for $i=1,\ldots , k$, in *reverse* order. Let $\ell_k$ be a line that passes through $a_k$ such that its slope is larger than the tangent of the parabola $\alpha$ at $a_k$. Let $b_k$ be a point in $\ell_k\cap D_k(\delta)$ above the parabola $\alpha$. Having defined line $\ell_j$ and point $b_j$ for all $j>i$, we choose $\ell_i$ and $b_i\in \ell_i \cap D_i(\delta)$ as follows: - let $\ell_i$ be a line passing through $a_i$ such that its slope is larger than that of $\ell_{i+1}$; - let $b_i\in \ell_i\cap D_i(\delta)$ be above the parabola $\alpha$; and - let $b_i$ be so close to $a_i$ that for every $j$, $i<j\leq k$, the supporting lines of segments $a_ia_j$ and $b_ib_j$ meet in the disk $D_i({\varepsilon})$. Write $B=\{b_i,\ldots , b_k\}$. We also ensure in each iteration that in the set $S=A\cup B$, (1) no three points are collinear; (2) no two lines determined by the points are parallel; and (3) no three lines determined by disjoint pairs of points are concurrent. Note that $S$ is not in strong general position: for instance, all points in $A$ lie on a parabola. (By strong general position it is meant here there is no nontrivial algebraic relation between the coordinates of the points.) In the last step of our construction, we slightly perturb the points in $S$. However, for the analysis of a covering path, we may ignore the perturbation. Let $\gamma$ be a noncrossing covering path for $S$. By perturbing the vertices of $\gamma$ if necessary, we may assume that every point in $S$ lies in the relative interior of a segment of $\gamma$. Denote by $s_0$, $s_1$ and $s_2$, respectively, the number of segments in $\gamma$ that contain 0, 1, and 2 points from $S$. We establish the following inequality. \[pp:s01\] $s_2\leq 4(s_0+s_1+1)$. Before the proof of Lemma \[pp:s01\], we show that it immediately implies Theorem \[thm:path-lower\]. Counting the number of points incident to the segments, we have $n=s_1+2s_2$. The number of segments in $\gamma$ is $s_0+s_1+s_2$. This number can be bounded from below by using Lemma \[pp:s01\] as follows. $$\begin{aligned} s_0+s_1+s_2&=& \frac{4(s_0+s_1+1)+5s_0+5s_1-4}{9}+s_2\nonumber\\ &\geq& \frac{s_2+5s_0+5s_1-4}{9}+s_2\nonumber\\ &\geq& \frac{5(s_1+2s_2)-4}{9} =\frac{5n-4}{9},\nonumber\end{aligned}$$ as claimed For the proof of Lemma \[pp:s01\], we introduce a charging scheme: each perfect segment is charged to either an endpoint of an imperfect segment, or one of the two endpoints of $\gamma$ such that every such endpoint is charged at most twice. The charges will be defined for maximal $x$-monotone chains of perfect segments. A subpath $\gamma'\subseteq \gamma$ is called *$x$-monotone*, if the intersection of $\gamma'$ with a vertical line is connected ([[i.e.]{}]{}, the empty set, a point, or a vertical segment). Recall that all points in $A=\{a_1,\ldots , a_k\}$ lie on the parabola $\alpha: x\rightarrow x^2$. Let $\beta$ be the graph of a strictly convex function that passes through the points $b_1,\ldots , b_k$, and lies strictly above $\alpha$ and below the curve $x\rightarrow x^2+\delta$. #### Properties of a noncrossing path covering $S$. We start by characterizing the perfect segments in $\gamma$. Note that if $pq$ is a perfect segment in $\gamma$, then $pq$ contains either a twin, or one point from each of two twins. First we make a few observations about perfect segments containing points from two twins. \[pp:perfect\] Let $pq$ be a perfect segment in $\gamma$ that contains one point from each of the twins $\{a_i,b_i\}$ and $\{a_j,b_j\}$, where $i<j$. Then $pq$ intersects both $D_i(\delta)$ and $D_j(\delta)$, and its endpoints lie below the curve $\beta$. The distance between any two twin points is less than $\delta$, so $pq$ intersects the $\delta$-neighborhood of $a_i$ and $a_j$ (even if $pq$ contains $b_i$ or $b_j$). The line $pq$ intersects the parabolas $\alpha:x\rightarrow x^2$ and $x\rightarrow x^2+\delta$ twice each. It also intersects $\beta$ exactly twice: at least twice, since $\beta$ is between the two parabolas; and at most twice since the region above $\beta$ is strictly convex. All points in $\{a_i,b_i\}$ and $\{a_j,b_j\}$ are on or below $\beta$; but $pq$ is above $\beta$ at some point between its intersections with $\{a_i,b_i\}$ and $\{a_j,b_j\}$, since $\delta\leq \delta_i$. Hence the endpoints of $pq$ are below $\beta$. \[pp:nearby\] Let $pq$ be a perfect segment of $\gamma$ that contains one point from each of the twins $\{a_i,b_i\}$ and $\{a_j,b_j\}$, where $i<j$. Assume that $p$ is the left endpoint of $pq$. Let $s$ be the segment of $\gamma$ containing the other point of the twin $\{a_i,b_i\}$. Then one of the following four cases occurs. - *Case 1:* $p$ is incident to an imperfect segment of $\gamma$, or $p$ is an endpoint of $\gamma$; - *Case 2:* $s$ is imperfect; - *Case 3:* $s$ is perfect, one of its endpoints $v$ lies in $D_i({\varepsilon})$, and $v$ is either incident to some imperfect segment or it is an endpoint of $\gamma$; - *Case 4:* $s$ is perfect and $p$ is the common left endpoint of segments $pq$ and $s$. =0.7 If $p$ is incident to an imperfect segment of $\gamma$, or $p$ is an endpoint of $\gamma$, then Case 1 occurs. Assume therefore that $p$ is incident to two perfect segments of $\gamma$, $pq$ and $pr$. If $pr=s$, then $p$ is the common left endpoint of two perfect segments, $pq$ and $s$, and Case 4 occurs. If $s$ is imperfect, then Case 2 occurs. Assume now that $pr\neq s$ and $s$ is perfect. We shall show that Case 3 occurs. We claim that the segment $pq$ traverses $D_i({\varepsilon})$. It is enough to show that $p$ and $q$ lie outside of $D_i({\varepsilon})$. Note that $pr$ does not contain any point from the twin $\{a_i,b_i\}$ (these points are covered by segments $pq$ and $s$). Since $pr$ is a perfect segment, it contains two points from $S\setminus \{a_i,b_i\}$. By construction, every line determined by $S\setminus \{a_i,b_i\}$ is disjoint from $D_i({\varepsilon})$, hence $pr$ (including $p$) is outside of $D_i({\varepsilon})$. Since $pq$ contains a point from $\{a_j,b_j\}$, $i<j$, point $q$ is also outside of $D_i({\varepsilon})$. Hence $pq$ traverses $D_i({\varepsilon})$. We also claim that $s$ cannot traverse $D_i({\varepsilon})$. Suppose, to the contrary, that $s$ traverses $D_i({\varepsilon})$. By Lemma \[lem:par\], the supporting lines of $pq$ and $s$ meet at an angle less than $\varphi$. By the choice of ${\varepsilon}$, the supporting line of $s$ can intersect the ${\varepsilon}$-neighborhoods of $a_i$ and $a_j$ only. However, by the choice of $b_i$, if $s$ contains one point from each of $\{a_i,b_i\}$ and $\{a_j,b_j\}$, then the supporting lines of $s$ and $pq$ intersect in $D_i({\varepsilon})$. This contradicts the fact that the segments of $\gamma$ do not cross, and proves the claim. Since $s$ does not traverse $D_i({\varepsilon})$, it has an endpoint $v$ in $D_i({\varepsilon})$. If $v$ is the endpoint of $\gamma$, then Case 3 occurs. If $v$ is incident to some other segment of $\gamma$, this segment cannot be perfect since every line intersects the ${\varepsilon}$-neighborhoods of at most two points in $A$. Hence $v$ is incident to an imperfect segment, and Case 3 occurs. We continue with two simple observations about perfect segments containing twins. \[pp:2vertical\] The supporting lines of any two twins intersect below $\alpha$. By construction, the supporting line of every twin has positive slope; and $a_ib_i$ has larger slope than $a_jb_j$ if $1\leq i<j\leq k$. Furthermore, the line $a_ib_i$ has larger slope than the tangent line of the parabola $x\rightarrow x^2$ at $a_i$, hence $a_i$ lies above the supporting line of $a_jb_j$ for $1\leq i<j\leq k$. It follows that the supporting lines of segments $a_ib_i$ and $a_jb_j$ intersect below $\alpha$. \[pp:vertical\] Let $pq$ be a perfect segment of $\gamma$ that contains a twin $\{a_i,b_i\}$, and let $q$ be the upper ([[i.e.]{}]{}, right) endpoint of $pq$. Then either $q$ is incident to an imperfect segment of $\gamma$ or $q$ is an endpoint of $\gamma$. Observe that $q$ lies above $\beta$. If $q$ is an endpoint of $\gamma$, then our proof is complete. Suppose that $q$ is incident to segments $pq$ and $qr$ of $\gamma$. By Lemma \[pp:2vertical\], $qr$ does not contain a twin. By Lemma \[pp:perfect\], $qr$ cannot contain one point from each of two twins, either, since then its endpoints would lie below $\beta$. It follows that $qr$ is an imperfect segment of $\gamma$, as required. Let $\Gamma'$ be the set of maximal $x$-monotone chains of perfect segments in $\gamma$. Consider a chain $\gamma'\in \Gamma'$. By Lemma \[pp:vertical\], only the rightmost segment of $\gamma'$ may contain a twin. It is possible that the leftmost segment of $\gamma'$ contains one point from each of two twins, and the left endpoint of $\gamma'$ is incident to another perfect segment, which is the left endpoints of another $x$-monotone chain in $\Gamma'$. Let $pq$ be a perfect segment of $\gamma$, and part of an $x$-monotone chain $\gamma'\in \Gamma'$. We charge $pq$ to a point $\sigma(pq)$ that is either an endpoint of some imperfect segment or an endpoint of $\gamma$. The point $\sigma(pq)$ is defined as follows. If $pq$ contains a twin, then charge $pq$ to the top vertex of $pq$, which is the endpoint of an imperfect segment or an endpoint of $\gamma$ by Lemma \[pp:vertical\]. Assume now that $pq$ does not contain a twin, its left endpoint is $p$, and it contains a point from each of the twins $\{a_i,b_i\}$ and $\{a_j,b_j\}$, with $i<j$. We consider the four cases presented in Lemma \[pp:nearby\]. In Case 1, charge $pq$ to $p$, which is the endpoint of an imperfect segment or an endpoint of $\gamma$. In Case 2, charge $pq$ to the left endpoint of the imperfect segment $s$ containing a point of the twin $\{a_i,b_i\}$. In Case 3, charge $pq$ to either an endpoint an imperfect segment or an endpoint of $\gamma$ located in $D_i({\varepsilon})$. So far, every endpoint of an imperfect segment and every endpoint of $\gamma$ is charged at most once. Now, consider Case 4 of Lemma \[pp:nearby\]. In this case, $pq$ is the leftmost segment of $\gamma'$. If $\gamma'$ contains exactly one perfect segment, namely $pq$, then charge $pq$ to its right endpoint, which is the endpoint of some imperfect segment or the endpoint of $\gamma$. If $\gamma'$ contains at least two perfect segments, then pick an arbitrary perfect segment $s$, $s \neq pq$, from $\gamma'$. Since $s$ is not the leftmost segment of $\gamma'$, the point $\sigma(s)$ is already defined, and we let $\sigma(pq)=\sigma(s)$. This completes the definition of $\sigma(pq)$. Each endpoint of $\gamma$ and each endpoint of every imperfect segment is now charged at most twice. Since $\gamma$ and every imperfect segment has two endpoints, we have $s_2\leq 4(s_0+s_1)+4$, as required. #### Remark. We do not know whether the lower bound $(5n-4)/9$ for the number of segments in a minimum noncrossing covering path is tight for the $n$-element point set $S$ we have constructed. The set $S$ certainly has a covering path with $5n/8+O(1)$ segments. Such a path is indicated in Fig. \[58\], where 5 consecutive segments (4 perfect and one imperfect) cover 4 consecutive twins. Noncrossing Covering Trees ========================== An upper bound $t(n) \leq \lceil (1-1/601080391)n\rceil$ for noncrossing covering trees follows from Theorem \[paththm\]. However, the argument can be greatly simplified while also improving the bound. Any set of 7 points with distinct $x$-coordinates contains a cap or cup of 4 points, say, $a,b,c,d$, from left to right. The $4$ points $a,b,c,d$, of a cap or cup admit a covering path with 2 segments, [[i.e.]{}]{}, a 2-edge star centered at the intersection point, say $v$, of the lines through $ab$ and $cd$, respectively. Augment this 2-edge star covering 4 points to a 5-edge star centered at $v$ and covering all 7 points. The star is contained in the vertical strip bounded by vertical lines incident to the leftmost and the rightmost point, respectively. We may assume, by rotating the point set if necessary, that no two points have the same $x$-coordinate. Let $p_1,p_2,\ldots,p_n$ be the points in $S$ listed in left to right order. Decompose $S$ into groups of 7 by drawing vertical lines incident to $p_{7+6i}$, $i=0,1,\ldots$. Any two consecutive groups in this decomposition share a point (the last point in group $i$ is also the first point in group $i+1$). Thus the stars covering the groups (using $5$ edges per group) are already connected in a tree covering all points, that yields the claimed bound. We prove a lower bound for $t(n)$ by analyzing noncrossing covering trees of the point set $S=\{a_1,\ldots , a_k, b_1,\ldots , b_k\}$, $n=2k$, defined in Section \[sec:cover2\] above. Let $\tau$ be a noncrossing covering tree for $S$. By perturbing the vertices of $\tau$ if necessary, we may assume that every point in $S$ lies in the relative interior of a segment of $\tau$. Let $s_0$, $s_1$ and $s_2$, respectively, denote the number of segments in $\tau$ that contain 0, 1, and 2 points from $S$; hence $n=s_1+2s_2$. We establish the following weaker version of Lemma \[pp:s01\]. \[lem:tree\] $s_2\leq 8(s_0+s_1)+4$ Before the proof of Lemma \[lem:tree\], we show that it directly implies Theorem \[thm:tree-lower\]. The total number of segments in $\tau$ is $$\begin{aligned} s_0+s_1+s_2&=& \frac{8(s_0+s_1)+4+9s_0+9s_1-4}{17}+s_2\nonumber\\ &\geq& \frac{s_2+9s_0+9s_1-4}{17}+s_2\nonumber\\ &\geq& \frac{9(s_1+2s_2)-4}{17} =\frac{9n-4}{17},\nonumber\end{aligned}$$ where we used Lemma \[lem:tree\] and the fact that the total number of points is $n=s_1+2s_2$. For the proof of Lemma \[lem:tree\], we set up a charging scheme, similar to the proof of Lemma \[pp:s01\]. Lemma \[pp:vertical\] continues to hold for $\tau$ in place of a noncrossing covering path $\gamma$ if we replace the “endpoints of $\gamma$” by the “leaves of $\tau$.” While the path $\gamma$ has exactly two endpoints, the tree $\tau$ may have arbitrarily many leaves. Therefore, the charging scheme has to be modified so that no perfect segment is charged to the leaves of $\tau$. Since $S$ is in general position, no three perfect segments have a common endpoint. Therefore, the perfect segments of $\tau$ form disjoint paths. Let $\Gamma$ be the set of maximal chains of perfect segments; and let $\Gamma_x$ denote the set of maximal $x$-monotone chains of perfect segments in $\tau$. Choose an arbitrary vertex $r_0$ in $\tau$ as a *root*, and direct all edges of $\tau$ towards $r_0$. Every chain in $\Gamma$ is incident to either vertex $r_0$, or to a unique outgoing imperfect edge. Since every chain in $\Gamma$ has exactly two endpoints, at most two vertices of a chain can have degree 1 in the tree $\tau$. In the proof of Lemma \[pp:s01\], we charged every perfect segment $pq$ of a covering path $\gamma$ to a point $\sigma(pq)$, which was an endpoint of an imperfect segment or the endpoint of $\gamma$. The function $\sigma$ relied on the properties established in Lemmas \[pp:perfect\]–\[pp:vertical\]. These Lemmas also hold for the covering tree $\tau$, if we replace the endpoints of $\gamma$ by the leaves ([[i.e.]{}]{}, vertices of degree 1) in $\tau$. With this interpretation, every perfect segment $pq$ is assigned to a point $\sigma(pq)$, which is either an endpoint of an imperfect segment of $\tau$ or an endpoint of a chain in $\Gamma$. We are now ready to define our charging scheme for $\tau$. Let $pq$ be a perfect segment of $\tau$. - If $\sigma(pq)$ is an endpoint of an imperfect segment of $\tau$, then charge $pq$ to $\sigma(pq)$. - Otherwise $\sigma(pq)$ is an end point of a chain $\gamma_{pq}\in \Gamma$. In this case, if $\gamma_{pq}$ is incident to $r_0$, then charge $pq$ to the root $r_0$ of $\tau$, else charge $pq$ to the outgoing imperfect edge of $\tau$ incident to $\gamma_{pq}$. Similarly to the proof of Lemma \[pp:s01\], each endpoint of an imperfect segment is charged at most twice. Since every imperfect segment has two endpoints, rule (i) is responsible for a total charge of at most $4(s_0+s_1)$. Each of the two endpoints of a chain $\gamma\in \Gamma$ is charged at most twice by $\sigma$, and so the root $r_0$ and every (directed) imperfect segment is charged at most four times. Thus rule (ii) is responsible for a total charge of at most $4(s_0+s_1+1)$. Altogether the total charge assigned by rules (i) and (ii) is $s_2\leq 8(s_0+s_1)+4$, as required. The case of small $n$ ($n \leq 4$) is easy to handle, so assume that $n \geq 5$. Given $S$, compute ${{\rm conv}}(S)$ and let $s_1$ be a segment extension of an arbitrary edge of ${{\rm conv}}(S)$; $s_1$ is long enough so that it intersects all non-parallel lines induced by pairs of points in $S':=S \setminus S \cap s_1$. For simplicity of exposition assume that $s_1$ is a vertical segment with all other points in $S'$ lying left of $s_1$. If ${{\rm conv}}(S')$ is a vertical segment, since $|S| \geq 5$, it is easy to find a $3$-segment covering tree for $S$. If ${{\rm conv}}(S')$ is not a vertical segment, select a non-vertical hull edge of ${{\rm conv}}(S')$ and extend it to the right until it hits $s_1$ and to the left until it hits all other non-parallel lines induced by pairs of points. Let $s_2$ be this segment extension. Continue in a similar way on the set of remaining points, $S'':=S' \setminus S' \cap s_2$, by choosing an arbitrary edge of ${{\rm conv}}(S'')$ and extending it until it hits $s_1$ or $s_2$. If a single point is left at the end, pick a segment incident to it and extend it until it hits the tree made from the previously chosen segments. Otherwise continue by extending an arbitrary hull edge of the remaining points until it hits the tree made from the previously chosen segments. Clearly the resulting tree covers all points and has at most $\lceil n/2 \rceil$ segments. For the lower bound, it is clear that $n$ points in general position require at least $\lceil n/2 \rceil$ segments in any covering tree. Bicolored Variants {#sec:two} ================== For the upper bound, we proceed as follows. Assume without loss of generality that no two points have the same $x$-coordinate (after a suitable rotation of the point set, if needed). We have $|B|+|R|=n$, and assume w.l.o.g. that $|B| \leq n/2 \leq |R|$. Cover the red points by an $x$-monotone spanning path $\pi_R$, which is clearly noncrossing. Let $B=B_1 \cup B_2$ be the partition of the blue points induced by $\pi_R$ into points above and below the red path (remaining points are partitioned arbitrarily). Cover the points in $B_1$ (above $\pi_R$) by an $x$-monotone covering path: for each consecutive pair of points in the $x$-order, extend two almost vertical rays that meet far above $\pi_R$ without crossing $\pi_R$. Proceed similarly for covering the points in $B_2$ (below $\pi_R$). Connect the two resulting blue covering paths for $B_1$ and $B_2$ by using at most $O(1)$ additional segments. The number of segments in the red path is $|R|-1$. The number of segments in the blue path is $2|B|+O(1)$. Consequently, since $|B| \leq n/2$, the two covering paths comprise at most $3n/2+O(1)$ segments. After sorting the red and blue points along a suitable direction, a pair of mutually noncrossing covering paths as above can be obtained in $O(n)$ time. So the entire procedure takes $O(n \log{n})$ time. For the lower bound, use a red and a blue copy of the point set constructed in the proof of Theorem \[thm:path-lower\], each with $n/2$ points, so that no three points are collinear. Since covering each copy requires at least $(5n/9-O(1))/2$ segments in any noncrossing covering path, the resulting $n$-element point set requires at least $5n/9 -O(1)$ segments in any pair of mutually noncrossing covering paths. For the lower bound we use two copies, red and blue, of the point-set from the proof of Theorem \[thm:tree-lower\] (which is the same as the point-set from the proof of Theorem \[thm:path-lower\]). It remains to show the upper bound. Assume without loss of generality that no two points have the same $x$-coordinate. Cover the blue points by a blue star with a center high above, and the red points by a red star with a center way below. Obviously, each star is non-crossing, and the distinct $x$-coordinates of the points suffice to guarantee that the two stars are mutually non-crossing for suitable center positions. The two centers can be easily computed after sorting the points in the above order. Computational Complexity {#sec:complexity} ======================== #### Proof of Theorem \[T3\]. We make a reduction from the sorting problem in the algebraic decision tree model of computation. Given $n$ distinct numbers, $x_1,\ldots,x_n$, we map them in $O(n)$ time to $n$ points on the parabola $y=x^2$: $x_i \to (x_i, x_i^2)$; similar reductions can be found in [@PS85]. Let $S$ denote this $n$-element point set. Since no 3 points are collinear, any covering path for $S$ has at least $\lceil n/2\rceil +1$ vertices. We show below that, given a noncrossing covering path of $S$ with $m=\Omega(n)$ vertices, the points in $S$ can be sorted in left to right order in $O(m)$ time; equivalently, given a noncrossing covering path with $m$ vertices, the $n=O(m)$ input numbers can be sorted in $O(m)$ time. Consequently, the $\Omega(n \log{n})$ lower bound is then implied. Thus it suffices to prove the following. > Given a noncrossing covering path $\gamma$ of $S$ with $m$ vertices, the points in $S$ can be sorted in left to right order in $O(m)$ time. The boundary of the convex hull of $\gamma$ is a closed polygonal curve, denoted $\partial {{\rm conv}}(\gamma)$. Melkman’s algorithm [@Me87] computes $\partial {{\rm conv}}(\gamma)$ in $O(m)$ time. (See [@Aloupis] for a review of convex hull algorithms for simple polygons, and [@BC06] for space-efficient variants). Triangulate all faces of the plane graph $\gamma \cup \partial {{\rm conv}}(\gamma)$ within $O(m)$ time [@Ch91], and let $T$ denote the triangulation. The parabola $y=x^2$ intersects the boundary of each triangle at most 6 times (at most twice per edge). The intersection points can be sorted in each triangle in $O(1)$ time. So we can trace the parabola $y=x^2$ from triangle to triangle through the entire triangulation, in $O(1)$ time per triangle, thus in $O(m)$ time overall. Since all points of $S$ are on the parabola, one can report the sorted order of the points within the same time. Conclusion {#sec:conclusion} ========== We conclude with a few (new or previously posed) questions and some remarks. 1. It seems unlikely that every point set with no three collinear points admits a covering path with $n/2 +O(1)$ segments. Can a lower bound of the form $f(n)=n/2 + \omega(1)$ be established? 2. It remains an open problem to close or narrow the gap between the lower and upper bounds for $g(n)$, $(5n-4)/9\leq g(n)\leq \lceil(1-1/601080391)n\rceil-1$; and for $t(n)$, $(9n-4)/17\leq t(n)\leq \lfloor 5n/6\rfloor$. 3. Let $p(n)$ denote the maximum integer such that every set of $n$ points in the plane has a perfect subset of size $p(n)$. As noticed by Welzl [@DO11; @We11], $p(n)=\Omega(\log{n})$ immediately follows from the theorem of Erdős and Szekeres [@ES35]. Any improvement in this lower bound would lead to a better upper bound on $f(n)$ in Theorem \[T1\], and thus to a smaller gap relative to the trivial lower bound $f(n) \geq n/2$. It is a challenging question whether Welzl’s lower bound $p(n)=\Omega(\log{n})$ can be improved; see also [@DO11]. 4. It is known that the minimum-link covering path problem is NP-complete for planar paths whose segments are unrestricted in orientation [@AMP03; @KKM94]. It is also NP-complete for axis-parallel paths in ${\mathbb{R}}^{10}$, as shown in [@J12]. Is the minimum-link covering path problem still NP-complete for axis-aligned paths in ${\mathbb{R}}^d$ for $2\leq d\leq 9$? It is known [@BBD+08] that a minimum-link axis-aligned covering path in the plane can be approximated with ratio $2$. Can the approximation ratio of $2$ be reduced? 5. Is the minimum-link covering path problem still NP-complete for points in general position and arbitrary oriented paths? 6. Is the minimum-link covering path problem still NP-complete for points in general position and arbitrary oriented noncrossing paths? 7. Given $n$ points ($n$ even), is it possible to compute a noncrossing perfect matching in $O(n)$ time? Observe that such a matching can be computed in $O(n \log{n})$ time by sorting the points along some direction. The same upper bound $O(n \log{n})$ holds for noncrossing covering paths and noncrossing spanning paths, and this is asymptotically optimal by Theorem \[T3\]. Observe finally that a noncrossing spanning tree can be computed in $O(n)$ time: indeed, just take a star rooted at an arbitrary point in the set. #### Acknowledgment. The authors are grateful to an anonymous reviewer for the running time improvement in Theorem \[T1\], and to the members of the MIT-Tufts Computational Geometry Research Group for stimulating discussions. [99]{} G. Aloupis, A history of linear-time convex hull algorithms for simple polygons, `http://cgm.cs.mcgill.ca/ athens/cs601/`. E. M. Arkin, J. S. B. Mitchell and C. D. Piatko, Minimum-link watchman tours, *Inf. Proc. Lett.* **86** (2003), 203–207. S. Bereg, P. Bose, A. Dumitrescu, F. Hurtado and P. Valtr, Traversing a set of points with a minimum number of turns, *Discrete Comput. Geom.* **41(4)** (2009), 513–532. H. Brönnimann and T. M. Chan, Space-efficient algorithms for computing the convex hull of a simple polygonal line in linear time, *Comput. Geom.* [**34**]{} (2006), 75–-82. B. Chazelle, Triangulating a simple polygon in linear time, *Discrete Comput. Geom.* **6** (1991), 485–524. V. Chvátal and G. Klincsek, Finding largest convex subsets, *Congressus Numerantium* **29** (1980), 453–460. M. J. Collins and M. E. Moret, Improved lower bounds for the link length of rectilinear spanning paths in grids, *Inf. Proc. Lett.* **68** (1998), 317–319. M. J. Collins, Covering a set of points with a minimum number of turns, *Internat. J. Comput. Geom. Appl.* **14(1-2)** (2004), 105–114. E.D. Demaine and J. O’Rourke, Open problems from CCCG 2010, in *Proc. 23rd Canadian Conf. Comput. Geom.*, 2011, Toronto, ON, pp. 153–156. P. Erdős and G. Szekeres, A combinatorial problem in geometry, *Compositio Mathematica* **2** (1935), 463–470. P. Erdős and G. Szekeres, On some extremum problems in elementary geometry, *Annales Univ. Sci. Budapest* **3-4** (1960–1961), 53–62. V. Estivill-Castro, A. Heednacram, and F. Suraweera, NP-completeness and FPT results for rectilinear covering problems, *Journal of Universal Computer Science* **15** (2010), 622–652. R. Fulek, B. Keszegh, F. Morić and I. Uljarević, On polygons excluding point sets, *Graphs and Combinatorics* (2012), in print. M. Jiang, On covering points with minimum turns, *Frontiers in Algorithmics and Algorithmic Aspects in Information and Management - Joint International Conference (FAW-AAIM)*, vol. 7285 of LNCS, Springer, 2012, pp. 58–69. E. Kranakis, D. Krizanc and L. Meertens, Link length of rectilinear Hamiltonian tours in grids, *Ars Combinatoria* **38** (1994), 177–192. S. Loyd, *Cyclopedia of Puzzles*, The Lamb Publishing Company, 1914. J. Matoušek, *Lectures on Discrete Geometry*, Springer, New York, 2002. A. Melkman, On-line construction of the convex hull of a simple polygon, *Inf. Proc. Lett.* **25** (1) (1987), 11–12. J. S. B. Mitchell, Geometric shortest paths and network optimization, in *Handbook of Computational Geometry (J-R. Sack and J. Urrutia, eds.)*, Chap. 15, Elsevier, 2000, pp. 633–701. F. Morić, Separating and covering points in the plane, communication at the Open Problem Session of *22nd Canadian Conf. Comput. Geom.*, Winnipeg, MB, 2010. F. Morić, Covering points by a polygonal line, in *Gremo’s Workshop on Open Problems*, Problem Booklet, Wergenstein, Switzerland, 2011. F. P. Preparata and M. I. Shamos, *Computational Geometry: An Introduction*, Springer, New York, 1985. C. Stein and D. P. Wagner, Approximation algorithms for the minimum bends traveling salesman problem, in *ICALP*, vol. 2081 of LNCS, 2001, pp. 406-421. E. Welzl, communication at the *9th Gremo’s Workshop on Open Problems*, Wergenstein, Switzerland, 2011. [^1]: Computer Science, University of Wisconsin–Milwaukee, USA. Email: `dumitres@uwm.edu`. Research was supported by the NSF grant DMS-1001667. [^2]: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary. Email: `gerbner.daniel@renyi.mta.hu`. Research was supported by OTKA, grant NK 78439 and by OTKA under EUROGIGA project GraDR 10-EuroGIGA-OP-003. [^3]: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary. Email: `keszegh.balazs@renyi.mta.hu`. Research was supported by OTKA, grant NK 78439 and by OTKA under EUROGIGA project GraDR 10-EuroGIGA-OP-003. [^4]: Department of Mathematics, California State University, Northridge, USA and University of Calgary, Canada. Email: `cdtoth@ucalgary.ca`. Research was supported by the NSERC grant RGPIN 35586, the NSF grant CCF-0830734, and the Fields Institute for Research in Mathematical Sciences, Toronto, Canada. [^5]: The first item was observed by the current authors during the Canadian Conference CCCG 2010 and was also communicated to the authors of [@DO11].

--- abstract: | We present late-time radio observations of 68 local Type Ibc supernovae, including six events with broad optical absorption lines (“hypernovae”). None of these objects exhibit radio emission attributable to off-axis gamma-ray burst jets spreading into our line-of-sight. Comparison with our afterglow models reveals the following conclusions: (1) Less than $\sim 10\%$ of Type Ibc supernovae are associated with typical gamma-ray bursts initially directed away from our line-of-sight; this places an empirical constraint on the GRB beaming factor of $\langle f_b^{-1}\rangle \lesssim 10^4$ corresponding to an average jet opening angle, $\theta_j \gtrsim 0.8$ degrees. (2) This holds in particular for the broad-lined supernovae (SNe 1997dq, 1997ef, 1998ey, 2002ap, 2002bl and 2003jd) which have been argued to host GRB jets. Our observations reveal no evidence for typical (or even sub-energetic) GRBs and rule out the scenario in which every broad-lined SN harbors a GRB at the $84\%$ confidence level. Their large photospheric velocities and asymmetric ejecta (inferred from spectropolarimetry and nebular spectroscopy) appear to be characteristic of the non-relativistic supernova explosion and do not necessarily imply the existence of associated GRB jets. author: - 'A. M. Soderberg , E. Nakar , E. Berger , S. R. Kulkarni' title: | Late-time Radio Observations of 68 Type Ibc Supernovae:\ Strong Constraints on Off-Axis Gamma-ray Bursts --- Introduction ============ It is now generally accepted that long duration gamma-ray bursts (GRBs) give rise to engine-driven relativistic jets as well as non-relativistic spherical supernova (SN) explosions. The first example of this GRB-SN connection came with the discovery of the Type Ic supernova, SN1998bw, associated with GRB980425 ($d\sim 36$ Mpc; @gvv+98 [@paa+00]). The unusually fast photospheric velocities and exceptionally bright radio emission of SN1998bw indicated $\sim 10^{52}$ erg of kinetic energy and mildly relativistic ejecta (bulk Lorentz factor, $\Gamma\sim 3$; @kfw+98 [@imn+98; @lc99; @wes99]). In comparison with other core-collapse events ($E_{KE}\sim 10^{51}$ erg and ejecta speeds, $v\lesssim 0.1c$), SN1998bw was considered a hyper-energetic supernova (“hypernova”; @imn+98). Broad optical absorption lines were also observed in the Type Ic SNe 2003dh and 2003lw, associated with the cosmological GRBs 030329 and 031203, indicative of comparably large photospheric velocities [@mgs+03; @mtc+04]. Together, these observations appear to suggest that broad spectral features are characteristic of GRB-associated SNe. In addition to events with prompt gamma-ray emission, the GRB-SN connection also implies the existence of “orphan” supernovae whose relativistic jets are initially beamed away from our line of sight [@rho99; @pac01]. Since the discovery of SN1998bw, several broad-lined SNe have been identified locally ($d\lesssim 100$ Mpc) and are currently estimated to represent $\sim 5\%$ of the Type Ibc supernova (SNe Ibc) population [@pmn+04]. Given their spectral similarity to the GRB-associated SNe, it has been argued that local broad-lined supernovae can be used as signposts for GRBs. Thus, associations with poorly-localized BATSE bursts have been invoked for the broad-lined SNe 1997cy, 1997ef and 1999E [@grs+00; @tsm+00; @ww98; @min00; @rtb+03]. In addition, association with off-axis GRBs have also been claimed. In the case of SN2002ap, broad optical absorption lines and evidence for mildly asymmetric ejecta (based on spectropolarimetry measurements) were interpreted to support an off-axis GRB jet (@kji+02 [@tot03], but see @lfc+02). More recently, an off-axis GRB model has been proposed for SN2003jd, for which photospheric velocities upward of 40,000 $\rm km~s^{-1}$ were measured at early time [@ffs03; @mck+03]. More intriguingly, late-time ($t\sim 400$ days) spectra showed double-peaked emission lines of light-elements, attributed to an asymmetric explosion [@kmd+04]. @mkm+05 argue that these observations can be understood if SN2003jd was accompanied by a highly collimated GRB jet initially directed $\sim 70$ degrees away from our line-of-sight. Regardless of viewing angle, however, strong afterglow emission eventually becomes visible as the decelerating GRB jets spread laterally and the emission becomes effectively isotropic. As the jets spread into our line-of-sight, a rapid increase of broadband synchrotron emission is observed on a timescale of a few weeks to several years. This late-time emission is most easily detected at long wavelengths [@pl98; @low+02; @wax04]. Targeting local Type Ibc supernovae with late-time radio observations has thus become the preferred method to search for evidence of off-axis GRBs [@svw+03; @sfw04]. Using early radio observations ($t\lesssim 100$ days) we have already limited the fraction of SNe Ibc harboring on-axis (or mildly off-axis) GRBs to be $\lesssim 3\%$ [@bkf+03]. In this paper, we present late-time ($t\sim 0.5$ to 20 yr) radio observations for 68 local Type Ibc supernovae, including SN2003jd and five additional broad-lined events, making this the most comprehensive study of late-time radio emission from SNe Ibc. We use these data to constrain the SN fraction associated with GRB jets regardless of viewing angle assumptions, constraining even those initially beamed perpendicular to our line-of-sight. Radio Observations {#sec:obs} ================== Type Ic SN2003jd ---------------- SN2003jd was discovered on 2003 October 25.2 UT within host galaxy MCG -01-59-021 ($d_L\sim 81$ Mpc; @bsl+03). In Table \[tab:SN2003jd\] we summarize our radio observations for SN2003jd, spanning $8-569$ days after the explosion. All observations were conducted with the Very Large Array (VLA) in the standard continuum mode with a bandwidth of $2\times 50$ MHz centered at 4.86, 8.46 or 22.5 GHz. We used 3C48 and 3C147 (J0137+331 and J0542+498) for flux calibration, while J2323-032 was used to monitor the phase. Data were reduced using standard packages within the Astronomical Image Processing System (AIPS). No radio emission was detected at the optical SN position during our early observations. Our radio limits imply that SN2003jd was a factor of $\gtrsim 100$ less luminous than SN1998bw on a comparable timescale. We conclude that SN2003jd, like the majority of SNe Ibc, did not produce relativistic ejecta along our line-of-sight. We re-observed SN2003jd at $t\sim 1.6$ yrs to search for radio emission from an off-axis GRB jet. No emission was detected, implying a limit of $F_{\nu} < 45~\mu$Jy ($3\sigma$) at 8.46 GHz. Late-time data on Local Type Ibc Supernovae ------------------------------------------- We supplement these data with late-time ($t\sim 0.5-20$ year) radio observations for 67 local ($d_L\lesssim 200$ Mpc) SNe Ibc, summarized in Table \[tab:vla\]. Eleven objects were observed at moderately late-time as part of our on-going VLA program to characterize the early ($t\lesssim 100$ days) radio emission from SNe Ibc (Soderberg [*et al.*]{}, in prep). The remaining 54 objects were observed on a later timescale ($t\gtrsim 1$ year) and were taken from the VLA archive. We note that five of these supernovae (SNe 1997dq, 1997ef, 1998ey, 2002ap, 2002bl) were spectroscopically observed to have broad optical absorption lines, similar to SN1998bw. All VLA observations were conducted at 8.46 GHz (except for SN1991D at 4.86 GHz) in the standard continuum mode with a bandwidth of $2\times 50$ MHz. Data were reduced using AIPS, and the resulting flux density measurements for this sample of SNe Ibc is given in Table \[tab:vla\]. With the exception of SN2001em, from which radio emission from the non-relativistic, spherical supernova ejecta is still detected at late-time (@sks+05 [@bb05], but see @gr04), none of the SNe Ibc show radio emission above our average detection limit of $\sim 0.15$ mJy ($3\sigma$). In comparison with SN1998bw, only SN2001em shows a comparable radio luminosity on this timescale. These results are consistent with the earlier report by @svw+03. In Figure \[fig:lum\_limits\_oa\] we plot the radio observations for this sample of SNe Ibc, in addition to late-time radio data for SN1954A [@ecb02] and SN1984L [@sfw04]. Off-Axis Models for Gamma-ray Bursts {#sec:model} ==================================== An Analytic Approach {#sec:waxman} -------------------- @wax04 present an analytic model for the late-time radio emission from a typical GRB viewed significantly away from the collimation axis. In this model, the GRB jet is initially characterized by a narrow opening angle, $\theta_j\sim$ few degrees, while the viewing angle is assumed to be large, $\theta_{\rm obs}\gtrsim 1$ radian. As the jet sweeps up circumstellar material (CSM) and decelerates, it eventually undergoes a dynamical transition to sub-relativistic expansion [@fwk00]. The timescale for this non-relativistic transition is estimated at $t_{NR}\approx 0.2 (E_{51}/n_0)^{1/3}$ yr ($\approx 0.3 E_{51}/A_*$ yr) in the case of a homogeneous (wind-stratified) medium [@wax04]. Here, $E_{51}$ is the beaming-corrected ejecta energy normalized to $10^{51}$ erg and $n_0$ is the circumstellar density of the homogeneous medium (interstellar medium; ISM) normalized to 1 particle cm$^{-3}$. For a wind-stratified medium, $A_*$ defines the circumstellar density in terms of the progenitor mass loss rate, $\dot{M}$, and wind velocity, $v_w$, such that $\dot{M}/4\pi v_w=5\times 10^{11} A_*~\rm g~cm^{-1}$, and thus $A_*=1$ for $\dot{M}=10^{-5}~M_{\odot}~\rm yr^{-1}$ and $v_w=10^{3}~\rm km~s^{-1}$ [@lc99]. Once sub-relativistic, the jets spread sideways, rapidly intersecting our line-of-sight as the ejecta approach spherical symmetry. At this point the afterglow emission is effectively isotropic and appears similar to both on-axis and off-axis observers. The broadband emission observed from the sub-relativistic ejecta is described by a standard synchrotron spectrum, characterized by three break frequencies: the synchrotron self-absorption frequency, $\nu_a$, the characteristic synchrotron frequency, $\nu_m$, and the synchrotron cooling frequency, $\nu_c$. On timescales comparable to the non-relativistic transition, $\nu_a$ and $\nu_m$ are typically below the radio band while $\nu_c$ is generally near the optical [@fwk00; @bkf04; @fsk+05]. Making the usual assumption that the kinetic energy is partitioned between relativistic electrons and magnetic fields ($\epsilon_e$ and $\epsilon_B$, respectively), and that these fractions are constant throughout the evolution of the jet, @wax04 estimate the radio luminosity of the sub-relativistic, isotropic emission to be $$\begin{aligned} L_{\nu} & \approx & 8.0\times 10^{29} \left(\frac{\epsilon_e}{0.1}\right) \left(\frac{\epsilon_B}{0.1}\right)^{3/4} n_0^{3/4} E_{51} \\ & & \times \left(\frac{\nu}{\rm 10 GHz}\right)^{-1/2} \left(\frac{t}{t_{\rm NR}}\right)^{-9/10}~\rm erg~s^{-1}~Hz^{-1} \nonumber\end{aligned}$$ for the ISM case, while for a wind-stratified medium $$\begin{aligned} L_{\nu} & \approx & 2.1\times 10^{29} \left(\frac{\epsilon_e}{0.1}\right) \left(\frac{\epsilon_B}{0.1}\right)^{3/4} A_*^{9/4} E_{51}^{-1/2} \\ & & \times \left(\frac{\nu}{\rm 10 GHz}\right)^{-(p-1)/2} \left(\frac{t}{t_{\rm NR}}\right)^{-3/2}~\rm erg~s^{-1}~Hz^{-1}. \nonumber\end{aligned}$$ Here it is assumed that the electrons are accelerated into a power-law distribution, $N(\gamma) \propto \gamma^{-p}$ with $p=2.0$. These equations reveal that the strength of the non-relativistic emission is strongly dependent on the density of the circumstellar medium (especially in the case of a wind) and is best probed at low frequencies. While this analytic model provides robust predictions for the afterglow emission at $t>t_{\rm NR}$, it does not describe the early evolution or the transition from relativistic to sub-relativistic expansion. At early time, the observed emission from an off-axis GRB is strongly dependent on the viewing angle and dynamics of the jet. To investigate this early afterglow evolution and the transition to sub-relativistic expansion, we developed a detailed semi-analytic model, described below. A Semi-analytic Model {#sec:our_model} --------------------- In modeling the afterglow emission from an off-axis GRB jet, we adopt the standard framework for a adiabatic blastwave expanding into either a uniform or wind stratified medium [@sar97; @gs02]. We assume a uniform, sharp-edged jet such that Lorentz factor and energy are constant over the jet surface. The hydrodynamic evolution of the jet is fully described in @onp04. As the bulk Lorentz factor of the ejecta approaches $\Gamma\sim 1$, the jets begin to spread laterally at the sound speed . Our off-axis light-curves are obtained by integrating the afterglow emission over equal arrival time surface. We note that these resulting light-curves are in broad agreement with Model 2 of @gpk+02 and are consistent with Waxman’s analytic model (§\[sec:waxman\]) on timescales, $t\gtrsim t_{\rm NR}$. Over-plotted in Figure \[fig:lum\_limits\_oa\] are our off-axis models calculated for both wind-stratified and homogeneous media at an observing frequency of $\nu_{\rm obs}=8.46$ GHz. We assume standard GRB parameters of $E_{51}=A_*=n=1$, $\epsilon_B=\epsilon_e=0.1$, $p=2.2$ and $\theta_j=5^{\rm o}$, consistent with the typical values inferred from broadband modeling of GRBs [@pk02; @yhs+03; @clf04]. We compute model light-curves for off-axis viewing angles between 30 and 90 degrees. As clearly shown in the figure, the majority of our late-time SNe Ibc limits are significantly fainter than [*all*]{} of the model light-curves, constraining even the extreme case where $\theta_{\rm obs}=90^{\rm o}$. SN2003jd: Constraints on the off-axis jet {#sec:SN2003jd} ========================================= Based on the double-peaked profiles observed for the nebular lines of neutral oxygen and magnesium, @mkm+05 argue that SN2003jd was an aspherical, axisymmetric explosion viewed near the equatorial plane. They suggest that this asymmetry may be explained if the SN explosion was accompanied by a tightly collimated and relativistic GRB jet, initially directed $\sim 70$ degrees from our line-of-sight. This hypothesis is consistent with the observed lack of prompt gamma-ray emission [@hcm+03] as well as the absence of strong radio and X-ray emission at early time [@skf03; @wpr+03]. Our radio observation of SN2003jd at $t\sim 1.6$ years imposes strong constraints on the putative off-axis GRB jet. While the early data constrain only mildly off-axis jets ($\theta_{\rm obs}\lesssim 30^{\rm o}$), our late-time epoch constrains even those jets initially directed perpendicular to our line-of-sight. As shown in Figure \[fig:lum\_limits\_oa\], our radio limit is a factor of $\gtrsim 200$ ($\gtrsim 20$) fainter than that predicted for a typical GRB expanding into a homogeneous (wind-stratified) medium, even in the extreme case where $\theta_{\rm obs}\sim 90^{\rm o}$. Given the assumption of typical GRB parameters, we conclude that our late-time radio limit is inconsistent with the presence of an off-axis GRB jet. We note that the model assumptions and physical parameters of our off-axis afterglow light-curves are identical to those adopted by @mkm+05. We next explore the range of parameters ruled out by our deep radio limits. As shown in Equations 1 and 2, the luminosity of the late-time emission is a function of the ejecta energy, the density of the circumstellar medium and the equipartition fractions. To investigate the effect of energy and density on the late-time radio luminosity, we fix the equipartition fractions to $\epsilon_e=\epsilon_B=0.1$, chosen to be consistent with the values typically inferred from afterglow modeling of cosmological GRBs [@pk02; @yhs+03]. In Figure \[fig:SN2003jd\_ed\], we illustrate how each radio epoch for SN2003jd maps to a curve within the two-dimensional parameter space of kinetic energy and circumstellar density for an off-axis GRB. Here we adopt our semi-analytic model (§\[sec:our\_model\]) for a wind-stratified medium, along with a typical electron index of $p=2.2$ and a viewing angle of $\theta_{\rm obs}=90^{\rm o}$; the faintest model for a given set of equipartition fractions. By comparing the luminosity limit for SN2003jd at a particular epoch with the off-axis model prediction for that time, we exclude the region of parameter space [ *rightward*]{} of the curve since this region produces a jet which is [*brighter*]{} than the observed limit. The union of these regions represents the total parameter space ruled out for an associated GRB. As shown in this figure, the total excluded parameter space extends from $A_*\gtrsim 0.03$ and $E \sim 10^{47}$ to $10^{52}$. We compare these constraints with the beaming-corrected kinetic energies and CSM densities for 18 cosmological GRBs (Table \[tab:grb\]). Here we make the rough approximation that $A_*\approx n_0$; a reasonable assumption for circumstellar radii near $\sim 10^{18}$ cm. As shown in Figure \[fig:SN2003jd\_ed\], these GRBs span the region of parameter space roughly bracketed by $A_*\sim 0.002$ to 100 and $E\sim 2\times 10^{49}$ to $4\times 10^{51}$. The majority of the bursts (13 out of 18) fall within the excluded region of parameter space for SN2003jd. We conclude that SN2003jd was not likely associated with a typical GRB at a confidence level of $\sim 72\%$. Local Type Ibc Supernovae: Further Constraints {#sec:SNe} ============================================== While physical parameters atypical of the cosmological GRB population can be invoked to hide an off-axis GRB for SN2003jd, it is exceedingly unlikely for atypical parameters to dominate a large statistical sample of SNe Ibc. Motivated thus, we searched for off-axis GRBs in the 67 local Type Ibc SNe for which we have compiled late-time ($t\sim 0.5-30$ yr) radio observations. Applying the method described in §\[sec:SN2003jd\] we produce exclusion regions in the $E_{51}-A_*$ parameter space for each SN. Figure \[fig:all\_ed\] shows the resulting contours for all 68 SNe, including SN2003jd and five broad-lined events. For the twenty SNe with early radio limits [@bkf+03; @bkc02] we combine late- and early-time data to provide further constraints. In Figure \[fig:confidence\_ed\] we compile all 68 exclusion regions to quantify the $E_{51}-A_*$ parameter space constrained by this statistical sample. Contours map the regions excluded by incremental fractions of our sample. As in the case of SN2003jd, all curves rule out bursts with $A_*\gtrsim 1$ and $E\gtrsim 10^{50}$ erg. Moreover, 50% exclude $A_* \gtrsim 0.1$ and $E \gtrsim 10^{49}$ erg. For comparison, the mean ejecta energy and CSM density values for cosmological GRBs are $E \approx 4.4\times 10^{50}$ erg and $A_*=n_0\approx 1.2$. Focusing on the subsample of broad-lined SNe, we emphasize that our deep limits rule out both putative GRB jets directed along our line-of-sight (e.g. SN1997ef) as well as those which are initially beamed off-axis (e.g. SN2002ap and SN2003jd). In particular, the large exclusion region for SN2002ap (see Figure \[fig:all\_ed\]) implies that an extremely low CSM density, less than $A_*\sim 3\times 10^{-3}$, is needed to suppress the emission from an associated GRB. This is a factor of $\sim 10$ below the density inferred from modeling of the early radio emission [@bkc02] and we therefore conclude that an off-axis GRB model is inconsistent with our late-time observations of SN2002ap. In Figure \[fig:confidence\_ed\] we show that this entire sample of six broad-lined SNe rule out bursts with energies $E\gtrsim 10^{49}$ erg, and 50% even rule out $E\sim 10^{47}$ erg (all assuming a typical $A_*=1$). We next address the limits on an association with GRBs defined by the cosmological sample (Table \[tab:grb\]). For each SN in our sample we calculate the fraction of observed GRBs that lie in its exclusion region. We then determine the probability of finding null-detections for our entire sample by calculating the product of the individual probabilities. We find that the probability that [*every*]{} Type Ibc supernova has an associated GRB is $1.1\times 10^{-10}$. We further rule out a scenario in which one in ten SNe Ibc is associated with a GRB at a confidence level of $\sim 90\%$. For the broad-lined events alone we rule out the scenario that every event is associated with a GRB at a confidence level of $\sim 84\%$. Confidence levels are shown as a function of GRB/SN fraction in Figure \[fig:binomial\_prob\]. Discussion and Conclusions {#sec:disc} ========================== We present late-time radio observations for 68 local Type Ibc supernovae, including six broad-lined SNe (“hypernovae”), making this the most comprehensive study of late-time radio emission from SNe Ibc. None of these objects show evidence for bright, late-time radio emission that could be attributed to off-axis jets coming into our line-of-sight. Comparison with our most conservative off-axis GRB afterglow models reveals the following conclusions: \(1) Less than $\sim 10\%$ of Type Ibc supernovae are associated with GRBs. These data impose an empirical constraint on the GRB beaming factor, $\langle f_b^{-1}\rangle$, where $f_b=(1-{\rm cos}~\theta_j)$. Assuming a local GRB rate of $\sim 0.5~\rm Gpc^{-3}~yr^{-1}$ [@sch01; @psf03; @gpw05] and an observed SNe Ibc rate of $\sim 4.8\times 10^{4}~\rm Gpc^{-3}~yr^{-1}$ [@mdp+98; @cet99; @frp+99], we constrain the GRB beaming factor to be $\langle f_b^{-1}\rangle\lesssim \times 10^{4}$. Adopting a lower limit of $\langle f_b^{-1}\rangle > 13$ [@low+02], the beaming factor is now observationally bound by $\langle f_b^{-1}\rangle\approx [13-10^4]$, consistent with the observed distribution of jet opening angles [@fks+01; @gpw05]. \(2) Despite predictions that most or all broad-lined SNe Ibc harbor GRB jets [@pmn+04], our radio observations for six broad-lined events (SNe 1997dq, 1997ef, 1998ey, 2002ap, 2002bl and 2003jd) reveal no evidence for association with typical (or even sub-energetic) GRBs. While unusual physical parameters can suppress the radio emission from off-axis jets in any one SN, it is unlikely that all six broad-lined events host atypical GRBs. We observationally rule out the scenario in which every broad-lined SN harbors GRB jets with a confidence level of $\sim 84\%$. \(3) While low CSM densities (e.g. $A_* \lesssim 0.1$) can suppress the emission from off-axis GRB jets, such values are inconsistent with the mass loss rates measured from local Wolf-Rayet stars ($0.6-9.5\times 10^{-5}~\rm M_{\odot}~yr^{-1}$; @cgv04), thought to be the progenitors of long-duration gamma-ray bursts. \(4) While we have so far considered only the signature from a highly collimated GRB jet, these late-time radio data also impose constraints on the presence of broader jets and/or jet cocoons. As demonstrated by GRBs 980425 and 030329, the fraction of energy coupled to mildly relativistic and mildly collimated ejecta can dominate the total relativistic energy budget [@kfw+98; @bkp+03]. Less sensitive to to the effects of beaming and viewing geometry, broad jets are more easily probed at early time ($t\sim 100$ days) when the emission is brightest. Still, we note that the majority of our late-time radio limits are significantly fainter than GRBs 980425 and 030329 on a comparable timescale, thus constraining even mildly relativistic ejecta. These conclusions, taken together with the broad spectral features observed for GRB-associated SNe 1998bw, 2003dh and 2003lw, motivate the question: what is the connection between GRBs and local Type Ibc supernovae? While current optical data suggest that all GRB-SNe are broad-lined, our late-time radio observations clearly show that the inverse is [*not*]{} true: broad optical absorption lines do not serve as a reliable proxy for relativistic ejecta. This suggests that their observed large photospheric velocities and asymmetric ejecta are often merely characteristics of the non-relativistic SN explosion and thus manifestations of the diversity within SNe Ibc. The authors thank Doug Leonard, Paolo Mazzali, Dale Frail, Brian Schmidt and Avishay Gal-Yam for helpful discussions. 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[lrrrr]{} 2003 Oct 29 & 8 & $\pm 52$ & $\pm 34$ & $\pm 58$\ 2003 Nov 4 & 14 & & $\pm 77$ &\ 2003 Nov 15 & 25 & & $\pm 74$ &\ 2005 May 12 & 569 & & $\pm 15$ &\ \[tab:SN2003jd\] [llrcclrr]{} 1983N & NGC 5236 & 7.2 & 1983 Jun 26 & 3835 & 2003 Oct 17 & 7416 & $\pm 124$\ 1985F & NGC 4618 & 7.7 & 1985 May 14 & 4042 & 2003 Oct 17 & 6730 & $\pm 37$\ 1987M & NGC 2715 & 18.9 & 1987 Aug 31 & 4451 & 2003 Oct 17 & 5891 & $\pm 34$\ 1990B & NGC 4568 & 32.0 & 1989 Dec 23-1990 Jan 20 & 4949 & 2003 Oct 17 & 5032 & $\pm 24$\ 1990U & NGC 7479 & 33.7 & 1990 Jun 28-Jul 27 & 5063 & 2003 Oct 17 & 4845 & $\pm 44$\ 1991A & IC 2973 & 45.5 & 1990 Dec 6-13 & 5178 & 2003 Oct 17 & 4695 & $\pm 35$\ 1991D & Anon. & 173 & 1991 Jan 16 & 5153 & 1992 Oct 11 & 633 & $\pm 49$\ 1991N & NGC 3310 & 14.0 & 1991 Feb 20-Mar 29 & 5227 & 2003 Oct 17 & 4604 & $\pm 59$\ 1991ar & IC 49 & 65.0 & 1991 Aug 3 & 5334 & 2003 Oct 17 & 4458 & $\pm 30$\ 1994I & NGC 5194 & 6.5 & 1994 Apr 1-2 & 5961 & 2003 Oct 17 & 3486 & $\pm 42$\ 1994ai & NGC 908 & 21.3 & 1994 Dec 8-16 & 6120 & 2003 Oct 17 & 3336 & $\pm 40$\ 1996D & NGC 1614 & 68.1 & 1996 Jan 28 & 6317 & 2003 Oct 15 & 2818 & $\pm 153$\ 1996N & NGC 1398 & 19.7 & 1996 Feb 16-Mar 13 & 6351 & 2003 Oct 15 & 2787 & $\pm 36$\ 1996aq & NGC 5584 & 23.2 & 1996 Jul 30 & 6465 & 2003 Oct 17 & 2635 & $\pm 32$\ 1997B & IC 438 & 44.3 & 1996 Dec 14 & 6535 & 2003 Oct 15 & 2498 & $\pm 33$\ 1997C & NGC 3160 & 99.2 & 1996 Dec 18-1997 Jan 14 & 6536 & 2003 Oct 17 & 2481 & $\pm 37$\ 1997X & NGC 4691 & 15.7 & 1997 Jan 16-Feb 2 & 6552 & 2003 Oct 17 & 2457 & $\pm 22$\ 1997dc & NGC 7678 & 49.6 & 1997 Jul 22 & 6715 & 2003 Oct 17 & 2278 & $\pm 36$\ 1997dq & NGC 3810 & 14.0 & 1997 Oct 16 & 6770 & 2003 Oct 17 & 2192 & $\pm 30$\ 1997ef & UGC 4107 & 49.8 & 1997 Nov 16-26 & 6778 & 2003 Oct 17 & 2156 & $\pm 33$\ 1998T & NGC 3690 & 44.3 & 1998 Feb 8-Mar 3 & 6830 & 2003 Oct 17 & 2066 & $\pm 482$\ 1998ey & NGC 7080 & 69.0 & 1998 Nov 1-Dec 5 & 6830 & 2003 Oct 17 & 1794 & $\pm 32$\ 1999bc & UGC 4433 & 90.2 & 1999 Jan 30 & 7133 & 2003 Oct 15 & 1721 & $\pm 25$\ 1999di & NGC 776 & 70.2 & 1999 Jul 6 & 7234 & 2003 Oct 17 & 1564 & $\pm 35$\ 1999dn & NGC 7714 & 39.7 & 1999 Aug 10-20 & 7241 & 2003 Oct 17 & 1524 & $\pm 42$\ 1999ec & NGC 2207 & 38.9 & 1999 Aug 24 & 7268 & 2003 Oct 15 & 1515 & $\pm 45$\ 1999eh & NGC 2770 & 27.6 & 1999 Jul 26 & 7282 & 2003 Oct 17 & 1544 & $\pm 19$\ 2000C & NGC 2415 & 53.8 & 1999 Dec 30-2000 Jan 4 & 7348 & 2003 Oct 17 & 1385 & $\pm 35$\ 2000F & IC 302 & 84.5 & 1999 Dec 30-2000 Jan 10 & 7353 & 2003 Oct 15 & 1383 & $\pm 26$\ 2000S & MCG -01-27-2 & 85.8 & 1999 Oct 9 & 7384 & 2003 Oct 17 & 1469 & $\pm 30$\ 2000de & NGC 4384 & 35.6 & 2000 Jul 13 & 7478 & 2003 Oct 17 & 1191 & $\pm 40$\ 2000ds & NGC 2768 & 19.4 & 2000 May 28 & 7507 & 2003 Oct 17 & 1237 & $\pm 34$\ 2000dv & UGC 4671 & 57.7 & 2000 Jul 4 & 7510 & 2003 Oct 17 & 1200 & $\pm 32$\ 2001B & IC 391 & 22.0 & 2000 Dec 25-2001 Jan 4 & 7555 & 2003 Oct 15 & 1021 & $\pm 30$\ 2001M & NGC 3240 & 50.6 & 2001 Jan 3-21 & 7568 & 2003 Oct 17 & 1008 & $\pm 54$\ 2001bb & IC 4319 & 66.3 & 2001 Apr 15-29 & 7614 & 2003 Oct 17 & 908 & $\pm 35$\ 2001ch & MCG -01-54-1 & 41.6 & 2001 Mar 23 & 7637 & 2003 Oct 17 & 938 & $\pm 32$\ 2001ci & NGC 3079 & 15.8 & 2001 Apr 17-25 & 7638 & 2003 Oct 17 & 909 & $\pm 188$\ 2001ef & IC 381 & 35.1 & 2001 Aug 29-Sep 9 & 7710 & 2003 Oct 17 & 774 & $\pm 33$\ 2001ej & UGC 3829 & 57.4 & 2001 Aug 30 & 7719 & 2003 Oct 15 & 778 & $\pm 36$\ 2001em & UGC 11794 & 83.6 & 2001 Sep 10-15 & 7722 & 2003 Oct 17 & 765 & $907\pm 58$\ 2001is & NGC 1961 & 56.0 & 2001 Dec 14-23 & 7782 & 2003 Oct 15 & 668 & $\pm 31$\ 2002J & NGC 3464 & 53.0 & 2002 Jan 15-21 & 7800 & 2003 Oct 17 & 637 & $\pm 23$\ 2002ap & NGC 628 & 9.3 & 2002 Jan 28-29 & 7810 & 2003 Oct 17 & 626 & $\pm 29$\ 2002bl & UGC 5499 & 67.8 & 2002 Jan 31 & 7845 & 2003 Oct 17 & 624 & $\pm 39$\ 2002bm & MCG -01-32-1 & 78.0 & 2002 Jan 16-Mar 6 & 7845 & 2003 Oct 17 & 630 & $\pm 39$\ 2002cp & NGC 3074 & 73.4 & 2002 Apr 11-28 & 7887 & 2003 Oct 17 & 547 & $\pm 29$\ 2002hf & MCG -05-3-20 & 80.2 & 2002 Oct 22-29 & 8004 & 2003 Oct 17 & 357 & $\pm 48$\ 2002ho & NGC 4210 & 38.8 & 2002 Sep 24 & 8011 & 2003 Oct 17 & 389 & $\pm 35$\ 2002hy & NGC 3464 & 53.0 & 2002 Oct 13-Nov 12 & 8016 & 2003 Oct 17 & 354 & $\pm 23$\ 2002hz & UGC 12044 & 77.8 & 2002 Nov 2-12 & 8017 & 2003 Oct 17 & 344 & $\pm 29$\ 2002ji & NGC 3655 & 20.8 & 2002 Oct 20 & 8025 & 2003 Oct 17 & 362 & $\pm 18$\ 2002jj & IC 340 & 60.1 & 2002 Oct 22 & 8026 & 2003 Oct 15 & 360 & $\pm 30$\ 2002jp & NGC 3313 & 52.7 & 2002 Oct 20 & 8031 & 2003 Oct 17 & 362 & $\pm 37$\ 2003A & UGC 5904 & 94.4 & 2002 Nov 23-2003 Jan 5 & 8041 & 2003 Jun 15 & 182 & $\pm 69$\ 2003I & IC 2481 & 76.0 & 2002 Dec 14 & 8046 & 2003 Jun 15 & 183 & $\pm 60$\ 2003aa & NGC 3367 & 43.1 & 2003 Jan 24-31 & 8063 & 2003 Jun 15 & 138 & $\pm 70$\ 2003bm & UGC 4226 & 113.7 & 2003 Feb 20-Mar 3 & 8086 & 2003 Jun 15 & 109 & $\pm 57$\ 2003bu & NGC 5393 & 86.1 & 2003 Feb 10-Mar 11 & 8092 & 2003 Jun 15 & 110 & $\pm 71$\ 2003cr & UGC 9639 & 155.2 & 2003 Feb 24-Mar 31 & 8103 & 2003 Nov 2 & 233 & $\pm 59$\ 2003dg & UGC 6934 & 79.1 & 2003 Mar 24-Apr 8 & 8113 & 2003 Nov 2 & 215 & $\pm 90$\ 2003dr & NGC 5714 & 31.7 & 2003 Apr 8-12 & 8117 & 2003 Nov 2 & 206 & $\pm 58$\ 2003ds & NGC 3191 & 132.9 & 2003 Mar 25-Apr 14 & 8120 & 2003 Nov 2 & 212 & $\pm 67$\ 2003el & NGC 5000 & 80.2 & 2003 Apr 19-May 22 & 8135 & 2003 Nov 2 & 180 & $\pm 65$\ 2003ev & Anonymous & 103.3 & 2003 Apr 9-Jun 1 & 8140 & 2003 Nov 2 & 180 & $\pm 61$\ 2003jd & MCG -01-59-21 & 80.8 & 2003 Oct 16-25 & 8232 & 2005 May 12 & 569 & $\pm 15$\ . . \[tab:vla\] [lrrrcc]{} 970508 & 0.835 & $2.4^{+1.4}_{-0.9}$ & $2.4^{+1.4}_{-0.9}$ & ISM & 1\ 980329 & $\lesssim 3.9$ & $1.1^{+0.26}_{-0.46}$ & $20^{+10}_{-10}$ & ISM & 2\ 980425 & 0.0085 & 0.012 & 0.04 & Wind & 3\ 980519 & 1 & $0.41^{+0.48}_{-0.14}$ & $0.14^{+0.48}_{-0.14}$ & ISM & 4\ 980703 & 0.966 & $3.5^{+1.26}_{-0.42}$ & $28^{+8}_{-6}$ & ISM & 2\ 990123 & 1.60 & $0.15^{+0.33}_{-0.04}$ & $0.0019^{+0.0005}_{-0.0015}$ & ISM & 4\ 990510 & 1.619 & $0.14^{0.49}_{-0.05}$ & $0.29^{+0.11}_{-0.15}$ & ISM & 4\ 991208 & 0.706 & $0.24^{+0.28}_{-0.22}$ & $18^{+22}_{-6}$ & ISM & 4\ 991216 & 1.02 & $0.11^{+0.1}_{-0.04}$ & $4.7^{+6.8}_{-1.8}$ & ISM & 4\ 000301c & 2.03 & $0.33^{+0.03}_{-0.05}$ & $27^{+5}_{-5}$ & ISM & 4\ 000418 & 1.118 & 3.4 & 0.02 & ISM & 5\ 000926 & 2.066 & $2.0^{+0.34}_{-0.2}$ & $16^{+6}_{-6}$ & ISM & 2\ 010222 & 1.477 & 0.51 & 1.7 & ISM & 4\ 011121 & 0.36 & 0.2 & 0.015 & Wind & 6\ 020405 & 0.69 & 0.3 & $\lesssim 0.07$ & Wind & 6\ 020903 & 0.251 & 0.4 & 100 & ISM & 7\ 030329 & 0.168 & 0.67 & 3.0 & ISM & 8\ 031203 & 0.105 & 0.017 & 0.6 & ISM & 9\ \[tab:grb\]

--- abstract: 'Deep surveys conducted during the past decades have shown that galaxies in the distant universe are generally of more irregular shapes, and are disky in appearance and in their star formation rate, compared to galaxies in similar environments in the nearby universe. Given that the merger rate between z=2 and the local universe is far from adequate to account for this observed morphological transformation rate, an internal mechanism for the morphological transformation of galaxies is to be sought, whose operation can be further aided by environmental factors. The secular evolution mechanism, especially with the discovery of a collisionless dissipation mechanism for stars within the secular evolution paradigm, has provided just such a framework for understanding the morphological evolution of galaxies across the Hubble time. In this paper we will summarize the past theoretical results on the dynamical mechanisms for secular evolution, and highlight new results in the analysis of the observational data, which confirmed that density waves in physical galaxies possess the kind of characteristics which could produce theáobserved rates of morphological transformation for both cluster and field galaxies.' author: - 'Xiaolei Zhang$^1$, Ronald J. Buta$^2$' date: 'September 5th, 2006' title: Secular Evolution and the Morphological Transformation of Cluster and Field Galaxies --- Introduction ============ It has been a commonly-held belief that the stellar orbit in a galaxy containing quasi-stationary density wave patterns does not exhibit secular decay or increase, and there is no wave and disk-star interaction except at the wave-particle resonances (Lynden-Bell & Kalnajs 1972). Zhang (1996, 1998, 1999) first demonstrated that secular orbital changes of stars across the [*entire galaxy disk*]{} are in fact possible due to a collective instability induced by the density wave patterns. The integral manifestation of this process is an azimuthal phase shift between the potential and density spirals, which results in a secular torque action between the wave pattern and the underlying disk matter. As a result, the matter inside the corotation radius loses angular momentum to the wave secularly, and sinks inward. The wave carries the angular momentum it receives from the inner disk matter to the outer disk and deposits it there, causing the matter in the outer disk to drift further out. ![[*Left*]{}: $K_s$-band image of NGC 1530 and the superimposed corotation circles determined using the phase shift method (Zhang & Buta 2006). [*Right*]{}: Radial mass accretion/excretion rate calculated for NGC 1530 from the $K_s$-band image.](zhang_se_fig01_left.ps "fig:"){height="1.8in" width="1.8in"} ![[*Left*]{}: $K_s$-band image of NGC 1530 and the superimposed corotation circles determined using the phase shift method (Zhang & Buta 2006). [*Right*]{}: Radial mass accretion/excretion rate calculated for NGC 1530 from the $K_s$-band image.](zhang_se_fig01_right.epsi "fig:"){height="1.8in" width="1.8in"} The secular morphological evolution process causes the Hubble type of an average galaxy to evolve from late to early (Zhang 1999). The rate of secular evolution is expected to be faster for cluster galaxies than for isolated field galaxies because it is proportional to the wave amplitude squared and pattern pitch-angle squared (Zhang 1998), and cluster galaxies are found to have large amplitude and open density wave patterns excited through the tidal interactions with neighboring galaxies and with the cluster potential as a whole. New Results on the Secular Mass Accretion Rates =============================================== We have found in this study that physical galaxies generally contain density waves strong enough to cause significant mass redistribution over their lifetime. In Figures 2, we show an image of the SBb galaxy NGC 1530 with superimposed corotation circles, and the resulting secular mass accretion/excretion rate calculated using the formulae given in Zhang (1998). The calculated mass accretion/excretion rates for NGC 1530, with peak value around 145M$_\odot$/year, are several orders of magnitude larger than for our Galaxy ($\sim$ 0.6M$_{\odot}$/year; Zhang 1999). This difference is due to the much larger density wave amplitude (average of 70%), larger pitch angle (65$^o$) and larger surface density (average of 100M$_{\odot}$/pc$^2$) of NGC 1530 compared to the Galaxy. We have calculated the same quantities for several other galaxies and obtained more moderate accretion rates, from a few to a few tens of M$_{\odot}$/year. In early type galaxies the central mass accretion rate can be much enhanced due to the presence of strong nuclear density wave patterns, even though the outer wave patterns generally become weaker as galaxies age. Thus density-wave induced radial mass accretion could serve as the main driver for the morphological transformation of both the field and cluster galaxies over the past Hubble time, as well as for fueling active galactic nuclei. Furthermore, the mass accretion of the stellar component (in addition to gas) also means that most of the [*stellar population*]{} of the bulge could form long before the [*shape*]{} of the bulge, maintaining a homogeneous and older stellar population of the bulge during the process of secular evolution. This helps to solve one of the main difficulties in the secular building of bulges from the accretion of the gas component alone, which leads to a much varied stellar population in the resulting bulges. The same phase-shift-induced mass redistribution process also works in galaxies which contain other skewed density distributions as observed in many high-z proto galaxies. XZ acknowledges the support of the Office of Naval Research. RB acknowledges the support of NSF grant AST 050-7140 to the University of Alabama. Lynden-Bell, D., & Kalnajs, A.J., 1972, *MNRAS*, 157, 1 , *ApJ*, 457, 125; [1998]{} *ApJ*, 499, 93; [1999]{} *ApJ*, 518, 613 Zhang, X. & Buta, R.J. 2006, *these proceedings*

--- abstract: | We construct small covers and quasitoric manifolds over $n$-dimensional simple polytopes which allow proper colorings of facets with $n$ colors. We calculate Stiefel-Whitney classes of these manifolds as obstructions to immersions and embeddings into Euclidean spaces. The largest dimension required for embedding is achieved in the case $n$ is a power of two. **2010 Mathematics Subject Classification**: 57N35, 52B12. **Keywords**: immersions, quasitoric manifolds, simple polytopes, colorings, Stiefel-Whitney classes. author: - | [ore Baralić]{}\ [Mathematical Institute of Serbian Academy of Sciences and Arts]{}\ [Belgrade, Serbia]{}\ [djbaralic@mi.sanu.ac.rs]{} - | Vladimir Grujić\ [Faculty of Mathematcs]{}\ [University of Belgrade, Serbia]{}\ [vgrujic@matf.bg.ac.rs]{} title: 'Quasitoric manifolds and Small covers over properly colored polytopes: Immersions and Embeddings' --- Introduction ============ Colorings of simple polytopes ----------------------------- An $n$-dimensional convex polytope is *simple* if the number of facets which are meeting at each vertex is equal to $n$. The *proper coloring* into $k$ colors of a simple polytope $P^n$ is a map $$h: \{F_1, \dots, F_m\}\rightarrow \{1,\ldots,k\}$$ of its set of facets such that $h (F_i)\neq h (F_j)$ for each two intersecting facets. The *chromatic number* $\chi (P^n)$ of a simple polytope $P^{n}$ is the least $k$ for which there exists a proper coloring of $P^n$ into $k$ colors. The chromatic numbers of the simplex $\Delta^n$, the cube $I^n$ and the permutahedron $\Pi^n$ (Figure \[bojenja\]) are $$\chi (\Delta^n)=n+1, \ \ \ \chi (I^n)=n, \ \ \ \chi (\Pi^n)=n.$$ ![The colorings of the cube and the permutahedron[]{data-label="bojenja"}](bojenja.pdf){width="\textwidth"} Obviously, $\chi (P^n)\geq n$ for any simple polytope $P^n$. The chromatic number of a polygon is clearly equal to $2$ or $3$, depending on the parity of the number of its faces. By famous Four Color Theorem chromatic numbers of $3$-dimensional simple polytopes are equal to $3$ or $4$. In general case, for $n\geq 4$ it does not hold $\chi (P^n)\leq n+1$. Moreover, there are simple polytopes such that their chromatic numbers are equal exactly to numbers of their facets. The polars of the cyclic polytopes with $m$ vertices $C^n (m)$ are examples of such type (see [@5 Example 0.6]). We consider the class of $n$-dimensional simple polytopes with the chromatic number equal to $n$. Denote this class by $\mathcal{C}$. The class $\mathcal{C}$ is closed with respect to products (see [@2 Construction 1.12]) and connected sums (see [@2 Construction 1.13]). Also from any given simple polytope $P^n$ by truncation over all its faces we obtain a simple polytope $Q^n$ which belongs to the class $\mathcal{C}$. The complete description of the class $\mathcal{C}$ is obtained in [@9]. A simple $n$-polytope $P^n$ admits a proper coloring in $n$ colors if and only if every its $2$-face has an even number of edges (see [@9 Theorem 16]). Small covers and quasitoric manifolds ------------------------------------- Quasitoric manifolds and they real analogues called small covers are introduced by Davis and Januszkiewicz in [@3]. Their geometric and algebraic topological properties are closely related to combinatorics of simple polytopes. The following definitions and notions are extensively elaborated in [@2]. Denote by $$G_d=\left\{\begin{array}{rl} S^{0}, & d=1,\\ S^{1}, & d=2,\end{array} \right. \ R_d=\left\{\begin{array}{rl} \mathbb{Z}_2, & d=1,\\ \mathbb{Z}, & d=2,\end{array}\right. \ \mathbb{K}_d=\left\{\begin{array}{rl} \mathbb{R}, & d=1,\\ \mathbb{C}, & d=2,\end{array} \right.$$ where $S^{0}=\{-1,+1\}$ and $S^{1}=\{z\mid |z|=1\}$ are multiplicative subgroups of real and complex numbers, $\mathbb{Z}$ is the ring of integers and $\mathbb{Z}_2=\{0,1\}$ is the ring of integers modulo $2$. The group $G_d^{n}$ acts standardly on $\mathbb{K}_d^{n}$ by $(t_1,\ldots,t_n)\cdot(x_1,\ldots,x_n)=(t_1x_1,\ldots,t_nx_n)$. The action of $G_d^{n}$ on a smooth $dn$-dimensional manifold $M^{dn}$ is locally standard if for any point of $M^{dn}$ there is a $G_d^{n}$-invariant neighborhood which is weakly equivariantly diffeomorphic to some open $G_d^{n}$-invariant subset of $\mathbb{K}_d^{n}$ with the standard action of $G_d^{n}$. Recall that two $G_d^{n}$-manifolds are weakly equivariantly diffeomorphic if there is an automorphism $\omega:G_d^n\rightarrow G_d^n$ and a diffeomorphism $f:M_1^{dn}\rightarrow M_2^{dn}$ such that $f(g\cdot x)=\omega(g)\cdot f(x)$ for any $g\in G_d^n$ and $x\in M_1^{dn}$. A smooth $dn$-dimensional $G_d^{n}$-manifold $M^{dn}$ with a locally standard action of the group $G_d^{n}$ such that the orbit space $M^{dn}/G_d^{n}$ is diffeomorphic, as a manifold with corners, to a simple $n$-dimensional polytope $P^{n}$ is called [*small cover*]{} over $P^{n}$ if $d=1$, correspondingly [*quasitoric manifolds*]{} over $P^{n}$ if $d=2$. Let $\pi:M^{dn}\rightarrow P^{n}$ be the projection map and $\{F_1,\ldots,F_m\}$ be the set of facets of the polytope $P^{n}$. The inverse images $M^{d(n-1)}_j=\pi^{-1}(F_j)$ are $G_d^{n}$-invariant submanifolds of codimension $d$ called [*characteristic submanifolds*]{}. To each characteristic submanifold $M_j^{d(n-1)}$ corresponds the isotropy subgroup $G(F_j)=G_{\lambda_j}$ of the rank one, where $G_{\lambda_j}=\{(1,\ldots,1),((-1)^{\lambda_{1j}},\ldots,(-1)^{\lambda_{nj}})\}$ for some $\lambda_j=(\lambda_{1j},\ldots,\lambda_{nj})\in\mathbb{Z}_2^{n}\setminus\{0\}$ if $d=1$ and $G_{\lambda_j}=\{(e^{2\pi i\lambda_{1j}t},\ldots,e^{2\pi i\lambda_{nj}t})\mid t\in\mathbb{R}\}$ for some primitive vector $\lambda_j=(\lambda_{1j},\ldots,\lambda_{nj})\in\mathbb{Z}^{n}$ defined up to the sign, if $d=2$. In this way, the action of the group $G_d^{n}$ on $M^{dn}$ defines the [*characteristic map*]{} $$l_d:\{F_1, \dots, F_m\}\rightarrow R^n_d,$$ which to a facet $F_j$ of the polytope $P^{n}$ assigns a primitive vector $\lambda_j=(\lambda_{1j},\ldots,\lambda_{nj})\in R^n_d$. Denote by $\Lambda= (\lambda_1, \dots, \lambda_m)$ the matrix formed by these vectors. Then $\det \Lambda_{{(V)}}=\pm 1$ for any vertex $V\in P^{n}$, where $\Lambda_{(V)}$ is the submatrix of the matrix $\Lambda$ formed by vectors corresponding to facets which are intersecting in that vertex. A pair $(P^{n}, \Lambda)$ satisfying this condition on submatrices is called a [*characteristic pair*]{}. The manifold $M^{dn}$ is reconstructible from the characteristic pair $(P^{n}, \Lambda)$ up to weak equivariant diffeomorphism in the following way (see [@3] and [@2 Construction 6.18]). Let $F_{i_1}\cap\ldots\cap F_{i_k}$ be the minimal face containing a point $q\in P^{n}$. To the point $q$ we associate the subgroup $G(q)=G_{\lambda_{i_1}}\times\cdots\times G_{\lambda_{i_k}}$. Define $M(P^{n}, \Lambda)=G_d^{n}\times P^{n}/\sim$, where $(t_1,p)\sim(t_2,q)$ if and only if $p=q$ and $t_1t_2^{-1}\in G(q)$. Though not all simple polytopes allow characteristic maps (see [@3 Example 1.15]), to considered polytopes $P^{n}\in\mathcal{C}$ in a simple way can be assigned a characteristic matrix $\Lambda$ if each color $i\in\{1,\ldots,n\}$ is identified with the vector $e_i$ of the standard basis in $R_d^{n}$. The cohomology ring of a small cover and a quasitoric manifold $M^{dn}, d=1,2$ is described in the following way. Let $v_j=D[M_j^{d(n-1)}], j=1,\ldots,m$ be cohomology classes which are Poincare duals to fundamental classes of characteristic manifolds. The characteristic matrix $\Lambda=(\lambda_1, \dots, \lambda_m)$ defines following linear forms $$\theta_i:=\sum_{j=1}^m \lambda_{i j} v_j, i=1,\ldots,n,$$ where $\lambda_j=(\lambda_{1 j}, \dots, \lambda_{n j})^t \in R_d^n, j=1,\ldots,m$. Let $\mathcal{J}$ be the ideal in $R_d[v_1, \dots, v_m]$ generated by elements $\theta_1,\ldots,\theta_n$ and $\mathcal{I}$ be the Stanley-Reisner ideal of the polytope $P^{n}$, which is generated by monomials $v_{i_1}\cdots v_{i_k}$ whenever $F_{i_1}\cap\ldots F_{i_k}=\emptyset$ in $P^{n}, i_1<\ldots<i_k$. Then for the cohomology ring the following isomorphism holds (see [@3]) $$\label{djf} H^\ast (M^{dn},R_d)\simeq R_d[v_1, \dots, v_m]/(\mathcal{I}+\mathcal{J}).$$ The same formula $(\ref{djf})$ holds with $\mathbb{Z}_2$ coefficients if $d=2$. The total Stiefel-Whitney class is determined by the following Davis-Januszkiewicz’s formula $$\label{djfswc} w (M^{dn})= \prod_{i=1}^m (1+v_i)\in H^\ast (M^{dn}; \mathbb{Z}_2),$$ where in the case $d=2$ all classes $v_i$ are regarded as $\mathbb{Z}_2$-restrictions of corresponding integral classes. Immersions and Embeddings ------------------------- Immersions and embeddings of manifolds are a classical topic in algebraic topology and differential topology. We consider immersions and embeddings in the smooth category. For a manifold $M^{n}$ define the numbers $imm (M^n)$ and $em(M^n)$ as the smallest dimensions of Euclidean spaces in which this manifold is immersed and embedded, correspondingly. In accordance with Whitney’s theorem for each smooth manifold $M^{n}$ is satisfied $imm(M^n)\leq 2n-1$ and $em(M^n)\leq 2n$. On the other hand, the dual Stiefel-Whitney classes $\overline{w}_i$ serve as obstructions to immersions and embeddings of manifolds in Euclidean spaces. Recall that dual Stiefel-Whitney classes $\overline{w}_i(M^{n})$ are characteristic classes of the stable normal bundle of a manifold $M^{n}$. \[imeem\] For a smooth manifold $M^{n}$ let $k:=\max \{i\mid {\overline{w}}_i(M^n)\neq 0\}$. Then ${\operatorname{imm}}(M^n)\geq n+k \, \mbox{\,and\,}\, {\operatorname{em}}(M^n)\geq n+k+1.$ The Stiefel-Whitney classes are also obstructions to so called totally skew embeddings introduced by Ghomi and Tabachnikov in [@6]. A manifold $M^{n}$ is totally skew embedded in the Euclidean space $\mathbb{R}^{N}$ if any two tangent lines in different points of $M^{n}$ are skew lines in $\mathbb{R}^{N}$. It is proved in the paper [@6] that the number $N(M^{n})$ defined as the smallest dimension of Euclidean spaces for which there is a totally skew embedding of $M^{n}$, satisfies $$2n+2\leq N(M^n)\leq 4n+1.$$ The better lower bound is obtained in [@7]. \[skewteo\] If $$k=\max \{i\mid \overline{w}_i(M^n)\neq 0\}$$ then $N(M^n)\geq 2n+2k+1.$ The immersions and embeddings of quasitoric manifolds over cubes are studied in [@1]. In this paper we prove the following results. \[main\] Let $n$ be a power of two and $P^n$ be a simple convex $n$-dimensional polytope which allows a proper coloring in $n$ colors. - There is a small cover $M^{n}$ over the polytope $P^{n}$ which satisfies following identities $imm(M^{n})=2n-1$ and $em(M^{n})=2n$. - There is a quasitoric manifold $M^{2n}$ over the polytope $P^{n}$ which satisfies following inequalities $imm(M^{2n})\geq 4n-2$ and $em(M^{2n})\geq 4n-1$. Moreover, for $n\geq 3$ in both relation the equality holds. We give an explicit construction of manifolds $M^{n}$ and $M^{2n}$ from Theorem \[main\]. By using combinatorial properties of the simple polytope $P^n$ and Davis-Januszkiewicz formula (\[djf\]) and (\[djfswc\]), we describe the cohomology rings $H^*(M^{dn}, \mathbb{Z}_2), d=1,2$ and calculate Stiefel-Whitney classes $w_k(M^{dn}), d=1,2$. When $n$ is a power of two, we prove that classes ${\overline{w}}_{d(n-1)}( M^{dn}), d=1,2$ are nontrivial, the claim that implies Theorem \[main\]. Also from this claim and Theorem \[skewteo\] for totally skew embeddings of constructed manifolds immediately follows If $n$ is a power of two then $$N(M^{n})\geq 4n-1, \ \ \ N(M^{2n})\geq 8n-3.$$ The obtained small covers $M^{n}$ are a new class of manifolds for which the numbers $N(M^{n})$ are equal to $4n-1, 4n,$ or $4n+1$. So far, only known examples with this property were real projective spaces (see [@7]). Cohen [@8] in 1985 resolved positively the famous [*Immersion Conjecture*]{}, by showing that each compact smooth $n$-manifold for $n>1$ can be immersed in $\mathbb{R}^{2n-\alpha(n)}$, where $\alpha(n)$ is the number of 1’s in the binary expansion of $n$. The products of real projective spaces are examples of manifolds that achieve the upper bounds. We construct new examples of this type in the class of small covers. \[exist\] For every positive integer $n$ there is a small cover $M^{n}$ and a quasitoric manifold $M^{2n}$ over some simple $n$-dimensional polytope which allows a proper coloring in $n$ colors such that $$\begin{aligned} imm(M^{n})=2n-\alpha(n), \ \ \ imm(M^{2n})\geq 4n-2\alpha(n) \\ em(M^{n})=2n-\alpha(n)+1, \ \ \ em(M^{2n})\geq 4n-2\alpha(n)+1, \\ N(M^{n})\geq 4n-2\alpha(n)+1, \ \ \ N(M^{n})\geq 8n-4\alpha(n)+1.\end{aligned}$$ We conjecture that the statement of Theorem \[exist\] is satisfied for any simple $n$-dimensional polytope which is properly colored by $n$ colors. \[conj1\] Let $P^n$ be a simple convex $n$-dimensional polytope which allows a proper coloring in $n$ colors. Then there exist a small cover $M^{n}$ and a quasitoric manifold $M^{2n}$ over the polytope $P^n$ which satisfy $$\begin{aligned} imm(M^{n})= 2n-\alpha(n),\;\;\;\;em(M^{n})= 2n-\alpha(n)+1,\\ imm(M^{2n})\geq 4n-2\alpha(n),\;\;\;\;em(M^{2n})\geq 4n-2 \alpha(n)+1.\end{aligned}$$ Manifolds $M^{dn}$ ================== Construction {#cons} ------------ Let $P^n$ be a simple polytope such that $\chi (P^n)=n$ and $h:\{F_1, \dots,F_m\}\rightarrow\{1,\ldots,n\}$ be its proper coloring. Denote by $\mathcal{F}_j$ the set $h^{-1} (j)$. Every vertex $V\in P^n$ is the intersection of $n$ differently colored facets. Take an arbitrary vertex $V=H_1\cap\cdots\cap H_n$, where the facet $H_i$ is colored by color $i$. Assign to each facet $H_i$ the vector $\lambda_i=(\underbrace{0,\dots,0}_{i-1},\underbrace{1,\dots,1}_{n-i+1})^t$ and vectors $\lambda_F=(\underbrace{0,\dots,0}_{i-1}, 1, \underbrace{0,\dots,0}_{n-i})^t$ to remaining facets $F\in \mathcal{F}_i\backslash \{H_i\}$. The corresponding matrix $\Lambda$ clearly induces the characteristic map, since $\det {\Lambda}_{(v)}=1$ for every vertex $V\in P^n$. Define $M^{dn}=M(P^{n}, \Lambda)$ as the manifold constructed from the characteristic pair $(P^{n}, \Lambda)$. Cohomology ring --------------- Let $u_1,\dots,u_n$ be Poincaré duals to characteristic submanifolds over the facets $H_1,\dots,H_n$ respectively. For every facet $F$ of $P^n$ distinct from $H_1,\dots,H_n$, let $v_F$ denotes the Poincaré dual to the characteristic submanifold over $F$. The cohomology ring of the manifold $M^{dn}$ is determined by the Stanley-Reisner ideal $\mathcal{I}$ of $P^n$ and the ideal $\mathcal{J}$ which is generated by linear forms $$\label{j1} \theta_i:=\sum_{j=1}^i u_j+\sum_{F\in {\mathcal{F}_i\setminus\{H_i\}}} v_F, i=1, \dots, n.$$ Recall that $$H^*(M^{dn}; \mathbb{Z}_2)=\mathbb{Z}_2[u_1, \dots, u_n, v_F |F\in {\mathcal{F}\setminus\{H_1,\dots, H_n\}}]/(\mathcal{I}+\mathcal{J}).$$ From the coloring of $P^n$ we easily deduce \[p1\] For facets $F$, $G\in {\mathcal{F}_i\setminus\{H_i\}}$, $F\neq G$ corresponding classes satisfy the following relation in the cohomology ring $H^*(M^{dn}; \mathbb{Z}_2)$ $$v_F v_G=u_i v_F=u_i v_G=0.$$ Proposition \[p1\] and (\[j1\]) together imply \[p2\] The following equalities hold in $H^*(M^{dn}; \mathbb{Z}_2)$ $$\label{r1} u_1^2=0, \, u_2^2=u_1 u_2, \dots, u_n^2=u_1 u_n+\cdots+u_{n-1} u_n.$$ Let $F$ be an arbitrary facet of $P^n$ and $v_F$ the corresponding class over $F$, even if $F$ is one of $H_1$, $\dots$, $H_n$. \[lema1\] Let $k$ be a positive integer and $h$ a proper coloring map of the polytope $P^n$. Then the class $v_F^k$ is either trivial or equal to the degree $k$ homogenous polynomial $Q_k^F(u_1,\ldots,u_{h(F)-1},v_F)$ whose monomials are square free. Let $h(F)=i$. We prove the claim by induction on $k$. For $k=1$ it is trivial. Assume that $v_F^k=Q_k^F (u_1,\ldots,u_{i-1}, v_F)$. By multiplying with $v_F^k$ the relation $$\sum_{j=1}^{i-1} u_j + \sum_{F\in {\mathcal{F}_i\setminus\{H_i\}}} v_{F}=0$$ and by using Proposition \[p1\] we get $$v_F^{k+1}=(u_1+\cdots+u_{i-1}) Q_k^F (u_1, \dots, u_{i-1}, v_F).$$ If $v_F^{k+1}=0,$ the claim follows directly. In the opposite case it is obvious that $(u_1+\cdots+u_{i-1}) Q_k^F (u_1, \dots, u_{i-1}, v_F)$ is the degree $k+1$ homogenous polynomial in variables $u_1$, $\dots$, $u_{i-1}$, $v_F$. However, from relations (\[r1\]) follows $(u_1+\cdots+u_{i-1}) Q_k^F (u_1, \dots, u_{i-1}, v_F)=Q_{k+1}^F (u_1, \dots, u_{i-1}, v_F)$, where all monomials of the polynomial $Q_{k+1}^{F}$ are square free. From lemma \[lema1\] immediately follows \[posl1\] Each class $v_{F_{i_1}}^{r_1}\cdots v_{F_{i_k}}^{r_k}$ is either trivial or equal to the degree $r_1+\dots+r_k$ homogenous polynomial in variables $u_1$, $\dots$, $u_{n}$, $v_{F_{i_1}}$, $\dots$, $v_{F_{i_k}}$ whose monomials are square free. \[p3\] The class $u_1 \dots u_n$ is the fundamental cohomology class in the ring $H^{dn} (M^{dn}; \mathbb{Z}_2)$. Denote by $\mathcal{F}_V$ the set of facets containing a vertex $V\in P^{n}$. Let $V^{\ast}=\prod_{F\in\mathcal{F}_V}v_F$. We show that $V^{\ast}=V'^{\ast}$ for each two vertices $V,V'\in P^{n}$. For this it is sufficient to suppose that vertices are connected by the edge $VV'$. In this case we have $\mathcal{F}_{V'}=\mathcal{F}_V\setminus\{G\}\cup\{G'\}$ for unique facets $G$ and $G'$ which are colored by the same color. If $h(G)=h(G')=i$, we multiply by $\prod_{F\in\mathcal{F}_V\cap\mathcal{F}_{V'}}v_F$ the relation $$\sum_{j=1}^{i-1} u_j + \sum_{F\in \mathcal{F}_i} v_{F}=0.$$ From Proposition \[p1\] we obtain $$(u_1+\ldots+u_{i-1})\prod_{F\in\mathcal{F}_V\cap\mathcal{F}_{V'}}v_F+V^{\ast}+V'^{\ast}=0.$$ In the case $i=1$ we get the required equality $V^{\ast}=V'^{\ast}$. If $i>1$, for each $1\leq j<i$, we have by Propositions \[p1\] and \[p2\] $$u_j\prod_{F\in\mathcal{F}_V\cap\mathcal{F}_{V'}}v_F=\left\{\begin{array}{cc} 0, &\mbox{if} \ H_j\notin\mathcal{F}_V\cap\mathcal{F}_{V'} \\ (u_1+\cdots u_{j-1})\prod_{F\in\mathcal{F}_V\cap\mathcal{F}_{V'}}v_F, & \mbox{if} \ H_j\in\mathcal{F}_V\cap\mathcal{F}_{V'}\end{array}\right..$$ Then by induction follows $u_j\prod_{F\in\mathcal{F}_V\cap\mathcal{F}_{V'}}v_F=0, j=1,\ldots,i-1$ which again gives $V^{\ast}=V'^{\ast}$. Suppose now that $u_1\cdots u_n=0$. From the so far proven part follows that $V^{\ast}=0$ for each vertex $V\in P^{n}$. Moreover in this case all $dn$-dimensional classes vanish since by Corollary \[posl1\] square free monomials of degree $n$ linearly generate $H^{dn} (M^{dn}; \mathbb{Z}_2)$. But it contradicts the known fact that $H^{dn} (M^{dn}; \mathbb{Z}_2)\simeq\mathbb{Z}_2$. Stiefel-Whitney classes ----------------------- Proposition \[p1\] implies that the total Stiefel-Whitney class could be expressed as $$w(M^{dn})=\prod_{i=1}^{n} \prod_{F\in \mathcal{F}_i} (1+v_F)=\prod_{i=1}^{n}(1+\sum_{F\in \mathcal{F}_i}v_F).$$ By applying (\[j1\]) we obtain $$w (M^{dn})=(1+u_1)(1+u_1+u_2)\cdots(1+u_1+u_2+\cdots+u_{n-1}).$$ In order to prove the main theorem \[main\], we are going to use another set of generators $t_1,\dots,t_n$ which are defined by $$\begin{aligned} \label{tgen} t_i=\sum_{j=1}^{i}u_j, i=1,\ldots,n.\end{aligned}$$ Consequently we have $$\label{sw3} w(M^{dn})=(1+t_1)\cdots(1+t_{n-1}).$$ By Proposition \[p2\] the classes $t_1^2$, $t_2^2+t_1 t_2,\ldots,t_n^2+t_{n-1}t_n$ vanish. Let $\mathcal{T}_n$ be the ideal generated by these classes. By Proposition \[p3\] it is easily seen that the class $t_1 t_2 \cdots t_n$ also represents the fundamental class. Proof of Theorem \[main\] ========================= In order to prove Theorem \[main\] it is sufficient to show that the top dual Stiefel-Whitney class $\overline{w}_{d(n-1)}(M^{dn})$ is nontrivial. Dual Stiefel-Whitney classes ---------------------------- Stiefel-Whitney classes and dual Stiefel-Whitney classes are related by $$w(M^{dn})\cdot \overline{w}(M^{dn})=1.$$ From the relation (\[sw3\]) we obtain \[swd\] The total Stiefel–Whitney class $\overline{w}(M^{dn})$ is expressed by $$\overline{w} (M^{dn})=(1+t_1)(1+t_2+t_2^2)\cdots(1+t_{n-1}+\cdots+t_{n-1}^{n-1}).$$ The statement follows from the fact that $t_k^{k+1}=0$ for all $k=1,\ldots,n-1$. In fact $t_k^{k+1}=(u_1+\cdots+u_k)^{k+1}$ is the sum of monomials of the form $\sum u_{i_1}^{r_1}\cdots u_{i_j}^{r_j},$ where $1\leq i_1<\cdots<i_j\leq k$ and $r_1+\cdots r_k=k+1$. In its turn, by Corollary \[posl1\] and Proposition \[p2\] this is a sum of homogeneous polnomials of the degree $k+1$ of the form $\sum Q_{k+1}(u_1,\ldots,u_k)$. Since monomials of each polynomial in the sum is square free, we have $Q_{k+1}(u_1,\ldots,u_k)=0$. We want to determine the highest nontrivial dual $\overline{w}_k (M^{dn})$. For small $n$, we could calculate $\overline{w} (M^{dn})$ directly. - $\overline{w} (M^{d \cdot 2})=1+t_1$, - $\overline{w} (M^{d\cdot 3})=1+(t_1+t_2)$, - $\overline{w} (M^{d\cdot 4})=1+(t_1+t_2+t_3)+t_1 t_3+t_1 t_2 t_3$, - $\overline{w} (M^{d\cdot 5})=1+(t_1+t_2+t_3+t_4)+(t_1 t_3+t_1 t_4+t_2 t_4)+(t_1 t_2 t_3+t_2 t_3 t_4)$. The class $\overline{w}_{d (n-1)}(M^{dn})$ ------------------------------------------ Consider arbitrary two manifolds $M^{dn}$ and $M^{d(n+1)}$ constructed as in subsection \[cons\] over simple polytopes $P_1^{n}$ and $P_2^{n+1}$, properly colored in $n$ and $n+1$ colors, respectively. \[sub\] The ring $\mathbb{Z}_2[t_1,\ldots,t_n]/\mathcal{T}_n$ is a subring of the cohomology ring $H^{\ast}(M^{dn}; \mathbb{Z}_2)$ which is generated by elements $t_1,\ldots,t_n$. It is necessary to prove that $\mathcal{T}_n$ is the ideal of all relations among elements $t_1,\ldots,t_n$ in $H^{\ast}(M^{dn}; \mathbb{Z}_2)$. The rings $\mathbb{Z}_2 [t_1, \dots, t_n]/\mathcal{T}_n$ and $\mathbb{Z}_2 [u_1, \dots, u_n]/\mathcal{U}_n$ are isomorphic, where $\mathcal{U}_n$ is the ideal generated by elements $u_1^{2}$ and $u_i^{2}+(u_1+\ldots+u_{i-1})u_i, i=2,\ldots,n$ from Proposition \[p2\]. It is sufficient to show that $\mathcal{U}_n$ is the ideal of all relations among $u_1,\ldots,u_n$ in $H^{\ast}(M^{dn}; \mathbb{Z}_2)$. By Corollary \[posl1\] any homogeneous polynomial in $H^{\ast}(M^{dn}; \mathbb{Z}_2)$ is expressed as a sum of square free monomials. So let we have a relation $$\sum_{j=1}^{m}u_{I_j}=0,$$ where $I_j=\{a_{j1}<\ldots<a_{jk}\}\subset\{1,\ldots,n\}, j=1,\ldots,m$ and $u_{I}=\prod_{i\in I}u_i$. It is easy to convince that $u_1\cdots u_{i-1}u_i^{2}=0, i=1,\ldots,n$. Let us order sets lexicographically $I_1<\ldots<I_m$. Let $I_m=\{a_1<\ldots<a_k\}$ and $d_i=a_i-a_{i-1}, i=1,\ldots,k$ where $a_0=0$. Define following elements $$U_i=\left\{\begin{array}{cc} u_{a_{i-1}+1}\cdots u_{a_i-1}, & d_i>1 \\ 1, & d_i=1\end{array}\right.,$$ where $i=1,\ldots,k$. By multiplying the relation with $U_1\cdots U_k,$ we get $u_1\cdots u_{a_k}=0,$ which contradicts the fact that $u_1\cdots u_n$ is the fundamental class. By abbreviating all terms in the identity in Lemma \[swd\] we obtain that each class $\overline{w}_{dk}(M^{dn})$ is expressed in the ring $\mathbb{Z}_2[t_1,\dots,t_n]/\mathcal{T}_n$ by square free homogeneous polynomials $\overline{W}_{k}(t_1,\dots,t_n)$ of the degree $k$. Note that the inclusion $$i:\mathbb{Z}_2[t_1, \dots, t_n]\rightarrow\mathbb{Z}_2[t_1, \dots, t_n, t_{n+1}]$$ induces a natural monomorphism $$i^*:\mathbb{Z}_2[t_1, \dots, t_n]/\mathcal{T}_n\rightarrow \mathbb{Z}_2[t_1, \dots, t_n, t_{n+1}]/\mathcal{T}_{n+1},$$ which by Lemma \[sub\] allows us to consider the total Stiefel-Whitney class $\overline{w}(M^{dn})$ as an element of the ring $H^*(M^{d(n+1)}; \mathbb{Z}_2)$. Thw total Stiefel-Whitney classes $\overline{w} (M^{dn})$ and $\overline{w} (M^{d(n+1)})$ satisfy the following relation in $H^*(M^{d(n+1)}; \mathbb{Z}_2)$ $$\overline{w} (M^{d(n+1)})=\overline{w}(M^{dn})(1+t_n+\dots+t_n^n).$$ Explicitly $$\label{jed2} \overline{w}_{dk}(M^{d(n+1)})=\overline{w}_{dk}(M^{dn})+t_n\overline{w}_{d(k-1)}(M^{dn})+\dots+t_n^k, k=0,\ldots, n.$$ Recall that $\overline{w}_{dn}(M^{dn})=0$ (see [@10]), which implies $$\label{jed3} \overline{w}_{dn}(M^{d(n+1)})= t_n \overline{w}_{d(n-1)}(M^{dn})+\cdots+t_n^n=t_n\overline{w}_{d(n-1)}(M^{d(n+1)}).$$ We use the same trick as in [@1]. Define numbers $\sigma^k_n, 0\leq k\leq n-1$ as follows $$\sigma^k_n=\overline{W}_{dk}(\underbrace{1,\dots,1}_n)\pmod 2.$$ By (\[jed2\]) and (\[jed3\]), we have $\sigma^k_{n+1}=\sum_{i=0}^{k} \sigma^i_n $ for every $k=1,\ldots,n-1$ and $\sigma^{n}_{n+1}=\sigma^{n-1}_{n+1}$. By definition of $\sigma^k_n$, if $\sigma^k_n=1$, then $\overline{w}_{dk}$ is the sum of an odd number of linearly independent square free monomials, which implies $\overline{w}_{dk}(M^{dn})\neq 0. $ An easy mathematical induction shows that $$\sigma^k_n\equiv{\tbinom}{n+k}{k}\pmod{2}.$$ Particularly, if $n=2^r$, we have $$\sigma^{n-1}_n \equiv{\tbinom}{2^r+(2^r-1)}{2^r-1}\equiv{\tbinom}{2^{r+1}-1}{2^r-1}\equiv 1\pmod{2}.$$ Consequently, $$\overline{w}_{d(n-1)}(M^{dn})=t_1 t_2\cdots t_{n-1}\neq 0.$$ Therefore by Theorem \[imeem\] we obtain the required bounds $$imm(M^{dn})\geq d(2n-1), em(M^{dn})\geq d(2n-1)+1.$$ For the small cover $M^{n}$, Whitney’s theorem implies $imm(M^{n})=2n-1$ and $em(M^{n})=2n$. The quasitoric manifold $M^{2n}$ is orientable, so it can be embedded into $\mathbb{R}^{4n-1}$. From Lemma \[swd\] follows $\overline{w}_2(M^{2n})=t_1+t_2+\cdots+t_{n-1},$ which implies that the characteristic class $\overline{w}_2(M^{dn})\cdot \overline{w}_{2n-2}(M^{2n})$ vanishes. If $n\geq 3$, the result of Massey [@11 Theorem V] yields $$imm(M^{2n})=4n-2 ,$$ which finishes the proof of Theorem \[main\]. Proof of Theorem \[exist\] ========================== Let $n=2^{r_1}+2^{r_2}+\dots+2^{r_t}$, $r_1>r_2>\dots>r_t\geq 0$ be the binary representation of $n$ and let $m_i=2^{r_i}$ for $i=1,\dots,t$ and $m_0=0$. Let $P^n$ be a simple $n$-polytope such that $$P^n= P_1^{2_{r_1}}\times \cdots\times P_t^{2^{r_t}},$$ where each $P_i^{2_{r_i}}$ is $2^{r_i}$-colored simple $2^{r_i}$-polytope. It is obvious that polytope $P^n$ is $n$-colored. In subsection \[cons\] we constructed manifolds $M^{d 2^{r_i}}$ over polytopes $P_i^{2_{r_i}}$. It follows from [@2 Proposition 4.7] that $M^{dn}=M^{d2^{r_1}}\times\dots\times M^{d 2^{r_t}}$ is a $G_d^{n}$-manifold over the polytope $P^n= P_1^{2_{r_1}}\times \cdots\times P_t^{2^{r_t}}$. The total Stiefel–Whitney class of the manifold $M^{dn}$ can be easily determined using the following formula (see [@4 pp.27,54]) $$w(M^{dn})=w(M^{d 2^{r_1}})\cdots w(M^{d 2^{r_t}})\in H^*(M^{dn})\cong H^*(M^{d 2^{r_1}})\otimes\dots\otimes H^*(M^{d 2^{r_t}}).$$ The corresponding dual total Stiefel-Whitney class is expressed as $$\label{dpr} \overline{w}(M^{dn})=\overline{w}(M^{d 2^{r_1}})\cdots \overline{w}(M^{d 2^{r_t}}).$$ Let ${\operatorname{rank}}w M):=\max \{k | w_k (M)\neq 0\}$. Thus, from formula (\[dpr\]) we have $$\mathrm{rank} \ \overline{w}(M^{dn})=\sum_{i=1}^t\mathrm{rank} \ \overline{w}(M^{d2^{r_i}})=\sum_{i=1}^t d(2^{r_i}-1)=nd -\alpha (n) d.$$ In this way, Theorem \[exist\] is a consequence of Theorems \[imeem\] and \[skewteo\]. Aknowledgements {#aknowledgements .unnumbered} =============== The authors express their special thanks to T. E. Panov and A. A. Gaifullin for attention to this work, as well as to reviewers for helpful comments. [99]{} G M. Ziegler, [*Lectures on polytopes*]{}, Springer, Graduate Texts in Math. $152, 1998$. V. Buchstaber and T. Panov. *Torus Actions and their applications in topology and combinatorics*, AMS University Lecture Series, volume 24, (2002). M. Joswig. Projectivities in simplicial complexes and colorings of simple polytopes. *Math. Z.* (2002) **240**:243–259. M. Davis and T. Januszkiewicz. Convex polytopes, Coxeter orbifolds and torus actions. *Duke Math. J.* **62** (1991), no. 2, 417–451. J.W. Milnor, J.D. Stasheff. *Characteristic Classes*. Princeton Univ. Press 1974. M. Ghomi, S. Tabachnikov. Totally skew embeddings of manifolds. *Math. Z.* (2008) **258**:499–512. Dj. Baralić, B. Prvulović, G. Stojanović, S. Vrećica and R. Živaljević. Topological Obstructions to Totally Skew Embeddings. *Transactions of American Mathematical Society.* AMS, (2012), vol. **364** no. **4**, 2213-2226. Dj. Baralić. Immersions and Embeddings of Quasitoric Manifolds over Cube. *Publications de l’Institut Mathematique* (2014) (N.S.) **95** (109), 63-71. R. L. Cohen. The immersion conjecture for differentiable manifolds. *Ann. Math.* **2** (1985), 237–328. W.S. Massey. On the Stiefel–Whitney classes of a manifold. *Am. J. Math.* **82** (1960), 92–102. W.S. Massey. Normal vector fields on manifolds. *Proc. Am. Math. Soc.* **12** (1961), 33–40.

--- abstract: | In this note, using the ideas from our recent article [@EM], we prove strong ill-posedness for the 2D Euler equations in $C^k$ spaces. This note provides a significantly shorter proof of many of the main results in [@BLi2]. In the case $k>1$ we show the existence of initial data for which the $kth$ derivative of the velocity field develops a logarithmic singularity immediately. The strong ill-posedness covers $C^{k-1,1}$ spaces as well. The ill-posedness comes from the pressure term in the Euler equation. We formulate the equation for $D^k u$ as: $$\partial_t D^k u=D^{k+1} p + l.o.t.$$ and then use the non-locality of the map $u\rightarrow p$ to get the ill-posedness. The real difficulty comes in how to deal with the “l.o.t.” terms which can be handled by special commutator estimates. author: - 'Tarek M. Elgindi and Nader Masmoudi' title: Note on The Euler equations in $C^k$ spaces --- introduction ============ In this note we give a short proof of the strong ill-posedness of the Euler equations for incompressible flow in $C^k$ spaces. Our proof works in the whole space case as well as the case of periodic boundary conditions and the bounded domain case. We use the non-locality of the map $u\rightarrow p$ to get growth in $C^k$ spaces (since Riesz transforms are unbounded on these spaces!). The proof uses classical commutator estimates which estimate the commutation of a Riesz transform (or a Calderón Zygmund operator) and composition by a bi-Lipschitz map. We prove the following theorem. The Euler equations are strongly ill-posed in $C^k$ spaces for $k\geq 1$. In other words, for every $\epsilon>0$ there exists initial data $u_0\in C^k$ such that the unique solution, $u(t)$, of the Euler equations with initial data $u_0$ leaves $C^k$ immediately. We note that very recently Bourgain and Li have proven the same result as above [@BLi2]. In their paper the authors cite our work [@EM] and note that it did not include ill-posedness $C^k$ spaces for $k>1$. In this note we show that the work in [@EM] can easily be extended to the case $k>1.$ Moreover we make clear the fact that strong ill-posedness in $C^k$ can be proven quite easily only using commutator estimates without having to rely upon very intricate constructions. The proof ========= Note that it suffices to consider the two dimensional Euler equations. Now consider the equation for $ D^k u$ which means $k$ spatial derivatives of $u,$ which we take to be a vector of many components. $\nabla D^{k-1} u$ satisfies the following equation: $$\partial_t \nabla D^{k-1} u + u \cdot \nabla D^k u +\sum_{j,l}^k Q(D^j u, D^l u) +D^{k-1} D^2 p=0$$ Recall that $$\Delta p= det (\nabla u)$$ so that $$(D^2 p)_{ij} =\big (R_{i}R_{j} det(\nabla u) \big )_{ij}$$ We rely upon the following commutator estimate in $L^p:$ Let $\Phi$ be a bi-Lipschitz measure preserving map. Let $K=\max\{|\Phi-Id|_{Lip}, |\Phi^{-1}-Id|_{Lip} \}.$ Let $R$ be a composition of Riesz transforms. Define the following commutator: $$[R,\Phi] \omega= R(\omega\circ\Phi)-R(\omega)\circ \Phi,$$ for $\omega\in L^p.$ Then, $$\|[R,\Phi]\|_{L^p\rightarrow L^p}\leq c_{p} K.$$ Moreover, $c_p\leq c p$ as $p\rightarrow \infty$. Now we recall the Lagrangian flow $$\dot{\Phi}(x,t)= u(\Phi(t,x),t)$$ $$\Phi(x,0)=x.$$ Because $u$ is divergence free, $\Phi$ is measure preserving. Furthermore, $\Phi(x,-t) = \Phi^{-1} (x,t).$ Now, we may write $$\Phi(x,t) = x +\int_{0}^{t} u(\Phi(x,\tau),\tau)d\tau.$$ Thus, $$\Phi(\cdot,t) - I = \int_{0}^{t} u(\Phi(\cdot,\tau),\tau)d\tau.$$ Consequently, $$|\Phi-I|_{\text{Lip}} \leq t | u|_{\text{Lip}}|\Phi|_{Lip}$$ and similarly for $\Phi^{-1} (\cdot,t)= \Phi(\cdot,-t).$ Furthermore, by Gronwall’s lemma, $$|\Phi|_{\text{Lip}} \leq \exp (t | u|_{\text{Lip}}).$$ In particular, $$|\Phi-I|_{\text{Lip}} \leq t | u|_{\text{Lip}}\exp(t|u|_{\text{Lip}}).$$ In particular, if $u$ is Lipschitz then the Lagrangian flow-map is controlled. Now, assume that $u$ remains Lipschitz (which in the case $k>0$ is trivial). Then we have: $$\partial_t(D^k u \circ \Phi) +\sum_{j,l}^k Q(D^j u, D^l u)\circ \Phi + \Big ( R_{i}R_{j}D^{k-1}det (\nabla u)\Big ) \circ \Phi=0$$ Now assume that the solution remains in $C^k.$ Now call $R_i R_j:=R$ for short. $$\partial_t(D^k u \circ \Phi) +\sum_{j,l}^k Q(D^j u, D^l u)\circ \Phi + \Big ( R D^{k-1}det (\nabla u)\Big ) \circ \Phi=0$$ $$\partial_t(D^k u \circ \Phi) +\sum_{j,l}^k Q(D^j u, D^l u)\circ \Phi + R \Big ( D^{k-1}det (\nabla u) \circ \Phi\Big )= [R,\Phi]D^{k-1}det (\nabla u)$$ We are going to take a special initial data $u_0\in C^k.$ Now, suppose that the solution remains in $C^k.$ Note that we can always solve the 2D Euler equations in $W^{k,p}$ and that the solution will be unique when $k>1$ (in the case k=1 we will also get a unique solution the class of velocity fields with bounded curl). Assume that the solution stays in $C^k$ with $|u|_{C^k}\leq M.$ Then we can solve the Euler equations formally using the Duhamel formula: $$|D^k u\circ \Phi|_{L^p} \geq |D^k u_0+tR(D^{k-1} det(\nabla u_0)|_{L^p}-\Big | \int_{0}^t e^{R(t-s)}[R,\Phi]D^{k-1} det(\nabla u) -Q(D^j u, D^l u)\circ \Phi ds \Big |_{L^p}$$ Now suppose that we construct compactly supported initial data $u_0$ such that $|RD^{k-1} det(\nabla u_0)|_{L^p}\geq cp$ as $p\rightarrow \infty.$ Then we see that $$|D^k u\circ \Phi|_{L^p} \geq tcp -C- t(1+tp) |[R,\Phi]|_{L^p\rightarrow L^p} C(M)+tC(M).$$ This is because $\|R\|_{L^p\rightarrow L^p}\lesssim p$ as $p\rightarrow \infty.$ Now using the commutator estimate in Lemma 2.1 we get: $$|D^k u\circ \Phi|_{L^p} \geq tcp -C- t(1+tp) tp \tilde{C}(M)+tC(M)$$ Now take $t$ very small depending upon $M$ then we get: $$|D^k u\circ \Phi|_{L^p} \geq t\tilde{c}p$$ for $t$ small and all $p$ large. This contradicts the fact that $|D^k u|_{L^\infty}$ remains bounded. Now, of course we relied upon the existence of an initial data $u_0$ such that $$|RD^{k-1} det(\nabla u_0)|_{L^p}\geq cp.$$ Constructing such initial data is not too difficult. We will copy the construction from our work [@EM] below in the case $k=1$. The higher order cases are similar. We are interested in showing that for some $i,j$ and for some divergence free $u,$ with $\nabla u \in L^\infty,$ $D^2 p= R_{i}R_{j} \text{det}(\nabla u)$ has a logarithmic singularity. Once that is shown, Lemma 8.2 will follow by a regularization argument. Take a harmonic polynomial, $Q,$ which is homogeneous of degree 4. In the two-dimensional case, we can take $$Q(x,y):=x^4+y^4-6x^2y^2,$$ $$\Delta P =0.$$ Define $$G(x,y):= Q(x,y) Log (x^2 +y^2).$$ Notice that $$\partial_{i}\partial_{j}\Delta G\in L^\infty (B_{1}(0)), i,j\in\{1,2\}.$$ Notice, on the other hand, that $$\partial_{xxyy} G=-24Log(x^2+y^2)+H(x,y),$$ with $H\in L^\infty(B_{1}(0)).$ In particular, $\partial_{xxyy}G$ has a logarithmic singularity at the origin–and the same can be said about $\partial_{xxxx}G$ and $\partial_{yyyy}G.$ Define $\tilde{u}=\nabla^\perp \Delta G.$ Then, by 8.12, $\nabla \tilde{u} \in L^\infty(B_{1}(0)).$ Moreover, by definition, $$R_{i}R_{j} \nabla \tilde{u}=\nabla\nabla^\perp\partial_{ij}G.$$ Thus, for example, $R_{1}R_2 \nabla\tilde{u}_{1x}=\partial_{xxyy}G$ has a logarithmic singularity in $B_{1}(0).$ Unfortunately, we are interested in showing that $R_{i}R_{j} \text{det}(\nabla u)$ has a logarithmic singularity for some $i,j,$ not $R_{i}R_{j}\nabla u.$ To rectify this, we choose $$u=\delta \nabla^\perp\Delta (\chi G)+\eta(2y\chi+y^2 \partial_y\chi,y\partial_x\chi),$$ where $\eta, \delta$ are small parameters which will be determined and $\chi$ is a smooth cut-off function with: $$\chi=1 \, \, \text{on} \, \, B_1(0),$$ $$\chi=0 \,\, \text{on} \, \, B_2(0)^c.$$ Note that $u$ is divergence free and $$u\equiv \delta\nabla^\perp \Delta G + \eta(y,0) \,\, \text{on} \,\, B_{1}(0).$$ Therefore, $$\nabla u= \delta \left[ {\begin{array}{cc} -\partial_{xy} \Delta G & -\partial_{yy}\Delta G \\ \partial_{xx}\Delta G & \partial_{xy}\Delta G \\\end{array} } \right] + \eta \left[ {\begin{array}{cc}0 & 1 \\0 & 0 \\\end{array} } \right] .$$ In particular, $$\text{det}(\nabla u) = \eta\delta\partial_{xx}\Delta G + \delta^2 J(x,y),$$ where $J$ is a bounded on $B_{1}(0)$. Now consider $R_2 R_2 \text{det}(\nabla u):$ $$R_2 R_2 \text{det}(\nabla u)= \eta \delta \partial_{xxyy}G +\delta^2 R_{2} R_2 J.$$ Now, by (8.13), we have $$R_2 R_2 \text{det}(\nabla u)= \eta \delta (-24Log(x^2+y^2)+H(x,y)) +\delta^2 R_{2} R_2 J,$$ with $H$ and $J$ bounded. Now, recall that $R_2R_2$ maps $L^\infty$ to BMO and that any BMO function can have at most a logarithmic singularity. Thus, $$|R_2 R_2 \text{det}(\nabla u)|\geq 24 \eta \delta Log(x^2+y^2) -C\delta^2 Log(x^2+y^2) -|H(x,y)|.$$ Choose $\delta < < \eta$ and we see that, near $(0,0)$ $$|R_2 R_2 \text{det}(\nabla u)|\geq 12 \delta^2 Log(x^2+y^2).$$ Taking $\delta\leq C$ small enough, we see that $|\nabla u| \leq 1$ but $|R_2 R_2 \text{det} (\nabla u)| \geq c Log(x^2 +y^2), $ for some small $c.$ This completes the construction. [2]{} J. Bourgain and D. Li. Strong illposedness of the incompressible Euler equation in integer $C^m$ spaces T.M. Elgindi and N. Masmoudi $L^\infty$ ill-posedness for a class of equations arising in hydrodynamics.

--- abstract: 'We propose probabilistic controlled-NOT and controlled-phase gates for qubits stored in the polarization of photons. The gates are composed of linear optics and photon detectors, and consume polarization entangled photon pairs. The fraction of the successful operation is only limited by the efficiency of the Bell-state measurement. The gates work correctly under the use of imperfect detectors and lossy transmission of photons. Combined with single-qubit gates, they can be used for producing arbitrary polarization states and for designing various quantum measurements.' address: | CREST Research Team for Interacting Carrier Electronics, School of Advanced Sciences,\ The Graduate University for Advanced Studies (SOKEN), Hayama, Kanagawa, 240-0193, Japan author: - 'Masato Koashi, Takashi Yamamoto, and Nobuyuki Imoto' title: Probabilistic manipulation of entangled photons --- [2]{} Entanglement plays an important role in various schemes of quantum information processing, such as quantum teleportation [@Bennett93], quantum dense coding [@Bennett92], certain types of quantum key distributions [@Ekert91], and quantum secret sharing[@Hillery99]. It is natural to expect that entanglement shared among many particles will be useful for more complicated applications including communication among many users. Among the physical systems that can be prepared in entangled states, photons are particularly suited for such applications because they can easily be transferred to remote places. Several schemes for creating multiparticle entanglement from a resource of lower numbers of entangled particles have been proposed[@anton97; @bose98], and experimentally a three-particle entangled state \[a Greenberger-Horne-Zeilinger (GHZ) state\] was created from two entangled photon pairs [@dik99]. In order to synthesize [*any*]{} states of $n$ photons on demand, the concept of quantum gates is useful. The universality of the set of the controlled-NOT gate and single-qubit gates[@sleator95; @barenco95] implies that you can create any states by making a quantum circuit using such gates. In addition to synthesizing quantum states, this scheme also enables general transformation and generalized measurement in the Hilbert space of $n$ photons. A difficulty in this strategy is how to make two photons interact with each other and realize two-qubit gates. One way to accomplish this is to implement conditional dynamics at the single-photon level through the strong coupling to the matter such as an atom, and a demonstration has been reported [@turchette95], which is a significant step toward this goal. On the other hand, if we restrict our tools to linear optical elements, a never-failing controlled-NOT gate is impossible, which is implied by the no-go theorem for Bell-state measurements [@lutkenhaus99]. It is, however, still possible to construct a “probabilistic gate”, which tells us whether the operation has been successful or not and do the desired operation faithfully for the successful cases. While the probabilistic nature hinders the use for the fast calculation of classical data that outruns classical computers, such a gate will still be a useful tool for the manipulation of quantum states of a modest number of photons, because no classical computer can be a substitute for this purpose. In this Rapid Communication, we propose probabilistic two-qubit gates for qubits stored in the polarization of photons. The gates are composed of photon detectors and linear optical components such as beam splitters and wave plates. As resources, the gates consume entangled photon pairs. When the detectors with quantum efficiency $\eta$ are used, the success probability of $\eta^4/4$ can be obtained, which is only limited by the efficiency of the Bell measurement used in the scheme. Combined with single-qubit gates that are easily implemented by linear optics, the proposed gates can build quantum circuits conducting arbitrary unitary operations with nonzero success probabilities. In Fig. \[f1\] we show the schematic of scheme I, the simplest of the schemes we propose in this paper. The gate requires two photons and a pair of photons in a Bell state as resources. Initially, they are in the states $|H\rangle_{2a}$, $|H\rangle_{2b}$, and $(|H\rangle_{3a}|V\rangle_{3b}-|V\rangle_{3a}|H\rangle_{3b})/\sqrt{2}$. The wave plate WP5 rotates the polarization of mode $3a$ by $45^\circ$, namely, $|H\rangle_{3a}\rightarrow (|H\rangle_{3^\prime a}+|V\rangle_{3^\prime a})/\sqrt{2}$ and $|V\rangle_{3a}\rightarrow (|H\rangle_{3^\prime a}-|V\rangle_{3^\prime a})/\sqrt{2}$. After WP5, the entangled photon pair becomes $$\begin{aligned} \frac{1}{2}&&( |V\rangle_{3^\prime a}|V\rangle_{3b} +|H\rangle_{3^\prime a}|V\rangle_{3b} \nonumber \\ &&+|V\rangle_{3^\prime a}|H\rangle_{3b} -|H\rangle_{3^\prime a}|H\rangle_{3b} ).\end{aligned}$$ The polarizing beam splitter PBS1 transmits $H$ photons and reflects $V$ photons. Combined with WP2, it gives the transformation $|H\rangle_{2a}|H\rangle_{3^\prime a} \rightarrow (|H\rangle_{4a}|H\rangle_{5a}+|V\rangle_{4a}|H\rangle_{4a})\sqrt{2}$ and $|H\rangle_{2a}|V\rangle_{3^\prime a} \rightarrow (|V\rangle_{4a}|V\rangle_{5a}+|H\rangle_{5a}|V\rangle_{5a})\sqrt{2}$. Photons in the $b$ modes are similarly transformed, and we obtain the state $$\begin{aligned} &&(\beta/2)\left( |V\rangle_{4a}|V\rangle_{4b}|V\rangle_{5a}|V\rangle_{5b} +|H\rangle_{4a}|V\rangle_{4b}|H\rangle_{5a}|V\rangle_{5b} \right. \nonumber \\ &&\left. +|V\rangle_{4a}|H\rangle_{4b}|V\rangle_{5a}|H\rangle_{5b} -|H\rangle_{4a}|H\rangle_{4b}|H\rangle_{5a}|H\rangle_{5b} \right) \nonumber \\ &&+\sqrt{1-\beta^2}|\phi\rangle \equiv \beta|\Psi\rangle+\sqrt{1-\beta^2}|\phi\rangle ,\end{aligned}$$ where $\beta=1/2$, and $|\phi\rangle$ is a normalized state in which the number of photons in mode $4a$ or mode $4b$ is not unity. [ \[f1\] ]{} The photon in the input mode $1a$ $(1b)$ passes through wave plate WP1 (WP4), which rotates its polarization by $90^\circ$, and is mixed with the photon in mode $4a$ $(4b)$ by a 50/50 polarization-independent beam splitter BS1 (BS2). After the beam splitters, the photon number of each mode and each polarization is measured by a photon counter. Let us assume that the state of the input qubits is $$\begin{aligned} \alpha_1&&|V\rangle_{1a}|V\rangle_{1b} +\alpha_2|V\rangle_{1a}|H\rangle_{1b} \nonumber \\ &&+\alpha_3|H\rangle_{1a}|V\rangle_{1b} +\alpha_4|H\rangle_{1a}|H\rangle_{1b}. \label{input}\end{aligned}$$ The total state after the beam splitter can be calculated straightforwardly, but we do not write down the whole since it is too lengthy. We focus on the terms in which one $H$ photon is found in modes $6a$ or $7a$, one $V$ photon is found in modes $6a$ or $7a$, and similar conditions hold for $b$ modes. There are 16 such combinations. For example, the terms including $|V\rangle_{6a}|H\rangle_{7a}|V\rangle_{6b}|H\rangle_{7b}$ are found to be $$\begin{aligned} -&&\frac{\beta}{8}|V\rangle_{6a}|H\rangle_{7a}|V\rangle_{6b}|H\rangle_{7b} (-\alpha_1|V\rangle_{5a}|V\rangle_{5b} \nonumber \\ &&+\alpha_2|V\rangle_{5a}|H\rangle_{5b} +\alpha_3|H\rangle_{5a}|V\rangle_{5b} +\alpha_4|H\rangle_{5a}|H\rangle_{5b}),\end{aligned}$$ and the terms including $|V\rangle_{6a}|H\rangle_{6a}|V\rangle_{7b}|H\rangle_{7b}$ are $$\begin{aligned} \frac{\beta}{8}&&|V\rangle_{6a}|H\rangle_{6a}|V\rangle_{7b}|H\rangle_{7b} (-\alpha_1|V\rangle_{5a}|V\rangle_{5b} \nonumber \\ &&-\alpha_2|V\rangle_{5a}|H\rangle_{5b} -\alpha_3|H\rangle_{5a}|V\rangle_{5b} +\alpha_4|H\rangle_{5a}|H\rangle_{5b}).\end{aligned}$$ As seen in these examples, the state in modes $5a$ and $5b$ depends on the photon distribution in modes 6 and 7. However, it is easy to check that this dependence is canceled if we introduce a phase shift by phase modulator PM1, $|H\rangle_{5a}\rightarrow |H\rangle_{8a}$ and $|V\rangle_{5a}\rightarrow -|V\rangle_{8a}$, only for the cases of $|V\rangle_{6a}|H\rangle_{6a}$ and $|V\rangle_{7a}|H\rangle_{7a}$, and similar operation for PM2. Then, for all 16 combinations, the state in modes $8a$ and $8b$ becomes $$\begin{aligned} -\alpha_1&&|V\rangle_{8a}|V\rangle_{8b} +\alpha_2|V\rangle_{8a}|H\rangle_{8b} \nonumber \\ &&+\alpha_3|H\rangle_{8a}|V\rangle_{8b} +\alpha_4|H\rangle_{8a}|H\rangle_{8b}. \label{output}\end{aligned}$$ The evolution from Eq. (\[input\]) to Eq. (\[output\]) shows that this scheme operates as a controlled-phase gate if we assign $|0\rangle=|H\rangle$ and $|1\rangle=|V\rangle$. The probability of obtaining these results is $\beta^2/4=1/16$. The factor of $1/4$ appearing here can be understood as due to the twofold use of Bell-state measurement schemes with 50% success probability, used in the dense coding experiment [@mattle96]. If we place two additional wave plates in modes $1b$ and $8b$, which rotate polarization by $45^\circ$ and $-45^\circ$, respectively, we obtain a probabilistic controlled-NOT gate. Next, we consider the effect of imperfect quantum efficiency of photon detectors. In order to characterize the behavior of the detector, we introduce the parameter $\eta_2$ in addition to the quantum efficiency $\eta$, in such a way that it detects two photons with probability $\eta^2\eta_2$ when two photons simultaneously arrive. For example, conventional avalanche photodiodes (APDs) have $\eta_2=0$ since they cannot distinguish two-photon events from one-photon events. Use of $N$ conventional APDs after beam splitting the input to $N$ branches leads to an effective value of $\eta_2=1-1/N$. Recently, a detector with high $\eta$ and with clearly distinguishable signals for one- and two-photon events was also demonstrated[@kim99]. There are two distinctive effects caused by the imperfect quantum efficiency. The first one is that the detectors report some successful events as false ones by overlooking incoming photons. The success probability of $1/16$ in the ideal case thus reduces to $p^{\rm (I)}_{\rm true}\equiv \eta^4/16$. The second effect is that the detectors report some failing events as successful ones. This may occur when two photons enter mode $4a$ or $4b$, hence the output mode $8a$ or $8b$ has no photon. After some simple algebra, the probability $p^{\rm (I)}_{\rm false}$ of this occurrence is obtained as $p^{\rm (I)}_{\rm false}=\eta^4(3-\kappa)(1-\kappa)/4$, with $\kappa\equiv \eta(1+\eta_2)/2$. Because of this effect, after discarding the failing events indicated by the results of the photon detection, the output of the gate still includes errors at probability $p^{\rm (I)}_{\rm err}\equiv p^{\rm (I)}_{\rm false}/ (p^{\rm (I)}_{\rm true}+p^{\rm (I)}_{\rm false})$. In the following, we describe two methods for removing these errors. The first method is the postselection that is applicable when every output qubit of the whole quantum circuit is eventually measured by photon detectors. As we have seen, the errors in the gate always accompany the loss of photons in the output. We also observe easily that if the input mode $1a$ $(1b)$ is initially in the vacuum state, the output mode $8a$ $(8b)$ has no photon whenever the detectors show successful outcomes. This implies that if one of the gates in the circuit causes errors, at least one photon is missing in the final state of the whole circuit. The errors can thus be discarded by postselecting the events of every detector at the end of the circuit registering a photon. This method also works when the Bell-state source fails to produce two photons reliably and emits fewer photons on occasion. The second method is to construct more reliable gates, using scheme I for the initialization processes, as shown in Fig. \[f2\](a). This method is advantageous when the Bell-state source is close to ideal and good optical delay lines are available. In scheme II, we use two Bell pairs of photons in the state $(|H\rangle|H\rangle-|V\rangle|V\rangle)/\sqrt{2}$, and operate conditional-phase gate (scheme I) on the two photons, one from each pair. If the operation fails, we discard everything and retry from the start. The initialization process is complete when the operation of scheme I is successful, and the outputs are sent to modes $4a$ and $4b$. The remaining photons of Bell pairs are sent to modes $5a$ and $5b$. At this point, the quantum state is prepared in the following mixed state, $$(1-p^{\rm (I)}_{\rm err}) |\Psi\rangle\langle\Psi |+ p^{\rm (I)}_{\rm err} \hat\rho,$$ where $\hat\rho$ is a normalized density operator representing a state in which no photon exists in mode $4a$ or $4b$. After this point, the operation of scheme II follows that of scheme I. The crucial difference from scheme I is that the state $\hat\rho$ has no chance to produce successful outcomes. This leads to $p^{\rm (II)}_{\rm false}=p^{\rm (II)}_{\rm err}=0$, namely, the faithful operation is obtained with the success probability of $p^{\rm (II)}_{\rm true}=(1-p^{\rm (I)}_{\rm err})\eta^4/4$. When one of the input modes is in the vacuum, this gate never reports the successful operation. This implies that errors caused by the loss of photons in the upstream circuit are detected and hence discarded. Using scheme II for the initializing process, we can enhance the success probability. Scheme III shown in Fig. \[f2\](b) is exactly the same as scheme II, except that scheme I inside is replaced by scheme II itself. When the initialization is completed, the gate inside produces exactly the state $|\Psi\rangle$. For this scheme, the probability of a successful operation is $p^{\rm (III)}_{\rm true}=\eta^4/4$, and in the ideal case it is $1/4$. This limitation stems from the success probability (50% each) of the two Bell-state measurements. It should be noted that the maximum of this probability is still an open question, and if a more efficient way of Bell measurement is discovered, it will be used in our scheme to enhance the success probability of the gate. Scheme III can be viewed as a particular implementation of the general scheme of constructing quantum gates using the concept of teleportation and Bell measurement[@gottesman], to the case of qubits stored in photons. In the general argument that considers the use of single-qubit gates, three-particle entangled states (GHZ states) are required as resources. What was shown here is that linear optical components for qubits made of photons have more functions than the single-qubit gates, and the resource requirement is further reduced to two-particle entanglement. [ (a) Scheme II. The gate uses scheme I inside. (b) Scheme III. The gate uses scheme II inside. \[f2\] ]{} Since the set of the controlled-NOT gate and single-qubit gates is universal[@sleator95; @barenco95], any unitary transformation can be realized with a nonzero success probability by quantum circuits composed of the proposed gates and linear optical components. For the tasks that take classical data as an input and return classical data as an output, the quantum circuits here will not surpass the conventional classical computers due to the probabilistic nature. But there are other applications in which either the input or the output includes quantum states. For instance, they can be used as a quantum-state synthesizer, which produces any quantum state on the polarization degree of freedom with a nonzero probability. They are also used as designing various types of quantum measurement. For any positive operator valued measure (POVM) [@peres] given by the set of positive operators $\{F_1,\ldots,F_n\}$ with $\sum_k F_k=\bbox{1}$, it is possible to realize a POVM given by $\{pF_1, \ldots, pF_n, (1-p)\bbox{1}\}$, where $p$ is a nonzero probability of success. As transformers of quantum states, they may be used for the purification protocol of entangled pairs[@BBPSSW]. This implies that if reliable resources of entangled photon pairs are realized, it may be possible to produce maximally entangled pairs shared by remote places connected only by noisy channels. Finally, we would like to mention the requirement on the property of the entangled-pair resources. While the mixing of fewer-photon states can be remedied as discussed before, the mixing of excess photons leads to errors that are difficult to correct. For example, the photon-pair source by parametric down-conversion of coherent light with a pair production probability of $\eta_{\rm PDC}$ emits two pairs with the probability of $O(\eta^2_{\rm PDC})$. This portion causes severe effects when the two or more gates are connected in series. The recent proposal for the regulated entangled photon pairs from a quantum dot [@benson00] seems to be a promising candidate for the resources of the proposed gates. [*Note added in proof*]{} — Recently, a proposal of quantum gates for photons, which is aimed at fast computation, was made by Knill [*et al.*]{} [@knill]. 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--- abstract: 'We give an example of quasiderivatives constructed by random time change, Girsanov’s Theorem and Levy’s Theorem. As an application, we investigate the smoothness and estimate the derivatives up to second order for the probabilistic solution to the Dirichlet problem for the linear degenerate elliptic partial differential equation of second order, under the assumption of non-degeneracy with respect to the normal to the boundary and an interior condition to control the moments of quasiderivatives, which is weaker than non-degeneracy.' address: '127 Vincent Hall, 206 Church St. SE, Minneapolis, MN 55455' author: - Wei Zhou title: Quasiderivative method for derivative estimates of solutions to degenerate elliptic equations --- Introduction and Background =========================== We consider the Dirichlet problem for the linear degenerate elliptic partial differential equation of second order $$\left\{\begin{array}{rcll} Lu(x)-c(x)u(x)+f(x)&=&0 &\text{in }D\\ u&=&g &\text{on }\partial D, \end{array} \right. \label{1a}$$ where $Lu(x):=a^{ij}(x)u_{x^ix^j}(x)+ b^i(x)u_{x^i}(x)$, with $a=(1/2)\sigma\sigma^*$, and summation convention is understood. The probabilistic solution of (\[1a\]) is known as $$u(x)=E\bigg[g\big(x_{\tau}(x)\big)e^{-\phi_\tau}+\int_0^{\tau}f\big(x_t(x)\big)e^{-\phi_t}dt\bigg], \label{1b}$$ $$\mbox{with }\phi_t=\int_0^tc(x_s(x))ds.$$ where $x_t(x)$ is the solution to the Itô equation $$\label{1aa} x_t=x+\int_0^t\sigma(x_s)dw_s+\int_0^tb(x_s)ds$$ and $\tau=\tau_D(x)$ is the first exit time of $x_t(x)$ from $D$. If we know a priori that $u\in C^2(D)\cap C(\bar D)$ and $u$ solves (\[1a\]), then $u$ satisfies (\[1b\]) via Itô’s formula, which implies the uniqueness of the solution of (\[1a\]) provided the uniqueness of the solution of (\[1aa\]). However, in general, $u$ defined by (\[1b\]) doesn’t necessarily have first and second derivatives in the differential operator $L$, especially when the diffusion term $a$ is degenerate, and the differential equation is understood in a generalized sense. We are interested in knowing under what conditions $u$ defined by (\[1b\]) is twice differentiable and does satisfy (\[1a\]). The accumulation of the research on the existence, uniqueness and regularity of degenerate elliptic or parabolic partial differential equations has become vast. See, for example, Hörmander [@MR0222474], Kohn-Nirenberg [@MR0234118] and Oleinik-Radkevich [@MR0457908], in which analysis techniques for PDEs are used. For probabilistic approaches, we refer to Freidlin [@MR833742] and Stroock-Varadhan [@MR0387812], to name a few. Our approach, quasiderivative method, is also probabilistic. The concept of quasiderivative was first introduced by N. V. Krylov in [@MR965890] (1988), in which this probabilistic technique is applied to find weaker and more flexible conditions on $\sigma$, $b$ and $c$ such that $u$ in (\[1b\]) is twice continuously differentiable in manifolds without boundary. Since then, this technique has been applied to investigate the smoothness of solutions of various elliptic and parabolic partial differential equations. The first derivatives of various linear elliptic and parabolic PDEs have been estimated under various conditions in Krylov [@MR1180379] (1992), [@MR1268004] (1993) and [@MR2144644] (2004), where each case was treated by its particular choice of quasiderivatives. In Krylov [@MR2082053] (2004), a unified quasiderivative method is presented, while $\sigma$ and $b$ are assumed to be constant. As far as the applications to nonlinear equations, for example, in Krylov [@MR992979] (1989), derivative estimates are obtained when controlled diffusion processes and consequently fully nonlinear elliptic equations are considered. Compared to the operators considered in [@MR965890; @MR1180379; @MR1268004; @MR2144644; @MR2082053], the differential equation in this article is more general. The differential operator $L$ in (\[1a\]) is the general linear elliptic differential operator, and $c$ and $f$ are non-trivial. Also, we estimate the derivatives up to the second order, not just the first order. More presicely, our main target is investigating first derivatives of $u$ if we only assume $f, g\in C^{0,1}(\bar D)$, as well as the second derivatives therein when assuming $f, g\in C^{1,1}(\bar D)$. Note that, in these cases, one cannot assert that the first and second derivatives of $u$ are bounded up to the boundary (for example, see Remark 1.0.2 and Example 4.2.1 in [@MR2144644]). One can only expect to prove that inside $D$ the derivatives of $u$ exist. We show that under our assumptions, the first and second derivatives of $u$ in (\[1b\]) exist almost everywhere in $D$, which implies the existence and uniqueness for the Dirichelet problem for the associated linear degenerate elliptic partial differential equation (\[1a\]) in our setting. We also obtain first and second derivative estimates. This article is organized as follows: In Section 2, we review the concept of quasiderivative and give an example of it. In Section 3, we take this approach to show the existence of, and then estimate, the first and second derivatives of $u$ in (\[1b\]), under the assumption of the non-degeneracy of $a$ with respect to the normal to the boundary and an interior condition to control the moments of quasiderivatives, which is weaker than the nondegeneracy of the diffusion term $a$ and necessary under the aforementioned assumption. To conclude this section, we introduce the notation: Above we have already defined $C^k(\bar{D}), k=1 \text{\ or\ } 2$, as the space of bounded continuous and $k$-times continuously differentiable functions in $\bar{D}$ with finite norm given by $$|g|_{1,D}=|g|_{0,D}+|g_x|_{0,D},\ \ |g|_{2,D}=|g|_{1,D}+|g_{xx}|_{0,D},$$ respectively, where $$|g|_{0,D}=\sup_{x\in D}|g(x)|,$$ $g_x$ is the gradient vector of $g$, and $g_{xx}$ is the Hessian matrix of $g$. For $\alpha\in(0,1]$, the Hölder spaces $C^{k,\alpha}(\bar D)$ are defined as the subspaces of $C^k(\bar D)$ consisting of functions with finite norm $$|g|_{k,\alpha,D}=|g|_{k,D}+[g]_{\alpha,D},\ \ \mbox{ where }[g]_{\alpha, D}=\sup_{x,y\in D}\frac{|g(x)-g(y)|}{|x-y|^\alpha}.$$ Throughout the article, the summation convention for repeated indices is assumed, and we always put the index in the superscript, since the subscript is for the time variable of stochastic processes. We let ${\mathbb{R}^d}$ is the $d$-dimensional Euclidean space with $x = (x^1, x^2, . . . , x^d )$ representing a typical point in ${\mathbb{R}^d}$, and $(x, y) = x^iy^i$ is the inner product for $x, y \in {\mathbb{R}^d}$. For $x,y,z\in{\mathbb{R}^d}$, set $$\begin{aligned} u_{(y)}=&u_{x^i}y^i,\ \ u_{(y)(z)}=u_{x^i x^j}y^i z^j,\ \ u_{(y)}^2=(u_{(y)})^2.\end{aligned}$$ For any matrix $\sigma=(\sigma^{ij})$, $$\|\sigma\|^2:=\mathrm{tr}(\sigma\sigma^*).$$ For any $s,t\in\mathbb R$, we define $$s\wedge t=\min(s,t),\ \ s\vee t=\max(s,t).$$ Constants $K, N$ and $\lambda$ appearing in inequalities are usually not indexed. They may differ even in the same chain of inequalities. Definition and Examples of Quasiderivative ========================================== In what follows, we consider the Itô stochastic equation $$x_t=x+\int_0^t \sigma^i(x_s)dw_s^i+\int_0^t b(x_s)ds \label{2a}$$ on a given complete probablity space $(\Omega,\mathcal{F},P)$, where $x\in{\mathbb{R}^d}$, $\sigma^i$ and $b$ are (nonrandom) ${\mathbb{R}^d}$-valued functions with bounded domain $D$ in ${\mathbb{R}^d}$, defined for $i=1,..., d_1$ with $d_1$ possibly different from $d$, and $w_t:=(w_t^1,..., w_t^{d_1})$ is a $d_1$-dimensional Wiener process with respect to a given increasing filtration $\{\mathcal{F}_t, t\ge 0\}$ of $\sigma$-algebras $\mathcal{F}_t\subset\mathcal{F}$, such that $\mathcal{F}_t$ contain all $P$-null sets. We denote by $\sigma$ the $d\times d_1$ matrix composed of the column-vectors $\sigma^i$, $i=1,..., d_1$. We also assume that $\sigma$ and $b$ are twice continuously differentiable in ${\mathbb{R}^d}$. Based on the assumptions above, for any $x\in D$, it is known that equation (\[2a\]) has a unique solution $x_t(x)$ on $[0,\tau(x))$, where $$\tau(x)=\inf\{t\ge0 : x_t(x)\notin D\} \qquad (\inf\{\varnothing\}:=\infty).$$ We write $$u\in\mathcal{M}^k(D,\sigma,b)$$ if $u$ is a real-valued $k$ times continuously differentiable function given on $\bar{D}$ such that the process $u(x_t(x))$ is a local $\{\mathcal{F}_t\}$-martingale on $[0,\tau(x))$ for any $x\in D$. We abbreviate $\mathcal{M}^k(D,\sigma,b)$ by $\mathcal{M}^k(D)$, or simply $\mathcal{M}^k$ when this will cause no confusion. \[2b\] Let $x\in D$, and let $\gamma$ be a stopping time, such that $\gamma\le\tau(x)$. Assume that $\xi\in\mathbb{R}^d$, $\xi_t$ and $\xi_t^0$ are adapted continuous processes defined on $[0,\gamma]\cap[0,\infty)$ with values in $\mathbb{R}^d$ and $\mathbb{R}$, respectively, such that $\xi_0=\xi$. We say that $\xi_t$ is a of $x_t$ in the direction of $\xi$ at point $x$ on $[0,\gamma]$ if for any $u\in\mathcal{M}^1(D,\sigma,b)$ the following process $$\label{2f} {u_{(\xi_t)}}(x_t(x))+\xi_t^0 u(x_t(x))$$ is a local martingale on $[0,\gamma]$. In this case the process $\xi_t^0$ is called a for $\xi_t$. If $\gamma=\tau(x)$ we simply say that $\xi_t$ is a first quasiderivative of $x_t(y)$ in D in the direction of $\xi$ at $x$. It is worth mentioning that the first adjoint process is not unique for the first quasiderivative in general. All of the first adjoint processes we consider in this article are local martingales with initial value 0. \[2bb\] Under the assumptions of Definition \[2b\], additionally assume that $\eta\in\mathbb{R}^d$, $\eta_t$ and $\eta_t^0$ are adapted continuous processes defined on $[0,\gamma]\cap[0,\infty)$ with values in $\mathbb{R}^d$ and $\mathbb{R}$, respectively, such that $\eta_0=\eta$. We say that $\eta_t$ is a of $x_t$ associated with $\xi_t$ and $\xi_t^0$ in the direction of $\eta$ at point $x$ on $[0,\gamma]$ if for any $u\in\mathcal{M}^2(D,\sigma,b)$ the following process $$\label{2g} {u_{(\xi_t)(\xi_t)}}(x_t(x))+ {u_{(\eta_t)}}(x_t(x)) + 2\xi_t^0 {u_{(\xi_t)}}(x_t(x))+\eta_t^0 u(x_t(x)),$$ $$\mbox{ where }\xi_t\mbox{ and }\xi_t^0\mbox{ are first quasiderivative and first adjoint process.}$$ is a local martingale on $[0,\tau)$. In this case the process $\eta_t^0$ is called a for $\eta_t$. If $\gamma=\tau(x)$ we simply say that $\eta_t$ is a second quasiderivative of $x_t(y)$ associated with $\xi_t$ in D in the direction of $\eta$ at $x$. Similarly, the second adjoint process is not unique for the second quasiderivative in general. All second adjoint processes we consider in this article are local martingales with initial value 0. Now let us consider $$u(x)=Eg\big(x_\tau(x)\big),$$ that is, we temporarily let $f=c=0$ in (\[1b\]). Based on the definitions above, if $u\in C^2(\bar{D})$, then the strong Markov property of $x_t(x)$ implies that $u\in\mathcal{M}^2(D)$, and the usual first and second “derivatives” with respect to $x$ of the process $x_t(x)$, which are defined as the solutions of the Itô equations $$\xi_t=\xi+\int_0^t{\sigma_{(\xi_s)}}^k(x_s)dw_s^k+\int_0^t{b_{(\xi_s)}}(x_s)ds$$ $$\eta_t=\eta+\int_0^t\Big[{\sigma_{(\xi_s)(\xi_s)}}^k(x_s)+{\sigma_{(\eta_s)}}^k(x_s)\Big]dw_s^k+\int_0^t\Big[{b_{(\xi_s)(\xi_s)}}(x_s)+{b_{(\eta_s)}}(x_s)\Big]ds$$ are first and second quasiderivatives with zero adjoint processes. This means, the “quasiderivative” of a given stochastic process is a generalization of the usual “derivative” of the stochastic process. Now we additionally assume that the domain $D$ is of class $C^2$ with $\partial D$ bounded, $\tau(x)<\infty$ (a.s.), and $g$ is twice continuously differentiable on $\partial D$. We abbreviate $\tau(x)$ to $\tau$. If the process (\[2f\]) is a uniformly integrable martingale on $[0,\tau]$ and $\xi_{\tau}$ is tangent to $\partial D$ at $x_{\tau}(x)$ (a.s.), then we have $$\label{uxi} u_{(\xi)}(x)=E[u_{(\xi_\tau)}(x_\tau)+\xi_\tau^0u(x_\tau)]=E[g_{(\xi_\tau)}(x_\tau)+\xi_\tau^0g(x_\tau)].$$ This shows how we can apply first quasiderivatives to get interior estimates of $u_{(\xi)}$ through $|g|_{1,D}$ or $|g|_{1,\partial D}$. As far as second derivatives are concerned, first notice that $$4u_{(\xi)(\zeta)}(x)=u_{(\xi+\zeta)(\xi+\zeta)}(x)-u_{(\xi-\zeta)(\xi-\zeta)}(x).$$ So to estimate $u_{(\xi)(\zeta)}(x), \forall\xi,\zeta\in{\mathbb{R}^d}$, it suffices to estimate $u_{(\xi)(\xi)}(x), \forall\xi\in{\mathbb{R}^d}$. Again, if the process (\[2g\]) is a uniformly integrable martingale on $[0,\tau]$, $\xi_{\tau}$ and $\eta_{\tau}$ are tangent to $\partial D$ at $x_{\tau}(x)$ (a.s.), then by letting $\eta=0$, we have $$\begin{aligned} u_{(\xi)(\xi)}(x)=&u_{(\xi)(\xi)}(x)+u_{(\eta)}(x)\\ =&E[u_{(\xi_\tau)(\xi_\tau)}(x_\tau)+ u_{(\eta_\tau)}(x_\tau)+2\xi_\tau^0 u_{(\xi_\tau)}(x_\tau)+\eta_\tau^0 u(x_\tau)]\\ =&E[g_{(\xi_\tau)(\xi_\tau)}(x_\tau)+u_{(n(x_\tau))}(x_\tau)\cdot h_{(\xi_\tau)(\xi_\tau)}(x_\tau)+ g_{(\eta_\tau)}(x_\tau)\\ &+2\xi_\tau^0 g_{(\xi_\tau)}(x_\tau)+\eta_\tau^0 g(x_\tau)],\end{aligned}$$ where $n(x)$ is the unit inward normal at $x\in\partial D$ and $h(x) : T_x(\partial D)\rightarrow\mathbb{R}$ is a local representation of $\partial D$ as a graph over tangent space of $\partial D$ at $x$. (Notice that it is different from the first order case that generally $u_{(\xi_\tau)(\xi_\tau)}(x_\tau)\ne g_{(\xi_\tau)(\xi_\tau)}(x_\tau)$.) Since $D$ is of class $C^2$ and $\partial D$ is bounded, $$h_{(\xi_\tau)(\xi_\tau)}(x_\tau)\le N|\xi_\tau|^2,$$ where $N$ is a positive constant depending on the domain $D$. This shows how we can apply second quasiderivatives to get interior estimates of $u_{(\xi)(\zeta)}$ through $|g|_{2,D}$, or even $|g|_{2,\partial D}$, provided that $u_{(n(y))}(y)$ can be estimated on $\partial D$ in terms of $|g|_{2,D}$ or $|g|_{2,\partial D}$. It is also worth mentioning that $\eta_{\tau}$ need not be tangent to $\partial D$ at $x_{\tau}(x)$, provided that we can control the moments of $\eta_{t\wedge\tau}$ and estimate the normal derivative of $u$, because we can represent $\eta_{\tau}$ as the sum of the tangential component and the normal component. The discussion above motivates us on attempting to construct as many quasiderivatives as possible. \[2c\] Let $r_t, \hat{r}_t,\pi_t, \hat{\pi}_t, P_t,\hat{P}_t$ be jointly measurable adapted processes with values in $\mathbb{R}$, $\mathbb{R}$, $\mathbb{R}^{d_1}$, $\mathbb{R}^{d_1}$, $\mathrm{Skew}(d_1,\mathbb{R})$, $\mathrm{Skew}(d_1,\mathbb{R})$, respectively, where $\mathrm{Skew}(d_1,\mathbb{R})$ denotes the set of $d_1\times d_1$ skew-symmetric real matrices. Assume that $$\int_0^T(|r_t|^4+|\hat{r}_t|^2+|\pi_t|^4+|\hat{\pi}_t|^2+|P_t|^4+|\hat{P}_t|^2)dt<\infty$$ for any $T\in [0,\infty)$. For $x\in D$, $\xi\in{\mathbb{R}^d}$ and $\eta\in{\mathbb{R}^d}$, on the time interval $[0,\infty)$, define the processes $\xi_t$ and $\eta_t$ as solutions of the following (linear) equations: $$\begin{aligned} \xi_t=\xi + \int_0^t\Big[&{\sigma_{(\xi_s)}}+r_s\sigma+\sigma P_s\Big]dw_s\label{2d}+ \int_0^t\Big[{b_{(\xi_s)}}+2r_sb-\sigma \pi_s\Big]ds,\end{aligned}$$ $$\label{2e} \begin{gathered} \eta_t=\eta + \int_0^t\Big[{\sigma_{(\eta_s)}}+\hat{r}_s\sigma+\sigma\hat{P}_s +{\sigma_{(\xi_s)(\xi_s)}}+ 2r_s{\sigma_{(\xi_s)}}\\ \qquad\qquad\ \ \ \ +2{\sigma_{(\xi_s)}}P_s +2r_s\sigma P_s-r_s^2\sigma+\sigma P_s^2\Big]dw_s\\ \qquad\ \ + \int_0^t\Big[{b_{(\eta_s)}}+2\hat{r}_s b-\sigma\hat{\pi}_s+{b_{(\xi_s)(\xi_s)}}+4r_s{b_{(\xi_s)}}\\ \qquad\ \ \ -2{\sigma_{(\xi_s)}}\pi_s-2r_s\sigma\pi_s-2\sigma P_s \pi_s\Big]ds, \end{gathered}$$ where in $\sigma, b$ and their derivatives we dropped the argument $x_s(x)$. Also define: $$\begin{aligned} \xi^0_t=&\int_0^t\pi_sdw_s, \label{2j}\\ \eta^0_t=&(\xi_t^0)^2-\langle\xi^0\rangle_t+\int_0^t\hat{\pi}_sdw_s. \label{2k}\end{aligned}$$ Then $\xi_t$ is a first quasiderivative of $x_t(y)$ in $D$ in the direction of $\xi$ at $x$ and $\xi_t^0$ is a first adjoint process for $\xi_t$, and $\eta_t$ is a second quasiderivative of $x_t(y)$ associated with $\xi_t$ in $D$ in the direction of $\eta$ at $x$ and $\eta_t^0$ is a second adjoint process for $\eta_t$. The processes $r_t$ and $\hat r_t$ come from random time change. The processes $\pi_t$ and $\hat\pi_t$ are due to Girsanov’s Theorem on changing the probability space, and the processes $P_t$ and $\hat P_t$ are based on changing the Wiener process based on Levy’s Theorem. Equations (\[2d\]) and (\[2e\]) give the most general forms of the first and second quasiderivatives known so far. On one hand, they contain various auxiliary processes, $r_t, \pi_t, P_t$, $\hat{r}_t, \hat{\pi}_t, \hat{P}_t,$ which supply us fruitful quasiderivatives for our applications. On the other hand, in specific applications, many of the auxiliary processes are defined to be zero (processes), which make the equations (\[2d\]) and (\[2e\]) shorter. Mimic the proof of Theorem 3.2.1 in [@MR2144644] by replacing $y_t({\varepsilon},x)$ as the solution to the Itô equation $$\begin{aligned} \nonumber dy_t=&\sqrt{1+2{\varepsilon}r_t+{\varepsilon}^2 \hat{r}_t}\sigma(y_t)e^{{\varepsilon}P_t}e^{\frac{1}{2} {\varepsilon}^2\hat{P}_t}dw_t +\Big[(1+2{\varepsilon}r_t+{\varepsilon}^2 \hat{r}_t)b(y_t)\\ &- \sqrt{1+2{\varepsilon}r_t+{\varepsilon}^2 \hat{r}_t}\sigma(y_t)e^{{\varepsilon}P_t}e^{\frac{1}{2} {\varepsilon}^2 \hat{P}_t} ({\varepsilon}\pi_t+\frac{1}{2}{\varepsilon}^2\hat{\pi}_t) \Big]dt\end{aligned}$$ with initial condition $y=x+{\varepsilon}\xi+\frac{1}{2}{\varepsilon}^2\eta$, and then differentiating the local martingale $$u(y_t({\varepsilon}, x))\exp\big(\int_0^t({\varepsilon}\pi_s+\frac{1}{2}{\varepsilon}^2\hat{\pi}_s)dw_s-\frac{1}{2}\int_0^t|{\varepsilon}\pi_s+\frac{1}{2}{\varepsilon}^2\hat{\pi}_s|^2ds\big)$$ twice which turns out to be a local martingale also. The auxiliary processes $r_t, \pi_t, P_t, \hat{r}_t, \hat{\pi}_t, \hat{P}_t$ are allowed to depend on $\xi_t$ and $\eta_t$. For instance, assume that $r(x,\xi), \pi(x,\xi)$ and $P(x,\xi)$ are locally bounded functions from $D\times{\mathbb{R}^d}$ to $\mathbb R$, $\mathbb R^{d_1}$ and $\mathrm{Skew}(d_1,\mathbb R)$, respectly, and they are linear with respect to $\xi$. We similarly assume that $\hat r(x,\xi,\eta), \hat\pi(x,\xi,\eta)$ and $\hat P(x,\xi,\eta)$ are locally bounded functions from $D\times{\mathbb{R}^d}\times{\mathbb{R}^d}$ to $\mathbb R$, $\mathbb R^{d_1}$ and $\mathrm{Skew}(d_1,\mathbb R)$, respectly, and they are linear with respect to $\eta$. If we define $$r_t=r(x_t,\xi_t),\qquad \pi_t=\pi(x,\xi), \qquad P_t=P(x_t,\xi_t),$$ $$\hat r_t=r(x_t,\xi_t,\eta_t),\qquad \hat\pi_t=\hat\pi(x,\xi,\eta), \qquad \hat P_t=\hat P(x_t,\xi_t,\eta_t),$$ then the Itô equations (\[2d\]) and (\[2e\]) have unique solutions, since the diffusion term and drift term in both Itô equations are linear with respect to $\xi_t$ and $\eta_t$, respectively. As a result, Theorem \[2c\] still holds. This is exactly how we construct the quasiderivatives in the next section. Before ending this section, we introduce two local martingales to be used in applications. \[2cc\] Let $c$, $f$, $g$ and $u$ be real-valued twice continuously differentiable functions in $D$. Suppose that $u$ satisfies . Take the processes $r_t, \hat{r}_t,\pi_t$, $\hat{\pi}_t, P_t,\hat{P}_t$, $\xi_t, \eta_t, \xi_t^0, \eta_t^0$ from Theorem \[2c\]. Then for any $x\in D$, the processes $$\label{Xi} X_t:=e^{-\phi_t}\Big[u_{(\xi_t)}(x_t)+\tilde{\xi}_t^0u(x_t)\Big] +\int_0^te^{-\phi_s}\Big[f_{(\xi_s)}(x_s)+\big(2r_s+\tilde{\xi}_s^0\big)f(x_s)\Big]ds,$$ $$\label{Zeta} \begin{gathered} Y_t:=e^{-\phi_t}\Big[u_{(\xi_t)(\xi_t)}(x_t)+u_{(\eta_t)}(x_t)+2\tilde{\xi}_t^0u_{(\xi_t)}(x_t)+\tilde{\eta}_t^0u(x_t)\Big]\\ \qquad+\int_0^te^{-\phi_s}\Big[f_{(\xi_s)(\xi_s)}(x_s)+f_{(\eta_s)}(x_s)+\big(4r_s+2\tilde{\xi}_s^0\big)f_{(\xi_s)}(x_s)\\ +\big(2\hat{r}_s+4\tilde{\xi}_s^0r_s+\tilde{\eta}_s^0\big)f(x_s)\Big]ds, \end{gathered}$$ with $$\begin{aligned} &\phi_t:=\int_0^tc(x_s)ds,\\ &\xi_t^{d+1}:=-\int_0^t\big[c_{(\xi_s)}(x_s)+2r_sc(x_s)\big]ds,\\ &\tilde{\xi}_t^0:=\xi_t^0+\xi_t^{d+1},\\ &\eta_t^{d+1}:=-\int_0^t\big[c_{(\xi_s)(\xi_s)}(x_s)+c_{(\eta_s)}(x_s)+4r_sc_{(\xi_s)}(x_s)+2\hat{r}_sc(x_s)\big]ds,\\ &\tilde{\eta}_t^0:=\eta_t^0+2\xi_t^0\xi_t^{d+1}+(\xi_t^{d+1})^2+\eta_t^{d+1},\end{aligned}$$ are local martingales on $[0,\tau_D(x))$. (We keep writing $x_t$ in place of $x_t(x)$ and drop this argument in many places.) Introduce two additional equations $$x_t^{d+1}=-\int_0^tc(x_s)ds, \ \ x_t^{d+2}=\int_0^t\exp(x_s^{d+1})f(x_s)ds.$$ For $\bar{x}=(x,x^{d+1},x^{d+2})\in D\times\mathbb{R}\times\mathbb{R}$, define $$\bar{u}(\bar{x})=\exp(x^{d+1})u(x)+x^{d+2}.$$ Itô’s formula and the assumption that $a^{ij}(x)u_{x^ix^j}+ b^i(x)u_{x^i}-c(x)u+f(x)=0$ in $D$ imply that $\bar{u}(\bar{x}_t(x,0,0))$ is a local martingale on $[0,\tau_D(x))$. That means, $\bar{u}(\bar{x}_t)\in \mathcal{M}^2$. According to definitions \[2b\] and \[2bb\], $$\bar{u}_{(\bar{\xi}_t)}(\bar{x}_t)+\xi_t^0\bar{u}(\bar{x}_t) \mbox{ and } \bar{u}_{(\bar{\eta}_t)}(\bar{x}_t)+\bar{u}_{(\bar{\xi}_t)(\bar{\xi}_t)}(\bar{x}_t)+2\xi_t^0\bar{u}_{(\bar{\xi}_t)}(\bar{x}_t)+\eta_t^0\bar{u}(\bar{x}_t)$$ are local martingales on $[0,\tau_D(x))$, where $\bar{\xi}_t=(\xi_t,\xi_t^{d+1},\xi_t^{d+2})$ and $\bar{\eta}_t=(\eta_t,\eta_t^{d+1},\eta_t^{d+2})$ are first and second quasiderivatives of $\bar{x}_t((x,0,0))$ in the directions of $\bar\xi=(\xi,0,0)$ and $\bar\eta=(\eta,0,0)$, respectively. Direct computation leads to $$\begin{aligned} &\bar{u}_{(\bar{\xi}_t)}(\bar{x}_t)=\exp(x_t^{d+1})\big[u_{(\xi_t)}(x_t)+\xi_t^{d+1}u(x_t)\big]+\xi_t^{d+2},\\ &\bar{u}_{(\bar{\xi}_t)(\bar{\xi}_t)}(\bar{x}_t)=\exp(x_t^{d+1})\big[u_{(\xi_t)(\xi_t)}(x_t)+2\xi_t^{d+1}u_{(\xi_t)}(x_t)+\big(\xi_t^{d+1}\big)^2u(x_t)\big],\\ &\bar{u}_{(\bar{\eta}_t)}(\bar{x}_t)=\exp(x_t^{d+1})\big[u_{(\eta_t)}(x_t)+\eta_t^{d+1}u(x_t)\big]+\eta_t^{d+2},\end{aligned}$$ with $$\begin{aligned} \xi_t^{d+2}=&\int_0^t\exp(x_s^{d+1})\big[f_{(\xi_s)}(x_s)+(\xi_s^{d+1}+2r_s)f(x_s)\big]ds,\\ \eta_t^{d+2}=&\int_0^t\exp(x_s^{d+1})\big[f_{(\xi_s)(\xi_s)}(x_s)+f_{(\eta_s)}(x_s)+(2\xi_s^{d+1}+4r_s)f_{(\xi_s)}(x_s)\\ &+\big((\xi_s^{d+1})^2+\eta_s^{d+1}+4r_s\xi_s^{d+1}+2\hat{r}_s\big)f(x_s)\big]ds.\end{aligned}$$ It remains to notice that $\xi^0_t$ and $\eta^0_t$ are local martingales, so by Lemma II.8.5(c) in [@MR1311478] $$\xi^0_tx^{d+2}_t-\int_0^t\xi^0_sdx^{d+2}_s, \xi^0_t\xi^{d+2}_t-\int_0^t\xi^0_sd\xi^{d+2}_s \mbox{ and }\eta^0_tx^{d+2}_t-\int_0^t\eta^0_sdx^{d+2}_s$$ are local martingales. Application of quasiderivatives to derivative estimates of non-homogeneous linear degenerate elliptic equations =============================================================================================================== In this section, we investigate the smoothness of $u$ given by (\[1b\]), which is the probabilistic solution of (\[1a\]). To be precise, let $\sigma$, $b$ and $c$ in (\[1b\]) and (\[1aa\]) be twice continuously differentiable in ${\mathbb{R}^d}$, and $c$ be non-negative. Let $D\in C^4$ be a bounded domain in ${\mathbb{R}^d}$, then there exists a function $\psi\in C^4$ satisfying $$\psi>0 \mbox{ in }D,\ \ \psi=0\mbox{ and } |\psi_x|\ge1 \mbox{ on }\partial D.$$ We also assume that $$\label{psi} L\psi:=a^{ij}(x)\psi_{x^ix^j}+b^i(x)\psi_{x^i}\le-1 \mbox{ in }D.$$ $$\label{3setting} |\sigma^{ij}|_{2,D}+|b^i|_{2,D}+|c|_{2,D}+|\psi|_{4,D}\le K_0,$$ with constant $K_0\in[1,\infty)$. Let $\mathfrak{B}$ be the set of all skew-symmetric $d_1\times d_1$ matrices. For any positive constant $\lambda$, define $$D_\lambda=\{x\in D: \psi(x)>\lambda\}.$$ \[nd\] (non-degeneracy along the normal to the boundary) $$\label{nondeg} (an,n)>0 \mbox{ on }\partial D,$$ where $n$ is the unit normal vector. \[ic\] (interior condition to control the moments of quasiderivatives, weaker than the non-degeneracy) There exist functions - $\rho(x): D\rightarrow{\mathbb{R}^d}$, bounded in $D_\lambda$ for all $\lambda>0$; - $Q(x,y): D\times{\mathbb{R}^d}\rightarrow\mathfrak{B}$, bounded with respect to $x$ in $D_\lambda$ for all $\lambda>0, y\in{\mathbb{R}^d}$ and linear in $y$; - $M(x): D\rightarrow \mathbb R$, bounded in $D_\lambda$ for all $\lambda>0$; such that for any $x\in D$ and $|y|=1$, $$\label{inequality} \begin{gathered} \big\|\sigma_{(y)}(x)+(\rho(x),y)\sigma(x)+\sigma(x)Q(x,y)\big\|^2+\\ \ 2\big(y,b_{(y)}(x)+2(\rho(x),y)b(x)\big)\le c(x)+M(x)\big(a(x)y,y\big). \end{gathered}$$ Our main result is the following: \[3c\] Define $u$ by , in which $x_t(x)$ is the solution of (\[1aa\]). Suppose that Assumption \[nd\] and Assumption \[ic\] are satisfied. 1. If $f, g\in C^{0,1}(\bar{D})$, then $u\in C^{0,1}_{loc}(D)$, and for any $\xi\in{\mathbb{R}^d}$, $$\big|u_{(\xi)}\big|\le N\bigg(|\xi|+\frac{|\psi_{(\xi)}|}{\psi^{\frac{1}{2}}}\bigg)\big(|f|_{0,1,D}+|g|_{0,1,D}\big)\ a.e.\mbox{ in }D,\label{3d}$$ where $N=N(K_0,d,d_1, D)$. 2. If $f, g\in C^{1,1}(\bar{D})$, then $u\in C^{1,1}_{loc}(D)$, and for any $\xi\in{\mathbb{R}^d}$, $$\big|u_{(\xi)(\xi)}\big|\le N\bigg(|\xi|^2+\frac{\psi_{(\xi)}^2}{\psi}\bigg)\big(|f|_{1,1,D}+|g|_{1,1,D}\big)\ a.e.\mbox{ in }D\label{3dd},$$ where $N=N(K_0,d,d_1, D)$. Furthermore, $u$ is the unique solution in $C_{loc}^{1,1}(D)\cap C^{0,1}(\bar D)$ of $$\label{solva} \left\{\begin{array}{rcll} Lu(x)-c(x)u(x)+f(x)&=&0 &\text{a.e. in }D\\ u&=&g &\text{on }\partial D. \end{array} \right.$$ The author doesn’t know whether the estimates (\[3d\]) and (\[3dd\]) are sharp. We give two examples to show that Assumption \[ic\] is necessary under Assumption \[nd\] and how to take advantage of the parameters $\rho, Q, M$ in (\[inequality\]), respectively. They are similar to Remark V.8.6 and Example VI.1.7 in [@MR1311478]. See Example V.8.3, Remark V.8.6, Example VI.1.2 and Example VI.1.7 in [@MR1311478] for more details. In the first example, we take $d=d_1=1$ and $D=(-2,2)$. Let $\sigma(x)=x, b(x)=\beta x$ in $[-2,2]$ and $c(x)=\nu, f(x)=0$ in $[-1,1]$, where $\nu>0, \beta\in\mathbb R$ are constants. Extend $c(x)$ and $f(x)$ outside $[-1,1]$ in such a way that $c(x)\ge\nu, f(x)>0$, and $c$ and $f$ are smooth on $[-2,2]$, bounded and have bounded derivatives up to second order. Let $g(x)=0$ on $\partial D=\{-2,2\}$. Define $$\tau_1(x)=\inf\{t\ge0:|x_t(x)|\ge1\}, \qquad \tau_2(x)=\inf\{t\ge0:|x_t(x)|\ge2\}.$$ Based on our construction, for all $t\in[0,\tau_2(x)]$ (a.s.), $$x_t(x)=xe^{w_t+(\beta-1/2)t}.$$ It follows that for any $x\in(0,1]$, $x_t(x)$ takes the value 1 at time $\tau_1(x)$ almost surely. Similarly, for any $x\in[-1,0)$, $x_{\tau_1(x)}(x)=-1$ (a.s.). Also, note that $$Ee^{-\nu\tau_1(x)}=x^\kappa, \mbox{ with }\kappa=[(\beta-1/2)^2+2\nu]^{1/2}-\beta+1/2.$$ Hence $$u(x)=\left\{\begin{array}{ll} Ee^{-\nu\tau_1(x)}u(x_{\tau_1(x)}(x))=u(1)x^\kappa&\mbox{ if }x\in(0,1],\\ Ee^{-\nu\tau_1(x)}u(x_{\tau_1(x)}(x))=u(-1)|x|^\kappa&\mbox{ if }x\in[-1,0),\\ 0&\mbox{ if }x=0. \end{array} \right.$$ Notice that $u(1)>0, u(-1)>0$, so $u(x)$ has Lipschitz continuous derivatives if and only if $\kappa\ge2$. It is equivalent to $1+2\beta\le\nu$, which is exactly (\[inequality\]) in which $\rho,Q,M$ are vanishing. This example shows that Assumption \[ic\] is necessary. Next, we discuss an advantage of the parameters $\rho, Q, M$ in (\[inequality\]). More precisely, we show that with the help of these parameters, based on some local information, Assumption \[ic\] holds. Assume that $d=d_1=1$ for the sake of simplicity, and for each $x\in D$ where $\sigma(x)=b(x)=0$, we have $$\label{bds} |\sigma'(x)|^2+2b'(x)< c(x).$$ With this local property, we claim that Assumption \[ic\] hold. Indeed, we observe that for $$\rho(x)=-nb(x),\qquad Q(x,y)=nb(x)y,\qquad M(x)=n,$$ the inequality (\[inequality\]) becomes $$\label{conv} |\sigma'(x)|^2+2b'(x)\le c(x)+n\sigma^2(x)+4nb^2(x).$$ Suppose that there exists $D_\lambda$, for any $n\in\{1,2,...\}$, there exists a point $x_n$ at which the inequality converse to (\[conv\]) holds. Then we can exact from the sequence $(\sigma(x_n),\sigma'(x_n),$ $ b(x_n), b'(x_n), c(x_n))$ a subsequence that converges to $(\sigma(x_0),\sigma'(x_0),$ $ b(x_0), b'(x_0), c(x_0))$ for some $x_0\in\bar D_\lambda$. It follows from (\[3setting\]) that $$n\sigma^2(x_n)+4nb^2(x_n)<|\sigma'(x_n)|^2+2b'(x_n)\le K_0, \forall n.$$ Therefore, $\sigma(x_0)=b(x_0)=0$ and $$|\sigma'(x_0)|^2+2b'(x_0)\ge c(x_0)$$ It is a contradiction to (\[bds\]), so for any $\lambda$, there exists $n_\lambda$, such that the inequality (\[conv\]) holds in $D_\lambda$ for $n_\lambda$. As a consequence, Assumption \[ic\] is indeed satisfied. The following two remarks are reductions of Theorem \[3c\]. \[reduction1\] Without loss of generality, we may assume that $c\ge1$ and replace condition (\[inequality\]) by $$\label{ineq} \begin{gathered} \big\|\sigma_{(y)}(x)+(\rho(x),y)\sigma(x)+\sigma(x)Q(x,y)\big\|^2+\\ \ 2\big(y,b_{(y)}(x)+2(\rho(x),y)b(x)\big)\le c(x)-1+M(x)\big(a(x)y,y\big). \end{gathered}$$ Indeed, letting $\displaystyle\tilde{u}=\frac{u}{\psi+1}$ in $D$, we have $$u_{x^i}=(\psi+1)\tilde{u}_{x^i}+\psi_{x^i}\tilde{u},\ \ u_{x^ix^j}=(\psi+1)\tilde{u}_{x^ix^j}+\psi_{x^j}\tilde{u}_{x^i}+\psi_{x^i}\tilde{u}_{x^j}+\psi_{x^ix^j}\tilde{u}$$ Hence (\[1a\]) turns into $$\left\{\begin{array}{rcll} \displaystyle\tilde a^{ij}(x)\tilde{u}_{x^ix^j}+\tilde b^i(x)\tilde{u}_{x^i}-\tilde{c}(x)\tilde{u}+f(x)&=&0,&\mbox{ in }D\\ \displaystyle\tilde{u}&=&\tilde g:=g/(1+\psi), &\mbox{ on }\partial D \end{array} \right.$$ with $$\tilde{a}^{ij}=(\psi+1)a^{ij},\ \ \tilde{b}^i=2a^{ij}\psi_{x^j}+(\psi+1)b^i,\ \ \tilde{c}=-L\psi+(1+\psi)c.$$ Notice that $\tilde{\sigma}^{ij}=\sqrt{\psi+1}\sigma^{ij}$. So a direct computation implies that $$|\tilde\sigma^{ij}|_{2,D}+|\tilde b^i|_{2,D}+|\tilde c|_{2,D}+|\psi|_{4,D}\le (d^2+2d+2)K_0^3,$$ which plays the same role as (\[3setting\]). Since $L\psi\le-1$ and $c\ge0$, $\tilde c\ge1$. We also have $(\tilde{a}n,n)>0$ on $\partial D$. Under the substitutions on $\sigma$, $b$ and $c$, by inequality (\[inequality\]), we have $$\begin{gathered} \frac{1}{\psi+1}\Big\|\tilde\sigma_{(y)}(x)-\frac{1}{2}\frac{\psi_{(y)}}{\psi+1}\tilde\sigma(x)+\big(\rho(x),y\big)\tilde\sigma(x)+\tilde\sigma(x)Q(x,y)\Big\|^2\\ +\frac{2}{\psi+1}\Big(y,\tilde{b}_{(y)}(x)-\frac{\psi_{(y)}}{\psi+1}\tilde{b}(x)+2\big(\rho(x),y\big)\tilde{b}(x)\Big)\\ \le \frac{\tilde{c}(x)+L\psi}{\psi+1}+M(x)\big(a(x)y,y\big)+2\bigg(y,\Big(\frac{2a\psi_x}{\psi+1}\Big)_{(y)}+2\big(\rho(x),y\big)\frac{2a\psi_x}{\psi+1}\bigg). \end{gathered}$$ Collecting similar terms and noticing that $L\psi\le-1$, we get $$\begin{gathered} \Big\|\tilde\sigma_{(y)}(x)+\big(\tilde \rho(x),y\big)\tilde\sigma(x)+\tilde\sigma(x)Q(x,y)\Big\|^2+2\Big(y,\tilde b_{(y)}(x)+2\big(\tilde \rho(x),y\big)\tilde b(x)\Big)\\ \le \tilde c(x)-1+\tilde M(x)\big(\tilde a(x)y,y\big)+4\big(\tilde a_{(y)}(x)\psi_x,y\big), \end{gathered}$$ with $$\tilde \rho(x):=\rho(x)-\frac{\psi_x}{2(\psi+1)},$$ and $\tilde M(x)$ is in terms of $M(x),K_0$ and $|\rho(x)|$. The term $4(\tilde a_{(y)}\psi_x,y)$ can not be bounded by $\tilde M(x)(\tilde a(x)y,y)$. However, notice that $$\tilde a_{(y)}(x)=\tilde\sigma(x)\tilde\sigma^*_{(y)}(x).$$ So $\tilde M(x)(\tilde a(x)y,y)+4(\tilde a_{(y)}\psi_x,y)$ can be rewritten in the form of $$\bigg(\tilde \sigma(x)\Big(\frac{\tilde M(x)}{2}\tilde\sigma^*(x)y+4\tilde\sigma^*_{(y)}(x)\psi_x\Big),y\bigg),$$ which can play the same role as that of $M(x)(a(x)y,y)$, which, in the proof, will be rewritten in the form of $$\Big(\sigma(x)\cdot\frac{M(x)}{2}\sigma^*(x)y,y\Big).$$ A direct computation shows that if $\tilde u$ satisfies estimates (\[3d\]) and (\[3dd\]), we have the same estimates for $u$. \[reduction2\] Without loss of generality, we may assume that $u\in C^1({D})$ and $f,g\in C^1(\bar D)$ when investigating first derivatives of $u$, and $u\in C^2({D})$ and $f,g\in C^2(\bar D)$ when investigating second derivatives of $u$. Let us take the first situation for example, in which $u, f, g$ can be assumed to be of class $C^1$. The second situation can be discussed by almost the same argument. We define the process $x_t^{\varepsilon}(x)$ to be the solution to the equation $$x_t=x_0+\int_0^t\sigma(x_s)dw_s+\int_0^t{\varepsilon}I d\tilde w_s+\int_0^tb(x_s)ds$$ where $\tilde w_t$ is a $d$-dimensional Wiener process independent of $w_t$ and $I$ is the identity matrix of size $d\times d$, and we define $\tau^{\varepsilon}(x)$ to be the first exit time of $x_t^{\varepsilon}(x)$ from $D$, then for the function $$u^{\varepsilon}(x):=E\bigg[g\big(x^{\varepsilon}_{\tau^{\varepsilon}(x)}(x)\big)e^{-\phi^{\varepsilon}_{\tau^{\varepsilon}(x)}}+\int_0^{\tau^{\varepsilon}(x)}f\big(x^{\varepsilon}_t(x)\big)e^{-\phi^{\varepsilon}_t}dt\bigg],$$ $$\mbox{with }\phi^{\varepsilon}_t:=\int_0^tc(x_t^{\varepsilon}(x))dt,$$ the relation $u^{\varepsilon}\rightarrow u$ holds as ${\varepsilon}\rightarrow0$. Indeed, notice that $$\begin{aligned} E|g(x^{\varepsilon}_{\tau^{\varepsilon}}(x))-g(x_\tau(x))| \le& KE\Big(|x^{\varepsilon}_{\tau^{\varepsilon}\wedge\tau}(x)-x_{\tau^{\varepsilon}\wedge\tau}(x)|\\ &+(\tau^{\varepsilon}\vee\tau-\tau^{\varepsilon}\wedge\tau)+(\tau^{\varepsilon}\vee\tau-\tau^{\varepsilon}\wedge\tau)^{1/2}\Big),\\ E|e^{-\phi^{\varepsilon}_{\tau^{\varepsilon}}}-e^{-\phi_\tau}|\le& Ee^{-\tau^{\varepsilon}\wedge\tau}|\phi^{\varepsilon}_{\tau^{\varepsilon}}-\phi_\tau|\\ \le& KEe^{-\tau^{\varepsilon}\wedge\tau}\Big(\tau^{\varepsilon}\wedge\tau\cdot\sup_{t\le\tau^{\varepsilon}\wedge\tau}|x_t^{\varepsilon}(x)-x_t(x)|\\ &+(\tau^{\varepsilon}\vee\tau-\tau^{\varepsilon}\wedge\tau)+(\tau^{\varepsilon}\vee\tau-\tau^{\varepsilon}\wedge\tau)^{1/2}\Big)\\ \le&KE\Big(\sup_{t\le\tau^{\varepsilon}\wedge\tau}|x_t^{\varepsilon}(x)-x_t(x)|\\ &+(\tau^{\varepsilon}\vee\tau-\tau^{\varepsilon}\wedge\tau)+(\tau^{\varepsilon}\vee\tau-\tau^{\varepsilon}\wedge\tau)^{1/2}\Big),\end{aligned}$$ and $$\begin{aligned} &E\Big|\int_0^{\tau^{\varepsilon}}f(x^{\varepsilon}_t(x))e^{-\phi_t^{\varepsilon}}dt-\int_0^\tau f(x_t(x))e^{-\phi_t}dt\Big|\\ \le&E\int_0^{\tau^{\varepsilon}\wedge\tau}|f(x^{\varepsilon}_t(x))e^{-\phi_t^{\varepsilon}}-f(x_t(x))e^{-\phi_t}|dt+KE(\tau^{\varepsilon}\vee\tau-\tau^{\varepsilon}\wedge\tau)\\ \le&E\int_0^{\tau^{\varepsilon}\wedge\tau}K\Big(|x_t^{\varepsilon}(x)-x_t(x)|+t\cdot\sup_{s\le t}|x_s^{\varepsilon}(x)-x_s(x)|\Big)e^{-t}dt\\ &+KE(\tau^{\varepsilon}\vee\tau-\tau^{\varepsilon}\wedge\tau)\\ \le&KE\Big(\sup_{t\le\tau^{\varepsilon}\wedge\tau}|x_t^{\varepsilon}(x)-x_t(x)|+(\tau^{\varepsilon}\vee\tau-\tau^{\varepsilon}\wedge\tau)\Big),\end{aligned}$$ where $K$ is a constant depending on $|g|_{0,1,D}, |f|_{0,1,D}$ and $K_0$. It follows that $$\begin{aligned} |u^{\varepsilon}(x)-u(x)|\le &KE\Big(\sup_{t\le\tau^{\varepsilon}\wedge\tau}|x_t^{\varepsilon}(x)-x_t(x)|\\ &+(\tau^{\varepsilon}\vee\tau-\tau^{\varepsilon}\wedge\tau)+(\tau^{\varepsilon}\vee\tau-\tau^{\varepsilon}\wedge\tau)^{1/2}\Big)\\ \le &K\Big(E\sup_{t\le\tau^{\varepsilon}\wedge\tau\wedge T}|x_t^{\varepsilon}(x)-x_t(x)|+KP(\tau>T)\\ &+EI_1+EI_2+\sqrt{EI_1}+\sqrt{EI_2}\Big),\end{aligned}$$ where $$I_1=(\tau^{\varepsilon}\vee\tau-\tau^{\varepsilon}\wedge\tau)I_{\tau>\tau^{\varepsilon}}=(\tau-\tau^{\varepsilon})I_{\tau>\tau^{\varepsilon}},$$ $$I_2=(\tau^{\varepsilon}\vee\tau-\tau^{\varepsilon}\wedge\tau)I_{\tau<\tau^{\varepsilon}}=(\tau^{\varepsilon}-\tau)I_{\tau<\tau^{\varepsilon}}.$$ It remains to notice that $$E\sup_{t\le \tau^{\varepsilon}\wedge\tau\wedge T}|x_t(x)-x^{\varepsilon}_t(x)|\le e^{KT}{\varepsilon}\rightarrow0, \mbox{ as }{\varepsilon}\rightarrow 0,$$ $$P(\tau>T)\le\frac{E\tau}{T}\le \frac{1}{T}E\int_0^\tau\Big(-L\psi\big(x_t(x)\big)\Big)dt=\frac{\psi(x)-\psi(x_\tau(x))}{T}\le\frac{K_0}{T},$$ and $$\begin{aligned} E(\tau-\tau^{\varepsilon})I_{\tau>\tau^{\varepsilon}}=&E\int_{\tau\wedge\tau^{\varepsilon}}^{\tau}1dt\\ \le&-E\int_{\tau\wedge\tau^{\varepsilon}}^{\tau}L\psi(x_t(x))dt\\ =&-E\Big(\psi\big(x_{\tau}(x)\big)-\psi\big(x_{\tau^{\varepsilon}}(x)\big)\Big)I_{\tau^{\varepsilon}<\tau}\\ =&E\psi\big(x_{\tau^{\varepsilon}(x)}(x)\big) I_{\tau^{\varepsilon}<\tau}\\ =&E\Big(\psi\big(x_{\tau^{\varepsilon}}(x)\big)-\psi\big(x^{\varepsilon}_{\tau^{\varepsilon}}(x)\big)\Big) I_{\tau^{\varepsilon}<\tau}\\ \le&E\Big(\psi\big(x_{\tau^{\varepsilon}}(x)\big)-\psi\big(x^{\varepsilon}_{\tau^{\varepsilon}}(x)\big)\Big) I_{\tau^{\varepsilon}<\tau\le T}+2K_0P(\tau>T)\\ \le&K_0E\sup_{t\le \tau^{\varepsilon}\wedge\tau\wedge T}|x_t(x)-x^{\varepsilon}_t(x)|+\frac{2K_0^2}{T}\\ E(\tau^{\varepsilon}-\tau)I_{\tau<\tau^{\varepsilon}}\le&-2E\int_{\tau\wedge\tau^{\varepsilon}}^{\tau^{\varepsilon}}L^{\varepsilon}\psi(x^{\varepsilon}_t(x))dt\\ \le&\cdots \le K_0E\sup_{t\le \tau^{\varepsilon}\wedge\tau\wedge T}|x_t(x)-x^{\varepsilon}_t(x)|+\frac{2K_0^2}{T}.\end{aligned}$$ Hence by first letting ${\varepsilon}\downarrow 0$ and then $T\uparrow\infty$, we conclude that $$|u^{\varepsilon}(x)-u(x)|\rightarrow0\mbox{ as }{\varepsilon}\rightarrow 0.$$ Moreover, for small ${\varepsilon}$ the condition (\[psi\]) holds for $2\psi$, taken instead of $\psi$ and $L^{\varepsilon}$ associated to the process $x_t^{\varepsilon}(x)$. The matrix $\sigma^{\varepsilon}$ corresponding to the process $x_t^{\varepsilon}(x)$ is obtained by attaching the identity matrix, multiplied by ${\varepsilon}$, to the right of the original matrix $\sigma$. In this connection we modify $P(x,y)$ by adding zero entries on the right and below to form a $(d_1+d)\times(d_1+d)$ matrix. Then the condition (\[ineq\]) corresponding to the process $x_t^{\varepsilon}(x)$ will differ from the original condition by the fact that the term ${\varepsilon}^2(\rho(x),y)^2d$ appears on the left, and $\frac{1}{2}M(x){\varepsilon}^2$ on the right. From this it is clear that the condition (\[ineq\]) for the process $x_t^{\varepsilon}(x)$ (for all ${\varepsilon}$) also holds when $M(x)$ is replaced by $M(x)+2|\rho(x)|^2d$. Finally, from analysis of PDE, we know that for ${\varepsilon}\ne0$ the nondegenerate elliptic equation $L^{\varepsilon}w=0$ in $D$ with the boundary condition $w=g$ on $\partial D$ has a solution that is continuous in $\bar D$ and twice continuously differentiable in $D$, and $u^{\varepsilon}=w$ in $D$ by Itô’s formula. From this it follows that it suffices to prove the theorem for small ${\varepsilon}\ne0$, the process $x^{\varepsilon}_t(x)$, and a function $u^{\varepsilon}$ that is continuously differentiable in $D$. Of course, we must be sure that the constants $N$ in (\[3d\]) is chosen to be independent of ${\varepsilon}$, which is true as we can see in the proof of the theorem. Observing further that for each fixed ${\varepsilon}\ne0$ the functions $f$ and $g$ can be uniformly approximated in $\bar D$ by infinitely differentiable functions, in such a way that the last factor in (\[3d\]) increases by at most a factor of two when $f$ and $g$ are replaced by the approximating functions, while for the latter the function $w$ (i.e., $u^{\varepsilon}$) has continuous and bounded first derivatives in $\bar D$, we conclude that we may assume $u$ has continuous first derivatives in $D$ and $f,g\in C^1(\bar D)$ when investigating first derivatives of $u$. Before proving the theorem, let us prove four lemmas. In Lemma \[lemma1\] we estimate the first exit time. It is a well-known result, but we still prove for the sake of completeness. Lemma \[lemma4\] concerns the estimate of the first derivative along the normal to the boundary, to be used when estimating the second derivatives. In Lemma \[3l1\] and Lemma \[3l2\], we construct two supermartingales, which will play the roles of barriers near the boundary and in the interior of the domain, respectively. \[lemma1\] Let $\tau_{D_0}(x)$ be the first exit time of $x_t(x)$ from $D_0$, which is a sub-domain of $D$ containing $x$. Then we have $$E\tau_{D_0}(x)\le E\tau_{D}(x)\le\psi(x)\le |\psi|_{0,D},$$ $$E\tau^2_{D_0}(x)\le E\tau^2_{D}(x)\le2|\psi|_{0,D}\psi(x)\le 2|\psi|_{0,D}^2.$$ The fact that $D_0\subset D$ implies $E\tau_{D_0}(x)\le E\tau_{D}(x)$ and $E\tau^2_{D_0}(x)\le E\tau^2_{D}(x)$. Now we abbreviate $\tau_{D}(x)$ by $\tau(x)$, or simply $\tau$ when this will cause no confusion. By (\[psi\]) and Itô’s formula, we have $$\begin{aligned} E\tau&=E\int_0^\tau1dt\le-E\int_0^\tau L\psi dt=\psi(x)-E\psi(x_\tau)=\psi(x),\\ E\tau^2&=2E\int_0^\infty(\tau-t)I_{\tau>t}dt=2E\int_0^\infty I_{\tau>t}E\tau(x_t)dt\\ &\le 2\sup_{y\in D}E\tau(y)\cdot E\int_0^\infty I_{\tau>t}dt=2\sup_{y\in D}E\tau(y)\cdot E\tau\le2|\psi|_{0,D}\psi(x).\end{aligned}$$ \[lemma4\] If $f, g\in C^2(\bar D)$, and $u\in C^1(\bar D)$, then for any $y\in\partial D$ we have $$|u_{(n)}(y)|\le K(|g|_{2,D}+|f|_{0,D})\label{normal},$$ where $n$ is the unit inward normal on $\partial D$ and the constant $K$ depends only on $K_0$. Fix a $y\in\partial D$, and choose ${\varepsilon}_0>0$ so that $y+{\varepsilon}n\in D$ as long as $0<{\varepsilon}\le{\varepsilon}_0$. Also, fix an ${\varepsilon}\in(0,{\varepsilon}_0]$ and let $x:=y+{\varepsilon}n$. By Itô’s formula, $$d(g(x_t)e^{-\phi_t})=e^{-\phi_t}g_{(\sigma^k)}(x_t)dw_t^k+e^{-\phi_t}(Lg(x_t)-c(x_t)g(x_t))dt.$$ Notice that $$E\int_0^\infty\Big(e^{-\phi_t}g_{(\sigma^k)}(x_t)\Big)^2I_{t\le\tau}dt\le N|g|_{1,D}^2E\tau<\infty.$$ The Wald identities hold: $$E\int_0^\tau e^{-\phi_t}g_{(\sigma^k)}(x_t)dw_t^k=0.$$ Thus $$Ee^{-\phi_\tau}g\big(x_\tau(x)\big)=g(x)+E\int_0^\tau e^{-\phi_t}\big(Lg(x_t)-c(x_t)g(x_t)\big)dt.$$ Together with (\[1b\]), we have $$\begin{aligned} u(x)=&g(x)+E\int_0^\tau e^{-\phi_t}(Lg(x_t(x))-c(x_t(x))g(x_t(x)))dt\\ &+E\int_0^{\tau}f\big(x_t(x)\big)e^{-\phi_t}dt\\ \le&g(x)+(|Lg|_{0,D}+|c|_{0,D}|g|_{0,D}+|f|_{0,D})E\tau\\ \le&g(x)+K(|g|_{2,D}+|f|_{0,D})\psi(x).\end{aligned}$$ Notice that $u(y)=g(y)$ and $\psi(y)=0$. So we have $$\frac{u(y+{\varepsilon}n)-u(y)}{{\varepsilon}}\le\frac{g(y+{\varepsilon}n)-g(y)}{{\varepsilon}}+K(|g|_{2,D}+|f|_{0,D})\frac{\psi(y+{\varepsilon}n)-\psi(y)}{{\varepsilon}}.$$ Letting ${\varepsilon}\downarrow0$, we get $$u_{(n)}(y)\le K(|g|_{2,D}+|f|_{0,D}).$$ Replacing $u$ with $-u$ yields the same estimate of $(-u)_{(n)}$ from above, which is an estimate of $u_{(n)}$ from below. Combining the estimates from above and from below leads to ($\ref{normal}$) and proves the lemma. For constants $\delta$ and $\lambda$, such that $0<\delta<\lambda^2<\lambda<1$, define $$\begin{aligned} D^\lambda&=\{x\in D: \psi<\lambda\},\\ D_\delta^\lambda&=\{x\in D: \delta<\psi<\lambda\}.\end{aligned}$$ Considering that the formulas of the quasiderivatives $\xi_t,\eta_t$ and the barrier functions $\mathrm B_1(x,\xi), \mathrm B_2(x,\xi)$ constructed in Lemmas \[3l1\] and \[3l2\] are complicated, and the proofs of Lemmas \[3l1\] and \[3l2\] are long and technical, we first make a remark on the motivation of these constructions. As discussed right after Definition \[2bb\] in section 2, when investigating the first derivative of $u$, the main difficulty comes from the term $Eu_{(\xi_\tau)}(x_\tau)$ in (\[uxi\]), and we should try to construct $\xi_t$ in such a way that $\xi_{\tau}$ is tangent to $\partial D$ at $x_{\tau}(x)$ almost surely. Considering that the diffusion process $x_t$ and domain $D$ are quite general in our setting, it is almost impossible, since there is no way to know when or where $x_t$ exits the domain. Therefore, what we actually try is constructing $\xi_t$ in such a way that either $\xi_{\tau}$ is tangent to $\partial D$ at $x_{\tau}(x)$ almost surely, or $|u_{(\xi_\tau)}(x_\tau)|$ is bounded by a nonnegative local supermartingale $\mathrm B(x_\tau,\xi_\tau)$. If we succeed, we will have $$E|u_{(\xi_\tau)}(x_\tau)| \left\{\ \begin{array}{ll} =E|g_{(\xi_\tau)}(x_\tau)|\le |g|_{1,\partial D}E|\xi_\tau|&\mbox{if $\xi_{\tau(x)}$ is tangent to $\partial D$}\\ \le E\mathrm B(x_\tau,\xi_\tau)\le \mathrm B(x,\xi) &\mbox{if $\xi_{\tau(x)}$ is not tangent to $\partial D$}. \end{array} \right.$$ As we will see in the following two lemmas, $\mathrm B(x,\xi)=\sqrt{\mathrm B_1(x,\xi)}$ near the boundary, while $B(x,\xi)=\sqrt{\mathrm B_2(x,\xi)}$ in the interior of the domain. \[3l1\] Introduce $$\varphi(x)=\lambda^2+\psi(1-\frac{1}{4\lambda}\psi), \ \ \mathrm{B}_1(x,\xi)=\big[\lambda+\sqrt{\psi}(1+\sqrt{\psi})\big]|\xi|^2+K_1\varphi^\frac{3}{2}{\frac{\psi_{(\xi)}^2}{\psi}},$$ where $K_1\in[1,\infty)$ is a constant depending only on $K_0$. In $D^\lambda$, if we construct first and second quasiderivatives by (\[2d\]) and (\[2e\]), in which $$\begin{aligned} &r(x,\xi):=\rho(x,\xi)+\frac{\psi_{(\xi)}}{\psi},\ \ r_t:=r(x_t,\xi_t),\\ &\mbox{ where }\rho(x,\xi):=-\frac{1}{A}\sum_{k=1}^{d_1}{\psi_{(\sigma^k)}}({\psi_{(\sigma^k)}})_{(\xi)},\mbox{ with }A:=\sum_{k=1}^{d_1}{\psi_{(\sigma^k)}}^2;\\ &\hat{r}(x,\xi):={\frac{\psi_{(\xi)}^2}{\psi^2}},\ \ \hat{r}_t:=\hat{r}(x_t,\xi_t);\\ &\pi^k(x,\xi):=\frac{2{\psi_{(\sigma^k)}}\psi_{(\xi)}}{\varphi\psi},\ \ k=1,...,d_1,\ \ \pi_t:=\pi(x_t,\xi_t);\\ &P^{ik}(x,\xi):=\frac{1}{A}\big[{\psi_{(\sigma^k)}}({\psi_{(\sigma^i)}})_{(\xi)}-{\psi_{(\sigma^i)}}({\psi_{(\sigma^k)}})_{(\xi)}\big],\ \ i,k=1,...,d_1,\ \ P_t:=P(x_t,\xi_t); \\ &\hat{\pi}_t^k=\hat{P}_t^{ik}=0,\ \ \forall i,k=1,...d_1,\forall t\in[0,\infty).\end{aligned}$$ Then for sufficiently small $\lambda$, when $x_0\in D_\delta^\lambda$, $\xi_0\in{\mathbb{R}^d}$ and $\eta_0=0$, we have 1. $\mathrm{B}_1(x_t,\xi_t)$ and $\sqrt{\mathrm{B}_1(x_t,\xi_t)}$ are local supermartingales on $[0,\tau_1^\delta]$, where $\tau_1^\delta=\tau_{D_\delta^\lambda}(x_0)$; 2. $\displaystyle{E\int_0^{\tau_1^\delta}|\xi_t|^2+{\frac{\psi_{(\xi_t)}^2}{\psi^2}}dt\le N\mathrm{B}_1(x_0,\xi_0)}$; 3. $\displaystyle{E\sup_{t\le\tau_1^\delta}|\xi_t|^2\le N\mathrm{B}_1(x_0,\xi_0)}$; 4. $\displaystyle{E|\eta_{\tau_1^\delta}|\le E\sup_{t\le\tau_1^\delta}|\eta_t|\le N\mathrm{B}_1(x_0,\xi_0)}$; 5. $\displaystyle{E\Big(\int_0^{\tau_1^\delta}|\eta_t|^2 dt\Big)^{\frac{1}{2}}\le N\mathrm{B}_1(x_0,\xi_0)}$; where $N$ is a constant depending on $K_0$ and $\lambda$. Throughout the proof, keep in mind that the constant $K$ depend only on $K_0$, while the constants $N\in[1,\infty)$ and $\lambda_0\in(0,1)$ depend on $K_0$ and $\lambda$. First, notice that, on $\partial D$, we have $$A=\sum_{k=1}^{d_1}{\psi_{(\sigma^k)}}^2=2(a\psi_x,\psi_x)=2|\psi_x|(an,n)\ge 2\delta,$$ where the constant $\delta>0$, because of the compactness of $\partial D$. Replacing $\psi$ by $\psi/2\delta$ if needed, we may, therefore, assume that $A\ge1$. By Itô’s formula, for $t<\tau_1^\delta$, we have $$d{\psi_{(\xi_t)}}=[({\psi_{(\sigma^i)}})_{(\xi_t)}+ r_t{\psi_{(\sigma^i)}}+{\psi_{(\sigma^k)}}P_t^{ki}]dw_t^i+[(L\psi)_{(\xi_t)}+2 r_tL\psi-{\psi_{(\sigma^i)}}\pi^i_t]dt.$$ A crucial fact about this equation is that owing to our choice of $r$ and $P$ $$({\psi_{(\sigma^i)}})_{(\xi_t)}+ r_t{\psi_{(\sigma^i)}}+{\psi_{(\sigma^k)}}P^{ki}_t=\frac{{\psi_{(\xi_t)}}}{\psi}{\psi_{(\sigma^i)}}.$$ Thus $$d{\psi_{(\xi_t)}}=\frac{{\psi_{(\xi_t)}}}{\psi}{\psi_{(\sigma^i)}}dw_t^i+[(L\psi)_{(\xi_t)}+2 r_tL\psi-{\psi_{(\sigma^i)}}\pi_t^i]dt.\label{4l}$$ Let $$\bar\sigma:=\sigma_{(\xi)}+ r\sigma+\sigma P,\qquad\bar b:=b_{(\xi)}+2 r b.$$ We have $$\label{sb1} \|\bar\sigma\|\le K(|\xi|+\frac{|{\psi_{(\xi)}}|}{\psi}),$$ $$\label{sb2} |\bar b|\le K(|\xi|+\frac{|{\psi_{(\xi)}}|}{\psi}).$$ By Itô’s formula, $$d\mathrm{B}_1(x_t, \xi_t)=\Gamma_1(x_t, \xi_t)dt+\Lambda_1^k(x_t,\xi_t) dw_t^k$$ with $$\begin{aligned} \Gamma_1(x,\xi)=I_1+I_2+...+I_{13}\end{aligned}$$ where $$\begin{aligned} &I_1=\lambda[2(\xi,\bar b)+\|\bar\sigma\|^2]\le\lambda K(|\xi|^2+{\frac{\psi_{(\xi)}^2}{\psi^2}})\le K\lambda^{\frac{5}{2}}\frac{|\xi|^2}{\psi^\frac{3}{2}}+K\varphi^{\frac{1}{2}}{\frac{\psi_{(\xi)}^2}{\psi^2}},\\ &\mbox{here we apply (\ref{sb1}), (\ref{sb2}) and } \lambda\le\varphi^{\frac{1}{2}},\\ &I_2=-\lambda2(\xi,\sigma^k)\pi^k\le \frac{K\lambda|\xi||\psi_{(\xi)}|}{\varphi\psi}\le\frac{\lambda^2|\xi|^2}{32\cdot2^{\frac{3}{2}}\varphi^{\frac{5}{2}}}+\frac{K\varphi^{\frac{1}{2}}\psi_{(\xi)}^2}{\psi^2}\\ &\qquad\le\frac{|\xi|^2}{32\cdot2^{\frac{3}{2}}\varphi^{\frac{3}{2}}}+\frac{K\varphi^{\frac{1}{2}}\psi_{(\xi)}^2}{\psi^2}\le\frac{|\xi|^2}{32\psi^\frac{3}{2}}+K\varphi^\frac{1}{2}{\frac{\psi_{(\xi)}^2}{\psi^2}},\\ &\mbox{here we apply }\lambda^2\le\varphi, \mbox{ and then observe that }\psi\le2\varphi,\\ &I_3=\sqrt{\psi}(1+\sqrt{\psi})2(\xi,\bar b)\le\sqrt{\psi}K|\xi|(|\xi|+\frac{|\psi_{(\xi)}|}{\psi})\le K\lambda\frac{|\xi|^2}{\psi^\frac{3}{2}},\\ &\mbox{ here we apply (\ref{sb2})},\\ &I_4=-\sqrt{\psi}(1+\sqrt{\psi})2(\xi,\sigma^k)\pi^k\le \frac{K\sqrt{\psi}|\xi||\psi_{(\xi)}|}{\varphi\psi}\le\frac{\psi|\xi|^2}{32\cdot2^{\frac{5}{2}}\varphi^{\frac{5}{2}}}+\frac{K\varphi^{\frac{1}{2}}\psi_{(\xi)}^2}{\psi^2}\\ &\qquad\le\frac{|\xi|^2}{32\psi^\frac{3}{2}}+K\varphi^\frac{1}{2}{\frac{\psi_{(\xi)}^2}{\psi^2}},\\ &\mbox{ here we observe that }\psi\le2\varphi,\\ &I_5=\sqrt{\psi}(1+\sqrt{\psi})\|\bar\sigma\|^2\le K\sqrt{\psi}(|\xi|^2+{\frac{\psi_{(\xi)}^2}{\psi^2}})\le K\lambda^2\frac{|\xi|^2}{\psi^\frac{3}{2}}+K\varphi^\frac{1}{2}{\frac{\psi_{(\xi)}^2}{\psi^2}},\\ &\mbox{ here we apply (\ref{sb1})},\\ &I_6=(1+2\sqrt{\psi})|\xi|^2[\frac{L\psi}{2\sqrt{\psi}}-\frac{A}{8\psi^\frac{3}{2}}]\le-\frac{|\xi|^2}{8\psi^\frac{3}{2}},\\ &I_7=\frac{A}{4\psi}|\xi|^2\le K\frac{|\xi|^2}{\psi}\le K\sqrt{\lambda}\frac{|\xi|^2}{\psi^\frac{3}{2}},\\ &I_8=(1+2\sqrt{\psi})\frac{{\psi_{(\sigma^k)}}}{\sqrt{\psi}}(\xi,\bar\sigma^k)\le K\frac{|\xi|}{\sqrt{\psi}}(|\xi|+\frac{|\psi_{(\xi)}|}{\psi})\le K\lambda\frac{|\xi|^2}{\psi^\frac{3}{2}}+\frac{|\xi|^2}{32\psi^\frac{3}{2}}+K\varphi^\frac{1}{2}\frac{\psi_{(\xi)}^2}{\psi^2},\\ &\mbox{ here we apply (\ref{sb1}) and }\psi\le2\varphi,\\ &I_9=K_1\frac{3}{2}\varphi^\frac{1}{2}\big[(1-\frac{\psi}{2\lambda})L\psi-\frac{A}{4\lambda}\big]{\frac{\psi_{(\xi)}^2}{\psi}}+K_1\varphi^{\frac{3}{2}}3{\frac{\psi_{(\xi)}^2}{\psi^2}}L\psi\le 0,\\ &I_{10}=K_1\frac{3}{8}\varphi^{-\frac{1}{2}}(1-\frac{\psi}{2\lambda})^2A{\frac{\psi_{(\xi)}^2}{\psi}}\le K_1\frac{3}{8}\frac{\psi}{\varphi^\frac{1}{2}}A{\frac{\psi_{(\xi)}^2}{\psi^2}}\le K_1\frac{3}{4}\varphi^\frac{1}{2}A{\frac{\psi_{(\xi)}^2}{\psi^2}},\\ &\mbox{ here we use }\psi\le2\varphi,\\ &I_{11}=K_1\varphi^{\frac{3}{2}}2\frac{\psi_{(\xi)}}{\psi}\big[(L\psi)_{(\xi)}+2\rho L\psi\big]\le K_1K\varphi^{\frac{3}{2}}\frac{|\psi_{(\xi)}|}{\psi}|\xi|\le K_1K\lambda\varphi^{\frac{1}{2}}\frac{|\psi_{(\xi)}|}{\psi}|\xi|\\ &\qquad\le K_1\lambda\varphi^\frac{1}{2}{\frac{\psi_{(\xi)}^2}{\psi^2}}+K_1K\lambda^3\frac{|\xi|^2}{\psi^\frac{3}{2}},\\ &\mbox{ here we first notice that }\varphi\le2\lambda,\mbox{ and then apply }\psi\le2\varphi,\\ &I_{12}=-K_1\varphi^\frac{3}{2}\frac{4}{\varphi}A{\frac{\psi_{(\xi)}^2}{\psi^2}}=-K_14\varphi^\frac{1}{2}A{\frac{\psi_{(\xi)}^2}{\psi^2}},\\ &I_{13}=K_1\frac{3}{2}\varphi^\frac{1}{2}(1-\frac{\psi}{2\lambda})A{\frac{\psi_{(\xi)}^2}{\psi^2}}\le K_1\frac{3}{2}\varphi^\frac{1}{2}A{\frac{\psi_{(\xi)}^2}{\psi^2}}.\end{aligned}$$ Collecting our estimates above we see that, when $x\in D_\delta^\lambda$, $$\begin{aligned} \Gamma_1(x,\xi)\le& \bigg[K(\lambda^{\frac{5}{2}}+\sqrt\lambda)+K_1K\lambda^3+\Big(\frac{3}{32}-\frac{1}{8}\Big)\bigg]\frac{|\xi|^2}{\psi^\frac{3}{2}}\\ &+\bigg[K+K_1\lambda+K_1A\Big(\frac{3}{4}+\frac{3}{2}-4\Big)\bigg]\varphi^\frac{1}{2}{\frac{\psi_{(\xi)}^2}{\psi^2}}.\end{aligned}$$ Recall that $K$ and $K_1$ depend only on $K_0$. By first choosing $K_1$ such that $K_1\ge K$, then letting $\lambda$ be sufficiently small, we get $$\Gamma_1(x,\xi)\le -\frac{1}{64}\frac{|\xi|^2}{\psi^\frac{3}{2}}-\frac{1}{2}\varphi^\frac{1}{2}{\frac{\psi_{(\xi)}^2}{\psi^2}}\le-\frac{1}{64\lambda^\frac{3}{2}}|\xi|^2-\frac{\lambda}{2}{\frac{\psi_{(\xi)}^2}{\psi^2}}\le0\label{3e}.$$ It follows that $\mathrm{B}_1(x_t,\xi_t)$ is a local supermartingale on $[0,\tau_1^\delta]$. Also, notice that $f(x)=\sqrt{x}$ is concave, so $\sqrt{\mathrm{B}_1(x_t,\xi_t)}$ is a local supermartingale on $[0,\tau_1^\delta]$. Thus (1) is proved. From (\[3e\]), there exists a sufficiently small positive $\lambda_0$, such that $$\Gamma_1(x,\xi)+\lambda_0(|\xi|^2+{\frac{\psi_{(\xi)}^2}{\psi^2}})\le 0,\forall x\in D_\delta^\lambda.$$ Therefore, $$\begin{aligned} \lambda_0 E\int_0^{\tau_1^\delta}\bigg(|\xi_t|^2+{\frac{\psi_{(\xi_t)}^2}{\psi^2}}\bigg)dt\le&-E\int_0^{\tau_1^\delta}\Gamma_1(x_t,\xi_t)\\ =&\mathrm{B}_1(x_0,\xi_0)-E\mathrm{B}_1(x_{\tau_1^\delta},\xi_{\tau_1^\delta})\le\mathrm{B}_1(x_0,\xi_0),\end{aligned}$$ which proves (2). Since $$|\xi_t|^2=|\xi_0|^2+\int_0^t2(\xi_s,\bar b)+\|\bar\sigma\|^2ds+\int_0^t2(\xi_s,\bar\sigma)dw_s,$$ by Burkholder-Davis-Gundy inequality, for $\tau_n=\tau_1^\delta\wedge\inf\{t\ge0:|\xi_t|\ge n\}$, we have, $$\begin{aligned} E\sup_{t\le\tau_n}|\xi_t|^2\le& |\xi_0|^2+\int_0^{\tau_n}\Big(2|\xi_t|\cdot|\bar b|+\|\bar\sigma\|^2\Big)dt+6E\Big(\int_0^{\tau_n}|(\xi_t,\bar\sigma)|^2dt\Big)^{\frac{1}{2}}\\ \le&|\xi_0|^2+NE\int_0^{\tau_n}\bigg(|\xi_t|^2+\frac{\psi_{(\xi_t)}^2}{\psi^2}\bigg)dt+E\Big(\int_0^{\tau_n} N|\xi_t|^2(|\xi_t|^2+\frac{\psi_{(\xi_t)}^2}{\psi^2})dt\Big)^{\frac{1}{2}}\\ \le&N\mathrm{B}_1(x_0,\xi_0)+E\Big[\sup_{t\le\tau_n}|\xi_t|\cdot\Big(\int_0^{\tau_n} N(|\xi_t|^2+\frac{\psi_{(\xi_t)}^2}{\psi^2})dt\Big)^{\frac{1}{2}}\Big]\\ \le&N\mathrm{B}_1(x_0,\xi_0)+\frac{1}{2}E\sup_{t\le\tau_n}|\xi_t|^2+\frac{1}{2}E\Big(\int_0^{\tau_n} N(|\xi_t|^2+\frac{\psi_{(\xi_t)}^2}{\psi^2})dt\Big)\\ \le&N\mathrm{B}_1(x_0,\xi_0)+\frac{1}{2}E\sup_{t\le\tau_n}|\xi_t|^2,\end{aligned}$$ which implies that $$E\sup_{t\le\tau_n}|\xi_t|^2\le N\mathrm{B}_1(x_0,\xi_0).$$ Now (3) is obtained by letting $n\rightarrow\infty$. Now we estimate the moments of second quasiderivative $\eta_t$. Based on our definition, we have $$d\eta_t=[\sigma_{(\eta_t)}+G(x_t,\xi_t)]dw_t+[b_{(\eta_t)}+H(x_t,\xi_t)]dt,$$ with $$\begin{aligned} &G(x,\xi)=\sigma_{(\xi)(\xi)}+2r\sigma_{(\xi)}+(2\sigma_{(\xi)}+2r\sigma+\sigma P)P+(\hat{r}-r^2)\sigma,\\ &H(x,\xi)=b_{(\xi)(\xi)}+4rb_{(\xi)}+2{\frac{\psi_{(\xi)}^2}{\psi^2}}b.\end{aligned}$$ Therefore, we have the estimates $$\|G\|\le N|\xi|(|\xi|+\frac{|\psi_{(\xi)}|}{\psi}),\qquad|H|\le N(|\xi|^2+{\frac{\psi_{(\xi)}^2}{\psi^2}}).$$ Itô’s formula implies $$\begin{aligned} &d(|\eta_t|^2e^{2\varphi})=\theta(x_t,\xi_t,\eta_t)dt+\mu^k(x_t,\xi_t,\eta_t)dw_t^k,\end{aligned}$$ where $$\begin{aligned} \theta(x,\xi,\eta)=e^{2\varphi}\Big\{&2|\eta|^2\big[(1-\frac{\psi}{2\lambda})L\psi-\frac{A}{4\lambda}+(1-\frac{\psi}{2\lambda})^2A\big]+\|\sigma_{(\eta)}+G(x,\xi)\|^2\\ &+2(\eta,b_{(\eta)}+H(x,\xi))+2(\eta,\sigma_{(\eta)}+G(x,\xi))[(1-\frac{\psi}{2\lambda}){\psi_{(\sigma^k)}}]\Big\}.\end{aligned}$$ It is not hard to see that, for any $x\in D_\delta^\lambda$, $$\begin{aligned} \theta(x,\xi,\eta)\le e^{2\varphi}\Big\{2(1-\frac{1}{4\lambda})A|\eta|^2+N\big[|\eta|^2+|\xi|^2(|\xi|^2+{\frac{\psi_{(\xi)}^2}{\psi^2}})+|\eta|(|\xi|^2+{\frac{\psi_{(\xi)}^2}{\psi^2}})\big]\Big\}.\end{aligned}$$ So for sufficiently small $\lambda$, we have $$\theta(x,\xi,\eta)+\lambda_0|\eta|^2\le Ne^{2\varphi}(|\xi|^2+|\eta|)(|\xi|^2+{\frac{\psi_{(\xi)}^2}{\psi^2}}).$$ Then for any bounded stopping time $\gamma$ with respect to $\{\mathcal{F}_t\}$, we have $$E(e^{2\varphi}|\eta_{\gamma}|^2)+\lambda_0 E\int_0^{\gamma}|\eta_t|^2dt\le E\int_0^{\gamma}Ne^{2\varphi}(|\xi_t|^2+|\eta_t|)(|\xi_t|^2+\frac{\psi_{(\xi_t)}^2}{\psi^2})dt.$$ Let $\tau_n=\tau_1^\delta\wedge\inf\{t\ge0:e^{\varphi}|\eta_t|\ge n\}$. Recall that $\eta_0=0$. By Theorem III.6.8 in [@MR1311478], we have $$\begin{aligned} &E\sup_{t\le\tau_n}(e^{\varphi}|\eta_t|)\\ \le& 3E\Big(\int_0^{\tau_n}Ne^{2\varphi}(|\xi_t|^2+|\eta_t|)(|\xi_t|^2+\frac{\psi_{(\xi_t)}^2}{\psi^2})dt\Big)^{\frac{1}{2}}\\ \le& E\Big[\Big(\int_0^{\tau_n}9Ne^{2\varphi}|\xi_t|^2(|\xi_t|^2+\frac{\psi_{(\xi_t)}^2}{\psi^2})dt\Big)^{\frac{1}{2}}+\Big(\int_0^{\tau_n}9Ne^{2\varphi}|\eta_t|(|\xi_t|^2+\frac{\psi_{(\xi_t)}^2}{\psi^2})dt\Big)^{\frac{1}{2}}\Big]\\ \le& E\Big[N\sup_{t\le\tau_n}|\xi_t|\cdot\Big(\int_0^{\tau_n}|\xi_t|^2+\frac{\psi_{(\xi_t)}^2}{\psi^2}dt\Big)^{\frac{1}{2}}+\sup_{t\le\tau_n}\sqrt{e^{\varphi}|\eta_t|}\cdot\Big(\int_0^{\tau_n}N(|\xi_t|^2+\frac{\psi_{(\xi_t)}^2}{\psi^2})dt\Big)^{\frac{1}{2}}\Big]\\ \le& NE\sup_{t\le\tau_n}|\xi_t|^2+NE\int_0^{\tau_n}\bigg(|\xi_t|^2+\frac{\psi_{(\xi_t)}^2}{\psi^2}\bigg)dt+\frac{1}{2}E\sup_{t\le\tau_n}(e^{\varphi}|\eta_t|)\\ \le&\frac{1}{2}E\sup_{t\le\tau_n}(e^{\varphi}|\eta_t|)+N\mathrm{B}_1(x_0,\xi_0),\end{aligned}$$ which implies that $$E\sup_{t\le\tau_n}|\eta_t|\le E\sup_{t\le\tau_n}(e^{\varphi}|\eta_t|)\le N\mathrm{B}_1(x_0,\xi_0),$$ $$E\Big(\int_0^{\tau_n}|\eta_t|^2 dt\Big)^\frac{1}{2}\le N\mathrm{B}_1(x_0,\xi_0).$$ Letting $n\rightarrow\infty$, we conclude that (4) and (5) are true. \[3l2\] Introduce $$\mathrm{B}_2(x,\xi)=\lambda^\frac{3}{4}|\xi|^2.$$ If we construct first and second quasiderivatives by (\[2d\]) and (\[2e\]), in which $$\begin{aligned} &r(x,y):=(\rho(x),y),\ \ r_t:=r(x_t,\xi_t),\ \ \hat{r}_t:=r(x_t,\eta_t),\\ &\pi(x,y):=\frac{M(x)}{2}\sigma^*(x)y,\ \ \pi_t:=\pi(x_t,\xi_t),\ \ \hat{\pi}_t:=\pi(x_t,\eta_t),\\ &P(x,y):=Q(x,y),\ \ P_t:=P(x_t,\xi_t),\ \ \hat{P}_t:=P(x_t,\eta_t).\end{aligned}$$ Then for sufficiently small $\lambda$, when $x_0\in D_{\lambda^2}$, $\xi_0\in{\mathbb{R}^d}$ and $\eta_0=0$, we have 1. $e^{-\phi_t}\mathrm{B}_2(x_t,\xi_t)$ and $\sqrt{e^{-\phi_t}\mathrm{B}_2(x_t,\xi_t)}$ are local supermartingales on $[0,\tau_2)$, where $\tau_2=\tau_{D_{\lambda^2}}(x)$; 2. $\displaystyle{E\int_0^{\tau_2} e^{-\phi_t}|\xi_t|^2 dt\le N\mathrm{B}_2(x_0,\xi_0)}$; 3. $\displaystyle{E\sup_{t\le\tau_2}e^{-\phi_t}|\xi_t|^2\le N\mathrm{B}_2(x_0,\xi_0)}$; 4. $\displaystyle{Ee^{-\phi_{\tau_2}}|\eta_{\tau_2}|\le E\sup_{t\le\tau_2}e^{-\phi_t}|\eta_t|\le N\mathrm{B}_2(x_0,\xi_0)}$; 5. $\displaystyle{E\Big(\int_0^{\tau_2} e^{-2\phi_t}|\eta_t|^2 dt\Big)^\frac{1}{2}\le N\mathrm{B}_2(x_0,\xi_0)}$; 6. The above inequalities are still all true if we replace $\phi_t$ by $\phi_t-\frac{1}{2}t$. More precisely, we have $$E\int_0^{\tau_2} e^{-\phi_t+\frac{1}{2}t}|\xi_t|^2 dt\le N\mathrm{B}_2(x_0,\xi_0),\ E\sup_{t\le\tau_2}e^{-\phi_t+\frac{1}{2}t}|\xi_t|^2\le N\mathrm{B}_2(x_0,\xi_0),$$ $$E\Big(\int_0^{\tau_2} e^{-2\phi_t+t}|\eta_t|^2 dt\Big)^\frac{1}{2}\le N\mathrm{B}_2(x_0,\xi_0),\ E\sup_{t\le\tau_2}e^{-\phi_t+\frac{1}{2}t}|\eta_t|\le N\mathrm{B}_2(x_0,\xi_0),$$ where $N$ is constant depending on $K_0$ and $\lambda$. First of all, replacing $K_0$ by $$\max\bigg\{K_0,\sup_{x\in D_{\lambda^2}}|\rho(x)|,\sup_{x\in D_{\lambda^2}, y\in {\mathbb{R}^d}}\frac{\|Q(x,y)\|}{|y|},\sup_{x\in D_{\lambda^2}}M(x)\bigg\},$$ we may assume that $$\sup_{x\in D_{\lambda^2}}|\rho(x)|\le K_0,\ \ \sup_{x\in D_{\lambda^2}}\|Q(x,y)\|\le K_0|y|,\forall y\in {\mathbb{R}^d},\ \ \sup_{x\in D_{\lambda^2}}M(x)\le K_0.$$ By Itô’s formula, for $t<\tau_2$, we have $$d|\xi_t|^2=\Lambda^k_2(x_t,\xi_t)dw_t^k+\Gamma_2(x_t,\xi_t)dt,$$ where $$\Lambda_2(x,\xi)=2(\xi_t,{\sigma_{(\xi_t)}}+r_t\sigma+\sigma P_t),$$ $$\Gamma_2(x,\xi)=\big[2(\xi,b_{(\xi)}+2rb-\sigma\pi)+\|\sigma_{(\xi)}+r\sigma+\sigma P\|^2\big]\le (c-1)|\xi|^2.$$ So $$\label{bd} \begin{gathered} d\big(e^{-\phi_t}|\xi_t|^2\big)=e^{-\phi_t}\big[\Gamma_2(x_t,\xi_t)-c(x_t)|\xi_t|^2\big]dt+dm_t\\ \le-e^{-\phi_t}|\xi_t|^2dt+dm_t, \end{gathered}$$ where $m_t$ is a local martingale. Thus $e^{-\phi_t}\mathrm{B}_2(x_t,\xi_t)$ is a local supermartingale on $[0,\tau_2)$. Also, notice that $f(x)=\sqrt{x}$ is concave, so $\sqrt{e^{-\phi_t}\mathrm{B}_2(x_t,\xi_t)}$ is a local supermartingale on $[0,\tau_2]$. (1) is proved. From (\[bd\]), we also have $$E\int_0^{\tau_2} e^{-\phi_t}|\xi_t|^2 dt=\mathrm{B}_2(x_0,\xi_0)-Ee^{-\phi_{\tau_2}}\mathrm{B}_2(x_{\tau_2},\xi_{\tau_2})\le\mathrm{B}_2(x_0,\xi_0),$$ which proves (2). Since $$e^{-\phi_t}|\xi_t|^2=|\xi_0|^2+\int_0^te^{-\phi_s}\big[2(\xi_s,\bar b)+\|\bar\sigma\|^2-c|\xi_t|^2\big]ds+\int_0^te^{-\phi_s}2(\xi_s,\bar\sigma)dw_s,$$ by Burkholder-Davis-Gundy inequality, for $\tau_n=\tau_2\wedge\inf\{t\ge0:|\xi_t|\ge n\}$, we have, $$\begin{aligned} E\sup_{t\le\tau_n}e^{-\phi_t}|\xi_t|^2\le& |\xi_0|^2+\int_0^{\tau_n}e^{-\phi_t}\big[2|\xi_t|\cdot|\bar b|+\|\bar\sigma\|^2+c|\xi_t|^2\big]dt\\ &+12E\Big(\int_0^{\tau_n}e^{-2\phi_t}|(\xi_t,\bar\sigma)|^2dt\Big)^{\frac{1}{2}}\\ \le&|\xi_0|^2+NE\int_0^{\tau_n}e^{-\phi_t}|\xi_t|^2dt+E\Big(\int_0^{\tau_n} Ne^{-2\phi_t}|\xi_t|^4dt\Big)^{\frac{1}{2}}\\ \le&N\mathrm{B}_2(x_0,\xi_0)+E\Big[\sup_{t\le\tau_n}e^{-\frac{1}{2}\phi_t}|\xi_t|\cdot\Big(\int_0^{\tau_n} Ne^{-\phi_t}|\xi_t|^2dt\Big)^{\frac{1}{2}}\Big]\\ \le&N\mathrm{B}_2(x_0,\xi_0)+\frac{1}{2}E\sup_{t\le\tau_n}e^{-\phi_t}|\xi_t|^2+\frac{1}{2}E\Big(\int_0^{\tau_n} Ne^{-\phi_t}|\xi_t|^2dt\Big)\\ \le&N\mathrm{B}_2(x_0,\xi_0)+\frac{1}{2}E\sup_{t\le\tau_n}e^{-\phi_t}|\xi_t|^2,\end{aligned}$$ which implies that $$E\sup_{t\le\tau_n}e^{-\phi_t}|\xi_t|^2\le N\mathrm{B}_2(x_0,\xi_0).$$ So (3) is true by letting $n\rightarrow\infty$. Now we estimate the moments of the second quasiderivative $\eta_t$. Based on our definition, we have $$d\eta_t=[\tilde\sigma+G]dw_t+[\tilde b+H]dt,$$ where $$\begin{aligned} &\tilde\sigma=\bar\sigma(x,\eta)=\sigma_{(\eta)}+\hat r\sigma+\sigma\hat{P},\\ &\tilde b=\bar b(x,\eta)=b_{(\eta)}+2\hat r b-\sigma\hat\pi,\\ &G=G(x,\xi)=\sigma_{(\xi)(\xi)}+2r\sigma_{(\xi)}-r^2\sigma+(2\sigma_{(\xi)}+2r\sigma+\sigma P)P,\\ &H=H(x,\xi)=b_{(\xi)(\xi)}+4rb_{(\xi)}-2(\sigma_{(\xi)}+r\sigma-\sigma P)\pi.\end{aligned}$$ From the expressions above, we have the estimates $$\|G\|\le N|\xi|^2,\qquad|H|\le N|\xi|^2.$$ Hence Itô’s formula implies $$d\big(e^{-2\phi_t}|\eta_t|^2\big)=e^{-2\phi_t}\big[2(\eta_t,\tilde b+H)+\|\tilde\sigma+G\|^2-2c|\eta_t|^2\big]dt+2e^{-2\phi_t}(\eta_t,\tilde\sigma+G)dw_t^k.$$ Notice that $$\begin{aligned} &2(\eta,\tilde b+H)+\|\tilde\sigma+G\|^2-2c|\eta|^2\\ =&2(\eta,\tilde b)+\|\tilde\sigma\|^2-2c|\eta|^2+2(\eta, H)+|H|^2+2(\tilde\sigma^k,G^k)\\ \le&(c-1)|\eta|^2-2c|\eta|^2+|\eta|^2+N|\xi|^4\\ \le&-|\eta|^2+N|\xi|^4.\end{aligned}$$ So for any bounded stopping time $\gamma$ with respect to $\{\mathcal{F}_t\}$, we have $$Ee^{-2\phi_\gamma}|\eta_\gamma|^2+E\int_0^{\gamma}e^{-2\phi_t}|\eta_t|^2dt\le E\int_0^{\gamma}Ne^{-2\phi_t}|\xi_t|^4dt.$$ Recall that $\eta_0=0$. By Theorem III.6.8 in [@MR1311478], we have $$\begin{aligned} E\sup_{t\le\tau_2}e^{-\phi_t}|\eta_t| \le& 3E\Big(\int_0^{\tau_2}Ne^{-2\phi_t}|\xi_t|^4dt\Big)^{\frac{1}{2}}\\ \le& E\Big[\sup_{t\le\tau_2}e^{-\frac{1}{2}\phi_t}|\xi_t|\cdot\Big(\int_0^{\tau_2}9Ne^{-\phi_t}|\xi_t|^2dt\Big)^{\frac{1}{2}}\Big]\\ \le& \frac{1}{2}E\sup_{t\le\tau_2}e^{-\phi_t}|\xi_t|^2+\frac{1}{2}E\int_0^{\tau_2}9Ne^{-\phi_t}|\xi_t|^2dt\\ \le&N\mathrm{B}_2(x_0,\xi_0),\end{aligned}$$ $$E\Big(\int_0^{\tau_2}e^{-2\phi_t}|\eta_t|^2dt\Big)^\frac{1}{2}\le3E\Big(\int_0^{\tau_2}Ne^{-2\phi_t}|\xi_t|^4dt\Big)^{\frac{1}{2}}\le N\mathrm{B}_2(x_0,\xi_0),$$ which implies that (4) and (5) are true. Finally, rewritting $c-1$ by $(c-\frac{1}{2})-\frac{1}{2}$ and repeating the argument above, we conclude that (6) is true. Now we are ready to prove the theorem. Denote $\tau_{D_\delta^\lambda}(x)$ and $\tau_{D_{\lambda^2}}(x)$ by $\tau_1^\delta$ and $\tau_2$, respectively. From (\[1b\]) we immediately have $$\label{u} |u|_{0,D}\le|g|_{0,D}+|f|_{0,D}E\int_0^\tau e^{-t}dt\le|g|_{0,D}+|f|_{0,D}.$$ When $x_0\in D_\delta^\lambda$, by Theorem \[2cc\], we have $$u_{(\xi_0)}(x_0)=X_0=EX_{\tau_1^\delta}.$$ So from (\[Xi\]) and (\[u\]), $$\begin{aligned} |u_{(\xi_0)}(x_0)|\le &E\Big|u_{(\xi_{\tau_1^\delta})}(x_{\tau_1^\delta})+(\xi_{\tau_1^\delta}^0+\xi_{\tau_1^\delta}^{d+1})u(x_{\tau_1^\delta})\Big|\\ &+|f|_{1,D}E\int_0^{\tau_1^\delta}e^{-s}\Big(|\xi_s|+2r_s+|\xi_s^0|+|\xi_s^{d+1}|\Big)ds\\ \le &E\Big|u_{(\xi_{\tau_1^\delta})}(x_{\tau_1^\delta})\Big|+\Big(|g|_{0,D}+|f|_{0,D}\Big)\Big(E|\xi_{\tau_1^\delta}^0|+E|\xi_{\tau_1^\delta}^{d+1}|\Big)\\ &+|f|_{1,D}\Big(E\int_0^{\tau_1^\delta}|\xi_s|+2r_sds+E\sup_{t\le\tau_1^\delta}|\xi_t^{0}|+E\sup_{t\le\tau_1^\delta}|\xi_t^{d+1}|\Big).\end{aligned}$$ By Lemma \[3l1\], Davis inequality and Hölder inequality, $$\begin{aligned} \nonumber E|u_{(\xi_{\tau_1^\delta})}(x_{\tau_1^\delta})|\le&\sup_{ x\in \partial D_\delta^\lambda}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}\cdot E\sqrt{\mathrm{B}_1(x_{\tau_1^\delta},\xi_{\tau_1^\delta})}\\ \nonumber\le&\sup_{x\in \partial D_\delta^\lambda}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}\cdot {\sqrt{\mathrm{B}_1(x_0,\xi_0)}},\\ E|\xi_{\tau_1^\delta}^0|\le E\sup_{t\le\tau_1^\delta}|\xi_t^{0}|\le&3E\langle\xi^0\rangle_{\tau_1^\delta}^\frac{1}{2}\le3\Big(E\langle\xi^0\rangle_{\tau_1^\delta}\Big)^\frac{1}{2}\\ \nonumber\le &N\bigg(E\int_0^{\tau_1^\delta}\frac{\psi_{(\xi_s)}^2}{\psi^2}ds\bigg)^\frac{1}{2}\le N{\sqrt{\mathrm{B}_1(x_0,\xi_0)}},\\ E|\xi_{\tau_1^\delta}^{d+1}|\le E\sup_{t\le\tau_1^\delta}|\xi_t^{d+1}|\le& NE\int_0^{\tau_1^\delta}|\xi_s|+\frac{|\psi_{(\xi_s)}|}{\psi}ds\\ \nonumber\le&NE\int_0^\infty I_{s\le\tau_1^\delta}\cdot I_{s\le\tau_1^\delta}\Big(|\xi_s|+\frac{|\psi_{(\xi_s)}|}{\psi}\Big)ds\\ \nonumber\le&N\bigg(E\tau_1^\delta\bigg)^{\frac{1}{2}}\bigg(E\int_0^{\tau_1^\delta}\Big(|\xi_s|^2+\frac{\psi_{(\xi_s)}^2}{\psi^2}\Big)ds\bigg)^\frac{1}{2}\\ \nonumber\le&N{\sqrt{\mathrm{B}_1(x_0,\xi_0)}},\\ E\int_0^{\tau_1^\delta}(|\xi_s|+2r_s)ds\le&NE\int_0^{\tau_1^\delta}\bigg(|\xi_s|+\frac{|\psi_{(\xi_s)}|}{\psi}\bigg)ds\le N{\sqrt{\mathrm{B}_1(x_0,\xi_0)}}.\end{aligned}$$ Collecting all estimates above, we conclude that $${|u_{(\xi_0)}(x_0)|}\le\sup_{x\in \partial D_\delta^\lambda}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}\cdot {\sqrt{\mathrm{B}_1(x_0,\xi_0)}}+N(|g|_{0,D}+|f|_{1,D}){\sqrt{\mathrm{B}_1(x_0,\xi_0)}}.$$ So for any $x_0\in D_\delta^\lambda$, $\xi_0\in{\mathbb{R}^d}\setminus\{0\}$, we have $$\label{3h} \frac{{|u_{(\xi_0)}(x_0)|}}{{\sqrt{\mathrm{B}_1(x_0,\xi_0)}}}\le\sup_{x\in \partial D_\delta^\lambda}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}+N_1,$$ with $$\label{N1} N_1=N(|g|_{1,D}+|f|_{1,D}).$$ Similarly, when $x_0\in D_{\lambda^2}$, by Theorem \[2cc\], we have $$u_{(\xi_0)}(x_0)=X_0=EX_{\tau_2}.$$ Again, from (\[Xi\]) and (\[u\]), $$\begin{aligned} |u_{(\xi_0)}(x_0)|\le &Ee^{-\phi_{\tau_2}}\Big|u_{(\xi_{\tau_2})}(x_{\tau_2})+(\xi^0_{\tau_2}+\xi_{\tau_2}^{d+1})u(x_{\tau_1^\delta})\Big|\\ &+|f|_{1,D}E\int_0^{\tau_2}e^{-\phi_s}\Big(|\xi_s|+2r_s+|\xi_s^0|+|\xi_s^{d+1}|\Big)ds\\ \le&Ee^{-\frac{1}{2}\phi_{\tau_2}}\Big|u_{(\xi_{\tau_2})}(x_{\tau_2})\Big|+\Big(|g|_{0,D}+|f|_{0,D}\Big)\Big(Ee^{-\frac{1}{2}\phi_{\tau_2}}|\xi_{\tau_2}^0|+Ee^{-\phi_{\tau_2}}|\xi_{\tau_2}^{d+1}|\Big)\\ &+|f|_{1,D}\bigg(E\int_0^{\tau_2}e^{-\phi_s}\big(|\xi_s|+2r_s\big)ds+4E\sup_{t\le\tau_2}e^{-\frac{3}{4}\phi_t}|\xi_t^{d+1}|+2E\sup_{s\le{\tau_2}}e^{-\frac{1}{2}\phi_{s}}|\xi_{s}^0|\bigg).\end{aligned}$$ By Lemma \[3l2\], Davis inequality and Hölder inequality, $$\begin{aligned} Ee^{-\frac{1}{2}\phi_{\tau_2}}|u_{(\xi_{\tau_2})}(x_{\tau_2})|\le&\sup_{ x\in \partial D_{\lambda^2}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_2(x,\xi)}}}\cdot E\sqrt{e^{-\phi_{\tau_2}}\mathrm{B}_2(x_{\tau_2},\xi_{\tau_2})}\\ \le&\sup_{x\in \partial D_{\lambda^2}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_2(x,\xi)}}}\cdot {\sqrt{\mathrm{B}_2(x_0,\xi_0)}},\\ Ee^{-\frac{1}{2}\phi_{\tau_2}}|\xi_{\tau_2}^0|\le E\sup_{s\le{\tau_2}}e^{-\frac{1}{2}\phi_{s}}|\xi_{s}^0|=&E\sup_{s\le\tau_2}\Big|\int_0^{s}e^{-\frac{1}{2}\phi_s}\pi_rdw_r\Big|\\ \le&3E\Big(\int_0^{\tau_2}e^{-\phi_{r}}|\pi_r|^2dr\Big)^{\frac{1}{2}}\\ \le&NE\Big(\int_0^{\tau_2}e^{-\phi_r}|\xi_r|^2dr\Big)^\frac{1}{2}\\ \le&N{\sqrt{\mathrm{B}_2(x_0,\xi_0)}},\\ Ee^{-\phi_{\tau_2}}|\xi_{\tau_2}^{d+1}|\le E\sup_{t\le\tau_2}e^{-\frac{3}{4}\phi_t}|\xi_t^{d+1}|\le&NE\int_0^{\tau_2}e^{-\frac{3}{4}\phi_s}|\xi_s|ds\\ \le&NE\Big(\int_0^{\tau_2}e^{-\frac{1}{2}\phi_s}ds\Big)^\frac{1}{2}\Big(\int_0^{\tau_2}e^{-\phi_s}|\xi_s|^2ds\Big)^\frac{1}{2}\\ \le&N\bigg(E\int_0^{\tau_2}e^{-\phi_s}|\xi_s|^2ds\bigg)^\frac{1}{2}\\ \le&N{\sqrt{\mathrm{B}_2(x_0,\xi_0)}},\\ E\int_0^{\tau_2}e^{-\phi_s}\big(|\xi_s|+2r_s\big)ds\le&NE\int_0^{\tau_2}e^{-\phi_s}|\xi_s|ds\le N{\sqrt{\mathrm{B}_2(x_0,\xi_0)}}.\end{aligned}$$ Collecting all estimates above, we conclude that $${|u_{(\xi_0)}(x_0)|}\le\sup_{x\in \partial D_{\lambda^2}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_2(x,\xi)}}}\cdot {\sqrt{\mathrm{B}_2(x_0,\xi_0)}}+N(|g|_{0,D}+|f|_{1,D}){\sqrt{\mathrm{B}_2(x_0,\xi_0)}}.$$ So for any $x_0\in D_{\lambda^2}$, $\xi_0\in{\mathbb{R}^d}\setminus\{0\}$, we have $$\label{3i} \frac{{|u_{(\xi_0)}(x_0)|}}{{\sqrt{\mathrm{B}_2(x_0,\xi_0)}}}\le\sup_{x\in \partial D_{\lambda^2}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_2(x,\xi)}}}+N_1,$$ with $N_1$ defined by (\[N1\]). Notice that $$B_1(x,\xi)\left\{ \begin{array}{ll} \ge\sqrt\psi(1+\sqrt\psi)|\xi|^2\ge\lambda^\frac{1}{2}|\xi|^2&\mbox{on }\{\psi=\lambda\}\\ \displaystyle\le\lambda(2+\lambda)|\xi|^2+K_1(2\lambda^2)^\frac{3}{2}\frac{\psi_{(\xi)}^2}{\lambda^2}\le K\lambda|\xi|^2&\mbox{on }\{\psi=\lambda^2\}. \end{array} \right.$$ Recall that $K$ doesn’t depend on $\lambda$. So for sufficiently small $\lambda$, we have $$B_1(x,\xi)\ge 4B_2(x,\xi)\mbox{ when }\psi=\lambda,\qquad 4B_1(x,\xi)\le B_2(x,\xi)\mbox{ when }\psi=\lambda^2.$$ Then on $\{x\in D:\psi(x)=\lambda\}$, we have $$\begin{aligned} {\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}\le&\frac{1}{2}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_2(x,\xi)}}}\\ \le&\frac{1}{2}(\sup_{\{\psi=\lambda^2\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_2(x,\xi)}}}+N_1)\\ \le&\frac{1}{4}\sup_{\{\psi=\lambda^2\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}+\frac{N_1}{2}\\ \le&\frac{1}{4}(\sup_{\{\psi=\lambda\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}+\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}+N_1)+\frac{N_1}{2}\\ =&\frac{1}{4}\sup_{\{\psi=\lambda\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}+\frac{1}{4}\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}+\frac{3N_1}{4},\end{aligned}$$ which implies that $$\label{3j} \sup_{\{\psi=\lambda\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}\le\frac{1}{3}\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}+N_1.$$ Meanwhile, on $\{x\in D:\psi(x)=\lambda^2\}$, we have $$\begin{aligned} {\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_2(x,\xi)}}}\le&\frac{1}{2}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}\\ \le&\frac{1}{2}(\sup_{\{\psi=\lambda\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}+\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}+N_1)\\ \le&\frac{1}{2}(\frac{1}{3}\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}+N_1)+\frac{1}{2}\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}+\frac{N_1}{2}\\ =&\frac{2}{3}\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}+N_1.\end{aligned}$$ Therefore, $$\label{3k} \sup_{\{\psi=\lambda^2\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_2(x,\xi)}}}\le\frac{2}{3}\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}+N_1.$$ Combining (\[3h\]) and (\[3j\]), we get, for any $x\in D_\delta^\lambda$, $\xi\in{\mathbb{R}^d}\setminus\{0\}$, $$\label{3l} {\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}\le\frac{4}{3}\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}+2N_1.$$ Combining (\[3i\]) and (\[3k\]), we get, for any $x\in D_{\lambda^2}$, $\xi\in{\mathbb{R}^d}\setminus\{0\}$, $$\label{3m} {\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_2(x,\xi)}}}\le\frac{2}{3}\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}+2N_1.$$ Thus it remains to estimate $$\varlimsup_{\delta\downarrow0}\Big(\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}\Big).$$ Notice that for each $\delta$, there exist $x(\delta)\in\{\psi=\delta\}$ and $\xi(\delta)\in\{\xi:|\xi|=1\}$, such that $$\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}=\frac{|u_{(\xi(\delta))}(x(\delta))|}{\sqrt{\mathrm{B}_1(x(\delta),\xi(\delta))}}.$$ A subsequence of $(x(\delta),\xi(\delta))$ converges to some $(y,\zeta)$, such that $y\in\partial D$ and $|\zeta|=1$. If $\psi_{(\zeta)}(y)\ne0$, then $\mathrm{B}_1(x(\delta),\xi(\delta))\rightarrow\infty$ as $\delta\downarrow0$. In this case, $$\varlimsup_{\delta\downarrow0}\Big(\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}\Big)=\varlimsup_{\delta\downarrow0}\frac{|u_{(\xi(\delta))}(x(\delta))|}{\sqrt{\mathrm{B}_1(x(\delta),\xi(\delta))}}=0.$$ If $\psi_{(\zeta)}(y)=0$, then $\zeta$ is tangential to $\partial D$ at $y$. In this case, $$\label{lims} \begin{gathered} \varlimsup_{\delta\downarrow0}\Big(\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}}\Big)=\varlimsup_{\delta\downarrow0}\frac{|u_{(\xi(\delta))}(x(\delta))|}{\sqrt{\mathrm{B}_1(x(\delta),\xi(\delta))}}\\ \qquad\qquad\qquad\qquad\qquad\ \ =\frac{|g_{(\zeta)}(y)|}{\sqrt\lambda}\le N\sup_{\partial D}|g_x|. \end{gathered}$$ From (\[3l\]), (\[3m\]) and (\[lims\]), we have $$\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_1(x,\xi)}}\le N(|f|_{1,D}+|g|_{1,D}), \mbox{ when }x\in D^\lambda;$$ $$\frac{|u_{(\xi)}(x)|}{\sqrt{\mathrm{B}_2(x,\xi)}}\le N(|f|_{1,D}+|g|_{1,D}), \mbox{ when }x\in D_{\lambda^2}.$$ Notice that $D^\lambda\cup D_{\lambda^2}=D$, and $$\sqrt{\mathrm{B}_1(x,\xi)}\le N(|\xi|+\frac{|\psi_{(\xi)}|}{\psi^{\frac{1}{2}}}), \mbox{ when }x\in D^\lambda;$$ $$\sqrt{\mathrm{B}_2(x,\xi)}\le N(|\xi|+\frac{|\psi_{(\xi)}|}{\psi^{\frac{1}{2}}}), \mbox{ when }x\in D_{\lambda^2}.$$ We conclude that, for any $x\in D$ and $\xi\in{\mathbb{R}^d}$, $$|u_{(\xi)}(x)|\le N(|\xi|+\frac{|\psi_{(\xi)}|}{\psi^{\frac{1}{2}}})(|f|_{1,D}+|g|_{1,D}).$$ The inequality (\[3d\]) is proved. The proof of (\[3dd\]) is similar. When $x_0\in D_\delta^\lambda$, by Theorem \[2cc\], we have $$u_{(\xi_0)(\xi_0)}(x_0)=u_{(\xi_0)(\xi_0)}(x_0)+u_{(\eta_0)}(x_0)=Y_0=EY_{\tau_1^\delta}.$$ From (\[Zeta\]) and (\[u\]), $$\begin{aligned} |u_{(\xi_0)(\xi_0)}(x_0)|\le &E|u_{(\xi_{\tau_1^\delta})(\xi_{\tau_1^\delta})}(x_{\tau_1^\delta})|+\sup_{x\in\partial D_\delta^\lambda,|\zeta|=1}|u_{(\zeta)}(x)|\cdot Ee^{-{\tau_1^\delta}}\Big(|\eta_{\tau_1^\delta}|+2|\tilde\xi^0_{\tau_1^\delta}||\xi_{\tau_1^\delta}|\Big)\\ +&\Big(|g|_{0,D}+|f|_{0,D}\Big)Ee^{-{\tau_1^\delta}}|\tilde\eta^0_{\tau_1^\delta}|+|f|_{2,D}E\int_0^{\tau_1^\delta} e^{-s}\Big[|\xi_s|^2+|\eta_s|\\ +&(4r_s+2|\tilde\xi^0_s|)|\xi_s|+2\hat r_s+4|\tilde\xi_s^0|r_s+|\tilde\eta^0_s|\Big]ds.\end{aligned}$$ Recall that in this case, $$\tilde\xi_t^0=\xi^0_t+\xi_t^{d+1},\qquad\tilde\eta^0_t=2\xi_t^0\xi_t^{d+1}+(\xi_t^{d+1})^2+\eta_t^{d+1}.$$ It follows that $$\begin{aligned} |u_{(\xi_0)(\xi_0)}(x_0)|\le &E|u_{(\xi_{\tau_1^\delta})(\xi_{\tau_1^\delta})}(x_{\tau_1^\delta})|+N\Big(|g|_{0,D}+|f|_{0,D}+\sup_{x\in\partial D_\delta^\lambda,|\zeta|=1}|u_{(\zeta)}(x)|\Big)\\ &\cdot Ee^{-{\tau_1^\delta}}\Big(|\eta_{\tau_1^\delta}|+|\xi_{\tau_1^\delta}|^2+|\xi^0_{\tau_1^\delta}|^2+|\xi_{\tau_1^\delta}^{d+1}|^2+|\eta_{\tau_1^\delta}^{d+1}|\Big)\\ +&N|f|_{2,D}E\int_0^{\tau_1^\delta} e^{-s}\Big[|\xi_s|^2+|\eta_s|+|\xi^0_{s}|^2+|\xi_{s}^{d+1}|^2+|\eta_{s}^{d+1}|+r_s^2+\hat r_s\Big]ds\\ \le&E|u_{(\xi_{\tau_1^\delta})(\xi_{\tau_1^\delta})}(x_{\tau_1^\delta})|+N\Big(|g|_{0,D}+|f|_{2,D}+\sup_{x\in\partial D_\delta^\lambda,|\zeta|=1}|u_{(\zeta)}(x)|\Big)\\ &\cdot\Big(E\sup_{t\le\tau_1^\delta}|\eta_t|+E\sup_{t\le\tau_1^\delta}|\xi_t|^2+E\sup_{t\le\tau_1^\delta}|\xi^0_t|^2+ E\sup_{t\le\tau_1^\delta}e^{-\frac{1}{2}t}|\xi_t^{d+1}|^2\\ &\ \ \ \ +E\sup_{t\le\tau_1^\delta}e^{-\frac{1}{2}t}|\eta_t^{d+1}|+E\int_0^{\tau_1^\delta}r_s^2+\hat{r}_sds\Big).\end{aligned}$$ By Lemma \[3l1\], Davis inequality and Hölder inequality, $$\begin{aligned} \nonumber E|u_{(\xi_{\tau_1^\delta})(\xi_{\tau_1^\delta})}(x_{\tau_1^\delta})|\le&\sup_{ x\in \partial D_\delta^\lambda}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}\cdot E\mathrm{B}_1(x_{\tau_1^\delta},\xi_{\tau_1^\delta})\\ \nonumber\le&\sup_{x\in \partial D_\delta^\lambda}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}\cdot {\mathrm{B}_1(x_0,\xi_0)},\\ E\sup_{t\le\tau_1^\delta}|\eta_t|\le&N{\mathrm{B}_1(x_0,\xi_0)},\\ E\sup_{t\le\tau_1^\delta}|\xi_t|^2\le&N{\mathrm{B}_1(x_0,\xi_0)},\\ E\sup_{t\le\tau_1^\delta}|\xi^0_t|^2\le&4E\langle\xi^0\rangle _{\tau_1^\delta}\le NE\int_0^{\tau_1^\delta}{\frac{\psi_{(\xi_t)}^2}{\psi^2}}dt\le N{\mathrm{B}_1(x_0,\xi_0)},\\ E\sup_{t\le\tau_1^\delta}e^{-\frac{1}{2}t}|\xi_t^{d+1}|^2\le&NE\sup_{t\le\tau_1^\delta}e^{-\frac{1}{2}t}\bigg(\int_0^t\Big(|\xi_s|+\frac{|\psi_{(\xi_s)}|}{\psi}\Big)ds\bigg)^2\\ \nonumber\le&NE\sup_{t\le\tau_1^\delta}e^{-\frac{1}{2}t}t\int_0^t\Big(|\xi_s|^2+\frac{\psi_{(\xi_s)}^2}{\psi^2} \Big)ds\\ \nonumber\le&NE\int_0^{\tau_1^\delta}\Big(|\xi_t|^2+{\frac{\psi_{(\xi_t)}^2}{\psi^2}}\Big) dt\\ \nonumber\le&N{\mathrm{B}_1(x_0,\xi_0)},\\ E\sup_{t\le\tau_1^\delta}e^{-\frac{1}{2}t}|\eta_t^{d+1}|\le&NE\sup_{t\le\tau_1^\delta}e^{-\frac{1}{2}t}\int_0^t\Big(|\xi_s|^2+\frac{\psi_{(\xi_s)}^2}{\psi^2}+|\eta_s|\Big)ds\\ \nonumber\le&NE\sup_{t\le\tau_1^\delta}e^{-\frac{1}{2}t}\bigg[\int_0^t\Big(|\xi_s|^2+\frac{\psi_{(\xi_s)}^2}{\psi^2}\Big)ds+\sqrt t\Big(\int_0^t|\eta_s|^2ds\Big)^\frac{1}{2}\bigg]\\ \nonumber\le&N\bigg[E\int_0^{\tau_1^\delta}\Big(|\xi_t|^2+{\frac{\psi_{(\xi_t)}^2}{\psi^2}}\Big) dt+E\Big(\int_0^{\tau_1^\delta}|\eta_t|^2dt\Big)^2\bigg]\\ \nonumber\le&N{\mathrm{B}_1(x_0,\xi_0)},\\ E\int_0^{\tau_1^\delta}(r_s^2+\hat{r}_s)ds\le&NE\int_0^{\tau_1^\delta}\Big(|\xi_t|^2+\frac{\psi_{(\xi_t)}^2}{\psi^2}\Big)dt\le N{\mathrm{B}_1(x_0,\xi_0)}.\end{aligned}$$ Collecting all estimates above, we conclude that $$\begin{aligned} {|u_{(\xi_0)(\xi_0)}(x_0)|}\le&\sup_{x\in \partial D_\delta^\lambda}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}\cdot {\mathrm{B}_1(x_0,\xi_0)}\\ &+N\Big(|g|_{0,D}+|f|_{2,D}+\sup_{x\in\partial D_\delta^\lambda,|\zeta|=1}|u_{(\zeta)}(x)|\Big){\mathrm{B}_1(x_0,\xi_0)}.\end{aligned}$$ So for any $x_0\in D_\delta^\lambda$, $\xi_0\in{\mathbb{R}^d}\setminus\{0\}$, we have $$\label{3o} \frac{{|u_{(\xi_0)(\xi_0)}(x_0)|}}{{\mathrm{B}_1(x_0,\xi_0)}}\le\sup_{x\in \partial D_\delta^\lambda}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}+N_2,$$ with $$\label{N2} N_2=N\Big(|g|_{2,D}+|f|_{2,D}+\sup_{x\in\partial D_\delta^\lambda,|\zeta|=1}|u_{(\zeta)}(x)|\Big).$$ When $x_0\in D_{\lambda^2}$, by Theorem \[2cc\], we have $$u_{(\xi_0)(\xi_0)}(x_0)=u_{(\xi_0)(\xi_0)}(x_0)+u_{(\eta_0)}(x_0)=Y_0=EY_{\tau_2}.$$ Again, from (\[Zeta\]) and (\[u\]), $$\begin{aligned} |u_{(\xi_0)(\xi_0)}(x_0)|\le &Ee^{-\phi_{\tau_2}}|u_{(\xi_{\tau_2})(\xi_{\tau_2})}(x_{\tau_2})|+\sup_{x\in\partial D_{\lambda^2},|\zeta|=1}|u_{(\zeta)}(x)|\cdot Ee^{-\phi_{\tau_2}}\Big(|\eta_{\tau_2}|+2|\tilde\xi^0_{\tau_2}||\xi_{\tau_2}|\Big)\\ +&\Big(|g|_{0,D}+|f|_{0,D}\Big)Ee^{-\phi_{\tau_2}}|\tilde\eta^0_{\tau_2}|+|f|_{2,D}E\int_0^{\tau_2} e^{-\phi_s}\Big[|\xi_s|^2+|\eta_s|\\ +&(4r_s+2|\tilde\xi^0_s|)|\xi_s|+2\hat r_s+4|\tilde\xi_s^0|r_s+|\tilde\eta^0_s|\Big]ds.\end{aligned}$$ Recall that in this case, $$\tilde\xi_t^0=\xi^0_t+\xi_t^{d+1},\qquad\tilde\eta^0_t=\eta_t^0+2\xi_t^0\xi_t^{d+1}+(\xi_t^{d+1})^2+\eta_t^{d+1}.$$ Also, notice that by (\[3d\]) $$\sup_{x\in\partial D_{\lambda^2},|\zeta|=1}|u_{(\zeta)}(x)|\le N\Big(1+\frac{|\psi|_{1,D}}{\lambda^2}\Big)\Big(|f|_{1,D}+|g|_{1,D}\Big)\le N\Big(|f|_{1,D}+|g|_{1,D}\Big).$$ Therefore, $$\begin{aligned} |u_{(\xi_0)(\xi_0)}(x_0)|\le &Ee^{-\phi_{\tau_2}}|u_{(\xi_{\tau_2})(\xi_{\tau_2})}(x_{\tau_2})|+N\Big(|g|_{1,D}+|f|_{1,D}\Big)\\ &\cdot Ee^{-\phi_{\tau_2}}\Big(|\eta_{\tau_2}|+|\xi_{\tau_2}|^2+|\xi^0_{\tau_2}|^2+|\xi_{\tau_2}^{d+1}|^2+|\eta_{\tau_2}^{d+1}|+|\eta^0_{\tau_2}|\Big)\\ +&N|f|_{2,D}E\int_0^{\tau_2} e^{-\phi_s}\Big[|\xi_s|^2+|\eta_s|+|\xi^0_{s}|^2+|\xi_{s}^{d+1}|^2+|\eta_{s}^{d+1}|+|\eta^0_s|+r_s^2+\hat r_s\Big]ds\\ \le&Ee^{-\phi_{\tau_2}}|u_{(\xi_{\tau_2})(\xi_{\tau_2})}(x_{\tau_2})|+N\Big(|g|_{1,D}+|f|_{2,D}\Big)\\ &\cdot\Big(E\sup_{t\le\tau_2}e^{-\phi_t+\frac{1}{2}t}|\eta_t|+E\sup_{t\le\tau_2}e^{-\phi_t+\frac{1}{2}t}|\xi_t|^2+E\sup_{t\le\tau_2}e^{-\phi_t+\frac{1}{2}t}|\xi^0_t|^2\\ &\ \ \ \ + E\sup_{t\le\tau_2}e^{-\phi_t+\frac{1}{4}t}|\xi_t^{d+1}|^2+E\sup_{t\le\tau_2}e^{-\phi_t+\frac{1}{4}t}|\eta_t^{d+1}|\\ &\ \ \ \ +E\sup_{t\le\tau_2}e^{-\phi_t+\frac{1}{2}t}|\eta^0_t|+E\int_0^{\tau_2}e^{-\phi_s}\big(r_s^2+\hat{r}_s\big)ds\Big).\end{aligned}$$ By Lemma \[3l2\], Davis inequality and Hölder inequality, $$\begin{aligned} Ee^{-\phi_{\tau_2}}|u_{(\xi_{\tau_2})(\xi_{\tau_2})}(x_{\tau_2})|\le&\sup_{ x\in \partial D_{\lambda^2}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_2(x,\xi)}}\cdot Ee^{-\phi_{\tau_2}}\mathrm{B}_2(x_{\tau_2},\xi_{\tau_2})\\ \le&\sup_{x\in \partial D_{\lambda^2}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_2(x,\xi)}}\cdot {\mathrm{B}_2(x_0,\xi_0)},\\ E\sup_{t\le\tau_2}e^{-\phi_t+\frac{1}{2}t}|\eta_t|\le&N{\mathrm{B}_2(x_0,\xi_0)},\\ E\sup_{t\le\tau_2}e^{-\phi_t+\frac{1}{2}t}|\xi_t|^2\le&N{\mathrm{B}_2(x_0,\xi_0)},\\ E\sup_{t\le\tau_2}e^{-\phi_t+\frac{1}{2}t}|\xi^0_t|^2=&E\sup_{t\le\tau_2}\bigg|\int_0^te^{-\frac{1}{2}\phi_t+\frac{1}{4}t}\pi_sdw_s\bigg|^2\\ \le&4E\int_0^{\tau_2}e^{-\phi_t+\frac{1}{2}t}|\pi_t|^2dt\\ \le&NE\int_0^{\tau_2}e^{-\phi_t+\frac{1}{2}t}|\xi_t|^2dt\\ \le&N{\mathrm{B}_2(x_0,\xi_0)},\\ E\sup_{t\le\tau_2}e^{-\phi_t+\frac{1}{4}t}|\xi_t^{d+1}|^2\le&NE\sup_{t\le\tau_2}e^{-\phi_t+\frac{1}{4}t}\bigg(\int_0^t|\xi_s|ds\bigg)^2\\ \le&NE\sup_{t\le\tau_2}e^{-\frac{1}{4}t}\bigg(\int_0^te^{-\frac{1}{2}\phi_s+\frac{1}{4}s}|\xi_s|ds\bigg)^2\\ \le&NE\sup_{t\le\tau_2}e^{-\frac{1}{4}t}\cdot t\int_0^te^{-\phi_s+\frac{1}{2}s}|\xi_s|^2ds\\ \le&NE\int_0^{\tau_2}e^{-\phi_s+\frac{1}{2}s}|\xi_s|^2ds\\ \le&N{\mathrm{B}_2(x_0,\xi_0)},\\ E\sup_{t\le\tau_2}e^{-\phi_t+\frac{1}{4}t}|\eta_t^{d+1}|\le&NE\sup_{t\le\tau_2}e^{-\phi_t+\frac{1}{4}t}\int_0^t|\xi_s|^2+|\eta_s|ds\\ \le&NE\int_0^{\tau_2}e^{-\phi_s+\frac{1}{4}s}|\xi_s|^2ds+NE\sup_{t\le\tau_2}e^{-\frac{1}{4}t}\int_0^te^{-\phi_s+\frac{1}{2}s}|\eta_s|ds\\ \le&N{\mathrm{B}_2(x_0,\xi_0)}+NE\sup_{t\le\tau_2}e^{-\frac{1}{4}t}\cdot\sqrt t\bigg(\int_0^te^{-2\phi_s+s}|\eta_s|^2ds\bigg)^\frac{1}{2}\\ \le&N{\mathrm{B}_2(x_0,\xi_0)}+NE\bigg(\int_0^{\tau_2}e^{-2\phi_s+s}|\eta_s|^2ds\bigg)^\frac{1}{2}\\ \le&N{\mathrm{B}_2(x_0,\xi_0)},\\ E\sup_{t\le\tau_2}e^{-\phi_t+\frac{1}{2}t}|\eta^0_t|\le&E\sup_{t\le\tau_2}\bigg(\Big|\int_0^te^{-\frac{1}{2}\phi_t+\frac{1}{4}t}\pi_sdw_s\Big|^2+\int_0^te^{-\phi_t+\frac{1}{2}t}|\pi_s|^2ds\\ &+\Big|\int_0^te^{-\phi_t+\frac{1}{2}t}\hat{\pi}_sdw_s\Big|\bigg)\\ \le&5E\int_0^{\tau_2}e^{-\phi_s+\frac{1}{2}s}|\pi_s|^2ds+3E\Big(\int_0^{\tau_2}e^{-2\phi_t+t}|\hat\pi_s|^2\Big)^\frac{1}{2}ds\\ \le&N{\mathrm{B}_2(x_0,\xi_0)},\\ E\int_0^{\tau_2}e^{-\phi_s}\big(r_s^2+\hat{r}_s\big)ds\le &NE\int_0^{\tau_2}e^{-\phi_s}\big(|\xi_s|^2+|\eta_s|\big)ds\\ \le& N{\mathrm{B}_2(x_0,\xi_0)}+N\bigg(\int_0^{\tau_2}e^{-s}ds\bigg)^\frac{1}{2}\bigg(\int_0^{\tau_2}e^{-2\phi_s+s}|\eta_s|^2ds\bigg)^\frac{1}{2}\\ \le &N{\mathrm{B}_2(x_0,\xi_0)}.\end{aligned}$$ Collecting all estimates above, we conclude that $${|u_{(\xi_0)(\xi_0)}(x_0)|}\le\sup_{x\in \partial D_{\lambda^2}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_2(x,\xi)}}\cdot {\mathrm{B}_2(x_0,\xi_0)}+N(|g|_{1,D}+|f|_{2,D}){\mathrm{B}_2(x_0,\xi_0)}.$$ So for any $x_0\in D_{\lambda^2}$, $\xi_0\in{\mathbb{R}^d}\setminus\{0\}$, we have $$\label{3p} \frac{{|u_{(\xi_0)(\xi_0)}(x_0)|}}{{\mathrm{B}_2(x_0,\xi_0)}}\le\sup_{x\in \partial D_{\lambda^2}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_2(x,\xi)}}+N_2,$$ with $N_2$ defined by (\[N2\]). Then on $\{x\in D:\psi(x)=\lambda\}$, we have $$\begin{aligned} {\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}\le&\frac{1}{4}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_2(x,\xi)}}\\ \le&\frac{1}{4}(\sup_{\{\psi=\lambda^2\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_2(x,\xi)}}+N_2)\\ \le&\frac{1}{16}\sup_{\{\psi=\lambda^2\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}+\frac{N_2}{4}\\ \le&\frac{1}{16}(\sup_{\{\psi=\lambda\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}+\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}+N_2)+\frac{N_2}{4}\\ =&\frac{1}{16}\sup_{\{\psi=\lambda\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}+\frac{1}{16}\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}+\frac{5N_2}{16},\end{aligned}$$ which implies that $$\label{3q} \sup_{\{\psi=\lambda\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}\le\frac{1}{15}\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}+\frac{N_2}{3}.$$ Meanwhile, on $\{x\in D:\psi(x)=\lambda^2\}$, we have $$\begin{aligned} {\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_2(x,\xi)}}\le&\frac{1}{4}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}\\ \le&\frac{1}{4}(\sup_{\{\psi=\lambda\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}+\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}+N_2)\\ \le&\frac{1}{4}(\frac{1}{15}\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}+\frac{N_2}{3})+\frac{1}{4}\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}+\frac{N_2}{4}\\ =&\frac{4}{15}\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}+\frac{N_2}{3}.\end{aligned}$$ Therefore, $$\label{3r} \sup_{\{\psi=\lambda^2\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_2(x,\xi)}}\le\frac{4}{15}\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}+\frac{N_2}{3}.$$ Combining (\[3o\]) and (\[3q\]), we get, for any $x\in D_\delta^\lambda$, $\xi\in{\mathbb{R}^d}\setminus\{0\}$, $$\label{3s} {\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}\le\frac{16}{15}\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}+\frac{4N_2}{3}.$$ Combining (\[3p\]) and (\[3r\]), we get, for any $x\in D_{\lambda^2}$, $\xi\in{\mathbb{R}^d}\setminus\{0\}$, $$\label{3t} {\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_2(x,\xi)}}\le\frac{4}{15}\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}+\frac{4N_2}{3}.$$ Thus it remains to estimate $$\varlimsup_{\delta\downarrow0}\Big(\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}\Big) \mbox{ and }\varlimsup_{\delta\downarrow0}\sup_{x\in\partial D_\delta^\lambda,|\zeta|=1}|u_{(\zeta)}(x)|.$$ First, notice that $$\begin{aligned} \varlimsup_{\delta\downarrow0}\sup_{x\in\partial D_\delta^\lambda,|\zeta|=1}|u_{(\zeta)}(x)|\le&\sup_{x\in\partial D,|\zeta|=1}|u_{(\zeta)}(x)|+\sup_{x\in\{\psi=\lambda\},|\zeta|=1}|u_{(\zeta)}(x)|\\ \le&\sup_{x\in\partial D,|l|=1,l\parallel \partial D}|u_{(l)}(x)|+\sup_{x\in\partial D,|n|=1,n\perp \partial D}|u_{(n)}(x)|\\ &+\sup_{x\in\{\psi=\lambda\},|\zeta|=1}|u_{(\zeta)}(x)|,\end{aligned}$$ Apply Lemma \[lemma4\] and first derivative estimate (\[3d\]), we get $$\begin{aligned} \varlimsup_{\delta\downarrow0}\sup_{x\in\partial D_\delta^\lambda,|\zeta|=1}|u_{(\zeta)}(x)|\le&\sup_{x\in\partial D,|l|=1,l\parallel \partial D}|g_{(l)}(x)|+N(|g|_{2,D}+|f|_{0,D})\\ &+N\Big(1+\frac{|\psi|_{1,D}}{\lambda}\Big)(|g|_{1,D}+|f|_{1,D})\\ \le&N(|g|_{2,D}+|f|_{1,D}).\end{aligned}$$ Second, notice that for each $\delta$, there exist $x(\delta)\in\{\psi=\delta\}$ and $\xi(\delta)\in\{\xi:|\xi|=1\}$, such that $$\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}=\frac{|u_{(\xi(\delta))(\xi(\delta))}(x(\delta))|}{\mathrm{B}_1(x(\delta),\xi(\delta))}.$$ A subsequence of $(x(\delta),\xi(\delta))$ converges to some $(y,\zeta)$, such that $y\in\partial D$ and $|\zeta|=1$. If $\psi_{(\zeta)}(y)\ne0$, then $\mathrm{B}_1(x(\delta),\xi(\delta))\rightarrow\infty$ as $\delta\downarrow0$. In this case, $$\varlimsup_{\delta\downarrow0}\Big(\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}\Big)=\varlimsup_{\delta\downarrow0}\frac{|u_{(\xi(\delta))(\xi(\delta))}(x(\delta))|}{\mathrm{B}_1(x(\delta),\xi(\delta))}=0.$$ If $\psi_{(\zeta)}(y)=0$, then $\zeta$ is tangential to $\partial D$ at $y$. In this case, $$\varlimsup_{\delta\downarrow0}\Big(\sup_{\{\psi=\delta\}}{\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}}\Big)=\varlimsup_{\delta\downarrow0}\frac{|u_{(\xi(\delta))(\xi(\delta))}(x(\delta))|}{\mathrm{B}_1(x(\delta),\xi(\delta))}=\frac{|g_{(\zeta)(\zeta)}(y)|+K|u_{(n)}(y)|}{\lambda}.$$ By Lemma (\[lemma4\]), we have $$\frac{|g_{(\zeta)(\zeta)}(y)|+K|u_{(n)}(y)|}{\lambda}\le N(|g|_{2,D}+|f|_{0,D}).$$ Therefore, we have $$\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_1(x,\xi)}\le N(|f|_{2,D}+|g|_{2,D}), \mbox{ when }x\in D^\lambda;$$ $$\frac{|u_{(\xi)(\xi)}(x)|}{\mathrm{B}_2(x,\xi)}\le N(|f|_{2,D}+|g|_{2,D}), \mbox{ when }x\in D_{\lambda^2}.$$ It follows that, for any $x\in D$ and $\xi\in{\mathbb{R}^d}$, $$|u_{(\xi)(\xi)}(x)|\le N(|\xi|^2+\frac{\psi_{(\xi)}^2}{\psi})(|f|_{2,D}+|g|_{2,D}).$$ The inequality (\[3dd\]) is proved. The fact that $u$ given by (\[1b\]) satisfies (\[solva\]) follows from Theorem 1.3 in [@MR617995]. To prove the uniqueness, assume that $u_1, u_2\in C_{loc}^{1,1}(D)\cap C^{0,1}(\bar D)$ are solutions of (\[solva\]). Let $\Lambda=|u_1|_{0,D}\vee|u_2|_{0,D}$. For constants $\delta$ and ${\varepsilon}$ satisfying $0<\delta<{\varepsilon}<1$, define $$\Psi(x,t)={\varepsilon}(1+\psi(x))\Lambda e^{-\delta t},\ U(x,t)=u(x) e^{-{\varepsilon}t}\mbox{ in }\bar D\times(0,\infty),$$ $$F[U]=U_t+LU-cU+f \mbox{ in } D\times (0,\infty).$$ Notice that a.e. in $D$, we have $$F[U_1-\Psi]=-{\varepsilon}e^{-{\varepsilon}t}u_1+\delta\Psi-{\varepsilon}\Lambda e^{-\delta t}L\psi+c\Psi\ge{\varepsilon}\Lambda(e^{-\delta t}-e^{-{\varepsilon}t})\ge0,$$ $$F[U_2+\Psi]={\varepsilon}e^{-{\varepsilon}t}u_2-\delta\Psi+{\varepsilon}\Lambda e^{-\delta t}L\psi-c\Psi\le{\varepsilon}\Lambda(e^{-{\varepsilon}t}-e^{-\delta t})\le0.$$ On $\partial D\times (0,\infty)$, we have $$U_1-U_2-2\Psi=-2\Psi\le0.$$ On $\bar D\times T$, where $T=T({\varepsilon},\delta)$ is a sufficiently large constant, we have $$U_1-U_2-2\Psi=(u_1-u_2)e^{-{\varepsilon}T}-2{\varepsilon}(1+\psi)\Lambda e^{-\delta T}\le2\Lambda(e^{-{\varepsilon}T}-{\varepsilon}e^{-\delta T})\le 0.$$ Applying Theorem 1.1 in [@MR538554], we get $$U_1-U_2-2\Psi\le 0 \mbox { a.e. in } \bar D\times(0,T).$$ It follows that $$u_1-u_2\le 2{\varepsilon}(1+\psi)\Lambda e\rightarrow0, \mbox{ as } {\varepsilon}\rightarrow0, \mbox{ a.e. in }D.$$ Similarly, $u_2-u_1\le0$ a.e. in $D$. The uniqueness is proved. Based on our proof, if we replace $\sigma(x), b(x), c(x), f(x)$ and $g(x)$ in (\[1aa\]) and (\[1b\]) by $\sigma(\omega,t,x),b(\omega,t,x),c(\omega,t,x), f(\omega,t,x)$ and $g(\omega,t,x)$, defined on $\Omega\times[0,\infty)\times D$, under appropriate measurable assumptions, the first and second derivative estimates (\[3d\]) and (\[3dd\]) are still true. Acknowledgements {#acknowledgements .unnumbered} ================ The author is sincerely grateful to his advisor, N.V. Krylov, for introducing this method to the author and giving many useful suggestions and comments on the improvements. The author also would like to thank the anonymous referees whose helpful comments improved the quality of this manuscript. [10]{} M. I. Fre[ĭ]{}dlin, *Functional integration and partial differential equations*, Annals of Mathematics Studies, vol. 109, Princeton University Press, Princeton, NJ, 1985. 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--- abstract: 'This article reviews progress in the theoretical modelling of the electronic structure of rotationally faulted multilayer graphenes. In these systems the crystallographic axes of neighboring layers are misaligned so that the layer stacking does not occur in the Bernal structure observed in three dimensional graphite and frequently found in exfoliated bilayer graphene. Notably, rotationally faulted graphenes are commonly found in other forms of multilayer graphene including epitaxial graphenes thermally grown on ${\rm SiC \, (000 \bar 1)}$, graphenes grown by chemical vapor deposition, folded mechanically exfoliated graphenes, and graphene flakes deposited on graphite. Rotational faults are experimentally associated with a strong reduction of the energy scale for coherent single particle interlayer motion. The microscopic basis for this reduction and its consequences have attracted significant theoretical attention from several groups that are highlighted in this review.' address: | Department of Physics and Astronomy\ David Rittenhouse Laboratory\ University of Pennsylvania\ Philadelphia, PA 19104 author: - 'E. J. Mele' title: Interlayer coupling in rotationally faulted multilayer graphenes --- Introduction ============ Coherent interlayer motion in multilayer graphenes play a crucial role in determining their low energy electronic properties. In single layer graphene the absence of the layer degree of freedom cleanly exposes the geometrical structure of its low energy electronic physics. This is controlled by single particle spectra containing linearly dispersing bands around singular points at its inequivalent zone corners, described by a pair of valley-polarized two dimensional massless Dirac Hamiltonians [@berrygeim; @berrykim]. The physics is very different for graphene bilayers that are stacked in the Bernal geometry with the “A" sublattice of one layer eclipsed with the “B" sublattice of its neighbor ($AB$ stacking). Here the effects of coherent interlayer coupling are quite strong and the low energy sector is described instead by a [*different*]{} class layer-coherent chiral fermions with a quadratic dispersion and a Berry’s phase of $2 \pi$ for reciprocal space orbits that encircle the point of degeneracy [@McCFal]. This physics is readily understood from the experimentally known strength of the interlayer tunneling amplitude at eclipsed sites and it can be generalized to describe the low energy physics of multi-layer graphenes where the crystallographic axes of neighboring layers are rotated by special angles $\theta = n \pi/3$ [@koshino; @minmac; @minmac2; @pablotrilayer]. Surprisingly, experimental work over the last five years has revealed a family of multilayer graphenes that show only weak (if any) effects of their interlayer coupling. This family includes graphenes that are grown epitaxially on the ${\rm SiC \, (000 \bar 1)}$ surface [@bergerepitax; @deheerreview; @haas], CVD grown graphenes [@reina] and some forms of exfoliated graphene [@schmidt; @STMgraphite; @graphenegraphite]. A common structural attribute of these systems is a rotational misorientation (a twist) of their neighboring layers at angles of $\theta \neq n \pi/3$. The layer decoupling has been inferred from the measurements of the magnetotransport [@bergerepitax; @deheerreview], of the Landau level spectra observed in scanning tunneling spectroscopy [@graphenegraphite; @miller] and perhaps most clearly in angle-resolved photoemission spectra [@sprinkle]. These experimental observations are attracting significant theoretical attention. The layer decoupling in twisted multilayers is frequently attributed to a kinematical effect whereby the layer projections of the zone corner crystal momenta are [*misaligned*]{} by the rotation, preventing momentum-conserving interlayer motion at sufficiently low energy [@JMLdS]. In this scenario the low energy theory is described by four separate valley- and layer- polarized Dirac cones. These fermions are then re-coupled at a crossover energy scale where the individual Dirac cones merge and hybridize thus changing the band topology [@vanHove]. The simplest version of this theory predicts that the residual low energy effect of the twist is to reduce the Fermi velocities by an angle dependent factor where the smallest velocities are expected for small rotation angles. These theoretical predictions have provided a taking off point for further investigations of this problem using a variety of methods ranging from microscopic atomistic calculations to continuum models designed to capture selected elements of the microscopic physics. Presently there is a lively discussion concerning the theoretical interpretation of the electronic physics in twisted graphenes: What is the appropriate long wavelength theory? How can one distinguish between the electronic physics for “small" and “large" rotation angles? How does the the interlayer coherence scale depend on the fault angle? What are the experimental consequences of the weak interlayer coherence? It is fair to say that the one-electron physics of these systems is proving to be unexpectedly rich and it has so far eluded a satisfactory (or at least complete) theoretical description. In this article we briefly highlight some recent theoretical progress on this problem and focus on some of the major unresolved issues. Section 2 presents a discussion of the geometric properties of rotationally faulted bilayers which are generally useful for analyzing their structural and electronic properties. The results presented here provide a foundation for a theoretical analysis we have presented earlier [@ejmrc] though these details have not been published previously. Sections 3-5 briefly review the existing theoretical approaches that have been developed for describing the electronic structure of these systems. Section 3 reviews the essential features of a long wavelength theory of layer that illustrates the physics of layer decoupling by “rotational mismatch" [@JMLdS]. Section 4 presents some highlights of microscopic atomistic calculations on these systems and Section 5 briefly reviews the content of several “second generation" continuum theories that refine the original theoretical proposal. Section 6 summarizes with a discussion of the connections of these theories to experiment and points to some interesting open problems. Geometrical Considerations ========================== Lattice Structures ------------------ A twisted graphene bilayer can be characterized by a relative rotation of the symmetry axes of its two layers through angle $\theta$ and a rigid translation $\vec \Delta$. Holding one layer fixed, a rotation about the point $\vec r_0$ maps coordinates $\vec r$ in the fixed layer to positions $\vec r'$ in the rotated layer in the manner $$\begin{aligned} \vec r' = {\cal R} (\theta) \cdot \left(\vec r - \vec r_0 \right) + \vec \Delta\end{aligned}$$ where ${\cal R}(\theta)$ is the two-dimensional rotation operator. For definiteness one can consider the situation where the rotation is taken about a lattice site and the relative shift $\Delta=0$. A commensurate rotation occurs when a lattice translation of the unrotated layer $\vec T_{mn} = m \vec a_1 + n \vec a_2$ spanned by its two primitive layer translations $\vec a_1$ and $\vec a_2$ and the mapping of an [*inequivalent*]{} translation $\vec T_{m'n'}$ (in the same star) are equal. This occurs only at discrete angles $\theta_{mn}$ that can be indexed by the two integers $$\begin{aligned} e^{i \theta_{mn}} = \frac{m e^{-i \pi/6} + n e^{i \pi / 6}}{n e^{-i \pi/6} + m e^{i \pi / 6}}\end{aligned}$$ Small angle rotations have large $m$ and $n=m+1$. These small angle faults describe large period superlattices where the atomic registry can be regarded as evolving smoothly between widely separated regions with locally Bernal-like and $AA$ like stacking. Complementary structures with large $m$ and $n=1$ correspond to small angular deviations from the $60^\circ$ rotated structure. Since the combination of a $60^\circ$ degree rotation and a translation by a nearest neighbor bond vector is a symmetry operation of the honeycomb lattice, a commensurate rotation near $60^\circ$ can be regarded as the superposition of a small angle rotation and a nonprimitive translation. The primitive translations $\vec A$ of a general $(m,n)$ commensuration supercell are $$\begin{aligned} \left( \begin{array}{c} \vec A_1\\ \vec A_2 \\ \end{array} \right) = \left( \begin{array}{cc} m & n \\ -n & m+n \\ \end{array} \right) \left( \begin{array}{c} \vec a_1\\ \vec a_2 \\ \end{array} \right)\end{aligned}$$ with length $|\vec A| = \sqrt{m^2 + n^2 + mn}$. Commensuration pairs at angles $\theta$ and $\bar \theta = 60^\circ - \theta$ are related. The simplest example of such a pair occurs trivially for $\theta = 0$ and $\bar \theta = 60^\circ$ which correspond to the smallest possible unit cells with $AB$ (Bernal) stacking and $AA$ (perfectly eclipsed) stacking. The unit cells of these structures have the same area but they have different sublattice symmetries. Importantly, all commensurate rotations share this property: they occur in partners where the sum of the rotation angles is $60^\circ$ and their unit cells have the same area. The commensuration indices $(m,n)$ and $(\bar m, \bar n)$ of the partners are related $$\begin{aligned} \left( \begin{array}{c} \bar m \\ \bar n \\ \end{array} \right) = \left( \begin{array}{cc} -1 & 1 \\ 2 & 1 \\ \end{array} \right) \left( \begin{array}{c} m\\ n \\ \end{array} \right)\end{aligned}$$ eliminating common divisors by 3 from the result. Figure 1 illustrates this situation where the structure in the left panel corresponds to $(m,n) = (1,3)$, $\theta = 32.204^\circ$ and on the right $(\bar m, \bar n) = (2,5)$, $\bar \theta = 27.796^\circ$. Partner commensurations can also be transformed into each other by a translation $\vec \Delta$ at a [*single*]{} rotation angle $\theta$ demonstrating the invariance of the primitive cell area. The structure at $\theta = \bar \theta = 30^\circ$ is its own commensuration partner and corresponds to an elementary two dimensional quasicrystalline lattice. Note also that the form of Eqn. 1 demonstrates that the indices $(m,n)$ generally provide a more useful specification of the structure than the fault angle $\theta$. Indeed nearby rotation angles can have very different fault indices and therefore describe crystalline structures with vastly different periods and different physical properties. A plot of the commensuration periods $|\vec A|$ as a function of rotation angle $\theta$ shows a complex distribution of allowed periods which is bounded from below. This lower bound has a smooth dependence on $\theta$, diverges as $\theta \rightarrow (0^\circ, 30^\circ, 60^\circ)$ and is symmetric around the self dual state at $30^\circ$. ![\[lattice\] Two lattice structures for rotationally faulted graphene bilayers at complementary rotation angles. Red and blue dots denote atomic positions in different layers. The highlighted rhombus is a primitive commensuration cell. The figure compares the stacking patterns for commensuration pairs that are related by $\bar{\theta} =60^\circ - \theta$. The structures are (left) $\theta=32.204^\circ$ ($m=1,n=3$) and (right) $27.796^\circ$ ($m=2,n=5$). The commensuration cells are the same for the partner structures but the point symmetry is different.](twolattices.jpg){width="100mm"} Commensuration partners are distinguished by their [*sublattice exchange parity*]{}. A commensuration is sublattice exchange “even" if the commensuration cell contains an $A$ and a $B$ sublattice site in each layer that are coincident with atomic sites in the neighboring layer. A commensuration is sublattice exchange “odd" if only one sublattice site in the commensuration cell is eclipsed. (Fixing the rotation center of the twist at an atom site guarantees that there will be at least one coincident site.) The sublattice exchange parity can be deduced from the translation indices $(m,n)$. It is convenient to label the eclipsed sites at the origin as the $A$ sublattice, a nearest neighbor bond vector $\vec \tau$ and its partner in the rotated layer $\tau'$. Then the condition for a second coincident site on the $B$ sublattice is $$\begin{aligned} \vec T + \vec \tau = \vec T' + \vec \tau'\end{aligned}$$ for some possible choice of $\vec T(\vec T')$ in the set of lattice translations in the reference(rotated) layers. Since the $\vec T$’s are both lattice translations, this requires integer $(p,q)$ solutions to $$\begin{aligned} e^{i \theta_{mn}} = \frac{ 1 + \sqrt{3} \left(p e^{i \pi/6} + q e^{-i \pi/6} \right)}{ 1 + \sqrt{3} \left(p e^{-i \pi/6} + q e^{i \pi/6} \right)} = \frac{ m e^{i \pi/6} + n e^{-i \pi/6}}{ m e^{-i \pi/6} + n e^{i \pi/6} }\end{aligned}$$ which can be expressed $$\begin{aligned} p = \frac{m - n + 3mq}{3n}\end{aligned}$$ and has integer solutions only when $m-n$ is divisible by $3$. When this is statisfied the coincident sites occur at special high symmetry points in the cell $\vec A_c = \pm \vec A_{mn}/3$ (with only one sign per structure) and correspond to high symmetry positions along the diagonal of the rhombus shown in the right hand panel of Figure 1. When $m-n$ is not divisible by $3$ the only coincident site occurs at the center of rotation and its supercell translates. Reciprocal Space ---------------- Similar considerations apply to the momentum space representation of the twisted bilayer for which Figure 2 gives a map illustrating the structure of its reciprocal space. The reciprocal lattice of the commensuration supercell can be treated as a conventional triangular lattice spanned by two primitive vectors $2 \pi (\hat e_z \times \vec A_i)/{\cal A}$ where ${\cal A} = |\vec A_1 \times \vec A_2|$ is the area of the commensuration supercell and $\hat e_z$ is the layer normal. However, it is often useful to observe that since the real space lattice translations of the supercell are coincident lattice translations of each of the layers, its reciprocal space can also be indexed by a reciprocal lattice spanned by momenta with [*four*]{} integer indices describing all linear combinations of the primitive reciprocal lattice translations of each of the layers, in the manner $$\begin{aligned} \vec {\cal G}_{p,q,p',q'} = p \vec G_1 + q \vec G_2 + p' \vec G_1' + q' \vec G_2'\end{aligned}$$ This demonstrates that the primitive $\vec G$’s and $\vec G'$’s and all possible combinations $\vec G + \vec G'$ are in the reciprocal lattice of the faulted structure. The smallest nonzero combinations of these vectors have length $4 \pi/\sqrt{3 (m^2 + mn +n^2)}$ and span the first star of reciprocal lattice vectors of the commensuration supercell. ![\[recipspace\] A reciprocal space map for a twisted graphene bilayer illustrating the rotation of the first star of reciprocal lattice vectors (open dots) to a star of rotated reciprocal lattice vectors (filled blue dots), and a corresponding rotation of the zone corner $K$ points (red dots). The offset points $K$ and $K(\theta)$ become coincident in the extended zone after translations by a particular pair of reciprocal lattice vectors.](recip1lines.jpg){width="40mm"} A critical question is whether the momentum offset $\vec K(\theta)- \vec K$ or $K(\theta) - \vec K'$ are [*also*]{} in the reciprocal lattice of the commensuration cell. For the former situation this is the question of whether $$\begin{aligned} \vec K(\theta) - \vec K = \vec {\cal G}_{p,q,p',q'} = p \vec G_1 + q \vec G_2 + p' \vec G_1' + q' \vec G_2' \, (?)\end{aligned}$$ for some choice of integers $(p,q,p',q')$. Representing these two dimensional vectors by complex numbers one finds that Eqn. 9 can be expressed $$\begin{aligned} e^{i \theta_{mn}} = \frac{ 1 + \sqrt{3} \left(p e^{i \pi/6} + q e^{-i \pi/6} \right)}{ 1 + \sqrt{3} \left(p' e^{-i \pi/6} + q' e^{i \pi/6} \right)}\end{aligned}$$ where $\theta_{mn}$ is given by Eqn. 2. Nontrivial solutions invert the indices $p'=q$ and $q'=p$ and lead to the matching condition $$\begin{aligned} p = \frac{m-n - 3mq}{3n}\end{aligned}$$ Thus $\vec K(\theta) - \vec K$ is in the reciprocal lattice only for [*supercommensurate*]{} structures where $m-n$ is a multiple of 3. Eqn. 11 is identical to Eqn. 7 that identifies the even sublattice exchange commensurations, so that sublattice “even" structures always allow intravalley interlayer coupling. For example, when $(m,n) = (2,5)$ we have $\theta = 27.796^{\circ}$ and the first integer solutions to Eqn. 10 occur for $q=3$ for which $p=1$ and $(p',q') = (3,1)$. The existence of this solution implies that these $K$ points are coincident in the extended zone after translations by $p \vec G_1 + q \vec G_2$ and $p' \vec G'_1 + q \vec G'_2$ as illustrated in the right panel of Fig. 3. One can also ask about the possibility of commensurability for intervalley momentum transfer $K(\theta) - K'$, namely $$\begin{aligned} \vec K(\theta) - \vec K' = \vec {\cal G}_{p,q,p',q'} = p \vec G_1 + q \vec G_2 + p' \vec G_1' + q' \vec G_2' \, (?)\end{aligned}$$ Following a similar line of analysis one finds a different set of commensurability conditions $$\begin{aligned} p &=& \frac{m(q+1) -nq}{m+2n} \nonumber\\ p' &=& \frac{nq -m(1+q)}{m+2n} \nonumber\\ q' &=& \frac{(2m+n)q + m}{m+2n}\end{aligned}$$ Thus for example, $m=1$, $n=3$ gives a rotation angle $\theta = 32.204^\circ$ which has its first integer solution when $\tilde q = q = 4$, giving $(p,q) = (-1,4)$ and $(p',q') = (1,3)$. Notice the asymmetry between the values of $(p,q)$ and $(p',q')$: the scattering between inequivalent Dirac cones requires [*different*]{} umklapp terms when indexed to the individual reciprocal lattices of the two layers. The indices would be reversed by considering $K \rightarrow K'(\theta)$ couplings. This matching rule implies that $\vec K(\theta)$ and $\vec K'$ are coincident in after translations by $p \vec G_1 + q \vec G_2$ and $p' \vec G'_1 + q \vec G'_2$ as illustrated in the left panel of Fig. 3. ![\[recipspace2\] Commensurability conditions in reciprocal space for partner commensurations at (left) $\theta = 21.787^\circ$ ($(m,n) = 1,3$) and (right) $\theta = 38.213^\circ$ ($(m,n) = (2,5)$). In the left panel the lattice structure is odd under sublattice exchange and the offset $K(\theta) - K'$ (red dots) is in the reciprocal lattice of the commensuration cell. Translation by particular layer reciprocal lattice vectors (spanned by the layer reciprocal lattice vectors and their rotated counterparts, shown as the blue and violet points, respectively) brings these two momenta into coincidence in the extended zone. In the right panel the lattice structure is even under sublattice exchange and the offset $K(\theta) - K$ is in the reciprocal lattice of the commensuration cell. Translation by different layer reciprocal lattice vectors brings these two momenta into coincidence in the extended zone. These two commensuration conditions are complementary and mutually exclusive.](newfig3.jpg){width="150mm"} One can prove that Eqns. 9 and 12 cannot be simultaneously satisfied for a common rotation angle. For example if $m=3 \mu + 1$ and $n = 3 \nu + 1$ then $m-n$ is a multiple of 3 and intravalley couplings are in the reciprocal lattice. In this situation Eqn. 13 requires that $$\begin{aligned} p = \frac{3(\mu q - \nu q + \mu) + 1}{3(\mu + 2 \nu +1)}\end{aligned}$$ which is impossible since the numerator is never divisible by 3 so that intervalley coupling is excluded. On the other hand, when $m=3 \mu \pm 1$ and $\nu = 3 \nu \mp 1$ intravalley are excluded and $$\begin{aligned} p = \frac{(3(\mu - \nu)q + \mu) + 2q +1}{3(\mu + 2 \nu) \mp 1 }\end{aligned}$$ which can be satisfied for integer $p$ by appropriate choice of integer $q$ so that intervalley coupling is allowed. These two possibilities are thus complementary and mutually exclusive: one or the other must occur if the rotation is commensurate. Using the indexing rule Eqn. 4, one can easily show that partner commensurations realize complementary commensuration conditions: one member admits only the intravalley interlayer coupling while the other allows only the analogous intervalley scattering. Thus the valley structure of the interlayer couplings are specified by the sublattice exchange symmetry of the structure. Layer Decoupling by Rotational Mismatch ======================================= Early work on the electronic properties of twisted graphene bilayers recognized that small angle rotational faults are inevitably described by large period commensuration cells that make atomistic calculations impractical. Instead it is useful to develop a long wavelength description that captures the effect of rotation on the low energy electronic structure. The essential physics in this treatment is the momentum offset of the Dirac nodes produced by the rotation [@JMLdS] as illustrated in Fig. 2. The starting point of the continuum theory is the long wavelength theory appropriate to [*single layer*]{} graphene. The effective mass theory for electrons in each valley introduces two Dirac Hamiltonians for their smoothly varying pseudo-spinor fields $$\begin{aligned} H_K = - i \hbar v_F \sigma \cdot \nabla; \,H_{K'} = \sigma_y H_K \sigma_y\end{aligned}$$ where the $\sigma$’s are $2\times 2$ Pauli matrices acting on the sublattice amplitudes. A small angle relative rotation of the crystallographic axes in the of the two layers offsets the crystal momenta of their closest Dirac nodes by $\Delta K = 2K \sin (\theta/2)$. This can be described by a pair of layer-polarized Dirac Hamiltonians parameterized by the momentum offset $\Delta \vec K = \vec K(\theta) - \vec K$ in the manner $$\begin{aligned} H_K &=& \hbar v_F \sigma \cdot (-i \nabla - \frac{\Delta \vec K}{2})\nonumber\\ H_{K(\theta)} &=& \hbar v_F \sigma^\theta \cdot (-i \nabla + \frac{\Delta \vec K}{2})\end{aligned}$$ where $\sigma_\mu^\theta = \exp(i \sigma_z \theta/2) \sigma_\mu \exp(-i \sigma_z \theta/2)$ because of the relative rotation of the two layers. Electrons in neighboring layers are coupled by a $\theta$-dependent interlayer coupling amplitude projected into the pseudospin basis. For small angle rotations these interlayer amplitudes vary smoothly in real space and one can focus on their lowest Fourier components. In the theory of Lopes dos Santos [*et al.*]{} [@JMLdS] the offset momentum $\Delta \vec K$ is not in the reciprocal lattice of the commensuration cell and these authors focus on the three momenta $\vec {\cal G}_i$ that leave the offset $|\vec K(\theta) - \vec K - \vec {\cal G}_i|$ invariant. These momenta can be expressed in terms of the offset $\Delta K$: in complex notation they are ${\cal G}_i = (0,{\cal G}_1 = \sqrt{3}e^{i \pi/6} \Delta K,{\cal G}_1 + {\cal G}_2 = \sqrt{3} e^{-i \pi/6} \Delta K)$. In the pseudospin basis, the interlayer coupling for each of these momentum transfers is characterized by a $2 \times 2$ matrix-valued tunneling coefficient $T({\cal G})$ whose elements have been estimated numerically using a tight binding model. This yields in the small angle limit $$\begin{aligned} T({\cal G}=0) = \tilde t_{\perp} \left(\begin{array}{cc} 1 & 1 \\ 1 & 1 \\ \end{array}\right) ; \,\,\, T({\cal G}= -{\cal G}_1) = \tilde t_{\perp} \left(\begin{array}{cc} z & 1 \\ \bar z & z \\ \end{array}\right) \nonumber\\ \,\,\,\,\,\, T({\cal G} = -({\cal G}_1 + {\cal G}_2) )= \tilde t_{\perp} \left(\begin{array}{cc} \bar z & 1 \\ z & \bar z \\ \end{array}\right)\end{aligned}$$ where $z=e^{2 \pi i /3}$, $\bar z = e^{-2 \pi i/3}$ and $\tilde t_{\perp} \sim 0.11 \, {\rm eV}$, approximately independent of the supercell period. The asymmetry in the set of selected $\vec {\cal G}$’s appearing in Eqn. 18 occurs because of the choice of the reference valley for the long wavelength expansion. Nevertheless, this approach explicitly preserves the threefold rotational symmetry of the commensuration cell. This is seen most clearly by observing that the $T$ matrices are off diagonal operators in the layer degree of freedom and one may therefore arbitrarily “shift" the interlayer coupling momenta by a layer-dependent $U(1)$ gauge transformation. In particular the gauge shift $e^{-i \Delta \vec K \cdot \vec r}$ in the rotated layer brings the two Dirac nodes into coincidence and shifts the three momentum transfers so that they form the three arms of a star generated by $Q=-\Delta K$ and its $\pm 2 \pi/3$-rotated partners. The negates of these three momenta occur in the reciprocal amplitudes describing the reverse tunneling processes. Thus the expansion about a [*single*]{} zone corner point preserves the full three fold symmetry of the commensuration cell, as required The essential features of this theory are (1) the existence of a crystal momentum offset due to the rotational fault, (2) the coupling of plane wave states in one layer to a triad of plane wave states in its neighbor and (3) the existence of a ${\cal G}=0$ term in the effective interlayer tunneling Hamiltonian. Feature (1) suggests that at sufficiently low energy the effect of the interlayer coupling can be treated perturbatively in the dimensionless coupling parameter $\Gamma = \tilde t_{\perp}/\hbar v_F \Delta K$. Feature (2) implies the perturbative effects of this coupling [*vanish by symmetry*]{} precisely at $E=0$ so that the coupled system preserves the Dirac nodes of its two (decoupled) layers. Perturbative effects of the coupling arise at linear order in the momentum differences $\vec k \pm \Delta \vec K/2$ and can be interpreted as a twist dependent renormalization of the Fermi velocity $$\begin{aligned} \frac{v^*_F}{v_F} = 1 - 9 \left( \frac{\tilde t_{\perp}}{\hbar v_F \Delta K} \right)^2\end{aligned}$$ Equation 19 needs to be applied with care since it breaks down both in the limit of small rotation angles due to a failure of the perturbation theory when $\Delta K \rightarrow 0$ and at large rotation angles when commensuration effects, neglected in this treatment, can intervene. Finally, feature (3) indicates that electron states in the two layers that have the [*same*]{} crystal momentum modulo $\vec {\cal G}$ are coupled through the interlayer Hamiltonian. In the low energy theory the layer-polarized Dirac cones degenerate in the planes that bisect lines connecting their nodes (above the crossover energy $\hbar v_F \Delta K/2$) and one expects the strongest interlayer mixing to occur in these planes. There are three such planes that bisect the lines along $\Delta K$ and its $\pm 2 \pi/3$ rotated counterparts. The onset of this mixing is associated with a change of topology of the bilayer bands, connecting a low energy sector with layer-decoupled Dirac cones to higher energy layer-coherent hyperbolic bands. In the lowest band this transition is associated with a saddle point in the electronic spectrum and a logarithmic van Hove singularity in the two dimensional density of states [@vanHove]. Atomistic Calculations ====================== The novel physics of rotationally-induced layer decoupling has stimulated theoretical work by several groups to explore this effect using various atomistic models. Ab initio calculations have been carried out for misaligned bilayer supercells containing up to $\sim 500$ atoms ($\theta \sim 5^\circ$) while tight binding methods have allowed workers to access larger systems of up to $15000$ atoms [@haas; @Latil; @GTdL; @Shallcross; @turbo]. These studies have examined the Fermi velocity renormalization, the form of the low energy electronic spectrum near the $K$ points and the spatial modulation of the electronic charge density. Much of the ab-initio work has understandably focused on the shortest period twisted structures, e.g. $\sqrt{7} \times \sqrt{7}$ and $\sqrt{13} \times \sqrt{13}$ commensurations [@haas; @Shallcross]. Calculations on these systems generally confirm a suppression of the interlayer coupling scale and a Fermi velocity near the $K$ point which is essentially indistinguishable from that of single layer graphene. The most thorough investigation of the Fermi velocity renormalization has been given by de Laissardi$\grave{\rm e}$re [*et al.*]{} [@GTdL] who suggest that the rotational faults are characterized by three different velocity renormalization regimes, determined by the fault angle: (a) $ 15^\circ < \theta < 30^\circ$ where the Fermi velocity is essentially the same as for single layer graphene, (b) $3^\circ < \theta < 15^\circ$ where a downward renormalization is found, well described by the perturbation theory of [@JMLdS], and a low angle regime $\theta < 3^\circ$ where the low energy bands are flattened and not described by the perturbative treatment. The small renormalization in the large fault angle regime (a) is at least qualitatively consistent with the continuum theory since the renormalization occurs via a virtual mixing of low energy states with states separated by an energy barrier $\hbar v_F \Delta K$. The breakdown of the perturbation theory for sufficiently small angle faults is similarly understandable since it involves an expansion in $\tilde t_{\perp}/\hbar v_F \Delta K$. Surprisingly, in this low angle regime de Laissardi$\grave{\rm e}$re [*et al.*]{} also report a pronounced spatial modulation of the low energy eigenstates that tends to localize their charge densities in spatial regions locally characterized by “AA" stacking, suggesting some form of multiband physics that is not captured by the truncated continuum model. The accuracy of the perturbation theory in the intermediate regime has been further questioned by the density functional calculations of Shallcross [*et al.*]{} [@Shallcross] who find that the bilayer $v_F$ is nearly equal to that of single layer graphene down to smallest angles ($\sim 9^\circ$) they were able to study. The calculations by Shallcross [*et al.*]{} [@Shallcross] also reveal features in the electronic spectra near the $K$ points that are not captured by the primitive continuum theory. Significantly, close to the zone corners the electronic bands are [*not*]{} linear but instead they are mixed, which requires an interlayer mass operator in the low energy Hamiltonian. Interestingly the spectral structure, and therefore the matrix structure of this mass term, is [*different*]{} for partner commensurations and therefore it cannot be determined solely by the size of the commensuration cell. The scale of the mixing is nevertheless small relative to its value for Bernal bilayers, e.g. the mixing scale for the $\theta = 30^\circ \pm 8.213^\circ$ structure is $\approx 7 \, {\rm meV}$ compared to $\approx 0.2 \, {\rm eV}$ for Bernal stacking [@McCFal; @koshino; @biasedbilayer]. Further, over the range of structures they studied this mass scale appears to a rapidly decreasing function of the commensuration cell period. But the existence of this mass matrix in the low energy theory presents a significant challenge to the interpretation of the electronic states even at energies [*above*]{} the mass scale. Notably, in order to match smoothly to these low energy eigenstates the bilayer eigenstates at higher energy must be (nearly) [*equal weight*]{} states coherently mixed between the two layers instead of the layer-polarized eigenstates that one would infer from the momentum space structure of the continuum theory . Second Generation Continuum Theories ==================================== There has been progress in the development of new long wavelength models that extend the physics identified in the original continuum formulation [@JMLdS]. These theories examine the effects of lattice commensuration [@ejmrc] and of multi-band mixing [@bismac1] on the low energy electronic structure. The former turn out to be most important for large angle faults while the latter are critical to the physics at small rotation angles. The new models are also formulated as continuum theories in order to circumvent the technical difficulty posed by fully microscopic atomistic treatments of large commensuration cells. Concurrently there has been an effort to distill the original continuum model to a simpler effective two band model [@saddledeGail; @saddleLL] in an effort to explore the effects of the novel band topology on the orbital quantization of its electronic states in a perpendicular magnetic field. We refer to all these new models as “second generation" continuum theories. Interlayer Matrix Elements -------------------------- A microscopic theory of the interlayer coupling can be formulated in the basis of Bloch orbitals $$\begin{aligned} \psi_\alpha (\vec k) = \frac{1}{\sqrt{N}} \sum_{\vec T} e^{i \vec k \cdot (\vec T + \vec \tau_\alpha)} \phi_\alpha (\vec T)\end{aligned}$$ where $\phi_{\alpha = (A,B)}$ are orbitals centered at positions $\vec T + \vec \tau_{\alpha}$ and $\vec T$ is a lattice translation. In this basis the interlayer Hamiltonian is $$\begin{aligned} \langle \psi_\beta(\vec k') | {\cal H} | \psi_\alpha(\vec k) \rangle = \frac{1}{N} \sum_{\vec T, \vec T'} \, e^{-i \vec k \cdot (\vec T' + \vec \tau_\beta)} \langle \phi_\beta (\vec T') | {\cal H} | \phi_\alpha (\vec T) \rangle e^{i \vec k \cdot (\vec T + \vec \tau_\alpha)}\end{aligned}$$ Assuming that the inter-site tunneling amplitude depends on the layer-projected difference coordinate, the matrix element can be expressed $$\begin{aligned} \langle \phi_\beta (\vec T') | {\cal H} | \phi_\alpha (\vec T) \rangle = \frac{1}{(2 \pi)^2} \int \, d^2 q \, f(\vec q) \, e^{i \vec q \cdot (\vec T' + \vec \tau'_\beta - \vec T - \vec \tau_\alpha)}\end{aligned}$$ Carrying out the lattice sums in Eqn. 21 and expressing the momenta in terms of their differences from the respective zone corners, $\vec k = \vec K + \vec q$, one obtains an expression for the interlayer tunneling amplitude in terms of sums over the reciprocal lattices of the reference ($\vec G$) and rotated ($\vec G'$) layers $$\begin{aligned} {\cal T}_{\beta \alpha} (\vec q',\vec q) = \frac{1}{\cal A} \sum_{\vec G, \vec G'} \, f(\vec q + \vec K + \vec G) \, e^{i \vec G' \cdot \vec \tau_\beta} e^{-i \vec G \cdot \tau_\alpha} \, \nonumber\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, \delta(\vec q' - \vec q + \Delta K + \vec G' - \vec G)\end{aligned}$$ where ${\cal A}$ is the area of the unit cell. When $q \ll G$ Eqn. 23 describes two distinct types of interlayer tunneling processes: (1) Direct interlayer terms conserve the [*crystal momentum*]{} $\vec k$ and occur when $\Delta K = |\vec K(\theta) - \vec K + \vec G' - \vec G|$. (Note that this occurs for $\vec G = \vec G' = 0$ and for all boosts by the reciprocal lattice vectors $\vec {\cal G} = \vec G' - \vec G$ that symmetrically shift the initial and final states to nearby valleys.) (2) Indirect interlayer terms conserve the [*Dirac momentum*]{} $\vec q$ and occur when $\vec K + \vec G = \vec K(\theta) + \vec G$. The matrix element for this latter process is dominated by the Fourier amplitude of the tunnelling potential at the first momentum $\vec K + \vec G_c$ in the extended zone where the zone corner points coincide. Processes (1) and (2) have very different character. In the Dirac language, process (1) requires $\vec q \neq \vec q'$ and provides a microscopic basis for the continuum formulation of Lopes dos Santos [*et al.*]{} [@JMLdS]. By contrast process (2) allows (indeed requires) $\vec q = \vec q'$ coupling and in particular it provides a mechanism for [*coupling between the tips of the Dirac cones in neighboring layers*]{}. It can be understood as an interlayer umklapp process whereby the scattering by a reciprocal lattice vector of the commensuration cell provides precisely the right momentum to bridge the momentum offset $\Delta K$. The ratio of the amplitudes for the indirect and direct couplings is approximately $f(|\vec K + \vec G|)/f(|\vec K|)$ so that the indirect term is generally weaker than the direct term. Superlattice Commensuration Effects ----------------------------------- As discussed in Section 2.2 either $K \rightarrow K(\theta)$ or $K' \rightarrow K(\theta)$ couplings are in the reciprocal lattice of the commensuration cell for a faulted bilayer, depending on the sublattice symmetry, and using Eqn. 23 they are allowed interlayer tunnelling processes. The low energy theory is fundamentally changed by these terms since they introduce an interlayer mass operator in the long wavelength Hamiltonian. Interestingly, the analytic structure of this mass matrix is determined solely by the sublattice symmetry of the commensuration. Thus one can define two complementary families of commensurate faults where all members of a common family have a common form for their low energy Hamiltonians. The energy scale of this mass operator depends on the period of the commensuration, and it is largest for low order commensurate rotations. The primitive stacked structures with $AB$ and $AA$ stacking are parent structures for this family behavior which give prototypical examples for the interlayer mixing possible for generic commensurate bilayers. The commensuration physics for these systems can be understood most easily by explicitly writing the layer Bloch states in real space in the “first star" approximation that retains only the three reciprocal lattice vectors that keep the combination $\vec K + \vec G$ to the first star of $K$ points $$\begin{aligned} \Psi(\vec r) = \sum_\alpha \, \Phi_\alpha (\vec r) u_\alpha (\vec r); \Phi_\alpha(\vec r) = \frac{1}{\sqrt{3}} \sum_{m=1}^3 \, e^{i \vec K_m \cdot (\vec r - \vec \tau_\alpha)}\end{aligned}$$ The coupling between layers is a functional of the Bloch fields $\Psi(\vec r)$ and its properties are captured by the local functional $$\begin{aligned} U=\frac{1}{2} \, \int \, d^2r \, T_{\ell} (\vec r) |\Psi_1(\vec r) - \Psi_2(\vec r)|^2\end{aligned}$$ where $T_{\ell}$ is a real modulated supercell-periodic function arising from the lattice structure of the commensuration cell. The coupling function acts to correlate the amplitudes and phases of the $\Psi$’s in the neighboring layers. The purely [*local*]{} coupling between layers in Eqn. 25 can be readily generalized to describe interlayer coupling with a finite range without substantially changing the physics. Equation 25 describes a coupled mode theory where the full Bloch waves $\Psi$ of the two layers (importantly these are [*not*]{} the Dirac envelope functions $u_\alpha(\vec r)$) are coupled by through a local spatially modulated potential. Although the exact form of the coupling function ${\cal T}_\ell$ is unknown its important properties are constrained by symmetry: it is a real supercell-periodic function with local maximum near aligned sites of the two layers and with minima for regions where atoms in neighboring layers are out of registry. A useful analytic model satisfying these constraints can be constructed from the elementary density waves in each of the layers $$\begin{aligned} n_{\mu=1,2} (\vec r) = \sum_{m \in [1]} \, \sum_{\alpha = A,B} e^{i \vec G_{\mu,m} \cdot (\vec r - \vec \tau_{\mu,\alpha})}\end{aligned}$$ summed over the first star of reciprocal lattice vectors in the $\mu$-th layer. Then, a nonlinear functional of the density fields that satisfies all the symmetry constraints is $$\begin{aligned} T_\ell (\vec r) = C_0 e^{C_1(n_1 (\vec r) + n_2(\vec r))}\end{aligned}$$ The sum of the layer density waves is a real function with the translational symmetry of the commensuration cell and no shorter. Eqn. 27 is maximized at special positions where the two density functions in each layer are separately maximized corresponding to aligned atomic sites, and it exhibits exponential suppression in regions where the density waves are out of registry. This ansatz for the coupling function has some important features. (1) It is a nonlinear function of the primitive reciprocal lattices vectors of each of the layers so that [*all*]{} the reciprocal lattice vectors of the commensuration cell are are represented in an expansion of the exponential in powers of its argument. (2) It is a separable function, constructed from a product of functions each of which is spanned by the [*separate*]{} reciprocal lattices of the two layers. (3) It is parameterized by two constants $C_0$ and $C_1$ which respectively describe the strength and range of the microscopic interlayer tunneling amplitudes. (Thus for example, very long range hopping is described by a small value of $C_1$.) The two constants $C_0$ and $C_1$ can be estimated from microscopic theory. Figure 4 gives a density plot of the local coupling function $T_\ell(\vec r)$ calculated for partner commensurations at $\theta = 21.787^\circ$ and $\theta = 38.213^\circ$. The former corresponds to a sublattice exchange “odd" structure and has a threefold rotational symmetry. Its partner is a sublattice exchange “even" structure and retains the full sixfold symmetry of the graphene layer, though on an inflated commensuration supercell. This illustrates a general property of all “odd“ and ”even" commensurate faults. The patterns shown in this density plot provide a real space image of the interlayer resonance pattern for a twisted bilayer. Interestingly, for large angle faults, one finds that the appearance of [*fivefold*]{} resonance rings (due to rotated misaligned hexagons) is a robust motif in the coupling function. For small angle faults the coupling function is described instead by the familiar Moire pattern that evolves smoothly between zones locally defined by by $AB$, $BA$ and $AA$ stacking. ![\[hopping\] The spatial dependence of the interlayer coupling function $T_\ell(\vec r)$ of Eqn. 27 is illustrated by these greyscale plots for commensurate faults at $21.787^\circ$ (left) and $38.213^\circ$ (right). The left structure has odd sublattice exchange parity,the right structure is even. The left pattern has a threefold rotational symmetry, the right pattern has the sixfold symmetry of an isolated graphene sheets. Both patterns contain combinations of fivefold resonance rings that are combined in clusters to form a periodic two dimensional pattern. Adapted from reference [@ejmrc].](partnerhops.jpg){width="80mm"} The couplings between the Dirac fields $u_\alpha$ in neighboring layers are obtained from the cross terms in Eqn. 25 after integrating out the lattice scale oscillations and are given by the Fourier transform of $T_\ell$ on the reciprocal lattice of the commensuration cell $t(\vec {\cal G})$. The $\vec {\cal G}= 0$ term describes the crystal momentum-conserving interlayer couplings discussed in the theory of Lopes dos Santos [*et al.*]{} [@JMLdS]. In addition there are umklapp terms involving $\vec {\cal G} \neq 0$ terms that express the symmetry allowed couplings between Dirac nodes $K_m \rightarrow K_{m'} (\theta)$. The geometrical considerations of Section 2.2 require that for any given commensuration there are couplings within two distinct pairs of Dirac nodes at the corners of their respective Brillouin zones. In the Bloch basis these matrix elements are spanned by a $3 \times 3$ matrix of finite momentum scattering amplitudes $\hat V_{\rm ps}$ which, using the threefold rotational symmetry, takes the form $$\begin{aligned} \hat V_{\rm ps} = \left(\begin{array}{ccc} V_0 & V_1 & V_2 \\ V_2 & V_0 & V_1 \\ V_1 & V_2 & V_0 \\ \end{array}\right)\end{aligned}$$ Here $V_0$ describes the scattering amplitude for a momentum transfer ${\cal G}=|\Delta \vec K|$ coupling the two layers while $V_1$ and $V_2$ describe scattering amplitudes with larger momentum transfers $\sim \vec G$. By projecting Eqn. 28 onto the sublattice (pseudospin) basis, one obtains the $2 \times 2$ interlayer mass matrices $\hat {\cal H}_{\rm int}$ that couple the Dirac fermions of the two layers. The low energy Hamiltonian for an even bilayer is thus expressed as a $4 \times 4$ matrix (acting on the two sublattice and two layer degrees of freedom) $$\begin{aligned} \hat {\cal H}_{\rm even} = \left( \begin{array}{cc} -i \hbar \tilde{v}_F \sigma_1 \cdot \nabla & \hat {\cal H}^+_{\rm int} \\ (\hat {\cal H}^+_{\rm int})^\dagger & -i \hbar \tilde{v}_F \sigma_2 \cdot \nabla \\ \end{array} \right)\end{aligned}$$ while for the odd bilayer $$\begin{aligned} \hat {\cal H}_{\rm odd} = \left( \begin{array}{cc} -i \hbar \tilde{v}_F \sigma_1 \cdot \nabla & \hat {\cal H}^-_{\rm int} \\ (\hat {\cal H}^-_{\rm int})^\dagger & i \hbar \tilde{v}_F \sigma_2^* \cdot \nabla \\ \end{array} \right)\end{aligned}$$ where $\sigma_n$ are Pauli matrices acting in the sublattice pseudospin basis of the $n-th$ layer and $\tilde{v}_F$ is the renormalized Fermi velocity. Note that for even parity faults the bilayer Hamiltonian couples nodes of the same chirality, whereas the odd parity faults introduce coupling between nodes of compensating chirality. In either case the spectrum for coupled system retains a two-valley character due to the two ways of matching nodes in either family of structures. The interlayer mass matrices $\hat {\cal H}^\pm_{\rm int}$ are $$\begin{aligned} \hat {\cal H}^+_{\rm int} = {\cal V} e^{i \vartheta} \left(\begin{array}{cc} e^{i \varphi/2} & 0 \\ 0 & e^{- i \varphi/2} \\ \end{array} \right) ,\,\,\, \hat {\cal H}^-_{\rm int} = {\cal V} e^{i \vartheta} \left(\begin{array}{cc} 1 & 0 \\ 0 & 0 \\ \end{array} \right)\end{aligned}$$ Eqns. 29 and 31 show that for sublattice “even" faults the mass term involves an $xy$ rotation of its pseudospin through angle $\varphi$. This angle can not be identified with the rotation angle $\theta$ but it results instead from the interference of the three complex scattering amplitudes $V_i$. By contrast in Eqns. 30 and 31 one finds that interlayer motion across an “odd" fault is mediated by the amplitude on its dominant eclipsed sublattice. In both mass matrices the overall phase of the operator $\vartheta$ can be removed by a gauge transformation. Eqn. 31 describes a coupling between Dirac waves in the neighboring layers that persists in the long wavelength $q \rightarrow 0$ limit, qualitatively changing the structure of the low energy spectra. Their effects are illustrated in Figure 5. For sublattice odd parity faults one pair of coupled bands are gapped on the interaction scale ${\cal V}$ leaving an $E=0$ contact point between a second pair of quadratically dispersing bands. For the even parity structures the $q=0$ spectrum contains a pair of coherence-split doublets. These two doublets are the symmetric and antisymmetric combinations of the original single layer Dirac modes; interestingly the layer-coupled states are topologically required to retain their Dirac character for small $q $, and disperse linearly away from the points of degeneracy. At finite momentum two branches undergo an avoided crossing which gaps the spectrum at its charge neutrality points. Thus these structures are generically [*fully gapped*]{} where the size of the gap is determined by the pseudospin rotation angle $\varphi$ in Eqn. 31 which allows these branches to hybridize. This gap degenerates to zero for the special case of an $AA$ stacked structure where $\varphi=0$ by symmetry. In both cases the residual effect of the mixing at high energy is to introduce a coherence splitting between two linearly dispersing layer-hybridized bands. ![\[massive\] The low energe electronic structure for the sublattice exchange odd parity $\theta = 21.787^\circ$ commensuration and the partner even parity $38.213^\circ$ structure are compared with the spectra for the parent Bernal ($AB$) and $AA$ stacked structures. All odd parity structures, as shown on the left, contain a pair of coherence split massive bands and a contact point between two quadratic bands at $E=0$. The even parity structures, shown on the right, feature a bonding/antibonding splitting, with a fully developed gap near the charge neutrality point which degenerates to a gapless state for the $AA$ stacked structure. Adapted from reference [@ejmrc]. ](massive.jpg){width="80mm"} Both these behaviors have precise analogs for the limiting cases of Bernal and $AA$ stacked bilayers which can be understood as the primitive parent structures of these two families. As shown in the lower left panel of Fig. 5 the Bernal spectra exhibit the mass structure expected for all sublattice exchange odd faults, though on an inflated energy scale ($\approx 0.2 \, {\rm eV}$) reflecting the full alignment of all sites on a single sublattice. Similarly the primitive $AA$ stacking features a coherence splitting of its bonding and antibonding layer-coupled states, but without the pseudospin rotation $\varphi$ so that the spectrum remains gapless and the zero energy states occur on a ring in reciprocal space. Nonlocal Potential Scattering ----------------------------- For small rotation angle the offset momentum $\Delta K \rightarrow 0$ and perturbation theory in the dimensionless parameter $\tilde t_{\perp}/\hbar v_F \Delta K$ fails. The breakdown of the perturbation theory occurs because of an incomplete treatment of multiple scattering processes involving the interlayer coupling operator. Recognizing this, Bistritzer and MacDonald (BM) [@bismac1] developed a theory that treats multiple scattering through the three fundamental interlayer amplitudes that describe a spatially modulated interlayer coupling with the period of the commensuration supercell. Their results show that the Fermi velocity renormalization of the perturbation theory presages more dramatic physics at small rotation angle which can be described as “velocity reversal," i.e. the Fermi velocity changes [*sign*]{} as a function of (small) rotation angle crossing through zero at a series of discrete magic angles. As a consequence the small angle regime is predicted to feature a manifold of nearly flat bands at low energy. The BM model is formulated as a two layer scattering theory: states with momentum $\vec k$ in one layer are scattered into states at momentum $\vec k + \vec Q$ in its neighbor. In the pseudospin basis the amplitudes for these processes are the $2 \times 2$ matrices given in Eqn. 18. The gauge transformation $e^{- i \Delta \vec K \cdot \vec r}$ on the rotated layer brings two Dirac nodes of neighboring layers into coincidence, and in this momentum shifted basis the three momentum transfers $\vec Q_{i(=0,\pm 1)}$ are $Q_0=-\Delta \vec K$ and two $\pm 2 \pi/3$-rotated partners $Q_{\pm 1}$ which form a threefold symmetric triad. Thus this construction considers a twofold layer-degenerate Dirac cone whose states are coupled through an off-diagonal nonlocal operator containing three possible momentum transfers $\vec Q_i$ in the interlayer hopping. In the long distance theory the single layer Hamiltonians are isotropic, so for [*arbitrary*]{} rotation angle $\theta$ the the theory is specified by its unrenormalized Fermi velocity, the rotation angle and the coupling strength labelled $w$ in BM [@bismac1], which combine to form a single dimensionless scaling parameter $\alpha = w/2 \hbar v_F \sin(\theta/2)$. Repeated action of the nonlocal interlayer operator generates a lattice of coupled momenta as shown in the top panel of Figure 6. The interlayer tunneling amplitudes are [*directed transitions*]{} in reciprocal space: the momenta $Q_i$ and their negates $-Q_i$ describe complementary processes that transport electrons to and from the rotated layer. Consequently, an even number of applications of the nonlocal operator to an initial single-layer Bloch state at wavevector $\vec k$ generates a Bloch state in the same layer with momentum $\vec k + \vec {\cal Q}$ where $\vec {\cal Q}$ is spanned by the primitive vectors $\vec Q_1 - \vec Q_0$ and $\vec Q_{-1} - \vec Q_0$. This defines a triangular reciprocal lattice whose six first star elements have magnitude $\sqrt{3}\, Q_0$, and are $90^\circ$ rotated with respect to the original $\pm \vec Q_i$’s. An odd number of applications of the operator transports the electron to the neighboring layer on the same reciprocal lattice, but offset by the momentum shift $\vec Q_0$. The combination of these two sets describes a honeycomb lattice where the alternating sites (momenta) occupy different layers as shown in Fig. 6. ![\[BMbands\] Top panel: A lattice of momenta is generated by repeated action of a nonlocal interlayer coupling operator on a Bloch state in a single layer. The nonlocal operator transports an electron between layers and boosts the momentum by a threefold symmetric triad of momentum transfers $Q_i$. An even number of applications of the operator generates a triangular lattice of momenta in the original layer (red), an odd number generates a triangular lattice offset by momentum $\Delta K$ (black). The combination forms a honeycomb lattice of coupled momenta. Bottom panel: Band structures obtained by numerical diagonalization of the continuum Hamiltonian in a truncated plane wave basis retaining kinetic energies of order the coupling strength $w$. The bands are plotted along the momentum space trajectory $ABCDA$ in the top figure. For small rotation angles the bands flatten and the Fermi velocity of the zero energy states is strongly suppressed. Adapted from reference [@bismac1].](Fig6.jpg){width="80mm"} BM studied this model by numerically diagonalizing a truncated Hamiltonian expanded in a plane wave basis and retaining plane waves with kinetic energies below the coherence scale $\sim w$. The effects of multiple scattering through the interlayer coupling terms is then encoded in the structure of the bilayer eigenstates which contain coherent superpositions of the single-layer Dirac modes. For large rotation angles ($\theta > 3^\circ$) the model reproduces the perturbation theory of Lopes dos Santos et al. By contrast, in the very small angle regime the bandwidth $\hbar v_F Q_0$ collapses, the number of elements in the low energy basis grows correspondingly and the electronic structure becomes spectrally congested as illustrated in Fig. 6. Thus the small angle regime is described by a strong coupled [*multiband*]{} theory that introduces physics inaccessible to a low order perturbation theory. BM find that the low energy spectra in this regime show a very substantial reduction of the Fermi velocity (typically $< 0.1$ of its single layer value) due to level repulsion among the coupled bands. Remarkably, the reduced velocity parameter [*oscillates*]{} as a function of the fault angle as shown in Figure 7 and crosses zero at a series of magic rotation angles. They suggest that this oscillation likely results from a $\theta$ dependence of the superpositions of single layer modes that contain velocities of opposite sign, though a complete theory of the velocity oscillations has yet to be developed. Thus the velocity renormalization found in the weak coupling limit represents just the first step towards a complete twist-induced reconstruction of the low energy spectrum! Two Band Models --------------- There has been interest in distilling the continuum theory to a simpler effective [*two band*]{} model that captures the topological structure of its low energy spectrum. The approach is similar in spirit to the theory of Bernal stacked bilayers [@McCFal; @biasedbilayer; @longbiasedbilayer] where one can integrate out its high energy degree of freedom to arrive at an effective theory for its low energy states. For the Bernal bilayer this procedure identifies a new class of layer-coherent chiral fermions that have quadratic low energy dispersion and a Berry’s phase of $2 \pi$. For the twisted bilayer, neglecting commensuration effects, the low energy spectrum contains two layer-polarized [*linear*]{} Dirac cones that are recoupled at an energy scale $\hbar v_F |\Delta \vec K|/2$ where they merge. The two band model attempts to provide a compact description of the topological transition of the band dispersion that connects the low energy “doubled cone" sector to its high energy layer-hybridized sector. For twisted bilayers the two band construction can be understood as a variant of the low energy theory for a Bernal bilayer that allows for a finite momentum offset $|\Delta \vec K|$ between its Dirac nodes. Thus the low energy theory for Bernal stacking is modified in the manner $$\begin{aligned} {\cal H}_K = -\frac{\hbar^2}{m} \left(\begin{array}{cc} 0 & \partial^2 \\ \bar \partial^2 & 0 \\ \end{array}\right) \rightarrow -\frac{\hbar^2}{m(\theta)} \left(\begin{array}{cc} 0 & \partial^2 - (\Delta K)^2\\ \bar \partial^2 - (\Delta \bar K)^2 & 0 \\ \end{array}\right)\end{aligned}$$ valid for very small fault angles where $\hbar v_F |\Delta K|\ll \tilde t_\perp$ and $m=\tilde t_\perp/v_F^2$. This expression can be derived by replacing the interlayer operators of Eqn. 18 by a simpler expression $\tilde t_\perp \sigma_-$ which physically describes an interlayer tunneling amplitude across a single sublattice in each layer. The spectrum of this Hamiltonian features a pair of Dirac cones, split by the momentum offset $|\Delta \vec K|$, that merge at a two dimensional saddle point at $q=0$ representing the topological transition of the band structure. Importantly, the single layer Dirac cones in this model have the [*same*]{} chirality so that annihilation of the Dirac points when they are coupled is topologically forbidden. Generically, this model does allow for an energy offset between the Dirac points of the coupled bilayer but this is believed to be small for physically reasonable coupling strengths. The Hamiltonian in Eqn. 32 has been used to study the orbital quantization of a twisted bilayer in the presence of a perpendicular magnetic field. By construction the limit $\Delta K=0$ describes the Landau quantization of Bernal bilayer graphene: a Berry’s phase of $2 \pi$ and quantized energy levels $\propto \sqrt{n(n-1)} B$. By contrast for finite rotation angles the low energy spectrum of the offset model is a “doubled" theory of single layer graphene: the fourfold degeneracy due to the spin and valley degrees of freedom is doubled by an approximate layer decoupling of its low energy eigenstates. An asymptotic analysis of the eigenvalues within this model demonstrates that splittings of the Landau level degeneracies due to interlayer coupling are exponentially suppressed as a function of the rotation angle in the low energy regime [@saddleLL]. The spectrum thus features a zero mode and Landau levels that disperse $\propto \sqrt{nB}$ [@saddledeGail; @saddleLL]. This twofold layer degeneracy is quickly eliminated as one passes through the crossover energy $\hbar v_F \Delta K/2$ where the Dirac cones merge and hybridize. Above this crossover the spectrum has a different character: layer degeneracies are removed and the quantized energies are $\propto (n+1/2)B$ as expected for a parabolic interlayer coherent band. Discussion ========== Rotational faults commonly occur in a several different forms of graphene and their electronic properties are actively studied experimentally. The rapidly growing experimental literature on this subject has not yet provided a unified picture of the effects of faults on the electronic behavior, possibly due to differences in the electronic properties of samples produced by different experimental methods. A significant point of [*agreement*]{} among the various experimental works is that the interlayer coherence scale is very small in these systems [@deheerreview; @haas; @graphenegraphite; @miller]. This can be deduced clearly from their Landau level spectra which have been measured by scanning tunneling spectroscopy (STS). These spectra show a scaling of the Landau level energies $E_n \propto \sqrt{nB}$ [@graphenegraphite; @miller] the signature of the Landau quantization of a massless Dirac band, as observed for single layer graphene and quite distinct from the level sequence observed for Bernal stacked bilayers [@McCFal]. Perhaps the strongest evidence for a reduction of the interlayer coupling scale comes from angle resolved photoemission experiments which directly measure the quasiparticle dispersion relation [@sprinkle; @hicks] and find spectra that follow the expected form for an isolated Dirac cone. These measurements have been interpreted as providing the first direct measurement of the Dirac dispersion relation in graphene, uncontaminated by substrate or other interlayer effects [@sprinkle]. Since the effects of the interlayer coupling in twisted multilayers are intrinsically weak, their study is posing a significant experimental challenge. It is here where different experiments carried out on different samples disagree. For example, the Fermi velocity can be deduced from the slope of the $\sqrt{nB}$ scaling relation for the Landau quantized energies. The strongest evidence for a twist-induced renormalization of $v_F$ comes from the Landau level spectra measured by scanning tunneling spectroscopy of CVD graphenes grown on Ni substrates [@STMvel]. This work reports that $v_F$ is not constant as a function of scanned position across a macroscopic sample, but instead it is found to vary in a range $0.87 \times 10^6 \, {\rm m/s} < v_F < 1.1 \times 10^6 \, {\rm m/s}$. Simultaneous measurement of the topography of the Moire superlattice period of these samples correlates the velocity reduction with the period and hence the rotation angle. The larger value, found for large angle rotations, agrees well with the $v_F$ for single layer graphene and the $20 \%$ reduced value is correlated with a small angle rotation $\sim 3^\circ$ as suggested by a perturbative analysis of the continuum theory [@JMLdS; @bismac1]. This contrasts with analogous STS measurements carried out for multilayer graphenes grown epitaxially on SiC ${\rm (000 \bar1)}$. These also show the $\sqrt{nB}$ scaling of the Landau quantized energies. However, for these materials the slope of the scaling relation yields a Fermi velocity $1.1 \times 10^6 \, {\rm m/s}$ for all samples studied down to a rotation angle of $1.4^\circ$ [@miller] completely spanning the range of rotation angles where a velocity renormalization is expected. A similar discrepancy arises in the spectroscopy of the van Hove singularity presumed to occur in the region where the momentum-offset Dirac cones of a twisted bilayer merge. Low energy STS on Ni/CVD grown graphene reveals low energy peaks whose energies disperse with their topographically measured rotation angles in the low angle regime $1.2 < \theta < 3.5^\circ$ [@vanHove] roughly consistent with the van Hove scenario. Yet these features are not seen at all in spectroscopy of the SiC epitaxial twisted graphenes regardless of the fault angle. Perhaps the strongest challenge to the idea of a twist-induced spectral reconstruction comes from angle resolved photoemission. These measurements directly measure the quasiparticle dispersion and clearly resolve the Dirac cone with a Fermi velocity that is indistinguishable from that of single layer graphene. Despite a careful search, no evidence is found in these measurements for any type of hybridization between Dirac cones in the spectral regions where they cross [@hicks]. The simplest interpretation of the ARPES data is that the first few graphene layers accessible to this spectroscopy are electronically floating, i.e. extremely weakly coupled to each other and to deeper layers in the film. An important goal for theory in this area is therefore to identify situations where the effects of the interlayer coupling across a rotational fault are manifested in their electronic behavior. There has been progress in this direction. Bistritzer and MacDonald have studied the effect of the rotation angle of a bilayer on its interlayer tunneling conductance, predicting dramatic enhancements of the vertical conductance at special rotation angles that can be identified with low order commensurate superlattices [@rafibilayer]. More recent work has pointed to nontrivial effects of a $\theta$-dependent interlayer coupling on the equilibrium charge redistribution across the bilayer in a perpendicular field [@brey]. Kindermann and I studied the Landau level spectra for weakly coupled bilayers and find that even a weak coherence splitting of bilayer bands at energies well above the mass scale produces a striking new effect, the Dirac comb [@mkejm]. Here small differences in the orbitally quantized states in two weakly coherence-split bands produces an amplitude modulation of the Landau level spectrum with a period that greatly exceeds the coherence scale, and should be observable by magnetotransport in the weak field regime. Small rotations angles can introduce long spatial Moire periods for twisted bilayers that can be made commensurate with the magnetic length $\sqrt{\hbar/eB}$ on accessible field scales, accessing Hofstadter commensuration physics in a new family of materials [@rafibutterflies]. The band flattening theoretically predicted for in the small twist angle regime will surely focus attention on many body effects in the low energy physics. 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--- abstract: 'The relations of antilinear maps, bipartite states and quantum channels is summarized. Antilinear maps are applied to describe bipartite states and entanglement. Teleportation is treated in this general formalism with an emphasis on conditional schemes applying partially entangled pure states. It is shown that in such schemes the entangled state shared by the parties, and those measured by the sender should “match” each other.' address: - | Department of Nonlinear and Quantum Optics,\ Research Institute for Solid State Physics and Optics,\ Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary - ' Institute of Physics, University of Pécs, Ifjúság út 6. H-7624 Pécs, Hungary' author: - 'Z. Kurucz' - 'M. Koniorczyk' - 'J. Janszky' title: Teleportation with partially entangled states --- Introduction {#sec:intro} ============ A fundamental part of quantum mechanics is the description of the evolution of quantum states. In the lack of measurements, the evolution of the quantum state of closed systems can be described by unitary operators. Measurements are described by projections onto the eigenstates of the measured quantity. In more general approach to the question of quantum state evolution, the system in argument is considered as a part of a larger subsystem. This approach gives rise to concepts of general quantum channels and generalized (POVM) measurements. In the context of quantum information and communication, quantum entanglement is a central issue. General quantum kinematics of multipartite systems harbors lots of secrets still. Quantum teleportation[@prl70_1895] is probably the most frequently quoted application of entanglement, and its experimental feasibility[@nature390_575; @prl80_1121; @science282_706] has further increased its relevance. The methods developed for representations of quantum channels can be used successfully in the description of entanglement and teleportation. The results in this paper are motivated by these kind of methods. Section \[sec:formalism\] summarizes some facts concerning the relations between quantum states of bipartite systems, channels, antilinear and antiunitary maps. Some of these have already found application in quantum information theory, but those use a fixed antilinear map, which is related to a specific maximally entangled state. We also present another possibility, namely we consider different antilinear maps, which is an alternative description of all pure bipartite states. This approach provides us with a very convenient description of quantum teleportation, including all schemes applying a pure (but not necessarily maximally entangled) resource. Popescu [@prl72_797] pointed out that teleportation is also possible using mixed states but with a fidelity less than 1. In a recent paper of Banaszek [@pra62_024301] a protocol using a partially entangled state was optimized for average fidelity. Horodecki [*et al*]{} [@pra60_1888] presented a formula for the fidelity of such imperfect teleportation schemes. In conditional schemes on the other hand, fidelity can be one but at the cost that the process sometimes fails, so the probability of successfulness (also called efficiency) is less than one. This is the case in conclusive teleportation [@9608005; @9906039], where partially entangled or even mixed states and generalized measurements (POVM) are considered. Using nonunitary transformation at Bob’s side also makes the process probabilistic [@pra61_034301]. The main part of our paper is based on the formalism summarized in Section \[sec:formalism\]. Our description is completely independent of the dimensions of the Hilbert-spaces involved, and we do not even need to fix a basis. In Section \[sec:teleport\] we give a general condition for conditional teleportation in terms of the applicable entangled states and joint measurements. In Section \[sec:matching\] we show that partially entangled states are capable of conditional teleportation with fidelity one, but only if the outcome of the measurement performed by Alice and the state shared by the parties “match” each other. Section \[sec:concl\] summarizes our results. States, channels and antilinear maps {#sec:formalism} ==================================== Consider a bipartite system with subsystems A and B. The subsystems are described by the Hilbert-spaces ${\cal H}_A$ and ${\cal H}_B$, thus the pure states of the system are in ${\cal H}_A\otimes {\cal H}_B$. Let $\dim {\cal H}_A=\dim {\cal H}_B=N$. Let $\{|i\rangle_A\}$ and $\{|i\rangle_B\}\ (i=0,\ldots ,N-1)$ denote the computational bases on ${\cal H}_A$ and ${\cal H}_B$, and let $$\label{eq:maxent} |\Psi^+\rangle_{AB}=\frac{1}{\sqrt{N}}\sum_{i=0}^{N-1}|i\rangle_A|i\rangle_B$$ be a maximally entangled state of the system. All other maximally entangled states of the system can be obtained from $|\Psi^+\rangle$ by local unitary transformations. The set of density matrices of system A will be denoted by ${\cal S}_A$. A quantum channel $\$ _A $ is a completely positive, trace-preserving, and hermiticity preserving ${\cal S}_A \rightarrow {\cal S}_A$ map. Keeping $|\Psi^{(+)}\rangle$ in mind, we may introduce the relative state representation [@pra54_2614] of states and operators on ${\cal H}_A$. Any pure state $|\Psi\rangle_A\in {\cal H}_A$ can be described by an (unnormalized) *index state* $|\Psi^*\rangle_B\in {\cal H}_B$ so that $$\label{eq:relrep_state} |\Psi\rangle_A=\, _B\langle \Psi^*|\Psi^+\rangle_{AB}.$$ The state in argument is obtained as a partial inner product of its index state and the maximally entangled state $|\Psi^+\rangle$. The mapping creating the index state from the original state, $$\label{eq:Lpsip} L_{|\Psi^+\rangle}: {\cal H}_A\rightarrow {\cal H}_B,\quad L_{|\Psi^+\rangle}|\Psi\rangle_A = |\Psi ^*\rangle_B$$ is antilinear, and in fact $\sqrt{N} L_{|\Psi^+\rangle}$ is antiunitary. Indeed, expanding an arbitrary $|\Psi\rangle_A$ on the computational basis, $$\label{eq:Lpsip_bas} L_{|\Psi^+\rangle}|\Psi\rangle_A=L_{|\Psi^+\rangle}\sum_i C_i|i\rangle_A= \frac{1}{\sqrt{N}} \sum_i C_i^*|i\rangle_B,$$ from which the above properties follow. The introduction of $L$ via $|\Psi^+\rangle$ is also useful in describing channels $\$ _A$. Let us have the compound system in the state $|\Psi ^+\rangle_{AB}$, and send subsystem A through the channel $\$ _A$ while doing nothing with subsystem B. The effect of the channel on any pure state $|\Psi\rangle_A$ of system A is then obtained by the partial inner product with the corresponding index state: $$\label{eq:Relrep_chann} \$_A\left(|\Psi\rangle_A\, _A\langle \Psi |\right)= N\, _B\langle \Psi ^*|(\$ _A\otimes I_B) \left (|\Psi^+\rangle _{AB} \,_{AB} \langle \Psi^+|\right) |\Psi ^* \rangle_B,$$ where $|\Psi^*\rangle_B=L_{|\Psi^+\rangle}|\Psi\rangle_A$, and $I_B$ stands for the identity operator. This is the so called relative state representation of channels, which is widely used to describe them. But even more can be stated [@pra60_1888]. An affine isomorphism between the set of all $\$ _A$ channels on ${\cal S}_A$, and the set of bipartite states $\varrho _{AB}\in {\cal S}_{{\cal H}_A\otimes{\cal H}_B}$ with maximally mixed partial trace, i.e. with the property $$\label{eq:parctrac} \mathop{\mbox{tr}}\nolimits _A \varrho_{AB}= \frac{1}{N} I_B,$$ can be found similarly to Eq. (\[eq:Relrep\_chann\]). The bipartite state corresponding to a channel can be obtained from $|\Psi^+\rangle$ by applying the channel on system A and doing nothing with system B: $$\label{eq:Relrep_izom} \varrho _{AB}=(\$ _A\otimes I_B) \left( |\Psi^+\rangle_{AB} \, _{AB}\langle \Psi^+| \right).$$ As $|\Psi^+\rangle_{AB}$ has a maximally mixed partial trace, and this property cannot be changed by local operations, (\[eq:parctrac\]) will also hold for the $\varrho _{AB}$ obtained in Eq. (\[eq:Relrep\_izom\]). The isomorphism has been found by the Horodecki [*et al.*]{} [@pra60_1888], who have discussed it in detail, and have used it for the description of teleportation channels. So far we have considered the antiunitary map $\sqrt{N}L_{|\Psi^+\rangle}$ arising from the maximally entangled state $|\Psi^+\rangle$. This is a useful tool in the description of channels and states. Let us follow the reverse way now. We may use the set of antilinear ${\cal H}_A\rightarrow {\cal H}_B $ maps in order to describe pure states in ${\cal H}_A\otimes {\cal H}_B$. As relative state representation is also based on $L$, changing this map can give rise to different relative state representations. Consider a bipartite pure state $|\Phi\rangle_{AB}\in {\cal H}_A\otimes {\cal H}_B $. We may write it on the computational basis as $|\Phi\rangle_{AB}=\sum_{ij} C_{ij} |i\rangle_A \otimes |j\rangle_B$. We define the ${\cal H}_A\rightarrow {\cal H}_B $ antilinear operator $L_{|\Phi\rangle}$ such that $L_{|\Phi\rangle}|i\rangle_A= \sum_j C_{ij} |j\rangle_{B}$. Thus we can write $$\label{eq:statdef} |\Phi\rangle_{AB}=\sum_i|i\rangle_A\otimes (L_{|\Phi\rangle}|i\rangle_A).$$ For any bipartite pure state $|\Phi\rangle_{AB}\in {\cal H}_A\otimes {\cal H}_B $, there uniquely exists an antilinear operator $L_{|\Phi\rangle}$ defined this way. Because of the antilinearity of $L_{|\Phi\rangle}$, (\[eq:statdef\]) is independent of the actual computational basis chosen on ${\cal H}_A$. Let ${{\mathcal C}}_{AB}$ denote the set of bounded antilinear operators $L \colon {\cal H}_A \to {\cal H}_B$ which have finite norm (that is, $\mathop{\mbox{tr}}(L^\dag L) < \infty$, where the adjoint of $L$ is defined by the relation $\langle f|Le \rangle = \langle L^\dag f|e \rangle^\ast$): $${{\mathcal C}}_{AB} = \big\{ L\colon {\cal H}_A \to {\cal H}_B\; \mbox{bound antilinear}\, \big| \mathop{\mbox{tr}}(L^\dag L) < \infty \big\}. \label{eq:CBC}$$ ${{\mathcal C}}_{AB}$ forms a Hilbert space (the scalar product is $(L,L') = \mathop{\mbox{tr}}(L'^\dag L)$, which is conjugate linear in the first argument). It is shown in Ref. [@jmp41_638] that (\[eq:statdef\]) establishes a unitary isomorphism between ${{\mathcal C}}_{AB}$ and ${\cal H}_A\otimes {\cal H}_B$ in a natural way. Every pure bipartite state $|\Phi\rangle_{AB}$ can uniquely be described by an antilinear operator $L_{|\Phi\rangle}\in {{\mathcal C}}_{AB}$ such that $\mathop{\mbox{tr}}(L_{|\Phi\rangle}^\dag L_{|\Phi\rangle})=1$. Conversely, every such $L$ describes a pure bipartite state. Now let us characterize maximally entangled states and possible relative state representations. By maximally entangled state we mean a pure bipartite state with maximally mixed partial trace (\[eq:parctrac\]). The partial traces of bipartite states over systems $A$ and $B$ are $LL^\dag$ and $L^\dag L$ respectively. Thus the state (\[eq:statdef\]) is maximally entangled if and only if $LL^\dag=N^{-1} I_B$ and $L^\dag L=N^{-1} I_A$. This is equivalent to that $\sqrt{N}L$ is antiunitary. On the other hand, the operator $L_{|\Phi\rangle}$ gives rise to a relative state representation if and only if $\{ L_{|\Phi\rangle}|i\rangle \}_{i=0,\ldots ,N-1}$ forms an orthogonal basis on ${\cal H}_B$, which means, that $\sqrt{N}L_{|\Phi\rangle}$ is antiunitary. Relative state representations can be defined via maximally entangled states. Probabilistic teleportation with partially entangled states {#sec:teleport} =========================================================== In this section we apply the antilinear description of bipartite states introduced in Section \[sec:formalism\] for quantum teleportation. Suppose that system A prepared in the unknown state $|\Phi\rangle_A$ is to be teleported, and systems B and C shared by the parties (Alice and Bob) are in a partially entangled state $|\sigma\rangle_{BC}$. We will call this *shared* state in what follows. The shared state is described by the antilinear map $L_{|\sigma\rangle}$. Systems A and B are located at Alice who performs a joint projective measurement on them. Suppose that its outcome corresponds to the projection onto the state $|\sigma_q\rangle_{AB}$. In the followings, we will regard only this outcome, thus our teleportation scheme will be probabilistic, conditional one. To have common computational bases in the description of the shared state and the state the measurement project onto, we expand $|\sigma_q\rangle$ in the following way: $$|\sigma_q\rangle = \sum_i (L_q |i\rangle) \otimes |i\rangle, \label{eq:measdef}$$ where $L_q \in {{\mathcal C}}_{BA}$. One can characterize the states corresponding to nondegenerate measurement outcomes by bounded antilinear operators $L_q \colon {{\mathcal H}}_B \to {{\mathcal H}}_A$ such that ${\mathrm{tr}}(L_q^\dag L_q)=1$. Note that $L_q$ is unique disregarding a unit complex phase factor. The projection resolution of every joint observable of $A$ and $B$, which has nondegenerate eigenvalues, is (up to phase factors) uniquely described by an orthonormal basis $L_q$ in ${{\mathcal C}}_{BA}$. Those measurements whose nondegenerate outcomes are represented by projections onto mutually orthogonal maximally entangled states, are called measurements of Bell type [@pla272_32]. Every Bell measurement can be described by an orthonormal basis $L_q$ in ${{\mathcal C}}_{BA}$ such that $(\dim{{\mathcal H}}_A)^{1/2} L_q$ is antiunitary for every $q$. Now we will calculate the teleportation channel. By this we mean a function $f_q \colon {{\mathcal H}}_A \rightarrowtail {{\mathcal H}}_C$ that relates the input state, and the state of system C after the measurement. Note that, although the reversing unitary transformation is usually also included in the definition of teleportation channel, our terminology is more convenient here, as we investigate linearity and reversibility. At the beginning, the three systems are in the state $|\Phi\rangle_A \otimes |\sigma\rangle_{BC}$. The probability of the outcome $q$ under consideration is given by $$\begin{aligned} p_q(|\Phi\rangle_A) &=& \left\| \strut [(|\sigma_q\rangle_{AB} \,_{AB}\langle\sigma_q|) \otimes I_C)] (|\Phi\rangle_A \otimes |\sigma\rangle_{BC}) \right\| ^2 = \left\| {\sum\limits_i} \big( {}_A \langle \Phi | L_q | i \rangle_B^\ast \big) L |i\rangle_B \right\|^2 \nonumber\\ &=& \left\| {\sum\limits_i} L( |i\rangle_B\,_B \langle i | L_q^\dag |\Phi \rangle_A) \right\|^2 = \left\| \strut LL_q^\dag |\Phi\rangle _A \right\|^2. \label{eq:p_q}\end{aligned}$$ On condition that the measurement yields the outcome $q$, the state of system $C$ can be written as $$\frac1{\sqrt{p_q(|\Phi\rangle_A)}} \sum_i \big( {}_{AB}\langle \sigma_q | \Phi \rangle_A |i\rangle_B \big) L | e_i\rangle_B = \frac1{\sqrt{p_q(|\Phi\rangle_A)}} LL_q^\dag |\Phi\rangle_A.$$ The teleportation channel for the outcome $q$ is $$f_q \colon {{\mathcal H}}_A \rightarrowtail {{\mathcal H}}_C, \quad f_q(|\Phi\rangle_A) = \frac{LL_q^\dag |\Phi\rangle_A}{\left\| LL_q^\dag |\Phi\rangle_A \right\|}. \label{eq:f_q}$$ If the input state is given by the density operator $\rho_{\mathrm{in}}$ then the probability of the outcome $q$ is $$p_q(\rho_{\mathrm{in}}) = {\mathrm{tr}}_A\left( L_q L^\dag L L_q^\dag \rho_{\mathrm{in}} \right) \label{eq:p_q(rho)}$$ and the output state is $$\rho_{\mathrm{out}} = \frac {L L_q^\dag \rho_{\mathrm{in}} L_q L^\dag} {{\mathrm{tr}}_A\big( L_q L^\dag L L_q^\dag \rho_{\mathrm{in}} \big)}. \label{eq:rhoout}$$ We have defined a special quantum operation based on the teleportation scheme of Ref. [@prl70_1895]. One can obtain from (\[eq:p\_q(rho)\]) that this operation is a generalized (POVM) measurement of the input state and the positive operator representing it is $L_q L^\dag L L_q^\dag$. The channel $f_q$ has to be reversible, so that we can obtain a teleported state identical to the original input state. We call the channel $f_q$ reversible, if it is injective, that is, for different input state $|\Phi\rangle_A$ ($\||\Phi\rangle_A\|=1$) the corresponding output state $f_q(|\Phi\rangle_A)$ is different. We remark, that the reversibility of teleportation channels has also been investigated in Ref. [@pra55_2547]. We adopt a more general definition here. Reversibility means that every input state can be recovered (theoretically) from the output state. One can easily verify that this condition is equivalent to that the linear operator $LL_q^\dag \colon {{\mathcal H}}_A \to {{\mathcal H}}_C$ is injective. It may be the case, however, that the channel $f_q$ is not linear. This way, the input state can be recovered from the output only using some sophisticated nonlinear transformations, which may not be realistic. Therefore, it is a natural requirement for the channel to be linear. We show that if the teleportation channel is reversible, then its linearity is equivalent to that the probability (\[eq:p\_q\]) of the outcome $q$ is independent of the input state $|\Phi\rangle_A$. Suppose that $|\Phi\rangle_1$ and $|\Phi\rangle_2$ are linearly independent, and let $(\alpha_1|\Phi\rangle_1 + \alpha_2|\Phi\rangle_2)$ be such that $\|\alpha_1|\Phi\rangle_1 + \alpha_2|\Phi\rangle_2\|=1$. From the linearity condition $f_q(\alpha_1 |\Phi\rangle_1 + \alpha_2 |\Phi\rangle_2) = \alpha_1 f_q(|\Phi\rangle_1) + \alpha_2 f_q(|\Phi\rangle_2)$, one can obtain: $$\begin{gathered} \alpha_1 \left( \frac1{\big\| LL_q^\dag (\alpha_1|\Phi\rangle_1 + \alpha_2|\Phi\rangle_2) \big\|} - \frac1{\big\| LL_q^\dag |\Phi\rangle_1 \big\|} \right) LL_q^\dag |\Phi\rangle_1 \\ + \alpha_2 \left( \frac1{\big\| LL_q^\dag (\alpha_1|\Phi\rangle_1 + \alpha_2|\Phi\rangle_2) \big\|} - \frac1{\big\| LL_q^\dag |\Phi\rangle_2 \big\|} \right) LL_q^\dag |\Phi\rangle_2 =0. \label{eq:linfq2}\end{gathered}$$ Since $f_q$ is injective, $LL_q^\dag |\Phi\rangle_1$ and $LL_q^\dag |\Phi\rangle_2$ are also linearly independent. Then (\[eq:linfq2\]) implies that their coefficients are zero, that is, the probability (\[eq:p\_q\]) of the outcome $q$ is independent of the input state $|\Phi\rangle_A$. Reversely, if (\[eq:p\_q\]) is independent of $|\Phi\rangle_A$, then $LL_q^\dag$ is injective and $f_q$ is linear. We can conclude that the condition that “the probability of the measurement outcome $q$ does not depend on the input state” (that is, Alice learns nothing about the input state due to the measurement) is equivalent to that the teleportation channel is linear. Moreover, it can be proven in a way not detailed here that the linearity of the channel is equivalent to its unitarity—therefore its unitary reversibility. Entanglement matching {#sec:matching} ===================== In this section we answer the question what measurements provide fidelity 1 teleportation for an arbitrarily given (even partially) entangled shared state. Suppose that the shared state $|\sigma\rangle$ is described by an invertible antilinear operator $L$. If Bob applies a unitary transformation $U_q\colon {{\mathcal H}}_C \to {{\mathcal H}}_C$ which may depend on the result $q$ of Alice’s measurement, then the final state of system $C$ reads $|\text{out}\rangle_C = p_q^{-1/2} U_q LL_q^\dag |\Phi\rangle_A$. Let $i_{AC}$ be a unitary isomorphism between ${{\mathcal H}}_A$ and ${{\mathcal H}}_C$ so that we can compare the states of systems $A$ and $C$. The teleportation condition is $$\frac1{\sqrt{p_q}} U_q LL_q^\dag = i_{AC}$$ which also guarantees that $p_q(|\Phi\rangle_A)$ is independent of $|\Phi\rangle_A$. From this we conclude that a measurement with an outcome described by the antilinear operator $$L_q = \sqrt{p_q} i_{AC}^{-1} U_q L^{-1}\strut^\dag \label{eq:L_q}$$ supports fidelity 1 conditional teleportation. The appropriate recovering unitary transformation applied by Bob is to be $U_q$. Although $p_q$ in (\[eq:p\_q\]) depends on $L_q$, this can be resolved by the fact that $L_q$ has a norm ${\mathrm{tr}}_B (L_q^\dag L_q)=1$. Then we obtain that the probability is $$p_q = \left[ {\mathrm{tr}}_B \left( (L^\dag L)^{-1} \right) \right]^{-1}. \label{eq:p_q2}$$ For an arbitrary entangled shared pair described by invertible $L$, the set of measurement outcomes providing fidelity 1 conditional teleportation is given by the set $${\mathcal M}_L = \left\{\, L_q = \frac{i_{AC}^{-1} U L^{-1}\strut^\dag} {\sqrt{{{\mathrm{tr}}}_B \big( \textstyle L^{-1} L^{-1}\strut^\dag \big)}} \, \Bigg| \, \mbox{$U$ is unitary} \,\right\}. \label{eq:M_L}$$ Thus not every possible measurement outcome allows teleportation, only those described by ${\mathcal M}_L$. The measurement and the shared state should be “matched” to each other. This can be regarded as a generalization of “entanglement matching” introduced in Ref. [@pra61_034301]. It is worth to note that (\[eq:p\_q2\]) is the same for every outcome $q$ that matches the shared state in the above sense. The probability of a successful outcome depends only on the shared state. Another important result is that the set ${\mathcal M}_L$ of matching outcomes is spanned by local unitary transformations: if one finds a measurement outcome which enables probabilistic teleportation, then every matching outcome can be obtained from it by a local unitary transformation on system $A$. We give an example for entanglement matching. Suppose that the antilinear operator $L$ describing the shared state $|\sigma\rangle_{BC}$ is given by the following matrix: $$L = \left( \begin{matrix} \alpha_1\cr & \ddots\cr &&\alpha_n\end{matrix} \right), \qquad |\sigma\rangle_{AB}=\sum_i \alpha_i | i \rangle_B | i \rangle_C,$$ where all $\alpha_i$ are nonzero (consider a Schmidt decomposition for example). Taking that the unitary transformation $U_q$ is identity, we obtain from (\[eq:L\_q\]) that a matching measurement outcome is given by $$\begin{aligned} L_1&=& \left( \sum_i \frac1{|\alpha_i|^2} \right)^{-1/2} \left( \begin{matrix}1/\alpha_1^\ast \cr & \ddots\cr && 1/\alpha_n^\ast\end{matrix} \right), \nonumber\\ |\sigma_1\rangle_{AB} &=& \left( \sum_i \frac1{|\alpha_i|^2} \right)^{-1/2} \sum_i \frac1{\alpha_i^\ast} | i \rangle_A | i \rangle_B.\end{aligned}$$ Thus if Alice measures an observable with an eigenstate equal to $|\sigma_1\rangle_{AB}$ then that measurement outcome implements a conditional teleportation. Conclusion {#sec:concl} ========== We have summarized the relations between quantum channels, bipartite states and antilinear operators, focusing the description of bipartite pure states with the latter. Applying this description, we have characterized all possible conditional teleportation schemes. We have found that the independence of the probability of a measurement outcome on the input state is a necessary and sufficient condition of the *linearity* of the transformation to be applied by the receiver. We have generalized the concept of “entanglement matching”, which means that in schemes under consideration the entangled state shared by the parties, and those measured by the sender should “match” each other. The results presented here show that this formalism is applicable of describing entanglement and quantum teleportation in a quite general way. This method may be also applicable for the treatment of entanglement between systems described by Hilbert-spaces of different dimensionality. It can also have consequences regarding the description of teleportation and related phenomena in the framework of quantum operations. This work was supported by the Research Fund of Hungary (OTKA) under contract No. T034484. One of the authors (M. K.) thanks Prof. Vladim' ir Bužek for useful discussions. [10]{} Bennett C. H., Brassard G., Crépeau C., Jozsa R., Peres A., and Wootters W. K., Phys. Rev. Lett. (1993) 1895 Bouwmeester D., Pan J.-W., Mattle K., Eibl M., Weinfurter H., and Zeilinger A., Nature (1997) 575 Boschi D., Branca S., Martini F. D., Hardy L., and Popescu S., Phys. Rev. Lett. (1998) 1121 Furusawa A., Sørensen J. L., Braunstein S. L., Fuchs C. A., Kimble H. J., and Polzik E. S., Science (1998) 706 Popescu S., Phys. Rev. Lett. (1994) 797 Banaszek K., Phys. Rev. A (2000) 024301 Horodecki M., Horodecki P., and Horodecki R., Phys. Rev. A (1999) 1888 Mor T., quant-ph/9608005 (1996) Mor T. and Horodecki P., quant-ph/9906039 (1999) Li W.-L., Li C.-F., and Guo G.-C., Phys. Rev. A (2000) 034301 Schumacher B., Phys. Rev. A (1996) 2614 Arens R. and Varadarajan V. S., J. Math. Phys. (2000) 638 D’Ariano G. M., Presti P. L., and Sacchi M. F., Phys. Lett. A (2000) 32 Nielsen M. A. and Caves C. M., Phys. Rev. A (1997) 2547

--- abstract: 'The vacuum polarization of a quark, when considered in terms of the external momentum $q^2$, is a function of the Stieltjes type. Consequently, the mathematical theory of Pade Approximants assures that the full function, at any finite value of $q^2$ away from the physical cut, can be reconstructed from its low-energy power expansion around $q^2=0$. We illustrate this point by applying this theory to the vacuum polarization of a heavy quark and obtain the value of the constant $K^{(2)}$ governing the threshold expansion at order $\mathcal{O}(\alpha_s^2)$.' --- [**Pade Theory applied to the vacuum polarization\ of a heavy quark**]{}\ [**P. Masjuan, S. Peris**]{}\ Grup de F[í]{}sica Te[ò]{}rica and IFAE\ Universitat Aut[ò]{}noma de Barcelona, 08193 Barcelona, Spain.\ Introduction ============ One interesting object to study in connection with the physics of heavy quarks is the vacuum polarization function of two electromagnetic currents. This requires high order perturbative calculations which, because of the obvious need to keep a nonzero mass $m$ for the quark, become extremely difficult to perform. This is why, while the $\mathcal{O}(\alpha_s^0)$ and $\mathcal{O}(\alpha_s^1)$ contributions have been known for a long time [@Kallen], state of the art calculations can only produce a result at $\mathcal{O}(\alpha_s^2)$ in the form of an expansion at low energies (i.e. $q^2=0$), at high energies (i.e. $q^2=\infty$) or at threshold (i.e. $q^2=4 m^2$), but not for the complete function, which is still out of calculational reach. In this circumstances, it would of course be very interesting to be able to reconstruct this function by some kind of interpolation between the former three expansions. After the work in refs. [@Broadhurst; @ChetyrkinMethod; @ChetyrkinH01], it has become customary to attempt this reconstruction of the vacuum polarization function with the help of Pade Approximants. Since these approximants are ratios of two polynomials in the variable $q^2$, they are very suitable for the matching onto the low-energy expansion. This is so because this low-$q^2$ expansion is truly an expansion in powers of $q^2$, as a consequence of the finite energy threshold[^1] starting at $4m^2$. However, it is clear that they cannot fully recover the nonanalytic terms which appear, e.g., in the form of logarithms of $q^2$ in the expansion at high energies (or at threshold, where there is also a squared root behavior). Therefore, what is really done in practice is to first subtract all these logarithmic pieces from the full function (impossible to match exactly with a Pade) with the help of a *guess* function with the appropriate threshold and high-energy behavior, and then apply Pades to the remaining regular expression[^2]. In this way, the authors of Ref. [@HoangMateu] were able to compute, e.g., the value of the constant $K^{(2)}$ appearing in the $\mathcal{O}(\alpha_s^2)$ expansion of the vacuum polarization at threshold, which has not yet been possible to obtain from a Feynman diagram calculation. Although this result is very interesting, the construction is not unique. As recognized in Ref. [@HoangMateu], some amount of educated guesswork is required in order to resolve the inherent ambiguity in the procedure. For instance, a certain number of unphysical poles are encountered, and some additional criteria have to be imposed in order to decide how to discard these poles. Since the resulting ambiguity leads to a systematic error which needs to be quantified, this error is then estimated by varying among several of the possible arbitrary choices in the construction. Although all these choices are made judiciously and in a physically motivated manner, it is very difficult to be confident of the error made in the result, which obviously has an impact on the value extracted for the constant $K^{(2)}$. In this short note we would like to point out that, regarding the vacuum polarization function, one can do away with all the above ambiguities. The vacuum polarization function belongs to a class of functions (the so-called Stieltjes functions) for which a well-known theorem assures that the diagonal (and paradiagonal) Pade Approximants converge. The result of this theorem together with the fantastic amount of information obtained on the Taylor expansion around $q^2=0$, for which 30 terms are known [@Maier], will allow us to predict a value for $K^{(2)}$. As it turns out, our result is very close to that of Ref. [@HoangMateu], although slightly smaller. The Theory of Pade Approximants is sufficiently developed to make these approximants a systematic mathematical tool. As we will see in this article, and has been exploited in Refs. [@Broadhurst; @ChetyrkinMethod; @ChetyrkinH01; @HoangMateu], they can be very useful for higher order calculations in perturbation theory. Furthermore, they can also be a very interesting conceptual tool for large-$N_c$ QCD [@PerisPades]. In this case this is true even when the function is not Stieltjes because large-$N_c$ QCD Green’s functions are meromorphic and there are powerful theorems in this case as well [@PerisMasjuan]. Finally, Pades can also be used for analyzing the experimental data by fitting to a rational function rather than the more commonly used polynomial fitting[@SC1]. They are also instrumental in discussing methods of unitarization [@SC2]. Pades and the vacuum polarization ================================= Let us start by defining the vacuum polarization function $\Pi(q^2)$ through the correlator of two electromagnetic currents $j^\mu(x)=\bar q(x)\gamma^\mu q(x)$, $$\begin{aligned} \label{pidef} \left(g_{\mu\nu}q^2-q_\mu q_\nu\right)\, \Pi(q^2) \, = \, \, - \,i \int\mathrm{d}x\, e^{iqx}\left\langle \,0\left|T\, j_\mu(x)j_\nu(0)\right|0\, \right\rangle \,,\end{aligned}$$ where $q^\mu$ is the external four-momentum. As it is known, due to the optical theorem, the $e^+e^-$ cross section is proportional to the imaginary part of $\Pi$. As a result, $\mbox{Im}\,\Pi$ is a positive definite function, i.e. $$\label{positive} \mbox{Im}\,\Pi(t+i \varepsilon)\geq 0 \ ,$$ a property which will become crucial in what follows. In perturbation theory $\Pi(q^2)$ may be decomposed to ${\mathcal O}(\alpha_s^2)$ as $$\begin{aligned} \label{Pi} \Pi(q^{2}) \, = \,&\, \Pi^{(0)}(q^{2}) \, + \,\left(\frac{\alpha_{s}}{\pi}\right)\, \Pi^{(1)}(q^{2})+\left(\frac{\alpha_{s}}{\pi}\right)^{2}\, \Pi^{(2)}(q^{2}) \, + {\mathcal O}(\alpha_s^3)\, .\end{aligned}$$ For definiteness, $\alpha_s$ denotes the strong coupling constant in the $\overline{\mathrm{MS}}$ scheme at the scale $\mu=m_{pole}$, but this is not important for the discussion which follows. Equation (\[Pi\]) will be understood in the on-shell normalization scheme where a subtraction at zero momentum has been made in such a way as to guarantee that $\Pi(0)=0$. As it is well known, the vacuum polarization in Eq. (\[Pi\]) satisfies a once subtracted dispersion relation, i.e. $$\label{disprel} \Pi(q^{2}) = q^2 \int_{0}^{\infty} \frac{dt}{t (t-q^2-i \varepsilon)}\ \frac{1}{\pi} \mbox{Im}\,\Pi(t+i \varepsilon)\ .$$ Since all diagrams with intermediate gluon states are absent up to $\mathcal{O}(\alpha_s^2)$, the lower limit for the dispersive integral (\[disprel\]) starts, in fact, at a finite value given by the threshold for pair production, i.e. $4m^2$. This fact only carries over to higher orders in $\alpha_s$ provided these intermediate gluon states are neglected. From now on, we will restrict ourselves to the vacuum polarization in Eq. (\[Pi\]) to $\mathcal{O}(\alpha_s^2)$, neglecting higher orders in $\alpha_s$. In terms of the more convenient variable $$\label{z} z \, \equiv \, \frac{q^2}{4 m^2}\ ,$$ one can rewrite Eq. (\[disprel\]), after redefining $u=4 m^2/t$, as[^3] $$\label{disprelz} \Pi(z) = z \int_{0}^{1} \frac{d u}{1- u z-i \varepsilon}\ \frac{1}{\pi} \mbox{Im}\,\Pi\left(4 m^2 u^{-1}+i \varepsilon\right)\ .$$ Recalling that a Stieltjes function is defined as[@Baker][^4] $$\label{Stieltjes} f(z)=\int_{0}^{1/R} \frac{d\phi(u)}{1-u z}$$ where $\phi(u)$ is any *nondecreasing* function, one sees that the identification $$\label{id} d\phi(u)=\frac{1}{\pi} \mbox{Im}\,\Pi\left(4 m^2 u^{-1}+i \varepsilon\right) \ du$$ allows one to recognize that the integral in Eq. (\[disprelz\]) defines the Stieltjes function $z^{-1}\Pi(z)$. As one can see, the positivity property Eq. (\[positive\]) is crucial for the identification (\[id\]) to be possible. The representation of the function $f(z)$ in Eq. (\[Stieltjes\]) clearly shows a cut in the $z$ complex plane on the positive real axis for $R\leq z <\infty$. For the physical function $\Pi(z)$, this of course corresponds to the physical cut in momentum for $4m^2\leq q^2< \infty$, i.e. the physical case corresponds to $R=1$ in Eqs. (\[Stieltjes\],\[id\]). Furthermore, just like the function $f(z)$ in Eq. (\[Stieltjes\]) has a power series expansion convergent in the disk $|z|<R$, so does the function $ \Pi(z)$ in Eq. (\[disprelz\]) have a power series expansion convergent in the disk $|z|<1$. A Pade Approximant to a function $f(z)$, which will be denoted by $P_N^M(z)$, is the ratio of two polynomials of degree $M$ and $N$ (respectively)[^5] such that its expansion in powers of $z$ about the origin matches the expansion of the original function up to and including the term of ${\mathcal O}(z^{M+N})$. When the original function $f(z)$ is Stieltjes with a finite radius of convergence about the origin, $R$, it is a well-known result in the theory of Pade Approximants that the sequence $P_N^{N+J}(z)$ (with $J\geq -1$) converges to the original function, as $N\rightarrow\infty$, on any *compact* set in the complex plane, excluding the cut at $R\leq z <\infty$ [@Baker]. This excludes , in the physical case, the cut at $4m^2\leq q^2< \infty$ (recall that $R=1$). The position of the poles in the Pade Approximant accumulate on the positive real axis starting at threshold, $q^2=4 m^2$, mimicking the presence of the physical cut in the original function. When Pades are applied to the vacuum polarization, this means, in particular, that there can be no spurious pole outside of the positive real axis in the $z$ plane and, consequently, no room for ambiguities. Furthermore, the convergence of the approximation (and the error) can be checked as a function of $N$, as we will see. Analysis ======== In Eq. (\[Pi\]), the full functions $\Pi^{(0,1)}(q^2)$ are known. They are given by the following expressions [@Kallen]: $$\begin{aligned} \label{eq:Pi01} \Pi^{(0)}(z) & \, = \, \frac{3}{16\pi^{2}}\left[\frac{20}{9}+\frac{4}{3z}-\frac{4(1-z)(1+2z)}{3z}G(z)\right], {\nonumber}\\[2mm] \Pi^{(1)}(z) & \, = \, \frac{3}{16\pi^{2}}\left[\frac{5}{6}+\frac{13}{6z}-\frac{(1-z)(3+2z)}{z}G(z)+ \frac{(1-z)(1-16z)}{6z}G^{\, 2}(z)\right. {\nonumber}\\ &\qquad \qquad \qquad -\,\left.\frac{(1+2z)}{6z}\left(1+2z(1-z)\frac{d}{dz}\right)\frac{I(z)}{z}\right] \quad ,\end{aligned}$$ where $$\begin{aligned} \label{eq:Gz} I(z) & \, = \, 6\Big[\zeta_{3}+4\,\mbox{Li}_{3}(-u)+2\,\mbox{Li}_{3}(u)\Big]- 8\Big[2\,\mbox{Li}_{2}(-u)+\mbox{Li}_{2}(u)\Big]\ln u{\nonumber}\\ &\qquad \qquad \qquad \qquad -2\Big[2\,\ln(1+u)+\ln(1-u)\Big]\ln^{2}u\,, {\nonumber}\\[2mm] G(z) & \, = \, \frac{2\, u\,\ln u}{u^{2}-1}\ , \quad \mbox{with}\quad u \, \equiv \, \frac{\sqrt{1-1/z}-1}{\sqrt{1-1/z}+1} \ .\end{aligned}$$ However, the situation with the function $\Pi^{(2)}(q^2)$ is different. In fact, $\Pi^{(2)}(q^2)$ is only partially known through its low-energy power series expansion around $q^2=0$, its high-energy expansion around $q^2=\infty$ and its threshold expansion around $q^2=4 m^2$, but the full function has not yet been computed. Unlike the latter two expansions, for which only a few terms are known, our knowledge of the expansion of $ \Pi^{(2)}(q^2)$ around $q^2=0$ is very impressive, after the work of Ref. [@Maier] where 30 terms of this expansion were computed. Although the full vacuum polarization function $\Pi(q^2)$ is Stieltjes, there is no reason why all the individual contributions $\Pi^{(0,1,2,...)}(q^2)$ should also have this property. Amusingly, however, this happens to be true both for $\Pi^{(0)}(q^2)$ and $\Pi^{(1)}(q^2)$ [@Broadhurst][^6]. As we will now show, this is no longer the case for $\Pi^{(2)}(q^2)$ because its power series expansion around $q^2=0$ does not satisfy certain determinantal conditions which hold for a Stieltjes function. Defining the power expansion around $z=0$ of the Stieltjes function $f(z)$ in Eq. (\[Stieltjes\]) as $$\label{exp} f(z)= \sum_{n=0}^{\infty} f_n z^n\ ,$$ the coefficients $f_n$ satisfy the following determinantal conditions. Let $D(m,n)$ be the determinant constructed with the Taylor coefficients $f_n$ $$\label{det} D(m,n)=\begin{vmatrix} f_m & f_{m+1} & \ldots & f_{m+n} \\ f_{m+1} & f_{m+2} & \ldots & f_{m+n+1} \\ \vdots & \vdots & & \vdots \\ f_{m+n} & f_{m+n+1} & \ldots & f_{m+2n} \\ \end{vmatrix} \quad .$$ A Stieltjes function must satisfy $D(m,n)>0$, for all $m,n$ [@Baker]. However, using the $f_n$ coefficients given in Ref. [@Maier] (in the on-shell scheme, with the number of light flavors $n_\ell=3$): $$\label{taylor} z^{-1} \Pi^{(2)}(z)\approx 0.631107 + 0.616294 \ z + 0.56596 \ z^2 + 0.520623 \ z^3+ \ldots \quad ,$$ one can immediately see that, e.g., $$\label{D11} D(0,1)=D(0,1)=\begin{vmatrix} 0.631107 &0.616294 \\ 0.616294 & 0.56596 \\ \end{vmatrix}= -0.0226376 < 0\quad .$$ This proves that the individual function $\Pi^{(2)}(q^2)$ is, all by itself, not a Stieltjes function, even though the combination $\Pi(q^2)$ in (\[Pi\]) is. Therefore, we will now focus on applying the Theory of Pade Approximants to the full combination $\Pi(q^2)$ in Eq. (\[Pi\]) in order to extract information on the individual term $\Pi^{(2)}(q^2)$. Since, as it is obvious from Eq. (\[Pi\]), the function $\Pi(q^2)$ depends on the value of $\alpha_s$, any Pade Approximant to it will also depend on the value of $\alpha_s$, i.e. $P_N^{N+J}(z;\alpha_s )$. This means that it is possible to construct a rational approximation to the three functions $ \Pi^{(0,1,2)}(q^2)$ from three different sequences of Pade Approximants to $\Pi(q^2)$ constructed at three arbitrary values of $\alpha_s$, let us say $\alpha_s=0, \pm \beta$, with $\beta$ sufficiently small so as to be able to neglect the terms of $\mathcal{O}(\alpha_s^3)$ in Eq. (\[Pi\]) . In this way one obtains $$\begin{aligned} \label{aprox} \Pi^{(0)}(z) &\approx & P_N^{N+J}(z;\alpha_s=0 ) {\nonumber}\\ \Pi^{(1)}(z) &\approx & \frac{\pi}{2 \beta}\left\{P_N^{N+J}(z;\alpha_s=\beta )-P_N^{N+J}(z;\alpha_s=-\beta ) \right\}{\nonumber}\\ \Pi^{(2)}(z) & \approx & \frac{\pi^2}{2 \beta^2}\left\{P_N^{N+J}(z;\alpha_s=\beta )+P_N^{N+J}(z;\alpha_s=-\beta )- 2 P_N^{N+J}(z;\alpha_s=0 )\right\}\, ,\end{aligned}$$ where $J\geq -1$ and $N\rightarrow \infty$. Since the value of $\beta$ chosen is arbitrary, the $N\rightarrow \infty$ limit should produce results which are independent of $\beta$, due to the convergence of the Pade Approximants to $\Pi(q^2)$. Therefore, one should see that the three combinations (\[aprox\]) are increasingly independent of $\beta$ as $N$ grows.[^7] This is indeed what happens. Furthermore, since we know the exact function $\Pi^{(1)}(z) $, we can compare it to the rational approximation on the right hand side of the second Eq. (\[aprox\]) in order to test the approximation. Figure \[precision\] shows the number of decimal places reproduced by this rational approximation in the interval $0.4 \leq z\leq 0.9$, when $N=14$ and $J=0$ (i.e. the diagonal Pade $P_{14}^{14}$), for values of $\beta$ in the interval $0.1 \leq \beta \leq 1$. As one can see, the dependence on $\beta$ cannot be distinguished, and the accuracy reaches, e.g., $\sim 10$ decimal places at $z=0.9$. ![Number of decimal places reproduced by the rational approximation in Eq. (\[aprox\]) to the function $\Pi^{(1)}(z) $ as a function of $z$, in the interval $0.4\leq z\leq 0.9$. []{data-label="precision"}](PadeoverP1.eps){width="3in"} The third Eq. (\[aprox\]) yields the desired approximation to $\Pi^{(2)}(z) $. To be precise, it gives us a rational approximation to $\Pi^{(2)}(z) $ in any compact set of the $z$ complex plane, away from the cut $1\leq z< \infty$. Since the threshold expansion at $z\approx 1$ can be written as [@HoangMateu; @Czarnecki] $$\begin{aligned} \label{threshold} \Pi^{(2)}_{\mathrm{th.}}(z) &= &\frac{1.72257}{\sqrt{1-z}} +\left[0.34375-0.0208333\ n_{\ell}\right] \ \ln^{2}(1-z){\nonumber}\\ &+& \left[0.0116822\ n_{\ell} + 1.64058 \right]\ \ln(1-z) + K^{(2)}{\nonumber}\\ &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+ \left[-0.721213 - 0.0972614\ n_\ell + 3.05433 \ \ln(1-z)\right] \sqrt{1-z} \, + \, {\mathcal O}(1-z)\ ,\end{aligned}$$ in terms of an unknown constant $K^{(2)}$, our Pade Approximation (\[aprox\]) may be used to determine this constant, as we will next discuss. In this threshold expansion we take $ n_{\ell}=3$ as the number of light flavors. Even though the numerical coefficients have been rounded off for simplicity, they may be extracted exactly from the results in Ref. [@Czarnecki]. Since Pades are not convergent on the physical cut, it is impossible to match the rational approximants (\[aprox\]) to the threshold expansion (\[threshold\]) as a function of $z$. This fact is obvious from the presence of logarithms and squared roots in Eq. (\[threshold\]). However, both approximations (\[aprox\]) and (\[threshold\]) are valid for values of $z$ at a finite distance from the cut and, in particular, in a certain window in the interval $0\leq z<1$. Within this window, a numerical matching of (\[aprox\]) and (\[threshold\]) is possible and will in fact allow us to determine the unknown constant $K^{(2)}$. ![Predicted value for $K^{(2)}$ from the sequence of diagonal, $P^{N}_{N}$ (left panel), and first paradiagonal, $P^{N-1}_{N}$ (right panel), Pade Approximants. This figure corresponds to $\beta=0.5$.[]{data-label="convergence"}](K2NN.eps "fig:"){width="3in"} ![Predicted value for $K^{(2)}$ from the sequence of diagonal, $P^{N}_{N}$ (left panel), and first paradiagonal, $P^{N-1}_{N}$ (right panel), Pade Approximants. This figure corresponds to $\beta=0.5$.[]{data-label="convergence"}](K2N1N.eps "fig:"){width="3in"} In order to determine this window, we make the following observations. First, although the rational approximation (\[aprox\]) is convergent as $N\rightarrow \infty$ in the interval $0\leq z< 1$, it is more accurate the closer one gets to $z=0$ in this interval, for a given value of $N$. On the other hand, the threshold expansion (\[threshold\]) is more accurate the closer one gets to the branching point at $z=1$. From these two competing effects it is possible to determine an optimal window in $z$ by minimizing a combined error function. We will call this error function $\mathcal{E}(z)$. The function $\mathcal{E}(z)$ has to take into account the error from the Pades as well as the error from the threshold expansion. To estimate the error from the Pades, we consider the difference between two consecutive elements in the sequence, i.e. $|P_{N}^{N+J}-P_{N-1}^{N-1+J} |$. As to the threshold expansion, we estimate its error as $|1-z|$, since the expression (\[threshold\]) is accurate up to terms of $\mathcal{O}(1-z)$. Therefore, in order to avoid possible accidental cancelations between the two errors, we define our combined error function as the following sum: $$\begin{aligned} \label{error} \!\!\!\! \mathcal{E}(z)&\!\!\!\!\!=&\!\!\!\!\!\left| \frac{\pi^2}{2 \beta^2}\Big\{P_N^{N+J}(z;\alpha_s=\beta )+P_N^{N+J}(z;\alpha_s=-\beta )- 2 P_N^{N+J}(z;\alpha_s=0 )\Big\}- \Big\{N\!\rightarrow \! N\!-\!1\Big\}\right|{\nonumber}\\ &&\qquad +\quad |1-z|\ . \end{aligned}$$ Minimizing $\mathcal{E}(z)$ with respect to $z$ in the interval $0 \leq z< 1$, for every given values of $N$ and $\beta$, we may determine a value of $z$ at the minimum, namely $z^*$. This $z^*$ is then the one used to determine the constant $K^{(2)}$ as $$\label{K} \!\!\! K^{(2)}\approx \frac{\pi^2}{2 \beta^2}\left\{P_N^{N+J}(z^*;\alpha_s=\beta )+P_N^{N+J}(z^*;\alpha_s=-\beta )- 2 P_N^{N+J}(z^*;\alpha_s=0 )\right\}- \widehat{\Pi}^{(2)}_{\mathrm{th.}}(z^*) \quad ,$$ for the given $N$ and $\beta$. In Eq. (\[K\]), $ \widehat{\Pi}^{(2)}_{\mathrm{th.}}(z^*) $ stands for the expression in Eq. (\[threshold\]) without the constant $K^{(2)}$ and, of course, without the term $\mathcal{O}(1-z)$, evaluated at $z=z^*$. The knowledge of 30 terms from the low-energy expansion gives us enough information to be able to construct up to the Pade $P_{14}^{14}$ from the diagonal sequence, and up to the Pade $P_{15}^{14}$ from the first paradiagonal sequence. This corresponds to $J=0$ and $J=-1$ in Eq. (\[K\]). In all cases considered we have varied $\beta$ in the generous range $0\leq \beta \leq 1$, but our results are insensitive to this variation within errors, as expected. ![Matching of the rational approximant in Eq. (\[aprox\]) (solid blue line) to the threshold expansion $\Pi^{(2)}_{\mathrm{th.}}(z)$ in (\[threshold\]) (solid-dashed red line) for $N=14, J=0$ and the value of $K^{(2)}$ in (\[result\]). This figure shows the result for different values of $\beta$ in the range $0\leq \beta\leq1$, but the dependence on $\beta$ is so small that cannot be discerned.[]{data-label="matching"}](plotmatching.eps){width="3in"} Result ====== Our results for the constant $K^{(2)}$ are shown in Fig. \[convergence\]. This figure shows the convergence of the diagonal sequence and the first paradiagonal sequence as a function of the order in the Pade. As one can see, we find a very nice convergence in both cases, with compatible results. Based on this analysis, we obtain the following value of $K^{(2)}$: $$\label{result} K^{(2)}=3.71\pm 0.03\ .$$ This result is very close to, although slightly smaller than, the value obtained in Ref. [@HoangMateu], i.e. $K^{(2)}=3.81\pm 0.02$. This is the main result of this work. The error bars shown on Fig. \[convergence\] have been calculated as $\pm \mathcal{E}(z^*)$. Looking at the figure we see that the change in the value of $K^{(2)}$ from one element of the sequence to the next is included in the errors shown, which we interpret as a sign that the estimate for the error we have made is rather accurate. Although we have taken symmetric errors for simplicity, it is also clear from the figure that the approach to the true value is made from below, so that a slightly more accurate determination could be achieved with the use of an asymmetric error. Apart from that, given the present knowledge of the expansions at low energy (\[taylor\]) and at threshold (\[threshold\]), we find it difficult to believe any error estimate which could significantly go below our figure in Eq. (\[result\]). Of course, should more terms in either expansion be known, a rerun of our analysis could immediately produce a more precise determination of $K^{(2)}$. Figure \[matching\] shows the matching of the rational approximant (i.e. the right hand side of the third of the Eqs. (\[aprox\])) to the threshold expansion given by $\Pi^{(2)}_{\mathrm{th.}}(z)$ in Eq. (\[threshold\]), for $N=14$ and $J=0$, i.e. with the Pade $P_{14}^{14}$, and for the value of $K^{(2)}$ we have obtained. As one can see, this Pade is able to reproduce, with high accuracy, the threshold expansion behavior in a window $0.92 \lesssim z < 1$. At $z=1$ and above, the two lines in Fig. \[matching\] will again diverge from each other, just as they do at low $z$. The value of $z^*$ minimizing the error function $\mathcal{E}(z)$ in (\[error\]) was found at $z^* \simeq 0.98$ in this particular case. For illustration, in Fig. \[poles\] we show the position of the poles in the Pade $P_{14}^{14}$. As one can see, all the poles are sitting on the positive real axis above $z=1$, as it should be. Notice how they accumulate in the region $z\gtrsim 1$. This is how Pades approximate the physical cut present in the original function. As ensured from Pade Theory, this behavior was found in all the Pades considered. ![Location of the poles in the pade $P_{14}^{14}$ in the complex plane. Notice the accumulation of poles at $z\gtrsim 1$, simulating the physical cut. []{data-label="poles"}](Den14Plot.eps){width="3in"} Finally, as a further test of our method, we have calculated the value of the constants $H^{(2)}_0$ and $H^{(2)}_1$ which appear in the large-$z$ expansion of the function $\Pi^{(2)}(z)$ (we take $n_f=n_{\ell}+1=4$ in the following expression): $$\begin{aligned} \label{Pihigh} &&\Pi_{High-z}^{(2)}(z) \,= \, \, (0.034829 - 0.0021109\ n_f)\ln^{2}(-4z) +(-0.050299 + 0.0029205\ n_f)\ln(-4z) {\nonumber}\\ & &+\quad H^{(2)}_0 + (0.18048 - 0.0063326\ n_f)\frac{\ln^{2}(-4z)}{z} + (-0.59843 + 0.027441\ n_f)\frac{\ln(-4z)}{z}{\nonumber}\\ & &+\quad \frac{H^{(2)}_1}{z} +\,{\mathcal O}\Big(z^{-3} \ln^3(-z)\Big)\ .\end{aligned}$$ Using our method, we find $H_0^{(2)}=-0.582 \pm 0.008$. This result is to be compared to the true value $H^{(2)}_0=-0.5857$ [@ChetyrkinH01]. If we now input this exact value of $H_0^{(2)}$, by a rerun of the method, we may then determine the value of $H^{(2)}_1$. In this way, we find $H_1^{(2)}=-0.194 \pm 0.033$, which is to be compared to the exact value $H^{(2)}_1=-0.1872$ [@ChetyrkinH01]. Again, we find this agreement rather reassuring. **Acknowledgements** We thank A. Pineda and J.J. Sanz-Cillero for discussions and A. Hoang and V. Mateu for comments on the manuscript. This work has been supported by CICYT-FEDER-FPA2008-01430, SGR2005-00916, the Spanish Consolider-Ingenio 2010 Program CPAN (CSD2007-00042) and by the EU Contract No. MRTN-CT-2006-035482, “FLAVIAnet”. [99]{} A. O. G. Kallen and A. Sabry, Kong. Dan. Vid. Sel. Mat. Fys. Med.  [**29N17**]{} (1955) 1. D. J. Broadhurst, J. Fleischer and O. V. Tarasov, Z. Phys.  C [**60**]{} (1993) 287 \[arXiv:hep-ph/9304303\]; J. Fleischer and O. V. Tarasov, Z. Phys.  C [**64**]{} (1994) 413 \[arXiv:hep-ph/9403230\]; P. A. Baikov and D. J. Broadhurst, arXiv:hep-ph/9504398. K. G. Chetyrkin, J. H. Kuhn and M. Steinhauser, Phys. Lett.  B [**371**]{} (1996) 93 \[arXiv:hep-ph/9511430\]. 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Press. 1996, chapter 5; C. Brezinski and J. Van Inseghem, [*Padé Approximations, Handbook of Numerical Analysis*]{}, P.G. Ciarlet and J.L. Lions (editors), North Holland, vol. III; see also, e.g., C. Diaz-Mendoza, P. Gonzalez-Vera and R. Orive, Appl. Num. Math. **53** (2005) 39 and references therein. For a very pedagogical summary, see C. Bender and S. Orszag, *Advanced Mathematical Methods for Scientists and Engineers I: asymptotic methods and perturbation theory*, Springer 1999, section 8.6 A. Czarnecki and K. Melnikov, Phys. Rev. Lett.  [**80**]{} (1998) 2531 \[arXiv:hep-ph/9712222\]. [^1]: In perturbation theory, this is true so long as purely gluonic intermediate states are not considered. Beyond perturbation theory, the threshold occurs at $4 m^2_{\pi}$, where $m_{\pi}$ is the pion mass. [^2]: In fact, this is done after a conformal mapping whereby all the (cut) complex plane is mapped into a circle of unit radius. [^3]: We are simplifying the notation by replacing $ \Pi(4 m^2 z) \rightarrow \Pi(z) $. [^4]: In Ref. [@Baker], the variable is chosen to be $-z$ rather than $z$. [^5]: Without loss of generality, the denominator polynomial of degree $N$ is chosen to be unity at $z=0$. [^6]: The case of $\Pi^{(0)}(q^2)$ is trivial as it coincides with the full vacuum polarization for $\alpha_s$=0. [^7]: This independence of $\beta$ in the case of $ \Pi^{(0)}(q^2)$ is trivially true.