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−m2⁢x−m−2⁢R+12⁢x−m−1⁢R′;𝑚2superscript𝑥𝑚2𝑅12superscript𝑥𝑚1superscript𝑅′\displaystyle-\frac{m}{2}x^{-m-2}R+\frac{1}{2}x^{-m-1}R^{\prime};- divide start_ARG italic_m end_ARG start_ARG 2 end_ARG italic_x start_POSTSUPERSCRIPT - italic_m - 2 end_POSTSUPERSCRIPT italic_R + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_x start_POSTSUPERSCRIPT - italic_m - 1 end_POSTSUPERSCRIPT italic_R start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ;

Rnm′′′⁢(x)superscriptsuperscriptsubscript𝑅𝑛𝑚′′′𝑥\displaystyle{R_{n}^{m}}^{\prime\prime\prime}(x)italic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT ( italic_x )

F′′superscript𝐹′′\displaystyle F^{\prime\prime}italic_F start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT

F′′′superscript𝐹′′′\displaystyle F^{\prime\prime\prime}italic_F start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT

(i) fast calculation of f′′/f′superscript𝑓′′superscript𝑓′f^{\prime\prime}/f^{\prime}italic_f start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT / italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT from f/f′𝑓superscript𝑓′f/f^{\prime}italic_f / italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT,

B

The nerve X𝑋Xitalic_X of a 2222-groupoid object X2⇛X1⇒X0⇛subscript𝑋2subscript𝑋1⇒subscript𝑋0X_{2}\Rrightarrow X_{1}\Rightarrow X_{0}italic_X start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⇛ italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⇒ italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT in (𝒞,𝒯)𝒞𝒯(\mathcal{C},\mathcal{T})( caligraphic_C , caligraphic_T ) as in

2222-groupoid objects in (𝒞,𝒯′′)𝒞superscript𝒯′′(\mathcal{C},\mathcal{T}^{\prime\prime})( caligraphic_C , caligraphic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) in the sense of Prop-Def. 2.17;

object in (𝒞,𝒯)𝒞𝒯(\mathcal{C},\mathcal{T})( caligraphic_C , caligraphic_T ), as we have demonstrated in the

Prop-Def. 2.17 is a 2222-groupoid object in (𝒞,𝒯)𝒞𝒯(\mathcal{C},\mathcal{T})( caligraphic_C , caligraphic_T ) as in Def. 1.3.

The first three layers of a 2222-groupoid object in (𝒞,𝒯)𝒞𝒯(\mathcal{C},\mathcal{T})( caligraphic_C , caligraphic_T ) as in Def. 1.3 is a 2222-groupoid object in (𝒞,𝒯)𝒞𝒯(\mathcal{C},\mathcal{T})( caligraphic_C , caligraphic_T ) as in Prop-Def. 2.17.

C

Then we define the quotient category of Q(mTT) over canonical isomorphisms in which we will interpret emTT. The equivalence relation generated by canonical isomorphisms coincides with extending the previous collection of canonical isomorphisms by including all identity morphisms between objects of Q(mTT) 888We still get a well defined quotient category, if we make canonical also all the morphisms between objects Q(mTT) which include the identity morphism between their supports as a representative (and hence the equivalence relations of the two objects are equivalent)..

We call Q(mTT)/≃similar-to-or-equals\simeq≃ the category obtained by quotienting Q(mTT) over canonical isomorphisms:

we can interpret emTT (and emTTdp) judgements in the category Q(mTT)/≃similar-to-or-equals\simeq≃ (Q(mTTdp)/≃similar-to-or-equals\simeq≃) by following the idea behind the naive interpretation of dependent types

emTT in a category Q(mTT)/≃similar-to-or-equals\simeq≃ obtained by quotienting Q(mTT) only over suitable

Indeed, if we take the category Q(mTT)/≃similar-to-or-equals\simeq≃  obtained from Q(mTT) by quotienting it over isomorphisms, then this category enjoys a unique choice

A

By construction, G/Γ𝐺ΓG/\Gammaitalic_G / roman_Γ is isomorphic to G𝐧0/Γ𝐧0subscript𝐺subscript𝐧0subscriptΓsubscript𝐧0G_{{\bf n}_{0}}/\Gamma_{{\bf n}_{0}}italic_G start_POSTSUBSCRIPT bold_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT / roman_Γ start_POSTSUBSCRIPT bold_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT for some 𝐧0subscript𝐧0{\bf n}_{0}bold_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, so we may assume without loss of generality that G/Γ=G𝐧0/Γ𝐧0𝐺Γsubscript𝐺subscript𝐧0subscriptΓsubscript𝐧0G/\Gamma=G_{{\bf n}_{0}}/\Gamma_{{\bf n}_{0}}italic_G / roman_Γ = italic_G start_POSTSUBSCRIPT bold_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT / roman_Γ start_POSTSUBSCRIPT bold_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT; since all smooth Riemannian metrics on a compact manifold are equivalent, we can also assume that dG/Γ=dG𝐧0/Γ𝐧0subscript𝑑𝐺Γsubscript𝑑subscript𝐺subscript𝐧0subscriptΓsubscript𝐧0d_{G/\Gamma}=d_{G_{{\bf n}_{0}}/\Gamma_{{\bf n}_{0}}}italic_d start_POSTSUBSCRIPT italic_G / roman_Γ end_POSTSUBSCRIPT = italic_d start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT bold_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT / roman_Γ start_POSTSUBSCRIPT bold_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT. We may also normalise F𝐹Fitalic_F to be bounded in magnitude by 1111. But this contradicts (5.1) for 𝐧𝐧{\bf n}bold_n sufficiently large, and the claim follows.

Thus, to establish Theorem 1.3, it will suffice to establish Conjecture 5.3 for s⩾3𝑠3s\geqslant 3italic_s ⩾ 3. This is the objective of the remainder of the paper.

The purpose of this paper is to establish the general case of a conjecture named the Inverse Conjecture for the Gowers norms by the first two authors in [23, Conjecture 8.3]. If N𝑁Nitalic_N is a (typically large) positive integer then we write [N]:={1,…,N}assigndelimited-[]𝑁1…𝑁[N]:=\{1,\dots,N\}[ italic_N ] := { 1 , … , italic_N }. For each integer s⩾1𝑠1s\geqslant 1italic_s ⩾ 1 the inverse conjecture GI⁡(s)GI𝑠{\operatorname{GI}}(s)roman_GI ( italic_s ), whose statement we recall shortly, describes the structure of 1111-bounded functions f:[N]→ℂ:𝑓→delimited-[]𝑁ℂf:[N]\rightarrow\mathbb{C}italic_f : [ italic_N ] → blackboard_C whose (s+1)stsuperscript𝑠1st(s+1)^{\operatorname{st}}( italic_s + 1 ) start_POSTSUPERSCRIPT roman_st end_POSTSUPERSCRIPT Gowers norm ‖f‖Us+1⁢[N]subscriptnorm𝑓superscript𝑈𝑠1delimited-[]𝑁\|f\|_{U^{s+1}[N]}∥ italic_f ∥ start_POSTSUBSCRIPT italic_U start_POSTSUPERSCRIPT italic_s + 1 end_POSTSUPERSCRIPT [ italic_N ] end_POSTSUBSCRIPT is large. These conjectures together with a good deal of motivation and background to them are discussed in [19, 21, 23]. The conjectures GI⁡(1)GI1{\operatorname{GI}}(1)roman_GI ( 1 ) and GI⁡(2)GI2{\operatorname{GI}}(2)roman_GI ( 2 ) have been known for some time, the former being a straightforward application of Fourier analysis, and the latter being the main result of [21] (see also [51] for the characteristic 2222 analogue). The case GI⁡(3)GI3{\operatorname{GI}}(3)roman_GI ( 3 ) was also recently established by the authors in [28]. The aim of the present paper is to establish the remaining cases GI⁡(s)GI𝑠{\operatorname{GI}}(s)roman_GI ( italic_s ) for s⩾3𝑠3s\geqslant 3italic_s ⩾ 3, in particular reestablishing the results in [28].

The inverse conjecture GI⁡(s)GI𝑠{\operatorname{GI}}(s)roman_GI ( italic_s ), Conjecture 1.2, has been formulated using linear nilsequences F⁢(gn⁢x⁢Γ)𝐹superscript𝑔𝑛𝑥ΓF(g^{n}x\Gamma)italic_F ( italic_g start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_x roman_Γ ). This is largely for compatibility with the earlier paper [23] of the first two authors on linear equations in primes, where this form of the conjecture was stated in precisely this form as Conjecture 8.3. Subsequently, however, it was discovered that it is more natural to deal with a somewhat more general class of object called a polynomial nilsequence F⁢(g⁢(n)⁢Γ)𝐹𝑔𝑛ΓF(g(n)\Gamma)italic_F ( italic_g ( italic_n ) roman_Γ ). This is particularly so when it comes to discussing the distributional properties of nilsequences, as was done in [24]. Thus, we shall now recast the inverse conjecture in terms of polynomial nilsequences, which is the formulation we will work with throughout the rest of the paper.

In our previous paper [28] it was already rather painful to keep proper track of such notions as “many” and “correlates with”. Here matters are even worse, and so to organise the above tasks it turns out to be quite convenient to first take an ultralimit of all objects being studied, effectively placing one in the setting of nonstandard analysis. This allows one to easily import results from infinitary mathematics, notably the theory of Lie groups and basic linear algebra, into the finitary setting of functions on [N]delimited-[]𝑁[N][ italic_N ]. In §5 and Appendix A we review the basic machinery of ultralimits that we will need here; we will not be exploiting any particularly advanced aspects of this framework. The reader does not really need to understand the ultrafilter language in order to comprehend the basic structure of the paper, provided that he/she is happy to deal with concepts like “dense” and “correlates with” in a somewhat informal way, resembling the way in which analysts actually talk about ideas with one another (and, in fact, analogous to the way we wrote this paper). It is possible to go through the paper and properly quantify all of these notions using appropriate parameters δ𝛿\deltaitalic_δ and (many) growth functions ℱℱ\mathcal{F}caligraphic_F. This would have the advantage of making the paper on some level comprehensible to the reader with an absolute distrust of ultrafilters, and it would also remove the dependence on the axiom of choice and in principle provide explicit but very poor bounds. However it would cause the argument to be significantly longer, and the notation would be much bulkier.

A

π0⁢π0¯=pmsubscript𝜋0¯subscript𝜋0superscript𝑝𝑚\pi_{0}\overline{\pi_{0}}=p^{m}italic_π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT over¯ start_ARG italic_π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG = italic_p start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT.

Let A~~𝐴\widetilde{A}over~ start_ARG italic_A end_ARG be an abelian variety over a number field k𝑘kitalic_k

A=((a,b⁢t~;c,d⁢t~)modN)𝐴modulo𝑎𝑏~𝑡𝑐𝑑~𝑡𝑁A=((a,b\widetilde{t};c,d\widetilde{t})\bmod{N})italic_A = ( ( italic_a , italic_b over~ start_ARG italic_t end_ARG ; italic_c , italic_d over~ start_ARG italic_t end_ARG ) roman_mod italic_N ).

an abelian variety A~~𝐴\widetilde{A}over~ start_ARG italic_A end_ARG in characteristic zero with a nice

that the endomorphisms of A~~𝐴\widetilde{A}over~ start_ARG italic_A end_ARG over k¯¯𝑘\overline{k}over¯ start_ARG italic_k end_ARG are defined over k𝑘kitalic_k.

A

\text{ }0 < italic_α ( italic_t ) ≤ italic_y start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_t , italic_ω ) ≤ italic_β ( italic_t ) .

solution set 𝒮b⁢(0).superscript𝒮𝑏0\mathcal{S}^{b}\left(0\right).caligraphic_S start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ( 0 ) . Observe that in Proposition 3, the convergence in law is rather weak. However, thanks to the

We have found the limiting solutions, i.e., 𝒮fb⁢(0)superscriptsubscript𝒮𝑓𝑏0\mathcal{S}_{f}^{b}\left(0\right)caligraphic_S start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ( 0 ). However, the following example shows that the probability

fixed ω∈Ω/𝒩,𝜔Ω𝒩\omega\in\Omega/\mathcal{N},italic_ω ∈ roman_Ω / caligraphic_N , we will explore what the limit of Xε⁢(⋅,ω)superscript𝑋𝜀⋅𝜔X^{\varepsilon}(\cdot,\omega)italic_X start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( ⋅ , italic_ω ) in 𝒮b⁢(0)superscript𝒮𝑏0\mathcal{S}^{b}\left(0\right)caligraphic_S start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ( 0 ) by

We now prove (21) holds. All the trajectories in 𝒮b⁢(0)superscript𝒮𝑏0\mathcal{S}^{b}\left(0\right)caligraphic_S start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ( 0 ) are analyzed and verified seriously in the following

D

Determining whether two given admissible trees are (topologically) equivalent is the problem we want to solve. The problem of determining whether two given admissible trees are isomorphic is combinatorial in nature and easy to solve algorithmically simply by enumerating all maps from the set of vertices of T1subscript𝑇1T_{1}italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT to that of T2subscript𝑇2T_{2}italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT until a suitable bijection is found (or not.)

There is an algorithm which to any admissible tree T𝑇Titalic_T associates a reduced admissible tree equivalent to T𝑇Titalic_T.

In this section, we simply call 2-automaton a planar topological 2-automaton. To any such 2-automaton 𝒳𝒳\mathcal{X}caligraphic_X, we shall associate a decorated graph G⁢(𝒳)𝐺𝒳G(\mathcal{X})italic_G ( caligraphic_X ) which contains enough information to recover the space of ends of the surface associated to 𝒳𝒳\mathcal{X}caligraphic_X. Then we show how to associate to a decorated graph G𝐺Gitalic_G a decorated tree of a special form, called an admissible tree, which also contains this information. This will allow us to reduce the topological classification of planar 2-automata to the classification of admissible trees up to some equivalence relation of topological nature. This latter classification problem will be solved in Section 3.

We now introduce three moves that can be used to simplify an admissible tree without changing its equivalence class.

We will show how to modify an admissible tree without changing its topological equivalence class, until it belongs to a special class of admissible trees, called reduced, for which topological equivalence will turn out to be equivalent to isomorphism. In order to motivate the construction, we first give some simple examples of pairs of admissible trees which are equivalent but fail to be isomorphic.

D

The collapsing of manifolds studied in this paper are closely related to multi-scale models in manifold learning where the intrinsic dimension of the data set is modelled by a function that depends on the scale, see [69].

For example, this occurs when the cryogenic electron-microscopy [57] is applied to image the structure of a large molecule which is connected to a small part of the molecule with a connection that allows a rotation (roughly

recall the counting function, NX⁢(E)subscript𝑁𝑋𝐸{{N}}_{X}(E)italic_N start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_E ) of X𝑋Xitalic_X, which is the number of the

The structure of collapsing in the moduli space 𝔐⁢(n,Λ,D)𝔐𝑛Λ𝐷{\mathfrak{M}}(n,\Lambda,D)fraktur_M ( italic_n , roman_Λ , italic_D ), with respect to the Gromov-Hausdorff distance,

speaking, the molecule has a moving ‘tail’). In this case, the problem of imaging the molecule is to find a manifold diffeomorphic to, e.g. S⁢O⁢(3)×S1⁢(ϵ)𝑆𝑂3superscript𝑆1italic-ϵSO(3)\times S^{1}(\epsilon)italic_S italic_O ( 3 ) × italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_ϵ ) which almost collapses to S⁢O⁢(3)𝑆𝑂3SO(3)italic_S italic_O ( 3 ) of lower dimension, where S1⁢(ϵ)superscript𝑆1italic-ϵS^{1}(\epsilon)italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_ϵ ) is a circle of small radius ϵitalic-ϵ\epsilonitalic_ϵ.

A

In 2011, Hong-Huang-Wang [7] studied a class of degenerate elliptic Monge-Ampère equation in a smooth, bounded and strictly convex domain

respect to the normal direction −D⁢u𝐷𝑢-Du- italic_D italic_u, we have the following formula on the m𝑚mitalic_m-th curvature of the level sets of the solution u𝑢uitalic_u,

Let uΩj,j=0,1,formulae-sequencesubscript𝑢subscriptΩ𝑗𝑗01u_{\Omega_{j}},j=0,1,italic_u start_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_j = 0 , 1 , be the solution to the problem

When they proved the existence of global smooth solutions to the homogeneous Dirichlet problem, they introduced the key auxiliary function ℋℋ\mathcal{H}caligraphic_H,

which is the product of curvature κ𝜅\kappaitalic_κ of the level line of u𝑢uitalic_u and the cubic of |D⁢u|𝐷𝑢|Du|| italic_D italic_u |, and got the uniformly lower bound of ℋℋ\mathcal{H}caligraphic_H

C

Now let d𝑑ditalic_d be even. The same results for the transvections t21⁢(ωℓ)subscript𝑡21superscript𝜔ℓt_{21}(\omega^{\ell})italic_t start_POSTSUBSCRIPT 21 end_POSTSUBSCRIPT ( italic_ω start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ) and t12⁢(ωℓ)subscript𝑡12superscript𝜔ℓt_{12}(\omega^{\ell})italic_t start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT ( italic_ω start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ) as for d𝑑ditalic_d odd can be obtained by replacing v𝑣vitalic_v by x𝑥xitalic_x in the formula for t21⁢(ωℓ)subscript𝑡21superscript𝜔ℓt_{21}(\omega^{\ell})italic_t start_POSTSUBSCRIPT 21 end_POSTSUBSCRIPT ( italic_ω start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ). It remains to compute t32⁢(ωℓ)subscript𝑡32superscript𝜔ℓt_{32}(\omega^{\ell})italic_t start_POSTSUBSCRIPT 32 end_POSTSUBSCRIPT ( italic_ω start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ) and t23⁢(ωℓ)subscript𝑡23superscript𝜔ℓt_{23}(\omega^{\ell})italic_t start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT ( italic_ω start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ) which can be done using Lemmas 3.2 and 3.6. First, we compute x⁢v−1𝑥superscript𝑣1xv^{-1}italic_x italic_v start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT and store it in the slot p⁢[2,3,1]𝑝231p[2,3,1]italic_p [ 2 , 3 , 1 ] for t23⁢(ω0)subscript𝑡23superscript𝜔0t_{23}(\omega^{0})italic_t start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT ( italic_ω start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ) which takes one operation. Then we compute t32⁢(ωℓ)=(x⁢v−1)⁢t21⁢(ωℓ)⁢(x⁢v−1)−1subscript𝑡32superscript𝜔ℓ𝑥superscript𝑣1subscript𝑡21superscript𝜔ℓsuperscript𝑥superscript𝑣11t_{32}(\omega^{\ell})=(xv^{-1})t_{21}(\omega^{\ell})(xv^{-1})^{-1}italic_t start_POSTSUBSCRIPT 32 end_POSTSUBSCRIPT ( italic_ω start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ) = ( italic_x italic_v start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) italic_t start_POSTSUBSCRIPT 21 end_POSTSUBSCRIPT ( italic_ω start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ) ( italic_x italic_v start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT for 0≤ℓ<f0ℓ𝑓0\leq\ell<f0 ≤ roman_ℓ < italic_f which needs three operations per transvection, and hence 3⁢f3𝑓3f3 italic_f operations overall. Lastly we compute s1=v⁢s⁢v−1subscript𝑠1𝑣𝑠superscript𝑣1s_{1}=vsv^{-1}italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_v italic_s italic_v start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT and store it in slot p⁢[2,3,f−1]𝑝23𝑓1p[2,3,f-1]italic_p [ 2 , 3 , italic_f - 1 ] which needs two operations and t23⁢(ωℓ)subscript𝑡23superscript𝜔ℓt_{23}(\omega^{\ell})italic_t start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT ( italic_ω start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ) for 0≤ℓ<f0ℓ𝑓0\leq\ell<f0 ≤ roman_ℓ < italic_f which needs 3⁢f3𝑓3f3 italic_f operations overall. This requires at most 16⁢f+716𝑓716f+716 italic_f + 7 operations.

as a word in the LGO standard generators of SL⁢(d,q)SL𝑑𝑞\textnormal{SL}(d,q)SL ( italic_d , italic_q ).

Our aim is to determine the length and memory quota for an MSLP for the Bruhat decomposition of an arbitrary matrix g∈SL⁢(d,q)𝑔SL𝑑𝑞g\in\textnormal{SL}(d,q)italic_g ∈ SL ( italic_d , italic_q ) via the above method, with the matrices u1subscript𝑢1u_{1}italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, u2subscript𝑢2u_{2}italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, w𝑤witalic_w returned as words in the LGO generators s,t,v,δ,x𝑠𝑡𝑣𝛿𝑥s,t,v,\delta,xitalic_s , italic_t , italic_v , italic_δ , italic_x of SL⁢(d,q)SL𝑑𝑞\textnormal{SL}(d,q)SL ( italic_d , italic_q ) given in Section 3.1.

We now compute upper bounds for the length and memory quota of an MSLP for expressing an arbitrary diagonal matrix h∈SL⁢(d,q)ℎSL𝑑𝑞h\in\textnormal{SL}(d,q)italic_h ∈ SL ( italic_d , italic_q ) as a word in the LGO generators, i.e. the computation phase of the algorithm.

Finally, we construct a second MSLP, described in Section 3.5, that writes a diagonal matrix h∈SL⁢(d,q)ℎSL𝑑𝑞h\in\textnormal{SL}(d,q)italic_h ∈ SL ( italic_d , italic_q ) as a word in the standard generators of SL⁢(d,q)SL𝑑𝑞\textnormal{SL}(d,q)SL ( italic_d , italic_q ) (when evaluated with these generators as input).

C

C=x+C−1⁢x−1+C−2⁢x−2+⋯with each C−i∈K⁢[y].𝐶𝑥subscript𝐶1superscript𝑥1subscript𝐶2superscript𝑥2⋯with each C−i∈K⁢[y].C=x+C_{-1}x^{-1}+C_{-2}x^{-2}+\cdots\qquad\text{with each $C_{-i}\in K[y]$.}italic_C = italic_x + italic_C start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT + italic_C start_POSTSUBSCRIPT - 2 end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT + ⋯ with each italic_C start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ∈ italic_K [ italic_y ] .

and, again by Proposition 1.7, there exist j∈ℤ𝑗ℤj\in\mathds{Z}italic_j ∈ blackboard_Z and λj∈K×subscript𝜆𝑗superscript𝐾\lambda_{j}\in K^{\times}italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ italic_K start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT

then there exists j∈ℤ𝑗ℤj\in\mathds{Z}italic_j ∈ blackboard_Z and λ∈K×𝜆superscript𝐾\lambda\in K^{\times}italic_λ ∈ italic_K start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT such that

ℤ/e⁢ℤℤ𝑒ℤ\mathds{Z}/e\mathds{Z}blackboard_Z / italic_e blackboard_Z acts on 𝒮0subscript𝒮0\mathcal{S}_{0}caligraphic_S start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. In fact, if

Since m+n>2𝑚𝑛2m+n>2italic_m + italic_n > 2, by Proposition 1.7 there exist j∈ℤ𝑗ℤj\in\mathds{Z}italic_j ∈ blackboard_Z and

D

In that case W𝑊Witalic_W can be identified with Xm×Ysuperscript𝑋𝑚𝑌X^{m}\times Yitalic_X start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT × italic_Y, where m𝑚mitalic_m is the order of H𝐻Hitalic_H, and is contractible as well with respect to the usual product topology.

The wreath product G≀H≀𝐺𝐻G\wr Hitalic_G ≀ italic_H acts from the left on W𝑊Witalic_W by the following rule: if (ϕ:H→G,h)∈G≀H=Map(H,G)⋊H({\phi}\colon H\to G,h)\in G\wr H=\mathrm{Map}(H,G)\rtimes H( italic_ϕ : italic_H → italic_G , italic_h ) ∈ italic_G ≀ italic_H = roman_Map ( italic_H , italic_G ) ⋊ italic_H and (α:H→X,y)∈W({{\alpha}}\colon H\to X,y)\in W( italic_α : italic_H → italic_X , italic_y ) ∈ italic_W, then

Moreover, the above action of G≀H≀𝐺𝐻G\wr Hitalic_G ≀ italic_H on W𝑊Witalic_W also turns out to be free.

If the actions of G𝐺Gitalic_G and H𝐻Hitalic_H are PD, then the action of G≀H≀𝐺𝐻G\wr Hitalic_G ≀ italic_H on W𝑊Witalic_W is PD as well.

If the actions of G𝐺Gitalic_G and H𝐻Hitalic_H are free, then the action of G≀H≀𝐺𝐻G\wr Hitalic_G ≀ italic_H is also free.

B

A~1subscript~𝐴1\tilde{A}_{1}over~ start_ARG italic_A end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT

64,510,416,320,220,110superscript64superscript510superscript416superscript320superscript220superscript1106^{4},5^{10},4^{16},3^{20},2^{20},1^{10}6 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , 5 start_POSTSUPERSCRIPT 10 end_POSTSUPERSCRIPT , 4 start_POSTSUPERSCRIPT 16 end_POSTSUPERSCRIPT , 3 start_POSTSUPERSCRIPT 20 end_POSTSUPERSCRIPT , 2 start_POSTSUPERSCRIPT 20 end_POSTSUPERSCRIPT , 1 start_POSTSUPERSCRIPT 10 end_POSTSUPERSCRIPT

220,116superscript220superscript1162^{20},1^{16}2 start_POSTSUPERSCRIPT 20 end_POSTSUPERSCRIPT , 1 start_POSTSUPERSCRIPT 16 end_POSTSUPERSCRIPT

216,120superscript216superscript1202^{16},1^{20}2 start_POSTSUPERSCRIPT 16 end_POSTSUPERSCRIPT , 1 start_POSTSUPERSCRIPT 20 end_POSTSUPERSCRIPT

216,120superscript216superscript1202^{16},1^{20}2 start_POSTSUPERSCRIPT 16 end_POSTSUPERSCRIPT , 1 start_POSTSUPERSCRIPT 20 end_POSTSUPERSCRIPT

C

Yksubscript𝑌𝑘Y_{k}italic_Y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. Suppose that the p𝑝pitalic_p-curvature of ψ𝜓\psiitalic_ψ is 00 for all

Concretely, this means that if Y¯¯𝑌\overline{Y}over¯ start_ARG italic_Y end_ARG is a smooth compactification

support equal to Lℂsubscript𝐿ℂL_{\mathbb{C}}italic_L start_POSTSUBSCRIPT blackboard_C end_POSTSUBSCRIPT means that ℳksubscriptℳ𝑘\mathcal{M}_{k}caligraphic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is

compactification which we will denote T∗⁢(X~k′)¯¯superscript𝑇subscript~𝑋superscript𝑘′\overline{T^{*}(\tilde{X}_{k^{\prime}})}over¯ start_ARG italic_T start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) end_ARG.

where L¯¯𝐿\overline{L}over¯ start_ARG italic_L end_ARG is a suitable compactification of L𝐿Litalic_L. The

A

(This remark was originally stated for an equidimensional scheme X.𝑋X.italic_X . We thank the referee for pointing out that the equidimentionality is unnecessary if we work with a Chow group indexed by dimension, not by codimension.)

For an arbitrary X∈Sch/k,𝑋Sch𝑘X\in\operatorname{Sch}/k,italic_X ∈ roman_Sch / italic_k , there is a pullback diagram by [ILO, Exposé 0, Theorem 3 (1)]

Suppose k𝑘kitalic_k is perfect and let X∈Sch/k.𝑋normal-Sch𝑘X\in\operatorname{Sch}/k.italic_X ∈ roman_Sch / italic_k . Then, there is an inclusion Γa⁢l⁢g⁢(M⁢(X))⊂H0⁢(X,ℤ)0.subscriptnormal-Γ𝑎𝑙𝑔𝑀𝑋subscript𝐻0superscript𝑋ℤ0\Gamma_{alg}(M(X))\subset H_{0}(X,{\mathbb{Z}})^{0}.roman_Γ start_POSTSUBSCRIPT italic_a italic_l italic_g end_POSTSUBSCRIPT ( italic_M ( italic_X ) ) ⊂ italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_X , blackboard_Z ) start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT .

If k𝑘kitalic_k is algebraically closed, then there exists a universal regular homomorphism for Γa⁢l⁢g⁢(M⁢(X))subscriptnormal-Γ𝑎𝑙𝑔𝑀𝑋\Gamma_{alg}(M(X))roman_Γ start_POSTSUBSCRIPT italic_a italic_l italic_g end_POSTSUBSCRIPT ( italic_M ( italic_X ) ) for any X∈Sch/k.𝑋normal-Sch𝑘X\in\operatorname{Sch}/k.italic_X ∈ roman_Sch / italic_k .

Let us assume the resolution of singularities. In this case, for an arbitrary X∈Sch/k,𝑋normal-Sch𝑘X\in\operatorname{Sch}/k,italic_X ∈ roman_Sch / italic_k , we have a comparison isomorphism between the

D

Let b∈s⁢Q⁢G4b𝑠𝑄subscript𝐺4\mathrm{b}\in sQG_{4}roman_b ∈ italic_s italic_Q italic_G start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT, T∈ℭ⁢𝔫⁢𝔱⁢(a4,b)Tℭ𝔫𝔱superscripta4b\mathrm{T}\in\mathfrak{Cnt}(\mathrm{a}^{4},\mathrm{b})roman_T ∈ fraktur_C fraktur_n fraktur_t ( roman_a start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , roman_b ) and

for any T,J∈MorB𝑇𝐽subscriptMor𝐵T,J\in\mathrm{Mor}_{B}italic_T , italic_J ∈ roman_Mor start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, such that (T,J)∈D⁢o⁢m⁢(∘)𝑇𝐽𝐷𝑜𝑚(T,J)\in Dom(\circ)( italic_T , italic_J ) ∈ italic_D italic_o italic_m ( ∘ ),

Let α𝛼\alphaitalic_α be a geodesic on M⁢(x,f)𝑀𝑥𝑓M(x,f)italic_M ( italic_x , italic_f ) such that 0∈d⁢o⁢m⁢(α)0𝑑𝑜𝑚𝛼0\in dom(\alpha)0 ∈ italic_d italic_o italic_m ( italic_α ) eventually by a reparametrization,

and an open neighbourhood K0subscript𝐾0K_{0}italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT of 00 in d⁢o⁢m⁢(α)𝑑𝑜𝑚𝛼dom(\alpha)italic_d italic_o italic_m ( italic_α ) such that

Assume that 0∈d⁢o⁢m⁢(α)0𝑑𝑜𝑚𝛼0\in dom(\alpha)0 ∈ italic_d italic_o italic_m ( italic_α ) eventually by a reparametrization,

D

On the other hand, note that any Siegel disk cannot contain a critical point. Hence the second question on the Siegel disk boundary is:

The answer is no. Ghys and Herman gave the first examples of polynomials having a Siegel disk whose boundary does not contain a critical point (see [Ghy84], [Her86] and [Dou87]).

On the other hand, note that any Siegel disk cannot contain a critical point. Hence the second question on the Siegel disk boundary is:

Suppose f𝑓fitalic_f is an analytic function which has a Siegel disk properly contained in the domain of holomorphy. Ghys proved that if the rotation number belongs to ℋℋ\mathcal{H}caligraphic_H and the boundary of the Siegel disk is a Jordan curve, then f𝑓fitalic_f has a critical point in the boundary of the Siegel disk [Ghy84]. Later, Herman generalized this result by dropping the topological condition on the Siegel disk boundary but requiring that the restriction of f𝑓fitalic_f on the Siegel disk boundary is injective [Her85] (see also [Pér97]). In particular, he proved that if a unicritical polynomial has a Siegel disk whose rotation number is contained in ℋℋ\mathcal{H}caligraphic_H, then the boundary of the Siegel disk contains a critical point. Recently, Chéritat and Roesch, Benini and Fagella, respectively, generalized this result to the polynomials with two critical values [CR16] and to a special class of transcendental entire functions with two singular values [BF18].

Suppose the closure of the Siegel disk of f𝑓fitalic_f is compactly contained in the domain of definition of f𝑓fitalic_f. One may wonder what phenomena near the boundary of a Siegel disk prevents f𝑓fitalic_f from having a larger linearization domain. Obviously, the presence of periodic cycles near the boundary is one of the reasons since any Siegel disk cannot contain periodic points except the center itself. It was proved by Avila and Cheraghi that under some condition on α𝛼\alphaitalic_α every neighborhood of the Siegel disk of Pαsubscript𝑃𝛼P_{\alpha}italic_P start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT contains infinitely many cycles [AC18], which is similar to the small cycle property that prevents linearization (see [Yoc88] and [Pér92]).

A

We will be interested in orbifolds X=Γ\ℍ𝑋\ΓℍX=\Gamma\backslash\mathbb{H}italic_X = roman_Γ \ blackboard_H (or X=Γ\𝔻𝑋\Γ𝔻X=\Gamma\backslash\mathbb{D}italic_X = roman_Γ \ blackboard_D, if we treat ΓΓ\Gammaroman_Γ as acting on 𝔻𝔻\mathbb{D}blackboard_D) for Fuchsian groups ΓΓ\Gammaroman_Γ of the first kind. Such an orbifold is a Riemann surface of genus g𝑔gitalic_g with n𝑛nitalic_n punctures and l𝑙litalic_l conical points of angles 2⁢πm1,…,2⁢πml2𝜋subscript𝑚1…2𝜋subscript𝑚𝑙\dfrac{2\pi}{m_{1}},\ldots,\dfrac{2\pi}{m_{l}}divide start_ARG 2 italic_π end_ARG start_ARG italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG , … , divide start_ARG 2 italic_π end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG. By a (p,q)𝑝𝑞(p,q)( italic_p , italic_q )-differential on the orbifold Riemann surface X=Γ\ℍ𝑋\ΓℍX=\Gamma\backslash\mathbb{H}italic_X = roman_Γ \ blackboard_H we understand a smooth function ϕitalic-ϕ\phiitalic_ϕ on ℍℍ\mathbb{H}blackboard_H that transforms according to the rule ϕ⁢(γ⁢z)⁢γ′⁢(z)p⁢γ′⁢(z)¯q=ϕ⁢(z)italic-ϕ𝛾𝑧superscript𝛾′superscript𝑧𝑝superscript¯superscript𝛾′𝑧𝑞italic-ϕ𝑧\phi(\gamma z)\gamma^{\prime}(z)^{p}\overline{\gamma^{\prime}(z)}^{q}=\phi(z)italic_ϕ ( italic_γ italic_z ) italic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_z ) start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT over¯ start_ARG italic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_z ) end_ARG start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT = italic_ϕ ( italic_z ). The space of harmonic (p,q)𝑝𝑞(p,q)( italic_p , italic_q )-differentials, square integrable with respect to the hyperbolic metric on X=Γ\ℍ𝑋\ΓℍX=\Gamma\backslash\mathbb{H}italic_X = roman_Γ \ blackboard_H, we denote by Ωp,q⁢(X)superscriptΩ𝑝𝑞𝑋\Omega^{p,q}(X)roman_Ω start_POSTSUPERSCRIPT italic_p , italic_q end_POSTSUPERSCRIPT ( italic_X ).

Here c1(λk,||⋅||kQ)c_{1}(\lambda_{k},||\cdot||_{k}^{Q})italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , | | ⋅ | | start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT ) is the first Chern form of the determinant line bundle λksubscript𝜆𝑘\lambda_{k}italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT of the vector bundle of square integrable meromorphic k𝑘kitalic_k-differentials, k≥1𝑘1k\geq 1italic_k ≥ 1, on

we show that the contribution to the local index formula from elliptic elements of Fuchsian groups is given by the symplectic form of a Kähler metric on the moduli space of orbisurfaces.

We proceed with the basics of the deformation theory of Fuchsian groups. Let ΓΓ\Gammaroman_Γ be a Fuchsian group of the first kind of signature (g;n;m1,…,ml)𝑔𝑛subscript𝑚1…subscript𝑚𝑙(g;n;m_{1},\ldots,m_{l})( italic_g ; italic_n ; italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ). Consider the space of quasiconformal mappings of the upper half-plane ℍℍ\mathbb{H}blackboard_H that fix 0, 1 and ∞\infty∞. Two quasiconformal mappings are equivalent if they coincide on the real axis. A mapping f𝑓fitalic_f is compatible with ΓΓ\Gammaroman_Γ if f−1∘γ∘f∈P⁢S⁢L⁢(2,ℝ)superscript𝑓1𝛾𝑓𝑃𝑆𝐿2ℝf^{-1}\circ\gamma\circ f\in PSL(2,\mathbb{R})italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∘ italic_γ ∘ italic_f ∈ italic_P italic_S italic_L ( 2 , blackboard_R ) for all γ∈Γ𝛾Γ\gamma\in\Gammaitalic_γ ∈ roman_Γ. The space of equivalence classes of ΓΓ\Gammaroman_Γ-compatible mappings is called the Teichmüller space of ΓΓ\Gammaroman_Γ and is denoted by T⁢(Γ)𝑇ΓT(\Gamma)italic_T ( roman_Γ ). The space T⁢(Γ)𝑇ΓT(\Gamma)italic_T ( roman_Γ ) is isomorphic to a bounded complex domain in ℂ3⁢g−3+n+lsuperscriptℂ3𝑔3𝑛𝑙\mathbb{C}^{3g-3+n+l}blackboard_C start_POSTSUPERSCRIPT 3 italic_g - 3 + italic_n + italic_l end_POSTSUPERSCRIPT. The Teichmüller modular group Mod⁢(Γ)ModΓ{\rm Mod}(\Gamma)roman_Mod ( roman_Γ ) acts on T⁢(Γ)𝑇ΓT(\Gamma)italic_T ( roman_Γ ) by complex isomorphisms. Denote by Mod0⁢(Γ)subscriptMod0Γ{\rm Mod}_{0}(\Gamma)roman_Mod start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( roman_Γ ) the subgroup of Mod⁢(Γ)ModΓ{\rm Mod}(\Gamma)roman_Mod ( roman_Γ ) consisting of pure mapping classes (i. e. those fixing the punctures and elliptic points on X𝑋Xitalic_X pointwise). The factor T⁢(Γ)/Mod0⁢(Γ)𝑇ΓsubscriptMod0ΓT(\Gamma)/{\rm Mod}_{0}(\Gamma)italic_T ( roman_Γ ) / roman_Mod start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( roman_Γ ) is isomorphic to the moduli space ℳg,n+lsubscriptℳ𝑔𝑛𝑙\mathcal{M}_{g,n+l}caligraphic_M start_POSTSUBSCRIPT italic_g , italic_n + italic_l end_POSTSUBSCRIPT of smooth complex algebraic curves of genus g𝑔gitalic_g with n+l𝑛𝑙n+litalic_n + italic_l labeled points.

The dimension of the space of square integrable meromorphic (with poles at punctures and conical points) k𝑘kitalic_k-differentials on X𝑋Xitalic_X, or cusp forms of weight 2⁢k2𝑘2k2 italic_k for ΓΓ\Gammaroman_Γ, is given by Riemann-Roch formula for orbifolds:

D

The above uncertainty principles agree exactly with the traditional Heisenberg uncertainty principle in the case of the QMHO since γ=−1𝛾1\gamma=-1italic_γ = - 1 and δ=1𝛿1\delta=1italic_δ = 1, giving an uncertainty bound of 1212\frac{1}{2}divide start_ARG 1 end_ARG start_ARG 2 end_ARG for each with the minimizers being Gaussians. For n>1𝑛1n>1italic_n > 1, the uncertainty product σ𝒳⁢σ𝒫subscript𝜎𝒳subscript𝜎𝒫\sigma_{\mathcal{X}}\sigma_{\mathcal{P}}italic_σ start_POSTSUBSCRIPT caligraphic_X end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT caligraphic_P end_POSTSUBSCRIPT has a lower bound of 1212\frac{1}{2}divide start_ARG 1 end_ARG start_ARG 2 end_ARG, just as in the Heisenberg uncertainty principle with minimizers exp⁡(−x2⁢n/2⁢n)superscript𝑥2𝑛2𝑛\exp(-x^{2n}/2n)roman_exp ( - italic_x start_POSTSUPERSCRIPT 2 italic_n end_POSTSUPERSCRIPT / 2 italic_n ), but the uncertainty products σℒ⁢σ𝒜subscript𝜎ℒsubscript𝜎𝒜\sigma_{\mathcal{L}}\sigma_{\mathcal{A}}italic_σ start_POSTSUBSCRIPT caligraphic_L end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT caligraphic_A end_POSTSUBSCRIPT and σℒ~⁢σ𝒜~subscript𝜎~ℒsubscript𝜎~𝒜\sigma_{\widetilde{\mathcal{L}}}\sigma_{\widetilde{\mathcal{A}}}italic_σ start_POSTSUBSCRIPT over~ start_ARG caligraphic_L end_ARG end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT over~ start_ARG caligraphic_A end_ARG end_POSTSUBSCRIPT are for n>1𝑛1n>1italic_n > 1 bounded by a larger constant since δ−γ=2⁢n𝛿𝛾2𝑛\delta-\gamma=2nitalic_δ - italic_γ = 2 italic_n.

By considering the QMHO in the context of SUSY, the coupled SUSY structure unifying the QMHO and SUSY was developed. Coupled SUSY has many of the desirable properties of both: true ladder operators exist, there are two sectors, and charge operators exist between the sectors. The existence of true ladder operators led to a richer theory for coherent states than what has existed in the past, namely applying a charge operator led to an intertwining in the coherent states structures in both sectors. Moreover, focusing on the case of unbroken coupled SUSY gave some results regarding the spectrum of coupled SUSY Hamiltonians as well as uncertainty principles which generalize the Heisenberg uncertainty principle. A strength of the theory is its background-independence, e.g., the above may apply to any L2⁢(Ω)superscript𝐿2ΩL^{2}(\Omega)italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ) space. ΩΩ\Omegaroman_Ω may simply be a locally compact group or even have a manifold structure. Unbroken coupled SUSYs for which −γ=δ𝛾𝛿-\gamma=\delta- italic_γ = italic_δ can be realized as harmonic oscillators and as such coupled SUSY may be a better model for elementary particles than traditional supersymmetric quantum mechanics. Moreover, coupled SUSY is inherently a multi-particle and multi-dimensional theory and does not suffer some of the pitfalls that traditional SUSY quantum mechanics has for multiple particles or multiple dimensions [21, 25, 22]. It has also been shown that spurious states exist for sector two tensor Hamiltonians for tensorial systems [9] In spite of these drawbacks, tensorial multidimensional SUSY quantum mechanics has been used to determine excited-state energies and wavefunctions using adiabatic switching [10] in addition to the construction of sector one states via imaginary time propagation [8].

The remainder of this paper is organized as follows. In Section 2, we develop the coupled SUSY structure which expands the relationship among the QMHO, SUSY, and the corresponding coupled SUSYs. In Section 3, we establish eigenvalues and eigenfunctions of the corresponding coupled SUSY. In Section 4, we establish the connections between coupled SUSY and other oscillator systems, namely Schwinger’s approach to the two-particle QMHO system, and the 𝔰⁢𝔲⁢(1,1)𝔰𝔲11\mathfrak{su}(1,1)fraktur_s fraktur_u ( 1 , 1 ) approach to the QMHO. In Section 5, we develop the coherent states for coupled SUSY. Due to the algebraic structure of coupled SUSY, a more complex coherent state structure exists than what has been found previously for either SUSY or the QMHO. In Section 6, we derive some uncertainty principles associated to coupled SUSY via ladder operators, generalizing the traditional Heisenberg uncertainty principle. In doing so, we discover new quantum operators corresponding to the classical Lagrangian and dilation. In Section 7, we show that harmonic oscillators can be realized as special classes of coupled SUSYs, suggesting that coupled SUSYs may have utitlity in quantum field theory and elementary particles.

Traditionally the QMHO is associated to the 1D Heisenberg-Weyl Lie algebra as this is the Lie algebra which corresponds to the canonical commutation relations which is reflected in the algebra generated by the ladder operators. This is not the only Lie algebra which may be associated to the QMHO. There are two other treatments of the QMHO: Schwinger’s “spinification” of the two-particle QMHO and the 𝔰⁢𝔲⁢(1,1)𝔰𝔲11\mathfrak{su}(1,1)fraktur_s fraktur_u ( 1 , 1 ) treatment of the QMHO. Coupled SUSY is to some degree a unification of the two treatments.

As previously noted, the quantum mechanical harmonic oscillator is a specific instance of a coupled supersymmetry, albeit a somewhat trivial case in which the two coupled SUSY equations are identical. This is not the only manner in which the two are connected. Indeed, a special class of coupled SUSYs may be realized as harmonic oscillator-like systems, i.e. they satisfy the same Lie algebra and by virtue of Stone-von Neumann, may be realized in some way as harmonic oscillators. If one takes γ=−δ𝛾𝛿\gamma=-\deltaitalic_γ = - italic_δ, then the coupled SUSY equations become

D

24242424;  H21⁢(2,1)subscriptsuperscript𝐻1221H^{1}_{2}(2,1)italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 2 , 1 ) 

3[1]⁢1[2]superscript3delimited-[]1superscript1delimited-[]23^{[1]}1^{[2]}3 start_POSTSUPERSCRIPT [ 1 ] end_POSTSUPERSCRIPT 1 start_POSTSUPERSCRIPT [ 2 ] end_POSTSUPERSCRIPT:

3[1]⁢1[2]superscript3delimited-[]1superscript1delimited-[]23^{[1]}1^{[2]}3 start_POSTSUPERSCRIPT [ 1 ] end_POSTSUPERSCRIPT 1 start_POSTSUPERSCRIPT [ 2 ] end_POSTSUPERSCRIPT:

(3[1],1[3])superscript3delimited-[]1superscript1delimited-[]3(3^{[1]},1^{[3]})( 3 start_POSTSUPERSCRIPT [ 1 ] end_POSTSUPERSCRIPT , 1 start_POSTSUPERSCRIPT [ 3 ] end_POSTSUPERSCRIPT )

3[1]⁢1[2]superscript3delimited-[]1superscript1delimited-[]23^{[1]}1^{[2]}3 start_POSTSUPERSCRIPT [ 1 ] end_POSTSUPERSCRIPT 1 start_POSTSUPERSCRIPT [ 2 ] end_POSTSUPERSCRIPT:

A

With the same notation as above, kerϕitalic-ϕ\phiitalic_ϕ is generated by the σi⁢jsubscript𝜎𝑖𝑗\sigma_{ij}italic_σ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT.

The paper is structured as follows. In Section 2, we collect basic notations, terminology, and results that will be used in the paper. The first syzygy of Hibi rings is discussed in Section 3. Explicit expression for the first Betti number for planar distributive lattices has been discussed in Section 4.

In this section, we find explicitly the generators of the first syzygy of Hibi ring for a planar distributive lattice and also find a condition for which the first syzygy linear.

For a planar distributive lattice ℒℒ\mathcal{L}caligraphic_L, the number of strip type generators of first syzygy of R⁢[ℒ]𝑅delimited-[]ℒR[\mathcal{L}]italic_R [ caligraphic_L ], we denote it by

The following theorem gives us the linear generators for the first syzygy of R⁢[ℒ]𝑅delimited-[]ℒR[\mathcal{L}]italic_R [ caligraphic_L ].

B

To cast this into the framework of regularity structures we first work with an abstract version of uεsubscript𝑢𝜀u_{\varepsilon}italic_u start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT, i.e., we write uεsubscript𝑢𝜀u_{\varepsilon}italic_u start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT as a generalised Taylor expansion. Here, the monomials may represent the classical Taylor monomials or they may represent abstract expressions that are functions of ξ𝜉\xiitalic_ξ. The control of these monomials typically amount to the control of explicit stochastic objects and heavily depend on the choice of the equation at hand.

The “Taylor coefficients” can be thought of as the “derivatives” of an abstract version of a 𝒞γsuperscript𝒞𝛾{\mathcal{C}}^{\gamma}caligraphic_C start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT-function and they are given by

To cast this into the framework of regularity structures we first work with an abstract version of uεsubscript𝑢𝜀u_{\varepsilon}italic_u start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT, i.e., we write uεsubscript𝑢𝜀u_{\varepsilon}italic_u start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT as a generalised Taylor expansion. Here, the monomials may represent the classical Taylor monomials or they may represent abstract expressions that are functions of ξ𝜉\xiitalic_ξ. The control of these monomials typically amount to the control of explicit stochastic objects and heavily depend on the choice of the equation at hand.

is a sequence of linear spaces 𝒳εsubscript𝒳𝜀{\mathcal{X}}_{\varepsilon}caligraphic_X start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT that can be viewed as subspaces of 𝒟′⁢(ℝd)superscript𝒟′superscriptℝ𝑑{\mathcal{D}}^{\prime}(\mathbb{R}^{d})caligraphic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ), the space of distributions, and

In particular we define an abstract notion of convolution 𝒦γεsuperscriptsubscript𝒦𝛾𝜀\mathcal{K}_{\gamma}^{\varepsilon}caligraphic_K start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT against the Green function Kεsuperscript𝐾𝜀K^{\varepsilon}italic_K start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT of ℒℒ{\mathcal{L}}caligraphic_L in (2) (which is an operator mapping into 𝒳εsubscript𝒳𝜀{\mathcal{X}}_{\varepsilon}caligraphic_X start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT!) and we show that 𝒦γεsuperscriptsubscript𝒦𝛾𝜀\mathcal{K}_{\gamma}^{\varepsilon}caligraphic_K start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT satisfies a certain Schauder estimate. It turns out that unlike in the continuous setting [Regularity] it is not necessarily true that the convolution operator 𝒦γεsuperscriptsubscript𝒦𝛾𝜀\mathcal{K}_{\gamma}^{\varepsilon}caligraphic_K start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT can be defined in a natural way so that

A

Since char⁢(K)=pchar𝐾𝑝\mathrm{char}(K)=proman_char ( italic_K ) = italic_p it is perfect and so v⁢K𝑣𝐾vKitalic_v italic_K is p𝑝pitalic_p-divisible.

Moreover, as K𝐾Kitalic_K is dependent it follows from the proof of [22, Proposition 5.3] that K⁢v𝐾𝑣Kvitalic_K italic_v is Artin-Schreier closed, and therefore infinite.

If K⁢v𝐾𝑣Kvitalic_K italic_v is separably closed and perfect it is algebraically closed and hence strongly dependent. If it is not perfect then by an argument of Scanlon’s [16, Proposition 3.7] v𝑣vitalic_v is definable in K𝐾Kitalic_K and hence (K,v)𝐾𝑣(K,v)( italic_K , italic_v ) is strongly dependent, so that K⁢v𝐾𝑣Kvitalic_K italic_v is perfect, a contradiction.

if [0,v⁢(p)]0𝑣𝑝[0,v(p)][ 0 , italic_v ( italic_p ) ] is infinite then K⁢v𝐾𝑣Kvitalic_K italic_v is infinite.

Keeping the same notation, assume that [0,v⁢(p)]0𝑣𝑝[0,v(p)][ 0 , italic_v ( italic_p ) ] is infinite. As before, Δ0subscriptΔ0\Delta_{0}roman_Δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is non trivial. The proof of [22, Proposition 5.3] shows that if K𝐾Kitalic_K is dependent and (K,v)𝐾𝑣(K,v)( italic_K , italic_v ) is a valued field of characteristic p>0𝑝0p>0italic_p > 0 then K⁢v𝐾𝑣Kvitalic_K italic_v is infinite. Applying this fact to the valuation v2:K2→Δ0:subscript𝑣2→subscript𝐾2subscriptΔ0v_{2}:K_{2}\to\Delta_{0}italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT → roman_Δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, whose residue field is K⁢v𝐾𝑣Kvitalic_K italic_v, finishes the proof.

A

Then, an R𝑅Ritalic_R-module ϕ∗⁢ℱsubscriptitalic-ϕℱ\phi_{*}\mathcal{F}italic_ϕ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT caligraphic_F is Cohen-Macaulay.

A reflexive R𝑅Ritalic_R-module M𝑀Mitalic_M is say to be a modifying module if EndR⁡(M)subscriptEnd𝑅𝑀\operatorname{End}_{R}(M)roman_End start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( italic_M ) is a (maximal) Cohen-Macaulay R𝑅Ritalic_R-module.

Let R𝑅Ritalic_R be a normal Cohen-Macaulay domain and M,N𝑀𝑁M,Nitalic_M , italic_N (maximal) Cohen-Macaulay R𝑅Ritalic_R-modules.

Then, an R𝑅Ritalic_R-module ϕ∗⁢ℱsubscriptitalic-ϕℱ\phi_{*}\mathcal{F}italic_ϕ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT caligraphic_F is Cohen-Macaulay.

Let R𝑅Ritalic_R be a normal Cohen-Macaulay domain and M𝑀Mitalic_M a (maximal) Cohen-Macaulay R𝑅Ritalic_R-module.

D

Here we study a related monoid, replacing continuous mappings with smooth ones and topological groupoids by Lie groupoids.

Let 𝒢=(G⇉M)𝒢⇉𝐺𝑀\mathcal{G}=(G\nobreak\rightrightarrows\nobreak M)caligraphic_G = ( italic_G ⇉ italic_M ) be a Lie groupoid.

Let 𝒢=(G⇉M)𝒢⇉𝐺𝑀\mathcal{G}=(G\nobreak\rightrightarrows\nobreak M)caligraphic_G = ( italic_G ⇉ italic_M ) be a Lie groupoid, then the map

Let 𝒢=(G⇉M)𝒢⇉𝐺𝑀\mathcal{G}=(G\nobreak\rightrightarrows\nobreak M)caligraphic_G = ( italic_G ⇉ italic_M ) be a Lie groupoid, then we define the set

Let 𝒢=(G⇉M)𝒢⇉𝐺𝑀\mathcal{G}=(G\nobreak\rightrightarrows\nobreak M)caligraphic_G = ( italic_G ⇉ italic_M ) be a Lie groupoid then we define

C

,H_{0},\phi_{0},n)}{C_{\mathit{lc}}(\{Z^{c}\}\cap\{A\},H_{0},\phi_{0},n)}\ .divide start_ARG italic_C start_POSTSUBSCRIPT italic_lc end_POSTSUBSCRIPT ( italic_Z ∩ { italic_A } , italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_n ) end_ARG start_ARG italic_C start_POSTSUBSCRIPT italic_lc end_POSTSUBSCRIPT ( italic_Z ∩ { italic_Z start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT } ∩ { italic_A } , italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_n ) end_ARG start_ARROW over≅ → end_ARROW divide start_ARG italic_C start_POSTSUBSCRIPT italic_lc end_POSTSUBSCRIPT ( { italic_A } , italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_n ) end_ARG start_ARG italic_C start_POSTSUBSCRIPT italic_lc end_POSTSUBSCRIPT ( { italic_Z start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT } ∩ { italic_A } , italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_n ) end_ARG .

Here we define the Roe algebra of a big family as the C∗superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-subalgebra of the Roe algebra of the ambient space generated by operators which are supported on members of the family.

The family big family {Zc}∩Zsuperscript𝑍𝑐𝑍\{Z^{c}\}\cap Z{ italic_Z start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT } ∩ italic_Z is the coarse geometric version of the boundary of Z𝑍Zitalic_Z.

The second part is the transition from the K𝐾Kitalic_K-theories of Roe algebras to the K𝐾Kitalic_K-theory of Roe categories which are in the background of the construction of the equivariant coarse K𝐾Kitalic_K-homology functor K⁢𝒳G𝐾superscript𝒳𝐺K\!\mathcal{X}^{G}italic_K caligraphic_X start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT.

We define the Roe algebra C𝑙𝑐⁢(𝒴,H0,ϕ0,n)subscript𝐶𝑙𝑐𝒴subscript𝐻0subscriptitalic-ϕ0𝑛C_{\mathit{lc}}({\mathcal{Y}},H_{0},\phi_{0},n)italic_C start_POSTSUBSCRIPT italic_lc end_POSTSUBSCRIPT ( caligraphic_Y , italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_n ) of a big family 𝒴𝒴{\mathcal{Y}}caligraphic_Y in M𝑀Mitalic_M as the C∗superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-subalgebra of the Roe algebra C𝑙𝑐⁢(M,H0,ϕ0,n)subscript𝐶𝑙𝑐𝑀subscript𝐻0subscriptitalic-ϕ0𝑛C_{\mathit{lc}}(M,H_{0},\phi_{0},n)italic_C start_POSTSUBSCRIPT italic_lc end_POSTSUBSCRIPT ( italic_M , italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_n ) generated by operators which are supported on members of the family 𝒴𝒴{\mathcal{Y}}caligraphic_Y. Recall that {A}𝐴\{A\}{ italic_A } denotes the big family of thickenings of the invariant subset A𝐴Aitalic_A of M𝑀Mitalic_M. In the ungraded case we have the equality C𝑙𝑐⁢({A},H0,ϕ0)=C∗⁢(A⊆M)subscript𝐶𝑙𝑐𝐴subscript𝐻0subscriptitalic-ϕ0superscript𝐶𝐴𝑀C_{\mathit{lc}}(\{A\},H_{0},\phi_{0})=C^{*}(A\subseteq M)italic_C start_POSTSUBSCRIPT italic_lc end_POSTSUBSCRIPT ( { italic_A } , italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_A ⊆ italic_M ), where the right-hand side is Roe’s notation [Roe16].

A

\boldsymbol{n}}^{\tau}italic_λ = caligraphic_A bold_∇ italic_u ⋅ bold_italic_n start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT for all elements τ𝜏\tauitalic_τ.

One difficulty that hinders the development of efficient methods is the presence of high-contrast coefficients [MR3800035, MR2684351, MR2753343, MR3704855, MR3225627, MR2861254]. When LOD or VMS methods are considered, high-contrast coefficients might slow down the exponential decay of the solutions, making the method not so practical. Here in this paper, in the presence of rough coefficients, spectral techniques are employed to overcome such hurdle, and by solving local eigenvalue problems we define a space where the exponential decay of solutions is insensitive to high-contrast coefficients. Additionally, the spectral techniques remove

In the spirit of the Multiscale Hybrid Methods [AHPV, HMV, HPV, MR3584539] and FETI methods [FETI, totalfeti, MR1285024, MR2282408, MR2104179], we consider the decomposition

It is hard to approximate such problem in its full generality using numerical methods, in particular because of the low regularity of the solution and its multiscale behavior. Most convergent proofs either assume extra regularity or special properties of the coefficients [AHPV, MR3050916, MR2306414, MR1286212, babuos85, MR1979846, MR2058933, HMV, MR1642758, MR3584539, MR2030161, MR2383203, vs1, vs2, MR2740478]. Some methods work even considering that the solution has low regularity [MR2801210, MR2753343, MR3225627, MR3177856, MR2861254]

mixed finite elements. We note the proposal in [CHUNG2018298] of generalized multiscale finite element methods based on eigenvalue problems inside the macro elements, with basis functions with support weakly dependent of the log of the contrast. Here, we propose eigenvalue problems based on edges of macro element removing the dependence

B

Moreover, τfsubscript𝜏𝑓\tau_{f}italic_τ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT does not have to be injective, an example is given in [23, 4.2].

Metric-minimizing discs include many well-studied maps from 𝔻𝔻\mathbb{D}blackboard_D to metric spaces.

Assume that a map f:𝔻→Y:𝑓→𝔻𝑌f\colon\mathbb{D}\to Yitalic_f : blackboard_D → italic_Y has no bubbles.

Let Y𝑌Yitalic_Y be a metric space and s:𝔻→Y:𝑠→𝔻𝑌s\colon\mathbb{D}\to Yitalic_s : blackboard_D → italic_Y be a metric-minimizing map.

However, for metric-minimizing discs f:𝔻→Y:𝑓→𝔻𝑌f\colon\mathbb{D}\to Yitalic_f : blackboard_D → italic_Y both statements hold true; see

D

(⋅,⋅)⋅⋅(\>\cdot\>,\>\cdot\>)( ⋅ , ⋅ ) renders C⁢(M)𝐶𝑀C(M)italic_C ( italic_M ) a complex Hilbert space what we shall

C⁢(M)𝐶𝑀C(M)italic_C ( italic_M ) we shall write A∈ℜ𝐴ℜA\in{\mathfrak{R}}italic_A ∈ fraktur_R but B^∈ℋ^𝐵ℋ\hat{B}\in{\mathscr{H}}over^ start_ARG italic_B end_ARG ∈ script_H from now on as usual.

𝔅⁢(ℋ)𝔅ℋ{\mathfrak{B}}({\mathscr{H}})fraktur_B ( script_H ). Multiplication in C⁢(M)𝐶𝑀C(M)italic_C ( italic_M ) from the left on

closure of the image of C⁢(M)𝐶𝑀C(M)italic_C ( italic_M ) under π𝜋\piitalic_π within 𝔅⁢(ℋ)𝔅ℋ{\mathfrak{B}}({\mathscr{H}})fraktur_B ( script_H ) or

\>\>.fraktur_R := ( italic_π ( italic_C ( italic_M ) ) ) start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ⊂ fraktur_B ( script_H ) .

B

Let H⁢(M)⊂P⁢(M)𝐻𝑀𝑃𝑀H(M)\subset P(M)italic_H ( italic_M ) ⊂ italic_P ( italic_M ) denote Waldhausen’s partitions that are also hℎhitalic_h-cobordisms. Using the diffeomorphism we may identify the pair ((𝒯2)kl,(ℋ2)kl)superscriptsubscriptsuperscript𝒯2𝑘𝑙superscriptsubscriptsuperscriptℋ2𝑘𝑙((\mathcal{T}^{2})_{k}^{l},(\mathcal{H}^{2})_{k}^{l})( ( caligraphic_T start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT , ( caligraphic_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT ) with the pair (H⁢(Tkl),H⁢(ℝk×Sl−1))𝐻superscriptsubscript𝑇𝑘𝑙𝐻superscriptℝ𝑘superscript𝑆𝑙1(H(T_{k}^{l}),H(\mathbb{R}^{k}\times S^{l-1}))( italic_H ( italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT ) , italic_H ( blackboard_R start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT × italic_S start_POSTSUPERSCRIPT italic_l - 1 end_POSTSUPERSCRIPT ) ). Note in particular that the conditions at infinity for f∈ℋk𝑓subscriptℋ𝑘f\in\mathcal{H}_{k}italic_f ∈ caligraphic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and the addition of (cf⊛q¯)⁢(∥y∥)⊛subscript𝑐𝑓¯𝑞delimited-∥∥𝑦(c_{f}\mbox{\kern 0.5pt{$\varoast$}\kern 0.5pt}\overline{q})(\lVert y\rVert)( italic_c start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ⊛ over¯ start_ARG italic_q end_ARG ) ( ∥ italic_y ∥ ) to bend down the positive direction is essentially that this cobordism is trivial over {s0}×Sl−1subscript𝑠0superscript𝑆𝑙1\{s_{0}\}\times S^{l-1}{ italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT } × italic_S start_POSTSUPERSCRIPT italic_l - 1 end_POSTSUPERSCRIPT.

As before, the inclusion of these partition spaces are highly connected when the inclusion inducing them are highly connected and the relative homotopy type is highly connected (in this case contractible).

The first inclusion is highly connected as this corresponds to an induced inclusion of Waldhausen’s partition spaces. Indeed, (𝒲′)klsuperscriptsubscriptsuperscript𝒲′𝑘𝑙(\mathcal{W}^{\prime})_{k}^{l}( caligraphic_W start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT correspond to a partitions in P⁢(Sk+l−1)k𝑃subscriptsuperscript𝑆𝑘𝑙1𝑘P(S^{k+l-1})_{k}italic_P ( italic_S start_POSTSUPERSCRIPT italic_k + italic_l - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. The subspace 𝒲klsuperscriptsubscript𝒲𝑘𝑙\mathcal{W}_{k}^{l}caligraphic_W start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT corresponds to the subspace P⁢(ℝk+l−1)k⊂P⁢(Sk+l−1)k𝑃subscriptsuperscriptℝ𝑘𝑙1𝑘𝑃subscriptsuperscript𝑆𝑘𝑙1𝑘P(\mathbb{R}^{k+l-1})_{k}\subset P(S^{k+l-1})_{k}italic_P ( blackboard_R start_POSTSUPERSCRIPT italic_k + italic_l - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⊂ italic_P ( italic_S start_POSTSUPERSCRIPT italic_k + italic_l - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. This inclusion is highly connected for large l𝑙litalic_l and k𝑘kitalic_k as the connectivity of the inclusion Dk+l−1⊂Sk+l−1superscript𝐷𝑘𝑙1superscript𝑆𝑘𝑙1D^{k+l-1}\subset S^{k+l-1}italic_D start_POSTSUPERSCRIPT italic_k + italic_l - 1 end_POSTSUPERSCRIPT ⊂ italic_S start_POSTSUPERSCRIPT italic_k + italic_l - 1 end_POSTSUPERSCRIPT is and the relative homotopy type is high. Indeed, given a partition in P⁢(Sk+l−1)𝑃superscript𝑆𝑘𝑙1P(S^{k+l-1})italic_P ( italic_S start_POSTSUPERSCRIPT italic_k + italic_l - 1 end_POSTSUPERSCRIPT ) it is a highly connected choice to pick a smoothly embedded path I→Sk+l−1×I→𝐼superscript𝑆𝑘𝑙1𝐼I\to S^{k+l-1}\times Iitalic_I → italic_S start_POSTSUPERSCRIPT italic_k + italic_l - 1 end_POSTSUPERSCRIPT × italic_I from s0×{0}subscript𝑠00s_{0}\times\{0\}italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT × { 0 } to s0×{1}subscript𝑠01s_{0}\times\{1\}italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT × { 1 } which only intersects the partition boundary once and transversely so. Picking an isotopy to the standard path and using isotopy extension proves the connectivity.

is a homotopy fibration sequence with this particular null homotopy. As the inclusion of these spaces into the original sequence are homotopy equivalences (Lemma 6.2) and the map to the homotopy fiber is again compatible with this inclusion, the result follows.

The second inclusion is highly connected as it is a highly connected choice to pick a point in M+subscript𝑀M_{+}italic_M start_POSTSUBSCRIPT + end_POSTSUBSCRIPT for M∈𝒯kl𝑀superscriptsubscript𝒯𝑘𝑙M\in\mathcal{T}_{k}^{l}italic_M ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT. Using such a point we may simply translate to make that point 00.

A

S=eJ+⁢eQ⁢(J−).𝑆superscript𝑒subscript𝐽superscript𝑒𝑄subscript𝐽S=e^{J_{+}}e^{Q(J_{-})}.italic_S = italic_e start_POSTSUPERSCRIPT italic_J start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_Q ( italic_J start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT .

The synergy between Lie theory and orthogonal polynomials is well-recognized, highlighted by pivotal references such as [19, 12, 6, 7, 8]. These polynomials notably emerge as matrix elements associated with the generators, paving the way for an algebraic framework that facilitates the derivation of generating functions, recurrence relations, and orthogonality properties of these polynomials.

The study of this operator not only extends the classical Meixner polynomials but also uncovers essential features such as generating function, recurrence relations, and differential equations through the lens of Barut-Girardello coherent states, as referenced in [4, 9, 22].

The paper is structured as follows: Section 2 revisits the foundational concepts and key findings regarding d𝑑ditalic_d-orthogonal polynomials. Section 3 introduces the 𝔰⁢𝔲⁢(1,1)𝔰𝔲11\mathfrak{su}(1,1)fraktur_s fraktur_u ( 1 , 1 ) algebra and its representations, detailing the Barut-Girardello coherent states and establishing significant identities within 𝔰⁢𝔲⁢(1,1)𝔰𝔲11\mathfrak{su}(1,1)fraktur_s fraktur_u ( 1 , 1 ). Section 4 presents an in-depth analysis of the operator S𝑆Sitalic_S and its related matrix elements expressed through d𝑑ditalic_d-OPS. This algebraic approach allows us to derive crucial properties, such as biorthogonality relations, recurrence formulas, generating functions, and the application of the lowering operator. Finally, Section 5 discusses our main results concerning a new family of Meixner-type d𝑑ditalic_d-OPS, explicitly defining a linear functional vector that confirms their d𝑑ditalic_d-orthogonality via biorthogonality relations.

Turning our attention to the operator S𝑆Sitalic_S, we examine its effect on the vectors |z,β⟩𝑧𝛽\lvert z,\beta\rangle| italic_z , italic_β ⟩. This analysis leads us to the generating function F⁢(z,k)𝐹𝑧𝑘F(z,k)italic_F ( italic_z , italic_k ), which encapsulates the properties of the normalized polynomials {P^n⁢(k)}n≥0subscriptsubscript^𝑃𝑛𝑘𝑛0\{\widehat{P}_{n}(k)\}_{n\geq 0}{ over^ start_ARG italic_P end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_k ) } start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT. This function provides a comprehensive view of the polynomial sequence through its representation as an infinite sum:

B

In particular, every two (closed) points of 𝒬̊⁢(β)̊𝒬𝛽\mathring{\mathscr{Q}}(\beta)over̊ start_ARG script_Q end_ARG ( italic_β ) are transferred to each other by the G⁢[[z]]𝐺delimited-[]delimited-[]𝑧G[\![z]\!]italic_G [ [ italic_z ] ]-action. In the same vein, every two points of 𝒬̊⁢(β,w)⊂𝕆⁢(w)̊𝒬𝛽𝑤𝕆𝑤\mathring{\mathscr{Q}}(\beta,w)\subset\mathbb{O}(w)over̊ start_ARG script_Q end_ARG ( italic_β , italic_w ) ⊂ blackboard_O ( italic_w ) (β∈Q+∨,w∈Wformulae-sequence𝛽subscriptsuperscript𝑄𝑤𝑊\beta\in Q^{\vee}_{+},w\in Witalic_β ∈ italic_Q start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w ∈ italic_W) are transferred to each other by the 𝐈𝐈{\mathbf{I}}bold_I-action.

By the dimension comparison using (4.3), we deduce that this embedding must be open dense in 𝒬⁢(β)=𝒬⁢(β,e,w0)𝒬𝛽𝒬𝛽𝑒subscript𝑤0\mathscr{Q}(\beta)=\mathscr{Q}(\beta,e,w_{0})script_Q ( italic_β ) = script_Q ( italic_β , italic_e , italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). The locus Y𝑌Yitalic_Y on which the singularity of 𝒬

We have ℬ=𝒬⁢(0)ℬ𝒬0\mathscr{B}=\mathscr{Q}(0)script_B = script_Q ( 0 ) by the Plücker embedding. By expanding the map

of 𝒵⁢(β)𝒵𝛽\mathscr{Z}(\beta)script_Z ( italic_β ). The map (4.5) can be also obtained as the formal completion of the map

For the first assertion, combine Theorem 2.12 and Corollary 3.13 to obtain the map Ψ−1∘ΦsuperscriptΨ1Φ\Psi^{-1}\circ\Phiroman_Ψ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∘ roman_Φ, that have dense image. Note that the both sides are rings and the identity [𝒪Gr0]delimited-[]subscript𝒪subscriptGr0[{\mathcal{O}}_{\mathrm{Gr}_{0}}][ caligraphic_O start_POSTSUBSCRIPT roman_Gr start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ] goes to the identity [𝒪ℬ]delimited-[]subscript𝒪ℬ[{\mathcal{O}}_{\mathscr{B}}][ caligraphic_O start_POSTSUBSCRIPT script_B end_POSTSUBSCRIPT ]. The map Ψ−1∘ΦsuperscriptΨ1Φ\Psi^{-1}\circ\Phiroman_Ψ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∘ roman_Φ commutes with the natural Q∨superscript𝑄Q^{\vee}italic_Q start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT-actions given by 𝚝γsubscript𝚝𝛾\mathtt{t}_{\gamma}typewriter_t start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT and Qγsuperscript𝑄𝛾Q^{\gamma}italic_Q start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT for each γ∈Q∨𝛾superscript𝑄\gamma\in Q^{\vee}italic_γ ∈ italic_Q start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT by Theorem 2.5 and Corollary 3.13. Moreover, the action of ΘisubscriptΘ𝑖\Theta_{i}roman_Θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT (see §2.3) and the quantum multiplication by [𝒪ℬ⁢(−ϖi)]delimited-[]subscript𝒪ℬsubscriptitalic-ϖ𝑖[{\mathcal{O}}_{\mathscr{B}}(-\varpi_{i})][ caligraphic_O start_POSTSUBSCRIPT script_B end_POSTSUBSCRIPT ( - italic_ϖ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ] corresponds for each i∈𝙸𝑖𝙸i\in\mathtt{I}italic_i ∈ typewriter_I (by Theorem 2.12 and Corollary 3.13). Therefore, the ⊙direct-product\odot⊙-multiplication by the element 𝐡isubscript𝐡𝑖{\mathbf{h}}_{i}bold_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and ⋆⋆\star⋆-multiplication by [𝒪ℬ⁢(si)]=([𝒪ℬ]−eϖi⁢[𝒪ℬ⁢(−ϖi)])delimited-[]subscript𝒪ℬsubscript𝑠𝑖delimited-[]subscript𝒪ℬsuperscript𝑒subscriptitalic-ϖ𝑖delimited-[]subscript𝒪ℬsubscriptitalic-ϖ𝑖[{\mathcal{O}}_{\mathscr{B}(s_{i})}]=([{\mathcal{O}}_{\mathscr{B}}]-e^{\varpi_%

B

A number of other interesting homological and homotopical finiteness properties have been studied in relation to monoids defined by complete rewriting systems; see [3, 55, 28].

on the behaviour of FPnsubscriptFP𝑛{\rm FP}_{n}roman_FP start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT under free products and certain rather restricted free products of monoids with amalgamation. The proofs in Cremanns and Otto are quite long and technical, as is often the case for results in this area. The results in this section demonstrate the type of result our topological methods were introduced to prove. They show that the topological approach may be used to prove more general results in a less technical and more conceptual way. Our results also generalise and simplify proofs of some results of Kobayashi [35] on preservation of left-, right- and bi-FPnsubscriptFP𝑛{\rm FP}_{n}roman_FP start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT under free products (see for example [35, Proposition 4.1]). There are no bi-FPnsubscriptFP𝑛{\rm FP}_{n}roman_FP start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT analogues in the literature of the two-sided results we obtain below on the behaviour of bi-FnsubscriptF𝑛{\rm F}_{n}roman_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT and geometric dimension for free products with amalgamation. Also, as far as we are aware, the results that we obtain here are the first to appear in the literature on cohomological dimension of amalgamated free products of monoids.

The Hochschild cohomological dimension of M𝑀Mitalic_M, written dimMdimension𝑀\dim Mroman_dim italic_M, is the length of a shortest projective resolution of ℤ⁢Mℤ𝑀\mathbb{Z}Mblackboard_Z italic_M as a ℤ⁢[M×Mo⁢p]ℤdelimited-[]𝑀superscript𝑀𝑜𝑝\mathbb{Z}[M\times M^{op}]blackboard_Z [ italic_M × italic_M start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT ]-module. The Hochschild cohomological dimension bounds both the left and right cohomological dimension and the geometric dimension bounds the Hochschild cohomological dimension. The geometric dimension also bounds both the left and right geometric dimensions because if X𝑋Xitalic_X is a bi-equivariant classifying space for M𝑀Mitalic_M of dimension n𝑛nitalic_n, then X/M𝑋𝑀X/Mitalic_X / italic_M is an equivariant classifying space of dimension n𝑛nitalic_n.

In general, the left- and right-cohomological dimensions of a monoid are not equal. In fact they are completely independent of each other; see [27]. One immediate corollary of the above result is that if M𝑀Mitalic_M is a finitely presented special monoid with left- and right-cohomological dimensions both at least equal to 2222, then the left cohomological dimension of M𝑀Mitalic_M is equal to its right cohomological dimension.

The cohomological dimension of monoids has also received attention in the literature; see for example [13, 27, 47].

D

Cheridito et al. (2006) considered dynamic coherent, convex monetary and monetary risk measures for discrete-time processes modelling the evolution of financial values. Acciaio et al. (2012) extended dynamic convex risk measures in Cheridito et al. to take the timing of cash flow into consideration. Sun and Hu (2018) introduced a new class of set-valued risk measures which satisfies cash sub-additivity and investigated dynamic set-valued cash sub-additive risk measures. Wang et al. (2021) constructed some general continuous-time equilibrium dynamic risk measures through using a adapted solution to a backward stochastic Volterra integral equation. Chen et al. (2018) and Sun et al. (2018) extended convex risk measures to loss-based cases. More recent research on dynamic risk measures reference in Chen et al. (2021), Chen and Feinstein (2022), Mastrogiacomo and Rosazza (2022), Yoshioka and Yoshioka (2024)

Nowadays, as the digital economy and cryptocurrencies develop rapidly, they have a great impact on the financial market. The volatility of cryptocurrencies is a distinctive characteristic defined by rapid and substantial price fluctuations within relatively short periods. Compared to traditional financial assets, cryptocurrencies such as Bitcoin and Ethereum are considered as asset which can be used for speculative purpose, hence it can lead to extreme volatility and bubbles (see Fry and Cheah, 2016). Factors contributing to this volatility include market sentiment, regulatory developments, technological advancements, and macroeconomic conditions. Apart from extreme volatility of cryptocurrencies, different orders of risk data are mixed in a short period of time can occur when multiple levels or types of risk factors are simultaneously influencing the financial market. For example, major economic events, such as financial crises, geopolitical tensions, or central bank policy announcements, can trigger rapid and varied responses across different asset classes. This simultaneous impact can result in mixed risk signals. In addition, sudden and unexpected shocks to the market, whether related to economic indicators, corporate news, or global events, can lead to a convergence of various risk factors that creating a mixed picture of risk data where Yang et al. (2023) found that an increase in economic policy uncertainty in the China and US exacerbates fluctuation in the global oil price, particularly during times of crisis. Besides, high correlations between different asset classes or markets can lead to a synchronization of risk data. For example, during periods of heightened risk aversion, equities, currencies, and commodities may all exhibit increased volatility simultaneously. Therefore, the need for comprehensive risk measures that can capture the complexity and increasing fluctuation of market volatility is significant not only for new financial assets but also for traditional financial market with rapidly changing financial environment and global landscape.

Although there are many different studies of risk measure, a common aspect of these studies is the space of financial positions was considered as the space or subspace of Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT. It is crucial to note that our focus diverges from this conventional space. As financial markets grow in complexity, traditional position spaces may prove inadequate in capturing market volatility. This has raised awareness of the urgent need for more appropriate risk measures under a financial systems with much greater volatility. It is necessary to explore a new position space that can depict the mixed occurrence of different orders of risk data. From our point of view, to better capture the increasing fluctuation of market volatility should base on risk measures with reasonable position spaces and these spaces should allow for higher volatility in the risk positions within them. A comprehensive study on risk measures in a novel space could address the issue of increasing fluctuation of market volatility and also it can provide some theoretic foundation for major importance to maintain stability of the financial systems. This paper aims to provide axiomatic theories of risk measures on a special space which reflects complexities of market volatility in the contemporary economic environment.

Recently, many external factors including changes in international situations, increase of war risk and significant environmental changes, all cause the financial market becomes much more volatile than before, and the financial market also has different orders of risk data mixed over a short period of time. Therefore, it is valuable to study the risk measures on variable exponent Bochner–Lebesgue spaces. Under this position space, the order risk position is no longer a fixed positive number, but a measurable function. The characteristics of this space are able to characterise risk positions in the above volatile financial market.

As economic landscapes evolve, traditional risk metrics may prove insufficient in capturing the complexities and nuances of contemporary risks. In particular, from the global financial crisis of 2008 to the more recent disruptions caused by geopolitical tensions and public health crises, which have underscored the need for robust tools and methodologies to assess and mitigate risks effectively. In addition, the financial markets continue to evolve, incorporating new technologies, new instruments, and regulatory frameworks, the demand for advanced risk measurement techniques has become even more pronounced. The traditional approaches which are foundational but are being complemented and, in some cases, supplanted by sophisticated models that account for non-linearities, tail risks, and the dynamic nature of volatility. Besides, it becomes increasingly evident that risk measures play a pivotal role not only in risk management but also in shaping investment strategies, influencing regulatory frameworks, and guiding decision-making across a spectrum of financial activities.

A

Observe that Im⁡(β)Im𝛽\operatorname{\mathrm{Im}}(\beta)roman_Im ( italic_β ) is 𝔤Γq⊕𝔤Γqdirect-sumsuperscript𝔤subscriptΓ𝑞superscript𝔤subscriptΓ𝑞\mathfrak{g}^{\Gamma_{q}}\oplus\mathfrak{g}^{\Gamma_{q}}fraktur_g start_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⊕ fraktur_g start_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUPERSCRIPT-stable under the component wise action of 𝔤Γq⊕𝔤Γqdirect-sumsuperscript𝔤subscriptΓ𝑞superscript𝔤subscriptΓ𝑞\mathfrak{g}^{\Gamma_{q}}\oplus\mathfrak{g}^{\Gamma_{q}}fraktur_g start_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⊕ fraktur_g start_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUPERSCRIPT on V⁢(μ∗)⊗V⁢(μ)tensor-product𝑉superscript𝜇𝑉𝜇V(\mu^{*})\otimes V(\mu)italic_V ( italic_μ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ⊗ italic_V ( italic_μ ) since Iμsubscript𝐼𝜇I_{\mu}italic_I start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT is 𝔤Γqsuperscript𝔤subscriptΓ𝑞\mathfrak{g}^{\Gamma_{q}}fraktur_g start_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUPERSCRIPT-invariant under the diagonal action of 𝔤Γqsuperscript𝔤subscriptΓ𝑞\mathfrak{g}^{\Gamma_{q}}fraktur_g start_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUPERSCRIPT. Moreover, V⁢(μ∗)⊗V⁢(μ)tensor-product𝑉superscript𝜇𝑉𝜇V(\mu^{*})\otimes V(\mu)italic_V ( italic_μ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ⊗ italic_V ( italic_μ ) is an irreducible 𝔤Γq⊕𝔤Γqdirect-sumsuperscript𝔤subscriptΓ𝑞superscript𝔤subscriptΓ𝑞\mathfrak{g}^{\Gamma_{q}}\oplus\mathfrak{g}^{\Gamma_{q}}fraktur_g start_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⊕ fraktur_g start_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUPERSCRIPT-module with highest weight (μ∗,μ)superscript𝜇𝜇(\mu^{*},\mu)( italic_μ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , italic_μ ); and Im⁡(β)Im𝛽\operatorname{\mathrm{Im}}(\beta)roman_Im ( italic_β ) has a nonzero component in each V⁢(μ∗)⊗V⁢(μ)tensor-product𝑉superscript𝜇𝑉𝜇V(\mu^{*})\otimes V(\mu)italic_V ( italic_μ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ⊗ italic_V ( italic_μ ). Thus, β𝛽\betaitalic_β is surjective.

By the Residue Theorem, β𝛽\betaitalic_β is indeed a Lie algebra embedding. (Observe that Lemma 7.3 has been used to show that β𝛽\betaitalic_β is an embedding.)

From the surjectivity of β𝛽\betaitalic_β, we get that the map F𝐹Fitalic_F is surjective by combining the equation

From the definition of β𝛽\betaitalic_β (cf. equation (48)), it is easy to see that, under the identification of V⁢(μ∗)⊗V⁢(μ)tensor-product𝑉superscript𝜇𝑉𝜇V(\mu^{*})\otimes V(\mu)italic_V ( italic_μ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ⊗ italic_V ( italic_μ ) with Endℂ⁡(V⁢(μ))subscriptEndℂ𝑉𝜇\operatorname{\mathrm{End}}_{\mathbb{C}}(V(\mu))roman_End start_POSTSUBSCRIPT blackboard_C end_POSTSUBSCRIPT ( italic_V ( italic_μ ) ),

Combining (132) and (133), we get λ≡0𝜆0\lambda\equiv 0italic_λ ≡ 0. This proves that σ˙˙𝜎\dot{\sigma}over˙ start_ARG italic_σ end_ARG is the identity map.

B

By Lemma 1.2, G⁢[C]𝐺delimited-[]𝐶G[C]italic_G [ italic_C ] is either a cycle in ℬℬ\mathcal{B}caligraphic_B, or one of the following biased subgraphs: a theta with no cycle in ℬℬ\mathcal{B}caligraphic_B, tight handcuffs, loose handcuffs, or a bracelet.

Then T𝑇Titalic_T is dependent in F⁢(G,ℬ)𝐹𝐺ℬF(G,\mathcal{B})italic_F ( italic_G , caligraphic_B ) and so dependent in M𝑀Mitalic_M, while every proper subset of T𝑇Titalic_T is independent in L⁢(G,ℬ)𝐿𝐺ℬL(G,\mathcal{B})italic_L ( italic_G , caligraphic_B ) and so independent in M𝑀Mitalic_M.

That is, M𝑀Mitalic_M is biased-graphic if there is a biased graph (G,ℬ)𝐺ℬ(G,\mathcal{B})( italic_G , caligraphic_B ) such that M𝑀Mitalic_M is intermediate between L⁢(G,ℬ)𝐿𝐺ℬL(G,\mathcal{B})italic_L ( italic_G , caligraphic_B ) and F⁢(G,ℬ)𝐹𝐺ℬF(G,\mathcal{B})italic_F ( italic_G , caligraphic_B ),

This implies that every dependent set of M𝑀Mitalic_M is dependent in L⁢(G,ℬ)𝐿𝐺ℬL(G,\mathcal{B})italic_L ( italic_G , caligraphic_B ).

By Lemma 2.7 every dependent set of F⁢(G,ℬ)𝐹𝐺ℬF(G,\mathcal{B})italic_F ( italic_G , caligraphic_B ) is dependent in M𝑀Mitalic_M.

C

Recall for a bosons, α=0𝛼0\alpha=0italic_α = 0; hence, a vacancy box for bosons can be occupied arbitrary number of times. For a fermion, however, α=1𝛼1\alpha=1italic_α = 1; hence, a vacancy box for fermions can be occupied at most once. Now we have (α~τ⁢τ¯D1)11=2subscriptsubscriptsuperscript~𝛼superscript𝐷1𝜏¯𝜏112(\tilde{\alpha}^{D^{1}}_{\tau\bar{\tau}})_{11}=2( over~ start_ARG italic_α end_ARG start_POSTSUPERSCRIPT italic_D start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_τ over¯ start_ARG italic_τ end_ARG end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT = 2; hence, not only a vacancy box of the pseudo-species can be occupied at most once but also the nearest neighbouring boxes in the same row cannot be occupied, as a generalization of the Pauli exclusion principle. We have briefly touched upon such exclusion rules earlier. The new ingredient here is that each pseudo-species anyon takes up two vacancy boxes, i.e., the two boxes in a column in Fig. 5. Remember also that the two vacancy boxes are not added at the same time, their configurations are independent. Thus, when a box in a column is occupied, it cannot exclude the other box in the same column being occupied because it takes the entire column to specify a single-particle state of the pseudo-species. As such, an occupied box can only expel the occupation of its nearest neighbouring boxes. The right most column is an exception because there we put in a duplicate box by hand.

Recall for a bosons, α=0𝛼0\alpha=0italic_α = 0; hence, a vacancy box for bosons can be occupied arbitrary number of times. For a fermion, however, α=1𝛼1\alpha=1italic_α = 1; hence, a vacancy box for fermions can be occupied at most once. Now we have (α~τ⁢τ¯D1)11=2subscriptsubscriptsuperscript~𝛼superscript𝐷1𝜏¯𝜏112(\tilde{\alpha}^{D^{1}}_{\tau\bar{\tau}})_{11}=2( over~ start_ARG italic_α end_ARG start_POSTSUPERSCRIPT italic_D start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_τ over¯ start_ARG italic_τ end_ARG end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT = 2; hence, not only a vacancy box of the pseudo-species can be occupied at most once but also the nearest neighbouring boxes in the same row cannot be occupied, as a generalization of the Pauli exclusion principle. We have briefly touched upon such exclusion rules earlier. The new ingredient here is that each pseudo-species anyon takes up two vacancy boxes, i.e., the two boxes in a column in Fig. 5. Remember also that the two vacancy boxes are not added at the same time, their configurations are independent. Thus, when a box in a column is occupied, it cannot exclude the other box in the same column being occupied because it takes the entire column to specify a single-particle state of the pseudo-species. As such, an occupied box can only expel the occupation of its nearest neighbouring boxes. The right most column is an exception because there we put in a duplicate box by hand.

In the exclusion rule above, we define a box as a vacancy and call it a vacancy box. A ■■\boxed{\blacksquare}■ is a vacancy box occupied, a ×\boxed{\times}× is a vacancy box that cannot be occupied (or unhabitable), and ✓✓\boxed{\checkmark}✓ is a vacancy box that may be occupied (or habitable). In other words, for the pseudo-species, if a vacancy box is occupied, the two neighbouring vacancy boxes become unhabitable. A vacancy box is analogous to an orbital of an electron in an atom, say, for example.

Here, □□\boxed{\square}□ is a wildcard, meaning the box is either occupied or unoccupied; the RHS of (c)𝑐(c)( italic_c ) forces the two boxes in the column to be both occupied (unoccupied) if the box on the LHS is occupied (unoccupied).

Now the question is: How we may decide which vacancy boxes in the disk basis Fig. 4(e) can be occupied and which cannot be? To find the answer, let us further simplify the basis into the form in Fig. 5 by removing the tree and the boundary but keeping the boxes only. This procedure is correct because the degrees of freedom of the Hilbert space being studied reside only in the vacancy boxes. In this simplified basis, a dashed box is added below the upper right box for reasons to be clear shortly. This new basis comprises of solely the pseudo-species vacancy boxes but keep in mind that between two neighbouring columns, there is a real τ⁢τ¯𝜏¯𝜏\tau\bar{\tau}italic_τ over¯ start_ARG italic_τ end_ARG inserted.

C

Following Deligne [Del90, 1.2], a tensor category 𝒯𝒯\mathcal{T}caligraphic_T over k𝑘kitalic_k is a k𝑘kitalic_k-linear

⊗tensor-product\otimes⊗-category which is ACU, rigid, and abelian with k=End⁢(𝟙)𝑘End1k=\mathrm{End}(\mathds{1})italic_k = roman_End ( blackboard_1 ). A category is

condition k=End⁢(𝟙)𝑘End1k=\mathrm{End}(\mathds{1})italic_k = roman_End ( blackboard_1 ) is superfluous.)

We start by noting that, in our particular case, the ring End⁢(𝟙)End1\mathrm{End}(\mathds{1})roman_End ( blackboard_1 ) is isomorphic to

For simplicity, in this work a tensor category 𝒯𝒯\mathcal{T}caligraphic_T over k𝑘kitalic_k is a k𝑘kitalic_k-linear ⊗tensor-product\otimes⊗-category which is ACU, rigid, and abelian with k=End⁢(𝟙)𝑘End1k=\mathrm{End}(\mathds{1})italic_k = roman_End ( blackboard_1 ). This amounts to say that ⊗tensor-product\otimes⊗ is a k𝑘kitalic_k-linear abelian category equipped with a bi-additive k𝑘kitalic_k-linear exact functor ⊗:𝒯×𝒯→𝒯\otimes:\mathcal{T}\times\mathcal{T}\to\mathcal{T}⊗ : caligraphic_T × caligraphic_T → caligraphic_T and a unit object 𝟙1\mathds{1}blackboard_1, satisfying the associativity, commutativity and unity axioms (cf. [DM82]). Moreover, we require the existence of the dual object to each object in 𝒯𝒯\mathcal{T}caligraphic_T. For each U∈𝒯𝑈𝒯U\in\mathcal{T}italic_U ∈ caligraphic_T, its dual will be denoted by U∨superscript𝑈U^{\vee}italic_U start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT and is uniquely determined by the isomorphism

A

For each β≤κ+𝛽superscript𝜅\beta\leq\kappa^{+}italic_β ≤ italic_κ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, ℚβsubscriptℚ𝛽{\mathbb{Q}}_{\beta}blackboard_Q start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT has the κ𝜅\kappaitalic_κ-chain condition.

As we will see, the following lemma will enable us to ease our path through the proof of Claim 4.7, in Section 4, in a significant way.

As we will see, the weak compactness of κ𝜅\kappaitalic_κ is used crucially in order to prove Lemma 4.1.

The preservation of all higher cardinals proceeds by showing that the construction has the κ𝜅\kappaitalic_κ-chain condition. For this, we use the weak compactness of κ𝜅\kappaitalic_κ in an essential way. The proof of the κ𝜅\kappaitalic_κ-c.c. of ℚβsubscriptℚ𝛽{\mathbb{Q}}_{\beta}blackboard_Q start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT, for each β<κ+𝛽superscript𝜅\beta<\kappa^{+}italic_β < italic_κ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, is modelled after the corresponding proof in [10]; in fact it is a natural adaptation, to the current setting, of the proof in [10] of the κ𝜅\kappaitalic_κ-c.c. of the main forcing in that paper. The fact that the length of our iteration is not greater than κ+superscript𝜅\kappa^{+}italic_κ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT seems to be needed in this proof.

Let ℱℱ\mathcal{F}caligraphic_F be the weak compactness filter on κ𝜅\kappaitalic_κ, i.e., the filter on κ𝜅\kappaitalic_κ generated by the sets

B

Now we claim that ℰc=0subscriptℰ𝑐0\mathcal{E}_{c}=0caligraphic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = 0 in our associated graded algebra.

According to [19], to define the RTT Yangian YV⁢(𝔤)subscript𝑌𝑉𝔤Y_{V}({\mathfrak{g}})italic_Y start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ( fraktur_g ), one can use any finite-dimensional representation V𝑉Vitalic_V of Y⁢(𝔤)𝑌𝔤Y({\mathfrak{g}})italic_Y ( fraktur_g ) which is not a sum of trivial representations, and Theorems 2.12 and 2.13 are still valid.

2.19. Filtration on YV⁢(𝔤)subscript𝑌𝑉𝔤Y_{V}({\mathfrak{g}})italic_Y start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ( fraktur_g )

It follows from relation [19, (5.10)] (which is still true for our filtration) and from [19, Lemma 4.13].

Note that from Theorem 2.4 it follows that this is indeed a filtration and gr⁢Yn⁢e⁢w⁢(𝔤)grsubscript𝑌𝑛𝑒𝑤𝔤{\rm gr}\,Y_{new}({\mathfrak{g}})roman_gr italic_Y start_POSTSUBSCRIPT italic_n italic_e italic_w end_POSTSUBSCRIPT ( fraktur_g ) is commutative.

C

We note that Dai, Mallick and Stoffregen have independently found examples of slice disks with large stabilization distance [DMSEquivariant]. Additionally, they use some of the techniques of this paper in their work to study equivariant knots.

and properly embedded surfaces in B4superscript𝐵4B^{4}italic_B start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT with boundary a knot K𝐾Kitalic_K with a type of metric

We proceed to define the stabilization distance of a pair of surfaces with boundary a given knot, which

of a pair of surfaces S𝑆Sitalic_S, S′∈Surf⁡(K)superscript𝑆′Surf𝐾S^{\prime}\in\operatorname{{Surf}}(K)italic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ roman_Surf ( italic_K ), which is similar to the stabilization distance,

We define the stabilization distance of the pair (S,S′)𝑆superscript𝑆′(S,S^{\prime})( italic_S , italic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ), for which we write

B

Both stacks are Pic⁡(Σ~b(1))Picsuperscriptsubscript~Σ𝑏1\operatorname{Pic}(\widetilde{\Sigma}_{b}^{(1)})roman_Pic ( over~ start_ARG roman_Σ end_ARG start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT )-torsors. Since τ~*superscript~𝜏\widetilde{\tau}^{*}over~ start_ARG italic_τ end_ARG start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT is compatible with the Pic⁡(Σ~b(1))Picsuperscriptsubscript~Σ𝑏1\operatorname{Pic}(\widetilde{\Sigma}_{b}^{(1)})roman_Pic ( over~ start_ARG roman_Σ end_ARG start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT )-actions, it induces an isomorphism between those two stacks.

Let k𝑘kitalic_k be an algebraically closed field. Let ℬℬ\mathcal{B}caligraphic_B be a k𝑘kitalic_k-scheme locally of finite type. Let Y𝑌Yitalic_Y be a stack locally of finite type over ℬℬ\mathcal{B}caligraphic_B. Let Y~⟶Y⟶~𝑌𝑌\widetilde{Y}\longrightarrow Yover~ start_ARG italic_Y end_ARG ⟶ italic_Y be a 𝔾msubscript𝔾𝑚\mathbb{G}_{m}blackboard_G start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT-gerbe over Y𝑌Yitalic_Y. We denote by QCoh⁡(Y~)QCoh~𝑌\operatorname{QCoh}(\widetilde{Y})roman_QCoh ( over~ start_ARG italic_Y end_ARG ) the category of quasi-coherent sheaves on Y~~𝑌\widetilde{Y}over~ start_ARG italic_Y end_ARG. We say Y~~𝑌\widetilde{Y}over~ start_ARG italic_Y end_ARG splits if there is an isomorphism Y~≅Y×B⁢𝔾m~𝑌𝑌𝐵subscript𝔾𝑚\widetilde{Y}\cong Y\times B\mathbb{G}_{m}over~ start_ARG italic_Y end_ARG ≅ italic_Y × italic_B blackboard_G start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT of 𝔾msubscript𝔾𝑚\mathbb{G}_{m}blackboard_G start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT-gerbes. In this case, there is a decomposition

In this subsection we review the Fourier-Mukai transforms on commutative group stacks, following [8]. Let k𝑘kitalic_k be an algebraically closed field. Let ℬℬ\mathcal{B}caligraphic_B be an irreducible k𝑘kitalic_k-scheme that is locally of finite type.

Let 𝒢𝒢\mathcal{G}caligraphic_G be a commutative group stack locally of finite type over ℬℬ\mathcal{B}caligraphic_B. The dual commutative group stack 𝒢∨superscript𝒢\mathcal{G}^{\vee}caligraphic_G start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT classifies 1-morphisms of group stacks from 𝒢𝒢\mathcal{G}caligraphic_G to B⁢𝔾m𝐵subscript𝔾𝑚B\mathbb{G}_{m}italic_B blackboard_G start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT. The main examples we are going to consider are:

Let S𝑆Sitalic_S be a k𝑘kitalic_k-scheme. Let b𝑏bitalic_b be an S𝑆Sitalic_S-point of (BP0)(1)superscriptsuperscriptsubscript𝐵𝑃01(B_{P}^{0})^{(1)}( italic_B start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT. Consider the following commutative diagram:

B

We can say more about (ii), and in D we give a sample result for slightly more abstract semigroups, for which we would not a priori know the associated equations.

This section is devoted to the proofs of the results of Section 3. We will prove them in a certain order to arrive at Corollaries 25 and 37, thus concluding by the optimality of the weight and the interpretation in terms of dual nonlinear semigroup. The proofs of Propositions

The proofs of Theorem 35 and Corollary 36 are given in Section 4.2, while Theorem 33 is proved in Section 4.5.

The rest of this paper is organized as follows. We recall basic facts in Section 2, we state our main results in Section 3, and prove them in Section 4. For completeness, some results for minimal discontinuous viscosity solutions are proved in A, a complete proof of well-posedness for L∞superscript𝐿L^{\infty}italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT entropy solutions is given in B, and further comments on our duality results are postponed to C and D.

In this section we precisely state our results: the weighted L1superscript𝐿1L^{1}italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT contraction estimate for (1) in Section 3.1, the optimality of the weight in Section 3.2, and the interpretation in terms of dual nonlinear semigroup in Section 3.3. Section 3.3 contains the L1superscript𝐿1L^{1}italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT theory for (2), and the long proofs are postponed

A

Throughout this paper function 𝒜⁢()𝒜\mbox{${{{\cal A}}}$}()caligraphic_A ( ) will denote the arithmetic mean of any number of variables. We will also use the notation 𝒜(ai:1≤i≤n)\mbox{${{{\cal A}}}$}(a_{i}:1\leq i\leq n)caligraphic_A ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : 1 ≤ italic_i ≤ italic_n ) for 𝒜⁢(a1,…,an)𝒜subscript𝑎1…subscript𝑎𝑛\mbox{${{{\cal A}}}$}(a_{1},\dots,a_{n})caligraphic_A ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ).

If H⊂ℝ𝐻ℝH\subset\mathbb{R}italic_H ⊂ blackboard_R is a finite set then 𝒜⁢(H)𝒜𝐻\mbox{${{{\cal A}}}$}(H)caligraphic_A ( italic_H ) denotes the arithmetic mean of its distinct points.

Let (an)subscript𝑎𝑛(a_{n})( italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) be a sequence and a≤b⁢(a,b∈ℝ)𝑎𝑏𝑎𝑏ℝa\leq b\ (a,b\in\mathbb{R})italic_a ≤ italic_b ( italic_a , italic_b ∈ blackboard_R ) are two of its accumulation points. Then [a,b]⊂A⁢A⁢R(an)𝑎𝑏𝐴𝐴subscript𝑅subscript𝑎𝑛[a,b]\subset AAR_{(a_{n})}[ italic_a , italic_b ] ⊂ italic_A italic_A italic_R start_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT.

Throughout this paper function 𝒜⁢()𝒜\mbox{${{{\cal A}}}$}()caligraphic_A ( ) will denote the arithmetic mean of any number of variables. We will also use the notation 𝒜(ai:1≤i≤n)\mbox{${{{\cal A}}}$}(a_{i}:1\leq i\leq n)caligraphic_A ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : 1 ≤ italic_i ≤ italic_n ) for 𝒜⁢(a1,…,an)𝒜subscript𝑎1…subscript𝑎𝑛\mbox{${{{\cal A}}}$}(a_{1},\dots,a_{n})caligraphic_A ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ).

Let (an)subscript𝑎𝑛(a_{n})( italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) be a sequence such that it has at least 4 accumulation points: a,b,−∞,+∞(a,b∈ℝ,a<b)a,b,-\infty,+\infty\ (a,b\in\mathbb{R},a<b)italic_a , italic_b , - ∞ , + ∞ ( italic_a , italic_b ∈ blackboard_R , italic_a < italic_b ). Let Z⊂[a,b]𝑍𝑎𝑏Z\subset[a,b]italic_Z ⊂ [ italic_a , italic_b ] be a closed set. Then there is a rearrangement (an′)subscriptsuperscript𝑎normal-′𝑛(a^{\prime}_{n})( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) of (an)subscript𝑎𝑛(a_{n})( italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) such that the accumulation points of the sequence (pn)subscript𝑝𝑛(p_{n})( italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) is exactly Z𝑍Zitalic_Z where pn=𝒜⁢(a1′,…,an′)subscript𝑝𝑛𝒜subscriptsuperscript𝑎normal-′1normal-…subscriptsuperscript𝑎normal-′𝑛p_{n}=\mbox{${{{\cal A}}}$}(a^{\prime}_{1},\dots,a^{\prime}_{n})italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = caligraphic_A ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ).

A

\frac{3}{8}-\frac{5}{4p},&\quad 4<p<6.\end{aligned}\right.italic_σ ( italic_p ) = { start_ROW start_CELL 0 , end_CELL start_CELL 2 < italic_p ⩽ 3 , end_CELL end_ROW start_ROW start_CELL divide start_ARG 1 end_ARG start_ARG 4 end_ARG - divide start_ARG 3 end_ARG start_ARG 4 italic_p end_ARG , end_CELL start_CELL 3 < italic_p ⩽ 4 , end_CELL end_ROW start_ROW start_CELL divide start_ARG 3 end_ARG start_ARG 8 end_ARG - divide start_ARG 5 end_ARG start_ARG 4 italic_p end_ARG , end_CELL start_CELL 4 < italic_p < 6 . end_CELL end_ROW

It is worth noting that the spacial support of the amplitude appearing in the right-hand side of (2.12) and (2.13) is slightly larger than that appearing in the left-hand side.

The results obtained above generalize its constant coefficient counterpart in [14, 12]. It is worth noting that the results in the case 2<p≤32𝑝32<p\leq 32 < italic_p ≤ 3 are sharp, except for possibly arbitrarily small regularity loss.

Very recently, Guth-Wang-Zhang [10] established the sharp square function estimate in the Euclidean case in 2+1212+12 + 1 dimensions. As a result, the corresponding local smoothing conjecture is resolved. For the variable coefficient setting, the Kakeya compression phenomena will happen which leads to the difference in the numerology of local smoothing conjecture between the variable and constant coefficient settings in n≥3𝑛3n\geq 3italic_n ≥ 3, see [1, 2, 9] for more details. This work provides additional methods and techniques toward handling the variable coefficient case which may help advance the research in this direction.

The following stability lemma makes the variable coefficient case and its constant counterpart comparable at sufficiently small scales.

B

+1}\|_{Z_{s}})\tilde{G}_{n}(u^{k+1}(s)))\,dW(s),~{}\mathbb{P}\textrm{-a.s..}+ ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_cos ( ( italic_k italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_s ) square-root start_ARG italic_A end_ARG ) ( italic_θ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( ∥ italic_u start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_Z start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) over~ start_ARG italic_G end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_u start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT ( italic_s ) ) ) italic_d italic_W ( italic_s ) , blackboard_P -a.s..

To prove the existence of a local mild solution to problem (5.4)-(5.5) in the sense of Definition 5.13.

Thus, in light of (6.43)-(6.44), in order to prove (6.42) it is sufficient to prove equality (6.46) below, i.e.,

In this section we will prove the deterministic Strichartz type estimate, see Theorem 3.2 below, which is a generalization of [42, Theorem 1.2] and is essential to tackle, both, the Dirichlet and the Neumann boundary case.

This section is devoted to prove a stochastic Strichartz inequality, which is sufficient to apply the Banach Fixed Point Theorem in the proof of a local well-posedness result for problem (1.2), see Theorem 7.1 in Section 7.

B

In constrast, in the S1superscript𝑆1S^{1}italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT-equivariant version we replace H⁢(X)𝐻𝑋H(X)italic_H ( italic_X ) by HS1⁢(X)≅H⁢(X)⊗H⁢(ℂ⁢ℙ∞)subscript𝐻superscript𝑆1𝑋tensor-product𝐻𝑋𝐻ℂsuperscriptℙH_{S^{1}}(X)\cong H(X)\otimes H(\mathbb{CP}^{\infty})italic_H start_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_X ) ≅ italic_H ( italic_X ) ⊗ italic_H ( blackboard_C blackboard_P start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ), giving rise to countably many independent classes even when X𝑋Xitalic_X is contractible.

This allows us to define capacities for larger class of domains which are not necessarily exact, and it also allows us to discuss Maurer–Cartan theory (see §4).

Before discussing our main results, we first motivate and illustrate the concept of “higher symplectic capacities” from the perspective of Floer theory, showing how to construct new capacities based on algebraic structures on symplectic cohomology.

Let us now observe that symplectic cohomology has various additional algebraic structures which we can try to exploit to define further capacities.

It is tempting to also use these non-equivariant structures to define symplectic capacities, but unfortunately

C

}}\Big{|}^{s+t}\Big{)}^{\frac{1}{s+t}}.start_ROW start_CELL ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT ( italic_q start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG italic_s + italic_t - 1 end_ARG start_ARG italic_s + italic_t end_ARG end_POSTSUPERSCRIPT ( italic_q start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_s + italic_t end_ARG end_POSTSUPERSCRIPT | divide start_ARG italic_g ( italic_b italic_q start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ) - italic_g ( italic_b italic_q start_POSTSUPERSCRIPT italic_j + 1 end_POSTSUPERSCRIPT ) end_ARG start_ARG italic_b italic_q start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT - italic_b italic_q start_POSTSUPERSCRIPT italic_j + 1 end_POSTSUPERSCRIPT end_ARG | ≤ ( ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT ( ( italic_q start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG italic_s + italic_t - 1 end_ARG start_ARG italic_s + italic_t end_ARG end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG italic_s + italic_t end_ARG start_ARG italic_s + italic_t - 1 end_ARG end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG italic_s + italic_t - 1 end_ARG start_ARG italic_s + italic_t end_ARG end_POSTSUPERSCRIPT × end_CELL end_ROW start_ROW start_CELL ( ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT italic_q start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT | divide start_ARG italic_g ( italic_b italic_q start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ) - italic_g ( italic_b italic_q start_POSTSUPERSCRIPT italic_j + 1 end_POSTSUPERSCRIPT ) end_ARG start_ARG italic_b italic_q start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT - italic_b italic_q start_POSTSUPERSCRIPT italic_j + 1 end_POSTSUPERSCRIPT end_ARG | start_POSTSUPERSCRIPT italic_s + italic_t end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_s + italic_t end_ARG end_POSTSUPERSCRIPT . end_CELL end_ROW

We multiply both inequalities by b⁢(1−q)𝑏1𝑞b(1-q)italic_b ( 1 - italic_q ), then raise them to the power s+t𝑠𝑡s+titalic_s + italic_t. Thus, we obtain

Multiplying this inequality by bp+1⁢(1−q)p+1superscript𝑏𝑝1superscript1𝑞𝑝1b^{p+1}(1-q)^{p+1}italic_b start_POSTSUPERSCRIPT italic_p + 1 end_POSTSUPERSCRIPT ( 1 - italic_q ) start_POSTSUPERSCRIPT italic_p + 1 end_POSTSUPERSCRIPT, and relying on the formula for the sum of the first n𝑛nitalic_n terms of the geometric series, we arrive at the inequality

in q𝑞qitalic_q-calculus was given. Here we eliminate some inaccuracies by simplifying and modifying the proofs of the theorems.

In view of f⁢(b⁢qn)=∑j=0n−1f⁢(b⁢qj+1)−f⁢(b⁢qj)𝑓𝑏superscript𝑞𝑛superscriptsubscript𝑗0𝑛1𝑓𝑏superscript𝑞𝑗1𝑓𝑏superscript𝑞𝑗f(bq^{n})=\sum\limits_{j=0}^{n-1}{f(bq^{j+1})-f(bq^{j})}italic_f ( italic_b italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT italic_f ( italic_b italic_q start_POSTSUPERSCRIPT italic_j + 1 end_POSTSUPERSCRIPT ) - italic_f ( italic_b italic_q start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ), we obtain

A

ψ∗⁢E:=N×ψEassignsuperscript𝜓∗𝐸subscript𝜓𝑁𝐸\psi^{\ast}E:=N\times_{\psi}Eitalic_ψ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_E := italic_N × start_POSTSUBSCRIPT italic_ψ end_POSTSUBSCRIPT italic_E over N𝑁Nitalic_N. In explicit terms

ψ∗⁢Esuperscript𝜓∗𝐸\psi^{\ast}Eitalic_ψ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_E over the open set ψ−1⁢(U)superscript𝜓1𝑈\psi^{-1}\left(U\right)italic_ψ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_U ). Then

∇ψsuperscript∇𝜓\nabla^{\psi}∇ start_POSTSUPERSCRIPT italic_ψ end_POSTSUPERSCRIPT over ψ∗⁢Esuperscript𝜓∗𝐸\psi^{\ast}Eitalic_ψ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_E satisfies the formula

ψ∗⁢Esuperscript𝜓∗𝐸\psi^{\ast}Eitalic_ψ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_E over ψ−1⁢(U)superscript𝜓1𝑈\psi^{-1}\left(U\right)italic_ψ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_U ). We compute the local

ψ∗⁢Esuperscript𝜓∗𝐸\displaystyle\psi^{\ast}Eitalic_ψ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_E

D

Remark: henceforth we will deal only with oriented elementary allowed paths, for this reason we will call them simply by allowed elementary paths, with certain abuse of language.

A disorientation of the allowed elementary (p+1)𝑝1(p+1)( italic_p + 1 )-paths of G𝐺Gitalic_G is a choice of orientation G+p+1subscriptsuperscript𝐺𝑝1G^{p+1}_{+}italic_G start_POSTSUPERSCRIPT italic_p + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT for these paths such that no pair of (p+1)𝑝1(p+1)( italic_p + 1 )-paths induces different orientations on any p𝑝pitalic_p-path.

We highlight that the notions of p𝑝pitalic_p-paths and of p𝑝pitalic_p-forms were introduced in grig without the notion of orientation. But such orientation will be essential to relate our Laplace operators to interesting Markov chains, as in its definition we have an alternating sum.

We close this paper with the consideration that we used the notion of up-adjacency to define our Markov chain. One may define a similar process using the notion of down-adjacency. But it is not trivial to relate this new walk to the spectrum of the Laplacian. If it can be done, one is one step closer to understand the relation of the irreducibility of this new Markov chain with the orientability of the digraph, in analogy to what is done in eidi2023irreducibility to simplicial complexes.

But, in higher orders, one can define to (p+1)𝑝1(p+1)( italic_p + 1 )-paths to be neighbors if they “intersect” in a p𝑝pitalic_p-path, as follow.

D

\mathord{\mbox{}\,\overline{\!y\!\!\;}\!\>}\mbox{}\rangle⟨ italic_x , italic_y ⟩ = ⟨ start_ID over¯ start_ARG italic_x end_ARG end_ID , start_ID over¯ start_ARG italic_y end_ARG end_ID ⟩.

For all x,y∈𝕆𝑥𝑦𝕆x,y\in\mathord{\mathbb{O}}italic_x , italic_y ∈ blackboard_O the following identities hold:

for all x,y∈S𝑥𝑦𝑆x,y\in Sitalic_x , italic_y ∈ italic_S and for all z∈𝕆𝑧𝕆z\in\mathord{\mathbb{O}}italic_z ∈ blackboard_O we have (x⁢y)⁢z=x⁢(y⁢z)𝑥𝑦𝑧𝑥𝑦𝑧(xy)z=x(yz)( italic_x italic_y ) italic_z = italic_x ( italic_y italic_z ).

For all x,y,z∈𝕆𝑥𝑦𝑧𝕆x,y,z\in\mathord{\mathbb{O}}italic_x , italic_y , italic_z ∈ blackboard_O, the following identities hold:

For all x,y,z∈𝕆𝑥𝑦𝑧𝕆x,y,z\in\mathord{\mathbb{O}}italic_x , italic_y , italic_z ∈ blackboard_O the following identities hold:

D

In [7], coarse proximity spaces were introduced to axiomatize the “at infinity” perspective of coarse geometry, providing general definitions of coarse neighborhoods (whose metric space specific definition was given by Dranishnikov in [3]), asymptotic disjointness, and closeness “at infinity.” Coarse proximity structures lie between metric spaces and coarse spaces (as defined by Roe in [12]) in a way similar to how proximity spaces relate to metric spaces and uniform spaces (see [8]). In [9], the authors construct a functor from the category of coarse proximity spaces to the category of compact Hausdorff spaces that assigns to each coarse proximity space a certain “boundary space.” This functor provides a common language for speaking of boundary spaces such as the Higson corona, the Gromov boundary, and other well-known boundary spaces. In this paper, we generalize the notion of asymptotic inductive dimension to all coarse proximity spaces (whose definition agrees with Dranishnokov’s definition for proper metric spaces) and investigate both of Dranishnikov’s questions in this more general context. In section 2, we review the necessary background information surrounding proximities as well as coarse proximities and their boundaries. In section 3, we define the asymptotic inductive dimension of coarse proximity spaces and show that it is an invariant within the category of coarse proximity spaces. We also show that the answer to Dranishnikov’s first question (“Does the asymptotic inductive dimension of the proper metric space coincide with the covering dimension of its Higson corona?”) generalized to this broader context is negative. In section 4, we describe two classes of completely traceable coarse proximity spaces in which the answer to the second of Dranishnokov’s questions (“Does the asymptotic inductive dimension of a proper metric space coincide with the large inductive dimension of its Higson corona?”) generalized to this broader context is positive. Specifically, these are locally compact Hausdorff spaces that admit metrizable compactification and spaces admitting compactifications whose boundaries are Z𝑍Zitalic_Z-sets. These classes include well-known boundaries such as the Gromov and visual boundaries, as well as the boundaries of the “coarse-compactification,” described in [5].

In [7], coarse proximity spaces were introduced to axiomatize the “at infinity” perspective of coarse geometry, providing general definitions of coarse neighborhoods (whose metric space specific definition was given by Dranishnikov in [3]), asymptotic disjointness, and closeness “at infinity.” Coarse proximity structures lie between metric spaces and coarse spaces (as defined by Roe in [12]) in a way similar to how proximity spaces relate to metric spaces and uniform spaces (see [8]). In [9], the authors construct a functor from the category of coarse proximity spaces to the category of compact Hausdorff spaces that assigns to each coarse proximity space a certain “boundary space.” This functor provides a common language for speaking of boundary spaces such as the Higson corona, the Gromov boundary, and other well-known boundary spaces. In this paper, we generalize the notion of asymptotic inductive dimension to all coarse proximity spaces (whose definition agrees with Dranishnokov’s definition for proper metric spaces) and investigate both of Dranishnikov’s questions in this more general context. In section 2, we review the necessary background information surrounding proximities as well as coarse proximities and their boundaries. In section 3, we define the asymptotic inductive dimension of coarse proximity spaces and show that it is an invariant within the category of coarse proximity spaces. We also show that the answer to Dranishnikov’s first question (“Does the asymptotic inductive dimension of the proper metric space coincide with the covering dimension of its Higson corona?”) generalized to this broader context is negative. In section 4, we describe two classes of completely traceable coarse proximity spaces in which the answer to the second of Dranishnokov’s questions (“Does the asymptotic inductive dimension of a proper metric space coincide with the large inductive dimension of its Higson corona?”) generalized to this broader context is positive. Specifically, these are locally compact Hausdorff spaces that admit metrizable compactification and spaces admitting compactifications whose boundaries are Z𝑍Zitalic_Z-sets. These classes include well-known boundaries such as the Gromov and visual boundaries, as well as the boundaries of the “coarse-compactification,” described in [5].

Every coarse proximity map between coarse proximity spaces induces a map between the boundaries of these coarse proximity spaces, as the following definition shows.

In this section, we provide the basic definitions and theorems surrounding proximity spaces and coarse proximity spaces. The definitions and theorems about proximity spaces come from [11], and the definitions and theorems about coarse proximity spaces come from [9].

The collection of coarse proximity spaces and closeness classes of coarse proximity maps makes up the category of coarse proximity spaces. For details, see [7].

C

(a), 9⁢μ9μ9\upmu9 roman_μs (b), 18⁢μ18μ18\upmu18 roman_μs (c), 45⁢μ45μ45\upmu45 roman_μs (d), 65⁢μ65μ65\upmu65 roman_μs

at 0⁢μ0μ0\upmu0 roman_μs (a), 9⁢μ9μ9\upmu9 roman_μs (b), 45⁢μ45μ45\upmu45 roman_μs (c), 65⁢μ65μ65\upmu65 roman_μs (d)

9⁢μ9μ9\upmu9 roman_μs (b), 18⁢μ18μ18\upmu18 roman_μs (c), 45⁢μ45μ45\upmu45 roman_μs (d), 65⁢μ65μ65\upmu65 roman_μs (e),

at 0⁢μ0μ0\upmu0 roman_μs (a), 9⁢μ9μ9\upmu9 roman_μs (b), 45⁢μ45μ45\upmu45 roman_μs (c), 65⁢μ65μ65\upmu65 roman_μs (d).

(a), 9⁢μ9μ9\upmu9 roman_μs (b), 18⁢μ18μ18\upmu18 roman_μs (c), 45⁢μ45μ45\upmu45 roman_μs (d), 65⁢μ65μ65\upmu65 roman_μs

B

_{\text{int},c}({\mathbb{C}}).over¯ start_ARG italic_π end_ARG start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT int , italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT : fraktur_G start_POSTSUBSCRIPT italic_R , over˙ start_ARG italic_R end_ARG , italic_σ end_POSTSUBSCRIPT ( blackboard_C ) → italic_G start_POSTSUBSCRIPT int , italic_c end_POSTSUBSCRIPT ( blackboard_C ) .

Proof. It immediately follows from [16, Lemma 3.2.32] and the explanation after its proof on Page 130. □□\Box□

By the above adjustment, the next proposition is a generalization of [16, Proposition 3.2.41]. Its proof is also similar to the proof

Proof. Let (A)𝐴(A)( italic_A ) denote the field of fractions of A𝐴Aitalic_A. The above relations hold

Proof. When R˙=An˙𝑅subscript𝐴𝑛\dot{R}=A_{n}over˙ start_ARG italic_R end_ARG = italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT (for n≥2𝑛2n\geq 2italic_n ≥ 2) this is [16, Proposition 3.3.1]. We now consider all other possible cases. Similar to the proof of [16, Proposition 3.3.1], it is clear

B

In particular, the ramification of L/K𝐿𝐾L/Kitalic_L / italic_K is non-logarithmically bounded by I𝐼Iitalic_I in our sense if and only if it is bounded by r𝑟ritalic_r in the sense of Abbes-Saito [AS02].

By Theorem 6.4 and by 2.9.2, the ramification of L/K𝐿𝐾L/Kitalic_L / italic_K is logarithmically bounded by I𝐼Iitalic_I if and only if

Gal(L/K)∗I={1}\operatorname{Gal}(L/K)_{*}^{I}=\{1\}roman_Gal ( italic_L / italic_K ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_I end_POSTSUPERSCRIPT = { 1 } if and only if the ramification of L/K𝐿𝐾L/Kitalic_L / italic_K is ∗*∗-bounded by I𝐼Iitalic_I.

In particular, the ramification of L/K𝐿𝐾L/Kitalic_L / italic_K is non-logarithmically bounded by I𝐼Iitalic_I in our sense if and only if it is bounded by r𝑟ritalic_r in the sense of Abbes-Saito [AS02].

By [Sa09, 1.2.6], this shows that the ramification of L/K𝐿𝐾L/Kitalic_L / italic_K is logarithmically bounded by I𝐼Iitalic_I in our sense if and only if it is logarithmically bounded by r𝑟ritalic_r in the sense of Abbes-Saito [AS02].

A

Explicitly, a set in g⁢(ℒr)𝑔subscriptℒ𝑟g(\mathcal{L}_{r})italic_g ( caligraphic_L start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) is a counter-example to a FD X⁢→⁡A𝑋→𝐴X\operatorname{\rightarrow}Aitalic_X → italic_A if it contains

counter-example in all realities where A⁢→⁡B𝐴→𝐵A\operatorname{\rightarrow}Bitalic_A → italic_B does not hold.

In particular, it satisfies C⁢→⁡A𝐶→𝐴C\operatorname{\rightarrow}Aitalic_C → italic_A and A⁢→⁡B𝐴→𝐵A\operatorname{\rightarrow}Bitalic_A → italic_B.

along with the abstract tuples which will be interpreted as counter-examples to A⁢→⁡B𝐴→𝐵A\operatorname{\rightarrow}Bitalic_A → italic_B or B⁢→⁡A𝐵→𝐴B\operatorname{\rightarrow}Aitalic_B → italic_A.

First, remark that both A⁢→⁡B𝐴→𝐵A\operatorname{\rightarrow}Bitalic_A → italic_B and B⁢→⁡A𝐵→𝐴B\operatorname{\rightarrow}Aitalic_B → italic_A are possible.

D

For edges e∈E∞𝑒subscript𝐸e\in E_{\infty}italic_e ∈ italic_E start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT, we define

ℐe={V0⊂V:the subgraph of (V,E0∪Eopen) induced by V0 makes e occupied}.subscriptℐ𝑒conditional-setsubscript𝑉0𝑉the subgraph of (V,E0∪Eopen) induced by V0 makes e occupied\mathcal{I}_{e}=\{V_{0}\subset V:\text{the subgraph of

G0subscript𝐺0G_{0}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and Gopensubscript𝐺openG_{\text{\rm open}}italic_G start_POSTSUBSCRIPT open end_POSTSUBSCRIPT with their subgraphs induced by Iesubscript𝐼𝑒I_{e}italic_I start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT, so that

induced by $V_{0}$ makes $e$ occupied}\}.caligraphic_I start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = { italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊂ italic_V : the subgraph of ( italic_V , italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∪ italic_E start_POSTSUBSCRIPT open end_POSTSUBSCRIPT ) induced by italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT makes italic_e occupied } .

an oriented subgraph G0=(V,E0)subscript𝐺0𝑉subscript𝐸0G_{0}=(V,E_{0})italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ( italic_V , italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). We then

A

Let ∥s∥2subscriptdelimited-∥∥𝑠2\lVert s\rVert_{2}∥ italic_s ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT be the input norm in [K23, Thm. 3.9] with respect to the Ohsawa measure d⁢V⁢[Ψ]Y𝑑𝑉subscriptdelimited-[]Ψ𝑌dV[\Psi]_{Y}italic_d italic_V [ roman_Ψ ] start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT. From Theorem 6.2, we have the following comparison of two L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT extension theorems from [K10] and [D15].

Let Y𝑌Yitalic_Y be a maximal lc center of (X,ψ)𝑋𝜓(X,\psi)( italic_X , italic_ψ ) with a unique lc place.

Now let Y𝑌Yitalic_Y be a maximal lc center of an lc pair (X,ψ)𝑋𝜓(X,\psi)( italic_X , italic_ψ ) as above. Let ΨΨ\Psiroman_Ψ be a quasi-psh function on X𝑋Xitalic_X with analytic singularities determined by the relation e−Ψ⁢h=e−ψsuperscript𝑒Ψℎsuperscript𝑒𝜓e^{-\Psi}h=e^{-\psi}italic_e start_POSTSUPERSCRIPT - roman_Ψ end_POSTSUPERSCRIPT italic_h = italic_e start_POSTSUPERSCRIPT - italic_ψ end_POSTSUPERSCRIPT where hℎhitalic_h is a smooth hermitian metric of L𝐿Litalic_L. In this setting, the Ohsawa measure d⁢V⁢[Ψ]Y𝑑𝑉subscriptdelimited-[]Ψ𝑌dV[\Psi]_{Y}italic_d italic_V [ roman_Ψ ] start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT is defined following the original definition of [O01], cf. [D15].

Let Y𝑌Yitalic_Y be a maximal lc center of (X,ψ)𝑋𝜓(X,\psi)( italic_X , italic_ψ ) with a unique lc place. If

A log canonical center (or an lc center) of (X,ψ)𝑋𝜓(X,\psi)( italic_X , italic_ψ ) is an irreducible subvariety Y⊂X𝑌𝑋Y\subset Xitalic_Y ⊂ italic_X that is the image of a prime divisor E𝐸Eitalic_E in a log resolution X′→X→superscript𝑋′𝑋X^{\prime}\to Xitalic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT → italic_X of the pair, with its discrepancy a⁢(E,X,ψ)𝑎𝐸𝑋𝜓a(E,X,\psi)italic_a ( italic_E , italic_X , italic_ψ ) equal to −11-1- 1 (i.e. the lowest possible value for an lc pair), cf. [Ko97, Def. 3.3], [K23, Def. 2.3]. We call such E𝐸Eitalic_E a log canonical place (or an lc place) of Y𝑌Yitalic_Y. When Y𝑌Yitalic_Y is an lc center and there is no other lc center Y1subscript𝑌1Y_{1}italic_Y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT such that Y1⊋Y𝑌subscript𝑌1Y_{1}\supsetneq Yitalic_Y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊋ italic_Y, we call Y𝑌Yitalic_Y a maximal lc center. The maximal lc centers are precisely the irreducible components of the non-klt locus of (X,ψ)𝑋𝜓(X,\psi)( italic_X , italic_ψ ). (We refer to [K23, Section 2] for more details on this setting.)

A

Define the operators999In fact we will prove stronger bounds where the low frequency inputs in (3.19) are replaced by vL1†,⋯⁢vL2⁢r†superscriptsubscript𝑣subscript𝐿1†⋯superscriptsubscript𝑣subscript𝐿2𝑟†v_{L_{1}}^{\dagger},\cdots v_{L_{2r}}^{\dagger}italic_v start_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT , ⋯ italic_v start_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT 2 italic_r end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT with max⁡(Lj)=Lsubscript𝐿𝑗𝐿\max(L_{j})=Lroman_max ( italic_L start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = italic_L, and similarly for (3.20). But for simplicity we will just write (3.19) and (3.20). (where 0≤l≤r0𝑙𝑟0\leq l\leq r0 ≤ italic_l ≤ italic_r)

𝒫−⁢(w)superscript𝒫𝑤\displaystyle\mathcal{P}^{-}(w)caligraphic_P start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_w )

Here the random averaging operator 𝒬=𝒫⁢(1−𝒫)−1𝒬𝒫superscript1𝒫1\mathcal{Q}=\mathcal{P}(1-\mathcal{P})^{-1}caligraphic_Q = caligraphic_P ( 1 - caligraphic_P ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, and 𝒫𝒫\mathcal{P}caligraphic_P has the form

\mathrm{HS}}).∥ caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT roman_OP end_POSTSUBSCRIPT ≤ ∥ caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT roman_OP end_POSTSUBSCRIPT ∥ caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT roman_OP end_POSTSUBSCRIPT , ∥ caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT roman_HS end_POSTSUBSCRIPT ≤ roman_min ( ∥ caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT roman_HS end_POSTSUBSCRIPT ∥ caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT roman_OP end_POSTSUBSCRIPT , ∥ caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT roman_OP end_POSTSUBSCRIPT ∥ caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT roman_HS end_POSTSUBSCRIPT ) .

𝒫+⁢(w)superscript𝒫𝑤\displaystyle\mathcal{P}^{+}(w)caligraphic_P start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_w )

D

We provide some applications of this in Section 4 before proving our main theorem in Section 5. The paper proper concludes with an examination of other circumstances under which domination implies isolation in Section 6.

This paper has an extensive Appendix that records a number of definitions and facts from basic geometric stability theory.

The results of this paper are concerned with non-locally modular regular types over models. It is natural to ask for other circumstances under which leaves are constructible. This is always possible if T𝑇Titalic_T is ω𝜔\omegaitalic_ω-stable, as such a theory has constructible models over any set.

We are grateful to the anonymous referee, as his/her thorough reading of this paper has led to many expositional improvements.

a witness to non-modularity over M′superscript𝑀′M^{\prime}italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. To see this, note that a⁢b⁢c⁢d⁢

C

After dropping nodes in 𝒱−superscript𝒱\mathcal{V}^{-}caligraphic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT and all their incident edges, the resulting graph is likely to be disconnected.

Therefore, we use a link construction procedure to obtain a connected graph supported by the nodes in 𝒱+superscript𝒱\mathcal{V}^{+}caligraphic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT.

By means of the rounding procedure in (5), the nodes in 𝒱𝒱\mathcal{V}caligraphic_V are partitioned in two sets, 𝒱+superscript𝒱\mathcal{V}^{+}caligraphic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT and 𝒱−=𝒱∖𝒱+superscript𝒱𝒱superscript𝒱\mathcal{V}^{-}=\mathcal{V}\setminus\mathcal{V}^{+}caligraphic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT = caligraphic_V ∖ caligraphic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, such that

The cost of inverting 𝐋𝒱−,𝒱−subscript𝐋superscript𝒱superscript𝒱{\mathbf{L}}_{\mathcal{V}^{-},\mathcal{V}^{-}}bold_L start_POSTSUBSCRIPT caligraphic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT , caligraphic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is 𝒪⁢(|𝒱−|3)𝒪superscriptsuperscript𝒱3\mathcal{O}(|\mathcal{V}^{-}|^{3})caligraphic_O ( | caligraphic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT | start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ), where |𝒱−|superscript𝒱|\mathcal{V}^{-}|| caligraphic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT | is the number of nodes that are dropped.

where 𝐋𝒱+,𝒱−subscript𝐋superscript𝒱superscript𝒱{\mathbf{L}}_{\mathcal{V}^{+},\mathcal{V}^{-}}bold_L start_POSTSUBSCRIPT caligraphic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT , caligraphic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_POSTSUBSCRIPT identifies a sub-matrix of 𝐋𝐋{\mathbf{L}}bold_L with rows (columns) corresponding to the nodes in 𝒱+superscript𝒱\mathcal{V}^{+}caligraphic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT (𝒱−superscript𝒱\mathcal{V}^{-}caligraphic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT).

A

By Lemma 6.3, s−a†≻1succeeds𝑠superscript𝑎†1s-a^{\dagger}\succ 1italic_s - italic_a start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ≻ 1 for all a∈L×𝑎superscript𝐿a\in L^{\times}italic_a ∈ italic_L start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT.

Giving L⁢(f)𝐿𝑓L(f)italic_L ( italic_f ) the valuation and ordering from Lemma 6.4 makes it a pre-H𝐻Hitalic_H-field extension of L𝐿Litalic_L with gap 00 of type (v)().

Then giving L⁢(f)𝐿𝑓L(f)italic_L ( italic_f ) the ordering and valuation from Lemma 6.5 makes it a pre-H𝐻Hitalic_H-field extension of L𝐿Litalic_L with gap 00 of type (vi)().

Let K𝐾Kitalic_K be a pre-H𝐻Hitalic_H-field and L𝐿Litalic_L be a pre-H𝐻Hitalic_H-field extension of K𝐾Kitalic_K with gap 00.

If K𝐾Kitalic_K is a pre-H𝐻Hitalic_H-field with gap 00 and L𝐿Litalic_L is moreover a differential field extension of K𝐾Kitalic_K with small derivation, then L𝐿Litalic_L is also a pre-H𝐻Hitalic_H-field with gap 00.

B

The associators in 𝒞𝒞\mathcal{C}caligraphic_C will descend to morphisms in 𝒞←←𝒞\underleftarrow{\mathcal{C}}under← start_ARG caligraphic_C end_ARG and still satisfy the pentagon equations. We have to convince ourselves that these morphisms define a natural isomorphism, with respect to the extra morphisms in the enriched hom-objects 𝒞←⁢(c,c′)←𝒞𝑐superscript𝑐′\underleftarrow{\mathcal{C}}(c,c^{\prime})under← start_ARG caligraphic_C end_ARG ( italic_c , italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) for c,c′∈𝒞𝑐superscript𝑐′𝒞c,c^{\prime}\in\mathcal{C}italic_c , italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_C. Looking at Definition 11, all these extra morphisms are just morphisms a⁢c→c′→𝑎𝑐superscript𝑐′ac\rightarrow c^{\prime}italic_a italic_c → italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT for some a∈𝒜𝑎𝒜a\in\mathcal{A}italic_a ∈ caligraphic_A. Using the pentagon equations on these morphisms, this means the associators from 𝒞𝒞\mathcal{C}caligraphic_C will also be natural for these extra morphisms.

Since we made a choice to use β𝛽\betaitalic_β rather than β−1superscript𝛽1\beta^{-1}italic_β start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT in Definition 17, we also have:

In defining the 2-category 𝐁𝐓𝐂⁢(𝒜)𝐁𝐓𝐂𝒜\mathbf{BTC}(\mathcal{A})bold_BTC ( caligraphic_A ) of braided tensor categories containing 𝒜𝒜\mathcal{A}caligraphic_A, there are several choices to be made, we use the following definition:

If β−2superscript𝛽2\beta^{-2}italic_β start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT is indeed a functor, then it is clearly invertible with inverse given by using the opposite crossings in Equation (23). So we need to check that β−2superscript𝛽2\beta^{-2}italic_β start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT preserves composition. That is, we need to check that the following diagram commutes:

the composite of the functor β−2superscript𝛽2\beta^{-2}italic_β start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT (Definition 21) with the symmetry in 𝒜𝒜\mathcal{A}caligraphic_A.

A

Let α=1+κ′𝛼1superscript𝜅′\alpha=1+\kappa^{\prime}italic_α = 1 + italic_κ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT for 0<κ′≪1,0superscript𝜅′much-less-than10<\kappa^{\prime}\ll 1,0 < italic_κ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≪ 1 ,then

converges in the 𝒳αsuperscript𝒳𝛼\mathcal{X}^{\alpha}caligraphic_X start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT topology, see e.g.

𝒳αsuperscript𝒳𝛼\mathcal{X}^{\alpha}caligraphic_X start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT norm of ΞΞ\Xiroman_Ξ (see Definition 3.5) it was

𝒳αsuperscript𝒳𝛼\mathcal{X}^{\alpha}caligraphic_X start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT in probability.

𝒳αsuperscript𝒳𝛼\displaystyle\mathcal{X}^{\alpha}caligraphic_X start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT

D

{gr}}}}_{2^{s}}))/2= ( italic_E start_POSTSUBSCRIPT 1 , 2 start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT end_POSTSUBSCRIPT + italic_e start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( start_OPFUNCTION over¯ start_ARG italic_gr end_ARG end_OPFUNCTION start_POSTSUBSCRIPT 2 start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ) / 2

≤v<1⁢(𝑔𝑟s)⁢ by (18)absentsubscript𝑣absent1subscript𝑔𝑟𝑠 by (18)\displaystyle\leq v_{<1}(\operatorname{\mathit{gr}}_{s})\text{ by \eqref{shade%

≤v<1⁢(𝑔𝑟3⋅2s)⁢ by (18)absentsubscript𝑣absent1subscript𝑔𝑟⋅3superscript2𝑠 by (18)\displaystyle\leq v_{<1}(\operatorname{\mathit{gr}}_{3\cdot 2^{s}})\text{ by %

≤(v<1⁢(𝑔𝑟2s)+eh⁢(𝑔𝑟¯2s))/2⁢ using (18).absentsubscript𝑣absent1subscript𝑔𝑟superscript2𝑠subscript𝑒ℎsubscript¯𝑔𝑟superscript2𝑠2 using (18)\displaystyle\leq(v_{<1}(\operatorname{\mathit{gr}}_{2^{s}})+e_{h}(%

=(E1,2s+eh⁢(𝑔𝑟¯2s))/2absentsubscript𝐸1superscript2𝑠subscript𝑒ℎsubscript¯𝑔𝑟superscript2𝑠2\displaystyle=(E_{1,2^{s}}+e_{h}(\operatorname{\overline{\operatorname{\mathit%

C

A linear parametrization using the Bott-Samelson coordinates for the finite dimensional Schubert cells (Proposition 4.2).

We obtain the defining equations for Kazhdan-Lusztig varieties by pulling back equations from (A) to coordinates determined by (B). Following the strategies developed in [WY12], we prove that these equations form a Gröbner basis of the Kazhdan-Lusztig ideals by an inductive argument on subword complexes. Our proof relies on a result of Knutson, stated in Theorem 5.2, that degenerate Kazhdan-Lusztig ideals expressed in Bott-Samelson coordinates to Stanley-Reisner ideals for subword complexes.

There is extensive literature on the geometry of Kazhdan-Lusztig varieties in the type A𝐴Aitalic_A setting. In [LY12b], Li and Yong studied Hilbert–Samuel multiplicity for points of Schubert varieties in the complete flag variety by Gröbner degeneration of Kazhdan–Lusztig ideals, and gave an explicit combinatorial interpretation in terms of subword complexes in the (co)vexillary case. This result was generalized by Anderson–Ikeda–Jeon–Kawago to classical types [AIJK23]. The potential relationship between the hℎhitalic_h-polynomials for Kazhdan-Lusztig varieties and the celebrated Kazhdan-Lusztig polynomials was explored in [LY12a]. Outside of type A𝐴Aitalic_A, the Gröbner geometry of Kazhdan–Lusztig varieties in type C𝐶Citalic_C has been studied by Escobar–Fink–Rajchgot–Woo [EFRW24]. We hope our work will provide useful tools for similar investigations in the affine type A𝐴Aitalic_A case.

4.2. Equations for Kazhdan-Lusztig varieties in F⁢l0⁢(V)𝐹subscript𝑙0𝑉Fl_{0}(V)italic_F italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_V )

In [WY12], Woo and Yong gave a Gröbner basis for Kazhdan-Lusztig ideals in the type A𝐴Aitalic_A flag variety, generalizing the Gröbner basis theorem [KM05, Theorem B] for Schubert determinantal ideals. In this paper, we present a Gröbner basis for Kazhdan-Lusztig ideals in the affine type A𝐴Aitalic_A flag variety, generalizing the main result of [WY12].

A

Since our identification will use a certain set of (framed) wild harmonic bundles, we remark that this set does not match any of the usual wild moduli spaces ℳHitsubscriptℳHit\mathcal{M}_{\text{Hit}}caligraphic_M start_POSTSUBSCRIPT Hit end_POSTSUBSCRIPT. In the usual story of moduli spaces of wild harmonic bundles over a punctured compact Riemann surface, one fixes the singular part of the Higgs field and the parabolic structure at the punctures. Under certain stability conditions, one obtains moduli spaces of these objects, with the natural hyperkähler metric [BB04]. On the other hand, in our set of wild harmonic bundles we will allow the simple pole of the Higgs field and the parabolic structure to vary. Furthermore, we will have the additional data of a “framing”. Hence our moduli space must a priori be different from the usual moduli spaces of wild harmonic bundles.

Since our identification will use a certain set of (framed) wild harmonic bundles, we remark that this set does not match any of the usual wild moduli spaces ℳHitsubscriptℳHit\mathcal{M}_{\text{Hit}}caligraphic_M start_POSTSUBSCRIPT Hit end_POSTSUBSCRIPT. In the usual story of moduli spaces of wild harmonic bundles over a punctured compact Riemann surface, one fixes the singular part of the Higgs field and the parabolic structure at the punctures. Under certain stability conditions, one obtains moduli spaces of these objects, with the natural hyperkähler metric [BB04]. On the other hand, in our set of wild harmonic bundles we will allow the simple pole of the Higgs field and the parabolic structure to vary. Furthermore, we will have the additional data of a “framing”. Hence our moduli space must a priori be different from the usual moduli spaces of wild harmonic bundles.

We hope that our interpretation of the Ooguri-Vafa space in terms of wild harmonic bundles serves as a first step to establish part of the conjectural picture of [GMN10, GMN13] and [Nei13, Section 7] mentioned above. For now, we leave the question of the specific relation between the Ooguri-Vafa metric and the hyperkähler metric of the Hitchin moduli spaces for future work. Independently of this problem, we also hope that our construction and methods can be generalized to produce hyperkähler structures for similar sets of wild harmonic bundles.

We remark that this proposed picture of the hyperkähler metric is very similar to the one given by Gross-Wilson for the hyperkähler metric of K3333 surfaces (see [GW00]). In the picture of [GW00], we have a generic elliptic fibration of a K3333 surface f:X→ℂ⁢P1:𝑓→𝑋ℂsuperscript𝑃1f:X\to\mathbb{C}P^{1}italic_f : italic_X → blackboard_C italic_P start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT with 24242424 singular nodal fibers; the hyperkähler metric of X𝑋Xitalic_X is then approximated by taking the semiflat metric away from the singular fibers, and gluing in the Ooguri-Vafa metric in a neighborhood of each singular fiber.

Motivated by these facts, our goal in this paper is to relate the Ooguri-Vafa space with the objects present in ℳHitsubscriptℳHit\mathcal{M}_{\text{Hit}}caligraphic_M start_POSTSUBSCRIPT Hit end_POSTSUBSCRIPT, namely harmonic bundles (E,∂¯E,θ,h)𝐸subscript¯𝐸𝜃ℎ(E,\overline{\partial}_{E},\theta,h)( italic_E , over¯ start_ARG ∂ end_ARG start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT , italic_θ , italic_h ). We will find an interpretation of the Ooguri-Vafa space in terms of a certain set of equivalence classes of framed wild harmonic bundles. Roughly speaking, the connection between the two is done as follows:

B

Applying Proposition 1 for the SBM X𝑋Xitalic_X and the function f:=fσassign𝑓subscript𝑓𝜎f:=f_{\sigma}italic_f := italic_f start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT and taking into account the latter equalities complete the proof.

We conclude the section with the lemma proving that the assumptions of Theorem 1 are stronger than the ones of Proposition 1, as stated in Remark 9.

Since the estimator proposed relies mostly on the behavior of the process around the threshold, it is slower than the ones based on quadratic variations and occupation times of the positive and negative part of the process proposed in [36] for OBM which exploit the entire trajectory.

The decomposition is introduced in Proposition 4, where it is stated that the discrete martingale satisfies the assumptions of Proposition 5 (a reformulation of a special case of Theorem 3.2 in [25]) which entails the limits in Theorem 1.

study of the number of crossings for a more general class of processes in Theorem 2, estimation of the skewness parameter of SBM in Theorem 3, and estimation of the parameters of OSBM in Proposition 3.

A

We study the sample efficiency of policy-based reinforcement learning in the episodic setting of linear MDPs with full-information feedback. We proposed an optimistic variant of the proximal policy optimization algorithm, dubbed as OPPO, which incorporates the principle of “optimism in the face of uncertainty” into policy optimization. When applied to the episodic MDP with unknown transition and adversarial reward, OPPO provably achieves a d2⁢H3⁢Tsuperscript𝑑2superscript𝐻3𝑇\sqrt{d^{2}H^{3}T}square-root start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_H start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_T end_ARG-regret up to logarithmic factors, which is near-optimal. To the best of our knowledge, OPPO is the first provably efficient policy optimization algorithm that explicitly incorporates exploration.

Theoretically, we establish the sample efficiency of OPPO in an episodic setting of Markov decision processes (MDPs) with full-information feedback, where the transition dynamics are linear in features (Yang and Wang, 2019b, a; Jin et al., 2019; Ayoub et al., 2020; Zhou et al., 2020). In particular, we allow the transition dynamics to be nonstationary within each episode. See also the work of Du et al. (2019a); Van Roy and Dong (2019); Lattimore and Szepesvari (2019) for a related discussion on the necessity of the linear representation. In detail, we prove that OPPO attains a d2⁢H3⁢Tsuperscript𝑑2superscript𝐻3𝑇\sqrt{d^{2}H^{3}T}square-root start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_H start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_T end_ARG-regret up to logarithmic factors, where d𝑑ditalic_d is the feature dimension, H𝐻Hitalic_H is the episode horizon, and T𝑇Titalic_T is the total number of steps taken by the agent. Note that such a regret does not depend on the numbers of states and actions, and therefore, allows them to be even infinite. In particular, OPPO attains such a regret without knowing the transition dynamics or accessing a “simulator”.

The authors would like to thank Lingxiao Wang, Wen Sun, and Sham Kakade for pointing out a technical issue in the first version regarding the covering number of value functions in the linear setting. This version has fixed the technical issue with a definition of the linear MDP different from the one in the first version. The authors would also like to thank Csaba Szepesvári, Lin F. Yang, Yining Wang, and Simon S. Du for helpful discussions. Zhaoran Wang acknowledges National Science Foundation (Awards 2048075, 2008827, 2015568, 1934931), Simons Institute (Theory of Reinforcement Learning), Amazon, J.P. Morgan, and Two Sigma for their supports.

We establish an upper bound of the regret of OPPO (Algorithm 1) in the following theorem. Recall that the regret is defined in (2.1) and T=H⁢K𝑇𝐻𝐾T=HKitalic_T = italic_H italic_K is the total number of steps taken by the agent, where H𝐻Hitalic_H is the length of each episode and K𝐾Kitalic_K is the total number of episodes. Also, |𝒜|𝒜|\mathcal{A}|| caligraphic_A | is the cardinality of 𝒜𝒜\mathcal{A}caligraphic_A and d𝑑ditalic_d is the dimension of the feature map ψ𝜓\psiitalic_ψ.

Although the variant of linear MDPs defined in Assumption 2.1 and the one studied by Yang and Wang (2019b); Jin et al. (2019) both cover the tabular setting and the one proposed by Yang and Wang (2019a) as special cases, they are two different definitions of linear MDPs as their feature maps ψ⁢(⋅,⋅,⋅)𝜓⋅⋅⋅\psi(\cdot,\cdot,\cdot)italic_ψ ( ⋅ , ⋅ , ⋅ ) and φ⁢(⋅,⋅)𝜑⋅⋅\varphi(\cdot,\cdot)italic_φ ( ⋅ , ⋅ ) are defined on different domains. It can be shown that the two variants are incomparable in the sense that one does not imply the other (Zhou et al., 2020).

B

The eigenvalues are (0,8,8,8,8,8,8)0888888(0,8,8,8,8,8,8)( 0 , 8 , 8 , 8 , 8 , 8 , 8 ) and K⁢(K8)=(ℤ/8⁢ℤ)6𝐾subscript𝐾8superscriptℤ8ℤ6K(K_{8})=(\mathbb{Z}/8\mathbb{Z})^{6}italic_K ( italic_K start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT ) = ( blackboard_Z / 8 blackboard_Z ) start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT.

Finally, we conclude with some remaining conjectures about the structure of K⁢(G)𝐾𝐺K(G)italic_K ( italic_G ) in Section 5.

In order to deal with the case p=2𝑝2p=2italic_p = 2, we adopt the approach of Benkart, Klivans, and Reiner [reinerquiver] that gives a natural ring structure on K⁢(G)𝐾𝐺K(G)italic_K ( italic_G ):

where K⁢(G)𝐾𝐺K(G)italic_K ( italic_G ) is a finite abelian group, known as the sandpile group (also critical group or Jacobian in the literature) of G𝐺Gitalic_G. Kirchhoff’s Matrix–Tree Theorem shows that the size of K⁢(G)𝐾𝐺K(G)italic_K ( italic_G ) is the number of spanning trees of G𝐺Gitalic_G. K⁢(G)𝐾𝐺K(G)italic_K ( italic_G ) is a subtle isomorphism invariant of a graph [SandpileBook, ChipBook], and is our main object of study. We are interested in understanding K⁢(G)𝐾𝐺K(G)italic_K ( italic_G ) for Cayley graphs of the group 𝔽2rsuperscriptsubscript𝔽2𝑟\mathbb{F}_{2}^{r}blackboard_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT, with an emphasis on their Sylow-2222 subgroups.

It is worth emphasizing that many of our arguments are based on the natural ring structure of K⁢(G)𝐾𝐺K(G)italic_K ( italic_G ) coming from representation theory [reinerquiver] and the corresponding polynomial algebra.

B

\frac{c_{2}}{L^{n}}.divide start_ARG italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_L start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG ≤ ∏ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ≤ divide start_ARG italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG italic_L start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG .

Next we estimate from below and above the measure and the cost of compression in the following sequel. We leverage the estimates established thus far to provide these estimates.

Next we present a second moment variant inequality of the unit sum of positive integers in the following statement.

In this section we lay down one striking and a stunning consequence of the entropy of compression. One could think of these applications as analogues of the Erdós type result for the unit sums of triples of the form (x1,x2,x3)subscript𝑥1subscript𝑥2subscript𝑥3(x_{1},x_{2},x_{3})( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ). We state two consequences of these estimates in the following sequel.

Next we present a second application of the estimates of the entropy of compression in the following sequel.

D

In this section, we give the definitions required to more precisely state the conjectural equivalence between Fano varieties up to deformation and rigid maximally mutable Laurent polynomials up to mutation (see, for example [1] and [15]).

For the definition of rigid maximally mutable Laurent polynomials, see Definition 2.3. A Fano variety X𝑋Xitalic_X is said to be mirror to f𝑓fitalic_f if the regularised quantum period of X𝑋Xitalic_X equals the classical period of f𝑓fitalic_f. The classical period of a Laurent polynomial (Definition 2.2) is mutation invariant.

On the A side (the Fano side) the main invariant is the quantum period. For a more detailed introduction to Gromov–Witten invariants, quantum cohomology, and the quantum period, see for example [7] (in particular, they record the formula for the quantum period of a Fano toric complete intersection).

As descendent invariants are deformation invariant, so is the quantum period. A closed form is known for smooth Fano toric complete intersections. The Abelian/non-Abelian correspondence [5, 26, 18] allows one to compute any number of terms of the quantum period of quiver flag zero loci.

The first 20 terms of the period sequence of the Laurent polynomial and the quantum period of the four dimensional Fano variety cut out of Gr⁡(6,3)Gr63\operatorname{\mathrm{Gr}}(6,3)roman_Gr ( 6 , 3 ) by a generic section of 𝒪⁢(1)⊕5𝒪superscript1direct-sum5\mathcal{O}(1)^{\oplus 5}caligraphic_O ( 1 ) start_POSTSUPERSCRIPT ⊕ 5 end_POSTSUPERSCRIPT agree. This is expected because under the Gelfand–Cetlin toric degeneration, this Fano subvariety degenerates to the complete intersection on the singular toric variety cut out by L1⊕5superscriptsubscript𝐿1direct-sum5L_{1}^{\oplus 5}italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊕ 5 end_POSTSUPERSCRIPT.

B

For the curvature equations in classical geometry, the existence of hypersurfaces with prescribed Weingarten curvature was studied by Pogorelov [40], Caffarelli-Nirenberg-Spruck [4, 5], Guan-Guan [18], Guan-Ma [19] and the later work by Sheng-Trudinger-Wang [42]. The Hessian equation on Riemannian manifolds was also studied by Y.Y. Li [29], Urbas [51] and Guan [17]. Hessian type equations also appear in conformal geometry, which started from Viaclovsky [53], Chang-Gursky-Yang [6]. In Kähler geometry, the Hessian equation was studied by Hou-Ma-Wu [20] and Dinew-Kolodziej [12].

The Yamabe problem with boundary is an important motivation for the study of the Neumann problems. The Yamabe problem on manifolds with boundary was first studied by Escobar, who shows in [13] that (almost) every compact Riemannian manifold (M,g)𝑀𝑔(M,g)( italic_M , italic_g ) is conformally equivalent to one of constant scalar curvature, whose boundary is minimal. The problem reduces to solving the semilinear elliptic critical Sobolev exponent equation with the Neumann boundary condition. It is naturally, the Neumann boundary value problem for Hessian type equations also appears in the fully nonlinear Yamabe problem for manifolds with boundary, which is to find a conformal metric such that the k𝑘kitalic_k-th elementary symmetric function of eigenvalues of Schouten tensor is constant and with the constant mean curvature on the boundary of manifold. See Jin-Li-Li [25], Chen [9] and Li-Luc [31], but in all these papers they need to impose the manifold are umbilic or total geodesic boundary for k≥2𝑘2k\geq 2italic_k ≥ 2, which are more like the condition in Trudinger [47] that the domain is ball.

In 1986, Lions-Trudinger-Urbas solved the Neumann problem of Monge-Ampère equations in the celebrated paper [35]. For related results on the Neumann or oblique derivative problem for some class of fully nonlinear elliptic equations can be found in Urbas [49]. Recently, the second author and G.H. Qiu [36] solved the the Neumann problem of k𝑘kitalic_k-Hessian equations, and then Chen-Zhang [8] generalized the above result to the the Neumann problem of Hessian quotient equations. Meanwhile, Jiang-Trudinger [23, 24] studied the general oblique boundary value problems for augmented Hessian equations with some regular condition and concavity condition. Motivated by the optimal transport Caffarelli [1] and Urbas [50] proved the existence of the Monge-Ampere equation with second boundary value problem, for the general convex cost function this second boundary value problem studied by Ma-Trudinger-Wang [37].

Meanwhile, the Neumann and oblique derivative problem of partial differential equations were widely studied. For a priori estimates and the existence theorem of Laplace equation with Neumann boundary condition, we refer to the book [15]. Also, we recommend the recent book written by Lieberman [33] for the Neumann and the oblique derivative problems of linear and quasilinear elliptic equations. Especially for the mean curvature equation with prescribed contact angle boundary value problem, Ural’tseva [52], Simon-Spruck [43] and Gerhardt [16] got the boundary gradient estimates and the corresponding existence theorem. Recently in [39], the second author and J.J. Xu got the boundary gradient estimates and the corresponding existence theorem for the Neumann boundary value problem on mean curvature equation.

These lemmas play an important role in the establishment of a priori estimates. Precisely, Lemma 2.5 is the key of the gradient estimates in Section 5, including the interior gradient estimate and the near boundary gradient estimate. Lemmas 2.4 and Lemma 2.6 are the keys of the lower and upper estimates of double normal second order derivatives on the boundary in Section 6, respectively.

C

})(b*\psi_{\delta t})](x)m(t)\frac{dt}{t}dx.∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( italic_ψ start_POSTSUBSCRIPT 3 , italic_t end_POSTSUBSCRIPT * italic_h ) ( italic_x ) [ ( italic_F ( ⋅ , italic_t ) * italic_φ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ( italic_b * italic_ψ start_POSTSUBSCRIPT italic_δ italic_t end_POSTSUBSCRIPT ) ] ( italic_x ) italic_m ( italic_t ) divide start_ARG italic_d italic_t end_ARG start_ARG italic_t end_ARG italic_d italic_x .

We next estimate the term B𝐵Bitalic_B: using the Cauchy-Schwartz inequality in x′superscript𝑥′x^{\prime}italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT we get that

convolutions together with the fact that for f∈L∞⁢(ℝ2)𝑓superscript𝐿superscriptℝ2f\in L^{\infty}(\mathbb{R}^{2})italic_f ∈ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )

Using the fact that m∈L∞𝑚superscript𝐿m\in L^{\infty}italic_m ∈ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT and the Cauchy-Schwarz inequality, this quantity can be

We note next that for x′∈[−H/2,H/2]superscript𝑥′𝐻2𝐻2x^{\prime}\in[-H/2,H/2]italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ [ - italic_H / 2 , italic_H / 2 ], by (16) and the fact that ‖θ‖L∞≤1subscriptnorm𝜃superscript𝐿1\|\theta\|_{L^{\infty}}\leq 1∥ italic_θ ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ≤ 1, one has

C

By [4, Theorem 7.4], we know that VRr⁢(𝕊1)subscriptVR𝑟superscript𝕊1\mathrm{VR}_{r}(\mathbb{S}^{1})roman_VR start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) is homotopy equivalent to 𝕊2⁢m+1superscript𝕊2𝑚1\mathbb{S}^{2m+1}blackboard_S start_POSTSUPERSCRIPT 2 italic_m + 1 end_POSTSUPERSCRIPT if r∈(2⁢π⁢m2⁢m+1,2⁢π⁢(m+1)2⁢m+3]𝑟2𝜋𝑚2𝑚12𝜋𝑚12𝑚3r\in\left(\frac{2\pi m}{2m+1},\frac{2\pi(m+1)}{2m+3}\right]italic_r ∈ ( divide start_ARG 2 italic_π italic_m end_ARG start_ARG 2 italic_m + 1 end_ARG , divide start_ARG 2 italic_π ( italic_m + 1 ) end_ARG start_ARG 2 italic_m + 3 end_ARG ] for m=0,1,⋯𝑚01⋯m=0,1,\cdotsitalic_m = 0 , 1 , ⋯. Observe that limm→∞2⁢π⁢m2⁢m+1=πsubscript→𝑚2𝜋𝑚2𝑚1𝜋\lim_{m\rightarrow\infty}\frac{2\pi m}{2m+1}=\piroman_lim start_POSTSUBSCRIPT italic_m → ∞ end_POSTSUBSCRIPT divide start_ARG 2 italic_π italic_m end_ARG start_ARG 2 italic_m + 1 end_ARG = italic_π. Therefore, VRπ⁢(𝕊1)subscriptVR𝜋superscript𝕊1\mathrm{VR}_{\pi}(\mathbb{S}^{1})roman_VR start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) cannot be homotopy equivalent to VRπ−ε⁢(𝕊1)subscriptVR𝜋𝜀superscript𝕊1\mathrm{VR}_{\pi-\varepsilon}(\mathbb{S}^{1})roman_VR start_POSTSUBSCRIPT italic_π - italic_ε end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) for all small enough ε𝜀\varepsilonitalic_ε, since for r𝑟ritalic_r in the interval [π−ε,π]𝜋𝜀𝜋[\pi-\varepsilon,\pi][ italic_π - italic_ε , italic_π ], VRr⁢(𝕊1)subscriptVR𝑟superscript𝕊1\mathrm{VR}_{r}(\mathbb{S}^{1})roman_VR start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) attains infinitely many different homotopy types.

Actually, we first expected the following conjecture to be true. Observe that, if true, the conjecture would imply Theorem 7. Also, it is obvious that this conjecture is true when X𝑋Xitalic_X is a finite metric space.

Then, one might now wonder whether the conjecture holds when we restrict the range of r𝑟ritalic_r to (0,diam⁢(X))0diam𝑋(0,\mathrm{diam}(X))( 0 , roman_diam ( italic_X ) ). But, again this new conjecture is false as the following example shows.

Since VRr⁢(X)subscriptVR𝑟𝑋\mathrm{VR}_{r}(X)roman_VR start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( italic_X ) is contractible for any r>diam⁢(X)𝑟diam𝑋r>\mathrm{diam}(X)italic_r > roman_diam ( italic_X ), it is clear that length⁢(I)≤diam⁢(X)length𝐼diam𝑋\mathrm{length}(I)\leq\mathrm{diam}(X)roman_length ( italic_I ) ≤ roman_diam ( italic_X ) in general. The corollary above improves this bound by a factor of 2323\frac{2}{3}divide start_ARG 2 end_ARG start_ARG 3 end_ARG when X𝑋Xitalic_X is geodesic.

After seeing the proof of Proposition 9.7, some readers might wonder whether one can prove a version of Hausmann’s theorem [50, Theorem 3.5] for compact ANR metric spaces. This leads to formulating the conjecture below.

B

Let u∈c1,α⁢(Ω,ℝ3)𝑢superscript𝑐1𝛼Ωsuperscriptℝ3u\in c^{1,\alpha}(\Omega,{\mathbb{R}}^{3})italic_u ∈ italic_c start_POSTSUPERSCRIPT 1 , italic_α end_POSTSUPERSCRIPT ( roman_Ω , blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) be an isometric immersion for 2/3≤α<123𝛼12/3\leq\alpha<12 / 3 ≤ italic_α < 1. In order to prove our main Theorem, and in view of Proposition 2.1, it is sufficient to show that ∇u∇𝑢\nabla u∇ italic_u satisfies condition (c). We first fix an open disk V𝑉Vitalic_V containing x𝑥xitalic_x and compactly contained in ΩΩ\Omegaroman_Ω and note the existence of the function v∈c1,α⁢(V)⊂c1,2/3⁢(V)𝑣superscript𝑐1𝛼𝑉superscript𝑐123𝑉v\in c^{1,\alpha}(V)\subset c^{1,2/3}(V)italic_v ∈ italic_c start_POSTSUPERSCRIPT 1 , italic_α end_POSTSUPERSCRIPT ( italic_V ) ⊂ italic_c start_POSTSUPERSCRIPT 1 , 2 / 3 end_POSTSUPERSCRIPT ( italic_V ) as defined in Proposition 4.5.

We then show the required regularity for v𝑣vitalic_v, and proceed to prove using the Gauss equation that v𝑣vitalic_v satisfies the very weak degenerate Monge-Ampère equation as required by the assumptions of Theorem 2. Finally, we need to prove that developability of v𝑣vitalic_v, as derived from Theorem 2, implies the developability of the isometric immersion u𝑢uitalic_u. Apart from Theorem 2, the 2/3232/32 / 3-Hölder exponent regularity is only required for proving that a weak version of Codazzi-Mainardi equations holds for isometric u∈c1,2/3𝑢superscript𝑐123u\in c^{1,2/3}italic_u ∈ italic_c start_POSTSUPERSCRIPT 1 , 2 / 3 end_POSTSUPERSCRIPT, i.e. when we need to show that A𝐴Aitalic_A is curl free.

Once again applying Lemma 2.11 and (5.4) implies that the Jacobian derivative ∇u∇𝑢\nabla u∇ italic_u of the isometric immersion u𝑢uitalic_u is constant along the segments generated by the vector field η→→𝜂\vec{\eta}over→ start_ARG italic_η end_ARG in B𝐵Bitalic_B. Therefore ∇u∇𝑢\nabla u∇ italic_u satisfies condition (c) of Proposition 2.1, which implies the developability of u|Vevaluated-at𝑢𝑉u|_{V}italic_u | start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT. As already mentioned, this part of the proof concludes the proof of Theorem 1 for u∈c1,α𝑢superscript𝑐1𝛼u\in c^{1,\alpha}italic_u ∈ italic_c start_POSTSUPERSCRIPT 1 , italic_α end_POSTSUPERSCRIPT, 2/3≤α<123𝛼12/3\leq\alpha<12 / 3 ≤ italic_α < 1.

The article is organized as follows: In Section 2, we will present and prove a few statements regarding the developability properties of C1superscript𝐶1C^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT mappings. In Section 3, we will gather a few analytical tools which deal with properties of Hölder continuous functions and with quadratic differential expressions in terms of functions of low regularity. In particular, our basic proposition 3.6 will allow us to define a second fundamental form for the immersions of Hölder type regularity. The subsequent section is dedicated to the definition and properties of this second fundamental form. In Section 5, building on the previous sections, we complete the proof of Theorem 1. Appendix I is dedicated to a side result (Proposition I.1) on the developability of each component of the immersion, which can be shown independently with a shorter proof. Finally, in Appendix II the proof of some standard facts regarding little Hölder spaces are presented.

We apply the key developability result Theorem 2 to v𝑣vitalic_v to obtain that ∇v∇𝑣\nabla v∇ italic_v satisfies any of the equivalent conditions of Proposition 2.1 in V𝑉Vitalic_V. The developability of u𝑢uitalic_u is a consequence of Corollory 2.2 and Proposition 5.1 below. The second conclusion follows from Corollary 3.5.

D

(b) and (c). Finally, to prove (f), from f𝑓fitalic_f we define f^=𝔦∘f∘π^𝑓𝔦𝑓𝜋\widehat{f}=\mathfrak{i}\circ f\circ\piover^ start_ARG italic_f end_ARG = fraktur_i ∘ italic_f ∘ italic_π and use

denote the principal piece of the cartesian product Xmsuperscript𝑋𝑚X^{m}italic_X start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT.

of the cartesian product Xmsuperscript𝑋𝑚X^{m}italic_X start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT.

Take X𝑋Xitalic_X a Hausdorff topological space, take m∈ℕ𝑚ℕm\in\mathbb{N}italic_m ∈ blackboard_N and Xmsuperscript𝑋𝑚X^{m}italic_X start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT endowed with the product topology, and

Using the partition of Xmsuperscript𝑋𝑚X^{m}italic_X start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT in big puzzles one may take care of the difference between the product topology

D

The size or length of a ladder is given by its number of rungs, i.e. the size of L𝐿Litalic_L is n𝑛nitalic_n.

With that end in view, from now on, a ladder is always supposed to be a subdivision of an elementary ladder with three rungs.

Let LAsubscript𝐿𝐴L_{A}italic_L start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT be an elementary ladder of length lAsubscript𝑙𝐴l_{A}italic_l start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and let LCsubscript𝐿𝐶L_{C}italic_L start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT be an elementary ladder of length lCsubscript𝑙𝐶l_{C}italic_l start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT.

In this section, we will prove that the elementary ladder with three rungs has the edge-Erdős-Pósa property.

A ladder is a subdivision of an elementary ladder. We adapt the notion of rungs, stringers and size for ladders from their counterparts in elementary ladders.

D

=s2+S2+𝐯2+𝐕2,absentsuperscript𝑠2superscript𝑆2superscript𝐯2superscript𝐕2\displaystyle=s^{2}+S^{2}+\mathbf{v}^{2}+\mathbf{V}^{2},= italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + bold_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + bold_V start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,

=2⁢(s⁢S+𝐯⋅𝐕),absent2𝑠𝑆⋅𝐯𝐕\displaystyle=2(sS+\mathbf{v}\cdot\mathbf{V}),= 2 ( italic_s italic_S + bold_v ⋅ bold_V ) ,

𝖠=s+𝐯+(S+𝐕)⁢I,𝖠𝑠𝐯𝑆𝐕𝐼\mathsf{A}=s+\mathbf{v}+(S+\mathbf{V})I,sansserif_A = italic_s + bold_v + ( italic_S + bold_V ) italic_I ,

=2⁢(s⁢S+𝐯⋅𝐕),absent2𝑠𝑆⋅𝐯𝐕\displaystyle=2(sS+\mathbf{v}\cdot\mathbf{V}),= 2 ( italic_s italic_S + bold_v ⋅ bold_V ) ,

=2⁢(s⁢S+𝐯⋅𝐕),absent2𝑠𝑆⋅𝐯𝐕\displaystyle=2(sS+\mathbf{v}\cdot\mathbf{V}),= 2 ( italic_s italic_S + bold_v ⋅ bold_V ) ,

A

{\star}}_{m},P^{k,1}_{m},P^{k,2}_{m}\big{)}\Big{]}.roman_inf start_POSTSUBSCRIPT italic_ϕ ∈ italic_U start_POSTSUPERSCRIPT italic_k , italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_E [ italic_ϱ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_Y start_POSTSUPERSCRIPT italic_k , italic_ϕ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_P start_POSTSUPERSCRIPT italic_k , 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_P start_POSTSUPERSCRIPT italic_k , 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ] = blackboard_E [ italic_ϱ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_Y start_POSTSUPERSCRIPT italic_k , italic_u start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_P start_POSTSUPERSCRIPT italic_k , 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_P start_POSTSUPERSCRIPT italic_k , 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ] .

Figure 1: Monte Carlo experiments for ck,⋆superscript𝑐𝑘⋆c^{k,\star}italic_c start_POSTSUPERSCRIPT italic_k , ⋆ end_POSTSUPERSCRIPT with ϵk=2−ksubscriptitalic-ϵ𝑘superscript2𝑘\epsilon_{k}=2^{-k}italic_ϵ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = 2 start_POSTSUPERSCRIPT - italic_k end_POSTSUPERSCRIPT

k,2}_{m}\right)\right]=0blackboard_E [ italic_ϱ start_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT italic_k , ⋆ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Y start_POSTSUPERSCRIPT italic_k , italic_u start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_P start_POSTSUPERSCRIPT italic_k , 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_P start_POSTSUPERSCRIPT italic_k , 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ] = 0. Table 1 presents a comparison between the true call option price c⋆superscript𝑐⋆c^{\star}italic_c start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT and the associated Monte Carlo price ck,⋆superscript𝑐𝑘⋆c^{k,\star}italic_c start_POSTSUPERSCRIPT italic_k , ⋆ end_POSTSUPERSCRIPT. Figure 1 presents the Monte Carlo experiments for ck,⋆superscript𝑐𝑘⋆c^{k,\star}italic_c start_POSTSUPERSCRIPT italic_k , ⋆ end_POSTSUPERSCRIPT with k=1,2𝑘12k=1,2italic_k = 1 , 2 and 3333. The number of Monte Carlo iterations in the experiment is 3×1043superscript1043\times 10^{4}3 × 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT.

The estimated value ck,⋆superscript𝑐𝑘⋆c^{k,\star}italic_c start_POSTSUPERSCRIPT italic_k , ⋆ end_POSTSUPERSCRIPT is computed according to

Table 1: Comparison between c⋆superscript𝑐⋆c^{\star}italic_c start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT and ck,⋆superscript𝑐𝑘⋆c^{k,\star}italic_c start_POSTSUPERSCRIPT italic_k , ⋆ end_POSTSUPERSCRIPT for ϵk=2−ksubscriptitalic-ϵ𝑘superscript2𝑘\epsilon_{k}=2^{-k}italic_ϵ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = 2 start_POSTSUPERSCRIPT - italic_k end_POSTSUPERSCRIPT

C

If a semigroup S𝑆Sitalic_S is not residually finite and embeds into a semigroup T𝑇Titalic_T, then T𝑇Titalic_T cannot be residually finite either.

While our main result significantly relaxes the hypothesis for showing that the free product of self-similar semigroups (or automaton semigroups) is self-similar (an automaton semigroup), it does not settle the underlying question whether these semigroup classes are closed under free product. It is possible that there is a different construction for the free product S⋆T⋆𝑆𝑇S\star Titalic_S ⋆ italic_T of two self-similar or automaton semigroup without the requirement of a homomorphism from one to the other and it is also possible that there is a pair of self-similar (or automaton) semigroups such that S⋆T⋆𝑆𝑇S\star Titalic_S ⋆ italic_T is not a self-similar (or an automaton semigroup). In this case, however, no homomorphism S→T→𝑆𝑇S\to Titalic_S → italic_T or T→S→𝑇𝑆T\to Sitalic_T → italic_S can exist. Thus, to make progress in either direction (towards a better construction or towards a counter-example), we need to look at pairs S,T𝑆𝑇S,Titalic_S , italic_T of self-similar (or even automaton) semigroups without a homomorphism from one to the other. However, it turns out that finding such a pair is not easy. In particular, neither S𝑆Sitalic_S nor T𝑇Titalic_T may contain an idempotent. Thus, we have to consider idempotent-free semigroups here. We will show, however, that we cannot find a pair of such semigroups in the class of finitely generated simple semigroups. More precisely, using results by Jones on idempotent-free semigroups [11], we show that finitely generated simple (or 00-simple) idempotent-free semigroups are not residually finite (Theorem 21) and, thus, not self-similar (and, in particular, not automaton semigroups; 22). We then conclude the paper with an example222The authors would like to thank Emanuele Rodaro for his help in finding this example. of a finitely generated residually finite semigroup (23) which has no homomorphism to its opposite semigroup (25). While this comes close to the sought pair S,T𝑆𝑇S,Titalic_S , italic_T, it is not clear whether the given semigroup is self-similar (26).

If the generating automaton is additionally complete, we speak of a completely self-similar semigroup or of a complete automaton semigroup.

Every complete automaton semigroup is residually finite [4, Proposition 3.2] and the argument can easily be extended to general self-similar semigroups.

The construction used to prove Theorem 6 can also be used to obtain results which are not immediate corollaries of the theorem (or its corollary for automaton semigroups in 8). As an example, we prove in the following theorem that it is possible to adjoin a free generator to every self-similar semigroup without losing the self-similarity property and that the analogous statement for automaton semigroups holds as well. The version for automaton semigroups does not follow directly from 8, as the free monogenic semigroup is not a complete automaton semigroup [4, Proposition 4.3] or even a (partial) automaton semigroup (see [8, Theorem 18] or [20, Theorem 1.2.1.4]).

C

φ~⁢(a)<a~𝜑𝑎𝑎\tilde{\varphi}(a)<aover~ start_ARG italic_φ end_ARG ( italic_a ) < italic_a for all a>1𝑎1a>1italic_a > 1.

It is important to note that this inequality does not hold in general. As it is informed by the spanning method, it only holds for functions that are right continuous and of bounded variation on intervals of the form [x,x+1)𝑥𝑥1[x,x+1)[ italic_x , italic_x + 1 ) for x≥1𝑥1x\geq 1italic_x ≥ 1 with x∈ℕ𝑥ℕx\in\mathbb{N}italic_x ∈ blackboard_N and additionally have left limits, generally known as cadlag functions. Indeed the challenge of approaching the Lehmer totient problem using the spanning method is to construct an appropriate cadlag function for the Euler totient function. The next section studies an extension of the Euler totient function.

We now state an analytic property of the fractional totient invariant function. In fact, the fractional totient invariant function can be seen as a slightly continuous analogue of the Euler totient function on subsets of the reals.

By the fractional Euler totient invariant function, we mean the function φ~:[1,∞)⟶ℝ:~𝜑⟶1ℝ\tilde{\varphi}:[1,\infty)\longrightarrow\mathbb{R}over~ start_ARG italic_φ end_ARG : [ 1 , ∞ ) ⟶ blackboard_R such that

The fractional Euler totient invariant function turns out to be an interesting function that in some way extends the Euler totient function to the reals. Even though the notion of co-primality in not well-defined on the entire real line, it captures the intrinsic property of the Euler totient function defined on the positive integers. In essence, the Euler totient function and the fractional totient invariant function coincides on the set of positive integers. Next, we examine some elementary properties of the fractional Euler totient invariant function in the following sequel.

B

\varepsilon}))=0.roman_lim start_POSTSUBSCRIPT italic_ε → 0 end_POSTSUBSCRIPT italic_d ( ( bold_0 , 0 ) , caligraphic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( caligraphic_K start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ) ) = 0 .

is closed by upper semicontinuity of density and Lemma 10.5. Thus, it suffices to prove that ℬℬ\mathcal{B}caligraphic_B is countable. Define

Everything but the last claim is proven above (in the rescaled setting). The last claim follows from the fact that all limit flows are convex so [HW20] applies.

The first claim follows immediately from Lemma 7.15. To prove the second claim, it suffices (by Lemma 7.5) to show that

Claim (1) follows by construction. Claim (2) follows from the fact101010Note that the simpler statement λ⁢(Σε)≤F⁢(Σ)+o⁢(1)𝜆subscriptΣ𝜀𝐹Σ𝑜1\lambda(\Sigma_{\varepsilon})\leq F(\Sigma)+o(1)italic_λ ( roman_Σ start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ) ≤ italic_F ( roman_Σ ) + italic_o ( 1 ) as ε→0→𝜀0\varepsilon\to 0italic_ε → 0 would suffice here. that λ⁢(Σε)≤F⁢(Σ)𝜆subscriptΣ𝜀𝐹Σ\lambda(\Sigma_{\varepsilon})\leq F(\Sigma)italic_λ ( roman_Σ start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ) ≤ italic_F ( roman_Σ ) proven in [BW17, Appendix C]. The claim (3) follows as in Lemma 7.5. We prove (4) below, but for now, we note that Corollary 7.14 immediately implies that ℱλ⁢(𝒦)⊂𝒦subscriptℱ𝜆𝒦𝒦\mathcal{F}_{\lambda}(\mathcal{K})\subset\mathcal{K}caligraphic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( caligraphic_K ) ⊂ caligraphic_K for λ∈(1,λ0)𝜆1subscript𝜆0\lambda\in(1,\lambda_{0})italic_λ ∈ ( 1 , italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). We will refer to this weaker property as (4’).

C

(α,β,γ,δ)=(2,1,2,2)𝛼𝛽𝛾𝛿2122(\alpha,\beta,\gamma,\delta)=(2,1,2,2)( italic_α , italic_β , italic_γ , italic_δ ) = ( 2 , 1 , 2 , 2 ):

Relevant indices: {i,j,n,ℓ}𝑖𝑗𝑛ℓ\{i,j,n,\ell\}{ italic_i , italic_j , italic_n , roman_ℓ }. Cases to consider:

Relevant indices: {i,j,k,n,ℓ}𝑖𝑗𝑘𝑛ℓ\{i,j,k,n,\ell\}{ italic_i , italic_j , italic_k , italic_n , roman_ℓ }. Since all the indices are relevant, we only have to consider cases where

#⁢{i,j,k,n,ℓ}⩾4#𝑖𝑗𝑘𝑛ℓ4\#\{i,j,k,n,\ell\}\geqslant 4# { italic_i , italic_j , italic_k , italic_n , roman_ℓ } ⩾ 4. The cases left to consider are

Relevant indices: {i,k,n,ℓ}𝑖𝑘𝑛ℓ\{i,k,n,\ell\}{ italic_i , italic_k , italic_n , roman_ℓ }. Cases to consider:

A

D1=ui⁢(i≥3)subscript𝐷1subscript𝑢𝑖𝑖3D_{1}=u_{i}(i\geq 3)italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_i ≥ 3 ), S1=v2subscript𝑆1subscript𝑣2S_{1}=v_{2}italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, and D3=ui⁢(1≤i≤m)subscript𝐷3subscript𝑢𝑖1𝑖𝑚D_{3}=u_{i}(1\leq i\leq m)italic_D start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( 1 ≤ italic_i ≤ italic_m ), in an MBD game on 𝔛msubscript𝔛𝑚\mathfrak{X}_{m}fraktur_X start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT.

Hence, Staller has a strategy such that Dominator needs at least m−1𝑚1m-1italic_m - 1 steps in order to win in an

the right. By induction, Dominator needs at least k−7𝑘7k-7italic_k - 7 steps to win. In total, he needs at least

Dominator needs at least m−5𝑚5m-5italic_m - 5 more steps to win. In total, he needs at least (m−5)+3=m−2𝑚53𝑚2(m-5)+3=m-2( italic_m - 5 ) + 3 = italic_m - 2 steps to win.

At the end of each branch, we explain at least how many steps Dominator would need in order to win the game. It covers

D

Given the initial scattering diagram 𝔇i⁢nsubscript𝔇𝑖𝑛\mathfrak{D}_{in}fraktur_D start_POSTSUBSCRIPT italic_i italic_n end_POSTSUBSCRIPT, there exists a unique complete scattering diagram 𝔇𝔇\mathfrak{D}fraktur_D such that θ𝔇,u=i⁢dsubscript𝜃𝔇𝑢𝑖𝑑\theta_{\mathfrak{D},u}=iditalic_θ start_POSTSUBSCRIPT fraktur_D , italic_u end_POSTSUBSCRIPT = italic_i italic_d for every u∈B0𝑢subscript𝐵0u\in B_{0}italic_u ∈ italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and l𝔡≠l𝔡′subscript𝑙𝔡subscript𝑙superscript𝔡′l_{\mathfrak{d}}\neq l_{\mathfrak{d}^{\prime}}italic_l start_POSTSUBSCRIPT fraktur_d end_POSTSUBSCRIPT ≠ italic_l start_POSTSUBSCRIPT fraktur_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT if 𝔡≠𝔡′𝔡superscript𝔡′\mathfrak{d}\neq\mathfrak{d}^{\prime}fraktur_d ≠ fraktur_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT.

Given a scattering diagram 𝔇𝔇\mathfrak{D}fraktur_D on B𝐵Bitalic_B, u∈B0𝑢subscript𝐵0u\in B_{0}italic_u ∈ italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and λ>0𝜆0\lambda>0italic_λ > 0. For a generic counterclockwise

A scattering diagram 𝔇𝔇\mathfrak{D}fraktur_D on B𝐵Bitalic_B is a set of 3333-tuples {(l𝔡,γ𝔡,f𝔡)}𝔡∈Isubscriptsubscript𝑙𝔡subscript𝛾𝔡subscript𝑓𝔡𝔡𝐼\{(l_{\mathfrak{d}},\gamma_{\mathfrak{d}},f_{\mathfrak{d}})\}_{\mathfrak{d}\in

The complete scattering diagram 𝔇𝔇\mathfrak{D}fraktur_D is closely related to tropical discs on B𝐵Bitalic_B.

Given the initial scattering diagram 𝔇i⁢nsubscript𝔇𝑖𝑛\mathfrak{D}_{in}fraktur_D start_POSTSUBSCRIPT italic_i italic_n end_POSTSUBSCRIPT, there exists a unique complete scattering diagram 𝔇𝔇\mathfrak{D}fraktur_D such that θ𝔇,u=i⁢dsubscript𝜃𝔇𝑢𝑖𝑑\theta_{\mathfrak{D},u}=iditalic_θ start_POSTSUBSCRIPT fraktur_D , italic_u end_POSTSUBSCRIPT = italic_i italic_d for every u∈B0𝑢subscript𝐵0u\in B_{0}italic_u ∈ italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and l𝔡≠l𝔡′subscript𝑙𝔡subscript𝑙superscript𝔡′l_{\mathfrak{d}}\neq l_{\mathfrak{d}^{\prime}}italic_l start_POSTSUBSCRIPT fraktur_d end_POSTSUBSCRIPT ≠ italic_l start_POSTSUBSCRIPT fraktur_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT if 𝔡≠𝔡′𝔡superscript𝔡′\mathfrak{d}\neq\mathfrak{d}^{\prime}fraktur_d ≠ fraktur_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT.

C

V𝑉Vitalic_V homothetic to an imaginary quadratic field different from ℚ⁢(−1)ℚ1\mathbb{Q}(\sqrt{-1})blackboard_Q ( square-root start_ARG - 1 end_ARG ) or ℚ⁢(−3)ℚ3\mathbb{Q}(\sqrt{-3})blackboard_Q ( square-root start_ARG - 3 end_ARG ).

V𝑉Vitalic_V homothetic to ℚ⁢(−3)ℚ3\mathbb{Q}(\sqrt{-3})blackboard_Q ( square-root start_ARG - 3 end_ARG ).

V𝑉Vitalic_V homothetic to ℚ⁢(−1)ℚ1\mathbb{Q}(\sqrt{-1})blackboard_Q ( square-root start_ARG - 1 end_ARG ).

}2\sqrt{3}≤ | italic_w | 2 square-root start_ARG 3 end_ARG ≤ ( square-root start_ARG 2 end_ARG italic_p start_POSTSUPERSCRIPT 2 italic_m - 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 1 / 3 end_POSTSUPERSCRIPT 2 square-root start_ARG 3 end_ARG

V𝑉Vitalic_V homothetic to an imaginary quadratic field different from ℚ⁢(−1)ℚ1\mathbb{Q}(\sqrt{-1})blackboard_Q ( square-root start_ARG - 1 end_ARG ) or ℚ⁢(−3)ℚ3\mathbb{Q}(\sqrt{-3})blackboard_Q ( square-root start_ARG - 3 end_ARG ).

B

∂t2u+(−Δ+a|x|2)⁢u=0superscriptsubscript𝑡2𝑢Δ𝑎superscript𝑥2𝑢0\partial_{t}^{2}u+\big{(}-\Delta+\tfrac{a}{|x|^{2}}\big{)}u=0∂ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u + ( - roman_Δ + divide start_ARG italic_a end_ARG start_ARG | italic_x | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) italic_u = 0. However, the scattering of the energy subcritical situation seems still open for us.

This paper is devoted to the study of scattering theory of solutions for (1.1) in sub-critical cases.

In this section, we shall prove a class of local Morawetz estimates for solutions in Proposition 2.5. Please note that these Morawetz estimates hold for non-radial solutions as well.

Fortunately, we can obtain scattering theory of radial finite energy solutions to the linear equation (1.6) in the appendix by the argument of [41] and a series of estimates. This fact shows us that the radial solutions of the energy critical nonlinear wave equations scatter to free waves, see [29].

for the proof of Theorem 1.1, we need to establish the decay estimate of the energy in the interior of light cones. Since the

A

X/\!\!/G).italic_Q italic_E italic_l italic_l start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_X ) ⊗ start_POSTSUBSCRIPT blackboard_Z [ italic_q start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT ] end_POSTSUBSCRIPT blackboard_Z ( ( italic_q ) ) ≅ italic_K start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_T italic_a italic_t italic_e end_POSTSUBSCRIPT ( italic_X / / italic_G ) .

In addition, we give an example computing quasi-elliptic cohomology, which is [Hua18, Example 3.3]. The conclusions in Example 2.6 is applied in Section 6.

In this section we review quasi-elliptic cohomology, the main reference of which is [Hua18]. It is a variant of Tate

In this section we give a loop space construction of twisted quasi-elliptic cohomology other than the twisted orbifold loop space in Example 5.3. This is a twisted version of the loop space in Definition 2.7.

In Section 2 we give a sketch of quasi-elliptic cohomology, including its definition, basic properties, and the loop space construction. In Section 3, we review Devoto’s equivariant elliptic cohomology. In Section 4, we recall the definition of twisted equivariant elliptic cohomology. In Section 5, we construct twisted quasi-elliptic cohomology. In Section 5.3, we define a model of twisted loop space, with which we can construct twisted quasi-elliptic cohomology. In Section 5.4.2, based on the Chern character of quasi-elliptic cohomology,

A

Thus, we require the number of edges is at least quartic in δ⁢(A,B)𝛿𝐴𝐵\delta(A,B)italic_δ ( italic_A , italic_B ) to ensure

such merging only increases the Eisubscript𝐸𝑖E_{i}italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT-degree of a vertex in U𝑈Uitalic_U by p𝑝pitalic_p.

Note that after the merging the total degree of each vertex increases by δ⁢(A0,B0)2𝛿superscriptsubscript𝐴0subscript𝐵02\delta(A_{0},B_{0})^{2}italic_δ ( italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT,

After the merging the total degree of each vertex increases by t⁢δ⁢(A0,B0)2𝑡𝛿superscriptsubscript𝐴0subscript𝐵02t\delta(A_{0},B_{0})^{2}italic_t italic_δ ( italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT.

Thus we can perform the merging in such a way that the total degree of each vertex in the real partition increases

D

Indeed, the latter operation indicates the existence of valuations on F𝐹Fitalic_F, which are strongly related to rigid elements.

To finish this section, let us see that from a 2222-henselian field, we can build a field with a non-trivial radical.

Finally, from valuation theory, we have pre-2222-henselian fields [9] as examples of fields with non-trivial Kaplansky radical, to be detailed in Section 4, Example 23.

Finally, in this section, we will see that rigid elements and a non-trivial Kaplansky radical usually do not exist together.

In this section, we describe the behavior of the Kaplansky radical under the basic operations (A) and (B) of Section 4, starting with free products. Since we focus on the radical, we can work only in the category of pro-2222 groups.

C

Q†⁢(x)=∫σ⁢(x;θ)⁢dν¯⁢(θ)superscript𝑄†𝑥𝜎𝑥𝜃differential-d¯𝜈𝜃Q^{\dagger}(x)=\int\sigma(x;\theta)\,{\mathrm{d}}\bar{\nu}(\theta)italic_Q start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_x ) = ∫ italic_σ ( italic_x ; italic_θ ) roman_d over¯ start_ARG italic_ν end_ARG ( italic_θ ). We assume that Dχ2⁢(ν¯∥ν0)<∞subscript𝐷superscript𝜒2conditional¯𝜈subscript𝜈0D_{\chi^{2}}(\bar{\nu}\,\|\,\nu_{0})<\inftyitalic_D start_POSTSUBSCRIPT italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( over¯ start_ARG italic_ν end_ARG ∥ italic_ν start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) < ∞ and ν¯⁢(θ)>0¯𝜈𝜃0\bar{\nu}(\theta)>0over¯ start_ARG italic_ν end_ARG ( italic_θ ) > 0 for any θ∈ℝD𝜃superscriptℝ𝐷\theta\in\mathbb{R}^{D}italic_θ ∈ blackboard_R start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT.

Under Assumptions 4.1 and 4.2, it holds for any k≤T/ϵ⁢(k∈ℕ)𝑘𝑇italic-ϵ𝑘ℕk\leq T/\epsilon\ (k\in\mathbb{N})italic_k ≤ italic_T / italic_ϵ ( italic_k ∈ blackboard_N ) that

Under Assumptions 4.1, 4.2, and 6.3, it holds for η=α−2𝜂superscript𝛼2\eta=\alpha^{-2}italic_η = italic_α start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT that

Under Assumptions 4.1, 4.2, and 6.1, it holds for η=α−2𝜂superscript𝛼2\eta=\alpha^{-2}italic_η = italic_α start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT that

Upon telescoping (5.5) and setting η=α−2𝜂superscript𝛼2\eta=\alpha^{-2}italic_η = italic_α start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT, we obtain that

C

As recalled above, simply dualizing the results for modules seen in [Lur17, 4.5.2.1] is not possible, as it would require totalizations to commute with the monoidal product, which is not the case in 𝒟≥0⁢(𝕜)superscript𝒟absent0𝕜\mathcal{D}^{\geq 0}(\Bbbk)caligraphic_D start_POSTSUPERSCRIPT ≥ 0 end_POSTSUPERSCRIPT ( roman_𝕜 ).

Theorem 1.1 allows us to bypass this difficulty and use an explicit model for the derived cotensor product.

We prove Theorem 1.1 in Section 3, and we apply this result to define the derived cotensor product of Theorem 1.2 in Section 4.

One of the main application of the above result is that it allows us to define a symmetric monoidal structure on the ∞\infty∞-category of C𝐶Citalic_C-comodules in 𝒟≥0⁢(𝕜)superscript𝒟absent0𝕜\mathcal{D}^{\geq 0}(\Bbbk)caligraphic_D start_POSTSUPERSCRIPT ≥ 0 end_POSTSUPERSCRIPT ( roman_𝕜 ). It is called the derived cotensor product and is given by a two-sided cobar resolution.

Throughout this paper, we work exclusively with 𝕜𝕜\Bbbkroman_𝕜 a commutative ring of global dimension zero, i.e. a finite product of fields. There are several reasons this condition is imposed. First, it induces that every object is cofibrant and fibrant in 𝖢𝗁𝕜subscript𝖢𝗁𝕜{\mathsf{Ch}_{\Bbbk}}sansserif_Ch start_POSTSUBSCRIPT roman_𝕜 end_POSTSUBSCRIPT, and the projective and injective model structures are equal. In particular, the model structure on comodules is left-induced from a nice monoidal model category. In [HKRS17], it was shown that the model structures for comodules are left-induced from injective model structures which are in general not monoidal model categories. This creates several issues to understand the induced homotopy theory on comodules. Moreover, as every module is flat, the tensor product preserves finite limits. This allows us to understand finite limits of comodules, which is essential for Theorem 1.1. Finally, the cotensor product of comodules is not a comodule without the flat setting, hence Theorem 1.2 cannot be true. Notice though that in [Pér20, 6.4.7], we proved an unbounded version of Theorem 1.1, where 𝕜𝕜\Bbbkroman_𝕜 is replaced by any commutative ring and C𝐶Citalic_C is a differential graded coalgebra that is perfect as a chain complex.

A

\right\}.italic_u start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_x ) < roman_min start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ caligraphic_S ( italic_x ) end_CELL end_ROW start_ROW start_CELL italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_CELL end_ROW end_ARG end_POSTSUBSCRIPT roman_max start_POSTSUBSCRIPT italic_i = 1 , … , italic_k end_POSTSUBSCRIPT { italic_u start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) } .

Thus, adding a small δ>0𝛿0\delta>0italic_δ > 0 to the left-hand side the previous inequality remains strict.

It remains to prove the claims about the coincidence set C⁢S⁢(g)𝐶𝑆𝑔CS(g)italic_C italic_S ( italic_g ).

then the previous idea can be applied to obtain the values of u𝑢uitalic_u at the nodes x𝑥xitalic_x such that

and choose δ>0𝛿0\delta>0italic_δ > 0 small enough such that adding δ𝛿\deltaitalic_δ to the left side

A

(b) It is important to note that the weak (and weak*) sequential notions can also be understood in a weaker and possibly more appropriate way. For example, one might call a vector x𝑥xitalic_x weakly sequentially hypercyclic for T𝑇Titalic_T if the smallest weakly sequentially closed set containing the orbit of x𝑥xitalic_x is the whole space; see [29], [5, Section 10.4]. We prefer to work here with the more naive definitions above.

On the other hand, the equivalence of (i) and (iv) was already obtained by Bernardes et al. [6, Theorems 9 and 26]; their proof even extends to all spaces in the setting of Theorem 4.1.

It was already noted by Chan and Seceleanu [15] that there is no general zero-one law for weak limit points in the bilateral case. Indeed, Chan and Sanders [13] had previously shown that any space ℓp⁢(ℤ)superscriptℓ𝑝ℤ\ell^{p}(\mathbb{Z})roman_ℓ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_Z ), 2≤p<∞2𝑝2\leq p<\infty2 ≤ italic_p < ∞, supports a weighted bilateral shift that is a weakly hypercyclic but not hypercyclic, see also [5, Corollary 10.28]. Interestingly, Shkarin [29] has shown that this phenomenon cannot arise if 1≤p<21𝑝21\leq p<21 ≤ italic_p < 2, see also [5, Proposition 10.20] and Theorem 3.14 below.

For the unilateral case, a corresponding extension has already been obtained by He, Huang and Yin [24, Lemma 5]. A generalization of the theorem of Chan and Seceleanu to the notion of ΓΓ\Gammaroman_Γ-supercyclicity was recently provided by Abakumov and Abbar [1].

In the context of linear dynamics, recurrence was first studied in detail by Costakis et al. [18], [17]; see [8], [10], [20], [11] and [21] for recent developments; weak (sequential) recurrence was recently introduced by Amouch et al. [3]. The consideration of limit points seems to be due to Chan and Seceleanu [28], [14], [15] and [16]; see also [12] for a survey.

D

Higher category theory aims to study more structured objects than categories. While categories consist of objects and morphisms between objects, higher categories also have higher morphisms. In this perspective, a 2222-category is obtained by also adding 2222-morphisms between the morphisms. A 2222-category can actually be seen as a category enriched in categories – its morphisms and 2222-morphisms between any pair of objects form a category. Another type of 2222-dimensional categories is given by internal categories to categories, called double categories. Such a structure has two types of morphisms between objects – horizontal and vertical morphisms – and its 2222-morphisms are squares. In particular, a 2222-category 𝒜𝒜\mathcal{A}caligraphic_A can be seen as a horizontal double category ℍ⁢𝒜ℍ𝒜\mathbb{H}\mathcal{A}blackboard_H caligraphic_A in which every vertical morphism is trivial; or equivalently, as an internal category to categories whose category of objects is discrete.

We first compute the space (ℕ⁢𝔸)0,0subscriptℕ𝔸00(\mathbb{N}\mathbb{A})_{0,0}( blackboard_N blackboard_A ) start_POSTSUBSCRIPT 0 , 0 end_POSTSUBSCRIPT, which is given by the space of objects. As expected from the completeness condition being in the horizontal direction, its 00-simplices are given by the objects, and its 1111-simplices by the horizontal adjoint equivalences.

Many aspects of 2222-category theory benefit from a passage to double categories. For example, a good notion of limit for 2222-categories is that of a 2222-limit, where the universal property is expressed by an isomorphism between hom-categories, rather than hom-sets. As clingman and the author show in [clingmanMoser], a 2222-limit cannot be characterized as a 2222-terminal object in the 2222-category of cones, but a passage to double categories allows such a characterization by results of Grandis and Paré [Grandis, GrandisPare]. Indeed, they show that the 2222-limit of a 2222-functor F𝐹Fitalic_F is double terminal in the double category of cones over the corresponding double functor ℍ⁢Fℍ𝐹\mathbb{H}Fblackboard_H italic_F. This result also holds in the more homotopical case of bi-limits, where the universal property is expressed by an equivalence of hom-categories, as clingman and the author show in [cM2].

These notions of categories, 2222-categories, and double categories are often too strict to accommodate many examples that appear in nature. In the perspective of generalizing categories, an (∞,1)1(\infty,1)( ∞ , 1 )-category is interpreted as a categorical structure that admits morphisms in all dimensions with all k𝑘kitalic_k-morphisms invertible for k>1𝑘1k>1italic_k > 1, where compositions are only associative and unital up to higher invertible morphisms. Such a higher structure should be thought of as a homotopical version of a category. Similarly to the strict case, we can then interpret an (∞,2)2(\infty,2)( ∞ , 2 )-category, as a “category enriched in (∞,1)1(\infty,1)( ∞ , 1 )-categories”, and a double (∞,1)1(\infty,1)( ∞ , 1 )-categories, as an “internal category to (∞,1)1(\infty,1)( ∞ , 1 )-categories”. A natural expectation is that (∞,2)2(\infty,2)( ∞ , 2 )-categories also admit a “horizontal embedding” into double (∞,1)1(\infty,1)( ∞ , 1 )-categories and that 2222-categories and double categories embed into their more homotopical versions, in such a way that the following diagram commutes (maybe only up to “homotopy”).

Higher category theory aims to study more structured objects than categories. While categories consist of objects and morphisms between objects, higher categories also have higher morphisms. In this perspective, a 2222-category is obtained by also adding 2222-morphisms between the morphisms. A 2222-category can actually be seen as a category enriched in categories – its morphisms and 2222-morphisms between any pair of objects form a category. Another type of 2222-dimensional categories is given by internal categories to categories, called double categories. Such a structure has two types of morphisms between objects – horizontal and vertical morphisms – and its 2222-morphisms are squares. In particular, a 2222-category 𝒜𝒜\mathcal{A}caligraphic_A can be seen as a horizontal double category ℍ⁢𝒜ℍ𝒜\mathbb{H}\mathcal{A}blackboard_H caligraphic_A in which every vertical morphism is trivial; or equivalently, as an internal category to categories whose category of objects is discrete.

B

Therefore, we have verified all the hypotheses in [Hov99, Theorem 2.1.19] and therefore the first statement that the structures

It may be important to point out that the term on the right is the quotient over SS\rm Sroman_S, i.e. the pushout of:

The left-properness may be established using the cellularity of the fixed point functors considered above. The right properness is clear since

are weak-equivalences. The first map is one since Q𝑄Qitalic_Q is the fibrant replacement functor considered above. The weak-equivalence

The left-properness is obvious, since the cofibrations and weak-equivalences are defined level-wise.

B

(ℰℰ\mathcal{E}caligraphic_E) is the equivariant sphere spectrum 𝕊Gsuperscript𝕊G\mathbb{S}^{\rm G}blackboard_S start_POSTSUPERSCRIPT roman_G end_POSTSUPERSCRIPT (the sphere spectrum Σ𝐓∞superscriptsubscriptΣ𝐓{\Sigma_{{\mathbf{T}}}^{\infty}}roman_Σ start_POSTSUBSCRIPT bold_T end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT, respectively), as proofs in the other cases follow

First we will consider (i). In this case we first observe that the homogeneous space G/TGT\rm G/\rm Troman_G / roman_T

Next we will consider (ii). One recalls the stable homotopy cofiber sequence (see [MV99, p. 115, Theorem 2.23])

First we will consider (i), namely the Mayer-Vietoris sequence. For this, one begins with the stable cofiber sequence

in Definition  4.1. This will also provide a second proof of the additivity and Mayer-Vietoris property

C

\mathcal{X}}(\mathbb{Q})\otimes\mathscr{L}^{-\dim(\mathcal{X})/2}.under¯ start_ARG caligraphic_I caligraphic_C end_ARG start_POSTSUBSCRIPT caligraphic_X end_POSTSUBSCRIPT ≔ caligraphic_I caligraphic_C start_POSTSUBSCRIPT caligraphic_X end_POSTSUBSCRIPT ( blackboard_Q ) ⊗ script_L start_POSTSUPERSCRIPT - roman_dim ( caligraphic_X ) / 2 end_POSTSUPERSCRIPT .

\widetilde{Q}))caligraphic_B caligraphic_P caligraphic_S start_POSTSUPERSCRIPT italic_G , roman_ζ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over~ start_ARG italic_Q end_ARG , over~ start_ARG italic_W end_ARG , roman_θ end_POSTSUBSCRIPT ∈ roman_MMHM start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT ( caligraphic_M start_POSTSUBSCRIPT roman_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_ζ start_OPFUNCTION - roman_ss end_OPFUNCTION end_POSTSUPERSCRIPT ( over~ start_ARG italic_Q end_ARG ) ) is a pure monodromic mixed Hodge module.

This complex has cohomology concentrated in degree 1, and is pure. Moreover there is an isomorphism (ℒ1/2)⊗2≅ℒsuperscriptsuperscriptℒ12tensor-productabsent2ℒ(\mathscr{L}^{1/2})^{\otimes 2}\cong\mathscr{L}( script_L start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT ⊗ 2 end_POSTSUPERSCRIPT ≅ script_L, justifying the notation. We define

The mixed Hodge module ℬ⁢𝒫⁢𝒮ΠQ,θζℬ𝒫subscriptsuperscript𝒮ζsubscriptΠ𝑄θ\mathcal{BPS}^{\upzeta}_{\Pi_{Q},\uptheta}caligraphic_B caligraphic_P caligraphic_S start_POSTSUPERSCRIPT roman_ζ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT , roman_θ end_POSTSUBSCRIPT is pure below if and only if it is pure above.

Since ℒ1/2superscriptℒ12\mathscr{L}^{1/2}script_L start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT is pure, this is a pure weight zero monodromic mixed Hodge module.

D

By Theorem 0.2, we know f=A⁢A*𝑓𝐴superscript𝐴f=AA^{*}italic_f = italic_A italic_A start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT for some osculating Möbius transformation A𝐴Aitalic_A from some circle pattern z𝑧zitalic_z to z~~𝑧\tilde{z}over~ start_ARG italic_z end_ARG. Then A−1superscript𝐴1A^{-1}italic_A start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT is the osculating Möbius transformation from z~~𝑧\tilde{z}over~ start_ARG italic_z end_ARG to z𝑧zitalic_z. It defines another discrete CMC-1 surface f~:=A−1⁢(A−1)*assign~𝑓superscript𝐴1superscriptsuperscript𝐴1\tilde{f}:=A^{-1}(A^{-1})^{*}over~ start_ARG italic_f end_ARG := italic_A start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_A start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT which satisfies Equation (22).

We prove a Weierstrass-type representation that every discrete CMC-1 surface corresponds to a pair of Delaunay circle patterns sharing the modulus of cross ratios. It makes use of the discrete osculating Möbius transformation.

Theorem 0.2 also holds for general circle patterns sharing the same modulus of cross ratios (not necessary Delaunay) if we broaden the definition of discrete CMC-1 surfaces as follows:

Hertrich-Jeromin [17] introduced discrete CMC-1 surfaces in the context of integrable systems. These surfaces are constructed from special meshes in the plane that admit a real-valued solution to a discrete Toda-type equation [9]. We show that such a solution induces a family of circle patterns sharing the same modulus of cross ratios and thus our discrete CMC-1 surfaces. In this way, our discrete CMC-1 surfaces generalize those from discrete integrable systems.

and hence the two circle patterns share the same modulus of cross ratios. On the other hand, the second condition in Definition 3.3 implies that

B