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A(y⋅x)𝐴⋅𝑦𝑥A(y\cdot x)italic_A ( italic_y ⋅ italic_x ) is stable. We would first like to clarify the relation between wide and generic sets under the assumption of stability (We thank the referee for pointing out that our original proof of Proposition 3.5 could be reformulated in a simpler context). | Let G𝐺Gitalic_G be a definable group with an ∅\emptyset∅-type-definable left-translation ideal ℐℐ\mathcal{I}caligraphic_I of definable subsets of G𝐺Gitalic_G. Assume that ℐℐ\mathcal{I}caligraphic_I is indiscernibly prime (see Definition 1.2). The following conditions are equivalent for every elementary submodel M𝑀Mitalic_M: | Let G𝐺Gitalic_G be an infinite group. Left-generic definable subsets are wide for every left translation invariant ideal ℐℐ\mathcal{I}caligraphic_I on the boolean algebra of all definable | Let G𝐺Gitalic_G be a definable group and ℐℐ\mathcal{I}caligraphic_I an ∅\emptyset∅-invariant indiscernibly | subsets of G𝐺Gitalic_G. Furthermore, if the definable subset A𝐴Aitalic_A is stable and wide for some indiscernibly prime left translation invariant ∅\emptyset∅-invariant ideal ℐℐ\mathcal{I}caligraphic_I on the boolean algebra of all definable subsets of G𝐺Gitalic_G, then A𝐴Aitalic_A is left-generic. Indeed, by [16, Remark 5.17 (i)], it suffices to show that the formula g⋅A⋅𝑔𝐴g\cdot Aitalic_g ⋅ italic_A does not fork over ∅\emptyset∅, which follows immediately from Remark 1.3. | B |
\gamma_{2}+\gamma_{3}-1}}{\log^{3}N}.∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_p start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_N end_CELL end_ROW start_ROW start_CELL italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_P ∩ blackboard_N start_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG end_POSTSUBSCRIPT 1 = ( 1 + italic_o ( 1 ) ) divide start_ARG italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT roman_Γ ( italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) roman_Γ ( italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) roman_Γ ( italic_γ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) end_ARG start_ARG roman_Γ ( italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_γ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) end_ARG divide start_ARG fraktur_S ( italic_N ) italic_N start_POSTSUPERSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_γ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT end_ARG start_ARG roman_log start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_N end_ARG . | In this subsection, we will prove Theorem 3.1 (ii) and Theorem 3.4 (ii) for θ∈𝔪𝜃𝔪\theta\in\mathfrak{m}italic_θ ∈ fraktur_m. Namely, we will study, for θ∈𝔪𝜃𝔪\theta\in\mathfrak{m}italic_θ ∈ fraktur_m, the exponential sum | For convenience, here we prove an weighted asymptotic formula, see e.g. Balog-Friedlander [2, (1.7)] and the weighted version implies the unweighted version, which was also stated in [2]. Namely, we will prove that for any A1>0subscript𝐴10A_{1}>0italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > 0, | By the weak Balog-Friedlander condition, for any sufficiently large A0>A1/(3−u)subscript𝐴0subscript𝐴13𝑢A_{0}>A_{1}/{(3-u)}italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT / ( 3 - italic_u ), we have | In this section, we will construct ρ+(n)superscript𝜌𝑛\rho^{+}(n)italic_ρ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_n ) in Theorem 3.4. We will first prove that Theorem 3.4 (iv) holds in the end of this section, and will prove that Theorem 3.4 (i) and (ii) hold in Section 7. | B |
Acknowledgements: I would like to thank the anonymous referee, whose comment led us to a correction of the original argument and to a strenghtening of our results. | In this section we show how to associate a polarized abelian surface to a cyclic cover of a genus two curve. We then use this abelian surface to study the differential of the Prym map and to prove Theorems A and B whenever the cover is not bielliptic. The construction of the abelian surface is quite standard, and can be found also in [BirkenhakeLange2004, RamananAmpleAbSurf1985], but we would like to outline the necessary steps nevertheless. | In the course of the proof of Theorem A, we will also show that the covering curve is never hyperelliptic, whenever d≥7𝑑7d\geq 7italic_d ≥ 7. This result was already observed by Ramanan in [RamananAmpleAbSurf1985, Section 3] for cyclic covers of degrees d≥3𝑑3d\geq 3italic_d ≥ 3, but we give here a proof along different lines. I thank Paweł Borówka for pointing me to Ramanan’s result and also for pointing out the similar result in [Borowka, Proposition 2.3] for cyclic covers of curves of higher degree. | The main ideas in this note were developed during my PhD at the Humboldt-Universität zu Berlin. I would like to thank my advisor Prof. Gavril Farkas for his guidance and the Berlin Mathematical School and the IRTG 1800 of the DFG for their support during my studies. I would also like to thank Angela Ortega for her useful comments on this manuscript. I thank Paweł Borówka for pointing out Ramanan’s result to me. | Acknowledgements: I would like to thank the anonymous referee, whose comment led us to a correction of the original argument and to a strenghtening of our results. | C |
Now, consider the auxiliary graph sequence {G~k}subscript~𝐺𝑘\{\tilde{G}_{k}\}{ over~ start_ARG italic_G end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }, given by adding one vertex to Gksubscript𝐺𝑘G_{k}italic_G start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and connecting it to every vertex in ∂Gksubscript𝐺𝑘\partial G_{k}∂ italic_G start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. No subsequene of this sequence can converge in the Benjamini–Schramm sense, as, with probability at least ε𝜀\varepsilonitalic_ε, a ball of radius 2r+22𝑟22r+22 italic_r + 2 around oksubscript𝑜𝑘o_{k}italic_o start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT has volume at least ε|V(Gk)|𝜀𝑉subscript𝐺𝑘\varepsilon|V(G_{k})|italic_ε | italic_V ( italic_G start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) |. Since {Gk}subscript𝐺𝑘\{G_{k}\}{ italic_G start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } is Følner with respect to {∂Gk}subscript𝐺𝑘\{\partial G_{k}\}{ ∂ italic_G start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }, |V(Gk)|→∞→𝑉subscript𝐺𝑘|V(G_{k})|\to\infty| italic_V ( italic_G start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) | → ∞, meaning B2r+2(ok)subscript𝐵2𝑟2subscript𝑜𝑘B_{2r+2}(o_{k})italic_B start_POSTSUBSCRIPT 2 italic_r + 2 end_POSTSUBSCRIPT ( italic_o start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) cannot converge to any finite graph with positive probability. However, since {Gk}subscript𝐺𝑘\{G_{k}\}{ italic_G start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } has uniformly integrable degrees and is Følner with respect to {∂Gk}subscript𝐺𝑘\{\partial G_{k}\}{ ∂ italic_G start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }, the sequence {G~k}subscript~𝐺𝑘\{\tilde{G}_{k}\}{ over~ start_ARG italic_G end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } must also have uniformly integrable degrees. This contradicts Lemma 4.8, as required. | There exists a p0>0subscript𝑝00p_{0}>0italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0 such that the following holds. Let {Gk}subscript𝐺𝑘\{G_{k}\}{ italic_G start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } be a sequence of finite simple planar graphs that converges to a limit (G,o)𝐺𝑜(G,o)( italic_G , italic_o ) in the sense of Benjamini–Schramm. Then | The final ingredient is a statement about Bernoulli site on Benjamini–Schramm limits of finite planar graphs. | This control, in turn, relies on recent bounds on the site percolation threshold of Benjamini–Schramm limits of finite planar graphs. | Instead of analyzing the Bernoulli site percolation process on an abstract finite planar graph, we will study these processes by considering an infinite planar graph that captures the properties of a sequence of graphs {Gk}subscript𝐺𝑘\{G_{k}\}{ italic_G start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } centered around a typical vertex. The formal framework for doing this is given by Benjamini–Schramm convergence [BS01]. In this section, we will recall its definition and collect some useful results about such limits. | B |
\right\}\leq e^{-\frac{r^{2}}{18T}},\qquad\forall r>0bold_P { roman_Δ ( italic_W start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) ≥ bold_E [ roman_Δ ( italic_W start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) ] + italic_r } ≤ italic_e start_POSTSUPERSCRIPT - divide start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 18 italic_T end_ARG end_POSTSUPERSCRIPT , ∀ italic_r > 0 | Since 𝐄[Δ(W1)]≤κβT𝐄delimited-[]Δsuperscript𝑊1𝜅𝛽𝑇{\mathbf{E}}[\Delta(W^{1})]\leq\kappa\beta Tbold_E [ roman_Δ ( italic_W start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) ] ≤ italic_κ italic_β italic_T, we see that (29) holds with probability at least 1−δ1𝛿1-\delta1 - italic_δ. | (Boucheron et al., 2013, Theorem 5.6). By the union bound, (28) holds with probability at least 1−δ1𝛿1-\delta1 - italic_δ. | holds with probability at least 1−δ1𝛿1-\delta1 - italic_δ. With a standard argument this can be converted to the expected L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT error bound, | Consider a synchronous coupling of the optimally controlled process (34) with the ideal Langevin dynamics (45) using the same Brownian motion and the same initial condition W0=W~0=0subscript𝑊0subscript~𝑊00W_{0}=\tilde{W}_{0}=0italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = over~ start_ARG italic_W end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 0. Then the following holds with probability at least 1−δ1𝛿1-\delta1 - italic_δ: | B |
^{0}_{y}=h_{1}]p_{\chi-\varphi^{0}}(h_{1})p_{\varphi^{0}}(h-h_{1})dh_{1}.start_ROW start_CELL italic_p start_POSTSUBSCRIPT italic_χ end_POSTSUBSCRIPT ( italic_h ) blackboard_P [ roman_Piv start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT , | italic_χ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT - italic_φ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT | ≤ italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | italic_χ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = italic_h ] end_CELL end_ROW start_ROW start_CELL = ∫ start_POSTSUBSCRIPT - italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT blackboard_P [ roman_Piv start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT | italic_χ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT - italic_φ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ] italic_p start_POSTSUBSCRIPT italic_χ - italic_φ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_p start_POSTSUBSCRIPT italic_φ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_h - italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_d italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT . end_CELL end_ROW | the level set {φ≥h}𝜑ℎ\{\varphi\geq h\}{ italic_φ ≥ italic_h } is simply an independent site percolation with certain inhomogeneous parameters 𝐩=(𝐩x)x∈ℤd𝐩subscriptsubscript𝐩𝑥𝑥superscriptℤ𝑑\mathbf{p}=(\mathbf{p}_{x})_{x\in\mathbb{Z}^{d}}bold_p = ( bold_p start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_x ∈ blackboard_Z start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT. Indeed, for every x∈ℤd𝑥superscriptℤ𝑑x\in\mathbb{Z}^{d}italic_x ∈ blackboard_Z start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT conditionally on ℱℱ\mathcal{F}caligraphic_F we know the precise value of φx−φx0subscript𝜑𝑥subscriptsuperscript𝜑0𝑥\varphi_{x}-\varphi^{0}_{x}italic_φ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT - italic_φ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT and that φx0subscriptsuperscript𝜑0𝑥\varphi^{0}_{x}italic_φ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT lies in a prescribed interval (depending only on the value of φx−φx0subscript𝜑𝑥subscriptsuperscript𝜑0𝑥\varphi_{x}-\varphi^{0}_{x}italic_φ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT - italic_φ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT and on whether x𝑥xitalic_x is in {φ≥h+mε},m=1,2formulae-sequence𝜑ℎ𝑚𝜀𝑚12\{\varphi\geq h+m\varepsilon\},\leavevmode\nobreak\ m=1,2{ italic_φ ≥ italic_h + italic_m italic_ε } , italic_m = 1 , 2, or not). Since φ0superscript𝜑0\varphi^{0}italic_φ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT is an i.i.d. field, the claim follows. Furthermore, on A𝐴Aitalic_A we know that 𝒞𝒞\mathscr{C}script_C is in {φ≥h+ε}𝜑ℎ𝜀\{\varphi\geq h+\varepsilon\}{ italic_φ ≥ italic_h + italic_ε } and |φx−φx0|≤Msubscript𝜑𝑥subscriptsuperscript𝜑0𝑥𝑀|\varphi_{x}-\varphi^{0}_{x}|\leq M| italic_φ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT - italic_φ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT | ≤ italic_M for every x∈𝒮𝑥𝒮x\in\mathscr{S}italic_x ∈ script_S, so we easily infer that | Since φ0superscript𝜑0\varphi^{0}italic_φ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT is a centered Gaussian variable, we have | Since φx−φxLsubscript𝜑𝑥subscriptsuperscript𝜑𝐿𝑥\varphi_{x}-\varphi^{L}_{x}italic_φ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT - italic_φ start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT is a centered Gaussian variable and, using (3.4),(3.7) and independence of ξxℓsubscriptsuperscript𝜉ℓ𝑥\xi^{\ell}_{x}italic_ξ start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT, ℓ≥0ℓ0\ell\geq 0roman_ℓ ≥ 0, in the first equality below and recalling the bound (3.5), its variance is bounded by | It will be important below that ℱℱ\mathcal{F}caligraphic_F does not completely determine φ0superscript𝜑0\varphi^{0}italic_φ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT, i.e. φ0superscript𝜑0\varphi^{0}italic_φ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT is not ℱℱ\mathcal{F}caligraphic_F-measurable. | B |
𝒪(ϵ−6)𝒪superscriptitalic-ϵ6\mathcal{O}({\epsilon}^{-6})caligraphic_O ( italic_ϵ start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT ) | 𝒪(ϵ−3)𝒪superscriptitalic-ϵ3\mathcal{O}({\epsilon}^{-3})caligraphic_O ( italic_ϵ start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT ) | 𝒪(ϵ−3)𝒪superscriptitalic-ϵ3\mathcal{O}({\epsilon}^{-3})caligraphic_O ( italic_ϵ start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT ) | 𝒪(ϵ−8)𝒪superscriptitalic-ϵ8\mathcal{O}({\epsilon}^{-8})caligraphic_O ( italic_ϵ start_POSTSUPERSCRIPT - 8 end_POSTSUPERSCRIPT ) | 𝒪(ϵ−8)𝒪superscriptitalic-ϵ8\mathcal{O}({\epsilon}^{-8})caligraphic_O ( italic_ϵ start_POSTSUPERSCRIPT - 8 end_POSTSUPERSCRIPT ) | C |
Note that the existence of hundreds of types and subtypes of centrality, coupled by the existence of infinite families of measures, highlights the need for powerful tools for comparing centrality indices and selecting the most appropriate ones. | At the initial step of the procedure, we are at the root of the tree. There are no other vertices in the tree yet, and no question is associated with the root. | We set t=1𝑡1t=1italic_t = 1 for the Forest, Heat, and Communicability kernels and t=(ρ(A)+1)−1𝑡superscript𝜌𝐴11t=(\rho(A)+1)^{-1}italic_t = ( italic_ρ ( italic_A ) + 1 ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT for the Walk kernel. | For more about these axiom, including the rationale behind choosing them, a necessary and sufficient condition for Self-consistency, and a sufficient condition for the Bridge axiom, see [44]. In particular, it is shown that in the presence of Monotonicity and Equivalence, these axioms are incompatible. | The authors thank Anna Khmelnitskaya and Konstantin Avrachenkov for helpful discussions and Matthew Jackson for an important remark at the final stage. We are grateful to the anonymous referees for their valuable comments. | D |
\right\}.italic_α ( italic_G ) = roman_max { ⟨ 1 start_POSTSUBSCRIPT roman_n × roman_n end_POSTSUBSCRIPT , roman_X ⟩ : roman_X ∈ roman_CTH start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_G ) , roman_X ∈ roman_STAB start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_G ) } . | Let (Pℰ)superscript𝑃ℰ(P^{\mathcal{E}})( italic_P start_POSTSUPERSCRIPT caligraphic_E end_POSTSUPERSCRIPT ) be the SDP on the | Let (P𝒞)superscript𝑃𝒞(P^{\mathcal{C}})( italic_P start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT ) be the SDP on the | In consequence, the objective function value of X𝑋Xitalic_X for (P𝒞)superscript𝑃𝒞(P^{\mathcal{C}})( italic_P start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT ) is | Furthermore, any feasible solution X𝑋Xitalic_X of (P𝒞)superscript𝑃𝒞(P^{\mathcal{C}})( italic_P start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT ) can be | B |
Therefore we conclude that, even though gauge invariant, the candidate Lagrangians (26) and (29) do not lead to local field equations after the partial gauge fixing (19); not even full gauge fixing is adequate. Thus, requiring invariance under (18) leads to a nonlocal Lagrangian for N≥3𝑁3N\geq 3italic_N ≥ 3. Allowing for nonlocality would indicate that one may just ignore the inverse boxes in the above Lagrangians. However, then the resulting field equations would be of order higher than two in derivatives and ghost instabilities are expected to arise. | Finally, one can easily see that both (26) and (28) are a priori gauge invariant under (18). The only thing that remains to be checked is whether they lead to local field equations upon imposing (19). The answer to this last question is negative and we will illustrate this in the simple example of a spin-4444 field in five dimensions. This is described by the irreducible 4444-partite tensor ω[1,1,1,1]subscript𝜔1111\omega_{[1,1,1,1]}italic_ω start_POSTSUBSCRIPT [ 1 , 1 , 1 , 1 ] end_POSTSUBSCRIPT, with its components being a fully symmetric 4444-tensor. The first candidate Galileon interaction term ℒ~(2)subscript~ℒ2\widetilde{\mathcal{L}}_{(2)}over~ start_ARG caligraphic_L end_ARG start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT has three field appearances and it reads as | Following a different strategy, one could attempt to avoid the above problem of nonlocality by considering Lagrangians that do not contain inverse box operators to start with. Rather than discussing this in general, let us look at a specific illustrative example. Consider an irreducible 4444-partite tensor ω[3,2,2,1]subscript𝜔3221\omega_{[3,2,2,1]}italic_ω start_POSTSUBSCRIPT [ 3 , 2 , 2 , 1 ] end_POSTSUBSCRIPT in seven dimensions. | This paper includes two results that do not appear in [6, 7]. The first is the already mentioned generalised kinetic term that unifies the kinetic and higher-derivative interacting sector of the bipartite tensor Lagrangians, see Eq. (23). Note that for a special class of bipartite tensors, when their partial degrees are equal, there are additional Galileon interactions that are not included in the generalised kinetic term. We describe these ones in the graded formalism separately. The second new result refers to spins greater than 2. In the formalism employed here, it is rather simple to construct candidate Lagrangians that would correspond to Galileons in this general case, although this appears to us as a formidable task otherwise. Note, however, that it turns out to be impossible to rescue both properties of full gauge invariance and local second order field equations. Although this sounds reasonable in view of the difficulty in constructing interacting theories for higher spins, it is interesting to see that certain types of higher derivative terms pass a number of tests. Specifically, we present two types of non-trivial (meaning that they are not total derivatives) terms. The first is manifestly gauge invariant, however it leads to non-local field equations, while the second is manifestly local, however it does not respect the full set of expected gauge symmetries, but only a subset of those. Finally, we mention some further results and our conclusions in section 4. | Physical theories that attempt to describe natural phenomena contain a diversity of fields representing their fundamental degrees of freedom. Typically, these fields can be categorised with respect to their spin. In this work we focus on bosonic fields: The spin-0 case corresponds to scalar fields, which play a fundamental role in both particle physics and cosmology. Spin-1 fields are usually differential 1-forms and they are the key players in gauge theories. Furthermore, spin-2 fields appear in gravity, where the mediator of the gravitational interaction is indeed such a field. However, apart from the above cherry picked cases, in general there exists a lot of motivation to consider further types of fields. For instance, higher degree differential forms and their corresponding higher gauge theories appear in certain frameworks, notably in string theory. Spin-2 fields different than the graviton may appear through duality in higher than four dimensions, for example the Curtright field of mixed symmetry (2,1)21(2,1)( 2 , 1 ) is dual to the graviton in five dimensions [1, 2]. Ultimately, spins larger than 2 can also be of physical interest and they appear in the spectrum of string theory. On the other hand, fermionic fields correspond to half-integer spins, however we do not discuss them further in this work. | B |
The fifth equation uses that N¯a=Nasubscript¯𝑁𝑎subscript𝑁𝑎\overline{N}_{a}=N_{a}over¯ start_ARG italic_N end_ARG start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT. | The third equation uses that we have tensored with ℤℤ\mathbb{Z}blackboard_Z over ℤπℤ𝜋\mathbb{Z}\piblackboard_Z italic_π. | where the last equation uses that 0=[(Na,1−b,0)]0delimited-[]subscript𝑁𝑎1𝑏00=[(N_{a},1-b,0)]0 = [ ( italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , 1 - italic_b , 0 ) ] in M𝑀Mitalic_M. | The sixth and final equation uses the second relation of M2subscript𝑀2M_{2}italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. | Using the description of M2subscript𝑀2M_{2}italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT in terms of generators and relations, | C |
I would like to thank my PhD supervisor F. E. A. Johnson whose suggestion that I read ‘Swan’s long paper’ [Swa83] was largely the inspiration for this work. I would also like to thank the referees for a number of helpful comments which improved the exposition of this article. This work was supported by the UK Engineering and Physical Sciences Research Council | Recall that mℍ(G)subscript𝑚ℍ𝐺m_{\mathbb{H}}(G)italic_m start_POSTSUBSCRIPT blackboard_H end_POSTSUBSCRIPT ( italic_G ) denotes the number of copies of ℍℍ\mathbb{H}blackboard_H in the Wedderburn decomposition of ℝGℝ𝐺\mathbb{R}Gblackboard_R italic_G for a finite group G𝐺Gitalic_G. This coincides with the number of irreducible quaternionic representations of G𝐺Gitalic_G of (quaternionic) dimension one. We say that a finite group G𝐺Gitalic_G is said to satisfy the Eichler condition if mℍ(G)=0subscript𝑚ℍ𝐺0m_{\mathbb{H}}(G)=0italic_m start_POSTSUBSCRIPT blackboard_H end_POSTSUBSCRIPT ( italic_G ) = 0. The following is [Nic21a, Proposition 3.3]. | The Sylow subgroups of G𝐺Gitalic_G are cyclic or generalised quaternionic Q2nsubscript𝑄superscript2𝑛Q_{2^{n}}italic_Q start_POSTSUBSCRIPT 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT. | Here mℍ(G)subscript𝑚ℍ𝐺m_{\mathbb{H}}(G)italic_m start_POSTSUBSCRIPT blackboard_H end_POSTSUBSCRIPT ( italic_G ) denotes the number of copies of the quaternions ℍ=M1(ℍ)ℍsubscript𝑀1ℍ\mathbb{H}=M_{1}(\mathbb{H})blackboard_H = italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( blackboard_H ) in the Wedderburn decomposition of ℝGℝ𝐺\mathbb{R}Gblackboard_R italic_G for a finite group G𝐺Gitalic_G (see [Nic21a, p318]). | Recall that a binary polyhedral group is a non-cyclic finite subgroup of ℍ×superscriptℍ\mathbb{H}^{\times}blackboard_H start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT where ℍℍ\mathbb{H}blackboard_H is the real quaternions. They are the generalised quaternion groups | D |
}\frac{{\mathrm{d}}t}{t}\,f_{|\phi|}(g)\,t^{s+1}\,|\Phi|_{B}(0).start_ROW start_CELL italic_( italic_) ≪ ∫ start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( blackboard_A ) start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_d start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT italic_g ∫ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG roman_d italic_t end_ARG start_ARG italic_t end_ARG italic_f start_POSTSUBSCRIPT | italic_ϕ | end_POSTSUBSCRIPT ( italic_g ) italic_t start_POSTSUPERSCRIPT italic_s + 1 end_POSTSUPERSCRIPT | roman_Φ | start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_t start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT italic_g ) end_CELL end_ROW start_ROW start_CELL + ∫ start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_F ) \ italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( blackboard_A ) start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_d start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT italic_g ∫ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG roman_d italic_t end_ARG start_ARG italic_t end_ARG italic_f start_POSTSUBSCRIPT | italic_ϕ | end_POSTSUBSCRIPT ( italic_g ) italic_t start_POSTSUPERSCRIPT italic_s - 1 end_POSTSUPERSCRIPT | roman_Φ | start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ( 0 ) end_CELL end_ROW start_ROW start_CELL + ∫ start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( blackboard_A ) start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_d start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT italic_g ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT divide start_ARG roman_d italic_t end_ARG start_ARG italic_t end_ARG italic_f start_POSTSUBSCRIPT | italic_ϕ | end_POSTSUBSCRIPT ( italic_g ) italic_t start_POSTSUPERSCRIPT italic_s - 1 end_POSTSUPERSCRIPT | roman_Φ | start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ( italic_t start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT italic_t end_FLOATSUPERSCRIPT italic_g start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) end_CELL end_ROW start_ROW start_CELL + ∫ start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_F ) \ italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( blackboard_A ) start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_d start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT italic_g ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT divide start_ARG roman_d italic_t end_ARG start_ARG italic_t end_ARG italic_f start_POSTSUBSCRIPT | italic_ϕ | end_POSTSUBSCRIPT ( italic_g ) italic_t start_POSTSUPERSCRIPT italic_s + 1 end_POSTSUPERSCRIPT | roman_Φ | start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( 0 ) . end_CELL end_ROW | and 𝒴3(s)subscript𝒴3𝑠\mathscr{Y}_{3}(s)script_Y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_s ) is convergent in the range −1<Re(s)<11Re𝑠1-1<{\mathop{\mathrm{Re}}}(s)<1- 1 < roman_Re ( italic_s ) < 1. | The zeta integral Z(Φ,ϕ,s)𝑍Φitalic-ϕ𝑠Z(\Phi,\phi,s)italic_Z ( roman_Φ , italic_ϕ , italic_s ) converges absolutely for sufficiently large Re(s)Re𝑠{\mathop{\mathrm{Re}}}(s)roman_Re ( italic_s ). | is absolutely convergent in the range −1<Re(s)<11Re𝑠1-1<{\mathop{\mathrm{Re}}}(s)<1- 1 < roman_Re ( italic_s ) < 1. | Hence, (3.3) converges for the range −1<Re(s)<11Re𝑠1-1<{\mathop{\mathrm{Re}}}(s)<1- 1 < roman_Re ( italic_s ) < 1. | D |
In this way, we have identified #∂+−1([P0])#superscriptsubscript1delimited-[]subscript𝑃0\#\partial_{+}^{-1}([P_{0}])# ∂ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( [ italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] ) with the counting of irreducible SO(3)𝑆𝑂3SO(3)italic_S italic_O ( 3 )-instantons over X0,1subscript𝑋01X_{0,1}italic_X start_POSTSUBSCRIPT 0 , 1 end_POSTSUBSCRIPT up to SU(2)𝑆𝑈2SU(2)italic_S italic_U ( 2 )-gauge transformations. On the other hand, Dw𝒯0(X0,1)subscriptsuperscript𝐷0subscript𝑤𝒯subscript𝑋01D^{0}_{w_{\mathcal{T}}}(X_{0,1})italic_D start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_X start_POSTSUBSCRIPT 0 , 1 end_POSTSUBSCRIPT ) counts irreducible SO(3)𝑆𝑂3SO(3)italic_S italic_O ( 3 )-instantons up to SO(3)𝑆𝑂3SO(3)italic_S italic_O ( 3 )-gauge transformations. To compare these groups 𝒢SU(2)(X0,1)superscript𝒢𝑆𝑈2subscript𝑋01\mathcal{G}^{SU(2)}(X_{0,1})caligraphic_G start_POSTSUPERSCRIPT italic_S italic_U ( 2 ) end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT 0 , 1 end_POSTSUBSCRIPT ) and 𝒢SO(3)(X0,1)superscript𝒢𝑆𝑂3subscript𝑋01\mathcal{G}^{SO(3)}(X_{0,1})caligraphic_G start_POSTSUPERSCRIPT italic_S italic_O ( 3 ) end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT 0 , 1 end_POSTSUBSCRIPT ), we note that there exists an exact sequence [AMR95, Page 66]: | Map(X0,1,ℤ/2)→𝒢SU(2)(X0,1)→𝛼𝒢SO(3)(X0,1)→H1(X0,1;ℤ/2)→1.→Mapsubscript𝑋01ℤ2superscript𝒢𝑆𝑈2subscript𝑋01𝛼→superscript𝒢𝑆𝑂3subscript𝑋01→superscript𝐻1subscript𝑋01ℤ2→1\operatorname{Map}(X_{0,1},\mathbb{Z}/2)\rightarrow\mathcal{G}^{SU(2)}(X_{0,1}% | where Dw𝒯0(X0,1)subscriptsuperscript𝐷0subscript𝑤𝒯subscript𝑋01D^{0}_{w_{\mathcal{T}}}(X_{0,1})italic_D start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_X start_POSTSUBSCRIPT 0 , 1 end_POSTSUBSCRIPT ) counts the irreducible anti-self-dual SO(3)𝑆𝑂3SO(3)italic_S italic_O ( 3 )-instantons on the ℝ3superscriptℝ3\mathbb{R}^{3}blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT-bundle E0→X0,1→subscript𝐸0subscript𝑋01E_{0}\rightarrow X_{0,1}italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → italic_X start_POSTSUBSCRIPT 0 , 1 end_POSTSUBSCRIPT characterized by | In this way, we have identified #∂+−1([P0])#superscriptsubscript1delimited-[]subscript𝑃0\#\partial_{+}^{-1}([P_{0}])# ∂ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( [ italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] ) with the counting of irreducible SO(3)𝑆𝑂3SO(3)italic_S italic_O ( 3 )-instantons over X0,1subscript𝑋01X_{0,1}italic_X start_POSTSUBSCRIPT 0 , 1 end_POSTSUBSCRIPT up to SU(2)𝑆𝑈2SU(2)italic_S italic_U ( 2 )-gauge transformations. On the other hand, Dw𝒯0(X0,1)subscriptsuperscript𝐷0subscript𝑤𝒯subscript𝑋01D^{0}_{w_{\mathcal{T}}}(X_{0,1})italic_D start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_X start_POSTSUBSCRIPT 0 , 1 end_POSTSUBSCRIPT ) counts irreducible SO(3)𝑆𝑂3SO(3)italic_S italic_O ( 3 )-instantons up to SO(3)𝑆𝑂3SO(3)italic_S italic_O ( 3 )-gauge transformations. To compare these groups 𝒢SU(2)(X0,1)superscript𝒢𝑆𝑈2subscript𝑋01\mathcal{G}^{SU(2)}(X_{0,1})caligraphic_G start_POSTSUPERSCRIPT italic_S italic_U ( 2 ) end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT 0 , 1 end_POSTSUBSCRIPT ) and 𝒢SO(3)(X0,1)superscript𝒢𝑆𝑂3subscript𝑋01\mathcal{G}^{SO(3)}(X_{0,1})caligraphic_G start_POSTSUPERSCRIPT italic_S italic_O ( 3 ) end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT 0 , 1 end_POSTSUBSCRIPT ), we note that there exists an exact sequence [AMR95, Page 66]: | Since H1(X0,1;ℤ/2)≃ℤ/2⊕ℤ/2similar-to-or-equalssuperscript𝐻1subscript𝑋01ℤ2direct-sumℤ2ℤ2H^{1}(X_{0,1};\mathbb{Z}/2)\simeq\mathbb{Z}/2\oplus\mathbb{Z}/2italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT 0 , 1 end_POSTSUBSCRIPT ; blackboard_Z / 2 ) ≃ blackboard_Z / 2 ⊕ blackboard_Z / 2, we conclude that [𝒢SO(3)(X0,1):imα]=4[\mathcal{G}^{SO(3)}(X_{0,1}):\operatorname{im}\alpha]=4[ caligraphic_G start_POSTSUPERSCRIPT italic_S italic_O ( 3 ) end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT 0 , 1 end_POSTSUBSCRIPT ) : roman_im italic_α ] = 4, which accounts for the factor ‘4’ in the statement. | A |
\mathbb{V}_{m}[(x_{1},x_{2},\ldots,x_{n})]}[(x_{1},x_{2},\ldots,x_{n})]over→ start_ARG italic_z end_ARG = ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ∈ caligraphic_B start_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG caligraphic_G ∘ blackboard_V start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT [ ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ] end_POSTSUBSCRIPT [ ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ] if it satisfies the inequality. | Theorem 2.11 tells us that points confined in certain balls induced under compression should by necessity have their induced ball under compression covered by these balls in which they are contained. | Next we prove that smaller balls induced by points should essentially be covered by the bigger balls in which they are embedded. We state and prove this statement in the following result. | Next we obtain an equivalent notion of the area of the circle induced by points under compression in the plane ℝ2superscriptℝ2\mathbb{R}^{2}blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT in the following result. | It is important to notice that the notion of admissible points of balls induced by points under compression encompasses points on the ball. These points in geometrical terms basically sit on the outer of the induced ball. Next we show that all balls can in principle be generated by their admissible points. | B |
Note that if ϕ,ϕ′italic-ϕsuperscriptitalic-ϕ′\phi,\phi^{\prime}italic_ϕ , italic_ϕ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT are two maps with the same constraint graph then ϕitalic-ϕ\phiitalic_ϕ and ϕ′superscriptitalic-ϕ′\phi^{\prime}italic_ϕ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT are the same up to permuting the indices in [n]delimited-[]𝑛[n][ italic_n ]. | Definition 2.18 (Labelings of H(α0,2k)𝐻subscript𝛼02𝑘H(\alpha_{0},2k)italic_H ( italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 2 italic_k )). | Definition 2.16 (Definition 3.8 of [AMP20]: Constraint graphs on H(α,2k)𝐻𝛼2𝑘H(\alpha,2k)italic_H ( italic_α , 2 italic_k )). | In this section, we show that the number of dominant constraint graphs on H(α,2k)𝐻𝛼2𝑘H(\alpha,2k)italic_H ( italic_α , 2 italic_k ) is Ck′subscriptsuperscript𝐶′𝑘C^{\prime}_{k}italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. | In [AMP20], this result was only proved for well-behaved constraint graphs (see Definition 3.19). That said, using the ideas in Appendix B of [AMP20], it can be shown for all constraint graphs C∈𝒞(α,2k)𝐶subscript𝒞𝛼2𝑘C\in\mathcal{C}_{(\alpha,2k)}italic_C ∈ caligraphic_C start_POSTSUBSCRIPT ( italic_α , 2 italic_k ) end_POSTSUBSCRIPT. | B |
∙∙\bullet∙ Case infn∈ℕ‖un‖=0subscriptinfimum𝑛ℕnormsubscript𝑢𝑛0\displaystyle\inf_{n\in\mathbb{N}}\|u_{n}\|=0roman_inf start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT ∥ italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∥ = 0. This means, either 00 is an accumulation point for real sequence {‖un‖}normsubscript𝑢𝑛\{\|u_{n}\|\}{ ∥ italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∥ } and hence there exists a subsequence of {un}subscript𝑢𝑛\{u_{n}\}{ italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } such that it strongly goes to 00. | In this section, we will establish the existence of arbitrarily many solutions to the problem (2). Prior to that let us define some useful tools to be used to guarantee the existence of solutions. | where λ>0𝜆0\lambda>0italic_λ > 0, 1<q<p<ps∗<∞1𝑞𝑝superscriptsubscript𝑝𝑠1<q<p<p_{s}^{*}<\infty1 < italic_q < italic_p < italic_p start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT < ∞. The authors in [23] have guaranteed the existence of infinitely many solutions. Such type of problems have led to the generalization of a few classical results for the case of 𝔐=1.𝔐1\mathfrak{M}=1.fraktur_M = 1 . In [42], the authors employed the Fountain and the dual Fountain theorem to guarantee the existence of infinitely many solutions for a symmetric subcritical Kirchhoff problem for a non-degenerate 𝔐𝔐\mathfrak{M}fraktur_M and p≥2𝑝2p\geq 2italic_p ≥ 2. In [13], the authors dealt with the case when p=2𝑝2p=2italic_p = 2 and used the notion of Krasnoselskii’s genus (refer [34]) to obtain the existence of infinitely many solutions. Further in [43] the authors had a similar conclusion but for a system of PDEs with subcritical degenerate Kirchhoff function. This is in no way a complete picture of the literature developed so far as it is vast. What we can do at this point is to direct the attention of the reader to the problem which prompted us to take up (1). The motivation of this problem was drawn from the results due to Azorero et al. [16], Khiddi-Sbai [23]. The literature consisting the study of infinitely many solution mainly deals with the concave-convex data, which may be both sub-linear as well as super-linear. Recently, in its first kind the study due to [19] guarantees the existence of infinitely solutions involving a singularity. | Motivated from the above studies, in this article, we prove that problem (1) possesses at least k∈ℕ𝑘ℕk\in\mathbb{N}italic_k ∈ blackboard_N solutions (for arbitrarily large k𝑘kitalic_k) within a finite range of λ𝜆\lambdaitalic_λ whose space norms converges to zero. It is worthy to mention here that the symmetric mountain pass theorem plays a key role to study the existence of infinitely many solutions to a PDE. The symmetric mountain pass theorem has two type of conclusions consisting a sequence of solutions. One is for sub-linear data in which the space norm of the solutions converges to zero another one is for the super-linear data which says the space norm of solutions goes to infinity. The major hurdles to us were to figure out a way to tackle the singular term as well as the critical exponent term, which is super-linear in the problem (1) and then to show that as k𝑘kitalic_k increases, the space norm of solutions decreases toward zero. To add to these issues, the functional also fails to be coercive. The main result proved in this article is the following. | The main hurdle with problems with critical exponent is the lack of compact embedding W0s,p(Ω)↪Lps∗(Ω)↪superscriptsubscript𝑊0𝑠𝑝Ωsuperscript𝐿subscriptsuperscript𝑝𝑠ΩW_{0}^{s,p}(\Omega)\hookrightarrow L^{p^{*}_{s}}(\Omega)italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( roman_Ω ) ↪ italic_L start_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( roman_Ω ). Such problems are tackled by the concentration-compactness principle introduced by Lions [27, 28] for the classical Sobolev spaces W1,p(Ω)superscript𝑊1𝑝ΩW^{1,p}(\Omega)italic_W start_POSTSUPERSCRIPT 1 , italic_p end_POSTSUPERSCRIPT ( roman_Ω ). The nonlocal version of this principle has also been developed and can be found in [44, Theorem 2.2]. The literature pertaining to these type of problems without singularity are so vast that it can’t be discussed here in this section completely. However, the readers may refer to the books [30, 32] and the references therein. Of late, existence and multiplicity of solutions to the Kirchhoff problem (1) have been investigated by many researchers. The reader may refer to [12, 14, 21, 40] and the references therein. In [14, 21], the authors have established the existence of at least two solutions by employing the Nehari manifold method. The authors in [40] investigated the existence of two solutions together with a Choquard term of a problem of the type (1). Fiscella [12] has employed the variational method in combination with a perturbation method to guarantee the existence of two solutions to the problem (1) even if the Kirchhoff function is degenerate. It is important to note that in all these studies the authors guaranteed the existence of two solutions by employing different variational tools. In this article, we establish the existence of arbitrarily many small solutions to the problem (1) by employing symmetric mountain pass theorem. The symmetric mountain pass theorem is mainly used to guarantee the existence of infinitely many solutions. Usually it is necessary to apply the symmetric mountain pass theorem to an elliptic PDE that the corresponding to a C1superscript𝐶1C^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT energy. This requirement fails to hold due to the singularity. We have tackled this delicate issue by employing a cut-off technique. | A |
Let G=(V,E)𝐺𝑉𝐸G=(V,E)italic_G = ( italic_V , italic_E ) be an ordered graph. The graphic matroid of | Let Gσsuperscript𝐺𝜎G^{\sigma}italic_G start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT be any orientation of an | Let Gσsuperscript𝐺𝜎G^{\sigma}italic_G start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT be any orientation of an | An oriented graph Gσsuperscript𝐺𝜎G^{\sigma}italic_G start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT is a directed | in Gσsuperscript𝐺𝜎G^{\sigma}italic_G start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT. In particular, | A |
It follows from the considerations in Section 2.5 and Section 5.1 that μj(e)superscript𝜇𝑗𝑒\mu^{j}(e)italic_μ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ( italic_e ) is the probability that a random layered spanning tree of the tropical curve does not contain the edge e𝑒eitalic_e (when sampled proportional to the product of the lengths of edges not present in the tree). | Figure 3. A layered graph (G,π)𝐺𝜋(G,\pi)( italic_G , italic_π ) with two layers and its four spanning trees T1,T2,T3,subscript𝑇1subscript𝑇2subscript𝑇3T_{1},T_{2},T_{3},italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , and T4subscript𝑇4T_{4}italic_T start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT. The underlying non-layered graph G𝐺Gitalic_G has five spanning trees, those of (G,π)𝐺𝜋(G,\pi)( italic_G , italic_π ) and the spanning tree T𝑇Titalic_T depicted in the figure. | Consider first the tropical curve of rank two with underlying layered graph from Example 3, and with edge lengths all equal to 1/2121/21 / 2. The corresponding probability distribution is uniform on four layered spanning trees. The resulting canonical measure is the Lebesgue measure. The total mass is two. | The integer r𝑟ritalic_r is called the rank of the layered graph G𝐺Gitalic_G, which thus equals the rank of the ordered partition π𝜋\piitalic_π. By an abuse of the notation, we use the same letter G𝐺Gitalic_G to denote the layered graph (V,E,π)𝑉𝐸𝜋(V,E,\pi)( italic_V , italic_E , italic_π ). The genus of a layered graph is defined as that of its underlying graph. | Consider now the same graph with ordered partition consisting of a single set, namely the set of all edges π1=Esubscript𝜋1𝐸\pi_{1}=Eitalic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_E, and with edge lengths all equal to 1/2121/21 / 2. The probability distribution on five spanning trees is uniform. In a random spanning tree, an edge is absent with probability either 2/5252/52 / 5 or 3/5353/53 / 5. In the associated canonical measure, two edges have a mass of 2/5252/52 / 5 and the other two a mass of 3/5353/53 / 5. The total mass is again two. | B |
In the following, we identify E(𝔲)𝐸𝔲E(\mathfrak{u})italic_E ( fraktur_u ) and the set of proper-edges of χ(𝔲)𝜒𝔲\chi(\mathfrak{u})italic_χ ( fraktur_u ) and view 𝔲̊̊𝔲\mathring{\mathfrak{u}}over̊ start_ARG fraktur_u end_ARG as a submap of χ(𝔲)𝜒𝔲\chi(\mathfrak{u})italic_χ ( fraktur_u ). | Note first that an oriented map admits at most one good labeling. If 𝔲𝔲\mathfrak{u}fraktur_u is a rooted unicellular well-oriented blossoming map, 𝔲𝔲\mathfrak{u}fraktur_u is well-labeled if and only if the labeling λ𝜆\lambdaitalic_λ of its corners defined in Section 3.2.2 is a good labeling. | We now present an alternative description of the closure, slightly more formal, which is built upon a labeling of the corners. Let 𝔲𝔲\mathfrak{u}fraktur_u be a rooted blossoming unicellular map. Recall that C(𝔲)𝐶𝔲C(\mathfrak{u})italic_C ( fraktur_u ) denotes the set of its corners. | Let 𝔲𝔲\mathfrak{u}fraktur_u be a rooted unicellular blossoming map, with the same number of leaves and buds. Then the closure of 𝔲𝔲\mathfrak{u}fraktur_u and the closure based on labels of 𝔲𝔲\mathfrak{u}fraktur_u coincide. | of a rooted unicellular blossoming map 𝔲𝔲\mathfrak{u}fraktur_u is the equivalence class of 𝔲𝔲\mathfrak{u}fraktur_u for root-equivalence. | B |
ℤ/2ℤ⊕ℤ/2N2ℤdirect-sumℤ2ℤℤ2subscript𝑁2ℤ\displaystyle\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2N_{2}\mathbb{Z}blackboard_Z / 2 blackboard_Z ⊕ blackboard_Z / 2 italic_N start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT blackboard_Z | 1≤N2≤4.1subscript𝑁24\displaystyle 1\leq N_{2}\leq 4.1 ≤ italic_N start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ 4 . | 1≤N2≤6,1subscript𝑁26\displaystyle 1\leq N_{2}\leq 6,1 ≤ italic_N start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ 6 , | 1≤N1≤12,N1≠11,formulae-sequence1subscript𝑁112subscript𝑁111\displaystyle 1\leq N_{1}\leq 12,N_{1}\neq 11,1 ≤ italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ 12 , italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≠ 11 , | N2=1,…,7.subscript𝑁21…7\displaystyle N_{2}=1,\dots,7.italic_N start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 1 , … , 7 . | B |
Since typically the amount of unlabeled data is large compared to the amount of labeled data, SSL methods are relevant when the performance of unsupervised learning is low, or when the cost of getting a large amount of labeled data for supervised learning is too high. | Moreover, concentration also occurs if we replace the adjacency matrix with the normalized Laplacian in Equation (3.7). In that case, we obtain a generalization of the Label Spreading algorithm [29], [11, Chapter 11]. | Semi-supervised learning (SSL) aims at achieving superior learning performance by combining unlabeled and labeled data. | Unfortunately, many standard SSL methods have been shown to not efficiently use the unlabeled data, leading to unsatisfactory or unstable performance [11, Chapter 4], [9, 12]. | Since typically the amount of unlabeled data is large compared to the amount of labeled data, SSL methods are relevant when the performance of unsupervised learning is low, or when the cost of getting a large amount of labeled data for supervised learning is too high. | C |
Anomalies in experimental data for the photodissociation of formaldehyde, H2COsubscriptH2CO{\rm H_{2}CO}roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_CO, [1] have led to new understanding of the dissociation dynamics of formaldehyde [2, 3, 4, 5, 6, 7] which has been referred to as the roaming mechanism for reaction dynamics. The widespread interest in this new reaction mechanism resulted in its discovery in a variety of reactions. Much of this work was described in the abundance of review papers that subsequently appeared [8, 9, 10, 11, 12, 13, 14]. | We find unstable periodic orbits, each of which is at the narrowest part of a different bottleneck on the surface of constant energy, that naturally divide the space into regions. One is at the edge of each potential well corresponding to the formation/breaking point of C-C and C-H bonds. The centrifugal barrier is located at approximately 22.922.922.922.9 a.u. and marks the location beyond which acetaldehyde has dissociated to radical products. These three orbits were also identified in Chesnavich’s phenomenological CH+4superscriptsubscriptabsent4{}_{4}^{+}start_FLOATSUBSCRIPT 4 end_FLOATSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT model [15, 28] and a similarly restricted formaldehyde [31]. In contrast to those systems, the restricted acetaldehyde has an additional unstable periodic orbit (and bottleneck) separating the two potential wells from what is usually referred to as the flat region. This bottleneck enables the reduced system to have trajectories that correspond to dissociation to molecular product over the potential saddle. Due to the absence of bottlenecks in the flat region, all roaming trajectories pass ‘near’ the centrifugal barrier while they roam, making the roaming mechanism in the restricted system equivalent to Chesnavich’s model and restricted formaldehyde. A detailed phase space analysis can be found in the Appendix. | Following the seminal studies on roaming in the dissociation of formaldehyde, attention was focused on roaming in acetaldehyde, CH3CHOsubscriptCH3CHO{\rm CH_{3}CHO}roman_CH start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT roman_CHO[16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Formaldehyde and acetaldehyde roaming dynamics present interesting contrasts. In formaldehyde the long range reorientational motion occurs between the HCO fragment and a hydrogen atom. In acetaldehyde the long range reorientation dynamics occurs between the HCO fragment and a methyl group, CH3subscriptCH3{\rm CH_{3}}roman_CH start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT. Both molecules possess competing dissociation channels to molecular and radical products. It was found that formaldehyde dissociation by roaming occurred much less frequently than dissociation to molecular products over a potential saddle point. In contrast, acetaldehyde prefers to dissociate to molecular products via roaming. | A reason for this could be the mass difference in the roaming fragment; H for formaldehyde and CH3subscriptCH3{\rm CH_{3}}roman_CH start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT for acetaldehyde. This question was investigated in [26] using a model that has been shown to exhibit the essential features of roaming, the Chesnavich model [27] for the reaction CH→4+{}_{4}^{+}\rightarrowstart_FLOATSUBSCRIPT 4 end_FLOATSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT → CH+3superscriptsubscriptabsent3{}_{3}^{+}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT + H [15, 28, 29]. An advantage of the Chesnavich model is that it is analytical and therefore the parameters that describe the reaction can be varied. In particular, the mass of the roaming fragment, H can be varied and we can study the effect of this variation on the roaming mechanism to dissociation. Following a detailed analysis of trajectories and phase space structures responsible for roaming, it was found that the variation of the mass of the roaming fragment had no significant effect on the quantity of trajectories dissociating via the roaming channel. Perhaps the simplicity of Chesnavich’s model is preventing a comparison of features mimicking formaldehyde and acetaldehyde. Recent work [30] raises questions about the nature and quantity of trajectories undergoing dissociation of acetaldehyde due to roaming at certain energies. | This begs the question, “what is roaming”? In [15] the authors noted that a dissociating molecule should possess two essential characteristics in order to label the reaction as “roaming”. In particular, the molecule should have competing dissociation channels, such as dissociation to molecular and radical products, and there should exist a long range attraction between fragments of the molecule. In a recent review article [14] Suits refines this definition of roaming even further by stating that “A roaming reaction is one that yields products via reorientational motion in the long-range (3−8383-83 - 8Å) region of the potential”. | D |
Ric44=Ric22=Ric24=0.subscriptRic44subscriptRic22subscriptRic240\operatorname{Ric}_{44}=\operatorname{Ric}_{22}=\operatorname{Ric}_{24}=0.roman_Ric start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT = roman_Ric start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT = roman_Ric start_POSTSUBSCRIPT 24 end_POSTSUBSCRIPT = 0 . | In this paper, we focus on the case when the underlying CR manifold has a Sasakian structure, i.e., it is strictly pseudoconvex and it admits an infinitesimal CR automorphism transversal to the distinguished distribution D𝐷Ditalic_D (see Section 2 for precise definitions). The main result of this article is Theorem 1.1, which is an analog to the theorem by Hill, Lewandowski and Nurowski, cited above. We remove the assumption of real analyticity and prove that a particular class of smooth 3-dimensional Sasakian manifolds admit lifts to quasi-Einstein Lorentzian manifolds with positive cosmological constant ΛΛ\Lambdaroman_Λ and of Petrov type II or D. Petrov type II or D means that the Weyl tensor is to some degree degenerate, that is certain components vanish and another certain component is different from 00. In what follows, we use (z=x+iy,w=u+iv)formulae-sequence𝑧𝑥i𝑦𝑤𝑢i𝑣(z=x+\mathrm{i}y,w=u+\mathrm{i}v)( italic_z = italic_x + roman_i italic_y , italic_w = italic_u + roman_i italic_v ) as the local coordinates of ℂ2superscriptℂ2\mathbb{C}^{2}blackboard_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. | Then the Goldberg-Sachs theorem [GoldSa, GHN] implies that the following components of the Weyl tensor vanish | for some function ρ𝜌\rhoitalic_ρ and for all vector fields v,w∈k⟂𝑣𝑤superscriptkperpendicular-tov,w\in\operatorname{k}^{\perp}italic_v , italic_w ∈ roman_k start_POSTSUPERSCRIPT ⟂ end_POSTSUPERSCRIPT. These integral curves are automatically geodesics and in the case of a 4-dimensional Lorentzian manifold they can be interpreted as light rays. The 4-dimensional case directly applies to General Relativity and has been intensively studied by many authors, see e.g. [DKS, RT1, LNT, Arman1, Arman2, Arman3, Rob, Traut1, Tafel1, RT1, Traut2, Traut3, Traut4, G, XF1, XF2]. The famous Goldberg-Sachs Theorem [GoldSa] (See also [GHN] for a generalisation of the Goldberg-Sachs theorem.) relates the existence of shearfree congruences to algebraically special solutions of the Einstein field equations for which the Weyl tensor satisfies certain degeneracy conditions, expressed through its so-called Petrov type. Of particular interest are the shearfree congruences with underlying Lorentzian metrics g𝑔gitalic_g that satisfy the Einstein condition | In the Theorem 3.2 below, proved in [HLN], we summarise a list of consequences for the functions 𝒫,𝒲𝒫𝒲\mathscr{P},\mathscr{W}script_P , script_W and ℋℋ\mathscr{H}script_H resulting from the vanishing of the complexified Ricci curvature restricted to the so-called α𝛼\alphaitalic_α-planes, that is the following components with respect to the frame (3.3) chosen above | B |
By Thm. 9.2 and Prohorov’s theorem, it is enough to show that any subsequential limit of the scaled KPZt line ensemble is the Airy line ensemble (after the parabolic shift). | A Brownian Gibbsian line ensemble is completely characterized by the finite-dimensional distributions of its top curve. | To this end, observe that by our main result the finite dimensional distributions of the top line of the scaled KPZt line ensemble, which is the the scaled time t𝑡titalic_t Hopf Cole solution to the narrow wedge initial data KPZ equation, converge to those of the top line of the Airy line ensemble. Hence the finite dimensional distributions of the top line of any subsequential limit of the scaled KPZt line ensemble match with those of the top line of the Airy line ensemble. Since any subsequential limit also has the Brownian Gibbs property, it follows from Thm. 9.3 that it is the Airy line ensemble (after the parabolic shift). | For t>0𝑡0t>0italic_t > 0 the scaled KPZt line ensemble is tight and any subsequential limit is a non-intersecting line ensemble with the Brownian Gibbs property. | By Thm. 9.2 and Prohorov’s theorem, it is enough to show that any subsequential limit of the scaled KPZt line ensemble is the Airy line ensemble (after the parabolic shift). | B |
Note that ΔwsubscriptΔ𝑤\Delta_{w}roman_Δ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT depends on the choice of S⊆G∖{e}𝑆𝐺𝑒S\subseteq G\setminus\{e\}italic_S ⊆ italic_G ∖ { italic_e }, and so does aW1,W,2subscript𝑎subscript𝑊1𝑊2a_{W_{1},W,2}italic_a start_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_W , 2 end_POSTSUBSCRIPT, bWsubscript𝑏𝑊b_{W}italic_b start_POSTSUBSCRIPT italic_W end_POSTSUBSCRIPT and cWsubscript𝑐𝑊c_{W}italic_c start_POSTSUBSCRIPT italic_W end_POSTSUBSCRIPT. However, once the the generic simplicity holds for S𝑆Sitalic_S, it follows from Theorem 1.1 that it also holds for every H⊆G∖{e}𝐻𝐺𝑒H\subseteq G\setminus\{e\}italic_H ⊆ italic_G ∖ { italic_e } containing S𝑆Sitalic_S. | To provide a deeper understanding of the theorem, we apply the established criterion in the following examples. These examples showcase situations where the irreducibility property holds true, as well as cases where it does not. By exploring both scenarios, we can gain a more nuanced understanding of the theorem and its range of applicability. | This paper is organized as follows. Section 2 introduces the fundamental definitions and properties of weighted Laplacians on Cayley graphs. In Section 3, we prove Theorem 1.1, the Schueth’s criterion for Cayley graphs, and we provide several examples to demonstrate its application. Lastly, in Section 4, we define a new operator, denoted as 𝙻wsubscript𝙻𝑤\operatorname{\mathtt{L}_{\mathnormal{w}}}typewriter_L start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT, that is similar to the weighted Laplacian. We then confirm that an equivalent criterion holds for the generic spectrum of 𝙻wsubscript𝙻𝑤\operatorname{\mathtt{L}_{\mathnormal{w}}}typewriter_L start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT. | The generic irreducibility question for operators on graphs has not received much atention. In our knowledge, up to now, Berkolaiko and Liu [2] is the only work in the setting of graphs with symmetries. They construct a family of combinatorial Schrodinger operators on symmetric graphs on which generic irreducibility fails. More precisely, they found a family of G𝐺Gitalic_G-invariant combinatorial Laplacians on graphs where the irreducibility of the eigenspace does not occur when G𝐺Gitalic_G is the tetrahedron symmetry group. In this paper our goal is to establish the generic situation of the spectrum of the Laplacian on finite Cayley graphs having the set of weights as parameter space. We point out that we provide families of graphs where the conjecture holds and does not hold. Our strategy consists in adapting Schueth’s method to Cayley graphs. | Zelditch [14] established the generic situation of spectrum of the Laplacian in the G𝐺Gitalic_G-manifolds with a finite group G𝐺Gitalic_G, under the hypothesis that dimension of M𝑀Mitalic_M is bigger than the degree of all orthogonal irreducible representation. The higher symmetry case occurs when the group G𝐺Gitalic_G acts transitively on the manifold, see [11, 8]. Schueth [11] provided a completely algebraic criterion for the existence of left invariant metrics compact Lie group G𝐺Gitalic_G such that each eigenspace of the Laplacian operator has an irreducible representation under the action of G𝐺Gitalic_G. In other words, the eigenspaces are G𝐺Gitalic_G-simple up to small perturbations on the set of left-invariant metrics when the criterion holds, i.e., the eigenvalues do not exhibit higher multiplicities than the ones prescribed by the symmetries. | A |
The operator Π−Π∗ΠsuperscriptΠ\Pi-\Pi^{*}roman_Π - roman_Π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is skew-adjoint, thus has purely imaginary spectrum. | Πo=Π∗∘(Id−Π+Π∗)−1.subscriptΠ𝑜superscriptΠsuperscriptIdΠsuperscriptΠ1\Pi_{o}=\Pi^{*}\circ(\textup{Id}-\Pi+\Pi^{*})^{-1}.roman_Π start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT = roman_Π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∘ ( Id - roman_Π + roman_Π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT . | Consequently, Id−Π+Π∗IdΠsuperscriptΠ\textup{Id}-\Pi+\Pi^{*}Id - roman_Π + roman_Π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is invertible and (7.18) follows. | Πo∘(Id−Π+Π∗)=Π∗.subscriptΠ𝑜IdΠsuperscriptΠsuperscriptΠ\Pi_{o}\circ(\textup{Id}-\Pi+\Pi^{*})=\Pi^{*}.roman_Π start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT ∘ ( Id - roman_Π + roman_Π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) = roman_Π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT . | such that P∘G=Id−Π𝑃𝐺IdΠP\circ G=\textup{Id}-\Piitalic_P ∘ italic_G = Id - roman_Π for a continuous projection ΠΠ\Piroman_Π in H2subscript𝐻2H_{2}italic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Then the operator Id−Π+Π∗IdΠsuperscriptΠ\textup{Id}-\Pi+\Pi^{*}Id - roman_Π + roman_Π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT in H2subscript𝐻2H_{2}italic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is invertible, and the | B |
Moreover, the cycles in 𝒞𝒞\mathcal{C}caligraphic_C are disjoint from P1(h)∪P3l+1(h)subscriptsuperscript𝑃ℎ1subscriptsuperscript𝑃ℎ3𝑙1P^{(h)}_{1}\cup P^{(h)}_{3l+1}italic_P start_POSTSUPERSCRIPT ( italic_h ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ italic_P start_POSTSUPERSCRIPT ( italic_h ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 italic_l + 1 end_POSTSUBSCRIPT. | Let (W,γ)𝑊𝛾(W,\gamma)( italic_W , italic_γ ) be a facially ΓΓ\Gammaroman_Γ-odd 3×2323\times 23 × 2-wall. | Let (W,γ)𝑊𝛾(W,\gamma)( italic_W , italic_γ ) be a strongly ΓΓ\Gammaroman_Γ-bipartite wall and let ℒℒ\mathcal{L}caligraphic_L be a pure ΓΓ\Gammaroman_Γ-odd linkage of (W,γ)𝑊𝛾(W,\gamma)( italic_W , italic_γ ) with |ℒ|=3lℒ3𝑙|\mathcal{L}|=3l| caligraphic_L | = 3 italic_l. | Let (W,γ)𝑊𝛾(W,\gamma)( italic_W , italic_γ ) be a facially ΓΓ\Gammaroman_Γ-odd 3l×(4k−1)3𝑙4𝑘13l\times(4k-1)3 italic_l × ( 4 italic_k - 1 )-wall. | Let (W,γ)𝑊𝛾(W,\gamma)( italic_W , italic_γ ) be a facially ΓΓ\Gammaroman_Γ-odd 3l×23𝑙23l\times 23 italic_l × 2-wall and let Pi(h)subscriptsuperscript𝑃ℎ𝑖P^{(h)}_{i}italic_P start_POSTSUPERSCRIPT ( italic_h ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT denote the i𝑖iitalic_i-th horizontal path of W𝑊Witalic_W for each i∈[3l+1]𝑖delimited-[]3𝑙1i\in[3l+1]italic_i ∈ [ 3 italic_l + 1 ]. | A |
{s})|\Big{]}{\rm d}s\leq Ct.blackboard_E start_POSTSUBSCRIPT blackboard_Q start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ ( italic_U start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT - italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] ≤ ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT blackboard_E start_POSTSUBSCRIPT blackboard_Q start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ | script_L start_POSTSUPERSCRIPT roman_dual end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_U start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) | ] roman_d italic_s ≤ italic_C italic_t . | On the other hand, there exists a (finite) constant C>0𝐶0C>0italic_C > 0 such that |ℒdualgx(1)(u)|≤Csuperscriptℒdualsuperscriptsubscript𝑔𝑥1𝑢𝐶|\mathscr{L}^{{\rm dual}}g_{x}^{(1)}(u)|\leq C| script_L start_POSTSUPERSCRIPT roman_dual end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_u ) | ≤ italic_C and |ℒdualgx(2)(u)|≤Cgx(1)(u)superscriptℒdualsuperscriptsubscript𝑔𝑥2𝑢𝐶superscriptsubscript𝑔𝑥1𝑢|\mathscr{L}^{{\rm dual}}g_{x}^{(2)}(u)|\leq Cg_{x}^{(1)}(u)| script_L start_POSTSUPERSCRIPT roman_dual end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ( italic_u ) | ≤ italic_C italic_g start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_u ), for u,x∈[0,1]𝑢𝑥01u,x\in[0,1]italic_u , italic_x ∈ [ 0 , 1 ]. Then, it follows from Proposition 1.1.5 in [8] that, for t≥0𝑡0t\geq 0italic_t ≥ 0 and x∈[0,1]𝑥01x\in[0,1]italic_x ∈ [ 0 , 1 ], | Another application of Proposition 1.1.5 in [8] and the previous inequality show that, for t≥0𝑡0t\geq 0italic_t ≥ 0 and x∈[0,1]𝑥01x\in[0,1]italic_x ∈ [ 0 , 1 ], | Then, the Markov property and the previous inequality imply that, for 0≤s<t0𝑠𝑡0\leq s<t0 ≤ italic_s < italic_t and u∈[0,1]𝑢01u\in[0,1]italic_u ∈ [ 0 , 1 ], | Theorem 1 (i) follows immediately from Proposition 6 (i) and Lemma 7. Theorem 1 (ii) follows from Proposition 6 (ii) and by using the coupling Ztc~,ρ≤Ztsubscriptsuperscript𝑍~𝑐𝜌𝑡subscript𝑍𝑡Z^{\tilde{c},\rho}_{t}\leq Z_{t}italic_Z start_POSTSUPERSCRIPT over~ start_ARG italic_c end_ARG , italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ≤ italic_Z start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT for all t≥0𝑡0t\geq 0italic_t ≥ 0. Finally, Theorem 1 (iii) follows Proposition 6 (i), Lemma 7, Lemma 8 and by using the coupling Zt≤Zt0,ρsubscript𝑍𝑡subscriptsuperscript𝑍0𝜌𝑡Z_{t}\leq Z^{0,\rho}_{t}italic_Z start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ≤ italic_Z start_POSTSUPERSCRIPT 0 , italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT for all t≥0𝑡0t\geq 0italic_t ≥ 0. | B |
We will see that there is a natural tangle associated to minimal counterexamples to Theorem 1.3, and we apply a structure theorem of [17] which roughly states that, given such a tangle, either there are many ΓΓ\Gammaroman_Γ-nonzero cycles distributed in one of three configurations, or there is a vertex set of bounded size intersecting every ΓΓ\Gammaroman_Γ-nonzero cycle in the highly connected part of the tangle. | Then, in directed ΓΓ\Gammaroman_Γ-labelled graphs, ΓΓ\Gammaroman_Γ-zero A𝐴Aitalic_A-paths satisfy the Erdős-Pósa property if and only if ΓΓ\Gammaroman_Γ is finite. | A ΓΓ\Gammaroman_Γ-labelled graph (G,γ)𝐺𝛾(G,\gamma)( italic_G , italic_γ ) is ΓΓ\Gammaroman_Γ-bipartite if every cycle in G𝐺Gitalic_G is ΓΓ\Gammaroman_Γ-zero. | In the first outcome we piece together the ΓΓ\Gammaroman_Γ-nonzero cycles to build many disjoint ΓΓ\Gammaroman_Γ-zero A𝐴Aitalic_A-paths. | We now use Lemma 4.9 to show that, in outcome (3) of Theorem 3.7, we can find many disjoint zero A𝐴Aitalic_A-paths. | C |
\gamma}(n))\,\geq\,1\,-\,\frac{O(1)}{r^{2}}.roman_lim start_POSTSUBSCRIPT italic_γ → - ∞ end_POSTSUBSCRIPT lim inf start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT italic_d start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ( italic_n ) ) ≥ 1 - divide start_ARG italic_O ( 1 ) end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG . | The structure of the proof is similar to the case of the cutoff phenomenon of the Glauber dynamics for the Curie-Weiss-Potts model in high temperature regime presented in [1]. The convergence to the stationary distribution is obtained by successively coalescing the proportion chains and the basket chains with the coupling methods. The major difference with the previous method is the construction of the appropriate couplings to deal with the asymmetric nature of the irreversible dynamics. They are based on the couplings introduced in [1], but more sophisticated constructions are needed in cases where symmetry is starting to break. | The following propositions bound the variance of the proportion chain value at time t𝑡titalic_t from the | the lower bound of the cutoff is then presented. Section 3 analyzes the coalescence of the proportion and basket chains. | [1], it is observed that the cutoff of the upper bound for the Glauber dynamics essentially follows from the precise bound on the | D |
In Sections 5 and 6 we consider more examples of spacer rank-two words and systems. In Section 5 we prove that all Sturmian words have a proper spacer rank-two construction, and that subshifts generated by Sturmian words have spacer rank two (in particular they do not have (spacer) rank one). In Section 6 we give additional examples of spacer rank-two systems and give a characterization of when a subshift generated by a spacer rank-two word has spacer rank two as a topological dynamical system. We end with a spacer rank-two system that has at least four orbit closures, in contrast to rank-one systems that have at most two orbit closures; hence this system cannot be topologically isomorphic to a rank-one system. | Starting from Section 4 we consider finite spacer rank subshifts as topological dynamical systems. More specifically, we consider subshifts defined by a one-sided infinite word and define the notion of spacer rank for such systems. | There are many questions regarding finite spacer rank words and subshifts that are left open by this paper. We hope that our results here will stimulate more research on this topic. | We conclude this section by showing that there are words with a spacer rank construction that do not have a finite spacer rank construction. | Note that Theorem 3.24 only shows that the words that are fixed points of proper constant-length substitutions are spacer rank-two, and that it does not show that the systems generated by those words are spacer rank-two. | B |
\right).italic_( italic_) ≤ 4 ( 2 italic_d - 1 ) start_POSTSUPERSCRIPT 2 italic_d end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT blackboard_P ( start_ARRAY start_ROW start_CELL each geodesic from italic_x to italic_y contains an edge end_CELL end_ROW start_ROW start_CELL italic_e ∈ over¯ start_ARG italic_R end_ARG ( italic_n ) with italic_t start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ≥ italic_M and italic_π is the only end_CELL end_ROW start_ROW start_CELL rung with at least divide start_ARG italic_n end_ARG start_ARG 2 end_ARG many edges, end_CELL end_ROW start_ROW start_CELL all whose edges italic_e have italic_t start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ≤ italic_r + divide start_ARG italic_δ end_ARG start_ARG 2 end_ARG end_CELL end_ROW end_ARRAY ) . | (4.27)≤4(2d−1)2dℙ(each geodesic from x to y contains an edge e∈R¯(n) with te≥M).italic-(4.27italic-)4superscript2𝑑12𝑑ℙeach geodesic from 𝑥 to 𝑦 contains an edge 𝑒¯𝑅𝑛 with subscript𝑡𝑒𝑀\eqref{eq: big_equation}\leq 4(2d-1)^{2d}\mathbb{P}(\text{each geodesic from }% | ℙ(each geodesic from x to y contains an edge e∈R¯(n) with te≥M)ℙeach geodesic from 𝑥 to 𝑦 contains an edge 𝑒¯𝑅𝑛 with subscript𝑡𝑒𝑀\displaystyle\mathbb{P}(\text{each geodesic from }x\text{ to }y\text{ contains% | an edge }e\in\overline{R}(n)\text{ with }t_{e}\geq M)blackboard_P ( each geodesic from italic_x to italic_y contains an edge italic_e ∈ over¯ start_ARG italic_R end_ARG ( italic_n ) with italic_t start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ≥ italic_M ) | (4.27)≤4(2d−1)2d∑πℙ(each geodesic from x to y contains an edgee∈R¯(n) with te≥M and π is the onlyrung with at least n2 many edges,all whose edges e have te≤r+δ2).italic-(4.27italic-)4superscript2𝑑12𝑑subscript𝜋ℙeach geodesic from 𝑥 to 𝑦 contains an edge𝑒¯𝑅𝑛 with subscript𝑡𝑒𝑀 and 𝜋 is the onlyrung with at least 𝑛2 many edges,all whose edges 𝑒 have subscript𝑡𝑒𝑟𝛿2\eqref{eq: big_equation}\leq 4(2d-1)^{2d}\sum_{\pi}\mathbb{P}\left(\begin{% | A |
\mathbf{x}_{\ast})}.∥ ∇ italic_f ( bold_x ) ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≥ square-root start_ARG divide start_ARG italic_α end_ARG start_ARG 2 end_ARG end_ARG square-root start_ARG italic_f ( bold_x ) - italic_f ( bold_x start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ) end_ARG . | Then, we introduce the definition of the strongly convex set, which has been well studied in offline optimization (Levitin and Polyak 1966; Demyanov and Rubinov 1970; Dunn 1979; Garber and Hazan 2015; Rector-Brooks, Wang, and Mozafari 2019). | Table 1: Comparisons of regret bounds for efficient projection-free online algorithms including OFW (Hazan and Kale 2012; Hazan 2016), LLOO-OCO (Garber and Hazan 2016), Fast OGD (Levy and Krause 2019), OSPF (Hazan and Minasyan 2020) and our algorithms. | More specifically, OFW is an online extension of an offline algorithm called Frank-Wolfe (FW) (Frank and Wolfe 1956; Jaggi 2013) that iteratively performs linear optimization steps to minimize a convex and smooth function. In each round, OFW updates the decision by utilizing a single step of FW to minimize a surrogate loss function, which implies that the approximation error caused by the single step of FW could be the main reason for the regret gap between projection-based algorithms and OFW. Recently, Garber and Hazan (2015) made a quadratic improvement in the convergence rate of FW for strongly convex and smooth offline optimization over strongly convex sets compared to the general case. They used a simple line search rule to choose the step-size of FW, which allows FW to converge faster even if the strong convexity of the decision set is unknown. It is therefore natural to ask whether the faster convergence of FW can be utilized to improve the regret of OFW. In this paper, we give an affirmative answer by improving OFW to achieve an regret bound of O(T2/3)𝑂superscript𝑇23O(T^{2/3})italic_O ( italic_T start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT ) over strongly convex sets, which is better than the original O(T3/4)𝑂superscript𝑇34O(T^{3/4})italic_O ( italic_T start_POSTSUPERSCRIPT 3 / 4 end_POSTSUPERSCRIPT ) regret bound. Inspired by Garber and Hazan (2015), the key idea is to refine the decaying step-size in the original OFW by a simple line search rule. | Furthermore, OFW has been extended to two practical scenarios. The first scenario is the bandit setting (Flaxman, Kalai, and McMahan 2005; Bubeck et al. 2015), where only the loss value is available to the player. Chen, Zhang, and Karbasi (2019) proposed the first bandit variant of OFW, and established an expected regret bound of O(T4/5)𝑂superscript𝑇45O(T^{4/5})italic_O ( italic_T start_POSTSUPERSCRIPT 4 / 5 end_POSTSUPERSCRIPT ). Later, two improved bandit variants of OFW were proposed to enjoy better expected regret bounds on the order of O(T3/4)𝑂superscript𝑇34O(T^{3/4})italic_O ( italic_T start_POSTSUPERSCRIPT 3 / 4 end_POSTSUPERSCRIPT ) for convex losses (Garber and Kretzu 2020a) and O~(T2/3)~𝑂superscript𝑇23\widetilde{O}(T^{2/3})over~ start_ARG italic_O end_ARG ( italic_T start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT ) for strongly convex losses (Garber and Kretzu 2020b). The second scenario is the distributed setting (Duchi, Agarwal, and Wainwright 2011; Hosseini, Chapman, and Mesbahi 2013), where many players are distributed over a network and each player can only access to the local loss function. The first projection-free algorithm for distributed OCO was proposed by Zhang et al. (2017), which requires the communication complexity of O(T)𝑂𝑇O(T)italic_O ( italic_T ). Recently, Wan, Tu, and Zhang (2020) further reduced the communication complexity from O(T)𝑂𝑇O(T)italic_O ( italic_T ) to O(T)𝑂𝑇O(\sqrt{T})italic_O ( square-root start_ARG italic_T end_ARG ). | A |
In particular, this representation allows us to write an explicit, integral-free time-dependent perturbation expansion for the time-evolution operator in the Schrödinger-picture enabling us to carry perturbation calculations without the need to switch to the interaction-picture. To showcase the utility of our formulation, we work out a number of examples, for which we explicitly calculate the first few terms of the series. | The main technical contribution of this work is an alternative, yet equivalent, formulation of the Dyson series wherein the summands are given in terms of efficiently computable divided differences of the exponential function. We argue that our novel representation, which is devoid of integrals and time ordering, makes it a very useful tool in the study of perturbation effects in many-body quantum systems — a fundamental branch of quantum physics. | We begin by a brief introduction of divided differences, followed by the derivation of the integral-free representation of the Dyson series. | Despite its fundamental role in quantum theory, one of the key challenges in using the Dyson series in practical applications remains the evaluation of the multi-dimensional integrals over products of time-ordered operators, making the calculation of the terms in the series an exceedingly complicated task Brouder et al. (2010). In this paper, we derive an analytical, closed-form expression for the summands of the Dyson series, by explicitly evaluating the Dyson integrals. We accomplish this by utilizing the machinery of ‘divided differences’ – a mathematical tool normally used for computing tables of logarithms and trigonometric functions and for calculating the coefficients in the interpolation polynomial in the Newton form Milne-Thomson (1933); Whittaker and Robinson (1967); de Boor (2005); Albash et al. (2017); Hen (2018); Gupta et al. (2019); Kalev and Hen (2021). | We derived an expansion that is equivalent to the usual Dyson series but which includes only sums of closed-form analytical simple-to-calculate expressions rather than the usual Dyson multi-dimensional time-ordered integrals. The terms at every expansion order in our new formulation coincide with those of the standard Dyson series, except that the Dyson integrals at every order replaced with finite sums. Therefore, both series share the same convergence criteria Dyson (1952); Thingna et al. (2014); Azam (2006); Avron and Kons (1999). However, our new formalism allowed us to write an integral-free perturbation expansion for the time-evolution operator. We illustrated the utility of our approach by working out a number of use cases and calculated the series coefficients for a number of examples for which the usual Dyson series calculation is cumbersome, demonstrating the functionality and practicality of our approach. | B |
Strong motivation for the study of quasiregular maps stems from works of Walsh [Wals] and Smale [Sm] (see Lemma 8.19), which establish a tight connection between the cofinite Hopf property of groups and open maps of CW-complexes. | Lemma 8.19 tells us that the results of our paper apply to open maps between finite CW-complexes whose fundamental groups satisfy the conditions of Theorem 1.3 and have trivial center. | If M𝑀Mitalic_M is a closed manifold whose fundamental group is given as in Theorem 1.3 and has trivial center, then every quasiregular map f:M→M:𝑓→𝑀𝑀f\colon M\to Mitalic_f : italic_M → italic_M is a homeomorphism. | Strong motivation for the study of quasiregular maps stems from works of Walsh [Wals] and Smale [Sm] (see Lemma 8.19), which establish a tight connection between the cofinite Hopf property of groups and open maps of CW-complexes. | Thus, Theorem 1.3 and its consequences apply to open maps between CW-complexes whose fundamental groups satisfy the conditions of Theorem 1.3 and have trivial center; see Remark 8.20 and Theorem 8.22. | D |
For example, we will show in the last section that the function f(x)=x1𝑓𝑥subscript𝑥1f(x)=\sqrt{x_{1}}italic_f ( italic_x ) = square-root start_ARG italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG can be extended to a weak slice regular on the slice-domain ΩΩ\Omegaroman_Ω defined by (9.1). However, according to Proposition 9.1, f𝑓fitalic_f cannot be extended to ΩΩ\Omegaroman_Ω as a strong slice regular function. We shall show, see Remark 5.3, that each strong slice regular extension is weak slice regular, but the converse does not hold, in general. | The class of weak slice regular and strong slice regular functions coincide on axially symmetric slice-domains. | the terminology in [Dou2020001]. This class of functions is defined by imposing the condition of being holomorphic on each slice ℂIsubscriptℂ𝐼\mathbb{C}_{I}blackboard_C start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT, I∈ℂ𝐼ℂI\in\mathbb{C}italic_I ∈ blackboard_C, instead of requiring that the stem functions are holomorphic. As it is well known, these two approaches give, in general, different classes of functions, unless one imposes additional conditions. The two classes are same on some axially symmetric sets. We now set | In this section, we prove that weak slice regular functions defined on axially symmetric slice-domains are slice functions. Hence they satisfy the | We prove that these two classes of functions coincide on axially symmetric domains in the slice-topology. | D |
Finally, observe that our sentinel functional J𝐽Jitalic_J is defined in the sense of L2−limit-fromsuperscript𝐿2L^{2}-italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT -norm. If we want to insensitize a functional with a norm greater than L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, for example, ∂xnusubscriptsuperscript𝑛𝑥𝑢\partial^{n}_{x}u∂ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_u, for n≥1𝑛1n\geq 1italic_n ≥ 1, then we need a system coupled in the second equation of (1.5) in the form ∂xn(1𝒪∂xnu)subscriptsuperscript𝑛𝑥subscript1𝒪subscriptsuperscript𝑛𝑥𝑢\partial^{n}_{x}(1_{\mathcal{O}}\partial^{n}_{x}u)∂ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( 1 start_POSTSUBSCRIPT caligraphic_O end_POSTSUBSCRIPT ∂ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_u ), this means, the coupling has twice as many derivatives. More details about this kind of problem will be given in Section 4. | In this spirit, there are many alternatives to define the sentinel functional related to the insensitizing control problems for 4NLS. Thus, we expect that these three works together with the results in this paper, open prospects to prove similar results considering a sentinel functional with the gradient of the solution. Moreover, since the Carleman estimate (2.3) has third-order terms, maybe it is possible, at some point, to adapt the arguments to consider a functional with the Laplacian of the solution of the 4NLS, but clearly, to prove it is necessary new arguments of those that were applied here, at least proving a new Carleman estimate for the fourth-order Schrödinger equation, as was done in [18, Theorem 1.1] for the Cahn-Hilliard type equation and as in [19] where a Carleman estimate for stochastic fourth order Schrödinger equation is showed. The readers are invited to read the recent and interesting work by Imanuvilov and Yamamoto [21], which proves a Carleman estimate for a fourth-order parabolic equation in general dimensions. | Let us now explain the ideas to prove the results introduced in the last subsection. The main strategy adopted is based on duality arguments (see, e.g. [15, 27]). Roughly speaking, we prove suitable observability inequalities for the solutions of an adjoint system, where the main tool is a new Carleman estimate. This Carleman estimate with the right-hand side in weight Sobolev spaces will be the key point to deal with the coupling terms of the linear system associated with (1.5). | As mentioned at the beginning of this work, the existence of insensitizing controls for (1.2) can be defined equivalently by means of a null controllability problem for a cascade type system similar to the initial (1.2). Indeed, this process can be systematized when, in defining the functional in (1.3), we study the condition given by (1.4). Precisely, by calculating the derivative in the sense of Gâteaux for the functional J𝐽Jitalic_J restricted to τ=0𝜏0\tau=0italic_τ = 0 and given that the functional is the localized L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT norm in 𝒪𝒪\mathcal{O}caligraphic_O of the solution u𝑢uitalic_u, the insensitizing condition (1.4) implies that we can reformulate this by a null controllability problem for a coupled system which is (1.5). Through these calculations, it can be proved that the left-hand side of the second equation in (1.5) is the adjoint state of the derivative of (1.2) with respect to τ𝜏\tauitalic_τ (at τ=0𝜏0\tau=0italic_τ = 0). Thus, the right-hand side couples this last equation with the localized state 1𝒪usubscript1𝒪𝑢1_{\mathcal{O}}u1 start_POSTSUBSCRIPT caligraphic_O end_POSTSUBSCRIPT italic_u. We remark that different definitions for the functional also imply different coupling terms (see [20], we also commented this in Section 4.3). After this, to ensure that the insensitivity condition of the functional is satisfied, i.e., to ensure (1.4), it is sufficient to ensure that v|t=0≡0evaluated-at𝑣𝑡00v|_{t=0}\equiv 0italic_v | start_POSTSUBSCRIPT italic_t = 0 end_POSTSUBSCRIPT ≡ 0 in ΩΩ\Omegaroman_Ω111See [4, 28] for a rigorous deduction of this fact and [34] for an explicit computation to obtain the cascade system with a general nonlinearity for a Ginzburg-Landau equation.. | Our work is outlined in the following way: Section 2 is devoted to presenting a new Carleman estimate which will be the key to proving the main result of this manuscript. In Section 3, we show the null controllability results, that is, the linear case (Theorem 1.3) and the nonlinear one (Theorem 1.2). Section 4, we present further comments and some open problems that seem to be of interest from the mathematical point of view. Finally, for | D |
In the Liu-Shen implementation of CEC based model, the Lagrange multiplier in the HS-estimator is chosen to be 20 and in the Liu-Shen estimator, it is fixed at 2000. They observed that for a refined velocity field it does not significantly affect the velocity profile in a range of 1000-20,000 except the peak velocity near the vortex cores in this flow. For the image sequences, the best result was obtained for the values α=100𝛼100\alpha=100italic_α = 100 and β=0.01𝛽0.01\beta=0.01italic_β = 0.01. It was also observed experimentally that the numerical scheme converges when the ratio β/α𝛽𝛼\beta/\alphaitalic_β / italic_α is less than or equal to 10−4superscript10410^{-4}10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT. | We have proposed a general framework for fluid motion estimation using a constraint-based refinement approach. We observed a surprising connection to the Cauchy-Riemann operator that diagonalizes the system leading to a diffusive phenomenon involving the divergence and the curl of the flow. For a particular choice of the additional constraint, we showed that our model closely approximates the continuity equation based model by a modified augmented Lagrangian approach. Additionally, we demonstrated that a flow-driven refinement process involving the curl of the flow outperforms the classical physics-based optical flow method without any additional assumptions on the image data. | In particular, our method uses a constraint-based refinement approach. As mentioned above, in the two-step technique, the first requirement is an initial flow estimate (u0,v0)subscript𝑢0subscript𝑣0(u_{0},v_{0})( italic_u start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) that obeys the classical optical flow principles like brightness constancy and pixel correspondence. This estimate may not be able to capture the underlying geometric features of the fluid flow. The main idea is to perform a refinement over this crude estimate to capture precise flow structures driven by additional constraints specific to applications. As a concrete example, we choose the initial estimate coming from the Horn and Schunck model [13]. It is well-known that this model is not well-suited for fluid motion estimation. In fact the global smoothness regularization damps both the divergence and vorticity of the motion field. We show in particular how this model can be adapted and refined through our approach. A special feature of our model is the diagonalization by the Cauchy-Riemann operator leading to a diffusion on the curl and a multiplicative perturbation of the laplacian on the divergence of the flow. | The paper is organized as follows. In Section 2, we give a detailed description of our model. Section 3 is devoted to the mathematical framework. Here we discuss the mathematical well-posedness and the regularity of the solutions. We also show the diagonalization process under the application of the Cauchy-Riemann operator. In Section 4, we show how for a particular choice of additional constraints, our model closely approximates the continuity equation model using a modified augmented Lagrangian formulation. We also employ the bounded constraint algorithm to show the convergence of the Uzawa iterates. Finally, in Section 5, we show our results on different datasets. | In Table (1) we have suggested two such choices for ψ𝜓\psiitalic_ψ, one penalizing the divergence of the flow and the other penalizing the curl. The operator ∇H:=(∂y,−∂x)assignsubscript∇𝐻subscript𝑦subscript𝑥\nabla_{H}:=(\partial_{y},-\partial_{x})∇ start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT := ( ∂ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT , - ∂ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) is called the orthogonal gradient, also referred to as the symplectic gradient in the literature [16]. This geometric constraint captures the rotational aspects of the flow better. In this work, we will particularly demonstrate that a flow-driven refinement process involving the curl of the flow outperforms the classical physics-based optical flow method without any additional assumptions on the image data. | A |
\Sigma).overroman_ˇ start_ARG italic_ρ end_ARG ( roman_Σ ) = roman_sup start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT roman_max start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT nth-root start_ARG italic_n end_ARG start_ARG ∥ roman_Σ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_i , italic_i end_POSTSUBSCRIPT end_ARG = italic_ρ ( roman_Σ ) . | The quantity ρˇ(Σ)ˇ𝜌Σ\check{\rho}(\Sigma)overroman_ˇ start_ARG italic_ρ end_ARG ( roman_Σ ) is called the generalized spectral radius, which may be different from the joint spectral radius in case of an unbounded set of matrices. | Note that ‖Σn‖i,isubscriptnormsuperscriptΣ𝑛𝑖𝑖\|\Sigma^{n}\|_{i,i}∥ roman_Σ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_i , italic_i end_POSTSUBSCRIPT is computed based on the computation of all |Σ|nsuperscriptΣ𝑛|\Sigma|^{n}| roman_Σ | start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT combinations. However, it is still reasonable since the problem of approximating the joint spectral radius is NP𝑁𝑃NPitalic_N italic_P-hard [10]. The theorem applies however very well if the set contains only one matrix, i.e. the case of the ordinary spectral radius. | The joint spectral radius is also independent of the matrix norm of choice like the ordinary spectral radius. In the text, both radii are denoted by ρ𝜌\rhoitalic_ρ, for which the radius to take depends on the argument (or one may simply relate ρ(A)=ρ({A})𝜌𝐴𝜌𝐴\rho(A)=\rho(\{A\})italic_ρ ( italic_A ) = italic_ρ ( { italic_A } )). | Note that (8) depends on (7), which is a special case of (3). However, (8) is independent of the joint spectral radius theorem. | D |
},\eta(r_{min})+\eta^{\prime}(r_{min})r_{min}\}italic_C start_POSTSUBSCRIPT italic_e italic_l end_POSTSUBSCRIPT = roman_min { italic_μ ( over~ start_ARG italic_s end_ARG start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT ) + 2 italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( over~ start_ARG italic_s end_ARG start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT ) over~ start_ARG italic_s end_ARG start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT , italic_η ( italic_r start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT ) + italic_η start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT ) italic_r start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT }. | 3 Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT regularity of the linearized system | Secondly, we linearize the resulting system and we show Lp−Lqsuperscript𝐿𝑝superscript𝐿𝑞L^{p}-L^{q}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT - italic_L start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT regularity property of a solution. This is discussed in Section 3 and it is obtained by means of the Weis multiplier theorem. | In this section we present a proof of Theorem 1.1. First, we transform system (1) into Lagrangian coordinates. Then a linearization of the transformed system is derived. Next, employing Theorem 3.1 we show that the solution operator to the linearized system is a contraction on a suitably chosen function space. Finally, by the Banach fixed point theorem we conclude the local-in-time well posedness of the transformed system and of (1) accordingly. | We omit the detailed presentation of the proof of the latter inequality in order to keep the length of the paper in a reasonable limit. By very similar manipulations we obtain for the last term of the difference in (55)2 that | C |
For our purposes, there is no risk of confusion because we will only need to work with the arch-with-membrane. | (some of which we mention shortly) that we can use to verify that the given 3333-manifolds are homeomorphic. | As with 2-3 moves, we will find it helpful to use a different visualisation of 0-2 moves in the special spine setting. | For our purposes, it will be helpful to use a different visualisation of 2-3 moves in the special spine setting. | Second, we use this terminology to paraphrase some key ideas from Matveev’s proof, since we will be reusing these ideas in Section 3.2. | D |
\Phi_{(\sigma,\tilde{\sigma})}:Y_{\sigma}\longrightarrow Y_{\eta}roman_Φ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_η , over~ start_ARG italic_η end_ARG ) end_POSTSUBSCRIPT ∘ roman_Ψ start_POSTSUBSCRIPT ( over~ start_ARG italic_σ end_ARG , over~ start_ARG italic_η end_ARG ) end_POSTSUBSCRIPT ∘ roman_Φ start_POSTSUBSCRIPT ( italic_σ , over~ start_ARG italic_σ end_ARG ) end_POSTSUBSCRIPT : italic_Y start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ⟶ italic_Y start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT | Fix ε>0𝜀0\varepsilon>0italic_ε > 0. The standard model semi-flat metric for an Iksubscript𝐼𝑘I_{k}italic_I start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT fiber relative to the holomorphic volume form ΩΩ\Omegaroman_Ω is the metric on the covering elliptic fibration | A point we are trying to emphasize is that the standard model semi-flat metric is defined on the universal cover of the model fibration π:Xmod→Δ∗:𝜋→subscript𝑋𝑚𝑜𝑑superscriptΔ\pi:X_{mod}\rightarrow\Delta^{*}italic_π : italic_X start_POSTSUBSCRIPT italic_m italic_o italic_d end_POSTSUBSCRIPT → roman_Δ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT, not on the fibration π:XΔ∗→Δ∗:𝜋→subscript𝑋superscriptΔsuperscriptΔ\pi:X_{\Delta^{*}}\rightarrow\Delta^{*}italic_π : italic_X start_POSTSUBSCRIPT roman_Δ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT → roman_Δ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT. A semi-flat metric relative to the section σ𝜎\sigmaitalic_σ, ωsf,σsubscript𝜔𝑠𝑓𝜎\omega_{sf,\sigma}italic_ω start_POSTSUBSCRIPT italic_s italic_f , italic_σ end_POSTSUBSCRIPT is then induced by using σ𝜎\sigmaitalic_σ to identify X𝑋Xitalic_X and Xmodsubscript𝑋𝑚𝑜𝑑X_{mod}italic_X start_POSTSUBSCRIPT italic_m italic_o italic_d end_POSTSUBSCRIPT. The model semi-flat metric is Ricci-flat on Xmodsubscript𝑋𝑚𝑜𝑑X_{mod}italic_X start_POSTSUBSCRIPT italic_m italic_o italic_d end_POSTSUBSCRIPT, and flat along the fibers of πmodsubscript𝜋𝑚𝑜𝑑\pi_{mod}italic_π start_POSTSUBSCRIPT italic_m italic_o italic_d end_POSTSUBSCRIPT (hence the title “semi-flat”), and thus the same holds for any induced semi-flat metric ωsf,σ,εsubscript𝜔𝑠𝑓𝜎𝜀\omega_{sf,\sigma,\varepsilon}italic_ω start_POSTSUBSCRIPT italic_s italic_f , italic_σ , italic_ε end_POSTSUBSCRIPT on XΔ∗subscript𝑋superscriptΔX_{\Delta^{*}}italic_X start_POSTSUBSCRIPT roman_Δ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT. | Let us describe a construction of semi-flat metrics which are not standard, in the sense of Definition 2.7. As before we pass to the universal cover of the model fibration, ℂ×ℋ>0ℂsubscriptℋabsent0\mathbb{C}\times\mathcal{H}_{>0}blackboard_C × caligraphic_H start_POSTSUBSCRIPT > 0 end_POSTSUBSCRIPT, with coordinates y=−log(z)𝑦𝑧y=-\log(z)italic_y = - roman_log ( italic_z ), and x𝑥xitalic_x the standard coordinate on ℂℂ\mathbb{C}blackboard_C, and let Ω=κ(e−y)dy∧dxΩ𝜅superscript𝑒𝑦𝑑𝑦𝑑𝑥\Omega=\kappa(e^{-y})dy\wedge dxroman_Ω = italic_κ ( italic_e start_POSTSUPERSCRIPT - italic_y end_POSTSUPERSCRIPT ) italic_d italic_y ∧ italic_d italic_x be the holomorphic (2,0)20(2,0)( 2 , 0 ) form on the universal cover ℂ×ℋ>0ℂsubscriptℋabsent0\mathbb{C}\times\mathcal{H}_{>0}blackboard_C × caligraphic_H start_POSTSUBSCRIPT > 0 end_POSTSUBSCRIPT. Let h(y)ℎ𝑦h(y)italic_h ( italic_y ) be a holomorphic function on ℋ>0subscriptℋabsent0\mathcal{H}_{>0}caligraphic_H start_POSTSUBSCRIPT > 0 end_POSTSUBSCRIPT and consider the translation map | To describe the standard semi-flat metric it is useful to pass to the universal cover of the model fibration. Therefore, we define a coordinate y=−log(z)𝑦𝑧y=-\log(z)italic_y = - roman_log ( italic_z ), and let | D |
3},\ldots,m_{1})|-2N^{2p-2}.= 4 | ∑ start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 2 italic_p - 3 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N - 1 end_POSTSUPERSCRIPT italic_j ( italic_n start_POSTSUBSCRIPT 2 italic_p - 1 end_POSTSUBSCRIPT , italic_n start_POSTSUBSCRIPT 2 italic_p - 2 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 2 italic_p - 3 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) | - 2 italic_N start_POSTSUPERSCRIPT 2 italic_p - 2 end_POSTSUPERSCRIPT . | If z∈\C∖[0,∞)𝑧\C0z\in\C\setminus[0,\infty)italic_z ∈ ∖ [ 0 , ∞ ), then we obtain the following functional equality: | We repeat the above calculation 2p−32𝑝32p-32 italic_p - 3 times. Then, we obtain the following inequality: | If z∈\C∖(−∞,0]𝑧\C0z\in\C\setminus(-\infty,0]italic_z ∈ ∖ ( - ∞ , 0 ], then we obtain the following functional equality: | We regard exp(2π−1ni/N)2𝜋1subscript𝑛𝑖𝑁\exp(2\pi\sqrt{-1}n_{i}/N)roman_exp ( 2 italic_π square-root start_ARG - 1 end_ARG italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT / italic_N ) as zisubscript𝑧𝑖z_{i}italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. We obtain the following equality: | B |
This list of previously known cases of topological recursion substantially intersects with Section 3.4, as in many cases where there exists an already known proof of the projection property, there also exists an already known proof of topological recursion. | It is important to stress that the present paper neatly combines all known topological recursion results for Hurwitz-type numbers into one framework, including the ones which require non-trivial ℏ2superscriptPlanck-constant-over-2-pi2\hbar^{2}roman_ℏ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-deformation, like the coefficients of the extended Ooguri-Vafa partition function and r𝑟ritalic_r-spin Hurwitz numbers, which previously were thought to be outliers (cf. [DBKPS23, Section 1.4], [DBKP+22, Introduction], and [ACEH18, ACEH20]. | There are many cases when the ℏ2superscriptPlanck-constant-over-2-pi2\hbar^{2}roman_ℏ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-deformation is not needed and we have ψ^(ℏ2,y)=ψ(z)^𝜓superscriptPlanck-constant-over-2-pi2𝑦𝜓𝑧\hat{\psi}(\hbar^{2},y)=\psi(z)over^ start_ARG italic_ψ end_ARG ( roman_ℏ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_y ) = italic_ψ ( italic_z ), y^(ℏ2,z)=y(z)^𝑦superscriptPlanck-constant-over-2-pi2𝑧𝑦𝑧\hat{y}(\hbar^{2},z)=y(z)over^ start_ARG italic_y end_ARG ( roman_ℏ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_z ) = italic_y ( italic_z ) [KZ15, ACEH20, KPS22, BDK+23], and there are also known cases where the ℏ2superscriptPlanck-constant-over-2-pi2\hbar^{2}roman_ℏ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-deformations are inevitable [DBKPS23, DBKP+22]. It is not known whether the General Principle, formulated this way, is actually working in full generality. But it works in all studied examples and can be even extended to the more general cases, for instance, there is a version of topological recursion that does not require the zeros of dx𝑑𝑥dxitalic_d italic_x to be simple [BE13]. | Up until now this has mostly been done on a case-by-case basis (with a notable, but still restricted, exception of the effort due to Alexandrov–Chapuy–Eynard–Harnad [ACEH18, ACEH20] which does not cover e.g. the important examples of the r𝑟ritalic_r-spin Hurwitz numbers and the Ooguri-Vafa partition function), with complicated separate proofs of topological recursion for many various cases of Hurwitz-type problems. | In the present paper, which is based on the results of our previous paper [BDBKS22], we give a new unified approach, which allows us to prove topological recursion for weighted Hurwitz numbers of very general type. | A |
The main motivation for proving such a type of adiabatic theorem stems from linear response theory and we refer to [9, 12] for a discussion of the problem of rigorously justifying linear response theory for gapped extended systems and the role of our version of the adiabatic theorem in it. In [9] we also announced the results of the present paper and put them into context. | Physically, this means that the adiabatic approximation breaks down when the adiabatic switching occurs at times that exceed the lifetime of the NEASS, an effect that has already been observed in adiabatic theory for resonances before, see, e.g., [1, 8]. | In order to keep the proofs manageable and transparent, we moved many technical arguments and some preliminaries to several appendices: in Appendix A we state the main result of [20] in the form we use it in our proof in Section 4; in Appendix B we adapt results of [18] on the existence of the infinite volume dynamics to our needs in order to define spaces of operators that ‘have a thermodynamic limit’; Appendix C briefly recalls a few facts about conditional expectations for fermionic operators; in Appendix D we prove the main new technical tools needed for our result, namely that the operations used in the construction of the super-adiabatic NEASS leave spaces of operators that have a thermodynamic limit invariant; | In order to describe fermionic system on the lattice ℤdsuperscriptℤ𝑑\mathbb{Z}^{d}blackboard_Z start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT in the thermodynamic limit, we consider sequences | Recently, adiabatic theorems for many-body lattice systems have been obtained with error estimates that are uniform in the size of the system, see [3, 14, 20]. While this uniformity clearly indicates that the corresponding adiabatic theorems should also apply to the infinite system dynamics, i.e. after a thermodynamic limit has been taken, on a technical level this is not a simple corollary. | D |
\mathop{\mathcal{A}\textrm{b}}\nolimits)Lex ( caligraphic_E start_POSTSUPERSCRIPT op end_POSTSUPERSCRIPT , start_BIGOP caligraphic_A b end_BIGOP ) and the functor iR:ℰ→𝒜r(ℰ):subscript𝑖𝑅→ℰsubscript𝒜𝑟ℰi_{R}\colon{\mathcal{E}}\to{\mathcal{A}}_{r}({\mathcal{E}})italic_i start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT : caligraphic_E → caligraphic_A start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( caligraphic_E ) is faithful (see Lemma 4.5). We don’t know whether it is always fully faithful, but when it is so, ℰℰ{\mathcal{E}}caligraphic_E is a fully exact subcategory of 𝒜r(ℰ)subscript𝒜𝑟ℰ{\mathcal{A}}_{r}({\mathcal{E}})caligraphic_A start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( caligraphic_E ) (see Lemma 4.8). | The main innovative tool of this paper are abelian envelopes of exact categories. We distinguish the right 𝒜r(ℰ)subscript𝒜𝑟ℰ{\mathcal{A}}_{r}({\mathcal{E}})caligraphic_A start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( caligraphic_E ) and left 𝒜l(ℰ)subscript𝒜𝑙ℰ{\mathcal{A}}_{l}({\mathcal{E}})caligraphic_A start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( caligraphic_E ) envelopes of an exact category ℰℰ{\mathcal{E}}caligraphic_E. One known way to associate an abelian category to an exact category is via the exact abelian hull. It is the universal abelian category among those which allow an exact functor from ℰℰ{\mathcal{E}}caligraphic_E into them. The left and right envelopes are similar universal abelian categories but for functors which are only right/left exact. The existence of the hull was proven in [Ade73, Ste14]. An arbitrary exact category does not need to have the right abelian envelope (see Example 4.7). We prove the existence of the right/left envelope under the condition that the exact category has enough projectives/injectives and the full subcategory of projectives/injectives has weak kernels/cokernels. We discuss further instances when the envelopes exist. For example, we show that the | The right abelian envelope extends to a monad on the 2-category of exact categories admitting the right abelian envelopes, see Section 4.3. | In Section 5 we focus on highest weight categories and show that they are abelian envelopes of thin categories. | In Section 4 we introduce the main tool of our paper, the machinery of abelian envelopes. The right abelian envelope for an exact category ℰℰ{\mathcal{E}}caligraphic_E is an abelian category 𝒜r(ℰ)subscript𝒜𝑟ℰ{\mathcal{A}}_{r}({\mathcal{E}})caligraphic_A start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( caligraphic_E ) together with a right exact functor iR:ℰ→𝒜r(ℰ):subscript𝑖𝑅→ℰsubscript𝒜𝑟ℰi_{R}\colon{\mathcal{E}}\to{\mathcal{A}}_{r}({\mathcal{E}})italic_i start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT : caligraphic_E → caligraphic_A start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( caligraphic_E ) which is universal for right exact functors ℰ→𝒜→ℰ𝒜{\mathcal{E}}\to{\mathcal{A}}caligraphic_E → caligraphic_A, with 𝒜𝒜{\mathcal{A}}caligraphic_A abelian. The left abelian envelope 𝒜l(ℰ)subscript𝒜𝑙ℰ{\mathcal{A}}_{l}({\mathcal{E}})caligraphic_A start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( caligraphic_E ) is defined analogously. Since 𝒜l(ℰop)≃𝒜r(ℰ)opsimilar-to-or-equalssubscript𝒜𝑙superscriptℰopsubscript𝒜𝑟superscriptℰop{\mathcal{A}}_{l}({\mathcal{E}}^{\mathop{\textrm{op}}\nolimits})\simeq{% | B |
\lambda_{h}^{T}(t)h_{x}(x(t))\right)y^{x,d}(t)}+ italic_λ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) ( italic_h start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_x ( italic_t ) ) italic_z ( italic_t ) ) start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT - italic_λ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( italic_t ) italic_h start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_x ( italic_t ) ) ) italic_y start_POSTSUPERSCRIPT italic_x , italic_d end_POSTSUPERSCRIPT ( italic_t ) | λhT(t)hx(x(t)))yx,d(t)+λfT(t)(fF)u(x(t),u(t))d(t)+λfT(t)hxT(x(t))yz,d(t)]dt.\displaystyle{\displaystyle\left.\left.\lambda_{h}^{T}(t)h_{x}(x(t))\right)y^{% | +λfT(t)(fF)u(x(t),u(t))d(t)+λfT(t)hxT(x(t))yz,d(t))dt\displaystyle{\displaystyle\left.+\lambda_{f}^{T}(t)(f_{F})_{u}(x(t),u(t))d(t)% | (−λfT(t)(fF)x(x(t),u(t))−λfT(t)(hxT(x(t))z(t))x\displaystyle\left(-\lambda_{f}^{T}(t)(f_{F})_{x}(x(t),u(t))-\lambda_{f}^{T}(t% | +λfT(t)(fF)u(x(t),u(t))d(t)+λfT(t)hxT(x(t))yz,d(t)+λfT(t)(hxT(x(t))z(t))xyx,d(t)superscriptsubscript𝜆𝑓𝑇𝑡subscriptsubscript𝑓𝐹𝑢𝑥𝑡𝑢𝑡𝑑𝑡superscriptsubscript𝜆𝑓𝑇𝑡superscriptsubscriptℎ𝑥𝑇𝑥𝑡superscript𝑦𝑧𝑑𝑡superscriptsubscript𝜆𝑓𝑇𝑡subscriptsuperscriptsubscriptℎ𝑥𝑇𝑥𝑡𝑧𝑡𝑥superscript𝑦𝑥𝑑𝑡\displaystyle{\displaystyle+\lambda_{f}^{T}(t)(f_{F})_{u}(x(t),u(t))d(t)+% | B |
Theorem 5.7 (Approximation of BqsI(Lτ(Ω))subscriptsuperscript𝐵subscript𝑠I𝑞superscript𝐿𝜏ΩB^{s_{\mathrm{I}}}_{q}(L^{\tau}(\Omega))italic_B start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT roman_I end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( italic_L start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ( roman_Ω ) ), ABqsA(Lτ(Ω))𝐴subscriptsuperscript𝐵subscript𝑠A𝑞superscript𝐿𝜏ΩAB^{s_{\mathrm{A}}}_{q}(L^{\tau}(\Omega))italic_A italic_B start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT roman_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( italic_L start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ( roman_Ω ) ) and MBqsM(Lτ(Ω))𝑀subscriptsuperscript𝐵subscript𝑠M𝑞superscript𝐿𝜏ΩMB^{s_{\mathrm{M}}}_{q}(L^{\tau}(\Omega))italic_M italic_B start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT roman_M end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( italic_L start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ( roman_Ω ) ) with Tensor Trains). | For Ω:=[0,1)DassignΩsuperscript01𝐷\Omega:=[0,1)^{D}roman_Ω := [ 0 , 1 ) start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT, | Let Ω:=[0,1)DassignΩsuperscript01𝐷\Omega:=[0,1)^{D}roman_Ω := [ 0 , 1 ) start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT and consider the approximation tools | Let Ω:=[0,1)DassignΩsuperscript01𝐷\Omega:=[0,1)^{D}roman_Ω := [ 0 , 1 ) start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT | Let Ω:=[0,1)DassignΩsuperscript01𝐷\Omega:=[0,1)^{D}roman_Ω := [ 0 , 1 ) start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT. | D |
Now let P𝑃Pitalic_P be the union of (the image of) all the periodic orbits of Z𝑍Zitalic_Z through x𝑥xitalic_x. By Lemma 4.3 it follows that, for every non-empty open set U⊆M𝑈𝑀U\subseteq Mitalic_U ⊆ italic_M, there is a periodic orbit segment γ𝛾\gammaitalic_γ that passes through x𝑥xitalic_x and intersects U𝑈Uitalic_U, hence a periodic orbit through x𝑥xitalic_x and U𝑈Uitalic_U which means that P∩U≠∅𝑃𝑈P\cap U\neq\emptysetitalic_P ∩ italic_U ≠ ∅, so P𝑃Pitalic_P is dense in M𝑀Mitalic_M. | We now take ΔΔ\Deltaroman_Δ as the union of all the points on periodic orbits as constructed in the preceding items. | For any non-empty open set U𝑈Uitalic_U, there are points x𝑥xitalic_x and y𝑦yitalic_y and orbits γxsubscript𝛾𝑥\gamma_{x}italic_γ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT and γysubscript𝛾𝑦\gamma_{y}italic_γ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT such that, following these orbits, the two points eventually become δ𝛿\deltaitalic_δ apart. The orbits are constructed by connecting the points to a sliding region and then following periodic orbits for some time. | Z𝑍Zitalic_Z is chaotic if it is topologically transitive, has sensitive dependence on initial conditions and the union of all Filippov periodic orbits is a dense set. | The above result we prove before proceeding to the proof of Theorem 2.1 and, as questioned before, should we get the set ΔΔ\Deltaroman_Δ to be a residual set? After the proof of Theorem 2.2 on Subsection 4.1 we make a discussion concerning the issues to get a residual set. | A |
A∩J2B=JA′.𝐴superscript𝐽2𝐵𝐽superscript𝐴′A\cap J^{2}B=JA^{\prime}.italic_A ∩ italic_J start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_B = italic_J italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT . | 0⟶A⟶B⟶𝜋C⟶0⟶0𝐴⟶𝐵𝜋⟶𝐶⟶00\longrightarrow A\longrightarrow B\overset{\pi}{\longrightarrow}C\longrightarrow | 0⟶A/JA′⟶B/J2B⟶C/J2C⟶0.⟶0𝐴𝐽superscript𝐴′⟶𝐵superscript𝐽2𝐵⟶𝐶superscript𝐽2𝐶⟶00\longrightarrow A/JA^{\prime}\longrightarrow B/J^{2}B\longrightarrow C/J^{2}C% | 0⟶A⟶B⟶C⟶0⟶0𝐴⟶𝐵⟶𝐶⟶00\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 00 ⟶ italic_A ⟶ italic_B ⟶ italic_C ⟶ 0 | 0⟶A⟶B⟶𝜋C⟶0⟶0𝐴⟶𝐵𝜋⟶𝐶⟶00\longrightarrow A\longrightarrow B\overset{\pi}{\longrightarrow}C\longrightarrow | B |
Now, we provide the notion of 𝐋𝐋\mathbf{L}bold_L-formulae in this new setting. We mimic the analogous notion given in Continuous Logic. | Given 𝐋𝐋\mathbf{L}bold_L a language based on a continuous structure, we define the notion of 𝐋𝐋\mathbf{L}bold_L-formula recursively, as follows: | We define the notion of satisfiability in an 𝐋𝐋\mathbf{L}bold_L-structure in an analogous ways as in Continuous Logic, by using the notion of L𝐿Litalic_L-conditions. | Now, we provide the notion of 𝐋𝐋\mathbf{L}bold_L-formulae in this new setting. We mimic the analogous notion given in Continuous Logic. | Given a language based on a continuous structure 𝐋𝐋{\bf L}bold_L, we define the notion of L-term recursively, as follows: | A |
If E⊂X𝐸𝑋E\subset Xitalic_E ⊂ italic_X is unbounded, then the diagonal diag(E)⊂Xndiag𝐸superscript𝑋𝑛\operatorname{diag}(E)\subset X^{n}roman_diag ( italic_E ) ⊂ italic_X start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is unbounded. | A good heuristic is that your favourite subcategory of PMet would satisfies axiom (A2) in the above sense if and only if it already has a classical operation which is called Cartesian product. | A subcategory S of PMet satisfies axiom (A2) if for any S-space X𝑋Xitalic_X and any integer n𝑛nitalic_n, there exists an S-object, called a n𝑛nitalic_nth Cartesian power of X𝑋Xitalic_X and written Xnsuperscript𝑋𝑛X^{n}italic_X start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, such that: | It implies in particular that a countable group has uncountable cofinality if and only if it is finitely generated. | We now turn our attention to properties that behave well under finite Cartesian products in the sense of axiom (A2). | A |
When using differential privacy in machine learning we typically use δ=1/N𝛿1𝑁\delta=1/Nitalic_δ = 1 / italic_N (or 1/(10N)110𝑁1/(10N)1 / ( 10 italic_N )) inversely proportional with the data set size N𝑁Nitalic_N. | In order to prevent data leakage from inference attacks in machine learning [24] such as the deep leakage from gradients attack | the added gradients need to be bounded by some constant C𝐶Citalic_C in order for the DP analysis to go through. | When we talk about accuracy (from Figure 7 and onward), we mean test accuracy defined as the fraction of samples from a test data set that get accurately labeled by the classifier (as a result of training on a training data set by minimizing a corresponding objective function). | Towards understanding how to a-priori set parameters for best utility and minimal privacy leakage, the tight f𝑓fitalic_f-DP formulation for DP-SGD can be translated into sharp privacy guarantees. However, | A |
I would like to express our gratitude to our research team in Aix-Marseille Université, especially to Dierk Schleicher who supported this project from the very beginning, Sergey Shemyakov who carefully proofread all drafts, as well as to Kostiantyn Drach, Mikhail Hlushchanka, Bernhard Reinke and Roman Chernov for uncountably many enjoyable and enlightening discussions of this project at different stages. I also want to thank Dzmitry Dudko for his multiple suggestions that helped to advance the project, Lasse Rempe for his long list of comments and relevant questions, and Adam Epstein for important discussions especially in the early stages of this project. | In this appendix we prove some properties of polynomials and of their compositions with the exponential, mainly those connecting the magnitude of the set of singular (or critical) values to the size of the coefficients. | We are finally ready to construct the invariant compact subset 𝒞fsubscript𝒞𝑓\mathcal{C}_{f}caligraphic_C start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT. In the first theorem we present the construction and prove the invariance. The statement that 𝒞fsubscript𝒞𝑓\mathcal{C}_{f}caligraphic_C start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT is compact will be the content of the second theorem afterwards. | I would like to express our gratitude to our research team in Aix-Marseille Université, especially to Dierk Schleicher who supported this project from the very beginning, Sergey Shemyakov who carefully proofread all drafts, as well as to Kostiantyn Drach, Mikhail Hlushchanka, Bernhard Reinke and Roman Chernov for uncountably many enjoyable and enlightening discussions of this project at different stages. I also want to thank Dzmitry Dudko for his multiple suggestions that helped to advance the project, Lasse Rempe for his long list of comments and relevant questions, and Adam Epstein for important discussions especially in the early stages of this project. | Finally, I am grateful to funding by the Deutsche Forschungsgemeinschaft and the European Research Council with the Advanced Grant “Hologram” (695621) whose support provided excellent conditions for the development of this project. | D |
Then we have 9 points which constitute the Gale diagram of a 5-polytope with 9 vertices. Its polar, which we will call P𝑃Pitalic_P, is a 5-polytope with 9 facets. The three planes contain the y𝑦yitalic_y-axis, so Theorem 5.1 ensures that P𝑃Pitalic_P is decomposable. Indeed there are no other lower dimensional cofacets, so the intersection of all the cofacets is just the intersection of the three triangular cofacets, which contains a segment in the y𝑦yitalic_y-axis. | If there are two different diameters, the intersection of these diameters will be the origin, then the intersection of all cofacets is the origin. And for any polytopes combinatorially equivalent to it, the two different diameters cannot coincide, so the intersection of all cofacets is the origin which is a combinatorial property, i.e. P𝑃Pitalic_P is combinatorially indecomposable. And if there are only identical diameters, those points remains colinear with respect to combinatorial equivalence, hence P𝑃Pitalic_P is combinatorially decomposable. | However if we tilt one the triangles slightly, the combinatorial structure of P𝑃Pitalic_P will not change, but the three planes, and hence the three triangles, will intersect only at the origin. This P𝑃Pitalic_P is conditionally decomposable. | Then we have 9 points which constitute the Gale diagram of a 5-polytope with 9 vertices. Its polar, which we will call P𝑃Pitalic_P, is a 5-polytope with 9 facets. The three planes contain the y𝑦yitalic_y-axis, so Theorem 5.1 ensures that P𝑃Pitalic_P is decomposable. Indeed there are no other lower dimensional cofacets, so the intersection of all the cofacets is just the intersection of the three triangular cofacets, which contains a segment in the y𝑦yitalic_y-axis. | Ad−1=(0,…,0), the origin,subscript𝐴𝑑10…0 the origin\displaystyle A_{d-1}=(0,\dots,0),\text{ the origin},italic_A start_POSTSUBSCRIPT italic_d - 1 end_POSTSUBSCRIPT = ( 0 , … , 0 ) , the origin , | B |
The hypothesized constraints on the Conley-Zehnder indices of x^^𝑥\hat{x}over^ start_ARG italic_x end_ARG and y^^𝑦\hat{y}over^ start_ARG italic_y end_ARG imply that μ(x^)∈{2k+1,2k}𝜇^𝑥2𝑘12𝑘\mu(\hat{x})\in\{2k+1,2k\}italic_μ ( over^ start_ARG italic_x end_ARG ) ∈ { 2 italic_k + 1 , 2 italic_k } and μ(y^)∈{2k,2k−1}𝜇^𝑦2𝑘2𝑘1\mu(\hat{y})\in\{2k,2k-1\}italic_μ ( over^ start_ARG italic_y end_ARG ) ∈ { 2 italic_k , 2 italic_k - 1 }, and so by Lemma 3.20 we deduce that b(x^)=−k=a(y^)𝑏^𝑥𝑘𝑎^𝑦b(\hat{x})=-k=a(\hat{y})italic_b ( over^ start_ARG italic_x end_ARG ) = - italic_k = italic_a ( over^ start_ARG italic_y end_ARG ), which contradicts the above inequality. Thus, Ev~~𝐸𝑣\widetilde{Ev}over~ start_ARG italic_E italic_v end_ARG is injective. | That Ev~~𝐸𝑣\widetilde{Ev}over~ start_ARG italic_E italic_v end_ARG is proper essentially follows from compactness results in Floer theory; if | πˇ:W~(x^,y^)→W(x^,y^):ˇ𝜋→~𝑊^𝑥^𝑦𝑊^𝑥^𝑦\check{\pi}:\widetilde{W}(\hat{x},\hat{y})\rightarrow W(\hat{x},\hat{y})overroman_ˇ start_ARG italic_π end_ARG : over~ start_ARG italic_W end_ARG ( over^ start_ARG italic_x end_ARG , over^ start_ARG italic_y end_ARG ) → italic_W ( over^ start_ARG italic_x end_ARG , over^ start_ARG italic_y end_ARG ), with fiber πˇ−1(t,p)={Ev~(s,t,u):u(s,t)=p}superscriptˇ𝜋1𝑡𝑝conditional-set~𝐸𝑣𝑠𝑡𝑢𝑢𝑠𝑡𝑝\check{\pi}^{-1}(t,p)=\{\widetilde{Ev}(s,t,u):u(s,t)=p\}overroman_ˇ start_ARG italic_π end_ARG start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_t , italic_p ) = { over~ start_ARG italic_E italic_v end_ARG ( italic_s , italic_t , italic_u ) : italic_u ( italic_s , italic_t ) = italic_p }, which under the hypotheses of Proposition 6.5 may be identified via Ev~~𝐸𝑣\widetilde{Ev}over~ start_ARG italic_E italic_v end_ARG with the orbit of any (s0,t,u0)∈ℝ×S1×ℳ~(x^,y^;H,J)subscript𝑠0𝑡subscript𝑢0ℝsuperscript𝑆1~ℳ^𝑥^𝑦𝐻𝐽(s_{0},t,u_{0})\in\mathbb{R}\times S^{1}\times\widetilde{\mathcal{M}}(\hat{x},% | It remains to show that Ev~~𝐸𝑣\widetilde{Ev}over~ start_ARG italic_E italic_v end_ARG is an immersion when μ(x^)−μ(y^)∈{1,2}𝜇^𝑥𝜇^𝑦12\mu(\hat{x})-\mu(\hat{y})\in\{1,2\}italic_μ ( over^ start_ARG italic_x end_ARG ) - italic_μ ( over^ start_ARG italic_y end_ARG ) ∈ { 1 , 2 }. We note that | We may suppose that ℳ~(x^,y^;H,J)≠∅~ℳ^𝑥^𝑦𝐻𝐽\widetilde{\mathcal{M}}(\hat{x},\hat{y};H,J)\neq\emptysetover~ start_ARG caligraphic_M end_ARG ( over^ start_ARG italic_x end_ARG , over^ start_ARG italic_y end_ARG ; italic_H , italic_J ) ≠ ∅, or else the proposition is vacuously true. Moreover, if x^=y^^𝑥^𝑦\hat{x}=\hat{y}over^ start_ARG italic_x end_ARG = over^ start_ARG italic_y end_ARG, then the statement is obvious. Thus, we may suppose that μ(x^)−μ(y^)∈{1,2}𝜇^𝑥𝜇^𝑦12\mu(\hat{x})-\mu(\hat{y})\in\{1,2\}italic_μ ( over^ start_ARG italic_x end_ARG ) - italic_μ ( over^ start_ARG italic_y end_ARG ) ∈ { 1 , 2 }. We will show that Ev~~𝐸𝑣\widetilde{Ev}over~ start_ARG italic_E italic_v end_ARG is a proper injective immersion. To see that Ev~~𝐸𝑣\widetilde{Ev}over~ start_ARG italic_E italic_v end_ARG is one-to-one, note that Ev~(s,t,u)=Ev~(s′,t′,v)~𝐸𝑣𝑠𝑡𝑢~𝐸𝑣superscript𝑠′superscript𝑡′𝑣\widetilde{Ev}(s,t,u)=\widetilde{Ev}(s^{\prime},t^{\prime},v)over~ start_ARG italic_E italic_v end_ARG ( italic_s , italic_t , italic_u ) = over~ start_ARG italic_E italic_v end_ARG ( italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v ) if and only if u≠v𝑢𝑣u\neq vitalic_u ≠ italic_v and the graphs of u~~𝑢\tilde{u}over~ start_ARG italic_u end_ARG and v~~𝑣\tilde{v}over~ start_ARG italic_v end_ARG intersect over (s,t)=(s′,t′)𝑠𝑡superscript𝑠′superscript𝑡′(s,t)=(s^{\prime},t^{\prime})( italic_s , italic_t ) = ( italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ). Thus, by Lemma 3.13 Ev~~𝐸𝑣\widetilde{Ev}over~ start_ARG italic_E italic_v end_ARG fails to be injective only if there exist u,v∈M~(x^,y^;H,J)𝑢𝑣~𝑀^𝑥^𝑦𝐻𝐽u,v\in\widetilde{M}(\hat{x},\hat{y};H,J)italic_u , italic_v ∈ over~ start_ARG italic_M end_ARG ( over^ start_ARG italic_x end_ARG , over^ start_ARG italic_y end_ARG ; italic_H , italic_J ), u≠v𝑢𝑣u\neq vitalic_u ≠ italic_v such that | A |
{a,b}×{e,t0}𝑎𝑏𝑒subscript𝑡0\displaystyle\{a,b\}\times\{e,t_{0}\}{ italic_a , italic_b } × { italic_e , italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT } | =ξa−babsentsuperscript𝜉𝑎𝑏\displaystyle=\xi^{a-b}= italic_ξ start_POSTSUPERSCRIPT italic_a - italic_b end_POSTSUPERSCRIPT | =ξa+babsentsuperscript𝜉𝑎𝑏\displaystyle=\xi^{a+b}= italic_ξ start_POSTSUPERSCRIPT italic_a + italic_b end_POSTSUPERSCRIPT | ,1,\sigma)(x)=y.( italic_ξ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT , italic_ξ start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT , 1 ) italic_x = italic_ξ start_POSTSUPERSCRIPT - italic_a end_POSTSUPERSCRIPT italic_x , ( italic_ξ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT , italic_ξ start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT , 1 ) italic_y = italic_ξ start_POSTSUPERSCRIPT - italic_b end_POSTSUPERSCRIPT italic_y , ( 1 , 1 , italic_σ ) ( italic_x ) = italic_y . | {a,b}2.superscript𝑎𝑏2\displaystyle\{a,b\}^{2}.{ italic_a , italic_b } start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . | D |
By the assumption for T𝑇Titalic_T, we have 2|qv−2564+qv−3964|<0.92superscriptsubscript𝑞𝑣2564superscriptsubscript𝑞𝑣39640.92\,\left|q_{v}^{-\frac{25}{64}}+q_{v}^{-\frac{39}{64}}\right|<0.92 | italic_q start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - divide start_ARG 25 end_ARG start_ARG 64 end_ARG end_POSTSUPERSCRIPT + italic_q start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - divide start_ARG 39 end_ARG start_ARG 64 end_ARG end_POSTSUPERSCRIPT | < 0.9 for any v∉T𝑣𝑇v\notin Titalic_v ∉ italic_T. | Hence, by Theorem 18 and the argument in the proof of Lemma 20, we find for v∉T𝑣𝑇v\notin Titalic_v ∉ italic_T | with some 0≤av≤20subscript𝑎𝑣20\leq a_{v}\leq 20 ≤ italic_a start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ≤ 2 and 0≤bv≤10subscript𝑏𝑣10\leq b_{v}\leq 10 ≤ italic_b start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ≤ 1 for v∉T𝑣𝑇v\notin Titalic_v ∉ italic_T. | Let Uvsubscript𝑈𝑣U_{v}italic_U start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT and ΦvsubscriptΦ𝑣\Phi_{v}roman_Φ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT be as in the proof of Lemma 26. | By the same argument as in the proof of Lemma 27, it suffices to show the absolute convergence at s=1𝑠1s=1italic_s = 1. | A |
The case n=3𝑛3n=3italic_n = 3 relies on a deep classification theorem for joinings by Einsiedler and Lindenstrauss [joiningsfinal]; effective versions of that theorem are well out of reach of current methods from homogeneous dynamics. | For k=1𝑘1k=1italic_k = 1, this oriented version already appears in the works [AES-dim3, AES-higherdim]. | We remark that, as it is written, [AES-dim3, AES-higherdim] treat only the case where Q𝑄Qitalic_Q is the sum of squares (that we will sometimes call the standard form), but the arguments carry over without major difficulties. | This has been studied extensively by the first named author with Einsiedler and Shapira in [AES-dim3, AES-higherdim] where the conjecture is settled for n≥6𝑛6n\geq 6italic_n ≥ 6 (i.e. n−k≥5𝑛𝑘5n-k\geq 5italic_n - italic_k ≥ 5), for n=4,5𝑛45n=4,5italic_n = 4 , 5 under a weak congruence condition and for n=3𝑛3n=3italic_n = 3 under a stronger congruence condition on D𝐷Ditalic_D. | Assuming the Generalized Riemann Hypothesis, Blomer and Brumley [BlomerBrumley] have recently removed the congruence condition in [AES-dim3]. | D |
\beta}}\zeta^{1-1/\beta}}{\zeta+\lambda t^{\beta}}.italic_F ( italic_ζ , italic_β ) = divide start_ARG 1 end_ARG start_ARG 2 italic_π italic_i italic_β italic_λ italic_t start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT end_ARG ⋅ divide start_ARG italic_e start_POSTSUPERSCRIPT italic_ζ start_POSTSUPERSCRIPT 1 / italic_β end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_ζ start_POSTSUPERSCRIPT 1 - 1 / italic_β end_POSTSUPERSCRIPT end_ARG start_ARG italic_ζ + italic_λ italic_t start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT end_ARG . | Let us denote by δ(θ)𝛿𝜃\delta(\theta)italic_δ ( italic_θ ) a contour traversed in the direction of non-decreasing argζ𝜁\arg\zetaroman_arg italic_ζ and consisting of the following parts: ray argζ=−θ𝜁𝜃\arg\zeta=-\thetaroman_arg italic_ζ = - italic_θ , |ζ|≥1𝜁1|\zeta|\geq 1| italic_ζ | ≥ 1, arc (−θ)≤argζ≤θ𝜃𝜁𝜃(-\theta)\leq\arg\zeta\leq\theta( - italic_θ ) ≤ roman_arg italic_ζ ≤ italic_θ, |ζ|=1𝜁1|\zeta|=1| italic_ζ | = 1 and ray argζ=θ𝜁𝜃\arg\zeta=\thetaroman_arg italic_ζ = italic_θ, |ζ|≥1𝜁1|\zeta|\geq 1| italic_ζ | ≥ 1. If 0<θ<π0𝜃𝜋0<\theta<\pi0 < italic_θ < italic_π, then the contour δ(θ)𝛿𝜃\delta(\theta)italic_δ ( italic_θ ) divides the entire complex ζ𝜁\zetaitalic_ζ - plane into two unbounded parts, namely G(−)(θ)superscript𝐺𝜃G^{(-)}(\theta)italic_G start_POSTSUPERSCRIPT ( - ) end_POSTSUPERSCRIPT ( italic_θ ) to the left of δ(θ)𝛿𝜃\delta(\theta)italic_δ ( italic_θ ) and G(+)(θ)superscript𝐺𝜃G^{(+)}(\theta)italic_G start_POSTSUPERSCRIPT ( + ) end_POSTSUPERSCRIPT ( italic_θ ) to the right of it. Contour δ(θ)𝛿𝜃\delta(\theta)italic_δ ( italic_θ ) | Note that the domain of integration δ(θ)𝛿𝜃\delta(\theta)italic_δ ( italic_θ ) also depends on β𝛽\betaitalic_β. To consider this when differentiating function | Note that the integrand in equation (43) is G(ζ,α)𝐺𝜁𝛼G(\zeta,\alpha)italic_G ( italic_ζ , italic_α ) (see (40)). By virtue of the choice of θ𝜃\thetaitalic_θ, along the contour δ(θ)𝛿𝜃\delta(\theta)italic_δ ( italic_θ ), the following relations hold | −λtβ∈G(−)(θ)𝜆superscript𝑡𝛽superscript𝐺𝜃-\lambda t^{\beta}\in G^{(-)}(\theta)- italic_λ italic_t start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT ∈ italic_G start_POSTSUPERSCRIPT ( - ) end_POSTSUPERSCRIPT ( italic_θ ) and by the definition of contour δ(θ)𝛿𝜃\delta(\theta)italic_δ ( italic_θ ), we have (see as6 , p.135) | B |
Then F𝐹Fitalic_F factors through AMsubscript𝐴𝑀A_{M}italic_A start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT and induces an isomorphism F:AM→A(VM):𝐹→subscript𝐴𝑀𝐴subscript𝑉𝑀F:A_{M}\rightarrow A(V_{M})italic_F : italic_A start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT → italic_A ( italic_V start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ). | By Lemma 2.2 in [DLM98(2)], for any a∈V0𝑎superscript𝑉0a\in V^{0}italic_a ∈ italic_V start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT and b∈V𝑏𝑉b\in Vitalic_b ∈ italic_V, we have | We first show that F(R)=0𝐹𝑅0F(R)=0italic_F ( italic_R ) = 0. Indeed, by (2.5), (4.3), and Lemma 4.5, we have | First, we show that emα+nβ∈C1(VM)superscript𝑒𝑚𝛼𝑛𝛽subscript𝐶1subscript𝑉𝑀e^{m\alpha+n\beta}\in C_{1}(V_{M})italic_e start_POSTSUPERSCRIPT italic_m italic_α + italic_n italic_β end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_V start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ) for any m≥n≥2𝑚𝑛2m\geq n\geq 2italic_m ≥ italic_n ≥ 2. Indeed, for any m≥n≥1𝑚𝑛1m\geq n\geq 1italic_m ≥ italic_n ≥ 1, since (α+β|mα+nβ)=m+n≥2𝛼conditional𝛽𝑚𝛼𝑛𝛽𝑚𝑛2(\alpha+\beta|m\alpha+n\beta)=m+n\geq 2( italic_α + italic_β | italic_m italic_α + italic_n italic_β ) = italic_m + italic_n ≥ 2, by the definition of lattice vertex operators, we have | For p<2l𝑝2𝑙p<2litalic_p < 2 italic_l, let FpAg,n(V):=0assignsubscript𝐹𝑝subscript𝐴𝑔𝑛𝑉0F_{p}A_{g,n}(V):=0italic_F start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_g , italic_n end_POSTSUBSCRIPT ( italic_V ) := 0. Clearly we have | B |
(B(a,b)t−c)(B(a,b)(t+1)−c)𝐵𝑎𝑏𝑡𝑐𝐵𝑎𝑏𝑡1𝑐\displaystyle(B(a,b)t-c)(B(a,b)(t+1)-c)( italic_B ( italic_a , italic_b ) italic_t - italic_c ) ( italic_B ( italic_a , italic_b ) ( italic_t + 1 ) - italic_c ) | |τi−τi+1|=(1B(a,b)ni−1B(a,b)ni+1)|c|⩽|c|B(a,b)ni⩽|c|2subscript𝜏𝑖subscript𝜏𝑖11𝐵superscript𝑎𝑏subscript𝑛𝑖1𝐵superscript𝑎𝑏subscript𝑛𝑖1𝑐𝑐𝐵superscript𝑎𝑏subscript𝑛𝑖𝑐2|\tau_{i}-\tau_{i+1}|=\left(\frac{1}{B(a,b)^{n_{i}}}-\frac{1}{B(a,b)^{n_{i+1}}% | σi=c+1−bB(a,b)ni−1B(a,b)niandτi=cB(a,b)ni(i=1,…,ν),subscript𝜎𝑖𝑐1𝑏𝐵superscript𝑎𝑏subscript𝑛𝑖1𝐵superscript𝑎𝑏subscript𝑛𝑖andsubscript𝜏𝑖𝑐𝐵superscript𝑎𝑏subscript𝑛𝑖𝑖1…𝜈\displaystyle\sigma_{i}=\frac{c+1-bB(a,b)^{n_{i-1}}}{B(a,b)^{n_{i}}}\ \text{% | ∑i=0ν(b−1)B(a,b)ni⩽(b−1)B(a,b)nν+1−1B(a,b)−1superscriptsubscript𝑖0𝜈𝑏1𝐵superscript𝑎𝑏subscript𝑛𝑖𝑏1𝐵superscript𝑎𝑏subscript𝑛𝜈11𝐵𝑎𝑏1\displaystyle\sum_{i=0}^{\nu}(b-1)B(a,b)^{n_{i}}\leqslant(b-1)\frac{B(a,b)^{n_% | ∏i=1ν(B(a,b)ni(t−iW)−c−1+bB(a,b)ni−1)(B(a,b)ni(t+1−iW)−c)superscriptsubscriptproduct𝑖1𝜈𝐵superscript𝑎𝑏subscript𝑛𝑖𝑡𝑖𝑊𝑐1𝑏𝐵superscript𝑎𝑏subscript𝑛𝑖1𝐵superscript𝑎𝑏subscript𝑛𝑖𝑡1𝑖𝑊𝑐\displaystyle\prod_{i=1}^{\nu}\left(B(a,b)^{n_{i}}(t-iW)-c-1+bB(a,b)^{n_{i-1}}% | D |
If M≥br−(d−g)𝑀subscript𝑏𝑟𝑑𝑔M\geq b_{r}-(d-g)italic_M ≥ italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT - ( italic_d - italic_g ) and N≥ar−(d−g)𝑁subscript𝑎𝑟𝑑𝑔N\geq a_{r}-(d-g)italic_N ≥ italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT - ( italic_d - italic_g ), then the permutation αNπαM−1subscript𝛼𝑁𝜋superscriptsubscript𝛼𝑀1\alpha_{N}\pi\alpha_{M}^{-1}italic_α start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT italic_π italic_α start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT restricts to a bijection from [0,M+N+d−g]0𝑀𝑁𝑑𝑔[0,M+N+d-g][ 0 , italic_M + italic_N + italic_d - italic_g ] to itself that is decreasing on [ar+M,∞)subscript𝑎𝑟𝑀[a_{r}+M,\infty)[ italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT + italic_M , ∞ ), and its inverse is decreasing on [br+N,∞)subscript𝑏𝑟𝑁[b_{r}+N,\infty)[ italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT + italic_N , ∞ ). | Figure 1. Examples of Brill-Noether degeneracy loci, with |Π|=3Π3|\Pi|=3| roman_Π | = 3. The shaded boxes indicate Ess(Π)EssΠ\mathrm{Ess}(\Pi)roman_Ess ( roman_Π ). | So Brill-Noether degeneracy loci are just degeneracy loci of Brill-Noether flags, hiding behind a twist. We can now use the versality theorem to prove the local statements from Theorem 1.7, including a stronger form of the smoothness statement. | We now have what we need to study Brill-Noether degeneracy loci as degeneracy loci of Brill-Noether flags. With an eye on Equation 19, we add a bit more notation to Situation 6.1. | This section analyzes the geometry of Brill-Noether degeneracy loci WdΠ(C,p,q)subscriptsuperscript𝑊Π𝑑𝐶𝑝𝑞W^{\Pi}_{d}(C,p,q)italic_W start_POSTSUPERSCRIPT roman_Π end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_C , italic_p , italic_q ) using the versality of Brill-Noether flags, and proves Theorem 1.7. Assume we are in the following situation. | C |
Each of the above definitions is stronger than the preceding one. Mind that even a smooth time function need not be temporal. Additionally, any time (temporal) function the level sets of which are Cauchy hypersurfaces is called a Cauchy time (temporal) function. | Finally, in globally hyperbolic spacetimes many more intersections of causal futures and pasts — not just those of the form J+(p)∩J−(q)superscript𝐽𝑝superscript𝐽𝑞J^{+}(p)\cap J^{-}(q)italic_J start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_p ) ∩ italic_J start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_q ), p,q∈ℳ𝑝𝑞ℳp,q\in\mathcal{M}italic_p , italic_q ∈ caligraphic_M — turn out to be compact. | The compact-open topology is by no means the only one the sets of causal curves can be endowed with. Standard choices include the so-called C0superscript𝐶0C^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT-topology [34, 35, 55] defined for compact causal curves (i.e. those with both endpoints) and the compact-open topology on the space of curves parametrized by their arc-length [20, 45] (see also [57, 58] for more details). In fact, one can show that the sequence (γn)⊂A𝒯[a,b]subscript𝛾𝑛superscriptsubscript𝐴𝒯𝑎𝑏(\gamma_{n})\subset A_{\mathcal{T}}^{[a,b]}( italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ⊂ italic_A start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT start_POSTSUPERSCRIPT [ italic_a , italic_b ] end_POSTSUPERSCRIPT converges in the compact-open topology iff it converges in the C0superscript𝐶0C^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT-topology [43, Theorem 3]. One can thus regard the former as a way of extending the latter to the case of noncompact causal curves — at least when a (not necessarily Cauchy) temporal function is available. | The plan of the paper is as follows. In Sec. 2 we (re)introduce Polish spaces of causal curves A𝒯Isuperscriptsubscript𝐴𝒯𝐼A_{\mathcal{T}}^{I}italic_A start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_I end_POSTSUPERSCRIPT endowed with the compact-open topology. We do so not only to make the paper self-contained, but, more importantly, to unravel the announced sensitivity of the employed topology to the curves’ differential properties. This is realized first by introducing the “weak-Hloc1subscriptsuperscript𝐻1locH^{1}_{\mathrm{loc}}italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_loc end_POSTSUBSCRIPT-topology” on A𝒯Isuperscriptsubscript𝐴𝒯𝐼A_{\mathcal{T}}^{I}italic_A start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_I end_POSTSUPERSCRIPT, obtained through embedding the latter into a suitable Hloc1subscriptsuperscript𝐻1locH^{1}_{\mathrm{loc}}italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_loc end_POSTSUBSCRIPT-type Sobolev space, and then by carefully proving that the weak topology pulled back on A𝒯Isuperscriptsubscript𝐴𝒯𝐼A_{\mathcal{T}}^{I}italic_A start_POSTSUBSCRIPT caligraphic_T end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_I end_POSTSUPERSCRIPT from that Sobolev space is in fact equal to the compact-open topology. In Sec. 3 we look more closely into the notion of a Llocpsubscriptsuperscript𝐿𝑝locL^{p}_{\mathrm{loc}}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_loc end_POSTSUBSCRIPT-vector field. Most importantly, we provide its alternative, equivalent definition that is much better suited for proving Theorem 6. The latter is done in Sec. 4 with the necessary tools and lemmas introduced or recalled when needed. Finally, Sec. 5 discusses how the descriptions (i)–(iii) transform between various observers and unveils the generally covariant objects behind them. The article is supplemented by three appendices. Appendix A contains several notational remarks. Appendix B summarizes the definitions and facts from Lorentzian causality theory needed in the current investigations. Finally, Appendix C provides a detailed exposition of the theory of Sobolev spaces of univariate functions, along with some technical lemmas used in the proofs throughout the paper. | Spacetimes can be classified according to their increasingly better causal features. Each of the levels of this “causal hierarchy” can be characterized in many equivalent ways. Here we recall only those notions which are needed in the current paper — the more complete exposition can be found e.g. in [47]. | D |
We will indicate for simplicity by Mω(τ1,τ2)subscript𝑀𝜔subscript𝜏1subscript𝜏2M_{\omega}(\tau_{1},\tau_{2})italic_M start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT ( italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) the right-hand side of the equation. | Notice that ∂X0subscript𝑋0\partial X_{0}∂ italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the disjoint union of three tori ∂D2×S1superscript𝐷2superscript𝑆1\partial D^{2}\times S^{1}∂ italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT × italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT and of the boundaries of tubular neighbourhoods of the three tangles’ closures: | In the case of coloured braids, we will show that Equation (2) provides exactly the multivariate extension of Equation (1): | We will indicate for simplicity by Mω(τ1,τ2)subscript𝑀𝜔subscript𝜏1subscript𝜏2M_{\omega}(\tau_{1},\tau_{2})italic_M start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT ( italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) the right-hand side of the equation. | Recall that the left-hand side of this equation is additive with respect to the disjoint union of tangles. | D |
It is important to note that the two above approximations dk(1)subscriptsuperscript𝑑1𝑘d^{(1)}_{k}italic_d start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and dk(2)subscriptsuperscript𝑑2𝑘d^{(2)}_{k}italic_d start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT lead to the same channel model, being the statistics of the noise in (73) and that in (62) the same. Therefore, they will lead to the same expression of the optimal estimation filter. | Figure 3: Normalized variance of the approximation error for the auxiliary channel models on which the design of the phase estimator is based. | In [8], a PN spectrum model which is very suitable for theoretical calculations is studied. The considered power spectral density (PSD) has the fundamental features typical of the PSD of practical oscillators, i.e., a −2020-20- 20 dB/decade slope, a flat part at low frequencies, representing the attenuation of the PN stabilized by means of a PLL, and another flat part at high frequencies. Moreover, it has the Wiener model as a special case. It is similar to other models in the literature [9, 10, 11], but it improves them since it allows us to describe the PN in terms of measurement parameters of practical oscillators and facilitates closed-form expressions of distortions and performance. Starting from the PN PSD, the PSD of the phasor can be derived and expressed in terms of the parameters of the PN PSD, that are related to the oscillator measurements. In this paper, the analysis is extended to the general case where the flat part of the PN PSD at high frequencies is not negligile and without resorting to the approximation of low PN as usually done in the literature. | In the following, the approximation error of the above models is studied. The exact model can be written as | For this reason, in the following we compute the variance of the residual PN error, which can be used to predict the performance of the considered system. | C |
In the situation presented in the previous result, it is well known that the pair ((𝒯⟂)⟂,𝒯⟂)({}^{\perp}(\mathcal{T}^{\perp}),\mathcal{T}^{\perp})( start_FLOATSUPERSCRIPT ⟂ end_FLOATSUPERSCRIPT ( caligraphic_T start_POSTSUPERSCRIPT ⟂ end_POSTSUPERSCRIPT ) , caligraphic_T start_POSTSUPERSCRIPT ⟂ end_POSTSUPERSCRIPT ) is complete in case S𝑆Sitalic_S is a set. That is, in such a case, the pair induces (𝒯⟂)⟂{}^{\perp}(\mathcal{T}^{\perp})start_FLOATSUPERSCRIPT ⟂ end_FLOATSUPERSCRIPT ( caligraphic_T start_POSTSUPERSCRIPT ⟂ end_POSTSUPERSCRIPT )-precovers and 𝒯⟂superscript𝒯perpendicular-to\mathcal{T}^{\perp}caligraphic_T start_POSTSUPERSCRIPT ⟂ end_POSTSUPERSCRIPT-preenvelopes. In the next section we will study these concepts. | and that will be useful for the development of the theory related with some special classes and properties of the relative homological dimensions. This treatment will be necessary for n𝑛nitalic_n-𝒳𝒳\mathcal{X}caligraphic_X-tilting theory in [4]. | Theorem 4.19, Theorem 4.20 and Corollary 4.21. These closure properties of the aforementioned relative classes play an important role in the study and development of n𝑛nitalic_n-𝒳𝒳\mathcal{X}caligraphic_X-tilting theory in the second forthcoming paper [4]. On the other hand, Theorem 4.4 shows the existence of the main approximations that we will be using in the case of relative coresolution classes. This theorem is a possible generalization of the dual | in ℬ∩𝒳.ℬ𝒳\mathcal{B}\cap\mathcal{X}.caligraphic_B ∩ caligraphic_X . Let us start by recalling the following | We will be using the following notation and vocabulary for approximations. We start this section by recalling the well known notions of precovers and preenvelopes. | D |
Among these works, [7, 32, 34] utilize the DPG formulation to investigate the effect of complex policy interventions under minimal assumptions. | The dynamical system (EV.1)–(EV.2) is derived by applying evolutionary dynamics to the payoffs of the equivalent population game F𝒢(d,π)superscript𝐹𝒢𝑑𝜋F^{\mathcal{G}}(d,\pi)italic_F start_POSTSUPERSCRIPT caligraphic_G end_POSTSUPERSCRIPT ( italic_d , italic_π ) given in Equations (9)–(10). | In all of the above works, the SNE is computed using the evolutionary dynamics (EV.1)–(EV.1), with Hτsubscript𝐻𝜏H_{\tau}italic_H start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT chosen as the perturbed best response dynamic [25]. | The evolutionary dynamics model (EV.1)–(EV.1) is used not only for SNE computation, but also as a model of the coupled evolution of infection states and population behaviors, which is needed to predict important off-equilibrium quantities such as the peak infections. | Theorem 1 moreover inspires the following evolutionary state-policy dynamics model for SNE computation | C |
Recall that Remark 3.4 provides an equivalence between explained and Weil entanglements for elliptic curves, and so the above description allows us to make group theoretic definitions of Weil and non-Weil entanglements for principally polarized abelian varieties A/ℚ𝐴ℚA/\mathbb{Q}italic_A / blackboard_Q as in [DM22, Definitions 3.7 & 3.9] where we replace the map det(⋅)⋅\det(\cdot)roman_det ( ⋅ ) with the similitude character ν(⋅)𝜈⋅\nu(\cdot)italic_ν ( ⋅ ). | In the remainder of the work, we study infinite families of Weil and Serre entanglements, describe how CM entanglements can be used to further analyze the adelic image of an elliptic curve with CM, and illustrate how group theoretic definitions of entanglements and our categorization of explained entanglements can be extended to principally polarized abelian varieties over ℚℚ\mathbb{Q}blackboard_Q. | The definition of entanglements in terms of intersection of division fields carries over for abelian varieties, however the group theoretic definitions require more care as the determinant of the mod n𝑛nitalic_n image of Galois associated to A𝐴Aitalic_A over ℚℚ\mathbb{Q}blackboard_Q need not be surjective. | While these group theoretic considerations take some care, the non-group theoretic definitions of abelian and Weil entanglements (Definitions 3.2 and 3.3) can be carried over mutatis mutandis. | With this construction, we can provide similar group theoretic definitions of explained and unexplained entanglements for principally polarized abelian varieties A/ℚ𝐴ℚA/\mathbb{Q}italic_A / blackboard_Q as follows. | C |
We can, however, imagine a situation where a notion of depth derived from a notion of graph centrality may work well. | Thus, on the one hand, notions of depth have been introduced in the context of point clouds, while on the other hand, notions of centrality have been proposed in the context of graphs and networks, and these two lines of work seem to have evolved completely separately, with no cross-pollination whatsoever, at least to our knowledge.444The only place where we found a hint of that is in the discussion of Aloupis (2006), who mentions a couple of “graph-based approach[es]” which seem to have been developed for the context of point clouds, although one of them — the method of Toussaint and | At this stage, we do not know whether graph-based notions of depth offer promise in regards to methodology. The notions that we study in the present article either lead to the likelihood, which does not appear to be particularly popular, or to somewhat peculiar functions which — as the expert in data depth will notice — do not define notions of depths which are affine invariant in general, while affine invariance is one of the desired properties of a depth function (Zuo and | In the context of multivariate analysis, a notion of depth is meant to provide an ordering of the space (Zuo and | For instance, graph-based notions of depth may adapt to the intrinsic geometry of a low-dimensional sampled manifold, a context — common in modern multivariate analysis — in which the standard notions of depth are all doomed. | D |
When 𝝍𝝍\boldsymbol{\mathbf{\psi}}bold_italic_ψ is non-redundantly spanned by r𝑟ritalic_r distinct KEFs, capturing the r𝑟ritalic_r concomitant Koopman modes necessitates a minimum model order of r𝑟ritalic_r. | This facilitates comparing the system complexity with the amount of training data (Definition 3.9) and grading the dynamical information contained in the trajectory starting at 𝝊1subscript𝝊1\boldsymbol{\mathbf{\upsilon}}_{1}bold_italic_υ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT (Definition 3.10). | Then, we may grade the (in)sufficiency of the training data, 𝐙𝐙\mathbf{Z}bold_Z, for imbibing the DMD model, 𝐜∗superscript𝐜\boldsymbol{\mathbf{c}}^{*}bold_c start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT, with generalizability as follows: | Now, by Definition 3.5, the model order, n𝑛nitalic_n, also quantifies the amount of training data available. | The hyper-parameter, n𝑛nitalic_n, which determines the sampling regime in Algorithms 1 and 2, also happens to be the associated model order (Definition 3.5). | C |
Indeed, it has been known since the 90s that mean-convex flows are unique. More recently, Hershkovits-White localized this result and showed that it is enough to assume that all singularities have a mean-convex neighborhood, and we exactly established this assumption. | One can think of this as a variational characterization of the mean curvature flow as the gradient flow of the area functional. | As another application, taking also into account Brendle’s classification of genus zero shrinkers, one can confirm a conjecture of White: | As an application of the mean-convex neighborhood theorem and the multiplicity-one theorem one can confirm the uniqueness conjecture for mean curvature flow through neck-singularities: | Brendle’s classification of genus zero shrinkers appeared in [Bre16]. The genericity conjecture has been proved by Chodosh-Choi-Mantoulidis-Schulze, first via a classification of ancient one-sided flows in [CCMS20, CCS23], and later via a simpler density drop argument in [CCMS21, CMS23]. The work on generic flows has been pioneered by Colding-Minicozzi [CM12], who also proved the foundational uniqueness result for cylindrical tangent flows [CM15], see also [CIM15]. The mean-convex neighborhood conjecture has been proved in my joint work with Choi and Hershkovits [CHH22], and the reformulation in terms of ancient asymptotically cylindrical flows is from our follow up paper with White [CHHW22]. Important prior classification results can be found in [Wan11, Has15, BW17, BC19, ADS20]. Finally, the Merle-Zaag ODE lemma is from [MZ98], and the Brendle-Choi neck-improvement is from [BC19]. | B |
By virtue of Tucker [Tuc99] and Ghys [Ghy07], the set of modular knots and the missing trefoil in S3superscript𝑆3S^{3}italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT is topologically equivalent to that of Lorenz knots, much of whose topological properties having been studied. | Every modular knot may be presented by some primitive hyperbolic γ𝛾\gammaitalic_γ, so that there is a natural surjection from the set of hyperbolic primitive elements to that of modular knots. | By virtue of Tucker [Tuc99] and Ghys [Ghy07], the set of modular knots and the missing trefoil in S3superscript𝑆3S^{3}italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT is topologically equivalent to that of Lorenz knots, much of whose topological properties having been studied. | Now let (Ki)subscript𝐾𝑖(K_{i})( italic_K start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) denote the sequence of modular knots and the missing trefoil ordered by length and suppose that Ln=∪i≤nKisubscript𝐿𝑛subscript𝑖𝑛subscript𝐾𝑖L_{n}=\cup_{i\leq n}K_{i}italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = ∪ start_POSTSUBSCRIPT italic_i ≤ italic_n end_POSTSUBSCRIPT italic_K start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT contains the missing trefoil K𝐾Kitalic_K. | Conversely, Bonatti–Pinsky [BP21] gives a parametrized family of Lorenz-like flows, and Pinsky [Pin23] points out that some of them contain a sublink that is topologically equivalent to the set of modular knots and the missing trefoil. | D |
For 0<ϵ<10italic-ϵ10<\epsilon<10 < italic_ϵ < 1, a condition Q𝑄Qitalic_Q has sample complexity n∗superscript𝑛n^{*}italic_n start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT for algorithms with (1,o(1/n))1𝑜1𝑛(1,o(1/n))( 1 , italic_o ( 1 / italic_n ) )-differential privacy (n∗superscript𝑛n^{*}italic_n start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is the smallest sample size that there exists an (1,o(1/n))1𝑜1𝑛(1,o(1/n))( 1 , italic_o ( 1 / italic_n ) )-differentially private algorithm 𝒜𝒜\mathcal{A}caligraphic_A which satisfies Q𝑄Qitalic_Q), if and only if it also has sample complexity Θ(n∗/ϵ)Θsuperscript𝑛italic-ϵ\Theta(n^{*}/\epsilon)roman_Θ ( italic_n start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT / italic_ϵ ) for algorithms with (ϵ,o(1/n))italic-ϵ𝑜1𝑛(\epsilon,o(1/n))( italic_ϵ , italic_o ( 1 / italic_n ) )-differential privacy. | We apply the group privacy technique in Steinke and Ullman (2016), based on the following technical lemma: | The second part is an improved lower bound in the constrained setting. For the lower bound construction, we use an ℓ∞subscriptℓ\ell_{\infty}roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ball as the domain and select the ℓ1subscriptℓ1\ell_{1}roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT loss function ℓ(θ;z)=|θ−z|1ℓ𝜃𝑧subscript𝜃𝑧1\ell(\theta;z)=|\theta-z|_{1}roman_ℓ ( italic_θ ; italic_z ) = | italic_θ - italic_z | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, and improve the previous lower bound via the group privacy technique. The choice of the norms on the domain and loss function makes it applicable for general ℓpsubscriptℓ𝑝\ell_{p}roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT geometry with p≥1𝑝1p\geq 1italic_p ≥ 1. | Previous works in the constrained case Bassily et al. (2014); Steinke and Ullman (2016) fail in the unconstrained and non-euclidean case for two reasons. | The following theorem presents the improved lower bound we obtain, which modifies and generalizes the techniques in Steinke and Ullman (2016); Bassily et al. (2014) to reach a tighter bound for the unconstrained case. | A |
A starting problem in the current paper is hence the construction of an appropriate non-degenerate subvariety. In the case of curves embedded in their Jacobians, by the quasi-finiteness of the Torelli map, we could restrict ourselves to consider subvarieties of an abelian scheme 𝒜→S→𝒜𝑆\mathcal{A}\rightarrow Scaligraphic_A → italic_S of maximal variation, that is, the moduli map from S𝑆Sitalic_S to the moduli space of abelian varieties is generically finite. In the current paper, we need a space parametrizing all subvarieties of a fixed degree in abelian varieties of a fixed dimension and polarization. While there is a natural candidate, the Hilbert scheme, the moduli map from it to the moduli space of abelian varieties has positive dimensional fibers. This makes the construction of the relevant non-degenerate subvarieties significantly harder than the case of curves. | The basic idea to prove this proposition is similar to the proof of Proposition 4.1, i.e. to put all (A,L)𝐴𝐿(A,L)( italic_A , italic_L ) and X𝑋Xitalic_X into (finitely many) families over the components of the restricted Hilbert scheme from (3.6). The core of this section is the result in family Proposition 5.2. | For our purpose and inspired by the second-named author’s [Ge21, Prop.3.4], we prove the following proposition. In fact, it is precisely [Ge21, Prop.3.4] adapted to our context of restricted Hilbert schemes, and the main idea of the proof is to apply the first-named author’s [Gao20, Thm.10.1] to the situation considered in Lemma 3.3. | A starting problem in the current paper is hence the construction of an appropriate non-degenerate subvariety. In the case of curves embedded in their Jacobians, by the quasi-finiteness of the Torelli map, we could restrict ourselves to consider subvarieties of an abelian scheme 𝒜→S→𝒜𝑆\mathcal{A}\rightarrow Scaligraphic_A → italic_S of maximal variation, that is, the moduli map from S𝑆Sitalic_S to the moduli space of abelian varieties is generically finite. In the current paper, we need a space parametrizing all subvarieties of a fixed degree in abelian varieties of a fixed dimension and polarization. While there is a natural candidate, the Hilbert scheme, the moduli map from it to the moduli space of abelian varieties has positive dimensional fibers. This makes the construction of the relevant non-degenerate subvarieties significantly harder than the case of curves. | We resolve this problem by showing that the moduli map on the total space, when restricted to the universal family over an open subset of the Hilbert scheme (3.6), becomes still generically finite after taking a high enough fibered power (Lemma 3.3), as inspired by the second-named author’s work [Ge21, §§\mathsection§3]. Then [Gao20, Thm. 10.1] gives us the desired non-degeneracy (Proposition 3.4). We expect the idea of this construction to be applicable in other settings, for example to study uniformity problems for semiabelian varieties, which would extend in particular the uniform results of Bombieri–Zannier [BZ95] on algebraic tori, and maybe even to study related problems in some families of dynamical systems. The need for non-degeneracy make it necessary to shift back and forth to fibered products. In the course of this, we have to control the exceptional sets appearing. For this purpose, we provide a technical key Lemma 4.3. | D |
It is reasonable to expect that the same heuristics should apply to 𝒥Ksubscript𝒥𝐾\mathcal{J}_{K}caligraphic_J start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT or 𝒥ℚsubscript𝒥ℚ\mathcal{J}_{\mathbb{Q}}caligraphic_J start_POSTSUBSCRIPT blackboard_Q end_POSTSUBSCRIPT, since the guiding principle is the same and independent of the extra conditions defining 𝒥Fsubscript𝒥𝐹\mathcal{J}_{F}caligraphic_J start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT. | If E∈ℰ𝐸ℰE\in\mathscr{E}italic_E ∈ script_E, i.e., rankℤE(ℚ)=0subscriptrankℤ𝐸ℚ0\mathrm{rank}_{\mathbb{Z}}E(\mathbb{Q})=0roman_rank start_POSTSUBSCRIPT blackboard_Z end_POSTSUBSCRIPT italic_E ( blackboard_Q ) = 0, then it is known that (E/ℚ)𝐸ℚ\Sh(E/\mathbb{Q})( italic_E / blackboard_Q ) is finite. | The extra p𝑝pitalic_p-ordinary and (†)†(\dagger)( † ) conditions defining 𝒥ℚsubscript𝒥ℚ\mathcal{J}_{\mathbb{Q}}caligraphic_J start_POSTSUBSCRIPT blackboard_Q end_POSTSUBSCRIPT as a subset of ℰℰ\mathscr{E}script_E should be independent of the heuristics for the variation of (E/ℚ)𝐸ℚ\Sh(E/\mathbb{Q})( italic_E / blackboard_Q ). | For x>0𝑥0x>0italic_x > 0, set ℰ(x)ℰ𝑥\mathscr{E}(x)script_E ( italic_x ) to be the subset of ℰℰ\mathscr{E}script_E consisting of E𝐸Eitalic_E such that H(E)≤x𝐻𝐸𝑥H(E)\leq xitalic_H ( italic_E ) ≤ italic_x. | First we discuss the variation of (E/ℚ)𝐸ℚ\Sh(E/\mathbb{Q})( italic_E / blackboard_Q ) as E𝐸Eitalic_E varies over ℰℰ\mathscr{E}script_E. | D |
An interpolating regression function is by construction linear in the labels, with the output being a weighted sum of training data labels. Furthermore, in the Hilbert kernel, the weights do not have any adjustable parameters. This provides a transparent route to credit assignment among different training data fragments. If an autoregressive generative model is used to create a time series xtsubscript𝑥𝑡x_{t}italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT (e.g., a musical wave form), then for any given time point t𝑡titalic_t, the contribution of each training data point i𝑖iitalic_i is given by the corresponding weight wisubscript𝑤𝑖w_{i}italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. For an entire generated piece, the weights of each training data fragment could be averaged to obtain the quantitative contribution of that data point (which could then be used, for example, to apportion a revenue share). Our considerations show that the weight distribution is expected to be long-tailed, with a few fragments contributing large weights and many fragments contributing small weights. A suitable cutoff could be used to truncate the number of contributing samples. This process would also show which fragments of the generated piece are truly innovative, since those fragments will not utilize training data fragments with large weights. | In this work, we have made the case, with rigorous mathematical proofs as well as in-depth applications in three widely different disciplines (behavioral biology, control engineering, and number theory), that a fundamental rethink of the current tool-set of artificial intelligence is possible, where highly parameterized network models are complemented by parameter-free data interpolation. In particular, in some AI applications where an ANN module is currently trained using paired input-output training data, the ANN module could be replaced by a Hilbert kernel. While it is out of the scope of the present work to explore the many ramifications of such an approach, it shows the potential for an entire field of research and applications with a parameter-free interpolation centered approach. | Examples of this procedure on simple stochastic processes are shown in the Supplementary Materials. We observed steady state, periodic (Extended Data Fig. 6), and occasionally intermittent behaviors, as would be expected from dynamical systems theory. In addition, we observed “copying” behavior, where the dynamics copying parts of the training data after an initial transient (Extended Data Fig. 5) or as part of periodic episodes (Extended Data Fig. 6). Such copying behavior has been reported in Large Language Models, in the context of extraction of snippets of training data using appropriate prompts [29], and is of interest in the context of “stochastic parrot” style phenomena, i.e. the idea that the machine learning algorithm is essentially repeating the content of the data with some added noise due to limitations of the algorithm [48]. The extent of copying can be monitored using the entropy of the predictive weights as a function of time. In the real data example considered in the next section, where the training data consists of several time series fragments, we observe short episodes of near-copying behavior, but this does not dominate the generated time series. | Generative AI architectures contain as core components ANNs that are subject to supervised learning using labeled examples (e.g., next token prediction in language models [45], or predicting added noise in diffusion-based image generation models [46]). These ANNs can be replaced by Hilbert kernel interpolators (other statistically consistent interpolating learners such as the wiNN algorithm [16] could be similarly used; in this paper we focus on the Hilbert kernel). This approach eliminates highly parameterized black boxes central to the generative architectures in favor of a parameter-free, pure-data approach, that brings the important benefits of transparency and ease of credit assignment. Here, we focus on generative or predictive modeling of time series, with applications to behavioral time series in animal behavior, feedback control of nonlinear dynamical systems, as well as series constructed from Zeta function zeros, as worked examples which we can examine in detail. Applying these ideas more broadly to generative AI is a rich area for future research. | A key notion in this regard is the phenomenon of statistically consistent interpolation (SCI) [18], i.e., regression function estimation that interpolates training data but also generalizes as well as possible by achieving the Bayes limit for expected generalization error (risk), when the sample size becomes large. The Hilbert kernel [20] is a Nadaraya-Watson style estimator [21, 44] with the unique property that it is fully parameter-free and hence does not have any bandwidth or scale parameter. It is global and uses all or batches of the data points for each estimate: the associated kernel is a power law and thus scale-free. Although statistical consistency of this estimator for d𝑑ditalic_d-dimensional input data was proven when it was proposed [20], there has been no systematic analysis of the associated convergence rates and asymptotic finite sample behavior, and it is generally not used in applications. We show that the Hilbert kernel is a member of a large class of SCI methods that have the same asymptotic convergence rates. | A |
Finally, a straightforward search tree argument shows that E𝐸Eitalic_E is also generically countably complete in 𝒲¯¯𝒲\bar{\mathcal{W}}over¯ start_ARG caligraphic_W end_ARG by absoluteness of well-foundedness. So E𝐸Eitalic_E must be added to ℳ¯¯ℳ\bar{\mathcal{M}}over¯ start_ARG caligraphic_M end_ARG, the result of an ℳℳ\mathcal{M}caligraphic_M-construction in 𝒲¯¯𝒲\bar{\mathcal{W}}over¯ start_ARG caligraphic_W end_ARG. But then we can argue that Eαℳ∗≠∅superscriptsubscript𝐸𝛼superscriptℳE_{\alpha}^{\mathcal{M}^{*}}\neq\emptysetitalic_E start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_M start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ≠ ∅. | In the case that E𝐸Eitalic_E is added by the fully backgrounded part of the ℳℳ\mathcal{M}caligraphic_M-construction, this follows as in the proof of the successor case in the proof of [12, Theorem 6.5] (see also Lemma 5.16 below). In the case that E𝐸Eitalic_E is added by the generically countably complete part of the construction, i.e., for the “moreover” part of the claim, the argument for (2) above shows that | As we are no longer working in the result of a fully backgrounded construction to obtain ℳ∞isuperscriptsubscriptℳ𝑖\mathcal{M}_{\infty}^{i}caligraphic_M start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT but in the result of an ℳℳ\mathcal{M}caligraphic_M-construction, we sketch how to modify the proof of the successor case in the proof of [12, Theorem 6.5] to obtain Lemma 5.11. We show the “moveover” part of the lemma parallelly to the first statement restricted to ℳ|δi+1conditionalℳsubscript𝛿𝑖1\mathcal{M}|\delta_{i+1}caligraphic_M | italic_δ start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT as most of the proof is the same. | This finishes the proof that the construction of ℳℳ\mathcal{M}caligraphic_M does not move in the comparison. | Let (ℳξ,𝒩ξ∣ξ≤Ω)subscriptℳ𝜉conditionalsubscript𝒩𝜉𝜉Ω(\mathcal{M}_{\xi},\mathcal{N}_{\xi}\mid\xi\leq\Omega)( caligraphic_M start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT , caligraphic_N start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ∣ italic_ξ ≤ roman_Ω ) be the models in the construction of ℳℳ\mathcal{M}caligraphic_M and suppose toward a contradiction that the construction of ℳℳ\mathcal{M}caligraphic_M does move in the comparison. | C |
W∘l=A∘ksuperscript𝑊absent𝑙superscript𝐴absent𝑘W^{\circ l}=A^{\circ k}italic_W start_POSTSUPERSCRIPT ∘ italic_l end_POSTSUPERSCRIPT = italic_A start_POSTSUPERSCRIPT ∘ italic_k end_POSTSUPERSCRIPT, and differentiating this equality at z0subscript𝑧0z_{0}italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT we conclude that μl=λk.superscript𝜇𝑙superscript𝜆𝑘\mu^{l}=\lambda^{k}.italic_μ start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT = italic_λ start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT . ∎ | Let A𝐴Aitalic_A be a non-special rational function of degree at least two, and z0subscript𝑧0z_{0}italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT its fixed point with the | Let A𝐴Aitalic_A, B𝐵Bitalic_B be polynomials of degree at least two, and X𝑋Xitalic_X a non-constant polynomial such that the diagram | Let A𝐴Aitalic_A be a rational function of degree at least two that is not a generalized Lattès map, and | Let A𝐴Aitalic_A, B𝐵Bitalic_B be rational functions of degree at least two, and X𝑋Xitalic_X a non-constant rational function | D |
A linear representation ρ:Γ→𝖯𝖦𝖫(E):𝜌→Γ𝖯𝖦𝖫𝐸\rho:\Gamma\to\mathsf{PGL}(E)italic_ρ : roman_Γ → sansserif_PGL ( italic_E ) is strongly irreducible if the restriction of ρ𝜌\rhoitalic_ρ to any finite index subgroup Γ′<ΓsuperscriptΓ′Γ\Gamma^{\prime}<\Gammaroman_Γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < roman_Γ is irreducible. | As a consequence we obtain the following hyperconvexity property of k𝑘kitalic_k-positive representations, which was proven by Labourie for Hitchin representations: | The boundary maps of (p,q,d−p−q)𝑝𝑞𝑑𝑝𝑞(p,q,d-p-q)( italic_p , italic_q , italic_d - italic_p - italic_q )–direct representations satisfy an important continuity property. | If the sequence is 1111–positively ratioed, in Corollary 4.2 Zariski density can be relaxed to irreducibility (Proposition 4.4). We verify that the limit satisfies condition (Tr) under weaker assumptions, which will be useful in other contexts as well. The following proposition includes irreducible representations, for which we let E2subscript𝐸2E_{2}italic_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT be the trivial subspace. | The next important property of strongly irreducible representations is due to Labourie. We give here a slightly modified argument that doesn’t require continuity, which will be useful for our purposes. | D |
{\mathbb{R}}roman_Trop ( italic_V ∩ blackboard_G start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( blackboard_C ) ) ⊂ italic_N start_POSTSUBSCRIPT blackboard_R end_POSTSUBSCRIPT of a semi-algebraic subset V𝑉Vitalic_V and showed that it is a finite union of rational polyhedral cones. | Supports of our weighted tropicalizations are contained in (the closure of) his tropicalizations. This follows from Lemma 8.1. | We also prove that weighted tropicalizations induces maps between ℚℚ\mathbb{Q}blackboard_Q-coefficient homology groups. | we will generalize weighted tropicalizations wtTrop(V)wtTrop𝑉\mathop{\mathrm{wtTrop}}\nolimits(V)roman_wtTrop ( italic_V ) of algebraic subvarieties to generic semi-algebraic subsets V𝑉Vitalic_V. | We also show that under a natural assumption, weighted tropicalizations give maps of tropical and the usual singular homology groups which are dual of Trop∗superscriptTrop\mathop{\mathrm{Trop}}\nolimits^{*}roman_Trop start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT. | A |
(\eta)\widehat{g^{2}}(\xi-\eta)\widehat{g^{3}}(-\xi)d\xi d\eta.italic_N ( italic_g start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , italic_g start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_g start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) = ∫ over~ start_ARG italic_σ end_ARG ( italic_ξ , italic_η ) over^ start_ARG italic_g start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_ARG ( italic_η ) over^ start_ARG italic_g start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( italic_ξ - italic_η ) over^ start_ARG italic_g start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG ( - italic_ξ ) italic_d italic_ξ italic_d italic_η . | For the next lemma we need to introduce some terminology. Consider a function F:(0,∞)→ℝ:𝐹→0ℝF:\,(0,\infty)\to\mathbb{R}italic_F : ( 0 , ∞ ) → blackboard_R. | We just need to bound N(g,f,f)𝑁𝑔𝑓𝑓N(g,f,f)italic_N ( italic_g , italic_f , italic_f ). By frequency localization, we have | to ensure |ξ|≪2jmuch-less-than𝜉superscript2𝑗|\xi|\ll 2^{j}| italic_ξ | ≪ 2 start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT we need to take J1subscript𝐽1J_{1}italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT sufficiently large) | We just need to consider Tσ(R⟂f≤J0+2,∇P>J0+4g[j−2,j+2])subscript𝑇𝜎superscript𝑅perpendicular-tosubscript𝑓absentsubscript𝐽02∇subscript𝑃absentsubscript𝐽04subscript𝑔𝑗2𝑗2T_{\sigma}(R^{\perp}f_{\leq J_{0}+2},\nabla P_{>J_{0}+4}g_{[j-2,j+2]})italic_T start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_R start_POSTSUPERSCRIPT ⟂ end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT ≤ italic_J start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + 2 end_POSTSUBSCRIPT , ∇ italic_P start_POSTSUBSCRIPT > italic_J start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + 4 end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT [ italic_j - 2 , italic_j + 2 ] end_POSTSUBSCRIPT ) with | B |
Cat∞/S×TsubscriptCatabsent𝑆𝑇{\rm Cat}_{\infty}\mbox{$\scriptstyle/S\times T$}roman_Cat start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT / italic_S × italic_T and Cat∞/Top×SopsubscriptCatabsentsuperscript𝑇opsuperscript𝑆op{\rm Cat}_{\infty}\mbox{$\scriptstyle/T^{\rm op}\times S^{\rm op}$}roman_Cat start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT / italic_T start_POSTSUPERSCRIPT roman_op end_POSTSUPERSCRIPT × italic_S start_POSTSUPERSCRIPT roman_op end_POSTSUPERSCRIPT. | (−)opsuperscriptop(-)^{\rm op}( - ) start_POSTSUPERSCRIPT roman_op end_POSTSUPERSCRIPT induces an equivalence | The functor (−)opsuperscriptop(-)^{\rm op}( - ) start_POSTSUPERSCRIPT roman_op end_POSTSUPERSCRIPT induces an equivalence | (−)opsuperscriptop(-)^{\rm op}( - ) start_POSTSUPERSCRIPT roman_op end_POSTSUPERSCRIPT induces an equivalence of ∞\infty∞-categories | Hence (−)opsuperscriptop(-)^{\rm op}( - ) start_POSTSUPERSCRIPT roman_op end_POSTSUPERSCRIPT induces an equivalence of ∞\infty∞-categories | C |
Table 1. Average running time and numerical error running Algorithm 2 on a mixture of 2222 Gaussians in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. The error is ϵ=∥v−v^∥2italic-ϵsubscriptdelimited-∥∥𝑣^𝑣2\epsilon=\lVert v-\hat{v}\rVert_{2}italic_ϵ = ∥ italic_v - over^ start_ARG italic_v end_ARG ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT where v∈ℝ4n+2𝑣superscriptℝ4𝑛2v\in\mathbb{R}^{4n+2}italic_v ∈ blackboard_R start_POSTSUPERSCRIPT 4 italic_n + 2 end_POSTSUPERSCRIPT is a vector of the true parameters and v^^𝑣\hat{v}over^ start_ARG italic_v end_ARG is a vector of the estimates. The normalized error is ϵ/(4n+2)italic-ϵ4𝑛2\epsilon/(4n+2)italic_ϵ / ( 4 italic_n + 2 ). | Other work has studied the global landscape of the EM algorithm and the structure of local optima in this setting [15, 53]. Further work has considered inference for Gaussian mixture models with known mixing coefficients and identity covariance matrices [39] and clustering analysis of the mixture of two Gaussians where the covariance matrices are equal and unknown [12]. When these covariance matrices are further assumed to be spherical, [44] gives polynomial time approximation schemes for (1.1) | Algorithm 1 is easily adaptable for this situation. If the mixing coefficients are not known, then using the block triangular structure highlighted above, one can solve for the mixing coefficients in the first step. | The rest of the algorithm then reduces to the case where the mixing coefficients are known. In our experimental results in Section 5.4 we take this approach and do not assume that the mixing coefficients are known. | We perform numerical experiments by running Algorithm 2 on randomly generated Gaussian mixture models with diagonal covariance matrices. We do not assume the mixing coefficients are known a priori, so we solve for them first and then run Algorithm 2. | D |
XP∘:=Pder\G.assignsuperscriptsubscript𝑋𝑃\superscript𝑃der𝐺\displaystyle X_{P}^{\circ}:=P^{\mathrm{der}}\backslash G.italic_X start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT := italic_P start_POSTSUPERSCRIPT roman_der end_POSTSUPERSCRIPT \ italic_G . | Here the quotient exists in the category of schemes and is constructed in the usual manner [Mil17, Appendix B]. | The left vertical arrow in (2.1.15) is the geometric quotient by the action of Mabsuperscript𝑀abM^{\mathrm{ab}}italic_M start_POSTSUPERSCRIPT roman_ab end_POSTSUPERSCRIPT, and the right vertical arrow is the geometric quotient by 𝔾mΔPsuperscriptsubscript𝔾𝑚subscriptΔ𝑃\mathbb{G}_{m}^{\Delta_{P}}blackboard_G start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_Δ start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT end_POSTSUPERSCRIPT. | where now σ′superscript𝜎′\sigma^{\prime}italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is sufficiently small. Here the support of the sum over χ𝜒\chiitalic_χ and s0subscript𝑠0s_{0}italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is finite by assumptions (2) or (3) in the number field case and the fact that E(I,fχs)𝐸𝐼subscript𝑓subscript𝜒𝑠E(I,f_{\chi_{s}})italic_E ( italic_I , italic_f start_POSTSUBSCRIPT italic_χ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) is rational in the sense of [MW95, IV.1.5] in the function field case [MW95, Proposition IV.1.12]. In the number field case the bound required to justify the contour shift is provided by Theorem 4.3. We now apply the functional equation of Theorem 4.2 and Mellin inversion to deduce the identity | be the partial closure of YPsubscript𝑌𝑃Y_{P}italic_Y start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT in XPsubscript𝑋𝑃X_{P}italic_X start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT constructed in (2.2.10) below. | A |
We say that a simplicial space X∙subscript𝑋∙X_{\bullet}italic_X start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT is a Kan groupoid if it is weakly right orthogonal to all horn inclusions Λkn→Δn→subscriptsuperscriptΛ𝑛𝑘superscriptΔ𝑛\Lambda^{n}_{k}\to\Delta^{n}roman_Λ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT → roman_Δ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT (0≤k≤n0𝑘𝑛0\leq k\leq n0 ≤ italic_k ≤ italic_n, n≥1𝑛1n\geq 1italic_n ≥ 1), that is if all maps of spaces Xn→Map(Λkn,X∙)→subscript𝑋𝑛MapsubscriptsuperscriptΛ𝑛𝑘subscript𝑋∙X_{n}\to{\textrm{Map}\!\left(\Lambda^{n}_{k},X_{\bullet}\right)}italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → Map ( roman_Λ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT ) are surjective in 𝒮𝒮\mathcal{S}caligraphic_S. | (in the model of 𝒮𝒮\mathcal{S}caligraphic_S with topological spaces, E𝐸Eitalic_E can be the set of points of Y𝑌Yitalic_Y; in the model with simplicial sets, E𝐸Eitalic_E can be the set of 0-simplices). | For an object X𝑋Xitalic_X in 𝒮𝒮\mathcal{S}caligraphic_S, a resolution of X𝑋Xitalic_X is defined as a simplicial diagram X∙subscript𝑋∙X_{\bullet}italic_X start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT equipped with a colimit cocone with apex X𝑋Xitalic_X. | More intrinsically, the category of cell-finite spaces can be defined as the smallest subcategory of 𝒮𝒮\mathcal{S}caligraphic_S containing ∅\emptyset∅ and 1 (or the whole of 𝒮etfin𝒮subscriptetfin{\mathrm{\mathcal{S}et}}_{\mathrm{fin}}caligraphic_S roman_et start_POSTSUBSCRIPT roman_fin end_POSTSUBSCRIPT) and closed under pushouts. | In the same way that Segal groupoid can be defined in any category with fiber products, Kan groupoids can be defined in any category with fibre products and a notion of surjection (e.g. a pretopos). | D |
The paper is mainly addressed to two audiences: 1) people from commutative algebra and 2) people from Poisson geometry and physics. We have resisted the temptation to write coordinates with upper indices to not irritate the former. However, we often use index notation for tensors and Einstein summation convention to make the presentation of the material clearer to the latter. Our article touches on three seemingly unrelated subjects that are named after J.-L. Koszul (1921–2018): the Koszul bracket, the Koszul complex, and Koszul algebras. | Finally we would like to address a peculiarity of the Koszul case. Let us assume that in the constructions of Theorem 6.4 we chose the resolvent R𝑅Ritalic_R to be a minimal model. Let us introduce the Euler derivation | for all f1,f2∈Ssubscript𝑓1subscript𝑓2𝑆f_{1},f_{2}\in Sitalic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_S and m∈M𝑚𝑀m\in Mitalic_m ∈ italic_M. Conversely, a bracket {,}:S×M→M\{\>,\>\}:S\times M\to M{ , } : italic_S × italic_M → italic_M with the above properties extends uniquely to a Poisson bracket on S⊕Mdirect-sum𝑆𝑀S\oplus Mitalic_S ⊕ italic_M that defines a Poisson module. Note that a free S𝑆Sitalic_S-module is in an obvious way a Poisson module by taking brackets coordinate-wise. In particular, this applies to the S𝑆Sitalic_S-module underlying the Koszul complex K∙(S,𝒇)subscriptK∙𝑆𝒇\operatorname{K}_{\bullet}(S,\boldsymbol{f})roman_K start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT ( italic_S , bold_italic_f ). | Acknowledgements. The authors would like to thank Daniel Levcovitz for sharing information on the cotangent complex. They are indebted to Srikanth Iyengar and Benjamin Briggs for their interest in this project and, in particular, for catching an error in an earlier draft. HCH would like to thank Martin Bordemann for indoctrination. | We would like to emphasize the following phenomenon. Let again 𝔞𝔞\mathfrak{a}fraktur_a be the ideal ker(κ)kernel𝜅\ker(\kappa)roman_ker ( italic_κ ) with κ:R→S:𝜅→𝑅𝑆\kappa:R\to Sitalic_κ : italic_R → italic_S. Then any bracket operation [,…,]m[\>,\dots,\>]_{m}[ , … , ] start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT, as well as the anchor ρmsubscript𝜌𝑚\rho_{m}italic_ρ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT, on the cotangent complex 𝕃A|𝒌subscript𝕃conditional𝐴𝒌\mathbb{L}_{A|\boldsymbol{k}}blackboard_L start_POSTSUBSCRIPT italic_A | bold_italic_k end_POSTSUBSCRIPT that corresponds to a πm∈𝔞2subscript𝜋𝑚superscript𝔞2\pi_{m}\in\mathfrak{a}^{2}italic_π start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ∈ fraktur_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT vanish. In the case of a complete intersection for m≥3𝑚3m\geq 3italic_m ≥ 3 we have πm∈𝔞2subscript𝜋𝑚superscript𝔞2\pi_{m}\in\mathfrak{a}^{2}italic_π start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ∈ fraktur_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT for degree reasons. This means that in the case of a complete intersection we get a dg Lie algebroid and we to recover Theorem 3.7. | C |
The contents of this paper are as follows. In the first section, we develop the general theory of Drinfeld doubles of orthogonal coideals. In the second section we introduce the ∗*∗-algebra Uq(𝔰𝔩(2,ℝ))subscript𝑈𝑞𝔰𝔩2ℝU_{q}(\mathfrak{sl}(2,\mathbb{R}))italic_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( fraktur_s fraktur_l ( 2 , blackboard_R ) ). In the third section, we classify its irreducible representations. In the fourth section, we give an outlook to further developments. | Acknowledgements: The work of K. De Commer and J. R. Dzokou Talla was supported by the FWO grant G032919N. We thank Christian Voigt and Bob Yuncken for their comments on the preliminary version of this article. We thank the organizers of IWOTA Lancaster UK 2021, hosted by the support of EPSRC grant EP/T007524/1, for the opportunity to present this work. We also thank an anonymous referee of an earlier version of this paper for valuable comments and remarks. | The contents of this paper are as follows. In the first section, we develop the general theory of Drinfeld doubles of orthogonal coideals. In the second section we introduce the ∗*∗-algebra Uq(𝔰𝔩(2,ℝ))subscript𝑈𝑞𝔰𝔩2ℝU_{q}(\mathfrak{sl}(2,\mathbb{R}))italic_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( fraktur_s fraktur_l ( 2 , blackboard_R ) ). In the third section, we classify its irreducible representations. In the fourth section, we give an outlook to further developments. | In this paper, we will show that a different quantization is possible for U(𝔩)𝑈𝔩U(\mathfrak{l})italic_U ( fraktur_l ), by quantizing in the direction of a maximally non-compact Cartan subalgebra. The upshot is that this approach will lead to an easier setting in which to consider the integration problem (see [DCDz21]), but the price we pay is that the resulting quantization Uq(𝔩)subscript𝑈𝑞𝔩U_{q}(\mathfrak{l})italic_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( fraktur_l ) will no longer be a Hopf ∗*∗-algebra but only a coideal ∗*∗-subalgebra. However, this does not seem to form a serious problem in practice, as the resulting ∗*∗-algebra has by itself still an interesting representation theory. This approach is motivated substantially by the work on coideal subalgebras of Uq(𝔤)subscript𝑈𝑞𝔤U_{q}(\mathfrak{g})italic_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( fraktur_g ) associated to symmetric pairs, as developed by G. Letzter [Let99] for general finite-dimensional 𝔤𝔤\mathfrak{g}fraktur_g and by S. Kolb in the Kac-Moody setting [Kol14]. We illustrate the above considerations in this paper for the case 𝔩=𝔰𝔩(2,ℝ)𝔩𝔰𝔩2ℝ\mathfrak{l}=\mathfrak{sl}(2,\mathbb{R})fraktur_l = fraktur_s fraktur_l ( 2 , blackboard_R ). In particular, we give a complete classification of all irreducible ∗*∗-representations of Uq(𝔰𝔩(2,ℝ))subscript𝑈𝑞𝔰𝔩2ℝU_{q}(\mathfrak{sl}(2,\mathbb{R}))italic_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( fraktur_s fraktur_l ( 2 , blackboard_R ) ). We also compare the resulting theory with the classical limit as q𝑞qitalic_q tends to 1111. Writing 𝔰𝔩(2,ℂ)ℝ𝔰𝔩subscript2ℂℝ\mathfrak{sl}(2,\mathbb{C})_{\mathbb{R}}fraktur_s fraktur_l ( 2 , blackboard_C ) start_POSTSUBSCRIPT blackboard_R end_POSTSUBSCRIPT for 𝔰𝔩(2,ℂ)𝔰𝔩2ℂ\mathfrak{sl}(2,\mathbb{C})fraktur_s fraktur_l ( 2 , blackboard_C ) considered as a real Lie algebra, we note that the corresponding problem for the Hopf ∗*∗-algebra Uq(𝔰𝔩(2,ℂ)ℝ)subscript𝑈𝑞𝔰𝔩subscript2ℂℝU_{q}(\mathfrak{sl}(2,\mathbb{C})_{\mathbb{R}})italic_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( fraktur_s fraktur_l ( 2 , blackboard_C ) start_POSTSUBSCRIPT blackboard_R end_POSTSUBSCRIPT ) has been considered in [Pus93, PW94], see also [BR99]. Note that there is also a different integrable quantization of U(𝔰𝔩(2,ℝ))𝑈𝔰𝔩2ℝU(\mathfrak{sl}(2,\mathbb{R}))italic_U ( fraktur_s fraktur_l ( 2 , blackboard_R ) ) (or rather its modular double) as a locally compact quantum group in the case of q𝑞qitalic_q unimodular [ByTe03, Ip13]. | In this section, we classify all admissible irreducible ∗*∗-representations of Uq(𝔰𝔩(2,ℝ)t)subscript𝑈𝑞𝔰𝔩subscript2ℝ𝑡U_{q}(\mathfrak{sl}(2,\mathbb{R})_{t})italic_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( fraktur_s fraktur_l ( 2 , blackboard_R ) start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ), in the sense of the following definition. We use the notation SL~(2,ℝ)~SL2ℝ\operatorname{\widetilde{\mathrm{SL}}}(2,\mathbb{R})start_OPFUNCTION over~ start_ARG roman_SL end_ARG end_OPFUNCTION ( 2 , blackboard_R ) to refer to the simply connected cover of SL(2,ℝ)𝑆𝐿2ℝSL(2,\mathbb{R})italic_S italic_L ( 2 , blackboard_R ). We also use again the notation as introduced in (2.1) and (2.6), and we fix t=[[a]]𝑡delimited-[]delimited-[]𝑎t=[[a]]italic_t = [ [ italic_a ] ]. | A |
The discretisation 𝖳𝗌𝖪ωsubscript𝖳𝗌𝖪𝜔{\sf TsK}_{\omega}sansserif_TsK start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT of 𝖳𝗌𝖪𝖳𝗌𝖪{\sf TsK}sansserif_TsK | ∫Yκ(x,dy)ι(y,C)subscript𝑌𝜅𝑥𝑑𝑦𝜄𝑦𝐶\displaystyle\int_{Y}\kappa(x,dy)\iota(y,C)∫ start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT italic_κ ( italic_x , italic_d italic_y ) italic_ι ( italic_y , italic_C ) | ι∘κ(x,C)=𝜄𝜅𝑥𝐶absent\displaystyle\iota\circ\kappa(x,C)=italic_ι ∘ italic_κ ( italic_x , italic_C ) = | ι∘κ(x,C)=𝜄𝜅𝑥𝐶absent\displaystyle\iota\circ\kappa(x,C)=italic_ι ∘ italic_κ ( italic_x , italic_C ) = | ∑y∈Yκ(x,{y})ι(y,C)subscript𝑦𝑌𝜅𝑥𝑦𝜄𝑦𝐶\displaystyle\sum_{y\in Y}\kappa(x,\{y\})\iota(y,C)∑ start_POSTSUBSCRIPT italic_y ∈ italic_Y end_POSTSUBSCRIPT italic_κ ( italic_x , { italic_y } ) italic_ι ( italic_y , italic_C ) | B |
If G𝐺Gitalic_G is a group acting two-transitively on a set ΩΩ\Omegaroman_Ω, and the restriction of the action to a subgroup H𝐻Hitalic_H of G𝐺Gitalic_G remains two-transitive on ΩΩ\Omegaroman_Ω, then the ASTs obtained from G𝐺Gitalic_G and from H𝐻Hitalic_H are equal if and only if the ASTs have the same number of relations. | Theorem 1.1 is obtained by determining the orbits of the groups’ two-point stabilizers. Indeed, the number of nontrivial relations of an AST obtained from a two-transitive group is equal to the number of orbits of a two-point stabilizer while the third valencies are the sizes of these orbits. | To further the study of ASTs, we extend the work in [9] by obtaining the sizes and third valencies of the ASTs obtained from the two-transitive permutation groups. These are given in Theorem 1.1. | Furthermore, by determining which elements of the underlying space can be sent to one another through certain elements of the acting group, we extend some of the results in [9] regarding the intersection numbers of ASTs obtained from PSL(2,n)𝑃𝑆𝐿2𝑛PSL(2,n)italic_P italic_S italic_L ( 2 , italic_n ) and AGL(1,n)𝐴𝐺𝐿1𝑛AGL(1,n)italic_A italic_G italic_L ( 1 , italic_n ) to the intersection numbers of ASTs obtained from PΓL(k,n)𝑃Γ𝐿𝑘𝑛P\Gamma L(k,n)italic_P roman_Γ italic_L ( italic_k , italic_n ) and AΓL(k,n)𝐴Γ𝐿𝑘𝑛A\Gamma L(k,n)italic_A roman_Γ italic_L ( italic_k , italic_n ). Additionally, we complete the intersection numbers of ASTs from PSL(2,n)𝑃𝑆𝐿2𝑛PSL(2,n)italic_P italic_S italic_L ( 2 , italic_n ) which were partially obtained in [9]. Through GAP 4.11.1 [5], we also determine the intersection numbers of the ASTs obtained from the sporadic two-transitive groups. These intersection numbers correct some errors in [9] and are given in Theorem 1.2. | We include here a lemma from [9] that is useful in determining the sizes of the ASTs obtained from two-transitive groups. | D |
We draw simulated data K=1,000𝐾1000K=1,000italic_K = 1 , 000 times and estimate β1((g1,g2))subscript𝛽1subscript𝑔1subscript𝑔2\beta_{1}((g_{1},g_{2}))italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( ( italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) on each simulated data set using three different estimators. We evaluate the performance of these estimators using two metrics: mean-squared error (MSE) and coverage rate. As shown in Table 2, our AIPW estimators consistently exhibit much smaller MSEs compared to both OLS and ORF, suggesting that our AIPW estimators can most accurately estimate β1((g1,g2))subscript𝛽1subscript𝑔1subscript𝑔2\beta_{1}((g_{1},g_{2}))italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( ( italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) for various interference functions f(𝐠)𝑓𝐠f(\mathbf{g})italic_f ( bold_g ), even under misspecifications of the outcome model. Moreover, ORF has a much smaller MSE than OLS, indicating that in the presence of interference, tree-based methods, with their greater flexibility, tend to perform better than the more restrictive regression-based methods. | Furthermore, as shown in Table 2, our AIPW estimators achieve the correct coverage rate (i.e., close to 95%) for various specifications of f(𝐠)𝑓𝐠f(\mathbf{g})italic_f ( bold_g ), while OLS and ORF do not. OLS tends to have a lower coverage rate when f(𝐠)𝑓𝐠f(\mathbf{g})italic_f ( bold_g ) is nonlinear (e.g., quadratic, reciprocal). This low coverage rate is likely due to its failure to obtain an accurate point estimate of the direct effects. On the other hand, ORF tends to have a higher coverage rate than the nominal rate. In fact, the coverage rate of ORF is very close to 100%, implying that the estimated confidence interval from tree-based methods can be too wide, making hypothesis tests using tree-based methods overly conservative. | Importantly, the outcome model in our AIPW estimators is misspecified when f((g1,g2))𝑓subscript𝑔1subscript𝑔2f((g_{1},g_{2}))italic_f ( ( italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) is not linear in g1subscript𝑔1g_{1}italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and g2subscript𝑔2g_{2}italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. However, even in such cases, our AIPW estimators can outperform OLS and ORF, thanks to the double robustness property of AIPW estimators. Therefore, we suggest that explicitly modeling the interference structure and using a doubly robust estimator can be crucial for accurate estimation and valid inference of treatment effects. | Table 2: Coverage rate and MSE of various estimators for β1((g1,g2))subscript𝛽1subscript𝑔1subscript𝑔2\beta_{1}((g_{1},g_{2}))italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( ( italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) | We draw simulated data K=1,000𝐾1000K=1,000italic_K = 1 , 000 times and estimate β1((g1,g2))subscript𝛽1subscript𝑔1subscript𝑔2\beta_{1}((g_{1},g_{2}))italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( ( italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) on each simulated data set using three different estimators. We evaluate the performance of these estimators using two metrics: mean-squared error (MSE) and coverage rate. As shown in Table 2, our AIPW estimators consistently exhibit much smaller MSEs compared to both OLS and ORF, suggesting that our AIPW estimators can most accurately estimate β1((g1,g2))subscript𝛽1subscript𝑔1subscript𝑔2\beta_{1}((g_{1},g_{2}))italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( ( italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) for various interference functions f(𝐠)𝑓𝐠f(\mathbf{g})italic_f ( bold_g ), even under misspecifications of the outcome model. Moreover, ORF has a much smaller MSE than OLS, indicating that in the presence of interference, tree-based methods, with their greater flexibility, tend to perform better than the more restrictive regression-based methods. | A |
Let ℕ={0,1,2…}ℕ012…\mathbb{N}=\{0,1,2\dots\}blackboard_N = { 0 , 1 , 2 … } and ℤ={…,−1,0,1,…}ℤ…101…\mathbb{Z}=\{\dots,-1,0,1,\dots\}blackboard_Z = { … , - 1 , 0 , 1 , … }. | We consider Turing machines that are deterministic, have a single bi-infinite tape with tape cells indexed by ℤℤ\mathbb{Z}blackboard_Z, | A Turing machine configuration is given by: the current state, the tape contents, and an integer tape head position. | Let ℕ={0,1,2…}ℕ012…\mathbb{N}=\{0,1,2\dots\}blackboard_N = { 0 , 1 , 2 … } and ℤ={…,−1,0,1,…}ℤ…101…\mathbb{Z}=\{\dots,-1,0,1,\dots\}blackboard_Z = { … , - 1 , 0 , 1 , … }. | Let M𝑀Mitalic_M and M′superscript𝑀′M^{\prime}italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT be single-tape Turing machines with alphabets ΣΣ\Sigmaroman_Σ and Σ′superscriptΣ′\Sigma^{\prime}roman_Σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. | A |
Let us denote the set of x∈𝕏𝑥𝕏x\in{\mathbb{X}}italic_x ∈ roman_𝕏 with historic behavior by HB(f)𝐻𝐵𝑓HB(f)italic_H italic_B ( italic_f ). | We say that a map f𝑓fitalic_f has abundance of historic behavior if HB(f)𝐻𝐵𝑓HB(f)italic_H italic_B ( italic_f ) is a fat set. | and Theorem D presents a dichotomy for those maps, a strongly transitive map is either uniquely ergodic or has abundance of historic behavior. | Nevertheless, it was shown in [ArP] that hyperbolicity may imply not only abundance of historic behavior, but also abundance of wild historic behavior. | Let us denote the set of x∈𝕏𝑥𝕏x\in{\mathbb{X}}italic_x ∈ roman_𝕏 with historic behavior by HB(f)𝐻𝐵𝑓HB(f)italic_H italic_B ( italic_f ). | A |
}\operatorname{span}L(\mathcal{I}^{k,n}_{+})roman_span italic_L ( caligraphic_I start_POSTSUPERSCRIPT italic_k , italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) = roman_res start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT roman_span italic_L ( caligraphic_I start_POSTSUPERSCRIPT italic_k , italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) | =resm(ℛk,n)resm(spanL(Gk,n))absentsubscriptres𝑚superscriptℛ𝑘𝑛subscriptres𝑚span𝐿subscript𝐺𝑘𝑛\displaystyle=\operatorname{res}_{m}(\mathcal{R}^{k,n})\operatorname{res}_{m}(% | =ℛk,mspanL(resm(Gk,n)).absentsuperscriptℛ𝑘𝑚span𝐿subscriptres𝑚subscript𝐺𝑘𝑛\displaystyle=\mathcal{R}^{k,m}\operatorname{span}L(\operatorname{res}_{m}(G_{% | =spanL(resm(ℐ+k,n))absentspan𝐿subscriptres𝑚subscriptsuperscriptℐ𝑘𝑛\displaystyle=\operatorname{span}L(\operatorname{res}_{m}(\mathcal{I}^{k,n}_{+% | =spanresm(L(ℐ+k,n))=resmspanL(ℐ+k,n).absentspansubscriptres𝑚𝐿subscriptsuperscriptℐ𝑘𝑛subscriptres𝑚span𝐿subscriptsuperscriptℐ𝑘𝑛\displaystyle=\operatorname{span}\operatorname{res}_{m}(L(\mathcal{I}^{k,n}_{+% | A |
With the notation of Definition 2.10, let G𝐺Gitalic_G be a deformation of g𝑔gitalic_g which is topologically trivial over a Milnor sphere. Let u∈B̊ρ𝑢subscript̊𝐵𝜌u\in\mathring{B}_{\rho}italic_u ∈ over̊ start_ARG italic_B end_ARG start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT and suppose that all the fibres of gusubscript𝑔𝑢g_{u}italic_g start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT are smooth or have isolated singularities except for one special fibre Xu≔gu−1(0)∩Bϵ≔subscript𝑋𝑢superscriptsubscript𝑔𝑢10subscript𝐵italic-ϵX_{u}\coloneqq g_{u}^{-1}(0)\cap B_{\epsilon}italic_X start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ≔ italic_g start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 0 ) ∩ italic_B start_POSTSUBSCRIPT italic_ϵ end_POSTSUBSCRIPT. Then Xusubscript𝑋𝑢X_{u}italic_X start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT is homotopy equivalent to a wedge of spheres of dimension n𝑛nitalic_n and its number is the sum of the Milnor numbers over all the fibres different from Xusubscript𝑋𝑢X_{u}italic_X start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT. | As usual, we would like to relate those algebraic 𝒜𝒜\mathscr{A}script_A-invariants with some invariants with a topological flavour. We study what happens when the dimension of the target is greater than the dimension of the source, especially when the difference is 1111. The image Milnor number μI(f)subscript𝜇𝐼𝑓\mu_{I}(f)italic_μ start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT ( italic_f ) of f:(ℂn,S)→(ℂn+1,0):𝑓→superscriptℂ𝑛𝑆superscriptℂ𝑛10f\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)italic_f : ( blackboard_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_S ) → ( blackboard_C start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT , 0 ) was introduced by Mond: the image of a stable perturbation has the homotopy type of a wedge of n𝑛nitalic_n-spheres and the number of such spheres is μI(f)subscript𝜇𝐼𝑓\mu_{I}(f)italic_μ start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT ( italic_f ) (see [27, Theorem 1.4]). Of course, one expects that, in the case where the source is an icis, the situation is similar. There is a hypothesis that simplifies many arguments because it gives extra structure to the objects we study, the corank one hypothesis: | of singularities of mappings defined on an isolated complete intersection singularity (icis). They also extended Damon’s results in [2], which related the 𝒜esubscript𝒜𝑒\mathscr{A}_{e}script_A start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT-versal unfolding of a map germ f𝑓fitalic_f with the 𝒦D(G)subscript𝒦𝐷𝐺\mathscr{K}_{D(G)}script_K start_POSTSUBSCRIPT italic_D ( italic_G ) end_POSTSUBSCRIPT-versal unfoldings of an associated map germ which induces f𝑓fitalic_f from a stable map G𝐺Gitalic_G. In particular, when the target has greater dimension than the source or both dimensions coincide, they proved that the discriminant Milnor number μΔ(X,f)subscript𝜇Δ𝑋𝑓\mu_{\Delta}(X,f)italic_μ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT ( italic_X , italic_f ) is greater than or equal to the 𝒜esubscript𝒜𝑒\mathscr{A}_{e}script_A start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT-codimension, with equality in the weighted homogeneous case. This is a generalisation of a theorem of Damon and Mond in the case of mappings between smooth manifolds (cf. [3]). Here, we study what happens when the dimension of the source is one less than the dimension of the target and we consider the image Milnor number μI(f)subscript𝜇𝐼𝑓\mu_{I}(f)italic_μ start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT ( italic_f ) instead of μΔ(f)subscript𝜇Δ𝑓\mu_{\Delta}(f)italic_μ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT ( italic_f ). | A desirable property of this topological 𝒜𝒜\mathscr{A}script_A-invariant is that it is conservative, as it was for the usual image Milnor number (see [8, Theorem 2.6]). The reasoning that proves the conservation of the usual image Milnor number can be applied verbatim for the general version, and is based as well on Theorem 2.11. Here, we give a sketch of the proof. | Finally, we give the main 𝒜𝒜\mathscr{A}script_A-invariant regarding this kind of germs, the image Milnor number, as Mond did in [27]. Proposition 2.12 below justifies the definition, which is nothing more than our version of [29, Theorem 2.4] (see also [3, pp. 219–220]). | D |
)^{2s}}\Gamma(s)^{2}.roman_Γ start_POSTSUBSCRIPT italic_F , italic_k end_POSTSUBSCRIPT ( italic_s ) := ( start_ROW start_CELL 2 italic_k - 2 end_CELL end_ROW start_ROW start_CELL italic_k - 1 end_CELL end_ROW ) divide start_ARG | italic_D start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT | caligraphic_O start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT | end_ARG start_ARG 4 ( 2 italic_π ) start_POSTSUPERSCRIPT 2 italic_s end_POSTSUPERSCRIPT end_ARG roman_Γ ( italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . | In this section, we obtain an integral representation of L(s,f,u)𝐿𝑠𝑓𝑢L(s,f,u)italic_L ( italic_s , italic_f , italic_u ) in a spirit of Hida [5, Section 7]. Let us denote t(u):=(1u01)∈GL2(𝔸F(∞))assign𝑡𝑢1𝑢01subscriptGL2superscriptsubscript𝔸𝐹t(u):=\left(\begin{smallmatrix}1&u\\ | Finally, we obtain the integral representation of L(s,f,u)𝐿𝑠𝑓𝑢L(s,f,u)italic_L ( italic_s , italic_f , italic_u ): | by the definition of L(s,f,u)𝐿𝑠𝑓𝑢L(s,f,u)italic_L ( italic_s , italic_f , italic_u ). From the Residue theorem, we have | For u∈R𝔮𝑢subscript𝑅𝔮u\in R_{\mathfrak{q}}italic_u ∈ italic_R start_POSTSUBSCRIPT fraktur_q end_POSTSUBSCRIPT, let us define a completed L-function Λ(s,f,u)Λ𝑠𝑓𝑢\Lambda(s,f,u)roman_Λ ( italic_s , italic_f , italic_u ) of L(s,f,u)𝐿𝑠𝑓𝑢L(s,f,u)italic_L ( italic_s , italic_f , italic_u ) by | B |
\rm{s}}\}}(\tau P_{\rm{tx}}H_{\alpha_{\rm{i}},\mu,p})^{2}}.italic_w start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = divide start_ARG ( italic_τ italic_P start_POSTSUBSCRIPT roman_tx end_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT , italic_μ , italic_p end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_κ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT roman_L end_POSTSUBSCRIPT ( italic_M - 2 ) / italic_M + ∑ start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT roman_i end_POSTSUBSCRIPT ∈ caligraphic_A ∖ { italic_α start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT } end_POSTSUBSCRIPT ( italic_τ italic_P start_POSTSUBSCRIPT roman_tx end_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT roman_i end_POSTSUBSCRIPT , italic_μ , italic_p end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG . | Based on (19), after the random rotation, the elevation angle of the p𝑝pitalic_p-th PD can be obtained as: | Based on the Shannon capacity, assuming electrical signals after optical to electrical conversion, the data rate of the μ𝜇\muitalic_μ-th UE on subcarrier m𝑚mitalic_m can be expressed as [29]: | where τ𝜏\tauitalic_τ is the optical-to-electrical conversion efficiency; Ptxsubscript𝑃txP_{\rm{tx}}italic_P start_POSTSUBSCRIPT roman_tx end_POSTSUBSCRIPT is the transmitted optical power of the AP; wpsubscript𝑤𝑝w_{p}italic_w start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT denotes the combining weight of PD p𝑝pitalic_p; Hαs,μ,psubscript𝐻subscript𝛼s𝜇𝑝H_{\alpha_{\rm{s}},\mu,p}italic_H start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT , italic_μ , italic_p end_POSTSUBSCRIPT is the overall DC channel gain between the PD p𝑝pitalic_p of user μ𝜇\muitalic_μ and the serving AP αssubscript𝛼s\alpha_{\rm{s}}italic_α start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT; κ𝜅\kappaitalic_κ is the ratio of DC optical power to the square root of electrical signal power; N0subscript𝑁0N_{0}italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT represents the noise power spectral density of the additive white Gaussian noise and BLsubscript𝐵LB_{\rm{L}}italic_B start_POSTSUBSCRIPT roman_L end_POSTSUBSCRIPT is the baseband modulation bandwidth; Hαi,μ,psubscript𝐻subscript𝛼i𝜇𝑝H_{\alpha_{\rm{i}},\mu,p}italic_H start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT roman_i end_POSTSUBSCRIPT , italic_μ , italic_p end_POSTSUBSCRIPT is the overall DC channel gain between the PD p𝑝pitalic_p of user μ𝜇\muitalic_μ and the interfering LiFi AP αisubscript𝛼i\alpha_{\rm{i}}italic_α start_POSTSUBSCRIPT roman_i end_POSTSUBSCRIPT. The serving AP assubscript𝑎sa_{\rm{s}}italic_a start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT for user μ𝜇\muitalic_μ is selected based on the signal strength strategy (SSS) where the UEs are connected to the APs providing the best received signal strength. Hence, the serving AP αssubscript𝛼s\alpha_{\rm{s}}italic_α start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT for user μ𝜇\muitalic_μ can be expressed as: | For an ADR, multiple PDs are receiving signals simultaneously. Thus, attention should be paid to the selection of the signal combing schemes. There are different combining schemes such as equal gain combining (EGC), SBC and MRC. An important metric to evaluate the link quality and capacity is the SINR. The SINR of user μ𝜇\muitalic_μ on subcarrier m𝑚mitalic_m can be obtained based on [15] and [28]: | B |
Oℙ(h2Ht)subscript𝑂ℙsuperscriptℎ2subscript𝐻𝑡O_{\mathbb{P}}(h^{2H_{t}})italic_O start_POSTSUBSCRIPT blackboard_P end_POSTSUBSCRIPT ( italic_h start_POSTSUPERSCRIPT 2 italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ). | functions are more regular than the trajectories. The reason is that, in some sense, our nonparametric estimators are close to the empirical mean and covariance, respectively, which are the ideal estimators if the trajectories were observed at any point without error. In the case where the mean and covariance functions are smoother than the trajectories, our penalized quadratic risks should be built using the regularity of the mean or covariance functions, instead the regularity of the trajectories. However, the estimation of the regularity of the mean or covariance function remains an open problem. | For illustration, we suppose here that the trajectories are not differentiable. The case of smooth paths is discussed in Section 6. On the bias part, by Cauchy-Schwarz inequality, we then have | It is well-known that the optimal rates, in the minimax sense, for estimating the mean and covariance functions, depend on the regularity of the paths. Moreover, the so-called sparse and dense regimes in functional data analysis, are defined using the regularity of the trajectories, which is usually supposed to be known. We therefore consider a simple estimator of the local regularity of the process and use it to build our penalized quadratic risks. Applied to real data sets, the local regularity estimator reveals that the regularity of the trajectories could be quite far from what is usually assumed in the literature. However, in some applications, assuming smooth trajectories seems reasonable. The mean and covariance functions estimation approach based on local regularity extends to such situations. In the case where the sample paths of X𝑋Xitalic_X admit derivatives up to the order, say α𝛼\alphaitalic_α, condition (10) has to be stated for the increments of the α𝛼\alphaitalic_αth derivative of the sample path. Golovkine et al., (2022, App. D) investigates this extension and proposes an estimator of α+Ht𝛼subscript𝐻𝑡\alpha+H_{t}italic_α + italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT, for which they derive a concentration bound. The mean and covariance functions can next be estimated using the estimates X^(i)tsubscriptsuperscript^𝑋𝑖𝑡{\widehat{X}^{(i)}}_{t}over^ start_ARG italic_X end_ARG start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT built with local polynomial weights Wm(i)(t)superscriptsubscript𝑊𝑚𝑖𝑡W_{m}^{(i)}(t)italic_W start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i ) end_POSTSUPERSCRIPT ( italic_t ). The risk bounds derived in Section 4 above can be extended to this case using standard arguments. See, for instance, Tsybakov, (2009). A discussion and an illustration of the performance of our adaptive estimation of the mean function with smooth sample paths is provided in Golovkine et al., (2024, Sec. LABEL:sub:case_of_differentiable_curves). | Gathering facts, we deduce that, in the case of non-differentiable trajectories, with the NW estimator and | D |
Let the circle act on a compact oriented manifold with two fixed points p𝑝pitalic_p and q𝑞qitalic_q. Then the weights at p𝑝pitalic_p and q𝑞qitalic_q are equal and ϵ(p)=−ϵ(q)italic-ϵ𝑝italic-ϵ𝑞\epsilon(p)=-\epsilon(q)italic_ϵ ( italic_p ) = - italic_ϵ ( italic_q ). | If a 6-dimensional compact oriented S1superscript𝑆1S^{1}italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT-manifold has exactly 2 fixed points, the two fixed points have the same multiset of weights and have different signs (see Theorem 2.7); by performing Operation (1) of Theorem 1.2 once to the fixed point data of the manifold, we reach the empty set. | If the multiset of weights at every fixed point are the same, the number of fixed points must be even and the signature of the manifold vanishes. | Let the circle act on a compact oriented manifold M𝑀Mitalic_M with a discrete fixed point set. Suppose that dimM≡2mod4dimension𝑀modulo24\dim M\equiv 2\mod 4roman_dim italic_M ≡ 2 roman_mod 4. Then the signature of M𝑀Mitalic_M vanishes, and the number of fixed points p𝑝pitalic_p with ϵ(p)=+1italic-ϵ𝑝1\epsilon(p)=+1italic_ϵ ( italic_p ) = + 1 and the number of fixed point p𝑝pitalic_p with ϵ(p)=−1italic-ϵ𝑝1\epsilon(p)=-1italic_ϵ ( italic_p ) = - 1 are equal. | It is well-known that if the number of fixed points is odd, the dimension of the manifold is a multiple of 4. | B |
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