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Hence, the amount of distance traveled between nodes in the tree is a metric that influences the running time. This can also be observed when running the academic MIP-solver SCIP [12]. | In addition to that, the procedure takes two other parameters ℒ,𝒰∈ℝℒ𝒰ℝ\mathcal{L},\mathcal{U}\in\mathbb{R}caligraphic_L , caligraphic_U ∈ blackboard_R as input, which correspond to (another) lower and upper bound on | We will first consider the running time induced by the base call itself, excluding any recursive subcalls. Note that all of this running time is incurred by the calls to the procedures DFS, Roots and GoodValues, plus the cost of moving to the corresponding node before each of these calls. In the base call, the procedure will only move between nodes that are among the ones with the n𝑛nitalic_n smallest values, or the nodes directly below them. For this reason, we can upper bound the cost of each movement action by a running time of O(n)𝑂𝑛O(n)italic_O ( italic_n ). | famous branch-and-bound algorithm and was originally proposed by Karp, Saks and Widgerson [14] to model node | Their result was later applied to prove an upper bound on the parallel running time of branch-and-bound [21]. | D |
(−1)k−2x+2(−1)k−1xsuperscript1𝑘2𝑥2superscript1𝑘1𝑥\displaystyle(-1)^{k-2}x+2(-1)^{k-1}x( - 1 ) start_POSTSUPERSCRIPT italic_k - 2 end_POSTSUPERSCRIPT italic_x + 2 ( - 1 ) start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT italic_x | -1)^{k-2}x+c_{k}^{{}^{\prime}}(-1)^{k}xitalic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( - 1 ) start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT italic_x + divide start_ARG ( - 1 ) start_POSTSUPERSCRIPT italic_k - 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_c start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT end_ARG italic_x + italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( - 1 ) start_POSTSUPERSCRIPT italic_k - 2 end_POSTSUPERSCRIPT italic_x + italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_x | (−1)k−2x+2(−1)k−1xsuperscript1𝑘2𝑥2superscript1𝑘1𝑥\displaystyle(-1)^{k-2}x+2(-1)^{k-1}x( - 1 ) start_POSTSUPERSCRIPT italic_k - 2 end_POSTSUPERSCRIPT italic_x + 2 ( - 1 ) start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT italic_x | (−1)k−1x.superscript1𝑘1𝑥\displaystyle(-1)^{k-1}x.( - 1 ) start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT italic_x . | x{xk−1F(x)}(k)𝑥superscriptsuperscript𝑥𝑘1𝐹𝑥𝑘\displaystyle x\{x^{k-1}F(x)\}^{(k)}italic_x { italic_x start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT italic_F ( italic_x ) } start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT | C |
Let C𝐶Citalic_C be a concordance between knots K1subscript𝐾1K_{1}italic_K start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and K2subscript𝐾2K_{2}italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. For any integer n>0𝑛0n>0italic_n > 0, consider the n𝑛nitalic_n-surgery (S3×I)n(C)subscriptsuperscript𝑆3𝐼𝑛𝐶(S^{3}\times I)_{n}(C)( italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT × italic_I ) start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_C ) along C𝐶Citalic_C, which is a homology cobordism from Sn3(K1)subscriptsuperscript𝑆3𝑛subscript𝐾1S^{3}_{n}(K_{1})italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_K start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) to Sn3(K2)subscriptsuperscript𝑆3𝑛subscript𝐾2S^{3}_{n}(K_{2})italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ). Then, when n𝑛nitalic_n is sufficiently large, the following diagram commutes. | HF−(Sn3(K1),[0])𝐻superscript𝐹subscriptsuperscript𝑆3𝑛subscript𝐾1delimited-[]0\textstyle{HF^{-}(S^{3}_{n}(K_{1}),[0])\ignorespaces\ignorespaces\ignorespaces% | \ignorespaces\ignorespaces\ignorespaces\ignorespaces}( - 1 , 5 )(−2,8)28\textstyle{(-2,8)}( - 2 , 8 )(−3,9)39\textstyle{(-3,9)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}( - 3 , 9 )(9,−3)93\textstyle{(9,-3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}( 9 , - 3 )(−2,−2)22\textstyle{(-2,-2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces% | \ignorespaces\ignorespaces\ignorespaces\ignorespaces}( - 2 , - 2 )(−1,−3)13\textstyle{(-1,-3)}( - 1 , - 3 )(−2,−2)22\textstyle{(-2,-2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces% | \ignorespaces\ignorespaces\ignorespaces\ignorespaces}( 3 , - 3 )(1,−3)13\textstyle{(1,-3)}( 1 , - 3 )(−2,−2)22\textstyle{(-2,-2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces% | A |
Moreover, if there exists an uncountable G𝐺Gitalic_G-Li-Yorke chaotic set S⊂X𝑆𝑋S\subset Xitalic_S ⊂ italic_X of f𝑓fitalic_f, f𝑓fitalic_f is said to be chaotic in the sense of G𝐺Gitalic_G-Li-Yorke. | The specific layout of the present paper is as follows. In Section 2, we introduce some preliminaries and definitions. In Section 3, we show G𝐺Gitalic_G-Li-Yorke chaos is iterable for equivariant maps. In Section 4, We study the relationship between Li-Yorke and G𝐺Gitalic_G-Li-Yorke chaos. In Section 5. We give a sufficient condition for G𝐺Gitalic_G-Li-Yorke chaos. In Section 6, We give a theorem which can be used to identify whether a map is G𝐺Gitalic_G-Li-Yorke chaotic or not. In Section 7, we summarize the conclusions. | Now, we give a sufficient condition for f𝑓fitalic_f to be chaotic in the sense of G𝐺Gitalic_G-Li-Yorke. | Now we define the notion of G𝐺Gitalic_G-recurrent point and G𝐺Gitalic_G-Li-Yorke chaos for maps on G𝐺Gitalic_G-spaces. | Before we show the iteration of G𝐺Gitalic_G-Li-Yorke chaos for equivariant maps, we give a well known lemma. | D |
δn,0δk,0subscript𝛿𝑛0subscript𝛿𝑘0\delta_{n,0}\delta_{k,0}italic_δ start_POSTSUBSCRIPT italic_n , 0 end_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT italic_k , 0 end_POSTSUBSCRIPT term is from the requirement that | An explicit formula for Sn,ksubscript𝑆𝑛𝑘S_{n,k}italic_S start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT in this case can be obtained in | choose to set B0=B0,0=1subscript𝐵0subscript𝐵001B_{0}=B_{0,0}=1italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_B start_POSTSUBSCRIPT 0 , 0 end_POSTSUBSCRIPT = 1. | B0,0=1subscript𝐵001B_{0,0}=1italic_B start_POSTSUBSCRIPT 0 , 0 end_POSTSUBSCRIPT = 1. This term arises in the sum when a particular metatile containing | βq+1subscript𝛽𝑞1\beta_{q+1}italic_β start_POSTSUBSCRIPT italic_q + 1 end_POSTSUBSCRIPT appears as a term in the sum over j𝑗jitalic_j for every m𝑚mitalic_m up | C |
To be precise, the class of measures considered in [HKL22] is different from 𝒞(α)𝒞𝛼\mathcal{C}(\alpha)caligraphic_C ( italic_α ) in that (1.10) is required for every r>0𝑟0r>0italic_r > 0, but this does not really change the problem since the local problem on B1×[1,2]subscript𝐵112B_{1}\times[1,2]italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × [ 1 , 2 ] is equivalent to the global problem on ℝn×[1,2]superscriptℝ𝑛12\mathbb{R}^{n}\times[1,2]blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT × [ 1 , 2 ] for estimating u(x,t)𝑢𝑥𝑡u(x,t)italic_u ( italic_x , italic_t ); see e.g. [HKL22, Lemma 2.6]. | We organize the paper as follows. We study the maximal functions 𝒩δsuperscript𝒩𝛿\mathcal{N}^{\delta}caligraphic_N start_POSTSUPERSCRIPT italic_δ end_POSTSUPERSCRIPT and 𝒮δsuperscript𝒮𝛿\mathcal{S}^{\delta}caligraphic_S start_POSTSUPERSCRIPT italic_δ end_POSTSUPERSCRIPT and prove Theorem 1.2(i) and Theorem 1.5(i) in Section 2 and Section 3, respectively. We prove Theorem 1.7 and Theorem 1.10 in Section 4. Section 4.4 contains a discussion on a geometric approach to fractal local smoothing estimates and the proof of Theorem 1.13. We prove lower bounds for maximal functions considered in this paper in Section 5. In Appendix A, we sketch the proof of Theorem 1.1 and in Appendix B, we prove volume bounds for the intersection of annuli used to study 𝒩δsuperscript𝒩𝛿\mathcal{N}^{\delta}caligraphic_N start_POSTSUPERSCRIPT italic_δ end_POSTSUPERSCRIPT and 𝒮δsuperscript𝒮𝛿\mathcal{S}^{\delta}caligraphic_S start_POSTSUPERSCRIPT italic_δ end_POSTSUPERSCRIPT. | We use results obtained in the papers [CHL17, HKL22] for the proof of Theorem 1.10, which is summarized in Theorem 4.11. | For the proof of Theorem 1.10, we additionally need the following estimates which is better than Theorem 4.11 for 1<α<6/51𝛼651<\alpha<6/51 < italic_α < 6 / 5 for n=2𝑛2n=2italic_n = 2 and Theorem 4.10 for α>(n+3)/2𝛼𝑛32\alpha>(n+3)/2italic_α > ( italic_n + 3 ) / 2 for n≥3𝑛3n\geq 3italic_n ≥ 3. | We prove Theorem 1.7 in this subsection. The approach here also provides a Fourier analytic proof of the L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-bound (2.3); see Theorem 4.7 below. We note that this subsection is independent of the remainder of this section and will not be used for the proof of Theorem 1.10. | B |
He then shows that this is independent of local coordinates. Our approach is more global in nature. Recall that Schwartz functions on a vector space are smooth functions f𝑓fitalic_f such that D(f)𝐷𝑓D(f)italic_D ( italic_f ) is bounded whenever D𝐷Ditalic_D is a differential operator with polynomial coefficients. Analogously, on the space 𝕋M𝕋𝑀\mathbb{T}Mblackboard_T italic_M, we define differential operators which play the role of differential operators with polynomial coefficients. We then define Schwartz functions to be functions whose derivative by such operators is bounded. | We define differential operators on noncommutative algebras to be maps which satisfy a Leibniz rule as in (2). | 1 Differential operators on C∗superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-algebras | In Section 1, we define differential operators on C∗superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-algebras and prove Theorem B. | To prove Theorem A we introduce differential operators on an arbitrary C∗superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-algebra A𝐴Aitalic_A. Recall that a derivation on A𝐴Aitalic_A is a map δ:𝒜→A:𝛿→𝒜𝐴\delta:\mathcal{A}\to Aitalic_δ : caligraphic_A → italic_A defined on a ∗*∗-subalgebra 𝒜⊆A𝒜𝐴\mathcal{A}\subseteq Acaligraphic_A ⊆ italic_A such that | C |
\alpha}{2}}}^{2}+\widetilde{c}_{1}|(\epsilon,\chi_{0})|^{2},( italic_L italic_ϵ , italic_ϵ ) ≥ divide start_ARG italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ∥ italic_ϵ ∥ start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT divide start_ARG italic_α end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + over~ start_ARG italic_c end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ( italic_ϵ , italic_χ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , | The inequality (4.38) is a consequence of the decay properties of Q𝑄Qitalic_Q and χ0subscript𝜒0\chi_{0}italic_χ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT in Lemmas 2.3 and 4.2, which imply | We also require some regularity and spatial decay properties of the function χ0subscript𝜒0\chi_{0}italic_χ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. | In what follows we need some additional spatial decay and regularity properties of the ground state, which we prove next. | The paper is organized as follows. In Section 2, we review the ground state and deduce some key properties concerning spatial decay and regularity of solutions of (1.9). | B |
0,&\text{otherwise.}\end{cases}italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_n ) = { start_ROW start_CELL italic_s italic_g italic_n ( italic_y ( italic_n ) ) , end_CELL start_CELL if italic_n ∈ italic_F start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL 0 , end_CELL start_CELL otherwise. end_CELL end_ROW | 6. Xℱsuperscript𝑋ℱX^{\mathcal{F}}italic_X start_POSTSUPERSCRIPT caligraphic_F end_POSTSUPERSCRIPT and the dual of Xℱsubscript𝑋ℱX_{\mathcal{F}}italic_X start_POSTSUBSCRIPT caligraphic_F end_POSTSUBSCRIPT | This is a vector of norm at most 1 in Xℱsuperscript𝑋ℱX^{\mathcal{F}}italic_X start_POSTSUPERSCRIPT caligraphic_F end_POSTSUPERSCRIPT and thus | that studying Xℱsuperscript𝑋ℱX^{\mathcal{F}}italic_X start_POSTSUPERSCRIPT caligraphic_F end_POSTSUPERSCRIPT is easier than Xℱ∗subscriptsuperscript𝑋ℱX^{*}_{\mathcal{F}}italic_X start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_F end_POSTSUBSCRIPT. This is why Xℱsuperscript𝑋ℱX^{\mathcal{F}}italic_X start_POSTSUPERSCRIPT caligraphic_F end_POSTSUPERSCRIPT may be helpful. | A vector y∈Xℱ𝑦superscript𝑋ℱy\in X^{\mathcal{F}}italic_y ∈ italic_X start_POSTSUPERSCRIPT caligraphic_F end_POSTSUPERSCRIPT is an extreme point of the unit ball of Xℱsuperscript𝑋ℱX^{\mathcal{F}}italic_X start_POSTSUPERSCRIPT caligraphic_F end_POSTSUPERSCRIPT if and only if it is of the form | B |
Contraction theory is a classical framework for studying dynamical systems where stability is defined incrementally between two arbitrary trajectories. Contracting systems feature desirable transient and asymptotic behaviors including i) forgetting their initial conditions, ii) exponential convergence to a single trajectory, and iii) input-to-state robustness with respect to disturbances and unmodelled dynamics. While the study of contracting systems can be traced back to the 1950s, many recent works have focused on infinitesimal frameworks [8] and Finsler-Lyapunov frameworks [9] for analysis of contracting systems. | It is well known that cooperative systems are amenable to efficient stability analysis using suitable Lyapunov functions [10], a feature that can be used to develop computationally efficient techniques for control design of large-scale cooperative systems. For cooperative dynamical systems over networks, several recent works have focused on construction of sum-separable or max-separable Lyapunov functions, i.e., Lyapunov functions that can be represented as sum or maximum of scalar-valued functions of each state component [11, 12]. | It turns out that, for cooperative systems, contractivity plays an essential role in the design of sum-separable and max-separable Lyapunov functions. In [13] contraction with respect to the ℓ∞subscriptℓ\ell_{\infty}roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT- and ℓ1subscriptℓ1\ell_{1}roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-norm has been used to establish existence of sum-separable and max-separable Lyapunov function for cooperative systems. In [14] contraction theory with respect to a Riemannian metric has been used to study sum-separable and max-separable Lyapunov functions for cooperative systems. | Nonetheless, certifying monotonicity of systems with respect to arbitrary cones is usually computationally complicated. Moreover, many techniques developed for cooperative systems, including those in [10, 11, 12, 13, 14], do not generalize easily or at all to the broader class of monotone systems. In particular, the connection between contraction theory and monotone system theory with respect to arbitrary cones and the existence of suitable Lyapunov functions for stability analysis of monotone systems with respect to arbitrary cones is not well understood or studied. Exceptions are [22], which considers searching for a polyhedral cone which makes a nonlinear system monotone, and [23], which studies incremental stability of monotone systems with respect to arbitrary or polyhedral cones. | It is known that monotone systems exhibit highly ordered dynamical behaviors [3] that can be used to establish stability of their interconnection [4], to develop computationally efficient techniques for their control synthesis [5, 6] and to perform reachability analysis to ensure their safety [7]. | A |
All known extreme examples for Conjecture 1 are obtained from those of the CHC, taking the color classes as stars. | This suggests looking at the case when the sets of edges are not stars, and trying to improve the upper bound on the girth. | a key ingredient in the proofs is a result by Bollobás and Szemerédi [4] on the girth of sparse graphs. | a set F𝐹Fitalic_F of edges is said to be rainbow for ℱℱ\mathcal{F}caligraphic_F if each of the edges in F𝐹Fitalic_F is taken from a distinct Fisubscript𝐹𝑖F_{i}italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT (if the sets Fisubscript𝐹𝑖F_{i}italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are disjoint, this means that |F∩Fi|≤1𝐹subscript𝐹𝑖1|F\cap F_{i}|\leq 1| italic_F ∩ italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | ≤ 1 for each i𝑖iitalic_i). The rainbow girth rgirth(ℱ)𝑟𝑔𝑖𝑟𝑡ℎℱrgirth(\mathcal{F})italic_r italic_g italic_i italic_r italic_t italic_h ( caligraphic_F ) of ℱℱ\mathcal{F}caligraphic_F is the minimal length of a rainbow cycle with respect to ℱℱ\mathcal{F}caligraphic_F. | To get the logarithmic in n𝑛nitalic_n bound on the rainbow girth, it is necessary to assume α>1/2𝛼12\alpha>1/2italic_α > 1 / 2 in Theorem 5 and Theorem 6. The following n𝑛nitalic_n-vertex graph F𝐹Fitalic_F with n/2𝑛2n/2italic_n / 2 matchings of size 2 in ℱMsubscriptℱ𝑀\mathcal{F}_{M}caligraphic_F start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT and n/2𝑛2n/2italic_n / 2 single edges in ℱEsubscriptℱ𝐸\mathcal{F}_{E}caligraphic_F start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT (so that α=1/2𝛼12\alpha=1/2italic_α = 1 / 2) has rainbow girth linear in n𝑛nitalic_n. | A |
2}\lVert\bm{x}-\bm{y}\rVert^{2}.prox start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ( bold_italic_y ) = start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT bold_italic_x end_POSTSUBSCRIPT italic_f ( bold_italic_x ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∥ bold_italic_x - bold_italic_y ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . | Intuitively, this seeks to shrink toward the current iterate 𝜽ksubscript𝜽𝑘\bm{\theta}_{k}bold_italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT when minimizing F𝐹Fitalic_F, where a parameter α𝛼\alphaitalic_α determines the strength of the shrinkage. Methods such as proximal gradient descent (Parikh et al., 2014; Li and Lin, 2015) consider composite objectives, alternating between proximal operations and gradient descent steps. | Stochastic versions of proximal methods have been studied to improve numerical instability. The stochastic proximal point algorithm (Bianchi, 2016; Asi and Duchi, 2019), also called implicit SGD (Toulis et al., 2014; Toulis and Airoldi, 2017; Lee et al., 2022), applies stochastic approximation to the proximal point algorithm. Subsampling one data point at each iteration, | Many powerful iterative algorithms for optimization involve the proximal operator (Bauschke et al., 2011; Parikh et al., 2014), defined as | A fundamental example is the proximal point algorithm (Rockafellar, 1976; Parikh et al., 2014), which minimizes an objective F𝐹Fitalic_F with successive proximal operations: | D |
1-\gamma_{11}}\bigg{)}=C(\Gamma).∑ start_POSTSUBSCRIPT italic_γ ∈ roman_Γ \ { italic_i italic_d } end_POSTSUBSCRIPT divide start_ARG roman_det italic_γ end_ARG start_ARG ( 1 - italic_γ ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG ( italic_γ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT - 3 italic_γ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT - 3 + divide start_ARG 4 italic_γ start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 21 end_POSTSUBSCRIPT end_ARG start_ARG 1 - italic_γ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_ARG ) = italic_C ( roman_Γ ) . | From the definition of ‘fixed point free’ above, one can prove that a unitary group ΓΓ\Gammaroman_Γ is ‘fixed point free’ if and only if any of the following equivalent conditions hold: (i) any non-identity element in ΓΓ\Gammaroman_Γ has no eigenvalue equal to 1; (ii) 0∈𝔹n0superscript𝔹𝑛0\in\mathbb{B}^{n}0 ∈ blackboard_B start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is the only fixed point for any non-identity element in ΓΓ\Gammaroman_Γ. | The representation of any of these groups above, in U(2)𝑈2U(2)italic_U ( 2 ), has to be faithful in this case, to obey the ‘fixed point free’ condition. Recall from Section 2 that a group Γ⊂Aut(𝔹2)Γ𝐴𝑢𝑡superscript𝔹2\Gamma\subset Aut(\mathbb{B}^{2})roman_Γ ⊂ italic_A italic_u italic_t ( blackboard_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) and its conjugate with an element in Aut(𝔹2)𝐴𝑢𝑡superscript𝔹2Aut(\mathbb{B}^{2})italic_A italic_u italic_t ( blackboard_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) produce biholomorphic ball quotients. So it is enough to consider the simplest unitary representations of the groups up to conjugation. | These groups above are sufficient to consider because the real sphere 𝕊3superscript𝕊3\mathbb{S}^{3}blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT forms the boundary of the two complex dimensional unit ball 𝔹2superscript𝔹2\mathbb{B}^{2}blackboard_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and we are interested in unitary action of a finite, fixed point free group on the unit ball which is a special case of orthogonal action (because U(2)𝑈2U(2)italic_U ( 2 ) is embeddable in SO(4)𝑆𝑂4SO(4)italic_S italic_O ( 4 )). | The following theorem by J. Milnor [Mil57] gives us all finite, fixed point free groups having a unitary action on ∂𝔹2superscript𝔹2\partial\mathbb{B}^{2}∂ blackboard_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. The classification of such finite, fixed point free, unitary groups in higher dimensions can be found in [Wol11] in the context of finding spaces of constant curvature. | D |
Let now M𝑀Mitalic_M be a matrix in GLn(ℤ)subscriptGL𝑛ℤ\operatorname{GL}_{n}(\mathbb{Z})roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_Z ) as before, and vMsubscript𝑣𝑀v_{M}italic_v start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT an eigenvector in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT for the eigenvalue ξ𝜉\xiitalic_ξ. | We now relate this geometric intersection form to the considerations on the ideal class associated to the conjugacy class of M𝑀Mitalic_M. | If ψ𝜓\psiitalic_ψ is conjugate to ϕitalic-ϕ\phiitalic_ϕ in the mapping class group of ΣΣ\Sigmaroman_Σ, then its associated lamination is the image of that of ϕitalic-ϕ\phiitalic_ϕ by a conjugating mapping class, and therefore, they define the same ideal class. | Since the intersection forms depend only on λ𝜆\lambdaitalic_λ, and not 𝒯𝒯\mathcal{T}caligraphic_T, we defined an ideal class of ℤ[ξ]ℤdelimited-[]𝜉\mathbb{Z}[\xi]blackboard_Z [ italic_ξ ] associated to ϕitalic-ϕ\phiitalic_ϕ. | We call the number of Aut(G)Aut𝐺{\mathrm{Aut}\left(G\right)}roman_Aut ( italic_G )-conjugacy classes in the commensurated-conjugacy class of ϕitalic-ϕ\phiitalic_ϕ, the class number of ϕitalic-ϕ\phiitalic_ϕ. This terminology is suggested and supported by our familiar example, as we explain now. | A |
Assume, without loss of generality, that AuL={1}subscriptsuperscript𝐴𝐿𝑢1A^{L}_{u}=\{1\}italic_A start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT = { 1 } and AvL={2}subscriptsuperscript𝐴𝐿𝑣2A^{L}_{v}=\{2\}italic_A start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = { 2 }. | Without loss of generality, assume that u𝑢uitalic_u lies strictly to the right of both L𝐿Litalic_L and L′superscript𝐿′L^{\prime}italic_L start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, and that v𝑣vitalic_v lies strictly to the left. | That is, assume that u𝑢uitalic_u lies strictly to the right of the line L𝐿Litalic_L and that v𝑣vitalic_v lies strictly to the left of the line L𝐿Litalic_L. | That is, u𝑢uitalic_u lies strictly to the right of the line L𝐿Litalic_L and v𝑣vitalic_v lies strictly to the left of the line L𝐿Litalic_L. | The case when u𝑢uitalic_u, v𝑣vitalic_v, w𝑤witalic_w lie to the right of L𝐿Litalic_L and x𝑥xitalic_x lies strictly to the left can be handled similarly. | C |
Euler-Maruyama discretization of (LD) (see, e.g., [54, 20, 4]). As a result, the convergence analysis of | Many modern learning tasks involve sampling from a high-dimensional density π∝e−fproportional-to𝜋superscript𝑒𝑓\pi\propto e^{-f}italic_π ∝ italic_e start_POSTSUPERSCRIPT - italic_f end_POSTSUPERSCRIPT, | To overcome the aforementioned challenges, our main contribution, from a high level, can be succinctly summarized as: | While, to our knowledge, our framework is significantly different from previous works on sampling, we | (Xt)t≥0subscriptsubscript𝑋𝑡𝑡0(X_{t})_{t\geq 0}( italic_X start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_t ≥ 0 end_POSTSUBSCRIPT. The discrete iterates {xk}k∈ℕsubscriptsubscript𝑥𝑘𝑘ℕ\{x_{k}\}_{k\in\mathbb{N}}{ italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_k ∈ blackboard_N end_POSTSUBSCRIPT can be converted to a | B |
\varepsilon^{\kappa}})∂ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT roman_Σ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_t ) ∩ ( italic_B start_POSTSUBSCRIPT 1 - italic_ε start_POSTSUPERSCRIPT italic_κ end_POSTSUPERSCRIPT / 2 end_POSTSUBSCRIPT \ italic_B start_POSTSUBSCRIPT 1 - 2 italic_ε start_POSTSUPERSCRIPT italic_κ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ). | Therefore, the choice of c∗subscript𝑐c_{*}italic_c start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT and (6.6) yield | By the assumption on cgsubscript𝑐𝑔c_{g}italic_c start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT, Lemma 2.10 and Corollary 4.2 yield that | \varepsilon^{\kappa}})∂ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT roman_Σ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_t ) ∩ ( italic_B start_POSTSUBSCRIPT 1 - italic_ε start_POSTSUPERSCRIPT italic_κ end_POSTSUPERSCRIPT / 2 end_POSTSUBSCRIPT \ italic_B start_POSTSUBSCRIPT 1 - 2 italic_ε start_POSTSUPERSCRIPT italic_κ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ), then Dahlberg’s lemma yields a dimensional constant c∗subscript𝑐c_{*}italic_c start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT such that | With this choice of c∗subscript𝑐c_{*}italic_c start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT, we finally define our barrier function by | A |
\scriptscriptstyle\bullet$}}}\hfil}}}S_{T}:\vartheta\in\Theta\}caligraphic_G = { italic_ϑ ∙ italic_S start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT : italic_ϑ ∈ roman_Θ } contains the terminal wealth distributions attainable by simple trading with zero initial wealth. The statement | condition (3.5) fails if there exists a fixed random variable in L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT that can be approximated arbitrarily well by elements of v+𝒢𝑣𝒢v+\mathcal{G}italic_v + caligraphic_G for two different values of v∈ℝ𝑣ℝv\in\mathbb{R}italic_v ∈ blackboard_R. | “the affine subspaces v+𝒢𝑣𝒢v+\mathcal{G}italic_v + caligraphic_G are disjoint for different values of v∈ℝ𝑣ℝv\in\mathbb{R}italic_v ∈ blackboard_R” | \scriptscriptstyle\bullet$}}}\hfil}}}S_{-})\right),italic_φ start_POSTSUPERSCRIPT ( italic_τ ) end_POSTSUPERSCRIPT ( italic_v , italic_H ) = blackboard_1 start_POSTSUBSCRIPT ⟧ italic_τ , italic_T ⟧ end_POSTSUBSCRIPT ( italic_ξ ( italic_H ) + italic_a ( italic_V start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ( italic_H ) - italic_v - italic_φ start_POSTSUPERSCRIPT ( italic_τ ) end_POSTSUPERSCRIPT ( italic_v , italic_H ) ∙ italic_S start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ) ) , | can be seen as the law of one price for simple wealth transfers between time 00 and time T𝑇Titalic_T. Indeed, the terminal wealths in v+𝒢𝑣𝒢v+\mathcal{G}italic_v + caligraphic_G are obtainable at the initial price v𝑣vitalic_v. If the same wealth is obtainable at two distinct initial prices, the law of one price no longer applies. Observe that (3.4) can be restated more compactly as 1∉𝒢1𝒢1\notin\mathcal{G}1 ∉ caligraphic_G. Since 𝒢𝒢\mathcal{G}caligraphic_G is not necessarily closed, one must strengthen this requirement to | B |
We say G𝐺Gitalic_G satisfies property (TWN+) if the first derivatives of the global normalization factors of the intertwining operators satisfy certain estimates; see Subsection 5.1. | We say G𝐺Gitalic_G satisfies property (BD) if the first derivatives of the local normalized intertwining operators satisfy certain estimates; see Subsection 5.2. | We say G𝐺Gitalic_G satisfies property (TWN+) if the first derivatives of the global normalization factors of the intertwining operators satisfy certain estimates; see Subsection 5.1. | In [19, Definition 1-2], the authors defined the Bounded Degree (BD) property of a reductive group. Property (BD) is a local property that concerns the size of the logarithmic derivative of RQ∣P(πv,s)subscript𝑅conditional𝑄𝑃subscript𝜋𝑣𝑠R_{Q\mid P}(\pi_{v},s)italic_R start_POSTSUBSCRIPT italic_Q ∣ italic_P end_POSTSUBSCRIPT ( italic_π start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT , italic_s ). Below we record an implication of property (BD) which we will use in this article. | First, we record the required estimates of the logarithmic derivatives of the normalizing factors and the normalized intertwining operators, denoted by properties (TWN+) and (BD), respectively. | A |
\mathrm{H}_{i_{n}}(A(\mathbb{C}),\mathbb{Z}),roman_H start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( italic_A start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( blackboard_C ) , blackboard_Z ) = ⨁ start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ⋯ + italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_r end_POSTSUBSCRIPT roman_H start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_A ( blackboard_C ) , blackboard_Z ) ⊗ ⋯ ⊗ roman_H start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_A ( blackboard_C ) , blackboard_Z ) , | The r𝑟ritalic_r-periods of an abelian variety are the values of a degree r𝑟ritalic_r polynomial evaluated in 1-periods. | \mathbb{Q})roman_H start_POSTSUBSCRIPT roman_H end_POSTSUBSCRIPT ( italic_A ) = ⊕ start_POSTSUBSCRIPT italic_r = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 italic_g end_POSTSUPERSCRIPT roman_H start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( italic_A ( blackboard_C ) , blackboard_Q ) we define the r𝑟ritalic_r-periods of A𝐴Aitalic_A (for r=0,…,2g𝑟0…2𝑔r=0,\dots,2gitalic_r = 0 , … , 2 italic_g) as the coefficients of the matrix which represents (with respect to K𝐾Kitalic_K-bases) the isomorphism between the cohomology groups HdRr(A)superscriptsubscriptHdR𝑟𝐴\mathrm{H}_{\mathrm{dR}}^{r}(A)roman_H start_POSTSUBSCRIPT roman_dR end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ( italic_A ) and Hr(A(ℂ),ℚ)⊗ℚKsubscripttensor-productℚsubscriptH𝑟𝐴ℂℚ𝐾\mathrm{H}_{r}(A(\mathbb{C}),\mathbb{Q})\otimes_{\mathbb{Q}}Kroman_H start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( italic_A ( blackboard_C ) , blackboard_Q ) ⊗ start_POSTSUBSCRIPT blackboard_Q end_POSTSUBSCRIPT italic_K which is given by the integration of differentials forms. In particular the coefficients of the matrix ΠAsubscriptΠ𝐴\Pi_{A}roman_Π start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT in (5.1) are the 1-periods of A𝐴Aitalic_A. | For 1-motives defined over ℚ¯¯ℚ\overline{\mathbb{Q}}over¯ start_ARG blackboard_Q end_ARG and of the type M=[u:ℤ→G],M=[u:\mathbb{Z}\to G],italic_M = [ italic_u : blackboard_Z → italic_G ] , with G𝐺Gitalic_G an extension of an elliptic curve ℰℰ\mathcal{E}caligraphic_E by 𝔾msubscript𝔾𝑚\mathbb{G}_{m}blackboard_G start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT, in Section 6 we list the polynomial relations between the periods of M𝑀Mitalic_M, which are dictated by its geometry. Using Grothendieck periods conjecture, | Observe that Legendre relation is not a polynomial relation between the 1-periods of ℰℰ\mathcal{E}caligraphic_E, since 2iπ2i𝜋2\mathrm{i}\pi2 roman_i italic_π is a 2-period of ℰ.ℰ\mathcal{E}.caligraphic_E . But as observed in Example 5.1, Legendre relation expresses the 2-period 2iπ2i𝜋2\mathrm{i}\pi2 roman_i italic_π as the value of a degree 2 polynomial evaluated in 1-periods (see Lemma 5.2). | A |
4. gr1L≠0subscriptgr1𝐿0\mathop{\mathrm{gr}}\nolimits_{1}L\neq 0roman_gr start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_L ≠ 0 (since L0¯subscript𝐿¯0L_{\bar{0}}italic_L start_POSTSUBSCRIPT over¯ start_ARG 0 end_ARG end_POSTSUBSCRIPT-action on L1¯subscript𝐿¯1L_{\bar{1}}italic_L start_POSTSUBSCRIPT over¯ start_ARG 1 end_ARG end_POSTSUBSCRIPT is reducible). | Theorem 4, p. 71 claims that any ℤℤ{\mathbb{Z}}blackboard_Z-graded Lie superalgebra satisfying the above conditions 1–4 is one of the following: | In both classifications, the ℤℤ{\mathbb{Z}}blackboard_Z-graded infinite-dimensional simple Lie algebras of vector fields over ℂℂ{\mathbb{C}}blackboard_C play a prominent role. | [Kac’s Main Theorem] ([K2, Theorem 5 on p. 74]) Any simple finite-dimensional Lie superalgebra 𝔤𝔤{\mathfrak{g}}fraktur_g over an algebraically closed field 𝕂𝕂{\mathbb{K}}blackboard_K of characteristic 00 is either isomorpic to a simple Lie algebra, or to one of the following Lie superalgebras: | It became more and more obvious with time that two digressions from the main goal (classification of simple and ℤℤ{\mathbb{Z}}blackboard_Z-graded objects) are important: classification of (a) deformations of ℤℤ{\mathbb{Z}}blackboard_Z-graded Lie algebras and (b) Lie algebras of derivations and central extensions of simple Lie algebras. | A |
We proceed by induction on the exponent of A𝐴Aitalic_A. Let Q𝑄Qitalic_Q be the subgroup of elements of order dividing p𝑝pitalic_p. Then Q𝑄Qitalic_Q is a finite abelian group of exponent p𝑝pitalic_p, and therefore it is the additive group of a finite-dimensional vector space over 𝔽psubscript𝔽𝑝\mathbb{F}_{p}blackboard_F start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT. Moreover, the image of QΓsuperscript𝑄ΓQ^{\Gamma}italic_Q start_POSTSUPERSCRIPT roman_Γ end_POSTSUPERSCRIPT under f𝑓fitalic_f also has exponent p𝑝pitalic_p, which implies that f𝑓fitalic_f restricts to an injective additive cellular automaton QΓ→QΓ→superscript𝑄Γsuperscript𝑄ΓQ^{\Gamma}\to Q^{\Gamma}italic_Q start_POSTSUPERSCRIPT roman_Γ end_POSTSUPERSCRIPT → italic_Q start_POSTSUPERSCRIPT roman_Γ end_POSTSUPERSCRIPT. This is automatically linear, and since Q𝑄Qitalic_Q is a finite-dimensional 𝔽psubscript𝔽𝑝\mathbb{F}_{p}blackboard_F start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT-vector space, by the assumption on L𝐿Litalic_L-surjunctivity we have f(QΓ)=QΓ𝑓superscript𝑄Γsuperscript𝑄Γf(Q^{\Gamma})=Q^{\Gamma}italic_f ( italic_Q start_POSTSUPERSCRIPT roman_Γ end_POSTSUPERSCRIPT ) = italic_Q start_POSTSUPERSCRIPT roman_Γ end_POSTSUPERSCRIPT, and moreover by injectivity f−1(QΓ)=QΓsuperscript𝑓1superscript𝑄Γsuperscript𝑄Γf^{-1}(Q^{\Gamma})=Q^{\Gamma}italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_Q start_POSTSUPERSCRIPT roman_Γ end_POSTSUPERSCRIPT ) = italic_Q start_POSTSUPERSCRIPT roman_Γ end_POSTSUPERSCRIPT as well. Since QΓsuperscript𝑄ΓQ^{\Gamma}italic_Q start_POSTSUPERSCRIPT roman_Γ end_POSTSUPERSCRIPT is closed and ΓΓ\Gammaroman_Γ-invariant, it follows that f𝑓fitalic_f induces an injective additive cellular automaton (A/Q)Γ→(A/Q)Γ→superscript𝐴𝑄Γsuperscript𝐴𝑄Γ(A/Q)^{\Gamma}\to(A/Q)^{\Gamma}( italic_A / italic_Q ) start_POSTSUPERSCRIPT roman_Γ end_POSTSUPERSCRIPT → ( italic_A / italic_Q ) start_POSTSUPERSCRIPT roman_Γ end_POSTSUPERSCRIPT. The exponent of A/Q𝐴𝑄A/Qitalic_A / italic_Q is strictly smaller than the exponent of A𝐴Aitalic_A, and thus we conclude by induction. | Let p𝑝pitalic_p be a prime. Kaplansky’s stable finiteness conjecture over some field of characteristic p𝑝pitalic_p implies Kaplansky’s stable finiteness conjecture over all fields of characteristic p𝑝pitalic_p. | Kaplansky’s direct finiteness conjecture is equivalent to Kaplansky’s stable finiteness conjecture by Theorem 3.3. | Therefore, Kaplansky’s direct finiteness conjecture is equivalent to Kaplansky’s stable finiteness conjecture. | Surjunctive groups satisfy Kaplansky’s stable finiteness conjecture. In particular, sofic groups satisfy Kaplansky’s stable finiteness conjecture. | D |
For functions f,g:ℕ→ℕ:𝑓𝑔→ℕℕf,g:\mathbb{N}\to\mathbb{N}italic_f , italic_g : blackboard_N → blackboard_N we write f⪯gprecedes-or-equals𝑓𝑔f\preceq gitalic_f ⪯ italic_g when there exists C>0𝐶0C>0italic_C > 0 such that f(n)≤Cg(Cn+C)+Cn+C𝑓𝑛𝐶𝑔𝐶𝑛𝐶𝐶𝑛𝐶f(n)\leq Cg(Cn+C)+Cn+Citalic_f ( italic_n ) ≤ italic_C italic_g ( italic_C italic_n + italic_C ) + italic_C italic_n + italic_C for all n𝑛nitalic_n. We write f≃gsimilar-to-or-equals𝑓𝑔f\simeq gitalic_f ≃ italic_g when f⪯gprecedes-or-equals𝑓𝑔f\preceq gitalic_f ⪯ italic_g and g⪯fprecedes-or-equals𝑔𝑓g\preceq fitalic_g ⪯ italic_f. | The most novel feature of the work here is the idea behind the exponential lower bound (as proved in Section 3). It relies on the observation that F2subscript𝐹2F_{2}italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and ℤ≀ℤ≀ℤℤ\mathbb{Z}\wr\mathbb{Z}blackboard_Z ≀ blackboard_Z admit height functions (homomorphisms onto ℤℤ\mathbb{Z}blackboard_Z) such that for all integers n>0𝑛0n>0italic_n > 0, there are pairs of height-00 elements a distance 2n2𝑛2n2 italic_n apart with the property that any path from one to the other travels up to height n𝑛nitalic_n en route—see Proposition 3.1. | Two finite generating sets for a group yield biLipschitz word metrics, with the constants reflecting the minimal length words required to express the elements of one generating set as words on the other. So, up to ≃similar-to-or-equals\simeq≃, the growth rate of a distortion function does not depend on the finite generating sets. | Suppose the lights at the elements e𝑒eitalic_e and xny−nsuperscript𝑥𝑛superscript𝑦𝑛x^{n}y^{-n}italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_y start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT of K𝐾Kitalic_K are set to 1111 and −11-1- 1, respectively, and all other lights are off (set to 00). How can a lamplighter turn all the lights off using x𝑥xitalic_x, y𝑦yitalic_y, σ𝜎\sigmaitalic_σ, and τ𝜏\tauitalic_τ? He has four types of moves at his disposal: he can navigate the Cayley graph of K𝐾Kitalic_K (by using x𝑥xitalic_x and y𝑦yitalic_y); because σ=[x,a]a=x−1a−1xa2𝜎𝑥𝑎𝑎superscript𝑥1superscript𝑎1𝑥superscript𝑎2\sigma=[x,a]a=x^{-1}a^{-1}xa^{2}italic_σ = [ italic_x , italic_a ] italic_a = italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, he can decrement by 1111 the lamp one step away in the x−1superscript𝑥1x^{-1}italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT-direction at the expense of incrementing the lamp where he stands by 2222; and likewise in the y−1superscript𝑦1y^{-1}italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT-direction using τ𝜏\tauitalic_τ. The answer is he sets the lamp at e𝑒eitalic_e to 00 at the expense of setting the lamp at x𝑥xitalic_x to 2222. Then he sets the lamp at x𝑥xitalic_x to 00 at the expense of setting the lamp at x2superscript𝑥2x^{2}italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT to 4444. And so on, until the lamp at xnsuperscript𝑥𝑛x^{n}italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is set to 2nsuperscript2𝑛2^{n}2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. He then sets that to 00 and, proceeding in the y−1superscript𝑦1y^{-1}italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT direction, sets the lamp at xny−1superscript𝑥𝑛superscript𝑦1x^{n}y^{-1}italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT to 2n−1superscript2𝑛12^{n-1}2 start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT. Continuing likewise in the y−1superscript𝑦1y^{-1}italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT-direction he sets the lamp at xny−(n−1)superscript𝑥𝑛superscript𝑦𝑛1x^{n}y^{-(n-1)}italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_y start_POSTSUPERSCRIPT - ( italic_n - 1 ) end_POSTSUPERSCRIPT to 2222. Finally, he adjusts the lamp at xny−(n−1)superscript𝑥𝑛superscript𝑦𝑛1x^{n}y^{-(n-1)}italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_y start_POSTSUPERSCRIPT - ( italic_n - 1 ) end_POSTSUPERSCRIPT to zero at the expense of changing the lamp at xny−nsuperscript𝑥𝑛superscript𝑦𝑛x^{n}y^{-n}italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_y start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT, but as that was initially set to −11-1- 1, this results in all lights being off, as required. | For a group G𝐺Gitalic_G with finite generating set S𝑆Sitalic_S, let |g|Ssubscript𝑔𝑆|g|_{S}| italic_g | start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT denote the length of a shortest word on S±1superscript𝑆plus-or-minus1S^{\pm 1}italic_S start_POSTSUPERSCRIPT ± 1 end_POSTSUPERSCRIPT representing g𝑔gitalic_g. The word metric dSsubscript𝑑𝑆d_{S}italic_d start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT on G𝐺Gitalic_G is dS(g,h)=|g−1h|Ssubscript𝑑𝑆𝑔ℎsubscriptsuperscript𝑔1ℎ𝑆d_{S}(g,h)=\left|g^{-1}h\right|_{S}italic_d start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( italic_g , italic_h ) = | italic_g start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_h | start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT. | B |
≤Chr^+t‖ψ‖r^,Γi,j,absent𝐶superscriptℎ^𝑟𝑡subscriptnorm𝜓^𝑟subscriptΓ𝑖𝑗\displaystyle\leq Ch^{\hat{r}+t}\|\psi\|_{\hat{r},\Gamma_{i,j}},≤ italic_C italic_h start_POSTSUPERSCRIPT over^ start_ARG italic_r end_ARG + italic_t end_POSTSUPERSCRIPT ∥ italic_ψ ∥ start_POSTSUBSCRIPT over^ start_ARG italic_r end_ARG , roman_Γ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT , | 0<t~≤r+1, 0<r~≤r+1,formulae-sequence0~𝑡𝑟1 0~𝑟𝑟1\displaystyle 0<\tilde{t}\leq r+1,\,0<\tilde{r}\leq r+1,0 < over~ start_ARG italic_t end_ARG ≤ italic_r + 1 , 0 < over~ start_ARG italic_r end_ARG ≤ italic_r + 1 , | ≤Chr^‖ζ‖r^,Ωi,absent𝐶superscriptℎ^𝑟subscriptnorm𝜁^𝑟subscriptΩ𝑖\displaystyle\leq Ch^{\hat{r}}\|\zeta\|_{\hat{r},\Omega_{i}},≤ italic_C italic_h start_POSTSUPERSCRIPT over^ start_ARG italic_r end_ARG end_POSTSUPERSCRIPT ∥ italic_ζ ∥ start_POSTSUBSCRIPT over^ start_ARG italic_r end_ARG , roman_Ω start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT , | 0≤r^≤r+1,0^𝑟𝑟1\displaystyle 0\leq\hat{r}\leq r+1,0 ≤ over^ start_ARG italic_r end_ARG ≤ italic_r + 1 , | ≤Chr^+t‖ψ‖r^,Γi,j,absent𝐶superscriptℎ^𝑟𝑡subscriptnorm𝜓^𝑟subscriptΓ𝑖𝑗\displaystyle\leq Ch^{\hat{r}+t}\|\psi\|_{\hat{r},\Gamma_{i,j}},≤ italic_C italic_h start_POSTSUPERSCRIPT over^ start_ARG italic_r end_ARG + italic_t end_POSTSUPERSCRIPT ∥ italic_ψ ∥ start_POSTSUBSCRIPT over^ start_ARG italic_r end_ARG , roman_Γ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT , | C |
E6subscript𝐸6E_{6}italic_E start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT with |ρ|=2𝜌2|\rho|=2| italic_ρ | = 2. | Let us now work towards proving Theorem 1.6. The irreducible ℓℓ\ellroman_ℓ-cuspidal finite reflection groups are classified in [Tow18, Table 2]. Of these cases, the types of W𝑊Witalic_W with non-trivial Coxeter diagram automorphisms are the following: | We will now introduce the preliminaries to prove Theorem 1.5. The ℓℓ\ellroman_ℓ-Sylow classes of parabolic and reflection subgroups were classified in [Tow20]. The normalizer of a parabolic subgroup P𝑃Pitalic_P in a finite complex reflection group W𝑊Witalic_W was described in [MT18]. Similarly to [How80], it is proven that there exists a semidirect product decomposition NW(P)=P⋊Usubscript𝑁𝑊𝑃right-normal-factor-semidirect-product𝑃𝑈N_{W}(P)=P\rtimes Uitalic_N start_POSTSUBSCRIPT italic_W end_POSTSUBSCRIPT ( italic_P ) = italic_P ⋊ italic_U, where the complement U𝑈Uitalic_U is often described as a stabilizer of a set of roots whose associated reflections generate P𝑃Pitalic_P. We call the complement U𝑈Uitalic_U the MT-complement. When W𝑊Witalic_W is the complexification of a finite real reflection group, the MT-complement of a parabolic subgroup agrees with the H-complement. In some of the strictly complex cases an ad-hoc choice of roots must be made. Furthermore, in some cases there is no choice of roots for the generating reflections whose stabilizer is a complement (see [MT18] for details). The MT-complement and other data can be calculated using the MAGMA code mentioned in [MT18] and found at https://www.maths.usyd.edu.au/u/don/software.html. We note that the roots and generating reflections used by MAGMA for finite complex reflection groups correspond to the defining relations and diagrams given in [BMR98, App. 2, Tables 1-5] except in cases G12subscript𝐺12G_{12}italic_G start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT and G22subscript𝐺22G_{22}italic_G start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT. Due to the ad-hoc nature of the roots stabilized by the MT-complement, relying on a generalization of Coxeter diagram automorphisms to prove Theorem 1.5 appears to be unfeasible. Hence, we instead consider the irreducible W𝑊Witalic_W that are not ℓℓ\ellroman_ℓ-cuspidal, which are deduced from [Tow20, Table 1], and check if there is a Sylow ℓℓ\ellroman_ℓ-subgroup normalized by the MT-complement. | These Coxeter diagram automorphisms will play an important role in proving Theorem 1.4 via their connection to the H-complement, which we will now describe. | We will now investigate a generalization of Theorem 1.6. Finite complex reflection groups do not have standard notion of simple roots or simple reflections. However, there is a commonly used set of generating reflections, which can be found in [BMR98, App. 2, Tables 1-5]. For a fixed W𝑊Witalic_W, let S𝑆Sitalic_S be the set of these commonly used generating reflections. We define a diagram automorphism of a complex reflection group W𝑊Witalic_W as a permutation of S𝑆Sitalic_S that induces a group automorphism. These diagram automorphisms permute the diagrams given in [BMR98, App. 2, Tables 1-5], generalizing the notion of Coxeter diagram automorphism. Proposition 6.2 is a classification of the non-trivial diagram automorphisms ρ𝜌\rhoitalic_ρ for irreducible W𝑊Witalic_W, where ρ𝜌\rhoitalic_ρ is written as a permutation of the generating reflections with respect to the ordering of the diagrams in Figure 2. | C |
So for every B1⊆Bsubscript𝐵1𝐵B_{1}\subseteq Bitalic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊆ italic_B, there is a b∈B𝑏𝐵b\in Bitalic_b ∈ italic_B with N(B1)=N(b).𝑁subscript𝐵1𝑁𝑏N(B_{1})=N(b).italic_N ( italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_N ( italic_b ) . | The graph G𝐺Gitalic_G being twin-free is equivalent to ℳℳ\mathcal{M}caligraphic_M having distinct rows and distinct columns. | the condition that the union of any set of columns of ℳℳ\mathcal{M}caligraphic_M is a column of ℳℳ\mathcal{M}caligraphic_M is equivalent with the condition that the union of any set of rows of ℳℳ\mathcal{M}caligraphic_M is a row of ℳℳ\mathcal{M}caligraphic_M. | The latter is the case if and only if this is true for every subset B1subscript𝐵1B_{1}italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT of size 2,22,2 , which is equivalent with the union of any 2222 rows being a row itself. | If there is a bipartition class, say B𝐵Bitalic_B, of G𝐺Gitalic_G for which the union of the neighbourhoods of some of its vertices, B1⊆Bsubscript𝐵1𝐵B_{1}\subseteq Bitalic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊆ italic_B (B1≠∅subscript𝐵1B_{1}\not=\emptysetitalic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≠ ∅), is not equal to the neighbourhood of a vertex in B𝐵Bitalic_B, then A∖N(B1)∪{u∈B∣N(u)⊆N(B1)}𝐴𝑁subscript𝐵1conditional-set𝑢𝐵𝑁𝑢𝑁subscript𝐵1A\setminus N(B_{1})\cup\{u\in B\mid N(u)\subseteq N(B_{1})\}italic_A ∖ italic_N ( italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∪ { italic_u ∈ italic_B ∣ italic_N ( italic_u ) ⊆ italic_N ( italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) } would be another maximal independent set not counted in the proof of Theorem 3. | C |
\times M^{0}italic_P start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT × italic_e start_POSTSUPERSCRIPT italic_s italic_c end_POSTSUPERSCRIPT italic_d italic_c × italic_M start_POSTSUPERSCRIPT italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT × italic_M start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT × italic_M start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT as desired. Here we discussed β1∈ℝ,β2=Q,β3=0formulae-sequencesubscript𝛽1ℝformulae-sequencesubscript𝛽2𝑄subscript𝛽30\beta_{1}\in\mathbb{R},\beta_{2}=Q,\beta_{3}=0italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R , italic_β start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_Q , italic_β start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0 to lighten notation, but the same argument applies for β1,β2,β3∈ℝsubscript𝛽1subscript𝛽2subscript𝛽3ℝ\beta_{1},\beta_{2},\beta_{3}\in\mathbb{R}italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_β start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_β start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∈ blackboard_R. We conclude that if ρ𝜌\rhoitalic_ρ is the uniform probability measure on {0}×[0,π]00𝜋\{0\}\times[0,\pi]{ 0 } × [ 0 , italic_π ] then Definition 2.28 agrees with Definition 2.8. | If h∼Pρsimilar-toℎsubscript𝑃𝜌h\sim P_{\rho}italic_h ∼ italic_P start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT then the law of hℎhitalic_h viewed as a distribution modulo additive constant does not depend on ρ𝜌\rhoitalic_ρ, so by the translation invariance of dc𝑑𝑐dcitalic_d italic_c, | Definition 2.28 does not depend on the choice of ρ𝜌\rhoitalic_ρ. Moreover, if β1,β2,β3∈ℝsubscript𝛽1subscript𝛽2subscript𝛽3ℝ\beta_{1},\beta_{2},\beta_{3}\in\mathbb{R}italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_β start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_β start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∈ blackboard_R, then the definition agrees with Definition 2.8. | Now let ρ𝜌\rhoitalic_ρ be arbitrary. It remains to verify that Definition 2.28 does not depend on the choice of ρ𝜌\rhoitalic_ρ. | the law of h+𝐜ℎ𝐜h+\mathbf{c}italic_h + bold_c does not depend on ρ𝜌\rhoitalic_ρ; since ϕitalic-ϕ\phiitalic_ϕ is a function of h+𝐜ℎ𝐜h+\mathbf{c}italic_h + bold_c and randomness independent of (h,𝐜)ℎ𝐜(h,\mathbf{c})( italic_h , bold_c ), the claim follows. | C |
We shall use the letter π𝜋\piitalic_π to denote both the bundle projection from TM𝑇𝑀TMitalic_T italic_M to M𝑀Mitalic_M and the bundle projection from TTM𝑇𝑇𝑀TTMitalic_T italic_T italic_M to TM𝑇𝑀TMitalic_T italic_M. Thus from TTM𝑇𝑇𝑀TTMitalic_T italic_T italic_M to TM𝑇𝑀TMitalic_T italic_M we have two canonical maps: the bundle projection π𝜋\piitalic_π, and the differential dπ𝑑𝜋d\piitalic_d italic_π of the bundle projection π:TM→M:𝜋→𝑇𝑀𝑀\pi:TM\to Mitalic_π : italic_T italic_M → italic_M. | The paper is organized as follows. In Section 2 we first recall some basic facts about Riemannian geometry of surfaces and about sprays. We then prove Proposition 2.15 regarding projective Finsler-metrizability of magnetic sprays. Finally, we give the necessary background and relevant results on needle decomposition. In Section 3 we introduce the notion of a nonnegatively curved weighted spray space, and give a characterization of such spaces in the case of a metric spray on a Riemannian surface. We also mention an analogue of the curvature-dimension condition from the theory of metric measure spaces [20, 12] in the setting of sprays on surfaces. In Section 4 we prove the equivalence between the nonnegative curvature condition and the Brunn-Minkowski inequality in the case of simple, proper metric sprays (and, more generally, in the case of projectively Finsler-metrizable sprays on surfaces). In Section 5 we provide some examples of weighted spray spaces satisfying the Brunn-Minkowski inequality. | The unit tangent bundle admits a natural global frame (E1,E2,V)subscript𝐸1subscript𝐸2𝑉(E_{1},E_{2},V)( italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_V ). The flow of the vector field E1subscript𝐸1E_{1}italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is the geodesic flow on SM𝑆𝑀SMitalic_S italic_M, the flow of V𝑉Vitalic_V restricts to rotation of each tangent circle (chosen according to the orientation of M𝑀Mitalic_M), and we have the commutation relations | Let (M,g)𝑀𝑔(M,g)( italic_M , italic_g ) be a two-dimensional oriented Riemannian manifold. Fix a family ΓΓ\Gammaroman_Γ of smooth unit-speed curves on M𝑀Mitalic_M, with the property that for any unit vector v∈SM𝑣𝑆𝑀v\in SMitalic_v ∈ italic_S italic_M, there is a unique curve γ∈Γ𝛾Γ\gamma\in\Gammaitalic_γ ∈ roman_Γ with γ˙(0)=v˙𝛾0𝑣\dot{\gamma}(0)=vover˙ start_ARG italic_γ end_ARG ( 0 ) = italic_v. Here SM𝑆𝑀SMitalic_S italic_M is the unit tangent bundle of M𝑀Mitalic_M. Equivalently, fix a function k:SM→ℝ:𝑘→𝑆𝑀ℝ{k}:SM\to\mathbb{R}italic_k : italic_S italic_M → blackboard_R and let ΓΓ\Gammaroman_Γ be the family of solutions to the ordinary differential equation | We recall some basic facts about Riemannian surfaces which can be found in [19]. The unit tangent bundle SM𝑆𝑀SMitalic_S italic_M of a Riemannian surface is the subbundle of TM𝑇𝑀TMitalic_T italic_M consisting of unit vectors, | D |
}v_{3}(n)=1.\end{cases}{ start_ROW start_CELL ( divide start_ARG italic_ρ end_ARG start_ARG 2 italic_n end_ARG ) start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT = 1 end_CELL start_CELL if italic_v start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_n ) = 0 , end_CELL end_ROW start_ROW start_CELL ( divide start_ARG italic_ρ end_ARG start_ARG 2 italic_n / 3 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∏ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ( divide start_ARG 3 end_ARG start_ARG italic_π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ) start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT = 1 end_CELL start_CELL if italic_v start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_n ) = 2 , end_CELL end_ROW start_ROW start_CELL ∏ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ( divide start_ARG 3 end_ARG start_ARG italic_π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ) start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT = 1 end_CELL start_CELL if italic_v start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_n ) = 1 . end_CELL end_ROW | formed from packaging the local solvability conditions of the cubic equations from (1.1) and (1.2). We will identify Selϕ^(E^n)subscriptSel^italic-ϕsubscript^𝐸𝑛\operatorname{Sel}_{\hat{\phi}}(\hat{E}_{n})roman_Sel start_POSTSUBSCRIPT over^ start_ARG italic_ϕ end_ARG end_POSTSUBSCRIPT ( over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) with the kernel of some ω1(n)+rsubscript𝜔1𝑛𝑟\omega_{1}(n)+ritalic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_n ) + italic_r by ω(n)+c𝜔𝑛𝑐\omega(n)+citalic_ω ( italic_n ) + italic_c matrix, where r,c∈{−1,0,1}𝑟𝑐101r,c\in\{-1,0,1\}italic_r , italic_c ∈ { - 1 , 0 , 1 }. Then the dimensions of the matrix guarantees the lower bound (1.6). Furthermore, as n𝑛nitalic_n grows, we expect that usually ω1(n)≈12loglognsubscript𝜔1𝑛12𝑛\omega_{1}(n)\approx\frac{1}{2}\log\log nitalic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_n ) ≈ divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_log roman_log italic_n and ω(n)≈loglogn𝜔𝑛𝑛\omega(n)\approx\log\log nitalic_ω ( italic_n ) ≈ roman_log roman_log italic_n in the sense of the Erdős–Kac theorem. Heuristically, viewing the entries of the matrix as random elements in 𝔽3subscript𝔽3\mathbb{F}_{3}blackboard_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT picked uniformly, the probability that the matrix takes the maximum possible rank tends to 1111 as n𝑛nitalic_n tends to infinity. | Applying properties of the cubic residue symbol [25, Theorem 7.8], the equations boil down to the local solvability conditions at the prime 3333 in [5, Lemma 6.11]. | In Heath-Brown’s work [13, 14], the solvability conditions that determines the elements in the 2222-Selmer group are in terms of quadratic residue symbols. In our case, the 3333-isogeny Selmer elements are controlled by conditions involving cubic residue symbols. To treat the sums, we adapt work of Klys [21] on the distribution of the ℓℓ\ellroman_ℓ-part of class groups of degree ℓℓ\ellroman_ℓ cyclic fields. Klys’ work is a generalization of a result by Fouvry–Klüners [8] on the distribution of the 4444-part of the class group of quadratic fields, which follows a line of attack similar to work of Heath-Brown [13, 14]. | which are precisely the local conditions for u𝑢uitalic_u to represent a class in Selϕ(En)subscriptSelitalic-ϕsubscript𝐸𝑛\operatorname{Sel}_{\phi}(E_{n})roman_Sel start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT ( italic_E start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) by [5, Section 6], except possibly at the prime 3333. In fact, the local conditions at 3333 are already encoded in the matrix. | B |
Putting all the regularizations together, our layerwise training process is shown in Algorithm 1. Algorithm 1 starts with training (minimizing the loss function (5)) a two-hidden layer network where the first layer is the (optional) upsampling/downsampling layer. Once, this network is trained, a new layer is added with the weights and biases initialized as zero (line 6 in Algorithm 1). Keeping parameters in the previous layers fixed, the newly added layer is trained (minimizing the cost function (6)). The procedure is repeated until a termination criteria is satisfied (line 4 in Algorithm 1). | Recall that promoting ε−δ𝜀𝛿\varepsilon-\deltaitalic_ε - italic_δ stability (Corollary 3.13) does not guarantee that the network classifies/regress correctly since training saturates for certain hyperparameter settings in Algorithm 1 (see Corollary 3.20). To overcome this issue, we incorporate a post-processing stage termed as the “sequential residual learning” step for further improving the predictions. The idea of sequential residual learning is to train a sequence of shallow neural networks to learn the residual (i.e lower the data loss ΦdsubscriptΦ𝑑\Phi_{d}roman_Φ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT while generalizing well) from Algorithm 1 where each network 𝒬isubscript𝒬𝑖{\mathcal{Q}}_{i}caligraphic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is trained for a limited epochs to prevent overfitting on the residuals. | As we shall show (see Corollary 3.20), even though Algorithm 1 proposed in this work prevents overfitting in early layers, training saturates after a certain critical layer. To overcome this issue, we add a post-processing stage for Algorithm 1 in order to further decrease the training loss without overfitting the data. | Note that even though we do not have infinitely many training samples as dictated by Proposition 3.10, the effect of manifold regularization is quite pronounced in this case. In order to further decrease the loss (promote approximate δ−limit-from𝛿\delta-italic_δ -robustness), we resort to Algorithm 2. The inputs for Algorithm 2 are provided in Appendix J. | In this paper, we presented a two-stage procedure for adaptive learning that promotes “robustness” and thereby generalizing well for a given data-set. The first stage (Algorithm 1) is designed to grow a neural network along the depth while promoting δ−limit-from𝛿\delta-italic_δ -stability for each hidden layer through the use of manifold regularization. In addition, non-important parameters are removed using a sparsity promoting regularization. We have proved that for certain hyperparameter settings of Algorithm 1, one faces with the training saturation problem where the newly added layer does not learn ( Corollary 3.20). In order to further improve the prediction accuracy, we designed Algorithm 2 as a post-processing stage where the main goal is to promote robustness (Definition 3.24). Besides several theoretical results, numerical results on prototype regression and classification tasks, including solving forward and inverse problems governed by elliptic PDEs, suggest that the proposed approach can outperform an ad-hoc baseline and other adaptation strategies in terms of generalization error. The proposed PIANN is perhaps the first PINN approaches that is capable of adapting the network architecture with theoretical results on stabilities. | B |
A good estimator should, at least in the asymptotic sense, be close to the true quantity that it wishes to estimate and we should be able to give uncertainty measure based on a finite sample size. An estimator with well-behaved asymptotic properties can help clinicians in many ways such as reducing the number of patients needed in a trial, cutting down the budget for toxicology studies and providing insightful findings for late phase trials. Suggested by Sr. Fisher [1], generations of statisticians have worked on the so-called "consistency" and "asymptotic normality" of estimators. The former is based on different versions of law of large numbers (LLN) and the later is based on various types of central limit theorems (CLT) [2]. In addition to these two main tools, statisticians also apply other important but less well-known results in probability theory and other mathematical fields. To name a few, extreme value theory for distributions of maxima and minima [3], convex analysis for checking the optimality of a statistical design [4], asymptotic relative efficiency (ARE) of an estimator [5], concentration inequalities for finite sample properties and selection consistency [6] and other non-normal limits , robustness and simultaneous confidence bands of common statistical estimators [7, 8]. | The definitive answer to consistency and asymptotic normality of estimates in GLM is given by Fahrmeier and Kaufman [17]. We use similar conditions below. | A good estimator should, at least in the asymptotic sense, be close to the true quantity that it wishes to estimate and we should be able to give uncertainty measure based on a finite sample size. An estimator with well-behaved asymptotic properties can help clinicians in many ways such as reducing the number of patients needed in a trial, cutting down the budget for toxicology studies and providing insightful findings for late phase trials. Suggested by Sr. Fisher [1], generations of statisticians have worked on the so-called "consistency" and "asymptotic normality" of estimators. The former is based on different versions of law of large numbers (LLN) and the later is based on various types of central limit theorems (CLT) [2]. In addition to these two main tools, statisticians also apply other important but less well-known results in probability theory and other mathematical fields. To name a few, extreme value theory for distributions of maxima and minima [3], convex analysis for checking the optimality of a statistical design [4], asymptotic relative efficiency (ARE) of an estimator [5], concentration inequalities for finite sample properties and selection consistency [6] and other non-normal limits , robustness and simultaneous confidence bands of common statistical estimators [7, 8]. | Despite of different properties, consistency and asymptotic normality are still the most celebrated and important properties of statistical estimators in either academia or industry. Hence, in the following, we present a roadmap to consistency and asymptotic normality. Then we provide their applications in toxicology studies and clinical trials using a COVID-19 dataset. | We illustrate the asymptotic results by fitting a Poisson regression model to a COVID-19 dataset. The dataset is available online [19]. To model the relation between the confirmed COVID-19 cases and the geographical location (e.g., location of the state in U.S.), we shall use three specific columns of the dataset: confirmed cases, latitude and longitude. We assume that the relation forms a Poisson model [20]. That is, let Y,x1,x2𝑌subscript𝑥1subscript𝑥2Y,x_{1},x_{2}italic_Y , italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT be the number of cases, longitude and latitude, respectively, assume | C |
Applied to the pre-braided homogeneous category Uℳ¯𝑈¯ℳU\bar{\mathcal{M}}italic_U over¯ start_ARG caligraphic_M end_ARG (see Proposition 3.3), Theorem A of [RWW17], applied to the constant coefficient system ℤℤ\mathbb{Z}blackboard_Z, says the following. Let A𝐴Aitalic_A and X𝑋Xitalic_X be two decorated surfaces (possibly of infinite type), each with exactly one boundary component. | Let ℳ¯¯ℳ\bar{\mathcal{M}}over¯ start_ARG caligraphic_M end_ARG be the braided monoidal groupoid decribed above. Then the Quillen bracket category Uℳ¯𝑈¯ℳU\bar{\mathcal{M}}italic_U over¯ start_ARG caligraphic_M end_ARG is pre-braided and homogeneous. | We note that we have not changed the isomorphisms of the category by passing from ℳ¯¯ℳ\bar{\mathcal{M}}over¯ start_ARG caligraphic_M end_ARG to its Quillen bracket construction Uℳ¯𝑈¯ℳU\bar{\mathcal{M}}italic_U over¯ start_ARG caligraphic_M end_ARG. | Theorem 3.5 ([RWW17, Theorem A applied to Uℳ¯𝑈¯ℳU\bar{\mathcal{M}}italic_U over¯ start_ARG caligraphic_M end_ARG]). | Applied to the pre-braided homogeneous category Uℳ¯𝑈¯ℳU\bar{\mathcal{M}}italic_U over¯ start_ARG caligraphic_M end_ARG (see Proposition 3.3), Theorem A of [RWW17], applied to the constant coefficient system ℤℤ\mathbb{Z}blackboard_Z, says the following. Let A𝐴Aitalic_A and X𝑋Xitalic_X be two decorated surfaces (possibly of infinite type), each with exactly one boundary component. | C |