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We establish the convergence of the unified two-timescale Reinforcement Learning (RL) algorithm presented by Angiuli et al. This algorithm provides solutions to Mean Field Game (MFG) or Mean Field Control (MFC) problems depending on the ratio of two learning rates, one for the value function and the other for the mean field term. We focus a setting with finite state and action spaces, discrete time and infinite horizon. The proof of convergence relies on a generalization of the two-timescale approach of Borkar. The accuracy of approximation to the true solutions depends on the smoothing of the policies. We then provide an numerical example illustrating the convergence. Last, we generalize our convergence result to a three-timescale RL algorithm introduced by Angiuli et al. to solve mixed Mean Field Control Games (MFCGs).

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Using the notion of integral distance to analytic functions ($\operatorname{IDA}$), we characterize the membership in the Schatten classes of Hankel operators acting on doubling Fock spaces $F^{2}_{\phi}$ on $\mathbb{C}$. This characterization will be used to show that for $f\in L^{\infty}$, $1<p<\infty$, $H_{f}$ is in the Schatten class $S_{p}(F^{2}_{\phi},L^{2}_{\phi})$ if and only if $H_{\bar{f}}$ is in the Schatten class. This property is known as the Berger-Coburn phenomenon. An application of a similar idea shows that for a bounded function $f$, $H_{f}:F^{2}_{\phi}\to L^{2}_{\phi}$ is compact if and only if $H_{\bar{f}}$ is compact. We also construct a counter example of the Berger-Coburn phenomenon for Schatten class Hankel operators on doubling Fock spaces when $0<p\leq 1$.

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This paper is the first part of the series "Spherical higher order Fourier analysis over finite fields", aiming to develop the higher order Fourier analysis method along spheres over finite fields, and to solve the geometric Ramsey conjecture in the finite field setting. In this paper, we prove a quantitative equidistribution theorem for polynomial sequences in a nilmanifold, where the average is taken along spheres instead of cubes. To be more precise, let $\Omega\subseteq\mathbb{F}_{p}^{d}$ be a sphere. We showed that if a polynomial sequence $(g(n)\Gamma)_{n\in\Omega}$ which is $p$-periodic along $\Omega$ is not equidistributed on a nilmanifold $G/\Gamma$, then there exists a nontrivial horizontal character $\eta$ of $G/\Gamma$ such that $\eta\circ g \mod \mathbb{Z}$ vanishes on $\Omega$. This result will serve as a fundamental tool in later parts of the series to proof the spherical Gowers inverse theorem and the geometric Ramsey conjecture.

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This paper is the second part of the series "Spherical higher order Fourier analysis over finite fields", aiming to develop the higher order Fourier analysis method along spheres over finite fields, and to solve the geometric Ramsey conjecture in the finite field setting. In this paper, we study additive combinatorial properties for shifted ideals, i.e. the structure of sets of the form $E\pm E$, where $E$ is a collection of shifted ideals of the polynomial ring $\mathbb{F}_{p}[x_{1},\dots,x_{d}]$ and we identify two ideals if their difference contains the zero polynomial. We show that under appropriate definitions, the set $E\pm E$ enjoys properties similar to the conventional setting where $E$ is a subset of an abelian group. In particular, among other results, we prove the Balog-Gowers-Szemerédi theorem, the Rusza's quasi triangle inequality and a weak form of the Plünnecke-Rusza theorem in the setting of shifted ideals. We also show that for a special class of maps $\xi$ from $\mathbb{F}_{p}^{d}$ to the collection of all shifted ideals of $\mathbb{F}_{p}[x_{1},\dots,x_{d}]$, if the set $\xi(\mathbb{F}_{p}^{d})+\xi(\mathbb{F}_{p}^{d})$ has large additive energy, then $\xi$ is an almost linear Freiman homomorphism. This result is the crucial additive combinatorial input we need to prove the spherical Gowers inverse theorem in later parts of the series.

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This paper is the fourth and the last part of the series "Spherical higher order Fourier analysis over finite fields", aiming to develop the higher order Fourier analysis method along spheres over finite fields, and to solve the Geometric Ramsey Conjecture in the finite field setting. In this paper, we proof a conjecture of Graham on the Remsey properties for spherical configurations in the finite field setting. To be more precise, we show that for any spherical configuration $X$ of $\mathbb{F}_{p}^{d}$ of complexity at most $C$ with $d$ being sufficiently large with respect to $C$ and $\vert X\vert$, and for some prime $p$ being sufficiently large with respect to $C$, $\vert X\vert$ and $\epsilon>0$, any set $E\subseteq \mathbb{F}_{p}^{d}$ with $\vert E\vert>\epsilon p^{d}$ contains at least $\gg_{C,\epsilon,\vert X\vert}p^{(k+1)d-(k+1)k/2}$ congruent copies of $X$, where $k$ is the dimension of $\text{span}_{\mathbb{F}_{p}}(X-X)$. The novelty of our approach is that we avoid the use of harmonic analysis, and replace it by the theory of spherical higher order Fourier analysis developed in previous parts of the series.

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This paper is the third part of the series "Spherical higher order Fourier analysis over finite fields", aiming to develop the higher order Fourier analysis method along spheres over finite fields, and to solve the geometric Ramsey conjecture in the finite field setting. In this paper, we prove an inverse theorem over finite field for spherical Gowers norms, i.e. a local Gowers norm supported on a sphere. We show that if the $(s+1)$-th spherical Gowers norm of a 1-bounded function $f\colon\mathbb{F}_{p}^{d}\to \mathbb{C}$ is at least $\epsilon$ and if $d$ is sufficiently large depending only on $s$, then $f$ correlates on the sphere with a $p$-periodic $s$-step nilsequence, where the bounds for the complexity and correlation depend only on $d$ and $\epsilon$. This result will be used in later parts of the series to prove the geometric Ramsey conjecture in the finite field setting.

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In this paper, we introduce and study the iterates of the following family of functions $\varphi_k$ defined on natural numbers which exhibits nice properties. $$\varphi_k(x)=\left\lbrace \begin{array}{ll} x+k, & \mbox{ if $x$ is prime;}\\ \mbox{largest prime divisor of $x$,} & \mbox{ if $x$ is composite;} \end{array} \right.$$ In particular, we study the periodic behaviour of the trajectories of these iterated functions. In some cases, we provide proofs of these properties and in some other cases we pose some open problems based on numerical evidences supported by heuristic arguments.

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We consider a certain class of infinitary rules of inference, called here restriction rules, using of which allows to deduce complete theories of given models. Such rules generalizing the $\omega$-rule were first considered by Henkin. Later on Barwise showed that within countable languages, for any countable model $\mathfrak M$, first-order logic expanded by the corresponding $\mathfrak M$-rule deduces from the diagram of $\mathfrak M$ all formulas that are true in all models that include $\mathfrak M$, in particular, it deduces the (relativized) complete theory of $\mathfrak M$. We show that, if the aim is only deducing the complete theory of a given model, these countability assumptions can be omitted. Moreover, similar facts hold for infinitary and higher-order logics, even if these logics are highly incomplete. Finally, we show that Barwise's theorem in its stronger form, for vocabularies of arbitrary cardinality, holds for infinitary logics $\mathscr L_{\kappa,\lambda}$ whenever $\kappa$ is supercompact. The note also contains brief historical remarks.

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In this paper, we test various models of wastewater infrastructure for risk analysis and compare their performance. While many representations are available, existing studies do not consider selection of the appropriate model for risk analysis. In this paper, we define two characteristics of wastewater models: the network granularity and the fidelity of the governing equations. We consider different combinations of these characteristics to determine 6 network representations that could be used as the foundation for risk analysis. We test the performance of each model as compared to predictions from the most detailed model, the full network with dynamic wave flow equations. We demonstrate the model selection for Seaside, Oregon. We conclude that the full network granularity is needed as compared to a coarse network representation. For the fidelity of the governing equations, connectivity analysis is reasonable if the primary goal is to determine the spatial distribution of hazard impact. To more accurately predict nodal performance measures, the dynamic wave equations are needed as they capture important physical phenomena.

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This paper is the first in a series of paper where we describe the differential operators on general nonlinear metric measure spaces, namely, the Finsler spaces. We try to propose a general method for gradient estimates of the positive solutions to nonlinear parabolic (or elliptic) equations on both compact and noncompact forward complete Finsler manifolds. The key of this approach is to find a valid comparison theorem, and particularly, to find related suitable curvature conditions. Actually, we define some non-Riemannian tensors and a generalization of the weighted Ricci curvature, called the mixed weighted Ricci curvature. With the assistance of these concepts, we prove a new kind of Laplacian comparison theorem. As an illustration of such method, we apply it to Finslerian Schrödinger equation. The Schrödinger equation is a typical class of parabolic equations, and the gradient estimation of their positive solution is representative. In this paper, based on the new Laplacian comparison theorem, we give global and local Li-Yau type gradient estimates for positive solutions to the Finslerian Schrödinger equation with respect to two different mixed weighted Ricci curvature conditions.

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Many well-studied problems in extremal combinatorics deal with the maximum possible size of a family of objects in which every pair of objects satisfies a given restriction. One problem of this type was recently raised by Alon, Gujgiczer, Körner, Milojević and Simonyi. They asked to determine the maximum size of a family $\mathcal{G}$ of graphs on $[n]$, such that for every two $G_1,G_2 \in \mathcal{G}$, the graphs $G_1 \setminus G_2$ and $G_2 \setminus G_1$ are isomorphic. We completely resolve this problem by showing that this maximum is exactly $2^{\frac{1}{2}\big(\binom{n}{2} - \lfloor \frac{n}{2}\rfloor\big)}$ and characterizing all the extremal constructions. We also prove an analogous result for $r$-uniform hypergraphs.

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For a random variable with a unimodal distribution and finite second moment Gauß\, (1823) proved a sharp bound on the probability of the random variable to be outside a symmetric interval around its mode. An alternative proof for it is given based on Khintchine's representation of unimodal random variables. Analogously, a sharp inequality is proved for the slightly broader class of unimodal distributions with finite first absolute moment, which might be called a Markovian Gauß inequality. For symmetric unimodal distributions with finite second moment Semenikhin (2019) generalized Gauß's inequality to arbitrary intervals. For the class of symmetric unimodal distributions with finite first absolute moment we construct a Markovian version of it. Related inequalities of Volkov (1969) and Sellke and Sellke (1997) will be discussed as well.

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Latent space models are powerful statistical tools for modeling and understanding network data. While the importance of accounting for uncertainty in network analysis has been well recognized, the current literature predominantly focuses on point estimation and prediction, leaving the statistical inference of latent space models an open question. This work aims to fill this gap by providing a general framework to analyze the theoretical properties of the maximum likelihood estimators. In particular, we establish the uniform consistency and asymptotic distribution results for the latent space models under different edge types and link functions. Furthermore, the proposed framework enables us to generalize our results to the dependent-edge and sparse scenarios. Our theories are supported by simulation studies and have the potential to be applied in downstream inferences, such as link prediction and network testing problems.

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Let $\Gamma$ be a Cayley graph, or a Cayley sum graph, or a twisted Cayley graph, or a twisted Cayley sum graph, or a vertex-transitive graph. Suppose $\Gamma$ is undirected and non-bipartite. Let $\mu$ (resp. $\mu_2$) denote the smallest (resp. the second largest) eigenvalue of the normalized adjacency operator of $\Gamma$, and $d$ denote the degree of $\Gamma$. We show that $1+ \mu = \Omega((1-\mu_2)/d)$ holds.

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Two isomorphic graphs can have inequivalent spatial embeddings in 3-space. In this way, an isomorphism class of graphs contains many spatial graph types. A common way to measure the complexity of a spatial graph type is to count the minimum number of straight sticks needed for its construction in 3-space. In this paper, we give estimates of this quantity by enumerating stick diagrams in a plane. In particular, we compute the planar stick indices of knotted graphs with low crossing numbers. We also show that if a bouquet graph or a theta-curve has the property that its proper subgraphs are all trivial, then the planar stick index must be at least seven.

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We introduce the centred and the uncentred triangular maximal operators $\mathcal T$ and $\mathcal U$, respectively, on any locally finite tree in which each vertex has at least three neighbours. We prove that both $\mathcal T$ and $\mathcal U$ are bounded on $L^p$ for every $p$ in $(1,\infty]$, that $\mathcal T$ is also bounded on $L^1(\mathfrak T)$, and that $\mathcal U$ is not of weak type $(1,1)$ on homogeneous trees. Our proof of the $L^p$ boundedness of $\mathcal U$ hinges on the geometric approach of A. Córdoba and R. Fefferman. We also establish $L^p$ bounds for some related maximal operators. Our results are in sharp contrast with the fact that the centred and the uncentred Hardy--Littlewood maximal operators (on balls) may be unbounded on $L^p$ for every $p<\infty$ even on some trees where the number of neighbours is uniformly bounded.

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Amazon Locker is a self-service delivery or pickup location where customers can pick up packages and drop off returns. A basic first-come-first-served policy for accepting package delivery requests to lockers results in lockers becoming full with standard shipping speed (3-5 day shipping) packages, and leaving no space left for expedited packages which are mostly Next-Day or Two-Day shipping. This paper proposes a solution to the problem of determining how much locker capacity to reserve for different ship-option packages. Yield management is a much researched field with popular applications in the airline, car rental, and hotel industries. However, Amazon Locker poses a unique challenge in this field since the number of days a package will wait in a locker (package dwell time) is, in general, unknown. The proposed solution combines machine learning techniques to predict locker demand and package dwell time, and linear programming to maximize throughput in lockers. The decision variables from this optimization provide optimal capacity reservation values for different ship options. This resulted in a year-over-year increase of 9% in Locker throughput worldwide during holiday season of 2018, impacting millions of customers.

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We show that any tangent cone of a singular shrinking Kähler-Ricci soliton is a normal affine algebraic variety. Moreover, the regular set of such a tangent cone in the metric sense coincides with the regular set as a variety. Along the way, we establish a parabolic proof of Hörmander's $L^{2}$ estimate, which can be used to solve the $\overline{\partial}$-equation on any singular shrinking Kähler-Ricci soliton.

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We generalize and simplify the constructions of \cite{DR2014} and \cite{Hsi2021} of an unbalanced triple product $p$-adic $L$-function $\mathscr{L}_p^f(\boldsymbol{f},\boldsymbol{g},\boldsymbol{h})$ attached to a triple $(\boldsymbol{f},\boldsymbol{g},\boldsymbol{h})$ of $p$-adic families of modular forms, allowing more flexibility for the choice of $\boldsymbol{g}$ and $\boldsymbol{h}$. Assuming that $\boldsymbol{g}$ and $\boldsymbol{h}$ are families of theta series of infinite $p$-slope, we prove a factorization of (an improvement of) $\mathscr{L}_p^f(\boldsymbol{f},\boldsymbol{g},\boldsymbol{h})$ in terms of two anticyclotomic $p$-adic $L$-functions. As a corollary, when $\boldsymbol{f}$ specializes in weight $2$ to the newform attached to an elliptic curve $E$ over $\mathbb{Q}$ with multiplicative reduction at $p$, we relate certain Heegner points on $E$ to certain $p$-adic partial derivatives of $\mathscr{L}_p^f(\boldsymbol{f},\boldsymbol{g},\boldsymbol{h})$ evaluated at the critical triple of weights $(2,1,1)$.

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Based on the success of a well-known method for solving higher order linear differential equations, a study of two of the most important mathematical features of that method, viz. the null spaces and commutativity of the product of unbounded linear operators on a Hilbert space is carried out. A principle is proved describing solutions for the product of such operators in terms of the solutions for each of the factors when the null spaces of those factors satisfy a certain geometric relation to one another. Another geometric principle equating commutativity of a closed densely defined operator and a projection to stability of the range of the projection under the closed operator is proved.

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In this work, we consider the problem of regularization in minimum mean-squared error (MMSE) linear filters. Exploiting the relationship with statistical machine learning methods, the regularization parameter is found from the observed signals in a simple and automatic manner. The proposed approach is illustrated through system identification examples, where the automatic regularization yields near-optimal results.

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Flexible antenna systems (FASs) can reconfigure their locations freely within a spatially continuous space. To keep favorable antenna positions, the channel state information (CSI) acquisition for FASs is essential. While some techniques have been proposed, most existing FAS channel estimators require several channel assumptions, such as slow variation and angular-domain sparsity. When these assumptions are not reasonable, the model mismatch may lead to unpredictable performance loss. In this paper, we propose the successive Bayesian reconstructor (S-BAR) as a general solution to estimate FAS channels. Unlike model-based estimators, the proposed S-BAR is prior-aided, which builds the experiential kernel for CSI acquisition. Inspired by Bayesian regression, the key idea of S-BAR is to model the FAS channels as a stochastic process, whose uncertainty can be successively eliminated by kernel-based sampling and regression. In this way, the predictive mean of the regressed stochastic process can be viewed as the maximum a posterior (MAP) estimator of FAS channels. Simulation results verify that, in both model-mismatched and model-matched cases, the proposed S-BAR can achieve higher estimation accuracy than the existing schemes.

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Given an irrational number $\alpha$, we study the asymptotic behaviour of the Sudler product denoted by $P_N(\alpha) = \prod_{r=1}^N 2\lvert \sin \pi r \alpha \rvert$. We show that $\liminf_{N \to \infty} P_N(\alpha) >0$ and $\limsup_{N \to \infty} P_N(\alpha)/N < \infty$ whenever the sequence of partial quotients in the continued fraction expansion of $\alpha$ exceeds 3 only finitely often, which confirms a conjecture of the second-named author and partially answers a question of J. Shallit. Furthermore, we show that the Hausdorff dimension of the set of those $\alpha$ that satisfy $\limsup_{N \to \infty} P_N(\alpha)/N < \infty,\liminf_{N \to \infty} P_N(\alpha) >0$ lies between $0.7056$ and $0.8677$, which makes significant progress in a question raised by Aistleitner, Technau, and Zafeiropoulos.

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In the late 1980s, Abrusci, Girard and van de Wiele defined a variant of Goodstein sequences: the so-called inverse Goodstein sequence. In their work, they show that it terminates precisely at the Bachmann-Howard ordinal. This reveals that a proof of this fact requires substantial consistency strength. Moreover, the authors could show that sequences of this kind terminate even if the hereditary base change at the heart of their construction is replaced by a generalization using arbitrary dilators. We show in a weak base system that this more general result is, in fact, equivalent to one of the most famous strong set existence principles from reverse mathematics: $\Pi^1_1$-comprehension. Moreover, the ordinal at which such sequences terminate is, in a fundamental way, isomorphic to the $1$-fixed point of their dilator, a new concept introduced by Freund and Rathjen. This yields explicit notation systems and a general method for specifying such ordinals. Also, using the notation systems provided by $1$-fixed points, we can reproduce the result that the Goodstein sequence terminates at the Bachmann-Howard ordinal in a weak system. Additionally, we perform a similar computation for a variant of Goodstein sequences, which terminates at a predicative ordinal.

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This paper introduces a probabilistic formulation for the isometric embedding of a Riemannian manifold $(M^n,g)$ into Euclidean space $\mathbb{R}^q$. We show that an embedding $u: M \to \mathbb{R}^q$ is isometric if and only if the intrinsic and extrinsic constructions of Brownian motion on $u(M)\subset \mathbb{R}^q$ yield processes with the same law. The equivalence is first established for smooth embeddings and it is followed by a renormalization procedure for $C^{1,\alpha}$ embeddings, $\alpha >\tfrac{1}{2}$. This formulation is based on a gedanken experiment that relates the intrinsic and extrinsic constructions of Brownian motion on an embedded manifold to the measurement of geodesic distance by observers in distinct frames of reference. This viewpoint provides a thermodynamic formalism for the isometric embedding problem that is suited to applications in geometric deep learning, stochastic optimization and turbulence.

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The Chambolle-Pock algorithm (CPA), also known as the primal-dual hybrid gradient method (PDHG), has surged in popularity in the last decade due to its success in solving convex/monotone structured problems. This work provides convergence results for problems with varying degrees of (non)monotonicity, quantified through a so-called oblique weak Minty condition on the associated primal-dual operator. Our results reveal novel stepsize and relaxation parameter ranges which do not only depend on the norm of the linear mapping, but also on its other singular values. In particular, in nonmonotone settings, in addition to the classical stepsize conditions for CPA, extra bounds on the stepsizes and relaxation parameters are required. On the other hand, in the strongly monotone setting, the relaxation parameter is allowed to exceed the classical upper bound of two. Moreover, sufficient convergence conditions are obtained when the individual operators belong to the recently introduced class of semimonotone operators. Since this class of operators encompasses many traditional operator classes including (hypo)- and co(hypo)monotone operators, this analysis recovers and extends existing results for CPA. Several examples are provided for the aforementioned problem classes to demonstrate and establish tightness of the proposed stepsize ranges.

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Given an arbitrary, finitely presented, residually finite group $\Gamma$, one can construct a finitely generated, residually finite, free-by-free group $M_\Gamma = F_\infty\rtimes F_4$ and an embedding $M_\Gamma \hookrightarrow (F_4\ast \Gamma)\times F_4$ that induces an isomorphism of profinite completions. In particular, there is a free-by-free group whose profinite completion contains $\widehat{\Gamma}$ as a retract.

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We prove a version of Weyl's Law for the basic spectrum of a closed singular Riemannian foliation $(M,\mathcal{F})$ with basic mean curvature. In the special case of $M=\mathbb{S}^n$, this gives an explicit formula for the volume of the leaf space $\mathbb{S}^n/\mathcal{F}$ in terms of the algebra of basic polynomials. In particular, $\operatorname{Vol}(\mathbb{S}^n/\mathcal{F})$ is a rational multiple of $\operatorname{Vol}(\mathbb{S}^m)$, where $m=\dim (\mathbb{S}^n/\mathcal{F})$.

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In view of a better understanding of the geometry of scalar flat Kähler metrics, this paper studies two families of scalar flat Kähler metrics constructed in [10] by A. D. Hwang and M. A. Singer on $\mathbb C^{n+1}$ and on $\mathcal O(-k)$. For the metrics in both the families, we prove the existence of an asymptotic expansion for their $\epsilon$-functions and we show that they can be approximated by a sequence of projectively induced Kähler metrics. Further, we show that the metrics on $\mathbb C^{n+1}$ are not projectively induced, and that the Burns-Simanca metric is characterized among the scalar flat metrics on $\mathcal O(-k)$ to be the only projectively induced one as well as the only one whose second coefficient in the asymptotic expansion of the $\epsilon$-function vanishes.

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In this paper we introduce the notion of generalized invertible 1-cocycle in a strict braided monoidal category C, and we prove that the category of Hopf trusses is equivalent to the category of generalized invertible 1-cocycles. On the other hand, we also introduce the notions of module for a Hopf truss and for a generalized invertible 1-cocycle. We prove some functorial results involving these categories of modules and we show that the category of modules associated to a generalized invertible 1-cocycle is equivalent to a category of modules associated to a suitable Hopf truss. Finally, assuming that in C we have equalizers, we introduce the notion of Hopf-module in the Hopf truss setting and we obtain the Fundamental Theorem of Hopf modules associated to a Hopf truss.

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Irregular repetition slotted Aloha (IRSA) has shown significant advantages as a modern technique for uncoordinated random access with massive number of users due to its capability of achieving theoretically a throughput of $1$ packet per slot. When the receiver has also the multi-packet reception of multi-user (MUD) detection property, by applying successive interference cancellation, IRSA also obtains very low packet loss probabilities at low traffic loads, but is unable in general to achieve a normalized throughput close to the $1$. In this paper, we reconsider the case of IRSA with $k$-MUD receivers and derive the general density evolution equations for the non-asymptotic analysis of the packet loss rate, for arbitrary frame lengths and two variants of the first slot used for transmission. Next, using the potential function, we give new capacity bounds on the capacity of the system, showing the threshold arrival rate for zero decoding error probability. Our numerical results illustrate performance in terms of throughput and average delay for $k$-MUD IRSA with finite memory at the receiver, and also with bounded maximum delay.

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Combining a variant of the Farkas lemma with the Flow Decomposition Theorem we show that the regions of any deformation of a graphical arrangement may be bijectively labeled with a set of weighted digraphs containing directed cycles of negative weight only. Bounded regions correspond to strongly connected digraphs. The study of the resulting labelings allows us to add the omitted details in Stanley's proof on the injectivity of the Stanley-Pak labeling of the regions of the extended Shi arrangement and to introduce a new labeling of the regions in the $a$-Catalan arrangement. We also point out that Athanasiadis-Linusson labelings may be used to directly count regions in a class of arrangements properly containing the extended Shi arrangement and the Fuss-Catalan arrangement.

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Existing asynchronous distributed optimization algorithms often use diminishing step-sizes that cause slow practical convergence, or use fixed step-sizes that depend on and decrease with an upper bound of the delays. Not only are such delay bounds hard to obtain in advance, but they also tend to be large and rarely attained, resulting in unnecessarily slow convergence. This paper develops asynchronous versions of two distributed algorithms, Prox-DGD and DGD-ATC, for solving consensus optimization problems over undirected networks. In contrast to alternatives, our algorithms can converge to the fixed point set of their synchronous counterparts using step-sizes that are independent of the delays. We establish convergence guarantees for strongly and weakly convex problems under both partial and total asynchrony. We also show that the convergence speed of the two asynchronous methods adapts to the actual level of asynchrony rather than being constrained by the worst-case. Numerical experiments demonstrate a strong practical performance of our asynchronous algorithms.

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In their seminal paper, Lubotzky, Phillips and Sarnak (LPS) defined the notion of regular Ramanujan graphs and gave an explicit construction of infinite families of $(p+1)$-regular Ramanujan Cayley graphs, for infinitely many primes $p$. In this paper we extend the work of LPS and its successors to bigraphs (biregular bipartite graphs), in several aspects: We compare the combinatorial properties of various generalizations of the notion of Ramanujan graphs, define a notion of Cayley bigraphs, and give explicit constructions of infinite families of $(p^3+1,p+1)$-regular Ramanujan Cayley bigraphs, for infinitely many primes $p$. Both the LPS graphs and our ones are arithmetic, arising as quotients of Bruhat-Tits trees by congruence subgroups of arithmetic lattices in a $p$-adic group, $PGL_2(\mathbb{Q}_p)$ for LPS and $PU_3(\mathbb{Q}_p)$ for us. In both cases the Ramanujan property relates to the Ramanujan Conjecture (RC), on the respective groups. But while for $PGL_2$ the RC holds unconditionally, this is not so in the case of $PU_3$. We find explicit cases where the RC does and does not hold, and use this to construct non-Ramanujan Cayley bigraphs as well, which nevertheless satisfy the Sarnak-Xue density hypothesis. On the combinatorial side, we present a pseudorandomness characterization of Ramanujan bigraphs, and a more general notion of biexpanders. We also show that the graphs we construct exhibit the cutoff phenomenon with bounded window size for the mixing time of non-backtracking random walk, either as a consequence of the Ramanujan property, or of the Sarnak-Xue density hypothesis. Finally, we present some other applications of our work: Golden and super golden gates for $PU(3)$, Ramanujan and non-Ramanujan complexes of type $\widetilde{A}_2$, optimal strong approximation for $p$-arithmetic subgroups of $PU_3$ and the vanishing Betti numbers of Picard modular surfaces.

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We present a family of non-CSS quantum stabilizer codes using the structure of duadic constacyclic codes over $\mathbb{F}_4$. Within this family, quantum codes can possess varying dimensions, and their minimum distances are bounded by a square root bound. For each fixed dimension, this allows us to construct an infinite sequence of binary quantum codes with a growing minimum distance. Additionally, we demonstrate that this quantum family includes an infinite subclass of degenerate codes with the mentioned properties. We also introduce a technique for extending splittings of duadic constacyclic codes, providing new insights into the minimum distance and minimum odd-like weight of specific duadic constacyclic codes. Finally, we establish that many best-known quantum codes belong to this family and provide numerical examples of quantum codes with short lengths within this family.

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In this paper, we address a time-dependent one-dimensional linear advection-diffusion equation with Dirichlet homogeneous boundary conditions. The equation is solved both analytically, using separation of variables, and numerically, employing the finite difference method. The computational output includes three dimensional (3D) plots for solutions, focusing on pollutants such as Ammonia, Carbon monoxide, Carbon dioxide, and Sulphur dioxide. Concentrations, along with their respective diffusivities, are analyzed through 3D plots and actual calculations. To comprehend the diffusivity-concentration relationship for predicting pollutant movement in the air, the domain is divided into two halves. The study explores the behavior of pollutants with higher diffusivity entering regions with lower diffusivity, and vice versa, using 2D and 3D plots. This task is crucial for effective pollution control strategies, and safeguarding the environment and public health.

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In this paper, we show examples of local cohomology modules over ramified regular local ring, having finite set of associated primes. In doing so we consider our ramified regular local ring as Eisenstein extension of an unramified regular local ring sitting inside it and we use the Mayer-Vietoris spectral sequence to show the finiteness of the set of associated primes. In ramified regular local ring for extended ideal (from the unramified one) set of associative primes of a local cohomology module is always finite. Using this result, for non extended ideal, we show example of local cohomology module which has finite set of associative primes and where associative primes also contains prime number $p$.

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We consider critical percolation on a supercritical Galton-Watson tree. We show that, when the offspring distribution is in the domain of attraction of an $\alpha$-stable law for some $\alpha \in (1,2)$, or has finite variance, several annealed properties also hold in a quenched setting. In particular, the following properties hold for the critical root cluster on almost every realisation of the tree: (1) the rescaled survival probabilities converge; (2) the Yaglom limit or its stable analogue hold - in particular, conditioned on survival, the number of vertices at generation $n$ that are connected to the root cluster rescale to a certain (explicit) random variable; (3) conditioned on initial survival, the sequence of generation sizes in the root cluster rescales to a continuous-state branching process. This strengthens some earlier results of Michelen (2019) who proved (1) and (2) in the case where the initial tree has an offspring distribution with all moments finite.

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We introduce the 3-alterfold topological quantum field theory (TQFT) by extending the quantum invariant of 3-alterfolds. The bases of the TQFT are explicitly characterized and the Levin-Wen model is naturally interpreted in 3-alterfold TQFT bases. By naturally considering the RT TQFT and TV TQFT as sub-TQFTs within the 3-alterfold TQFT, we establish their equivalence. The 3-alterfold TQFT is unitary when the input fusion category is unitary. Additionally, we extend the 3-alterfold TQFT to the Morita context and demonstrate that Morita equivalent fusion categories yield equivalent TV TQFTs. We also provide a simple pictorial proof of complete positivity criteria for unitary categorization when the 3-alterfold TQFT is unitary. Expanding our scope to high-genus surfaces by replacing the torus, we introduce the high genus topological indicators and proving the equivariance under the mapping class group actions.

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It is a long-standing conjecture that all symplectic capacities which are equal to the Gromov width for ellipsoids coincide on convex domains in $\mathbb{R}^{2n}$. It is known that they coincide for monotone toric domains in all dimensions. In this paper, we study whether requiring a capacity to be equal to the $k^{th}$ Ekeland-Hofer capacity for all ellipsoids can characterize it on a class of domains. We prove that for $k=n=2$, this holds for convex toric domains, but not for all monotone toric domains. We also prove that for $k=n\ge 3$, this does not hold even for convex toric domains.

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We show $\textit{pathwise uniqueness}$ of multiplicative SDEs, in arbitrary dimensions, driven by fractional Brownian motion with Hurst parameter $H\in (1/3,1)$ with volatility coefficient $\sigma$ that is at least $\gamma$-Hölder continuous for $\gamma > \frac{1}{2H} \vee \frac{1-H}{H}$. This improves upon the long-standing results of Lyons (94, 98) and Davie (08) which cover the same regime but require $\sigma$ to be at least $\frac{1}{H}$-Hölder continuous. Our central innovation is to combine stochastic averaging estimates with refined versions of the stochastic sewing lemma, due to Lê (20), Gerencsér (22) and Matsuda and Perkowski (22).

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We consider a variational problem modeling transition between flat and wrinkled region in a thin elastic sheet, and identify the $\Gamma$-limit as the sheet thickness goes to 0, thus extending the previous work of the first author [Bella, ARMA 2015]. The limiting problem is scalar and convex, but constrained and posed for measures. For the $\Gamma$-liminf inequality we first pass to quadratic variables so that the constraint becomes linear, and then obtain the lower bound using Reshetnyak's theorem. The construction of the recovery sequence for the $\Gamma$- limsup inequality relies on mollification of quadratic variables, and careful choice of multiple construction parameters. Eventually for the limiting problem we show existence of a minimizer and equipartition of the energy for each frequency.

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This paper addresses the challenging task of guide wire navigation in cardiovascular interventions, focusing on the parameter estimation of a guide wire system using Ensemble Kalman Inversion (EKI) with a subsampling technique. The EKI uses an ensemble of particles to estimate the unknown quantities. However since the data misfit has to be computed for each particle in each iteration, the EKI may become computationally infeasible in the case of high-dimensional data, e.g. high-resolution images. This issue can been addressed by randomised algorithms that utilize only a random subset of the data in each iteration. We introduce and analyse a subsampling technique for the EKI, which is based on a continuous-time representation of stochastic gradient methods and apply it to on the parameter estimation of our guide wire system. Numerical experiments with real data from a simplified test setting demonstrate the potential of the method.

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By restricting the set of diameters in the coverings in the usual definition of the Hausdorff dimension we introduce a new family of dimensions. Among others we prove that this family contains continuum many distinct dimensions and all of them share most of the properties of the Hausdorff dimension, which answers negatively a question of Fraser. On the other hand, we also prove that among these new dimensions only the Hausdorff dimension behaves nicely with respect to Hölder functions, which supports a conjecture posed by Banaji obtained as a natural modification of the question of Fraser. We also consider the supremum of these new dimensions, which turns out to be an other interesting notion of dimension. We prove that among those bilipschitz invariant, monotone dimensions on the compact subsets of $\mathbb{R}^n$ that agree with the similarity dimension for the simplest self-similar sets, the modified lower dimension is the smallest and when $n=1$ the Assouad dimension is the greatest, and this latter statement is false for $n>1$. This answers a question of Rutar.

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We show that under minimal assumption on a class of functions $\mathcal{H}$ defined on a probability space $(\mathcal{X},\mu)$, there is a threshold $\Delta_0$ satisfying the following: for every $\Delta\geq\Delta_0$, with probability at least $1-2\exp(-c\Delta m)$ with respect to $\mu^{\otimes m}$, \[ \sup_{h\in\mathcal{H}} \sup_{t\in\mathbb{R}} \left| \mathbb{P}(h(X)\leq t) - \frac{1}{m}\sum_{i=1}^m 1_{(-\infty,t]}(h(X_i)) \right| \leq \sqrt{\Delta};\] here $X$ is distributed according to $\mu$ and $(X_i)_{i=1}^m$ are independent copies of $X$. The value of $\Delta_0$ is determined by an unexpected complexity parameter of the class $\mathcal{H}$ that captures the set's geometry (Talagrand's $\gamma_1$-functional). The bound, the probability estimate and the value of $\Delta_0$ are all optimal up to a logarithmic factor.

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Let $P$ be a pointed, closed convex cone in $\mathbb{R}^d$. We prove that for two pure isometric representations $V^{(1)}$ and $V^{(2)}$ of $P$, the associated CAR flows $\beta^{V^{(1)}}$ and $\beta^{V^{(2)}}$ are cocycle conjugate if and only if $V^{(1)}$ and $V^{(2)}$ are unitarily equivalent. We also give a complete description of pure isometric representations of $P$ with commuting range projections that give rise to type I CAR flows. We show that such an isometric representation is completely reducible with each irreducible component being a pullback of the shift semigroup $\{S_t\}_{t \geq 0}$ on $L^2[0,\infty)$. We also compute the index and the gauge group of the associated CAR flows and show that the action of the gauge group on the set of normalised units need not be transitive.

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We give a new method to calculate the universal cohomology classes of coincident root loci. We show a polynomial behavior of them and apply this result to prove that generalized Plücker formulas are polynomials in the degree, just as the classical Plücker formulas counting the bitangents and flexes of a degree $d$ generic plane curve. We calculate the leading term of these polynomials to determine the asymptotic behavior of the Plücker formulas. We believe that the paper is understandable without detailed knowledge of equivariant cohomology. It may serve as a demonstration of the use of equivariant cohomology in enumerative geometry through the examples of coincident root strata. We also explain how the equivariant method can be "translated" into the traditional non-equivariant method of resolutions.

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We present a general framework for preconditioning Hermitian positive definite linear systems based on the Bregman log determinant divergence. This divergence provides a measure of discrepancy between a preconditioner and a target matrix. Given an approximate factorisation of a target matrix, the proposed framework tells us how to construct a low-rank approximation of the typically indefinite factorisation error. The resulting preconditioner is therefore a sum of a Hermitian positive definite matrix given by an approximate factorisation plus a low-rank matrix. Notably, the low-rank term is not generally obtained as a truncated singular value decomposition. This framework leads to a new truncation where principal directions are not based on the magnitude of the singular values. We describe a procedure for determining these \emph{Bregman directions} and prove that preconditioners constructed in this way are minimisers of the aforementioned divergence. Finally, we demonstrate using several numerical examples how the proposed preconditioner performs in terms of convergence of the preconditioned conjugate gradient method (PCG). For the examples we consider, an incomplete Cholesky preconditioner can be greatly improved in this way, and in some cases only a modest low-rank compensation term is required to obtain a considerable improvement in convergence. We also consider matrices arising from interior point methods for linear programming that do not admit such an incomplete factorisation by default, and present a robust incomplete Cholesky preconditioner based on the proposed methodology. The results highlight that the choice of truncation is critical for ill-conditioned matrices. We show numerous examples where PCG converges to a small tolerance by using the proposed preconditioner, whereas PCG with a SVD-based preconditioner fails to do so.

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In this paper we set up a theory of two-matrix weighted little $\mathrm{BMO}$ in two parameters. We prove that being a member of this class is equivalent to belonging uniformly in each variable to two-matrix weighted (one-parameter) $\mathrm{BMO}$, a class studied extensively by J. Isralowitz, S. Pott, S. Treil and others. Using this equivalence, we deduce lower and upper bounds in terms of the two-matrix weighted little $\mathrm{BMO}$ norm of the symbol for the norm of commutators with Journé operators. Moreover, we set up the foundations of a theory of two-matrix weighted product $\mathrm{BMO}$ in two parameters. Our main result here is an appropriate version of $H^1$-$\mathrm{BMO}$ duality in this setting for any Lebesgue exponent $1<p<\infty$, extending results of I. Holmes, S. Petermichl and B. Wick in the biparameter, scalar weighted setting and J. Isralowitz in the one-parameter, two-matrix weighted setting.

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We prove the necessary and sufficient condition for the removability of the fundamental singularity, and equivalently for the unique solvability of the singular Dirichlet problem for the heat equation. In the measure-theoretical context the criterion determines whether the $h$-parabolic measure of the singularity point is null or positive. From the probabilistic point of view, the criterion presents an asymptotic law for conditional Brownian motion.

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In this paper, we carry out in-depth research centering around the Harnack inequality for positive solutions to nonlinear heat equation on Finsler metric measure manifolds with weighted Ricci curvature ${\rm Ric}_{\infty}$ bounded below. Aim on this topic, we first give a volume comparison theorem of Bishop-Gromov type. Then we prove a weighted Poincaré inequality by using Whitney-type coverings technique and give a local uniform Sobolev inequality. Further, we obtain two mean value inequalities for positive subsolutions and supersolutions of a class of parabolic differential equations. From the mean value inequality, we also derive a new local gradient estimate for positive solutions to heat equation. Finally, as the application of the mean value inequalities and weighted Poincaré inequality, we get the desired Harnack inequality for positive solutions to heat equation.

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Geometrically regular weighted shifts (in short, GRWS) are those with weights $\alpha (N,D)$ given by $\alpha_n (N,D) = \sqrt{\frac{p^n + N}{p^n + D}}$, where $p > 1$ and $(N,D)$ is fixed in the open unit square $ (-1, 1)\times (-1, 1)$. We study here the zone of pairs $ (M,P)$ for which the weight $\frac{\alpha (N,D) }{ \alpha (M,P) }$ gives rise to a moment infinitely divisible ($ \mathcal {MID}$) or a subnormal weighted shift, and deduce immediately the analogous results for product weights $\alpha (N,D) \alpha (M,P)$, instead of quotients. Useful tools introduced for this study are a pair of partial orders on the GRWS.

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Shell structures with a high stiffness-to-weight ratio are desirable in various engineering applications. In such scenarios, topology optimization serves as a popular and effective tool for shell structures design. Among the topology optimization methods, solid isotropic material with penalization method(SIMP) is often chosen due to its simplicity and convenience. However, SIMP method is typically integrated with conventional finite element analysis(FEA) which has limitations in computational accuracy. Achieving high accuracy with FEA needs a substantial number of elements, leading to computational burdens. In addition, the discrete representation of the material distribution may result in rough boundaries and checkerboard structures. To overcome these challenges, this paper proposes an isogeometric analysis(IGA) based SIMP method for optimizing the topology of shell structures based on Reissner-Mindlin theory. We use NURBS to represent both the shell structure and the material distribution function with the same basis functions, allowing for higher accuracy and smoother boundaries. The optimization model takes compliance as the objective function with a volume fraction constraint and the coefficients of the density function as design variables. The Method of Moving Asymptotes is employed to solve the optimization problem, resulting in an optimized shell structure defined by the material distribution function. To obtain fairing boundaries in the optimized shell structure, further process is conducted by fitting the boundaries with fair B-spline curves automatically. Furthermore, the IGA-SIMP framework is applied to generate porous shell structures by imposing different local volume fraction constraints. Numerical examples are provided to demonstrate the feasibility and efficiency of the IGA-SIMP method, showing that it outperforms the FEA-SIMP method and produces smoother boundaries.

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On a Riemannian manifold of dimension $2 \leq d \leq 6$, with or without boundary, and whether bounded or unbounded, we consider a semilinear wave (or Klein-Gordon) equation with a subcritical nonlinearity, either defocusing or focusing. We establish local controllability around a partially analytic solution, under the geometric control condition. Specifically, some blow-up solutions can be controlled. In the case of a Klein-Gordon equation on a non-trapping exterior domain, we prove the null-controllability of scattering solutions. The proof is based on local energy decay and global-in-time Strichartz estimates. Some corollaries are given, including the null-controllability of a solution starting near the ground state in certain focusing cases, and exact controllability in certain defocusing cases.

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For integers $n \geq k \geq 1$, the {\em Kneser graph} $K(n, k)$ is the graph with vertex-set consisting of all the $k$-element subsets of $\{1,2,\ldots,n\}$, where two $k$-element sets are adjacent in $K(n,k)$ if they are disjoint. We show that if $(n,k,s) \in \mathbb{N}^3$ with $n > 10000 k s^5$ and $\mathcal{F}$ is set of vertices of $K(n,k)$ of size larger than $\{A \subset \{1,2,\ldots,n\}:\ |A|=k,\ A \cap \{1,2,\ldots,s\} \neq \varnothing\}$, then the subgraph of $K(n,k)$ induced by $\mathcal{F}$ has maximum degree at least \[ \left(1 - O\left(\sqrt{s^3 k/n}\right)\right)\frac{s}{s+1} \cdot {n-k \choose k} \cdot \frac{|\mathcal{F}|}{\binom{n}{k}}.\] This is sharp up to the behaviour of the error term $O(\sqrt{s^3 k/n})$. In particular, if the triple of integers $(n, k, s)$ satisfies the condition above, then the minimum maximum degree does not increase `continuously' with $|\mathcal{F}|$. Instead, it has $s$ jumps, one at each time when $|\mathcal{F}|$ becomes just larger than the union of $i$ stars, for $i = 1, 2, \ldots, s$. An appealing special case of the above result is that if $\mathcal{F}$ is a family of $k$-element subsets of $\{1,2,\ldots,n\}$ with $|\mathcal{F}| = {n-1 \choose k-1}+1$, then there exists $A \in \mathcal{F}$ such that $\mathcal{F}$ is disjoint from at least $$\left(1/2-O\left(\sqrt{k/n}\right)\right){n-k-1 \choose k-1}$$ of the other sets in $\mathcal{F}$; this is asymptotically sharp if $k=o(n)$. Frankl and Kupavskii, using different methods, have recently proven closely related results under the hypothesis that $n$ is at least quadratic in $k$.

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This paper introduces a time-domain combined field integral equation for electromagnetic scattering by a perfect electric conductor. The new equation is obtained by leveraging the quasi-Helmholtz projectors, which separate both the unknown and the source fields into solenoidal and irrotational components. These two components are then appropriately rescaled to cure the solution from a loss of accuracy occurring when the time step is large. Yukawa-type integral operators of a purely imaginary wave number are also used as a Calderon preconditioner to eliminate the ill-conditioning of matrix systems. The stabilized time-domain electric and magnetic field integral equations are linearly combined in a Calderon-like fashion, then temporally discretized using a proper pair of trial functions, resulting in a marching-on-in-time linear system. The novel formulation is immune to spurious resonances, dense discretization breakdown, large-time step breakdown and dc instabilities stemming from non-trivial kernels. Numerical results for both simply-connected and multiply-connected scatterers corroborate the theoretical analysis.

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In order to minimize a differentiable geodesically convex function, we study a second-order dynamical system on Riemannian manifolds with an asymptotically vanishing damping term of the form $\alpha/t$. For positive values of $\alpha$, convergence rates for the objective values and convergence of trajectory is derived. We emphasize the crucial role of the curvature of the manifold for the distinction of the modes of convergence. There is a clear correspondence to the results that are known in the Euclidean case. When $\alpha$ is larger than a certain constant that depends on the curvature of the manifold, we improve the convergence rate of objective values compared to the previously known rate and prove the convergence of the trajectory of the dynamical system to an element of the set of minimizers. For $\alpha$ smaller than this curvature-dependent constant, the best known sub-optimal rates for the objective values and the trajectory are transferred to the Riemannian setting. We present computational experiments that corroborate our theoretical results.

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Real-time hybrid testing is a method in which a substructure of the system is realised experimentally and the rest numerically. The two parts interact in real time to emulate the dynamics of the full system. Such experiments however are often difficult to realise as the actuators and sensors, needed to ensure compatibility and force-equilibrium conditions at the interface, can seriously affect the predicted dynamics of the system and result in stability and fidelity issues. The traditional approach of using feedback control to overcome the additional unwanted dynamics is challenging due to the presence of an outer feedback loop, passing interface displacements or forces to the numerical substructure. We, therefore, advocate for an alternative approach, removing the problematic interface dynamics with an iterative scheme to minimise interface errors, thus, capturing the response of the true assembly. The technique is examined by hybrid testing of a bench-top four-storey building with different interface configurations, where using conventional hybrid measurement techniques is very challenging. A case where the physical part exhibits nonlinear restoring force characteristics is also considered. These tests show that the iterative approach is capable of handling even scenarios which are theoretically infeasible with feedback control.

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We give a description of the Picard group of a reductive group over a number field as an abelianized Galois cohomology group. It gives another approach of a result due to Labesse.

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In this article, we consider the multiarrangements whose underlying arrangements are the Coxeter arrangement of type $B_2$. For some special multiplicities, we give an explicit description of bases for the derivation modules. As an application, we also describe the lower derivations of bases for the derivation modules of some Coxeter multiarrangements of type $A_2$, which are different from ones given by Wakamiko.

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Octonions are 8-dimensional hypercomplex numbers which form the biggest normed division algebras over the real numbers. Motivated by applications in theoretical physics, continuous octonionic analysis has become an area of active research in recent year. Looking at possible practical applications, it is beneficial to work directly with discrete structures, rather than approximate continuous objects. Therefore, in previous papers, we have proposed some ideas towards the discrete octonionic analysis. It is well known, that there are several possibilities to discretise the continuous setting, and the Weyl calculus approach, which is typically used in the discrete Clifford analysis, to octonions has not been studied yet. Therefore, in this paper, we close this gap by presenting the discretisation of octonionic analysis based on the Weyl calculus.

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In this paper, we derive sufficient conditions ensuring the existence of a weak solution $u$ for a tempered fractional Euler-Lagrange equations $$ \frac{\partial L}{\partial x}(u,{^C}\mathbb{D}_{a^+}^{\alpha, \sigma} u, t) + \mathbb{D}_{b^-}^{\alpha, \sigma}\left(\frac{\partial L}{\partial y}(u, {^C}\mathbb{D}_{a^+}^{\alpha, \sigma}u, t) \right) = 0 $$ on a real interval $[a,b]$ and ${^C}\mathbb{D}_{a^+}^{\alpha, \sigma}, \mathbb{D}_{b^-}^{\alpha, \sigma}$ are the left and right Caputo and Riemann-Liouville tempered fractional derivatives respectively of order $\alpha$. Furthermore, we study a fractional tempered version of Noether theorem and we provide a very explicit expression of a constant of motion in terms of symmetry group and Lagrangian for fractional problems of calculus of variations. Finally we study a mountain pass type solution of the cited problem.

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We show trilinear Strichartz estimates in one and two dimensions on frequency-dependent time intervals. These improve on the corresponding linear estimates of periodic solutions to the Schrödinger equation. The proof combines decoupling iterations with bilinear short-time Strichartz estimates. Secondly, we use decoupling to show new linear Strichartz estimates on frequency dependent time intervals. We apply these in case of the Airy propagator to obtain the sharp Sobolev regularity for the existence of solutions to the modified Korteweg-de Vries equation.

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We show that certain smooth tori with group $\mathbb{Z}$ in $S^4$ have exteriors with standard equivariant intersection forms, and so are topologically unknotted. These include the turned 1-twist-spun tori in the 4-sphere constructed by Boyle, the union of the genus one Seifert surface of Cochran and Davis, that has no slice derivative, with a ribbon disc, and any torus with precisely four critical points. This gives evidence towards the conjecture that all $\mathbb{Z}$-surfaces in $S^4$ are topologically unknotted, which is open for genus one and two. It is unclear whether these tori are smoothly unknotted. The double cover of $S^4$ branched along any of these surfaces is a potentially exotic copy of $S^2 \times S^2$ that cannot be distinguished from $S^2 \times S^2$ using Seiberg-Witten invariants.

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Toroidal Lie algebras are $n$ variable generalizations of affine Kac-Moody Lie algebras. Full toroidal Lie algebra is the semidirect product of derived Lie algebra of toroidal Lie algebra and Witt algebra, also it can be thought of $n$-variable generalization of Affine-Virasoro algebras. Let $\tilde{\mathfrak{h}}$ be a Cartan subalgebra of a toroidal Lie algebra as well as full toroidal Lie algebra without containing the zero-degree central elements. In this paper, we classify the module structure on $U(\tilde{\mathfrak{h}})$ for all toroidal Lie algebras as well as full toroidal Lie algebras which are free $U(\tilde{\mathfrak{h}})$-modules of rank 1. These modules exist only for type $A_l (l\geq 1)$, $C_l (l\geq2)$ toroidal Lie algebras and the same is true for full toroidal Lie algebras. Also, we determined the irreducibility condition for these classes of modules for both the Lie algebras.

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We describe the $p$-divisibility transposition for the Fourier coefficients of Hermitian modular forms. The results show that the same phenomenon as that for Siegel modular forms holds for Hermitian modular forms.

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For a system of wave equations, we prove that boundary exact controllability at one level of regularity is equivalent to boundary exact controllability at all levels of regularity. We construct the solutions of the wave equation for both homogeneous and inhomogeneous Dirichlet boundary conditions at every level of regularity, and we check that exact controllability and observability are equivalent. Then, we prove that we can shift the regularity of an observability inequality in a natural way. The main ingredient of the proof is the ellipticity of the time-derivative on normal derivatives of solutions, which can be proved using microlocal analysis.

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We construct a new Yang-Milles 3D-solution on the space of negative scalar curvarure. We discuss a problem of non-abelian gauge symmetry is broken with the assumption that a scalar curvature of the domain is a negative small parameter. In this case we use the following fact: a geometrical scale related with Vassiliev's discriminant of magnetic lines coincids with a phisical Kolmogorov scale. This gives an estimation of $\alpha$-effect by the dispersion of the asymptotic ergodic Hopf invariant in the assimptotic limite with a negative scalar curvature parameter.

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In this work, we present and analyze a system of PDEs, which models tumor growth by considering chemotaxis, active transport, and random effects. The stochasticity of the system is modelled by random initial data and Wiener noises that appear in the tumor and nutrient equations. The volume fraction of the tumor is governed by a stochastic phase-field equation of Cahn-Hilliard type, and the mass density of the nutrients is modelled by a stochastic reaction-diffusion equation. We allow a variable mobility function and non-increasing growth functions, such as logistic and Gompertzian growth. Via approximation and stochastic compactness arguments, we prove the existence of a probabilistic weak solution and, in the case of constant mobilities, the well-posedness of the model in the strong probabilistic sense. Lastly, we propose a numerical approximation based on the Galerkin finite element method in space and the semi-implicit Euler-Maruyama scheme in time. We illustrate the effects of the stochastic forcing in the tumor growth in several numerical simulations.

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We present the theory of linear rank-metric codes from the point of view of their fundamental parameters. These are: the minimum rank distance, the rank distribution, the maximum rank, the covering radius, and the field size. The focus of this chapter is on the interplay among these parameters and on their significance for the code's (combinatorial) structure. The results covered in this chapter span from the theory of optimal codes and anticodes to very recent developments on the asymptotic density of MRD codes.

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Singularly perturbed boundary value problems pose a significant challenge for their numerical approximations because of the presence of sharp boundary layers. These sharp boundary layers are responsible for the stiffness of solutions, which leads to large computational errors, if not properly handled. It is well-known that the classical numerical methods as well as the Physics-Informed Neural Networks (PINNs) require some special treatments near the boundary, e.g., using extensive mesh refinements or finer collocation points, in order to obtain an accurate approximate solution especially inside of the stiff boundary layer. In this article, we modify the PINNs and construct our new semi-analytic SL-PINNs suitable for singularly perturbed boundary value problems. Performing the boundary layer analysis, we first find the corrector functions describing the singular behavior of the stiff solutions inside boundary layers. Then we obtain the SL-PINN approximations of the singularly perturbed problems by embedding the explicit correctors in the structure of PINNs or by training the correctors together with the PINN approximations. Our numerical experiments confirm that our new SL-PINN methods produce stable and accurate approximations for stiff solutions.

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Some variations of $\pi$-regular and nil clean rings were recently introduced in [5, 7, 6], respectively. In this paper, we examine the structure and relationships between these classes of rings. Specifically, we prove that (m, n)-regularly nil clean rings are left-right symmetric and also show that the inclusions (D-regularly nil clean) $\subseteq$(regularly nil clean) $\subseteq$ ((m, n)-regularly nil clean) hold, as well as we answer Questions 1, 2 and 3 posed in [7].

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In this paper we develop a general `analytic' splitting principle for RCD spaces: we show that if there is a function with suitable Laplacian and Hessian, then the space is (isomorphic to) a warped product. Our result covers most of the splitting-like results currently available in the literature about RCD spaces. We then apply it to extend to the non-smooth category some structural property of Riemannian manifolds obtained by Li and Wang.

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We show existence of a non-trivial phase transition for the contact process, a simple model for infection without immunity, on a network which reacts dynamically to the infection trying to prevent an epidemic. This network initially has the distribution of an Erdős-Rényi graph, but is made adaptive via updating in only the infected neighbourhoods, at constant rate. Under these graph dynamics, the presence of infection can help to prevent the spread and so many monotonicity-based techniques fail. Instead, we approximate the infection by an idealised model, the contact process on an evolving forest (CPEF). In this model updating events of a vertex in a tree correspond to the creation of a new tree with that vertex as the root. By seeing the newly created trees as offspring, the CPEF generates a Galton-Watson tree and we use that it survives if and only if its offspring number exceeds one. It turns out that the phase transition in the adaptive model occurs at a different infection rate when compared to the non-adaptive model.

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We prove a structure theorem for ergodic homological rotation sets of homeomorphisms isotopic to the identity on a closed orientable hyperbolic surface: this set is made of a finite number of pieces that are either one-dimensional or almost convex. The latter ones give birth to horseshoes; in the case of a zero-entropy homeomorphism we show that there exists a geodesic lamination containing the directions in which generic orbits with respect to ergodic invariant probabilities turn around the surface under iterations of the homeomorphism. The proof is based on the idea of $\textit{geodesic tracking}$ of orbits that are typical for some invariant measure by geodesics on the surface, that allows to get links between the dynamics of such points and the one of the geodesic flow on some invariant subset of the unit tangent bundle of the surface.

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This paper is a study of the set of rational numbers of the form 1 < a^q /b^p < a with a and b co-prime integers. The set F (a,b) of these numbers, with an appropriate binary law, is a monoid isomorphic to (N, +, 0). We identify the sequences of minimum and maximum record holders in F (a,b) and prove that the first one converges to 1 while the second one converges to a. We conclude that F (a,b) is dense in the set of the real numbers comprise between 1 and a.

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In [6], Geroch, Kronheimer and Penrose introduced a way to attach ideal points to a spacetime M , defining the causal completion of M. They established that this is a topological space which is Hausdorff when M is globally hyperbolic. In this paper, we prove that if, in addition, M is simply-connected and conformally flat, its causal completion is a topological manifold with boundary homeomorphic to S x [0, 1] where S is a Cauchy hypersurface of M. We also introduce three remarkable families of globally hyperbolic conformally flat spacetimes and provide a description of their causal completions.

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It is known that monoidal categories have a finite definition, whereas multicategories have an infinite (albeit finitary) definition. Since monoidal categories correspond to representable multicategories, it goes without saying that representable multicategories should also admit a finite description. With this in mind, we give a new finite definition of a structure called a short multicategory, which only has multimaps of dimension at most four, and show that under certain representability conditions short multicategories correspond to various flavours of representable multicategories. This is done in both the classical and skew settings.

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In this paper, we investigate the Gibbs measures associated with the focusing nonlinear Schr{ö}dinger equation with an anharmonic potential. We establish a dichotomy for normalizability and non-normalizability of the Gibbs measures in one dimension and higher dimensions with radial data. This extends a recent result of the third and fourth authors with Robert and Seong (2022), where the focusing Gibbs measures with a harmonic potential were addressed. Notably, in the case of a subharmonic potential, we identify a novel critical nonlinearity (below the usual mass-critical exponent) for which the Gibbs measures exhibit a phase transition. The primary challenge emerges from the limited understanding of eigenvalues and eigenfunctions of the Schr{ö}dinger operator with an anharmonic potential. We overcome the difficulty by employing techniques related to a recent work of the first two authors (2022).

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We prove central and local limit theorems for random walks on the Poincar{é} hyperbolic space of dimension n {\v e} 2. To this end we use the ball model and describe the walk therein through the M{ö}bius addition and multiplication. This also allows to derive a corresponding law of large numbers.

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As a sequel to [14], in this article we first introduce a so-called duplex Hecke algebras of type B which is a Q(q)-algebra associated with the Weyl group W (B) of type B, and symmetric groups S_l for l = 0, 1, . . . ,m, satisfying some Hecke relations. This notion originates from the degenerate duplex Hecke algebra arising from the course of study of a kind of Schur-Weyl duality of Levi-type, extending the duplex Hecke algebra of type A arising from the related q-Schur-Weyl duality of Levi-type. A duplex Hecke algebra of type B admits natural representations on certain tensor spaces. We then establish a Levi-type q-Schur-Weyl duality of type B, which reveals the double centralizer property between such duplex Hecke algebras and ıquantum groups studied by Bao-Wang in [1].

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Kurtosis minus squared skewness is bounded from below by 1, but for unimodal distributions this parameter is bounded by 189/125. In some applications it is natural to compare distributions by comparing their kurtosis-minus-squared-skewness parameters. The asymptotic behavior of the empirical version of this parameter is studied here for i.i.d. random variables. The result may be used to test the hypothesis of unimodality against the alternative that the kurtosis-minus-squared-skewness parameter is less than 189/125. However, such a test has to be applied with care, since this parameter can take arbitrarily large values, also for multimodal distributions. Numerical results are presented and for three classes of distributions the skewness-kurtosis sets are described in detail.

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In this paper, we investigate the numerical solution of the two-dimensional fractional Laplacian wave equations. After splitting out the Riesz fractional derivatives from the fractional Laplacian, we treat the Riesz fractional derivatives with an implicit scheme while solving the rest part explicitly. Thanks to the tensor structure of the Riesz fractional derivatives, a splitting alternative direction implicit (S-ADI) scheme is proposed by incorporating an ADI remainder. Then the Gohberg-Semencul formula, combined with fast Fourier transform, is proposed to solve the derived Toeplitz linear systems at each time integration. Theoretically, we demonstrate that the S-ADI scheme is unconditionally stable and possesses second-order accuracy. Finally, numerical experiments are performed to demonstrate the accuracy and efficiency of the S-ADI scheme.

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In this paper, we study the lossless analog compression for i.i.d. nonsingular signals via the polarization-based framework. We prove that for nonsingular source, the error probability of maximum a posteriori (MAP) estimation polarizes under the Hadamard transform. Building on this insight, we propose partial Hadamard compression and develop the corresponding analog successive cancellation (SC) decoder. The proposed scheme consists of deterministic measurement matrices and non-iterative reconstruction algorithm, providing benefits in both space and computational complexity. Using the polarization of error probability, we prove that our approach achieves the information-theoretical limit for lossless analog compression developed by Wu and Verdu.

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We study the iterative methods for large moment systems derived from the linearized Boltzmann equation. By Fourier analysis, it is shown that the direct application of the block symmetric Gauss-Seidel (BSGS) method has slower convergence for smaller Knudsen numbers. Better convergence rates for dense flows are then achieved by coupling the BSGS method with the micro-macro decomposition, which treats the moment equations as a coupled system with a microscopic part and a macroscopic part. Since the macroscopic part contains only a small number of equations, it can be solved accurately during the iteration with a relatively small computational cost, which accelerates the overall iteration. The method is further generalized to the multiscale decomposition which splits the moment system into many subsystems with different orders of magnitude. Both one- and two-dimensional numerical tests are carried out to examine the performances of these methods. Possible issues regarding the efficiency and convergence are discussed in the conclusion.

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We investigate the phase retrieval problem perturbed by dense bounded noise and sparse outliers that can change an adversarially chosen $s$-fraction of the measurement vector. The adversarial sparse outliers may exhibit dependence on both the observation and the measurement. We demonstrate that the nonlinear least absolute deviation based on amplitude measurement can tolerate adversarial outliers at a fraction of $s^{*,1}\approx0.2043$, while the intensity-based model can tolerate a fraction of $s^{*,2}\approx0.1185$. Furthermore, we construct adaptive counterexamples to show that the thresholds are theoretically sharp, thereby showing the presentation of phase transition in the adversarial phase retrieval problem when the corruption fraction exceeds the sharp thresholds. This implies that the amplitude-based model exhibits superior adversarial robustness in comparison with the intensity-based model. Corresponding experimental results are presented to further illustrate our theoretical findings. To the best of our knowledge, our results provide the first theoretical examination of the distinction in robustness performance between amplitude and intensity measurement. A crucial point of our analysis is that we explore the exact distribution of some combination of two non-independent Gaussian random variables and present the novel probability density functions to derive the sharp thresholds.

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In this article, we prove $K$-stability for a family of $C^*$-algebras, which are generated by a finite set of unitaries and isometries satisfying twisted commutation relations. This family includes the $C^*$-algebra of doubly non-commuting isometries and free twist of isometries. Next, we consider the $C^*$-algebra $A_{\mathcal{V}}$ generated by an $n$-tuple of $\mathcal{U}$-twisted isometries $\mathcal{V}$ with respect to a fixed $n\choose 2$-tuple $\mathcal{U}=\{U_{ij}:1\leq i<j \leq n\}$ of commuting unitaries (see \cite{NarJaySur-2022aa}). Under the assumption that the spectrum of the commutative $C^*$-algebra generated by $(\{U_{ij}:1\leq i<j \leq n\})$ does not contain any element of finite order in the torus group $\bbbt^{n\choose 2}$, we show that $A_{\mathcal{V}}$ is $K$-stable. Finally, we prove the same result for the $C^*$-algebra generated by a tuple of free $\mathcal{U}$-twisted isometries.

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We study continuous-time Markov chains on the non-negative integers under mild regularity conditions (in particular, the set of jump vectors is finite and both forward and backward jumps are possible). Based on the so-called flux balance equation, we derive an iterative formula for calculating stationary measures. Specifically, a stationary measure $\pi(x)$ evaluated at $x\in\mathbb{N}_0$ is represented as a linear combination of a few generating terms, similarly to the characterization of a stationary measure of a birth-death process, where there is only one generating term, $\pi(0)$. The coefficients of the linear combination are recursively determined in terms of the transition rates of the Markov chain. For the class of Markov chains we consider, there is always at least one stationary measure (up to a scaling constant). We give various results pertaining to uniqueness and non-uniqueness of stationary measures, and show that the dimension of the linear space of signed invariant measures is equal to the number of generating terms. A minimization problem is constructed in order to compute stationary measures numerically. Moreover, a heuristic linear approximation scheme is suggested for the same purpose by first approximating the generating terms. The correctness of the linear approximation scheme is justified in some special cases. Furthermore, a decomposition of the state space into different types of states (open and closed irreducible classes, and trapping, escaping and neutral states) is presented. Applications to stochastic reaction networks are well illustrated.

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We study the spectral properties of flipped Toeplitz matrices of the form $H_n(f)=Y_nT_n(f)$, where $T_n(f)$ is the $n\times n$ Toeplitz matrix generated by the function $f$ and $Y_n$ is the $n\times n$ exchange (or flip) matrix having $1$ on the main anti-diagonal and $0$ elsewhere. In particular, under suitable assumptions on $f$, we establish an alternating sign relationship between the eigenvalues of $H_n(f)$, the eigenvalues of $T_n(f)$, and the quasi-uniform samples of $f$. Moreover, after fine-tuning a few known theorems on Toeplitz matrices, we use them to provide localization results for the eigenvalues of $H_n(f)$. Our study is motivated by the convergence analysis of the minimal residual (MINRES) method for the solution of real non-symmetric Toeplitz linear systems of the form $T_n(f)\mathbf x=\mathbf b$ after pre-multiplication of both sides by $Y_n$, as suggested by Pestana and Wathen.

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We obtain an upper bound on the lowest magnetic Neumann eigenvalue of a bounded, convex, smooth, planar domain with moderate intensity of the homogeneous magnetic field. This bound is given as a product of a purely geometric factor expressed in terms of the torsion function and of the lowest magnetic Neumann eigenvalue of the disk having the same maximal value of the torsion function as the domain. The bound is sharp in the sense that equality is attained for disks. Furthermore, we derive from our upper bound that the lowest magnetic Neumann eigenvalue with the homogeneous magnetic field is maximized by the disk among all ellipses of fixed area provided that the intensity of the magnetic field does not exceed an explicit constant dependent only on the fixed area.

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Let $X$ be a toric Calabi-Yau 3-fold and let $L\subset X$ be an Aganagic-Vafa outer brane. We prove two versions of open WDVV equations for the open Gromov-Witten theory of $(X,L)$. The first version of the open WDVV equation leads to the construction of a semi-simple (formal) Frobenius manifold and the second version leads to the construction of a (formal) $F$-manifold.

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A marked strongly invertible knot is a triple $(K,h,\delta)$ of a knot $K$ in $S^3$, a strong inversion $h$ of $K$, and a subarc $\delta \subset \operatorname{Fix}(h)\cong S^1$ bounded by $\operatorname{Fix}(h)\cap K\cong S^0$. An invariant Seifert surface for $(K,h,\delta)$ is an $h$-invariant Seifert surface for $K$ that intersects $\operatorname{Fix}(h)$ in the arc $\delta$. In this paper, we completely determine the equivariant genus (the minimum of the genera of invariant Seifert surfaces for $(K,h,\delta)$) of every marked strongly invertible knot $(K,h,\delta)$ with $K$ a $2$-bridge knot.

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Conformal field theory (CFT) has become an active area of research beyond its origins in statistical physics and attracted much attention due to its intrinsic mathematical interest, which reveals deep connections with other diverse branches of mathematics. We study a specific subclass of CFTs that involve either uncoupled or coupled free fermions. Coupled free fermion CFTs arise from parafermion CFTs and lattice constructions. We analyse their representation spaces and reveal the exclusion statistics of coupled free fermions with universal chiral partition functions under specific bases. We explicitly decompose their modules into Virasoro modules of minimal models. We reveal an unexpected connection that integrates the coset construction, lattice construction, and orbifold construction, which is supported by proving a range of character identities within the context of coupled free fermions. Simultaneously, we obtain explicit expressions of certain string functions in terms of Dedekind eta functions.

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In this paper, we give matrix formulae for non-orientable surfaces that provide the Laurent expansion for quasi-cluster variables, generalizing the orientable surface matrix formulae by Musiker-Williams. We additionally use our matrix formulas to prove the skein relations for the elements in the quasi-cluster algebra associated to curves on the non-orientable surface.

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This paper investigates the functional calculus of the harmonic oscillator on each Moyal-Groenewold plane, the noncommutative phase space which is a fundamental object in quantum mechanics. Specifically, we show that the harmonic oscillator admits a bounded $\mathrm{H}^\infty(\Sigma_\omega)$ functional calculus for any angle $0 < \omega < \frac{\pi}{2}$ and even a bounded Hörmander functional calculus on the associated noncommutative $\mathrm{L}^p$-spaces, where $\Sigma_\omega=\{ z \in \mathbb{C}^*: |\arg z| <\omega \}$. To achieve these results, we develop a connection with the theory of 2-step nilpotent Lie groups by introducing a notion of twisted Weyl tuple and connecting it to some semigroups of operators previously investigated by Robinson via group representations. Along the way, we demonstrate that $\mathrm{L}^p$-square-max decompositions lead to new insights between noncommutative ergodic theory and $R$-boundedness, and we prove a twisted transference principle, which is of independent interest. Our approach accommodate the presence of a constant magnetic field and they are indeed new even in the framework of magnetic Weyl calculus on classical $\mathrm{L}^p$-spaces. Our results contribute to the understanding of functional calculi on noncommutative spaces and have implications for the maximal regularity of the most basic evolution equations associated to the harmonic oscillator.

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Due to uncertainty in demand, sometimes management may delay their planning of shifts later in the horizon when they have more information. We consider a novel dynamic and flexible system for scheduling personnel for work shifts planned at the last minute called 'on-call' shifts. The flexibility in the system allows employees to freely select or reject shifts offered to them. Senior employees may also replace or bump junior employees from the current schedule if no other shift of their choice is available provided certain conditions are met. The operator seeks to quickly assign all available shifts to employees while ensuring minimum total bumps. We develop several easy-to-implement policy approximations as well as deep reinforcement learning-based policies to efficiently operate the system. A comparison of all the developed policies is made using real data from our industrial partner. We identify the best policies under different settings and significantly improve upon the current practices.

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We consider partition functions on the $N\times N$ square lattice with the local Boltzmann weights given by the $R$-matrix of the $U_{t}(\widehat{sl}(n+1|m))$ quantum algebra. We identify boundary states such that the square lattice can be viewed on a conic surface. The partition function $Z_N$ on this lattice computes the weighted sum over all possible closed coloured lattice paths with $n+m$ different colours: $n$ ``bosonic'' colours and $m$ ``fermionic'' colours. Each bosonic (fermionic) path of colour $i$ contributes a factor of $z_i$ ($w_i$) to the weight of the configuration. We show the following. (i) $Z_N$ is a symmetric function in the spectral parameters $x_1\dots x_N$ and generates basis elements of the commutative trigonometric Feigin--Odesskii shuffle algebra. The generating function of $Z_N$ admits a shuffle-exponential formula analogous to the Macdonald Cauchy kernel. (ii) $Z_N$ is a symmetric function in two alphabets $(z_1\dots z_n)$ and $(w_1\dots w_m)$. When $x_1\dots x_N$ are set to be equal to the box content of a skew Young diagram $\mu/\nu$ with $N$ boxes the partition function $Z_N$ reproduces the skew Macdonald function $P_{\mu/\nu}\left[w-z\right]$.

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We give explicit expressions for MacMahon's generalized sums-of-divisors $q$-series $A_r$ and $C_r$ by relating them to (odd) multiple Eisenstein series. Recently, these sums-of-divisors have been studied in the context of quasimodular forms, vertex algebras, $N=4$ $SU(N)$ Super-Yang-Mills theory, and the study of congruences of partitions. We relate them to a broader mathematical framework and give explicit expressions for both $q$-series in terms of Eisenstein series and their odd variants.

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In this paper, we establish some results concerning the vanishing of specific Ext modules and the finiteness of the Gorenstein dimension of the dual of a certain module, aiming to characterize the freeness of finitely generated modules.

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In this paper, we shall study the weak solution to the supercritical deformed Hermitian-Yang-Mills equation in the boundary case.

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