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### Sort: (click the keys to login/signup to enable post sorting) &lt;p&gt;The aim of the paper is to study the group scheme $G:=\operatorname{SL}(2, A)$, it's comodules and a virtual Clebsch-Gordan formula. Here $A$ is a field or a Dedekind domain. If $V:=A\{e_1,e_2\}$ is the free rank $2$ module on $A$ and if we give $V$ the "standard" structure as comodule on $G$, we may form the symmetric powers $\operatorname{Sym}^n(V)$ for $n \geq 1$ an integer. If $A$ is a field of characteristic zero, there is a direct sum decomposition of the tensor product $\operatorname{Sym}^n(V) \otimes \operatorname{Sym}^m(V)$ into irreducible $G$-comodules and the main aim of the paper is to investigate if similar resu…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;As quantum theory allows for information processing and computing tasks that otherwise are not possible with classical systems, there is a need and use of quantum Internet beyond existing network systems. At the same time, the realization of a desirably functional quantum Internet is hindered by fundamental and practical challenges such as high loss during transmission of quantum systems, decoherence due to interaction with the environment, fragility of quantum states, etc. We study the implications of these constraints by analyzing the limitations on the scaling and robustness of quantum Internet. Considering quantum networks, we present p…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;Given a square pencil $A+ \lambda B$, where $A$ and $B$ are complex matrices, we consider the problem of finding the singular pencil nearest to it in the Frobenius distance. This problem is known to be very difficult, and the few algorithms available in the literature can only deal efficiently with pencils of very small size. We show that the problem is equivalent to minimizing a certain objective function over the Riemannian manifold $SU(n) \times SU(n)$, where $SU(n)$ denotes the special unitary group. With minor modifications, the same approach extends to the case of finding a nearest singular pencil with a specified minimal index. This …
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;In our study, we conduct magnetization and heat capacity measurements to investigate field-induced magnetic phase transitions within the newly synthesized compound K2Ni2(SeO3)3, a spin-1 dimer system arranged on a triangular lattice. The Ni-Ni dimers exhibit a ferromagnetic intra-dimer interaction, effectively behaving as an ensemble with a total spin of S=2. In contrast, antiferromagnetic interactions manifest between these dimers on the triangular lattice. The trigonal distortion of the NiO6 octahedra introduces easy-axis magnetic anisotropy, accounting for the distinct magnetic phase diagrams observed when applying c-axis directional and…
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Latest: Sept. 21, 2023, 7:30 a.m. &lt;p&gt;In this article, we present a novel data assimilation strategy in pore-scale imaging and demonstrate that this makes it possible to robustly address reactive inverse problems incorporating Uncertainty Quantification (UQ). Pore-scale modeling of reactive flow offers a valuable opportunity to investigate the evolution of macro-scale properties subject to dynamic processes. Yet, they suffer from imaging limitations arising from the associated X-ray microtomography (X-ray microCT) process, which induces discrepancies in the properties estimates. Assessment of the kinetic parameters also raises challenges, as reactive coefficients are critical p…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;In this paper, we present a fast divergence-free spectral algorithm (FDSA) for the curl-curl problem. Divergence-free bases in two and three dimensions are constructed by using the generalized Jacobi polynomials. An accurate spectral method with exact preservation of the divergence-free constraint point-wisely is then proposed, and its corresponding error estimate is established. We then present a highly efficient solution algorithm based on a combination of matrix-free preconditioned Krylov subspace iterative method and a fully diagonalizable auxiliary problem, which is derived from the spectral discretisations of generalized eigenvalue pr…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;The increasing trend in adopting electric vehicles (EVs) will significantly impact the residential electricity demand, which results in an increased risk of transformer overload in the distribution grid. To mitigate such risks, there are urgent needs to develop effective EV charging controllers. Currently, the majority of the EV charge controllers are based on a centralized approach for managing individual EVs or a group of EVs. In this paper, we introduce a decentralized Multi-agent Reinforcement Learning (MARL) charging framework that prioritizes the preservation of privacy for EV owners. We employ the Centralized Training Decentralized E…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;Decision-making problems can be represented as mathematical optimization models, finding wide applications in fields such as economics, engineering and manufacturing, transportation, and health care. Optimization models are mathematical abstractions of the problem of making the best decision while satisfying a set of requirements or constraints. One of the primary barriers to deploying these models in practice is the challenge of helping practitioners understand and interpret such models, particularly when they are infeasible, meaning no decision satisfies all the constraints. Existing methods for diagnosing infeasible optimization models o…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;We show that the range of a critical branching random walk conditioned to survive forever and the Minkowski sum of two independent simple random walk ranges are intersection-equivalent in any dimension $d\ge 5$, in the sense that they hit any finite set with comparable probability, as their common starting point is sufficiently far away from the set to be hit. Furthermore, we extend a discrete version of Kesten, Spitzer and Whitman's result on the law of large numbers for the volume of a Wiener sausage. Here, the sausage is made of the Minkowski sum of $N$ independent simple random walk ranges in $\mathbb{Z}^d$, with $d&gt;2N$, and…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;In the article, we investigate the average behaviour of normalised Hecke eigenvalues over certain polynomials and establish an estimate for the power moments of the normalised Hecke eigenvalues of a normalised Hecke eigenform of weight $k \ge 2$ for the full modular group $SL_2(\mathbb{Z})$ over certain polynomial, given by a sum of triangular numbers with certain positive coefficients. More precisely, for each $r \in \mathbb{N}$, we obtain an asymptotic for the following sum \begin{equation*} \begin{split} \displaystyle{\sideset{}{^{\flat }}\sum_{ \alpha(\underline{x}))+1 \le X \atop \underline{x} \in {\mathbb Z}^{4}} } \lambda_{f}^{r}(\al…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;We develop a novel asymptotic theory for local polynomial (quasi-) maximum-likelihood estimators of time-varying parameters in a broad class of nonlinear time series models. Under weak regularity conditions, we show the proposed estimators are consistent and follow normal distributions in large samples. Our conditions impose weaker smoothness and moment conditions on the data-generating process and its likelihood compared to existing theories. Furthermore, the bias terms of the estimators take a simpler form. We demonstrate the usefulness of our general results by applying our theory to local (quasi-)maximum-likelihood estimators of a time-…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;Consider the following prediction problem. Assume that there is a block box that produces bits according to some unknown computable distribution on the binary tree. We know first $n$ bits $x_1 x_2 \ldots x_n$. We want to know the probability of the event that that the next bit is equal to $1$. Solomonoff suggested to use universal semimeasure $m$ for solving this task. He proved that for every computable distribution $P$ and for every $b \in \{0,1\}$ the following holds: $$\sum_{n=1}^{\infty}\sum_{x: l(x)=n} P(x) (P(b | x) - m(b | x))^2 &lt; \infty\ .$$ However, Solomonoff's method has a negative aspect: Hutter and Muchnik proved t…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;Given a monoid $S$ with $E$ any non-empty subset of its idempotents, we present a novel one-sided version of idempotent completion we call left $E$-completion. In general, the construction yields a one-sided variant of a small category called a constellation by Gould and Hollings. Under certain conditions, this constellation is inductive, meaning that its partial multiplication may be extended to give a left restriction semigroup, a type of unary semigroup whose unary operation models domain. We study the properties of those pairs $S,E$ for which this happens, and characterise those left restriction semigroups that arise as such left $E$-co…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;This paper shows that immersed totally geodesic $m$-dimensional suborbifolds of $n$-dimensional arithmetic hyperbolic orbifolds correspond to finite subgroups of the commensurator whenever $m \geqslant \lfloor \frac{n}{2} \rfloor$. We call such totally geodesic suborbifolds finite centraliser subspaces (or fc-subspaces) and use them to formulate an arithmeticity criterion for hyperbolic lattices. &lt;/p&gt; &lt;p&gt;We show that a hyperbolic orbifold $M$ is arithmetic if and only if it has infinitely many fc-subspaces, exhibiting examples of non-arithmetic orbifolds that contain non-fc subspaces of codimension one. We provid…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;We revisit Vafa-Witten theory in the more general setting whereby the underlying moduli space is not that of instantons, but of the full Vafa-Witten equations. We physically derive (i) a novel Vafa-Witten four-manifold invariant associated with this moduli space, (ii) their relation to Gromov-Witten invariants, (iii) a novel Vafa-Witten Floer homology assigned to three-manifold boundaries, (iv) a novel Vafa-Witten Atiyah-Floer correspondence, (v) a proof and generalization of a conjecture by Abouzaid-Manolescu in  about the hypercohomology of a perverse sheaf of vanishing cycles, (vi) a Langlands duality of these invariants, Floer homolo…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;In this paper, we study min-max optimization problems on Riemannian manifolds. We introduce a Riemannian Hamiltonian function, minimization of which serves as a proxy for solving the original min-max problems. Under the Riemannian Polyak--{\L}ojasiewicz condition on the Hamiltonian function, its minimizer corresponds to the desired min-max saddle point. We also provide cases where this condition is satisfied. For geodesic-bilinear optimization in particular, solving the proxy problem leads to the correct search direction towards global optimality, which becomes challenging with the min-max formulation. To minimize the Hamiltonian function, …
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;The classical Ham Sandwich theorem states that any $d$ point sets in $\mathbb{R}^d$ can be simultaneously bisected by a single affine hyperplane. A generalization of Dolnikov asserts that any $d$ families of pairwise intersecting compact, convex sets in $\mathbb{R}^d$ admit a common hyperplane transversal. We extend Dolnikov's theorem by showing that families of compact convex sets satisfying more general non-disjointness conditions admit common transversals by multiple hyperplanes. In particular, these generalize all known optimal results to the long-standing Gr\"unbaum--Hadwiger--Ramos measure equipartition problem in the case o…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;We work over a perfect field. Recent work of the third-named author established a Derived Auslander Correspondence that relates finite-dimensional self-injective algebras that are twisted $3$-periodic to algebraic triangulated categories of finite type. Moreover, the aforementioned work also shows that the latter triangulated categories admit a unique differential graded enhancement. In this article we prove a higher-dimensional version of this result that, given an integer $d\geq1$, relates twisted $(d+2)$-periodic algebras to algebraic triangulated categories with a $d\mathbb{Z}$-cluster tilting object. We also show that the latter triang…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;The optimal stopping problem is one of the core problems in financial markets, with broad applications such as pricing American and Bermudan options. The deep BSDE method [Han, Jentzen and E, PNAS, 115(34):8505-8510, 2018] has shown great power in solving high-dimensional forward-backward stochastic differential equations (FBSDEs), and inspired many applications. However, the method solves backward stochastic differential equations (BSDEs) in a forward manner, which can not be used for optimal stopping problems that in general require running BSDE backwardly. To overcome this difficulty, a recent paper [Wang, Chen, Sudjianto, Liu and Shen, …
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;Geometric quantiles are location parameters which extend classical univariate quantiles to normed spaces (possibly infinite-dimensional) and which include the geometric median as a special case. The infinite-dimensional setting is highly relevant in the modeling and analysis of functional data, as well as for kernel methods. &lt;/p&gt; &lt;p&gt;We begin by providing new results on the existence and uniqueness of geometric quantiles. Estimation is then performed with an approximate M-estimator and we investigate its large-sample properties in infinite dimension. &lt;/p&gt; &lt;p&gt;When the population quantile…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;Copy-move forgery on speech (CMF), coupled with post-processing techniques, presents a great challenge to the forensic detection and localization of tampered areas. Most of the existing CMF detection approaches necessitate pre-segmentation of speech to facilitate similarity calculations among these segments. However, these approaches usually suffer from the problems of uncontrollable computational complexity and sensitivity to the presence of a word that is read multiple times within a speech recording. To address these issues, we propose a local feature tensors-based CMF detection algorithm that can transform duplicate detection and locali…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;Recently, [Phys. Rev. Lett. 130, 221501 (2023)] Jacobson and Visser calculated the quantum partition function of a fixed, finite volume of a region with the topology of a ball in the saddle point approximation within the context of Einstein's gravity with or without a cosmological constant. The result can be interpreted as the dimension of Hilbert space of the theory. Here we extend their computation to a theory defined in principle with infinitely many powers of curvature in three dimensions. We confirm their result: The partition function of a spatial region in the leading saddle point approximation is given as the exponential of the…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;The Hopf formula for Hamilton-Jacobi reachability (HJR) analysis has been proposed to solve high-dimensional differential games, producing the set of initial states and corresponding controller required to reach (or avoid) a target despite bounded disturbances. As a space-parallelizable method, the Hopf formula avoids the curse of dimensionality that afflicts standard dynamic-programming HJR, but is restricted to linear time-varying systems. To compute reachable sets for high-dimensional nonlinear systems, we pair the Hopf solution with Koopman theory for global linearization. By first lifting a nonlinear system to a linear space and then s…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;In this paper, we consider stochastic monotone Nash games where each player's strategy set is characterized by possibly a large number of explicit convex constraint inequalities. Notably, the functional constraints of each player may depend on the strategies of other players, allowing for capturing a subclass of generalized Nash equilibrium problems (GNEP). While there is limited work that provide guarantees for this class of stochastic GNEPs, even when the functional constraints of the players are independent of each other, the majority of the existing methods rely on employing projected stochastic approximation (SA) methods. However,…
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Latest: Aug. 26, 2023, 7:32 a.m. &lt;p&gt;Recently, [Phys. Rev. Lett. 130, 221501 (2023)] Jacobson and Visser calculated the quantum partition function of a fixed, finite volume of a region with the topology of a ball in the saddle point approximation within the context of Einstein's gravity with or without a cosmological constant. The result can be interpreted as the dimension of Hilbert space of the theory. Here we extend their computation to a theory defined in principle with infinitely many powers of curvature in three dimensions. We confirm their result: The partition function of a spatial region in the leading saddle point approximation is given as the exponential of the…
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Latest: Aug. 26, 2023, 7:32 a.m.
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