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arXiv:2311.11190v2 Announce Type: replace Abstract: The set of partial partitions of $\{1,\ldots,n\}$, ordered by containment, forms an abstract simplicial complex $D_n$ whose vertices are the nonempty subsets of $\{1,\ldots,n\}$ and whose simplices are collections of pairwise disjoint subsets. We prove that $D_n$ is vertex-decomposable, give an explicit nonpure shelling, and use it to compute the reduced homology: for $1 \le j \le n$, the..
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Interpolation and Extrapolation Statements equivalent to the Riemann Hypothesis
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arxiv.org
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2 days ago
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eng
arXiv:2312.00211v3 Announce Type: replace Abstract: The goal of this paper is twofold; on one hand we wish to present some statements that can be formulated in terms of Interpolation theory which are equivalent to the truth or the falseness of the Riemann Hypothesis, on the other hand we will use a key result of the Jawerth-Milman extrapolation to improve on a well-known criterion by Beurling and Nyman for the Riemann Hypothesis giving sha..
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Optimal lower bound for the variance of hitting times for simple random walks on graphs
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arxiv.org
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2 days ago
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eng
arXiv:2312.07726v3 Announce Type: replace Abstract: We study hitting times in simple random walks on graphs, which measure the time required to reach specific target vertices. Our main result establishes a sharp lower bound for the variance of hitting times. For a simple random walk on a graph with $n$ vertices, we prove that the variance of the hitting time from a vertex $x$ to a vertex $y$, denoted $\tau_y$, is at least of the order $\ma..
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Novel approaches for the reliable and efficient numerical evaluation of Landau-type operators
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arxiv.org
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2 days ago
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eng
arXiv:2402.02247v3 Announce Type: replace Abstract: Numerical approximations of Landau-type operators represent fundamental components of time integration methods for demanding problems such as inhomogeneous Vlasov-Landau-type equations. Substantial computational issues arise from the treatment of the physically most relevant three-dimensional case with Coulomb-type interaction. This work is concerned with the introduction and numerical co....
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arXiv:2402.03150v3 Announce Type: replace Abstract: Chv\'{a}tal conjectured that amongst the largest intersecting subfamiles of a finite subset-closed family of sets is a star. Kleitman later strengthened Chv\'{a}tal's conjecture, defining a partial ordering on the vector space freely generated by $2^{[n]}$ and suggesting that the vector of every maximal intersecting subfamily of $2^{[n]}$ is bigger than a convex combination of stars. We r..
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New geometric structures on 3-manifolds: surgery and generalized geometry
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arxiv.org
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2 days ago
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eng
arXiv:2402.12471v5 Announce Type: replace Abstract: Cosymplectic and normal almost contact structures are analogues of symplectic and complex structures that can be defined on 3-manifolds. Their existence imposes strong topological constraints. Generalized geometry offers a natural common generalization of these two structures: $B_3$-generalized complex structures. We prove that any closed orientable 3-manifold admits such a structure, whi..
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arXiv:2402.14969v2 Announce Type: replace Abstract: In a recent paper, the author and St\"ohr established a bound on the number of iterated Frobenius pullbacks needed to transform a non-smooth purely inseparable point on a regular geometrically integral curve into a rational point. In this paper we improve this result, by establishing a new bound that is sharp in every characteristic $p>0$.
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Simple closed curves contained in~$\varepsilon$-boundaries of planar sets
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arxiv.org
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2 days ago
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eng
arXiv:2403.20166v2 Announce Type: replace Abstract: The $\varepsilon$-boundary of a set ${A}\subseteq\mathbb{R}^2$ is the set $\{{p}\in\mathbb{R}^2:\rho({p},{A})=\varepsilon\}$, where $\rho$ is the Euclidean distance. We prove that if ${A},{B}\subseteq\mathbb{R}^2$ are nonempty, connected sets, ${A}$ is bounded, and $0<\varepsilon<\rho({A},{B})$, then the $\varepsilon$-boundary of ${A}$ contains a simple closed curve (aka a Jordan curve) t....
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arXiv:2404.03950v3 Announce Type: replace Abstract: Given a matching $M$ in the hypercube $Q^n$, the \emph{profile} of $M$ is the vector $\boldsymbol{x}=(x_1,\ldots, x_n) \in \mathbb{N}^n$ such that $M$ contains $x_i$ edges whose endpoints differ in the $i$th coordinate. If $M$ is a perfect matching, then it is clear that $||\boldsymbol{x}||_1 = 2^{n-1}$ and it is easy to show that each $x_i$ must be even. Verifying a special case of a con..
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arXiv:2404.10184v2 Announce Type: replace Abstract: We prove an accessibility theorem for finite-index splittings of groups. Given a finitely presented group G there is a number n(G) such that, for every reduced locally finite G-tree T with finitely generated stabilizers, T/G has at most n(G) vertices and edges. We also show that deformation spaces of locally finite trees (with finitely generated stabilizers) are maximal in the partial ord..
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arXiv:2406.03860v3 Announce Type: replace Abstract: We introduce a new definition of a model for a formal mathematical system. The definition is based upon the substitution in the formal systems, which allows a purely algebraic approach to model theory. This is very suitable for applications due to a general syntax used in the formal systems. For our models we present a new proof of the downward L\"owenheim-Skolem Theorem for elementary su..
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arXiv:2407.06865v4 Announce Type: replace Abstract: We provide a geometric realization of the quasi-split affine $\imath$quantum group of type AIII$_{2n-1}^{(\tau)}$ in terms of equivariant K-groups of non-connected Steinberg varieties of type C. This uses a new Drinfeld type presentation of this affine $\imath$quantum group which admits very nontrivial Serre relations. We then construct \`a la Springer a family of finite-dimensional stand..
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arXiv:2407.14048v3 Announce Type: replace Abstract: We introduce a new family of higher-rank graphs, whose construction was inspired by the graphical techniques of Lambek \cite{Lambek} and Johnstone \cite{Johnstone} used for monoid and category emedding results. We show that they are planar $k$-trees for $2 \le k \le 4$. We also show that higher-rank trees differ from $1$-trees by giving examples of higher-rank trees having properties whic..
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A martingale-type characterisation of the Gaussian free field and fractional Gaussian free fields
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arxiv.org
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2 days ago
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eng
arXiv:2407.16261v3 Announce Type: replace Abstract: We establish a martingale-type characterisations for the continuum Gaussian free field (GFF) and for fractional Gaussian free fields (FGFs), using their connection to the stochastic heat equation and to fractional stochastic heat equations. The main theorem on the GFF generalizes previous results of similar flavour and the characterisation theorems on the FGFs are new. The proof strategy ..
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A Two Stepsize SQP Method for Nonlinear Equality Constrained Stochastic Optimization
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arxiv.org
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2 days ago
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eng
arXiv:2408.16656v2 Announce Type: replace Abstract: We develop a Sequential Quadratic Optimization (SQP) algorithm for minimizing a stochastic objective function subject to deterministic equality constraints. The method utilizes two different stepsizes, one which exclusively scales the component of the step corrupted by the variance of the stochastic gradient estimates and a second which scales the entire step. We prove that this stepsize ....
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A theory of generalised coordinates for stochastic differential equations
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arxiv.org
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2 days ago
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eng
arXiv:2409.15532v3 Announce Type: replace Abstract: Stochastic differential equations are ubiquitous modelling tools in physics and the sciences. In most modelling scenarios, random fluctuations driving dynamics or motion have some non-trivial temporal correlation structure, which renders the SDE non-Markovian; a phenomenon commonly known as ``colored'' noise. Thus, an important objective is to develop effective tools for mathematically an....
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On the cover time of Brownian motion on the Brownian continuum random tree
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arxiv.org
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2 days ago
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eng
arXiv:2410.03922v2 Announce Type: replace Abstract: Upon almost-every realisation of the Brownian continuum random tree (CRT), it is possible to define a canonical diffusion process or `Brownian motion'. The main result of this article establishes that the cover time of the Brownian motion on the Brownian CRT (i.e.\ the time taken by the process in question to visit the entire state space) is equal to the infimum over the times at which th....
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Conditioned random walks on linear groups I: construction of the target harmonic measure
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arxiv.org
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2 days ago
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eng
arXiv:2410.05812v2 Announce Type: replace Abstract: Our objective is to explore random walks on the general linear group, constrained to a specific domain, with a primary focus on establishing the conditioned local limit theorem. This paper represents the first step toward achieving this goal, specifically entailing the construction of a novel entity -- the target harmonic measure. This measure, together with the harmonic function, serves ..
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Auslander-Reiten combinatorics and $q$-characters of representations of affine quantum groups
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arxiv.org
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2 days ago
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eng
arXiv:2410.06046v3 Announce Type: replace Abstract: For each simple Lie algebra $\mathfrak{g}$ of simply-laced type, Hernandez and Leclerc introduced a certain category $\mathcal{C}_{\mathbb{Z}}$ of finite-dimensional representations of the quantum affine algebra of $\mathfrak{g}$, as well as certain subcategories $\mathcal{C}_{\mathbb{Z}}^{\leq \xi}$ depending on a choice of height function adapted to an orientation of the Dynkin graph of....
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Almost sure convergence of differentially positive systems on a globally orderable manifold
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arxiv.org
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2 days ago
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eng
arXiv:2410.11895v3 Announce Type: replace Abstract: Differentially positive systems are nonlinear systems whose linearization along trajectories preserves a cone field on a smooth Riemannian manifold. The structures of cone field come from general relativity and Lie theory. We prove that on a globally orderable manifold, the set of convergent points has full Riemann-Lebesgue measure, thus establishing almost sure convergence. This result t..
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arXiv:2410.22549v2 Announce Type: replace Abstract: In this paper we introduce a multiparameter version of the quantum universal enveloping superalgebras introduced by Yamane in [H. Yamane, "Quantized enveloping algebras associated to simple Lie superalgebras and their universal $R$-matrices", Publ. Res. Inst. Math. Sci. 30 (1994), no. 1, 15-87]. For these objects we consider: - (1) their deformations by twist and by 2-cocycle (both of "t..
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Refined $L^p$ restriction estimate for eigenfunctions on Riemannian surfaces
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arxiv.org
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2 days ago
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eng
arXiv:2411.01577v2 Announce Type: replace Abstract: We refine the $L^p$ restriction estimates for Laplace eigenfunctions on a Riemannian surface, originally established by Burq, G\'erard, and Tzvetkov. First, we establish estimates for the restriction of eigenfunctions to arbitrary Borel sets on the surface, following the formulation of Eswarathasan and Pramanik. We achieve this by proving a variable coefficient version of a weighted Fouri..
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arXiv:2411.03383v3 Announce Type: replace Abstract: How hard is it to estimate a discrete-time signal $(x_{1}, ..., x_{n}) \in \mathbb{C}^n$ satisfying an unknown linear recurrence relation of order $s$ and observed in i.i.d. complex Gaussian noise? The class of all such signals is parametric but extremely rich: it contains all exponential polynomials over $\mathbb{C}$ with total degree $s$, including harmonic oscillations with $s$ arbitra....
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Solver-in-the-Loop joint operator learning: fractional Laplace-Beltrami features for interface reconstruction
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arxiv.org
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2 days ago
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eng
arXiv:2411.05341v2 Announce Type: replace Abstract: In this work, we propose a joint operator learning method for reconstructing images of conductivity coefficients from boundary data. Inspired by the idea of employing partial differential equation (PDE) solvers as preconditioners for this inverse problem, we investigate a ``solver-in-the-loop'' training mechanism. It allows the interaction of learnable parameters integrated in a PDE solve....
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The constructive inverse Galois problem via Hilbert modular forms: realizing the transitive group 17T7
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arxiv.org
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2 days ago
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eng
arXiv:2411.07857v4 Announce Type: replace Abstract: We show how Hilbert modular forms can be used in the constructive inverse Galois problem over the rationals. In particular, we prove that the transitive permutation group 17T7, isomorphic to a split extension of C_2 by PSL_2(FF_16), is a Galois group over the rationals and exhibit an explicit degree 17 polynomial with this Galois group. The group arises from the field of definition of the..
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Stochastic Optimal Linear Quadratic Regulation Control of Discrete-time Systems with Delay and Quadratic Constraints
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arxiv.org
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2 days ago
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eng
arXiv:2411.10795v2 Announce Type: replace Abstract: This article explores the discrete-time stochastic optimal LQR control with delay and quadratic constraints. The inclusion of delay, compared to delay-free optimal LQR control with quadratic constraints, significantly increases the complexity of the problem. Using Lagrangian duality, the optimal control is obtained by solving the Riccati-ZXL equation in conjunction with a gradient ascent ..
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The Polymatroid Representation of a Greedoid, and Associated Galois Connections
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arxiv.org
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2 days ago
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eng
arXiv:2411.15363v4 Announce Type: replace Abstract: A greedoid is a generalization of a matroid allowing for more flexible analyses and modeling of combinatorial optimization problems. However, these structures decimate many matroid properties contributing to their pervasive nature. A polymatroid greedoid [KL85] presents an interesting middle ground, so we further develop this class. First we prove every local poset greedoid for which the ....
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Wonderful Compactification of a Cartan Subalgebra of a Semisimple Lie Algebra
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arxiv.org
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2 days ago
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eng
arXiv:2411.19936v3 Announce Type: replace Abstract: Let $\mathfrak h$ be a Cartan subalgebra of a complex semisimple Lie algebra $\mathfrak g.$ We define a compactification $\bar {\mathfrak h}$ of $\mathfrak h$, which is analogous to the closure $\bar H$ of the corresponding maximal torus $H$ in the adjoint group of $\mathfrak g$ in its wonderful compactification, which was introduced and studied by De Concini and Procesi \cite{DCP}. We ob....
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Acceleration by Random Stepsizes: Hedging, Equalization, and the Arcsine Stepsize Schedule
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arxiv.org
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2 days ago
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eng
arXiv:2412.05790v2 Announce Type: replace Abstract: We show that for separable convex optimization, random stepsizes fully accelerate Gradient Descent. Specifically, using inverse stepsizes i.i.d. from the Arcsine distribution improves the convergence rate from $O(k)$ to $O(\sqrt{k})$, where $k$ is the condition number. No momentum or other algorithmic modifications are required. Our starting point is a remarkable "equalization property" o..
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On the role of semismoothness in the implicit programming approach to selected nonsmooth optimization problems
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arxiv.org
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2 days ago
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eng
arXiv:2412.05953v2 Announce Type: replace Abstract: The paper deals with the implicit programming approach to a class of Mathematical Programs with Equilibrium Constraints (MPECs) and bilevel programs in the case when the corresponding reduced problems are solved using a bundle method of nonsmooth optimization. The obtained results allow us to supply the bundle algorithm with suitable, easily computable ``pseudosubgradients'', ensuring con..
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Highest Posterior Density Intervals of Unimodal Distributions As Analogues to Profile Likelihood Ratio Confidence Intervals
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arxiv.org
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2 days ago
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eng
arXiv:2412.06528v5 Announce Type: replace Abstract: In Bayesian statistics, the highest posterior density (HPD) interval is often used to describe properties of a posterior distribution. As a method for estimating confidence intervals (CIs), the HPD has two main desirable properties. Firstly, it is the shortest interval to have a specified coverage probability. Secondly, every point inside the HPD interval has a density greater than every ..
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arXiv:2412.11086v3 Announce Type: replace Abstract: We study the behavior of perturbations in a compressible one-dimensional inviscid gas with an ambient state consisting of constant pressure and periodically-varying density. We show through asymptotic analysis that long-wavelength perturbations approximately obey a system of dispersive nonlinear wave equations. Computational experiments demonstrate that solutions of the 1D Euler equations..
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Rellich type theorem and unique continuation property for discrete Maxwell operators
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arxiv.org
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2 days ago
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eng
arXiv:2412.11568v2 Announce Type: replace Abstract: We study the Rellich type theorem (RT) for the Maxwell operator __ D = D____0 on Z3 in a constant anisotropic medium, i.e., the permittivity and permeability of which are constant non-scalar diagonal matrices. We also prove the unique continuation property (UCP) in the exterior of a compact convex set Kint $\subset$ Z3 for the perturbed Maxwell operator __ Dp = D__p __0 on Z3 for which th..
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arXiv:2412.11975v4 Announce Type: replace Abstract: The Nielsen-Thomsen sequence plays a pivotal role in refining invariants for C$^*$-algebras beyond the Elliott classification framework. This paper revisits the sequence, introducing the concepts of Nielsen-Thomsen bases, rotation maps and diagonalisable morphisms, to better understand its unnatural splitting. These insights enable novel comparison methods for *-homomorphisms at the level..
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arXiv:2501.00710v2 Announce Type: replace Abstract: Given a triangulated category $\mathcal{C}$, we construct a partial compactification, denoted $\mathcal{A}\mathrm{Stab}(\mathcal{C})$, of the quotient of its stability manifold by $\mathbb{C}$. The purpose of $\mathcal{A}\mathrm{Stab}(\mathcal{C})$ is to shed light on the structure of semiorthogonal decompositions of $\mathcal{C}$. A point of $\mathcal{A}\mathrm{Stab}(\mathcal{C})$, calle....
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Existence of weak solutions for nonlinear drift-diffusion equations with measure data
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arxiv.org
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2 days ago
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eng
arXiv:2501.07847v2 Announce Type: replace Abstract: We consider nonlinear drift-diffusion equations (both porous medium equations and fast diffusion equations) with measure data. We establish the existence of nonnegative weak solutions satisfying gradient estimates, provided that the drift term belongs to a sub-scaling class relevant to the $L^1$ space. When the drift is divergence-free, this requirement can be relaxed: the drift may belon....
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A Min-Max Relation on Dicuts and Dijoins in Weighted Chordal Digraphs
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arxiv.org
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2 days ago
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eng
arXiv:2501.10918v2 Announce Type: replace Abstract: In a digraph, a dicut is a cut where all the arcs cross in one direction. A dijoin is a subset of arcs that intersects every dicut. Edmonds and Giles conjectured that in a weighted digraph, the minimum weight of a dicut is equal to the maximum size of a packing of dijoins. This has been disproved. However, the unweighted version conjectured by Woodall remains open. We prove that the Edmon..
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Kink dynamics for the Yang-Mills field in an extremal Reissner-Nordstr\"om black hole
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arxiv.org
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2 days ago
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eng
arXiv:2501.12790v4 Announce Type: replace Abstract: Considered in this work is the Yang-Mills field in an extremal Reissner-Nordstr\"om black hole, a physically motivated mathematical model introduced by Bizo\'n and Kahl. The kink is a fundamental, strongly unstable stationary solution in this non-perturbative, variable coefficients model, with a polynomial tail and no explicit form. In this paper, we introduce and extend several virial te..
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Stabilization-Free H(curl) and H(div)-Conforming Virtual Element Method
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arxiv.org
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2 days ago
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eng
arXiv:2501.15168v2 Announce Type: replace Abstract: Standard Virtual Element Method (VEM) requires stabilization terms that significantly affect the numerical computation performance. In this work, we propose a stabilization-free VEM for general order \(\mathbf{H}(\operatorname{\mathbf{curl}})\) and \(\mathbf{H}(\operatorname{div})\)-conforming spaces by constructing novel serendipity projectors and corresponding serendipity spaces with mi..
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arXiv:2502.00753v4 Announce Type: replace Abstract: Smoothness is crucial for attaining fast rates in first-order optimization. However, many optimization problems in modern machine learning involve non-smooth objectives. Recent studies relax the smoothness assumption by allowing the Lipschitz constant of the gradient to grow with respect to the gradient norm, which accommodates a broad range of objectives in practice. Despite this progres....
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arXiv:2502.02108v3 Announce Type: replace Abstract: We study the Schur algebra counterpart of a vast class of quantum wreath products. This is achieved by developing a theory of twisted convolution algebras, inspired by geometric intuition. In parallel, we provide an algebraic Schurification via a Kashiwara-Miwa-Stern-type action on a tensor space. We give a uniform proof of Schur duality, and construct explicit bases of the new Schur alge..
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Error estimates for full discretization by an almost mass conservation technique for Cahn--Hilliard systems with dynamic boundary conditions
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arxiv.org
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2 days ago
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eng
arXiv:2502.03847v2 Announce Type: replace Abstract: A proof of optimal-order error estimates is given for the full discretization of the bulk--surface Cahn--Hilliard system with dynamic boundary conditions in a smooth domain. The numerical method combines a linear bulk--surface finite element discretization in space and linearly implicit backward difference formulae of order one to five in time. The error estimates are obtained by a consis..
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arXiv:2503.13157v4 Announce Type: replace Abstract: We introduce and study the notion of ramification ideals in higher ramification theory. After general results on their computation for finite extensions, we discuss their connection with the possibly nontrivial defect of the extensions. We compute them for Artin-Schreier extensions and Kummer extensions of prime degree equal to the residue characteristic, which may or may not have nontriv..
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arXiv:2503.15046v2 Announce Type: replace Abstract: We address the enumeration of Eulerian orientations of 4-valent planar maps according to three parameters: the number of vertices, the number of alternating vertices (having in/out/in/out incident edges), and the number of clockwise oriented faces. This is a refinement of the six vertex model studied by Kostov, then Zinn-Justin and Elvey Price, where one only considers the first two param....
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