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e4dd00312eea9d09d829e25a46c89593e511b3aed749ac022358ea15513d0c82
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2026-01-23T00:00:00-05:00
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The distinguishing number of complete bipartite and crown graphs
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arXiv:2601.15913v1 Announce Type: new Abstract: The distinguishing number of a permutation group $G\leqslant\Sym(\Omega)$ is the minimum number of colours needed to colour $\Omega$ in such a way that the only colour preserving element of $G$ is the identity. The distinguishing number of a graph is the distinguishing number of its automorphism group (as a permutation group on vertices). We determine the distinguishing number of the complete bipartite graphs $K_{n,n}$ and the crown graphs $K_{n,n}-nK_2$, as well as the distinguishing number of some `large' subgroups of their automorphism groups, that is, the subgroups that are vertex- and edge-transitive and such that the induced action on each bipart is $\Alt(n)$ or $\Sym(n)$. We show that, if $G$ is a `large' group of automorphisms of $K_{n,n}$, then $n-1\leqslant D(G) \leqslant n+1$. Similarly, if $G$ is a `large' group of automorphisms of a crown graph, then $\lceil \sqrt{n-1}\rceil \leqslant D(G)\leqslant \lfloor \sqrt{n}\rfloor+1$. \smallskip \textit{Keywords:} complete bipartite graph; crown graph; distinguishing number; symmetric group; alternating group
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https://arxiv.org/abs/2601.15913
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Academic Papers
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47d9ee2345db260cd90200f0b465888388974bdb4ce0e7086f618e6e8b6a44da
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2026-01-23T00:00:00-05:00
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Mutation Of Matrices Over Group Rings
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arXiv:2601.15920v1 Announce Type: new Abstract: We give a precise definition of mutation of skew symmetrizable matrices over group rings and relate it to folding and mutation of quivers with symmetries. These matrices can have non-zero diagonal entries and we explain a mutation rule in some of these cases as well. This new rule comes from a notion of a generalized mutation of an entire quiver or sub-quiver.
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https://arxiv.org/abs/2601.15920
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7ccc32f827be2beccfd98c35d7db2077a2af2aef82bf7e98fb45ad0d5109165e
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2026-01-23T00:00:00-05:00
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A new proof of unboundedness of Riesz operator in $L^\infty$ and applications to mild ill-posedness in $W^{1,\infty}$ of the Euler type equations
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arXiv:2601.15922v1 Announce Type: new Abstract: In this paper, we first present a new and simple proof of unboundedness of Riesz operator in $L^\infty$ and then establish the mild ill-posedness in $W^{1,\infty}$ of 3D rotating Euler equations and 2D Euler equations with partial damping. To the best of our knowledge, our work is the first one addressing the ill-posedness issue on the rotating Euler equations in $W^{1,\infty}$ without the vorticity formulation. As a further application, we prove the instability of perturbations for the 2D surface quasi-geostrophic equation and porous medium system in $W^{1,\infty}$.
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https://arxiv.org/abs/2601.15922
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db26fa9460d56668b9718dcb46dbfbc51e9e7e64704e6f87b9812657462f7fdb
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2026-01-23T00:00:00-05:00
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Higher-dimensional Heegaard Floer homology and spectral networks
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arXiv:2601.15923v1 Announce Type: new Abstract: Given a closed surface $C$ and a real exact Lagrangian $\Sigma \subset T^*C$ associated to a spectral curve, we construct a homomorphism $\operatorname{BSk}_\kappa(C)\to\operatorname{Mat}(N^{\kappa},\operatorname{BSk}_\kappa(\Sigma))$ from the braid skein algebra of $C$ to the matrix-valued braid skein algebra of $\Sigma$ using Floer theory and in particular higher-dimensional Heegaard Floer homology (HDHF). We sketch a proof that this map coincides with a hybrid Floer-Morse approach which counts HDHF-type holomorphic curves coupled with certain Morse gradient graphs -- called fold\-ed Morse trees -- using a variant of the adiabatic limit theorems of Fukaya-Oh and Ekholm, which compares holomorphic curves and Morse flow trees.
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https://arxiv.org/abs/2601.15923
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1814c83547cf2d0600c05f6d4753f603eb8f68a8ab84717499af22e5121b004f
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2026-01-23T00:00:00-05:00
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The representations of the Lie superalgebra p(3) in prime characteristic
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arXiv:2601.15932v1 Announce Type: new Abstract: Let g be the Lie superalgebra p(3) of rank 2 over an algebraically closed field K of characteristic p > 3. We classify all irreducible modules of g, and give the character formulae for irreducible modules.
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https://arxiv.org/abs/2601.15932
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faa052a01f53775eb81b10ba09a459f6b52f37c09e523c887b4e3d3a220ce82f
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2026-01-23T00:00:00-05:00
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Extreme Score Distributions in Countable-Outcome Round-Robin Tournaments of Equally Strong Players
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arXiv:2601.15950v1 Announce Type: new Abstract: We consider a general class of round-robin tournament models of equally strong players. In these models, each of the $n$ players competes against every other player exactly once. For each match between two players, the outcome is a value from a countable subset of the unit interval, and the scores of the two players in a match sum to one. The final score of each player is defined as the sum of the scores obtained in matches against all other players. We study the distribution of extreme scores, including the maximum, second maximum, and lower-order extremes. Since the exact distribution is computationally intractable even for small values of $n$, we derive asymptotic results as the number of players $n$ tends to infinity, including limiting distributions, and rates of convergence.
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https://arxiv.org/abs/2601.15950
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b5b3f5e278386afbb465289417e7b06326d86e68eeab174b1e69299d5214010c
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2026-01-23T00:00:00-05:00
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Geometry of spherical spin glasses
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arXiv:2601.15966v1 Announce Type: new Abstract: Spherical spin glasses are canonical models for smooth random functions in high dimensions. In this review, we survey several interrelated lines of research on their geometric structure. We begin with results concerning critical points and their relationship to the Gibbs measure. For the pure models, the measure concentrates on spherical bands around critical points that approximately maximize the energy at a particular radius. Next, we present another approach in which a similar picture is derived for general mixed models. At the core of this approach is a free energy functional computed over bands using multiple orthogonal replicas, satisfying a strong concentration of measure. We discuss several implications of this method for a generalized Thouless-Anderson-Palmer (TAP) approach. Finally, we explain how these geometric insights inform optimization algorithms, and briefly relate them to Smale's 17th problem over the real numbers.
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https://arxiv.org/abs/2601.15966
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84d34af1c031ad8efa018a1f9c8c15e034fdbba7f4386276cdea7d89b480f09c
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2026-01-23T00:00:00-05:00
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Iteration complexity of the Difference-of-Convex Algorithm for unconstrained optimization: a simple proof
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arXiv:2601.15970v1 Announce Type: new Abstract: We propose a simple proof of the worst-case iteration complexity for the Difference of Convex functions Algorithm (DCA) for unconstrained minimization, showing that the global rate of convergence of the norm of the objective function's gradients at the iterates converge to zero like o(1/k). A small example is also provided indicating that the rate cannot be improved.
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https://arxiv.org/abs/2601.15970
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95a9de52034de3f285323516eb2a5a5642ed9d4bffd29300b36035b3f60437d4
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2026-01-23T00:00:00-05:00
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On maximal rank properties for symmetric polynomials in an equigenerated monomial complete intersection
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arXiv:2601.15978v1 Announce Type: new Abstract: It is well known that a monomial complete intersection has the strong Lefschetz property in characteristic zero. This property is equivalent to the statement that any power of the sum of the variables is a maximal rank element on the complete intersection. In this paper, we investigate what happens when this element is replaced by another symmetric polynomial, in an equigenerated complete intersection. We answer the question completely for the power sum symmetric polynomial using a grading technique, and for any Schur polynomial in the case of two variables by deriving a closed formula for the determinants of a family of Toeplitz matrices. Further, we obtain partial results in three or more variables for the elementary and the complete homogeneous symmetric polynomials and pose several open questions.
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https://arxiv.org/abs/2601.15978
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4d0132de95e88b0f1a769326f68e0d8a73966f3ab20903821bf495a934e594e6
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2026-01-23T00:00:00-05:00
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Primes and The Field of Values of Characters
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arXiv:2601.15987v1 Announce Type: new Abstract: Let $p$ be a prime. For $p=2$, the fields of values of the complex irreducible characters of finite groups whose degrees are not divisible by $p$ have been classified; for odd primes $p$, a conjectural classification has been proposed. In this work, we extend this conjecture to characters whose degrees are divisible by arbitrary powers of $p$, and we provide some evidence supporting its validity.
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https://arxiv.org/abs/2601.15987
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1a157b58fed7b6c7391fdf8bcc3374ca828131d8b3ce64b159cf2eb99ff421ff
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2026-01-23T00:00:00-05:00
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Rank of elliptic curves and class groups of real quadratic fields
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arXiv:2601.15988v1 Announce Type: new Abstract: In this paper, we are going to prove the relation between rank of elliptic curves and the non-triviality of class groups of infinitely many real quadratic fields.
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https://arxiv.org/abs/2601.15988
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9fee75bf11a8636270781a68e7476e98f97909c6020956c127737bac5f432241
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2026-01-23T00:00:00-05:00
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A sharp criterion and complete classification of global-in-time solutions and finite time blow-up of solutions to a chemotaxis system in supercritical dimensions
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arXiv:2601.15990v1 Announce Type: new Abstract: We consider the chemotaxis system with indirect signal production in the whole space, \begin{equation}\label{abst:p}\tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u\nabla v),\\ 0 = \Delta v + w,\\ w_t = \Delta w + u \end{cases} \end{equation} with emphasis on supercritical dimensions. In contrast to the classical parabolic-elliptic Keller--Segel system, where the analysis can be reduced to a single equation, the above system is essentially parabolic-parabolic and does not admit such a reduction. In this paper, we establish a sharp threshold phenomenon separating global-in-time existence from finite time blow-up in terms of scaling-critical Morrey norms of the initial data. In particular, we prove the existence of singular stationary solutions and show that their Morrey norm values serve as the critical thresholds determining the long-time behavior of solutions. Consequently, we identify new critical exponents at which the long-time behavior of solutions changes. This yields a complete classification of the long-time behavior of solutions, providing the first such results for the essentially parabolic-parabolic chemotaxis system \eqref{abst:p} in supercritical dimensions.
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https://arxiv.org/abs/2601.15990
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0a992f9e64d956891ab623db504f1c7a03f0954c5e4d3f877de91f89de61a453
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2026-01-23T00:00:00-05:00
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Counting Saddle Connections on Hyperelliptic Translation Surfaces with a Slit
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arXiv:2601.15993v1 Announce Type: new Abstract: We consider saddle connections on a translation surface in a hyperelliptic connected component of a stratum that do not intersect the interior of a distinguished saddle connection. For this restricted set of saddle connections, we show that it satisfies an $L (\log L)^{d-2}$ growth rate, where $d$ is the complex dimension of the hyperelliptic stratum. The upper bound holds for all translation surfaces in the hyperelliptic stratum while the lower bound holds for almost every surface in the hyperelliptic stratum. The proof of the lower bound uses horocycle renormalization.
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https://arxiv.org/abs/2601.15993
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7e49abf2c8d09bd7df482cf2b76fa1bebd7c1896b1cd582eacd197c7999b03ba
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2026-01-23T00:00:00-05:00
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Minimax-optimal Halpern iterations for Lipschitz maps
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arXiv:2601.15996v1 Announce Type: new Abstract: This paper investigates the minimax-optimality of Halpern fixed-point iterations for Lipschitz maps in general normed spaces. Starting from an a priori bound on the orbit of iterates, we derive non-asymptotic estimates for the fixed-point residuals. These bounds are tight, meaning that they are attained by a suitable Lipschitz map and an associated Halpern sequence. By minimizing these tight bounds we identify the minimax-optimal Halpern scheme. For contractions, the optimal iteration exhibits a transition from an initial Halpern phase to the classical Banach-Picard iteration and, as the Lipschitz constant approaches one, we recover the known convergence rate for nonexpansive maps. For expansive maps, the algorithm is purely Halpern with no Banach-Picard phase; moreover, on bounded domains, the residual estimates converge to the minimal displacement bound. Inspired by the minimax-optimal iteration, we design an adaptive scheme whose residuals are uniformly smaller than the minimax-optimal bounds, and can be significantly sharper in practice. Finally, we extend the analysis by introducing alternative bounds based on the distance to a fixed point, which allow us to handle mappings on unbounded domains; including the case of affine maps for which we also identify the minimax-optimal iteration.
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https://arxiv.org/abs/2601.15996
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b99dfa8503d16d2456c369620af2e323762845e162aa341b22fdd7075cbf316b
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2026-01-23T00:00:00-05:00
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The Recovery of Semilinear Potentials Satisfying Null Conditions From Scattering Data
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arXiv:2601.15997v1 Announce Type: new Abstract: We construct oscillatory solutions of fully semilinear wave equations in Minkowski space satisfying a null condition of the form $$\square u:=(-\partial_{x_0}^2 +\sum_{j=1}^n \partial_{x_j}^2 )u= q(x,u)((\partial_{x_0}u)^2-|\nabla_{x'}u|^2),$$ $$x=(x_0,x'), \;\ x'=(x_1,\ldots, x_n) \text{ and } x_0=t \text{ is the time variable,}$$ on an interval $x_0\in [-T,T]$, $T<\infty$ arbitrary, which consist of the superposition of a non-oscillatory background solution and a single phase train of highly oscillatory waves of wave length $h\ll1$ and amplitudes given by powers of $h$; the waves interact with the nonlinearity and we measure the response $u(x_0,x')|_{x_0=T'}$ at a fixed time $x_0=T'<T$. We show that the coefficient of amplitude $h$ of the oscillatory part of the nonlinear geometric optics expansion of the solution determines the light-ray transform of a vector field associated with $q(x,u)$, which determines $q(x,u)$ uniquely in the maximal region determined by the data. Our methods also work for systems of semilinear wave equations satisfying null conditions, but in this paper we focus on the scalar case.
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https://arxiv.org/abs/2601.15997
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9cf7cf5ade8719615cf7170e32f4cf317d804c51f35ea0046fdd77d29fd48e56
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2026-01-23T00:00:00-05:00
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A brief note about p-curvature on graphs
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arXiv:2601.16010v1 Announce Type: new Abstract: In this paper, we consider Wang's $CD_p(m,K)$ condition on graphs, which depends on the $p$-Laplacian $\Delta_p$ for $p>1$ and is an extension of the classical Bakry-\'Emery $CD(m,K)$ curvature dimension condition. We calculate several examples including paths, cycles and star graphs, and we show that the $p$-curvature is non-negative at some vertices in the case $p\geq 2$, while it approaches to $-\infty$ in the case of $1 2$. As a consequence, an analogous proof that non-negative curvature is preserved under taking Cartesian products is not possible for $p > 2$.
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https://arxiv.org/abs/2601.16010
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5987d58e0732b1e72455fa3fbace49b76b875a670096398d59bbcdb255319093
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2026-01-23T00:00:00-05:00
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Infinite random graphs
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arXiv:2601.16013v1 Announce Type: new Abstract: We study countable graphs that -- up to isomorphism and with probability one -- arise from a random process, in a similar fashion as the Rado graph. Unlike in the classical case, we do not require that probabilities assigned to pairs of points are all equal. We give examples of such generalized random graphs, and show that the class of graphs under consideration has a two-element basis.
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https://arxiv.org/abs/2601.16013
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a5ac5064d27d250a3d5e1603a557451cc46e35e29f83e94e14e9e0a4651201d5
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2026-01-23T00:00:00-05:00
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The T-tensor of spherically symmetric Finsler metrics
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arXiv:2601.16021v1 Announce Type: new Abstract: This paper is devoted to the study of the T-tensor associated with a spherically symmetric Finsler metric $F=u\phi(r,s)$ on \(\mathbb{R}^n\). We derive a general expression for the T-tensor in terms of the scalar function \(\phi(r, s)\) and its partial derivatives. Furthermore, we characterize all spherically symmetric Finsler metrics satisfying the so-called T-condition, that is, those for which the T-tensor vanishes. In addition, we obtain the formula for the mean Cartan tensor and demonstrate that all spherically symmetric Finsler metrics of dimension $n \geq 3$, with a non-zero mean Cartan tensor are quasi-C-reducible.
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https://arxiv.org/abs/2601.16021
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8fa08d35db02d364f3771d148cb16d471d4c0a175a2d43e6de3b3ac44932fcda
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2026-01-23T00:00:00-05:00
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A Second-Order Dynamical System for Solving Generalized Inverse Mixed Variational Inequality problems
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arXiv:2601.16043v1 Announce Type: new Abstract: In this paper, we study a class of generalized inverse mixed variational inequality problems (GIMVIPs). We propose a novel projection-based second-order time-varying dynamical system for solving GIMVIPs. Under the assumptions that the underlying operators are strongly monotone and Lipschitz continuous, we establish the existence and uniqueness of solution trajectories and prove their global exponential convergence to the unique solution of the GIMVIP. Furthermore, a discrete-time realization of the continuous dynamical system is developed, resulting in an inertial projection algorithm. We show that the proposed algorithm achieves linear convergence under suitable choices of parameters. Finally, numerical experiments are presented to illustrate the effectiveness and convergence behavior of the proposed method in solving GIMVIPs.
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https://arxiv.org/abs/2601.16043
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a8ae53476f7cd75b1e4d5e3e626d72f5045bd5de213a99a39bac044f9f2ba0e6
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2026-01-23T00:00:00-05:00
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On the Identification of Elliptic Curves That Admit Infinitely Many Twists Satisfying the Birch-Swinnerton-Dyer Conjecture
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arXiv:2601.16044v1 Announce Type: new Abstract: Recent work of Burungale-Skinner-Tian-Wan established the first infinite families of quadratic twists of non-CM elliptic curves over $\mathbb{Q}$ for which the strong Birch-Swinnerton-Dyer (BSD) conjecture holds. Building on their results, we encode the required hypotheses into an explicit algorithm and apply it to the database of elliptic curves in the $L$-functions and Modular Forms Database (LMFDB), identifying all elliptic curves $E$ of conductor at most $500{,}000$ that admit infinitely many quadratic twists satisfying the strong BSD conjecture. Our computations provide certain numerical evidence for a conjecture of Radziwi{\l}{\l} and Soundararajan predicting Gaussian behavior in the analytic order of the Shafarevich-Tate group, while also observing a systematic positive bias within the BSD-satisfying subfamily.
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https://arxiv.org/abs/2601.16044
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201a1445d69e9a66b290b731facd3598a79be5e707c1ce82e90fe897bfd9c9f6
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2026-01-23T00:00:00-05:00
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Fujita exponents on quantum Euclidean spaces
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arXiv:2601.16053v1 Announce Type: new Abstract: We study the well-posedness of a non-linear heat equation with power nonlinearity with positive initial data on quantum Euclidean spaces. We prove a noncommutative analogue of the classical Fujita theorem by identifying the critical exponent separating finite-time blow-up from global existence for small initial data. Moreover, we establish a fundamental inequality in general semifinite von Neumann algebras that is of independent interest and plays a crucial role in the study of global existence and local well-posedness of solutions of nonlinear equations in noncommutative setting.
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https://arxiv.org/abs/2601.16053
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d5fc4bac5bb8e096dcb3294c4c4cfa3e3b5c6e4ebcb0ba043402eacdc6284744
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2026-01-23T00:00:00-05:00
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Fully Functional Weighted Testing for Abrupt and Gradual Location Changes in Functional Time Series
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arXiv:2601.16058v1 Announce Type: new Abstract: Change point tests for abrupt changes in the mean of functional data, i.e., random elements in infinite-dimensional Hilbert spaces, are either based on dimension reduction techniques, e.g., based on principal components, or directly based on a functional CUSUM (cumulative sum) statistic. The former have often been criticized as not being fully functional and losing too much information. On the other hand, unlike the latter, they take the covariance structure of the data into account by weighting the CUSUM statistics obtained after dimension reduction with the inverse covariance matrix. In this paper, as a middle ground between these two approaches, we propose an alternative statistic that includes the covariance structure with an offset parameter to produce a scale-invariant test procedure and to increase power when the change is not aligned with the first components. We obtain the asymptotic distribution under the null hypothesis for this new test statistic, allowing for time dependence of the data. Furthermore, we introduce versions of all three test statistics for gradual change situations, which have not been previously considered for functional data, and derive their limit distribution. Further results shed light on the asymptotic power behavior for all test statistics under various ground truths for the alternatives.
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https://arxiv.org/abs/2601.16058
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ffddf38c57428641a8086050b6041ce7b75a0792a61d7565d9fdc30286e4b900
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2026-01-23T00:00:00-05:00
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Bivariate topological complexity: a framework for coordinated motion planning
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arXiv:2601.16059v1 Announce Type: new Abstract: We introduce a bivariate version of topological complexity, $\mathrm{TC}(f,g)$, associated with two continuous maps $f\colon X\to Z$ and $g\colon Y\to Z$. This invariant measures the minimal number of continuous motion planning rules required to coordinate trajectories in $X$ and $Y$ through a shared target space $Z$. It recovers Farber's classical topological complexity when $f=g=\mathrm{id}_X$ and Pave\v{s}i\'c's map-based invariant when one of the maps is the identity. We develop a structural theory for $\mathrm{TC}(f,g)$, including symmetry, product inequalities, stability properties, and a collaboration principle showing that, when one of the maps is a fibration, the complexity of synchronization is controlled by the other. We also introduce a homotopy-invariant bivariate complexity $\mathrm{TC}_H(f,g)$ of Scott type, defined via homotopic distance, and study its relationship with the strict invariant. Concrete examples reveal rigidity phenomena with no analogue in the classical case, including strict gaps between $\mathrm{TC}(f,g)$ and $\mathrm{TC}_H(f,g)$ and situations where synchronization becomes impossible. Cohomological estimates provide computable obstructions in both the strict and homotopy-invariant settings.
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https://arxiv.org/abs/2601.16059
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f80d6250a40d9132f324a5c8bc5e1571d473dc16691d353d4405b3f860b7eee2
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2026-01-23T00:00:00-05:00
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Continuum limit of hypergraph $p$-Laplacian equations on point clouds
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arXiv:2601.16063v1 Announce Type: new Abstract: This paper studies a class of $p$-Laplacian equations on point clouds that arise from hypergraph learning in a semi-supervised setting. Under the assumption that the point clouds consist of independent random samples drawn from a bounded domain $\Omega\subset\mathbb{R}^d$, we investigate the asymptotic behavior of the solutions as the number of data points tends to infinity, with the number of labeled points remains fixed. We show, for any $p>d$ in the viscosity solution framework, that the continuum limit is a weighted $p$-Laplacian equation subject to mixed Dirichlet and Neumann boundary conditions. The result provides a new discretization of the $p$-Laplacian on point clouds.
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https://arxiv.org/abs/2601.16063
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a460fd36126db4619584c42971784b087577f9f529f32fa147e36c7aeef00a3c
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2026-01-23T00:00:00-05:00
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On the Stable Euclidean Distance Degree of Algebraic Layers
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arXiv:2601.16071v1 Announce Type: new Abstract: We study the projective geometry of algebraic neural layers, namely families of maps induced by a polynomial activation function, with particular emphasis on the generic Euclidean Distance degree ($\mathrm{gED}$). This invariant is projective in nature and measures the number of optimal approximations of a general point in the ambient space with respect to a general metric. For a fixed architecture (i.e. fixed width and activation polynomial), we prove that the $\mathrm{gED}$ is stably polynomial in the dimensions of the input and output spaces. Moreover, we show that this stable polynomial depends only on the degree of the activation function. Our approach relies on standard intersection theory on the Nash blow-up, which allows us to express the $\gED$ as an intersection number over products of Grassmannians. Stable polynomiality is deduced via equivariant localization, while the reduction to the monomial case follows from an explicit Schubert calculus computation on Grassmannians.
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https://arxiv.org/abs/2601.16071
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85f911d3d2ea4ff58e4691f36c989a00cef8713dd287623e719e981ea88f51cb
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2026-01-23T00:00:00-05:00
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The Hyperrigidity Conjecture for Spectrahedra
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arXiv:2601.16075v1 Announce Type: new Abstract: We show that if K is a compact spectrahedron whose set of extreme points is closed, then the operator system of continuous affine functions on K is hyperrigid in the C*-algebra C(ex(K)).
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https://arxiv.org/abs/2601.16075
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ba8594566d2d6e4dbcdaf8ef6613e6e7f2889c35db64373af56e6579f96601f8
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2026-01-23T00:00:00-05:00
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Rainbow spanning structures in strongly edge-colored graphs
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arXiv:2601.16084v1 Announce Type: new Abstract: An edge-colored graph is a graph in which each edge is assigned a color. Such a graph is called strongly edge-colored if each color class forms an induced matching, and called rainbow if all edges receive pairwise distinct colors. In this paper, by establishing a connection with $\mu n$-bounded graphs, we prove that for all sufficiently large integers $n$, every strongly edge-colored graph $G$ on $n$ vertices with minimum degree at least $\frac{n+1}{2}$ contains a rainbow Hamilton cycle. We also characterize all strongly edge-colored graphs on $n$ vertices with minimum degree exactly $\frac{n}{2}$ that do not contain a rainbow Hamilton cycle. As an application, we determine the optimal minimum degree conditions for the existence of rainbow Hamilton paths and rainbow perfect matchings in strongly edge-colored graphs. Together, these results verify three conjectures concerning strongly edge-colored graphs for sufficiently large $n$.
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https://arxiv.org/abs/2601.16084
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e30f7365705bf603ee9f319e78a716d990f095b7bfc5aa199cbbb2159fcb3181
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2026-01-23T00:00:00-05:00
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Birational automorphism groups in families of hyper-K\"ahler manifolds
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arXiv:2601.16090v1 Announce Type: new Abstract: We study the behavior of birational automorphism groups in families of projective hyper-K\"ahler manifolds.
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https://arxiv.org/abs/2601.16090
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d33f8a04836868bede8877f06407180bfe2c0425632e40761db892c5c51723aa
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2026-01-23T00:00:00-05:00
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Monoidal adjunctions and abelian envelopes
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arXiv:2601.16092v1 Announce Type: new Abstract: We show how monoidal adjunctions can be used to prove the existence of monoidal abelian envelopes of pseudo-tensor categories, in particular, those admitting a combinatorial description with certain properties. We derive concrete general criteria that we demonstrate by giving relatively simple combinatorial proofs of the existence of new abelian envelopes for interpolation categories of the hyperoctahedral and of the modified symmetric groups.
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https://arxiv.org/abs/2601.16092
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276b2fb57d1623b3bd3dea8477095bd3f39fc506769e5a9a2ca8e54839872120
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2026-01-23T00:00:00-05:00
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On Seshadri constants of adjoint divisors on surfaces and threefolds in arbitrary characteristic
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arXiv:2601.16094v1 Announce Type: new Abstract: We develop a new approach towards obtaining lower bounds of the Seshadri constants of ample adjoint divisors on smooth projective varieties $X$ in arbitrary characteristic. Let $x\in X$ be a closed point and $A$ an ample divisor on $X$. If $X$ is a surface, we recover some known lower bounds by proving, e.g., that $\varepsilon(K_X+4A;x)\geq 3/4$. If $X$ is a threefold, we prove that for all $\delta>0$ and all but finitely many curves $C$ through $x$, we have $\frac{(K_X+6A).C}{\operatorname{mult}_x C}\geq\frac{1}{2\sqrt{2}}-\delta$. In particular, if $\varepsilon(K_X+6A;x)<1/(2\sqrt{2})$, then $\varepsilon(K_X+6A;x)$ is a rational number, attained by a Seshadri curve $C$.
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https://arxiv.org/abs/2601.16094
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88acc2666284c3b7d5de4a1bec827c23e5123310dcc8227a7052adb9bb7d3cf6
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2026-01-23T00:00:00-05:00
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Intersections of Convex Hulls of Polynomial Shifts and Critical Points
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arXiv:2601.16102v1 Announce Type: new Abstract: Let $p(z)$ be a complex polynomial of degree $n\ge 2$. For each $c\in\mathbb{C}$, let $K_c$ denote the convex hull of the zeros of $p(z)+c$, and let $K'$ denote the convex hull of the zeros of $p'(z)$. We prove that $$\bigcap_{c\in\mathbb{C}} K_c = K',$$ by combining a strict separating hyperplane argument with a half-plane non-surjectivity theorem for polynomials without critical points (proved via analytic continuation, the monodromy theorem and Liouville's Theorem). We also characterize when $K_0=K'$ in terms of the multiplicities of the zeros of $p(z)$ that form the vertices of $K_0$. As an application, we obtain a partial result toward the Schmeisser's conjecture: if all zeros of $p$ lie in the closed unit disk, then for every $\zeta\in K'$ the disk $|z-\zeta|\le \sqrt{1-|\zeta|^2}$ contains a critical point of $p(z)$. Finally, we refine a recent barycentric bound in \cite{Zha26+} by showing that there is always a critical point within distance $\sqrt{\frac{n-2}{n-1}}\sqrt{1-|G|^2}$ of the centroid $G$ of the zeros.
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https://arxiv.org/abs/2601.16102
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f0c4097bd20390539fb412c982a0b348cf755ff1c88d1eb382fd91fb1721aa0a
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2026-01-23T00:00:00-05:00
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The Eisenbud-Goto conjecture for projectively normal varieties with mild singularities
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arXiv:2601.16103v1 Announce Type: new Abstract: For a nondegenerate projective variety $X$, the Eisenbud-Goto conjecture asserts that $\operatorname{reg}X\leq\operatorname{deg}X-\operatorname{codim}X+1$. Despite the existence of counterexamples, identifying the classes of varieties for which the conjecture holds remains a major open problem. In this paper, we prove that the Eisenbud-Goto conjecture holds for $2$-very ample projectively normal varieties with mild singularities.
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https://arxiv.org/abs/2601.16103
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b69b92ab2f0cb420902a7ab05e486e5ed862f44e42af0242091a68227ce4fb1b
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2026-01-23T00:00:00-05:00
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A Linear Bound on the Rich Flow Number for Graphs with a Given Maximum Degree
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arXiv:2601.16104v1 Announce Type: new Abstract: A rich $k$-flow is a nowhere-zero $k$-flow $\phi$ such that, for every pair of adjacent edges $e$ and $f$, $|\phi(e)| \neq |\phi(f)|$. A graph is rich flow admissible if it admits a rich $k$-flow for some integer $k$. In this paper, we prove that if $G$ is a rich flow admissible graph with maximum degree $\Delta$, then $G$ admits a rich $(264\Delta - 445)$-flow.
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https://arxiv.org/abs/2601.16104
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7f512ed8330aca7881c91ccfc66b321bcda121b86c970352916478d569af7bc6
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2026-01-23T00:00:00-05:00
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Stability and Decay for the 2D Anisotropic Navier-Stokes Equations with Fractional Horizontal Dissipation on $\mathbb{R}^2$
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arXiv:2601.16110v1 Announce Type: new Abstract: The stability problem for the 2D Navier-Stokes equations with dissipation in only one direction on $\mathbb R^2$ is not fully understood. This dissipation is in the intermediate regime between the fully dissipative Navier-Stokes and the inviscid Euler. Navier-Stokes solutions in $\mathbb R^2$ decay algebraically in time while Euler solutions can grow rather rapidly in time. This paper solves the fundamental stability and large-time behavior problem on the anisotropic Navier-Stokes with fractional dissipation $\Lambda_1^{2s}$ for all $0\leq s<\frac{11}{12}$, and $\frac{11}{12} \leq s <1$. The final range is the most difficult case, for which we introduce the spatial polynomial $A_2$ weights and exploit the boundedness of Riesz transforms on weighted $L^2$-spaces.
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https://arxiv.org/abs/2601.16110
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c26d663e749a5d7127a21f2a0f69c5fd9657008c413d98ce2e9e42379b7c123c
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2026-01-23T00:00:00-05:00
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Equivariant Morse-Bott cohomology through stabilization
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arXiv:2601.16119v1 Announce Type: new Abstract: For closed manifolds with compact Lie group actions, we study Austin-Braam's Morse-theoretic construction of Borel equivariant cohomology using the technique of stabilization. We show that a $C^1$-small equivariant perturbation produces stable invariant Morse-Bott functions. This allows us to realize the equivariant transversality and orientability assumptions in Austin-Braam's framework by choosing generic invariant Riemannian metrics.
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https://arxiv.org/abs/2601.16119
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36fbd7deeb2886797e060f988b64d8289838a6128015ad542a11a8b11d8351d7
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2026-01-23T00:00:00-05:00
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Proximity Operator of the $\ell_1$ over $\ell_2$ Function
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arXiv:2601.16128v1 Announce Type: new Abstract: We study the proximity operator of the nonconvex, scale-invariant ratio $h(\vx)=\|\vx\|_{1}/\|\vx\|_{2}$ and show it can be computed exactly in any dimension. By expressing $\vx=r\vu$ and exploiting sign and permutation invariance, we reduce the proximal step to a smooth optimization of a rank-one quadratic over the nonnegative orthant of the unit sphere. We prove that every proximal point arises from a finite candidate set indexed by $k\in\{1,\dots,n\}$: the active subvector is a local, but nonglobal, minimizer on $\mathbb{S}^{k-1}$ characterized by the roots of an explicit quartic. This yields closed-form candidates, an exact selection rule, and a necessary and sufficient existence test. Building on these characterizations, we develop practical algorithms, including an $O(n)$ implementation via prefix sums and a pruning criterion that avoids unnecessary quartic solves. The method returns all proximal points when the prox is non-unique, and in experiments it attains strictly lower objective values than approaches that guess sparsity or rely on sphere projections with limited scalability.
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https://arxiv.org/abs/2601.16128
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6f36073eb4e1711e74b5fb729335a367c7ace8e76af6bdde56da640db8113b6d
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2026-01-23T00:00:00-05:00
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A pseudo-bosonic Klein-Gordon field with finite two-points function
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arXiv:2601.16131v1 Announce Type: new Abstract: We introduce a class of pseudo-bosonic Klein-Gordon fields in 1+1 dimensions and we discuss some of their properties. This work originates from non Hermitian quantum mechanics and deformed canonical commutation relations. We show that, within this class of fields, there exist a specific subclass with the interesting feature of having finite equal space-time two-points function, contrarily to what happens for {\em standard} Klein-Gordon fields. This, in our opinion, is a relevant aspect of our proposal which is a good motivation to undertake a deeper analysis of this (and related) quantum fields.
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https://arxiv.org/abs/2601.16131
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c985ee6fcc3b8ae0b89cc4ab07f2ae07829920e064318a0a2d529c351a7f7f1c
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2026-01-23T00:00:00-05:00
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Modular Weil representation and compatibility of cuspidals with congruences
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arXiv:2601.16132v1 Announce Type: new Abstract: Let $F$ be a non-archimedean local field of characteristic different from $2$ and of residual characteristic $p$. We generalise the theory of the Weil representation over $F$ with complex coefficients to $\ell$-modular representations \textit{i.e.} when the complex coefficients are replaced by a coefficient field $R$ of characteristic $\ell \neq p$. We obtain along the way a generalisation of the Stone-von Neumann theorem to the $\ell$-modular setting, together with the Weil representation with coefficients in $R$ on the $R$-metaplectic group. Surprisingly enough, the latter $R$-metaplectic group happens to be split over the symplectic group if $\ell = 2$. The theory also makes sense when $F$ is a finite field of odd characteristic. We also establish the irreducibility of the theta lift in the cuspidal case as long as $\ell$ does not divide the pro-orders of the groups at stake and we provide a compatibility to congruences in this setting via an integral version of the theta lift.
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https://arxiv.org/abs/2601.16132
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8c3f9116408d95fb2f749c1c830131a14b112610980be54bb9aef5c3846cc389
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2026-01-23T00:00:00-05:00
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Pointwise Ergodic Averages Along the Omega Function in Number Fields
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arXiv:2601.16136v1 Announce Type: new Abstract: We show the failure of the pointwise convergence of averages along the Omega function in a number field. As a consequence, we show, for instance, that the averages \[ \frac{1}{N^2}\sum_{1\leq m,n \leq N} f(T^{\Omega(m^2+n^2)}x)\] do not converge pointwise in ergodic systems, addressing a question posed by Le, Moreira, Sun, and the second author. On the other hand, using number-theoretic methods, we establish the pointwise convergence of averages along the $\Omega$ function defined on the ideals of a number field in uniquely ergodic systems. Using this dynamical framework, we also derive several natural number-theoretic consequences of independent interest.
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https://arxiv.org/abs/2601.16136
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c356ef752d4ab2e75b4b7f7b654777c6f39609f4494f454a0dee8b8709abc287
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2026-01-23T00:00:00-05:00
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On the rationality of the Weil Representation and the local theta correspondence
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arXiv:2601.16141v1 Announce Type: new Abstract: We prove that the Weil representation over a non-archimedean local field can be realised with coefficients in a number field. We give an explicit descent argument to describe precisely which number field the Weil representation descends to. Our methods also apply over more general coefficient fields, such as $\ell$-modular coefficient fields, as well as coefficient rings such as rings of integers i.e. in families. We also prove that the theta correspondence over a perfect field is valid if and only if it is valid over the algebraic closure of this perfect field. These two results together show that the classical local theta correspondence is rational.
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https://arxiv.org/abs/2601.16141
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07396532e300a7928fcd30c71d2a65e79aedfe7545fd629d728e3ed6c64a2cd4
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2026-01-23T00:00:00-05:00
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High-Degree Polynomial Approximations for Solving Linear Integral, Integro-Differential, and Ordinary Differential Equations
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arXiv:2601.16143v1 Announce Type: new Abstract: This paper presents a universal numerical scheme tailored for tackling linear integral, integro-differential, and both initial and boundary value problems of ordinary differential equations. The numerical scheme is readily adapted for resolving ill-posed problems. Central to our approach is high-degree piecewise-polynomial approximation to the exact solution. We illustrate the accuracy and stability of our numerical solutions in the presence of noise through illustrative examples. Additionally, we demonstrate that proposed regularization being applied to high-degree interpolation, effectively eliminates Runge's phenomenon.
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https://arxiv.org/abs/2601.16143
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4da1973afc58462e5cfef0fc8cba1b86105010bb1235e32bebc236a7e1a6234c
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2026-01-23T00:00:00-05:00
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On the Ginzburg-Landau approximation for quasilinear pattern forming reaction-diffusion-advection systems
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arXiv:2601.16145v1 Announce Type: new Abstract: We prove that the Ginzburg-Landau equation correctly predicts the dynamics of quasilinear pattern-forming reaction-diffusion-advection systems, close to the first instability. We present a simple theorem which is easily applicable for such systems and relies on key maximal regularity results. The theorem is applied to the Gray-Scott-Klausmeier vegetation-water interaction model and its application to general reaction-diffusion-advection systems is discussed.
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https://arxiv.org/abs/2601.16145
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0128408acf80e9ebafc91e3623aec1f2004fa30e5dc9bba265e64ecb2e47e933
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2026-01-23T00:00:00-05:00
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On the structural properties of Lie algebras via associated labeled directed graphs
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arXiv:2601.16161v1 Announce Type: new Abstract: We present a method for associating labeled directed graphs to finite-dimensional Lie algebras, thereby enabling rapid identification of key structural algebraic features. To formalize this approach, we introduce the concept of graph-admissible Lie algebras and analyze properties of valid graphs given the antisymmetry property of the Lie bracket as well as the Jacobi identity. Based on these foundations, we develop graph-theoretic criteria for solvability, nilpotency, presence of ideals, simplicity, semisimplicity, and reductiveness of an algebra. Practical algorithms are provided for constructing such graphs and those associated with the lower central series and derived series via an iterative pruning procedure. This visual framework allows for an intuitive understanding of Lie algebraic structures that goes beyond purely visual advantages, since it enables a simpler and swifter grasping of the algebras of interest beyond computational-heavy approaches. Examples, which include the Schr\"odinger and Lorentz algebra, illustrate the applicability of these tools to physically relevant cases. We further explore applications in physics, where the method facilitates computation of similtude relations essential for determining quantum mechanical time evolution via the Lie algebraic factorization method. Extensions to graded Lie algebras and related conjectures are discussed. Our approach bridges algebraic and combinatorial perspectives, offering both theoretical insights and computational tools into this area of mathematical physics.
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https://arxiv.org/abs/2601.16161
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f13e7e94b81a7f1dae4427ab768688dba6fb7ff886fb388ad323476774dd3c0a
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2026-01-23T00:00:00-05:00
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Maximal toroids and Cartan subgroups of algebraic groups
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arXiv:2601.16162v1 Announce Type: new Abstract: We introduce a unified theory of Cartan subgroups and maximal toroids - defined as connected multiplicative type subgroups that are maximal amongst all such subgroups - which holds for all affine algebraic groups over a field, regardless of smoothness. For instance we show that maximal toroids always exist, that they are invariant under base change, and that they are in natural 1-1 correspondence with Cartan subgroups. Our results generalise known results for Cartan subgroups and maximal tori of smooth affine algebraic groups, as well as their analogues for restricted Lie algebras. We conclude with some applications to, and a brief discussion of, some generation problems for algebraic groups.
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https://arxiv.org/abs/2601.16162
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bae5025882d1026f54402ec931e64ca0bc95248a8300050558d6420bdb8aff2f
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2026-01-23T00:00:00-05:00
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On the dimension drop for harmonic measure on uniformly non-flat Ahlfors-David regular boundaries
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arXiv:2601.16167v1 Announce Type: new Abstract: We extend earlier results of Azzam on the dimension drop of the harmonic measure for a domain $\Omega\subset \R^{n}$ with $n\geq 3$, with dimensional Ahlfors regular boundary $\partial\Omega$ of dimension $s$ with $n-1-\delta_0 \leq s\leq n-1$, that is uniformly non flat. Here $\delta_0$ is a small positive constant dependent on the parameters of the problem. Our novel construction relies on elementary geometric and potential theoretic considerations. We avoid the use of Riesz transforms and compactness arguments, and also give quantitative bounds on the $\delta_0$ parameter.
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https://arxiv.org/abs/2601.16167
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864fb2cf012ff3700d6a115470b40108760c29a9b24ffea7c9712f3a868bab4c
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2026-01-23T00:00:00-05:00
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Fixed-point proportion of geometric iterated Galois groups
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arXiv:2601.16173v1 Announce Type: new Abstract: In 1980, Odoni initiated the study of the fixed-point proportion of iterated Galois groups of polynomials motivated by prime density problems in arithmetic dynamics. The main goal of the present paper is to completely settle the longstanding open problem of computing the fixed-point proportion of geometric iterated Galois groups of polynomials. Indeed, we confirm the well-known conjecture that Chebyshev polynomials are the only complex polynomials whose geometric iterated Galois groups have positive fixed-point proportion. Our proof relies on methods from group theory, ergodic theory, martingale theory and complex dynamics. This result has direct applications to the proportion of periodic points of polynomials over finite fields. The general framework developed in this paper applies more generally to rational functions over arbitrary fields and generalizes, via a unified approach, previous partial results, which have all been proved with very different methods.
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https://arxiv.org/abs/2601.16173
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0516cb5d9184d66d7cdbbd0f0e83998ef8522aa80eacb9c4d785ae581844a32a
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2026-01-23T00:00:00-05:00
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Metric-uniform spectral inequality for the Laplacian on manifolds with bounded sectional curvature
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arXiv:2601.16176v1 Announce Type: new Abstract: Given a Riemannian manifold $M$ endowed with a smooth metric $g$ satisfying upper and lower sectional curvature bounds, we show an equivalence property between the $\mathrm{L}^2$ norm on $M$ and the $\mathrm{L}^2$ norm on subsets $\omega$ satisfying a thickness condition, for functions in the range of a spectral projector. The thickness condition is known to be optimal in this setting. The constant appearing in the equivalence of norms property depends only on the dimension of the manifold, curvature bounds, and frequency threshold of the spectral cutoff, but, crucially, not on the injectivity radius.
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https://arxiv.org/abs/2601.16176
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81b76d043e541d3b9c27b13f2db21193e88680dd96e95cb69d5fc87540135b32
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2026-01-23T00:00:00-05:00
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Mild Solutions for Path-Dependent Parabolic PDEs with Neumann Boundary Conditions via Generalized BSDEs
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arXiv:2601.16178v1 Announce Type: new Abstract: We study a system of Forward-Backward Stochastic Differential Equations (FBSDEs) with time-delayed generators. The forward process includes a reflection component expressed via a Stieltjes integral, while the backward process takes the form of a Generalized BSDE. We establish the connection between this FBSDE system and non-linear path-dependent PDEs with Neumann boundary conditions by deriving a representation formula.
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https://arxiv.org/abs/2601.16178
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2fef10926f266295d9a1e0cc05d21a0d9a21a307553fe518ccf1518ffe97366c
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2026-01-23T00:00:00-05:00
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Gaussian maps on trigonal curves
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arXiv:2601.16183v1 Announce Type: new Abstract: In this paper we study higher even Gaussian maps of the canonical bundle for cyclic trigonal curves. More precisely, we study suitable restrictions of these maps determining a lower bound for the rank, and more generally, a lower bound for the rank for the general trigonal curve. We also manage to give the explicit description of the kernel of the second Gaussian map. Finally, we use these results to show the non existence of "extra" asymptotic directions for cyclic trigonal curves in some spaces generated by higher Schiffer variations.
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https://arxiv.org/abs/2601.16183
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cfa9c3919aae4ec147c711a20883f4df1f9bb7f6b4f23b7464290fa7cc6478f4
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2026-01-23T00:00:00-05:00
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The Pohozaev identity for the Spectral Fractional Laplacian
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arXiv:2601.16185v1 Announce Type: new Abstract: In this paper, we prove a Pohozaev identity for the Spectral Fractional Laplacian (SFL). This identity allows us to establish non-existence results for the semilinear Dirichlet problem $(-\Delta|_{\Omega})^su = f(u)$ in star-shaped domains. The first such identity for non-local operators was established by Ros-Oton and Serra in 2014 for the Restricted Fractional Laplacian (RFL). However, the SFL differs fundamentally from the RFL, and the integration by parts strategy of Ros-Oton and Serra cannot be applied. Instead, we develop a novel spectral approach that exploits the underlying quadratic structure. Our main result expresses the identity as a Schur product of the classical Pohozaev quadratic form and a transition matrix that depends on the eigenvalues of the Laplacian and the fractional exponent.
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https://arxiv.org/abs/2601.16185
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8f899e1d2cf8f2ecd07a0bfcabcb41df2596cf9896269d74ea641ccb4578e973
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2026-01-23T00:00:00-05:00
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Inversion problem in algebras of integrable functions with summable Fourier transforms
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arXiv:2601.16186v1 Announce Type: new Abstract: In this paper, we study the norm-controlled inversion problem in two classes of algebras of integrable functions. In contrast of the classical case of $L^{1}(G)$, we prove that this problem has a positive solution in our setting without any additional restrictions.
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https://arxiv.org/abs/2601.16186
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9d51c102180f2ce37aa5416ef593330335f195b2e2f53064ffa770f629aee20d
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2026-01-23T00:00:00-05:00
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Ergodic averages for commutative transformations along return times
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arXiv:2601.16188v1 Announce Type: new Abstract: In this paper, we extend recent results on the convergence of ergodic averages along sequences generated by return times to shrinking targets in rapidly mixing systems, partially answering questions posed by the first author, Maass and the third author in [6]. In particular, for a fixed parameter $a\in (0,1)$ and for generic $y\in [0,1]$, we establish both $L^2$ and pointwise convergence for single averages and multiple averages for commuting transformations along the sequences $(a_n(y))_{n\in \mathbb{N}}$, obtained by arranging the set $$\Big\{n\in\mathbb{N}: 0<2^ny \mod{1}<n^{-a} \Big\}$$ in an increasing order. We also obtain new results for semi-random ergodic averages along sequences of similar type.
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https://arxiv.org/abs/2601.16188
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61db48fb4e44dcc49de67016038a0de3c2ab761996471b9ad38f4cf876421e68
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2026-01-23T00:00:00-05:00
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Density-based structural frameworks for prime numbers, prime gaps, and Euler products
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arXiv:2601.16193v1 Announce Type: new Abstract: We develop a unified density-based framework for primality, coprimality, and prime pairs, and introduce an intrinsic normalized model for prime gaps constrained by the Prime Number Theorem. Within this setting, a structural tension between Hardy-Littlewood, Cramer, and PNT predictions emerges, leading to quantitative estimates on the rarity of extreme gaps. Additive representations of even integers are reformulated as local density problems, yielding non-conjectural upper and lower bounds compatible with Hardy-Littlewood heuristics. Finally, the Riemann zeta function is analyzed via truncated Euler products, whose stability and oscillatory structure provide a coherent interpretation of the critical line and prime-based numerical criteria for the localization of non-trivial zeros.
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https://arxiv.org/abs/2601.16193
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402b26da03c3a5e76199771adeb9fd3e126e1b766566f1ec80843b22bb0433e8
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2026-01-23T00:00:00-05:00
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Generalized Bassian Modules over Non-primitive Dedekind Prime Rings
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arXiv:2601.16201v1 Announce Type: new Abstract: A right $A$-module $M$ is said to be generalized bassian if the existence of an injective homomorphism $M\to M/N$ for some submodule $N$ of $M$ implies that $N$ is a direct summand of $M$. We describe singular generalized bassian modules over non-primitive Dedekind prime rings.\\ The study is supported by grant of Russian Science Foundation.
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https://arxiv.org/abs/2601.16201
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2ecb58b5c79baacfdb3dcba3cfe5bb5b99a7e55ba2aaddac03ffc28b82c24afd
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2026-01-23T00:00:00-05:00
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Pairwise Beats All-at-Once: Behavioral Gains from Sequential Choice Presentation
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arXiv:2601.15332v1 Announce Type: cross Abstract: This paper presents the Sequential Rationality Hypothesis, which argues that consumers are better able to make utility-maximizing decisions when products appear in sequential pairwise comparisons rather than in simultaneous multi-option displays. Although this involves higher cognitive costs than the all-at-once format, the current digital market, with its diverse products listed by review ratings, pricing, and paid products, often creates inconsistent choices. The present work shows that preparing the list sequentially supports more rational choice, as the consumer tries to minimize cognitive costs and may otherwise make an irrational decision. If the decision remains the same on both offers, then that is a consistent preference. The platform uses this approach by reducing cognitive costs while still providing the list in an all-at-once format rather than sequentially. To show how sequential exposure reduces cognitive overload and prevents context-dependent errors, we develop a bounded attention model and extend the monotonic attention rule of the random attention model to theorize the sequential rational hypothesis. Using a theoretical design with common consumer goods, we test these hypotheses. This theoretical model helps policymakers in digital market laws, behavioral economics, marketing, and digital platform design consider how choice architectures may improve consumer choices and encourage rational decision-making.
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https://arxiv.org/abs/2601.15332
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2469d3b6bd178139d82fcface8205cbfdb417914cf6ca37777a57de65fcdba48
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2026-01-23T00:00:00-05:00
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Exactly solvable topological phase transition in a quantum dimer model
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arXiv:2601.15377v1 Announce Type: cross Abstract: We introduce a family of generalized Rokhsar-Kivelson (RK) Hamiltonians, which are reverse-engineered to have an arbitrary edge-weighted superposition of dimer coverings as their exact ground state at the RK point. We then focus on a quantum dimer model on the triangular lattice, with doubly-periodic edge weights. For simplicity we consider a $2\times1$ periodic model in which all weights are set to one except for a tunable horizontal edge weight labeled $\alpha$. We analytically show that the model exhibits a continuous quantum phase transition at $\alpha=3$, changing from a topological $\mathbb{Z}_2$ quantum spin liquid ($\alpha3$). The dimer-dimer correlator decays exponentially on both sides of the transition with the correlation length $\xi\propto1/|\alpha-3|$ and as a power-law at criticality. The vison correlator exhibits an exponential decay in the spin liquid phase, but becomes a constant in the ordered phase. We explain the constant vison correlator in terms of loops statistics of the double-dimer model. Using finite-size scaling of the vison correlator, we extract critical exponents consistent with the 2D Ising universality class.
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https://arxiv.org/abs/2601.15377
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33ffefc75929e61fc1a5a95568d457fd32b953acd5923f50a3fe6e4267165de2
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2026-01-23T00:00:00-05:00
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Quadratic tensors as a unification of Clifford, Gaussian, and free-fermion physics
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arXiv:2601.15396v1 Announce Type: cross Abstract: Certain families of quantum mechanical models can be described and solved efficiently on a classical computer, including qubit or qudit Clifford circuits and stabilizer codes, free-boson or free-fermion models, and certain rotor and GKP codes. We show that all of these families can be described as instances of the same algebraic structure, namely quadratic functions over abelian groups, or more generally over (super) Hopf algebras. Different kinds of degrees of freedom correspond to different "elementary" abelian groups or Hopf algebras: $\mathbb{Z}_2$ for qubits, $\mathbb{Z}_d$ for qudits, $\mathbb{R}$ for continuous variables, both $\mathbb{Z}$ and $\mathbb{R}/\mathbb{Z}$ for rotors, and a super Hopf algebra $\mathcal F$ for fermionic modes. Objects such as states, operators, superoperators, or projection-operator valued measures, etc, are tensors. For the solvable models above, these tensors are quadratic tensors based on quadratic functions. Quadratic tensors with $n$ degrees of freedom are fully specified by only $O(n^2)$ coefficients. Tensor networks of quadratic tensors can be contracted efficiently on the level of these coefficients, using an operation reminiscent of the Schur complement. Our formalism naturally includes models with mixed degrees of freedom, such as qudits of different dimensions. We also use quadratic functions to define generalized stabilizer codes and Clifford gates for arbitrary abelian groups. Finally, we give a generalization from quadratic (or 2nd order) to $i$th order tensors, which are specified by $O(n^i)$ coefficients but cannot be contracted efficiently in general.
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https://arxiv.org/abs/2601.15396
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d881d565eea8a9b6d8d9fb6db6124b6a415812aa328c6d401215f4a9cae9563b
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2026-01-23T00:00:00-05:00
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Dynamic Mean Field Theories for Nonlinear Noise in Recurrent Neuronal Networks
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arXiv:2601.15462v1 Announce Type: cross Abstract: Strong, correlated noise in recurrent neural circuits often passes through nonlinear transfer functions, complicating dynamical mean-field analyses of complex phenomena such as transients and bifurcations. We introduce a method that replaces nonlinear functions of Ornstein-Uhlenbeck (OU) noise with a Gaussian-equivalent process matched in mean and covariance, and combine this with a lognormal moment closure for expansive nonlinearities to derive a closed dynamical mean-field theory for recurrent neuronal networks. The resulting theory captures order-one transients, fixed points, and noise-induced shifts of bifurcation structure, and outperforms standard linearization-based approximations in the strong-fluctuation regime. More broadly, the approach applies whenever dynamics depend smoothly on OU processes via nonlinear transformations, offering a tractable route to noise-dependent phase diagrams in computational neuroscience models.
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https://arxiv.org/abs/2601.15462
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86255efdff01faf091d8cfcd31bc9635a4832e1379fe273d8a1f5af7bacd8da7
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2026-01-23T00:00:00-05:00
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A Modified Center-of-Mass Conservation Law in Finite-Domain Simulations of the Zakharov--Kuznetsov Equation
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arXiv:2601.15573v1 Announce Type: cross Abstract: We investigate conservation laws of the two-dimensional Zakharov-Kuznetsov (ZK) equation, a natural higher-dimensional and non-integrable extension of the Korteweg--de Vries equation. The ZK equation admits three scalar conserved quantities -- mass, momentum, and energy -- represented as $I_1$, $I_2$, and $I_3$, as well as a vector-valued quantity $\bm{I}_4$. In high-accuracy numerical simulations on a finite double-periodic domain, most of these quantities are well preserved, while a systematic temporal drift is observed only in the $x$-component $I_{4x}$. We show that the nontrivial evolution of $I_{4x}$ originates from an explicit boundary-flux contribution, which is induced by fluctuations of the solution and its spatial derivatives at the domain boundaries. We successfully identify the source of the inaccuracy in the numerical solutions. Motivated by this analysis, we define a modified center-of-mass quantity $I_{4x}^{\mathrm{mod}}$ and demonstrate its conservation numerically for single-pulse configurations. The modified quantity thus provides a consistent conservation law for the ZK equation and yields an appropriate description of center-of-mass motion in finite-domain numerical simulations.
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https://arxiv.org/abs/2601.15573
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2659ecb296fc92aaddd9e6b655cc63785577a42997094e95f536e5b0cb9e228d
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2026-01-23T00:00:00-05:00
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A two-sample pseudo-observation-based regression approach for the relative treatment effect
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arXiv:2601.15880v1 Announce Type: cross Abstract: The relative treatment effect is an effect measure for the order of two sample-specific outcome variables. It has the interpretation of a probability and also a connection to the area under the ROC curve. In the literature it has been considered for both ordinal or right-censored time-to-event outcomes. For both cases, the present paper introduces a distribution-free regression model that relates the relative treatment effect to a linear combination of covariates. To fit the model, we develop a pseudo-observation-based procedure yielding consistent and asymptotically normal coefficient estimates. In addition, we propose bootstrap-based hypothesis tests to infer the effects of the covariates on the relative treatment effect. A simulation study compares the novel method to Cox regression, demonstrating that the proposed hypothesis tests have high power and keep up with the z-test of the Cox model even in scenarios where the latter is specified correctly. The new methods are used to re-analyze data from the SUCCESS-A trial for progression-free survival of breast cancer patients.
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https://arxiv.org/abs/2601.15880
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337aba6d2a24fdfc3c2be941c9774a8c7db7ca462ae119ca7e14a39d24c56070
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2026-01-23T00:00:00-05:00
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On the spherical cardioid distribution and its goodness-of-fit
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arXiv:2601.16095v1 Announce Type: cross Abstract: In this paper, we study the spherical cardioid distribution, a higher-dimensional and higher-order generalization of the circular cardioid distribution. This distribution is rotationally symmetric and generates unimodal, multimodal, axial, and girdle-like densities. We show several characteristics of the spherical cardioid that make it highly tractable: simple density evaluation, closedness under convolution, explicit expressions for vectorized moments, and efficient simulation. The moments of the spherical cardioid up to a given order coincide with those of the uniform distribution on the sphere, highlighting its closeness to the latter. We derive estimators by the method of moments and maximum likelihood, their asymptotic distributions, and their asymptotic relative efficiencies. We give the machinery for a bootstrap goodness-of-fit test based on the projected-ecdf approach, including the projected distribution and closed-form expressions for test statistics. An application to modeling the orbits of long-period comets shows the usefulness of the spherical cardioid distribution in real data analyses.
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https://arxiv.org/abs/2601.16095
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218169404e549b70876df658c39d0e407ff10294153a5b15ec57b099361144b7
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2026-01-23T00:00:00-05:00
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Exceptional points in Gaussian channels: diffusion gauging and drift-governed spectrum
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arXiv:2601.16121v1 Announce Type: cross Abstract: McDonald and Clerk [Phys.\ Rev.\ Research 5, 033107 (2023)] showed that for linear open quantum systems the Liouvillian spectrum is independent of the noise strength. We first make this noise-independence principle precise in continuous time for multimode bosonic Gaussian Markov semigroups: for Hurwitz drift, a time-independent Gaussian similarity fixed by the Lyapunov equation gauges away diffusion for all times, so eigenvalues and non-diagonalizability are controlled entirely by the drift, while diffusion determines steady states and the structure of eigenoperators. We then extend the same separation to discrete time for general stable multimode bosonic Gaussian channels: for any stable Gaussian channel, we construct an explicit Gaussian similarity transformation that gauges away diffusion at the level of the channel parametrization. We illustrate the method with a single-mode squeezed-reservoir Lindbladian and with a non-Markovian family of single-mode Gaussian channels, where the exceptional-point manifolds and the associated gauging covariances can be obtained analytically.
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https://arxiv.org/abs/2601.16121
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927869583f65ffc05962dccb77f983018eabf9a47f75c51acfce19bb612d8688
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2026-01-23T00:00:00-05:00
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Gauge Theory and Skein Modules
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arXiv:2601.16213v1 Announce Type: cross Abstract: We study skein modules of 3-manifolds by embedding them into the Hilbert spaces of 4d ${\cal N}=4$ super-Yang-Mills theories. When the 3-manifold has reduced holonomy, we present an algorithm to determine the dimension and the list of generators of the skein module with a general gauge group. The analysis uses a deformation preserving ${\cal N}=1$ supersymmetry to express the dimension as a sum over nilpotent orbits. We find that the dimensions often differ between Langlands-dual pairs beyond the A-series, for which we provide a physical explanation involving chiral symmetry breaking and 't Hooft operators. We also relate our results to the structure of $\mathbb{C}^*$-fixed loci in the moduli space of Higgs bundles. This approach helps to clarify the relation between the gauge-theoretic framework of Kapustin and Witten with other versions of the geometric Langlands program, explains why the dimensions of skein modules do not exhibit a TQFT-like behavior, and provides a physical interpretation of the skein-valued curve counting of Ekholm and Shende.
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https://arxiv.org/abs/2601.16213
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60d5071fea2a37856e35db0f30db1fb95c7d8db0bb6d8811f8a624b492ecae3d
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2026-01-23T00:00:00-05:00
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Deterministic Structures in the Stopping Time Dynamics of the 3x+1 Problem
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arXiv:1709.03385v5 Announce Type: replace Abstract: The $3x+1$ problem concerns the iteration of the map $T:\mathbb{Z}\to\mathbb{Z}$ defined by $T(x)=x/2$ for even $x$ and $T(x)=(3x+1)/2$ for odd $x$. This paper investigates the stopping time dynamics associated with $T$ within a deterministic and algebraic framework. By relating the parity vectors of Collatz trajectories to exponential Diophantine equations, we construct a recursively generated tree of congruence classes $\bmod\, 2^{\sigma_n}$ that characterizes the stopping time classes $\sigma(x)=\sigma_n$. We demonstrate that the generation of these classes follows an explicit deterministic recursion and derive arithmetic transition rules between neighboring congruence classes, based on the differences of the associated Diophantine sums. Finally, we prove that the union of stopping time congruence classes generated up to a fixed order $N$ is periodic, establishing a computable finite-range coverage bound.
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https://arxiv.org/abs/1709.03385
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3dc871e473d2291df9001be7d603440d780313819621f147dc01b9150559ed47
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2026-01-23T00:00:00-05:00
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Unbounded field operators in categorical extensions of conformal nets
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arXiv:2001.03095v5 Announce Type: replace Abstract: We prove the equivalence of VOA tensor categories and conformal net tensor categories for the following examples: all WZW models; all lattice VOAs; all unitary parafermion VOAs; type $ADE$ discrete series $W$-algebras; their tensor products; their regular cosets. A new proof of the complete rationality of conformal nets is also given.
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https://arxiv.org/abs/2001.03095
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89a110910d6a112ad9d127b799349d1f8741cce2b92f10fd10ea0dafb7c8b270
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2026-01-23T00:00:00-05:00
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On rationality for $C_2$-cofinite vertex operator algebras
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arXiv:2108.01898v3 Announce Type: replace Abstract: Let $V$ be an $\mathbb{N}$-graded, simple, self-contragredient, $C_2$-cofinite vertex operator algebra. We show that if the $S$-transformation of the character of $V$ is a linear combination of characters of $V$-modules, then the category $\mathcal{C}$ of grading-restricted generalized $V$-modules is a rigid tensor category. We further show, without any assumption on the character of $V$ but assuming that $\mathcal{C}$ is rigid, that $\mathcal{C}$ is a factorizable finite ribbon category, that is, a not-necessarily-semisimple modular tensor category. As a consequence, we show that if the Zhu algebra of $V$ is semisimple, then $\mathcal{C}$ is semisimple and thus $V$ is rational. The proofs of these theorems use techniques and results from tensor categories together with the method of Moore-Seiberg and Huang for deriving identities of two-point genus-one correlation functions associated to $V$. We give two main applications. First, we prove the conjecture of Kac-Wakimoto and Arakawa that $C_2$-cofinite affine $W$-algebras obtained via quantum Drinfeld-Sokolov reduction of admissible-level affine vertex algebras are strongly rational. The proof uses the recent result of Arakawa and van Ekeren that such $W$-algebras have semisimple (Ramond twisted) Zhu algebras. Second, we use our rigidity results to reduce the "coset rationality problem" to the problem of $C_2$-cofiniteness for the coset. That is, given a vertex operator algebra inclusion $U\otimes V\hookrightarrow A$ with $A$, $U$ strongly rational and $U$, $V$ a pair of mutual commutant subalgebras in $A$, we show that $V$ is also strongly rational provided it is $C_2$-cofinite.
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https://arxiv.org/abs/2108.01898
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d55a3a590c969669e514fa127ff5f2da308103afdd85388177513e790b58ba2d
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2026-01-23T00:00:00-05:00
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Parameterising the effect of a continuous treatment using average derivative effects
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arXiv:2109.13124v2 Announce Type: replace Abstract: The average treatment effect (ATE) is commonly used to quantify the main effect of a binary treatment on an outcome. Extensions to continuous treatments are usually based on the dose-response curve or shift interventions, but both require strong overlap conditions and the resulting curves may be difficult to summarise. We focus instead on average derivative effects (ADEs) that are scalar estimands related to infinitesimal shift interventions requiring only local overlap assumptions. ADEs, however, are rarely used in practice because their estimation usually requires estimating conditional density functions. By characterising the Riesz representers of weighted ADEs, we propose a new class of estimands that provides a unified view of weighted ADEs/ATEs when the treatment is continuous/binary. We derive the estimand in our class that minimises the nonparametric efficiency bound, thereby extending optimal weighting results from the binary treatment literature to the continuous setting. We develop efficient estimators for two weighted ADEs that avoid density estimation and are amenable to modern machine learning methods, which we evaluate in simulations and an applied analysis of Warfarin dosage effects.
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https://arxiv.org/abs/2109.13124
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dadcfc43c1b28e1f866dda026c6de379945dbcd83870a5449d38da3a4789a3e5
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2026-01-23T00:00:00-05:00
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Uni-width subgroups, universal elements, and lambda number of finite groups
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arXiv:2202.09818v2 Announce Type: replace Abstract: A cyclic subgroup $N$ of a finite group $G$ is called a uni-width subgroup of $G$ if $N$ is the unique cyclic subgroup of $G$ of order $|N|$. In this article, we prove that a finite group $G$ admits a unique largest uni-width subgroup denoted by $U(1;G)$. We then show that the prime factors of the order of $U(1;G)$ influence the structure decomposition of its Fitting subgroup ${\mathrm{Fit}}(G)$. A power graph $\Gamma_G$ of a finite group is defined by $G$ being its set of vertices, and a pair of distinct elements $x,y \in G$ are connected by an edge if either $x \in \langle y \rangle$ or $y \in \langle x \rangle$. A universal element of a graph is a vertex that is adjacent to each of the remaining vertices. Our following result shows that a power graph $\Gamma_G$ of a finite non-trivial group admits a non-identity universal element if and only if it is either cyclic or a generalized quaternion $2$-group. The lambda number $\lambda(G)$ of a finite group $G$ is a measure of the least number of colors required for an $L(2,1)$-type of vertex coloring on $\Gamma_G$, which is known to be $\geq |G|$. Generalizing an earlier result, we then derive a necessary condition on a finite group $G$ such that $\lambda(G) = |G|$. Finally, we show that this result is best possible by exhibiting a family of groups without the necessary condition for which $\lambda(G) > |G|$.
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https://arxiv.org/abs/2202.09818
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4807e2a6ec07a7349492edb4d62a6d829276b757fa7e12507b65c234c3b93d8b
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2026-01-23T00:00:00-05:00
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Proper harmonic embeddings of open Riemann surfaces into $\mathbb{R}^4$
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arXiv:2206.03566v2 Announce Type: replace Abstract: We prove that every open Riemann surface admits a proper embedding into $\mathbb{R}^4$ by harmonic functions. This reduces by one the previously known embedding dimension in this framework, dating back to a theorem by Greene and Wu from 1975.
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https://arxiv.org/abs/2206.03566
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1eae3fb23ecce039ee422972c82edd88730e572bae0213e20758611057d76f4d
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2026-01-23T00:00:00-05:00
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On the additivity of Newton-Okounkov bodies
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arXiv:2207.09229v3 Announce Type: replace Abstract: We study the additivity of Newton-Okounkov bodies. Our main result states that on two-dimensional subcones of the ample cone the Newto-Okounkov body associated to an appropriate flag acts additively. We prove this by induction relying on the slice formula for Newton-Okounkov bodies. Moreover, we discuss a necessary condition for the additivity showing that our result is optimal in general situations. As an application, we deduce an inequality between intersection numbers of nef line bundles.
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https://arxiv.org/abs/2207.09229
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0866786a232eecf7dfc5102ca21febb97989b4af192356d03b86bdc96a35fd9f
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2026-01-23T00:00:00-05:00
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Mixing times of a Burnside process Markov chain on set partitions
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arXiv:2207.14269v3 Announce Type: replace Abstract: Let $X$ be a finite set and let $G$ be a finite group acting on $X$. The group action splits $X$ into disjoint orbits. The Burnside process is a Markov chain on $X$ which has a uniform stationary distribution when the chain is lumped to orbits. We consider the case where $X = [k]^n$ with $k \geq n$ and $G = S_k$ is the symmetric group on $[k]$, such that $G$ acts on $X$ by permuting the value of each coordinate. The resulting Burnside process gives a novel algorithm for sampling a set partition of $[n]$ uniformly at random. We obtain bounds on the mixing time and show that the chain is rapidly mixing. For the case $k < n$, the algorithm corresponds to sampling a set partition of $[n]$ with at most $k$ blocks, and we obtain a mixing time bound which is independent of $n$. Along the way, we obtain explicit formulas for the transition probabilities and bounds on the second largest eigenvalue for both the original process and the lumped chain.
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https://arxiv.org/abs/2207.14269
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d4819a338e4b1377efe9873b22b5e8fd5db0421d8c4d519cf87c8bd7560aa72c
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2026-01-23T00:00:00-05:00
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Conformal and extrinsic upper bounds for the harmonic mean of Neumann and Steklov eigenvalues
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arXiv:2208.13959v5 Announce Type: replace Abstract: Let $M$ be an $m$-dimensional compact Riemannian manifold with boundary. We obtain the upper bound of the harmonic mean of the first $m$ nonzero Neumann eigenvalues and Steklov eigenvalues involving the conformal volume and relative conformal volume, respectively. We also give an optimal sharp extrinsic upper bound for closed submanifolds in space forms. These extend the previous related results for the first nonzero eigenvalues.
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https://arxiv.org/abs/2208.13959
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1a49c758fc6713947825940f3b78a619fd5386932b63d2d5ff30f42121bd6c8e
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2026-01-23T00:00:00-05:00
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An analogue of Bonami's Lemma for functions on spaces of linear maps, and 2-2 Games
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arXiv:2209.04243v2 Announce Type: replace Abstract: We prove an analogue of Bonami's (hypercontractive) lemma for complex-valued functions on $\mathcal{L}(V,W)$, where $V$ and $W$ are vector spaces over a finite field. This inequality is useful for functions on $\mathcal{L}(V,W)$ whose `generalised influences' are small, in an appropriate sense. It leads to a significant shortening of the proof of a recent seminal result by Khot, Minzer and Safra that pseudorandom sets in Grassmann graphs have near-perfect expansion, which (in combination with the work of Dinur, Khot, Kindler, Minzer and Safra) implies the 2-2 Games conjecture (the variant, that is, with imperfect completeness).
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https://arxiv.org/abs/2209.04243
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7c1dc9f9ceebd3b4e470be9b513a05b88514699199545f34fa02081351ac6c59
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2026-01-23T00:00:00-05:00
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Fibrantly-transferred model structures
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arXiv:2301.07801v2 Announce Type: replace Abstract: We develop new techniques for constructing model structures from a given class of cofibrations, together with a class of fibrant objects and a choice of weak equivalences between them. As a special case, we obtain a more flexible version of the classical right-transfer theorem in the presence of an adjunction. Namely, instead of lifting the classes of fibrations and weak equivalences through the right adjoint, we now only do so between fibrant objects, which allows for a wider class of applications.
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https://arxiv.org/abs/2301.07801
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91307ac24559888f1a3408ac24bc69711238e8d99bf3f7bc1b6d535b26244b84
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2026-01-23T00:00:00-05:00
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Determinantally equivalent nonzero functions
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arXiv:2302.02471v4 Announce Type: replace Abstract: We study the problem raised in [Marco Stevens, Equivalent symmetric kernels of determinantal point processes, RMTA, 10(03):2150027, 2021] concerning the extension of its main result to the more general (potentially non-symmetric) setting. We construct a counterexample disproving the conjecture proposed in the paper, and subsequently solve it under some additional minor assumptions that preclude such counterexamples. The problem is plainly stated as follows: Let $\Lambda$ be a set and $\mathbb{F}$ a field, and suppose that $K,Q:\Lambda^2\to\mathbb{F}$ are two functions such that for any $n\in\mathbb{N}$ and $x_1,x_2,\ldots,x_n\in\Lambda$, the determinants of matrices $(K(x_i,x_j))_{1\leq i,j\leq n}$ and $(Q(x_i,x_j))_{1\leq i,j\leq n}$ agree. What are all the possible transformations that transform $Q$ into $K$? In [Marco Stevens, Equivalent symmetric kernels of determinantal point processes, RMTA, 10(03):2150027, 2021] the following two were conjectured: $(Tf)(x,y)=f(y,x)$; and $(Tf)(x,y)=g(x)g(y)^{-1}f(x,y)$ for some nowhere-zero function $g$. In the same paper, this conjectured classification is verified in the case of symmetric functions $K$ and $Q$. By extending the graph-theoretic techniques of the paper, we show that under some surprisingly simple and natural conditions the conjecture remains valid even with the symmetry constraints relaxed. By taking $\Lambda$ finite, the above problem, furthermore, reduces to that between two square matrices investigated in [Raphael Loewy, Principal minors and diagonal similarity of matrices, Linear Algebra and its Applications 78 (1986), 23--64]. Hence, our paper presents a simple non-linear-algebraic proof that uses only some elementary combinatorics and three simple algebraic identities involving $3$-cycles and $4$-cycles.
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https://arxiv.org/abs/2302.02471
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84274be276f825ea350d40891aa69d7affbde5b8215e5653ef3a5d83ff03ee30
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2026-01-23T00:00:00-05:00
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Hopf 2-algebras and Braided Monoidal 2-Categories
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arXiv:2304.07398v4 Announce Type: replace Abstract: Following the theory of principal $\infty$-bundles of Niklaus-Schreiber-Steveson, we develop a homotopy categorification of Hopf algebras, which model quantum groups. We study their higher-representation theory in the setting of $\mathsf{2Vect}^{hBC}$, which is a homotopy refinement of the notion of 2-vector spaces due to Baez-Crans that allows for higher coherence data. We construct in particular the 2-quantum double as a homotopy double crossed product, and prove its duality and factorization properties. We also define and characterize "2-$R$-matrices", which can be seen as an extension of the usual notion of $R$-matrix in an ordinary Hopf algebra. We found that the 2-Yang-Baxter equations describe the braiding of extended defects in 4d, distinct from but not unlike the Zamolodchikov tetrahedron equations. The main results we prove in this paper is that the 2-representation 2-category of a weak 2-bialgebra is braided monoidal if it is equipped with a universal 2-$R$-matrix, and that our homotopy quantization admits the theory of Lie 2-bialgebras as a semiclassical limit.
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https://arxiv.org/abs/2304.07398
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2cca507e186812f375bf2ed45d5fcc377ee95bf3f8f17c80000f2c2458f46102
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2026-01-23T00:00:00-05:00
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Nijenhuis operators on 2D pre-Lie algebras and 3D associative algebras
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arXiv:2308.12121v3 Announce Type: replace Abstract: In this paper, we describe all Nijenhuis operators on 2-dimensional complex pre-Lie algebras and 3-dimensional complex associative algebras. As an application, using these operators, we obtain solutions of the classical Yang-Baxter equation on the corresponding sub-adjacent Lie algebras.
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https://arxiv.org/abs/2308.12121
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6995d2e125a0f7b56bfba46745d5e70253fba1a09a374ab07683d41c03d99226
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2026-01-23T00:00:00-05:00
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K\"{a}hler Solitons, Contact Structures, and Isoparametric Functions
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arXiv:2310.11328v3 Announce Type: replace Abstract: All known examples of simply-connected gradient K\"{a}hler-Ricci soliton in real dimension four are toric, and the symmetry is intrinsically related to the potential function $f$ and the scalar curvature $\SS$. In this article, we consider the case that $f$ and $\SS$ are functionally dependent and deduce a complete classification, while the independence case is addressed elsewhere. The main theorem recovers all known examples of cohomogeneity one symmetry. We also discover a connection to the theory of isoparametric functions and contact geometry. Indeed, a key ingredient is a new characterization for a deformed Sasakian structure generalizing a classical result.
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https://arxiv.org/abs/2310.11328
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0e7a959c074b596c8d55547493d224c5e755e75d1b017383ae5345def16fab4c
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2026-01-23T00:00:00-05:00
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Sharp quantitative stability for the fractional Sobolev trace inequality
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arXiv:2312.01766v3 Announce Type: replace Abstract: In this paper, we study the stability of fractional Sobolev trace inequality within both the functional and critical point settings. In the functional setting, we establish the following sharp estimate: $$C_{\mathrm{BE}}(n,m,\alpha)\inf_{v\in\mathcal{M}_{n,m,\alpha}}\left\Vert f-v\right\Vert_{D_\alpha(\mathbb{R}^n)}^2 \leq \left\Vert f\right\Vert_{D_\alpha(\mathbb{R}^n)}^2 - S(n,m,\alpha) \left\Vert\tau_mf\right\Vert_{L^{q}(\mathbb{R}^{n-m})}^2,$$ where $0\leq m< n$, $\frac{m}{2}<\alpha<\frac{n}{2}, q=\frac{2(n-m)}{n-2\alpha}$ and $\mathcal{M}_{n,m,\alpha}$ denotes the manifold of extremal functions. Additionally, We find an explicit bound for the stability constant $C_{\mathrm{BE}}$ and establish a compactness result ensuring the existence of minimizers. In the critical point setting, we investigate the validity of a sharp quantitative profile decomposition related to the Escobar trace inequality and establish a qualitative profile decomposition for the critical elliptic equation \begin{equation*} \Delta u= 0 \quad\text{in }\mathbb{R}_+^n,\quad\frac{\partial u}{\partial t}=-|u|^{\frac{2}{n-2}}u \quad\text{on }\partial\mathbb{R}_+^n. \end{equation*} We then derive the sharp stability estimate: $$ C_{\mathrm{CP}}(n,\nu)d(u,\mathcal{M}_{\mathrm{E}}^{\nu})\leq \left\Vert \Delta u +|u|^{\frac{2}{n-2}}u\right\Vert_{H^{-1}(\mathbb{R}_+^n)}, $$ where $\nu=1,n\geq 3$ or $\nu\geq2,n=3$ and $\mathcal{M}_{\mathrm{E}}^\nu$ represents the manifold consisting of $\nu$ weak-interacting Escobar bubbles. Through some refined estimates, we also give a strict upper bound for $C_{\mathrm{CP}}(n,1)$, which is $\frac{2}{n+2}$.
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https://arxiv.org/abs/2312.01766
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5a25d7f64f86499870887ec90f37791113898190d22d52fc731ceffe49e6abbe
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2026-01-23T00:00:00-05:00
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Classification of positive solutions to the H\'enon-Sobolev critical systems
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arXiv:2312.01784v2 Announce Type: replace Abstract: In this paper, we investigate positive solutions to the following H\'enon-Sobolev critical system: $$ -\mathrm{div}(|x|^{-2a}\nabla u)=|x|^{-bp}|u|^{p-2}u+\nu\alpha|x|^{-bp}|u|^{\alpha-2}|v|^{\beta}u\quad\text{in }\mathbb{R}^n,$$ $$ -\mathrm{div}(|x|^{-2a}\nabla v)=|x|^{-bp}|v|^{p-2}v+\nu\beta|x|^{-bp}|u|^{\alpha}|v|^{\beta-2}v\quad\text{in }\mathbb{R}^n,$$ $$u,v\in D_a^{1,2}(\mathbb{R}^n),$$ where $n\geq 3,-\infty0$ and $\alpha>1,\beta>1$ satisfying $\alpha+\beta=p$. Our findings are divided into two parts, according to the sign of the parameter $a$. For $a\geq 0$, we demonstrate that any positive solution $(u,v)$ is synchronized, indicating that $u$ and $v$ are constant multiples of positive solutions to the decoupled H\'enon equation: \begin{equation*} -\mathrm{div}(|x|^{-2a}\nabla w)=|x|^{-bp}|w|^{p-2}w. \end{equation*} For $aa$, we characterize all nonnegative ground states. Additionally, we study the nondegeneracy of nonnegative synchronized solutions. This work also delves into some general $k$-coupled H\'enon-Sobolev critical systems.
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https://arxiv.org/abs/2312.01784
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3ba1f8e61e33f8472ba6088968ce163bee4048253faf3ce945aab44062a84926
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2026-01-23T00:00:00-05:00
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Differential operators on the base affine space of $SL_n$ and quantized Coulomb branches
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arXiv:2312.10278v2 Announce Type: replace Abstract: We show that the algebra $D_\hbar(SL_n/U)$ of differential operators on the base affine space of $SL_n$ is the quantized Coulomb branch of a certain 3d $\mathcal{N} = 4$ quiver gauge theory. In the semiclassical limit this proves a conjecture of Dancer-Hanany-Kirwan about the universal hyperk\"ahler implosion of $SL_n$. We also formulate and prove a generalization identifying the Hamiltonian reduction of $T^* SL_n$ with respect to an arbitrary unipotent character as a Coulomb branch. As an application of our results, we provide a new interpretation of the Gelfand-Graev symmetric group action on $D_\hbar(SL_n/U)$.
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https://arxiv.org/abs/2312.10278
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a32a8923e7697669d49425b0f5cd5c85f88004a80aa5df21f8aed521a547eb88
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2026-01-23T00:00:00-05:00
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The stability on the Caffarelli-Kohn-Nirenberg and Hardy-type inequalities and beyond
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arXiv:2312.15735v3 Announce Type: replace Abstract: In this paper, we establish several improved Caffarelli-Kohn-Nirenberg and Hardy-type inequalities. Our main results are divided into two parts. In the first part, we consider the following Caffarelli-Kohn-Nirenberg inequality: \begin{equation*} \left(\int_{\mathbb{R}^n}|x|^{-pa}|\nabla u|^pdx\right)^{\frac{1}{p}}\geq S(p,a,b)\left(\int_{\mathbb{R}^n}|x|^{-qb}|u|^qdx\right)^{\frac{1}{q}},\quad\forall\; u\in D_a^p(\mathbb{R}^n), \end{equation*} We establish gradient stability of this inequality in both functional and critical settings, and we derive some functional properties of the stability constant. Building on the gradient stability, we also obtain several refined Sobolev-type embeddings involving weak Lebesgue norms for functions supported in general domains. In the second part, we focus on various classical Hardy-type inequalities, including the standard Hardy inequality, the $L^p$-logarithmic Sobolev inequality with weights, the logarithmic Hardy inequality, the Hardy-Morrey inequality, the Hardy-Sobolev interpolation inequality, and the interpolated Caffarelli-Kohn-Nirenberg inequality. We investigate their weighted versions and derive corresponding extremal functions, refinements, new remaining terms and stability constants.
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https://arxiv.org/abs/2312.15735
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498a896b88ebb8044899c3dd7d0f49177dd05b49158d963cc3b9de4f53ba45e2
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2026-01-23T00:00:00-05:00
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Recursion relations and BPS-expansions in the HOMFLY-PT skein of the solid torus
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arXiv:2401.10730v2 Announce Type: replace Abstract: Inspired by the skein valued open Gromov-Witten theory of Ekholm and Shende and the Gopakumar-Vafa formula, we associate to each pair of non-negative integers $(g,l)$ a formal power series with values in the HOMFLY-PT skein of a disjoint union of $l$ solid tori. The formal power series can be thought of as open BPS-states of genus $g$ with $l$ boundary components and reduces to the contribution of a single BPS state of genus $g$ for $l=0$. Using skein theoretic methods we show that the formal power series satisfy gluing identities and multi-cover skein relations corresponding to an elliptic boundary node of the underlying curves. For $(g,l)=(0,1)$ we prove a crossing formula which is the multi-cover skein relation corresponding to a hyperbolic boundary node, also known as the pentagon identity.
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https://arxiv.org/abs/2401.10730
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e4dcfcc168c5a96d6d78bf707e7b3aafd2dc13b31814508a8e5ace8bea0df5b0
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2026-01-23T00:00:00-05:00
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Operational Methods Applied to the Spherical Mean and X-Ray Transform
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arXiv:2402.10272v4 Announce Type: replace Abstract: We employ the framework of operational calculus to derive the operators associated with the spherical mean and a class of related averaging means of a function in $n$-dimensional space. Beginning with the classical definition of the spherical mean, we obtain a compact operator representation in terms of confluent hypergeometric functions of the Laplacian. This operator-based formulation provides a straightforward approach to the analysis of spherical means, allowing us to determine their power series expansions, construct series solutions to the corresponding inversion problems, derive the partial differential equations they satisfy, and give meaning to iterated and fractional spherical means. Finally, we apply the spherical mean operator to derive the inversion formula for the X-ray transform in an operational manner.
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https://arxiv.org/abs/2402.10272
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ea2743ada965213e1103f5206279edef28d5cd5317ae293ff4149f8fe44625be
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2026-01-23T00:00:00-05:00
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Toughness and A{\alpha}-spectral radius in graphs
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arXiv:2402.17421v2 Announce Type: replace Abstract: Let $\alpha\in[0,1)$, and let $G$ be a connected graph of order $n$ with $n\geq f(\alpha)$, where $f(\alpha)=6$ for $\alpha\in[0,\frac{2}{3}]$ and $f(\alpha)=\frac{4}{1-\alpha}$ for $\alpha\in(\frac{2}{3},1)$. A graph $G$ is said to be $t$-tough if $|S|\geq tc(G-S)$ for each subset $S$ of $V(G)$ with $c(G-S)\geq2$, where $c(G-S)$ is the number of connected components in $G-S$. The $A_{\alpha}$-spectral radius of $G$ is denoted by $\rho_{\alpha}(G)$. In this paper, it is verified that $G$ is a 1-tough graph unless $G=K_1\vee(K_{n-2}\cup K_1)$ if $\rho_{\alpha}(G)\geq\rho_{\alpha}(K_1\vee(K_{n-2}\cup K_1))$, where $\rho_{\alpha}(K_1\vee(K_{n-2}\cup K_1))$ equals the largest root of $x^{3}-((\alpha+1)n+\alpha-3)x^{2}+(\alpha n^{2}+(\alpha^{2}-\alpha-1)n-2\alpha+1)x-\alpha^{2}n^{2}+(3\alpha^{2}-\alpha+1)n-4\alpha^{2}+5\alpha-3=0$. Further, we present an $A_{\alpha}$-spectral radius condition for a graph to be a $t$-tough graph.
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https://arxiv.org/abs/2402.17421
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4bc3b48d4b22b1fcde32058475087af44aba26b10befe440cf78ce7e8d0b078e
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2026-01-23T00:00:00-05:00
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Unipotent normal subgroups of algebraic groups
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arXiv:2404.12221v2 Announce Type: replace Abstract: Let $G$ be an affine algebraic group scheme over a field $k$. We show there exists a unipotent normal subgroup of $G$ which contains all other such subgroups; we call it the restricted unipotent radical $\mathrm{Rad}_u(G)$ of $G$. We investigate some properties of $\mathrm{Rad}_u(G)$, and study those $G$ for which $\mathrm{Rad}_u(G)$ is trivial. In particular, we relate these notions to their well-known analogues for smooth connected affine $k$-groups.
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https://arxiv.org/abs/2404.12221
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867be02539a8e9e05683fec1f39be1b8ba8e829569ca592e8e5285854bb0a19a
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2026-01-23T00:00:00-05:00
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Fujita-Kato Solutions and Optimal Time Decay for the Vlasov-Navier-Stokes System in the Whole Space
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arXiv:2405.09937v2 Announce Type: replace Abstract: We are concerned with the construction of global-in-time strong solutions for the incompressible Vlasov-Navier-Stokes system in the whole three-dimensional space. One of our goals is to establish that small initial velocities with critical Sobolev regularity and sufficiently well localized initial kinetic distribution functions give rise to global and unique solutions. This constitutes an extension of the celebrated result for the incompressible Navier-Stokes equations (NS) that has been established in 1964 by Fujita and Kato. If in addition the initial velocity is integrable, we establish that the total energy of the system decays to 0 with the optimal rate t^{-3/2}, like for the weak solutions of (NS). Our results partly rely on the use of a higher order energy functional that controls the regularity $H^1$ of the velocity and seems to have been first introduced by Li, Shou and Zhang in the context of nonhomogeneous Vlasov-Navier-Stokes system. In the small data case, we show that this energy functional decays with the rate t^{-5/2}.
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https://arxiv.org/abs/2405.09937
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7c13e044a1f13b062049183ceb31b81ecfdee2aa6396da8890275d144c08602a
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2026-01-23T00:00:00-05:00
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A combinatorial interpretation of the Bernstein degree of unitary highest weight modules
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arXiv:2405.18766v2 Announce Type: replace Abstract: Consider the $(\mathfrak{g}, K)$-modules $L_{\lambda}$ for unitary highest weight representations of the real reductive group $G_{\mathbb{R}} = \operatorname{U}(p,q)$, $\operatorname{Mp}(2n, \mathbb{R})$, or $\operatorname{O}^*(2n)$, where $\operatorname{Mp}(2n,\mathbb{R})$ denotes the metaplectic double cover of $\operatorname{Sp}(2n,\mathbb{R})$. Let $k$ be a positive integer. Corresponding to $G_{\mathbb{R}}$ via Howe duality is the compact group $\operatorname{U}(k)$, $\operatorname{O}(k)$, or $\operatorname{Sp}(k)$, respectively, for which every irreducible representation $\sigma$ corresponds to a unique $L_{\lambda} = L_{\lambda(\sigma)}$. Nishiyama-Ochiai-Taniguchi (2001) expressed the Bernstein degree $\operatorname{Deg} L_{\lambda(\sigma)}$ as the product of $\dim \sigma$ and the degree of the associated variety of $L_{\lambda(\sigma)}$; this result is valid when $k \leq r :=$ the real rank of $G_{\mathbb{R}}$. In this paper, for arbitrary $k$, we give a new combinatorial interpretation $\operatorname{Deg} L_{\lambda(\sigma)} = \#(\mathcal{Q}_k(\sigma) \times \mathcal{P}_k)$, where $\mathcal{Q}_k(\sigma)$ is a certain set of semistandard tableaux, whose cardinality (for $k \geq r$) interpolates between $\dim \sigma$ and the dimension of the simple $K$-module with highest weight $\lambda(\sigma)$. The set $\mathcal{P}_k$ consists of certain plane partitions that encode the Hilbert series of the associated variety. We exhibit analogous sets $\mathcal{P}_k$ of plane partitions for all real reductive groups of Hermitian type, including the exceptional groups.
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https://arxiv.org/abs/2405.18766
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0d0a061047bdf657645b0f53c9b08ff2c3c621b14c8d52da84f128f624ec3475
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2026-01-23T00:00:00-05:00
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Crossed product splitting of intermediate operator algebras via 2-cocycles
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arXiv:2406.00304v2 Announce Type: replace Abstract: We investigate the C*-algebra inclusions $B \subset A \rtimes_{\rm r} \Gamma$ arising from inclusions $B \subset A$ of $\Gamma$-C*-algebras. The main result shows that, when $B \subset A$ is C*-irreducible in the sense of R{\o}rdam, and is centrally $\Gamma$-free in the sense of the author, then after tensoring with the Cuntz algebra $\mathcal{O}_2$, all intermediate C*-algebras $B \subset C\subset A \rtimes_{\rm r} \Gamma$ enjoy a natural crossed product splitting \[\mathcal{O}_2\otimes C=(\mathcal{O}_2 \otimes D) \rtimes_{{\rm r}, \gamma, \mathfrak{w}} \Lambda\] for $D:= C \cap A$, some $\Lambda<\Gamma$, and a subsystem $(\gamma, \mathfrak{w})$ of a unitary perturbed cocycle action $\Lambda \curvearrowright \mathcal{O}_2\otimes A$. As an application, we give a new Galois's type theorem for the Bisch--Haagerup type inclusions \[A^K \subset A\rtimes_{\rm r} \Gamma\] for actions of compact-by-discrete groups $K \rtimes \Gamma$ on simple C*-algebras. Due to a K-theoretical obstruction, the operation $\mathcal{O}_2\otimes -$ is necessary to obtain the clean splitting. Also, in general 2-cocycles $\mathfrak{w}$ appearing in the splitting cannot be removed even further tensoring with any unital (cocycle) action. We show them by examples, which further show that $\mathcal{O}_2$ is a minimal possible choice. We also establish a von Neumann algebra analogue, where $\mathcal{O}_2$ is replaced by the type I factor $\mathbb{B}(\ell^2(\mathbb{N}))$.
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https://arxiv.org/abs/2406.00304
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1e9b23ed79f112b2261f7ce1ae382f229aa88432e7186b94f67d08c8ed966a1b
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2026-01-23T00:00:00-05:00
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Surface groups among cubulated hyperbolic and one-relator groups
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arXiv:2406.02121v3 Announce Type: replace Abstract: Let $X$ be a non-positively curved cube complex with hyperbolic fundamental group. We prove that $\pi_1(X)$ has a non-free subgroup of infinite index unless $\pi_1(X)$ is either free or a surface group, answering questions of Gromov and Whyte (in a special case) and Wise. A similar result for one-relator groups follows, answering a question posed by several authors. The proof relies on a careful analysis of free and cyclic splittings of cubulated groups.
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https://arxiv.org/abs/2406.02121
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67aab8bc7b5b786b0ec039c8755da705ac7060635bb164633b7fa1dbd6c395fc
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2026-01-23T00:00:00-05:00
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Exact worst-case convergence rates of gradient descent: a complete analysis for all constant stepsizes over nonconvex and convex functions
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arXiv:2406.17506v2 Announce Type: replace Abstract: We consider gradient descent with constant stepsizes and derive exact worst-case convergence rates on the minimum gradient norm of the iterates. Our analysis covers all possible stepsizes and arbitrary upper/lower bounds on the curvature of the objective function, thus including convex, strongly convex and weakly convex (hypoconvex) objective functions. Among the challenging parts of the analysis, we note the necessity to exploit dependencies between non-consecutive iterates. While this complicates the proofs to some extent, it enables us to achieve an exact full-range analysis of gradient descent for any constant stepsize (covering, in particular, normalized stepsizes greater than one), whereas the literature contained only conjectured rates of this type. In the nonconvex case, allowing arbitrary bounds on upper and lower curvatures extends existing partial results that are valid only for gradient Lipschitz functions (i.e., where lower and upper bounds on curvature are equal), leading to improved rates for weakly convex functions. From our exact worst-case performance bounds, we deduce the optimal constant stepsize for gradient descent. Leveraging our analysis, we also introduce a new variant of gradient descent based on a unique, fixed sequence of variable stepsizes, demonstrating its superiority in the worst-case over any constant stepsize schedule.
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https://arxiv.org/abs/2406.17506
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bba13ed0be86d12b3eec01b1d833debcb91bb09994c5b739b2c27a931a5e6173
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2026-01-23T00:00:00-05:00
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The convergence and uniqueness of a discrete-time nonlinear Markov chain
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arXiv:2407.00314v2 Announce Type: replace Abstract: In this paper, we prove the convergence and uniqueness of a general discrete-time nonlinear Markov chain with specific conditions. The results have important applications in discrete differential geometry. First, we prove the discrete-time Ollivier Ricci curvature flow $d_{n+1}:=(1-\alpha\kappa_{d_{n}})d_{n}$ converges to a constant curvature metric on a finite weighted graph. As shown in \cite[Theorem 5.1]{M23}, a Laplacian separation principle holds on a locally finite graph with nonnegative Ollivier curvature. We further prove that the Laplacian separation flow converges to the constant Laplacian solution and generalize the result to nonlinear $p$-Laplace operators. Moreover, our results can also be applied to study the long-time behavior in the nonlinear Dirichlet forms theory and nonlinear Perron-Frobenius theory. Finally, we define the Ollivier Ricci curvature of the nonlinear Markov chain which is consistent with the classical Ollivier Ricci curvature, sectional curvature \cite{CMS24}, coarse Ricci curvature on hypergraphs \cite{IKTU21} and the modified Ollivier Ricci curvature for $p$-Laplace. We also establish the convergence results for the nonlinear Markov chain with nonnegative Ollivier Ricci curvature.
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https://arxiv.org/abs/2407.00314
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12905aff395719eab222d17dd7fc595a47a9958219253b089e731eab4abdc86f
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2026-01-23T00:00:00-05:00
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Degenerate stability of critical points of the Caffarelli-Kohn-Nirenberg inequality along the Felli-Schneider curve
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arXiv:2407.10849v2 Announce Type: replace Abstract: In this paper, we investigate the validity of a quantitative version of stability for the critical Hardy-H\'enon equation \begin{equation*} H(u):=\div(|x|^{-2a}\nabla u)+|x|^{-pb}|u|^{p-2}u=0,\quad u\in D_a^{1,2}(\R^n), \end{equation*} \begin{equation*} n\geq 2,\quad a<\frac{n-2}{2},\quad p=\frac{2n}{n-2+2(b-a)}, \end{equation*} which is well known as the Euler-Lagrange equation of the classical Caffarelli-Kohn-Nirenberg inequality. Establishing quantitative stability for this equation amounts to finding a nonnegative function $F$ such that the estimate \begin{equation*} \inf_{\substack{U_i\in\mathcal{M} 1\leq i\leq\nu}}\norm*{u-\sum_{i=1}^\nu U_i}_{D_a^{1,2}(\R^n)}\leq C(a,b,n)F(\norm*{H(u)}_{D_a^{-1,2}(\R^n)}) \end{equation*} holds for any nonnegative function $u$ satisfying \begin{equation*} \left(\nu-\frac{1}{2}\right)S(a,b,n)^{\frac{p}{p-2}}\leq\int_{\R^n}|x|^{-2a}|\nabla u|^2\mathrm{d}x\leq \left(\nu+\frac{1}{2}\right)S(a,b,n)^{\frac{p}{p-2}}. \end{equation*} Here $\nu\in\N_+$ and $\mathcal{M}$ denotes the set of positive solutions to this equation. When $(a,b)$ falls above the Felli-Schneider curve, Wei and Wu \cite{Wei} found an optimal $F$. Their proof relies heavily on the fact that $\mathcal{M}$ is non-degenerate. When $(a,b)$ falls on the Felli-Schneider curve, due to the absence of the non-degeneracy condition, it becomes complicated and technical to find a suitable $F$. In this paper, we focus on this case. When $\nu=1$, we obtain an optimal $F$. When $\nu\geq2$ and $u$ is not too degenerate, we also derive an optimal $F$. To our knowledge, the results in this paper provide the first instance of degenerate stability in the critical point setting. We believe that our methods will be useful in other works on degenerate stability.
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https://arxiv.org/abs/2407.10849
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57d996392544a3134e60f99c6c482a2478cd3d1a2b59671df009c061e3a4590c
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2026-01-23T00:00:00-05:00
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Polynomials Counting Group Colorings in Graphs
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arXiv:2409.12404v5 Announce Type: replace Abstract: Jaeger et al. in 1992 introduced group coloring as the dual concept to group connectivity in graphs. Let $A$ be an additive Abelian group, $ f: E(G)\to A$ and $D$ an orientation of a graph $G$. A vertex coloring $c:V(G)\to A$ is an $(A, f)$-coloring if $c(v)-c(u)\ne f(e)$ for each oriented edge $e=uv$ from $u$ to $v$ under $D$. Kochol recently introduced the assigning polynomial to count nowhere-zero chains in graphs--nonhomogeneous analogues of nowhere-zero flows in \cite{Kochol2022}, and later extended the approach to regular matroids in \cite{Kochol2024}. Motivated by Kochol's work, we define the $\alpha$-compatible graph and the cycle-assigning polynomial $P(G, \alpha; k)$ at $k$ in terms of $\alpha$-compatible spanning subgraphs, where $\alpha$ is an assigning of $G$ from its cycles to $\{0,1\}$. We prove that $P(G,\alpha;k)$ evaluates the number of $(A,f)$-colorings of $G$ for any Abelian group $A$ of order $k$ and $f:E(G)\to A$ such that the assigning $\alpha_{D,f}$ given by $f$ equals $\alpha$. Such an assigning is admissible. Based on Kochol's work, we derive that $k^{-c(G)}P(G,\alpha;k)$ is a polynomial enumerating $(A,f)$-tensions and counting specific nowhere-zero chains. Furthermore, by extending Whitney's broken cycle concept to broken compatible cycles, we show that the absolute value of the coefficient of $k^{|V(G)|-i}$ in $P(G,\alpha;k)$ associated with admissible assignings $\alpha$ equals the number of $\alpha$-compatible spanning subgraphs that have $i$ edges and contain no broken $\alpha$-compatible cycles. According to the combinatorial explanation, we establish a unified order-preserving relation from admissible assignings to cycle-assigning polynomials, and further show that for any admissible assigning $\alpha$ of $G$ with $\alpha(e)=1$ for every loop $e$, the coefficients of $P(G,\alpha;k)$ are nonzero and alternate in sign.
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https://arxiv.org/abs/2409.12404
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9df6e0be5126fa6b83b081ce346f007678db66469d0020eabcbf09ac99447bf4
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2026-01-23T00:00:00-05:00
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The Hilbert scheme of points on a threefold: broken Gorenstein structures and linkage
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arXiv:2409.17009v2 Announce Type: replace Abstract: We investigate the Hilbert scheme of points on a smooth threefold. We introduce a notion of broken Gorenstein structure for finite schemes, and show that its existence guarantees smoothness on the Hilbert scheme. Moreover, we conjecture that it is exhaustive: every smooth point admits a broken Gorenstein structure. We give an explicit characterization of the smooth points on the Hilbert scheme of A^3 corresponding to monomial ideals. We investigate the nature of the singular points, and prove several conjectures by Hu. Along the way, we obtain a number of additional results, related to linkage classes, nested Hilbert schemes, and a bundle on the Hilbert scheme of a surface.
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https://arxiv.org/abs/2409.17009
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c33ece82a4af351865836097a13a58f8833fefacf052d7349636b94c8012890e
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2026-01-23T00:00:00-05:00
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On Ulam widths of finitely presented infinite simple groups
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arXiv:2410.07512v2 Announce Type: replace Abstract: A fundamental notion in group theory, which originates in an article of Ulam and von Neumann from $1947$ is uniform simplicity. A group $G$ is said to be $n$-uniformly simple for $n \in \mathbf{N}$ if for every $f,g\in G\setminus \{id\}$, there is a product of no more than $n$ conjugates of $g$ and $g^{-1}$ that equals $f$. Then $G$ is uniformly simple if it is $n$-uniformly simple for some $n \in \mathbf{N}$, and we refer to the smallest such $n$ as the Ulam width, denoted as $\mathcal{R}(G)$. If $G$ is simple but not uniformly simple, one declares $\mathcal{R}(G)=\infty$. In this article, we construct for each $n\in \mathbf{N}$, a finitely presented infinite simple group $G$ such that $n<\mathcal{R}(G)<\infty$. These are the first such examples among the class of finitely presented infinite simple groups. For the class of finitely generated (but not finitely presentable) infinite simple groups, the existence of such examples was settled in the work of Muranov. However, this had remained open for the class of finitely presented infinite simple groups. Our examples are also of type $F_{\infty}$, which means that they are fundamental groups of aspherical CW complexes with finitely many cells in each dimension. Uniformly simple groups are in particular uniformly perfect: there is an $n\in \mathbf{N}$ such that every element of the group can be expressed as a product of at most $n$ commutators of elements in the group. We also show that the analogous notion of width for uniform perfection is unbounded for our family of finitely presented infinite simple groups. To our knowledge, this is also the first such family.
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https://arxiv.org/abs/2410.07512
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cb7d04200ee1e088276b707783217bfa9249d8d8c50ff482213014c08de292d6
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2026-01-23T00:00:00-05:00
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Sobolev estimates for the Keller-Segel system and applications to the JKO scheme
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arXiv:2410.15095v3 Announce Type: replace Abstract: We prove $L^{\infty}_{t}W^{1,p}_{x}$ Sobolev estimates in the Keller-Segel system with linear diffusion in any dimensionby proving a functional inequality, inspired by the Brezis-Gallou\"et-Wainger inequality. These estimates are also valid at the discrete level in the Jordan-Kinderlehrer-Otto (JKO) scheme. By coupling this result with the diffusion properties of a functional according to Bakry-Emery theory, we deduce the $L^2_t H^{2}_{x}$ convergence of the scheme, thereby extending the recent result of Santambrogio and Toshpulatov in the context of the Fokker-Planck equation to the Keller-Segel system.
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https://arxiv.org/abs/2410.15095
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be1d61daa00b27981f7885aaa24fec7f99a5e42f486cef03d7d8a324a93020a6
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2026-01-23T00:00:00-05:00
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The Calder\'on problem for third order nonlocal wave equations with time-dependent nonlinearities and potentials
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arXiv:2411.08657v3 Announce Type: replace Abstract: In this article, we study the Calder\'on problem for nonlocal generalizations of the semilinear Moore--Gibson--Thompson (MGT) equation and the Jordan--Moore--Gibson--Thompson (JMGT) equation of Westervelt-type. These partial differential equations are third order wave equations that appear in nonlinear acoustics, describe the propagation of high-intensity sound waves and exhibit finite speed of propagation. For semilinear MGT equations with nonlinearity $g$ and potential $q$, we show the following uniqueness properties of the Dirichlet to Neumann (DN) map $\Lambda_{q,g}$: (i) If $g$ is a polynomial-type nonlinearity whose $m$-th order derivative is bounded, then $\Lambda_{q,g}$ uniquely determines $q$ and $(\partial^{\ell}_\tau g(x,t,0))_{2\leq \ell \leq m}$. (ii) If $g$ is a polyhomogeneous nonlinearity of finite order $L$, then $\Lambda_{q,g}$ uniquely determines $q$ and $g$. The uniqueness proof for polynomial-type nonlinearities is based on a higher order linearization scheme, while the proof for polyhomogeneous nonlinearities only uses a first order linearization. Finally, we demonstrate that a first linearization suffices to uniquely determine Westervelt-type nonlinearities from the related DN maps. We also remark that all the unknowns, which we wish to recover from the DN data, are allowed to depend on time.
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https://arxiv.org/abs/2411.08657
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e933765a38ae678f784d069207e7feec8947d73f01f4636fa097de4c338aaa7d
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2026-01-23T00:00:00-05:00
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Berkovich Motives
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arXiv:2412.03382v3 Announce Type: replace Abstract: We construct a theory of (etale) Berkovich motives. This is closely related to Ayoub's theory of rigid-analytic motives, but works uniformly in the archimedean and nonarchimedean setting. We aim for a self-contained treatment, not relying on previous work on algebraic or analytic motives. Applying the theory to discrete fields, one still recovers the etale version of Voevodsky's theory. Two notable features of our setting which do not hold in other settings are that over any base, the cancellation theorem holds true, and under only minor assumptions on the base, the stable $\infty$-category of motivic sheaves is rigid dualizable.
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https://arxiv.org/abs/2412.03382
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Academic Papers
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6836316dec9c30d7cd438494c4f1d02965e88f721ca6151a0fb7bb8584e8373c
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2026-01-23T00:00:00-05:00
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On a class of Nonlinear Grushin equations
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arXiv:2412.08039v2 Announce Type: replace Abstract: In this paper, we study two kinds of nonlinear degenerate elliptic equations containing the Grushin operator. First, we prove radial symmetry and a decay rate at infinity of solutions to such a Grushin equation by using the moving plane method in combination with suitable integral inequalities. Applying similar methods, we obtain nonexistence results for solutions to a second type of Grushin equation in Euclidean half space. Finally, we derive a priori estimates and the existence for positive solutions to more general types of Grushin equations by employing blow up analysis and topological degree methods, respectively.
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https://arxiv.org/abs/2412.08039
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Academic Papers
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