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4cc8315040ce6330fceb0a499e4acc898a4046702ca7983551ade9aa0e465d7f
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2026-01-21T00:00:00-05:00
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2-Adic Obstructions to Presburger-Definable Characterizations of Collatz Cycles
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arXiv:2601.12772v1 Announce Type: new Abstract: I investigate structural limitations of Presburger-arithmetic-based approaches to the Collatz problem. I show that the Collatz cycle equation admits a unique solution in the $2$-adic integers, which I term a \emph{ghost cycle}. These ghost cycles are shown to be genuine periodic orbits of the $2$-adic Collatz map, satisfying all local parity constraints. I prove unconditionally that the divisibility predicate $\mathcal{D}_y = \{(x, C) \in \mathbb{N}^2: (2^x - 3^y) \mid C\}$, which acts as the algebraic necessary condition for integrality, is not semilinear for any fixed number of odd steps $y \ge 1$. This result is established by demonstrating that the fibers of $\mathcal{D}_y$ exhibit unbounded periods, an obstruction to Presburger definability. Consequently, strategies relying solely on Presburger arithmetic or finite automata to define the integrality constraint cannot capture the distinction between ghost cycles and genuine integer cycles. I conclude with a heuristic argument suggesting that because ghost cycles satisfy the algebraic cycle equation, the non-existence of integer cycles cannot be proven solely through algebraic manipulation of the cycle equation itself.
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https://arxiv.org/abs/2601.12772
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cd6b497c0ca71110265138d4ebfd5dd26c4cb3167b7b16ce5f43ceeacd498804
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2026-01-21T00:00:00-05:00
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Probabilistic degenerate logarithm and heterogeneous stirling numbers
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arXiv:2601.12794v1 Announce Type: new Abstract: Let Y be a random variable whose moment-generating function exists in some neighborhood of the origin. While probabilistic Stirling numbers of the first and second kind have been introduced, early definitions often failed to satisfy fundamental orthogonality and inverse relations or lacked consistency with classical forms in the case when Y = 1. This paper addresses these limitations by utilizing redefined probabilistic Stirling numbers of the first kind and the second kind alongside their degenerate counterparts. Our primary objective is twofold: first,to introduce the probabilistic (degenerate) logarithm associated with Y, providing explicit expressions for various random variables and defining new probabilistic degenerate Daehee and Cauchy numbers; and second, to investigate probabilistic heterogeneous Stirling numbers and establish a probabilistic degenerate version of the Schlomilch formula, demonstrating that these new frameworks maintain the essential algebraic properties of their classical counterparts.
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https://arxiv.org/abs/2601.12794
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e4285ec8d046762082673d9daa1026b37456afcae29db0d7bf4ea964ea9d41ac
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2026-01-21T00:00:00-05:00
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Llarull's theorem on noncompact manifolds with boundary
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arXiv:2601.12803v1 Announce Type: new Abstract: Recently, Zhang \cite{Zh20} and Li-Su-Wang-Zhang \cite{LSWZ24+} generalized Llarull's theorem to the noncompact complete spin manifold. In this paper, we further extend their results to the noncompact manifold with compact boundary.
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https://arxiv.org/abs/2601.12803
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8ddf5fbac075cbd414676a5f5e8133dccbee8c99b127fd136031d0833a2bccc7
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2026-01-21T00:00:00-05:00
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Optimal bounds for the boundary control cost of one-dimensional fractional Schr\"odinger and heat equations
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arXiv:2601.12810v1 Announce Type: new Abstract: We derive sharp bounds for the boundary control cost of the one-dimensional fractional Schr\"odinger and heat equations. The analysis of the lower bound is based on the study of the control cost of a related singular boundary control problem in finite time, using tools from complex analysis. The analysis of the upper bound relies on the moment method, involving estimates of the Fourier transform of a class of compactly supported functions.
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https://arxiv.org/abs/2601.12810
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4582a75590bde22501d1fb731e55c9eaa2324925aa5bade8f8111bad304ece7d
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2026-01-21T00:00:00-05:00
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Perfect codes in weakly metric association schemes
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arXiv:2601.12818v1 Announce Type: new Abstract: The Lloyd Theorem of (Sol\'e, 1989) is combined with the Schwartz-Zippel Lemma of theoretical computer science to derive non-existence results for perfect codes in the Lee metric, NRT metric, mixed Hamming metric, and for the sum-rank distance. The proofs are based on asymptotic enumeration of integer partitions. The framework is the new concept of {\em polynomial} weakly metric association schemes. A connection between this notion and the recent theory of multivariate P-polynomial schemes of ( Bannai et al. 2025) and of $m$-distance regular graphs ( Bernard et al 2025) is pointed out.
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https://arxiv.org/abs/2601.12818
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b6b44d43f7986d2d20f49b44f50cc8236748b192f9ce71e54258e0dd4960afc7
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2026-01-21T00:00:00-05:00
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Sub-wavelength resonances in two-dimensional multi-layer elastic media
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arXiv:2601.12821v1 Announce Type: new Abstract: In this paper, we focus on the sub-wavelength resonances in two-dimensional elastic media characterized by high contrasts in both Lam\'e parameters and density. Our contributions are fourfold. First, it is proved that the operator $\hat{\mathbf{S}}_{\partial D}^{\omega}$, which serves as a leading order approximation to $\mathbf{S}_{\partial D}^{\omega}$ as $\omega\rightarrow0$, is invertible in the space $\mathcal{L}(L^{2}\left(\partial D)^{2},H^{1}(\partial D)^{2}\right)$. Second, based on layer potential techniques in combination with asymptotic analysis, we derive an original formula for the leading-order terms of sub-wavelength resonance frequencies, which are controlled by the determinant of the $3N \times 3N$ matrices. Specifically, there are $3N$ resonance frequencies within an $N$-nested layer structure. In addition, the scattering field exhibits an enhancement coefficient on the order of $\mathcal{O}(\omega^{-2})$ as the incident frequency $\omega$ approaches the resonance frequency. Third, by applying spectral properties to solve the corresponding eigenvalue problem, we compute the quantitative expressions for sub-wavelength resonance frequencies within a disk. Finally, some numerical experiments are provided to illustrate theoretical results and demonstrate the existence of the sub-wavelength resonance modes.
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https://arxiv.org/abs/2601.12821
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795455902587e3509b712a4099d6b206e1d363fca94ca13c059808f76f66ad3d
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2026-01-21T00:00:00-05:00
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A converse of Berndtsson's theorem on the positivity of direct images
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arXiv:2601.12825v1 Announce Type: new Abstract: Berndtsson's famous theorem asserts that, for a compact K\"ahler fibration $p:X\to Y$, the direct image bundle $p_*(K_{X/Y}\otimes L)$ of a semi-positive Hermitian holomorphic line bundle $L\to X$ is Nakano semi-positive. As a continuation of our previous work, we prove a converse of Berndtsson's theorem in the case of a projective fibration: if $p_*(K_{X/Y}\otimes L\otimes E)$ is Griffiths semi-positive for every semi-positive Hermitian holomorphic line bundle $E\to X$, then the curvature of $L$ must be semi-positive.
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https://arxiv.org/abs/2601.12825
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9f5ca3d0db57fa37a3b48b37bd09de3ff16665f217f3b4dab1cddf47ae7ace8d
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2026-01-21T00:00:00-05:00
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Cofiniteness and $P(z)$-tensor product bifunctors in orbifold theories associated to abelian but not-necessarily-finite groups
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arXiv:2601.12834v1 Announce Type: new Abstract: Let $V$ be a M\"{o}bius vertex algebra and $G$ an abelian group of automorphisms of $V$. We construct $P(z)$-tensor product bifunctors for the category of $C_{n}$-cofinite grading-restricted generalized $g$-twisted $V$-modules (without $g$-actions) for $g\in G$ and the category of $C_{n}$-cofinite grading-restricted generalized $g$-twisted $V$-modules with $G$-actions for $g\in G$. In this paper, an automorphism $g$ of $V$ can be of infinite order and does not have to act semisimply on $V$, and the group $G$ can be an infinite abelian group containing nonsemisimple automorphisms of $V$.
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https://arxiv.org/abs/2601.12834
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c96b35ea5f30dfae487ac9921ff54f4242fda18a5b69273de8852243919ab452
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2026-01-21T00:00:00-05:00
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A diagrammatic approach to the three-page index
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arXiv:2601.12846v1 Announce Type: new Abstract: The three-page index $\alpha_3(L)$ is an invariant that measures the complexity of representing a link $L$ in a three-page book. It is known that $\alpha_3(L)$ admits a linear upper bound in terms of the crossing number, with equality realized by the Hopf link. In this paper, we investigate the equality case of this bound from a diagrammatic viewpoint. Starting from a reduced link diagram, we construct three-page presentations via binding circles arising as boundaries of suitable contractible subcomplexes of the induced cell decomposition of the $2$-sphere. This approach allows a refined control of the number of arcs in the resulting three-page presentation. As a consequence, we prove that for any non-split, nontrivial link $L$ other than the Hopf link, \[ \alpha_3(L)\le 3c(L)-1, \] and hence characterize completely the links for which $\alpha_3(L)=3c(L)$.
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https://arxiv.org/abs/2601.12846
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cd5b5d439cce4b1d85b8aef1a3ceb27bf0439786dedf5d99881410fce569c073
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2026-01-21T00:00:00-05:00
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Rankin-Cohen Bracket for Vector-Valued Modular Forms
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arXiv:2601.12860v1 Announce Type: new Abstract: In this paper, we explore the relationship between Rankin-Cohen brackets for vector-valued modular forms and Petersson's inner products, deriving an explicit description of the adjoint map for the bracket operator. The study extends to the cases of Jacobi forms and skew-holomorphic Jacobi forms, establishing connections between their respective Rankin-Cohen brackets and those defined for vector-valued modular forms through an isomorphism. Adjoint maps for these extended bracket operators are also examined.
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https://arxiv.org/abs/2601.12860
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415c489c40b1e9a0e3a0a991974a0a6c28f7d8cd66cc5c4469f5a88c6490b0e7
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2026-01-21T00:00:00-05:00
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Traveling waves for monostable reaction-diffusion-convection equations with discontinuous density-dependent coefficients
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arXiv:2601.12869v1 Announce Type: new Abstract: This paper concerns wave propagation in a class of scalar reaction-diffusion-convection equations with $p$-Laplacian-type diffusion and monostable reaction. We introduce a new concept of a non-smooth traveling wave profile, which allows us to treat discontinuous diffusion with possible degenerations and singularities at 0 and 1, as well as only piecewise continuous convective velocity. Our approach is based on comparison arguments for an equivalent non-Lipschitz first-order ODE. We formulate sufficient conditions for the existence and non-existence of these generalized solutions and discuss how the convective velocity affects the minimal wave speed compared to the problem without convection. We also provide brief asymptotic analysis of the profiles, for which we need to assume power-type behavior of the diffusion and reaction terms.
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https://arxiv.org/abs/2601.12869
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be8727085b8f42760d16d8b0f8c59f33a904ee60b6284b5fe1d443caf4fa12de
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2026-01-21T00:00:00-05:00
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Residues and Infinitesimal Torelli for Equisingular Curves
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arXiv:2601.12873v1 Announce Type: new Abstract: We study infinitesimal Torelli problems and infinitesimal variations of Hodge structure for families of curves arising in singular and extrinsically constrained geometric settings. Motivated by the Green--Voisin philosophy, we develop an explicit approach based on Poincar\'e residue calculus, allowing a uniform treatment of smooth, singular, and equisingular situations. In particular, we prove infinitesimal Torelli theorems for general equisingular plane curves of sufficiently high degree and construct relative IVHS exact sequences for curves lying on smooth projective threefolds. Our results show that maximal infinitesimal variation of Hodge structure persists even after imposing strong extrinsic conditions, such as fixed degree and prescribed singularities, and in the presence of isolated planar singularities. The methods presented here provide a concrete and geometric realization of Jacobian-type constructions and extend the Green--Voisin philosophy to singular and equisingular settings and provide a unified residue--theoretic framework for Torelli--type problems across dimensions and codimensions.
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https://arxiv.org/abs/2601.12873
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ca7e81ab7d18e3bae8f0e6a56637e0f8f284f8042323d746dc919fac9c6fc14e
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2026-01-21T00:00:00-05:00
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Hunting The Poles in the Staircases
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arXiv:2601.12881v1 Announce Type: new Abstract: Motivated by applications to the fractional quantum Hall effect and, in particular, to the Bernevig-Haldane conjectures, we investigates the behavior of Macdonald polynomials under specializations of the form q a t b = 1. Our main focus is to explain, in a simple and purely combinatorial way, why certain nonsymmetric Macdonald polynomials indexed by staircase vectors with steps of height a and width b remain regular at the specialization q a t b+1 = 1, despite the presence of potential poles in their rational coefficients. To this end, we introduce a set of combinatorial tools that track how poles are created or cancelled along paths in the Yang-Baxter graph. By carefully constructing paths from the zero vector to the staircase and analyzing the resulting denominators, we show that the absence of certain poles follows from intrinsic symmetries and cancellations encoded in the Yang-Baxter graph.
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https://arxiv.org/abs/2601.12881
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979ae89ad5348189312e97f309bec6fb4420a87a022d4c52c02a93f35c17a9b0
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2026-01-21T00:00:00-05:00
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A formula for the local Heun solution
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arXiv:2601.12888v1 Announce Type: new Abstract: The local Heun solution is the unique solution to Heun's equation which is analytic in the unit disk centered at $0\in\mathbb{C}$ and taking the value $1$ at the center of the disk. In this paper, as an application of the theory of orthogonal polynomials, we are able to express the coefficients in the corresponding power series as finite multiple sums. In addition, the obtained formula can be used to derive an explicit estimate on the coefficients giving a hint on their asymptotic behavior for large indices.
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https://arxiv.org/abs/2601.12888
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36efb05ab1d6f1d7e6358af5cd16f48d262b8c802d64c1ffac2026e09c9f8cf1
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2026-01-21T00:00:00-05:00
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Bi-Lipschitz invariance of Newton polygons along gradient canyons
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arXiv:2601.12897v1 Announce Type: new Abstract: We study bi-Lipschitz right-equivalence of holomorphic function germs $f:(\mathbb{C}^2,0)\to(\mathbb{C},0)$ via polar arcs and gradient canyons. For a polar arc $\gamma$ we consider the Newton polygon of $f_x(X+\gamma(Y),Y)$ and define its augmentation by adjoining the point $(0,\text{ord } f(\gamma(y),y)-1)$. We prove that the resulting augmented Newton polygon is constant along each gradient canyon of degree $>1$ and is invariant under bi-Lipschitz right-equivalence. Moreover, its compact edges decompose into a topological part and a Lipschitz part: the latter encodes, through simple intercept relations, the second-level Henry-Parusi\'nski type invariants. As an application we introduce the polar multiplicity of a canyon and identify it with the horizontal length of the top edge of the augmented polygon, yielding a new discrete bi-Lipschitz invariant.
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https://arxiv.org/abs/2601.12897
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3fc28bc3153d3cd75946b6c874f8781d8a09b0529fec3410674377eaa3d951d9
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2026-01-21T00:00:00-05:00
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On the number of spanning trees of bicirculant graphs
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arXiv:2601.12899v1 Announce Type: new Abstract: A bi-Cayley graph over a cyclic group $\mathbb{Z}_n$ is called a bicirculant graph. Let $\Gamma=BC(\mathbb{Z}_n; R,T,S)$ be a bicirculant graph with $R=R^{-1}\subseteq \mathbb{Z}_n\setminus \{0\}$ and $T=T^{-1}\subseteq \mathbb{Z}_n\setminus \{0\}$ and $S\subseteq \mathbb{Z}_n$. In this paper, using Chebyshev polynomials, we obtain a closed formula for the number of spanning trees of bicirculant graph $\Gamma$, investigate some arithmetic properties of the number of spanning trees of $\Gamma$, and find its asymptotic behaviour as $n$ tends infinity. In addition, we show that $F(x)=\sum_{n=1}^{\infty}\tau(\Gamma)x^n$ is a rational function with integer coefficients.
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https://arxiv.org/abs/2601.12899
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0767a0eb33e8b9d7f0dbd59c12f71b8fd16c1fd3f090b0f6efd78b27a8746a53
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2026-01-21T00:00:00-05:00
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Gender and assessment in mathematics: a comparative study of managing assessment episodes
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arXiv:2601.12908v1 Announce Type: new Abstract: The article focuses on the differences in mathematics performance between girls and boys visible from the first four months of compulsory schooling in the French education system. The influence of gender stereotypes in the evaluation practices of teachers and the threat of the gender stereotype on student performance are questioned. To obtain answers, a comparative study of management of evaluative episodes is proposed based on different theoretical tools.
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https://arxiv.org/abs/2601.12908
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4774bfbc0f0378cb86a88495eed22d3417a96d119b926e3692d7d1f9ef1541aa
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2026-01-21T00:00:00-05:00
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A functional inequalities approach for the field-road diffusion model with (symmetric) nonlinear exchanges
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arXiv:2601.12909v1 Announce Type: new Abstract: In this note, we consider the so-called field-road diffusion model in a bounded domain, consisting of two parabolic PDEs posed on sets of different dimensions and coupled through (symmetric) nonlinear exchange terms. We propose a new and rather direct functional inequalities approach to prove the exponential decay of a relative entropy, and thus the convergence of the solution towards the stationary state selected by the total mass of the initial datum.
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https://arxiv.org/abs/2601.12909
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08e0c6a708bf356ed4d241c7c7ff0b65fc9b732fbb0f8c1300187a289b01afe1
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2026-01-21T00:00:00-05:00
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Sharp lower bound for the Monge-Amp\`ere torsion on convex sets
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arXiv:2601.12915v1 Announce Type: new Abstract: The \emph{Monge-Amp\`ere} torsion deficit of an open, bounded convex set $\Omega\subset\R^n$ of class $C^2$ is the normalized gap between the value of the torsion functional evaluated on $\Omega$ and its value on the ball with the same $(n-1)$-quermassintegral as $\Omega$. Using the technique of the \emph{shape derivative}, we prove that the ratio between this deficit and to a geometric deficit arising from the \emph{Alexandrov-Fenchel inequality}, for any given family of open, bounded convex sets of $\R^n$ ($n\geq2$) of class $C^2$, smoothly converging to a ball, is bounded from below by a dimensional constant. We also show that this ratio is always bounded from above by a constant.
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https://arxiv.org/abs/2601.12915
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4c372dd6103403436f0f3351dd600c9a036472c5b2278bee84acd2e903d68ea6
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2026-01-21T00:00:00-05:00
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On Kippenhahn curves of low rank partial isometries
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arXiv:2601.12923v1 Announce Type: new Abstract: Conditions are established for rank three partial isometries to have circular components contained in their Kippenhahn curves. In particular, such matrices with circular numerical ranges are described. It is also established that the Gau-Wang-Wu conjecture holds for matrices under consideration.
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https://arxiv.org/abs/2601.12923
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f9c748cecfd78c158e33f4b3eae59a5c9971c93d35b9be36e43d0101d8e663f2
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2026-01-21T00:00:00-05:00
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Stone Duality for Preordered Topological Spaces
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arXiv:2601.12932v1 Announce Type: new Abstract: A preordered topological space is a topological space with a preordering. We exhibit a Stone-like duality for preordered topological spaces, Inspired by a similar duality for bitopological spaces, due to Jung-Moshier and Jakl, and by a duality for preordered sets due to Bonsangue, Jacobs and Kok.
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https://arxiv.org/abs/2601.12932
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baa498e9a777d9463f253ebf00dc193d9470b48f635db445b14cc0df1bc2e7f6
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2026-01-21T00:00:00-05:00
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Random tree Besov priors: Data-driven regularisation parameter selection
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arXiv:2601.12957v1 Announce Type: new Abstract: We develop a data-driven algorithm for automatically selecting the regularisation parameter in Bayesian inversion under random tree Besov priors. One of the key challenges in Bayesian inversion is the construction of priors that are both expressive and computationally feasible. Random tree Besov priors, introduced in Kekkonen et al. (2023), provide a flexible framework for capturing local regularity properties and sparsity patterns in a wavelet basis. In this paper, we extend this approach by introducing a hierarchical model that enables data-driven selection of the wavelet density parameter, allowing the regularisation strength to adapt across scales while retaining computational efficiency. We focus on nonparametric regression and also present preliminary plug-and-play results for a deconvolution problem.
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https://arxiv.org/abs/2601.12957
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c97104cb5cae3a559428efed2962c7d65d3ede36bbbe859fb582d8999faa0e3a
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2026-01-21T00:00:00-05:00
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On cohomological dimensions of totally disconnected locally compact groups
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arXiv:2601.12958v1 Announce Type: new Abstract: In this paper, we introduce Mackey functors for a t.d.l.c. group and define the cohomological dimension of this group over the Mackey category. We then compare this dimension to the rational discrete cohomological dimension defined by Castellano and Weigel, as well as to the Bredon cohomological dimension of that t.d.l.c. group with respect to the family of compact open subgroups. We also extend results about the geometric dimension of a t.d.l.c. group.
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https://arxiv.org/abs/2601.12958
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ad702582a7de535f7bbf6094165f2a89198eb26e14169b225d743ea230d14699
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2026-01-21T00:00:00-05:00
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Counting Irreducible polynomials with coefficients from thin subgroups
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arXiv:2601.12968v1 Announce Type: new Abstract: L. Bary-Soroker and R. Shmueli (2026) have given an asymptotic formula for the number of irreducible polynomials over the finite fields $\mathbb F_q$ of $q$ elements, such that their coefficients are perfect squares in $\mathbb F_q$ and also extended this to classes of polynomials with coefficients described by finitely many unions of intersections of polynomial images. Here we use a different approach, which allows us to obtain another generalisation of this result to polynomials with coefficients from small subgroups of $\mathbb F_q^*$. As a demonstration of the power of our approach, we also use it to count such irreducible polynomials with an additional condition, namely, with a prescribed value of their discriminant. This generalisation seems to be unachievable via the approach of L. Bary-Soroker and R. Shmueli (2026).
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https://arxiv.org/abs/2601.12968
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638f7f28f25c2c15581baa8d0e956ce6979ddd218ee49edcb731916f1436e39e
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2026-01-21T00:00:00-05:00
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Bernstein type gradient estimate for system of weighted local heat equations with potential term
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arXiv:2601.12992v1 Announce Type: new Abstract: In this article we provide Bernstein type gradient estimates for two system of local weighted heat type equations with potentials on a weighted Riemannian manifold. We derive all possible cases considering linear potential, exponential potential, combining with static manifold and evolving manifold. This work partially resolved the problem raised by Bhattacharyya et al. in \cite{SB-1}.
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https://arxiv.org/abs/2601.12992
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3936ebdbdf2af390a96ec53cb6ecf3bc186c5786b5bdabbec8b8ca272ff8359b
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2026-01-21T00:00:00-05:00
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The distribution of the ratio of products of independent zero mean normal random variables
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arXiv:2601.12997v1 Announce Type: new Abstract: Let $X_1,\ldots,X_M$ and $Y_1,\ldots,Y_N$ be independent zero mean normal random variables with variances $\sigma_{X_i}^2$, $i=1,\ldots,M$, and $\sigma_{Y_j}^2$, $j=1,\ldots,N$, respectively, and let $X=X_1\cdots X_M$ and $Y=Y_1\cdots Y_N$. In this paper, we derive the exact probability density function of the ratio $X/Y$. We apply this formula to derive exact formulas for the cumulative distribution function and the characteristic function. We also obtain further distributional properties, including asymptotic approximations for the probability density function, tail probabilities and the quantile function.
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https://arxiv.org/abs/2601.12997
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d545c2c1e4d4b979771d63e9b09f705fd1524b62ec373e6f66c1dec10e643012
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2026-01-21T00:00:00-05:00
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Isomorphism relations on classes of c.e. algebras
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arXiv:2601.13005v1 Announce Type: new Abstract: We investigate the complexity of isomorphism relations for classes of finitely generated and n-generated computably enumerable (c.e.) algebras, presented via c.e. presentations -- that is, as quotients of term algebras over decidable sets of generators by c.e. congruences. Our goal is to develop a systematic framework for analyzing such isomorphism problems from a computability-theoretic perspective. To compare their complexity, we employ the notion of computable reducibility, measuring these relations against canonical benchmarks on c.e. sets, such as =^{ce}, E_0^{ce}, and the ordinal-indexed family E_min(\alpha). A central insight of our work is the interplay between the algebraic structure and the algorithmic complexity: we show that if every algebra in a class satisfies the ascending chain condition on its congruence lattice, then the corresponding isomorphism relation is computably reducible to =^{ce}. We also apply this framework to a range of concrete cases. In particular, we analyze the isomorphism relations for finitely generated commutative semigroups, monoids, and groups, positioning them within the broader landscape of classification problems.
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https://arxiv.org/abs/2601.13005
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c518418cf4d8664bc9f91805f1c4a3ccd36242748db89718e8456b5983db75b2
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2026-01-21T00:00:00-05:00
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Semi-infinite Lakshmibai--Seshadri paths and level-zero extremal weight modules over twisted quantum affine algebras
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arXiv:2601.13016v1 Announce Type: new Abstract: In this paper, we study level-zero extremal weight modules over twisted quantum affine algebras. To this end, we introduce semi-infinite Lakshmibai--Seshadri paths associated with a level-zero dominant integral weight $\lambda$. We then show that the set $\tfrac{\infty}{2}\mathrm{LS}(\lambda)$ of semi-infinite LS paths of shape $\lambda$ is isomorphic, as a crystal, to the crystal basis $\mathcal{B}(\lambda)$ of the corresponding level-zero extremal weight module $V(\lambda)$.
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https://arxiv.org/abs/2601.13016
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65227e385b0628d2c9433677a78cd51958b353106b3db9e0b53cdd6d9a61d665
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2026-01-21T00:00:00-05:00
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Characterization of eigenfunctions of the Laplacian having exponential growth
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arXiv:2601.13017v1 Announce Type: new Abstract: In 1993, Robert Strichartz proved a characterization for the bounded eigenfunctions of Laplacian $\Delta=-\sum_{j=1}^d \frac{\partial^2}{\partial x_j^2} $ on $\mathbb{R}^d$: If $\left\{f_k \right\}_{k\in \mathbb{Z}}$ be a doubly infinite sequence of functions on $\mathbb{R}^d$ such that $\Delta f_k=f_{k+1}$ and $ \|f_k\|_{L^{\infty}(\mathbb{R}^d)} \leq C$ for all $ k \in \mathbb{Z}$, for some $C>0$, then $f_0$ is an eigenfunction of $\Delta$. Observing the existence of unbounded eigenfunctions of the Laplacian, Howard and Reese generalized Strichartz's theorem to characterize eigenfunctions of the Laplacian having at most polynomial growth. In this article, we shall prove an extended version of Strichartz's theorem to characterize eigenfunctions of the Laplacian having exponential growth.
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https://arxiv.org/abs/2601.13017
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9af6508bea74c268c3dfeede9cd1087ae852f9971a67fd83f5856c464d776728
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2026-01-21T00:00:00-05:00
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The Reduced Phase Space of $N=1, D=4$ Supergravity in the BV-BFV formalism
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arXiv:2601.13025v1 Announce Type: new Abstract: This paper describes the reduced phase space of $N=1$, $D=4$ supergravity in the fully off-shell Palatini--Cartan formalism. This is achieved through the KT construction, allowing an explicit description of first-class constraints on the boundary. The corresponding BFV description is obtained, and its relation with the BV one in the bulk is described by employing the BV pushforward in the particular example of a cylindrical spacetime.
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https://arxiv.org/abs/2601.13025
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2eae668d5de97c3570718f9a24750449de1c1798c4acac2bdf8d4d025625a2d4
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2026-01-21T00:00:00-05:00
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Multi-gear bandits, partial conservation laws, and indexability
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arXiv:2601.13026v1 Announce Type: new Abstract: This paper considers what we propose to call multi-gear bandits, which are Markov decision processes modeling a generic dynamic and stochastic project fueled by a single resource and which admit multiple actions representing gears of operation naturally ordered by their increasing resource consumption. The optimal operation of a multi-gear bandit aims to strike a balance between project performance costs or rewards and resource usage costs, which depend on the resource price. A computationally convenient and intuitive optimal solution is available when such a model is indexable, meaning that its optimal policies are characterized by a dynamic allocation index (DAI), a function of state--action pairs representing critical resource prices. Motivated by the lack of general indexability conditions and efficient index-computing schemes, and focusing on the infinite-horizon finite-state and -action discounted case, we present a verification theorem ensuring that, if a model satisfies two proposed PCL-indexability conditions with respect to a postulated family of structured policies, then it is indexable and such policies are optimal, with its DAI being given by a marginal productivity index computed by a downshift adaptive-greedy algorithm in $A N$ steps, with $A+1$ actions and $N$ states. The DAI is further used as the basis of a new index policy for the multi-armed multi-gear bandit problem.
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https://arxiv.org/abs/2601.13026
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db5071468faadfb14b9159018f7432314992f5bde4cb165ab09a95702add018a
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2026-01-21T00:00:00-05:00
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Optimality Conditions for Sparse Bilinear Least Squares Problems
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arXiv:2601.13027v1 Announce Type: new Abstract: The first-order optimality conditions of sparse bilinear least squares problems are studied. The so-called T-type and N-type stationary points for this problem are characterized in terms of tangent cone and normal cone in Bouligand and Clarke senses, and another stationarity concept called the coordinate-wise minima is introduced and discussed. Moreover, the L-like stationary point for this problem is introduced and analyzed through the newly introduced concept of like-projection, and the M-stationary point is also investigated via a complementarity-type reformulation of the problem. The relationship between these stationary points is discussed as well. It turns out that all stationary points discussed in this work satisfy the necessary optimality conditions for the sparse bilinear least squares problem.
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https://arxiv.org/abs/2601.13027
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7a1b90ce1a3785718739f4905181c3c96e9dfea14f5e04082a42e5cc3e5ed524
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2026-01-21T00:00:00-05:00
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Generalized MICZ-Kepler systems on three-dimensional sphere and hyperboloid
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arXiv:2601.13028v1 Announce Type: new Abstract: We propose analogs of the generalized MICZ-Kepler system on the three-dimensional sphere and (two-sheet) hyperboloid. We then construct their energy spectra and normalized wave functions, concluding that the suggested systems are minimally superintegrable.
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https://arxiv.org/abs/2601.13028
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e897da308a98a10b5a3bf601758572198be14e7517d388e555d789707a560378
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2026-01-21T00:00:00-05:00
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Complete orbit equivalence relation and non-universal Polish groups
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arXiv:2601.13030v1 Announce Type: new Abstract: We show that a non-universal Polish group can induce a complete orbit equivalence relation, which answers a question of Sabok from \cite{OPENPROBLEMS}.
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https://arxiv.org/abs/2601.13030
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024299edbedda03a66e6b8b22d5f37c3db3d2c50e487870cb13e392ec4ddc6fe
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2026-01-21T00:00:00-05:00
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On the additive index of the Diffie-Hellman mapping and the discrete logarithm
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arXiv:2601.13034v1 Announce Type: new Abstract: Several complexity measures such as degree, sparsity and multiplicative index for cryptographic functions including the Diffie-Hellman mapping and the discrete logarithm in a finite field have been studied in the literature. In 2022, Reis and Wang introduced another complexity measure, the additive index, of a self-mapping of a finite field. In this paper, under certain conditions, we determine lower bounds on the additive index of the univariate Diffie-Hellman mapping and a self-mapping of $\mathbb{F}_q$ which can be identified with the discrete logarithm in a finite field.
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https://arxiv.org/abs/2601.13034
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275c4730d95ac4f8e80e21746387cd17b07ae1b3f243c5f9c08fb28f36a244e1
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2026-01-21T00:00:00-05:00
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Classification of quaternionic skew-Hermitian symmetric spaces
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arXiv:2601.13036v1 Announce Type: new Abstract: We provide a complete classification of quaternionic skew-Hermitian symmetric spaces, namely symmetric spaces that admit a torsion-free ${\rm SO}^{*}(2n){\rm Sp}(1)$-structure for arbitrary $n>1$. Moreover, we prove that any homogeneous quaternionic skew-Hermitian manifold is necessarily a symmetric space.
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https://arxiv.org/abs/2601.13036
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e4d47e65005052ac7914becfbbdabb0fb043e40e06054b5de859e4e32d13568b
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2026-01-21T00:00:00-05:00
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Markovian restless bandits and index policies: A review
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arXiv:2601.13045v1 Announce Type: new Abstract: The restless multi-armed bandit problem is a paradigmatic modeling framework for optimal dynamic priority allocation in stochastic models of wide-ranging applications that has been widely investigated and applied since its inception in a seminal paper by Whittle in the late 1980s. The problem has generated a vast and fast-growing literature from which a significant sample is thematically organized and reviewed in this paper. While the main focus is on priority-index policies due to their intuitive appeal, tractability, asymptotic optimality properties, and often strong empirical performance, other lines of work are also reviewed. Theoretical and algorithmic developments are discussed, along with diverse applications. The main goals are to highlight the remarkable breadth of work that has been carried out on the topic and to stimulate further research in the field.
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https://arxiv.org/abs/2601.13045
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10765c5f72466b46c18f56ce6ce58e1cbcec80db54dbcfbfa9246229d9376df9
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2026-01-21T00:00:00-05:00
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Optimal existence of weak solutions for the generalised Navier-Stokes-Voigt equations
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arXiv:2601.13051v1 Announce Type: new Abstract: In this study, we investigate the incompressible generalised Navier-Stokes-Voigt equations within a bounded domain $\Omega \subset \mathbb{R}^d$, where $d \geq 2$. The governing momentum equation is expressed as: \begin{align*} \partial_t(\boldsymbol{v} - \kappa \Delta \boldsymbol{v}) + \nabla \cdot (\boldsymbol{v} \otimes \boldsymbol{v}) + \nabla \pi - \nu \nabla \cdot \big( |\mathbf{D}(\boldsymbol{v})|^{p-2} \mathbf{D}(\boldsymbol{v}) \big) = \boldsymbol{f}. \end{align*} Here, for $d \in \{2,3\}$, $\boldsymbol{v}$ represents the velocity field, $\pi$ denotes the pressure, and $\boldsymbol{f}$ is the external forcing term. The constants $\kappa$ and $\nu$ correspond to the relaxation time and kinematic viscosity, respectively. The parameter $p \in (1, \infty)$ characterizes the fluid's flow behavior, and $\mathbf{D}(\boldsymbol{v})$ denotes the symmetric part of the velocity gradient $\nabla \boldsymbol{v}$. For the power-law exponent $p \in \big( \frac{2d}{d+2}, \infty \big)$, we establish the existence of a weak solution to the generalised Navier-Stokes-Voigt equations. Furthermore, we demonstrate that the weak solution is unique for the same range of the exponent $p$. The optimality of our results lies in the framework's use of a Gelfand triple, which allows the Aubin-Dubinskii lemma to yield strong convergence of approximate solutions, essential for existence and valid precisely for $p > \frac{2d}{d+2}$.
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https://arxiv.org/abs/2601.13051
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1b34c6df8087ce2d28f7c91d82850d6cba226eaaeae1725ea09c5afd62c10c85
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2026-01-21T00:00:00-05:00
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Period growth and co-context-free groups
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arXiv:2601.13058v1 Announce Type: new Abstract: We study period growth in co-context-free groups, giving general results and looking at specific examples such as Thompson groups $T$ and $V$ and the Houghton groups $H_m$. Along the way, we give a refined upper bound on the word metric in Thompson $V$, as well as efficient algorithms to determine if elements of $V$ are torsion, and compute their order. We also adapt our algorithm to compute the rotation number of elements of $T$ and answer a question of D. Calegari.
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https://arxiv.org/abs/2601.13058
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f3026924d32ef4a1e41e39e0db341b79ce6cbd5aecee7bc48f6d59c228736120
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2026-01-21T00:00:00-05:00
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Generalized Reproducing Kernel Banach Spaces: A Functional Analytic Framework for Abstract Neural Networks
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arXiv:2601.13062v1 Announce Type: new Abstract: In this paper, we introduce a generalization of Reproducing Kernel Banach Spaces (RKBS), which we term \emph{Generalized Reproducing Kernel Banach Spaces} (GRKBS). The motivation stems from recent results showing that classical fully connected neural networks can be understood as finite-dimensional subspaces of RKBS. Our generalization extends this perspective to settings with Banach-valued codomains, allowing the construction of \emph{abstract neural networks} (AbsNN) as compositions of GRKBS. This framework provides a natural pathway to model neural architectures that go beyond classical machine learning paradigms, including physically-informed structures governed by differential equations. We establish a unified definition of GRKBS, prove structural uniqueness results, and analyze the existence of sparse minimizers for the corresponding abstract training problem. This contributes to bridging functional analytic theory and the design of new neural architectures with applications in both approximation theory and mathematical modeling.
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https://arxiv.org/abs/2601.13062
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064fc0be8626c79d1fdd86ad1919df4685a1be15c588ba9bb3fe3a0ee0a57be4
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2026-01-21T00:00:00-05:00
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Non-abelian Hodge correspondence over singular K\"ahler spaces
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arXiv:2601.13071v1 Announce Type: new Abstract: In this paper, we establish the non-abelian Hodge correspondence over compact K\"ahler spaces with Kawamata log terminal (klt) singularities as well as over their regular loci, thereby extending the result of Greb-Kebekus-Peternell-Taji for projective klt varieties to the context of compact K\"ahler klt spaces. The proof relies on two key ingredients: first, we establish an equivalence over the regular loci-via harmonic bundles-between polystable Higgs bundles with vanishing orbifold Chern numbers and semi-simple flat bundles; second, we prove a descent theorem for semistable Higgs bundles with vanishing Chern classes along resolutions of singularities. As an application of our framework, we obtain a quasi-uniformization theorem for projective klt varieties with big canonical divisor that satisfy the orbifold Miyaoka-Yau equality.
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https://arxiv.org/abs/2601.13071
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9072236477770046764215ade45996867fbf86b86a0e919476f7861b47a89f77
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2026-01-21T00:00:00-05:00
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Faster 3-colouring algorithm for graphs of diameter 3
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arXiv:2601.13072v1 Announce Type: new Abstract: We show that given an $n$-vertex graph $G$ of diameter 3 we can decide if $G$ is $3$-colourable in time $2^{O(n^{2/3-\varepsilon})}$ for any $\varepsilon < 1/33$. This improves on the previous best algorithm of $2^{O((n\log n)^{2/3})}$ from D\k{e}bski, Piecyk and Rz\k{a}\.zewski [Faster 3-coloring of small-diameter graphs, ESA 2021].
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https://arxiv.org/abs/2601.13072
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7cdaacb32591203247ba56c6343fbab9deb05f71b5f8280288eccc23c8be2191
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2026-01-21T00:00:00-05:00
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Wasserstein geometry of nonnegative measures on finite Markov chains I: Gradient flow
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arXiv:2601.13073v1 Announce Type: new Abstract: We investigate a Benamou--Brenier type transportation metric for nonnegative measures on a finite reversible Markov chain, which endows the space of measures with a Riemannian structure. Using this geometric framework, we identify a generalized heat equation with source as the gradient flow of the discrete entropy. Moreover, by means of a local \L{}ojasiewicz inequality, we prove exponential convergence of the flow to a unique equilibrium. Our results clarify the role of the Benamou--Brenier formulation in discrete optimal transport for nonnegative measures and provide a coherent geometric interpretation of generalized diffusion equations with source terms.
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https://arxiv.org/abs/2601.13073
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37b8cb942dd499ba05cd82d0e620f433c79f6f242553efbb3a3b9f7ca45403bd
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2026-01-21T00:00:00-05:00
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Some results on the $\mathfrak{g}$-stability of surfaces with boundary
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arXiv:2601.13077v1 Announce Type: new Abstract: In this paper, we investigate the geometric properties associated with the $\mathfrak{g}$-stability of surfaces with boundary whose null expansion satisfies $\Theta^{+} = h \geq 0$. First, we show that a $\mathfrak{g}$-stable hypersurface with free boundary admits a metric of positive scalar curvature with minimal boundary under suitable conditions. Second, for $\mathfrak{g}$-stable surfaces with free boundary, we derive an area estimate and determine the topology of the surface. Finally, we extend our free boundary results to the case of capillary boundary.
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https://arxiv.org/abs/2601.13077
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1cd9885b6f18e54dbf3da26ab7d85a4219d9ec8496c820c094482a4c3484185d
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2026-01-21T00:00:00-05:00
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Wasserstein geometry of nonnegative measures on finite Markov chains II: Geodesic and duality formulae
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arXiv:2601.13080v1 Announce Type: new Abstract: In this paper, we investigate the geodesic structure and the associated Kantorovich-type duality for a Benamou-Brenier-type transportation metric defined on the space of nonnegative measures over a finite reversible Markov chain. The metric is introduced through a dynamic formulation that combines transport and source costs along solutions of a nonconservative continuity equation, where mass variation is constrained to occur along a fixed strictly positive reference direction. We show that geodesics associated with this metric exhibit a non-locality property: almost every time, they are supported on the whole state space, independently of the choice of endpoints. Moreover, along optimal curves, the source term displays a characteristic temporal profile, with mass creation occurring at early times and subsequent decay as the curve approaches the target measure. As an application of this property, we compare our metric with the shift-transport distance and prove that the latter is always bounded above by our metric. Finally, we establish a Kantorovich-type duality formula in terms of Hamilton-Jacobi subsolutions, which provides a characterization of the metric and highlights the role of the momentum associated with geodesic curves.
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https://arxiv.org/abs/2601.13080
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993848bb3d0e5bbf133334f53f86396d1aa83e3dcc3b0d489743ad130bc88ccd
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2026-01-21T00:00:00-05:00
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Further progress on Wojda's conjecture
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arXiv:2601.13085v1 Announce Type: new Abstract: Two digraphs of order $n$ are said to pack if they can be found as edge-disjoint subgraphs of the complete digraph of order $n$. It is well established that if the sum of the sizes of the two digraphs is at most $2n-2$, then they pack, with this bound being sharp. However, it is sufficient for the size of the smaller digraph to be only slightly below $n$ for the sum of their sizes to significantly exceed this threshold while still guaranteeing the existence of a packing. In 1985, Wojda conjectured that for any $2 \leq m \leq n/2$, if one digraph has size at most $n - m$ and the other has size less than $2n - \lfloor n/m \rfloor$, then the two digraphs pack. It was previously known that this conjecture holds for $m = \Omega(\sqrt{n})$. In this paper, we confirm it for $m \geq 93$ and $n \geq 31m$.
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https://arxiv.org/abs/2601.13085
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769123457e421562ed5116066e9f2219a6cd9e5d9ed738bf68c0f9542e744592
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2026-01-21T00:00:00-05:00
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Brownian Loops and the Selberg Zeta Function
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arXiv:2601.13086v1 Announce Type: new Abstract: We study the Brownian loop measure on hyperbolic surfaces for Brownian motion with a constant killing rate. We compute the mass of Brownian loops with killing in a free homotopy class and then relate the total mass of loops in all essential homotopy classes to the Selberg zeta function when the surface is geometrically finite. As an application, we provide a probabilistic interpretation of different notions of regularized determinants of Laplacian, in both the compact and infinite-area cases.
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https://arxiv.org/abs/2601.13086
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f7f1b3a856b0023b62655e8a08532a4a5ae1ed0b6a8aef9b9a94f8edd8530eab
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2026-01-21T00:00:00-05:00
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Dynamical boundaries of affine buildings: C*-simplicity and Poisson boundaries
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arXiv:2601.13092v1 Announce Type: new Abstract: We investigate a class of groups acting on possibly exotic affine buildings $X$ and possessing good proximal properties. Such groups are termed of general type, and their dynamics is analyzed through their flag limit sets in the space of chambers at infinity of $X$. For a group $G$ of general type, we prove C*-simplicity by showing that its flag limit set $\Lambda_{\mathcal F}(G)$ is topologically free, minimal, and strongly proximal. When $\Lambda_{\mathcal F}(G)$ intersects all Schubert cells relative to a limit chamber, then it is a mean proximal space, in the sense that it carries a unique proximal stationary measure for any admissible probability measure on the acting group. Lattices are established as examples of groups of general type, and their Poisson boundaries are identified. The arguments rely on constructing an equivariant barycenter map from triples of chambers in generic position to the affine building.
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https://arxiv.org/abs/2601.13092
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28e8b1e8693a86890d8d91dbeac74699e7d24fc4303c68bf433a98513aa5286e
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2026-01-21T00:00:00-05:00
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Quasi-maximal ideals and ring extensions
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arXiv:2601.13093v1 Announce Type: new Abstract: Alan and al. defined and studied quasi-maximal ideals. We add a comprehensive characterization of these ideals, introducing submaximal ideals. The conductor of a finite minimal extension $R\subset S$ is quasi-maximal in $S$. This allows us to give a new characterization of these extensions. We also examine the links between quasi-maximal ideals and Badawi 2-absorbing ideals.
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https://arxiv.org/abs/2601.13093
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93c27a25c24719f56862ac8778cff088a8011b5b07638e9f9b4c1c477dd080f6
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2026-01-21T00:00:00-05:00
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Equiprojective polytopes in higher dimension
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arXiv:2601.13095v1 Announce Type: new Abstract: A 3-dimensional polytope is called k-equiprojective if every planar projection along a direction non-parallel to any facet is a k-gon. In this article, we generalise equiprojectivity to higher dimensions and give a lower bound on the number of combinatorial types of equiprojective polytopes. We also establish the pathwise connectedness of a subset of the Grassmannian in the case of (d-2)-dimensional spaces with conditions on the explicit path. This makes it possible to extend the Hasan--Lubiw characterisation of equiprojectivity to higher dimensions. Equiprojectivity provides cases relevant to the study of the Shadow Vertex algorithm, showing there is no hope minimising the complexity of the projection. It also offers a reverse point of view on the usual study of planar projections of polytopes as the projections have a fixed size.
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https://arxiv.org/abs/2601.13095
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a3ad0e43281dc5117b56bad0511ff09c256400e933079926cf6a31d9306357b9
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2026-01-21T00:00:00-05:00
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Parallel mean curvature surfaces with constant contact angle along free boundaries
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arXiv:2601.13101v1 Announce Type: new Abstract: We classify branched immersed disks in space forms with non-zero parallel mean curvature vector and non-orthogonal constant contact angle along the boundary in 4-dimensional space form. For higher codimensional case, we prove a codimension reduction theorem for branched immersed bordered Riemann surfaces of higher genus with multiple boundary components under the same parallel mean curvature and constant contact angle assumptions. Furthermore, we construct a family of explicit examples of branched minimal immersions satisfying the non-orthonormal constant contact angle free boundary condition, which demonstrate the sharpness of both the classification result and the codimension reduction result.
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https://arxiv.org/abs/2601.13101
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17bd9ee89d13c9ab2b0ea36eeb16b541d3d1425a1b224668bf1e1f00e4176e25
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2026-01-21T00:00:00-05:00
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Full characterization of core for nonlinear optimization games
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arXiv:2601.13124v1 Announce Type: new Abstract: We fully characterize the core of a broad class of nonlinear games by identifying a suitable relaxation for inherent nonlinearity, directly generalizing the linear frameworks in the literature. This characterization significantly expands the scope of cooperative games that can be analyzed and contributes to the literature on games induced from optimization models. We apply these insights to not only establish connections with and provide new insights on classical models but also solve new games untamed in the existing literature, including combinatorial quadratic and ratio games such as portfolio, maximum cut, matching, and assortment games. These results are further extended to more general models and also the approximate core.
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https://arxiv.org/abs/2601.13124
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59be1d79800030befb262696ac9256c3e8ed805554de42dfdd77492b211c207a
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2026-01-21T00:00:00-05:00
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An algebro-geometric perspective on the topology of moduli spaces of differentials
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arXiv:2601.13127v1 Announce Type: new Abstract: Differentials on Riemann surfaces correspond to translation surfaces with conical singularities, and affine transformations acting on them preserve the orders of these singularities. This viewpoint allows the moduli spaces of differentials to appear in various guises across many areas, including algebraic geometry, dynamical systems, combinatorial enumeration, and mathematical physics. Over the past few decades, remarkable progress has been made in computing invariants of these moduli spaces, classifying linear subvarieties, understanding degenerations and compactifications, and developing intersection theory on these spaces. Despite these advances, our understanding of the topology of moduli spaces of differentials remains limited, and many fundamental questions are still open. In this survey, we aim to present, from an algebro-geometric perspective, the known results and open problems concerning the topology of moduli spaces of differentials, as well as their connections to other aspects of the field, with the hope of inspiring further developments in the coming decade.
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https://arxiv.org/abs/2601.13127
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fd8e1cf576cfd07a3fa8b5a411c17d22c7533ec5eb3499b54b072b09f98a9760
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2026-01-21T00:00:00-05:00
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On the splitting of Neumann eigenvalues in perforated domains
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arXiv:2601.13129v1 Announce Type: new Abstract: We address the problem of splitting of eigenvalues of the Neumann Laplacian under singular domain perturbations. We consider a domain perturbed by the excision of a small spherical hole shrinking to an interior point. Our main result establishes that the splitting of multiple eigenvalues is a generic property: if the center of the hole is located outside a set of Hausdorff dimension $N-1$ and the radius is sufficiently small, multiple eigenvalues split into branches of lower multiplicity. The proof relies on the validity of an asymptotic expansion for the perturbed eigenvalues in terms of the scaling parameter. Such an asymptotic formula is of independent interest and generalizes previous results; notably, in dimension $N\geq 3$, it is valid for holes of arbitrary shape.
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https://arxiv.org/abs/2601.13129
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6d2425344de4134604a211e98241935181bcfda097e9ac38d3d4e9a80a5ce18d
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2026-01-21T00:00:00-05:00
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Age of information cost minimization with no buffers, random arrivals and unreliable channels: A PCL-indexability analysis
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arXiv:2601.13130v1 Announce Type: new Abstract: Over the last decade, the Age of Information has emerged as a key concept and metric for applications where the freshness of sensor-provided data is critical. Limited transmission capacity has motivated research on the design of tractable policies for scheduling information updates to minimize Age of Information cost based on Markov decision models, in particular on the restless multi-armed bandit problem (RMABP). This allows the use of Whittle's popular index policy, which is often nearly optimal, provided indexability (index existence) is proven, which has been recently accomplished in some models. We aim to extend the application scope of Whittle's index policy in a broader AoI scheduling model. We address a model with no buffers incorporating random packet arrivals, unreliable channels, and nondecreasing AoI costs. We use sufficient indexability conditions based on partial conservation laws previously introduced by the author to establish the model's indexability and evaluate its Whittle index in closed form under discounted and average cost criteria. We further use the index formulae to draw insights on how scheduling priority depends on model parameters.
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https://arxiv.org/abs/2601.13130
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71d9d50e63eea26be2f4e7f793d957f088e9b169ef1d69ecf438968352408d18
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2026-01-21T00:00:00-05:00
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The descriptive complexity of the set of arc-connected compact subsets of the plane
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arXiv:2601.13135v1 Announce Type: new Abstract: We compute the exact complexity of the set of all arc-connected compact subsets of $\boldmath R^2$, which turns out to be strictly higher than the classical $\boldmath \Sigma^1_1$ and $\boldmath \Pi^1_1$ classes of analytic and coanalytic sets, but stricly lower than the class $\boldmath \Pi^1_2$ which is the exact descriptive class of the set of all arc-connected compact subsets of $\boldmath R^3$.
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https://arxiv.org/abs/2601.13135
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5dedd82eaf179be2f58a6183df3b0734c381d4e1695143e883cae02551964296
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2026-01-21T00:00:00-05:00
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Blackwell optimality in risk-sensitive stochastic control
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arXiv:2601.13136v1 Announce Type: new Abstract: In this paper, we consider a discrete-time Markov Decision Process (MDP) on a finite state-action space with a long-run risk-sensitive criterion used as the objective function. We discuss the concept of Blackwell optimality and comment on intricacies which arise when the risk-neutral expectation is replaced by the risk-sensitive entropy. Also, we show the relation between the Blackwell optimality and ultimate stationarity and provide an illustrative example that helps to better understand the structural difference between these two concepts.
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https://arxiv.org/abs/2601.13136
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f461dd3918b03e4ffb362761c87225e1de09cbbaff6e4b3ae6d700fb404ad5e0
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2026-01-21T00:00:00-05:00
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Classical Optimal Designs for Stationary Diffusion with Multiple Phases
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arXiv:2601.13149v1 Announce Type: new Abstract: We study optimal design problems for stationary diffusion involving one or more state equations and mixtures of an arbitrary number of anisotropic materials. Since such problems typically do not admit classical solutions, we adopt a homogenization-based relaxation framework. The objective considered is the maximization of a weighted sum of the energies associated with each state equation, with particular emphasis on identifying cases in which the optimal design is classical, that is, of bang-bang type, composed solely of the original pure materials. Such cases provide valuable benchmarks for numerical methods in optimal design. A simplified optimization problem expressed in terms of local material proportions is analyzed through a dual formulation in terms of fluxes. Using a saddle-point characterization, we establish a complete description of its optimal solutions. The proposed approach is applied in detail to spherically symmetric problems. In the case of a ball, the method yields explicit classical solutions of the homogenization-based relaxation problem.
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https://arxiv.org/abs/2601.13149
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09a99c13e9d80e5d1d5e760d6ed0bdd7362402fd94e02e132d1f3adac46d01f5
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2026-01-21T00:00:00-05:00
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Factoriality of normal projective varieties
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arXiv:2601.13151v1 Announce Type: new Abstract: For a normal projective variety $X$, the $\bf Q$-factoriality defect $\sigma(X)$ is defined to be the rank of the quotient of the group of Weil divisors by the subgroup of Cartier ones. We prove a slight improvement of a topological formula of S.G. Park and M. Popa asserting that $\sigma(X)=h^{2n-2}(X)-h^2(X)$ by assuming only 2-semi-rationality, that is, $R^k\pi_*{\mathcal O}_{\widetilde{X}}=0$ for $k=1,2$, instead of rational singularities for $X$, where $\pi:\widetilde{X}\to X$ is a desingularization with $h^k(X):=\dim H^k(X,{\bf Q})$ and $n:=\dim X>2$. Our proof generalizes the one by Y. Namikawa and J.H.M. Steenbrink for the case $n=3$ with isolated hypersurface singularities. We also give a proof of the assertion that $\bf Q$-factoriality implies factoriality if $X$ is a local complete intersection whose singular locus has at least codimension three. (This seems to be known to specialists in the case $X$ has only isolated hypersurface singularities with $n=3$ using Milnor's Bouquet theorem.) These imply another proof of Grothendieck's theorem in the projective case asserting that $X$ is factorial if $X$ is a local complete intersection whose singular locus has at least codimension four. We can also prove a variant with factorial and local complete intersection replaced respectively by $\bf Q$-factorial and Cohen-Macaulay, where $\bf Q$-factorial cannot be replaced by factorial.
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https://arxiv.org/abs/2601.13151
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c4a803e3ce9e34c18b4462c0eebb22f00b849dfc319fbf78cfc38d8658a614f8
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2026-01-21T00:00:00-05:00
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Character degrees in $2$-blocks of $\mathfrak{S}_n$ and $\mathfrak{A}_n$
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arXiv:2601.13152v1 Announce Type: new Abstract: Let $p$ be an odd prime. We show that for sufficiently large $n$, every $2$-block of $\mathfrak{S}_n$ and $\mathfrak{A}_n$ contains an ordinary irreducible character of degree divisible by $p$. For almost all $2$-blocks of $\mathfrak{A}_n$, we classify whether it contains a rational valued ordinary irreducible character of degree divisible by $p$.
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https://arxiv.org/abs/2601.13152
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40876afe9a2dcc3131255bd5f1a30f63165aa9eea6ae6e22fd3f2ee589a47918
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2026-01-21T00:00:00-05:00
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On the characteristic function of the asymmetric Student's $t$-distribution and an integral involving the sine function
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arXiv:2601.13158v1 Announce Type: new Abstract: We obtain a new closed-form formula for the characteristic function of the asymmetric Student's $t$-distribution. As part of our analysis, we derive a new closed-form formula for the integral $\int_0^\infty \sin(ax)/(b^2+x^2)^n\,\mathrm{d}x$, for $a,b>0$, $n\in\mathbb{Z}^+$, expressed in terms of the exponential integral function. As a consequence of our integral formula, we deduce a closed-form formula for the limit $\lim_{\nu\rightarrow n} \{I_{\nu-1/2}(x)-\mathbf{L}_{1/2-\nu}(x)\}/\sin(\pi\nu)$, for $n\in\mathbb{Z}^+$, $x>0$.
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https://arxiv.org/abs/2601.13158
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375a0da3d16b7cbdc240c048b8bb45af6ef0bf470697b697c3804435bd76ef33
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2026-01-21T00:00:00-05:00
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On the discrete logarithmic Minkowski problem in the plane
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arXiv:2601.13159v1 Announce Type: new Abstract: The paper characterizes the convex hull of the closure of the cone-volume set $C_\cv(U)$, consisting of all cone-volume vectors of polygons with outer unit normals vectors contained in $U$, for any finite set $U \subseteq \R^2, \pos(U) = \R^2$. We prove that this convex hull has finitely many extreme points by providing both a vertex representation as well as a half space representation. As a consequence, we derive new necessary conditions, which depend on $U$, for the existence of solutions to the logarithmic Minkowski problem in $\R^2$.
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https://arxiv.org/abs/2601.13159
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aceed42217ae5c7c0dec16d69a1f6b2e9f3ba1a942b978c92920dc8543e56a77
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2026-01-21T00:00:00-05:00
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A new notion of dimension for dynamical systems and shift embeddability
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arXiv:2601.13161v1 Announce Type: new Abstract: A dynamical system $(X,T)$ is \emph{shift embaddable} if $(X,T)$ embeds continuously and equivariantly in the shift over $[0,1]^d$ for some finite $d$. Refuting a major conjecture in the field, in a recent result of Dranishnikov and Levin it was shown that Gromov's mean dimension and Lebesgue covering dimension of finite orbits are not the only obstructions for shift embaddability. We present a new notion of dimension for dynamical systems over any countable group. We show that this new notion of dimension accounts for all known obstructions for shift embaddability.
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https://arxiv.org/abs/2601.13161
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f9a7ac057b31c74fdcb2a3da6d43cf59f7fffa10c8efbc0710349e7d752e25cf
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2026-01-21T00:00:00-05:00
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Nash approximation of differentiable semialgebraic maps
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arXiv:2601.13164v1 Announce Type: new Abstract: Let $T\subset{\mathbb R}^n$ be a semialgebraic set and let $\mu\ge0$ be a non-negative integer. We say that $T$ is a {\em Nash $\mu$-approximation target space} (or a $({\mathcal N},\mu)$-${\tt ats}$ for short) if it has the following universal approximation property: {\em For each $m\in{\mathbb N}$ and each locally compact semialgebraic subset $S\subset{\mathbb R}^m$, the subspace of Nash maps ${\mathcal N}(S,T)$ is dense in the space ${\mathcal S}^\mu(S,T)$ of ${\mathcal C}^\mu$ semialgebraic maps between $S$ and $T$}. A necessary condition to be a $({\mathcal N},\mu)$-${\tt ats}$ is that $T$ is locally connected by analytic paths. In this paper we show: {\em Nash manifolds with corners are $({\mathcal N},\mu)$-${\tt ats}$ for each $\mu\geq0$}. As an application of a stronger version of the previous statement, we show that if two Nash maps $f,g:S\to Q$, where $S$ is a locally compact semialgebraic set of ${\mathbb R}^m$ and $Q$ is a Nash manifold with corners, are close enough in the (strong) Whitney's semialgebraic topology of ${\mathcal S}^0(S,T)$ (and consequently they are (continuous) semialgebraically homotopic), then $f,g$ are Nash homotopic.
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https://arxiv.org/abs/2601.13164
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7897fcc604a47af1a65c9efb3fcc50216fe00c22174135407d65b4fbd4fa4487
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2026-01-21T00:00:00-05:00
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PDE aspects of the dynamical optimal transport in the Lorentzian setting
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arXiv:2601.13167v1 Announce Type: new Abstract: One of the crucial features of optimal transport on Riemannian manifolds is the equivalence of the `static', original, formulation of the problem and of the `dynamic' one, based on the study of the continuity equation. This furnishes the key link between Wasserstein geometry and PDEs that has found so many applications in the last 20 years. In this paper we investigate this kind of equivalence on spacetimes. At the PDE level, this requires to transition from the continuity equation to a suitable `continuity inequality', to which we shall refer to as `causal continuity inequality'. As a direct consequence of our findings we obtain a Lorentzian version of the celebrated Benamou--Brenier formula.
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https://arxiv.org/abs/2601.13167
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992818d4a4caf8345c2cbff32da37980b702d8b5e5cbf63c271c5fbefd7ae3cb
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2026-01-21T00:00:00-05:00
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Calculating The Local Ideal Class Monoid and Gekeler Ratios
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arXiv:2601.13184v1 Announce Type: new Abstract: Let $A = \mathbb{F}_q[T]$, $\mathfrak{p} \subset A$ prime, $f(x) \in A[x]$ irreducible and set $R = A[x]/f(x)$. Denote its completion by $R_\mathfrak{p}$. The ideal class monoid $\text{ICM}(R_\mathfrak{p})$ is the set of fractional $R_\mathfrak{p}$ ideals modulo the principal $R_\mathfrak{p}$ ideals. We provide an algorithm to compute $\text{ICM}(R_\mathfrak{p})$. In the process we also get algorithms to compute the overorders and weak equivalence classes of $R_\mathfrak{p}$. We then use the algorithms to compute the product of local Gekeler ratios $\prod_{\mathfrak{p} \subset A} v_\mathfrak{p}(f) = \prod_{\mathfrak{p} \subset A} \lim_{n \rightarrow \infty} \frac{|\{M \in \text{Mat}_r(A/\mathfrak{p}^n)\mid \text{charpoly}(M)=f\}}{|\text{SL}_r(A/\mathfrak{p}^n)|/|\mathfrak{p}|^{n(r-1)}}$. This provides part of an algorithm to compute the weighted size of an isogeny class of Drinfeld modules.
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https://arxiv.org/abs/2601.13184
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70dd50e9614d9ac03b60fecdac6b9c448121888f7f965d11ca24726bea4a4a7e
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2026-01-21T00:00:00-05:00
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Radicals of Lie-solvable Novikov algebras
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arXiv:2601.13185v1 Announce Type: new Abstract: We prove that in a Lie-solvable Novikov algebra, the Baer radical coincides with the set of all right-nilpotent elements, and the Andrunakievich radical coincides with the largest left-quasiregular ideal. We investigate the stability of some properties of commutative algebras with derivation after applying the Gelfand-Dorfman construction.
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https://arxiv.org/abs/2601.13185
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c14228e96905a53d2d839722e1ff93729141a46c9b50dcc3b3b61039d1e7403b
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2026-01-21T00:00:00-05:00
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Onsager's Mean Field Theory of Vortex Flows with Singular Sources: Blow-Up and Concentration without Quantization
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arXiv:2601.13192v1 Announce Type: new Abstract: Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we make a first step in the generalization of the mean field theory in [Caglioti, Lions, Marchioro, Pulvirenti; Comm. Math. Phys. (1995)]. On one side we prove the equivalence of statistical ensembles, on the other side we are bound to the analysis of a new blow up phenomenon, which we call "blow up and concentration without quantization", where the mass associated with the concentration is allowed to take values in a full interval of real numbers. This singular behavior may be regarded as lying between the classical blow up-concentration-quantization and the blow up without concentration phenomenon first proposed in [Lin, Tarantello; C.R. Math. Acad. Sci. Paris (2016)]. A careful analysis is needed to generalize known pointwise estimates in this non standard context, resulting in a complete description of the allowed asymptotic profiles.
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https://arxiv.org/abs/2601.13192
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5cd9de52116dd25204edcfd5fbb379d31ee26fd04f4ef29000feeb63c5e6d0a2
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2026-01-21T00:00:00-05:00
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Asymptotic Stability of Rarefaction Waves for the Hyperbolized Navier-Stokes-Fourier System
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arXiv:2601.13193v1 Announce Type: new Abstract: This paper investigates the asymptotic stability of rarefaction waves for a one-dimensional compressible fluid system, where the Newton's law of viscosity and Fourier's law of heat conduction are replaced by Maxwell's law and Cattaneo's law, respectively. The system, which generalizes the classical Navier-Stokes-Fourier equations, features finite signal propagation speeds. We consider the Cauchy problem in Lagrangian coordinates with initial data connecting two different constant states via a rarefaction wave of the corresponding Euler system. Our main result proves that, provided the initial perturbation and wave strength are sufficiently small, the relaxation system admits a unique global solution. Furthermore, this solution converges uniformly to the background rarefaction wave as time approaches infinity. The proof is established through a combination of the relative entropy method and usual energy estimates.
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https://arxiv.org/abs/2601.13193
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180355a01125456592effb0d959ac5f9a0960eedd9fbd32f014fd340115e4c11
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2026-01-21T00:00:00-05:00
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A Lower Bound on the Expected Number of Distinct Patterns in a Random Permutation
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arXiv:2601.13194v1 Announce Type: new Abstract: Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. The authors of [2] showed that the expected number of distinct consecutive patterns of all lengths $k\in\{1,2,\ldots,n\}$ in $\pi_n$ was $\frac{n^2}{2}(1-o(1))$ as $n\to\infty$, exhibiting the fact that random permutations pack consecutive patterns near-perfectly. A conjecture was made in [11] that the same is true for non-consecutive patterns, i.e., that there are $2^n(1-o(1))$ distinct non-consecutive patterns expected in a random permutation. This conjecture is false, but, in this paper, we prove that a random permutation contains an expected number of at least $2^{n-1}(1+o(1))$ distinct permutations; this number is half of the range of the number of distinct permutations.
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https://arxiv.org/abs/2601.13194
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59654c33e1aae851af16c05a4446d896e0c12ad604edb025bb41290af3fc4534
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2026-01-21T00:00:00-05:00
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AI for Mathematics: Progress, Challenges, and Prospects
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arXiv:2601.13209v1 Announce Type: new Abstract: AI for Mathematics (AI4Math) has emerged as a distinct field that leverages machine learning to navigate mathematical landscapes historically intractable for early symbolic systems. While mid-20th-century symbolic approaches successfully automated formal logic, they faced severe scalability limitations due to the combinatorial explosion of the search space. The recent integration of data-driven approaches has revitalized this pursuit. In this review, we provide a systematic overview of AI4Math, highlighting its primary focus on developing AI models to support mathematical research. Crucially, we emphasize that this is not merely the application of AI to mathematical activities; it also encompasses the development of stronger AI systems where the rigorous nature of mathematics serves as a premier testbed for advancing general reasoning capabilities. We categorize existing research into two complementary directions: problem-specific modeling, involving the design of specialized architectures for distinct mathematical tasks, and general-purpose modeling, focusing on foundation models capable of broader reasoning, retrieval, and exploratory workflows. We conclude by discussing key challenges and prospects, advocating for AI systems that go beyond facilitating formal correctness to enabling the discovery of meaningful results and unified theories, recognizing that the true value of a proof lies in the insights and tools it offers to the broader mathematical landscape.
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https://arxiv.org/abs/2601.13209
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dadc97f03274d221d9f21400c911081fefbe67f417f3d758bef0cce4aa152dc4
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2026-01-21T00:00:00-05:00
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A Harnack-type inequality for a perturbed singular Liouville Equation
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arXiv:2601.13212v1 Announce Type: new Abstract: Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we obtain a Harnack-type inequality for sequences of solutions of the following perturbed Liouville equation, \begin{equation}\nonumber -\Delta v_n=\left({\epsilon_n^2+|x|^2}\right)^{\alpha_n}V_n(x)e^{\displaystyle v_n} \qquad\text{in} \,\,\, \Omega, \end{equation} where $\epsilon_n\to0^+$, $\alpha_n\to\alpha_\infty\in(-1,1)$, $\Omega$ is a bounded domain in $\mathbb{R}^2$ containing the origin and $V_n$ satisfies, \begin{equation}\nonumber 0<+\infty, \,\, V_n\in C^{0}(\Omega), \,\,V_n\to V \,\, \text{locally uniformly in}\,\,{\Omega}. \end{equation}
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https://arxiv.org/abs/2601.13212
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678418897ecb4c3821370e383dd7f2093505e008a93919f3f01504d10572bd43
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2026-01-21T00:00:00-05:00
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Volume polynomials
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arXiv:2601.13249v1 Announce Type: new Abstract: Volume polynomials form a distinguished class of log-concave polynomials with remarkable analytic and combinatorial properties. I will survey realization problems related to them, review fundamental inequalities they satisfy, and discuss applications to the combinatorics of algebraic matroids. These notes are based on lectures given at the 2025 Summer Research Institute in Algebraic Geometry at Colorado State University.
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https://arxiv.org/abs/2601.13249
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3fbc10cc58d293294c25a58682b3dd3e363d470946cc8b0d9cd0f7d9d9cac64a
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2026-01-21T00:00:00-05:00
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Inverting the Fisher information operator in non-linear models
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arXiv:2601.13254v1 Announce Type: new Abstract: We consider non-linear regression models corrupted by generic noise when the regression functions form a non-linear subspace of L^2, relevant in non-linear PDE inverse problems and data assimilation. We show that when the score of the model is injective, the Fisher information operator is automatically invertible between well-identified Hilbert spaces, and we provide an operational characterization of these spaces. This allows us to construct in broad generality the efficient Gaussian involved in the classical minimax and convolution theorems to establish information lower bounds, that are typically achieved by Bayesian algorithms thus showing optimality of these methods. We illustrate our results on time-evolution PDE models for reaction-diffusion and Navier-Stokes equations.
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https://arxiv.org/abs/2601.13254
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aa0ddca20aab7e45d725f34e0086acd30a5f3aebf97d9bcb220a09284c5b412b
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2026-01-21T00:00:00-05:00
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Entropy-Wasserstein regularization, defective local concentration and a cutoff criterion beyond non-negative curvature
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arXiv:2601.13259v1 Announce Type: new Abstract: Notions of positive curvature have been shown to imply many remarkable properties for Markov processes, in terms, e.g., of regularization effects, functional inequalities, mixing time bounds and, more recently, the cutoff phenomenon. In this work, we are interested in a relaxed variant of Ollivier's coarse Ricci curvature, where a Markov kernel $P$ satisfies only a weaker Wasserstein bound $W_p(\mu P, \nu P) \leq K W_p(\mu,\nu)+M$ for constants $M\ge 0, K\in [0,1], p \ge 1$. Under appropriate additional assumptions on the one-step transition measures $\delta_x P$, we establish (i) a form of local concentration, given by a defective Talagrand inequality, and (ii) an entropy-transport regularization effect. We consider as illustrative examples the Langevin dynamics and the Proximal Sampler when the target measure is a log-Lipschitz perturbation of a log-concave measure. As an application of the above results, we derive criteria for the occurrence of the cutoff phenomenon in some negatively curved settings.
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https://arxiv.org/abs/2601.13259
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155fe0d9e1e7c6257d99340c034b10a3bf46faaf9d700132d4b9502507e9a13e
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2026-01-21T00:00:00-05:00
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Covariant tomography of fields
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arXiv:2601.13261v1 Announce Type: new Abstract: This paper addresses the Inverse Boundary Value Problem (IBVP) for classical fields, specifically focusing on the recovery of parallelly transformed fields within a region based on known boundary data. We introduce a local solution framework, termed "covariant tomography," that uses geometric decomposition to reconstruct interior fields and currents within star-shaped open subsets. The core of our approach involves decomposing differential forms into exact and antiexact components, enabling the formulation of the parallel transport equation via a homotopy operator. We examine three primary extension techniques - radial, heat equation, and harmonic - to map boundary values into the interior, noting that the choice of extension directly influences the regularity of the resulting currents. The proposed methodology provides a systematic way to identify the realizability of boundary values and offers solutions for both current and gauge field tomography. Finally, we demonstrate the utility of this framework through illustrative examples in low-dimensional spaces and electromagnetic potential reconstruction in $\mathbb{R}^{3}$.
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https://arxiv.org/abs/2601.13261
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e67a9e8de21faae91184161f339ccf7e2c3896551cad0671eb23290b6357cbad
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2026-01-21T00:00:00-05:00
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On surfaces with smooth projective models over $\mathbb{Z}$
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arXiv:2601.13277v1 Announce Type: new Abstract: In this expository article, we prove a birational classification of smooth projective models of surfaces with negative Kodaira dimension over $\mathbb{Z}$ and over more general rings of integers $\mathcal{O}_K$, depending on their arithmetic and cohomological invariants. Along the way we collect some results on smooth projective models of surfaces over Dedekind domains.
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https://arxiv.org/abs/2601.13277
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607b75626eb8a60039e7d5c26c91a6ee177657ccd8cc06ceab2e36ec0041450f
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2026-01-21T00:00:00-05:00
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On constructing topology from algebra
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arXiv:2601.13279v1 Announce Type: new Abstract: In this thesis we explore natural procedures through which topological structure can be constructed from specific semigroups. We will do this in two ways: 1) we equip the semigroup object itself with a topological structure, and 2) we find a topological space for the semigroup to act on continuously. We discuss various minimum/maximum topologies which one can define on an arbitrary semigroup (given some topological restrictions). We give explicit descriptions of each these topologies for the monoids of binary relations, partial transformations, transformations, and partial bijections on a countable set. Using similar methods we determine whether or not each of these semigroups admits a unique Polish semigroup topology. We also do this for the various other semigroups, provide a proof of Rubin's theorem, and give a description of the automorphism groups of the Brin-Thompson groups. The thesis also contains many background results.
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https://arxiv.org/abs/2601.13279
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3ef06b79d5e22dec6e4c94268f293131520dbbd78fdc02589c2855bfb80e5953
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2026-01-21T00:00:00-05:00
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Total curvature of convex hypersurfaces in Cartan-Hadamard manifolds
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arXiv:2601.13280v1 Announce Type: new Abstract: We show that if the curvature of a Cartan-Hadamard $n$-manifold is constant near a convex hypersurface $\Gamma$, then the total Gauss-Kronecker curvature $\mathcal{G}(\Gamma)$ is not less than that of any convex hypersurface nested inside $\Gamma$. This extends Borb\'{e}ly's monotonicity theorem in hyperbolic space. It follows that $\mathcal{G}(\Gamma)$ is bounded below by the volume of the unit sphere in Euclidean space $\mathbf{R}^n$.
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https://arxiv.org/abs/2601.13280
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ee3e26659cd6593c61f8bd8959c9d38cc6e2885aa54aa5aa938c219c8b4f6a2a
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2026-01-21T00:00:00-05:00
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Second order periodic boundary value problems with reflection and piecewise constant arguments
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arXiv:2601.13291v1 Announce Type: new Abstract: In this paper, we analyze a second-order differential equation with a piecewise constant argument and reflection coupled to periodic boundary conditions. Our main contribution is the construction of the related Green's function and a detailed analysis of its properties. In particular, we determine the region in which the Green's function has constant sign, depending on the parameters $m$ and $M$ on which it depends. In some cases, we are able to characterize these parameter values in terms of the first eigenvalue related to suitable Dirichlet problems. Building in these results, we apply the Krasnosel'skii method to establish the existence of solutions for different nonlinear problems, and prove the existence of a positive solution of a perturbed Schrodinger equation.
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https://arxiv.org/abs/2601.13291
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7f0b5a70f83680c733be83ebb952d590fd15991e86c325dd577a37c0f7a2f3a6
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2026-01-21T00:00:00-05:00
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Long-time behavior of solutions to fluid dynamic shape optimization problems via phase-field method
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arXiv:2601.13293v1 Announce Type: new Abstract: We investigate the long time behavior of solutions to a shape and topology optimization problem with respect to the time-dependent Navier--Stokes equations. The sought topology is represented by a stationary phase-field that represents a smooth indicator function. The fluid equations are approximated by a porous media approach and are time-dependent. In the latter aspect, the considered problem formulation extends earlier work. We prove that if the time horizon tends to infinity, minima of the time-dependent problem converge towards minima of the corresponding stationary problem. To do so, a convergence rate with respect to the time horizon, of the values of the objective functional, is analytically derived. This allowed us to prove that the solution to the time-dependent problem converges to a phase-field, as the time horizon goes to infinity, which is proven to be a minimizer for the stationary problem. We validate our results by numerical investigation.
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https://arxiv.org/abs/2601.13293
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b3ff3ecb5b9fc0144717dc1233f750639173610cb2b274f933600e865482aefa
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2026-01-21T00:00:00-05:00
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Limit Theorems for $\theta$-expansions and the Failure of the Strong Law
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arXiv:2601.13296v1 Announce Type: new Abstract: The paper presents fundamental metrical theorems for a class of continued fraction-like expansions known as $\theta$-expansions. We first prove Khinchine's Weak Law of Large Numbers for the sum of digits, followed by the Diamond-Vaaler Strong Law for the sum of digits minus the largest one. Our main result is a general theorem on the failure of the strong law, showing that no regular norming sequence can yield a finite, non-zero almost sure limit. This result extends a classical theorem of Philipp to the $\theta$-expansion setting. The proofs leverage the system's explicit invariant measure and a detailed analysis of its mixing properties.
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https://arxiv.org/abs/2601.13296
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6ac60e91145df2ce1ffdcd7188bb0868c3faa9999be1b94f690497b0db061ea3
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2026-01-21T00:00:00-05:00
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Locally involutive semigroups
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arXiv:2601.13301v1 Announce Type: new Abstract: We introduce locally involutive semigroups and embed them into the category of ordered groupoids. This embedding restricts to a correspondence between quasi-involutive semigroups and ordered groupoids with mediator, extending the classical ESN-correspondence between inverse semigroups and inductive groupoids. An important subcategory of locally involutive semigroups is formed by left involutive semigroups because the classifying topos of an inverse semigroup S is equivalent to the category of left involutive semigroups \'etale over S [4]. We recover this equivalence from a general adjointness and use the latter to determine when a left involutive semigroup \'etale over S is actually an involutive semigroup. Any left involutive semigroup \'etale over S embeds into an involutive S-algebra as we call it. The underlying semigroup of this algebra is involutive.
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https://arxiv.org/abs/2601.13301
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0c30ad50684e52cadb8855f622f93d1a8fe2ecea84e0bd3df66cb70680835ac7
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2026-01-21T00:00:00-05:00
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Structured eigenbases and pair state transfer on threshold graphs
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arXiv:2601.13318v1 Announce Type: new Abstract: Recently, Macharete, Del-Vecchio, Teixeira and de Lima showed that a star and any threshold graph on the same number of vertices share the same eigenbasis relative to the Laplacian matrix. We use this fact to establish two main results in this paper. The first one is a characterization of threshold graphs that are \textit{simply structured}, i.e., their associated Laplacian matrices have eigenbases consisting of vectors with entries from the set $\{-1,0,1\}$. Then, we provide sufficient conditions such that a simply structured threshold graph is weakly Hadamard diagonalizable (WHD). This allows us to list all connected simply structured threshold graphs on at most 20 vertices, and identify those that are WHD. Second, we characterize Laplacian pair state transfer on threshold graphs. In particular, we show that the existence of Laplacian vertex state transfer and Laplacian pair state transfer on a threshold graph are equivalent if and only if it is not a join of a complete graph and an empty graph of certain sizes.
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https://arxiv.org/abs/2601.13318
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Academic Papers
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56b09cc88b559c0543e7a06d22d974a4e01239922a9442670182f497c229d7af
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2026-01-21T00:00:00-05:00
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On the problem of generalized measures: an impossibility result
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arXiv:2601.13321v1 Announce Type: new Abstract: This paper investigates the problem of extending measure theory to non-separable structures, from generalized descriptive set theory to a broader class of spaces beyond this framework. While various notions, such as the ideal of measure zero sets, have been generalized, the question of whether a satisfactory notion of $\lambda^+$-measure could be defined in generalized descriptive set theory has remained open. We introduce a broad class of $\lambda^+$-measures as functions taking values in arbitrary positively totally ordered monoids equipped with an infinitary sum. This definition relies on minimal assumptions and captures most natural generalizations of measures to this context. We then prove that, under certain cardinal assumptions, no continuous $\lambda^+$-measure of this kind exists on ${}^\kappa\lambda$, nor on any $\lambda^+$-Borel space or $T_0$ topological space of weight at most $\lambda$. We also show the optimality of these cardinal assumptions.
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https://arxiv.org/abs/2601.13321
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Academic Papers
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9bbed82d6f7d078a57c38edac7f399d46b0b2c58c764d4cf9729d9a9731794fb
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2026-01-21T00:00:00-05:00
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Ribbon complexes for the 0-Hecke algebra
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arXiv:2601.13324v1 Announce Type: new Abstract: We construct explicit tableau-level maps between indecomposable projective modules for the type A 0-Hecke algebra that assemble into canonical split short exact sequences lifting the basic ribbon product rule in NSym via concatenation and near-concatenation. Iterating these maps yields cochain complexes indexed by generalized ribbons; we prove these complexes are acyclic in positive degrees and that their zeroth cohomology is the projective module indexed by full concatenation. We apply these complexes, together with VandeBogert's ribbon Schur module criterion, to prove Koszulness for a naturally defined internally graded algebra object built from the 0-Hecke tower. Finally, we define skew projective modules whose noncommutative Frobenius characteristics realize skewing by fundamental quasisymmetric functions on NSym.
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https://arxiv.org/abs/2601.13324
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Academic Papers
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3a45c4e1039ab2f72bdbd0ddcb42809e44b1edab0d2070d0e579f0e8776b0168
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2026-01-21T00:00:00-05:00
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Domino tilings of black-and-white Temperleyan cylinders
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arXiv:2601.13332v1 Announce Type: new Abstract: We consider the dimer model in cylindrical domains $\Omega_\delta$ on square grids of mesh size $\delta$ with two Temperleyan boundary components of different colors. Assuming that the $\Omega_\delta$ approximate a cylindrical domain $\Omega$ as $\delta\to 0$, we prove the convergence of height fluctuations to the Gaussian Free Field in $\Omega$ plus an independent discrete Gaussian multiple of the harmonic measure of one of the boundary components. The limit of the dimer coupling functions on $\Omega_\delta$ is holomorphic in $\Omega$ but not conformally covariant. Given this, we determine the limiting structure of height fluctuations from general principles rather than from explicit computations. In particular, our analysis justifies the inevitable appearance of the discrete Gaussian distribution in the doubly connected setup.
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https://arxiv.org/abs/2601.13332
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Academic Papers
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f518a83bf3cd655a07f3bc1d90b5b4166c732faed2f7ff6f94cf42a5a75479dc
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2026-01-21T00:00:00-05:00
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$D$-affinity of Quadrics Revisited
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arXiv:2601.13340v1 Announce Type: new Abstract: Let $K$ be aa algebraically closed field of characteristic $p\geq3$ and let $Q_{n}\subset\mathbb{P}^{n+1}_{K}$ be a smooth quadric hypersurface. We show that if $n=2m\geq4$ then $Q_{n}$ is not $D$-affine. In particular, we show the grassmannian ${Gr}(2,4)$ is not $D$-affine, which gives an example of a non $D$-affine flag variety of minimal possible dimension in characteristic $p\geq3$. Our result complements previous work of A. Langer, who showed that if $p\geq n=2m+1$ then $Q_{n}$ is $D$-affine.
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https://arxiv.org/abs/2601.13340
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Academic Papers
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0cdffc859d9193dcb28925404e1b4118e5fcc091d0c4434ca86c661dd422a2d7
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2026-01-21T00:00:00-05:00
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Existence and uniqueness of invariant measures for non-Feller Markov semigroups
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arXiv:2601.13354v1 Announce Type: new Abstract: We study existence and uniqueness of invariant probability measures for continuous-time Markov processes on general state spaces. Existence is obtained from tightness of time averages under a weak regularity assumption inspired by quasi-Feller semigroups, allowing for discontinuous and non-Feller dynamics. Our main contribution concerns uniqueness. Under a natural $\psi$-irreducibility assumption, we show that the normalized resolvent kernel satisfies a domination property with respect to a reference measure. As a consequence, every invariant probability measure charges this reference measure. Since distinct ergodic invariant measures are mutually singular on standard Borel spaces, this domination property implies uniqueness whenever an invariant probability measure exists. The argument is purely measure-theoretic and does not rely on Harris recurrence, return-time estimates, or Foster--Lyapunov conditions, and applies in particular to jump processes and hybrid models with discontinuous dynamics.
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https://arxiv.org/abs/2601.13354
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Academic Papers
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33f48d1e09525507cf635daa0f77c247035d49b8575aed5ecf1e481e5ed37697
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2026-01-21T00:00:00-05:00
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Center of distances of ultrametric spaces generated by labeled trees
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arXiv:2601.13363v1 Announce Type: new Abstract: The center of distances of a metric space $(X,d)$ is the set $C(X) $ of all $t\in\mathbb{R}^+$ for which the equation $d(x,p)=t$ has a solution for each $p\in X$. We prove that the equalities $ C(X)=\{0\} $ or $C(X)=\{\operatorname{diam} X,0\} $ hold if $(X,d)$ is an ultrametric space generated by labeled trees. The necessary and sufficient conditions under which $\operatorname{diam} X\in C(X) $ are found.
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https://arxiv.org/abs/2601.13363
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Academic Papers
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93d3b704830b9a56be5e77da3fcd8e3aec23d83247fe32b0ab1d38b8a208f5bd
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2026-01-21T00:00:00-05:00
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Linear relations on star coefficients of the chromatic symmetric function
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arXiv:2601.13390v1 Announce Type: new Abstract: We prove that the coefficient of the star $\mathfrak{st}_{21^{n-2}}$ in the chromatic symmetric function $X_G$ determines whether a connected graph $G$ is $2$-connected. We also prove new linear relations on other star coefficients of chromatic symmetric functions. This allows us to find new bases for certain spans of chromatic symmetric functions. Finally, we relate the coefficient of the star $\mathfrak{st}_n$ to acyclic orientations.
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https://arxiv.org/abs/2601.13390
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Academic Papers
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23087600c4b6b3528e94ea05ecd2a8e2d45c59da4dc6a5f6ba7e12eb6655d938
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2026-01-21T00:00:00-05:00
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Discrete-Time Optimal Control of Species Augmentation for Predator-Prey Model
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arXiv:2601.13394v1 Announce Type: new Abstract: Species augmentation is one of the methods used to promote biodiversity and prevent endangered species loss and extinction. The current work applies discrete-time optimal control theory to two models of species augmentation for predator-prey relationships. In discrete-time models, the order in which events occur can give different qualitative results. Two models representing different orders of events of optimal augmentation timing are considered. In one model, the population grows and predator-prey action occurs before the translocation of reserve species for augmentation. In the second model, the augmentation happens first and is followed by growth and then predator-prey action. The reserve and target populations are subjected to strong Allee effects. The optimal augmentation models employed in this work aim to maximize the prey (target population) and reserve population at the final time and minimize the associated cost at each time step. Numerical simulations in the two models are conducted using the discrete version of the forward-backward sweep method and the sequential quadratic programming iterative method, respectively. The simulation results show different population levels in the two models under varying parameter scenarios. Objective functional values showing percentage increases with optimal controls are calculated for each simulation. Different optimal augmentation strategies for the two orders of events are discussed. This work represents the first optimal augmentation results for models incorporating the predator-prey relationship with discrete events.
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https://arxiv.org/abs/2601.13394
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Academic Papers
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4c3e48e8f8a84b5cb23fba3e7c49a40f46f4466893bd4d6d2f9685cebb708384
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2026-01-21T00:00:00-05:00
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Generalized Adjoint Method
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arXiv:2601.13395v1 Announce Type: new Abstract: The adjoint method is an efficient way to numerically compute gradients in optimization problems with constraints, but is only formulated to differentiable cost and constraint functions on real variables. With the introduction of complex variables, which occur often in many inverse problems in electromagnetism and signal processing problems, both the cost and constraint can become non-holomorphic and hence non-differentiable in the standard definitions. Using the notion of CR-calculus, a generalized adjoint method is introduced that can compute the direction of steepest ascent for the cost function while enforcing the constraint even if both are non-holomorphic.
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https://arxiv.org/abs/2601.13395
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Academic Papers
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e145ed089e767c1347e49e92734ee1e7a9b7d909cf7bcce4561575bce59c6e71
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2026-01-21T00:00:00-05:00
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Analytic spectral perturbation theory for a high-contrast Maxwell operator
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arXiv:2601.13408v1 Announce Type: new Abstract: We study analytic spectral perturbation theory for the time-harmonic Maxwell operator in a perfectly electrically conducting cavity containing a high-contrast core--shell structure. The dielectric permittivity equals $1$ in a bounded inclusion and a small complex parameter $\delta$ in the surrounding shell. The limit $\delta \to 0$ corresponds to an infinite-contrast regime and leads to a degenerate Maxwell system. Despite this degeneracy, we develop a detailed spectral theory for the limiting problem for general Lipschitz inclusions and shells. Using a novel operator-theoretic reformulation, we prove complex-analytic dependence of the spectrum on $\delta$ in a neighborhood of $\delta = 0$. When the inclusion is a ball, we analyze the asymptotic expansion of eigenvalues and identify conditions under which the leading-order term is independent of the geometry of the surrounding shell. We also construct examples of resonances for which the leading-order asymptotics depend sensitively on the shell geometry, even in this symmetric setting. These results clarify the mechanisms underlying geometry-invariance of resonances in high-contrast Maxwell systems and explain their robustness under small complex perturbations.
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https://arxiv.org/abs/2601.13408
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Academic Papers
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c99f74451f7b10fb3c28d790c5c2235a2cf98f80a39a322754ae60032fea9104
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2026-01-21T00:00:00-05:00
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A uniformity principle for spatial matching
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arXiv:2601.13426v1 Announce Type: new Abstract: Platforms matching spatially distributed supply to demand face a fundamental design choice: given a fixed total budget of service range, how should it be allocated across supply nodes ex ante, i.e. before supply and demand locations are realized, to maximize fulfilled demand? We model this problem using bipartite random geometric graphs where $n$ supply and $m$ demand nodes are uniformly distributed on $[0,1]^k$ ($k \ge 1$), and edges form when demand falls within a supply node's service region, the volume of which is determined by its service range. Since each supply node serves at most one demand, platform performance is determined by the expected size of a maximum matching. We establish a uniformity principle: whenever one service range allocation is more uniform than the other, the more uniform allocation yields a larger expected matching. This principle emerges from diminishing marginal returns to range expanding service range, and limited interference between supply nodes due to bounded ranges naturally fragmenting the graph. For $k=1$, we further characterize the expected matching size through a Markov chain embedding and derive closed-form expressions for special cases. Our results provide theoretical guidance for optimizing service range allocation and designing incentive structures in ride-hailing, on-demand labor markets, and drone delivery networks.
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https://arxiv.org/abs/2601.13426
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Academic Papers
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52b4c5ef8160c54d2f6dfbd42924d2cd9a131a00dfce134c0e0d804dbe17d153
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2026-01-21T00:00:00-05:00
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Stabilization of an incompressible fluid-elastic structure system using a vacuum bubble
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arXiv:2601.13430v1 Announce Type: new Abstract: We prove a priori estimates for the system of partial differential equations modeling the interaction between an elastic body and an incompressible fluid in a 3D curved domain. The fluid is governed by the incompressible Navier-Stokes equations and contains a bubble whose interior is a vacuum. The elastic body is described by a damped wave equation, and interaction with the fluid takes place along a free interface whose initial domain is curved. We show that the presence of the vacuum bubble stabilizes the system in the sense that it provides control of the average of the pressure function, and hence allows global existence and exponential decay of smooth solutions for small data.
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https://arxiv.org/abs/2601.13430
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Academic Papers
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4876d7c3dbce55b738b27a84cf4d7378937dcc1b1b303fbf5a3973f7802f2c4c
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2026-01-21T00:00:00-05:00
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Independence complexes of generalized Mycielskian graphs
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arXiv:2601.13432v1 Announce Type: new Abstract: We show that the homotopy type of the independence complex of the generalized Mycielskian of a graph $G$ is determined by the homotopy type of the independence complex of $G$ and the homotopy type of the independence complex of the Kronecker double cover of $G$. As an application we calculate the homotopy type for paths, cycles and the categorical product of two complete graphs.
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https://arxiv.org/abs/2601.13432
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Academic Papers
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71ae18df4cb64ff7f9f9dc48cea7469cf4190c8b97ea6d1cf9c70ac6947570f0
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2026-01-21T00:00:00-05:00
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A priori estimates and exact solvability for non-coercive stochastic control equations
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arXiv:2601.13444v1 Announce Type: new Abstract: We establish, for the first time, explicit a priori and regularity estimates for solutions of the Dirichlet problem for Hamilton-Jacobi-Bellman operators from stochastic control, whose principal half-eigenvalues have opposite signs. In addition, if the negative eigenvalue is not too negative, the problem can have exactly two, one or zero solutions, depending on the valuation function. This is a novel exact multiplicity result for fully nonlinear equations, which also yields a generalization of the Ambrosetti-Prodi theorem to such equations.
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https://arxiv.org/abs/2601.13444
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Academic Papers
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6071cc5a24f7d586b781f0ff77c73c72d62e6788fe57075d6eba435152d4e274
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2026-01-21T00:00:00-05:00
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Monge-Ampere type equations on compact Hermitian manifolds with bounded mass property
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arXiv:2601.13446v1 Announce Type: new Abstract: In this paper, we study possibly non-closed big (1, 1)-forms on a compact Hermitian manifold satisfying the bounded mass property. We propose several criteria for the existence of rooftop envelopes. As applications, we establish the existence of solutions to complex Monge-Ampere type equations with prescribed singularities, allowing for non-pluripolar measures on the right-hand side. We also obtain stability results when singularity types vary, by extending the Darvas-Di Nezza-Lu distance to the Hermitian context.
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https://arxiv.org/abs/2601.13446
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Academic Papers
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svg
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b58ea9a346f0537cd2826745952f9a82743d349cbf39477835cdc0c1bbc37024
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2026-01-21T00:00:00-05:00
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From multiplicative to additive geometry: Deformation theory and 2D TQFT
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arXiv:2601.13455v1 Announce Type: new Abstract: In this paper, we present a theory of Poisson deformation of Hamiltonian quasi-Poisson manifolds to Hamiltonian Poisson manifolds that include degenerate cases. More significantly, this theory extends to singular cases arising from symplectic implosion: we introduce a generalized Hamiltonian deformation theory and we show that the imploded cross section of the double $D(G)_\imp$ deforms to the implosion of the cotangent bundle $T^*G_\imp$ with applications to the master moduli space of $G$-flat connections.\\ In parallel, we construct a topological quantum field theory $\N: \text{Cob}_{2}\to \mathbf{QHam}$, where $\mathbf{QHam}$ is the category of quasi-Hamiltonian manifolds. To each cobordism $\Sigma$, we associate a quasi-Hamiltonian space $\N(\Sigma)$ built from the fusion product of copies of the double $D(G).$ We show that these spaces are invariant under the \emph{quiver homotopy} and that the composition of cobordisms corresponds to a quasi-Hamiltonian reduction. This provides a multiplicative version of the 2D Hamiltonian TQFT of Maiza-Mayrand.
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https://arxiv.org/abs/2601.13455
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Academic Papers
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