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75f251c503f789f588e23c9413c20b52607d2b8c4bbc4443c0ac767e63effeba
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2026-02-02T00:00:00-05:00
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The Riemann Hypothesis in Oaxaca
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arXiv:2601.22413v1 Announce Type: new Abstract: An equivalence of the Riemann Hypothesis (RH) enables a direct bridge to the Young lattice. In specific, the classical threshold $\lim_{n\to\infty} \sigma(n)/(n \log\log n) = e^{\gamma} \approx 1.78107$, derived from the asymptotic behavior of the sum-of-divisors function, can be realized combinatorially via limiting proportions associated to specific families of integer partitions.
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https://arxiv.org/abs/2601.22413
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d6138a6896d1970ee091d2340ab89d0e49669c357b37ad22e3ab15dfe8198068
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2026-02-02T00:00:00-05:00
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Leader-Follower Linear-Quadratic Stochastic Graphon Games
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arXiv:2601.22429v1 Announce Type: new Abstract: This paper investigates leader-follower linear-quadratic stochastic graphon games, which consist of a single leader and a continuum of followers. The state equations of the followers interact through graphon coupling terms, with their diffusion coefficients depending on the state, the graphon aggregation term, and the control variables. The diffusion term of the leader's state equation depends on its state and control variables. Within this framework, a hierarchical decision-making structure is established: for any strategy adopted by the leader, the followers compete to attain a Nash equilibrium, while the leader optimizes its own cost functional by anticipating the followers' equilibrium response. This work develops a rigorous mathematical model for the game, proves the existence and uniqueness of solutions to the system's state equations under admissible control sets, and constructs a Stackelberg-Nash equilibrium for the continuum follower game. By employing the continuity method, we establish the existence, uniqueness, and stability of solutions to the associated forward-backward stochastic differential equation with a graphon aggregation term.
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https://arxiv.org/abs/2601.22429
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813fc3a7af98bfa534ccb89bc32b41b5df609418e4a7683095d95595a03a09e3
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2026-02-02T00:00:00-05:00
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Selective Adaptation of Beliefs and Communication on Cellular Sheaves
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arXiv:2601.22431v1 Announce Type: new Abstract: We extend opinion dynamics on discourse sheaves to incorporate "directional stubbornness": agents may hold fixed positions in specified directions of their opinion stalk while remaining flexible in others. This converts the equilibrium problem from harmonic extension to a forced sheaf equation: the free-opinion component satisfies a sheaf Poisson equation with forcing induced by the clamped directions. We develop a parallel theory for "selective learning" of expression policies. When only a designated subset of incidence maps may adapt, the resulting gradient flow is sheaf diffusion on an auxiliary structure sheaf whose global sections correspond to sheaf structures making a fixed opinion profile publicly consistent. For joint evolution of beliefs and expressions, we give conditions (and regularized variants) guaranteeing convergence to nondegenerate equilibria, excluding spurious agreement via vanishing opinions or trivialized communication maps. Finally, we derive stagnation bounds in terms of the rate ratio between opinion updating and structural adaptation, quantifying when rapid rhetorical accommodation masks nearly unchanged beliefs, and conversely when flexible beliefs conform to rigid communication norms.
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https://arxiv.org/abs/2601.22431
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e8dea9d63d4813a1bab40ad445f61703b57512116740c54580bd0fea696956da
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2026-02-02T00:00:00-05:00
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Divergence Identity for the scalar curvature and Rigidity of Codazzi Tensors
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arXiv:2601.22437v1 Announce Type: new Abstract: We introduce a local vector field on an $n$-dimensional Riemannian manifold, defined as the sum of the covariant derivatives of a local orthonormal frame, and derive an explicit identity for its divergence, decomposed into a scalar curvature term and an auxiliary term involving connection coefficients. This result is applied to rigidity problems for Codazzi symmetric tensors. In particular, we give a new proof of a Tang-Yan theorem, which states that on a closed $n$-dimensional manifold with nonnegative scalar curvature, a smooth Codazzi symmetric tensor whose trace invariants up to order $n-1$ are constant must have constant eigenvalues. We also obtain further rigidity results under assumptions on elementary symmetric functions of the eigenvalues, with applications to the isoparametric rigidity of closed hypersurfaces in the unit sphere.
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https://arxiv.org/abs/2601.22437
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2c06396950c9e97be2f5611bfb484de7345cedebe618d5bdc55b3b43535ee2bd
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2026-02-02T00:00:00-05:00
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On Monogeneity of reciprocal polynomials
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arXiv:2601.22453v1 Announce Type: new Abstract: Let $\mathbb{Z}_K$ denote the ring of integers of the number field $K = \mathbb{Q}(\theta)$, where $\theta$ is a root of the monic irreducible polynomial $f(x) \in \mathbb{Z}[x]$. We say that $f(x)$ is monogenic if $\mathbb{Z}_K = \mathbb{Z}[\theta]$. A polynomial $f(x) \in \mathbb{Z}[x]$ is called reciprocal if $f(x) = x^{\operatorname{deg}(f)} f(1/x)$. In this article, we derive sufficient conditions for the monogeneity of even degree reciprocal polynomials. By employing properties of the discriminant of reciprocal polynomials, we partially prove a conjecture proposed by Jones in $2021$. Furthermore, we establish a lower bound on the number of certain sextic monogenic reciprocal polynomials.
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https://arxiv.org/abs/2601.22453
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ccfb3736a5090e90888e780a63973d0e2200c88408cbabbc01cd3c753a9ae535
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2026-02-02T00:00:00-05:00
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Compactification of Reductive Group Schemes
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arXiv:2601.22462v1 Announce Type: new Abstract: Let $\mathrm G$ be an isotrivial reductive group over a scheme $S$. We construct a smooth projective $S$-scheme containing $\mathrm G$ as a fiberwise-dense open subscheme equipped with left and right actions of $\mathrm G$ which extend the translation actions of $\mathrm G$ on itself. This verifies a conjecture of \v{C}esnavi\v{c}ius (arXiv:2201.06424). When $\mathrm G$ is adjoint, we recover fiberwise the wonderful compactification. Finally, we give an example of a non-isotrivial torus admitting no equivariant compactification.
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https://arxiv.org/abs/2601.22462
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218ee5635fb522ee6c34dbac21c5eebbbe15783a6f302fe3156263e9d9ad3f19
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2026-02-02T00:00:00-05:00
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The isomorphism problem for reduced finitary power monoids
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arXiv:2601.22469v1 Announce Type: new Abstract: Let $H$ be a multiplicatively written monoid with identity $1_H$ and let $\mathcal{P}_{\text{fin},1}(H)$ denote the reduced finitary power monoid of $H$, that is, the monoid consisting of all finite subsets of $H$ containing $1_H$ with set multiplication as operation. Building on work of Tringali and Yan, we give a complete description of pairs of commutative and cancellative monoids $H,K$ for which $\mathcal{P}_{\text{fin},1}(H)$ and $\mathcal{P}_{\text{fin},1}(K)$ are isomorphic.
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https://arxiv.org/abs/2601.22469
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b547777ba0e917b9b856b9c0c1fc96a6cb7d3fc91971fd80bc93989227749802
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2026-02-02T00:00:00-05:00
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Tangents to Lipschitz and Sobolev images
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arXiv:2601.22473v1 Announce Type: new Abstract: We develop geometric versions of Rademacher and Calderon type differentiability theorems in two categories. A special case of our results is that for any Lipschitz or continuous $W^{1,p}$ Sobolev map $f$ from $[0,1]^n$ into a Euclidean space with $p>n$, the image $f([0,1]^n)$ has a unique tangent set (Attouch-Wets convergence) at almost every point with respect to the $n$-dimensional Hausdorff measure. In the analogous case when $f$ is a continuous $N^{1,p}$ map from $[0,1]^n$ into a metric space, we show that the image $f([0,1]^n)$ has a unique metric tangent (Gromov-Hausdorff convergence) almost everywhere. These results complement, but are distinct from Federer's theorem on existence and uniqueness of approximate tangents of $n$-rectifiable sets in $\mathbb{R}^d$. We show that approximate tangents to Sobolev images can be upgraded to Attouch-Wets or Gromov-Hausdorff tangents by first proving that the $n$-packing content of Sobolev images is finite, then proving that the inability to upgrade on a set of positive measure implies infinite packing content.
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https://arxiv.org/abs/2601.22473
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453e49a04329109d27a6767569d884b19b230013f8291df1e1d68acd927391b4
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2026-02-02T00:00:00-05:00
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Grothendieck rigidity and virtual retraction of higher-rank GBS groups
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arXiv:2601.22477v1 Announce Type: new Abstract: A rank $n$ generalized Baumslag-Solitar group ($GBS_n$ group) is a group that splits as a finite graph of groups such that all vertex and edge groups are isomorphic to $\mathbb{Z}^n$. This paper investigates Grothendieck rigidity and virtual retraction properties of $GBS_n$ groups. We show that every residually finite $GBS_n$ group is Grothendieck rigid. Further, we characterize when a $GBS_n$ group satisfies property (VRC), showing that it holds precisely when the monodromy is trivial.
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https://arxiv.org/abs/2601.22477
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2a992f982efeb0c919e2b3f5a4c7e3084584acd07a2eb71ff67a8c82c4ccf8f5
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2026-02-02T00:00:00-05:00
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Note on Euler characteristic of a toric vector bundle
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arXiv:2601.22514v1 Announce Type: new Abstract: A convex chain is a finite integer linear combination of indicator functions of convex polytopes. Khovanskii-Pukhlikov extend the Ehrhart theory of convex lattice polytopes to the setting of convex chains. Extending the relationship between equivariant line bundles on projective toric varieties and virtual lattice polytopes, we associate a lattice convex chain to a torus equivariant vector bundle on a toric variety and show that sum of values of this convex chain on lattice points gives the Euler characteristic of the bundle.
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https://arxiv.org/abs/2601.22514
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c4e868b579a6b344e90a58c8f73122544b59505a10e9c903bca4089bd149c3b7
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2026-02-02T00:00:00-05:00
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Local existence and nonexistence of solutions to the Hardy parabolic equation with general nonlinearity
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arXiv:2601.22520v1 Announce Type: new Abstract: In this paper, we consider the Cauchy problem for the Hardy parabolic equation with general nonlinearity and establish the local existence and nonexistence results. Our results provide the optimal integrability conditions on initial function for the existence of a local-in-time nonnegative solution. The proof of the existence result is based on the supersolution method.
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https://arxiv.org/abs/2601.22520
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9b43697d4cb7a7dc998c509b388c5167b480af2afb1d0da9379a50095a256d1d
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2026-02-02T00:00:00-05:00
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Transmission and Reflection coefficients for Schr\"odinger Operators with Truncated Periodic Potentials that support defect states
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arXiv:2601.22544v1 Announce Type: new Abstract: We consider scattering waves through truncated periodic potentials with perturbations that support localized gap eigenstates. In a small complex neighborhood around an assumed positive bound state of the model operator, we prove the existence of a distinct zero reflection state, or transmission resonance. We compare its location to a previously found scattering resonance and use the properties of solutions near these interesting points to analyze the behavior of transmission and reflection coefficients of scattering solutions near the assumed bound state. By example, we also discuss the truncated simple harmonic oscillator and compare the analysis to the crystalline case.
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https://arxiv.org/abs/2601.22544
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ba159aa15dc371bc2dd6ba446ce5f42a47506ab226713f5a9362ebcc9f2c30ce
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2026-02-02T00:00:00-05:00
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Corrigendum: Strong $\mathbb A^1$-invariance of $\mathbb A^1$-connected components of reductive algebraic groups (J. Topol. 16 (2023), no. 2, 634--649.)
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arXiv:2601.22555v1 Announce Type: new Abstract: The proof of Lemma 5.1 in the paper Strong $\mathbb A^1$-invariance of $\mathbb A^1$-connected components of reductive algebraic groups (J. Topol. 16 (2023), no. 2, 634--649) is incomplete as it relies on some results of Choudhury-Hagadi, the proof of which contains a gap. The goal of this note is to give a complete and self-contained proof of this lemma.
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https://arxiv.org/abs/2601.22555
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f3d49577f9361abedb6986b11c1ca64e11928e30d281f1618400ba53c91f1dce
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2026-02-02T00:00:00-05:00
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Generalized Zalcman Conjecture for Starlike Mappings in Several Complex Variables
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arXiv:2601.22558v1 Announce Type: new Abstract: Generalizing the Zalcman conjecture given by $\vert a_n^2 - a_{2n-1}\vert \leq (n-1)^2$, Ma proposed and proved that the inequality $$\vert a_n a_m-a_{n+m-1}\vert \leq (n-1)(m-1), \quad m,n \in \mathbb{N},$$ holds for functions $f(z)=z+a_2z^2 +a_3 z^3 +\cdots\in \mathcal{S}^*$, the class of starlike functions in the open unit disk. In this work, we extend this problem to several complex variables for $m=2$ and $n=3$, considering the class of starlike mappings defined on the unit ball in a complex Banach space and on bounded starlike circular domains in $\mathbb{C}^n$.
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https://arxiv.org/abs/2601.22558
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62e760a3826abbbe82eb11c7692b6db24d15815912d8802abee4ba2bbec759a4
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2026-02-02T00:00:00-05:00
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Kernels of Arithmetic Jet Spaces and Frobenius Morphism
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arXiv:2601.22591v1 Announce Type: new Abstract: For any $\pi$-formal group scheme $G$, the Frobenius morphism between arithmetic jet spaces restricts to generalized kernels of the projection map. Using the functorial properties of such kernels of arithmetic jet spaces, we show that this morphism is indeed induced by a natural ring map between shifted $\pi$-typical Witt vectors. In the special case when $G = \hat{\mathbb{G}}_a$, the arithmetic jet space, as well as the generalized kernels are affine $\pi$-formal planes with Witt vector addition as the group law. In that case the above morphism is the multiplication by $\pi$ map on Witt vector schemes. In fact, the system of arithmetic jet spaces and generalized kernels of any $\pi$-formal group scheme $G$ along with their maps and identitites satisfied among them are a generalization of the case of the Witt vector scheme with the system of maps such as the Frobenius, Verschiebung and multiplication by $\pi$.
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https://arxiv.org/abs/2601.22591
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d71103c71e537917ab5c4ee9374e1742331ddc2364cf30362775b77ad2f35f1a
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2026-02-02T00:00:00-05:00
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A spectral approach for online covariance change point detection
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arXiv:2601.22602v1 Announce Type: new Abstract: Change point detection in covariance structures is a fundamental and crucial problem for sequential data. Under the high-dimensional setting, most of the existing research has focused on identifying change points in historical data. However, there is a significant lack of studies on the practically relevant online change point problem, which means promptly detecting change points as they occur. In this paper, applying the limiting theory of linear spectral statistics for random matrices, we propose a class of spectrum based CUSUM-type statistic. We first construct a martingale from the difference of linear spectral statistics of sequential sample Fisher matrices, which converges to a Brownian motion. Our CUSUM-type statistic is then defined as the maximum of a variant of this process. Finally, we develop our detection procedure based on the invariance principle. Simulation results show that our detection method is highly sensitive to the occurrence of change point and is able to identify it shortly after they arise, outperforming the existing approaches.
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https://arxiv.org/abs/2601.22602
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a802dee35eb1336312802690c2a0d15713c84276424baa9f4fda4b126bfa7a79
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2026-02-02T00:00:00-05:00
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Spectral properties and bound states of the Dirac equation on periodic quantum graphs
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arXiv:2601.22603v1 Announce Type: new Abstract: We investigate nonlinear Dirac equations on a periodic quantum graph $G$ and develop a variational approach to the existence and multiplicity of bound states. After introducing the Dirac operator on $G$ with a $\mathbb Z^{d}$-periodic potential, we describe its spectral decomposition and work in the natural energy space. Under asymptotically linear or superquadratic assumptions on the nonlinearity, we establish the required linking geometry and a Cerami-type compactness property modulo $\mathbb Z^{d}$-translations. As a consequence, we prove the existence of at least one bound state and, when the nonlinearity is even, infinitely many geometrically distinct bound states.
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https://arxiv.org/abs/2601.22603
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287cbf6e099c7593a5cb20f35fc769e3e8185938dcdee6090f1c5998e6a74f7d
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2026-02-02T00:00:00-05:00
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Weighted estimates for Hodge-Maxwell systems
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arXiv:2601.22604v1 Announce Type: new Abstract: We establish up to the boundary regularity estimates in weighted $L^{p}$ spaces with Muckenhoupt weights $A_{p}$ for weak solutions to the Hodge systems \begin{align*} d^{\ast}\left(Ad\omega\right) + B^{\intercal}dd^{\ast}\left(B\omega\right) = \lambda B\omega + f \quad \text{ in } \Omega \end{align*} with either $\nu \wedge \omega $ and $\nu \wedge d^{\ast}\left(B\omega\right)$ or $\nu \lrcorner B\omega$ and $\nu \lrcorner Ad\omega$ prescribed on $\partial\Omega.$ As a consequence, we prove the solvability of Hodge-Maxwell systems and derive Hodge decomposition theorems in weighted Lebesgue spaces. Our proof avoids potential theory, does not rely on representation formulas and instead uses decay estimates in the spirit of `Campanato method' to establish weighted $L^{p}$ estimates.
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https://arxiv.org/abs/2601.22604
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55d47d0985e72b8fd583db2e9f36753cdd3da39470835cf56e4f63c79f263b7d
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2026-02-02T00:00:00-05:00
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Local controllability of the Cahn-Hilliard-Burgers' equation around certain steady states
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arXiv:2601.22611v1 Announce Type: new Abstract: In this article we study the local controllability of the one-dimensional Cahn-Hilliard-Navier-Stokes equation, that is Cahn-Hilliard-Burgers' equation, around a certain steady state using a localized interior control acting only in the concentration equation. To do it, we first linearize the nonlinear equation around the steady state. The linearized system turns out to be a system coupled between second order and fourth order parabolic equations and the control acts in the fourth order parabolic equation. The null controllability of the linearized system is obtained by a duality argument proving an observability inequality. To prove the observability inequality, a new Carleman inequality for the coupled system is derived. Next, using the source term method, it is shown that the null controllability of the linearized system with non-homogeneous terms persists provided the non-homogeneous terms satisfy certain estimates in a suitable weighted space. Finally, using a Banach fixed point theorem in a suitable weighted space, the local controllability of the nonlinear system is obtained.
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https://arxiv.org/abs/2601.22611
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a5862d8693beb1162d099423b482eb008f8506ca02f6dbfe54cea2e78c41d46c
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2026-02-02T00:00:00-05:00
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Sequence entropy of rank one systems
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arXiv:2601.22626v1 Announce Type: new Abstract: We study the sequence entropy of rank one measure-preserving systems along subexponential sequences. We prove that the sequence entropy along a large class of sequences can be infinite using Ornstein's probabilistic constructions. Moreover, we show that sequence entropy necessarily vanishes for subexponential sequences if the growth of tower heights remains below certain growth rates, and obtain a flexibility result for polynomial sequences at this critical threshold.
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https://arxiv.org/abs/2601.22626
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234f6ec37787277e4ef382efcbae5147a9b77d29762290823a1f6253ea603bf0
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2026-02-02T00:00:00-05:00
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Maximal Prikry Sequences
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arXiv:2601.22643v1 Announce Type: new Abstract: In this paper we investigate the covering machinery of the Mitchell-Steel core model $K$, under the hypothesis that there is no inner model with a Woodin cardinal. In an earlier work, Mitchell and the first author showed that if $\nu>\omega_2$ is a regular cardinal in $K$ but a singular ordinal in $V$, then $\nu$ is a measurable cardinal in $K$. In this article, we further show that under certain circumstances, there exists a maximal Prikry sequence $C$ for a measure on $\nu$ in $K$. The first author shows that the anti-large cardinal hypothesis is necessary. In a more restrictive setting, we prove that every subset of $\nu$ with size $<|\nu|$ can be covered by a set in $K[C]$ with size $<|\nu|$. Benhemou and the first author show that the result is optimal.
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https://arxiv.org/abs/2601.22643
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323b42e3f3acaef40a852a499e93d8ec3ef3f1ab5dca44173a9f6a791b6dc2b8
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2026-02-02T00:00:00-05:00
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On the BSE and BED properties of the Beurling algebra $L^1(G,\omega)$
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arXiv:2601.22646v1 Announce Type: new Abstract: Let $G$ be a locally compact abelian group, and let $\omega:G \to [1,\infty)$ be a weight, i.e., $\omega$ is measurable, $\omega$ is locally bounded and $\omega(s+t)\leq \omega(s)\omega(t)$ for all $s, t \in G$. If $\omega^{-1}$ is vanishing at infinity, then we show that the Beurling algebra $L^1(G,\omega)$ is both BSE- algebra and BED- algebra.
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https://arxiv.org/abs/2601.22646
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ea69d75ca64eef94705f42a2388e20b6aca6a689e30d2d06d8f5c088a80d4259
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2026-02-02T00:00:00-05:00
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Multisets of finite intervals and a universal category of poset representations
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arXiv:2601.22649v1 Announce Type: new Abstract: For any finite totally ordered set, the multisets of intervals form an abelian category. Various classes of subcategories admit natural combinatorial descriptions, and counting them yields familiar integer sequences. Surprisingly, in some cases new integer sequences arise. The formulation of this counting problem leads to a universal construction which assigns to any poset a finitely cocomplete additive category; it is abelian when the poset is finite and does not depend on the choice of any ring of coefficients. For a general poset the universal category of representations is abelian if and only if for the lattice of ideals the meet of two compact elements is again compact.
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https://arxiv.org/abs/2601.22649
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8a870ef543ac9c7dd4ab6fa348bb9c02a50ff08591cb86373851d5e6c699847d
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2026-02-02T00:00:00-05:00
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Sharp thresholds for Escobar and Gagliardo-Nirenberg functionals: the Escobar-Willmore mass, geometric selection, and compactness trichotomy
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arXiv:2601.22665v1 Announce Type: new Abstract: We develop a unified quantitative framework for sharp threshold phenomena in boundary-critical variational problems on compact Riemannian manifolds, covering the Escobar quotient and Gagliardo-Nirenberg inequalities. Via transfer-stability-reduction, we obtain attainment-versus-bubbling alternatives, $H^1$-compactness, and finite-dimensional reductions. Geometric selection is governed by mean curvature $H_g$ and a Willmore-type anisotropy from $|\mathring{\mathrm{II}}|^2$. At hemisphere threshold $S_\ast=C^*_{\mathrm{Esc}}(\mathbb S^n_+)$ for $n\ge5$ on $H_g\equiv0$, we identify a renormalized boundary mass $\mathfrak R_g=\kappa_1(n)\,\mathrm{Ric}_g(\nu,\nu)+\kappa_2(n)\,\mathrm{Scal}_{g|\partial M}+\kappa_3(n)\,|\mathring{\mathrm{II}}|^2$, $\kappa_3(n)<0$, yielding one-bubble expansions and energy-only estimators. Threshold dichotomy: if the first nonvanishing coefficient among $\{\rho_n^{\mathrm{conf}}H_g,\mathfrak R_g,\Theta_g\}$ is negative somewhere, then $C^*_{\mathrm{Esc}}(M,g)0$ at all critical points, no bubbling occurs. In multi-bubble regime ($n\ge5$), dynamics governed by $\mathcal W_k=\sum_{i=1}^k\mathfrak R_g(x_i)$ produce $k$-bubble critical points at levels $k^{1/(n-1)}S_\ast$. In the degenerate case we obtain conformal hemispherical rigidity. The GN track yields analogous dichotomies and resolves a question of Christianson et al.: the sharp constant with small Dirichlet windows diverges at optimal capacitary rate, relating threshold to spectral/isoperimetric invariants. Applications include entropy inequalities for fast diffusion, curvature-driven NLS ground states, and (in $n=2$) Euler characteristic recovery from GN measurements.
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https://arxiv.org/abs/2601.22665
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0773e7a51abe0f41844b16c0660540aa69cc21abf8c0468382992c557b6f7dc9
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2026-02-02T00:00:00-05:00
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Classification of horospherical invariant measures in higher rank: The Full Story
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arXiv:2601.22668v1 Announce Type: new Abstract: In this paper, we classify horospherical invariant Radon measures for Anosov subgroups of arbitrary semisimple real algebraic groups. This generalizes the works of Burger and Roblin in rank one to higher ranks. At the same time, this extends the works of Furstenberg, Veech, and Dani, and a special case of Ratner's theorem for finite-volume homogeneous spaces to infinite-volume Anosov homogeneous spaces. Especially, this resolves the open problems proposed by Landesberg--Lee--Lindenstrauss--Oh and by Oh. Our measure classification is in fact for a more general class of discrete subgroups, including relatively Anosov subgroups with respect to any parabolic subgroups, not necessarily minimal. Our method is rather geometric, not relying on continuous flows or ergodic theorems.
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https://arxiv.org/abs/2601.22668
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a822f3b8d3735431ce42556013bb0401950fd995553028dc5ad54fdd4155aa8a
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2026-02-02T00:00:00-05:00
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Wall singularity of spaces with an upper curvature bound
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arXiv:2601.22673v1 Announce Type: new Abstract: We study typical wall singularity of codimension one for locally compact geodesically complete metric spaces with an upper curvature bound. We provide a geometric structure theorem of codimension one singularity, and a geometric characterization of codimension two regularity. These give us necessary and sufficient conditions for singular sets to be of codimension at least two.
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https://arxiv.org/abs/2601.22673
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802fdae76528efb87aac0db675c14861e08a00e8927ea591ed7652adbcec0634
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2026-02-02T00:00:00-05:00
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Anisotropic Minkowski Content for Countably $\mathcal{H}^k$-rectifiable Sets
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arXiv:2601.22681v1 Announce Type: new Abstract: This paper investigates the existence of the anisotropic lower-dimensional Minkowski content. We establish that the $C$-anisotropic $k$-dimensional Minkowski content of a $k$-rectifiable compact set always exists and coincides with a specific functional that depends naturally on $C$. We further show that the same conclusion holds for countably $\mathcal{H}^k$-rectifiable compact sets, provided that the so-called \emph{AFP-condition} is satisfied. In addition, we discuss how the existence of the $C$-anisotropic $k$-dimensional Minkowski content for a countably $\mathcal{H}^k$-rectifiable compact set depends on the choice of $C$.
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https://arxiv.org/abs/2601.22681
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39090b340344e635ce11be70db2bff955f12bdd2ab31df1e86f2c5719e5ad0f5
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2026-02-02T00:00:00-05:00
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SUN-DSBO: A Structured Unified Framework for Nonconvex Decentralized Stochastic Bilevel Optimization
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arXiv:2601.22682v1 Announce Type: new Abstract: Decentralized stochastic bilevel optimization (DSBO) is a powerful tool for various machine learning tasks, including decentralized meta-learning and hyperparameter tuning. Existing DSBO methods primarily address problems with strongly convex lower-level objective functions. However, nonconvex objective functions are increasingly prevalent in modern deep learning. In this work, we introduce SUN-DSBO, a Structured Unified framework for Nonconvex DSBO, in which both the upper- and lower-level objective functions may be nonconvex. Notably, SUN-DSBO offers the flexibility to incorporate decentralized stochastic gradient descent or various techniques for mitigating data heterogeneity, such as gradient tracking (GT). We demonstrate that SUN-DSBO-GT, an adaptation of the GT technique within our framework, achieves a linear speedup with respect to the number of agents. This is accomplished without relying on restrictive assumptions, such as gradient boundedness or any specific assumptions regarding gradient heterogeneity. Numerical experiments validate the effectiveness of our method.
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https://arxiv.org/abs/2601.22682
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Academic Papers
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a72021efd506c71b4d8ee28f0313fc5065ac1da3373aed0d5eb8722998d045c4
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2026-02-02T00:00:00-05:00
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Enhancing Exploration in Global Optimization by Noise Injection in the Probability Measures Space
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arXiv:2601.22753v1 Announce Type: new Abstract: McKean-Vlasov (MKV) systems provide a unifying framework for recent state-of-the-art particlebased methods for global optimization. While individual particles follow stochastic trajectories, the probability law evolves deterministically in the mean-field limit, potentially limiting exploration in multimodal landscapes. We introduce two principled approaches to inject noise directly into the probability law dynamics: a perturbative method based on conditional MKV theory, and a geometric approach leveraging tangent space structure. While these approaches are of independent interest, the aim of this work is to apply them to global optimization. Our framework applies generically to any method that can be formulated as a MKV system. Extensive experiments on multimodal objective functions demonstrate that both our noise injection strategies enhance consistently the exploration and convergence across different configurations of dynamics, such as Langevin, Consensus-Based Optimization, and Stein Boltzmann Sampling, providing a versatile toolkit for global optimization.
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https://arxiv.org/abs/2601.22753
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11e039ebff9bb5a1846b2fc41615ea4275c651ac42388909c4fc36f6dc7adf46
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2026-02-02T00:00:00-05:00
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Characterization of $n$-Lie Derivations on Generalized Matrix Algebras
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arXiv:2601.22774v1 Announce Type: new Abstract: The principal objective of this paper is to determine the structure of $n$-Lie derivations ($n\geq 3$) on generalized matrix algebras.It is shown that under certain mild assumptions, every $n$-Lie derivation can be decomposed into the sum of an extremal $n$-derivation and an $n$-linear centrally-valued mapping. As direct applications, we provide complete characterizations of $n$-Lie derivations on both full matrix algebras and triangular algebras.
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https://arxiv.org/abs/2601.22774
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9fd199ee513fda2086fcc962c32375843489ffdaf77d8aa502073688b6cc8f87
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2026-02-02T00:00:00-05:00
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The Symplectic-to-Contact Dictionary
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arXiv:2601.22775v1 Announce Type: new Abstract: Contact Geometry is an odd dimensional analogue of Symplectic Geometry. This vague idea can actually be formalized in a rather precise way by means of a Symplectic-to-Contact Dictionary. The aim of this review paper is discussing the basic entries in this dictionary. Surprisingly, the dictionary can also be applied to apparently far away situations like complex and $G$-structures, to get old and new interesting geometries.
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https://arxiv.org/abs/2601.22775
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97ad03cc61b666581f6c97618f3ebf84185c0d571b2311983e853fbc73430736
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2026-02-02T00:00:00-05:00
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Generators for automorphisms of special groups
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arXiv:2601.22789v1 Announce Type: new Abstract: Let $G$ be a (compact) special group in the sense of Haglund and Wise. We show that ${\rm Out}(G)$ is finitely generated, and provide a virtual generating set consisting of Dehn twists and ``pseudo-twists''. We exhibit instances where Dehn twists alone do not suffice and completely characterise this phenomenon: it is caused by certain abelian subgroups of $G$, called ``poison subgroups'', which can be removed by replacing $G$ with a finite-index subgroup. Similar results hold for coarse-median preserving automorphisms, without the pathologies: For every special group $G$, the coarse-median preserving subgroups ${\rm Out}(G,[\mu])\leq{\rm Out}(G)$ are virtually generated by finitely many Dehn twists with respect to splittings of $G$ over centralisers. Proofs are based on a novel, hierarchical version of Rips and Sela's shortening argument.
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https://arxiv.org/abs/2601.22789
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b3340318ad0a9ffd619b5f82d61be965f73dffd276afec0d27f8fc1aa36529b2
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2026-02-02T00:00:00-05:00
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Convergence of Multi-Level Markov Chain Monte Carlo Adaptive Stochastic Gradient Algorithms
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arXiv:2601.22799v1 Announce Type: new Abstract: Stochastic optimization in learning and inference often relies on Markov chain Monte Carlo (MCMC) to approximate gradients when exact computation is intractable. However, finite-time MCMC estimators are biased, and reducing this bias typically comes at a higher computational cost. We propose a multilevel Monte Carlo gradient estimator whose bias decays as $O(T_{n}^{-1} )$ while its expected computational cost grows only as $O(log T_n )$, where $T_n$ is the maximal truncation level at iteration n. Building on this approach, we introduce a multilevel MCMC framework for adaptive stochastic gradient methods, leading to new multilevel variants of Adagrad and AMSGrad algorithms. Under conditions controlling the estimator bias and its second and third moments, we establish a convergence rate of order $O(n^{-1/2} )$ up to logarithmic factors. Finally, we illustrate these results on Importance-Weighted Autoencoders trained with the proposed multilevel adaptive methods.
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https://arxiv.org/abs/2601.22799
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89a21e83d11477add0a6cc82f1fb658be9194edae4a4d8c69e7ce252058723fb
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2026-02-02T00:00:00-05:00
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Rapid stabilizability of delayed infinite-dimensional control systems
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arXiv:2601.22819v1 Announce Type: new Abstract: In this paper, the rapid stabilizability of linear infinite-dimensional control system with constant-valued delay is studied. Under assumptions that the state operator generates an immediately compact semigroup and the coefficient of the delay term is constant, we mainly prove the following two results: (i) the delay does not affect rapid stabilizability of the control system; (ii) from the perspective of observation-feedback, it is not necessary to use historical information to stabilize the control system when the system is rapidly stabilizable. Some applications are given.
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https://arxiv.org/abs/2601.22819
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a2e5460bebd7776cc50e2070a9bf9bf713f62b97de43f25c2f9a76447a2d8859
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2026-02-02T00:00:00-05:00
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On the average number of representations of an integer as a sum of polynomials computed at prime values
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arXiv:2601.22822v1 Announce Type: new Abstract: We study the average number of representations of an integer $n$ as $n = \phi(n_{1}) + \dots + \phi(n_{j})$, for polynomials $\phi \in \mathbb{Z}[n]$ with $\partial\phi = k\ge 1$, $\operatorname{lead}(\phi) = 1$, $j \ge k$, where $n_{i}$ is a prime power for each $i \in \{1, \dots, j\}$. We extend the results of Languasco and Zaccagnini (2019), for $k=3$ and $j=4$, and of Cantarini, Gambini and Zaccagnini (2020), where they focused on monomials $\phi(n) = n^k$, $k\ge 2$ and $j=k, k + 1$.
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https://arxiv.org/abs/2601.22822
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5255b72ee5a6d02234476abbab90bf01ba7c58d7a3c4069f5ba626406574774c
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2026-02-02T00:00:00-05:00
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Simplicity of eigenvalues for elliptic problems with mixed Steklov-Robin boundary condition
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arXiv:2601.22829v1 Announce Type: new Abstract: This paper investigates the spectral properties of two classes of elliptic problems characterized by mixed Steklov-Robin boundary conditions. Our main objective is to prove that, for a generic domain, all the eigenvalues are simple. This result is established by employing domain perturbation techniques and analyzing the transversality of the associated operators.
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https://arxiv.org/abs/2601.22829
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b9a696afbe0a864797ff84c5f02220fb572a3adbd2b026c44d5f5ef6b822c068
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2026-02-02T00:00:00-05:00
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Asymmetric conformal prediction with penalized kernel sum-of-squares
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arXiv:2601.22834v1 Announce Type: new Abstract: Conformal prediction (CP) is a distribution-free method to construct reliable prediction intervals that has gained significant attention in recent years. Despite its success and various proposed extensions, a significant practical feature which has been overlooked in previous research is the potential skewed nature of the noise, or of the residuals when the predictive model exhibits bias. In this work, we leverage recent developments in CP to propose a new asymmetric procedure that bridges the gap between skewed and non-skewed noise distributions, while still maintaining adaptivity of the prediction intervals. We introduce a new statistical learning problem to construct adaptive and asymmetric prediction bands, with a unique feature based on a penalty which promotes symmetry: when its intensity varies, the intervals smoothly change from symmetric to asymmetric ones. This learning problem is based on reproducing kernel Hilbert spaces and the recently introduced kernel sum-of-squares framework. First, we establish representer theorems to make our problem tractable in practice, and derive dual formulations which are essential for scalability to larger datasets. Second, the intensity of the penalty is chosen using a novel data-driven method which automatically identifies the symmetric nature of the noise. We show that consenting to some asymmetry can let the learned prediction bands better adapt to small sample regimes or biased predictive models.
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https://arxiv.org/abs/2601.22834
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cc5c710c9183ef9523bcbdd22d712ac03769b417e4ed19966c2e2e7cda8a0671
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2026-02-02T00:00:00-05:00
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Vidinli algebras
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arXiv:2601.22839v1 Announce Type: new Abstract: A new class of nonassociative algebras, Vidinli algebras, is defined based on recent work of Co\c{s}kun and Eden. These algebras are conic (or quadratic) algebras with the extra restriction that the commutator of any two elements is a scalar multiple of the unity. Over fields of characteristic not 2, Vidinli algebras may be considered as generalizations of the Jordan algebras of Clifford type. However, in characteristic 2, the class of Vidinli algebras is much larger and include the unitizations of anticommutative algebras.
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https://arxiv.org/abs/2601.22839
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0e4b4aaae61472149e713a8031f75b67fb465687b461870da0c12e4283ef532c
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2026-02-02T00:00:00-05:00
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Distance Optimization in the Grassmannian of Lines
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arXiv:2601.22843v1 Announce Type: new Abstract: The square of a skew-symmetric matrix is a symmetric matrix whose eigenvalues have even multiplicities. When the matrices have rank two, they represent the Grassmannian of lines, and the squaring operation takes Pl\"ucker coordinates to projection coordinates. We develop metric algebraic geometry for varieties of lines in this linear algebra setting. The Grassmann distance (GD) degree is introduced as a new invariant for subvarieties of a Grassmannian. We study the GD degree for Schubert varieties and other models.
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https://arxiv.org/abs/2601.22843
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4d2425d4c025d03f957ec3ae1b7228ab08c3f470b24364904cbc6f0a8b8aa5f4
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2026-02-02T00:00:00-05:00
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Unconditional well-posedness of the master equation for monotone mean field games of controls
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arXiv:2601.22845v1 Announce Type: new Abstract: We establish the first unconditional well-posedness result for the master equation associated with a general class of mean field games of controls. Our analysis covers games with displacement monotone or Lasry--Lions monotone data, as well as those with a small time horizon. By unconditional, we mean that all assumptions are imposed solely at the level of the Lagrangian and the terminal cost. In particular, we do not require any a priori regularity or structural assumptions on the additional fixed-point mappings arising from the control interactions; instead we show that these fixed-point mappings are well-behaved as a consequence of the regularity and the monotonicity of the data. Our approach is bottom-up in nature, unlike most previous results which rely on a generalized method of characteristics. In particular, we build a classical solution of the master equation by showing that the solutions of the corresponding $N$-player Nash systems are compact, in an appropriate sense, and that their subsequential limit points must be solutions to the master equation. Compactness is obtained via uniform-in-$N$ decay estimates for derivatives of the $N$-player value functions. The underlying games are driven by non-degenerate idiosyncratic Brownian noise, and our results allow for the presence of common noise with constant intensity.
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https://arxiv.org/abs/2601.22845
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01363877c58454102d95cc4b44ceb0340227b5ddb78c377e3b517a49c4cb5e05
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2026-02-02T00:00:00-05:00
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Relative Kazhdan Lusztig isomorphism for $GL_{2n}/Sp_{2n}$
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arXiv:2601.22846v1 Announce Type: new Abstract: The Kazhdan Lusztig isomorphism, relating the affine Hecke algebra of a $p$-adic group to the equivariant $K$ theory of the Steinberg variety of its Langlands dual, played a key role in the proof of the Deligne Langlands conjectures concerning the classification of smooth irreducible representations with an Iwahori fixed vector. In this work we state and prove a relative version of the Kazhdan Lusztig isomorphism for the symmetric pair $(GL_{2n},Sp_{2n})$. The relative isomorphism is an isomorphism between the module of compactly supported Iwahori invariant functions on $X=GL_{2n}/Sp_{2n}$ and another module over the affine Hecke algebra constructed using equivariant $K$ theory and the relative Langlands duality. We use this isomorphism to give a new proof of a condition on $X$ distinguished representations.
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https://arxiv.org/abs/2601.22846
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92b54211a4ba8f733c8e4d9ab55e001662ffaba275582606894b7a4b0a8f7ad7
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2026-02-02T00:00:00-05:00
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Existence of a solution of the TV Wasserstein gradient flow
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arXiv:2601.22847v1 Announce Type: new Abstract: On the flat torus in any dimension we prove existence of a solution to the TV Wasserstein gradient flow equation, only assuming that the initial density $\rho_0$ is bounded from below and above by strictly positive constants. This solution preserves upper and lower bounds of the densities, and shows a certain decay of the BV norm (of the order of $t^{-1/3}$ for $t\to 0$ -- if $\rho_0\notin BV$, otherwise the BV norm is of course bounded -- and of the order of $t^{-1}$ as $t\to\infty$). This generalizes a previous result by Carlier and Poon, who only gave a full proof in one dimension of space and did not consider the case $\rho_0\notin BV$. The main tool consists in considering an approximated TV-JKO scheme which artificially imposes a lower bound on the density and allows to find a continuous-in-time solution regular enough to prove that the lower bounds of the initial datum propagates in time, and study on this approximated equation the decay of the BV norm.
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https://arxiv.org/abs/2601.22847
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411964c531e4ffed2acec85d0c9edd96eadbb8a3729d9f6e5e8d7aa11790a966
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2026-02-02T00:00:00-05:00
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Convergence Rates for the Alternating Minimization Algorithm in Structured Nonsmooth and Nonconvex Optimization
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arXiv:2601.22850v1 Announce Type: new Abstract: This paper is devoted to developing the alternating minimization algorithm for problems of structured nonconvex optimization proposed by Attouch, Bolt\'e, Redont, and Soubeyran in 2010. Our main result provides significant improvements of the convergence rate of the algorithm, especially under the low exponent Polyak-{\L}ojasiewicz-Kurdyka condition when we establish either finite termination of this algorithm or its superlinear convergence rate instead of the previously known linear convergence. We also investigate the PLK exponent calculus and discuss applications to noncooperative games and behavioral science.
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https://arxiv.org/abs/2601.22850
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e5d7fc554d4aee2c73d4070d0783099ea6c18033d7b5bb3d5333943a61d41d4c
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2026-02-02T00:00:00-05:00
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The two-nest ants process on triangle-series-parallel graphs
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arXiv:2601.22855v1 Announce Type: new Abstract: The ants process is a stochastic process introduced by Kious, Mailler and Schapira as a model for the phenomenon of ants finding shortest paths between their nest and a source of food (seen as two marked nodes in a finite graph), with no other means of communications besides the pheromones they lay behind them as they explore their environment. The ants process relies on a reinforcement learning mechanism. In this paper, we modify the ants process by having more than one ants nest (and still one source of food). For technical reasons, we restrict ourselves to the case when there are two nests, and when the graph is a triangle between the two nests and the source of food, whose edges have been replaced by series-parallel graphs. In this setting, using stochastic approximation techniques, comparison with P\'olya urns, and combinatorial arguments, we are able to prove that the ants process converges and to describe its limit.
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https://arxiv.org/abs/2601.22855
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f819c7d4c4f8775bd2b8243da55f50e6029e5075174976d7b22164f885f2e192
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2026-02-02T00:00:00-05:00
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Global Well-posedness of Strong Solutions to the Cauchy Problem of 2D Nonhomogeneous Navier-Stokes Equations with Density-Dependent Viscosity and Vacuum
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arXiv:2601.22877v1 Announce Type: new Abstract: This paper is concerned with the Cauchy problem for the modified two-dimensional (2D) nonhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity. By fully using the structure of the system, we can obtain the key estimates of $\|\nabla \rho\|_{L_t^\infty L_x^q},q>2$ without any smallness asuumption on the initial data, and thus establish the global existence of the strong solutions with the far-field density being either vacuum or nonvacuum. Notably, the initial data can be arbitrarily large and the initial density is allowed to vanish. Furthermore, the large-time asymptotic behavior of the gradients of the velocity and the pressure is also established.
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https://arxiv.org/abs/2601.22877
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1f029d8a6730a43dfa7e641d7d115e46637a3ff4b5cef95cff9b0d0c0cee55e4
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2026-02-02T00:00:00-05:00
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Arbitrary harmonic functions as Bose--Einstein condensates
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arXiv:2601.22883v1 Announce Type: new Abstract: We show that a suitable choice of boundary conditions for the Laplacian allows for the appearance of an an arbitrary number of condensates, described by arbitrary harmonic functions, in the thermodynamic limit of an ideal Bose gas.
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https://arxiv.org/abs/2601.22883
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aa6fdf6895ee70d6df0ac8e75790919dfba654ace08f1e7dc6ab9ee00333894f
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2026-02-02T00:00:00-05:00
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Uncoupled Dirac-Yang-Mills Pairs on Closed Riemannian Spin Manifolds
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arXiv:2601.22886v1 Announce Type: new Abstract: We study the Dirac-Yang-Mills equations on closed spin manifolds with a focus on uncoupled solutions, i.e. solutions for which the connection form satisfies the Yang-Mills equation. Such solutions require the Dirac current, a quadratic form on the spinor bundle, to vanish. We study the condition that this current vanishes on all harmonic spinors using perturbation theory and obtain a classification of the connection forms for which this holds, which we show contains an open and dense subset of connections. This has several implications for the generic dimension of the kernel of the Dirac operator. We further establish existence results for uncoupled solutions, in particular in dimension $4$ using the index theorem. Finally we generalize a construction method for twisted harmonic spinors to construct explicit uncoupled solutions on $4$-manifolds admitting twistor spinors and on spin manifolds of any dimension admitting parallel spinors.
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https://arxiv.org/abs/2601.22886
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d2de9cdb114a86f7a43fc9ec0cf19db6012857e24788817ac4dfffaa29ce1d7a
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2026-02-02T00:00:00-05:00
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Grassmannian Geometry and Global Convergence of Variable Projection for Neural Networks
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arXiv:2601.22897v1 Announce Type: new Abstract: Training deep neural networks and Physics-Informed Neural Networks (PINNs) often leads to ill-conditioned and stiff optimization problems. A key structural feature of these models is that they are linear in the output-layer parameters and nonlinear in the hiddenlayer parameters, yielding a separable nonlinear least-squares formulation. In this work, we study the classical variable projection (VarPro) method for such problems in the context of deep neural networks. We provide a geometric formulation on the Grassmannian and analyze the structure of critical points and convergence properties of the reduced problem. When the feature map is parametrized by a neural network, we show that these properties persist except in rank-deficient regimes, which we address via a regularized Grassmannian framework. Numerical experiments for regression and PINNs, including an efficient solver for the heat equation, illustrate the practical effectiveness of the approach.
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https://arxiv.org/abs/2601.22897
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a2c1058c7355b8b283ba6039618cdc596e2bd31d701efd8cacc2e5b114ba01a7
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2026-02-02T00:00:00-05:00
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Rigidity of circle polyhedra and hyperideal polyhedra: the tangency case
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arXiv:2601.22903v1 Announce Type: new Abstract: We prove the global rigidity of proper triangulated convex hyperbolic circle polyhedra on the sphere $\mathbb{S}^2$. These circle polyhedra correspond to proper triangulated convex hyperbolic polyhedra in the Beltrami-Klein model $\mathbb{B}^{3}$ of hyperbolic space with hyperideal vertices whose faces meet $\mathbb{B}^{3}$. Although the vertices of these polyhedra lie outside $\mathbb{B}^{3} \cup \mathbb{S}^{2}$ and the faces meet $\mathbb{B}^{3}$, the edges may miss $\mathbb{B}^{3}$ entirely, meet $\mathbb{B}^{3}$, or, more importantly, lie tangent to $\mathbb{B}^{3}$ at ideal points on the boundary $\partial \mathbb{B}^{3} = \mathbb{S}^{2}$. The latter case is new and generalizes the global rigidity results of both Bao-Bonahon and arXiv:1703.09338. This result also generalizes the uniqueness part of the celebrated Koebe-Andre'ev-Thurston theorem to the case where adjacent circles need not touch.
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https://arxiv.org/abs/2601.22903
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6b953b623243ce0dc0dbd55cc749d11c781fa6cc5fa55737eed6a5ce153b766b
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2026-02-02T00:00:00-05:00
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Examples of finitely presented groups with strong fixed point properties and property (T)
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arXiv:2601.22907v1 Announce Type: new Abstract: We construct a finitely presented group with property (T) which can not act on on reasonable spaces. Such group is constructed using an generalization of Hall embedding theorem, where property (T) is added at the expense of weakening the simplicity requirement.
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https://arxiv.org/abs/2601.22907
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a9772e942d7470c0ff36e93f5c4033fbeb5b4db4d301d48e2ee25275fdb1d803
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2026-02-02T00:00:00-05:00
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Reducibility of self-maps in monoid and its related invariants
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arXiv:2601.22908v1 Announce Type: new Abstract: Given a positive integer $k$, we investigate the $k$-redcibility of self-maps in the monoid $\AA^k(X\vee Y)$, consisting of self-maps that induce isomorphisms on homology groups up to degree $k$. In general, verifying $k$-reducibility is a subtle problem. We show that the $k$-reducibility of a self-map is determine through its induced endomorphisms on homology or cohomology groups. Moreover, under the k-reducibility assumption, the computation of the homology self-closeness number of the wedge sum of spaces essentially reduces to the computation of the homology self-closeness numbers of the individual wedge summands. We generalize the notion of an atomic space to that of an $n$-atomic space and establish some of its fundamental properties. We show that the $k$-reducibility criteria for self-maps in a monoid $\AA^k(X)$ is satisfied when the space $X$ decomposes as a wedge sum of distinct $n$-atomic spaces. Finally, we determine the homology self-closeness numbers of wedge sums of distinct $n$-atomic spaces.
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https://arxiv.org/abs/2601.22908
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c8a34ece507e6e83c447e5bd879a12a8cb763ea547dbce4a96fe70cbd1b814e0
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2026-02-02T00:00:00-05:00
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Left Ehresmann monoids with a proper basis
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arXiv:2601.22923v1 Announce Type: new Abstract: This article gives an abstract characterisation of a class of left Ehresmann monoids possessing certain universal properties. It is known that every left Ehresmann monoid has a cover, that is, a projection separating preimage, of the form $\mathcal{P}_{\ell}(T,X)$, where $\mathcal{P}_{\ell}(T,X)$ is a left Ehresmann monoid constructed from a monoid $T$ and an order-preserving action of $T$ on a semilattice $X$ with identity. We introduce the class of $*$-left Ehresmann monoids and show that each $\mathcal{P}_{\ell}(T,X)$ belongs to this class; in particular so does any free left Ehresmann monoid. Further, we present the notion of a proper basis, and show that $\mathcal{P}_{\ell}(T,X)$ possesses a proper basis. Next, we exhibit a class of subsemigroups $\mathcal{Q}_{\ell}(T,X,Y)$ (properly, biunary monoid subsemigroups) of the monoids $\mathcal{P}_{\ell}(T,X)$ which are also $*$-left Ehresmann with a proper basis, and prove that up to isomorphism they form exactly the class of all such monoids. Our results can be regarded as being analogous to those for proper inverse semigroups, due to McAlister and O'Carroll, the $\mathcal{Q}_{\ell}(T,X,Y)$ playing the role of the $P$-semigroups and the $\mathcal{P}_{\ell}(T,X)$ the role of the semidirect products of a semilattice by a group.
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https://arxiv.org/abs/2601.22923
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a4fa07a47839f0ca74231c35e6b90cc13594ddfff48e9f6f96e0ace96edf8773
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2026-02-02T00:00:00-05:00
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Poset modules of the $0$-Hecke algebras of type $B$
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arXiv:2601.22926v1 Announce Type: new Abstract: In 2001, Chow developed the theory of the $B_n$ posets $P$ and the type $B$ $P$-partition enumerators $K^B_P$. To provide a representation-theoretic interpretation of $K^B_P$, we define the poset modules $M^B_P$ of the 0-Hecke algebra $H_n^B(0)$ of type $B$ by endowing the set of type-$B$ linear extensions of $P$ with an $H_n^B(0)$-action. We then show that the Grothendieck group of the category associated to type-$B$ poset modules is isomorphic to the space of type $B$ quasisymmetric functions as both a $\mathrm{QSym}$-module and comodule, where $\mathrm{QSym}$ denotes the Hopf algebra of quasisymmetric functions. Considering an equivalence relation on $B_n$ posets, where two posets are equivalent if they share the same set of type-$B$ linear extensions, we identify a natural representative of each equivalence class, which we call a distinguished poset. We further characterize the distinguished posets whose sets of type-$B$ linear extensions form intervals in the right weak Bruhat order on the the hyperoctahedral groups. Finally, we discuss the relationship among the categories associated to type-$B$ weak Bruhat interval modules, $B_n$ poset modules, and finite-dimensional $H_n^B(0)$-modules.
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https://arxiv.org/abs/2601.22926
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93275aa3417a3ac4c0ee5c5e52d4cb387176b8d5d2996db02fe59c5d1d1147a4
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2026-02-02T00:00:00-05:00
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Prescribed $T$-curvature flow on the four-dimensional unit ball
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arXiv:2601.22934v1 Announce Type: new Abstract: In this paper, we study the prescribed $T$-curvature problem on the unit ball $\mathbb{B}^4$ of $\mathbb{R} ^4$ via the $T$-curvature flow approach. By combining Ache-Chang's inequality with the Morse-theoretic approach of Malchiodi-Struwe, we establish existence results under strong Morse-type inequalities at infinity. As a byproduct of our argument, we also prove the exponential convergence of the $T$-curvature flow on $\mathbb{B}^4$, starting from a $Q$-flat and minimal metric conformal to the standard Euclidean metric, to an extremal metric of Ache-Chang's inequality whose explicit expression was derived by Ndiaye-Sun.
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https://arxiv.org/abs/2601.22934
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f73ce1d5c8f96c9c461dc1129bcc3e2bdbcea93d9ff6cbb88a590643ee84b2bb
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2026-02-02T00:00:00-05:00
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Local Well-posedness and Blow-up for the Restricted Fourth-Order Prandtl Equation
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arXiv:2601.22940v1 Announce Type: new Abstract: We prove local well-posedness and finite-time blow-up for a restricted fourth-order Prandtl equation posed on the half-line with clamped boundary conditions. The equation arises from a two-dimensional fourth-order Prandtl system via an ansatz reduction, and its nonlinearity involves a nonlocal integral term. To close a Duhamel fixed-point argument, we need uniform $L^1$ bounds for the associated half-line biharmonic heat kernel. We establish uniform $L^1$ estimates for the kernel and its derivatives, and we show that the semigroup preserves spatial regularity under appropriate compatibility conditions, using an alternative representation derived by integration by parts. These kernel estimates yield local existence and uniqueness for the restricted model and allow us to construct solutions that blow-up in finite time.
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https://arxiv.org/abs/2601.22940
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197102a0958480f215b681fe70aad1b87a54c0be5adbb1cea70d43b83e25ee66
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2026-02-02T00:00:00-05:00
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Compact group Rohlin actions and $G$-kernels on von Neumann algebras
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arXiv:2601.22941v1 Announce Type: new Abstract: We provide a new construction of a topological group model for the string group of a compact, simple, and simply-connected Lie group, by solving the obstruction realization problem for compact group $G$-kernels on full factors. Furthermore, we introduce the Rohlin property for actions and cocycle actions of compact groups in order to establish cohomology vanishing theorems.
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https://arxiv.org/abs/2601.22941
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25e3e70f4073c4ba44b747383d7156f8ec590a05bcdfca2272f8aa6616675603
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2026-02-02T00:00:00-05:00
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On polynomial functors and polynomial comonads over infinity groupoids
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arXiv:2601.22968v1 Announce Type: new Abstract: We show that single-variable polynomial functors over the category $\mathcal{S}$ of infinity groupoids, as defined by Gepner-Haugseng-Kock, are exactly colimits of representable copresheaves indexed by infinity groupoid. This allows us to establish certain categorical properties of the $\infty$-category $Poly_{\mathcal{S}}$, in parallel with the case of the ordinary category $Poly$. We define the notion of polynomial comonad under the monoidal structure of $Poly_{\mathcal{S}}$ induced by composition of polynomials, and describe a construction toward exploring the connection between polynomial comonads and complete Segal spaces. This construction partially generalizes the classical one given in the proof of a theorem of Ahman-Uustalu.
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https://arxiv.org/abs/2601.22968
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5e89fdb3633c968d0e400665e73a165231b1a05dd0a686da2f061c093237f254
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2026-02-02T00:00:00-05:00
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Periods of Ehrhart coefficients of rational polytopes
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arXiv:2601.22992v1 Announce Type: new Abstract: Let $\mathcal{P} \subseteq \mathbb{R}^{n}$ be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the $k$th dilate of $\mathcal{P}$ ($k$ a positive integer) is a quasi-polynomial function of $k$ -- that is, a "polynomial" in which the coefficients are themselves periodic functions of $k$. It is an open problem to determine which quasi-polynomials are the Ehrhart quasi-polynomials of rational polytopes. As partial progress on this problem, we construct families of polytopes in which the periods of the coefficient functions take on various prescribed values.
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https://arxiv.org/abs/2601.22992
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ebc2cec96172d9a2bd72f2cf916e74ff9757ebbc15755744b3dbb878efa17a07
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2026-02-02T00:00:00-05:00
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A Remark on Stability Conditions on Smooth Projective Varieties
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arXiv:2601.22994v1 Announce Type: new Abstract: Let $X$ be a smooth projective variety over $\mathbb C$. In this paper, we prove that $\mathrm{D}^b(X)$, the bounded derived category of coherent sheaves on $X$, always admits stability conditions in the sense of Bridgeland.
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https://arxiv.org/abs/2601.22994
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6e24479a594ca75101db3a36390ecd57b7fdf0f9ea08be856b9989ccf895d1c3
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2026-02-02T00:00:00-05:00
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Baire-type properties of topological vector spaces
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arXiv:2601.23008v1 Announce Type: new Abstract: Burzyk, Kli\'{s} and Lipecki proved that every topological vector space (tvs) $E$ with the property $(K)$ is a Baire space. K\c{a}kol and S\'{a}nchez Ruiz proved that every sequentially complete Fr\'{e}chet--Urysohn locally convex space (lcs) is Baire. Being motivated by the property $(K)$ and the notion of a Mackey null sequence we introduce a property $(MK)$ which is strictly weaker than the property $(K)$, and show that any locally complete lcs has the property $(MK)$. We prove that any $\kappa$-Fr\'{e}chet--Urysohn tvs with the property $(MK)$ is a Baire space; consequently, each locally complete $\kappa$-Fr\'{e}chet--Urysohn lcs is a Baire space. This generalizes both the aforementioned results. We construct a feral Baire space $E$ with the property $(K)$ and which is not $\kappa$-Fr\'{e}chet--Urysohn. Although a $\kappa$-Fr\'{e}chet--Urysohn lcs $E$ can be not a Baire space, we show that $E$ is always $b$-Baire-like in the sense of Ruess. Applications to spaces of Baire functions and $C_k$-spaces are given.
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https://arxiv.org/abs/2601.23008
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2436671508f17a55de0300da1f8da1964e3ad2859dbd0f9515378947342e4ef4
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2026-02-02T00:00:00-05:00
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On the finiteness of prime trees and their relation to modular forms
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arXiv:2601.23016v1 Announce Type: new Abstract: In this paper, we introduce the prime trees associated with a finite subset $P$ of the set of all prime numbers, and provide conditions under which the tree is of finite type. Moreover, we compute the density of finite-type subsets $P$. As an application, we show that for weight $k \ge 2$ and levels $N = N'\prod_{p \in P} p^{a_p}$, where $N'$ is squarefree and $a_{p} \geq 2$, every cusp form $f \in \mathcal{S}_k(\Gamma_0(N))$ can be expressed as a linear combination of products of two specific Eisenstein series whenever $P$ is of finite type.
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https://arxiv.org/abs/2601.23016
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93d16fc663623231986c3bce0bcd20cacf44935ef83fe61b0e3cb027d8d99817
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2026-02-02T00:00:00-05:00
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The uniqueness theorem for Kasparov theory
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arXiv:2601.23029v1 Announce Type: new Abstract: Answering a question of Carri\'on et al in their recent landmark paper on C*-algebra classification, we prove a general uniqueness theorem for $KK$-theory. Given arbitrary separable C*-algebras $A$ and $B$ and a Cuntz pair consisting of two absorbing representations $\varphi,\psi: A\to\mathcal{M}(B\otimes\mathcal{K})$, the induced element of $KK(A,B)$ vanishes if and only if $\varphi$ and $\psi$ are strongly asymptotically unitarily equivalent. This improves upon the Lin-Dadarlat-Eilers stable uniqueness theorem. The conclusion is deduced by first showing the $K_1$-injectivity of an auxiliary C*-algebra associated to the C*-pair $(A,B)$, which is sometimes called the Paschke dual algebra in the literature. Most of the article is concerned with the treatment of an umbrella theorem, which yields such a uniqueness theorem for other variants of $KK$-theory. This encompasses nuclear $KK$-theory, ideal-related $KK$-theory, equivariant $KK$-theory, or any combinations thereof.
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https://arxiv.org/abs/2601.23029
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accc9c7c9c217ab5ee676d4d5258f432145ba103d13cd370b699b1dfac19dfb4
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2026-02-02T00:00:00-05:00
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Breaking the Stochasticity Barrier: An Adaptive Variance-Reduced Method for Variational Inequalities
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arXiv:2601.23034v1 Announce Type: new Abstract: Stochastic non-convex non-concave optimization, formally characterized as Stochastic Variational Inequalities (SVIs), presents unique challenges due to rotational dynamics and the absence of a global merit function. While adaptive step-size methods (like Armijo line-search) have revolutionized convex minimization, their application to this setting is hindered by the Stochasticity Barrier: the noise in gradient estimation masks the true operator curvature, triggering erroneously large steps that destabilize convergence. In this work, we propose VR-SDA-A (Variance-Reduced Stochastic Descent-Ascent with Armijo), a novel algorithm that integrates recursive momentum (STORM) with a rigorous Same-Batch Curvature Verification mechanism. We introduce a theoretical framework based on a Lyapunov potential tracking the Operator Norm, proving that VR- SDA-A achieves an oracle complexity of O(epsilon -3) for finding an epsilon-stationary point in general Lipschitz continuous operators. This matches the optimal rate for non-convex minimization while uniquely enabling automated step-size adaptation in the saddle-point setting. We validate our approach on canonical rotational benchmarks and non-convex robust regression tasks, demonstrating that our method effectively suppresses limit cycles and accelerates convergence with reduced dependence on manual learning rate scheduling.
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https://arxiv.org/abs/2601.23034
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69ea95c212db4114bee41583b55bfd37740bc575db2e6f5a2dd8ca91f3ffaaac
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2026-02-02T00:00:00-05:00
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Accelerated Inertial Gradient Algorithms with Vanishing Tikhonov Regularization
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arXiv:2601.23035v1 Announce Type: new Abstract: In this paper, we study an explicit Tikhonov-regularized inertial gradient algorithm for smooth convex minimization with Lipschitz continuous gradient. The method is derived via an explicit time discretization of a damped inertial system with vanishing Tikhonov regularization. Under appropriate control of the decay rate of the Tikhonov term, we establish accelerated convergence of the objective values to the minimum together with strong convergence of the iterates to the minimum-norm minimizer. In particular, for polynomial schedules $\varepsilon_k = k^{-p}$ with $0<2$, we prove strong convergence to the minimum-norm solution while preserving fast objective decay. In the critical case $p=2$, we still obtain fast rates for the objective values, while our analysis does not guarantee strong convergence to the minimum-norm minimizer. Furthermore, we provide a thorough theoretical analysis for several choices of Tikhonov schedules. Numerical experiments on synthetic, benchmark, and real datasets illustrate the practical performance of the proposed algorithm.
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https://arxiv.org/abs/2601.23035
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bb181f5065e8658763ad96431bdfa6a2cde9bbc3a3c0acccea8b3e5123d3abe5
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2026-02-02T00:00:00-05:00
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Stationary Mean-Field singular control of an Ornstein-Uhlenbeck process
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arXiv:2601.23036v1 Announce Type: new Abstract: Motivated by continuous-time optimal inventory management, we study a class of stationary mean-field control problems with singular controls. The dynamics are modeled by a mean-reverting Ornstein-Uhlenbeck process, and the performance criterion is given by a quadratic long-time average expected cost functional. The mean-field dependence is through the stationary mean of the controlled process itself, which enters the ergodic cost functional. We characterize the solution to the stationary mean-field control problem in terms of the equilibria of an associated stationary mean-field game, showing that solutions of the control problem are in bijection with the equilibria of this mean-field game. Finally, we solve the stationary mean-field game explicitly, thereby providing a solution to the original stationary mean-field control problem.
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https://arxiv.org/abs/2601.23036
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a50ac2236ab421d90fd73852b6d2ed221da87544435cb982eced8506713a9f80
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2026-02-02T00:00:00-05:00
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Instability of two-dimensional Taylor-Green Vortices
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arXiv:2601.23040v1 Announce Type: new Abstract: For a wide class of linear Hamiltonian operators we develop a general criterion that characterizes the unstable eigenvalues as the zeros of a holomorphic function given by the determinant of a finite-dimensional matrix. We apply the latter result to prove the spectral instability of the Taylor-Green vortex in two-dimensional ideal fluids. The linearized Euler operator at this steady state possesses different invariant subspaces, within which we apply our criterion to rule out or detect instabilities. We show linear stability of odd perturbations, for which the unstable spectrum can appear only on the real axis. We exclude this possibility by applying our stability criterion. Real instabilities, instead, exist and can be detected with the same criterion if we consider suitable rescalings of the Taylor-Green vortex. In the subspace of functions even in both variables, the problem is reduced to finding a single complex root of our stability function. We successfully locate this value by combining our general criterion with a rigorous computer-assisted argument. As a consequence, we fully characterize the unstable spectrum of the Taylor-Green vortex.
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https://arxiv.org/abs/2601.23040
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77ba48df79fcc68ed54fdc873c07224cdbfa10bfc1466ac093223be1f84d5dba
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2026-02-02T00:00:00-05:00
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On two-dimensional Dirac operators with critical delta-shell interactions
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arXiv:2601.23053v1 Announce Type: new Abstract: We study two-dimensional Dirac operators with singular interactions of electrostatic and Lorentzscalar type, supported either on a straight line or a circle. For certain critical values of the interaction strengths, the essential spectrum of such operators comprises an isolated point lying within the mass gap. We clarify the nature of this point in both geometries. For the straight line model, this point is known to be an eigenvalue of infinite multiplicity, and we provide a detailed analysis of the corresponding eigenfunctions. By contrast, in the case of a circle, we show that the said point is not itself an eigenvalue, but rather an accumulation point of a double sequence of simple eigenvalues. In view of the high degree of symmetry of the configurations under analysis, this behavior is unexpected and our findings lead us to formulate some conjectures concerning critical singular interactions supported on generic smooth curves.
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https://arxiv.org/abs/2601.23053
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677660a6a6d60d92bb7bcba89204d4fe318523968996d15ba63a3fbb021d809c
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2026-02-02T00:00:00-05:00
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Some elementary amenable subgroups of interval exchange transformations
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arXiv:2601.23054v1 Announce Type: new Abstract: In this paper, we study a family of finitely generated elementary amenable iet-groups. These groups are generated by finitely many rationals iets and rotations. For them, we state criteria for not virtual nilpotency or solvability, and we give conditions to ensure that they are not virtually solvable. We precise their abelianizations, we determine when they are isomorphic to certain lamplighter groups and we provide non isomorphic cases among them. As consequences, in the class of infinite finitely generated subgroups of iets up to isomorphism, we exhibit infinitely many non virtually solvable and non linear groups, and infinitely many solvable groups of arbitrary derived length.
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https://arxiv.org/abs/2601.23054
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166212226dca438f440c41060b076fd1fb07ddb6f0935a4617551a320a143ec7
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2026-02-02T00:00:00-05:00
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Spectrum of bidual uniform algebras
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arXiv:2601.23055v1 Announce Type: new Abstract: We obtain a description of the spectrum of bidual algebra $A^{**}$ of a uniform algebra $A$. This spectrum turns out to be a quotient space of the hyper-Stonean envelope of the spectrum of $A$.
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https://arxiv.org/abs/2601.23055
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d042c4ec2e85f07d3c1571663f626afb6f1edb6ef66d789d2b17d0dc581b455a
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2026-02-02T00:00:00-05:00
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Some notes on plump ordinals
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arXiv:2601.23070v1 Announce Type: new Abstract: In this exposition, we attempt to formalise a treatment of Paul Taylor's notion of plump ordinals in weak intuitionistic axiomatic set theories such as IKP. We will explore basic properties of plump ordinals, especially in relation to G\"odel's constructible universe $L$ and incomparable codings. As a quick application, we explain at the end how plump ordinals can be used to build a Heyting-valued model $V^\mathbb{H}$ from a classical $V \vDash \mathrm{ZFC}$ such that for some arbitrary, fixed $x \in V$ we have $V^\mathbb{H} \vDash \mathcal{P}{\left(\check{x}\right)} \in L$.
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https://arxiv.org/abs/2601.23070
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b17b73dcbbd01fb2b13a478b0e374f840bb0a44940ef2360253db2ebd514f53b
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2026-02-02T00:00:00-05:00
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The $L^p$- regularity problem for the Bergman projection of two-dimensional Rudin ball quotients
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arXiv:2601.23074v1 Announce Type: new Abstract: We solve the $L^p$-regularity problem of the Bergman projection of two-dimensional Rudin ball quotients.
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https://arxiv.org/abs/2601.23074
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1603ef686632936a59f4bace58e43d52d7b5e18094595c8956e40bf509b26f63
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2026-02-02T00:00:00-05:00
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Mermin-Wagner theorems for quantum systems with multipole symmetries
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arXiv:2601.23078v1 Announce Type: new Abstract: We prove Mermin-Wagner-type theorems for quantum lattice systems in the presence of multipole symmetries. These theorems show that the presence of higher-order symmetries protects against the breaking of lower-order ones. In particular, we prove that the critical dimension in which the charge symmetry can be broken increases if the system admits higher multipole symmetries, e.g. $ d = 4 $ on the regular lattice $ \mathbb{Z}^d $ in the presence of dipole symmetry.
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https://arxiv.org/abs/2601.23078
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de7d446ddf212c714a47787aee3820cd78318db6583ebf79c16c4e5e0e37bf40
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2026-02-02T00:00:00-05:00
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Lifting property for finite groups
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arXiv:2601.23089v1 Announce Type: new Abstract: We classify all finite groups that have lifting property of mod $p$ representations to mod $p^2$ representations for all prime $p$.
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https://arxiv.org/abs/2601.23089
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965f49260025a403cf6d2f6e699aaff1d0acddd02ae00db1cefd40d1ed912683
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2026-02-02T00:00:00-05:00
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Existence of Traveling Waves in Infinite Range FPUT Lattices
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arXiv:2601.23091v1 Announce Type: new Abstract: We prove the existence of solitary waves in a lattice where all particles interact with each other by pair-wise repulsive forces that decay with distance. The variational existence proof is based on constrained optimization and provides a one-parameter family of unimodal solutions. We also describe the asymptotic behavior of large, fast, high-energy waves.
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https://arxiv.org/abs/2601.23091
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022baac651615030d28fc0bf61f874cfaa43f47f9a78039ec8b45f00abe4ded2
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2026-02-02T00:00:00-05:00
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Seminoetherian Modules over Non-Primitive HNP rings
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arXiv:2601.23099v1 Announce Type: new Abstract: We study the structure of seminoetherian modules. Seminoetherian modules over non-primitive hereditary noetherian prime rings are completely described.
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https://arxiv.org/abs/2601.23099
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c2e4548749cfbbef13e4c0d1e7526fde37b6abe38aa09227a34ac959bf38fcac
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2026-02-02T00:00:00-05:00
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Bipartite Graphs Are Not Well-Ordered by Bipartite Minors
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arXiv:2601.23101v1 Announce Type: new Abstract: In "Bipartite minors," Chudnovsky etal. introduced the bipartite minor relation, a partial order on the set of bipartite graphs somewhat analogous the minor relation on general graphs and asked whether it is a well-order. We answer this question negatively by giving an infinite set of $2$-connected bipartite graphs that are pairwise incomparable with respect to the bipartite minor relation. We additionally give two sets of infinitely many pairs of bipartite graphs: one set of pairs $G,H$ such that $H$ is a bipartite minor, but not a minor, of $G$, and one set of pairs $G,H$ such that $H$ is a minor, but not a bipartite minor, of $G$.
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https://arxiv.org/abs/2601.23101
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97dc247f260c7334b85fd1ed838678a91e8bb981c9f67b0e224e1814aa84fa46
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2026-02-02T00:00:00-05:00
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Series-Parallel and Planar Graphs for Efficient Broadcasting
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arXiv:2601.23104v1 Announce Type: new Abstract: The broadcasting problem concerns the efficient dissemination of information in graphs. In classical broadcasting, a single originator vertex initially has a message to be transmitted to all vertices. Every vertex which has received the message informs at most one uninformed neighbor at each discrete time unit. In this paper, we introduce infinite families of series-parallel graphs with efficient broadcast times: graphs on $n$ vertices with broadcast time at most $\lceil\log_2 n \rceil + 1$ for any $n$, graphs on $n$ vertices with broadcast time $\lfloor \frac{3 \lceil \log_2 n \rceil}{2} \rfloor$ and maximum degree $\lceil \log_2 n \rceil - 1$ for any $n$, and broadcast graphs on up to $2^{k-1} + 2^{\lfloor \frac{k}{2} \rfloor }$ vertices with broadcast time $k$ for any $k$. We also introduce an infinite family of planar broadcast graphs on up to $2^{k-1} + 2^{\lfloor \frac{3k}{4} \rfloor - 1}$ vertices with broadcast time $k$ for any $k$, which improves the known lower bound on the maximum number of vertices in a planar broadcast graph.
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https://arxiv.org/abs/2601.23104
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83d0838d4930cd432770b68ef3bec436a8130aeb9a8551dd54c403f81c922bc8
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2026-02-02T00:00:00-05:00
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Lifts of endomorphisms of Weyl algebras modulo $p^2$
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arXiv:2601.23110v1 Announce Type: new Abstract: Let $\varphi$ denote a $k$-algebra endomorphism of the $n$-th Weyl algebra $A_n(k)$ over a perfect field $k$ of positive characteristic $p$. We prove that $\varphi$ can be lifted to an endomorphism of the Weyl algebra $A_n(W_2(k))$ over the Witt vectors $W_2(k)$ of length two over $k$ if and only if $\varphi$ induces a Poisson morphism of the center of $A_n(k)$. Furthermore, we improve a result of Tsuchimoto, which enables us to conclude that these equivalent statements hold at least when ${\rm deg}(\varphi) < p$. In particular, we conclude that $\varphi$ is injective if ${\rm deg}(\varphi) < p$.
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https://arxiv.org/abs/2601.23110
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fd15db3c8f94e2c3120ca4ab88b1810d9be7c40527c3669643ddcecf4f8409e9
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2026-02-02T00:00:00-05:00
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The Coxeter Flag Variety
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arXiv:2601.23111v1 Announce Type: new Abstract: For a Coxeter element $c$ in a Weyl group $W$, we define the $c$-Coxeter flag variety $\operatorname{CFl}_c\subset G/B$ as the union of left-translated Richardson varieties $w^{-1}X^{wc}_w$. This is a complex of toric varieties whose geometry is governed by the lattice $\operatorname{NC}(W,c)$ of $c$-noncrossing partitions. We show that $\operatorname{CFl}_c$ is the common vanishing locus of the generalized Pl\"ucker coordinates indexed by $W\setminus\operatorname{NC}(W,c)$. We also construct an explicit affine paving of $\operatorname{CFl}_c$ and identify the $T$-weights of each cell in terms of $c$-clusters. This paving gives a GKM description of $H^\bullet(\operatorname{CFl}_c)$ and $H^\bullet_{T_{ad}}(\operatorname{CFl}_c)$ in terms of the induced Cayley subgraph on $\operatorname{NC}(W,c)$, and we show these rings are naturally isomorphic for different choices of $c$. In type $\mathrm{A}$, this recovers the quasisymmetric flag variety for a special $c$, and for general $c$ we show the cohomology ring has a presentation as permuted quasisymmetric coinvariants.
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https://arxiv.org/abs/2601.23111
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5461f5483feb1fbf945c79a0b2e98b418bd8fc0ba01c17407a08ff13593733b3
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2026-02-02T00:00:00-05:00
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Graded Lie superalgebras from embedding tensors
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arXiv:2601.23113v1 Announce Type: new Abstract: We show how various constructions of $\mathbb{Z}$-graded Lie superalgebras are related to each other. These Lie superalgebras have a Lie algebra $\mathfrak{g}$ as the subalgebra at degree 0, an odd $\mathfrak{g}$-module V as the subspace at degree 1, and an embedding tensor as an element at degree -1. This is a linear map from V to $\mathfrak{g}$ satisfying a quadratic constraint, which equips V with the structure of a Leibniz algebra.
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https://arxiv.org/abs/2601.23113
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74013151098ed1cb5fc9dc5de24d01aa47ef492f81c24f9b37ce52e6cba032b9
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2026-02-02T00:00:00-05:00
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Nonlinear Schr\"odinger Equation with magnetic potential on metric graphs
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arXiv:2601.23115v1 Announce Type: new Abstract: In this manuscript, we shall investigate the Nonlinear Magnetic Schr\"odinger Equation on noncompact metric graphs, focusing on the existence of ground states. We prove that the magnetic Hamiltonian is variationally equivalent to a non-magnetic operator with additional repulsive potentials supported on the graph's cycles. This effective potential is strictly determined by the Aharonov-Bohm flux through the topological loops. Leveraging this reduction, we extend classical existence criteria to the magnetic setting. As a key application, we characterize the ground state structure on the tadpole graph, revealing a mass-dependent phase transition. The ground states exist for sufficiently small repulsion in an intermediate regime of masses while sufficiently strong flux prevents the formation of ground states.
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https://arxiv.org/abs/2601.23115
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dd7f87e2e596ae2fdcc02ad23748265a4a69da9e12a1eb69b7a8ac45b708cc00
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2026-02-02T00:00:00-05:00
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Log canonical thresholds at infinity
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arXiv:2601.23118v1 Announce Type: new Abstract: The paper considers a global version of the notion of log canonical threshold for plurisubharmonic functions $u$ of logarithmic growth in $\mathbb{C}^n$, aiming at description of the range of all $p>0$ such that $e^{-u}\in L^p(\mathbb{C}^n)$. Explicit formulas are obtained in the toric case. By considering Bergman functions of corresponding weighted Hilbert spaces, a new polynomial approximation of plurisubharmonic functions of logarithmic growth with control over its singularities and behavior at infinity (a global version of Demailly's approximation theorem) is established. Some application to tame polynomial maps are given.
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https://arxiv.org/abs/2601.23118
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82ffd8f8136517125cfeafeaa18f6adc3142bba7503223971a5658586b5235a3
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2026-02-02T00:00:00-05:00
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A General Tikhonov Regularized Second-Order Dynamical System for Convex-Concave Bilinear Saddle Point Problems
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arXiv:2601.23120v1 Announce Type: new Abstract: In this paper, we propose a general Tikhonov regularized second-order dynamical system with viscous damping, time scaling and extrapolation coefficients for the convex-concave bilinear saddle point problem. By the Lyapunov function approach, we show that the convergence properties of the proposed dynamical system depend on the choice of the Tikhonov regularization parameter. Specifically, when the Tikhonov regularization parameter tends to zero rapidly, the convergence rate of the primal-dual gap along the generated trajectory is O(1 over t squared times beta(t)); when the Tikhonov regularization parameter tends to zero slowly, the convergence rate of the primal-dual gap is o(1 over beta(t)). We also prove the strong convergence property of the trajectory generated by the Tikhonov regularized dynamical system to the minimum-norm solution of the convex-concave bilinear saddle point problem, and derive several integral estimates. In addition, the effectiveness of the proposed dynamical system is verified through a series of numerical experiments.
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https://arxiv.org/abs/2601.23120
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Academic Papers
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75d4627e19ca783985b506ae0ebb7791a2cdee880fd47893eb030feaece69a20
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2026-02-02T00:00:00-05:00
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Learning and Teaching Calculus Through Its History
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arXiv:2601.23122v1 Announce Type: new Abstract: This paper frames calculus as a global, centuries-long development rather than a subject that began only with Newton and Leibniz. Drawing on ideas from Greek, Indian, Islamic, and later European mathematics, it highlights how concepts like infinity, area, motion, and continuous change slowly evolved through solving problems and cultural exchange. I argue that bringing this history into the classroom helps students see calculus as more than a set of procedures: it becomes a story of human creativity and persistence. By revisiting the questions early mathematicians struggled with, students can better appreciate and better understand the core ideas behind the formulas they use today.
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https://arxiv.org/abs/2601.23122
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Academic Papers
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b6b03d0a6b87243c8b3ad5288658b8ab61b2fe76f56acd1851de70c5dc3bd12c
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2026-02-02T00:00:00-05:00
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Semi-knockoffs: a model-agnostic conditional independence testing method with finite-sample guarantees
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arXiv:2601.23124v1 Announce Type: new Abstract: Conditional independence testing (CIT) is essential for reliable scientific discovery. It prevents spurious findings and enables controlled feature selection. Recent CIT methods have used machine learning (ML) models as surrogates of the underlying distribution. However, model-agnostic approaches require a train-test split, which reduces statistical power. We introduce Semi-knockoffs, a CIT method that can accommodate any pre-trained model, avoids this split, and provides valid p-values and false discovery rate (FDR) control for high-dimensional settings. Unlike methods that rely on the model-$X$ assumption (known input distribution), Semi-knockoffs only require conditional expectations for continuous variables. This makes the procedure less restrictive and more practical for machine learning integration. To ensure validity when estimating these expectations, we present two new theoretical results of independent interest: (i) stability for regularized models trained with a null feature and (ii) the double-robustness property.
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https://arxiv.org/abs/2601.23124
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Academic Papers
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620197f62f5f92f32281bd0f8e054f944181cf641da27d35d4dc73027eda4381
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2026-02-02T00:00:00-05:00
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On the b-function with respect to weights of annihilating ideals in the Weyl algebra
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arXiv:2601.23125v1 Announce Type: new Abstract: Given a polynomial $f\in\mathbb{C}[x_1,\ldots,x_n]$ and an integer $\ell\in\mathbb{Z}$, we study some properties of the b-function with respect to weights of the annihilating ideal Ann$(f^\ell)$. In some particular cases the expression of the b-function is given explicitly.
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https://arxiv.org/abs/2601.23125
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Academic Papers
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241b156f63f863d3f79363e0237bae77284ccd5ce6b39dfaf129a45abc3830e9
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2026-02-02T00:00:00-05:00
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Hyperbolic partial differential equations with complex characteristics on Fourier Lebesgue spaces
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arXiv:2601.23138v1 Announce Type: new Abstract: The aim of this paper is to establish well-posedness properties for hyperbolic PDEs on Fourier Lebesgue spaces. We consider hyperbolic operators with complex characteristics. Since our approach comes from harmonic analysis, we establish boundedness properties of Fourier integral operators with complex-valued phase functions on Fourier Lebesgue spaces, Besov spaces and Triebel-Lizorkin spaces. Indeed, these classes of operators serve as propagators of the considered PDE problems. In terms of the boundedness properties, we prove new results in the case where the canonical relation of the operator is assumed to satisfy the {\it spatial smooth factorization condition}
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https://arxiv.org/abs/2601.23138
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Academic Papers
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760742a05dc9e07725b8e3a57d04c9e7596fead34a78c85146abba7f73606094
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2026-02-02T00:00:00-05:00
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2-covering numbers of some finite solvable groups
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arXiv:2601.23144v1 Announce Type: new Abstract: A 2-covering for a finite group $G$ is a set of proper subgroups of $G$ such that every pair of elements of $G$ is contained in at least one subgroup in the set. The minimal number of subgroups needed to 2-cover a group $G$ is called the 2-covering number and denoted by $\sigma_2(G).$ In \cite{gk} it is conjectured that if $G$ is solvable and not 2-generated, then $\sigma_2(G)=1+q+q^2,$ where $q$ is a prime power. We disprove this conjecture.
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https://arxiv.org/abs/2601.23144
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Academic Papers
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b5fa863629ea39248e5eaf7c7310af3ce515139953c402daf7c7d2b6dae953fa
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2026-02-02T00:00:00-05:00
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Some series representing the zeta function for $\Re s>1$
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arXiv:2601.23158v1 Announce Type: new Abstract: We present series converging to the Riemann zeta function in its half-plane of convergence, and possessing remainders whose sizes decrease geometrically. They are easy to implement numerically, using only polynomial and power functions, and are efficient for obtaining dozens or hundreds of digits (when the imaginary part is not too large). They may prove less suited to very high precision (tens of thousands of digits), due to a linear cost for each new term. One can express the coefficients as linear combinations of Bernoulli numbers, but this is not advantageous numerically. The method is a development of tools introduced by the author for the evaluation of harmonic series with restricted digits in a given radix.
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https://arxiv.org/abs/2601.23158
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Academic Papers
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1b4323a399eb3b73bc8988517f8508b00c1a9cdcb2d9965e6a175c0e0c5fe6a1
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2026-02-02T00:00:00-05:00
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Class choice and the surprising weakness of Kelley-Morse set theory
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arXiv:2601.23165v1 Announce Type: new Abstract: Kelley-Morse set theory KM is weaker than generally supposed and fails to prove several principles that may be desirable in a foundational second-order set theory. Even though KM includes the global choice principle, for example, (i) KM does not prove the class choice scheme, asserting that whenever every set $x$ admits a class $X$ with $\varphi(x,X)$, then there is a class $Z\subseteq V\times V$ for which $\varphi(x,Z_x)$ on every section. This scheme can fail with KM even in low-complexity first-order instances $\varphi$ and even when only a set of indices $x$ are relevant. For closely related reasons, (ii) the theory KM does not prove the {\L}o\'s theorem scheme for internal second-order ultrapowers, even for large cardinal ultrapowers, such as the ultrapower by a normal measure on a measurable cardinal. Indeed, the theory KM itself is not generally preserved by internal ultrapowers. Finally, (iii) KM does not prove that the $\Sigma^1_n$ logical complexity is invariant under first-order quantifiers, even bounded first-order quantifiers. For example, $\forall \alpha{<}\delta\ \psi(\alpha,X)$ is not always provably equivalent to a $\Sigma^1_1$ assertion when $\psi$ is. Nevertheless, these various weaknesses in KM are addressed by augmenting it with the class choice scheme, thereby forming the theory KM+, which we propose as a robust KM alternative for the foundations of second-order set theory.
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https://arxiv.org/abs/2601.23165
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Academic Papers
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96ceb93605daa781e194002a89b0458e2d1f24d526d69dbd016d506483ec2360
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2026-02-02T00:00:00-05:00
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The Total Chromatic Quasisymmetric Functions of a Graph
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arXiv:2601.23170v1 Announce Type: new Abstract: In this paper, we introduce and study two variants of the chromatic quasisymmetric function of a graph: the total chromatic quasisymmetric function via vertex labeling and via acyclic orientations. The original definition of the chromatic quasisymmetric function of a graph by Shareshian and Wachs depends on a labeling of the vertices of the graph, which directly affects the properties of the coefficients appearing in the decomposition of the chromatic quasisymmetric function of a graph into different bases. Motivated by this, we construct the first variant of the chromatic quasisymmetric function of a graph by normalizing it with respect to all the labelings of the vertices. The second variant is motivated by the \emph{tree isomorphism conjecture} and is constructed in terms of acyclic orientations. We investigate the properties of the coefficients in the expansion in the monomial quasisymmetric basis for both variants and provide a comparative analysis. Furthermore, we derive explicit formulas for the coefficients in the monomial decomposition of the two variants for the star graph. For the labeling-based variant, these coefficients arise from a binomial identity for which we provide a combinatorial proof.
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https://arxiv.org/abs/2601.23170
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Academic Papers
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9b203cd0d2d9f046e6e1620515dc911f78fd4495e2d97c2e228a491b8575a1e6
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2026-02-02T00:00:00-05:00
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Interacting dynamical systems on networks and fractals: discrete and continuous models, mean-field limit, and convergence rates
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arXiv:2601.23175v1 Announce Type: new Abstract: We develop a continuum limit and mean-field theory for interacting particle systems (IPS) on self-similar networks, a new class of discrete models whose large-scale behavior gives rise to nonlocal evolution equations on fractal domains. This work extends the graphon-based framework for IPS, used to derive continuum and mean-field limits in the non-exchangeable setting, to situations where the spatial domain is fractal rather than Euclidean. The motivation arises from both physical models naturally formulated on fractals and real-world networks exhibiting hierarchical or quasi-self-similar structure. Our analysis relies on tools from fractal geometry, including Iterated Function Systems and self-similar measures. A central result is an explicit isomorphism between self-similar IPS and graphon IPS, which allows us to justify the continuum and mean-field limits in the self-similar setting. This connection reveals that macroscopic dynamics on fractal domains emerge naturally as limits of dynamics on appropriate discretizations of fractal sets. Another contribution of the paper is the derivation of optimal convergence rates for the discrete self-similar models. We introduce a scale of generalized Lipschitz spaces on fractals, extending the Nikolskii-Besov spaces used in the Euclidean setting, and obtain convergence estimates for discontinuous Galerkin approximations of nonlocal equations posed on fractal domains. These results apply to kernels with minimal regularity addressing models relevant in applications.
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https://arxiv.org/abs/2601.23175
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Academic Papers
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aa0972bcbe819cf580c11f61ad85f6718bd9094f8c456e1be9dfd2ada168560a
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2026-02-02T00:00:00-05:00
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Noetherianity for powers of algebraic representations
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arXiv:2601.23186v1 Announce Type: new Abstract: Powers of a polynomial $\operatorname{GL}$-representation are topologically Noetherian under the action of $\operatorname{Sym} \times \operatorname{GL}$. We show that this result extends to powers of algebraic representations of the orthogonal and the symplectic groups. This work is a natural follow-up to arXiv:2212.05790 and to arXiv:1708.06420.
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https://arxiv.org/abs/2601.23186
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Academic Papers
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e58902b6995ae4a1c449cab34ae9a93a4780550faae020c42fab8316da9ca0cb
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2026-02-02T00:00:00-05:00
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General Optimal Stopping without Time Consistency
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arXiv:2601.23187v1 Announce Type: new Abstract: In this paper, we propose a new framework for solving a general dynamic optimal stopping problem without time consistency. A sophisticated solution is proposed and is well-defined for any time setting with general flows of objectives. A backward iteration is proposed to find the solution. The iteration works with an additional condition, which holds in interesting cases including the time inconsistency arising from non-exponential discounting. Even if the iteration does not work, the equilibrium solution can still be studied by a forward definition.
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https://arxiv.org/abs/2601.23187
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Academic Papers
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5cd6b3c747112a56fe8b190dddf21ca7d0d7849544743eb55785b6b0a72342e3
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2026-02-02T00:00:00-05:00
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Schopieray's Galois-modular extension conjecture
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arXiv:2601.23192v1 Announce Type: new Abstract: Plavnik, Schopieray, Yu, and Zhang have drawn attention to those (automatically premodular) fusion subcategories of modular fusion categories which are submodules for the Galois action on the ambient category. In particular, they showed that a subcategory is a Galois submodule if and only if its centralizer is integral. In the other direction, Schopieray has conjectured that every premodular fusion category can be embedded as a Galois-closed subcategory of a modular category; Schopieray calls such an embedding a "Galois-modular extension." We prove Schopieray's conjecture for pseudounitary categories. Along the way we record some general comments about the minimal nondegenerate extension problem for braided fusion categories.
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https://arxiv.org/abs/2601.23192
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Academic Papers
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25da02c7f868e0f12aecbefeec4e359bf5458e6014ac6883dbc0898250d7bd01
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2026-02-02T00:00:00-05:00
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Non-uniformly elliptic variational problems on BV
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arXiv:2601.23195v1 Announce Type: new Abstract: We establish $\mathrm{W}^{1,1}$-regularity and higher gradient integrability for relaxed minimizers of convex integral functionals on $\mathrm{BV}$. Unlike classical examples such as the minimal surface integrand, we only require linear growth from below but not necessarily from above. This typically comes with a non-uniformly degenerate elliptic behaviour, for which our results extend the presently available bounds from the superlinear growth case in a sharp way.
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https://arxiv.org/abs/2601.23195
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Academic Papers
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b6bca2c37bac1529e13d235675666517d1aa33db3a90a870cc94869c5b5b6506
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2026-02-02T00:00:00-05:00
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Vector-valued Gelfand-Kazhdan criterion
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arXiv:2601.23199v1 Announce Type: new Abstract: The Gelfand-Kazhdan criterion is a fundamental tool for studying multiplicity-one properties of local periods of representations. However, it does not apply to many cases arising in the relative Langlands program. Generalizing the usual Gelfand-Kazhdan criterion, we formulate and prove a vector-valued Gelfand-Kazhdan criterion that fits into the general framework of the relative Langlands program. As an illustration of its effectiveness, we establish the multiplicity-one property for the local Asai Rankin-Selberg periods.
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https://arxiv.org/abs/2601.23199
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Academic Papers
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0319fe64157c7f9ebd48252b50bd3e116c8b8992a002513d08a6b4d1f1913210
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2026-02-02T00:00:00-05:00
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Eigenweights for arithmetic Hirzebruch Proportionality
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arXiv:2601.23245v1 Announce Type: new Abstract: Prior work of Feng--Yun--Zhang established a (Higher) Arithmetic Hirzebruch Proportionality Principle, expressing the arithmetic volumes of moduli stacks of shtukas in terms of differential operators applied to $L$-functions. This formula involves certain "eigenweights" which were calculated in simple cases by Feng--Yun--Zhang, but not in general. We document work of a (custom) AI Agent built upon Gemini Deep Think, which employs tools from algebraic combinatorics to connect these eigenweights to the representation theory of symmetric groups, and then determines them for all classical groups.
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https://arxiv.org/abs/2601.23245
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Academic Papers
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2725291ae8f3abe620b5adc62a18038ad27567bd3264174e5469b96029145309
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2026-02-02T00:00:00-05:00
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Radicals and Nilpotents in Equivariant Algebra
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arXiv:2601.23247v1 Announce Type: new Abstract: Associated to each Tambara functor $T$ is its Nakaoka spectrum $\mathrm{Spec}(T)$, analogous to the Zariski spectrum of a commutative ring. We establish that this topological space is spectral. This result follows from an analysis of the notion of nilpotence in Tamabra functors. We prove that the nilradical of a Tambara functor $T$ (the intersection of all of its prime ideals) is computed levelwise, i.e. consists precisely of the nilpotent elements in $T$. In contrast to ordinary commutative algebra, the nilpotents of $T$ are not the same as the elements $x$ such that $T[1/x] = 0$; we therefore also give a classification of these elements. As a corollary, we observe that the set of these elements in $\pi_\star^s$ (the equivariant stable stems, viewed as an $\mathrm{RO}(G)$-graded Tambara functor) forms an ideal.
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https://arxiv.org/abs/2601.23247
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Academic Papers
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c2a5421b38de1b21373e28ea10ceaf326d298b0e208bc17ecfe41cfaf803acf0
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2026-02-02T00:00:00-05:00
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Theoretical Challenges in Learning for Branch-and-Cut
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arXiv:2601.23249v1 Announce Type: new Abstract: Machine learning is increasingly used to guide branch-and-cut (B&C) for mixed-integer linear programming by learning score-based policies for selecting branching variables and cutting planes. Many approaches train on local signals from lookahead heuristics such as strong branching, and linear programming (LP) bound improvement for cut selection. Training and evaluation of the learned models often focus on local score accuracy. We show that such local score-based methods can lead to search trees exponentially larger than optimal tree sizes, by identifying two sources of this gap. The first is that these widely used expert signals can be misaligned with overall tree size. LP bound improvement can select a root cut set that yields an exponentially larger strong branching tree than selecting cuts by a simple proxy score, and strong branching itself can be exponentially suboptimal (Dey et al., 2024). The second is that small discrepancies can be amplified by the branch-and-bound recursion. An arbitrarily small perturbation of the right-hand sides in a root cut set can change the minimum tree size from a single node to exponentially many. For branching, arbitrarily small score discrepancies, and differences only in tie-breaking, can produce trees of exponentially different sizes, and even a small number of decision differences along a trajectory can incur exponential growth. These results show that branch-and-cut policies trained and learned using local expert scores do not guarantee small trees, thus motivating the study of data-driven methods that produce policies better aligned with tree size rather than only accuracy on expert scores.
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https://arxiv.org/abs/2601.23249
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Academic Papers
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