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7200f48f4876a1f9c3d1ea1fa31b722c1a6dc33f06a3135e20fce494d9c19d5e
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2026-01-13T00:00:00-05:00
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A Note on NBUE and NWBUE Classes of Life Distributions
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arXiv:2601.06760v1 Announce Type: new Abstract: Non-monotonic ageing notions are looked upon as an extension of the corresponding monotonic ageing notions in this work. In particular, the New Better than Used in Expectation (NBUE) and the corresponding non-monotonic analogue New Worse then Better than Used in Expectation (NWBUE) classes of life distributions is considered. Some additional results for the NBUE class are obtained. While many properties of the NBUE class carry over in an analogous way to the NWBUE class, it is shown by means of counterexamples that the moment bounds do not. Some corrective results with respect to popular notions of the NWBUE class are also presented.
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https://arxiv.org/abs/2601.06760
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6807e39408f962bfa9ac3a52b4b4d70d7a342bc48d9519e446408232236bb714
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2026-01-13T00:00:00-05:00
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Massively Parallel Reductions in Multivariate Polynomial Systems: Bridging the Symbolic Preprocessing Gap on GPGPU Architectures
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arXiv:2601.06765v1 Announce Type: new Abstract: Gr\"obner basis computation over multivariate polynomial rings remains one of the most powerful yet computationally hostile primitives in symbolic computation. While modern algorithms (Faug\`ere-type F4 and signature-based F5) reduce many instances to large sparse linear algebra over finite fields, their dominant cost is not merely elimination but the symbolic preprocessing that constructs Macaulay-style matrices whose rows encode shifted reducers. This phase is characterized by dynamic combinatorics (monomial discovery, sparse row assembly, and deduplication) and is typically memory-latency bound, resisting naive parallelization. This article develops a rigorous synthesis that reframes S-polynomial reduction as syzygy discovery: row construction is a structured map from module relations to the kernel of a massive, sparse, highly non-random Macaulay matrix A over Fp. Building on this viewpoint, we propose a GPU-targeted architecture that (i) converts dynamic symbolic data structures into static, two-pass allocations via prefix-sum planning; (ii) enforces coalesced memory access through structure-of-arrays polynomial layouts and sorted monomial dictionaries; and (iii) integrates finite-field arithmetic kernels (Montgomery/Barrett-style reduction) at register granularity. On the linear-algebra side, we explore the transition from classical Gaussian elimination to parallel structured Gaussian elimination (PSGE) and to Krylov-type kernel solvers (Block Wiedemann/Lanczos) that better match GPU throughput while controlling fill-in. The result is a principled bridge between algebraic syzygy theory and SIMT hardware constraints, isolating the true bottleneck and providing a pathway to massively parallel reductions for multivariate polynomial systems.
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https://arxiv.org/abs/2601.06765
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43e6c95223711dcc53fdb62395edc0383e579ddd110a223f2496b5fef860a528
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2026-01-13T00:00:00-05:00
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On the conformal preimage decay exponent of the Julia sets of rational graph-directed Markov systems
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arXiv:2601.06785v1 Announce Type: new Abstract: We define and investigate the conformal preimage decay exponent of the Julia sets of rational graph-directed Markov systems. We show that this exponent coincides with the difference between the topological entropy and upper sequential capacity topological pressure for the rational skew product map associated with the system $S$. Here, upper sequential capacity topological pressure is a slight generalisation of upper capacity topological pressure given in \cite{MR969568}.
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https://arxiv.org/abs/2601.06785
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d8ea9b0d9b35b7708206df91435c71a8a3f692067cea0a81e1eddf0ed6eb55b2
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2026-01-13T00:00:00-05:00
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Generalized Space-Fractional Poisson Process via Variable-Order Stable Subordinator
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arXiv:2601.06808v1 Announce Type: new Abstract: This paper introduces a variable-order stable subordinator (VOSS) $S^{\alpha(t)}(t)$ with index $\alpha(t)\in(0,1)$, where $\alpha(t)$ is a right-continuous piecewise constant function. We drive the Generalized Space-Fractional Poisson Process via Variable-Order Stable Subordinator (GSFPP-VO) defined by $\{N(S^{\alpha(t)}(t))\}_{t \geq 0}$, obtained by time-changing a homogeneous Poisson process $\{N(t,\lambda)\}_{t\geq 0}$ with rate parameter $\lambda>0$ by an independent VOSS. Explicit expressions for the Laplace transform, probability generating function, probability mass function, and moment generating function of the GSFPP-VO are derived, and these quantities are shown to satisfy partial differential equations. Finally, we establish the associated generalized distributions, analyze the hitting-time properties, and characterize the L\'evy measures of the GSFPP-VO.
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https://arxiv.org/abs/2601.06808
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0c124a70d0510417835579e9c28e4adb47acacbcc4db28b86dc0914da937eaa9
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2026-01-13T00:00:00-05:00
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Moment Summation Methods and Non-Homogeneous Carleman Classes
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arXiv:2601.06812v1 Announce Type: new Abstract: We extend the classical theorems of F. Nevanlinna and Beurling by characterizing the image of various spaces of smooth functions under the generalized Laplace transform. To achieve this, we introduce and analyze novel non-homogeneous Carleman classes, which generalize the traditional homogeneous definitions. This characterization allows us to derive necessary and sufficient conditions for the applicability of moment summation methods within a given class of functions. Furthermore, we establish an extension of \'{E}calle's concept of quasianalytic continuation and apply these results to the theory of multi-summability and Euler-type differential equations.
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https://arxiv.org/abs/2601.06812
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c9a9be42a6dacff822075b390aa1431dfbc6b694ae01d1ada7b13291e768c4ed
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2026-01-13T00:00:00-05:00
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Algebraic topology of the Lagrange inversion
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arXiv:2601.06814v1 Announce Type: new Abstract: The Lagrange inversion formula for power series is one of the classical formulas from analysis and combinatorics. A nice geometric interpretation of this formula in terms of the Stasheff polytopes was discovered by Loday. We show that it also admits a natural topological interpretation in terms of the Chern numbers of the complex projective space. The proof is based on our earlier work on the Chern-Dold character in complex cobordism theory and leads to a new derivation of the Lagrange inversion formula. We provide a similar interpretation of the multiplicative inversion formulas in terms of Chern numbers of the smooth theta divisors. We discuss also the general related problem when all Chern numbers of an algebraic variety are divisible by its Euler characteristic.
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https://arxiv.org/abs/2601.06814
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387447686083113abb07e07c836285e533d3cf080515c2d2fd47e4196c78902e
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2026-01-13T00:00:00-05:00
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Unimodular Equivalence of Integral Simplices
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arXiv:2601.06819v1 Announce Type: new Abstract: Testing the unimodular equivalence of two full-dimensional integral simplices can be reduced to testing unimodular permutation (UP) equivalence of two nonsingular matrices. We conduct a systematic study of UP-equivalence, which leads to the first average-case quasi-polynomial time algorithm, called \texttt{HEM}, for deciding the unimodular equivalence of $d$-dimensional integral simplices, as well as achieving a polynomial-time complexity with a failure probability less than $2.5 \times 10^{-7}$. A key ingredient is the introduction of the \emph{permuted Hermite normal form} and its associated \emph{pattern group}, which streamlines the UP-equivalence test by comparing canonical forms derived from induced coset representatives. We also present an acceleration strategy based on Smith normal forms. As a theoretical by-product, we prove that two full-dimensional integral simplices are unimodularly equivalent if and only if their $n$-dimensional pyramids are unimodularly equivalent. This resolves an open question posed by Abney-McPeek et al.
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https://arxiv.org/abs/2601.06819
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038a0e1350442ff2bab5a31e6ba55d16cbd143d2049a4e067beb4bf5226568d0
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2026-01-13T00:00:00-05:00
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Classical elliptic ${\rm BC}_1$ Ruijsenaars-van Diejen model: relation to Zhukovsky-Volterra gyrostat and 1-site classical XYZ model with boundaries
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arXiv:2601.06826v1 Announce Type: new Abstract: We present a description of the classical elliptic ${\rm BC}_1$ Ruijsenaars-van Diejen model with 8 independent coupling constants through a pair of ${\rm BC}_1$ type classical Sklyanin algebras generated by the (classical) quadratic reflection equation with non-dynamical XYZ $r$-matrix. For this purpose, we consider the classical version of the $L$-operator for the Ruijsenaars-van Diejen model proposed by O. Chalykh. In ${\rm BC}_1$ case it is factorized to the product of two Lax matrices depending on 4 constants. Then we apply an IRF-Vertex type gauge transformation and obtain a product of the Lax matrices for the Zhukovsky-Volterra gyrostats. Each of them is described by the ${\rm BC}_1$ version of the classical Sklyanin algebra. In particular case, when 4 pairs of constants coincide, the ${\rm BC}_1$ Ruijsenaars-van Diejen model exactly coincides with the relativistic Zhukovsky-Volterra gyrostat. Explicit change of variables is obtained. We also consider another special case of the ${\rm BC}_1$ Ruijsenaars-van Diejen model with 7 independent constants. We show that it can be reproduced by considering the transfer matrix of the classical 1-site XYZ chain with boundaries. In the end of the paper, using another gauge transformation we represent the Chalykh's Lax matrix in a form depending on the Sklyanin's generators.
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https://arxiv.org/abs/2601.06826
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b5a9b100fd4ce2e4a2d5e7ef197c4c9b7e1ef5731209bf68dbeea0a7d053ad00
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2026-01-13T00:00:00-05:00
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Homogenization of L\'evy-type operators: operator estimates with correctors
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arXiv:2601.06832v1 Announce Type: new Abstract: The goal of the paper is to study in $L_2(\R^d)$ a self-adjoint operator ${\mathbb A}_\eps$, $\eps >0$, of the form $$ ({\mathbb A}_\eps u) (\x) = \int_{\R^d} \mu(\x/\eps, \y/\eps) \frac{\left( u(\x) - u(\y) \right)}{|\x - \y|^{d+\alpha}}\,d\y $$ with $1< \alpha < 2$; here the function $\mu(\x,\y)$ is $\Z^d$-periodic in the both variables, satisfies the symmetry relation $\mu(\x,\y) = \mu(\y,\x)$ and the estimates $0< \mu_- \leqslant \mu(\x,\y) \leqslant \mu_+< \infty$. The rigorous definition of the operator ${\mathbb A}_\eps$ is given in terms of the corresponding quadratic form. In the previous work of the authors it was shown that the resolvent $({\mathbb A}_\eps + I)^{-1}$ converges, as $\eps\to0$, in the operator norm in $L_2(\mathbb R^d)$ to the resolvent of the effective operator $A^0$, and the estimate $\|({\mathbb A}_\eps + I)^{-1} - (\A^0 + I)^{-1} \| = O(\eps^{2-\alpha})$ holds. In the present work we achieve a more accurate approximation of the resolvent of ${\mathbb A}_\eps$ which takes into account the correctors. Namely, for $N\in\mathbb N$ such that $2-1/N < \alpha \le 2-1/(N+1)$, we obtain $$ \bigl\|({\mathbb A}_\eps + I)^{-1} - (\A^0 + I)^{-1} - \sum_{m=1}^N \eps^{m(2-\alpha)} \mathbb{K}_m \bigr\| = O(\eps). $$
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https://arxiv.org/abs/2601.06832
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1be38beaf62185a779154e610dbe66ee2bcbc7c68719cdd662288180358470bb
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2026-01-13T00:00:00-05:00
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Polyominoes with maximal number of deep holes
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arXiv:2601.06840v1 Announce Type: new Abstract: In this paper, we study the extremal behaviour of deep holes in polyominoes. We determine the maximum number, $h_n$ of deep holes that an $n$-omino can enclose, ensuring that the boundary of each hole is disjoint from the boundaries of any other hole and from the outer boundary of the $n$-tile. Using the versatile application of Pick's theorem, we establish the lower and the upper bound for $h_n$, and show that $h_n=\frac{n}{3}+o(n)$ asymptotically. To further develop these results, we compute $h_n$ as a function of $n$ for an infinite subset of positive integers.
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https://arxiv.org/abs/2601.06840
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5e37db15d43b865cbd7174d7445c2c8ce33d29320231ce4908efbc83a76cc918
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2026-01-13T00:00:00-05:00
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Explosion and non-explosion in pure birth Crump--Mode--Jagers branching processes
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arXiv:2601.06850v1 Announce Type: new Abstract: In this short note, we provide an explicit sufficient condition for non-explosion of Crump--Mode--Jagers branching processes with pure birth reproduction. It shows that the standard sufficient condition for explosion, namely the convergence of the series of reciprocals of the birth rates, is -- at least for rate sequences without excessive oscillations -- remarkably close to being necessary. At the same time, it is not necessary in full generality: we construct a counterexample which also yields a general preferential attachment tree without fitness with an infinite path and no vertices of infinite degree, thereby answering an open question previously raised in the literature.
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https://arxiv.org/abs/2601.06850
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4dc39e8ab3ba9f172a6448ee4e3893b29761488cdd5703811e37241f22bec97f
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2026-01-13T00:00:00-05:00
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Fixed points and holomorphic structures on line bundles over the quantum projective line $\mathbb{C}\mathrm{P}_q^1$
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arXiv:2601.06852v1 Announce Type: new Abstract: It has recently been observed that, in contrast to the classical case, holomorphic structures on line bundles over the quantum projective line $\mathbb{C}\mathrm{P}_q^1$ are not uniquely determined by degree. In this work, we develop a fixed-point-theoretic framework for the analysis of flat $\overline\partial$-connections that define holomorphic structures on line bundles over the quantum projective line. Within this framework, we establish sufficient conditions ensuring the gauge equivalence of holomorphic connections. Furthermore, we obtain a necessary and sufficient criterion characterising when two such holomorphic connections are gauge equivalent. This criterion is formulated in terms of the existence of fixed points, lying in the open unit ball, of certain nonlinear maps acting on an appropriate Banach space.
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https://arxiv.org/abs/2601.06852
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b070b752effe77fd6bbcd0ef81b180bb5da8c83e5e43b012ed8fa55d863cba4f
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2026-01-13T00:00:00-05:00
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Complex Analysis and Riemann Surfaces: A Graduate Path to Algebraic Geometry
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arXiv:2601.06868v1 Announce Type: new Abstract: These lecture notes present a computation driven pathway from classical complex analysis to the theory of compact Riemann surfaces and their connections to algebraic geometry. The exposition follows a compute first then abstract philosophy, in which analytic and geometric structures are introduced through explicit calculations and local models before being organized into conceptual frameworks. The notes begin with the foundations of complex analysis, including holomorphic functions, Cauchy theory, power series, residues, and contour integration, with an emphasis on hands on techniques such as Laurent expansions, residue calculus, and branch cut methods. These analytic tools are then used to construct Riemann surfaces explicitly via branched coverings and gluing constructions, which serve as recurring test cases throughout the text. Differential forms, Stokes theorem, curvature, and the Gauss Bonnet theorem provide the geometric bridge to Hodge theory, culminating in a detailed and self contained treatment of the Hodge Weyl theorem on compact Riemann surfaces, including weak formulations, regularity, and concrete examples. The algebraic geometric core develops holomorphic line bundles, divisors, the Picard group, and sheaves, followed by Cech and sheaf cohomology, the exponential sequence, and de Rham and Dolbeault theories, all treated with explicit computations. The Riemann Roch theorem is presented with full proofs and applications, leading to the construction of the Jacobian, Abel Jacobi theory, theta functions, and the correspondence between Riemann surfaces, algebraic curves, and Galois coverings. Originating from collaborative study groups associated with the Enjoying Math community, these notes are intended for graduate students seeking a concrete and unified route from complex analysis to algebraic geometry.
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https://arxiv.org/abs/2601.06868
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c8e7b3d00ea9e9a290e055e087d90736d0a6300bda61b072aabb2e8a4df314fc
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2026-01-13T00:00:00-05:00
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A note on Bohr chaos and hyperbolic sets
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arXiv:2601.06869v1 Announce Type: new Abstract: This paper studies the relationship between shadowing phenomena and Bohr chaos in dynamical systems. We provide sufficient conditions for Bohr chaos in terms of shadowing. By combining those conditions with the shadowing lemma, we obtain some results on Bohr chaos and hyperbolic sets. Our results also highlight some simple but non-trivial structural properties of hyperbolic sets.
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https://arxiv.org/abs/2601.06869
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fc6faec4aec9d57a7d509c2351e3c5709952ad82bb96f5f3886d58a994283548
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2026-01-13T00:00:00-05:00
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Nearly Erd\H{o}s-Ko-Rado theorems
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arXiv:2601.06871v1 Announce Type: new Abstract: If a family $\mathcal{F}$ of $k$-element subsets of an $n$-element set is pairwise intersecting, $2k\leq n$ then $|\mathcal{F}|\leq {n-1\choose k-1}$ holds by the celebrated Erd\H{o}s-Ko-Rado theorem. But an intersecting family obviously satisfies the condition $${\ell \choose 2}\leq \sum_{1\leq i<j\leq \ell}|F_i\cap F_j| $$ for any $\ell$ distinct members of the family. It has been proved in [5] that even if ${\ell \choose 2}$ is replaced by ${\ell -1 \choose 2}+1$ the conclusion $|\mathcal{F}|\leq {n-1\choose k-1}$ remains valid for large $n$. However the 1 cannot be omitted, because there is a larger family satisfying that weaker condition. In the present paper we determine the largest size of the family under this weaker condition when $n$ is sufficiently large. All of these are treated in the more general setting of $t$-intersecting families.
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https://arxiv.org/abs/2601.06871
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32fc5a526b4f02c763928ca10db93dc081e0e7f9308a579aa606ed2510fe1c32
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2026-01-13T00:00:00-05:00
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The Brauer-Manin obstruction of Symmetric products
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arXiv:2601.06872v1 Announce Type: new Abstract: This article focuses on smooth, projective, and geometrically integral varieties $X$ over a field $k$, whose geometric Picard group $Pic(X_{\overline{k}})$ is torsion-free. We establish an isomorphism$Br(X)/Br(k) \simeq Br_{nr}\bigl(Sym_{X/k}^{n}\bigr)/Br(k)$, where $Sym_{X/k}^{n}$ denotes the $n$-th symmetric product. Using this isomorphism, we investigate the relationship between the Brauer--Manin obstruction to the Hasse principle and weak approximation for rational points on the smooth proejctive model $Sym_{X/k}^{n,sm}$, and the corresponding obstruction for $0$-cycles of degree $n$ on $X$.
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https://arxiv.org/abs/2601.06872
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5a1e755fa898c9df67d259c329f405bb756c38742bfb6a7bb6eca47e7a552f1f
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2026-01-13T00:00:00-05:00
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The second Hochschild cohomology and deformations of Brauer graph algebras
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arXiv:2601.06888v1 Announce Type: new Abstract: In this paper, we give an explicit description about the second Hochschild cohomology groups of bipartite Brauer graph algebras with trivial grading. Based on this, we provide geometric interpretations of deformations associated to some standard cocycles in terms of the surface models of Brauer graph algebras.
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https://arxiv.org/abs/2601.06888
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f29b595dbf349fa752fcf61b77544c96a20ddb381f6d6d48299a2b512fad853b
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2026-01-13T00:00:00-05:00
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Global regularity and sharp decay to the 2D Hypo-Viscous compressible Navier-Stokes equations
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arXiv:2601.06889v1 Announce Type: new Abstract: In this paper, we consider the global regularity and the optimal time decay rate for the 2D isentropic hypo-viscous compressible Navier-Stokes equations. Firstly, we prove that there exists a global strong solution with the small initial data are close to the constant equilibrium state in $H^s$ framework with $s>1$. Furthermore, by virtue of improved Fourier splitting method and the Littlewood-Paley decomposition theory, we then establish the optimal time decay rate for low regularity data.
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https://arxiv.org/abs/2601.06889
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afd8d3e39177945b316d8882c254e874150dc4ca7210e623c0dacf411234b007
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2026-01-13T00:00:00-05:00
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Absorption times for discrete Whittaker processes and non-intersecting Brownian bridges
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arXiv:2601.06893v1 Announce Type: new Abstract: We present evidence for a conjectural relationship between absorption times for discrete Whittaker processes and maximal heights of non-intersecting Brownian bridges.
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https://arxiv.org/abs/2601.06893
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138caae7382bb2732f44ae9aff684260100a474172fa19fd268e38e53ada70b9
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2026-01-13T00:00:00-05:00
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Explicit Evaluations of Euler Sums Involving Harmonic Numbers with Rational Arguments
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arXiv:2601.06895v1 Announce Type: new Abstract: This study presents explicit evaluations of the series \begin{equation*} \sum_{k=1}^\infty \frac{H_{k/n}^{(p)}}{k^q} \quad \text{and} \quad \sum_{k=1}^\infty \frac{(-1)^k H_{k/2n}^{(p)}}{k^q}, \quad p,q,n \in \mathbb{Z}_{\ge 1},\; q \ne 1, \end{equation*} for odd values of $p+q$. These explicit evaluations are expressed in terms of the Riemann zeta function and the Hurwitz zeta function.
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https://arxiv.org/abs/2601.06895
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0b0faa8b92730b499e6ac64f1188929d7189404c23b7708d608edcaf1fbf593d
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2026-01-13T00:00:00-05:00
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Elimination ideals of Pl\"ucker ideals and algebras with straightening laws
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arXiv:2601.06897v1 Announce Type: new Abstract: It is well known that the Pl\"ucker ideal defining the Grassmannian is generated by quadratic Pl\"ucker relations. These relations form a reverse lexicographic Gr\"obner basis and endow the Pl\"ucker algebra with the structure of an algebra with straightening laws (ASL). In this paper, we study quadratically generated projections of the Grassmannian of lines $\mathrm{Gr}(2,n)$. We then combinatorially characterize the Gorenstein ASL subalgebras of the Pl\"ucker algebra of $\mathrm{Gr}(2,n)$.
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https://arxiv.org/abs/2601.06897
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83d8aecc343f2469693eac78a47ea72171b7520b9b21a7c37a247267a9f8d1eb
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2026-01-13T00:00:00-05:00
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Existence results for a non-relativistic Chern-Simons model with purely mutual interaction
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arXiv:2601.06901v1 Announce Type: new Abstract: We are concerned with a skew-symmetric singular Liouville system arising in non-relativistic Chern-Simons theory. Based on its variational structure, we establish existence and multiplicity results. Since the energy functional is indefinite, standard variational approaches do not apply directly. We overcome this difficulty by introducing a suitable constrained problem and implementing a Morse-theoretical argument
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https://arxiv.org/abs/2601.06901
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c93741727b4f8f7d78e4ae9a85d158852fc4a014a7907e2685658ac4f93019e5
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2026-01-13T00:00:00-05:00
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The edge-isoperimetric inequality for powers of cycles
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arXiv:2601.06912v1 Announce Type: new Abstract: This note provides a complete solution to a certain version of the edge-isoperimetric problem for powers of a cycle graph. Namely, it shows that the maximum number of edges inside a vertex subset of $C_n^s$ of size $k$ is achieved by a set of $k$ consecutive vertices.
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https://arxiv.org/abs/2601.06912
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6081ab8d2c2f8af7faf5816b8c3b21be1b15b0f7c3f535ba80d65dc09d2159c8
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2026-01-13T00:00:00-05:00
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Infinite sumsets in $U^k(\Phi)$-uniform sets
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arXiv:2601.06915v1 Announce Type: new Abstract: Extending recent developments of Kra, Moreira, Richter and Roberson, we study infinite sumset patterns in $U^k(\Phi)$-uniform subsets of the integers, defined via the local uniformity seminorms introduced by Host and Kra. The main result relates the degree $k$ of a $U^k(\Phi)$-uniform set to the existence of sumset patterns along prescribed vertices of $\ell$-dimensional parallelepipeds, for $k \leq \ell$. The proof relies on a dynamical analysis of return-time sets to neighborhoods of points lying over pronilfactor fibers. We then derive higher-order parity obstructions for sumset patterns and consequences in topological dynamics.
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https://arxiv.org/abs/2601.06915
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9a1f58d6c96f2379bdeacea36a3558a2ea1bd40e2c97dc31802de66ea94e8f58
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2026-01-13T00:00:00-05:00
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Asymptotic formulas for phase recovering from phaseless data of biharmonic waves at a fixed frequency
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arXiv:2601.06917v1 Announce Type: new Abstract: This paper focuses on phase retrieval from phaseless total-field data in biharmonic scattering problems. We prove that a phased biharmonic wave can be uniquely determined by the modulus of the total biharmonic wave within a nonempty domain. As a direct corollary, the uniqueness for the inverse biharmonic scattering problem with phaseless total-field data is established. Moreover, using the Atkinson-type asymptotic expansion, we derive explicit asymptotic formulas for the problem of phase retrieval.
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https://arxiv.org/abs/2601.06917
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164dc1c38b7999a7bcb8f9422de3a21fd85b24f5aa7efe89bdf2200be632d8b4
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2026-01-13T00:00:00-05:00
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On the zero-free region for the chromatic polynomial of claw-free graphs with and without induced square and induced diamond
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arXiv:2601.06918v1 Announce Type: new Abstract: Given a claw-free graph $G=(V,E)$ with maximum degree $\Delta$, we define the parameter $\kappa\in [0,1]$ as $\kappa={\max_{v\in V}|I_v|\over \lfloor\Delta^2/4\rfloor}$ where $I_v$ is the set of all independent pairs in the neighborhood of $v$. We refer to $\kappa$ as the pair independence ratio of $G$. We prove that for any claw-free graph $G$ with pair independence ratio at most $\kappa$ the zeros of its chromatic polynomial $P_G(q)$ lie inside the disk $D=\{q\in \mathbb{C}:~|q|< C_\kappa^0\Delta\}$, where $C_\kappa^0$ is an increasing function of $\kappa\in [0,1]$. If $G$ is also square-free and diamond free, the function $C_\kappa^0$ can be replaced by a sharper function $C_\kappa^1$. These bounds constitute an improvement upon results recently given by Bencs and Regts in ''Improved bounds on the zeros of the chromatic polynomial of graphs and claw-free graphs''.
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https://arxiv.org/abs/2601.06918
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584814b8f67486af48a1becfa81dda293d486d36a1a9ee28a2571aa91a678f22
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2026-01-13T00:00:00-05:00
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The structure of Morse flows and co-dimension one gradient flows on the sphere with holes
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arXiv:2601.06926v1 Announce Type: new Abstract: We describe all possible topological structures of typical one-parameter bifurcations of gradient flows on the 2-sphere with holes in the case that the number of singular point of flows is at most six. To describe structures, we separatrix diagrams of flows. The saddle-node singularity is specified by selecting a separatrix in the diagram of the flow befor the bifurcation and the saddle connection is specified by a separatrix, which conect two saddles.
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https://arxiv.org/abs/2601.06926
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b6e2aad1e57565a537f4bdddcd0ce44d3060ef950534410a5b6ae63b50f814bc
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2026-01-13T00:00:00-05:00
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Deep level Deligne--Lusztig induction for tamely ramified tori
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arXiv:2601.06929v1 Announce Type: new Abstract: Deep level Deligne--Lusztig representations, which are natural analogues of classical Deligne--Lusztig representations, recently play an important role in geometrization of irreducible supercuspidals of $p$-adic groups. In this paper, we propose a construction of deep level Deligne--Lusztig varieties/representations in the tamely ramified case, extending previous constructions in the unramified case. As an application, under a mild assumption on the residue field, we show that each regular irreducible supercuspidal is the compact induction of a deep level Deligne--Lusztig representation, and generally, each irreducible supercuspidal is a direct summand of the compact induction of the cohomology of a deep level Deligne--Lusztig variety.
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https://arxiv.org/abs/2601.06929
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5e154cf0702efcc2cbba39b1c1d6e7d6cb3e0b20aa333b1e494e10289812f5e1
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2026-01-13T00:00:00-05:00
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The symplectic left companion of a Littlewood-Richardson-Sundaram tableau and the Kwon property
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arXiv:2601.06930v1 Announce Type: new Abstract: As a consequence of the Littlewood-Richardson (LR) commuters coincidence and the Kumar-Torres branching model via Kushwaha-Raghavan-Viswanath flagged hives, we have solved the Lecouvey-Lenart conjecture on the bijections between the Kwon and Sundaram branching models for the pair $({GL}_{2n}(\mathbb{C}), {Sp}_{2n}(\mathbb{C})) $ consisting of the general linear group ${GL}_{2n}(\mathbb{C})$ and the symplectic group ${Sp}_{2n}(\mathbb{C})$. In particular, thanks to the Henriques-Kamnitzer $gl_n$-crystal commuter, we have recognized that the left companion of an LR-Sundaram tableau is characterized by the Kwon symplectic condition. We now show that the construction of the left companion tableau of an LR-Sundaram tableau exhibits in fact the Kwon symplectic property.
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https://arxiv.org/abs/2601.06930
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53edbcbe2cd45e5f1732a630f6444902d1077f0824510a3042c399e4dd2a3d8d
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2026-01-13T00:00:00-05:00
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Profinite genus of HNN-extensions with finite associated subgroups
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arXiv:2601.06934v1 Announce Type: new Abstract: We study the profinite genus of HNN-extensions whose associated subgroups are finite. We give precise formulas for the number of isomorphism classes of HNN(G,H,K,t,f) and of its profinite completion and compute the profinite genus of such an HNN-extension HNN(G,H,K,t,f). We also list various situations when HNN(G,H,K,t,f) is determined by its profinite completion.
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https://arxiv.org/abs/2601.06934
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485ccbf84db5fae4be06791a29392f2c3735d58b6514e5339fc3f0c61c72b6ee
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2026-01-13T00:00:00-05:00
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Wave packet systems and connections to spectral analysis of limiting operators
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arXiv:2601.06945v1 Announce Type: new Abstract: We discuss the design of ``wave packet systems'' that admit strong concentration properties in phase space. We make a connection between this problem and topics in signal processing related to the spectral behavior of spatial and frequency-limiting operators. The results have engineering applications in medical imaging, geophysics, and astronomy.
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https://arxiv.org/abs/2601.06945
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33e0a0aef507e69b6188c1da31d3332e5dd4983411444af71ca44fb3608c9aec
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2026-01-13T00:00:00-05:00
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Normalized Rank- and Determinant-Preserving Mappings of Locally Matrix Algebras
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arXiv:2601.06950v1 Announce Type: new Abstract: Let $A$ be a unital locally matrix algebra. Among the examples of such algebras are: (1) an infinite tensor product $\otimes M_{n_i}(\mathbb{F})$ of matrix algebras over a field $\mathbb{F}$, and (2) the Clifford algebra of a nondegenerate quadratic form on an infinite-dimensional vector space over an algebraically closed field of characteristic different from $2$. We describe linear mappings $A \to B$ between unital locally matrix algebras that preserve the normalized rank. When $\mathbb{F}$ is a field of real or complex numbers, we also describe linear mappings $A \to A$ that preserve the normalized determinant.
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https://arxiv.org/abs/2601.06950
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9c0d79cf7bc42d1675a41fa14d3e37a0e4272a14483a7d376a719ef57cc3583c
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2026-01-13T00:00:00-05:00
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A note on the lower bounds of the first nonzero Steklov eigenvalue on compact manifolds
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arXiv:2601.06951v1 Announce Type: new Abstract: Let $(\Omega^{n+1}, g)$ be an $(n+1)$-dimensional smooth compact connected Riemannian manifold with smooth boundary $\Sigma$, satisfying that ${\text{Ric}_{\Omega}}\ge 0$ and $\Sigma$ is strictly convex, more precisely, its second fundamental form $h\ge cg_{\Sigma}$ for some positive constant $c$. Escobar {\cite{escobar1997geometry}} considered the first nonzero Steklov eigenvalue $\sigma_1$ of $(\Omega^{n+1}, g)$ and proved that $\sigma_1\geq c$ when $n=1$ and $\sigma_1>{\frac{c}{2}}$ when $n \geq 2$. He then conjectured {\cite{escobar1999isoperimetric}} that the first nonzero Steklov eigenvalue $\sigma_1\ge c$. Very recently, Xia and Xiong {\cite{xia2023escobar}} confirmed Escobar's conjecture in the case that $\Omega$ has nonnegative sectional curvature, by constructing a weight function and using appropriate integral identities. In this paper, we construct a new weight function under certain sectional curvature assumptions and provide some new lower bounds for the first nonzero Steklov eigenvalue, which can be considered as generalizations of the results of Escobar and Xia-Xiong. As an application of the weight function, we also consider lower bound estimate of the first nonzero Steklov eigenvalue under conformal transformations.
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https://arxiv.org/abs/2601.06951
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3c24d377c83d8c9532a21ad06133ce54de0b8b6d4a6a0304715366b50e3018ba
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2026-01-13T00:00:00-05:00
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Splitting Proximal Point Algorithms for the Sum of Prox-Convex Functions
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arXiv:2601.06970v1 Announce Type: new Abstract: This paper addresses the minimization of a finite sum of prox-convex functions under Lipschitz continuity of each component. We propose two variants of the splitting proximal point algorithms proposed in \cite{Bacak,Bertsekas}: one deterministic with a fixed update order, and one stochastic with random sampling, and we extend them from convex to prox-convex functions. We prove global convergence for both methods under standard stepsize a\-ssump\-tions, with almost sure convergence for the stochastic variant via supermartingale theory. Numerical experiments with nonconvex quadratic functions illustrate the efficiency of the proposed methods and support the theoretical results.
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https://arxiv.org/abs/2601.06970
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246c79652e832830c04ed1e54017a7962f814220c25746c4dbaa446dfb1be0af
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2026-01-13T00:00:00-05:00
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A belief-state restless bandit model for treatment adherence: Whittle indexability via partial conservation laws
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arXiv:2601.06976v1 Announce Type: new Abstract: We study capacity-constrained treatment-adherence outreach via a belief-state restless multi-armed bandit model where patients are a partially observed two-state (adherent/nonadherent) Markov processes and interventions induce reset-type belief dynamics. Using partial conservation laws (PCLs), we establish Whittle indexability of the single-patient problem and derive a closed-form Whittle (marginal productivity) index, together with closed-form reward/work performance metrics under threshold policies and an explicit optimal threshold map. This yields an analytic Lagrangian relaxation: the single-patient Lagrangian value is a piecewise-affine convex function of the intervention price, enabling efficient computation of multi-patient dual bounds and certified relative optimality gaps. We also analyze how the Whittle index depends on the lapse and spontaneous-recovery parameters, providing qualitative insights on intervention priorities. In a large-scale numerical study over heterogeneous two-type populations, we compare Whittle's index policy with a myopic index rule and simple baselines; while myopic is highly competitive on most instances, Whittle's policy yields substantial gains in tight-capacity regimes with a fragile minority, reaching up to about $26\%$ higher reward and markedly smaller relative optimality gaps.
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https://arxiv.org/abs/2601.06976
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32bb4a3bf1e333b5c96f6f1af161b3ab6b54628d47520b6154242d471e2f9147
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2026-01-13T00:00:00-05:00
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Alternating Direction Method of Multipliers for nonlinear constrained convex problems and applications to distributed resource allocation and constrained machine learning
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arXiv:2601.06977v1 Announce Type: new Abstract: We study a class of structured convex optimization problems, which have a two-block separable objective and nonlinear functional constraints as well as affine constraints that couple the two block variables. Such problems naturally arise from distributed resource allocation and constrained machine learning. To achieve high communication efficiency for the distributed applications, we propose a nonlinear alternating direction method of multipliers (NL-ADMM) that preserves the classical splitting structure while accommodating general convex functional constraints. Unlike existing ADMM variants for nonconvex constrained problems, the proposed method does not require smoothness of the objective functions or differentiability of the constraint mapping, by leveraging convexity of the considered problem. We establish global convergence and an ergodic $O(1/k)$ convergence rate of NL-ADMM by assuming the existence of a KKT solution. The results extend those of ADMM for linearly constrained convex problems. Numerical experiments are conducted on two representative distributed tasks. The results on numerous instances demonstrate that NL-ADMM can achieve (in many cases) 100x higher communication efficiency than the classic augmented Lagrangian method and nearly 2x higher than the Douglas-Rachford operator splitting method, making the new method well suited for large-scale distributed learning systems.
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https://arxiv.org/abs/2601.06977
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961fb7ed04e8e213505e4c1bf5ca3435e1196c8edee766b3206555eb57ab1391
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2026-01-13T00:00:00-05:00
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On globally invariant Euler--Lagrange equations for curves
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arXiv:2601.06985v1 Announce Type: new Abstract: Invariant Lagrangians yield invariant Euler-Lagrange equations, and it was discussed in the literature how to compute those using various local methods. The focus of this paper is on global algebraic differential invariants. In this case the computation can be modified in several aspects. We will discuss relations with previous approaches and some foundational aspects. The theory of invariant Euler-Lagrange equations was applied to curves with respect to the motion group in the Euclidean plane and space. We expand those computations to the next dimension four (Minkowski spacetime), which already exhibits computational challenges. We also provide formulas for other examples, namely the projective and conformal (M\"obius) groups and relate to some recent applications.
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https://arxiv.org/abs/2601.06985
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461709cfea5e4a149a4212ad9a08a05d2936be78342dc9a5dd3746569557a4f7
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2026-01-13T00:00:00-05:00
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Single conflict coloring and palette sparsification of uniform hypergraphs
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arXiv:2601.06990v1 Announce Type: new Abstract: We introduce and investigate single conflict coloring in the setting of r-uniform hypergraphs. We establish some basic properties of this hypergraph coloring model and study a probabilistic model of single conflict coloring where the conflicts for each edge are chosen randomly; in particular, we prove a sharp threshold-type result for complete graphs and establish a sufficient condition for single conflict colorability of r-uniform hypergraphs in this model. Furthermore, we obtain a related palette sparsification-type result for general list coloring of linear uniform hypergraphs (i.e. uniform hypergraphs where any two edges share at most one common vertex). Throughout the paper we pose several questions and conjectures
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https://arxiv.org/abs/2601.06990
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255d27dd1263bfa204021b019665fa05d3c6c1848ba30b88626d67b789317c95
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2026-01-13T00:00:00-05:00
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Policy stability and ultimate stationarity in discounted risk-sensitive stochastic control
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arXiv:2601.06998v1 Announce Type: new Abstract: We study discrete-time Markov Decision Processes (MDPs) on finite state-action spaces and analyze the stability of optimal policies and value functions in the long-run discounted risk-sensitive objective setting. Our analysis addresses robustness with respect to perturbations of the risk-aversion parameter and the discount factor, the emergence of ultimate stationarity, and the interaction between discounted and averaged formulations under suitable mixing assumptions. We further investigate limiting regimes associated with vanishing discount and vanishing risk sensitivity, and discuss the role of Blackwell-type stability properties in the discounted setting. Finally, we provide numerical illustrations that highlight the intrinsic non-stationarity of optimal discounted risk-sensitive policies.
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https://arxiv.org/abs/2601.06998
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99e917afb8ab79e06f60efb8c86b6c91ad2ccac8b1702067c7a40ace0e846211
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2026-01-13T00:00:00-05:00
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Product representations of perfect powers
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arXiv:2601.07000v1 Announce Type: new Abstract: Let $\rho_k(N)$ denote the maximum size of a set $A\subseteq \{1,2,\dots,N\}$ such that no product of $k$ distinct elements of $A$ is a perfect $d$-th power. In this short note, we prove that $\rho _d(N)=\sum\limits_{k=1}^{d-1}\pi\left( \frac{N}{k} \right) +O_d(\pi (N^{1/2}))$, furthermore, for prime power $d$ and sufficiently large $N$ we have $\rho _d(N)=\sum\limits_{k=1}^{d-1}\pi\left( \frac{N}{k} \right)$. This answers a question of Verstra\"ete.
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https://arxiv.org/abs/2601.07000
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aca643e11bd02a2ea3fd72b3024456131abe2bf8ba5e416f673334838294f75e
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2026-01-13T00:00:00-05:00
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The G\"unt\"urk-Thao theorem revisited: polyhedral cones and limiting examples
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arXiv:2601.07002v1 Announce Type: new Abstract: In 2023, G\"unt\"urk and Thao proved that the sequence $(x^{(n)})_{n\in\mathbb{N}}$ generated by random (relaxed) projections drawn from a finite collection of innately regular closed subspaces in a real Hilbert space satisfies $\sum_{n\in\mathbb{N}} \|x^{(n)}-x^{(n+1)}\|^\gamma 0$. We extend their result to a finite collection of polyhedral cones. Moreover, we construct examples showing the tightness of our extension: indeed, the result fails for a line and a convex set in $\mathbb{R}^2$, and for a plane and a non-polyhedral cone in $\mathbb{R}^3$.
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https://arxiv.org/abs/2601.07002
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297c08b828d87ff0e1cfbe09e7c6273b56d02334d31147af615eefb36c1a26c5
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2026-01-13T00:00:00-05:00
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Higher order Petri Loci
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arXiv:2601.07026v1 Announce Type: new Abstract: Denote by ${\mathcal P}_{g,d}^{r,k}$ the subset of the moduli space of curves of genus g consisting of those curves that have a linear series of degree d and dimension r for which the Petri map has kernel of dimension at least k. We show the existence of codimension k components of ${\mathcal P}_{g,d}^{r,k}$.
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https://arxiv.org/abs/2601.07026
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262c82fba56e9484411d9ee02b72b8d0d2ab020ea57d3151556aeca64ef798de
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2026-01-13T00:00:00-05:00
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Characterizations of $G$-ANR spaces and inverse limits
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arXiv:2601.07027v1 Announce Type: new Abstract: In this paper we prove that, for a compact group $G$, a metrizable $G$-space is a $G$-ANR under the following asumptions: (1) if it dominates a $G$-ANR space through a fine $G$-homotopy equivalence; (2) if it is $G$-homotopy dense in a $G$-ANR; (3) if it contains a $G$-ANR as a $G$-homotopy dense subset; (4) if it is the inverse limit of an inverse sequence of $G$-ANR spaces with bonding maps that are fine $G$-homotopy equivalences.
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https://arxiv.org/abs/2601.07027
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50806b5d5445ddf3c5330926ca7e494e12bec856bc14b0806c5c60ecb4b12274
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2026-01-13T00:00:00-05:00
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Quantitative convergence rates for extended mean field games with volatility control
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arXiv:2601.07028v1 Announce Type: new Abstract: We investigate the convergence of symmetric stochastic differential games with interactions via control, where the volatility terms of both idiosyncratic and common noises are controlled. We apply the stochastic maximum principle, following the approach of Lauri\`{e}re and Tangpi, to reduce the convergence analysis to the study of forward-backward propagation of chaos. Under the standard monotonicity conditions, we derive quantitative convergence rates for open-loop Nash equilibria of $N$-player stochastic differential games toward the corresponding mean field equilibrium. As a prerequisite, we also establish the well-posedness of the conditional McKean--Vlasov forward-backward stochastic differential equations by the method of continuation. Moreover, we analyze a specific class of linear-quadratic settings to demonstrate the applicability of our main result.
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https://arxiv.org/abs/2601.07028
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854d31bc652224ccd9b2f863185cd167cab15194859e8b1a22fb74c9775e637b
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2026-01-13T00:00:00-05:00
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On families of monic polynomials
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arXiv:2601.07029v1 Announce Type: new Abstract: In this paper we derive generalizations of different properties of monic polynomial families of binomial type, i.e. families of monic polynomials, for which the binomial theorem holds $$ p_n(\alpha+\beta)=\sum_{k=0}^n \left(\vphantom{\bigg|}\genfrac{}{}{0pt}{0}{n}{k}\right) p_k(\alpha)p_{n-k}(\beta) $$ Some trivial representations of general ''multiplication'' and ''derivative'' operators are derived. In addition we derive a formula for the logarithmic derivative of general monic polynomial $p_n(x)$ which reduces to the formula $$ \frac{1}{n}\frac{p_n'(x)}{p_n(x)} =\left(x+\frac{1}{\varphi'(y)}\left(\frac{d}{dy}-n\mathrm{L}\right)\right)^{-1}\cdot\left.\frac{\varphi(y)}{y\varphi'(y)}~\right|_{y=0} $$ derived by the author in binomial case, when the generating function of $p_n(x)$ equals to $e^{x\varphi(y)}$.
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https://arxiv.org/abs/2601.07029
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83f600ee6a2c625b21d8291155b4c8b79439e5a11816112a4e9d342fef04c120
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2026-01-13T00:00:00-05:00
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Special $L$-values of certain CM weight three Hecke eigenforms
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arXiv:2601.07030v1 Announce Type: new Abstract: Ramanujan's theories of elliptic functions to alternative bases for modular forms connect hypergeometric series with modular forms and have led to applications such as the modularity of certain hypergeometric Galois representations. In this paper, we relate special values of $L$-function of certain CM cusp forms to Ramanujan's alternative bases via the modularity of hypergeometric Galois representations arising from CM elliptic curves over real quadratic fields. We also give a complete classification of these representations.
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https://arxiv.org/abs/2601.07030
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e4f277b56f01ce151c40089ee65ebd9bf331f1a5186c43d18e3fa02459adfdae
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2026-01-13T00:00:00-05:00
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A PDE approach for the invariant measure of stochastic oscillators with hysteresis
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arXiv:2601.07039v1 Announce Type: new Abstract: This paper presents a PDE approach as an alternative to Monte Carlo simulations for computing the invariant measure of a white-noise-driven bilinear oscillator with hysteresis. This model is widely used in engineering to represent highly nonlinear dynamics, such as the Bauschinger effect. The study extends the stochastic elasto-plastic framework of Bensoussan et al. [SIAM J. Numer. Anal. 47 (2009), pp. 3374--3396] from the two-dimensional elasto-perfectly-plastic oscillator to the three-dimensional bilinear elasto-plastic oscillator. By constructing an appropriate Lyapunov function, the existence of an invariant measure is established. This extension thus enables the modelling of richer hysteretic behavior and broadens the scope of PDE alternatives to Monte Carlo methods. Two applications demonstrate the method's efficiency: calculating the oscillator's threshold crossing frequency (providing an alternative to Rice's formula) and probability of serviceability.
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https://arxiv.org/abs/2601.07039
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906189b54b7947053915f64e22c6c65d14d859c6c60b0fd1d97d860eadfc2025
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2026-01-13T00:00:00-05:00
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The topological and smooth Hausmann-Weinberger invariants disagree
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arXiv:2601.07040v1 Announce Type: new Abstract: For $\pi$ a finitely presented group, Hausmann and Weinberger defined $q(\pi) \in \mathbb Z$ to be the minimum Euler characteristic over all closed, oriented $4$-manifolds with fundamental group $\pi$. This short note establishes that this minimum value in general differs depending on whether one minimizes over topological manifolds or only those admitting a smooth structure.
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https://arxiv.org/abs/2601.07040
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2886b3c378e1d21dcb00cb01aebf368e15281103f2b834375a61cb511a7392a7
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2026-01-13T00:00:00-05:00
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A Tannakian description of the local Kaletha gerbe
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arXiv:2601.07042v1 Announce Type: new Abstract: We construct, for a $p$-adic field $F$, an explicit semisimple Tannakian category $\text{RigIsoc}_{F}$ whose category of fiber functors recovers Kaletha's Galois gerbe $\mathcal{E}_{\text{Kal}}$. We then classify and write down the simple objects in $\text{RigIsoc}_{F}$, all of which come from elliptic twisted Levi subgroups of $\mathrm{GL}_{n}$.
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https://arxiv.org/abs/2601.07042
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1c0c486a405cf9910ae3b2a710a23072fc91079686f37a020942862569e2b093
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2026-01-13T00:00:00-05:00
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From Continuous to Discrete: a No-U-Turn Sampler for Permutations
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arXiv:2601.07045v1 Announce Type: new Abstract: We introduce a discrete-space analogue of the No-U-Turn sampler on the symmetric group $S_n$, yielding a locally adaptive and reversible Markov chain Monte Carlo method for $\mathrm{Mallows}(d,\sigma_0)$. Here $d:S_n\times S_n\to[0,\infty)$ is any fixed distance on $S_n$, $\sigma_0\in S_n$ is a fixed reference permutation, and the target distribution on $S_n$ has mass function $\pi(\sigma)\propto e^{-\beta d(\sigma,\sigma_0)}$ where $\beta>0$ is the inverse temperature. The construction replaces Hamiltonian trajectories with measure-preserving group-orbit exploration. A randomized dyadic expansion is used to explore a one-dimensional orbit until a probabilistic \emph{no-underrun} criterion is met, after which the next state is sampled from the explored orbit with probability proportional to the target weights. On the theory side, embedding this transition within the Gibbs self-tuning (GIST) framework provides a concise proof of reversibility. Moreover, we construct a \emph{shift coupling} for orbit segments and prove an explicit edge-wise contraction in the Cayley distance under a mild Lipschitz condition on the energy $E(\sigma)=d(\sigma,\sigma_0)$. A path-coupling argument then yields an $O(n^2\log n)$ total-variation mixing-time bound.
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https://arxiv.org/abs/2601.07045
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2026-01-13T00:00:00-05:00
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Idempotents and Powers of Ideals in Quandle Rings
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arXiv:2601.07057v1 Announce Type: new Abstract: This article addresses two central problems in the theory of quandle rings. First, motivated by Conjecture 3.10 in Internat. J. Math. 34 (2023), no. 3, Paper No. 2350011: for a semi-latin quandle $X$, every nonzero idempotent in the integral quandle ring $\mathbb{Z}[X]$ necessarily corresponds to an element of $X$, we investigate idempotents in quandle rings of semi-latin quandles. Precisely, we prove that if the ground ring is an integral domain with unity, then the quandle ring of Core($\mathbb{Z}$) admits only trivial idempotents. Second, powers of augmentation ideals in quandle rings have only been computed in few cases previously. We extend the computations to include dihedral quandles and commutative quandles. Finally, we examine idempotents in quandle rings of $2$-almost latin quandles and apply these results to compute the automorphism groups of their integral quandle rings.
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https://arxiv.org/abs/2601.07057
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4057698d4b4eeb79b1f91c91b87ec65c9a973ba4bbb4384bda891b12a7a30796
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2026-01-13T00:00:00-05:00
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Lebesgue points of measures and non tangential convergence of Poisson-Hermite integrals
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arXiv:2601.07063v1 Announce Type: new Abstract: We study differentiability conditions on a complex measure $\nu$ at a point $x_0\in\mathbb{R}^d$, in relation with the boundary convergence at that point of the Poisson-type integral $P_t\nu=e^{-t\sqrt L}\nu$, where $L=-\Delta+|x|^2$ is the Hermite operator. In particular, we show that $x_0$ is a Lebesgue point for $\nu$ iff a slightly stronger notion than non-tangential convergence holds for $P_t\nu$ at $x_0$. We also show non-tangential convergence when $x_0$ is a $\sigma$-point of $\nu$, a weaker notion than Lebesgue point, which for $d=1$ coincides with the classical Fatou condition.
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https://arxiv.org/abs/2601.07063
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a901e6b7156199fde1d8c1328f3cafb6667bce55c0f58bd56c4939c21c1043cc
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2026-01-13T00:00:00-05:00
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Varieties of associative algebras with an identity of third degree
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arXiv:2601.07066v1 Announce Type: new Abstract: We give a complete description of the varieties of associative algebras over a field of characteristic zero which satisfy a polynomial identity of third degree.
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https://arxiv.org/abs/2601.07066
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41d147970e0807f38e2860bae4c6671dcf08a98c94feaf6237d5eb2b322c8bb7
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2026-01-13T00:00:00-05:00
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Greenberg's conjecture and Iwasawa module of Real biquadratic fields II
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arXiv:2601.07067v1 Announce Type: new Abstract: In this paper we are interested in the stability of the $2$-rank of the class group in the cyclotomic $\mathbb{Z}_2$-extension of real biquadratic fields. In fact, we give several families of real biquadratic fields $K$ such that $ rank(A(K)) =rank(A_\infty(K))$ and $rank(A(K))\leq 3$, where $A(K)$ and $A_\infty(K)$ are the $2$-class group and the $2$-Iwasawa module of $K$ respectively. Moreover, Greenberg's conjecture is verified for some new families of number fields; in particular, we determine the complete list of all real biquadratic fields with trivial $2$-Iwasawa module. This work is a continuation of M. M. Chems-Eddin, Greenberg's conjecture and Iwasawa module of real biquadratic fields I, J. Number Theory, 281 (2026), 224-266.
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https://arxiv.org/abs/2601.07067
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b36d6ff92efbd0b152f908088407191d3a758d34eb406a7d69b297f1d0c34466
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2026-01-13T00:00:00-05:00
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The Greedy Algorithm for Dissociated Sets
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arXiv:2601.07068v1 Announce Type: new Abstract: A set $\mathcal S\subset \mathbb N$ is said to be a \textit{subset-sum-distinct} or \textit{dissociated} if all of its finite subsets have different sums. Alternately, an equivalent classification is if any equality of the form $$\sum_{s\in \mathcal S} \varepsilon_s \cdot s =0$$ where $\varepsilon_s \in \{-1,0,+1\}$ implies that all the $\varepsilon_s$'s are $0$. For a dissociated set $\mathcal S$, we prove that for $c_\ast = \frac 12 \log_2 \left(\frac \pi 2\right)$ and any $c_\ast-1<c_\ast$, we have $$\mathcal S(n) \,:=\, \mathcal S\cap [1,n] \,\le\, \log_2 n +\frac 12 \log_2\log_2 n + C$$ for all $n\in \mathcal N_C$ with asymptotic density $\mathbf d\left(\mathcal N_C\right)=2-2^{c_\ast-C}$. Further, we consider the greedy algorithm for generating these sets and prove that this algorithm always eventually doubles. Finally, we also consider some generalizations of dissociated sets and prove similar results about them.
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https://arxiv.org/abs/2601.07068
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Academic Papers
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a2eae85cd74b8524a9e62a33e523970091f89341c09f4a2e4189cd360d586ff8
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2026-01-13T00:00:00-05:00
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Diffraction by a Right-Angle Penetrable Wedge: Closed-Form Solution for General Refractive Index
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arXiv:2601.07070v1 Announce Type: new Abstract: We consider the two-dimensional time-harmonic transmission problem for an impedance-matched ($\rho=1$) right-angle penetrable wedge at general refractive index ratio $\nu>1$. Starting from Sommerfeld spectral representations, the transmission conditions on the two wedge faces yield a closed spectral functional system whose unknowns live on the Snell surface $\Sigma_\nu: Y^2=\nu^2 t^4+2(\nu^2-2)t^2+\nu^2$. We uniformize $\Sigma_\nu$ by Jacobi/Weierstrass elliptic functions on a torus $\mathbb{C}/\Lambda$ and solve the resulting $2\times 2$ genus-one Riemann--Hilbert problem in closed form for general finite forcing data. The Sommerfeld radiation condition and the Meixner edge condition are enforced by a simple residue-sum constraint. The construction extends the special lemniscatic case $\nu^2=2$ treated in arXiv:2601.04183 and yields a practical evaluation recipe expressed in theta/sigma products and explicit triangular factor matrices. We include the complete Sommerfeld representation connecting the spectral solution to the physical field, explicit forcing data for plane wave incidence, and a worked symbolic example at $\nu=3/2$.
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https://arxiv.org/abs/2601.07070
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cef97b4c8b4cfd8027d44ba35e34d564930bc89524bc7b7023ddf54051e93ac7
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2026-01-13T00:00:00-05:00
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Volume of the domain bounded by a Hermitian quadric in complex projective space
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arXiv:2601.07077v1 Announce Type: new Abstract: We compute explicitly the Riemannian volume, with respect to the Fubini-Study metric, of a domain bounded by a Hermitian quadric in complex projective space. The volume is a rational function of the eigenvalues of the defining quadratic form.
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https://arxiv.org/abs/2601.07077
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897b49d888fdb36c9aa949a0f18fbf3096e1653c82693f351870c0587ffc9b52
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2026-01-13T00:00:00-05:00
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An Inverse Almost Periodic Problem for a Semilinear Strongly Damped Wave Equation
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arXiv:2601.07081v1 Announce Type: new Abstract: The paper investigates an inverse boundary value problem for a semilinear strongly damped wave equation with the Dirichlet boundary condition in Sobolev spaces of bounded (in particular, almost periodic and periodic) functions. In addition to finding a weak solution, we also determine a source coefficient in the right-hand side of the differential equation. To make the problem well-posed, an integral-type overdetermination condition is imposed. After reducing the inverse problem to a direct one, we solve the latter in several steps. First, we prove the existence and uniqueness of a weak solution to the corresponding initial-boundary value problem on a finite time interval. Next, we show that this solution can be extended in a bounded way to the semiaxis $t\ge 0$. In the following step, we further extend this bounded solution to all $t\in R$. Finally, we establish that if the data of the original problem are almost periodic (or periodic), then the resulting bounded weak solution is itself almost periodic (or periodic).
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https://arxiv.org/abs/2601.07081
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142adfadd846b9e220dff9342ea807da8473d784c27c00cce6b7727f9f7ac0d6
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2026-01-13T00:00:00-05:00
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The infinitude of square-free palindromes
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arXiv:2601.07097v1 Announce Type: new Abstract: We settle an open problem regarding palindromes; that is, positive integers which are the same when written forwards and backwards. In particular, we prove that for any fixed base $b\geq 2$, there exist infinitely many square-free palindromes in base $b$. We also provide an asymptotic expression for the number of such integers $\leq x$. The core of our proof utilises a hybrid $p$-adic/Archimedean van der Corput process, used in conjunction with an equidistribution estimate of Tuxanidy and Panario, as well as an elementary argument of Cilleruelo, Luca and Shparlinski.
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https://arxiv.org/abs/2601.07097
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7bbdfcff4b6c2c16d8b2cb8dc684a6c5651a176bea77a3bbaa78e384fbd30d61
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2026-01-13T00:00:00-05:00
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A Dichotomy for Inverse-Semigroup Crossed Products via Dynamical Cuntz Semigroups
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arXiv:2601.07100v1 Announce Type: new Abstract: We characterise stable finiteness and pure infiniteness of the essential crossed product of a C*-algebra by an action of an inverse semigroup. Under additional assumptions, we prove a stably finite / purely infinite dichotomy. Our main technique is the development, using an induced action, of a ''dynamical Cuntz semigroup'' that is a subquotient of the usual Cuntz semigroup. We prove that the essential crossed product is stably finite / purely infinite if and only if the dynamical Cuntz semigroup admits / does not admit a nontrivial state. Indeed, a retract of our dynamical Cuntz semigroup suffices to prove the dichotomy. Our results generalise those by Rainone on crossed products of groups acting by automorphisms of a C*-algebra, and we recover results by Kwa\'sniewski--Meyer--Prasad on C*-algebras of non-Hausdorff groupoids.
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https://arxiv.org/abs/2601.07100
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ee94ee13a176b9faee4f7e0d2b55690241809400f2c188e380473ec0fb1875ad
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2026-01-13T00:00:00-05:00
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Inverse problems for history-enriched linear model reduction
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arXiv:2601.07101v1 Announce Type: new Abstract: Standard projection-based model reduction for dynamical systems incurs closure error because it only accounts for instantaneous dependence on the resolved state. From the Mori-Zwanzig (MZ) perspective, projecting the full dynamics onto a low-dimensional resolved subspace induces additional noise and memory terms arising from the dynamics of the unresolved component in the orthogonal complement. The memory term makes the resolved dynamics explicitly history dependent. In this work, based on the MZ identity, we derive exact, history-enriched models for the resolved dynamics of linear driven dynamical systems and formulate inverse problems to learn model operators from discrete snapshot data via least-squares regression. We propose a greedy time-marching scheme to solve the inverse problems efficiently and analyze operator identifiability under full and partial observation data availability. For full observation data, we show that, under mild assumptions, the operators are identifiable even when the full-state dynamics are governed by a general time-varying linear operator, whereas with partial observation data the inverse problem has a unique solution only when the full-state operator is time-invariant. To address the resulting non-uniqueness in the time-varying case, we introduce a time-smoothing Tikhonov regularization. Numerical results demonstrate that the operators can be faithfully reconstructed from both full and partial observation data and that the learned history-enriched MZ models yield accurate trajectories of the resolved state.
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https://arxiv.org/abs/2601.07101
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c533699862fd46e74860bd3637a05e3ae79d4d43a7bb3b578f28f39482818dc1
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2026-01-13T00:00:00-05:00
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Large Deviations for the d'Arcais Numbers
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arXiv:2601.07103v1 Announce Type: new Abstract: The d'Arcais polynomials $P_n(z)$ for $n\in\{0,1,\dots\}$ are defined as $\sum_{n=0}^{\infty} P_n(z) q^n = \exp(-z\ln((q;q)_{\infty}))$ where the $q$-Pochhammer symbol is $(q;q)_{\infty} = \prod_{k=1}^{\infty} (1-q^k)$ for $|q|<1$. Denoting the coefficients for $n \in \mathbb{N}$ by the formula $P_n(z) = \sum_{k=1}^{n} A(2,n,k) z^k/n!$, we prove that $k_n! A(2,n,k_n)/n!$ satisfies a Bahadur-Rao type large deviation formula in the limit $n \to \infty$ with $k_n/n \to \kappa \in [0,1)$ as long as $k_n \to \infty$. The large deviation rate function is the Legendre-Fenchel transform $g^*(-\kappa)$ where $g(\kappa) = f^{-1}(\kappa)$ for the function $f : (0,\infty) \to \mathbb{R}$ given by $f(y)= \ln(-\ln((e^{-y};e^{-y})_{\infty}))$. We relate this fact to information about the abundancy index.
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https://arxiv.org/abs/2601.07103
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e2e5c0778578ec324dd83cd1f52d53f70a84ecb337bef5dd7493fdf5fdbfb4db
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2026-01-13T00:00:00-05:00
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A sharp point-sphere incidence bound for $(u, s)$-Salem sets
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arXiv:2601.07105v1 Announce Type: new Abstract: We establish a sharp point-sphere incidence bound in finite fields for point sets exhibiting controlled additive structure. Working in the framework of \((4,s)\)-Salem sets, which quantify pseudorandomness via fourth-order additive energy, we prove that if \(P\subset \mathbb{F}_q^d\) is a \((4,s)\)-Salem set with \(s\in \big( \frac{1}{4}, \frac{1}{2} \big]\) and \(|P|\ll q^{ \frac{d}{4s}}\), then for any finite family \(S\) of spheres in \(\mathbb{F}_q^d\), \[ \bigg| I(P,S)-\frac{|P||S| }{q} \bigg| \ll q^{\frac{d}{4}}\,|P|^{1-s}\,|S|^{\frac{3}{4}}. \] This estimate improves the classical point-sphere incidence bounds for arbitrary point sets across a broad parameter range. The proof combines additive energy estimates with a lifting argument that converts point-sphere incidences into point-hyperplane incidences in one higher dimension while preserving the \((4,s)\)-Salem property. As applications, we derive refined bounds for unit distances, dot-product configurations, and sum-product type phenomena, and we extend the method to \((u,s)\)-Salem sets for even moments \(u\ge4\).
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https://arxiv.org/abs/2601.07105
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Academic Papers
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47ac092d2cb23691eb1302469f2e76183e3879c953886cc62c1cf3e60014a266
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2026-01-13T00:00:00-05:00
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Center-freeness of finite-step solvable groups arising from anabelian geometry
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arXiv:2601.07112v1 Announce Type: new Abstract: Anabelian geometry suggests that, for suitably geometric objects, their \'etale fundamental group determines the object up to isomorphism. From a group-theoretic viewpoint, this philosophy requires rigidity properties of the associated \'etale fundamental groups, which often follow from their center-freeness. In fact, some profinite groups arising from anabelian geometry are center-free. In the present paper, we investigate how such center-freeness behaves when passing to maximal $m$-step solvable quotients for any integer $m\geq 2$. In particular, we show that the maximal $m$-step solvable quotient of the geometric \'etale fundamental group of a hyperbolic curve over a field of characteristic $0$ is center-free. Furthermore, we show that this implies the injectivity statement, i.e., the rigidity property, of the $m$-step solvable Grothendieck conjecture.
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https://arxiv.org/abs/2601.07112
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62aec9128f22c40c72edb7d401fa557deaa4e21da657c59790d621b90828f9ab
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2026-01-13T00:00:00-05:00
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Asymptotic values of solutions to a periodic linear difference equation modeling discrimination training
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arXiv:2601.07113v1 Announce Type: new Abstract: This work is concerned with the study of $w(mT)$ as $m$ goes to infinity, where $w(t)$ evolves according to $w(t)-w(t-1)=F(t)-A(t)w(t-1)$, and where $T$ is the period of the vector $F(t)$ and the matrix $A(t)$. Motivated by applications to associative learning, particularly to discrimination training, extra conditions are imposed on $F(t)$ and $A(t)$, one of them relating $A(t)$ to a symmetric non-negative definite matrix $K$ relevant to mathematical models of associative learning. Structural relationships between the matrices imply an identity satisfied by the Floquet multipliers driving the dynamics of $w(mT)$ from which follows that the unstable subspace is $\ker K$. Then, the limit of $w(mT)$ is explicitly identified when $K$ is invertible, while the limit of $Kw(mT)$ is established otherwise. Given that divergence of $w(mT)$ can happen when $K$ is singular, while $Kw(mT)$ is the psychologically relevant quantity, the result can be considered optimal.
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https://arxiv.org/abs/2601.07113
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630f09bc3729838515933ed599dbb1d385dbc77d93b28e2adcb0a20db0247d88
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2026-01-13T00:00:00-05:00
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Cyclic Modulation Control of Multi-Conflict Connected Automated Traffic
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arXiv:2601.07114v1 Announce Type: new Abstract: Multi-conflict traffic is ubiquitous. Connected Automated Vehicles (CAVs) offer unprecedented opportunities to enhance safety, reduce emissions, and increase throughput through precise coordination and automation. However, existing CAV strategies remain confined to specialized scenarios, such as highway on-ramp merging or single-lane roundabouts, and traditional traffic signals sacrifice efficiency for safety via rigid phasing and all-red intervals. In this paper, we present Cyclic Modulation Control of Multi-Conflict Connected Automated Traffic (CMAT), a unified, geometry-agnostic framework that embeds each conflict point into a repeating sequence of "micro-phases". Vehicles dynamically form platoons with demand-responsive sizes and negotiate time slots for occupying conflict points, enabling collision-free traversal and high intersection utilization. CMAT aims to minimize delay, guarantee safety, and accommodate arbitrary merging, diverging, and crossing patterns without manual retuning. We formalize CMAT as a mixed-integer linear programming model constructed on a directed graph abstracted from the physical intersection layout. The performance of CMAT is evaluated across a suite of multi-conflict tests, including simple two-way crossings, four-leg intersections, complex connected intersections. The results demonstrate substantial reductions in delay and significant throughput improvements compared with state-of-the-art CAV coordination methods and traditional signal timing strategies.
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https://arxiv.org/abs/2601.07114
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fa6eaa79bacf0290c82864e2ea1ab08a2739cd5040bfd08f4139ea5677b650c7
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2026-01-13T00:00:00-05:00
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The Interval $[\mathsf{V}(S_7),\mathsf{V}(B_2^1)]$ of Semiring Varieties Has the Cardinality of the Continuum
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arXiv:2601.07116v1 Announce Type: new Abstract: We prove that the interval $[\mathsf{V}(S_7),\mathsf{V}(B_2^1)]$ in the lattice of additively idempotent semiring (ai-semiring) varieties has the cardinality of the continuum,where $S_7$ is the smallest nonfinitely based ai-semiring (a three-element algebra), and $B_2^1$ is the ai-semiring whose multiplicative reduct is the six-element Brandt monoid.
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https://arxiv.org/abs/2601.07116
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3d1fed458d742961418903a657d4b198735cae1c76a8c1856ec496cf9081ab70
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2026-01-13T00:00:00-05:00
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Minimum and extremal process for a branching random walk outside the boundary case
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arXiv:2601.07129v1 Announce Type: new Abstract: This work extends the studies on the minimum and extremal process of a supercritical branching random walk outside the boundary case which cannot be reduced to the boundary case. We study here the situation where the log-generating function explodes at $1$ and the random walk associated to the spine possesses a stretched exponential tail with exponent $b\in(0,\frac12)$. Under suitable conditions, we confirm the conjecture of Barral, Hu and Madaule [Bernoulli 24(2) 2018 801-841], and obtain the weak convergence for the minimum and the extremal process. We also establish an a.s. infimum result over all infinity rays of this system.
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https://arxiv.org/abs/2601.07129
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260a7d950107828bdc187cb3ac8e2f4ad2549741f7ec65b2224cfe1b4a916e7c
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2026-01-13T00:00:00-05:00
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Decompositions for Cyclic Groups with 3 Prime Factors
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arXiv:2601.07135v1 Announce Type: new Abstract: In this paper, we characterize the direct sum decompositions of the cyclic group $\mathbb{Z}_{(pqr)^2}$, where $p$, $q$, and $r$ are distinct primes. We show that if $A \oplus B = \mathbb{Z}_{(pqr)^2}$ with $|A| = |B| = pqr$, then Sands' conjecture fails to hold, in other words, neither $A$ nor $B$ is contained in a proper subgroup of $\mathbb{Z}_{(pqr)^2}$, if and only if the sets $A, B$ form a Szab\'{o} pair.
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https://arxiv.org/abs/2601.07135
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30faf56bc93e605d2ae9b6bc09c07030b377616c7f2dc9cc915da8b9d3193e46
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2026-01-13T00:00:00-05:00
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Local and global $C^{1,\beta}$-regularity for uniformly elliptic quasilinear equations of $p$-Laplace and Orlicz-Laplace type
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arXiv:2601.07140v1 Announce Type: new Abstract: We establish gradient H\"older continuity for solutions to quasilinear, uniformly elliptic equations, including $p$-Laplace and Orlicz-Laplace type operators. We revisit and improve upon the results existing in the literature, proving gradient regularity both in the interior and up to the boundary, under Dirichlet or Neumann boundary conditions.
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https://arxiv.org/abs/2601.07140
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f61f2c92aa8ccde35c81372d466610159449f38330a83a0d3b98be9b4f5a858d
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2026-01-13T00:00:00-05:00
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Constructing left-continuous triangular norms on complete lattices
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arXiv:2601.07146v1 Announce Type: new Abstract: This article focuses on the construction of left-continuous t-norms on complete lattices. The concepts of $\mathfrak{f}$-mappings and weak $\mathfrak{f}$-mappings on complete lattices are first introduced, respectively. They are then applied to establish the following key results: weak $\mathfrak{f}$-mappings are used to induce left-continuous t-subnorms; $\mathfrak{f}$-mappings are used to generate left-continuous t-norms whenever the top element $1$ of the complete lattice is a completely join-irreducible element. Finally, some necessary and sufficient conditions are provided for an operator constructed by the ordinal sum of a series of annihilating binary operators being a left-continuous t-norm on a complete lattice.
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https://arxiv.org/abs/2601.07146
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Academic Papers
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11cb96084c0d682b4b0e19d5bf270afbb22434ed79938a68065350b2b6fcd359
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2026-01-13T00:00:00-05:00
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Blues for Alice: The Interplay of Neo-Riemannian and Cadential Viewpoints
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arXiv:2601.07161v1 Announce Type: new Abstract: We extend a property of Mazzola's theory of cadential sets in relation to the modulation between minor and major tonalities from triadic to tetradic harmony, using the PLRQ group of Cannas et al. (2017) as the analogue of the classical PLR group. While the PLR group connects triadic cadential sets via the relative morphism $R$, the tetradic case reveals a richer structure: two pairs of cadential sets connected by distinct morphisms forming a "prism" in the slice category over the tonic seventh chord, and a single pair for those that allow quantized modulations. We demonstrate this structure through analysis of Charlie Parker's "Blues for Alice" (1951) and Ray Noble's "Cherokee" (1938), showing how the prism morphism, PLRQ transformations and quantized modulations organize harmonic navigation in bebop. The categorical framework captures what syntactic approaches miss: the transformational v\'{e}cu that musicians actually experience when navigating between cadential regions.
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https://arxiv.org/abs/2601.07161
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d444f9abded6372735575c1ba8a42d1125aee62569ba198d4e30958e2b767456
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2026-01-13T00:00:00-05:00
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Jordan decompositions in Lie algebras and their duals
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arXiv:2601.07168v1 Announce Type: new Abstract: We provide a discussion of Jordan decompositions in the Lie algebra, and the dual Lie algebra, of a reductive group in as uniform a way as possible. We give a counterexample to the claim that Jordan decompositions on the dual Lie algebra are unique, and state an upper bound on how non-unique they can be. We also prove some Chevalley-restriction-type claims about GIT quotients for the adjoint and co-adjoint actions of $G$.
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https://arxiv.org/abs/2601.07168
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a7c60927e5e6196c13ca73cd0bf8373385636055295a43cd3641b0c0cd780f08
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2026-01-13T00:00:00-05:00
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The spinor type number formula for totally definite quaternion orders
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arXiv:2601.07171v1 Announce Type: new Abstract: Let $D$ be a totally definite quaternion algebra over a totally real number field $F$, and $\mathcal{O}$ be an $O_F$-order (of full rank) in $D$. The type number $t(\mathcal{O})$ is an important arithmetic invariant of $\mathcal{O}$ that counts the number of isomorphism classes of orders belonging to the same genus as $\mathcal{O}$ (i.e. locally isomorphic to $\mathcal{O}$ at every finite place $\mathfrak{p}$ of $F$). The type number formula has been studied by Eichler, Peters, Pizer, Vigneras, K\"orner and many others. As the genus of $\mathcal{O}$ further divides into spinor genera, one naturally seeks a finer type number formula for the number of isomorphism classes of orders belonging to the same spinor genus of $\mathcal{O}$. The main goal of this paper is to provide such a refinement for a large class of quaternion $O_F$-orders $\mathcal{O}$ that includes all Eichler orders. This enables us to prove that $t(\mathcal{O})$ is divisible by the order of a quotient group $\mathrm{WSG}(\mathcal{O})$ of the Gauss genus group $\mathrm{Cl}^+(O_F)/\mathrm{Cl}^+(O_F)^2$ naturally attached to $\mathcal{O}$. Similarly, we show that the trace of the $\mathfrak{n}$-Brandt matrix $\mathfrak{B}(\mathcal{O}, \mathfrak{n})$ is divisible by the class number $h(F)$ for any nonzero integral $O_F$-ideal $\mathfrak{n}$. In particular, the class number $h(\mathcal{O})=\mathrm{Tr}(\mathfrak{B}(\mathcal{O}, O_F))$ is always divisible by $h(F)$ for such quaternion orders. This generalizes the divisibility result of $h(\mathcal{O})$ proved in a different way by Chia-Fu Yu and the second named author [Indiana Univ. Math. J., Vol. 70, No. 2 (2021)] in the case when $\mathcal{O}$ is a maximal $O_F$-order in a totally definite quaternion algebra unramified at all the finite places.
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https://arxiv.org/abs/2601.07171
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c3d9e8e869489effdaf1e498a515f421ae3825e398369e343e1eb3f7e5bd8801
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2026-01-13T00:00:00-05:00
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Optimal Equivariant Matchings on the 6-Cube: With an Application to the King Wen Sequence
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arXiv:2601.07175v1 Announce Type: new Abstract: We characterize perfect matchings on the Boolean hypercube {0,1}^n that are equivariant under the Klein four-group K_4 generated by bitwise complement and reversal. For n = 6, we prove there exists a unique K_4-equivariant matching minimizing total Hamming cost among matchings using only comp or rev pairings, achieving cost 120 versus 192 for the complement-only matching. The optimal matching is determined by a simple "reverse-priority rule": pair each element with its reversal unless it is a palindrome, in which case pair with its complement. We verify that the historically significant King Wen sequence of the I Ching is isomorphic to this optimal matching. Notably, allowing comp(rev) pairings yields lower cost (96), but the King Wen sequence follows the structurally simpler rule. All results are formally verified in Lean 4 with the Mathlib library.
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https://arxiv.org/abs/2601.07175
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89bdf1a858c3c2e0cff308ab92eccf9d8008764ac78fed66dcbcac96fcb2c56f
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2026-01-13T00:00:00-05:00
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Local linearization for the nonlinear damped stochastic Klein-Gordon equation
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arXiv:2601.07176v1 Announce Type: new Abstract: For the $1+1$ dimensional nonlinear damped stochastic Klein-Gordon equation driven by space-time white noise, we prove that the second-order increments of the solution can be approximated, after scaling with the diffusion coefficient, by those of the corresponding linearized stochastic Klein-Gordon equation. This extends the result of Huang et al. \cite{HOO2024} for the stochastic wave equation. A key difficulty arises from the more complex structure of the Green function, which we overcome by means of subtle analytical estimates. As applications, we analyze the quadratic variation of the solution and construct a consistent estimator for the diffusion parameter.
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https://arxiv.org/abs/2601.07176
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c1905faffaeb468e0062887d033faa5e56e1414afd9eec0de6210b30fd34b59e
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2026-01-13T00:00:00-05:00
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Lu's conjecture for minimal surfaces
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arXiv:2601.07194v1 Announce Type: new Abstract: After Chern's conjecture on the discreteness of the constant scalar curvatures of compact minimal submanifolds $M^n$ in unit spheres $\mathbb{S}^{n+q}$, Z. Q. Lu proposed a conjecture regarding the second gap, based on his ingenious refinement of the known first gap theorem. This refinement unifies Simons' first gap theorem for hypersurfaces with the corresponding theorems for high-codimensional submanifolds established by Yau, Shen, Li and Li, among others. In this paper, for arbitrary codimension, we prove Lu's conjecture for minimal 2-spheres, and for any minimal surfaces under some slight inequality conditions about the normal scalar curvature.
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https://arxiv.org/abs/2601.07194
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2742c0e0e5310ca7d43e0abd3e675cdbebe1f381b0318e0d91d88ee50fcffc61
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2026-01-13T00:00:00-05:00
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On symmetric pattern avoidance sets
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arXiv:2601.07195v1 Announce Type: new Abstract: For a set of permutations $S\subseteq S_n$, consider the quasisymmetric generating function $$Q(S): = \sum_{w\in S}F_{n, \mathrm{Des}(w)},$$ where $\mathrm{Des}(w) := \{i\mid w(i)> w(i+1)\}$ is the descent set of $w$ and $F_{n, \mathrm{Des}(w)}$ is Gessel's fundamental quasisymmetric function. A set of permutations is said to be symmetric (respectively, Schur-positive) if its quasisymmetric generating function is symmetric (respectively, Schur-positive). Given a set $\Pi$ of permutations, let $S_n(\Pi)$ denote the set of permutations in $S_n$ that avoid all patterns in $\Pi.$ A set $\Pi$ is said to be symmetrically avoided (respectively, Schur-positively avoided) if $S_n(\Pi)$ is symmetric (respectively, Schur-positive) for all $n.$ Marmor proved in 2025 that for $n\ge 5$, a symmetric set $S\subseteq S_n$ has size at least $n-1$ unless $S\subseteq \{12\cdots n, n\cdots 21\}$ and asked for a general classification of the possible sizes of symmetric sets not containing the monotone elements $12\cdots n $ and $n\cdots 21$. We give a complete answer to this question for $n\ge 52.$ We also give a classification of symmetric sets of size at most $n-1$, thereby showing that they are actually Schur-positive, resolving a conjecture of Marmor. Finally, we give a classification of symmetrically avoided sets of size at most $n-1$, thereby showing that they are actually Schur-positively avoided.
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https://arxiv.org/abs/2601.07195
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Academic Papers
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b03e36929b6058edd0bc6b405b3ff9873749478a4201d1388e5a778279d574e0
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2026-01-13T00:00:00-05:00
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Posinormality and the Root Problem
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arXiv:2601.07203v1 Announce Type: new Abstract: The paper extends three results regarding the nth root problem by embedding classes of Hilbert-space operators into the class of posinormal operators. For instance, it is shown that (i) for coposinormal operators, if T is paranormal and T^n is quasinormal, then T is normal, and (ii) for posinormal operators, if T is k-quasiparanormal and T^n is normal, then T is normal. Moreover, (iii) it is also shown that the latter result is not conditioned to the separability of the underlying Hilbert space, even if T is not posinormal, where, in such a case, T is the direct sum of a normal operator with a nilpotent one.
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https://arxiv.org/abs/2601.07203
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Academic Papers
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3042141cc1c7623b8a97875a320ef2a2109b16f142e1cdd85644c3a41da6c440
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2026-01-13T00:00:00-05:00
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The motivic class of the space of genus $0$ maps to the flag variety
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arXiv:2601.07222v1 Announce Type: new Abstract: Let $\operatorname{Fl}_{n+1}$ be the variety of complete flags in $\mathbb{A}^{n+1}$ and let $\Omega^{2}_{\beta}(\operatorname{Fl}_{n+1})$ be the space of based maps $f:\mathbb{P}^{1}\to \operatorname{Fl}_{n+1}$ in the class $f_{*}[\mathbb{P}^{1}]=\beta$. We show that under a mild positivity condition on $\beta$, the class of $\Omega^{2}_{\beta}(\operatorname{Fl}_{n+1})$ in $K_{0}(\operatorname{Var})$, the Grothendieck group of varieties, is given by \[ [\Omega^{2}_{\beta}(\operatorname{Fl}_{n+1})] = [\operatorname{GL}_{n}\times \mathbb{A}^{a}]. \] The proof of this result was obtained in conjunction with Google Gemini and related tools. We briefly discuss this research interaction, which may be of independent interest. However, the treatment in this paper is entirely human-authored (aside from excerpts in an appendix which are clearly marked as such).
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https://arxiv.org/abs/2601.07222
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547b56f8712d91c62735be8a707adba90250fc6acc08751b1a1f16db745906a1
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2026-01-13T00:00:00-05:00
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Wasserstein Concentration of Empirical Measures for Dependent Data via the Method of Moments
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arXiv:2601.07228v1 Announce Type: new Abstract: We establish a general concentration result for the 1-Wasserstein distance between the empirical measure of a sequence of random variables and its expectation. Unlike standard results that rely on independence (e.g., Sanov's theorem) or specific mixing conditions, our result requires only two conditions: (1) control over the variance of the empirical moments, and (2) a flexible tail condition we term $\Psi_{r_n}$-sub-Gaussianity. This approach allows for significant dependencies between variables, provided their algebraic moments behave predictably. The proof uses the method of moments combined with a polynomial approximation of Lipschitz functions via Jackson kernels, allowing us to translate moment concentration into topological concentration.
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https://arxiv.org/abs/2601.07228
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Academic Papers
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48f3f02febd0015c270717d78812ece68069cbba465ac6f11f3fb4b64d09968c
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2026-01-13T00:00:00-05:00
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Configured locally smooth cohomology and $\mathbb{Q}/\mathbb{Z}$-torsion in $H_3$ of diffeomorphism groups
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arXiv:2601.07230v1 Announce Type: new Abstract: We introduce configured group cohomology, a variant of locally smooth cohomology built from well-configured tuples and geometric fillings. This framework yields explicit locally smooth $\R/\Z$-valued $3$-cocycles of Chern--Simons type on diffeomorphism groups preserving geometric structures. As an application we show that, for several such groups, the third group homology contains a subgroup isomorphic to $\Q/\Z$.
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https://arxiv.org/abs/2601.07230
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1f964860bab59f737c4ffd3aa042fa9684fedb05f231521d34e8f3b735fe25e7
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2026-01-13T00:00:00-05:00
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Integrable Stochastic Processes Associated with the $D_2$ Algebra
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arXiv:2601.07265v1 Announce Type: new Abstract: We introduce novel integrable stochastic processes associated with the $D_2$ quantum group, which can be decomposed into two XXX spin chains (or two symmetric simple exclusion processes). We establish the integrability of the model under three types of boundary conditions (periodic, twisted, and open boundaries), and present its exact solution, including the spectrum, eigenstates, and some observables.
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https://arxiv.org/abs/2601.07265
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4f9cba4ae69b4e0eacf5b73765450717d874953970f0d0299b4ac6ff9396b13d
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2026-01-13T00:00:00-05:00
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Homogeneous spaces with geodesic orbit Riemannian metrics and with integrable invariant distributions
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arXiv:2601.07277v1 Announce Type: new Abstract: We consider homogeneous spaces of Lie groups with compact stabilizer subgroups of two types: those with integrable invariant distributions and those with geodesic orbit invariant Riemannian metrics. The latter means that for an arbitrary invariant Riemannian metric on the space, every geodesic is an orbit of a 1-parameter subgroup of the isometry group. We found several homogeneous spaces of the first type that are not spaces of the second type. Among them there are several homogeneous spaces that admit invariant Einstein metrics.
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https://arxiv.org/abs/2601.07277
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037922ad2342a26f19caff9c037203a696f7426fc3adbf911f1eddb6614f755f
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2026-01-13T00:00:00-05:00
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On faithfulness and DP-transformations generated by arithmetic Cantor series expansions
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arXiv:2601.07285v1 Announce Type: new Abstract: The paper is devoted to the study of conditions for the Hausdorff-Besicovitch faithfulness of the family of cylinders generated by Cantor series expansions. We show that there exist subgeometric Cantor series expansions for which the corresponding families of cylinders are not faithful for the Hausdorff-Besicovitch dimension on the unit interval. On the other hand we found a rather wide subfamily of subgeometric Cantor series expansions generating faithful families of cylinders. We also study conditions for the Hausdorff-Besicovitch dimension preservation on [0;1] by probability distribution functions of random variables with independent symbols of arithmetic Cantor series expansions.
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https://arxiv.org/abs/2601.07285
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4cba773fb1e9da816db1af9fdd64c6530da4fa568afa7f4cff12c332fa8badd8
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2026-01-13T00:00:00-05:00
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Weak majorization inequalities for the cubic and quartic coefficients of $e^{(A+B)t}$ versus $e^{At}e^{Bt}$
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arXiv:2601.07286v1 Announce Type: new Abstract: Let $A,B\in\mathbb{H}_n$ and set $H=A+B$. For each integer $k\ge 1$ define $$ Q_k:=\sum_{p=0}^k \binom{k}{p} A^pB^{k-p}, R_k:=\Re\,Q_k=\frac{Q_k+Q_k^*}{2}. $$ Then $H^k=\left.\frac{d^k}{dt^k}e^{Ht}\right|_{t=0}$ and $Q_k=\left.\frac{d^k}{dt^k}(e^{At}e^{Bt})\right|_{t=0}$. We prove that, for $k=3,4,$ $$ \lambda(H^k)\prec_w \sigma(Q_k). $$ Equivalently, the eigenvalues of the cubic and quartic Taylor coefficients of $e^{(A+B)t}$ are weakly majorized by the singular values of the corresponding coefficients of the Golden--Thompson product $e^{At}e^{Bt}$. Our argument combines Ky Fan variational principles with explicit commutator identitiesfor $R_k-H^k$ at orders $k=3,4$, reducing the problem to the positivity of certain double-commutator trace forms tested against Ky Fan maximizing projections. We also record a general sufficient condition for higher orders based on commutator decompositions.
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https://arxiv.org/abs/2601.07286
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6385647b03fa93fd8c0f1da5b0a6a408b66a507465eb38b3c15a0724bce3599d
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2026-01-13T00:00:00-05:00
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Irregularities of special C-pairs
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arXiv:2601.07318v1 Announce Type: new Abstract: This paper studies irregularity-type invariants of special C-pairs, or "geometric orbifolds" in the sense of Campana. Under mild assumptions on the singularities, we show that the augmented irregularity of a C-pair (X,D) is bounded by its dimension. This generalizes earlier results of Campana, and strengthens known results even in the classic case where X is a projective manifold and D = 0. The proof builds on new extension results for adapted forms, analysis of foliations on Albanese varieties, and constructions of Bogomolov sheaves using strict wedge subspaces of adapted forms.
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https://arxiv.org/abs/2601.07318
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Academic Papers
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46dd73f74b2f3073d211f36b5b5a2581fa47dbdd819a317abd8ffa1c1a02c5a2
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2026-01-13T00:00:00-05:00
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Finslerian geometrodynamics
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arXiv:2601.07321v1 Announce Type: new Abstract: We construct a unified framework of geometrodynamics based on the Finsler geometry to reveal the relationship between spacetime and dynamics.The Lagrangian of electron in electromagnetic field as the Finsler function gives the Finslerian metric, which modifies spacetime metric in the Finsler-Randers space. The geodesic equation gives the effective mass, forces, and effective (or geometric) fields. Using the Chern connection, we construct the generalized Einstein-Maxwell equations. In the local natural basis, we give generalized Maxwell equations and wave equations. We find that the geometric field couples with electromagnetic field and gives effective charges and currents. We analyze several typical cases, such as flat spacetime, vacuum and Berwald structure. We find that the electromagnetic field vanishes, but there still exists the magnetic potential in the Berwald space. These results provide some hints to understand some puzzles, such as axion and dark energy. These formulations stimulate some clues to explore deeper geometric structures behind physical phenomena.
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https://arxiv.org/abs/2601.07321
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Academic Papers
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220539e24a98e24ad2de82c299b3c9d700314d4bbc28465ac99298de0fc5d766
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2026-01-13T00:00:00-05:00
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On polynomial equations over split-octonions: the arbitrary field case
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arXiv:2601.07332v1 Announce Type: new Abstract: Over the split-octonion algebra defined over an arbitrary field, we solve all polynomial equations whose coefficients are scalar except for the constant term. As an application, we determine the square and cubic roots of an octonion.
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https://arxiv.org/abs/2601.07332
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Academic Papers
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75e6b8462e32285939e04fe63ef50764ae3146ded0839a00b5d2fb480cddb800
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2026-01-13T00:00:00-05:00
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Uniform bounds for Neumann heat kernels and their traces in convex sets
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arXiv:2601.07341v1 Announce Type: new Abstract: We prove a bound on the heat trace of the Neumann Laplacian on a convex domain that captures the first two terms in its small-time expansion, but is valid for all times and depends on the underlying domain only through very simple geometric characteristics. This is proved via a precise and uniform expansion of the on-diagonal heat kernel close to the boundary. Most of our results are valid without the convexity assumption and we also consider two-term asymptotics for the heat trace for Lipschitz domains.
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https://arxiv.org/abs/2601.07341
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Academic Papers
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dcd177f0dcf34ebbee2acb981d0dfeedb12f92d46f8cd8b99e809105e4248542
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2026-01-13T00:00:00-05:00
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A proposal for the algebra of a novel noncommutative spacetime
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arXiv:2601.07350v1 Announce Type: new Abstract: We investigate the quantum structure of spacetime at fundamental scales via a novel, Lorentz-invariant noncommutative coordinate framework. Building on insights from noncommutative geometry, spectral theory, and algebraic quantum field theory, we systematically construct a quantum spacetime algebra whose geometric and causal properties are derived from first principles. Using the Weyl algebra formalism and the Gelfand--Naimark--Segal (GNS) construction, we rigorously define operator-valued coordinates that respect Lorentz symmetry and encode quantum gravitational effects through nontrivial commutation relations. We show how the emergent quantum spacetime exhibits minimal length effects, which deliver both classical Minkowski distances and quantum corrections proportional to the Planck length squared. Furthermore, we establish that noncommutativity respects a fuzzy form of causality, where the quantum causal structure gives back the light cone in the classical limit, vanishing for spacelike separations and encoding a time orientation for timelike intervals.
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https://arxiv.org/abs/2601.07350
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Academic Papers
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8f14d1c35c618e1cdcb510eb3540216eb5b31ffcabbddcc00f61563f63f669c1
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2026-01-13T00:00:00-05:00
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Homotopy categories of admissible model structures on extriangulated categories
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arXiv:2601.07352v1 Announce Type: new Abstract: The extriangulated category is a simultaneous generalization of exact categories and triangulated categories. H. Nakaoka and Y. Palu have proved that the homotopy category of an admissible model structure on a weakly idempotent complete extriangulated category is a triangulated category. Using the classic construction of distinguished triangles given by A. Heller and D. Happel, this paper provides an alternative proof of Nakaoka - Palu Theorem. In fact, the class $\Delta$ of distinguished triangles in the present paper and the class $\widetilde{\Delta}$ of distinguished triangles in \cite{NP} have the relation $\Delta = - \widetilde{\Delta}$, and hence the two triangulated structures on the homotopy category are isomorphic.
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https://arxiv.org/abs/2601.07352
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Academic Papers
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79e754320b01531386457464cc73e78a31eff77e58938169cd33fe774dba6cb4
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2026-01-13T00:00:00-05:00
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Pro-\'etale motives and solid rigidity
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arXiv:2601.07358v1 Announce Type: new Abstract: We introduce coefficient systems of pro-\'etale motives and pro-\'etale motivic spectra with coefficients in any condensed ring spectrum and show that they afford the six operations. Over locally \'etale bounded schemes, \'etale motivic spectra embed into pro-\'etale motivic spectra. We then use the framework of condensed category theory to define a solidification process for any $\widehat{\mathbb{Z}}$-linear condensed category. Pro-\'etale motives naturally enhance to a condensed category and we show that their solidification is very close to the category of solid sheaves defined by Fargues-Scholze, suitably modified to work on schemes: this is a rigidity result. As a consequence, we obtain that in contrast with the rigid-analytic setting, solid sheaves on schemes afford the six operations, and we obtain a solid realization functor of motives, extending the $\ell$-adic realization functor. The solid realization functor is compatible with change of coefficients, which allows one to recover the $\mathbb{Q}_\ell$-adic realization functor while remaining in a setting of presentable categories.
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https://arxiv.org/abs/2601.07358
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Academic Papers
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f0490f20405ce76a28f59755c8238b03b80a7be70a8d4b37a56d36aa00e72f9b
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2026-01-13T00:00:00-05:00
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On the geometry of generalised Koch snowflakes
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arXiv:2601.07371v1 Announce Type: new Abstract: We consider the geometry of a class of fractal sets in $\mathbb{R}^{2}$ that generalise the famous Koch curve and Koch snowflake. While the classical Koch curve is defined by an iterative process that divides a line segment into three parts and replaces the middle part by the legs of an isosceles triangle 'above' the line segment, in this more general setting, a choice can be made at each iteration as to whether to place this triangle 'above' or 'below' the line segment. The resulting fractals bear a striking visual resemblance to curves appearing in nature, such as coastlines and snowflakes. While these fractals can be generated by a random process that flips a coin each time to decide the orientation of the triangle, leading to 'almost sure' results for their geometrical properties, we define and study them deterministically to provide exact results. In particular, we show, using the theory of non-integer expansions, that the set of all possible values for the area enclosed by these generalised Koch curves is a closed interval. Moreover, we prove that the union of all these generalised snowflakes does not contain an open set, and has zero $2$-dimensional Lebesgue measure. Complementing these results, using arguments from calculus and fractal geometry, namely properties of geometric series and Frostman's Lemma, we show that each generalised Koch curve has infinite length and the same Hausdorff dimension as its classical counterpart. Further, we also give a classification for when a generalised Koch curve is a quasicircle.
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https://arxiv.org/abs/2601.07371
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a7a02d329018eb93242c97b1a5503e53360c658914847051a2000fde78d987f0
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2026-01-13T00:00:00-05:00
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On the Enumeration of Generalized Cospectral Mates of Graphs
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arXiv:2601.07373v1 Announce Type: new Abstract: This paper investigates the enumeration of generalized cospectral mates of simple graphs, where the generalized spectrum consists of the spectra of a graph and its complement. Moving beyond the classical problem of identifying graphs determined by their generalized spectrum, we address the more quantitative question of how many non-isomorphic graphs can share the same generalized spectrum. Our approach is based on arithmetic constraints derived from the Smith Normal Form (SNF) of the walk matrix $W(G)$, which lead to a tight upper bound on the number of generalized cospectral mates of a graph. Our upper bound applies to a much broader class of graphs than those previously shown to have no generalized cospectral mates (determined by generalized spectrum). Consequently, this work extends the family of graphs for which strong and informative spectral uniqueness
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https://arxiv.org/abs/2601.07373
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Academic Papers
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b2104d06b476d7f6557ef14741d42a208392dcc245a660b961e33e65ce606970
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2026-01-13T00:00:00-05:00
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Igusa stacks for certain abelian-type Shimura varieties
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arXiv:2601.07383v1 Announce Type: new Abstract: We construct Igusa stacks for the good reduction locus of a class of abelian-type Shimura varieties that can be defined in terms of a PEL datum, under the assumption that it is of type (A even) or (C) and unramified at a prime p.
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https://arxiv.org/abs/2601.07383
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Academic Papers
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0a5c779bdda08bb16b0b5cfb0365458cf2f9b8cd15e50ae2f6f887c179a43a22
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2026-01-13T00:00:00-05:00
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A Coherent Version of Geometric Satake Equivalence for Type A
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arXiv:2601.07390v1 Announce Type: new Abstract: In this paper we prove a coherent version of geometric Satake equivalence proposed in Cautis-Williams' work arXiv:2306.03023 for type A. In their work, they studied an abelian version of the classical limit Satake category, namely, the Koszul perverse heart of the categorified Coulomb branch for adjoint representations. In this paper we study a subcategory generated by a collection of simple objects. We endow this subcategory with a neutral Tannakian structure and identify it with the finite dimensional representation category $\mathrm{Rep}({\check{G}})$ for the Langlands dual group ${\check{G}}$. Our method uses tools in Cautis-Williams theory and a Hodge module description of the coherent IC extensions of differential sheaves in Xin's work arXiv:2503.14890.
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https://arxiv.org/abs/2601.07390
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Academic Papers
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d97befd57a97b7e90c4c91dff5c0fa889f4ebef8e0082dc8597d15447dd51c1d
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2026-01-13T00:00:00-05:00
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Stationary internal waves in a two-dimensional aquarium at low viscosity
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arXiv:2601.07391v1 Announce Type: new Abstract: We prove the uniform solvability of a stationary problem associated to internal waves equation with small viscosity in a two dimensional aquarium with real-analytic boundary, under a Morse--Smale dynamical assumption. This is achieved by using complex deformations of the aquarium, on which the inviscid stationary internal wave operator is invertible.
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https://arxiv.org/abs/2601.07391
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d4ece6d652d00202a566340ebe9f9ae103cecb1da7533d50e66657658b94528e
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2026-01-13T00:00:00-05:00
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Modeling and analysis of a novel two-strain dengue epidemics model considering secondary infections with increased mortality
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arXiv:2601.07403v1 Announce Type: new Abstract: In this study, we develop and analyze a deterministic two-strain host--vector model for dengue transmission that incorporates key immuno-epidemiological mechanisms, including temporary cross-immunity, antibody-dependent enhancement (ADE), disease-induced mortality during secondary infections, and explicit vector co-infection. The human population is divided into compartments for primary and secondary infections, while the mosquito population includes single- and co-infected classes. ADE is modeled through distinct primary ($\alpha$) and secondary ($\sigma$) transmission rates. Using the next-generation matrix method, we derive the basic reproduction number $R_0$ and establish the local stability of the disease-free equilibrium for $R_0 \alpha$), allowing invasion by a heterologous strain. Employing center-manifold theory and numerical continuation (COCO), we demonstrate the occurrence of backward bifurcation, bistability between disease-free and endemic states, and Hopf-induced oscillations. Numerical simulations confirm transitions among disease-free, endemic, and periodic regimes as key parameters vary. The model highlights how ADE, waning cross-immunity, and vector co-infection interact to generate complex dengue dynamics and provides insights useful for designing effective control and vaccination strategies in dengue-endemic regions.
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https://arxiv.org/abs/2601.07403
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Academic Papers
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b120d39a7fb80cf5159fd4a02b8cdfe20c87d9772570cab553f54fcbcdd79f86
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2026-01-13T00:00:00-05:00
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Bipartite Tur\'an problem on cographs
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arXiv:2601.07406v1 Announce Type: new Abstract: A cograph is a graph that contains no induced path $P_4$ on four vertices or equivalently a graph that can be constructed from vertices by sum and product operations. We study the bipartite Tur\'an problem restricted to cographs: for fixed integers $s \leq t$, what is the maximum number of edges in an $n$-vertex cograph that does not contain $K_{s,t}$ as a subgraph? This problem falls within the framework of induced Tur\'an numbers $\text{ex}(n, \{K_{s,t}, P_4\text{-ind}\})$ introduced by Loh, Tait, Timmons, and Zhou. Our main result is a Pumping Theorem: for every $s\le t$ there exists a period $R$ and core cographs such that for all sufficiently large $n$ an extremal cograph is obtained by repeatedly pumping one designated pumping component inside the appropriate core (depending on $n\bmod R$). We determine the linear coefficient of $\text{ex}(n, \{K_{s,t}, P_4\text{-ind}\})$ to be $s-1 + \frac{t-1}{2}$. Moreover, the pumping components are $(t-1)$-regular and have $s-1$ common neighbours in the respecitve core graphs, giving the extremal cographs a particularly rigid extremal star-like shape. Motivated by the rarity of complete classification of extremal configurations, we completely classify all $K_{3,3}$-free extremal cographs by proof. We also develop a dynamic programming algorithm for enumerating extremal cographs for small $n$.
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https://arxiv.org/abs/2601.07406
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Academic Papers
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