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0061641704cdd8028d86fd961af16a3f7c18c7cffb2395acdc63bb1aa4f296ba
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2026-01-16T00:00:00-05:00
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On refinements of two-term Machin-like formulas
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arXiv:2601.10300v1 Announce Type: new Abstract: We develop a refinement process for two-term Machin-like formulas: $a_0 \arctan{u_0} + a_1 \arctan{u_1} = \frac{\pi}{4}$ (where $a_0 , a_1 \in \mathbb{Z}$, $u_0 , u_1 \in \mathbb{Q}_+^*$, $u_0 > u_1$) by exploiting the continued fraction expansion of the ratio $\alpha := \frac{\arctan{u_0}}{\arctan{u_1}}$. This construction yields a sequence of derived two-term Machin-like formulas: $a_{- n} \arctan{u_n} + a_{- n + 1} \arctan{u_{n + 1}} = \frac{\pi}{4}$ ($n \in \mathbb{N}$) with positive rational arguments $u_n$ decreasing to zero and corresponding integer coefficients $a_{- n}$. We derive closed forms and estimates for $a_{-n}$ and $u_n$ in terms of the convergents of $\alpha$ and prove that the associated rational sequence $(a_{- n} u_n + a_{- n + 1} u_{n + 1})_n$ converges to $\pi/4$ with geometric decay. The method is illustrated using Euler's two-term Machin-like formula : $\arctan(1/2) + \arctan(1/3) = \pi/4$.
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https://arxiv.org/abs/2601.10300
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7e2597ec4336c754a45941ec7b8aef5627b2d0ba1349d7ab67361da340171bb7
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2026-01-16T00:00:00-05:00
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On Force Interactions for Electrodynamics-Like Theories
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arXiv:2601.10308v1 Announce Type: new Abstract: A framework for premetric p-form electrodynamics is proposed. Independently of particular constitutive relations, the corresponding Maxwell equations are derived as a special case of stress theory in geometric continuum mechanics. Expressions for the potential energy of a charged region in spacetime, as well as expressions for the force and stress interactions on the region, are presented. The expression for the force distribution is obtained by computing the rate of change of the proposed potential energy under a virtual motion of the region. These expressions differ from those appearing in the standard references. The cases of electrostatics and magnetostatics in R^3 are presented as examples.
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https://arxiv.org/abs/2601.10308
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cf71a0d38f999ef16c32528d01a28dd330d82d187a01f2e4d0e4206694daf644
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2026-01-16T00:00:00-05:00
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Deformations of Chow groups via cyclic homology
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arXiv:2601.10309v1 Announce Type: new Abstract: Let $X$ be a smooth projective variety over an arbitrary field $k$ of characteristic zero. We explore infinitesimal deformations of the Chow group $CH^{p}(X)$ via its formal completion $\widehat{CH}^{p}$, a functor defined on the category of local augmented Artinian $k$-algebras. Under a natural vanishing condition on Hodge cohomology groups, for certain augmented graded Artinian $k$-algebras $A$ with the maximal ideal $m_{A}$, we prove that \[ \widehat{CH}^{p}(A) \cong H^{p}(X, \Omega^{p-1}_{X/ k})\otimes_{k}m_{A}. \]This extends earlier results of Bloch and others from the case where $k$ is algebraic over $\mathbb{Q}$ to arbitrary fields of characteristic zero,and gives a partial affirmative answer to a general question linking the pro-representability of Chow groups to a specific set of Hodge-theoretic vanishing conditions.
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https://arxiv.org/abs/2601.10309
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6d69adc2ddd2a2afb1826bde473595d31e487c558b5725866e7a1e7fd550d406
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2026-01-16T00:00:00-05:00
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On the characterization of geometric distance-regular graphs
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arXiv:2601.10330v1 Announce Type: new Abstract: In 2010, Koolen and Bang proposed the following conjecture: For a fixed integer $m \geq 2$, any geometric distance-regular graph with smallest eigenvalue $-m$, diameter $D \geq 3$ and $c_2 \geq 2$ is either a Johnson graph, a Grassmann graph, a Hamming graph, a bilinear forms graph, or the number of vertices is bounded above by a function of $m$. In this paper, we obtain some partial results towards this conjecture.
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https://arxiv.org/abs/2601.10330
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54aa5a1d1a86063ff4fe9726d8fe884d307b28a20b406d6fed54a68b4662d78d
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2026-01-16T00:00:00-05:00
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Polymultiplicative maps associated with the algebra of Iterated Laurent series and the higher-dimensional Contou-Carrere Symbol
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arXiv:2601.10335v1 Announce Type: new Abstract: We study functorial polymultiplicative maps from the multiplicative group of the algebra of $n$-times iterated Laurent series over a commutative ring in $n+1$ variables into the multiplicative group of the ring. It is proven that if such a map is invariant under continuous automorphisms of this algebra, then it coincides, up to a sign, with an integer power of the $n$-dimensional Contou-Carr\`ere symbol.
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https://arxiv.org/abs/2601.10335
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2ed603673a094b9238d957918453c5e38eace3d1bcd3b7a87de11801bdaf4276
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2026-01-16T00:00:00-05:00
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Triggered urn models for frequently asked questions (FAQ)
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arXiv:2601.10337v1 Announce Type: new Abstract: We investigate a nonclassic urn model with triggers that increase the number of colors. The scheme has emerged as a model for web services that set up frequently asked questions (FAQ). We present a thorough asymptotic analysis of the FAQ urn scheme in generality that covers a large number of special cases, such as Simon urn. For instance, we consider time dependent triggering probabilities. We identify regularity conditions on these probabilities that classify the schemes into those where the number of colors in the urn remains almost surely finite or increases to infinity and conditions that tell us whether all the existing colors are observed infinitely often or not. We determine the rank curve, too. In view of the broad generality of the trigger probabilities, a spectrum of limit distributions appears, from central limit theorems to Poisson approximation, to power-laws, revealing connections to Heap's exponent and Zipf's law. A combinatorial approach to the Simon urn is presented to indicate the possibility of such exact analysis, which is important for short-term predictions. Extensive simulations on real datasets (from Amazon sales) as well as computer-generated data clearly indicate that the asymptotic and exact theory developed agrees with practice.
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https://arxiv.org/abs/2601.10337
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8e9ba160f9aae67e693c2b371dfbe40ef854d600b4d5a2ab443f6b5529b626b0
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2026-01-16T00:00:00-05:00
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Characteristic free Galois rings and generalized Weyl algebras
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arXiv:2601.10346v1 Announce Type: new Abstract: This paper develops from scratch a theory of Galois rings and orders over arbitrary fields. Our approach is different from others in the literature in that there is no non-modularity assumption. We prove, when the field is algebraically closed, the analogue of the Main Theorem of the representation theory of Galois orders by V. Futorny and S. Ovsienko. Then we develop a theory of infinite rank generalized Weyl algebras, which was never explicitly introduced in the literature before, and prove its basic properties. We expect their representation theory to be of interest for future works. Finally we show that under very mild assumptions, the invariants of generalized Weyl algebras under the action of non-exceptional irreducible complex reflection groups are a principal Galois orders, greatly generalizing, in an elementary fashion, results obtained previously for the Weyl algebras.
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https://arxiv.org/abs/2601.10346
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395f5a0614df277db960083278647f15624218dc5fedafdc5e2c31396d6350ed
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2026-01-16T00:00:00-05:00
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Waring's problem for pseudo-polynomials
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arXiv:2601.10351v1 Announce Type: new Abstract: Waring's problem has a long history in additive number theory. In its original form it deals with the representability of every positive integer as sum of $k$-th powers with integer $k$. Instead of these powers we deal with pseudo-polynomials in this paper. A pseudo-polynomial is a ``polynomial'' with at least one exponent not being an integer. Our work extends earlier results on the related problem of Waring for arbitrary real powers $k>12$ by Deshouillers and Arkhipov and Zhitkov.
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https://arxiv.org/abs/2601.10351
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41f1f5e214015b241082d0e0e805b0e2ecfe80dda13f7522e090a09505c9664f
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2026-01-16T00:00:00-05:00
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An It\^o Formula via Predictable Projection for Non-Semimartingale Processes
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arXiv:2601.10359v1 Announce Type: new Abstract: We derive an It\^o-type change-of-variables formula for a class of adapted stochastic processes that do not necessarily admit semimartingale structure. The formulation is based on an intrinsic Hilbert-space derivative together with a predictable projection operator, allowing stochastic integrals to be expressed without reliance on quadratic variation or anticipative calculus. The resulting formula replaces the classical quadratic variation term with a computable second-order contribution expressed as a norm of the projected derivative. In the semimartingale case, the formula reduces to the classical It\^o formula. The approach applies naturally to processes with memory and non-Markovian dependence, providing a unified and intrinsic framework for stochastic calculus beyond the semimartingale setting.
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https://arxiv.org/abs/2601.10359
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1b1433a3c0a00e106b62f0cc6faba120326dd26cbc3949699b12981b52213a7d
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2026-01-16T00:00:00-05:00
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On UC-multipliers for multiple trigonometric systems
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arXiv:2601.10360v1 Announce Type: new Abstract: We investigate the class of sequences $w(n)$ that can serve as almost-everywhere convergence Weyl multipliers for all rearrangements of multiple trigonometric systems. We show that any such sequence must satisfy the bounds $\log n\lesssim w(n)\lesssim\log^2 n$. Our main result establishes a general equivalence principle between one-dimensional and multidimensional trigonometric systems, which allows one to extend certain estimates known for the one-dimensional case to higher dimensions.
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https://arxiv.org/abs/2601.10360
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99c1e59f25a048f7502e9e5bcf191a25e6f8a4af02012798f1fbc135924dea5a
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2026-01-16T00:00:00-05:00
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A two-step inertial method with a new step-size rule for variational inequalities in hilbert spaces
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arXiv:2601.10370v1 Announce Type: new Abstract: In this paper, a two-step inertial Tseng extragradient method involving self-adaptive and Armijo-like step sizes is introduced for solving variational inequalities with a quasimonotone cost function in the setting of a real Hilbert space. Weak convergence of the sequence generated by the proposed algorithm is proved without assuming the Lipschitz condition. An interesting feature of the proposed algorithm is its ability to select the better step size between the self-adaptive and Armijo-like options at each iteration step. Moreover, removing the requirement for the Lipschitz condition on the cost function broadens the applicability of the proposed method. Finally, the algorithm accelerates and complements several existing iterative algorithms for solving variational inequalities in Hilbert spaces.
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https://arxiv.org/abs/2601.10370
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27f5d1dd9fb3d327e1874e50d166053a84172fee690b666cc29e2b59905a9c31
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2026-01-16T00:00:00-05:00
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On surgeries from lens space $L(p,1)$ to $L(q,2)$
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arXiv:2601.10377v1 Announce Type: new Abstract: We mainly use the d-invariant surgery formula established by Wu and Yang \cite{wu2025surgerieslensspacestype} to study the distance one surgeries along a homologically essential knot between lens spaces of the form $L(p,1)$ and $L(q,2)$ where $p,q$ are odd integers.
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https://arxiv.org/abs/2601.10377
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eccdc59c373f90ca6b93594d62ec363b2494be26fbf4bcdb1782c2e7367ab18d
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2026-01-16T00:00:00-05:00
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Phase Space structure on Clifford Algebras
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arXiv:2601.10381v1 Announce Type: new Abstract: I argue that the Hodge structure on a Euclidean Clifford algebra $Cl(n)$ provides a way to generalise K\"ahler structure to higher dimensions, in the sense that the paired variables are now associated with $k-$ and $(n-k)-$ dimensional subspaces rather than with vectors. This puts a phase space structure on Clifford algebras, and so allows us to construct a Hamiltonian dynamics on these multilinear variables. This construction shows that alternating pairs of subspaces obey commuting and anticommuting dynamics, hinting that this construction is indeed a natural one, with interesting new behaviour.
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https://arxiv.org/abs/2601.10381
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fdcb9a04128ec134ca820dbb2f46a7fa1022396d41a4494e501579613fc611ab
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2026-01-16T00:00:00-05:00
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Algebraic Farkas Lemma and Strong Duality for Perturbed Conic Linear Programming
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arXiv:2601.10390v1 Announce Type: new Abstract: This paper addresses the study of algebraic versions of Farkas lemma and strong duality results in the very broad setting of infinite-dimensional conic linear programming in dual pairs of vector spaces. To this end, purely algebraic properties of perturbed optimal value functions of both primal and dual problems and their corresponding hypergraph/epigraph are investigated. The newly developed hypergraphical/epigraphical sets, inspired by Kretschmer's closedness conditions \cite{Kretschmer61}, together with their novel convex separation-type characterizations, give rise to various perturbed Farkas-type lemmas which allow us to derive complete characterizations of ``zero duality gap''. Principally, when certain structures of algebraic or topological duals are imposed, illuminating implications of the developed condition are also explored.
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https://arxiv.org/abs/2601.10390
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549b9b1b051995b88573bfcf9b5e3acf8e56bddef5a8a7a322d94a17d7112424
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2026-01-16T00:00:00-05:00
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A proof of Alexander's conjecture on an inequality of Cassels
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arXiv:2601.10411v1 Announce Type: new Abstract: Let $z_1,\dots,z_n$ be complex numbers with $|z_j|\le \rho$, where $\rho>1$. Cassels proved that, under an additional restriction on $\rho$, the inequality \[ \prod_{j\ne k}\bigl|1-\overline{z_j}z_k\bigr| \le \left(\frac{\rho^{2n}-1}{\rho^2-1}\right)^{\!n} \] holds. In a subsequent note, Alexander conjectured that this inequality is in fact valid without any restriction on $\rho$. In this paper, we confirm Alexander's conjecture.
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https://arxiv.org/abs/2601.10411
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742093bd9826980d2014414fd2b02f209634456cf3d7b73c76bf4c975e262440
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2026-01-16T00:00:00-05:00
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Optimality in nonlocal time-dependent obstacle problems
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arXiv:2601.10417v1 Announce Type: new Abstract: This paper showcases the effectiveness of the quasiconvexity property in addressing the optimal regularity of the temporal derivative and establishes conditions for its continuity in nonlocal time-dependent obstacle problems.
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https://arxiv.org/abs/2601.10417
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2738d2ad413493e298a62bc8583c9d2fa3af14d7c08dea881f3707534fdeafb4
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2026-01-16T00:00:00-05:00
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On the Canonical Construction of Simple Lie Superalgebras
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arXiv:2601.10419v1 Announce Type: new Abstract: Axioms for the generalization of root systems were defined and classified (irreducible) by V. Serganova, which precisely correspond to the root systems of basic classical Lie Superalgebras. Here, we present a unified method for constructing simple Lie Superalgebras from the abstract root system, with the choice of base having the minimal number of isotropic roots.
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https://arxiv.org/abs/2601.10419
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1e85dfa6da74a40b77aceed80a6b3f1297417acfa0e5f53debd38cbf0c647f6c
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2026-01-16T00:00:00-05:00
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Positivity of Schur forms for Griffiths positive vector bundles of rank three over complex threefolds
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arXiv:2601.10424v1 Announce Type: new Abstract: In this paper, we prove the positivity of the double mixed discriminant associated with a positive linear map between spaces of \(3\times 3\) complex matrices, thereby settling the three-dimensional case of Finski's open problem. As an application, we show that all Schur forms are weakly positive for Griffiths positive Hermitian holomorphic vector bundles of rank three over complex threefolds. This yields a complete affirmative answer, in the case where both the rank and the dimension are three, to the question posed by Griffiths in 1969.
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https://arxiv.org/abs/2601.10424
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3f35afd10fffa6679b2dc63fdd7543568276cd006dc448744e22a20e6a181171
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2026-01-16T00:00:00-05:00
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Algebraic functional equation for big Galois representations over multiple $\mathbb{Z}_p$-extensions
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arXiv:2601.10426v1 Announce Type: new Abstract: We present a general approach to establish algebraic functional equations for big Galois representations over multiple $\mathbb{Z}_p$-extensions. Our result is formulated in both Selmer group and Selmer complex settings, and encompasses a broad range of Iwasawa-theoretic scenarios. In particular, our result applies to the triple product of Hida families in both balanced and unbalanced cases, as well as the half-ordinary Rankin-Selberg universal deformations recently studied by the first named author and Loeffler. Our result also significantly generalizes many previously known cases of algebraic functional equations and answers a question of Greenberg.
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https://arxiv.org/abs/2601.10426
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317e85d5256e6503213dfda194c914b1d44b619464f1bbea40bbdeb75c863c5a
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2026-01-16T00:00:00-05:00
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Geometric characterization of frictional impacts by means of breakable kinetic constraints
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arXiv:2601.10432v1 Announce Type: new Abstract: In the context of geometric Impulsive Mechanics of systems with a finite number of degrees of freedom, we model the roughness of a unilateral constraint ${\mathcal S\/}$ by introducing a suitable instantaneous kinetic constraint ${\mathcal B\/}\subset {\mathcal S\/}$. A constitutive characterization of ${\mathcal B\/}$ based only on the geometric properties of the setup and on the dry friction laws can then be introduced to model the frictional behavior of ${\mathcal S\/}$ in an impact of the system. Such a model restores determinism and avoids the analysis of frictional forces in the contact point, with all its associated theoretical problems of causality. Three examples of increasing complexity, showing a natural stick--slip behavior of the impact, are presented.
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https://arxiv.org/abs/2601.10432
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bc07f90e13f545b2035976586bef941b7c0930900b37381f960323ce46d1c375
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2026-01-16T00:00:00-05:00
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Linear identities for partition pairs with $4$-cores
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arXiv:2601.10438v1 Announce Type: new Abstract: We determine an infinite family of linear identities for the number $A_4(n)$ of partition pairs of $n$ with $4$-cores by employing elementary $q$-series techniques and certain $3$-dissection formulas. We then discover an infinite family of congruences for $A_4(n)$ as a consequence of these linear identities.
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https://arxiv.org/abs/2601.10438
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90751abfec7901f88c1a9b17d88ebcf20e1c54d965159f5c8182601508d4bda6
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2026-01-16T00:00:00-05:00
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Non-Intrusive Hyperreduction by a Physics-Augmented Neural Network with Second-Order Sobolev Training
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arXiv:2601.10442v1 Announce Type: new Abstract: The finite element method is an indispensable tool in engineering, but its computational complexity prevents applications for control or at system-level. Model order reduction bridges this gap, creating highly efficient yet accurate surrogate models. Reducing nonlinear setups additionally requires hyperreduction. Compatibility with commercial finite element software requires non-intrusive methods based on data. Methods include the trajectory piecewise linear approach, or regression, typically via neural networks. Important aspects for these methods are accuracy, efficiency, generalization, including desired physical and mathematical properties, and extrapolation. Especially the last two aspects are problematic for neural networks. Therefore, several studies investigated how to incorporate physical knowledge or desirable properties. A promising approach from constitutive modeling is physics augmented neural networks. This concept has been elegantly transferred to hyperreduction by Fleres et al. in 2025 and guarantees several desired properties, incorporates physics, can include parameters, and results in smaller architectures. We augment this reference work by second-order Sobolev training, i.e., using a function and its first two derivatives. These are conveniently accessible and promise improved performance. Further modifications are proposed and studied. While Sobolev training does not meet expectations, several minor changes improve accuracy by up to an order of magnitude. Eventually, our best model is compared to reference work and the trajectory piecewise linear approach. The comparison relies on the same numerical case study as the reference work and additionally emphasizes extrapolation due to its critical role in typical applications. Our results indicate quick divergence of physics-augmented neural networks for extrapolation, preventing its deployment.
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https://arxiv.org/abs/2601.10442
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0cc470d4ddce693c170b6b67bfb89e8f1ebe66d24a863074c00b25b7f348cb48
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2026-01-16T00:00:00-05:00
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Umbral theory and the algebra of formal power series
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arXiv:2601.10443v1 Announce Type: new Abstract: Umbral theory, formulated in its modern version by S. Roman and G.~C. Rota, has been reconsidered in more recent times by G. Dattoli and collaborators with the aim of devising a working computational tool in the framework of special function theory. Concepts like umbral image and umbral vacuum have been introduced as pivotal elements of the discussion, which, albeit effective, lacks of generality. This article is directed towards endowing the formalism with a rigorous formulation within the context of the formal power series with complex coefficients $(\mathbb{C}[[ t ]], \partial)$. The new formulation is founded on the definition of the umbral operator $\operatorname{\mathfrak{u}}$ as a functional in the "umbral ground state" subalgebra of analytically convergent formal series $\varphi \in \mathbb{C}\{t\}$. We consider in detail some specific classes of umbral ground states $\varphi$ and analyse the conditions for analytic convergence of the corresponding umbral identities, defined as formal series resulting from the action on $\varphi$ of operators of the form $f(\zeta \operatorname{\mathfrak{u}}^\mu)$ with $f \in \mathbb{C}\{t\}$ and $\mu, \zeta \in \mathbb{C}$. For these umbral states, we exploit the Gevrey classification of formal power series to establish a connection with the theory of Borel-Laplace resummation, enabling to make rigorous sense of a large class of -- even divergent -- umbral identities. As an application of the proposed theoretical framework, we introduce and investigate the properties of new umbral images for the Gaussian trigonometric functions, which emphasise the trigonometric-like nature of these functions and enable to define the concept of "Gaussian Fourier transform", a potentially powerful tool for applications.
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https://arxiv.org/abs/2601.10443
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f3da51b5bb798904ae58c4af87a01aded61816950619f5c0779899f1a9481cc0
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2026-01-16T00:00:00-05:00
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Symmetric spaces, non-formal star products and Drinfel'd twists
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arXiv:2601.10456v1 Announce Type: new Abstract: These notes refer to a minicourse I gave at the occasion of the conference meeting ``Applications of Noncommutative Geometry to Gauge Theories, Field Theories, and Quantum Space-Time'' to be held from 7 April to 11 April 2025 at the Centre International de Rencontres Math\'ematiques in Luminy. They consist in a review of a long standing work of mine and collaborators (see references therein) in the field of non-formal deformation quantization admitting a large group of symmetries. But they also contain new material and results. More precisely, in a first part, I present a method (called the Retract Method) to define quantizations/symbolic calculi and associated operator symbol composition formulae (non-formal deformations/star products) of symplectic symmetric spaces such as the hyperbolic plane (Kahler) or symmetric co-adjoint orbits of the Poincar\'e group (non-metric). In a second part, I explain how to derive non-formal Drinfel'd twists for actions of non-Abelian solvable Lie groups (non-Abelian Universal Deformation Formulae) on or Fr \'echet algebras from the non-formal noncommutative symmetric spaces defined in the first part.
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https://arxiv.org/abs/2601.10456
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cad366d4697305030ea2012a18b51b52a4f6c3ab8436313b9e3b1ee15a08fc38
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2026-01-16T00:00:00-05:00
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The Wiener Wintner and Return Times Theorem Along the Primes
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arXiv:2601.10459v1 Announce Type: new Abstract: We prove the following Return Times Theorem along the sequence of prime times, the first extension of the Return Times Theorem to arithmetic sequences: For every probability space, $(\Omega,\nu)$, equipped with a measure-preserving transformation, $T \colon \Omega \to \Omega$, and every $f \in L^\infty(\Omega)$, there exists a set of full probability, $\Omega_f \subset \Omega$ with $\nu(\Omega_f) =1$, so that for all $\omega \in \Omega_f$, for any other probability space $(X,\mu)$, equipped with a measure-preserving transformation $S : X \to X$, for any $g \in L^{\infty}(X)$, \begin{align} \frac{1}{N} \sum_{n \leq N} f(T^{p_n} \omega) g(S^{p_n} \cdot) \end{align} converges $\mu$-almost surely; above, $\{ 2=p_1 < p_2 < \dots \}$ are an enumeration of the primes. The Wiener-Wintner theorem along the primes is an immediate corollary. Our proof lives at the interface of classical Fourier analysis, combinatorial number theory, higher order Fourier analysis, and pointwise ergodic theory, with $U^3$ theory playing an important role; our $U^3$-estimates for \emph{Heath-Brown} models of the von Mangoldt function may be of independent interest.
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https://arxiv.org/abs/2601.10459
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757405181544972dfdcda3e0fa53748dd5e4c83d4a0e2ce1d94d6276cd5fd85f
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2026-01-16T00:00:00-05:00
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On the projective dimension of some deformations of Weyl arrangements
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arXiv:2601.10466v1 Announce Type: new Abstract: We show that the logarithmic derivation module of (the cone of) the deformation A of a Weyl arrangement associated with a root system of simply laced type has projective dimension one if the deforming parameter ranges from -j to j+2. In addition, we give an explicit minimal free resolution when the root system is of type A3 and B2. Moreover, in the second case, we determine the jumping lines of maximal jumping order of the associated vector bundle. When the deforming parameter of A (respectively A') ranges from -k to k+j (respectively, from -k' to k'+j), with k different from k' and j at least 3, this allows to distinguish D0(A) from D0(A') shifted by 4(k'-k), even though these modules have the same graded Betti numbers.
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https://arxiv.org/abs/2601.10466
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ccac594129363bfe9910efe22981636c77f81ab8b112bbddf7b3e4c607fb839e
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2026-01-16T00:00:00-05:00
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Positive Damping Region: A Graphic Tool for Passivization Analysis with Passivity Index
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arXiv:2601.10475v1 Announce Type: new Abstract: This paper presents a geometric framework for analyzing output-feedback and input-feedforward passivization of linear time-invariant systems. We reveal that a system is passivizable with a given passivity index when the Nyquist plot for SISO systems or the Rayleigh quotient of the transfer function for MIMO systems lies within a specific, index-dependent region in the complex plane, termed the positive damping region. The criteria enable a convenient graphic tool for analyzing the passivization, the associated frequency bands, the maximum achievable passivity index, and the waterbed effect between them. Additionally, the tool can be encoded into classical tools such as the Nyquist plot, the Nichols plot, and the generalized KYP lemma to aid control design. Finally, we demonstrate its application in passivity-based power system stability analysis and discuss its implications for electrical engineers regarding device controller design trade-offs.
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https://arxiv.org/abs/2601.10475
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209a8c1a3bc655c7920bacc4a1bed808f5a9c518181f217b4c24f80ae11613c4
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2026-01-16T00:00:00-05:00
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On Generalized Strong and Norm Resolvent Convergence
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arXiv:2601.10476v1 Announce Type: new Abstract: We present a streamlined approach for generalized strong and norm convergence of self-adjoint operators in different Hilbert spaces. In particular, we establish convergence of associated (semi-)groups, (essential) spectra and spectral projections. In addition, we give some applications to Sturm-Liouville operators.
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https://arxiv.org/abs/2601.10476
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6c575e6ea54686b023f6f6f496a567514d21017b9a695ec22b6df71ffd9c41a9
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2026-01-16T00:00:00-05:00
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A Riemannian Autocorrelation Function and its Application to Non-Local Isoperimetric Energies
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arXiv:2601.10481v1 Announce Type: new Abstract: We study a family of non-local isoperimetric energies $E_{\gamma,\varepsilon}$ on the round sphere $M = S^n$, where the non-local interaction kernel $K_\varepsilon$ is the fundamental solution of the Helmholtz operator $1 - \varepsilon^2 \Delta$. To analyse these energies, we introduce a Riemannian autocorrelation function $c_\Omega$ associated to a measurable set $\Omega\subset M$, defined on any compact, connected, oriented Riemannian manifold without boundary $(M^n,g)$ of dimension $n\ge2$. This function is intimately linked to Matheron's set covariogram from convex geometry. By establishing a characterisation of functions of bounded variation $BV(M)$ in terms of geodesic difference quotients, we show that $\Omega$ has finite perimeter if and only if $c_\Omega$ is Lipschitz, and we relate the Lipschitz constant to the perimeter of $\Omega$. We show that on the round sphere $E_{\gamma,\varepsilon}$ admits a reformulation in terms of $c_\Omega$, which allows us to compute the limit as $\varepsilon \to 0$ in a variational sense, that is, in the framework of $\Gamma$-convergence.
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https://arxiv.org/abs/2601.10481
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c6ca2c6fb1013ac80c7de98a6c740ec1b3227ca271101a8c36a0716b3381c241
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2026-01-16T00:00:00-05:00
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High-Dimensional Analysis of Gradient Flow for Extensive-Width Quadratic Neural Networks
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arXiv:2601.10483v1 Announce Type: new Abstract: We study the high-dimensional training dynamics of a shallow neural network with quadratic activation in a teacher-student setup. We focus on the extensive-width regime, where the teacher and student network widths scale proportionally with the input dimension, and the sample size grows quadratically. This scaling aims to describe overparameterized neural networks in which feature learning still plays a central role. In the high-dimensional limit, we derive a dynamical characterization of the gradient flow, in the spirit of dynamical mean-field theory (DMFT). Under l2-regularization, we analyze these equations at long times and characterize the performance and spectral properties of the resulting estimator. This result provides a quantitative understanding of the effect of overparameterization on learning and generalization, and reveals a double descent phenomenon in the presence of label noise, where generalization improves beyond interpolation. In the small regularization limit, we obtain an exact expression for the perfect recovery threshold as a function of the network widths, providing a precise characterization of how overparameterization influences recovery.
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https://arxiv.org/abs/2601.10483
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b7be0d900981b79825357d963e9364be291a954f1c4baf5af013e1c620c4882f
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2026-01-16T00:00:00-05:00
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Finite lattice kinetic equations for bosons, fermions, and discrete NLS
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arXiv:2601.10486v1 Announce Type: new Abstract: We introduce and study finite lattice kinetic equations for bosons, fermions, and discrete NLS. For each model this closed evolution equation provides an approximate description for the evolution of the appropriate covariance function in the system. It is obtained by truncating the cumulant hierarchy and dropping the higher order cumulants in the usual manner. To have such a reference solution should simplify controlling the full hierarchy and thus allow estimating the error from the truncation. The harmonic part is given by nearest neighbour hopping, with arbitrary symmetric interaction potential of coupling strength $\lambda>0$. We consider the well-posedness of the resulting evolution equation up to finite kinetic times on a finite but large enough lattice. We obtain decay of the solutions and upper bounds that are independent of $\lambda$ and depend on the lattice size only via some Sobolev type norms of the interaction potential and initial data. We prove that the solutions are not sensitive to how the energy conservation delta function is approximated.
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https://arxiv.org/abs/2601.10486
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d1b71b3648bcfe5811730da510f891728bfce8f8e565f13c1bf9f498968dd596
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2026-01-16T00:00:00-05:00
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A proof of the soliton resolution conjecture for the Benjamin--Ono equation
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arXiv:2601.10488v1 Announce Type: new Abstract: We give a proof of the soliton resolution conjecture for the Benjamin--Ono equation, namely every solution with sufficiently regular and decaying initial data can be written as a finite sum of soliton solutions with different velocities up to a radiative remainder term in the long--time asymptotics. We provide a detailed correspondence between the spectral theory of the Lax operator associated to the initial data and the different terms of the soliton resolution expansion. The proof is based on a new use of a representation formula of the solution due to the second author, and on a detailed analysis of the distorted Fourier transform associated to the Lax operator.
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https://arxiv.org/abs/2601.10488
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1ee554b2a2c69f961f3d3e7d11d0a54a4f717a8ae10255eb8b12481dc1c8d87f
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2026-01-16T00:00:00-05:00
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Malliavin Calculus for the stochastic Cahn-Hilliard equation driven by fractional noise
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arXiv:2601.10490v1 Announce Type: new Abstract: The stochastic partial differential equation analyzed in this work is the Cahn-Hilliard equation perturbed by an additive fractional white noise (fractional in time and white in space). We work in the case of one spatial dimension and apply Malliavin calculus to investigate the existence of a density for the stochastic solution $u$. In particular, we show that $u$ admits continuous paths almost surely and construct a localizing sequence through which we prove that its Malliavin derivative exists locally, and that its law is absolutely continuous with respect to the Lebesgue measure on $\bf R$, establishing thus that a density exists. A key contribution of this work is the analysis of the stochastic integral appearing in the mild formulation: we derive sharp estimates for the expectation of the $p$-th power ($p \geq 2$) of the $L^{\infty}(D)$-norm of this stochastic integral as well as for the integral involving the $L^{\infty}(D)$-norm of the operator associated with the kernel appearing in the integral representation of the fractional noise, all of which are essential for this study.
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https://arxiv.org/abs/2601.10490
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bf6ab6b01fed5c36889d80beb2c47c3a6eb890a201e74ef32030beedb2c03f21
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2026-01-16T00:00:00-05:00
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Some Eigenvalue Inequalities for the Schr\"odinger Operator on Integer Lattices
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arXiv:2601.10523v1 Announce Type: new Abstract: In this paper, we establish analogues of the Payne-P\'olya-Weinberger, Hile-Protter, and Yang eigenvalue inequalities for the Schr\"odinger operator on arbitrary finite subsets of the integer lattice $\mathbb{Z}^n$. The results extend known inequalities for the discrete Laplacian to a more general class of Schr\"odinger operators with nonnegative potentials and weighted eigenvalue problems.
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https://arxiv.org/abs/2601.10523
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7782e8179da444d27c84430b74216cabca82a7b39736dcdf4055b0e707647f7c
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2026-01-16T00:00:00-05:00
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Three realization problems about univariate polynomials
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arXiv:2601.10529v1 Announce Type: new Abstract: We consider three realization problems about monic real univariate polynomials without vanishing coefficients. Such a polynomial $P:=\sum_{j=0}^db_jx^j$ defines the sign pattern $\sigma (P):=({\rm sgn}(b_d)$, $\ldots$, ${\rm sgn}(b_0))$. The numbers $p_d$ and $n_d$ of positive and negative roots of $P$ (counted with multiplicity) satisfy the Descartes' rule of signs. Problem~1 asks for which couples $C$ of the form (sign pattern $\sigma$, pair $(p_d,n_d)$ compatible with $\sigma$ in the sense of Descartes' rule of signs), there exist polynomials $P$ defining these couples. Problem~2 asks for which $d$-tuples of pairs $T:=((p_d,n_d)$, $\ldots$, $(p_1,n_1))$, there exist polynomials $P$ such that $P^{(d-j)}$ has $p_j$ positive and $n_j$ negative roots. A $d$-tuple $T$ determines the sign pattern $\sigma (P)$, but the inverse is false. We show by an example that $6$ is the smallest value of $d$ for which there exist non-realizable tuples $T$ for which the corresponding couples $C$ are realizable. The third problem concerns polynomials with all roots real. We give a geometric interpretation of the three problems in the context of degree $4$ polynomials.
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https://arxiv.org/abs/2601.10529
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2e9b4604924ec56911cd69de6baa56b5cff2a151ce03ab79ab7d7e736f3de027
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2026-01-16T00:00:00-05:00
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Smoothness of martingale observables and generalized Feynman-Kac formulas
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arXiv:2601.10539v1 Announce Type: new Abstract: We prove that, under the H\"ormander criterion on an It\^{o} process, all its martingale observables are smooth. As a consequence, we also obtain a generalized Feynman-Kac formula providing smooth solutions to certain PDE boundary-value problems, while allowing for degenerate diffusions as well as boundary stopping (under very mild boundary regularity assumptions). We also highlight an application to a question posed on Schramm-Loewner evolutions, by making certain Girsanov transform martingales accessible via It\^{o} calculus.
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https://arxiv.org/abs/2601.10539
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38f95ab36868d983d854ba19990a5f6aba212d262875fbe83cbe681b7529d520
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2026-01-16T00:00:00-05:00
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Linear independence properties of the signature components of time-augmented stochastic processes
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arXiv:2601.10545v1 Announce Type: new Abstract: The addition of the running time as a component of a path before computing its signature is a widespread approach to ensure the one-to-one property between them and leads to universal approximation theorems (Cuchiero, Primavera and Svaluto-Ferro, 2023). However, this also leads to the linear dependence of the components of the terminal value of the signature of the time-augmented path. More precisely, for a given natural number $N$, the signature components associated with words of length $N$ have the same linear span as the signature components associated with words of length not greater than $N$. We generalize this result by exhibiting other subfamilies of signature components with the same spanning properties. In particular we recover the result of Dupire and Tissot-Daguette which states that the spanning of the iterated integrals with the last integrator different from the time variable is the same as the spanning of all iterated integrals. We check that this choice leads to the minimal computation time when the terms of the signature are calculated using Chen's relation in a backward way. The same optimal computation time is symmetrically achieved in a forward way for the iterated integrals with the first integrator different from the time variable. Building on these results, we derive several results regarding the linear independence of the signature components of a time-augmented stochastic process. We show that if the stochastic process we consider is solution to some SDE with additive Brownian noise then any subfamily of components proposed previously is linearly independent. We also prove that the linear independence of these subfamilies of components is still true when we consider the discretization of the sample paths of this stochastic process on a grid with a sufficiently small discretization time step.
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https://arxiv.org/abs/2601.10545
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05c85d53da7f433798e47dfc907a57cae9a9c461675d5e3bd2dac2ad4d623177
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2026-01-16T00:00:00-05:00
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The inducibility of Tur\'an graphs
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arXiv:2601.10548v1 Announce Type: new Abstract: Let $I(F,n)$ denote the maximum number of induced copies of a graph $F$ in an $n$-vertex graph. The inducibility of $F$, defined as $i(F)=\lim_{n\to \infty} I(F,n)/\binom{n}{v(F)}$, is a central problem in extremal graph theory. In this work, we investigate the inducibility of Tur\'an graphs $F$. This topic has been extensively studied in the literature, including works of Pippenger--Golumbic, Brown--Sidorenko, Bollob\'as--Egawa--Harris--Jin, Mubayi, Reiher, and the first author, and Yuster. Broadly speaking, these results resolve or asymptotically resolve the problem when the part sizes of $F$ are either sufficiently large or sufficiently small (at most four). We complete this picture by proving that for every Tur\'an graph $F$ and sufficiently large $n$, the value $I(F,n)$ is attained uniquely by the $m$-partite Tur\'an graph on $n$ vertices, where $m$ is given explicitly in terms of the number of parts and vertices of $F$. This confirms a conjecture of Bollob\'as--Egawa--Harris--Jin from 1995, and we also establish the corresponding stability theorem. Moreover, we prove an asymptotic analogue for $I_{k+1}(F,n)$, the maximum number of induced copies of $F$ in an $n$-vertex $K_{k+1}$-free graph, thereby completely resolving a recent problem of Yuster. Finally, our results extend to a broader class of complete multipartite graphs in which the largest and smallest part sizes differ by at most on the order of the square root of the smallest part size.
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https://arxiv.org/abs/2601.10548
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6f317b4d4675d27b13566a834196b42b1e9ed27e523d346bfcf70a7291b9da23
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2026-01-16T00:00:00-05:00
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(a,b)-Fibonacci-Legendre Cordial Graphs and k-Pisano-Legendre Primes
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arXiv:2601.10561v1 Announce Type: new Abstract: Let $p$ be an odd prime and let $F_i$ be the $i$th $(a,b)$-Fibonacci number with initial values $F_0=a$ and $F_1=b$. For a simple connected graph $G=(V,E)$, define a bijective function $f:V(G)\to \{0,1,\ldots,|V|-1\}$. If the induced function $f_p^*:E(G)\to \{0,1\}$, defined by $f_p^*(uv)=\frac{1+([F_{f(u)}+F_{f(v)}]/p)}{2}$ whenever $F_{f(u)}+F_{f(v)}\not\equiv 0\pmod{p}$ and $f_p^*(uv)=0$ whenever $F_{f(u)}+F_{f(v)}\equiv 0\pmod{p}$, satisfies the condition $|e_{f_p^*}(0)-e_{f_p^*}(1)|\leq 1$ where $e_{f_p^*}(i)$ is the number of edges labeled $i$ ($i=0,1$), then $f$ is called $(a,b)$-Fibonacci-Legendre cordial labeling modulo $p$. In this paper, the $(a,b)$-Fibonacci-Legendre cordial labeling of path graphs, star graphs, wheel graphs, and graphs under the operations join, corona, lexicographic product, cartesian product, tensor product, and strong product is explored in relation to $k$-Pisano-Legendre primes relative to $(a,b)$. We also present some properties of $k$-Pisano-Legendre primes relative to $(a,b)$ and numerical observations on its distribution, leading to several conjectures concerning their density and growth behavior.
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https://arxiv.org/abs/2601.10561
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9edc3209dcf46a24ff72b994c9863dd0af089f4f8139124b4536e9aed50d64d0
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2026-01-16T00:00:00-05:00
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Hydrodynamic Limit with a Weierstrass-type result
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arXiv:2601.10568v1 Announce Type: new Abstract: We show that any positive, continuous, and bounded function can be realised as the diffusion coefficient of an evolution equation associated with a gradient interacting particle system. The proof relies on the construction of an appropriate model and on the entropy method.
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https://arxiv.org/abs/2601.10568
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6dcecfe3f776ef151978a1a44f1ab9f37530a38188a17a68c1b1d647c80f231d
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2026-01-16T00:00:00-05:00
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Mind the gap: A real-valued distance on combinatorial games
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arXiv:2601.10574v1 Announce Type: new Abstract: We define a real-valued distance metric $wd$ on the space $\mathcal{C}$ of short combinatorial games in canonical form. We demonstrate the existence of Cauchy sequences informed by sidling sequences, find limit points, and investigate the closure $\overline{\mathcal{C}}$, which is shown to partition the set of loopy games in a non-trivial way. Stoppers, enders, and non-stopper-sided loopy games are explored, as well as the topological properties of $(\mathcal{C},wd)$.
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https://arxiv.org/abs/2601.10574
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231985af0a4cad57e47d9155b3a2c166a56723481a076bb579d9116f003bba05
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2026-01-16T00:00:00-05:00
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On Zalcman's and Bieberbach conjectures
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arXiv:2601.10584v1 Announce Type: new Abstract: The well-known Zalcman conjecture, which implies the Bieberbach conjecture, states that the coefficients of univalent functions $f(z) = z + \sum\limits_2^{\infty} a_n z^n$ on the unit disk satisfy $|a_n^2 - a_{2n-1}| \le (n-1)^2$ for all $n > 2$, with equality only for the Koebe function and its rotations. The conjecture was proved by the author for $n \le 6$ (using geometric arguments related to the Ahlfors-Schwarz lemma) and remains open for $n \ge 7$. The main theorem of this paper states that these conjectures are equivalent and provides their simultaneous proof for all $n \ge 3$ combining the indicated geometric arguments with a new author's approach to extremal problems for holomorphic functions based on lifting the rotationally homogeneous coefficient functionals to the Bers fiber space over universal Teichmuller space.
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https://arxiv.org/abs/2601.10584
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e26c402b309091a8d6aedf1dc3802a2269b007ee0f52a4c956c43cd500f14504
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2026-01-16T00:00:00-05:00
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Comparison of viscosity solutions for a class of non-linear PDEs on the space of finite nonnegative measures
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arXiv:2601.10586v1 Announce Type: new Abstract: We establish a comparison principle for viscosity solutions of a class of nonlinear partial differential equations posed on the space of nonnegative finite measures, thereby extending recent results for PDEs defined on the Wasserstein space of probability measures. As an application, we study a controlled branching McKean-Vlasov diffusion and characterize the associated value function as the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. This yields a PDE-based approach to the optimal control of branching processes.
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https://arxiv.org/abs/2601.10586
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d9e0f2128990b7511b24eae2b0f72ae86dab815c2d194609b99f1142f821826b
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2026-01-16T00:00:00-05:00
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Schur--Horn type inequalities for hyperbolic polynomials
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arXiv:2601.10602v1 Announce Type: new Abstract: We establish a Schur--Horn type inequality for symmetric hyperbolic polynomials. As an immediate consequence, we resolve a conjecture of Nam Q. Le on Hadamard-type inequalities for hyperbolic polynomials. Our argument is based on the Schur--Horn theorem, the Birkhoff theorem, and G{\aa}rding's concavity theorem for hyperbolicity cones. Beyond the eigenvalue level, we develop a symmetrization principle on hyperbolicity cones: if a hyperbolic polynomial is invariant under a finite group action, then its value increases under the associated Reynolds operator (group averaging). Applied to the sign-flip symmetries of linear principal minor polynomials introduced by Blekherman et al., this yields a short proof of the hyperbolic Fischer--Hadamard inequalities for PSD-stable lpm polynomials.
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https://arxiv.org/abs/2601.10602
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99594a09beadc1b6f52a1ad4466c8ed0ff93cd4591c99c97c5a8c842b9c5eae8
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2026-01-16T00:00:00-05:00
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Local times and excursions for self-similar Markov trees
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arXiv:2601.10610v1 Announce Type: new Abstract: This work builds upon the recent monograph [5] on self-similar Markov trees. A self-similar Markov tree is a random real tree equipped with a function from the tree to $[0,\infty)$ that we call the decoration. Here, we construct local time measures $L(x,dt)$ at every level $x>0$ of the decoration for a large class of self-similar Markov trees. This enables us to mark at random a typical point in the tree at which the decoration is $x$. We identify the law of the decoration along the branch from the root to this tagged point in terms of a remarkable (positive) self-similar Markov process. We also show that after a proper normalization, $L(x,dt)$ converges as $x\to 0+$ to the harmonic measure $\mu$ on the tree. Finally, we point out that using a local time measure instead of the usual length measure $\lambda$ to compute distances on the tree turn the latter into a continuous branching tree. This is relevant to analyze the excusions of the decoration away from a given level. Many results of the present work shall be compared with the recent ones in [22,23] about local times and excursions of a Markov process indexed by L\'evy tree.
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https://arxiv.org/abs/2601.10610
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80c843a2284512df241764de8ec3231000ebe4f6cf3229577cbda9156670589c
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2026-01-16T00:00:00-05:00
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Malcev classification for the variety of left-symmetric algebras
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arXiv:2601.10613v1 Announce Type: new Abstract: In this paper, we study three classes of subvarieties inside the variety of left-symmetric algebras. We show that these subvarieties are naturally related to some well-known varieties, such as alternative, assosymmetric and Zinbiel algebras. For certain subvarieties of the varieties of alternative and assosymmetric algebras, we explicitly construct bases of the corresponding free algebras. We then define the commutator and anti-commutator operations on these algebras and derive a number of identities satisfied by these operations in all degrees up to $4$.
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https://arxiv.org/abs/2601.10613
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0e6885b63cf7779ec22130edad3a40a3b1158683a38293a51a450c7ac68de45a
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2026-01-16T00:00:00-05:00
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The directedness of the Rudin-Keisler order at measurable cardinals
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arXiv:2601.10614v1 Announce Type: new Abstract: The manuscript is concerned with the Rudin-Keisler order of ultrafilters on measurable cardinals. The main theorem proved read as follows: Given regular cardinals $\lambda\leq \kappa$, the following theories are equiconsistent modulo ZFC: (1) $\kappa$ is a measurable cardinal with $o(\kappa)=\lambda^+$ (resp. $o(\kappa)=\kappa$). (2) The Rudin-Keisler order restricted to the set of $\kappa$-complete (non-principal) ultrafilters on $\kappa$ is $\lambda^+$-directed (resp. $\kappa^+$-directed). The theorem reported here is proved after bridging the directedness of the RK-order with the $\lambda$-Gluing Property introduced by the authors in \cite{HP}. Our result provides what seems to be the first example of a compactness-type property at the level of measurable cardinals whose consistency strength is much lower than the existence of a strong cardinal. As part of our analysis we also answer a question of Gitik by showing that the $\aleph_0$-Gluing Property fails in his classical model from ''Changing cofinalities and the nonstationary ideal". As a consequence of this, in Gitik's model the Rudin-Keisler order fails to be $\aleph_1$-directed.
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https://arxiv.org/abs/2601.10614
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ecc15c7616ff1f73b243c6ca809572e2ae0508b76d6d9ea78e89c4ea5b54d73d
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2026-01-16T00:00:00-05:00
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Quantitative surgery and total mean curvature
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arXiv:2601.10617v1 Announce Type: new Abstract: We develop quantitative surgery, which extends the classical constructions of Gromov--Lawson and Lawson--Michelsohn. As an application, we prove a conjecture of Gromov on the total mean curvature of fill-ins.
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https://arxiv.org/abs/2601.10617
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1731488bd6dd34cb72919e82e3e7c928490cf5057bf4fa0d349ad5e1a99aab51
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2026-01-16T00:00:00-05:00
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A universal Bochner formula for scalar curvature
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arXiv:2601.10618v1 Announce Type: new Abstract: We introduce a universal Bochner formula for scalar curvature that contains, as special cases, the stability inequality for minimal slicings, a Schr\"odinger-Lichnerowicz-type formula, and a higher-dimensional version of Stern's level-set identity.
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https://arxiv.org/abs/2601.10618
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4fdad00deba237436cded6f8985600c65bb006f75a5a0697965fc42e22b13654
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2026-01-16T00:00:00-05:00
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Source localisation in simple random walks
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arXiv:2601.10624v1 Announce Type: new Abstract: We consider the problem of locating the source (starting vertex) of a simple random walk, given a snapshot of the set of edges (or vertices) visited in the first $n$ steps. Considering lattices $\mathbb{Z}^d$, in dimensions $d \geq 5$, we show that the source can be identified (a) with probability bounded away from $0$ using one guess, and (b) with probability arbitrarily close to $1$ using a constant number of guesses. On the other hand, for dimensions $d \leq 2$, we show that one cannot locate the source with positive constant probability. Our arguments apply more generally to strongly transient and recurrent simple random walks on vertex-transitive graphs.
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https://arxiv.org/abs/2601.10624
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b8e615968c0036f657d7be9909bbf700b18b3705656ee917136a98a2c36f7a3e
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2026-01-16T00:00:00-05:00
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Discrete-time maximally superintegrable systems and deformed symmetry algebras: the Calogero-Moser case
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arXiv:2601.10625v1 Announce Type: new Abstract: We determine the complete structure of the symmetry algebras associated with the N-body Calogero-Moser system and its maximally superintegrable discretization. We prove that the discretization naturally leads to a nontrivial deformation of the continuous symmetry algebra, with the discretization parameter playing the r\^ole of a deformation parameter. This phenomenon illustrates how discrete superintegrable systems can be viewed as natural sources of deformed polynomial algebraic structures. As a byproduct of these results, we also reveal a connection between the above-mentioned symmetry algebras and the Bell polynomials, as a consequence of the trace properties.
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https://arxiv.org/abs/2601.10625
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3b0bdcd6c371fd638839438d3b92c61730d04f3396287e33a4e6db39c55b120c
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2026-01-16T00:00:00-05:00
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On subradically sifted sums related to Alladi's higher order duality between prime factors
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arXiv:2601.10636v1 Announce Type: new Abstract: In this paper, I utilize a variant of the Selberg--Delange method to find quantitative estimates of the sums \[M_{k,\omega}(x,y)=\sum_{\substack{p_{1}(n)> y\\ n\leq x} } \mu(n) {\omega(n)-1\choose k-1},\] where $y$ can grow with $x$ but we must have $y\leq Y_0\exp(\mathscr{p}\frac{\log x}{(\log\log (x+1))^{1+\epsilon}})$ with $Y_0,\mathscr{p},\epsilon>0$. Moreover, I give preliminary upper bounds for the general range $1.9\leq y\leq x^{\frac{1}{k}}$. In addition, I formalize the notions of subradical and radical dominance and discuss their relevance to the analytic approach of the study of arithmetic functions. Lastly, I give a fascinating formula related to the derivatives of the gamma function and the Hankel contour, which should be relevant for those employing the Selberg--Delange method to obtain higher-order terms.
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https://arxiv.org/abs/2601.10636
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874a4324bc42d38d538614c3fe486dbecc913c8db649a595af28c8e088270241
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2026-01-16T00:00:00-05:00
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Michael-Simon inequality for anisotropic energies close to the area via multilinear Kakeya-type bounds
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arXiv:2601.10647v1 Announce Type: new Abstract: Given an anisotropic integrand $F:\text{Gr}_k(\mathbb R^n)\to(0,\infty)$, we can generalize the classical isotropic area by looking at the functional $$\mathcal{F}(\Sigma^k):=\int_\Sigma F(T_x\Sigma)\,d\mathcal{H}^k.$$ While a monotonicity formula is not available for critical points, when $k=2$ and $n=3$ we show that the Michael-Simon inequality holds if $F$ is convex and close to $1$ (in $C^1$), meaning that $\mathcal{F}$ is close to the usual area. Our argument is partly based on some key ideas of Almgren, who proved this result in an unpublished manuscript, but we largely simplify his original proof by showing a new functional inequality for vector fields on the plane, which can be seen as a quantitative version of Alberti's rank-one theorem. As another byproduct, we also show Michael-Simon for another class of integrands which includes the $\ell^p$ norms for $p\in(1,\infty)$. For a general $F$ satisfying the atomic condition, we also show that the validity of Michael-Simon is equivalent to compactness of rectifiable varifolds.
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https://arxiv.org/abs/2601.10647
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ce48bcaa8818cc4e71deb3331c0b5f9b7a52a05ea02d89bd1bba00116473745e
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2026-01-16T00:00:00-05:00
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Uniform stability of the inverse Sturm-Liouville problem on a star-shaped graph
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arXiv:2601.10652v1 Announce Type: new Abstract: In this paper, we study the inverse spectral problem for the Sturm-Liouville operators on a star-shaped graph, which consists in the recovery of the potentials from specral data or several spectra. The uniform stability of these inverse problems on the whole graph is proved.
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https://arxiv.org/abs/2601.10652
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6a2e04c132aee5eabe22a4a5018f1e2bad935055e1f842a231493337773eef1c
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2026-01-16T00:00:00-05:00
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Symmetries of Borcherds algebras
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arXiv:2601.10653v1 Announce Type: new Abstract: We give an overview of the construction of Borcherds algebras, particularly the Monstrous Lie algebras $\mathfrak m_g$ constructed by Carnahan, where $g$ is an element of the Monster finite simple group. When $g$ is the identity element, $\mathfrak m_g$ is the Monster Lie algebra of Borcherds. We discuss the appearance of the $\mathfrak m_g$ in compactified models of the Heterotic String. We also summarize recent work on associating Lie group analogs to the Lie algebras $\mathfrak m_g$. We include a discussion of some open problems.
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https://arxiv.org/abs/2601.10653
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e6b1cfed012a4daafa3068486bd692102d8e0ff6a1700278ff1bdc8051f70162
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2026-01-16T00:00:00-05:00
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A note on strong similarity and the Connes embedding problem
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arXiv:2601.10654v1 Announce Type: new Abstract: We show that there exists a completely bounded (c.b. in short) homomorphism $u$ from a $C^*$-algebra $C$ with the lifting property (in short LP) into a QWEP von Neumann algebra $N$ that is not strongly similar to a $*$-homomorphism, i.e. the similarities that ``orthogonalize" $u$ (which exist since $u$ is c.b.) cannot belong to the von Neumann algebra $N$. Moreover, the map $u$ does not admit any c.b. lifting up into the WEP $C^*$-algebra of which $N$ is a quotient. We can take $C=C^*(G)$ (full $C^*$-algebra) where $G$ is any nonabelian free group and $N= B(H)\bar \otimes M$ where $M$ is the von Neumann algebra generated by the reduced $C^*$-algebra of $G$.
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https://arxiv.org/abs/2601.10654
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9ab98bb6e143de887d9f072cb86efcc1beaaa008628b297cc9e1c2e68a4cf967
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2026-01-16T00:00:00-05:00
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Hyperk\"ahler Degenerations from Parabolic $\mathrm{SL}(2,\mathbb{C})$-Higgs Bundles Moduli Spaces on the Punctured Sphere to Hyperpolygon Spaces
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arXiv:2601.10656v1 Announce Type: new Abstract: Complete hyperk\"ahler 4-manifolds of finite energy are grouped into ALE, ALF, ALG$^{(*)}$, ALH$^{(*)}$, each of these being further classified according to the Dynkin type of their noncompact end. A family of ALG-$D_4$ spaces are modeled by certain moduli spaces of strongly parabolic $\mathrm{SL}(2,\mathbb{C})$-Higgs bundles on the Riemann sphere with $n=4$ punctures. Meanwhile, a family of ALE-$D_4$ spaces are modeled by certain Nakajima quiver varieties known as $n=4$ hyperpolygon spaces. There is a map from hyperpolygon space to the moduli space of strong parabolic $\mathrm{SL}(2,\mathbb{C})$-Higgs bundles that is a diffeomorphism onto its open and dense image. We show that under a fine-tuned degenerate limit, the pullback of a family of ALG-$D_4$ metrics parameterized by $R$ converges pointwise to the ALE-$D_4$ metric as $R \to 0$. While the connection to gravitational instantons occurs in the $n=4$ case, we prove our result for any finite $n$.
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https://arxiv.org/abs/2601.10656
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b075eeeaab358741eac97c1b8cc18d19082f615fddd02fca84d5ee32cea6e171
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2026-01-16T00:00:00-05:00
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On Necessary and Sufficient Conditions for Fixed Point Convergence: A Contractive Iteration Principle
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arXiv:2601.10669v1 Announce Type: new Abstract: While numerous extensions of Banach's fixed point theorem typically offer only sufficient conditions for the existence and uniqueness of a fixed point and the convergence of iterative sequences, this study introduces a generalization grounded in the iterative contraction principle in complete metric spaces. This generalization establishes both the necessary and sufficient conditions for the existence of a unique fixed point to which all iterative sequences converge, along with an accurate error estimate. Furthermore, we present and prove an additional theorem that characterizes the convergence of all iterative sequences to fixed points that may not be unique. Several examples are provided to illustrate the practical application of these results, including a case where the traditional and well-known generalizations of Banach's theorem, such as those by Banach, Kannan, Chatterjea, Hardy-Rogers, Meir-Keeler, and Guseman, are inapplicable.
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https://arxiv.org/abs/2601.10669
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Academic Papers
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003267ad06ec24de3c241d8cb5f8c94a96a1da72bd6a26f520b1110d666568ce
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2026-01-16T00:00:00-05:00
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Real characters and real classes of $\mathrm{GL}_2$ and $\mathrm{GU}_2$ over discrete valuation rings
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arXiv:2601.10670v1 Announce Type: new Abstract: Let $\mathfrak{o}$ be the ring of integers of a non-archimedean local field with residue field of odd characteristic, $\mathfrak{p}$ be its maximal ideal and let $\mathfrak{o}_\ell = \mathfrak{o}/\mathfrak{p}^\ell$ for $\ell\ge 2$. In this article, we study real-valued characters and real representations of the finite groups $\mathrm{GL}_2(\mathfrak{o}_\ell)$ and $\mathrm{GU}_2(\mathfrak{o}_\ell)$. We give a complete classification of real and strongly real classes of these groups and characterize the real-valued irreducible complex characters. We prove that every real-valued irreducible complex character of $\mathrm{GL}_2(\mathfrak{o}_\ell)$ is afforded by a representation over $\mathbb{R}$. In contrast, we show that $\mathrm{GU}_2(\mathfrak{o}_\ell)$ admits real-valued irreducible characters that are not realizable over $\mathbb{R}$. These results extend the parallel known phenomena for the finite groups $\mathrm{GL}_n(\mathbb{F}_q)$ and $\mathrm{GU}_n(\mathbb{F}_q)$.
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https://arxiv.org/abs/2601.10670
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b25d91b7d75fc431b151ed71bd6296df92d90bf0956fa01b13ff73723bce40d7
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2026-01-16T00:00:00-05:00
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Vertex operator algebra bundles on modular curves and their associated modular forms
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arXiv:2601.10686v1 Announce Type: new Abstract: This paper describes the vector bundle on the elliptic modular curve that is associated to a vertex operator algebra $V$ (VOA) or more generally a quasi-vertex operator algebra (QVOA), with a view towards future applications aimed at studying the characters of VOAs. We explain how the modes of sections of $V$ give rise naturally to $V$-valued quasi-modular forms. The space $Q(V)$ of $V$-valued quasi-modular forms is endowed with the structure of a doubled QVOA, and in particular the algebra $Q$ of quasi-modular forms is itself a doubled QVOA. $Q(V)$ also admits a natural derivative operator arising from the connection on the bundle defined by $V$ and the modular derivative, which we call the raising operator. We introduce an associated lowering operator $\Lambda$ on $Q(V)$ having the property that the $V$-valued modular forms $M(V)\subseteq Q(V)$ are the kernel of $\Lambda$. This extends the classical theory of scalar-valued quasi-modular forms. We exhibit an explicit isomorphism of $M(V)$ with $M \otimes V$. Finally, the coordinate invariance of vertex operators implies that $M(V)$ has a natural Hecke theory, and we use this isomorphism to fully describe the Hecke eigensystems: they are the same as the systems of eigenvalues that arise from scalar-valued quasi-modular forms.
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https://arxiv.org/abs/2601.10686
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d56fda378c6c2111e0ecfdc8e5be083360a85e83fc33bd4d9b3853818d8b27db
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2026-01-16T00:00:00-05:00
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Unbounded symbols, heat flow, and Toeplitz operators
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arXiv:2601.10711v1 Announce Type: new Abstract: We disprove the natural domain extension of the Berger--Coburn heat-flow conjecture for Toeplitz operators on the Bargmann space and identify the failure mechanism as a gap between pointwise and uniform control of a Gaussian averaging of the squared modulus of the symbol, a gap that is invisible to the linear form $T_g$. We establish that the form-defined operator $T_g$ and the natural-domain operator $U_g$ decouple in the unbounded symbols regime: while $T_g$ is governed by linear averaging, $U_g$ is controlled by the quadratic intensity of $|g|^2$. We construct a smooth, nonnegative radial symbol $g$ satisfying the coherent-state admissibility hypothesis with bounded heat transforms for all time $t>0$; for this symbol, $T_g$ is bounded, yet $U_g$ is unbounded. This is a strictly global phenomenon: under the coherent-state hypothesis, local singularities are insufficient to cause unboundedness, leaving the ``geometry at infinity'' as the sole obstruction. Boundedness of $U_g$ is equivalent to the condition that $|g|^2 d\mu$ is a Fock--Carleson measure, a condition strictly stronger than the linear average $g d\mu$ governing $T_g$. Finally, regarding the gap between the known sub-critical sufficiency condition and the critical heat time, we prove that heat-flow regularity is irreversible in this context and show that bootstrapping strategies cannot resolve the gap between sufficiency and critical time.
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https://arxiv.org/abs/2601.10711
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1035a23735f98433931a925ab0704c6080e9080fe0d9f733c5b8e404f1e281bb
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2026-01-16T00:00:00-05:00
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Absorption and fixation times for evolutionary processes on graphs
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arXiv:2601.09737v1 Announce Type: cross Abstract: In this paper, we study the absorption and fixation times for evolutionary processes on graphs, under different updating rules. While in Moran process a single neighbour is randomly chosen to be replaced, in proliferation processes other neighbours can be replaced using Bernoulli or binomial draws depending on $0 p_c$ or $p < p_c$. We clarify the role of symmetries for computing the fixation time in Moran process. We show that the Maruyama-Kimura symmetry depend on the graph structure induced in each state, implying asymmetry for all graphs except cliques and cycles. There is a fitness value, not necessarily $1$, beyond which the fixation time decreases monotonically. We apply Harris' graphical method to prove that the fixation time decreases monotonically depending on $p$. Thus there exists another value $p_t$ for which the proliferation is advantageous or disadvantageous in terms of time. However, at the critical level $p=p_c$, the proliferation is highly advantageous when $r \to +\infty$.
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https://arxiv.org/abs/2601.09737
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d4d91901abcf5e2b2168df715ae5ff633ed74af734f4f24dfaea97b2cf825aa5
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2026-01-16T00:00:00-05:00
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Topological Percolation in Urban Dengue Transmission: A Multi-Scale Analysis of Spatial Connectivity
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arXiv:2601.09747v1 Announce Type: cross Abstract: We investigate the spatial organization of dengue cases in the city of Recife, Brazil, from 2015 to 2024, using tools from statistical physics and topological data analysis. Reported cases are modeled as point clouds in a metric space, and their spatial connectivity is studied through Vietoris-Rips filtrations and zero-dimensional persistent homology, which captures the emergence and collapse of connected components across spatial scales. By parametrizing the filtration using percentiles of the empirical distance distribution, we identify critical percolation thresholds associated with abrupt growth of the largest connected component. These thresholds define distinct geometric regimes, ranging from fragmented spatial patterns to highly concentrated, percolated structures. Remarkably, years with similar incidence levels exhibit qualitatively different percolation behavior, demonstrating that case counts alone do not determine the spatial organization of transmission. Our analysis further reveals pronounced temporal heterogeneity in the percolation properties of dengue spread, including a structural rupture in 2020 characterized by delayed or absent spatial percolation. These findings highlight percolation-based topological observables as physically interpretable and sensitive descriptors of urban epidemic structure, offering a complementary perspective to traditional spatial and epidemiological analyses.
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https://arxiv.org/abs/2601.09747
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bdcce7212e10148a52bc25485b871de56a0d2cba7ed9a3a1784cb81d2eff06ec
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2026-01-16T00:00:00-05:00
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Localization of quantum states within subspaces
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arXiv:2601.09817v1 Announce Type: cross Abstract: A precise definition is proposed for the localization probability of a quantum state within a given subspace of the full Hilbert space of a quantum system. The corresponding localized component of the state is explicitly identified, and several mathematical properties are established. Applications and interpretations in the context of quantum information are also discussed.
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https://arxiv.org/abs/2601.09817
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Academic Papers
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a895adf792577b8c0cafd18b8f15513c10e6efd40695f454a6f927e7594402d9
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2026-01-16T00:00:00-05:00
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Forecasting Seasonal Peaks of Pediatric Respiratory Infections Using an Alert-Based Model Combining SIR Dynamics and Historical Trends in Santiago, Chile
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arXiv:2601.09821v1 Announce Type: cross Abstract: Acute respiratory infections (ARI) are a major cause of pediatric hospitalization in Chile, producing marked winter increases in demand that challenge hospital planning. This study presents an alert-based forecasting model to predict the timing and magnitude of ARI hospitalization peaks in Santiago. The approach integrates a seasonal SIR model with a historical mobile predictor, activated by a derivative-based alert system that detects early epidemic growth. Daily hospitalization data from DEIS were smoothed using a 15-day moving average and Savitzky-Golay filtering, and parameters were estimated using a penalized loss function to reduce sensitivity to noise. Retrospective evaluation and real-world implementation in major Santiago pediatric hospitals during 2023 and 2024 show that peak date can be anticipated about one month before the event and predicted with high accuracy two weeks in advance. Peak magnitude becomes informative roughly ten days before the peak and stabilizes one week prior. The model provides a practical and interpretable tool for hospital preparedness.
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https://arxiv.org/abs/2601.09821
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866a3454af784f37d9daa8359c997c3079a60444275d9af16324fc6556cc7e5b
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2026-01-16T00:00:00-05:00
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The formation of periodic three-body orbits for Newtonian systems
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arXiv:2601.09843v1 Announce Type: cross Abstract: Braids are periodic solutions to the general N-body problem in gravitational dynamics. These solutions seem special and unique, but they may result from rather usual encounters between four bodies. We aim at understanding the existence of braids in the Galaxy by reverse engineering the interactions in which they formed. We simulate self-gravitating systems of N particles, starting with the constructing of a specific braid, and bombard it with a single object. We study how frequently the bombarded braid dissolves in four singles, a triple and a single, a binary and 2 singles, or 2 binaries. The relative proportion of those events gives us insight into how easy it is to generate a braid through the reverse process. It turns out that braids are easily generated from encounters between 2 binaries, or a triple with a single object, independent on the braid's stability. We find that 3 of the explored braids are linearly stable against small perturbations, whereas one is unstable and short-lived. The shortest-lived braid appears the least stable and the most chaotic. nonplanar encounters also lead to braid formation, which, in our experiments, themselves are planar. The parameter space in azimuth and polar angle that lead to braid formation via binary-binary or triple-single encounters is anisotropic, and the distribution has a low fractal dimension. Since a substantial fraction of ~9% of our calculations lead to periodic 3-body systems, braids may be more common than expected. They could form in multi-body interactions. We do not expect many to exist for time, but they may be common as transients, as they survive for tens to hundreds of periodic orbits. We argue that braids are common in relatively shallow-potential background fields, such as the Oort cloud or the Galactic halo. If composed of compact objects, they potentially form interesting targets for gravitational wave detectors.
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https://arxiv.org/abs/2601.09843
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7deaa63c6b72b85d5f880146e03ce119de07d2c4ca148aaa3600269e973f0abd
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2026-01-16T00:00:00-05:00
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Model selection by cross-validation in an expectile linear regression
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arXiv:2601.09874v1 Announce Type: cross Abstract: For linear models that may have asymmetric errors, we study variable selection by cross-validation. The data are split into training and validation sets, with the number of observations in the validation set much larger than in the training set. For the model coefficients, the expectile or adaptive LASSO expectile estimators are calculated on the training set. These estimators will be used to calculate the cross-validation mean score (CVS) on the validation set. We show that the model that minimizes CVS is consistent in two cases: when the number of explanatory variables is fixed or when it depends on the number of observations. Monte Carlo simulations confirm the theoretical results and demonstrate the superiority of our estimation method compared to two others in the literature. The usefulness of the CV expectile model selection technique is illustrated by applying it to real data sets.
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https://arxiv.org/abs/2601.09874
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f5818e2bced675c0a604a16d75074d9c211e63dec8b64d7e65cd214e1e6e8715
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2026-01-16T00:00:00-05:00
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Learning about Treatment Effects with Prior Studies: A Bayesian Model Averaging Approach
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arXiv:2601.09888v1 Announce Type: cross Abstract: We establish concentration rates for estimation of treatment effects in experiments that incorporate prior sources of information -- such as past pilots, related studies, or expert assessments -- whose external validity is uncertain. Each source is modeled as a Gaussian prior with its own mean and precision, and sources are combined using Bayesian model averaging (BMA), allowing data from the new experiment to update posterior weights. To capture empirically relevant settings in which prior studies may be as informative as the current experiment, we introduce a nonstandard asymptotic framework in which prior precisions grow with the experiment's sample size. In this regime, posterior weights are governed by an external-validity index that depends jointly on a source's bias and information content: biased sources are exponentially downweighted, while unbiased sources dominate. When at least one source is unbiased, our procedure concentrates on the unbiased set and achieves faster convergence than relying on new data alone. When all sources are biased, including a deliberately conservative (diffuse) prior guarantees robustness and recovers the standard convergence rate.
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https://arxiv.org/abs/2601.09888
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faaccc0e0c451f678d7403c93a59715066e896515664a8ebbf8c74188a80b87d
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2026-01-16T00:00:00-05:00
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Learning and Equilibrium under Model Misspecification
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arXiv:2601.09891v1 Announce Type: cross Abstract: This chapter develops a unified framework for studying misspecified learning situations in which agents optimize and update beliefs within an incorrect model of their environment. We review the statistical foundations of learning from misspecified models and extend these insights to environments with endogenous, action-dependent data, including both single agent and strategic settings.
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https://arxiv.org/abs/2601.09891
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b88eecc666688a82af6abfd4eb8cc39c957a328b11513f72eae406721398ece3
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2026-01-16T00:00:00-05:00
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Analytic approach to boundary integrability with application to mixed-flux $AdS_3 \times S^3$
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arXiv:2601.09935v1 Announce Type: cross Abstract: Boundary integrability provides rare analytic control over field theories in the presence of an interface, from quantum impurity problems to open string dynamics. We develop an analytic framework for integrable boundaries in two-dimensional sigma-models that determines admissible reflection maps directly from the meromorphic Lax connection. Applying it to open strings on $AdS_3\times S^3$ with mixed NSNS and RR flux, we find two branches of integrable boundary conditions. One branch admits D-branes wrapping twisted conjugacy classes on $SU(1,1)\times SU(2)$, with the mixed-flux deformation encoded entirely into dynamical boundary data. At the exactly solvable WZW point these coincide with the conformal D-branes, providing a natural link to conformal perturbation theory.
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https://arxiv.org/abs/2601.09935
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ba3d69cd514e0cb6fdfabaaddc265a74e415dc6838912b9fd71d915a18bafc77
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2026-01-16T00:00:00-05:00
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Model-Agnostic and Uncertainty-Aware Dimensionality Reduction in Supervised Learning
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arXiv:2601.10357v1 Announce Type: cross Abstract: Dimension reduction is a fundamental tool for analyzing high-dimensional data in supervised learning. Traditional methods for estimating intrinsic order often prioritize model-specific structural assumptions over predictive utility. This paper introduces predictive order determination (POD), a model-agnostic framework that determines the minimal predictively sufficient dimension by directly evaluating out-of-sample predictiveness. POD quantifies uncertainty via error bounds for over- and underestimation and achieves consistency under mild conditions. By unifying dimension reduction with predictive performance, POD applies flexibly across diverse reduction tasks and supervised learners. Simulations and real-data analyses show that POD delivers accurate, uncertainty-aware order estimates, making it a versatile component for prediction-centric pipelines.
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https://arxiv.org/abs/2601.10357
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a649ebf292b759b25360e899e199f8b30741947ebb665afcab17bfcbd0521916
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2026-01-16T00:00:00-05:00
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Gene genealogies in diploid populations evolving according to sweepstakes reproduction
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arXiv:2601.10364v1 Announce Type: cross Abstract: Recruitment dynamics, or the distribution of the number of offspring among individuals, is central for understanding ecology and evolution. Sweepstakes reproduction (heavy right-tailed offspring number distribution) is central for understanding the ecology and evolution of highly fecund natural populations. Sweepstakes reproduction can induce jumps in type frequencies and multiple mergers in gene genealogies of sampled gene copies. We take sweepstakes reproduction to be skewed offspring number distribution due to mechanisms not involving natural selection, such as in chance matching of broadcast spawning with favourable environmental conditions. Here, we consider population genetic models of sweepstakes reproduction in a diploid panmictic populations absent selfing and evolving in a random environment. Our main results are {\it (i)} continuous-time Beta and Poisson-Dirichlet coalescents, when combining the results the skewness parameter $\alpha$ of the Beta-coalescent ranges from $0$ to $2$, and the Beta-coalescents may be incomplete due to an upper bound on the number of potential offspring produced by any pair of parents; {\it (ii)} in large populations time is measured in units proportional to either $N/\log N$ or $N$ generations (where $2N$ is the population size when constant); {\it (iii)} it follows that incorporating population size changes leads to time-changed coalescents with the time-change independent of $\alpha$; {\it (iv)} using simulations we show that the ancestral process is not well approximated by the corresponding coalescent (as measured through certain functionals of the processes); {\it (v)} whenever the skewness of the offspring number distribution is increased the conditional (conditioned on the population ancestry) and the unconditional ancestral processes are not in good agreement.
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https://arxiv.org/abs/2601.10364
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f89050cfad88b2cfb0f0b37baa0961b904d64f2e448b55a030198098551aa820
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2026-01-16T00:00:00-05:00
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Dynamic reinsurance via martingale transport
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arXiv:2601.10375v1 Announce Type: cross Abstract: We formulate a dynamic reinsurance problem in which the insurer seeks to control the terminal distribution of its surplus while minimizing the L2-norm of the ceded risk. Using techniques from martingale optimal transport, we show that, under suitable assumptions, the problem admits a tractable solution analogous to the Bass martingale. We first consider the case where the insurer wants to match a given terminal distribution of the surplus process, and then relax this condition by only requiring certain moment or risk-based constraints.
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https://arxiv.org/abs/2601.10375
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6ed2ac1aa93ea307b3a3fa2d25cab6036964c36aa0b9d4254f4bfba61c24fa2c
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2026-01-16T00:00:00-05:00
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A Predictive Model for Synergistic Oncolytic Virotherapy: Unveiling the Ping-Pong Mechanism and Optimal Timing of Combined Vesicular Stomatitis and Vaccinia Viruses
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arXiv:2601.10405v1 Announce Type: cross Abstract: We present a mathematical model that describes the synergistic mechanism of combined Vesicular Stomatitis Virus (VSV) and Vaccinia Virus (VV). The model captures the dynamic interplay between tumor cells, viral replication, and the interferon-mediated immune response, revealing a `ping-pong' synergy where VV-infected cells produce B18R protein that neutralizes interferon-$\alpha$, thereby enhancing VSV replication within the tumor. Numerical simulations demonstrate that this combination achieves complete tumor clearance in approximately 50 days, representing an 11\% acceleration compared to VV monotherapy (56 days), while VSV alone fails to eradicate tumors. Through bifurcation analysis, we identify critical thresholds for viral burst size and B18R inhibition, while sensitivity analysis highlights infection rates and burst sizes as the most influential parameters for treatment efficacy. Temporal optimization reveals that therapeutic outcomes are maximized through immediate VSV administration followed by delayed VV injection within a 1-19 day window, offering a strategic approach to overcome the timing and dosing challenges inherent in OVT.
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https://arxiv.org/abs/2601.10405
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56cd781bb582c20cfddd1f1d78591f6546704d9ffc4c18c93b4af1614cf16385
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2026-01-16T00:00:00-05:00
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The eigenvalues and eigenvectors of finite-rank normal perturbations of large rotationally invariant non-Hermitian matrices
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arXiv:2601.10427v1 Announce Type: cross Abstract: We study finite-rank normal deformations of rotationally invariant non-Hermitian random matrices. Extending the classical Baik-Ben Arous-P\'ech\'e (BBP) framework, we characterize the emergence and fluctuations of outlier eigenvalues in models of the form $\mathbf{A} + \mathbf{T}$, where $\mathbf{A}$ is a large rotationally invariant non-Hermitian random matrix and $\mathbf{T}$ is a finite-rank normal perturbation. We also describe the corresponding eigenvector behavior. Our results provide a unified framework encompassing both Hermitian and non-Hermitian settings, thereby generalizing several known cases.
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https://arxiv.org/abs/2601.10427
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69a7c34aff3a707b1f710bdb0aa8bbd4bae97e34db8689eb52d583cfe6b1a7fb
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2026-01-16T00:00:00-05:00
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Twisted Cherednik spectrum as a $q,t$-deformation
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arXiv:2601.10500v1 Announce Type: cross Abstract: The common eigenfunctions of the twisted Cherednik operators can be first analyzed in the limit of $q\longrightarrow 1$. Then, the polynomial eigenfunctions form a simple set originating from the symmetric ground state of non-vanishing degree and excitations over it, described by non-symmetric polynomials of higher degrees and enumerated by weak compositions. This pattern is inherited by the full spectrum at $q\neq 1$, which can be considered as a deformation. The whole story looks like a typical NP problem: the Cherednik equations are difficult to solve, but easy to check the solution once it is somehow found.
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https://arxiv.org/abs/2601.10500
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c4aa690c5f8db7f49e5af473c165c0278e01d12de2a662f19456199f6150e2fc
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2026-01-16T00:00:00-05:00
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Madelung hydrodynamics of spin-orbit coupling: action principles, currents, and correlations
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arXiv:2601.10698v1 Announce Type: cross Abstract: We exploit the variational and Hamiltonian structures of quantum hydrodynamics with spin to unfold the correlation and torque mechanisms accompanying spin-orbit coupling (SOC) in electronic motion. Using Hamilton's action principle for the Pauli equation, we isolate SOC-induced quantum forces that act on the orbital Madelung--Bohm trajectories and complement the usual force terms known to appear in quantum hydrodynamics with spin. While the latter spin-hydrodynamic forces relate to the quantum geometric tensor (QGT), SOC-induced orbital forces originate from a particular current operator that contributes prominently to the spin current and whose contribution was overlooked in the past. The distinction between different force terms reveals two fundamentally different mechanisms generating quantum spin-orbit correlations. Leveraging the Hamiltonian structure of the hydrodynamic system, we also elucidate spin transport features such as the current shift in the spin Hall effect and the correlation-induced quantum torques. Finally, we illustrate the framework via the Madelung--Rashba equations for planar SOC configurations and propose a particle-based scheme for numerical implementation.
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https://arxiv.org/abs/2601.10698
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daa5b86d3ea836874316a65c6813956a44a43fcf5e25af8f672015110d66db01
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2026-01-16T00:00:00-05:00
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$\mathfrak{B}$-free integers in number fields and dynamics
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arXiv:1507.00855v2 Announce Type: replace Abstract: In 2010, Sarnak initiated the study of the dynamics of the system determined by the square of the M\"obius function (the characteristic function of the square-free integers). We deal with his program in the more general context of $\mathfrak{B}$-free integers in number fields, suggested 5 years later by Baake and Huck. This setting encompasses the classical square-free case and its generalizations. Given a number field $K$, let $\mathfrak{B}$ be a family of pairwise coprime ideals in its ring of integers $\mathcal{O}_K$, such that $\sum_{\mathfrak{b}\in\mathfrak{B}}1/|\mathcal{O}_K / \mathfrak{b}|<\infty$. We study the dynamical system determined by the set $\mathcal{F}_\mathfrak{B}=\mathcal{O}_K\setminus \bigcup_{\mathfrak{b}\in\mathfrak{B}}\mathfrak{b}$ of $\mathfrak{B}$-free integers in $\mathcal{O}_K$. We show that the characteristic function $\mathbb{1}_{\mathcal{F}_\mathfrak{B}}$ of $\mathcal{F}_\mathfrak{B}$ is generic along the natural F\o{}lner sequence for a probability measure on $\{0,1\}^{\mathcal{O}_K}$, invariant under the multidimensional shift. The corresponding measure-theoretical dynamical system is proved to be isomorphic to an ergodic rotation on a compact Abelian group. In particular, it is of zero Kolmogorov entropy. Moreover, we provide a description of ``patterns'' appearing in $\mathcal{F}_\mathfrak{B}$ and compute the topological entropy of the orbit closure of $\mathbb{1}_{\mathcal{F}_\mathfrak{B}}$. Finally, we show that this topological dynamical system has a non-trivial topological joining with an ergodic rotation on a compact Abelian group.
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https://arxiv.org/abs/1507.00855
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7e4d86c294e9ad5b1d004fa5cbd52ed56db7a783d719f36c7ab7df3172def988
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2026-01-16T00:00:00-05:00
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An invitation to Alexandrov geometry: CAT(0) spaces
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arXiv:1701.03483v4 Announce Type: replace Abstract: Our goal is to show the beauty and power of Alexandrov geometry by reaching interesting applications and theorems with a minimum of preparation. The topics include 1. Reshetnyak's gluing theorem, 2. Estimates on the number of collisions in billiards, 3. Reshetnyak's majorization theorem, 4. Hadamard--Cartan globalization theorem, 5. Polyhedral spaces, 6. Construction of exotic aspherical manifolds, 7. The geometry of two-convex sets in Euclidean space, 8. Barycenters and dimension theory.
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https://arxiv.org/abs/1701.03483
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908f2cd85e54d16624aa6114db928ddf43e1e791ab8db462da7bd51a4bf12737
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2026-01-16T00:00:00-05:00
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Meta-nilpotent knot invariants and symplectic automorphism groups of free nilpotent groups
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arXiv:2105.14414v4 Announce Type: replace Abstract: We develop nilpotently $p$-localization of knot groups in terms of the (symplectic) automorphism groups of free nilpotent groups. We show that any map from the set of conjugacy classes of the outer automorphism groups yields a knot invariant. We also investigate the automorphism groups and compute the resulting knot invariants.
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https://arxiv.org/abs/2105.14414
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7ed0f69b6560480c7ca629234420f2b3c14cd25c82566719f11595e729880f06
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2026-01-16T00:00:00-05:00
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Decreasing subsequences and Viennot for oscillating tableaux
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arXiv:2108.11528v2 Announce Type: replace Abstract: We establish an extension of Viennot's geometric (shadow line) construction to the setting of oscillating tableaux. We then use this to give a new proof of the Type $C$ analogue of Schensted's theorem on longest decreasing subsequences. This pairs with our results from arXiv:2103.14997v1 [math.RT] on Type $C$ webs to give a direct proof of a result of Sundaram and Stanley: that the dimension of the space of invariant vectors in a $2k$-fold tensor product of the vector representation of $\mathfrak{sp}_{2n}$ equals the number of $(n+1)$-avoiding matchings of $2k$ points.
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https://arxiv.org/abs/2108.11528
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e92ac7e69a75df1e5c3d155f95e2176f31ed4f4b6aff4acbc687453f53b4b7c4
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2026-01-16T00:00:00-05:00
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Steenrod Lengths and a Problem of Vakil
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arXiv:2110.01672v3 Announce Type: replace Abstract: We give an explicit combinatorial description of the function $f(n)$ governing the Steenrod length of real projective spaces $\mathbb{RP}^n$. This function arises in stable homotopy theory through the action of Steenrod squares on mod-$2$ cohomology and is closely related to the ghost length, which measures the minimal number of spheres required to construct a space up to homotopy. Building on the directed graphs $T_n$ introduced by Vakil to encode degree constraints for Steenrod operations, we interpret $f(n)$ as the length of the longest directed path starting at $n$. Using this framework, we resolve a question posed by Vakil by deriving concrete combinatorial formulas for $f(n)$ in terms of binary classes and a distinguished family of integers, which we call Vakil numbers.
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https://arxiv.org/abs/2110.01672
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1fcd03452e427ed568424cb94d86d8f7add686308e2561a0bb5b9702dcec28e9
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2026-01-16T00:00:00-05:00
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Surface slices and homology spheres
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arXiv:2202.02696v3 Announce Type: replace Abstract: We develop the theory of the diagrammatics of surface cross sections to prove that there are an infinite number of homology 3-spheres smoothly embeddable in a homology 4-sphere but not in a homotopy 4-sphere. Our primary obstruction comes from work of Taubes.
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https://arxiv.org/abs/2202.02696
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c2654fa8d6dd62f023a5eca3721547faaa01c4cedd4647535ca7f17c7f89a853
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2026-01-16T00:00:00-05:00
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On The large Time Asymptotics of Schr\"odinger type equations with General Data
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arXiv:2203.00724v5 Announce Type: replace Abstract: For the Schr\"odinger equation with a general interaction term, which may be linear or nonlinear, time dependent and including charge transfer potentials, we prove the global solutions are asymptotically given by the sum of a free wave and a weakly localized part. The proof is based on constructing in a new way the Free Channel Wave Operator, and further tools from the recent works \cite{Liu-Sof1,Liu-Sof2,SW2020}. This work generalizes the results of the first part of \cite{Liu-Sof1,Liu-Sof2} to arbitrary dimension, and non-radial data.
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https://arxiv.org/abs/2203.00724
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ed1b04e4f2f6e06f0b454416151f990d9c4033e92abfeaefaf225fd1504a8721
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2026-01-16T00:00:00-05:00
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A Modern Theory for High-dimensional Cox Regression Models
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arXiv:2204.01161v2 Announce Type: replace Abstract: The proportional hazards model has been extensively used in many fields such as biomedicine to estimate and perform statistical significance testing on the effects of covariates influencing the survival time of patients. The classical theory of maximum partial-likelihood estimation (MPLE) is used by most software packages to produce inference, e.g., the coxph function in R and the PHREG procedure in SAS. In this paper, we investigate the asymptotic behavior of the MPLE in the regime in which the number of parameters p is of the same order as the number of samples n. The main results are (i) existence of the MPLE undergoes a sharp 'phase transition'; (ii) the classical MPLE theory leads to invalid inference in the high-dimensional regime. We show that the asymptotic behavior of the MPLE is governed by a new asymptotic theory. These findings are further corroborated through numerical studies. The main technical tool in our proofs is the Convex Gaussian Min-max Theorem (CGMT), which has not been previously used in the analysis of partial likelihood. Our results thus extend the scope of CGMT and shed new light on the use of CGMT for examining the existence of MPLE and non-separable objective functions.
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https://arxiv.org/abs/2204.01161
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0006f3826b8cee5ec9cdcdfbb387d6c218ec208667a705cf348ebbe866b5420e
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2026-01-16T00:00:00-05:00
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Hamiltonicity in generalized quasi-dihedral groups
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arXiv:2204.05484v2 Announce Type: replace Abstract: Witte Morris showed in [21] that every connected Cayley graph of a finite (generalized) dihedral group has a Hamiltonian path. The infinite dihedral group is defined as the free product with amalgamation $\mathbb Z_2 \ast \mathbb Z_2$. We show that every connected Cayley graph of the infinite dihedral group has both a Hamiltonian double ray, and extend this result to all two-ended generalized quasi-dihedral groups.
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https://arxiv.org/abs/2204.05484
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66cfd310717a4ddf96963c8a88c88e25cd9f26d156151e8cfb900ee47fab0bdc
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2026-01-16T00:00:00-05:00
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Extension of Lipschitz maps definable in Hensel minimal structures
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arXiv:2204.05900v4 Announce Type: replace Abstract: In this paper, we establish a theorem on extension of Lipschitz maps $f$ definable in Hensel minimal, non-trivially valued fields $K$ of equicharacteristic zero. This may be regarded as a definable, non-Archimedean, non-locally compact version of Kirszbraun's extension theorem. We proceed with double induction with respect to the dimensions of the ambient space and of the domain of $f$. To this end we introduce the concept of a definable open cell package with a skeleton, which along with the concept of a risometry plays a key role in our induction procedure.
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https://arxiv.org/abs/2204.05900
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31a34ee74795de947a276088bf7a3b4f87aed601b0380ae2d01b04c5b7a3e824
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2026-01-16T00:00:00-05:00
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Planar site percolation via tree embeddings
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arXiv:2304.00923v4 Announce Type: replace Abstract: We prove that if $G$ is an infinite, connected, planar graph properly embedded in $\mathbb{R}^2$ with minimum degree at least $7$, then i.i.d.\ Bernoulli$(p)$ site percolation on $G$ almost surely has infinitely many infinite open (1-)clusters for every \[ p \in \bigl(p_c^{\mathrm{site}},\, 1-p_c^{\mathrm{site}}\bigr). \] Moreover, we show that $p_c^{\mathrm{site}}<\tfrac12$, so this non-uniqueness interval is nonempty. This verifies Conjecture~7 of Benjamini and Schramm~\cite{bs96} for this class of properly embedded planar graphs. Our proof introduces a new construction of embedded trees in $G$. These trees yield infinitely many infinite clusters for percolation parameters near $\tfrac12$, and they also enable exponential decay of two-point connection probabilities by partitioning $G$ using infinitely many disjoint trees. Variants of this approach were later used in~\cite{ZL26} to construct a counterexample to Conjecture~7 of~\cite{bs96} for planar graphs with uncountably many ends. Finally, the methods developed here have further applications: in~\cite{perc24} they are used to prove a vertex-cut characterization of $p_c^{\mathrm{site}}$ (conjectured by Kahn in~\cite{JK03}) and to refute an edge-cut characterization proposed by Lyons and Peres~\cite{LP16} and Tang (\cite{Tang2023}).
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https://arxiv.org/abs/2304.00923
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abfaa1e20dad3ec927441b9a28bf46e0e20a96a065079546d58db459d182f8cd
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2026-01-16T00:00:00-05:00
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Differential characterization of quadratic surfaces
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arXiv:2304.08073v3 Announce Type: replace Abstract: Let $f\in W^{3,1}_{\mathrm{loc}}(\Omega)$ be a function defined on a connected open subset $\Omega\subseteq\mathbb R^2$. We will show that its graph is contained in a quadratic surface if and only if $f$ is a weak solution to a certain system of third-order partial differential equations unless the Hessian determinant of $f$ is non-positive everywhere on $\Omega$. Moreover, we will prove that the system is, in a sense, the simplest possible in a wide class of differential equations, which will lead to the classification of all polynomial partial differential equations satisfied by parametrizations of generic quadratic surfaces. Although we will mainly use the tools of linear and commutative algebra, the theorem itself is also somewhat related to holomorphic functions.
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https://arxiv.org/abs/2304.08073
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ab2b5e636db95c0ba6e50c38f57cc9d161421113cb71a0c31f0b7c735c095953
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2026-01-16T00:00:00-05:00
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Higher Lie theory in positive characteristic
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arXiv:2306.07829v4 Announce Type: replace Abstract: The main goal of this article is to develop integration theory for absolute partition $L_\infty$-algebras, which are point-set models for the (spectral) partition Lie algebras of Brantner-Mathew where infinite sums of operations are well-defined by definition. We construct a Quillen adjunction between absolute partition $L_\infty$-algebras and simplicial sets, and show that the right adjoint is a well-behaved integration functor. Points in this simplicial set are given by solutions to a Maurer-Cartan equation, and we give explicit formulas for gauge equivalences between them. We construct the analogue of the Baker-Campbell-Hausdorff formula in this setting and show it produces an isomorphic group to the classical one over a characteristic zero field. We apply these constructions to show that absolute partition $L_\infty$-algebras encode the $p$-adic homotopy types of pointed connected finite nilpotent spaces, up to certain equivalences which we describe by explicit formulas. In particular, these formulas also allow us to give a combinatorial description of the homotopy groups of the $p$-completed spheres as solutions to a certain equation in a given degree, up to an equivalence relation imposed by elements one degree above. Finally, we construct absolute partition $L_\infty$ models for $p$-adic mapping spaces, which combined with the description of the homotopy groups gives an algebraic description of the homotopy type of these $p$-adic mapping spaces parallel to the unstable Adams spectral sequence.
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https://arxiv.org/abs/2306.07829
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3810f2d784e72ca4ecb812902eb47c9c2ad78962abe24175c3ea432b7693e430
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2026-01-16T00:00:00-05:00
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The cohomology of $BPU(p^m)$ and invariant polynomials
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arXiv:2306.17599v5 Announce Type: replace Abstract: Let $p$ be an odd prime. For a compact Lie group $G$ and an elementary abelian $p$-group $A$ of $G$, one may define the Weyl group $W_A$ of $A$ in a similar fashion as defining the Weyl group of a maximal torus, such that $W_A$ acts on $H^*(BA;R)$ for any coefficient ring $R$, and the image of the restriction $H^*(BG;R)\to H^*(BA;R)$ lies in $H^*(BA;R)^{W_A}$, the sub-algebra of $H^*(BA:R)$ of $W_A$-invariant elements. In this paper, we consider the projective unitary group $PU(p^m)$ and one of its maximal elementary abelian $p$-subgroup $A_m$, of which the Weyl group is isomorphic to $Sp_{2m}(\mathbb{F}_p)$. Then the theory of $Sp_{2m}(\mathbb{F}_p)$-invariant polynomials over $\mathbb{F}_p$ may be applied to study the cohomology of $BPU(p^m)$, the classifying space of $PU(p^m)$. Following a theorem by Quillen, we deduce several theorems on $H^*(BPU(p^m);\mathbb{F}_p)$ modulo the nilradical from results on invariant polynomials.
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https://arxiv.org/abs/2306.17599
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0d02b8409ab64e53ef9b4adf2cc05bfc7fe60fa9ea0b5cd6d62b7c7dc185b2e5
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2026-01-16T00:00:00-05:00
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A non-Archimedean Arens--Eells isometric embedding theorem on valued fields
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arXiv:2309.06704v2 Announce Type: replace Abstract: In 1959, Arens and Eells proved that every metric space can be isometrically embedded into a normed linear space as a closed subset. In later years, in the paper on a short proof of the Arens--Eells theorem, Michael implicitly pointed out that the Arens--Eells theorem follows from the statement that every metric space can be isometrically embedded into a normed linear space as a linearly independent subset. In this paper, we prove a non-Archimedean analogue of the Arens--Eells isometric embedding theorem, which states that for every non-Archimedean valued field $K$, every ultrametric space can be isometrically embedded into a non-Archimedean valued field that is a valued field extension of $K$ such that the image of the embedding is algebraically independent over $K$.
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https://arxiv.org/abs/2309.06704
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f8a6fa34afbadd631f95c2a4b46353060bc97f88b313bfb7dd269c5233390928
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2026-01-16T00:00:00-05:00
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Reformulation of the stable Adams conjecture
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arXiv:2310.14425v3 Announce Type: replace Abstract: We revisit methods of proof of the Adams Conjecture in order to correct and supplement earlier efforts to prove analogous conjectures in the stable homotopy category. We utilize simplicial schemes over an algebraically closed field of positive characteristic and a rigid version of Artin-Mazur \'etale homotopy theory. Consideration of special $\mathcal F$-spaces and together with Bousfield-Kan $\mathbb Z/\ell$-completion enables us to employ an "\'etale functor" which commutes up to homotopy with products of simplicial schemes. In order to prove the Stable Adams Conjecture, we construct the universal $\mathbb Z/\ell$-completed $X$-fibrations for various pointed simplicial sets $X$. Thus, two maps from a given $\mathcal F$-space $\underline{\mathcal B}$ to the base $\mathcal F$-space of the universal $\mathbb Z/\ell$-completed $X$-fibration $\pi_{X,\ell}: \underline {\mathcal B} (G_\ell(X),X_\ell) \to \underline {\mathcal B} G_\ell(X)$ determine homotopy equivalent maps of spectra if and only they correspond via pull-back of $\pi_{X,\ell}$ to fiber homotopy equivalent $\mathbb Z/\ell$-completed $X$-fibrations over $\underline {\mathcal B}$. For the proof of the Stable Adams Conjecture, we consider maps of $\mathcal F$-spaces $\underline {\mathcal B }\to \underline {\mathcal B} G_\ell(S^2)$ where $\underline {\mathcal B}$ is an $\mathcal F$-space model of connective $\ell$-completed connective $K$-theory.
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https://arxiv.org/abs/2310.14425
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cce697505b37e2e3e23350a96c3f1d1a575fb6b2e40c77c27f2e0ec0ba1db3f9
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2026-01-16T00:00:00-05:00
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Distribution-uniform anytime-valid sequential inference and the Robbins-Siegmund distributions
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arXiv:2311.03343v3 Announce Type: replace Abstract: This paper develops a theory of distribution- and time-uniform asymptotics, culminating in the first large-sample anytime-valid inference procedures that are shown to be uniformly valid in a rich class of distributions. Historically, anytime-valid methods -- including confidence sequences, anytime $p$-values, and sequential hypothesis tests -- have been justified nonasymptotically. By contrast, large-sample inference procedures such as those based on the central limit theorem occupy an important part of statistical toolbox due to their simplicity, universality, and the weak assumptions they make. While recent work has derived asymptotic analogues of anytime-valid methods, they were not distribution-uniform (also called \emph{honest}), meaning that their type-I errors may not be uniformly upper-bounded by the desired level in the limit. The theory and methods we outline resolve this tension, and they do so without imposing assumptions that are any stronger than the distribution-uniform fixed-$n$ (non-anytime-valid) counterparts or distribution-pointwise anytime-valid special cases. It is shown that certain ``Robbins-Siegmund'' probability distributions play roles in anytime-valid asymptotics analogous to those played by Gaussian distributions in standard asymptotics. As an application, we derive the first anytime-valid test of conditional independence without the Model-X assumption.
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https://arxiv.org/abs/2311.03343
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434af48b87a060a6ca86392e258a23670ed1e2ef10d8c79652d9f11226de857c
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2026-01-16T00:00:00-05:00
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Euler Product Asymptotics for $L$-functions of Elliptic Curves
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arXiv:2312.05236v4 Announce Type: replace Abstract: Let $E/\mathbb Q$ be an elliptic curve and for each prime $p$, let $N_p$ denote the number of points of $E$ modulo $p$. The original version of the Birch and Swinnerton-Dyer conjecture asserts that $\prod \limits _{p \leq x} \frac{N_p}{p} \sim C (\log x) ^{\text{rank}(E(\mathbb Q))}$ as $x \to \infty$. Goldfeld (1982) showed that this conjecture implies both the Riemann Hypothesis for $L(E, s)$ and the modern formulation of the conjecture i.e. that $\text{ord}_{s=1} L(E, s)= \text{rank}(E(\mathbb Q))$. In this paper, we prove that if we let $r=\text{ord} _{s=1}L(E, s)$, then under the assumption of the Riemann Hypothesis for $L(E, s)$, we have that $\prod \limits _{p \leq x} \frac{N_p}{p} \sim C (\log x)^r$ for all $x$ outside a set of finite logarithmic measure. As corollaries, we recover not only Goldfeld's result, but we also prove a result in the direction of the converse. Our method of proof is based on establishing the asymptotic behaviour of partial Euler products of $L(E, s)$ in the right-half of the critical strip.
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https://arxiv.org/abs/2312.05236
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a673f6d10f3bab4a628c3349840841073a0cec276bce20cd08391cfc35ba14e2
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2026-01-16T00:00:00-05:00
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Mixed-Integer Linear Optimization for Semi-Supervised Optimal Classification Trees
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arXiv:2401.09848v2 Announce Type: replace Abstract: Decision trees are one of the most popular methods for solving classification problems, mainly because of their good interpretability properties. Moreover, due to advances in recent years in mixed-integer optimization, several models have been proposed to formulate the problem of computing optimal classification trees. The goal is, given a set of labeled points, to split the feature spacewith hyperplanes and assign a class to each part of the resulting partition. In certain scenarios, however, labels are only available for a subset of the given points. Additionally, this subset may be non-representative, such as in the case of self-selection in a survey. Semi-supervised decision trees tackle the setting of labeled and unlabeled data and often contribute to enhancing the reliability of the results. Furthermore, undisclosed sources may provide extra information about the size of the classes. We propose a mixed-integer linear optimization model for computing semi-supervised optimal classification trees that cover the setting of labeled and unlabeled data points as well as the overall number of points in each class for a binary classification. Our numerical results show that our approach leads to a better accuracy and a better Matthews correlation coefficient for biased samples compared to other optimal classification trees, even if onlyfew labeled points are available.
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https://arxiv.org/abs/2401.09848
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fb16d13416b0745fe86b901c202f6f37c3672cbb9127e6864266d1b734d26440
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2026-01-16T00:00:00-05:00
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Higher-dimensional multifractal analysis for the cusp winding process on hyperbolic surfaces
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arXiv:2402.16418v2 Announce Type: replace Abstract: We perform a multifractal analysis of the growth rate of the number of cusp windings for the geodesic flow on hyperbolic surfaces with $m \geq 1$ cusps. Our main theorem establishes a conditional variational principle for the Hausdorff dimension spectrum of the multi-cusp winding process. Moreover, we show that the dimension spectrum defined on $\mathbb{R}_{>0}^m$ is real analytic. To prove the main theorem we use a countable Markov shift with a finitely primitive transition matrix and thermodynamic formalism.
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https://arxiv.org/abs/2402.16418
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a4dc52bd3be0bb36729a05405df6485eb7c1bbeb95707ccb1465fa930bb25edc
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2026-01-16T00:00:00-05:00
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A refinement of the Ewens sampling formula
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arXiv:2403.05077v3 Announce Type: replace Abstract: We consider an infinitely-many neutral allelic model of population genetics where all alleles are divided into a finite number of classes, and each class is characterized by its own mutation rate. For this model the allelic composition of a sample taken from a very large population of genes is characterized by a random matrix, and the problem is to describe the joint distribution of the matrix entries. The answer is given by a new generalization of the classical Ewens sampling formula called the refined Ewens sampling formula in the present paper. We discuss a Poisson approximation for the refined Ewens sampling formula, and present its derivation by several methods. As an application we obtain limit theorems for the numbers of alleles in different asymptotic regimes.
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https://arxiv.org/abs/2403.05077
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2eb78856cc959200049e43913f0917e847612e2ff5bef87fa05828286122c591
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2026-01-16T00:00:00-05:00
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Collapsing regular Riemannian foliations with flat leaves
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arXiv:2403.11602v2 Announce Type: replace Abstract: In this manuscript we present how to collapse a manifold equipped with a closed flat regular Riemannian foliation with leaves of positive dimension, while keeping the sectional curvature uniformly bounded from above and below. From this deformation, we show that in the case when the manifold is compact and simply connected the foliation is given by torus actions. This gives a geometric characterization of aspherical regular Riemannian foliations given by torus actions
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https://arxiv.org/abs/2403.11602
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ec6b2c04e15cd19ab3bf3e4c8f6552cbab2415e71ae346756539a7e0d1b46b3f
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2026-01-16T00:00:00-05:00
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Large sieve inequalities for exceptional Maass forms and the greatest prime factor of $n^2+1$
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arXiv:2404.04239v3 Announce Type: replace Abstract: We prove new large sieve inequalities for the Fourier coefficients $\rho_{j\mathfrak{a}}(n)$ of exceptional Maass forms of a given level, weighted by sequences $(a_n)$ with sparse Fourier transforms - including two key types of sequences that arise in the dispersion method. These give the first savings in the exceptional spectrum for the critical case of sequences as long as the level, and lead to improved bounds for various multilinear forms of Kloosterman sums. As an application, we show that the greatest prime factor of $n^2+1$ is infinitely often greater than $n^{1.3}$, improving Merikoski's previous threshold of $n^{1.279}$. We also announce applications to the exponents of distribution of primes and smooth numbers in arithmetic progressions.
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https://arxiv.org/abs/2404.04239
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