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650c7cda2d81187e6e724d2ae7a4a050362e8e0d5fbe51c8fedd27b24321010b
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2026-01-13T00:00:00-05:00
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Classification of single-bubble blow-up solutions for Calogero--Moser derivative nonlinear Schr\"odinger equation
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arXiv:2601.07410v1 Announce Type: new Abstract: We study the Calogero--Moser derivative nonlinear Schr\"odinger equation (CM-DNLS), a mass-critical and completely integrable dispersive model. Recent works established finite-time blow-up constructions and soliton resolution, describing the asymptotic behaviors of blow-up solutions. In this paper, we go beyond soliton resolution and provide a sharp classification of finite-time blow-up dynamics in the \textit{single-bubble} regime. Assuming that a solution blows up at time $0<\infty$ with a single-soliton profile, we determine all possible blow-up rates. For initial data in $H^{2L+1}(\mathbb{R})$ with $L\ge1$, we prove a dichotomy: either the solution lies in a \emph{quantized regime}, where the scaling parameter satisfies \[ \lambda(t)\sim (T-t)^{2k},\qquad 1\le k\le L, \] with convergent phase and translation parameters, or it lies in an \emph{exotic regime}, where the blow-up rate satisfies $\lambda(t)\lesssim (T-t)^{2L+\frac 32}$. To our knowledge, this is the first classification result for quantized blow-up dynamics in the class of dispersive models. We provide a framework for identifying the quantized blow-up rates in classification problems. The proof relies on a modulation analysis combined with the hierarchy of conservation laws provided by the complete integrability of (CM-DNLS). However, it does not use \emph{more refined integrability-based techniques}, such as the inverse scattering method, the method of commuting flows, or the explicit formula. As a result, our analysis applies beyond the chiral solutions.
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https://arxiv.org/abs/2601.07410
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8badc34980695207f86612577e116d7efce8e3d601bbd676806c082d26eaa541
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2026-01-13T00:00:00-05:00
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Critical points of solutions of elliptic equations in divergence form in planar non simply connected domains with smooth or nonsmooth boundary
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arXiv:2601.07412v1 Announce Type: new Abstract: We study the critical points of the solution of second elliptic equations in divergence and diagonal form with a bounded and positive definite coefficient, under the assumption that the statement of the Hopf lemma holds (sign assumptions on its normal derivatives) along the boundary. The proof combines the argument principle introduced in [1] for elliptic equations with the representation formula (using quasi-conformal mappings) for operators in divergence form in simply connected domains [2]. The case of a degenerate coefficient is also treated where we combine the level lines technique and the maximum principle with the argument principle. Finally, some numerical experiments on illustrative examples are presented. [1] G. Alessandrini and R. Magnanini. The index of isolated critical points and solutions of elliptic equations in the plane. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 19(4):567-589, 1992 [2] G. Alessandrini and R. Magnanini. Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions. SIAM J. Math. Anal., 25(5):1259-1268, 1994
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https://arxiv.org/abs/2601.07412
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9eaff1f23e7af0341dda75fb8863d0733f13b6ab8577395255ce266554681288
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2026-01-13T00:00:00-05:00
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Resolution of Erd\H{o}s Problem #728: a writeup of Aristotle's Lean proof
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arXiv:2601.07421v1 Announce Type: new Abstract: We provide a writeup of a resolution of Erd\H{o}s Problem #728; this is the first Erdos problem (a problem proposed by Paul Erd\H{o}s which has been collected in the Erdos Problems website) regarded as fully resolved autonomously by an AI system. The system in question is a combination of GPT-5.2 Pro by OpenAI and Aristotle by Harmonic, operated by Kevin Barreto. The final result of the system is a formal proof written in Lean, which we translate to informal mathematics in the present writeup for wider accessibility. The proved result is as follows. We show a logarithmic-gap phenomenon regarding factorial divisibility: For any constants $0< a+b-n < C_2\log n. \] The argument reduces this to a binomial divisibility $\binom{m+k}{k}\mid\binom{2m}{m}$ and studies it prime-by-prime. By Kummer's theorem, $\nu_p\binom{2m}{m}$ translates into a carry count for doubling $m$ in base $p$. We then employ a counting argument to find, in each scale $[M,2M]$, an integer $m$ whose base-$p$ expansions simultaneously force many carries when doubling $m$, for every prime $p\le 2k$, while avoiding the rare event that one of $m+1,\dots,m+k$ is divisible by an unusually high power of $p$. These "carry-rich but spike-free" choices of $m$ force the needed $p$-adic inequalities and the divisibility. The overall strategy is similar to results regarding divisors of $\binom{2n}{n}$ studied earlier by Erd\H{o}s and by Pomerance.
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https://arxiv.org/abs/2601.07421
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2f00fcf9bb81c68879d9abcdf6950be6104f6d3402f8e66a79222e0f63d36f89
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2026-01-13T00:00:00-05:00
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Log-concavity of solutions of parabolic equations related to the Ornstein-Uhlenbeck operator and applications
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arXiv:2601.07426v1 Announce Type: new Abstract: In this paper, we investigate the log-concavity of the kernel for the parabolic Ornstein-Uhlenbeck operator in a bounded, convex domain. Consequently, we get the preservation of the log-concavity of the initial datum by the related flow. As an application, we give another proof of a Brunn-Minkowski type inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and of the log-concavity of the related first eigenfunction (both results have been proved in [9], by different methods).
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https://arxiv.org/abs/2601.07426
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a5de3ebb41f2b302ea24e806cae9245f8ce248be5bc9ca2adfeffc72f3edbb22
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2026-01-13T00:00:00-05:00
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Cardinal invariants of idealized Miller null sets
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arXiv:2601.07428v1 Announce Type: new Abstract: This paper provides an extensive study of the $\mathscr{I}$-Miller null ideals $M_\mathscr{I}$, $\sigma$-ideals on the Baire space parametrized by ideals $\mathscr{I}$ on countable sets. These $\sigma$-ideals are associated to the idealized versions of Miller forcing in the same way that the meager ideal is associated to Cohen forcing. We compute the cardinal invariants of $M_\mathscr{I}$ for typical examples of Borel ideals $\mathscr{I}$ and show that Cicho\'{n}'s Maximum can be extended by adding the uniformity and covering numbers of $M_\mathscr{I}$ for different ideals $\mathscr{I}$.
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https://arxiv.org/abs/2601.07428
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b87df97a2f10a7e5f82a3bb0ccd85531932c5d0167fcd41401a3d3238f56cec4
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2026-01-13T00:00:00-05:00
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Design of Optimal Controls in Acausal LQG Problems
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arXiv:2601.07433v1 Announce Type: new Abstract: In control theory, a system which has output depending only on the present and past values of the input is said to be causal (or nonanticipative). Respectively, a system is acausal (or non-causal) if its output depends on future inputs as well. Overall majority of literature in stochastic control theory discusses causal systems. Only a few sources indirectly concern acausal systems. In this paper, we systemize these results under main idea of acausality and present a background for designing optimal controls in acausal LQG problems.
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https://arxiv.org/abs/2601.07433
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1f2cd76d2ab4a5124e5529f47d4c039c55cabd8931368dd87ae048a851a91cba
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2026-01-13T00:00:00-05:00
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On the weak coupling limit of the periodic quantum Lorentz gas
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arXiv:2601.07453v1 Announce Type: new Abstract: We report partial progress on the weak coupling limit behavior of observables for the periodic quantum Lorentz gas. Our results indicate that for certain observables, the limit behavior is trivial and can be described via a transport equation, while for other observables, the existence of the limit hinges on the regularity properties at resonant momenta of a certain Bloch-Wigner transform. We are currently unable to prove or disprove this regularity property, and so the weak coupling limit for these observables remains an open question. A novelty of this work is the use of the sewing lemma in the derivation of the kinetic scaling limit for almost every mometum.
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https://arxiv.org/abs/2601.07453
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aa568802830b143bff0d3a00012d9d5f6733fffde1a4c0caf15b4a24c1aeb38a
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2026-01-13T00:00:00-05:00
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Frobenius Number Of Almost Symmetric Numerical Generalized Almost Arithmetic Semigroups
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arXiv:2601.07467v1 Announce Type: new Abstract: Let a, k, h, c be positive integers and d a non zero integer. Recall that a numerical generalized almost arithmetic semigroup S is a semigroup minimally generated by relatively prime positive integers a, ha + d, ha + 2d, . . . , ha + kd, c, that is its embedding dimension is k + 2. In a previous work, the authors described the Ap{\'e}ry set and a Gr{\"o}bner basis of the ideal defining S under one technical assumption, the complete version will be published in a forthcoming paper. In this paper we continue with this assumption and we describe the Pseudo Frobenius set. As a consequence we give a complete description of S when it is symmetric or almost symmetric as well as generalize and extend the previous results of Ignacio Garc{\'i}a-Marco, J. L. Ram{\'i}rez Alfons{\'i}n and O. J. R{{\o}}dseth; we also find a quadratic formula for its Frobenius number that generalizes some results of J.C. Rosales, and P.A. Garc{\'i}a-S{\'a}nchez. Moreover, for given numbers a, d, k, h, c, a simple algorithm allows us to determine if S is almost symmetric or not and furthermore to find its type and Frobenius number.
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https://arxiv.org/abs/2601.07467
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e81618860cad3050a2828161e21042c0ba5ecc865e2ea07105c991ba3903dc95
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2026-01-13T00:00:00-05:00
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Global renormalized solutions to Boltzmann systems modeling mixture gases of monatomic and polyatomic species
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arXiv:2601.07480v1 Announce Type: new Abstract: Inspired by DiPerna-Lions' work \cite{Diperna-Lions}, we study the renormalized solutions to the large-data Cauchy problem of the Boltzmann systems modeling mixture gases of monatomic and polyatomic species, in which the distribution functions $f_\alpha$ characterized the polyatomic species contain the continuous internal energy variable $I \in \mathbb{R}_+$. We first construct the smooth approximated problem and establish the corresponding uniform and physically natural bounds. Then, by employing the averaged velocity (-internal energy) lemma, we can show that the weak $L^1$ limit of the approximated solution is exactly a renormalized solution what we required. Moreover, we also justify that the constructed renormalized solution subjects to the entropy inequality.
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https://arxiv.org/abs/2601.07480
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eb17c955cdb766c53a98ba64fda84abd2dca629d056e299b54e8924e9488fa18
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2026-01-13T00:00:00-05:00
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Online Markov Decision Processes with Terminal Law Constraints
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arXiv:2601.07492v1 Announce Type: new Abstract: Traditional reinforcement learning usually assumes either episodic interactions with resets or continuous operation to minimize average or cumulative loss. While episodic settings have many theoretical results, resets are often unrealistic in practice. The infinite-horizon setting avoids this issue but lacks non-asymptotic guarantees in online scenarios with unknown dynamics. In this work, we move towards closing this gap by introducing a reset-free framework called the periodic framework, where the goal is to find periodic policies: policies that not only minimize cumulative loss but also return the agents to their initial state distribution after a fixed number of steps. We formalize the problem of finding optimal periodic policies and identify sufficient conditions under which it is well-defined for tabular Markov decision processes. To evaluate algorithms in this framework, we introduce the periodic regret, a measure that balances cumulative loss with the terminal law constraint. We then propose the first algorithms for computing periodic policies in two multi-agent settings and show they achieve sublinear periodic regret of order $\tilde O(T^{3/4})$. This provides the first non-asymptotic guarantees for reset-free learning in the setting of $M$ homogeneous agents, for $M > 1$.
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https://arxiv.org/abs/2601.07492
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3269005d6e9d9b125bce2d81c1268b71e94653eb01d496c73fe634aab85cc01e
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2026-01-13T00:00:00-05:00
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On eigenvalues of the Landau Hamiltonian with a periodic electric potential
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arXiv:2601.07495v1 Announce Type: new Abstract: We consider the Landau Hamiltonian $\widehat H_B+V$ on $L^2({\mathbb R}^2)$ with a periodic electric potential $V$. For every $m\in {\mathbb N}$ we prove that there exist nonconstant periodic electric potentials $V\in C^{\infty }({\mathbb R}^2;{\mathbb R})$ with zero mean values that analytically depend on a small parameter $\varepsilon \in {\mathbb R}$ such that the Landau level $(2m+1)B$ is an eigenvalue of the Hamiltonian (of infinite multiplicity) where $B>0$ is a strength of a homogeneous magnetic field.
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https://arxiv.org/abs/2601.07495
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f7dc27ebafeec883675128c32eaaa07b1721f43ade3886c0ca7e2a8c8de232bc
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2026-01-13T00:00:00-05:00
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Phase-field approximation of sharp-interface energies accounting for lattice symmetry
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arXiv:2601.07497v1 Announce Type: new Abstract: We present a phase-field approximation of sharp-interface energies defined on partitions, designed for modeling grain boundaries in polycrystals. The independent variable takes values in the orthogonal group $\mathrm{O}(d)$ modulo a lattice point group $\mathcal{G}$, reflecting the crystallographic symmetries of the underlying lattice. In the sharp-interface limit, the surface energy exhibits a Read-Shockley-type behavior for small misorientation angles, scaling as $\theta|\log\theta|$. The regularized functionals are applicable to grain growth simulation and the reconstruction of grain boundaries from imaging data.
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https://arxiv.org/abs/2601.07497
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d37013fb3c7437f2b7d143fe8b3582ff8be726b0d82681ac5f7501a2bebebc4b
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2026-01-13T00:00:00-05:00
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On Limiting Behaviour of Moves in Multidimensional Elephant Random Walk with Stops
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arXiv:2601.07502v1 Announce Type: new Abstract: In this paper, we discuss the number of moves in multidimensional elephant random walk with stops (MERWS). We obtain the conditional mean increments of the number of moves. Using this conditional result, a recursive relation for the expected number of moves of MERWS is derived and a multiplicative martingale is constructed. The asymptotic behaviour of the solution of this recursive relation is discussed. Later, we discuss several convergence results for the number of moves that include the law of large numbers and law of iterated logarithm.
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https://arxiv.org/abs/2601.07502
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39c8ff3db37900da2e82c291a747eec54be3c1b6f13f4d8b941ddf968894b612
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2026-01-13T00:00:00-05:00
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Gold standard process Markovian poisoning: a semiparametric approach
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arXiv:2601.07503v1 Announce Type: new Abstract: We consider in this paper a stochastic process that mixes in time, according to a nonobserved stationary Markov selection process, two separate sources of randomness: i) a stationary process which distribution is accessible (gold standard); ii) a pure i.i.d. sequence which distribution is unknown (poisoning process). In this framework we propose to estimate, with two different approaches, the transition of the hidden Markov selection process along with the distribution, not supposed to belong to any parametric family, of the unknown i.i.d. sequence, under minimal (identifiability, stationarity and dependence in time) conditions. We show that both estimators provide consistent estimations of the Euclidean transition parameter, and also prove that one of them, which is $\sqrt$ n-consistent, allows to establish a functional central limit theorem about the unknown poisoning sequence cumulative distribution function. The numerical performances of our estimators are illustrated through various challenging examples.
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https://arxiv.org/abs/2601.07503
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f82efa0d15a774118445bd2137ea8ac3d9e61cd01627f9c0f1c27c9af7d32323
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2026-01-13T00:00:00-05:00
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A categorical perspective on extended metric-topological spaces
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arXiv:2601.07505v1 Announce Type: new Abstract: Motivated by the analysis and geometry of metric-measure structures in infinite dimensions, we study the category of extended metric-topological spaces, along with many of its distinguished subcategories (such as the one of compact spaces). One of the main achievements is the proof of the bicompleteness (i.e. of the existence of all small limits and colimits) of the aforementioned categories.
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https://arxiv.org/abs/2601.07505
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9f12ddef10b5605711fde24d0df1e5e43ee4c7bc39d825b97cf6ac389be27359
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2026-01-13T00:00:00-05:00
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Dually cone-boundedness of a set and applications
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arXiv:2601.07517v1 Announce Type: new Abstract: We introduce and study a generalized concept of boundedness of a subset of a normed vector space with respect to a cone, which is defined as lower boundedness of the images of the underlying set through all the positive functionals of the cone. We show that this is a weaker notion when compared to other similar ones and we explore several links with the existing literature. We subsequently demonstrate that this concept furnishes the properties required to obtain various generalizations of important results and techniques, including conic cancellation rules and the R{\aa}dstr\"om embedding procedure.
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https://arxiv.org/abs/2601.07517
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1c96f039b6249a96fbd9f0016283bb8f3eb92e9a31b533b8387f87bd4122c3d9
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2026-01-13T00:00:00-05:00
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Factoriality and Class Groups of Upper Cluster Algebras and Finite Laurent Intersection Rings: A Computational Approach
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arXiv:2601.07520v1 Announce Type: new Abstract: We study factoriality and the class groups of locally acyclic cluster algebras. To do so, we introduce a new class of rings called finite Laurent intersection rings (FLIRs), which includes locally acyclic cluster algebras, full-rank upper cluster algebras, and certain generalized upper cluster algebras and Laurent phenomenon algebras. Our main results are algorithms to compute the class group of an explicit FLIR, to determine factoriality, and to compute all factorizations of a given element. The algorithms are based on multivariate polynomial factorizations, avoiding computationally expensive Gr\"obner basis calculations.
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https://arxiv.org/abs/2601.07520
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ed57019b7c631a6b4e4f919b744814eabda1aa5a6d6c2d09dda2f8795b318b12
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2026-01-13T00:00:00-05:00
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On a Sobolev critical problem for the superposition of a local and nonlocal operator with the "wrong sign''
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arXiv:2601.07521v1 Announce Type: new Abstract: We study a critical problem for an operator of mixed order obtained by the superposition of a Laplacian with a fractional Laplacian. The main novelty is that we consider a mixed operator of the form $-\Delta- \gamma(-\Delta)^s$, namely we suppose that the fractional Laplacian has the ``wrong sign'' and can be seen as a nonlocal perturbation of the purely local case, which is needed to produce a nontrivial solution of the critical problem.
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https://arxiv.org/abs/2601.07521
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de9a192f7118ab7187ca3207438e723211ee7950a97c69d8d521ff5f78bec603
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2026-01-13T00:00:00-05:00
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Non-commutative cluster Lagrangians
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arXiv:2601.07538v1 Announce Type: new Abstract: The space Loc(m,S) of rank m flat bundles on a closed surface S is K_2-symplectic. A threefold M bounding S gives rise a K_2-Lagrangian in Loc(m,S) given by the flat bundles on S extending to M. We generalize this, replacing the zero section in the cotangent bundle to M by certain singular Lagrangians. First, we introduce Q-diagrams in threefolds. They are collections Q of smooth cooriented surfaces, intersecting transversally everywhere but in a finite set of quadruple crossing points. We require that shifting any surface of the collection from such a point in the direction of its coorientation creates a simplex with the cooriented out faces. The Q-diagrams are 3d analogs of bipartite ribbon graphs. Let L be the Lagrangian in the cotangent bundle to M given by the union of the zero section and the conormal bundles to the cooriented surfaces of Q. Let X(L) be the stack of admissible dg-sheaves on M with the microlocal support in L, whose microlocalization at the conormal bundle to each cooriented surface of Q is a rank one local system. We introduce the boundary dL of L. It is a singular Lagrangian in a symplectic space, providing a symplectic stack X(dL), and a restriction functor from X(L) to X(dL). The image of the latter is Lagrangian. We show that, under mild conditions on Q, this Lagrangian has a cluster description, and so it is a K_2-Lagrangian. It also has a simple description in the non-commutative setting.
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https://arxiv.org/abs/2601.07538
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d7922a6bdd5b7e11e6aff59b967ff5affe865abb3e19b561f0d5c1ef496983af
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2026-01-13T00:00:00-05:00
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The holonomy group of a locally symmetric space
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arXiv:2601.07541v1 Announce Type: new Abstract: We show that the holonomy group of a connected Riemannian locally symmetric space (not necessarily complete) without local flat factor is compact and has finite index in its normalizer in the orthogonal group.
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https://arxiv.org/abs/2601.07541
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4896a6d1161d324dcaf83a9915a3a2855a2f6e3434ac140288d286d39c3f2191
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2026-01-13T00:00:00-05:00
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Enumeration of weighted plane trees by a permutation model
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arXiv:2601.07544v1 Announce Type: new Abstract: This work addresses an enumeration problem on weighted bi-colored plane trees with prescribed vertex data, with all vertices labeled distinctly. We give a bijection proof of the enumeration formula originally due to Kochetkov, hence affirmatively answer a question of Adrianov-Pakovich-Zvonkin. The argument is purely combinatorial and totally constructive, remaining valid for real-valued edge weights. A central process is a geometric construction that directly encodes each tree as a permutation. We also exhibit algebraic relationships between the enumeration problem, the partial order on partitions of vertices and the Stirling numbers of the second kind. Some computation examples are presented as appendices.
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https://arxiv.org/abs/2601.07544
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aa8d7de8fb4cd50068e3ffa8cb91d433cffb856ab11e8c6b74271d850fd34d2a
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2026-01-13T00:00:00-05:00
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$A_3$-formality for Demushkin groups at odd primes
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arXiv:2601.07551v1 Announce Type: new Abstract: We study a weak form of formality for differential graded algebras, called $A_3$-formality, for the cohomology of pro-p Demushkin groups at odd primes p. We show that the differential graded $\mathbb{F}_p$-algebras of continuous cochains of Demushkin groups with q-invariant not equal 3 are $A_3$-formal, whereas Demushkin groups with q-invariant 3 are not $A_3$-formal. We prove these results by an explicit computation of the Benson-Krause-Schwede canonical class in Hochschild cohomology.
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https://arxiv.org/abs/2601.07551
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12f79ad36f5aea7f95777cecf2efd9039aa5bbdb6207ffc2ac382a4cb8ecb474
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2026-01-13T00:00:00-05:00
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An introduction to Coxeter polyhedra
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arXiv:2601.07552v1 Announce Type: new Abstract: This paper is an introduction to Coxeter polyhedra in spherical, Euclidean, and hyperbolic geometries. It consists of essentially two parts that could be read independently. In the first we introduce non-obtuse polyhedra in the spherical, Euclidean, and hyperbolic spaces, and prove various fundamental theorems originated from Andreev, Coxeter, and Vinberg. In the second we introduce Coxeter polyhedra and use them to describe regular, semiregular, and uniform polyhedra and tessellations, mostly via the Wythoff construction.
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https://arxiv.org/abs/2601.07552
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f98e03ee7a251902ddbbffc4a0e66360092a4a030ca3e3318080ea77d045117c
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2026-01-13T00:00:00-05:00
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Dynamics of the translation semigroup on directed metric trees
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arXiv:2601.07561v1 Announce Type: new Abstract: The dynamics of the left translation semigroup $\{T_t\}_{t \geq 0}$ on weighted $L^p$ spaces over a directed metric tree $L(G)$ is investigated. Necessary and sufficient conditions on the weight family $\rho$ for the strong continuity of the semigroup are provided. Furthermore, hypercyclicity and weak mixing properties are characterized in terms of the asymptotic decay of $\rho$ along the tree structure. These results generalize classical $L^p$ translation semigroup dynamics to a graph setting.
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https://arxiv.org/abs/2601.07561
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457c993200e1bd1785d126df7276d72ee411cf08c111d208c0d456f3e6802ca8
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2026-01-13T00:00:00-05:00
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Inhomogeneous almost symmetric submanifolds
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arXiv:2601.07564v1 Announce Type: new Abstract: We completely describe inhomogeneous properly embedded almost symmetric submanifolds of Euclidean space as certain unions of parallel symmetric submanifolds of the ambient Euclidean space.
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https://arxiv.org/abs/2601.07564
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de87f50c2116c2d333b13461ac62b31ff21d14da0eb87ce1ac998ff225dd5f6e
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2026-01-13T00:00:00-05:00
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Low-like basis theorems for Ramsey's theorem for pairs in first-order arithmetic
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arXiv:2601.07569v1 Announce Type: new Abstract: We construct an $\ll^2$-solution (also known as a weakly low solution) to ${\mathrm{D}^2}$ within ${\mathrm{B}\Sigma^0_{3}}$ and prove the $\ll^2$-basis theorem for $\mathrm{RT}^2$ over ${\mathrm{B}\Sigma^0_{3}}$. The $\ll^2$-basis theorem is a variant of the low basis theorem, which has recently received focus in the context of the first-order part of Ramsey type theorems. For the construction, we use Mathias forcing in an effectively coded $\omega$-model of $\mathsf{WKL_0}$ to ensure sufficient computability under the system with weaker induction. Using a similar method, we also show the $\ll^2$-basis theorem for $\mathrm{RT}^2_2$ and $\mathrm{EM}_{<\infty}$, a version of Erd\H{o}s-Moser principle, within $\mathrm{I}\Sigma^0_{2}$. These results provide simpler proofs of known results on the $\Pi^1_1$-conservativities of $\mathrm{RT}^2, \mathrm{RT}^2_2$ and $\mathrm{EM}_{<\infty}$ as corollaries.
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https://arxiv.org/abs/2601.07569
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92b4be620a0c372f1bc6cad8473036f6218d3f6d67b73102bb066a2de660ee93
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2026-01-13T00:00:00-05:00
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Exact Fell bundles with the approximation property over inverse semigroups
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arXiv:2601.07572v1 Announce Type: new Abstract: We prove that the reduced cross-sectional algebra of a Fell bundle with the approximation property over an inverse semigroup is exact if and only if the unit fiber of the Fell bundle is exact. This generalizes a recent result of the first-named author for actions of second countable locally compact Hausdorff groupoids on separable $C^*$-algebras. Along the way, we reprove some results of Kwa\'sniewski--Meyer on Fell bundle ideals.
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https://arxiv.org/abs/2601.07572
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d0bfbc403b5351c251d657e7e4b584b2f252c897be4306a68caff3a590d22b85
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2026-01-13T00:00:00-05:00
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The random stable roommates problem typically has no solution
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arXiv:2601.07612v1 Announce Type: new Abstract: Assume that $n = 2k$ potential roommates each have an ordered preference of the $n-1$ others. A stable matching is a perfect matching of the $n$ roommates in which no two unmatched people prefer each other to their matched partners. In their seminal 1962 stable marriage paper, Gale and Shapley noted that not every instance of the stable roommates problem admits a stable matching. In the case when the preferences are chosen uniformly at random, Gusfield and Irving predicted in 1989 that there is no stable matching with high probability for large $n$. We prove this conjecture and show that for $n$ sufficiently large, the probability there is a stable matching is at most $n^{-1/17}$.
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https://arxiv.org/abs/2601.07612
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eb4a4286dd6a61c5dfd44778da7c95550b1af16d369f413f9c020cee90c21ce9
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2026-01-13T00:00:00-05:00
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Preservation of some topological properties under forcing
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arXiv:2601.07624v1 Announce Type: new Abstract: We add to the theory of preservation of topological properties under forcing. In particular, we answer a question of Gilton and Holshouser in a strong sense, showing that if player II has a winning strategy in the strong countable fan tightness game of a space at a point, then this continues to hold in every set forcing extension of the universe. The same is also true for the Rothberger game, but not for the countable fan tightness or Menger games.
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https://arxiv.org/abs/2601.07624
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935b761fc5ff4127d67a022896e77a5514fdbf6a29c967e4b675985bcecfe081
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2026-01-13T00:00:00-05:00
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Rotation of a polytope in another one
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arXiv:2601.07627v1 Announce Type: new Abstract: We are interested in the naive problem whether we can move a solid object in a solid box or not. We restrict move to rotation. In the case we can, the centre and the ``direction'' of rotation may be restricted. Simplifying, we consider possibility of rotation of a polytope within another one of the same dimension and give a criterion for the possibility. Consider the particular case of simplices of the same dimension assuming that the vertices of the inner simplex are contained in different facets of the outer one. Premising further that simplices are even dimensional, rotation is possible in a very general situation. However, in dimension 3, the possible case is not not general. Even in these elementary phenomena, the parity of the dimension seems to yield difference.
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https://arxiv.org/abs/2601.07627
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d9ca08c78a18333481c72e49e95a7ad4b67efeff4c9a12d3744d26b8e6486f03
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2026-01-13T00:00:00-05:00
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Beyond Single-GPU: Scaling PDLP to Distributed Multi-GPU Systems
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arXiv:2601.07628v1 Announce Type: new Abstract: In this work, we present a distributed implementation of the Primal-Dual Hybrid Gradient (PDHG) algorithm for solving massive-scale linear programming (LP) problems. Although PDHG-based solvers have shown strong performance on single-node GPU architectures, their applicability to industrial-scale instances is often limited by GPU memory capacity and computational throughput. To overcome these challenges, we extend the PDHG framework to a distributed-memory setting via a practical two-dimensional grid partitioning of the constraint matrix, enabling scalable execution across multiple GPUs. Our implementation leverages the NCCL communication backend to efficiently synchronize primal-dual updates across devices. To improve load balance and computational efficiency, we introduce a block-wise random shuffling strategy combined with nonzero-aware data distribution, and further accelerate computation through fused CUDA kernels. By distributing both memory and computation, the proposed framework not only overcomes the single-GPU memory bottleneck but also achieves substantial speedups by exploiting multi-GPU parallelism with relatively low communication overhead. Extensive experiments on standard LP benchmarks, including MIPLIB and Hans' instances, as well as large-scale real-world datasets, show that our distributed implementation, built upon cuPDLPx, achieves strong scalability and high performance while preserving full FP64 numerical accuracy.
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https://arxiv.org/abs/2601.07628
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c273999376c85c8f8f939572153064f8ffbd13033fa0945b2a5afd15efca64d2
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2026-01-13T00:00:00-05:00
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Pluripotential theory on algebraic curves
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arXiv:2601.07639v1 Announce Type: new Abstract: In previous works, the second author defined directional Robin constants associated to a compact, nonpolar subset $K$ of an algebraic curve $A$ in $\mathbb{C}^N$ and related these to a natural class of Chebyshev constants for $K$. We define a second class of Chebyshev constants for $K$; relate these two classes; and utilize each of them to define two families of extremal-like functions which can be used to recover the Siciak-Zaharjuta extremal function for $K$.
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https://arxiv.org/abs/2601.07639
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ef30a19cb3494f36f3c2d5ebb4f515a3e600c760e298ff3a26a6bea21864258c
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2026-01-13T00:00:00-05:00
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A new family of hyperbolic slits in the Gabor frame set of B-spline generators
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arXiv:2601.07642v1 Announce Type: new Abstract: We exhibit a new infinite family of hyperbolic curves in the complement of the frame set of Gabor systems with B-spline generators. The proof technique is a combination of an approach by Gr\"ochenig [Partitions of unity and new obstructions for Gabor frames, arXiv:1507.08432, 2015] and a partly partition of unity argument by Nielsen and the author [Counterexamples to the B-spline conjecture for Gabor frames, J. Fourier Anal. Appl., 22(6):1440-1451, 2016]. We relate the new hyperbolic obstructions to the "right bow tie" of the so-called Janssen tie [Zak transforms with few zeros and the tie, In Advances in Gabor analysis, Birkh\"auser, 2003].
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https://arxiv.org/abs/2601.07642
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fc55ada0715eb738e79a6f4de0b34c241854c110132af0c777943e5f481a3568
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2026-01-13T00:00:00-05:00
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On the number of antichains in $\{0,1,2\}^n$
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arXiv:2601.07650v1 Announce Type: new Abstract: We provide precise asymptotics for the number of antichains in the poset $\{0,1,2\}^n$, answering a question of Sapozhenko. Finding improved estimates for this number was also a problem suggested by Noel, Scott, and Sudakov, who obtained asymptotics for the logarithm of the number. Key ingredients for the proof include a graph-container lemma to bound the number of expanding sets in a class of irregular graphs, isoperimetric inequalities for generalizations of the Boolean lattice, and methods from statistical physics based on the cluster expansion.
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https://arxiv.org/abs/2601.07650
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b26142a4c924fcd8ef5e47e93306d43ee8c002be4eabdca5b3eb8bd7e55d0ceb
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2026-01-13T00:00:00-05:00
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To report or not to report: Optimal claim reporting in a bonus-malus system
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arXiv:2601.07655v1 Announce Type: new Abstract: We study an optimal claim reporting problem in a bonus-malus setting. We assume, that the insurance contract consists of two regimes, where reporting a claim leads to a transition to a higher-premium regime, whereas remaining claim-free for a prespecified time period results in a shift to the lower premium regime. The insured can decide whether or not to report an occurred claim. We formulate this as an optimal control problem, where the policyholder follows a barrier-type reporting strategy, with the goal of maximizing the expected value of a function of their terminal wealth. We show that the associated value function is the unique viscosity solution to a system of Hamilton-Jacobi-Bellman equations. This characterization allows us to compute numerical approximations of the optimal barrier strategies.
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https://arxiv.org/abs/2601.07655
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9c5e951b2bdc0826270920dc498df513c8f1497ca8d6aa70e14cf68c848d5144
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2026-01-13T00:00:00-05:00
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Geometry of low nonnegative rank matrix completion
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arXiv:2601.07658v1 Announce Type: new Abstract: We study completion of partial matrices with nonnegative entries to matrices of nonnegative rank at most $r$ for some $r \in \mathbb{N}$. Most of our results are for $r \leq 3$. We show that a partial matrix with nonnegative entries has a nonnegative rank-1 completion if and only if it has a rank-1 completion. This is not true in general when $r \geq 2$. For $3 \times 3$ matrices, we characterize all the patterns of observed entries when having a rank-2 completion is equivalent to having a nonnegative rank-2 completion. If a partial matrix with nonnegative entries has a rank-$r$ completion that is nonnegative, where $r \in \{1,2\}$, then it has a nonnegative rank-$r$ completion. We will demonstrate examples for $r=3$ where this is not true. We do this by introducing a geometric characterization for nonnegative rank-$r$ completion employing families of nested polytopes which generalizes the geometric characterization for nonnegative rank introduced by Cohen and Rothblum (1993).
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https://arxiv.org/abs/2601.07658
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2c32d480fa868d049bb444d11d01a0b8b7fdbc3255cd95145e3d971d29a2e9fc
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2026-01-13T00:00:00-05:00
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Coupled continuity equations for constant scalar curvature K\"ahler metrics
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arXiv:2601.07677v1 Announce Type: new Abstract: Inspired by a parabolic system of Li-Yuan-Zhang and the continuity equation of La Nave-Tian, we study a system of elliptic equations for a K\"ahler metric $\omega$ and a closed $(1, 1)$-form $\alpha$. Assuming a uniform estimate for $\omega$, we prove higher order estimates and smooth convergence to a cscK metric coupled to a harmonic $(1, 1)$-form. A simplification of the system is used to recover existence results for K\"ahler-Einstein metrics when $c_1(X) < 0$. On Riemann surfaces with genus at least $2$, we show smooth convergence to the unique K\"ahler-Einstein metric from a large class of initial data.
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https://arxiv.org/abs/2601.07677
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cd55916293bfbf5cdd974971a588c55d04dec6b5e18731718c4b569145bf7723
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2026-01-13T00:00:00-05:00
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Cross-intersecting families with covering number constraints
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arXiv:2601.07679v1 Announce Type: new Abstract: Two families $\mathcal{F}$ and $\mathcal{G}$ are cross-intersecting if every set in $\mathcal{F}$ intersects every set in $\mathcal{G}$. The covering number $\tau(\mathcal{F})$ of a family $\mathcal{F}$ is the minimum size of a set that intersects every member of $\mathcal{F}$. In 1992, Frankl and Tokushige determined the maximum of $|\mathcal{F}| + |\mathcal{G}|$ for cross-intersecting families $\mathcal{F} \subset \binom{[n]}{a}$ and $\mathcal{G} \subset \binom{[n]}{b}$ that are non-empty (covering number at least 1) and also characterized the extremal configurations. This seminar result was recently extended by Frankl (2024) and Frankl and Wang (2025) to cases where both families are non-trivial (covering number at least 2), and where one is non-empty and the other non-trivial, respectively. In this paper, we establish a unified stability hierarchy for cross-intersecting families under general covering number constraints. We determine the maximum of $|\mathcal{F}| + |\mathcal{G}|$ for cross-intersecting families $\mathcal{F} \subset \binom{[n]}{a}$ and $\mathcal{G} \subset \binom{[n]}{b}$ with the following covering number constraints: (1) $\tau(\mathcal{F}) \geq s$ and $\tau(\mathcal{G}) \geq t$; (2) $\tau(\mathcal{F}) = s$ and $\tau(\mathcal{G}) \geq t \geq 2$; (3) $\tau(\mathcal{F}) \geq s$ and $\tau(\mathcal{G}) = t$; (4) $\tau(\mathcal{F}) = s$ and $\tau(\mathcal{G}) = t$; provided $a \geq b + t - 1$ and $n \geq \max\{a + b, bt\}$. The corresponding extremal families achieving the upper bounds are also characterized.
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https://arxiv.org/abs/2601.07679
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758b26505b19d9a34e4f8d05f78208adf23b8fb224a2c2431a86def50e91cb4e
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2026-01-13T00:00:00-05:00
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Rigidity of the escaping set of polynomial automorphisms of $\mathbb{C}^2$
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arXiv:2601.07681v1 Announce Type: new Abstract: Let $H$ be a polynomial automorphism of $\mathbb{C}^2$ of positive entropy and degree $d \ge 2$. We prove that the escaping set $U^+$ (or equivalently, the non-escaping set $K^+$), of $H$ is rigid under the action of holomorphic automorphisms of $\mathbb{C}^2$. Specifically, every holomorphic automorphism of $\mathbb{C}^2$ that preserves $U^+$ takes the form $L \circ H^s$ where $s \in \mathbb{Z}$ and $L$ belongs to a finite cyclic group of affine maps that preserve the escaping set. Second, note that the sub-level sets $\{G^+ 0$, of the Greens function $G^+$ associated with the map $H$ are canonical examples of Short $\mathbb{C}^2$s. As a consequence of the above theorem, we show that the holomorphic automorphisms of these Short $\mathbb{C}^2$s are affine automorphisms of $\mathbb{C}^2$ preserving the escaping set $U^+$. Hence, the automorphism group of these Short $\mathbb{C}^2$s are the same for every $c>0$ and is a finite cyclic group.
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https://arxiv.org/abs/2601.07681
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27b38515c1e4016f2fd62da52c740722c3f42f9c34967bca95038b759b1e2b11
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2026-01-13T00:00:00-05:00
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Three Combinatorial Algorithms for the Cave Polynomial of a Polymatroid
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arXiv:2601.07697v1 Announce Type: new Abstract: The cave polynomial of a polymatroid was recently introduced and used to study the syzygies of polymatroidal ideals. We study the combinatorial relationships between three formulas for the cave polynomial. As an application, we interpret the Snapper polynomial in terms of these three formulas.
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https://arxiv.org/abs/2601.07697
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6bde888d0484e1dab82ca5ea20e57420b92ee55715b3908677c55dd1035ffbae
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2026-01-13T00:00:00-05:00
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Asymptotic-M\"obius maps
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arXiv:2601.07702v1 Announce Type: new Abstract: We introduce asymptotic-M\"obius (AM) maps, a large-scale analogue of quasi-M\"obius maps tailored to geometric group theory. AM-maps capture coarse cross-ratio behavior for configurations of points that lie far apart, providing a notion of "conformality at infinity" that is stable under quasi-isometries, compatible with scaling limits, and rigid enough to yield structural consequences absent from Pansu's notion of large-scale conformality. We establish basic properties of AM-maps, give several sources of examples, including quasi-isometries, sublinear bi-Lipschitz equivalences, snowflaking, and Assouad embeddings, and apply the theory to large-scale dimension and metric cotype. As applications we obtain dimension-monotonicity results for nilpotent groups and CAT(0) spaces, and new obstructions to the existence of AM-maps arising from metric cotype.
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https://arxiv.org/abs/2601.07702
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511b7708f25f2342772d8caf8e2646b4884e454b1efe30fc27183c54ddac7215
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2026-01-13T00:00:00-05:00
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Topology of domains of discontinuity for Anosov representations via circle actions
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arXiv:2601.07705v1 Announce Type: new Abstract: Among the remarkable properties shared with convex cocompact representations, Anosov representations admit cocompact domains of discontinuity in flag varieties. For representations produced by embedding Fuchsian representations into higher rank Lie groups, these domains are known to admit fiber bundle structures and the structure group is $\operatorname{SO}(2)$. In this article, we determine the equivariant diffeomorphism type of the fiber for these bundles when the domain lives inside a $3$-dimensional complex flag variety. In order to do so, we explicitly work out a smooth version of Fintushel's classification theorem for smooth $S^1$-actions on $4$-manifolds. We show that, in each case, the action on the fiber is equivalent to a circle action on a Hirzebruch surface (or an equivariant connected sum of such actions).
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https://arxiv.org/abs/2601.07705
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76136d5cdecbf9d89ffaddabe3f8cd9369d1473e63ac7cb80bd8ea7b5aa78b99
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2026-01-13T00:00:00-05:00
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A note on extensions of $p$-adic representations of $\mathrm{GL}_2(\mathbb{Q}_p)$
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arXiv:2601.07707v1 Announce Type: new Abstract: We compute extension groups in the category of duals of $p$-adic Banach space representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. Focusing on representations arising from the $p$-adic local Langlands correspondence for generic Galois representations, we classify these extensions completely. These results are then applied to prove the vanishing of extensions between the duals of reducible representations and supercuspidal isotypic components of the \`etale cohomology of the finite level Drinfeld spaces.
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https://arxiv.org/abs/2601.07707
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9dc572d0177b2bfd13defde43a7f3e68c1ad2e59937c400b4004c46bef34fa11
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2026-01-13T00:00:00-05:00
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A note on somewhere positive loops of contactomorphisms
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arXiv:2601.07714v1 Announce Type: new Abstract: In this note, we consider contractible loops of contactomorphisms that are positive over some non-empty closed subset of a contact manifold. Such closed subsets are called immaterial. We argue that the complement of a Reeb-invariant immaterial subset can be seen as big in contact geometric terms. This is supported by two results: one regarding symplectic homology of the filling and the other regarding recently introduced contact quasi-measures.
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https://arxiv.org/abs/2601.07714
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61f7b4b08b5cac2a8cbfa8755efa0279f7120e02c1d17c1a9ea68a6d43970296
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2026-01-13T00:00:00-05:00
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Principal bundles on toroidal compactifications of integral canonical models of abelian-type Shimura varieties
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arXiv:2601.07720v1 Announce Type: new Abstract: In this paper, we construct canonical extensions of principal $\mathcal{G}^c$- (and $M^c$-)bundles on toroidal compactifications of integral canonical models of abelian-type Shimura varieties with hyperspecial levels.
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https://arxiv.org/abs/2601.07720
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3884c6fd02f64c34541000daca57eaf98c88fc7b5fceadb46f49a817af49e96f
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2026-01-13T00:00:00-05:00
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Semisimple algebraic groups over real closed fields
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arXiv:2601.07732v1 Announce Type: new Abstract: We give a self-contained introduction to linear algebraic and semialgebraic groups over real closed fields, and we generalize several key results about semisimple Lie groups to algebraic and semialgebraic groups over real closed fields. We prove that a torus in a semisimple algebraic group is maximal $\mathbb{R}$-split if and only if it is maximal $\mathbb{F}$-split for real closed fields $\mathbb{F}$. For the $\mathbb{F}$-points we formulate and prove the Iwasawa-decomposition $KAU$, the Cartan-decomposition $KAK$ and the Bruhat-decomposition $BWB$. For unipotent subgroups we prove the Baker-Campbell-Hausdorff formula, facilitating the analysis of root groups. We give a proof of the Jacobson-Morozov Lemma about subgroups whose Lie algebra is isomorphic to $\mathfrak{sl}_2$ for algebraic groups and a version for the $\mathbb{F}$-points, when the root system is reduced. We describe the rank 1 subgroups which are the semisimple parts of Levi-subgroups. We prove a semialgebraic version of Kostant's convexity theorem. The main tool used is a model theoretic transfer principle that follows from the Tarski-Seidenberg theorem.
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https://arxiv.org/abs/2601.07732
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d5d65165360cbf90fd30f2247d8e8617b761773d7693ba0eff7263fee2e865ab
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2026-01-13T00:00:00-05:00
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Backward Reconstruction of the Chafee--Infante Equation via Physics-Informed WGAN-GP
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arXiv:2601.07733v1 Announce Type: new Abstract: We present a physics-informed Wasserstein GAN with gradient penalty (WGAN-GP) for solving the inverse Chafee--Infante problem on two-dimensional domains with Dirichlet boundary conditions. The objective is to reconstruct an unknown initial condition from a near-equilibrium state obtained after 100 explicit forward Euler iterations of the reaction-diffusion equation \[ u_t - \gamma\Delta u + \kappa\left(u^3 - u\right)=0. \] Because this mapping strongly damps high-frequency content, the inverse problem is severely ill-posed and sensitive to noise. Our approach integrates a U-Net generator, a PatchGAN critic with spectral normalization, Wasserstein loss with gradient penalty, and several physics-informed auxiliary terms, including Lyapunov energy matching, distributional statistics, and a crucial forward-simulation penalty. This penalty enforces consistency between the predicted initial condition and its forward evolution under the \emph{same} forward Euler discretization used for dataset generation. Earlier experiments employing an Eyre-type semi-implicit solver were not compatible with this residual mechanism due to the cost and instability of Newton iterations within batched GPU training. On a dataset of 50k training and 10k testing pairs on $128\times128$ grids (with natural $[-1,1]$ amplitude scaling), the best trained model attains a mean absolute error (MAE) of approximately \textbf{0.23988159} on the full test set, with a sample-wise standard deviation of about \textbf{0.00266345}. The results demonstrate stable inversion, accurate recovery of interfacial structure, and robustness to high-frequency noise in the initial data.
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https://arxiv.org/abs/2601.07733
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d77432b462a149716bd37d31b8d01ba7e27511106781be31821b09be08dd5dd9
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2026-01-13T00:00:00-05:00
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Anticoncentration of random spanning trees in almost regular graphs
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arXiv:2601.07740v1 Announce Type: new Abstract: The celebrated formula of Otter \emph{[Ann. of Math. (2) 49 (1948), 583--599]} asserts that the complete graph contains exponentially many non-isomorphic spanning trees. In this paper, we show that every connected almost regular graph with sufficiently large degree already contains exponentially many non-isomorphic spanning trees. Indeed, we prove a stronger statement: for every fixed $n$-vertex tree $T$, $$ \Pr\bigl[\mathcal{T} \simeq_{\mathrm{iso}} T\bigr] = e^{-\Omega(n)}, $$ where $\mathcal{T}$ is a uniformly random spanning tree of a connected $n$-vertex almost regular graph with sufficiently large degree. To prove this, we introduce a graph-theoretic variant of the classical balls--into--bins model, which may be of independent interest.
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https://arxiv.org/abs/2601.07740
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a74a2885f52b1672bc5fa50c0bd425ce05022b73eb89315a04a35e8b30cad5a6
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2026-01-13T00:00:00-05:00
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Subprincipal Control of Pseudospectral Quasimodes, II
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arXiv:2601.07743v1 Announce Type: new Abstract: In this paper, we continue the analysis of the effects of semiclassical sub principal controlled quasimodes, approximate solutions to P(h)u(h,b), depending on the subprincipal symbol b, which can give spectral insta bility (pseudospectrum). We consider a pseudodifferential operator, which has double zeros for the principal symbol, p. This means that p = dp = 0 in a small neighborhood. In the first paper in this series, we considered operators with transversal inter sections of bicharacteristics. Now we study operators with tangential in tersections of bicharacteristics, as well as with double characteristics for p. We put the pseudodifferential operator on normal form microlocally, and use a model operator, P(h) to test for quasimodes. We demonstrate two cases where this happens. We shall also continue with more advanced cases, when the operators are factorable to P(h) = P2(h)P1(h,B), thus annihilating the subprincipal control over the quasimodes.
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https://arxiv.org/abs/2601.07743
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c19f1bf820de9c49d792a5317cbf0e1040980633b7de81dd3b889c972b05e909
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2026-01-13T00:00:00-05:00
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MinDist is less than 7
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arXiv:2601.07746v1 Announce Type: new Abstract: The metric MinDist, introduced recently to quantify the distance of an arbitrary Rummy hand from a valid declaration, plays a central role in algorithmic hand evaluation and optimal play. Existing results show that the MinDist of any $13$-card Rummy hand from a single deck is bounded above by $9$. In this paper, we sharpen this bound and prove that the MinDist of any hand is at most $7$. We further show that this bound is tight by explicitly exhibiting a hand whose MinDist equals $7$ for a suitable choice of wildcard joker. The proof combines elementary combinatorial arguments with structural properties of card partitions across suits and resolves the gap between the previously known upper bound and the true extremal value.
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https://arxiv.org/abs/2601.07746
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928a8896d905c9d264bd8fc52dd9fa7a70adcf07cc6ee154402719f6b5003adf
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2026-01-13T00:00:00-05:00
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Monotonicity and a Taylor approximation theorem for transseries
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arXiv:2601.07747v1 Announce Type: new Abstract: We show that the composition of omega-series by surreal numbers, or more generally by elements of any confluent field of transseries, is monotonic in its second argument. In particular, omega-series and LE-series interpreted as functions on positive infinite omega-series, or respectively LE-series, have the intermediate value property. We also deduce a Taylor approximation theorem for omega-series with maximal radius of validity.
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https://arxiv.org/abs/2601.07747
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9aa74245d834e189666c6a6d6ce887c25d0bfbb51eea140f864636cbb7904229
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2026-01-13T00:00:00-05:00
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Central polynomials of minimal degree for matrices
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arXiv:2601.07750v1 Announce Type: new Abstract: Formanek made the conjecture that the minimal degree of the central polynomials for the $n\times n$ matrix algebra over a field of characteristic 0 is $(n^2+3n-2)/2$ and this is true for $n\leq 3$. For $n=4$ there are examples of central polynomials of degree $13=(4^2+3\cdot 4-2)/2$ and we do not know whether there are central polynomials of lower degree. In this paper we discuss methods for searching for central polynomials of low degree and prove that the algebra of $4\times 4$ matrices does not have central polynomials in two variables of degree $\leq 12$. As a byproduct of our computations we obtain that this algebra does not have also polynomial identities in two variables of degree $\leq 12$.
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https://arxiv.org/abs/2601.07750
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b81cc18a7f5c1a10281306c063ab9fee60876631c532cb73ee791be1285f4fab
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2026-01-13T00:00:00-05:00
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Real critical points of $T$-polynomials that are sums of squared monomials and topology of $T$-hypersurfaces
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arXiv:2601.07751v1 Announce Type: new Abstract: We study the topology of the real algebraic hypersurfaces in $\mathbb{P}^n$ that can be constructed via combinatorial patchworking using triangulations that are dilations by two of other triangulations. By examining the real critical points of the polynomials that define such hypersurfaces, we find some asymptotical upper bounds on various sums of their Betti numbers. We then discuss the sharpness of those bounds.
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https://arxiv.org/abs/2601.07751
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899d2c5d2a2d9af35e7aa5f10c6e40fac48e50c6b59a70651375c61257223bde
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2026-01-13T00:00:00-05:00
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The value of random zero-sum games
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arXiv:2601.07759v1 Announce Type: new Abstract: We study the value of a two-player zero-sum game on a random matrix $M\in \mathbb{R}^{n\times m}$, defined by $v(M) = \min_{x\in\Delta_n}\max_{y\in \Delta_m}x^T M y$. In the setting where $n=m$ and $M$ has i.i.d. standard Gaussian entries, we prove that the standard deviation of $v(M)$ is of order $\frac{1}{n}$. This confirms an experimental conjecture dating back to the 1980s. We also investigate the case where $M$ is a rectangular Gaussian matrix with $m = n+\lambda\sqrt{n}$, showing that the expected value of the game is of order $\frac{\lambda}{n}$, as well as the case where $M$ is a random orthogonal matrix. Our techniques are based on probabilistic arguments and convex geometry. We argue that the study of random games could shed new light on various problems in theoretical computer science.
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https://arxiv.org/abs/2601.07759
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e8196daa31de3c23f87d6b6cae5ebcea60f923bb36d542e2e54a47d29d193175
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2026-01-13T00:00:00-05:00
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Comparing three learn-then-test paradigms in a multivariate normal means problem
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arXiv:2601.07764v1 Announce Type: new Abstract: Many modern procedures use the data to learn a structure and then leverage it to test many hypotheses. If the entire data is used at both stages, analytical or computational corrections for selection bias are required to ensure validity (post-learning adjustment). Alternatively, one can learn and/or test on masked versions of the data to avoid selection bias, either via information splitting or null augmentation}. Choosing among these three learn-then-test paradigms, and how much masking to employ for the latter two, are critical decisions impacting power that currently lack theoretical guidance. In a multivariate normal means model, we derive asymptotic power formulas for prototypical methods from each paradigm -- variants of sample splitting, conformal-style null augmentation, and resampling-based post-learning adjustment -- quantifying the power losses incurred by masking at each stage. For these paradigm representatives, we find that post-learning adjustment is most powerful, followed by null augmentation, and then information splitting. Moreover, null augmentation can be nearly as powerful as post-learning adjustment, while avoiding its challenges: the power of the former approaches that of the latter if the number of nulls used for augmentation is a vanishing fraction of the number of hypotheses. We also prove for a tractable proxy that the optimal number of nulls scales as the square root of the number of hypotheses, challenging existing heuristics. Finally, we characterize optimal tuning for information splitting by identifying an optimal split fraction and tying it to the difficulty of the learning problem. These results establish a theoretical foundation for key decisions in the deployment of learn-then-test methods.
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https://arxiv.org/abs/2601.07764
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c6cdcb8b9c9c54096b2b03a33dd6a4c902b16d631989fe98787cd43c07ec022a
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2026-01-13T00:00:00-05:00
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On the well-posedness of the initial value problem for the MMT model
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arXiv:2601.07771v1 Announce Type: new Abstract: This work investigates the initial value problem (IVP) for the two-parameter family of dispersive wave equations known as the Majda-McLaughlin-Tabak (MMT) model, which arises in the weak turbulence theory of random waves. The MMT model can be viewed as a derivative nonlinear Schr\"odinger (dNLS) equation where both the nonlinearity and dispersion involve nonlocal fractional derivatives. The purpose of this study is twofold: first, to establish a sharp well-posedness theory for the MMT model; and second, to identify the critical threshold for the derivative in the nonlinearity relative to the dispersive order required to ensure well-posedness. As a by- product, we establish sharp well-posedness for non-local fractional dNLS equations; notably, our results resolve the regularity endpoint left open in https://www.aimsciences.org/article/doi/10.3934/dcdsb.2022039 .
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https://arxiv.org/abs/2601.07771
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415323e18d02c2b7179e3285cf8bbdc264525454097a75e89c5358b87c0ec593
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2026-01-13T00:00:00-05:00
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On wall-crossing coordinates in Cerf theory
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arXiv:2601.07776v1 Announce Type: new Abstract: We relate Bruhat numbers in real Morse theory to cluster variables in braid varieties. This provides instances of wall-crossing coordinates in the study of Cerf diagrams.
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https://arxiv.org/abs/2601.07776
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866fcd2706cbe563b759ce19200b78453860fb1dd7789501a2403f696eb98f8c
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2026-01-13T00:00:00-05:00
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Combinatorial invariance for the coefficient of $q$ in Kazhdan-Lusztig polynomials
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arXiv:2601.07793v1 Announce Type: new Abstract: We prove the combinatorial invariance of the coefficient of $q$ in Kazhdan-Lusztig polynomials for arbitrary Coxeter groups. As a result, we obtain the Combinatorial Invariance Conjecture for Bruhat intervals of length at most $6$. We also prove the Gabber-Joseph conjecture for the second-highest $\mathrm{Ext}$ group of a pair of Verma modules, as well as the combinatorial invariance of the dimension of this group.
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https://arxiv.org/abs/2601.07793
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0f9331585013a8575e63a094a6fc0c52852d2f31fe8ae8810b938b2215c42611
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2026-01-13T00:00:00-05:00
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Critical level-set percolation on finite graphs and spectral gap
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arXiv:2601.07802v1 Announce Type: new Abstract: We study the bond percolation on finite graphs induced by the level-sets of zero-average Gaussian free field on the associated metric graph above a given height (level) parameter $h \in \mathbb{R}$. We characterize the near- and off-critical phases of this model for any expanders family $\mathcal{G}_n = (V_n, E_n)$ with uniformly bounded degrees. In particular, we show that the volume of the largest open cluster at level $h_n$ is of the order $|V_n|^{\frac23}$ when $h_n$ lies in the corresponding critical window which we identify as $|h_n| = O(|V_n|^{-\frac13})$. Outside this window, the volume starts to deviate from $\Theta(|V_n|^{\frac23})$ culminating into a linear order in the supercritical phase $h_n = h 0$. We deduce these from effective estimates on tail probabilities for the maximum volume of an open cluster at any level $h$ for a generic base graph $\mathcal{G}$. The estimates depend on $\mathcal{G}$ only through its size and upper and lower bounds on its degrees and spectral gap respectively. To the best of our knowledge, this is the first instance where a mean-field critical behavior is derived under such general setup for finite graphs. The generality of these estimates preclude any local approximation of $\mathcal{G}$ by regular infinite trees -- a standard approach in the area. Instead, our methods rely on exploiting the connection between spectral gap of the graph $\mathcal{G}$ and its connection to the level-sets of zero-average Gaussian free field mediated via a set function we call the zero-average capacity.
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https://arxiv.org/abs/2601.07802
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0264f2a547c46669dad745017d52b398eba60d6f12ad4b8c31c0f4efa89a1d5f
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2026-01-13T00:00:00-05:00
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Adventures of Harish-Chandra in $\mathbb Z_2 \times \mathbb Z_2$-graded world
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arXiv:2601.07803v1 Announce Type: new Abstract: We study $\mathbb Z_2\times\mathbb Z_2$ bi-graded Lie algebras. We describe their properties in relation to Lie superalgebras with some compatible structures. Then we focus on the approach to the Lie group--algebra correspondence based on Harish-Chandra pairs and provide some examples of application of it in the bi-graded setting.
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https://arxiv.org/abs/2601.07803
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57890af4dfd0d5a64dc7ee1306a949c0f3f6cc4aab660fb4a85dc6c128dde91c
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2026-01-13T00:00:00-05:00
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Foundations of local iterated function systems
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arXiv:2601.07804v1 Announce Type: new Abstract: In this paper we present a systematic study of continuous local iterated function systems. We prove local iterated function systems admit compact attractors and, under a contractivity assumption, construct their code space and present an extended shift that describes admissible compositions. In particular, the possible combinatorial structure of a local iterated function system is in bijection with the space of invariant subsets of the full shift. Nevertheless, these objects reveal a degree of unexpectedness relative to the classical framework, as we build examples of local iterated function systems which are not modeled by subshifts of finite type and give rise to non self-similar attractors. We also prove that all attractors of graph-directed IFSs are obtained from local IFSs on an enriched compact metric space. We provide several classes of examples illustrating the scope of our results, emphasizing both their contrasts and connections with the classical theory of iterated function systems.
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https://arxiv.org/abs/2601.07804
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82d64b20ce9750646e621bc38748a9aa7d1f48eb9f87434b429cb1368710651b
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2026-01-13T00:00:00-05:00
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Double Categorical Approaches to AQFT I: Axiomatic Setup
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arXiv:2601.07807v1 Announce Type: new Abstract: In operator-algebraic AQFT one routinely moves back and forth between two kinds of structure: inclusions of local algebras coming from inclusions of regions, and bimodules/intertwiners that implement the standard $L^2$-based constructions used to compare and compose observables. The obstruction to making this interplay genuinely functorial is that there are two independent compositions (restriction along inclusions and fusion/transport along bimodules) and they must be compatible on commuting spacetime diagrams, which is exactly the situation a double category is designed to encode. Part I resolves this by building a spacetime double category and a von Neumann algebra double category inspired by previous work by Orendain, and by packaging an AQFT input as a pseudo double functor whose vertical part is the Haag-Kastler net and whose squares record the required compatibilities in a well-typed way forced by commutativity. We formulate the Haag-Kastler axioms in this setup, establish the coherence needed for the construction, and work out representative examples.
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https://arxiv.org/abs/2601.07807
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ea0e8b745f11dada376d656da967f6da834750ae02c1072da45c56747ab94cf3
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2026-01-13T00:00:00-05:00
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Supercritical long-range percolation on graphs of polynomial growth: the truncated one-arm exponent
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arXiv:2601.07808v1 Announce Type: new Abstract: We consider supercritical long-range percolation on transitive graphs of polynomial growth. In this model, any two vertices $x$ and $y$ of the underlying graph $G$ connect by a direct edge with probability $1-\exp(-\beta J(x,y))$, where $J(x,y)$ is a function that is invariant under the automorphism group of $G$, and we assume that $J$ decays polynomially with the graph distance between $x$ and $y$. We give up-to-constant bounds on the decay of the radius of finite cluster for $\beta > \beta_c$. In the same setting, we also give upper and lower bounds on the tail volume of finite clusters. The upper and lower bounds are of matching order, conjecturally on sharp volume bounds for spheres in transitive graphs of polynomial growth. As a corollary, we obtain a lower bound on the anchored isoperimetric dimension of the infinite component.
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https://arxiv.org/abs/2601.07808
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a21ec53471f6111f0d21e47acb3d9dd271729b30995310373b097bbe9725494e
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2026-01-13T00:00:00-05:00
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On curves of degree 10 with 12 triple points
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arXiv:2601.07809v1 Announce Type: new Abstract: We construct an irreducible rational curve of degree 10 in $CP^2$ which has 12 triple points. This gives a counter-example to a conjecture by Dimca, Harbourne, and Sticlaru. We also prove that there exists an analytic family $C_u$ of such curves which converges as $u\to 0$ to the union of the dual Hesse arrangement of lines (9 lines with 12 triple points) with an additional line. We hope that our approach to the proof of the latter fact could be of independent interest.
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https://arxiv.org/abs/2601.07809
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2aafddfb3bb36e85e89a2b3a334c4a7ca4160a4482c4b9eae68b287b3eac8641
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2026-01-13T00:00:00-05:00
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Finiteness of complete intersection dimensions of RHom and Ext modules
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arXiv:2601.07811v1 Announce Type: new Abstract: In this paper, we explore the implications of the finiteness of complete intersection dimensions for RHom complexes and Ext modules. We prove various stability results and criteria for detecting finite complete intersection homological dimension of complexes and modules. In addition, we introduce and explore the concept of CI-perfect modules. We also study the vanishing of Ext when certain Hom module have finite complete intersection homological dimension. In this direction, we improve a result by Ghosh and Samanta, prove the Auslander-Reiten conjecture for finitely generated modules $M$ over a Noetherian local ring $R$ such that $\operatorname{Hom}_R(M,R)$ or $\operatorname{Hom}_R(M,M)$ has finite complete intersection injective dimension, and provide Gorenstein criteria.
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https://arxiv.org/abs/2601.07811
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fa8c74a6a027bdc19eb25936e8ac1cfe7801da5a9facd261d6ebd0d632a4caa6
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2026-01-13T00:00:00-05:00
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Convergence and turnpike properties of linear-quadratic mean field control problems with common noise
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arXiv:2601.07815v1 Announce Type: new Abstract: We investigate convergence and turnpike properties for linear-quadratic mean field control problems with common noise. Within a unified framework, we analyze a finite-horizon social optimization problem, its mean field control limit, and the corresponding ergodic mean field control problem. The finite-horizon problems are characterized by coupled Riccati differential equations, whereas the ergodic problem is addressed via a Bellman equation on the Wasserstein space, which reduces to a system of stabilizing algebraic Riccati equations. By deriving estimates for these Riccati systems, we establish a turnpike property for the finite-horizon mean field control problem and obtain quantitative convergence results from the social optimization problem to its mean field limit and the associated ergodic control problem.
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https://arxiv.org/abs/2601.07815
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01b5a8a9a5fff58dda1092670b468bd44de73689c5e079091cc29ae715002e49
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2026-01-13T00:00:00-05:00
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Local Density of Activated Random Walk on $\mathbb{Z}$
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arXiv:2601.07816v1 Announce Type: new Abstract: We consider one-dimensional activated random walk (ARW) on $\mathbb{Z}$ started from a `point source' initial condition, with many particles at the origin and no other particles. We prove that, uniformly throughout a macroscopic window around the source, the probability that a site contains a sleeping particle after the configuration is stabilized is approximately the critical density. This represents a first step towards understanding the local structure of the critical stationary measure for ARW.
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https://arxiv.org/abs/2601.07816
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1623b69e0f2cf1c79395fe2fcb00d1caf9826d5ba256148084cc99e681af31f6
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2026-01-13T00:00:00-05:00
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Counting Square-full Solutions to $x+y=z$
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arXiv:2601.07817v1 Announce Type: new Abstract: We show that there are $O(B^{3/5-3/1555+\ep})$ triples $(x,y,z)$ of square-full integesr up to $B$ satisfying the equation $x+y=z$ for any fixed $\ep>0$. This is the first improvement over the `easy' exponent $3/5$, given by Browning and Van Valckenborgh. One new tool is a strong uniform bound for the counting function for equations $aX^3+bY^3=cZ^3$.
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https://arxiv.org/abs/2601.07817
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137188b27f58e4e0a57bdb514f7272e2be50503a2a500606c1183dd886f81449
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2026-01-13T00:00:00-05:00
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On the range of two-distance graphs
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arXiv:2601.07828v1 Announce Type: new Abstract: The topic of this paper is related to the well-known notion of unit distance graphs. Take a graph with its edges coloured red and blue such that for some $d$ it can be mapped into the plane with all vertices going to distinct points, the red edges to segments of length $1$ and the blue edges to segments of length $d$. We define the range of this graph to be the set of such numbers $d$. It is easy to show that the range of any edge-bicoloured graph consists of finitely many intervals with algebraic endpoints, and we now prove that any such set with a finite positive upper and lower bound is the range of a suitable graph.
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https://arxiv.org/abs/2601.07828
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ae7f0232417a635a4a5c14da19b369f3fdb8955f3dec3e18cc06456976ab7885
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2026-01-13T00:00:00-05:00
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A Complete Decomposition of Stochastic Differential Equations
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arXiv:2601.07834v1 Announce Type: new Abstract: We show that any stochastic differential equation with prescribed time-dependent marginal distributions admits a decomposition into three components: a unique scalar field governing marginal evolution, a symmetric positive-semidefinite diffusion matrix field and a skew-symmetric matrix field.
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https://arxiv.org/abs/2601.07834
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4e4d349b9f170061c424d1085fffa53f514267cf348167ea938d3b83729caf1d
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2026-01-13T00:00:00-05:00
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Integrable Systems
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arXiv:2601.04077v1 Announce Type: cross Abstract: These notes are based on lecture courses I gave to third year mathematics students at Cambridge. They could form a basis of an elementary one--term lecture course on integrable systems covering the Arnold-Liouville theorem, inverse scattering transform, Hamiltonian methods in soliton theory and Lie point symmetries. No knowledge beyond basic calculus and ordinary differential equation is assumed.
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https://arxiv.org/abs/2601.04077
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25c11403bf40a1323f14ed2e2d147fb632849218ffe2da038fdb8c028ae10b2d
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2026-01-13T00:00:00-05:00
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A Clarifying Note on Long-Horizon Investment and Dollar-Cost Averaging: An Effective Investment Exposure Perspective
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arXiv:2601.06074v1 Announce Type: cross Abstract: It is widely claimed in investment education and practice that extending the investment horizon reduces risk, and that diversifying investment timing, for example through dollar-cost averaging (DCA), further mitigates investment risk. Although such claims are intuitively appealing, they are often stated without precise definitions of risk or a clear separation between risk and uncertainty. This paper revisits these two beliefs within a unified probabilistic framework. We define risk at the expectation level as a property of the generating distribution of cumulative investment outcomes, and distinguish it from uncertainty, understood as the dispersion of realized outcomes across possible paths. To enable meaningful comparisons across horizons and investment schedules, we introduce the notion of effective investment exposure, defined as time-integrated invested capital. Under stationary return processes with finite variance, we show that extending the investment horizon does not alter expected risk, expected return, or the risk-return ratio on a per-unit-exposure basis. In contrast, different investment timing strategies can induce distinct exposure profiles over time. As a result, lump-sum investment and dollar-cost averaging may differ not only in uncertainty but also in expected risk when compared at equal return exposure, although the resulting risk differences are of constant order and do not grow with the investment horizon. These results clarify why common narratives surrounding long-horizon investment and dollar-cost averaging are conceptually misleading, while also explaining why adopting such strategies under budgetary or timing constraints need not be regarded as irrational.
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https://arxiv.org/abs/2601.06074
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15833a4366e5e7296c9154a2d1089b6a44487b4def3b435fe47cdc66f45c9cba
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2026-01-13T00:00:00-05:00
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Bipartitioning of Graph States for Distributed Measurement-Based Quantum Computing
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arXiv:2601.06332v1 Announce Type: cross Abstract: Measurement-Based Quantum Computing (MBQC) is inherently well-suited for Distributed Quantum Computing (DQC): once a resource state is prepared and distributed across a network of quantum nodes, computation proceeds through local measurements coordinated by classical communication. However, since non-local gates acting on different Quantum Processing Units (QPUs) are a bottleneck, it is crucial to optimize the qubit assignment to minimize inter-node entanglement of the shared resource. For graph state resources shared across two QPUs, this task reduces to finding bipartitions with minimal cut rank. We introduce a simulated annealing-based algorithm that efficiently updates the cut rank when two vertices swap sides across a bipartition, such that computing the new cut rank from scratch, which would be much more expensive, is not necessary. We show that the approach is highly effective for determining qubit assignments in distributed MBQC by testing it on grid graphs and the measurement-based Quantum Approximate Optimization Algorithm (QAOA).
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https://arxiv.org/abs/2601.06332
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8be35e71381150a377d29636832b68efa9bf198054a60c2f204d30cf4748c760
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2026-01-13T00:00:00-05:00
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Designing a Resilient Allee-Ornstein-Uhlenbeck model
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arXiv:2601.06354v1 Announce Type: cross Abstract: In stochastic population dynamics, stochastic wandering can produce transition to an absorbing state. In particular, under Allee effects, low densities amplify the possibility of population collapse. We investigate this in an Allee-Ornstein-Uhlenbeck (Allee-OU) model, that couples a bistable Allee growth equation, with demographic noise, and environmental fluctuations modeled as an Ornstein-Uhlenbeck process. This process replaces the bifurcation parameter of the deterministic Allee effect equation. In the model, small noise may induce escape from the safe basin around the positive equilibrium toward extinction. We construct a stochastic control, altering the process to have a stationary distribution. We enable tractable control design, approximating the process by one with a stationary distribution. Two controlled models are developed, one acting directly on population size and another also modulating the environment. A threshold-based implementation minimizes the frequency of interventions while maximizing safe time. Simulations demonstrate that the control stabilizes fluctuations around the equilibrium.
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https://arxiv.org/abs/2601.06354
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75cca391ea3519defa22939f5463a6d082f1b2e9633ef3e0669b2c4f65d88312
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2026-01-13T00:00:00-05:00
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Exact Solutions for Compact Support Parabolic and Landau Barriers
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arXiv:2601.06369v1 Announce Type: cross Abstract: We derive exact solutions to the one-dimensional Schr\"odinger equation for compact support parabolic and hyperbolic secant potential barriers, along with combinations of these types of potential barriers. We give the expressions for transmission and reflection coefficients and calculate some dwell times of interest
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https://arxiv.org/abs/2601.06369
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534f9b9d0af42fb101b88e6b5a32ff913cd2375dcb730f4941ff4fbee5c53e2b
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2026-01-13T00:00:00-05:00
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Even Odd Splitting of the Gaussian Quantum Fisher Information: From Symplectic Geometry to Metrology
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arXiv:2601.06513v1 Announce Type: cross Abstract: We introduce a canonical decomposition of the quantum Fisher information (QFI) for centered multimode Gaussian states into two additive pieces: an even part that captures changes in the symplectic spectrum and an odd part associated with correlation-generating dynamics. On the pure-state manifold, the even contribution vanishes identically, while the odd contribution coincides with the QFI derived from the natural metric on the Siegel upper half-space, revealing a direct geometric underpinning of pure-Gaussian metrology. This also provides a link between the graphical representation of pure Gaussian states and an explicit expression for the QFI in terms of graphical parameters. For evolutions completely generated by passive Gaussian unitaries (orthogonal symplectics), the odd QFI vanishes, while thermometric parameters contribute purely to the even sector with a simple spectral form; we also derive a state-dependent lower bound on the even QFI in terms of the purity-change rate. We extend the construction to the full QFI matrix, obtaining an additive even odd sector decomposition that clarifies when cross-parameter information vanishes. Applications to unitary sensing (beam splitter versus two-mode squeezing) and to Gaussian channels (loss and phase-insensitive amplification), including joint phase loss estimation, demonstrate how the decomposition cleanly separates resources associated with spectrum versus correlations. The framework supplies practical design rules for continuous-variable sensors and provides a geometric lens for benchmarking probes and channels in Gaussian quantum metrology.
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https://arxiv.org/abs/2601.06513
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e758e8e254d11d7847992a4a5a83a7cf79777f56a03ea4e92e41340c21538790
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2026-01-13T00:00:00-05:00
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A Lindblad and Non-Hermitian Spectral Framework for Fragmentation Dynamics and Particle Size Distributions
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arXiv:2601.06638v1 Announce Type: cross Abstract: Population balance equations (PBEs) for pure fragmentation describe how particle size distributions (PSDs) evolve under breakage and fragment redistribution. We map a self-similar fragmentation class to: a conservative pure-jump master equation in log-size space; an exact Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) dilation whose diagonal sector reproduces that master equation; and a controlled small-jump limit where the dynamics reduce to a Fokker-Planck operator that can be symmetrized into a Schrodinger-type spectral problem. Two points ensure correctness and applicability. First, the Lindblad embedding is exact when the daughter kernel is interpreted in mass-weighted form (equivalently, when z*kappa(z) is a probability measure). Second, for genuinely non-Hermitian dynamics the stationary PSD is naturally a biorthogonal product of left and right ground states, not a naive modulus square; the usual modulus squared of psi appears only in pseudo-Hermitian or effectively dephased regimes. We then give a spectral dictionary linking typical PSD shapes to low-dimensional potential families in log-size space and outline inverse routes to infer effective potentials from data: parametric fitting with time-resolved PSDs, a direct steady-state inversion from a smoothed PSD, and an outlook toward inverse spectral ideas. A synthetic example demonstrates forward simulation and inverse parameter recovery in an Airy-type half-line model.
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https://arxiv.org/abs/2601.06638
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972c8f4bf8200294055ebb14554743b9dffce968e8babdc5d206af7f2a33f3f7
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2026-01-13T00:00:00-05:00
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Censored Graphical Horseshoe: Bayesian sparse precision matrix estimation with censored and missing data
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arXiv:2601.06671v1 Announce Type: cross Abstract: Gaussian graphical models provide a powerful framework for studying conditional dependencies in multivariate data, with widespread applications spanning biomedical, environmental sciences, and other data-rich scientific domains. While the Graphical Horseshoe (GHS) method has emerged as a state-of-the-art Bayesian method for sparse precision matrix estimation, existing approaches assume fully observed data and thus fail in the presence of censoring or missingness, which are pervasive in real-world studies. In this paper, we develop the Censored Graphical Horseshoe (CGHS), a novel Bayesian framework that extends the GHS to censored and arbitrarily missing Gaussian data. By introducing a latent-variable representation, CGHS accommodates incomplete observations while retaining the adaptive global-local shrinkage properties of the Horseshoe prior. We derive efficient Gibbs samplers for posterior computation and establish new theoretical results on posterior behavior under censoring and missingness, filling a gap not addressed by frequentist Lasso-based methods. Through extensive simulations, we demonstrate that CGHS consistently improves estimation accuracy compared to penalized likelihood approaches. Our methods are implemented in the package GHScenmis available on Github: https://github.com/tienmt/ghscenmis .
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https://arxiv.org/abs/2601.06671
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be8218bc5763048a8e7e9a38844e934eef7b94e8c0fad41666d55536c84c2de3
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2026-01-13T00:00:00-05:00
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Extensions of the solidarity principle of the spectral gap for Gibbs samplers to their blocked and collapsed variants
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arXiv:2601.06745v1 Announce Type: cross Abstract: Connections of a spectral nature are formed between Gibbs samplers and their blocked and collapsed variants. The solidarity principle of the spectral gap for full Gibbs samplers is generalized to different cycles and mixtures of Gibbs steps. This generalized solidarity principle is employed to establish that every cycle and mixture of Gibbs steps, which includes blocked Gibbs samplers and collapsed Gibbs samplers, inherits a spectral gap from a full Gibbs sampler. Exact relations between the spectra corresponding to blocked and collapsed variants of a Gibbs sampler are also established. An example is given to show that a blocked or collapsed Gibbs sampler does not in general inherit geometric ergodicity or a spectral gap from another blocked or collapsed Gibbs sampler.
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https://arxiv.org/abs/2601.06745
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b365ca76f927609bcc2de557d9e8602e292a5ab606597bf4cddfe05cf2079673
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2026-01-13T00:00:00-05:00
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Introduction of Probability Density Alternation Method for Inverse Analyses of Integral Equations in Surface Science
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arXiv:2601.06797v1 Announce Type: cross Abstract: Integral equations frequently arise in surface science, and in some cases, they must be treated as inverse problems. In our previous work on optical tweezers, atomic force microscopy, and surface force measurement apparatus, we performed inverse calculations to obtain the pressure between parallel plates from measured interaction forces. These inverse analyses were used to reconstruct solvation structures near solid surfaces and density distribution profiles of colloidal particles. In the course of these studies, we developed a method that enables inverse analyses through a unified and systematic procedure, hereafter referred to as the Probability Density Alternation (PDA) method. The central idea of this method is to reformulate a given integral equation in terms of probability density functions. In this letter, we demonstrate the validity of the PDA method both analytically and numerically. While the PDA method is less advantageous for single integral equations, it becomes a convenient and powerful approach for inverse analyses involving double or higher-order integral equations.
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https://arxiv.org/abs/2601.06797
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df1c74ed0c2c1041087ff702a232573a326deecaac3716e0b044d0a0ad87baf1
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2026-01-13T00:00:00-05:00
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Obstructions to Unitary Hamiltonians in Non-Unitary String-Net Models
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arXiv:2601.06821v1 Announce Type: cross Abstract: The Levin-Wen string-net formalism provides a canonical mapping from spherical fusion categories to local Hamiltonians defining Topological Quantum Field Theories (TQFTs). While the topological invariance of the ground state is guaranteed by the pentagon identity, the realization of the model on a physical Hilbert space requires the category to be unitary. In this work, we investigate the obstructions arising when this construction is applied to non-unitary spherical categories, specifically the Yang-Lee model (the non-unitary minimal model $\mathcal{M}(2,5)$). We first validate our framework by explicitly constructing and verifying the Hamiltonians for rank-3 ($\text{Rep}(D_3)$), rank-5 ($\text{TY}(\mathbb{Z}_4)$), and Abelian ($\mathbb{Z}_7$) unitary categories. We then apply this machinery to the non-unitary Yang-Lee model. Using a custom gradient-descent optimization algorithm on the manifold of $F$-symbols, we demonstrate that the Yang-Lee fusion rules admit no unitary solution to the pentagon equations. We explain this failure analytically by proving that negative quantum dimensions impose an indefinite metric on the string-net space, realizing a Krein space rather than a Hilbert space. Finally, we invoke the theory of $\mathcal{PT}$-symmetric quantum mechanics to interpret the non-Hermitian Hamiltonian, establishing that the obstruction is intrinsic to the fusion ring and cannot be removed by unitary gauge transformations.
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https://arxiv.org/abs/2601.06821
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8736f4ed982fa7b4a101fe7478941ba00c269a6db84240ab21741df40ba63fdb
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2026-01-13T00:00:00-05:00
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The past stability of Kasner singularities for the $(3+1)$-dimensional Einstein vacuum spacetime under polarized $U(1)$-symmetry
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arXiv:2601.06957v1 Announce Type: cross Abstract: In this paper, we give a new proof to a past stability result established in Fournodavlos-Rodnianski-Speck (arXiv:2012.05888), for Kasner solutions of the $(3+1)$-dimensional Einstein vacuum equations under polarized $U(1)$-symmetry. Our method, inspired by Beyer-Oliynyk-Olvera-Santamar{\'\i}a-Zheng (arXiv:1907.04071, arXiv:2502.09210), relies on a newly developed $(2+1)$ orthonormal-frame decomposition and a careful symmetrization argument, after which the Fuchsian techniques can be applied. We show that the perturbed solutions are asymptotically pointwise Kasner, geodesically incomplete and crushing at the Big Bang singularity. They are achieved by reducing the $(3+1)$ Einstein vacuum equations to a Fuchsian system coupled with several constraint equations, with the symmetry assumption playing an important role in the reduction. Using Fuchsian theory together with finite speed of constraints propagation, we obtain global existence and precise asymptotics of the solutions up to the singularities.
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https://arxiv.org/abs/2601.06957
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762287c373b8a6cc860d9fc8ee815488693c9bfd28051392a0fa2524284b4249
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2026-01-13T00:00:00-05:00
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Subspace Selected Variational Quantum Configuration Interaction with a Partial Walsh Series
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arXiv:2601.07037v1 Announce Type: cross Abstract: Estimating the ground-state energy of a quantum system is one of the most promising applications for quantum algorithms. Here we propose a variational quantum eigensolver (VQE) \emph{Ansatz} for finding ground state configuration interaction (CI) wavefunctions. We map CI for fermions to a quantum circuit using a subspace superposition, then apply diagonal Walsh operators to encode the wavefunction. The algorithm can be used to solve both full CI and selected CI wavefunctions, resuling in exact and near-exact solutions for electronic ground states. Both the subspace selection and wavefunction \emph{Ansatz} can be applied to any Hamiltonian that can be written in a qubit basis. The algorithm bypasses costly classical matrix diagonalizations, which is advantageous for large-scale applications. We demonstrate results for several molecules using quantum simulators and hardware.
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https://arxiv.org/abs/2601.07037
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ccb437e8a963221c973e117e9f9193c50917bbfb876348272ac2ef162a69ad89
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2026-01-13T00:00:00-05:00
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Neuronal Spike Trains as Functional-Analytic Distributions: Representation, Analysis, and Significance
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arXiv:2601.07215v1 Announce Type: cross Abstract: The action potential constitutes the digital component of the signaling dynamics of neurons. But the biophysical nature of the full time course of the action potential associated with changes in membrane potential is fundamentally and mathematically distinct from its representation as a discrete set of events that encode when action potentials triggered in a collection spike trains. In this paper, we rigorously explore from first principles the transition and modeling from the standard biophysical picture of a single action potential to its representation as a spike in a spike train. In particular, we adopt a functional-analytic framework, using Schwartz distribution theory to represent spike trains as generalized Dirac delta functions acting on smooth test functions. We then show how and why this representation transcends a purely descriptive formalism to support deep downstream analysis and modeling of spike train neural dynamics in a mathematically consistent way.
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https://arxiv.org/abs/2601.07215
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e04fa01f6a24d6c6bb17abffb092fda6957552edb1d0a46564a3e9827e429965
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2026-01-13T00:00:00-05:00
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Characterization of multi-way binary tables with uniform margins and fixed correlations
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arXiv:2601.07369v1 Announce Type: cross Abstract: In many applications involving binary variables, only pairwise dependence measures, such as correlations, are available. However, for multi-way tables involving more than two variables, these quantities do not uniquely determine the joint distribution, but instead define a family of admissible distributions that share the same pairwise dependence while potentially differing in higher-order interactions. In this paper, we introduce a geometric framework to describe the entire feasible set of such joint distributions with uniform margins. We show that this admissible set forms a convex polytope, analyze its symmetry properties, and characterize its extreme rays. These extremal distributions provide fundamental insights into how higher-order dependence structures may vary while preserving the prescribed pairwise information. Unlike traditional methods for table generation, which return a single table, our framework makes it possible to explore and understand the full admissible space of dependence structures, enabling more flexible choices for modeling and simulation. We illustrate the usefulness of our theoretical results through examples and a real case study on rater agreement.
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https://arxiv.org/abs/2601.07369
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688f9fcc14138a4897c98fb799fe0458de5a2f3951607cf348d38b2d82e3e1ea
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2026-01-13T00:00:00-05:00
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Refined Invariants and Quantum Curves from Supersymmetric Localization
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arXiv:2601.07662v1 Announce Type: cross Abstract: We study an Aganagic-Vafa brane supported on a special Lagrangian submanifold $\mathcal{L}$ in a non-compact toric Calabi-Yau threefold $\mathcal{X}$. From the perspective of geometric engineering, the Aganagic-Vafa branes give rise to a special class of half-BPS codimension-two defects in 5d $\mathcal{N}=1$ supersymmetric field theories in the presence of $\Omega$-background. We propose that the defect partition functions give generating functions of refined, non-negative, integral open BPS invariants of the pair $(\mathcal{X},\mathcal{L})$, across different K\"{a}hler moduli chambers they are expanded in. In the Nekrasov-Shatashvili limit, the partition function provides a partially resummed solution to a $q$-difference equation that quantizes the mirror curve of $\mathcal{X}$ in an unambiguous fashion, in a polarization determined by the discrete labels of the Aganagic-Vafa brane. We demonstrate our method at examples of $\mathbb{C}^3$, resolved conifold, resolved $A_1$-singularity, local $F_0$, and local $F_1$.
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https://arxiv.org/abs/2601.07662
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411f9710722365c19ee8748ce4f986e79f2fbc06626bc2928708982890f1b7a0
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2026-01-13T00:00:00-05:00
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Quantum information and statistical complexity of hydrogen-like ions in Dunkl-Schr\"odinger system
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arXiv:2601.07683v1 Announce Type: cross Abstract: In this work, we present analytical solutions of Schr\"odinger equation for Coulomb potential in presence of a Dunkl reflection operator. Expressions are offered for eigenvalues, eigenfunctions and radial densities for H-isoelectronic series (Z=1-3). The degeneracy in energy in absence and presence of the reflection has been discussed. The standard deviation, Shannon entropy, R\'enyi entropy in position space have been derived for arbitrary quantum states. Then several important complexity measures like L\'opez-Ruiz-Mancini-Calbet (LMC), Shape-R\'enyi complexity (SRC), Generalized R\'enyi complexity (GRC), R\'enyi complexity ratio (RCR) are considered in the analytical framework. Representative results are given for three one-electron atomic ions in tabular and graphical format. Changes in these measures with respect to parity and Dunkl parameter have been given in detail. Most of these results are offered here for the first time.
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https://arxiv.org/abs/2601.07683
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bef5d01699be728646461cadc2343e3211c2832f34ae5b681615c5d77a6f511c
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2026-01-13T00:00:00-05:00
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Assembly to Quantum Compiler
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arXiv:2601.07706v1 Announce Type: cross Abstract: This research presents a novel approach in quantum computing by transforming ARM assembly instructions for use in quantum algorithms. The core achievement is the development of a method to directly map the ARM assembly language, a staple in classical computing, to quantum computing paradigms. The practical application of this methodology is demonstrated through the computation of the Fibonacci sequence. This example serves to validate the approach and underscores its potential in simplifying quantum algorithms. Grover's Algorithm was realized through the use of quantum-specific instructions. These transformations were developed as part of an open-source assembly-to-quantum compiler (github.com/arhaverly/AssemblyToQuantumCompiler). This effort introduces a novel approach to utilizing classical instruction sets in quantum computing and offers insight into potential future developments in the field. The AssemblyToQuantumCompiler streamlines quantum programming and enables computer scientists to transition more easily from classical to quantum computer programming.
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https://arxiv.org/abs/2601.07706
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f16c6d90e05985187eaff76957825897d5aeae8bcc343f01f47bbd53fe83d33b
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2026-01-13T00:00:00-05:00
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Riesz Representer Fitting under Bregman Divergence: A Unified Framework for Debiased Machine Learning
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arXiv:2601.07752v1 Announce Type: cross Abstract: Estimating the Riesz representer is a central problem in debiased machine learning for causal and structural parameter estimation. Various methods for Riesz representer estimation have been proposed, including Riesz regression and covariate balancing. This study unifies these methods within a single framework. Our framework fits a Riesz representer model to the true Riesz representer under a Bregman divergence, which includes the squared loss and the Kullback--Leibler (KL) divergence as special cases. We show that the squared loss corresponds to Riesz regression, and the KL divergence corresponds to tailored loss minimization, where the dual solutions correspond to stable balancing weights and entropy balancing weights, respectively, under specific model specifications. We refer to our method as generalized Riesz regression, and we refer to the associated duality as automatic covariate balancing. Our framework also generalizes density ratio fitting under a Bregman divergence to Riesz representer estimation, and it includes various applications beyond density ratio estimation. We also provide a convergence analysis for both cases where the model class is a reproducing kernel Hilbert space (RKHS) and where it is a neural network.
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https://arxiv.org/abs/2601.07752
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fec54a0e0d2207ee55006d3b692cc3bfab75f0aa83d3ce6016f5435cfb0d3b03
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2026-01-13T00:00:00-05:00
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Spacetime Quasicrystals
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arXiv:2601.07769v1 Announce Type: cross Abstract: Self-similar quasicrystals (like the famous Penrose and Ammann-Beenker tilings) are exceptional geometric structures in which long-range order, quasiperiodicity, non-crystallographic orientational symmetry, and discrete scale invariance are tightly interwoven in a beautiful way. In this paper, we show how such structures may be generalized from Euclidean space to Minkowski spacetime. We construct the first examples of such Lorentzian quasicrystals (the spacetime analogues of the Penrose or Ammann-Beenker tilings), and point out key novel features of these structures (compared to their Euclidean cousins). We end with some (speculative) ideas about how such spacetime quasicrystals might relate to reality. This includes an intriguing scenario in which our infinite $(3+1)$D universe is embedded (like one of our spacetime quasicrystal examples) in a particularly symmetric $(9+1)$D torus $T^{9,1}$ (which was previously found to yield the most symmetric toroidal compactification of the superstring). We suggest how this picture might help explain the mysterious seesaw relationship $M_{\rm Pl}M_{\rm vac}\approx M_{\rm EW}^{2}$ between the Planck, vacuum energy, and electroweak scales ($M_{\rm Pl}$, $M_{\rm vac}$, $M_{\rm EW}$).
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https://arxiv.org/abs/2601.07769
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6d622b1e11cdb041219d705cd6bc112fc817afb6e8aea6f20241692457193cd2
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2026-01-13T00:00:00-05:00
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Serial vs parallel recall in the Blume-Every-Griffiths neural networks
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arXiv:2601.07777v1 Announce Type: cross Abstract: Fully connected Blume-Emery-Griffiths neural networks performing pattern recognition and associative memory have been heuristically studied in the past (mainly via the replica trick and under the replica symmetric assumption) as generalization of the standard Hopfield reference. In these notes, at first, by relying upon Guerra interpolation, we re-obtain the existing picture rigorously. Next we show that, due to dilution in the patterns, these networks are able to switch from serial recall (where one pattern is retrieved per time) to parallel recall (where several patterns are retrieved at once) and the larger the dilution, the stronger this emerging multi-tasking capability. In particular, we inspect the regimes of mild dilution (where solely a low storage of pattern can be enabled) and extreme dilution (where a medium storage of patterns can be sustained) separately as they give rise to different outcomes: the former displays hierarchical recall (distributing the amplitudes of the retrieved signals with different amplitudes), the latter -instead- performs a equal-strength recall (where a O(1) fraction of all the patterns is simultaneously retrieved with the same amplitude per pattern). Finally, in order to implement graded responses in the neurons, variations on theme obtained by enlarging the possible values of neural activity these neurons may sustain are also discussed generalizing the Ghatak-Sherrington model for inverse freezing in Hebbian terms.
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https://arxiv.org/abs/2601.07777
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4219714642871f0ce9b1c0554cf64521bd85eb945a0b69f30e8ca1f8f7076e15
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2026-01-13T00:00:00-05:00
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A proof of Newman's conjecture for the extended Selberg class
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arXiv:2005.05142v2 Announce Type: replace Abstract: Newman's conjecture (proved by Rodgers and Tao in 2018) concerns a certain family of deformations $\{\xi_t(s)\}_{t \in \mathbb{R}}$ of the Riemann xi function for which there exists an associated constant $\Lambda \in \mathbb{R}$ (called the de Bruijn-Newman constant) such that all the zeros of $\xi_t$ lie on the critical line if and only if $t \geq \Lambda$. The Riemann hypothesis is equivalent to the statement that $\Lambda \leq 0$, and Newman's conjecture states that $\Lambda \geq 0$. In this paper we give a new proof of Newman's conjecture which avoids many of the complications in the proof of Rodgers and Tao. Unlike the previous best methods for bounding $\Lambda$, our approach does not require any information about the zeros of the zeta function, and it can be readily be applied to a wide variety of $L$-functions. In particular, we establish that any $L$-function in the extended Selberg class has an associated de Bruijn-Newman constant and that all of these constants are nonnegative. Stated in the Riemann xi function case, our argument proceeds by showing that for every $t < 0$ the function $\xi_t$ can be approximated in terms of a Dirichlet series $\zeta_t(s)=\sum_{n=1}^{\infty}\exp(\frac{t}{4} \log^2 n)n^{-s}$ whose zeros then provide infinitely many zeros of $\xi_t$ off the critical line.
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https://arxiv.org/abs/2005.05142
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dd099e243244fec5aa5a4f40d57ce258f6e328067c0171680a47410c3227a52a
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2026-01-13T00:00:00-05:00
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Moduli of vector bundles on primitive multiple schemes
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arXiv:2202.12569v3 Announce Type: replace Abstract: A primitive multiple scheme is a Cohen-Macaulay scheme $Y$ such that the associated reduced scheme $X=Y_{red}$ is smooth, irreducible, and that $Y$ can be locally embedded in a smooth variety of dimension $\dim(X)+1$. If $n$ is the multiplicity of $Y$, there is a canonical filtration $X=X_1\subset X_2\subset\cdots\subset X_n=Y$, such that $X_i$ is a primitive multiple scheme of multiplicity $i$. The simplest example is the trivial primitive multiple scheme of multiplicity $n$ associated to a line bundle $L$ on $X$: it is the $n$-th infinitesimal neighborhood of $X$, embedded in the line bundle $L^*$ by the zero section. The main subject of this paper is the construction and properties of fine moduli spaces of vector bundles on primitive multiple schemes. Suppose that $Y=X_n$ is of multiplicity $n$, and can be extended to $X_{n+1}$ of multiplicity $n+1$, and let $M_n$ a fine moduli space of vector bundles on $X_n$. With suitable hypotheses, we construct a fine moduli space $M_{n+1}$ for the vector bundles on $X_{n+1}$ whose restriction to $X_n$ belongs to $M_n$. It is an affine bundle over the subvariety $N_n\subset M_n$ of bundles that can be extended to $X_{n+1}$. In general this affine bundle is not banal. This applies in particular to Picard groups. We give also many new examples of primitive multiple schemes $Y$ such that the dualizing sheaf $\omega_Y$ is trivial.
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https://arxiv.org/abs/2202.12569
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42ce4cae049c9833c43509881609b2601fc007a4edb34bcd4d57a8af9326337f
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2026-01-13T00:00:00-05:00
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Approximations of $SL(3,\mathbb{Z})$ Hecke-Maass $L$-Functions by short Dirichlet polynomials
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arXiv:2204.12612v5 Announce Type: replace Abstract: We study averages of $L$-functions associated with Hecke-Maass cusp forms for $SL(3,\mathbb{Z})$, multiplied by Dirichlet polynomials built from the Fourier coefficients of the cusp forms. To prove this, we employ a variant of the Kuznetsov trace formula. In particular, we show that the reciprocals of these $L$-functions can be approximated by very short Dirichlet polynomials, on average over $t$ and over the forms.
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https://arxiv.org/abs/2204.12612
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7ab1a951a4e6aa93d11bb44d6df71d04273249f8d2aa323eb6cb3015122c1bd7
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2026-01-13T00:00:00-05:00
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Homotopy type of spaces of locally convex curves in the sphere S^3
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arXiv:2205.10928v3 Announce Type: replace Abstract: Locally convex (or nondegenerate) curves in the sphere $S^n$ have been studied for several reasons, including the study of linear ordinary differential equations of order $n+1$. Taking Frenet frames allows us to obtain corresponding curves $\Gamma$ in the group $Spin_{n+1}$. Let $L_n(z_0;z_1)$ be the space of such curves $\Gamma$ with prescribed endpoints $\Gamma(0) = z_0$, $\Gamma(1) = z_1$. The aim of this paper is to determine the homotopy type of the spaces $L_3(z_0;z_1)$ for all $z_0, z_1 \in Spin_4$. As a corollary, we obtain the homotopy type of the space of closed locally convex curves in either $S^3$ or $P^3$. There are many previous papers addressing related questions. An early paper solves the corresponding problem for curves in $S^2$. Another previous result (with B. Shapiro) reduces the problem to $z_0 = 1$ and $z_1 \in Quat_4$ where $Quat_4 \subset Spin_4$ is a finite group of order $16$. A more recent paper shows that for $z_1 \in Quat_4 \smallsetminus Z(Quat_4)$ we have a homotopy equivalence $L_3(1;z_1) \approx \Omega Spin_4$. In this paper we compute the homotopy type of $L_3(1;z_1)$ for $z_1 \in Z(Quat_4)$: it is equivalent to the wedge of $\Omega Spin_4$ with an infinite countable family of spheres (as for the case $n = 2$). The structure of the proof can be compared to that of the case $n = 2$ but some of the steps require the creation of new theories, involving algebra and combinatorics. We construct explicit subsets $Y \subset L_n(z_0;z_1)$ for which the inclusion $Y \subset \Omega Spin_{n+1}(z_0;z_1)$ is a homotopy equivalence. For $n = 2$, there is a simple geometric description of $Y$; for $n = 3$, the far less natural construction is based on the theory of itineraries of such curves. The itinerary of a curve in $L_n(1;z_1)$ is a finite word in the alphabet $S_{n+1} \smallsetminus \{e\}$ of nontrivial permutations.
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https://arxiv.org/abs/2205.10928
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a304f84fe43ba2770af0b1c6fe7436470748337762a4cdf0690362ac85dc975d
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2026-01-13T00:00:00-05:00
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Algebraic Relations among Special Gamma Values and the Chowla-Selberg Phenomenon over Function Fields
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arXiv:2207.01165v2 Announce Type: replace Abstract: The aim of this paper is to determine all algebraic relations among various special gamma values over function fields, and prove a Chowla-Selberg-type formula for quasi-periods of CM abelian $t$-modules. Our results are based on the intrinsic relations between gamma values in question and periods of CM dual $t$-motives, which are interpreted in terms of their "distributions". This also enables us to derive an analogue of the Deligne-Gross period conjecture for CM Hodge-Pink structures.
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https://arxiv.org/abs/2207.01165
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a91dba775c73758b4f5e30976031a448f103a00703f237d76facfa710cf9f59f
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2026-01-13T00:00:00-05:00
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Quantum loop groups for symmetric Cartan matrices
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arXiv:2207.05504v4 Announce Type: replace Abstract: We introduce a quantum loop group associated to a general symmetric Cartan matrix, by imposing just enough relations between the usual generators $\{e_{i,k}, f_{i,k}\}_{i \in I, k \in \mathbb{Z}}$ in order for the natural Hopf pairing between the positive and negative halves of the quantum loop group to be perfect. As an application, we describe the localized K-theoretic Hall algebra of any quiver without loops, endowed with a particularly important $\mathbb{C}^*$ action.
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https://arxiv.org/abs/2207.05504
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b9670ae26bac74a2a8c89057d4164248c9425557f2171ffe68d77ad7709911ae
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2026-01-13T00:00:00-05:00
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Hereditarily indecomposable continua as generic mathematical structures
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arXiv:2208.06886v4 Announce Type: replace Abstract: We characterize the pseudo-arc as well as P-adic pseudo-solenoids (for a set of primes P) as generic structures, arising from a natural game in which two players alternate in building an inverse sequence of surjections. The second player wins if the limit of this sequence is homeomorphic to a concrete (fixed in advance) space, called generic whenever the second player has a winning strategy. For this purpose, we develop a new robust approximate Fra\"iss\'e theory in the context of MU-categories, a generalization of metric-enriched categories, suitable for working directly with continuous maps between metrizable compacta. Our framework extends both the classical and projective Fra\"iss\'e theories. We reprove the Fra\"iss\'e-theoretic characterization of the pseudo-arc and we realize every P-adic pseudo-solenoid as a Fra\"iss\'e limit of a suitable category of continuous surjections on the circle. Moreover, we show that, when playing the game with continuous surjections between non-degenerate Peano continua, the pseudo-arc is always generic, while the universal pseudo-solenoid is generic over all surjections between circle-like continua. This gives a complete classification of generic continua over full non-trivial subcategories of connected polyhedra with continuous surjections.
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https://arxiv.org/abs/2208.06886
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34649bdbf47aa6941f328a0165a135a8427aa7d9887cf547a02e1ba6e5c424e4
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2026-01-13T00:00:00-05:00
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A Nomizu-van Est theorem in Ekedahl's derived $\ell$-adic setting
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arXiv:2209.09744v2 Announce Type: replace Abstract: A theorem of Nomizu and van Est computes the cohomology of a compact nilmanifold, or equivalently the group cohomology of an arithmetic subgroup of a unipotent linear algebraic group over $\mathbb{Q}$. We prove a similar result for the cohomology of a compact open subgroup of a unipotent linear algebraic group over $\mathbb{Q}_{\ell}$ with coefficients in a complex of continuous $\ell$-adic representations. We work with the triangulated categories defined by Ekedahl which play the role of ``derived categories of continuous $\ell$-adic representations''. This is motivated by Pink's formula computing the derived direct image of an $\ell$-adic local system on a Shimura variety in its minimal compactification, and its application to automorphic perverse sheaves on Shimura varieties. The key technical result is the computation of the cohomology with coefficients in a unipotent representation with torsion coefficients by an explicit complex of polynomial cochains which is of finite type.
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https://arxiv.org/abs/2209.09744
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d2089553469d6586e0a10de709414818350a608744283a5393ed883f21fbf430
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2026-01-13T00:00:00-05:00
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Strongly rigid metrics in spaces of metrics
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arXiv:2210.02170v5 Announce Type: replace Abstract: A metric space is said to be strongly rigid if no positive distance is taken twice by the metric. In 1972, Janos proved that a separable metrizable space has a strongly rigid metric if and only if it is zero-dimensional. In this paper, we shall develop this result for the theory of space of metrics. For a strongly zero-dimensional metrizable space, we prove that the set of all strongly rigid metrics is dense in the space of metics. Moreover, if the space is the union of countable compact subspaces, then that set is comeager. As its consequence, we show that for a strongly zero-dimensional metrizable space, the set of all metrics possessing no nontrivial (bijective) self-isometry is comeager in the space of metrics.
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https://arxiv.org/abs/2210.02170
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