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96766d3dacabf84d0649335198353d233097b0896e236e6e9c8cfaf1991f2f48
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2026-01-07T00:00:00-05:00
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A Context for Manifold Calculus
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arXiv:2403.03321v5 Announce Type: replace Abstract: We develop Weiss's manifold calculus in the setting of $\infty$-categories, where we allow the target $\infty$-category to be any $\infty$-category with small limits. We will establish the connection between polynomial functors, Kan extensions, and Weiss sheaves, and will classify homogeneous functors. We will also generalize Weiss and Boavida de Brito's theorem to functors taking values in arbitrary $\infty$-categories with small limits.
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https://arxiv.org/abs/2403.03321
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Academic Papers
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f760fc2aa02a4a7df47bdaba8649f0fcfcc284ed30e8f02eb066de2c39e4329a
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2026-01-07T00:00:00-05:00
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The global Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups I: Coarse expansions of the relative trace formulae
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arXiv:2404.07342v3 Announce Type: replace Abstract: This is the first of a series of three papers where we prove the Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups and an Ichino--Ikeda type refinement. Our strategy is based on the comparison of relative trace formulae formulated by Liu. The goal of this first paper is to introduce the relative trace formulae and establish the coarse expansions.
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https://arxiv.org/abs/2404.07342
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821848ed4f5a497ec2c8186f6f0621853ea0b069dbf3acafc90b9951986a6f82
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2026-01-07T00:00:00-05:00
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Real plane separating (M-2)-curves of degree d and totally real pencils of degree d-3
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arXiv:2404.09671v4 Announce Type: replace Abstract: It is well known that a non-singular real plane projective curve of degree five with five connected components is separating if and only if its ovals are in non-convex position. In this article, this property is set into a different context and generalised to all real plane separating (M-2)-curves.
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https://arxiv.org/abs/2404.09671
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Academic Papers
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9483d87f0f5c12798d10c93cb03624dcf3afa2cf76f39de288c74a6a1870b2b7
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2026-01-07T00:00:00-05:00
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Master equations with indefinite nonlinearities
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arXiv:2405.02091v2 Announce Type: replace Abstract: In this paper, we consider the following indefinite fully fractional heat equation involving the master operator \begin{equation} (\partial_t -\Delta)^{s} u(x,t) = x_1u^p(x,t)\ \ \mbox{in}\ \R^n\times\R , \end{equation} where $s\in(0,1)$, and $-\infty < p < \infty$. Under mild conditions, we prove that there is no positive bounded solutions. To this end, we first show that the solutions are strictly increasing along $x_1$ direction by employing the direct method of moving planes. Then by constructing an unbounded sub-solution, we derive the nonexistence of bounded solutions. To circumvent the difficulties caused by the fully fractional master operator, we introduced some new ideas and novel approaches that, as we believe, will become useful tool in studying a variety of other fractional elliptic and parabolic problems.
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https://arxiv.org/abs/2405.02091
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08cd68ae82b7a62962658a808905b3807068ed91f851efcf690d6c20363c89cb
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2026-01-07T00:00:00-05:00
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Common Noise by Random Measures: Constructing Mean-Field Equilibria for Competitive Investment and Hedging
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arXiv:2408.01175v2 Announce Type: replace Abstract: We construct Nash-equilibria in mean-field portfolio games of optimal investment and hedging under relative performance concerns with exponential (CARA) utility preferences. Common noise dynamics are modeled by integer-valued random measures, for instance Poisson random measures, in addition to Brownian motions. Agents differ in individual risk aversions, competition weights, and initial capital endowments, while their contingent claim liabilities depend on both common and idiosyncratic risk factors. Mean-field equilibria are characterized by solutions to McKean-Vlasov backward stochastic differential equations with jumps, for which we prove existence and uniqueness of solutions, without assuming mean field interaction to be small. Moreover, we show how the equilibrium can be constructed from the optimal strategy of a single-agent optimization problem (without mean-field interaction) via an appropriate projection. Using successive changes of measure, our derivation provides a decomposition of the equilibrium strategy into three components with clear interpretations. Finally, we show how a limiting mean-field game of quadratic (instead of utility-based) hedging with relative performance concerns arises for vanishing risk aversion.
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https://arxiv.org/abs/2408.01175
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d3153a4eed35f0b8e47fb019fbfe68ee3daa5ed418ff70ef9f475e69bd182523
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2026-01-07T00:00:00-05:00
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Monodromy and vanishing cycles for complete intersection curves
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arXiv:2408.06479v3 Announce Type: replace Abstract: We compute the topological monodromy of every family of complete intersection curves. Like in the case of plane curves previously treated by the second-named author, we find the answer is given by the $r$-spin mapping class group associated to the maximal root of the adjoint line bundle. Our main innovation is a suite of tools for studying the monodromy of sections of a tensor product of very ample line bundles in terms of the monodromy of sections of the factors, allowing for an induction on (multi-)degree.
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https://arxiv.org/abs/2408.06479
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d682f7444a363686ace5beceb04d2e07d09ee8fdfca1e2753a0208f8fb21d447
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2026-01-07T00:00:00-05:00
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Instability of Legendrian knottedness, and non-regular Lagrangian concordances of knots
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arXiv:2409.00290v2 Announce Type: replace Abstract: We show that the family of smoothly non-isotopic Legendrian pretzel knots from the work of Cornwell-Ng-Sivek that all have the same Legendrian invariants as the standard unknot have front-spuns that are Legendrian isotopic to the front-spun of the unknot. Besides that, we construct the first examples of Lagrangian concordances between Legendrian knots that are not regular, and hence not decomposable. Finally, we show that the relation of Lagrangian concordance between Legendrian knots is not anti-symmetric, and hence does not define a partial order. The latter two results are based upon a new type of flexibility for Lagrangian concordances with stabilised Legendrian ends.
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https://arxiv.org/abs/2409.00290
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ded1859347adf67ec9a4b1ea26eba0e39e774db0c2c6d13f61cbd557fad93abe
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2026-01-07T00:00:00-05:00
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A combination theorem for hierarchically quasiconvex subgroups, and application to geometric subgroups of mapping class groups
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arXiv:2409.03602v2 Announce Type: replace Abstract: We provide sufficient conditions for two subgroups of a hierarchically hyperbolic group to generate an amalgamated free product over their intersection. The result applies in particular to certain geometric subgroups of mapping class groups of finite-type surfaces, that is, those subgroups coming from the embeddings of closed subsurfaces. In the second half of the paper, we study under which hypotheses our amalgamation procedure preserves several notions of convexity, such as hierarchical quasiconvexity (as introduced by Behrstock, Hagen, and Sisto) and strong quasiconvexity (every quasigeodesic with endpoints on the subset lies in a uniform neighbourhood). This answers a question of Russell, Spriano, and Tran.
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https://arxiv.org/abs/2409.03602
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64f32b7dd527f4d7c88499d85a6b80373510f26862e8fff8026c6313dc4d77eb
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2026-01-07T00:00:00-05:00
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Approximability of deep equilibria
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arXiv:2409.06064v4 Announce Type: replace Abstract: We introduce a structural framework for computations involving floating-point operations.Informed by real-valued logic, we introduce deep computations (ultracomputations) and deep iterates, formalizing the ideas of "asymptotic limit" of computations and compositional iterates, respectively. As an application of this framework, we prove the existence of deep equilibria, which hitherto have been found only empirically (yielding remarkable memory savings in deep learning). Our proof of existence of deep equilibria is based on the concept of idempotent ultrafilter from combinatorics and inspired by the notion of indiscernibility from model theory. We study and characterize deep computations (and hence deep equilibria) that are bona fide computable, i.e., uniformly approximable by a priori given computable primitive real-valued functions. Informed by model theory of real-valued structures, as well as Cp-theory from topology, we use a classical result of Grothendieck to characterize computability of deep computations in terms of continuous extendibility. Our framework does not impose a priori uniform/global bounds on real-valued quantities; therefore, our structures yield non-compact types spaces. Such type spaces require a more nuanced topological treatment than compact ones arising in model theory of [0,1]-valued structures.
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https://arxiv.org/abs/2409.06064
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7c248f0d93de8813c95220fb13b4e3ee61321954b0b6ae37f1eca201fc4fb3cb
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2026-01-07T00:00:00-05:00
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The leftmost particle of branching subordinators
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arXiv:2409.16617v2 Announce Type: replace Abstract: We define a family of continuous-time branching particle systems on the non-negative real line, called branching subordinators, where particles move as independent subordinators. Each particle can also split (at possibly infinite rate) into several children (possibly infinitely many) whose positions relative to the position of the parent are random. These particle systems are in the continuity of branching L\'evy processes introduced by Bertoin and Mallein [Ann. Probab. 47(3): 1619-1652 (2019)]. We pay a particular attention to the asymptotic behavior of the leftmost particle of branching subordinators. It turns out that, under some assumptions, the rate of growth of the position of the leftmost particle is of order $t^{\gamma}$ where $\gamma \in (0,1)$ depends explicitly on the parameters. This sub-linear growth is significantly different from the classical linear growth observed for regular branching random walks with non-negative displacements.
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https://arxiv.org/abs/2409.16617
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242a97be78bec354d15f4f0cbad6150f5b8e91f064f42a88838081d1ce70f29f
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2026-01-07T00:00:00-05:00
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Stability of reverse isoperimetric inequalities in the plane: area, Cheeger, and inradius
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arXiv:2410.06096v2 Announce Type: replace Abstract: In this paper, we present sharp stability results for various reverse isoperimetric problems in $\mathbb R^2$. Specifically, we prove the stability of the reverse isoperimetric inequality for $\lambda$-convex bodies -- convex bodies with the property that each of their boundary points $p$ supports a ball of radius $1/\lambda$ so that the body lies inside the ball in a neighborhood of $p$. For convex bodies with smooth boundaries, $\lambda$-convexity is equivalent to having the curvature of the boundary bounded below by $\lambda > 0$. Additionally, within this class of convex bodies, we establish stability for the reverse inradius inequality and the reverse Cheeger inequality. Even without its stability version, the sharp reverse Cheeger inequality is new in dimension $2$.
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https://arxiv.org/abs/2410.06096
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32c3cccabc520ab1725ab6cd9f5aa0defe81826c423c56a5d57cfb9fab160c27
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2026-01-07T00:00:00-05:00
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Mapping class group orbit closures for Deroin-Tholozan representations
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arXiv:2411.10269v2 Announce Type: replace Abstract: We prove that infinite mapping class group orbits are dense in the character variety of Deroin-Tholozan representations. In other words, the action is minimal except for finite orbits. Our arguments rely on the symplectic structure of the character variety, emphasizing this geometric perspective over its algebraic properties.
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https://arxiv.org/abs/2411.10269
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bb6da10529879f1c2d381de10e43aaee948f892a942455893da5e01f78ce7490
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2026-01-07T00:00:00-05:00
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Information geometric regularization of unidimensional pressureless Euler equations yields global strong solutions
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arXiv:2411.15121v2 Announce Type: replace Abstract: Partial differential equations describing compressible fluids are prone to the formation of shock singularities, arising from faster upstream fluid particles catching up to slower, downstream ones. In geometric terms, this causes the deformation map to leave the manifold of diffeomorphisms. Information geometric regularization addresses this issue by changing the manifold geometry to make it geodesically complete. Empirical evidence suggests that this results in smooth solutions without adding artificial viscosity. This work makes a first step towards understanding this phenomenon rigorously, in the setting of the unidimensional pressureless Euler equations. It shows that their information geometric regularization has smooth global solutions. By establishing $\Gamma$-convergence of its variational description, it proves convergence of these solutions to entropy solutions of the nominal problem, in the limit of vanishing regularization parameter. A consequence of these results is that manifolds of unidimensional diffeomorphisms with information geometric regularization are geodesically complete.
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https://arxiv.org/abs/2411.15121
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8bc3157e68019d3e414b897dbb37de3ae48a50318a4116a91e3ef53ac08d6d1c
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2026-01-07T00:00:00-05:00
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Dg-separable dg-extensions
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arXiv:2412.06526v2 Announce Type: replace Abstract: We define and characterise completely dg-separable dg-extensions $\varphi:(A,d_A)\rightarrow (B,d_B)$. We completely characterise the case of graded commutative dg-division algebras in characteristic different from $2$. We prove that for a dg-separable extension a short exact sequence of dg-modules over $(B,d_B)$ splits if and only if the restriction to $(A,d_A)$ splits.
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https://arxiv.org/abs/2412.06526
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9a4f2b90f5254fbddecb05848009d85537c7e08b5cfee546682d10c54200c98c
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2026-01-07T00:00:00-05:00
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Regularity of solutions for fully fractional parabolic equations
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arXiv:2502.07530v2 Announce Type: replace Abstract: In this paper, we study the fully fractional heat equation involving the master operator: $$ (\partial_t -\Delta)^{s} u(x,t) = f(x,t)\ \ \mbox{in}\ \mathbb{R}^n\times\mathbb{R} , $$ where $s\in(0,1)$ and $f(x,t) \geq 0$. First we derive H\"{o}lder and Schauder estimates for nonnegative solutions of this equation. Due to the {\em nonlocality} of the master operator, existing results (cf. \cite{ST}) rely on global bounds of the solutions $u$ to control their higher local norms. However, such results are inadequate for blow-up and rescaling analysis aimed at obtaining a priori estimates for solutions to {\em nonlocal } equations on unbounded domains, as the global norms of the rescaled functions may diverge. This limitation raises to a natural and challenging question: {\em Can local bounds of solutions replace global bounds to control their higher local norms?} Here, we provide an affirmative answer to this question for nonnegative solutions. To achieve this, we introduced several new ideas and novel techniques. One of the key innovations is to use a {\em directional perturbation average} to derive an important estimate for the fully fractional heat kernel, as stated in Lemma \ref{key0}. We believe this estimate, along with other new techniques introduced here, will serve as powerful tools in regularity estimates for a wide range of nonlocal equations. Building on this breakthrough, we employ the blow-up and rescaling arguments to establish a priori estimates for solutions to a broader class of nonlocal equations in unbounded domains, such as $$(\partial_t -\Delta)^{s} u(x,t) = b(x,t) |\nabla_x u (x,t)|^q + f(x, u(x,t))\ \ \mbox{in}\ \ \mathbb{R}^n\times\mathbb{R}.$$ Under appropriate conditions, we prove that all nonnegative solutions, along with their spatial gradients, are uniformly bounded.
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https://arxiv.org/abs/2502.07530
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267817652f0af233b32351bdf9a99c040bc16e56c74b7e7b19ba4f6586f5b160
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2026-01-07T00:00:00-05:00
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Lie algebras of quotient groups
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arXiv:2502.10260v2 Announce Type: replace Abstract: We give conditions on a diffeological group $G$ and a normal subgroup $H$ under which the quotient group $G/H$ differentiates to a Lie algebra for which $\operatorname{Lie}(G/H) \cong \operatorname{Lie}(G)/\operatorname{Lie}(H)$. Our Lie functor is instantiated by the tangent structure on elastic diffeological spaces introduced by Blohmann. The requisite conditions on $G$ and $H$ hold, for example, when $G$ is a convenient infinite-dimensional Lie group and $H$ is countable, or when $G$ is finite-dimensional and $H$ is arbitrary. To recognize that convenient infinite-dimensional manifolds are elastic diffeological spaces, we give a characterization of convenience in terms of the diffeological tangent functor: a separated and bornological locally convex topological vector space $E$ is convenient if and only if the natural map $E \times E \to TE$ is an isomorphism of diffeological spaces. As an application, we integrate some classically non-integrable Banach-Lie algebras to diffeological groups.
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https://arxiv.org/abs/2502.10260
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be10f49bdc527614d9fbf10db6ef29d9266f799a0a810806f439c2b5881a5708
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2026-01-07T00:00:00-05:00
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The shift-homological spectrum and parametrising kernels of rank functions
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arXiv:2502.11939v2 Announce Type: replace Abstract: For any compactly generated triangulated category we introduce two topological spaces, the shift-spectrum and the shift-homological spectrum. We use them to parametrise a family of thick subcategories of the compact objects, which we call radical. These spaces can be viewed as non-monoidal analogues of the Balmer and homological spectra arising in tensor-triangular geometry: we prove that for monogenic tensor-triangulated categories the Balmer spectrum is a subspace of the shift-spectrum. To construct these analogues we utilise quotients of the module category, rather than the lattice theoretic methods which have been adopted in other approaches. We characterise radical thick subcategories and show in certain cases, such as the perfect derived categories of tame hereditary algebras or monogenic tensor-triangulated categories, that every thick subcategory is radical. We establish a close relationship between the shift-homological spectrum and the set of irreducible integral rank functions, and provide necessary and sufficient conditions for every radical thick subcategory to be given by an intersection of kernels of rank functions. In order to facilitate these results, we prove that both spaces we introduce may equivalently be described in terms of the Ziegler spectrum.
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https://arxiv.org/abs/2502.11939
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4c5d6edbf798150030e62cd58a83c7efbfc45ca73cb67de44e0a0c1118c7d23c
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2026-01-07T00:00:00-05:00
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A Note on the Phragmen-Lindelof Theorem
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arXiv:2502.13282v3 Announce Type: replace Abstract: We provide a generalization of the Phragm\'en-Lindel\"of principal of Rademacher with the aim of correcting, or at least provide a pathway to correcting, several errors appearing in the literature.
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https://arxiv.org/abs/2502.13282
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09f40396eb81fa3cf1f690c917c669f845de6c21086b0982f285b47f9faed997
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2026-01-07T00:00:00-05:00
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Characterizations of Tilt-Stable Local Minimizers of a Class of Matrix Optimization Problems
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arXiv:2503.03217v3 Announce Type: replace Abstract: Tilt stability plays a pivotal role in understanding how local solutions of an optimization problem respond to small, targeted perturbations of the objective. Although quadratic bundles are a powerful tool for capturing second-order variational behavior, their characterization remains incomplete beyond well-known polyhedral and certain specialized nonpolyhedral settings. To help bridge this gap, we propose a new point-based criterion for tilt stability in prox-regular, subdifferentially continuous functions by exploiting the notion of minimal quadratic bundles. Furthermore, we derive an explicit formula for the minimal quadratic bundle associated with a broad class of general spectral functions, thus providing a practical and unifying framework that significantly extends existing results and offers broader applicability in matrix optimization problems.
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https://arxiv.org/abs/2503.03217
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7f50b396c19e4891ffce73b16e49cc39b737114db9be14f0b47bf94e9beafcd9
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2026-01-07T00:00:00-05:00
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On action rate admissibility criteria
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arXiv:2503.03491v5 Announce Type: replace Abstract: We formulate new admissibility criteria for initial value problems motivated by the least action principle. These are applied to a two-dimensional Riemann initial value problem for the isentropic compressible Euler fluid flow. It is shown that the criterion prefers the 2-shock solution to solutions obtained by convex integration by Chiodaroli and Kreml or to the hybrid solutions recently constructed by Markfelder and Pellhammer.
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https://arxiv.org/abs/2503.03491
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523eb7a22de60e27aa31c23e91e3d2b04c38a64ffa169df3a3eedc14c79ad2e5
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2026-01-07T00:00:00-05:00
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The large sieve for square moduli, revisited
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arXiv:2503.18009v4 Announce Type: replace Abstract: We revisit the large sieve for square moduli and obtain conditional improvements under hypotheses on higher additive energies of modular square roots.
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https://arxiv.org/abs/2503.18009
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5b42100a22b77a92b737fd31ecf34290fdd489c93affc4e9df0750449a126715
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2026-01-07T00:00:00-05:00
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Secant varieties of Segre-Veronese varieties $\mathbb{P}^m\times\mathbb{P}^n$ embedded by $\mathcal{O}(1,2)$ are non-defective for $n\gg m^3$, $m\geq3$
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arXiv:2503.21972v2 Announce Type: replace Abstract: We prove that for any $m\geq3$, $n\gg m^3$, all secant varieties of the Segre-Veronese variety $\mathbb{P}^m\times\mathbb{P}^n$ have the expected dimension. This was already proved by Abo and Brambilla in the subabundant case, hence we focus on the superabundant case. We generalize an approach due to Brambilla and Ottaviani into a construction we call the inductant. With this, the proof of non-defectivity reduces to checking a finite collection of base cases, which we verify using a computer-assisted proof.
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https://arxiv.org/abs/2503.21972
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3d52dc13786c17e081ecc11c7b37110befe7ac2a9900cd1cea817a964e3a648e
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2026-01-07T00:00:00-05:00
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Bounded domains on Kobayashi hyperbolic manifolds covering compact complex manifolds
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arXiv:2504.03935v2 Announce Type: replace Abstract: The lack of a uniformization theorem in several complex variables leads to a desire to classify all of the simply connected domains. We use established computational methods and a localization technique to generalize a recently-published classification. In particular, we show that if a domain with $C^{1,1}$ boundary on a Kobayashi hyperbolic complex manifold contains a totally real boundary point and covers a compact manifold, then its universal cover must be the Euclidean ball.
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https://arxiv.org/abs/2504.03935
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26e402afe1314e5945d3545ad2af28fb81f83df0442941bd6f10818fba8998c2
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2026-01-07T00:00:00-05:00
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Hanf numbers for poset games
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arXiv:2504.07317v3 Announce Type: replace Abstract: Given two well partial orders $(P;\leq_P)$ and $(T;\leq_T)$, each with a minimum element, we study the following question: which player has a winning strategy for Chomp on the poset $(P\times T;\leq_{P\times T})$? Here, $(P\times T;\leq_{P\times T})$ denotes the poset obtained as the Cartesian product of $P$ and $T$, equipped with the corresponding lexicographic order. The answer to this very natural question depends strongly on the specific choice of $(P;\leq_P)$ and $(T;\leq_T)$. For this reason, we restrict our attention to classes of posets given by powers of a fixed poset: $\{(P^\sigma;\leq_{P^\sigma})\mid \sigma\in\mathrm{Ord}\}$. A fundamental fact about these classes of structures is that, if the second player does not have a winning strategy for all the posets in $\{(P^\sigma;\leq_{P^\sigma})\mid \sigma\in\mathrm{Ord}\}$, there exists an ordinal $\xi$ such that the second player has a winning strategy on $(P^{\xi};\leq_{P^{\xi}})$ but not on $(P^{\gamma};\leq_{P^{\gamma}})$ for all $\gamma\geq\xi+1$. Determining the corresponding ordinal for this Hanf number-style property constitutes the main objective of this work. Inspired by results of Garc\'ia-Marco and Knauer, we focus on classes of posets with a purely algebraic definition. These posets arise from submonoids (with respect to the natural sum, or Hessenberg sum) of ordinals of the form $\omega^\sigma$ and are generated by sets of ordinals. In the process, we provide a test to determine whether a finite set $\Gamma$ of ordinals indeed yields well partial orders, and, using set-theoretic techniques, we establish an upper bound for the ordinal $\xi$: if $\Gamma\subset\omega$, then $\xi<\omega_1$, and otherwise $\xi<|\bigcup\Gamma|^+$.
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https://arxiv.org/abs/2504.07317
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daa68e69e7fe7d4cfcababa12f5a9f849867df3065065fdf1f7295ab5433489d
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2026-01-07T00:00:00-05:00
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On Minimal generating sets of splitting field, Cluster towers and Multiple transitivity of Galois groups
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arXiv:2505.00672v3 Announce Type: replace Abstract: A natural generating set for a Galois extension regarded as the splitting field of an irreducible polynomial is introduced and investigated here. Minimal generating sets arising in this context throw many surprises compared to the analogous concept in linear algebra: they can be of different cardinalities. In fact we establish that for a certain family of polynomials over the rationals, we have minimal generating sets of all cardinalities in a certain range and that these are the only possible cardinalities for minimal generating set for such a polynomial. We also study how minimal generating sets behave under multiple transitivity of the Galois group and consequently prove the existence of polynomials with all minimal generating sets of uniformly same cardinality. We also connect minimal generating sets with the concept of root cluster tower of an irreducible polynomial introduced by the second author and Krithika in [8].
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https://arxiv.org/abs/2505.00672
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7a79ab0ae427fd3f5a39808ead8cf89a543cb4cde814f8b813c6bbe89e0853cc
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2026-01-07T00:00:00-05:00
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Minimal Simplicial Degree $d$ Maps from Genus $g$ Surfaces to the Torus
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arXiv:2505.02386v2 Announce Type: replace Abstract: The degree of a map between orientable manifolds is a fundamental concept in topology, offering deep insights into the structure of the manifolds and the nature of the corresponding maps. This concept has been extensively studied, particularly in the context of simplicial maps between orientable triangulable spaces. In 1982, Gromov proved that if degree $d$ maps exist from a genus $g$ orientable surface to a genus $h$ orientable surface for every $d \in \mathbb{Z}$, then $h$ must be 0 or 1. Recently, degree $d$ self-maps on spheres, particularly on genus 0 surfaces, have been investigated. In this paper, we focus on the unique minimal 7-vertex triangulation of the torus. We construct simplicial degree $d$ maps from a triangulation of a genus $g$ surface to the 7-vertex triangulation of the torus for $g \geq 1$. Our construction of degree $d$ maps is minimal for every $d$ when $g = 1,2$. If $g \geq 3$, then our construction remains minimal for $|d| \geq 2g - 1$. We believe that this concept will be highly useful in combinatorial topology, as it leads to several intriguing open research problems. In the final section, we propose some of these open research problems.
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https://arxiv.org/abs/2505.02386
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9cbfc6bffedc7926f6ba4039690db27b632abf91c9bcb815488875d7690b2a4b
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2026-01-07T00:00:00-05:00
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Bounded cohomology, quotient extensions, and hierarchical hyperbolicity
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arXiv:2505.20462v3 Announce Type: replace Abstract: We call a central extension bounded if its Euler class is represented by a bounded cocycle. We prove that a bounded central extension of a hierarchically hyperbolic group (HHG) is still a HHG; conversely if a central extension is a HHG, then the extension is bounded, and under a further mild assumption the quotient is commensurable to a HHG. Motivated by questions on hierarchical hyperbolicity of quotients of mapping class groups, we therefore consider the general problem of determining when a quotient of a bounded central extension is still bounded, which we prove to be equivalent to an extendability problem for quasihomomorphisms. Finally, we show that quotients of the 4-strands braid group by suitable powers of a pseudo-Anosov are HHG, and in fact bounded central extensions of some HHG. We also speculate on how to extend the previous result to all mapping class groups.
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https://arxiv.org/abs/2505.20462
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44cb2008414a675c066b519dd2b49eca1a9ae0b56c865213ebf616571ce1b295
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2026-01-07T00:00:00-05:00
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Frostman and Fourier characterisations of fractal dimensions
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arXiv:2505.21217v2 Announce Type: replace Abstract: We examine Frostman-type characterisations and other extremal measure criteria for a range of fractal dimensions of sets. In particular we derive properties of the less familiar modified lower box dimension and upper correlation dimension. We also express a number of fractal dimensions in terms of Fourier properties of measures.
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https://arxiv.org/abs/2505.21217
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7e16977e1fa01e8720f937c421767945036cbfccbe9f6b4437889d774e8114d3
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2026-01-07T00:00:00-05:00
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Enumerating log rational curves on some toric varieties
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arXiv:2506.13975v3 Announce Type: replace Abstract: The genus 0, fixed-domain log Gromov-Witten invariants of a smooth, projective toric variety X enumerate maps from a general pointed rational curve to a smooth, projective toric variety passing through the maximal number of general points and with prescribed multiplicities along the toric boundary. We determine these invariants completely for the projective bundle X=P_{P^r}(O^s+O(-a)), proving a conjecture of Cela--Iribar L\'opez. A different conjecture when X is the blow-up of P^r at r points is disproven. Whereas the conjectures were predicted using tropical methods, we give direct intersection-theoretic calculations on moduli spaces of "naive log quasimaps."
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https://arxiv.org/abs/2506.13975
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9e349d69eb8909160f3fa28968a26ac21c84a46001de960082cb2cb3a879a9e5
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2026-01-07T00:00:00-05:00
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Maximal transitivity of the cactus group on standard Young tableaux
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arXiv:2506.16561v2 Announce Type: replace Abstract: The action of the cactus group $C_n$ on Young tableaux of a given shape $\lambda$ goes back to Berenstein and Kirillov and arises naturally in the study of crystal bases and quantum integrable systems. We show that this action is $2$-transitive on standard Young tableaux of the shape $\lambda$ if and only if $\lambda$ is not self-transpose and not a single hook. Moreover, we show that in these cases, the image of the cactus group in the permutation group of standard Young tableaux is either the whole permutation group or the alternating group, and prove that both cases are possible for infinitely many $\lambda$ (though the alternating group is more frequent). As an application, this implies that the Galois group of solutions to the Bethe ansatz in the Gaudin model attached to the Lie group $GL_d$ is, in many cases, at least the alternating group. This also extends the results of Sottile and White on the multiple transitivity of the Galois group of Schubert calculus problems in Grassmannians to many new cases.
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https://arxiv.org/abs/2506.16561
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5f842cb852966689cd1af1af3aefda84fad840055fceac773dbb36b9c5b755b3
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2026-01-07T00:00:00-05:00
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Non-unique equilibrium measures and freezing phase transitions for matrix cocycles for negative $t$
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arXiv:2507.01148v2 Announce Type: replace Abstract: We consider a one-step matrix cocycle generated by a pair of non-negative parabolic matrices and study the equilibrium measures for $t\log \|\mathcal A\|$ as $t$ runs over the reals. We show that there is a freezing first order phase transition at some parameter value $t_c$ so that for $tt_c$, the equilibrium measure is unique, non-atomic and fully supported. The phase transition closely resembles the classical Hofbauer example. In particular, our example shows that there may be non-unique equilibrium measures for negative $t$ even if the cocycle is strongly irreducible and proximal.
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https://arxiv.org/abs/2507.01148
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e906344e2f17a0dfee4dc142529604523f7d12463b52a108c851ed8b37aa7a2f
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2026-01-07T00:00:00-05:00
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Matrix Fej\'er-Riesz type theorem for a union of an interval and a point
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arXiv:2507.01357v2 Announce Type: replace Abstract: The matrix Fej\'er-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line. In the previous work of the second-named author this was extended to the characterization on arbitrary closed semialgebraic sets $K$ in $\mathbb{R}$ by using matrix quadratic modules from real algebraic geometry. In the compact case there is a denominator-free characterization, while in the non-compact case denominators are needed except when $K$ is the whole line, an unbounded interval, a union of two unbounded intervals, and it was conjectured also when $K$ is a union of an unbounded interval and a point or a union of two unbounded intervals and a point. In this paper, we confirm this conjecture by solving the truncated matrix-valued moment problem (TMMP) on a union of a bounded interval and a point. The presented technique for solving the corresponding TMMP can potentially be used to determine degree bounds in the positivity certificates for matrix polynomials on compact sets $K$.
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https://arxiv.org/abs/2507.01357
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73d7337c929c572861e2b0cd9098877710707c1843f7b1be56dfceb35c085739
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2026-01-07T00:00:00-05:00
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Bounded diameter monochromatic component covers
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arXiv:2507.05842v2 Announce Type: replace Abstract: Ryser conjectured that every $r$-edge-coloured complete graph can be covered by $r-1$ monochromatic trees. Motivated by a question of Austin in analysis, Mili\'cevi\'c predicted something stronger -- that every $r$-edge-coloured complete graph can be covered by $r-1$ monochromatic trees \emph{of bounded diameter}. Here we show that the two conjectures are equivalent. As immediate corollaries we obtain new results about Mili\'cevi\'c's Conjecture, most notably that it is true for $r=5$. We also obtain several new cases of a generalization of Mili\'cevi\'c's Conjecture to non-complete graphs due to DeBiasio-Kamel-McCourt-Sheats.
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https://arxiv.org/abs/2507.05842
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54345a285064d7ecc98d24e66f2fd3c1ef3d565bc92a3b6240543ee820038d36
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2026-01-07T00:00:00-05:00
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Classifying Nakayama algebras with a braid group action on $\tau$-exceptional sequences
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arXiv:2507.07608v2 Announce Type: replace Abstract: We characterise those basic and connected Nakayama algebras $\Lambda$ for which the mutation of $\tau$-exceptional sequences respects the braid group relations. We show that this is the case if and only if $\Lambda$ is hereditary or all indecomposable projective $\Lambda$-modules have length at least $|\Lambda|$.
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https://arxiv.org/abs/2507.07608
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3b77bf5264c8ebc7953b04b3e8a4a9e7b24202e9b6c8377f57cc51876ad5ea8a
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2026-01-07T00:00:00-05:00
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Donaldson-Thomas invariants of $[\mathbb C^4/\mathbb Z_r]$
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arXiv:2507.21582v2 Announce Type: replace Abstract: We compute the zero-dimensional Donaldson-Thomas invariants of the quotient stack $[\mathbb{C}^4/\mathbb{Z}_r]$, confirming a conjecture of Cao-Kool-Monavari. Our main theorem is established through an orbifold analogue of Cao-Zhao-Zhou's degeneration formula combined with the zero-dimensional Donaldson-Thomas invariants for $\mathcal{A}_{r-1}\times\mathbb{C}^2$ and an explicit determination of orientations of Hilbert schemes of points on $[\mathbb{C}^4/\mathbb{Z}_r]$.
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https://arxiv.org/abs/2507.21582
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24fb90617092abd70a26a4662f010a05fda8c0d552c56115de8ff63c9bdb8a8f
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2026-01-07T00:00:00-05:00
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Strong Feller Regularisation of 1-d Nonlinear Transport by Reflected Ornstein-Uhlenbeck Noise
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arXiv:2508.01355v2 Announce Type: replace Abstract: We consider equations of nonlinear transport on the circle with regular self interactions appearing in aggregation models and deterministic mean field dynamics. We introduce a random perturbation of such systems through a stochastic orientation preserving flow, which is given as an integrated infinite dimensional periodic Ornstein- Uhlenbeck process with reflection. As our main result we show that the induced stochastic dynamics yields a measure valued Markov process on a class of regular measures. Moreover, we show that this process is strong Feller in the corresponding topology. This is interpreted as a qualitative regularisation by noise phenomenon.
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https://arxiv.org/abs/2508.01355
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32dede0f968f746ba9fb9733718602d5a4564486db7fb7f5e9a96e85429c6d23
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2026-01-07T00:00:00-05:00
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Joint distribution of Hecke eigenforms on $\mathbb{H}^3$
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arXiv:2508.06331v2 Announce Type: replace Abstract: We prove a joint value equidistribution statement for Hecke-Maa{\ss} cusp forms on the hyperbolic three-space $\mathbb{H}^3$. This supports the conjectural statistical independence of orthogonal cusp forms.
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https://arxiv.org/abs/2508.06331
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40977c81dc122a95c33a132b08ca334bc331fff85c8a59e851b605b509103fee
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2026-01-07T00:00:00-05:00
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K-promotion on m-packed labelings of posets
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arXiv:2508.09305v2 Announce Type: replace Abstract: Schutzenberger's promotion operator, pro, is a fundamental map in dynamical algebraic combinatorics. At first, its action was mainly considered on standard Young tableaux. But pro was subsequently shown to have interesting properties when applied to natural labelings of other posets. Pechenik defined a K-theoretic version of promotion, pro_K, on m-packed labelings of tableaux. The operator pro was then extended to increasing labelings of other posets. The purpose of the current work is to show that the original action of pro_K on m-packed labelings yields interesting results when applied to partially ordered sets in general, and to rooted trees in particular. We show that under certain conditions, the sizes of the orbits and order of pro_K exhibit nice divisibility properties. We also completely determine, for certain values of m, the orbit sizes for the action on various types of rooted trees such as extended stars, combs, zippers, and a type of three-leaved tree.
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https://arxiv.org/abs/2508.09305
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0f0f59b7c6640d231dc6dea024eaadadf6cd6aecce4b9ee30d0fffe6f255366b
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2026-01-07T00:00:00-05:00
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On a Grassmann odd analogue of Carrollian Manifolds
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arXiv:2508.14240v3 Announce Type: replace Abstract: We define a Grassmann odd analogue of a Carrollian manifold as a supermanifold of dimension $n|1$ with an even degenerate metric such that the kernel is generated by a non-singular odd vector field that is a supersymmetry generator. Alongside other results, we establish that the reduced manifold is a pseudo-Riemannian manifold, and show that compatible affine connections always exist, albeit they must carry torsion. As a physically relevant example, we examine an In\"on\"u--Wigner contraction of the supertranslation algebra on standard superspace $\mathbb{R}^{4|4}$.
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https://arxiv.org/abs/2508.14240
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70ddd53305e5a300ad6af2e170fadd72b0fbc32e5efbd74ef4ff7384ddd01eaa
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2026-01-07T00:00:00-05:00
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Mutations of quivers with 2-cycles
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arXiv:2508.15022v2 Announce Type: replace Abstract: We develop a mutation theory for quivers with oriented 2-cycles using a structure called a homotopy, defined as a normal subgroupoid of the quiver's fundamental groupoid. This framework extends Fomin-Zelevinsky mutations of 2-acyclic quivers and yields involutive mutations that preserve the fundamental groupoid quotient by the homotopy. It generalizes orbit mutations arising from quiver coverings and allows for infinite mutation sequences even when orbit mutations are obstructed. We further construct quivers with homotopies from triangulations of marked surfaces with colored punctures, and prove that flips correspond to mutations, extending the Fomin-Shapiro-Thurston model to the setting with 2-cycles.
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https://arxiv.org/abs/2508.15022
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4af3d762162acc56320f815450faa94d8192f8df525d1ea0621fa15192b81ac2
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2026-01-07T00:00:00-05:00
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Symmetric Poisson geometry, totally geodesic foliations and Jacobi-Jordan algebras
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arXiv:2508.15890v3 Announce Type: replace Abstract: We introduce symmetric Poisson structures as pairs consisting of a symmetric bivector field and a torsion-free connection satisfying an integrability condition analogous to that in usual Poisson geometry. Equivalent conditions in Poisson geometry have inequivalent analogues in symmetric Poisson geometry and we distinguish between symmetric and strong symmetric Poisson structures. We prove that symmetric Poisson structures correspond to locally geodesically invariant distributions together with a characteristic metric, whereas strong symmetric Poisson structures correspond to totally geodesic foliations together with a leaf metric and a leaf connection. We introduce, using the Patterson-Walker metric, a dynamics on the cotangent bundle and show its connection to symmetric Poisson geometry, the parallel transport equation and the Newtonian equation for conservative systems. Finally, we prove that linear symmetric Poisson structures are in correspondence with Jacobi-Jordan algebras, whereas strong symmetric correspond to those that are moreover associative.
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https://arxiv.org/abs/2508.15890
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f2bc9eb25ada5f4a7f273241b6a0ab61b4a3b4992d9583889bd1e0b8134ced33
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2026-01-07T00:00:00-05:00
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Minimal ${A}_{\infty}$-algebras of endomorphisms: The case of $d\mathbb{Z}$-cluster tilting objects
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arXiv:2508.18852v2 Announce Type: replace Abstract: The Derived Auslander--Iyama Corresponence, a recent result of the authors, provides a classification up to quasi-isomorphism of the derived endomorphism algebras of basic $d\mathbb{Z}$-cluster tilting objects in $\operatorname{Hom}$-finite algebraic triangulated categories in terms of a small amount of algebraic data. In this note we highlight the role of minimal $A_\infty$-algebra structures in the proof of this result, as well as the crucial role of the enhanced $A_\infty$-obstruction theory developed by the second-named author.
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https://arxiv.org/abs/2508.18852
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b057c2b4f230750c23700d6fa2a4c695af8afcb97b146b71519bb914499a52ff
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2026-01-07T00:00:00-05:00
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Lipschitz-free spaces and Bossard's reduction argument
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arXiv:2509.00722v2 Announce Type: replace Abstract: We set up a descriptive set-theoretic framework to study Lipschitz-free spaces and use the reduction argument of Bossard to prove several results. We prove two universality results: if a separable Banach space is isomorphically universal for the class of Lipschitz-free spaces over the countable complete discrete metric spaces then it is isomorphically universal for the class of separable Banach spaces, and if a complete separable metric space is Lipschitz universal for the same class of metric spaces then it is Lipschitz universal for all separable metric spaces. We also show that there exist countable complete discrete metric spaces whose Lipschitz-free spaces fail the bounded approximation property and are thus not isomorphic to any dual Banach space. Finally, we calculate the descriptive complexity of the classes of separable Banach spaces and separable Lipschitz-free spaces having the approximation property.
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https://arxiv.org/abs/2509.00722
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da2103a0e9730732aad94231a4d2b1ba5a2e41acb45f51994928172afa4e2473
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2026-01-07T00:00:00-05:00
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Global existence of the irrotational Euler-Nordstr\"om equations with a positive cosmological constant: The gravitational field equation
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arXiv:2509.02023v4 Announce Type: replace Abstract: Our aim is to establish the global existence of classical solutions to the nonlinear irrotational Euler--Nordstr\"om system, which incorporates a linear equation of state and a cosmological constant. In this setting, gravitation is described by a single scalar field satisfying a specific semilinear wave equation. We restrict attention to spatially periodic perturbations of the background metric and therefore study this equation on the three-dimensional torus $\mathbb{T}^3$, working within the Sobolev spaces $H^m(\mathbb{T}^3)$. We begin by analysing the Nordstr\"om equation in isolation, with a source term generated by an irrotational fluid obeying a linear equation of state. This separation is motivated by the fact that such a fluid produces a source term containing a nonlinear contribution of fractional order. To obtain a global solution for the gravitational field, the fractional-order nonlinearity $(1+u)^\mu$, with $\mu\in\mathbb{R}$, must remain smooth throughout the evolution. This condition, in turn, requires that $u$ remain small for all time. We ensure this by introducing a suitably chosen energy functional. We also prove that, asymptotically, the solutions tend to a constant.
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https://arxiv.org/abs/2509.02023
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01c1e9c6a8534335c6f948a9939a778042abf3906048704fbff9616e6dc72680
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2026-01-07T00:00:00-05:00
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Concentration Inequalities for Branching Random Walk
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arXiv:2509.05860v2 Announce Type: replace Abstract: While classical concentration inequalities are typically restricted to two special cases -- independence and martingale difference sequences -- we extend concentration inequalities to a much broader class of stochastic processes by relaxing these foundational conditions. %\vspace{0.2\baselineskip} Specifically, heuristically and in the language of calculus, while independence and the martingale difference property correspond to \[ \displaystyle \frac { \partial y } {\partial t}= \text{constant}, \quad \displaystyle \frac { \partial y } {\partial t} = 0 \] respectively, %\vspace{0.3\baselineskip} we relax these conditions to %\[ \left| \frac { \partial^2 y } {\partial u_i \, \partial t} \right| \le L, \] %thereby allowing the drift $\displaystyle\frac { \partial y } {\partial t}$ to vary with past state $u_i$. \vspace{0.3\baselineskip} a general setting that requires only the existence of a drift $\displaystyle\frac { \partial y } {\partial t}$ which is allowed to vary with the past state. \vspace{0.3\baselineskip} Furthermore, concentration inequalities are established for branching random walks.
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https://arxiv.org/abs/2509.05860
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1919851ce01006dd4e39ce8e2655bf1f554a72181845fbff15729c33f92ef209
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2026-01-07T00:00:00-05:00
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Quantization of bounded symplectic domains associated with compact Lie groups
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arXiv:2509.05931v2 Announce Type: replace Abstract: We present a systematic quantization scheme for bounded symplectic domains of the form $D \times G \subset T^\ast G$, where $D \subset \mathfrak{g}^\ast$ is a bounded region defined by algebraic inequalities and $G$ is a compact Lie group with Lie algebra $\mathfrak{g}$. The finiteness of the symplectic volume implies that quantization yields a finite-dimensional Hilbert space, with observables represented by Hermitian matrices, for which we provide an explicit realization. Boundary effects necessitate modifications of the standard von Neumann and Dirac conditions, which usually underlie the correspondence principle. Physically, the compact group $G$ plays the role of momentum space, while $\mathfrak{g}^\ast$ corresponds to the (noncommutative) position space of a particle. The assumption of compact momentum space has profound physical consequences, including the supertunneling phenomenon and the emergence of a maximal fermion density.
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https://arxiv.org/abs/2509.05931
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8e750d93e50dddee6241278e56b634b5ea24196ffcfa24a0d8886b8a6426582a
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2026-01-07T00:00:00-05:00
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K-theoretic Hikita conjecture for quiver gauge theories
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arXiv:2509.06226v2 Announce Type: replace Abstract: We study variants of Hikita conjecture for Nakajima quiver varieties and corresponding Coulomb branches. First, we derive the equivariant version of the conjecture from the non-equivariant one for a set of gauge theories. Second, we suggest a variant of the conjecture, with K-theoretic Coulomb branches involved. We show that this version follows from the usual (homological) one for a set of theories. We apply this result to prove the conjecture in finite ADE types. In the course of the proof, we show that appropriate completions of K-theoretic and homological (quantized) Coulomb branches are isomorphic.
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https://arxiv.org/abs/2509.06226
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2e57e4a45969a9b9f3ddc7d704034d41751c441e6747cc2bba80cc5778c331fc
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2026-01-07T00:00:00-05:00
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$HS$-tensional maps and $HM$-tensional maps
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arXiv:2509.08564v2 Announce Type: replace Abstract: Let $\psi: (M,g)\longrightarrow (N,h)$ be a smooth map between Riemannian manifolds. The tension field of $\psi$ can be regarded as a map from $(M,g)$ into the Riemannian vector bundle $\psi^{-1}TN$, equipped with the Sasaki metric $G_{S}$. In this paper, we study certain aspects of two types of maps: those whose tension fields are harmonic maps (called $HM$-tensional maps) and those whose tension fields are harmonic sections (called $HS$-tensional maps).
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https://arxiv.org/abs/2509.08564
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5cff0179772a1d1ad3fa450aca61b1b73f1f47c85263ffc7c67a621ff3a53a1e
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2026-01-07T00:00:00-05:00
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Augmentations, reduced ideal point gluings and compact type degenerations of curves
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arXiv:2509.12429v2 Announce Type: replace Abstract: In this note we demonstrate some unexpected properties that simple gluings of the simplest derived categories may have. We consider two special cases: the first is an augmented curve, i.e., the gluing of the derived categories of a point and a curve with the gluing bimodule given by the structure sheaf of the curve; the second is an ideal point gluing of curves, i.e., the gluing of the derived categories of two curves with the gluing bimodule given by the ideal sheaf of a point in the product of the curves. We construct unexpected exceptional objects contained in these categories and discuss their orthogonal complements. We also show that the simplest example of compact type degeneration of curves, a flat family of curves with a smooth general fiber and a 1-nodal reducible central fiber, gives rise to a smooth and proper family of triangulated categories with the general fiber an augmented curve and the central fiber the orthogonal complement of the exotic exceptional object in the ideal point gluing of curves, called the reduced ideal point gluing of curves.
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https://arxiv.org/abs/2509.12429
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6100280458bc9df90903e4b19f9f50baea8c8f27d79780ee405b241aadb6f65c
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2026-01-07T00:00:00-05:00
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The Complexity of Arc-Connectedness Relation in the Plane
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arXiv:2509.24596v2 Announce Type: replace Abstract: In this paper, we show that the arc-connectedness equivalence relation on a Polish subspace of the real plane is an essentially hyperfinite Borel equivalence relation. This result provides the optimal upper bound for such a Borel equivalence relation.
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https://arxiv.org/abs/2509.24596
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c9f3b8fcce9bd6fc34de6ebde589339783c6fa80ba8fdb2d78f4facb11520e76
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2026-01-07T00:00:00-05:00
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Impact of memory on clustering in spontaneous particle aggregation
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arXiv:2510.15335v2 Announce Type: replace Abstract: The effect of short-term and long-term memory on spontaneous aggregation of organisms is investigated using a stochastic agent-based model. Each individual modulates the amplitude of its random motion according to the perceived local density of neighbors. Memory is introduced via a chain of $K$~internal variables that allow agents to retain information about previously encountered densities. The parameter $K$ controls the effective length of memory. A formal mean-field limit yields a macroscopic Fokker--Planck equation, which provides a continuum description of the system in the large-population limit. Steady states of this equation are characterized to interpret the emergence and morphology of clusters. Systematic stochastic simulations in one- and two-dimensional spatial domains reveal that short- or moderate-term memory promotes coarsening, resulting in a smaller number of larger clusters, whereas long-term memory inhibits aggregation and increases the proportion of isolated individuals. Statistical analysis demonstrates that extended memory reduces the agents' responsiveness to environmental stimuli, explaining the transition from aggregation to dispersion as $K$ increases. These findings identify memory as a key factor controlling the collective organization of self-driven agents and provide a bridge between individual-level dynamics and emergent spatial patterns.
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https://arxiv.org/abs/2510.15335
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4adcb2af839612dca6f7f316906c90940b2eb91e494bd07239bc7eba93d1e84c
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2026-01-07T00:00:00-05:00
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A Linear Representation for Functions on Finite Sets
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arXiv:2510.20167v5 Announce Type: replace Abstract: We demonstrate that any function $f$ from a finite set $Y$ to itself can be represented linearly. Specifically, we prove the existence of an injective map $j$ from $Y$ into a modular ring $\mathbb{Z}/m\mathbb{Z}$ and a constant $a \in \mathbb{Z}/m\mathbb{Z}$ such that $j(f(y)) = a \cdot j(y)$ in $\mathbb{Z}/m\mathbb{Z}$ holds for all $y \in Y$. This result is established by analyzing the algebraic properties of the adjugate of the characteristic matrix associated with the function's digraph. The proof is constructive, providing a method for finding the embedding $j$, the modulus $m$, and the linear multiplier $a$.
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https://arxiv.org/abs/2510.20167
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2026-01-07T00:00:00-05:00
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Heterochromatic two-arm probabilities for metric graph Gaussian free fields
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arXiv:2510.20492v2 Announce Type: replace Abstract: For the Gaussian free field on the metric graph of $\mathbb{Z}^d$ ($d\ge 3$), we consider the heterochromatic two-arm probability, i.e., the probability that two points $v$ and $v'$ are contained in distinct clusters of opposite signs with diameter at least $N$. For all $d\ge 3$ except the critical dimension $d_c=6$, we prove that this probability is asymptotically proportional to $N^{-[(\frac{d}{2}+1)\land 4]}$. Furthermore, we prove that conditioned on this two-arm event, the volume growth of each involved cluster is comparable to that of a typical (unconditioned) cluster; precisely, each cluster has a volume of order $M^{(\frac{d}{2}+1)\land 4}$ within a box of size $M$.
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https://arxiv.org/abs/2510.20492
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017c86601974026c78a59bf08e6c4cffae86dd7068441b8576b171a8579550c2
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2026-01-07T00:00:00-05:00
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Separation and cut edge in macroscopic clusters for metric graph Gaussian free fields
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arXiv:2510.20516v2 Announce Type: replace Abstract: We prove that for the Gaussian free field (GFF) on the metric graph of $\mathbb{Z}^d$ (for all $d\ge 3$ except the critical dimension $d_c=6$), with uniformly positive probability there exist two distinct sign clusters of diameter at least $cN$ within a box of size $N$ such that their graph distance is less than $N^{-[(d-2)\vee (2d-8)]}$. This phenomenon contrasts sharply with the two-dimensional case, where the distance between two macroscopic clusters is typically on the order of their diameters, following from the basic property of the scaling limit ``conformal loop ensembles'' $\mathrm{CLE}_4$ (Sheffield-Werner'2001). As a byproduct, we derive that the number of pivotal edges for the one-arm event (i.e., the sign cluster containing the origin has diameter at least $N$) is typically of order $N^{(\frac{d}{2}-1)\land 2}$. This immediately implies that for the incipient infinite cluster (IIC) of the metric graph GFF, the dimension of cut edges (i.e., edges whose removal disconnects the IIC) equals $(\frac{d}{2}-1)\land 2$. Translated in the language of critical loop soups (whose clusters, by the isomorphism theorem, have the same distribution as GFF sign clusters), this leads to the analogous estimates where the counterpart of a pivotal edge is a pivotal loop at scale $1$. This result hints at the new and possibly surprising idea that already in dimension $3$, microscopic loops (even those at scale $1$) play a crucial role in the construction of macroscopic loop clusters.
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https://arxiv.org/abs/2510.20516
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23b6fa82ad07a4237cfaa3be217448ccf96d9e7bbd383fe82a205419ea41c9fb
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2026-01-07T00:00:00-05:00
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On the gap between cluster dimensions of loop soups on $\mathbb{R}^3$ and the metric graph of $\mathbb{Z}^3$
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arXiv:2510.20526v2 Announce Type: replace Abstract: The question of understanding the scaling limit of metric graph critical loop soup clusters and its relation to loop soups in the continuum appears to be one of the subtle cases that reveal interesting new scenarios about scaling limits, with a mixture of macroscopic and microscopic randomness. In the present paper, we show that in three dimensions, scaling limits of the metric graph clusters are strictly larger than the clusters of the limiting continuum Brownian loop soup. We actually show that the upper box counting dimension of the latter clusters is strictly smaller than $5/2$, while that of the former is $5/2$.
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https://arxiv.org/abs/2510.20526
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a0a3498710275f703a6ddb303008267f3c3662afce0e606cc59893db9ec6a369
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2026-01-07T00:00:00-05:00
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The power of trees
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arXiv:2510.26419v2 Announce Type: replace Abstract: We give two consistent constructions of trees $T$ whose finite power $T^{n+1}$ is sharply different from $T^n$: 1. An $\aleph_1$-tree $T$ whose interval topology $X_T$ is perfectly normal, but $(X_T)^2$ is not even countably metacompact. 2. For an inaccessible $\kappa$ and a positive integer $n$, a $\kappa$-tree such that all of its $n$-derived trees are Souslin and all of its $(n+1)$-derived trees are special.
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https://arxiv.org/abs/2510.26419
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31a74710a24bb498af81298306caa0f29b23d51a1d916b0481887295c18613d9
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2026-01-07T00:00:00-05:00
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Arithmetic Properties of Several Generalized-Constant Sequences, with Implications for $\Gamma^{\left(n\right)}\left(1\right)$
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arXiv:2511.01849v4 Announce Type: replace Abstract: Neither the Euler-Mascheroni constant, $\gamma=0.577215...$, nor the Euler-Gompertz constant, $\delta=0.596347...$, is currently known to be irrational. However, it has been proved that at least one of them is transcendental. The two constants are related by a well-known equation of Hardy, equivalent to $\gamma+\delta/e=\mathrm{Ein}(1)$, which recently has been generalized to $\gamma^{(n)}+\delta^{(n)}/e=\eta^{(n)}$, $n\ge0$ for sequences of constants $\gamma^{(n)}$, $\delta^{(n)}$, and $\eta^{(n)}$ (given respectively by raw, conditional, and partial moments of the Gumbel(0,1) probability distribution). Investigating the $\gamma^{(n)}$ through recurrence relations (where $\gamma^{(0)}=1$ and $\gamma^{(1)}=\gamma$), we find that at least one of the pair {$\gamma,\gamma^{(2)}$} and -- conditional on a realistic conjecture verified for $2\leq n\leq26$ -- at least two of each set {$\gamma,\gamma^{(n)},\gamma^{(n+1)},\ldots,\gamma^{(2n)}$} are transcendental, implying that the $\gamma^{(n)}$ are transcendental infinitely often (with analogous results for the sequence $\Gamma^{(n)}(1)=\left(-1\right)^{n}\gamma^{\left(n\right)}$). We then show, via a theorem of Shidlovskii, that the $\eta^{(n)}$ are algebraically independent, and therefore transcendental, for all $n\ge0$, implying that at least one of each pair, {$\gamma^{(n)},\delta^{(n)}/e$} and {$\gamma^{(n)},\delta^{(n)}$}, and at least two of the triple {$\gamma^{(n)},\delta^{(n)}/e,\delta^{(n)}$}, are transcendental for all $n\ge1$. Further analysis of the $\gamma^{(n)}$ and $\eta^{(n)}$ reveals that the values $\delta^{(n)}/e$ are transcendental infinitely often with positive asymptotic density. Finally, we provide parallel results for the sequences $\tilde{\delta}^{(n)}$ and $\tilde{\eta}^{(n)}$ satisfying the "non-alternating analogue" equation $\gamma^{(n)}+\tilde{\delta}^{(n)}/e=\tilde{\eta}^{(n)}$.
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https://arxiv.org/abs/2511.01849
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36591f1ad2b508f1db56bae235f57e010602e606fa921a03ff90aca05601b983
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2026-01-07T00:00:00-05:00
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Completeness conditions for spacetimes with low-regularity metrics
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arXiv:2511.07867v2 Announce Type: replace Abstract: We extend Beem's three completeness notions -- finite compactness, timelike Cauchy completeness, and Condition A -- originally defined for spacetimes, to Lorentzian length spaces and study their relationships. We prove that finite compactness implies timelike Cauchy completeness and that timelike Cauchy completeness implies Condition A for globally hyperbolic Lorentzian length spaces. Furthermore, for globally hyperbolic $C^{1}$-spacetimes, we establish the equivalence of the three conditions assuming the causally non-branching and non-intertwining conditions, which in fact imply the continuity of the causal exponential map. These results can be regarded as a Hopf-Rinow type theorem for low-regularity Lorentzian geometry. The appendix presents examples of $C^{1}$-spacetimes -- where geodesic uniqueness may fail -- in which causal geodesics nevertheless behave well, illustrating the scope of our results.
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https://arxiv.org/abs/2511.07867
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db98b8cea3bec52421fb16b6b94600806a356b26fb68fdbeb9829a260ea9ac2d
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2026-01-07T00:00:00-05:00
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Weak optimal transport with moment constraints: constraint qualification, dual attainment and entropic regularization
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arXiv:2511.16211v2 Announce Type: replace Abstract: We consider weak optimal problems (possibly entropically penalized) incorporating both soft and hard (including the case of the martingale condition) moment constraints. Even in the special case of the martingale optimal transport problem, existence of Lagrange multipliers corresponding to the martingale constraint is notoriously hard (and may fail unless some specific additional assumptions are made). We identify a condition of qualification of the hard moment constraints (which in the martingale case is implied by well-known conditions in the literature) under which general dual attainment results are established. We also analyze the convergence of entropically regularized schemes combined with penalization of the moment constraint and illustrate our theoretical findings by numerically solving in dimension one, the Brenier-Strassen problem of Gozlan and Juillet and a family of problems which interpolates between monotone transport and left-curtain martingale coupling of Beiglb\"{o}ck and Juillet.
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https://arxiv.org/abs/2511.16211
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8efcf4248bdbe11edc854c73543e52391b1e50ab51b9e13d75ea47de684d9671
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2026-01-07T00:00:00-05:00
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Constancy of an Infinite Cyclotomic Product via Ramanujan Sums
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arXiv:2511.16975v2 Announce Type: replace Abstract: We show that the infinite product defined by \[ P(z) = -\prod_{n=1}^{\infty} (\Phi_n(z))^{-1/n}, \] where \( \Phi_n(z) \) is the \( n \)-th cyclotomic polynomial, is constant inside the unit disk. The proof translates a result of Ramanujan on Ramanujan sums, equivalent to the prime number theorem, to the setting of infinite products. We also show that similar identities proved by Ramanujan lead to additional results on infinite cyclotomic products.
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https://arxiv.org/abs/2511.16975
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17e8d4e3b4ed0785856d66e8be318fd5cdc39b14af9f554493c5b9fbdbc7ba99
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2026-01-07T00:00:00-05:00
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Boundary regularity and a priori estimates for fractional equations on unbounded domains
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arXiv:2511.17325v2 Announce Type: replace Abstract: In this paper, we study the boundary H\"older regularity for solutions to the fractional Dirichlet problem in unbounded domains with boundary \begin{equation*} \begin{cases} (-\Delta)^s u(x) = g(x),&\text{in } \Omega, u(x)=0, &\text{in } \Omega^c. \end{cases} \end{equation*} Existing results rely on the global $L^{\infty}$ norm of solutions to control their boundary $C^s$ norm, which is insufficient for blow-up and rescaling analysis to obtain a priori estimates in unbounded domains. To overcome this limitation, we first derive a local version of boundary H\"older regularity for nonnegative solutions in which we replace the global $L^{\infty}$ norm by only a local $L^{\infty}$ norm. Then as an important application, we establish a priori estimates for nonnegative solutions to a family of nonlinear equations on unbounded domains with boundaries.
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https://arxiv.org/abs/2511.17325
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c802a2ac566453d587de87c2cc7014c4a1b6be66bdc7d4efffd4e3d13b98b86e
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2026-01-07T00:00:00-05:00
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More on the sum-product problem for integers with few prime factors
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arXiv:2512.04931v2 Announce Type: replace Abstract: We show that if $A\subset \mathbb{Z}$ is a finite set of integers in which every integer is divisible by $O(1)$ many primes then \[\max(\lvert A+A\rvert,\lvert AA\rvert) \geq \lvert A\rvert^{12/7-o(1)}\] and, for any $m\geq 2$, \[\max(\lvert mA\rvert, \lvert A^{(m)}\rvert) \geq \lvert A\rvert^{\frac{2}{3}m+\frac{1}{3}-o(1)}.\] Finally, we show that if $A\subset \mathbb{Q}$ is a finite set of rationals in which the numerator and denominator of every $x\in A$ is divisible by $O(1)$ many primes then $\lvert A+AA\rvert \geq \lvert A\rvert^{2-o(1)}$.
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https://arxiv.org/abs/2512.04931
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67b3d08608605eddb3c39ab0a0b0840091d1a78c0a05888b2d57296f79989a0a
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2026-01-07T00:00:00-05:00
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Singularity of the loops within a cable-graph loop-soup conditioned by its occupation time
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arXiv:2512.05086v2 Announce Type: replace Abstract: In this note, we show the following feature of the relation between Brownian loop-soups on cable-graphs and their total occupation time-field $\Lambda$: When conditioned on $\Lambda$, the conditional law of individual loops becomes singular with respect to that of unconditioned loops. The idea of the proof is to see that some type of fast points on the curve $\Lambda$ impose an exceptional behaviour of all the loops when they go through these points.
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https://arxiv.org/abs/2512.05086
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2dac483a32c9f34940d3c697b4fc28bff4a806b0cdaaa2124dfd68f563b5eeef
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2026-01-07T00:00:00-05:00
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Critical behaviour of the fully packed loop-$O(n)$ model on planar triangulations
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arXiv:2512.05867v2 Announce Type: replace Abstract: We study the fully packed loop-$O(n)$ model on planar triangulations. This model is also bijectively equivalent to the Fortuin--Kasteleyn model of planar maps with parameter $q\in (0,4)$ at its self-dual point. These have been traditionally studied using either techniques from analytic combinatorics (based in particular on the gasket decomposition of Borot, Bouttier and Guitter arXiv:1106.0153) or probabilistic arguments (based on Sheffield's hamburger-cheeseburger bijection arXiv:1108.2241). In this paper we establish a dictionary relating quantities of interest in both approaches. This has several consequences. First, we derive an exact expression for the partition function of the fully packed loop-$O(n)$ model on triangulations, as a function of the outer boundary length. This confirms predictions by Gaudin and Kostov. In particular, this model exhibits critical behaviour, in the sense that the partition function exhibits a power-law decay characteristic of the critical regime at this self-dual point. Finally, we derive precise asymptotics for geometric features of the FK model of planar maps when $0 < q <4$, such as the exact tail behaviour of the perimeters of clusters and loops. This sharpens previous results of arXiv:1502.00450 and arXiv:1502.00546. A key step is to use the above dictionary and the probabilistic results to justify rigorously an ansatz commonly assumed in the analytic combinatorics literature.
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https://arxiv.org/abs/2512.05867
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25bc565f681d6aab8abf01b5bc394bad6d1fcef49f48dd1b7e072248b5c74b94
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2026-01-07T00:00:00-05:00
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Low-degree mod 2 cohomology of classifying spaces of $G_2$-gauge groups
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arXiv:2512.12195v2 Announce Type: replace Abstract: Let $G$ be a simply connected compact simple Lie group and let $\mathcal{G}_k$ denote the gauge group of a principal $G$--bundle over $S^4$ with second Chern class $k\in \pi_4(BG)\cong \mathbb Z$. For $G=G_2$, the $p$--local homotopy types of the gauge groups have been completely classified by Kishimoto--Theriault--Tsutaya and Kameko in terms of the order of the fundamental Samelson product $\langle i_3,1\rangle\in [\Sigma^3G_2,G_2]$. In this paper, we begin a complementary study of the mod $2$ cohomology of the classifying spaces $B\mathcal{G}_k(G_2)$. Our goal is to understand the structure of $H^*(B\mathcal{G}_k;\mathbb{F}_2)$ as an unstable module over the mod~$2$ Steenrod algebra in a low range of degrees. Using the evaluation fibration \[ \Omega_0^3 G_2 \longrightarrow B\mathcal{G}_k \xrightarrow{\;\mathrm{ev}\;} BG_2 \] together with Serre and Eilenberg--Moore spectral sequences, we study the Serre spectral sequence \[ H^s(BG_2;H^t(\Omega^3_0G_2)) \Longrightarrow H^{s+t}(B\mathcal{G}_k) \] in total degree $\le 10$. A careful analysis of the homotopy groups of $G_2$ shows that \[ H^j(\Omega^3_0G_2;\mathbb{F}_2)=0\quad\text{for }1\le j\le 4, \qquad H^5(\Omega^3_0G_2;\mathbb{F}_2)\neq 0, \] so the first positive-degree generator of the fibre cohomology occurs in degree $5$. As a consequence, there is a distinguished class \[ u_5\in H^5(\Omega^3_0G_2;\mathbb{F}_2) \] whose only possible Serre differential in total degree $\le 10$ is a $d_6$--differential \[ d_6(u_5) = \epsilon(k)\,x_6 \] from $u_5$ to the degree-$6$ generator $x_6\in H^6(BG_2;\mathbb{F}_2)$, for a scalar $\epsilon(k)\in\mathbb{F}_2$ encoding the low-degree effect of the bundle class. In addition, $2$--locally we prove that $\epsilon(k)$ is $4$--periodic in $k$ (i.e. it depends only on $k\bmod 4$) and that $\epsilon(k)=0$ for all $k\equiv 0\pmod 4$.
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https://arxiv.org/abs/2512.12195
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2b58382cc23672e899380aa52018b2dd8f23f931349e16f57663d88c0eb9c136
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2026-01-07T00:00:00-05:00
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Determining subgroups via stationary measures
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arXiv:2512.12966v2 Announce Type: replace Abstract: In this paper, we consider random walks on the isometry groups of general metric spaces. Under some mild conditions, we show that if two non-elementary random walks on a discrete subgroup of the isometry group have non-singular stationary measures, then subgroups generated by the random walks are commensurable. This result in particular applies to separable Gromov hyperbolic spaces and Teichm\"uller spaces. As a specific application, we prove singularity between stationary measures associated to random walks on different fiber subgroups of the fundamental group of a hyperbolic 3-manifold fibering over the circle.
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https://arxiv.org/abs/2512.12966
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4ae1a91aecb7725ffd96834bf35078b4866304679f6a90fecd661d889bc5fefd
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2026-01-07T00:00:00-05:00
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Asymptotics of the graph Laplace operator near an isolated singularity
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arXiv:2512.13314v3 Announce Type: replace Abstract: In this paper, we investigate asymptotics of the continuous graph Laplace operator on a smooth Riemannian manifold $(M,g)$ admitting an isolated singularity $x$. We show that if the curvature function $\kappa$ doesn't grow too fast near $x$, then the graph Laplace operator at $x$ converges to the weighted Laplace-Beltrami operator as the bandwidth $t\downarrow 0.$ On the other hand, we also prove that if one locally modifies a given Riemannian metric across $x$ by a non-constant \textit{purely angular }conformal factor, then $\kappa$ grows too fast and the graph Laplace operator behaves like $O(\frac{1}{\sqrt{t}})$ near $x$, as $t\downarrow 0$, given a mild condition on the angular conformal factor. We provide the Taylor expansion of the graph Laplace operator as $t\downarrow 0$ in specific cases. Numerical simulations at the end illustrate our results.
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https://arxiv.org/abs/2512.13314
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7d921d3d03f8c1136a66d534704722a1b7f2d9503648da9077bf18eb8d12ab9d
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2026-01-07T00:00:00-05:00
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Graph Sensitivity under Join and Decomposition
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arXiv:2512.19915v2 Announce Type: replace Abstract: The sensitivity, $\sigma(G)$, of a finite undirected simple graph $G$ is the smallest maximum degree of an induced subgraph on more than the maximum number of independent vertices. Call an indexed family of graphs $G_n$ with maximum degree $\Delta(G_n) \to \infty$ as $n \to \infty$ sensitive if $\sigma(G_n) \to \infty$, and insensitive otherwise. We describe sensitivity under the join operation and decomposition into stable blocks and construct sensitive and insensitive, primarily non-regular, graph families. We determine the sensitivity explicitly for numerous singly- and doubly-indexed graph families, including certain generalized joins - e.g., complete multipartite graphs and some generalized windmill graphs; general rooted products; and families of corona graphs.
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https://arxiv.org/abs/2512.19915
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968c6a6e2cc9553971fe4b1d532c40a45a63e5dfce8017590aa680b23f30edb3
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2026-01-07T00:00:00-05:00
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Regularization methods for solving hierarchical variational inequalities with complexity guarantees
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arXiv:2512.20772v2 Announce Type: replace Abstract: We consider hierarchical variational inequality problems, or more generally, variational inequalities defined over the set of zeros of a monotone operator. This framework includes convex optimization over equilibrium constraints and equilibrium selection problems. In a real Hilbert space setting, we combine a Tikhonov regularization and a proximal penalization to develop a flexible double-loop method for which we prove asymptotic convergence and provide rate statements in terms of gap functions. Our method is flexible, and effectively accommodates a large class of structured operator splitting formulations for which fixed-point encodings are available. Finally, we validate our findings numerically on various examples.
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https://arxiv.org/abs/2512.20772
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96f87c9ba338ebe4350b0eb89d12dc6b8af776ce1b39da34aeed53b16d04ee70
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2026-01-07T00:00:00-05:00
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Trisections and Lefschetz fibrations with $(-n)$-sections
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arXiv:2512.21001v2 Announce Type: replace Abstract: Castro and Ozbagci constructed a trisection of a closed 4-manifold admitting a Lefschetz fibration with a $(-1)$-section such that the corresponding trisection diagram can be explicitly constructed from a monodromy of the Lefschetz fibration. In this paper, for a closed 4-manifold $X$ admitting an achiral Lefschetz fibration with a $(-n)$-section, we construct a trisection of $X \# n\mathbb{C}P^2$ if $n$ is positive and $X \# (-n)\overline{\mathbb{C}P^2}$ if $n$ is negative such that the corresponding trisection diagram can be explicitly constructed from a monodromy of the Lefschetz fibration. We also construct a trisection of the fiber sum of two achiral Lefschetz fibrations with $n$- and $(-n)$-sections such that the corresponding trisection diagram can be explicitly constructed from monodromies of the Lefschetz fibrations.
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https://arxiv.org/abs/2512.21001
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d6097e9e493669d079bef6b65f7f8f54b7cab3c7121b9faab3193a3e56acf90c
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2026-01-07T00:00:00-05:00
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Non-finite generatedness of the congruences defined by tropical varieties
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arXiv:2512.21565v2 Announce Type: replace Abstract: In tropical geometry, there are several important classes of ideals and congruences such as tropical ideals, bend congruences, and the congruences of the form $\mathbf E(Z)$. Although they are analogues of the concept of ideals of rings, it is not well known whether they are finitely generated. In this paper, we study whether the congruences of the form $\mathbf E(Z)$ are finitely generated. In particular, we show that when $Z$ is the support of a tropical variety, $\mathbf E(Z)$ is not finitely generated except for a few specific cases. In addition, we give an explicit minimal generating set of $\mathbf E(|L|)$ for the tropical standard line $L$.
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https://arxiv.org/abs/2512.21565
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9ab58c0d95d59dba2ba026dd3b60d96f294a865990c409853ac87f7257d63083
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2026-01-07T00:00:00-05:00
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MSO logic of the real order with the set quantifier ranging over Borel sets
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arXiv:2512.23003v3 Announce Type: replace Abstract: A celebrated 1969 theorem of Michael Rabin is that the MSO theory of the real order where the monadic quantifier is allowed only to range over the sets of rational numbers, is decidable. In 1975 Saharon Shelah proved that if the monadic quantifier is allowed to range over all subsets of the reals, the resulting MSO theory is undecidable. He conjectured that when we allow the monadic quantifier to range over the Borel subsets of the reals, the resulting MSO theory is decidable. We confirm this conjecture. Namely, the MSO theory of the real order where the set quantifier is allowed to range over the Borel sets, is decidable.
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https://arxiv.org/abs/2512.23003
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4bc30543c81c2c5808310012c417526031844d245b18a32e249393e1c9b3997c
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2026-01-07T00:00:00-05:00
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Structure preservation and emergent dissipation in stochastic wave equations with transport noise
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arXiv:2512.23309v2 Announce Type: replace Abstract: We study nonlinear wave equations perturbed by transport noise acting either on the displacement or on the velocity. Such noise models random advection and, under suitable scaling of space covariance, may generate an effective dissipative term. We establish well-posedness in both cases and analyse the associated scaling limits. When the noise acts on the displacement, the system preserves its original structure and converges to the deterministic nonlinear wave equation, whereas if it acts on the velocity, the rescaled dynamics produce an additional Laplacian damping term, leading to a stochastic derivation of a Westervelt-type acoustic model.
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https://arxiv.org/abs/2512.23309
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2f95f08c951e7d123288bac3d4f5e3e05c6e4bbb642b1db2abf6ffe641bda8f3
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2026-01-07T00:00:00-05:00
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The Lagrangian and symplectic structures of the Kuramoto oscillator model
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arXiv:2601.00113v2 Announce Type: replace Abstract: Despite being under intense scrutiny for 50 years, the Kuramoto oscillator model has remained a quintessential representative of non-equilibrium phase transitions. One of the reasons for its enduring relevance is the apparent lack of an optimization formulation, due to the fact that (superficially), the equations of motion seem to not be compatible with a Lagrangian structure. We show that, as a mean-field classical (twisted) spin model on $S^2$, the Kuramoto model can be described variationaly. Based on this result perturbation analysis around (unstable) Kuramoto equilibria are shown to be equivalent to low-energy fluctuations of mean-field Heisenberg spin models. Intriguingly, off-plane perturbations around these equilibria configurations turn out to be described by a semiclassical Gaudin model, pointing to the fact that oscillator synchronization maps to the spin pairing mechanism investigated by Richardson and subsequently by others.
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https://arxiv.org/abs/2601.00113
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0f859699c91211260041e4250f1102683b7b43116747081f052dea3a5dd38721
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2026-01-07T00:00:00-05:00
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Bilinear forms with Kloosterman fractions and applications
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arXiv:2601.00292v2 Announce Type: replace Abstract: We establish improved bounds for bilinear forms with Kloosterman fractions of the form ${\sum\sum}_{m,n} \alpha_m \beta_n e(a\overline{m}/(bn))$ with $M< n \le 2N$ and $(m,n)=1$. Our approach works directly with arbitrary coefficient sequences $(\alpha_m), (\beta_n) \in \mathbb{C}$, avoiding the temporary restriction to squarefree support used in prior work. While this requires handling additional arithmetic complexity, it yields strictly stronger bounds that improve upon the estimates of Duke, Friedlander, and Iwaniec \cite{DFI} and Bettin-Chandee \cite{BC}; in the balanced case $M \approx N$, the new saving over the trivial bound is $1/12$%, compared to $1/48$ in \cite{DFI} . As an application, we prove a generalized asymptotic formula for the twisted second moment of the Riemann zeta-function with Dirichlet polynomials of length $T^{1/2+\delta}$ for $\delta = 1/46$, extending beyond the previously limiting $\theta = 1/2$ barrier established by Bettin, Chandee, and Radziwi{\l}{\l} \cite{BCR}. We also establish bounds for related Hermitian sums involving Sali\'{e}-type exponential phases and develop techniques for more general bilinear forms with Kloosterman fractions.
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https://arxiv.org/abs/2601.00292
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9e3b4d790b01808115a54b0cd7087d463ca5f50b92792ccd24db4099d936caa0
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2026-01-07T00:00:00-05:00
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Hadamard-type formulas for real eigenvalues of canonically symplectic operators
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arXiv:2601.00520v2 Announce Type: replace Abstract: We give first-order asymptotic expansions for the resolvent and Hadamard-type formulas for the eigenvalue curves of one-parameter families of canonically symplectic operators. We allow for parameter dependence in the boundary conditions, bounded perturbations and trace operators associated with each off-diagonal operator, and give formulas for derivatives of eigenvalue curves emanating from the discrete eigenvalue of the unperturbed operator in terms of Maslov crossing forms. We derive the Hadamard-type formulas using two different methods: via a symplectic resolvent difference formula and asymptotic expansions of the resolvent, and using Lyapunov-Schmidt reduction and the implicit function theorem. The latter approach facilitates derivative formulas when the eigenvalue curves are viewed as functions of the spectral parameter. We apply our abstract results to derive a spectral index theorem for the linearised operator associated with a standing wave in the nonlinear Schr\"odinger equation on a compact star graph.
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https://arxiv.org/abs/2601.00520
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927a748509a27d377b40b96896a1dcc1838698cf7f12b36bef48c06482192204
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2026-01-07T00:00:00-05:00
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Study of Composition Operators in Certain Functional Spaces
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arXiv:2601.00801v2 Announce Type: replace Abstract: In this thesis we study three problems. The first is the superposition of the operators and their proprities, such as boundedness,continuity,regularity and the inequalities of the norms of the composition of functions in some functional spaces. The second is to generalize some results of the composition of more than two functions, and the third is to give a generalization of Peetre's theorem.
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https://arxiv.org/abs/2601.00801
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3403e4e7ab2790e0769c4d757afdfa7d3d7e1078c9d70043bb0cc3bc41bd2b2d
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2026-01-07T00:00:00-05:00
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The Maximum of the Volume of a Part of a Cevian Simplex
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arXiv:2601.00876v2 Announce Type: replace Abstract: The cevians passing through a point in a simplex create a cevian simplex, which is divided by these cevians into smaller simplices. We consider the problem about the maximum of the ratio of the sum of the volumes of some of these smaller simplices by the volume of the reference simplex. The special case of tetrahedron is given as an example.
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https://arxiv.org/abs/2601.00876
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Academic Papers
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dd715d14eb088e40151f43911cbeebb61a82972c303459b9da58efe6535d2e4d
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2026-01-07T00:00:00-05:00
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On the universal curve with unordered marked points in positive characteristic
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arXiv:2601.01336v2 Announce Type: replace Abstract: We study the relative pro-$\ell$ and continuous relative completions of the algebraic fundamental groups of universal curves over the moduli stack of curves with unordered marked points in positive characteristic. Using specialization and homotopy exact sequences, we compare the ordered and unordered settings and prove that the natural projection from the relative completion of the universal curve over the unordered moduli stack admits no section in positive characteristic. This yields a non-splitting result for the corresponding projection on algebraic fundamental groups. The present paper is a sequel to our earlier work in characteristic zero.
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https://arxiv.org/abs/2601.01336
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Academic Papers
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7acffe08cf7847f779d94daf965bd43ab05e432c646063040fb7507d1f1e0de0
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2026-01-07T00:00:00-05:00
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Iterating PP-packages without Choice: A Cohen symmetric seed and a localization framework
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arXiv:2601.01855v2 Announce Type: replace Abstract: The Partition Principle $\mathsf{PP}$ asserts that whenever there is a surjection $A\twoheadrightarrow B$, there is an injection $B\hookrightarrow A$. Russell conjectured in 1906 that $\mathsf{PP}$ is equivalent to the Axiom of Choice $\mathsf{AC}$; while $\mathsf{AC}\Rightarrow \mathsf{PP}$ is immediate, the converse has remained open. We show that $\mathsf{PP}$ does not imply $\mathsf{AC}$ by constructing a transitive model of $\mathsf{ZF}+\mathsf{DC}+\mathsf{PP}+\neg\mathsf{AC}$. Starting from a Cohen symmetric model $\mathcal{N}$ of $\mathrm{Add}(\omega,\omega_1)$ built with a countable-support symmetry filter, we fix parameters $S:=A^\omega$ and $T:=\mathcal{P}(S)$ and perform a class-length countable-support symmetric iteration. At successor stages we use orbit-symmetrized packages that split targeted surjections, yielding $\mathsf{PP}\!\restriction T$ and $\mathsf{AC}_{\mathsf{WO}}$, while preserving $\mathsf{DC}$ and ensuring that $A$ remains non-well-orderable. A diagonal-cancellation/diagonal-lift infrastructure supplies a proper $\omega_1$-complete normal filter at limit stages. Finally, Ryan--Smith localization shows that under $\mathsf{SVC}^+(T)$, $\mathsf{PP}$ is equivalent to $\mathsf{PP}\!\restriction T \wedge \mathsf{AC}_{\mathsf{WO}}$, so the final model satisfies $\mathsf{PP}$ but not $\mathsf{AC}$.
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https://arxiv.org/abs/2601.01855
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1c9ffa91d7bac26f160ac23fa2c69f155d37ab58f3eab1ac1f3634b560f57079
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2026-01-07T00:00:00-05:00
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A Perturbed DCA for Computing d-Stationary Points of Nonsmooth DC Programs
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arXiv:2601.02084v2 Announce Type: replace Abstract: This paper introduces an efficient perturbed difference-of-convex algorithm (pDCA) for computing d-stationary points of an important class of structured nonsmooth difference-of-convex problems. Compared to the principal algorithms introduced in [J.-S. Pang, M. Razaviyayn, and A. Alvarado, Math. Oper. Res. 42(1):95--118 (2017)], which may require solving several subproblems for a one-step update, pDCA only requires solving a single subproblem. Therefore, the computational cost of pDCA for one-step update is comparable to the widely used difference-of-convex algorithm (DCA) introduced in [D. T. Pham and H. A. Le Thi, Acta Math. Vietnam. 22(1):289--355 (1997)] for computing a critical point. Importantly, under practical assumptions, we prove that every accumulation point of the sequence generated by pDCA is a d-stationary point almost surely. Numerical experiment results on several important examples of nonsmooth DC programs demonstrate the efficiency of pDCA for computing d-stationary points.
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https://arxiv.org/abs/2601.02084
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afe53a696e4e8a469f34f3a886e66f2b7d14b08dccb14c815dff074fd0f80b1f
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2026-01-07T00:00:00-05:00
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Square integrability of regular representations on reductive homogeneous spaces
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arXiv:2601.02188v2 Announce Type: replace Abstract: Let $G$ be a real reductive Lie group and $H$ a reductive subgroup of $G$. Benoist-Kobayashi studied when $L^2(G/H)$ is a tempered representation of $G$ and in particular they gave a necessary and sufficient condition for the temperedness in terms of certain functions on Lie algebras. In this paper, we consider when $L^2(G/H)$ is equivalent to a unitary subrepresentation of $L^2(G)$ and we will give a sufficient condition for this in terms of functions introduced by Benoist-Kobayashi. As a corollary, we prove the non-existence of discrete series for homogeneous spaces $G/H$ satisfying certain conditions.
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https://arxiv.org/abs/2601.02188
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ab0cd21ad003cfa7cbd15306a8dee26c71d774cfafe421e1ce86cdd2ccf4ec67
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2026-01-07T00:00:00-05:00
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Integrable systems on multiplicative quiver varieties from cyclic quivers
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arXiv:2108.02496v3 Announce Type: replace-cross Abstract: We consider a class of complex manifolds constructed as multiplicative quiver varieties associated with a cyclic quiver extended by an arbitrary number of arrows starting at a new vertex. Such varieties admit a Poisson structure, which is obtained by quasi-Hamiltonian reduction. We construct several families of Poisson subalgebras inside the coordinate ring of these spaces, which we use to obtain degenerately integrable systems. We also extend the Poisson centre of these algebras to maximal abelian Poisson algebras, hence defining Liouville integrable systems. By considering a suitable set of local coordinates on the multiplicative quiver varieties, we can derive the local Poisson structure explicitly. This allows us to interpret the integrable systems that we have constructed as new generalisations of the spin Ruijsenaars-Schneider system with several types of spin variables.
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https://arxiv.org/abs/2108.02496
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77c06da0edae26a6f5232d5913fc236999a8ad78f998dc5fbb764e465888bfd9
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2026-01-07T00:00:00-05:00
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Using prior information to boost power in correlation structure support recovery
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arXiv:2111.11278v2 Announce Type: replace-cross Abstract: Hypothesis testing of structure in correlation and covariance matrices is of broad interest in many application areas. In high dimensions and/or small to moderate sample sizes, high error rates in testing is a substantial concern. This article focuses on increasing power through a frequentist assisted by Bayes (FAB) procedure. This FAB approach boosts power by including prior information on the correlation parameters. In particular, we suppose there is one of two sources of prior information: (i) a prior dataset that is distinct from the current data but related enough that it may contain valuable information about the correlation structure in the current data; and (ii) knowledge about a tendency for the correlations in different parameters to be similar so that it is appropriate to consider a hierarchical model. When the prior information is relevant, the proposed FAB approach can have significant gains in power. A divide-and-conquer algorithm is developed to reduce computational complexity in massive testing dimensions. We show improvements in power for detecting correlated gene pairs in genomic studies while maintaining control of Type I error or false discover rate (FDR).
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https://arxiv.org/abs/2111.11278
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070725a79b0b9eeeb1b271d3a4a355f309570692aa3232fc65f1c5aac276debd
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2026-01-07T00:00:00-05:00
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On the Structure of Wave Functions in Complex Chern-Simons Theory
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arXiv:2312.00624v2 Announce Type: replace-cross Abstract: We study the structure of wave functions in complex Chern-Simons theory on the complement of a hyperbolic knot, emphasizing the similarities with the topological string/spectral theory correspondence. We first conjecture a hidden integrality structure in the holomorphic blocks and show that this structure guarantees the cancellation of potential singularities in the full non-perturbative wave function at rational values of the coupling constant. We then develop various techniques to determine the wave function at such rational points. Finally, we illustrate our conjectures and obtain explicit results in the examples of the figure-eight and three-twist knots. In the case of the figure-eight knot, we also perform a direct evaluation of the state integral in the rational case and observe that the resulting wave function has the features of the ground state for a quantum mirror curve.
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https://arxiv.org/abs/2312.00624
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6f8fa73484b7c87c18f41b7eead68f0e2da616496752a0aa0923c48283499ed2
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2026-01-07T00:00:00-05:00
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Semiparametric fiducial inference for Cox models
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arXiv:2404.18779v2 Announce Type: replace-cross Abstract: R. A. Fisher introduced the concept of fiducial as a potential replacement for the Bayesian posterior distribution in the 1930s. During the past century, fiducial approaches have been explored in various parametric and nonparametric settings. However, to the best of our knowledge, no fiducial inference has been developed in the realm of semiparametric statistics. In this paper, we propose a novel fiducial approach for semiparametric models. To streamline our presentation, we use the Cox proportional hazards model, which is the most popular model for the analysis of survival data, as a running example. Other models and extensions are also discussed. In our experiments, we find our method to perform well especially in situations when the maximum likelihood estimator fails.
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https://arxiv.org/abs/2404.18779
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Academic Papers
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4efbf64cac263173b76e771473ba19c1d773af2bccc8bc144854ebaf91e12510
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2026-01-07T00:00:00-05:00
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Bell-CHSH inequality and unitary transformations in Quantum Field Theory
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arXiv:2412.03840v3 Announce Type: replace-cross Abstract: Unitary transformations are employed to enhance the violations of the Bell-CHSH inequality in relativistic Quantum Field Theory. The case of the scalar field in $1+1$ Minkowski space-time is scrutinized by relying on the Tomita-Takesaki modular theory. The example of the bounded Hermitian operator $sign(\varphi(f))$, where $\varphi(f)$ stands for the smeared scalar field, is worked out. It is shown that unitary deformations enable for violations of the Bell-CHSH inequality. The setup is generalized to the Proca vector field by means of its equivalence with the scalar theory.
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https://arxiv.org/abs/2412.03840
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e2be221752c2d7ba6af7c943991f8092083e09900a13f31f06ac5d830608af79
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2026-01-07T00:00:00-05:00
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Robust Quantum Control for Bragg Pulse Design in Atom Interferometry
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arXiv:2502.04618v3 Announce Type: replace-cross Abstract: We formulate a robust optimal control algorithm to synthesize minimum energy pulses that can transfer a cold atom system into various momentum states. The algorithm uses adaptive linearization of the evolution operator and sequential quadratic programming to iterate the control towards a minimum energy pulse that achieves optimal target state fidelity. Robustness to parameter variation is achieved using Legendre polynomial approximation over the domain of variation. The method is applied to optimize the Bragg beamsplitting operation in ultra-cold atom interferometry. Even in the presence of 10-40% variability in the initial momentum dispersion of the atomic cloud and the intensity of the optical pulse, the algorithm reliably converges to a control protocol that robustly achieves unprecedented momentum levels with high fidelity for a single-frequency multi-photon Bragg diffraction scheme (e.g. $|\pm 40\hbar k\rangle$). We show the advantages of our method by comparison to stochastic optimization using sampled parameter values, provide detailed sensitivity analyses, and performance of the designed pulses is verified in laboratory experiments.
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https://arxiv.org/abs/2502.04618
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Academic Papers
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939f58aa5056ca5175e31f4c8db395e1457fcb09117d7d991e66ab794bb8646e
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2026-01-07T00:00:00-05:00
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Shift orbifolds, decompactification limits, and lattices
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arXiv:2502.18453v2 Announce Type: replace-cross Abstract: We describe the general shift orbifold of a Narain CFT and use this to investigate decompactification limits in the heterotic Narain moduli space. We also comment on higher rank theories and describe some applications to the CFT based on the Leech lattice and its shift orbifolds.
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https://arxiv.org/abs/2502.18453
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Academic Papers
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4c08dce512be047038916cb9b08dd5697106c9fb3df7ac531ce1af7a3b70f28d
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2026-01-07T00:00:00-05:00
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Toward a Complexity Classification of High-Temperature Bosons: Computational Tractability and Power-Law Clustering
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arXiv:2509.25572v2 Announce Type: replace-cross Abstract: Determining when quantum many-body systems admit simple, efficiently simulable structure is a central problem. High-temperature thermal states are a natural candidate for such simplicity, yet for bosons, the unbounded local Hilbert space and energy invalidate the usual expectation that large $T$ guarantees tractability. Here we investigate the resulting complexity boundary for interacting lattice bosons and show that the repulsive Bose--Hubbard class lies on the ``simple'' side. For a family with long-range hopping decaying as $r^{-\alpha}$, we prove convergence of a controlled cluster expansion, which implies (above an explicit temperature threshold) an efficient classical algorithm to approximate the partition function and a rigorous power-law clustering bound for connected correlations. More broadly, our results provide a first step toward charting complexity boundaries for high-temperature bosons and suggest the repulsive Bose--Hubbard class as a natural candidate cusp.
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https://arxiv.org/abs/2509.25572
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Academic Papers
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66af70461a653db28c2c456955bbb6e1aefe8a1b28e926cb9625a49bd94191da
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2026-01-07T00:00:00-05:00
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A new super integrable hierarchy and a generalized super-AKNS hierarchy
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arXiv:2509.25995v2 Announce Type: replace-cross Abstract: In this paper, we investigate a non-isospectral problem on the loop algebra of the Lie superalgebra osp(1,6), and construct an super integrable system using the supertrace identity. The resulting super integrable system can be reduced to the super-AKNS hierarchy under certain conditions. By reconsidering a new (2 + 1)-dimensional non-isospectral problem with spectral matrices satisfying these conditions, we obtain a (2+1)-dimensional generalization of the superAKNS hierarchy.
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https://arxiv.org/abs/2509.25995
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4f77b41d0fcd3bb9fb39910493a4f668337147e147bc7f8692896ff8a11362bb
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2026-01-07T00:00:00-05:00
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Exact State Evolution and Energy Spectrum in Solvable Bosonic Models
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arXiv:2510.20046v2 Announce Type: replace-cross Abstract: Solvable bosonic models provide a fundamental framework for describing light propagation in nonlinear media, including optical down-conversion processes that generate squeezed states of light and their higher-order generalizations. In quantum optics a central objective is to determine the time evolution of a given initial state. Exact analytic solution to the state-evolution problem is presented, applicable to a broad class of solvable bosonic models and arbitrary initial states. Moreover, the characteristic equation governing the energy spectrum is derived and the eigenstates are found in the form of continued fractions and as the principal minors of the associated Jacobi matrix. The results provide a solid analytical framework for discussion of exactly solvable bosonic models.
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https://arxiv.org/abs/2510.20046
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Academic Papers
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042b20960ae63d1e91f2191f3a2514e9444f851211696e944038d120dc2068de
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2026-01-07T00:00:00-05:00
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Boundary-driven quantum systems near the Zeno limit: steady states and long-time behavior
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arXiv:2512.12825v2 Announce Type: replace-cross Abstract: We study composite open quantum systems with a finite-dimensional state space ${\mathcal H}_A\otimes {\mathcal H}_B$ governed by a Lindblad equation $\rho'(t) = {\mathcal L}_\gamma \rho(t)$ where ${\mathcal L}_\gamma\rho = -i[H,\rho] + \gamma {\mathcal D} \rho$, and ${\mathcal D}$ is a dissipator ${\mathcal D}_A\otimes I$ acting non-trivially only on part $A$ of the system, which can be thought of as the boundary, and $\gamma$ is a parameter. It is known that the dynamics simplifies for large $\gamma$: after a time of order $\gamma^{-1}$, $\rho(t)$ is well approximated for times small compared to $\gamma^2$ by $\pi_A\otimes R(t)$ where $\pi_A$ is a steady state of ${\mathcal D}_A$, and $R(t)$ is a solution of $\frac{{\rm d}}{{\rm d}t}R(t) = {\mathcal L}_{P,\gamma}R(t)$ where ${\mathcal L}_{P,\gamma} R := -i[H_P,R] + \gamma^{-1} {\mathcal D}_P R$ with $H_P$ being a Hamiltonian on ${\mathcal H}_B$ and ${\mathcal D}_P$ being a Lindblad generator over ${\mathcal H}_B$. We prove this assuming only that ${\mathcal D}_A$ is ergodic and gapped. In order to better control the long time behavior, and study the steady states $\bar\rho_\gamma$, we introduce a third Lindblad generator ${\mathcal D}_P^\sharp$ that does not involve $\gamma$, but still closely related to ${\mathcal L}_\gamma$. We show that if ${\mathcal D}_P^\sharp$ is ergodic and gapped, then so is ${\mathcal L}_\gamma$ for all large $\gamma$, and if $\bar\rho_\gamma$ denotes the unique steady state for ${\mathcal L}_\gamma$, then $\lim_{\gamma\to\infty}\bar\rho_\gamma = \pi_A\otimes \bar R$ where $\bar R$ is the unique steady state for ${\mathcal D}_P^\sharp$. We show that there is a convergent expansion $\bar\rho_\gamma = \pi_A\otimes\bar R +\gamma^{-1} \sum_{k=0}^\infty \gamma^{-k} \bar n_k$ where, defining $\bar n_{-1} := \pi_A\otimes\bar R$, ${\mathcal D} \bar n_k = -i[H,\bar n_{k-1}]$ for all $k\geq 0$.
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https://arxiv.org/abs/2512.12825
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Academic Papers
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fdd9fbc00290319359f60896fa68dfb5e19892f729deb1aa73e4204de4c02112
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2026-01-07T00:00:00-05:00
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When Indemnity Insurance Fails: Parametric Coverage under Binding Budget and Risk Constraints
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arXiv:2512.21973v2 Announce Type: replace-cross Abstract: In high-risk environments, traditional indemnity insurance is often unaffordable or ineffective, despite its well-known optimality under expected utility. We compare excess-of-loss indemnity insurance with parametric insurance within a common mean-variance framework, allowing for fixed costs, heterogeneous premium loadings, and binding budget constraints. We show that, once these realistic frictions are introduced, parametric insurance can yield higher welfare for risk-averse individuals, even under the same utility objective. The welfare advantage arises precisely when indemnity insurance becomes impractical, and disappears once both contracts are unconstrained. Our results help reconcile classical insurance theory with the growing use of parametric risk transfer in high-risk settings.
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https://arxiv.org/abs/2512.21973
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Academic Papers
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594f29c02323815e5bccca21f5874a25c008cd1b0a962ffbb3aae41de5521bc7
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2026-01-07T00:00:00-05:00
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Exact inference via quasi-conjugacy in two-parameter Poisson-Dirichlet hidden Markov models
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arXiv:2512.22098v2 Announce Type: replace-cross Abstract: We introduce a nonparametric model for time-evolving, unobserved probability distributions from discrete-time data consisting of unlabelled partitions. The latent process is a two-parameter Poisson-Dirichlet diffusion, and observations arise via exchangeable sampling. Applications include social and genetic data where only aggregate clustering summaries are observed. To address the intractable likelihood, we develop a tractable inferential framework that avoids label enumeration and direct simulation of the latent state. We exploit a duality between the diffusion and a pure-death process on partitions, together with coagulation operators that encode the effect of new data. These yield closed-form, recursive updates for forward and backward inference. We compute exact posterior distributions of the latent state at arbitrary times and predictive distributions of future or interpolated partitions. This enables online and offline inference and forecasting with full uncertainty quantification, bypassing MCMC and sequential Monte Carlo. Compared to particle filtering, our method achieves higher accuracy, lower variance, and substantial computational gains. We illustrate the methodology with synthetic experiments and a social network application, recovering interpretable patterns in time-varying heterozygosity.
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https://arxiv.org/abs/2512.22098
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Academic Papers
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106658b423ef947ff93fdd47c6d66d2a578326ac9aa89a908408c1ee7c991359
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2026-01-07T00:00:00-05:00
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A Novel Multiple Imputation Approach For Parameter Estimation in Observation-Driven Time Series Models With Missing Data
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arXiv:2601.01259v2 Announce Type: replace-cross Abstract: Handling missing data in time series is a complex problem due to the presence of temporal dependence. General-purpose imputation methods, while widely used, often distort key statistical properties of the data, such as variance and dependence structure, leading to biased estimation and misleading inference. These issues become more pronounced in models that explicitly rely on capturing serial dependence, as standard imputation techniques fail to preserve the underlying dynamics. This paper proposes a novel multiple imputation method specifically designed for parameter estimation in observation-driven models (ODM). The approach takes advantage of the iterative nature of the systematic component in ODM to propagate the dependence structure through missing data, minimizing its impact on estimation. Unlike traditional imputation techniques, the proposed method accommodates continuous, discrete, and mixed-type data while preserving key distributional and dependence properties. We evaluate its performance through Monte Carlo simulations in the context of GARMA models, considering time series with up to 70\% missing data. An application to the proportion of stocked energy stored in South Brazil further demonstrates its practical utility.
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https://arxiv.org/abs/2601.01259
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Academic Papers
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6dd476ed14c17543f5fb5a01e25ae7232540c61f0e2ddaf125128508c6b12eca
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2026-01-07T00:00:00-05:00
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Predictability of bursts of a recurrent nova using topological data analysis and machine learning
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arXiv:2601.02403v1 Announce Type: new Abstract: RS Oph is a recurrent nova, a kind of cataclismic variable that shows bursts in a period approximately shorter than a century. Persistent homology, a technique from topological data analysis, studies the evolution of topological features of a simplicial complex composed of the data points or an embedding of them, as some distance parameter is varied. For this work I trained a supervised learning model based on several featurizations, namely persistence landscapes, Carlsson coordinates, persistent images, and template functions, of the persistence diagrams of sections of the lightcurve of RS Oph. A tenfold cross validation of the model based on one of the featurizations, persistence landscapes, consistently shows high recalls and accuracies. This method serves the purpose of predicting whether RS Oph is bursting within a year.
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https://arxiv.org/abs/2601.02403
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Academic Papers
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cb318910849980e5e9f9f881c66990e38ac62dee5e0979f612740054bcfc0edf
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2026-01-07T00:00:00-05:00
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Exactly solved model of a one dimensional self gravitating system
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arXiv:2601.02423v1 Announce Type: new Abstract: A model one-dimensional self consistent steady state collisionless self-gravitating system in which all the particles have the same energy is presented. This has the remarkable property that the position and velocity of the particles orbiting in their own self consistent potential are given exactly, in terms of time, by the truncations of sine and cosine functions to the first two terms in their respective Taylor series. The potential and density also have simple analytic expressions in terms of time as parameter. It is not being claimed that this system has any direct astronomical application. However, it does motivate a conjecture about the behaviour of the density, potential, and orbits near caustics in simulations of cold collisionless dark matter. It is a rather surprising result which might interest practitioners of stellar dynamics and serve as an elementary example in teaching the subject.
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https://arxiv.org/abs/2601.02423
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8e85dbea1bff1db06d3ec656da17c35c66ccb27b33127536182184032eed3fdb
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2026-01-07T00:00:00-05:00
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Experiments in binary evolution
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arXiv:2601.02448v1 Announce Type: new Abstract: The majority of stars more massive than the Sun is found in binary or multiple star systems and many of them will interact during their evolution. Specific interactions, where progenitors and post-mass transfer (MT) systems are clearly linked, can provide yet missing observational constraints. Volume-complete samples of progenitor and post-MT systems are well suited to study those processes. To compile them, we need to determine the parameters of thousands of binary systems with periods spanning several orders of magnitude. The bottleneck are the orbital parameters, because accurate determinations require a good coverage of the orbital phases. The next generation of time-resolved spectroscopic surveys should be optimized to follow-up and solve whole populations of binary systems in an efficient way. To achieve this, a scheduler predicting the best times of the next observation for any given target in real time should be combined with a flexibly schedulable multi-object spectrograph or ideally a network of independent telescopes.
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https://arxiv.org/abs/2601.02448
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Academic Papers
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80e72f0b1648f065678b8d938a7b051175b2e2e15005958155a7b0cb1ba34cd7
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2026-01-07T00:00:00-05:00
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Validation of Satellite Lifetime Predictions at Leonid Space
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arXiv:2601.02453v1 Announce Type: new Abstract: We validate Leonid Space's satellite lifetime prediction pipeline through comprehensive backtesting against 934 non-maneuvering satellites that deorbited from LEO between 1961 and 2024. This represents the first large-scale validation of lifetime prediction tooling using forecasted space weather conditions rather than historical hindsight. Our toolchain combines ballistic coefficient estimation from on-orbit data with probabilistic orbit propagation under varying environmental conditions. Using TLE data and space weather records spanning six solar cycles, our three-stage validation approach progressively removes hindsight bias to arrive at fully predictive operational conditions. We achieve 1-year prediction accuracy (median continuously ranked probability score) of 6.0 days (1.6%) under perfect knowledge conditions, 18.6 days (5.1%) with estimated ballistic coefficients and known space weather, and 45.5 days (12.4%) under fully predictive conditions. Comparison against ESA's standard DRAMA & DISCOS toolchain demonstrates a 4x improvement in state-of-the-art accuracy for well-characterized satellites. A custom semianalytic propagator provides a >3500x speedup over Orekit and 4.5x speedup over DRAMA, enabling rapid Monte Carlo analysis across large satellite populations. Our analysis reveals that solar cycle forecasting dominates error budgets after ballistic coefficient estimation, with higher-fidelity propagators and atmosphere models providing marginal benefit. These results establish a validated performance baseline for operational lifetime prediction services supporting LEO mission planning and regulatory compliance.
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https://arxiv.org/abs/2601.02453
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