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ff5e69c27ecb25b3c3ce887aff0fb756df14134291473395d22cbc9d72833d21
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2026-01-01T00:00:00-05:00
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Insights on the homogeneous $3$-local representations of the twin groups
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arXiv:2512.24874v1 Announce Type: new Abstract: We provide a complete classification of the homogeneous $3$-local representations of the twin group $T_n$, the virtual twin group $VT_n$, and the welded twin group $WT_n$, for all $n\geq 4$. Beyond this classification, we examine the main characteristics of these representations, particularly their irreducibility and faithfulness. More deeply, we show that all such representations are reducible, and most of them are unfaithful. Also, we find necessary and sufficient conditions of the first two types of the classified representations of $T_n$ to be irreducible in the case $n=4$. The obtained results provide insights into the algebraic structure of these three groups.
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https://arxiv.org/abs/2512.24874
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3f707166663c6302047d1e101221c85d152d8108f0a2e8b5e0ec5fc1bd1d847f
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2026-01-01T00:00:00-05:00
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Totally compatible structures on the radical of an incidence algebra
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arXiv:2512.24881v1 Announce Type: new Abstract: We describe totally compatible structures on the Jacobson radical of the incidence algebra of a finite poset over a field. We show that such structures are in general non-proper.
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https://arxiv.org/abs/2512.24881
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1f50a1b765a2747a79f9a5428f760dd3f6095437547d9900dc8440c68bc2896c
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2026-01-01T00:00:00-05:00
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Coherent span-valued 2D TQFTs
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arXiv:2512.24887v1 Announce Type: new Abstract: We consider commutative Frobenius pseudomonoids in the bicategory of spans, and we show that they are in correspondence with 2-Segal cosymmetric sets. Such a structure can be interpreted as a coherent 2-dimensional topological quantum field theory taking values in the bicategory of spans. We also describe a construction that produces a 2-Segal cosymmetric set from any partial monoid equipped with a distinguished element.
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https://arxiv.org/abs/2512.24887
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9d3ef7783a720fd7e77279913c6c30a8542ed731aaa66c3a236b67015776e90e
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2026-01-01T00:00:00-05:00
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Adaptive Clutter Suppression via Convex Optimization
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arXiv:2512.24889v1 Announce Type: new Abstract: Passive and bistatic radar systems are often limited by strong clutter and direct-path interference that mask weak moving targets. Conventional cancellation methods such as the extensive cancellation algorithm require careful tuning and can distort the delay-Doppler response. This paper introduces a convex optimization framework that adaptively synthesizes per-cell delay-Doppler filters to suppress clutter while preserving the canonical cross-ambiguity function (CAF). The approach formulates a quadratic program that minimizes distortion of the CAF surface subject to linear clutter-suppression constraints, eliminating the need for a separate cancellation stage. Monte Carlo simulations using common communication waveforms demonstrate strong clutter suppression, accurate CFAR calibration, and major detection-rate gains over the classical CAF. The results highlight a scalable, CAF-faithful method for adaptive clutter mitigation in passive radar.
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https://arxiv.org/abs/2512.24889
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8fc8471a5c7c245066b1b3b0fbf482a52bbd7c06e3a1a57e8b03546488a9aad5
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2026-01-01T00:00:00-05:00
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Global boundedness and absorbing sets in two-dimensional chemotaxis-Navier-Stokes systems with weakly singular sensitivity and a sub-logistic source
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arXiv:2512.24892v1 Announce Type: new Abstract: This paper studies the following chemotaxis-fluid system in a two-dimensional bounded domain $\Omega$: \begin{equation*} \begin{cases} n_t + u \cdot \nabla n &= \Delta n - \chi \nabla \cdot \left (n \frac{\nabla c}{c^k} \right ) + r n - \frac{\mu n^2}{\log^\eta(n+e)}, c_t + u \cdot \nabla c &= \Delta c - \alpha c + \beta n, u_t + u \cdot \nabla u &= \Delta u - \nabla P + n \nabla \phi + f, \nabla \cdot u &= 0, \end{cases} \end{equation*} where $r, \mu, \alpha, \beta, \chi$ are positive parameters, $k, \eta \in (0,1)$, $\phi \in W^{2,\infty}(\Omega)$, and $f \in C^1\left(\bar{\Omega}\times [0, \infty)\right) \cap L^\infty\left(\Omega \times (0, \infty)\right)$. We show that, under suitable conditions on the initial data and with no-flux/no-flux/Dirichlet boundary conditions, this system admits a globally bounded classical solution. Furthermore, the system possesses an absorbing set in the topology of $C^0(\bar{\Omega}) \times W^{1, \infty}(\Omega) \times C^0(\bar{\Omega}; \mathbb{R}^2)$.
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https://arxiv.org/abs/2512.24892
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a279df32891161287f69ad09b31f55628e9f0e6a660ed3c2aec5c43f478ac3de
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2026-01-01T00:00:00-05:00
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Self-Supervised Amortized Neural Operators for Optimal Control: Scaling Laws and Applications
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arXiv:2512.24897v1 Announce Type: new Abstract: Optimal control provides a principled framework for transforming dynamical system models into intelligent decision-making, yet classical computational approaches are often too expensive for real-time deployment in dynamic or uncertain environments. In this work, we propose a method based on self-supervised neural operators for open-loop optimal control problems. It offers a new paradigm by directly approximating the mapping from system conditions to optimal control strategies, enabling instantaneous inference across diverse scenarios once trained. We further extend this framework to more complex settings, including dynamic or partially observed environments, by integrating the learned solution operator with Model Predictive Control (MPC). This yields a solution-operator learning method for closed-loop control, in which the learned operator supplies rapid predictions that replace the potentially time-consuming optimization step in conventional MPC. This acceleration comes with a quantifiable price to pay. Theoretically, we derive scaling laws that relate generalization error and sample/model complexity to the intrinsic dimension of the problem and the regularity of the optimal control function. Numerically, case studies show efficient, accurate real-time performance in low-intrinsic-dimension regimes, while accuracy degrades as problem complexity increases. Together, these results provide a balanced perspective: neural operators are a powerful novel tool for high-performance control when hidden low-dimensional structure can be exploited, yet they remain fundamentally constrained by the intrinsic dimensional complexity in more challenging settings.
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https://arxiv.org/abs/2512.24897
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44cef96163407ccddec4545b8d242f811bdfd37d99b0ddf6bae65fd891bb31bf
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2026-01-01T00:00:00-05:00
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Anomalous cw-expansive homeomorphisms on compact surfaces of higher genus
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arXiv:2512.24904v1 Announce Type: new Abstract: In this paper, we construct cw-expansive homeomorphisms on compact surfaces of genus greater than or equal to zero with a fixed point whose local stable set is connected but not locally connected. This provides an affirmative answer to question posed by Artigue [3]. To achieve this, we generalize the construction from the example of Artigue, Pacifico and Vieitez [6], obtaining examples of homeomorphisms on compact surfaces of genus greater than or equal to two that are 2-expansive but not expansive. On the sphere and the torus, we construct new examples of cw2-expansive homeomorphisms that are not N -expansive for all N greater than or equal to one.
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https://arxiv.org/abs/2512.24904
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082091d6127087bd1b89f4db61744055c38eb9b9072d565989bce8eff286a4bb
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2026-01-01T00:00:00-05:00
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Polynomial $\chi$-boundedness for excluding $P_5$
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arXiv:2512.24907v1 Announce Type: new Abstract: We obtain some $d\ge2$ such that every graph $G$ with no induced copy of the five-vertex path $P_5$ has chromatic number at most $\omega(G)^d$, thereby resolving an open problem of Gy\'arf\'as from 1985. The proof consists of three main ingredients: $\bullet$ an analogue of R\"odl's theorem for the chromatic number of $P_5$-free graphs, proved via the ``Gy\'arf\'as path'' argument; $\bullet$ a decomposition argument for $P_5$-free graphs that allows one to grow high-chromatic anticomplete pairs indefinitely or to capture a polynomially chromatic-dense induced subgraph; and $\bullet$ a ``chromatic density increment'' argument that uses the Erd\H{o}s-Hajnal result for $P_5$ as a black box.
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https://arxiv.org/abs/2512.24907
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29f27c1a5d6b068b467e9d08a1409ad1a3b6d4cf5608763bcd86a6ae63382ec6
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2026-01-01T00:00:00-05:00
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A Liouville-Weierstrass correspondence for Spacelike and Timelike Minimal Surfaces in $\mathbb{L}^3$
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arXiv:2512.24908v1 Announce Type: new Abstract: We investigate a correspondence between solutions $\lambda(x,y)$ of the Liouville equation \[ \Delta \lambda = -\varepsilon e^{-4\lambda}, \] and the Weierstrass representations of spacelike ($\varepsilon = 1$) and timelike ($\varepsilon = -1$) minimal surfaces with diagonalizable Weingarten map in the three-dimensional Lorentz--Minkowski space $\mathbb{L}^3$. Using complex and paracomplex analysis, we provide a unified treatment of both causal types. We study the action of pseudo-isometries of $\mathbb{L}^3$ on minimal surfaces via M\"obius-type transformations, establishing a correspondence between these transformations and rotations in the special orthochronous Lorentz group. Furthermore, we show how local solutions of the Liouville equation determine the Gauss map and the associated Weierstrass data. Finally, we present explicit examples of spacelike and timelike minimal surfaces in $\mathbb{L}^3$ arising from solutions of the Liouville equation.
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https://arxiv.org/abs/2512.24908
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8e5c8d96e664089693c850797c400b8f98ac1d0005fd8c9dd064b74e585f1ed5
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2026-01-01T00:00:00-05:00
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Gibbs conditioning principle for log-concave independent random variables
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arXiv:2512.24910v1 Announce Type: new Abstract: Let $\nu_1,\nu_2,\dots$ be a sequence of probabilities on the nonnegative integers, and $X=(X_1,X_2, \dots)$ be a sequence of independent random variables $X_i$ with law $\nu_i$. For $\lambda>0$ denote $Z^\lambda_i:= \sum_x \lambda^x\nu_i(x)$ and $\lambda^{\max}:= \sup\{\lambda>0: Z^\lambda_i1$. For $\lambdaR^*_n)$ converges weakly to the law of $X^{\lambda^*}$, as $n\to\infty$. We prove the GCP for log-concave $\nu_i$'s, meaning $\nu_i(x+1)\,\nu_i(x-1) \le ( \nu_i(x))^2$, subject to a technical condition that prevents condensation. The canonical measures are the distributions of the first $n$ variables, conditioned on their sum being $k$. Efron's theorem states that for log-concave $\nu_i$'s, the canonical measures are stochastically ordered with respect to $k$. This, in turn, leads to the ordering of the conditioned tilted measures $P(X^\lambda\in\cdot|S^\lambda_n>R^*_n)$ in terms of $\lambda$. This ordering is a fundamental component of our proof.
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https://arxiv.org/abs/2512.24910
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fdeed806eb7aad5e0083cb6c25a9aa008f8c15a079679f676ee1c447391ab929
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2026-01-01T00:00:00-05:00
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The Lyapunov Exponents of Hyperbolic Measures for $C^1$ Vector Fields with Dominated Splitting
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arXiv:2512.24911v1 Announce Type: new Abstract: In this paper, we prove that for every $C^1$ vector field preserving an ergodic hyperbolic invariant measure which is not supported on singularities, if the Oseledec splitting of the ergodic hyperbolic invariant measure is a dominated splitting, then the ergodic hyperbolic invariant measure can be approximated by periodic measures, and the Lyapunov exponents of the ergodic hyperbolic invariant measure can also be approximated by the Lyapunov exponents of those periodic measures.
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https://arxiv.org/abs/2512.24911
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4358f7f3c484d529c2431c06c23a13804f741bf70aa3e2919beed4b6e8af6f4e
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2026-01-01T00:00:00-05:00
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On Maps that Preserve the Lie Products Equal to Fixed Elements
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arXiv:2512.24912v1 Announce Type: new Abstract: This work characterizes the general form of a bijective linear map $\Psi:\mathscr{M}_n(\mathbb{C}) \to \mathscr{M}_n(\mathbb{C})$ such that $[\Psi(A_1),~\Psi(A_2)]=D_2$ whenever $[A_1,~A_2]=D_1$ where $D_1~\text{and}~D_2$ are fixed matrices. Additionally, let $\mathscr{H}_1$ and $\mathscr{H}_2$ be the infinite-dimensional complex Hilbert spaces. We characterize the bijective linear map $\Psi: \mathscr{B}(\mathscr{H}_1) \to \mathscr{B}(\mathscr{H}_2)$ where $\Psi(A_1) \circ ~\Psi(A_2)=D_2$ whenever $A_1\circ ~A_2=D_1$ and $D_1~\text{and}~D_2$ are fixed operators.
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https://arxiv.org/abs/2512.24912
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00d4fe5cf5138071e823ca241a21436fd866ccebd9704a73885761b460a1e463
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2026-01-01T00:00:00-05:00
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On Diophantine exponents of lattices
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arXiv:2512.24913v1 Announce Type: new Abstract: We describe the spectrum of ordinary Diophantine exponents for $d$-dimensional lattices. The result reduces the problem to two-dimensional case and uses argument of metric theory.
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https://arxiv.org/abs/2512.24913
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6110b81bc69688785001ad65f0a12a6245799f0fb8a6bc063b081b715cc1507f
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2026-01-01T00:00:00-05:00
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Existence, uniqueness, and approximability of solutions to the classical Melan equation in suspension bridges
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arXiv:2512.24915v1 Announce Type: new Abstract: The classical Melan equation modeling suspension bridges is considered. We first study the explicit expression and the uniform positivity of the analytical solution for the simplified ``less stiff'' model, based on which we develop a monotone iterative technique of lower and upper solutions to investigate the existence, uniqueness and approximability of the solution for the original classical Melan equation.The applicability and the efficiency of the monotone iterative technique for engineering design calculations are discussed by verifying some examples of actual bridges. Some open problems are suggested.
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https://arxiv.org/abs/2512.24915
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93ba0492bbaac9188141bb35c6b4e933fd0f6f98759f494014a1b21de861ae46
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2026-01-01T00:00:00-05:00
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A Pontryagin Maximum Principle on the Belief Space for Continuous-Time Optimal Control with Discrete Observations
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arXiv:2512.24916v1 Announce Type: new Abstract: We study a continuous time stochastic optimal control problem under partial observations that are available only at discrete time instants. This hybrid setting, with continuous dynamics and intermittent noisy measurements, arises in applications ranging from robotic exploration and target tracking to epidemic control. We formulate the problem on the space of beliefs (information states), treating the controller's posterior distribution of the state as the state variable for decision making. On this belief space we derive a Pontryagin maximum principle that provides necessary conditions for optimality. The analysis carefully tracks both the continuous evolution of the state between observation times and the Bayesian jump updates of the belief at observation instants. A key insight is a relationship between the adjoint process in our maximum principle and the gradient of the value functional on the belief space, which links the optimality conditions to the dynamic programming approach on the space of probability measures. The resulting optimality system has a prediction and update structure that is closely related to the unnormalised Zakai equation and the normalised Kushner-Stratonovich equation in nonlinear filtering. Building on this analysis, we design a particle based numerical scheme to approximate the coupled forward (filter) and backward (adjoint) system. The scheme uses particle filtering to represent the evolving belief and regression techniques to approximate the adjoint, which yields a practical algorithm for computing near optimal controls under partial information. The effectiveness of the approach is illustrated on both linear and nonlinear examples and highlights in particular the benefits of actively controlling the observation process.
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https://arxiv.org/abs/2512.24916
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c83b6180e4e50d186cf3c8afba8b601bee660aa7e62e4c7d95b67836dd20d25c
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2026-01-01T00:00:00-05:00
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Property (T) and Poincar\'e duality in dimension three
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arXiv:2512.24919v1 Announce Type: new Abstract: We use a recent result of Bader and Sauer on coboundary expansion to prove residually finite three-dimensional Poincar\'e duality groups never have property (T). This implies such groups are never K\"ahler. The argument applies to fundamental groups of (possibly non-aspherical) compact 3-manifolds, giving a new proof of a theorem of Fujiwara that states if the fundamental group of a compact 3-manifold has property (T), then that group is finite. The only consequence of geometrization needed in the proof is that 3-manifold groups are residually finite.
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https://arxiv.org/abs/2512.24919
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1feceb7f9d5b269db85986fd7da3428474c5eef9ad5c23ac25f77dc03c82e955
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2026-01-01T00:00:00-05:00
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Transgression in the primitive cohomology
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arXiv:2512.24920v1 Announce Type: new Abstract: We study the Chern-Weil theory for the primitive cohomology of a symplectic manifold. First, given a symplectic manifold, we review the superbundle-valued forms on this manifold and prove a primitive version of the Bianchi identity. Second, as the main result, we prove a transgression formula associated with the boundary map of the primitive cohomology. Third, as an application of the main result, we introduce the concept of primitive characteristic classes and point out a further direction.
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https://arxiv.org/abs/2512.24920
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b745c429bf3c4ccaf0f6b462d38e8165cad8d08e0e59a022fc126a1f46eb8981
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2026-01-01T00:00:00-05:00
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On a new filtration of the variational bicomplex
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arXiv:2512.24931v1 Announce Type: new Abstract: We define a filtration on the variational bicomplex according to jet order. The filtration is preserved by the interior Euler operator, which is not a module homomorphism with respect to the ring of smooth functions on the jet space. However, the induced maps on the graded components of this filtration are. Furthermore, the space of functional forms in the image of the interior Euler operator inherits a filtration. Though the filtered subspaces are not submodules either, the graded components are isomorphic to linear spaces which do have module structures. This works for any fixed degree of the functional forms. In this way, the condition that a functional form vanishes can be stated concisely with a module basis. We work out explicitly two examples: one for functional forms of degree two in relation to the Helmholtz conditions and the other of arbitrary degree but with jet order one.
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https://arxiv.org/abs/2512.24931
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3a66bb8f0ff5a70faab84eff6575574c7275e5b692066dcd7ce206a80e5f203d
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2026-01-01T00:00:00-05:00
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Generalised Hermite-Einstein Fibre Metrics and Slope Stability for Holomorphic Vector Bundles
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arXiv:2512.24932v1 Announce Type: new Abstract: Let $X$ be a compact complex manifold of dimension $n$ and let $m$ be a positive integer with $m\leq n$. Assume that $X$ admits a K\"ahler metric $\omega$ and a weakly positive, $\partial\bar\partial$-closed, smooth $(n-m,\,n-m)$-form $\Omega$. We introduce the notions of $(\omega,\,\Omega)$-Hermite-Einstein holomorphic vector bundles and $(\omega,\,\Omega)$(-semi)-stable coherent sheaves on $X$ by generalising the classical definitions depending only on $\omega$. We then prove that the $(\omega,\,\Omega)$-Hermite-Einstein condition implies the $(\omega,\,\Omega)$-semi-stability of a holomorphic vector bundle and its splitting into $(\omega,\,\Omega)$-stable subbundles. This extends a classical result by Kobayashi and L\"ubke to our generalised setting.
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https://arxiv.org/abs/2512.24932
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19a4fee00fcc63e8f6358d26bcb57385d488b828337964f4dad6771dc1144cb5
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2026-01-01T00:00:00-05:00
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Green's function on the Tate curve
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arXiv:2512.24935v1 Announce Type: new Abstract: Motivated by the question of defining a $p$-adic string worldsheet action in genus one, we define a Laplacian operator on the Tate curve, and study its Green's function. We show that the Green's function exists. We provide an explicit formula for the Green's function, which turns out to be a non-Archimedean counterpart of the Archimedean Green's function on a flat torus.
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https://arxiv.org/abs/2512.24935
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1e7c208b8d7b6974e5c9b0b1b141b493b23d6846ca4a3ec24f93e578c6397865
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2026-01-01T00:00:00-05:00
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Data-Driven Spectral Analysis Through Pseudo-Resolvent Koopman Operator in Dynamical Systems
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arXiv:2512.24953v1 Announce Type: new Abstract: We present a data-driven method for spectral analysis of the Koopman operator based on direct construction of the pseudo-resolvent from time-series data. Finite-dimensional approximation of the Koopman operator, such as those obtained from Extended Dynamic Mode Decomposition, are known to suffer from spectral pollution. To address this issue, we construct the pseudo-resolvent operator using the Sherman-Morrison-Woodbury identity whose norm serves as a spectral indicator, and pseudoeigenfunctions are extracted as directions of maximal amplification. We establish convergence of the approximate spectrum to the true spectrum in the Hausdorff metric for isolated eigenvalues, with preservation of algebraic multiplicities, and derive error bounds for eigenvalue approximation. Numerical experiments on pendulum, Lorenz, and coupled oscillator systems demonstrate that the method effectively suppresses spectral pollution and resolves closely spaced spectral components.
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https://arxiv.org/abs/2512.24953
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816530fe3d4e43a42501fbe31392031eba48a84575b8cf2bdfc2b5bc57ef60c0
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2026-01-01T00:00:00-05:00
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Numerical study of solitary waves in Dirac--Klein--Gordon system
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arXiv:2512.24954v1 Announce Type: new Abstract: We use numerics to construct solitary waves in Dirac--Klein--Gordon (in one and three spatial dimensions) and study the dependence of energy and charge on $\omega$. For the construction, we use the iterative procedure, starting from solitary waves of nonlinear Dirac equation, computing the corresponding scalar field, and adjusting the coupling constant. We also consider the case of massless scalar field, when the iteration procedure could be compared with the shooting method. We use the virial identities to control the error of simulations. We also discuss possible implications from the obtained results for the spectral stability of solitary waves.
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https://arxiv.org/abs/2512.24954
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8912586da8fa204810a2aad5b1aadf169f15c6e9be06f15fc531d6918925ec53
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2026-01-01T00:00:00-05:00
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The least prime with a given cycle type
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arXiv:2512.24963v1 Announce Type: new Abstract: Let $G$ be a finite group. Let $K/k$ be a Galois extension of number fields with Galois group isomorphic to $G$, and let $C \subseteq \mathrm{Gal}(K/k) \simeq G$ be a conjugacy invariant subset. It is well known that there exists an unramified prime ideal $\mathfrak{p}$ of $k$ with Frobenius element lying in $C$ and norm satisfying $\mathrm{N}\mathfrak{p} \ll |\mathrm{Disc}(K)|^{\alpha}$ for some constant $\alpha = \alpha(G,C)$. There is a rich literature establishing unconditional admissible values for $\alpha$, with most approaches proceeding by studying the zeros of $L$-functions. We give an alternative approach, not relying on zeros, that often substantially improves this exponent $\alpha$ for any fixed finite group $G$, provided $C$ is a union of rational equivalence classes. As a particularly striking example, we prove that there exist absolute constants $c_1,c_2 > 0$ such that for any $n\geq 2$ and any conjugacy class $C \subset S_n$, one may take $\alpha(S_n,C) = c_1 \exp(-c_2n)$. Our approach reduces the core problem to a question in character theory.
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https://arxiv.org/abs/2512.24963
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57c6a01f9421c4cf7df7966d5c69a0813b2877250cb3ef2dc2ffb5772a0200cd
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2026-01-01T00:00:00-05:00
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Cartier duality for gerbes of vector bundles
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arXiv:2512.24967v1 Announce Type: new Abstract: We prove a Cartier duality for gerbes of algebraic and analytic vector bundles as an anti-equivalence of Hopf algebras in the category of kernels of analytic stacks. As an application, we prove that the category of solid quasi-coherent sheaves on the Hodge-Tate stack of a smooth rigid variety over an algebraically closed field $C$ of mixed characteristic $(0,p)$ is equivalent to the category of weight $1$ sheaves on Bhatt-Zhang's Simpson gerbe.
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https://arxiv.org/abs/2512.24967
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b2a541d4da58489d6c689ac2736ce2c8d46c4f143b8c6207c5ad2847d57faab6
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2026-01-01T00:00:00-05:00
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From Complex-Analytic Models to Sparse Domination: A Dyadic Approach of Hypersingular Operators via Bourgain's Interpolation Method
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arXiv:2512.24972v1 Announce Type: new Abstract: Motivated by the work of Cheng--Fang--Wang--Yu on the hypersingular Bergman projection, we develop a real-variable and dyadic framework for hypersingular operators in regimes where strong-type estimates fail at the critical line. The main new input is a hypersingular sparse domination principle combined with Bourgain's interpolation method, which provides a flexible mechanism to establish critical-line (and endpoint) estimates. In the unit disc setting with $1<3/2$, we obtain a full characterization of the $(p,q)$ mapping theory for the dyadic hypersingular maximal operator $\mathcal M_t^{\mathcal D}$, in particular including estimates on the critical line $1/q-1/p=2t-2$ and a weighted endpoint criterion in the radial setting. We also prove endpoint estimates for the hypersingular Bergman projection \[ K_{2t}f(z)=\int_{\mathbb D}\frac{f(w)}{(1-z\overline w)^{2t}}\,dA(w), \] including a restricted weak-type bound at $(p,q)=\bigl(\tfrac{1}{3-2t},1\bigr)$. Finally, we introduce a class of hypersingular cousin of sparse operators in $\mathbb R^n$ associated with \emph{graded} sparse families, quantified by the sparseness $\eta$ and a new structural parameter (the \emph{degree}) $K_{\mathcal S}$, and we characterize the corresponding strong/weak/restricted weak-type regimes in terms of $(n,t,\eta,K_{\mathcal S})$. Our real-variable perspective addresses to an inquiry raised by Cheng--Fang--Wang--Yu on developing effective real-analytic tools in the hypersingular regime for $K_{2t}$, and it also provides a new route toward the critical-line analysis of Forelli--Rudin type operators and related hypersingular operators in both real and complex settings.
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https://arxiv.org/abs/2512.24972
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ae6328935b8fb23a8e006018dc620cc62e688647402831961f36c6c2e21738e4
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2026-01-01T00:00:00-05:00
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A guide to the $2$-generated axial algebras of Monster type
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arXiv:2512.24987v1 Announce Type: new Abstract: Axial algebras of Monster type are a class of non-associative algebras which generalise the Griess algebra, whose automorphism group is the largest sporadic simple group, the Monster. The $2$-generated algebras, which are the building blocks from which all algebras in this class can be constructed, have recently been classified by Yabe; Franchi and Mainardis; and Franchi, Mainardis and McInroy. There are twelve infinite families of examples as well as the exceptional Highwater algebra and its cover, however their properties are not well understood. In this paper, we detail the properties of each of these families, describing their ideals and quotients, subalgebras and idempotents in all characteristics. We also describe all exceptional isomorphisms between them. We give new bases for several of the algebras which better exhibit their axial features and provide code for others to work with them.
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https://arxiv.org/abs/2512.24987
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516e7ea82088fcb34ea9b63e0cf5ed310fdcc107bc921d63af39d82109515c87
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2026-01-01T00:00:00-05:00
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The Fourier extension conjecture for the paraboloid
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arXiv:2512.24990v1 Announce Type: new Abstract: We give a proof of Fourier extension conjecture on the paraboloid in all dimensions bigger than 2 that begins with a decomposition suggested in Sawyer [Saw8] of writing a smooth Alpert projection as a sum of pieces whose Fourier extensions are localized. This is then used in the case d=3 to establish the trilinear equivalence of the Fourier extension conjecture given in C. Rios and E. Sawyer [RiSa1] and [RiSa3]. A key aspect of the proof is that the trilinear equivalence only requires an averaging over grids, which converts a difficult exponential sum into an oscillatory integral with periodic amplitude, that is then used to prove the localization on the Fourier side. Finally, we extend this argument to all dimensions bigger than 2 using bilinear analogues of the smooth Alpert trilinear inequalities, which generalize those in Tao, Vargas and Vega [TaVaVe].
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https://arxiv.org/abs/2512.24990
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d00bcc01f560a1bb5b34381244e3bc3d9d8eddf50bfeb7f15531299901b0dd19
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2026-01-01T00:00:00-05:00
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Manifold classification from the descriptive viewpoint
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arXiv:2512.24996v1 Announce Type: new Abstract: We consider classification problems for manifolds and discrete subgroups of Lie groups from a descriptive set-theoretic point of view. This work is largely foundational in conception and character, recording both a framework for general study and Borel complexity computations for some of the most fundamental classes of manifolds. We show, for example, that for all $n\geq 0$, the homeomorphism problem for compact topological $n$-manifolds is Borel equivalent to the relation $=_{\mathbb{N}}$ of equality on the natural numbers, while the homeomorphism problem for noncompact topological $2$-manifolds is of maximal complexity among equivalence relations classifiable by countable structures. A nontrivial step in the latter consists of proving Borel measurable formulations of the Jordan--Schoenflies and surface triangulation theorems. Turning our attention to groups and geometric structures, we show, strengthening results of Stuck--Zimmer and Andretta--Camerlo--Hjorth, that the conjugacy relation on discrete subgroups of any noncompact semisimple Lie group is essentially countable universal. So too, as a corollary, is the isometry relation for complete hyperbolic $n$-manifolds for any $n\geq 2$, generalizing a result of Hjorth--Kechris. We then show that the isometry relation for complete hyperbolic $n$-manifolds with finitely generated fundamental group is, in contrast, Borel equivalent to the equality relation $=_{\mathbb{R}}$ on the real numbers when $n=2$, but that it is not concretely classifiable when $n=3$; thus there exists no Borel assignment of numerical complete invariants to finitely generated Kleinian groups up to conjugacy. We close with a survey of the most immediate open questions.
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https://arxiv.org/abs/2512.24996
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1fd97f9622dd2f4a54c0a78167c8f448b6095fe9122df92a57a5fa2d1c575463
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2026-01-01T00:00:00-05:00
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The local limit of weighted spanning trees on balanced networks
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arXiv:2512.25001v1 Announce Type: new Abstract: We prove that the local limit of the weighted spanning trees on any simple connected high degree almost regular sequence of electric networks is the Poisson(1) branching process conditioned to survive forever, by generalizing [NP22] and closing a gap in their proof. We also study the local statistics of the WST's on high degree almost balanced sequences, which is interesting even for the uniform spanning trees. Our motivation comes from studying an interpolation $\{\mathsf{WST}^{\beta}(G)\}_{\beta\in [0, \infty)}$ between UST(G) and MST(G) by WST's on a one-parameter family of random environments. This model has recently been introduced in [MSS24, K\'us24], and the phases of several properties have been determined on the complete graphs. We show a phase transition of $\mathsf{WST}^{\beta_n}(G_n)$ regarding the local limit and expected edge overlaps for high degree almost balanced graph sequences $G_n$, without any structural assumptions on the graphs; while the expected total length is sensitive to the global structure of the graphs. Our general framework results in a better understanding even in the case of complete graphs, where it narrows the window of the phase transition of [Mak24].
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https://arxiv.org/abs/2512.25001
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33389be22b1848332e966b792a789b3e4eda336777d541921dde5c8884c4b793
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2026-01-01T00:00:00-05:00
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Uniqueness for stochastic differential equations in Hilbert spaces with irregular drift
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arXiv:2512.25003v1 Announce Type: new Abstract: We present a versatile framework to study strong existence and uniqueness for stochastic differential equations (SDEs) in Hilbert spaces with irregular drift. We consider an SDE in a separable Hilbert space $H$ \begin{equation*} dX_t= (A X_t + b(X_t))dt +(-A)^{-\gamma/2}dW_t,\quad X_0=x_0 \in H, \end{equation*} where $A$ is a self-adjoint negative definite operator with purely atomic spectrum, $W$ is a cylindrical Wiener process, $b$ is $\alpha$-H\"older continuous function $H\to H$, and a nonnegative parameter $\gamma$ such that the stochastic convolution takes values in $H$. We show that this equation has a unique strong solution provided that $\alpha > 2\gamma/(1+\gamma)$. This substantially extends the seminal work of Da Prato and Flandoli (2010) as no structural assumption on $b$ is imposed. To obtain this result, we do not use infinite-dimensional Kolmogorov equations but instead develop a new technique combining L\^e's theory of stochastic sewing in Hilbert spaces, Gaussian analysis, and a method of Lasry and Lions for approximation in Hilbert spaces.
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https://arxiv.org/abs/2512.25003
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660bbb1a0da55f6d563b1ec51978f2704ee19b4f651d50de745526fd6bf77ac2
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2026-01-01T00:00:00-05:00
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Limit Theorems for Fixed Point Biased Pattern Avoiding Involutions
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arXiv:2512.25006v1 Announce Type: new Abstract: We study fixed point biased involutions that avoid a pattern. For every pattern of length three we obtain limit theorems for the asymptotic distribution of the (appropriately centered and scaled) number of fixed points of a random fixed point biased involution avoiding that pattern. When the pattern being avoided is either $321$, $132$, or $213$, we find a phase transition depending on the strength of the bias. We also obtain a limit theorem for distribution of fixed points when the pattern is $123\cdots k(k+1)$ for any $k$ and partial results when the pattern is $(k+1)k\cdots 321$.
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https://arxiv.org/abs/2512.25006
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e5eb0c5b33ed087c3f9cada765938533ac3ff7686e280ea2f6720bc0688d67d9
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2026-01-01T00:00:00-05:00
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The splitting field and generators of the elliptic surface $Y^2=X^3 +t^{360} +1$
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arXiv:2512.25009v1 Announce Type: new Abstract: The splitting field of an elliptic surface $\mathcal{E}/\mathbb{Q}(t)$ is the smallest finite extension $\mathcal{K} \subset \mathbb{C}$ such that all $\mathbb{C}(t)$-rational points are defined over $\mathcal{K}(t)$. In this paper, we provide a symbolic algorithmic approach to determine the splitting field and a set of $68$ linearly independent generators for the Mordell--Weil lattice of Shioda's elliptic surface $Y^2=X^3 +t^{360} +1$. This surface is noted for having the largest known rank 68 for an elliptic curve over $\mathbb{C}(t)$. Our methodology utilizes the known decomposition of the Mordell-Weil Lattice of this surface into Lattices of ten rational elliptic surfaces and one $K3$ surface. We explicitly compute the defining polynomials of the splitting field, which reach degrees of 1728 and 5760, and verify the results via height pairing matrices and specialized symbolic software packages.
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https://arxiv.org/abs/2512.25009
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0505c4c96533faf8474f46a527e7f3ff1793f42f7fd40c2c3b062d130a4e1c8c
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2026-01-01T00:00:00-05:00
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Bounding regularity of $\mathrm{VI}^m$-modules
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arXiv:2512.25010v1 Announce Type: new Abstract: Fix a finite field $\mathbb{F}$. Let $\mathrm{VI}$ be a skeleton of the category of finite dimensional $\mathbb{F}$-vector spaces and injective $\mathbb{F}$-linear maps. We study $\mathrm{VI}^m$-modules over a noetherian commutative ring in the nondescribing characteristic case. We prove that if a finitely generated $\mathrm{VI}^m$-module is generated in degree $\leqslant d$ and related in degree $\leqslant r$, then its regularity is bounded above by a function of $m$, $d$, and $r$. A key ingredient of the proof is a shift theorem for finitely generated $\mathrm{VI}^m$-modules.
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https://arxiv.org/abs/2512.25010
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816ade7a9ab7b83b1d3baaf7bf5f14d2489dd9efbd98d9461ffc1a56bf95d963
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2026-01-01T00:00:00-05:00
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A note on semistable unitary operators on $L^2(\mathbb{R})$
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arXiv:2512.25013v1 Announce Type: new Abstract: In this note, we present a characterization of semistable unitary operators on $L^2(\mathbb{R})$, under the assumption that the operator is (i) translation-invariant, (ii) symmetric, and (iii) locally uniformly continuous (LUC) under dilation. As a consequence, we characterize one-parameter groups formed by such operators, which are of the form $e^{i\beta t|{d}/{dx}|^\alpha}$, with $\alpha,\beta\in\mathbb R$.
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https://arxiv.org/abs/2512.25013
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62e2efcb58f56cff23f19e74b240c451b7703bf97f3498c7b6e0a0e28c35571a
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2026-01-01T00:00:00-05:00
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Strengthening Dual Bounds for Multicommodity Capacitated Network Design with Unsplittable Flow Constraints
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arXiv:2512.25018v1 Announce Type: new Abstract: Multicommodity capacitated network design (MCND) models can be used to optimize the consolidation of shipments within e-commerce fulfillment networks. In practice, fulfillment networks require that shipments with the same origin and destination follow the same transfer path. This unsplittable flow requirement complicates the MCND problem, requiring integer programming (IP) formulations in which binary variables replace continuous flow variables. To enhance the solvability of this variant of the MCND problem for large-scale logistics networks, this work focuses on strengthening dual bounds. We investigate the polyhedra of arc-set relaxations, and we introduce two new classes of valid inequalities that can be implemented within solution approaches. We develop one approach that dynamically adds valid inequalities to the root node of a reformulation of the MCND IP with additional valid metric inequalities. We show the effectiveness of our ideas with a comprehensive computational study using path-based fulfillment instances, constructed from data provided by a large U.S.-based e-commerce company, and the well-known arc-based Canad instances. Experiments show that our best solution approach for a practical path-based model reduces the IP gap by an average of 26.5% and 22.5% for the two largest instance groups, compared to solving the reformulation alone, demonstrating its effectiveness in improving the dual bound. In addition, experiments using only the arc-based relaxation highlight the strength of our new valid inequalities relative to the linear programming relaxation (LPR), yielding an IP-gap reduction of more than 85%.
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https://arxiv.org/abs/2512.25018
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37830ac65c02db5db21c9369dd1d9334860e243b6d963d10a4f22fa7465eefac
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2026-01-01T00:00:00-05:00
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Real Riemann Surfaces: Smooth and Discrete
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arXiv:2512.25022v1 Announce Type: new Abstract: This paper develops a discrete theory of real Riemann surfaces based on quadrilateral cellular decompositions (quad-graphs) and a linear discretization of the Cauchy-Riemann equations. We construct a discrete analogue of an antiholomorphic involution and classify the topological types of discrete real Riemann surfaces, recovering the classical results on the number of real ovals and the separation of the surface. Central to our approach is the construction of a symplectic homology basis adapted to the discrete involution. Using this basis, we prove that the discrete period matrix admits the same canonical decomposition $\Pi = \frac{1}{2} H + i T$ as in the smooth setting, where $H$ encodes the topological type and $T$ is purely imaginary. This structural result bridges the gap between combinatorial models and the classical theory of real algebraic curves.
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https://arxiv.org/abs/2512.25022
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59ffba010bed7f7b73512aeaf5854e3c044f39ed3c73e8ca0c5d6fd31752ccce
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2026-01-01T00:00:00-05:00
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Mod $p$ Poincar\'e duality for $p$-adic period domains
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arXiv:2512.25029v1 Announce Type: new Abstract: In this article, we introduce a new class of smooth partially proper rigid analytic varieties over a $p$-adic field that satisfy Poincar\'e duality for \'etale cohomology with mod $p$-coefficients : the varieties satisfying "primitive comparison with compact support". We show that almost proper varieties, as well as p-adic (weakly admissible) period domains in the sense of Rappoport-Zink belong to this class. In particular, we recover Poincar\'e duality for almost proper varieties as first established by Li-Reinecke-Zavyalov, and we compute the \'etale cohomology with $\mathbb{F}_p$-coefficients of p-adic period domains, generalizing a computation of Colmez-Dospinescu-Niziol for Drinfeld's symmetric spaces. The arguments used in this paper rely crucially on Mann's six functors formalism for solid $\mathcal{O}^{+,a}/\pi$ coefficients.
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https://arxiv.org/abs/2512.25029
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709b5b68cb22072ce2858090631760d82bd674c19bd3caafc6f7e2a1452c28ee
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2026-01-01T00:00:00-05:00
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Multivariate Generalized Counting Process via Gamma Subordination
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arXiv:2512.25030v1 Announce Type: new Abstract: In this paper, we study a multivariate gamma subordinator whose components are independent gamma processes subject to a random time governed by an independent negative binomial process. We derive the explicit expressions for its joint Laplace-Stieltjes transform, its probability density function and the associated governing differential equations. Also, we study a time-changed variant of the multivariate generalized counting process where the time is changed by an independent multivariate gamma subordinator. For this time-changed process, we obtain the corresponding L\'evy measure and probability mass function. Later, we discuss an application of the time-changed multivariate generalized counting process to a shock model.
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https://arxiv.org/abs/2512.25030
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ebb5c89f927d79384dd3b424b5332d36b04ec564c7112441ce03e0564007f75c
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2026-01-01T00:00:00-05:00
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Universal polar dual pairs of spherical codes found in $E_8$ and $\Lambda_{24}$
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arXiv:2512.25037v1 Announce Type: new Abstract: We identify universal polar dual pairs of spherical codes $C$ and $D$ such that for a large class of potential functions $h$ the minima of the discrete $h$-potential of $C$ on the sphere occur at the points of $D$ and vice versa. Moreover, the minimal values of their normalized potentials are equal. These codes arise from the known sharp codes embedded in the even unimodular extremal lattices $E_8$ and $\Lambda_{24}$ (Leech lattice). This embedding allows us to use the lattices' properties to find new universal polar dual pairs. In the process we extensively utilize the interplay between the binary Golay codes and the Leech lattice. As a byproduct of our analysis, we identify a new universally optimal (in the sense of energy) code in the projective space $\mathbb{RP}^{21}$ with $1408$ points (lines). Furthermore, we extend the Delsarte-Goethals-Seidel definition of derived codes from their seminal $1977$ paper and generalize their Theorem 8.2 to show that if a $\tau$-design is enclosed in $k\leq \tau$ parallel hyperplanes, then each of the hyperplane's sub-code is a $(\tau+1-k)$-design in the ambient subspace.
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https://arxiv.org/abs/2512.25037
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5dad628f6580d2d6df2d30f216c8c77128a02404f498de75eeeca07ac5ee0b1e
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2026-01-01T00:00:00-05:00
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The Hochschild homology of a noncommutative symmetric quotient stack
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arXiv:2512.25039v1 Announce Type: new Abstract: We prove an orbifold type decomposition theorem for the Hochschild homology of the symmetric powers of a small DG category $\mathcal{A}$. In noncommutative geometry, these can be viewed as the noncommutative symmetric quotient stacks of $\mathcal{A}$. We use this decomposition to show that the total Hochschild homology of the symmetric powers of $\mathcal{A}$ is isomorphic to the symmetric algebra $S^*(\mathrm{HH}_\bullet(\mathcal{A}) \otimes t \mathbb{k}[t])$. Our methods are explicit - we construct mutually inverse homotopy equivalences of the standard Hochschild complexes involved. These explicit maps are then used to induce from the symmetric algebra onto the total Hochschild homology the structures of the Fock space for the Heisenberg algebra of $\mathcal{A}$, of a Hopf algebra, and of a free $\lambda$-ring generated by $\mathrm{HH}_\bullet(\mathcal{A})$.
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https://arxiv.org/abs/2512.25039
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18181120d0527ead9a9c98e4c90a28e7938eacb21ab7f32d5ea468e61371b587
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2026-01-01T00:00:00-05:00
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On exact Observability for Compactly perturbed infinite dimension system
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arXiv:2512.25041v1 Announce Type: new Abstract: In this paper, we study the observability of compactly perturbed infinite dimensional systems. Assuming that a given infinite-dimensional system with self-adjoint generator is exactly observable we derive sufficient conditions on a compact self adjoint perturbation to guarantee that the perturbed system stays exactly observable. The analysis is based on a careful asymptotic estimation of the spectral elements of the perturbed unbounded operator in terms of the compact perturbation. These intermediate results are of importance themselves.
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https://arxiv.org/abs/2512.25041
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f56dbc8b1da7d439eefcd1bb14b0f3082e05c23caf3132cf4a407796def125b5
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2026-01-01T00:00:00-05:00
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The PDE-ODI principle and cylindrical mean curvature flows
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arXiv:2512.25050v1 Announce Type: new Abstract: We introduce a new approach for analyzing ancient solutions and singularities of mean curvature flow that are locally modeled on a cylinder. Its key ingredient is a general mechanism, called the \emph{PDE--ODI principle}, which converts a broad class of parabolic differential equations into systems of ordinary differential inequalities. This principle bypasses many delicate analytic estimates used in previous work, and yields asymptotic expansions to arbitrarily high order. As an application, we establish the uniqueness of the bowl soliton times a Euclidean factor among ancient, cylindrical flows with dominant linear mode. This extends previous results on this problem to the most general setting and is made possible by the stronger asymptotic control provided by our analysis. In the other case, when the quadratic mode dominates, we obtain a complete asymptotic expansion to arbitrary polynomial order, which will form the basis for a subsequent paper. Our framework also recovers and unifies several classical results. In particular, we give new proofs of the uniqueness of tangent flows (due to Colding-Minicozzi) and the rigidity of cylinders among shrinkers (due to Colding-Ilmanen-Minicozzi) by reducing both problems to a single ordinary differential inequality, without using the {\L}ojasiewicz-Simon inequality. Our approach is independent of prior work and the paper is largely self-contained.
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https://arxiv.org/abs/2512.25050
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8d4fd795650bd126f1d3054a068a2c36869721d81e0a72ed535deb00f546bd1d
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2026-01-01T00:00:00-05:00
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Bilinear tau forms of quantum Painlev\'e equations and $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations in SUSY gauge theories
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arXiv:2512.25051v1 Announce Type: new Abstract: We derive bilinear tau forms of the canonically quantized Painlev\'e equations, thereby relating them to those previously obtained from the $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations for the $\mathcal{N}=2$ supersymmetric gauge theory partition functions on a general $\Omega$-background. We fully fix the refined Painlev\'e/gauge theory dictionary by formulating the proper equations for the quantum nonautonomous Painlev\'e Hamiltonians. We also describe the symmetry structure of the quantum Painlev\'e tau functions and, as a byproduct of this analysis, obtain the $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations in the nontrivial holonomy sector of the gauge theory.
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https://arxiv.org/abs/2512.25051
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7f5603a21743cfe5aaf60c436c0711363a998b82edc539a19977324c2416c7a6
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2026-01-01T00:00:00-05:00
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The variety of orthogonal frames
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arXiv:2512.25058v1 Announce Type: new Abstract: An orthogonal n-frame is an ordered set of n pairwise orthogonal vectors. The set of all orthogonal n-frames in a d-dimensional quadratic vector space is an algebraic variety V(d,n). In this paper, we investigate the variety V(d,n) as well as the quadratic ideal I(d,n) generated by the orthogonality relations, which cuts out V(d,n). We classify the irreducible components of V(d,n), give criteria for the ideal I(d,n) to be prime or a complete intersection, and for the variety V(d,n) to be normal. We also give near-equivalent conditions for V(d,n) to be factorial. Applications are given to the theory of Lov\'asz-Saks-Schrijver ideals.
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https://arxiv.org/abs/2512.25058
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1338c3d5c50897528d116827b66a263a263bd2a7f2d15f8faf3619455c230f5b
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2026-01-01T00:00:00-05:00
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Overflow-Avoiding Memory AMP
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arXiv:2407.03898v1 Announce Type: cross Abstract: Approximate Message Passing (AMP) type algorithms are widely used for signal recovery in high-dimensional noisy linear systems. Recently, a principle called Memory AMP (MAMP) was proposed. Leveraging this principle, the gradient descent MAMP (GD-MAMP) algorithm was designed, inheriting the strengths of AMP and OAMP/VAMP. In this paper, we first provide an overflow-avoiding GD-MAMP (OA-GD-MAMP) to address the overflow problem that arises from some intermediate variables exceeding the range of floating point numbers. Second, we develop a complexity-reduced GD-MAMP (CR-GD-MAMP) to reduce the number of matrix-vector products per iteration by 1/3 (from 3 to 2) with little to no impact on the convergence speed.
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https://arxiv.org/abs/2407.03898
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c2cde6b4787258144fd4293bd2ce3e2fc782d6419713643195d276ab0056e848
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2026-01-01T00:00:00-05:00
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A Course in Ring Theory
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arXiv:2512.22133v1 Announce Type: cross Abstract: An introductory textbook on ring theory, including ideals and homomorphisms, Euclidean domains, PIDs, and UFDs, with examples and exercises.
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https://arxiv.org/abs/2512.22133
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46567e6894cfd1c9cc4cd8ab210eb7c547d9aeae1be0a29a54eeb6168e9fc1f2
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2026-01-01T00:00:00-05:00
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Hawksmoor's Ceiling, Mercator's Projection and the Roman Pantheon
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arXiv:2512.22327v1 Announce Type: cross Abstract: The ceiling of the Buttery in All Souls College, Oxford, designed by the English Baroque architect Nicholas Hawksmoor, has a vaulted form on an oval base. It is coffered with an array of approximately square sunken lacunaria, whose sizes and positions vary so as to accommodate the constraints of the curved surface and its boundaries. A similar design appears in the dome of the Roman Pantheon. Using methods of differential geometry, we hypothesise that these cofferings should be the images under conformal mappings of regular square tilings of a rectangle or finite cylinder. This guarantees that the coffer ribs meet exactly at right angles and the coffers are close to being square. These mappings are simply the inverse of Mercator's projection of the curved surface onto a plane. For a ceiling which is a general surface of revolution, we derive formulae for the dimensions and location of each coffer. Our results, taking into account camera distortion, are in excellent agreement with photographs of the Hawksmoor ceiling and the Pantheon dome, as well as with recent direct measurements of the latter. We also describe a protocol by which Hawksmoor's ceiling might have been constructed without advanced mathematics.
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https://arxiv.org/abs/2512.22327
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e17f20ff282cbd075400c6e00bf52b6b94c08193b84beb9663d1139d87aa26ac
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2026-01-01T00:00:00-05:00
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Machine Learning Invariants of Tensors
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arXiv:2512.23750v1 Announce Type: cross Abstract: We propose a data-driven approach to identifying the functionally independent invariants that can be constructed from a tensor with a given symmetry structure. Our algorithm proceeds by first enumerating graphs, or tensor networks, that represent inequivalent contractions of a product of tensors, computing instances of these scalars using randomly generated data, and then seeking linear relations between invariants using numerical linear algebra. Such relations yield syzygies, or functional dependencies relating different invariants. We apply this approach in an extended case study of the independent invariants that can be constructed from an antisymmetric $3$-form $H_{\mu \nu \rho}$ in six dimensions, finding five independent invariants. This result confirms that the most general Lagrangian for such a $3$-form, which depends on $H_{\mu \nu \rho}$ but not its derivatives, is an arbitrary function of five variables, and we give explicit formulas relating other invariants to the five independent scalars in this generating set.
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https://arxiv.org/abs/2512.23750
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5a0600e568e94532d94b948406a887765c0e2364cadeab395c3653204d683b3f
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2026-01-01T00:00:00-05:00
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Marked point processes intensity estimation using sparse group Lasso method applied to locations of lucrative and cooperative banks in mainland France
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arXiv:2512.23772v1 Announce Type: cross Abstract: In this paper, we model the locations of five major banks in mainland France, two lucrative and three cooperative institutions based on socio-economic considerations. Locations of banks are collected using web scrapping and constitute a bivariate spatial point process for which we estimate nonparametrically summary functions (intensity, Ripley and cross-Ripley's K functions). This shows that the pattern is highly inhomogenenous and exhibits a clustering effect especially at small scales, and thus a significant departure to the bivariate (inhomogeneous) Poisson point process is pointed out. We also collect socio-economic datasets (at the living area level) from INSEE and propose a parametric modelling of the intensity function using these covariates. We propose a group-penalized bivariate composite likelihood method to estimate the model parameters, and we establish its asymptotic properties. The application of the methodology to the banking dataset provides new insights into the specificity of the cooperative model within the sector, particularly in relation to the theories of institutional isomorphism.
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https://arxiv.org/abs/2512.23772
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53e342f2aad4be9c875152e037ef1309811f58b5b7ac8e8743fb438b0be78a80
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2026-01-01T00:00:00-05:00
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5D AGT conjecture for circular quivers
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arXiv:2512.23878v1 Announce Type: cross Abstract: The best way to represent generic conformal blocks is provided by the free-field formalism, where they acquire a form of multiple Dotsenko-Fateev-like integrals of the screening operators. Degenerate conformal blocks can be described by the same integrals with special choice of parameters. Integrals satisfy various recurrent relations, which for the special choice of parameters reduce to closed equations. This setting is widely used in explaining the AGT relation, because similar integral representations exist also for Nekrasov functions. We extend this approach to the case of q-Virasoro conformal blocks on elliptic surface -- generic and degenerate. For the generic case, we check equivalence with instanton partition function of a 5d circular quiver gauge theory. For the degenerate case, we check equivalence with partition function of a defect in the same theory, also known as the Shiraishi function. We find agreement in both cases. This opens a way to re-derive the sophisticated equation for the Shiraishi function as the equation for the corresponding integral, what seems straightforward, but remains technically involved and is left for the future.
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https://arxiv.org/abs/2512.23878
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7d5c9741a256ae697e6270782de2ac26fde659ebf7f149ce1e68c7e90efb5167
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2026-01-01T00:00:00-05:00
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Kinks in composite scalar field theories
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arXiv:2512.23890v1 Announce Type: cross Abstract: In this work, families of kinks are analytically identified in multifield theories with either polynomial or deformed sine-Gordon-type potentials. The underlying procedure not only allows us to obtain analytical solutions for these models, but also provides a framework for constructing more general families of field theories that inherit certain analytical information about their solutions. Specifically, this method combines two known field theories into a new composite field theory whose target space is the product of the original target spaces. By suitably coupling the fields through a superpotential defined on the product space, the dynamics in the subspaces become entangled while preserving original kinks as boundary kinks. Different composite field theories are studied, including extensions of well-known models to wider target spaces.
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https://arxiv.org/abs/2512.23890
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a05fa1ec42b3a0a01f4448f99f7f3242b77c481b68bef82123a7016713a40d61
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2026-01-01T00:00:00-05:00
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Coulomb Branches of Noncotangent Type: a Physics Perspective
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arXiv:2512.23908v1 Announce Type: cross Abstract: We study the Coulomb-branch sector of 3D $\mathcal{N}=4$ gauge theories with half-hypermultiplets in general pseudoreal representations $\mathbf{R}$ ("noncotangent" theories). This yields (short) quantization of the Coulomb branch and correlators of the Coulomb branch operators captured by the 1d topological sector. This is done by extending the hemisphere partition function approach to noncotangent matter. In this setting one must first cancel the parity anomaly, and overcome an obstacle that $(2,2)$ boundary conditions for half-hypers are generically incompatible with gauge symmetry. Using the Dirichlet boundary conditions for the gauge fields and a careful treatment of half-hypermultiplet boundary data, we describe the resulting shift/difference operators implementing monopole insertions (including bubbling effects) on $HS^3$, and use the $HS^3$ partition function as a natural module on which the Coulomb-branch operator algebra $\mathcal{A}_C$ is represented. As applications we derive generators and relations of $\mathcal{A}_C$ for $SU(2)$ theories with general matter (including half-integer spin representations), analyze theories with Coulomb branch $y^2=z(x^2-1)$, compute the Coulomb branch of an $A_n$ quiver with spin-$\frac32$ half-hypers, and check consistency of a general monopole-antimonopole two-point function.
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https://arxiv.org/abs/2512.23908
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1d36923a3d5160d14b601b7c800784ac4ac439a0f75b50bc58d654572c7cf04f
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2026-01-01T00:00:00-05:00
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Completing and studentising Spearman's correlation in the presence of ties
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arXiv:2512.23993v1 Announce Type: cross Abstract: Non-parametric correlation coefficients have been widely used for analysing arbitrary random variables upon common populations, when requiring an explicit error distribution to be known is an unacceptable assumption. We examine an \(\ell_{2}\) representation of a correlation coefficient (Emond and Mason, 2002) from the perspective of a statistical estimator upon random variables, and verify a number of interesting and highly desirable mathematical properties, mathematically similar to the Whitney embedding of a Hilbert space into the \(\ell_{2}\)-norm space. In particular, we show here that, in comparison to the traditional Spearman (1904) \(\rho\), the proposed Kemeny \(\rho_{\kappa}\) correlation coefficient satisfies Gauss-Markov conditions in the presence or absence of ties, thereby allowing both discrete and continuous marginal random variables. We also prove under standard regularity conditions a number of desirable scenarios, including the construction of a null hypothesis distribution which is Student-t distributed, parallel to standard practice with Pearson's r, but without requiring either continuous random variables nor particular Gaussian errors. Simulations in particular focus upon highly kurtotic data, with highly nominal empirical coverage consistent with theoretical expectation.
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https://arxiv.org/abs/2512.23993
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8bcb7b92a49f9a15c846f8b83750e2302643cd9039d47425f71250c33931c8ce
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2026-01-01T00:00:00-05:00
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An exact unbiased semi-parametric maximum quasi-likelihood framework which is complete in the presence of ties
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arXiv:2512.24009v1 Announce Type: cross Abstract: This paper introduces a novel quasi-likelihood extension of the generalised Kendall \(\tau_{a}\) estimator, together with an extension of the Kemeny metric and its associated covariance and correlation forms. The central contribution is to show that the U-statistic structure of the proposed coefficient \(\tau_{\kappa}\) naturally induces a quasi-maximum likelihood estimation (QMLE) framework, yielding consistent Wald and likelihood ratio test statistics. The development builds on the uncentred correlation inner-product (Hilbert space) formulation of Emond and Mason (2002) and resolves the associated sub-Gaussian likelihood optimisation problem under the \(\ell_{2}\)-norm via an Edgeworth expansion of higher-order moments. The Kemeny covariance coefficient \(\tau_{\kappa}\) is derived within a novel likelihood framework for pairwise comparison-continuous random variables, enabling direct inference on population-level correlation between ranked or weakly ordered datasets. Unlike existing approaches that focus on marginal or pairwise summaries, the proposed framework supports sample-observed weak orderings and accommodates ties without information loss. Drawing parallels with Thurstone's Case V latent ordering model, we derive a quasi-likelihood-based tie model with analytic standard errors, generalising classical U-statistics. The framework applies to general continuous and discrete random variables and establishes formal equivalence to Bradley-Terry and Thurstone models, yielding a uniquely identified linear representation with both analytic and likelihood-based estimators.
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https://arxiv.org/abs/2512.24009
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dfeaef8b955ae8c4e922b748acfe17006dbd3eacfe9ec09bebe1781c7a6db895
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2026-01-01T00:00:00-05:00
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Solving the initial value problem for cellular automata by pattern decomposition
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arXiv:2512.24337v1 Announce Type: cross Abstract: For many cellular automata, it is possible to express the state of a given cell after $n$ iterations as an explicit function of the initial configuration. We say that for such rules the solution of the initial value problem can be obtained. In some cases, one can construct the solution formula for the initial value problem by analyzing the spatiotemporal pattern generated by the rule and decomposing it into simpler segments which one can then describe algebraically. We show an example of a rule when such approach is successful, namely elementary rule 156. Solution of the initial value problem for this rule is constructed and then used to compute the density of ones after $n$ iterations, starting from a random initial condition. We also show how to obtain probabilities of occurrence of longer blocks of symbols.
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https://arxiv.org/abs/2512.24337
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7a597500cb5532cc172d00eab51c5932ade8195d493d6e0a6c45b6d4a181c81f
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2026-01-01T00:00:00-05:00
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Relativistic Lindblad description of the electron's radiative dynamics
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arXiv:2512.24341v1 Announce Type: cross Abstract: An effective model for describing the relativistic quantum dynamics of a radiating electron is developed via a relativistic generalization of the Lindblad master equation. By incorporating both radiation reaction and vacuum fluctuations into the Dirac equation within an open quantum system framework, our approach captures the Zitterbewegung of the electron, ensuing noncommutativity of its effective spatial coordinates, and provides the quantum analogue of the Landau-Lifshitz (LL) classical equation of motion with radiation reaction. We develop the corresponding phase-space representation via the relativistic Wigner function and derive the semiclassical limit through a Foldy-Wouthuysen transformation. The latter elucidates the signature of quantum vacuum fluctuations in the LL equation, and shows its relationship with the corrected Sokolov equation. Our results offer a robust framework for investigating quantum radiation reaction effects in ultrastrong laser fields.
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https://arxiv.org/abs/2512.24341
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22a6f3fbdf569cc58df77b91e48b0c8682ea2b0ac030089071525dfd3f26365d
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2026-01-01T00:00:00-05:00
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Les Houches Lectures Notes on Tensor Networks
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arXiv:2512.24390v1 Announce Type: cross Abstract: Tensor networks provide a powerful new framework for classifying and simulating correlated and topological phases of quantum matter. Their central premise is that strongly correlated matter can only be understood by studying the underlying entanglement structure and its associated (generalised) symmetries. In essence, tensor networks provide a compressed, holographic description of the complicated vacuum fluctuations in strongly correlated systems, and as such they break down the infamous many-body exponential wall. These lecture notes provide a concise overview of the most important conceptual, computational and mathematical aspects of this theory.
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https://arxiv.org/abs/2512.24390
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ba43890edaf21bb11939d10574f7dbff0b2501345b2e5b6c4b6c77097906aa5e
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2026-01-01T00:00:00-05:00
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Exact finite mixture representations for species sampling processes
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arXiv:2512.24414v1 Announce Type: cross Abstract: Random probability measures, together with their constructions, representations, and associated algorithms, play a central role in modern Bayesian inference. A key class is that of proper species sampling processes, which offer a relatively simple yet versatile framework that extends naturally to non-exchangeable settings. We revisit this class from a computational perspective and show that they admit exact finite mixture representations. In particular, we prove that any proper species sampling process can be written, at the prior level, as a finite mixture with a latent truncation variable and reweighted atoms, while preserving its distributional features exactly. These finite formulations can be used as drop-in replacements in Bayesian mixture models, recasting posterior computation in terms of familiar finite-mixture machinery. This yields straightforward MCMC implementations and tractable expressions, while avoiding ad hoc truncations and model-specific constructions. The resulting representation preserves the full generality of the original infinite-dimensional priors while enabling practical gains in algorithm design and implementation.
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https://arxiv.org/abs/2512.24414
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55ad2829c207068855120d2cb4b1e880fbebf36de44a5dd9b146f8c8171618bf
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2026-01-01T00:00:00-05:00
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Generalized Level-Rank Duality, Holomorphic Conformal Field Theory, and Non-Invertible Anyon Condensation
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arXiv:2512.24419v1 Announce Type: cross Abstract: We study the interplay between holomorphic conformal field theory and dualities of 3D topological quantum field theories generalizing the paradigm of level-rank duality. A holomorphic conformal field theory with a Kac-Moody subalgebra implies a topological interface between Chern-Simons gauge theories. Upon condensing a suitable set of anyons, such an interface yields a duality between topological field theories. We illustrate this idea using the $c=24$ holomorphic theories classified by Schellekens, which leads to a list of novel sporadic dualities. Some of these dualities necessarily involve condensation of anyons with non-abelian statistics, i.e. gauging non-invertible one-form global symmetries. Several of the examples we discover generalize from $c=24$ to an infinite series. This includes the fact that Spin$(n^{2})_{2}$ is dual to a twisted dihedral group gauge theory. Finally, if $-1$ is a quadratic residue modulo $k$, we deduce the existence of a sequence of holomorphic CFTs at central charge $c=2(k-1)$ with fusion category symmetry given by $\mathrm{Spin}(k)_{2}$ or equivalently, the $\mathbb{Z}_{2}$-equivariantization of a Tambara-Yamagami fusion category.
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https://arxiv.org/abs/2512.24419
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1872d6132d7cd5ab4c825ab72c6eb96360e4831550d32c66f4e56994c86fb254
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2026-01-01T00:00:00-05:00
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Multidimensional derivative-free optimization. A case study on minimization of Hartree-Fock-Roothaan energy functionals
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arXiv:2512.24509v1 Announce Type: cross Abstract: This study presents an evaluation of derivative-free optimization algorithms for the direct minimization of Hartree-Fock-Roothaan energy functionals involving nonlinear orbital parameters and quantum numbers with noninteger order. The analysis focuses on atomic calculations employing noninteger Slater-type orbitals. Analytic derivatives of the energy functional are not readily available for these orbitals. Four methods are investigated under identical numerical conditions: Powell's conjugate-direction method, the Nelder-Mead simplex algorithm, coordinate-based pattern search, and a model-based algorithm utilizing radial basis functions for surrogate-model construction. Performance benchmarking is first performed using the Powell singular function, a well-established test case exhibiting challenging properties including Hessian singularity at the global minimum. The algorithms are then applied to Hartree-Fock-Roothaan self-consistent-field energy functionals, which define a highly non-convex optimization landscape due to the nonlinear coupling of orbital parameters. Illustrative examples are provided for closed$-$shell atomic configurations, specifically the He, Be isoelectronic series, with calculations performed for energy functionals involving up to eight nonlinear parameters.
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https://arxiv.org/abs/2512.24509
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0ffd22d62cd928a1af1804641318eef9e9a170e06e67dc8e7bec2bd0bb213fed
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2026-01-01T00:00:00-05:00
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${\cal N}=8$ supersymmetric mechanics with spin variables from indecomposable multiplets
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arXiv:2512.24692v1 Announce Type: cross Abstract: We define two new indecomposable (not fully reducible) ${\cal N}=8$, $d=1$ off-shell multiplets and consider the corresponding models of ${\cal N}=8$ supersymmetric mechanics with spin variables. Each multiplet is described off shell by a scalar superfield which is a nonlinear deformation of the standard scalar superfield $X$ carrying the $d=1$ multiplet ${\bf (1,8,7)}$. Deformed systems involve, as invariant subsets, two different off-shell versions of the irreducible multiplet ${\bf (8,8,0)}$. For both systems we present the manifestly ${\cal N}=8$ supersymmetric superfield constraints, as well as the component off- and on-shell invariant actions, which for one version exactly match those given in arXiv:2402.00539 [hep-th]. The two models differ off shell, but prove to be equivalent to each other on shell, with the spin variables sitting in the adjoint representation of the maximal $R$-symmetry group ${\rm SO}(8)$.
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https://arxiv.org/abs/2512.24692
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dca0ff08aabf4bd6db475bc5427b75e8c7e04d4180e9a3aea397395072903467
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2026-01-01T00:00:00-05:00
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Fragile Topological Phases and Topological Order of 2D Crystalline Chern Insulators
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arXiv:2512.24709v1 Announce Type: cross Abstract: We apply methods of equivariant homotopy theory, which may not previously have found due attention in condensed matter physics, to classify first the fragile/unstable topological phases of 2D crystalline Chern insulator materials, and second the possible topological order of their fractional cousins. We highlight that the phases are given by the equivariant 2-Cohomotopy of the Brillouin torus of crystal momenta (with respect to wallpaper point group actions) -- which, despite the attention devoted to crystalline Chern insulators, seems not to have been considered before. Arguing then that any topological order must be reflected in the adiabatic monodromy of gapped quantum ground states over the covariantized space of these band topologies, we compute the latter in examples where this group is non-abelian, showing that any potential FQAH anyons must be localized in momentum space. We close with an outlook on the relevance for the search for topological quantum computing hardware. Mathematical details are spelled out in a supplement.
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https://arxiv.org/abs/2512.24709
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3f926f1c0eff770752fb40383d1262c3936947e0bae7823bce0eb57f53251cff
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2026-01-01T00:00:00-05:00
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T-duality for toric manifolds in $\mathcal{N}=(2, 2)$ superspace
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arXiv:2512.24726v1 Announce Type: cross Abstract: We study the situation when the T-dual of a toric K\"ahler geometry is a generalized K\"ahler geometry involving semi-chiral fields. We explain that this situation is generic for polycylinders, tori and related geometries. Gauging multiple isometries in this case requires the introduction of semi-chiral gauge fields on top of the standard ones. We then apply this technology to the generalized K\"ahler geometry of the $\eta$-deformed $\mathbb{CP}^{n-1}$ model, relating it to the K\"ahler geometry of its T-dual.
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https://arxiv.org/abs/2512.24726
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9e7eaeaf49abc180a32739ccb1f2b6d52295a571946ccdd64f881b01e9437b74
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2026-01-01T00:00:00-05:00
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Twisted Cherednik systems and non-symmetric Macdonald polynomials
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arXiv:2512.24811v1 Announce Type: cross Abstract: We consider eigenfunctions of many-body system Hamiltonians associated with generalized (a-twisted) Cherednik operators used in construction of other Hamiltonians: those arising from commutative subalgebras of the Ding-Iohara-Miki (DIM) algebra. The simplest example of these eigenfunctions is provided by non-symmetric Macdonald polynomials, while generally they are constructed basing on the ground state eigenfunction coinciding with the twisted Baker-Akhiezer function being a peculiar (symmetric) eigenfunction of the DIM Hamiltonians. Moreover, the eigenfunctions admit an expansion with universal coefficients so that the dependence on the twist $a$ is hidden only in these ground state eigenfunctions, and we suggest a general formula that allows one to construct these eigenfunctions from non-symmetric Macdonald polynomials. This gives a new twist in theory of integrable systems, which usually puts an accent on symmetric polynomials, and provides a new dimension to the {\it triad} made from the symmetric Macdonald polynomials, untwisted Baker-Akhiezer functions and Noumi-Shiraishi series.
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https://arxiv.org/abs/2512.24811
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a9d76ffe1d5a15c19eccf19e233d92092da9896d449e93648a0c902bf29103b7
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2026-01-01T00:00:00-05:00
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Classical integrability in 2D and asymptotic symmetries
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arXiv:2512.24833v1 Announce Type: cross Abstract: These lecture notes are a contribution to the proceedings of the school "Geometric, Algebraic and Topological Methods for Quantum Field Theory", held in Villa de Leyva, Colombia, from 31st of July to 9th of August 2023. Its intention is to put together several basic tools of classical integrability and contrast them with those available in the formulation of asymptotic symmetries and the definition of canonical charges in gauge theories. We consider as a working example the Chern-Simons theory in 3D dimensions, motivated by its various applications in condensed matter physics, gravity, and black hole physics. We review basic aspects of the canonical formulation, symplectic geometry, Liouville integrability, and Lax Pairs. We define the Hamiltonian formulation of the Chern-Simons action and the canonical generators of the gauge symmetries, which are surface integrals that subject to non-trivial boundary conditions, realize transformations that do change the physical state, namely large (or improper gauge transformations). We propose asymptotic conditions that realize an infinite set of abelian conserved charges associated with integral models. We review two different cases: the Korteweg-de Vries equation for its connection with the Virasoro algebra and fluid dynamics, and the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, as it embeds an infinite class of non-linear notable integrable evolution equations. We propose a concrete example for gravity in 3D with $\Lambda<0$, where we find a near-horizon asymptotic dynamics. We finalize offering some insights on the initial value problem, its connection with integrable systems and flat connections. We study some properties of the Monodromy matrix and recover the infinite KdV charges from the trace invariants extracted from the Monodromy evolution equation that can be written in a Lax form.
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https://arxiv.org/abs/2512.24833
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bf9163f660128f746a2c0f91f988ece300113d3bf9252784e43c1d59d3ee3d3e
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2026-01-01T00:00:00-05:00
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Stochastic factors can matter: improving robust growth under ergodicity
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arXiv:2512.24906v1 Announce Type: cross Abstract: Drifts of asset returns are notoriously difficult to model accurately and, yet, trading strategies obtained from portfolio optimization are very sensitive to them. To mitigate this well-known phenomenon we study robust growth-optimization in a high-dimensional incomplete market under drift uncertainty of the asset price process $X$, under an additional ergodicity assumption, which constrains but does not fully specify the drift in general. The class of admissible models allows $X$ to depend on a multivariate stochastic factor $Y$ and fixes (a) their joint volatility structure, (b) their long-term joint ergodic density and (c) the dynamics of the stochastic factor process $Y$. A principal motivation of this framework comes from pairs trading, where $X$ is the spread process and models with the above characteristics are commonplace. Our main results determine the robust optimal growth rate, construct a worst-case admissible model and characterize the robust growth-optimal strategy via a solution to a certain partial differential equation (PDE). We demonstrate that utilizing the stochastic factor leads to improvement in robust growth complementing the conclusions of the previous study by Itkin et. al. (arXiv:2211.15628 [q-fin.MF], forthcoming in $\textit{Finance and Stochastics}$), which additionally robustified the dynamics of the stochastic factor leading to $Y$-independent optimal strategies. Our analysis leads to new financial insights, quantifying the improvement in growth the investor can achieve by optimally incorporating stochastic factors into their trading decisions. We illustrate our theoretical results on several numerical examples including an application to pairs trading.
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https://arxiv.org/abs/2512.24906
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e4e2a01a9f342a8b0f8b03e6d7f0cf94ab4ead775515567e7d14b7cb68ffe6d6
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2026-01-01T00:00:00-05:00
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Dynamic response phenotypes and model discrimination in systems and synthetic biology
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arXiv:2512.24945v1 Announce Type: cross Abstract: Biological systems encode function not primarily in steady states, but in the structure of transient responses elicited by time-varying stimuli. Overshoots, biphasic dynamics, adaptation kinetics, fold-change detection, entrainment, and cumulative exposure effects often determine phenotypic outcomes, yet are poorly captured by classical steady-state or dose-response analyses. This paper develops an input-output perspective on such "dynamic phenotypes," emphasizing how qualitative features of transient behavior constrain underlying network architectures independently of detailed parameter values. A central theme is the role of sign structure and interconnection logic, particularly the contrast between monotone systems and architectures containing antagonistic pathways. We show how incoherent feedforward (IFF) motifs provide a simple and recurrent mechanism for generating non-monotonic and adaptive responses across multiple levels of biological organization, from molecular signaling to immune regulation and population dynamics. Conversely, monotonicity imposes sharp impossibility results that can be used to falsify entire classes of models from transient data alone. Beyond step inputs, we highlight how periodic forcing, ramps, and integral-type readouts such as cumulative dose responses offer powerful experimental probes that reveal otherwise hidden structure, separate competing motifs, and expose invariances such as fold-change detection. Throughout, we illustrate how control-theoretic concepts, including monotonicity, equivariance, and input-output analysis, can be used not as engineering metaphors, but as precise mathematical tools for biological model discrimination. Thus we argue for a shift in emphasis from asymptotic behavior to transient and input-driven dynamics as a primary lens for understanding, testing, and reverse-engineering biological networks.
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https://arxiv.org/abs/2512.24945
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b2fbda4ce91c13a54e162b298ea649827e52d19fbf2964f266a60657712861fd
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2026-01-01T00:00:00-05:00
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Wall crossing, string networks and quantum toroidal algebras
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arXiv:2512.24988v1 Announce Type: cross Abstract: We investigate BPS states in 4d N=4 supersymmetric Yang-Mills theory and the corresponding (p, q) string networks in Type IIB string theory. We propose a new interpretation of the algebra of line operators in this theory as a tensor product of vector representations of a quantum toroidal algebra, which determines protected spin characters of all framed BPS states. We identify the SL(2,Z)-noninvariant choice of the coproduct in the quantum toroidal algebra with the choice of supersymmetry subalgebra preserved by the BPS states and interpret wall crossing operators as Drinfeld twists of the coproduct. Kontsevich-Soibelman spectrum generator is then identified with Khoroshkin-Tolstoy universal R-matrix.
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https://arxiv.org/abs/2512.24988
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a014ba40d2a554cebd8b9b38a585b25f5092a081a8dcd9735993d9d3675a562d
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2026-01-01T00:00:00-05:00
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Grassmannian Geometries for Non-Planar On-Shell Diagrams
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arXiv:2512.25005v1 Announce Type: cross Abstract: On-shell diagrams are gauge invariant quantities which play an important role in the description of scattering amplitudes. Based on the principles of generalized unitarity, they are given by products of elementary three-point amplitudes where the kinematics of internal on-shell legs are determined by cut conditions. In the ${\cal N}=4$ Super Yang-Mills (SYM) theory, the dual formulation for on-shell diagrams produces the same quantities as canonical forms on the Grassmannian $G(k,n)$. Most of the work in this direction has been devoted to the planar diagrams, which dominate in the large $N$ limit of gauge theories. On the mathematical side, planar on-shell diagrams correspond to cells of the positive Grassmannian $G_+(k,n)$ which have been very extensively studied in the literature in the past 20 years. In this paper, we focus on the non-planar on-shell diagrams which are relevant at finite $N$. In particular, we use the triplet formulation of Maximal-Helicity-Violating (MHV) on-shell diagrams to obtain certain regions in the Grassmannian $G(2,n)$. These regions are unions of positive Grassmannians with different orderings (referred to as oriented regions). We explore the features of these unions, and show that they are pseudo-positive geometries, in contrast to positive geometry of a single oriented region. For all non-planar diagrams which are \emph{internally planar} there always exists a strongly connected geometry, and for those that are \emph{irreducible}, there exists a geometry with no spurious facets. We also prove that the already known identity moves, square and sphere moves, form the complete set of identity moves for all MHV on-shell diagrams.
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https://arxiv.org/abs/2512.25005
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fd9832965a61e61aa9ec553400c333e68ff03d1ded23cae8a7f833ef93ca9e90
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2026-01-01T00:00:00-05:00
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Modewise Additive Factor Model for Matrix Time Series
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arXiv:2512.25025v1 Announce Type: cross Abstract: We introduce a Modewise Additive Factor Model (MAFM) for matrix-valued time series that captures row-specific and column-specific latent effects through an additive structure, offering greater flexibility than multiplicative frameworks such as Tucker and CP factor models. In MAFM, each observation decomposes into a row-factor component, a column-factor component, and noise, allowing distinct sources of variation along different modes to be modeled separately. We develop a computationally efficient two-stage estimation procedure: Modewise Inner-product Eigendecomposition (MINE) for initialization, followed by Complement-Projected Alternating Subspace Estimation (COMPAS) for iterative refinement. The key methodological innovation is that orthogonal complement projections completely eliminate cross-modal interference when estimating each loading space. We establish convergence rates for the estimated factor loading matrices under proper conditions. We further derive asymptotic distributions for the loading matrix estimators and develop consistent covariance estimators, yielding a data-driven inference framework that enables confidence interval construction and hypothesis testing. As a technical contribution of independent interest, we establish matrix Bernstein inequalities for quadratic forms of dependent matrix time series. Numerical experiments on synthetic and real data demonstrate the advantages of the proposed method over existing approaches.
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https://arxiv.org/abs/2512.25025
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a5a84d17e01f6a83a3f29557ceacddd5f1f6fb45197bfaf95c69489cf531dcc5
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2026-01-01T00:00:00-05:00
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Testing Monotonicity in a Finite Population
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arXiv:2512.25032v1 Announce Type: cross Abstract: We consider the extent to which we can learn from a completely randomized experiment whether everyone has treatment effects that are weakly of the same sign, a condition we call monotonicity. From a classical sampling perspective, it is well-known that monotonicity is untestable. By contrast, we show from the design-based perspective -- in which the units in the population are fixed and only treatment assignment is stochastic -- that the distribution of treatment effects in the finite population (and hence whether monotonicity holds) is formally identified. We argue, however, that the usual definition of identification is unnatural in the design-based setting because it imagines knowing the distribution of outcomes over different treatment assignments for the same units. We thus evaluate the informativeness of the data by the extent to which it enables frequentist testing and Bayesian updating. We show that frequentist tests can have nontrivial power against some alternatives, but power is generically limited. Likewise, we show that there exist (non-degenerate) Bayesian priors that never update about whether monotonicity holds. We conclude that, despite the formal identification result, the ability to learn about monotonicity from data in practice is severely limited.
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https://arxiv.org/abs/2512.25032
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7038ad80f4ab38726d49b8973ac75d5da4ce08ff8343f16827129ef921b7e58b
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2026-01-01T00:00:00-05:00
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Compound Estimation for Binomials
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arXiv:2512.25042v1 Announce Type: cross Abstract: Many applications involve estimating the mean of multiple binomial outcomes as a common problem -- assessing intergenerational mobility of census tracts, estimating prevalence of infectious diseases across countries, and measuring click-through rates for different demographic groups. The most standard approach is to report the plain average of each outcome. Despite simplicity, the estimates are noisy when the sample sizes or mean parameters are small. In contrast, the Empirical Bayes (EB) methods are able to boost the average accuracy by borrowing information across tasks. Nevertheless, the EB methods require a Bayesian model where the parameters are sampled from a prior distribution which, unlike the commonly-studied Gaussian case, is unidentified due to discreteness of binomial measurements. Even if the prior distribution is known, the computation is difficult when the sample sizes are heterogeneous as there is no simple joint conjugate prior for the sample size and mean parameter. In this paper, we consider the compound decision framework which treats the sample size and mean parameters as fixed quantities. We develop an approximate Stein's Unbiased Risk Estimator (SURE) for the average mean squared error given any class of estimators. For a class of machine learning-assisted linear shrinkage estimators, we establish asymptotic optimality, regret bounds, and valid inference. Unlike existing work, we work with the binomials directly without resorting to Gaussian approximations. This allows us to work with small sample sizes and/or mean parameters in both one-sample and two-sample settings. We demonstrate our approach using three datasets on firm discrimination, education outcomes, and innovation rates.
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https://arxiv.org/abs/2512.25042
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eafc6346f18b0b50d9a05461a865cea5a85913124a32e37851edf8e9dc5f548c
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2026-01-01T00:00:00-05:00
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Fluid dynamics as intersection problem
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arXiv:2512.25053v1 Announce Type: cross Abstract: We formulate the covariant hydrodynamics equations describing the fluid dynamics as the problem of intersection theory on the infinite dimensional symplectic manifold associated with spacetime. This point of view separates the structures related to the equation of state, the geometry of spacetime, and structures related to the (differential) topology of spacetime. We point out a five-dimensional origin of the formalism of Lichnerowicz and Carter. Our formalism also incorporates the chiral anomaly and Onsager quantization. We clarify the relation between the canonical velocity and Landau $4$-velocity, the meaning of Kelvin's theorem. Finally, we discuss some connections to topological strings, Poisson sigma models, and topological field theories in various dimensions.
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https://arxiv.org/abs/2512.25053
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962e626af4d45a3e3420eec08ccb63b99e55f649d1fc5c3c0c7fc51b8ba236ce
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2026-01-01T00:00:00-05:00
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Roller Coaster Permutations and Partition Numbers
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arXiv:1703.08735v2 Announce Type: replace Abstract: This paper explores the partition properties of roller coaster permutations, a class of permutations characterized by maximizing the number of alternating runs in all subsequences. We establish a connection between the structure of these permutations and their partition numbers, defined as the minimum number of monotonic subsequences required to cover the permutation. Our main result provides a theoretical upper bound for the partition number of a roller coaster permutation of length $n$, given by $P_{max}(n) \le \lfloor\frac{\lceil\frac{n-2}{2}\rceil}{2}\rfloor + 2$. We further present experimental data for $n < 15$ that suggests this bound is nearly sharp.
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https://arxiv.org/abs/1703.08735
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bb527de461e8e1710412704a19449fd5bad1be41518320d671a7c2132d7ddfc9
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2026-01-01T00:00:00-05:00
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Nonisothermal Richards flow in porous media with cross diffusion
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arXiv:2102.00455v3 Announce Type: replace Abstract: The existence of large-data weak entropy solutions to a nonisothermal immiscible compressible two-phase unsaturated flow model in porous media is proved. The model is thermodynamically consistent and includes temperature gradients and cross-diffusion effects. Due to the fact that some terms from the total energy balance are non-integrable in the classical weak sense, we consider so-called variational entropy solutions. A priori estimates are derived from the entropy balance and the total energy balance. The compactness is achieved by using the Div-Curl lemma.
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https://arxiv.org/abs/2102.00455
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b9690c1a06d91d118c2c1a88470829d79e78790bf870695b73c06a9446fdd9d3
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2026-01-01T00:00:00-05:00
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The Birch--Swinnerton-Dyer exact formula for quadratic twists of elliptic curves
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arXiv:2102.11798v2 Announce Type: replace Abstract: In the present paper, we obtain a general lower bound for the $2$-adic valuation of the algebraic part of the central value of the complex $L$-series for the quadratic twists of any elliptic curve over $\mathbb{Q}$, showing that when the $2$-part of the product of Tamagawa factors grows, the $2$-part of the algebraic central $L$-value grows as well, in accordance with the Birch--Swinnerton-Dyer exact formula. This generalises a result of Coates--Kim--Liang--Zhao to all elliptic curves defined over $\mathbb{Q}$. We also prove the existence of an explicit infinite family of quadratic twists with analytic rank $0$ for a large family of elliptic curves.
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https://arxiv.org/abs/2102.11798
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31c0034fb991f1f072eb27c4bf7d2cfef5f4ca2aaf2927780c562203ca84e9c8
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2026-01-01T00:00:00-05:00
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P\'olya-Ostrowski Group and Unit Index in Real Biquadratic Fields
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arXiv:2108.01904v4 Announce Type: replace Abstract: The P\'olya-Ostrowski group of a Galois number field $K$, is the subgroup $Po(K)$ of the ideal class group $Cl(K)$ of $K$ generated by the classes of all the strongly ambiguous ideals of $K$. The number field $K$ is called a P\'olya field, whenever $Po(K)$ is trivial. In this paper, using some results of Bennett Setzer \cite{Bennett} and Zantema \cite{Zantema}, we give an explicit relation between the order of P\'olya groups and the Hasse unit indices in real biquadratic fields. As an application, we refine Zantema's upper bound on the number of ramified primes in P\'olya real biquadratic fields.
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https://arxiv.org/abs/2108.01904
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25ef64a1fe512142e2354acd9d2d6ea6ab16fe7e89cb62ba2a2040557b5705a3
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2026-01-01T00:00:00-05:00
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Eisenstein series on arithmetic quotients of rank 2 Kac--Moody groups over finite fields
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arXiv:2108.02919v4 Announce Type: replace Abstract: Let $G$ be an affine or hyperbolic rank 2 Kac--Moody group over a finite field $\mathbb F_q$. Let $X=X_{q+1}$ be the Tits building of $G$, the $(q+1)$--homogeneous tree, and let $\Gamma$ be a non-uniform lattice in $G$. When $\Gamma$ is a standard parabolic subgroup for the negative $BN$--pair, we define Eisenstein series on $\Gamma \backslash X$ and prove its convergence in a half space using Iwasawa decomposition of the Haar measure on $G$. A crucial tool is a description of the vertices of $X$ in terms of Iwasawa cells. We also prove meromorphic continuation of the Eisenstein series. This requires us to construct an integral operator on the Tits building $X$ and a truncation operator for the Eisenstein series. We also develop the functional analytic framework necessary for proving meromorphic continuation in our setting, by refining and extending Bernstein's Continuation Principle.
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https://arxiv.org/abs/2108.02919
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271c029bacd01a6d674aaeb1728d6e759f745c16bd1246fe3d3bd7605a3ff8cf
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2026-01-01T00:00:00-05:00
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Towards a mirror theorem for GLSMs
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arXiv:2108.12360v4 Announce Type: replace Abstract: We propose a method for computing generating functions of genus-zero invariants of a gauged linear sigma model $(V, G, \theta, w)$. We show that certain derivatives of $I$-functions of quasimap invariants of $[V //_\theta G]$ produce $I$-functions (appropriately defined) of the GLSM. When $G$ is an algebraic torus we obtain an explicit formula for an $I$-function, and check that it agrees with previously computed $I$-functions in known special cases. Our approach is based on a new construction of GLSM invariants which applies whenever the evaluation maps from the moduli space are proper, and includes insertions from light marked points.
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https://arxiv.org/abs/2108.12360
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b919ab48fe683ceea67d92698ab366f46ca97d6cff89ab1c74ca2e23ee4a3af5
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2026-01-01T00:00:00-05:00
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The \'etale cohomology ring of a punctured arithmetic curve
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arXiv:2110.01597v5 Announce Type: replace Abstract: We compute the cohomology ring $H^*(U,\mathbb{Z}/n\mathbb{Z})$ for $U=X\setminus S$ where $X$ is the spectrum of the ring of integers of a number field $K$ and $S$ is a finite set of finite primes. As a consequence, we obtain an efficient way to compute presentations of $Q_2(G_S)$, where $G_S$ is Galois group of the maximal extension of $K$ unramified outside of a finite set of primes $S$, for varying $K$. This includes the following cases (for $p$ any prime dividing $n$): $\mu_p(\overline{K}) \not\subseteq K$; $S$ does not contain the primes above $p$; and $p=2$ with $K$ admitting real archimedean places. We also show how to recover the classical reciprocity law of the Legendre symbol from the graded commutativity of the cup product.
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https://arxiv.org/abs/2110.01597
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2618429a6b0884013205fc35e480d4463c49b7100605e58eae20a991a8d59396
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2026-01-01T00:00:00-05:00
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On the local existence of solutions to the Navier-Stokes-wave system with a free interface
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arXiv:2202.02707v2 Announce Type: replace Abstract: We address a system of equations modeling a compressible fluid interacting with an elastic body in dimension three. We prove the local existence and uniqueness of a strong solution when the initial velocity belongs to the space $H^{2+\epsilon}$ and the initial structure velocity is in $H^{1.5+\epsilon}$ , where $\epsilon \in (0, 1/2)$.
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https://arxiv.org/abs/2202.02707
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df9f1bccd4e7323953c5b1cd66f1763186ca59896fb51ea05fee7f34b668dee0
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2026-01-01T00:00:00-05:00
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On the regularity of axisymmetric, swirl-free solutions of the Euler equation in four and higher dimensions
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arXiv:2204.13406v3 Announce Type: replace Abstract: In this paper, we consider axisymmetric, swirl-free solutions of the Euler equation in four and higher dimensions. We show that in dimension $d\geq 4$, axisymmetric, swirl-free solutions of the Euler equation have properties which could allow finite-time singularity formation of a form that is excluded when $d=3$, and we prove a conditional blowup result for axisymmetric, swirl-free solutions of the Euler equation in dimension $d\geq 4$. The condition which must be imposed on the solution in order to imply blowup becomes weaker as $d\to +\infty$, suggesting the dynamics are becoming much more singular as the dimension increases.
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https://arxiv.org/abs/2204.13406
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6ced5e0c4dd7eaea3faef42fc642fbc639ace491354e065589e1054069323a2b
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2026-01-01T00:00:00-05:00
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Small resolutions of moduli spaces of scaled curves
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arXiv:2208.03721v5 Announce Type: replace Abstract: We construct small resolutions of the moduli space $\overline{Q}_n$ of stable scaled $n$-marked lines of Ziltener and Ma'u--Woodward and of the moduli space $\overline{P}_n$ of stable $n$-marked ${\mathbb G}_a$-rational trees introduced in earlier work. The resolution of $\overline{P}_n$ is the augmented wonderful variety corresponding to the graphic matroid of the complete graph. The resolution of $\overline{Q}_n$ is a further blowup, also a wonderful model of an arrangement in ${\mathbb P}^{n-1}$.
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https://arxiv.org/abs/2208.03721
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f766acbc75019c64d6d2bed8e98ee9dd09070933ac98b9d3bbb81fe0756688a9
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2026-01-01T00:00:00-05:00
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Convex Hulls of Dragon Curves
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arXiv:2208.12434v2 Announce Type: replace Abstract: The fundamental geometry of self-similar sets becomes significantly more complex when the generating contractive maps include non-trivial rotational components. A well-known family exemplifying this complexity is that of the dragon curves in the plane. In this paper, we prove that every dragon curve has a polygonal convex hull. Moreover, we completely characterize their convex hulls.
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https://arxiv.org/abs/2208.12434
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291cba21ef66d33175bee76f609c721c833e8f861e0dbff94a2eec1f6941d1dc
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2026-01-01T00:00:00-05:00
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Progressive Optimal Path Sampling for Closed-Loop Optimal Control Design with Deep Neural Networks
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arXiv:2209.04078v3 Announce Type: replace Abstract: Closed-loop optimal control design for high-dimensional nonlinear systems has been a long-standing challenge. Traditional methods, such as solving the associated Hamilton-Jacobi-Bellman equation, suffer from the curse of dimensionality. Recent literature proposed a new promising approach based on supervised learning, by leveraging powerful open-loop optimal control solvers to generate training data and neural networks as efficient high-dimensional function approximators to fit the closed-loop optimal control. This approach successfully handles certain high-dimensional optimal control problems but still performs poorly on more challenging problems. One of the crucial reasons for the failure is the so-called distribution mismatch phenomenon brought by the controlled dynamics. In this paper, we investigate this phenomenon and propose the Progressive Optimal Path Sampling (POPS) method to mitigate this problem. We theoretically prove that this enhanced sampling strategy outperforms both the vanilla approach and the widely used Dataset Aggregation (DAgger) method on the classical linear-quadratic regulator by a factor proportional to the total time duration. We further numerically demonstrate that the proposed sampling strategy significantly improves the performance on tested control problems, including the optimal landing problem of a quadrotor and the optimal reaching problem of a 7 DoF manipulator.
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https://arxiv.org/abs/2209.04078
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2eae1408a5625ba4794bb54c3021071e29e51d91569822763b30d10f3092cc48
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2026-01-01T00:00:00-05:00
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Knot surgery formulae for instanton Floer homology II: applications
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arXiv:2209.11018v2 Announce Type: replace Abstract: This is a companion paper to earlier work of the authors, which proved an integral surgery formula for framed instanton homology. First, we present an enhancement of the large surgery formula, a rational surgery formula for null-homologous knots in any 3-manifold, and a formula encoding a large portion of $I^\sharp(S^3_0(K))$. Second, we use the integral surgery formula to study the framed instanton homology of many 3-manifolds: Seifert fibered spaces with nonzero orbifold degrees, especially nontrivial circle bundles over any orientable surface, surgeries on a family of alternating knots and all twisted Whitehead doubles, and splicings with twist knots. Finally, we use the previous techniques and computations to study almost L-space knots, ${\it i.e.}$, the knots $K\subset S^3$ with $\dim I^\sharp(S_n^3(K))=n+2$ for some $n\in\mathbb{N}_+$. We show that an almost L-space knot of genus at least $2$ is fibered and strongly quasi-positive, and a genus-one almost L-space knot must be either the figure eight or the mirror of the $5_2$ knot in Rolfsen's knot table.
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https://arxiv.org/abs/2209.11018
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9387e4f0eb79a335160d7aeb8c5c2290e62dcee74fd9905ca4ad5e1c8cab708f
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2026-01-01T00:00:00-05:00
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Incidence Estimates for Tubes in Complex Space
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arXiv:2210.05477v3 Announce Type: replace Abstract: In this paper, we prove a complex version of the incidence estimate of Guth, Solomon and Wang for tubes obeying certain strong spacing conditions, and we use one of our new estimates to resolve a discretized variant of Falconer's distance set problem in $\mathbb{C}^2$.
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https://arxiv.org/abs/2210.05477
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e69b0b3417286ef76b9079620117c7d69a75193b25cd13959dff5e5d00bd80fc
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2026-01-01T00:00:00-05:00
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Pullback of quantum principal bundles
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arXiv:2212.06946v2 Announce Type: replace Abstract: We introduce an abstract framework of Cartesian squares beyond the context of fiber products, and use it to extend the notion of pullback from classical to compact quantum principal bundles. Based only on our abstract notion of a Cartesian square, we extend key concepts of Equivariant Topology, such as the pullback of a family of group actions, orbit spaces, slices and global sections, change of base and structure group, free actions, and the groupoid of compact principal bundles. Finally, we embed the thus extended Equivariant Topology inside the 2-category of Grothendieck categories in such a way that our notion of a Cartesian square becomes the appropriate Beck-Chevalley condition.
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https://arxiv.org/abs/2212.06946
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cafc08e01298a37ff0ac9df72eb43b86e78c68648e782e272752665bab3e5f9b
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2026-01-01T00:00:00-05:00
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Gromov width of the disk cotangent bundle of spheres of revolution
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arXiv:2301.08528v2 Announce Type: replace Abstract: Inspired by work of the first and second author, this paper studies the Gromov width of the disk cotangent bundle of spheroids and Zoll spheres of revolution. This is achieved with the use of techniques from integrable systems and embedded contact homology capacities.
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https://arxiv.org/abs/2301.08528
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3261102ff3d8e7c9f183520d67d33b6daf2b40a1eee5e622e92f0a6cf6f6d2b7
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2026-01-01T00:00:00-05:00
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Time-Domain Moment Matching for Second-Order Systems
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arXiv:2305.01254v2 Announce Type: replace Abstract: The paper develops a second-order time-domain moment matching framework for the structure-preserving model reduction of second-order dynamical systems of high dimension, avoiding the first-order double-sized equivalent system. The moments of a second-order system are defined based on the solutions of second-order Sylvester equations, leading to families of parameterized second-order reduced models that match the moments of an original second-order system at selected interpolation points. Furthermore, a two-sided moment matching problem is addressed, providing a unique second-order reduced system that matches two distinct sets of interpolation points. We also construct the reduced second-order systems that match the moments of both the zero and first-order derivatives of the transfer function of the original second-order system. Finally, the Loewner framework is extended to second-order systems, where two parameterized families of models are presented that retain the second-order structure and interpolate sets of tangential data. The theory of the second-order time-domain moment matching is illustrated on vibrating systems.
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https://arxiv.org/abs/2305.01254
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54cd4fad241393d02c8d970f1c60c6832f7e19c5fe55741d473eb4d311697fc0
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2026-01-01T00:00:00-05:00
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Topological Hochschild homology of the image of j
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arXiv:2307.04248v2 Announce Type: replace Abstract: We compute the mod $(p,v_1)$ and mod $(2,\eta,v_1)$ $\mathrm{THH}$ of many variants of the image-of-$J$ spectrum. In particular, we do this for $j_{\zeta}$, whose $\mathrm{TC}$ is closely related to the $K$-theory of the $K(1)$-local sphere. We find in particular that the failure for $\mathrm{THH}$ to satisfy $\mathbb{Z}_p$-Galois descent for the extension $j_{\zeta} \to \ell_p$ corresponds to the failure of the $p$-adic circle to be its own free loop space. For $p>2$, we also prove the Segal conjecture for $j_{\zeta}$, and we compute the $K$-theory of the $K(1)$-local sphere in degrees $\leq 4p-6$.
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https://arxiv.org/abs/2307.04248
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8d298dcd46904f4d204be43e3ebe7b359d613169cfac79de284fe8150e27a918
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2026-01-01T00:00:00-05:00
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MLC at Feigenbaum points
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arXiv:2309.02107v3 Announce Type: replace Abstract: We prove a priori bounds for Feigenbaum quadratic polynomials, i.e., infinitely renormalizable polynomials $f_c: z\mapsto z^2+c$ of bounded type. It implies local connectivity of the corresponding Julia sets $J(f_c)$ and MLC (local connectivity of the Mandelbrot set) at the corresponding parameters $c$. It also yields the scaling Universality, dynamical and parameter, for the corresponding combinatorics. The MLC Conjecture was open for the most classical period-doubling Feigenbaum parameter as well as for the complex tripling renormalizations. Universality for the latter was conjectured by Goldberg-Khanin-Sinai in the early 1980s.
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https://arxiv.org/abs/2309.02107
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186950d86fc510212b679a007030db28e9551717c2a55c534802d7176beec801
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2026-01-01T00:00:00-05:00
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GeNIOS: an (almost) second-order operator-splitting solver for large-scale convex optimization
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arXiv:2310.08333v2 Announce Type: replace Abstract: We introduce the GEneralized Newton Inexact Operator Splitting solver (GeNIOS) for large-scale convex optimization. GeNIOS speeds up ADMM by approximately solving approximate subproblems: it uses a second-order approximation to the most challenging ADMM subproblem and solves it inexactly with a fast randomized solver. Despite these approximations, GeNIOS retains the convergence rate of classic ADMM and can detect primal and dual infeasibility from the algorithm iterates. At each iteration, the algorithm solves a positive-definite linear system that arises from a second-order approximation of the first subproblem and computes an approximate proximal operator. GeNIOS solves the linear system using an indirect solver with a randomized preconditioner, making it particularly useful for large-scale problems with dense data. Our high-performance open-source implementation in Julia allows users to specify convex optimization problems directly (with or without conic reformulation) and allows extensive customization. We illustrate GeNIOS's performance on a variety of problem types. Notably, GeNIOS is up to ten times faster than existing solvers on large-scale, dense problems.
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https://arxiv.org/abs/2310.08333
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4e9093b9c9ac39321ddd27c0d68bcf6d5e0a364bc39bfee338882400590aa9ce
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2026-01-01T00:00:00-05:00
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The L'vov-Kaplansky Conjecture for Polynomials of Degree Three
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arXiv:2310.15600v2 Announce Type: replace Abstract: The L'vov-Kaplansky conjecture states that the image of a multilinear noncommutative polynomial $f$ in the matrix algebra $M_n(K)$ is a vector space for every $n \in {\mathbb N}$. We prove this conjecture for the case where $f$ has degree $3$ and $K$ is an algebraically closed field of characteristic $0$.
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https://arxiv.org/abs/2310.15600
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7809199241f8d856ce6227052dff8d2813a7979bb92deb830b48c47eb84ca4d3
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2026-01-01T00:00:00-05:00
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Paley-Wiener Theorem for Probabilistic Frames
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arXiv:2310.17830v3 Announce Type: replace Abstract: This paper establishes Paley-Wiener perturbation theorems for probabilistic frames. The classical Paley-Wiener perturbation theorem shows that if a sequence is close to a basis in a Banach space, then this sequence is also a basis. Similar perturbation results have been established for frames in Hilbert spaces. In this work, we show that if a probability measure is sufficiently close to a probabilistic frame in an appropriate sense, then this probability measure is also a probabilistic frame. Moreover, we obtain explicit frame bounds for such probability measures that are close to a given probabilistic frame in the $2$-Wasserstein metric. This yields an alternative proof of the fact that the set of probabilistic frames is open in $\mathcal{P}_2(\mathbb{R}^n)$ under the $2$-Wasserstein topology.
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https://arxiv.org/abs/2310.17830
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7e5abaeba7197ca61d06bec11592011b7dd12f7134cca03485d5560b4e6e004d
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2026-01-01T00:00:00-05:00
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On the complexity of meander-like diagrams of knots
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arXiv:2312.05014v2 Announce Type: replace Abstract: It is known that each knot has a semimeander diagram (i. e. a diagram composed of two smooth simple arcs), however the number of crossings in such a diagram can only be roughly estimated. In the present paper we provide a new estimate of the complexity of the semimeander diagrams. We prove that for each knot $K$ with more than 10 crossings, there exists a semimeander diagram with no more than $0.31 \cdot 1.558^{\operatorname{cr}(K)}$ crossings, where $\operatorname{cr}(K)$ is the crossing number of $K$. As a corollary, we provide new estimates of the complexity of other meander-like types of knot diagrams, such as meander diagrams and potholders. We also describe an efficient algorithm for constructing a semimeander diagram from a given one.
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https://arxiv.org/abs/2312.05014
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098cf9832142bbf36e6f467e8b4a5fa256a579aeb98ea2394435b7cb9529c83e
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2026-01-01T00:00:00-05:00
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A Rank-Dependent Theory for Decision under Risk and Ambiguity
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arXiv:2312.05977v3 Announce Type: replace Abstract: This paper axiomatizes, in a two-stage setup, a new theory for decision under risk and ambiguity. The axiomatized preference relation $\succeq$ on the space $\tilde{V}$ of random variables induces an ambiguity index $c$ on the space $\Delta$ of probabilities, a probability weighting function $\psi$, generating the measure $\nu_{\psi}$ by transforming an objective probability measure, and a utility function $\phi$, such that, for all $\tilde{v},\tilde{u}\in\tilde{V}$, \begin{align*} \tilde{v}\succeq\tilde{u} \Leftrightarrow \min_{Q \in \Delta} \left\{\mathbb{E}_Q\left[\int\phi\left(\tilde{v}^{\centerdot}\right)\,\mathrm{d}\nu_{\psi}\right]+c(Q)\right\} \geq \min_{Q \in \Delta} \left\{\mathbb{E}_Q\left[\int\phi\left(\tilde{u}^{\centerdot}\right)\,\mathrm{d}\nu_{\psi}\right]+c(Q)\right\}. \end{align*} Our theory extends the rank-dependent utility model of Quiggin (1982) for decision under risk to risk and ambiguity, reduces to the variational preferences model when $\psi$ is the identity, and is dual to variational preferences when $\phi$ is affine in the same way as the theory of Yaari (1987) is dual to expected utility. As a special case, we obtain a preference axiomatization of a decision theory that is a rank-dependent generalization of the popular maxmin expected utility theory. We characterize ambiguity aversion in our theory.
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https://arxiv.org/abs/2312.05977
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b97b6c2c4d01700d871eed113a1f09f9bdeb6618a0fb70dc2b56f05a68056472
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2026-01-01T00:00:00-05:00
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Some frustrating questions on dimensions of products of posets
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arXiv:2312.12615v4 Announce Type: replace Abstract: For $P$ a poset, the dimension of $P$ is defined to be the least cardinal $\kappa$ such that $P$ is embeddable in a direct product of $\kappa$ totally ordered sets. We study the behavior of this function on finite-dimensional (not necessarily finite) posets. In general, the dimension dim($P$ x $Q$) of a product of two posets can be smaller than dim($P$) + dim($Q$), though no cases are known where the discrepancy is greater than 2. We obtain a result that gives upper bounds on the dimensions of certain products of posets, including cases where the discrepancy 2 is achieved. But the paper is mainly devoted to stating questions, old and new, about dimensions of product posets, noting implications among their possible answers, and introducing some related concepts that might be helpful in tackling these questions.
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https://arxiv.org/abs/2312.12615
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9ddc6a8b30a6ae64d194fa1043150f93fc2c0438443fcf3d3059824f9f9c0ee5
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2026-01-01T00:00:00-05:00
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Categorical absorption of a non-isolated singularity
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arXiv:2402.18513v2 Announce Type: replace Abstract: We study an example of a projective threefold with a non-isolated singularity and its derived category. The singular locus can be locally described as a line of surface nodes compounded with a threefold node at the origin. We construct a semiorthogonal decomposition where one component absorbs the singularity in the sense of Kuznetsov--Shinder, and the other components are equivalent to the derived categories of smooth projective varieties. The absorbing category is seen to be closely related to the absorbing category constructed for nodal varieties by Kuznetsov--Shinder, reflecting the geometry of the singularity. We further show that the semiorthogonal decomposition is induced by one on a geometric resolution, and briefly consider the properties of the absorbing category under smoothing.
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https://arxiv.org/abs/2402.18513
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0eeaecd536eb5a27b18cf7380176fd409cf36881f2b440ffdf5343b237dea6bf
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2026-01-01T00:00:00-05:00
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Growth of root multiplicities along imaginary root strings in Kac--Moody algebras
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arXiv:2403.01687v3 Announce Type: replace Abstract: Let $\mathfrak{g}$ be a symmetrizable Kac--Moody algebra. Given a root $\alpha$ and a real root $\beta$ of $\mathfrak{g}$, it is known that the $\beta$-string through $\alpha$, denoted $R_\alpha(\beta)$, is finite. Given an imaginary root $\beta$, we show that $R_\alpha(\beta)=\{\beta\}$ or $R_\alpha(\beta)$ is infinite. If $(\beta,\beta)<0$, we also show that the multiplicity of the root ${\alpha+n\beta}$ grows at least exponentially as $n\to\infty$. If $(\beta,\beta)=(\alpha, \beta) = 0$, we show that $R_\alpha(\beta)$ is bi-infinite and the multiplicities of $\alpha+n\beta$ are bounded. If $(\beta,\beta)=0$ and $(\alpha, \beta) \neq 0$, we show that $R_\alpha(\beta)$ is semi-infinite and the muliplicity of $\alpha+n\beta$ or $\alpha-n\beta$ grows faster than every polynomial as $n\to\infty$. We also prove that $\dim \mathfrak{g}_{\alpha+\beta} \geq \dim \mathfrak{g}_\alpha + \dim \mathfrak{g}_\beta -1$ whenever $\alpha \neq \beta$ with $(\alpha, \beta)<0$.
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https://arxiv.org/abs/2403.01687
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Academic Papers
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