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Abstract: Abstract We give a simple, down-to-earth, proof that Kottwitz’s (resp. Rapoport–Zink) PEL type integral models of Shimura varieties admit closed embeddings into Siegel integral models. We also show that Rapoport’s and Kottwitz’s integral models agree with Kisin’s integral models for relevant Shimura data. Our result also applies to certain exotic PEL type integral models not covered by Kottwitz (or Kisin, Kisin–Pappas etc). Moreover, combined with a result of Lan’s, we obtain closed embeddings for toroidal compactifications of integral models. PubDate: 2024-08-10

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Abstract: Abstract We establish a strong converse inequality by a K-functional of the rate of approximation by the Kantorovich operators in variable exponent Lebesgue spaces. It shows that a recently obtained direct inequality is optimal. A Voronovskaya inequality is also proved. The approach applied heavily relies on the boundedness of the Hardy-Littlewood maximal operator. PubDate: 2024-08-10

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Abstract: Abstract The optimal mass transport problem was formulated centuries ago, but only recently there has been a surge in its applications, particularly in functional inequalities, geometry, stochastic analysis, and numerical solutions for partial differential equations. Quantum optimal transport aims to extend this success story to non-commutative systems, where density operators replace probability measures. This brief review paper aims to describe the latest approaches, highlighting their advantages, disadvantages, and open mathematical problems relevant to applications. PubDate: 2024-08-09

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Abstract: Abstract We survey some recent recent results whose proofs depend in an essential way on the study of Galois representations. We discuss in particular the scarcity of rational points on ramified covers of abelian varieties, the problem of algorithmically computing endomorphism rings of abelian varieties over number fields, and a version of Kummer theory for commutative algebraic groups. PubDate: 2024-08-03

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Abstract: Abstract Every compact Riemann surface X admits a natural projective structure \(p_u\) as a consequence of the uniformization theorem. In this work we describe the construction of another natural projective structure on X, namely the Hodge projective structure \(p_h\) , related to the second fundamental form of the period map. We then describe how projective structures correspond to (1, 1)-differential forms on the moduli space of projective curves and, from this correspondence, we deduce that \(p_u\) and \(p_h\) are not the same structure. PubDate: 2024-07-30

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Abstract: Abstract We consider circles of common centre and increasing radius on a compact hyperbolic surface and, more generally, on its unit tangent bundle. We establish a precise asymptotics for their rate of equidistribution. Our result holds for translates of any circle arc by arbitrary elements of \({{\,\textrm{SL}\,}}_2({{\mathbb {R}}})\) . Our proof relies on a spectral method originating in Harish-Chandra’s study of the asymptotics of matrix coefficients of representations of semisimple Lie groups; the method was subsequently employed by Ratner, and further developed by Burger, in the study of geodesic and horocycle flows. We further derive statistical limit theorems, with compactly supported limiting distribution, for appropriately rescaled circle averages of sufficient regular observables. Finally, we discuss a classical application to the circle problem in the hyperbolic plane, following the approach of Duke-Rudnick-Sarnak and Eskin-McMullen. PubDate: 2024-07-27

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Abstract: Abstract We review some concepts and basic features of high-order polynomial interpolation of fields in space. The interpolation, from weights, of Lagrange and Hermite types for physical fields, intended as differential forms, is analyzed on an interval. PubDate: 2024-07-27

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Abstract: Abstract Let m and n be two positive integers such that \(m \le n\) and \(n \ge 3\) . In this article, by the unstable K-theory method, we will study the homotopy types of gauge groups of the principal SU(n)-bundles over \(\mathbb {C}P^3\) . Let \(\mathcal {G}_{l,k}(\mathbb {C}P^3)\) be the gauge groups of the principal SU(n)-bundles over \(\mathbb {C}P^3\) , we will partially classify the homotopy types of \(\mathcal {G}_{0,k}(\mathbb {C}P^3)\) by showing that if there is a homotopy equivalence \(\mathcal {G}_{0,k}(\mathbb {C}P^3)\simeq \mathcal {G}_{0,k'}(\mathbb {C}P^3)\) then we have \((\frac{1}{2}(n-1)n(n+1)(n + 2), k)=(\frac{1}{2}(n-1)n(n+1)(n+2), k')\) , when n is odd and \((\frac{1}{4}(n-1)n(n+1)(n + 2), k) = (\frac{1}{4}(n-1)n(n+1)(n+2), k')\) , when n is even. PubDate: 2024-07-26

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Abstract: Abstract In view of the recent proofs of the P=W conjecture, the present paper reviews and relates the latest results in the field, with a view on how P=W phenomena appear in multiple areas of algebraic geometry. As an application, we give a detailed sketch of the proof of P=W by Maulik, Shen and Yin. PubDate: 2024-07-25

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Abstract: Abstract Perturbation theory is a very useful tool to investigate the dynamics of models in space science. We start by presenting some results obtained implementing classical perturbation theory to investigate the motion of space debris, which are objects that populate the sky around the Earth after a satellite break-up event. When dealing with two or more break-up events, a clusterization of the fragments can be computed using machine learning techniques. We also present the celebrated KAM theory for symplectic and conformally symplectic systems. We recall several computer-assisted results in Celestial Mechanics in conservative and dissipative settings. Finally, we consider the spin-orbit problem and we show how machine learning methods can be conveniently used to classify regular and chaotic motions. PubDate: 2024-07-23

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Abstract: Abstract A Hermitian metric on a complex manifold is said to be pluriclosed or SKT if the torsion of the associated Bismut connection is closed, and it is called balanced if its fundamental form is co-closed. In the paper we give an overview of recent results on pluriclosed and balanced metrics, provide new constructions of compact non-Kähler manifolds and also present a few open problems. PubDate: 2024-07-20

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Abstract: Abstract This article presents a Kantorovich-type modification of the generalization of Lupaş operators. We investigate the approximation properties of the defined operators. We obtain the value of operators at test functions and based on that, we prove the well-established Korovkin-type theorem. For the Local approximation properties, we use moduli of smoothness and Peter’s K - functional to prove the results on the rate of convergence. We also discuss the Voronovskaja-type theorem to study the asymptotic behavior of defined operators. PubDate: 2024-07-07

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Abstract: Abstract We show how Hamkins’ Gap Forcing Theorem of Hamkins (Israel J Math 125:237–252, 2001, Bull Symb Logic 5: 264–272, 1999) can be used to give an alternate construction of models for the level by level equivalence between strong compactness and supercompactness when forcing over models of ZFC containing one supercompact cardinal in which no cardinal is supercompact up to a measurable cardinal. As an application of our methods, we also show that starting from such a model V which in addition satisfies Goldberg’s Ultrapower Axiom UA, it is possible to force and construct a model M with a supercompact cardinal \(\kappa \) satisfying level by level equivalence having certain additional properties. In particular, in M, no cardinal is supercompact up to a measurable cardinal. there is a stationary subset of measurable cardinals \(A \subseteq \kappa \) such that for every \(\delta \in A\) , \((o(\delta ))^V < \delta ^{++}\) , \((o(\delta ))^V = (o(\delta ))^M\) , and the Mitchell ordering of normal measures over \(\delta \) is linear. M can contain inaccessible and Mahlo cardinals above \(\kappa \) . It is currently unknown whether a model of ZFC with a supercompact cardinal and M’s properties can also satisfy UA. PubDate: 2024-07-02

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Abstract: Abstract Let G be a finite abelian group acting faithfully on \({\mathbb C}{\mathbb P}^1\) via holomorphic automorphisms. In [2] the G–equivariant algebraic vector bundles on G–invariant affine open subsets of \({\mathbb C}{\mathbb P}^1\) were classified. We classify the G–equivariant algebraic vector bundles on \({\mathbb C}{\mathbb P}^1\) . PubDate: 2024-06-11

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Abstract: Abstract In this paper, we study a problem to determine the forms of a non-constant entire function f when it shares a set S of 4 distinct elements with its k-th derivatives. This work is motivated by the work of Chang et al. (Arch Math 89: 561–569, 2007) which pertains to a set of three elements. We have comprehensively extended the same result, as this type of extension has not been addressed earlier. PubDate: 2024-06-05 DOI: 10.1007/s40574-024-00416-9

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Abstract: Abstract We present some results about the irreducible representations appearing in the exterior algebra \(\Lambda \mathfrak {g}\) , where \(\mathfrak {g}\) is a simple Lie algebra over \({\mathbb {C}}\) . For Lie algebras of type B, C or D we prove that certain irreducible representations, associated to weights characterized in a combinatorial way, appear as irreducible components of \(\Lambda \mathfrak {g}\) . Moreover, we propose an analogue of a conjecture of Kostant, about irreducibles appearing in the exterior algebra of the little adjoint representation. Finally, we give some closed expressions, in type B, C and D, for generalized exponents of small representations that are fundamental representations and we propose a generalization of some results of De Concini, Möseneder Frajria, Procesi and Papi about the module of special covariants of adjoint and little adjoint type. PubDate: 2024-06-01 DOI: 10.1007/s40574-023-00390-8

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Abstract: Abstract This paper investigates G. Fano’s contributions to the study of three-dimensional varieties—the so called Fano threefolds—starting from unpublished manuscripts kept in Turin’s Special Mathematical Library. The main aspects examined are the evolution of the research path, the placing into the wake of the Italian school of algebraic geometry, Fano’s links with the English geometers and his legacy on later studies. PubDate: 2024-06-01 DOI: 10.1007/s40574-023-00374-8

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Abstract: Abstract In this paper, we provide a different approach to the Alt–Caffarelli–Friedman monotonicity formula, reducing the problem to test the monotone increasing behavior of the mean value of a function involving the gradient’s norm. In particular, we show that our argument holds in the general framework of Carnot groups. PubDate: 2024-06-01 DOI: 10.1007/s40574-023-00393-5

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Abstract: Abstract Subgradient methods are the natural extension to the non-smooth case of the classical gradient descent for regular convex optimization problems. However, in general, they are characterized by slow convergence rates, and they require decreasing step-sizes to converge. In this paper we propose a subgradient method with constant step-size for composite convex objectives with \(\ell _1\) -regularization. If the smooth term is strongly convex, we can establish a linear convergence result for the function values. This fact relies on an accurate choice of the element of the subdifferential used for the update, and on proper actions adopted when non-differentiability regions are crossed. Then, we propose an accelerated version of the algorithm, based on conservative inertial dynamics and on an adaptive restart strategy, that is guaranteed to achieve a linear convergence rate in the strongly convex case. Finally, we test the performances of our algorithms on some strongly and non-strongly convex examples. PubDate: 2024-06-01 DOI: 10.1007/s40574-023-00389-1

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