In 2012, Taelman proved a class formula for -series associated to Drinfeld -modules and considered it as a function field analogue of the Birch and Swinnerton-Dyer conjecture. Since then, Taelman's class formula has been generalized to the setting of Anderson -modules. Let be a monic irreducible polynomial of , we define the -adic -series associated with Anderson -modules and prove a -adic class formula & agrave; la Taelman linking a -adic regulator, the class module and a local factor at . Next, we study the vanishing of the -adic -series and give some applications to Drinfeld modules defined over itself. Finally, we extend this result to the multi-variable setting & agrave; la Pellarin.
Let be a hyperplane arrangement in . We define a quadratic form on that is entirely determined by the intersection poset of . Using the Bogomolov-Gieseker inequality for parabolic bundles, we show that if is such that the weighted arrangement is stable, then . As an application, we consider the symmetric case where all the weights are equal. The inequality gives a lower bound for the total sum of multiplicities of codimension intersection subspaces of . The lower bound is attained when every intersects all the other members of along codimension subspaces; extending from to higher dimensions a condition found by Hirzebruch for line arrangements in the complex projective plane.
For time-dependent compressible Euler flows passing around a fixed solid body in three-dimensional space, there may exist an infinitesimally thin layer of concentrated mass, momentum and energy, wherein all particles impacting the body move along the body's windward boundary surface. By proposing a concept of Radon measure-valued solutions for initial-boundary-value problems of the unsteady compressible Euler equations, which captures both the large-scale three-dimensional distributions of the surrounding flows and the small-scale motions of particles on the two-dimensional boundary surfaces, we derive the governing partial differential equations for the concentration boundary layer-an unsteady (pressureless) compressible Euler system defined on the boundary surface with appropriate source terms. This down-scaling approach can be further generalized to incorporate skin-frictions and phase-transitions within the concentration boundary layer. It constitutes a novel methodology for addressing the complex fluid-solid-heat coupling problems encountered in fluid dynamics. Illustrative examples are presented to demonstrate the applicability of the proposed method to several specific problems, including the derivation for the most general case the Newtonian-Busemann pressure law of hypersonic aerodynamics.
We prove that among 1 and the odd zeta values , , , , at least are linearly independent over the rationals, for any sufficiently large odd integer . This is the first asymptotic improvement on the lower bound, logarithmic in , obtained by Ball-Rivoal in 2001. The proof is based on Siegel's lemma to construct non-explicit linear forms in values at odd integers of the Riemann zeta function, instead of using explicit well-poised hypergeometric series. A new refinement of Siegel's linear independence criterion is applied, together with a multiplicity estimate (namely a generalization of Shidlovsky's lemma). The result is also adapted to deal with values of the first polylogarithms at a fixed algebraic point in the unit disk, improving bounds of Rivoal and Marcovecchio.
Let and be the Witt Lie algebras. Clearly, is a proper subalegbra of . Surprisingly, we prove that simple smooth modules over are exactly the simple modules over studied by Rudakov (no need to take completion). Then, we find an easy and elementary way to classify all simple smooth modules over . When the height or , any nontrivial simple smooth -module is isomorphic to an induced module from a simple smooth -module . When and , any such module is the unique simple quotient of the tensor module for some simple -module , where is a particular simple module over the Weyl algebra . We further show that a simple -module is a smooth module if and only if the action of each of particular vectors in is locally finite on .
Let be a holomorphic Hecke cusp form of weight for , and let denote its sequence of normalised Hecke eigenvalues. We compute the first and second moments of the sums , on average over forms of large weight . In the range , the size of the second moment lies between and . This is in sharp contrast to the regime , where the second moment was shown in preceding work (part I) to be of size .
Heilbronn's triangle problem is a classical question in discrete geometry. It asks to determine the smallest number Delta=Delta(N) for which every collection in N points in the unit square spans a triangle with area at most Delta. We outline old and new developments around this problem, discuss related questions and highlight connections to recent developments in combinatorics and analysis.
The discrete data encoded in the power moments of a positive measure, fast decaying at infinity on Euclidean space, are incomplete for recovery, leading to the concept of moment indeterminateness. On the other hand, classical integral transforms (Fourier-Laplace, Fantappi & egrave;, Poisson) of such measures are complete, often invertible via an effective inverse operation. The gap between the two non-uniqueness/uniqueness phenomena is manifest in the dual picture, when trying to extend the measure, regarded as a positive linear functional, from the polynomial algebra to the full space of continuous functions. This point of view was advocated by Marcel Riesz a century ago, in the single real variable setting. Notable advances in functional analysis have their root in Riesz's celebrated four notes devoted to the moment problem. A key technical ingredient being there the monotone approximation by polynomials of kernels of integral transforms. With inherent new obstacles, we reappraise in the context of several real variables M. Riesz's variational principle. The result is an array of necessary and sufficient moment indeterminateness criteria, some raising real algebra questions, as well as others involving intriguing analytic problems, all gravitating around the concept of moment separating function.
We study the derivative of the characteristic polynomial of N & times;N Haar-distributed unitary matrices. We obtain new explicit formulae for complex-valued moments when the spectral variable is inside the unit disc, in the limit N ->infinity. These formulae are expressed in terms of the confluent hypergeometric function of the first kind. We explore the connection between these moments and those of the derivative of the Riemann zeta function away from the critical line. Under the Lindel & ouml;f hypothesis, we prove that all positive integer moments agree with our random matrix results up to a well-known arithmetic factor. Inspired by this finding, we propose a conjecture on the asymptotics of noninteger moments of the derivative of the Riemann zeta function off the critical line. Within random matrix theory, we also investigate the microscopic regime where the spectral variable z satisfies |z|(2)=1-c/N for a fixed constant c.We obtain an asymptotic formula for the moments in this regime as a determinant involving the finite temperature Bessel kernel, which reduces to the Bessel kernel when c=0.
We investigate a class of second-order difference equations featuring operator-valued coefficients with the aim of approaching problems of stationary scattering theory. We focus on various compact perturbations of the discrete Laplacian given in a Hilbert space of bi-infinite square-summable sequences with entries from a fixed Hilbert space. This work includes a detailed spectral analysis of the perturbed Laplacian and the construction and study of the corresponding objects pertaining to scattering theory, including the entries of the scattering matrix.
This paper establishes the global existence of classical solutions with large initial energy and vacuum to the isentropic compressible Navier-Stokes equations under slip boundary conditions in a three-dimensional (3D) exterior domain. For a near-isothermal fluid (the adiabatic exponent gamma>1 sufficiently close to 1) with zero far-field density (rho(infinity)=0), the solutions are proved to be global. This extends the classical one-dimensional result of Nishida and Smoller in 1973 (Comm. Pure Appl. Math. 26 (1973), 183-200). It provides the first global existence result for large-energy solutions with vacuum in a 3D exterior domain with a physical boundary. As a byproduct, for the whole space R-3 with rho(infinity)>0, a global result is obtained under a general smallness condition on rho(infinity), relaxing the previous restriction in Hong-Hou-Peng-Zhu (Math. Ann. 388 (2024), no. 2, 2163-2194).
This paper establishes a unified framework for analyzing the far-field convergence rates and structural stability of subsonic Euler flows with arbitrarily large vorticity and characteristic discontinuities in infinitely long nozzles. The main approach is an Euler-Lagrange transformation that flattens the discontinuities and reformulates the flow as a quasilinear elliptic equation in divergence form, combined with weighted L-2 local average energy estimates. This method addresses challenges arising from free boundaries of discontinuities and strong vorticity effects. It is shown that the convergence is governed by the slower of an intrinsic exponential decay and the boundary-induced decay, without the smallness or convexity assumptions. This yields the first such results for flows with characteristic discontinuities. Structural stability is established for both smooth and discontinuous flows, which have linear dependence respect to the finite boundary perturbations, thereby removing the need for a small-perturbation assumption. Both results extend naturally to incompressible flows.
Number theory for positive characteristic contains analogues of the special values that were introduced by Carlitz; these include the Carlitz gamma values and Carlitz zeta values. These values were further developed to the arithmetic gamma values and multiple zeta values by Goss and Thakur, respectively. In this work, by generalizing a result attained by Chang et al., we obtain the algebraic independence of arithmetic gamma values, positive characteristic multiple zeta values of restricted indices, and the hyperderivatives of their deformations. We prove this by using Chang-Papanikolas-Yu's derivation, Maurichat's prolongation, Namoijam's formula, and Papanikolas' theory of the t-motivic Galois group.
We discuss the papers P. Erd & odblac;s and P. Tur & aacute;n, On some sequences of integers, J. London Math. Soc. (1) 11 (1936), 261-264 and K. F. Roth, On certain sets of integers, J. London Math. Soc. (1) 28 (1953), 104-109, both foundational papers in the study of arithmetic progressions in sets of integers, and their subsequent influence.
This is the first of two articles on the strength of m-Sigma(3)(0)-determinacy for m is an element of N, the strongest theories of determinacy contained in Hilbert's second-order arithmetic (Z(2)). In this article, we refute two natural conjectures on the strength of these principles in terms of inductive definability, strengthening the well-known Montalban-Shore theorem. Our results show that m-Sigma(0)(3)-determinacy does not coincide in strength with any subsystem of Z(2 )previously considered in the literature. In the process, we establish connections between these determinacy principles, the complexity of sets definable by analytical inductive definitions, and nonstandard admissible sets.
We prove that for every integer d >= 2, every nonnegative integer k and every finite field F there exists an integer C(d,k,|F|) such that every order-d tensor with slice rank k over F admits at most C(d,k,|F|) decompositions with length k, up to a class of transformations that can be easily described. A key result in the proof asserts that if an order-d tensor admits d+1 slice rank decompositions and the linear subspaces spanned by their one-variable functions constitute a sunflower for each choice of special coordinate, then the tensor admits a decomposition where these linear subspaces are contained in the centers of these respective sunflowers.
This paper will consider combinatorial properties related to coding a cardinal by its bounded subsets. These properties have traditionally been studied in the context of very large cardinals and variations of these properties either reach the level of Kunen inconsistency or are very close to it. Within the descriptive set-theoretic framework with determinacy or partition properties, these combinatorial properties are quite robust and have numerous natural examples. Let kappa be a cardinal, & varepsilon; <= kappa, and X subset of kappa. BI kappa(& varepsilon;, X) is the set of all subsets of X of ordertype & varepsilon; which are bounded below kappa. BI kappa(< & varepsilon;, X) is the set of all subsets of X of ordertype less than & varepsilon; bounded below kappa. The following will be shown which answer or address several questions of Ben-Neria and Garti. center dot Let mu(1) (omega 1) be the club filter on omega 1. Assume omega(1) ->& lowast; (omega(1)) (omega 1) <(omega 1) and j mu 1 omega(1) (omega(1))=omega(2). For any function Phi & ratio; BI omega omega (< omega(1), omega(omega))->omega(omega), there is an X subset of omega omega with |X| = |omega(omega)| so that Phi[BI omega(omega) (< omega(1), X)] not equal omega(omega). center dot Let mu(1) (kappa) be the omega-club filter on kappa. If AC (& Pscr;)(omega)(kappa) and kappa ->& lowast; (kappa)(2)( kappa, there is an X is an element of mu(1) (kappa) so that Phi[BI kappa(< & varepsilon;, X)] not equal kappa. Let Theta be the supremum of the ordinal onto which & Ropf; surjects. For any cardinal kappa with omega(1) <= kappa kappa so that for all X is an element of mu(1) (kappa) , Phi[BI kappa(omega & sdot; omega, X)] = kappa. center dot Assume AD and DC & Ropf;. For any uniform countably complete filter F on omega(1), there is a function Phi & ratio; BI omega 1 (omega & sdot; omega, omega(1) )->omega(1) so that for all X is an element of F, Phi[BI omega 1 (omega & sdot; omega, X)] = omega(1). center dot Assume AD. Let delta(1) (omega) = sup{delta(1) (n) & ratio; n is an element of omega} be the supremum of the projective ordinals. For any & varepsilon;delta(1) (omega), there is an X subset of delta(1) (omega) with |X| = |delta(1) (omega)| so that Phi[BI delta(1) (omega) (< & varepsilon;, X)] not equal delta(1) (omega). There is also a uniform filter F on delta(1) (omega) so that for all & varepsilon;delta(1) (omega), there is an X is an element of F so that Phi[BI delta(1) (omega) (< & varepsilon;, X)] not equal delta(1) (omega).
Total positivity of matrices plays an important role in various branches of mathematics. In this paper, we present some criteria for the coefficientwise total positivity of Riordan arrays and the coefficientwise Hankel-total positivity of the row-generating polynomial sequence of Riordan arrays. In addition, we also give some criteria for the coefficientwise Hankel-total positivity of sequences of coefficients of compositional functions and some results for triangular convolutions preserving Stieltjes moment properties. As applications, we first obtain the coefficientwise Hankel-total positivity of two kinds of multivariate Fuss-Narayana-Riordan polynomials, which implies those of two kinds of multivariate Fuss-Narayana polynomials proved by P & eacute;tr & eacute;olle, Sokal and Zhu (Mem. Amer. Math. Soc., 2023). Then, we also derive the total positivity of Fuss-Catalan Riordan arrays, unifying total positivity results of a few well-known combinatorial triangles.
We prove an asymptotic local-global theorem on the heights of point orbits of thin subgroups of Bianchi groups in H-3.
To each weakly holomorphic modular function f 0 for SL(2, Z), which is nonnegative on the geodesic arc {e(it) : pi/3 <= t <= 2 pi/3}, we attach a GL(2, Z)-invariant map Lambda(f ): P-1 (R) -> R that generalizes the Lyapunov exponent function introduced by Spalding and Veselov. We prove that it takes every value between 0 and Lambda(f ) (1+root 5/2) and it gives an increasing convex function 2 on the Markov irrationalities when ordered using their parameterization by Farey fractions in [0, 1/2]. In the case of quadratic irrationals w with purely periodic continued fraction expansion, the value Lambda(f )(omega) equals the real part of the cycle integral off along the associated geodesic C(omega )on the modular surface, normalized with the word length of the associated hyperbolic matrix Aw as a word in the generators T =((1 1) (0 1) )and V = ((1 0)( 1 1)) These results are related to conjectures of Kaneko who observed several similar behavior for the cycle integrals of the modular j function when normalized by the hyperbolic length of the geodesic C-omega.
