Mathematika最新文献

Inspired by the work of Karamata, we consider an extremization problem associated with the probability of intersecting two random chords inside a circle of radius $$, where the endpoints of the chords are drawn according to a given probability distribution on $$. We show that, for $$, the problem is degenerated in the sense that any continuous measure is an extremizer, and that, for $$ sufficiently close to 1, the desired maximal value is strictly below the one for $$ by a polynomial factor in $$. Finally, we prove, by considering the auxiliary problem of drawing a single random chord, that the desired maximum is $$ for $$. Connections with other variational problems and energy minimization problems are also presented.

Extending results of Szalay, Bennett, Bugeaud and Mignotte in this paper, we prove finiteness results concerning perfect powers having two or three digits in their representation in a canonical number system of the equation order of an algebraic number field.

We consider the magnetic Steklov eigenvalue problem on compact Riemannian manifolds with boundary for generic magnetic potentials and establish various results concerning the spectrum. We provide equivalent characterizations of magnetic Steklov operators which are unitarily equivalent to the classical Steklov operator and study bounds for the smallest eigenvalue. We prove a Cheeger–Jammes-type lower bound for the first eigenvalue by introducing magnetic Cheeger constants. We also obtain an analogue of an upper bound for the first magnetic Neumann eigenvalue due to Colbois, El Soufi, Ilias, and Savo. In addition, we compute the full spectrum in the case of the Euclidean 2-ball and 4-ball for a particular choice of magnetic potential given by Killing vector fields, and discuss the behavior. Finally, we establish a comparison result for the magnetic Steklov operator associated with the manifold and the square root of the magnetic Laplacian on the boundary, which generalizes the uniform geometric upper bounds for the difference of the corresponding eigenvalues in the nonmagnetic case due to Colbois, Girouard, and Hassannezhad.

Suppose $$ is a finite abelian group, $$ is not contained in any strict coset in $$, and $$ are dense subsets of $$ such that the sumset $$ avoids $$. We show that $$ and $$ are almost entirely contained in sets defined by a bounded number of coordinates, that is, sets $$ and $$, where the size of $$ is non-zero and independent of $$, and $$ are subsets of $$ such that $$ avoids $$. Furthermore, we show that this result extends to any finite group and $$ summands for any $$.

Let $$, and let $$ be an expanding piecewise linear map sending each interval of linearity to [0,1]. For $$, $$, and $$, we consider the recurrence counting function

We investigate when the local Lipschitz property of the real-valued function $$ implies the global Lipschitz property of the mapping $$ between the metric spaces $$ and $$. Here, $$ denotes the distance of $$ from the non-empty set $$. As a consequence, we find that an analytic function on a uniform domain of a normed space belongs to the Lipschitz class if and only if its modulus satisfies the same condition; in the case of the unit disk this result is proved by Dyakonov. We use the recently established version of a classical theorem by Hardy and Littlewood for mappings between metric spaces. This paper is a continuation of the recent article by the author [Marković, J. Geom. Anal. 34 (2024), https://doi.org/10.48550/arXiv.2405.11509].

A regularized Petersson inner product on the space of Jacobi forms is defined and the regularized Petersson norms of Jacobi–Eisenstein series are computed. We use this result to establish Gross–Kohnen–Zagier's formula for Eisenstein series. In addition, we give an answer to the question raised by Böcherer and Das asking whether the regularized norm of Jacobi–Eisenstein series defined by them is non-zero. In the Supporting Information, we compute the Fourier coefficients of a suitable “new” basis of the space of Jacobi–Eisenstein series and give a remark on the proportional constant of the inner product formula in the theory of Jacobi forms.

Conrey, Ghosh and Gonek studied the first moment of the derivative of the Riemann zeta function evaluated at the non-trivial zeros of the zeta function, resolving a problem known as Shanks' conjecture. Conrey and Ghosh studied the second moment of the Riemann zeta function evaluated at its local extrema along the critical line to leading order. In this paper, we combine the two results, evaluating the first moment of the zeta function and its derivatives at the local extrema of zeta along the critical line, giving a full asymptotic. We also consider the factor from the functional equation for the zeta function at these extrema.

It has recently been conjectured by Bogosel, Henrot, and Michetti that the second positive eigenvalue of the Neumann Laplacian is maximized, among all planar convex domains of fixed perimeter, by the rectangle with one edge length equal to twice the other. In this note, we prove that this conjecture is true within the class of parallelogram domains.

In this paper, we establish different sharp forms of Mahler's conjecture for $$-concave even functions in dimensions $$, for $$ and 2, for $$, thus generalizing our previous results in Fradelizi and Nakhle (Int. Math. Res. Not. 12 (2023), 10067–10097) on log-concave even functions in dimension 2, which corresponds to the case $$. The functional volume product of an even $$-concave function $$ is