Mathematika最新文献

Given an integer $��b�⩾�2�$ and a set $�P$ of prime numbers, the set $�T_P$ of Toeplitz numbers comprises all elements of [0, b[ whose digits $�{�\left(�a_n�\right)�}_{�n�⩾�1�}$ in the base-b expansion satisfy $��a_n�=�a_{�p�n�}�$ for all $��p�\in �P�$ and $��n�⩾�1�$. Using a completely additive arithmetical function, we construct a number in $�T_P$ that is simply Borel normal if, and only if, $���{\sum }_{�p�\notin �}$

We study the density of solutions to Diophantine inequalities involving non-singular ternary forms, or equivalently, the density of rational points close to non-singular plane algebraic curves.

In this paper, we obtain the functional Orlicz–Brunn–Minkowski inequality and the functional Orlicz–Minkowski inequality for q-torsional rigidity in the smooth category. Furthermore, using an approximation method, we give the general functional Orlicz–Brunn–Minkowski inequality for q-torsional rigidity. As a corollary, we reveal that the functional Orlicz–Brunn–Minkowski inequality is equivalent to the functional Orlicz–Minkowski inequality for q-torsional rigidity in the smooth category. We also give some applications with respect to these two inequalities.

We consider random symmetric matrices with independent entries distributed according to the Haar measure on $�Z_p$ for odd primes p and derive the distribution of their canonical form with respect to several equivalence relations. We give a few examples of applications including an alternative proof for the result of Bhargava, Cremona, Fisher, Jones and Keating on the probability that a random quadratic form over $�Z_p$ has a non-trivial zero.

In this paper, we study upper bounds for the degrees of polynomials with only rational roots. First, we assume that the coefficients are bounded. In the second theorem, we suppose that the primes 2 and 3 do not divide any coefficient. The third theorem concerns the case that all coefficients are composed of primes from a fixed finite set.

In the present paper, the author adopts a pretentious approach and recovers an estimate obtained by Linnik for the sums of the von Mangoldt function Λ on arithmetic progressions. It is the analogue of an estimate that Linnik established in his attempt to prove his celebrated theorem concerning the size of the smallest prime number of an arithmetic progression. Our work builds on ideas coming from the pretentious large sieve of Granville, Harper, and Soundararajan and it also borrows insights from the treatment of Koukoulopoulos on multiplicative functions with small averages.

We develop an information-theoretic approach to study the Kneser–Poulsen conjecture in discrete geometry. This leads us to a broad question regarding whether Rényi entropies of independent sums decrease when one of the summands is contracted by a 1-Lipschitz map. We answer this question affirmatively in various cases.

We prove several dimension formulas for spaces of scalar-valued Siegel modular forms of degree 2 with respect to certain congruence subgroups of level 4. In case of cusp forms, all modular forms considered originate from cuspidal automorphic representations of $��\mathrm{GSp}�(�4�,�A�)�$ whose local component at $��p�=�2�$ admits nonzero fixed vectors under the principal congruence subgroup of level 2. Using known dimension formulas combined with dimensions of spaces of fixed vectors in local representations at $��p�=�2�$, we obtain formulas for the number of relevant automorphic representations. These, in turn, lead to new dimension formulas, in particular for Siegel modular forms with respect to the Klingen congruence subgroup of level 4.

Using the twisted fourth moment of the Riemann zeta-function, we study large gaps between consecutive zeros of the derivatives of Hardy's function $��Z�\left(�t�\right)�$, improving upon previous results of Conrey and Ghosh (J. Lond. Math. Soc. 32 (1985) 193–202), and of the second named author (Acta Arith. 111 (2004) 125–140). We also exhibit small distances between the zeros of $��Z�\left(�t�\right)�$ and the zeros of $��Z^{�\left(�2�k�\right)�}��\left(�t�\right)��$ for every $��k�\in �N�$, in support of our numerical observation that the zeros of $��Z^{�\left(�k�\right)�}��\left(�t�\right)��$ and

In this article, we present new gradient estimates for positive solutions to a class of non-linear elliptic equations  involving the f-Laplacian on a smooth metric measure space. The gradient estimates of interest are of Souplet–Zhang and Hamilton types, respectively, and are established under natural lower bounds on the generalised Bakry–Émery Ricci curvature tensor. From these estimates, we derive amongst other things Harnack inequalities and general global constancy and Liouville-type theorems. The results and approach undertaken here provide a unified treatment and extend and improve various existing results in the literature. Some implications and applications are presented and discussed.