Mathematika最新文献

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.

In this article, we study the distribution of values of Dirichlet L-functions, the distribution of values of the random models for Dirichlet L-functions, and the discrepancy between these two kinds of distributions. For each question, we consider the cases of and $��Re�s�=�1�$ separately.

Let $��A�\subset �R_{�>�0�}�$ be a finite set of distances, and let $��G_A��\left(�R^n�\right)��$ be the graph with vertex set $�R^n$ and edge set $��\{��(�x�,�y�)��\in �R^n�:��\parallel �x�-�y�{\parallel }_2�\in �A�\}�$, and let $��\chi ��(�R^n�,�A�)��=�\chi ��(�G_A��(�{}^{}$

We prove the existence of the curvature measures for a class of $�U_{\mathrm{WDC}}$ sets, which is a direct generalisation of $�U_{�P��R�}$ sets studied by Rataj and Zähle. Moreover, we provide a simple characterisation of $�U_{\mathrm{WDC}}$ sets in $�R^2$ and prove that in $�R^2$, the class of $�U_{\mathrm{WDC}}$ sets contains essentially all classes of sets known to admit curvature measures.

Motivated by a problem in additive Ramsey theory, we extend Todorčević's partitions of three-dimensional combinatorial cubes to handle additional three-dimensional objects. As a corollary, we get that if the continuum hypothesis fails, then for every Abelian group G of size ℵ₂, there exists a coloring $��c�:�G�\to �Z�$ such that for every uncountable $��X�\subseteq �G�$ and every integer k, there are three distinct elements $��x�,�y�,�z�$ of X such that $��c�(�x�+�y�+�z�)�=�k�$.

Let $��G�=�(�V�,�E�)�$ be a finite, connected graph. We consider a greedy selection of vertices: given a list of vertices $��x_1�,�\cdots �,�x_k�$, take $�x_{�k�+�1�}$ to be any vertex maximizing the sum of distances to the vertices already chosen and iterate, keep adding the “most remote” vertex. The frequency with which the vertices of the graph appear in this sequence converges to a set of probability measures with nice properties. The support of these measures is, generically, given by a rather small number of vertices $��m�\ll �\left|�V�\right|�$. We prove that this suggests that the graph G is, in a suitable sense, “m-dimensional” by exhibiting an explicit 1-Lipschitz embedding $��\varphi �:�V�\to �{\ell }^1��\left(�R^m�\right)��$ with good properties.

We investigate games played between Maker and Breaker on an infinite complete graph whose vertices are coloured with colours from a given set, each colour appearing infinitely often. The players alternately claim edges, Maker's aim being to claim all edges of a sufficiently colourful infinite complete subgraph and Breaker's aim being to prevent this. We show that if there are only finitely many colours, then Maker can obtain a complete subgraph in which all colours appear infinitely often, but that Breaker can prevent this if there are infinitely many colours. Even when there are infinitely many colours, we show that Maker can obtain a complete subgraph in which infinitely many of the colours each appear infinitely often.

There has recently been some interest in optimizing the error term in the asymptotic for the fourth moment of Dirichlet L-functions and a closely related mixed moment of L-functions involving automorphic L-functions twisted by Dirichlet characters. We obtain an improvement for the error term of the latter.