狄利克雷判别法(数值级数)条件的说明

狄利克雷判别法(数值级数)的条件不但是充分条件,而且是必要条件。     

关于“拟线性抛物型方程解的爆破时间的界”的一个注解（英文）

本文研究在空间维数是一维和二维情形下,一类拟线性抛物型方程在狄利克雷边界条件下的初边值问题.我们获得爆破速度和爆破时间的估计.     

关于四元二次型表整数个数的均值估计

利用数论函数的可乘性和狄里克雷卷积方法,研究了用四元二次型表整数个数的均值估计,并且得到了相应的渐近公式,推广了相关结果.     

黎曼流形上一类算子的特征值不等式（英文）

本文研究了等距浸入欧氏空间的黎曼流形、容许特殊函数的黎曼流形上的一类椭圆算子的加权狄利克雷特征值问题.我们建立了该问题的一些万有特征值不等式.同时,作为应用,我们获得了拉普拉斯算子的二次多项式算子的加权狄利克雷问题的一些结果.     

几何对称法在求取某些区域格林函数中的应用

格林函数法是数学物理方程中一种常用的方法,适用于求解狄利克雷问题.针对几种特殊区域上的上狄利克雷问题,采用几何对称法求取这些区域对应的格林函数,该方法对于该区域上格林函数的求解是有效的.     

分数阶朗之万方程的非局部狄利克雷边值问题解的存在性

讨论了分数阶Langevin方程的非局部狄利克雷边值问题,利用Leray-Schauder's和压缩映像原理,分别得到了方程的解的存在及唯一性.     

带类p(x)-拉普拉斯算子的双非局部问题的无穷多解

该文运用变分方法研究一类带类P(x)-拉普拉斯算子的双非局部狄利克雷问题.利用喷泉定理和对称山路定理,得到了此类问题一列高能量和低能量解的存在性.     

关于齐次四元Monge-Ampère方程的一些注记北大核心CSCD

本文考虑了四元数空间Hn中齐次四元Monge-Ampère方程的狄利克雷问题解的正则性.首先,当区域是边界为C1,1的强拟凸域时,作者给出了解的Lipschitz估计.其次,考虑了四元MongeAmpère算子的收敛性.最后,讨论了齐次四元Monge-Ampère方程的粘性次解与F-次调和函数之间的关系.     

关于具有不连续边界条件的椭圆型方程的一些注记

§1.引言在本文中,作者用“斯蒂阶型积分方程”方法把[1]中处理椭圆型方程的狄利克雷问题与牛孟问题的弗雷特霍姆积分方程法推广到具有不连续边界条件的情形,研究了“广义狄利克雷问题”与“广义牛孟问题”,即求m维空间的有界区域Ω上方程(1)的正规解u(p),并在Ω的边界     

p-级数域重排特征系统加权极大函数

通过研究狄利克雷核的一般性质,讨论P-级数域重排特征系统的加权极大狄利克雷核函数的积分情况,并给出加权极大函数可积的充要条件．     

On Scattering of Poles for Schr?dinger Operator from the Point of View of Dirichlet Series

In this article, we are concerned with the scattering problem of Schr?dinger operators with compactly supported potentials on the real line. We aim at combining the theory of Dirichlet series with scattering theory. New estimate on the number of poles is obtained under the situation that the growth of power series which is related to the potential is not too fast by using a classical result of Littlewood. We propose a new approach of Dirichlet series such that significant upper bounds and lower bounds on the number of poles are obtained. The results obtained in this paper improve and extend some related conclusions on this topic.     

Dirichlet series obtained from the error term in the Dirichlet divisor problem

In this paper we consider the Dirichlet series obtained from the error term in the Dirichlet divisor problem. We shall prove their analytic continuations and determine the locations of poles. We shall also discuss the relation between our results and the conjecture of Lau and Tsang on the mean square estimate of the error term.     

On L-functions with poles satisfying Maass's functional equation

We prove a generalisation of the Converse Theorem of Maass for Dirichlet series with a finite number of poles.     

Lower bounds on the number of scattering poles under lines parallel to the axis

We provide lower bounds in slab domains on the number of scattering poles, generated by two different types of periodic non-hyperbolic rays. For the first one all eigenvalues of the corresponding Poincaré map are equal to one and for the second one two of the eigenvalues are equal to one and two are different from one. In the second case we find also the closest to the real axis “line of poles”     

Transmission Eigenvalues and the Riemann Zeta Function in Scattering Theory for Automorphic Forms on Fuchsian Groups of Type I

We introduce the concept of transmission eigenvalues in scattering theory for automorphic forms on fundamental domains generated by discrete groups acting on the hyperbolic upper half complex plane. In particular, we consider Fuchsian groups of type Ⅰ. Transmission eigenvalues are related to those eigen-parameters for which one can send an incident wave that produces no scattering. The notion of transmission eigenvalues, or non-scattering energies, is well studied in the Euclidean geometry, where in some cases these eigenvalues appear as zeros of the scattering matrix. As opposed to scattering poles, in hyperbolic geometry such a connection between zeros of the scattering matrix and non-scattering energies is not studied, and the goal of this paper is to do just this for particular arithmetic groups. For such groups, using existing deep results from analytic number theory, we reveal that the zeros of the scattering matrix, consequently non-scattering energies, are directly expressed in terms of the zeros of the Riemann zeta function. Weyl's asymptotic laws are provided for the eigenvalues in those cases along with estimates on their location in the complex plane.     

Local properties of Hilbert spaces of Dirichlet series

We show that the asymptotic behavior of the partial sums of a sequence of positive numbers determine the local behavior of the Hilbert space of Dirichlet series defined using these as weights. This extends results recently obtained describing the local behavior of Dirichlet series with square summable coefficients in terms of local integrability, boundary behavior, Carleson measures and interpolating sequences. As these spaces can be identified with functions spaces on the infinite-dimensional polydisk, this gives new results on the Dirichlet and Bergman spaces on the infinite-dimensional polydisk, as well as the scale of Besov-Sobolev spaces containing the Drury-Arveson space on the infinite-dimensional unit ball. We use both techniques from the theory of sampling in Paley-Wiener spaces, and classical results from analytic number theory.     

Enumerating Quartic Dihedral Extensions of ℚ

We give an explicit Dirichlet series for the generating function of the discriminants of quartic dihedral extensions of . From this series we deduce an asymptotic formula for the number of isomorphism classes of such quartic extensions with discriminant up to a given bound. On the other hand, by using essentially classical results of genus theory combined with elementary analytical methods such as the method of the hyperbola, we show how to compute exactly this number up to quite large bounds, and we give a table of selected values.     

Low Frequency Scattering by Spheroids and Disks 2. Neumann Problem for a Prolate Spheroid

The problem of scattering of a scalar plane wave by a prolatespheroid is solved for Neumann boundary condition, arbitrarymajor to minor axis ratio, and arbitrary incident direction.The solution is obtained by using an iterative method appliedto solutions of the corresponding potential problem and is expressedas a series of products of Legendre and trigonometric functionsand ascending powers of wave number. A recursion relation forthe coefficients in this series is derived. These results andthe corresponding results for the Dirichlet case are employedto calculate scattering cross-sections for 2: 1, 5: 1 and 10:1 prolate spheroids.     

On solutions with power-law singularities of the homogeneous Dirichlet problem for the Laplace equation in domains with biquadratic boundaries

Properties of the Dirichlet problem for the Laplace equation in a bounded plane domain of a special type are studied in a certain class of solutions with power-law singularities. We prove that if a harmonic function is allowed to have a finite number of poles, then it can satisfy the trivial Dirichlet condition on certain curves of the studied family. The specified curves are selected, and it is shown that the set of those curves is dense (in a certain sense) in the studied family.     

Low Frequency Scattering by Spheroids and Disks 1. Dirichlet Problem for a Prolate Spheroid

The problem of scattering of a scalar plane wave by a prolatespheroid of revolution is solved for Dirichlet boundary condition,arbitrary major to minor axis ratio, and arbitrary incidentdirection. The solution is obtained by using an iterative methodapplied to solutions of the corresponding potential problemand is expressed as a series of products of Legendre and trigonometricfunctions, and ascending powers of wave number. A recursionrelation for the coefficients in this series is derived.