Spectral analysis of the finite Hankel transform and circular prolate spheroidal wave functions

In this paper, we develop two practical methods for the computation of the eigenvalues as well as the eigenfunctions of the finite Hankel transform operator. These different eigenfunctions are called circular prolate spheroidal wave functions (CPSWFs). This work is motivated by the potential applications of the CPSWFs as well as the development of practical methods for computing their values. Also, in this work, we should prove that the CPSWFs form an orthonormal basis of the space of Hankel band-limited functions, an orthogonal basis of L²([0,1]) and an orthonormal system of L²([0,+∞[). Our computation of the CPSWFs and their associated eigenvalues is done by the use of two different methods. The first method is based on a suitable matrix representation of the finite Hankel transform operator. The second method is based on the use of an efficient quadrature method based on a special family of orthogonal polynomials. Also, we give two Maple programs that implement the previous two methods. Finally, we present some numerical results that illustrate the results of this work.     

共形复Finsler流形上的交换公式

设M为n维复流形,M^-～=T~（1,0）M-{0},F为M上的强拟凸复Finsler度量, F^-=e^σF为F的共形变换。本文得到定义在M^-上的各种Hermitian张量场分别关于复Finsler流形（M,F）和（M,F^-）的复Rund联络求共变微分的各种交换公式。     

Homogenization of dynamic laminates

This paper addresses the study of the homogenization problem associated with propagation of long wave disturbances in materials whose properties exhibit not only spacial but also temporal inhomogeneities (called dynamic materials). The study was initiated by Lurie in his pioneering work of 1997. Homogenization theory is employed to replace an equation with oscillating coefficients by a homogenized equation. Two typical examples of periodic homogenization are considered: the wave equation and Maxwell's system coefficients oscillating rapidly not only in space but also in time. Conditions that generate applicability of the homogenization procedure to dynamic materials composites are developed. In particular, we examine a cell problem for periodic composites as well as the laminate formulae. The effective tensors of rank-one laminates for one-dimensional wave equation and the full Maxwell's system are computed explicitly. We also note some dramatic differences between the hyperbolic and the elliptic cases.     

梯度液相色谱中任意等度、线性和阶梯梯度组合下的保留时间公式

基于线性溶剂强度模型,应用特征线分析的方法求解梯度洗脱模式下的理想液相色谱模型。在考虑到梯度延迟时间会对溶质的保留时间造成影响的情况下,得到适合于梯度液相色谱中任意等度、线性和阶梯梯度组合条件下的保留时间推导公式。应用这些公式计算任意的梯度条件下的保留时间,并将得到的结果与数值计f算的结果进行比较,二者完全一致,从而验证了推导得到的保留时间公式的正确性。由于这些公式具有形式简单、适用范围广等优点,因此可方便地应用于实际应用中,具有较高的实用价值。     

Existence and Regularity for a Class of Nonlinear Hyperbolic Boundary Value Problems

The regularity of the solution of the telegraph system with nonlinear monotone boundary conditions is investigated by two methods. The first one is based on D'Alembert-type representation formulae for the solution. In the second method the telegraph system is reduced to a linear Cauchy problem with a locally Lipschitzian functional perturbation; then regularity results are established by appealing to the theory of linear semigroups.     

Certain Modular Functions Similar to the Dedekind eta Function

In this paper, we study certain modular functions η_ψ(z) similar to the Dedekind eta function η(z). The functions are given by an analogue of Borcherds type liftings. It turns out that the functions η_ψ(z) have some good properties, similarly as in the case of the Dedekind eta function.     

Constructing fully symmetric cubature formulae for the sphere

We construct symmetric cubature formulae of degrees in the 13-39 range for the surface measure on the unit sphere. We exploit a recently published correspondence between cubature formulae on the sphere and on the triangle. Specifically, a fully symmetric cubature formula for the surface measure on the unit sphere corresponds to a symmetric cubature formula for the triangle with weight function , where , , and are homogeneous coordinates.

     

The orthogonality property of the Lommel polynomials and a twofold infinity of relations between Rayleigh''s σ-sums

Making use of a remarkable theorem which expresses a relationship between a certain type of infinite continued fractions and systems of orthogonal polynomials, it is proven that the known infinite continued fraction development of the ratio of Bessel functions J_v?1(z)/J_v(z) gives rise to an orthogonality property of the Lommel polynomials $\text{{R}{}_{\mathrm{m,v}}\text{(}\text{1}\text{z}\text{)|m?}\text{N}\text{}}$ when v is real and positive. The corresponding weight function which appears to be non-negative in the interval of definition, is obtained by the application of two successive integral transforms. It consists of an infinite series of Dirac δ-functions whose singularities are distributed symmetrically around the origin on the real axis in such a manner that the origin is their limit point on both sides. For any positive v, the Lommel polynomials form a system of so-called orthogonal polynomials of a discrete variable. The orthogonality property may also be conveniently expressed by means of a Stieltjes integral. One of its corollaries is a twofold infinity of linear relations between the sums σ_v^(r) defined by σ_v^(r)=Σ_n=1^+∞1/j_v,n^2r, with $\text{v+1Σ}\text{R}{}_0{}^{\mathrm{+}}\text{, rΣ}\text{N}{}_0\text{,}\text{in which}\text{j}{}_{\mathrm{v,n}}\text{represents the}\text{n}\text{th positive zero of}\text{J}{}_{\mathrm{v}}\text{(z)}$.Another by-product consists of a complement to a theorem of Hurwitz concerning the nature and the position of the zeros of the Lommel polynomials written as g_m,v(z) in the modified notation of the mentioned author. From this study also result two interesting approximations of j_v,1 applicable for v?]?1, +1].