The Eigenvalues of the Angular Spheroidal Wave Equation

Approximate relations are obtained between the eigenvalues λ and the ellipticity parameter c² of the angular spheroidal wave equation. Although based on WKBJ methods and the assumption that λ is large, the relations are useful throughout the complex c²-plane. They are exact at c² = 0, and reproduce the standard asymptotic formulas for λ when c² is large. At intermediate values of c², they provide approximations for the square-root branch points of the multivalued function λ(c²) in the complex c²-plane at which adjacent eigenvalues of the same class become equal in pairs. These branch points lie on an infinite sequence of distorted circular rings. Their exact locations have been computed for the first four rings for angular wavenumbers m = 0,…,4.     

Asymptotic formulae for eigenvalues and eigenfunctions of q‐Sturm‐Liouville problems

We investigate the asymptotic behavior of the eigenvalues and the eigenfunctions of q‐Sturm‐Liouville eigenvalue problems. For this aim we study the asymptotic behavior of q‐trigonometric functions as well as fundamental sets of solutions of the associated second order q‐difference equation. As in classical Sturm‐Liouville theory, the eigenvalues behave like zeros of q‐trigonometric functions and the eigenfunctions behave like q‐trigonometric functions. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

The Haar condition and multiplicity of zeros

Summary In this paper the problem is investigated of how to take the (possibly noninteger) multiplicity of zeros into account in the Haar condition for a linear function space on a given interval. Therefore, a distinction is made between regular and singular points of the interval, and a notion of geometric multiplicity, which always is a positive integer, is introduced. It is pointed out that, for regular zeros (i.e., zeros situate at regular points), aq-fold zero (in the sense that its geometric multiplicity equalsq), counts forq distinct zeros in the Haar condition. For singular zeros (i.e., zeros situated at singular points), this geometric multiplicity has to be diminished by some well-determinable integer.     

ASYMPTOTIC PROPERTIES OF ELEMENTS IN GROUP RINGS OF FINITE GROUPS

A relation between algebraic degrees of eigenvalues of an element in the group ring Z G of a finite group G and their multiplicities is obtained. This result can be regarded as a theorem on asymptotic behavior of parameters in group rings of finite groups which is a parallel result of several well known theorems concerning the relationships of parameters in finite groups.     

On the Asymptotic Expansion of the Spheroidal Wave Function and Its Eigenvalues for Complex Size Parameter

We provide a rapid and accurate method for calculating the prolate and oblate spheroidal wave functions (PSWFs and OSWFs),   S_mn ( c , η)  , and their eigenvalues,  λ _mn   , for arbitrary complex size parameter c in the asymptotic regime of large  | c |  , m and n fixed. The ability to calculate these SWFs for large and complex size parameters is important for many applications in mathematics, engineering, and physics. For arbitrary  arg( c )  , the PSWFs and their eigenvalues are accurately expressed by established prolate -type or oblate -type asymptotic expansions. However, determining the proper expansion type is dependent upon finding spheroidal branch points,   c ^mn _{○; r}   , in the complex c -plane where the PSWF alternates expansion type due to analytic continuation. We implement a numerical search method for tabulating these branch points as a function of spheroidal parameters m , n , and  arg( c )  . The resulting table allows rapid determination of the appropriate asymptotic expansion type of the SWFs. Normalizations, which are dependent on c , are derived for both the prolate - and oblate -type asymptotic expansions and for both  ( n − m )  even and odd. The ordering for these expansions is different from the original ordering of the SWFs and is dictated by the location of   c ^mn _{○; r}   . We document this ordering for the specific case of  arg( c ) =π/4  , which occurs for the diffusion equation in spheroidal coordinates. Some representative values of  λ _mn   and   S_mn ( c , η)  for large, complex c are also given.     

A reconstruction method for a two-dimensional inverse eigenvalue problem

In this paper we describe a method for constructing approximate solutions of a two-dimensional inverse eigenvalue problem. Here we consider the problem of recovering a functionq(x, y) from the eigenvalues of — +q(x, y) on a rectangle with Dirichlet boundary conditions. The potentialq(x, y) is assumed to be symmetric with respect to the midlines of the rectangle. Our method is a generalization of an algorithm Hald presented for the construction of symmetric potentials in the one-dimensional inverse Sturm-Liouville problem. Using a projection method, the inverse spectral problem is reduced to an inverse eigenvalue problem for a matrix. We show that if the given eigenvalues are small perturbations of simple eigenvalues ofq=0, then the matrix problem has a solution. This solution is used to construct a functionq which has the same lowest eigenvalues as the unknownq, and several numerical examples are given to illustrate the methods.     

Stability of collocation methods for delay differential equations with vanishing delays

We analyze the asymptotic stability of collocation solutions in spaces of globally continuous piecewise polynomials on uniform meshes for linear delay differential equations with vanishing proportional delay qt (0<q<1) (pantograph DDEs). It is shown that if the collocation points are such that the analogous collocation solution for ODEs is A-stable, then this asymptotic behaviour is inherited by the collocation solution for the pantograph DDE.     

A numerical algorithm for solving the generalized eigenvalue problem for completely continuous self-adjoint operators with nonlinear spectral parameter

We construct and justify a numerical algorithm for finding the generalized eigenvalues and eigenfunctions for self-adjoint positive semidefinite linear operators with nonlinear spectral parameter. We give an example of its application to finding the branch points of solutions of a class of nonlinear integral equations of Hammerstein type. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, Vol. 40, No. 1, 1997, pp. 146–150.     

Generalized Lyapunov–Schmidt Reduction Method and Normal Forms for the Study of Bifurcations of Periodic Points in Families of Reversible Diffeomorphisms

We introduce a general reduction method for the study of periodic points near a fixed point in a family of reversible diffeomorphisms. We impose no restrictions on the linearization at the fixed point except invertibility, allowing higher multiplicities. It is shown that the problem reduces to a similar problem for a reduced family of diffeomorphisms, which is itself reversible, but also has an additional ? ^q -symmetry. The reversibility in combination with the ? ^q -symmetry translates to a 𝕋 ^q -symmetry for the problem, which allows to write down the bifurcation equations. Moreover, the reduced family can be calculated up to any order by a normal form reduction on the original system. The method of proof combines normal forms with the Lyapunov–Schmidt method, and makes repetitive use of the Implicit Function Theorem. As an application we analyze the branching of periodic points near a fixed point in a family of reversible mappings, when for a critical value of the parameters the linearization at the fixed point has either a pair of simple purely imaginary eigenvalues that are roots of unity or a pair of non-semisimple purely imaginary eigenvalues that are roots of unity with algebraic multiplicity 2 and geometric multiplicity 1.     

The pair correlation of zeros of DirichletL-functions and primes in arithmetic progressions

We define a function which correlates the zeros of two DirichletL-functions to the modulusq and we prove an asymptotic estimate for averages of the pair correlation functions over all pairs of characters to (modq). An analogue of Montgomery’s pair correlation conjecture is formulated as to how this estimate can be extended to a greater domain for the parameters that are involved. Based on this conjecture we obtain results about the distribution of primes in an arithmetic progression (to a prime modulusq) and gaps between such primes.     

Asymptotics of zeros of basic sine and cosine functions

We give an elementary calculus proof of the asymptotic formulas for the zeros of the q-sine and cosine functions which have been recently found numerically by Gosper and Suslov. Monotone convergent sequences of the lower and upper bounds for these zeros are constructed as an extension of our method. Improved asymptotics are found by a different method using the Lagrange inversion formula. Asymptotic formulas for the points of inflection of the basic sine and cosine functions are conjectured. Analytic continuation of the q-zeta function is discussed as an application. An interpretation of the zeros is given.     

Skew codes of prescribed distance or rank

In this paper we generalize the notion of cyclic code and construct codes as ideals in finite quotients of non-commutative polynomial rings, so called skew polynomial rings of automorphism type. We propose a method to construct block codes of prescribed rank and a method to construct block codes of prescribed distance. Since there is no unique factorization in skew polynomial rings, there are much more ideals and therefore much more codes than in the commutative case. In particular we obtain a [40, 23, 10]₄ code by imposing a distance and a [42,14,21]₈ code by imposing a rank, which both improve by one the minimum distance of the previously best known linear codes of equal length and dimension over those fields. There is a strong connection with linear difference operators and with linearized polynomials (or q-polynomials) reviewed in the first section.      

Derivation of q-difference equation from connection matrix for selberg type jackson integrals

Selberg type Jackson integrals satisfy q-difference equations of Birkhoff type (generally called q Knizhnik Zamolodchikov equations). In one variable case, these equation are explicitly derived in matrix form of Gauss decomposition, by limit procedure q→0 from the connection matrices, by solving Riemann-Hilbert problem. The latter are evaluated from the asymptotic behaviours of Jackson integrals and the connection formulas given in our previous paper.     

Basis property of eigenfunctions of nonselfadjoint operator pencils generated by the equation of nonhomogeneous damped string

We consider a class of nonselfadjoint quadratic operator pencils generated by the equation, which governs the vibrations of a string with nonconstant bounded density subject to viscous damping with a nonconstant damping coefficient. These pencils depend on a complex parameterh, which enters the boundary conditions. Depending on the values ofh, the eigenvalues of the above pencils may describe the resonances in the scattering of elastic waves on an infinite string or the eingenmodes of a finite string. We obtain the 7asymptotic representations for these eigenvalues. Assuming that the proper multiplicity of each eigenvalue is equal to one, we prove that the eigenfunctions of these pencils form Riesz bases in the weightedL ²-space, whose weight function is exactly the density of the string. The general case of multiple eigenvalues will be treated in another paper, based on the results of the present work.     

Reduction of XXZ Model with Generalized Periodic Boundary Conditions

We examine the XXZ model with generalized periodic boundary conditions and identify conditions for the truncation of the functional relations of the transfer-matrix fusion. After the truncation, the fusion relations become a closed system of functional equations. The energy spectrum can be obtained by solving these equations. We obtain the explicit form of the Hamiltonian eigenvalues for the special case where the anisotropy parameter q ⁴ = –1.     

Smith Normal Form and acyclic matrices

An approach, based on the Smith Normal Form, is introduced to study the spectra of symmetric matrices with a given graph. The approach serves well to explain how the path cover number (resp. diameter of a tree T) is related to the maximal multiplicity MaxMult(T) occurring for an eigenvalue of a symmetric matrix whose graph is T (resp. the minimal number q(T) of distinct eigenvalues over the symmetric matrices whose graphs are T). The approach is also applied to a more general class of connected graphs G, not necessarily trees, in order to establish a lower bound on q(G).     

Inverse Sturm–Liouville problems with the potential known on an interior subinterval

The uniqueness problem of inverse Sturm–Liouville problems with the potential known on an interior subinterval is considered. We prove that the potential on the entire interval and boundary conditions are uniquely determined in terms of the known eigenvalues and some information on the eigenfunctions at some interior point (interior spectral data). Moreover, we also concern with the situation where the potential q is C^2k-smoothness at some given points.     

Asymptotics of Multivariate Randomness Statistics

Kiefer considered the asymptotics of q-sample Cramer-Von Mises statistics for a fixed q and sample sizes tending to infinity. For univariate observations, McDonald proved the asymptotic normality of these statistics when q goes to infinity while the sample sizes stay fixed. Here we define a class of multivariate randomness statistics that generalizes the class considered by McDonald. We also prove the asymptotic normality of such statistics when the sample sizes stay fixed while q tends to infinity.     

Branch duplication for the construction of multiple eigenvalues in an Hermitian matrix whose graph is a tree

Suppose that the eigenvalues of an Hermitian matrix A whose graph is a tree T are known, as well as the eigenvalues of the principal submatrix of A corresponding to a certain branch of T. A method for constructing a larger tree T?', in which the branch is ‘`duplicated’', and an Hermitian matrix A′ whose graph is T?' is described. The eigenvalues of A' are all of those of A, together with those corresponding to the branch, including multiplicities. This idea is applied (1) to give a solution to the inverse eigenvalue problem for stars, (2) to prove that the known diameter lower bound, for the minimum number of distinct eigenvalues among Hermitian matrices with a given graph, is best possible for trees of bounded diameter, and (3) to increase the list of trees for which all possible lists for the possible spectra are know. A generalization of the basic branch duplication method is presented.     

Multifractal formalism for self-affine measures with overlaps

We prove that for a class of self-affine measures defined by an expanding matrix whose eigenvalues have the same modulus, the L^q-spectrum τ(q) is differentiable for all q > 0. Furthermore, we prove that the multifractal formalism holds in the region corresponding to q > 0. Received: 4 August 2008, Revised: 19 January 2009