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.     

The Lebesgue constants for Fourier-Bessel series,mean convergence,and the order of singularity of a Sturm-Liouville equation

Questions of mean convergence of classical orthogonal expansions and rates of divergence of their Lebesgue constants are dealt with, under two aspects. The first aspect is that the known results for Jacobi, Laguerre, and Fourier-Bessel series can be seen to be closely related to each other with respect to the kind of singularities in their Sturm-Liouville equations. The second, and the main aspect is to show that the rate of divergence of the Lebesgue constants for Fourier-Bessel series, which were unknown so far, fits well into this interpretation. For the latter purpose we use the Hankel translation in order to reduce the kernel of the Fourier-Bessel partial sum to a function of one variable, a representation of which is derived by the residue calculus. This method of proof is also discussed in connection with the methods used for the other orthogonal systems and with possible generalizations to more general eigenfunction expansions.     

Locally finite graphs with ends: A topological approach, I. Basic theory

This paper is the first of three parts of a comprehensive survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. The first two parts of the survey together provide a suitable entry point to this field for new readers; they are available in combined form from the ArXiv [18]. They are complemented by a third part [28], which looks at the theory from an algebraic-topological point of view.The topological approach indicated above has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. While the second part of this survey [19] will concentrate on those applications, this first part explores the new theory as such: it introduces the basic concepts and facts, describes some of the proof techniques that have emerged over the past 10 years (as well as some of the pitfalls these proofs have in stall for the naive explorer), and establishes connections to neighbouring fields such as algebraic topology and infinite matroids. Numerous open problems are suggested.     

A Simulation-free Nonlinear Model Order-reduction Approach and Comparison Study

In this paper, a new approach to the model order reduction of nonlinear systems is presented. This approach does not need a simulation of the original system, and therefore, it is suitable for large systems. By separating the linear and nonlinear parts of the original nonlinear model, the idea is to consider the nonlinearities of the resulting system as additional inputs. Based on the linear system from the last step, a known order-reduction method can be applied to find the coefficients of the nonlinear and the linear parts of a reduced-order model. Two different methods from linear-order reduction (balancing and truncation and Eitelberg's method with some modification) are used for this purpose, and their advantages and disadvantages are discussed. For comparison with some known methods in order reduction of nonlinear systems, three other methods are discussed briefly. Finally, a technical nonlinear system is reduced, and different methods are compared.     

Locally finite graphs with ends: A topological approach, II. Applications

This paper is the second of three parts of a comprehensive survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. The first two parts of the survey together provide a suitable entry point to this field for new readers; they are available in combined form from the ArXiv [20]. They are complemented by a third part [31], which looks at the theory from an algebraic-topological point of view.The topological approach indicated above has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. This second part of the survey concentrates on these applications, many of which solve problems or extend earlier work of Thomassen on infinite graphs. Numerous new problems are suggested.     

Expansions of ordered fields without definable gaps

In this paper we are concerned with definably, with or without parameters, (Dedekind) complete expansions of ordered fields, i. e. those with no definable gaps. We present several axiomatizations, like being definably connected, in each of the two cases. As a corollary, when parameters are allowed, expansions of ordered fields are o‐minimal if and only if all their definable subsets are finite disjoint unions of definably connected (definable) subsets. We pay attention to how simply (in terms of the quantifier complexity and/or usage of parameters) a definable gap in an expansion is so. Next we prove that over parametrically definably complete expansions of ordered fields, all one‐to‐one definable (with parameters) continuous functions are monotone and open. Moreover, in both parameter and parameter‐free cases again, definably complete expansions of ordered fields satisfy definable versions of the Heine‐Borel and Extreme Value theorems and also Bounded Intersection Property for definable families of closed bounded subsets.     

Prolate spheroidal wave functions, an introduction to the Slepian series and its properties

For decades mathematicians, physicists, and engineers have relied on various orthogonal expansions such as Fourier, Legendre, and Chebyschev to solve a variety of problems. In this paper we exploit the orthogonal properties of prolate spheroidal wave functions (PSWF) in the form of a new orthogonal expansion which we have named the Slepian series. We empirically show that the Slepian series is potentially optimal over more conventional orthogonal expansions for discontinuous functions such as the square wave among others. With regards to interpolation, we explore the connections the Slepian series has to the Shannon sampling theorem. By utilizing Euler's equation, a relationship between the even and odd ordered PSWFs is investigated. We also establish several other key advantages the Slepian series has such as the presence of a free tunable bandwidth parameter.     

Evaluation codes and plane valuations

We apply tools coming from singularity theory, as Hamburger–Noether expansions, and from valuation theory, as generating sequences, to explicitly describe order functions given by valuations of 2-dimensional function fields. We show that these order functions are simple when their ordered domains are isomorphic to the value semigroup algebra of the corresponding valuation. Otherwise, we provide parametric equations to compute them. In the first case, we construct, for each order function, families of error correcting codes which can be decodified by the Berlekamp–Massey–Sakata algorithm and we give bounds for their minimum distance depending on minimal sets of generators for the above value semigroup.     

A Generalized Chebyshev Method for Non-linear Parabolic Equations in One Space Variable

The derivation and implementation of a generalized Chebyshevmethod is described for the numerical solution of non-linearparabolic equations in one space dimension. The solution isobtained by using the method of lines and is approximated inthe space variable by piecewise Chebyshev polynomial expansions.These expansions are normally few in number and of high order.It is shown that the method can be derived from a perturbedform of the original equation. A numerical example is givento illustrate its performance compared with the finite elementand finite difference method. A comparison of various Chebyshev methods is made by applyingthem to two-point eigenproblems. It is shown by analysis andnumerical examples that the approach used to derive the generalizedChebyshev method is comparable, in terms of the accuracy obtained,with existing Chebyshev methods.     

求解双材料裂纹结构全域应力场的扩展边界元法

在线弹性理论中,复合材料裂纹尖端具有多重应力奇异性,常规数值方法不易求解.该文建立的扩展边界元法(XBEM)对围绕尖端区域位移函数采用自尖端径向距离r的渐近级数展开式表达,其幅值系数作为基本未知量,而尖端外部区域采用常规边界元法离散方程.两方程联立求解可获得裂纹结构完整的位移和应力场.对两相材料裂纹结构尖端的两个材料域分别采用合理的应力特征对,然后对其进行计算,通过计算结果的对比分析,表明了扩展边界元法求解两相材料裂纹结构全域应力场的准确性和有效性.     

Rational approximation to the Fermi–Dirac function with applications in density functional theory

We are interested in computing the Fermi–Dirac matrix function in which the matrix argument is the Hamiltonian matrix arising from density functional theory (DFT) applications. More precisely, we are really interested in the diagonal of this matrix function. We discuss rational approximation methods to the problem, specifically the rational Chebyshev approximation and the continued fraction representation. These schemes are further decomposed into their partial fraction expansions, leading ultimately to computing the diagonal of the inverse of a shifted matrix over a series of shifts. We describe Lanczos and sparse direct methods to address these systems. Each approach has advantages and disadvantages that are illustrated with experiments.     

On asymptotic expansions in nonlinear,singularly perturbed difference equations ∗

The present paper deals with the problem of constructing and proving asymptotic expansions for nonlinear, singularly perturbed difference equations. New methods for the construction of asymptotic expansions are presented and compared with well-known ones. For the proof of their validity, fundamental principles for the treatment of nonlinear singular perturbation problems are applied, based on the concepts of e-stability, formal asymptotic expansions, matching and asymptotic expansions. The results are derived from a general theory of asymptotic expansions of nonlinear operator equations that has been developed recently by the author.     

Covariant quantization: spectral analysis versus deformation theory

Formal deformation or rather symbolic calculus? To which extent do these approaches complete each other in the study of symmetry-preserving quantization procedures for homogeneous spaces? The representation theory of underlying Lie groups shows that the answer is much more delicate than initially thought and that it cannot be always reduced to asymptotic expansions with respect to some Planck’s constant. The main goal of this survey is to give hints regarding the aims of each approach and, on the domain where these intersect, to compare the answers they lead to.      

Low-frequency dipolar electromagnetic scattering by a solid ellipsoid in lossless environment

Electromagnetic wave scattering phenomena for target identification are important in many applications related to fundamental science and engineering. Here, we present an analytical formulation for the calculation of the magnetic and electric fields that scatter off a highly conductive ellipsoidal body, located within an otherwise homogeneous and isotropic lossless medium. The primary excitation source assumes a time-harmonic magnetic dipole, precisely fixed and arbitrarily orientated that operates at low frequencies and produces the incident fields. The scattering problem itself is modeled with respect to rigorous expansions of the electromagnetic fields at the low-frequency regime in terms of positive integral powers of the real wave number of the ambient. Obviously, the Rayleigh static term and a few dynamic terms are sufficient for the purpose of the present work, as the additional terms are neglected due to their minor contribution. Therein, the classical Maxwell's theory is suitably modified, leading to intertwined either Laplace's or Poisson's equations, accompanied by the impenetrable boundary conditions for the total fields and the limiting behavior at infinity. On the other hand, the complete spatial anisotropy of the three-dimensional space is secured via the introduction of the genuine ellipsoidal coordinate system, being appropriate for tackling incrementally such scattering boundary value problems. The nonaxisymmetric fields are obtained via infinite series expansions in terms of ellipsoidal harmonic eigenfunctions, providing handy closed-form solutions in a compact fashion, whose validity is verified by a straightforward reduction to simpler geometries of the metal object. The main idea is to demonstrate an efficient methodology, according to which the constructed analytical formulae can offer the appropriate environment for a fast numerical estimation of the scattered electromagnetic fields that could be useful for real data inversion.     

Rough sets for pattern classification using pairwise-comparison-based tables

Rough set theory is a useful mathematical tool to deal with vagueness and uncertainty in available information. The results of a rough set approach are usually presented in the form of a set of decision rules derived from a decision table. Because using the original decision table is not the only way to implement a rough set approach, it could be interesting to investigate possible improvement in classification performance by replacing the original table with an alternative table obtained by pairwise comparisons among patterns. In this paper, a decision table based on pairwise comparisons is generated using the preference relation as in the Preference Ranking Organization Methods for Enrichment Evaluations (PROMETHEE) methods, to gauges the intensity of preference for one pattern over another pattern on each criterion before classification. The rough-set-based rule classifier (RSRC) provided by the well-known library for the Rough Set Exploration System (RSES) running under Windows as been successfully used to generate decision rules by using the pairwise-comparisons-based tables. Specifically, parameters related to the preference function on each criterion have been determined using a genetic-algorithm-based approach. Computer simulations involving several real-world data sets have revealed that of the proposed classification method performs well compared to other well-known classification methods and to RSRC using the original tables.     

Integrability of Generalized (Matrix) Ernst Equations in String Theory

We elucidate the integrability structures of the matrix generalizations of the Ernst equation for Hermitian or complex symmetric (d×d)-matrix Ernst potentials. These equations arise in string theory as the equations of motion for the truncated bosonic parts of the low-energy effective action for the respective dilaton and d×d matrix of moduli fields or for a string gravity model with a scalar (dilaton) field, a U(1) gauge vector field, and an antisymmetric 3-form field, all depending on only two space-time coordinates. We construct the corresponding spectral problems based on the overdetermined 2d×2d linear systems with a spectral parameter and the universal (i.e., solution-independent) structures of the canonical Jordan forms of their matrix coefficients. The additionally imposed existence conditions for each of these systems of two matrix integrals with appropriate symmetries provide specific (coset) structures of the related matrix variables. We prove that these spectral problems are equivalent to the original field equations, and we envisage an approach for constructing multiparametric families of their solutions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 214–225, August, 2005.     

Complexity Theory for Lie-Group Solvers

Commencing with a brief survey of Lie-group theory and differential equations evolving on Lie groups, we describe a number of numerical algorithms designed to respect Lie-group structure: Runge–Kutta–Munthe-Kaas schemes, Fer and Magnus expansions. This is followed by derivation of the computational cost of Fer and Magnus expansions, whose conclusion is that for order four, six, and eight an appropriately discretized Magnus method is always cheaper than a Fer method of the same order. Each Lie-group method of the kind surveyed in this paper requires the computation of a matrix exponential. Classical methods, e.g., Krylov-subspace and rational approximants, may fail to map elements in a Lie algebra to a Lie group. Therefore we survey a number of approximants based on the splitting approach and demonstrate that their cost is compatible (and often superior) to classical methods.     

A Volterra integral formulation for determining the periodic solutions of some autonomous, nonlinear, third-order ordinary differential equations

A Volterra integral formulation based on the introduction of a term proportional to the velocity times the square of the (unknown) frequency of oscillation, a new independent variable equal to the original one times the (unknown) frequency of oscillation, the method of variation of parameters and series expansions of both the solution and the frequency of oscillation, is used to determine the periodic solutions to three nonlinear, autonomous, third-order, ordinary differential equations. It is shown that the first term of the series expansion of the frequency of oscillation coincides with that determined from a first-order harmonic balance procedure, whereas the two-term approximation to the frequency of oscillation is shown to be more accurate than that of a parameter perturbation procedure for the second equation considered in this paper. For the third equation, it is shown that the two-term approximation presented in this paper is more accurate than the corresponding one for one of the parameter perturbation methods, and for initial velocities less than one, for the other parameter perturbation approach.     

Uniform Asymptotic Expansions of Symmetric Elliptic Integrals

Symmetric standard elliptic integrals are considered when two or more parameters are larger than the others. The distributional approach is used to derive seven expansions of these integrals in inverse powers of the asymptotic parameters. Some of these expansions also involve logarithmic terms in the asymptotic variables. These expansions are uniformly convergent when the asymptotic parameters are greater than the remaining ones. The coefficients of six of these expansions involve hypergeometric functions with less parameters than the original integrals. The coefficients of the seventh expansion again involve elliptic integrals, but with less parameters than the original integrals. The convergence speed of any of these expansions increases for an increasing difference between the asymptotic variables and the remaining ones. All the expansions are accompanied by an error bound at any order of the approximation. January 31, 2000. Date revised: May 18, 2000. Date accepted: August 4, 2000.     

Innovative solution of a 2D elastic transmission problem¶

This article is concerned with a boundary-field equation approach to a class of boundary value problems exterior to a thin domain. A prototype of this kind of problems is the interaction problem with a thin elastic structure. We are interested in the asymptotic behavior of the solution when the thickness of the elastic structure approaches to zero. In particular, formal asymptotic expansions will be developed, and their rigorous justification will be considered. As will be seen, the construction of these formal expansions hinges on the solutions of a sequence of exterior Dirichlet problems, which can be treated by employing boundary element methods. On the other hand, the justification of the corresponding formal procedure requires an independence on the thickness of the thin domain for the constant in the Korn inequality. It is shown that in spite of the reduction of the dimensionality of the domain under consideration, this class of problems are, in general, not singular perturbation problems, because of appropriate interface conditions.