Numerical generation of transparent boundary conditions on the side surface of a vertical transverse isotropic layer

We consider elastodynamics in transversely isotropic media with vertical symmetry axis. The governing equations are the two-dimensional second-order system for displacements. A numerical method for generating transparent boundary conditions on the cylindrical surface is proposed. The correspondent operator is non-local in both z-direction and time: it handles low-frequency spatial harmonics of the solution convolving their Fourier coefficients with sums-of-exponentials kernels with respect to time. Test calculations show high accuracy, efficiency, and stability of the proposed non-reflecting conditions even for those media parameters where PML fails.     

On the integrable discrete versions of the Leznov lattice: Determinant solutions and pfaffianization

In this paper, we present the Casorati form of the N-soliton solution for an integrable fully-discrete version and two integrable semi-discrete versions of the Leznov lattice, which arise from the integrable discretization of the two-dimensional Leznov lattice. By using the pfaffianization procedure of Hirota and Ohta, a new integrable coupled system is generated from the semi-discrete version of the Leznov lattice in the y-direction.     

Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point

We study the microlocal kernel of h-pseudodifferential operators Oph(p)−z, where z belongs to some neighborhood of size O(h) of a critical value of its principal symbol p₀(x,ξ). We suppose that this critical value corresponds to a hyperbolic fixed point of the Hamiltonian flow Hp₀. First we describe propagation of singularities at such a hyperbolic fixed point, both in the analytic and in the C^∞ category. In both cases, we show that the null solution is the only element of this microlocal kernel which vanishes on the stable incoming manifold, but for energies z in some discrete set. For energies z out of this set, we build the element of the microlocal kernel with given data on the incoming manifold. We describe completely the operator which associate the value of this null solution on the outgoing manifold to the initial data on the incoming one. In particular it appears to be a semiclassical Fourier integral operator associated to some natural canonical relation.     

Singularities of Jacobi series on C2 and the Poisson process equation

A classical theorem of Gabor Szego relates the singularities of real zonal harmonic expansions with those of associated analytic functions of a single complex variable. Zeev Nehari developed the counterpart for Legendre series on the C-plane by generalizing Szego's theorem. This paper function theretically identifies the singularities of analytic symmetric Jacobi series on C² with those of analytic functions on the C-plane. One feature is that information about the singularities of solutions of Solomon Bochner's Poisson process equation flow from the expansion coefficients. Others are that the Szego and Nehari theorems appear on characteristic subspaces. And, that this PDE, unlike those normally encountered in function theory, is hyperbolic in the real domain.     

Fundamental solutions of singular SPDEs

This paper deals with some models of mathematical physics, where random fluctuations are modeled by white noise or other singular Gaussian generalized processes. White noise, as the distributional derivative od Brownian motion, which is the most important case of a Lévy process, is defined in the framework of Hida distribution spaces. The Fourier transformation in the framework of singular generalized stochastic processes is introduced and its applications to solving stochastic differential equations involving Wick products and singularities such as the Dirac delta distribution are presented. Explicit solutions are obtained in form of a chaos expansion in the Kondratiev white noise space, while the coefficients of the expansion are tempered distributions. Stochastic differential equations of the form P(ω, D) ◊ u(x, ω) = A(x, ω) are considered, where A is a singular generalized stochastic process and P(ω, D) is a partial differential operator with random coefficients. We introduce the Wick-convolution operator which enables us to express the solution as u = sA ◊ I^◊(−1), where s denotes the fundamental solution and I is the unit random variable. In particular, the stochastic Helmholtz equation is solved, which in physical interpretation describes waves propagating with a random speed from randomly appearing point sources.     

On semi-infinite spectral elements for Poisson problems with re-entrant boundary singularities

A spectral element method is described which enables Poisson problems defined in irregular infinite domains to be solved as a set of coupled problems over semi-infinite rectangular regions. Two choices of trial functions are considered, namely the eigenfunctions of the differential operator and Chebyshev polynomials. The coefficients in the series expansions are obtained by imposing weak C¹ matching conditions across element interfaces. Singularities at re-entrant corners are treated by a post-processing technique which makes use of the known asymptotic behaviour of the solution at the singular point. Accurate approximations are obtained with few degrees of freedom.     

A generalized Green's formula for elliptic problems in domains with edges

The usual Green's formula connected with the operator of a boundary-value problem fails when both of the solutions u and v that occur in it have singularities that are too strong at a conic point or at an edge on the boundary of the domain. We deduce a generalized Green's formula that acquires an additional bilinear form in u and v and is determined by the coefficients in the expansion of solutions near singularities of the boundary. We obtain improved asymptotic representations of solutions in a neighborhood of an edge of positive dimension, which together with the generalized Green's formula makes it possible, for example, to describe the infinite-dimensional kernel of the operator of an elliptic problem in a domain with edge. Bibliography: 14 titles. Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 106–147.     

A tensor product generalized ADI method for elliptic problems on cylindrical domains with holes

We consider solving second order linear elliptic partial differential equations together with Dirichlet boundary conditions in three dimensions on cylindrical domains (nonrectangular in x and y) with holes.We approximate the partial differential operators by standard partial difference operators. If the partial differential operator separates into two terms, one depending on x and y, and one depending on z, then the discrete elliptic problem may be written in tensor product form as (T_z⊗I + I⊗A_xy) U=F. We consider a specific implementation which uses a Method of Planes approach with unequally spaced finite differences in the xy direction and symmetric finite difference in the z direction. We establish the convergence of the Tensor Product Generalized Alternating Direction Implicit iterative method applied to such discrete problems. We show that this method gives a fast and memory efficient scheme for solving a large class of elliptic problems.     

Oblique derivative and interface problems on polygonal domains and networks

We Investigate oblique derivative problems associated to the Laplace operator on a polygon and we extend our study to "polygonal interface problems" which are an extension to networks of the prevlous ones. We focus on the non variational character of such problems. We obtain index formulae, a calculus of the dimension of the kernel, an expansion of the 'semi-variational" (or weak) solutions into regular and singular parts and formulae for the coefficients of the singularities In such expanslons.     

K. Saito's Conjecture for nonnegative eta products and analogous results for other infinite products

We prove that the Fourier coefficients of a certain general eta product considered by K. Saito are nonnegative. The proof is elementary and depends on a multidimensional theta function identity. The z=1 case is an identity for the generating function for p-cores due to Klyachko [A.A. Klyachko, Modular forms and representations of symmetric groups, J. Soviet Math. 26 (1984) 1879-1887] and Garvan, Kim and Stanton [F. Garvan, D. Kim, D. Stanton, Cranks and t-cores, Invent. Math. 101 (1990) 1-17]. A number of other infinite products are shown to have nonnegative coefficients. In the process a new generalization of the quintuple product identity is derived.     

Bezoutians and projections

With each polynomial p of degree n whose roots lie inside the unit disc we may associate the n-dimensional space of all solutions of the recurrence relation whose coefficients are those of p (considered as a subspace of 1²). The main result consists in establishing a close relation between the Bezoutian of two such polynomials (of the same degree) and the projection operator onto one of the corresponding spaces along the complement of the other. The note forms a loose continuation of the author's investigations of the infinite companion matrix—the generating function of the infinite companion matrix of a polynomial p appears thus as a particular case; the corresponding Bezoutian is that of the pair p and zⁿ.     

The {\bar{\partial}} -Neumann problem on product domains in {\mathbb{C}^{n}}

Let ${\Omega=\Omega_{1}\times\cdots\times\Omega_{n}\subset\mathbb{C}^{n}}$ , where ${\Omega_{j}\subset\mathbb{C}}$ is a bounded domain with smooth boundary. We study the solution operator to the ${\overline\partial}$ -Neumann problem for (0,1)-forms on Ω. In particular, we construct singular functions which describe the singular behavior of the solution. As a corollary our results carry over to the ${\overline\partial}$ -Neumann problem for (0,q)-forms. Despite the singularities, we show that the canonical solution to the ${\overline\partial}$ -equation, obtained from the Neumann operator, does not exhibit singularities when given smooth data.     

The Maximum Principle for Holomorphic Operator Functions

We show that if an operator-valued analytic function f of a complex variable attains its maximum modulus at z ₀, then the coefficients of the nonconstant terms in the power series expansion about z ₀ cannot be invertible, provided a complex uniform convexity condition holds. One application is that the norm of the resolvent of an operator on a complex uniformly convex space cannot have a local maximum.     

A Jacobi-collocation method for solving second kind Fredholm integral equations with weakly singular kernels

In this work, we propose a Jacobi-collocation method to solve the second kind linear Fredholm integral equations with weakly singular kernels. Particularly, we consider the case when the underlying solutions are sufficiently smooth. In this case, the proposed method leads to a fully discrete linear system. We show that the fully discrete integral operator is stable in both infinite and weighted square norms. Furthermore, we establish that the approximate solution arrives at an optimal convergence order under the two norms. Finally, we give some numerical examples, which confirm the theoretical prediction of the exponential rate of convergence.     

Scattering and local absorption for the Schrödinger operator

We investigate the problem of local absorption for the Schrödinger operator H = ?Δ + V with potential V singular on a compact set ∑ of measure zero but sufficiently regular outside. In dimension n = 3 and for V?L² + L^∞ outside of ∑, Pearson proved that the subspace of absolute continuity of H can be decomposed as the direct sum of the subspace of scattering states and of the subspace of states locally absorbed on ∑. We extend this result to arbitrary dimension and to potentials that are only locally semibounded with respect to Δ in a suitable sense away from ∑ (in particular they may be strongly oscillating away from ∑ and have arbitrary behavior at infinity). As a by-product, we prove that certain types of local singularities do not interfere with the question of asymptotic completeness, thereby generalizing previous results by Deift and Simon.     

Tau-functions,Grassmannians and rank one conditions

In integrable systems, specifically the KP hierarchy, there are functions known as “tau-functions”, closely related to the Schur polynomials in terms of which they are often written. Although they are generally viewed as the solutions to a collection of nonlinear PDEs, in this note they will equivalently be characterized by a quadratic difference equation. Sato's theorem associates tau-functions to the points of a Grassmann manifold. To make that amazing theorem clear to non-experts, we will first show an analogous (but easily understood) example of a linear ODE and its solution from a flow on the xy-plane. In each case the solution is created via a flow generated by a certain linear operator. The question we pose is this: “What other operators could have been used to generate solutions in the same way?” Although the answer is well known in the ODE case, the question in the nonlinear case is the main result of our new paper. We will state the result and discuss its relationship to the “trend” of writing tau-functions in terms of matrices satisfying certain rank one conditions. The elucidation of a geometric interpretation of the Hirota bilinear difference equation (HBDE) is a key feature of the proof and will be briefly described.     

Optimal reconstruction of the solution of the wave equation from inaccurate initial data

In the present paper, we consider the problem of the optimal reconstruction of the solution of the wave equation from the approximate values of the Fourier coefficients of the function specifying the initial form of the string. For an operator defined on the weight space of vectors from l ₂, we present the solution of the more general problem of reconstruction from the approximate values of the coordinates of these vectors.     

On solution of Lamé equations in axisymmetric domains with conical points

Partial Fourier series expansion is applied to the Dirichlet problem for the Lamé equations in axisymmetric domains ??³ with conical points on the rotation axis. This leads to dimension reduction of the three‐dimensional boundary value problem resulting to an infinite sequence of two‐dimensional boundary value problems on the plane meridian domain Ω_a?? of with solutions u _n(n=0,1,2,…) being the Fourier coefficients of the solution û of the 3D BVP. The asymptotic behaviour of the Fourier coefficients u _n (n=0,1,2,…) near the angular points of the meridian domain Ω_a is fully described by singular vector‐functions which are related to the zeros α_n of some transcendental equations involving Legendre functions of the first kind. Equations which determine the values of α_n are given and a numerical algorithm for the computation of α_n is proposed with some plots of values obtained presented. The singular vector functions for the solution of the 3D BVP is obtained by Fourier synthesis. Copyright © 2004 John Wiley & Sons, Ltd     

Singular values and Schmidt pairs of composition operators on the Hardy space

We find the singular values and corresponding Schmidt pairs of a compact composition operator Cφ induced by φ(z)=az+b, where |a|+|b|<1, on the classical Hardy space. We do so by solving a functional equation that is a generalization of Schröder's equation: find a function f, holomorphic on the open unit disc, and a complex number λ such that G(z)f(ψ(z))=λf(ψ(z)), where ψ is a holomorphic self-map of the open unit disc with an interior fixed point and G is a bounded holomorphic function on the open unit disc. In addition, we find the spectrum of the weighted composition operator MGCψ.