Second-order Linear Differential Operator Research Articles

Reciprocal Diffusions and Symmetries of Parabolic PDE: The Nonflat Case

We introduce a new set of Reciprocal Characteristics for the class of reciprocal diffusions naturally associated to a general parabolic second-order linear differential operator. All the coefficients of this operator, including the diffusion matrix, depend on time. This set of reciprocal characteristics is provided by the study of the symmetries of the differential operator. The Riemannian metric defined by the diffusion matrix is of central importance. Our reciprocal characteristics are the natural extension of Ovsiannikov's differential invariants to the time dependent parabolic case. We also show that the symmetries of the PDE coincide with the one parameter families of transformations which leave the usual stochastic Lagrangian as well as a modified Onsager–Machlup Lagrangian invariant.

Two-dimensional Krall-Sheffer polynomials and integrable systems

Two-dimensional Krall-Sheffer polynomials are analogues of the classical orthogonal polynomial. They are eigenfunctions of second-order linear partial differential operators and moreover satisfy orthogonality conditions. We show that all Krall-Sheffer polynomials are connected with two-dimensional superintegrable systems on spaces with constant curvature.

Timing microstructure in Schumann's “Träumerei” as an expression of harmony, rhythm, and motivic structure in music performance

We give statistical evidence that specific rhythmic, melodic, and harmonic analyses are dominating shaping structures for the agogical streams from 28 performances of Schumann's “Träumerei”, as measured by Repp [1]. These analyses are provided by the Rubato analysis and performance workstation. The statistical model is based on regression analysis and realizes shaping of agogics by a second-order linear differential operator. More precisely, the logarithm of the tempo function T( E) at onset time E can be expressed as a linear function of a sequence of particular smooth “weight” functions W i ( E) and their first and second derivatives. These numerical functions are generated by Rubato's harmonic, melodic, and rhythmical analyses and include an averaging process over a natural grouping hierarchy defined by Schumann's score.

Projectively Equivariant Quantization Map

Let M be a manifold endowed with a symmetric affine connection Γ. The aim of this Letter is to describe a quantization map between the space of second-order polynomials on the cotangent bundle T* M and the space of second-order linear differential operators, both viewed as modules over the group of diffeomorphisms and the Lie algebra of vector fields on M. This map is an isomorphism, for almost all values of certain constants, and it depends only on the projective class of the affine connection Γ.

Partial Differential Equations and Bivariate Orthogonal Polynomials

In 1929, S. Bochner identified the families of polynomials which are eigenfunctions of a second-order linear differential operator. What is the appropriate generalization of this result to bivariate polynomials? One approach, due to Krall and Sheffer in 1967 and pursued by others, is to determine which linear partial differential operators have orthogonal polynomial solutions with all the polynomials in the family of the same degree sharing the same eigenvalue. In fact, such an operator only determines a multi-dimensional eigenspace associated with each eigenvalue; it does not determine the individual polynomials, even up to a multiplicative constant. In contrast, our approach is to seek pairs of linear differential operators which have joint eigenfunctions that comprise a family of bivariate orthogonal polynomials. This approach entails the addition of some “normalizing" or “regularity" conditions which allow determination of a unique family of orthogonal polynomials. In this article we formulate and solve such a problem and show with the help of Mathematica that the only solutions are disk polynomials. Applications are given to product formulas and hypergroup measure algebras.

On the General Structure of Small on Large Problems for Elastic Deformations of Varga Materials I: Plane Strain Deformations

In three recent papers [6–8], the present authors show that both plane strain and axially symmetric deformations of perfectly elastic incompressible Varga materials admit certain first integrals, which means that solutions for finite elastic deformations can be determined from a second order partial differential equation, rather than a fourth order one. For plane strain deformations there are three such integrals, while for axially symmetric deformations there are two. The purpose of the present papers is to present the general equations for small deformations which are superimposed upon a large deformation, which is assumed to satisfy one of the previously obtained first integrals. The governing partial differential equations for the small superimposed deformations are linear but highly nonhomogeneous, and we present here the precise structure of these equations in terms of a second-order linear differential operator D2, which is first defined by examining solutions of the known integrals. The results obtained are illustrated with reference to a number of specific large deformations which are known solutions of the first integrals. For deformations of limited magnitude, the Varga strain-energy function has been established as a reasonable prototype for both natural rubber vulcanizates and styrene-butadiene vulcanizates. Plane strain deformations are examined in this present part while axially symmetric deformations are considered in Part II [16].

Space of Second-Order Linear Differential Operators as a Module over the Lie Algebra of Vector Fields

The space of linear differential operators on a smooth manifoldMhas a natural one-parameter family of Diff(M)- (and Vect(M)-) module structures, defined by their action on the space of tensor densities. It is shown that, in the case of second-order differential operators, the Vect(M)-module structures are equivalent, for any degree of tensor densities except for three critical values; {0,12,1}. A second-order analogue of the Lie derivative appears as an intertwining operator between the spaces of second-order differential operators on tensor densities.

A Loewner-type lemma for weighted biharmonic operators

In the present note we give a simpler proof of the recent result of Hedenmalm that the Green function for the weighted biharmonic operator ∆|z|2α∆, α > −1, on the unit disc D with the Dirichlet boundary conditions is positive. The main ingredient, which in the special case of the unweighted biharmonic operator ∆ is due to Loewner and which is of an independent interest, is a lemma characterizing, for a positive C weight function w, the second-order linear differential operators which take any function u satisfying ∆w−1∆u = 0 into a harmonic function. Another application of this lemma concerning positivity of the Poisson kernels for the biharmonic operator ∆ is also given.

Accurate and truncated three-point variational schemes for the Sturm-Liouville problem with boundary conditions of the second or third kind

On the basis of a variational approach for the eigenvalue problem in the case of a second-order linear differential operator with boundary conditions of the second or third kind, accurate and truncated threepoint schemes of high order of accuracy are constructed. Error estimates of the approximate solution are obtained. The results of the numerical experiment are presented.

A unified approach to optimal image interpolation problems based on linear partial differential equation models

The unified approach to optimal image interpolation problems presented provides a constructive procedure for finding explicit and closed-form optimal solutions to image interpolation problems when the type of interpolation can be either spatial or temporal-spatial. The unknown image is reconstructed from a finite set of sampled data in such a way that a mean-square error is minimized by first expressing the solution in terms of the reproducing kernel of a related Hilbert space, and then constructing this kernel using the fundamental solution of an induced linear partial differential equation, or the Green's function of the corresponding self-adjoint operator. It is proved that in most cases, closed-form fundamental solutions (or Green's functions) for the corresponding linear partial differential operators can be found in the general image reconstruction problem described by a first- or second-order linear partial differential operator. An efficient method for obtaining the corresponding closed-form fundamental solutions (or Green's functions) of the operators is presented. A computer simulation demonstrates the reconstruction procedure.

A note on the symmetry group and perturbation algebra of a parabolic partial differential equation

In this paper, the symmetry group of the parabolic partial differential equation ∂tf=Af is focused on (A is an analytic second-order elliptic linear differential operator on Rn). Following Olver [Applications of Lie Groups to Differential Equations (Springer, New York, 1986)], Ovsjannikov [Group Analysis of Differential Equations (Academic, New York, 1982)], and Rosencrans [J. Math. Anal. Appl. 56, 317 (1976); 61, 537 (1977)], the Lie algebra of infinitesimal symmetries is studied. It will be proven that it is finite dimensional (putting apart the trivial symmetries due to the linearity of the partial differential equation) by relating it to the Lie algebra of infinitesimal homothetic transformations of a Riemannian manifold (Rn,a), thus generalizing results of Ovsjannikov on elliptic equations and making use of the same Riemannian geometric techniques. Thanks to these techniques, the perturbation algebra of A, introduced by Rosencrans, is also shown to be related to geometric characteristics of (Rn,a).

On a method for computing eigenvalues and eigenfunctions of linear differential operators

In this paper a simple method is presented to compute the eigenvalues and the eigenfunctions of second-order linear differential operators, with homogeneous boundary conditions, both in a finite interval and on the semi-line. Our technique overcomes the drawbacks of the method proposed by Calogero to compute the eigenvalues of Sturm-Liouville problems in a finite interval. An estimate of the convergence for the eigenvalues is given in the finite case and numerical tests are performed, exhibiting a very fast rate of convergence for the eigenvalues both for the finite interval and the semi-line cases. An excellent convergence for the eigenfunctions is also obtained in both cases.

Conformally convariant equations on differential forms

Let M be a pseudo-Riemannian manifold o f dimension n≧3. A second-order linear differential operator , which is the sum of a variant of the Laplace-Beltrami operator on k-forms (obtained by weighting the d∗d and dd∗ terms differently) and a zeroth order operator depending on t he Ricci tensor of M, has remarkable conformal quasi-invariance properties. Specifically, intertwines two mu1tip1ier representations o f the conformal group of M. The ] are natura1 generalizations of the quasi -invariant shift of the Laplace-Beltrami operator on functions by a multiple of the scalar curvature of M, and the Maxwell operator on “vector potentials “ -forms).. The conformal quasi-invariance of the operator on functions was treated by Orsted in [9]-[11], and in-directly by differential geometers studying the “Yamabe problem” of prescribing the scalar curvature on a compact manifold (see [7] for a bibliography) . It also seems to be known to physicists.

On orthogonal polynomials and second-order linear differential operators

On orthogonal polynomials and second-order linear differential operators