Construction of Separation Variables for Finite-Dimensional Integrable Systems

We consider a class of integrable systems such that solutions of the corresponding Hamilton–Jacobi equation depend on n+m arbitrary parameters and are represented as products of flat curves. The first n parameters are identified with the values of the integrals of motion. The remaining parameters enter the definition of the integrals of motion as arbitrary constants (charges) and can be used to find separation variables. We show that on the coadjoint orbits of Lie groups, the Casimir operators not only generate a family of integrals but also allow constructing separation variables.     

Conformally Invariant Operators,Differential Forms,Cohomology and a Generalisation of Q-Curvature

On conformal manifolds of even dimension n ≥ 4 we construct a family of new conformally invariant differential complexes, each containing one coboundary operator of order greater than 1. Each bundle in each of these complexes appears either in the de Rham complex or in its dual (which is a different complex in the non-orientable case). Each of the new complexes is elliptic in case the conformal structure has Riemannian signature. We also construct gauge companion operators which (for differential forms of order k ≤ n/2) complete the exterior derivative to a conformally invariant and (in the case of Riemannian signature) elliptically coercive system. These (operator, gauge) pairs are used to define finite dimensional conformally stable form subspaces which are are candidates for spaces of conformal harmonics. This generalizes the n/2-form and 0-form cases, in which the harmonics are given by conformally invariant systems. These constructions are based on a family of operators on closed forms which generalize in a natural way Branson's Q-curvature. We give a universal construction of these new operators and show that they yield new conformally invariant global pairings between differential form bundles. Finally we give a geometric construction of a family of conformally invariant differential operators between density-valued differential form bundles and develop their properties (including their ellipticity type in the case of definite conformal signature). The construction is based on the ambient metric of Fefferman and Graham, and its relationship to the tractor bundles for the Cartan normal conformal connection. For each form order, our derivation yields an operator of every even order in odd dimensions, and even order operators up to order n in even dimension n. In the case of unweighted (or true) forms as domain, these operators are the natural form analogues of the critical order conformal Laplacian of Graham et al., and are key ingredients in the new differential complexes mentioned above.     

Differential operators for Siegel-Jacobi forms

For any positive integers n and m, H_(n,m):= H_n× C~(m,n) is called the Siegel-Jacobi space, with the Jacobi group acting on it. The Jacobi forms are defined on this space. We compute the Chern connection of the Siegel-Jacobi space and use it to obtain derivations of Jacobi forms. Using these results, we construct a series of invariant differential operators for Siegel-Jacobi forms. Also two kinds of Maass-Shimura type differential operators for H_(n,m) are obtained.     

Comparative asymptotics of solutions and trace formulas for a class of difference equations

Properties of Jacobi operators generated by Markov functions are studied. The main results refer to the case where the support of the corresponding spectral measure µ consists of several intervals of the real line. In this class of operators, a comparative asymptotic formula for two solutions of the corresponding difference equation, polynomials orthogonal with respect to the measure µ and functions of the second kind (Weyl solutions) is found. Asymptotic trace formulas for the coefficients a _n and b _n in this difference equation are obtained. The derivation of the asymptotic formulas is based on standard techniques for studying the asymptotic properties of polynomials orthogonal on several intervals and consists in reducing the asymptotic problem to investigating properties of solutions to the Nuttall singular integral equation.     

Heckman-Opdam's Jacobi polynomials for the BCn root system and generalized spherical functions

We prove that the radial part of the Laplacian on the space of generalized spherical functions on the symmetric space GL(m+n)/GL(m)×GL(n) is the Sutherland differential operator for the root system BC_n and the radial parts of the differential operators corresponding to the higher Casimirs yield the integrals of the quantum Calogero-Moser system. It allows us to give a representation theoretical construction for the three parameter family of Heckman-Opdam's Jacobi polynomials for the BC_n root system.     

Compatibility,Multi-brackets and Integrability of Systems of PDEs

We establish an efficient compatibility criterion for a system of generalized complete intersection type in terms of certain multi-brackets of differential operators. These multi-brackets generalize the higher Jacobi-Mayer brackets, important in the study of evolutionary equations and the integrability problem. We also calculate Spencer δ-cohomology of generalized complete intersections and evaluate the formal functional dimension of the solutions space. The results are used to establish new integration methods.     

Spectral properties of Jacobi matrices by asymptotic analysis

We consider two classes of Jacobi matrix operators in l² with zero diagonals and with weights of the form n^α+c_n for 0<α1 or of the form n^α+c_nn^α−1 for α>1, where {c_n} is periodic. We study spectral properties of these operators (especially for even periods), and we find asymptotics of some of their generalized eigensolutions. This analysis is based on some discrete versions of the Levinson theorem, which are also proved in the paper and may be of independent interest.     

Existence theory for generalized nonlinear complementarity problems

The nonlinear complementarity problem is generalized by replacing the usual nonnegative ordering ofR ⁿ by an ordering generated by a convex cone. Two new classes of operators are introduced, each of which is used to guarantee the existence of a solution to the generalized problem. The new classes can be seen to be broader than previously studied classes. In addition, conditions are presented under which the solution set of the generalized linear complementarity problem is shown to have at most a finite number of solutions.This research was partially supported by National Science Foundation, Grant No. GP-16293, and constitutes part of the junior author's doctoral thesis. The authors are indebted to Dr. Carlton E. Lemke for many helpful discussions.     

Solvability criterion and representation of solutions of <Emphasis Type="Italic">n</Emphasis>-normal and <Emphasis Type="Italic">d</Emphasis>-normal linear operator equations in a Banach space

On the basis of a generalization of the well-known Schmidt lemma to the case of n-normal and d-normal linear bounded operators in a Banach space, we propose constructions of generalized inverse operators. We obtain criteria for the solvability of linear equations with these operators and formulas for the representation of solutions of these equations.     

Higher-dimensional Osserman metrics with non-nilpotent Jacobi operators

We exhibit Osserman metrics with non-nilpotent Jacobi operators and with non-trivial Jordan normal form in neutral signature (n, n) for any n ≥ 3. These examples admit a natural almost para-Hermitian structure and are semi para-complex Osserman with non-trivial Jordan normal form as well; they neither satisfy the third Gray identity nor are they integrable.     

Generalized Bernstein–Durrmeyer Operators and the Associated Limit Semigroup

The aim of this paper is the study of a new sequence of positive linear approximation operators M_n, λ on C([0, 1]) which generalize the classical Bernstein–Durrmeyer operators. After proving a Voronovskaja-type result, we show that there exists a strongly continuous positive contraction semigroup on C([0, 1]) which may be expressed in terms of powers of these operators. As a direct consequence, we are able to represent explicitly the solutions of the Cauchy problems associated with a particular class of second order differential operators.     

A Jacobi spectral Galerkin method for the integrated forms of fourth‐order elliptic differential equations

This article analyzes the solution of the integrated forms of fourth‐order elliptic differential equations on a rectilinear domain using a spectral Galerkin method. The spatial approximation is based on Jacobi polynomials P (x), with α, β ∈ (?1, ∞) and n the polynomial degree. For α = β, one recovers the ultraspherical polynomials (symmetric Jacobi polynomials) and for α = β = ?½, α = β = 0, the Chebyshev of the first and second kinds and Legendre polynomials respectively; and for the nonsymmetric Jacobi polynomials, the two important special cases α = ?β = ±½ (Chebyshev polynomials of the third and fourth kinds) are also recovered. The two‐dimensional version of the approximations is obtained by tensor products of the one‐dimensional bases. The various matrix systems resulting from these discretizations are carefully investigated, especially their condition number. An algebraic preconditioning yields a condition number of O(N), N being the polynomial degree of approximation, which is an improvement with respect to the well‐known condition number O(N⁸) of spectral methods for biharmonic elliptic operators. The numerical complexity of the solver is proportional to N^d+1 for a d‐dimensional problem. This operational count is the best one can achieve with a spectral method. The numerical results illustrate the theory and constitute a convincing argument for the feasibility of the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

New exact Jacobi elliptic functions solutions for the generalized coupled Hirota-Satsuma KdV system

In this paper, an generalized Jacobi elliptic functions expansion method with computerized symbolic computation is used for constructing more new exact Jacobi elliptic functions solutions of the generalized coupled Hirota-Satsuma KdV system. As a result, eight families of new doubly periodic solutions are obtained by using this method, some of these solutions are degenerated to solitary wave solutions and triangular functions solutions in the limit cases when the modulus of the Jacobi elliptic functions m → 1 or 0, which shows that the applied method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.     

Generalized Virasoro Operators

Abstract

We define generalized Virasoro operators acting on a Fock space V(Γ). These generalize the standard construction of Virasoro operators. By using the Jacobi Identity we compute the commutators of these operators. These operators result in an abelian extension of the toroidal Lie algebra. We explicitly describe the abelian extension.     

New Jacobi elliptic functions solutions for the variable-coefficient mKdV equation

In this work, a new generalized Jacobi elliptic functions expansion method based upon four new Jacobi elliptic functions is described and abundant new Jacobi-like elliptic functions solutions for the variable-coefficient mKdV equation are obtained by using this method, some of these solutions are degenerated to solitary-like solutions and triangular-like functions solutions in the limit cases when the modulus of the Jacobi elliptic functions m→1 or 0, which shows that the new method can be also used to solve other nonlinear partial differential equations in mathematical physics.     

Spectral properties of some Jacobi matrices with double weights

We study the spectral properties of Jacobi matrices with the weights satisfying λ_2n−1=λ_2n=n^a or λ_2n=λ_2n+1=n^a, a>0. We show that for a=1 these are cases of spectral phase transitions in a. We use a new method of estimating transfer matrix products to describe the absolutely continuous part of these operators. For a=1 the existence of a spectral gap is proved. We also show how the results for double weights can be used for the spectral analysis of the Jacobi matrices related to some birth and death processes, previously studied by Janas and Naboko.     

Sharp Weighted Multidimensional Integral Inequalities for Monotone Functions

We prove sharp weighted inequalities for general integral operators acting on monotone functions of several variables. We extend previous results in one dimension, and also those in higher dimension for particular choices of the weights (power weights, etc.). We introduce a new kind of conditions, which take into account the more complicated structure of monotone functions in dimension n > 1, and give an example that shows how intervals are not enough to characterize the boundedness of the operators (contrary to what happens for n = 1). We also give several applications of our techniques.     

On the Jacobi series

Summary The Jacobi series of a functionf is an expansion in a series of ascending powers of a prescribed polynomialP of degreen in which the coefficients are polynomials of lesser degree. These coefficients are usually expressed as contour integrals or are determined by their interpolatory properties. We show how they may be expressed as generalized derivatives off with respect toP. In so doing we also show how the Jacobi series may be expressed (in yet another way) as a generalized Taylor series. In addition, we obtain a number of interesting relations among the generalized derivatives.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.