Schur function identities, their t-analogs, and k-Schur irreducibility

We obtain general identities for the product of two Schur functions in the case where one of the functions is indexed by a rectangular partition, and give their t-analogs using vertex operators. We study subspaces forming a filtration for the symmetric function space that lends itself to generalizing the theory of Schur functions and also provides a convenient environment for studying the Macdonald polynomials. We use our identities to prove that the vertex operators leave such subspaces invariant. We finish by showing that these operators act trivially on the k-Schur functions, thus leading to a concept of irreducibility for these functions.     

The Higher Spin Laplace Operator

This paper deals with a certain class of second-order conformally invariant operators acting on functions taking values in particular (finite-dimensional) irreducible representations of the orthogonal group. These operators can be seen as a generalisation of the Laplace operator to higher spin as well as a second-order analogue of the Rarita-Schwinger operator. To construct these operators, we will use the framework of Clifford analysis, a multivariate function theory in which arbitrary irreducible representations for the orthogonal group can be realised in terms of polynomials satisfying a system of differential equations. As a consequence, the functions on which this particular class of operators act are functions taking values in the space of harmonics homogeneous of degree k. We prove the ellipticity of these operators and use this to investigate their kernel, focusing on polynomial solutions. Finally, we will also construct the fundamental solution using the theory of Riesz potentials.     

The correlation functions of vertex operators and Macdonald polynomials

The n-point correlation functions introduced by Bloch and Okounkov have already found several geometric connections and algebraic generalizations. In this note we formulate a q,t-deformation of this n-point function. The key operator used in our formulation arises from the theory of Macdonald polynomials and affords a vertex operator interpretation. We obtain closed formulas for the n-point functions when n = 1,2 in terms of the basic hypergeometric functions. We further generalize the q,t-deformed n-point function to more general vertex operators.     

Hypergeometric Functions as Infinite-Soliton Tau Functions

It is known that resonant multisoliton solutions depend on higher times and a set of parameters (integrals of motion). We show that soliton tau functions of the Toda lattice (and of the multicomponent Toda lattice) are tau functions of a dual hierarchy, where the higher times and the parameters (integrals of motion) exchange roles. The multisoliton solutions turn out to be rational solutions of the dual hierarchy, and the infinite-soliton tau functions turn out to be hypergeometric-type tau functions of the dual hierarchy. The variables in the dual hierarchies exchange roles. Soliton momenta are related to the Frobenius coordinates of partitions in the decomposition of rational solutions with respect to Schur functions. As an example, we consider partition functions of matrix models: their perturbation series is, on one hand, a hypergeometric tau function and, on the other hand, can be interpreted as an infinite-soliton solution. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 222–250, February, 2006.     

Green's Function of a Hartree-Type Equation with a Quadratic Potential

Using the complex WKB–Maslov method, we consider a solution of the Cauchy problem for a Hartree-type equation with a quadratic potential in the class of semiclassically supported functions. In this class, we obtain the evolution operator explicitly. We find parametric families of symmetry operators of the Hartree-type equation. Using the symmetry operators, we construct a family of exact solutions of this equation.     

Local approximation properties of certain class of linear positive operators via <Emphasis Type="Italic">I</Emphasis>—convergence

In this study, we obtain a local approximation theorems for a certain family of positive linear operators via I—convergence by using the first and the second modulus of continuities and the elements of Lipschitz class functions. We also give an example to show that the classical Korovkin Theory does not work but the theory works in I—convergence sense.      

Jack vertex operators and realization of Jack functions

We give an iterative method to realize general Jack functions using vertex operators. We first prove some cases of Stanley’s conjecture on positivity of the Littlewood–Richardson coefficients, and then use this method to give a new realization of Jack functions. We also show in general that the images of coefficients of products of Jack vertex operators form a basis of symmetric functions. In particular, this gives a new proof of linear independence for the rectangular and marked rectangular Jack vertex operators. Finally, a generalized Frobenius formula for Jack functions is given and used for evaluation of Dyson integrals and even powers of Vandermonde determinants.     

The Liouville Field Theory Zero-Mode Problem

We quantize the canonical free-field zero modes p, q on the half-plane p > 0 for both Liouville field theory and its reduced Liouville particle dynamics. We describe the particle dynamics in detail, calculate one-point functions of particle vertex operators, deduce their zero-mode realization on the half-plane, and prove that the particle vertex operators act self-adjointly on the Hilbert space L ²(₊) because of symmetries generated by the S-matrix. Similarly, we obtain the self-adjointness of the corresponding Liouville field theory vertex operator in the zero-mode sector by applying the Liouville reflection amplitude, which is derived by the operator method.     

Soliton equations,vertex operators,and simple singularities

We prove the equivalence of two hierarchies of soliton equations associated to a simply-laced finite Dynkin diagram. The first was defined by Kac and Wakimoto (Proc. Symp. Pure Math. 48:138–177, 1989) using the principal realization of the basic representations of the corresponding affine Kac–Moody algebra. The second was defined in Givental and Milanov (The Breadth of Symplectic and Poisson Geometry, Progress in Mathematics, vol. 232, pp. 173–201, Birkhäuser, Basel, 2005) using the Frobenius structure on the local ring of the corresponding simple singularity. We also obtain a deformation of the principal realization of the basic representation over the space of miniversal deformations of the corresponding singularity. As a by-product, we compute the operator product expansions of pairs of vertex operators defined in terms of Picard–Lefschetz periods for more general singularities. Thus, we establish a surprising link between twisted vertex operators and deformation theory of singularities.     

Matrix elements of vertex operators of the deformed W A n -algebra and the Harish-Chandra solutions to Macdonald's difference equations

In this paper we prove that certain matrix elements of vertex operators of the deformed W A _n -algebra satisfy Macdonald's difference equations and form a natural (n + 1)!-dimensional space of solutions. These solutions are the analogues of the Harish-Chandra solutions of the radial parts of the Laplace-Casimir operators on noncompact Riemannian symmetric spaces G/K with prescribed asymptotic behavior. We obtain formulas for analytic continuation of our Harish-Chandra type solutions as a consequence of braiding properties (obtained earlier by Y. Asai, M. Jimbo, T. Miwa, and Y. Pugay) of certain vertex operators of the deformed W A _n -algebra.     

Algebraic geometry and soliton dynamics

Algebro-geometric sectors of solutions of the KP hierarchy are described in terms of τ-functions and vertex operators. Some useful identities involving theta functions and prime forms on Riemann surfaces are provided which are applied to obtain explicit solutions in the bilinear formalism. By using a dressing method for τ-functions the soliton dynamics against the background of quasiperiodic solutions is characterized. Furthermore, a formula for the soliton shifts in terms of prime forms on Riemann surfaces is obtained.     

Large‐time behavior of a two‐scale semilinear reaction–diffusion system for concrete sulfatation

We study the large‐time behavior of (weak) solutions to a two‐scale reaction–diffusion system coupled with a nonlinear ordinary differential equations modeling the partly dissipative corrosion of concrete (or cement)‐based materials with sulfates. We prove that as t → ∞ , the solution to the original two‐scale system converges to the corresponding two‐scale stationary system. To obtain the main result, we make use essentially of the theory of evolution equations governed by subdifferential operators of time‐dependent convex functions developed combined with a series of two‐scale energy‐like time‐independent estimates. Copyright © 2014 John Wiley & Sons, Ltd.     

Operational,umbral methods,Borel transform and negative derivative operator techniques

ABSTRACT

Differintegral methods, namely those techniques using differential and integral operators on the same footing, currently exploited in calculus, provide a fairly unexhausted source of tools to be applied to a wide class of problems involving the theory of special functions and not only. The use of integral transforms of Borel type and the associated formalism will be shown to be an effective means, allowing a link between umbral and operational methods. We merge these two points of view to get a new and efficient method to obtain integrals of special functions and the summation of the associated generating functions as well.     

Fundamental solutions and Liouville type theorems for nonlinear integral operators

In this article we study basic properties for a class of nonlinear integral operators related to their fundamental solutions. Our goal is to establish Liouville type theorems: non-existence theorems for positive entire solutions for Iu?0 and for Iu+up?0, p>1.We prove the existence of fundamental solutions and use them, via comparison principle, to prove the theorems for entire solutions. The non-local nature of the operators poses various difficulties in the use of comparison techniques, since usual values of the functions at the boundary of the domain are replaced here by values in the complement of the domain. In particular, we are not able to prove the Hadamard Three Spheres Theorem, but we still obtain some of its consequences that are sufficient for the arguments.     

Certain Norm Related Quantities of Unbounded Linear Operators

Certain norm related functions of linear operators are considered in the very general setting of not necessarily continuous linear operators in normed spaces. These are shown to be closely related to the theory of precompact, strictly cosingular and a class of Φ_? type operators in addition to having applications to perturbation theory. We also obtain some basic continuity and precompactness properties of linear operators in normed spaces which are expressed in terms of the functions under consideration.     

q-Hypergeometric Series and Macdonald Functions

We derive a duality formula for two-row Macdonald functions by studying their relation with basic hypergeometric functions. We introduce two parameter vertex operators to construct a family of symmetric functions generalizing Hall-Littlewood functions. Their relation with Macdonald functions is governed by a very well-poised q-hypergeometric functions of type ₄₃, for which we obtain linear transformation formulas in terms of the Jacobi theta function and the q-Gamma function. The transformation formulas are then used to give the duality formula and a new formula for two-row Macdonald functions in terms of the vertex operators. The Jack polynomials are also treated accordingly.     

Polar coordinates in Carnot groups

We describe a procedure for constructing ”polar coordinates” in a certain class of Carnot groups. We show that our construction can be carried out in groups of Heisenberg type and we give explicit formulas for the polar coordinate decomposition in that setting. The construction makes use of nonlinear potential theory, specifically, fundamental solutions for the p-sub-Laplace operators. As applications of this result we obtain exact capacity estimates, representation formulas and an explicit sharp constant for the Moser-Trudinger inequality. We also obtain topological and measure-theoretic consequences for quasiregular mappings. Received: 26 June 2001; in final form: 14 January 2002/Published online: 5 September 2002     

The best approximation of Laplace operator by linear bounded operators in the space <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We obtain close two-sided estimates for the best approximation of Laplace operator by linear bounded operators on the class of functions for which the square of the Laplace operator belongs to the L _p -space. We estimate the best constant in the corresponding Kolmogorov inequality and the error of the optimal recovery of values of the Laplace operator on functions from this class defined with an error. In a particular case (p = 2) we solve all three problems exactly.     

Pseudoinvexity,optimality conditions and efficiency in multiobjective problems; duality

In this paper, we establish characterizations for efficient solutions to multiobjective programming problems, which generalize the characterization of established results for optimal solutions to scalar programming problems. So, we prove that in order for Kuhn–Tucker points to be efficient solutions it is necessary and sufficient that the multiobjective problem functions belong to a new class of functions, which we introduce. Similarly, we obtain characterizations for efficient solutions by using Fritz–John optimality conditions. Some examples are proposed to illustrate these classes of functions and optimality results. We study the dual problem and establish weak, strong and converse duality results.     

First Order Differential Operators Associated to the Space of Monogenic Functions in Parameter-Depending Clifford Algebras

This paper is aimed at finding all linear first order partial differential operators \({\mathcal{F}}\) with parameter-depending Clifford-algebra-valued coefficients, that are associated to the generalized Cauchy-Riemann operator. In order to obtain the conditions on the coefficients of \({\mathcal{F}}\), a Leibniz rule for functions with values in a more general Clifford-type algebra is proved. Using the theory of associated spaces, we show the construction of solutions of initial value problems involving these operators.