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.     

Matrix Pearson Equations Satisfied by Koornwinder Weights in Two Variables

We consider Koornwinder’s method for constructing orthogonal polynomials in two variables from orthogonal polynomials in one variable. If semiclassical orthogonal polynomials in one variable are used, then Koornwinder’s construction generates semiclassical orthogonal polynomials in two variables. We consider two methods for deducing matrix Pearson equations for weight functions associated with these polynomials, and consequently, we deduce the second order linear partial differential operators for classical Koornwinder polynomials.     

Computational matrix representation modules for linear operators with explicit constructions for a class of lie operators

A linear mapping from a finite-dimensional linear space to another has a matrix representation. Certain multilinear functions are also matrix-representable. Using these representations, symbolic computations can be done numerically and hence more efficiently. This paper presents an organized procedure for constructing matrix representations for a class of linear operators on finite-dimensional spaces. First we present serial number functions for locating basis monomials in the linear space of homogeneous polynomials of fixed degree, ordered under structured lexicographies. Next basic lemmas describing the modular structure of matrix representations for operators constructed canonically from elementary operators are presented. Using these results, explicit matrix representations are then given for the Lie derivative and Lie-Poisson bracket operators defined on spaces of homogeneous polynomials. In particular, they are comprised of blocks obtained as Kronecker sums of modular components, each corresponding to specific Jordan blocks. At an implementation level, recursive programming is applied to construct these modular components explicitly. The results are also applied to computing power series approximations for the center manifold of a dynamical system. In this setting, the linear operator of interest is parameterized by two matrices, a generalization of the Lie-Poission bracket.     

Dirichlet空间上Toeplitz算子与Berezin型变换相关的一些性质

本文研究了单位圆盘D 的Dirichlet 空间上Toeplitz 算子和小Hankel 算子. 利用Berezin 型变换讨论了Toeplitz 算子的不变子空间问题, 具有Berezin 型符号的Toeplitz 算子的渐进可乘性以及Toeplitz 算子的Riccati 方程的可解性. 应用Berezin 变换得到了Toeplitz 算子和小Hankel 算子可逆的充分条件. 此外, 还利用Hankel 算子和Berezin 变换刻画了算子2T_uv-T_uT_v-T_vT_u 的紧性, 其中函数u,v ∈ L^2,1.     

Minimal Cuntz-Krieger dilations and representations of Cuntz-Krieger algebras

Given a contractive tuple of Hilbert space operators satisfying certainA-relations we show that there exists a unique minimal dilation to generators of Cuntz-Krieger algebras or its extension by compact operators. This Cuntz-Krieger dilation can be obtained from the classical minimal isometric dilation as a certain maximalA-relation piece. We define a maximal piece more generally for a finite set of polynomials inn noncommuting variables. We classify all representations of Cuntz-Krieger algebrasO _A obtained from dilations of commuting tuples satisfyingA-relations. The universal properties of the minimal Cuntz-Krieger dilation and the WOT-closed algebra generated by it is studied in terms of invariant subspaces.     

Total Rarita-Schwinger operators in Clifford analysis

Rarita-Schwinger operators in Clifford analysis can be realized as first-order differential operators acting on functions f(x, u) taking values in the vector space of homogeneous monogenic polynomials. In this paper, the Scasimir operator for the orthosymplectic Lie superalgebra will be used to construct an invariant operator which acts on the full space of functions in two vector variables and therefore has more invariance properties. Also the fundamental solution for this operator will be constructed.     

Nonlinear Commutation Relations: Representations by Point-Supported Operators

We present a class of non-Lie commutation relations admitting representations by point-supported operators (i.e., by operators whose integral kernels are generalized point-supported functions). For such relations we construct all operator-irreducible representations (up to equivalence). Each representation is realized by point-supported operators in the Hilbert space of antiholomorphic functions. We show that the reproducing kernels of these spaces can be represented via hypergeometric series and the theta function, as well as via their modifications. We construct coherent states that intertwine abstract representations with irreducible representations.     

A new class of polynomials associated with Bernstein and beta polynomials

The purpose of this paper is to define a new class polynomials. Special cases of these polynomials give many famous family of the Bernstein type polynomials and beta polynomials. We also construct generating functions for these polynomials. We investigate some fundamental properties of these functions and polynomials. Using functional equations and generating functions, we derive various identities related to theses polynomials. We also construct interpolation function that interpolates these polynomials at negative integers. Finally, we give a matrix representations of these polynomials. Copyright © 2013 John Wiley & Sons, Ltd.     

Approximating characteristic equations for autonomous systems of differential equations with aftereffect

In this paper we construct approximating polynomial characteristic equations for a linear autonomous system with aftereffect. The procedures for constructing approximating characteristic equations use analytic representations of resolvents of infinitesimal operators and the theory of characteristic determinants and perturbation determinants in a separable Hilbert space.     

The skew Schubert polynomials

We obtain a tableau definition of the skew Schubert polynomials named by Lascoux, which are defined as flagged double skew Schur functions. These polynomials are in fact Schubert polynomials in two sets of variables indexed by 321-avoiding permutations. From the divided difference definition of the skew Schubert polynomials, we construct a lattice path interpretation based on the Chen–Li–Louck pairing lemma. The lattice path explanation immediately leads to the determinantal definition and the tableau definition of the skew Schubert polynomials. For the case of a single variable set, the skew Schubert polynomials reduce to flagged skew Schur functions as studied by Wachs and by Billey, Jockusch, and Stanley. We also present a lattice path interpretation for the isobaric divided difference operators, and derive an expression of the flagged Schur function in terms of isobaric operators acting on a monomial. Moreover, we find lattice path interpretations for the Giambelli identity and the Lascoux–Pragacz identity for super-Schur functions. For the super-Lascoux–Pragacz identity, the lattice path construction is related to the code of the partition which determines the directions of the lines parallel to the y-axis in the lattice.     

Degree of Approximation for Bivariate Chlodowsky–Szasz–Charlier Type Operators

We consider a combination of Chlodowsky polynomials with generalized Szasz operators involving Charlier polynomials. We give the degree of approximation for these bivariate operators by means of the complete and partial modulus of continuity, and also by using weighted modulus of continuity. Furthermore, we construct a GBS (Generalized Boolean Sum) operator of bivariate Chlodowsky–Szasz–Charlier type and estimate the order of approximation in terms of mixed modulus of continuity.     

Bilinear Generating Functions for Orthogonal Polynomials

Using realizations of the positive discrete series representations of the Lie algebra su(1,1) in terms of Meixner—Pollaczek polynomials, the action of su(1,1) on Poisson kernels of these polynomials is considered. In the tensor product of two such representations, two sets of eigenfunctions of a certain operator can be considered and they are shown to be related through continuous Hahn polynomials. As a result, a bilinear generating function for continuous Hahn polynomials is obtained involving the Poisson kernel of Meixner—Pollaczek polynomials; this result is also known as the Burchnall—Chaundy formula. For the positive discrete series representations of the quantized universal enveloping algebra U _q (su(1,1)) a similar analysis is performed and leads to a bilinear generating function for Askey—Wilson polynomials involving the Poisson kernel of Al-Salam and Chihara polynomials. July 6, 1997. Date accepted: September 23, 1998.     

Automorphisms of the solution spaces of special double-confluent Heun equations

Two new linear operators determining automorphisms of the solution space of a special double-confluent Heun equation in the general case are obtained. This equation has two singular points, both of which are irregular. The obtained result is applied to solve the nonlinear equation of the resistively shunted junction model for an overdamped Josephson junction in superconductors. The new operators are explicitly expressed in terms of structural polynomials, for which recursive computational algorithms are constructed. Two functional equations for the solutions of the special double-confluent Heun equation are found.     

Simplification of symbolic polynomials on non-commutative variables

We address the issue of simplifying symbolic polynomials on non-commutative variables. The problem is motivated by applications in optimization and various problems in systems and control. We develop theory for polynomials which are linear in a subset of the variables and develop algorithms to produce representations which have the minimal possible number of terms. The results can handle polynomial matrices as well as block-matrix variables.     

Darboux transformations and recursion operators for differential-difference equations

We review two concepts directly related to the Lax representations of integrable systems: Darboux transformations and recursion operators. We present an extensive list of integrable differential-difference equations with their Hamiltonian structures, recursion operators, nontrivial generalized symmetries, and Darboux-Lax representations. The new results include multi-Hamiltonian structures and recursion operators for integrable Volterra-type equations and integrable discretizations of derivative nonlinear Schrödinger equations such as the Kaup-Newell, Chen-Lee-Liu, and Ablowitz-Ramani-Segur (Gerdjikov-Ivanov) lattices. We also compute the weakly nonlocal inverse recursion operators.     

Unitary Representations of the Wigner Group ISL(2, ℂ) and A Two-Spinor Description of Massive Particles With An Arbitrary Spin

Based on Wigner unitary representations for the covering group ISL(2,?) of the Poincaré group, we obtain spin-tensor wave functions of free massive particles with an arbitrary spin that satisfy the Dirac–Pauli–Fierz equations. In the framework of a two-spinor formalism, we construct spin-polarization vectors and obtain conditions that fix the corresponding density matrices (the Behrends–Fronsdal projection operators) determining the numerators in the propagators of the fields of such particles. Using these conditions, we find explicit expressions for the particle density matrices with integer (Behrends–Fronsdal projection operators) and half-integer spin. We obtain a generalization of the Behrends–Fronsdal projection operators to the case of an arbitrary number D of space–time dimensions.     

On Local Smoothness Classes of Periodic Functions

We obtain a characterization of local Besov spaces of periodic functions in terms of trigonometric polynomial operators. We construct a sequence of operators whose values are (global) trigonometric polynomials, and yet their behavior at different points reflects the behavior of the target function near each of these points. In addition to being localized, our operators preserve trigonometric polynomials of degree commensurate with the degree of polynomials given by the operators. Our constructions are “universal;” i.e., they do not require an a priori knowledge about the smoothness of the target functions. Several numerical examples are discussed, including applications to the problem of direction finding in phased array antennas and finding the location of jump discontinuities of derivatives of different order.     

A Theory of Quantized Fields Based on Orthogonal and Symplectic Clifford Algebras

The transition from a classical to quantum theory is investigated within the context of orthogonal and symplectic Clifford algebras, first for particles, and then for fields. It is shown that the generators of Clifford algebras have the role of quantum mechanical operators that satisfy the Heisenberg equations of motion. For quadratic Hamiltonians, the latter equations are obtained from the classical equations of motion, rewritten in terms of the phase space coordinates and the corresponding basis vectors. Then, assuming that such equations hold for arbitrary path, i.e., that coordinates and momenta are undetermined, we arrive at the equations that contains basis vectors and their time derivatives only. According to this approach, quantization of a classical theory, formulated in phase space, is replacement of the phase space variables with the corresponding basis vectors (operators). The basis vectors, transformed into the Witt basis, satisfy the bosonic or fermionic (anti)commutation relations, and can create spinor states of all minimal left ideals of the corresponding Clifford algebra. We consider some specific actions for point particles and fields, formulated in terms of commuting and/or anticommuting phase space variables, together with the corresponding symplectic or orthogonal basis vectors. Finally we discuss why such approach could be useful for grand unification and quantum gravity.     

Hecke Operators on Linear Ordinary Differential Equations

We construct Hecke operators acting on the space of certain linear ordinary differential equations, and describe a Hermitian inner product on the space of such differential equations. We also determine the adjoint of the Hecke operator with respect to this inner product, and prove that the space of ordinary differential equations associated to an automorphic form for a certain discrete subgroup of SL(2, R) has a basis consisting of common eigenvectors of a class of Hecke operators.     

Bilinear Summation Formulas from Quantum Algebra Representations

The tensor product of a positive and a negative discrete series representation of the quantum algebra U_q(su(1,1)) decomposes as a direct integral over the principal unitary series representations. Discrete terms can appear, and these terms are a finite number of discrete series representations, or one complementary series representation. From the interpretation as overlap coefficients of little q-Jacobi functions and Al-Salam and Chihara polynomials in base q and base q^–1, two closely related bilinear summation formulas for the Al-Salam and Chihara polynomials are derived. The formulas involve Askey-Wilson polynomials, continuous dual q-Hahn polynomials and little q-Jacobi functions. The realization of the discrete series as q-difference operators on the spaces of holomorphic and anti-holomorphic functions, leads to a bilinear generating function for a certain type of ₂₁-series, which can be considered as a special case of the dual transmutation kernel for little q-Jacobi functions.