This short note presents four examples of compactly supported symmetric refinable componentwise polynomial functions: (i) a componentwise constant interpolatory continuous refinable function and its derived symmetric tight wavelet frame; (ii) a componentwise constant continuous orthonormal and interpolatory refinable function and its associated symmetric orthonormal wavelet basis; (iii) a diffe...

We give two equivalent conditions of the P -polynomial property of a symmetric association scheme. The first equivalent condition shows that the P -polynomial property is determined only by the first and second eigenmatrices of the symmetric association scheme. The second equivalent condition is another expression of the first using predistance polynomials.

Given a polynomial f with coefficients from a field F, is it easier to compute f over an extension ring R than over F? We address this question, and show the following. For every polynomial f , there is a noncommutative extension ring R such that F is in the center of R and f has a polynomial-size formula over R. On the other hand, if F is algebraically closed, no commutative extension ring R c...

The classification of complex wave functions of infinite variables is an important problem since it is related to the classification of possible quantum states of matter. In this paper, we propose a way to classify symmetric polynomials of infinite variables using the pattern of zeros of the polynomials. Such a classification leads to a construction of a class of simple non-Abelian quantum Hall...

LetW be a finite dimensional representation of a linearly reductive group G over a field k. Motivated by their work on classical rings of invariants, Levasseur and Stafford asked whether the ring of invariants under G of the symmetric algebra of W has a simple ring of differential operators. In this paper, we show that this is true in prime characteristic. Indeed, if R is a graded subring of a ...

Let g = k + p be a complexified Cartan decomposition of a complex semisimple Lie algebra g and let K be the subgroup of the adjoint group of g corresponding to k. If H is an irreducible Harish-Chandra module of U (g), then H is completely determined by the finite-dimensional action of the centralizer U (g) K on any one fixed primary k component in H. This original approach of Harish-Chandra to ...

The effectiveness of Savitzky–Golay type symmetric polynomial smoothers is known to be strongly dependent on the window size. Many authors note that selection of the appropriate window size is essential for achieving the correct trade-off between noise reduction and avoiding the introduction of bias. However, it is often overlooked that, in the case of non-stationary signals, the optimal window...

1.1 Let V be a finite dimensional vector space and G a group of linear automorphisms of V , Then G acts by transposition on the dual V ∗ of V and hence on the space of regular functions R[V ∗] on V ∗, which may be identified with the symmetric algebra S(V ) of V , that is to say, the polynomial algebra generated by a basis of V . A basic problem of algebraic invariant theory is to determine S(V...

Let X be an irreducible Hermitian symmetric space of noncompact type and rank r. Let p ∈ X and let K be the isotropy group of p in the group of biholomorphic transformations. Let S denote the symmetric algebra in the holomorphic tangent space to X at p. Then S is multiplicity free as a representation of K and the irreducible constituents are parametrized by r-tuples, (m1, . . . ,mr) with m1 ≥ ·...

Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called p, q -Fibonacci polynomials. We obtain combinatorial identities and by using Riordanmethodwe get factorizations of Pascal matrix involvin...