In this paper, we consider some cubic near-Hamiltonian systems obtained from perturbing the symmetric cubic Hamiltonian system with two symmetric singular points by cubic polynomials. First, following Han [2012] we develop a method to study the analytical property of the Melnikov function near the origin for near-Hamiltonian system having the origin as its elementary center or nilpotent center....

We consider symmetric (as well as multi-symmetric) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of fixed degrees. We give polynomial (in the dimension of the ambient space) bounds on the number of irreducible representations of the symmetric group which acts on these sets, as well as their multip...

In this note, we study the notion of structured pseudospectra. We prove that for Toeplitz, circulant and symmetric structures, the structured pseudospectrum equals the unstructured pseudospectrum. We show that this is false for Hermitian and skew-Hermitian structures. We generalize the result to pseudospectra of matrix polynomials. Indeed, we prove that the structured pseudospectrum equals the ...

A q-analogue of the type A Dunkl operator and integral kernel We introduce the q-analogue of the type A Dunkl operators, which are a set of degree–lowering operators on the space of polynomials in n variables. This allows the construction of raising/lowering operators with a simple action on non-symmetric Macdonald polynomials. A bilinear series of non-symmetric Macdonald polynomials is introdu...

In this paper, we define a new form of Carlitz’s type degenerate twisted (p,q)-Euler numbers and polynomials by generalizing the Euler polynomials, q-Euler polynomials. Some interesting identities, explicit formulas, symmetric properties, connection with are obtained. Finally, investigate zeros using computer.

We present an approach to the theory of Schubert polynomials, corresponding symmetric functions, and their generalizations that is based on exponential solutions of the Yang-Baxter equation. In the case of the solution related to the nilCoxeter algebra of the symmetric group, we recover the Schubert polynomials of Lascoux and Schiitzenberger, and provide simplified proofs of their basic propert...

The main objective of this article is to construct a totally new, effective algorithm, for the calculations of all series of elementary symmetric polynomials at once for the arbitrary order of the general polynomial. We proved that the effectiveness of the advanced algorithm proposed in this article is by the linear term better than the algorithms existing in the literature so far. Moreover, we...

An explicit expansion for the Macdonald polynomials of the form H (32 a 1 b) [X; q, t] and H (41 k) [X; q, t] in terms of Hall-Littlewood symmetric functions is presented. The expansion gives a proof of a symmetric function operator that adds a row of size 3 to Macdonald polynomials of the form H (2 a 1 b) [X; q, t] and another operator that adds a row of size 4 to Macdonald polynomials of only...

Let f(x) ∈ Z[x] be a totally real polynomial with roots α1 ≤ . . . ≤ αd. The span of f(x) is defined to be αd − α1. Monic irreducible f(x) of span less than 4 are special. In this paper we give a complete classification of those small-span polynomials which arise as characteristic polynomials of integer symmetric matrices. As one application, we find some low-degree polynomials that do not aris...

Polynomials with values in an irreducible module of the symmetric group can be given the structure of a module for the rational Cherednik algebra, called a standard module. This algebra has one free parameter and is generated by differential-difference (“Dunkl”) operators, multiplication by coordinate functions and the group algebra. By specializing Griffeth’s (arχiv:0707.0251) results for the ...