Lipschitz symmetric functions on Banach spaces with symmetric bases

We investigate Lipschitz symmetric functions on a Banach space $X$ with a symmetric basis. We consider power symmetric polynomials on $\ell_1$ and show that they are Lipschitz on the unbounded subset consisting of vectors $x\in \ell_1$ such that $|x_n|\le 1.$ Using functions $\max$ and $\min$ and tropical polynomials of several variables, we constructed a large family of Lipschitz symmetric functions on the Banach space $c_0$ which can be described as a semiring of compositions of tropical polynomials over $c_0$.

Return words of linear involutions and fundamental groups

We investigate the shifts associated with natural codings of linear involutions. We deduce, from the geometric representation of linear involutions as Poincaré maps of measured foliations, a suitable definition of return words which yields that the set of return words to a given word is a symmetric basis of the free group on the underlying alphabet,$A$. The set of return words with respect to a subgroup of finite index$G$of the free group on$A$is also proved to be a symmetric basis of$G$.

The Trigonometric Polynomial Like Bernstein Polynomial

A symmetric basis of trigonometric polynomial space is presented. Based on the basis, symmetric trigonometric polynomial approximants like Bernstein polynomials are constructed. Two kinds of nodes are given to show that the trigonometric polynomial sequence is uniformly convergent. The convergence of the derivative of the trigonometric polynomials is shown. Trigonometric quasi-interpolants of reproducing one degree of trigonometric polynomials are constructed. Some interesting properties of the trigonometric polynomials are given.

A CLASS OF ASSOCIATION SCHEMES IN WHICH ALL BASIS RELATIONS HAVE VALENCY 1 OR λ > 2

In this paper, we classify the commutative association schemes with both the property that each basis relation has valency either 1 or λ > 2 for some fixed λ and the property that there is a connected, symmetric basis relation R for which R2 contains exactly two basis relations. We show that up to isomorphism, the unique association scheme with these properties is the direct product of the trivial association scheme of order 2 and the trivial association scheme of order λ + 1.