Probabilistic G-Metric space and some fixed point results

In this note we introduce the notions of generalized probabilistic metric spaces and generalized Menger probabilistic metric spaces. After making our elementary observations and proving some basic properties of these spaces, we are going to prove some fixed point result in these spaces

A Note on the Symmetric Hit Problem

The symmetric hit problem was introduced for the first time by the author in his thesis ([5]). The aim of this paper is to solve an important open problem posed in ([7]), in an special case, which is one of the fundamental results in the studies of the symmetric hit problem

Generating H*(BO(3), F2) as a module over the Steenrod algebra

This paper continues the investigation of the hit problem, started in [5], for the algebra of symmetric polynomials B(n) viewed as a left ${\cal A}$-module graded by degree, where ${\cal A}$ denotes the Steenrod algebra over the field of two elements {\bb F}_2. We recall that a homogeneous element f of grading d in a graded left ${\cal A}$-module M is hit if there is a hit equation in the form of a finite sum $f=\sum_{k>0}Sq^{k}(h_k)$, where the homogeneous elements hk in M have grading less than d and the Sqk are the Steenrod squares, which generate ${\cal A}$. We denote by Q = Q(M) = {\bb F}_2\otimes_{\cal A} M the quotient of the module M by the hit elements, where {\bb F}_2 is here viewed as a right ${\cal A}$-module concentrated in grading 0. Then Q is a graded vector space over {\bb F}_2 and a basis for Q lifts to a minimal generating set for M as a module over ${\cal A}$. The hit problem is to find minimal generating sets for M and criteria for elements to be hit. We recall that B(n) = {\bb F}_{2}\[\sigma_1,\ldots,\sigma_n\] is the polynomial subalgebra of P(n) = {\bb F}_{2}\[x_1,\ldots,x_n\] generated by the elementary symmetric functions $\sigma_i$ in the variables xj . In particular, $\sigma_n=x_1\cdots x_n$. The algebras P(n) and B(n) realize respectively the cohomology of the product of n copies of infinite real projective space and the cohomology of the classifying space BO(n) of the orthogonal group O(n) over {\bb F}_2, where the usual grading in cohomology corresponds to degree in the polynomial algebra. The ideal M(n) in B(n), generated by $\sigma_n$, can be identified with the cohomology H*(MO(n), {\bb F}_2) of the Thom space MO(n) in positive dimensions. It is also convenient to introduce the notation L(n) for the polynomials in P(n) divisible by $\sigma_n$. Topologically, L(n) corresponds in positive degrees to the cohomology of the n-fold smash product of infinite real projective space. From the topological point of view, ${\cal A}$ is the algebra of universal stable operations in ordinary cohomology with {\bb F}_2 coefficients and this explains the action of ${\cal A}$ on P(n), L(n), B(n) and M(n). However, the whole subject may be treated in a purely algebraic fashion [5, 11]

The hit problem for symmetric polynomials over the Steenrod algebra

We cite [18] for references to work on the hit problem for the polynomial algebra P(n) = [open face F]2[x1, ;…, xn] = [oplus B: plus sign in circle]d[gt-or-equal, slanted]0 Pd(n), viewed as a graded left module over the Steenrod algebra [script A] at the prime 2. The grading is by the homogeneous polynomials Pd(n) of degree d in the n variables x1, …, xn of grading 1. The present article investigates the hit problem for the [script A]-submodule of symmetric polynomials B(n) = P(n)[sum L: summation operator]n , where [sum L: summation operator]n denotes the symmetric group on n letters acting on the right of P(n). Among the main results is the symmetric version of the well-known Peterson conjecture. For a positive integer d, let [mu](d) denote the smallest value of k for which d = [sum L: summation operator]ki=1(2[lambda]i[minus sign]1), where [lambda]i [gt-or-equal, slanted] 0