‎Let $mathcal{A}_p$ be the mod $p$ Steenrod algebra‎, ‎where $p$ is an odd prime‎, ‎and let $mathcal{A}$ be the‎ subalgebra $mathcal{A}$ of $mathcal{A}_p$ generated by the Steenrod $p$th powers‎. ‎We generalize the $X$-basis in $mathcal{A}$ to $mathcal{A}_p$‎.

We show that the universal Gröbner basis and the Graver basis of a binomial edge ideal coincide. We provide a description for this basis set in terms of certain paths in the underlying graph. We conjecture a similar result for a parity binomial edge ideal and prove this conjecture for the case when the underlying graph is the complete graph.

This paper proves the convergence of applying the radial basis functions as a global spatial approximation method for solving the option pricing models. The computational advantage of this method is illustrated by giving numerical examples on solving both the European and American options pricing models whereas the latter is a free boundary value problem.

Linear codes over any finite field can be associated to binomial ideals. Focussing on two specific instances, called the code ideal and the generalized code ideal, we present how these binomial ideals can be computed from toric ideals by substituting some variables. Drawing on this result we further show how their Graver bases as well as their universal Gröbner bases can be computed.

In this paper we apply normalized radial basis function networks to function learning and in nonparametric classification. A simple parameter learning technique is proposed and convergence and the rates of convergence of the empirically trained networks are studied theoretically and in computer experiments.