‎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 $mathcal{A}_p$‎.

The basis number of a graph G is defined to be the least integer d such that there is a basis B of the cycle space of G such that each edge of G is contained in at most d members of B. In this paper we give an upper bound of the basis number of the strong product of a graph with a bipartite graph and we show that this upper bound is the best possible.

A minimum cycle basis of a weighted undirected graph G is a basis of the cycle space of G such that the total weight of the cycles in this basis is minimized. If G is a planar graph with non-negative edge weights, such a basis can be found in O(n2) time and space, where n is the size of G. We show that this is optimal if an explicit representation of the basis is required. We then present an O(...

Let P be an n-dimensional polytope admitting a finite reflection group G as its symmetry group. Consider the set ( P(k) of all continuous functions on Rn satisfying the mean value property with respect to the k-skeleton P(k) of P, as well as the set ( G of all G-harmonic functions. Then a necessary and sufficient condition for the equality ( P(k) ̄( G is given in terms of a distinguished invaria...

The present work is devoted to the analysis of the G-quadruplex DNA structure using the bioinformatics method. The interest towards quadruplex DNAs is determined by their involvement in the functioning of telomeres and onco-promoters as well as by the possibility to create on their basis aptamers and nanostructures. Here, we present an algorithm for a general analysis of the polymorphism of the...

G-Frames in Hilbert spaces are a redundant set of operators which yield a representation for each vector in the space. In this paper we investigate the connection between g-frames, g-orthonormal bases and g-Riesz bases. We show that a family of bounded operators is a g-Bessel sequences if and only if the Gram matrix associated to its denes a bounded operator.

We have calculated position and dot intracules for a series of atomic and molecular systems, starting from an unrestricted Hartree-Fock wave function, expanded using the STO-3G, 6-31G, 6-311G, 6-311++G, 6-311++G(d,p), 6-311++G(3d,3p), and 6-311++G(3df,3pd) basis sets as well as the nonpolarized part of Dunning's cc-pV5Z basis. We find that the basis set effects on the intracules are small and t...

A group G is called a group with basis property if there exists a basis (minimal generating set) for every subgroup H of G and every two bases are equivalent. A group G is called a group with exchange property, if x / ∈ 〈X〉 ∧ x ∈ 〈X ∪ {y}〉 , then y ∈ 〈X ∪ {x}〉, for all x, y ∈ G and for every subset X ⊆ G. In this research, we proved the following: Every polycyclic group satisfies the basis prop...

.00 .00 .00 .00 .17 .3 .3 1.5 RGBC MYW (if 3 extra basis were introduced: W=min(R,G,B) C=min(B,G) – W M=min(R,B) – W Y=min(R,G) – W .00 .00 .00 .2 .3 .3 1.5 RGBW (if 4th base was introduced as w=min(RGB) ) .04 .11 2.7 4.9 9.1 51 RGB (if only 3 basis was used) B8 B7 B6 B5 B4 B3 B2 B1 Number of Basis Vectors PCA can be used to determine how many basis are needed to predict the tristimulus values ...