This paper is an investigation of $L$-dual frames with respect to a function-valued inner product, the so called $L$-bracket product on $L^{2}(G)$, where G is a locally compact abelian group with a uniform lattice $L$. We show that several well known theorems for dual frames and dual Riesz bases in a Hilbert space remain valid for $L$-dual frames and $L$-dual Riesz bases in $L^{2}(G)$.

One of the problems modern discrete mathematics is R. Dedekind problem on number monotone boolean functions. For other precomplete classes, general formulas for functions classes had been found, but it has not found so far class Within framework this problem, there are a lower level. them absence formula intersection $MS$ two --- and self-dual In paper, new bounds proposed estimating cardinalit...

We introduce the notions of T-dual Rickart and strongly T-dual Rickart modules. We provide several characterizations and investigate properties of each of these concepts. It is shown that every free (resp. finitely generated free) $R$-module is T-dual Rickart if and only if $overline{Z}^2(R)$ is a  direct summand of $R$ and End$(overline{Z}^2(R))$ is a semisimple (resp. regular) ring. It is sho...

background: this study set out to evaluate the utility of cerebrovascular virtual non-contrast (vnc) scans. materials and methods: conventional non-contrast (cnc) and dual-energy computed tomography angiography (de-cta) head scans were conducted on 100 subjects, of which 46 were normal, 15 had parenchymal hematomas of the brain, 13 had ischemic infarction, 22 had tumors, and 4 had calcified les...

gerbera jamesoniiis of commercial significance and fifth most used cut flower in the world today. a pot experiment was performed to see the effect of co-inoculation of arbuscular mycorrhizal fungi (amf) i.e. (glomus mosseae andacaulospora laevis) with phosphate solubilizing bacteria pseudomonas fluorescensin the presence of different doses of superphosphate (low, medium, high) on growth establi...