Abstract Let $${\mathcal {T}}_\infty (F)$$ T∞(F) be the algebra of all $${\mathbb {N}}\times {\mathbb {N}}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">N</mml:...

suppose $gamma$ is a graph with $v(gamma) = { 1,2, cdots, p}$and $ mathcal{f} = {gamma_1,cdots, gamma_p} $ is a family ofgraphs such that $n_j = |v(gamma_j)|$, $1 leq j leq p$. define$lambda = gamma[gamma_1,cdots, gamma_p]$ to be a graph withvertex set $ v(lambda)=bigcup_{j=1}^pv(gamma_j)$ and edge set$e(lambda)=big(bigcup_{j=1}^pe(gamma_j)big)cupbig(bigcup_{ijine(gamma)}{uv;uin v(gamma_i),vin ...

Abstract Let R be a finite commutative ring. The set $${{\mathcal{F}}}(R)$$ F ( R ) of polynomial functions on is ring with pointwise operations. Its group units $${{\mathcal{F}}}(R)^\times $$ × j...

Given a suitable extension $ F'/F of algebraic function fields over finite field \mathbb{F}_q $, we introduce the conorm code \operatorname{Con}_{F'/F}( \mathcal{C}) defined F' which is constructed from an geometry \mathcal{C} F $. We study parameters in terms ramification behavior places used to define and genus In case unramified extensions prove that \mathcal{C})^\perp = \mathcal{C}^\perp) w...

We prove that if a family of compact connected sets in the plane has property every three members it are intersected by line, then there lines intersecting all family. This answers question Eckhoff [Discrete Comput. Geom., 9 (1993), pp. 203--214], who proved that, under same condition, four sets. In fact, we colorful version this result weakened conditions on Three $A,B,C$ form tight triple $\t...

A suitability of a nanostructure for metal-enhanced fluorescence (MEF) is usually assessed from the value maximal enhancement factor, $\mathcal{F}$, which it can generate. However, $\mathcal{F}$ an ambiguous quantity may dramatically depend on intrinsic quantum yield, $q_0$, emitter in free space. Here we suggest $\mathcal{F}=\mathcal{F}_{q_0=1}$ taken limiting case $q_0=1$ as proper figure mer...

We survey some classical norm inequalities of Hardy, Kallman, Kato, Kolmogorov, Landau, Littlewood, and Rota the type \[ \|A f\|_{\mathcal{X}}^2 \leq C \|f\|_{\mathcal{X}} \big\|A^2 f\big\|_{\mathcal{X}}, \quad f \in dom\big(A^2\big), \] recall that under exceedingly stronger hypotheses on operator $A$ and/or Banach space $\mathcal{X}$, optimal constant $C$ in these diminishes from $4$ (e.g., w...