A quadrangular embedding of a graph in surface Σ, also known as quadrangulation is cellular which every face bounded by 4-cycle. Σ minimal if there no (simple) smaller order Σ. In this paper we determine n(Σ), the for all surfaces, both orientable and nonorientable. Letting S0 denote sphere N2 Klein bottle, prove that n(S0)=4,n(N2)=6, n(Σ)=⌈(5+25−16χ(Σ))/2⌉ other surfaces where χ(Σ) Euler chara...

In this lecture, we describe two proofs of a central theorem of mathematics , namely the central limit theorem. One will be using cumulants, and the other using moments. Actually, our proofs won't be entirely formal, but we will explain how to make them formal. What it the central limit theorem? The theorem says that under rather general circumstances, if you sum independent random variables an...

Here, we describe a systematic derivation of the general form optical helicity density ellipticaly polarized paraxial Laguerre–Gaussian modes L mathvariant="normal">G ℓ , <m...

Let X be a locally convex Hausdorff space (briefly, lcHs) and L(X) denote the space of all continuous linear operators of X into itself. The space L(X) is denoted by Ls(X) when it is equipped with the topology of pointwise convergence in X (i.e. the strong operator topology). By a spectral measure in X is meant a σ-additive map P : Σ → Ls(X), defined on a σ-algebra Σ of subsets of some set Ω, w...

Quandle homology theory has been developed and cocycles have used to define invariants of oriented classical or surface links. We introduce a shifting chain map σ \sigma</mml...

Introduction . The availability of an objective assessment the microcirculatory bed is possible when examining vessels conjunctiva. This especially true in individuals who have had COVID-19 and diseases associated with endothelial dysfunction. Aim To study microcirculation conjunctiva after a new coronavirus infection caused by SARS-CoV-2 virus. Materials methods 83 patients history were examin...

Generalized least-squares (GLS ) regression extends ordinary least-squares (OLS) estimation of the normal linear model by providing for possibly unequal error variances and for correlations between different errors. A common application of GLS estimation is to time-series regression, in which it is generally implausible to assume that errors are independent. This appendix to Fox and Weisberg (2...