We develop the formulation of turbulence in terms of the functional integral over the phase space configurations of the vortex cells. The phase space consists of Clebsch coordinates at the surface of the vortex cells plus the Lagrange coordinates of this surface plus the conformal metric. Using the Hamiltonian dynamics we find an invariant probability distribution which satisfies the Liouville ...

Many non-Abelian finite subgroups of SU(3) have been used to explain the flavor structure of the Standard Model. In order to systematize and classify successful models, a detailed knowledge of their mathematical structure is necessary. In this paper we shall therefore look closely at the series of finite non-Abelian groups known as ∆(6n), its smallest members being S3 (n = 1) and S4 (n = 2). Fo...

Hurwitz loci have played a basic role in the study of themoduli space of curves at least since 1872 when Clebsch proved that Mg is irreducible by showing that a certain Hurwitz space is connected. Hurwitz cycles on Mg are essential in the work of Harris and Mumford [HM] on the Kodaira dimension of Mg and are expected to govern the length of minimal affine stratifications of Mg. Faber and Pandha...

Abstract The notion of classical r-matrix is re-examined, and a definition suitable to differential (-difference) Lie algebras, – where the standard definitions are shown to be deficient, – is proposed, the notion of an O-operator. This notion has all the natural properties one would expect form it, but lacks those which are artifacts of finite-dimensional isomorpisms such as not true in differ...

We investigate helicity amplitudes (HAs) of A→BC-type decays for arbitrary spin towards the kinematic endpoint. show that HAs are proportional to product Clebsch-Gordan coefficients (CGC) and velocity a non-negative power. The latter can be zero in which case HA is non-vanishing at At endpoint explicit breaking rotational symmetry, through external momenta, restored findings interpreted as spec...

a graph is called textit{circulant} if it is a cayley graph on a cyclic group, i.e. its adjacency matrix is circulant. let $d$ be a set of positive, proper divisors of the integer $n>1$. the integral circulant graph $icg_{n}(d)$ has the vertex set $mathbb{z}_{n}$ and the edge set e$(icg_{n}(d))= {{a,b}; gcd(a-b,n)in d }$. let $n=p_{1}p_{2}cdots p_{k}m$, where $p_{1},p_{2},cdots,p_{k}$ are disti...

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in this paper we study the coprime graph of a group $g$. the coprime graph of a group $g$, is a graph whose vertices are elements of $g$ and two distinct vertices $x$ and $y$ are adjacent iff $(|x|,|y|)=1$. in this paper we classify all the groups which the coprime graph is a complete r-partite graph or a planar graph. also we study the automorphism group of the coprime graph.