Here V is the tangent to M and A is the tangent to ∂M . (By the tangent, we mean, e.g., that V (X) is the unit positively oriented pseudoscalar in the tangent algebra to M at X.) Recall the important relationship V A = N , where N is the unit outward normal to M [4, p. 319]. The relationships dV = |dV |V and dA = |dA|A define the integrals componentwise as Lebesgue integrals on M and ∂M [4, p. ...

Let F be a totally real number field of odd degree. We prove several purely local criteria for the asymptotic Fermat's Last Theorem to hold over F, and also non-existence solutions unit equation F. For example, if 2 ramifies 3 splits completely in then holds

in this paper, after reviewing some results in minimal space, some new results in this setting are given. we prove a generalized form of the fan-kkm typetheorem in minimal vector spaces. as some applications, the open type of matching theorem and generalized form of the classical kkm theorem in minimal vector spaces are given.

Given any simple connected graph G =< V,E > such that each edge e ∈ E has unit resistance, Foster’s Resistance Theorem states that the summation of effective resistances over all edges equals n− 1. That is, ∑ e∈E Re = n− 1 Further, given a general graph with resistances re corresponding to edges e ∈ E, the theorem states: ∑ e∈E Re re = n− 1 Tetali[1] proved Foster’s Resistance Theorem by consid...

The classical Julia-Wolff-Carathéodory Theorem is one of the main tools to study the boundary behavior of holomorphic self-maps of the unit disc of C. In this paper we prove a Julia-Wolff-Carathéodory’s type theorem in the case of the polydisc of Cn. The Busemann functions are used to define a class of “generalized horospheres” for the polydisc and to extend the notion of non-tangential limit. ...