This paper further develops the theory of arbitrary semigroups acting on trees via elliptic mappings. A key tool is the Lyndon-Chiswell length function L for the semigroup S which allows one to construct a tree T and an action of S on T via elliptic maps. Improving on previous results, the length function of the action will also be L.

‎We study the existence of soliton solutions for a class of‎ ‎quasilinear elliptic equation in $mathbb{textbf{R}}^2$ with critical exponential growth‎. ‎This model has been proposed in the self-channeling of a‎ ‎high-power ultra short laser in matter‎.

In this paper, we consider two dimensional nonlinear elliptic equations of the form $ -{rm div}(a(u,nabla u)) = f $. Then, in order to solve these equations on rectangular domains, we propose a numerical method based on Sinc-Galerkin method. Finally, the presented method is tested on some examples. Numerical results show the accuracy and reliability of the proposed method.

in this paper we study properties of symbols such that these belong to class of symbols sitting insidesm ρ,φ that we shall introduce as the following. so for because hypoelliptic pseudodifferential operatorsplays a key role in quantum mechanics we will investigate some properties of m−hypoelliptic pseudodifferential operators for which define base on this class of symbols. also we consider maxi...

x.1. Elliptic Surfaces Deenition. An elliptic surface consists of a smooth (projective) surface E, a smooth (projective) curve C, and a morphism : E ?! C such that almost all bers E t = ?1 (t) are (smooth projective) curves of genus 1. In addition, we will generally assume that our elliptic surfaces come equipped with an identity section 0 : C ?! E which serves as the identity element of the gr...

Elliptic curve cryptosystems, proposed by Koblitz ((11]) and Miller ((15]), can be constructed over a smaller eld of deenition than the ElGamal cryptosystems ((5]) or the RSA cryptosystems ((19]). This is why elliptic curve cryptosystems have begun to attract notice. In this paper, we investigate eecient elliptic curve exponentiation. We propose a new coordinate system and a new mixed coordinat...

We describe a new algebraically completely integrable system, whose integral manifolds are co-elliptic subvarieties of Jacobian varieties. This is a multi-periodic extension of the Krichever-Treibich-Verdier system, which consists of elliptic solitons. The goal of this work is to generalize the theory of elliptic solitons, which was developed by A. Treibich and J.-L. Verdier based on earlier wo...

Recently, Wang et al. [1] proposed a new method for constructing pairingfriendly elliptic curves of embedding degree 1. Authors claim that this method significantly improves the efficiency of generating elliptic curves. In this paper, we give the arithmetic of pairing-friendly elliptic curves of embedding degree 1. We prove that conventional classification of pairings into Type 1, 2, 3 and 4 is...