Affine Geometry of Space Curves

This paper is devoted to the complete classification of space curves under affine transformations in the view of Cartans theorem. Spivak has introduced the method but has not found the invariants. Furthermore, for the first time, we propound a necessary and sufficient condition for the invariants. Then, we study the shapes of space curves with constant curvatures in detail and suggest their applications in physics, computer vision and image processing.

Lie Symmetries and Solutions of KdV Equation

Symmetries of a differential equations is one of the most important concepts in theory of differential equations and physics. One of the most prominent equations is KdV (Kortwege-de Vries) equation with application in shallow water theory. In this paper we are going to explain a particular method for finding symmetries of KdV equation, which is called Harrison method. Our tools in this method are Lie derivatives and differential forms, which will be discussed in the first section more precisely. In second chapter we will have some analysis on the solutions of KdV equation and we give a method, which is called first integral method for finding the solutions of KdV equation.

Symmetries of 2nd order ODE: y + G(x)y + H(x)y = 0

This paper is devoted to study the Lie algebra of linear symmetries of a homogenous 2nd order ODE, by the method of Kushner, Lychagin and Robstov.

Symmetry of Quadratic Homogeneous Differential Systems

In this paper, the symmetry group of a differential system of n quadratic homogeneous first order ODEs of n variables is studied. For this purpose, we consider the action of both point and contact transformations to signify the corresponding Lie algebras. We also find the independent differential invariants of these actions.

Differential Invariants of SL(2) and SL(3)-ACTIONS on R^2

The main purpose of this paper is calculation of differential invariants which arise from prolonged actions of two Lie groups SL(2) and SL(3) on the $n$th jet space of $R^2$. It is necessary to calculate $n$th prolonged infenitesimal generators of the action.

Affine Classification of n-Curves

Classification of curves up to affine transformation in a finite dimensional space was studied by some different methods. In this paper, we achieve the exact formulas of affine invariants via the equivalence problem and in the view of Cartans lemma and then, state a necessary and sufficient condition for classification of n--Curves.

Self-equivalence 3rd order ODEs by time-fixed transformations

Let y = f(x, y, y, y) be a 3rd order ODE. By Cartan equivalence method, we will study the local equivalence problem under the transformations group of time-fixed coordinates.

The special linear representations of compact Lie groups

The special linear representation of a compact Lie group G is a kind of linear representation of compact Lie group G with special properties. It is possible to define the integral of linear representation and extend this concept to special linear representation for next using.

The Generalized Classical Time-Space

The newest model for space-time is based on sub-Riemannian geometry. In this paper, we use a combination of Lorentzian and sub-Riemannian geometry, the suggest a new model which likes to its ancestors, but with the most efficient in application. In continuation, we try to show a new connection which calls generalized connection, and prove some its properties.

Galilean Classification of Curves

In this paper, we classify space-time curves up to Galilean group of transformations with Cartans method of equivalence. As an aim, we elicit invariats from action of special Galilean group on space-time curves, that are, in fact, conservation laws in physics. We also state a necessary and sufficient condition for equivalent Galilean motions.