Sharif Digital Repository

At first the killing vector fields will be investigated. Conditions are introduced for the hypersurface of a Riemannian manifold with a killing vector field to be equipped with the same killing vector field. Then 2-killing vector field is studied and its relation with killing vector fields and monotone vector fields is presented. After that conformal vector fields are discussed and conditions are introduced in order that the Riemannian manifold equipped with a conformal vector field, isisometric to n-dimensional sphere with constant curvature. Finally we will present the conditions which conformal vector field is a 2-killing vector field. Then we will present the results in which the... 

In this thesis we study one of the applications of algebraic topology to computer science and define some tools to analyze concurrent programs. For this purpose a geometric space is corresponded to every concurrent program and a partial order is considered on this space, so every path on it corresponds to an execution of the program. Some of these paths which are dihomotop (homotop in partial ordered spaces) with each other induce the same execution of the program. By defining “Fundamental Category” in which objects are points of the space and morphisms are dihomotopy classes, the space that should be analyzed shrinks to a smaller one. By defining component category in the next step, without... 

Simultaneous triangulation of matrices is a subject with a rich literature. There are many well known theorems available, such as McCoy theorem or Burnsides. In the nite dimensional case since the all the topologies on vector spaces are the same, there is a little bit diculty and most of the arguments are from linear algebra. In this thesis we study the simultaneous triangulation of sub algebras of K(X),with X a innite dimensional Banach space. We will give a denition of simultaneous triangulation which is independent of the notion of Basis and totally relies on Invariant subspaces. This denition coincides with the denition of simultaneous triangulation in nite dimensional case. Then we will... 

A number of questions from a variety of areas of mathematics lead one to the problem of analyzing the topology of a simplicial complex. However, there are few general techniques available to aid us in this study. On the other hand, some very general theories have been developed for the study of smooth manifolds. One of the most powerful, and useful, of these theories is Morse Theory. We present a combinatorial adaptation of Morse Theory, which we call discrete Morse theory that may be applied to any simplicial complex (or more general cell complex). Our goal is to present an overview of the subject of discrete Morse Theory that is sufficient both to understand the major applications of the... 

In this thesis, we have shown that unique subgraphs of a graph have a key role in structure of the graph. Using unique subgraph which is called “anchor” here, the reconstruction of graphs is explained. Using anchor, we have shown that almost every n-vertex graph is determined by its 3log(n)-vertex subgraphs. In the second part of the thesis, a novel randomized algorithm is proposed for the graph isomorphism problem which is very simple and fast. It solves this problem with running time O(n^{2.373} \log(n)) for any pair of $n$-vertex graphs whose adjacency matrices are not strongly co-det. Strongly co-det pair of matrices have very special symmetric structure which can be disarranged to be... 

In this thesis, we review the proof to standardness of Einstein solvamanifolds which is based on some results from Geometric Invariant Theory and stratification of topological spaces. Standardness is a very simple and yet powerful algebraic condition on the lie algebra of a solvmanifold which yields to remarkable existence and uniqueness and obstruction results  

The main goal of this thesis is to classify minimal translation surfaces of three-dimensional Euclidean space. In pursuing that, a method will be introduced that constructs explicit examples. A translation surface is the sum of two regular curves α and β. A minimal surface is a surface, with zero mean curvature. Will be shown that besides the know examples (plane and surfaces of Scherk type) any minimal translation surfaces can be described Ψ(s, t) = α(s)+α(t) , where α is the unit speed curve and its curvature κα is a positive solution of (y ′ ) 2 + y 4 + c3y 2 + c 2 1 y −2 + c1c2 = 0 and its torsion is τ (s) = c1/κ(s) 2 . the above coefficients and their relations will be described  

There are different approaches to the ideal closure of geodesic metric space with non-positive curvature in the sense of Busemann. We established relations between them with Andreev theorem. In this theorem we introduced a continuous surjection which coincides with identity mapping of Idx onto X. Due to we studied metric spaces, geodesics and their angles and also CATk domains. Then with asymptotic rays we introduced metric boundary and we produced compact metric space. Then we proved Andreev theorem with some theorems and lemmas. The Andreev theorem is correct for Alexandrov space. Finally we construct the counterexample showing that Busemann ideal closure can differ from the geodesic... 

A surfaceMin the Euclidean space is called a translation surfaceif it is given by the graph z(s,t)=f(s)+g(t), where f and gare smooth functions on some interval of R. These surfaces are called translation surfaces since its parameterization X(s,t)=(s,t,f(s)+g(t) ) can be written as the sum of two curves (translation), namely , X(s,t)=(s,0,f(s) )+(0,t,g(t) )
In this work , Minimal surfaces in Sol_3 and Nil_3have been studied,where Sol_3and Nil_3are two model geometry of the eight geometries of Thurston. We propose a similar problem in Sol_3 and Nil_3 changing the additive + in the Euclidean space by the group operation * of Sol_3 and Nil_3, such that we have X(s,t)=α(s)*β(t), where α... 

In this work, we study spacelike surfaces in Minkowski space E3 1 foliated by pieces of circles that satisfy a linear Weingarten condition of type aH + bK=c, where a, b and c are constants and H and K denote the mean curvature and the Gauss curvature respectively. We show that such surfaces must be surfaces of revolution or surfaces with constant mean curvature H=0 or surfaces with constant Gauss curvature K=0  

In this thesis we study three versions of K−theory. The most well-known vesrsion is topological K−theory, a generalization of Grothendieck works on algebraic varieties to the topological setting by Atyiah and Hirzebruch. Since its birth it has been an indespensible tool in topology,differential geometry and index theory. In the early 1970s C∗−algebraic version of K−theory introduced through associating two abelian groups,K0(A)and K1(A)to a C∗−algebra like A. These functors proved to be a powerful machine, making it possible to calculate the K−theory of a great many C∗−algebras. At last,algebraic K−theory is dealig with linear algebra over a ring R by associating it, a sequence of abelian... 

In Twentieth century, W.P. Thurston formulated a geometric conjecture for three dimensional manifolds, namely every compact orientablethree-manifold admits a canonical decomposition into pieces, each of them having a canonical geometric structure from the following eight maximal and simply connected homogeneous Riemannian spaces among Sol spaces.A surface in homogeneous space Sol is said to be an invariant surface if it is invariant under some of the two one-parameter groups of isometries of the ambient space whose fix point sets are totally geodesic surfaces. In this work we study invariant surfaces that satisfy a certain condition on their curvatures. We classify invariant surfaces with... 

In this thesis, we study a rigidity problem for a 2-step nilmanifold such as Γ by some information about its geodesic flows, where is a simply connected 2-step nilpotent Lie group with a left invariant metric, and Γ is a discrete cocompact subgroup of . For the solution to this problem, first, we consider an algebraic aspect of it; since isometry groups of simply connected Riemannian manifolds can be characterized in a purely algebraic way, i.e., normalizers. Also, as we will show, proper and smooth actions of Lie groups and closed subgroups of isometries for smooth Riemannian structures can be regarded as the same topic. Then, in a generalized setting, when passing from the case of... 

One of the topics that have recently been given in science and the topics of optimality have been raised and expanding in such fields, is the topic of classified algorithms. These algorithms are used to solve optimization problems, because probability bases are used in them, and some non-deterministic problems with the above can be answered to a suitable extent or a relatively optimal solution can be found for these problems. These algorithms can sometimes perform multiple solutions to the decision problems they have and generate their answers. We will examine the types of segmentation algorithms and we will get to know these algorithms to a good extent, and also we have reached the problems... 

In this thesis, the aim is to provide lower or upper bounds for the entropy of the geodesic flow on Riemannian manifolds. There are well-known results in this area derived from the works of mathematicians such as Sarnak, Osserman, Mane, and others. Our focus is primarily on geometric bounds that are expressed in terms of Riemannian curvature. We will attempt to examine these results in detail and consider some of their applications