Sharif Digital Repository

Persistent homology is one of the main tools in Topological Data Analysis. Indeed, to deal with a huge dataset while noise sensitivity is important, persistent homology can reflect some information about data in the form of persistent homology groups and persistence diagrams. Note that statistical or linear algebraic tools are not suitable to work with huge datasets with very high dimensions. In this thesis, we discuss the concept of persistent homology and investigate some of its properties such as the stability of the persistence diagrams. Indeed, persistence diagrams are obtained from the generating sets of the persistent homology groups. Further, we discuss an application of persistent... 

Meaningful gesture recognition, whose purpose is to interpret human movements, plays a crucial role in various fields such as human and computer interaction, sign language recognition, robot control, and medical applications. Sign language recognition is regarded as the most significant use of gesture recognition by many researchers. Sign languages are the natural medium of communication for millions of deaf people all over the world, and the existence of a sign language recognition system has significantly aided in facilitating communication between deaf individuals and others. Despite numerous studies conducted in this field in recent years, there are still many challenges to continuous... 

One of the important questions in mathematics is generalized. In case of metric spaces, one of the questions is: Can we extend notions of Riemannian geometry to arbitrary metric spaces smoothly? Ricci curvature is one of the most important concepts in differential geometry. Ricci curvature is defined for the Riemannian manifold and has many applications in mathematics and physics like Einstein’s equation in relativity theory. There is a good notion for a metric space having “sectional curvature bounded below by K” or “sectional curvature bounded above by K”, due to Alexandrov. Can we define the concept of Ricci curvature in metric space? A motivation for this question comes from Gromov’s... 

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In this thesis, it has been introduced Hardy inequality and it’s extends. The fractional Hardy inequality for Ω ⊆ Rn and f ∈ C ∞ (Ω) is:1f (x) − f ( y)2∫α +n2dx dy ≥ kn,α ∫f (x)dx( )αΩ×Ωx − yΩ M α x In this thesis we are going to introduce the fractional and derivative forms of Hardy inequality then the Hardy inequality will be proved to fractional form on Euclidean and Hyperbolic domains, and finally we will get right into Hardy inequality with remainder.
 

In this thesis, we present some extension of Burton and Kirk fixed point theorem for the sum of two nonlinear operators, which one of them is compact and the other one is strict contraction; and we investigate the existence of fixed point where these two operators don’t need to be weakly continuous. Next, we will check the necessity of some conditions, like being strict contraction or compactness, in some specific Banach spaces, such as reflective spaces or Banach spaces equipped with uniformly convex norm.Finally, we analysis the existence of the solution for integral equations in L1 space by using the expressed theorems  

The so called T(b) theorem for boundedness of generalized CalderonZygmund singular integrals is investigated. This theorem provides a criterion, in which boundedness of the operator is equivalent to a special condition for the image of some function under action of that operator.Then we discuss some applications of the theorem in analysis and partial differential equations  

For a Banach space E and a compact metric space (X, d), a function F : X → E is a Lipschitz function if there exists k > 0 such that ∥F (x) − F (y)∥ ≤ kd(x, y) for all x, y ∈ X.The smallest such k is called the Lipschitz constant L(F ) for F . The space Lip(X, E) of all Lipschitz functions from X to E is a Banach space under the norm defined by ∥F ∥ = max{L(F ), ∥F ∥∞}, where ∥F ∥∞ = sup{∥F (x)∥ : x ∈ X}.Recent results characterizing isometries on these vector-valued Lipschitz spaces require the Banach space E to be strictly convex. We inves- tigate the nature of the extreme points of the dual ball for Lip(X, E) and use the information to describe the surjective isometries on Lip(X, E) under... 

Let C(S) and C(T) denote the sup-normed Banach spaces of real or complex-valued continuous functions on the compact Hausdorff spaces S and T, respectively. A linear map H:C(T)→C(S) is called separating if, for x,y∈C(T), xy≡0 implies HxHy≡0. In the second chapter of this thesis, we will show that any continuous separating map is a continuous multiple of a composition map. Moreover, it will be proved that any linear separating isomorphism of C(T) onto C(S) is continuous. We will also define separating and biseparating maps on the rings of continuous functions equipped with the compact-open topology. In addition, vector-valued separating maps will be investigated. For example, assume that C(T)... 

Monge’s problem was solved by Brenier in 1990. In general, the problem remained unresolved for a long time. some of its cases were solved by assumptions, but the general case and its analytical solution for the first time by Ian Bernier The French mathematician was introduced. He wrote his famous article Solved this nonlinear problem with technical assumptions. With the help of convex analysis and fundamental theorems in functions with vector values, he proved the existence and unity of this nonlinear problem. Monge’s problem, also known as optimal transport, suggests whether a stable­sized mapping can be done by having two probabilistic spaces and one cost function. (Inverted image of any... 

In this thesis we consider dynamical aspects of fractals. More precisely, answering questions like how heat diffuses on fractals and how does a material with fractal structure vibrates? To give an answer to these questions we need a PDE theory on fractals. Since fractals do not have smooth structures, defining differential operators like Laplacian is not possible from a classical viewpoint of analysis, to overcome such a difficulty we also need a theory of analysis on fractals. So as a good instance of analysis on fractals we first define Laplacian on Sierpinsky gasket and we try to extend the concept on other finitely ramified self-similar fractals. We also construct Dirichlet forms,... 

In this thesis, we introduce and prove the inequalities and theorems of Smulian which are about derivatives of convex functions and optimization of linear functionals. Then we show that some old ideas of Smulian can be used to give another proof of theorems of Bourgain. The first theorem of Bourgain associates to any bounded, convex, closed and dentable subset like D of a Banach space X a G-delta subset of the dual space and the second theorem of Bourgain investigates the optimization in Banach spaces. Finally, we characterize subsets of Banach spaces having the Radon-Nikodym property by means of optimization results.
 

This millennium could be named as a world of relationship and connection which these two terms impacts on each of our business and daily life dimensions, in the world of marketing , consumer relationship management (CRM) was the notion that introduced in 1995 and emphasize on the importance of 1 to 1 interaction. wide perspective and golden target of CRM makes this concept more desirable for all organizations which everyone hope to be achieved it and this is a gains of not just more sales but more profit, in fact CRM focused on the emotional interaction between to part of deals and tries to locked the consumer in the system by creation of sentimental dependence in consumer which drives the... 

The development of intrinsic theories for area-minimization problems was motivated in the 1950 by the diffichlty to prove, existence for the Plateau problem for surfaces in Euclidean spaces of dimension higher than two. After the pioneering work of R. Caccioppoli and E. De Giorgi on set with finite perimeter, W. H. Fleming and H. Federer developed the theory of currents, which leads to existence results for the Plateau problem for oriented surfaces of any dimension and codimension. The aim of this paper is to develop an existence of the Federer-Fleming theory to spaces without a differentiable structure, and virtually to any complete metric space. The starting point of our research has been... 

In this thesis, vve generali :ed the notion of (upper) gradient to arbitrary func- tions on a. metric measure space a.nd under some conditions -vve study deriva- t.ives of Lipschitz functions and their differential properties. Finally, we consider metric measure spaces that satisfy certa.in additiona.l conditions and give more detailed study of these notions  

Suppose $X$ is a locally compcat Hausdorf‌f space. We denote by $C_0(X)$ the Banach space of all continuous functions real or complex-valued def‌ined on $X$ which vanish at inf‌inity, equipped with the supremum norm. Suppose linear subspace $A$ of $C_0(X)$ is strongly separating, that is, for all pair of distinct points $x_1,x_2 \in X$, there exists $f \in A$ such that $|f(x_1)|\neq |f(x_2)|$. In this thesis, we show that Shilov boundary of $A$ exists and is closure Choquet boundary of $A$. Furthermore, \ we show that a linear isometry $T$ of $A$ onto such a subspace $B$ of $C_0(Y)$ induces a homeomorphism $h$ between two certain subspaces of the Shilov boundaries of $B$ and $A$, sending the... 

Nowadays, implementing a complex system, which executes various applications with different levels of assurance, is a growing trend in modern embedded real- time systems to meet cost, space, timing, and power consumption requirements. Medical devices, automotive, and avionics industries are the most common safety- critical applications, exploiting these systems known as Mixed-Criticality (MC) systems. MC applications are real-time, and to ensure the correctness of these applications, it is essential to meet strict timing requirements as well as functional specifications. The correct design of such MC systems requires a thorough understanding of the system’s functions and their importance to... 

We prove that if an n-dimensional complete minimal submanifold M in hyperbolic space has sufficiently small total scalar curvature then M has only one end. We also prove that for such M there exist no nontrivial L^2 harmonic 1-forms on M