Sharif Digital Repository

In this essay we focus on the question of whether a 3-manifold (possibly with bound- ary) like M supports a codimension-1 transversely oriented foliation like F , such that F is transverse to ∂M and does not have Reeb components. If such foliation exists, then ∂M is necessarily a (possibly
empty) union of tori and (based on the works of Novikov, Reeb and Rosenberg) M is either S2 × S1(with F being the prod- uct foliation) or M is irreducible. In a paper by David Gabai, which we discuss, the sufficiency of these conditions is proved in case of non-trivial second homology group.Furthermore, from the works of Thurston it is concluded that compact leaves of such foliation are norm minimizing... 

Given a principal bundle a characteristic class is an element of the cohomology al- gebra of the classifying space of structure group of the bundle with coefficients in a commutative ring with unit ,which have a functorial property .When the structure groupe of the bundle is a Lie group and the coefficient ring is the real or complex num- bers , the Chern–Weil approach provides a geometric construction of char-acteristic classes. Classical Chern–Weil formalism relates geometry to topology, assigning to the cur- vature of a connection, de Rham cohomology classes of the underlying manifold.This theory developped in the 40’s by Shiing-ShenChern and Andre Weil which can be seen as a generalisation... 

In this thesis, the aim is to stating some of the noncommutative geometry(NCG) applications in formulating and describing the gravity. So, in the first chapter, we will work on some of metric aspects of NCG. In this chapter, we will review elements of the spin geometry, specifically we will define the Dirac operator which is the most important object in this thesis. Then, we will try to find counterparts of some of Riemannian geometric notions like integral and distance in the NCG framework. We will define some notions such as noncommutative infinitesimals, Dixmier trace, Wodzicki residue, noncommutative integral and spectral triples. Then, in the second chapter, we will work on some of the... 

In the present study, the Ricci curvature is defined on the graphs by using two methods and the lower bounds are obtained. In the first method, which is related to Bakry and Emery, a definition for Ricci curvature and lower bounds would be obtained by using Curvature-Dimension inequalities.The second method which is related to Ollivier, is based on the fact that in Riemannian Geometry,the Ricci Curvature controls the velocity of convergence and divergence of emanated geodesics from a same source. Next, the more Curvature-Dimension inequalities for Ricci curvature would be obtained by using local clustering. Finally, the Bonnet-Myers theorem would be proved on the graphs.
 

In this thesis, a discrete exterior calculus is studied. Discrete exterior calculus is an attempt to create a discrete analogy of differential geometry and topology which is potentially applicable in computer science and physics. At first, simplicial complexes are presented as approximation of smooth manifolds. Then discrete forms and discrete exterior derivative are defined. We will observe that Stokes, theorem can be stated and proved in discrete analogy too. We will obtain Hodge operator by using dual complex and after that discrete Laplacian and Heat kernel will be calculated by using Hodge operator. In addition, we will define discrete vector field independent of discrete forms and then... 

Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising model, self-avoiding polymers, . . .This has led to numerous exact (but non-rigorous) predictions of their scaling exponents and dimensions. We will discuss how to prove the conformal invariance conjectures, especially in relation to Schramm-Loewner Evolution  

One of the most important applications of artificial intelligence, and especially deep learning is image captioning. Given an image, the task is to automatically produce a sentence, describing the image. Image captioning has several real world applications like helping the blind understanding the images, generating automatic captions for the social media, etc. In the past, several different methods for image captioning have been used, but after the emergence of deep learning, like many other areas, image captioning algorithms have been improved significantly. In this thesis, I talk about a specific method for image captioning, called ”Deep Compositional Captioning. In this method, at first... 

In this thesis, we first define the pseudo-distance p on the group of Hamiltonian diffeomorphisms.Using the concept of displacement energy, we show that the pseudo-distance p is degenerate and if the manifold is closed, p will be zero for each p = 1; 2; 3; : : : Then, we introduce Lagrangian submanifolds and prove that if L R2n is a rational Lagrangian submanifold, we have the following inequality e(L) ≥ 1γ(L). : Finally, using the above inequality and the concept of displacement energy, for M = R2n we prove that 1 is non-degenerate. Therefore, the hypothesis for 1 to be a metric, are satisfied. This metric is called Hofer’s metric  

Similar to the classical Poincare-Hopf index formula for the Euler characteristic, Zhang gave a counting formula for the Kervaire semi-characteristic of a 4q+1 manifold. We first study Witten deformation techniques in the proof of the Poincare-Hopf formula. Then we will represent Zhang’s formula which uses similar analysis. Both these formulas establish a connection between topological and geometrical data on a manifold. Both proofs will be analytical. Index theory will play the ’bridge’ between the topology and analysis. We begin with an analytic interpretation of the topologically defined quantities as indices of differential operators. We deform the operators with respect to appropriate... 

Hyperbolic branching Brownian motion is a branching diffusion process in which individual particles follow independent Brownian paths in the hyperbolic plane H2, and undergo binary fission(s) at rate λ > 0. It is shown that there is a phase transition in λ : For λ ≤ 1/8 the number
of particles in any compact region of H2 is eventually 0, w.p.1, but for λ > 1/8 the number of particles in any open set grows to ∞ w.p.1. In the subcritical case (λ ≤ 1/8) the set Λ of all limit points in ∂H2 (the boundary circle at ∞) of particle trails is a Cantor set, while in the supercritical case (λ > 1/8) the set Λ has full Lebesgue measure. For λ ≤ 1/8 it is shown that w.p.1 the Hausdorff dimension of Λ... 

Non-overlapping Cameras in multi-camera systems have become prevalent in robotics and computer vision research; therefore, it is possible to cover the wide field of view, and researches have been done for computing extrinsic parameters of cameras. These cameras do not have any overlap in their views, so obtaining the corresponding point in their images is somehow impossible. Light and shadow geometry is analogous to Structure from Motion problem. In this thesis,we study Structure from Motion problem and have tried to propose an approach for estimating extrinsic parameters of non-overlapping cameras in Multi-camera systems. We formulate the problem by using light and shadow geometry and... 

Fatigue life prediction of a mechanical part is one of issues which a group of engineers are engaged with it and always they try to design the parts with the maximum of lifetime. Although many researches have been done in this field but yet we can see that predicted life are different from that happens in the reality because there are some uncertainties in the phenomena. Our effort in this project is creating an algorithm design so that the parts are designed by it, have the maximum fatigue life and the minimum uncertainty in prediction. In this project we have considered geometrical, material and random loading uncertainties as error resources. Older methods those are presented in this... 

In this thesis we are going to explore a new formalism in micro domain which describes quantum phenomena in a realistic casual fashion by introducing an unified concept of matter, energy, and Information. In this theory, a micro-entity, e.g. an electron, is a combination of a particle and its associated field integrated to a new entity called particle-field. The dynamics of such system will be analyzed. Then, some quantum examples which have controversial aspects will be discussed within the new theory. In a generalized form of formalism, the non-linear Schrödinger Equation is introduced, which enables us to predict some experimental data for common Quantum Mechanical stationary systems....