Sharif Digital Repository

Cohomology groups of complex algebraic varieties with coefficients in ℂ, can be considered more than just a vector space and they can equipped with various and rich linear algebra structures which are functorial with respect to morphisms between them. In this thesis we first review classic Hodge theory, Hodge decomposition and Lefschetz decomposition theorems which enable us to introduce the concept of pure Hodge structure on cohomology group H^n (X,C) of a compact Kähler manifold X. Then we define Frölicher spectral sequence for a complex manifold X and show that it will degenerate at E_1 when X is compact and Kähler. For generalization of pure Hodge structures to smooth non... 

In 2002 Baskaran and Jafari published an article that was about a spin-1 collective mode. Similarity between carbon nanotubes and graphene was a motivation for working on this project, in order to find another spin mode by considering random phase approximation as a simple way of finding collective motion of a physical systems. Moreover rolling graphene sheet induce diffrent physical properties as a result of chiral axis . In this project we focus only on achiral nanotubes and at last find two collective modes in this structures  

For more than a century, physicists have recognized the contrast between the asymmetry of thermodynamic phenomena and T-symmetry of the fundamental laws, and they have referred to it as the arrow of time. In this thesis we tried to give an overview of the past and present status of this problem, using various theories. It is shown that the problem becomes a question about the boundary conditions of thermodynamic systems, and then a question about cosmology, and finally it becomes the problem of initial conditions of universe. An analysis of initial condition theories is presented for knowing how each of them tries to solve the problem of arrow of time. Different views about the relationship... 

Computational complexity of graph problems is an important branch in the-oretical computer science. We introduce to some of ideas for computing the complexity of graph problems with some kind and beautiful examples. Next, we show hardness and inapproximability of some problems. Representation number of graphs has been introduce by Pavel Erdos by Number theory. We prove n1−ϵ inapproximability of that. Lucky number η has been studied by Grytczuk et.al . We show for planar and 3-colorable graphs, it is NP-Complete whether η = 2. Note that since a conjecture, for those graphs, 2 ≤ η ≤ 3. Also for each k ≥ 2, we show NP-completeness of η ≤ k for the graphs. Proper orientation number −→ is a... 

In this thesis we review several arithmetical questions about the diophantine equation φ[x]=q where φ is a nondegenerate symmetric bilinear form on a finite dimensional vector space, φ[x] is the corresponding quadratic form and q is a nonzero element of the ground field. We obtain class number and mass formulae for the orthogonal group associated to φ in a number field utilizing the concept of quadratic forms, lattices and adelic language  

Jets originating from bottom (b) quarks, play an important role in the study of Standard Model processes together with the search for new physics. In the CMS experiment a considerable e?ort is dedicated to the development and the performance study of so-called b-tagging algorithms. On the other hand, the top quark which is produced in pair (tt¯) with a high rate at the LHC, decays near 99% of the time to a b-quark. Hence it provides a rich source of b-quark jets, suitable for b-jet identi?cation studies. In this thesis, a fully data-driven method to measure the b-tagging e?ciency, using top quark events in its semi-electron ?nal state, is developed. The results of the application of the... 

In this thesis, first we will have a thorough review on problems and models presented in the field of contagion in networks (spread of diseases, spread of computer viruses,etc.). Then we will investigate the basic model of virus inoculation game with purely selfish players and the modified version of it which takes into account a factor of friendship. We will discuss the results of these models including computing and comparing Nash Equilibria  

The Mordell–Weil theorem states that for an abelian variety A over a number field k, the group A(k) of k-rational points of A is a finitely-generated abelian group, called the Mordell-Weil group. The case with A an elliptic curve E and k the rational number field Q is Mordell’s theorem, answering a question apparently posed by Poincare around 1908; it was proved by Louis Mordell in 1922. The tangent-chord process (one form of addition theorem on a cubic curve) had been known as far back as the seventeenth century. The process of infinite descent of Fermat was well known, but Mordell succeeded in establishing the finiteness of the quotient group E(Q)=2E(Q) which forms a major step in the... 

Let G be a simple graph. A harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number, h(G), is the least number of colors in such a coloring. In this project, first we discuss about complexity of harmonious coloring. Then we find some bounds for harmonious chromatic number  

Nurse scheduling problem has received a great amount of attention during recent years. In this problem number of needed nurses from different skill levels is given for each Location-shift during planning horizon, and the goal is to allocate each nurse to shifts, such that the demand of each location-shift is satisfied. Many researches have been done with the aim of minimizing the number of needed nurses to satisfy the demand of locationshifts or maximizing nurses preferences to work in desired shifts. In this research an integer programming model is proposed to minimize the number of needed nurses and maximize nurses preferences simultaneously. The proposed mathematical model is appropriate... 

This project aims to study a situation in which several projects must be selected. The decision makers should select a subset of these projects taking into account the limitation of resources and scheduling. Project selection models generally do not consider schedule of project activities as part of selection process. On the other hand, except the cases where only one project is under process in each period, prioritizing the selected projects is not optimal not considering their scheduling. Projects scheduling in project activities level may increase the complexity of decision making for portfolio selection and expand the area of search for selection of projects portfolio. Therefore, all the... 

We consider a network creation game in which, each player(vertex) has bounded budget to draw edges to other vertices. Each player wants to minimize its local diameter in MAX version, and its sum of distances to other vertices in SUM version.
In this thesis, we first prove that the problem of finding the best response for each player in this network creation game, is NP-Hard. But we show that always there exists a Pure Nash Equilibria for each of the versions. Then we focus on the diameter of equilibrium graphs. In Unit Budget case (when each player’s budget is one) we show that the diameter of each equilibrium graph in O(1), and always there exists a Mixed Nash Equilibirum. For each n,... 

Numerical study of the interaction mechanism of the premixed flame and the acoustic field is performed in this work. The geometry similar to those used in experiments, is a two-dimensional tube in which flame travels form open end towards the closed end. The numerical study is done using PPM method. Results show that the shape and speed of laminar flame is strongly affected by the induced acoustic field. Stronger interaction is observed in the closed-end tube with no slip boundary conditions. The effect of the Lewis number on the interaction mechanism is also investigated. Premixed flame- accelerated flame – interaction of flame and acoustic field  

” The story of the "Weil conjectures" is a marvelous example of mathematical imagination, and one of the most striking instances exhibiting the fundamental unity of mathematics.”In 1949,Andre Weil stated some conjectures on the zeta function of Algebraic varieties over finite fields .These conjectures were analogue of the properties of Riemann zeta function ,in particular Riemann hypothesis.In fact ,Weil built a bridge between Diophantine structure on varieties over finite fields (Counting of rational points on varieties) and cohomological structure of them over the field of complex numbers(topology of variety).In this thesis, first we state Weil’s motivations for these conjectures and state... 

This thesis is about one of the newest attractive theories in mathematics and theoretical physics; Topological Quantum Field Theory (TQFT) has some basic algebraic requirements called tensor categories, after introducing them and describing axioms of the theory which has suggested by Atiyah in 1988, we will see that the category of 2d-TQFT is equivalent to the category of commutative frobenius algebras over an arbitrary field .In the last part we focus on some topological aspects of theory in dimension 3 and some invariants of 3d manifolds , we will see these invariants are the main tool for construction a 3d-TQFT  

Blood is a perishable product and can only be stored for a limited time. Reducing the level of wastage and shortage in supply chain of blood products is the main challenge of supply chain management. Wastage occurs whenever hospitals and blood banks order excessive blood while ordering insufficient amount results in shortage. Wastage can increase the costs in supply chain while shortage can cause important operation to be cancelled.This thesis presents an integer programming model for improving blood related operations in hospitals and blood banks. The presented model determines optimal order levels for each period which minimize total cost, shortage and wastage levels. The model is... 

In this thesis we have been mainly interested in studying Single Impurity Anderson model (SIAM) in Dirac fermions. Although this model seems to be simple, the rich physics of SIAM can not be underestimated. In this thesis we were interested in Dirac materials, systems which their low energy excitations are described by Dirac equation.Therefore, considering Dirac material in two dimensions(2D), we briefly review the distinct features of one atom thick layer of carbon. In addition, we will explain the effective Hamiltonian of Bismuth near L point which is considered a three dimensional(3D) Dirac material. Because of very strong spin-orbit interaction, bismuth is a key element in topological...