Sharif Digital Repository

Markov chains are memoryless stochastic processes that undergoes transitions from one state to another state on a state space having the property that, given the present,the future is conditionally independent of the past. Under general conditions, the markov chain has a stationary distribution and the probability distribution of the markov chain, independent of the staring state, converges to it’s stationary distribution.
We use this fact to construct markov chain monte carlo, which are a class of algorithms for sampling from probability distributions based on constructing a markov chain that has the desired distribution as its stationary distribution. The state of a chain after a large... 

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  

With Cramer’s model we have a probability measure on the power set of N. This probability measure is concentrated on the set that its elements are that subsets of N which number of their elements up to a certain natural number is asymptotically equal with the number of primes up to the same number. Let Pc be a sample obtained from this probability measure and consider 8n 2 N, an counts the number of ways that ncan be represented as a multiplication of some elements of Pc, such that changing the arrangement of factors in a representation does not introduce a new one. In this thesis, we prove that limn!1 a1++an n almost surely exists and is positive  

Given two stationary random measures on Rd, we study transport kernels between them that are translation-covariant. We will provide an algorithm that constructs such a transport kernel, with some assumption on their intensities. As a result, this algorithm can be used to construct the Palm version of an ergodic random measure by simply applying a (random) translation and vice versa, to reconstruct the distribution of the random measure from its Palm distribution. Given realizations of the two random measures, our algorithm provides its result in a deterministic way. The existence of such transport kernels is proved in [18] in an abstract way. Nevertheless, there has been tremendous interest... 

We can note to at least four very different interpretations of probability that have become popular in the twentieth century. These interpretations are as follows:1. Logical theory: According to this theory, the probability of each event will be equal to the degree of rational belief in that event. In addition, these rational beliefs are assumed to be equal for each individual in the same situation. This is the difference between logical theory and subjective theory.2. Subjective theory: as mentioned, in Subjective theory like Logical theory, probability of an event equals to degree of rational belief in that event, except that Subjective theory allows the differences between the beliefs... 

Why firms does not submit several steps in their bids curve? In this paper, we present a mathematical model for the behavior of power plants in a Pay-as-Bid auction market. According to this model, in the case that the power plant has a fixed marginal cost, the optimal behavior for him is to offer only one price, but in the case of linear marginal cost, bidding one price is not an optimal price, and with increasing the slope of this function, the number of steps and their price level will increase. We also show that increasing the level of competition in the market will force the plants to lower the price level and the number of steps. The model of this paper also predicts the factors... 

Suppose that A is a large subset of N. It is interesting to think about the arithmetic progressions in A.In 1936, Erdos and Turan conjectured that for > 0 and k 2 N, there exists N = N(k; ) that for all subsets A {1; 2; : : : ;N}, if lAl N, A has a nontrivial arithmetic progression of length k. Roth proved the conjecture for k = 3 in 1953. In 1969, Szemeredi proved the case k = 4 and in 1975, he gave a combinatorial proof for the general case. In 1977, using ergodic theory, Furstenberg gave a different proof for the Erdos-Turan conjecture (or Szemeredi Theorem!) and finally Gowers found another proof for the Szemeredi theorem, which was an elegant generalization of the Roth’s proof for k =... 

The purpose of this dissertation is to study issues in the field of graphical models.At the beginning, we will mention the main concepts of graphical models. Then we describe algorithms in exact inference. These algorithms are used to solve inferential issues and when the graph is related to the tree graph modeling. We also describe how these algorithms apply to non-tree graphs. In addition, we recall definitions such as cumulative function and set of mean parameters and important theorems applied in graphical models. Finally, we describe the important algorithms that are used to estimate the parameters in graphical models  

Deep Neural Networks, are predictive models in Machine Learning, that during the last decade they've had a great success. However being in an over-parametrized and highly non-convex regime, the analytical examinations of these models is quite a challenging task to do. The empirical developments of Neural Networks, and their distinguishing performance in prediction problems, has motivated researchers, to formalize a theoretical foundations for these models and provide us with a framework, in which one can explain and justify their behavior and properties. this framework is of great importance because it would help us to come to a better understanding of how these models work and also enables... 

Cancer can be defined as a stochastic phenomenon. Thus, the tumor growth can be defined as a stochastic process. A Cancer tumor can be analyzed by its geometric shape. Furthermore, computing fractal dimension of shapes is a useful technique to analyze chaosity and complexity of shapes. This approach can be used to compare results from laboratory with simulated results of a mathematical model. A stochastic model of tumor growth will be presented. And some geometrical properties will be analyzed through computer simulation of the model  

The problem of learning using high volume of data as stream, has attracted much attention recently. In this thesis, the problem is modeled and analized using Online Convex Optimization tools [1], [2]. General performance bounds are stated and clarified in this framework [8]. Using the practical experience in Online Decision Making (e.g., predicting price in Stock Market), the need for a more flexible model, which adapts to changes in problem, is presented. In this thesis, after reviewing the literature and online convex optimization framework, we will define ”Concept Drift”, which describes changes in the dynamics of the problem and the statistical tools to detect it [13], [5]. And finally,... 

Random Polytopes, the first occurrence of which dates back to the famous Sylvester’s four points problem in the 1860s, is a branch of geometric probability, typically concerning the convex hull of some random points chosen from a convex subset of Rd. In this thesis we have studied some special kind of random polytopes; the one that is the convex hull of some independent random points chosen from a convex body (a convex, compact set with interior point) according to the uniform distribution. It was a new approach from A. Rényi and R. Sulanke in 1963 to consider this type when the number of random points tends to infinity.This thesis consists of three main parts: The first part is devoted to... 

“How can we generate a random point with uniform distribution over a convex body ?” According to it’s applications, it’s important for a solution to this problem to be applicable in high dimensions. Here, we are interested in algorithms with polynomial order with respect to the dimension. All existing methods for dealing with this problem are based on the Markov chain Monte Carlo method, i.e. a random walk is constructed in such that its stationary distribution is the uniform distribution over. Then, after simulating “enough” steps of this random walk, the distribution of the resulting point is “approximately” uniform. The real problem in Monte Carlo method is analyzing its “mixing time”,... 

The spherical ensemble is a well-studied determinantal process with a fixed number of points on the sphere. The points of this process correspond to the generalized eigenvalues of two appropriately chosen random matrices, mapped to the surface of the sphere by stereographic projection. This model can be considered as a spherical analogue for other random matrix models on the unit circle and complex plane such as the circular unitary ensemble or the Ginibre ensemble, and is one of the most natural constructions of a (statistically) rotation invariant point process with repelling property on the sphere. In this dissertation we study the spherical ensemble and its local repelling property by... 

Determinantal processes are a special family of stochastic processes that arise in physics (fermions), random matrices (eigenvalues), and in combinatorics (random spanning trees and non-intersecting paths). These processes have repelling property (points close to each other are chosen with low probability). Because of this repelling property, determinantal processes are approporiat for modeling some physical quantities (e.g. the position of electrons). Their probabilistic structure is described by operators on complex vector spaces and their eigenvalues. Determinantal processes have interesting properties, e.g. number of points in a region is a sum of independent Bernoulli random variables.... 

Application of optimal control in cancer modeling is studied through both linear and nonlinear modeling of the dynamics in ordinary differential equations. At the outset, a fairly straight-forward analysis of a linear model in presented. Through comparably simple machinery, this seminal work published at early 2000s covers some of most important techniques previously developed. The model here is infinite- dimensional, taking different number of gene amplifications into account. Thereafter by surveying recently published papers, the literature is reviewed and different lines of progress is followed, culminating in detailed study of a specific approach which is theoretically of interest.... 

The focus of this thesis is the computation of the discrepancy for any given distribution of points on S2. The problem of the distribution of points on the sphere has a long history and Thomson’s Problem, inspired by early atomic theory dating back to 1904, was a landmark. While Thomson’s Problem is based on the Coulomb potential, the discrepancy measures the deviation of the number of points in a set from the expected one. The Polar Coordinates method was introduced in the context of Thomson’s problem. In this thesis the order of the growth of the discrepancy for this method is investigated and a modification of it is shown to lead to the best known results. In addition a new algorithm is... 

Percolation is a simple probabilistic model which exhibits a phase transition. Here, we study this critical model from properties of random curves which in the scaling limit, appear as features seen on the macroscopic scale, in situations where the microscopic scale is taken to zero. Among the principal questions are the construction of the scaling limit, and the discription of some of the emergent properties, in particular the behavior under conformal maps Over the past few years, SLE has been developed as a valuable new tool to study the random paths of the scaling limit of two-dimensional critical models, and it is believed that SLE is the conformally invariant scaling limit of these... 

In this thesis, we are going to prove the fundamental theorem of asset pricing and then define option and use the binomial option pricing model to for pricing the option. Afterwards, we explain the recovery theorem which gives the relationship between real and risk neutral measure. Moreover, we present an introduction to financial mathematics and state the generalized Black-Scholes model for option pricing.Then, we prove a theorem for options which reveals a relationship between option prices and real measure  

This thesis is devoted to introduce coherent risk measures on general probability spaces. After studying their properties, we also will characterize them using functional analysis tools. First we describe some related economic concepts such as risk concept, risk management and risk measures. Then we will study Value at Risk (VaR) as an applicable risk measure and determine its advantages and disadvantages. The motivation for studying risk measures in an axiomatic point of view and also introducing coherent risk measures was that VaR doesn’t have the diversification property. In chapter 2 and 3, we introduced coherent risk measures comprehensively. We began the second chapter by the...