Abstract It is common to think of our universe according to the “block universe” idea, which says that spacetime consists of many “stacked” 3-surfaces varied as a function of some kind of proper time τ . Standard ideas do not distinguish past and future, but Ellis’ “evolving block universe” tries to make a fundamental distinction. One proposal for this proper time is the proper time measured along the timelike Ricci eigenlines, starting from the big bang. The main idea of this ...
Abstract In this thesis, we develop the 1 + 1 + 2 formalism, a technique originally devised for General Relativity, to treat spherically symmetric spacetimes in for fourth order theories of gravity. Using this formalism, we derive equations for a static and spherically symmetric spacetime for general f(R) gravity. We apply these master eqautions to derive some exact solutions, which are used to gain insight on Birkhoff's theorem in this framework. Additionally, we derive a covariant form of t...
Abstract The non-life insurance pricing consists of establishing a premium or a tariff paid by the insured to the insurance company in exchange for the risk transfer. A key factor in doing that is properly estimating the distribution that the claim and frequency of claim follows. This thesis aim at having a deep knowledge of loss function and their estimation, several concept from Measure Theory, Probability Theory and Statistics were combined in the study of loss function and estimating the...
Let us first introduce some keywords that will enable us to specify our principal objective. Given a nonempty set X and a function f : X → R which is bounded below, computing the number
Abstract Let X be a uniformly convex and uniformly smooth real Banach space with dual space X∗ . Let F : X → X∗ and K : X∗ → X be bounded monotone mappings such that the Hammerstein equation u + KF u = 0 has a solution in X. An explicit iteration sequence is constructed and proved to converge strongly to a solution of the equation. This is achieved by combining geometric properties of uniformly convex and uniformly smooth real Banach spaces recently introduced by Alber with our met...
This project is mainly focused on the theory of Monotone (increasing) Operators and its applications. Monotone operators play an important role in many branches of Mathematics such as Convex Analysis, Optimization Theory, Evolution Equations Theory, Variational Methods and Variational Inequalities.
ABSTRACT The study of variational inequalities frequently deals with a mapping F from a vector space X or a convex subset of X into its dual X 0 . Let H be a real Hilbert space and a(u, v) be a real bilinear form on H. Assume that the linear and continuous mapping A : H −→ H 0 determines a bilinear form via the pairing a(u, v) = hAu, vi. Given K ⊂ H and f ∈ H 0 . Then, Variational inequality(VI) is the problem of finding u ∈ K such that a(u, v − u) ≥ hf, v − ui, for all v ∈ ...
This body of work introduces exterior calculus in Euclidean spaces and subsequently implements classical results from standard Riemannian geometry to analyze certain differential forms on a manifold of reference, which here is a symmetric ellipsoid in R n . We focus on the foundations of the theory of differential forms in a progressive approach to present the relevant classical theorems of Green and Stokes and establish volume (length, area or volume) formulas. Furthermore, we introduce the ...
Abstract This thesis presents an overview on the theory of stopping times, martingales and Brownian motion which are the foundations of stochastic modeling. We started with a detailed study of discrete stopping times and their properties. Next, we reviewed the theory of martingales and saw an application to solving the problem of "extinction of populations". After that, we studied stopping times in the continuous case and finally, we treated extensively the concepts of Brownian motion and th...
ABSTRACT Let E be a 2-uniformly convex and uniformly smooth real Banach space with dual space E ∗ . Let A : C → E ∗ be a monotone and Lipschitz continuous mapping and U : C → C be relatively nonexpansive. An algorithm for approximating the common elements of the set of fixed points of a relatively nonexpansive map U and the set of solutions of a variational inequality problem for the monotone and Lipschitz continuous map A in E is constructed and proved to converge strongly.
ABSTRACT Compact operators are linear operators on Banach spaces that maps bounded set to relatively compact sets. In the case of Hilbert space H it is an extension of the concept of matrix acting on a finite dimensional vector space. In Hilbert space, compact operators are the closure of the finite rank operators in the topology induced by the operator norm. In general, operators on infinite dimensional spaces feature properties that do not appear in the finite dimension case; i.e matrices. ...
The scope of Quadratic Form Theory is historically wide although it usually appears almost as an afterthought when needed to solve a variety of problems such as the classification of Hessian matrices in finite dimensional Calculus [1], [2], [3], the finding of invariants that fully describe the equivalence class of a given form in Algebraic Geometry and Number Theory [4], the use of Rayleigh-Ristz methods for finding eigenvalues of real symmetric matrices in Linear Algebra [5], [6], the secon...
ABSTRACT Algorithms for single-valued and multi-valued nonexpansive-type mappings have continued to attract a lot of attentions because of their remarkable utility and wide applicability in modern mathematics and other reasearch areas,(most notably medical image reconstruction, game theory and market economy). The first part of this thesis presents contributions to some crucial new concepts and techniques for a systematic discussion of questions on algorithms for singlevalued and multi-valued...
The most popular method for studying stability of nonlinear systems is introduced by a Russian Mathematician named Alexander Mikhailovich Lyapunov. His work ”The General Problem of Motion Stability ” published in 1892 includes two methods: Linearization Method, and Direct Method. His work was then introduced by other scientists like Poincare and LaSalle . In chapter one of this work, we focussed on the basic concepts of the ordinary differential equations. Also, we emphasized on relevant ...
ABSTRACT In this thesis, a hybrid extragradient-like iteration algorithm for approximating a common element of the set of solutions of a variational inequality problem for a monotone, k-Lipschitz map and common fixed points of a countable family of relatively nonexpansive maps in a uniformly smooth and 2-uniformly convex real Banach space is introduced. A strong convergence theorem for the sequence generated by this algorithm is proved. The theorem obtained is a significant improvement of the...