Department of Mathematics
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Item Martingale Hardy-Amalgam Spaces(University of Ghana, 2022-07) Bansah, J.S.In this work, we introduce the new spaces, Hs p;q; HS p;q; H_ p;q; Qp;q; Pp;q; called the martingale Hardy-amalgam spaces. We study some of the properties of these newly introduced spaces; two de_nitions of atoms are given and hence two atomic decompositions are given, dualities of these spaces are characterized and the martingale inequalities and embeddings of these spaces are also discussed. It is proved that the dual of Hs p;q; (0 < p _ q _ 1); is a Campanato-type space and the dual of Hs p;q; (1 < p _ q < 1); is Hs p0;q0 where (p; p0); (q; q0) are conjugate pairs. The variation integrable space Gp;q is also introduced and it is established that the jump bounded space BDp;q is the dual of Gp;q: To be able to characterize this duality, a larger space, which we denote by K(Lp;q; `r); is introduced, such that Gp;q can be embedded into. The classical Doob's martingale inequality is also extended from the classical martingale Hardy spaces to the newly introduced martingale Hardy-amalgam spaces. The Burkholder-Davis-Gundy inequality is also extended from the classical martingale Hardy spaces to the martingale Hardy-amalgam spaces as well as the convexity inequality and the concavity inequalities involving measurable functions. The classical martingale Hardy space embeddings are also extended to the martingale Hardy-amalgam spaces. The Davis decompositions of martingales in the classical martingale Hardy spaces are also extended to the martingale Hardy-amalgam spaces. As an application of the Davis decomposition and the Garsia space, a duality theorem for H_ p;q (1 _ p; q _ 2) is provided. Finally, the boundedness of martingale transforms between the martingale Hardy-amalgam spaces are also discussed. No data was collected for this study as the methodology used is purely theoretical in nature.Item Iterative Methods for Large Scale Convex Optimization(University of Ghana, 2017-07) Katsekpor, T.This thesis presents a detailed description and analysis of Bregman’s iterative method for convex programming with linear constraints. Row and block action methods for large scale problems are adopted for convex feasibility problems. This motivates Bregman type methods for optimization. A new simultaneous version of the Bregman’s method for the optimization of Bregman function subject to linear constraints is presented and an extension of the method and its application to solving convex optimization problems is also made. Closed-form formulae are known for Bregman’s method for the particular cases of entropy maximization like Shannon and Burg’s entropies. The algorithms such as the Multiplicative Algebraic Reconstruction Technique (MART) and the related methods use closed-form formulae in their iterations. We present a generalization of these closed-form formulae of Bregman’s method when the objective function variables are separated and analyze its convergence. We also analyze the algorithm MART when the problem is inconsistent and give some convergence results.Item On The Existence of Prime Numbers in Polynomial Sequences, And Odd Perfect Numbers(University Of Ghana, 2015-06) Acquaah, P.; Adu-Gyam, D.; McIntyre, M.; University Of Ghana, College of Basic and Applied Sciences, School of Physical and Mathematical Sciences, Department of MathematicsIt is known that certain polynomials of degree one, with integer coefficients, admit in nitely-many primes. In this thesis, we provide an alternative proof of Dirichlets theorem concerning primes in arithmetic progressions, without applying methods involving Dirichlet characters or the Riemann Zeta func- tion. A more general result concerning multiples of primes in short-intervals is also provided. This thesis also considers problems concerning the existence of odd perfect numbers. The main contribution is a good upper-bound on the largest prime divisor of an odd perfect number. In addition, we show how new results concerning odd perfect numbers or k - perfect numbers can be obtained by applying a property of completely-multiplicative functions.