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Item New Existence Results on Group Divisible Designs and 3-GDDs(Addis Ababa University, 2024-06) Zebene Girma; Dinesh G. Sarvate; Samuel Asefa (Professor)Combinatorial design theory offers a framework for efficiently and systematically arranging objects with given criterion and studying these arrangement (combinatorial designs) and their applications is various fields. For example, combinatorial designs are used in statistical design theory, computer science, cryptography and coding theory. Two specific combinatorial designs are group divisible designs (GDDs) and t-GDDs. The aim of our study is to find new existence results on GDDs and t-GDDs when t = 3. In particular, we gave some new constructions and necessary conditions for GDDs with two groups and block size four. An extension of the concepts of group divisible designs is called t-GDD. It is obtained by combining the concepts of group divisible designs and t-design. So for example, a 3-GDD(n, 2, k, λ1, λ2) is defined by combining the concepts of GDDs and a 3-design. 3-GDDs with 2 groups and block size 4 have been studied in the literature. We obtain new results on 3-GDDs with three groups and block size five, as well as with four groups and block size five. In this work, certain required necessary conditions for these GDDs to exist are designed, several new constructions are given, and specific instances of non-existence are proved.Item New Existence Results on Group Divisible Designs and 3-GDDs(Addis Ababa University, 2024-06) Zebene Girma; Dinesh G. Sarvate; Samuel AssefaCombinatorial design theory offers a framework for efficiently and systematically arranging objects with given criterion and studying these arrangement (combinatorial designs) and their applications is various fields. For example, combinatorial designs are used in statistical design theory, computer science, cryptography and coding theory. Two specific combinatorial designs are group divisible designs (GDDs) and t-GDDs. The aim of our study is to find new existence results on GDDs and t-GDDs when t = 3. In particular, we gave some new constructions and necessary conditions for GDDs with two groups and block size four. An extension of the concepts of group divisible designs is called t-GDD. It is obtained by combining the concepts of group divisible designs and t-design. So for example, a 3-GDD(n, 2, k, λ1, λ2) is defined by combining the concepts of GDDs and a 3-design. 3-GDDs with 2 groups and block size 4 have been studied in the literature. We obtain new results on 3-GDDs with three groups and block size five, as well as with four groups and block size five. In this work, certain required necessary conditions for these GDDs to exist are designed, several new constructions are given, and specific instances of non-existence are proved.Item On the Method of Green's Functions(Addis Ababa University, 2024-06) Lechisa Gashew; Tadesse AbdiIn this thesis, the construction of Green's function for IVP and BVP is discussed. We established continuity and symmetric properties of Green's function for ODEs, especially for Sturm-Liouville problems. We demonstrated the method of Green's function in conjunction with the Fourier transform to solve PDEs.Item Sturm-Liouville Boundary Value Problems and General Solutions to 2D Liouville Equations(Addis Ababa University, 2024-07-24) Icon Abebe; Tesfa BisetThe thesis focuses on Sturm-Liouville equations, boundary value problems, and the general solutions to 2D Liouville Equations. It covers computations of eigenvalues and eigenfunctions, types of Sturm-Liouville boundary value problems, solving methods and solutions, inner products of eigenfunctions, and orthogonality of eigenfunctions with weighted functions. It also covers applications of Strum-Liouville Boundary Value problems and ways to solve them. Additionally, the thesis covers several characteristics of complex-valued functions and derivations emphasizing the significance of the Liouville Equation in resolving Elliptic as well as Hyperbolic 2D differential equations. Furthermore, it provides a short but detailed analysis aimed at enhancing the understanding and application of SLP, even at its most complex levels in science and engineering, thereby promoting general solution techniques for elaborate mathematical problems.Item Further More on the theory of BH-Lattices(Addis Ababa University, 2024-08) Mekonnen Mamo; Kolluru VenkateswarluIn this dissertation, we study further properties of BH-lattices, which is a subclass of BH-monoids. We furnish certain examples of BH-monoids that are not BH-lattices. We give a characterization of BH-lattice in terms of bounded BH-lattice and commutative l-group. Also, we prove that every BH-lattice is a direct product of Heyting algebra and commutative l-group under certain conditions. Further, we obtain the decomposition theorem in terms of Boolean algebra and a commutative l-group. Moreover, we introduce the concept of filters in BH-lattices and furnish certain examples. We obtain certain basic properties of BH-lattices. Also, we characterize the filter generated by a given subset of a BH-lattice. Besides these, we prove that the set of all filters with set inclusion forms a Heyting algebra Furthermore, we define the congruence relation on BH-lattices and obtain a one-toone correspondence between the set of congruences and the filter of BH-lattices, which gives more insight for constructing quotient algebra. Also, we prove that the quotient algebra is a BH-lattice. Finally, we introduce different types of filters in BH-lattices, furnish examples, and prove certain properties of each type of filter, their interrelation, and state some open problems for further study in the area.Item On the Structural Properties of Ordered Weak Idempotent Rings(Addis Ababa University, 2024-08) Tamiru Abera; Kolluru Venkateswarlu; Yibeltal YitayewThe notion of weak idempotent rings is a new concept and has recently been introduced. It is a ring R of characteristic two, and a4 = a2 for each a in R. Very few studies have been done on its structural development and the structure of submaximal ideals of weak idempotent rings. So, there are a lot of gaps in its structural development. One of these gaps is the introduction of ordering in it. In this dissertation, we introduce the concept of partial ordering in a commutative weak idempotent ring R with unity and prove that the introduced partial ordering coincides with the one in Boolean rings whenever it is restricted to the idempotent parts of R. Besides, we introduce the concept of an atom in R and obtain certain results concerning an atom. Further, we study the structure of primary submaximal ideals. To achieve this, we de_ne partial ordering and an atom in a commutative weak idempotent ring with unity and use the concepts to obtain various results. Here we introduced the notion of partial ordering in a commutative weak idempotent ring R with unity, studied the role of an atom in the direct product decomposition of R and further on the structure of primary submaximal ideals of R. We recommend that researchers further study these in detail, shortly prove theorem 4.2.11, and extend the concepts to non-commutative weak idempotent rings with or without unity, and study its potential application in the real world.Item Varational Formulation of Elliptic Partial Differential Equation(Addis Ababa University, 2024-07) Ali Eshetu; Tsegaye GedifThe focus of this thesis is to examine weak or variational formulations of various elliptic boundary value problems and determine whether or not they are well-posed. The weak version of the homogeneous Dirichlet boundary value problem for the Poisson equation is first derived. A weak formulation can be thought of as an operator equation in its abstract form. Additionally, we offer some broad conclusions on the existence and uniqueness of linear operator equations.Item Colombeau Generalized Functions(Addis Ababa University, 2024-08-12) Tesfaye Kassa; Tsegaye GedifSome outcomes regarding products of distributions are presented in this thesis. The results are obtained in the most applicable algebra for nonlinear problems related to Schwartz distributions, which is the Colombeau algebra of generalized functions.Item Critical Curve and Singular Zeros of Harmonic Quadrinomial(Addis Ababa University, 2024-10) Leyla Abdulkadir; Hunduma LegesseIn this thesis, we show that the critical curve of the quadrinomial Pb(z) = bzk + 1 k1 zk1 + 1 n zn + kb n1 zn1 is the part of the unit circle. Moreover, we prove that there are at most two singular zeros for the quadrinomial Pb(z) = bzk+ 1 k1 zk1+ 1 n zn+ kb n1 zn1 we are considering.Item Coefficients Estimate for Log-Harmonic Mappings(Addis Ababa University, 2024-09) Namomsa Tafasa; Hunduma LegeseThis thesis explores coe_cient estimates for close-to-starlike log-harmonic mappings, On the unit disk, a subclass of univalent log-harmonic functions is de_ned..The results contribute to the understanding of log-harmonic map-pings and propose a Log-Harmonic Coe_cient Conjecture, which parallels classical conjectures in complex analysis. This work lays the foundation for future research aimed at further exploring and validating these conjec- tures and their broader implications. Topics discussed include log-harmonic polynomials, subclasses of log-harmonic mappings, we _rst give a general un- derstanding of how to construct log-harmonic Koebe mappings. additionally, For some special subclasses of log-harmonic mappings, growth and distortion theorems are examined.Item On Reduced Submodules of Finite Dimensional Modules and a Generalization of Torsion Functor(Addis Ababa University, 2025-03-10) Teklemichael Worku; David SsevviiriLet k be a field with characteristic zero, R be the ring k[x1, ・ ・ ・ , xn] and P be a monomial ideal of R. We study the Artinian local algebra RP when considered as an R-module M. We show that the largest reduced submodule of M,R(M), coincides with both the socle of M and the k-submodule of M generated by all outside corner elements of the Young diagram associated with M. we further study properties of reduced submodules, in particular R(M). Let R be an associative Noetherian unital noncommutative ring. We introduce the functor PΓP over the category of R-modules and use it to characterize P-semiprime. We also show that the Greenless-May type Duality (GM) and Matlis Greenless-May Equality(MGM) hold over the full subcategory of R-Mod consisting of P-semiprime and P-semisecond modules. Finally, we generate a one-sided right ideal PΓP (R), which gives an equivalent formulation to solve K¨othe conjecture.Item Thesis on Mathematical Analysis of the Transmission Dynamics of Norovirus Infection Model(Addis Ababa University, 2024-08-30) Desalegn Kitil; Manalebish DebalikeThe mathematical modeling of the transmission dynamics of norovirus in a population is designed and rigorously analyzed using some dynamical system theories and techniques. The model was used to gain insight into the disease dynamics and to evaluate control strategies of the norovirustransmission . The model has both global asymptotic stability and local asymptotic stability whenever the reproduction number is less than one. Numerical simulations of the model show that the use of basic control measure strategy is more effective than the treatment strategy while the combination of the two control strategies has a good effective impact on the community in controlling the transmission of the virus .Item Solution Methods of Monotone Inclusion Problems, Equilibrium Problems and Fixed Point Problems in Banach Spaces(Addis Ababa University, 2024-09) Solomon Bekele; Mengistu Goa; Habtu Zegeye; Sebsibe TeferiIn this dissertation, we discuss four fixed point approximation methods that can be applied to solve optimization problems, differential equations, variational inequalities and equilibrium problems. In the first main result of the dissertation, we propose an inertial algorithm for solving split equality of monotone inclusion and fixed point of Bregman relatively f-nonexpansive mapping problems in reflexive real Banach spaces and established strong convergence theorems for the algorithm. Secondly, we establish a strong convergence theorem for approximating a common element of sets of solutions of a finite family of generalized mixed equilibrium problem, sets of semi-fixed points of a finite family of continuous semi-pseudocontractive mappings and sets of solutions of a finite family of variational inequality for a finite family of monotone and L-Lipschitz mappings in Banach spaces. Thirdly, we constructed and proved a strong convergence of an algorithm for approximating a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of f-fixed points of a finite family of f-pseudocontractive mappings and the set of solutions of a finite family of variational inequality problems for Lipschitz monotone mappings in real reflexive Banach spaces. In the fourth main result of the dissertation, we introduce an iterative process which converges strongly to a common point of sets of solutions of a finite family of generalized equilibrium problems, fixed points of a finite family of continuous asymptotically quasi-ϕ-nonexpansive mapping in intermediate sense, and zeros of a finite family of γ-inverse strongly monotone operators in uniformly convex and uniformly smooth real Banach space. We give numerical examples to demonstrate the behavior of the convergence of the algorithms analysed in each of the four main results of the thesis.Item Solution Methods of Monotone Inclusion Problems, Equilibrium Problems and Fixed Point Problems in Banach Spaces(Addis Ababa University, 2024-09) Solomon Bekele; Mengistu Goa; Habtu Zegeye; Sebsibe TeferiIn this dissertation, we discuss four fixed point approximation methods that can be applied to solve optimization problems, differential equations, variational inequalities and equilibrium problems. In the first main result of the dissertation, we propose an inertial algorithm for solving split equality of monotone inclusion and fixed point of Bregman relatively f-nonexpansive mapping problems in reflexive real Banach spaces and established strong convergence theorems for the algorithm. Secondly, we establish a strong convergence theorem for approximating a common element of sets of solutions of a finite family of generalized mixed equilibrium problem, sets of semi-fixed points of a finite family of continuous semi-pseudocontractive mappings and sets of solutions of a finite family of variational inequality for a finite family of monotone and L-Lipschitz mappings in Banach spaces. Thirdly, we constructed and proved a strong convergence of an algorithm for approximating a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of f-fixed points of a finite family of f-pseudocontractive mappings and the set of solutions of a finite family of variational inequality problems for Lipschitz monotone mappings in real reflexive Banach spaces. In the fourth main result of the dissertation, we introduce an iterative process which converges strongly to a common point of sets of solutions of a finite family of generalized equilibrium problems, fixed points of a finite family of continuous asymptotically quasi-ϕ-nonexpansive mapping in intermediate sense, and zeros of a finite family of γ-inverse strongly monotone operators in uniformly convex and uniformly smooth real Banach space. We give numerical examples to demonstrate the behavior of the convergence of the algorithms analysed in each of the four main results of the thesis.Item Autometrized Lattice Ordered Monoids(Addis Ababa University, 2025-06) Tekalign Regasa; Kolluru VenkateswarluIn this dissertation, we introduce autometrized lattice ordered monoids (AL-monoids), a new generalization of dually residuated lattice ordered semigroups (DRl-semigroups). We explore various algebraic properties of AL-monoids and investigate isometries within this structure. A key finding is that the set of invertible elements in an AL-monoid forms an l-group, while the set of complemented elements constitutes a Boolean algebra. We further establish that an AL-monoid A with a unity element is a Boolean algebra if the mapping x 7→ a ∗ x is an isometry for every a ∈ A. The geometric aspects of AL-monoids are examined through the introduction of metric betweenness and related concepts, including B-linearity, D-linearity, lattice betweenness, segments, and equilateral triangles. Notably, we prove that equilateral triangles cannot exist in AL-monoids, subsume the geometric properties of commutative DRlsemigroups. Moreover, we define various types of ideals within AL-monoids, such as polar ideals, regular ideals, prime ideals, and annihilators, and elucidate their interconnections. We introduce the value of an element for an ideal I(A) and characterize regular ideals based on this value. Finally, we provide characterizations of polar ideals concerning minimal prime ideals, annihilators, and maximal polar ideals, contributing to a deeper understanding of the structure and relationships within AL-monoids.Item The Study of Soft Groups Based on Soft Binary Operations(Addis Ababa University, 2025-06-18) Tesfaye Degife; Berhanu Bekele (PhD)In this thesis, we propose a new de nition for soft groups based on soft binary operations. The idea is to bring the archetype of 'softness' into the spectrum of algebraic structures using soft binary operations parametrized by a given set of suitable parameters. One of our achievement is that we obtain an ordinary group model representing our soft group. The existing classical group serves as a model to describe and characterize the overall internal properties of our soft groups. In this vein, we further investigate the soft subgroups (respectively, normal soft subgroups) and proved some structural theorems. In this thesis, we also study soft homomorphisms on soft groups and investigate their properties. Given a soft mapping hf;Ai from G to G0, we obtain an ordinary map ^f from the set SEA(G) of soft elements of G to the set SEA(G0) of soft elements of G0, and show that hf;Ai is a soft homomorphism (respectively, soft isomorphism) if and only if ^f is an ordinary group homomorphism (respectively, isomorphism). We apply this concept to study soft isomorphism theorems on soft groups. In addition, we study those soft automorphisms of soft groups and the particular class of soft inner automorphisms. Moreover, we study a few soft group-related ndings based on soft binary operations, including soft orbits, soft stabilizers, and the action of a soft group on a set. Given a soft mapping h ;Ai from G X to X; we obtain an ordinary map b from the set SEA(G) SEA(X) to the set SEA(X) and show that h ;Ai is a soft action if and only if b is an ordinary action. Finally, we present the fundamental ideas and characteristics of normal fuzzy soft subgroups.Item New Contributions to Hardy-type Inequalities and Boundary-Domain Integral Equations(Addis Ababa University, 2025-06) Bizuneh Minda; Sorina Barza (Professor)For a large class of operators acting between weighted `1 spaces, exact formulas are obtained for their norms and the norms of their restrictions to the cones of nonnegative sequences and nonnegative monotone sequences. The weights involved are arbitrary nonnegative sequences and may differ in the domain and codomain spaces. The results are applied to the Ces`aro and Copson operators, giving their norms and their distances to the identity operator on the whole space and on the cones. Simplifications of these formulas are derived in the case of these operators acting on power-weighted `1. As an application, best constants are given for inequalities relating the weighted `1 norms of the Ces`aro and Copson operators both for general weights and for power weights. Moreover, we characterize the optimal non-absolute domain for the Hardy operator (and its dual) minus the identity, in the Lebesgue space Lp(0;1), 1 _ p _ 1. For variable coefficient Helmholtz equation, using appropriate parametrix, we formulate boundary-domain integral equations (BDIEs) for the Dirichlet and mixed (Dirichlet-Neumann) boundary value problems (BVPs) in a twodimensional bounded domain. The Dirichlet BVP is reduced to two different BDIE systems, depending on whether the trace or co-normal derivative of the third Green identity is employed on the boundary. On the other hand, the mixed BVP is reduced to four different BDIE systems, depending on whether the trace or co-normal derivative of the third Green identity is employed on the Dirichlet and Neumann boundaries. It is not clear in advance which of them will be more suitable for particular applications and for numerical implementation, and hence we analyzed all the BDIE systems. The equivalence between the BVPs and the formulated BDIE systems are shown. Fredholm properties, invertibility and unique solvability of BDIE systems are investigated in appropriate Sobolev spaces.Item Common Fixed Point Results for Some Class of Generalized Nonexpansive Mappings in Banach Spaces(Addis Ababa University, 2025-06) Gezahegn Anberber; Mengistu Goa (PhD)This study explores generalizations of nonexpansive mappings, focusing on two commuting mappings that satisfy the Bγ,μ condition. We used some algorithm for approximating a common fixed point on some class of generalized nonexpansive mappings and proved its strong convergence to a common _xed point. Our _ndings extend and enhance several recent results in the literature. We discussed the properties of generalized nonexpansive mappings, particularly emphasizing a sequence of commuting mappings that satisfy the B;_ condition. We proposed iterative algorithms for approximating a common _xed point for these sequences, demonstrating their convergence under mild assumptions on the parameters. Additionally, we introduced a pair of some class of generalized nonexpansive mappings and investigated the convergence and existence of common _xed points within this class. We applied the three-step iteration process of Abbas-Nazir for a pair mappings satisfying some class of generalized nonexpansive on a nonempty subset of a Banach space. This approach yielded results related to both strong and weak convergence, leading to the identi_cation of the common _xed point of the two mappings. Finally, we provided an example illustrating two mappings that satisfy the speci_ed conditions.Item Mathematical Modeling of Treatment Switch for People Living With HIV(Addis Ababa University, 2025-06) Helen Girma; Manalebish Debalike (PhD)This thesis develops and analyzes a deterministic compartmental model to optimize treatment switching strategies for HIV/AIDS Antiretroviral Therapy (ART). Motivated by the critical need to mitigate drug resistance and prolong treatment efficacy, we extend the classical Susceptible-Infected-Treatment (SIT) framework to incorporate three distinct ART cocktail classes. The model explicitly allows strategic switching of patients between these classes to manage viral load and suppress resistance emergence. We formulate a system of nonlinear ordinary differential equations capturing population dynamics between susceptible (S), infected (I), and three treated compartments (T1, T2, T3), with bidirectional switching rates between T −classes. The parameters include drug efficacy, switching rates, and resistance development thresholds. Analytical methods establish the basic reproduction number ( R0) and equilibrium stability, while sensitivity analysis identifies dominant control parameters. Numerical simulations evaluate optimal switching protocols under varying epidemiological scenarios. The results show that structured switching between cocktails significantly delays treatment failure and reduces cumulative resistance compared to static regimens. An adaptive ”cycling” strategy, guided by viral load thresholds, is shown to be most effective in sustaining long-term treatment viability. The model provides quantitative evidence for personalized rotation-based ART policies, providing practitioners with a framework to maximize therapeutic outcomes while conserving limited drug options.Item Maximum Flow Problem(Addis Ababa University, 2022-09) Asnake Abrham; Berhanu Guta (PhD)In this thesis, We address about the Concept of Maximum flow Problem and the Solution Methods to Solve the Problem. In this Work the Generic Augmenting Path Algorithm and Ford-Fulkerson Labeling Method are Presented to find the Maximum flow in a Network flow Problem.