# Mathematics

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Item Maximum Flow Problem(Addis Ababa University, 2022-09) Asnake Abrham; Berhanu Guta (PhD)Show more 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.Show more Item Quadratic Optimal Control Problems with Shooting Method(Addis Ababa University, 2022-11-25) Mamo, Asrat; Guta, Berhanu (PhD)Show more Control theory is an area of applied mathematics that deals with principles,laws, and desire of dynamic systems. Optimal control problems are generalized form of variation problems. A very important tool in variational calculus is the notion of Gateaux-differentiability. It is the basis of the development of necessary optimality conditions. ELDE is a necessary optimality condition to solve variational problems. The solution of ELDE is an extremal function of a variational problem. Characterizing theorem of convex optimization is the necessary and sufficient condition of many convex problems. i.e Let (P) be given, S is convex set, f is convex function and x0 ∈ S. Then x0 ∈ M(f, S) if and only if f′(x0, x−x0) ≥ 0, ∀x ∈ S. In an OCP our aim is to find the optimal state function x∗(t) and the optimal control function u∗(t) which optimize the objective functional in t ∈ [a, b] by using necessary and sufficient optimality conditions. The necessary optimality conditions for (x∗(t), u∗(t)) to be extremal solutions of optimal control problem is the validity of :- Pontryagin minimum principle, of ELDE with TR, and ODE conditions. To determine whether the extremals are optimal solutions of OCP or not; sufficient optimality conditions are required; (e.g checking the convexity of the objective functional and the convexity of the feasible set). QOCP is a non linear optimization where the cost function is quadratic but the differential equation is linear. In quadratic control problem since the objective function is convex then the extremals are the optimal solution of the problem. Linear QOCP is an important type of quadratic control problem that simples the work of feed back control system. OCP can be solved by different methods depending on the type of the problem. This paper mainly considers solving QOCP by using the method of lagrange multiplier and shooting method.Show more Item Hierarchical Multilevel Multi-Leader Multi-Follower Problem Multi-Parametric Solution Approach(Addis Ababa University, 2022-04-07) Belete, Addis; Mitiku, Semu (Professor)Show more Hierarchical multilevel multi-leader multi-follower games are non-cooperative decision problems in which multiple decision-makers of equal status in the upperlevel andmultiple decision-makers of equal status are involved at each of the lowerlevels of the hierarchy. Much of solution methods proposed so far on the topic are either model specific which may work only for a particular sub-class of problems or are based on some strong assumptions and mainly for two level cases. In this dissertation we have proposed a multi-parametric programming based solution approach for hierarchical multilevel multi-leader multi-follower games in which the objective functions contain separable and non-separable terms (but the non-separable terms can be written as a factor of two functions, a function which depends on other level decision variables and a function which is common to all objectives across the same level) and shared constraint. The proposed solution approach transforms a hierarchical multilevel multi-leader multi-follower game into multilevel game involving a single decision maker at each level of the hierarchy. In addition, a solution algorithm for bilevel optimization problems whose lower-level problem involves convex nonlinear constraints is also developed. The solution algorithm recasts the lower-level problem as a multi-parametric problem and employs an equivalent barrier problem reformulation. The solution obtained with this method is shown to be exact if the lower-level problem and the nonlinear constraints can be expressed by a polynomial of utmost degree three with followers’ variable and upto quadratic in the variable of the leader.Show more Item Graduate Seminar Report on Dynamic Construction of Approximate Subdiff Erntials: Dual form of Bundle Methods(Addis Ababa University, 2003-07) Aba-oli, Zinab; Deumlich, R. (Prof.)Show more Item Graduate Seminar Report on Perth Following Methods for linear, Quadratic and Convex Programming(Addis Ababa University, 2006) Teklie, Yoseph; Mitiku, Semu (PhD)Show more Item A Graduate Semi Nar Report on Generating Functions and Some of Their Applications(Addis Ababa University, 2004-07) Tadesse, Yohannes; Gemeda, Demissu (PhD)Show more Item Graduate Seminar Report on Hyperbolic Equation and Flow in Pipelines(Addis Ababa University, 2004-07) Negussie, Yibeltal; Gedif, Tsegaye (PhD)Show more Item Graduate Seminar Report Some Remarks on Secant Method of Approximation for the Solutions of Systems of Non Linear Equations(Addis Ababa University, 2003-06) Abdela, Yasin; R. Deumlich, rer.nat.habil. (Prof.)Show more Item Seminar Report on: Positive Solution for Higher Order Nonlinear Eigen Value Problem(Addis Ababa University, 2007-07) Fikadu, Yared; Abdi, Tadesse (PhD)Show more Item Iterated Norms in Nikol'skii-Besov spaces with Generalized Smoothness(Addis Ababa University, 2002-06) Woubante, Wondimu; Gedif, Tsegaye (PhD)Show more Item Some Remarks on Conjugacy in Convex Analysis a Graduate Seminar Report(Addis Ababa University, 2003) Tolu, Tesfaye; R. Deumlich, rer.nat.habil. (Prof.)Show more Item Graduate Seminar Report on Generating Function Operator(Addis Ababa University, 2004-06) Berhane, Tesfahun; Tsegaye, Yirgalem (PhD)Show more Item Some Remarks on Convex Sets(Addis Ababa University, 2002-06) Yigrem, Tesfa; R.Deumlich, rer.nat.habil. (Prof.)Show more Item Graduate Seminar Report on Eigenvalue Problems(Addis Ababa University, 2004-06) Gemechu, Tekle; Murthy, S.N. (Prof.)Show more Item Graduate Seminar Report on Techniques for Finding Critical Path of a Project Network Model.(Addis Ababa University, 2005-06) Tilahun, Tadesse; Ahmad, Mobin (PhD)Show more Item Graduate Seminar Report on Singular Perturbation Theory(Addis Ababa University, 2003-06) Haregewoin, Solomon; Murthy, S.N. (Prof.)Show more Item Graduate Seminar Report on the Finite Element Method and its Application in two Dimensional Spaces(Addis Ababa University, 2003-06) Guadie, Solomon; Murthy, S.N. (Prof.)Show more Item Graduate Seminar Report on Harmonic Functions(Addis Ababa University, 2005-06) Geysa, Sidise; Mohammed, Seid (PhD)Show more Item A Graduate Seminar Report for Mathematics on Stability Theory for Differential Equations(Addis Ababa University, 2004-06) Abeje, Shumetie; Gedif, Tsegaye (PhD)Show more Item Spectrum of Prime Fuzzy Ideals of A Semigroup(Addis Ababa University, 2005-06) Jemal, Saadu; Kumbhojkar, H.V. (Prof)Show more This paper has three parts: The first part deals with preliminaries on semi groups, spectrum of prime ideals of a semi group and preliminaries on fuzzy sets that are going to be used for the works in the following parts. In the second part primary fuzzy ideals, semi prime fuzzy ideals and nil and prime radical of a fuzzy ideal are discussed and their relationship with prime fuzzy ideals is investigated. It is also proved that the nil and prime ideals coincide when the grade membership lattice is totally ordered. In the last, a topological space called spectrum of fuzzy ideals, FSpec(S), is obtained by introducing a topology on the set of prime fuzzy ideals of a semi group S. It is also proved that FSpec(S) is compact and the non-fuzzy prime spectrum, Spec(S) , is dense in FSpec(S). It is shown that if S and S' are isomorphic, then FSpec(S) and FSpec(S') are homeomorphic. Key words: Fuzzy ideal, Prime fuzzy ideal, Semi prime fuzzy ideal, Primary fuzzy ideal, Radical of a fuzzy ideal, Compact topological space.Show more