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Browsing Mathematics by Author "Abebaw Tilahun (PhD)"
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Item Automatic Geometric Theorem Proving(Addis Ababa University, 2016-08) Lemma Hiwot; Abebaw Tilahun (PhD)For several decades algebraic method have been successfully used in automated deduction in Geometry objects in Euclidean geometry are relations between them are expressed as polynomials and algebraic method e.g.Groebner bases are used over that set of polynomials we describe aformalization of an algorithm that accepts a term representation of a geometry construction and returns a corresponding set of polynomials our furtherwork will be to use the method of Groebner bases on the generated polynomials , in order to implement a formallyverified algebraic prover for geometry .Item “Clifford semigroups and seminear-rings of endomorphism”(Addis Ababa University, 2017-09) Hunegnaw Dejene; Abebaw Tilahun (PhD)In this project we consider the structure of semi groups of self mapping of a semi group S under pointwise composition, generated by the endomorphisms of S. It is shown that,ifS is Clifford semi groups with underlying semilattice ∧, the endomorphisms of S generate a Clifford semigroup E+ (S) whose underlying semilattice is the set of endomorphisms of ∧.These results contribute to the wider theory of seminear-rings of endomorphisms, Since E+ (S) has a natural structure as distributively generated seminear-ring.Item Dimensions of Affine Varieties(Addis Ababa University, 2011-06) Tsegay Hailegebriel; Abebaw Tilahun (PhD)In this Project, we introduced the affine varieties, which are curves and surfaces (and higher-dimensional objects) defined by Polynomial Equations. By considering ideals in the polynomial ring 1,…], we also understood the dimension of an affine variety and the affine Hilbert Function of an ideal, which is a function on the non-negative integers Given an affine variety which is the union of a finite number of linear subspaces of the affine space , we defined the dimension of in terms of the dimensions of the subspaces. We also defined the dimension of in terms of the degree of the affine Hilbert Polynomial of the corresponding ideal . Finally, we stated several basic properties of dimension over an infinite field using the degree of affine Hilbert PolynomialsItem Numerical Method for Heat Transfer under Chance Constrained State Variables(Addis Ababa University, 2017-11-06) Amare Getinet; Abebaw Tilahun (PhD); Sani MohammedStochastic chance-constrained programming is mainly concerned with the problem that the decision maker must give his solution before the random variables come true. In this problem, the probability of decision satisfying the constraints cannot be less than some given probability level, or reliability level or con dence level . There are two main di culties with such chance-constrained problems. First, checking feasibility of a given candidate solution exactly is impossible in general. Second, the feasible region induced by chance constraints is, in general, non-convex leading to severe optimization challenges. Chance constrained optimization problems in engineering applications possess highly non-linear process models and non-convex structures. As a result, solving a non-linear non-convex chance constrained optimization (CCOPT) problem remains as a challenging task. The major di culty lies in the evaluation of probability values and gradients of inequality constraints which are non-linear functions of stochastic variables. This thesis will focus on Inner-Outer smooth analytic approximation to improve tractability of non-convex chance constraints. Also this thesis is devoted to an example of optimization problems that include PDEs constraint in the case of heat transfer by implementing the Inner-Outer approximation scheme.Item On Decomposition of D-modules and Bernstein-Sato polynomials for Hyperplane Arrangements(Addis Ababa University, 2016-06) Atikaw Sebsibew; Abebaw Tilahun (PhD). This thesis discusses the relationship between Bernstein-Sato ideals of = xy(a3x + y):::(amx + y); ai 2 C; ai 6= aj ;m _ 3 and the decomposition of the D2-module M_ Chx; y; @x; @yi___ over the Weyl algebra Chx; y; @x; @yi, where for each i 2 f1; 2; :::;mg, m ; _i 2 C and _1 := x; _2 = y; _i := aix + y; (3 _ i _ m) are linear forms on C2. The thesis starts by summarizing the de_nition, properties and the results on the number of decomposition factors of M_ Then it continues with the de_nition and basic properties of univariate Bernstein-Sato polynomials, and collects what is known of Bernstein-Sato polynomials for hyperplane arrangements. A variation of the idea are the multivariate Bernstein-Sato polynomials and ideals. Main new results in the thesis are on the description of di_erent types of Bernstein- Sato ideals of = xy Qm i=3(aix + y) (in chapter 4) and on the use of these ideals in the decomposition of the D2-module M_ (in chapterItem On Polynomial Functions on a Variety(Addis Ababa University, 2011-01) Abebe Legesse; Abebaw Tilahun (PhD)Let be a field and given a polynomials inK(x1,x2,...Xn)2 K(X1X2......NK)], we can define an affine varieties in and ideals in a polynomials ring1 ,2 ,. . .]. This project considers the polynomial functions on a variety. The algebraic properties of polynomial functions on a variety yield many insights in to the geometric properties of the variety. The collection of polynomial functions from the variety to the field (or the coordinate ring ]) has the sum and product operations constructed using the sum and product operations in . The construction of the coordinate ring ] is a special case of the quotient ring In particular, we relate the quotient ring 2 ,…,)⁄ to the ring] of polynomial functions on . And the relation between two isomorphic varieties and two coordinate rings of an affine varieties are consideredItem On Structure and Commutativity of Near-Rings(Addis Ababa University, 2018-08-12) Ayansa Muluneh; Abebaw Tilahun (PhD)The aim of this paper is to generalize Certain Near-rings are rings. Here we are interested in two problems concerning certain classes of Near-rings satisfying the following polynomial identities:- (* ) For each in x,y a near-ring N, there exist positive integers t=t ( x,y) >1 S=S and (x,y ) >1 such that .xy +Ys xt ( **) For each x,y in a near-ring N,there exist positive integers t=t ( x,y) >1 and S=S (x,y ) >1 such that xy=+ Xt YsItem Quotient Subtraction Algebra(Addis Ababa University, 2016-07) Lakew Zenebe; Abebaw Tilahun (PhD)Certain techniques fundamental to the study of algebra. One of such techniques is the construction of a quotient set of an algebraic structure by a means of an equivalence relation on the given set. Based on this, we construct a quotient subtraction algebra, which plays a crucial role in the study of algebra. In this project work, we will introduce the notion of subtraction algebra, which is an algebraic structure on a set with a binary operation, usually called subtraction. To define subtraction algebra, we use the the concept of equivalence relation, subtraction algebra and ideal of subtraction algebra to define a quotient subtraction algebras and study its properties. The idea of isomorphism theorems of other algebraic structures is used to define isomorphism theorems of quotient subtraction algebras.