In this thesis, we wanted to stress the tendency to the geometry of data. This should be applicable in almost every branch of science, where data are of great importance, and also in every kind of industry, economy, medicine etc. Since machine&#039 / s hard-disk capacities which is used for storing datas and the amount of data you can reach through internet is increasing day by day, there should be a need to turn this information into knowledge. This is one of the reasons for studying formal concept analysis. We wanted to point out how this application is related with algebra and logic. The beginning of the first chapter emphasis the relation between closure systems, galois connections, lattice theory as a mathematical structure and concept analysis. Then it describes the basic step in the formalization: An elementary form of the representation of data is defined mathematically. Second chapter explains the logic of formal concept analysis. It also shows how implications, which can be regard as special formulas on a set,between attributes can be shown by fewer implications, so called generating set for implications. These mathematical tools are then used in the last chapter, in order to describe complex &#039 / concept&#039 / lattices by means of decomposition methods in examples.

The present thesis aims to develop an analogy between the methods for recognizing a black box group and the classification of the finite simple groups. We propose a uniform approach for recognizing simple groups of Lie type which can be viewed as the computational version of the classification of the finite simple groups. Similar to the inductive argument on centralizers of involutions which plays a crucial role in the classification project, our approach is based on a recursive construction of the centralizers of involutions in black box groups. We present an algorithm which constructs a long root SL_2(q)-subgroup in a finite simple group of Lie type of odd characteristic $p$ extended possibly by a p-group. Following this construction, we take the Aschbacher&#039 / s ``Classical Involution Theorem&#039 / &#039 / as a model in the final recognition algorithm and we propose an algorithm which constructs all root SL_2(q)-subgroups corresponding to the nodes in the extended Dynkin diagram, that is, our approach is the construction of the the extended Curtis - Phan - Tits presentation of the finite simple groups of Lie type of odd characteristic which further yields the construction of all subsystem subgroups which can be read from the extended Dynkin diagram. In this thesis, we present this algorithm for the groups PSL_n(q) and PSU_n(q). We also present an algorithm which determines whether the p-core (or ``unipotent radical&#039 / &#039 / ) O_p(G) of a black box group G is trivial or not where G/O_p(G) is a finite simple classical group of Lie type of odd characteristic p answering a well-known question of Babai and Shalev. The algorithms presented in this thesis have been implemented extensively in the computer algebra system GAP.

A group G is called a barely transitive group if it acts transitively and faithfully on an infinite set and every orbit of every proper subgroup is finite. A subgroup H of a group G is called a permutable subgroup, if H commutes with every subgroup of G. We showed that if an infinitely generated barely transitive group G has a permutable point stabilizer, then G is locally finite. We proved that if a barely transitive group G has an abelian point stabilizer H, then G is isomorphic to one of the followings: (i) G is a metabelian locally finite p-group, (ii) G is a finitely generated quasi-finite group (in particular H is finite), (iii) G is a finitely generated group with a maximal normal subgroup N where N is a locally finite metabelian group. In particular, G=N is a quasi-finite simple group. In all of the three cases, G is periodic.

This thesis is devoted to the analysis of relations between two major conjectures in the theory of groups of finite Morley rank. One of them is the Cherlin-Zil&#039 / ber Algebraicity Conjecture which states that infinite simple groups of finite Morley rank are isomorphic to simple algebraic groups over algebraically closed fields. The other conjecture is due to Hrushovski and it states that a generic automorphism of a simple group of finite Morley rank has pseudofinite group of fixed points. Hrushovski showed that the Cherlin-Zil&#039 / ber Conjecture implies his conjecture. Proving his Conjecture and reversing the implication would provide a new efficient approach to prove the Cherlin-Zil&#039 / ber Conjecture. This thesis proposes an approach to derive a proof of the Cherlin-Zil&#039 / ber Conjecture from Hrushovski&#039 / s Conjecture and contains a proof of a step in that direction. Firstly, we show that John S. Wilson&#039 / s classification theorem for simple pseudofinite groups can be adapted for definably simple non-abelian pseudofinite groups of finite centralizer dimension. Combining this result with recent related developments, we identify definably simple non-abelian pseudofinite groups with Chevalley or twisted Chevalley groups over pseudofinite fields. After that in the context of Hrushovski&#039 / s Conjecture, in a purely algebraic set-up, we show that the pseudofinite group of fixed points of a generic automorphism is actually an extension of a Chevalley group or a twisted Chevalley group over a pseudofinite field.

A group G is called locally finite if every finitely generated subgroup of G is finite. In this thesis we study the centralizers of subgroups in simple locally finite groups. Hartley proved that in a linear simple locally finite group, the fixed point of every semisimple automorphism contains infinitely many elements of distinct prime orders. In the first part of this thesis, centralizers of finite abelian subgroups of linear simple locally finite groups are studied and the following result is proved: If G is a linear simple locally finite group and A is a finite d-abelian subgroup consisting of semisimple elements of G, then C_G(A) has an infinite abelian subgroup isomorphic to the direct product of cyclic groups of order p_i for infinitely many distinct primes pi. Hartley asked the following question: Let G be a non-linear simple locally finite group and F be any subgroup of G. Is CG(F) necessarily infinite? In the second part of this thesis, the following problem is studied: Determine the nonlinear simple locally finite groups G and their finite subgroups F such that C_G(F) contains an infinite abelian subgroup which is isomorphic to the direct product of cyclic groups of order pi for infinitely many distinct primes p_i. We prove the following: Let G be a non-linear simple locally finite group with a split Kegel cover K and F be any finite subgroup consisting of K-semisimple elements of G. Then the centralizer C_G(F) contains an infinite abelian subgroup isomorphic to the direct product of cyclic groups of order p_i for infinitely many distinct primes p_i.

The notion of &rdquo / Planar functions&rdquo / goes back to Dembowski and Ostrom, who introduced it in 1968 first time to describe projective planes with special properties in finite geometry. Recently, they attracted an interest from cryptography because of having an optimal resistance to differential cryptanalysis.This thesis is based on the paper &rdquo / New semifields, PN and APN functions&rdquo / by J&uuml / rgen Bierbrauer. The whole purpose of this thesis is to understand and present a detailed description of the results of the paper of Bierbrauer about planar functions. Here and throughout this thesis &rdquo / new&rdquo / means &rdquo / new&rdquo / in the paper of Bierbrauer. In particular we have no new constructions here and we only explain the results of Bierbrauer.

This thesis is a contribution to determine the absolute irreducibility of polynomials via their Newton polytopes. For any field F / a polynomial f in F[x1, x2,..., xk] can be associated with a polytope, called its Newton polytope. If the polynomial f has integrally indecomposable Newton polytope, in the sense of Minkowski sum, then it is absolutely irreducible over F / i.e. irreducible over every algebraic extension of F. We present some new results giving integrally indecomposable classes of polytopes. Consequently, we have some new criteria giving infinitely many types of absolutely irreducible polynomials over arbitrary fields.

In this thesis, we give two simple and effective methods for constructing Kummer extensions of algebraic function fields over finite fields with many rational places. Some explicit examples are obtained after a practical search. We also study fibre products of Kummer extensions over a finite field and determine the exact number of rational places. We obtain explicit examples with many rational places by a practical search. We have a record (i.e the lower bound is improved) and a new entry for the table of van der Geer and van der Vlugt.

In this thesis, it is aimed to make a study on Primary Schools of Rum Minority in Istanbul, and in light of this study, to define general principles for conservation studies on these schools. Rum Minority had an important part in social and cultural life in Istanbul and in late 19th Century, their impact in the city increased with their financial power, especially in Beyoglu. Increase in number of schools they built also occurred in the same timeline. Schools built in and after this term by Rum Minority were built as important public buildings of a minority group and possess strong authenticity, technical-artistic, socio-cultural and economical values. In order to decide which values, problems and potentials these buildings bear, a site survey study is done for this thesis. Primary Schools of Rum Minority in Istanbul are among strongest solid evidences of cultural diversity in Istanbul, and Turkey. While conserving these cultural assets, considering all their values, problems, and potentialities is vital. In addition to conservation studies, interpretation and presentation are also necessary steps for reintegration of these buildings to the city. This thesis performs as an effort made to document features and current state of these schools, decide their value, problems, and potentials / and defining general conservation principles for them.

abstract INERT SUBGROUPS AND CENTRALIZERS OF INVOLUTIONS IN LOCALLY FINITE SIMPLE GROUPS &uml / Ozyurt, Erdal Ph. D., Department of Mathematics Supervisor: Prof. Dr. Mahmut Kuzucuo&amp / #728 / glu September 2003, 68 pages A subgroup H of a group G is called inert if [H : H Hg] is finite for all g 2 G. A group is called totally inert if every subgroup is inert. Among the basic properties of inert subgroups, we prove the following. Let M be a maximal subgroup of a locally finite group G. If M is inert and abelian, then G is soluble with derived length at most 3. In particular, the given properties impose a strong restriction on the derived length of G. We also prove that, if the centralizer of every involution is inert in an infinite locally finite simple group G, then every finite set of elements of G can not be contained in a finite simple group. In a special case, this generalizes a Theorem of Belyaev&amp / #8211 / Kuzucuo&amp / #728 / glu&amp / #8211 / Se&cedil / ckin, which proves that there exists no infinite locally finite totally inert simple group.