A Prime Factor Theorem for a Generalized Direct Product

We introduce the concept of neighborhood systems as a generalization of directed, reflexive graphs and show that the prime factorization of neighborhood systems with respect to the the direct product is unique under the condition that they satisfy an appropriate notion of thinness

Combining drift analysis and generalized schema theory to design efficient hybrid and/or mixed strategy EAs

Hybrid and mixed strategy EAs have become rather popular for tackling various complex and NP-hard optimization problems. While empirical evidence suggests that such algorithms are successful in practice, rather little theoretical support for their success is available, not mentioning a solid mathematical foundation that would provide guidance towards an efficient design of this type of EAs. In the current paper we develop a rigorous mathematical framework that suggests such designs based on generalized schema theory, fitness levels and drift analysis. An example-application for tackling one of the classical NP-hard problems, the "single-machine scheduling problem" is presented

A Unifying View on Recombination Spaces and Abstract Convex Evolutionary Search

This is the author accepted manuscript. The final version is available from Springer via the DOI in this record.Proceedings of EvoCOP 2019 - 19th European Conference on Evolutionary Computation, 24-26 April 2019, Leipzig, GermanyPrevious work proposed to unify an algebraic theory of fitness landscapes and a geometric framework of evolutionary algorithms (EAs). One of the main goals behind this unification is to develop an analytical method that verifies if a problem's landscape belongs to certain abstract convex landscapes classes, where certain recombination-based EAs (without mutation) have polynomial runtime performance. This paper advances such unification by showing that: (a) crossovers can be formally classified according to geometric or algebraic axiomatic properties; and (b) the population behaviour induced by certain crossovers in recombination-based EAs can be formalised in the geometric and algebraic theories. These results make a significant contribution to the basis of an integrated geometric-algebraic framework with which analyse recombination spaces and recombination-based EAs

Saddles and Barrier in Landscapes of Generalized Search Operators

Barrier trees are a convenient way of representing the structure of complex combinatorial landscapes over graphs. Here we generalize the concept of barrier trees to landscapes defined over general multi-parent search operators based on a suitable notion of topological connectedness that depends explicitly on the search operator. We show that in the case of recombination spaces, path-connectedness coincides with connectedness as defined by the mutation operator alone. In contrast, topological connectedness is more general and depends on the details of the recombination operators as well. Barrier trees can be meaningfully defined for both concepts of connectedness

New Generalized Definitions of Rough Membership Relations and Functions from Topological Point of View

In this paper, we shall integrate some ideas in terms of concepts in topology. In fact, we introduce two different views to define generalized membership relations and functions as mathematical tools to classify the sets and help for measuring exactness and roughness of sets. Moreover, we define several types of fuzzy sets. Comparisons between the induced operations were discussed. Finally, many results, examples and counter examples to indicate connections are investigated

The Topology of Evolutionary Biology

Central notions in evolutionary biology are intrinsically topological. This claim is maybe most obvious for the discontinuities associated with punctuated equilibria. Recently, a mathematical framework has been developed that derives the concepts of phenotypic characters and homology from the topological structure of the phenotype space. This structure in turn is determined by the genetic operators and their interplay with the properties of the genotype-phenotype map

Quasi-Independence, Homology and the Unity of Type: A Topological Theory of Characters

In this paper Lewontin’s notion of “quasi-independence” of characters is formalized as the assumption that a region of the phenotype space can be represented by a product space of orthogonal factors. In this picture each character corresponds to a factor of a region of the phenotype space. We consider any region of the phenotype space that has a given factorization as a “type”, i.e. as a set of phenotypes that share the same set of phenotypic characters. Using the notion of local factorizations we develop a theory of character identity based on the continuation of common factors among different regions of the phenotype space. We also consider the topological constraints on evolutionary transitions among regions with different regional factorizations, i.e. for the evolution of new types or body plans. It is shown that direct transition between different “types” is only possible if the transitional forms have all the characters that the ancestral and the derived types have and are thus compatible with the factorization of both types. Transitional forms thus have to go over a “complexity hump” where they have more quasi-independent characters than either the ancestral as well as the derived type. The only logical, but biologically unlikely, alternative is a “hopeful monster” that transforms in a single step from the ancestral type to the derived type. Topological considerations also suggest a new factor that may contribute to the evolutionary stability of “types”. It is shown that if the type is decomposable into factors which are vertex irregular (i.e. have states that are more or less preferred in a random walk), the region of phenotypes representing the type contains islands of strongly preferred states. In other words types have a statistical tendency of retaining evolutionary trajectories within their interior and thus add to the evolutionary persistence of types

Landscapes and Effective Fitness

The concept of a fitness landscape arose in theoretical biology, while that of effective fitness has its origin in evolutionary computation. Both have emerged as useful conceptual tools with which to understand the dynamics of evolutionary processes, especially in the presence of complex genotype-phenotype relations. In this contribution we attempt to provide a unified discussion of these two approaches, discussing both their advantages and disadvantages in the context of some simple models. We also discuss how fitness and effective fitness change under various transformations of the configuration space of the underlying genetic model, concentrating on coarse-graining transformations and on a particular coordinate transformation that provides an appropriate basis for illuminating the structure and consequences of recombination

Bi-izotonik uzaylar ve ayırma aksiyomları

06.03.2018 tarihli ve 30352 sayılı Resmi Gazetede yayımlanan “Yükseköğretim Kanunu İle Bazı Kanun Ve Kanun Hükmünde Kararnamelerde Değişiklik Yapılması Hakkında Kanun” ile 18.06.2018 tarihli “Lisansüstü Tezlerin Elektronik Ortamda Toplanması, Düzenlenmesi ve Erişime Açılmasına İlişkin Yönerge” gereğince tam metin erişime açılmıştır.Bu tez dört bölümden oluşmaktadır. Birinci bölüm tez konusuna ilişkin ayrıntılı literatür bilgisi içermektedir. İkinci bölümde izotonik uzayları tanımlamak üzere kapanış operatörü ve özellikleri verilmiştir. Ayrıca bu operatör yardımıyla iç ve komşuluk operatörleri tanımlanmış ve ilgili teoremler ifade ve ispat edilmiştir. Daha sonra bu operatörler göz önüne alarak bir uzayın izotonik uzay olması için gerek ve yeter koşulları belirtilmiştir. Ayrıca izotonik uzaylar arasında tanımlı dönüşümün sürekliliği ve izotonik uzaylarda ayırma aksiyomları ile ilgili tanım ve karakterizasyonlar verilmiştir. Üçüncü bölümde bitopolojik uzaylarının temel tanımları ve bitopolojik uzaylarda dönüşümlerin sürekliliği ve bitopolojik uzaylarda ayırma aksiyomlarının genellemelerine yer verilmiştir. Dördüncü bölüm bu çalışmanın orijinal kısmını oluşturmaktadır. Bu tezin ikinci ve üçüncü bölümde verilen izotonik uzaylar ve bitopolojik uzaylara ilişkin temel bilgiler ışığında bi-izotonik uzayları tanımlanmış ve temel karakterizasyonlar verilmiştir. Akabinde bi-izotonik uzaylar arasında i-sürekli ve bisürekli dönüşümler tanımlanarak ilgili teoremler ifade edilmiştir. Son olarak bi-izotonik uzaylarda ayırma aksiyomları tanımlanmış ve ilgili teoremler ifade ve ispat edilmiştir. Beşinci bölümde bu tez çalışmasında elde edilen sonuçlar özetlenmiş ve bundan sonra yapılacak araştırmalara yönelik öneride bulunulmuştur.This thesis consists of five chapters. The first chapter is devoted to detailed literature knowledge related to the subject of the thesis. In the second chapter, the closure operator and its properties are given to define the closure spaces. In addition, by the aid of this function the interior and neighborhood operators are defined. The related theorems are stated and proved. Afterwards, by considering these operators, the necessary and sufficient conditions for a space to be an isotonic space are expressed. Moreover, the definitions and characterizations of the continuity of mappings between the isotonic spaces and the separation axioms in isotonic spaces are given. In the third chapter, the fundamental definitions of bitopological spaces and the continuity of mappings between bitopological spaces and the generalizations of separation axioms in bitopological spaces are represented. The fourth chapter is the original part of this study. In the light of basic information on isotonic spaces and bitopological spaces given in the second and third chapters of this thesis, bi-isotonic spaces are introduced and fundamental characterizations are given Subsequently, by defining i-continuous and bicontinuous mappings between bi-isotonic spaces the corresponding theorems are expressed and proved. Finally, separation axioms are described in bi-isotonic spaces and the relevant theorems are expressed and proved. In the fifth chapter of this thesis, a brief summary of this study is given and some suggestions are proposed for new investigations