Discrete Mereotopology

PublishedWhereas mereology, in the strict sense, is concerned solely with the part–whole relation, mereotopology extends mereology by including also the notion of connection, enabling one to distinguish, for example, between internal and peripheral parts, and between contact and separation. Mereotopology has been developed particularly within the Qualitative Spatial Reasoning research community, where it has been applied to, amongst other areas, geographical information science and image analysis. Most research in mereotopology has assumed that the entities being studied may be subdivided without limit, but a number of researchers have investigated mereotopological structures based on discrete spaces in which entities are built up from atomic elements that are not themselves subdivisible. This chapter presents an introductory treatment of mereotopology and its discrete variant, and provides references for readers interested in pursuing this subject in further detail

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

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

Qualitative Spatial Reasoning about Relative Orientation --- A Question of Consistency ---

Abstract. After the emergence of Allen s Interval Algebra Qualitative Spatial Reasoning has evolved into a fruitful field of research in artificial intelligence with possible applications in geographic information systems (GIS) and robot navigation Qualitative Spatial Reasoning abstracts from the detailed metric description of space using rich mathematical theories and restricts its language to a finite, often rather small, set of relations that fulfill certain properties. This approach is often deemed to be cognitively adequate . A major question in qualitative spatial reasoning is whether a description of a spatial situation given as a constraint network is consistent. The problem becomes a hard one since the domain of space (often R2 ) is infinite. In contrast many of the interesting problems for constraint satisfaction have a finite domain on which backtracking methods can be used. But because of the infinity of its domains these methods are generally not applicable to Qualitative Spatial Reasoning. Anyhow the method of path consistency or rather its generalization algebraic closure turned out to be helpful to a certain degree for many qualitative spatial calculi. The problem regarding this method is that it depends on the existence of a composition table, and calculating this table is not an easy task. For example the dipole calculus (operating on oriented dipoles) DRAf has 72 base relations and binary composition, hence its composition table has 5184 entries. Finding all these entries by hand is a hard, long and error-prone task. Finding them using a computer is also not easy, since the semantics of DRAf in the Euclidean Plane, its natural domain, rely on non-linear inequalities. This is not a special problem of the DRAf calculus. In fact, all calculi dealing with relative orientation share the property of having semantics based on non-linear inequalities in the Euclidean plane. This not only makes it hard to find a composition table, it also makes it particularly hard to decide consistency for these calculi. As shown in [79] algebraic closure is always just an approximation to consistency for these calculi, but it is the only method that works fast. Methods like Gröbner reasoning can decide consistency for these calculi but only for small constraint networks. Still finding a composition table for DRAf is a fruitful task, since we can use it analyze the properties of composition based reasoning for such a calculus and it is a starting point for the investigation of the quality of the approximation of consistency for this calculus. We utilize a new approach for calculating the composition table for DRAf using condensed semantics, i.e. the domain of the calculus is compressed in such a way that only finitely many possible configurations need to be investigated. In fact, only the configurations need to be investigated that turn out to represent special characteristics for the placement of three lines in the plane. This method turns out to be highly efficient for calculating the composition table of the calculus. Another method of obtaining a composition table is borrowing it via a suitable morphism. Hence, we investigate morphisms between qualitative spatial calculi. Having the composition table is not the end but rather the beginning of the problem. With that table we can compute algebraically closed refinements of constraint networks, but how meaningful is this process? We know that all constraint networks for which such a refinement does not exist are inconsistent, but what about the rest? In fact, they may be consistent or not. If they are all consistent, then we can be happy, since algebraic closure would decide consistency for the calculus at hand. We investigate LR, DRAf and DRAfp and show that for all these calculi algebraic closure does not decide consistency. In fact, for the LR calculus algebraic closure is an extremely bad approximation of consistency. For this calculus we introduce a new method for the approximation of consistency based on triangles, that performs far better than algebraic closure. A major weak spot of the field of Qualitative Spatial Reasoning is the area of applications. It is hard to refute the accusation of qualitative spatial calculi having only few applications so far. As a step into the direction of scrutinizing the applicability of these calculi, we examine the performance of DRA and OPRA in the issue of describing and navigating street networks based on local observations. Especially for OPRA we investigate a factorization of the base relations that is deemed cognitively adequate . Whenever possible we use real-world data in these investigations obtained from OpenStreetMap