433 research outputs found
Dual Logic Concepts based on Mathematical Morphology in Stratified Institutions: Applications to Spatial Reasoning
Several logical operators are defined as dual pairs, in different types of
logics. Such dual pairs of operators also occur in other algebraic theories,
such as mathematical morphology. Based on this observation, this paper proposes
to define, at the abstract level of institutions, a pair of abstract dual and
logical operators as morphological erosion and dilation. Standard quantifiers
and modalities are then derived from these two abstract logical operators.
These operators are studied both on sets of states and sets of models. To cope
with the lack of explicit set of states in institutions, the proposed abstract
logical dual operators are defined in an extension of institutions, the
stratified institutions, which take into account the notion of open sentences,
the satisfaction of which is parametrized by sets of states. A hint on the
potential interest of the proposed framework for spatial reasoning is also
provided.Comment: 36 page
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Shape theory and mathematical design of a general geometric kernel through regular stratified objects
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This dissertation focuses on the mathematical design of a unified shape kernel for geometric computing, with possible applications to computer aided design (CAM) and manufacturing (CAM), solid geometric modelling, free-form modelling of curves and surfaces, feature-based modelling, finite element meshing, computer animation, etc.
The generality of such a unified shape kernel grounds on a shape theory for objects in some Euclidean space. Shape does not mean herein only geometry as usual in geometric modelling, but has been extended to other contexts, e. g. topology, homotopy, convexity theory, etc. This shape theory has enabled to make a shape analysis of the current geometric kernels. Significant deficiencies have been then identified in how these geometric kernels represent shapes from different applications.
This thesis concludes that it is possible to construct a general shape kernel capable of representing and manipulating general specifications of shape for objects even in higher-dimensional Euclidean spaces, regardless whether such objects are implicitly or parametrically defined, they have âincomplete boundariesâ or not, they are structured with more or less detail or subcomplexes, which design sequence has been followed in a modelling session, etc. For this end, the basic constituents of such a general geometric kernel, say a combinatorial data structure and respective Euler operators for n-dimensional regular stratified objects, have been introduced and discussed
Dual attachment pairs in categorically-algebraic topology
[EN] The paper is a continuation of our study on developing a new approach to (lattice-valued) topological structures, which relies on category theory and universal algebra, and which is called categorically-algebraic (catalg) topology. The new framework is used to build a topological setting, based in a catalg extension of the set-theoretic membership relation "e" called dual attachment, thereby dualizing the notion of attachment introduced by the authors earlier. Following the recent interest of the fuzzy community in topological systems of S. Vickers, we clarify completely relationships between these structures and (dual) attachment, showing that unlike the former, the latter have no inherent topology, but are capable of providing a natural transformation between two topological theories. We also outline a more general setting for developing the attachment theory, motivated by the concept of (L,M)-fuzzy topological space of T. Kubiak and A. Sostak.This research was partially supported by the ESF Project of the University of Latvia No. 2009/0223/1DP/1.1.1.2.0/09/APIA/VIAA/008.Frascella, A.; Guido, C.; Solovyov, SA. (2011). Dual attachment pairs in categorically-algebraic topology. Applied General Topology. 12(2):101-134. doi:10.4995/agt.2011.1646.SWORD10113412
Fuzzy Sets, Fuzzy Logic and Their Applications
The present book contains 20 articles collected from amongst the 53 total submitted manuscripts for the Special Issue âFuzzy Sets, Fuzzy Loigic and Their Applicationsâ of the MDPI journal Mathematics. The articles, which appear in the book in the series in which they were accepted, published in Volumes 7 (2019) and 8 (2020) of the journal, cover a wide range of topics connected to the theory and applications of fuzzy systems and their extensions and generalizations. This range includes, among others, management of the uncertainty in a fuzzy environment; fuzzy assessment methods of human-machine performance; fuzzy graphs; fuzzy topological and convergence spaces; bipolar fuzzy relations; type-2 fuzzy; and intuitionistic, interval-valued, complex, picture, and Pythagorean fuzzy sets, soft sets and algebras, etc. The applications presented are oriented to finance, fuzzy analytic hierarchy, green supply chain industries, smart health practice, and hotel selection. This wide range of topics makes the book interesting for all those working in the wider area of Fuzzy sets and systems and of fuzzy logic and for those who have the proper mathematical background who wish to become familiar with recent advances in fuzzy mathematics, which has entered to almost all sectors of human life and activity
Functorial comparisons of bitopology with topology and the case for redundancy of bitopology in lattice-valued mathematics
[EN] This paper studies various functors between (lattice-valued) topology and (lattice-valued) bitopology, including the expected âdoublingâ functor Ed : L-Top â L-BiTop and the âcrossâ functor EĂ : L-BiTop â L2-Top introduced in this paper, both of which are extremely well-behaved strict, concrete, full embeddings. Given the greater simplicity of lattice-valued topology vis-a-vis lattice-valued bitopology and the fact that the class of L2-Topâs is strictly smaller than the class of L-Topâs encompassing fixed-basis topology, the class of EĂâs makes the case that lattice-valued bitopology is categorically redundant. As a special application, traditional bitopology as represented by BiTop is (isomorphic in an extremely well-behaved way to) a strict subcategory of 4-Top, where 4 is the four element Boolean algebra; this makes the case that traditional bitopology is a special case of a much simpler fixed-basis topology.Support of Youngstown State University via a sabbatical for the 2005â2006 academic year is gratefully acknowledged.Rodabaugh, S. (2008). Functorial comparisons of bitopology with topology and the case for redundancy of bitopology in lattice-valued mathematics. Applied General Topology. 9(1):77-108. doi:10.4995/agt.2008.1871.SWORD771089
New Challenges in Neutrosophic Theory and Applications
Neutrosophic theory has representatives on all continents and, therefore, it can be said to be a universal theory. On the other hand, according to the three volumes of âThe Encyclopedia of Neutrosophic Researchersâ (2016, 2018, 2019), plus numerous others not yet included in Encyclopedia book series, about 1200 researchers from 73 countries have applied both the neutrosophic theory and method. Neutrosophic theory was founded by Professor Florentin Smarandache in 1998; it constitutes further generalization of fuzzy and intuitionistic fuzzy theories. The key distinction between the neutrosophic set/logic and other types of sets/logics lies in the introduction of the degree of indeterminacy/neutrality (I) as an independent component in the neutrosophic set. Thus, neutrosophic theory involves the degree of membership-truth (T), the degree of indeterminacy (I), and the degree of non-membership-falsehood (F). In recent years, the field of neutrosophic set, logic, measure, probability and statistics, precalculus and calculus, etc., and their applications in multiple fields have been extended and applied in various fields, such as communication, management, and information technology. We believe that this book serves as useful guidance for learning about the current progress in neutrosophic theories. In total, 22 studies have been presented and reflect the call of the thematic vision. The contents of each study included in the volume are briefly described as follows. The first contribution, authored by Wadei Al-Omeri and Saeid Jafari, addresses the concept of generalized neutrosophic pre-closed sets and generalized neutrosophic pre-open sets in neutrosophic topological spaces. In the article âDesign of Fuzzy Sampling Plan Using the Birnbaum-Saunders Distributionâ, the authors Muhammad Zahir Khan, Muhammad Farid Khan, Muhammad Aslam, and Abdur Razzaque Mughal discuss the use of probability distribution function of BirnbaumâSaunders distribution as a proportion of defective items and the acceptance probability in a fuzzy environment. Further, the authors Derya Bakbak, Vakkas Ulucžay, and Memet Sžahin present the âNeutrosophic Soft Expert Multiset and Their Application to Multiple Criteria Decision Makingâ together with several operations defined for them and their important algebraic properties. In âNeutrosophic Multigroups and Applicationsâ, Vakkas Ulucžay and Memet Sžahin propose an algebraic structure on neutrosophic multisets called neutrosophic multigroups, deriving their basic properties and giving some applications to group theory. Changxing Fan, Jun Ye, Sheng Feng, En Fan, and Keli Hu introduce the âMulti-Criteria Decision-Making Method Using Heronian Mean Operators under a Bipolar Neutrosophic Environmentâ and test the effectiveness of their new methods. Another decision-making study upon an everyday life issue which empowered us to organize the key objective of the industry developing is given in âNeutrosophic Cubic Einstein Hybrid Geometric Aggregation Operators with Application in Prioritization Using Multiple Attribute Decision-Making Methodâ written by Khaleed Alhazaymeh, Muhammad Gulistan, Majid Khan, and Seifedine Kadry
Subgroup discovery for structured target concepts
The main object of study in this thesis is subgroup discovery, a theoretical framework for finding subgroups in dataâi.e., named sub-populationsâ whose behaviour with respect to a specified target concept is exceptional when compared to the rest of the dataset. This is a powerful tool that conveys crucial information to a human audience, but despite past advances has been limited to simple target concepts. In this work we propose algorithms that bring this framework to novel application domains. We introduce the concept of representative subgroups, which we use not only to ensure the fairness of a sub-population with regard to a sensitive trait, such as race or gender, but also to go beyond known trends in the data. For entities with additional relational information that can be encoded as a graph, we introduce a novel measure of robust connectedness which improves on established alternative measures of density; we then provide a method that uses this measure to discover which named sub-populations are more well-connected. Our contributions within subgroup discovery crescent with the introduction of kernelised subgroup discovery: a novel framework that enables the discovery of subgroups on i.i.d. target concepts with virtually any kind of structure. Importantly, our framework additionally provides a concrete and efficient tool that works out-of-the-box without any modification, apart from specifying the Gramian of a positive definite kernel. To use within kernelised subgroup discovery, but also on any other kind of kernel method, we additionally introduce a novel random walk graph kernel. Our kernel allows the fine tuning of the alignment between the vertices of the two compared graphs, during the count of the random walks, while we also propose meaningful structure-aware vertex labels to utilise this new capability. With these contributions we thoroughly extend the applicability of subgroup discovery and ultimately re-define it as a kernel method.Der Hauptgegenstand dieser Arbeit ist die Subgruppenentdeckung (Subgroup Discovery), ein theoretischer Rahmen fĂŒr das Auffinden von Subgruppen in Datenâd. h. benannte Teilpopulationenâderen Verhalten in Bezug auf ein bestimmtes Targetkonzept im Vergleich zum Rest des Datensatzes auĂergewöhnlich ist. Es handelt sich hierbei um ein leistungsfĂ€higes Instrument, das einem menschlichen Publikum wichtige Informationen vermittelt. Allerdings ist es trotz bisherigen Fortschritte auf einfache Targetkonzepte beschrĂ€nkt. In dieser Arbeit schlagen wir Algorithmen vor, die diesen Rahmen auf neuartige Anwendungsbereiche ĂŒbertragen. Wir fĂŒhren das Konzept der reprĂ€sentativen Untergruppen ein, mit dem wir nicht nur die Fairness einer Teilpopulation in Bezug auf ein sensibles Merkmal wie Rasse oder Geschlecht sicherstellen, sondern auch ĂŒber bekannte Trends in den Daten hinausgehen können. FĂŒr EntitĂ€ten mit zusĂ€tzlicher relationalen Information, die als Graph kodiert werden kann, fĂŒhren wir ein neuartiges MaĂ fĂŒr robuste Verbundenheit ein, das die etablierten alternativen DichtemaĂe verbessert; anschlieĂend stellen wir eine Methode bereit, die dieses MaĂ verwendet, um herauszufinden, welche benannte Teilpopulationen besser verbunden sind. Unsere BeitrĂ€ge in diesem Rahmen gipfeln in der EinfĂŒhrung der kernelisierten Subgruppenentdeckung: ein neuartiger Rahmen, der die Entdeckung von Subgruppen fĂŒr u.i.v. Targetkonzepten mit praktisch jeder Art von Struktur ermöglicht. Wichtigerweise, unser Rahmen bereitstellt zusĂ€tzlich ein konkretes und effizientes Werkzeug, das ohne jegliche Modifikation funktioniert, abgesehen von der Angabe des Gramian eines positiv definitiven Kernels. FĂŒr den Einsatz innerhalb der kernelisierten Subgruppentdeckung, aber auch fĂŒr jede andere Art von Kernel-Methode, fĂŒhren wir zusĂ€tzlich einen neuartigen Random-Walk-Graph-Kernel ein. Unser Kernel ermöglicht die Feinabstimmung der Ausrichtung zwischen den Eckpunkten der beiden unter-Vergleich-gestelltenen Graphen wĂ€hrend der ZĂ€hlung der Random Walks, wĂ€hrend wir auch sinnvolle strukturbewusste Vertex-Labels vorschlagen, um diese neue FĂ€higkeit zu nutzen. Mit diesen BeitrĂ€gen erweitern wir die Anwendbarkeit der Subgruppentdeckung grĂŒndlich und definieren wir sie im Endeffekt als Kernel-Methode neu
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