Tense logic based on finite orthomodular posets

It is widely accepted that the logic of quantum mechanics is based on orthomodular posets. However, such a logic is not dynamic in the sense that it does not incorporate time dimension. To fill this gap, we introduce certain tense operators on such a logic in an inexact way, but still satisfying requirements asked on tense operators in the classical logic based on Boolean algebras or in various non-classical logics. Our construction of tense operators works perfectly when the orthomodular poset in question is finite. We investigate the behaviour of these tense operators, e.g. we show that some of them form a dynamic pair. Moreover, we prove that if the tense operators preserve one of the inexact connectives conjunction or implication as defined by the authors recently in another paper, then they also preserve the other one. Finally, we show how to construct the binary relation of time preference on a given time set provided the tense operators are given, up to equivalence induced by natural quasiorders

Fuzzy Mathematics

This book provides a timely overview of topics in fuzzy mathematics. It lays the foundation for further research and applications in a broad range of areas. It contains break-through analysis on how results from the many variations and extensions of fuzzy set theory can be obtained from known results of traditional fuzzy set theory. The book contains not only theoretical results, but a wide range of applications in areas such as decision analysis, optimal allocation in possibilistics and mixed models, pattern classification, credibility measures, algorithms for modeling uncertain data, and numerical methods for solving fuzzy linear systems. The book offers an excellent reference for advanced undergraduate and graduate students in applied and theoretical fuzzy mathematics. Researchers and referees in fuzzy set theory will find the book to be of extreme value

Cyber Security of Critical Infrastructures

Critical infrastructures are vital assets for public safety, economic welfare, and the national security of countries. The vulnerabilities of critical infrastructures have increased with the widespread use of information technologies. As Critical National Infrastructures are becoming more vulnerable to cyber-attacks, their protection becomes a significant issue for organizations as well as nations. The risks to continued operations, from failing to upgrade aging infrastructure or not meeting mandated regulatory regimes, are considered highly significant, given the demonstrable impact of such circumstances. Due to the rapid increase of sophisticated cyber threats targeting critical infrastructures with significant destructive effects, the cybersecurity of critical infrastructures has become an agenda item for academics, practitioners, and policy makers. A holistic view which covers technical, policy, human, and behavioural aspects is essential to handle cyber security of critical infrastructures effectively. Moreover, the ability to attribute crimes to criminals is a vital element of avoiding impunity in cyberspace. In this book, both research and practical aspects of cyber security considerations in critical infrastructures are presented. Aligned with the interdisciplinary nature of cyber security, authors from academia, government, and industry have contributed 13 chapters. The issues that are discussed and analysed include cybersecurity training, maturity assessment frameworks, malware analysis techniques, ransomware attacks, security solutions for industrial control systems, and privacy preservation methods

Essays on the economics of networks

Networks (collections of nodes or vertices and graphs capturing their linkages) are a common object of study across a range of fields includ- ing economics, statistics and computer science. Network analysis is often based around capturing the overall structure of the network by some reduced set of parameters. Canonically, this has focused on the notion of centrality. There are many measures of centrality, mostly based around statistical analysis of the linkages between nodes on the network. However, another common approach has been through the use of eigenfunction analysis of the centrality matrix. My the- sis focuses on eigencentrality as a property, paying particular focus to equilibrium behaviour when the network structure is fixed. This occurs when nodes are either passive, such as for web-searches or queueing models or when they represent active optimizing agents in network games. The major contribution of my thesis is in the applica- tion of relatively recent innovations in matrix derivatives to centrality measurements and equilibria within games that are function of those measurements. I present a series of new results on the stability of eigencentrality measures and provide some examples of applications to a number of real world examples

A simple descriptive method for multidimensional item response theory based on stochastic dominance

In this paper we develop a descriptive concept of a (partially) ordinal joint scaling of items and persons in the context of (dichotomous) item response analysis. The developed method has to be understood as a purely descriptive method describing relations among the data observed in a given item response data set, it is not intended to directly measure some presumed underlying latent traits. We establish a hierarchy of pairs of item difficulty and person ability orderings that empirically support each other. The ordering principles we use for the construction are essentially related to the concept of first order stochastic dominance. Our method is able to avoid a paradoxical result of multidimensional item response theory models described in \cite{hooker2009paradoxical}. We introduce our concepts in the language of formal concept analysis. This is due to the fact that our method has some similarities with formal concept analysis and knowledge space theory: Both our methods as well as descriptive techniques used in knowledge space theory (concretely, item tree analysis) could be seen as two different stochastic generalizations of formal implications from formal concept analysis

A simple descriptive method for multidimensional item response theory based on stochastic dominance

In this paper we develop a descriptive concept of a (partially) ordinal joint scaling of items and persons in the context of (dichotomous) item response analysis. The developed method has to be understood as a purely descriptive method describing relations among the data observed in a given item response data set, it is not intended to directly measure some presumed underlying latent traits. We establish a hierarchy of pairs of item difficulty and person ability orderings that empirically support each other. The ordering principles we use for the construction are essentially related to the concept of first order stochastic dominance. Our method is able to avoid a paradoxical result of multidimensional item response theory models described in \cite{hooker2009paradoxical}. We introduce our concepts in the language of formal concept analysis. This is due to the fact that our method has some similarities with formal concept analysis and knowledge space theory: Both our methods as well as descriptive techniques used in knowledge space theory (concretely, item tree analysis) could be seen as two different stochastic generalizations of formal implications from formal concept analysis

Ordem supervisionada baseada em valores fuzzy para morfologia matemática multivalorada

Orientador: Marcos Eduardo Ribeiro do Valle MesquitaDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Morfologia Matemática foi concebida como uma ferramenta para a análise e processamento de imagens binárias e foi subsequentemente generalizada para o uso em imagens em tons de cinza e imagens multivaloradas. Reticulados completos, que são conjuntos parcialmente ordenados em que todo subconjunto tem extremos bem definidos, servem como a base matemática para uma definição geral de morfologia matemática. Em contraste a imagens em tons de cinza, imagens multivaloradas não possuem uma ordem não-ambígua. Essa dissertação trata das chamadas ordens reduzidas para imagens multivaloradas. Ordens reduzidas são definidas por meio de uma relação binária que ordena os elementos de acordo com uma função h do conjunto de valores em um reticulado completo. Ordens reduzidas podem ser classificadas em ordens não-supervisionadas e ordens supervisionadas. Numa ordem supervisionada, o função de ordenação h depende de conjuntos de treinamento de valores de foreground e de background. Nesta dissertação, estudamos ordens supervisionadas da literatura. Também propomos uma ordem supervisionada baseada em valores fuzzy. Valores fuzzy generalizam cores fuzzy - conjuntos fuzzy que modelam o modo que humanos percebem as cores - para imagens multivaloradas. Em particular, revemos como construir o mapa de ordenação baseado em conjuntos fuzzy para o foreground e para o background. Também introduzimos uma função de pertinência baseada numa estrutura neuro-fuzzy e generalizamos a função de pertinência baseada no diagrama de Voronoi. Por fim, as ordens supervisionadas são avaliadas num experimento de segmentação de imagens hiperespectrais baseado num perfil morfológico modificadoAbstract: Mathematical morphology has been conceived initially as a tool for the analysis and processing of binary images and has been later generalized to grayscale and multivalued images. Complete lattices, which are partially ordered sets in whose every subset has well defined extrema, serve as the mathematical background for a general definition of mathematical morphology. In contrast to gray-scale images, however, there is no unambiguous ordering for multivalued images. This dissertation addresses the so-called reduced orderings for multi-valued images. Reduced orderings are defined by means of a binary relation which ranks elements according to a mapping h from the value set into a complete lattice. Reduced orderings can be classified as unsupervised and supervised ordering. In a supervised ordering, the mapping h depends on training sets of foreground and background values. In this dissertation, we study some relevant supervised orderings from the literature. We also propose a supervised ordering based on fuzzy values. Fuzzy values are a generalization of fuzzy colors - fuzzy sets that model how humans perceive colors - to multivalued images other than color images. In particular, we review how to construct the fuzzy ordering mapping based on fuzzy sets that model the foreground and the background. Also, we introduce a membership function based on a neuro-fuzzy framework and generalize the membership function based on Voronoi diagrams. The supervised orderings are evaluated in an experiment of hyperspectral image segmentation based on a modified morphological profileMestradoMatematica AplicadaMestre em Matemática Aplicada131635/2018-2CNP

Abstraktní studium úplnosti pro infinitární logiky

V této dizertační práci se zabýváme studiem vlastností úplnosti infinitárních výrokových logik z pohledu abstraktní algebraické logiky. Cílem práce je pochopit, jak lze základní nástroj v důkazech uplnosti, tzv. Lindenbaumovo lemma, zobecnit za hranici finitárních logik. Za tímto účelem studujeme vlastnosti úzce související s Lindenbaumovým lemmatem (a v důsledku také s vlastnostmi úplnosti). Uvidíme, že na základě těchto vlastností lze vystavět novou hierarchii infinitárních výrokových logik. Také se zabýváme studiem těchto vlastností v případě, kdy naše logika má nějaké (případně hodně obecně definované) spojky implikace, disjunkce a negace. Mimo jiné uvidíme, že přítomnost daných spojek může zajist platnost Lindenbaumova lemmatu. Keywords: abstraktní algebraická logika, infinitární logiky, Lindenbau- movo lemma, disjunkce, implikace, negaceIn this thesis we study completeness properties of infinitary propositional logics from the perspective of abstract algebraic logic. The goal is to under- stand how the basic tool in proofs of completeness, the so called Linden- baum lemma, generalizes beyond finitary logics. To this end, we study few properties closely related to the Lindenbaum lemma (and hence to com- pleteness properties). We will see that these properties give rise to a new hierarchy of infinitary propositional logic. We also study these properties in scenarios when a given logic has some (possibly very generally defined) connectives of implication, disjunction, and negation. Among others, we will see that presence of these connectives can ensure provability of the Lin- denbaum lemma. Keywords: abstract algebraic logic, infinitary logics, Lindenbaum lemma, disjunction, implication, negationKatedra logikyDepartment of LogicFaculty of ArtsFilozofická fakult