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 “Fuzzy Sets, Fuzzy Loigic and Their Applications” 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

Approximation Theory and Related Applications

In recent years, we have seen a growing interest in various aspects of approximation theory. This happened due to the increasing complexity of mathematical models that require computer calculations and the development of the theoretical foundations of the approximation theory. Approximation theory has broad and important applications in many areas of mathematics, including functional analysis, differential equations, dynamical systems theory, mathematical physics, control theory, probability theory and mathematical statistics, and others. Approximation theory is also of great practical importance, as approximate methods and estimation of approximation errors are used in physics, economics, chemistry, signal theory, neural networks and many other areas. This book presents the works published in the Special Issue "Approximation Theory and Related Applications". The research of the world’s leading scientists presented in this book reflect new trends in approximation theory and related topics

Fitting aggregation operators to data

Theoretical advances in modelling aggregation of information produced a wide range of aggregation operators, applicable to almost every practical problem. The most important classes of aggregation operators include triangular norms, uninorms, generalised means and OWA operators.With such a variety, an important practical problem has emerged: how to fit the parameters/ weights of these families of aggregation operators to observed data? How to estimate quantitatively whether a given class of operators is suitable as a model in a given practical setting? Aggregation operators are rather special classes of functions, and thus they require specialised regression techniques, which would enforce important theoretical properties, like commutativity or associativity. My presentation will address this issue in detail, and will discuss various regression methods applicable specifically to t-norms, uninorms and generalised means. I will also demonstrate software implementing these regression techniques, which would allow practitioners to paste their data and obtain optimal parameters of the chosen family of operators.<br /

Mathematics in Software Reliability and Quality Assurance

This monograph concerns the mathematical aspects of software reliability and quality assurance and consists of 11 technical papers in this emerging area. Included are the latest research results related to formal methods and design, automatic software testing, software verification and validation, coalgebra theory, automata theory, hybrid system and software reliability modeling and assessment

VECTOR LATTICES, POLYHEDRAL GEOMETRY, AND VALUATIONS

The present thesis explores the connections between Hadwiger\u2019s Characterization Theorem for valuations, vector lattices, and MV-algebras. The study of valuations can be seen as a precursor to the measure theory of modern probability. For this reason, valuations are one of the most important topics of geometric probability. One of the topics that turns out to be central is the study of measures on polyconvex sets (i.e., finite unions of compact convex sets) in Euclidean spaces of arbitrary finite dimension, that are invariant under the group of Euclidean motions. Hadwiger\u2019s Characterization Theorem states that the linear space of such invariant measures is of dimension n + 1, if the ambient has dimension n. Moreover, its proof shows that the Euler-Poincar\ue9 characteristic is a basic invariant measure. The Euler-Poincar\ue9 characteristic, indeed, is the unique such measure that assigns value one to each compact and convex set. The main topic of the first part of our work is the characterization of the Euler-Poinca\ue9 characteristic as a valuation on finitely presented unital vector lattices. By the Baker-Baynon duality, we represent each finitely presented unital vector lattice as the lattice of continuous and piecewise linear real-valued functions on a suitable polyhedron in the Euclidean space. Then we define vl-Schauder hats, that are special elements of the vector lattice with a \u201cpyramidal shape\u201d, and that can be used to generate the vector lattice, via addition and products by real scalars. On the positive cone of every finitely presented vector lattice V we define a pc-valuation as a valuation (in the usual classical sense) that is insensitive to addition. The (Euler-Poincar\ue9) number \u3c7(f) of any function f in the positive cone of V is next defined as the Euler-Poincar\ue9 characteristic of the support of f. Pc-valuations uniquely extend to a suitable kind of valuations over V, called vl-valuations. We then prove a Hadwiger-like theorem, to the effect that our \u3c7 is the only vl-valuation assigning 1 to each vl-Shauder hat of V. In the second part of the thesis we explore two different ways to associate continuous and piecewise linear functions (and hence elements of vector lattices) to geometric objects. First we use the notion of support function to establish a correspondence between a suitable subset of the free vector lattice on n generators and the set of polytopes of the n-dimensional euclidean space. This special set of algebraic objects generates the whole free vector lattice via finite meets. We call it the set of support elements. Then we consider valuations on the free vector lattice that are also additive on the set of support elements. By the Volland-Groemer Extension Theorem, we prove that such valuations are in a one-to-one correspondence with the valuations on the lattice of polyconvex sets that are additive on the subset of convex objects. Next we proceed in a similar way, using gauge functions. In this case, the first correspondence that we prove is between the positive cone of the vector lattice of continuous and positively homogeneous real-valued functions of the n-dimensional euclidean space and a lattice, equipped with appropriate vector space operations, of a new kind of geometric objects. We call these sets star-shaped objects. Then we define a geometric notion of good sequence (a tool introduced by D.Mundici in the study of MV-algebras), and we prove a representation theorem for star-shaped objects, in terms of good sequences. Imposing a polyhedral condition on our star-shaped objects, we obtain a correspondence between them and the elements of the positive cone of the free vector lattice on n generators. Finally, we specialize the result obtained for good sequences to these polyhedral star-shaped objects

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group