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

Matrix-product-state based studies of bosonic flux ladders

Flux ladders constitute the minimal setup enabling a systematic understanding of the rich physics of interacting particles subjected simultaneously to a strong magnetic field and a lattice potential. The recent realization of flux ladders in ultracold quantum gases with artificial magnetic fields has attracted great interest. In this thesis, we study various aspects of interacting bosonic flux ladders using extensive matrix-product-state based calculations. Specifically, the numerical techniques include the variational ground-state optimization by means of the density-matrix renormalization-group method, a purification approach for the study of finite-temperature states, as well as time-evolution methods for the simulation of quench dynamics. In an introductory part, we recapitulate key features and important ground-state phases of the flux-ladder model and discuss the numerical methods. Subsequently, the main results are presented as follows. First, the emphasis is put on model parameters which are envisioned to be realized in a future quantum gas experiment exploiting the internal states of potassium atoms as a synthetic dimension. Considering a particle filling of one boson per rung, we map out the ground-state phase diagram and report on a Mott-insulating Meissner phase as well as on biased-ladder phases, which might exist on top of superfluids and Mott insulators. Moreover, we demonstrate that quantum quenches of suitably chosen initial states can be used to probe the equilibrium properties in the transient dynamics. Concretely, we consider the instantaneous turning on of particle hopping along the rungs or legs in the synthetic flux-ladder model, with different initial particle distributions. We show that clear signatures of the biased-ladder phase and the Meissner phase can be observed in the transient dynamics. Second, we study the effect of finite temperatures in flux ladders. So far, most of the theoretical work on these systems has concentrated on zero-temperature phases while the finite-temperature regime remains largely unexplored. However, the question if and up to which temperature characteristic features of the ground-state phases persist is relevant in experimental realizations. In order to explore the finite-temperature regime, a matrix-product-state based purification approach for the simulation of strongly interacting bosons has been implemented. Our study is focused on chiral currents and momentum-distribution functions, which are key observables in ultracold quantum gases, and our main results concern the most prominent vortex liquid-to-Meissner crossover. We demonstrate that signatures of the vortex-liquid phase can still be detected at elevated temperatures from characteristic finite-momentum maxima in the momentum-distribution functions, while the vortex-liquid phase leaves weaker fingerprints in the chiral current. In order to determine the range of temperatures over which these signatures can be observed, we introduce a suitable measure for the contrast of these maxima. The results are condensed into a finite-temperature crossover diagram. Third, we investigate the Hall response in bosonic flux ladders. While flux ladders are the most simple lattice models giving rise to the Hall effect, the theoretical description of the many-body ground-state Hall response in these systems remains a tricky problem and an active line of research. In view of current quantum gas experiments, we discuss feasible schemes to extend measurements of the Hall polarization to a study of the Hall voltage, allowing for direct comparison with solid state systems. Most importantly, we report on characteristic zero crossings and a remarkable robustness of the Hall voltage with respect to interaction strengths, particle fillings, and ladder geometries, which is unobservable in the Hall polarization. Moreover, we investigate the site-resolved Hall response in spatially inhomogeneous quantum phases using a semiclassical approach. In conclusion, we present a brief summary of our work and touch on possible follow-up studies which are directly connected to the contents of this thesis.Flussleitermodelle beschreiben das komplexe Zusammenspiel von wechselwirkenden quantenmechanischen Teilchen, die sich unter dem Einfluss von effektiven Magnetfeldern in quasi-eindimensionalen Gittern bewegen. In den letzten Jahren konnten diese Modelle in optischen Gittern durch die Erzeugung von künstlichen Magnetfeldern für kalte Atome experimentell realisiert werden. Flussleitermodelle sind aufgrund ihrer formalen Einfachheit, ihrer reichhaltigen Phasendiagramme und gegenwärtiger Quantengasexperimente von großem Interesse. Die vorliegende Arbeit befasst sich mit der theoretischen Untersuchung von bosonischen Flussleitermodellen unter Verwendung numerischer Methoden, die auf Matrixproduktzuständen basieren. Wir erkunden Grundzustandsphasendiagramme mit dem Verfahren der Dichtematrix-Renormierungsgruppe. Darüber hinaus untersuchen wir thermische Zustände sowie dynamische Vielteilchenprobleme in Flussleitern mithilfe moderner Zeitentwicklungsmethoden. In einem einleitenden Teil dieser Arbeit wird das zentrale bosonische Flussleitermodell in den breiteren Forschungskontext eingeordnet. Wir diskutieren dessen wesentliche Eigenschaften und stellen die in dieser Arbeit verwendeten numerischen Methoden vor. Im Anschluss präsentieren wir die gewonnenen Forschungsergebnisse wie folgt. Zunächst liegt der Fokus auf Modellparametern, die durch ein angedachtes Experiment motiviert sind. In dem Experiment soll eine zweibeinige bosonische Flussleiter unter der Ausnutzung interner Spinzustände von kalten bosonischen Kaliumatomen realisiert werden. Wir zeigen, dass das zugehörige Grundzustandsphasendiagramm eine Mott-isolierende Meissner-Phase sowie superfluide und Mott-isolierende Biased-Ladder-Phasen aufweist. Mithilfe zeitabhängiger Simulationen demonstrieren wir, dass realistische Quantenquenchprotokolle es erlauben, Gleichgewichtseigenschaften der relevanten Grundzustandsphasen in der transienten Vielteilchendynamik zu beobachten und zu quantifizieren. Im Weiteren untersuchen wir die Quantenzustände von Flussleitern bei endlichen Temperaturen. Während die Nulltemperaturphasen von Flussleitermodellen im Zentrum zahlreicher theoretischer Arbeiten stehen, bleibt der Einfluss von Temperatureffekten auf die charakteristischen Grundzustandseigenschaften weitestgehend unerforscht. Dieser Einfluss spielt in Experimenten allerdings eine wichtige Rolle. Um die bei endlichen Temperaturen angenommenen Quantenzustände zu untersuchen bedienen wir uns einer Matrixproduktzustandsmethode, die im Rahmen dieser Arbeit implementiert wurde. Unsere Studie konzentriert sich auf chirale Randströme und charakteristische Quasiimpuls-Verteilungen, die in gegenwärtigen Quantengasexperimenten gemessen werden können. Für stark wechselwirkende Bosonen und ausgehend von dem Quantenphasenübergang von einer Vortex-Phase zu einer Meissner-Phase erarbeiten wir das zugehörige Crossoverdiagramm bei endlichen Temperaturen. Darüber hinaus untersuchen wir die Hall-Antwort bosonischer Flussleitermodelle. Flussleitern sind die minimalsten Gittermodelle, in denen sich ein Hall-Effekt untersuchen lässt. Dessen ungeachtet ist die Frage nach der Hall-Antwort in Quantenphasen, die auf Vielteilcheneffekten beruhen, theoretisch schwierig und Gegenstand aktueller Forschung. Vor dem Hintergrund gegenwärtiger Quantengasexperimente berichten wir über zeitabhängige Protokolle, mit denen sich Messungen der Hall-Polarisation auf Messungen der Hall-Spannung erweitern lassen. Durch umfangreiche numerische Simulationen zeigen wir, dass die Hall-Spannung in verschiedenen Quantenphasen eine große Robustheit im Hinblick auf die Wechselwirkungsstärke und die Teilchenfüllung aufweist. Diese Robustheit lässt sich in der Hall-Polarisation nicht beobachten. Wir untermauern unsere numerischen Ergebnisse mit semiklassischen Rechnungen und diskutieren die lokal aufgelöste Hall-Antwort in räumlich inhomogenen Vortexgitter-Phasen. Abschließend fassen wir die gewonnenen Ergebnisse kurz zusammen und erwähnen Folgestudien, die unmittelbar mit der vorliegenden Arbeit in Verbindung stehen

A modular genetic programming system

Genetic Programming (GP) is an evolutionary algorithm for the automatic discovery of symbolic expressions, e.g. computer programs or mathematical formulae, that encode solutions to a user-defined task. Recent advances in GP systems and computer performance made it possible to successfully apply this algorithm to real-world applications. This work offers three main contributions to the state-of-the art in GP systems: (I) The documentation of RGP, a state-of-the art GP software implemented as an extension package to the popular R environment for statistical computation and graphics. GP and RPG are introduced both formally and with a series of tutorial examples. As R itself, RGP is available under an open source license. (II) A comprehensive empirical analysis of modern GP heuristics based on the methodology of Sequential Parameter Optimization. The effects and interactions of the most important GP algorithm parameters are analyzed and recommendations for good parameter settings are given. (III) Two extensive case studies based on real-world industrial applications. The first application involves process control models in steel production, while the second is about meta-model-based optimization of cyclone dust separators. A comparison with traditional and modern regression methods reveals that GP offers equal or superior performance in both applications, with the additional benefit of understandable and easy to deploy models. Main motivation of this work is the advancement of GP in real-world application areas. The focus lies on a subset of application areas that are known to be practical for GP, first of all symbolic regression and classification. It has been written with practitioners from academia and industry in mind

Semantic Systems. The Power of AI and Knowledge Graphs

This open access book constitutes the refereed proceedings of the 15th International Conference on Semantic Systems, SEMANTiCS 2019, held in Karlsruhe, Germany, in September 2019. The 20 full papers and 8 short papers presented in this volume were carefully reviewed and selected from 88 submissions. They cover topics such as: web semantics and linked (open) data; machine learning and deep learning techniques; semantic information management and knowledge integration; terminology, thesaurus and ontology management; data mining and knowledge discovery; semantics in blockchain and distributed ledger technologies

Pretopological operators for gray-level image analysis

International audienceThis paper deals with new operators for gray-level image analysis. These operators are based on concepts of pretopology and they extend mathematical morphology operators. Instead of using one structuring element, these new operators use a basis of several structuring elements. If this basis is composed of only one element, these operators are equivalent to mathematical morphology ones. This article presents the pretopological representation space and four pretopological structures of operators. Relations between these new operators and the corresponding morphological operators are described and compared. Properties and examples are displayed

General Adaptive Neighborhood-Based Pretopological Image Filtering

International audienceThis paper introduces pretopological image filtering in the context of the General Adaptive Neighborhood Image Processing (GANIP) approach. Pretopological filters act on gray level image while satisfying some topological properties. The GANIP approach enables to get an image representation and mathematical structure for adaptive image processing and analysis. Then, the combination of pretopology and GANIP leads to efficient image operators. They enable to process images while preserving region structures without damaging image transitions. More precisely, GAN-based pretopological filters and GAN-based viscous pretopological filters are proposed in this paper. The viscous notion enables to adjust the filtering activity to the image gray levels. These adaptive filters are evaluated through several experiments highlighting their efficiency with respect to the classical operators. They are practically applied in both the biomedical and material application areas for image restoration, image background subtraction and image enhancement