Type reduction operators for interval type–2 defuzzification

Fuzzy sets are an important approach to model uncertainty. Defuzzification maps fuzzy sets to non–fuzzy (crisp) values. Type–2 fuzzy sets model uncertainty in the degree of membership in a fuzzy set. Type–2 defuzzification maps type–2 fuzzy sets to non–fuzzy values. Type reduction maps type–2 fuzzy sets to type–1 fuzzy sets, in order to make type–2 defuzzification easier and to implement more efficient type–2 defuzzification algorithms. This paper is a first step towards a theoretical foundation of the emerging field of type reduction. Five mathematical properties of type reduction are defined, and two existing type reduction methods (Nie–Tan and uncertainty weight) are examined with respect to our five properties. Furthermore, two new type reduction methods are proposed: consistent linear type reduction and consistent quadratic type reduction. All our five properties are satisfied by consistent quadratic type reduction

A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping "pictures". We hope that this paper will contribute to the appreciation and apprehension of a stream of current research directions in image understanding.Comment: 65 pages, 33 figures, 303 reference

Loop Quantum Cosmology: A Status Report

The goal of this article is to provide an overview of the current state of the art in loop quantum cosmology for three sets of audiences: young researchers interested in entering this area; the quantum gravity community in general; and, cosmologists who wish to apply loop quantum cosmology to probe modifications in the standard paradigm of the early universe. An effort has been made to streamline the material so that, as described at the end of section I, each of these communities can read only the sections they are most interested in, without a loss of continuity.Comment: 138 pages, 15 figures. Invited Topical Review, To appear in Classical and Quantum Gravity. Typos corrected, clarifications and references adde

Quantum Cosmology

Within the second half of the last century, quantum cosmology concretely became one of the main research lines within gravitational theory and cosmology. Substantial progress has been made. Furthermore, quantum cosmology can become a domain that will gradually develop further over the next handful of decades, perhaps assisted by technological developments. Indications for new physics (i.e., beyond the standard model of particle physics or general relativity) could emerge and then the observable universe would surely be seen from quite a new perspective. This motivates bringing quantum cosmology to more research groups and individuals.This Special Issue (SI) aims to provide a wide set of reviews, ranging from foundational issues to (very) recent advancing discussions. Concretely, we want to inspire new work proposing observational tests, providing an aggregated set of contributions, covering several lines, some of which are thoroughly explored, some allowing progress, and others much unexplored. The aim of this SI is motivate new researchers to employ and further develop quantum cosmology over the forthcoming decades. Textbooks and reviews exist on the present subject, and this SI will complementarily assist in offering open access to a set of wide-ranging reviews. Hopefully, this will assist new interested researchers, in having a single open access online volume, with reviews that can help. In particular, this will help in selecting what to explore, what to read in more detail, where to proceed, and what to investigate further within quantum cosmology

Aspects of Emergent Geometry, Strings, and Branes in Gauge / Gravity Duality

We explore the emergence of locality and geometry in string theories from the perspective of gauge theories using gauge / gravity duality.First, we explicitly construct open strings stretched between giant gravitons in N=4 SYM. We find that these strings satisfy a relativistic dispersion relation up to three-loop order and conjecture that this should hold to all loop orders. We find the explicit dual solution to the string sigma model and find exact agreement with the geometric nature of the SYM operator and dispersion relation. Using these open strings as probes, we explore the local field theory on the worldvolume of the giant gravitons.Second, we use classical configurations in holographic matrix models to understand the emergence of geometry from matrix coordinates. We construct an effective Hamiltonian for a probe brane that observes the geometry in a background matrix configuration from which we can construct membranes embedded in three dimensional space. Adding angular momentum to these configurations we are able to observe continuous topology changes. We also study the classical evolution of holographic matrix models to generate a microcanonical ensemble of configurations and study their thermal and chaotic behavior. We argue that these thermal configurations are dual to black holes