On density theorems, connectedness results and error bounds in vector optimization.

Yung Hon-wai.Thesis (M.Phil.)--Chinese University of Hong Kong, 2001.Includes bibliographical references (leaves 133-139).Abstracts in English and Chinese.Chapter 0 --- Introduction --- p.1Chapter 1 --- Density Theorems in Vector Optimization --- p.7Chapter 1.1 --- Preliminary --- p.7Chapter 1.2 --- The Arrow-Barankin-Blackwell Theorem in Normed Spaces --- p.14Chapter 1.3 --- The Arrow-Barankin-Blackwell Theorem in Topolog- ical Vector Spaces --- p.27Chapter 1.4 --- Density Results in Dual Space Setting --- p.32Chapter 2 --- Density Theorem for Super Efficiency --- p.45Chapter 2.1 --- Definition and Criteria for Super Efficiency --- p.45Chapter 2.2 --- Henig Proper Efficiency --- p.53Chapter 2.3 --- Density Theorem for Super Efficiency --- p.58Chapter 3 --- Connectedness Results in Vector Optimization --- p.63Chapter 3.1 --- Set-valued Maps --- p.64Chapter 3.2 --- The Contractibility of the Efficient Point Sets --- p.67Chapter 3.3 --- Connectedness Results in Vector Optimization Prob- lems --- p.83Chapter 4 --- Error Bounds In Normed Spaces --- p.90Chapter 4.1 --- Error Bounds of Lower Semicontinuous Functionsin Normed Spaces --- p.91Chapter 4.2 --- Error Bounds of Lower Semicontinuous Convex Func- tions in Reflexive Banach Spaces --- p.100Chapter 4.3 --- Error Bounds with Fractional Exponents --- p.105Chapter 4.4 --- An Application to Quadratic Functions --- p.114Bibliography --- p.13

Compression bounds for Lipschitz maps from the Heisenberg group to L1L_1

We prove a quantitative bi-Lipschitz nonembedding theorem for the Heisenberg group with its Carnot-Carath\'eodory metric and apply it to give a lower bound on the integrality gap of the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem

The Encyclopedia of Neutrosophic Researchers - vol. 1

This is the first volume of the Encyclopedia of Neutrosophic Researchers, edited from materials offered by the authors who responded to the editor’s invitation. The authors are listed alphabetically. The introduction contains a short history of neutrosophics, together with links to the main papers and books. Neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics, neutrosophic measure, neutrosophic precalculus, neutrosophic calculus and so on are gaining significant attention in solving many real life problems that involve uncertainty, impreciseness, vagueness, incompleteness, inconsistent, and indeterminacy. In the past years the fields of neutrosophics have been extended and applied in various fields, such as: artificial intelligence, data mining, soft computing, decision making in incomplete / indeterminate / inconsistent information systems, image processing, computational modelling, robotics, medical diagnosis, biomedical engineering, investment problems, economic forecasting, social science, humanistic and practical achievements

Residual Networks as Flows of Diffeomorphisms

International audienceThis paper addresses the understanding and characterization of residual networks (ResNet), which are among the state-of-the-art deep learning architectures for a variety of supervised learning problems. We focus on the mapping component of ResNets, which map the embedding space towards a new unknown space where the prediction or classification can be stated according to linear criteria. We show that this mapping component can be regarded as the numerical implementation of continuous flows of diffeomorphisms governed by ordinary differential equations. Especially, ResNets with shared weights are fully characterized as numerical approximation of exponential diffeomorphic operators. We stress both theoretically and numerically the relevance of the enforcement of diffeormorphic properties and the importance of numerical issues to make consistent the continuous formulation and the discretized ResNet implementation. We further discuss the resulting theoretical and computational insights on ResNet architectures

Bayesian Optimisation for Planning under Uncertainty

Under an increasing demand for data to understand critical processes in our world, robots have become powerful tools to automatically gather data and interact with their environments. In this context, this thesis addresses planning problems where limited prior information leads to uncertainty about the outcomes of a robot's decisions. The methods are based on Bayesian optimisation (BO), which provides a framework to solve planning problems under uncertainty by means of probabilistic modelling. As a first contribution, the thesis provides a method to find energy-efficient paths over unknown terrains. The method applies a Gaussian process (GP) model to learn online how a robot's power consumption varies as a function of its configuration while moving over the terrain. BO is applied to optimise trajectories over the GP model being learnt so that they are informative and energetically efficient. The method was tested in experiments on simulated and physical environments. A second contribution addresses the problem of policy search in high-dimensional parameter spaces. To deal with high dimensionality the method combines BO with a coordinate-descent scheme that greatly improves BO's performance when compared to conventional approaches. The method was applied to optimise a control policy for a race car in a simulated environment and shown to outperform other optimisation approaches. Finally, the thesis provides two methods to address planning problems involving uncertainty in the inputs space. The first method is applied to actively learn terrain roughness models via proprioceptive sensing with a mobile robot under localisation uncertainty. Experiments demonstrate the method's performance in both simulations and a physical environment. The second method is derived for more general optimisation problems. In particular, this method is provided with theoretical guarantees and empirical performance comparisons against other approaches in simulated environments

The Necessary Structure of the All-pervading Aether: Discrete or Continuous? Simple or Symmetric?

In this book I investigate the necessary structure of the aether – the stuff that fills the whole universe. Some of my conclusions are. 1. There is an enormous variety of structures that the aether might, for all we know, have. 2. Probably the aether is point-free. 3. In that case, it should be distinguished from Space-time, which is either a fiction or a construct. 4. Even if the aether has points, we should reject the orthodoxy that all regions are grounded in points by summation. 5. If the aether is point-free but not continuous, its most likely structure has extended atoms that are not simples. 6. Space-time is symmetric if and only if the aether is continuous. 7. If the aether is continuous, we should reject the standard interpretation of General Relativity, in which geometry determines gravity. 8. Contemporary physics undermines an objection to discrete aether based on scale invariance, but does not offer much positive support

Fuzzy Sets, Fuzzy Logic and Their Applications 2020

The present book contains the 24 total articles accepted and published in the Special Issue “Fuzzy Sets, Fuzzy Logic and Their Applications, 2020” of the MDPI Mathematics journal, which covers a wide range of topics connected to the theory and applications of fuzzy sets and systems of fuzzy logic and their extensions/generalizations. These topics include, among others, elements from fuzzy graphs; fuzzy numbers; fuzzy equations; fuzzy linear spaces; intuitionistic fuzzy sets; soft sets; type-2 fuzzy sets, bipolar fuzzy sets, plithogenic sets, fuzzy decision making, fuzzy governance, fuzzy models in mathematics of finance, a philosophical treatise on the connection of the scientific reasoning with fuzzy logic, etc. It is hoped that the book will be interesting and useful for those working in the area of fuzzy sets, fuzzy systems and fuzzy logic, as well as for those with the proper mathematical background and willing to become familiar with recent advances in fuzzy mathematics, which has become prevalent in almost all sectors of the human life and activity