Search results for "Vectorization"
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Set-valued Optimization
Set-valued optimization is a vibrant and expanding branch of mathematics that deals with optimization problems where the objective map and/or the constraints maps are set-valued maps acting between certain spaces. Since set-valued maps subsumes single valued maps, set-valued optimization provides an important extension and unification of the scalar as well as the vector optimization problems. Therefore this relatively new discipline has justifiably attracted a great deal of attention in recent years. This book presents, in a unified framework, basic properties on ordering relations, solution concepts for set-valued optimization problems, a detailed description of convex set-valued maps, most recent developments in separation theorems, scalarization techniques, variational principles, tangent cones of first and higher order, sub-differential of set-valued maps, generalized derivatives of set-valued maps, sensitivity analysis, optimality conditions, duality and applications in economics among other things.
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Topological Vector Spaces II
In the preface to Volume One I promised a second volume which would contain the theory of linear mappings and special classes of spaces im portant in analysis. It took me nearly twenty years to fulfill this promise, at least to some extent. To the six chapters of Volume One I added two new chapters, one on linear mappings and duality (Chapter Seven), the second on spaces of linear mappings (Chapter Eight). A glance at the Contents and the short introductions to the two new chapters will give a fair impression of the material included in this volume. I regret that I had to give up my intention to write a third chapter on nuclear spaces. It seemed impossible to include the recent deep results in this field without creating a great further delay. A substantial part of this book grew out of lectures I held at the Mathematics Department of the University of Maryland· during the academic years 1963-1964, 1967-1968, and 1971-1972. I would like to express my gratitude to my colleagues J. BRACE, S. GOLDBERG, J. HORVATH, and G. MALTESE for many stimulating and helpful discussions during these years. I am particularly indebted to H. JARCHOW (Ziirich) and D. KEIM (Frankfurt) for many suggestions and corrections. Both have read the whole manuscript. N. ADASCH (Frankfurt), V. EBERHARDT (Miinchen), H. MEISE (Diisseldorf), and R. HOLLSTEIN (Paderborn) helped with important observations.
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Random And Vector Measures
The book is devoted to the structural analysis of vector and random (or both) valued countably additive measures, and used for integral representations of random fields. The spaces can be Banach or Frechet types. Special attention is given to Bochner's boundedness principle and Grothendieck's representation unifying and simplyfying stochastic integrations. Several stationary aspects, extensions and random currents as well as related multilinear forms are analyzed, whilst numerous new procedures and results are included, and many research areas are opened up which also display the geometric aspects in multi dimensions.
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Introduction to Multivariable Analysis from Vector to Manifold
Multivariable analysis is of interest to pure and applied mathematicians, physicists, electrical, mechanical and systems engineers, mathematical economists, biologists, and statisticians. This book takes the student and researcher on a journey through the core topics of the subject. Systematic exposition, with numerous examples and exercises from the computational to the theoretical, makes difficult ideas as concrete as possible. Good bibliography and index.
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Fixed Point Theory In P-vector Spaces
This monograph provides an updated development of fixed point theory under a unified framework of the 'best approximation approach' in p-vector spaces, a core component of nonlinear analysis in mathematics, where p∊(0,1] (the same for p below unless specified). This book exposes some important contents of the new fixed point theory, with highlights in four parts.Specifically, the book focuses on the development of general new fixed point theory for both single-valued and set-valued mappings under the framework of p-vector and locally convex spaces for p∊(0,1], including topological vector spaces and locally convex spaces as special cases. It provides affirmative answers to the Schauder conjecture under the general setting of p-vector spaces and locally p-convex spaces. The book establishes best approximation results for upper semicontinuous and 1-set contractive set-valued mappings, which are used as tools to establish new fixed point theorems for non-self set-valued mappings with either inward or outward set conditions under various situations. These results improve or unify corresponding results in the existing literature for nonlinear analysis and lay the foundation for the development of fixed point theorems in topological vector spaces since Schauder's conjecture was raised in 1930. In addition, this book demonstrates the power of the fixed point theorem by showing the equivalence among the Ekeland variational principle, Takahashi minimization theorem, Oettli-Théra theorem, Caristi-Kirk type fixed point theorem, and related principles in nonlinear functional analysis.Overall, this book provides an accessible way to establish the new theory in the development of fixed point theorems and results. It is designed to be understandable for senior undergraduate students majoring in mathematics, physical sciences, social sciences, and related fields. We expect that this monograph will serve as a staple textbook for undergraduate and postgraduate students, a reference book for researchers in the field of fixed point theory in nonlinear functional analysis, and an accessible resource for general readers in mathematics and related disciplines.
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Operator Algebras Generated by Commuting Projections: A Vector Measure Approach
This book presents a systematic investigation of the theory of those commutative, unital subalgebras (of bounded linear operators acting in a Banach space) which are closed for some given topology and are generated by a uniformly bounded Boolean algebra of projections. One of the main aims is to employ the methods of vector measures and integration as a unifying theme throughout. This yields proofs of several classical results which are quite different to the classical ones. This book is directed to both those wishing to learn this topic for the first time and to current experts in the field.
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Arthropods as Vectors of Emerging Diseases
Global warming and globalization are the buzzwords of our time. They have nearly reached a religious status and those who deny their existence are considered modern heretics. Nevertheless, the earth has become an overcrowded village, traversable within a single day. Thus it is hardly surprising that besides persons and goods also agents of disease are easily transported daily from one end of the world to the other, threatening the health and lives of billions of humans and their animals. Agents of diseases (prions, viruses, bacteria, fungi and parasites) are not only transmitted by body contact or direct exchange of bodily fluids, but also by means of vectors which belong to the groups of licking or blood-sucking arthropods (mites, ticks, insects) that live close to humans and their houses. Without a doubt the recently accelerating globalization supports the import of agents of disease into countries where they never had been or where they had long since been eradicated, leading to a false sense of living on a “safe island.” These newly imported or reintroduced diseases – called “emerging diseases” – may lead to severe outbreaks in cases where the countries are not prepared to combat them, or in cases where viruses are introduced that cannot be controlled by medications or vaccines. Arthropods are well known vectors for the spread of diseases. Thus their invasion from foreign countries and their spreading close to human dwellings must be blocked everywhere (in donor and receptor countries) using safe and effective measures. This book presents reviews on examples of such arthropod-borne emerging diseases that lurk on the fringes of our crowded megacities. The following topics show that there is an ongoing invasion of potential vectors and that control measures must be used now in order to avoid disastrous outbreaks of mass diseases.
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Graph Classification And Clustering Based On Vector Space Embedding
This book is concerned with a fundamentally novel approach to graph-based pattern recognition based on vector space embedding of graphs. It aims at condensing the high representational power of graphs into a computationally efficient and mathematically convenient feature vector.This volume utilizes the dissimilarity space representation originally proposed by Duin and Pekalska to embed graphs in real vector spaces. Such an embedding gives one access to all algorithms developed in the past for feature vectors, which has been the predominant representation formalism in pattern recognition and related areas for a long time.
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Nonstandard Analysis and Vector Lattices
This volume collects applications of nonstandard methods to the theory of vector lattices. Primary attention is paid to combining infinitesimal and Boolean-valued constructions of use in the classical problems of representing abstract analytical objects, such as Banach-Kantorovich spaces, vector measures, and dominated and integral operators. The book is a complement to Volume 358 of "MIA Vector Lattices and Integral Operators", printed in 1996.
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Graph-Based Clustering and Data Visualization Algorithms
This work presents a data visualization technique that combines graph-based topology representation and dimensionality reduction methods to visualize the intrinsic data structure in a low-dimensional vector space. The application of graphs in clustering and visualization has several advantages. A graph of important edges (where edges characterize relations and weights represent similarities or distances) provides a compact representation of the entire complex data set. This text describes clustering and visualization methods that are able to utilize information hidden in these graphs, based on the synergistic combination of clustering, graph-theory, neural networks, data visualization, dimensionality reduction, fuzzy methods, and topology learning. The work contains numerous examples to aid in the understanding and implementation of the proposed algorithms, supported by a MATLAB toolbox available at an associated website.
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Knowledge Discovery with Support Vector Machines
An easy-to-follow introduction to support vector machines This book provides an in-depth, easy-to-follow introduction to support vector machines drawing only from minimal, carefully motivated technical and mathematical background material. It begins with a cohesive discussion of machine learning and goes on to cover: Knowledge discovery environments Describing data mathematically Linear decision surfaces and functions Perceptron learning Maximum margin classifiers Support vector machines Elements of statistical learning theory Multi-class classification Regression with support vector machines Novelty detection Complemented with hands-on exercises, algorithm descriptions, and data sets, Knowledge Discovery with Support Vector Machines is an invaluable textbook for advanced undergraduate and graduate courses. It is also an excellent tutorial on support vector machines for professionals who are pursuing research in machine learning and related areas.
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Bio-Carrier Vectors
This detailed book explores techniques used in research efforts to adopt and combine the best capabilities from natural and artificial vector systems to assemble improved delivery technologies aimed for specific applications. The collection discusses vectors other than the traditional viral and non-viral systems and explores ideas for blending the best features of bacteria, nanoparticles, peptides, and hybrid systems for the delivery of biomaterials into cells, as well as a chapter on the application of shotgun proteomics and mass spectrometry as a tool to analyze the proteomic profile changes in cells that result from these interventions. Written in the highly successful Methods in Molecular Biology format, chapters include introductions to their respective topics, lists of the necessary materials and reagents, step-by-step, readily reproducible laboratory protocols, and tips on troubleshooting and avoiding known pitfalls. Authoritative and practical, Bio-Carrier Vectors: Methods and Protocols serves as an ideal guide for researchers working toward harnessing the power of multiple vectors in gene and drug delivery.
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Geometric Invariant Theory, Holomorphic Vector Bundles and the Harder-Narasimhan Filtration
This book introduces key topics on Geometric Invariant Theory, a technique to obtaining quotients in algebraic geometry with a good set of properties, through various examples. It starts from the classical Hilbert classification of binary forms, advancing to the construction of the moduli space of semistable holomorphic vector bundles, and to Hitchin’s theory on Higgs bundles. The relationship between the notion of stability between algebraic, differential and symplectic geometry settings is also covered.Unstable objects in moduli problems -- a result of the construction of moduli spaces -- get specific attention in this work. The notion of the Harder-Narasimhan filtration as a tool to handle them, and its relationship with GIT quotients, provide instigating new calculations in several problems. Applications include a survey of research results on correspondences between Harder-Narasimhan filtrations with the GIT picture and stratifications of the moduli space of Higgs bundles.Graduate students and researchers who want to approach Geometric Invariant Theory in moduli constructions can greatly benefit from this reading, whose key prerequisites are general courses on algebraic geometry and differential geometry.
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Vector Databases
The AI revolution is here, and at its core lies a game-changing technology that most developers haven't fully explored: vector databases. From powering semantic search to enabling large language models (LLMs) and generative AI, vector databases are reshaping how we build applications with unstructured data like text, images, and audio. But how do you go from curious to capable with this vital technology? That's where this book comes in.In this hands-on guide, author Nitin Borwankar takes you through the "why, what, and how" of vector databases, starting with the basic theory behind vector embeddings and progressing to building applications with real-world tools. You'll learn about Word2vec, how to convert open source SQL databases like SQLite3 and PostgreSQL into vector databases, and integrate them into retrieval-augmented generation (RAG) applications. Whether you're a Python developer, data engineer, or ML practitioner, this book gives you the foundation to leverage vector databases confidently in your AI projects.Understand the connection between vector databases, embeddings, and LLMsLearn practical approaches for transforming SQL databases into vector databasesBuild RAG applications for both personal and enterprise useApply vector databases to solve real-world AI challengesLearn how to use vector databases with LLMs to build applications
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High Performance Computing on Vector Systems 2006
The book presents the state-of-the-art in high performance computing and simulation on modern supercomputer architectures. It covers trends in high performance application software development in general and specifically for parallel vector architectures. The contributions cover among others the field of computational fluid dynamics, physics, chemistry, and meteorology. Innovative application fields like reactive flow simulations and nano technology are presented.
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Lectures on Vector Bundles
This work consists of two courses on the moduli spaces of vector bundles. The first part tackles the classification of vector bundles on algebraic curves. The construction and elementary properties of the moduli spaces of stable bundles are also discussed. In particular, Hilbert-Grothendieck schemes of vector bundles are constructed, and Mumford's geometric invariant theory is succinctly treated. The second part centres on the structure of the moduli space of semi-stable sheaves on the projective plane. Existence conditions for sheaves of given rank and Chern Class and construction ideas are sketched in the general context of projective algebraic surfaces. Professor Le Potier has provided a treatment of vector bundles that will be welcomed by experienced algebraic geometers and novices alike.
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Vectored Propulsion, Supermaneuverability and Robot Aircraft
This book is designed to fill a professional vacuum in the new field of advance, high-angle, vectored stealth aircraft. The subject matter presented in the volume has never before been investigated and presented as a unified field of study because it covers entirely new fields and because specialized fragments of this unified field are scattered throughout literature in specific problems. The book is of interest to aeronautical and mechanical engineers, electrical and control engineers, aerospace industry, USAF, US Navy, NASA, pilots and instructors.
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Finite Elements in Vector Lattices
The book is the first systematical treatment of the theory of finite elements in Archimedean vector lattices and contains the results known on this topic up to the year 2013. It joins all important contributions achieved by a series of mathematicians that can only be found in scattered in literature.
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Cohomology of Vector Bundles and Syzygies
The central theme of this book is an exposition of the geometric technique of calculating syzygies. It is written from a point of view of commutative algebra, and without assuming any knowledge of representation theory the calculation of syzygies of determinantal varieties is explained. The starting point is a definition of Schur functors, and these are discussed from both an algebraic and geometric point of view. Then a chapter on various versions of Bott's Theorem leads on to a careful explanation of the technique itself, based on a description of the direct image of a Koszul complex. Applications to determinantal varieties follow, plus there are also chapters on applications of the technique to rank varieties for symmetric and skew symmetric tensors of arbitrary degree, closures of conjugacy classes of nilpotent matrices, discriminants and resultants. Numerous exercises are included to give the reader insight into how to apply this important method.
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