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A Geometric Approach to the Unification of Symbolic Structures and Neural Networks (Studies in Computational Intelligence #910)
by Tiansi DongThe unification of symbolist and connectionist models is a major trend in AI. The key is to keep the symbolic semantics unchanged. Unfortunately, present embedding approaches cannot. The approach in this book makes the unification possible. It is indeed a new and promising approach in AI. -Bo Zhang, Director of AI Institute, TsinghuaIt is indeed wonderful to see the reviving of the important theme Nural Symbolic Model. Given the popularity and prevalence of deep learning, symbolic processing is often neglected or downplayed. This book confronts this old issue head on, with a historical look, incorporating recent advances and new perspectives, thus leading to promising new methods and approaches. -Ron Sun (RPI), on Governing Board of Cognitive Science SocietyBoth for language and humor, approaches like those described in this book are the way to snickerdoodle wombats. -Christian F. Hempelmann (Texas A&M-Commerce) on Executive Board of International Society for Humor Studies
Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 2020-2022 (Lecture Notes in Mathematics #2327)
by Ronen Eldan Bo’az Klartag Alexander Litvak Emanuel MilmanThis book reflects general trends in the study of geometric aspects of functional analysis, understood in a broad sense. A classical theme in the local theory of Banach spaces is the study of probability measures in high dimension and the concentration of measure phenomenon. Here this phenomenon is approached from different angles, including through analysis on the Hamming cube, and via quantitative estimates in the Central Limit Theorem under thin-shell and related assumptions. Classical convexity theory plays a central role in this volume, as well as the study of geometric inequalities. These inequalities, which are somewhat in spirit of the Brunn-Minkowski inequality, in turn shed light on convexity and on the geometry of Euclidean space. Probability measures with convexity or curvature properties, such as log-concave distributions, occupy an equally central role and arise in the study of Gaussian measures and non-trivial properties of the heat flow in Euclidean spaces. Also discussed are interactions of this circle of ideas with linear programming and sampling algorithms, including the solution of a question in online learning algorithms using a classical convexity construction from the 19th century.
Geometric Aspects of Functional Analysis
by Bo'Az Klartag Shahar Mendelson Vitali D. MilmanThis collection of original papers related to the Israeli GAFA seminar (on Geometric Aspects of Functional Analysis) from the years 2006 to 2011 continues the long tradition of the previous volumes, which reflect the general trends of Asymptotic Geometric Analysis, understood in a broad sense, and are a source of inspiration for new research. Most of the papers deal with various aspects of the theory, including classical topics in the geometry of convex bodies, inequalities involving volumes of such bodies or more generally, logarithmically-concave measures, valuation theory, probabilistic and isoperimetric problems in the combinatorial setting, volume distribution on high-dimensional spaces and characterization of classical constructions in Geometry and Analysis (like the Legendre and Fourier transforms, derivation and others). All the papers here are original research papers.
Geometric Aspects of Functional Analysis
by Bo'Az Klartag Emanuel MilmanAs in the previous Seminar Notes, the current volume reflects general trends in the study of Geometric Aspects of Functional Analysis. Most of the papers deal with different aspects of Asymptotic Geometric Analysis, understood in a broad sense; many continue the study of geometric and volumetric properties of convex bodies and log-concave measures in high-dimensions and in particular the mean-norm, mean-width, metric entropy, spectral-gap, thin-shell and slicing parameters, with applications to Dvoretzky and Central-Limit-type results. The study of spectral properties of various systems, matrices, operators and potentials is another central theme in this volume. As expected, probabilistic tools play a significant role and probabilistic questions regarding Gaussian noise stability, the Gaussian Free Field and First Passage Percolation are also addressed. The historical connection to the field of Classical Convexity is also well represented with new properties and applications of mixed-volumes. The interplay between the real convex and complex pluri-subharmonic settings continues to manifest itself in several additional articles. All contributions are original research papers and were subject to the usual refereeing standards.
Geometric Aspects of Functional Analysis
by Bo'Az Klartag Emanuel MilmanAs in the previous Seminar Notes, the current volume reflects general trends in the study of Geometric Aspects of Functional Analysis, understood in a broad sense. A classical theme in the Local Theory of Banach Spaces which is well represented in this volume is the identification of lower-dimensional structures in high-dimensional objects. More recent applications of high-dimensionality are manifested by contributions in Random Matrix Theory, Concentration of Measure and Empirical Processes. Naturally, the Gaussian measure plays a central role in many of these topics, and is also studied in this volume; in particular, the recent breakthrough proof of the Gaussian Correlation Conjecture is revisited. The interplay of the theory with Harmonic and Spectral Analysis is also well apparent in several contributions. The classical relation to both the primal and dual Brunn-Minkowski theories is also well represented, and related algebraic structures pertaining to valuations and valent functions are discussed. All contributions are original research papers and were subject to the usual refereeing standards.
Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 2017-2019 Volume I (Lecture Notes in Mathematics #2256)
by Bo’az Klartag Emanuel MilmanContinuing the theme of the previous volumes, these seminar notes reflect general trends in the study of Geometric Aspects of Functional Analysis, understood in a broad sense. Two classical topics represented are the Concentration of Measure Phenomenon in the Local Theory of Banach Spaces, which has recently had triumphs in Random Matrix Theory, and the Central Limit Theorem, one of the earliest examples of regularity and order in high dimensions. Central to the text is the study of the Poincaré and log-Sobolev functional inequalities, their reverses, and other inequalities, in which a crucial role is often played by convexity assumptions such as Log-Concavity. The concept and properties of Entropy form an important subject, with Bourgain's slicing problem and its variants drawing much attention. Constructions related to Convexity Theory are proposed and revisited, as well as inequalities that go beyond the Brunn–Minkowski theory. One of the major current research directions addressed is the identification of lower-dimensional structures with remarkable properties in rather arbitrary high-dimensional objects. In addition to functional analytic results, connections to Computer Science and to Differential Geometry are also discussed.
Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 2017-2019 Volume II (Lecture Notes in Mathematics #2266)
by Bo’az Klartag Emanuel MilmanContinuing the theme of the previous volumes, these seminar notes reflect general trends in the study of Geometric Aspects of Functional Analysis, understood in a broad sense. Two classical topics represented are the Concentration of Measure Phenomenon in the Local Theory of Banach Spaces, which has recently had triumphs in Random Matrix Theory, and the Central Limit Theorem, one of the earliest examples of regularity and order in high dimensions. Central to the text is the study of the Poincaré and log-Sobolev functional inequalities, their reverses, and other inequalities, in which a crucial role is often played by convexity assumptions such as Log-Concavity. The concept and properties of Entropy form an important subject, with Bourgain's slicing problem and its variants drawing much attention. Constructions related to Convexity Theory are proposed and revisited, as well as inequalities that go beyond the Brunn–Minkowski theory. One of the major current research directions addressed is the identification of lower-dimensional structures with remarkable properties in rather arbitrary high-dimensional objects. In addition to functional analytic results, connections to Computer Science and to Differential Geometry are also discussed.
Geometric Aspects of General Topology
by Katsuro SakaiThis book is designed for graduate students to acquire knowledge of dimension theory, ANR theory (theory of retracts), and related topics. These two theories are connected with various fields in geometric topology and in general topology as well. Hence, for students who wish to research subjects in general and geometric topology, understanding these theories will be valuable. Many proofs are illustrated by figures or diagrams, making it easier to understand the ideas of those proofs. Although exercises as such are not included, some results are given with only a sketch of their proofs. Completing the proofs in detail provides good exercise and training for graduate students and will be useful in graduate classes or seminars. Researchers should also find this book very helpful, because it contains many subjects that are not presented in usual textbooks, e.g., dim X × I = dim X + 1 for a metrizable space X; the difference between the small and large inductive dimensions; a hereditarily infinite-dimensional space; the ANR-ness of locally contractible countable-dimensional metrizable spaces; an infinite-dimensional space with finite cohomological dimension; a dimension raising cell-like map; and a non-AR metric linear space. The final chapter enables students to understand how deeply related the two theories are. Simplicial complexes are very useful in topology and are indispensable for studying the theories of both dimension and ANRs. There are many textbooks from which some knowledge of these subjects can be obtained, but no textbook discusses non-locally finite simplicial complexes in detail. So, when we encounter them, we have to refer to the original papers. For instance, J.H.C. Whitehead's theorem on small subdivisions is very important, but its proof cannot be found in any textbook. The homotopy type of simplicial complexes is discussed in textbooks on algebraic topology using CW complexes, but geometrical arguments using simplicial complexes are rather easy.
Geometric Aspects of the Trace Formula (Simons Symposia)
by Werner Müller Sug Woo Shin Nicolas TemplierThe second of three volumes devoted to the study of the trace formula, these proceedings focus on automorphic representations of higher rank groups. Based on research presented at the 2016 Simons Symposium on Geometric Aspects of the Trace Formula that took place in Schloss Elmau, Germany, the volume contains both original research articles and articles that synthesize current knowledge and future directions in the field. The articles discuss topics such as the classification problem of representations of reductive groups, the structure of Langlands and Arthur packets, interactions with geometric representation theory, and conjectures on the global automorphic spectrum. Suitable for both graduate students and researchers, this volume presents the latest research in the field. Readers of the first volume Families of Automorphic Forms and the Trace Formula will find this a natural continuation of the study of the trace formula.
Geometric Complex Analysis: In Honor of Kang-Tae Kim’s 60th Birthday, Gyeongju, Korea, 2017 (Springer Proceedings in Mathematics & Statistics #246)
by Jisoo Byun Hong Rae Cho Sung Yeon Kim Kang-Hyurk Lee Jong-Do ParkThe KSCV Symposium, the Korean Conference on Several Complex Variables, started in 1997 in an effort to promote the study of complex analysis and geometry. Since then, the conference met semi-regularly for about 10 years and then settled on being held biannually. The sixth and tenth conferences were held in 2002 and 2014 as satellite conferences to the Beijing International Congress of Mathematicians (ICM) and the Seoul ICM, respectively. The purpose of the KSCV Symposium is to organize the research talks of many leading scholars in the world, to provide an opportunity for communication, and to promote new researchers in this field.
Geometric Concepts for Geometric Design
by null Hartmut Prautzsch null Wolfgang BoehmThis book is a comprehensive tool both for self-study and for use as a text in classical geometry. It explains the concepts that form the basis for computer-aided geometric design.
Geometric Continuum Mechanics (Advances in Mechanics and Mathematics #42)
by Marcelo Epstein Reuven SegevThis contributed volume explores the applications of various topics in modern differential geometry to the foundations of continuum mechanics. In particular, the contributors use notions from areas such as global analysis, algebraic topology, and geometric measure theory. Chapter authors are experts in their respective areas, and provide important insights from the most recent research. Organized into two parts, the book first covers kinematics, forces, and stress theory, and then addresses defects, uniformity, and homogeneity. Specific topics covered include:Global stress and hyper-stress theoriesApplications of de Rham currents to singular dislocationsManifolds of mappings for continuum mechanicsKinematics of defects in solid crystalsGeometric Continuum Mechanics will appeal to graduate students and researchers in the fields of mechanics, physics, and engineering who seek a more rigorous mathematical understanding of the area. Mathematicians interested in applications of analysis and geometry will also find the topics covered here of interest.
Geometric Control of Fracture and Topological Metamaterials (Springer Theses)
by Noah MitchellThis thesis reports a rare combination of experiment and theory on the role of geometry in materials science. It is built on two significant findings: that curvature can be used to guide crack paths in a predictive way, and that protected topological order can exist in amorphous materials. In each, the underlying geometry controls the elastic behavior of quasi-2D materials, enabling the control of crack propagation in elastic sheets and the control of unidirectional waves traveling at the boundary of metamaterials. The thesis examines the consequences of this geometric control in a range of materials spanning many orders of magnitude in length scale, from amorphous macroscopic networks and elastic continua to nanoscale lattices.
Geometric Control Theory and Sub-Riemannian Geometry
by Gianna Stefani Ugo Boscain Jean-Paul Gauthier Andrey Sarychev Mario SigalottiHonoring Andrei Agrachev's 60th birthday, this volume presents recent advances in the interaction between Geometric Control Theory and sub-Riemannian geometry. On the one hand, Geometric Control Theory used the differential geometric and Lie algebraic language for studying controllability, motion planning, stabilizability and optimality for control systems. The geometric approach turned out to be fruitful in applications to robotics, vision modeling, mathematical physics etc. On the other hand, Riemannian geometry and its generalizations, such as sub-Riemannian, Finslerian geometry etc. , have been actively adopting methods developed in the scope of geometric control. Application of these methods has led to important results regarding geometry of sub-Riemannian spaces, regularity of sub-Riemannian distances, properties of the group of diffeomorphisms of sub-Riemannian manifolds, local geometry and equivalence of distributions and sub-Riemannian structures, regularity of the Hausdorff volume, etc.
Geometric Flows and the Geometry of Space-time (Tutorials, Schools, and Workshops in the Mathematical Sciences)
by Vicente Cortés Klaus Kröncke Jan LouisThis book consists of two lecture notes on geometric flow equations (O. Schnürer) and Lorentzian geometry - holonomy, spinors and Cauchy Problems (H. Baum and T. Leistner) written by leading experts in these fields. It grew out of the summer school “Geometric flows and the geometry of space-time” held in Hamburg (2016) and provides an excellent introduction for students of mathematics and theoretical physics to important themes of current research in global analysis, differential geometry and mathematical physics
Geometric Flows on Planar Lattices (Pathways in Mathematics)
by Andrea Braides Margherita SolciThis book introduces the reader to important concepts in modern applied analysis, such as homogenization, gradient flows on metric spaces, geometric evolution, Gamma-convergence tools, applications of geometric measure theory, properties of interfacial energies, etc. This is done by tackling a prototypical problem of interfacial evolution in heterogeneous media, where these concepts are introduced and elaborated in a natural and constructive way. At the same time, the analysis introduces open issues of a general and fundamental nature, at the core of important applications. The focus on two-dimensional lattices as a prototype of heterogeneous media allows visual descriptions of concepts and methods through a large amount of illustrations.
Geometric Folding Algorithms
by Erik D. Demaine Joseph O'RourkeDid you know that any straight-line drawing on paper can be folded so that the complete drawing can be cut out with one straight scissors cut? That there is a planar linkage that can trace out any algebraic curve, or even 'sign your name'? Or that a 'Latin cross' unfolding of a cube can be refolded to 23 different convex polyhedra? Over the past decade, there has been a surge of interest in such problems, with applications ranging from robotics to protein folding. With an emphasis on algorithmic or computational aspects, this treatment gives hundreds of results and over 60 unsolved 'open problems' to inspire further research. The authors cover one-dimensional (1D) objects (linkages), 2D objects (paper), and 3D objects (polyhedra). Aimed at advanced undergraduate and graduate students in mathematics or computer science, this lavishly illustrated book will fascinate a broad audience, from school students to researchers.
Geometric Function Theory: A Second Course in Complex Analysis (Springer Undergraduate Mathematics Series)
by Tom CarrollThis textbook provides a second course in complex analysis with a focus on geometric aspects. It covers topics such as the spherical geometry of the extended complex plane, the hyperbolic geometry of the Poincaré disk, conformal mappings, the Riemann Mapping Theorem and uniformisation of planar domains, characterisations of simply connected domains, the convergence of Riemann maps in terms of Carathéodory convergence of the image domains, normal families and Picard's theorems on value distribution, as well as the fundamentals of univalent function theory. Throughout the text, the synergy between analysis and geometry is emphasised, with proofs chosen for their directness. The textbook is self-contained, requiring only a first undergraduate course in complex analysis. The minimal topology needed is introduced as necessary. While primarily aimed at upper-level undergraduates, the book also serves as a concise reference for graduates working in complex analysis.
Geometric Function Theory in Higher Dimension (Springer INdAM #26)
by Filippo BracciThe book collects the most relevant outcomes from the INdAM Workshop “Geometric Function Theory in Higher Dimension” held in Cortona on September 5-9, 2016. The Workshop was mainly devoted to discussions of basic open problems in the area, and this volume follows the same line. In particular, it offers a selection of original contributions on Loewner theory in one and higher dimensions, semigroups theory, iteration theory and related topics. Written by experts in geometric function theory in one and several complex variables, it focuses on new research frontiers in this area and on challenging open problems. The book is intended for graduate students and researchers working in complex analysis, several complex variables and geometric function theory.
Geometric Function Theory in One and Higher Dimensions
by Ian Graham Gabriela KohrThis reference details valuable results that lead to improvements in existence theorems for the Loewner differential equation in higher dimensions, discusses the compactness of the analog of the Caratheodory class in several variables, and studies various classes of univalent mappings according to their geometrical definitions. It introduces the in
Geometric Integration Theory
by Hassler WhitneyGeared toward upper-level undergraduates and graduate students, this treatment of geometric integration theory consists of three parts: an introduction to classical theory, a postulational approach to general theory, and a final section that continues the general study with Lebesgue theory.The introductory chapter shows how the simplest hypotheses lead to the employment of basic tools. The opening third of the treatment, an examination of classical theory, leads to the theory of the Riemann integral and includes a study of smooth (i.e., differentiable) manifolds. The second part, on general theory, explores abstract integration theory, some relations between chains and functions, general properties of chains and cochains, and chains and cochains in open sets. The third and final section surveys Lebesgue theory in terms of flat cochains and differential forms, Lipschitz mappings, and chains and additive set functions. Appendixes on vector and linear spaces, geometric and topological preliminaries, and analytical preliminaries, along with indexes of symbols and terms, conclude the text.
Geometric Invariant Theory for Polarized Curves
by Gilberto Bini Fabio Felici Margarida Melo Filippo VivianiWe investigate GIT quotients of polarized curves. More specifically, we study the GIT problem for the Hilbert and Chow schemes of curves of degree d and genus g in a projective space of dimension d-g, as d decreases with respect to g. We prove that the first three values of d at which the GIT quotients change are given by d=a(2g-2) where a=2, 3. 5, 4. We show that, for a>4, L. Caporaso's results hold true for both Hilbert and Chow semistability. If 3. 5
Geometric Invariant Theory, Holomorphic Vector Bundles and the Harder-Narasimhan Filtration (SpringerBriefs in Mathematics)
by Alfonso Zamora Saiz Ronald A. Zúñiga-RojasThis 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.
Geometric Magic Squares: A Challenging New Twist Using Colored Shapes Instead of Numbers (Dover Recreational Math)
by Lee C.F. SallowsTraditional magic squares employ a chessboard-like arrangement of numbers in which the total of all rows, columns, and diagonals add up to the same number. This innovative approach by a British engineer shines a new light on the centuries-old amusement, challenging puzzlists to think two dimensionally in solving dozens of creative puzzles with vivid geometric shapes instead of numbers. Attempt The Twelve Pentominoes, a challenge consisting of the formation of a dozen different shapes from five-unit squares. Squeeze together the 16 pieces of a 4 x 4 self-interlocking square to form an empty 4 x 4 array. Take aim at 48 targets featuring astrological signs in a self-interlocking square. Suitable for puzzle lovers ages 12 and up, these and other colorful variations on conventional magic squares offer hours of captivating entertainment.
Geometric Measure Theory and Free Boundary Problems: Cetraro, Italy 2019 (Lecture Notes in Mathematics #2284)
by Guido De Philippis Xavier Ros-Oton Georg S. WeissThis volume covers contemporary aspects of geometric measure theory with a focus on applications to partial differential equations, free boundary problems and water waves. It is based on lectures given at the 2019 CIME summer school “Geometric Measure Theory and Applications – From Geometric Analysis to Free Boundary Problems” which took place in Cetraro, Italy, under the scientific direction of Matteo Focardi and Emanuele Spadaro. Providing a description of the structure of measures satisfying certain differential constraints, and covering regularity theory for Bernoulli type free boundary problems and water waves as well as regularity theory for the obstacle problems and the developments leading to applications to the Stefan problem, this volume will be of interest to students and researchers in mathematical analysis and its applications.