The Volume of Vector Fields on Riemannian Manifolds
This book focuses on the study of the volume of vector fields on Riemannian manifolds. Providing a thorough overview of research on vector fields defining minimal submanifolds, and on the existence and characterization of volume minimizers, it includes proofs of the most significant results obtained since the subject's introduction in 1986. Aiming to inspire further research, it also highlights a selection of intriguing open problems, and exhibits some previously unpublished results. The presentation is direct and deviates substantially from the usual approaches found in the literature, requiring a significant revision of definitions, statements, and proofs. A wide range of topics is covered, including: a discussion on the conditions for a vector field on a Riemannian manifold to determine a minimal submanifold within its tangent bundle with the Sasaki metric; numerous examples of minimal vector fields (including those of constant length on punctured spheres); athorough analysis of Hopf vector fields on odd-dimensional spheres and their quotients; and a description of volume-minimizing vector fields of constant length on spherical space forms of dimension three. Each chapter concludes with an up-to-date survey which offers supplementary information and provides valuable insights into the material, enhancing the reader's understanding of the subject. Requiring a solid understanding of the fundamental concepts of Riemannian geometry, the book will be useful for researchers and PhD students with an interest in geometric analysis.
Applications of Holomorphic Functions in Geometry
This book expounds on the recent developments in applications of holomorphic functions in the theory of hypercomplex and anti-Hermitian manifolds as well as in the geometry of bundles. It provides detailed information about holomorphic functions in algebras and discusses some of the areas in geometry with applications. The book proves the existence of a one-to-one correspondence between hyper-complex anti-K瓣hler manifolds and anti-Hermitian manifolds with holomorphic metrics, and also a deformed lifting to bundles. Researchers and students of geometry, algebra, topology and physics may find the book useful as a self-study guide.
Complex Geometry of Slant Submanifolds
This book contains an up-to-date survey and self-contained chapters on complex slant submanifolds and geometry, authored by internationally renowned researchers. The book discusses a wide range of topics, including slant surfaces, slant submersions, nearly Kaehler, locally conformal Kaehler, and quaternion Kaehler manifolds. It provides several classification results of minimal slant surfaces, quasi-minimal slant surfaces, slant surfaces with parallel mean curvature vector, pseudo-umbilical slant surfaces, and biharmonic and quasi biharmonic slant surfaces in Lorentzian complex space forms. Furthermore, this book includes new results on slant submanifolds of para-Hermitian manifolds. This book also includes recent results on slant lightlike submanifolds of indefinite Hermitian manifolds, which are of extensive use in general theory of relativity and potential applications in radiation and electromagnetic fields. Various open problems and conjectures on slant surfaces in complex space forms are also included in the book. It presents detailed information on the most recent advances in the area, making it valuable for scientists, educators and graduate students.
Practical solution of selected geometrical problems by the iterative approximation method
Scientific Essay from the year 2023 in the subject Mathematics - Geometry, language: English, abstract: The book presents practical solutions for geometric problems using iterative approximation. But what is a geometric problem, and in particular its solution by iterative approximation? I will not discuss or define the basic principles of Euclidean geometry, nor its origins, which are lost in the darkness of history, nor will I discuss its development. I merely want the reader to find a practical way to solve geometrical problems. At the same time, geometry should not only be considered a technician's, engineer's or scientist's field of work but as a toy that opens the way to an intellectually active mind, at the highest level.
Heritage
The study of geometry can play an important role in stimulating mathematical imagination and intuition, particularly in its relation to algebra. The author of this book is convinced that the two are but different sides of the same coin, and that they should be presented together to a student of mathematics as soon as the curriculum will permit. Certainly, no graduate of an Honours course should miss at least a brief exposure to these stimulating ideas.When the first edition of this book appeared, a reviewer in the American Mathematical Monthly commented: 'If this book had a subtitle, it might well have been "The Theorems of Desargues and Pappus," for these two theorems play a central role throughout the text; their dependence and independence of various axioms is continually studied. Nowhere before, in English, has the importance of these two theorems been so carefully demonstrated...this is a most excellent book, and it will inspire many students of mathematics.'Largely as a result of the work of Ruth Moufang, the interplay between algebraic and geometrical ideas has been further investigated of recent years. Many interesting problems have come to light, some of which have been solved. In this fourth edition, an Appendix has been added which attempts to summarize a part of this work and provide the reader with references so that he may learn more of these interesting developments.
Geometry, Geodesics, and the Universe
The story of the development of geometry is told as it emerged from the concepts of the ancient Greeks, familiar from high school, to the four-dimensional space-time that is central to our modern vision of the universe. The reader is first reacquainted with the geometric system compiled by Euclid with its postulates thought to be self-evident truths. A particular focus is on Euclid's fifth postulate, the Parallel Postulate and the many efforts to improve Euclid's system over hundreds of years by proving it from the first four postulates. Two thousand years after Euclid, in the process that would reveal the Parallel Postulate as an independent postulate, a new geometry was discovered that changed the understanding of geometry and mathematics, while paving the way for Einstein's General Relativity. The mathematics to describe the non-Euclidean geometries and the geometric universe of General Relativity is initiated in the language of mathematics available to a general audience. The story is told as a mathematical narrative, bringing the reader along step by step with all the background needed in analytic geometry, the calculus, vectors, and Newton's laws to allow the reader to move forward to the revolutionary extension of geometry by Riemann that would supply Einstein with the language needed to overthrow Newton's universe. Using the mathematics acquired for Riemannian geometry, the principles behind Einstein's General Relativity are described and their realization in the Field Equations is presented. From the Field Equations, it is shown how they govern the curved paths of light and that of planets along the geodesics formed from the geometry of space-time, and how they provide a picture of the universe's birth, expansion, and future. Thus, Euclid's geometry while no longer thought to spring from perceived absolute truths as the ancients believed, ultimately provided the seed for a new understanding of geometry that in its infinite variety became central to the description of the universe, marking mathematics as a one of the great modes of human expression.
Learn Math Fast System Volume III
This is the black and white edition of the Learn Math Fast System Volume 3. This book teaches pre-algebra in just a month or two. The older you are the faster you'll learn. Topics include Order of Operations, solving for x, solving for x in terms of y, Ratios, Proportions, Combining Like Terms, Exponents, Square Root, Distributive Property, Slopes, Graphing, Y-Intercept, Linear Equations. The Learn Math Fast System is being used in all 50 States and a dozen other countries with TREMENDOUS success! Read the reviews, it's the one math program that REALLY WORKS! Everything is included; lessons, worksheets, tests, and answers with full solutions.
Empowering Novel Geometric Algebra for Graphics and Engineering
This book constitutes the proceedings of the Workshop Empowering Novel Geometric Algebra for Graphics and Engineering, ENGAGE 2022, held in conjunction with Computer Graphics International conference, CGI 2022, which took place virtually, in September 2022. The 10 full papers included in this volume were carefully reviewed and selected from 12 submissions. The workshop focused specifically on important aspects of geometric algebra including algebraic foundations, digitized transformations, orientation, conic fitting, protein modelling, digital twinning, and multidimensional signal processing.
Geometry, Geodesics, and the Universe
The story of the development of geometry is told as it emerged from the concepts of the ancient Greeks, familiar from high school, to the four-dimensional space-time that is central to our modern vision of the universe. The reader is first reacquainted with the geometric system compiled by Euclid with its postulates thought to be self-evident truths. A particular focus is on Euclid's fifth postulate, the Parallel Postulate and the many efforts to improve Euclid's system over hundreds of years by proving it from the first four postulates. Two thousand years after Euclid, in the process that would reveal the Parallel Postulate as an independent postulate, a new geometry was discovered that changed the understanding of geometry and mathematics, while paving the way for Einstein's General Relativity. The mathematics to describe the non-Euclidean geometries and the geometric universe of General Relativity is initiated in the language of mathematics available to a general audience. The story is told as a mathematical narrative, bringing the reader along step by step with all the background needed in analytic geometry, the calculus, vectors, and Newton's laws to allow the reader to move forward to the revolutionary extension of geometry by Riemann that would supply Einstein with the language needed to overthrow Newton's universe. Using the mathematics acquired for Riemannian geometry, the principles behind Einstein's General Relativity are described and their realization in the Field Equations is presented. From the Field Equations, it is shown how they govern the curved paths of light and that of planets along the geodesics formed from the geometry of space-time, and how they provide a picture of the universe's birth, expansion, and future. Thus, Euclid's geometry while no longer thought to spring from perceived absolute truths as the ancients believed, ultimately provided the seed for a new understanding of geometry that in its infinite variety became central to the description of the universe, marking mathematics as a one of the great modes of human expression.
A Book for Thinkers!
The idea for this book came to the author when he was talking with some men and women who had been looking at and thinking about how to better understand some of today's problems. They said they would appreciate and enjoy activities to help them understand new issues and arrive at better and valid conclusions, but they added to please keep it simple.
A Book for Thinkers!
The idea for this book came to the author when he was talking with some men and women who had been looking at and thinking about how to better understand some of today's problems. They said they would appreciate and enjoy activities to help them understand new issues and arrive at better and valid conclusions, but they added to please keep it simple.
An Introduction to Optimization on Smooth Manifolds
Optimization on Riemannian manifolds-the result of smooth geometry and optimization merging into one elegant modern framework-spans many areas of science and engineering, including machine learning, computer vision, signal processing, dynamical systems and scientific computing. This text introduces the differential geometry and Riemannian geometry concepts that will help students and researchers in applied mathematics, computer science and engineering gain a firm mathematical grounding to use these tools confidently in their research. Its charts-last approach will prove more intuitive from an optimizer's viewpoint, and all definitions and theorems are motivated to build time-tested optimization algorithms. Starting from first principles, the text goes on to cover current research on topics including worst-case complexity and geodesic convexity. Readers will appreciate the tricks of the trade for conducting research and for numerical implementations sprinkled throughout the book.
An Introduction to Optimization on Smooth Manifolds
Optimization on Riemannian manifolds-the result of smooth geometry and optimization merging into one elegant modern framework-spans many areas of science and engineering, including machine learning, computer vision, signal processing, dynamical systems and scientific computing. This text introduces the differential geometry and Riemannian geometry concepts that will help students and researchers in applied mathematics, computer science and engineering gain a firm mathematical grounding to use these tools confidently in their research. Its charts-last approach will prove more intuitive from an optimizer's viewpoint, and all definitions and theorems are motivated to build time-tested optimization algorithms. Starting from first principles, the text goes on to cover current research on topics including worst-case complexity and geodesic convexity. Readers will appreciate the tricks of the trade for conducting research and for numerical implementations sprinkled throughout the book.
Associative Algebraic Geometry
Classical Deformation Theory is used for determining the completions of local rings of an eventual moduli space. When a moduli variety exists, the main result explored in the book is that the local ring in a closed point can be explicitly computed as an algebraization of the pro-representing hull, called the local formal moduli, of the deformation functor for the corresponding closed point.The book gives explicit computational methods and includes the most necessary prerequisites for understanding associative algebraic geometry. It focuses on the meaning and the place of deformation theory, resulting in a complete theory applicable to moduli theory. It answers the question 'why moduli theory', and gives examples in mathematical physics by looking at the universe as a moduli of molecules, thereby giving a meaning to most noncommutative theories.The book contains the first explicit definition of a noncommutative scheme, not necessarily covered by commutative rings. This definition does not contradict any previous abstract definitions of noncommutative algebraic geometry, but sheds interesting light on other theories, which is left for further investigation.
Differential Geometry and Its Visualization
Differential Geometry and Its Visualization is suitable for graduate level courses in differential geometry, serving both students and teachers. It can also be used as a supplementary reference for research in mathematics and the natural and engineering sciences.Differential geometry is the study of geometric objects and their properties using the methods of mathematical analysis. The classical theory of curves and surfaces in three-dimensional Euclidean space is presented in the first three chapters. The abstract and modern topics of tensor algebra, Riemannian spaces and tensor analysis are studied in the last two chapters. A great number of illustrating examples, visualizations and genuine figures created by the authors' own software are included to support the understanding of the presented concepts and results, and to develop an adequate perception of the shapes of geometric objects, their properties and the relations between them.Features Extensive, full colour visualisations Numerous exercises Self-contained and comprehensive treatment of the topic
A Full Axiomatic Development of High School Geometry
This textbook provides a full and complete axiomatic development of exactly that part of plane Euclidean geometry that forms the standard content of high school geometry. It begins with a set of points, a measure of distance between pairs of points and ten simple axioms. From there the notions of length, area and angle measure, along with congruence and similarity, are carefully defined and their properties proven as theorems. It concludes with a proof of the consistency of the axioms used and a full description of their models. It is provided in guided inquiry (inquiry-based) format with the intention that students will be active learners, proving the theorems and presenting their proofs to their class with the instructor as a mentor and a guide.The book is written for graduate and advanced undergraduate students interested in teaching secondary school mathematics, for pure math majors interested in learning about the foundations of geometry, for faculty preparing future secondary school teachers and as a reference for any professional mathematician. It is written with the hope of anchoring K-12 geometry in solid modern mathematics, thereby fortifying the teaching of secondary and tertiary geometry with a deep understanding of the subject.
Directed Quantities in Electrodynamics
This monograph explores classical electrodynamics from a geometrical perspective with a clear visual presentation throughout. Featuring over 200 figures, readers will delve into the definitions, properties, and uses of directed quantities in classical field theory. With an emphasis on both mathematical and electrodynamic concepts, the author's illustrative approach will help readers understand the critical role directed quantities play in physics and mathematics. Chapters are organized so that they gradually scale in complexity, and carefully guide readers through important topics. The first three chapters introduce directed quantities in three dimensions with and without the metric, as well as the development of the algebra and analysis of directed quantities. Chapters four through seven then focus on electrodynamics without the metric, such as the premetric case, waves, and fully covariant four-dimensional electrodynamics. Complementing the book's careful structure, exercises are included throughout for readers seeking further opportunities to practice the material. Directed Quantities in Electrodynamics will appeal to students, lecturers, and researchers of electromagnetism. It is particularly suitable as a supplement to standard textbooks on electrodynamics.
Differential Geometry
This book combines the classical and contemporary approaches to differential geometry. An introduction to the Riemannian geometry of manifolds is preceded by a detailed discussion of properties of curves and surfaces.The chapter on the differential geometry of plane curves considers local and global properties of curves, evolutes and involutes, and affine and projective differential geometry. Various approaches to Gaussian curvature for surfaces are discussed. The curvature tensor, conjugate points, and the Laplace-Beltrami operator are first considered in detail for two-dimensional surfaces, which facilitates studying them in the many-dimensional case. A separate chapter is devoted to the differential geometry of Lie groups.
The Mathematical Principles of Natural Philosophy
Mathematical Principles of Natural Philosophy, often referred to as simply the Principia, is a work in three books by Isaac Newton, in Latin, first published 5 July 1687. After annotating and correcting his personal copy of the first edition, Newton published two further editions, in 1713 and 1726. In Principia, Isaac Newton laid out in mathematical terms the principles of time, force, and motion that have guided the development of modern physical science. Even after more than three centuries and the revolutions of Einsteinian relativity and quantum mechanics, Newtonian physics continues to account for many of the phenomena of the observed world, and Newtonian celestial dynamics is used to determine the orbits of our space vehicles.
Research Directions in Symplectic and Contact Geometry and Topology
This book highlights a number of recent research advances in the field of symplectic and contact geometry and topology, and related areas in low-dimensional topology. This field has experienced significant and exciting growth in the past few decades, and this volume provides an accessible introduction into many active research problems in this area. The papers were written with a broad audience in mind so as to reach a wide range of mathematicians at various levels. Aside from teaching readers about developing research areas, this book will inspire researchers to ask further questions to continue to advance the field.The volume contains both original results and survey articles, presenting the results of collaborative research on a wide range of topics. These projects began at the Research Collaboration Conference for Women in Symplectic and Contact Geometry and Topology (WiSCon) in July 2019 at ICERM, Brown University. Each group of authors includedfemale and nonbinary mathematicians at different career levels in mathematics and with varying areas of expertise. This paved the way for new connections between mathematicians at all career levels, spanning multiple continents, and resulted in the new collaborations and directions that are featured in this work.
A History of Mathematical Impossibility
Many of the most famous results in mathematics are impossibility theorems stating that something cannot be done. Good examples include the quadrature of the circle by ruler and compass, the solution of the quintic equation by radicals, Fermat's last theorem, and the impossibility of proving the parallel postulate from the other axioms of Euclidean geometry. This book tells the history of these and many other impossibility theorems starting with the ancient Greek proof of the incommensurability of the side and the diagonal in a square. L羹tzen argues that the role of impossibility results have changed over time. At first, they were considered rather unimportant meta-statements concerning mathematics but gradually they obtained the role of important proper mathematical results that can and should be proved. While mathematical impossibility proofs are more rigorous than impossibility arguments in other areas of life, mathematicians have employed great ingenuity to circumvent impossibilities by changing the rules of the game. For example, complex numbers were invented in order to make impossible equations solvable. In this way, impossibilities have been a strong creative force in the development of mathematics, mathematical physics, and social science.
Visual Complex Analysis
Complex Analysis is the powerful fusion of the complex numbers (involving the 'imaginary' square root of -1) with ordinary calculus, resulting in a tool that has been of central importance to science for more than 200 years. This book brings this majestic and powerful subject to life by consistently using geometry (not calculation) as the means of explanation. The 501 diagrams of the original edition embodied geometrical arguments that (for the first time) replaced the long and often opaque computations of the standard approach, in force for the previous 200 years, providing direct, intuitive, visual access to the underlying mathematical reality. This new 25th Anniversary Edition introduces brand-new captions that fully explain the geometrical reasoning, making it possible to read the work in an entirely new way--as a highbrow comic book!
Finite Geometries
Finite Geometries stands out from recent textbooks about the subject of finite geometries by having a broader scope. The authors thoroughly explain how the subject of finite geometries is a central part of discrete mathematics. The text is suitable for undergraduate and graduate courses. Additionally, it can be used as reference material on recent works. The authors examine how finite geometries' applicable nature led to solutions of open problems in different fields, such as design theory, cryptography and extremal combinatorics. Other areas covered include proof techniques using polynomials in case of Desarguesian planes, and applications in extremal combinatorics, plus, recent material and developments. Features: Includes exercise sets for possible use in a graduate course Discusses applications to graph theory and extremal combinatorics Covers coding theory and cryptography Translated and revised text from the Hungarian published version
Geometric Transformations
This textbook teaches the transformations of plane Euclidean geometry through problems, offering a transformation-based perspective on problems that have appeared in recent years at mathematics competitions around the globe, as well as on some classical examples and theorems. It is based on the combined teaching experience of the authors (coaches of several Mathematical Olympiad teams in Brazil, Romania and the USA) and presents comprehensive theoretical discussions of isometries, homotheties and spiral similarities, and inversions, all illustrated by examples and followed by myriad problems left for the reader to solve. These problems were carefully selected and arranged to introduce students to the topics by gradually moving from basic to expert level. Most of them have appeared in competitions such as Mathematical Olympiads or in mathematical journals aimed at an audience interested in mathematics competitions, while some are fundamental facts of mathematics discussed in the framework of geometric transformations. The book offers a global view of the geometric content of today's mathematics competitions, bringing many new methods and ideas to the attention of the public.Talented high school and middle school students seeking to improve their problem-solving skills can benefit from this book, as well as high school and college instructors who want to add nonstandard questions to their courses. People who enjoy solving elementary math problems as a hobby will also enjoy this work.
One-Cocycles and Knot Invariants
One-Cocycles and Knot Invariants is about classical knots, i.e., smooth oriented knots in 3-space. It introduces discrete combinatorial analysis in knot theory in order to solve a global tetrahedron equation. This new technique is then used to construct combinatorial 1-cocycles in a certain moduli space of knot diagrams. The construction of the moduli space makes use of the meridian and the longitude of the knot. The combinatorial 1-cocycles are therefore lifts of the well-known Conway polynomial of knots, and they can be calculated in polynomial time. The 1-cocycles can distinguish loops consisting of knot diagrams in the moduli space up to homology. They give knot invariants when they are evaluated on canonical loops in the connected components of the moduli space. They are a first candidate for numerical knot invariants which can perhaps distinguish the orientation of knots.
Geometry of Black Holes
Black holes present one of the most fascinating predictions of Einstein's general theory of relativity. There is strong evidence of their existence through observation of active galactic nuclei, including the centre of our galaxy, observations of gravitational waves, and others. There exists a large scientific literature on black holes, including many excellent textbooks at various levels. However, most of these steer clear from the mathematical niceties needed to make the theory of black holes a mathematical theory. Those which maintain a high mathematical standard are either focused on specific topics, or skip many details. The objective of this book is to fill this gap and present a detailed, mathematically oriented, extended introduction to the subject. The book provides a wide background to the current research on all mathematical aspects of the geometry of black hole spacetimes.
The Geometrical Lectures of Isaac Barrow
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A System of the Mathematics
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Analytic Geometry
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Famous Problems of Elementary Geometry
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Inventional Geometry
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Inventional Geometry
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Graphical Statics
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A Short Practical Treatise on Spherical Trigonometry
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Inventional Geometry
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Structural Details of hip and Valley Rafters
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The Geometrical Lectures of Isaac Barrow
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Elements of Descriptive Geometry
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A Course of Plane Geometry for Advanced Students, Part II
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The Elements of Solid Geometry
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A Geometry for Beginners
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The Elements of Descriptive Geometry, Shadows and Perspective
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Le癟ons de G矇om矇trie ?l矇mentaire
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Structural Details of hip and Valley Rafters
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Euclid, Book V
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Geometry
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