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Math and Art
Math and Art: An Introduction to Visual Mathematics explores the potential of mathematics to generate visually appealing objects and reveals some of the beauty of mathematics. It includes numerous illustrations, computer-generated graphics, photographs, and art reproductions to demonstrate how mathematics can inspire or generate art. Focusing on accessible, visually interesting, and mathematically relevant topics, the text unifies mathematics subjects through their visual and conceptual beauty. Sequentially organized according to mathematical maturity level, each chapter covers a cross section of mathematics, from fundamental Euclidean geometry, tilings, and fractals to hyperbolic geometry, platonic solids, and topology. For art students, the book stresses an understanding of the mathematical background of relatively complicated yet intriguing visual objects. For science students, it presents various elegant mathematical theories and notions. Features Provides an accessible introduction to mathematics in art Supports the narrative with a self-contained mathematical theory, with complete proofs of the main results (including the classification theorem for similarities) Presents hundreds of figures, illustrations, computer-generated graphics, designs, photographs, and art reproductions, mainly presented in full color Includes 21 projects and approximately 280 exercises, about half of which are fully solved Covers Euclidean geometry, golden section, Fibonacci numbers, symmetries, tilings, similarities, fractals, cellular automata, inversion, hyperbolic geometry, perspective drawing, Platonic and Archimedean solids, and topology New to the Second Edition New exercises, projects and artworks Revised, reorganized and expanded chapters More use of color throughout
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Lessons in Experimental and Practical Geometry [microform]
This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it.This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface.We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
Arithmetica Universalis
This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it.This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface.We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
Foundations of Geometry
This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it.This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface.We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
A Treatise on Conic Sections
This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it.This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface.We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
Catechism of the Third Order of St. Francis / by Fr. Ferdinand ...
This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it.This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface.We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
On Metric Lattices ...
This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it.This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface.We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
Lectures on Algebraic Topology
Algebraic Topology and basic homotopy theory form a fundamental building block for much of modern mathematics. These lecture notes represent a culmination of many years of leading a two-semester course in this subject at MIT. The style is engaging and student-friendly, but precise. Every lecture is accompanied by exercises. It begins slowly in order to gather up students with a variety of backgrounds, but gains pace as the course progresses, and by the end the student has a command of all the basic techniques of classical homotopy theory.
Lectures on Algebraic Topology
Algebraic Topology and basic homotopy theory form a fundamental building block for much of modern mathematics. These lecture notes represent a culmination of many years of leading a two-semester course in this subject at MIT. The style is engaging and student-friendly, but precise. Every lecture is accompanied by exercises. It begins slowly in order to gather up students with a variety of backgrounds, but gains pace as the course progresses, and by the end the student has a command of all the basic techniques of classical homotopy theory.
Differential Geometry in Physics
Differential Geometry in Physics is a treatment of the mathematical foundations of the theory of general relativity and gauge theory of quantum fields. The material is intended to help bridge the gap that often exists between theoretical physics and applied mathematics.The approach is to carve an optimal path to learning this challenging field by appealing to the much more accessible theory of curves and surfaces. The transition from classical differential geometry as developed by Gauss, Riemann and other giants, to the modern approach, is facilitated by a very intuitive approach that sacrifices some mathematical rigor for the sake of understanding the physics. The book features numerous examples of beautiful curves and surfaces often reflected in nature, plus more advanced computations of trajectory of particles in black holes. Also embedded in the later chapters is a detailed description of the famous Dirac monopole and instantons.Features of this book: * Chapters 1-4 and chapter 5 comprise the content of a one-semester course taught by the author for many years.* The material in the other chapters has served as the foundation for many master's thesis at University of North Carolina Wilmington for students seeking doctoral degrees.* An open access ebook edition is available at Open UNC (https: //openunc.org)* The book contains over 80 illustrations, including a large array of surfaces related to the theory of soliton waves that does not commonly appear in standard mathematical texts on differential geometry.
Parametric Geometry of Curves and Surfaces
This textbook provides a thorough introduction to the differential geometry of parametrized curves and surfaces, along with a wealth of applications to specific architectural elements. Geometric elements in architecture respond to practical, physical and aesthetic needs. Proper understanding of the mathematics underlying the geometry provides control over the construction. This book relates the classical mathematical theory of parametrized curves and surfaces to multiple applications in architecture. The presentation is mathematically complete with numerous figures and animations illustrating the theory, and special attention is given to some of the recent trends in the field. Solved exercises are provided to see the theory in practice.Intended as a textbook for lecture courses, Parametric Geometry of Curves and Surfaces is suitable for mathematically-inclined students in engineering, architecture and related fields, and can also serve as a textbook for traditional differential geometry courses to mathematics students. Researchers interested in the mathematics of architecture or computer-aided design will also value its combination of precise mathematics and architectural examples.
Differential Geometry, Algebra, and Analysis
This book is a collection of selected research papers, some of which were presented at the International Conference on Differential Geometry, Algebra and Analysis (ICDGAA 2016), held at the Department of Mathematics, Jamia Millia Islamia, New Delhi, from 15-17 November 2016. It covers a wide range of topics--geometry of submanifolds, geometry of statistical submanifolds, ring theory, module theory, optimization theory, and approximation theory--which exhibit new ideas and methodologies for current research in differential geometry, algebra and analysis. Providing new results with rigorous proofs, this book is, therefore, of much interest to readers who wish to learn new techniques in these areas of mathematics.
Smooth Manifolds and Observables
This textbook demonstrates how differential calculus, smooth manifolds, and commutative algebra constitute a unified whole, despite having arisen at different times and under different circumstances. Motivating this synthesis is the mathematical formalization of the process of observation from classical physics. A broad audience will appreciate this unique approach for the insight it gives into the underlying connections between geometry, physics, and commutative algebra.The main objective of this book is to explain how differential calculus is a natural part of commutative algebra. This is achieved by studying the corresponding algebras of smooth functions that result in a general construction of the differential calculus on various categories of modules over the given commutative algebra. It is shown in detail that the ordinary differential calculus and differential geometry on smooth manifolds turns out to be precisely the particular case that corresponds to the category of geometric modules over smooth algebras. This approach opens the way to numerous applications, ranging from delicate questions of algebraic geometry to the theory of elementary particles.Smooth Manifolds and Observables is intended for advanced undergraduates, graduate students, and researchers in mathematics and physics. This second edition adds ten new chapters to further develop the notion of differential calculus over commutative algebras, showing it to be a generalization of the differential calculus on smooth manifolds. Applications to diverse areas, such as symplectic manifolds, de Rham cohomology, and Poisson brackets are explored. Additional examples of the basic functors of the theory are presented alongside numerous new exercises, providing readers with many more opportunities to practice these concepts.
Basic Lessons on Isometries, Similarities and Inversions in the Euclidean Plane
The aim of this book is to provide a complete synthetic exposition of plane isometries, similarities and inversions to readers who are interested in studying, teaching, and using this material.The topics developed in this book can provide new proofs and solutions to many results and problems of classical geometry, which are presented with different proofs in the literature. Their applications are numerous and some, such as the Steiner Chains and Point, are useful to engineers.The book contains many good examples, important applications and numerous exercises of various level and difficulty, which are classified in the three groups of: general exercises, geometrical constructions, and geometrical loci. Some lengthy exercises or groups of related exercises can be viewed as projects. On the basis of the above, this book, besides Classical Geometry, is an important addition to Mathematics Education.
Basic Lessons on Isometries, Similarities and Inversions in the Euclidean Plane
The aim of this book is to provide a complete synthetic exposition of plane isometries, similarities and inversions to readers who are interested in studying, teaching, and using this material.The topics developed in this book can provide new proofs and solutions to many results and problems of classical geometry, which are presented with different proofs in the literature. Their applications are numerous and some, such as the Steiner Chains and Point, are useful to engineers.The book contains many good examples, important applications and numerous exercises of various level and difficulty, which are classified in the three groups of: general exercises, geometrical constructions, and geometrical loci. Some lengthy exercises or groups of related exercises can be viewed as projects. On the basis of the above, this book, besides Classical Geometry, is an important addition to Mathematics Education.
Differential Geometry in Physics
Differential Geometry in Physics is a treatment of the mathematical foundations of the theory of general relativity and gauge theory of quantum fields. The material is intended to help bridge the gap that often exists between theoretical physics and applied mathematics.The approach is to carve an optimal path to learning this challenging field by appealing to the much more accessible theory of curves and surfaces. The transition from classical differential geometry as developed by Gauss, Riemann and other giants, to the modern approach, is facilitated by a very intuitive approach that sacrifices some mathematical rigor for the sake of understanding the physics. The book features numerous examples of beautiful curves and surfaces often reflected in nature, plus more advanced computations of trajectory of particles in black holes. Also embedded in the later chapters is a detailed description of the famous Dirac monopole and instantons.Features of this book: * Chapters 1-4 and chapter 5 comprise the content of a one-semester course taught by the author for many years.* The material in the other chapters has served as the foundation for many master's thesis at University of North Carolina Wilmington for students seeking doctoral degrees.* An open access ebook edition is available at Open UNC (https: //openunc.org)* The book contains over 80 illustrations, including a large array of surfaces related to the theory of soliton waves that does not commonly appear in standard mathematical texts on differential geometry.
Dimensionstheorie
Dieses Essential gibt eine kompakte Einf羹hrung in die Dimensionstheorie. Die topologische Dimension und mehrere fraktale Dimensionen werden definiert und anhand von Beispielen erl瓣utert. Lesende lernen grundlegende S瓣tze 羹ber die Dimension von kartesischen Produkten, Projektionen, Schnitten und arithmetischen Summen kennen. Weiterhin wird eine Vielfalt von Anwendungen der Dimensionstheorie in der Zahlentheorie, der Geometrie, der Analysis, den dynamischen Systemen und der Stochastik vorgestellt.
Differential Geometry and Lie Groups
This textbook explores advanced topics in differential geometry, chosen for their particular relevance to modern geometry processing. Analytic and algebraic perspectives augment core topics, with the authors taking care to motivate each new concept. Whether working toward theoretical or applied questions, readers will appreciate this accessible exploration of the mathematical concepts behind many modern applications. Beginning with an in-depth study of tensors and differential forms, the authors go on to explore a selection of topics that showcase these tools. An analytic theme unites the early chapters, which cover distributions, integration on manifolds and Lie groups, spherical harmonics, and operators on Riemannian manifolds. An exploration of bundles follows, from definitions to connections and curvature in vector bundles, culminating in a glimpse of Pontrjagin and Chern classes. The final chapter on Clifford algebras and Clifford groups draws the book to an algebraic conclusion, which can be seen as a generalized viewpoint of the quaternions. Differential Geometry and Lie Groups: A Second Course captures the mathematical theory needed for advanced study in differential geometry with a view to furthering geometry processing capabilities. Suited to classroom use or independent study, the text will appeal to students and professionals alike. A first course in differential geometry is assumed; the authors' companion volume Differential Geometry and Lie Groups: A Computational Perspective provides the ideal preparation.
General Expressions for the 5-Dimensional Riemann-Christoffel, Ricci, and Einstein Curvature Tensors and Riemann Curvature Scalar Allowing for Non-Vanishing Torsion and Arbitrary Functional Dependence
This book presents general expressions for the 5-dimensional Riemann-Christoffel, Ricci, and Einstein curvature tensors and Riemann curvature scalar allowing for non-vanishing torsion and arbitrary functional dependence on the fifth dimension.
Gauge Theory in Higher Dimensions
This book is an introduction to gauge theory in dimensions greater than . It is organised as a reader's guide to Gang Tian's landmark article Gauge theory and calibrated geometry, I, published in the Annals of Mathematics in 2000. Tian carried out to substantial lengths the programme outlined by Simon Donaldson and Richard Thomas in their seminal paper, by extending the fundamental notion of instanton in Yang-Mills theory to higher dimensional special geometries. Moreover, his bubbling theorem relates instanton compactness to the theory of calibrated submanifolds. This work expresses our admiration for all of these key authors.The text is aimed at graduate students and advanced undergraduates, as well as specialists in other areas of Mathematics and Physics, with an interest in modern Differential Geometry. We adopt a fast-paced but self-contained exposition of the background on connections and curvature on bundles, special geometric structures, analysis on manifolds and geometric measure theory, alongside the proof of Tian's bubbling theorem. We also highlight some links with other important works in contemporary Geometry and Topology.This version was commissioned as supporting material to an advanced course by the authors at the 32 Brazilian Colloquium of Mathematics, hosted by the Institute for Pure and Applied Mathematics (IMPA) from July 29th to August 2nd, 2019.We hope that the reader will share in our enthusiasm for this beautiful and fast-evolving subject.
Ebene Elementar-Geometrie, Trigonometrie, Theilungslehre
Keine ausf羹hrliche Beschreibung f羹r "Ebene Elementar-Geometrie, Trigonometrie, Theilungslehre" verf羹gbar.
The Brauer-Grothendieck Group
This monograph provides a systematic treatment of the Brauer group of schemes, from the foundational work of Grothendieck to recent applications in arithmetic and algebraic geometry. The importance of the cohomological Brauer group for applications to Diophantine equations and algebraic geometry was discovered soon after this group was introduced by Grothendieck. The Brauer-Manin obstruction plays a crucial role in the study of rational points on varieties over global fields. The birational invariance of the Brauer group was recently used in a novel way to establish the irrationality of many new classes of algebraic varieties. The book covers the vast theory underpinning these and other applications. Intended as an introduction to cohomological methods in algebraic geometry, most of the book is accessible to readers with a knowledge of algebra, algebraic geometry and algebraic number theory at graduate level. Much of the more advanced material is not readily available inbook form elsewhere; notably, de Jong's proof of Gabber's theorem, the specialisation method and applications of the Brauer group to rationality questions, an in-depth study of the Brauer-Manin obstruction, and proof of the finiteness theorem for the Brauer group of abelian varieties and K3 surfaces over finitely generated fields. The book surveys recent work but also gives detailed proofs of basic theorems, maintaining a balance between general theory and concrete examples. Over half a century after Grothendieck's foundational seminars on the topic, The Brauer-Grothendieck Group is a treatise that fills a longstanding gap in the literature, providing researchers, including research students, with a valuable reference on a central object of algebraic and arithmetic geometry.
An Introduction to Geometric Algebra and Geometric Calculus
This is an introduction to geometric algebra in n-dimensional Euclidean space and its application to manifolds and to calculus on manifolds. The treatment is moderately rigorous and is suitable for advanced undergraduates and beginning graduate students in mathematics though it should also be accessible to well-prepared students in physics, engineering, computer science, statistics, etc. Preparation in linear algebra and multivariable analysis as encountered in calculus as well as a modest amount of mathematical maturity should be sufficient.
Lectures on Convex Geometry
This book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops Brunn-Minkowski theory, with an exposition of mixed volumes, the Brunn-Minkowski inequality, and some of its consequences, including the isoperimetric inequality. Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations. Finally, an introduction to integral-geometric formulas in Euclidean space is provided. The numerous exercises and the supplementary material at the end of each section form an essential part of the book. Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry. Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.
Mastering Essential Math Skills
This is the new Geometry 2nd Edition, a completely edited version including a handy glossary and resource center.Geometry is one of the most useful topics in math. If you look around, you will see geometry in houses, cars, furniture, toys, roads and freeways, and much more. To design and build things, a great deal of geometry is used.Working through this book will provide you with a strong, basic understanding of geometry. Enjoy the journey.Perfect for students who are math challenged.Lessons are presented in a format that everyone can understand.Each lesson flows smoothly and logically to the next.Each lesson is short, concise, and to the point.Lots of examples with step-by-step solutions.Each lesson includes a valuable Helpful Hints section.Review is built into each lesson. Students will retain what they have learned!Each lesson includes Problem Solving. This ensures that students will learn to apply their math skills to real-life situations.Includes free access to video tutorials. Just go to www.mathessentials.net and click on the Videos button.Topics: Points, Lines, and PlanesParallel & Intersecting LinesTypes of AnglesMeasuring & Constructing AnglesPolygonsClassification of TrianglesPerimeter of PolygonsProblem Solving Involving Perimeter of PolygonsCirclesCircumference of CirclesProblem Solving Involving Circumference of CirclesThe Pythagorean TheoremArea of PolygonsArea of CirclesProblem Solving Involving AreaSolid FiguresSurface AreaVolumeFinal Review and TestsSolutionsGlossary
The Fractal Geometry of Nature
The Essential Guide that Introduced Fractals to the WorldExplore the wondrously complex repeating shapes of the natural world in The Fractal Geometry of Nature. Written in a style that is accessible to a wide audience, computer scientist, professor, mathematician, economist, and visionary Benoit B Mandelbrot's fascinating work has inspired popular interest in the geometry inherent in the natural world.Unlike the squares, circles, spheres, and cones of fundamental geometry, nature has rough edges and no straight lines or perfect curves. Mandelbrot observed that, even with this roughness, there still exists a kind of symmetry, which he dedicated his work to document and study. This became the basis for his development of a new kind of geometry; indeed, he coined the term "fractal."Mandelbrot spent 35 years with IBM, which allowed him access to the level of computing power that would enable him to manipulate computer-generated images and develop his theory of a geometry found throughout our natural environment. He was among the first to use computer graphics to illustrate and test these kinds of concepts, demonstrating that natural phenomena which appear to be rough or chaotic actually have a certain degree of order and predictability. This definitive overview builds on Mandelbrot's 1977 work, Fractals: Form, Chance and Dimension (also published by Echo Point Books), revealing an in-depth look at this still-emerging field. Richly illustrated and presented in an engaging manner which embraces geometric and visual dimensions interspersed with aspects of theory, this book will inspire curiosity and wonder in artists, mathematicians and naturalists alike.This book is also available from Echo Point Books in paperback (ISBN 1648370411).Be sure to check out Benoit Mandelbrot's other definitive work, also availabe from Echo Point books: Fractals: Form, Chance and Dimension (use the web address https: //www.amazon.com/dp/1635619025/).
Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles Over Arithmetic Curves
This book presents the most up-to-date and sophisticated account of the theory of Euclidean lattices and sequences of Euclidean lattices, in the framework of Arakelov geometry, where Euclidean lattices are considered as vector bundles over arithmetic curves. It contains a complete description of the theta invariants which give rise to a closer parallel with the geometric case. The author then unfolds his theory of infinite Hermitian vector bundles over arithmetic curves and their theta invariants, which provides a conceptual framework to deal with the sequences of lattices occurring in many diophantine constructions. The book contains many interesting original insights and ties to other theories. It is written with extreme care, with a clear and pleasant style, and never sacrifices accessibility to sophistication.
Elementary Topology and Applications
The textbook is a very good start into the mathematical field of topology. A variety of topological concepts with some elementary applications are introduced. It is organized in such a way that the reader gets to significant applications quickly.This revised version corrects the many discrepancies in the earlier edition. The emphasis is on the geometric understanding and the use of new concepts, indicating that topology is really the language of modern mathematics.
de Rham Cohomology of Differential Modules on Algebraic Varieties
This is the revised second edition of the well-received book by the first two authors. It offers a systematic treatment of the theory of vector bundles with integrable connection on smooth algebraic varieties over a field of characteristic 0. Special attention is paid to singularities along divisors at infinity, and to the corresponding distinction between regular and irregular singularities. The topic is first discussed in detail in dimension 1, with a wealth of examples, and then in higher dimension using the method of restriction to transversal curves. The authors develop a new approach to classical algebraic/analytic comparison theorems in De Rham cohomology, and provide a unified discussion of the complex and the p-adic situations while avoiding the resolution of singularities. They conclude with a proof of a conjecture by Baldassarri to the effect that algebraic and p-adic analytic De Rham cohomologies coincide, under an arithmetic condition on exponents. As used in this text, the term "De Rham cohomology" refers to the hypercohomology of the De Rham complex of a connection with respect to a smooth morphism of algebraic varieties, equipped with the Gauss-Manin connection. This simplified approach suffices to establish the stability of crucial properties of connections based on higher direct images. The main technical tools used include: Artin local decomposition of a smooth morphism in towers of elementary fibrations, and spectral sequences associated with affine coverings and with composite functors.
Mathematical Theory of Optics
Mathematical Theory of Optics by Dr. Rudolf K. Luneburg stands as a monumental work in the field of optical theory, offering a rigorous and innovative approach to understanding instrumental optics through the lens of Maxwell's electromagnetic equations. First presented as a lecture series at Brown University in 1944, and later refined into these notes, Luneburg's work connects geometrical optics and scalar diffraction optics as systematic extensions of electromagnetic theory. This unification was groundbreaking, as these disciplines were previously treated as distinct fields. Luneburg's formulation of geometrical optics as exact solutions to Maxwell's equations, rather than simply as asymptotic approximations, redefined the theoretical landscape and laid the groundwork for future developments in optics. The volume meticulously explores topics such as Hamilton's theory of geometrical optics, the mathematical treatment of optical aberrations, and the now-famous Luneburg lens, which has become a key component in microwave antenna technology. Later chapters delve into diffraction theory, introducing Luneburg diffraction integrals that extend classical results and address complex problems such as resolution improvements. With additional appendices and notes by prominent scholars, the book not only consolidates Luneburg's profound insights but also enriches the field with practical applications and theoretical advancements. Recognized as one of the most significant contributions to optical theory, this volume remains a foundational text for physicists and engineers engaged in advanced optical research. This title is part of UC Press's Voices Revived program, which commemorates University of California Press's mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1964.
Mathematical Theory of Optics
Mathematical Theory of Optics by Dr. Rudolf K. Luneburg stands as a monumental work in the field of optical theory, offering a rigorous and innovative approach to understanding instrumental optics through the lens of Maxwell's electromagnetic equations. First presented as a lecture series at Brown University in 1944, and later refined into these notes, Luneburg's work connects geometrical optics and scalar diffraction optics as systematic extensions of electromagnetic theory. This unification was groundbreaking, as these disciplines were previously treated as distinct fields. Luneburg's formulation of geometrical optics as exact solutions to Maxwell's equations, rather than simply as asymptotic approximations, redefined the theoretical landscape and laid the groundwork for future developments in optics. The volume meticulously explores topics such as Hamilton's theory of geometrical optics, the mathematical treatment of optical aberrations, and the now-famous Luneburg lens, which has become a key component in microwave antenna technology. Later chapters delve into diffraction theory, introducing Luneburg diffraction integrals that extend classical results and address complex problems such as resolution improvements. With additional appendices and notes by prominent scholars, the book not only consolidates Luneburg's profound insights but also enriches the field with practical applications and theoretical advancements. Recognized as one of the most significant contributions to optical theory, this volume remains a foundational text for physicists and engineers engaged in advanced optical research. This title is part of UC Press's Voices Revived program, which commemorates University of California Press's mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1964.
Thinking, Drawing, Modelling
This book presents a selection of papers from the International Conference Geometrias'17, which was hosted by the Department of Architecture at the University of Coimbra from 16 to 18 June 2017. The Geometrias conferences, organized by Aproged (the Portuguese Geometry and Drawing Teachers' Association), foster debate and exchange on practical and theoretical research in mathematics, architecture, the arts, engineering, and related fields.Geometrias'17, with the leitmotif "Thinking, Drawing, Modelling", brought together a group of recognized experts to discuss the importance of geometric literacy and the science of representation for the development of scientific and technological research and professional practices. The 12 peer-reviewed papers gathered here show how geometry, drawing, stereotomy, and the science of representation are still at the core of every act leading to the conception and materialization of form, and highlight their continuing relevance for scholars andprofessionals in the fields of architecture, engineering, and applied mathematics.
Introduction to Riemannian Manifolds
​This textbook is designed for a one or two semester graduate course on Riemannian geometry for students who are familiar with topological and differentiable manifolds. The second edition has been adapted, expanded, and aptly retitled from Lee's earlier book, Riemannian Manifolds: An Introduction to Curvature. Numerous exercises and problem sets provide the student with opportunities to practice and develop skills; appendices contain a brief review of essential background material.While demonstrating the uses of most of the main technical tools needed for a careful study of Riemannian manifolds, this text focuses on ensuring that the student develops an intimate acquaintance with the geometric meaning of curvature. The reasonably broad coverage begins with a treatment of indispensable tools for working with Riemannian metrics such as connections and geodesics. Several topics have been added, including an expanded treatment of pseudo-Riemannianmetrics, a more detailed treatment of homogeneous spaces and invariant metrics, a completely revamped treatment of comparison theory based on Riccati equations, and a handful of new local-to-global theorems, to name just a few highlights.Reviews of the first edition: Arguments and proofs are written down precisely and clearly. The expertise of the author is reflected in many valuable comments and remarks on the recent developments of the subjects. Serious readers would have the challenges of solving the exercises and problems. The book is probably one of the most easily accessible introductions to Riemannian geometry. (M.C. Leung, MathReview) The book's aim is to develop tools and intuition for studying the central unifying theme in Riemannian geometry, which is the notion of curvature and its relation with topology. The main ideas of the subject, motivated as in the original papers, are introduced here in an intuitive and accessible way...The book is an excellent introduction designed for a one-semester graduate course, containing exercises and problems which encourage students to practice working with the new notions and develop skills for later use. By citing suitable references for detailed study, the reader is stimulated to inquire into further research. (C.-L. Bejan, zBMATH)
Noncommutative Geometry and Particle Physics
This book provides an introduction to noncommutative geometry and presents a number of its recent applications to particle physics. It is intended for graduate students in mathematics/theoretical physics who are new to the field of noncommutative geometry, as well as for researchers in mathematics/theoretical physics with an interest in the physical applications of noncommutative geometry. In the first part, we introduce the main concepts and techniques by studying finite noncommutative spaces, providing a "light" approach to noncommutative geometry. We then proceed with the general framework by defining and analyzing noncommutative spin manifolds and deriving some main results on them, such as the local index formula. In the second part, we show how noncommutative spin manifolds naturally give rise to gauge theories, applying this principle to specific examples. We subsequently geometrically derive abelian and non-abelian Yang-Mills gauge theories, and eventually the full Standard Model of particle physics, and conclude by explaining how noncommutative geometry might indicate how to proceed beyond the Standard Model.
Non-Euclidean Laguerre Geometry and Incircular Nets
This textbook is a comprehensive and yet accessible introduction to non-Euclidean Laguerre geometry, for which there exists no previous systematic presentation in the literature. Moreover, we present new results by demonstrating all essential features of Laguerre geometry on the example of checkerboard incircular nets.Classical (Euclidean) Laguerre geometry studies oriented hyperplanes, oriented hyperspheres, and their oriented contact in Euclidean space. We describe how this can be generalized to arbitrary Cayley-Klein spaces, in particular hyperbolic and elliptic space, and study the corresponding groups of Laguerre transformations. We give an introduction to Lie geometry and describe how these Laguerre geometries can be obtained as subgeometries. As an application of two-dimensional Lie and Laguerre geometry we study the properties of checkerboard incircular nets.
Numerical Solutions of Boundary Value Problems of Non-Linear Differential Equations
The book presents in comprehensive detail numerical solutions to boundary value problems of a number of non-linear differential equations. Replacing derivatives by finite difference approximations in these differential equations leads to a system of non-linear algebraic equations which we have solved using Newton's iterative method. In each case, we have also obtained Euler solutions and ascertained that the iterations converge to Euler solutions. We find that, except for the boundary values, initial values of the 1st iteration need not be anything close to the final convergent values of the numerical solution. Programs in Mathematica 6.0 were written to obtain the numerical solutions.
Visual Differential Geometry and Forms
An inviting, intuitive, and visual exploration of differential geometry and forms Visual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry. Using 235 hand-drawn diagrams, Needham deploys Newton's geometrical methods to provide geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to differential forms that treats advanced topics in an intuitive and geometrical manner. Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss's famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n-manifold; and a detailed geometrical treatment of Einstein's field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology. The final act elucidates such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell's equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan's method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms. Six of the seven chapters of Act V can be read completely independently from the rest of the book. Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be considered and taught.
Visual Differential Geometry and Forms
An inviting, intuitive, and visual exploration of differential geometry and forms Visual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry. Using 235 hand-drawn diagrams, Needham deploys Newton's geometrical methods to provide geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to differential forms that treats advanced topics in an intuitive and geometrical manner. Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss's famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n-manifold; and a detailed geometrical treatment of Einstein's field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology. The final act elucidates such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell's equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan's method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms. Six of the seven chapters of Act V can be read completely independently from the rest of the book. Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be considered and taught.
Ethereum
The first difference is that ethereum allows cryptocurrency transaction records called 'blocks' to be created faster than it is currently done using bitcoin. Although bitcoin is now widely accepted by online merchants and is even being adopted by physical stores, the fans of ethereum believe the efficiency of this cryptocurrency makes for better transactions. In this book, you will learn: - What cryptocurrencies are- What ethereum is- How to mine for ethereum- What blockhain is and how it works- The pros and cons of cryptocurrency- The technology behind it all- How to determine mining profitability- The different types of cryptocurrencies- The most profitable types of cryptocurrenciesTоdау, ѕmаrt contracts саn run on the рubliс ethеrеum blockchain, a diѕtributеd lеdgеr tесhnоlоgу thаt is uѕеd tо kеер trасk of аll rеlаtеd trаnѕасtiоnѕ аnd аgrееmеntѕ.The ѕmаrt соntrасtѕ that run оn itѕ blосkсhаin соuld hаvе widеѕрrеаd applications, as dеvеlореrѕ соuld uѕе thеm tо сrеаtе markets, еxесutе transactions based on agreements сrеаtеd lоng аgо аnd kеер track оf рlеdgеѕ mаdе bу diffеrеnt counterparties.
Ethereum
Ethereum is a decentralized platform that aims to offer its users even more services than bitcoin. In fact, the ethereum technology doesn't only powers a digital currency, ether, but also provides the technology for smart contracts, a cheaper and revolutionary contract solution based on the blockchain technology.This book will discuss everything that you need to know about ethereum, so that you can make an informed decision for your investments. You'll discover the technology behind the ethereum platform, advantages and possible problems you may run into, how smart contracts work and how to program your own smart contract, how to buy and mine ether for profit. This book will introduce you to the basics of investing in ethereum. You will learn: - The fundamentals of ethereum- The pros and cons of investing in ethereum- The process of investing in ethereum- Understanding smart contracts- And much more...Being the next biggest cryptocurrency to bitcoin, comparisons are inevitable between ethereum and its big brother. There have also been rumors and claims of ethereum being a bubble because of its rapid rise in value and fame. But advocates of this fast-rising cryptocurrency state some advantages compared to bitcoin that are worth taking a closer look at.
Ethereum for Beginners
Having said that, it's impossible to jump on the crypto wagon if you don't really understand what cryptocurrency really is and how it works. In this book, you will learn everything you need to know to quickly and effortlessly get a massive return on your ethereum investment by putting the basics to work. You'll learn: - What is ethereum and how it works- A step by step guide to buy ether today- Pros and cons of the ethereum platform- How to find a secure wallet to safely store your coins- The ethereum virtual machine revolution- Interesting future developments of ethereum- The most important tips to buy ethereum safely- A step by step guide to mining ether for profit- How to program your own smart contract- Myths most people believe about smart contracts- The ethereum ecosystem and how to take advantage of it- And much, much moreImagine a world where your box of important documents is nonexistent. Instead of keeping track of physical paper documents, you're able to contractually engage with others over the internet in an environment that is safe and can be transparently monitored at all times. Additionally, you're able to exchange currency on a digital network. In essence, your physical life is able to become more digitally recognized and reputable.
Ethereum for Beginners
Understanding how the world is shifting when it comes to digital currency is really important, as these cryptos are currently being used and traded as you are reading. This is a primary reason as to why you need to learn all you need to know, to ensure you can take advantage of the opportunities that are present. Learn: - What ethereum is- What ether is- The technology behind ethereum- Different ways to mine ether- The mining process- Different uses of ethereum- Financial history of ether- How to buy and sell ether for profit- How to store ether- Why you should invest in ether- The future of ethereum- And so much more!Anyone looking to get into the world of cryptocurrency has already given some careful consideration to how to go about it. If you are one of those who has seen the potential of ethereum and want to take advantage of the growing profits, then you need some basic guidelines on how best to invest.
Ethereum
The aim of this book is to give you a deep insight into the intricacies of how ethereum works and how to develop your first decentralized application from scratch on the ethereum blockchain.With the aid of vivid examples, you can navigate smoothly and gain smooth understanding. The book also include how ethereum keen agreement works and proper understanding of how to develop your first decentralized application from scratch on the ethereum blockchain. Have you asked yourself questions such as...- What is ethereum?- How is it different to bitcoin?- How can i use ethereum?- How is ether mined?- What are smart contracts?- What does the future hold for ethereum?Mining is the process of sorting through blocks in the blockchain, and approving of transactions. In an earlier chapter, we described how it's essentially the act of solving difficult code puzzles. Blocks in ethereum don't actually represent currency, they represent anything of value, so they take the form of smart contracts. In normal systems, there's one central authority that does this work, but because ethereum is a decentralized platform, it's up to all the nodes to make sure the record-keeping is accurate.
Geometron
This book describes a new way of thinking about media, technology, and civilization. We propose building a new civilization on the broken pieces of the old, one built entirely from trash and in equilibrium with the ecosystem we live in. This book documents a social network based on web servers we carry around with us from place to place, networking out in the physical world. The network also contains a version of the book itself, on self-replicating code which installs itself on new servers as they are built. Each book tells you how to make more servers which copy the book and so on. It is an open book, released into the Public Domain to be shared, replicated, and edited by anyone any time anywhere.
Surfaces
The mathematics of surfaces is a fascinating subject - lying as it does at the heart of much architecture, art and maths, engineering and science. John Sharp's innovative use of sliceforms to explore the properties of surfaces is here produced in a systematic way, providing the reader the tools to build surfaces from paper to explore their mathematical properties. The extensive commentary explains the mathematics behind particular surfaces. Chapters on using computers to generate slices and sample templates complete the book. An invaluable exercise in practical geometry that will stimulate ideas for the student, professionals involved in modelling surfaces and the enthusiast. Contents: - Introduction and Basic Techniques - Contours and profiles - Properties and Types of Surfaces - Surfaces of Revolution - Ruled Surfaces - Surfaces from Equations: Algebraic surfaces - Polyhedra - Variations and explorations - Using the Computer - Templates and further resources
Lectures and Surveys on G2-Manifolds and Related Topics
This book, one of the first on G2 manifolds in decades, collects introductory lectures and survey articles largely based on talks given at a workshop held at the Fields Institute in August 2017, as part of the major thematic program on geometric analysis. It provides an accessible introduction to various aspects of the geometry of G2 manifolds, including the construction of examples, as well as the intimate relations with calibrated geometry, Yang-Mills gauge theory, and geometric flows. It also features the inclusion of a survey on the new topological and analytic invariants of G2 manifolds that have been recently discovered. The first half of the book, consisting of several introductory lectures, is aimed at experienced graduate students or early career researchers in geometry and topology who wish to familiarize themselves with this burgeoning field. The second half, consisting of numerous survey articles, is intended to be useful to both beginners and experts in the field.
Extrinsic Geometry of Foliations
This book is devoted to geometric problems of foliation theory, in particular those related to extrinsic geometry, modern branch of Riemannian Geometry. The concept of mixed curvature is central to the discussion, and a version of the deep problem of the Ricci curvature for the case of mixed curvature of foliations is examined. The book is divided into five chapters that deal with integral and variation formulas and curvature and dynamics of foliations. Different approaches and methods (local and global, regular and singular) in solving the problems are described using integral and variation formulas, extrinsic geometric flows, generalizations of the Ricci and scalar curvatures, pseudo-Riemannian and metric-affine geometries, and 'computable' Finsler metrics.The book presents the state of the art in geometric and analytical theory of foliations as a continuation of the authors' life-long work in extrinsic geometry. It is designed for newcomers to the field as well as experienced geometers working in Riemannian geometry, foliation theory, differential topology, and a wide range of researchers in differential equations and their applications. It may also be a useful supplement to postgraduate level work and can inspire new interesting topics to explore.
Encounters with Euclid
A sweeping cultural history of one of the most influential mathematical books ever written Euclid's Elements of Geometry is one of the fountainheads of mathematics--and of culture. Written around 300 BCE, it has traveled widely across the centuries, generating countless new ideas and inspiring such figures as Isaac Newton, Bertrand Russell, Abraham Lincoln, and Albert Einstein. Encounters with Euclid tells the story of this incomparable mathematical masterpiece, taking readers from its origins in the ancient world to its continuing influence today. In this lively and informative book, Benjamin Wardhaugh explains how Euclid's text journeyed from antiquity to the Renaissance, introducing some of the many readers, copyists, and editors who left their mark on the Elements before handing it on. He shows how some read the book as a work of philosophy, while others viewed it as a practical guide to life. He examines the many different contexts in which Euclid's book and his geometry were put to use, from the Neoplatonic school at Athens and the artisans' studios of medieval Baghdad to the Jesuit mission in China and the workshops of Restoration London. Wardhaugh shows how the Elements inspired ideas in theology, art, and music, and how the book has acquired new relevance to the strange geometries of dark matter and curved space. Encounters with Euclid traces the life and afterlives of one of the most remarkable works of mathematics ever written, revealing its lasting role in the timeless search for order and reason in an unruly world.
