Geometri

I anledning af den seneste reform aflæreruddannelsen er Matematik for lærerstuderende nu blevet målrettet de nye krav. Geometri 4.-10. klasse præsenterer fagligt og fagdidaktisk materiale samt tilhørende arbejdsopgaver svarende til ét modul efter LU13. De I anledning af den seneste reform aflæreruddannelsen er Matematik for lærerstuderende nu blevet målrettet de nye krav. matematikfaglige emner strækker sig fra geometriske eksperimenter og argumenter over måling og areal til flytninger, tegning og symmetriske mønstre. Den såkaldte stofdidaktik, der er knyttet til bogens faglige emner, findes i særlige kapitler, mens det fagdidaktiske stof, der er fælles for al matematikundervisning, skal søges i to andre bøger i systemet: Fagdidaktikken findes i Delta, og bogen My dækker elever med særlige behov.

I anledning af den seneste reform aflæreruddannelsen er Matematik for lærerstuderende nu blevet målrettet de nye krav. Geometri 1.-6. klasse præsenterer fagligt og fagdidaktisk materiale samt tilhørende arbejdsopgaver svarende til ét modul efter LU13. De I anledning af den seneste reform aflæreruddannelsen er Matematik for lærerstuderende nu blevet målrettet de nye krav. matematikfaglige emner strækker sig fra geometriske eksperimenter og argumenter over måling og areal til flytninger, tegning og symmetriske mønstre. Den såkaldte stofdidaktik, der er knyttet til bogens faglige emner, findes i særlige kapitler, mens det fagdidaktiske stof, der er fælles for al matematikundervisning, skal søges i to andre bøger i systemet: Fagdidaktikken findes i Delta, og bogen My dækker elever med særlige behov.

Euklids ELEMENTER er et hovedværk ikke alene i matematikken, men i hele den vestlige civilisation. Det er det ældste aksiomatisk opbyggede værk fra 300 f.Kr. med linjer tilbage til Platons og Aristoteles’ filosofi og fremad mod Archimedes og den senere matematik.Dette er den første danske oversættelse af Euklids plangeometri samlet i hans ELEMENTER I-VI siden Thyra Eibes oversættelse fra 1897-1912.Den nye oversættelse er udsprunget af et ønske om at vise Euklids værk i lyset af den moderne forståelse af græsk matematik som en udløber af samtidens filosofiske retninger; Platon og Aristoteles citeres derfor flittigt i den lange indledning. På samme måde er bogen udstyret med en redegørelse for Euklids sprog og dets mangel på symboler af enhver art - i modsætning til den hidtidige oversættelse, der i overensstemmelse med praksis i det sene 19. århundrede gør brug af symboler fra samtidens matematiske sprog.Desuden er der i appendices vigtige matematikhistoriske tekster og emner (Proklos' oversigt over geometrier, Hilberts aksiomer, Hippias' Quadratrix) og andre hjælpemidler, som ikke findes andetsteds.Bogen henvender sig til en bred gruppe læsere: professionelle matematikere fx i gymnasiet og på universitetet, hvor der undervises i matematikkens historie og videnskabsteori, filosofi- og antikstuderende sammesteds og endelig den ret store gruppe mennesker, der interesserer sig for antik filosofi og matematik.Euklids ELEMENTER består af 13 mindre bøger. Gyldendal samler dem i fire bind. Dette første bind indeholder Bog I-VI.  Euklids ELEMENTER I-VI er indledt, oversat og forsynet med bemærkninger af Claus Glunk, Hanne Eggert Strand, Chr. Marinus Taisbak og Chr. Gorm Tortzen.

Endeavors to explain Einstein's general theory of relativity, beginning with the equivalence principle and covering the necessary mathematics of Riemannian spaces and tensor calculus, offering readers a deeper understanding of the universe's real structure.

A-infinity structure was introduced by Stasheff in the 1960s in his homotopy characterization of based loop space, which was the culmination of earlier works of Sugawara's homotopy characterization of H-spaces and loop spaces. At the beginning of the 1990s, a similar structure was introduced by Fukaya in his categorification of Floer homology in symplectic topology. This structure plays a fundamental role in the celebrated homological mirror symmetry proposal by Kontsevich and in more recent developments of symplectic topology.A detailed construction of A-infinity algebra structure attached to a closed Lagrangian submanifold is given in Fukaya, Oh, Ohta, and Ono's two-volume monograph Lagrangian Intersection Floer Theory (AMS-IP series 46 I & II), using the theory of Kuranishi structures¿a theory that has been regarded as being not easily accessible to researchers in general. The present lecture note is provided by one of the main contributors to the Lagrangian Floer theory and is intended to provide a quick, reader-friendly explanation of the geometric part of the construction. Discussion of the Kuranishi structures is minimized, with more focus on the calculations and applications emphasizing the relevant homological algebra in the filtered context.The book starts with a quick explanation of Stasheff polytopes and their two realizations¿one by the rooted metric ribbon trees and the other by the genus-zero moduli space of open Riemann surfaces¿and an explanation of the A-infinity structure on the motivating example of the based loop space. It then provides a description of the moduli space of genus-zero bordered stable maps and continues with the construction of the (curved) A-infinity structure and its canonical models. Included in the explanation are the (LandaüGinzburg) potential functions associated with compact Lagrangian submanifolds constructed by Fukaya, Oh, Ohta, and Ono. The book explains calculations of potential functions for toric fibers in detail and reviews several explicit calculations in the literature of potential functions with bulk as well as their applications to problems in symplectic topology via the critical point theory thereof. In the Appendix, the book also provides rapid summaries of various background materials such as the stable map topology, Kuranishi structures, and orbifold Lagrangian Floer theory.

T¿puna Rock is a tale of two teenage siblings raised in Canada by their M¿ori mother and Canadian father. While on vacation in Aotearoa New Zealand, their borrowed sloop is disabled in a storm with only the two teens aboard. Having lost their navigation gear, after six days adrift with no sense of direction, they beach their damaged boat on a desolate 'rock'. With no idea where they are and their boat damaged beyond repair, the teens recognize that they cannot save themselves unless they listen to the spirits of their Polynesian ancestors. Recalling knowledge shared by their M¿ori grandparents and basic science and mathematics, the protagonists reinvent enough traditional non-instrument navigation to determine where they are, design and build a boat and plot a course to sail one thousand kilometres back to Aotearoa after more than three months on their 'rock'.T¿puna Rock is written to appeal to a broad range of ages, including secondary-school students among whom engagement in science and mathematics is often declining, particularly among indigenous students. The reader becomes invested in the protagonists and their situation on a purely human level in the novel's first two chapters. In subsequent chapters the protagonists come to understand and value ancestral knowledge in the context of modern science and mathematics. In the process, the teens teach each other, and the reader.

"This volume contains nine survey articles by the invited speakers of the 30th British Combinatorial Conference, one of the major international events in combinatorics. Written by leading experts in the field, these articles provide a snapshot of current developments in combinatorics for researchers and graduate students in discrete mathematics"--

Unravel the intricate relationship between quantum mechanics and highly-curved spacetime in this groundbreaking book by Dr. Kuro Aksara. Through advanced mathematical analysis, Aksara explores the fusion of quantum mechanics and general relativity, and delves into the quantum dynamics of black holes, quantum cosmology, entanglement, quantum gravity, quantum technology, and more. Prepare to embark on a quantum voyage into the unknown, where the boundaries of our understanding are pushed to their limits.

This is a comprehensive two-volumes text on plane and space geometry, transformations and conics, using a synthetic approach. The first volume focuses on Euclidean Geometry of the plane, and the second volume on Circle measurement, Transformations, Space geometry, Conics.The book is based on lecture notes from more than 30 courses which have been taught over the last 25 years. Using a synthetic approach, it discusses topics in Euclidean geometry ranging from the elementary (axioms and their first consequences), to the complex (the famous theorems of Pappus, Ptolemy, Euler, Steiner, Fermat, Morley, etc.). Through its coverage of a wealth of general and specialized subjects, it provides a comprehensive account of the theory, with chapters devoted to basic properties of simple planar and spatial shapes, transformations of the plane and space, and conic sections. As a result of repeated exposure of the material to students, it answers many frequently asked questions. Particular attention has been given to the didactic method; the text is accompanied by a plethora of figures (more than 2000) and exercises (more than 1400), most of them with solutions or expanded hints. Each chapter also includes numerous references to alternative approaches and specialized literature. The book is mainly addressed to students in mathematics, physics, engineering, school teachers in these areas, as well as, amateurs and lovers of geometry. Offering a sound and self-sufficient basis for the study of any possible problem in Euclidean geometry, the book can be used to support lectures to the most advanced level, or for self-study.

Voronoi diagrams, named after the Russian mathematician Georgy Feodosievych Voronoy, are geometric structures that have been extensively studied in Computational Geometry. A Voronoi diagram can be constructed for a set of geometric objects called sites. The geometric objects can be points, lines, circles, curves, surfaces, spheres, etc. A Voronoi diagram divides the space into regions called Voronoi cells such that there is a Voronoi cell corresponding to each object in the input set. A Voronoi cell is a set of points closer to the corresponding object than any other object in the input set. Applications of the Voronoi diagram can be seen in various ¿elds of science and engineering. Voronoi diagram is used in the motion planning and collision avoidance for robotic motion. Candeloro used the Voronoi diagram of a point set in a 3D space to develop a path-planning system for Unmanned Underwater Vehicles (UUVs). In town planning, the Voronoi diagram can be used for the e¿cient allocation of resources. Image segmentation and feature extraction are other areas where properties of the Voronoi diagram are utilized. similar applications can be seen in disease research and diagnosis. Voronoi diagram of a set of circles is used for sensor deployment where the radius of a circle represents the range of a sensor. It is also used in cable packing of electric wires to ¿nd the enclosing circle with a minimum radius.

This is a comprehensive two-volumes text on plane and space geometry, transformations and conics, using a synthetic approach. The first volume focuses on Euclidean Geometry of the plane, and the second volume on Circle measurement, Transformations, Space geometry, Conics.The book is based on lecture notes from more than 30 courses which have been taught over the last 25 years. Using a synthetic approach, it discusses topics in Euclidean geometry ranging from the elementary (axioms and their first consequences), to the complex (the famous theorems of Pappus, Ptolemy, Euler, Steiner, Fermat, Morley, etc.). Through its coverage of a wealth of general and specialized subjects, it provides a comprehensive account of the theory, with chapters devoted to basic properties of simple planar and spatial shapes, transformations of the plane and space, and conic sections. As a result of repeated exposure of the material to students, it answers many frequently asked questions. Particular attention has been given to the didactic method; the text is accompanied by a plethora of figures (more than 2000) and exercises (more than 1400), most of them with solutions or expanded hints. Each chapter also includes numerous references to alternative approaches and specialized literature. The book is mainly addressed to students in mathematics, physics, engineering, school teachers in these areas, as well as, amateurs and lovers of geometry. Offering a sound and self-sufficient basis for the study of any possible problem in Euclidean geometry, the book can be used to support lectures to the most advanced level, or for self-study.

In der Arbeit des Künstlers Titus Schade geht es um Malerei und um ihre visuelle Räumlichkeit. In seinen an Kulissen erinnernden Umgebungen entwickelt er Orte, die mal wie ein Modell, mal wie eine Theaterbühne wirken. So erfindet Schade in seinen Bildern eine Reihe verschiedener Architekturen und Ausstattungsstücke, die den Betrachter in private Räume führen. Dabei versucht er nicht, die Wirklichkeit abzubilden. Vielmehr arbeitet er mit Ersatzelementen, die er in einer in sich geschlossenen Umgebung anordnet. Seine Formen und die meist architektonischen Raumstrukturen werden auf geradezu barocke Art beleuchtet. Klassische Landschaften begegnen geometrischen Formen, deren Zeitlosigkeit eine universelle Lesart erlaubt. TITUS SCHADE (*1984, Leipzig), einer der bedeutendsten Künstler der jungen deutschen Malerei, war Schüler der Leipziger Akademie der Bildenden Künste, unter anderem bei Neo Rauch. Er lebt und arbeitet in Leipzig. Seine Arbeit wurde in zahlreichen Einzel- und Gruppenausstellungen in Deutschland gezeigt. Titus Schade wird vertreten durch die Galerie EIGEN + ART, Leipzig / Berlin.

Il presente libro trae origine dalle lezioni del corso di Geometria e vuole essere un utile strumento per la preparazione agli esami presenti in diversi corsi di laurea triennale, quali, Architettura e Ingegneria. Gli esercizi scelti, prima di tutto, suggeriscono percorsi per approfondimenti e riflessioni, personali, sulle nozioni teoriche da studiare per gli esami. Inoltre, sono stati elaborati in maniera tale da indurre il lettore a moderare l'uso dei procedimenti in serie, ripetitivi, applicati in maniera acritica, offrendo strategie per trovare soluzioni più dirette ed soprattutto ad affinare la capacità di pensiero e ragionamento.

The finite generation theorem is a major achievement of modern algebraic geometry. Based on the minimal model theory, it states that the canonical ring of an algebraic variety defined over a field of characteristic zero is a finitely generated graded ring. This graduate-level text is the first to explain this proof. It covers the progress on the minimal model theory over the last 30 years, culminating in the landmark paper on finite generation by Birkar-Cascini-Hacon-McKernan. Building up to this proof, the author presents important results and techniques that are now part of the standard toolbox of birational geometry, including Mori's bend and break method, vanishing theorems, positivity theorems and Siu's analysis on multiplier ideal sheaves. Assuming only the basics in algebraic geometry, the text keeps prerequisites to a minimum with self-contained explanations of terminology and theorems.

This book intends to focus exclusively on anamorphic experiments in contemporary art and design, leaving an in-depth historical examination of its Baroque season to other studies. Themes, languages and fields of application of anamorphosis in contemporary culture are critically analyzed to make the reader aware of the communicative potentiality of this kind of geometrical technique. The book also has the aim to teach the reader the most appropriate geometric techniques for each of them, in order to achieve the designed illusion. Each typology of anamorphosis is accompanied in this book by contemporary installations, a geometrical explanation by means of 3D models and didactic experiments carried on in collaboration with the students of the Department of Architecture in Naples.

This book contains a self-consistent treatment of a geometric averaging technique, induced by the Ricci flow, that allows comparing a given (generalized) Einstein initial data set with another distinct Einstein initial data set, both supported on a given closed n-dimensional manifold. This is a case study where two vibrant areas of research in geometric analysis, Ricci flow and Einstein constraints theory, interact in a quite remarkable way. The interaction is of great relevance for applications in relativistic cosmology, allowing a mathematically rigorous approach to the initial data set averaging problem, at least when data sets are given on a closed space-like hypersurface. The book does not assume an a priori knowledge of Ricci flow theory, and considerable space is left for introducing the necessary techniques. These introductory parts gently evolve to a detailed discussion of the more advanced results concerning a Fourier-mode expansion and a sophisticated heat kernel representation of the Ricci flow, both of which are of independent interest in Ricci flow theory. This work is intended for advanced students in mathematical physics and researchers alike.

This concise book reviews methods used for gluing space-time manifolds together. It is therefore relevant to theorists working on branes, walls, domain walls, concepts frequently used in theoretical cosmology, astrophysics, and gravity theory. Nowadays, applications are also in theoretical condensed matter physics where Riemannian geometry appears. The book also reviews the history of matching conditions between two space-time manifolds from the early times of general relativity up to now.

This volume celebrates the 100th birthday of Professor Chen-Ning Frank Yang (Nobel 1957), one of the giants of modern science and a living legend. Starting with reminiscences of Yang's time at the research centre for theoretical physics at Stonybrook (now named C. N. Yang Institute) by his successor Peter van Nieuwenhuizen, the book is a collection of articles by world-renowned mathematicians and theoretical physicists. This emphasizes the Dialogue Between Physics and Mathematics that has been a central theme of Professor Yang's contributions to contemporary science. Fittingly, the contributions to this volume range from experimental physics to pure mathematics, via mathematical physics. On the physics side, the contributions are from Sir Anthony Leggett (Nobel 2003), Jian-Wei Pan (Willis E. Lamb Award 2018), Alexander Polyakov (Breakthrough Prize 2013), Gerard 't Hooft (Nobel 1999), Frank Wilczek (Nobel 2004), Qikun Xue (Fritz London Prize 2020), and Zhongxian Zhao (Bernd T. Matthias Prize 2015), covering an array of topics from superconductivity to the foundations of quantum mechanics. In mathematical physics there are contributions by Sir Roger Penrose (Nobel 2022) and Edward Witten (Fields Medal 1990) on quantum twistors and quantum field theory, respectively. On the mathematics side, the contributions by Vladimir Drinfeld (Fields Medal 1990), Louis Kauffman (Wiener Gold Medal 2014), and Yuri Manin (Cantor Medal 2002) offer novel ideas from knot theory to arithmetic geometry.Inspired by the original ideas of C. N. Yang, this unique collection of papers b masters of physics and mathematics provides, at the highest level, contemporary research directions for graduate students and experts alike.

Over the course of his distinguished career, Claude Viterbo has made a number of groundbreaking contributions in the development of symplectic geometry/topology and Hamiltonian dynamics.  The chapters in this volume - compiled on the occasion of his 60th birthday - are written by distinguished mathematicians and pay tribute to his many significant and lasting achievements. 

This book offers an up-to-date, comprehensive account of determinantal rings and varieties, presenting a multitude of methods used in their study, with tools from combinatorics, algebra, representation theory and geometry.After a concise introduction to Grobner and Sagbi bases, determinantal ideals are studied via the standard monomial theory and the straightening law. This opens the door for representation theoretic methods, such as the Robinson-Schensted-Knuth correspondence, which provide a description of the Grobner bases of determinantal ideals, yielding homological and enumerative theorems on determinantal rings. Sagbi bases then lead to the introduction of toric methods. In positive characteristic, the Frobenius functor is used to study properties of singularities, such as F-regularity and F-rationality. Castelnuovo-Mumford regularity, an important complexity measure in commutative algebra and algebraic geometry, is introduced in the general setting of a Noetherian base ring and then applied to powers and products of ideals. The remainder of the book focuses on algebraic geometry, where general vanishing results for the cohomology of line bundles on flag varieties are presented and used to obtain asymptotic values of the regularity of symbolic powers of determinantal ideals. In characteristic zero, the Borel-Weil-Bott theorem provides sharper results for GL-invariant ideals. The book concludes with a computation of cohomology with support in determinantal ideals and a survey of their free resolutions.Determinants, Grobner Bases and Cohomology provides a unique reference for the theory of determinantal ideals and varieties, as well as an introduction to the beautiful mathematics developed in their study. Accessible to graduate students with basic grounding in commutative algebra and algebraic geometry, it can be used alongside general texts to illustrate the theory with a particularly interesting and important class of varieties.

This book completes the comprehensive introduction to modern algebraic geometry which was started with the introductory volume Algebraic Geometry I: Schemes.It begins by discussing in detail the notions of smooth, unramified and étale morphisms including the étale fundamental group. The main part is dedicated to the cohomology of quasi-coherent sheaves. The treatment is based on the formalism of derived categories which allows an efficient and conceptual treatment of the theory, which is of crucial importance in all areas of algebraic geometry. After the foundations are set up, several more advanced topics are studied, such as numerical intersection theory, an abstract version of the Theorem of Grothendieck-Riemann-Roch, the Theorem on Formal Functions, Grothendieck's algebraization results and a very general version of Grothendieck duality. The book concludes with chapters on curves and on abelian schemes, which serve to develop the basics of the theory of these two important classes of schemes on an advanced level, and at the same time to illustrate the power of the techniques introduced previously.The text contains many exercises that allow the reader to check their comprehension of the text, present further examples or give an outlook on further results.

This book provides a comprehensive introduction to the Calculus of Variations and its use in modelling mechanics and physics problems. Presenting a geometric approach to the subject, it progressively guides the reader through this very active branch of mathematics, accompanying key statements with a huge variety of exercises, some of them solved. Stressing the need to overcome limitations of the initial point of view, and emphasising the interconnectivity of various branches of mathematics (algebra, analysis and geometry), the book includes some advanced material to challenge the most motivated students. Systematic, short historical notes provide details on the subject's odyssey, and how new tools have been developed over the last two centuries. This English translation updates a set of notes for a course first given at the Ecole polytechnique in 1987. It will be accessible to graduate students and advanced undergraduates.

This new edition has been thoroughly revised, expanded and contain some updates function of the novel results and shift of scientific interest in the topics. The book has a Foreword by Jerry L. Bona and Hongqiu Chen. The book is an introduction to nonlinear waves and soliton theory in the special environment of compact spaces such a closed curves and surfaces and other domain contours. It assumes familiarity with basic soliton theory and nonlinear dynamical systems.The first part of the book introduces the mathematical concept required for treating the manifolds considered, providing relevant notions from topology and differential geometry. An introduction to the theory of motion of curves and surfaces - as part of the emerging field of contour dynamics - is given.The second and third parts discuss the modeling of various physical solitons on compact systems, such as filaments, loops and drops made of almost incompressible materials thereby intersecting with a large number of physical disciplines from hydrodynamics to compact object astrophysics.This book is intended for graduate students and researchers in mathematics, physics and engineering.

Euclid's Book on Divisions of Figures is a classic work of mathematics that explores the principles of geometry. This edition is restored based on Woepcke's text and on the Practica Geometriae of Leonardo Pisano, and is an essential resource for mathematicians and students 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. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.