|
Books > Science & Mathematics > Mathematics > Geometry
The book presents a comprehensive overview of various aspects of
three-dimensional geometry that can be experienced on a daily
basis. By covering the wide range of topics - from the psychology
of spatial perception to the principles of 3D modelling and
printing, from the invention of perspective by Renaissance artists
to the art of Origami, from polyhedral shapes to the theory of
knots, from patterns in space to the problem of optimal packing,
and from the problems of cartography to the geometry of solar and
lunar eclipses - this book provides deep insight into phenomena
related to the geometry of space and exposes incredible nuances
that can enrich our lives.The book is aimed at the general
readership and provides more than 420 color illustrations that
support the explanations and replace formal mathematical arguments
with clear graphical representations.
In the study of the structure of substances in recent decades,
phenomena in the higher dimension was discovered that was
previously unknown. These include spontaneous zooming (scaling
processes), discovery of crystals with the absence of translational
symmetry in three-dimensional space, detection of the fractal
nature of matter, hierarchical filling of space with polytopes of
higher dimension, and the highest dimension of most molecules of
chemical compounds. This forces research to expand the formulation
of the question of constructing n-dimensional spaces, posed by
David Hilbert in 1900, and to abandon the methods of considering
the construction of spaces by geometric figures that do not take
into account the accumulated discoveries in the physics of the
structure of substances. There is a need for research that accounts
for the new paradigm of the discrete world and provides a solution
to Hilbert's 18th problem of constructing spaces of higher
dimension using congruent figures. Normal Partitions and
Hierarchical Fillings of N-Dimensional Spaces aims to consider the
construction of spaces of various dimensions from two to any finite
dimension n, taking into account the indicated conditions,
including zooming in on shapes, properties of geometric figures of
higher dimensions, which have no analogue in three-dimensional
space. This book considers the conditions of existence of polytopes
of higher dimension, clusters of chemical compounds as polytopes of
the highest dimension, higher dimensions in the theory of heredity,
the geometric structure of the product of polytopes, the products
of polytopes on clusters and molecules, parallelohedron and
stereohedron of Delaunay, parallelohedron of higher dimension and
partition of n-dimensional spaces, hierarchical filling of
n-dimensional spaces, joint normal partitions, and hierarchical
fillings of n-dimensional spaces. In addition, it pays considerable
attention to biological problems. This book is a valuable reference
tool for practitioners, stakeholders, researchers, academicians,
and students who are interested in learning more about the latest
research on normal partitions and hierarchical fillings of
n-dimensional spaces.
This comprehensive reference begins with a review of the basics
followed by a presentation of flag varieties and finite- and
infinite-dimensional representations in classical types and
subvarieties of flag varieties and their singularities. Associated
varieties and characteristic cycles are covered as well and
Kazhdan-Lusztig polynomials are treated. The coverage concludes
with a discussion of pattern avoidance and singularities and some
recent results on Springer fibers.
This book surveys the mathematical and computational properties of
finite sets of points in the plane, covering recent breakthroughs
on important problems in discrete geometry, and listing many open
problems. It unifies these mathematical and computational views
using forbidden configurations, which are patterns that cannot
appear in sets with a given property, and explores the implications
of this unified view. Written with minimal prerequisites and
featuring plenty of figures, this engaging book will be of interest
to undergraduate students and researchers in mathematics and
computer science. Most topics are introduced with a related puzzle
or brain-teaser. The topics range from abstract issues of
collinearity, convexity, and general position to more applied areas
including robust statistical estimation and network visualization,
with connections to related areas of mathematics including number
theory, graph theory, and the theory of permutation patterns.
Pseudocode is included for many algorithms that compute properties
of point sets.
The term "stereotype space" was introduced in 1995 and denotes a
category of locally convex spaces with surprisingly elegant
properties. Its study gives an unexpected point of view on
functional analysis that brings this fi eld closer to other main
branches of mathematics, namely, to algebra and geometry. This
volume contains the foundations of the theory of stereotype spaces,
with accurate definitions, formulations, proofs, and numerous
examples illustrating the interaction of this discipline with the
category theory, the theory of Hopf algebras, and the four big
geometric disciplines: topology, differential geometry, complex
geometry, and algebraic geometry.
Over the last number of years powerful new methods in analysis and
topology have led to the development of the modern global theory of
symplectic topology, including several striking and important
results. The first edition of Introduction to Symplectic Topology
was published in 1995. The book was the first comprehensive
introduction to the subject and became a key text in the area. A
significantly revised second edition was published in 1998
introducing new sections and updates on the fast-developing area.
This new third edition includes updates and new material to bring
the book right up-to-date.
Noncommutative geometry combines themes from algebra, analysis and
geometry and has significant applications to physics. This book
focuses on cyclic theory, and is based upon the lecture courses by
Daniel G. Quillen at the University of Oxford from 1988-92, which
developed his own approach to the subject. The basic definitions,
examples and exercises provided here allow non-specialists and
students with a background in elementary functional analysis,
commutative algebra and differential geometry to get to grips with
the subject. Quillen's development of cyclic theory emphasizes
analogies between commutative and noncommutative theories, in which
he reinterpreted classical results of Hamiltonian mechanics,
operator algebras and differential graded algebras into a new
formalism. In this book, cyclic theory is developed from motivating
examples and background towards general results. Themes covered are
relevant to current research, including homomorphisms modulo powers
of ideals, traces on noncommutative differential forms, quasi-free
algebras and Chern characters on connections.
This contributed volume is a follow-up to the 2013 volume of the
same title, published in honor of noted Algebraist David Eisenbud's
65th birthday. It brings together the highest quality expository
papers written by leaders and talented junior mathematicians in the
field of Commutative Algebra. Contributions cover a very wide range
of topics, including core areas in Commutative Algebra and also
relations to Algebraic Geometry, Category Theory, Combinatorics,
Computational Algebra, Homological Algebra, Hyperplane
Arrangements, and Non-commutative Algebra. The book aims to
showcase the area and aid junior mathematicians and researchers who
are new to the field in broadening their background and gaining a
deeper understanding of the current research in this area. Exciting
developments are surveyed and many open problems are discussed with
the aspiration to inspire the readers and foster further research.
MESH ist ein mathematisches Video ber vielfl chige Netzwerke und
ihre Rolle in der Geometrie, der Numerik und der Computergraphik.
Der unter Anwendung der neuesten Technologie vollst ndig
computergenierte Film spannt einen Bogen von der antiken
griechischen Mathematik zum Gebiet der heutigen geometrischen
Modellierung. MESH hat zahlreiche wissenschaftliche Preise weltweit
gewonnen. Die Autoren sind Konrad Polthier, ein Professor der
Mathematik, und Beau Janzen, ein professioneller Filmdirektor.
Der Film ist ein ausgezeichnetes Lehrmittel f r Kurse in
Geometrie, Visualisierung, wissenschaftlichem Rechnen und
geometrischer Modellierung an Universit ten, Zentren f r
wissenschaftliches Rechnen, kann jedoch auch an Schulen genutzt
werden.
|
|