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Books > Science & Mathematics > Mathematics > Geometry
The study of the geometry of structures that arise in a variety of
specific natural systems, such as chemical, physical, biological,
and geological, revealed the existence of a wide range of types of
polytopes of the highest dimension that were unknown in classical
geometry. At the same time, new properties of polytopes were
discovered as well as the geometric patterns to which they obey.
There is a need to classify these types of polytopes of the highest
dimension by listing their properties and formulating the laws to
which they obey. The Classes of Higher Dimensional Polytopes in
Chemical, Physical, and Biological Systems explains the meaning of
higher dimensions and systematically generalizes the results of
geometric research in various fields of knowledge. This book is
useful both for the fundamental development of geometry and for the
development of branches of science related to human activities. It
builds upon previous books published by the author on this topic.
Covering areas such as heredity, geometry, and dimensions, this
reference work is ideal for researchers, scholars, academicians,
practitioners, industry professionals, instructors, and students.
Symmetry is all around us. Of fundamental significance to the
way we interpret the world, this unique, pervasive phenomenon
indicates a dynamic relationship between objects. Combining a rich
historical narrative with his own personal journey as a
mathematician, Marcus du Sautoy takes a unique look into the
mathematical mind as he explores deep conjectures about symmetry
and brings us face-to-face with the oddball mathematicians, both
past and present, who have battled to understand symmetry's elusive
qualities.
Classical Deformation Theory is used for determining the
completions of the local rings of an eventual moduli space. When a
moduli variety exists, a main result in the book is that the local
ring in a closed point can be explicitly computed as an
algebraization of the pro-representing hull (therefore, called the
local formal moduli) of the deformation functor for the
corresponding closed point.The book gives explicit computational
methods and includes the most necessary prerequisites. It focuses
on the meaning and the place of deformation theory, resulting in a
complete theory applicable to moduli theory. It answers the
question 'why moduli theory' and it give examples in mathematical
physics by looking at the universe as a moduli of molecules.
Thereby giving a meaning to most noncommutative theories.The book
contains the first explicit definition of a noncommutative scheme,
covered by not necessarily commutative rings. This definition does
not contradict any of the previous abstract definitions of
noncommutative algebraic geometry, but rather gives interesting
relations to other theories which is left for further
investigation.
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.
This book is devoted to group-theoretic aspects of topological
dynamics such as studying groups using their actions on topological
spaces, using group theory to study symbolic dynamics, and other
connections between group theory and dynamical systems. One of the
main applications of this approach to group theory is the study of
asymptotic properties of groups such as growth and amenability. The
book presents recently developed techniques of studying groups of
dynamical origin using the structure of their orbits and associated
groupoids of germs, applications of the iterated monodromy groups
to hyperbolic dynamical systems, topological full groups and their
properties, amenable groups, groups of intermediate growth, and
other topics. The book is suitable for graduate students and
researchers interested in group theory, transformations defined by
automata, topological and holomorphic dynamics, and theory of
topological groupoids. Each chapter is supplemented by exercises of
various levels of complexity.
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 book is the second of a three-volume set of books on the
theory of algebras, a study that provides a consistent framework
for understanding algebraic systems, including groups, rings,
modules, semigroups and lattices. Volume I, first published in the
1980s, built the foundations of the theory and is considered to be
a classic in this field. The long-awaited volumes II and III are
now available. Taken together, the three volumes provide a
comprehensive picture of the state of art in general algebra today,
and serve as a valuable resource for anyone working in the general
theory of algebraic systems or in related fields. The two new
volumes are arranged around six themes first introduced in Volume
I. Volume II covers the Classification of Varieties, Equational
Logic, and Rudiments of Model Theory, and Volume III covers Finite
Algebras and their Clones, Abstract Clone Theory, and the
Commutator. These topics are presented in six chapters with
independent expositions, but are linked by themes and motifs that
run through all three volumes.
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.
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