Series: Studies in Fuzziness and Soft Computing , Vol. 206
2006, XVI, 358 p., 60 illus., Hardcover
ISBN-10: 3-540-31756-2
ISBN-13: 978-3-540-31756-2
About this book
This carefully edited book presents a focused debate on the
mathematics and physics of chaos, nonlinearity and complexity in
nature. It explores the role of non-extensive statistical
mechanics in non-equilibrium thermodynamics, and presents an
overview of the strong nonlinearity of chaos and complexity in
natural systems that draws on the relevant mathematics from
topology, measure-theory, inverse and ill-posed problems, set-valued
analysis, and nonlinear functional analysis. It presents a self-contained
scientific theory of complexity and complex systems as the steady
state of non-equilibrium systems, denoting a homeostatic dynamic
equilibrium between stabilizing order and destabilizing disorder.
Table of contents
Chaos, Periodicity and Complexity on Dynamical Systems.-
Foundations of Nonextensive Statistical Mechanics.- Critical
Attractors and the Physical Realm of q-statistics.- Non-Boltzmannian
Entropies for Complex Classical Systems, Quantum Coherent States
and Black Holes.- Power Law and Tsallis Entropy: Network Traffic
and Applications.- The Role of Chaos and Resonances in Brownian
Motion.- Models of Finite Bath and Generalised Thermodynamics.-
Quantum Black Hole Thermodynamics.- Complexity in Organizations:
A Paradigm Shift.- Chaos, Nonlinearity, Complexity: A Unified
Perspective.
Series: Fundamental Theories of Physics , Vol. 152
2006, XIII, 229 p., Hardcover
ISBN-10: 1-4020-5253-7
ISBN-13: 978-1-4020-5253-8
About this book
Reading Bohr: Physics and Philosophy offers a new perspective on
Niels Bohr's interpretation of quantum mechanics as
complementarity, and on the relationships between physics and
philosophy in Bohr's work, which has had momentous significance
for our understanding of quantum theory and of the nature of
knowledge in general. Philosophically, the book reassesses Bohr's
place in the Western philosophical tradition, from Kant and Hegel
on. Physically, it reconsiders the main issues at stake in the
Bohr-Einstein confrontation and in the ongoing debates concerning
quantum physics. It also devotes greater attention than in most
commentaries on Bohr to the key developments and transformations
of his thinking concerning complementarity.
Most significant among them were those that occurred, first,
under the impact of Bohr's exchanges with Einstein and, second,
under the impact of developments in quantum theory itself, both
quantum mechanics and quantum field theory. The importance of
quantum field theory for Bohr's thinking has not been adequately
addressed in the literature on Bohr, to the considerable
detriment to our understanding of the history of quantum physics.
Filling this lacuna is one of the main contributions of the book,
which also enables us to show why quantum field theory compels us
to move beyond Bohr without, however, simply leaving him behind.
Table of contents
Series: Lecture Notes in Computational Science and Engineering
, Vol. 57
2007, VIII, 312 p., 165 illus., 9 in colour, Softcover
ISBN-10: 3-540-46214-7
ISBN-13: 978-3-540-46214-9
About this book
Meshfree methods for the numerical solution of partial
differential equations are becoming more and more mainstream in
many areas of applications. Their flexiblity and wide
applicability are attracting engineers, scientists, and
mathematicians to this very dynamic research area. This volume
represents the state of the art in meshfree methods. It consists
of articles which address the different meshfree techniques,
their mathematical properties and their application in applied
mathematics, physics and engineering.
Written for:
Researchers and graduate students in Mathematics, Engineering and
Physics
Keywords:
element-free Galerkin methods
engineering applications
kernel particle methods
meshfree discretizations
meshless methods
partial differential equations
partition of unity method
smoothed particle hydrodynamics
stochastic particle methods
2007, Approx. 445 p., Softcover
ISBN-10: 1-4020-5494-7
ISBN-13: 978-1-4020-5494-5
About this textbook
Linear algebra is a living, active branch of mathematics which is
central to almost all other areas of mathematics, both pure and
applied, as well as computer science, the physical and social
sciences, and engineering. It entails an extensive corpus of
theoretical results as well as a large body of computational
techniques. Unfortunately, in recent years the content of the
linear algebra courses required to complete an undergraduate
degree in mathematics has been depleted to the extent that they
fail to provide a sufficient theoretical or computational
background. Students are not only less able to formulate or even
follow mathematical proofs, they are also less able to understand
the mathematics of the numerical algorithms they need for
applications. Certainly, the material presented in the average
undergraduate linear algebra course is insufficient for graduate
study. This book is intended to fill this gap by providing enough
material "theoretical and computational" to allow the
student to work independently or in advanced courses.
Table of contents
1. Notation and terminology.- 2. Fields.- 3. Vector spaces over a
field.- 4. Algebras over a field.- 5. Linear Dependence and
Dimension.- 6. Linear Transformations.- 7. The endomorphism
algebra of a vector space.- 8. Representation of linear
transformations by matrices.- 9. The algebra of square matrices.-
10. Systems of linear equations.- 11.Determinants.- 12.
Eigenvalues and eigenvectors.- 13. Krylov subspaces.- 14. The
dual space.- 15. Inner product spaces.- 16. Orthogonality.- 17.
Selfadjoint endomorphisms.- 18. Unitary and normal endomorphisms.-
19. Moore-Penrose pseudoinverses.- 20. Bilinear transformations
and forms.
Series: Algorithms and Computation in Mathematics , Vol. 20
2007, Approx. 240 p., 17 illus., Hardcover
ISBN-10: 3-540-46367-4
ISBN-13: 978-3-540-46367-2
About this book
The book deals with algorithmic problems related to binary
quadratic forms, such as finding the representations of an
integer by a form with integer coefficients, finding the minimum
of a form with real coefficients and deciding equivalence of two
forms. In order to solve those problems, the book introduces the
reader to important areas of number theory such as diophantine
equations, reduction theory of quadratic forms, geometry of
numbers and algebraic number theory. The book explains
applications to cryptography. It requires only basic mathematical
knowledge.
Table of contents
1 Binary Quadratic Forms.- 2 Equivalence of Forms.- 3
Constructing Forms.- 4 Forms, Bases, Points, and Lattices.- 5
Reduction of Positive Definite Forms.- 6 Reduction of Indefinite
Forms.- 7 Multiplicative Lattices.- 8 Quadratic Number Fields.- 9
Class Groups.- 10 Infrastructure.- 11 Subexponential Algorithms.-
12 Cryptographic Applications.- Appendix - Bibliography -
References - Subject Index - Index.
Series: Fundamental Theories of Physics , Vol. 154
2006, XIX, 275 p., Hardcover
ISBN-10: 1-4020-5168-9
ISBN-13: 978-1-4020-5168-5
About this book
This work presents a "Clean Quantum Theory of the Electron",
based on Dirac’s equation. "Clean" in the sense of a
complete mathematical explanation of the well known paradoxes of
Dirac’s theory, and a connection to classical theory, including
the motion of a magnetic moment (spin) in the given field, all
for a charged particle (of spin ?) moving in a given
electromagnetic field.
This theory is relativistically covariant, and it may be regarded
as a mathematically consistent quantum-mechanical generalization
of the classical motion of such a particle, a la Newton and
Einstein. Normally, our fields are time-independent, but also
discussed is the time-dependent case, where slightly different
features prevail. A "Schroedinger particle", such as a
light quantum, experiences a very different (time-dependent)
"Precise Predictablity of Observables". An attempt is
made to compare both cases.
There is not the Heisenberg uncertainty of location and momentum;
rather, location alone possesses a built-in uncertainty of
measurement.
Mathematically, our tools consist of the study of a pseudo-differential
operator (i.e. an "observable") under conjugation with
the Dirac propagator: such an operator has a "symbol"
approximately propagating along classical orbits, while taking
its "spin" along. This is correct only if the operator
is "precisely predictable", that is, it must
approximately commute with the Dirac Hamiltonian, and, in a
sense, will preserve the subspaces of electronic and positronic
states of the underlying Hilbert space.
Table of contents
Series: Springer Undergraduate Mathematics Series
2007, CCCLXXVI, 24 p., 68 illus., Softcover
ISBN-10: 1-84628-632-8
ISBN-13: 978-1-84628-632-2
About this textbook
Worlds out of Nothing is the first book to provide a course on
the history of geometry in the 19th century. Based on the latest
historical research, the book is aimed primarily at undergraduate
and graduate students in mathematics but will also appeal to the
reader with a general interest in the history of mathematics.
Emphasis is placed on understanding the historical significance
of the new mathematics: Why was it done? How - if at all - was it
appreciated? What new questions did it generate?
Topics covered in the first part of the book are projective
geometry, especially the concept of duality, and non-Euclidean
geometry. The book then moves on to the study of the singular
points of algebraic curves (Plucker’s equations) and their role
in resolving a paradox in the theory of duality; to Riemann’s
work on differential geometry; and to Beltrami’s role in
successfully establishing non-Euclidean geometry as a rigorous
mathematical subject. The final part of the book considers how
projective geometry, as exemplified by Klein’s Erlangen
Program, rose to prominence, and looks at Poincare’s ideas
about non-Euclidean geometry and their physical and philosophical
significance. It then concludes with discussions on geometry and
formalism, examining the Italian contribution and Hilbert’s
Foundations of Geometry; geometry and physics, with a look at
some of Einstein’s ideas; and geometry and truth.
Three chapters are devoted to writing and assessing work in the
history of mathematics, with examples of sample questions in the
subject, advice on how to write essays, and comments on what
instructors should be looking for.
Jeremy Gray is Professor of the History of Mathematics and
Director of the Centre for the History of the Mathematical
Sciences at the Open University in England, and is an Honorary
Professor in the Mathematics Department at the University of
Warwick. He is the author, co-author, or editor of 14 books on
the history of mathematics in the 19th and 20th Centuries.
Table of contents
Mathematics in the French Revolution.- Poncelet (and Pole and
Polar).- Theorems in Projective Geometry.- Poncelet’s Traite.-
Duality and the Duality Controversy.- Poncelet and Chasles.-
Lambert and Legendre.- Gauss.- Janos Bolyai.- Lobachevskii.- To
1855.- Writing.- Mobius.- The Duality ParadoxNon-Euclidean .- The
Plucker Formulae.- Higher Plane Curves.- Complex Curves.- Riemann.-
Differential Geometry of Surfaces.- Geometry Accepted- Summary:
.- Writing.- Fundamental Geometry.- Hilbert.- Italian Foundations.-
The Disc Model.- The Geometry of Space.Geometry to Bibliography.-
1900.- The Formal Side.- The Physical Side.- Is Geometry True?-
Writing.- Appendix: Von Staudt and his Influence.- Index.