Series: SpringerBriefs in Physics
1st Edition., 2011, VIII, 114 p. 28 illus.
Softcover, ISBN 978-3-642-17976-1
This book provides an original introduction to the geometry of Minkowski space-time. A hundred years after the space-time formulation of special relativity by Hermann Minkowski, it is shown that the kinematical consequences of special relativity are merely a manifestation of space-time geometry.
The book is written with the intention of providing students (and teachers) of the first years of University courses with a tool which is easy to be applied and allows the solution of any problem of relativistic kinematics at the same time. The book treats in a rigorous way, but using a non-sophisticated mathematics, the Kinematics of Special Relativity. As an example, the famous "Twin Paradox" is completely solved for all kinds of motions.
The novelty of the presentation in this book consists in the extensive use of hyperbolic numbers, the simplest extension of complex numbers, for a complete formalization of the kinematics in the Minkowski space-time.
Moreover, from this formalization the understanding of gravity comes as a manifestation of curvature of space-time, suggesting new research fields.
1st Edition., 2011, 350 p.
Hardcover, ISBN 978-3-642-20745-7
Due: June 30, 2011
This research monograph deals with a modeling theory of the system of Navier-Stokes-Fourier equations for a Newtonian fluid governing a compressible viscous and heat conducting flows. The main objective is threefold. First , to 'deconstruct' this Navier-Stokes-Fourier system in order to unify the puzzle of the various partial simplified approximate models used in Newtonian Classical Fluid Dynamics and this, first facet, have obviously a challenging approach and a very important pedagogic impact on the university education.
The second facet of the main objective is to outline a rational consistent asymptotic/mathematical theory of the of fluid flows modeling on the basis of a typical Navier-Stokes-Fourier initial and boundary value problem. The third facet is devoted to an illustration of our rational asymptotic/mathematical modeling theory for various technological and geophysical stiff problems from: aerodynamics, thermal and thermocapillary convections and also meteofluid dynamics.
Content Level ≫ Research
Keywords ≫ Deconstruction approach - Euler Equations - Newtonian fluids - Rational Asymptotic modeling- viscous fluids
Related subjects ≫ Classical Continuum Physics - Computational Intelligence and Complexity - Dynamical Systems & Differential Equations - Mechanics - Meteorology & Climatology
Some Preliminary Comments.- From Euler and Navier Equations to NS-F Full Unsready Equations.- Dimensionless NS-F Equations and Parameters.- The Mathematics of the Rational Asymptotic Modelling.- A Deconstruction Approach for an Unsteady NS-F Fluid Flow at Large Reynolds Number.- Three RAM Applications in Aerodynamics.- The RAM Approach of Benard Problem.- Two RAM Applications for Atmospheric Motions.
Series: Applied Mathematical Sciences, Vol. 180
2011, X, 400 p. 20 illus., 10 in color.
Hardcover, ISBN 978-1-4614-0486-6
Due: December 28, 2011
This book offers an analytical rather than measure-theoretical approach to the derivation of the partial differential equations of nonlinear filtering theory. The basis for this approach is the discrete numerical scheme used in Monte-Carlo simulations of stochastic differential equations and Wiener's associated path integral representation of the transition probability density. Furthermore, it presents analytical methods for constructing asymptotic approximations to their solution and for synthesizing asymptotically optimal filters. It also offers a new approach to the phase tracking problem, based on optimizing the mean time to loss of lock. The book is based on lecture notes from a one-semester special topics course on stochastic processes and their applications that the author taught many times to graduate students of mathematics, applied mathematics, physics, chemistry, computer science, electrical engineering, and other disciplines. The book contains exercises and worked-out examples aimed at illustrating the methods of mathematical modeling and performance analysis of phase trackers.
Content Level ≫ Graduate
Keywords ≫ Markov processes - Optimal filtering - Stochastic Differential Equations - Stochastic processes - Stochastic stability
Related subjects ≫ Dynamical Systems & Differential Equations - Probability Theory and Stochastic Processes - Theoretical, Mathematical & Computational Physics
Diffusion and Stochastic Differential Equations.- Euler's Simulation Scheme and Wiener's Measure.- Nonlinear Filtering and Smoothing of Diffusions.- Small Noise Analysis of Zakai's Equation.- Loss of Lock in Phase Trackers.- Loss of Lock in RADAR and Synchronization.- Phase Tracking with Optimal Lock Time.- Bibliography.- Index
Series: Universitext
2011, XIV, 376 p. 37 illus., 1 in color.
Softcover, ISBN 978-3-642-21865-1
Due: September 30, 2011
The book gives a streamlined introduction to quantum mechanics while describing the basic mathematical structures underpinning this discipline.
Starting with an overview of key physical experiments illustrating the origin of the physical foundations, the book proceeds with a description of the basic notions of quantum mechanics and their mathematical content.
It then makes its way to topics of current interest, specifically those in which mathematics plays an important role. The more advanced topics presented include many-body systems, modern perturbation theory, path integrals, the theory of resonances, quantum statistics, mean-field theory, second quantization, the theory of radiation (non-relativistic quantum electrodynamics), and the renormalization group.
With different selections of chapters, the book can serve as a text for an introductory, intermediate, or advanced course in quantum mechanics. The last four chapters could also serve as an introductory course in quantum field theory.
Content Level ≫ Graduate
Keywords ≫ Schrodinger equation - quantization - quantum mechanics - spectral theory
Related subjects ≫ Analysis - Quantum Physics - Theoretical, Mathematical & Computational Physics
1 Physical Background.- 2 Dynamics.- 3 Observables.- 4 Quantization.- 5 Uncertainty Principle and Stability of Atoms and Molecules.- 6 Spectrum and Dynamics.- 7 Special Cases.- 8 Bound States and Variational Principle.- 9 Scattering States.- Existence of Atoms and Molecules.- 11 Perturbation Theory: Feshbach-Schur Method.- 12 General Theory of Many-particle Systems.- 13 Self-consistent Approximations.- 14 The Feynman Path Integral.- 15 Quasi-classical Analysis.- 16 Resonances.- 17 Quantum Statistics.- 18 The Second Quantization.- 19 Quantum Electro-Magnetic Field * Photons.- 20 Standard Model of Non-relativistic Matter and Radiation.- 21 Theory of Radiation.- 22 Renormalization Group.- 23 Mathematical Supplement: Spectral Analysis.- 24 Mathematical Supplement: The Calculus of Variations.- 25 Comments on Literature, and Further Reading.- References.- Index
Series: Publications of the Scuola Normale Superiore, Vol. 15
Subseries: Theses (Scuola Normale Superiore)
2011, Approx. 250 p.
Softcover, ISBN 978-88-7642-380-2
Defining and computing a greatest common divisor of two polynomials with inexact coefficients is a classical problem in symbolic-numeric computation. The first part of this book reviews the main results that have been proposed so far in the literature. As usual with polynomial computations, the polynomial GCD problem can be expressed in matrix form: the second part of the book focuses on this point of view and analyses the structure of the relevant matrices, such as Toeplitz, Toepliz-block and displacement structures. New algorithms for the computation of approximate polynomial GCD are presented, along with extensive numerical tests. The use of matrix structure allows, in particular, to lower the asymptotic computational cost from cubic to quadratic order with respect to polynomial degree.
Content Level ≫ Research
Keywords ≫ displacement structured matrices - polynomial computation - structured numerical linear algebra
Related subjects ≫ Scuola Normale Superiore
i. Introduction.- ii. Notation.- 1. Approximate polynomial GCD.- 2. Structured and resultant matrices.- 3. The Euclidean algorithm.- 4. Matrix factorization and approximate GCDs.- 5. Optimization approach.- 6. New factorization-based methods.- 7. A fast GCD algorithm.- 8. Numerical tests.- 9. Generalizations and further work.- 10. Appendix A: Distances and norms.- 11. Appendix B: Special matrices.- 12. Bibliography.- 13. Index.
Series: Publications of the Scuola Normale Superiore, Vol. 16
Subseries: Theses (Scuola Normale Superiore)
2011, 250 p.
Softcover, ISBN 978-88-7642-383-3
Due: August 26, 2011
This book is devoted to studying algorithms for the solution of a class of quadratic matrix and vector equations. These equations appear, in different forms, in several practical applications, especially in applied probability and control theory. The equations are first presented using a novel unifying approach; then, specific numerical methods are presented for the cases most relevant for applications, and new algorithms and theoretical results developed by the author are presented. The book focuses on “matrix multiplication-rich” iterations such as cyclic reduction and the structured doubling algorithm (SDA) and contains a variety of new research results which, as of today, are only available in articles or preprints.
Content Level ≫ Research
Keywords ≫ applied probability - control theory - numerical linear algebra
Related subjects ≫ Scuola Normale Superiore
Linear algebra preliminaries.* Quadratic vector equations.* A Perron vector iteration for QVEs.* Unilateral quadratic matrix equations.* Nonsymmetric algebraic Riccati equations.* Transforming NAREs into UQMEs.* Storage optimal algorithms for Cauchy-like matrices.* Newton method for rank-structured algebraic Riccati equations.* Lur'e equations.* Generalized SDA.* An effective matrix geometric mean.* Constructing other matrix geometric means.