Numerical and computational methods play a major role in modelling compressible flow and are important tools in solving fluid dynamical problems faced in many areas of science and technology. This book thoroughly surveys and analyzes up-to-date methods, while reviewing the basic theoretical mathematical analysis.

This book addresses the mechanism of enrichment of heavy elements in galaxies, a long standing problem in astronomy. It mainly focuses on explaining the origin of heavy elements by performing state-of-the-art, high-resolution hydrodynamic simulations of dwarf galaxies. In this book, the author successfully develops a model of galactic chemodynamical evolution by means of which the neutron star mergers can be used to explain the observed abundance pattern of the heavy elements synthesized by the rapid neutron capture process, such as europium, gold, and uranium in the Local Group dwarf galaxies. The book argues that heavy elements are significant indicators of the evolutionary history of the early galaxies, and presents theoretical findings that open new avenues to understanding the formation and evolution of galaxies based on the abundance of heavy elements in metal-poor stars.

"Difference Equations, Second Edition," presents a practical introduction to this important field of solutions for engineering and the physical sciences. Topic coverage includes numerical analysis, numerical methods, differential equations, combinatorics and discrete modeling. A hallmark of this revision is the diverse application to many subfields of mathematics.

* Phase plane analysis for systems of two linear equations
* Use of equations of variation to approximate solutions
* Fundamental matrices and Floquet theory for periodic systems
* LaSalle invariance theorem
* Additional applications: secant line method, Bison problem, juvenile-adult population model, probability theory
* Appendix on the use of "Mathematica" for analyzing difference equaitons
* Exponential generating functions
* Many new examples and exercises

This multi-volume handbook is the most up-to-date and comprehensive reference work in the field of fractional calculus and its numerous applications. This eighth volume collects authoritative chapters covering several applications of fractional calculus in engineering, life and social sciences, including applications in signal and image analysis, and chaos.

This thesis describes the structures of six-dimensional (6d) superconformal field theories and its torus compactifications. The first half summarizes various aspects of 6d field theories, while the latter half investigates torus compactifications of these theories, and relates them to four-dimensional superconformal field theories in the class, called class S. It is known that compactifications of 6d conformal field theories with maximal supersymmetries provide numerous insights into four-dimensional superconformal field theories. This thesis generalizes the story to the theories with smaller supersymmetry, constructing those six-dimensional theories as brane configurations in the M-theory, and highlighting the importance of fractionalization of M5-branes. This result establishes new dualities between the theories with eight supercharges.

Containing an extensive illustration of the use of finite difference method in solving boundary value problem numerically, a wide class of differential equations have been numerically solved in this book.

This book provides a comprehensive review of complex networks from three different domains, presents novel methods for analyzing them, and highlights applications with accompanying case studies. Special emphasis is placed on three specific kinds of complex networks of high technological and scientific importance: software networks extracted from the source code of computer programs, ontology networks describing semantic web ontologies, and co-authorship networks reflecting collaboration in science. The book is primarily intended for researchers, teachers and students interested in complex networks and network data analysis. However, it will also be valuable for researchers dealing with software engineering, ontology engineering and scientometrics, as it demonstrates how complex network analysis can be used to address important research issues in these three disciplines.

This book features a selection of articles based on the XXXV Bialowieza Workshop on Geometric Methods in Physics, 2016. The series of Bialowieza workshops, attended by a community of experts at the crossroads of mathematics and physics, is a major annual event in the field. The works in this book, based on presentations given at the workshop, are previously unpublished, at the cutting edge of current research, typically grounded in geometry and analysis, and with applications to classical and quantum physics. In 2016 the special session "Integrability and Geometry" in particular attracted pioneers and leading specialists in the field. Traditionally, the Bialowieza Workshop is followed by a School on Geometry and Physics, for advanced graduate students and early-career researchers, and the book also includes extended abstracts of the lecture series.

Interesting to anybody who wants to unearth the real sense and nature of solitary waves, and the relevant mathematical tools to use for effective investigation and analysis of these phenomena, the text focuses on numerical analysis of solitons. The integrability and multidimensionality of solitons is inextricably bound up with the approach of investigation and, as the more physical systems are not fully integrable, even in one dimension, numerical analysis is the main tool to investigate and understand the pertinent physical mechanisms.

Quantum mechanics is arguably one of the most successful scientific theories ever and its applications to chemistry, optics, and information theory are innumerable. This book provides the reader with a rigorous treatment of the main mathematical tools from harmonic analysis which play an essential role in the modern formulation of quantum mechanics. This allows us at the same time to suggest some new ideas and methods, with a special focus on topics such as the Wigner phase space formalism and its applications to the theory of the density operator and its entanglement properties. This book can be used with profit by advanced undergraduate students in mathematics and physics, as well as by confirmed researchers.

This book presents simple interdisciplinary stochastic models meant as a gentle introduction to the field of non-equilibrium statistical physics. It focuses on the analysis of two-state models with cooperative effects and explores a variety of mathematical techniques to solve the master equations that govern these models. The models discussed are at the confluence of nanophysics, biology, mathematics and the social science, and they provide a pedagogical path toward understanding the complex dynamics of particle self-assembly with the tools of statistical physics.

This multi-volume handbook is the most up-to-date and comprehensive reference work in the field of fractional calculus and its numerous applications. This seventh volume collects authoritative chapters covering several applications of fractional calculus in in engineering, life, and social sciences, including applications in biology and medicine, mechanics of complex media, economy, and electrical devices.

Analysis and Control of Polynomial Dynamic Models with Biological Applications synthesizes three mathematical background areas (graphs, matrices and optimization) to solve problems in the biological sciences (in particular, dynamic analysis and controller design of QP and polynomial systems arising from predator-prey and biochemical models). The book puts a significant emphasis on applications, focusing on quasi-polynomial (QP, or generalized Lotka-Volterra) and kinetic systems (also called biochemical reaction networks or simply CRNs) since they are universal descriptors for smooth nonlinear systems and can represent all important dynamical phenomena that are present in biological (and also in general) dynamical systems.

Explore real-world applications of selected mathematical theory, concepts, and methods

Exploring related methods that can be utilized in various fields of practice from science and engineering to business, A First Course in Applied Mathematics details how applied mathematics involves predictions, interpretations, analysis, and mathematical modeling to solve real-world problems.

Written at a level that is accessible to readers from a wide range of scientific and engineering fields, the book masterfully blends standard topics with modern areas of application and provides the needed foundation for transitioning to more advanced subjects. The author utilizes MATLAB(R) to showcase the presented theory and illustrate interesting real-world applications to Google's web page ranking algorithm, image compression, cryptography, chaos, and waste management systems. Additional topics covered include:

Linear algebra

Ranking web pages

Matrix factorizations

Least squares

Image compression

Ordinary differential equations

Dynamical systems

Mathematical models

Throughout the book, theoretical and applications-oriented problems and exercises allow readers to test their comprehension of the presented material. An accompanying website features related MATLAB(R) code and additional resources.

A First Course in Applied Mathematics is an ideal book for mathematics, computer science, and engineering courses at the upper-undergraduate level. The book also serves as a valuable reference for practitioners working with mathematical modeling, computational methods, and the applications of mathematics in their everyday work.

This book uses art photography as a point of departure for learning about physics, while also using physics as a point of departure for asking fundamental questions about the nature of photography as an art. Although not a how-to manual, the topics center around hands-on applications, sometimes illustrated by photographic processes that are inexpensive and easily accessible to students (including a versatile new process developed by the author, and first described in print in this series). A central theme is the connection between the physical interaction of light and matter on the one hand, and the artistry of the photographic processes and their results on the other. One half of Energy and Color focuses on the physics of energy, power, illuminance, and intensity of light, and how these relate to the photographic exposure, including a detailed example that follows the emission of light from the sun all the way through to the formation of the image in the camera. These concepts are described in both their traditional manner, but also using very-low sensitivity photography as an example, which brings the physical concepts to the fore in a visible way, whereas they are often hidden with ordinary high-speed photographic detectors. Energy and Color also considers color in terms of the spectrum of light, how it interacts with the subject, and how the camera's light detector interacts with the image focused upon it. But of equal concern is the only partially-understood and sometimes unexpected ways in which the human eye/brain interprets this spectral stimulus as color. The volume covers basic photographic subjects such as shutter, aperture, ISO, metering and exposure value, but also given their relations to the larger themes of the book less familiar topics such as the Jones-Condit equation, Lambertian versus isotropic reflections, reflection and response curves, and the opponent-process model of color perception. Although written at a beginning undergraduate level, the topics are chosen for their role in a more general discussion of the relation between science and art that is of interest to readers of all backgrounds and levels of expertise.

This thesis focuses on an unresolved problem in particle and nuclear physics: the relation between two important non-perturbative phenomena in quantum chromodynamics (QCD) - quark confinement and chiral symmetry breaking. The author develops a new analysis method in the lattice QCD, and derives a number of analytical formulae to express the order parameters for quark confinement, such as the Polyakov loop, its fluctuations, and the Wilson loop in terms of the Dirac eigenmodes closely related to chiral symmetry breaking. Based on the analytical formulae, the author analytically as well as numerically shows that at finite temperatures there is no direct one-to-one correspondence between them. The thesis describes this extraordinary achievement using the first-principle analysis, and proposes a possible new phase in which quarks are confined and chiral symmetry is restored.

This book provides a comprehensive and systematic approach to the study of the qualitative theory of boundedness, periodicity, and stability of Volterra difference equations. The book bridges together the theoretical aspects of Volterra difference equations with its applications to population dynamics. Applications to real-world problems and open-ended problems are included throughout. This book will be of use as a primary reference to researchers and graduate students who are interested in the study of boundedness of solutions, the stability of the zero solution, or in the existence of periodic solutions using Lyapunov functionals and the notion of fixed point theory.

In any linear system the input and the output are connected by means of a linear operator. When the input can be notionally represented by a function that is null valued everywhere except at a specific location in space-time, the corresponding output is called the Green function in field theories. Dyadic Green functions are commonplace in electromagnetics, because both the input and the output are vector functions of space and time. This book provides a survey of the state-of-the-art knowledge of infinite-space dyadic Green functions.

The self-avoiding walk is a classical model in statistical mechanics, probability theory and mathematical physics. It is also a simple model of polymer entropy which is useful in modelling phase behaviour in polymers. This monograph provides an authoritative examination of interacting self-avoiding walks, presenting aspects of the thermodynamic limit, phase behaviour, scaling and critical exponents for lattice polygons, lattice animals and surfaces. It also includes a comprehensive account of constructive methods in models of adsorbing, collapsing, and pulled walks, animals and networks, and for models of walks in confined geometries. Additional topics include scaling, knotting in lattice polygons, generating function methods for directed models of walks and polygons, and an introduction to the Edwards model. This essential second edition includes recent breakthroughs in the field, as well as maintaining the older but still relevant topics. New chapters include an expanded presentation of directed models, an exploration of methods and results for the hexagonal lattice, and a chapter devoted to the Monte Carlo methods.

This volume emphasises studies related to
classical Stefan problems. The term "Stefan problem" is
generally used for heat transfer problems with
phase-changes such
as from the liquid to the solid. Stefan problems have some
characteristics that are typical of them, but certain problems
arising in fields such as mathematical physics and engineering
also exhibit characteristics similar to them. The term
classical" distinguishes the formulation of these problems from
their weak formulation, in which the solution need not possess
classical derivatives. Under suitable assumptions, a weak solution
could be as good as a classical solution. In hyperbolic Stefan
problems, the characteristic features of Stefan problems are
present but unlike in Stefan problems, discontinuous solutions are
allowed because of the hyperbolic nature of the heat equation. The
numerical solutions of inverse Stefan problems, and the analysis of
direct Stefan problems are so integrated that it is difficult to
discuss one without referring to the other. So no strict line of
demarcation can be identified between a classical Stefan problem
and other similar problems. On the other hand, including every
related problem in the domain of classical Stefan problem would
require several volumes for their description. A suitable
compromise has to be made.
The basic concepts, modelling, and analysis of the classical
Stefan problems have been extensively investigated and there seems
to be a need to report the results at one place. This book
attempts to answer that need. Within the framework of the
classical Stefan problem with the emphasis on the basic concepts,
modelling and analysis, it tries to include some weak
solutions and analytical and numerical solutions also. The main
considerations behind this are the continuity and the clarity of
exposition. For example, the description of some phase-field
models in Chapter 4 arose out of this need for a smooth transition
between topics. In the mathematical formulation of Stefan
problems, the curvature effects and the kinetic condition are
incorporated with the help of the modified Gibbs-Thomson relation.
On the basis of some thermodynamical and metallurgical
considerations, the modified Gibbs-Thomson relation can be
derived, as has been done in the text, but the rigorous
mathematical justification comes from the fact that this relation
can be obtained by taking appropriate limits of phase-field
models. Because of the unacceptability of some phase-field models
due their so-called thermodynamical inconsistency, some consistent
models have also been described. This completes the discussion of
phase-field models in the present context.
Making this volume self-contained would require reporting and
deriving several results from tensor analysis, differential
geometry, non-equilibrium thermodynamics, physics and functional
analysis. The text is enriched with appropriate
references so as not to enlarge the scope of the book. The proofs
of propositions and theorems are often lengthy and different from
one another. Presenting them in a condensed way may not be of much
help to the reader. Therefore only the main features of proofs
and a few results have been presented to suggest the essential
flavour of the theme of investigation. However at each place,
appropriate references have been cited so that inquisitive
readers can follow them on their own.
Each chapter begins with basic concepts, objectives and the
directions in which the subject matter has grown. This is followed
by reviews - in some cases quite detailed - of published works. In a
work of this type, the author has to make a suitable compromise
between length restrictions and understandability.

Lissajous figures are produced by combining two oscillations at right angles to each other. The figures, drawn by mechanical devices called harmonographs, have scientific uses, but are also enjoyed for their own beauty. The author has been working with harmonographs since his undergraduate days, building several of them, lecturing on them and has written articles about them. This book is intended for people who enjoy physics or art or both.

This multi-volume handbook is the most up-to-date and comprehensive reference work in the field of fractional calculus and its numerous applications. This sixth volume collects authoritative chapters covering several applications of fractional calculus in control theory, including fractional controllers, design methods and toolboxes, and a large number of engineering applications of control.

Reliability is a fundamental criterium in engineering systems. This book shows innovative concepts and applications of mathematics in solving reliability problems. The contents address in particular the interaction between engineers and mathematicians, as well as the cross-fertilization in the advancement of science and technology. It bridges the gap between theory and practice to aid in practical problem-solving in various contexts.