This book highlights new developments in the wide and growing field of partial differential equations (PDE)-constrained optimization. Optimization problems where the dynamics evolve according to a system of PDEs arise in science, engineering, and economic applications and they can take the form of inverse problems, optimal control problems or optimal design problems. This book covers new theoretical, computational as well as implementation aspects for PDE-constrained optimization problems under uncertainty, in shape optimization, and in feedback control, and it illustrates the new developments on representative problems from a variety of applications.

The results of renormalized perturbation theory, in QCD and other quantum field theories, are ambiguous at any finite order, due to renormalization-scheme dependence. The perturbative results depend upon extraneous scheme variables, including the renormalization scale, that the exact result cannot depend on. Such 'non-invariant approximations' occur in many other areas of physics, too. The sensible strategy is to find where the approximant is stationary under small variations of the extraneous variables. This general principle is explained and illustrated with various examples. Also dimensional transmutation, RG equations, the essence of renormalization and the origin of its ambiguities are explained in simple terms, assuming little or no background in quantum field theory. The minimal-sensitivity approach leads to 'optimized perturbation theory,' which is developed in detail. Applications to Re+e-, the infrared limit, and to the optimization of factorized quantities, are also discussed thoroughly.

Using the familiar software Microsoft ® Excel, this book examines the applications of complex variables. Implementation of the included problems in Excel eliminates the “black box” nature of more advanced computer software and programming languages and therefore the reader has the chance to become more familiar with the underlying mathematics of the complex variable problems. This book consists of two parts. In Part I, several topics are covered that one would expect to find in an introductory text on complex variables. These topics include an overview of complex numbers, functions of a complex variable, and the Cauchy integral formula. In particular, attention is given to the study of analytic complex variable functions. This attention is warranted because of the property that the real and imaginary parts of an analytic complex variable function can be used to solve the Laplace partial differential equation (PDE). Laplace's equation is ubiquitous throughout science and engineering as it can be used to model the steady-state conditions of several important transport processes including heat transfer, soil-water flow, electrostatics, and ideal fluid flow, among others. In Part II, a specialty application of complex variables known as the Complex Variable Boundary Element Method (CVBEM) is examined. CVBEM is a numerical method used for solving boundary value problems governed by Laplace's equation. This part contains a detailed description of the CVBEM and a guide through each step of constructing two CVBEM programs in Excel. The writing of these programs is the culminating event of the book. Students of complex variables and anyone with interest in a novel method for approximating potential functions using the principles of complex variables are the intended audience for this book. The Microsoft Excel applications (including simple programs as well as the CVBEM program) covered will also be of interest in the industry, as these programs are accessible to anybody with Microsoft Office.

Quantum mechanics is one of the most fascinating, and at the same time most controversial, branches of contemporary science. Disputes have accompanied this science since its birth and have not ceased to this day.

"Uncommon Paths in Quantum Physics" allows the reader to contemplate deeply some ideas and methods that are seldom met in the contemporary literature. Instead of widespread recipes of mathematical physics, based on the solutions of integro-differential equations, the book follows logical and partly intuitional derivations of non-commutative algebra. Readers can directly penetrate the abstract world of quantum mechanics.
First book in the market that treats this newly developed area of theoretical physics; the book will thus provide a fascinating overview of the prospective applications of this area, strongly founded on the theories and methods that it describes.Provides a solid foundation for the application of quantum theory to current physical problems arising in the interpretation of molecular spectra and important effects in quantum field theory.New insight into the physics of anharmonic vibrations, more feasible calculations with improved precision.

This book uses a hands-on approach to nonlinear dynamics using commonly available software, including the free dynamical systems software Xppaut, Matlab (or its free cousin, Octave) and the Maple symbolic algebra system. Detailed instructions for various common procedures, including bifurcation analysis using the version of AUTO embedded in Xppaut, are provided. This book also provides a survey that can be taught in a single academic term covering a greater variety of dynamical systems (discrete versus continuous time, finite versus infinite-dimensional, dissipative versus conservative) than is normally seen in introductory texts. Numerical computation and linear stability analysis are used as unifying themes throughout the book. Despite the emphasis on computer calculations, theory is not neglected, and fundamental concepts from the field of nonlinear dynamics such as solution maps and invariant manifolds are presented.

Ideal for college students in intermediate finance courses, this book uniquely applies mathematical formulas to teach the underpinnings of financial and lending decisions, covering common applications in real estate, capital budgeting, and commercial loans. An updated and expanded version of the time-honored classic text on financial math, this book provides, in one place, a complete and practical treatment of the four primary venues for finance: commercial lending, financial formulas, mortgage lending, and resource allocation or capital budgeting techniques. With an emphasis on understanding the principles involved rather than blind reliance on formulas, the book provides rigorous and thorough explanations of the mathematical calculations used in determining the time value of money, valuation of loans by commercial banks, valuation of mortgages, and the cost of capital and capital budgeting techniques for single as well as mutually exclusive projects. This new edition devotes an entire chapter to a method of evaluating mutually exclusive projects without resorting to any imposed conditions. Two chapters not found in the previous edition address special topics in finance, including a novel and innovative way to approach amortization tables and the time value of money for cash flows when they increase geometrically or arithmetically. This new edition also features helpful how-to sections on Excel applications at the end of each appropriate chapter. Lays the foundation of all the topics that are typically covered in a financial management textbook or class Demonstrates how the mastery of a few basic concepts-such as the time value of money under all possible situations-allows for a precise understanding of more complex topics in finance Describes how all advanced capital budgeting techniques can be reduced to the simplest technique-the payback period method Examines traditional financial techniques using simple interest rate and accounting rate of return methods to conclusively show how these practices are now defunct

Since the earliest days of human existence, the clash of thunder and trembling of the hills has struck fear into the hearts of seasoned warriors and tribal villagers alike. Great gods, demi-gods, and heroes were created to explain the awesome, mysterious, and incomprehensibly powerful forces of Nature in a feeble attempt to make sense of the world around them. To our advanced scientific minds today, these explanations seem childish and ridiculous; however, the power to flatten thousands of square miles of ancient forest, create massive holes in the Earth itself, and cause mountains to tremble to their very roots are more than enough reason to believe. Indeed, perhaps our scientific advancement has caused us to not fully or completely appreciate the awesome scale and power that Nature can wield against us. The study of shock wave formation and dynamics begins with a study of waves themselves. Simple harmonic motion is used to analyze the physical mechanisms of wave generation and propagation, and the principle of superposition is used to mathematically generate constructive and destructive interference. Further development leads to the shock singularity where a single wave of immense magnitude propagates and decays through various media. Correlations with the fields of thermodynamics, meteorology, crater formation, and acoustics are made, as well as a few special applications. Direct correlation is made to events in Arizona, Siberia, and others. The mathematical requirement for this text includes trigonometry, differential equations, and large series summations, which should be accessible to most beginning and advanced university students. This text should serve well as supplementary material in a course covering discrete wave dynamics, applied thermodynamics, or extreme acoustics.

Feynman path integrals are ubiquitous in quantum physics, even if a large part of the scientific community still considers them as a heuristic tool that lacks a sound mathematical definition. Our book aims to refute this prejudice, providing an extensive and self-contained description of the mathematical theory of Feynman path integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics.This second edition presents a detailed discussion of the general theory of complex integration on infinite dimensional spaces, providing on one hand a unified view of the various existing approaches to the mathematical construction of Feynman path integrals and on the other hand a connection with the classical theory of stochastic processes. Moreover, new chapters containing recent applications to several dynamical systems have been added.This book bridges between the realms of stochastic analysis and the theory of Feynman path integration. It is accessible to both mathematicians and physicists.

For various scientific and engineering problems, how to deal with variables and parameters of uncertain value is an important issue. Full analysis of the specific errors in measurement, observations, experiments, and applications are vital in dealing with the parameters taken to simplify the problem. Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems aims to provide the reader with basic concepts for soft computing and other methods for various means of uncertainty in handling solutions, analysis, and applications. This book is an essential reference work for students, scholars, practitioners and researchers in the assorted fields of engineering and applied mathematics interested in a model for uncertain physical problems.

Hulchul: The Common Ingredient of MotionMotionMotionMotion and Time Author, Sohan Jain, proposes the following in the book: Instants of Motion, Instants of Time and Time Outage: Just as time has instants of time, motion has instants of motion, too. Instants of time and motion can be divided into three classes: pure instants of time, pure instants of motion, and composite instants of time and motion. The sequences of the three types of instants are interspersed into a single sequence of their occurrences. A body does not experience time during pure instants of motion, a phenomenon we will call time outage -the cause of time dilation. Time outage is not continuous; it is intermittent. Internal and external motion of a body and their inheritance: Each body has, generally, two kinds of motions: internal motion and external motion. A body goes, wherever its outer bodies go. An inner body inherits external motion of its outer bodies. An outer body inherits internal motion of its inner bodies. Photons and light do not inherit motion; may be, this is why their motions are independent of their sources. Prime ticks, the building blocks of time and any motion: Motion of a common body is not continuous; it is intermittent. Any kind of motion is perceived to be made of discrete, indivisible tiny movements, called prime ticks (p-ticks). P-ticks are to motion what elementary particles are to matter or what photons are to light. There is time only because there is motion. Prime ticks are events and imply motion. Events have concurrency, which implies time. Total concurrency hulchul, a universal constant: Concurrency events of external and internal p-ticks of a body are precisely the instants of motion and time. The sum of the two is called the total concurrency hulchul (c-hulchul). Total c-hulchul is the same for all bodies. The proposed theory possibly explains: Why a particle accelerator works. Why atoms have compartmentalized internal structure. Why lighter bodies, such as elementary particles and photons, have wavy straight motion rather than straight motion. The theory predicts: The sharing of an electron by two atoms is not continuous; it alternates between the two atoms.

The interface between Physics and Mathematics has been increasingly spotlighted by the discovery of algebraic, geometric, and topological properties in physical phenomena. A profound example is the relation of noncommutative geometry, arising from algebras in mathematics, to the so-called quantum groups in the physical viewpoint. Two apparently unrelated puzzles - the solubility of some lattice models in statistical mechanics and the integrability of differential equations for special problems - are encoded in a common algebraic condition, the Yang-Baxter equation. This backdrop motivates the subject of this book, which reveals Knot Theory as a highly intuitive formalism that is intimately connected to Quantum Field Theory and serves as a basis to String Theory.This book presents a didactic approach to knots, braids, links, and polynomial invariants which are powerful and developing techniques that rise up to the challenges in String Theory, Quantum Field Theory, and Statistical Physics. It introduces readers to Knot Theory and its applications through formal and practical (computational) methods, with clarity, completeness, and minimal demand of requisite knowledge on the subject. As a result, advanced undergraduates in Physics, Mathematics, or Engineering, will find this book an excellent and self-contained guide to the algebraic, geometric, and topological tools for advanced studies in theoretical physics and mathematics.

The interface between Physics and Mathematics has been increasingly spotlighted by the discovery of algebraic, geometric, and topological properties in physical phenomena. A profound example is the relation of noncommutative geometry, arising from algebras in mathematics, to the so-called quantum groups in the physical viewpoint. Two apparently unrelated puzzles - the solubility of some lattice models in statistical mechanics and the integrability of differential equations for special problems - are encoded in a common algebraic condition, the Yang-Baxter equation. This backdrop motivates the subject of this book, which reveals Knot Theory as a highly intuitive formalism that is intimately connected to Quantum Field Theory and serves as a basis to String Theory.This book presents a didactic approach to knots, braids, links, and polynomial invariants which are powerful and developing techniques that rise up to the challenges in String Theory, Quantum Field Theory, and Statistical Physics. It introduces readers to Knot Theory and its applications through formal and practical (computational) methods, with clarity, completeness, and minimal demand of requisite knowledge on the subject. As a result, advanced undergraduates in Physics, Mathematics, or Engineering, will find this book an excellent and self-contained guide to the algebraic, geometric, and topological tools for advanced studies in theoretical physics and mathematics.

The book systematically introduces smart power system design and its infrastructure, platform and operating standards. It focuses on multi-objective optimization and illustrates where the intelligence of the system lies. With abundant project data, this book is a practical guideline for engineers and researchers in electrical engineering, as well as power network designers and managers in administration.

Susanna Epp's DISCRETE MATHEMATICS WITH APPLICATIONS, 4e, International Edition provides a clear introduction to discrete mathematics. Renowned for her lucid, accessible prose, Epp explains complex, abstract concepts with clarity and precision. This book presents not only the major themes of discrete mathematics, but also the reasoning that underlies mathematical thought. Students develop the ability to think abstractly as they study the ideas of logic and proof. While learning about such concepts as logic circuits and computer addition, algorithm analysis, recursive thinking, computability, automata, cryptography, and combinatorics, students discover that the ideas of discrete mathematics underlie and are essential to the science and technology of the computer age. Overall, Epp's emphasis on reasoning provides students with a strong foundation for computer science and upper-level mathematics courses.

It is not uncommon for the Principle of Complementarity to be invoked in either Science or Philosophy, viz. the ancient oriental philosophy of Yin and Yang whose symbolic representation is portrayed on the cover of the book. Or Niels Bohr's use of it as the basis for the so-called Copenhagen interpretation of Quantum Mechanics. This book arose as an outgrowth of the author's previous book entitled 'Knots, Braids and Moebius Strips,' published by World Scientific in 2015, wherein the Principle itself was discovered to be expressible as a simple 2x2 matrix that summarizes the algebraic essence of both the well-known Microbiology of DNA and the author's version of the elementary particles of physics. At that point, the possibility of an even wider utilization of that expression of Complementarity arose.The current book, features Complementarity, in which the matrix algebra is extended to characterize not only DNA itself but the well-known process of its replication, a most gratifying outcome. The book then goes on to explore Complementarity, with and without its matrix expression, as it occurs, not only in much of physics but in its extension to cosmology as well.

Magic squares are among the more popular mathematical recreations. Over the last 50 years, many generalizations of "magic" ideas have been applied to graphs. Recently there has been a resurgence of interest in "magic labelings" due to a number of results that have applications to the problem of decomposing graphs into trees.

Key features of this second edition include:

. a new chapter on magic labeling of directed graphs

. applications of theorems from graph theory and interesting counting arguments

. new research problems and exercises covering a range of difficulties

. a fully updated bibliography and index

This concise, self-contained exposition is unique in its focus on the theory of magic graphs/labelings. It may serve as a graduate or advanced undergraduate text for courses in mathematics or computer science, and as reference for the researcher."