Quartic anharmonic oscillator with potential V(x)= x(2) + g(2)x4 was the first non-exactly-solvable problem tackled by the newly-written Schroedinger equation in 1926. Since that time thousands of articles have been published on the subject, mostly about the domain of small g(2) (weak coupling regime), although physics corresponds to g(2) ~ 1, and they were mostly about energies.This book is focused on studying eigenfunctions as a primary object for any g(2). Perturbation theory in g(2) for the logarithm of the wavefunction is matched to the true semiclassical expansion in powers of : it leads to locally-highly-accurate, uniform approximation valid for any g(2) [0, ) for eigenfunctions and even more accurate results for eigenvalues. This method of matching can be easily extended to the general anharmonic oscillator as well as to the radial oscillators. Quartic, sextic and cubic (for radial case) oscillators are considered in detail as well as quartic double-well potential.

This book and its prequel (Theories of Matter, Space, and Time: Classical Theories) grew out of courses that are taught by the authors on the undergraduate degree program in physics at Southampton University, UK. The authors aim to guide the full MPhys undergraduate cohort through some of the trickier areas of theoretical physics that undergraduates are expected to master. To move beyond the initial courses in classical mechanics, special relativity, electromagnetism and quantum theory to more sophisticated views of these subjects and their interdependence. This approach keeps the analysis as concise and physical as possible whilst revealing the key elegance in each subject discussed.This second book of the pair looks at ideas to the arena of Quantum Mechanics. First quickly reviewing the basics of quantum mechanics which should be familiar to the reader from a first course, it then links the Schrodinger equation to the Principle of Least Action introducing Feynman's path integral methods. Next, it presents the relativistic wave equations of Klein, Gordon and Dirac. Finally, Maxwell's equations of electromagnetism are converted to a wave equation for photons and make contact with Quantum Electrodynamics (QED) at a first quantized level. Between the two volumes the authors hope to move a student's understanding from their first courses to a place where they are ready to embark on graduate level courses on quantum field theory.

This book provides a concise introduction to both the special theory of relativity and the general theory of relativity. The format is chosen to provide the basis for a single semester course which can take the students all the way from the foundations of special relativity to the core results of general relativity: the Einstein equation and the equations of motion for particles and light in curved spacetime. To facilitate access to the topics of special and general relativity for science and engineering students without prior training in relativity or geometry, the relevant geometric notions are also introduced and developed from the ground up. Students in physics, mathematics or engineering with an interest to learn Einstein's theories of relativity should be able to use this book already in the second semester of their third year. The book could also be used as the basis of a graduate level introduction to relativity for students who did not learn relativity as part of their undergraduate training.

Recent discoveries in astronomy and relativistic astrophysics as well as experiments on particle and nuclear physics have blurred the traditional boundaries of physics. It is believed that at the birth of the Universe, a whirlwind of matter and antimatter, of quarks and exotic leptons, briefly appeared and merged into a sea of energy. The new phenomena and new states of matter in the Universe revealed the deep connection between quarks and the Cosmos. Motivated by these themes, this book discusses different topics: gravitational waves, dark matter, dark energy, exotic contents of compact stars, high-energy and gamma-ray astrophysics, heavy ion collisions and the formation of the quark-gluon plasma in the early Universe. The book presents some of the latest researches on these fascinating themes and is useful for experts and students in the field.

The clinical use of Artificial Intelligence (AI) in radiation oncology is in its infancy. However, it is certain that AI is capable of making radiation oncology more precise and personalized with improved outcomes. Radiation oncology deploys an array of state-of-the-art technologies for imaging, treatment, planning, simulation, targeting, and quality assurance while managing the massive amount of data involving therapists, dosimetrists, physicists, nurses, technologists, and managers. AI consists of many powerful tools which can process a huge amount of inter-related data to improve accuracy, productivity, and automation in complex operations such as radiation oncology.This book offers an array of AI scientific concepts, and AI technology tools with selected examples of current applications to serve as a one-stop AI resource for the radiation oncology community. The clinical adoption, beyond research, will require ethical considerations and a framework for an overall assessment of AI as a set of powerful tools.30 renowned experts contributed to sixteen chapters organized into six sections: Define the Future, Strategy, AI Tools, AI Applications, and Assessment and Outcomes. The future is defined from a clinical and a technical perspective and the strategy discusses lessons learned from radiology experience in AI and the role of open access data to enhance the performance of AI tools. The AI tools include radiomics, segmentation, knowledge representation, and natural language processing. The AI applications discuss knowledge-based treatment planning and automation, AI-based treatment planning, prediction of radiotherapy toxicity, radiomics in cancer prognostication and treatment response, and the use of AI for mitigation of error propagation. The sixth section elucidates two critical issues in the clinical adoption: ethical issues and the evaluation of AI as a transformative technology.

Edited by internationally recognized authorities in the field, this expanded and updated new edition of the bestselling Handbook, containing many new articles, is aimed at the design and operation of modern particle accelerators. It is intended as a vade mecum for professional engineers and physicists engaged in these subjects. With a collection of more than 2000 equations, 300 illustrations and 500 graphs and tables, here one will find, in addition to common formulae of previous compilations, hard to find, specialized formulae, recipes and material data pooled from the lifetime experience of many of the world's most able practioners of the art and science of accelerators.The seven chapters include both theoretical and practical matters as well as an extensive glossary of accelerator types. Chapters on beam dynamics and electromagnetic and nuclear interactions deal with linear and nonlinear single particle and collective effects including spin motion, beam-environment, beam-beam, beam-electron, beam-ion and intrabeam interactions. The impedance concept and related calculations are dealt with at length as are the instabilities with the various interactions mentioned. A chapter on operational considerations including discussions on the assessment and correction of orbit and optics errors, realtime feedbacks, generation of short photon pulses, bunch compression, tuning of normal and superconducting linacs, energy recovery linacs, free electron lasers, cryogenic vacuum systems, steady state microbuching , cooling, space-charge compensation, brightness of light sources, collider luminosity optimization and collision schemes, machine learning, multiple frequency rf systems, fel seeding. Chapters on mechanical and electrical considerations present material data and important aspects of component design including heat transfer and refrigeration. Hardware systems for particle sources, feedback systems, confinement, including undulators, and acceleration (both normal and superconducting) receive detailed treatment in a sub-systems chapter, beam measurement and apparatus being treated therein as well.A detailed name and subject index is provided together with reliable references to the literature where the most detailed information available on all subjects treated can be found.

Discontinuous (first-order) phase transitions constitute the most fundamental and widespread type of structural transitions existing in Nature, forming a large majority of the transitions found in elemental crystals, alloys, inorganic compounds, minerals and complex fluids. Nevertheless, only a small part of them, namely, weakly discontinuous transformations, were considered by phenomenological theories, leaving aside the most interesting from a theoretical point of view and the most important for application cases. Discontinuous Phase Transitions in Condensed Matter introduces a density-wave approach to phase transitions which results in a unified, symmetry-based, model-free theory of the weak crystallization of molecular mixtures to liquid-crystalline mesophases, strongly discontinuous crystallization from molten metals and alloys to conventional, fully segregated crystals, to aperiodic, quasi-crystalline structures. Assembly of aperiodic closed virus capsids with non-crystallographic symmetry also falls into the domain of applicability of the density-wave approach.The book also considers the applicability domains of the symmetry-based approach in physics of low-dimensional systems. It includes comparisons of stability of different surface superstructures and metal monoatomic coverage structures on the surface of single-crystalline substrates. The example of the twisted graphene bilayer demonstrates how parametrization in the spirit of an advanced phenomenological approach can establish symmetry-controlled, and therefore model-free, links between geometrical parameters of the twisted bilayer structure and reconstruction of its Brillouin zone and energy bands.

This book is a short introduction to classical field theory, most suitable for undergraduate students who have had at least intermediate-level courses in electromagnetism and classical mechanics. The main theme of the book is showcasing role of fields in mediating action-at-a-distance interactions. Suitable technical machinery is developed to explore at least some aspect of each of the four known fundamental forces in nature. Beginning with the physically-motivated introduction to field theory, the text covers the relativistic formulation of electromagnetism in great detail so that aspects of gravity and the nuclear interaction not usually encountered at the undergraduate level can be covered by using analogies with familiar electromagentism. Special topics such as the behavior of gravity in extra, compactified dimensions, magnetic monopoles and electromagnetic duality, and the Higgs mechanism are also briefly considered.

This book provides a set of theoretical and numerical tools useful for the study of wave propagation in metamaterials and photonic crystals. While concentrating on electromagnetic waves, most of the material can be used for acoustic (or quantum) waves. For each presented numerical method, numerical code written in MATLAB (R) is presented. The codes are limited to 2D problems and can be easily translated in Python or Scilab, and used directly with Octave as well.

Nonlinear Time Series Analysis with R provides a practical guide to emerging empirical techniques allowing practitioners to diagnose whether highly fluctuating and random appearing data are most likely driven by random or deterministic dynamic forces. It joins the chorus of voices recommending 'getting to know your data' as an essential preliminary evidentiary step in modelling. Time series are often highly fluctuating with a random appearance. Observed volatility is commonly attributed to exogenous random shocks to stable real-world systems. However, breakthroughs in nonlinear dynamics raise another possibility: highly complex dynamics can emerge endogenously from astoundingly parsimonious deterministic nonlinear models. Nonlinear Time Series Analysis (NLTS) is a collection of empirical tools designed to aid practitioners detect whether stochastic or deterministic dynamics most likely drive observed complexity. Practitioners become 'data detectives' accumulating hard empirical evidence supporting their modelling approach. This book is targeted to professionals and graduate students in engineering and the biophysical and social sciences. Its major objectives are to help non-mathematicians - with limited knowledge of nonlinear dynamics - to become operational in NLTS; and in this way to pave the way for NLTS to be adopted in the conventional empirical toolbox and core coursework of the targeted disciplines. Consistent with modern trends in university instruction, the book makes readers active learners with hands-on computer experiments in R code directing them through NLTS methods and helping them understand the underlying logic (please see www.marco.bittelli.com). The computer code is explained in detail so that readers can adjust it for use in their own work. The book also provides readers with an explicit framework - condensed from sound empirical practices recommended in the literature - that details a step-by-step procedure for applying NLTS in real-world data diagnostics.

The burning of fossil fuels and emission of greenhouse gasses critically impacts the global environment. By utilizing better techniques and process, businesses can aid in the journey to an economic, sustainable, and environmentally-friendly future for generations to come. Business Models for Renewable Energy Initiatives: Emerging Research and Opportunities is an essential reference source for the latest scholarly perspectives on present and future business models in the renewable energy sector. Featuring coverage on a range of perspectives and topics such as techno-economics, decentralized power systems, and risk assessment, this book is designed for academicians, students, and researchers seeking current scholarly research on green business opportunities for renewable energy.

Extremum Seeking through Delays and PDEs, the first book on the topic, expands the scope of applicability of the extremum seeking method, from static and finite-dimensional systems to infinite-dimensional systems. Readers will find: Numerous algorithms for model-free real-time optimization are developed and their convergence guaranteed. Extensions from single-player optimization to noncooperative games, under delays and pdes, are provided. The delays and pdes are compensated in the control designs using the pde backstepping approach, and stability is ensured using infinite-dimensional versions of averaging theory. Accessible and powerful tools for analysis. This book is intended for control engineers in all disciplines (electrical, mechanical, aerospace, chemical), mathematicians, physicists, biologists, and economists. It is appropriate for graduate students, researchers, and industrial users.

This book contains an extensive illustration of use of finite difference method in solving the boundary value problem numerically. A wide class of differential equations has been numerically solved in this book. Starting with differential equations of elementary functions like hyperbolic, sine and cosine, we have solved those of special functions like Hermite, Laguerre and Legendre. Those of Airy function, of stationary localised wavepacket, of the quantum mechanical problem of a particle in a 1D box, and the polar equation of motion under gravitational interaction have also been solved. Mathematica 6.0 has been used to solve the system of linear equations that we encountered and to plot the numerical data. Comparison with known analytic solutions showed nearly perfect agreement in every case. On reading this book, readers will become adept in using the method.

This book is written in a lucid and systematic way for advanced postgraduates and researchers studying applied mathematics, plasma physics, nonlinear differential equations, nonlinear optics, and other engineering branches where nonlinear wave phenomena is essential.In sequential order of the book's development, readers will understand basic plasmas with elementary definitions of magnetized and unmagnetized plasmas, plasma modeling, dusty plasma and quantum plasma. Following which, the book describes linear and nonlinear waves, solitons, shocks and other wave phenomena, while solutions to common nonlinear wave equations are derived via standard techniques. Readers are introduced to elementary perturbation and non-perturbation methods. They will discover several evolution equations in different plasma situations as well as the properties of solitons in those environments. Pertaining to those equations, readers will learn about their higher order corrections, as well as their different forms and solutions in non-planar geometry. The book offers further studies on different types of collisions between solitons in plasma environment, phenomena of soliton turbulence as a consequence of multi-soliton interactions, properties of large amplitude solitary waves which are discovered via non-perturbative Sagdeev's Pseudopotential Approach, as well as the speed and shape of solitons. Finally, the book reveals possible future developments of research in this rich field.