The heart of the book is the development of a short-time asymptotic expansion for the heat kernel. This is explained in detail and explicit examples of some advanced calculations are given. In addition some advanced methods and extensions, including path integrals, jump diffusion and others are presented. The book consists of four parts: Analysis, Geometry, Perturbations and Applications. The first part shortly reviews of some background material and gives an introduction to PDEs. The second part is devoted to a short introduction to various aspects of differential geometry that will be needed later. The third part and heart of the book presents a systematic development of effective methods for various approximation schemes for parabolic differential equations. The last part is devoted to applications in financial mathematics, in particular, stochastic differential equations. Although this book is intended for advanced undergraduate or beginning graduate students in, it should also provide a useful reference for professional physicists, applied mathematicians as well as quantitative analysts with an interest in PDEs.

This book presents a major step forward in experimentally understanding the behavior of muon neutrinos and antineutrinos. Apart from providing the world's first measurement of these interactions in a mostly unexplored energy region, the data presented advances the neutrino community's preparedness to search for an asymmetry between matter and anti-matter that may very well provide the physical mechanism for the existence of our universe. The details of these measurements are preceded by brief summaries of the history of the neutrino, the phenomenon of neutrino oscillations, and a description of their interactions. Also provided are details of the experimental setup for the measurements and the muon antineutrino cross-section measurement which motivates the need for dedicated in situ background constraints. The world's first measurement of the neutrino component of an antineutrino beam using a non-magnetized detector, as well as other crucial background constraints, are also presented in the book. By exploiting correlated systematic uncertainties, combined measurements of the muon neutrino and antineutrino cross sections described in the book maximize the precision of the extracted information from both results.

In this thesis, the author explains the background of problems in quantum estimation, the necessary conditions required for estimation precision benchmarks that are applicable and meaningful for evaluating data in quantum information experiments, and provides examples of such benchmarks. The author develops mathematical methods in quantum estimation theory and analyzes the benchmarks in tests of Bell-type correlation and quantum tomography with those methods. Above all, a set of explicit formulae for evaluating the estimation precision in quantum tomography with finite data sets is derived, in contrast to the standard quantum estimation theory, which can deal only with infinite samples. This is the first result directly applicable to the evaluation of estimation errors in quantum tomography experiments, allowing experimentalists to guarantee estimation precision and verify quantitatively that their preparation is reliable.

Studying and using light or "photons" to image and then to control and transmit molecular information is among the most challenging and significant research fields to emerge in recent years. One of the fastest growing areas involves research in the temporal imaging of quantum phenomena, ranging from molecular dynamics in the femto (10-15s) time regime for atomic motion to the atto (10-18s) time scale of electron motion. In fact, the attosecond "revolution" is now recognized as one of the most important recent breakthroughs and innovations in the science of the 21st century. A major participant in the development of ultrafast femto and attosecond temporal imaging of molecular quantum phenomena has been theory and numerical simulation of the nonlinear, non-perturbative response of atoms and molecules to ultrashort laser pulses. Therefore, imaging quantum dynamics is a new frontier of science requiring advanced mathematical approaches for analyzing and solving spatial and temporal multidimensional partial differential equations such as Time-Dependent Schroedinger Equations (TDSE) andTime-Dependent Dirac equations (TDDEs for relativistic phenomena). These equations are also coupled to the photons in Maxwell's equations for collective propagation effects. Inversion of the experimental imaging data of quantum dynamics presents new mathematical challenges in the imaging of quantum wave coherences on subatomic (subnanometer) spatial dimensions and multiple timescales from atto to femto and even nanoseconds.In "Quantum Dynamic Imaging: Theoretical and Numerical Methods," leading researchers discuss these exciting state-of-the-art developments and theirimplications for R&D in view of the promise of quantum dynamic imagingscience as the essential tool for controlling matter at the molecular level."

Magnetic and spintronic materials are ubiquitous in modern technological applications, e.g. in electric motors, power generators, sensors and actuators, not to mention information storage and processing. Medical technology has also greatly benefited from magnetic materials - especially magnetic nanoparticles - for therapy and diagnostics methods. All of the above-mentioned applications rely on the properties of the materials used. These properties in turn depend on intrinsic and extrinsic material parameters. The former are related to the actual elements used and their properties, e.g. atomic magnetic moment and exchange interaction between atoms; the latter are related to the structural and microstructural properties of the materials used, e.g. their crystal structure, grain size, and grain boundary phases. Focusing on state-of-the-art magnetic and spintronic materials, this book will introduce readers to a range of related topics in Physics and Materials Science. Phenomena and processes at the nanoscale are of particular importance in this context; accordingly, much of the book addresses such topics.

Many technological applications exploit a variety of magnetic structures, or magnetic phases, to produce and optimise solid-state functionality. However, most research advances are restricted to a reduced number of phases owing to computational and resource constraints. This thesis presents an ab-initio theory to efficiently describe complex magnetic phases and their temperature-dependent properties. The central assumption is that magnetic phases evolve slowly compared with the underlying electronic structure from which they emerge. By describing how the electronic structure adapts to the type and extent of magnetic order, a theory able to describe multi-spin correlations and their effect on the magnetism at finite temperature is obtained. It is shown that multi-spin correlations are behind the temperature and magnetic field dependence of the diverse magnetism in the heavy rare earth elements. Magnetically frustrated Mn-based materials and the effect of strain are also investigated. These studies demonstrate that the performance of solid-state refrigeration can be enhanced by multi-spin effects.

This book uses less strict yet still formal mathematical language to clarify a variety of concepts in Quantum Field Theory that remain somewhat "fuzzy" in many books designed for undergraduates and fresh graduates. The aim is not to replace formal books on Quantum Field Theory, but rather to offer a helpful complementary tool for beginners in the field. Features include a reader-friendly introduction to tensor calculus and the concept of manifolds; a simple and robust treatment for dimensional regularization; a consistent explanation of the renormalization procedure, step by step and in a transparent manner at all orders, using the QED Lagrangian; and extensive treatment of infrared as well as ultraviolet divergences. The most general (Lorentz invariant) form of Noether's theorem is presented and applied to a few simple yet relevant examples in Quantum Field Theory. These and further interesting topics are addressed in a way that will be accessible for the target readership. Some familiarity with basic notions of Quantum Field Theory and the basics of Special Relativity is assumed.

This thesis offers a comprehensive introduction to surface acoustic waves in the quantum regime. It addresses two of the most significant technological challenges in developing a scalable quantum information processor based on spins in quantum dots: (i) decoherence of the electronic spin qubit due to the surrounding nuclear spin bath, and (ii) long-range spin-spin coupling between remote qubits. Electron spins confined in quantum dots (QDs) are among the leading contenders for implementing quantum information processing. To this end, the author pursues novel strategies that turn the unavoidable coupling to the solid-state environment (in particular, nuclear spins and phonons) into a valuable asset rather than a liability.

At a level accessible to advanced undergraduates, this textbook explains the fundamental role of quantum mechanics in determining the structure, dynamics, and other properties of molecules. Readers will come to understand the quantum-mechanical basis for harmonic oscillators, angular momenta and scattering processes. Exercises are provided to help readers deepen their grasp of the essential phenomena.

This is the seventh volume in a series on the general topics of supersymmetry, supergravity, black objects (including black holes) and the attractor mechanism. The present volume is based on lectures held in March 2013 at the INFN-Laboratori Nazionali di Frascati during the Breaking of supersymmetry and Ultraviolet Divergences in extended Supergravity Workshop (BUDS 2013), organized by Stefano Bellucci, with the participation of prestigious speakers including P. Aschieri, E. Bergshoeff, M. Cederwall, T. Dennen, P. Di Vecchia, S. Ferrara, R. Kallosh, A. Karlsson, M. Koehn, B. Ovrut, A. Van Proeyen, G. Ruppeiner. Special attention is devoted to discussing topics related to the cancellation of ultraviolet divergences in extended supergravity and Born-Infeld-like actions. All talks were followed by extensive discussions and subsequent reworking of the various contributions a feature which is reflected in the unique "flavor" of this volume.

The book provides readers with an understanding of the mutual conditioning of spacetime and interactions and matter. The spacetime manifold will be looked at to be a reservoir for the parametrization of operation Lie groups or subgroup classes of Lie groups. With basic operation groups or Lie algebras, all physical structures can be interpreted in terms of corresponding realizations or representations. Physical properties are related eigenvalues or invariants. As an explicit example of operational spacetime is proposed, called electroweak spacetime, parametrizing the classes of the internal hypercharge - isospin group in the general linear group in two complex dimensions, i.e., the Lorentz cover group, extended by the casual (dilation) and phase group. Its representations and invariants will be investigated with the aim to connect them, qualitatively and numerically, with the properties of interactions and particles as arising in the representations of its tangent Minkowski spaces.

With this thesis the author contributes to the development of a non-mainstream but long-standing approach to electroweak symmetry breaking based on an analogy with superconductivity. Electroweak symmetry breaking is assumed to be caused by dynamically generated masses of typical fermions, i.e., of quarks and leptons, which in turn assumes a new dynamics between quarks and leptons. Primarily it is designed to generate fermion masses and electroweak symmetry breaking is an automatic consequence.

After the summary of the topic, the first main part of the thesis addresses the question as to whether the masses of known quarks and leptons provide sufficiently strong sources of electroweak symmetry breaking. It is demonstrated that neutrino masses subject tothe seesaw mechanism are indispensable ingredients. The other two parts of the thesis are dedicated to the presentation of two particular models: The first model is based on the new strong Yukawa dynamics and serves as a platform for studying the ability to reproduce fermion masses. The second, more realistic model introduces a flavor gauge dynamics and its phenomenological consequences are studied.

Even though, in the past, this type of models has already been of some interest, following the discovery of the Standard-Model-like Higgs particle, it is regaining its relevance."

For many years, scientists have attempted to unite the four fundamental forces-the strong and weak nuclear forces, gravity, and electromagnetism. Many have tried uniting known theories, such as general relativity, with quantum mechanics, string theory, and even the standard model. These theories differ, and it seems difficult to find a link to connect them.

In The Theory of Everything, Solved author and researcher Lawrence J. Wippler explains a new theory and provides an alternate understanding of the workings of the atom. He found that the four fundamental forces of nature can be united by just three particles-the north and south magnetic monopoles and a particle of matter that represents an element. He describes how these particles interact with each other and how they are able to create all forms of energy, including magnetism and gravity.

Setting aside the presently known theories and laws of physics and attacking the problem from a different perspective, Wippler kept his assumptions simple when developing the three-particle theory. In The Theory of Everything, Solved Wippler shows that the north and south monopoles and a particle of matter are the building blocks of the universe.

This thesis sheds new light on the fascinating properties of composite quantum systems. Quantum systems of different sizes, ranging from small bipartite systems to large many-body ensembles, can be studied with the help of modern quantum optical experiments. These experiments make it possible to observe a broad variety of striking features, including nonclassical correlations, complex dynamics and quantum phase transitions. By adopting the complementary perspectives of quantum information theory, quantum chemistry and many-body theory, the thesis develops new methods for the efficient characterization and description of interacting, composite quantum systems.

Focusing on spectroscopic properties of molecular systems, Quantum Modeling of Molecular Materials presents the state-of-the-art methods in theoretical chemistry that are used to determine molecular properties relevant to different spectroscopies. This timely reference gives a basic presentation of response theory and its application to the simulation of spectroscopic properties of molecular materials. This in-depth presentation of time-dependent response theory and its applications in spectroscopy provides an important advance towards a modern vision of theoretical tools for researchers in academia and industry and postgraduate students.

A ground-breaking and practical treatment of probability and stochastic processes

"A Modern Theory of Random Variation" is a new and radical re-formulation of the mathematical underpinnings of subjects as diverse as investment, communication engineering, and quantum mechanics. Setting aside the classical theory of probability measure spaces, the book utilizes a mathematically rigorous version of the theory of random variation that bases itself exclusively on finitely additive probability distribution functions.

In place of twentieth century Lebesgue integration and measure theory, the author uses the simpler concept of Riemann sums, and the non-absolute Riemann-type integration of Henstock. Readers are supplied with an accessible approach to standard elements of probability theory such as the central limmit theorem and Brownian motion as well as remarkable, new results on Feynman diagrams and stochastic integrals.

Throughout the book, detailed numerical demonstrations accompany the discussions of abstract mathematical theory, from the simplest elements of the subject to the most complex. In addition, an array of numerical examples and vivid illustrations showcase how the presented methods and applications can be undertaken at various levels of complexity.

"A Modern Theory of Random Variation" is a suitable book for courses on mathematical analysis, probability theory, and mathematical finance at the upper-undergraduate and graduate levels. The book is also an indispensible resource for researchers and practitioners who are seeking new concepts, techniques and methodologies in data analysis, numerical calculation, and financial asset valuation.

Patrick Muldowney, PhD, served as lecturer at the Magee Business School of the UNiversity of Ulster for over twenty years. Dr. Muldowney has published extensively in his areas of research, including integration theory, financial mathematics, and random variation.

In his rich and varied career as a mathematician, computer scientist, and educator, Jacob T. Schwartz wrote seminal works in analysis, mathematical economics, programming languages, algorithmics, and computational geometry. In this volume of essays, his friends, students, and collaborators at the Courant Institute of Mathematical Sciences present recent results in some of the fields that Schwartz explored: quantum theory, the theory and practice of programming, program correctness and decision procedures, dextrous manipulation in Robotics, motion planning, and genomics. In addition to presenting recent results in these fields, these essays illuminate the astonishingly productive trajectory of a brilliant and original scientist and thinker.

Device-independent quantum cryptography is a method for exchanging secret messages over potentially insecure quantum communication channels, such as optical fibers. In contrast to conventional quantum cryptography, security is guaranteed even if the devices used by the communication partners, such as photon sources and detectors, deviate from their theoretical specifications. This is of high practical relevance, for attacks to current implementations of quantum cryptography exploit exactly such deviations. Device-independent cryptography is however technologically so demanding that it looked as if experimental realizations are out of reach. In her thesis, Rotem Arnon-Friedman presents powerful information-theoretic methods to prove the security of device-independent quantum cryptography. Based on them, she is able to establish security in a parameter regime that may be experimentally achievable in the near future. Rotem Arnon-Friedman's thesis thus provides the theoretical foundations for an experimental demonstration of device-independent quantum cryptography.

The production of heavy quarks in high-energy experiments offers a rich field to study, both experimentally and theoretically. Due to the additional quark mass, the description of these processes in the framework of perturbative QCD is much more demanding than it is for those involving only massless partons. In the last two decades, a large amount of precision data has been collected by the deep inelastic HERA experiment. In order to make full use of these data, a more precise theoretical description of charm quark production in deep inelastic scattering is needed. This work deals with the first calculation of fixed moments of the NNLO heavy flavor corrections to the proton structure function F2 in the limit of a small charm-quark mass. The correct treatment of these terms will allow not only a more precise analysis of the HERA data, but starting from there also a more precise determination of the parton distribution functions and the strong coupling constant, which is an essential input for LHC physics. The complexity of this calculation requires the application and development of technical and mathematical methods, which are also explained here in detail.

Quantum mechanics is one of the great success stories of modern physics, making sense of the very small just as Einstein's theory of relativity made sense of the very large. But, for most students, the ideas that make quantum mechanics powerful can be confusing and counterintuitive. This volume in the Greenwood Guides to Great Ideas in Science series provides a history of quantum mechanics from the early breakthroughs of Planck and Einstein, at the beginning of the 20th century, to the present frontiers of quantum computing and quantum gravity. The approach is entirely non-technical, and is aimed at the general reader who may not have much mathematical background but who has a strong curiosity about some of the most important developments in modern science. Quantum Mechanics: A Historical Perspective traces the history of this powerful theory, including: BLThe early discoveries by Max Planck and Albert Einstein regarding the quantization of radiation BLThe "early quantum theory," including Neils Bohr's theory of the atom BLThe birth of modern quantum mechanics through the work of Heisenberg, Schrodinger, Born, Dirac and others BLApplications of quantum mechanics in chemistry, nuclear physics, electronics, and many other areas BLRecent work in quantum computation and quantum information theory The book emphasizes the fact that despite the great success of quantum mechanics, many exciting intellectual frontiers remain open for further researchers to explore. It includes a glossary, a timeline, and a bibliography of accessible resources for further research.

This thesis presents a study of the scalar sector in the standard model (SM), as well as various searches for an extended scalar sector in theories beyond the SM (BSM). The first part of the thesis details the search for an SM Higgs boson decaying to taus, and produced by gluon fusion, vector boson fusion, or associated production with a vector boson, leading to evidence for decays of the Higgs boson to taus. In turn, the second part highlights several searches for an extended scalar sector, with scalar boson decays to taus. In all of the analyses presented, at least one scalar boson decays to a pair of taus. The results draw on data collected by the Compact Muon Solenoid (CMS) detector during proton-proton collisions with a center-of-mass energy of 7 or 8 TeV.

This book addresses electron spin-qubit based quantum computing and quantum information processing with a strong focus on the background and applications to EPR/ESR technique and spectroscopy. It explores a broad spectrum of topics including quantum computing, information processing, quantum effects in electron-nuclear coupled molecular spin systems, adiabatic quantum computing, heat bath algorithmic cooling with spins, and gateway schemes of quantum control for spin networks to NMR quantum information. The organization of the book places emphasis on relevant molecular qubit spectroscopy. These revolutionary concepts have never before been included in a comprehensive volume that covers theory, physical basis, technological basis, applications, and new advances in this emerging field. Electron Spin Resonance (ESR) Based Quantum Computing, co-edited by leading and renowned researchers Takeji Takui, Graeme Hanson and Lawrence J Berliner, is an ideal resource for students and researchers in the fields of EPR/ESR, NMR and quantum computing. This book also * Explores methods of harnessing quantum effects in electron-nuclear coupled molecular spin systems * Expertly discusses applications of optimal control theory in quantum computing * Broadens the readers' understanding of NMR quantum information processing

This thesis exploits the ability of the linear-scaling quantum mechanical code ONETEP to analyze systems containing many thousands of atoms. By implementing an electron transport capability to the code, it also investigates a range of phenomena associated with electrical conduction by nanotubes and, in particular, the process of transport electrons between tubes. Extensive work has been done on the conductivity of single carbon nanotubes. However, any realistic wire made of nanotubes will consist of a large number of tubes of finite length. The conductance of the resulting wire is expected to be limited by the process of transferring electrons from one tube to another.These quantum mechanical calculations on very large systems have revealed a number of incorrect claims made previously in the literature. Conduction processes that have never before been studied at this level of theory are also investigated.

The fundamental concept of quantum coherence plays a central role in quantum physics, cutting across disciplines of quantum optics, atomic and condensed matter physics. Quantum coherence represents a universal property of the quantum s- tems that applies both to light and matter thereby tying together materials and p- nomena. Moreover, the optical coherence can be transferred to the medium through the light-matter interactions. Since the early days of quantum mechanics there has been a desire to control dynamics of quantum systems. The generation and c- trol of quantum coherence in matter by optical means, in particular, represents a viable way to achieve this longstanding goal and semiconductor nanostructures are the most promising candidates for controllable quantum systems. Optical generation and control of coherent light-matter states in semiconductor quantum nanostructures is precisely the scope of the present book. Recently, there has been a great deal of interest in the subject of quantum coh- ence. We are currently witnessing parallel growth of activities in different physical systems that are all built around the central concept of manipulation of quantum coherence. The burgeoning activities in solid-state systems, and semiconductors in particular, have been strongly driven by the unprecedented control of coherence that previously has been demonstrated in quantum optics of atoms and molecules, and is now taking advantage of the remarkable advances in semiconductor fabrication technologies. A recent impetus to exploit the coherent quantum phenomena comes from the emergence of the quantum information paradigm.

This book is aimed at those readers who already have some knowledge of mathematical methods and have also been introduced to the basic ideas of quantum optics. It should be attractive to students who have already explored one of the more introductory texts such as Loudon's The quantum theory of light (2/e, 1983, OUP) and are seeking to acquire the mathematical skills used in real problems. This book is not primarily about the physics of quantum optics but rather presents the mathematical methods widely used by workers in this field. There is no comparable book which covers either the range or the depth of mathematical techniques.