This monograph is a unified presentation of several theories of finding explicit formulas for heat kernels for both elliptic and sub-elliptic operators. These kernels are important in the theory of parabolic operators because they describe the distribution of heat on a given manifold as well as evolution phenomena and diffusion processes.
Heat Kernels for Elliptic and Sub-elliptic Operators is an ideal reference for graduate students, researchers in pure and applied mathematics, and theoretical physicists interested in understanding different ways of approaching evolution operators.

This book provides the mathematical foundations of the theory of hyperhamiltonian dynamics, together with a discussion of physical applications. In addition, some open problems are discussed. Hyperhamiltonian mechanics represents a generalization of Hamiltonian mechanics, in which the role of the symplectic structure is taken by a hyperkahler one (thus there are three Kahler/symplectic forms satisfying quaternionic relations). This has proved to be of use in the description of physical systems with spin, including those which do not admit a Hamiltonian formulation. The book is the first monograph on the subject, which has previously been treated only in research papers.

This book is about synergy in computational intelligence (CI). It is a c- lection of chapters that covers a rich and diverse variety of computer-based techniques, all involving some aspect of computational intelligence, but each one taking a somewhat pragmatic view. Many complex problems in the real world require the application of some form of what we loosely call "intel- gence"fortheirsolution. Fewcanbesolvedbythenaiveapplicationofasingle technique, however good it is. Authors in this collection recognize the li- tations of individual paradigms, and propose some practical and novel ways in which di?erent CI techniques can be combined with each other, or with more traditional computational techniques, to produce powerful probl- solving environments which exhibit synergy, i. e., systems in which the whole 1 is greater than the sum of the parts . Computational intelligence is a relatively new term, and there is some d- agreement as to its precise de?nition. Some practitioners limit its scope to schemes involving evolutionary algorithms, neural networks, fuzzy logic, or hybrids of these. For others, the de?nition is a little more ?exible, and will include paradigms such as Bayesian belief networks, multi-agent systems, case-based reasoning and so on. Generally, the term has a similar meaning to the well-known phrase "Arti?cial Intelligence" (AI), although CI is p- ceived moreas a "bottom up" approachfrom which intelligent behaviour can emerge, whereasAItendstobestudiedfromthe"topdown,"andderivefrom pondering upon the "meaning of intelligence." (These and other key issues will be discussed in more detail in Chapter 1.

The book comprises contributions by some of the most respected scientists in the field of mathematical modeling and numerical simulation of the human cardiocirculatory system. It covers a wide range of topics, from the assimilation of clinical data to the development of mathematical and computational models, including with parameters, as well as their efficient numerical solution, and both in-vivo and in-vitro validation. It also considers applications of relevant clinical interest. This book is intended for graduate students and researchers in the field of bioengineering, applied mathematics, computer, computational and data science, and medicine wishing to become involved in the highly fascinating task of modeling the cardiovascular system.

This book provides a generalised approach to fractal dimension theory from the standpoint of asymmetric topology by employing the concept of a fractal structure. The fractal dimension is the main invariant of a fractal set, and provides useful information regarding the irregularities it presents when examined at a suitable level of detail. New theoretical models for calculating the fractal dimension of any subset with respect to a fractal structure are posed to generalise both the Hausdorff and box-counting dimensions. Some specific results for self-similar sets are also proved. Unlike classical fractal dimensions, these new models can be used with empirical applications of fractal dimension including non-Euclidean contexts. In addition, the book applies these fractal dimensions to explore long-memory in financial markets. In particular, novel results linking both fractal dimension and the Hurst exponent are provided. As such, the book provides a number of algorithms for properly calculating the self-similarity exponent of a wide range of processes, including (fractional) Brownian motion and Levy stable processes. The algorithms also make it possible to analyse long-memory in real stocks and international indexes. This book is addressed to those researchers interested in fractal geometry, self-similarity patterns, and computational applications involving fractal dimension and Hurst exponent.

An original motivation for algebraic geometry was to understand curves and surfaces in three dimensions. Recent theoretical and technological advances in areas such as robotics, computer vision, computer-aided geometric design and molecular biology, together with the increased availability of computational resources, have brought these original questions once more into the forefront of research. One particular challenge is to combine applicable methods from algebraic geometry with proven techniques from piecewise-linear computational geometry (such as Voronoi diagrams and hyperplane arrangements) to develop tools for treating curved objects. These research efforts may be summarized under the term nonlinear computational geometry. This volume grew out of an IMA workshop on Nonlinear Computational Geometry in May/June 2007 (organized by I.Z. Emiris, R. Goldman, F. Sottile, T. Theobald) which gathered leading experts in this emerging field. The research and expository articles in the volume are intended to provide an overview of nonlinear computational geometry. Since the topic involves computational geometry, algebraic geometry, and geometric modeling, the volume has contributions from all of these areas. By addressing a broad range of issues from purely theoretical and algorithmic problems, to implementation and practical applications this volume conveys the spirit of the IMA workshop.

The Pacific Symposium on Biocomputing (PSB) 2011 is an international, multidisciplinary conference for the presentation and discussion of current research in the theory and application of computational methods in problems of biological significance. Presentations are rigorously peer reviewed and are published in an archival proceedings volume. PSB 2011 will be held on January 3 - 7, 2011 in Kohala Coast, Hawaii. Tutorials and workshops will be offered prior to the start of the conference.PSB 2011 will bring together top researchers from the US, Asia Pacific, and around the world to exchange research results and address pertinent issues in all aspects of computational biology. It is a forum for the presentation of work in databases, algorithms, interfaces, visualization, modeling, and other computational methods, as applied to biological problems, with emphasis on applications in data-rich areas of molecular biology.The PSB has been designed to be responsive to the need for critical mass in sub-disciplines within biocomputing. For that reason, it is the only meeting whose sessions are defined dynamically each year in response to specific proposals. PSB sessions are organized by leaders of research in biocomputing's "hot topics". In this way, the meeting provides an early forum for serious examination of emerging methods and approaches in this rapidly evolving field.

During the last four decades there has been a remarkable development in optimization and optimal control. Due to its wide variety of applications, many scientists and researchers have paid attention to fields of optimization and optimal control. A huge number of new theoretical, algorithmic, and computational results have been observed in the last few years. This book gives the latest advances, and due to the rapid development of these fields, there are no other recent publications on the same topics.

Key features:

Provides a collection of selected contributions giving a state-of-the-art account of recent developments in the field

Covers a broad range of topics in optimization and optimal control, including unique applications

Includes chapters written by experts in their respective disciplines

Appeals to a broad audience of researchers, practitioners, and advanced graduate students in applied mathematics and engineering

This book covers the proceedings from the 2016 International Symposium on Chaos, Complexity and Leadership, and reflects current research results of chaos and complexity studies and their applications in various fields. Included are research papers in the fields of applied nonlinear methods, modeling of data and simulations, as well as theoretical achievements of chaos and complex systems. Also discussed are leadership and management applications of chaos and complexity theory.

This contributed volume is based on talks given at the August 2016 summer school "Fluids Under Pressure," held in Prague as part of the "Prague-Sum" series. Written by experts in their respective fields, chapters explore the complex role that pressure plays in physics, mathematical modeling, and fluid flow analysis. Specific topics covered include: Oceanic and atmospheric dynamics Incompressible flows Viscous compressible flows Well-posedness of the Navier-Stokes equations Weak solutions to the Navier-Stokes equations Fluids Under Pressure will be a valuable resource for graduate students and researchers studying fluid flow dynamics.

This book brings together 12 chapters on a new stream of research examining complex phenomena in nonlinear systems-including engineering, physics, and social science. Complex Motions and Chaos in Nonlinear Systems provides readers a particular vantage of the nature and nonlinear phenomena in nonlinear dynamics that can develop the corresponding mathematical theory and apply nonlinear design to practical engineering as well as the study of other complex phenomena including those investigated within social science.

This resource has been developed to fully cover unit AS 2: Applied Mathematics of the CCEA specification, addressing both mechanics and statistics. For each topic, the book begins with a logical explanation of the theory, examples to reinforce the explanation, and any key words and definitions that are required. Examples and definitions are clearly differentiated to ease revision and progression through the book. The material then flows into exercises, before introducing the next topic. In this way, the student is guided through the subject. The book contains a large number of exercises in order to provide teachers with as much flexibility as possible for their students. Answers to the questions are included at the back of the book. Contents: 1 Concepts in Mechanics; 2 Kinematics; Constant Acceleration; 3 Motion Graphs; 4 Forces; 5 Newton's Laws; 6 Friction; 7 Connected Bodies; 8 Statistical Sampling; 9 Data Presentation and Interpretation; 10 Central Tendency and Variation; 11 Correlation and Regression; 12 Data Cleaning; 13 Probability; 14 Binomial Distribution

Improved geospatial instrumentation and technology such as in laser scanning has now resulted in millions of data being collected, e.g., point clouds. It is in realization that such huge amount of data requires efficient and robust mathematical solutions that this third edition of the book extends the second edition by introducing three new chapters: Robust parameter estimation, Multiobjective optimization and Symbolic regression. Furthermore, the linear homotopy chapter is expanded to include nonlinear homotopy. These disciplines are discussed first in the theoretical part of the book before illustrating their geospatial applications in the applications chapters where numerous numerical examples are presented. The renewed electronic supplement contains these new theoretical and practical topics, with the corresponding Mathematica statements and functions supporting their computations introduced and applied. This third edition is renamed in light of these technological advancements.

The book answers long-standing questions on scientific modeling and inference across multiple perspectives and disciplines, including logic, mathematics, physics and medicine. The different chapters cover a variety of issues, such as the role models play in scientific practice; the way science shapes our concept of models; ways of modeling the pursuit of scientific knowledge; the relationship between our concept of models and our concept of science. The book also discusses models and scientific explanations; models in the semantic view of theories; the applicability of mathematical models to the real world and their effectiveness; the links between models and inferences; and models as a means for acquiring new knowledge. It analyzes different examples of models in physics, biology, mathematics and engineering. Written for researchers and graduate students, it provides a cross-disciplinary reference guide to the notion and the use of models and inferences in science.

The analysis of PDEs is a prominent discipline in mathematics research, both in terms of its theoretical aspects and its relevance in applications. In recent years, the geometric properties of linear and nonlinear second order PDEs of elliptic and parabolic type have been extensively studied by many outstanding researchers. This book collects contributions from a selected group of leading experts who took part in the INdAM meeting "Geometric methods in PDEs", on the occasion of the 70th birthday of Ermanno Lanconelli. They describe a number of new achievements and/or the state of the art in their discipline of research, providing readers an overview of recent progress and future research trends in PDEs. In particular, the volume collects significant results for sub-elliptic equations, potential theory and diffusion equations, with an emphasis on comparing different methodologies and on their implications for theory and applications.

Neutrinos continue to be the most mysterious and, arguably, the most fascinating particles of the Standard Model as their intrinsic properties such as absolute mass scale and CP properties are unknown. The open question of the absolute neutrino mass scale will be addressed with unprecedented accuracy by the Karlsruhe Tritium Neutrino (KATRIN) experiment, currently under construction. This thesis focusses on the spectrometer part of KATRIN and background processes therein. Various background sources such as small Penning traps, as well as nuclear decays from single radon atoms are fully characterized here for the first time. Most importantly, however, it was possible to reduce the background in the spectrometer by more than five orders of magnitude by eliminating Penning traps and by developing a completely new background reduction method by stochastically heating trapped electrons using electron cyclotron resonance (ECR). The work beautifully demonstrates that the obstacles and challenges in measuring the absolute mass scale of neutrinos can be met successfully if novel experimental tools (ECR) and novel computing methods (KASSIOPEIA) are combined to allow almost background-free tritium ss-spectroscopy.

Fluid turbulence is often referred to as `the unsolved problem of classical physics'. Yet, paradoxically, its mathematical description resembles quantum field theory. The present book addresses the idealised problem posed by homogeneous, isotropic turbulence, in order to concentrate on the fundamental aspects of the general problem. It is written from the perspective of a theoretical physicist, but is designed to be accessible to all researchers in turbulence, both theoretical and experimental, and from all disciplines. The book is in three parts, and begins with a very simple overview of the basic statistical closure problem, along with a summary of current theoretical approaches. This is followed by a precise formulation of the statistical problem, along with a complete set of mathematical tools (as needed in the rest of the book), and a summary of the generally accepted phenomenology of the subject. Part 2 deals with current issues in phenomenology, including the role of Galilean invariance, the physics of energy transfer, and the fundamental problems inherent in numerical simulation. Part 3 deals with renormalization methods, with an emphasis on the taxonomy of the subject, rather than on lengthy mathematical derivations. The book concludes with some discussion of current lines of research and is supplemented by three appendices containing detailed mathematical treatments of the effect of isotropy on correlations, the properties of Gaussian distributions, and the evaluation of coefficients in statistical theories.

This book, along with its companion volume, Nonlinear Dynamics New Directions: Models and Applications, covers topics ranging from fractal analysis to very specific applications of the theory of dynamical systems to biology. This first volume is devoted to fundamental aspects and includes a number of important new contributions as well as some review articles that emphasize new development prospects. The second volume contains mostly new applications of the theory of dynamical systems to both engineering and biology. The topics addressed in the two volumes include a rigorous treatment of fluctuations in dynamical systems, topics in fractal analysis, studies of the transient dynamics in biological networks, synchronization in lasers, and control of chaotic systems, among others. This book also: * Presents a rigorous treatment of fluctuations in dynamical systems and explores a range of topics in fractal analysis, among other fundamental topics * Features recent developments on large deviations for higher-dimensional maps, a study of measures resisting multifractal analysis and a overview of complex Kleninan groups * Includes thorough review of recent findings that emphasize new development prospects

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.

This book is the first to present the application of the hybrid system theory to systems with EPCA (equations with piecewise continuous arguments). The hybrid system paradigm is a valuable modeling tool for describing a wide range of real-world applications. Moreover, although new technology has produced, and continues to produce highly hierarchical sophisticated machinery that cannot be analyzed as a whole system, hybrid system representation can be used to reduce the structural complexity of these systems. That is to say, hybrid systems have become a modeling priority, which in turn has led to the creation of a promising research field with several application areas. As such, the book explores recent developments in the area of deterministic and stochastic hybrid systems using the Lyapunov and Razumikhin-Lyapunov methods to investigate the systems' properties. It also describes properties such as stability, stabilization, reliable control, H-infinity optimal control, input-to-state stability (ISS)/stabilization, state estimation, and large-scale singularly perturbed systems.

This monograph presents some theoretical and computational aspects of the parameterization method for invariant manifolds, focusing on the following contexts: invariant manifolds associated with fixed points, invariant tori in quasi-periodically forced systems, invariant tori in Hamiltonian systems and normally hyperbolic invariant manifolds. This book provides algorithms of computation and some practical details of their implementation. The methodology is illustrated with 12 detailed examples, many of them well known in the literature of numerical computation in dynamical systems. A public version of the software used for some of the examples is available online. The book is aimed at mathematicians, scientists and engineers interested in the theory and applications of computational dynamical systems.

Chemical Modelling: Applications and Theory comprises critical literature reviews of all aspects of molecular modelling. Molecular modelling in this context refers to modelliing the structure, properties and reactions of atoms, molecules and materials. Each chapter provides a selective review of recent literature, incorporating sufficient historical perspective for the non-specialist to gain an understanding. With chemical modelling covering such a wide range of subjects, this Specialist Periodical Report serves as the first port of call to any chemist, biochemist, materials scientist or molecular physicist needing to acquaint themselves with major developments in the area.

Examining recent mathematical developments in the study of Fredholm operators, spectral theory and block operator matrices, with a rigorous treatment of classical Riesz theory of polynomially-compact operators, this volume covers both abstract and applied developments in the study of spectral theory. These topics are intimately related to the stability of underlying physical systems and play a crucial role in many branches of mathematics as well as numerous interdisciplinary applications. By studying classical Riesz theory of polynomially compact operators in order to establish the existence results of the second kind operator equations, this volume will assist the reader working to describe the spectrum, multiplicities and localization of the eigenvalues of polynomially-compact operators.

Simulation based on mathematical models plays a major role in computer aided design of integrated circuits (ICs). Decreasing structure sizes, increasing packing densities and driving frequencies require the use of refined mathematical models, and to take into account secondary, parasitic effects. This leads to very high dimensional problems which nowadays require simulation times too large for the short time-to-market demands in industry. Modern Model Order Reduction (MOR) techniques present a way out of this dilemma in providing surrogate models which keep the main characteristics of the device while requiring a significantly lower simulation time than the full model. With Model Reduction for Circuit Simulation we survey the state of the art in the challenging research field of MOR for ICs, and also address its future research directions. Special emphasis is taken on aspects stemming from miniturisations to the nano scale. Contributions cover complexity reduction using e.g., balanced truncation, Krylov-techniques or POD approaches. For semiconductor applications a focus is on generalising current techniques to differential-algebraic equations, on including design parameters, on preserving stability, and on including nonlinearity by means of piecewise linearisations along solution trajectories (TPWL) and interpolation techniques for nonlinear parts. Furthermore the influence of interconnects and power grids on the physical properties of the device is considered, and also top-down system design approaches in which detailed block descriptions are combined with behavioral models. Further topics consider MOR and the combination of approaches from optimisation and statistics, and the inclusion of PDE models with emphasis on MOR for the resulting partial differential algebraic systems. The methods which currently are being developed have also relevance in other application areas such as mechanical multibody systems, and systems arising in chemistry and to biology. The current number of books in the area of MOR for ICs is very limited, so that this volume helps to fill a gap in providing the state of the art material, and to stimulate further research in this area of MOR. Model Reduction for Circuit Simulation also reflects and documents the vivid interaction between three active research projects in this area, namely the EU-Marie Curie Action ToK project O-MOORE-NICE (members in Belgium, The Netherlands and Germany), the EU-Marie Curie Action RTN-project COMSON (members in The Netherlands, Italy, Germany, and Romania), and the German federal project System reduction in nano-electronics (SyreNe).

This new approach to real analysis stresses the use of the subject with respect to applications, i.e., how the principles and theory of real analysis can be applied in a variety of settings in subjects ranging from Fourier series and polynomial approximation to discrete dynamical systems and nonlinear optimization. Users will be prepared for more intensive work in each topic through these applications and their accompanying exercises. This book is appropriate for math enthusiasts with a prior knowledge of both calculus and linear algebra.