Metaharmonic Lattice Point Theory covers interrelated methods and tools of spherically oriented geomathematics and periodically reflected analytic number theory. The book establishes multi-dimensional Euler and Poisson summation formulas corresponding to elliptic operators for the adaptive determination and calculation of formulas and identities of weighted lattice point numbers, in particular the non-uniform distribution of lattice points. The author explains how to obtain multi-dimensional generalizations of the Euler summation formula by interpreting classical Bernoulli polynomials as Green's functions and linking them to Zeta and Theta functions. To generate multi-dimensional Euler summation formulas on arbitrary lattices, the Helmholtz wave equation must be converted into an associated integral equation using Green's functions as bridging tools. After doing this, the weighted sums of functional values for a prescribed system of lattice points can be compared with the corresponding integral over the function. Exploring special function systems of Laplace and Helmholtz equations, this book focuses on the analytic theory of numbers in Euclidean spaces based on methods and procedures of mathematical physics. It shows how these fundamental techniques are used in geomathematical research areas, including gravitation, magnetics, and geothermal.

As the Earth`s surface deviates from its spherical shape by less than 0.4 percent of its radius and today's satellite missions collect their gravitational and magnetic data on nearly spherical orbits, sphere-oriented mathematical methods and tools play important roles in studying the Earth's gravitational and magnetic field. Geomathematically Oriented Potential Theory presents the principles of space and surface potential theory involving Euclidean and spherical concepts. The authors offer new insight on how to mathematically handle gravitation and geomagnetism for the relevant observables and how to solve the resulting potential problems in a systematic, mathematically rigorous framework. The book begins with notational material and the necessary mathematical background. The authors then build the foundation of potential theory in three-dimensional Euclidean space and its application to gravitation and geomagnetism. They also discuss surface potential theory on the unit sphere along with corresponding applications. Focusing on the state of the art, this book breaks new geomathematical grounds in gravitation and geomagnetism. It explores modern sphere-oriented potential theoretic methods as well as classical space potential theory.

Metaharmonic Lattice Point Theory covers interrelated methods and tools of spherically oriented geomathematics and periodically reflected analytic number theory. The book establishes multi-dimensional Euler and Poisson summation formulas corresponding to elliptic operators for the adaptive determination and calculation of formulas and identities of weighted lattice point numbers, in particular the non-uniform distribution of lattice points.

The author explains how to obtain multi-dimensional generalizations of the Euler summation formula by interpreting classical Bernoulli polynomials as Green's functions and linking them to Zeta and Theta functions. To generate multi-dimensional Euler summation formulas on arbitrary lattices, the Helmholtz wave equation must be converted into an associated integral equation using Green's functions as bridging tools. After doing this, the weighted sums of functional values for a prescribed system of lattice points can be compared with the corresponding integral over the function.

Exploring special function systems of Laplace and Helmholtz equations, this book focuses on the analytic theory of numbers in Euclidean spaces based on methods and procedures of mathematical physics. It shows how these fundamental techniques are used in geomathematical research areas, including gravitation, magnetics, and geothermal.

Lattice Point Identities and Shannon-Type Sampling demonstrates that significant roots of many recent facets of Shannon's sampling theorem for multivariate signals rest on basic number-theoretic results. This book leads the reader through a research excursion, beginning from the Gaussian circle problem of the early nineteenth century, via the classical Hardy-Landau lattice point identity and the Hardy conjecture of the first half of the twentieth century, and the Shannon sampling theorem (its variants, generalizations and the fascinating stories about the cardinal series) of the second half of the twentieth century. The authors demonstrate how all these facets have resulted in new multivariate extensions of lattice point identities and Shannon-type sampling procedures of high practical applicability, thereby also providing a general reproducing kernel Hilbert space structure of an associated Paley-Wiener theory over (potato-like) bounded regions (cf. the cover illustration of the geoid), as well as the whole Euclidean space. All in all, the context of this book represents the fruits of cross-fertilization of various subjects, namely elliptic partial differential equations, Fourier inversion theory, constructive approximation involving Euler and Poisson summation formulas, inverse problems reflecting the multivariate antenna problem, and aspects of analytic and geometric number theory. Features: New convergence criteria for alternating series in multi-dimensional analysis Self-contained development of lattice point identities of analytic number theory Innovative lattice point approach to Shannon sampling theory Useful for students of multivariate constructive approximation, and indeed anyone interested in the applicability of signal processing to inverse problems.

In industry and economics, the most common solutions of partial differential equations involving multivariate numerical integration over cuboids include techniques of iterated one-dimensional approximate integration. In geosciences, however, the integrals are extended over potato-like volumes (such as the ball, ellipsoid, geoid, or the Earth) and their boundary surfaces which require specific multi-variate approximate integration methods. Integration and Cubature Methods: A Geomathematically Oriented Course provides a basic foundation for students, researchers, and practitioners interested in precisely these areas, as well as breaking new ground in integration and cubature in geomathematics.

This monograph presents the geoscientific context arising in decorrelative gravitational exploration to determine the mass density distribution inside the Earth. First, an insight into the current state of research is given by reducing gravimetry to mathematically accessible, and thus calculable, decorrelated models. In this way, the various unresolved questions and problems of gravimetry are made available to a broad scientific audience and the exploration industry. New theoretical developments will be given, and innovative ways of modeling geologic layers and faults by mollifier regularization techniques are shown. This book is dedicated to surface as well as volume geology with potential data primarily of terrestrial origin. For deep geology, the geomathematical decorrelation methods are to be designed in such a way that depth information (e.g., in boreholes) may be canonically entered. Bridging several different geo-disciplines, this book leads in a cycle from the potential measurements made by geoengineers, to the cleansing of data by geophysicists and geoengineers, to the subsequent theory and model formation, computer-based implementation, and numerical calculation and simulations made by geomathematicians, to interpretation by geologists, and, if necessary, back. It therefore spans the spectrum from geoengineering, especially geodesy, via geophysics to geomathematics and geology, and back. Using the German Saarland area for methodological tests, important new fields of application are opened, particularly for regions with mining-related cavities or dense development in today's geo-exploration.

This monograph presents the geoscientific context arising in decorrelative geomagnetic exploration. First, an insight into the current state of research is given by reducing magnetometry to mathematically accessible, and thus calculable, decorrelated models. In this way, various questions and problems of magnetometry are made available to a broad scientific audience and the exploration industry. New stimuli are given, and innovative ways of modeling geologic strata by mollifier magnetometric techniques are shown. Potential data sets primarily of terrestrial origin constitute the main data basis in the book. For deep geology, the geomathematical decorrelation methods are designed in such a way that depth information (e.g., in boreholes) may be canonically entered. Overall, this book provides pioneering and ground-breaking innovative mathematical knowledge as a transfer methodology from the "reality space" of magnetometric measurements into the "virtual space" of mathematical-numerical modeling structures and mollifier solutions with novel geological application areas. It pursues a double goal: On the one hand, it represents a geoscientific set of rules for today's geoengineering, interested in the application of innovative modelling and simulation techniques to promising data sets and structures occurring in geomagnetics. On the other hand, the book serves as a collection of current material in Applied Mathematics to offer alternative methodologies in the theory of inverse problems.

This monograph presents the geoscientific context arising in decorrelative gravitational exploration to determine the mass density distribution inside the Earth. First, an insight into the current state of research is given by reducing gravimetry to mathematically accessible, and thus calculable, decorrelated models. In this way, the various unresolved questions and problems of gravimetry are made available to a broad scientific audience and the exploration industry. New theoretical developments will be given, and innovative ways of modeling geologic layers and faults by mollifier regularization techniques are shown. This book is dedicated to surface as well as volume geology with potential data primarily of terrestrial origin. For deep geology, the geomathematical decorrelation methods are to be designed in such a way that depth information (e.g., in boreholes) may be canonically entered. Bridging several different geo-disciplines, this book leads in a cycle from the potential measurements made by geoengineers, to the cleansing of data by geophysicists and geoengineers, to the subsequent theory and model formation, computer-based implementation, and numerical calculation and simulations made by geomathematicians, to interpretation by geologists, and, if necessary, back. It therefore spans the spectrum from geoengineering, especially geodesy, via geophysics to geomathematics and geology, and back. Using the German Saarland area for methodological tests, important new fields of application are opened, particularly for regions with mining-related cavities or dense development in today's geo-exploration.

This book presents, in a consistent and unified overview, results and developments in the field of todays spherical sampling, particularly arising in mathematical geosciences. Although the book often refers to original contributions, the authors made them accessible to (graduate) students and scientists not only from mathematics but also from geosciences and geoengineering. Building a library of topics in spherical sampling theory it shows how advances in this theory lead to new discoveries in mathematical, geodetic, geophysical as well as other scientific branches like neuro-medicine. A must-to-read for everybody working in the area of spherical sampling.

Written by leading experts, this book provides a clear and comprehensive survey of the "status quo" of the interrelating process and cross-fertilization of structures and methods in mathematical geodesy. Starting with a foundation of functional analysis, potential theory, constructive approximation, special function theory, and inverse problems, readers are subsequently introduced to today's least squares approximation, spherical harmonics reflected spline and wavelet concepts, boundary value problems, Runge-Walsh framework, geodetic observables, geoidal modeling, ill-posed problems and regularizations, inverse gravimetry, and satellite gravity gradiometry. All chapters are self-contained and can be studied individually, making the book an ideal resource for both graduate students and active researchers who want to acquaint themselves with the mathematical aspects of modern geodesy.

Written by leading experts, this book provides a clear and comprehensive survey of the "status quo" of the interrelating process and cross-fertilization of structures and methods in mathematical geodesy. Starting with a foundation of functional analysis, potential theory, constructive approximation, special function theory, and inverse problems, readers are subsequently introduced to today's least squares approximation, spherical harmonics reflected spline and wavelet concepts, boundary value problems, Runge-Walsh framework, geodetic observables, geoidal modeling, ill-posed problems and regularizations, inverse gravimetry, and satellite gravity gradiometry. All chapters are self-contained and can be studied individually, making the book an ideal resource for both graduate students and active researchers who want to acquaint themselves with the mathematical aspects of modern geodesy.

Special functions enable us to formulate a scientific problem by reduction such that a new, more concrete problem can be attacked within a well-structured framework, usually in the context of differential equations. A good understanding of special functions provides the capacity to recognize the causality between the abstractness of the mathematical concept and both the impact on and cross-sectional importance to the scientific reality. The special functions to be discussed in this monograph vary greatly, depending on the measurement parameters examined (gravitation, electric and magnetic fields, deformation, climate observables, fluid flow, etc.) and on the respective field characteristic (potential field, diffusion field, wave field). The differential equation under consideration determines the type of special functions that are needed in the desired reduction process. Each chapter closes with exercises that reflect significant topics, mostly in computational applications. As a result, readers are not only directly confronted with the specific contents of each chapter, but also with additional knowledge on mathematical fields of research, where special functions are essential to application. All in all, the book is an equally valuable resource for education in geomathematics and the study of applied and harmonic analysis. Students who wish to continue with further studies should consult the literature given as supplements for each topic covered in the exercises.

This self-contained text/reference provides a basic foundation for practitioners, researchers, and students interested in any of the diverse areas of multiscale (geo)potential theory. New mathematical methods are developed enabling the gravitational potential of a planetary body to be modeled using a continuous flow of observations from land or satellite devices. Harmonic wavelets methods are introduced, as well as fast computational schemes and various numerical test examples. Presented are multiscale approaches for numerous geoscientific problems, including geoidal determination, magnetic field reconstruction, deformation analysis, and density variation modelling With exercises at the end of each chapter, the book may be used as a textbook for graduate-level courses in geomathematics, applied mathematics, and geophysics. The work is also an up-to-date reference text for geoscientists, applied mathematicians, and engineers.

Duringthelastdecades,geosciencesand-engineeringwerein?uencedbytwo essentialscenarios. First, thetechnologicalprogresshaschangedcompletely the observational and measurement techniques. Modern high speed c- puters and satellite-based techniques are entering more and more all (geo) disciplines. Second, there is a growing public concern about the future of our planet, its climate, its environment, and about an expected shortage of natural resources. Obviously, both aspects, viz. (i) e?cient strategies of protection against threats of a changing Earth and (ii) the exceptional s- uation of getting terrestrial, airborne as well as spaceborne, data of better and better quality explain the strong need for new mathematical structures, tools, and methods. In consequence, mathematics concerned with geosci- ti?c problems, i.e., geomathematics, is becoming more and more important. Nowadays, geomathematics may be regarded as the key technology to build the bridge between real Earth processes and their scienti?c understanding. In fact, it is the intrinsic and indispensable means to handle geoscient- cally relevant data sets of high quality within high accuracy and to improve signi?cantly modeling capabilities in Earth system research.

This self-contained book provides a basic foundation for students, practitioners, and researchers interested in some of the diverse new areas of multiscale (geo)potential theory. New mathematical methods are developed enabling the gravitational potential of a planetary body to be modeled and analyzed using a continuous flow of observations from land or satellite devices. Harmonic wavelet methods are introduced, as well as fast computational schemes and various numerical test examples.Topic and key features: - Comprehensive coverage of topics which, thus far, are only scattered in journal articles and conference proceedings- Important applications and developments for future satellite scenarios; new modelling techniques involving low-orbiting satellites- Multiscale approaches for numerous geoscientific problems, including geoidal determination, magnetic field reconstruction, deformation analysis, and density variation modelling- Exercises at the end of each chapter and an appendix with hints to their solutions

This book is an enlarged second edition of a monograph published in the Springer AGEM2-Series, 2009. It presents, in a consistent and unified overview, a setup of the theory of spherical functions of mathematical (geo-)sciences. The content shows a twofold transition: First, the natural transition from scalar to vectorial and tensorial theory of spherical harmonics is given in a coordinate-free context, based on variants of the addition theorem, Funk-Hecke formulas, and Helmholtz as well as Hardy-Hodge decompositions. Second, the canonical transition from spherical harmonics via zonal (kernel) functions to the Dirac kernel is given in close orientation to an uncertainty principle classifying the space/frequency (momentum) behavior of the functions for purposes of data analysis and (geo-)application. The whole palette of spherical functions is collected in a well-structured form for modeling and simulating the phenomena and processes occurring in the Earth's system. The result is a work which, while reflecting the present state of knowledge in a time-related manner, claims to be of largely timeless significance in (geo-)mathematical research and teaching.

This book presents, in a consistent and unified overview, results and developments in the field of todays spherical sampling, particularly arising in mathematical geosciences. Although the book often refers to original contributions, the authors made them accessible to (graduate) students and scientists not only from mathematics but also from geosciences and geoengineering. Building a library of topics in spherical sampling theory it shows how advances in this theory lead to new discoveries in mathematical, geodetic, geophysical as well as other scientific branches like neuro-medicine. A must-to-read for everybody working in the area of spherical sampling.

Special functions enable us to formulate a scientific problem by reduction such that a new, more concrete problem can be attacked within a well-structured framework, usually in the context of differential equations. A good understanding of special functions provides the capacity to recognize the causality between the abstractness of the mathematical concept and both the impact on and cross-sectional importance to the scientific reality. The special functions to be discussed in this monograph vary greatly, depending on the measurement parameters examined (gravitation, electric and magnetic fields, deformation, climate observables, fluid flow, etc.) and on the respective field characteristic (potential field, diffusion field, wave field). The differential equation under consideration determines the type of special functions that are needed in the desired reduction process. Each chapter closes with exercises that reflect significant topics, mostly in computational applications. As a result, readers are not only directly confronted with the specific contents of each chapter, but also with additional knowledge on mathematical fields of research, where special functions are essential to application. All in all, the book is an equally valuable resource for education in geomathematics and the study of applied and harmonic analysis. Students who wish to continue with further studies should consult the literature given as supplements for each topic covered in the exercises.

The authors introduce geomathematics as an active research area to a wider audience. Chapter 1 presents an introduction to the Earth as a system to apply scientific methods. Emphasis is laid on transfers from virtual models to reality and vice versa. In the second chapter geomathematics is introduced as a new scientific area which nevertheless has its roots in antiquity. The modern conception of geomathematics is outlined from different points of view and its challenging nature is described as well as its interdisciplinarity. Geomathematics is shown as the bridge between the real world and the virtual world. The complex mathematical tools are shown from a variety of fields necessary to tackle geoscientific problems in the mathematical language. Chapter 3 contains some exemplary applications as novel exploration methods. Particular importance is laid on the change of language when it comes to translate measurements to mathematical models. New solution methods like the multiscale mollifier technique are presented. Further applications discussed are aspects of reflection seismics. Chapter 4 is devoted to the short description of recent activities in geomathematics. The Appendix (Chapter 5) is devoted to the GEM - International Journal on Geomathematics founded ten years ago. Besides a detailed structural analysis of the editorial goals an index of all papers published in former issues is given.

Die durch das Reaktorungluck in Fukushima forcierte Energiewende hin zu einer Stromversorgung mit primar erneuerbaren Energien konzentriert sich in der aktuellen Wahrnehmung nur auf den Ausbau von Solar und Windkraftenergie. Dabei wird vergessen, dass aufgrund fehlender Stromspeichertechnologien und UEberlandstromtrassen eine erneuerbare Energie benoetigt wird, die konstant Strom liefern und so Erzeugungsschwankungen bei Solar- und Windkraftenergie ausgleichen kann. Tiefe Geothermie, also Energie, die aus der Erde kommt, kann diese Aufgabe leisten, da sie die einzige erneuerbare Energie ist, die nicht klimatischen oder wetterbedingten Schwankungen unterliegt. Mit einem durch Wissenschaftlern und Praktikern erstellten Normenkatalog fur tiefengeothermische Bohrungen, wurde hier ein hoechstmoeglichen Sicherheitsstandard erreicht, und die wirtschaftlichen wie geologischen Risiken jedes Projektes minimiert werden.

Das vorliegende Handbuch stellt erstmalig in deutscher Sprache umfassend alle Themen im Bereich Tiefe Geothermie vor: Prospektion, Exploration, Realisierung und Nutzung. In zwei abschliessenden Kapiteln werden Zukunft und Perspektiven von Tiefer Geothermie vorgestellt und diskutiert. Die von anerkannten Spezialisten verfassten Artikel sind so konzipiert und geschrieben, dass alle an der Konzeption und Erschliessung tiefengeothermischer Ressourcen Beteiligten einen direkten Nutzen fur ihre Arbeit daraus ziehen koennen.

Frontiers in Entropy Across the Disciplines presents a panorama of entropy emphasizing mathematical theory, physical and scientific significance, computational methods, and applications in mathematics, physics, statistics, engineering, biomedical signals, and signal processing.In the last century classical concepts of entropy were introduced in the areas of thermodynamics, information theory, probability theory, statistics, dynamical systems, and ergodic theory. During the past 50 years, dozens of new concepts of entropy have been introduced and studied in many disciplines. This volume captures significant developments in this arena. It features expository, review, and research papers by distinguished mathematicians and scientists from many disciplines. The level of mathematics ranges from intermediate level to research level. Each chapter contains a comprehensive list of references. Topics include entropy and society, entropy and time, Souriau entropy on symplectic model of statistical physics, new definitions of entropy, geometric theory of heat and information, maximum entropy in Bayesian networks, maximum entropy methods, entropy analysis of biomedical signals (review and comparison of methods), spectral entropy and its application to video coding and speech coding, a comprehensive review of 50 years of entropy in dynamics, a comprehensive review on entropy, entropy-like quantities and applications, topological entropy of multimodal maps, entropy production in complex systems, entropy production and convergence to equilibrium, reversibility and irreversibility in entropy, nonequilibrium entropy, index of various entropy, entropy and the greatest blunder ever.

Die Entwicklung immer leistungsfahigerer absoluter wie auch relativer Gravimeter mit deutlich verbesserter Messgenauigkeit ermoeglicht es, dass sich kunftig nicht nur pragnante Schwereanomalien (wie z. B. die eines Salzstocks), sondern auch schwachere Signaturen erfassen und modellieren lassen. Mehr noch, die rasante Entwicklung der Computer fuhrt zu neuartigen Methoden der Datendekomposition, wie z. B. Waveletdekorrelation. Dekorrelative Gravimetrie ist somit eine neue Explorationstechnik, die als kanonische Innovation aus der Verbindung neuartiger Mess- und Modellierungstechniken resultiert. Dekorrelative Gravimetrie dient der Reduzierung des Fundigkeitsrisikos von Aquiferen sowie von Gas- und OEllagerstatten, auch durch Vergleich und Zusammenschau alternativer, aber strukturell ahnlich gelagerter, dekorrelativer Verfahren wie etwa Magnetometrie und Seismik. Hier setzt dieses Buch mit einem exemplarischen UEberblick uber die neuartige Dekorrelationsmethoden der heutigen Geomathematik mit Hauptgewicht fur den Fall der Gravimetrie an. Wesentliches mathematisches Hilfsmittel ist die Regularisierung des Newtonschen Volumenintegrals durch taylorisierte Mollifier-Varianten des Newton-Kerns. Ziel des vorliegenden Buches ist somit die Vermittlung des Grundverstandnisses, dass Zooming-In Mollifier-Potentialmethoden wie etwa dekorrelative Gravimetrie neue wichtige Anwendungsfelder in der heutigen Geoexploration eroeffnen, insbesondere fur Gebiete mit bergbaubedingten Hohlraumen oder sehr dichter Bebauung wie etwa das Saarland oder Sachsen, die den Einsatz von reflexionsseismischen Messungen erschweren oder sogar unmoeglich machen. Zusammenfassend lasst sich fur dieses Buchprojekt festhalten, dass es einen Einblick in den aktuellen Stand gravimetrischer Multiskalenforschung vermittelt. Als wesentliches Resultat ergibt sich, dass die Schlusseltechnologie Geomathematik in der Tat in der Lage ist, die Gravimetrie auf einfache, fur Explorationszwecke zugangliche und somit rechenbare Dekorrelationsmodelle zu reduzieren. Mehr noch, das Buch macht auf diese Weise ein breites Publikum mit den vielfaltigen Fragen und Problemen der heutigen Gravimetrie vertraut und setzt Denkanstoesse fur eine erfolgreiche Weiterentwicklung und eine adaquate praxisrelevante Anwendbarkeit von Potentialmethoden in der Exploration in Gang.

Selbstkonsistente Darstellung von Schlussel- und Transfermethodologien vom Realitatsraum geodatischer Messungen und Beobachtungen in den Modellraum mathematischer Strukturen und Loesungen und zuruck, neue Perspektiven und Forschungstrends im Bereich Mathematischer Geodasie.

Das Handbuch vermittelt die Grundlagen wie das thermische Regime der Erde, die oberflachennahe Geologie, geologisch-geophysikalische Grundlagen, zugrunde liegende mathematische Methoden, Risikomanagement und Bohrtechniken. In den angewandten Kapiteln geht es konkret um Geothermieprojekte aus der Sicht eines Bauherren, u.a. um Heizlastberechnung und die hydraulische Abgleichung, daneben um das Projektmanagement, zu berucksichtigende Umweltaspekte, Finanzierung und Foerdermoeglichkeiten und die Dimensionierung von Anlagen. Auch Verfahrenstechnische Grundlagen, die Maschinentechnik, die Qualitatssicherung und Fragen der Kommunikation und Akzeptanz wie auch der Arbeitssicherheit und des Gesundheitsschutzes werden behandelt. Auch wenn der Schwerpunkt des Handbuches auf Deutschland, OEsterreich und der Schweiz liegt, so wird auch auf die Oberflachennahe Geothermiebranche weltweit eingegangen und Herausforderungen wie auch Chancen und Perspektiven aufgezeigt.