These proceedings were prepared in connection with the international conference Approximation Theory XIII, which was held March 7-10, 2010 in San Antonio, Texas. The conference was the thirteenth in a series of meetings in Approximation Theory held at various locations in the United States, and was attended by 144 participants. Previous conferences in the series were held in Austin, Texas (1973, 1976, 1980, 1992), College Station, Texas (1983, 1986, 1989, 1995), Nashville, Tennessee (1998), St. Louis, Missouri (2001), Gatlinburg, Tennessee (2004), and San Antonio, Texas (2007). Along with the many plenary speakers, the contributors to this proceedings provided inspiring talks and set a high standard of exposition in their descriptions of new directions for research. Many relevant topics in approximation theory are included in this book, such as abstract approximation, approximation with constraints, interpolation and smoothing, wavelets and frames, shearlets, orthogonal polynomials, univariate and multivariate splines, and complex approximation.

This book contains selected papers of NSC08, the 2nd Conference on Nonlinear Science and Complexity, held 28-31 July, 2008, Porto, Portugal. It focuses on fundamental theories and principles, analytical and symbolic approaches, computational techniques in nonlinear physics and mathematics. Topics treated include * Chaotic Dynamics and Transport in Classic and Quantum Systems * Complexity and Nonlinearity in Molecular Dynamics and Nano-Science * Complexity and Fractals in Nonlinear Biological Physics and Social Systems * Lie Group Analysis and Applications in Nonlinear Science * Nonlinear Hydrodynamics and Turbulence * Bifurcation and Stability in Nonlinear Dynamic Systems * Nonlinear Oscillations and Control with Applications * Celestial Physics and Deep Space Exploration * Nonlinear Mechanics and Nonlinear Structural Dynamics * Non-smooth Systems and Hybrid Systems * Fractional dynamical systems

Line and hyperplane location problems play an important role not only in operations research and location theory, but also in computational geometry and robust statistics. This book provides a survey on line and hyperplane location combining analytical and geometrical methods. The major portion of the text presents new results on this topic, including the extension of some special cases to all distances derived from norms and a discussion of restricted problems in the plane. Almost all results are proven in the text and most of them are illustrated by examples. Furthermore, relations to classical facility location and to problems in computational geometry are pointed out. Audience: The book is suitable for researchers, lecturers, and graduate students working in the fields of location theory or computational geometry.

Along with finite differences and finite elements, spectral methods are one of the three main methodologies for solving partial differential equations on computers. This book provides a detailed presentation of basic spectral algorithms, as well as a systematical presentation of basic convergence theory and error analysis for spectral methods. Readers of this book will be exposed to a unified framework for designing and analyzing spectral algorithms for a variety of problems, including in particular high-order differential equations and problems in unbounded domains. The book contains a large number of figures which are designed to illustrate various concepts stressed in the book. A set of basic matlab codes has been made available online to help the readers to develop their own spectral codes for their specific applications.

Spontaneous potential (SP) well-logging is one of the most common and useful well-logging techniques in petroleum exploitation. This monograph is the first of its kind on the mathematical model of spontaneous potential well-logging and its numerical solutions. The mathematical model established in this book shows the necessity of introducing Sobolev spaces with fractional power, which seriously increases the difficulty of proving the well-posedness and proposing numerical solution schemes. In this book, in the axisymmetric situation the well-posedness of the corresponding mathematical model is proved and three efficient schemes of numerical solution are proposed, supported by a number of numerical examples to meet practical computation needs.

When the DFG (Deutsche Forschungsgemeinschaft) launched its collabora tive research centre or SFB (Sonderforschungsbereich) 438 "Mathematical Modelling, Simulation, and Verification in Material-Oriented Processes and Intelligent Systems" in July 1997 at the Technische Vniversitat Munchen and at the Vniversitat Augsburg, southern Bavaria got its second nucleus of the still young discipline scientific computing. Whereas the first and older one, FORTWIHR, the Bavarian Consortium for High Performance Scientific Com puting, had put its main emphasis on the supercomputing aspect, this new initiative was now expected to focus on the mathematical part. Consequently, throughout all of the five main research topics (A) adaptive materials and thin layers, (B) adaptive materials in medicine, (C) robotics, aeronautics, and automobile technology, (D) microstructured devices and systems, and (E) transport processes in flows, mathematical aspects play a predominant role. The formation of the SFB 438 and its scientific program are inextricably linked with the name of Karl-Heinz Hoffmann. As full professor for applied mathematics in Augsburg (1981-1991) and in Munchen (since 1992) and as dean of the faculty of mathematics at the TV Munchen, he was the driv ing force of this fascinating, but not always easy-to-realize idea of bringing together scientists from mathematics, physics, engineering, informatics, and medicine for joint efforts in modern applied mathematics. However, scarcely work had begun when the successful captain was called to take command on a bigger boat."

These days, the nature of services and the volume of demand in the telecommu nication industry is changing radically, with the replacement of analog transmis sion and traditional copper cables by digital technology and fiber optic transmis sion equipment. Moreover, we see an increasing competition among providers of telecommunication services, and the development of a broad range of new services for users, combining voice, data, graphics and video. Telecommunication network planning has thus become an important problem area for developing and applying optimization models. Telephone companies have initiated extensive modeling and planning efforts to expand and upgrade their transmission facilities, which are, for most national telecommunication networks, divided in three main levels (see Balakrishnan et al. [5]), namely, l. the long-distance or backbone network that typically connects city pairs through gateway nodes; 2. the inter-office or switching center network within each city, that interconnects switching centers in different subdivisions (clusters of customers) and provides access to the gateway(s) node(s); 1 2 DESIGN OF SURVNABLE NETWORKS WITH BOUNDED RINGS 3. the local access network that connects individual subscribers belonging to a cluster to the corresponding switching center. These three levels differ in several ways including their design criteria. Ideally, the design of a telecommunication network should simultaneously account for these three levels. However, to simplify the planning task, the overall planning problem is decomposed by considering each level separately.

Everything should be made as simple as possible, but not simpler. (Albert Einstein, Readers Digest, 1977) The modern practice of creating technical systems and technological processes of high effi.ciency besides the employment of new principles, new materials, new physical effects and other new solutions ( which is very traditional and plays the key role in the selection of the general structure of the object to be designed) also includes the choice of the best combination for the set of parameters (geometrical sizes, electrical and strength characteristics, etc.) concretizing this general structure, because the Variation of these parameters ( with the structure or linkage being already set defined) can essentially affect the objective performance indexes. The mathematical tools for choosing these best combinations are exactly what is this book about. With the advent of computers and the computer-aided design the pro bations of the selected variants are usually performed not for the real examples ( this may require some very expensive building of sample op tions and of the special installations to test them ), but by the analysis of the corresponding mathematical models. The sophistication of the mathematical models for the objects to be designed, which is the natu ral consequence of the raising complexity of these objects, greatly com plicates the objective performance analysis. Today, the main (and very often the only) available instrument for such an analysis is computer aided simulation of an object's behavior, based on numerical experiments with its mathematical model.

This book is intended to provide economists with mathematical tools necessary to handle the concepts of evolution under uncertainty and adaption arising in economics, pursuing the Arrow-Debreu-Hahn legacy. It applies the techniques of viability theory to the study of economic systems evolving under contingent uncertainty, faced with scarcity constraints, and obeying various implementation of the inertia principle. The book illustrates how new tools can be used to move from static analysis, built on concepts of optima, equilibria and attractors to a contingent dynamic framework.

This book is a compilation of the entire research work on the topic of Complex Binary Number System (CBNS) carried out by the author as the principal investigator and members of his research groups at various universities during the years 2000-2012. Pursuant to these efforts spanning several years, the realization of CBNS as a viable alternative to represent complex numbers in an "all-in-one" binary number format has become possible and efforts are underway to build computer hardware based on this unique number system.
It is hoped that this work will be of interest to anyone involved in computer arithmetic and digital logic design and kindle renewed enthusiasm among the engineers working in the areas of digital signal and image processing for developing newer and efficient algorithms and techniques incorporating CBNS.

Developed over a period of two years at the University of Utah Department of Computer Science, this course has been designed to encourage the integration of computation into the science and engineering curricula. Intended as an introductory course in computing expressly for science and engineering students, the course was created to satisfy the standard programming requirement, while preparing students to immediately exploit the broad power of modern computing in their science and engineering courses.

Algebraic, differential, and integral equations are used in the applied sciences, en gineering, economics, and the social sciences to characterize the current state of a physical, economic, or social system and forecast its evolution in time. Generally, the coefficients of and/or the input to these equations are not precisely known be cause of insufficient information, limited understanding of some underlying phe nomena, and inherent randonmess. For example, the orientation of the atomic lattice in the grains of a polycrystal varies randomly from grain to grain, the spa tial distribution of a phase of a composite material is not known precisely for a particular specimen, bone properties needed to develop reliable artificial joints vary significantly with individual and age, forces acting on a plane from takeoff to landing depend in a complex manner on the environmental conditions and flight pattern, and stock prices and their evolution in time depend on a large number of factors that cannot be described by deterministic models. Problems that can be defined by algebraic, differential, and integral equations with random coefficients and/or input are referred to as stochastic problems. The main objective of this book is the solution of stochastic problems, that is, the determination of the probability law, moments, and/or other probabilistic properties of the state of a physical, economic, or social system. It is assumed that the operators and inputs defining a stochastic problem are specified."

This book takes readers on a tour through modern methods in image analysis and reconstruction based on level set and PDE techniques, the major focus being on morphological and geometric structures in images. The aspects covered include edge-sharpening image reconstruction and denoising, segmentation and shape analysis in images, and image matching. For each, the lecture notes provide insights into the basic analysis of modern variational and PDE-based techniques, as well as computational aspects and applications.

Algorithms for the numerical computation of definite integrals have been proposed for more than 300 years, but practical considerations have led to problems of ever-increasing complexity, so that, even with current computing speeds, numerical integration may be a difficult task. High dimension and complicated structure of the region of integration and singularities of the integrand are the main sources of difficulties.

This book offers a mathematical update of the state of the art of the research in the field of mathematical and numerical models of the circulatory system. It is structured into different chapters, written by outstanding experts in the field. Many fundamental issues are considered, such as: the mathematical representation of vascular geometries extracted from medical images, modelling blood rheology and the complex multilayer structure of the vascular tissue, and its possible pathologies, the mechanical and chemical interaction between blood and vascular walls, and the different scales coupling local and systemic dynamics. All of these topics introduce challenging mathematical and numerical problems, demanding for advanced analysis and efficient simulation techniques, and pay constant attention to applications of relevant clinical interest. This book is addressed to graduate students and researchers in the field of bioengineering, applied mathematics and medicine, wishing to engage themselves in the fascinating task of modeling the cardiovascular system or, more broadly, physiological flows.

Of the many different approaches to solving partial differential equations numerically, this book studies difference methods. Written for the beginning graduate student in applied mathematics and engineering, this text offers a means of coming out of a course with a large number of methods that provide both theoretical knowledge and numerical experience. The reader will learn that numerical experimentation is a part of the subject of numerical solution of partial differential equations, and will be shown some uses and taught some techniques of numerical experimentation.
Prerequisites suggested for using this book in a course might include at least one semester of partial differential equations and some programming capability. The author stresses the use of technology throughout the text, allowing the student to utilize it as much as possible. The use of graphics for both illustration and analysis is emphasized, and algebraic manipulators are used when convenient. This is the second volume of a two-part book.

In this book we analyze the error caused by numerical schemes for the approximation of semilinear stochastic evolution equations (SEEq) in a Hilbert space-valued setting. The numerical schemes considered combine Galerkin finite element methods with Euler-type temporal approximations. Starting from a precise analysis of the spatio-temporal regularity of the mild solution to the SEEq, we derive and prove optimal error estimates of the strong error of convergence in the first part of the book. The second part deals with a new approach to the so-called weak error of convergence, which measures the distance between the law of the numerical solution and the law of the exact solution. This approach is based on Bismut's integration by parts formula and the Malliavin calculus for infinite dimensional stochastic processes. These techniques are developed and explained in a separate chapter, before the weak convergence is proven for linear SEEq.

The focus from most Virtual Reality (VR) systems lies mainly on the visual immersion of the user. But the emphasis only on the visual perception is insufficient for some applications as the user is limited in his interactions within the VR. Therefore the textbook presents the principles and theoretical background to develop a VR system that is able to create a link between physical simulations and haptic rendering which requires update rates of 1\, kHz for the force feedback. Special attention is given to the modeling and computation of contact forces in a two-finger grasp of textiles. Addressing further the perception of small scale surface properties like roughness, novel algorithms are presented that are not only able to consider the highly dynamic behaviour of textiles but also capable of computing the small forces needed for the tactile rendering at the contact point. Final analysis of the entire VR system is being made showing the problems and the solutions found in the work

'Et moi, ..., si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point aile.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded n- sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Semi-infinite optimization is a vivid field of active research. Recently semi infinite optimization in a general form has attracted a lot of attention, not only because of its surprising structural aspects, but also due to the large number of applications which can be formulated as general semi-infinite programs. The aim of this book is to highlight structural aspects of general semi-infinite programming, to formulate optimality conditions which take this structure into account, and to give a conceptually new solution method. In fact, under certain assumptions general semi-infinite programs can be solved efficiently when their bi-Ievel structure is exploited appropriately. After a brief introduction with some historical background in Chapter 1 we be gin our presentation by a motivation for the appearance of standard and general semi-infinite optimization problems in applications. Chapter 2 lists a number of problems from engineering and economics which give rise to semi-infinite models, including (reverse) Chebyshev approximation, minimax problems, ro bust optimization, design centering, defect minimization problems for operator equations, and disjunctive programming."

Elasticity theory is a classical discipline. The mathematical theory of elasticity in mechanics, especially the linearized theory, is quite mature, and is one of the foundations of several engineering sciences. In the last twenty years, there has been significant progress in several areas closely related to this classical field, this applies in particular to the following two areas. First, progress has been made in numerical methods, especially the development of the finite element method. The finite element method, which was independently created and developed in different ways by sci entists both in China and in the West, is a kind of systematic and modern numerical method for solving partial differential equations, especially el liptic equations. Experience has shown that the finite element method is efficient enough to solve problems in an extremely wide range of applica tions of elastic mechanics. In particular, the finite element method is very suitable for highly complicated problems. One of the authors (Feng) of this book had the good fortune to participate in the work of creating and establishing the theoretical basis of the finite element method. He thought in the early sixties that the method could be used to solve computational problems of solid mechanics by computers. Later practice justified and still continues to justify this point of view. The authors believe that it is now time to include the finite element method as an important part of the content of a textbook of modern elastic mechanics."

hereafter calledvolume the of In a volume study previous (H6non 1997, I), the restricted initiated. families in problem (We generating three body was recallthat families defined asthe limits offamilies of are periodic generating determinationof orbitsfor Themain wasfoundto lieinthe 4 problem p 0.) bifurcation wheretwo the betweenthebranches ata ormore orbit, junctions A solutionto this was familiesof orbits intersect. partial problem generating and sidesof theuseofinvariants: Manysimple symmetries passage. givenby In the evolution of the bifurcations can be solved in this way. particular, orbits be described almost nine natural families of can completely. periodic become i.e.when thenumber of asthe bifurcations morecomplex, However, fails. the bifurcation orbit themethod families increases, passingthrough of This volume describes another to the a approach problem, consisting in of bifurcation ofthe families the a analysis vicinity detailed, quantitative used in Vol. I. orbit. This moreworkthan the requires qualitativeapproach in at to deter it has the of least, However, advantage allowing us, principle branches Infact it morethanthat: minein allcaseshowthe are joined. gives almost all the first order we will see in asymptotic approxima that, cases, the families in the ofthe bifurcation can be derived. tion of neighbourhood found in with This a comparison numerically allows, particular, quantitative families. and The 11 dealswiththerelevant definitions Chapter generalequations. of describedin 12 16.The ofbifurcations 1 is Chaps. study type quantitative it is described in 17 23. 3 of 2 ismore Chaps. Type analysis type involved; its hadnot been at thetime of isevenmore completed complex; analysis yet writing.

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Everybody is current in a world surrounded by computer. Computers determine our professional activity and penetrate increasingly deeper into our everyday life. Therein we also need increasingly refined c- puter technology. Sometimes we think that the next generation of c- puter will satisfy all our dreams, giving us hope that most of our urgent problems will be solved very soon. However, the future comes and il- sions dissipate. This phenomenon occurs and vanishes sporadically, and, possibly, is a fundamental law of our life. Experience shows that indeed 'systematically remaining' problems are mainly of a complex tech- logical nature (the creation of new generation of especially perfect - croschemes, elements of memory, etc. ). But let us note that amongst these problems there are always ones solved by our purely intellectual efforts alone. Progress in this direction does not require the invention of any 'superchip' or other similar elements. It is important to note that the results obtained in this way very often turn out to be more significant than the 'fruits' of relevant technological progress. The hierarchical asymptotic analytical-numerical methods can be - garded as results of such 'purely intellectual efforts'. Their application allows us to simplify essentially computer calculational procedures and, consequently, to reduce the calculational time required. It is obvious that this circumstance is very attractive to any computer user.

Computer-aided-design (CAD) of semiconductor microtransducers is relatively new in contrast to their counterparts in the integrated circuit world. Integrated silicon microtransducers are realized using microfabrication techniques similar to those for standard integrated circuits (ICs). Unlike IC devices, however, microtransducers must interact with their environment, so their numerical simulation is considerably more complex. While the design of ICs aims at suppressing "parasitic effects, microtransducers thrive on optimizing the one or the other such effect. The challenging quest for physical models and simulation tools enabling microtransducer CAD is the topic of this book. It is intended as a text for graduate students in Electrical Engineering and Physics and as a reference for CAD engineers in the microsystems industry. This text evolved from a series of courses offered to graduate students from Electrical Engineering and Physics. Much of the material in the book can be presented in about 40 hours of lecture time. The book starts with an illustrative example which highlights the goals and benefits of microtransducer CAD. This follows with a summary of model equations describing electrical transport in semiconductor devices and microtransducers in the absence of external fields. Models treating the effects of the external radiant, magnetic, thermal, and mechanical fields on electrical transport are then systematically introduced. To enable a smooth transition into modeling of mechanical systems, an abridged version of solid structural and fluid mechanics is presented, whereby the focus is on pertinent model equations and boundary conditions. This follows with model equations and boundary conditions relevant to various types of mechanical microactuators including electrostatic, thermal, magnetic, piezoelectric, and electroacoustic. The book concludes with a glimpse into SPICE simulation of the mixed-signal microsystem, i.e., microtransducer plus circuitry. Where possible, the model equations are supplemented with tables and/or graphs of process-dependent material data to enable the CAD engineer to carry out simulations even when reliable material models are not available. IVZ LANG: Introduction: Modeling and Simulation of Microtransducers; Illustrative Example; Progress in Microtransducer Modeling; References.- Basic Electronic Transport: Poisson s Equation; Continuity Equations; Carrier Transport in Crystalline Materials and Isothermal Behavior; Electrical Conductivity and Isothermal Behavior in Polycrystalline Materials; Electrical Conductivity and Isothermal Behavior in Metals; Boundary and Interface Conditions; The External Fields What Do They Influence?; References.- Radiation Effects on Carrier Transport: Reflection and Transmission of Optical Signals; Modeling Optical Absorption in Intrinsic Semiconductors; Absorption in Heavily-Doped Semiconductors; Optical Generation Rate and Quantum Efficiency; Low Energy Interactions with Insulators and Metals; High Energy Interactions and Monte Carlo Simulations; Model Equations for Radiant Sensor Simulation; Illustrative Simulation Example Color Sensor; References.- Magnetic-Field Effects on Carrier Transport: Galvanomagnetic Transport Equation; Galvanomagnetic Transport Coefficients; Equations and Boundary Conditions for Magnetic Sensor Simulation; Illustrative Simulation Example Micromachined Magnetic Vector Probe; References.- Thermal Non-Uniformity Effects on Carrier Transport: Non-Isothermal Effects; Electrothermal Transport Model; Electrical and Thermal Transport Coefficients; Electro-Thermo-Magnetic Interactions; Heat Transfer in Thermal Microstructures; Summary of Equations and Computational Procedure; Illustrative Simulation Example Micro Pirani Gauge; References.- Mechanical Effects on Carrier Transport: Piezoresistive Effect; Strain and Electron Transport; Strain and Hole Transport; Piezojunction Effect; Effects of Stress Gradients; Galvano-Piezo-Magnetic Effects; The Piezo Drift-Diffusion Transport Model; Illustrative Simulation Example Stress Effects on Hall Sensors; References.- Mechanical and Fluidic Signals: Definitions; Model Equations for Mechanical Analysis; Model Equations for Analysis of Fluid Transport; Illustrative Simulation Example Analysis of Flow Channels; References.- Micro-Actuation: Transduction Principles; State-of-the-Art and Preview; Electrostatic Actuation; Thermal Actuation; Magnetic Actuation; Piezoelectric Actuation; Electroacoustic Transducers; Computational Procedure and Coupling; Illustrative Example CMOS Micromirror.- Microsystem Simulation: Electrical Analogues for Mixed-Signals and Historical Developments; Circuit Modeling and Implementation Considerations; Lumped Analysis: Illustrative Example Electrostatic Micromirror; Distributed Analysis: Illustrative Example Flow Microsensor; References.- Subject Index."