Applied Mathematical Sciences Research Articles

Hellenic Studies in History, Philosophy of Science and Science Teaching: New Perspectives

This volume contains selected papers from the Third Hellenic Conference on ‘‘History, Philosophy of Science and Science Teaching’’ which was held in the central Amphitheatre of the National Hellenic Research Foundation September 19–22, 2005. The Conference was endorsed by the Hellenic Physical Society and was organized jointly by the Department of Education, University of Athens, and by the Program ‘‘History and Philosophy of Science’’ of the National Hellenic Research Foundation. International speakers were Fabio Bevilacqua and Michael Matthews, while invited speakers from Greek universities were: Stathis Psillos (Philosophy of Science, University of Athens), Efthymios Papademitriou (Philosophy of Nature, University of Ioannina), Menelaos Gkivalos (Epistemology, University of Athens) and Theodoros Christides (History and Philosophy of Science, University of Thessaly). The year 2005 was celebrated worldwide as the centennial anniversary of the ‘‘annus mirabilis’’ 1905, the year of publication of the Principle of Relativity by Albert Einstein. The year 2005 was celebrated worldwide as the world year of Physics and hundreds of ceremonials were organized to commemorate the event. In the framework of the 3rd Hellenic Conference two events were organized. First, a round-table discussion ‘‘Physicists discuss Einstein’’ with four prominent Physics teachers of secondary education, namely: Andreas Cassetas, Andreas Valadakis, Napoleon Papadopoulos and Panayiotis Skountzos. Second, a presentation of the CERN Educational Program ‘‘Master Classes in Particle Physics’’ by Prof. Nicholas Trakas of the School of Applied Mathematical and Physical Sciences of the National Technical University of Athens. The CERN Educational Program is a European Initiative for popularizing Physics and aims to bring Physics closer to the general public and especially to young students in secondary schools in order to encourage them to acquire a positive attitude towards science. More than 120 participants attended the Conference and 45 papers were presented in two parallel sessions. Six papers were given by invited speakers while the rest 39 papers can be classified as follows: On the contribution of History of Science to Science Teaching (8 papers), on the Contribution of Philosophy of Science to the Educational Theory and

Enabling technologies for petascale electromagnetic accelerator simulation

The SciDAC2 accelerator project at SLAC aims to simulate an entire three-cryomodule radio frequency (RF) unit of the International Linear Collider (ILC) main Linac. Petascale computing resources supported by advances in Applied Mathematics (AM) and Computer Science (CS) and INCITE Program are essential to enable such very large-scale electromagnetic accelerator simulations required by the ILC Global Design Effort. This poster presents the recent advances and achievements in the areas of CS/AM through collaborations.

The Applied Mathematics and Computer Science Schism

The Applied Mathematics and Computer Science Schism

The Applied Mathematics and Computer Science Schism

Isolating applied mathematics from computer science harms the profession and those who depend on it.

Bifurcation Theory: An Introduction With Applications to PBEs (Applied Mathematical Sciences 156 Series)

9R3. Bifurcation Theory: An Introduction With Applications to PBEs (Applied Mathematical Sciences 156 Series). - Edited by H Kielhofer (Inst for Math, Univ of Augsburg, Universitatsstr 14, Raum 2011, Augsburg D-86135 Germany). Springer-Verlag, New York. 2004. 346 pp. ISBN 0-387-40401-5. $69.95.Reviewed by HW Haslach Jr (Dept of Mech Eng, Univ. of Maryland, College Park MD 20742-3035).This unified exposition of the single-parameter bifurcation response of an operator on infinite dimensional space is organized into three sections: an abstract development of local and then global bifurcation and applications to elliptic and parabolic partial differential equations. The local bifurcation theory, taking up about half the book, depends on the breakdown of the implicit function theorem and is based on the Lyapunov-Schmidt reduction for infinite dimensional spaces. The case that the Fre´chet derivative has a one-dimensional kernel includes the saddle-node and the various types of pitchfork bifurcations. A principle of exchange of stability is proved. A Hopf bifurcation theorem for periodic solutions is first proved for ordinary differential equations and then for retarded functional differential equations. A center manifold theorem is used for Hopf bifurcation of Hamiltonian, reversible, and conservative systems. A principle of exchange of stability for Hopf bifurcations is proved, and the stability of global continuation of solutions is examined. A principle of exchange of stability for period-doubling bifurcations is developed. The Newton polygon method is described for problems with a one-dimensional kernel in both the degenerate and nondegenerate cases. Floquet exponents are used to create a principle of exchange of stability for degenerate Hopf bifurcations. Some results are given for higher dimensional kernels of the Fre´chet derivative and multiparameter bifurcation. The global bifurcation theory, subject of the second section, is based on the local index of the operator F, obtained from the Leray-Schauder degree which takes the place of the Brouwer degree of finite-dimensional problems. The crossing number, which is related to a change in Morse index, is defined from the index of the operator. Then if DxF0,λ has an odd crossing number for a particular value of the bifurcation parameter λ, that value of the parameter gives a bifurcation point. A degree is defined for a class of Fredholm operators that includes some classes of elliptic operators and for potential operators, and global bifurcation theorems proved, often using Rabinowitz’s theorem. The applications forming the third section begin with results on elliptic operators of second order acting on scalar functions. An example using the degenerate Newton polygon is developed. The particular case of the Cahn-Hilliard energy that describes a binary alloy is solved by taking the mass as the bifurcation parameter and applying a symmetry constraint to obtain a one-dimensional kernel. Then local bifurcations of the free nonlinear vibrations described by the nonlinear wave equation with the period as bifurcation parameter are examined in the one-dimensional case and some extensions are given to higher dimensions. Results are also obtained for the wave equation on the unit sphere. A Hopf bifurcation is analyzed for a nonlinear parabolic problem in which the principle of exchange of stability holds. Finally global bifurcation and continuation results are given for quasi-linear elliptic problems and are applied to the Euler-Lagrange equation for the Cahn-Hilliard energy problem. This mathematics book requires substantial mathematical background. The book is directed to advanced readers rather than beginners so that some basic ideas are not defined. The author has separated the abstract description in the first two sections from the applications with the goal of clarity and allowing the results to be available for any application, but at the risk of making it harder to understand the significance of some of the ideas. In many applications to models of physical behavior, the hypotheses of the various theorems are quite difficult to verify. The book is very useful as a reference because it collects and organizes the bifurcation analysis of infinite-dimensional operators. It could also be used as a text in an advanced course on bifurcation theory with an emphasis on partial differential equations.

QUALITY OF SERVICE IN NETWORKS WITH PRIORITISED TRAFFIC INVOLVING THE MULTI-PRIORITY DUAL QUEUE

International Journal of Computational Engineering ScienceVol. 05, No. 02, pp. 303-313 (2004) No AccessQUALITY OF SERVICE IN NETWORKS WITH PRIORITISED TRAFFIC INVOLVING THE MULTI-PRIORITY DUAL QUEUEA. BEDFORD and P. ZEEPHONGSEKULA. BEDFORDDepartment of Mathematics and Statistics, Bundoora East Campus, RMIT University, Plenty Road, Bundoora, Vic., Australia, 3083, AustraliaCorresponding author. Search for more papers by this author and P. ZEEPHONGSEKULDepartment of Mathematics and Statistics, City Campus, RMIT University, GPO Box 2476V, Melbourne, Vic., Australia, 3001, Australia Search for more papers by this author https://doi.org/10.1142/S1465876304002393Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractThe Multi Priority Dual Queue (MPDQ) is a new scheduling regime shown to reduce congestion for multi-class traffic over conventional scheduling disciplines in isolation. As the MPDQ improves traffic congestion, most notably in communication systems, it is not necessary to include it at every node, or service centre, within a network. To gain insight into the advantages of a mixed MPDQ / First In First Out network with prioritised traffic, simulations are performed with and without a dual queue and various Quality of Service (QoS) criteria are obtained. Queueing networks containing differentiated traffic, also known as multi-class networks, are complicated to solve analytically using existing queueing theory techniques. We discuss the complications of exact solutions and describe how we used simulation to obtain performance statistics. We also look at high-class traffic under different offered loads, and provide a comparison of the delay characteristics. These findings provide communication service providers valuable information in determining the improvement in QoS to differentiated networks with the MPDQ. They also highlight the importance of simulation as a tool of evaluation of networks, and the best MPDQ network scenario.Keywords:Quality of Service (QoS)QueueingNetworksMulti Priority Dual QueueDelaySimulationAMSC: 60K20, 90B18, 65C60, 68M20 References M. Ogawa, T. Sueoka and T. Hattoti, Priority Based Wireless Packet Communication with Admission and Throughput Control, Proc. of the 51st IEEE Conference on Vehicular Technology, pp. 370-374 (2000) . Google Scholar D. A. Hayes, M. Rumsewicz, and L. L. H. Andrew, Quality of service driven packet scheduling disciplines for real-time applications: looking beyond fairness, Proc. IEEE Infocom'99, 1, pp. 405-412 (1999) . Google Scholar J. Wang, Y. Levy, "Managing Performance Using Weighted Round Robin", 5th IEEE Symposium on Computers & Communications, pp. 785-792 (2000) . Google Scholar Bedford and P. Zeephongsekul, Topics in Applied and Theoretical Mathematics and Computer Science, eds. V. V. Kluev and N. E. Mastorakis (WSEAS Press, 2001) pp. 220–225. Google Scholar Bedford and P. Zeephongsekul, Topics in Applied and Theoretical Mathematics and Computer Science, eds. V. V. Kluev and N. E. Mastorakis (WSEAS Press, 2001) pp. 226–231. Google Scholar R. S. Ranasinghe, L. L. H. Andrew, D. A. Hayes, D. Everitt, Scheduling disciplines for multimedia WLANs: embedded round robin and wireless dual queue, Proc. IEEE Int. Conf. Commun., pp. 1243-1248 (2001) . Google Scholar Bedford and P. Zeephongsekul, "Simulation Studies of a Multi-Priority Dual Queue (MPDQ) with Preemptive and Non-preemptive Scheduling", Lecture Notes in Computer Science, Vol. 2659, Springer-Verlag, 2003, pp. 179-189 . Google Scholar Bedford and P. Zeephongsekul, "On A Dual Queueing System with Preemptive Priority Service Discipline", European Journal of Operational Research, Vol 161, Issue 1, pp. 224-239 (2005) . Google Scholar M. Neuts , Matrix-Geometric Solutions in Stochastic Matrices, An Algorithmic Approach ( John Hopkins University Press , 1981 ) . Google Scholar Odlyzko, "Paris Metro Pricing On The Internet", Proc. ACM Conference on Electronic Commerce, ACM, 1999, 140-147 . Google Scholar P. J. Burke, "The Output of a Queueing System", Operations research, Vol 4, 699-704, 1966 . Google Scholar W. D. Kelton , R. P. Sadowski and D. A. Sadowski , Simulation with Arena , 2nd edn. ( McGraw-Hill , 2002 ) . Google Scholar FiguresReferencesRelatedDetails Recommended Vol. 05, No. 02 Metrics History KeywordsQuality of Service (QoS)QueueingNetworksMulti Priority Dual QueueDelaySimulationPDF download

A Conversation with James O. Berger

James O. Berger was born on April 6, 1950, in Minneapolis, Minnesota, to Orvis and Thelma Berger. He earned his AB, MA and Ph.D. degrees in mathematics from Cornell University in 1971, 1973 and 1974, respectively. He served at Purdue University 1974–1997, as the Richard M. Brumfield Distinguished Professor of Statistics 1986–1997, and at Duke University as the Arts and Sciences Professor of Statistics in the Institute of Statistics and Decision Sciences (ISDS) from 1997 to the present. He is also the Director of the Statistical and Applied Mathematical Sciences Institute (SAMSI), one of the U.S. National Science Foundation’s national institutes in the mathematical and statistical sciences. Berger served as President of the Institute of Mathematical Statistics (IMS) 1995–1996 and Chair of the Section on Bayesian Statistical Science (SBSS) of the American Statistical Association (ASA) in 1995; he is President of the International Society for Bayesian Analysis (ISBA) in 2004. He has served as Co-Editor of The Annals of Statistics and on the editorial boards of the Springer Series in Statistics, of the Journal of Statistical Planning and Inference, of Statistics and Decisions, of the International Statistical Review and of Test. He has organized or been on the organizing committees of twenty-eight conferences, including five of the Purdue Symposia on Statistical Decision Theory and Related Topics and four of the Valencia International Meetings on Bayesian Statistics. Berger is the recipient of a host of honors. He is an elected fellow of the AAAS, the ASA and the IMS and an Elected Member of the International Statistical Institute (ISI). He has earned Guggenheim and Sloan Fellowships and was the 1985 winner of the COPSS Presidents’ Award ( joint from IMS, ASA, ENAR, WNAR and CSS) given to an outstanding statistician below forty years of age (it is particularly noteworthy that he won that award at the tender age of 35), and was selected as the COPSS Fisher Lecturer in 2001. He was elected as a foreign member of the Spanish Real Academia de Ciencias in 2002, and to membership in the U.S. National Academy of Sciences in 2003. Berger has been the advisor of 30 Ph.D. students, has written or edited thirteen books and has numerous other statistical publications. A listing of these publications can be found at his Web site, http://www.stat.duke.edu/~berger/. Berger married Ann Louise Duer (whom he first met when they were in the seventh grade together) in 1970, and they have two children, Jill Berger, who is married to Sascha Hallstein and works as an optical scientist in Silicon Valley, and Julie Gish, who is married to Ryan Gish and works as a consultant in Chicago.

On the Structure of Solutions to the Periodic Hunter--Saxton Equation

We prove the local existence of strong solutions of the periodic Hunter--Saxton equation, and we show that all strong solutions except space-independent solutions blow up in finite time.

A coupled level set projection method applied to ink jet simulation

A finite difference level set-projection method on rectangular grid is developed for piezoelectric ink jet simulation. The model is based on the Navier-Stokes equations for incompressible two-phase flows in the presence of surface tension and density jump across the interface separating ink and air, coupled to an electric circuit model which describes the driving mechanism behind the process, and a macroscopic contact model which describes the air-ink-wall dynamics. We simulate the axisymmetric flow using a combination of second order projection methods to solve the fluid equations and level set methods to track the air/ink interface. The numerical method can be used to analyze the motion of the interface, breakoff and formation of satellites, and effect of nozzle geometry on droplet size and motion. We focus on close comparison of our numerical ink jet simulation with experimental data. The corresponding author. This author was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract Number DE-AC03-76SF00098, and the Division of Mathematical Sciences of the National Science Foundation.

Random Perturbation Methods with Applications in Science and Engineering. Applied Mathematical Sciences, Vol 150

7R3. Random Perturbation Methods with Applications in Science and Engineering. Applied Mathematical Sciences, Vol 150. - AV Skorokhod (Inst of Math, Ukrainian Acad of Sci, 3 Tereshchenkivska St, Kiev, 01601, Ukraine), FC Hoppensteadt (Syst Sci and Eng Res Center, Arizona State Univ, Tempe AZ 85287-7606), H Salehi (Dept of Stat and Probab, Michigan State Univ, E Lansing MI 48824). Springer-Verlag, New York. 2002. 488 pp. ISBN 0-387-95427-9. $79.95.Reviewed by VD Radulescu (Dept of Math, Univ of Craiova, 13, St AI Cuza, Craiova, 1100, Romania).The authors attempt to describe the basic methods and general principles of the theory of random perturbation, as well as some of the main applications of this theory in Mechanics, Engineering, Genetics, and Population Biology. This monograph contains a well-written rigorous mathematical treatment of this subject and deals with modern concepts such as dynamical systems, stochastic processes, stability, Markov chains, population biology, etc. The volume consists of 12 chapters, but the structure of the book is roughly in two parts. The first seven chapters develop several mathematical methods that are useful in the study of random perturbations of dynamical systems. The second part (Chs 8–12) presents nonrandom problems in a variety of important applications. The purpose of Chapter 1 is to describe some basic ergodic theorems (including Birkhoff’s Classical Ergodic Theorem) from a mathematical point of view and to recall some results that will be used in the book. There are also described discrete-time and continuous-time Markov processes, and continuous-time stationary processes. Chapter 2 is devoted to some convergence properties of stochastic processes. Technical conditions that are sufficient for various kinds of convergence are described. The results contained in this chapter are developed mostly in the framework of continuous-time processes. In Chapter 3 the authors develop averaging methods for random perturbations of Volterra integral equations, of differential equations and of difference equations. In each of these cases, the remainder term is considered in greater detail in Chapter 4. The main results in this part of the book establish that the deviation tends to zero in a prescribed sense if an averaging theorem is true. Chapter 5 deals with randomly perturbed systems of differential and difference equations whose averaged systems are static. The main result contained in this chapter establishes that, under some reasonable assumptions, the stochastic process becomes, asymptotically, a diffusion process. Chapter 6 gives an outlook to a variety of stability problems for differential and difference equations when they are perturbed by random noise. The authors also describe the growth of solutions to certain randomly perturbed convolution equations. The abstract results are illustrated by several examples of stability phenomena. Chapter 7 deals with Markov chains having a finite state space, but random transition probabilities. The basic hypothesis is that the transition probabilities of the Markov chain are close to those of a homogeneous Markov chain having nonrandom transition probabilities. Under this assumption, the authors establish the asymptotic behavior of the chain and its transition probabilities. In Chapter 8 it is described a method for studying various kinds of bifurcations for two-dimensional mechanical systems. There are also discussed random perturbations of oscillatory linear systems and of rigid-body motions. In Chapter 9 the authors consider a dynamical system on a two-dimensional torus. The main results involve either flows that have non random elements or random perturbations of problems of this type. In Chapter 10 an important electronic circuit, namely the phase-locked loop is described. It first investigates the circuit’s dynamics without and with random perturbations, but in the absence of external forcing. Next, the response of the circuit to noisy external signals is analyzed. Interesting applications are also given in Chapter 11, in which the authors consider random perturbation of ecological systems, of epidemic disease processes, and of demographic models. Chapter 12 provides several concrete problems from genetics. In both chapters, the authors describe classical models for these phenomena and related results concerning their asymptotic behavior for large time. Appendix A recalls some basic notions of probability theory, and Appendix B contains historical comments and further remarks. In summary, this reviewer would definitely recommend Random Perturbation Methods with Applications in Science and Engineering to those researchers and graduate students in Science and Engineering who require tools to investigate stochastic systems. The book qualifies to be a reference work that certainly would be a valuable addition in libraries of universities and laboratories pursuing research in Applied Mathematics.

Appendix 3: Report of study group on computational physics

The great improvement of algorithms and computing hardware in the last few years must be ranked as one of the most important turning points in the history of multiphase flow research. After a brief review of some of this recent progress, it is pointed out that, besides its application to solving actual problems, computational physics plays other key roles: (1) As a tool to develop and understand basic physics and as a guide toward asking more penetrating questions; (2) As an aid in closing the averaged equations; (3) As a means to learn to compute better. Roadblocks toward greater effectiveness are the huge complexity of many of the necessary computational tasks but also, at a more practical level, the transmission of “computational knowledge” from one researcher to another, much in the same way as experimentalists can rely on readily available equipment (e.g., lasers, etc.), without having to build each item themselves. The solution to this problem will require a cultural shift––from a “cottage industry” to a “big science” mentality––which can be aided by a different attitude on the part of the funding agencies. Great synergism can be achieved by a closer integration of the multiphase computational physics enterprise with both Applied Mathematics and Computer Science.

A stability estimate of an inverse problem in financial prospection

In many option pricing models, there is a need to derive a stability estimate of their solutions. This paper is concerned with the stability estimate of the regularized solution arising in the inverse problem of option pricing. This kind of inverse problems, where one looks for causes for observed or desired effects, are usually “ill-posed”, i.e., their solution is not unique or unstable with respect to data perturbations. Stability estimate is needed for stable solution of these ill-posed problems. If one cannot guarantee the stability of a solution to these problems, then the problem does not make sense and there is no hope of handling them numerically. It is shown here that one can achieve stability by using so called stabilizing functional suggested by Tikhonov [Linear Integral Equation, Applied Mathematical Sciences, Springer, New york, 1989].

Imperfect Bifurcation in Structures and Materials: Engineering Use of Group-Theoretic Bifurcation Theory. Applied Mathematical Sciences, Vol 149

5R19. Imperfect Bifurcation in Structures and Materials: Engineering Use of Group-Theoretic Bifurcation Theory. Applied Mathematical Sciences, Vol 149. - K Ikeda (Dept of Civil Eng, Tohoku Univ, Aoba Sendai, 980-8579, Japan) and K Murota (Dept of Math Informatics, Univ of Tokyo, Tokyo, 113-0033, Japan). Springer-Verlag, New York. 2002. 411 pp. ISBN 0-387-95409-0. $69.95.Reviewed by J Petrolito (Sch of Sci and Eng, La Trobe Univ, PO Box 199, Bendigo, Vic 3550, Australia).Stability theory is of fundamental importance in structural engineering, and there is a large body of literature in the field. Much of the theory is directed toward the prediction of buckling or bifurcation loads for ideal structures. However, the role of imperfections is crucial for real structures, particularly for those that are sensitive to these effects. The current book is a graduate-level text that presents an overview of imperfections and the prediction of the initial post-buckling response of a system. The treatment is predominately analytical, rather than numerical. Hence, the problems treated are relatively simple since the analysis of most practical systems requires numerical techniques such as the finite element method. The book is divided into three parts and 15 chapters. Each chapter includes a range of problems and a summary to consolidate the material. The theory is complemented by nearly 200 references from the field. The first chapter provides an overview of the problems that are considered in the book. This chapter, although brief, could be used as an upper-undergraduate level introduction to the field. The first part of the book introduces the basic theory and derives the imperfection sensitivity laws for simple critical points. The theory is also linked to the classification system from chaos theory. The discussion includes procedures to identify critical imperfections and the role of probability theory for modeling random imperfections. Part 1 concludes with a chapter that links the theory to the behavior of real structures and materials. Two restrictions of the theory should be noted. Firstly, the loading is assumed to be described by one parameter only. Hence, the theory is not applicable to non-proportional loading, which occurs frequently in practice. Secondly, only the initial post-buckling response is considered. Other techniques need to be used if the complete load-deflection response of the system is required. Part 2 extends the theory to the analysis of systems with multiple critical points. This is inherently more difficult and the mathematical demands on the reader increase considerably. In particular, extensive use is made of group theory. Although the basics of this theory are covered, readers will probably need to supplement this with material from standard texts in the field to provide an adequate background to follow this part. Part 3 is primarily focused on modeling bifurcation in materials, including both metals and soils. This part of the books also relies on group theory. In summary, Imperfect Bifurcation in Structures and Materials provides an extensive range of material on the role of imperfections in stability theory. It would be suitable for a graduate-level course on the subject or as a reference to research workers in the field.

Shape Optimization by the Homogenization Method. Applied Mathematical Sciences, Vol 146

3R25. Shape Optimization by the Homogenization Method. Applied Mathematical Sciences, Vol 146. - G Allaire (Center of Appl Math, Ecole Polytechnic, Paliseau Cedex, 91128, France). Springer-Verlag, New York. 2002. 456 pp. ISBN 0-387-95298-5. $79.95.Reviewed by MS Qatu (Ford Motor Co, EVB Bldg, MD X8, 20800 Oakwood Blvd, Dearborn MI 48124).This is mainly a research monograph on the theory and application of the homogenization method in shape optimization problems. Applications given in the book are in the fields of composite materials, conductivity, and elasticity with emphasis on the minimization of compliance. The audience for this book includes researchers and advanced graduate students in applied mathematics and engineers interested in the design of shapes of structures for conductive and elastic purposes. It also fits very well for material scientists with inclination towards theoretical aspects of design and analysis of composite materials. Based on the theoretical content and examples in the book, the word shape in the title should rather be understood as topology. In some engineering fields, shape is more related to boundary perturbation, however, the methods presented in this book are designed to define the boundary, rather than to perturb it. This concept of defining the boundary is closer to the idea of topology optimization as a material distribution problem. The book is divided into five chapters. The first chapter covers the subject of homogenization in quite details. Periodicity is first assumed and used to derive homogenized operators in elasticity by means of the two-scale asymptotic expansion. Definitions and proofs of H- and G-convergence are then presented in a rigorous way. The second chapter is titled mathematical modeling of composite materials. In this chapter, Allaire explains carefully the G-closure problem and its interpretations in the mechanical setting. Finding bounds on elastic properties of bi-phase materials is an old problem in mechanics that can be explained and treated mathematically with the help of the G-closure theory. Rank laminated materials are used to demonstrate the bounds derived in the book. Optimal design of materials under electric conductivity considerations is addressed in the third chapter. With an extensive number of proofs, this chapter explains the theory behind the computation of optimal topology (shape) using homogenization theory as a mean to relax the optimization problem. Composite materials are used to relax the problem, and homogenization of conductivity tensors is then used to go from an ill-posed problem to a well-posed one. Structural optimization has been an important subject of research since the days of Lagrange and Saint Venant. This chapter is the counterpart of the previous one, this time, in the field of elastic structures. Composite materials are again introduced to relax the optimization problem and homogenized elastic properties are computed in order to attain a solution to the problem. The minimization of compliance (maximization of stiffness) is addressed intensively in this chapter. The fifth and last chapter is titled Numerical Algorithms. Academic two-dimensional rectangular domains using quadrilateral elements in regular meshes are used to exemplify the theory presented in the previous chapters. Examples include minimization of compliance and maximization of eigenvalues. Multiple loadings and optimum topologies (shapes) of complaint mechanisms were considered. In summary, Shape Optimization by the Homogenization Method is an excellent book that complements very well the extensive literature in the field, including Sanchez-Palencia and Zaoui [1], Bendsøe [2], Cherkaev [3], and Bendsøe and Sigmund [4].

Spectral Methods for Incompressible Viscous Flow. Applied Mathematical Sciences, Vol 148

1R40. Spectral Methods for Incompressible Viscous Flow. Applied Mathematical Sciences, Vol 148. - Edited by R Peyret (Lab JA Dieudonne, Univ de Nice-Sophia Antipolis, Parc Valrose, Nice, 06000, France). Springer-Verlag, New York. 2002. 432 pp. ISBN 0-387-95221-7. $59.95.Reviewed by G de Vahl Davis (Sch of Mech and Manuf Eng, Univ of New S Wales, Sydney, 2052 NSW, Australia).The foundations of spectral methods are not new. They lie in the use of series expansions, typically a Fourier series, to attack problems in mathematical physics, and especially fluid mechanics, and have so done since the 19th century. However, computational limitations in the BC (before computer) era—and even in the early AC years—limited their application. It was not until the 1970s, as Peyret points out, that a revival of the Fourier method was applied to the direct simulation of turbulence. Historically, spectral methods have formed the cornerstone of numerical methods used to study the physics of turbulence. Their resurgence can be attributed to the continually increasing power of computers and to the development of the fast Fourier transform. Three principal methods are available for the solution of a system of coupled nonlinear partial differential equations such as those which describe fluid motion: finite difference (FD), finite element (FE), and spectral. FD and FE methods are based on a subdivision of the solution region into a number of small elements or control volumes, within which local representations of functions, usually by low-order polynomials, are made. Spectral methods, on the other hand, use global representations—now, typically, Fourier series and Chebyshev polynomials—which cover the entire computational domain. FDM and FEM generally require fine grids and, therefore, a greater computational effort than spectral methods to achieve a given accuracy. To increase accuracy, both FDM and FEM use global or local grid refinement. Spectral methods (as well as some FEM) achieve increased accuracy by using higher order polynomials. FDM and FEM lead to matrix equations which are sparse because of the local nature of the basis functions. Spectral methods lead to algebraic equations with full matrices. FDM and FEM have little difficulty in handling conditions at the boundaries of the solution region. Spectral methods, in contrast, often suffer from stability problems which demand much smaller time steps to overcome, especially in two and three dimensions. They are most useful when the geometry of the solution region is fairly smooth and regular. Spectral methods are usually more difficult to program, certainly in comparison with FDM. They are more costly, per degree of freedom, and they accommodate irregular geometries less happily. They thus enjoy some advantages, and suffer from some disadvantages, in comparison with FD and FE methods. Trefethen has stated that “difficulties with boundaries… are probably the primary reason why spectral methods have not replaced their lower-accuracy competition in most applications.” In summary, spectral methods constitute a class of highly accurate numerical techniques generally only suitable for simple geometries. In this book, the author sets out “to provide a comprehensive discussion of Fourier and Chebyshev spectral methods for the computation of incompressible viscous flows, based on the Navier-Stokes equations.” He has succeeded admirably. The book encompasses the necessary mathematical background, provides a clear exposition of the associated computational techniques, and gives information on the implementation of these techniques. This book, like Peyret’s Gaul, is divided into three parts. The first part (four chapters and almost 150 pages) discusses basic spectral methods: fundamentals, the Fourier method, the Chebyshev method, and time-dependent equations. Part II (three chapters and about 140 pages) covers the Navier-Stokes equations (in both velocity-pressure and vorticity-streamfunction formulations) and spectral methods for their solution in each formulation. Details such as the Boussinesq approximation and possible treatments of a semi-infinite domain are examined. Examples are given of the solution of Rayleigh-Be´nard convection, axisymmetric flow in a rotating annulus and three-dimensional flow in a rotating annulus. Part III (90 pages) has chapters on stiff and singular problems, and on domain decomposition (or spectral element methods). Short appendices cover formulas on Chebyshev polynomials, an algorithm for the solution of a quasi-tridiagonal system, and theorems on the zeros of a polynomial. There are about 300 references; Peyret is author or co-author of some 10% of these, unquestionably establishing his credentials on this subject. Spectral Methods for Incompressible Viscous Flow is an advanced text. It will appeal to applied mathematicians and CFD-oriented engineers at the post-graduate level and to anyone teaching or undertaking research on problems described by the Navier-Stokes equations. This book is highly recommended.

A stability estimate for convolution equation

In this study we obtain an estimate for the solution of a convolution equation. These types of equations arise in the integral equations theory [Linear Integral Equation, Applied Mathematical Sciences, Springer, Berlin, 1989].

Stability and Transition in Shear Flows. Applied Mathematical Sciences, Vol. 142

Stability and Transition in Shear Flows. Applied Mathematical Sciences, Vol. 142

Optimal Controls of 3-Dimensional Navier--Stokes Equations with State Constraints

This work is concerned with the maximum principles for optimal control problems governed by 3-dimensional Navier--Stokes equations. Some types of state constraints (time variables) are considered.

A Newton method for American option pricing

The variational inequality formulation provides a mechanism to determine both the option value and the early exercise curve implicitly [17]. Standard finite difference approximation typically leads to linear complementarity problems with tridiagonal coefficient matrices. The second order upwind finite difference formulation gives rise to finite dimensional linear complementarity problems with nontridiagonal matrices, whereas the upstream weighting finite difference approach with the van Leer flux limiter for the convection term [19, 22] yields nonlinear complementarity problems. We propose a Newton type interior-point method for solving discretized complementarity/variational inequality problems that arise in the American option valuation. We illustrate that the proposed method on average solves a discretized problem in 2 ∼ 5 iterations to an appropriate accuracy. More importantly, the average number of iterations required does not seem to depend on the number of discretization points in the spatial dimension; the average number of iterations actually decreases as the time discretization becomes finer. The arbitrage condition for the fair value of the American option requires that the delta hedge factor be continuous. We investigate continuity of the delta factor approximation using the complementarity approach, the binomial method, and the explicit payoff method. We illustrate that, while the (implicit finite difference) complementarity approach yields continuous delta hedge factors, both the binomial method and the explicit payoff method (with the implicit finite difference) yield discontinuous delta approximations. Hence the early exercise curve computed from the binomial method and the explicit payoff method can be inaccurate. In addition, it is demonstrated that the delta factor computed using the Crank-Nicolson method with complementarity approach oscillates around the early exercise curve. ∗ Computer Science Department and Cornell Theory Center, Cornell University, Ithaca, NY 14850. Research partially supported by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under grant DE-FG02-90ER25013.A000, NSF through grant DMS9805602 and ONR through grant N00014-96-1-0050, and the Finance Industry Solutions Center, an SGI/Cornell joint venture in New York.

Professor Ernest Czogala Memorial Issue Part 2

After the publication of Professor Ernest Czogala Memorial Issue Part 1, I found 2 obituaries, one in the International Journal of Applied Mathematics and Computer Science, Vo1.9 No.1, published by the Technical University Press, Poland, by Maria Drelichowska and Lucjan Karwan of the Technical University, Silesia, Gliwice, Poland. The other was published"Fuzzy Systems in Medicine" (P.S. Szczepaniak eds.), Physica-Verlag, published in January 2000, which included my paper,"Fuzzy and Crisp Logical Rule Extraction Methods in Application to Medical Data," which I wrote in collaboration with W. Duch et al. This issue includes the final paper,"A Classifier Based on Neurofuzzy Inference," by Ernest Czogala, Jacek Leski, and Yoichi Hayashi, which is why this edition has been published later than planned. I would like include a comment by Samuel Ullman, written in Birmingham, Alabama, where Ernest's cousin and researcher James J. Buckley works: "Youth is not a time of life; it is a state of mind... Nobody grows old merely by years. We grow old by deserting our ideals." The Samuel Ullman Museum at the University of Alabama is at 2150 15th Avenue South, Birmingham, Alabama 35294-1150 (phone:001(205)934-5634), thanks to citizens of Japan and the United States. We thank Drs. Toshio Fukuda and Kaoru Hirota, Editors in Chief of the JACI, for accepting my proposal for this special issue. Special thanks also go to the referees for their cooperation, devotion, and review. We also thank Fuji Technology Press President Keiji Hayashi for his dedicated work.