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2006

Volume 48, Issue 3, pp. 437-626

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Survey and Review

Fadil Santosa, Section Editor

SIAM Rev. 48, pp. 437-437 (1 page)

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Consider the fact that you can put your entire music collection on your portable mp3 player and you begin to realize what a tremendous amount of data is being stored in its tiny disk drive. How is it possible that data can be so efficiently stored in a magnetic storage device? The answer is ferromagnetism. Progress in storage technology follows Kryder's law, which is akin to the more familiar Moore's law, governing the performance of CPUs. The storage capacity of disk drives, measured by areal density, has increased exponentially in the past 20 or so years. At the time of this writing, areal density of 110 Gbits/square inch is common. A new storage format, called perpendicular format, is capable of areal density of 245 Gbits/square inch.
Ferromagnetism, the subject of the article by Martin Kružík and Andreas Prohl, is modeled by a continuum theory called micromagnetics. The mathematics of micromagnetics is complex and beautiful, exhibiting rich and surprising behavior. The article introduces the reader to the models of micromagnetics, their mathematical analyses, and simulation methods for them. The authors show that despite great progress in this field, much still needs to be understood. Perhaps the key to quantifying the theoretical limit of magnetic storage capacity lies in the mathematics of micromagnetics. Mathematics might also provide guidance on how one can achieve this limit in practical devices.

Recent Developments in the Modeling, Analysis, and Numerics of Ferromagnetism

Martin Kruzík and Andreas Prohl

SIAM Rev. 48, pp. 439-483 (45 pages) | Cited 21 times

Online Publication Date: August 03, 2006

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Micromagnetics is a continuum variational theory describing magnetization patterns in ferromagnetic media. Its multiscale nature due to different inherent spatiotemporal physical and geometric scales, together with nonlocal phenomena and a nonconvex side-constraint, leads to rich behavior and pattern formation. This variety of effects is also the reason for severe problems in analysis, model validation, reductions, and numerics, which have only recently been accessed and are reviewed in this work.
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Problems and Techniques

Ilse Ipsen, Section Editor

SIAM Rev. 48, pp. 485-485 (1 page)

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This issue's Problems and Techniques section contains, as usual, three papers on very different subjects: integration of singular functions by quasi-Monte Carlo methods; functions on the unit sphere with limited frequency range and maximal spatial concentration; and an approximation for a “cardiac restitution curve” that may aid in predicting the occurrence of heart disease.
The first paper, by Art Owen, is concerned with integrating real-valued functions $f(x)$ over the $d$-dimensional unit cube, that is, the computation of $$ I=\int_{[0,1]^d}{f(x) dx}. $$ This is done by means of quasi-Monte Carlo integration, which approximates the integral I by a finite-dimensional sum $$ S={1\over n}\sum_{i=1}^nf(x_i), $$ where the $x_i$ are points in the cube $[0,1]^d$. If $f(x)$ and its partial derivatives are appropriately bounded (“bounded variation”) on the unit cube, and if the points $x_i$ are appropriately chosen, then the approximation $S$ converges to the integral $I$, in the sense that the asymptotic error $|S-I|\sim 1/n^{1-\epsilon}$ as $n\rightarrow\infty$ for all $\epsilon>0$.
However, if $f(x)$ has a singularity and goes to $\pm\infty$ at a boundary of the unit cube, as happens with certain functions in computational finance, then the approximation $S$ diverges—unless one can manage to keep the points $x_i$ away from the boundary. Art Owen shows that this is possible for functions $f(x)$ with a singularity at the origin, provided their partial derivatives do not grow too fast near the boundary. In this case particular sequences of points $x_i$, so-called Halton sequences, avoid the origin (hence the title of the paper) and give rise to converging approximations $S$ whose errors $|S-I|$ can be bounded in the same form as above. The approach is based on defining a well-behaved (“low-variation”) extension of $f(x)$ close to the origin.
The second paper, by Frederik Simons, Anthony Dahlen, and Mark Wieczorek, is based on doing the FFT on a sphere. This is important in a variety of applications, such as geophysics, where locally flat approximations are not adequate due to the curvature of the earth.
Let's start with the one-dimensional version of the time-frequency concentration problem: Find a function $g(t)$ whose FFT $G(\omega)$ vanishes for all frequencies $\omega$ outside the interval $[-W,W]$ (such a $g$ is called band-limited) but is optimally concentrated in the time interval $[-T,T]$. “Optimally concentrated” here means that $g(t)$ has minimal energy outside $[-T,T]$, i.e., $$ \lambda=\int_{-T}^{T}{g^2(t)dt}/\int_{-\infty}^{\infty}{g^2(t)dt} \quad\mathrm{is~maximal}. $$ The ratio $\lambda$, which satisfies $0<\lambda<1$, measures the spatial concentration of the function $g$.
Now let's look at the sphere. The counterpart of the FFT is a decomposition into spherical harmonics, “frequencies” are now quantities associated with longitude and latitude, and time is replaced by a region on the sphere. The problem is to find band-limited functions that are optimally concentrated in space. Again, “optimally concentrated” is defined by a variational problem as the one above for $\lambda$. The authors show that $\lambda$ can be expressed as the largest eigenvalue of a symmetric positive-definite eigenvalue problem, and that the eigenfunction associated with $\lambda$ represents a band-limited function with the the best spatial concentration.
In the third paper, John Cain and David Schaeffer derive an approximation for a function that may help in predicting abnormal heartbeats.
The main pumping action of the heart comes from the contraction of the left ventricle, which is stimulated by an electrical pulse, called the action potential. In the simplest model of heart dynamics, the electrical voltage behavior is segregated into two kinds of time intervals (think of an EEG). The first time interval is the action potential duration, during which the voltage rises rapidly above a threshold. The second time interval is the diastolic interval, when the voltage returns to the resting level. During this time the tissue is less sensitive to stimuli, so that the duration and size of the next action potential depend on this delay. The dependence of an action potential duration on the previous diastolic interval is described by the so-called restitution curve. The slopes of this curve may be useful in predicting ventricular fibrillation and sudden cardiac death.
The authors derive an asymptotic approximation for the restitution curve in the context of a “two-current ionic model” which is described by two ordinary differential equations.

Halton Sequences Avoid the Origin

Art B. Owen

SIAM Rev. 48, pp. 487-503 (17 pages) | Cited 1 time

Online Publication Date: August 03, 2006

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The nth point of the Halton sequence in [0,1]d is shown to have components whose product is larger than Cn-1, where C > 0 depends on d. This property makes the Halton sequence very well suited to quasi-Monte Carlo (QMC) integration of some singular functions that become unbounded as the argument approaches the origin. The Halton sequence avoids a similarly shaped (though differently sized) region around every corner of the unit cube, making it suitable for functions with singularities at all corners. Convergence rates are established for QMC integration based on two assumptions: a growth condition on the integrand, and a measure of how the sample points avoid the boundary. In some settings the error is O(n-1 + epsilon), while in others the error diverges to infinity. Star discrepancy does not suffice to distinguish the cases.

Spatiospectral Concentration on a Sphere

Frederik J. Simons, F. A. Dahlen, and Mark A. Wieczorek

SIAM Rev. 48, pp. 504-536 (33 pages) | Cited 43 times

Online Publication Date: August 03, 2006

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We pose and solve the analogue of Slepian's time-frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere or, alternatively, of strictly spacelimited functions that are optimally concentrated in the spherical harmonic domain. Such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications, as well as in medical imaging, computer science, cosmology, and numerical analysis. The spherical Slepian functions can be found by solving either an algebraic eigenvalue problem in the spectral domain or a Fredholm integral equation in the spatial domain. The associated eigenvalues are a measure of the spatiospectral concentration. When the concentration region is an axisymmetric polar cap, the spatiospectral projection operator commutes with a Sturm--Liouville operator; this enables the eigenfunctions to be computed extremely accurately and efficiently, even when their area-bandwidth product, or Shannon number, is large. In the asymptotic limit of a small spatial region and a large spherical harmonic bandwidth, the spherical concentration problem reduces to its planar equivalent, which exhibits self-similarity when the Shannon number is kept invariant.

Two-Term Asymptotic Approximation of a Cardiac Restitution Curve

John W. Cain and David G. Schaeffer

SIAM Rev. 48, pp. 537-546 (10 pages) | Cited 4 times

Online Publication Date: August 03, 2006

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If spatial extent is neglected, ionic models of cardiac cells consist of systems of ordinary differential equations (ODEs) which have the property of excitability, i.e., a brief stimulus produces a prolonged evolution (called an action potential in the cardiac context) before the eventual return to equilibrium. Under repeated stimulation, or pacing, cardiac tissue exhibits electrical restitution: the steady-state action potential duration (APD) at a given pacing period B shortens as B is decreased. Independent of ionic models, restitution is often modeled phenomenologically by a one-dimensional mapping of the form APDnext = f(B - APDprevious). Under some circumstances, a restitution function f can be derived as an asymptotic approximation to the behavior of an ionic model.
In this paper, extending previous work, we derive the next term in such an asymptotic approximation for a particular ionic model consisting of two ODEs. The two-term approximation exhibits excellent quantitative agreement with the actual restitution curve, whereas the leading-order approximation significantly underestimates actual APD values.
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SIGEST

The Editors

SIAM Rev. 48, pp. 547-547 (1 page)

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It is not terribly common for a paper published in a SIAM journal to be referred to as “classical”—many of our topics and areas are too young for that term to apply. However, several of the nominators of this issue's SIGEST paper, “Some New Aspects of the Coupon Collectors Problem,” by Amy Myers and Herbert Wilf, used that term in a very complimentary sense. Indeed they found the paper to be classical in two regards: in the problem it addresses, and in the mathematical techniques it uses. These characteristics, along with the paper's excellent results, broad appeal, and clear exposition, are the reasons it is the SIGEST selection from the SIAM Journal on Discrete Mathematics.
The classical coupon collector's problem is the analysis of how many trials it will take for one collector to obtain a full set of d coupons if at each trial the collector has an equal chance of selecting any one of the d possible coupons. More precisely, it determines the probability of the collector first assembling a full set of d coupons after n trials, for each $\mbox{\textit{n}} \geq \mbox{\textit{d}}$. The paper by Myers and Wilf extends this problem to one in which two people each are collecting the same d coupons simultaneously and independently. It asks and answers a number of questions related to this scenario, including the probability that each person first collects a full set of d coupons after the same particular number n of trials, and the probability that the person who first collects a full set of coupons is always ahead (or tied) in terms of the number of different coupons collected, up until and including the time that the full set is obtained.
One of the interesting and appealing aspects of this paper is that the solution techniques are fairly direct and accessible. The key is the utilization of classical lattice path techniques. Another interesting aspect is that while many probabilities are initially expressed as infinite sums, the authors show that they can be evaluated in finite terms. Indeed, the paper shows that this phenomenon is true for a general set of infinite series of the type that occur in their analyses.
We hope that this paper will serve to familiarize a broad range of SIAM readers with a very interesting, enumerative problem area.

Some New Aspects of the Coupon Collector's Problem

Amy N. Myers and Herbert S. Wilf

SIAM Rev. 48, pp. 549-565 (17 pages)

Online Publication Date: August 03, 2006

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We extend the classical coupon collector's problem to one in which two collectors are simultaneously and independently seeking collections of d coupons. We find, in finite terms, the probability that the two collectors finish at the same trial, and we find, using the methods of Gessel and Viennot, the probability that the game has the following "ballot-like" character: the two collectors are tied with each other for some initial number of steps, and after that the player who first gains the lead remains ahead throughout the game. As a by-product we obtain the evaluation in finite terms of certain infinite series whose coefficients are powers and products of Stirling numbers of the second kind.
We study the variant of the original coupon collector's problem in which a single collector wants to obtain at least h copies of each coupon. Here we give a simpler derivation of results of Newman and Shepp and extend those results. Finally, we obtain the distribution of the number of coupons that have been obtained exactly once ("singletons") at the conclusion of a successful coupon collecting sequence.
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Education

Andrew J. Bernoff, Section Editor

SIAM Rev. 48, pp. 567-567 (1 page)

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When Google debuted in the late 1990s and began its meteoric rise to become the Internet's zeitgeist at the turn of the millennium, the subtext for mathematics could have been that linear algebra matters. At its core, Google is built on the PageRank algorithm, which is a clever way of ranking the importance of a page by how it links to other pages. Amazingly, armed with a basic understanding of directed graphs and eigenvalues problems, the PageRank concept is easily accessible to students in an introductory linear algebra course.
This issue's Education section article,“The $25,000,000,000 Eigenvector: The Linear Algebra behind Google" by Kurt Bryan and Tanya Leise, provides just such an introduction to the ideas behind PageRank. The authors describe the algorithm intuitively, translate the problem into mathematics, and then give a quick tutorial on its solution. Along the way, they illustrate the link between graph theory and linear algebra, introduce the reader to some examples of analysis problems that arise in these subjects, and give a glimmer of the numerical algorithm used to make the actual calculation. The paper is liberally strewn with exercises suitable for linear algebra novices.
The ideas in the paper could easily be adapted as an applications lecture or two in a first linear algebra course. It also is written at a level where it should make a nice supplement or basis for a project in either linear algebra or graph theory. Finally, it provides a definitive answer to why the aspiring business student really should pay attention in math class; the resulting payday could literally be a few billion dollars.

The $25,000,000,000 Eigenvector: The Linear Algebra behind Google

Kurt Bryan and Tanya Leise

SIAM Rev. 48, pp. 569-581 (13 pages) | Cited 8 times

Online Publication Date: August 03, 2006

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Google's success derives in large part from its PageRank algorithm, which ranks the importance of web pages according to an eigenvector of a weighted link matrix. Analysis of the PageRank formula provides a wonderful applied topic for a linear algebra course. Instructors may assign this article as a project to more advanced students or spend one or two lectures presenting the material with assigned homework from the exercises. This material also complements the discussion of Markov chains in matrix algebra. Maple and Mathematica files supporting this material can be found at www.rose-hulman.edu/~bryan.
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Book Reviews

Bob O'Malley, Section Editor

SIAM Rev. 48, pp. 585-626 (42 pages)

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One might think it a dreary chore to write a lively featured review of the Oxford Users' Guide to Mathematics, but that depends on the author. Jonathan Borwein certainly made a colorful job of it, suggesting that the forthcoming Collins Dictionary might also be a worthwhile one to review, perhaps by Eberhard Zeidler.
A number of other significant books in pure and applied mathematics are also reviewed in this issue by a diverse variety of mathematicians. We thank them all for providing us opinions about this new literature.
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