OPAC - Pencarian Artikel Jurnal & Majalah Library USD
Menampilkan semua artikel (Halaman 692 dari 34166, Total: 341652 data)
Strings, Chains, and Ropes
klik untuk membaca file PDF
| Pengarang | : | Edmond Levy |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 48 (No. 4) |
| Halaman | : | 771-781 |
| Abstrak | : | Following Antman [Amer. Math. Mon., 87 (1980), pp. 359–370], we advocate a more physically realistic and systematic derivation of the wave equation suitable for a typical undergraduate course in partial differential equations. To demonstrate the utility of this derivation, three applications that follow naturally are described: strings, hanging chains, and jump ropes. |
Dynamical Scaling in Smoluchowski’s Coagulation Equations: Uniform Convergence
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 48 (No. 4) |
| Halaman | : | 745-768 |
| Abstrak | : | Smoluchowski’s coagulation equation is a fundamental mean?field model of clustering dynamics. We consider the approach to self?similarity (or dynamical scaling) of the cluster size distribution for the “solvable” rate kernels , and . In the case of continuous cluster size distributions, we prove uniform convergence of densities to a self?similar solution with exponential tail, under the regularity hypothesis that a suitable moment have an integrable Fourier transform. For discrete size distributions, we prove uniform convergence under optimal moment hypotheses. Our results are completely analogous to classical local convergence theorems for the normal law in probability theory. The proofs rely on the Fourier inversion formula and the solution for the Laplace transform by the method of characteristics in the complex plane. |
Monte Carlo Tennis
| Pengarang | : | Paul K. Newton |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 48 (No. 4) |
| Halaman | : | 722-742 |
| Abstrak | : | The probability of winning a game, set, match, or single elimination tournament in tennis is computed using Monte Carlo simulations based on each player’s probability of winning a point on serve, which can be held constant or varied from point to point, game to game, or match to match. The theory, described in Newton and Keller [Stud. Appl. Math., 114 (2005), pp. 241–269], is based on the assumption that points in tennis are independent, identically distributed (i.i.d.) random variables. This is used as a baseline to compare with the simulations, which under similar circumstances are shown to converge quickly to the analytical curves in accordance with the weak law of large numbers. The concept of the importance of a point, game, and set to winning a match is described based on conditional probabilities and is used as a starting point to model non?i.i.d.effects, allowing each player to vary, from point to point, his or her probability of winning on serve. Several non?i.i.d.models are investigated, including the “hot?hand?effect,” in which we increase each player’s probability of winning a point on serve on the next point after a point is won. The “back?to?the?wall” effect is modeled by increasing each player’s probability of winning a point on serve on the next point after a point is lost. In all cases, we find that the results provided by the theoretical curves based on the i.i.d.assumption are remarkably robust and accurate, even when relatively strong non?i.i.d.effects are introduced. We end by showing examples of tournament predictions from the 2002 men’s and women’s U.S. Open draws based on the Monte Carlo simulations. We also describe Arrow’s impossibility theorem and discuss its relevance with regard to sports ranking systems, and we argue for the development of probability?based ranking systems as a way to soften its consequences. |
Globalization Techniques for Newton–Krylov Methods and Applications to the Fully Coupled Solution of the Navier–Stokes Equations
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 48 (No. 4) |
| Halaman | : | 700-721 |
| Abstrak | : | A Newton–Krylov method is an implementation of Newton’s method in which a Krylov subspace method is used to solve approximately the linear subproblems that determine Newton steps. To enhance robustness when good initial approximate solutions are not available, these methods are usually globalized, i.e., augmented with auxiliary procedures (globalizations) that improve the likelihood of convergence from a starting point that is not near a solution. In recent years, globalized Newton–Krylov methods have been used increasingly for the fully coupled solution of large?scale problems. In this paper, we review several representative globalizations, discuss their properties, and report on a numerical study aimed at evaluating their relative merits on large?scale two? and three?dimensional problems involving the steady?state Navier–Stokes equations. |
The Fastest Mixing Markov Process on a Graph and a Connection to a Maximum Variance Unfolding Problem
| Pengarang | : | Jun Sun |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 48 (No. 4) |
| Halaman | : | 681-699 |
| Abstrak | : | We consider a Markov process on a connected graph, with edges labeled with transition rates between the adjacent vertices. The distribution of the Markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue of the Laplacian of the weighted graph. In this paper we consider the problem of assigning transition rates to the edges so as to maximize subject to a linear constraint on the rates. This is the problem of finding the fastest mixing Markov process (FMMP) on the graph. We show that the FMMP problem is a convex optimization problem, which can in turn be expressed as a semidefinite program, and therefore effectively solved numerically. We formulate a dual of the FMMP problem and show that it has a natural geometric interpretation as a maximum variance unfolding (MVU) problem, the problem of choosing a set of points to be as far apart as possible, measured by their variance, while respecting local distance constraints. This MVU problem is closely related to a problem recently proposed by Weinberger and Saul as a method for “unfolding” high?dimensional data that lies on a low?dimensional manifold. The duality between the FMMP and MVU problems sheds light on both problems, and allows us to characterize and, in some cases, find optimal solutions. |
Mode-Locked Soliton Lasers
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 48 (No. 4) |
| Halaman | : | 629-678 |
| Abstrak | : | A comprehensive treatment is given for the formation of mode?locked soliton pulses in optical fiber and solid state lasers. The pulse dynamics is dominated by the interaction of the cubic Kerr nonlinearity and chromatic dispersion with an intensity?dependent perturbation provided by the mode?locking element in the laser cavity. The intensity?dependent perturbation preferentially attenuates low intensity electromagnetic radiation which makes the mode?locked pulses attractors of the laser cavity. A review of the broad spectrum of mode?locked laser models, both qualitative and quantitative, is considered with the basic pulse formation phenomena highlighted. The strengths and weaknesses of each model are considered with two key instabilities studied in detail: Q?switching and harmonic mode?locking. Although the numerous mode?locking models are considerably different, they are unified by the fact that stable solitons are exhibited in each case due to the intensity discrimination perturbation in the laser cavity. |
The $25,000,000,000 Eigenvector: The Linear Algebra behind Google
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 48 (No. 3) |
| Halaman | : | 569-581 |
| Abstrak | : | 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. |
Some New Aspects of the Coupon Collector's Problem
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 48 (No. 3) |
| Halaman | : | 549-565 |
| Abstrak | : | 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. |
Two-Term Asymptotic Approximation of a Cardiac Restitution Curve
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 48 (No. 3) |
| Halaman | : | 537-546 |
| Abstrak | : | 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. |
Spatiospectral Concentration on a Sphere
| Pengarang | : | Frederik J. Simons |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 48 (No. 3) |
| Halaman | : | 504-536 |
| Abstrak | : | 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. |