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Studies the problems caused in skiving, in relation to the paper industry. Indepth look at the skiving process; What are the problems caused in this process; Indication that the software packaging MAJIQTIM solves the problems.

Examines the nonlinear elasticity of a paper sheet. Formulation of a free boundary value problem for the quasi-static evolution of one-half period of a fold; Dimensional analysis of paper sheet folding; Characteristic of free-standing fold.

The article highlights two papers published within the issue titled "Divide-and-Conquer: A Proportional, Minimal-Envy Cake-Cutting Algorithm," by Steven Brams, Michael Jones and Christian Klamler, and "Impossibility of Fast Stable Approximation of Analytic Functions from Equispaced Samples," by Rodrigo Platte, Lloyd Trefethen and Arno Kuijlaars. The first paper discussed the mathematical problem of proportional division. The second one talked about potential theory, matrix iterations in the form of Krylov space methods and quadrature formulae.

The article discusses papers published within the issue, including "On the Block Triangular Form of Symmetric Matrices," by Iain Duff and Bora Ucar and "Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization," by Benjamin Recht, Maryann Fazel and Pablo Parrilo.

It is shown that bubbling on wine bottle labels is due to absorption of water from the glue, with subsequent hygroscopic expansion. Contrary to popular belief, most of the glue's water must be lost to the atmosphere rather than to the paper. A simple lubrication model is developed for spreading glue piles in the pressure chamber of the labeling machine. This model predicts a maximum rate for application of labels. Buckling theory shows that the current arrangement of periodic glue strips can indeed accommodate paper expansion. This project provides interesting applications of various areas of undergraduate mathematics, such as trigonometry, Maclaurin series, dimensional analysis, and fluid mechanics. It illustrates that simple mathematical modeling may provide insight into complicated real-world problem. [ABSTRACT FROM AUTHOR]

Just when modern computers (digital, electronic, and programmable) were being invented, John von Neumann and Herman Goldstine wrote a paper to illustrate the mathematical analyses that they believed would be needed to use the new machines effectively and to guide the development of still faster computers. Their foresight and the congruence of historical events made their work the first modern paper in numerical analysis. Von Neumann once remarked that to found a mathematical theory one had to prove the first theorem, which he and Goldstine did for the accuracy of mechanized Gaussian elimination--but their paper was about more than that. Von Neumann and Goldstine described what they surmised would be the significant questions once computers became available for computational science, and they suggested enduring ways to answer them. [ABSTRACT FROM AUTHOR]

In recent years, a nonoverlapping Robin-type domain decomposition method (DDM) for the finite element discretization systems of the second order elliptic equations, which is based on using Robin-type boundary conditions as information transmission conditions on the subdomain interfaces, has been developed and analyzed since it was first proposed by P. L. Lions in [On the Schwarz alternating method III: A variant for nonoverlapping subdomains, in Proceedings of the 3rd International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, PA, 1990, pp. 202-223]. However, the convergence rate of this DDM with many subdomains remains open when the lower term of equations vanishes. This open problem will be considered in this paper. The convergence rate is almost 1 — O(h¹/²H-1/²) in certain cases—for example, the case of a small number of subdomains, where h is the mesh size and H is the size of subdomain. In order to get the desirous convergence results, two mathematics skills are introduced in this paper; one is complexification of real linear space and the other is the spectral radius formula. [ABSTRACT FROM AUTHOR]

Introduces a series of articles on industrial and applied mathematics.

This paper is devoted to mathematical modeling in photoacoustic imaging of small absorbers. We propose a new method for reconstructing small absorbing regions inside a bounded domain from boundary measurements of the induced acoustic signal. We also show the focusing property of the back-propagated acoustic signal. Indeed, we provide two different methods for locating a targeted optical absorber from boundary measurements of the induced acoustic signal. The first method consists of a Multiple Signal Classification (MUSIC)-type algorithm and the second one uses a multifrequency approach. We also show results of computational experiments to demonstrate efficiency of the algorithms. [ABSTRACT FROM AUTHOR]

This paper presents a Lie-Trotter splitting for inertial Langevin equations (geometric Langevin algorithm) and analyzes its long-time statistical properties. The splitting is defined as a composition of a variational integrator with an Ornstein-Uhlenbeck flow. Assuming that the exact solution and the splitting are geometrically ergodic, the paper proves the discrete invariant measure of the splitting approximates the invariant measure of inertial Langevin equations to within the accuracy of the variational integrator in representing the Hamiltonian. In particular, if the variational integrator admits no energy error, then the method samples the invariant measure of inertial Langevin equations without error. Numerical validation is provided using explicit variational integrators with first-, second-, and fourth-order accuracy. [ABSTRACT FROM AUTHOR]

The present paper uses results from algebraic geometry to study the classical model describing the equilibrium frequencies of the four gametic types in the two-locus, two-allele genetic model under the assumption that the fitnesses are constant and that the cis- and trans- fitnesses are equal. It shows that there is a finite process for determining an upper bound on the maximum number of equilibrium frequencies for almost all choices of recombination and fitness parameters, and that this number is finite. It also shows that the solutions are locally continuous for almost all choices of parameters. The paper studies the equilibrium frequencies as functions of the recombination parameter, with attention to the cases when the recombination becomes infinite and when one of the equilibrium frequencies becomes infinite. [ABSTRACT FROM AUTHOR]

This paper studies L² norm error estimates for the div least-squares method for which the associated homogeneous least-squares functional is equivalent to the H(div) × H¹ norm for the respective dual and primal variables. Least-squares of this type for the second-order elliptic equations, elasticity, and the Stokes equations are an active area of research, and error estimates in the H(div) × H¹ norm were previously established. In this paper, we establish optimal L² norm error estimates for the primal variable under the minimum regularity requirement through a refined duality argument. [ABSTRACT FROM AUTHOR]

This is the second paper in the series A Mixture Theory for the Genesis of Residual Stresses in Growing Tissues. While the first paper in the series elaborated a general formulation for such a theory, the present paper develops a simple biphasic model of residual stress evolution in a growing multicell spheroid comprising a linear-elastic cellular phase and an inviscid interstitial fluid. Both isotropic and anisotropic growth are considered in this study, highlighting the necessity to incorporate stress relaxation in order to predict an evolution of stresses over a period of growth. The solutions to the biphasic equations are juxtaposed with the corresponding solutions to the single phase equations, illuminating the approximate nature of the single phase formulation for growing tissues. Moreover, the analysis demonstrates the significance of both interphase drag and the stress-relaxation characteristics of the solid phase in distinguishing between the single phase and multiphase paradigms. [ABSTRACT FROM AUTHOR]

Pseudo-transient continuation is a Newton-like iterative method for computing steady-state solutions of differential equations in cases where the initial data are far from a steady state. The iteration mimics a temporal integration scheme, with the time step being increased as steady state is approached. The iteration is an inexact Newton iteration in the terminal phase. In this paper we show how steady-state solutions to certain ordinary and differential algebraic equations with nonsmooth dynamics can be computed with the method of pseudo-transient continuation. An example of such a case is a discretized PDE with a Lipschitz continuous, but nondifferentiable, constitutive relation as part of the nonlinearity. In this case we can approximate a generalized derivative with a difference quotient. The existing theory for pseudo-transient continuation requires Lipschitz continuity of the Jacobian. Newton-like methods for nonsmooth equations have been globalized by trust-region methods, smooth approximations, and splitting methods in the past, but these approaches are not designed to find steady-state solutions of time-dependent problems. The method in this paper synthesizes the ideas from nonsmooth calculus and the method of pseudo-transient continuation. [ABSTRACT FROM AUTHOR]

Hashing with linear probing dates back to the 1950s and is among the most studied algorithms for storing (key, value) pairs. In recent years it has become one of the most important hash table organizations since it uses the cache of modern computers very well. Unfortunately, previous analyses rely either on complicated and space consuming hash functions, or on the unrealistic assumption of free access to a hash function with random and independent function values. Carter and Wegman, in their seminal paper on universal hashing, raised the question of extending their analysis to linear probing. However, we show in this paper that linear probing using a 2-wise independent hash function may have expected logarithmic cost per operation. Recently, Păatraşcu and Thorup have shown that 3- and 4-wise independent hash functions may also give rise to logarithmic expected query time. On the positive side, we show that 5-wise independence is enough to ensure constant expected time per operation. This resolves the question of finding a space and time efficient hash function that provably ensures good performance for hashing with linear probing. [ABSTRACT FROM AUTHOR]

This paper presents a review and critical analysis of the mathematical literature concerning the modeling of vehicular traffic and crowd phenomena. The survey of models deals with the representation scales and the mathematical frameworks that are used for the modeling approach. The paper also considers the challenging objective of modeling complex systems consisting of large systems of individuals interacting in a nonlinear manner, where one of the modeling difficulties is the fact that these systems are difficult to model at a global level when based only on the description of the dynamics of individual elements. The review is concluded with a critical analysis focused on research perspectives that consider the development of a unified modeling strategy. [ABSTRACT FROM AUTHOR]

In this paper, we develop a C0 interior penalty method for a fourth order singular perturbation elliptic problem in two dimensions on polygonal domains. Using some a posteriori error analysis techniques, we are able to show that the method converges in the energy norm uniformly with respect to the perturbation parameter under minimal regularity assumptions. In addition, we analyze the convergence of the numerical solution to the unperturbed second order problem. Finally, we perform some numerical experiments that back up the theoretical results. [ABSTRACT FROM AUTHOR]

The article discusses the paper "Some New Aspects of the Coupon Collectors Problem," by researchers Amy Myers and Herbert Wilf. It mentions that the number of trials that 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 is the problem of the classical coupon collector.

In this paper, we propose a uniaxial perfectly matched layer (PML) method for solving the tilne-harnlonic scattering problems ill two-layered media. The exterior region of the scatterer is divided into two half spaces by all infinite plane, on two sides of which the wave number takes different vahms. We surround the conlputational donmin where the scattering field is interested by a PML with the uniaxial medium property. By imposing homogeneous boundary condition on the outer boundary of the PML, we show that the solution of the PML problem converges exponentially to the solution of the original scattering problem in the computational dolnain as either the PML absorbing coefficient or the thickness of the PML tends to infinity. [ABSTRACT FROM AUTHOR]

This paper is concerned with the mathematical analysis of emissions markets. We review the existing quantitative analyses on the subject and introduce some of the mathematical challenges posed by the implementation of the new phase of the European Union Emissions Trading Scheme as well as the cap-and-trade schemes touted by the U.S., Canada, Australia, and Japan. From a practical point of view, the main thrust of the paper is the design and numerical analysis of new cap-and-trade schemes for the control and reduction of atmospheric pollution. We develop tools intended to help policy makers and regulators understand the pros and cons of the emissions markets. We propose a model for an economy where risk neutral firms produce goods to satisfy an inelastic demand and are endowed with permits in order to offset their pollution at compliance time and avoid having to pay a penalty. Firms that can easily reduce emissions do so, while those for which it is harder buy permits from firms that anticipate they will not need all their permits, creating a financial market for pollution credits. Our equilibrium model elucidates the joint price formation for goods and pollution allowances, capturing most of the features of the first phase of the European Union Emissions Trading Scheme. We show existence of an equilibrium and uniqueness of emissions credit prices. We also characterize the equilibrium prices of goods and the optimal production and trading strategies of the firms. We use the electricity market in Texas to numerically illustrate the qualitative properties of these cap-and-trade schemes. Comparing the numerical implications of cap-and-trade schemes to the business-as-usual benchmark, we show that our numerical results match those observed during the implementation of the first phase of the European Union cap-and-trade CO2 emissions scheme. In particular, we confirm the presence of windfall profits criticized by the opponents of these markets. We also demonstrate the shortcomings of tax and subsidy alternatives. Finally we introduce a relative allocation scheme which, while easy to implement, leads to smaller windfall profits than the standard scheme. [ABSTRACT FROM AUTHOR]

This paper continues the convergence analysis in [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 882-904] of discrete approximations to the linearized and adjoint equations arising from an unsteady one-dimensional hyperbolic equation with a convex flux function. We consider a simple modified Lax-Friedrichs discretization on a uniform grid, and a key point is that the numerical smoothing increases the number of points across the nonlinear discontinuity as the grid is refined. It is proved that there is convergence in the discrete approximation of linearized output functionals even for Dirac initial perturbations and pointwise convergence almost everywhere for the solution of the adjoint discrete equations. In particular, the adjoint approximation converges to the correct uniform value in the region in which characteristics propagate into the discontinuity. Moreover, it is shown that the results of [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 882-904] and the present paper hold also for quite general nonlinear initial data which contain multiple shocks and for which shocks form at a later time and/or merge. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is the (2+1)-dimensional dispersive long wave system, which describes the hydrodynamics of wide channels or open seas of finite depth. Its bilinear form is derived with the generalized Bell polynomials. New analytic solutions are obtained via symbolic computation, based on which the propagation of the water waves is analyzed graphically and the different phenomena of fission and fusion are revealed. Additionally, the bilinear auto-Bäcklund transformation is obtained by the generalized Bell polynomials. [ABSTRACT FROM AUTHOR]

This paper studies a new recovery-based a posteriori error estimator for the conforming linear finite element approximation to elliptic interface problems. Instead of recovering the gradient in the continuous finite element space, the flux is recovered through a weighted L2 projection onto H(div) conforming finite element spaces. The resulting error estimator is analyzed by establishing the reliability and efficiency bounds and is supported by numerical results. This paper also proposes an adaptive finite element method based on either the recovery-based estimators or the edge estimator through local mesh refinement and establishes its convergence. In particular, it is shown that the reliability and efficiency constants as well as the convergence rate of the adaptive method are independent of the size of jumps. [ABSTRACT FROM AUTHOR]

A full discrete two-level postprocessing Fourier Galerkin scheme for the unsteady Navier-Stokes equations with a periodic boundary condition is proposed in this paper. By defining a new projection, the interaction between the large and small eddies is reflected by the associated space splitting to some extent. Therefore, a weakly coupled system of the large and small eddies is obtained. Stability and error estimates for the weakly coupled two-level scheme are established in the paper. The proposed scheme is an effective algorithm because nonlinearity is treated only in the coarse-level subspace by solving the coarse-level standard Galerkin equation, and the matrix of the linear algebraic equations arising in each fine-level time stepping is a sparse matrix, especially for large scale computations. As a result, the new scheme saves a lot of CPU time and memory in comparison with the standard Fourier Galerkin method for deriving an approximation of prescribed accuracy. [ABSTRACT FROM AUTHOR]

In this paper we consider numerical approximations of a constraint minimization problem, where the object function is a quadratic Dirichlet functional for vector fields and the interior constraint is given by a convex function. The solutions of this problem are usually referred to as harmonic maps. The solution is characterized by a nonlinear saddle point problem, and the corresponding linearized problem is well-posed near strict local minima. The main contribution of the present paper is to establish a corresponding result for a proper finite element discretization in the case of two space dimensions. Iterative schemes of Newton type for the discrete nonlinear saddle point problems are investigated, and mesh independent preconditioners for the iterative methods are proposed. [ABSTRACT FROM AUTHOR]

This paper studies mixed finite element approximations of the viscosity solution to the Dirichlet problem for the fully nonlinear Monge-Ampère equation det(D²u0) = f (> 0) based on the vanishing moment method which was proposed recently by the authors in [X. Feng and M. Neilan, J. Scient. Comp., DOI 10.1007/s10915-008-9221-9, 2008]. In this approach, the second-order fully nonlinear Monge-Amp`ere equation is approximated by the fourth order quasilinear equation -ϵΔ²uϵ + detD²uϵ = f. It was proved in [X. Feng, Trans. AMS, submitted] that the solution ue converges to the unique convex viscosity solution u0 of the Dirichlet problem for the Monge-Ampère equation. This result then opens a door for constructing convergent finite element methods for the fully nonlinear second-order equations, a task which has been impracticable before. The goal of this paper is threefold. First, we develop a family of Hermann-Miyoshi-type mixed finite element methods for approximating the solution uϵ of the regularized fourth-order problem, which computes simultaneously ue and the moment tensor σϵ := D²uϵ . Second, we derive error estimates, which track explicitly the dependence of the error constants on the parameter ϵ, for the errors uϵ - uϵ h and σ0 - σϵ h . Finally, we present a detailed numerical study on the rates of convergence in terms of powers of e for the error u0 -uϵ h and σϵ -σϵh , and numerically examine what is the "best" mesh size h in relation to e in order to achieve these rates. Due to the strong nonlinearity of the underlying equation, the standard perturbation argument for error analysis of finite element approximations of nonlinear problems does not work for the problem. To overcome the difficulty, we employ a fixed point technique which strongly relies on the stability of the linearized problem and its mixed finite element approximations. [ABSTRACT FROM AUTHOR]

The present paper examines the effect of vertical harmonic vibration on the onset of convection in an infinite horizontal layer of a binary fluid mixture saturating a porous medium. A constant temperature and concentration distribution are assigned on the rigid boundaries, so that there exist vertical temperature and concentration gradients. The mathematical model is described by equations of filtration convection in the Darcy-Oberbeck-Boussinesq approximation. The linear stability analysis for the quasi-equilibrium solution is performed using Floquet theory. Employing the method of continued fractions allows derivation of the dispersion equation for the Floquet exponent in an explicit form. The neutral curves of the Rayleigh number versus horizontal wave number are constructed for the three types of instability modes: synchronous, subharmonic, and complex conjugate. Asymptotic formulas for these curves are derived for large values of vibration frequency using the method of averaging. It is shown that, at some finite frequencies of vibration, there exist regions of parametric instability. Investigations carried out in the paper demonstrate that, depending on the governing parameters of the problem, vertical vibration can significantly affect the stability of the system by increasing or decreasing its susceptibility to convection. In addition, even in the presence of vibration, the onset of convection in the system is affected by variations in the concentration of the solute in the mixture. [ABSTRACT FROM AUTHOR]

In their 2006 paper, Chernyshenko et al. [J. Math. Phys., 48 (2007), 065204, 15 pp]. prove that a sufficiently smooth strong solution of the 3D Navier-Stokes equations is robust with respect to small enough changes in initial conditions and forcing function. They also show that if a regular enough strong solution exists, then Galerkin approximations converge to it. They then use these results to conclude that the existence of a sufficiently regular strong solution can be verified using sufficiently refined numerical computations. In this paper we study the strong solutions with less regularity than those considered in Chernyshenko et al. [J. Math. Phys., 48 (2007), 065204, 15 pp]. We prove a similar robustness result and show the validity of the results relating convergent numerical computations and the existence of the strong solutions. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of sensitivity analysis in calculus of variations. A perturbation technique is applied to derive the boundary value problem and the system of equations that allow us to obtain the partial derivatives (sensitivities) of the objective function value and the primal and dual optimal solutions with respect to all parameters. Two examples of applications, a simple mathematical problem and a slope stability analysis problem, are used to illustrate the proposed method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the convergence properties of iterative algorithms of conjugate gradient type applied to the computation of an extreme eigenvalue of a Hermitian operator via the optimization of the Rayleigh quotient functional. The algorithms in focus employ the line search for the extremum of the Rayleigh quotient functional in the direction that is a linear combination of the gradient and the previous search direction. An asymptotically explicit equation for the reduction in the eigenvalue error after the line search step as a function of the search direction is derived and applied to the analysis of the local convergence properties (i.e., the error reduction after an iteration) of the algorithms in question. The local (stepwise) asymptotic equivalence of various conjugate gradient algorithms and the generalized Davidson method is proved, and a new convergence estimate for conjugate gradient iterations is derived, showing the reduction in the eigenvalue error after any two consecutive iterations. The paper's analysis extensively employs remarkable properties of the operator of the so-called Jacobi orthogonal complement correction equation. [ABSTRACT FROM AUTHOR]

It is known that some traditional shock-capturing schemes, including Godunov- or Roe-type schemes, generate downstream oscillatory wave patterns that cannot be effectively dampened by the dissipation methods they provide. The theme of this paper is to understand the formation and behavior at these upstream/downstream patterns from the point view of the symmetry inherent in the PDE. This paper investigates symmetry breaking and spatial oscillations in finite difference solutions for the inviscid Burgers and one-dimensional Navier-Stokes shock equations using the Lax-Wendroff, TVD, and nonoscillatory, nonfree parameter dissipative schemes. Symmetry breaking is considered as a candidate for the inherent cause of the nonphysical oscillations in the vicinity of the shock. [ABSTRACT FROM AUTHOR]

The classic atomic mix approximation for particle transport in a stochastic spatial medium is accurate when the material chunks in the medium are small compared to a mean free path. In this paper, we show that for charged particle transport in a stochastic medium, the atomic mix approximation is accurate when the chunk sizes are small compared to a transport mean free path. For charged particle transport, the transport mean free path is generally several orders of magnitude larger than the mean free path. Therefore, the result obtained in this paper greatly extends the known range of applicability of the atomic mix approximation. Numerical results are given that validate the asymptotic theory, and an application of the theory to a practical problem in radiation oncology is discussed. [ABSTRACT FROM AUTHOR]

This paper deals with the qualitative analysis of the solutions to a model that refers to the competition between the immune system and an aggressive host such as a tumor. The model which describes this competition is governed by a system of differential equations with one delay. It is shown that the dynamics depends crucially on the time delay parameter. By using the time delay as a parameter of bifurcation, the analysis is focused on the Hopf bifurcation problem to predict the occurrence of a limit cycle bifurcating from the nontrivial steady state. The obtained results depict the oscillations, given by simulations (see [M. Galach, Int. J. Appl. Math. Comput. Sci., 13 (2003), pp. 395–406]), which are observed in reality (see [D. Kirschner and J. C. Panetta, J. Math. Biol., 37 (1998), pp. 235–252]). It is suggested to examine by laboratory experiments how to employ these results for control of tumor growth. [ABSTRACT FROM AUTHOR]

The paper is concerned with convergence estimates for the preconditioned steepest descent method for the computation of the smallest eigenvalue of a Hermitian operator. Available estimates are reviewed and new estimates are introduced that improve on the known ones in certain respects. In addition to the estimates for the error reduction after one iteration, we consider estimates for the so-called asymptotic convergence factor defined as the upper limit of the average error reduction per iteration. The paper focuses on sharp estimates, i.e., those that cannot be improved without using additional information. [ABSTRACT FROM AUTHOR]

The aim of this paper is to analyze explicit exponential Runge--Kutta methods for the time integration of semilinear parabolic problems. The analysis is performed in an abstract Banach space framework of sectorial operators and locally Lipschitz continuous nonlinearities. We commence by giving a new and short derivation of the classical (nonstiff) order conditions for exponential Runge--Kutta methods, but the main interest of our paper lies in the stiff case. By expanding the errors of the numerical method in terms of the solution, we derive new order conditions that form the basis of our error bounds for parabolic problems. We show convergence for methods up to order four, and we analyze methods that were recently presented in the literature. These methods have classical order four, but they do not satisfy some of the new conditions. Therefore, an order reduction is expected. We present numerical experiments which show that this order reduction in fact arises in practical examples. Based on our new conditions, we finally construct methods that do not suffer from order reduction. [ABSTRACT FROM AUTHOR]

Over the last few years a great effort has been made to develop monotone high order explicit Runge--Kutta methods by means of their Shu--Osher representations. In this context, the stepsize restriction to obtain numerical monotonicity is normally computed using the optimal representation. In this paper we extend the Shu--Osher representations for any Runge--Kutta method giving sufficient conditions for monotonicity. We show how optimal Shu--Osher representations can be constructed from the Butcher tableau of a Runge--Kutta method. The optimum stepsize restriction for monotonicity is given by the radius of absolute monotonicity of the Runge--Kutta method [L. Ferracina and M. N. Spijker, SIAM J. Numer. Anal., 42 (2004), pp. 1073--1093], and hence if this radius is zero, the method is not monotone. In the Shu--Osher representation, methods with zero radius require negative coefficients, and to deal with them, an extra associate problem is considered. In this paper we interpret these schemes as representations of perturbed Runge--Kutta methods. We extend the concept of radius of absolute monotonicity and give sufficient conditions for monotonicity. Optimal representations can be constructed from the Butcher tableau of a perturbed Runge--Kutta method. [ABSTRACT FROM AUTHOR]

We consider the scalar conservation law with flux function discontinuous in the space variable, i.e., \begin{eqnarray} \label{eq1} u_t+(H(x)f(u)+(1-H(x))g(u))_{x} &=& 0 \quad \mbox{in } \R \times \R_{+}, \nonumber \\ u(0, x) &=& u_{0}(x) \quad \mbox{in } \R, \label{0.1} \end{eqnarray} where $H$ is the Heaviside function and $f$ and $g$ are smooth with the assumptions that either $f$ is convex and $g$ is concave or $f$ is concave and $g$ is convex. The existence of a weak solution of (\ref{eq1}) is proved by showing that upwind finite difference schemes of Godunov and Enquist--Osher type converge to a weak solution. Uniqueness follows from a Kruzkhov-type entropy condition. We also provide explicit solutions to the Riemann problem for (\ref{eq1}). At the level of numerics, we give easy-to-implement numerical schemes of Godunov and Enquist--Osher type. The central feature of this paper is the modification of the singular mapping technique (the main analytical tool for these types of equations) which allows us to show that the numerical schemes converge. Equations of type (\ref{eq1}) with the above hypothesis on the flux may occur when considering the following scalar conservation law with discontinuous flux: \begin{equation} \label{eq2} \begin{array}{r@{\;}l} u_t + (k (x) f (u))_x &= 0, \\ u (0, x) &= u_0 (x), \end{array} \end{equation} with $f$ convex and $k$ of indefinite sign. [ABSTRACT FROM AUTHOR]

This is the third part of a series of papers on least-squares Galerkin methods for parabolic initial-boundary value problems. These methods are based on the minimization of a least-squares functional for an equivalent first-order system over space and time with respect to suitable discrete spaces. This paper presents the derivation and analysis of one-step methods for semidiscretization in time from least-squares principles for semilinear parabolic problems. One of the most important features of the least-squares methodology is a built-in a posteriori estimate for the approximation error. For the presentation in this paper, we focus our attention on the specific combination of piecewise linear, not necessarily continuous, functions in time with continuous piecewise linear for the flux and scalar variables, respectively. For the resulting method, a convergence result is shown for the scalar variable. [ABSTRACT FROM AUTHOR]

This paper provides a theoretical analysis of a higher-order, FFT-based integral equation method introduced recently [IEEE Trans. Antennas and Propagation, 48 (2000), pp. 1862-1864] for the evaluation of transverse electric-polarized electromagnetic scattering from a bounded, penetrable inhomogeneity in two-dimensional space. Roughly speaking, this method is based on Fourier smoothing of the integral operator and the refractive index n (x). Here we prove that the solution of the resulting integral equation approximates the solution of the exact integral equation with higher-order accuracy, even when n (x) is a discontinuous function-as suggested by the numerical experiments contained in the paper mentioned above. In detail, we relate the convergence rates of the computed interior and exterior fields to the regularity of the scatterer, and we demonstrate, with a few numerical examples, that the predicted convergence rates are achieved in practice. [ABSTRACT FROM AUTHOR]

The paper presents a new affine invariant theory on asymptotic mesh independence of Newton's method for discretized nonlinear operator equations. Compared to earlier attempts, the new approach is both much simpler and more intuitive from the algorithmic point of view. The theory is exemplified at finite element methods for elliptic PDE problems. [ABSTRACT FROM AUTHOR]

This paper presents the Cholesky factor-alternating direction implicit (CF-ADI) algorithm, which generates a low-rank approximation to the solution X of the Lyapunov equation AX + XAT = -BBT. The coefficient matrix A is assumed to be large, and the rank of the right-hand side -BBT is assumed to be much smaller than the size of A. The CF-ADI algorithm requires only matrix-vector products and matrix-vector solves by shifts of A. Hence, it enables one to take advantage of any sparsity or structure in A. This paper also discusses the approximation of the dominant invariant subspace of the solution X. We characterize a group of spanning sets for the range of X. A connection is made between the approximation of the dominant invariant subspace of X and the generation of various low-order Krylov and rational Krylov subspaces. It is shown by numerical examples that the rational Krylov subspace generated by the CF-ADI algorithm, where the shifts are obtained as the solution of a rational minimax problem, often gives the best approximation to the dominant invariant subspace of X. [ABSTRACT FROM AUTHOR]

This paper is a continuation of recent attempts to understand, via mathematical modeling, the dynamics of marine bacteriophage infections. Previous authors have proposed systems of ordinary differential delay equations with delay dependent coefficients. In this paper we continue these studies in two respects. First, we show that the dynamics is sensitive to the phage mortality function, and in particular to the parameter we use to measure the density dependent phage mortality rate. Second, we incorporate spatial effects by deriving, in one spatial dimension, a delay reaction- diffusion model in which the delay term is rigorously derived by solving a von Foerster equation. Using this model, we formally compute the speed at which the viral infection spreads through the domain and investigate how this speed depends on the system parameters. Numerical simulations suggest that the minimum speed according to linear theory is the asymptotic speed of propagation. [ABSTRACT FROM AUTHOR]

We consider processes in which a focused laser beam is used to induce the melting of silicium. The first goal of this paper is to propose a simple three-dimensional (3D) model of this melting process. Our model is partly based on an energy balance equation. This model leads to a nontrivial ODE describing the evolution in time of the dimension of the melt region. The second goal of this paper is to obtain approximate analytical solutions of this ODE. After using basic solution methods, we propose an original geometrical method to derive asymptotic solutions for time → ∞. These solutions turn out to be the most useful for the description of this process. [ABSTRACT FROM AUTHOR]

In this paper, a local minimax-Newton method is developed to solve for multiple saddle points. The local minimax method [SIAM I Sci. Comput., 23 (2001), pp. 840-865]. is used to locate an initial guess and a version of the generalized Newton method is used to speed up convergence. When a problem possesses a symmetry, the local minimax method is invariant to the symmetry. Thus the symmetry can be used to greatly enhance the efficiency and stability of the local minimax method. But such an invariance is sensitive to numerical error and the Haar projection has been used to enforce the symmetry [SIAM I Numer. Anal., submitted]. In this paper, we prove that the Newton method is invariant to symmetries and that such an invariance is insensitive to numerical error. When a symmetric degeneracy takes place, it is proved that the Newton direction can be easily solved in an invariant subspace. Thus the Newton method can be used not only to speed up convergence but also to avoid using the Haar projection if the symmetric degeneracy is removable by a discretization. Finally, numerical examples are presented to illustrate the theory. [ABSTRACT FROM AUTHOR]

This paper continues the analysis started in the first part of this article (cf. [J. J. L. Velázquez, SIAM J. Appl. Math., 64 (2004), pp. 1198-1223]). It was seen there, using the method of matched asymptotics, that a regularized version of the Keller-Segel system admits, for a suitable asymptotic limit, solutions with some regions of high concentrations for the cell density. This paper considers the relation between the phenomenon of blow-up for the limit problem and the dynamics of the concentration regions described in [J. J. L. Velázquez, SIAM J. Appl. Math., 64 (2004), pp. 1198- 1223]. In particular, this paper analyzes the precise way in which the regularization introduced in the Keller-Segel system stops the aggregation process and yields the formation of concentration regions. [ABSTRACT FROM AUTHOR]

The goal of the paper is to study asymptotic behavior of the number of lost messages. Long messages are assumed to be divided into a random number of packets which are transmitted independently of one another. An error in transmission of a packet results in the loss of the entire message. Messages arrive to the M/GI/1 finite buffer model and can be lost in two cases as either at least one of its packets is corrupted or the buffer is overflowed. With the parameters of the system typical for models of information transmission in real networks, we obtain theorems on asymptotic behavior of the number of lost messages. We also study how the loss probability changes if redundant packets are added. Our asymptotic analysis approach is based on Tauberian theorems with remainder. [ABSTRACT FROM AUTHOR]

Modern cancer treatment relies upon microscopic tissue examination to classify tumors according to anatomical site of origin. This approach is effective but subjective and variable even among experienced clinicians and pathologists. Recently, DNA microarray-generated gene expression data has been used to build molecular cancer classifiers. Previous work from our group and others demonstrated methods for solving pairwise classification problems using such global gene expression patterns. However, classification across multiple primary tumor classes poses new methodological and computational challenges. In this paper we describe a computational methodology for multiclass prediction that combines class-specific (one vs. all) binary support vector machines. We apply this methodology to the diagnosis of multiple common adult malignancies using DNA microarray data from a collection of 198 tumor samples, spanning 14 of the most common tumor types. Overall classification accuracy is 78%, far exceeding the expected accuracy for random classification. In a large subset of the samples (80%), the algorithm attains 90% accuracy. The methodology described in this paper both demonstrates that accurate gene expression-based multiclass cancer diagnosis is possible and highlights some of the analytic challenges inherent in applying such strategies to biomedical research. [ABSTRACT FROM AUTHOR]

We consider a problem of eliminating the unwanted time-harmonic noise on a predetermined region of interest. The desired objective is achieved by active means, i.e., by introducing additional sources of sound called control sources, which generate the appropriate annihilating acoustic signal (antisound). A general solution for the control sources has been obtained previously in both continuous and discrete formulation of the problem. In the current paper, we focus on optimizing the overall absolute acoustic source strength of the control sources. Mathematically, this amounts to the minimization of multivariable complex-valued functions in the sense of L[SUB1] with conical constraints, which are only "marginally" convex. The corresponding numerical optimization problem appears very challenging even for the most sophisticated state-of-the-art methodologies, and even when the dimension of the grid is small and the waves are long. Our central result is that the global L[SUB1]-optimal solution can, in fact, be obtained without solving the numerical optimization problem. This solution is given by a special layer of monopole sources on the perimeter of the protected region. We provide a rigorous proof of global L[SUB1] minimality for both continuous and discrete optimization problems in the one-dimensional case. We also provide numerical evidence that corroborates our result in the two-dimensional case, when the protected domain is a cylinder. Even though we cannot fully justify it, we believe that the same result holds in the general case, i.e., for multidimensional settings and domains of arbitrary shape. We formulate this notion as a conjecture at the end of the paper. [ABSTRACT FROM AUTHOR]