Abstract: Rigid-body dynamics with unilateral contact is a good approximation for a wide range of everyday phenomena, from the operation of car brakes to walking to rock slides. It is also of vital importance for simulating robots, virtual reality, and realistic animation. However, correctly modeling rigid-body dynamics with friction is difficult due to a number of discontinuities in the behavior of rigid bodies and the discontinuities inherent in the Coulomb friction law. This is particularly crucial for handling situations with large coefficients of friction, which can result in paradoxical results known at least since Painleve [C. R. Acad. Sci. Paris, 121 (1895), pp. 112--115]. This single example has been a counterexample and cause of controversy ever since, and only recently have there been rigorous mathematical results that show the existence of solutions to his example. The new mathematical developments in rigid-body dynamics have come from several sources: "sweeping processes" and the measure differential inclusions of Moreau in the 1970s and 1980s, the variational inequality approaches of Duvaut and J.-L. Lions in the 1970s, and the use of complementarity problems to formulate frictional contact problems by Lotstedt in the early 1980s. However, it wasn't until much more recently that these tools were finally able to produce rigorous results about rigid-body dynamics with Coulomb friction and impulses.

Abstract: A review is presented of recent results on front propagation in reaction-diffusion-advection equations in homogeneous and heterogeneous media. Formal asymptotic expansions and heuristic ideas are used to motivate the results wherever possible. The fronts include constant-speed monotone traveling fronts in homogeneous media, periodically varying traveling fronts in periodic media, and fluctuating and fractal fronts in random media. These fronts arise in a wide range of applications such as chemical kinetics, combustion, biology, transport in porous media, and industrial deposition processes. Open problems are briefly discussed along the way.

Abstract: In this paper we propose a new method for the iterative computation of a few of the extremal eigenvalues of a symmetric matrix and their associated eigenvectors. The method is based on an old and almost unknown method of Jacobi. Jacobi's approach, combined with Davidson's method, leads to a new method that has improved convergence properties and that may be used for general matrices. We also propose a variant of the new method that may be useful for the computation of nonextremal eigenvalues as well.

Abstract: Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that uses an adjoint PDE to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest. Numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy. Numerical results are presented for the Poisson equation in one and two dimensions and for the nonlinear quasi-one-dimensional Euler equations. The theory is equally applicable to nonlinear equations in complex multi-dimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations.

Abstract: Matrix factorization in numerical linear algebra (NLA) typically serves the purpose of restating some given problem in such a way that it can be solved more readily; for example, one major application is in the solution of a linear system of equations. In contrast, within applied statistics/psychometrics (AS/P), a much more common use for matrix factorization is in presenting, possibly spatially, the structure that may be inherent in a given data matrix obtained on a collection of objects observed over a set of variables. The actual components of a factorization are now of prime importance and not just as a mechanism for solving another problem. We review some connections between NLA and AS/P and their respective concerns with matrix factorization and the subsequent rank reduction of a matrix. We note in particular that several results available for many decades in AS/P were more recently (re)discovered in the NLA literature. Two other distinctions between NLA and AS/P are also discussed briefly: how a generalized singular value decomposition might be defined, and the differing uses for the (newer) methods of optimization based on cyclic or iterative projections.

Abstract: This paper intends to serve as an educational introduction to sampling theory. Basically, sampling theory deals with the reconstruction of functions (signals) through their values (samples) on an appropriate sequence of points by means of sampling expansions involving these values. In order to obtain such sampling expansions in a unified way, we propose an inductive procedure leading to various orthogonal formulas. This procedure, which we illustrate with a number of examples, closely parallels the theory of orthonormal bases in a Hilbert space. All intermediate steps will be described in detail, so that the presentation is self-contained. The required mathematical background is a basic knowledge of Hilbert space theory. Finally, despite the introductory level, some hints are given on more advanced problems in sampling theory, which we motivate through the examples.

Abstract: A hierarchy of models for type-II superconductors is presented. Through appropriate asymptotic limits we pass from the mesoscopic Ginzburg--Landau model to the London model with isolated superconducting vortices as line singularities, to vortex-density models, and finally to macroscopic critical-state models.

Abstract: Let \[ H=\pmatrix{H_{11}&H_{12}\cr H_{12}^*&H_{22}} \] be an $n\times n$ positive semidefinite matrix, where $H_{11}$ is $k\times k$ with $1 \le k < n$. The {\it generalized Schur complement} of $H_{11}$ in $H$ is defined as $$ S(H) = H_{22} - H_{12}^*H_{11}^{\dag}H_{12}, $$ where $H_{11}^{\dag}$ is the Moore\,--Penrose generalized inverse of $H_{11}$. It has the extremal characterizations \[ S(H) = \max\left\{W: H - \pmatrix{0_k & 0 \cr 0 & W\cr} \ge 0, W \hbox{ is } (n-k) \times (n-k) \hbox{ Hermitian} \right\} \] and\rule[-4.75pt]{0pt}{0pt} $$ S(H) = \min \left\{[Z|I_{n-k}] H [Z|I_{n-k}]^*: Z \ \hbox{ is } \ (n-k) \times k \right\}. $$ oindent These characterizations are used to deduce many old and new inequalities for Schur complements of positive semidefinite matrices. In many cases, stronger statements and shorter proofs can be obtained using the extremal characterizations.

Abstract: In 1775, J. F. de Tuschis a Fagnano observed that in every acute triangle, the orthoptic triangle represents a periodic billiard trajectory, but to the present day it is not known whether or not in every obtuse triangle a periodic billiard trajectory exists. The limiting case of right triangles was settled in 1993 by F. Holt, who proved that all right triangles possess periodic trajectories. The same result had appeared independently in the Russian literature in 1991, namely in the work of G. A. Gal'perin, A. M. Stepin, and Y. B. Vorobets. The latter authors discovered in 1992 a class of obtuse triangles which contain particular periodic billiard paths. In this article, we review the above-mentioned results and some of the techniques used in the proofs and at the same time show for an extended class of obtuse triangles that they contain periodic billiard trajectories.

Abstract: Maps that show the area of all regions of the earth's surface in their correct proportions, appropriately called equal area maps, are often used by cartographers to display area-based data, such as the extent of rain forests, the range of butterfly migrations, or the access of people in various regions to medical facilities. In this article, we will examine the step-by-step construction of an equal area map first announced by Karl Brandan Mollweide (1774-1825) and commonly used in atlases today. Aesthetically, Mollweide's map, which represents the whole world in an ellipse whose axes are in a 2:1 ratio, reflects the essentially round character of the earth better than rectangular maps. The mathematics involved in the construction requires mainly high school algebra and trigonometry with only a bit of calculus (which can plausibly be avoided if one so desires). All of this material, except for the calculus, was included in a first-year liberal arts math/geography course taught at Villanova University.

Abstract: Applications of mass, momentum, and energy balances are central to the teaching of fluid dynamics. In this study, a model to determine the trajectory of a water rocket is given that is far simpler than the system of coupled partial differential equations that typically results in modern hydrodynamic problems of interest. This makes the problem an excellent choice for a student project---it can reasonably be completed with a day or two of effort. In addition to the fundamental mathematics, this problem offers opportunities in scale analysis, numerical methods for IVPs, balance principles in accelerated frames of reference, and the collection and assessment of flight test data.

Abstract: We propose a globally convergent method for the numerical solution of a class of continuous nonlinear knapsack problems arising, e.g., in chemical production service facilities.

Abstract: An alternative example of the method of multiple scales is presented. This example arises in the study of the classical heat equation with a slowly varying flux imposed at one end. The module presents introductory ideas about dimensionless variables, multiple-scale expansions, and scaling of the dependent variable. The necessarily obfuscating algebraic computation is less than that for more familiar multiple-scale examples, such as the perturbed oscillator. The results are analyzed for both their physical and mathematical importance.

Abstract: Many students first learn of the law of reflection when they are told about the reflective properties of conics. Typically students are exposed to a curve or surface, and then its reflective properties are derived or verified. In this paper we explore reflection by starting from a desired property of an unknown reflector and proceeding to find a reflector with those properties. This is done in the context of finding a reflector that produces uniform illumination of nearby objects. The reflectors considered are those whose analysis can be reduced to studying curves in two dimensions.

Abstract: Free material design deals with the question of finding the stiffest structure with respect to one or more given loads which can be made when both the distribution of material and the material itself can freely vary The case of one single load has been discussed in several recent papers, and an efficient numerical approach was presented in [M Kovara, M Zibulevsky, and J Zowe, RAIRO Model Math Anal Numer, 32 (1998), pp 255--281] We attack here the multiload situation (understood in the worst-case sense), which is of much more interest for applications but also significantly more challenging from both the theoretical and the numerical points of view After a series of transformation steps we reach a problem formulation for which we can prove existence of a solution; a suitable discretization leads to a semidefinite programming problem for which modern polynomial time algorithms of interior point type are available A number of numerical examples demonstrate the efficiency of our approach

Abstract: This paper presents several practical applications of the univariate contouring problem, and it demonstrates how these problems can be solved accurately and reliably as numerical solutions of a particular differential algebraic equation derived for that purpose.

Abstract: We prove possible blowup in finite time of the solutions to reaction-diffusion systems which preserve nonnegativity and for which the total mass of the components is nonincreasing in time (two natural properties in applications) This is done by presenting explicit counterexamples constructed with the help of formal computation software Several partial results of global existence had been obtained previously in the literature Our counterexamples explain a posteriori why extra conditions were needed Negative results of independent interest are also provided as a by-product for linear parabolic equations in nondivergence form and with discontinuous coefficients and for nonlinear Hamilton--Jacobi evolution equations

Abstract: We review our recent results on fingering, bamboo waves, and drop breakup in low Reynolds number flows composed of two viscous liquids under shear.

Abstract: We extend Schoenberg's family of polynomial splines with uniform knots to all fractional degrees $\alpha>-1$. These splines, which involve linear combinations of the one-sided power functions $x_{+}^{\alpha}=\max(0,x)^{\alpha}$, are $\alpha$-Holder continuous for $\alpha>0$. We construct the corresponding B-splines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains. We show that these functions satisfy most of the properties of the traditional B-splines, including the convolution property, and a generalized fractional differentiation rule that involves finite differences only. We characterize the decay of the B-splines that are not compactly supported for nonintegral $\alpha$'s. Their most astonishing feature (in reference to the Strang--Fix theory) is that they have a fractional order of approximation $\alpha+1$ while they reproduce the polynomials of degree $\lceil\alpha\rceil$. For $\alpha>-\frac{1}{2}$, they satisfy all the requirements for a multiresolution analysis of $\LL^{2}$ (Riesz bounds, two-scale relation) and may therefore be used to build new families of wavelet bases with a continuously varying order parameter. Our construction also yields symmetrized fractional B-splines which provide the connection with Duchon's general theory of radial $(m,s)$-splines (including thin-plate splines). In particular, we show that the symmetric version of our splines can be obtained as the solution of a variational problem involving the norm of a fractional derivative.

Abstract: We classify the possible normal forms of quadratic Hamiltonians in 2 dimensions. Then we give a method to reduce by one the number of degrees of freedom of an arbitrary polynomial Hamiltonian whose principal part is quadratic in positions and moments. The reduction procedure is based on the extension of an integral of the unperturbed part to the whole system, up to a certain order. The corresponding reduced phase spaces have dimension 2 and are described by means of the set of invariants associated to the reduction.

Abstract: The Herglotz algorithm for constructing autonomous canonical transformations that was presented by Guenther, Gottsch, and Kramer in [SIAM Rev., 38 (1996), pp. 287--293] is extended to include nonautonomous canonical transformations and a remainder function ${\cal R}^*$ that is related to a generating function by ${\cal R}^* = \partial F/\partial t$.

Abstract: We survey some of the highlights of inverse scattering theory as it has developed over the last 15 years, with emphasis on uniqueness theorems and reconstruction algorithms for time harmonic acoustic waves. Included in our presentation are numerical experiments using real data and numerical examples of the use of inverse scattering methods to detect buried objects.

Abstract: Trajectories of Hooke's law in the complex plane, which are conic sections, are mapped onto trajectories of Newton's law of gravitation via the transformation $z \rightarrow z^2$. Newton's law of ellipses (objects attracted to a center by a force inversely proportional to the square of the distance travel in conic sections) follows from a geometric analysis of this map. An extension of this approach reveals a similar relation between more general pairs of power laws of centripetal attraction. The implications of these relations are discussed and a Matlab program is provided for their numerical study. This material is suitable for an undergraduate complex analysis class.

Abstract: This paper chronicles the wide dispersal of Perron's 1907 result on positive matrices into many fields of science. The many proofs given during the last 93 years are categorized and critiqued (including Perron's original two proofs), and a more natural proof is presented. This simple-to-understand result of Perron provides a unequaled vehicle for taking students on a tour of many applied areas with some depth.