Topological spaces and quasi-varieties

We describe Top ^op and Sob ^op as quasi-varieties by means of suitable schizophrenic objects.Research of the first author supported by grants from the NSERC of Canada and the FCAR du Québec. Research of the second author supported by the Topology grant 40% and by the NATO grant CRG 941330.     

Nonlinear stability of oscillatory core-annular flow: A generalized Kuramoto-Sivashinsky equation with time periodic coefficients

In this paper the nonlinear stability of two-phase core-annular flow in a pipe is examined when the acting pressure gradient is modulated by time harmonic oscillations and viscosity stratification and interfacial tension is present. An exact solution of the Navier-Stokes equations is used as the background state to develop an asymptotic theory valid for thin annular layers, which leads to a novel nonlinear equation describing the spatio-temporal evolution of the interface. The evolution equation is an extension of the equation found for constant pressure gradients and generalizes the Kuramoto-Sivashinsky equation with dispersive effects found by Papageorgiou, Maldarelli and Rumschitzki, Phys. Fluids A2(3), 340–352 (1990), to a similar system with time periodic coefficients. The distinct regimes of slow and moderate flow are considered and the corresponding evolution is derived. Certain solutions are described analytically in the neighborhood of the first bifurcation point by use of multiple scales asymptotics. Extensive numerical experiments, using dynamical systems ideas, are carried out in order to evaluate the effect of the oscillatory pressure gradient on the solutions in the presence of a constant pressure gradient.Research supported by NATO grant CRG 920097.Research was supported by the National Aeronautics and Space Administration under NASA Contract Nos. NAS1-19480 and NAS1-18605 while the author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681; also by grants NATO CRG 920097, and SBR NJIT-93.     

Kazhdan constants and the dual space topology

Supported by NATO collaborative research grant CRG 900029     

Minimum residual methods for augmented systems

For large systems of linear equations, iterative methods provide attractive solution techniques. We describe the applicability and convergence of iterative methods of Krylov subspace type for an important class of symmetric and indefinite matrix problems, namely augmented (or KKT) systems. Specifically, we consider preconditioned minimum residual methods and discuss indefinite versus positive definite preconditioning. For a natural choice of starting vector we prove that when the definite and indenfinite preconditioners are related in the obvious way, MINRES (which is applicable in the case of positive definite preconditioning) and full GMRES (which is applicable in the case of indefinite preconditioning) give residual vectors with identical Euclidean norm at each iteration. Moreover, we show that the convergence of both methods is related to a system of normal equations for which the LSQR algorithm can be employed. As a side result, we give a rare example of a non-trivial normal(1) matrix where the corresponding inner product is explicitly known: a conjugate gradient method therefore exists and can be employed in this case. This work was supported by British Council/German Academic Exchange Service Research Collaboration Project 465 and NATO Collaborative Research Grant CRG 960782     

Block band Toeplitz preconditioners derived from generating function approximations: analysis and applications

We are concerned with the study and the design of optimal preconditioners for ill-conditioned Toeplitz systems that arise from a priori known real-valued nonnegative generating functions f(x,y) having roots of even multiplicities. Our preconditioned matrix is constructed by using a trigonometric polynomial θ(x,y) obtained from Fourier/kernel approximations or from the use of a proper interpolation scheme. Both of the above techniques produce a trigonometric polynomial θ(x,y) which approximates the generating function f(x,y), and hence the preconditioned matrix is forced to have clustered spectrum. As θ(x,y) is chosen to be a trigonometric polynomial, the preconditioner is a block band Toeplitz matrix with Toeplitz blocks, and therefore its inversion does not increase the total complexity of the PCG method. Preconditioning by block Toeplitz matrices has been treated in the literature in several papers. We compare our method with their results and we show the efficiency of our proposal through various numerical experiments.This research was co-funded by the European Union in the framework of the program “Pythagoras I” of the “Operational Program for Education and Initial Vocational Training” of the 3rd Community Support Framework of the Hellenic Ministry of Education, funded by national sources (25%) and by the European Social Fund - ESF (75%). The work of the second and of the third author was partially supported by MIUR (Italian Ministry of University and Research), grant number 2004015437.     

A numerical study of optimized sparse preconditioners

Preconditioning strategies based on incomplete factorizations and polynomial approximations are studied through extensive numerical experiments. We are concerned with the question of the optimal rate of convergence that can be achieved for these classes of preconditioners.Our conclusion is that the well-known Modified Incomplete Cholesky factorization (MIC), cf. e.g., Gustafsson [20], and the polynomial preconditioning based on the Chebyshev polynomials, cf. Johnson, Micchelli and Paul [22], have optimal order of convergence as applied to matrix systems derived by discretization of the Poisson equation. Thus for the discrete two-dimensional Poisson equation withn unknowns,O(n ^1/4) andO(n ^1/2) seem to be the optimal rates of convergence for the Conjugate Gradient (CG) method using incomplete factorizations and polynomial preconditioners, respectively. The results obtained for polynomial preconditioners are in agreement with the basic theory of CG, which implies that such preconditioners can not lead to improvement of the asymptotic convergence rate.By optimizing the preconditioners with respect to certain criteria, we observe a reduction of the number of CG iterations, but the rates of convergence remain unchanged.Supported by The Norwegian Research Council for Science and the Humanities (NAVF) under grants no. 413.90/002 and 412.93/005.Supported by The Royal Norwegian Council for Scientific and Industrial Research (NTNF) through program no. STP.28402: Toolkits in industrial mathematics.     

Adaptive polynomial preconditioning for hermitian indefinite linear systems

This paper explores the use of polynomial preconditioned CG methods for hermitian indefinite linear systems,Ax=b. Polynomial preconditioning is attractive for several reasons. First, it is well-suited to vector and/or parallel architectures. It is also easy to employ, requiring only matrix-vector multiplication and vector addition. To obtain an optimum polynomial preconditioner we solve a minimax approximation problem. The preconditioning polynomial,C(), is optimum in that it minimizes a bound on the condition number of the preconditioned matrix,C(A)A. We also characterize the behavior of this minimax polynomial, which makes possible a thorough understanding of the associated CG methods. This characterization is also essential to the development of an adaptive procedure for dynamically determining the optimum polynomial preconditioner. Finally, we demonstrate the effectiveness of polynomial preconditioning in a variety of numerical experiments on a Cray X-MP/48. Our results suggest that high degree (20–50) polynomials are usually best.This research was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Dept. of Energy, by Lawrence Livermore National Laboratory under contract W-7405-ENG-48.This research was supported in part by the Dept. of Energy and the National Science Foundation under grant DMS 8704169.This research was supported in part by U.S. Dept. of Energy grant DEFG02-87ER25026 and by National Science Foundation grant DMS 8703226.     

Compactly supported box-spline wavelets

A general procedure for constructing multivariate non-tensor-product wavelets that generate an orthogonal decomposition ofL ²(R)^s,s s≥1, is described and applied to yield explicit formulas for compactly supported spline-wavelets based on the multiresolution analysis ofL ²(R)^s 1≤s≤3, generated by any box spline whose direction set constitutes a unimodular matrix. In particular, when univariate cardinal B-splines are considered, the minimally supported cardinal spline-wavelets of Chui and Wang are recovered. A refined computational scheme for the orthogonalization of spaces with compactly supported wavelets is given. A recursive approximation scheme for “truncated” decomposition sequences is developed and a sharp error bound is included. A condition on the symmetry or anti-symmetry of the wavelets is applied to yield symmetric box-spline wavelets. Partially supported by ARO Grant DAAL 03-90-G-0091 Partially supported by NSF Grant DMS 89-0-01345 Partially supported by NATO Grant CRG 900158.     

Pseudozero Set of Real Multivariate Polynomials

The pseudozero set of a system P of polynomials in n variables is the subset of C ⁿ consisting of the union of the zeros of all polynomial systems Q that are near to P in a suitable sense. This concept arises naturally in Scientific Computing where data often have a limited accuracy. When the polynomials of the system are polynomials with complex coefficients, the pseudozero set has already been studied. In this paper, we focus on the case where the polynomials of the system have real coefficients and such that all the polynomials in all the perturbed polynomial systems have real coefficients as well. We provide an explicit definition to compute this pseudozero set. At last, we analyze different methods to visualize this set.      

On the Hierarchy of screw systems

LetE(3) be the Lie group of proper rigid motions of Euclidean 3-space. The adjoint action ofE(3) on its Lie algebrae(3) induces an action on the Grassmannian of subspaces of given dimensiond. Projectively, these subspaces are the screw systems of classical kinematics. The authors show that existing classifications of screw systems give rise to Whitney regular stratifications of the Grassmannians, and establish diagrams of specialisations for the strata. A list is given of the screw systems which can appear generically for motions of 3-space with at most three degrees of freedom.     

On max-min linear inequalities and Coalitional Resource Games with sharable resources

In a Coalitional Resource Game (CRG for brief), agents form coalitions to pool their resources in order to achieve certain goals, requiring the expenditure of these resources. A particular coalition is said to be successful, if the common resources of its members enable to achieve a set of goals that satisfies all members of the coalition. It is known that when resources are consumable it is NP-complete to decide whether a given coalition is successful. In this paper, we show a connection of CRGs with sharable resources and max-min linear systems of inequalities. This correspondence leads to polynomial algorithms for checking whether a given CRG admits a successful coalition and for several other problems whose counterparts for CRGs with consumable resources are hard. On the other hand, we prove that some problems concerning the structure of successful coalitions are hard also in the case of sharable resources.     

A non-analytic proof of the newman—znám result for disjoint covering systems

A direct combinatorial proof is given to a generalization of the fact that the largest modulusN of a disjoint covering system appears at leastp times in the system, wherep is the smallest prime dividingN. The method is based on geometric properties of lattice parallelotopes. This research was supported by grant 85-00368 from the United States-Is rael Binational Science Foundation, Jerusalem, Israel.     

Differential form methods in the theory of variational systems and Lagrangian field theories

This paper will attempt to unify diverse material from physics and engineering in terms of differential forms on manifolds. A variational system will be defined by means of a scalar-valued differential form on a manifold and an ideal in the Grassmann algebra of differential forms on that manifold to serve as constraints. Two types of extremal submanifolds will be defined. The first-called the Euler-Lagrange extremals-will be defined by a method that is the generalization of the classical methods in the calculus of variations. The second—a generalization of a method used by Cartan in his treatise Leçons sur les invariants intégraux-will define extremals as integral submanifolds of an exterior differential system invariently attached to the variational system. As examples, the variational systems attached to string theories in Riemannian manifolds and Yang-Mills fields will be discussed from this differential form point of view. Finally, as application, the differential geometric properties and definition of energy will be presented from the differential form point of view.This work was supported by a grant from the Applied Mathematics program of the National Science Foundation.     

Weighted median algorithms for L1 approximation

The weighted median problem arises as a subproblem in certain multivariate optimization problems, includingL ₁ approximation. Three algorithms for the weighted median problem are presented and the relationships between them are discussed. We report on computational experience with these algorithms and on their use in the context of multivariateL ₁ approximation.This work was supported in part by National Science Foundation Grant CCR-8713893 and in part by a grant from The City University of New York PSC-CUNY Research Award program.     

Continuity of the spectrum on closed similarity orbits

We present a useful case when the spectral radius of a norm limit of operator similar to a fixed operatorT still equals that ofT.This work was supported in part by grants from the National Science Foundation DMS-8811084, ECS-9001371, ECS-9122106, by the Air Force Office of Scientific Research AFOSR-90-0024 and AFOSR-90-0053, and by the Army Research Office DAAL03-91-G-0019.     

Uniform boundedness of level structures on abelian varieties over complex function fields

Let X = Ω/Γ be a smooth quotient of a bounded symmetric domain Ω by an arithmetic subgroup . We prove the following generalization of Nadel's result: for any non-negative integer g, there exists a finite étale cover X_g = Ω/Γ(g) of X determined by a subgroup depending only on g, such that for any compact Riemann surface R of genus g and any non-constant holomorphic map f : R → X_g^* from R into the Satake-Baily-Borel compactification X_g^* of X_g, the image f(R) lies in the boundary ∂X_g: = X^*_g\X_g. Nadel proved it for g = 0 or 1. Moreover, for any positive integer n and any non-negative integer g≥0, we show that there exists a positive number a(n,g) depending only on n and g with the following property: a principally polarized non-isotrivial n-dimensional abelian variety over a complex function field of genus g does not have a level-N structure for N≥a(n,g). This was proved by Nadel for g = 0 or 1, and by Noguchi for arbitrary g under the additional hypothesis that the abelian variety has non-empty singular fibers.     

On the convergence of the multigrid method for a hypersingular integral equation of the first kind

Summary We present a multigrid method to solve linear systems arising from Galerkin schemes for a hypersingular boundary integral equation governing three dimensional Neumann problems for the Laplacian. Our algorithm uses damped Jacobi iteration, Gauss-Seidel iteration or SOR as preand post-smoothers. If the integral equation holds on a closed, Lipschitz surface we prove convergence ofV- andW-cycles with any number of smoothing steps. If the integral equation holds on an open, Lipschitz surface (covering crack problems) we show convergence of theW-cycle. Numerical experiments are given which underline the theoretical results.     

Uniqueness and minimality of large face-width embeddings of graphs

LetG be a graph embedded in a surface of genusg. It is shown that if the face-width of the embedding is at leastclog(g)/loglog(g), then such an embedding is unique up to Whitney equivalence. If the face-width is at leastclog(g), then every embedding ofG which is not Whitney equivalent to our embedding has strictly smaller Euler characteristic.Supported in part by the Ministry of Science and Technology of Slovenia, Research Project P1-0210-101-94.