Lower bounds for nonoverlapping domain decomposition preconditioners in two dimensions

Lower bounds for the condition numbers of the preconditioned systems are obtained for the Bramble-Pasciak-Schatz substructuring preconditioner and the Neumann-Neumann preconditioner in two dimensions. They show that the known upper bounds are sharp.

     

Lower bounds for dimensions of representation varieties

The set of -dimensional complex representations of a finitely generated group form a complex affine variety . Suppose that is such a representation and consider the associated representation on complex matrices obtained by following with conjugation of matrices. Then it is shown that the dimension of at is at least the difference of the complex dimensions of and . It is further shown that in the latter cohomology may be replaced by various proalgebraic groups associated to and .

     

Anisotropic error bounds of Lagrange interpolation with any order in two and three dimensions

In this paper, using the Newton’s formula of Lagrange interpolation, we present a new proof of the anisotropic error bounds for Lagrange interpolation of any order on the triangle, rectangle, tetrahedron and cube in a unified way.     

Planforms in two and three dimensions

Summary When solving systems of PDE with two space dimensions it is often assumed that the solution is spatially doubly periodic. This assumption is usually made in systems such as the Boussinesq equation or reaction-diffusion equations where the equations have Euclidean invariance. In this article we use group theoretic techniques to determine a large class of spatially doubly periodic solutions that are forced to existence near a steady-state bifurcation from a translation-invariant equilibrium.This type of bifurcation problem has been considered by many authors when studying a number of different systems of PDE. Typically, these studies focus at the beginning on equilibria that are spatially periodic with respect to a fixed planar lattice type-such as square or hexagonal. Our focus is different in that we attempt to find all spatially periodic equilibria that bifurcate on all lattices. This point of view leads to some technical simplifications such as being able to restrict to translation free irreducible representations.Of course, many of the types of solutions that we find are well-known-such as hexagon and roll solutions on a hexagonal lattice. This coordinated group theoretic approach does lead, however, to solutions which seem not to have been discussed previously (antisquare solutions on a square lattice) as well as to a more complete classification of the symmetry types of possible solutions. Moreover, our methods extend to triply periodic solutions of PDE with three spatial variables. Some of these results, namely those concerned with primitive cubic lattices, are presented here. The complete results on triply periodic solutions may be found in [6, 7].In honor of Klaus Kirchgässner on the occasion of his sixtieth birthdayResearch supported in part by NSF/DARPA (DMS-8700897) and by the Texas Advanced Research Program (ARP-1100).     

Lower bounds of the life span of classical solutions to a system of semilinear wave equations in two space dimensions

In this paper we consider a system of semilinear wave equations in two space dimensions and with propagation speeds possibly different from one. Under smallness assumptions on the data, we show lower bounds for the life span of classical solutions.     

Composites and quasiconformal mappings: new optimal bounds in two dimensions

We prove new bounds for the homogenized tensor of two dimensional multiphase conducting composites. The bounds are optimal for a large class of composites. In physical terms these are mixtures of one polycrystal and several isotropic phases, with prescribed volume fractions. Optimality is understood in the strongest possible sense of exact microgeometries. The techniques to prove the bounds for composites are based on variational methods and results from quasiconformal mappings. We need to refine the quasiconformal area distortion theorem due to the first author and prove new distortion results with weigths. These distortion theorems are of independent interest for PDEs and quasiconformal mappings. They imply e.g. the following surprising theorem on integrability of derivatives at the borderline case: For K > 1, if is K-quasiregular, if is measurable and bounded and if a.e. in E, then Received: 20 January 2001, Accepted: 25 October 2001, Published online: 6 June 2003Mathematics Subject Classification (2000): 74Q20, 35B27, 30C62     

Extremal problems on triangle areas in two and three dimensions

The study of extremal problems on triangle areas was initiated in a series of papers by Erd?s and Purdy in the early 1970s. In this paper we present new results on such problems, concerning the number of triangles of the same area that are spanned by finite point sets in the plane and in 3-space, and the number of distinct areas determined by the triangles.In the plane, our main result is an O(n44/19)=O(n^2.3158) upper bound on the number of unit-area triangles spanned by n points, which is the first breakthrough improving the classical bound of O(n7/3) from 1992. We also make progress in a number of important special cases. We show that: (i) For points in convex position, there exist n-element point sets that span Ω(nlogn) triangles of unit area. (ii) The number of triangles of minimum (nonzero) area determined by n points is at most ; there exist n-element point sets (for arbitrarily large n) that span (6/π²−o(1))n² minimum-area triangles. (iii) The number of acute triangles of minimum area determined by n points is O(n); this is asymptotically tight. (iv) For n points in convex position, the number of triangles of minimum area is O(n); this is asymptotically tight. (v) If no three points are allowed to be collinear, there are n-element point sets that span Ω(nlogn) minimum-area triangles (in contrast to (ii), where collinearities are allowed and a quadratic lower bound holds).In 3-space we prove an O(n17/7β(n))=O(n^2.4286) upper bound on the number of unit-area triangles spanned by n points, where β(n) is an extremely slowly growing function related to the inverse Ackermann function. The best previous bound, O(n8/3), is an old result of Erd?s and Purdy from 1971. We further show, for point sets in 3-space: (i) The number of minimum nonzero area triangles is at most n²+O(n), and this is worst-case optimal, up to a constant factor. (ii) There are n-element point sets that span Ω(n4/3) triangles of maximum area, all incident to a common point. In any n-element point set, the maximum number of maximum-area triangles incident to a common point is O(n4/3+ε), for any ε>0. (iii) Every set of n points, not all on a line, determines at least Ω(n2/3/β(n)) triangles of distinct areas, which share a common side.     

Classification of gradient-like flows on dimensions two and three

We consider here gradient-like flows of classC ^r ( ^r ≥1) on a closed manifoldM of classC ^r+1 and dimension two or three. We study the classification of these flows by the relation of topological equivalence. In this sense, the flows which are more relevant are the polar flows (only one source and only one sink).     

The Schrödinger equation with a moving point interaction in three dimensions

In the case of a single point interaction we improve, by using different methods, the existence theorem for the unitary evolution generated by a Schrödinger operator with moving point interactions obtained by Dell'Antonio, Figari and Teta.

     

A discrete Korn's inequality in two and three dimensions

In this paper, we present a simple and general proof for Korn's inequality for nonconforming elements, like Wilson's Element and Carey's Element.     

On free Zp-torus actions in dimensions two and three

We confirm the Halperin-Carlsson conjecture for free Z_p-torus actions(p is a prime) on 2-dimensional finite CW-complexes and free Z_2-torus actions on closed 3-manifolds.     

Almost tight upper bounds for the single cell and zone problems in three dimensions

We consider the problem of bounding the combinatorial complexity of a single cell in an arrangement ofn low-degree algebraic surface patches in 3-space. We show that this complexity isO(n ^2+ε), for any ε>0, where the constant of proportionality depends on ε and on the maximum degree of the given surfaces and of their boundaries. This extends several previous results, almost settles a 9-year-old open problem, and has applications to motion planning of general robot systems with three degrees of freedom. As a corollary of the above result, we show that the overall complexity of all the three-dimensional cells of an arrangement ofn low-degree algebraic surface patches, intersected by an additional low-degree algebraic surface patch σ (the so-calledzone of σ in the arrangement) isO(n ^2+ε), for any ε>0, where the constant of proportionality depends on ε and on the maximum degree of the given surfaces and of their boundaries. Work on this paper by the first author has been supported by a Rothschild Postdoctoral Fellowship, by a grant from the Stanford Integrated Manufacturing Association (SIMA), by NSF/ARPA Grant IRI-9306544, and by NSF Grant CCR-9215219. Work on this paper by the second author has been supported by NSF Grants CCR-91-22103 and CCR-93-111327, and by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F., the German-Israeli Foundation for Scientific Research and Development, and the Israel Science Fund administered by the Israeli Academy of Sciences.     

Lower bounds on the cover-index of a graph

In this paper we give some new lower bounds for the cover-index of graphs with multiple edges permitted. The results are analogous to upper bounds for the chromatic index. We show that a simple graph with cover-index different from the minimum degree has at least three vertices of minimum degree. This implies that almost all simple graphs have cover-index equal to the minimum degree.