Compact,convex sets in R n and a new banach lattice. I.- Theory.

The present paper shows that compact, non-empty convex sets in R ⁿ form a wedge in a well-defined Banach lattice, which turns out to be isometrically Riesz-isomorphic to the continuous functions in S ^n–1, the unit sphere of R ⁿ . Among other results, we obtain Dini-like convergence results for sets, linking order- and norm-convergence.     

What do multistep methods approximate?

Summary Standard analysis of multistep methods for ODE's assumes the application of an initialization routine that generates the starting points. Here ak-step method is considered directly as a mappingR ^kn R ⁿ . It is shown to approximate a mapping which is expressible directly in terms of the flow of the vector field. Some useful properties of that mapping are shown and for strictly stable methods these are applied to the question of invariant circles near a hyperbolic periodic solution.     

Correlated equilibria of games with many players

Let G _m,n be the class of strategic games with n players, where each player has m≥2 pure strategies. We are interested in the structure of the set of correlated equilibria of games in G _m,n when n→∞. As the number of equilibrium constraints grows slower than the number of pure strategy profiles, it might be conjectured that the set of correlated equilibria becomes large. In this paper, we show that (1) the average relative measure of the set of correlated equilibria is smaller than 2⁻ⁿ; and (2) for each 1<c<m, the solution set contains c ⁿ correlated equilibria having disjoint supports with a probability going to 1 as n grows large. The proof of the second result hinges on the following inequality: Let c ₁, …, c _l be independent and symmetric random vectors in R ^k, l≥k. Then the probability that the convex hull of c ₁, …, c _l intersects R ^k ₊ is greater than or equal to . Received: December 1998/Final version: March 2000     

Discrete isoperimetric and Poincaré-type inequalities

We study some discrete isoperimetric and Poincaré-type inequalities for product probability measures μ ⁿ on the discrete cube {0, 1} ⁿ and on the lattice Z ⁿ . In particular we prove sharp lower estimates for the product measures of boundaries of arbitrary sets in the discrete cube. More generally, we characterize those probability distributions μ on Z which satisfy these inequalities on Z ⁿ . The class of these distributions can be described by a certain class of monotone transforms of the two-sided exponential measure. A similar characterization of distributions on R which satisfy Poincaré inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes. Received: 30 April 1997 / Revised version: 5 June 1998     

Embedding a polytope in a lattice

We present a special similarity ofR ⁴ⁿ which maps lattice points into lattice points. Applying this similarity, we prove that if a (4n−1)-polytope is similar to a lattice polytope (a polytope whose vertices are all lattice points) inR ⁴ⁿ , then it is similar to a lattice polytope inR ⁴ⁿ⁻¹, generalizing a result of Schoenberg [4]. We also prove that ann-polytope is similar to a lattice polytope in someR ^N if and only if it is similar to a lattice polytope inR ²ⁿ⁺¹, and if and only if sin²(<ABC) is rational for any three verticesA, B, C of the polytope.     

An Optimal Inequality and Extremal Classes of Affine Spheres in Centroaffine Geometry

A hypersurface f : M → Rⁿ⁺¹ in an affine (n+1)-space is called centroaffine if its position vector is always transversal to f_*(TM) in Rⁿ⁺¹. In this paper, we establish a general optimal inequality for definite centroaffine hypersurfaces in Rⁿ⁺¹ involving the Tchebychev vector field. We also completely classify the hypersurfaces which verify the equality case of the inequality.     

Cohomogeneity one Alexandrov spaces

We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and 4. As a corollary we obtain the classification of closed, n-dimensional, cohomogeneity one Alexandrov spaces admitting an isometric T ⁿ⁻¹ action. In contrast to the one- and two-dimensional cases, where it is known that an Alexandrov space is a topological manifold, in dimension 3 the classification contains, in addition to the known cohomogeneity one manifolds, the spherical suspension of \mathbbRP² \mathbb{R}{P^2} , which is not a manifold.     

Smooth Quasiregular Maps with Branching in R n

According to a theorem of Martio, Rickman and Väisälä, all nonconstant C^n/(n-2)-smooth quasiregular maps in Rⁿ, n≥3, are local homeomorphisms. Bonk and Heinonen proved that the order of smoothness is sharp in R³. We prove that the order of smoothness is sharp in R⁴. For each n≥5 we construct a C^1+ε(n)-smooth quasiregular map in Rⁿ with nonempty branch set.     

The natural functions on the cotangent bundle of higher order vector tangent bundles over fibered manifolds

For natural numbers r,s,q,m,n with s≥r≤q we determine all natural functions g: T ^*(J ^(r,s,q)(Y, R ^1,1)₀)^*→R for any fibered manifold Y with m-dimensional base and n-dimensional fibers. For natural numbers r,s,m,n with s≥r we determine all natural functions g: T ^*(J ^(r,s) (Y, R)₀)^*→R for any Y as above.     

Focal points and support functions in affine differential geometry

The notions of focal point and support function are considered for a nondegenerate hypersurfaceM ⁿ in affine spaceR ⁿ⁺¹ equipped with an equiaffine transversal field. IfM ⁿ is locally strictly convex, these two concepts are related via an Index theorem concerning the critical points of the support functions onM ⁿ . This is used to obtain characterizations of spheres and ellipsoids in terms of the critical point behavior of certain classes of affine support functions.Research supported by NSF Grant No. DMS-9101961.     

On the division by R n

Summary In this paper we prove that, ifS ×R ⁿ is homeomorphic toR ^{n + 1}, thenS is homeomorphic toR.     

Unbounded quadrature domains inR n (n≥3)

As a result of the powerful tools of complex analysis a lot of problems have been solved in the theory of Q.D.s (quadrature domain) inR ². These problems are almost untouched inR ⁿ (n≥3). To study Q.D.s inR ⁿ , one has to supply the subject with new techniques. This is the goal of papers by Shapiro, Khavinson and Shapiro, Sakai, and Gustafsson, where the authors approach the subject inR ⁿ by different methods. The main purpose of this paper is to generalize some of the ideas (already known inR ²) toR ⁿ (n>-3), and we merely work with unbounded Q.D.     

Witt equivalence of local and global fields of characteristic 2

ABSTRACT

In this note it is proved that certain level sets of some real proper polynomial maps are nothing but spheres. As an application of this, we provide new proofs of Theorems 1.1, 1.2 and of the fundamental theorem of algebra. In addition, we show that every strictly convex (concave) polynomial map is proper. The latter implies that every real polynomial map g(x): R ⁿ  → R ⁿ , whose Jacobian matrix is symmetric and has nonzero eigenvalues of the same sign, is a homeomorphism of R ⁿ onto R ⁿ .     

Degenerate principal series and local theta correspondence II

Following our previous paper [LZ] which deals with the groupU(n, n), we study the structure of certain Howe quotients Ω ^p,q and Ω ^p,q (1) which are natural Sp(2n,R) modules arising from the Oscillator representation associated with the dual pair (O(p, q), Sp(2n,R)), by embedding them into the degenerate principal series representations of Sp(2n,R) studied in [L2].     

Singularity of mean curvature flow of Lagrangian submanifolds

In this article we study the tangent cones at first time singularity of a Lagrangian mean curvature flow. If the initial compact submanifold ₀ is Lagrangian and almost calibrated by Re in a Calabi-Yau n-fold (M,), and T>0 is the first blow-up time of the mean curvature flow, then the tangent cone of the mean curvature flow at a singular point (X₀,T) is a stationary Lagrangian integer multiplicity current in R²ⁿ with volume density greater than one at X₀. When n=2, the tangent cone is a finite union of at least two 2-planes in R⁴ which are complex in a complex structure on R⁴.     

Global existence and uniform decay for the coupled Klein-Gordon-Schr?dinger equations

This paper is concerned to the existence, uniqueness and uniform decay for the solutions of the coupled Klein-Gordon-Schr?dinger damped equations where ω is a bounded domain of R ⁿ , n≤ 3, F : R ²→R is a C ¹-function; γ, β; θ are constants such that γ, β > 0 and 1 ≤ 2θ≤ 2. Received January 1999 – Accepted October 1999     

Fast Thermoviscous Convection

This paper studies the asymptotic structure of convection in an infinite Prandtl number fluid with strongly temperature-dependent viscosity, in the limit where the dimensionless activation energy 1/ε is large, and the Rayleigh number R, defined (essentially) with the basal viscosity and the prescribed temperature drop, is also large. We find that the Nusselt number N is given by N~CεR^1/5, where C depends on the aspect ratio a. The relative error in this result is O(R^?1/10ε^?1/4, ε^1/2, R^?2/5ε^?2, R^?2/20ε^?1/24), so that we cannot hope to find accurate confirmation of this result at moderate Rayleigh numbers, though it should serve as a useful indicator of the relative importance of R and ε. For the above result to be valid, we require R ? 1/ε⁵ ?1. More important, however, is the asymptotic structure of the flow: there is a cold (hence rigid) lid with sloping base, beneath which a rapid, essentially isoviscous, convection takes place. This convection is driven by plumes at the sides, which generate vorticity due to thermal buoyancy, as in the constant viscosity case (Roberts, 1979). However, the slope of the lid base is sufficient to cause a large shear stress to be generated in the thermal boundary layer which joins the lid to the isoviscous region underneath (though a large velocity is not generated); consequently, the layer does not “see” the shear stress exerted by the interior flow (at leading order), and therefore the thermal boundary layer structure is totally self-determined: it even has a similarity structure (as a consequence). This fact makes it easy to analyse the problem, since the boundary layer uncouples from the rest of the flow. In addition, we find an alternative scaling (in which the lid base is “almost” flat), but it seems that the resulting boundary layer equations have no solution, though this is certainly open to debate: the results quoted above are not for this case. When a free slip boundary condition is applied at the top surface, one finds that there exists a thin “skin” at the top of the lid which is a stress boundary layer. The shear stress changes rapidly to zero, and there exists a huge longitudinal stress (compressive/tensile) in this skin. For earthlike parameters, this stress far exceeds the fracture strength of silicate rocks.     

Spectral convergence for vibrating systems containing a part with negligible mass

We consider a set of Neumann (mixed, respectively) eigenvalue problems for the Laplace operator. Each problem is posed in a bounded domain Ω_R of ?ⁿ, with n=2,3, which contains a fixed bounded domain B where the density takes the value 1 and 0 outside. Ω_R has a diameter depending on a parameter R, with R?1, diam(Ω_R) →∞ as R→∞ and the union of these sets is the whole space ?ⁿ (the half space {x∈?ⁿ/x_n<0}, respectively). Depending on the dimension of the space n, and on the boundary conditions, we describe the asymptotic behaviour of the eigenelements as R→∞. We apply these asymptotics in order to derive important spectral properties for vibrating systems with concentrated masses. Copyright © 2005 John Wiley & Sons, Ltd.     

The Bernstein problem for affine maximal hypersurfaces

In this paper, we prove the validity of the Chern conjecture in affine geometry [18], namely that an affine maximal graph of a smooth, locally uniformly convex function on two dimensional Euclidean space, R ², must be a paraboloid. More generally, we shall consider the n-dimensional case, R ⁿ , showing that the corresponding result holds in higher dimensions provided that a uniform, “strict convexity” condition holds. We also extend the notion of “affine maximal” to non-smooth convex graphs and produce a counterexample showing that the Bernstein result does not hold in this generality for dimension n≥10. Oblatum 16-IV-1999 & 4-XI-1999?Published online: 21 February 2000