Typical ℤn-actions can be inserted only in injective ℝn-actions

We study actions of the groups ?ⁿ and ?ⁿ by Lebesgue space automorphisms. We prove that a typical ?ⁿ-action can be inserted only in injective actions of ?ⁿ, n ∈ ?. We give a simple proof of the fact that a typical ?²-action cannot be inserted in an ?-action.     

On the Discrete Poincaré–Friedrichs Inequalities for Nonconforming Approximations of the Sobolev Space H 1

We present a direct proof of the discrete Poincaré–Friedrichs inequalities for a class of nonconforming approximations of the Sobolev space H ¹(Ω), indicate optimal values of the constants in these inequalities, and extend the discrete Friedrichs inequality onto domains only bounded in one direction. We consider a polygonal domain Ω in two or three space dimensions and its shape-regular simplicial triangulation. The nonconforming approximations of H ¹(Ω) consist of functions from H ¹ on each element such that the mean values of their traces on interelement boundaries coincide. The key idea is to extend the proof of the discrete Poincaré–Friedrichs inequalities for piecewise constant functions used in the finite volume method. The results have applications in the analysis of nonconforming numerical methods, such as nonconforming finite element or discontinuous Galerkin methods.     

A fixed point theorem for the infinite-dimensional simplex

We define the infinite-dimensional simplex to be the closure of the convex hull of the standard basis vectors in R^∞, and prove that this space has the fixed point property: any continuous function from the space into itself has a fixed point. Our proof is constructive, in the sense that it can be used to find an approximate fixed point; the proof relies on elementary analysis and Sperner's lemma. The fixed point theorem is shown to imply Schauder's fixed point theorem on infinite-dimensional compact convex subsets of normed spaces.     

An ergodic theorem for Schlögl models with small migration

Summary We consider a class of reaction-diffusion processes with state space N^Zd. The reaction part is described by a birth and death process where the rates are given by certain polynomials. The diffusion part is an irreducible symmetric random walk. We prove ergodicity in the case of a sufficiently small migration rate. For the proof we couple two processes and show that the density of the discrepancies goes to zero.     

A linear-time algorithm for edge-disjoint paths in planar graphs

In this paper we discuss the problem of finding edge-disjoint paths in a planar, undirected graph such that each path connects two specified vertices on the boundary of the graph. We will focus on the “classical” case where an instance additionally fulfills the so-calledevenness-condition. The fastest algorithm for this problem known from the literature requiresO (n ^5/3(loglogn)^1/3) time, wheren denotes the number of vertices. In this paper now, we introduce a new approach to this problem, which results in anO(n) algorithm. The proof of correctness immediately yields an alternative proof of the Theorem of Okamura and Seymour, which states a necessary and sufficient condition for solvability.     

On the Generalized Ray-Singer Conjecture for Covering Spaces

We find a relation between L² -combinatorial torsion and analytic torsion associated with a small spectrum of the Witten-type Laplacian on a covering space. The result is used as one crucial step in a proof of the generalized Ray-Singer conjecture for covering spaces.     

Conformal uniformization and packings

A new short proof is given for Brandt and Harrington’s theorem about conformal uniformizations of planar finitely connected domains as domains with boundary components of specified shapes. This method of proof generalizes to periodic domains. Letting the uniformized domains degenerate in a controlled manner, we deduce the existence of packings of specified shapes and with specified combinatorics. The shapes can be arbitrary smooth disks specified up to homothety, for example. The combinatorics of the packing is described by the contacts graph, which can be specified to be any finite planar graph whose vertices correspond to the shapes. This is in the spirit of Koebe’s proof of the Circle Packing Theorem as a consequence of his uniformization by circle domains. The author thankfully acknowledges support of NSF grant DMS-9112150.     

A Filtered Version of the Bipolar Theorem of Brannath and Schachermayer

We extend the Bipolar Theorem of Kramkov and Schachermayer⁽¹²⁾ to the space of nonnegative càdlàg supermartingales on a filtered probability space. We formulate the notion of fork-convexity as an analogue to convexity in this setting. As an intermediate step in the proof of our main result we establish a conditional version of the Bipolar theorem. In an application to mathematical finance we describe the structure of the set of dual processes of the utility maximization problem of Kramkov and Schachermayer⁽¹²⁾ and give a budget-constraint characterization of admissible consumption processes in an incomplete semimartingale market.     

The universal partition theorem for oriented matroids

We consider realization spaces of a family of oriented matroids of rank three as point configurations in the affine plane. The fundamental problem arises as to which way these realization spaces partition their embedding space. The Universal Partition Theorem roughly states that such a partition can be as complicated as any partition of ℝ ⁿ into elementary semialgebraic sets induced by an arbitrary finite set of polynomials in ℤ[X]. We present the first proof of the Universal Partition Theorem. In particular, it includes the first complete proof of the so-called Universality Theorem. This work was supported by the Deutsche Forschungsgemeinschaft, Graduiertenkolleg “Analyse und Konstruktion in der Mathematik”.     

Optimal regularity for the Signorini problem

We prove under general assumptions that solutions of the thin obstacle or Signorini problem in any space dimension achieve the optimal regularity C ^1,1/2. This improves the known optimal regularity results by allowing the thin obstacle to be defined in an arbitrary C ^1,β hypersurface, β > 1/2, additionally, our proof covers any linear elliptic operator in divergence form with smooth coefficients. The main ingredients of the proof are a version of Almgren’s monotonicity formula and the optimal regularity of global solutions.     

Hausdorff's moment problem and expansions in Legendre polynomials

A new proof is given for Hausdorff's condition on a set of moments which determines when the function generating these moments is in L². The proof uses Legendre polynomials and their discrete extensions found by Tchebychef. Then an extension is given to a weighted L² space using Jacobi polynomials and their discrete extensions.     

Two-dimensional quotients of C<Superscript>n</Superscript> are isomorphic to C<Superscript>2</Superscript>/Γ

We will prove that any two-dimensional quotient of an affine space modulo a reductive algebraic group is isomorphic to a quotient of C² modulo a finite group. The proof uses some new results due to Koras and Russell on contractible surfaces with at most quotient singularities and also several results about reductive group actions on affine varieties.     

The theorem of Mather on generic projections in the setting of algebraic geometry

We present the proof of the theorem of Mather on generic projections, stated in the setting of algebraic geometry. The main tools used are the Thom-Boardman singularities in the jet space. This theorem has been applied in the study of codimension two submanifold ofP ⁿ and it seems that it could have further applications. Work partially supported by MURST     

Existence of Weak Solutions to a Class of Fourth Order Partial Differential Equations with Wasserstein Gradient Structure

We prove the global-in-time existence of nonnegative weak solutions to a class of fourth order partial differential equations on a convex bounded domain in arbitrary spatial dimensions. Our proof relies on the formal gradient flow structure of the equation with respect to the L²-Wasserstein distance on the space of probability measures. We construct a weak solution by approximation via the time-discrete minimizing movement scheme; necessary compactness estimates are derived by entropy-dissipation methods. Our theory essentially comprises the thin film and Derrida-Lebowitz-Speer-Spohn equations.     

Stationary solutions to the drift–diffusion model in the whole spaces

We study the stationary problem in the whole space ?ⁿ for the drift–diffusion model arising in semiconductor device simulation and plasma physics. We prove the existence and uniqueness of stationary solutions in the weighted L^p spaces. The proof is based on a fixed point theorem of the Leray–Schauder type. Copyright © 2008 John Wiley & Sons, Ltd.     

Line congruences as surfaces in the space of lines

The space of lines in R³ can be viewed as a four dimensional homogeneous space of the group of Euclidean motions, E(3). Line congruences arise in the classical method of transforming one surface to another by lines. These transformations are particularly interesting if some geometric property of the original surface is preserved. Line congruences, then, are two parameter families of lines and can be studied as surfaces in the space of lines. In this paper, we use the method of moving frames to study line congruences. We calculate the first order invariants of line congruences for which there are two real focal surfaces, and give the geometric meaning of these invariants. We look specifically at the case where the two first order invariants are constant and give a simple proof of Bäcklund's Theorem which relates to the transformation of one pseudospherical surface, a surface of constant negative Gaussian curvature, to another. These transformations are of interest since pseudospherical surfaces correspond to solutions to the sine-Gordon equation. We also give a proof of Bianchi's permutability theorem for pseudospherical surfaces in this context. Finally, we use the results of these theorems to generate some pseudospherical surfaces. All of these concepts and results are understood in terms of the structure equations of the line congruence.     

Spacelike Hypersurfaces with Constant Scalar Curvature in the Lorentz–Minkowski Space

We show that the only compact spacelike hypersurfaces in the Lorentz–Minkowski space Lⁿ⁺¹ having nonzero constant scalar curvature and spherical boundary are the hyperbolic caps (with negative constant scalar curvature). One key ingredient in our proof will be an integral formula for the n-dimensional volume enclosed by the boundary of a compact spacelike hypersurface, in the case where the boundary is contained in a hyperplane of Lⁿ⁺¹. As a direct application of that integral formula we also derive an interesting result for the volume of spacelike hypersurfaces.     

Adjoints of rationally induced composition operators

We give an elementary proof of a formula recently obtained by Hammond, Moorhouse, and Robbins for the adjoint of a rationally induced composition operator on the Hardy space H² [Christopher Hammond, Jennifer Moorhouse, Marian E. Robbins, Adjoints of composition operators with rational symbol, J. Math. Anal. Appl. 341 (2008) 626-639]. We discuss some variants and implications of this formula, and use it to provide a sufficient condition for a rationally induced composition operator adjoint to be a compact perturbation of a weighted composition operator.     

Extrinsic isoperimetry and compactification of minimal surfaces in Euclidean and hyperbolic spaces

We study the topology of (properly) immersed complete minimal surfaces P ² in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces (see [10]). We present an alternative and unified proof of the Chern-Osserman inequality satisfied by these minimal surfaces (in ? ⁿ and in ? ⁿ (b)), based in the isoperimetric analysis mentioned above. Finally, we show a Chern-Osserman-type equality attained by complete minimal surfaces in the Hyperbolic space with finite total extrinsic curvature.     

The Limiting Absorption Principle for Elastic Wave Propagation Problems in Perturbed Stratified Media ℝ3

We consider the self-adjoint operator governing the propagation of elastic waves in perturbed stratified media ℝ³ with free boundary–interface conditions. In this paper we establish the limiting absorption principle for this self-adjoint operator in appropriate Hilbert space. The proof of the limiting absorption principle is based on the division theorem which is proved by means of eigenfunction expansions for the self-adjoint operator governing the propagation of elastic waves in unperturbed stratified media ℝ³.