Nonsingularity and symmetry for linear normal maps

Normal maps are single-valued, generally nonsmooth functions expressing conditions for the solution of variational problems such as those of optimization or equilibrium. Normal maps arising from linear transformations are particularly important, both in their own right and as predictors of the behavior of related nonlinear normal maps. They are called (locally or globally)nonsingular if the functions appearing in them are (local or global) homeomorphisms satisfying a Lipschitz condition. We show here that when the linear transformation giving rise to such a normal map has a certain symmetry property, the necessary and sufficient condition for nonsingularity takes a particularly simple and convenient form, being simply a positive definiteness condition on a certain subspace.This paper in dedicated to Phil Wolfe on the occasion of his 65th birthday.The research reported here was sponsored by the National Science Foundation under Grant CCR-9109345, by the Air Force Systems Command, USAF, under Grant AFOSR-91-0089, by the U.S. Army Research Office under Contract No. DAAL03-89-K-0149, and by the U.S. Army Strategic Defense Command under Contract DASG60-91-C-0144. The US Government has certain rights in this material, and is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.     

Ovaloids and their symmetry types

A convex body B of dimension n in Euclidean n-space Eⁿ (called hereafter an n-body or simply a body) must have at least n+1 extreme points, this minimum number being attained when B is a simplex. On the other hand, it may happen thatevery point of the frontier B of B in Eⁿ is an extreme point. In this case, we call B anovaloid or n-ovaloid. We study the symmetry classification of n-ovaloids in relation to the symmetry classification of convex n-bodies in general. We show first that the set Oⁿ of n-ovaloids is a dense subset of the space Bⁿ of n-bodies. We show also that for any n-body X there is an ovaloid B with the same symmetry group.While two n-bodies may have conjugate symmetry groups without being symmetry equivalent, we prove that this is not so for n-ovaloids: two n-ovaloids with conjugate symmetry groups are necessarily symmetry equivalent. Consequently, the hierarchy of symmetry types of n-ovaloids conforms exactly to the lattice of conjugacy classes of compact subgroups of O(n) that occur as the symmetry group of some n-body. In particular, the spherical n-ball is the only perfect n-ovaloid.     

Noninvertible maps and their embedding into higher dimensional invertible maps

The first part is devoted to a presentation of specific features of noninvertible maps with respect to the invertible ones. When embedded into a three-dimensional invertible map, the specific dynamical features of a plane noninvertible map are the germ of the three-dimensional dynamics, at least for sufficiently small absolute values of the embedding parameter. The form of the paper, as well as its contents, is approached from a non abstract point of view, in an elementary form from a simple class of examples.     

Exponentially harmonic maps and their properties

We show that the associated quadratic differentials of exponentially harmonic maps are holomorphic under certain circumstance. We study the sufficient and necessary conditions for axially symmetric maps which are exponentially harmonic. We investigate exponentially harmonic equations for rotationally symmetric maps between rotationally symmetric manifolds of low dimensions.     

On biharmonic maps and their generalizations

We give a new proof of regularity of biharmonic maps from four-dimensional domains into spheres, showing first that the biharmonic map system is equivalent to a set of bilinear identities in divergence form. The method of reverse Hölder inequalities is used next to prove continuity of solutions and higher integrability of their second order derivatives. As a byproduct, we also prove that a weak limit of biharmonic maps into a sphere is again biharmonic. The proof of regularity can be adapted to biharmonic maps on the Heisenberg group, and to other functionals leading to fourth order elliptic equations with critical nonlinearities in lower order derivatives.Received: 6 February 2003, Accepted: 12 March 2003, Published online: 16 May 2003Mathematics Subject Classification (2000): 35J60, 35H20Pawel Strzelecki: Current address (till September 2003): Mathematisches Institut der Universität Bonn, Beringstr. 1, 53115 Bonn, Germany (email: strzelec@math.uni-bonn.de). The author is partially supported by KBN grant no. 2-PO3A-028-22;he gratefully acknowledgesthe hospitality of his colleagues from Bonn,and the generosity of Humboldt Foundation.     

Square turning maps and their compactifications

In this paper we introduce some infinite rectangle exchange transformations which are based on the simultaneous turning of the squares within a sequence of square grids. We will show that such noncompact systems have higher dimensional dynamical compactifications. In good cases, these compactifications are polytope exchange transformations based on pairs of Euclidean lattices. In each dimension \(8m+4\) there is a \(4m+2\) dimensional family of them. Here \(m=0,1,2,\ldots \) We studied the case \(m=0\) in depth in Schwartz (The octagonal PETs, research monograph, 2012).     

On the symmetry of images of word maps in groups

Word maps on a group are defined by substitution of formal words. Lubotzky gave a characterization of the images of word maps in finite simple groups, and a consequence of his characterization is the existence of a group G such that the image of some word map on G is not closed under inversion. We show that there are only two groups with order less than 108 with the property that there is a word map with image not closed under inversion. We also study this behavior in nilpotent groups.     

The 24 symmetry pairings of self-dual maps on the sphere

Given a self-dual map on the sphere, the collection of its self-dual permutations generates a transformation group in which the map automorphism group appears as a subgroup of index two. A careful examination of this pairing yields direct constructions of self-dual maps and provides a classification of self-dual maps.     

Reversible linear systems and their versal deformations

Let R be a fixed linear involution (R ²=id) of the spaceR ⁿ . A linear operator M is said to bereversible with respect to R if RM R=M^–1 and infinitesimally reversible with respect to R if M R=–RM. A linear differential equation dx/dt=B(t)x is said to be reversible with respect to R if V(t)R –RV(–t). We construct normal forms and versal deformations for reversible and infinitesimally reversible operators. The results are applied to describe the homotopy classes of strongly stable reversible linear differential equations with periodic coefficients. The analogous theory for linear Hamiltonian systems was developed by J. Williamson, M. G. Krein, I.M. Gel'fand, V. B. Lidskii, D. M. Galin, and H. Koçak.Translated fromTrudy Seminara imeni I. G. Petrovskogo, No. 15, pp. 33–54, 1991. Original article submitted April 27, 1988.     

Singular Lagrangian manifolds and their Lagrangian maps

The paper examines the singularity theory of Lagrangian manifolds and its connection with variational calculus, classification of Coxeter groups, and symplectic topology. We consider the application of the theory to the problem of going past an obstacle, to partial differential equations, and to the analysis of singularities of ray systems.Translated from Itogi Nauki i Tekhniki, Seriya Sovermennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 33, pp. 55–112, 1988.     

Link maps and the geometry of their invariants

The geometry of two types of link homotopy invariants of a link map f:S^pS^qS^m is discussed. The first one is the -invariant which greatly generalizes the classical notion of linking number. The second one, the -invariant, is closely related to the linking behaviour of f|s^p with only the double point set of f|S^q, and therefore measures (to some extend) the obstruction to embedding S^q. These invariants are related by a Hopf invariant homomorphism. In many cases link maps are classified up to link homotopy here, and a setting is provided e.g. for future injectivity results for . Also the image of is studied, yielding an interesting double filtration of stable homotopy groups of spheres.     

Virtually stable maps and their fixed point sets

We introduce the concept of virtually stable selfmaps of Hausdorff spaces, which generalizes virtually nonexpansive selfmaps of metric spaces introduced in the previous work by the first author, and explore various properties of their convergence sets and fixed point sets. We also prove that the fixed point set of a virtually stable selfmap satisfying a certain kind of homogeneity is always star-convex.     

Projective ring lines and their generalisations

We give a survey on projective ring lines and some of their substructures which in turn are more general than a projective line over a ring.     

Reconstructing birational maps from their face functions

We first prove that any birational map, from an affine space of dimension ≥ 2 to itself, is not determined by its face functions. On the other hand, we prove that a birational map with irreducibly polynomial inverse is completely determined, within the class of all birational maps with irreducibly polynomial inverses, by its face functions. We show also how to effectively reconstruct such a map from its face functions. Supported partly by the Centre Interuniversitaireen Calcul Mathématique Algébrique.     

On essential spectra of operator-matrices and their Feshbach maps

A connection between the essential spectrum of certain operator-matrices and essential spectra of the corresponding “Feshbach maps” is discussed and applied to some concrete rational operator-valued functions.     

Higher dimensional extensions of substitutions and their dual maps

Given a substitution σ ond letters, we define itsk-dimensional extension,E _k (σ), for 0≤k≤d. Thek-dimensional extension acts on the set ofk-dimensional faces of unit cubes inR ^d with integer vertices. The extensions of a substitution satisfy a commutation relation with the natural boundary operator: the boundary of the image is the image of the boundary. We say that a substitution is unimodular (resp. hyperbolic) if the matrix associated to the substitution by abelianization is unimodular (resp. hyperbolic). In the case where the substitution is unimodular, we also define dual substitutions which satisfy a similar coboundary condition. We use these constructions to build self-similar sets on the expanding and contracting space for an hyperbolic substitution.     

CR submanifolds in a sphere and their Gauss maps

The relationship between CR submanifolds in a sphere and their Gauss maps are investigated.Let V be the image of a sphere by a rational holomorphic map F with degree two in another sphere.It is show that the Gauss map of V is degenerate if and only if F is linear fractional.