Random walks generated by affine mappings

This article is concerned with Markov chains on ^m constructed by randomly choosing an affine map at each stage, and then making the transition from the current point to its image under this map. The distribution of the random affine map can depend on the current point (i.e., state of the chain). Sufficient conditions are given under which this chain is ergodic.     

On the Tikhonov regularization of affine pseudomonotone mappings

The pseudomonotonicity of affine mappings on polyhedral convex sets is characterized in the one-dimensional case and in a higher-dimensional setting. The obtained results allow us to investigate the pseudomonotonicity of the regularized mappings (in the sense of Tikhonov regularization). Among other things, it is shown that there exists a pseudomonotone affine variational inequality problem VI( $K,F$ ) with a nonempty solution set for which the regularized problem VI( $K,F_\varepsilon $ ) is not pseudomonotone for every $\varepsilon \in (0,\frac{1}{2})$ . In addition, we prove that the feasibility of a pseudomonotone linear complementarity problem implies the solution uniqueness of the regularized problem.     

Efficient algorithms for the recognition of topologically conjugate gradient-like diffeomorhisms

It is well known that the topological classification of structurally stable flows on surfaces as well as the topological classification of some multidimensional gradient-like systems can be reduced to a combinatorial problem of distinguishing graphs up to isomorphism. The isomorphism problem of general graphs obviously can be solved by a standard enumeration algorithm. However, an efficient algorithm (i. e., polynomial in the number of vertices) has not yet been developed for it, and the problem has not been proved to be intractable (i. e., NPcomplete). We give polynomial-time algorithms for recognition of the corresponding graphs for two gradient-like systems. Moreover, we present efficient algorithms for determining the orientability and the genus of the ambient surface. This result, in particular, sheds light on the classification of configurations that arise from simple, point-source potential-field models in efforts to determine the nature of the quiet-Sun magnetic field.     

On strictly ergodic models which are not almost topologically conjugate

Answering a question raised by Glasner and Rudolph (1984) we construct uncountably many strictly ergodic topological systems which are metrically isomorphic to a given ergodic system (X, ℬ,μ, T) but not almost topologically conjugate to it. This paper is part of the second author’s Ph.D. thesis, written under the supervision of Professor A. Bellow of the Department of Mathematics, Northwestern University. The author is grateful for her encouragement and advice. We acknowledge B. Weiss for helpful comments.     

Regular functions in a semiplane which are topologically equivalent to quasi-isometric mappings

Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 6, pp. 1305–1313, November–December, 1994.     

Weak compactness and fixed point property for affine mappings

It is shown that a closed convex bounded subset of a Banach space is weakly compact if and only if it has the generic fixed point property for continuous affine mappings. The class of continuous affine mappings can be replaced by the class of affine mappings which are uniformly Lipschitzian with some constant M>1 in the case of c₀, the class of affine mappings which are uniformly Lipschitzian with some constant in the case of quasi-reflexive James’ space J and the class of nonexpansive affine mappings in the case of L-embedded spaces.     

Classification of smooth affine spherical varieties

Let G be a complex reductive group. A normal G-variety X is called spherical if a Borel subgroup of G has a dense orbit in X. Of particular interest are spherical varieties which are smooth and affine since they form local models for multiplicity free Hamiltonian K-manifolds, K a maximal compact subgroup of G. In this paper, we classify all smooth affine spherical varieties up to coverings, central tori, and -fibrations.     

Divisibility of multilinear mappings with applications to affine differential geometry

For real finite-dimensional vector spaces V, W we call a bilinear symmetric mapping h?:?V?×?V?→?W non-degenerate if the components of h with respect to a certain basis are linearly independent and non-degenerate. We say that a symmetric trilinear mapping C?:?V?×?V?×?V?→?W is divisible by h if there exists a linear form α such that C(v,?v,?v)?=?α(v)h(v,?v) for every v?∈?V. In affine differential geometry of affine immersions h is the second fundamental form and C – the cubic form of the immersion. The immersion has pointwise planar normal sections if h(v,?v)?∧?C(v,?v,?v)?=?0 for every tangent vector v. We prove that it implies that C is divisible by h if h is non-degenerate and the codimension is greater than two. For immersions with Wiehe's or Sasaki's choice of transversal bundles divisibility of C by h implies vanishing of C.     

Harmonicity of gradient mappings of level surfaces in a real affine space

In our previous paper [4] we have investigated level surfaces of a non-degenerate function in a real affine space A ⁿ⁺¹ by using the gradient vector field . We gave characterizations of by means of the shape operatorS, the transversal connection , and studied the difference between and the affine normal. In particular we showed that a graph defined by a non-degenerate function satisfiesS=0 and =0. In this paper we consider harmonic gradient mappings of level surfaces and apply these results to a certain problem which is similar to the affine Bernstein problem conjectured by S. S. Chern [3].     

Classification of affine symmetry groups of orbit polytopes

Let G be a finite group acting linearly on a vector space V. We consider the linear symmetry groups \({\text {GL}}(Gv)\) of orbits \(Gv\subseteq V\), where the linear symmetry group \({\text {GL}}(S)\) of a subset \(S\subseteq V\) is defined as the set of all linear maps of the linear span of S which permute S. We assume that V is the linear span of at least one orbit Gv. We define a set of generic points in V, which is Zariski open in V, and show that the groups \({\text {GL}}(Gv)\) for v generic are all isomorphic, and isomorphic to a subgroup of every symmetry group \({\text {GL}}(Gw)\) such that V is the linear span of Gw. If the underlying characteristic is zero, “isomorphic” can be replaced by “conjugate in \({\text {GL}}(V)\).” Moreover, in the characteristic zero case, we show how the character of G on V determines this generic symmetry group. We apply our theory to classify all affine symmetry groups of vertex-transitive polytopes, thereby answering a question of Babai (Geom Dedicata 6(3):331–337, 1977.  https://doi.org/10.1007/BF02429904).     

球面上具有相对仿射共形Gauss映照的超曲面

用Moebius不变量刻画了单位球面上的子流形的共形Gauss映照为相对仿射映照的充要条件,给出了单位球面上具有相对仿射共形 Gauss映照的所有超曲面的分类.     

Canonical decompositions of piecewise affine mappings,polyhedra-traces,and geometrical variational problems

In this paper, canonical decompositions of arbitrary piecewise affine mappings are constructed. Then the equivalence of these mappings is introduced and the concept of polyhedron-trace is defined as an equivalence class. Finally, the concepts of the volume and the deformation of polyhedra-traces are introduced, the continuity of the volume is proved, and the formula of first variation is obtained. These concepts give an analog of the Plateau principles. This research was partially supported by RF President’s grants NSh-1988.2003.1 and MD-263.2003.01. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 1, pp. 57–94, 2006.     

A fixed point theorem for affine mappings and its application to elasticity theory

In this paper we obtain a general fixed point theorem for an affine mapping in Banach space. As an application of this theorem we study existence of periodic solutions to the equations of the linear elasticity theory.     

Classification of harmonic mappings of constant energy density into spheres

We classify all eigenmaps and isometric minimal immersions of a flat torus into the unit sphere using the parametrization theorem (cf. [2], [14]) for range-equivalence classes of all eigenmaps of an arbitrary compact homogeneous Riemannian manifold into the unit sphere.The second author would like to thank the Mathematical Sciences Research Institute, Berkeley, for its financial support and hospitality during his stay.