On projective-differential properties of point correspondences between three hypersurfaces

Point correspondences between three hypersurfaces of projective spaces are studied on the base of G. F. Laptev??s invariant methods [1] (Proc. of Moscow Math. Soc., 2, 275?C382, 1953). We establish the basic equations and geometrical objects of the point correspondences in question. We construct invariant normalizations of the hypersurfaces, single out the basic tensors of the correspondences, and establish a connection between the geometry of correspondences and the theory of multidimensional 3-webs.     

Properties of point correspondences between three multidimensional surfaces of projective spaces

For a correspondence in question, we establish a sequence of fundamental geometric objects, construct invariant normalizations of the surfaces, and single out basic tensors. We also establish a connection between the geometry of correspondences in question and the theory of multidimensional three-webs.     

Nonlinear generalized sections and vector bundle homomorphisms in Colombeau spaces of generalized functions

We define and characterize spaces of manifold‐valued generalized functions and generalized vector bundle homomorphisms in the setting of the full diffeomorphism‐invariant vector‐valued Colombeau algebra. Furthermore, we establish point value characterizations for these spaces.     

Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product

We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the null spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares (?) solutions to a multilinear system and establish the relationship between the minimum-norm (N) least-squares (?) solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.     

On canonical almost geodesic mappings which preserve the weyl projective tensor

We study a partial case of canonical almost geodesic mappings of the first type of spaces with affine connection that preserve Weyl projective curvature tensor and certain other tensors. Main equations under consideration are reduced to a closed Cauchy system type in covariant derivatives. Therefore a general solution to these equations depends on a finite number of constants. We submit an example of above mappings between flat spaces. We establish that projective Euclidean and equiaffine spaces form closed classes of spaces with respect to these mappings.     

Banach operator pairs and common fixed points in hyperconvex metric spaces

The purpose of this paper is to establish DeMarr’s well-known theorem for an arbitrary family of symmetric Banach operator pairs in hyperconvex metric spaces without the compactness assumption. We also give necessary and sufficient criteria for the existence of a common fixed point of a semigroup of isometric mappings. As an application, several results on the invariant best approximation are proved.     

Some remarks on analytic continuation in weighted backward shift invariant subspaces

By a famous result of Douglas, Shapiro, and Shields, functions in backward shift invariant subspaces in Hardy spaces are characterized by the fact that they admit a pseudocontinuation outside the closed unit disk. More can be said when the spectrum of the associated inner function has holes on \mathbb T{{\mathbb T}}. Then the functions of the invariant subspaces even extend analytically through these holes. Here we will be interested in weighted backward shift invariant subspaces which appear naturally in the context of kernels of Toeplitz operators. Note that such kernels are special cases of so-called nearly invariant subspaces. In our setting a result by Aleksandrov allows to deduce analytic continuation properties which we will then apply to consider embeddings of weighted invariant subspaces into their unweighted companions. We hope that this connection might shed some new light on known results. We will also establish a link between the spectrum of the inner function and the approximate point spectrum of the backward shift in the weighted situation in the spirit of results by Aleman, Richter, and Ross.     

Symmetric tensor fields of bounded deformation

We introduce and study spaces of symmetric tensor fields of bounded deformation for tensors of arbitrary order, i.e., where the symmetrized derivative is still a Radon measure. A Sobolev–Korn type estimate, a boundary trace theorem and continuous as well as compact embedding properties into Lebesgue spaces are obtained, showing that these spaces can be regarded as a natural generalization of the spaces of bounded deformation to higher-order symmetric tensors.     

The Study of Abstract Economies with Two Constraint Correspondences

We apply the Himmelberg fixed-point theorem to establish existence theorems of equilibria for generalized abstract economies with two constraint correspondences in which the strategic spaces may not be compact and the set of players may not be countable. Our results have applications in other sciences or disciplines (e.g. game theory, optimization theory and economics) in our subsequent paper. This research was supported by the National Science Council of Republic of China.     

The Harnack inequality for second-order parabolic equations with divergence-free drifts of low regularity

We establish the Harnack inequality for advection-diffusion equations with divergence-free drifts by adapting the classical Moser technique to parabolic equations with drifts with regularity lower than the scale invariant spaces.     

Convex analysis and non-trivial invariant subspaces

Positivity - We establish sufficient conditions for the existence of invariant subspaces for operators on real Banach spaces, and we investigate the behaviour of operators without such subspaces....     

Cardinality bounds via covers by compact sets

Acta Mathematica Hungarica - We establish results concerning covers of spaces by compact and related sets. Several cardinality bounds follow as corollaries. Introducing the cardinal invariant...     

Applications of fixed point theorems to invariant approximation

We prove some fixed point theorems. As applications we obtain Brosowski-Meinardus type theorems on invariant approximations on a class of nonconvex sets in locally bounded topological vector spaces.     

Dilations of Some VH-Spaces Operator Valued Invariant Kernels

We investigate VH-spaces (Vector Hilbert spaces, or Loynes spaces) operator valued Hermitian kernels that are invariant under actions of *-semigroups from the point of view of generation of *-representations, linearizations (Kolmogorov decompositions), and reproducing kernel spaces. We obtain a general dilation theorem in both Kolmogorov and reproducing kernel space representations, that unifies many dilation results, in particular B. Sz.-Nagy??s and Stinesprings?? dilation type theorems.     

The parameterization method for invariant manifolds III: overview and applications

We describe a method to establish existence and regularity of invariant manifolds and, at the same time to find simple maps which are conjugated to the dynamics on them. The method establishes several invariant manifold theorems. For instance, it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in Banach spaces. We also present several other applications of the method.