A local extension theorem for proper holomorphic mappings in ℂ<Superscript>2</Superscript>

Let ƒ: D → D′ be a proper holomorphic mapping between bounded domains D, D′ in ℂ².Let M, M′ be open pieces on δD, δD′, respectively that are smooth, real analytic and of finite type. Suppose that the cluster set of M under ƒ is contained in M′. It is shown that ƒ extends holomorphically across M. This can be viewed as a local version of the Diederich-Pinchuk extension result for proper mappings in ℂ².     

ℂ-differentiability of mappings of Hilbert spaces

The process of differentiation is studied for a class of mappings of regions of a Hilbert space. We introduce the concept of Cl-differentiability of a mapping at a point and establish sufficient conditions of Cl-differentiability.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 3, pp. 424–428, March, 1995.     

Distortion theorems for subclasses of starlike mappings along a unit direction in ℂ

In this paper, we obtain a distortion theorem of Jacobian matrix for biholomorphic subclasses of starlike mappings along a unit direction on the unit polydisc. These results extend the classical distortion theorem of starlike mappings to higher dimensions. We then give an upper bound estimate for biholomorphic subclasses of starlike mappings along a unit direction in a complex Banach space.     

On the ℂ-differentiability of mappings of Banach spaces

We study the monogeneity conditions for -differentiable mappings of domains of a Banach space and establish criteria of -differentiability of a mapping at a point.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1336–1342, October, 1994.     

Pluriharmonic mappings and linearly connected domains in ℂ n

In this paper we obtain certain sufficient conditions for the univalence of pluriharmonic mappings defined in the unit ball \(\mathbb{B}^n \) of ? ⁿ . The results are generalizations of conditions of Chuaqui and Hernández that relate the univalence of planar harmonic mappings with linearly connected domains, and show how such domains can play a role in questions regarding injectivity in higher dimensions. In addition, we extend recent work of Hernández and Martín on a shear type construction for planar harmonic mappings, by adapting the concept of stable univalence to pluriharmonic mappings of the unit ball \(\mathbb{B}^n \) into ? ⁿ .     

A Strong Approximation Theorem for Quasi-associated Sequences

By combining the Csorgo-Révész quantile transform methods and the Skorohod-Strassen martingale embedding theorem, we prove a strong approximation theorem for quasi-associated random variables with mean zero and finite （2 ＋ δ）th moment under polynomial decay rate. As a consequence, the decay rate for a strong approximation theorem of associated sequences of Yu （1996） is weakened.     

A Cantor-Bernstein Theorem for σ-Complete MV-Algebras

The Cantor-Bernstein theorem was extended to -complete boolean algebras by Sikorski and Tarski. Chang's MV-algebras are a nontrivial generalization of boolean algebras: they stand to the infinite-valued calculus of Lukasiewicz as boolean algebras stand to the classical two-valued calculus. In this paper we further generalize the Cantor-Bernstein theorem to -complete MV-algebras, and compare it to a related result proved by Jakubík for certain complete MV-algebras.     

A New Proof for the Interpolating Theorem

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An Analog of the Cameron--Johnson Theorem for Linear ℂ-Analytic Equations in Hilbert Space

The well-known Cameron--Johnson theorem asserts that the equation $\dot x = \mathcal{A}\left( t \right)x$ with a recurrent (Bohr almost periodic) matrix $\mathcal{A}\left( t \right)$ can be reduced by a Lyapunov transformation to the equation $\dot y = \mathcal{B}\left( t \right)y$ with a skew-symmetric matrix $\mathcal{B}\left( t \right)$ , provided that all solutions of the equation $\dot x = \mathcal{A}\left( t \right)x$ and of all its limit equations are bounded on the whole line. In the note, a generalization of this result to linear $\mathbb{C}$ -analytic equations in a Hilbert space is presented.     

A Böchner technique for harmonic mappings from a 3-manifold to a surface

Let be a harmonic mapping from a Riemannian 3-manifold to a Riemannian 2-manifold. A smooth function on M is associated to , derived from the eigenvalues of the first fundamental form, the vanishing of which is equivalent to being a harmonic morphism. The Laplacian of this function is computed and a maximum principle applied to derive criteria when a harmonic map must be a harmonic morphism.     

A Frobenius–Schur Theorem for Hopf Algebras

We prove a version of the Frobenius–Schur theorem for a finite-dimensional semisimple Hopf algebra H over an algebraically closed field; if the field has characteristic p not 0, H is also assumed to be cosemisimple. Then for each irreducible representation V of H, we define a Schur indicator for V, which reduces to the classical Schur indicator when H is the group algebra of a finite group. We prove that this indicator is 0 if and only if V is not self-dual. If V is self dual, then the indicator is positive (respectively, negative) if and only if V admits a nondegenerate bilinear symmetric (resp., skew-symmetric) H-invariant form. A more general result is proved for algebras with involution.     

A Pinching Theorem for Minimal Submanifolds in Sphere

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On Quillen?s Theorem A for posets

A theorem of McCord of 1966 and Quillen?s Theorem A of 1973 provide sufficient conditions for a map between two posets to be a homotopy equivalence at the level of complexes. We give an alternative elementary proof of this result and we deduce also a stronger statement: under the hypotheses of the theorem, the map is not only a homotopy equivalence but a simple homotopy equivalence. This leads then to stronger formulations of the simplicial version of Quillen?s Theorem A, the Nerve Lemma and other known results. In particular we establish a conjecture of Kozlov on the simple homotopy type of the crosscut complex and we improve a well-known result of Cohen on contractible mappings.     

Invariant fibrations for some birational maps of ℂ2

Abstract

In this article, we extract and study the zero entropy subfamilies of a certain family of birational maps of the plane. We find these zero entropy mappings and give the invariant fibrations associated to them.     

A direct theorem for strictly convex domains in ℂn

For a strictly covex C²-smooth domain Ω??ⁿ and a function f ? Λ^α(Ω) holomorphic in Ω, we construct polynomials p_N, deg p_N<-N, such that $$\left| {f(z) - p_N (z)} \right| \leqslant CN^{ - \alpha } ,z \in \bar \Omega .$$ Bibliography: 12 titles.     

A Decomposition Theorem for the Solutions to the Interface Problems of Quasi-Linear Elliptic Equations

Interface problems for second order quasi-linear elliptic partial differential equations in a two-dimensional space are studied. We prove that each weak solution can be decomposed into two parts near singular points, one of which is a finite sum of functions of the form cr^a log^m rφ(θ), where the coefficients c depend on the H^1-norm of the solution, the C^(0,δ) -norm of the solution, and the equation only; and the other one of which is a regular one, the norm of which is also estimated.     

Operator conditions for ℂ-differentiability

The concepts of dilation and rotation operators are introduced for maps of domains of infinite-dimensional Hibert spaces. Sufficient conditions for -differentiability along subspaces are established and theorems on operator conditions for IR-differentiable maps to be monogenic and holomorphic are proved.Translated from Ukrainskii Matematicheskii Zhurnal,42, No. 11, pp. 1448–1454, November, 1990.