Convexity properties of holomorphic mappings in

Not many convex mappings on the unit ball in for are known. We introduce two families of mappings, which we believe are actually identical, that both contain the convex mappings. These families which we have named the ``Quasi-Convex Mappings, Types A and B' seem to be natural generalizations of the convex mappings in the plane. It is much easier to check whether a function is in one of these classes than to check for convexity. We show that the upper and lower bounds on the growth rate of such mappings is the same as for the convex mappings.

     

On the Existence of Continuous Selections in Subset Spaces with Ochan-Type Topologies

We prove the existence of a continuous selection in the space (X) (the nonempty subsets of a topologic space X) equipped with the Ochan-type -topology in the case where X can be imbedded into the Tikhonov product of a countable family of left spaces (Theorem 1). We also prove the existence of a continuous selection on the exponent of a complete metric space in the Ochan-type -topology (Theorem 2).     

-energy integral and harmonic mappings

In this paper, we discuss harmonic mappings on the unit disk with respect to any metric by using the -energy integral that was first introduced by Li in 1997 to treat quasiconformal harmonic mappings on the Poincaré disk, instead of the total energy integral. Some basic properties of harmonic mappings are given. Moreover, we give a new proof of the uniqueness theorem of Markovic and Mateljevic, which is more explicit and natural.

     

The Boy–Gauss–Bonnet Theorems for C ∞-Singular Surfaces with Limiting Tangent Bundle

Using C -singularity theory, we describe natural conditions on nonimmersivemappings of a compact orientable 2-manifold into Euclidean 3-space (e.g., theexistence of a global limiting tangent bundle). These conditions are weaker thanthose of caustic theory and imply that the image is a CW-complex. In general, the domainmanifold is not homeomorphic to the image CW-complex. We then adapt classical methods to this setting. (This adaptation is simplified by freely moving betweenthe Gauss, Cartan, and Kozul formulations for surface theory.) As an application,we show that part of the extrinsic topology of the image is determined by the firstfundamental form, i.e., a global `theorem egregium'.     

Extensions of l.s.c. Mappings into Reflexive Banach Spaces

Every lower semi-continuous closed-and-convex valued mapping : X2 ^Y where X is a collectionwise normal and countably paracompact space and Y is a reflexive Banach space can be extended to the Dieudonné completion (X) of X. Generalizations and applications to the theory of selections of set-valued mappings are presented.     

On four-sheeted polynomial mappings of ℂ2. I. The case of an irreducible ramification curve

The paper is devoted to the Jacobian Conjecture: a polynomial mappingf²² with a constant nonzero Jacobian is polynomially invertible. The main result of the paper is as follows. There is no four-sheeted polynomial mapping whose Jacobian is a nonzero constant such that after the resolution of the indeterminacy points at infinity there is only one added curve whose image is not a point and does not belong to infinity.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 847–862, December, 1998.The authors are grateful to A. G. Vitushkin and P. Cassou-Nogues for useful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01218. The work of the second author was done under the financial support of DGICYT (Spain), grant No. SAB95-0502.     

An efficient and novel numerical method for quasiconformal mappings of doubly connected domains

A numerical method for quasiconformal mapping of doubly connected domains onto annuli is presented. The ratio R of the radii of the annulus is not known a priori and is determined as part of the solution procedure. The numerical method presented in this paper requires solving iteratively a sequence of inhomogeneous Beltrami equations, each for a different R. R is updated using a procedure based on the bisection method. The new method is an extension of Daripas method for the quasiconformal mapping of the exterior of simply connected domains onto the interior of unit disks [15]. It uses fast and accurate algorithms for evaluating certain singular integrals and is, thus, very efficient and accurate. Its performance is demonstrated for several doubly connected domains.     

Lipschitz images with fractal boundaries and their small surface wrapping

Assume and is a Lipschitz -mapping; and denote the volume and the surface area of . We verify that there exists a figure with , and, of course, , where depends only on the dimension and on . We also give an example when is a square and ; in fact, the boundary of can contain a fractal of Hausdorff dimension exceeding one.

     

Behavior of the effective fields for a regular ising model on the Cayley tree

A regular Ising model with nearest-neighbor interactions ofJ and–J(J>0) on a Cayley tree of coordination number 3 is investigated for the behavior of effective fields in a uniform external field. The effective fields show periodic and also aperiodic structures in the temperature-field plane. At absolute zero temperature, the equations determining effective fields are reduced to a nonlinear, one-dimensional, iterative equation. Arithmetic furcations of period and a screening of the furcations are observed.     

Quantitative Bulk and Trace Element X-Ray Mapping Using Multiple Detectors

X-ray mapping using energy dispersive spectroscopy or wavelength dispersive spectroscopy is a very popular characterisation tool for determining the elemental distribution in materials. Furthermore, quantitative X-ray mapping has become a very powerful technique enabling reliable quantitative results that can be an order of magnitude better than traditional analysis. Quantitative X-ray mapping is also far superior to regions of interest X-ray maps where low levels of an element or elemental overlaps are present. The one major drawback with X-ray mapping is the time required to obtain a high resolution X-ray map with good statistics at low levels of concentration. The use of multi-detectors, and just developed dual turret detectors for X-ray mapping, allows improvement in performance at low levels without compromising quantification quality and precision of traces, even in the presence of overlaps. However, for quantitative X-ray mapping to work properly, the characteristics of each detector must be accurately determined so that the final quantification of the individual detectors can be summed. To accomplish this effectively, the full spectrum at each pixel for each energy dispersive detector should be saved. As a final check for consistency between detectors, a technique was developed that involves assigning a different red-green-blue colour for each detector for the same element. By doing this, when we combine the three maps of the same element, we should obtain a grey scale map that indicates total correlation between the three detectors at the most critical final stage of quantification. To reduce contrast noise and further improve the quality of quantitative X-ray mapping images, a filter referred to as a “speckle filter” has been developed that allows the eye to see a more correct elemental concentration relationship.