Nonlinear Cauchy-Riemann operators in

This paper has arisen from an effort to provide a comprehensive and unifying development of the -theory of quasiconformal mappings in . The governing equations for these mappings form nonlinear differential systems of the first order, analogous in many respects to the Cauchy-Riemann equations in the complex plane. This approach demands that one must work out certain variational integrals involving the Jacobian determinant. Guided by such integrals, we introduce two nonlinear differential operators, denoted by and , which act on weakly differentiable deformations of a domain .

Solutions to the so-called Cauchy-Riemann equations and are simply conformal deformations preserving and reversing orientation, respectively. These operators, though genuinely nonlinear, possess the important feature of being rank-one convex. Among the many desirable properties, we give the fundamental -estimate

In quest of the best constant , we are faced with fascinating problems regarding quasiconvexity of some related variational functionals. Applications to quasiconformal mappings are indicated.

     

Functional differential equations

The method of quasilinearization is a well-known technique for obtaining approximate solutions of nonlinear differential equations. In this paper we apply this technique to functional differential problems. It is shown that linear iterations converge to the unique solution and this convergence is superlinear.     

Degradation of DH lasers caused by growth of dislocation networks

The features of the process of rapid degradation of (GaAl) As-GaAs DH lasers are discussed. The existing models for dislocation networks growth during the device operation are critically analysed. A new approach to the mechanism of degradation is proposed according to which the dislocation dipoles in the active layer of the device develop by conservative climb due to the pipe diffusion along helical dislocations.     

Copper(II) and nickel(II) complexes of a neutral pentadentate Schiff base

A new pentadentate Schiff base 2,6,10-triaza-1,11-bis(2′-aminophenyl)-undeca-1,10-diene, abaDPT, and its complexes of general formula M(abaDPT)X₂ where M = Cu(II), Ni(II), X = Cl, Br, I, NO₃ and ClO₄, have been prepared. The complexes have been characterized by electronic and IR spectra, EPR, magnetic moments, molar conductances, and elemental analysis. IR data show an interaction between halide anion of the outer coordination sphere and the complexed amino group. EPR and spectrophotometric data of most of the copper compounds are consistent with a distorted square pyramidal geometry. Single crystal EPR studies of Cu(abaDPT)(NO₃)₂ and Cu(abaDPT)Br₂ revealed that copper atoms in the former compound occupy two magnetically inequivalent places in the lattice while copper atoms in the latter compound take identical sites. Principal g tensor axes of the two compounds have been determined.     

Critical behavior in vacuum gravitational collapse in 4 + 1 dimensions

We show that the (4 + 1)-dimensional vacuum Einstein equations admit gravitational waves with radial symmetry. The dynamical degrees of freedom correspond to deformations of the three-sphere orthogonal to the (t,r) plane. Gravitational collapse of such waves is studied numerically and shown to exhibit discretely self-similar type II critical behavior at the threshold of black hole formation.     

Polymeric complexes of cornstarch and waxy cornstarch phosphates with milk casein and their performance as biodegradable materials

Complexes of phosphated cornstarch and waxy cornstarch with casein were prepared and characterised. They were prepared from casein in defatted milk and corn and waxy corn starches phosphated to degree of substitution values (DS) of 0.0637 and 0.0968, respectively. The components were blended in starch to casein ratios of 2:1, 1:1, and 1:2, then precipitated with hydrochloric acid. Aqueous solubility, water binding capacity, IR spectra, and thermal analysis (thermogravimetry, TG, and differential thermogravimetry, DTG) of the precipitates revealed that they were not simple physical mixtures of the components. The components interact with one another electrostatically with involvement of the starch phosphate groups and the peptide bonds of casein as documented by the IR spectra. Because of their insolubility in 7 M aqueous urea solution they might also be considered as complexes in which the components were chemically bound. Enzymatic studies showed that they are biodegradable materials.     

Exceptionally long-range self-assembly of hexa-peri-hexabenzocoronene with dove-tailed alkyl substituents

The substitution of a hexa-peri-hexabenzocoronene by bulky, space-demanding, 2-decyltetradecyl side chains proved to be an effective procedure to influence the thermal and self-aggregation behavior. The extremely large steric requirement of the introduced side chains modulated the aggregation and resulted in a dramatic lowering of the isotropization temperature and a higher solubility. As an additional consequence of the exceptional, long-range self-aggregation of the discotic molecule, spherulite formation was observed during crystallization by polarized light microscopy. Macroscopic self-assembly was induced by zone crystallization of the material, leading to aligned columnar superstructures as revealed by 2D-WAXS experiments. These extraordinary properties have not yet been reported for a discotic columnar material, making this compound very promising for application in electronic devices.     

Contents of Volume 26 (2004)

Volume Contents

Contents of Volume 26 (2004)     

Three-dimensional reconstruction of biological objects' internal structure heterogeneity from the set of ultrasonic tomograms

The paper presents the method of the three-dimensional reconstruction of biological objects' internal structure heterogeneity based on the ultrasonic examination of a woman's breast biopsy phantom. The phantom is made of quasi-homogeneous dense gel in which drops of lesions, characterized by fixed sizes and two different acoustic impedances, were dipped at random. For the purpose of this research a special measurement setup was elaborated, enabling a non-invasive in vitro imaging of biological objects' internal structure in cross-sections for fixed levels, by means of ultrasound transmission tomography (UTT) using the parallel-ray projection geometry of scanning. The two-dimensional images of the local values of ultrasonic wave's propagation velocity in the phantom's internal structure (ultrasonic tomograms) were reconstructed for fixed levels (by using the convolution and back-projection algorithm) from the measurements of average values of ultrasonic signals' runtime propagated from many directions around the object dipped in water. Analyzing the values of particular pixels and using an appropriate image processing technique, in effect the three-dimensional image of heterogeneity boundaries in the examined phantom's internal structure was computer-reconstructed. The obtained results are compatible with the specification provided by the phantom's producer in terms of sizes and acoustic parameters of lesions, which can simulate pathological changes and of the gel imitating the healthy tissue. It means that the method presented, after an appropriate modification and development of the measurement setup with an aim to accelerate the object scanning process and thus provide an opportunity for non-invasive in vivo examinations, could be applied for detecting and diagnosing tumors in women's breasts.