Bergman Theory Research Articles

Bergman theory for the inhomogeneous Cimmino system

We first prove a Cauchy's integral theorem and a Cauchy-type formula for certain inhomogeneous Cimmino system from quaternionic analysis perspective. The second part of the paper directs the attention towards some applications of the mentioned results, dealing in particular with four kinds of weighted Bergman spaces, reproducing kernels, projection and conformal invariant properties.

On a left-α-ψ-hyperholomorphic Bergman space

Given a structural set , some papers [M. Shapiro, V. Vasilevski, (Complex Var Theory Appl. 1995;27:17–46 and 1995;27:67–96)] study the quaternionic ψ-Fueter operator calculus, where the Fueter-type operator is The aim of this paper is to present the function theory and the Bergman theory induced by the ψ-Fueter-type operator: where a is a quaternionic constant and f is a continuously differentiable function. Particularly, this work shows the Stokes and Borel–Pompieu formulas associated to and presents several results of the Bergman theory such as the existence of the Bergman kernel, Bergman projection and their conformal covariant properties.

Retrieval of dielectric and structural properties of amorphous SiO2 nanopowder based on optical measurements and Bergman's spectral representation theory

The dielectric and structural properties of an amorphous SiO2 nanopowder have been characterized in the framework of Bergman's spectral representation. The methodology used in this paper is based on the acquisition of reflectivity spectra from a set of nanopowder pellets with different levels of porosity and the measurement of the transmittance spectra of KBr pellets containing low levels of silica nanoparticles. The fit of the spectra was performed with a dielectric function model through the solution of a nonlinear optimization problem including all the experimental data. In addition to the optical properties, the complete analysis also provided an accurate estimate of the total porosity of pellets and evaluated the percolation of the silica agglomerates in the porous matrix. A comparison with the predictions obtained by other mixing formulas clearly demonstrates that Bergman's theory is the most appropriate for treating this type of powder.

Bergman theory of certain generalized Hartogs triangles

The Bergman theory of domains $\{ |{z_{1} |^{\gamma}} < |{z_{2}} | < 1 \}$ in $\mathbb{C}^2$ is studied for certain values of $\gamma$, including all positive integers. For such $\gamma$, we obtain a closed form expression for the Bergman kernel, $\mathbb{B}_{\gamma}$. With these formulas, we make new observations relating to the Lu Qi-Keng problem and analyze the boundary behavior of $\mathbb{B}_{\gamma}(z,z)$.

Further properties of the Bergman spaces of slice regular functions

Abstract We continue the study of Bergman theory for the class of slice regular functions. In the slice regular setting there are two possibilities to introduce the Bergman spaces, that are called of the first and of the second kind. In this paperwe mainly consider the Bergman theory of the second kind, by providing an explicit description of the Bergman kernel in the case of the unit ball and of the half space. In the case of the unit ball, we study the Bergman-Sce transform. We also show that the two Bergman theories can be compared only if suitableweights are taken into account. Finally,we use the Schwarz reflection principle to relate the Bergman kernel with its values on a complex half plane.

Theoretical Analyses of Cellular Transmembrane Voltage in Suspensions Induced by High-frequency Fields

A change of the transmembrane voltage is considered to cause biophysical and biochemical responses in cells. The present study focuses on the cellular transmembrane voltage (Δφ) induced by external fields. We detail analytical equations for the transmembrane voltage induced by external high-frequency (above the relaxation frequency of the cell membrane) fields on cells of a spherical shape in suspensions and layers. At direct current (DC) and low frequencies, the cell membrane was assumed to be non-conductive under physiologic conditions. However, with increasing frequency, the permittivity of the cytoplasm/extracellular medium and conductivity of the membrane must be accounted for. Our main work is to extend application of the analytical solution of Δφ to the high-frequency range. We first introduce the transmembrane voltage generated by DC and low-frequency exposures on a single cell. Then, we focus on cell suspensions exposed to high-frequency fields. Using the effective medium theory and the reasonable assumption, the approximate analytical solution of Δφ on cells in suspensions and layers can be derived. Phenomenological effective medium theory equations cannot be used to calculate the local electric field of cell suspensions, so we raised a possible solution based on the Bergman theory.

On slice biregular functions and isomorphisms of Bergman spaces

In this article we deepen the study of the Bergman theory in the slice regular setting by providing a description of the Bergman projection and by considering a subclass of slice regular functions which generate isometric isomorphisms of Bergman spaces. As a tool to show our result, we further study the multiplication of slice regular functions and their composition.

The Bergman-Sce transform for slice monogenic functions

In a recent paper, we showed that the classical Bergman theory admits two possible formulations for the class of slice regular functions with quaternionic values. In the so called formulation of the first kind, we provide a Bergman kernel which is defined on and is a reproducing kernel. In the so called formulation of the second kind, we use the Representation Formula for slice regular functions to define a second Bergman kernel; this time the kernel is still defined on U, but the integral representation of f is based on an integral computed only on and the integral does not depend on , (here denotes the sphere unit of purely imaginary quaternions, and represents the complex plane with imaginary unit I). In this paper, we extend the second formulation of the Bergman theory to the case of slice monogenic functions and we focus our attention on the so-called Bergman–Sce transform. This integral transform is defined by using the Bergman kernel and the Sce mapping theorem and associates to every slice monogenic function f, an axially monogenic function . Copyright © 2011 John Wiley & Sons, Ltd.

PS‐NH2 + PMMA‐COOH blend: A promising substrate material for the deposition of densely packed gold nanoparticles

AbstractDensely packed gold nanoparticles with narrow size distribution were fabricated by pulsed laser deposition using a blend of amino‐terminated polystyrene (PS‐NH2) and carboxyl‐terminated polymethylmethacrylate (PMMA‐COOH) as a deposition substrate. Transmission electron microscopy showed spherical Au nanoparticles with an average size of 5 nm. UV–vis absorption spectra of the Au/blend nanocomposite films exhibited a strong resonance band at 658 nm. The analyses of the spectrum using the effective medium Bergman theory showed that the nanocomposite is comprised of isolated and randomly arranged nanoparticles. (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

On the Bergman Theory for Solenoidal and Irrotational Vector Fields. II. Conformal Covariance and Invariance of the Main Objects

This is a continuation of our work (Gonzalez-Cervantes et al. in On the Bergman theory for solenoidal and irrotational vector fields. I. General theory. Operator theory: advances and applications. Birkhauser, accepted) where for solenoidal and irrotational vector fields theory as well as for the Moisil–Theodoresco quaternionic analysis we introduced the notions of the Bergman space and the Bergman reproducing kernel and studied their main properties. In particular, we described the behavior of the Bergman theory for a given domain whenever the domain is transformed by a conformal map. The formulas obtained hint that the corresponding objects (spaces, operators, etc.) can be characterized as conformally covariant or invariant, and in the present paper we construct a series of categories and functors which allow us to give such characterizations in precise terms.

RAS Studies of Laterally Nanostructured Surfaces

With Pt(100) and Ag(110) as examples we discuss the application of Bergman theory to describe optical anisotropy of laterally nanostructured metal surfaces. In both cases the surfaces are characterized by scanning tunneling microscopy. The clean Pt(100) surface shows a pseudo-hexagonal reconstruction. The incommensurability of the top layer with the underlying cubic structure leads to a corrugated surface. The observed reflection-anisotropy spectrum (RAS) can be accounted for by a simple choice for the surface spectral density function. In the Ag(110) case we measure and discuss the spectra when the surface step edges are parallel with the in-plane [110] direction, and when there is a statistically isotropic distribution of steps and terraces. The spectra are interpreted in terms of local-field calculations where the screened dipole–dipole interaction coefficients are modified by surface steps. Step-induced coupling to surface plasmons is proposed as an additional mechanism. The influence of both effects on the experimental reflection-anisotropy curve increases with decreasing correlation length. The plasmon-based mechanism takes place already at lengths of order 102 nm, whereas the cut-off in the dipole–dipole interaction needs correlation lengths that are almost one magnitude lower to be important.

The Bloch space for the minimal ball

We introduce the Bloch space for the minimal ball and we prove that this space can be identified with the dual of a certain analytic space which is strongly related to the Bergman theory on the minimal ball.

On the theory of class p operators and generalizations

Motivated by integral-free representations of solutions of complex second-order partial differential equations Lu = 0 obtained from class P Bergman integral operators, we introduce a new type of infinite series (“P-series”) and show that their theory is an analog of the usual Bergman theory in integral form, of the same generality and in certain applications of greater utility. A basic spinoff of this theory is a novel kind of criterion for L [d] class P

On class of elliptic partial differential equations in four variables

where A(r), C(r) are analytic functions of the real variable r = x^Xμ. [μ = 1, 2, 3] (Repeated indices mean the summation convention is used.) In this paper we shall investigate the four variable analogue of this equation, TA[W] = 0, and show that many of Bergman's results carry over to this case. Here, we need in many instances, the methods of several complex variables in order to find the natural generalizations. In Bergman's theory, the integral operator B3 [/] plays an important role in studying the solutions of (1.1). In our case, there is an analogous operator [7]~[1O], which is a four-variable analogue to