Random holomorphic iterations and degenerate subdomains of the unit disk

Given a random sequence of holomorphic maps of the unit disk to a subdomain , we consider the compositions

The sequence is called the iterated function system coming from the sequence We prove that a sufficient condition on the domain for all limit functions of any to be constant is also necessary. We prove that the condition is a quasiconformal invariant. Finally, we address the question of uniqueness of limit functions.

     

Markov interlacing property for exponential polynomials

Let Un be an extended Tchebycheff system on the real line. Given a point , where x₁<?<xn, we denote by the polynomial from Un, which has zeros x₁,…,xn. (It is uniquely determined up to multiplication by a constant.) The system Un has the Markov interlacing property (M) if the assumption that and interlace implies that the zeros of and interlace strictly, unless . We formulate a general condition which ensures the validity of the property (M) for polynomials from Un. We also prove that the condition is satisfied for some known systems, including exponential polynomials and . As a corollary we obtain that property (M) holds true for Müntz polynomials , too.     

Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron

Mixed-mode dynamics is a complex type of dynamical behavior that has been observed both numerically and experimentally in numerous prototypical systems in the natural sciences. The compartmental Wilson-Callaway model for the dopaminergic neuron is an example of a system that exhibits a wide variety of mixed-mode patterns upon variation of a control parameter. One characteristic feature of this system is the presence of multiple time scales. In this article, we study the Wilson-Callaway model from a geometric point of view. We show that the observed mixed-mode dynamics is caused by a slowly varying canard structure. By appropriately transforming the model equations, we reduce them to an underlying three-dimensional canonical form that can be analyzed via a slight adaptation of the approach developed by M. Krupa, N. Popovic, and N. Kopell (unpublished).     

Stability properties of autonomous homogeneous polynomial differential systems

A geometrical approach is used to derive a generalized characteristic value problem for dynamic systems described by homogeneous polynomials. It is shown that a nonlinear homogeneous polynomial system possesses eigenvectors and eigenvalues, quantities normally associated with a linear system. These quantities are then employed in studying stability properties. The necessary and sufficient conditions for all forms of stabilities characteristic of a two-dimensional system are provided. This result, together with the classical theorem of Frommer, completes a stability analysis for a two-dimensional homogeneous polynomial system.     

Attraction of nonlocal dark optical solitons

We study the formation and interaction of spatial dark optical solitons in materials with a nonlocal nonlinear response. We show that unlike in local materials, where dark solitons typically repel, the nonlocal nonlinearity leads to a long-range attraction and formation of stable bound states of dark solitons.     

Travelling Fronts and Entire Solutions¶of the Fisher-KPP Equation in ℝN

This paper is devoted to time-global solutions of the Fisher-KPP equation in ℝ ^N :

Adaptive microphone array for unknown desired speaker's transfer function

The main drawback of minimum variance distortionless response (MVDR) beamformer is the cancellation of the desired speech signal and its degradation in multi-path wave propagation environment. To make the adaptive algorithm robust against room reverberation and to prevent desired signal cancellation an estimation of unknown desired speaker's transfer function was proposed. The estimation is based on the signal and the interference covariance matrices. The estimated transfer function is then applied to the MVDR beamformer. The proposed algorithm was tested on a simulated room with reverberation. The results showed better quality of the restored speech compared to some typical adaptive algorithms.     

Water-soluble polymer carriers. Ionically bonded substances with physiological activity

The preparation of water-soluble polymers with physiological activity via ion bonding of low molecular weight physiologically active substances to a polymer carrier has been studied. The conditions for obtaining polymer salts of physiologically active substances with acid properties and amino-containing polymer carrier have been established. The maximum conversion degree has been determined. The polymer salts have been characterized by potentiometric titration, ³¹P-NMR and IR spectroscopy, and elemental analysis. The stability of the salts and the immobilization rate of the low molecular weight physiologically active substances have been investigated. © 1997 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 35 : 1933–1938, 1997