Full measure of a set of singular continuous measures

On the space of structurally similar measures, we construct a nontrivial measure m such that the subclass of all purely singular continuous measures is a set of full m-measure. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 83–91, January, 2009.     

Matrix random products with singular harmonic measure

Any Zariski dense countable subgroup of SL(d, mathbb R){SL(d, mathbb {R})} is shown to carry a non-degenerate finitely supported symmetric random walk such that its harmonic measure on the flag space is singular. The main ingredients of the proof are: (1) a new upper estimate for the Hausdorff dimension of the projections of the harmonic measure onto Grassmannians in mathbb R^d{mathbb {R}^d} in terms of the associated differential entropies and differences between the Lyapunov exponents; (2) an explicit construction of random walks with uniformly bounded entropy and arbitrarily long Lyapunov vector.     

A singular measure on the Cantor group

Let Ω=N{−1,1} and {ωj} be independent random variables taking values in {−1,1} with equal probability. Endowed with the product topology and under the operation of pointwise product, Ω is a compact Abelian group, the so-called Cantor group. Let a,b,c be real numbers with 1+a+b+c>0, 1+a−b−c>0, 1−a+b−c>0 and 1−a−b+c>0. Finite products on Ω,

     

Fourier transformation and singular integrals on self-similar measure

This paper serves two purposes. One is to modify Strichartz's results with respect to the asymptotic averages of the Fourier transform of μ onR, self-similar measure defined by Hutchinson. Another purpose is to consider a singular integral operator on μ and show that this operator is of type (p,p)(1<p<∞).     

A measure of nonlinearity in singular nonlinear regression models

The method used in nonlinear regression is most of the time the nonlinear squares. If the nonlinearity of the model is not too strong, it is possible to linearize the model and to use the linear statistical theory which is more simple. The question is under which condition this can be done. If the linearized model is regular such criteria were found. To find a solution for singular models is the aim of the paper.     

A sharp bound on the Hausdorff dimension of the singular set of a uniform measure

The study of the geometry of n-uniform measures in \(\mathbb {R}^{d}\) has been an important question in many fields of analysis since Preiss’ seminal proof of the rectifiability of measures with positive and finite density. The classification of uniform measures remains an open question to this day. In fact there is only one known example of a non-trivial uniform measure, namely 3-Hausdorff measure restricted to the Kowalski–Preiss cone. Using this cone one can construct an n-uniform measure whose singular set has Hausdorff dimension \(n-3\). In this paper, we prove that this is the largest the singular set can be. Namely, the Hausdorff dimension of the singular set of any n-uniform measure is at most \(n-3\).     

Construction of approximations for a stationary solution of a system of singular ordinary differential equations

We propose a procedure for the construction of successive approximations of a stationary solution of a system of nonlinear ordinary differential equations with a small parameter with the derivative. We present sufficient conditions for the convergence of constructed approximations to the required stationary solution.     

Construction of invariant curves for singular holomorphic vector fields

Camacho and Sad proved the existence of invariant analytic curves for germs of singular holomorphic foliations over a two dimensional complex analytic variety . Their proof is only of existential nature. Here we provide a simple constructive proof by giving criteria to choose a singular point at each blowing-up that follows an analytic invariant curve.

     

Construction of a Pfaff system of Fuchs type with three singular surfaces on a complex projective space

We construct a multidimensional Pfaff equation of Fuchs type for a given monodromy group on CP ⁿ . The monodromy group is generated by three noncommutating matrices.