Related papers: Notes on the Szego minimum problem. II. Singular m…

This paper takes a new step in the direction of proving the Duffin-Schaeffer Conjecture for measures arbitrarily close to Lebesgue. The main result is that under a mild `extra divergence' hypothesis, the conjecture is true.

The present paper consists of two parts. In the first part, we prove a noncommutative analogue of the Riesz(-Markov-Kakutani) theorem on representation of functionals on an algebra of continuous functions by regular measures on the…

We extend Szemeredi's Regularity Lemma (SRL) to abstract measure spaces. Our main aim is to find general conditions under which the original proof of Szemeredi still works. To illustrate that our approach has some merit, we outline several…

We prove an extension of the Moore-Schmidt theorem on the triviality of the first cohomology class of cocycles for the action of an arbitrary discrete group on an arbitrary measure space and for cocycles with values in an arbitrary compact…

We establish the second part of Milnor's conjecture on the volume of simplexes in hyperbolic and spherical spaces. A characterization of the closure of the space of the angle Gram matrices of simplexes is also obtained.

Let mu be a probability measure on the Borelian sigma-algebra of the unit circle. Then we associate a Schur function theta in the unit disk with mu and give characterizations of the case that mu is a Helson-Szeg\"o measure in terms of the…

Let $(G,+)$ be a finite abelian group. Then, $\so(G)$ and $\eta(G)$ denote the smallest integer $\ell$ such that each sequence over $G$ of length at least $\ell$ has a subsequence whose terms sum to $0$ and whose length is equal to and at…

In this article, we get properties for singular (dual) Nakano semi-positivity and obtain singular type vanishing theorem involving $L^2$-subsheaves on weakly pseudoconvex manifolds by $L^2$-estimates and $L^2$-type Dolbeault isomorphisms.…

We give a new proof of a classical uniqueness theorem of Alexandrov using the weak uniqueness continuation theorem of Bers-Nirenberg. We prove a version of this theorem with the minimal regularity assumption: the spherical hessians of the…

We consider the simplest optimal control problem with one nonregular mixed inequality constraint, i.e. when its gradient in the control can vanish on the zero surface. Using the Dubovitskii--Milyutin theorem on the approximate separation of…

We consider random walks on semisimple Lie groups where the support of the step distribution generates (as a group) a Zariski dense discrete subgroup of infinite covolume. When the semisimple Lie group has property (T), we show that the…

Let $\mathcal{C}$ be the family of compact convex subsets $S$ of the hemisphere in $\rn$ with the property that $S$ contains its dual $S^*;$ let $u\in S^*$, and let $ \Phi(S,u)=\frac{2}{\omega_n}\int_{S}\ < \theta, u \ > \,\,…

We generalize the derivation of Leggett-Garg inequalities to systematically treat a larger class of experimental situations by allowing multi-particle correlations, invasive detection, and ambiguous detector results. Furthermore, we show…

We consider the Stefan problem, firstly with regular data and secondly with irregular data. In both cases is given a proof for the convergence of an approximation obtained by regularising the problem. These proofs are based on weak…

In this paper, we obtain two new lower bounds for the smallest singular value of nonsingular matrices which is better than the bound presented by zou \cite{zou2012lower}, Lin, Minghua and Xie, Mengyan \cite{lin2021some} under certain…

The aim of this note is to present some recent results on the structure of the singular part of measures satisfying a PDE constraint and to describe some applications.

We develop techniques for classifying the nonnegatively curved left-invariant metrics on a compact Lie group G. We prove rigidity theorems for general G and a partial classification for G=SO(4). Our approach is to reduce the general…

The first Szego limit theorem has been extended by Bump-Diaconis and Tracy-Widom to limits of other minors of Toeplitz matrices. We extend their results still further to allow more general measures and more general determinants. We also…

In this paper, we study the Skorokhod problem with two constraints, where the constraints are in a nonlinear fashion. We prove the existence and uniqueness of the solution and also provide the explicit construction for the solution. In…

We prove theorems of the following form: if $A\subseteq {\mathbb R}^2$ is a big set, then there exists a big set $P\subseteq {\mathbb R}$ and a perfect set $Q\subseteq {\mathbb R}$ such that $P\times Q\subseteq A$. We discuss cases where…