Proximality and equidistribution on the Furstenberg boundary

Let G be a connected semisimple Lie group with finite center and without compact factors, P a minimal parabolic subgroup of G, and \Gamma a lattice in G. We prove that every \Gamma-orbits in the Furstenberg boundary G/P is equidistributed for the averages over Riemannian balls. The proof is based on the proximality of the action of \Gamma on G/P

Power-free values of polynomials on symmetric varieties

Given a symmetric variety Y defined over the rationals and a non-zero polynomial with integer coefficients, we use techniques from homogeneous dynamics to establish conditions under which the polynomial can be made r-free for a Zariski dense set of integral points on Y. We also establish an asymptotic counting formula for this set. In the special case that Y is a quadric hypersurface, we give explicit bounds on the size of r by combining the argument with a uniform upper bound for the density of integral points on general affine quadrics.Comment: 47 pages; accepted versio

Limit theorems for rank-one Lie groups

We investigate asymptotic behaviour of averaging operators for actions of simple rank-one Lie groups. It was previously known that these averaging operators converge almost everywhere, and we establish a more precise asymptotic formula that describes their deviations from the limit

Strong wavefront lemma and counting lattice points in sectors

We compute the asymptotics of the number of integral quadratic forms with prescribed orthogonal decompositions and, more generally, the asymptotics of the number of lattice points lying in sectors of affine symmetric spaces. A new key ingredient in this article is the strong wavefront lemma, which shows that the generalized Cartan decomposition associated to a symmetric space is uniformly Lipschitz

Khinchin theorem for integral points on quadratic varieties

We prove an analogue the Khinchin theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that in this system, generic trajectories visit a family of shrinking subsets infinitely often.Comment: 19 page

The main directions for pharmacological correction (combinations of drugs for general anesthesia) of neurological and cognitive disorders in patients with neoplasms of the central nervous system

The aim of the study was to develop a goal-oriented combination of drugs for general anesthesia, based on a retrospective assessment of the baseline level of neurological and cognitive disorders in adults and children at the stage of preparation for surgery for neoplasms of the central nervous system (sub- and supratentorial neoplasms - SubTNN and SupraTNN), and a prospective evaluation of complications in the postoperative perio

Ultrametric Logarithm Laws, II

We prove positive characteristic versions of the logarithm laws of Sullivan and Kleinbock-Margulis and obtain related results in Metric Diophantine Approximation.Comment: submitted to Montasefte Fur Mathemati

Pharmacological correction of intrarenal hemodynamic disorders in acute kidney injury (part 2)

Evaluate the possibilities of individual pharmacological correction and intensive care of patients with acute kidney injury of different origin. A prospective nonrandomized study. Inclusion criteria: patients with prerenal, renal and subrenal AKI module in stage of oligoanuria and restoration of diuresis. Exclusion criteria: AKI in patients after cardiosurgery and operations on large vessels. Individual pharmacological and non-pharmacological correction (renoprotection) was performed in 250 ICU patients with prerenal (130), renal (81) and subrenal (39) AK

The Asymptotic distribution of circles in the orbits of Kleinian groups

Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which describes the asymptotic distribution of small circles in P, assuming that either the critical exponent of Gamma is strictly bigger than 1 or P does not contain an infinite bouquet of tangent circles glued at a parabolic fixed point of Gamma. Our construction also works for P invariant under a geometrically infinite group Gamma, provided Gamma admits a finite Bowen-Margulis-Sullivan measure and the Gamma-skinning size of P is finite. Some concrete circle packings to which our result applies include Apollonian circle packings, Sierpinski curves, Schottky dances, etc.Comment: 31 pages, 8 figures. Final version. To appear in Inventiones Mat