Genetic polymorphism and content of some milk protein fractions in Polish cattle

International audienc

Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive

We investigate the set a) of positive, trace preserving maps acting on density matrices of size N, and a sequence of its nested subsets: the sets of maps which are b) decomposable, c) completely positive, d) extended by identity impose positive partial transpose and e) are superpositive. Working with the Hilbert-Schmidt (Euclidean) measure we derive tight explicit two-sided bounds for the volumes of all five sets. A sample consequence is the fact that, as N increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamiolkowski isomorphism, the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets (such as states with positive partial transpose or separable states) or supersets. Our approach depends on systematic use of duality to derive quantitative estimates, and on various tools of classical convexity, high-dimensional probability and geometry of Banach spaces, some of which are not standard.Comment: 34 pages in Latex including 3 figures in eps, ver 2: minor revision

Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres

We establish an asymptotic formula for the number of lattice points in the sets Sh1,h2,h3(λ):={x∈Z+3:⌊h1(x1)⌋+⌊h2(x2)⌋+⌊h3(x3)⌋=λ} with λ∈Z+; where functions h1, h2, h3 are constant multiples of regularly varying functions of the form h(x) : = xcℓh(x) , where the exponent c> 1 (but close to 1) and a function ℓh(x) is taken from a certain wide class of slowly varying functions. Taking h1(x) = h2(x) = h3(x) = xc we will also derive an asymptotic formula for the number of lattice points in the sets Sc3(λ):={x∈Z3:⌊|x1|c⌋+⌊|x2|c⌋+⌊|x3|c⌋=λ}withλ∈Z+;which can be thought of as a perturbation of the classical Waring problem in three variables. We will use the latter asymptotic formula to study, the main results of this paper, norm and pointwise convergence of the ergodic averages 1#Sc3(λ)∑n∈Sc3(λ)f(T1n1T2n2T3n3x)asλ→∞;where T1, T2, T3: X→ X are commuting invertible and measure-preserving transformations of a σ-finite measure space (X, ν) for any function f∈ Lp(X) with p>11-4c11-7c. Finally, we will study the equidistribution problem corresponding to the spheres Sc3(λ).Foundation for Polish Science via the START Scholarship, the Juan de la Cierva Incorporaci´on 2019, grant number IJC2019-039661-I, the Agencia Estatal de Investigaci´on, grant PID2020-113156GB-I00/AEI/10.13039/501100011033, the Basque Government through the BERC 2022-2025 program, and by the Spanish Ministry of Sciences, Innovation and Universities: BCAM Severo Ochoa accreditation SEV-2017-0718

On the Hausdorff dimension of invariant measures of weakly contracting on average measurable IFS

We consider measures which are invariant under a measurable iterated function system with positive, place-dependent probabilities in a separable metric space. We provide an upper bound of the Hausdorff dimension of such a measure if it is ergodic. We also prove that it is ergodic iff the related skew product is.Comment: 16 pages; to appear in Journal of Stat. Phy

Polynomial averages and pointwise ergodic theorems on nilpotent groups

We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on $\sigma$-finite measure spaces. We also establish corresponding maximal inequalities on $L^p$ for $1<p\leq \infty$ and $\rho$-variational inequalities on $L^2$ for $2<\rho<\infty$. This gives an affirmative answer to the Furstenberg--Bergelson--Leibman conjecture in the linear case for all polynomial ergodic averages in discrete nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fourier transform tools, which are not available in the non-commutative, nilpotent setting. In particular, we develop what we call a \textit{nilpotent circle method} that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.Juan de la Cierva Incorporación 2019, Grant Number IJC2019-039661-I BERC 2022-2025 progra

A regional model of interprofessional education

This paper describes the innovative features of the first regional model of interprofessional education (IPE) in the US, developed by The Commonwealth Medical College, Scranton, PA, USA, as a new, independent, community-based medical school in northeastern Pennsylvania. Essential educational components include collaborative care seminars, interprofessional sessions, simulations, live web-based seminars and newly innovative virtual environment interactive exercises. All of these elements are being integrated into the curricula of 14 undergraduate and allied professional schools, and three graduate medical education programs located in the region. Activities incorporate simulation, standardized patients, student leadership, and faculty and student facilitation. As this new regional model of interprofessional education is fully implemented, its impact will be assessed using both quantitative and qualitative outcomes measurements. Appropriate ongoing modifications to the model will be made to ensure improvement and further applicability to collaborative learning