1 Probability basics A probability space is a pair (Ω,Pr) where Pr is a normalized measure on Ω, i.e. Pr(Ω) = 1. In combinatorics, it’s mostly sufficient to work with finite probability spaces, so we can avoid a lot of the technicalities of measure theory. We can assume that Ω is a finite set and each elementary event ω ∈ Ω has a certain probability Pr[ω] ∈ [0, 1]; ω∈Ω Pr[ω] = 1. Any subset A ⊆...

The main objective of this paper is to investigate the distribution Andrews-Garvan-Dyson cranks partitions. Let M(m,n) denote number partitions n with crank m, we show that sequence {M(m,n)}|m|≤n−1 unimodal for n≥44. It turns out unimodality related monotonicity properties two partition functions pr(n) and ppr(n). parts taken from {2,3,…,r} let ppr(n) pairs (α,β) partitions, where α a counted b...

Background: Matrix MetalloProteinases (MMPs) degrade extracellular matrix components to provide normal remodeling and contribute to pathological tissue destruction and cell migration in endometriosis. It is accepted that MMPs are resistant to suppression by progesterone in endometriotic tissues. The physiological effects of progesterone are mediated by its two progesterone receptor (PR) isoform...

Introduction Patients with COPD who exhibit elevated levels of anxiety and/or depression are typically less able to improve symptoms and physical activity (PA) following a programme pulmonary rehabilitation (PR). Objective To provide proof concept that offering an intervention comprising cognitive behavioural therapy (CBT) alongside PA modification strategies (BPA) during PR is more effective i...

Regulation of gene transcription by the progesterone receptor (PR) in cooperation with coactivator/corepressor complexes coordinates crucial processes in female reproduction. To investigate functional relationships between PR and steroid receptor coactivators (SRCs) in distinct cell types of uterine tissue during gene transcription, we generated a new transgenic mouse model utilizing a Progeste...

The ovarian steroid hormones estradiol (E) and progesterone (P) play important roles in reproductive physiology and behavior. The effects of E and P are mediated by their respective receptors, estrogen receptor (ER) and progestin receptor (PR) which function as transcription factors. PR is expressed as two isoforms, full length PR-B and the N-terminally truncated PR-A, by the induction of E in ...

We study incompressible magnetohydrodynamic turbulence in both two and three dimensions, with an emphasis on the number of degrees of freedom N. This number is estimated in terms of the magnetic Prandtl number Pr, kinetic Reynolds number Re, and magnetic Reynolds number Rm. Here Re and Rm are dynamic in nature, defined in terms of the kinetic and magnetic energy dissipation rates (or averages o...