Log-concavity and strong log-concavity: a review

We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on $\mathbb{R}$ under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron's theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.Comment: 67 pages, 1 figur

Domain walls in a chiral d-wave superconductor on the honeycomb lattice

We perform a fully self-consistent study of domain walls between different chiral domains in chiral $d_{x^2-y^2} \pm id_{xy}-wave superconductors with an underlying honeycomb lattice structure. We investigate domain walls along all possible armchair and zigzag directions and with a finite global phase shift across the domain wall, in addition to the change of chirality. For armchair domain walls we find the lowest domain wall energy at zero global phase shift, while the most favorable zigzag domain wall has a finite global phase shift dependent on the doping level. Below the van Hove singularity the armchair domain wall is most favorable, while at even higher doping the zigzag domain wall has the lowest energy. The domain wall causes a local suppression of the superconducting order parameter, with the superconducting recovery length following a universal curve for all domain walls. Moreover, we always find four subgap states crossing zero energy and well localized to the domain wall. However, the details of their energy spectrum vary notably, especially with the global phase shift across the domain wall.Comment: 8 pages, 8 figure

The Study of Administration Extract Kebar\u27s Grass on Number Leydig Cells and Sertoli Cells in Rat Rattus Norvegicus That Exposed with Cigarette Smoke

This research aimed to study on the microscopic changes of number leydig cells and sertoli cells. A factorial CRD with periode of treatment and sample collection was applied in this study. An exposure of cigarette smoke was carried out at 10 cigarettes/rat/day for 2.5 hours in a smoking chamber. Extract kebar\u27s grass given was 0.0945 mg/g body weight/day. Twenty seven (27) of male rats were divided into four groups. N group was untreated animals, 20h group is group that was expose to cigarette smoke for 20 days then given administration extract kebar\u27s grass for 20 days and stopping treatment for 20 days and 60h group is group that was expose to cigarette smoke for 60 days then given administration extract kebar\u27s grass 60 days and stopping treatment for 60 days. Data collection was carried out at twice that after cigarette smoke exposure (T0), second after administration extract kebar\u27s grass (T1) and third after stopping treatment (T2). The parameters measured were the number of leydig cells and sertoli cells. That increasing number leydig cells and sertoli cells of rat exposed to cigarette smoke after giving seaweed extract kebar\u27s grass for 20 days and 60 day