On the Volume of Boolean expressions of Large Congruent Balls

We consider the volume of a Boolean expression of some congruent balls about a given system of centers in the d-dimensional Euclidean space. When the radius r of the balls is large, this volume can be approximated by a polynomial of r, which will be computed up to an O(r^{d−3}) error term. We study how the top coefficients of this polynomial depend on the set of the centers. It is known that in the case of the union of the balls, the top coefficients are some constant multiples of the intrinsic volumes of the convex hull of the centers. Thus, the coefficients in the general case lead to generalizations of the intrinsic volumes, in particular, to a generalization of the mean width of a set. Some known results on the mean width, along with the theorem on its monotonicity under contractions are extended to the "Boolean analogues" of the mean width

Transcriptome Sequencing in the Preoptic Region of Rat Dams Reveals a Role of Androgen Receptor in the Control of Maternal Behavior

(1) Background: Preoptic region of hypothalamus is responsible to control maternal behavior, which was hypothesized to be associated with gene expressional changes. (2) Methods: Transcriptome sequencing was first applied in the preoptic region of rat dams in comparison to a control group of mothers whose pups were taken away immediately after parturition and did not exhibit caring behavior 10 days later. (3) Results: Differentially expressed genes were found and validated by quantitative RT-PCR, among them NACHT and WD repeat domain containing 1 (Nwd1) is known to control androgen receptor (AR) protein levels. The distribution of Nwd1 mRNA and AR was similar in the preoptic area. Therefore, we focused on this steroid hormone receptor and found its reduced protein level in rat dams. To establish the function of AR in maternal behavior, its antagonist was administered intracerebroventricularly into mother rats and increased pup-directed behavior of the animals. (4) Conclusions: AR levels are suppressed in the preoptic area of mothers possibly mediated by altered Nwd1 expression in order to allow sustained high-level care for the pups. Thus, our study first implicated the AR in the control of maternal behaviors

Locked and Unlocked Chains of Planar Shapes

We extend linkage unfolding results from the well-studied case of polygonal linkages to the more general case of linkages of polygons. More precisely, we consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are hinged together sequentially at rotatable joints. Our goal is to characterize the families of planar shapes that admit locked chains, where some configurations cannot be reached by continuous reconfiguration without self-intersection, and which families of planar shapes guarantee universal foldability, where every chain is guaranteed to have a connected configuration space. Previously, only obtuse triangles were known to admit locked shapes, and only line segments were known to guarantee universal foldability. We show that a surprisingly general family of planar shapes, called slender adornments, guarantees universal foldability: roughly, the distance from each edge along the path along the boundary of the slender adornment to each hinge should be monotone. In contrast, we show that isosceles triangles with any desired apex angle less than 90 degrees admit locked chains, which is precisely the threshold beyond which the inward-normal property no longer holds.Comment: 23 pages, 25 figures, Latex; full journal version with all proof details. (Fixed crash-induced bugs in the abstract.