We show that the formalism of overpartitions gives a simple involution for the product definition of the Gaussian coefficients. In the formulation of our involution, an overline in the representation of an overpartition is endowed with a weight.

We introduce a generalization of the Euclidean algorithm for rings equipped with an involution, and completely enumerate all isomorphism classes orders over definite, rational quaternion algebras orthogonal involution that admit such algorithm. give two applications: first, any order admits has class number 1; second, we show how existence relates to problem constructing explicit Dirichlet doma...

Elevated mammographic density (MD) is an established breast cancer risk factor. Reduced involution of terminal duct lobular units (TDLU), the histologic source of most breast cancers, has been associated with higher MD and breast cancer risk. We investigated relationships of TDLU involution with area and volumetric MD, measured throughout the breast and surrounding biopsy targets (perilesional)...

Tsetse flies are important vectors of human and animal trypanosomiasis. Ability to reduce tsetse populations is an effective means of disease control. Lactation is an essential component of tsetse's viviparous reproductive physiology and requires a dramatic increase in the expression and synthesis of milk proteins by the milk gland organ in order to nurture larval growth. In between each gonotr...

Extracellular matrix (ECM) plays an important role in the maintenance of mammary epithelial differentiation in culture. We asked whether changes in mouse mammary specific function in vivo correlate with changes in the ECM. We showed, using expression of beta-casein as a marker, that the temporal expression of ECM-degrading proteinases and their inhibitors during lactation and involution are inv...

We describe an involution on a set of sequences associated with lattice paths with north or east steps constrained to lie between two arbitrary boundaries. This involution yields recursions (from which determinantal formulas can be derived) for the number and area enumerator of such paths. An analogous involution can be defined for parking functions with arbitrary lower and upper bounds. From t...

Xenopus provides a well-studied model of vertebrate gastrulation, but a central feature, the movement of the mesoderm to the interior of the embryo, has received little attention. Here, we analyze mesoderm involution at the Xenopus dorsal blastopore lip. We show that a phase of rapid involution – peak involution – is intimately linked to an early stage of convergent extension, which involves di...

On a Lebesgue measure space with measure element do, and total measure finite or infinite, we consider the complex-valued measurable functions f, y, so, ., each determined only a.e., and each belonging to all L,-classes simultaneously, 1<r< a. We donote (i) by F = {f} a ring of functions with complex constants and ordinary multiplication and closed under the involution: if f -= f + if2 e F then...

Androgens control growth of prostate epithelial cells and androgen deprivation induces apoptosis, leading to prostate involution. We investigated the effects of surgical stress on prostate involution induced by androgen ablation and determined the underlying mechanisms. Androgen ablation in mice was induced by surgical castration and administration of the anti-androgenic drugs bicalutamide and ...