Let (W,S) be a Coxeter system. An element in W and an ordered list of elements in S give us a subword complex, as defined by Knutson and Miller. We define the “fish product” between an involution in W and a generator in S, which is also discussed by Hultman. This “fish product” always produces an involution. The structure of the involution subword complex ∆̂(Q,w) behaves very similarly to the re...

Postlactational involution of the mammary gland is characterized by two distinct physiological events: apoptosis of the secretory, epithelial cells undergoing programmed cell death, and proteolytic degradation of the mammary gland basement membrane. We examined the spatial and temporal patterns of apoptotic cells in relation to those of proteinases during involution of the BALB/c mouse mammary ...

In this paper we generalize the notion of an ι-symmetric word, from an antimorphic involution, to an arbitrary involution ι as follows: a nonempty word w is said to be ι-symmetric if w = αβ = ι(βα) for some words α, β . We propose the notion of ιtwin-roots (x, y) of an ι-symmetric wordw. We prove the existence and uniqueness of the ι-twin-roots of an ι-symmetric word, and show that the left fac...

This paper follows upon the study of the Airy bispectral involution made in [KR]. There we gave an analogue, for arbitrary rank, of the rank-one bispectral involution developed by Wilson [W1]. Recently, [W2], Wilson has established a relationship between the rank-one bispectral involution and the complex analogue of the CalogeroMoser phase space. This relationship leads to explicit formulae for...

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...

Let (G; ) be given. Here G is an algebraic group and an involution in Out(G) determining an inner class of real forms. Fix a Cartan subgroup and a BorelH B G and a set of root vectors fX g. This determines a unique distinguished involution of G mapping to and …xing the pinning (H;B; fX g). Form the extended group G = Go , where = f1; g is the Galois group of C over R, and acts on G by the invol...

We construct a uniform model for highest weight crystals and $B(\infty)$ generalized Kac--Moody algebras using rigged configurations. also show an explicit description of the $\ast$-involution on configurations $B(\infty)$: that interchanges rigging corigging. do this by giving recognition theorem $\ast$-involution. As consequence, we characterize $B(\lambda)$ as subcrystal

Mammary gland involution represents one of the most dramatic examples of programmed cell death/apoptosis and tissue regression during development, yet large gaps still exist in the understanding of the mechanisms involved, and the key factors that trigger involution, are not yet identified. With the focus on identifying "novel" genes associated with mammary gland regression, we used microarray ...

It truly is a remarkable event when traditional pathologic observations lead to new ideas about the prevention of cancer. In this issue of the Journal , Milanese et al. (1) , through a histologic review of breast biopsy specimens, show that the extent of age-related lobular involution is strongly associated with a reduced risk of breast cancer. Breast cancer risk decreased with increasing exten...

The method of forming complex algebras of finite ternary relations will produce all finite nonassociative relation algebras. We use this method to construct many interesting algebras. Every atomic NA determines an involution on atoms, which in turn determines both the cycles of the algebra and also the missing, or forbidden cycles. So we begin by studying the cycles of an arbitrary involution. ...