In this issue of the Journal, Ghosh et al. (1) describe the application of two visual assessments to predict breast cancer risk—lobular involution, a histological observation, and mammographic breast density (MBD), a radiological assessment. Previously, this group has demonstrated that more extensive lobular involution was inversely related to both MBD (2) and breast cancer risk (3) in a cohort...

Given a mixing shift of finite type X , we consider which subshifts of finite type Y ⊂ X can be realized as the fixed point shift of an inert involution of X . We establish a condition on the periodic points of X and Y that is necessary for Y to be the fixed point shift of an inert involution of X . We show that this condition is sufficient to realize Y as the fixed point shift of an involution...

This paper introduces a generalization of the operation of catenation: u[k]lv, the left-k-insertion, is the set of all words obtained by inserting v into u in positions that are at most k letters away from the left extremity of the word u. We define k-suffix codes using the left-kinsertion operation and extend the concept of k-prefix and k-suffix codes to involution k-prefix and involution k-su...

In 2000, Babson and Steingrímsson generalized the notion of permutation patterns to the so-called vincular patterns, and they showed that many Mahonian statistics can be expressed as sums of vincular pattern occurrence statistics. STAT is one of such Mahonian statistics discoverd by them. In 2016, Kitaev and the third author introduced a words analogue of STAT and proved a joint equidistributio...

An extremal element in a Lie algebra g over field of characteristic not 2 is an x ∈ such that [ , ] contained the linear span . The element, called point, inner ideal i.e. subspace I satisfying ≤ We show different from 3 geometry with point set points and as lines minimal ideals containing at least two Moufang spherical building, or case there are no set. This last result on sets obtained by co...

Proper selection of material in manufacturing firms is a vital role of designer depending upon the different era of application. The material selection problem is very complex and challenging task today. Erroneous cull of material frequently leads to astronomically immense cost involution, and finally drives towards unfortunate component or product breakdown. Thus, the designer necessitates dis...

Theorem 1.1. Let A be a semisimple complex Banach algebra with the following properties: (i) for every closed right ideal R in A, there exists a closed left ideal L such that R∩L= {0} and R+ L= A (each a∈ A can be written in the form a= a1 + a2 with a1 ∈ R, a2 ∈ L); (ii) if a,b in A are such that ab = ba= 0, then ‖a+ b‖2 = ‖a‖2 +‖b‖2. Then A is a commutative proper H∗-algebra [1]. It is easy to...

Morphogenetic cell movements mould the shape of the developing organism. These movements have been extensively studied during Drosophila embryonic development. In particular live imaging analyses have revealed the complexities underlying the comparatively simple processes of germband extension, retraction and dorsal closure. I am applying live imaging techniques to an apparently complex morphog...

Morphogenetic cell movements mould the shape of the developing organism. These movements have been extensively studied during Drosophila embryonic development. In particular live imaging analyses have revealed the complexities underlying the comparatively simple processes of germband extension, retraction and dorsal closure. I am applying live imaging techniques to an apparently complex morphog...

Let $X$ be a finitistic space with the mod $2$ cohomology of product projective and $4$-sphere. Assume that admits free involution. In this paper we study algebra quotient by action involution derive some consequences regarding existence $\mathbb{Z}\_2$-equivariant maps between such an $n$-sphere.