We define an equivariant K0-theory for Yetter-Drinfeld algebras over a Hopf algebra with an invertible antipode. We show that there exists a pairing, generalizing Connes’ pairing, between this theory and a suitably defined Hopf algebra equivariant cyclic cohomology theory.

We demonstrate high-resolution fluorescence microscopy based on a cyclic sequential multiphoton (CSM) process, which gives rise to fluorescence emission following a sequence of cyclic transitions between the bright and dark states of a fluorophore induced by pump and reverse light. By temporally modulating the reverse intensity, we can extract the fluorescence signal generated through the CSM p...

Our goal in this paper is to define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called “module algebras” (Definition 2.1). Our motivation lies in the following problem: for an algebra A which admits a module structure over an arbitrary bialgebra B compatible with its product structure, the Hochschild or the cyclic ...

the submodules with the property of the title ( a submodule $n$ of an $r$-module $m$ is called strongly dense in $m$, denoted by $nleq_{sd}m$, if for any index set $i$, $prod _{i}nleq_{d}prod _{i}m$) are introduced and fully investigated. it is shown that for each submodule $n$ of $m$ there exists the smallest subset $d'subseteq m$ such that $n+d'$ is a strongly dense submodule of $m$...

Throughout this paper, R will denote a commutative ring with identity and M is a unitary R- module and Z will denote the ring of integers. We introduce the graph Ω(M) of module M with the set of vertices contain all nontrivial non-essential submodules of M. We investigate the interplay between graph-theoretic properties of Ω(M) and algebraic properties of M. Also, we assign the values of natura...

in this paper we characterize the radical of an arbitrary‎ ‎submodule $n$ of a finitely generated free module $f$ over a‎ ‎commutatitve ring $r$ with identity‎. ‎also we study submodules of‎ ‎$f$ which satisfy the radical formula‎. ‎finally we derive‎ ‎necessary and sufficient conditions for $r$ to be a‎ ‎pr$ddot{mbox{u}}$fer domain‎, ‎in terms of the radical of a‎ ‎cyclic submodule in $rbigopl...

In this paper, we have studied cyclic codes over the ring R = Z4 +uZ4, u = 0. We have considered cyclic codes of odd lengths. A sufficient condition for a cyclic code over R to be a Z4-free module is presented. We have provided the general form of the generators of a cyclic code over R and determined a formula for the ranks of such codes. In this paper we have mainly focused on principally gene...

The purpose of this paper is to analysis the viscoelastic models under dynamic loading. A five-parameter model is chosen for study exhibits elastic, viscous, and retarded elastic response to shearing stress. The viscoelastic specimen is chosen which closely approximates the actual behavior of a polymer. The module of elasticity and viscosity coefficients are assumed to be space dependent i.e. f...

If R is a commutative ring, then we prove that every finitely generated R-module has a pure-composition series with indecomposable cyclic factors and any two such series are isomorphic if and only if R is a Bézout ring and a CF-ring. When R is a such ring, the length of a pure-composition series of a finitely generated R-module M is compared with its Goldie dimension and we prove that these num...