Relative to a hereditary torsion theory $tau$ we introduce a dimension for a module $M$, called {em $tau$-rank of} $M$, which coincides with the reduced rank of $M$ whenever $tau$ is the Goldie torsion theory. It is shown that the $tau$-rank of $M$ is measured by the length of certain decompositions of the $tau$-injective hull of $M$. Moreover, some relations between the $tau$-rank of $M$ and c...

‎Let $R$ be a $*$-prime ring with center‎ ‎$Z(R)$‎, ‎$d$ a non-zero $(sigma,tau)$-derivation of $R$ with associated‎ ‎automorphisms $sigma$ and $tau$ of $R$‎, ‎such that $sigma$‎, ‎$tau$‎ ‎and $d$ commute with $'*'$‎. ‎Suppose that $U$ is an ideal of $R$ such that $U^*=U$‎, ‎and $C_{sigma,tau}={cin‎ ‎R~|~csigma(x)=tau(x)c~mbox{for~all}~xin R}.$ In the present paper‎, ‎it is shown that if charac...

 A topoframe, denoted by $L_{ tau}$,  is a pair $(L, tau)$ consisting of a frame $L$ and a subframe $ tau $ all of whose elements are complementary elements in $L$. In this paper, we define and study the notions of a $tau $-real-continuous function on a frame $L$ and the set of real continuous functions $mathcal{R}L_tau $ as an $f$-ring. We show that $mathcal{R}L_{ tau}$ is actually a generali...

Alzheimer's disease (AD) is a progressive neurodegenerative disease that is pathologically characterized by extracellular deposition of beta-amyloid peptide (Abeta) and intraneuronal accumulation of hyperphosphorylated tau protein. Because cerebrospinal fluid (CSF) is in direct contact with the extracellular space of the brain, it provides a reflection of the biochemical changes in the brain in...

Tau is one of several types of microtubule-associated proteins (MAPs), responsible for the assembly and stability of microtubule networks that is present only in neurons and predominantly localized in axons which its functions are tightly regulated by phosphorylation. Via as yet unknown mechanisms, tau becomes hyperphosphorylated and accompanies with neuronal degeneration, loss of synapses...

A topoframe, denoted by $L_{ tau}$, is a pair $(L, tau)$ consisting of a frame $L$ and a subframe $ tau $ all of whose elements are complementary elements in$L$. $f$-ring $mathcal{R}(L_{ tau})$ is equal to the set $${fin Frm(mathcal{P}(mathbb R), L): f(mathfrak{O}(mathbb R))subseteq tau} .$$ In this paper, for every complemented element $ain L$ with $a, a'...

Conflicting evidence supports a role for tau as an essential neuronal cytoskeletal protein or as a redundant protein whose function can be fulfilled by other microtubule-associated proteins. To investigate the function of tau in axonogenesis, we created tau deficient mice by disrupting the TAU gene. The engineered mice do not express the tau protein, appear physically normal and are able to rep...

We first characterize $tau$-complemented modules with relative (pre)-covers. We also introduce an extending module relative to $tau$-pure submodules on a hereditary torsion theory $tau$ and give its relationship with $tau$-complemented modules.

‎In this paper‎, ‎a spectral Tau method for solving fractional Riccati‎ ‎differential equations is considered‎. ‎This technique describes‎ ‎converting of a given fractional Riccati differential equation to a‎ ‎system of nonlinear algebraic equations by using some simple‎ ‎matrices‎. ‎We use fractional derivatives in the Caputo form‎. ‎Convergence analysis of the proposed method is given an...

we first characterize $tau$-complemented modules with relative (pre)-covers. we also introduce an extending module relative to $tau$-pure submodules on a hereditary torsion theory $tau$ and give its relationship with $tau$-complemented modules.