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'...

Abnormal accumulation of the microtubule-interacting protein tau is associated with neurodegenerative diseases including Alzheimer's disease (AD). β-amyloid (Aβ) lies upstream of abnormal tau behavior, including detachment from microtubules, phosphorylation at several disease-specific sites, and self-aggregation into toxic tau species in AD brains. To prevent the cascade of events leading to ne...

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.

We examine the enumeration problem for essential closed sets of a formal context. Essential closed sets are sets that can be written as the closure of a pseudo-intent. The results for enumeration of essential closed sets are similar to existing results for pseudo-intents, albeit some differences exist. For example, while it is possible to compute the lectically first pseudo-intent in polynomial...

We consider the class of biorthogonal polynomials that are used to solve the inverse spectral problem associated to elementary co-adjoint orbits of the Borel group of upper triangular matrices; these orbits are the phase space of generalized integrable lattices of Toda type. Such polynomials naturally interpolate between the theory of orthogonal polynomials on the line and orthogonal polynomial...

Microtubule-associated Tau proteins are the basic component of intraneuronal and glial inclusions observed in many neurological disorders, the so-called tauopathies. Many etiological factors, phosphorylation, splicing, and mutations, relate Tau proteins to neurodegeneration. Molecular analysis has revealed that hyperphosphorylation and abnormal phosphorylation might be one of the important even...

Axonal microtubules are essential for transport of materials to the synapse. Compromised microtubules and synaptic loss have been demonstrated in AD (Alzheimer's disease), which is believed to contribute to cognitive dysfunction before neuronal death in the early stages of the disease. The mechanism by which hyperphosphorylated tau, the building block of neurofibrillary tangles, one of the path...

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.

The microtubule-associated protein tau regulates diverse and essential microtubule functions, from the nucleation and promotion of microtubule polymerization to the regulation of microtubule polarity and dynamics, as well as the spacing and bundling of axonal microtubules. Thermodynamic studies show that tau interacts with microtubules in the low- to mid-nanomolar range, implying moderate bindi...