Let $R$ be a commutative ring with identity and $M$ be a finitely generated unital $R$-module. In this paper, first we give necessary and sufficient conditions that a finitely generated module to be a multiplication module. Moreover, we investigate some conditions which imply that the module $M$ is the direct sum of some cyclic modules and free modules. Then some properties of Fitting ideals o...

We study rigidity properties of lattices in Isom(H) SOn,1(R), n ≥ 3, and of surface groups in Isom(H2) SL2(R) in the context of integrable measure equivalence. The results for lattices in Isom(H), n ≥ 3, are generalizations of Mostow rigidity; they include a cocycle version of strong rigidity and an integrable measure equivalence classification. Despite the lack of Mostow rigidity for n = 2 we ...

Given a CAT(0) cube complex X, we show that if Aut(X) 6= Isom(X) then there exists a full subcomplex of X which decomposes as a product with R. As applications, we prove that ifX is δ-hyperbolic, cocompact and 1-ended, then Aut(X) = Isom(X) unless X is quasi-isometric to H, and extend the rank-rigidity result of Caprace–Sageev to any lattice Γ ≤ Isom(X).

We study the twisted index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group. We apply these results to obtain qualitative results on real and complex hyperbolic spaces in 2 and 4 dimensions, related to generalizations of the Bethe-Sommerfeld conjecture and the Ten Martini Problem, ...

In this paper we show that a variation of a technique of Mǐskin and Tall yields a cocompact completely regular Moore space that is Scott-domain-representable and has a closed Gδ-subspace that is not Scott-domain-representable. This clarifies the general topology of Scott-domain-representable spaces and raises additional questions about Scott-domain representability in Moore spaces. 2000 AMS Cla...

Let M be a complete Riemannian manifold with a free cocompact Z-action. Let k(t, m1, m2) be the heat kernel on M . We compute the asymptotics of k(t, m1, m2) in the limit in which t → ∞ and d(m1, m2) ∼ √ t. We show that in this limit, the heat diffusion is governed by an effective Euclidean metric on R coming from the Hodge inner product on H(M/Z;R).

$r$-module. in this paper, we explore more properties of $max$-injective modules and we study some conditions under which the maximal spectrum of $m$ is a $max$-spectral space for its zariski topology.

Neurodegenerative tauopathies characterized by hyperphosphorylated tau include frontotemporal dementia and Parkinsonism linked to chromosome 17 (FTDP-17) and Alzheimer's disease (AD). Reducing tau levels improves cognitive function in mouse models of AD and FTDP-17, but the mechanisms regulating the turnover of pathogenic tau are unknown. We found that tau is acetylated and that tau acetylation...

Microtubule (MT) associated protein tau is abnormally hyperphosphorylated and aggregated into paired helical filaments (PHFs), which manifest as neurofibrillary tangles (NFTs) in the brains of individuals with Alzheimer's disease (AD) and related tauopathies. Hyperphosphorylation and truncation of tau have been linked to the progression of the disease. However, the nature of phosphorylation and...