Let $\tau(n)$ be Ramanujan's tau function, defined by the discriminant modular form \[ \Delta(z) = q\prod_{j=1}^{\infty}(1-q^{j})^{24}\ =\ \sum_{n=1}^{\infty}\tau(n) q^n \,,q=e^{2\pi i z} \] (this is unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer's conjecture asserts that $\tau(n)\neq 0$ for all $n\geq 1$; since multiplicative, it suffices to study primes $p$ w...

In Alzheimer's disease (AD), an extensive accumulation of extracellular amyloid plaques and intraneuronal tau tangles, along with neuronal loss, is evident in distinct brain regions. Staging of tau pathology by postmortem analysis of AD subjects suggests a sequence of initiation and subsequent spread of neurofibrillary tau tangles along defined brain anatomical pathways. Further, the severity o...

 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...

Abstract Background Decreased amyloid β 1‐42 (Aβ42) and increased tau (t‐Tau p‐Tau) concentrations in cerebrospinal fluid (CSF) reflect the pathological changes of Alzheimer’s disease (AD) aid diagnosis AD dementia. In United States, ADmark® assay has been almost exclusively used for measurement these biomarkers clinical practice. Recently, our institution switched from to Roche Elecsys assays....

INTRODUCTION Guidelines for the use of cerebrospinal fluid (CSF) biomarkers in the diagnosis of Alzheimer's disease (AD) establish that each laboratory must use internally qualified cutoff values. We determined the concentrations of biomarkers that discriminate cases from controls and combinations that predict the progression to dementia in a Brazilian cohort. METHODS Concentrations of amyloi...

BACKGROUND Despite an extensive evaluation program, patients may remain diagnostically unresolved with regard to the etiology of their cognitive dysfunction. Cerebrospinal fluid neuroinflammation and Alzheimer disease (AD) biomarkers may act as indicators of neurodegenerative disorders in diagnostically unresolved patients. METHODS Data on 348 patients were retrospectively evaluated. All part...

Little is known about the biological mechanisms that shape the distribution of intervals between the completion of RNA molecules (T(p)RNA) , and thus transcriptional noise. We characterize numerically and analytically how the promoter open complex delay (tau(P)) and the transcription initiation rate (k(t)) shape T(p)RNA. From this, we assess the noise and mean of transcript levels and show that...

Hippocampal atrophy as assessed by magnetic resonance imaging (MRI) and abnormal cerebrospinal fluid (CSF) biomarkers are supportive features for the diagnosis of Alzheimer's disease (AD) and are assumed to be indirect pathological markers of the disease. In AD patients, antemortem MRI hippocampal volumes (HVs) correlate with the density of neurofibrillary tangles (but not with senile plaques) ...

BACKGROUND The preclinical patterns of biological markers for Alzheimer's disease (AD) in vivo need further exploration. The aim of this study was therefore to investigate CSF biomarkers, regional cerebral blood flow (rCBF) and cognitive performance in cognitively healthy older individuals. METHOD Within a 2-week period, 32 cognitively healthy older individuals underwent CSF analysis, rCBF me...