The rrth moment of the divisor function: an elementary approach

Let $\tau(n)$ be the number of divisors of $n$. We give an elementary proof of the fact that $$\sum_{n\le x} \tau(n)^r =xC_{r} (\log x)^{2^r-1}+O(x(\log x)^{2^r-2}),$$ for any integer $r\ge 2$. Here, $$C_{r}=\frac{1}{(2^r-1)!} \prod_{p\ge 2}\left( \left(1-\frac{1}{p}\right)^{2^r} \left(\sum_{\alpha\ge 0} \frac{(\alpha+1)^r}{p^{\alpha}}\right)\right).$$Comment: 8 pages, revise

Entanglement and Extreme Spin Squeezing for a Fluctuating Number of Indistinguishable Particles

We extend the criteria for $k$-particle entanglement from the spin squeezing parameter presented in [A.S. S{\o}rensen and K. M{\o}lmer, Phys. Rev. Lett. {\bf 86}, 4431 (2001)] to systems with a fluctating number of particles. We also discuss how other spin squeezing inequalities can be generalized to this situation. Further, we give an operational meaning to the bounds for cases where the individual particles cannot be addressed. As a by-product, this allows us to show that in spin squeezing experiments with cold gases the particles are typically distinguishable in practise. Our results justify the application of the S{\o}rensen-M{\o}lmer bounds in recent experiments on spin squeezing in Bose-Einstein condensates

Single Mild Traumatic Brain Injury Induces Persistent Disruption of the Blood-Brain Barrier, Neuroinflammation and Cognitive Decline in Hypertensive Rats

Traumatic brain injury (TBI) induces blood-brain barrier (BBB) disruption, which contributes to secondary injury of brain tissue and development of chronic cognitive decline. However, single mild (m)TBI, the most frequent form of brain trauma disrupts the BBB only transiently. We hypothesized, that co-morbid conditions exacerbate persistent BBB disruption after mTBI leading to long term cognitive dysfunction. Since hypertension is the most important cerebrovascular risk factor in populations prone to mild brain trauma, we induced mTBI in normotensive Wistar and spontaneously hypertensive rats (SHR) and we assessed BBB permeability, extravasation of blood-borne substances, neuroinflammation and cognitive function two weeks after trauma. We found that mTBI induced a significant BBB disruption two weeks after trauma in SHRs but not in normotensive Wistar rats, which was associated with a significant accumulation of fibrin and increased neuronal expression of inflammatory cytokines TNFα, IL-1β and IL-6 in the cortex and hippocampus. SHRs showed impaired learning and memory two weeks after mild TBI, whereas cognitive function of normotensive Wistar rats remained intact. Future studies should establish the mechanisms through which hypertension and mild TBI interact to promote persistent BBB disruption, neuroinflammation and cognitive decline to provide neuroprotection and improve cognitive function in patients with mTBI

Orientational properties of nematic disclinations

Topological defects play a pivotal role in the physics of liquid crystals and represent one of the most prominent and well studied aspects of mesophases. While in two-dimensional nematics, disclinations are traditionally treated as point-like objects, recent experimental studies on active nematics have suggested that half-strength disclinations might in fact possess a polar structure. In this article, we provide a precise definition of polarity for half-strength nematic disclinations, we introduce a simple and robust method to calculate this quantity from experimental and numerical data and we investigate how the orientational properties of half-strength disclinations affect their relaxational dynamics.Comment: 6 pages, 5 figures, supplementary movies at http://wwwhome.lorentz.leidenuniv.nl/~giomi/sup_mat/20150720

Age-dependent changes at the blood-brain barrier. A Comparative structural and functional study in young adult and middle aged rats

Abstract Decreased beta-amyloid clearance in Alzheimer’s disease and increased blood-brain barrier permeability in aged subjects have been reported in several articles. However, morphological and functional characterization of blood-brain barrier and its membrane transporter activity have not been described in physiological aging yet. The aim of our study was to explore the structural changes in the brain microvessels and possible functional alterations of P-glycoprotein at the blood-brain barrier with aging. Our approach included MR imaging for anatomical orientation in middle aged rats, electronmicroscopy and immunohistochemistry to analyse the alterations at cellular level, dual or triple-probe microdialysis and SPECT to test P-glycoprotein functionality in young and middle aged rats. Our results indicate that the thickness of basal lamina increases, the number of tight junctions decreases and the size of astrocyte endfeet extends with advanced age. On the basis of microdialysis and SPECT results the P-gp function is reduced in old rats. With our multiparametric approach a complex regulation can be suggested which includes elements leading to increased permeability of blood-brain barrier by enhanced paracellular and transcellular transport, and factors working against it. To verify the role of P-gp pumps in brain aging further studies are warranted