The Decomposition of 4(xp-1)/(x-1). II

If 4X = 4(xp - 1) / (x - 1) where p is an odd prime then 4X= Y2 - (- 1) p-1 / 2 pZ2, Y and Z being polynomials in x with integral coefficients. These decompositions for 100\u3c p\u3c 200 were given by the author in the Proceedings of the Iowa Academy of Science, 43: 255-262. This work has now been extended to values of p\u3c 225. The decompositions are given herewith. For all decompositions Y is a polynomial of degree (p - 1) / 2 and Z a polynomial of degree (p- 3) / 2. The coefficients only are given in each case first for Y, then for Z

Interactions Between Amyloid-beta and Microglial Cells

Alzheimer’s disease (AD), the most common cause of dementia, is a neurodegenerative condition characterized by loss of memory and intellectual abilities. Intracellular plaques of aggregated amyloid-beta (Ab) protein are a well-known pathology associated with AD. Although symptoms usually appear late in life, the accumulation of Ab begins decades earlier and causes activation of microglia, the brain’s immune cells. The ensuing inflammation contributes significantly to neurodegeneration. Determination of the particular form of Ab that causes the most damage in the brain is one of the major questions in the AD field. My research focused on the interactions of microglia with monomers, protofibrils, and fibrils of Ab. I found that protofibrils, not monomers or fibrils, bind to microglial surfaces, and I confirmed earlier reports that protofibrils elicit a proinflammatory response from microglia. These results were consistent regardless of changing conditions such as temperature, incubation time and Ab concentrations. Another aspect of my research was to investigate how microglia internalize different forms of Ab. The distinction between monomers and protofibrils may have physiological significance in AD, yet there are few reports in the literature in which these two forms of Ab are examined separately. Monomers and protofibrils were carefully separated by size exclusion chromatography before cell treatments, which sets apart this work from research done in other labs. Multiple conditions and strategies, including a novel quantitation method for internalized Ab, demonstrated that microglia favor internalization of protofibrils over monomers. Further experiments determined that microglia are capable of internalizing protofibrils in high amounts without degradation. A significant amount of Ab protofibrils remain in the cytoplasm and are not routed to lysosomes, contradicting reports in the literature. A third research objective involves the study of microvesicles released from microglia. Microvesicles may have a role in AD by transporting Ab within the brain. I conducted experiments in which microglia were stimulated to produce microvesicles, and carried out assays to both confirm the presence of and visualize microvesicles. The studies described here contribute to the understanding of the interactions between microglia and Ab, potentially leading to a possible treatment or cure for AD

The Decomposition of 4(x^p-1)/(x-1)

If 4X = 4(xP-1)/(x-1) where p is an odd prime then 4X = Y2-(-1)(P-1)/2 pZ2, where Y and Z are polynomials in x with integral coefficients. For p = 37 we find the decomposition cited in Recherches sur la theorie des nombres by M. Kraitchik (1924) p. 126. For 37 ≤ p ≤ 61 the decomposition is given by Pocklington in Nature, VoL 107 (1921) pp. 456 and 587. For 67 ≤ p ≤ 97 the results are given by Gouwens in The Mathematical Monthly, Vol. 43, (1936) page 283. Herewith are presented the results for 101 ≤ p ≤ 199. For all decompositions Y is a polynomial of degree (p-1)/2 and Z is a polynomial of degree (p-3)/2. Y is listed first in each case, then Z

Acute control of the sleep switch in Drosophila reveals a role for gap junctions in regulating behavioral responsiveness

Sleep is a dynamic process in most animals, involving distinct stages that probably perform multiple functions for the brain. Before sleep functions can be initiated, it is likely that behavioral responsiveness to the outside world needs to be reduced, even while the animal is still awake. Recent work in Drosophila has uncovered a sleep switch in the dorsal fan-shaped body (dFB) of the fly’s central brain, but it is not known whether these sleep-promoting neurons also govern the acute need to ignore salient stimuli in the environment during sleep transitions. We found that optogenetic activation of the sleep switch suppressed behavioral responsiveness to mechanical stimuli, even in awake flies, indicating a broader role for these neurons in regulating arousal. The dFB-mediated suppression mechanism and its associated neural correlates requires innexin6 expression, suggesting that the acute need to reduce sensory perception when flies fall asleep is mediated in part by electrical synapses. We thank Leonie Kirszenblat for help and comments on the manuscript. We thank Eleni Notaras for help with behavioral experiments. We also thank Chia-Lin Wu for the INX6 antibody. This work was supported by an NIH grant RO1 NS076980-01 to PJS and BVS, and by an NHMRC grant GNT1065713 to BVS. The authors declare no conflicts of interest