Votes and a half-binomial

AbstractIn two party elections with popular vote ratio pq, 12≤p=1 −q, a theoretical model suggests replacing the so-called MacMahon cube law approximation (pq)3, for the ratio PQ of candidates elected, by the ratio ƒk(p)ƒk(q) of the two half sums in the binomial expansion of (p+q)2k+1 for some k. This ratio is nearly (pq)3 when k = 6. The success probability gk(p)=(pa(pa+qa) for the power law (pq)a≐PQ is shown to so closely approximate ƒk(p)=Σ0k(r2k+1)p2k+1−rqr, if we choose a = ak=(2k+1)!4kk!k!, that 1≤ƒk(p)gk(p)≤1.01884086 for k≥1 if 12≤p≤1. Computationally, we avoid large binomial coefficients in computing ƒk(p) for k>22 by expressing 2ƒk(p)−1 as the sum (p−q) Σ0k(4pq)sas(2s+1), whose terms decrease by the factors (4pq)(1−12s). Setting K = 4k+3, we compute ak for the large k using a continued fraction πak2=K+12(2K+32(2K+52(2K+…))) derived from the ratio of π to the finite Wallis product approximation

Some Combinatorial Properties of Hook Lengths, Contents, and Parts of Partitions

This paper proves a generalization of a conjecture of Guoniu Han, inspired originally by an identity of Nekrasov and Okounkov. The main result states that certain sums over partitions p of n, involving symmetric functions of the squares of the hook lengths of p, are polynomial functions of n. A similar result is obtained for symmetric functions of the contents and shifted parts of n.Comment: 20 pages. Correction of some inaccuracies, and a new Theorem 4.

FGF/heparin differentially regulates Schwann cell and olfactory ensheathing cell interactions with astrocytes: a role in astrocytosis

After injury, the CNS undergoes an astrocyte stress response characterized by reactive astrocytosis/proliferation, boundary formation, and increased glial fibrillary acidic protein (GFAP) and chondroitin sulfate proteoglycan (CSPG) expression. Previously, we showed that in vitro astrocytes exhibit this stress response when in contact with Schwann cells but not olfactory ensheathing cells (OECs). In this study, we confirm this finding in vivo by demonstrating that astrocytes mingle with OECs but not Schwann cells after injection into normal spinal cord. We show that Schwann cell-conditioned media (SCM) induces proliferation in monocultures of astrocytes and increases CSPG expression in a fibroblast growth factor receptor 1 (FGFR1)-independent manner. However, SCM added to OEC/astrocyte cocultures induces reactive astrocytosis and boundary formation, which, although sensitive to FGFR1 inhibition, was not induced by FGF2 alone. Addition of heparin to OEC/astrocyte cultures induces boundary formation, whereas heparinase or chlorate treatment of Schwann cell/astrocyte cultures reduces it, suggesting that heparan sulfate proteoglycans (HSPGs) are modulating this activity. In vivo, FGF2 and FGFR1 immunoreactivity was increased over grafted OECs and Schwann cells compared with the surrounding tissue, and HSPG immunoreactivity is increased over reactive astrocytes bordering the Schwann cell graft. These data suggest that components of the astrocyte stress response, including boundary formation, astrocyte hypertrophy, and GFAP expression, are mediated by an FGF family member, whereas proliferation and CSPG expression are not. Furthermore, after cell transplantation, HSPGs may be important for mediating the stress response in astrocytes via FGF2. Identification of factors secreted by Schwann cells that induce this negative response in astrocytes would further our ability to manipulate the inhibitory environment induced after injury to promote regeneration

Evidence of Color Coherence Effects in W+jets Events from ppbar Collisions at sqrt(s) = 1.8 TeV

We report the results of a study of color coherence effects in ppbar collisions based on data collected by the D0 detector during the 1994-1995 run of the Fermilab Tevatron Collider, at a center of mass energy sqrt(s) = 1.8 TeV. Initial-to-final state color interference effects are studied by examining particle distribution patterns in events with a W boson and at least one jet. The data are compared to Monte Carlo simulations with different color coherence implementations and to an analytic modified-leading-logarithm perturbative calculation based on the local parton-hadron duality hypothesis.Comment: 13 pages, 6 figures. Submitted to Physics Letters

Search for electroweak production of single top quarks in ppˉp\bar{p} collisions.

We present a search for electroweak production of single top quarks in the electron+jets and muon+jets decay channels. The measurements use ~90 pb^-1 of data from Run 1 of the Fermilab Tevatron collider, collected at 1.8 TeV with the DZero detector between 1992 and 1995. We use events that include a tagging muon, implying the presence of a b jet, to set an upper limit at the 95% confidence level on the cross section for the s-channel process ppbar->tb+X of 39 pb. The upper limit for the t-channel process ppbar->tqb+X is 58 pb. (arXiv