Bounds on the number of real solutions to polynomial equations

We use Gale duality for polynomial complete intersections and adapt the proof of the fewnomial bound for positive solutions to obtain the bound (e^4+3) 2^(k choose 2) n^k/4 for the number of non-zero real solutions to a system of n polynomials in n variables having n+k+1 monomials whose exponent vectors generate a subgroup of Z^n of odd index. This bound exceeds the bound for positive solutions only by the constant factor (e^4+3)/(e^2+3) and it is asymptotically sharp for k fixed and n large.Comment: 5 page

An inequality of Kostka numbers and Galois groups of Schubert problems

We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. Using a criterion of Vakil and a special position argument due to Schubert, this follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, an easy combinatorial injection proves the inequality. For the remaining cases, we use that these Kostka numbers appear in tensor product decompositions of sl_2(C)-modules. Interpreting the tensor product as the action of certain commuting Toeplitz matrices and using a spectral analysis and Fourier series rewrites the inequality as the positivity of an integral. We establish the inequality by estimating this integral.Comment: Extended abstract for FPSAC 201

Analysis of Interactive Audio in Journey

https://remix.berklee.edu/graduate-studies-scoring/1063/thumbnail.jp

Dynamic distribution and stem cell characteristics of Sox1-expressing cells in the cerebellar cortex

Bergmann glia cells are a discrete radial glia population surrounding Purkinje cells in the cerebellar cortex. Although Bergmann glia are essential for the development and correct arborization of Purkinje cells, little is known about the regulation of this cell population after the developmental phase. In an effort to characterize this population at the molecular level, we have analyzed marker expression and established that adult Bergmann glia express Sox1, Sox2 and Sox9, a feature otherwise associated with neural stem cells (NSCs). In the present study, we have further analyzed the developmental pattern of Sox1-expressing cells in the developing cerebellum. We report that before becoming restricted to the Purkinje cell layer, Sox1-positive cells are present throughout the immature tissue, and that these cells show characteristics of Bergmann glia progenitors. Our study shows that these progenitors express Sox1, Sox2 and Sox9, a signature maintained throughout cerebellar maturation into adulthood. When isolated in culture, the Sox1-expressing cerebellar population exhibited neurosphere-forming ability, NSC-marker characteristics, and demonstrated multipotency at the clonal level. Our results show that the Bergmann glia population expresses Sox1 during cerebellar development, and that these cells can be isolated and show stem cell characteristics in vitro, suggesting that they could hold a broader potential than previously thought. © 2009 IBCB, SIBS, CAS. All rights reserved