568 research outputs found
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
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