1,839 research outputs found
Markov basis and Groebner basis of Segre-Veronese configuration for testing independence in group-wise selections
We consider testing independence in group-wise selections with some
restrictions on combinations of choices. We present models for frequency data
of selections for which it is easy to perform conditional tests by Markov chain
Monte Carlo (MCMC) methods. When the restrictions on the combinations can be
described in terms of a Segre-Veronese configuration, an explicit form of a
Gr\"obner basis consisting of moves of degree two is readily available for
performing a Markov chain. We illustrate our setting with the National Center
Test for university entrance examinations in Japan. We also apply our method to
testing independence hypotheses involving genotypes at more than one locus or
haplotypes of alleles on the same chromosome.Comment: 25 pages, 5 figure
On positivity of Ehrhart polynomials
Ehrhart discovered that the function that counts the number of lattice points
in dilations of an integral polytope is a polynomial. We call the coefficients
of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive
if all Ehrhart coefficients are positive (which is not true for all integral
polytopes). The main purpose of this article is to survey interesting families
of polytopes that are known to be Ehrhart positive and discuss the reasons from
which their Ehrhart positivity follows. We also include examples of polytopes
that have negative Ehrhart coefficients and polytopes that are conjectured to
be Ehrhart positive, as well as pose a few relevant questions.Comment: 40 pages, 7 figures. To appear in in Recent Trends in Algebraic
Combinatorics, a volume of the Association for Women in Mathematics Series,
Springer International Publishin
Contribution of <i>sox9b</i> to pigment cell formation in medaka fish
SoxE-type transcription factors, Sox10 and Sox9, are key regulators of the development of neural crest cells. Sox10 specifies pigment cell, glial, and neuronal lineages, whereas Sox9 is reportedly closely associated with skeletogenic lineages in the head, but its involvement in pigment cell formation has not been investigated genetically. Thus, it is not fully understood whether or how distinctly these genes as well as their paralogs in teleosts are subfunctionalized. We have previously shown using the medaka fish Oryzias latipes that pigment cell formation is severely affected by the loss of sox10a, yet unaffected by the loss of sox10b. Here we aimed to determine whether Sox9 is involved in the specification of pigment cell lineage. The sox9b homozygous mutation did not affect pigment cell formation, despite lethality at the early larval stages. By using sox10a, sox10b, and sox9b mutations, compound mutants were established for the sox9b and sox10 genes and pigment cell phenotypes were analyzed. Simultaneous loss of sox9b and sox10a resulted in the complete absence of melanophores and xanthophores from hatchlings and severely defective iridophore formation, as has been previously shown for sox10a −/−; sox10b −/− double mutants, indicating that Sox9b as well as Sox10b functions redundantly with Sox10a in pigment cell development. Notably, leucophores were present in sox9b −/−; sox10a −/− and sox10a −/−; sox10b −/− double mutants, but their numbers were significantly reduced in the sox9b −/−; sox10a −/− mutants. These findings highlight that Sox9b is involved in pigment cell formation, and plays a more critical role in leucophore development than Sox10b.</p
On Witten multiple zeta-functions associated with semisimple Lie algebras IV
In our previous work, we established the theory of multi-variable Witten
zeta-functions, which are called the zeta-functions of root systems. We have
already considered the cases of types , , , and . In
this paper, we consider the case of -type. We define certain analogues of
Bernoulli polynomials of -type and study the generating functions of them
to determine the coefficients of Witten's volume formulas of -type. Next
we consider the meromorphic continuation of the zeta-function of -type and
determine its possible singularities. Finally, by using our previous method, we
give explicit functional relations for them which include Witten's volume
formulas.Comment: 22 pag
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