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 $A_2$, $A_3$, $B_2$, $B_3$ and $C_3$. In this paper, we consider the case of $G_2$-type. We define certain analogues of Bernoulli polynomials of $G_2$-type and study the generating functions of them to determine the coefficients of Witten's volume formulas of $G_2$-type. Next we consider the meromorphic continuation of the zeta-function of $G_2$-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

Regularity of Edge Ideals and Their Powers

We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{ reg } I(G)$ and the asymptotic linear function $\text{ reg } I(G)^q$, for $q \geq 1,$ in terms of combinatorial data of the given graph $G.$Comment: 31 pages, 15 figure