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

Methylation pattern of CDH13 gene in digestive tract cancers

Recently, the loss of CDH13 (T-cadherin, H-cadherin) gene expression accompanied by CDH13 promoter methylation was identified in colon cancers. We examined CDH13 methylation in oesophageal and gastric carcinomas. Five of 37 oesophageal cancers (14%) and 23 of 66 gastric cancers (35%) demonstrated abnormal methylation of the CDH13 promoter. Abnormal methylation was frequently found in gastric cancers of patients at all clinical stages just as in E-cadherin, another of the cadherin family, suggesting that these cancers could be methylated at an early stage. These results suggested that CDH13 might play a variety of roles depending on the tissue type

Unimodality Problems in Ehrhart Theory

Ehrhart theory is the study of sequences recording the number of integer points in non-negative integral dilates of rational polytopes. For a given lattice polytope, this sequence is encoded in a finite vector called the Ehrhart $h^*$-vector. Ehrhart $h^*$-vectors have connections to many areas of mathematics, including commutative algebra and enumerative combinatorics. In this survey we discuss what is known about unimodality for Ehrhart $h^*$-vectors and highlight open questions and problems.Comment: Published in Recent Trends in Combinatorics, Beveridge, A., et al. (eds), Springer, 2016, pp 687-711, doi 10.1007/978-3-319-24298-9_27. This version updated October 2017 to correct an error in the original versio

A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes

The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In particular, we show that the normalized volume of the free sum of full dimensional polytopes is precisely the product of the normalized volumes of the summands.Comment: Published in the proceedings of 2017 Southern Regional Algebra Conferenc

Crown Rust Resistance in Transgenic Italian Ryegrass (\u3cem\u3eL. Multiflorum\u3c/em\u3e) Expressing a Rice Chitinase Gene and Crosses with Cytoplasmic Male Sterile Hybrid Ryegrass

Italian ryegrass (Lolium multiflorum Lam.) is one of the most important forage grasses in the temperate region. In ryegrasses, crown rust (Puccinia coronata) is the most serious foliar fungal disease and brings about a reduction of herbage yield and loss of palatability to grass-eating domestic animals. In this study, we tried to increase tolerance to the pathogen by introducing a rice chitinase gene using particle bombardment

Backbone and side-chain 1H, 15N and 13C resonance assignments of S18Y mutant of ubiquitin carboxy-terminal hydrolase L1

Ubiquitin carboxy-terminal hydrolase L1 (UCH-L1), also known as PGP9.5, is a protein of 223 amino acids. Although it was originally characterized as a deubiquitinating enzyme, recent studies indicate that it also functions as a ubiquitin (Ub) ligase and a mono-Ub stabilizer. It is highly abundant in brain, constituting up to 2% of total brain proteins. Down-regulation and extensive oxidative modification of UCH-L1 have been observed in the brains of Alzheimer’s disease (AD) and Parkinson’s disease (PD) patients. Mutations in the UCH-L1 gene have been reported to be linked to Parkinson’s disease, in particular, the I93 M variant is associated with a higher susceptibility of PD in contrast to a higher protection against PD for the S18Y variant. Hence, the structure of UCH-L1 and the underlying effects of disease associated mutations on the structure and function of UCH-L1 are of considerable interest. Here, we report the NMR spectral assignments of the S18Y human UCH-L1 mutant with the aim to obtain better understanding about the risk of Parkinson’s disease against structural and dynamical changes induced by this mutation on UCH-L1

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

Monomial ideals whose depth function has any given number of strict local maxima

We construct monomial ideals with the property that their depth function has any given number of strict local maxima