Obstructions to shellability, partitionability, and sequential Cohen-Macaulayness

For a property $\cal P$ of simplicial complexes, a simplicial complex $\Gamma$ is an obstruction to $\cal P$ if $\Gamma$ itself does not satisfy $\cal P$ but all of its proper restrictions satisfy $\cal P$. In this paper, we determine all obstructions to shellability of dimension $\le 2$, refining the previous work by Wachs. As a consequence we obtain that the set of obstructions to shellability, that to partitionability and that to sequential Cohen-Macaulayness all coincide for dimensions $\le 2$. We also show that these three sets of obstructions coincide in the class of flag complexes. These results show that the three properties, hereditary-shellability, hereditary-partitionability, and hereditary-sequential Cohen-Macaulayness are equivalent for these classes

Nonadditive Set Functions on a Finite Set and Linear Inequalities

AbstractA set function is a function whose domain is the power set of a set, which is assumed to be finite in this paper. We treat a possibly nonadditive set function, i.e., a set function which does not satisfy necessarily additivity, ϕ(A)+ϕ(B)=ϕ(A∪B) forA∩B=∅, as an element of the linear space on the power set. Then some of the famous classes of set functions are polyhedral in that linear space, i.e., expressed by a finite number of linear inequalities. We specify the sets of the coefficients of the linear inequalities for some classes of set functions. Then we consider the following three problems: (a) the domain extension problem for nonadditive set functions, (b) the sandwich problem for nonadditive set functions, and (c) the representation problem of a binary relation by a nonadditive set function, i.e., the problem of nonadditive comparative probabilities

Envelopes and clutters

AbstractIn this paper, a set function ϕ defined on a finite set Ω is said to be an upper envelope if there exists a set {pi} of nonnegative vectors on Ω such that ϕ(G)=max{p1(G),…,pn(G)} for all G⊂Ω. All upper envelopes form a convex cone. We give a necessary and sufficient condition for an upper envelope to be extremal in the cone of all upper envelopes in terms of its representation. Furthermore we study the upper envelopes represented by clutters. We show that a clutter is extremal in the cone of the upper envelopes if and only if it satisfies some kind of connectivity

Fast Lattice Basis Reduction Suitable for Massive Parallelization and Its Application to the Shortest Vector Problem

The hardness of the shortest vector problem for lattices is a fundamental assumption underpinning the security of many lattice-based cryptosystems, and therefore, it is important to evaluate its difficulty. Here, recent advances in studying the hardness of problems in large-scale lattice computing have pointed to need to study the design and methodology for exploiting the performance of massive parallel computing environments. In this paper, we propose a lattice basis reduction algorithm suitable for massive parallelization. Our parallelization strategy is an extension of the Fukase-Kashiwabara algorithm~(J. Information Processing, Vol. 23, No. 1, 2015). In our algorithm, given a lattice basis as input, variants of the lattice basis are generated, and then each process reduces its lattice basis; at this time, the processes cooperate and share auxiliary information with each other to accelerate lattice basis reduction. In addition, we propose a new strategy based on our evaluation function of a lattice basis in order to decrease the sum of squared lengths of orthogonal basis vectors. We applied our algorithm to problem instances from the SVP Challenge. We solved a 150-dimension problem instance in about 394 days by using large clusters, and we also solved problem instances of dimensions 134, 138, 140, 142, 144, 146, and 148. Since the previous world record is the problem of dimension 132, these results demonstrate the effectiveness of our proposal

Tumor Induction by Azoxymethane (AOM) and 2-Amino-l-methyl-6-phenylimidazo[4,5-b]pyridine (PhIP) in F344 Rat Gastric Mucosa Featuring Intestinal Metaplasia Caused by X-irradiation

Male F344 5-week-old rats were X-irradiated, and 16 weeks after the first dose. azoxymethane (AOM) was injected or 2-amino-l-methyl-6-phenylimidazo [4,5-b]pyridine (PhIP) was given by intragastric intubation. Tumors in the pylorus of the glandular stomach were observed in 4 out of the 29 animals receiving X-rays + AOM and in 4 out of the 25 animals receiving X-rays + PhIP, 12 months after administration. No such lesions were found in the chemical or X-ray alone groups. Intestinal metaplasia and some induced tumors were positive for CDX2. It was concluded that the presence of intestinal metaplasia may increase sensitivity to the induction of gastric tumors by colon carcinogens

GATA2 Mediates Thyrotropin-Releasing Hormone-Induced Transcriptional Activation of the Thyrotropin β Gene

Thyrotropin-releasing hormone (TRH) activates not only the secretion of thyrotropin (TSH) but also the transcription of TSHβ and α-glycoprotein (αGSU) subunit genes. TSHβ expression is maintained by two transcription factors, Pit1 and GATA2, and is negatively regulated by thyroid hormone (T3). Our prior studies suggest that the main activator of the TSHβ gene is GATA2, not Pit1 or unliganded T3 receptor (TR). In previous studies on the mechanism of TRH-induced activation of the TSHβ gene, the involvements of Pit1 and TR have been investigated, but the role of GATA2 has not been clarified. Using kidney-derived CV1 cells and pituitary-derived GH3 and TαT1 cells, we demonstrate here that TRH signaling enhances GATA2-dependent activation of the TSHβ promoter and that TRH-induced activity is abolished by amino acid substitution in the GATA2-Zn finger domain or mutation of GATA-responsive element in the TSHβ gene. In CV1 cells transfected with TRH receptor expression plasmid, GATA2-dependent transactivation of αGSU and endothelin-1 promoters was enhanced by TRH. In the gel shift assay, TRH signal potentiated the DNA-binding capacity of GATA2. While inhibition by T3 is dominant over TRH-induced activation, unliganded TR or the putative negative T3-responsive element are not required for TRH-induced stimulation. Studies using GH3 cells showed that TRH-induced activity of the TSHβ promoter depends on protein kinase C but not the mitogen-activated protein kinase, suggesting that the signaling pathway is different from that in the prolactin gene. These results indicate that GATA2 is the principal mediator of the TRH signaling pathway in TSHβ expression