Ranking patterns of the unfolding model and arrangements

In the unidimensional unfolding model, given m objects in general position there arise 1+m(m-1)/2 rankings. The set of rankings is called the ranking pattern of the m given objects. By changing these m objects, we can generate various ranking patterns. It is natural to ask how many ranking patterns can be generated and what is the probability of each ranking pattern when the objects are randomly chosen? These problems are studied by introducing a new type of arrangement called mid-hyperplane arrangement and by counting cells in its complement.Comment: 29 pages, 2 figure

Periodicity of hyperplane arrangements with integral coefficients modulo positive integers

We study central hyperplane arrangements with integral coefficients modulo positive integers $q$. We prove that the cardinality of the complement of the hyperplanes is a quasi-polynomial in two ways, first via the theory of elementary divisors and then via the theory of the Ehrhart quasi-polynomials. This result is useful for determining the characteristic polynomial of the corresponding real arrangement. With the former approach, we also prove that intersection lattices modulo $q$ are periodic except for a finite number of $q$'s

Periodicity of non-central integral arrangements modulo positive integers

An integral coefficient matrix determines an integral arrangement of hyperplanes in R^m. After modulo q reduction, the same matrix determines an arrangement A_q of "hyperplanes" in Z^m. In the special case of central arrangements, Kamiya, Takemura and Terao [J. Algebraic Combin., to appear] showed that the cardinality of the complement of A_q in Z_q^m is a quasi-polynomial in q. Moreover, they proved in the central case that the intersection lattice of A_q is periodic from some q on. The present paper generalizes these results to the case of non-central arrangements. The paper also studies the arrangement B_m^{[0,a]} of Athanasiadis [J. Algebraic Combin. Vol.10 (1999), 207-225] to illustrate our results

Characterization of rankings generated by linear discriminant analysis

AbstractPairwise linear discriminant analysis can be regarded as a process to generate rankings of the populations. But in general, not all rankings are generated. We give a characterization of generated rankings. We also derive some basic properties of this model

Hierarchical orbital decompositions and extended decomposable distributions

Elliptically contoured distributions can be considered to be the distributions for which the contours of the density functions are proportional ellipsoids. Kamiya, Takemura and Kuriki (2006) generalized the elliptically contoured distributions to star-shaped distributions, for which the contours are allowed to be arbitrary proportional star-shaped sets. This was achieved by considering the so-called orbital decomposition of the sample space in the general framework of group invariance. In the present paper, we extend their results by conducting the orbital decompositions in steps and obtaining a further, hierarchical decomposition of the sample space. This allows us to construct probability models and distributions with further independence structures. The general results are applied to the star-shaped distributions with a certain symmetric structure, the distributions related to the two-sample Wishart problem and the distributions of preference rankings

Asymmetric nature of two subunits of RAD18, a RING-type ubiquitin ligase E3, in the human RAD6A–RAD18 ternary complex

RAD18, a RING-type ubiquitin ligase (E3) that plays an essential role in post-replication repair, possesses distinct domains named RING, UBZ, SAP and the RAD6-binding domain (R6BD) and forms a dimer. RAD6, an ubiquitin-conjugating enzyme (E2), stably associates with R6BD in the C-terminal portion. In this study, we established a method to distinguish between the two subunits of RAD18 by introduction of different tags, and analyzed mutant complexes. Our results, surprisingly, demonstrate that RAD6A and RAD18 form a ternary complex, RAD6A–(RAD18)2 and the presence of only one R6BD in the two RAD18 subunits is sufficient for ternary complex formation and the ligase activity. Interestingly, ligase activity of a mutant dimer lacking both R6BDs is not restored even with large amounts of RAD6A added in solution, suggesting a requirement for precise juxtaposition via interaction with R6BD. We further show that mutations in both subunits of either RING or SAP, but not UBZ, strongly reduce ligase activity, although inactivation in only one of two subunits is without effect. These results suggest an asymmetric nature of the two RAD18 subunits in the complex

A Novel ATM/TP53/p21-Mediated Checkpoint Only Activated by Chronic γ-Irradiation

Different levels or types of DNA damage activate distinct signaling pathways that elicit various cellular responses, including cell-cycle arrest, DNA repair, senescence, and apoptosis. Whereas a range of DNA-damage responses have been characterized, mechanisms underlying subsequent cell-fate decision remain elusive. Here we exposed cultured cells and mice to different doses and dose rates of γ-irradiation, which revealed cell-type-specific sensitivities to chronic, but not acute, γ-irradiation. Among tested cell types, human fibroblasts were associated with the highest levels of growth inhibition in response to chronic γ-irradiation. In this context, fibroblasts exhibited a reversible G1 cell-cycle arrest or an irreversible senescence-like growth arrest, depending on the irradiation dose rate or the rate of DNA damage. Remarkably, when the same dose of γ-irradiation was delivered chronically or acutely, chronic delivery induced considerably more cellular senescence. A similar effect was observed with primary cells isolated from irradiated mice. We demonstrate a critical role for the ataxia telangiectasia mutated (ATM)/tumor protein p53 (TP53)/p21 pathway in regulating DNA-damage-associated cell fate. Indeed, blocking the ATM/TP53/p21 pathway deregulated DNA damage responses, leading to micronucleus formation in chronically irradiated cells. Together these results provide insights into the mechanisms governing cell-fate determination in response to different rates of DNA damage