Pseudodeterminants and perfect square spanning tree counts

The pseudodeterminant $\textrm{pdet}(M)$ of a square matrix is the last nonzero coefficient in its characteristic polynomial; for a nonsingular matrix, this is just the determinant. If $\partial$ is a symmetric or skew-symmetric matrix then $\textrm{pdet}(\partial\partial^t)=\textrm{pdet}(\partial)^2$. Whenever $\partial$ is the $k^{th}$ boundary map of a self-dual CW-complex $X$, this linear-algebraic identity implies that the torsion-weighted generating function for cellular $k$-trees in $X$ is a perfect square. In the case that $X$ is an \emph{antipodally} self-dual CW-sphere of odd dimension, the pseudodeterminant of its $k$th cellular boundary map can be interpreted directly as a torsion-weighted generating function both for $k$-trees and for $(k-1)$-trees, complementing the analogous result for even-dimensional spheres given by the second author. The argument relies on the topological fact that any self-dual even-dimensional CW-ball can be oriented so that its middle boundary map is skew-symmetric.Comment: Final version; minor revisions. To appear in Journal of Combinatoric

Selected Challenges From Spatial Statistics For Spatial Econometricians

Griffith and Paelinck (2011) present selected non-standard spatial statistics and spatial econometrics topics that address issues associated with spatial econometric methodology. This paper addresses the following challenges posed by spatial autocorrelation alluded to and/or derived from the spatial statistics topics of this book: the Gaussian random variable Jacobian term for massive datasets; topological features of georeferenced data; eigenvector spatial filtering-based georeferenced data generating mechanisms; and, interpreting random effects.Artykuł prezentuje wybrane, niestandardowe statystyki przestrzenne oraz zagadnienia ekonometrii przestrzennej. Rozważania teoretyczne koncentrują się na wyzwaniach wynikających z autokorelacji przestrzennej, nawiązując do pojęć Gaussowskiej zmiennej losowej, topologicznych cech danych georeferencyjnych, wektorów własnych, filtrów przestrzennych, georeferencyjnych mechanizmów generowania danych oraz interpretacji efektów losowych

Robust Principal Component Analysis for Compositional Tables

A data table which is arranged according to two factors can often be considered as a compositional table. An example is the number of unemployed people, split according to gender and age classes. Analyzed as compositions, the relevant information would consist of ratios between different cells of such a table. This is particularly useful when analyzing several compositional tables jointly, where the absolute numbers are in very different ranges, e.g. if unemployment data are considered from different countries. Within the framework of the logratio methodology, compositional tables can be decomposed into independent and interactive parts, and orthonormal coordinates can be assigned to these parts. However, these coordinates usually require some prior knowledge about the data, and they are not easy to handle for exploring the relationships between the given factors. Here we propose a special choice of coordinates with a direct relation to centered logratio (clr) coefficients, which are particularly useful for an interpretation in terms of the original cells of the tables. With these coordinates, robust principal component analysis (PCA) is performed for dimension reduction, allowing to investigate the relationships between the factors. The link between orthonormal coordinates and clr coefficients enables to apply robust PCA, which would otherwise suffer from the singularity of clr coefficients.Comment: 20 pages, 2 figure

Localized structures in Kagome lattices

We investigate the existence and stability of gap vortices and multi-pole gap solitons in a Kagome lattice with a defocusing nonlinearity both in a discrete case and in a continuum one with periodic external modulation. In particular, predictions are made based on expansion around a simple and analytically tractable anti-continuum (zero coupling) limit. These predictions are then confirmed for a continuum model of an optically-induced Kagome lattice in a photorefractive crystal obtained by a continuous transformation of a honeycomb lattice

Spatial and temporal cellular responses to single-strand breaks in human cells

DNA single-strand breaks (SSB) are one of the most frequent DNA lesions produced by reactive oxygen species and during DNA metabolism, but the analysis of cellular responses to SSB remains difficult due to the lack of an experimental method to produce SSB alone in cells. By using human cells expressing a foreign UV damage endonuclease (UVDE) and irradiating the cells with UV through tiny pores in membrane filters, we created SSB in restricted areas in the nucleus by the immediate action of UVDE on UV-induced DNA lesions. Cellular responses to the SSB were characterized by using antibodies and fluorescence microscopy. Upon UV irradiation, poly(ADP-ribose) synthesis occurred immediately in the irradiated area. Simultaneously, but dependent on poly(ADP-ribosyl)ation, XRCC1 was translocated from throughout the nucleus, including nucleoli, to the SSB. The BRCT1 domain of XRCC1 protein was indispensable for its poly(ADP-ribose)-dependent recruitment to the SSB. Proliferating cell nuclear antigen and the p150 subunit of chromatin assembly factor 1 also accumulated at the SSB in a detergent-resistant form, which was significantly reduced by inhibition of poly(ADP-ribose) synthesis. Our results show the importance of poly(ADP-ribosyl)ation in sequential cellular responses to SSB