Tau functions as Widom constants

We define a tau function for a generic Riemann-Hilbert problem posed on a union of non-intersecting smooth closed curves with jump matrices analytic in their neighborhood. The tau function depends on parameters of the jumps and is expressed as the Fredholm determinant of an integral operator with block integrable kernel constructed in terms of elementary parametrices. Its logarithmic derivatives with respect to parameters are given by contour integrals involving these parametrices and the solution of the Riemann-Hilbert problem. In the case of one circle, the tau function coincides with Widom's determinant arising in the asymptotics of block Toeplitz matrices. Our construction gives the Jimbo-Miwa-Ueno tau function for Riemann-Hilbert problems of isomonodromic origin (Painlev\'e VI, V, III, Garnier system, etc) and the Sato-Segal-Wilson tau function for integrable hierarchies such as Gelfand-Dickey and Drinfeld-Sokolov.Comment: 26 pages, 6 figure

Riordan graphs I : structural properties

In this paper, we use the theory of Riordan matrices to introduce the notion of a Riordan graph. The Riordan graphs are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other fami- lies of graphs. The Riordan graphs are proved to have a number of interesting (fractal) properties, which can be useful in creating computer networks with certain desirable features, or in obtaining useful information when designing algorithms to compute values of graph invariants. The main focus in this paper is the study of structural properties of families of Riordan graphs obtained from infinite Riordan graphs, which includes a fundamental decomposition theorem and certain conditions on Riordan graphs to have an Eulerian trail/cycle or a Hamiltonian cycle. We will study spectral properties of the Riordan graphs in a follow up paper

Riordan graphs II : spectral properties

The authors of this paper have used the theory of Riordan matrices to introduce the notion of a Riordan graph in [3]. Riordan graphs are proved to have a number of interesting (fractal) properties, and they are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other families of graphs. The main focus in [3] is the study of structural properties of families of Riordan graphs obtained from certain infinite Riordan graphs. In this paper, we use a number of results in [3] to study spectral properties of Riordan graphs. Our studies include, but are not limited to the spectral graph invariants for Riordan graphs such as the adjacency eigenvalues, (signless) Laplacian eigenvalues, nullity, positive and negative inertia indices, and rank. We also study determinants of Riordan graphs, in particular, giving results about determinants of Catalan graphs

Transcriptome and proteome analyses of adaptive responses to methyl methanesulfonate in Escherichia coli K-12 and ada mutant strains

<p>Abstract</p> <p>Background</p> <p>The Ada-dependent adaptive response system in <it>Escherichia coli </it>is important for increasing resistance to alkylation damage. However, the global transcriptional and translational changes during this response have not been reported. Here we present time-dependent global gene and protein expression profiles following treatment with methyl methanesulfonate (MMS) in <it>E. coli </it>W3110 and its <it>ada </it>mutant strains.</p> <p>Results</p> <p>Transcriptome profiling showed that 1138 and 2177 genes were differentially expressed in response to MMS treatment in the wild-type and mutant strains, respectively. A total of 81 protein spots representing 76 nonredundant proteins differentially expressed were identified using 2-DE and LC-MS/MS. In the wild-type strain, many genes were differentially expressed upon long-exposure to MMS, due to both adaptive responses and stationary phase responses. In the <it>ada </it>mutant strain, the genes involved in DNA replication, recombination, modification and repair were up-regulated 0.5 h after MMS treatment, indicating its connection to the SOS and other DNA repair systems. Interestingly, expression of the genes involved in flagellar biosynthesis, chemotaxis, and two-component regulatory systems related to drug or antibiotic resistance, was found to be controlled by Ada.</p> <p>Conclusion</p> <p>These results show in detail the regulatory components and pathways controlling adaptive response and how the related genes including the Ada regulon are expressed with this response.</p