Forcing highly connected subgraphs

A theorem of Mader states that highly connected subgraphs can be forced in finite graphs by assuming a high minimum degree. We extend this result to infinite graphs. Here, it is necessary to require not only high degree for the vertices but also high vertex‐degree (or multiplicity) for the ends of the graph, that is, a large number of disjoint rays in each end. We give a lower bound on the degree of vertices and the vertex‐degree of the ends which is quadratic in k, the connectedness of the desired subgraph. In fact, this is not far from best possible: we exhibit a family of graphs with a degree of order 2^k at the vertices and a vertex‐degree of order k log k at the ends which have no k‐connected subgraphs. Furthermore, if in addition to the high degrees at the vertices, we only require high edge‐degree for the ends (which is defined as the maximum number of edge‐disjoint rays in an end), Mader's theorem does not extend to infinite graphs, not even to locally finite ones. We give a counterexample in this respect. But, assuming a lower bound of at least 2k for the edge‐degree at the ends and the degree at the vertices does suffice to ensure the existence (k + 1)‐edge‐connected subgraphs in arbitrary graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 331–349, 2007     

Maturation of the lantibiotic subtilin: matrix-assisted laser desorption/ionization time-of-flight mass spectrometry to monitor precursors and their proteolytic processing in crude bacterial cultures.

Bacillus subtilis synthesizes the lanthionine containing 32-amino-acid peptide antibiotic (lanti-biotic) subtilin from a ribosomally generated 56-amino-acid precursor pre-propeptide by extensive posttranslational modifications. Matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOFMS) was used to monitor the production of matured subtilin within crude samples taken from B. subtilis culture media without prior fractionation. The processing reaction of subtilin was blocked with the serine protease inhibitor phenylmethylsulfonyl fluoride and different subtilin precursor peptides in the molecular mass range up to 6220 were observed. Two of these species were isolated by reversed-phase high-performance liquid chromatography (HPLC) and structurally analyzed by post-source decay MALDI-TOFMS. We provide evidence that the precursor species comprise the posttranslational modified C-terminal part of subtilin to which leader peptide moieties with different chain lengths are attached. These antimicrobial-inactive species could be processed to antibiotic-active subtilin by incubation with culture media of different subtilin-nonproducing B. subtilis strains as indicated by a combination of antimicrobial growth assays and MALDI-TOFMS analyses. These achievements are strong evidence for the sensitivity of MALDI-TOFMS methodology that allows straightforward investigations of analytes even in complex mixtures without time-consuming sample preparations.     

The harmonic consistency index for the analytic hierarchy process

A new consistency measure, the harmonic consistency index, is obtained for any positive reciprocal matrix in the analytic hierarchy process. We show how this index varies with changes in any matrix element. A tight upper bound is provided for this new consistency measure when the entries of matrix are at most 9, as is often recommended. Using simulation, the harmonic consistency index is shown to give numerical values similar to the standard consistency index but it is easier to compute and interpret. In addition, new properties of the column sums of reciprocal matrices are obtained.     

Elliott-Morse measures and Kakutani's dichotomy theorem

Dedicated to the memory of John Oxtoby     

Bayesian Object Identification: Variants

We present a Bayesian theory of object identification. Here, identifying an object means selecting a particular observation from a group of observations (variants), this observation (the regular variant) being characterized by a distributional model. In this sense, object identification means assigning a given model to one of several observations. Often, it is the statistical model of the regular variant, only, that is known. We study an estimator which relies essentially on this model and not on the characteristics of the “irregular” variants. In particular, we investigate under what conditions this variant selector is optimal. It turns out that there is a close relationship with exchangeability and Markovian reversibility. We finally apply our theory to the case of irregular variants generated from the regular variant by a Gaussian linear model.