On the forced matching numbers of bipartite graphs

Let G be a graph that admits a perfect matching. A forcing set for a perfect matching M of G is a subset S of M, such that S is contained in no other perfect matching of G. This notion has arisen in the study of finding resonance structures of a given molecule in chemistry. Similar concepts have been studied for block designs and graph colorings under the name defining set, and for Latin squares under the name critical set. There is some study of forcing sets of hexagonal systems in the context of chemistry, but only a few other classes of graphs have been considered. For the hypercubes Q(n), it turns out to be a very interesting notion which includes many challenging problems. In this paper we study the computational complexity of finding the forcing number of graphs, and we give some results on the possible values of forcing number for different matchings of the hypercube Q(n). Also we show an application to critical sets in back circulant Latin rectangles. (C) 2003 Elsevier B.V. All rights reserved.

Complex numbers and symmetries in quantum mechanics, and a nonlinear superposition principle for Wigner functions

Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert space, whereas in phase space they are described by real, true representations. Equivalence of the formulations requires that the former representations can be obtained from the latter and vice versa. Examples are given. Equivalence of the two formulations also requires that complex superpositions of state vectors can be described in the phase space formulation, and it is shown that this leads to a nonlinear superposition principle for orthogonal, pure-state Wigner functions. It is concluded that the use of complex numbers in quantum mechanics can be regarded as a computational device to simplify calculations, as in all other applications of mathematics to physical phenomena.

Experimental validation of the regulated expression of large numbers of non-coding RNAs from the mouse genome

Recent large-scale analyses of mainly full-length cDNA libraries generated from a variety of mouse tissues indicated that almost half of all representative cloned sequences did flat contain ail apparent protein-coding sequence, and were putatively derived from non-protein-coding RNA (ncRNA) genes. However, many of these clones were singletons and the majority were unspliced, raising the possibility that they may be derived from genomic DNA or unprocessed pre-rnRNA contamination during library construction, or alternatively represent nonspecific transcriptional noise. Here we Show, using reverse transcriptase-dependent PCR, microarray, and Northern blot analyses, that many of these clones were derived from genuine transcripts Of unknown function whose expression appears to be regulated. The ncRNA transcripts have larger exons and fewer introns than protein-coding transcripts. Analysis of the genomic landscape around these sequences indicates that some cDNA clones were produced not from terminal poly(A) tracts but internal priming sites within longer transcripts, only a minority of which is encompassed by known genes. A significant proportion of these transcripts exhibit tissue-specific expression patterns, as well as dynamic changes in their expression in macrophages following lipopolysaccharide Stimulation. Taken together, the data provide strong support for the conclusion that ncRNAs are an important, regulated component of the mammalian transcriptome.