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.

Tacit collusion, firm asymmetries and numbers:evidence from EC merger cases

This paper estimates the implicit model, especially the roles of size asymmetries and firm numbers, used by the European Commission to identify mergers with coordinated effects. This subset of cases offers an opportunity to shed empirical light on the conditions where a Competition Authority believes tacit collusion is most likely to arise. We find that, for the Commission, tacit collusion is a rare phenomenon, largely confined to markets of two, more or less symmetric, players. This is consistent with recent experimental literature, but contrasts with the facts on ‘hard-core’ collusion in which firm numbers and asymmetries are often much larger.

Tacit vs. overt collusion firm asymmetries and numbers:what's the evidence?

It is conventional wisdom that collusion is more likely the fewer firms there are in a market and the more symmetric they are. This is often theoretically justified in terms of a repeated non-cooperative game. Although that model fits more easily with tacit than overt collusion, the impression sometimes given is that ‘one model fits all’. Moreover, the empirical literature offers few stylized facts on the most simple of questions—how few are few and how symmetric is symmetric? This paper attempts to fill this gap while also exploring the interface of tacit and overt collusion, albeit in an indirect way. First, it identifies the empirical model of tacit collusion that the European Commission appears to have employed in coordinated effects merger cases—apparently only fairly symmetric duopolies fit the bill. Second, it shows that, intriguingly, the same story emerges from the quite different experimental literature on tacit collusion. This offers a stark contrast with the findings for a sample of prosecuted cartels; on average, these involve six members (often more) and size asymmetries among members are often considerable. The indirect nature of this ‘evidence’ cautions against definitive conclusions; nevertheless, the contrast offers little comfort for those who believe that the same model does, more or less, fit all.

Tacit collusion, firm asymmetries and numbers:evidence from EC merger cases

The purpose of this paper is to identify empirically the implicit structural model, especially the roles of size asymmetries and concentration, used by the European Commission to identify mergers with coordinated effects (i.e. collective dominance). Apart from its obvious policy-relevance, the paper is designed to shed empirical light on the conditions under which tacit collusion is most likely. We construct a database relating to 62 candidate mergers and find that, in the eyes of the Commission, tacit collusion in this context virtually never involves more than two firms and requires close symmetry in the market shares of the two firms.

Random walks and random numbers from supercontinuum generation

We report a numerical study showing how the random intensity and phase fluctuations across the bandwidth of a broadband optical supercontinuum can be interpreted in terms of the random processes of random walks and Lévy flights. We also describe how the intensity fluctuations can be applied to physical random number generation. We conclude that the optical supercontinuum provides a highly versatile means of studying and generating a wide class of random processes at optical wavelengths. © 2012 Optical Society of America.

Transition of planar Couette flow at infinite Reynolds numbers

An outline of the state space of planar Couette flow at high Reynolds numbers (Re<105) is investigated via a variety of efficient numerical techniques. It is verified from nonlinear analysis that the lower branch of the hairpin vortex state (HVS) asymptotically approaches the primary (laminar) state with increasing Re. It is also predicted that the lower branch of the HVS at high Re belongs to the stability boundary that initiates a transition to turbulence, and that one of the unstable manifolds of the lower branch of HVS lies on the boundary. These facts suggest HVS may provide a criterion to estimate a minimum perturbation arising transition to turbulent states at the infinite Re limit. © 2013 American Physical Society.

Cyberstalking:a matter for community safety but the numbers do not add up

The advent of the internet has led to the new phenomenon of ‘cyberstalking’. This paper examines the extent of this problem. It is argued that all of the estimates commonly cited are flawed because they rely on inaccurate or outdated information or are based on a number of unproven assumptions. Although some estimates suggest that there may be as many as 10 million victims in the United States and Canada alone, available evidence supports a far smaller estimate. It is concluded that additional research is needed to arrive at more accurate and reliable estimates