Some minimization problems for the free analogue of the Fisher information

We consider the free non-commutative analogue Phi^*, introduced by D. Voiculescu, of the concept of Fisher information for random variables. We determine the minimal possible value of Phi^*(a,a^*), if a is a non-commutative random variable subject to the constraint that the distribution of aa^* is prescribed. More generally, we obtain the minimal possible value of Phi^*({a_{ij},a_{ij}^*), if {a_{ij}} is a family of non-commutative random variables such that the distribution of AA^* is prescribed, where A is the matrix (a_{ij}). The d*d-generalization is obtained from the case d=1 via a result of independent interest, concerning the minimal value of Phi^*({a_{ij},a_{ij}^*), when the matrix A=(a_{ij}) and its adjoint have a given joint distribution. We then show how the minimization results obtained for Phi^* lead to maximization results concerning the free entropy chi^*, also defined by Voiculescu.Comment: 31 pages, Late

Random matrix theory for CPA: Generalization of Wegner's nn--orbital model

We introduce a generalization of Wegner's $n$-orbital model for the description of randomly disordered systems by replacing his ensemble of Gaussian random matrices by an ensemble of randomly rotated matrices. We calculate the one- and two-particle Green's functions and the conductivity exactly in the limit $n\to\infty$. Our solution solves the CPA-equation of the $(n=1)$-Anderson model for arbitrarily distributed disorder. We show how the Lloyd model is included in our model.Comment: 3 pages, Rev-Te

Seven-fluorochrome mouse M-FISH for high-resolution analysis of interchromosomal rearrangements

The mouse has evolved to be the primary mammalian genetic model organism. Important applications include the modeling of human cancer and cloning experiments. In both settings, a detailed analysis of the mouse genome is essential. Multicolor karyotyping technologies have emerged to be invaluable tools for the identification of mouse chromosomes and for the deciphering of complex rearrangements. With the increasing use of these multicolor technologies resolution limits are critical. However, the traditionally used probe sets, which employ 5 different fluorochromes, have significant limitations. Here, we introduce an improved labeling strategy. Using 7 fluorochromes we increased the sensitivity for the detection of small interchromosomal rearrangements (700 kb or less) to virtually 100%. Our approach should be important to unravel small interchromosomal rearrangements in mouse models for DNA repair defects and chromosomal instability. Copyright (C) 2003 S. Karger AG, Basel

Detection of amplified DNA sequences by reverse chromosome painting using genomic tumor DNA as probe

A modification of reverse chromosome painting was carried out using genomic DNA from tumor cells as a complex probe for chromosomal in situ suppression hybridization to normal metaphase chromsome spreads. Amplified DNA sequences contained in such probes showed specific signals, revealing the normal chromosome positions from which these sequences were derived. As a model system, genomic DNAs were analyzed from three tumor cell lines with amplification units including the proto-oncogene c-myc. The smallest amplification unit was about 90 kb and was present in 16–24 copies; the largest unit was bigger than 600 kb and was present in 16–32 copies. Specific signals that co-localized with a differently labeled c-myc probe on chromosome band 8q24 were obtained with genomic DNA from each cell line. In further experiments, genomic DNA derived from primary tumor material was used in the case of a male patient with glioblastoma multiforme (GBM). Southern blot analysis using an epidermal growth factor receptor gene (EGFR) probe that maps to 7p13 indicated the amplification of sequences from this gene. Using reverse chromosome painting, signals were found both on band 7p13 and bands 12q13–q15. Notably, the signal on 12q13–q15 was consistently stronger. The weaker 7p13 signal showed co-localization with the major signal of the differently labeled EGFR probe. A minor signal of this probe was seen on 12q13, suggesting cross-hybridization to ERB3 sequences homologous to EGFR. The results indicate co-amplification of sequences from bands 12q13–q15, in addition to sequences from band 7p13. Several oncogenes map to 12q13–q15 providing candidate genes for a tumor-associated proto-oncogene amplification. Although the nature of the amplified sequences needs to be clarified, this experiment demonstrates the potential of reverse chromosome painting with genomic tumor DNA for rapidly mapping the normal chromosomal localization of the DNA from which the amplified sequences were derived. In addition, a weaker staining of chromosomes 10 and X was consistently observed indicating that these chromosomes were present in only one copy in the GBM genome. This rapid approach can be used to analyze cases where no metaphase spreads from the tumor material are available. It does not require any preknowledge of amplified sequences and can be applied to screen large numbers of tumors

Rigorous mean field model for CPA: Anderson model with free random variables

A model of a randomly disordered system with site-diagonal random energy fluctuations is introduced. It is an extension of Wegner's $n$-orbital model to arbitrary eigenvalue distribution in the electronic level space. The new feature is that the random energy values are not assumed to be independent at different sites but free. Freeness of random variables is an analogue of the concept of independence for non-commuting random operators. A possible realization is the ensemble of at different lattice-sites randomly rotated matrices. The one- and two-particle Green functions of the proposed hamiltonian are calculated exactly. The eigenstates are extended and the conductivity is nonvanishing everywhere inside the band. The long-range behaviour and the zero-frequency limit of the two-particle Green function are universal with respect to the eigenvalue distribution in the electronic level space. The solutions solve the CPA-equation for the one- and two-particle Green function of the corresponding Anderson model. Thus our (multi-site) model is a rigorous mean field model for the (single-site) CPA. We show how the Llyod model is included in our model and treat various kinds of noises.Comment: 24 pages, 2 diagrams, Rev-Tex. Diagrams are available from the authors upon reques

The Free Quon Gas Suffers Gibbs' Paradox

We consider the Statistical Mechanics of systems of particles satisfying the $q$-commutation relations recently proposed by Greenberg and others. We show that although the commutation relations approach Bose (resp.\ Fermi) relations for $q\to1$ (resp.\ $q\to-1$), the partition functions of free gases are independent of $q$ in the range $-1<q<1$. The partition functions exhibit Gibbs' Paradox in the same way as a classical gas without a correction factor $1/N!$ for the statistical weight of the $N$-particle phase space, i.e.\ the Statistical Mechanics does not describe a material for which entropy, free energy, and particle number are extensive thermodynamical quantities.Comment: number-of-pages, LaTeX with REVTE