Trotter-Kato product formulae in Dixmier ideal

It is shown that for a certain class of the Kato functions the Trotter-Kato product formulae converge in Dixmier ideal C 1,$\infty$ in topology, which is defined by the $\times$ 1,$\infty$-norm. Moreover, the rate of convergence in this topology inherits the error-bound estimate for the corresponding operator-norm convergence. 1 since [24], [14]. Note that a subtle point of this program is the question about the rate of convergence in the corresponding topology. Since the limit of the Trotter-Kato product formula is a strongly continuous semigroup, for the von Neumann-Schatten ideals this topology is the trace-norm $\times$ 1 on the trace-class ideal C 1 (H). In this case the limit is a Gibbs semigroup [25]. For self-adjoint Gibbs semigroups the rate of convergence was estimated for the first time in [7] and [9]. The authors considered the case of the Gibbs-Schr{\"o}dinger semigroups. They scrutinised in these papers a dependence of the rate of convergence for the (exponential) Trotter formula on the smoothness of the potential in the Schr{\"o}dinger generator. The first abstract result in this direction was due to [19]. In this paper a general scheme of lifting the operator-norm rate convergence for the Trotter-Kato product formulae was proposed and advocated for estimation the rate of the trace-nor

Scattering Theory for Open Quantum Systems

Quantum systems which interact with their environment are often modeled by maximal dissipative operators or so-called Pseudo-Hamiltonians. In this paper the scattering theory for such open systems is considered. First it is assumed that a single maximal dissipative operator $A_D$ in a Hilbert space \sH is used to describe an open quantum system. In this case the minimal self-adjoint dilation $\widetilde K$ of $A_D$ can be regarded as the Hamiltonian of a closed system which contains the open system \{A_D,\sH\}, but since $\widetilde K$ is necessarily not semibounded from below, this model is difficult to interpret from a physical point of view. In the second part of the paper an open quantum system is modeled with a family $\{A(\mu)\}$ of maximal dissipative operators depending on energy $\mu$, and it is shown that the open system can be embedded into a closed system where the Hamiltonian is semibounded. Surprisingly it turns out that the corresponding scattering matrix can be completely recovered from scattering matrices of single Pseudo-Hamiltonians as in the first part of the paper. The general results are applied to a class of Sturm-Liouville operators arising in dissipative and quantum transmitting Schr\"{o}dinger-Poisson systems

Mean field approaches to the totally asymmetric exclusion process with quenched disorder and large particles

The process of protein synthesis in biological systems resembles a one dimensional driven lattice gas in which the particles (ribosomes) have spatial extent, covering more than one lattice site. Realistic, nonuniform gene sequences lead to quenched disorder in the particle hopping rates. We study the totally asymmetric exclusion process with large particles and quenched disorder via several mean field approaches and compare the mean field results with Monte Carlo simulations. Mean field equations obtained from the literature are found to be reasonably effective in describing this system. A numerical technique is developed for computing the particle current rapidly. The mean field approach is extended to include two-point correlations between adjacent sites. The two-point results are found to match Monte Carlo simulations more closely

The molecular basis of human retinal and vitreoretinal diseases

During the last two to three decades, a large body of work has revealed the molecular basis of many human disorders, including retinal and vitreoretinal degenerations and dysfunctions. Although belonging to the group of orphan diseases, they affect probably more than two million people worldwide. Most excitingly, treatment of a particular form of congenital retinal degeneration is now possible. A major advantage for treatment is the unique structure and accessibility of the eye and its different components, including the vitreous and retina. Knowledge of the many different eye diseases affecting retinal structure and function (night and color blindness, retinitis pigmentosa, cone and cone rod dystrophies, photoreceptor dysfunctions, as well as vitreoretinal traits) is critical for future therapeutic development. We have attempted to present a comprehensive picture of these disorders, including clinical, genetic and molecular information. The structural organization of the review leads the reader through non-syndromic and syndromic forms of (i) rod dominated diseases, (ii) cone dominated diseases, (iii) generalized retinal degenerations and (iv) vitreoretinal disorders, caused by mutations in more than 165 genes. Clinical variability and genetic heterogeneity have an important impact on genetic testing and counselling of affected families. As phenotypes do not always correlate with the respective genotypes, it is of utmost importance that clinicians, geneticists, counsellors, diagnostic laboratories and basic researchers understand the relationships between phenotypic manifestations and specific genes, as well as mutations and pathophysiologic mechanisms. We discuss future perspectives

Virale und nichtvirale Gentherapieansätze zur Behandlung von Netzhauterkrankungen

Zusammenfassung: Für die Behandlung von Netzhauterkrankungen eröffnet der Einsatz der Gentherapie neue Perspektiven. Die Verwendung von verschiedenartigen Oligonukleotiden oder viralen Expressionsvektoren erlaubt die Entwicklung von neuen Heilungsstrategien für Neovaskularisierungskrankheiten und retinale Degeneration. Therapeutische Oligonukleotide ("Antisense"-Oligonukleotide, Aptamere und siRNA) können den gezielten Abbau von Transkripten und damit die Konzentrationsabnahme eines an der Pathogenese beteiligten Proteins induzieren. Dagegen wird mit viralen Vektoren (rAAV und Lentivirus) häufig die Funktion eines defekten Gens durch die eines gesunden ersetzt und so die Ursache der Krankheit bekämpft. Die an Tiermodellen erfolgreich angewandten Gentherapien führten bereits zur Entwicklung von Medikamenten, und weitere werden zurzeit klinisch erprob

Circuit architecture explains functional similarity of bacterial heat shock responses

Heat shock response is a stress response to temperature changes and a consecutive increase in amounts of unfolded proteins. To restore homeostasis, cells upregulate chaperones facilitating protein folding by means of transcription factors (TF). We here investigate two heat shock systems: one characteristic to gram negative bacteria, mediated by transcriptional activator sigma32 in E. coli, and another characteristic to gram positive bacteria, mediated by transcriptional repressor HrcA in L. lactis. We construct simple mathematical model of the two systems focusing on the negative feedbacks, where free chaperons suppress sigma32 activation in the former, while they activate HrcA repression in the latter. We demonstrate that both systems, in spite of the difference at the TF regulation level, are capable of showing very similar heat shock dynamics. We find that differences in regulation impose distinct constrains on chaperone-TF binding affinities: the binding constant of free sigma32 to chaperon DnaK, known to be in 100 nM range, set the lower limit of amount of free chaperon that the system can sense the change at the heat shock, while the binding affinity of HrcA to chaperon GroE set the upper limit and have to be rather large extending into the micromolar range.Comment: 17 pages, 5 figure

Sufficient conditions for the anti-Zeno effect

The ideal anti-Zeno effect means that a perpetual observation leads to an immediate disappearance of the unstable system. We present a straightforward way to derive sufficient conditions under which such a situation occurs expressed in terms of the decaying states and spectral properties of the Hamiltonian. They show, in particular, that the gap between Zeno and anti-Zeno effects is in fact very narrow.Comment: LatEx2e, 9 pages; a revised text, to appear in J. Phys. A: Math. Ge

A Markov Chain based method for generating long-range dependence

This paper describes a model for generating time series which exhibit the statistical phenomenon known as long-range dependence (LRD). A Markov Modulated Process based upon an infinite Markov chain is described. The work described is motivated by applications in telecommunications where LRD is a known property of time-series measured on the internet. The process can generate a time series exhibiting LRD with known parameters and is particularly suitable for modelling internet traffic since the time series is in terms of ones and zeros which can be interpreted as data packets and inter-packet gaps. The method is extremely simple computationally and analytically and could prove more tractable than other methods described in the literatureComment: 8 pages, 2 figure

Instabilities in complex mixtures with a large number of components

Inside living cells are complex mixtures of thousands of components. It is hopeless to try to characterise all the individual interactions in these mixtures. Thus, we develop a statistical approach to approximating them, and examine the conditions under which the mixtures phase separate. The approach approximates the matrix of second virial coefficients of the mixture by a random matrix, and determines the stability of the mixture from the spectrum of such random matrices.Comment: 4 pages, uses RevTeX 4.