Lie-series for orbital elements -- I. The planar case

Lie-integration is one of the most efficient algorithms for numerical integration of ordinary differential equations if high precision is needed for longer terms. The method is based on the computation of the Taylor-coefficients of the solution as a set of recurrence relations. In this paper we present these recurrence formulae for orbital elements and other integrals of motion for the planar $N$-body problem. We show that if the reference frame is fixed to one of the bodies -- for instance to the Sun in the case of the Solar System --, the higher order coefficients for all orbital elements and integrals of motion depend only on the mutual terms corresponding to the orbiting bodies.Comment: Accepted for publication in CeMDA, 10 page

Kulcsfontosságú gének genomikai előrejelzése: In Silico megközelítés = Genomic prediction of essential genes: in silico approach

Kulcsfontosságú gének bioinformatikai elemzése: Csoportunk számos számos olyan sajátságot ismertek fel, melyek segítségével jellemezni lehet az esszenciális vagy a géndózis változására érzékeny géneket. Ezek közül a génduplikációt, az alternatív anyagcsereútvonalak jelenlétét, a génkifejeződés mértékét és a gén genomon belüli pozícióját érdemes megemlíteni. Rendszerbiológiai modellek alapján kulcsfontosságú metabolikus gének előrejelzése: Előzetesen leírt módszerekre alapozva, részletes vizsgálatnak vetettük alá a sörélesztő rekonstruált metabolikus hálózatát, majd megvizsgáltuk, hogyan viselkedik a rendszer ha egy-egy enzim működésképtelen. Módszerünk sikeresen jelzi előre az esszenciális gének 85%-át. Ez a siker lehetővé tette, hogy a biológia olyan kulcskérdéseire keressünk választ, mint a mutációkkal szembeni robusztusság háttere, a biológiai hálózatok evolúciós változása vagy a minimál genomok természete. Genetikai interakciók rendszerbiológiai és kísérleti vizsgálata: Anyagcserehálózat rendszerbiológiai modellünk komoly lehetőséget biztosít a genetikai interakciók mélyebb megértéséhez. A modell sikeresen képes előrejelezni speciális genetikai interakciók jelenlétét. Számos érvünk szól amelett, hogy a mutációkkal szembeni robusztusság a különböző környezeti feltételekhez való alkalmazkodás mellékterméke. | Bioinformatics analyses of essential genes: We identified several cellular and genomic features that enable reliable characterization of essential and dosage sensitive genes: Gene duplication, alternative metabolic pathways, gene expression level and genomic position all have some effect on gene dispensability. In silico prediction of essential metabolic genes using systems biological models: We have employed and further developed a previously elaborated metabolic network model of yeast. Our method predicts gene essentiality with about 85% accuracy. These methods have enabled us to study several key issues in evolutionary biology, such as the nature of mutational robustness and minimal genomes or the driving forces in the evolution of metabolic networks. Computational and experimental analyses of genetic interactions: The computational model described above paves the way for gaining novel insights into the nature of genetic interactions. The current model is able to predict the presence of genetic interactions in the metabolic networks of yeast with nearly 50% accuracy, while only approximately 0.5% would be expected by chance. Along with other arguments, our findings suggest that apparent robustness against harmful mutations is not a directly selected trait, but it's rather a by-product of organismal adaptation to varying environments

Hysteretic optimization for the Sherrington-Kirkpatrick spin glass

Hysteretic optimization is a heuristic optimization method based on the observation that magnetic samples are driven into a low energy state when demagnetized by an oscillating magnetic field of decreasing amplitude. We show that hysteretic optimization is very good for finding ground states of Sherrington-Kirkpatrick spin glass systems. With this method it is possible to get good statistics for ground state energies for large samples of systems consisting of up to about 2000 spins. The way we estimate error rates may be useful for some other optimization methods as well. Our results show that both the average and the width of the ground state energy distribution converges faster with increasing size than expected from earlier studies.Comment: Physica A, accepte

Lie-series for orbital elements -- II. The spatial case

If one has to attain high accuracy over long timescales during the numerical computation of the N-body problem, the method called Lie-integration is one of the most effective algorithms. In this paper we present a set of recurrence relations with which the coefficients needed by the Lie-integration of the orbital elements related to the spatial N-body problem can be derived up to arbitrary order. Similarly to the planar case, these formulae yields identically zero series in the case of no perturbations. In addition, the derivation of the formulae has two stages, analogously to the planar problem. Namely, the formulae are obtained to the first order, and then, higher order relations are expanded by involving directly the multilinear and fractional properties of the Lie-operator.Comment: Accepted for publication in CeMDA, in press, 12 page

A contribution to the Aleksandrov conservative distance problem in two dimensions

Let $E$ be a two-dimensional real normed space. In this paper we show that if the unit circle of $E$ does not contain any line segment such that the distance between its endpoints is greater than 1, then every transformation $\phi\colon E\to E$ which preserves the unit distance is automatically an affine isometry. In particular, this condition is satisfied when the norm is strictly convex.Comment: 8 pages, 3 figure

Random walks in compact groups

Let X_1,X_2,... be independent identically distributed random elements of a compact group G. We discuss the speed of convergence of the law of the product X_l*...*X_1 to the Haar measure. We give poly-log estimates for certain finite groups and for compact semi-simple Lie groups. We improve earlier results of Solovay, Kitaev, Gamburd, Shahshahani and Dinai.Comment: 35 pages, no figures, revision based on referee's report, results and proofs unchange