2,615 research outputs found

    Selection theorem for systems with inheritance

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    The problem of finite-dimensional asymptotics of infinite-dimensional dynamic systems is studied. A non-linear kinetic system with conservation of supports for distributions has generically finite-dimensional asymptotics. Such systems are apparent in many areas of biology, physics (the theory of parametric wave interaction), chemistry and economics. This conservation of support has a biological interpretation: inheritance. The finite-dimensional asymptotics demonstrates effects of "natural" selection. Estimations of the asymptotic dimension are presented. After some initial time, solution of a kinetic equation with conservation of support becomes a finite set of narrow peaks that become increasingly narrow over time and move increasingly slowly. It is possible that these peaks do not tend to fixed positions, and the path covered tends to infinity as t goes to infinity. The drift equations for peak motion are obtained. Various types of distribution stability are studied: internal stability (stability with respect to perturbations that do not extend the support), external stability or uninvadability (stability with respect to strongly small perturbations that extend the support), and stable realizability (stability with respect to small shifts and extensions of the density peaks). Models of self-synchronization of cell division are studied, as an example of selection in systems with additional symmetry. Appropriate construction of the notion of typicalness in infinite-dimensional space is discussed, and the notion of "completely thin" sets is introduced. Key words: Dynamics; Attractor; Evolution; Entropy; Natural selectionComment: 46 pages, the final journal versio

    The density-matrix renormalization group

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    The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly acquired the status of method of choice for numerical studies of one-dimensional quantum systems. Its applications to the calculation of static, dynamic and thermodynamic quantities in such systems are reviewed. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and non-equilibrium statistical physics, and time-dependent phenomena is discussed. This review also considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by DMRG.Comment: accepted by Rev. Mod. Phys. in July 2004; scheduled to appear in the January 2005 issu

    One- to Two-Exciton Transitions in Perylene Bisimide Dimer Revealed by Two-Dimensional Electronic Spectroscopy

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    The excited-state energy levels of molecular dimers and aggregates play a critical role in their photophysical behavior and an understanding of the photodynamics in such structures is important for developing applications such as photovoltaics and optoelectronic devices. Here, exciton transitions in two different covalently bound PBI dimers are studied by two-dimensional electronic spectroscopy (2DES), a powerful spectroscopic method, providing the most complete picture of vibronic transitions in molecular systems. The data are accurately reproduced using the equation of motion-phase matching approach. The unambiguous presence of one-exciton to two-exciton transitions are captured in our results and described in terms of a molecular exciton energy level scheme based on the Kasha model. Furthermore, the results are supported by comparative measurements with the PBI monomer and another dimer in which the interchromophore distance is increased

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    Lattice-gas cellular automata for the analysis of cancer invasion

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    Cancer cells display characteristic traits acquired in a step-wise manner during carcinogenesis. Some of these traits are autonomous growth, induction of angiogenesis, invasion and metastasis. In this thesis, the focus is on one of the latest stages of tumor progression, tumor invasion. Tumor invasion emerges from the combined effect of tumor cell-cell and cell-microenvironment interactions, which can be studied with the help of mathematical analysis. Cellular automata (CA) can be viewed as simple models of self-organizing complex systems in which collective behavior can emerge out of an ensemble of many interacting "simple" components. In particular, we focus on an important class of CA, the so-called lattice-gas cellular automata (LGCA). In contrast to traditional CA, LGCA provide a straightforward and intuitive implementation of particle transport and interactions. Additionally, the structure of LGCA facilitates the mathematical analysis of their behavior. Here, the principal tools of mathematical analysis of LGCA are the mean-field approximation and the corresponding Lattice Boltzmann equation. The main objective of this thesis is to investigate important aspects of tumor invasion, under the microscope of mathematical modeling and analysis: Impact of the tumor environment: We introduce a LGCA as a microscopic model of tumor cell migration together with a mathematical description of different tumor environments. We study the impact of the various tumor environments (such as extracellular matrix) on tumor cell migration by estimating the tumor cell dispersion speed for a given environment. Effect of tumor cell proliferation and migration: We study the effect of tumor cell proliferation and migration on the tumor’s invasive behavior by developing a simplified LGCA model of tumor growth. In particular, we derive the corresponding macroscopic dynamics and we calculate the tumor’s invasion speed in terms of tumor cell proliferation and migration rates. Moreover, we calculate the width of the invasive zone, where the majority of mitotic activity is concentrated, and it is found to be proportional to the invasion speed. Mechanisms of tumor invasion emergence: We investigate the mechanisms for the emergence of tumor invasion in the course of cancer progression. We conclude that the response of a microscopic intracellular mechanism (migration/proliferation dichotomy) to oxygen shortage, i.e. hypoxia, maybe responsible for the transition from a benign (proliferative) to a malignant (invasive) tumor. Computing in vivo tumor invasion: Finally, we propose an evolutionary algorithm that estimates the parameters of a tumor growth LGCA model based on time-series of patient medical data (in particular Magnetic Resonance and Diffusion Tensor Imaging data). These parameters may allow to reproduce clinically relevant tumor growth scenarios for a specific patient, providing a prediction of the tumor growth at a later time stage.Krebszellen zeigen charakteristische Merkmale, die sie in einem schrittweisen Vorgang während der Karzinogenese erworben haben. Einige dieser Merkmale sind autonomes Wachstum, die Induktion von Angiogenese, Invasion und Metastasis. Der Schwerpunkt dieser Arbeit liegt auf der Tumorinvasion, einer der letzten Phasen der Tumorprogression. Die Tumorinvasion ensteht aus der kombinierten Wirkung von den Wechselwirkungen Tumorzelle-Zelle und Zelle-Mikroumgebung, die mit die Hilfe von mathematischer Analyse untersucht werden können. Zelluläre Automaten (CA) können als einfache Modelle von selbst-organisierenden komplexen Systemen betrachtet werden, in denen kollektives Verhalten aus einer Kombination von vielen interagierenden "einfachen" Komponenten entstehen kann. Insbesondere konzentrieren wir uns auf eine wichtige CA-Klasse, die sogenannten Zelluläre Gitter-Gas Automaten (LGCA). Im Gegensatz zu traditionellen CA bieten LGCA eine einfache und intuitive Umsetzung der Teilchen und Wechselwirkungen. Zusätzlich erleichtert die Struktur der LGCA die mathematische Analyse ihres Verhaltens. Die wichtigsten Werkzeuge der mathematischen Analyse der LGCA sind hier die Mean-field Approximation und die entsprechende Lattice - Boltzmann - Gleichung. Das wichtigste Ziel dieser Arbeit ist es, wichtige Aspekte der Tumorinvasion unter dem Mikroskop der mathematischen Modellierung und Analyse zu erforschen: Auswirkungen der Tumorumgebung: Wir stellen einen LGCA als mikroskopisches Modell der Tumorzellen-Migration in Verbindung mit einer mathematischen Beschreibung der verschiedenen Tumorumgebungen vor. Wir untersuchen die Auswirkungen der verschiedenen Tumorumgebungen (z. B. extrazellulären Matrix) auf die Migration von Tumorzellen dürch Schätzung der Tumorzellen-Dispersionsgeschwindigkeit in einem gegebenen Umfeld. Wirkung von Tumor-Zellenproliferation und Migration: Wir untersuchen die Wirkung von Tumorzellenproliferation und Migration auf das invasive Verhalten der Tumorzellen durch die Entwicklung eines vereinfachten LGCA Tumorwachstumsmodells. Wir leiten die entsprechende makroskopische Dynamik und berechnen die Tumorinvasionsgeschwindigkeit im Hinblick auf die Tumorzellenproliferation- und Migrationswerte. Darüber hinaus berechnen wir die Breite der invasiven Zone, wo die Mehrheit der mitotischer Aktivität konzentriert ist, und es wird festgestellt, dass diese proportional zu den Invasionsgeschwindigkeit ist. Mechanismen der Tumorinvasion Entstehung: Wir untersuchen Mechanismen, die für die Entstehung von Tumorinvasion im Verlauf des Krebs zuständig sind. Wir kommen zu dem Schluss, dass die Reaktion eines mikroskopischen intrazellulären Mechanismus (Migration/Proliferation Dichotomie) zu Sauerstoffmangel, d.h. Hypoxie, möglicheweise für den Übergang von einem gutartigen (proliferative) zu einer bösartigen (invasive) Tumor verantwortlich ist. Berechnung der in-vivo Tumorinvasion: Schließlich schlagen wir einen evolutionären Algorithmus vor, der die Parameter eines LGCA Modells von Tumorwachstum auf der Grundlage von medizinischen Daten des Patienten für mehrere Zeitpunkte (insbesondere die Magnet-Resonanz-und Diffusion Tensor Imaging Daten) ermöglicht. Diese Parameter erlauben Szenarien für einen klinisch relevanten Tumorwachstum für einen bestimmten Patienten zu reproduzieren, die eine Vorhersage des Tumorwachstums zu einem späteren Zeitpunkt möglich machen
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