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Mathematical Institute,
"... There may be late changes and amendments to this Lecture List. For an uptodate version, please check the Mathematical Institute Website: ..."
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There may be late changes and amendments to this Lecture List. For an uptodate version, please check the Mathematical Institute Website:
Mathematical Institute,
"... In this talk, we study the structure of the approximate spectra of analytic elementary operators and characterize the solvability of several types of operator equation. Moreover, we prove the spectral mapping theorems for the approximate spectra of bounded linear operators on Banach spaces. It has b ..."
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In this talk, we study the structure of the approximate spectra of analytic elementary operators and characterize the solvability of several types of operator equation. Moreover, we prove the spectral mapping theorems for the approximate spectra of bounded linear operators on Banach spaces. It has been aproblem of essential importance to study the spectral properties of elementary operators. Let $x $ be acomplex Banach space and $\mathcal{L}(X) $ the Banach algebra of all bounded linear operators on X. We denote the spectrum of an operator $T\in \mathcal{L}(\mathrm{X}) $ by $\sigma(T) $ , that is, the set of all complex numbers Asuch that $\lambda IT $ fails to be invertible, where I stands for the identity operator on I. An elementary operator on $\mathcal{L}(X) $ is defined by $\Phi_{\mathrm{A},\mathrm{B}}(X):=\sum_{j=1}^{n}$AjXBj $(X\in \mathcal{L}(X)) $, where $\mathrm{A}=(A_{1}, \ldots, A_{n}) $ and $\mathrm{B}=(B_{1}, \ldots, B_{n}) $ are both $n$tuples of mutually commuting operators in $\mathcal{L}(X) $. $\Phi_{\mathrm{A},\mathrm{B}} $ is abounded linear operator on $\mathcal{L}(X) $ (i.e., $\Phi_{\mathrm{A},\mathrm{B}}\in \mathcal{L}(\mathcal{L}(X)) $ ) and this operation was first introduced in order to solve the following tyPe of operator equation: $A_{1}XB_{1}+A_{2}XB_{2}+\cdots+A_{n}XB_{n}=\mathrm{Y}$. (0.1)
Mathematical Institute,
, 2007
"... ∗ Partially supported by the Alexander von HumboldtFoundation 1 1 ..."
Mathematical Institute,
, 2013
"... Received (to be inserted by publisher) In this paper we show that a generalized form of Parrondo’s paradoxical game can be applied to discrete systems, working out the logistic map as a concrete example, to generate stable orbits. Written in Parrondos ’ terms, this reads: chaos1 + chaos2 +:::+ chaos ..."
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Received (to be inserted by publisher) In this paper we show that a generalized form of Parrondo’s paradoxical game can be applied to discrete systems, working out the logistic map as a concrete example, to generate stable orbits. Written in Parrondos ’ terms, this reads: chaos1 + chaos2 +:::+ chaosN = order, where chaosi, i = 1; 2;:::; N, are denoted the chaotic behaviors generated by N values of the parameter control, and by order one understands some stable behavior. The numerical results are sustained by quantitative dynamics generated by Parrondo’s game. The implementation of the generalized Parrondo’s game is realized here via the parameter switching (PS) algorithm for continuoustime systems [Danca, 2013] adapted to the logistic map. Some related results for more general maps on averaging, which represent discrete analogies of the PS method for ODE, are also presented and discussed.
Mathematical Institute,
"... Limit order books are used to match buyers and sellers in more than half of the world’s financial markets, and have been studied extensively in several disciplines during the past decade. This survey highlights the many insights from the wealth of empricial and theoretical studies that have been con ..."
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Limit order books are used to match buyers and sellers in more than half of the world’s financial markets, and have been studied extensively in several disciplines during the past decade. This survey highlights the many insights from the wealth of empricial and theoretical studies that have been conducted, and the numerous unsolved problems that remain. We illustrate the differences between observations from empirical studies of limit order books and the models that attempt to replicate them. In particular, many modelling assumptions are poorly supported by data and several wellestablished empirical facts have yet to be reproduced satisfactorily by models. By examining existing models of limit order books, we identify some key unresolved questions and difficultes currently facing researchers of limit order trading.
Mathematical Institute,
"... Variational aspects of the exisntence of L ∞ bounds for global solutions of ..."
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Variational aspects of the exisntence of L ∞ bounds for global solutions of
Mathematical Institute,
"... The subject of this report is the semiclassical distribution of eigenvalues for the Schr\"odinger equation $h^{2}\Delta u+V(x)u=Eu$. “Semiclassical distribution ” means the asymptotics with respect to $h $ as $h$ tends to 0, while the energy $E $ is restrained in aneighborhood of afixed real e ..."
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The subject of this report is the semiclassical distribution of eigenvalues for the Schr\"odinger equation $h^{2}\Delta u+V(x)u=Eu$. “Semiclassical distribution ” means the asymptotics with respect to $h $ as $h$ tends to 0, while the energy $E $ is restrained in aneighborhood of afixed real energy $E_{0} $. In this report, we restrict ourselves to the $\mathrm{I}_{}^{\mathfrak{n}}.\mathrm{o}\mathrm{s}\mathrm{t} $ fundamental problem of asimple well potential in one dimension: $h^{2} \frac{d^{2}u}{dx^{2}}+V(x)u=Eu$, (0.1) where the potential $V(x) $ is arealvalued analytic function on $\mathbb{R} $ and the classically allowed region $\{x\in \mathbb{R};V(x)\leq E_{0}\} $ is aconnected interval $[\alpha, \beta]$ $(\infty<\alpha<\beta<+\infty) $. We assume moreover that $V’(\alpha)<0 $ , $V’(\beta)>0 $. For $E\in(E_{0}\epsilon, E_{0}+\epsilon) $ with sufficiently small $\epsilon $ , the classically allowed region is still connected interval $[\alpha(E), \beta(E)] $. It is well known that the eigenvalues near $E_{0} $ are given by the s0called BohrSommerfeld quantization condition:
Mathematical Institute,
, 2010
"... We provide and analyse a model for the growth of bacterial biofilms based on the concept of extracellular polymeric substance (EPS) as a polymer solution, whose viscoelastic rheology is described by the classical FloryHuggins theory. We show that onedimensional solutions exist, which take the for ..."
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We provide and analyse a model for the growth of bacterial biofilms based on the concept of extracellular polymeric substance (EPS) as a polymer solution, whose viscoelastic rheology is described by the classical FloryHuggins theory. We show that onedimensional solutions exist, which take the form at large times of travelling waves, and we characterise their form and speed in terms of the describing parameters of the problem. Numerical solutions of the timedependent problem converge to the travelling wave solutions.
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