Fachbereich Mathematik
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Aufgrund der vernetzten Strukturen und Wirkungszusammenhänge dynamischer Systeme werden die zugrundeliegenden mathematischen Modelle meist sehr komplex und erfordern ein hohes mathematisches Verständnis und Geschick. Bei Verwendung von spezieller Software können jedoch auch ohne tiefgehende mathematische oder informatorische Fachkenntnisse komplexe Wirkungsnetze dynamischer Systeme interaktiv erstellt werden. Als Beispiel wollen wir schrittweise das Modell einer Miniwelt entwerfen und Aussagen bezüglich ihrer Bevölkerungsentwicklung treffen.
Die Akustik liefert einen interessanten Hintergrund, interdisziplinären und fächerverbindenen Unterricht zwischen Mathematik, Physik und Musik durchzuführen. SchülerInnen können hierbei beispielsweise experimentell tätig sein, indem sie Audioaufnahmen selbst erzeugen und sich mit Computersoftware Frequenzspektren erzeugen lassen. Genauso können die Schüler auch Frequenzspektren vorgeben und daraus Klänge erzeugen. Dies kann beispielsweise dazu dienen, den Begriff der Obertöne im Musikunterricht physikalisch oder mathematisch greifbar zu machen oder in der Harmonielehre Frequenzverhältnisse von Intervallen und Dreiklängen näher zu untersuchen.
Der Computer ist hier ein sehr nützliches Hilfsmittel, da der mathematische Hintergrund dieser Aufgabe -- das Wechseln zwischen Audioaufnahme und ihrem Frequenzbild -- sich in der Fourier-Analysis findet, die für SchülerInnen äußerst anspruchsvoll ist. Indem man jedoch die Fouriertransformation als numerisches Hilfsmittel einführt, das nicht im Detail verstanden werden muss, lässt sich an anderer Stelle interessante Mathematik betreiben und die Zusammenhänge zwischen Akustik und Musik können spielerisch erfahren werden.
Im folgenden Beitrag wird eine Herangehensweise geschildert, wie wir sie bereits bei der Felix-Klein-Modellierungswoche umgesetzt haben: Die SchülerInnen haben den Auftrag erhalten, einen Synthesizer zu entwickeln, mit dem verschiedene Musikinstrumente nachgeahmt werden können. Als Hilfsmittel haben sie eine kurze Einführung in die Eigenschaften der Fouriertransformation erhalten, sowie Audioaufnahmen verschiedener Instrumente.
With this article we first like to give a brief review on wavelet thresholding methods in non-Gaussian and non-i.i.d. situations, respectively. Many of these applications are based on Gaussian approximations of the empirical coefficients. For regression and density estimation with independent observations, we establish joint asymptotic normality of the empirical coefficients by means of strong approximations. Then we describe how one can prove asymptotic normality under mixing conditions on the observations by cumulant techniques.; In the second part, we apply these non-linear adaptive shrinking schemes to spectral estimation problems for both a stationary and a non-stationary time series setup. For the latter one, in a model of Dahlhaus on the evolutionary spectrum of a locally stationary time series, we present two different approaches. Moreover, we show that in classes of anisotropic function spaces an appropriately chosen wavelet basis automatically adapts to possibly different degrees of regularity for the different directions. The resulting fully-adaptive spectral estimator attains the rate that is optimal in the idealized Gaussian white noise model up to a logarithmic factor.
We derive minimax rates for estimation in anisotropic smoothness classes. This rate is attained by a coordinatewise thresholded wavelet estimator based on a tensor product basis with separate scale parameter for every dimension. It is shown that this basis is superior to its one-scale multiresolution analog, if different degrees of smoothness in different directions are present.; As an important application we introduce a new adaptive wavelet estimator of the time-dependent spectrum of a locally stationary time series. Using this model which was resently developed by Dahlhaus, we show that the resulting estimator attains nearly the rate, which is optimal in Gaussian white noise, simultaneously over a wide range of smoothness classes. Moreover, by our new approach we overcome the difficulty of how to choose the right amount of smoothing, i.e. how to adapt to the appropriate resolution, for reconstructing the local structure of the evolutionary spectrum in the time-frequency plane.
We consider wavelet estimation of the time-dependent (evolutionary) power spectrum of a locally stationary time series. Allowing for departures from stationary proves useful for modelling, e.g., transient phenomena, quasi-oscillating behaviour or spectrum modulation. In our work wavelets are used to provide an adaptive local smoothing of a short-time periodogram in the time-freqeuncy plane. For this, in contrast to classical nonparametric (linear) approaches we use nonlinear thresholding of the empirical wavelet coefficients of the evolutionary spectrum. We show how these techniques allow for both adaptively reconstructing the local structure in the time-frequency plane and for denoising the resulting estimates. To this end a threshold choice is derived which is motivated by minimax properties w.r.t. the integrated mean squared error. Our approach is based on a 2-d orthogonal wavelet transform modified by using a cardinal Lagrange interpolation function on the finest scale. As an example, we apply our procedure to a time-varying spectrum motivated from mobile radio propagation.
The following two norms for holomorphic functions \(F\), defined on the right complex half-plane \(\{z \in C:\Re(z)\gt 0\}\) with values in a Banach space \(X\), are equivalent:
\[\begin{eqnarray*} \lVert F \rVert _{H_p(C_+)} &=& \sup_{a\gt0}\left( \int_{-\infty}^\infty \lVert F(a+ib) \rVert ^p \ db \right)^{1/p}
\mbox{, and} \\ \lVert F \rVert_{H_p(\Sigma_{\pi/2})} &=& \sup_{\lvert \theta \lvert \lt \pi/2}\left( \int_0^\infty \left \lVert F(re^{i \theta}) \right \rVert ^p\ dr \right)^{1/p}.\end{eqnarray*}\] As a consequence, we derive a description of boundary values ofsectorial holomorphic functions, and a theorem of Paley-Wiener typefor sectorial holomorphic functions.
We consider a variant of the generalized assignment problem (GAP) where the amount of space used in each bin is restricted to be either zero (if the bin is not opened) or above a given lower bound (a minimum quantity). We provide several complexity results for different versions of the problem and give polynomial time exact algorithms and approximation algorithms for restricted cases.
For the most general version of the problem, we show that it does not admit a polynomial time approximation algorithm (unless P=NP), even for the case of a single bin. This motivates to study dual approximation algorithms that compute solutions violating the bin capacities and minimum quantities by a constant factor. When the number of bins is fixed and the minimum quantity of each bin is at least a factor \(\delta>1\) larger than the largest size of an item in the bin, we show how to obtain a polynomial time dual approximation algorithm that computes a solution violating the minimum quantities and bin capacities by at most a factor \(1-\frac{1}{\delta}\) and \(1+\frac{1}{\delta}\), respectively, and whose profit is at least as large as the profit of the best solution that satisfies the minimum quantities and bin capacities strictly.
In particular, for \(\delta=2\), we obtain a polynomial time (1,2)-approximation algorithm.
This paper is devoted to the mathematica l description of the solution of the so-called rainflow reconstruction problem, i.e. the problem of constructing a time series with an a priori given rainflow m atrix. The algorithm we present is mathematically exact in the sense that no app roximations or heuristics are involved. Furthermore it generates a uniform distr ibution of all possible reconstructions and thus an optimal randomization of the reconstructed series. The algorithm is a genuine on-line scheme. It is easy adj ustable to all variants of rainflow such as sysmmetric and asymmetric versions a nd different residue techniques.
We prove that there exists a positive \(\alpha\) such thatfor any integer \(\mbox{$d\ge 3$}\) and any topological types \(\mbox{$S_1,\dots,S_n$}\) of plane curve singularities, satisfying \(\mbox{$\mu(S_1)+\dots+\mu(S_n)\le\alpha d^2$}\), there exists a reduced irreducible plane curve of degree \(d\) with exactly \(n\) singular points of types \(\mbox{$S_1,\dots,S_n$}\), respectively. This estimate is optimal with respect to theexponent of \(d\). In particular, we prove that for any topological type \(S\) there exists an irreducible polynomial of degree \(\mbox{$d\le 14\sqrt{\mu(S)}$}\) having a singular point of type \(S\).
