Mathematik und Statistik
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daquino_212588.pdf (273.1Kb)D'Aquino, Paola; Kuhlmann, Salma; Lange, Karen (2013), Preprint[more][less]
Zusammenfassung: We give a valuation theoretic characterization for a real closed field to be recursively saturated. Our result extends the characterization of Harnik and Ressayre \cite{hr} for a divisible ordered abelian group to be recursively saturated. Dateien zu dieser Publikation: 1
daquino_212588.pdf (273.1Kb) -
kuhlmann_212594.pdf (176.9Kb)Kuhlmann, Salma; Matusinski, Mickael; Point, Francoise (2013), Preprint[more][less]
Zusammenfassung: There are several equivalent characterizations of the valuation rank of an ordered field (endowed with its natural valuation). In this paper, we extend the theory to the case of an ordered difference field and introduce the notion of {\it difference rank}. We characterize the difference rank as the quotient modulo the equivalence relation naturally induced by the automorphism (which encodes its growth rate). In analogy to the theory of convex valuations, we prove that any linearly ordered set can be realized as the difference rank of an ordered difference field. Dateien zu dieser Publikation: 1
kuhlmann_212594.pdf (176.9Kb) -
Makowski, Matthias (2013), Artikel[more][less]
Erschienen in: Calculus of Variations and Partial Differential Equations ; 46 (2013), 1-2. - S. 213-252 Zusammenfassung: Let $N$ be a $(n+1)$-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface $\mathcal{S}_0$ and $F$ a curvature function, either the mean curvature $H$, the root of the second symmetric polynomial $\si_2 = \sqrt{H_2}$ or a curvature function of class $(K^*)$, a class of curvature functions which includes the $n$-th root of the Gaussian curvature $\si_n = K^{\frac{1}{n}}$. We consider curvature flows with curvature function $F$ and a volume preserving term and prove long time existence of the flow and exponential convergence of the corresponding graphs in the $C^\infty$-topology to a hypersurface of constant $F$-curvature, provided there are barriers. Furthermore we examine stability properties and foliations of constant $F$-curvature hypersurfaces. Dateien zu dieser Publikation: 0
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Ghasemi, Mehdi; Kuhlmann, Salma; Samei, Ebrahim (2013), Artikel[more][less]
Erschienen in: Archiv der Mathematik ; 100 (2013), 1. - S. 43-53 Zusammenfassung: Let τ be a locally convex topology on the countable dimensional polynomial ${\mathbb{R}}$ -algebra ${\mathbb{R} [\underline{X}] := \mathbb{R} [X_1, \ldots, X_{n}]}$ . Let K be a closed subset of ${\mathbb{R} ^{n}}$ , and let ${M := M_{\{g_1, \ldots, g_s\}}}$ be a finitely generated quadratic module in ${\mathbb{R} [\underline{X}]}$ . We investigate the following question: When is the cone Psd(K) (of polynomials nonnegative on K) included in the closure of M? We give an interpretation of this inclusion with respect to representing continuous linear functionals by measures. We discuss several examples; we compute the closure of ${M = \sum \mathbb{R} [\underline{X}]^{2}}$ with respect to weighted norm-p topologies. We show that this closure coincides with the cone Psd(K) where K is a certain convex compact polyhedron. Dateien zu dieser Publikation: 0
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Ghasemi, Mehdi; Kuhlmann, Salma (2013), Artikel[more][less]
Erschienen in: Journal of Functional Analysis ; 264 (2013), 1. - S. 413-427 Zusammenfassung: Let R be a unitary commutative R-algebra and K⊆X(R)=Hom(R,R), closed with respect to the product topology. We consider R endowed with the topology T(K), induced by the family of seminorms ρα(a):=|α(a)|, for α∈K and a∈R. In case K is compact, we also consider the topology induced by ‖a‖K:=supα∈K|α(a)| for a∈R. If K is Zariski dense, then those topologies are Hausdorff. In this paper we prove that the closure of the cone of sums of 2d-powers, ∑R2d, with respect to those two topologies is equal to Psd(K):={a∈R:α(a)⩾0, for all α∈K}. In particular, any continuous linear functional L on the polynomial ring View the MathML source with L(h2d)⩾0 for each View the MathML source is integration with respect to a positive Borel measure supported on K. Finally we give necessary and sufficient conditions to ensure the continuity of a linear functional with respect to those two topologies. Dateien zu dieser Publikation: 0
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313_Borgmeyer.pdf (355.8Kb)Borgmeyer, Karin; Quintanilla, Ramón; Racke, Reinhard (2013), Working Paper / Technical Report
Konstanzer Schriften in Mathematik -Nr. 313 [more][less]Zusammenfassung: In two recent papers the authors have studied conditions on the relaxation parameters in order to guarantee the stability or instability of solutions for the Taylor approximations to dual-phase-lag and three-phase-lag heat conduction equations. However, for several limit cases relating to the parameters the kind of stability was unclear. Here we analyze these limit cases and clarify whether we can expect exponential or slow decay for the solutions. Moreover, rather general well-posedness results for three-phase-lag models are presented. Finally, the exponential stability expected by spectral analysis is rigorously proved exemplarily. Dateien zu dieser Publikation: 1
313_Borgmeyer.pdf (355.8Kb) -
Carl_219570.pdf (437.7Kb)Carl, Merlin (2013), Preprint[more][less]
Zusammenfassung: It is frequently claimed that the formalization of a mathematical proof requires a quality of understanding that subsumes all necessary acts for checking the proof and that, consequently, automatic proof checking cannot lead to an epistemic gain about a proof. We present a project developing what is sometimes called a 'fortified formalism' and argue taking a phenomenological look at proof understanding, that proofs can be (and often are) given in a way that allows a formalization sufficient for producing an automatically checkable writeup, but does not subsume checking. Dateien zu dieser Publikation: 1
Carl_219570.pdf (437.7Kb) -
Carl, Merlin; Moroz, Boris (2013), Artikel[more][less]
Erschienen in: Bulletin of the Belgian Mathematical Society - Simon Stevin ; 20 (2013), 1. - S. 181-187 Dateien zu dieser Publikation: 0
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Fehm, Arno; Paran, Elad (2013), Artikel[more][less]
Erschienen in: Journal für die reine und angewandte Mathematik ; 2013 (2013), 676. - S. 213-225 Zusammenfassung: The quotient field of a generalized Krull domain of dimension exceeding one is Hilbertian by a theorem of Weissauer. Building on work of R. Klein we generalize this criterion for Hilbertianity to a wider class of domains. This allows us to extend recent results on Hilbertianity of fields of power series and obtain new Hilbertian fields. Dateien zu dieser Publikation: 0
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Diss_Weiershäuser.pdf (11.96Mb)Weiershäuser, Arno (2012), Dissertation[more][less]
Zusammenfassung: In this work we deal with the mathematical analysis and application of piecewise (or segmented) polynomial regression. Motivated by an application in neurobiology we allow the residual processes of our model to exhibit long memory, short memory or antipersistence.<br /><br /> As a solid biological background is essential for understanding the application in this work, we start with an introduction to neurobiology and the related experimental techniques. We conclude this introduction with a sample of data sets by means of which we illustrate piecewise polynomial regression.<br /><br /> Thereafter, we discuss least squares estimation with piecewise polynomials when the residuals exhibit antipersistence, short memory or long memory. This purely mathematical discussion is completely detached from the initial biological application. We start with an introduction to the related mathematical foundations and then discuss consistency and the asymptotic properties of the least squares estimator. In addition to the usual least squares estimator we treat the weighted least squares estimator as well. The asymptotic distribution is represented as a stochastic integral with respect to a fractional Brownian motion (in the case of antipersistence, short memory or long memory) or as a stochastic integral with respect to a Hermite process (in the case of long memory). We derive our results by means of fractional calculus which allows us to state a unifying formula of the asymptotic covariance matrix which covers all three correlation structures. In the case of an unknown number of segments we show that an information criterion can be used to estimated this unknown number. However, as the precise normalisation of the information criterion depends on the underlying correlation structure of the residuals, the latter results is only of theoretical interest.<br /><br /> We conclude this work by applying the derived methods on a large biological data sets. In this analysis, we apply piecewise polynomials to estimate the trend function of temporal response patterns. These estimates serve then as an input for an errors-in-variables regression model. Dateien zu dieser Publikation: 1
Diss_Weiershäuser.pdf (11.96Mb)
