Number | Date | Author | Title | Abstract/MSC |
---|---|---|---|---|

2431 (opens in new tab) | 20.12.2005 |
Farwig, Reinhard Ri Myong-Hwan |
Resolvent Estimates and Maximal Regularity in Weighted $L^q$-spaces of the Stokes Operator in an Infinite Cylinder |
MSC: 35Q30; 76D07 Let $\Omega= \Sigma \times R$ be an infinite cylinder of $R^n, n \geq 3,$ with a bounded cross-section $\Sigma \subset R^{n-1}$ of $C^{1,1}$-class. We study resolvent estimates and maximal regularity of the Stokes operator in $L^q(R;L^r_\omega(\Sigma))$ for $1 < q,r < \infty$ and for arbitrary Muckenhoupt weights $\omega\in A_r$ with respect to $x' \in \Sigma$. The proofs use an operator-valued Fourier multiplier theorem and techniques of unconditional Schauder decompositions based on the ${\mathcal R}$-boundedness of the family of solution operators for a system in $\Sigma$ parametrized by the phase variable of the one-dimensional partial Fourier transform. |

2430 (opens in new tab) | 20.12.2005 |
Farwig, Reinhard Kozono, Hideo Sohr, Hermann |
The Helmholtz Decomposition in Arbitrary Unbounded Domains -- A Theory Beyond $L^2$ |
MSC: 35Q35; 35Q30; 76D07 It is well known that the usual $L^q$-theory of the Stokes operator valid for bounded or exterior domains cannot be extended to arbitrary unbounded domains $\Omega\subset R^n$ when $q\neq 2$. One reason is given by the Helmholtz projection which fails to exist for certain unbounded smooth planar domains unless $q=2.$ However, as recently shown, the Helmholtz projection does exist for general unbounded domains in $R^3$ if we replace the space $L^q, 1 < q < \infty,$ by $L^2\cap L^q$ for $q>2$ and by $L^q+L^2$ for $1 < q < 2.$ In this paper, we generalize this new approach from the three-dimensional case to the $n$-dimensional case, $n\geq 2.$ |

2429 | 12.12.2005 |
Rabinovich, Vladimir Roch, Steffen |
Exact and numerical inversion of pseudodifferential operators and applications to signal processing |
MSC: 35S05; 65N30; 94A12 A large class of time-varying filters can be described via pseudodifferential operators belonging to the Hörmander class $OPS_{0, ¸ 0}^0$. The questions whether and how an input signal can be reconstructed from a known output lead to the problems of invertibility of pseudodifferential operators in that class and of (at least, numerical) solution of pseudodifferential equations. We are going to derive effective conditions for the invertibility for pseudodifferential operators with globally slowly varying symbols as well as for causal pseudodifferential operators, and we study the stability of the finite sections method with respect to time and frequency for these operators. |

2428 (opens in new tab) | 15.12.2005 | Zahn, Peter | On the Use of Hypotheses in Cumulative Type Theory (Enlarged Version) |
MSC: 03B15; 03B45 Given a language of ramified cumulative type theory as introduced in (Zahn 2004). We shall construct and investigate an extension, $\cal{L}$, of it, which is a language of the same sort, but also containes sentences which express that certain sentences of $\cal{L}$ are deducible from others (hypotheses) by given rules. To this we introduce `names' of terms and formulas of $\cal{L}$ and include them in $\cal{L}$. So in $\cal{L}$ we can not only use but also `speak about' sentences of that language. Especially, by means of first order sentences we can speak about higher order sentences. Despite this possibility of `reduction' of order, all sentences of $\cal{L}$ are non-circular. The considered deducibility-relations of sentences from others correspond to systems of labelled modal logic of types $K_4$ and $G$. |

2427 | 01.11.2005 | Mäurer, Helmut | Harmonische Involutionen und Kreisspiegelungen in Möbiusebenen | |

2426 | 01.11.2005 |
Neeb, Karl-Hermann Wagemann, Friedrich |
The second cohomology of current algebras of general Lie algebras |
MSC: 17B56; 17B65 Let $A$ be a unital commutative associative algebra over a field of characteristic zero, $\k$ be a Lie algebra, and $\z$ a vector space, considered as a trivial module of the Lie algebra $\g := A \otimes \k$. In this paper we give a description of the cohomology space $H^2(\g,\z)$ in terms of well accessible data associated to $A$ and $\k$. We also discuss the topological situation, where $A$ and $\k$ are locally convex algebras. |

2425 | 10.11.2005 | Neeb, Karl-Hermann | On the classification of rational quantum tori and their automorphism groups |
MSC: 16S35 An n$-dimensional quantum torus is a twisted group algebra of the group $\Z^n$. It is called rational if all invertible commutators are roots of unity. In the present note we classify all rational n-dimensional quantum tori over any field. Moreover, we show that for $ = 2$ the natural exact sequence describing the automorphism group of the quantum torus splits over any field. |

2324 (opens in new tab) | 31.10.2005 |
Michael Joswig Thilo Schröder |
Neighborly Cubical Polytopes and Spheres |
MSC: 52B12; 53A03 We prove that the neighborly cubical polytopes studied by Günter M. Ziegler and the first named author arise as a special case of the neighborly cubical spheres constructed by Babson, Billera, and Chan. By relating the two constructions we obtain an explicit description of a non-polytopal neighborly cubical sphere and, further, a new proof of the fact that the cubical equivelar surfaces of McMullen, Schulz, and Wills can be embedded into ${\bf R}^3$. |

2423 (opens in new tab) | 10.11.2005 |
Farwig, Reinhard Neustupa, Jiri |
On the Spectrum of a Stokes--Type Operator Arising from Flow around a Rotating Body |
MSC: 35Q35; 35P99; 76D07 We present the description of the spectrum of a linear perturbed Stokes--type operator which arises from equations of motion of a viscous incompressible fluid in the exterior of a rotating compact body. Considering the operator in the function space $L^2_{\sigma}(\Omega)$ we prove that the essential spectrum consists of a set of equally spaced half lines parallel to the negative real half line in the complex plane. Our approach is based on a reduction to invariant closed subspaces of $L^2_{\sigma}(\Omega)$ and on a Fourier series expansion with respect to an angular variable in a cylindrical coordinate system attached to the axis of rotation. Moreover, we show that the operator is normal if and only if the body is axially symmetric about this axis. |

2422 (opens in new tab) | 10.11.2005 |
Farwig, Reinhard Krbec, Miroslav Necasova, Sarka |
A Weighted $L^q$-approach to Stokes Flow Around a Rotating Body |
MSC: 76D05; 35Q30; 42B25; 46E35 Considering time-periodic Stokes flow around a rotating body in $R^2$ or $R^3$ we prove weighted a priori estimates in $L^q$-spaces for the whole space problem. After a time-dependent change of coordinates the problem is reduced to a stationary Stokes equation with the additional term $(\omega\times x)\cdot\nabla u$ in the equation of momentum, where $\omega$ denotes the angular velocity. In cylindrical coordinates attached to the rotating body we allow for Muckenhoupt weights which may be anisotropic or even depend on the angular variable and prove weighted $L^q$-estimates using the weighted theory of Littlewood-Paley decomposition and of maximal operators. |

2421 | 02.11.2005 |
Bartuzel, Grzegorz Fryszkowski, Andrzej |
Filippov Lemma for beam differential |
MSC: 26A24; 28A15; 46G05; 39A05; 28A05 In the paper we deal with beam differential inclusion \begin{equation*} \mathcal{D}y=y^{\prime \prime \prime \prime }+4k^{4}y\in F\left( x,y\right) \end{equation*}% defined on $\left[ -1,1\right] $\ with boundary conditions \begin{equation*} y\left( -1\right) =y\left( 1\right) =y^{\prime }\left( -1\right) =y^{\prime }\left( 1\right) =0. |

2420 (opens in new tab) | 25.10.2005 | Heil, Erhard | Trefoil Knots With Tritangent Planes |
MSC: 2000; 53A04 As we know, there are examples of trefoil knots without tritangent planes. Here we show that trefoil knots of the familiar shape always have such planes. We relate this result to the search for the minimum number of vertices. |

2419 (opens in new tab) | 19.10.2005 | Rößler | Multi--colored rooted tree analysis for Runge--Kutta methods for the weak approximation of stochastic differential equations |
MSC: 65C30; 65L06; 60H35; 60H10 A general class of stochastic Runge--Kutta methods for Itô and Stratonovich stochastic differential equation systems with a multi--dimensional Wiener process is considered. The multi--colored rooted tree analysis is applied to calculate order conditions for the coefficients of explicit and implicit stochastic Runge--Kutta methods assuring convergence in the weak sense with a prescribed order. Especially, order conditions and some coefficients for stochastic Runge--Kutta schemes of weak order two are calculated explicitly. |

2418 | 18.10.2005 | V. Rabinovich, S. Roch | Reconstruction of input signals in time-varying filters |
MSC: 47G20; 47N99; 65J10 We consider the problem of reconstruction of input signals $u$ from output signals of time-varying filters of the form \[ (Au)(x) = \sum_{j \in \mathbb{Z}} a_j(x) u(x-j), \quad x \in \mathbb{Z}, \] under the assumption that $\sum_{j \in \mathbb{Z}} \|a_j\|_\infty < \infty$. The proposed algorithm of reconstruction of signals is based on the theory of band-dominated and pseudodifference operators as presented in the recent monograph \cite{RRSB} and on the finite sections method. The following classes of filters are considered this paper: slowly time-varying filters, perturbations of periodic time-varying filters, causal time-varying filters, and finite filters acting on signals with a finite number of values. |

2417 | 18.10.2005 | Vladimir S. Rabinovich, Steffen Roch | The Fredholm index of locally compact band-dominated operators on $L^p (\sR)$ |
MSC: 47A53; 45E10; 47G10 We establish a necessary and sufficient criterion for the Fredholmness of a general locally compact band-dominated operator $A$ on $L^p(\sR)$ and derive a formula for its Fredholm index in terms of the limit operators of $A$. The results are applied to operators of convolution type with almost periodic symbol. |

2416 (opens in new tab) | 16.09.2005 |
Teleaga, Ioan Seaïd, Mohammed Gasser, Ingenuin Klar, Axel Struckmeier, Jens |
Radiation Models for Thermal Flows at Low Mach Number |
MSC: 76V05; 80A25; 76M20; 76M45 Simplified approximate models for radiation are proposed to study thermal effects in low Mach flow in open tunnels. The governing equations for fluid dynamics are derived by applying a low-Mach asymptotic in the compressible Navier-Stokes problem. Based on an asymptotic analysis we show that the integro-differential equation for radiative transfer can be replaced by a set of differential equations which are independent of angle variable and easy to solve using standard numerical discretizations. As an application we consider the situation of fires in vehicular tunnels. The results presented in this paper show that the proposed models are able to predict temperature in the tunnels accurately with low computational cost. |

2415 (opens in new tab) | 05.09.2005 |
Farwig, Reinhard Ri Myong-Hwan |
An $L^q(L^2)$-Theory of the Generalized Stokes Resolvent System in Infinite Cylinders |
MSC: 35Q30; 76D07 Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinder $\Omega=\Sigma\times R$ with $\Sigma\subset R^{n-1}$, a bounded domain of $C^{1,1}$-class, are obtained in the space $L^q(R;L^2(\Sigma))$, $q\in (1,\infty)$. For the preparation, spectral decompositions of vector-valued homogeneous Sobolev spaces are studied. The main theorem is proved using the techniques of Schauder decompositions, operator-valued multiplier functions and $R$-boundedness of operator families. |

2414 (opens in new tab) | 02.09.2005 |
Neff, Patrizio Muench, Ingo |
Simple glide for a nonlinear elastic Cosserat model: analytical and computational results with induced microstructure |
MSC: 74A35; 74A30; 74C05; 74C10 We study the static simple glide problem for a geometrically exact, generalized continua of micropolar type.In contrast to linear micropolar elasticity, where the unique solution is available in closed form we exhibit a multitude of solutions to the nonlinear problem, even if the two fields of deformations $\varphi$ and microrotations $\overline{R}$ can remain homogeneous. This motivates a search for new conditions on the microrotations $\overline{R}$ which single out a unique, physically acceptable, response. The influence of material parameters, notably the Cosserat couple modulus $\mu_c$ and the length scale $L_c$ on the response is also studied. For small Cosserat couple modulus $\mu_c>0$ we observe a pitchfork bifurcation of the homogeneous response and for vanishing internal length and zero Cosserat couple modulus $\mu_c=0$ the Cosserat model may show highly oscillating "microstructure» solutions which are energetically better than the homogeneous response.The numerical results show that even for $\mu>0$ the nonlinear Cosserat model has a quite different qualitative response than the linear model. |

2413 | 13.07.2005 |
Haller-Dintelmann, Robert Wiedl, Julian |
Kolmogorov kernel estimates for the Ornstein-Uhlenbeck operator |
MSC: 47D06; 35K20 Replacing the Gaussian semigroup in the heat kernel estimates by the Ornstein-Uhlenbeck semigroup on ${\mathbb{R}}^d$, we define the notion of Kolmogorov kernel estimates. This allows us to show that under Dirichlet boundary conditions Ornstein-Uhlenbeck operators are generators of consistent, positive, (quasi-)contractive $C_0$-semigroups on $L^p(\Omega)$ for all $1 \le p < \infty$ and for every domain $\Omega \subseteq {\mathbb{R}}^d$. For exterior domains with sufficiently smooth boundary a result on the location of the spectrum of these operators is also given. |

2412 | 22.08.2005 |
Neff, Patrizio Chelminski, Krzysztof |
Well-posedness of dynamic Cosserat plasticity. |
MSC: 74A35; 74A30; 74C05; 74H20; 74H25 We investigate the regularizing properties of generalized continua of micropolar type for dynamic elasto-plasticity. To this end we propose an extension of classical infinitesimal elasto-plasticity to include consistently non-dissipative micropolar effects and we show that the dynamic model allows for a unique, global in-time solution of the corresponding rate-independent initial boundary value problem. The method of choice are the Yosida-approximation and a passage to the limit. |

2411 (opens in new tab) | 16.08.2005 |
Küpper, Dominique Lehn, Jürgen Rößler, Andreas |
A step size control algorithm for the weak approximation of stochastic differential equations |
MSC: 60H10; 65C30; 60H35; 65L06; 65L50; 34F05 A variable step size control algorithm for the weak approximation of stochastic differential equations is introduced. The algorithm is based on embedded Runge-Kutta methods which yield two approximations of different orders with a negligible additional computational effort. The difference of these two approximations is used as an estimator for the local error of the less precise approximation. We prove the convergence of the proposed method with step size control by means of rooted tree analysis. Furthermore, some numerical results are presented to demonstrate the effectiveness of the introduced step size control method. |

2410 (opens in new tab) | 17.08.2005 |
Farwig, Reinhard Ri Myong-Hwan |
Stokes Resolvent Systems in an Infinite Cylinder |
MSC: 35Q30; 76D07 In an infinite cylinder $\Omega= \Sigma \times \mathbb R$, where $\Sigma \subset \mathbb R^{n-1}, n \geq 3$, is a bounded domain of $C^{1,1}$ class, we study the unique solvability of Stokes resolvent systems in $L^q({\mathbb R}; L^2(\Sigma))$ for $1< q <\infty$ and in vector-valued homogeneous Besov spaces $\dot{{\cal B}}^s_{pq}({\mathbb R}; L^r(\Sigma))$ for $1\leq p,q\leq\infty,¸¸ s\in{\mathbb R},¸¸ 1< r <\infty$. By a partial Fourier transform along the axis of the cylinder $\Omega$ the given system is reduced to a parametrized system on $\Sigma$, for which parameter independent estimates are proved. For further applications we obtain even parameter independent estimates in $L^r(\Sigma),1< r <\infty,$ in the non-solenoidal case prescribing an arbitrary divergence $g= div \;u$. Then operator-valued multiplier theorems are used for the final estimates of the Stokes resolvent systems in $\Omega$. |

2409 | 25.07.2005 | Neff, Patrizio | The Cosserat couple modulus for continuous solids is zero viz the linearized Cauchy-stress tensor is symmetric. |
MSC: 74A35 We investigate weaker than usual constitutive assumptions in linear Cosserat theory still providing for existence and uniqueness and show a continuous dependence result for Cosserat couple modulus $\mu_c\to 0$. This result is needed when using Cosserat elasticity not as a physical model but as a numerical regularization device. Thereafter it is shown that the usually adopted material restrictions of uniform positivity for a linear Cosserat model cannot be consistent with experimental findings for continuous solids. The analytical solutions for both the torsion and the bending problem in general predict an unbounded stiffness for ever thinner samples. This unphysical behaviour can only be avoided for specific choices of parameters in the curvature energy expression. However, these choices do not satisfy the usual constitutive restrictions. We show that the possibly remaining linear elastic Cosserat problem is nevertheless well-posed but that it is impossible to determine the appearing curvature modulus independent of boundary conditions. This puts a serious doubt on the use of the linear elastic Cosserat model (or the geometrically exact model with $\mu_c>0$) for the physically consistent description of continuous solids like polycrystals in the framework of elasto-plasticity. The problem can be avoided in geometrically exact Cosserat models if the Cosserat couple modulus $\mu_c$ is set to zero. |

2408 | 21.07.2005 |
Harth, Tobias Lehn, Juergen |
Identification of Material Parameters for Inelastic Constitutive Models Using Stochastic Methods |
MSC: 62P30 The parameters of a constitutive model are usually identified by minimization of the distance between model response and experimental data. However, measurement failures and differences in the specimens lead to deviations in the determined parameters. In this article we present our results of a study of these uncertainties for two constitutive models of Chaboche-type. The models differ only by a kinematic hardening variable. It turns out, that the kinematic hardening variable proposed by Haupt, Kamlah, and Tsakmakis yields a better description quality than the one of Armstrong and Frederick. For the parameter optimization as well as for the study of the deviations of the fitted parameters we apply stochastic methods. The available test data result from creep tests, tension-relaxation tests and cyclic tests performed on AINSI SS316 stainless steel at 600$\grad$. Since the amount of test data is too small for a proper statistical analysis we apply a stochastic simulation technique to generate artificial data which exhibit the same stochastic behaviour as the experimental data. |

2406 | 21.07.2005 |
Wille, Rudolf Renate Wille-Henning |
Beurteilung von Musikstücken durch Adjektive: Eine begriffsanalytische Exploration |
MSC: 03B Die Beurteilung von Musikstücken durch Adjektive hat sich in der Musikpsychologie als eine Methode entwickelt, musikalisches Erleben sprachlich zu beschreiben und zu analysieren. Allerdings werden die dabei eingesetzten statistischen Verfahren nachhaltig kritisiert. Deswegen wird mit diesem Beitrag angeregt, die Formale Begriffsanalyse als datenanalytisches Verfahren zu verwenden. Wie das konkret aussehen kann, wird anhand der begriffsanalytischen Merkmalexploration demonstriert. |

2405 | 12.07.2005 | Fröhlich, Steffen | Über zweidimensionale Immersionen im $\mathbb R^n$ |
MSC: 53A10; 53A07; 53C42 Am Beispiel minimaler Graphen im $\mathbb R^4$ stellen wir ausgewählte analytische Methoden zur Gewinnung geometrischer Eigenschaften zweidimensionaler Immersionen in Räumen höherer Kodimension vor. |

2404 | 29.06.2005 | Fröhlich, Steffen | A note on two-dimensional minimal surface graphs in $\mathbb R^n$ and a theorem of Bernstein-Liouville type |
MSC: 53A10; 53A07; 53C42 Using Schauder's theory for linear elliptic partial differential equations in two independent variables and fundamental estimates for univalent mappings due to E. Heinz we establish an upper bound of the Gaussian curvature of two-dimensional minimal surface graphs in $\mathbb R^n.$ This leads us to a theorem of Bernstein-Liouville type. |

2403 (opens in new tab) | 12.07.2005 |
Farwig, Reinhard Galdi, Giovanni Paolo Sohr, Hermann |
Large Existence, Uniqueness and Regularity Classes of Stationary Navier-Stokes Equations in Bounded Domains of $R^n$ |
MSC: 76D05; 35J55; 35J65; 35Q30; 76D07 Using the notion of very weak solutions we obtain a new and very large uniqueness class for solutions of the inhomogeneous Navier-Stokes system $-\Delta u + u \cdot \nabla u + \nabla p=f$, $div\; u = k$, $u_{\partial\Omega}=g$ with $u \in L^q(\Omega)$, $q \geq n$, and very general data classes for $f,k,g$ such that $u$ may have no differentiability property. If the data are sufficiently smooth we get a large class of unique and regular solutions extending the well known classical solution classes and generalize a regularity result of C. Gerhardt. |

2402 | 06.07.2005 |
Reif, Ulrich Peters, Jörg |
Structural Analysis of Subdivision Surfaces -- A Summary |
MSC: 65D17; 65D07; 68U07 This paper summarizes the structure and analysis of subdivision surfaces and characterizes the inherent similarities and differences to parametric spline surfaces. Besides presenting well known results in a unified way, we introduce new ideas for analyzing schemes with a linearly dependent generating system, and a significantly simplified test for the injectivity of the characteristic map. |

2401 (opens in new tab) | 01.07.2005 | Zahn, Peter | Assertion Games to Justify Classical Reasoning |
MSC: 2000; 03A05 To reduce the problem of beginning reasoning, we introduce a 'primary game' of assertion by stipulating certain rules to restrict assertions. But we also agree that assertions must not be restricted besides. So we can see that both the stipulated rules for compound sentences and their inverses may be used as inference rules. By liberalizing the primary game, we establish a 'classical game' of assertion which permits to apply classical logic, and serves certain purposes of reasoning most favorably. Furthes themes: Arithmetical induction. A ramified cumulative type theory, in which 'type-free' equations x = y are definable. A combination of substitutional and objectual quantification. |

2400 | 16.06.2005 | Rudolf Wille | Conceptual Knowledge Processing in the Field of Economics |
MSC: 03B; 91B Conceptual Knowledge Processing is obliged to a pragmatic understanding of knowledge according to which human knowledge is obtained and supported in a process of human thinking, reasoning, and communicating. It is based on a mathematical theory of concepts directed toward an interaction of formal and material thoughts. How this theoretical conception enables effects in the economics practice is explained in this paper, guided by the key processes of the organizational knowledge management. These key processes are knowledge identification, knowledge acquisition, knowledge development, knowledge distribution and sharing, knowledge usage and knowledge preservation. For each key process, an example demontrates the use of specific methos of Conceptual Knowledge Processing. Finally, ojectives and evaluation of knowledge management are included into the discussion. |

2399 | 07.06.2005 |
Didenko, Victor Lee, Seng Luan Roch, Steffen Silbermann, Bernd |
Approximate foveated images and reconstruction of their uniform pre-images |
MSC: 65R20; 45P05 Approximate foveated images can be obtained from uniform images via the approximation of some integral operators. In this paper it is shown that these operators belong to a well studied operator algebra, and the problem of restoration of the approximate uniform pre-images is considered. Under common assumptions on smoothness of the integral operator kernels, necessary and sufficient conditions are established for such procedure to be feasible. |

2398 | 02.06.2005 |
Vladimir Rabinovich Steffen Roch Bernd Silbermann |
Finite sections of band-dominated operators with almost periodic coefficients |
MSC: 65J10; 47B36 We consider the sequence of the finite sections $R_nAR_n$ of a band-dominated operator $A$ on $l^2(\sZ)$ with almost periodic coefficients. Our main result says that if the compressions of $A$ onto $\sZ^+$ and $\sZ^-$ are invertible, then there is a distinguished subsequence of $(R_n A R_n)$ which is stable. Moreover, this subsequence proves to be fractal, which allows us to establish the convergence in the Hausdorff metric of the singular values and pseudoeigenvalues of the finite section matrices. |

2397 | 31.05.2005 | Wille, Rudolf | Allgemeine Wissenschaft und transdisziplinäre Methodologie |
MSC: 03A; 06B Allgemeine Wissenschaft und Transdisziplinarität sind aufs Engste miteinander verbunden. Deshalb kann Allgemeine Wissenschaft die Entwicklung transdisziplinärer Methodologien unterstützen. Wie das geschehen kann, wird aufgezeigt anhand der Formalen Begriffsanalyse, bei deren Anwendungen ein transdisziplinärer Übergang vom mathematischen zum logischen Denken grundlegend ist. |

2396 (opens in new tab) | 12.05.2005 |
Farwig, Reinhard Galdi, Giovanni, P.: Sohr, Hermann |
Very Weak Solutions and Large Uniqueness Classes of Stationary Navier-Stokes Equations in Bounded Domains of $\R^2$ |
MSC: 76D05; 35J55; 35J65; 35Q30; 76D07 Extending the notion of very weak solutions, developed recently in the three-dimensional case, to bounded domains $\Omega\subset \R^2$ we obtain a new class of unique solutions $u$ in $L^q(\Omega)$, $q > 2$, to the stationary Navier-Stokes system $-\Delta u + u \cdot \nabla u + \nabla p = f$, $\div u = k$, $u|_{\partial \Omega} = g$ with data $f,k,g$ of low regularity. As a main consequence we obtain a new uniqueness class also for classical weak or strong solutions. Indeed, such a solution is unique if its $L^q$--norm is sufficiently small or the data satisfy the uniqueness condition of a very weak solution. |

2395 | 27.04.2005 | Wille, Rudolf | Contextual Logic and Aristotle's Syllogistic |
MSC: 03B This paper is concerned with incorporating negational relationships into the semantics of Contextual Logic by mathematizing Aristotle's Syllogistic in contextual-logic terms. For preparing this, a short sketch of Aristotle's Syllogistic is presented. Then it is shown how a contextual semantics for Aristotle's syllogisms can be developed on the basis of so-called syllogistic contexts. This semantic approach is used to determine implication bases for elementary judgments within syllogistic contexts. Finally, directions of further research are mentioned. |

2394 (opens in new tab) | 19.04.2005 | Wockel, Christoph | Manifolds with Corners and the Topology of Gauge Groups |
MSC: 81R10; 22E65; 58A05; 58B10; 55Q52 In this paper we develop an effective way to access the topology of the gauge group for a smooth $K$-principal bundle $\mathcal{P}=(K,\pi,P,M)$ with possibly infinite-dimensional structure group $K$ over a finite-dimensional manifold with corners $M$. In the first section we establish the notion smooth mappings for (not necessarily open) subsets of a locally convex space, a corresponding concept of a manifold with corners (which is a generalisation of a manifold with boundary) and link this to the existing notion of smooth mappings on not necessarily open subsets of $\mathbb{R}^{n}$. This enables us in the second section to see the gauge group Gau$(\mathcal{P})$, with a natural topology on it, as an infinite-dimensional Lie group if $M$ is compact and $K$ is locally exponential. In the last section we discuss some applications. We show that the inclusion Gau$(\mathcal{P})\hookrightarrow \text{Gauc}_c(\mathcal{P})$ of smooth into continuous gauge transformations is a weak homotopy equivalence, apply this result to the calculation of $\pi_{n}($Gau$(\cP))$ and answer the question which locally smooth maps occur as pullbacks from gauge transformations. |

2393 | 18.04.2005 | Glöckner, Helge | Locally compact groups built up from $p$-adic Lie groups, for $p$ in a given set of primes |
MSC: 22D05; 22D45; 22E20; 22E35 We analyze the structure of locally compact groups which can be built up from $p$-adic Lie groups, for $p$ in a given set of primes. In particular, we calculate the scale function and determine tidy subgroups for such groups, and use them to recover the primes needed to build up the group. |

2392 | 14.04.2005 | Neeb, Karl-Hermann | Non-abelian extensions of infinite-dimensional Lie groups |
MSC: 22E65; 57T10; 22E15 In this paper we study non-abelian extensions of a Lie group $G$ modeled on a locally convex space by a Lie group $N$. The equivalence classes of such extension are grouped into those corresponding to a class of so-called smooth outer actions $S$ of $G$ on $N$. If $S$ is given, we show that the corresponding set $\Ext(G,N)_S$ of extension classes is a principal homogeneous space of the locally smooth cohomology group $H^2_{ss}(G,Z(N))_S$. To each $S$ a locally smooth obstruction class $\chi(S)$ in a suitably defined cohomology group $H^3_{ss}(G,Z(N))_S$ is defined. It vanishes if and only if there is a corresponding extension of $G$ by $N$. A central point is that we reduce many problems concerning extensions by non-abelian groups to questions on extensions by abelian groups, which have been dealt with in previous work. An important tool is a Lie theoretic concept of a smooth crossed module $\alpha \: H \to G$, which we view as a central extension of a normal subgroup of $G$. |

2391 | 01.03.2005 | Rudolf Wille | Mathematik präsentieren, reflektieren, beurteilen |
MSC: 00A05 Um zu verstehen, was es heißt, Mathematik zu präsentieren, zu reflektieren und zu beurteilen, wird die Mathematik zunächst im Zusammenhang der Peirceschen Klassifikation der forschenden Wissenschaften charakterisiert. Durch die dreifache Sicht der universalen Kategorien von Peirce wird nahe gelegt, die Ausführungen zum Präsentieren am Selbstverständnis der Mathematik (als einem Ersten), zum Reflektieren am Bezug der Mathematik zur realen Welt (als einem Zweiten) und zum Beurteilen an Sinn, Bedeutung und Zusammenhang von Mathematik (als einem Dritten) zu orientieren. Diese Ausführungen stützen, wie abschließend erläutert wird, die These: Sinn und Bedeutung von Mathematik liegen letzlich darin, dass Mathematik die rationale Kommunikation von Menschen wirksam zu unterstützen vermag. |

2390 | 18.03.2005 | Glöckner, Helge | Discontinuous non-linear mappings on locally convex direct limits |
MSC: 46F05; 46T20; 46A13; 46M40 We show that the self-map $C^\infty_c(R)\to C^\infty_c(R)$, $\gamma \mapsto \gamma \circ \gamma – \gamma(0)$ of the space of real-valued test functions on the line is discontinuous, although its restriction to the space $C^\infty_K(R)$ of functions supported in $K$ is smooth (and hence continuous), for each compact subset $K$ of $R$. More generally, for each non-compact, finite-dimensional manifold of positive dimension and each locally convex space $E\not={0}$, we describe discontinuous mappings $C^\infty_c(M,E)\to C^\infty_c(M, R)$ whose restriction to $C^\infty_K(M,E)$ is smooth for each compact subset $K$ of $M$. The construction also applies to spaces of compactly supported smooth sections in vector bundles. |

2389 | 11.03.2005 |
Gwiazda Piotr Malek Josef Swierczewska Agnieszka |
On flows of an incompressible fluid with discontinuous power-law-like rheology |
MSC: 35Q35; 76A05 We consider the model for blood flow, which takes into account the platelets activation. Platelets are very sensitive to chemical and mechanical inputs, thus the viscosity of a material may change very rapidly. This phenomenon can be described with help of discontinuos Cauchy stress tensor. We will formulate the problem also in terms of maximal monotone operators. |

2388 | 11.03.2005 | Swierczewska Agnieszka | Dynamical approach to Large Eddy Simulation of turbulent flows. Existence and compactness. |
MSC: 76F65; 35Q35 We consider the system of equations coming from turbulence modelled by Large Eddy Simulation (LES) technique. The idea of this approach bases on decomposing the velocity into a part containing large flow structures and a part consisting of small scales. The equations for large scale quantities are derived from the Navier Stokes equations with an additional constitutive relation for a contribution of small eddies into the flow. The difficulties focus on the nonlinear and nonlocal |

2387 | 04.03.2005 | Glöckner, Helge | $Diff(R^n)$ as a Milnor-Lie group |
MSC: 58D05; 22E65; 46F05; 46T20 We describe a construction of the Lie group structure on the diffeomorphism group of $R^n$, modelled on the space of $R^n$-valued test functions on $R^n$, in John Milnor's setting of infinite-dimensional Lie groups. New tools are introduced to simplify this task. |

2386 | 21.02.2005 |
Clerc, Jean-Louis Neeb, Karl-Hermann |
Orbits of triples in the Shilov boundary of a bounded symmetric domain |
MSC: 32M15; 53D12; 22F30 Let ${\cal D}$ be a bounded symmetric domain of tube type, $S$ its Shilov boundary, and $G$ its group of biholomorphic automorphisms. We classify the orbits of the identity component $G$ of the group of biholomorphic maps of ${\cal D}$ in the set $S\times S\times S$. |

2385 | 14.02.2005 | Rabinovich, V. S., Roch, S. | The Fredholm property of pseudodifferential operators with non-smooth symbols on modulation spaces |
MSC: 47G30; 35S05; 47L80 The aim of the paper is to study the Fredholm property of pseudodifferential operators in the Sjöstrand class $OPS_w$ where we consider these operators as acting on the modulation spaces $M^{2, ¸ p} (\sR^N)$. These spaces are introduced by means of a time-frequency partition of unity. The symbol class $S_w$ does not involve any assumptions on the smoothness of its elements. In terms of their limit operators, we will derive necessary and sufficient conditions for operators in $OPS_w$ to be Fredholm. In particular, it will be shown that the Fredholm property and, thus, the essential spectra of operators in this class are independent of the modulation space parameter $p \in (1, ¸ \infty)$. |

2384 | 09.02.2005 | Wille, Rudolf | Formal Concept Analysis as Mathematical Theory of Concepts and Concept Hierarchies |
MSC: 03B; 06A Formal Concept Analysis has orginally been developed as a subfield of Applied Mathematics based on the mathematization of concept and concept hierarchy. Only after more than a decade of development, the connections to the philosophical logic of human thought became clearer and even later the connections to Piaget's cognitive structuralism which Thomas Bernhard Seiler convincingly elaborated to a comprehensive theory of concepts in his recent book [Se01]. It is the main concern of this paper to show the surprisingly rich correspondence between Seiler's multifarious aspects of concepts in the human mind and the structural properties and relationships of formal concepts in Formal Concept Analysis. These correspondences make understandable, what has been experienced in a great multitude of applications, that Formal Concept Analysis may function in the sense of transdisciplinary mathematics, i.e., it allows mathematical thought to aggregate with other ways of thinking and thereby to support human thought and action. |

2383 | 01.02.2005 |
Farwig, Reinhard Kozono, Hideo Sohr, Hermann |
An $L^q$--approach to Stokes and Navier-Stokes Equations in General Domains |
MSC: 76D05; 35Q30; 76D07 It is known by counter-examples that the usual $L^q$-approach to the Stokes equations, well known e.g. for bounded and exterior domains, cannot be extended to general domains $\Omega\seq R^3$ without any modification for $q\neq 2$. In the present paper we will show that important properties like Helmholtz decomposition, analyticity of the Stokes semigroup, and the maximal regularity estimate of the nonstationary Stokes equations remain valid for general domains even for $q \neq 2$ if we replace the space $L^q$ for $2 \leq q \leq \infty$ by the intersection $L^2 \cap L^q$ and for $1 < q < 2$ by the sum space $ L^2 + L^q$. As an application we prove the existence of a (suitable) weak solution $u$ of the Navier-Stokes equations with pressure term $\nabla p\in L_{loc}^{5/4}$, conjectured by Caffarelli-Kohn-Nirenberg and satisfying both the local and strong energy inequality. |

2382 | 01.02.2005 |
Farwig, Reinhard Kozono, Hideo Sohr, Hermann |
Very Weak Solutions of the Navier-Stokes Equations in Exterior Domains with Nonhomogeneous Data |
MSC: 76D05; 35J25; 35J65; 35K60 We investigate the nonstationary Navier-Stokes equations in an exterior domain $\Omega\subset \R^3$ in a solution class $L^s\big( 0,T;L^q(\Omega)\big)$ of very low regularity in space and time satisfying Serrin's condition $\frac{2}{s} + \frac{3}{q} = 1$ but not necessarily any differentiability property. The nonhomogeneous boundary data $u|_{\partial\Omega} = g\in L^s\big( 0,T;W^{-1/q,q}(\partial\Omega)\big)$ are the weakest possible beyond the notion of usual trace theorems; moreover, we prescribe a divergence $k=\div u \in L^s big((0,T;L^r(\Omega)\big)$, where $\frac{1}{3} + \frac{1}{q} = \frac{1}{r}$. |

2381 | 01.02.2005 | Neuenkirch, Andreas | Optimal approximation of SDE's with additive fractional noise |
MSC: 60H35; 60H07; 60H10; 65C30 We study pathwise approximation of scalar stochastic differential equations with additive fractional Brownian noise of Hurst parameter $H>1/2$, considering the mean square $L^{2}$-error criterion. By means of the Malliavin calculus we derive the exact rate of convergence of the Euler scheme, also for non-equidistant discretizations. Moreover, we establish a sharp lower error bound that holds for arbitrary methods, which use a fixed number of bounded linear functionals of the driving fractional Brownian motion. The Euler scheme based on a discretization, which reflects the local smoothness properties of the equation, matches this lower error bound up to the factor 1.39. |

2380 | 01.02.2005 | Neeb, Karl-Hermann | Derivations of locally simple Lie algebras |
MSC: 17B65; 7B20; 17B56 Let $\g$ be a locally finite Lie algebra over a field of characteristic zero which is a direct limit of finite-dimensional simple ones. In this note we study the Lie algebra of derivations of $\g$. The main results are that each invariant symmetric bilinear form on $\g$ is invariant under all derivations and that each such form defines a natural embedding $\der\g \into \g^*$. The latter embedding is used to determine $\der \g$ explicitly for all locally finite split simple Lie algebras. |

2379 | 18.01.2005 | Niese, Birgit | A Martingale Characterization of Pólya-Lundberg Processes |
MSC: 60Gxx We study exponential families within the class of counting processes and show that a mixed Poisson process belongs to an exponential family if and only if it is either a Poisson process or it has a Gamma structure distribution. This property can be expressed via exponential martingales. |

2378 (opens in new tab) | 15.01.2005 | Zahn, Peter | On the Use of Hypotheses in Cumulative Type Theory |
MSC: 03B15; 03B45 Given a language of ramified cumulative type theory as introduced in (Zahn 2004). We shall construct and investigate an extension, L, of it, which is a language of the same sort, but also contains sentences which express that certain sentences of L are deducible from others (hypotheses) by given rules. To this we introduce `names' of terms and formulas of L and include them in L. So in L we cannot only use, but also `speak about' sentences of that language. Especially, by means of first order sentences we can speak about higher order sentences. Despite of this possibility of `reduction' of order, all sentences of L are non-circular. The considered deducibility-relations of sentences from others correspond to systems of labelled modal logic of types K4 and G. |

2377 (opens in new tab) | 03.01.2005 | H.-D. Alber, P. Zhu |
Solutions to a Model with Nonuniformly Parabolic Terms for Phase Evolution Driven by Configurational Forces |
MSC: 74N20; 35Q72 We prove existence of solutions global in time to an initial-boundary value problem for a system of partial differential equations, which consists of the equations of linear elasticity and a nonlinear, non-uniformly parabolic equation of second order. This problem models the behavior of material phases, whose evolution in time is driven by configurational forces. The model is obtained by inserting a parabolic, regularizing term into an original model with hyperbolic character, which is derived in previous papers of the authors by transforming a well known sharp interface model for the evolution of a surface of strain discontinuity. Our existence proof, which contributes to the verification of the model, is only valid in one space dimension. |

Number | Date | Author | Title | Abstract/MSC |
---|---|---|---|---|

2376 | 21.12.2004 |
Haller-Dintelmann, Robert Hieber, Matthias |
$H^\infty$-calculus for products of non-commuting operators |
MSC: 47A60; 34G10; 47D06 It is shown that the product of two sectorial operators $A$ and $B$ admits a bounded $H^\infty$-calculus on a Banach space $X$ provided suitable commutator estimates and Kalton-Weis type assumptions on $A$ and $B$ are satisfied. |

2375 | 20.12.2004 |
Haller-Dintelmann, Robert Heck, Horst Hieber, Matthias |
$L^p$-$L^q$-estimates for parabolic systems in non-divergence form with VMO-coefficients |
MSC: 35J45; 35K55 Consider a parabolic $N \times N$-system of order $m$ on ${\mathbb{R}}^n$ with top-order coefficients $a_\alpha \in {\mathrm{VMO}} \cap L^\infty$. Let $1 < p,q < \infty$ and let $\omega$ be a Muckenhoupt weight. It ist proved that systems of this kind possess a unique solution $u$ satisfying \[ \| u' \|_{L^q(J; L^p_\omega({\mathbb{R}}^n)^N)} + \| {\mathcal{A}} u \|_{L^q(J; L^p_\omega({\mathbb{R}}^n)^N)} \le C \| f \|_{L^q(J; L^p_\omega({\mathbb{R}}^n)^N)}, \] where ${\mathcal{A}} u = \sum_{|\alpha| \le m} a_\alpha D^\alpha u$ and $J = [0, \infty)$. In particular, chosing $\omega = 1$, the realization of ${\mathcal{A}}$ in $L^p({\matbb{R}}^n)^N has maximal $L^p$-$L^q$-regularity. |

2374 | 02.12.2004 |
Cameron, Peter Knarr, Norbert |
Tubes in $PG(3,q)$ |
MSC: 51E20 A tube (resp. an oval tube) in $PG(3,q)$ is a pair ${\cal T} = {L, {\cal L}}$, where ${L} \cup {\cal L}$ is a collection of mutually disjoint lines of $PG(3,q)$ such that for each plane $\pi$ of $PG(3,q)$ containing $L$ the intersection of $\pi$ with the lines of ${\cal L}$ is a hyperoval (resp. an oval). The line $L$ is called the axis of ${\cal T}$. We show that every tube for $q$ even and every oval tube for $q$ odd can be naturally embedded into a regular spread and hence admits a group of automorphisms which fixes every element of ${\cal T}$ and acts regularly on each of them. For $q$ odd we obtain a classification of oval tubes up to projective equivalence. Furthermore, we characterize the reguli in $PG(3,q), q$ odd, as oval tubes which admit more than one axis. |

2373 | 01.11.2004 |
Neff, Patrizio Forest, Samuel |
A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. |
MSC: 74A30; 74G65 We investigate a geometrically exact generalized continua of micromorphic type in the sense of Eringen for the phenomenological description of metallic foams. The two-field problem for the macrodeformation $\varphi$ and the “affine microdeformation” $\overline{P}\in\GL^+(3,\R)$ in the quasistatic, conservative elastic case is investigated in a variational form. The elastic stress-strain relation is taken for simplicity as physically linear. Depending on material constants different mathematical existence theorems in Sobolev-spaces are given for the resulting nonlinear boundary value problems. These results extend existence results obtained by the first author for the micro-incompressible case $\overline{P}\in\SL(3,\R)$ and the micropolar case $\overline{P}\in\SO(3,\R)$. In order to mathematically treat external loads for large deformations a new condition, called bounded external work, has to be included, overcoming the conditional coercivity of the formulation. The observed possible lack of coercivity is related to fracture of the substructure of the metallic foam. We identify the relevant effective material parameters by comparison with the linear micromorphic model and its classical response for large scale samples. We corroborate the performance of the micromorphic model by presenting numerical calculations based on a linearized version of the finite-strain model and comparing the predictions to experimental results showing a marked size effect. |

2372 | 15.11.2004 | Ralf Gramlich | Defining amalgams of compact Lie groups | For $n \geq 2$ let $\Delta$ be a Dynkin diagram of rank $n$ and let $I = { 1, \ldots, n }$ be the set of labels of $\Delta$. A group $G$ admits a weak Phan system of type $\Delta$ over $\mathbb{C}$ if $G$ is generated by subgroups $U_i$, $i \in I$, which are central quotients of simply connected compact semisimple Lie groups of rank one, and contains subgroups $U_{i,j} = \langle U_i,U_j \rangle$, $i \neq j \in I$, which are central quotients of simply connected compact semisimple Lie groups of rank two such that $U_i$ and $U_j$ are rank one subgroups of $U_{i,j}$ corresponding to a choice of a maximal torus and a fundamental system of roots for $U_{i,j}$. It is shown in this article that $G$ then is a central quotient of the simply connected compact semisimple Lie group whose complexification is the simply connected complex semisimple Lie group of type $\Delta$. |

2371 | 01.11.2004 | Karl-Hermann Neeb | Non-abelian extensions of topological Lie algebras |
MSC: 17B65; 17B55; 17B40; 17B66 In this paper we extend and adapt several results on extensions of Lie algebras to topological Lie algebras over topological fields of characteristic zero. In particular we describe the set of equivalence classes of extensions of the Lie algebra ${\g}$ by the Lie algebra ${\n}$ as a disjoint union of affine spaces with translation group ${ H^2(\g,\z(\n))_{[S]}}$, where ${[S]}$ denotes the equivalence class of the continuous outer action ${ S \: \g \to \der \n}$. We also discuss topological crossed modules and explain how they are related to extensions of Lie algebras by showing that any continuous outer action gives rise to a crossed module whose obstruction class in ${H^3(\g,\z(\n))_S}$ is the characteristic class of the corresponding crossed module. The correspondence between crossed modules and extensions further leads to a description of ${\n}$-extensions of ${\g}$ in terms of certain ${\z(\n)}$-extensions of a Lie algebra which is an extension of ${\g}$ by ${\n/\z(\n)}$. We discuss several types of examples, describe applications to Lie algebras of vector fields on principal bundles, and in two appendices we describe the set of automorphisms and derivations of topological Lie algebra extensions. |

2370 | 01.10.2004 | Neff, Patrizio | On material constants for micromorphic continua. |
MSC: 74A35; 74A30; 74N15 I investigate a geometrically exact generalized isotropic continua of micromorphic type in the sense of Eringen. The two-field problem for the macrodeformation $\varphi$ and the “affine microdeformation” $\overline{P}\in\GL^+(3,\R)$ in the quasistatic, elastic case is presented in a variational form. The relative elastic stress-strain relation is taken for simplicity as physically linear. The corresponding infinitesimal strain problem obtained by linearization is also presented. Focus of attention is shifted to the interpretation of the appearing material constants. I derive simple homogenization-like formulas which relate the Lamé constants of the substructure and the classical Lamé constants obtained for arbitrarily large samples with the effective parameters in the micromorphic model. The relation of the thus obtained model to the intrinsically linear representation of Mindlin and Eringen is also established. The results should be useful for finite-element simulations of micromorphic continua. |

2369 | 25.10.2004 |
Gramlich, Ralf Van Maldeghem, Hendrik |
Intransitive geometries | In this paper we extend the existing covering theory for flag-transitive geometries to intransitive geometries. Furthermore, we apply our new theory to two intransitive geometries, the geometry of nondegenerate subspaces of a finite orthogonal space and a geometry for $G_2(3)$ useful for a construction of the Thompson sporadic simple group. |

2368 (opens in new tab) | 18.10.2004 | Rößler, Andreas | Rooted Tree Analysis for Order Conditions of Stochastic Runge-Kutta Methods for the Weak Approximation of Stochastic Differential Equations |
MSC: 65C30; 60H35; 65L05; 60H10; 34F05 A general class of stochastic Runge-Kutta methods for the weak approximation of Itô and Stratonovich stochastic differential equations with a multi-dimensional Wiener process is introduced. Colored rooted trees are used to derive an expansion of the solution process and of the approximation process calculated with the stochastic Runge-Kutta method. A theorem on general order conditions for the coefficients and the random variables of the stochastic Runge-Kutta method is proved by rooted tree analysis. This theorem can be applied for the derivation of stochastic Runge-Kutta methods converging with an arbitrarily high order. |

2367 | 11.10.2004 |
Geißert, Matthias Heck, Horst Hieber, Matthias |
$L^p$-Theory of the Navier-Stokes Flow in the Exterior of a Moving or Rotating Obstacle | Consider the equations of Navier-Stokes in the exterior of a rotating domain. It is shown that, after rewriting the problem on a fixed domain $\Omega$, the solution of the corresponding Stokes equation is governed by a $C_0$-semigroup on $L^p_\sigma(\Omega)$, $1<p<\infty$, with generator $Au = P(\Delta u +Mx \cdot \nabla u -Mu )$. Moreover, for $p \ge n$ and initial data $u_0 \in L^p_\sigma(\Omega)$, we prove the existence of a unique local mild solution to the Navier-Stokes problem. |

2366 | 07.10.2004 |
Neeb, Karl-Hermann Oersted, Bent |
A topological Maslov index for 3-graded Lie groups |
MSC: 22E65; 17C65; 17C30; 17C37 Motivated by the generalization of the Maslov index to tube domains and by numerous applications of related index function in infinite-dimensional situations, we describe in this paper a topologically oriented approach to an index function generalizing the Maslov index for bounded symmetric domains of tube type to a variety of infinite-dimensional situations containing in particular the class of all bounded symmetric domains of tube type in Banach spaces. The framework is that of 3-graded Banach--Lie groups and corresponding Jordan triple systems. |

2365 | 06.10.2004 |
Neff, Patrizio Chelminski, Krzysztof |
A geometrically exact Cosserat shell-model includingsize effects, avoiding degeneracy in the thin shell limit. Rigourous justification via $\Gamma$-convergence for the elastic plate. |
MSC: 74K20; 74K25; 74B20; 74D10; 74A35; 74E05; 74G65; 74N15; 74K35 We are concerned with the derivation of the $\Gamma$-limit to a three-dimensional geometrically exact Cosserat model as the relative thickness $h>0$ of a flat domain tends to zero. The Cosserat bulk model involves exact rotations as a second independent field. It is shown that the $\Gamma$-limit based on a natural scaling assumption consists of a membrane like energy contribution and a homogenized transverse shear energy both scaling with $h$, augmented by an additional curvature stiffness due to the underlying Cosserat bulk formulation, also scaling with $h$. No specific bending term appears in the dimensional homogenization process. The formulation exhibits an internal length scale $L_c$ which survives the homogenization process. A major technical difficulty, which we encounter in applying the $\Gamma$-convergence arguments, is to establish equi-coercivity of the sequence of functionals as the relative thickness $h$ tends to zero. Usually, equi-coercivity follows from a local coerciveness assumption. While the three-dimensional problem is well-posed for the Cosserat couple modulus $\mu_c\ge 0$, equi-coercivity forces us to assume a strictly positive Cosserat couple modulus $\mu_c>0$. The $\Gamma$-limit model determines the midsurface deformation $m\in H^{1,2}(\omega,\R^3)$. For the case of zero Cosserat couple modulus $\mu_c=0$ we obtain an estimate of the $\Gamma-\liminf$ and $\Gamma-\limsup$, without equi-coercivity which is then strenghtened to a $\Gamma$-limit result for zero Cosserat couple modulus. The classical linear Reissner-Mindlin model is “almost” the linearization of the $\Gamma$-limit for $\mu_c=0$ apart from a stabilizing shear energy term. |

2364 | 06.10.2004 | Neff, Patrizio | Local existence and uniqueness for a geometrically exact membrane-plate with viscoleastic transverse shear resistance. |
MSC: 74K15; 74K20; 74G65 We prove the local existence and uniqueness to a geometrically exact, observer-invariant membrane-plate model introduced by the author. The model consists of an elliptic partial differential system of equations describing the equilibrium response of the membrane which is nonlinearly coupled with a viscoelastic evolution equation for exact rotations, taking on the role of an orthonormal triad of directors. This coupling introduces a viscoelastic transverse shear resistance. Refined elliptic regularity results together with a new extended Korn's first inequality for plates and shells allow to proceed by a fixed point argument in appropriately chosen Sobolev-spaces in order to prove existence and uniqueness. |

2363 | 04.10.2004 |
Abels, Helmut Wiegner, Michael |
Resolvent Estimates for the Stokes Operator on an Infinite Layer |
MSC: 35Q30; 76D07 In this paper we prove unique solvability of the generalized Stokes resolvent equations in an infinite layer $\Omega=\R^{n-1}\times (-1,1)$, $n\geq 2$, in $L^q$-Sobolev spaces, $1<q<\infty$, with non-slip boundary condition $u|_{\partial\Omega}=0$. The unique solvability is proved for every $\lambda\in \C\setminus (-\infty,-\pi^2/4]$, where $-\frac{\pi^2}4$ is the least upper bound of the spectrum of Dirichlet realization of the Laplacian and the Stokes operator in $\Omega$. Moreover, we provide uniform estimates of the solutions for large spectral parameter $\lambda$ as well as $\lambda$ close to $-\frac{\pi^2}4$. Because of the special geometry of the domain, partial Fourier transformation is used to calculate the solution explicitly. Then Fourier multiplier theorems are used to estimate the solution operator. |

2362 | 30.09.2004 | Abels, Helmut | Bounded Imaginary Powers and $H_\infty$-Calculus of the Stokes Operator in Unbounded Domains |
MSC: 35Q30; 76D07; 47A60; 47F05 In the present contribution we study the Stokes operator $A_q=-P_q\Delta$ on $L^q_\sigma(\Omega)$, $1<q<\infty$, where $\Omega$ is a suitable bounded or unbounded domain in $\Rn$, $n\geq 2$, with $C^{1,1}$-boundary. We present some conditions on $\Omega$ and the related function space and basic equations which guarantee that $c+A_q$ for suitable $c\in\R$ is of positive type and admits a bounded $H_\infty$-calculus. This implies the existence of bounded imaginary powers of $c+A_q$. Most domains studied in the theory of Navier-Stokes like e.g. bounded, exterior, and aperture domains as well as asymptotically flat layers satisfy the conditions. The proof is done by constructing an approximate resolvent based on the results of [3], which were obtained by applying the calculus of pseudodifferential boundary value problems. Finally, the result is used to proof the existence of a bounded $H_\infty$-calculus of the Stokes operator $A_q$ on an aperture domain. |

2361 | 23.09.2004 | Froehlich, Steffen | Remarks on Nitsche's functional: The rotationally symmetric case |
MSC: 49Q10; 34L30; 53C42 We investigate existence and stability of rotationally symmetric critical immersions of variational problems of higher order which were considered by Nitsche. |

2360 | 01.09.2004 | Froehlich, Steffen | Ueber das Friedmannsche kosmologische Modell |
MSC: 83-01; 83C15; 83F05 Es wird uebersichtsartig das Friedmannsche kosmologische Modell in der Allgemeinen Relativitaetstheorie vorgestellt. |

2359 | 06.09.2004 | Neff, Patrizio | Local existence and uniqueness for quasistatic finite plasticity with grain boundary relaxation. |
MSC: 74A35; 74C05; 74C10; 74C20; 74D10 This paper is concerned with a phenomenological model of initially isotropic finite-strain multiplicative elasto-plasticity for polycrystals with grain boundary relaxation (Neff, Cont.Mech.Thermo.,2003). We prove a local in time existence and uniqueness result of the corresponding initial boundary value problem in the quasistatic rate-dependent case. Use is made of a generalized Korn's first inequality (Neff, Proc.Roy.Soc.Edinb.A,2002) taking into account the incompatibility of the plastic deformation $F_p$. This is the first rigorous result concerning classical solutions in geometrically exact nonlinear finite visco-plasticity for polycrystals. Global existence is not proved and cannot be expected due to the natural possibility of material degradation in time. |

2358 (opens in new tab) | 01.09.2004 | M. El-Kyal, D. Esselaoui, A. Machmoum and M. Seaïd | Characteristics method for the problem of viscoelastic fluid flow of PTT model | We formulate and analyze a characteristics finite element approximation of a class of flows in viscoelastic fluids described by the Phan-Thien-Tanner model. Compared to the classical Oldroyd model, the considered model presents further difficulties due to the presence of nonlinear terms of exponential type in the constitutive equation. In this paper, we propose a characteristics-based method to treat the transport part of the equations. The stress, velocity and pressure approximations are $P_1$ discontinuous, $P_2$ continuous and $P_1$ continuous finite element, respectively. By assuming that the continuous problem admits a sufficiently smooth and sufficiently small solution, and using a fixed point method, we show existence of solution to the approximate problem. We also give an error bound for the numerical solution. |

2357 | 06.09.2004 | Neff, Patrizio | A geometrically exact Cosserat shell-model including size effects, avoiding degeneracy in the thin shell limit. Existence of minimizers for zero Cosserat couple modulus. |
MSC: 74K20; 74K25; 74A35; 74B20 This paper establishes the existence of minimizers to a finite-strain, geometrically exact Cosserat plate model. The membrane energy of the investigated model is a quadratic, uniformly Legendre-Hadamard elliptic energy in contrast to classical approaches. The bending contribution is augmented by a curvature term representing an additional stiffness of the Cosserat theory and the corresponding nonlinear system of balance equations remains of second order. The lateral boundary conditions corresponding to simple support are non-standard. The model includes size effects, transverse shear resistance, drilling degrees of freedom and accounts implicitly for thickness extension and asymmetric shift of the midsurface. The formal thin shell “membrane” limit without classical $h^3$-bending term is non-degenerate due to the additional Cosserat curvature stiffness and control of drill rotations. In this formulation, the drill-rotations are strictly related to the size-effects of the Cosserat bulk model and not introduced artificially for numerical convenience. Upon linearization with zero Cosserat couple modulus $\mu_c=0$ exclusively, we recover the well known infinitesimal-displacement Reissner-Mindlin model without size-effects and without drill-rotations. It is shown that this new finite-strain Cosserat plate formulation is well-posed for $\mu_c=0$ by means of the direct methods of variations. The midsurface deformation $m$ is found in $H^1(\omega,\R^3)$. Decisive use is made of a dimensionally reduced version of an extended Korn's first inequality proved by the author. |

2338 | 01.09.2004 | Abels, Helmut | Bounded Imaginary Powers and $H_\infty$-Calculus of the Stokes Operator in Two-Dimensional Exterior Domains |
MSC: 35Q30; 76D07; 47A60; 47F05 The present contribution deals with the Stokes operator $A_q$ on $L^q_\sigma(\Omega)$, $1<q<\infty$, where $\Omega$ is an exterior domain in $\R^2$ of class $C^2$. It is proved that $A_q$ admits a bounded $H_\infty$-calculus.% for every $\delta\in (0,\pi)$. This implies the existence of bounded imaginary powers of $A_q$, which has several important applications. -- So far this property was only known for exterior domains in $\Rn$, $n\geq 3$. -- In particular, this shows that $A_q$ has maximal regularity on $L^q_\sigma(\Omega)$. For the proof the resolvent $(\lambda+A_q)^{-1}$ has to be analyzed for $|\lambda|\to\infty$ and $\lambda \to 0$. For large $\lambda$ this is done using an approximate resolvent based on the results of \cite{HInftyInLayer}, which were obtained by applying the calculus of pseudodifferential boundary value problems. For small $\lambda$ we analyze the representation of the resolvent developed in \cite{BorchersVarnhorn} by a potential theoretical method. |

2337 | 08.09.2004 | Neff, Patrizio | A geometrically exact viscoplastic membrane-shell with viscoelastic transverse shear resistance avoiding degeneracy in the thin-shell limit. Part I: The viscoelastic membrane-plate. |
MSC: 74K15; 74K20; 74G65 We reduce a viscoelastic finite-strain continuum model to a two-dimensional membrane-plate. The reduction is based on assumed kinematics, analytical integration through the thickness and physically motivated simplifications. The resulting formulation is observer-invariant and accounts for thickness stretch and finite rotations. The membrane energy is a quadratic, uniformly Legendre-Hadamard elliptic, first order energy in contrast to classical membrane models and the corresponding system of balance equations remains of second order. An evolution equation for some independent rotation is appended (already in the bulk-model) introducing viscoelastic transverse shear resistance. It can be shown that this reduced membrane formulation is locally well-posed. Use is made of a dimensionally reduced version of an extended Korn's first inequality. In the equilibrium relaxation limit an intrinsic membrane-plate formulation is obtained similar to the proposal of Fox/Simo, which is, however, non-elliptic. Nevertheless, the linearization of this last equilibrium model coincides with the classical linear membrane-plate model. In this sense, the new viscoelastic membrane-plate model regularizes the occurring loss of ellipticity in classical finite-strain membrane models. |

2356 (opens in new tab) | 01.08.2004 | Glöckner, Helge | Lie groups over non-discrete topological fields |
MSC: 22E65; 22E67; 58D05; 26E30; 26E15; 26E20; 46A16; 46S10; 58C20 We generalize the classical construction principles of infinite-dimensional real (and complex) Lie groups to the case of Lie groups over non-discrete topological fields. In particular, we discuss linear Lie groups, mapping groups, test function groups, diffeomorphism groups, and weak direct products of Lie groups. The specific tools of differential calculus required for the Lie group constructions are developed. Notably, we establish differentiability properties of composition and evaluation, as well as exponential laws for function spaces. We also present techniques to deal with the subtle differentiability and continuity properties of non-linear mappings between spaces of test functions. Most of the results are independent of any specific properties of the topological vector spaces involved; in particular, we can deal with real and complex Lie groups modeled on non-locally convex spaces. |

2355 | 29.07.2004 | Roch, Steffen | Finite sections of band-dominated operators |
MSC: 65J10; 46N40; 65F05; 65F15; 65F40 The goal of this paper is to review recent advances and to present new results in the numerical analysis of the finite sections method for general band and band-dominated operators. The main topics are the stability of the finite section method and the asymptotic behavior of singular values. The latter topic is closely related with compactness and Fredholm properties of approximation sequences, and the paper can also serve as an introduction into this remarkable field of numerical analysis. Further we discuss the behavior of approximation numbers, determinants, essential spectra and essential pseudospectra as well as the localization of pseudomodes of finite sections of band-dominated operators. |

2354 (opens in new tab) | 01.07.2004 | Rößler, Andreas | Runge-Kutta methods for Itô stochastic differential equations with scalar noise |
MSC: 65C30; 65L06; 60H35; 60H10 A class of explicit stochastic Runge-Kutta methods for non-autonomous Itô stochastic differential equation systems w.r.t.\ a one-dimensional Wiener process is developed. General conditions for the coefficients of stochastic Runge-Kutta schemes ensuring convergence with order two in the weak sense are derived. As a solution of these conditions, coefficients for new stochastic Runge-Kutta schemes with three stages are given explicitly. Some results of a simulation study reveal their good performance in comparison with some other known schemes. |

2353 | 07.07.2004 | Holzer, Richard | Knowledge acquisition under incomplete knowledge using methods from formal concept analysis Part II |
MSC: 68R99; 68P99 Formal contexts with unknown entries can be represented by three-valued contexts $K = (G, M, { \times, o, ? }, I)$, where a questionmark indicates that it is not known whether the object $g \in G$ has the attribute $m \in M$. To describe logical formulas between columns of such incomplete contexts the Kripke-semantics are used for propositional formulas over the set $M$ of attributes. Attribute implications are considered as special propositional formulas. If a context is too large to be fully represented, an interactive computer algorithm may help the user to get maximal information (with respect to his knowledge) about the valid attribute implications of the unknown context. This computer algorithm is called «attribute exploration». |

2352 | 07.07.2004 | Holzer, Richard | Knowledge acquisition under incomplete knowledge using methods from formal concept analysis Part I |
MSC: 68R99; 68P99 Formal contexts with unknown entries can be represented by three-valued contexts $K = (G, M, { \times, o, ? }, I)$, where a questionmark indicates that it is not known whether the object $g \in G$ has the attribute $m \in M$. To describe logical formulas between columns of such incomplete contexts the Kripke-semantics are used for propositional formulas over the set $M$ of attributes. Attribute implications are considered as special propositional formulas. If a context is too large to be fully represented, an interactive computer algorithm may help the user to get maximal information (with respect to his knowledge) about the valid attribute implications of the unknown context. This computer algorithm is called «attribute exploration». |

2351 | 24.06.2004 |
Sascha Karl Dörflein Rudolf Wille |
Coherence Networks of Concept Lattices: The Basic Theorem |
MSC: 06B05 For representing different views and their connections, networks of formal contexts are considered which are coded by so-called {\em multicontexts}. The coincidences between the network contexts of a multicontext give rise to a {\em coherence network of concept lattices}. It is the aim of this paper to state and to prove {\em Basic Theorem on Coherence Networks of Concept Lattices}as an extension of the Basic Theorem on Concept Lattices. |

2350 (opens in new tab) | 28.06.2004 |
Fügenschuh, Armin Herty, Michael Klar, Axel Martin, Alexander |
Combinatorial and Continuous Models for the Optimization of Traffic Flows on Networks | A hierachy of simplified models for traffic flow on networks is derived from continuous traffic flow models based on partial differential equations. The hierachy contains nonlinear and linear combinatorial models with and without dynamics. Optimization problems are treated for all models and numerical results and algorithms are compared. |

2349 | 21.06.2004 |
Farwig, R. Galdi, G.P. Sohr, H. |
Very weak solutions of stationary and instationary Navier-Stokes equations with nonhomogeneous data |
MSC: 35Q30; 76D05; 35J25; 35J65; 35K60 We investigate several aspects of {\it very weak solutions} $u$ to stationary and nonstationary Navier-Stokes equations in a bounded domain $\Om\subset\R^3$. This notion was introduced by Amann for the nonstationary case with nonhomogeneous boundary data $u|_{\partial\Omega}=g$ leading to a new and very large solution class in which uniqueness, but no regularity in space or energy estimate holds. Here we are mainly interested to investigate the »largest possible« class for the more general problem with arbitrary divergence $k = \div u$, boundary data $g = u|_{\partial\Omega}$ and an external force $f$, as weak as possible. In principle, we will follow Amann's approach. |

2348 (opens in new tab) | 15.06.2004 |
Fügenschuh, Armin Martin, Alexander |
A Multicriterial Approach for Optimizing Bus Schedules and School Starting Times |
MSC: 90C11; 90B90 A successful optimization of public mass transit in rural areas concentrates on the traffic caused by pupils on their ways to school, for they are the largest group of customers. Besides a change in the schedules of the buses and the starting times of the trips, also the school starting time may become an integrated part of the planning process. We discuss the legal framework for this optimization problem in German states and counties, and present a multi-objective mixed-integer programming formulation for the simultaneous rectification of school and trip starting times. For its solution, we develop a two-stage decomposition heuristic, and apply it to real-world data sets from three different counties. |

2347 | 15.06.2004 | Biller, Harald | Integral representations of positive functionals |
MSC: 46K10; 46K15; 36J25; 46A55 For positive linear functionals on complex commutative $^*$-algebras, we prove abstract Bochner and Plancherel Theorems without any hypothesis of non-degeneracy. A central positive functional on a $^*$-algebra is decomposed as the sum of a non-degenerate and a totally degenerate positive linear functional by relating the non-degenerate part to the natural trace of an associated Hilbert algebra. |

2346 | 15.06.2004 | Biller, Harald | Continuous inverse algebras with involution |
MSC: 46K05; 46H20; 46H40; 47L40; 46H30 A large part of the theory of Banach $^*$-algebras is developed and generalized to continuous inverse $^*$-algebras (ie complex locally convex unital $^*$-algebras with open unit group and continuous inversion) which are (Mackey) complete. If the involution is continuous, the closed unit ball with respect to the greatest C$^*$-semi-norm is the closed convex hull of the unitary elements. (This is originally due to Palmer.) For hermitian continuous inverse $^*$-algebras, we generalize characterizations due to Ra{\u\i}kov, Pták, and Palmer, we prove the Shirali--Ford Theorem, and we show that closed subalgebras are equispectrally embedded. |

2345 | 23.05.2004 |
Gramlich, Ralf Hoffman, Corneliu Nickel, Werner Shpectorov, Sergey |
A revision of Phan's theorem of type $D_n$ | The purpose of the present paper is to reprove and extend Phan's theorem characterizing the groups $Spin^\pm(2n,q)$. |

2344 (opens in new tab) | 01.05.2004 |
Fügenschuh, Armin Martin, Alexander Stöveken, Peter |
An mTSP-CTW Model for Optimising School Starting Times and Public Bus Services | MSC: 90C11; 90B90 |

2343 | 19.05.2004 | Abels, Helmut | Pseudodifferential Boundary Value Problems with Non-Smooth Coefficients |
MSC: 35S15; 35J55 In this contribution we establish a calculus of pseudodifferential boundary value problems with Hölder continuous coefficients. It is a generalization of the calculus of pseudodifferential boundary value problems introduced by Boutet de Monvel. We discuss their mapping properties in Bessel potential and certain Besov spaces. Although having non-smooth coefficients and the operator classes being not closed under composition, we will prove that the composition of Green operators $a_1(x,D_x)a_2(x,D_x)$ coincides with a Green operator $a(x,D_x)$ up to order $m_1+m_2-\theta$, where $\theta\in (0,\tau_2)$ is arbitrary, $a_j(x,\xi)$ is in $C^{\tau_j}(\Rn)$ w.r.t. $x$, and $m_j$ is the order of $a_j(x,D_x)$, $j=1,2$. Moreover, $a(x,D_x)$ is obtained by the asymptotic expansion formula of the smooth coefficient case leaving out all terms of order less than $m_1+m_2-\theta$. This result is used to construct a parametrix of a uniformly elliptic Green operator $a(x,D_x)$. |

2342 | 06.05.2004 | Otto,Martin | Elementary Proof of the van Benthem-Rosen Characterisation Theorem |
MSC: 03-01; 03C13; 03C07 This note presents an elementary proof of the well-known characterisation theorem that associates propositional modal logic with the bisimulation invariant fragment of first-order logic. The classical version of this theorem is due to Johann van Benthem, its finite model theory analogue to Eric Rosen. The present proof of the van Benthem/Rosen characterisation theorem is uniformly applicable in both the classical and in the finite model theory scenario. While it is broadly based on Rosen's proof, it reduces the technical input from classical logic and the model theory of modal logics strictly to the use of Ehrenfeucht-Fraïssé games (for first-order, and for the modal variant). Furthermore the proof is constructive and the model constructions and accompanying analysis of games in the expressive completeness argument yield an optimal bound on the modal nesting depth in terms of the first-order quantifier rank. |

2340 | 01.05.2004 |
Martin,Alexander Möller,Markus |
Cutting Planes for the Optimization of Gas Networks | This paper presents cutting planes which are useful or potentially useful for solving mixed integer programs that arise in the optimization of gas networks. We consider polyhedra that are defining essential parts of the model and give an polynomial algorithm for the calculation of the set of vertices of such polyhedra. So a separation algorithm for the convex hull of the polyhedra can be developed. |

2339 | 30.04.2004 |
Gramlich, Ralf Hofmann, Georg Neeb, Karl-Hermann |
Semi-edges, reflections and Coxeter groups | We study coverings of graphs and characterize Coxeter groups by their action via reflections on their Cayley graph. |

2336 | 29.04.2004 | Wille, Rudolf | Implicational Concept Graps |
MSC: 03B60; 03B70 This paper introduces {\em implicational concept graphs} as special existential concept graphs of power context families and shows how such implicational concept graphs give rise to a {\em mathematical semantics of implications}. The advantage of the offered mathematical semantics is that it opens the door to mathematical theory with its structural insights and patterns of argumentations. As a consequence, it could be proved that the {\implicational theory} of implicational concept graphs is equivalent (in the main) to the theory of attribute implications of formal contexts. This result could even be generalized to an analogue result for {\em clausal concept graphs}. |

2335 | 28.04.2004 |
Altmann, Kristina Gramlich, Ralf |
On the geometry on the nondegenerate subspaces of orthogonal space | We study the Phan-theoretic flipflop geometries related to the flip induced by a nondegenerate orthogonal form on a vector space over an arbitrary field of characteristic distinct from two. |

2334 | 01.04.2004 | Komech, Sergei A. | Geometric Interpretation of Entropy in the theory of Random Processes | Calculation of the entropy for symbolic sdynamical systems is considered basing on a geometric approach. An analytical expression for the measure-theoretical entropy is obtained depending on measures of simple geometrical objects like balls and their shift-transformations. The convergence of the estimator is proved. |

2333 | 22.04.2004 | Biller, Harald | Holomorphic generation of continuous inverse algebras |
MSC: 46H30; 32A38; 32E30; 41A20; 46A20 We study complex commutative Banach algebras, and more generally continuous inverse algebras, in which the holomorphic functions of a fixed $n$-tuple of elements are dense. In particular, we characterize the compact subsets of $\mathbb{C}^n$ which appear as joint spectra of such $n$-tuples. The characterization is compared to several established notions of holomorphic convexity by means of approximation conditions. |

2332 | 22.04.2004 | Biller, Harald | Analyticity and naturality of the multi-variable functional calculus |
MSC: 46H30; 32A38; 41A20; 46G20; 58B12 Mackey-complete complex commutative continuous inverse algebras generalize complex commutative Banach algebras. After constructing the Gelfand transform for these algebras, we develop the functional calculus for holomorphic functions on neighbourhoods of the joint spectra of finitely many elements and for holomorphic functions on neighbourhoods of the Gelfand spectrum. To this end, we study the algebra of holomorphic germs in weak$^*$-compact subsets of the dual. We emphasize the simultaneous analyticity of the functional calculus in both the function and its arguments and its naturality. Finally, we treat systems of analytic equations in these algebras. |

2331 | 22.04.2004 | Biller, Harald | Algebras of complex analytic germs |
MSC: 22E65; 46G20; 46J40 Let $X$ be a metrizable complex analytic manifold modelled on a locally convex space $E$, and let $K \subseteq X$ be compact. Let $A$ be a normed unital algebra over $\C$. Let $\mathcal{O}(K,A)$ be the algebra of germs of complex analytic $A$-valued functions in $K$, topologized as the locally convex direct limit of the normed algebras of bounded complex analytic $A$-valued functions on open neighbourhoods of $K$ in $X$. Then $\mathcal{O}(K, A)$ is a locally $m$-convex Hausdorff algebra. If the unit group of $A$ is open then so is the unit group of $\mathcal{O}(K,A)$. If $A$ has finite dimension then $\mathcal{O}(K, A)$ is complete. |

2330 (opens in new tab) | 20.04.2004 | Glöckner, Helge | Conveniently Hölder Homomorphisms are Smooth in the Convenient Sense |
MSC: 22E65; 25E15; 26E20; 46T20; 58C20 We show that every «conveniently Hölder» homomorphism between Lie groups in the sense of convenient differential calculus is smooth (in the convenient sense). In particular, every $Lip^0$-homomorphism is smooth. |

2329 | 14.04.2004 |
Alber, Hans-Dieter Zhu Peicheng |
Evolution of phase boundaries by configurational forces |
MSC: 74N20; 35Q75 In this article an initial-boundary value problem modeling the evolution of a surface of strain discontinuity driven by configurational forces is studied. Starting from a sharp interface model the problem is transformed into a problem with an evolution equation for the order parameter, which has similarities with a hyperbolic balance law. It is proved that in one space dimension global solutions exist. The method of transformation suggests that solutions of this evolution equation are approximated by solutions of a viscous Hamilton-Jacobi equation. If the approximation is valid then the initial-boundary value problem to this Hamilton-Jacobi eqution is a phase field model regularizing the sharp interface model. |

2325 | 04.11.2010 | Farwig, Reinhard | An $L^q$-analysis of viscous fluid flow past a rotating obstacle |
MSC: 35C15; 35Q35; 76D05; 76D99; 76U05 Consider the problem of time-periodic strong solutions of the Stokes and Navier-Stokes system modelling viscous incompressible fluid flow past or around a rotating obstacle in $R^3$. Introducing a rotating coordinate system attached to the body a linearization yields a system of partial differential equations of second order involving an angular derivative not subordinate to the Laplacian. In this paper we find an explicit solution for the linear whole space problem when the axis of rotation is parallel to the velocity of the fluid at infinity. For the analysis of this solution in $L^q$-spaces, 1 < q < ∞, we will use tools from harmonic analysis and a special maximal operator reflecting paths of fluid particles past or around the obstacle. |

2328 (opens in new tab) | 01.03.2004 | Kindler, Jürgen | Helly and Klee type intersection theorems for finitary connected paved spaces | In the present paper, the concept of $n-$ary and finitary connectedness is introduced, where $1-$ary connectedness coincides with the usual notion of (abstract) connectedness. Relationships between ($n$ary) connectedness and an abstract concept of separation are studied. As applications, the classical intersection theorems of Helly, Klee, and others are obtained from the previous results by showing that the paving of closed convex resp. open convex subsets of a topological vector space are finitary connected. Based on a general minimax theorem, an abstract separation theorem is proved, generalizing the classical separation theorem for convex compact subsets of a locally compact topological vector space. This theorem and other results on abstract separation can be used to derive fairly general results on finitary connectedness which can be applied to various types of (convex) topological spaces. |

2327 (opens in new tab) | 01.03.2004 | Glöckner, Helge | Hölder continuous homomorphisms between infinite-dimensional Lie groups are smooth |
MSC: 22E65; 26E15; 26E20; 26E30; 46T20; 58C20 It is an open problem whether every continuous homomorphism between infinite-dimensional Lie groups is smooth. In this article, we show that every Hölder continuous homomorphism between infinite-dimensional Lie groups is smooth. |

2326 | 01.03.2004 | Roch, Steffen | Band-dominated operators on $l^p$-spaces: Fredholm indices and finite sections |
MSC: 47A53; 65J10 We derive an index formula for band-dominated operators on $l^p(\sZ)$ when $1 < p < \infty$ in terms of local indices of their limit operators. This formula is applied to verify the stability of the finite section method for invertible band-dominated operators with slowly oscillating coefficients. Hilbert space versions of these results (partially under further restrictions) were obtained in \cite{RRR1} and \cite{LRR1}, respectively. |

2324 (opens in new tab) | 04.03.2004 | Glöckner, Helge | Fundamentals of direct limit Lie theory |
MSC: 22E65; 46T05; 57N40; 58B10; 58B25 We show that every countable direct system of finite-dimensional real or complex Lie groups has a direct limit in the category of Lie groups modelled on locally convex spaces. This enables us to push all basic constructions of finite-dimensional Lie theory to the case of direct limit groups. In particular, we obtain an analogue of Lie's third theorem: Every countable-dimensional locally finite real or complex Lie algebra is enlargible, i.e., it is the Lie algebra of some regular Lie group (a suitable direct limit group). |

2323 | 01.03.2004 |
Bergner, Matthias Froehlich, Steffen |
On two-dimensional immersions of prescribed mean curvature in $\mathbb R^n$ |
MSC: 53C42; 35J60; 53A05 We consider two-dimensional immersions of disc-type in $\mathbb R^n.$ We focus well known classical concepts and study the nonlinear elliptic systems of such mappings. Using an Osserman-type condition we give a priori-estimates of the principle curvatures for certain graphs in $\mathbb R^4$ with prescribed mean curvature. |

2322 | 23.02.2004 | Steffen Froehlich | Steffen Froehlich |
MSC: 49-01; 34A34; 53A05 Das klassische Variationsproblem fuer das Katenoid wird auf weitere Klassen parametrischer Funktionale erweitert. Insbesondere betrachten wir Variationsprobleme mit Volumennebenbedingungen, kristalline Funktionale und Variationsprobleme hoeherer Ordnung. |

2321 | 27.02.2004 |
Kumar, Shrawan Neeb, Karl-Hermann |
Extensions of Algebraic Groups |
MSC: 20G12; 20G10 Let $G$ be a connected complex algebraic group and $A$ a connected abelian algebraic group endowed with an algebraic action of $G$ by group automorphisms. In the present note we describe the abelian group $\Ext_{alg}(G,A)$ of algebraic group extensions of $G$ by $A$ in terms of a short exact sequence relating it to a relative second Lie algebra cohomology space and the fundamental group of the commutator group. Our second main result is an analog of the Van-Est Theorem for algebraic group cohomology, saying that for an algebraic $G$ module $\a$ and $p \geq 0$ the algebraic group cohomology $H^p_{alg}(G,\a)$ is given by the relative cohomology of its Lie algebra with respect to the Lie algebra of a maximal reductive subgroup. |

2320 | 18.01.2004 | Neeb, Karl-Hermann | Abelian extensions of infinite-dimensional Lie groups |
MSC: 22E65; 57T10; 58B25 In the present paper we study abelian extensions of connected Lie groups $G$ modeled on locally convex spaces by smooth $G$-modules $A$. We parametrize the extension classes by a suitable cohomology group $H^2_s(G,A)$ defined by locally smooth cochains and construct an exact sequence that describes the difference between $H^2_s(G,A)$ and the corresponding continuous Lie algebra cohomology space $H^2_c(\g,\a)$. The obstructions for the integrability of a Lie algebra extensions to a Lie group extension are described in terms of period and flux homomorphisms. We also characterize the extensions with global smooth sections resp. those given by global smooth cocycles. Finally we apply the general theory to extensions of several types of diffeomorphism groups. |

2318 | 10.02.2004 | Neff, Patrizio | A geometrically exact micromorphic elastic solid. Modellling and existence of minimizers. |
MSC: 74A35; 74B20; 74G65 We investigate geometrically exact generalized continua of micromorphic type in the sense of Eringen. The two-field problem for the macrodeformation $\varphi$ and the microdeformation $\overline{P}\in\SL(3,\R)$ in the quasistatic, conservative case is investigated in a variational form. Depending on material constants different mathematical existence theorems in Sobolev-spaces are given for the resulting nonlinear boundary value problems including as a special case an existence theorem for a geometrically exact Cosserat micropolar model. These are the first results known to the author for geometrically exact microcontinuum formulations. In order to treat external loads a new condition, called bounded external work, has to be included, overcoming the conditional coercivity of the formulation. The mathematical analysis heavily uses an extended Korn's first inequality (Neff, Proc.Roy.Soc.Edinb.A, 2002) discovered by the author recently. The methods of choice are the direct methods of the calculus of variations. |

2317 (opens in new tab) | 01.01.2004 | Dau, Frithjof | Query Graphs with Cuts: Mathematical Foundations |
MSC: 68T27; 68T30 Query graphs with cuts are inspired by Sowa's conceptual graphs, which are in turn based on Peirce's existential graphs. In my thesis `The Logic System of Concept Graphs with Negations', conceptual graphs are elaborated mathematically, and the cuts of existential graphs are added to them. This yields the system of concept graphs with cuts. These graphs correspond to the closed formulas of first order predicate logic. Particularly, concept graphs are propositions which are evaluated to truth-values. In this paper, concept graphs are extended to so-called query graphs, which are evaluated to relations instead. As the truth-values TRUE and FALSE can be understood as the two 0-ary relations, query graphs extend the expres- siveness of concept graphs. Query graphs can be used to elaborate the logic of relations. In this sense, they bridge the gap between concept graphs and the Peircean Algebraic Logic, as it is described in Burch's book 'A Peircean Reduction Thesis'. But in this paper, we focus on deduction procedures on query graphs, instead of operations on relations, which is the focus in PAL. Particularly, it is investigated how the adequate calculus of concept graphs can be transferred to query graphs. |

2316 (opens in new tab) | 01.01.2004 | Dau, Frithjof | Background Knowledge in Concept Graphs. |
MSC: 68T27; 68T30 Traditional logic can be understood as the investigation of the three main essential functions of thinking concepts, judgements and conclusions. In the last years, in a new research field termed Contextual Logic, a mathematical theory of this logic is elaborated. Concepts have already been mathematically elaborated by Formal Concept Analysis. Judgements and Conclusions can be expressed by so-called Concept Graphs, which are built upon families of formal contexts. There are two approaches to concept graphs: A semantical approach, which investigates the theory of concept graphs in an algebraic manner, and a logical approach, which focuses on derivation rules for concept graphs, relying on a separation between syntax and semantics. In [26], Wille introduced two forms of complex implications (object implications and concept implications) to the semantical approach. In this paper it is investigated how these implications can be incorporated into the logical approach. |

2315 (opens in new tab) | 12.02.2004 |
Hofmann, Karl H. Morris, Sidney A. |
The Structure of Abelian Pro-Lie Groups |
MSC: 22B; 22 E A pro-Lie group is a projective limit of a projective system of finite dimensional Lie groups. A prodiscrete group is a complete abelian topological group in which the open normal subgroups form a basis of the filter of identity neighborhoods. It is shown here that an abelian pro-Lie group is a product of (in general infinitely many) copies of the additive topological group of reals and of an abelian pro-Lie group of a special type; this last factor has a compact connected component, and a characteristic closed subgroup which is a union of all compact subgroups; the factor group modulo this subgroup is pro-discrete and free of nonsingleton compact subgroups. Accordingly, a connected abelian pro-Lie group is a product of a family of copies of the reals and a compact connected abelian group. A topological group is called compactly generated if it is algebraically generated by a compact subset, and a group is called almost connected if the factor group modulo its identity component is compact. It is further shown that a compactly generated abelian pro-Lie group has a characteristic almost connected locally compact subgroup which is a product of a finite number of copies of the reals and a compact abelian group such that the factor group modulo this characteristic subgroup is a compactly generated prodiscrete group without nontrivial compact subgroups. |

2314 | 02.02.2004 | Swierczewska, Agnieszka | Large Eddy Simulation. Existence of Stationary Solutions to a Dynamical Model | We consider the existence of stationary solutions to the Germano Model – equations describing turbulent flow of fluids. The model comes from Large Eddy Simulation techniques yielding modified Navier-Stokes Equations with an additional nonlocal term. On one hand this nonlocalness disturbs monotonicity, but on the other hand it is helpful for compactness arguments. Thus we combine the methods of monotone operators and smoothing properties of convolutions in passing to the limit in the approximate problem. |

2313 (opens in new tab) | 01.01.2004 |
Banda, Mapundi K. Seaid, Mohammed |
Higher-Order Relaxation Schemes for Hyperbolic Systems of Conservation Laws | The accuracy and efficiency of several lower and higher order time integration schemes in Eulerian-Lagrangian computations are investigated for solution of advection diffusion problems with nonlinear reaction terms. The implementation of these schemes differ from their Eulerian counterparts in the fact that they are applied during each time step, along the characteristic curves rather than in the time direction. The major focus is to examine the computational characteristics of a class of implicit, explicit, and implicit-explicit time marching methods combined to Eulerian-Lagrangian procedure. The obtained results for several benchmark problems are considered to be representative, and might be helpful for a fair rating of solution schemes, particularly in long time computations. |

2312 (opens in new tab) | 01.01.2004 | Mohammed Seaid | Time Integration Schemes in Eulerian-Lagrangian Computations for Advection-Diffusion Reaction Problems | The accuracy and efficiency of several lower and higher order time integration schemes in Eulerian-Lagrangian computations are investigated for solution of advection diffusion problems with nonlinear reaction terms. The implementation of these schemes differ from their Eulerian counterparts in the fact that they are applied during each time step, along the characteristic curves rather than in the time direction. The major focus is to examine the computational characteristics of a class of implicit, explicit, and implicit-explicit time marching methods combined to Eulerian-Lagrangian procedure. The obtained results for several benchmark problems are considered to be representative, and might be helpful for a fair rating of solution schemes, particularly in long time computations. |

2311 | 16.01.2004 |
Bertram, Wolfgang Neeb, Karl-Hermann |
Projective completions of Jordan pairs Part II. Manifold structures and symmetric spaces |
MSC: 17C36; 46H70; 17C30; 17C65 We define {\it symmetric spaces} in arbitrary dimension and over arbitrary non-discrete topological fields $\K$, and we construct manifolds and symmetric spaces associated to topological {\it continuous quasi-inverse Jordan pairs} and {\it -triple systems}. This class of spaces, called {\it smooth generalized projective geometries}, generalizes the well-known (finite or infinite-dimensional) bounded symmetric domains as well as their «compact-like» duals. An interpretation of such geometries as models of Quantum Mechanics is proposed, and particular attention is paid to geometries that might be considered as «standard models" -- they are associated to {\it associative continuous inverse algebras} and to {\it Jordan algebras of hermitian elements} in such an algebra. |

Number | Date | Author | Title | Abstract/MSC |
---|---|---|---|---|

2310 | 05.12.2003 | Wille, Rudolf | Preconcept Algebras and Generalized Double Boolean Algebras |
MSC: 03B07; 06E25 {\it Boolean Concept Logic} as an integrated generalization of Contextual Object Logic and Contextual Attribute Logic can be substantially developed on the basis of preconcept algebras. The main results reported in this paper are the {\it Basic Theorem on Preconcept Algebras} and the theorem characterizing the equational class generated by all preconcept algebras by the equational axioms of the {\it generalized double Boolean algebras}. |

2309 | 05.12.2003 |
Wille, Rudolf Björn Vormbrock |
Semiconcept and Protoconcept Algebras: The Basic Theorems |
MSC: 03B07; 06E25 The concern of this paper is to elaborate a basic understanding of semiconcepts and protoconcepts as notions of Formal Concept Analysis. First, semiconcepts and protoconcepts are motivated by their use for effectively describing formal concepts. It is shown that one can naturally operate with those description units, namely with operations which constitute semiconcept and protoconcept algebras as so-called {\it double Boolean algebras}. The main results of this paper are the two basic theorems which characterize {\it semiconcept res. protoconcept algebras} as pure resp. fully contextual double Boolean algebras whose related Boolean algebras are complete and atomic. Those theorems may, for instance, be applied to check whether line diagram representations of semiconcept and protoconcept algebras are correct. |

2308 | 25.11.2003 | Puzarenko, Vadim G. | On computability on $\mathcal I$-minimal models |
MSC: 03D60; 03D20; 03D30 A description of the computable principles on $\mathcal I$-minimal admissible sets is given. It is shown that the reduction and total extension properties do not hold, and that the properties of separation and existence of a universal function are preserved from ideals. |

2307 (opens in new tab) | 05.12.2003 | Glöckner, Helge | Every smooth $p$-adic Lie group admits a compatible analytic structure |
MSC: 22E20; 22E65; 22D05; 22E35 We show that every finite-dimensional $p$-adic Lie group of class $C^k$ (where $k$ is a positive integer or $k=\infty$) admits a $C^k$-compatible analytic Lie group structure. We also construct an exponential map for every $k+1$ times strictly differentiable ultrametric $p$-adic Banach-Lie group, which is a diffeomorphism and admits Taylor expansions of all finite orders $\leq k$. |

2306 | 26.11.2003 |
Rabinovich, Vladimir S. Roch, Steffen |
Pseudodifference operators on weighted spaces, and applications to discrete Schrödinger operators |
MSC: 35S05; 47G30; 47A53 We study pseudodifference operators on ${\mathbb Z}^N$ with symbols which are bounded on ${\mathbb Z}^N \times {\mathbb T}^N$ together with their derivatives with respect to the second variable. In the same way as partial differential operators on ${\mathbb R}^N$ are included in an algebra of pseudodifferential operators, difference operators on ${\mathbb Z}^N$ are included in an algebra of pseudodifference operators. Particular attention is paid to the Fredholm properties of pseudodifference operators on general exponentially weighted spaces $l_w^p ({\mathbb Z}^N)$ and to Phragmen-Lindelöf type theorems on the exponential decay at infinity of solutions to pseudodifference equations. The results are applied to describe the essential spectrum of discrete Schrödinger operators and the decay of their eigenfunctions at infinity. |

2305 | 01.11.2003 | Ralf Gramlich | Simple connectedness of the geometry of nondegenerate subspaces of a symplectic space over arbitrary fields | K.M. Das has shown that the geometry of nondegenerate subspaces of a symplectic space over a finite field is simply connected. The purpose of the present article is to provide short, direct and general proof of that result for arbitrary fields. |

2304 | 10.11.2003 |
Fischer, Tom Roehrl, Armin |
Risk and performance optimization for portfolios of bonds and stocks |
MSC:. 91B28; 91B70; 90C06; 90C60 We explain how to optimize portfolios of bonds and stocks with respect to the Expected Shortfall (ES), respectively RORC or RORAC based on ES. In a pragmatic approach we combine and correlate a stock market model with geometric brownian motions with a two-factor Cox-Ingersoll-Ross (CIR-2) model for the interest rates/bonds. We use recent results from the theory of risk capital allocation, performance measurement and Swarm Intelligence for optimization. Examples for German market data as well as an analysis of the scalability of the solution to assure fast run-times on clusters of computers for large real-life portfolios are given. |

2303 | 27.10.2003 | Rößler, Andreas | Stochastic Taylor expansions for the expectation of functionals of diffusion processes |
MSC: 60H10; 65C30; 60J60; 41A58 Stochastic Taylor expansions of the expectation of functionals applied to diffusion processes which are solution of a stochastic differential equation system are introduced. Taylor formulas w.r.t.\ increments of the time are presented for both, Itô and Stratonovich stochastic differential equation systems with multi-dimensional Wiener processes. Due to the very complex formulas arising for higher order expansions, an advantageous graphical representation by colored trees is developed. The convergence of truncated formulas is analyzed and estimates for the truncation error are calculated. |

2302 | 22.10.2003 |
Rabinovich, V. S. Roch, S. |
Wiener algebras of operators, and applications to pseudodifferential operators |
MSC: 35S05; 35P05; 47G30 We introduce a Wiener algebra of operators on $L^2(\sR^N)$ which contains, for example, all pseudodifferential operators in the Hörmander class $OPS^0_{0,0}$. A discretization based on the action of the discrete Heisenberg group associates to each operator in this algebra a band-dominated operator in a Wiener algebra of operators on $l^2(\sZ^{2N}, ¸ L^2(\sR^N))$. The (generalized) Fredholmness of these discretized operators can be expressed by the invertibility of their limit operators. This implies a criterion for the Fredholmness on $L^2(\sR^N)$ of pseudodifferential operators in $OPS^0_{0,0}$ in terms of their limit operators. Applications to Schrödinger operators with continuous potential and other partial differential operators are given. |

2301 | 16.10.2003 | Neff, Patrizio | A geometrically exact derived Cosserat-plate including size effects, avoiding degeneracy in the thin plate limit. Modelling and mathematical analysis |
MSC: 74K20; 74K25; 74K35; 74G65; 74B20 This contribution is concerned with the consistent dimensional reduction of a previously introduced finite three-dimensional Cosserat micropolar elasticity model to the two-dimensional situation of thin plates and shells. The resulting membrane energy turns out to be a quadratic, elliptic, first order, non degenerate energy in contrast to classical approaches, the standard bending contribution is augmented with a term representing an additional stiffness of the Cosserat model and the corresponding system of balance equations remains of second order. The model includes size effects, transverse shear resistance, thickness stretch and drilling degrees of freedom. The thin shell limit is non-degenerate due to the additional Cosserat bending stiffness. It is shown that the dimensionally reduced formulation is well-posed along the same line of argument which showed the well posedness of the three-dimensional model. Decisive use is made of a dimensionally reduced version of an extended Korn's first inequality recently proved by the author. |

2300 | 01.10.2003 |
A. Lew P. Neff D. Sulsky M. Ortiz |
Optimal BV estimates for a discontinuous Galerkin method in linear elasticity |
MSC: 65N12; 65N15; 65N30 We analyze a discontinuous Galerkin method for linear elasticity. The discrete formulation derives from the Hellinger-Reissner variational principle with the addition of stabilization terms analoguous to those previously considered by others for the Navier-Stokes equations and a scalar Poisson equation. For our formulation we first obtain convergence in a mesh-dependent norm and in the natural mesh-independent BD norm. We then prove a generalization of Korn's second inequality w |

2299 (opens in new tab) | 01.09.2003 | Mohammed Seaid | Multidimensional hyperpolic conservation laws by relaxation approximations | We construct and implement a new nonoscillatory relaxation scheme for multidimensional hyperbolic systems of conservation laws. The method transforms the nonlinear hyperbolic system to a semilinear model with a relaxation source term and linear characteristics which can be solved numerically without using either Riemann solver or linear iterations. To discretize the relaxation system we consider a high-resolution reconstruction in space and a TVD Runge-Kutta time integration. Detailed formulation of the scheme is given for problems in three space dimensions and numerical experiments are implemented in both scalar and system cases to show the effectiveness of the method. |

2298 (opens in new tab) | 01.09.2003 |
Dau, Frithjof Hereth Correia, Joachim |
Nested Concept Graphs: Applications for Databases and Mathematical Foundations |
MSC: 68T; 27; 30 While the basic idea of using Conceptual Graphs as query interface to relational databases has already been stated very early in \cite{So84}, no approach so far has covered the full expressiveness of modern database query languages. Especially negation and the so-called aggregating functions have not been treated. In this paper, we present \emph{Nested Concept Graphs with Cuts} which extend the syntactical Concept Graph (which mathematize Conceptual Graphs) to treat simultaneously negation and nesting. With these extensions they have the expressiveness of database query languages (e.¸g. SQL), which will be exemplified by selected queries. |

2297 | 05.09.2003 | Neff, Patrizio | Finite multiplicative elastic-plastic Cosserat micropolar theory for polycrystals with grain rotations including fracture. Modelling and mathematical analysis. |
MSC: 74A35; 74C20; 74D10; 74E15; 74G65; 74N15 We investigate geometrically exact generalized continua of Cosserat micropolar type. The variational form of these models is introduced and consistently extended to cover finite elasto-plasticity based on the multiplicative decomposition of the deformation gradient only. The decisive stress is the Eshelby energy momentum tensor. It is motivated that the traditional Cosserat couple modulus $\mu_c$ can and should be set to zero for macroscopic specimens liable to fracture in shear, still leading to a complete consistent Cosserat theory with independent rotations in the geometrically exact finite case in contrast to the infinitesimal, linearized model. Depending on material constants different mathematical existence theorems in Sobolev-spaces are given for the resulting nonlinear boundary value problems in the elastic case. These are the first such results known to the author. Various assumptions on the magnitude of deformations and microrotations lead to simplified models which are all analysed mathematically. Partial focus is set to the possible regularization properties of micropolar models compared to classical continuum models in the macroscopic case of materials failing in shear. The mathematical analysis heavily uses an extended Korn's first inequality (Neff, Proc.Roy.Soc.Edinb.A, 2002) discovered by the author recently. The methods of choice are the direct methods of the calculus of variations. |

2296 | 26.08.2003 | Pompe, Waldemar | Korn's First Inequality with Variable Coefficients and its generalization |
MSC: 35F15; 35J55 \def\Om{\Omega} \def\Ga{\Gamma} \def\pa{\partial} \def\grad{\nabla} \def\gr{\nabla} \newfont{\zupka}{msbm9 scaled\magstep1} \def\r{\mbox{\zupka \char'122}} \def\nwp#1#2{|\hspace*{-1.3pt}|#1|\hspace*{-1.3pt}|_{W^{1,p}{(#2)}}} \def\H{{\cal H}} If $\Om\subset\r^n$ is a bounded domain with Lipschitz boundary $\pa\Om$ and $\Gamma$ is an open subset of $\pa\Om$, we prove that the following inequality $$\biggl(\int_\Om|A(x)\gr u(x)|^p¸dx\biggr)^{1/p}+ \biggl(\int_\Gamma|u(x)|^p¸d\H^{n-1}(x)\biggr)^{1/p}\geq c¸\nwp{u}{\Om}$$ holds for all $u\in W^{1,p}(\Om;\r^m)$ and $1<p<\infty$, where $$(A(x)\gr u(x))_k=\sum_{i=1}^m\sum_{j=1}^n¸ a_k^{ij}(x)¸\frac{\pa u_i}{\pa x_j}(x) \quad(k=1,2,\ldots,r;\ r\geq m)$$ defines an elliptic differential operator of first order with continuous continuous coefficients on $\overline\Om$. As a special case we obtain $$\int_{\Om}\bigl|\grad u(x)F(x)+(\grad u(x)F(x))^T\bigr|^p¸dx\geq c\int_{\Om}|\grad u(x)|^p¸dx¸,\leqno(*)$$ for all $u\in W^{1,p}(\Om;\r^n)$ vanishing on $\Ga$, where $F¸\colon¸\overline{\Om}\to M^{n\times n}(\r)$ is a continuous mapping with $\det F(x)\geq \mu>0$. Next we show that $(*)$ is not valid if $n\geq3$, $F\in L^\infty(\Om)$ and $\det F(x)=1$, but does hold if $p=2$, $\Gamma=\pa$ and $F(x)$ is symmetric and positive definite in $\Om$. |

2295 | 21.08.2003 | Abels, Helmut | The Initial Value Problem for the Navier-Stokes Equations with a Free Surface in $L^q$-Sobolev Spaces |
MSC: 35Q30; 76D07; 35R35 We prove small-time existence of strong solutions of a free boundary value problem, which describes the motion of an incompressible viscous fluid occupying a semi-infinite domain bounded above by a free surface. This problem was studied by Beale [6] and others in $L^2$-Sobolev spaces. In contrast to the latter contribution we study solutions in $L^q$-Sobolev spaces for $q>n$ in space dimension $n\geq 2$. This approach has the advantage that the regularity assumptions can be reduced in comparison to [6]. In order to solve the linearized system, we reduce the system to the instationary reduced Stokes equations with a mixed boundary conditions and use the maximal regularity of the associated reduced Stokes operator. |

2294 | 20.08.2003 | Abels, Helmut | Reduced and Generalized Stokes Resolvent Equations in Asymptotically Flat Layers, Part II: {$H_{\infty}$}-Calculus |
MSC: 35Q30; 76D07; 35R35; 35S15 We study the generalized Stokes equations in asymptotically flat layers, which can be considered as compact perturbations of an infinite (flat) layer $\Omega_0=\R^{n-1}\times (-1,1)$. Besides standard non-slip boundary conditions, we consider a mixture of slip and non-slip boundary conditions on the upper and lower boundary, respectively. In the first part we prove the unique solvability in $L^q$-Sobolev spaces, $1<q<\infty$, by extending the known results in the case of an infinite layer $\Omega_0$ via an perturbation argument to asymptotically flat layers which are sufficiently close to $\Omega_0$. Combining this result with standard cut-off techniques and the parametrix constructed in the second part, we prove the unique solvability for an arbitrary asymptotically flat layer. Moreover, we show equivalence of unique solvability of the generalized and the reduced Stokes resolvent equations, which is essential for the second part of this contribution. |

2293 | 20.08.2003 | Abels, Helmut | Reduced and Generalized Stokes Resolvent Equations in Asymptotically Flat Layers, Part I: Unique Solvability |
MSC: 35Q30; 76D07; 35R35; 35S15 We study the generalized Stokes equations in asymptotically flat layers, which can be considered as compact perturbations of an infinite (flat) layer $\Omega_0=\R^{n-1}\times (-1,1)$. Besides standard non-slip boundary conditions, we consider a mixture of slip and non-slip boundary conditions on the upper and lower boundary, respectively. In the first part we prove the unique solvability in $L^q$-Sobolev spaces, $1<q<\infty$, by extending the known results in the case of an infinite layer $\Omega_0$ via an perturbation argument to asymptotically flat layers which are sufficiently close to $\Omega_0$. Combining this result with standard cut-off techniques and the parametrix constructed in the second part, we prove the unique solvability for an arbitrary asymptotically flat layer. Moreover, we show equivalence of unique solvability of the generalized and the reduced Stokes resolvent equations, which is essential for the second part of this contribution. |

2292 | 07.08.2003 | Abels, Helmut | Generalized Stokes Resolvent Equations in an Infinite Layer with Mixed Boundary Conditions |
MSC: 35Q30; 76D07; 35R35; 35S15 In this paper we prove unique solvability of the generalized Stokes resolvent equations in an infinite layer $\Omega_0=\R^{n-1}\times (-1,1)$, $n\geq 2$, in $L^q$-Sobolev spaces, $1<q<\infty$, with slip boundary condition on the «upper boundary» $\partial\Omega^+= \R^{n-1}\times{1}$ and non-slip boundary condition on the «lower boundary» $\partial\Omega^-= \R^{n-1}\times{-1}$. The solution operator to the Stokes system will be expressed with the aid of the solution operators of the Laplace resolvent equation and a Mikhlin multiplier operator acting on the boundary. The present result is the first step to establish an $L^q$-theory for the free boundary value problem studied by Beale and Sylvester in $L^2$-spaces. |

2291 | 01.08.2003 | Lengnink, Katja | Situative Vorstellungswelten von Lernenden und mathematische Grundvorstellungen: Auf dem Weg zu mathematischer Mündigkeit | Schülerinnen und Schüler bringen lebensweltliche Vorstellungen in den Mathematikunterricht mit. Diese haben zumindest implizit Einfluss auf den Lernprozess, selbst wenn sie nicht explizit zum Thema gemacht werden. In dieser Arbeit wird basierend auf dem Lebensweltbegriff von Habermas der Begriff der situativen Vorstellungswelt einer Lerngruppe entwickelt. Mit diesem Begriff werden die tatsächlichen Vorstellungen innerhalb einer Lerngruppe bezogen auf einen situativen Handlungskontext erfasst. Darauf aufbauend wird ein didaktisches Konzept ausgearbeitet, das explizite Auseinandersetzungen im Spannungsfeld situativer Vorstellungswelten und mathematischer Grundvorstellungen als wichtigen Bestandteil mathematischer Bildungsprozesse ansieht. An einem Projekt zur Einführung funktionaler Abhängigkeit wird erläutert, wie dieses Konzept für Mathematikunterricht nutzbar gemacht werden kann und in wie weit eine solche unterrichtliche Umsetzung zur mathematischen Mündigkeit der Lernenden beitragen kann. |

2290 | 28.07.2003 |
Neff, Patrizio Chelminski, Krzysztof |
Infinitesimal elastic-plastic Cosserat micropolar theory. Modelling and global existence in the rate-independent case. |
MSC: 74A35; 74A30 In this contribution we investigate the regularizing properties of generalized continua of Cosserat micropolar type in the elasto-plastic case. We propose an extension of classical infinitesimal elasto-plasticity to include consistently non-dissipative micropolar effects. It is shown that the new model is thermodynamically admissible and allows for unique, global in-time solution of the corresponding rate-independent initial boundary value problem. The method of choice are the Yosida-approximation and a passage to the limit. |

2288 | 10.07.2003 | Okihiro Sawada | The Navier-Stokes flow with linearly growing initial velocity in the whole space |
MSC: 35Q30 In this paper, the locally-in-time unique classical solution to the Navier-Stokes equations in the whole space is constructed, provided that the initial velocity grows linearly at infinity. The initial velocity can be chosen as Mx+u_0(x) for some constant matrix M and some function u_0. The perturbation u_0 is taken in some homogeneous Besov spaces, which contain some nondecaying functions at space infinity, typically, some almost periodic functions. |

2287 | 10.07.2003 | Vormbrock, Björn | Congruence Relations on Double Boolean Algebras |
MSC: 03G25; 06E99; 06F99; 68T30 Double Boolean algebras form the variety generated by protoconcept algebras which are one of the fundamental structures of Contextual Logic. Every double Boolean algebra contains two Boolean algebras. In this paper it is shown that congruence relations on pure double Boolean algebras may be characterized by pairs consisting of an ideal in one Boolean algebra and a filter in the other. We explain how this characterization can be generalized for double Boolean algebras. Moreover, these results are applied to protoconcept algebras in order to obtain a direct decomposition in simple protoconcept algebras. Finally, it is shown that every finite subdirectly irreducible double Boolean algebra is simple. |

2286 (opens in new tab) | 01.07.2003 | Glöckner, Helge | Tensor products in the category of topological vector spaces are not associative |
MSC: 46A16; 46A32; 22A05 We show by example that the associative law does not hold for tensor products in the category of general (not necessarily locally convex) topological vector spaces. The same pathology occurs for tensor products of Hausdorff abelian topological groups. |

2285 | 03.07.2003 |
Rabinovich, Vladimir S. Roch, Steffen Roe, John |
Fredholm indices of band-dominated operators |
MSC: 47A53; 46L80 The Fredholmness of a band-dominated operator on $l^2(\sZ)$ is closely related with the invertibility of its limit operators: the operator is Fredholm if and only if each of its limit operators is invertible and if the norms of their inverses are uniformly bounded. The goal of the present note is to show how the Fredholm index of a Fredholm band-dominated operator can be determined in terms of its limit operators. |

2284 | 26.06.2003 | Burmeister, Peter | Formal Concept Analysis with ConImp: Introduction to the Basic Features |
MSC: 03B05; 06A; 68-01; 68-04; 68N; 68 P; 68 T In this article we give an introduction into the basic concepts of Formal Concept Analysis as far as they are needed to understand the features of the program «ConImp» («{\bf Con}text and {\bf Imp}lications») – a fundamental program in Formal Concept Analysis. Several examples are discussed. At the same time we present a description of the main features and subroutines of «ConImp» , and the keystrokes necessary to start a subroutine under consideration are always given. Thus this note can also be seen as a rough program description explaining the necessary concepts in order to be able to understand most of what «ConImp» can do. |

2283 | 01.06.2003 | Scheffold, Egon | Von inversen Operatoren erzeugte Einparameter-Halbgruppen | Let $E$ be a Banach space and let $T$ be a singular bounded linear operator on $E$ with $(- \infty, 0) \subseteq \varrho (T)$ and $\| R (\alpha, T)^n¸ T^n\| \leq C$ for all $\alpha < 0$ and $n \in \N$¸.<br /> We show that $T$ is injective on ${\overline{T(E)}}$ and that the operator $-T^{-1}$ generates a $C_0$-semigroup on ${\overline{T(E)}}$¸.<br /> The following examples are considered: \begin{enumerate} \item Normal operators $T$ on Hilbert spaces with ${\rm Re} (\sigma (T)) \subseteq [0, \infty]$¸. \item Positive multiplication operators and averaging Markov operators on $C(K)$¸. \item Certain positive integral opertors on $C[a,b]$¸. \end{enumerate} |

2282 | 17.06.2003 | Burmeister, Peter | Algebraic Theory of Quasivarieties of Heterogeneous Partial Algebras |
MSC: 08A55; 08A68; 08C15 Based on existence equations, quasivarieties of heterogeneous partial algebras have the same algebraic description as those of total algebras. Because of the restrictions of the valuations to the free variables of a formula – the usual reference to the needed variables e.g. for identities (in order to get useful and manageable results) is essentially replaced here by the use of the “`logical Craig projections'” – already varieties of heterogeneous partial algebras behave to some extent rather like quasivarieties than having the properties known from varieties of total homogeneous algebras. It is one of the main aims of this note to make this more explicit. On the other hand we want to list several results known for quasivarieties of heterogeneous partial algebras – and adopt them to the extended signature – after having recalled the language and the main concepts necessary for the understanding of the results. |

2281 | 01.06.2003 | Maciej Maczynski and Egon Scheffold | On elementary operators of length 1 and some order intervals in ${\cal B}_s(H)$ | In the physical papers [1]--[4] there are some interesting propositions about operators acting in Hilbert spaces. For example in [3] there were considered all bijective mappings on $\cB_s(H)$, which preserve the order in both directions. These mappings have been characterized with the help of elementary operators of length 1. In the present paper we will generalize these results. |

2280 | 11.06.2003 | Holzer, Richard | Describing fields by implications between strong equations |
MSC: 08A55; 12E99 In this note it will be shown that the class of all fields can be described by implications between strong equations. The signature is extended by the logical Craig projection to get more expressive power for the strong equations. |

2279 | 27.05.2003 | Wille, Rudolf | Dyadic Mathematics-Abstractions from Logical Thought |
MSC: 06A15; 06B23; 03B; 08A; 051D Mathematics finally aims at supporting thought and action of human beings. For fulfilling this aim, mathematical abstractions of logical thought are essential. Because human logical reasoning is based on concepts as the basic units of thought, the dyadic mathematization of concepts performed in Formal Concept Analysis is such an abstraction. The dyadic nature of concepts is grasped through the notion of a formal context with its object-attribute-relation and its corresponding Galois connection. It is outlined how this dyadic foundation may lead to dyadic conceptions and results in order and lattice theory, contextual logic, algebra and geometry, and how that supports the development of a human-oriented mathematics. |

2278 | 19.05.2003 | Wille, Rudolf | Sind unsere Vorstellungen von Raum und Zeit richtig? oder: Besteht ein Kontinuum aus Punkten? |
MSC: 03G25; 06E99; 06F99; 68T30 Double Boolean algebras form the variety generated by protoconcept algebras which are one of the fundamental structures of Contextual Logic. Every double Boolean algebra contains two Boolean algebras. In this paper it is shown that congruence relations on pure double Boolean algebras may be characterized by pairs consisting of an ideal in one Boolean algebra and a filter in the other. We explain how this characterization can be generalized for double Boolean algebras. Moreover, these results are applied to protoconcept algebras in order to obtain a direct decomposition in simple protoconcept algebras. Finally, it is shown that every finite subdirectly irreducible double Boolean algebra is simple. |

2277 | 01.05.2003 |
Neeb, Karl-Hermann Wagemann, Friedrich |
The Universal Central Extension of the Holomorphic Current Algebra |
MSC: 17B65; 17B67 In the present paper we determine the universal central extension of the Lie algebra ${\cal O}(X,\k)$ of holomorphic functions of a complex manifold $X$ which is a Riemannian domain over a Stein manifold with values in a finite-dimensional complex simple Lie algebra $\k$. In view of the abstract description of the universal central extensions, this amounts to determine the universal differential module of the Fréchet algebra ${\cal O}(X)$ of holomorphic functions on $X$. We show that the de Rham differential into the space $\Omega^1(X)$ of holomorphic $1$-forms on $X$ is universal. Therefore the kernel of the universal central extension is the quotient space $\Omega^1(X)/d{\cal O}(X)$. |

2276 (opens in new tab) | 01.05.2003 | Fröhlich, Steffen | On twodimensional immersions that are stable for parametric functionals of constant mean curvature type |
MSC: 35J60; 53A10; 53C42 We consider twodimensional immersions in Euclidean $3$-space that are stable for parametric functionals of constant mean curvature type. We develop analytical and geometric concepts to give a perturbation result to estimate the principle curvatures of such mappings via uniformization. |

2275 | 29.04.2003 |
Wille Rudolf |
Conceptual Contents as Information – Basics for Contextual Judgement Logic |
MSC: 03B; 03B42 In Contextual Judgement Logic, Sowa´s conceptual graphs (understood as graphically structured judgements) are made mathematically explicit as concept graphs which represent information formally based on a power context family and rhetorically structured by relational graphs. The conceptual content of a concept graph is viewed as the information directly represented by the graph together with the information deducible from the direct information by object and concept implications coded in the power context family. The main result of this paper is that the conceptual contents can be derived as extents of the so-called conceptual information context of the corresponding power context family. In short, the conceptual contents of judgments are formally derivable as concept extents. |

2274 | 15.04.2003 |
Klinger, Julia Vormbrock, Björn |
Contextual Boolean Logic: How did it develop? |
MSC: 68T27; 03G25; 68T30 The aim of this paper is to explain the fundamental notions of Boolean Concept Logic and, based on this theory, to compare various approaches to Contextual Judgment Logic. First we motivate the basic definitions of semiconcept algebras and show how semiconcept algebras generalize to protoconcept algebras and double Boolean algebras. This enables us to give a brief presentation of the theories of concept graphs, semiconcept graphs, protoconcept graphs and concept graphs with cuts as specific approaches to a mathematical judgment logic. In addition, differences and common grounds of these contributions are discussed. |

2273 | 28.03.2003 |
Schoolmann, Lars Wille, Rudolf |
Concept Graphs with subdivision: A Semantic Approach |
MSC: 03B; 03B45 The semantic approach defines a concept graph with subdivision as a mathematical structure derived from triadic power context family. The aim of introducing concept graphs with subdivision is to represent modal information mathematically. Based on the notion of the conceptual content of a concept graph with subdivision, we can show that the content graphs with subdivision of a triadic power context family form a complete lattice with respect to the information order. Finally, our approach is extended to existential concept graphs with subdivision. |

2272 | 25.03.2003 | Glöckner, Helge | Examples of differentiable mappings into non-locally convex spaces |
MSC: 58C20; 26E20; 46A16; 46G20 Examples of differentiable mappings into real or complex topological vector spaces with specific properties are given, which illustrate the differences between differential calculus in the locally convex and the non-locally convex case. In particular, for a suitable non-locally convex space $E$, we describe a smooth injection $R \to E$ whose derivative vanishes identically; we present a complex $C^\infty$-map on the complex field $C$ which is not given locally by its Taylor series, around any point; we present a complex $C^1$-map from $C$ to a complete, non-locally convex topological vector space which is not $C^2$; and we present a compactly supported, non-zero, complex $C^\infty$-map from $C$ to a suitable non-locally convex space. |

2271 | 25.03.2003 | Glöckner, Helge | Implicit Functions from Topological Vector Spaces to Banach Spaces |
MSC: 58C20; 26E20; 46A16; 46G20 Examples of differentiable mappings into real or complex topological vector spaces with specific properties are given, which illustrate the differences between differential calculus in the locally convex and the non-locally convex case. In particular, for a suitable non-locally convex space $E$, we describe a smooth injection $R \to E$ whose derivative vanishes identically; we present a complex $C^\infty$-map on the complex field $C$ which is not given locally by its Taylor series, around any point; we present a complex $C^1$-map from $C$ to a complete, non-locally convex topological vector space which is not $C^2$; and we present a compactly supported, non-zero, complex $C^\infty$-map from $C$ to a suitable non-locally convex space. |

2270 | 01.03.2003 |
Bertram, Wolfgang Glöckner, Helge Neeb, Karl-Hermann |
Differential Calculus, Manifolds and Lie Groups over Arbitrary Infinite Fields |
MSC: 58C20; 22E65; 26E30; 26E15; 26E20; 46T05 We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed. Special attention is paid to the case of mappings between topological vector spaces over non-discrete topological fields, in particular ultrametric fields or the fields of real and complex numbers. In the latter case, a theory of differentiable mappings between general, not necessarily locally convex spaces is obtained, which in the locally convex case is equivalent to Keller's $C^k_c$-theory. |

2269 | 26.03.2003 |
Neeb, Karl-Hermann Penkov, Ivan |
Cartan subalgebras of gl_\infty |
MSC: 17B65; 17B20 Let ${\scriptstyle V}$ be a vector space over a field ${\scriptstyle \K}$ of characteristic zero and ${\scriptstyle V_* }$ be a space of linear functionals on ${ \scriptstyle V}$ which separate the points of ${\scriptstyle V}$. We consider ${\scriptstyle V \otimes V_*}$ as a Lie algebra of finite rank operators on ${ \scriptstyle V}$, and set ${ \scriptstyle \gl(V,V_*):= V \otimes V_*}$. We define a Cartan subalgebra of ${\scriptstyle \gl(V,V_*)}$ as the centralizer of a maximal subalgebra every element of which is semisimple, and then give the following description of all Cartan subalgebras of ${\scriptstyle \gl(V,V_*)}$ under the assumption that ${\scriptstyle \K}$ is algebraically closed. %Let ${\scriptstyle V}$ be the defining representation of %${\scriptstyle \gl_\infty}$ and ${\scriptstyle V_*}$ its restricted dual. A subalgebra of ${\scriptstyle \gl(V,V_*)}$ is a Cartan subalgebra if and only if it equals ${\scriptstyle \oplus_{j } (V_j \otimes (V_j)_*) \oplus (V^0 \otimes V_*^0)}$ for some one-dimensional subspaces ${\scriptstyle V_j \subeq V}$ and ${\scriptstyle (V_j)_* \subeq V_*}$ with ${\scriptstyle (V_i)_*(V_j) = \delta_{ij} \K}$ and such that the spaces ${\scriptstyle V_*^0 = \cap_{j } (V_j)^\bot \subeq V_*}$ and ${\scriptstyle V^0 = \cap_{j } ((V_j)_*)^\bot \subeq V}$ satisfy ${\scriptstyle V_*^0(V^0) = {0}}$. We then discuss explicit constructions of subspaces ${\scriptstyle V_j}$ and ${\scriptstyle(V_j)_*}$ as above. Our second main result claims that a Cartan subalgebra of ${\scriptstyle \gl(V,V_*)}$ can be described alternatively as a locally nilpotent self-normalizing subalgebra whose adjoint representation is locally finite, or as a subalgebra ${\scriptstyle \h}$ which coincides with the maximal locally nilpotent ${\scriptstyle \h}$-submodule of ${\scriptstyle \gl(V,V_*)}$, and such that the adjoint representation of ${\scriptstyle \h}$ is locally finite. |

2268 | 17.03.2003 | Fischer, Tom | An axiomatic approach to valuation in life insurance |
MSC: 91B24; 91B28; 91B30 The classical Principle of Equivalence ensures that a life insurance company can accomplish that the mean balance per contract converges to zero almost surely for an increasing number of clients. In an axiomatic approach, this idea is adapted to the general case of stochastic financial markets. In accordance with existing results, the implied minimum fair price of general life insurance products is then uniquely determined by the product of the given equivalent martingale measure of the financial market with the probability measure of the biometric state space. A detailed historical example concerning contract pricing and valuation is given. |

2267 | 15.03.2003 | Wille, Rudolf | Truncated Distributive Lattices: Conceptual Structures of Simply Implicational Theories |
MSC: 06D; 03B The logic relationships in the everyday human thought are predominantly inferences with one-element premise. This becomes apparent in the practice of Formal Concept Analysis by the frequent occurence of truncated distributive lattices as concept lattices. This paper gives a mathematization of the underlying everyday theories of logical relationships and elaborates useful mathematical results, in particular about algorithmically drawing concept lattices which correspond to the everyday logical theories. |

2266 | 01.02.2003 | W. Krabs | A General Predator-Prey Model. | We consider n >= 2 populations of animals or plants that are living in mutual predator-prey relations or are pairwise neutral to each other. We assume the temporal development of the population densities to be described by a system of differential equations which has an equilibrium state solution. We at first give sufficient conditions for this equilibrium state to be asymptotically stable by linearizing the systems around it. Then we derive sufficient conditions for asymptot ic stability by Lyapunov's method. Finally we investigate a discretization of the Volterra-Lotka model and determine its coefficients from discrete data. |

2265 (opens in new tab) | 01.02.2003 |
Bruder, Regina Lengnink, Katja, Prediger, Susanne |
Ein Instrumentarium zur Erfassung subjektiver Theorien über Mathematikaufgaben | In diesem Aufsatz wird eine Technik zur Erfassung subjektiver Theorien über Mathematik-aufgaben vorgestellt, die sich der Methode des Repertory Grid bedient (s. Kapitel 2). Dieses Instrumentarium wurde entwickelt, um die Reflexions- und Sprachebenen von Mathematik-lehrkräften über Aufgaben differenziert dokumentieren zu können. Auf dieser Grundlage wird es u.a. möglich, individuelle Lernfortschrittsbeschreibungen in der Lehreraus- und fortbildung vorzunehmen. Erste Anwendung fand die Repertory Grid-Technik im Rahmen einer fachdidaktischen Lehrveranstaltung mit Lehramtsstudierenden, aus der Fallbeispiele vorgestellt und diskutiert werden. Unser Ansatz ordnet sich innerhalb der Fachdidaktik Mathematik in den noch wenig bearbeiteten Bereich der qualitativen Evaluationsforschung zu Lehr- und Lernkonzepten für Mathematik ein. Der Terminus “subjektive Theorie” wird als Sammelbezeichnung für individuelle Einstellungen, Vorstellungen und Kenntnisse zu einem Sachverhalt verwendet – hier also zu Mathematikaufgaben. |

2264 | 04.02.2003 |
Rabinovich, Vladimir. S. Roch, Steffen |
Fredholmness of convolution type operators |
MSC: 45E10; 45A05 We study the Fredholmness on $L^p(D)$ of operators of convolution type. Here $D$ is an unbounded measurable domain in $\sR^N$, and an operator $A$ on $L^p(D)$ is of convolution type if it is constituted by operators of the form $a C(k) bI$ where $C(k)$ is the operator of convolution by the $L^1(\sR^N)$-function $k$ and where $a$ and $b$ are bounded and uniformly continuous functions. The domains under consideration include, for example, curved layers, curved cylinders, and cones with angular or cuspidal edges. The criterion for the Fredholmness of the operator $A$ is formulated in terms of limit operators of $A$. |

2263 | 28.01.2003 |
Fischer, Tom May, Angelika Walther, Brigitte |
Anpassung eines CIR-$k$-Modells mit Hilfe der Kalman-Filter-Methode |
MSC: 62P05; 62M20; 62M05; 91B70 In dieser Arbeit wird erläutert, wie die Theorie des Kalman-Filters für die Parameterschätzung eines Cox-Ingersoll-Ross-Modells mit $k$ Faktoren genutzt werden kann. Die zunächst theoretische Ausführung der Vorgehensweise wird an Hand des Deutschen Rentenmarkts konkretisiert. Schätzwerte für die Fälle $k = 1, 2$ und $3$ werden angegeben und die Cox-Ingersoll-Ross-Modelle für die ermittelten Parameterwerte verglichen. Als Anwendung wird auf die Erzeugung von Szenarien mit Hilfe stochastischer Simulation eingegangen. |

2262 | 09.01.2003 | M. V. Semenova | Sublattices of Suborder Lattices |
MSC: 06B23; 06B05; 06B15; 06B35; 06C05 We represent various types of lattices as sublattices of suborder lattices of posets possessing certain properties. In particular, we prove that the class of sublattices of suborder lattices of posets of height at most $n$ is a variety, for all $n < \omega$. |

2261 | 01.01.2003 |
Bertram, Wolfgang Neeb, Karl-Hermann |
Projective completions of Jordan pairs Part I. The generalized projective geometry of a Lie algebra |
MSC: 17C30; 17C37 We prove that the projective completion $(X^+,X^-)$ of the Jordan pair $(\g_1,\g_{-1})$ corresponding to a $3$-graded Lie algebra $\g=\g_1 \oplus \g_0 \oplus \g_{-1}$ can be realized inside the space $\cal F$ of inner $3$-filtrations of $\g$ in such a way that the remoteness relation on $X^+ \times X^-$ corresponds to transversality of flags. This realization is used to give geometric proofs of structure results which will be used in Part II of this work in order to define on $X^+$ and $X^-$ the structure of a smooth manifold (in arbitrary dimension and over general base fields or -rings). |

2260 | 08.01.2003 | Gramlich, Ralf | Weak Phan systems of type $C_n$ | This paper provides a classification of Phan amalgams of type $C_n$ and of shape $S \supseteq S_2$ over finite fields of odd square order. This is the final step in the proof of a new Phan-type theorem for the group $Sp_{2n}(q)$ for $n \geq 3$ and $q \geq 8$. |

2259 (opens in new tab) | 01.01.2003 |
Farwig, Reinhard Müller, Detlef Hishida, Toshiaki |
$L^q$--Theory of a Singular »Winding» Integral Operator Arising from Fluid Dynamics |
MSC: 35; 42; 76; 35Q35; 42B20; 42B25; 76D07 We analyze in classical $L^q({\Bbb R}^n)$--spaces, $n = 2$ or $n=3$, $1 < q <\infty$, a singular integral operator arising from the linearization of a hydrodynamical problem with a rotating obstacle. The corresponding system of partial differential equations of second order involves an angular derivative which is not subordinate to the Laplacian. The main tools are Littlewood-Paley theory and a decomposition of the singular kernel in Fourier space. |

Number | Date | Author | Title | Abstract/MSC |
---|---|---|---|---|

2257 | 01.12.2002 | Glöckner, Helge | Lie Groups of Measurable Mappings |
MSC: 22E65; 46E40; 46E30; 22E67; 46T20; 46T25 We describe new construction principles for infinite-dimensional Lie groups. In particular, given any measure space $(X,\Sigma,\mu)$ and (possibly infinite-dimensional) Lie group $G$, we construct a Lie group $L^\infty(X,G)$, which is a Fréchet-Lie group if so is $G$. We also show that the weak direct product $\prod^*_{i\in I} G_i$ of an arbitrary family $(G_i)_{i\in I}$ of Lie groups can be made a Lie group, modelled on the locally convex direct sum $\bigoplus_{i\in I} L(G_i)$. |

2256 | 01.12.2002 | Glöckner, Helge | Remarks on Holomorphic Families of Operators |
MSC: 47A56; 47A05; 47A07 Let $E$, $F$ and $G$ be complex locally convex spaces, and $U\supseteq {\mathbb C}$ be open. We show that, for any holomorphic families of operators $f : U \to L(E,F)$ and $g : U \to L(F,G)$, the composed family $U \to L(E,G)$, $z \mapsto g(z) \circ f(z)$ is holomorphic. It is also shown that the family $U \to L(F',E')$, $z \mapsto f(z)'$ of adjoints is holomorphic, provided the evaluation homomorphism $\eta_F : F \to F»$ is continuous. |

2255 | 01.12.2002 | Neeb, Karl-Hermann | Locally convex root graded Lie algebras |
MSC: 17B65; 17B60; 17B70 In the present paper we start to build a bridge from the algebraic theory of root graded Lie algebras to the global Lie theory of infinite-dimensional Lie groups by showing how root graded Lie algebras can be defined and analyzed in the context of locally convex Lie algebras. Our main results concern the description of locally convex root graded Lie algebras in terms of a locally convex coordinate algebra and its universal covering algebra, which has to be defined appropriately in the topological context. Although the structure of the isogeny classes is much more complicated in the topological context, we give an explicit description of the universal covering Lie algebra which implies in particular that it depends only on the root system and the coordinate algebra. Not every root graded locally convex Lie algebra is integrable in the sense that it is the Lie algebra of a Lie group. |

2254 | 04.12.2002 | N.Ya. Medvedev | Ideals in free vector lattices and free Abelian $l$-groups |
MSC: 06F15; 06F20; 20B27; 20F60 We examine and classify the ideal structur of free vector lattices and free Abelian lattice-ordered groups with two and three generators. We also examine the vector lattice of all continious piecewise polynomial functions with pointwise ordering and classify its ideals. |

2253 | 01.12.2002 |
Martin,Alexander Möller,Markus |
Cutting Planes for the Optimization of Gas Networks | This paper presents cutting planes which are useful or potentially useful for solving mixed integer programs that arise in the optimization of gas networks. We consider polyhedra that are defining essential parts of the model and give an polynomial algorithm for the calculation of the set of vertices of such polyhedra. So a separation algorithm for the convex hull of the polyhedra can be developed. |

2252 | 14.11.2002 |
Lang, Jens Cao, Weiming Huang, Weizhang Russell, Robert D. |
A Two--dimensional Moving Finite Element Method with Local Refinement Based on A Posteriori Error Estimates |
MSC: 65M60 In this paper, we consider the numerical solution of time--dependent PDEs using a finite element method based upon rh--adaptivity. An adaptive horizontal method of lines strategy equipped with a posteriori error estimates to control the discretization through variable time steps and spatial grid adaptations is used. Our approach combines an r--refinement method based upon solving so--called moving mesh PDEs with h--refinement. Numerical results are presented to demonstrate the capabilities and benefits of combining mesh movement and local refinement. |

2251 (opens in new tab) | 08.08.2002 |
Bokowski, Jürgen Martin, Alexander |
Egoisten schaden sich selbst | MSC: 90C90 |

2250 | 01.10.2002 | Glöckner, Helge | Lie groups of germs of analytic mappings |
MSC: 22E65; 22E67; 46H05; 46T25 Let $X$ be a metrizable topological vector space over ${\mathbb K} \in { {\mathbb R}, {\mathbb C} }$, $K \subseteq X$ be a non-empty compact subset, and $G$ be a Banach-Lie group over ${\mathbb K}$. Then $\Gamma(K,G)$ denotes the group of germs $[\gamma]$ around $K$ of ${\mathbb K}$-analytic $G$-valued mappings $\gamma : U \to G$ defined on open neighbourhoods $U$ of $K$ in $X$. We show that $\Gamma(K,G)$ can be made a ${\mathbb K}$-analytic Baker-Campbell-Hausdorff (BCH-) Lie group in a natural way. We also show that if $A$ is a unital Banach algebra over ${\mathbb K}$, then the natural locally convex direct limit topology on $\Gamma(K,A)$ makes the latter a unital topological ${\mathbb K}$-algebra with open unit group and continuous inversion. The unit group $\Gamma(K,A)^\times \cong \Gamma(K,A^\times)$ is in fact a ${\mathbb K}$-analytic BCH-Lie group. |

2249 | 01.10.2002 | Scheffold, Egon | Befreundete Zahlentripel und die Formeln von Barlow und Abel | Peter Barlow und Niels Henrik Abel haben versucht, die Unmöglichkeit der Gleichung $a^n + b^n = c^n $ in ganzen Zahlen zu beweisen, wenn $n $ größer als $2$ ist (s. [1] und [3]). Bei ihrem Versuch haben sie eine formelmäßige Darstellung der Zahlen $a,b$ und $c$ angegeben. In dieser Arbeit möchte ich mit Hilfe der von mir definierten befreundeten Zahlentripel diese Formeln untersuchen. |

2248 | 29.10.2002 |
Arjeh M. Cohen Hans Cuypers Ralf Gramlich |
Local recognition of non-incident point-hyperplane graphs | Let $\mathbb{P}$ be a projective space. By $\mathbf{H}(\mathbb{P})$ we denote the graph whose vertices are the non-incident point-hyperplane pairs of $\mathbb{P}$, two vertices $(p,H)$ and $(q,I)$ being adjacent if and only if $p \in I$ and $q \in H$. In this paper we give a characterization of the graph $\mathbf{H}(\mathbb{P})$ (as well as of some related graphs) by its local structure. Group-theoretic applications are given as well. |

2247 | 17.10.2002 | Ralf Gramlich | Line-hyperline pairs of projective spaces and fundamental subgroups of linear groups | This articles provides a self-contained, purely combinatorial local recognition of the graph on the non-intersecting line-hyperline pairs of the projective space $\mathbb{P}_n(\mathbb{F})$ for $n \geq 8$ and $\mathbb{F}$ a division ring with the exception of the case $n=8$ and $\mathbb{F} = \mathbb{F}_2$. Consequences of that result are a characterization of the hyperbolic root group geometry of $SL_{n+1}(\mathbb{F})$, $\mathbb{F}$ a division ring, and a local recognition of certain groups containing a central extension of $PSL_{n+1}(\mathbb{F})$, $\mathbb{F}$ a field, using centralizers of $p$-elements. |

2246 | 17.10.2002 | Ralf Gramlich | On the hyperbolic symplectic geometry | The present article provides a new characterization of the geometry on the points and hyperbolic lines of a nondegenerate symplectic polar space. This characterization is accomplished by studying the family of subspaces obtained when considering the polars of all hyperbolic lines. |

2245 | 17.10.2002 |
Ralf Gramlich Corneliu Hoffman Sergey Shpectorov |
A Phan-type theorem for $Sp(2n,q)$ | The authors extend Phan's theory on defining amalgams of (twisted) Chevalley groups belonging to a simply-laced Dynkin diagram to the diagram $C_n$. |

2244 | 17.10.2002 |
Curtis Bennett Ralf Gramlich Corneliu Hoffman Sergey Shpectorov |
Curtis-Phan-Tits theory | In the present article the authors demonstrate that there is a relation between the Curtis-Tits theorem and Phan's theorems that goes well beyond the similarity in appearance. In particular, the authors present a geometric construction connecting those theorems and suggesting that Phan's theorems can be thought of a «twisted versions» of the Curtis-Tits theorem. The construction itself further suggests that Phan's theorems are only some of many possible such theorems. The authors make this explicit by presenting a new Phan-type theorem for the symplectic groups. The work discussed in this article began as an attempt to provide a complete and clear proof of Phan's first theorem, together with a desire for a more geometric proof of the Curtis-Tits theorem. The authors were surprised, however, to find that their construction led to a unifying point of view on these two theorems, and unexpected bonus. Another remarkable observation is that the geometric constructions do not seem to depend on the finiteness of the field or the sphericality of the diagram. So the present article may be of interest not only to finite geometers and finite group-theorists but also to people interested in nonspherical twin buildings and Kac-Moody groups. |

2243 | 23.10.2002 | SCHAPPACHER, Norbert | Politisches in der Mathematik – Versuch einer Spurensicherung |
MSC: 00A30 In diesem Aufsatz geht es in historischer Sicht um zwei Gegensatzpaare bezüglich mathematischer Erkenntnis: Im ersten Gegensatzpaar geht es um die Frage, ob Mathematik eine reine Schöpfung des menschlichen Geistes ist und, losgelöst von der empirischen Realität, nur im transzendentalen Sinne existiert, oder ob sie ein ideeller Bestandteil unserer empirisch erfahrbaren Welt ist. Im zweiten Gegensatzpaar geht es um die Frage, of die Mathematik ihre Erkenntnis primär durch Intuition erwirbt, oder ob sie ein Bestandteil der Logik ist. Das erste Gegensatzpaar wird widergespiegelt in der Auffassung Kants von der Konstruktion der Mathematik einerseits und der Platonschen Auffassung von der Entdeckung der Mathematik andererseits. das zweite Gegensatzpaar findet seinen Niederschlag in der Frage, ob mathematische Sätze synthetisch oder analytisch sind. |

2242 | 01.10.2002 | Spellucci, Peter | Solving QP problems by penalization and smoothing |
MSC: 90C30; 65K05 In this paper we describe a new technique for solving QP problems with general linear constraints. It is intended to solve very large scale problems, where active set methods become impractical. It solves the problems iteratively, and, of course, approximately only. There are known lots of methods of this type. However, the successful ones known so far either deal with bound constrained problems only or belong to the class of interior point methods. The latter are quite successf$ in the convex case but get trouble otherwise. Contrary, the method used here is able to deal with nonconvex cases and aims in finding a point satisfying the second order necessary optimality conditions. It works without using a modification of the Hessian. It is based on the well known exact l1-penalty function, smoothing of abs(.) and min(.) and solving the resulting unconstrained or bound constrained problem by a variant of the Lanczos method.<br /> |

2241 | 01.10.2002 | Krabs, Werner | Mathematische Erkenntnis von Kant bis Gödel |
MSC: 00A30 In diesem Aufsatz geht es in historischer Sicht um zwei Gegensatzpaare bezüglich mathematischer Erkenntnis: Im ersten Gegensatzpaar geht es um die Frage, ob Mathematik eine reine Schöpfung des menschlichen Geistes ist und, losgelöst von der empirischen Realität, nur im transzendentalen Sinne existiert, oder ob sie ein ideeller Bestandteil unserer empirisch erfahrbaren Welt ist. Im zweiten Gegensatzpaar geht es um die Frage, of die Mathematik ihre Erkenntnis primär durch Intuition erwirbt, oder ob sie ein Bestandteil der Logik ist. Das erste Gegensatzpaar wird widergespiegelt in der Auffassung Kants von der Konstruktion der Mathematik einerseits und der Platonschen Auffassung von der Entdeckung der Mathematik andererseits. das zweite Gegensatzpaar findet seinen Niederschlag in der Frage, ob mathematische Sätze synthetisch oder analytisch sind. |

2240 | 11.10.2002 | Neeb, Karl-Hermann | Current groups for non-compact manifolds and their central extensions |
MSC: 22E65; 58D15; 57T20 In this paper we study two types of groups of smooth maps from a non-compact manifold ${\scriptstyle M}$ into a Lie group ${\scriptstyle K}$ which may be infinite-dimensional: the group ${\scriptstyle C_c^\infty(M,K)}$ of compactly supported maps and for a compact manifold ${\scriptstyle M}$ and a closed subset ${\scriptstyle S}$ the group ${\scriptstyle C^\infty(M,S;K)}$ of those maps which vanish on ${\scriptstyle S}$, together with all their derivatives. We study central extensions of these groups associated to Lie algebra cocycles of the form ${\scriptstyle \omega(\xi, \eta) = [\kappa(\xi, d\eta)]}$, where ${\scriptstyle \kappa \: \k \times \k \to Y}$ is a symmetric invariant bilinear map on the Lie algebra ${\scriptstyle \k}$ of ${\scriptstyle K}$ and the values of ${\scriptstyle \omega}$ lie in ${\scriptstyle \Omega^1(M;Y)/dC^\infty(M;Y)}$. For such cocycles we show that a corresponding central Lie group extension exists if and only if this is the case for ${\scriptstyle M = \SS^1}$. If ${\scriptstyle K}$ is finite-dimensional semisimple, this implies the existence of a universal central Lie group extension of the identity component of the current groups. |

2239 | 28.09.2002 |
Bokowski, Juergen King, Simon Mock, Susanne Streinu, Ileana |
A Topological Representation Theorem for Oriented Matroids |
MSC: 52C40; 57N50; 57N45 We present a new direct proof of a topological representation theorem for oriented matroids in the general rank case. Our proof is based on an earlier rank 3 version. It uses hyperline sequences and the generalized Schönflies theorem. As an application, we show that one can read off oriented matroids from arrangements of embedded spheres of codimension one, even if wild spheres are involved. |

2238 | 01.10.2002 | Rößler, Andreas | Stochastic Runge-Kutta Methods for Stochastic Differential Equation Systems with Commutative Noise |
MSC: 60H35; 60C30; 68U20 A class of explicit stochastic Runge-Kutta methods for Stratonovich stochastic differential equation systems w.r.t. $m$-dimensional Wiener processes satisfying a commutativity condition is developed. General conditions for the coefficients of the stochastic Runge-Kutta scheme converging with order two in the weak sense are presented. Due to the commutativity condition, no correlated random variables have to be generated for these Runge-Kutta methods. |

2237 | 01.09.2002 | Holzer, Richard | On subdirectly irreducible OMAs |
MSC: 08C15; 06F99 In this paper some properties of epi-representations and Schmidt-congruence relations of orthomodular partial algebras are investigated and an infinite list of OMA-epi subdirectly irreducible orthomodular partial algebras will be constructed. |

2236 | 10.09.2002 |
H{ö}llig, Klaus Reif, Ulrich |
Nonuniform WEB-Splines |
MSC: 41A15; 65N30 The construction of weighted extended B-splines (web-splines), as recently introduced by the authors and J. Wipper for uniform knot sequences, is generalized to the non-uniform case. We show that web-splines form a stable basis for splines on arbitrary domains which provides optimal approximation power. Moreover, homogeneous boundary conditions, as encountered frequently in finite element applications, can be satisfied exactly by using an appropriate weight function. To illustrate the performance of the method, it is applied to a scattered data fitting problem and a finite element approximation of an elliptic boundary value problem. |

2235 (opens in new tab) | 04.09.2002 |
Achterberg, Tobias Koch, Thorsten Martin Alexander |
Branching on History Information |
MSC: 90C10; 90C11 Mixed integer programs (MIPs) are commonly solved with branch and bound algorithms based on linear programming. The success and the speed of the algorithm strongly depends on the strategy used to select the branching variables. Today's state-of-the-art strategy is called pseudocost branching and uses information of previous branchings to determine the current branching. We propose a modification of pseudocost branching which we call history branching. This strategy has been implemented in SIP, a state-of-the-art MIP solver. We give computational results that show the superiority of the new strategy. |

2234 | 01.09.2002 | Helmerich, Markus | Begriffliche Informationskarten – Orientierungs- und Navigationshilfe in Lernumgebungen auf kontextuell-logischer Grundlage |
MSC: 97D20; 68U35; 05C90 Die Gestaltung von computer-gestützten Lernumgebungen bietet die Chance zu neuen Bildungsmöglichkeiten. Damit diese Lernumgebungen dem Anspruch, exploratives Lernen zu ermöglichen, gerecht werden, bedürfen sie einer nach besonderen Gesichtspunkten gestaltete Orientierungs- und Navigationshilfe. Im Folgenden wird beschrieben,welche Anforderungen sich für das Design einer solchen Orientierungs- und Navigationshilfe ergeben, welche Möglichkeiten die Mathematik zur Unterstützung einer Mensch-Maschine-Kommunikation bietet und welche mathematischen Methoden und graphischen Darstellung zur Erschließung einer Lernumgebung helfen können. |

2233 | 26.08.2002 |
Wille, Rudolf Wille, Uta |
Restructuring General Geometry: Measurement and Visualization of Spatial Structures |
MSC: 51-01 An approach of restructuring general geometry based on the idea of measurement and visualization is proposed. The approach is guided by the purpose of making realities graphic, intelligible, and workable. |

2232 | 01.08.2002 | Mohammed Seaid | Notes on Numerical Methods for Two-Dimensional Neutron Transport Equation |
MSC: 65R20; 65N06; 65F10; 82D75 Detailed numerical methods for two-dimensional neutron transport equation are presented. Using the discrete ordinates for angle collocation and the Diamond differencing for space discretization, the neutron transport equation is transformed to a system of sparse matrices. To solve the final system we formulate the source iteration, a full BICGSTAB and GMRES algorithms. Additionally, the diffusion limit and the diffusion synthetic acceleration are included in these notes. The robustness, efficiency and convergence rates of these methods are illustrated by two numerical examples. |

2231 (opens in new tab) | 28.08.2002 | Alber, Hans-Dieter | Justification of homogenized models for viscoplastic material behavior |
MSC: 74Q10; 74C05; 74C10; 35Q72 We justify the formal homogenization of the quasistatic initial boundary value problem with internal variables, called the microscopic problem, which models the deformation behavior of viscoplastic bodies. To this end it is first shown that the formally derived homogenized initial-boundary value problem has a solution. >From this solution an asymptotic solution of the microscopic problem is constructed, and it is shown that the difference of the exact solution and the asymptotic solution tends to zero if the lengthscale of the microstructure converges to zero. Our results are proved for viscoplastic material behavior that can be modeled by constitutive equations of monotone type with linear hardening terms. For technical reasons we are only able to prove the convergence result locally in time and for smooth data. |

2230 (opens in new tab) | 17.08.2002 | Nesenenko, Sergiy | Asymptotic Expansions for Linear Differential-Algebraic Equations |
MSC: 34E05 Asymptotic expansions for solutions of linear differential-algebraic equation with variable matrix coefficients are considered. The solution is being sought in the form of a formal power series. The coefficients of this series satisfies linear infinite-dimensional system of the algebraic equations with triangular matrix of coefficients. Existence and uniqueness theorem is proved for such equations and initial manifolds are described. The Drazin inverse matrices are used to demonstrate the existence of asymptotic expansion. |

2229 | 01.07.2002 | Wille, Rudolf | Kommunikative Rationalität und Mathematik |
MSC: 00A30 In dieser Arbeit wird folgende These erläutert und begründet: Sinn und Bedeutung der Mathematik liegen letztlich darin, daß Mathematik die rationale Kommunikation von Menschen wirksam zu unterstützen vermag. Kern der Argumentation ist, daß die wirksame Unterstützung rationaler Kommunikation möglich wird durch die enge Verbindung von Mathematik und Logik (im Sinne der Peirceschen Spätphilosophie), durch die wiederum die kommunikative Rationalität (im Sinne von Jürgen Habermas) aktiviert werden kann. Wie eine derartige Unterstützung konkret aussehen kann, wird beispielgebend an einer Zusammenarbeit von Politikwissenschaft und Mathematik verdeutlicht. |

2228 | 01.07.2002 | Farwig, Reinhard | Existence of Avalanching Flows |
MSC: 35; 70; 73 Avalanches, landslides and debris flows are devastingly powerful natural phenomena that are far too little understood. These granular matters are mixtures of solid particles and of an interstitial fluid and are easily modelled on the microscopic level by the laws of classical mechanics. On mesoscopic and macroscopic levels the different scales of the influence of the particles, the fluid and their interaction lead to various models of avalanching flows. In this survey we consider several models of granular materials characterized by height and in case also by momentum, discuss the existence of similarity solutions, existence of arbitrary solutions and particle segregation. The main part concerns the Savage-Hutter equations for dense flow avalanches. |

2227 | 01.07.2002 | Wille, Rudolf | Begriffsanalytische Mathematisierung logischer Strukturen |
MSC: 00A30 Anliegen dieses Textes ist aufzuzeigen, wie die begriffsanalytische Mathematisierung logischer Strukturen die Begriffs- und Wissensforschung zu unterstützen vermag. Das wird vor allem an dem von Helmut Spinner entwickelten Karlsruher Ansatz der integrierten Wissensforschung exemplifiziert. |

2226 | 01.06.2002 |
Fischer, Tom May, Angelika Walther, Brigitte |
Simulation of the Yield Curve: Checking a Cox-Ingersoll-Ross Model |
MSC: 62P05; 62M02; 62M05; 91B70 We give a complete description of a simulation of future bond prices by a one-factor Cox-Ingersoll-Ross (CIR) interest rate model. Explicit methods and formulas are provided, the time series service of the German Federal Reserve is used as data source. Several model checks are developed and applied to the CIR model. As a result, the model has to be rejected for the German debt securities market. |

2225 | 01.06.2002 | Fröhlich, Andreas | Solutions of the Navier-Stokes Initial Value Problem in weighted $L^q$-spaces |
MSC: 76D05; 35Q30; 35D05 The problem of strong solvability of the nonstationary Navier-Stokes equations is considered in weighted $L^q$-spaces $\lw(\Omega),$ where the domain $\Omega\subset\R^n$ is equal to the half space $\rp$ or to a bounded domain with boundary of class $C^{1,1}$ and the weight $\omega$ belongs to the Muckenhoupt class $A_q$. We give general conditions on the weight function ensuring the existence of a unique strong solution at least locally in time. In particular, these conditions admit weight functions $\omega\in A_q,$ which become singular at the boundary or, in the case $\Omega=\rp$, grow for $|x|\rightarrow\infty.$ |

2224 (opens in new tab) | 01.05.2002 | J. He, Karl H. Hofmann, S. M. Miller, and D. A. Robbie | Compact Semigroups and Suitable Sets |
MSC: 22A05; 54D20; 22A15 A suitable set $A$ in a topological semigroup $S$ is a subset of $S$ which contains no idempotents, any limit points of $A$ in $S$ are idempotents, and $A$, together with all idempotents of $S$, generates a dense subsemigroup of $S$. Following work of Hofmann and Morris, who showed that every compact Hausdorff topological group has such a suitable set, this paper extends that result to several classes of compact semigroups all of whose members satisfy $S^2=S$. In particular all compact simple semigroups are shown to have a suitable set. Cartesian products of compact monoids each with a suitable set have suitable sets as do continuous homomorphic images of compact semigroups with suitable sets. It is shown that certain classes of $\cal H $-chain semigroups have suitable sets. The class of irreducible semigroups falls into two classes, where the members of one class always have a suitable set and in the other class a semigroup which contains no suitable set is constructed. It is shown that compactifications of subsemigroups of Lie groups tend to have suitable sets; these include the `triangle semigroup' as a typical test case. If $S$ is compact, connected, and $S^2\ne S$, then $S$ cannot have a suitable set. |

2223 | 01.05.2002 | Fischer, Tom | Risk Capital Allocation by Coherent Risk Measures Based on One-Sided Moments |
MSC: 91B30; 91A80; 91B28; 91B32 In this paper we propose differentiability properties for positively homogeneous risk measures. These properties ensure that the gradient can be applied for reasonable risk capital allocation on non-trivial portfolios. We show that the differentiability properties are fulfilled for a wide class of coherent risk measures based on the mean and the one-sided moments of a risky payoff. In contrast to quantile-based risk measures like Value-at-Risk, risk measures of this class allow allocation in portfolios of very general distributions, e.g. discrete ones. In an example we show how a particular risk measure of this class can be chosen by adapting it to the VaR of a certain portfolio. As a consequence, the risk capital corresponding to the VaR can be allocated by the gradient due to the adapted risk measure. |

2222 | 01.05.2002 | Hofmann, Karl H. | A Category of Topological Groups Suitable for a Structure Theory of Locally Compact Groups |
MSC: 22D; 22E This survey outlines an approach to a projected monograph on the «Structure of pro-Lie groups and Locally Compact Groups» [K. H. Hofmann and S. A. Morris The Structure of Pro-Lie Groups and Locally Compact Groups, in preparation, visit\hfill\break {\tt http://www.ballarat.edu.au/\hskip-3pt$\sim$\hskip-1pt % smorris/loccocont.pdf}] which may be considered a sequel to the book «The Structure of Compact Groups» [Hofmann, K. H., and S. A. Morris, De Gruyter Berlin, 1998, xvii+835pp]. In the focus is the category of projective limits of finite dimensional Lie groups. In a nontrivial fashion, every member $G$ of this category has a special presentation as a projective limit of quotients $G/N$ which are finite dimensional Lie groups. The category of these limits is complete, and all of its objects have a good Lie theory in terms of certain Lie algebras which are well behaved projective limits of finite dimensional ones. |

2221 | 01.05.2002 |
Fügenschuh, Armin Martin, Alexander |
Computational Integer Programming and Cutting Planes |
MSC: 90C10; 90C11 The purpose of this paper is to describe the main ingredients of todays (commercial or research oriented) solvers for mixed integer programs. It includes modelling aspects, preprocessing techniques, relaxation methods as well as branch-and-bound issues. |

2220 (opens in new tab) | 01.05.2002 | Froehlich, Steffen | A note on $\mu$-stable surfaces with prescribed constant mean curvature |
MSC: 53A10; 53C42; 53A10 Using a generalized stability condition we give an upper bound of the principle curvatures of certain constant mean curvature surfaces which implies a theorem of Bernstein type. |

2219 | 01.05.2002 | Jörg Hagspiel, Angelika May, Alexander Szimayer | Testing for Conditional Heteroscedasticity: Studying the Power Function | In this paper, we study the power function of Likelihood Ratio (LR) tests for testing AR-ARCH-type models for reduction to white noise. We briefly summarize the asymptotic distribution of the pseudo-log-likelihood ratio statistics for these tests. The power function is examined for the ARCH(1) and the AR(1)-ARCH(1) model by a simulation study. Primarily, the simulations are carried out for Gaussian innovations and in a second step, we replace the Gaussian distribution by the heavy tailed t-distribution. With this, we point out the loss of power caused by heavy tailed innovations. The relevance of the results on financial time-series modeling is shown in the context of Value-at-Risk (VaR) calculation. For the sample size of 500 days, we show that in most cases we are not able to find significant conditional heteroscedasticity effects, i.e. the empirical LR statistics suggests to reject the null hypothesis of white noise, but not with suitable power. |

2216 | 01.05.2002 | Fröhlich, Andreas | Global in time $L^p$-estimates for the instationary Stokes- and Navier-Stokes flow through an aperture |
MSC: 35Q30; 35D05; 47D06 Using a characterisation of maximal $L^p$-regularity by ${\cal R}$-bounded operator families we prove global in time estimates in $L^p(\R_+;L^q(\Omega)),¸ 1<p,¸q<\infty,$ for solutions of the instationary Stokes system in an aperture domain $\Omega\subset\R^n,¸ n\ge 3,$ with $\partial\Omega\in C^{1,1}.$ The results are applied to obtain new global in time estimates for weak solutions of the Navier-Stokes equations with nonvanishing flux through the aperture. |

2215 | 01.05.2002 |
Neeb, Karl-Hermann Vizman, Cornelia |
Flux homomorphisms and principal bundles over infinite dimensional manifolds |
MSC: 22E65; 58B20; 53D50 Flux homomorphisms for closed vector-valued differential forms on infinite dimensional manifolds are defined. We extend the relation between the kernel of the flux for a ${\scriptstyle 2}$-form ${\scriptstyle \omega}$ and Kostant's exact sequence associated to a principal bundle with curvature $\omega$ to the context of infinite-dimensional fiber and base space. We then use these results to construct central extensions of infinite dimensional Lie groups. |

2214 (opens in new tab) | 01.04.2002 |
Lengnink, Katja Prediger, Susanne |
Mathematik öffnen: Bildung zum mathematikverständigen Bürger | Gegenwärtig wird viel über offene Aufgaben, den Erwerb von Problemlösefähigkeiten und über eine Öffnung der Unterrichtskultur im Mathematikunterricht diskutiert. Mit unserem Text wollen wir dafür werben, im Rahmen dieser wichtigen inhaltlichen und methodischen Neufindung nicht vor der Mathematik selbst halt zu machen. Exemplarisch arbeiten wir an einem Unterrichtsprojekt heraus, wie sich gerade im Öffnen von zunächst geschlossenen Mathematisierungen in einer für die Lernenden relevanten Entscheidungssituation Bildungschancen ergeben. In diesem Sinne kann eine Öffnung auch der mathematischen Fachinhalte im Unterricht ein anderes Bild von Mathematik bei den Lernenden fördern und so einen wichtigen Beitrag zu demokratischer Erziehung leisten. |

2213 | 01.04.2002 | Rabinovich, Vladimir S., Roch, Steffen | Integral operators with shifts on homogeneous groups |
MSC: 45A05; 47G10 We study Fredholm properties of integral operators with shifts on homogeneous groups. This investigation is based on the limit operators method which allows us to reduce the problem of Fredholmness of convolution operators with variable coefficients and with variable shifts to the problem of invertibility of convolution operators with constant coefficients and constant shifts. For the invertibility of these operators, methods of harmonic analysis on noncommutative groups are available. |

2212 (opens in new tab) | 01.04.2002 | Zahn, Peter | Nicht-restriktive Vermeidung einiger Paradoxien in einem Behauptungsspiel | Ein Ziel ist es, zu zeigen, dass die im Folgenden besprochenen Paradoxien und die Erforderlichkeit ihrer Vermeidung kein Hindernis dafür bilden, in einer Sprache über diese selbst zu reden, etwas über Sprache im Allgemeinen auszusagen und z.B. in sprachwissenschaftliche Untersuchungen diese selbst einzuschließen. |

2211 | 01.04.2002 | Biller, Harald | Characterizations of Proper Actions |
MSC: 54H15; 19K35; 19L47; 46L80; 57S05 Three kinds of proper actions of increasing strength are defined. We prove that the three definitions specialize to the definitions by Bourbaki, by Palais, and by Baum, Connes, and Higson in their respective settings. The third of these, which thus turns out to be the strongest, originally only concerns actions of second countable locally compact groups on metrizable spaces. In this situation, it is shown to coincide with the other two definitions if the total space locally has the Lindelöf property and the orbit space is regular. |

2210 | 01.04.2002 |
Wille Rudolf |
Existential Concept Graphs of Power Context Families |
MSC: 00 The aim of this paper is to show how \em {existential concept graphs} may be introduced on the semantic level. For this the «free extension» of a power context family \vec{\mathbb{K}} by a given set X of variables is constructed as a power context family «freely» enlarged by X. Then, an existential concept graph of \vec{\mathbb{K}} can be appropriately defined as a concept graph of the free extension of \vec{\mathbb{K}} that can be projected onto a concept graph of \vec{\mathbb{K}} by some mapping induced by an interpretation of the variables of X by basic objects of \vec{\mathbb{K}}. The introduced \em {conceptual content} of existential concept graphs allows a simple description of the generalization order between those graphs. All this can be generalized to \em {existential protoconcept graphs}for also including negations. In this way, the actual development of Contextual Judgement logic disposes of (implicit) existential quantifiers as well as negations and negating inversions. |

2209 | 01.04.2002 | Farwig, Reinhard | Weighted $L^q$--Helmholtz Decompositions in Infinite Cylinders and in Infinite Layers | |

2208 (opens in new tab) | 01.04.2002 | Hereth, Joachim | Relational Scaling and Databases |
MSC: 00 More than 20 years of theoretical development and practical experience in the field of Conceptual Information Systems have made available a wide variety of structure and procedures to gain new knowledge from data or to present it in a user-friendly way, by restructuring the data in a conceptual way to help the user interpret and understand the meaning. Even longer, Database Theory has helped develop highly efficient database systems, processing daily huge amounts of data. However, both theories can profit from a cooperation: on the one hand, data and database modeling methodologies could be applied to the building of Conceptual Information System, the connection between the presented conceptual structures and the original data can be clarified. On the other hand, database theory may profit from the experience and ideas for more user-centered interfaces to the stored data, as well as profit from the translation of theoretical results. In this paper, we present the first necessary steps to perform a translation between the languages used in both domains. For this purpose, we introduce basic notions from Database Theory with a focus on the operations, which are basic for a first application: a more formal way to describe the process of Relational Scaling and the transformation of data for Conceptual Information Systems in general. Conversely, we present an approach for a standard problem of database theory by using methods from Formal Concept Analysis. Finally, we discuss the next steps needed for the integration of these two theories. |

2207 | 01.04.2002 |
Maier, Peter Neeb, Karl-Hermann |
Central extensions of current groups |
MSC: 22E65; 58D15; 22E67; 58B05 In this paper we study central extensions of the identity component ${\scriptstyle G}$ of the Lie group ${\scriptstyle C^\infty(M,K)}$ of smooth maps from a compact manifold ${\scriptstyle M}$ into a Lie group ${\scriptstyle K}$ which might be infinite-dimensional. We restrict our attention to Lie algebra cocycles of the form ${\scriptstyle \omega(\xi, \eta) = [\kappa(\xi, d\eta)]}$, where ${\scriptstyle \kappa \: \k \times \k \to Y}$ is a symmetric invariant bilinear map on the Lie algebra ${\scriptstyle \k}$ of ${\scriptstyle K}$ and the values of ${\scriptstyle \omega}$ lie in ${\scriptstyle \Omega^1(M,Y)/dC^\infty(M,Y)}$. For such cocycles we show that a corresponding central Lie group extension exists if and only if this is the case for ${\scriptstyle M = \SS^1}$. If ${\scriptstyle K}$ is finite-dimensional semisimple, this implies the existence of a universal central Lie group extension ${\scriptstyle \hat G}$ of ${\scriptstyle G}$. The groups ${\scriptstyle \Diff(M)}$ and ${\scriptstyle C^\infty(M,K)}$ act naturally on ${\scriptstyle G}$ by automorphisms. We also show that these smooth actions can be lifted to smooth actions on the central extension ${\scriptstyle \hat G}$ if it also is a central extension of the universal covering group ${\scriptstyle \tilde G}$ of ${\scriptstyle G}$. |

2206 | 01.04.2002 | Neeb, Karl-Hermann | Lectures at the European School in Group Theory: Infinite-Dimensional Groups and Their Representations |
MSC: 22E65; 17B65; 2201; 22E45 These lecture notes provide an introduction to the representation theory of Banach--Lie groups of operators on Hilbert spaces, where our main focus lies on highest weight representations and their geometric realization as spaces of holomorphic sections of a complex line bundle. After discussing the finite-dimensional case in Section I, we describe the algebraic side of the theory in Sections II and III. Then we turn in Sections IV and V to Banach--Lie groups and holomorphic representations of complex classical ones. The geometry of the coadjoint action is discussed in Section VI, and in the concluding Section VII all threads lead to a full discussion of the theory for the group ${\scriptstyle U_2(H)}$ of unitary operators ${\scriptstyle u}$ on a Hilbert space ${\scriptstyle H}$ for which ${\scriptstyle u – \1}$ is Hilbert--Schmidt. |

2205 (opens in new tab) | 01.04.2002 |
Eisenblätter Andreas Fügenschuh Armin Koch, Thorsten Koster, Arie Martin, Alexander Pfender, Tobias Wegel, Oliver Wessäly, Roland |
Modelling Feasible Network Configurations for UMTS |
MSC: 90C11 A model for the optimisation of the location and configuration of base stations in a UMTS network is described. The focus is primarily on modelling the configuration problem sufficiently accurate using mixed-integer variables and (essentially) linear constraints. These constraints reflect the limited downlink code capacity in each cell, the interference limitations for successful up- and downlink transmissions, the need for sufficiently strong (cell) pilot signals, and the potential gain for mobiles from being in soft(er) hand-over. It is also explained how to use the model as a basis for rating network configurations. |

2204 | 01.04.2002 |
Pollandt, Silke Wille, Rudolf |
Functorial Scaling of Ordinal Data |
MSC: 06A06; 06B23; 08A05; 18A99 In this paper we investigate how methods of deriving conceptual structures from ordinal data can be justified structurally. For the investigation we choose a categorical approach which is elaborated for suitable functors from the category of all ordinal structures to the category of all closure structures. The main result yields that the best functors are those which correspond to contra-ordinal scaling, a method developed in Formal Concept Analysis. |

2203 | 01.03.2002 | Neeb, Karl-Hermann | Nancy Lectures on Infinite-Dimensional Lie Groups |
MSC: 22E65; 2201; 58D05; 58D15 These are lecture note of a course given in Februaru and March 2002 in Nancy. The main purpose of this course was to present some of the main ideas of infinite-dimensional Lie theory and to explain how it differs from the finite-dimensional classical theory. After the introduction where we present some of the main types of infinite-dimensional Lie groups: lineare Lie groups associated to continuous inverse algebras, groups of maps and diffeomorphism groups, we turn in more detail to manifolds modeled on locally convex spaces. In Section III we present some of the basic Lie theory of locally convex Lie groups, including a discussion of the exponential function and the non-existence of groups for Lie algebras. In the final Section IV we discuss the topology of the main classes of infinite-dimensional Lie groups with an emphasis on their homotopy groups. |

2202 | 01.03.2002 | Wille, Rudolf | Wissensmanagement im universitären Bereich – eine systematische Ordnung - |
MSC: 00 Ei´ne umfassende Gestaltung eines universitären Wissensmanagements ist bisher kaum angegangen worden, obwohl sie sehr wünschenswert wäre. Wie ein Wissensmanagement im universitären Bereich systematisch aufgebaut werden könnte, wird in Anlehnung an Vorschläge für das Wissensmanagement in wirtschaftlichen Unternehmen anhand der sogenannten Kernprozesse des Wissensmanagement erörtert. |

2201 | 01.03.2002 | Wille, Rudolf | Kommunikative Rationalität, Logik und Mathematik |
MSC: 00 In dieser Arbeit wird folgende These erläutert und begründet: {\em Sinn und Bedeutung von Mathematik liegen letztlich darin, dass Mathematik die rationale Kommunikation von Menschen wirksam zu unterstützen vermag}. Dazu werden die Begriffe der kommunikativen Rationalität (im Sinne von Jürgen Habermas), der kommunikativen Logik (im Sinne der Peirceschen Spätphilosophie) und der kommunikativen Mathematik (im Sinne der Allgemeinen Wissenschaft) ausführlich expliziert. Mit diesen Begriffen wird argumentiert, dass die Mathematik über ihre enge Verbindung zur Logik kommunikative Rationalität aktivieren kann, durch die sie die rationale Kommunikation von Menschen wirksam zu unterstützen vermag. |

2200 | 01.03.2002 | Wille, Rudolf | Transdisziplinarität und allgemeine Wissenschaft |
MSC: 00 Erläutert und begründet wird die These: Die Disziplinen können die Forderung nach Transdiszplinarität am besten erfüllen, wenn sie ihren Teil an Allgemeiner Wissenschaft in möglichst großer Breite entwickeln, pflegen und aktivieren. |

2199 | 01.03.2002 | Wille, Rudolf | Begriffliche Wissensverarbeitung in der Wirtschaft |
MSC: 00 Begriffliche Wissensverarbeitung ist einem pragmatischem Wissensverständnis verpflichtet, nach dem menschliches Wissen in einem offenen Prozeß meschlichen Denkens, Argumentierens und Kommunizierens entsteht und weiterlebt. Sie gründet sich auf eine mathematische Begriffstheorie, die auf das wechselseitige Zusammenwirken von Formalem und Inhaltlichem ausgerichtet ist. Wie diese theoretische Konzeption in der wirtschaftlichen Praxis zur Wirkung kommt, wird erläutert anhand der Kernprozesse des organisationalen Wissensmanagement, d.h. nach G. Probst et al. anhand von Wissensidentifikation, Wissenserwerb, Wissensentwicklung, Wissens(ver)teilung, Wissensnutzung und Wissensbewahrung; jeweils an einem Beispiel wird der einsatz spezifischer Methoden der Begrifflichen Wissensverarbeitung demonstriert. Abschließend wird auf den prozeßhaften Wirkungszusammenhang von wissenszielen und Wissensbewertungen mit den kernprozessen aus Sicht der Begrifflichen Wissensverarbeitung eingegangen. |

2198 (opens in new tab) | 01.03.2002 | Alber, Hans-Dieter | Mathematical analysis of constitutive equations: Existence and collapse of solutions |
MSC: 35Q72; 35F25; 74C10; 74C05 In this article we review a new existence result for quasistatic initial-boundary value problems with interval variables, which model the viscoelastic or viscoplastic behavior of solids at small strain. The constitutive models considered belong to the class of constitutive equations of monotone type, which include classical models like the Prandtl-Reuss law and the Norton-Hoff law, but can be much more general. In particular, nonlinear hardening can be included. Also, the relation between collapse of solutions and a coercivity condition is discussed. |

2197 (opens in new tab) | 01.02.2002 | Prediger, Susanne | “Was bedeutet das eigentlich, wenn ich zwei Gleichungen addiere, um eine Variable weg zu kriegen?” Ein Dialog von der geometrischen Deutung eines Lösungsverfahrens für Lineare Gleichungssysteme bis zum “Sinn des sinnlosen Umformens” | Ausgangspunkt des im Artikel beschriebenen Dialogs ist die Frage eines Schülers nach der geometrischen Interpretation des Additionsverfahrens für Lineare Gleichungssysteme. Die Klärung dieser Frage eröffnet das Feld für die Diskussion einer wichtigen Grundidee der Mathematik, der Idee der Beschreibungswechsel, hier zwischen Geometrie und Algebra. Von da aus ist es im Dialog mit dem Lernenden kein weiter Weg mehr zum Gespräch über den “Sinn des sinnlosen Umformens”, einer Leitidee der Algebra. Mit dem Dialog soll gezeigt werden, dass im ernsthaften und tiefgehenden Verfolgen von Schülerfragen interessante Lernchancen stecken, die wir nicht verschenken sollten. |

2196 (opens in new tab) | 01.02.2002 | Prediger, Susanne | Kommunikationsbarrieren beim Mathematiklernen – Analysen aus kulturalistischer Sicht | Ausgangsthese des Beitrags ist, dass jede Kommunikation über Mathematik mit Lernenden interkulturelle Kommunikation ist, bei der die Lehrkraft als Vertreterin der Kultur Mathematik fungiert. Auftretende Kommunikationsbarrieren sind oft kulturell bedingt, wie an Beispielen erläutert wird. Als Vorschlag zum Umgang mit Kommunikationsbarrieren wird das Konzept des virtuellen interkulturellen Diskurses zur Klärung von kulturbedingten Konflikten vorgestellt. |

2195 | 01.02.2002 | Karl H. Hofmann, Detlev Poguntke, and Sidney A. Morris | The exponential function of locally connected compact abelian groups |
MSC: 22B05 It is shown that the following four conditions are equivalent for a compact connected abelian group $G$: (i) the exponential function of $G$ is open onto its image; (ii) $G$ has arbitrarily small connected direct summands $N$ such that $G/N$ is a finite dimensional torus; (iii) the arc component $G_a$ of the identity is locally arcwise connected; (iv) the character group $\hat G$ is a torsion free group in which every finite rank pure subgroup is free and is a direct summand. \bye |

2194 | 01.02.2002 | Karl H. Hofmann and Sidney A. Morris | Projective limits of finite dimensional Lie groups |
MSC: 22E65 MSC: 22A05 For a topological group $G$ we define ${\cal N}(G)$ to be the set of all normal subgroups $N$ of $G$ such that $G/N$ is a finite dimensional Lie group. Then $G$ is said to be a {\it pro-Lie group} if, firstly, $G$ is complete, secondly, ${\cal N}(G)$ is a filter basis, and thirdly, every identity neighborhood of $G$ contains some $N\in{\cal N}(G)$. It is easy to see that every pro-Lie group $G$ is a projective limit $\lim_{N\in{\cal N}G}G/N$. The converse emerges as a difficult question, but it is shown here that any projective limit of finite dimensional Lie groups is a pro-Lie group. It is also shown that a closed subgroup $H$ of a pro-Lie group $G$ is a pro-Lie group, and that for any closed normal subgroup $N$ of a pro-Lie group $G$, for any one parameter subgroup $Y\colon{\bf R}\to G/N$ there is a one parameter subgroup $X\colon{\bf R}\to G$ such that $X(t)N=Y(t)$ for $t\in{\bf R}$. It is proved that the category of all pro-Lie groups and continuous group homomorphisms between them is closed under the formation of {\it all} limits, and that the Lie algebra functor preserves limits and quotients. \bye |

2193 | 01.02.2002 | Karl H. Hofmann and Sidney A. Morris | Compact groups with large abelian subgroups |
MSC: 22D05 In this paper we formulate a new conjecture: \noindent Conjecture.\quad \rm(The Abelian Subgroup Conjecture) \it Every infinite compact group $G$ has an abelian subgroup $A$ of weight $w(A)=w(G)$.\rm Groups for which the conjecture holds are called LAS-groups. We introduce methods to identify large classes of LAS-groups. \noindent \bf Added in proof.\quad\rm In the meantime, Wolfgang Herfort pointed out (W. Herfort, The Abelian Subgroup Conjecture: A Counter Example, J. of Lie Theory {\bf 12} (2002), 305--308) that the free profinite $p$-group $F_p(X)$ on any infinite set $X$ converging to 1 has weight card$(X)$ and has the property that all of its closed subgroups are free. Thus the only nondegenerate abelian closed subgroups are isomorphic to the additive group $p$-adic integers, and thus all abelian subgroups have a countable weight. Hence $F_p(X)$ is a counterexample to the Abelian Subgroup Conjecture for any uncountable set $X$. The results of this paper remain valid and are a challenge to find further sufficient, perhaps even necessary and sufficient conditions for a profinite group to be an LAS-group. \bye |

2192 | 01.02.2002 |
Hofmann, Norbert Müller-Gronbach, Thomas |
On the global error of Itô-Taylor schemes for strong approximation of scalar stochastic differential equations |
MSC: 65C30; 60H10 We analyze the global error of Itô-Taylor schemes for pathwise approximation of scalar stochastic differential equations on the interval $[0,1]$. The error of an approximation is defined by its expected $L_p$-distance to the solution, and the number $n$ of multiple Itô integrals that are evaluated is used as a rough measure of its computational cost. We show that the optimal order of convergence is $n^{-1/2}$ for $1\le p<\infty$ and $(n/\ln n)^{-1/2}$ for $p=\infty$. Consequently, there are no Itô-Taylor methods of higher order with respect to the global error on $[0,1]$. These results are in sharp contrast to the corresponding well known result for the error at the discretization points where arbitrary high orders can be achieved. |

2191 | 01.02.2002 | Chelminski, Krzysztof | Mathematical Analysis of the Armstrong – Frederick Model from the Theory of Inelastic Deformations of Metals. First Results and Open Problems |
MSC: 74C05; 74C10; 35Q72 This paper tries to analyse, from the mathematical point of view, the system of equations proposed by P.J. Armstrong and C.O. Frederick and used in the theory of inelastic deformations of metals. We present global in time existence result for so called «weak-type admissible solutions» for the quasistatic Armstrong-Frederick model. This nonmonotone model is here written as a model of pre-monotone type with a nonassociated flow rule. Moreover, a monotone model is presented, which has a structure that is very similar to the Armstrong-Frederick model. |

2190 (opens in new tab) | 01.01.2002 |
Alber, Hans-Dieter Chelminski, Krzysztof |
Quasistatic problems in viscoplasticity theory |
MSC: 74C10 We study the existence theory to quasistatic initial-boundary value problems with internal variables, which model the viscoelastic or viscoplastic behavior of solids at small strain. In these problems a system of linear partial differential equations coupled with a nonlinear system of differential equations or differential inclusions must be solved. The solution theory is based on monotonicity properties of the differential equations or differential inclusions. The article gives an essentially complete account of the recent progress. |

2189 | 01.01.2002 |
André Noll Jürgen Saal |
$H^\infty$-calculus for the Stokes operator on $L_q$-spaces |
MSC: 35Q30; 47A60 It is proved that the Stokes operator on a bounded domain, an exterior domain, or a perturbed \HS{} $\Omega$ admits a bounded {\HIC} on $L_q(\Omega)$ if $q\in(1,\infty)$. |

2188 | 01.01.2002 | King, Simon | Crossing number of links formed by edges of a triangulation |
MSC:.57M25; 57Q15; 52C45; 52B22 We study the crossing number of links that are formed by edges of a triangulation $T$ of $S^3$ with $n$ tetrahedra. We show that the crossing number is bounded from above by an exponential function of $n^2$. In general, this bound can not be replaced by a subexponential bound. However, if $T$ is polytopal (resp. shellable) then there is a quadratic (resp. biquadratic) upper bound in $n$ for the crossing number. In our proof, we use a numerical invariant $p(T)$, called polytopality, that we have introduced in a previous paper. |

Number | Date | Author | Title | Abstract/MSC |
---|---|---|---|---|

2187 | 01.12.2001 | Fröhlich, Andreas | Maximal Regulartity for the Instationary Stokes System in an Aperture Domain |
MSC: 35Q30; 35D05; 47D06 We prove estimates in $L^s(0,T;\lw(\Omega))$ for the solution of the instationary Stokes system in an aperture domain, where $1<s,q<\infty$ and the weight function $\omega$ is in the Muckenhoupt class $A_q.$ The result is archieved by combining a characterisation of maximal regularity by ${\cal R}$-bounded operator families with the fact that ${\cal R}$-boundedness follows from weighted estimates for Muckenhoupt weights. |

2186 | 01.12.2001 | Rabinovich, V. S., Roch, S. | Algebras of approximation sequences: Spectral and pseudospectral approximation of band-dominated operators |
MSC: 65J10; 65F15 This paper is devoted to relations between the spectrum (or certain kinds of a generalized spectrum) of a band-dominated operator $A$ and of the spectra of its approximations $A_n = P_n A P_n$, obtained by compressing $A$ onto the ranges of the orthogonal projections $P_n$. Particular attention is paid to the asymptotic behaviour of the spectra (or its generalizations) of the operators $A_n$. These results will appear as special cases of some general theorems on spectral approximation. |

2185 (opens in new tab) | 01.12.2001 | Zahn, Peter | Bemerkungen zu Standardlogiken und zur `Kulturalistischen Logikbegründung` Dirk Hartmanns | We maintain the advantage for scientific argumentations of intuitionistic and especially of classical logic as compared with the `Kulturalistischen Relevanzlogik` of Dirk Hartmann. To this we start with the aim to introduce linguistic means such that we can write $\Lambda x [A_1(x) \wedge A_2(x) \rightarrow B(x)]$, e.g., to mean that, for all values $c$ of $x$, if both $A_1(c)$ and $A_2(c)$ may be asserted, then $B(c)$ may be asserted too. For a precision of those linguistic means, the use of them must be restricted by proper rules. -- A discussion of some paradoxes, with partially result of a colloquial or conversational understanding of subjunction or its context. -- Which benefits could be derived from the `Kulturalistischen Relevanzlogik`? -- Differences between standard subjunctions and colloquial subjunctions. |

2184 | 01.12.2001 | Hishida, Toshiaki | The nonstationary Stokes and Navier-Stokes flows through an aperture |
MSC: 35Q30; 76D05 We consider the nonstationary Stokes and Navier-Stokes flows in aperture domains $\Omega\subset\mathbb R^n, n\geq 3$. We develop the $L^q$-$L^r$ estimates of the Stokes semigroup and apply them to the Navier-Stokes initial value problem. As a result, we obtain the global existence of a unique strong solution, which satisfies the vanishing flux condition through the aperture and some sharp decay properties as $t\to\infty$, when the initial velocity is sufficiently small in the $L^n$ space. Such a global existence theorem is up to now well known in the cases of the whole and half spaces, bounded and exterior domains. |

2183 | 01.11.2001 | Biller, Harald | Fixed Points of Pro-Tori in Cohomology Spheres |
MSC: 57S10; 57S25; 57P05 Essential results from the theory of torus actions, initiated by P. A. Smith, are generalized to actions of pro-tori, i.e. compact connected abelian groups. We show that the fixed point set in a (rational cohomology) manifold, resp. sphere, is a rational cohomology manifold, resp. sphere, of even codimension. Borel's dimension formula for the fixed spheres of codimension one subgroups is proved for actions of pro-tori on (cohomology) spheres. This yields a sharp upper bound for the group dimension. |

2182 | 01.11.2001 | Biller, Harald | Proper actions on cohomology manifolds |
MSC: 57S10; 57S20; 57P05 Essential results about actions of compact Lie groups on manifolds are generalized to proper actions of arbitrary groups on connected cohomology manifolds. Slices are replaced by certain fibre bundle structures on orbit neighbourhoods. The group dimension is shown to be effectively finite. The orbits of maximal dimension form a dense open connected subset. If some orbit has codimension at most 2 then the group is effectively a Lie group. |

2181 | 01.11.2001 |
Bartuzel, Grzegorz Zabielski, Leszek |
Lebesgue points and local properties of integrable functions | |

2180 | 01.11.2001 |
Pinnau, René Thömmes, Guido |
Optimal Boundary Control of Glass Cooling Processes |
MSC: 49K20; 35K55; 80A20 An optimal control problem for glass cooling processes is considered. We model glass cooling using the $SP_1$ approximations to the radiative heat transfer equations. The control variable is the temperature at the boundary of the domain. This results in a boundary control problem for a parabolic/elliptic system which is treated by a constrained optimization approach. We consider several cost functionals of tracking type, define the corresponding Lagrange functionals and derive the first-order optimality system. We investigate several numerical methods based on the adjoint variables and present results of numerical simulations illustrating the feasibility and performance of the different approaches. |

2179 | 01.10.2001 |
Thömmes, G. Pinnau, R. Seaïd, M. Götz, T. Klar, A. |
Numerical Methods and Optimal Control for Glass Cooling Processes | In this paper, we discuss numerical and analytical approximations of radiative heat transfer equations used to model cooling processes of molten glass. Simplified diffusion type approximations are discussed and investigated numerically. These approximations are also used to develop acceleration methods for the iterative solution of the full radiative heat transfer problem. Moreover, applications of the above diffusion type approximations to optimal control problems for glass cooling processes are discussed. |

2178 | 01.10.2001 |
May, Angelika Szimayer, Alexander |
Testing for Conditional Heteroscedasticity in Financial Time-Series | |

2177 (opens in new tab) | 01.10.2001 | Zahn, Peter | Ein normatives Erklärungsmodell für die Eignung klassischer Schlussweisen | We introduce “meaningful” (i.e. not merely formal) languages, the use and the semantics of which are determined by “elementary external facts” on the one hand and rules of assertion on the other. Within those languages, classical reasoning will be justified. |

2176 | 01.09.2001 | Wüstner, Michael | $G=({\rm exp}{\mathfrak g})^2$ | It is shown that every real Lie group is equal to the square of the exponential image of its Lie algebra. |

2175 | 01.09.2001 | Kindler, Jürgen | Klee's intersection theorem and the Euler characteristic for abstract paved spaces | |

2174 | 01.09.2001 | Ebenfeld, Stefan | Remarks on the Quasistatic Problem of Viscoelasticity -- Existence, Uniqueness and Homogenization |
MSC: 74C10; 74Q10; 35Q72 This paper is devoted to the quasistatic problem of viscoelasticity. We rewrite the problem as an abstract evolution equation with a maximal monotone operator. For the abstract problem we prove existence, uniqueness and stability of solution w.r.t. the data. Next we apply our abstract theory to the quasistatic problem of viscoelasticity. We prove existence and uniqueness for the n--dimensional problem, and statistic homogenization for the 1--dimensional problem. We close our discussion with a remark on homogenization of the n--dimensional problem. |

2173 | 01.09.2001 | Fröhlich, Andreas | The Stokes operator in weighted $L^q$-spaces II: Weighted resolvent estimates and maximal $L^p$-regularity |
MSC: 35Q30; 35D05; 47D06; 46E25 In this paper we establish a general weighted $L^q$-theory of the Stokes operator ${\cal A}_{q,\omega}$ in the whole space, the half space and a bounded domain for general Muckenhoupt weights $\omega\in A_q$. We show weighted $L^q$-estimates for the Stokes resolvent system in bounded domains for general Muckenhoupt weights. These weighted resolvent estimates imply not only that the Stokes operator ${\cal A}_{q,\omega}$ generates a bounded analytic semigroup but even yield the maximal $L^p$-regularity of ${\cal A}_{q,\omega}$ in the respective weighted $L^q$-spaces for arbitrary Muckenhoupt weights $\omega\in A_q$. This conclusion is archived by combining a recent characterisation of maximal $L^p$-regularity by ${\cal R}$-bounded families due to L. Weis \cite{weis1} with the fact that for $L^q$-spaces ${\cal R}$-boundedness is implied by weighted estimates. |

2172 | 01.09.2001 | Fröhlich, Andreas | The Stokes operator in weighted $L^q$-spaces I: Weighted estimates for the Stokes resolvent problem in a half space |
MSC: 35Q30; 35D05; 46E25 In this paper we derive weighted $L^q$-estimates for the Stokes resolvent system in the half space for weights of Muckenhoupt class, on which a new approach to maximal $L^p$-regularity of the Stokes operator for the half space and a bounded domain in weighted $L^q$-spaces in the forthcoming part II is based. We stress that our results hold for general Muckenhoupt weights. In particular, the weights may tend to zero or become singular at the boundary of the domain. |

2171 | 01.09.2001 | Arai, Natsuko | Restriktion- und Fortsetzungssätze für Hankelmultiplikatoren |
MSC: 42A45; 42C10 Es werden Einbettungen zwischen Räumen von Hankelmultiplikatoren verschiedener Ordnungen gezeigt. Für Fouriermultiplikatoren sind Einbettungen dieser Art von deLeeuw, sowie Coifman und Weiss bekannt. Diese Einbettungen werden mit den von D. Müller und A. Seeger für Fouriermultiplikatoren entwickelten Methoden zunächst für lokale Hankelmultiplikatoren, d.h. Multiplikatoren, deren Träger kompakt ist, gezeigt, und dann global fortgesetzt. |

2170 | 01.09.2001 | Burmeister, Peter | On some types of Galois connections arising in the theory of partial algebras |
MSC: 08A55 In connection with partial algebras one has much more relevant polarities (i.e.\ Galois connections induced by binary relations) than in the case of total algebras. On one side there are many different subsets of the set of first order formulas, which one wants to use as a concept of \emph{identity} in some special context, and where one is interested in the closure operators induced by restricting the \emph{validity of first order formulas} to this special subset. On the other hand the polarity induced by the \emph{reflection of formulas by mappings} allows to keep track on many interesting properties of homomorphisms between partial algebras, while others can be related to these via \emph{factorization systems} --- which can be considered as special pairs of corresponding closed classes (in Formal Concept Analysis one would call such pairs «formal concepts») of the polarity induced by the \emph{(unique) diagonal-fill-in property} on the class of all homomorphisms. --- Moreover, having an interesting set of properties of homomorphisms, the relation «a homomorphism has a property» can be used to apply the method of \emph{attribute exploration} from Formal Concept Analysis in order to elaborate a basis for all implications among these properties and on the other hand a small but «complete» set of counterexamples against all non-valid implications. In this note we want to describe some of such polarities or corresponding pairs of interest in them, and we shall present them in the context of \emph{many-sorted} partial algebras, since this context seems to be less known. Moreover, we want to give an example of an attribute exploration as mentioned above. |

2169 | 01.09.2001 |
Rabinovich, V. S. Roch, S. |
Local theory of the Fredholmness of band-dominated operators with slowly oscillating coefficients |
MSC: 47A11; 47B37; 47D25 A band-dominated operators on an $l^p$-space of vector-valued functions is an (in a generalized sense) Fredholm operator if and only if all of its limit operators are invertible and if their inverses are uniformly bounded (see \cite{RRS2}). We show that the limit operators approach is also compatible with the local Fredholmness of band-dominated operators with respect to localization over the maximal ideal space of the algebra of the slowly oscillating scalar-valued functions. A corollary of this result is that the uniform boundedness condition is redundant for band-dominated operators with slowly oscillating operator-valued coefficients. |

2168 | 01.08.2001 | René Pinnau | A Review on the Quantum Drift Diffusion Model |
MSC: 76Y05; 35J55; 35J60; 35K35; 65M12; 65M20 We consider the quantum drift diffusion model for semiconductor devices and collect recent results on the stationary and transient equations. The stationary model including generation--recombination terms is studied for bipolar devices and the transient equations are considered in the unipolar case. We cover several topics, such as existence and uniqueness of solutions, asymptotic limits and convergence of a nonlinear iteration scheme in the stationary case as well convergence of a positivity preserving semidiscretization of the transient equations and the linear stability of stationary states. |

2167 | 01.08.2001 | Abels, Helmut | Boundedness of Imaginary Powers of the Stokes Operator in an Infinite Layer |
MSC: 35Q30; 67D07 In this article we prove the existence of bounded purely imaginary powers of the Stokes operator $A_q$, which is defined on the space of solenoidal vector fields $J_q(\Omega)$, $1<q<\infty$, where $\Omega=\R^{n-1}\times (-1,1)$ is an infinite layer. It is a consequence of a special representation of the resolvent of the Stokes operator in terms of the Stokes operator on $\Rn$, a composition of a trace and a Poisson operator -- a singular Green operator -- and a negligible part. |

2166 | 01.08.2001 | Martin, Alexander | Large Scale Optimization |
MSC: 90C06; 90C10; 90C11; 90C57 In this article we present classical methods for the solution of large scale integer optimization problems. Among them are LP relaxations and cutting planes, Lagrangean relaxation, Dantzig-Wolfe and Benders' decomposition as well as ideas from lifting and projection. We also discuss some modeling issues and discuss their influence on the solvability of some large scale problems. |

2165 | 01.07.2001 | Holzer, Richard | Greechie diagrams of orthomodular partial algebras |
MSC: 08A55; 06F99 Greechie diagrams are a well known graphical representation of orthomodular partial algebras, orthomodular posets and orthomodular lattices. Kalmbach and Dichtl gave some characterisations of Greechie diagrams of orthomodular posets and of orthomodular lattices under some assumptions, for example, that the family of blocks is pasted, or that the intersection of each pair of blocks contains less or equal than four elements. In this paper I present a generalisation of these characterisations for orthomodular partial algebras (or equivalently orthomodular posets). Here we consider arbitrary hypergraphs with finite lines. A Greechie diagram can be seen as a special hypergraph: Different points of the hypergraph have different interpretations in the corresponding partial algebra of type (2,1,0) and each line in the hypergraph has a maximal Boolean subalgebra as interpretation, in which the points are the atoms. A diagram is complete if each maximal Boolean subalgebra is induced by a line of the hypergraph. Every nontrivial orthomodular partial algebra with finite blocks is the interpretation of a Greechie diagram. The characterisation theorems in this paper provide conditions to check, whether a hypergraph is a complete diagram of an orthomodular partial algebra. This poperty can be checked without having to compute the interpretation. We just have to consider the lines in the hypergraph. |

2164 | 01.06.2001 | Chelminski, Krzysztof | Coercive and self-controlling quasistatic models of gradient type with convex composite inelastic constitutive equations | |

2163 | 01.06.2001 | Dau, Frithjof | Implications of Properties Concerning Complementation in Finite Lattices (Contributions to General Algebra 12, Proceedings of the Vienna Conference, June 3 – 6, 1999, Verlag Johannes Heyn, Klagenfurt 2000) | |

2162 | 01.06.2001 |
Dau, Frithjof Wille, Rudolf |
On the Modal Understanding of Triadic Contexts | |

2161 | 01.06.2001 | Wille, Rudolf | Mensch und Mathematik: Logisches und mathematisches Denken | |

2160 | 01.06.2001 | Wille, Rudolf | The Contextual-Logic Structure of Distinctive Judgments | |

2159 | 01.06.2001 |
Alpay, Safak Uyar, Ayse |
On Local Approximation of Positive Operators | |

2158 | 01.06.2001 | Wille, Rudolf | Why Can Concept Lattices Support Knowledge Discovery in Databases? | |

2157 | 01.06.2001 | Pollandt, Silke | Relational Constructions on Semiconcept Graphs | The aim of the paper is to develop a logic of relations on semiconcept graphs corresponding to the Contextual Logic of Relations on power context families. Semiconcept graphs allow the representation of negations. The operations from Peircean Algebraic Logic (i.e., the operations of relation algebras of power context families) are used to generate compound semiconcepts (or relations, resp.). For an arbitrary (semi-)concept graph, most specific semiconcept graphs are constructed where a compound semiconcept is assigned to each of the edges, i.e.\ compound semiconcepts are constructed directly on semiconcept graphs independent of the corresponding power context family. |

2156 | 01.06.2001 | Robert Denk, Matthias Hieber, Jan Prüss | Relational Constructions on Semiconcept Graphs | |

2155 | 01.06.2001 | Gwiazda, Piotr | An Existence Result for a Model of Granular Material with Non-constant Density |
MSC: 35L65; 35Q72; 73Q05 We consider a model developed by Savage and Hutter which describes the flow of granular avalanches down a smoothly varying slope. The system consists of two nonstrictly hyperbolic equations for height and momentum. The existence of entropy solutions to this model is proved using the vanishing viscosity method, where we make extensive use of a generalised version of the invariant region theorem in order to prove a priori estimates. Since the model has a discontinuous source term, a new definition of entropy solution must be introduced. |

2154 | 01.05.2001 | Abels, Helmut | $L_q$-$L_r$-Estimates for the Non-Stationary Stokes Equations in an Aperture Domain |
MSC: 35Q30; 76D07; 35B40; 35C20 The article deals with asymptotic estimates of strong solutions of Stokes equations in aperture domain. An aperture domain is a domain, which outside a bounded set is identical to two half spaces separated by a wall and connected inside the bounded set by one or more holes in the wall. It is known that the corresponding Stokes operator generates a bounded analytic semigroup in the closed subspace $J_q(\Omega)$ of divergence free vector fields of $L_q(\Omega)^n$. We deal with $L_q-L_r$-estimates for the semigroup, which are known for $\Rn$, the half space and exterior domains. |

2153 | 01.05.2001 |
Rabinovich, V. S. Roch, S. |
An axiomatic approach to the limit operators method |
MSC: 47A53; 35S05; 47G10 We propose an axiomatic approach for the application of the limit operators method. This approach will be applied to operators in a $C^*$-algebra which is generated by operators of right convolution on a homogeneous non-commutative group $\mathbf{X}$ and by operators of multiplication by functions in $L_\infty(\mathbf{X)}$. In terms of limit operators, we derive necessary and sufficient conditions for these operators to be semi-Fredholm or Fredholm. As another application, we obtain necessary and sufficient conditions for the semi-Fredholmness and Fredholmness of pseudodifferential operators with double symbols in the class $OPS_{0,0,0}^{0}$. |

2152 | 01.05.2001 | Martin, Alexander | General Mixed Integer Programming: Computational Issues for Branch-and-Cut Algorithms |
MSC: 90C10; 90C11 In this paper we survey the basic features of state-of-the-art branch-and-cut algorithms for the solution of general mixed integer programming problems. In particular we focus on preprocessing techniques, branch-and-bound issues and cutting plane generation. |

2151 | 01.05.2001 |
Jüngel, Ansgar Pinnau, René |
Convergent Semidiscretization of a Nonlinear Fourth Order Parabolic System |
MSC: 35K35; 65M12; 65M15; 65M20; 76Y05 A semidiscretization in time of a fourth order nonlinear parabolic system in several space dimensions arising in quantum semiconductor modelling is studied. The system is numerically treated by introducing an additional nonlinear potential. The resulting sequence of nonlinear second order elliptic systems admits at each time level strictly positive solutions as long as the lattice temperature is sufficiently large. Exploiting the stability of the discretization, convergence is shown in the multi-dimensional case. Under some assumptions on the regularity of the solution the rate of convergence proves to be optimal. |

2150 | 01.05.2001 | Neeb, Karl-Hermann | Universal central extensions of Lie groups |
MSC: 22E65; 17E65 We call a central {$\scriptstyle Z$}-extension of a group {$\scriptstyle G$} weakly universal for an abelian group {$\scriptstyle A$} if the correspondence assigning to a homomorphism {$\scriptstyle Z \to A$} the corresponding {$\scriptstyle A$}-extension yields a bijection of extension classes. The main problem discussed in this paper is the existence of central Lie group extensions of a connected Fréchet--Lie group {$\scriptstyle G$} which is weakly universal for all abelian Fréchet--Lie groups whose identity components are quotients of vector spaces by discrete subgroups. We call these abelian groups regular. In the first part of the paper we deal with the corresponding question in the context of topological, Fréchet-, and Banach--Lie algebras, and in the second part we turn to the groups. Here we start with a discussion of the weak universality for discrete abelian groups and then turn to regular Fréchet--Lie groups {$\scriptstyle A$}. The main results are a Recognition- and a Characterization Theorem for weakly universal central extensions. |

2149 | 01.05.2001 | Ebenfeld, Stefan | Nonlinear Initial Boundary Value Problems of Hyperbolic--Parabolic Type / A General Investigation of Admissible Couplings between Systems of Higher Order |
MSC: 35G20; 35M10 In this article we investigate nonlinear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between the single systems. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. |

2148 | 01.04.2001 | Wille, Rudolf | Boolean Judgement Logic | |

2147 | 01.04.2001 | Prediger, Susanne | Mathematik als kulturelles Produkt menschlicher Denktätigkeit und ihr Bezug zum Individuum |
MSC: 00A30 Ausgehend von dem für das Mathematiklernen fundamentalen Problem des (oft fehlenden) Subjektbezuges soll das Verhältnis von Mensch und Mathematik analysiert werden. Dazu wird die Auffassung von Mathematik als kulturelles Produkt menschlicher Denktätigkeit vorgestellt und begründet. So wird das eigentümliche Spannungsverhältnis verstehbar, dass die Mathematik einerseits zwar durch die Menschheit gestaltet ist, sie dem einzelnen andererseits aber als objektiv Gegebenes gegenübertritt. Erklärungsbedürftig ist in dieser Auffassung die hohe Kohärenz der mathematischen Theorien und der große Konsens unter Mathematikern. Eine Erklärung wird unter Rückgriff auf historische Untersuchungen und wissenschaftstheoretische Überlegungen angeboten. |

2146 | 01.04.2001 |
Glöckner, Helge Neeb, Karl-Hermann |
Banach-Lie Quotients, Enlargibility, and Universal Complexifications |
MSC: 22E65; 22E15 We characterize those real Banach-Lie groups which admit universal complexifications, and present examples of Banach-Lie groups which have none. To achieve these goals, we prove new results concerning the enlargibility of Banach-Lie algebras, and derive a necessary and sufficient condition for the existence of Lie group structures on quotients of Banach-Lie groups. |

2145 | 01.04.2001 | Neeb, Karl-Hermann | Highest weight representations and infinite-dimensional Kaehler manifolds |
MSC: 22E65; 57T20 The geometry of unitary highest weight representations and the corresponding coadjoint orbits has many infinite-dimensional relatives. This becomes apparent from a geometric approach to unitary highest weight representations. In this note we discuss such representations for the unitary group of a ${\scriptstyle C^*}$-algebra and for groups related to ${\scriptstyle L^*}$-groups. |

2144 | 01.04.2001 |
Amster, Pablo Pinnau, René |
Convergent Iterative Schemes for a Non-isentropic Hydrodynamic Model for Semiconductors |
MSC: 65J15; 76N10 Two iterative schemes for the solution of the one-dimensional stationary full hydrodynamic model for semiconductor devices are studied. This model consists of a system of balance equations for the electron density, temperature and the electric field. The first iterative scheme relies on a decoupling of the equations in the spirit of the well-known Gummel-iteration for the standard drift diffusion model. Convergence is proven in the case of small deviations from the equilibrium state and high lattice temperature. Secondly, a full Newton-iteration is analyzed and its local second order convergence is proven. |

2143 | 01.03.2001 | Maier, Peter | Central Extensions of Topological Current Algebras |
MSC: 17B65; 17B56; 17B99 In this note we describe universal central extensions of certain Fréchet current algebras, which in our context are algebras of type $A\otimes\mathfrak{g}$, where $\mathfrak{g}$ is a finite dimensional semisimple real Lie algebra and $A$ a commutative associative Fréchet algebra. |

2142 | 01.03.2001 | Neeb, Karl-Hermann | Classical Hilbert-Lie groups, their extensions and their homotopy groups | Let {$\scriptstyle H$} be a complex Hilbert space and {$\scriptstyle D$} a hermitian operator on {$\scriptstyle H$} with finite spectrum. Then the operators for which the commutator with {$\scriptstyle D$} is of Schatten class {$\scriptstyle p$} form a Banach algebra {$\scriptstyle B_p(H,D)$}. In the present paper we study groups {$\scriptstyle \GL_p(H,D)$} associated to this kind of Lie algebra and also groups {$\scriptstyle \GL_p(H,I,D)$} associated to the sub Lie algebras {$\scriptstyle B_p(H,I,D) := {x \in B_p(H,D) \: Ix^*I^{-1} =- x}$}, where {$\scriptstyle I$} is an antilinear isometry with {$\scriptstyle I^2 \in {\pm \1}$}. For {$\scriptstyle p = 2$} we determine the full second continuous cohomology for these Lie algebras and for the groups we compute all homotopy groups. These results then lead to a direct description of universal central extensions of the groups {$\scriptstyle \GL_2(H,D), \GL_2(H,I,D)$} and some of their real forms. In particular we obtain the infinite-dimensional metaplectic and the metagonal group as special examples. In a last section we discuss associated complex flag manifolds and show that the unitary forms of the complex groups act transitively. |

2141 | 01.03.2001 | Schindler, Werner | Nonlinear Feedback Shift Registers with Embedded Short Cycles | |

2140 | 01.02.2001 | Prediger, Susanne | Mathematiklernen als interkulturelles Lernen – Entwurf für einen didaktischen Ansatz | Ausgehend von der Transfer-Problematik wird ein Ansatz entworfen, Mathematiklernen als interkulturelles Lernen aufzufassen. Dabei wird Mathematik als Kultur formalen Denkens betrachtet, die dem individuellen Alltagsdenken entgegentritt und dann integriert werden muss. Pädagogische Konzepte zum interkulturellen Lernen sind hilfreich, diesen Prozess zu analysieren und Ideen für seine Unterstützung zu entwickeln. Grundlage für eine solche Perspektive auf das Mathematiklernen ist die in der neueren Philosophie der Mathematik vertretene kulturalistische Sicht auf Mathematik. Starting with the problem of transfer, an approach is developed to consider mathematics learning as intercultural learning. For this, mathematics is understood as a culture of formal thinking, that shall be integrated into the general thinking of each individual learner. In order to analyse this process and to give ideas how to support it, pedagogical conceptions about intercultural learning are activated. The basis of such an approach to mathematics learning is given by newer tendencies in the philosophy of mathematics where a culturalistic viewpoint becomes more and more common. |

2139 | 01.02.2001 | Franzke, Martin | Strong $L^q$-Theory of the Navier-Stokes Equations in Aperture Domains |
MSC: 35Q30; 76D07 This article deals with the Stokes equations in aperture domains. Geometrically spoken, such domains consist of two halfspaces separated by a wall, but connected by a hole within that wall. It is well known that in these domains the solution of the Stokes equations is not unique, but that an additional boundary condition has to be imposed: This can be either the flux through the hole or the pressure drop between the two halfspaces. In this paper suitable Stokes operators are constructed for both cases, which are shown to generate bounded analytic semigroups. This is used to prove the existence and uniqueness of strong solutions of the Stokes and Navier-Stokes equations subject to one of the additional boundary conditions. |

2138 | 01.01.2001 | Neff, Patrizio | Formulation of visco-plastic approximations in finite plasticity for a model of small elastic strains. Part II: Local existence and uniqueness results. | |

2137 | 01.01.2001 | Spellucci, Peter | Nonlinear (local) Optimization The State of the Art |
MSC: 90C30; 65K05 Nonlinear Optimization came into live about fifty years ago with the seminal paper of Kuhn and Tucker, very shortly only after the birth of LP praxis through Dantzig's pioneering work on the simplex algorithm. Since then it has grown to a mathematical discipline in its own right, with deep interconnections with nonlinear analysis and numerical analysis. The appearance of more and more powerful computer equipment and more and more involved numerical solution techniques made the solution of nonlinear models a routine job, which forty years ago were beyond any possible consideration. This is true at least for the solution of problems involving some hundred variables and some thousand constraints, solution meant in a local and weak sense, that means in the sense of identifying points which satisfy certain necessary optimality conditions. The research in this area came to some maturity about ten years ago and a large amount of software is now available even in the public domain, see e.g. http://plato.la.asu.edu/guide.html. During the past ten years many working groups throughout the world aimed at solution techniques also capable of coping with true large-scale nonlinear optimization problems. Several overview papers about this subject appeared in recent years, e.g. by Conn, Gould and Toint, Murray and Gould and Toint. In this overview article we give a short introduction into {\sl NLP} theory first and then review some of the most promising solution techniques. Whereas convex problems can be dealt with also in very high dimension successfully already, the treatment of nonconvex cases offers resistance to a satisfactory solution approach, since obviously methods which worked well for medium large problems cannot be transfered to very high dimensions. |